3. Kinematics

 

 

 

 

3.1 Basic Assumptions

 

Continuum mechanics is a combination of mathematics and physical laws that approximate the large-scale behavior of matter that is subjected to mechanical loading.   It is a generalization of Newtonian particle dynamics, and starts with the same physical assumptions inherent to Newtonian mechanics; and adds further assumptions that describe the structure of matter.  Specifically:

 

* The Newtonian reference frame: In classical continuum mechanics, the world is idealized as a three dimensional Euclidean space (a vector space consisting of all triads of real numbers ( x 1 , x 2 , x 3 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG4bWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamiE [email protected]@  ).  A point in space is identified by a unique set of three real numbers.   A Euclidean space is endowed with a metric, which defines the distance between points: d= x i x i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqGH9aqpdaGcaaqaaiaadIhada WgaaWcbaGaamyAaaqabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaaqa [email protected]@ .  Vectors can be expressed as components in a basis - { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz [email protected]@  of mutually perpendicular unit vectors.  Physical quantities such as force, velocity, acceleration are expressed as vectors in this space.  A Cartesian Coordinate Frame is a fixed point O together with a basis.   A Newtonian reference frame is a particular choice of Cartesian coordinate frame in which Newton’s laws of motion hold.  

 

  * The Continuum:  Matter is idealized as a continuum, which has two properties: (i) it is infinitely divisible (you can subdivide some region of the solid as many times as you wish); and (ii) it is locally homogeneous [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  in other words if you subdivide it sufficiently many times, all sub-divisions have identical properties (eg mass density).  A continuum can be thought of as an infinite set of vanishingly small particles, connected together.

 

Both the existence of a Newtonian reference frame, and the concept of a continuum, are mathematical idealizations.   Experimental evidence suggest that the laws of motion based on these assumptions accurately approximate the behavior of most solid and fluid materials at length scales of order mm-km or so in engineering applications.  In some cases continuum models can also approximate behavior at much shorter length scales (for volumes of material containing a few 1000 atoms), but models at these length scales often require different relations between internal forces deformation measures in the solid to those used to model larger volumes.

 

 

 

3.2 Reference and deformed configuration of a solid

 

The configuration of a solid is a region of space occupied (filled) by the solid.   When we describe motion, we normally choose some convenient configuration of the solid to use as reference  - this is often the initial, undeformed solid, but it can be any convenient region that could be occupied by the solid.   The material changes its shape under the action of external loads, and at some time t occupies a new region which is called the deformed or current configuration of the solid. 

 

For some applications (fluids, problems with growth or evolving microstructures) a fixed reference configuration can’t be identified [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzagaeaa [email protected]@  in this case we usually use the deformed material as the reference configuration.

 

Mathematically, we describe a deformation as a 1:1 mapping which transforms points from the reference configuration of a solid to the deformed configuration.  Specifically, let ξ i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe67a4naaBaaaleaacaWGPbaabeaaaa [email protected]@  be three numbers specifying the position of some point in the undeformed solid (these could be the three components of position vector in a Cartesian coordinate system, or they could be a more general ccoordinate system, such as polar coordinates).  As the solid deforms, each the values of the coordinates change to different numbers.  We can write this in general form as η i = f i ( ξ k ,t) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeE7aOnaaBaaaleaacaWGPbaabeaaki abg2da9iaadAgadaWgaaWcbaGaamyAaaqabaGccaGGOaGaeqOVdG3a [email protected]@ .  This is called a deformation mapping.

 

To be a physically admissible deformation

(i)     The coordinates must specify positions in a Newtonian reference frame.  This means that it must be possible to find some coordinate transformation x i ( ξ k ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaGcca [email protected]@ , such that x i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  are components in an orthogonal basis, which is taken to be ‘stationary’ in the sense of Newtonian dynamics.

(ii)   The functions f i ( ξ k ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgadaWgaaWcbaGaamyAaaqabaGcca [email protected]@  must be 1:1 on the full set of real numbers; and f i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  must be invertible

(iii)   f i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  must be continuous and continuously differentiable (we occasionally relax these two assumptions, but this has to be dealt with on a case-by-case basis)

(iv)  The mapping must satisfy det( η i / ξ j )>0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacsgacaGGLbGaaiiDaiaacIcacqGHci ITcqaH3oaAdaWgaaWcbaGaamyAaaqabaGccaGGVaGaeyOaIyRaeqOV [email protected]@ .

 

To begin with, we will describe all motions and deformations by expressing positions of points in both undeformed and deformed solids as components in a Cartesian reference frame (which is also taken to be an inertial frame).  Thus x i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  will denote components of the position vector of a materal particle before deformation, and y i ( x k ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGcca [email protected]@  will be components of its position vector after deformation.  

 

 

 

 

3.3 The Displacement and Velocity Fields

 

The displacement vector u(x,t) describes the motion of each point in the solid. To make this precise, visualize a solid deforming under external loads.  Every point in the solid moves as the load is applied: for example, a point at position x in the undeformed solid might move to a new position y at time t.  The displacement vector is defined as

y=x+u(x,t) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH5bGaeyypa0JaaCiEaiabgUcaRi [email protected]@

We could also express this formula using index notation, as

y i = x i + u i ( x 1 , x 2 , x 3 ,t) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaamiEamaaBaaaleaacaWGPbaabeaakiabgUcaRiaadwha daWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEamaaBaaaleaacaaIXa aabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa [email protected]@

Here, the subscript i  has  values 1,2, or 3, and (for example) y i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaa [email protected]@  represents the three Cartesian components of the vector y.

 

The displacement field completely specifies the change in shape of the solid. The velocity field would describe its motion, as

v i ( x k ,t)= y i t = u i ( x k ,t) t | x k =const [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG2bWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaam4AaaqabaGccaGGSaGaamiDaiaa cMcacqGH9aqpdaWcaaqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaa qabaaakeaacqGHciITcaWG0baaaiabg2da9maaeiaabaWaaSaaaeaa cqGHciITcaWG1bWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhada WgaaWcbaGaam4AaaqabaGccaGGSaGaamiDaiaacMcaaeaacqGHciIT caWG0baaaaGaayjcSdWaaSbaaSqaaiaadIhadaWgaaadbaGaam4Aaa qabaWccqGH9aqpcaqGJbGaae4Baiaab6gacaqGZbGaaeiDaaqabaaa [email protected]@

We also define the acceleration field

a i ( x k ,t)= 2 y i t 2 = v i ( x k ,t) t | x k =const [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaam4AaaqabaGccaGGSaGaamiDaiaa cMcacqGH9aqpdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaaki aadMhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG0bWaaWba aSqabeaacaaIYaaaaaaakiabg2da9maaeiaabaWaaSaaaeaacqGHci ITcaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadIhadaWgaaWc baGaam4AaaqabaGccaGGSaGaamiDaiaacMcaaeaacqGHciITcaWG0b aaaaGaayjcSdWaaSbaaSqaaiaadIhadaWgaaadbaGaam4AaaqabaWc [email protected]@

 

 

 

Examples of some simple deformations

 

 Volume preserving uniaxial extension

y 1 =λ x 1 y 2 = x 2 / λ y 3 = x 3 / λ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadMhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcqaH7oaBcaWG4bWaaSbaaSqaaiaaigdaaeqaaaGc baGaamyEamaaBaaaleaacaaIYaaabeaakiabg2da9iaadIhadaWgaa WcbaGaaGOmaaqabaGccaGGVaWaaOaaaeaacqaH7oaBaSqabaaakeaa caWG5bWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaamiEamaaBaaale [email protected]@

 

 

 Simple shear

y 1 = x 1 +tanθ x 2 y 2 = x 2 y 3 = x 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadMhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIa ciiDaiaacggacaGGUbGaeqiUdeNaaGPaVlaadIhadaWgaaWcbaGaaG OmaaqabaaakeaacaWG5bWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Ja amiEamaaBaaaleaacaaIYaaabeaaaOqaaiaadMhadaWgaaWcbaGaaG [email protected]@

 

 

 Rigid rotation through angle θ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  about e 3 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa [email protected]@  axis

y 1 = x 1 cosθ x 2 sinθ y 2 = x 2 cosθ+ x 1 sinθ y 3 = x 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadMhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaci4yaiaa c+gacaGGZbGaeqiUdeNaeyOeI0IaamiEamaaBaaaleaacaaIYaaabe aakiGacohacaGGPbGaaiOBaiabeI7aXbqaaiaadMhadaWgaaWcbaGa aGOmaaqabaGccqGH9aqpcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaci 4yaiaac+gacaGGZbGaeqiUdeNaey4kaSIaamiEamaaBaaaleaacaaI XaaabeaakiGacohacaGGPbGaaiOBaiabeI7aXbqaaiaadMhadaWgaa WcbaGaaG4maaqabaGccqGH9aqpcaWG4bWaaSbaaSqaaiaaiodaaeqa [email protected]@

 

 General rigid rotation about the origin

y=Rxor y i = R ij x j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH5bGaeyypa0JaaCOuaiabgwSixl aahIhacaaMc8UaaGPaVlaaykW7caaMc8Uaae4BaiaabkhacaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG5bWaaSbaaSqaaiaadM gaaeqaaOGaeyypa0JaamOuamaaBaaaleaacaWGPbGaamOAaaqabaGc [email protected]@

where R must satisfy R R T = R T R=I [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHsbGaeyyXICTaaCOuamaaCaaale qabaGaamivaaaakiabg2da9iaahkfadaahaaWcbeqaaiaadsfaaaGc [email protected]@ , det(R)>0. (i.e. R is proper orthogonal). I  is the identity tensor with components δ ik ={ 1,i=k 0,ik [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazdaWgaaWcbaGaamyAaiaadU gaaeqaaOGaeyypa0ZaaiqaaqaabeqaaiaaigdacaGGSaGaaGPaVlaa ykW7caaMc8UaamyAaiabg2da9iaadUgaaeaacaaIWaGaaiilaiaayk [email protected]@

 

Alternatively, a rigid rotation through angle θ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  (with right hand screw convention) about an axis through the origin that is parallel to a unit vector n can be written as

y=cosθx+(1cosθ)( nx )n+sinθ( n×x ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH5bGaeyypa0Jaci4yaiaac+gaca GGZbGaeqiUdeNaaCiEaiabgUcaRiaacIcacaaIXaGaeyOeI0Iaci4y aiaac+gacaGGZbGaeqiUdeNaaiykamaabmaabaGaaCOBaiabgwSixl aahIhaaiaawIcacaGLPaaacaWHUbGaey4kaSIaci4CaiaacMgacaGG UbGaeqiUde3aaeWaaeaacaWHUbGaey41aqRaaCiEaaGaayjkaiaawM [email protected]@

The components of R are thus

R ij =cosθ δ ij +(1cosθ) n i n j +sinθ ikj n k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iGacogacaGGVbGaai4CaiabeI7aXjabes7aKnaa BaaaleaacaWGPbGaamOAaaqabaGccqGHRaWkcaGGOaGaaGymaiabgk HiTiGacogacaGGVbGaai4CaiabeI7aXjaacMcacaWGUbWaaSbaaSqa aiaadMgaaeqaaOGaamOBamaaBaaaleaacaWGQbaabeaakiabgUcaRi GacohacaGGPbGaaiOBaiabeI7aXjabgIGiopaaBaaaleaacaWGPbGa [email protected]@

 where ijk [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHiiIZdaWgaaWcbaGaamyAaiaadQ [email protected]@  is the permutation symbol, satisfying

ijk ={ 1i,j,k=1,2,3;2,3,1or3,1,2 1i,j,k=3,2,1;2,1,3or1,3,2 0otherwise [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgIGiopaaBaaaleaacaWGPbGaamOAai aadUgaaeqaaOGaeyypa0ZaaiqaaqaabeqaaiaaigdacaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGPb GaaiilaiaadQgacaGGSaGaam4Aaiabg2da9iaaigdacaGGSaGaaGOm aiaacYcacaaIZaGaai4oaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGOmaiaacYcacaaIZaGaaiilaiaaigdacaaMc8UaaGPaVlaaykW7 caqGVbGaaeOCaiaaykW7caaMc8UaaGPaVlaaiodacaGGSaGaaGymai aacYcacaaIYaaabaGaeyOeI0IaaGymaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaamyAaiaacYcacaWGQbGaaiilaiaadUgacqGH9aqpca aMc8UaaG4maiaacYcacaaIYaGaaiilaiaaigdacaGG7aGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaIYaGaaiilaiaaigdacaGGSaGaaG 4maiaaykW7caaMc8UaaGPaVlaaykW7caqGVbGaaeOCaiaaykW7caaM c8UaaGPaVlaabgdacaqGSaGaae4maiaabYcacaqGYaaabaGaaeimai aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caqGVbGa aeiDaiaabIgacaqGLbGaaeOCaiaabEhacaqGPbGaae4Caiaabwgaaa [email protected]@

 General homogeneous deformation

y 1 = A 11 x 1 + A 12 x 2 + A 13 x 3 + c 1 y 2 = A 21 x 1 + A 22 x 2 + A 23 x 3 + c 2 y 3 = A 31 x 1 + A 32 x 2 + A 3 x 3 + c 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadMhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcaWGbbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaa dIhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGbbWaaSbaaSqaai aaigdacaaIYaaabeaakiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGH RaWkcaWGbbWaaSbaaSqaaiaaigdacaaIZaaabeaakiaadIhadaWgaa WcbaGaaG4maaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaaigdaaeqa aaGcbaGaamyEamaaBaaaleaacaaIYaaabeaakiabg2da9iaadgeada WgaaWcbaGaaGOmaiaaigdaaeqaaOGaamiEamaaBaaaleaacaaIXaaa beaakiabgUcaRiaadgeadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaam iEamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadgeadaWgaaWcbaGa aGOmaiaaiodaaeqaaOGaamiEamaaBaaaleaacaaIZaaabeaakiabgU caRiaadogadaWgaaWcbaGaaGOmaaqabaaakeaacaWG5bWaaSbaaSqa aiaaiodaaeqaaOGaeyypa0JaamyqamaaBaaaleaacaaIZaGaaGymaa qabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyqamaa BaaaleaacaaIZaGaaGOmaaqabaGccaWG4bWaaSbaaSqaaiaaikdaae qaaOGaey4kaSIaamyqamaaBaaaleaacaaIZaaabeaakiaadIhadaWg aaWcbaGaaG4maaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaaiodaae [email protected]@

or

y=Ax+c y i = A ij x j + c i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH5bGaeyypa0JaaCyqaiabgwSixl aahIhacqGHRaWkcaWHJbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaamyEamaaBaaaleaacaWGPbaabeaakiabg2da9iaadgeada WgaaWcbaGaamyAaiaadQgaaeqaaOGaamiEamaaBaaaleaacaWGQbaa [email protected]@

where A ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbWaaSbaaSqaaiaadMgacaWGQb [email protected]@  are constants.

The physical significance of a homogeneous deformation is that all straight lines in the solid remain straight under the deformation.  Thus, every point in the solid experiences the same shape change.  All the deformations listed above are examples of homogeneous deformations.

 

 

 

 

 

 

 

3.4 Eulerian and Lagrangian descriptions of motion and deformation.

 

The displacement and velocity are vector valued functions.   In any application, we have a choice of writing the vectors as functions of the position of material particles before deformation x i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@

y i = x i + u i ( x j ,t) y i t | x i =const = u i t = v i ( x j ,t) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamyDamaa BaaaleaacaWGPbaabeaakiaacIcacaWG4bWaaSbaaSqaaiaadQgaae qaaOGaaiilaiaadshacaGGPaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7daabcaqaamaalaaabaGaeyOaIyRaamyEamaaBaaaleaa caWGPbaabeaaaOqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcba GaamiEamaaBaaameaacaWGPbaabeaaliabg2da9iaabogacaqGVbGa aeOBaiaabohacaqG0baabeaakiabg2da9maalaaabaGaeyOaIyRaam yDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadshaaaGaeyyp a0JaamODamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bWaaSbaaS [email protected]@

This is called the lagrangean description of motion.  It is usually the easiest way to visualize a deformation.

 

But in some applications (eg fluid flow problems, where it’s hard to identify a reference configuration) it is preferable to write the displacement, velocity and acceleration vectors as functions of the deformed position of particles.  

y i = x i + u i ( y j ,t) y i t | x i =const = v i ( y j ,t) 2 y i t 2 | x i =const = a i ( y j ,t) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamyDamaa BaaaleaacaWGPbaabeaakiaacIcacaWG5bWaaSbaaSqaaiaadQgaae qaaOGaaiilaiaadshacaGGPaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daabcaqaam aalaaabaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabeaaaOqaaiab gkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaamiEamaaBaaameaaca WGPbaabeaaliabg2da9iaabogacaqGVbGaaeOBaiaabohacaqG0baa beaakiabg2da9iaadAhadaWgaaWcbaGaamyAaaqabaGccaGGOaGaam yEamaaBaaaleaacaWGQbaabeaakiaacYcacaWG0bGaaiykaiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaqGaaeaadaWcaaqaaiab gkGi2oaaCaaaleqabaGaaGOmaaaakiaadMhadaWgaaWcbaGaamyAaa qabaaakeaacqGHciITcaWG0bWaaWbaaSqabeaacaaIYaaaaaaaaOGa ayjcSdWaaSbaaSqaaiaadIhadaWgaaadbaGaamyAaaqabaWccqGH9a qpcaqGJbGaae4Baiaab6gacaqGZbGaaeiDaaqabaGccqGH9aqpcaWG HbWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadMhadaWgaaWcbaGaam [email protected]@

These express displacement, velocity and displacement as functions of a particular point in space (visualize describing air flow, for example).  This is called the Eulerian description of motion. Of course the functions of x i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  and y i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  are not the same [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzagaeaa [email protected]@  we just run out of symbols if we introduce different variables in the Lagrangian and Eulerian descriptions.      

 

The relationships between displacement, velocity, and acceleration are somewhat more complicated in the Eulerian description.  In the laws of motion, we normally are interested in the velocity and acceleration of a particular material particle, rather the rate of change of displacement and velocity at a particular point in space.  When computing the time derivatives, it is necessary to take into account that y i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  is a function of time.  Thus, displacement, velocity and acceleration are related by

( δ ij u i y k ) y k t | x i =const = u i t | y i =const 2 y i t 2 | x i =const = a i ( y j ,t)= v i t | y i =const + v k ( y j ,t) v i y k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaGaaGPaVpaabmaabaGaeqiTdq 2aaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTmaalaaabaGaeyOa IyRaamyDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadMhada WgaaWcbaGaam4AaaqabaaaaaGccaGLOaGaayzkaaWaaSaaaeaacqGH ciITcaWG5bWaaSbaaSqaaiaadUgaaeqaaaGcbaGaeyOaIyRaamiDaa aaaiaawIa7amaaBaaaleaacaWG4bWaaSbaaWqaaiaadMgaaeqaaSGa eyypa0Jaae4yaiaab+gacaqGUbGaae4CaiaabshaaeqaaOGaeyypa0 ZaaqGaaeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaaqa baaakeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaadMhada WgaaadbaGaamyAaaqabaWccqGH9aqpcaqGJbGaae4Baiaab6gacaqG ZbGaaeiDaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVpaaeiaabaWaaSaaaeaacqGHciITdaahaa WcbeqaaiaaikdaaaGccaWG5bWaaSbaaSqaaiaadMgaaeqaaaGcbaGa eyOaIyRaamiDamaaCaaaleqabaGaaGOmaaaaaaaakiaawIa7amaaBa aaleaacaWG4bWaaSbaaWqaaiaadMgaaeqaaSGaeyypa0Jaae4yaiaa b+gacaqGUbGaae4CaiaabshaaeqaaOGaeyypa0JaamyyamaaBaaale aacaWGPbaabeaakiaacIcacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGa aiilaiaadshacaGGPaGaeyypa0ZaaqGaaeaadaWcaaqaaiabgkGi2k aadAhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG0baaaaGa ayjcSdWaaSbaaSqaaiaadMhadaWgaaadbaGaamyAaaqabaWccqGH9a qpcaqGJbGaae4Baiaab6gacaqGZbGaaeiDaaqabaGccqGHRaWkcaWG 2bWaaSbaaSqaaiaadUgaaeqaaOGaaiikaiaadMhadaWgaaWcbaGaam OAaaqabaGccaGGSaGaamiDaiaacMcadaWcaaqaaiabgkGi2kaadAha daWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaai [email protected]@

You can derive these results by a simple application of the chain rule.

 

 

3.5 The Displacement gradient and Deformation gradient tensors

 

These quantities are defined by

 Displacement Gradient Tensor:   u [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  is a tensor with components u i x k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaaykW7caWG1b WaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaa [email protected]@

 Deformation Gradient Tensor: F=I+uor in Cartesian components  F ik = δ ik + u i x k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCysaiabgUcaRi abgEGirlaahwhacaaMc8UaaGPaVlaaykW7caaMc8Uaae4Baiaabkha caqGGaGaaeyAaiaab6gacaqGGaGaae4qaiaabggacaqGYbGaaeiDai aabwgacaqGZbGaaeyAaiaabggacaqGUbGaaeiiaiaabogacaqGVbGa aeyBaiaabchacaqGVbGaaeOBaiaabwgacaqGUbGaaeiDaiaabohaca qGGaGaaGPaVlaaykW7caWGgbWaaSbaaSqaaiaadMgacaWGRbaabeaa kiabg2da9iabes7aKnaaBaaaleaacaWGPbGaam4AaaqabaGccqGHRa WkdaWcaaqaaiabgkGi2kaaykW7caWG1bWaaSbaaSqaaiaadMgaaeqa aaGcbaGaeyOaIyRaaGPaVlaadIhadaWgaaWcbaGaam4Aaaqabaaaaa [email protected]@

where I  is the identity tensor, with components described by the Kronekor delta symbol:

δ ik ={ 1,i=k 0,ik [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazdaWgaaWcbaGaamyAaiaadU gaaeqaaOGaeyypa0ZaaiqaaqaabeqaaiaaigdacaGGSaGaaGPaVlaa ykW7caaMc8UaamyAaiabg2da9iaadUgaaeaacaaIWaGaaiilaiaayk [email protected]@

and [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  represents the gradient operator. Formally, the gradient of a vector field u(x) is defined so that

[ u ]n= Lim α0 u(x+αn)u(x) α [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaGaey4bIeTaaCyDaaGaay5wai aaw2faaiabgwSixlaah6gacqGH9aqpdaWfqaqaaiaabYeacaqGPbGa aeyBaaWcbaGaeqySdeMaeyOKH4QaaeimaaqabaGcdaWcaaqaaiaahw hacaGGOaGaaCiEaiabgUcaRiabeg7aHjaah6gacaGGPaGaeyOeI0Ia [email protected]@

 but in practice the component formula u i / x j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaa qabaGccaG[email protected]@  is more useful.

 

Note also that

y=( x+u(x) )=I+F or    y i x j = x j ( x i + u i )= δ ij + u i x j = F ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaaceqaaiabgEGirlaahMhacqGH9aqpcq GHhis0daqadaqaaiaahIhacqGHRaWkcaWH1bGaaiikaiaahIhacaGG PaaacaGLOaGaayzkaaGaeyypa0JaaCysaiabgUcaRiaahAeaaeaaca qGVbGaaeOCaiaabccacaqGGaGaaeiiamaalaaabaGaeyOaIyRaamyE amaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcba GaamOAaaqabaaaaOGaeyypa0ZaaSaaaeaacqGHciITaeaacqGHciIT caWG4bWaaSbaaSqaaiaadQgaaeqaaaaakmaabmaabaGaamiEamaaBa aaleaacaWGPbaabeaakiabgUcaRiaadwhadaWgaaWcbaGaamyAaaqa baaakiaawIcacaGLPaaacqGH9aqpcqaH0oazdaWgaaWcbaGaamyAai aadQgaaeqaaOGaey4kaSYaaSaaaeaacqGHciITcaaMc8UaamyDamaa BaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaam OAaaqabaaaaOGaeyypa0JaamOramaaBaaaleaacaWGPbGaamOAaaqa [email protected]@

 

The concepts of displacement gradient and deformation gradient are introduced to quantify the change in shape of infinitesimal line elements in a solid body. To see this, imagine drawing a straight line on the undeformed configuration of a solid, as shown in the figure.  The line would be mapped to a smooth curve on the deformed configuration.  However, suppose we focus attention on a line segment dx, much shorter than the radius of curvature of this curve, as shown.  The segment would be straight in the undeformed configuration, and would also be (almost) straight in the deformed configuration.  Thus, no matter how complex a deformation we impose on a solid, infinitesimal line segments are merely stretched and rotated by a deformation.The infinitesimal line segments dx and dy are related by

dy=Fdxor   d y i = F ik d x k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCyEaiabg2da9iaahAeacq GHflY1caWGKbGaaCiEaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua ae4BaiaabkhacaqGGaGaaeiiaiaabccacaWGKbGaamyEamaaBaaale aacaWGPbaabeaakiabg2da9iaadAeadaWgaaWcbaGaamyAaiaadUga [email protected]@

Written out as a matrix equation, we have

[ d y 1 d y 2 d y 3 ]=[ 1+ u 1 x 1 u 1 x 2 u 1 x 3 u 2 x 1 1+ u 2 x 2 u 2 x 3 u 3 x 1 u 3 x 2 1+ u 3 x 3 ][ d x 1 d x 2 d x 3 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadeaaaeaacaWGKb GaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG5bWaaSba aSqaaiaaikdaaeqaaaGcbaGaamizaiaadMhadaWgaaWcbaGaaG4maa qabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaamWaaeaafaqabeWadaaa baGaaGymaiabgUcaRmaalaaabaGaeyOaIyRaamyDamaaBaaaleaaca aIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaa aaGcbaWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaigdaaeqaaa GcbaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaaakeaadaWc aaqaaiabgkGi2kaadwhadaWgaaWcbaGaaGymaaqabaaakeaacqGHci ITcaWG4bWaaSbaaSqaaiaaiodaaeqaaaaaaOqaamaalaaabaGaeyOa IyRaamyDamaaBaaaleaacaaIYaaabeaaaOqaaiabgkGi2kaadIhada WgaaWcbaGaaGymaaqabaaaaaGcbaGaaGymaiabgUcaRmaalaaabaGa eyOaIyRaamyDamaaBaaaleaacaaIYaaabeaaaOqaaiabgkGi2kaadI hadaWgaaWcbaGaaGOmaaqabaaaaaGcbaWaaSaaaeaacqGHciITcaWG 1bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaale aacaaIZaaabeaaaaaakeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWc baGaaG4maaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaae qaaaaaaOqaamaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaIZaaa beaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaaGcba GaaGymaiabgUcaRmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaI ZaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaG4maaqabaaaaa aaaOGaay5waiaaw2faamaadmaabaqbaeqabmqaaaqaaiaadsgacaWG 4bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamizaiaadIhadaWgaaWcba GaaGOmaaqabaaakeaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaa [email protected]@

 

To derive this result, consider an infinitesimal line element dx in a deforming solid.  When the solid is deformed, this line element is stretched and rotated to a deformed line element dy.  If we know the displacement field in the solid, we can compute dy=[x+dx+u(x+dx)]-[x+u(x)] from the position vectors of its two end points

d y i = x i +d x i + u i ( x k +d x k )( x i + u i ( x k )) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyEamaaBaaaleaacaWGPb aabeaakiabg2da9iaadIhadaWgaaWcbaGaamyAaaqabaGccqGHRaWk caWGKbGaamiEamaaBaaaleaacaWGPbaabeaakiabgUcaRiaadwhada WgaaWcbaGaamyAaaqabaGccaGGOaGaamiEamaaBaaaleaacaWGRbaa beaakiabgUcaRiaadsgacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaai ykaiabgkHiTiaacIcacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaey4k aSIaamyDamaaBaaaleaacaWGPbaabeaakiaacIcacaWG4bWaaSbaaS [email protected]@

Expand u i ( x k +d x k ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcaWGKbGa [email protected]@  as a Taylor series

u i ( x k +d x k ) u i ( x k )+ u i x k d x k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcaWGKbGa amiEamaaBaaaleaacaWGRbaabeaakiaacMcacqGHijYUcaWG1bWaaS baaSqaaiaadMgaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaam4Aaaqa baGccaGGPaGaey4kaSYaaSaaaeaacqGHciITcaaMc8UaamyDamaaBa aaleaacaWGPbaabeaaaOqaaiabgkGi2kaaykW7caWG4bWaaSbaaSqa aiaadUgaaeqaaaaakiaadsgacaWG4bWaaSbaaSqaaiaadUgaaeqaaa [email protected]@

so that

d y i =d x i + u i x k d x k =( δ ik + u i x k )d x k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyEamaaBaaaleaacaWGPb aabeaakiabg2da9iaadsgacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGa ey4kaSYaaSaaaeaacqGHciITcaaMc8UaamyDamaaBaaaleaacaWGPb aabeaaaOqaaiabgkGi2kaaykW7caWG4bWaaSbaaSqaaiaadUgaaeqa aaaakiaadsgacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaeyypa0Zaae WaaeaacqaH0oazdaWgaaWcbaGaamyAaiaadUgaaeqaaOGaey4kaSYa aSaaaeaacqGHciITcaaMc8UaamyDamaaBaaaleaacaWGPbaabeaaaO qaaiabgkGi2kaaykW7caWG4bWaaSbaaSqaaiaadUgaaeqaaaaaaOGa [email protected]@

We identify the term in parentheses as the deformation gradient, so

d y i = F ik d x k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyEamaaBaaaleaacaWGPb aabeaakiabg2da9iaadAeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGa [email protected]@

 

The inverse of the deformation gradient F 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbWaaWbaaSqabeaacqGHsislca [email protected]@  arises in many calculations.  It is defined through

d x i = F ik 1 d y k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamiEamaaBaaaleaacaWGPb aabeaakiabg2da9iaadAeadaqhaaWcbaGaamyAaiaadUgaaeaacqGH [email protected]@

or alternatively

F ij 1 = x i y j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaa0baaSqaaiaadMgacaWGQb aabaGaeyOeI0IaaGymaaaakiabg2da9maalaaabaGaeyOaIyRaamiE amaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcba [email protected]@

 

 

 

3.6 Deformation gradient resulting from two successive deformations

 

Suppose that two successive deformations are applied to a solid, as shown.  Let

dy= F (1) dxdz= F (2) dyord y i = F ij (1) d x j d z i = F ij (2) d y j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCyEaiabg2da9iaahAeada ahaaWcbeqaaiaacIcacaaIXaGaaiykaaaakiabgwSixlaadsgacaWH 4bGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caWGKbGaaCOEaiabg2da9iaahAeadaahaaWcbeqaaiaacIcacaaI YaGaaiykaaaakiabgwSixlaadsgacaWH5bGaaGPaVlaaykW7caaMc8 UaaGPaVlaab+gacaqGYbGaaGPaVlaaykW7caaMc8UaamizaiaadMha daWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGgbWaa0baaSqaaiaadM gacaWGQbaabaGaaiikaiaaigdacaGGPaaaaOGaamizaiaadIhadaWg aaWcbaGaamOAaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamizaiaadQhadaWgaaWc baGaamyAaaqabaGccqGH9aqpcaWGgbWaa0baaSqaaiaadMgacaWGQb aabaGaaiikaiaaikdacaGGPaaaaOGaamizaiaadMhadaWgaaWcbaGa [email protected]@

map infinitesimal line elements from the original configuration to the first deformed shape, and from the first deformed shape to the second, respectively, with

F (1) =y x F (2) =z y or F ij (1) = y i x j F ij (2) = z i y j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbWaaWbaaSqabeaacaGGOaGaaG ymaiaacMcaaaGccqGH9aqpcaWH5bGaey4LIqSaey4bIe9aaSbaaSqa aiaahIhaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWHgb WaaWbaaSqabeaacaGGOaGaaGOmaiaacMcaaaGccqGH9aqpcaWH6bGa ey4LIqSaey4bIe9aaSbaaSqaaiaahMhaaeqaaOGaaGPaVlaaykW7ca aMc8Uaae4BaiaabkhacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWGgbWaa0baaSqaaiaadMgacaWGQbaabaGaaiikaiaaigdaca GGPaaaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG5bWaaSbaaSqaaiaa dMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaa GccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caWGgbWaa0baaSqaaiaadMgacaWGQbaabaGaaiikaiaaik dacaGGPaaaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG6bWaaSbaaSqa aiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGQbaabe [email protected]@

The deformation gradient that maps infinitesimal line elements from the original configuration directly to the second deformed shape then follows as

dz=Fdxwith  F= F (2) F (1) ord z i = F ij d x j F ij = F ik (2) F kj (1) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCOEaiabg2da9iaahAeacq GHflY1caWGKbGaaCiEaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua ae4DaiaabMgacaqG0bGaaeiAaiaabccacaqGGaGaaCOraiabg2da9i aahAeadaahaaWcbeqaaiaacIcacaaIYaGaaiykaaaakiabgwSixlaa hAeadaahaaWcbeqaaiaacIcacaaIXaGaaiykaaaakiaaykW7caaMc8 UaaGPaVlaab+gacaqGYbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaamizaiaadQhadaWgaaWcbaGaamyAaaqabaGccqGH9aqpca WGgbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaadsgacaWG4bWaaSba aSqaaiaadQgaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaa kiabg2da9iaadAeadaqhaaWcbaGaamyAaiaadUgaaeaacaGGOaGaaG OmaiaacMcaaaGccaWGgbWaa0baaSqaaiaadUgacaWGQbaabaGaaiik [email protected]@

Thus, the cumulative deformation gradient due to two successive deformations follows by multiplying their individual deformation gradients.

 

To see this, write the cumulative mapping as z i ( y j ( x k ) ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG6bWaaSbaaSqaaiaadMgaaeqaaO WaaeWaaeaacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGaaiikaiaadIha [email protected]@  and apply the chain rule

d z i = z i y j y j x k d x k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamOEamaaBaaaleaacaWGPb aabeaakiabg2da9maalaaabaGaeyOaIyRaamOEamaaBaaaleaacaWG PbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaaaO WaaSaaaeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaaGcbaGa eyOaIyRaamiEamaaBaaaleaacaWGRbaabeaaaaGccaWGKbGaamiEam [email protected]@

 

 

 

  

 

3.7 The Jacobian of the deformation gradient [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa [email protected]@  change of volume

 

The Jacobian is defined as

J=det( F )=det( δ ij + u i x j ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeyypa0Jaciizaiaacwgaca GG0bWaaeWaaeaacaWHgbaacaGLOaGaayzkaaGaeyypa0Jaciizaiaa cwgacaGG0bWaaeWaaeaacqaH0oazdaWgaaWcbaGaamyAaiaadQgaae qaaOGaey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaadMga aeqaaaGcbaGaeyOaIyRaaGPaVlaadIhadaWgaaWcbaGaamOAaaqaba [email protected]@

It is a measure of the volume change produced by a deformation.  To see this, consider the infinitessimal volume element shown with sides dx, dy, and dz in the figure above. The original volume of the element is

d V 0 =dz(dx×dy)= ijk d z i d x j d y k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamOvamaaBaaaleaacaaIWa aabeaakiabg2da9iaadsgacaWH6bGaeyyXICTaaiikaiaadsgacaWH 4bGaey41aqRaamizaiaahMhacaGGPaGaeyypa0JaeyicI48aaSbaaS qaaiaadMgacaWGQbGaam4AaaqabaGccaWGKbGaamOEamaaBaaaleaa caWGPbaabeaakiaadsgacaWG4bWaaSbaaSqaaiaadQgaaeqaaOGaam [email protected]@

Here, ijk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIGiopaaBa [email protected]@  is the permutation symbol. The element is mapped to a paralellepiped with sides dr, dv, and dw with volume given by

dV= ijk d w i d r j d v k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamOvaiabg2da9iabgIGiop aaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaamizaiaadEhadaWg aaWcbaGaamyAaaqabaGccaWGKbGaamOCamaaBaaaleaacaWGQbaabe [email protected]@

Recall that

d r i = F il d x l ,d v j = F jm d y m ,d w k = F kn d z n [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamOCamaaBaaaleaacaWGPb aabeaakiabg2da9iaadAeadaWgaaWcbaGaamyAaiaadYgaaeqaaOGa amizaiaadIhadaWgaaWcbaGaamiBaaqabaGccaGGSaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caWGKbGaamODamaaBaaaleaacaWGQbaa beaakiabg2da9iaadAeadaWgaaWcbaGaamOAaiaad2gaaeqaaOGaam izaiaadMhadaWgaaWcbaGaamyBaaqabaGccaGGSaGaaGPaVlaaykW7 caaMc8UaamizaiaadEhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpca WGgbWaaSbaaSqaaiaadUgacaWGUbaabeaakiaadsgacaWG6bWaaSba [email protected]@

so that

dV= ijk F il d x l F jm d y m F kn d z n = ijk F il F jm F kn d x l d y m d z n [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamOvaiabg2da9iabgIGiop aaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaamOramaaBaaaleaa caWGPbGaamiBaaqabaGccaWGKbGaamiEamaaBaaaleaacaWGSbaabe aakiaadAeadaWgaaWcbaGaamOAaiaad2gaaeqaaOGaamizaiaadMha daWgaaWcbaGaamyBaaqabaGccaWGgbWaaSbaaSqaaiaadUgacaWGUb aabeaakiaadsgacaWG6bWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0Ja eyicI48aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGccaWGgbWaaS baaSqaaiaadMgacaWGSbaabeaakiaadAeadaWgaaWcbaGaamOAaiaa d2gaaeqaaOGaamOramaaBaaaleaacaWGRbGaamOBaaqabaGccaWGKb GaamiEamaaBaaaleaacaWGSbaabeaakiaadsgacaWG5bWaaSbaaSqa [email protected]@

Recall that

ijk A il A jm A kn = lmn det(A) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHiiIZdaWgaaWcbaGaamyAaiaadQ gacaWGRbaabeaakiaadgeadaWgaaWcbaGaamyAaiaadYgaaeqaaOGa amyqamaaBaaaleaacaWGQbGaamyBaaqabaGccaWGbbWaaSbaaSqaai aadUgacaWGUbaabeaakiabg2da9iabgIGiopaaBaaaleaacaWGSbGa amyBaiaad6gaaeqaaOGaciizaiaacwgacaGG0bGaaiikaiaahgeaca [email protected]@

so that

dV=det(F) lmn d x l d y m d z n =det( F )d V 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamOvaiabg2da9iGacsgaca GGLbGaaiiDaiaacIcacaWHgbGaaiykaiabgIGiopaaBaaaleaacaWG SbGaamyBaiaad6gaaeqaaOGaamizaiaadIhadaWgaaWcbaGaamiBaa qabaGccaWGKbGaamyEamaaBaaaleaacaWGTbaabeaakiaadsgacaWG 6bWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaciizaiaacwgacaGG0b WaaeWaaeaacaWHgbaacaGLOaGaayzkaaGaamizaiaadAfadaWgaaWc [email protected]@

Hence

dV d V 0 =det( F )=J [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWGwbaabaGaam izaiaadAfadaWgaaWcbaGaaGimaaqabaaaaOGaeyypa0Jaciizaiaa cwgacaGG0bWaaeWaaeaacaWHgbaacaGLOaGaayzkaaGaeyypa0Jaam [email protected]@

Observe that

 For any physically admissible deformation, the volume of the deformed element must be positive (no matter how much you deform a solid, you can’t make material disappear).  Therefore, all physically admissible displacement fields must satisfy J>0

 If a material is incompressible, its volume remains constant.  This requires J=1.

 If the mass density of the material at a point in the undeformed solid is ρ 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba [email protected]@ , its mass density in the deformed solid is ρ= ρ 0 /J [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCcqGH9aqpcqaHbpGCdaWgaa [email protected]@

 

Derivatives of J. When working with constitutive equations, it is occasionally necessary to evaluate derivatives of J with respect to the components of F.  The following result (which can be proved e.g. by expanding the Jacobian using index notation [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  see HW1, problem 7, eg) is extremely useful

J F ij =J F ji 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadQeaaeaacq GHciITcaWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaaaaGccqGH9aqp caWGkbGaamOramaaDaaaleaacaWGQbGaamyAaaqaaiabgkHiTiaaig [email protected]@

 

 

 

 

 

3.8 Transformation of internal surface area elements

 

 

When we deal with internal forces in a solid, we need to work with forces acting on internal surfaces in a solid.  An important question arises in this treatment: if we identify an element of area d A 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaaicdaae [email protected]@  with normal n 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  in the reference configuration, and then what are the area of dA [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  and normal n [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  of this area element in the deformed solid?

 

The two are related through

dAn=J F T d A 0 n 0 dA n i =J F ki 1 n k 0 d A 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbGaaCOBaiabg2da9iaadQ eacaWHgbWaaWbaaSqabeaacqGHsislcaWGubaaaOGaeyyXICTaamiz aiaadgeadaWgaaWcbaGaaGimaaqabaGccaWHUbWaaSbaaSqaaiaaic daaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadsgacaWGbb GaamOBamaaDaaaleaacaWGPbaabaaaaOGaeyypa0JaamOsaiaadAea daqhaaWcbaGaam4AaiaadMgaaeaacqGHsislcaaIXaaaaOGaamOBam aaDaaaleaacaWGRbaabaGaaGimaaaakiaadsgacaWGbbWaaSbaaSqa [email protected]@

 

To see this,

1.      let d v i 0 ,d w j 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG2bWaa0baaSqaaiaadMgaae aacaaIWaaaaOGaaiilaiaadsgacaWG3bWaa0baaSqaaiaadQgaaeaa [email protected]@  be two infinitesimal material fibers with different orientations at some point in the reference configuration.  These fibers bound a parallelapiped with area and normal

d A 0 n i 0 = ijk d w j 0 d v k 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaaicdaae qaaOGaamOBamaaDaaaleaacaWGPbaabaGaaGimaaaakiabg2da9iab gIGiopaaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaamizaiaadE hadaqhaaWcbaGaamOAaaqaaiaaicdaaaGccaWGKbGaamODamaaDaaa [email protected]@

2.      The vectors map to d v j = F jq d v q 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG2bWaaSbaaSqaaiaadQgaae qaaOGaeyypa0JaamOramaaBaaaleaacaWGQbGaamyCaaqabaGccaWG [email protected]@ d w k = F kn d w n 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG3bWaaSbaaSqaaiaadUgaae qaaOGaeyypa0JaamOramaaBaaaleaacaWGRbGaamOBaaqabaGccaWG [email protected]@  in the deformed solid

3.      In the deformed solid the area element is thus

dA n i = ijk F jq d v q 0 F kn d w n 0 = F ml F li 1 mjk F jq F kn d v q 0 d w n 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbGaamOBamaaBaaaleaaca WGPbaabeaakiabg2da9iabgIGiopaaBaaaleaacaWGPbGaamOAaiaa dUgaaeqaaOGaamOramaaBaaaleaacaWGQbGaamyCaaqabaGccaWGKb GaamODamaaDaaaleaacaWGXbaabaGaaGimaaaakiaadAeadaWgaaWc baGaam4Aaiaad6gaaeqaaOGaamizaiaadEhadaqhaaWcbaGaamOBaa qaaiaaicdaaaGccqGH9aqpcaWGgbWaaSbaaSqaaiaad2gacaWGSbaa beaakiaadAeadaqhaaWcbaGaamiBaiaadMgaaeaacqGHsislcaaIXa aaaOGaeyicI48aaSbaaSqaaiaad2gacaWGQbGaam4AaaqabaGccaWG gbWaaSbaaSqaaiaadQgacaWGXbaabeaakiaadAeadaWgaaWcbaGaam 4Aaiaad6gaaeqaaOGaamizaiaadAhadaqhaaWcbaGaamyCaaqaaiaa icdaaaGccaWGKbGaam4DamaaDaaaleaacaWGUbaabaGaaGimaaaaaa [email protected]@

4.      Recall the identity lmn det(A)= ijk A il A jm A kn [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHiiIZdaWgaaWcbaGaamiBaiaad2 gacaWGUbaabeaakiGacsgacaGGLbGaaiiDaiaacIcacaWHbbGaaiyk aiaaykW7caaMc8Uaeyypa0JaaGPaVlaaykW7caaMc8UaeyicI48aaS baaSqaaiaadMgacaWGQbGaam4AaaqabaGccaWGbbWaaSbaaSqaaiaa dMgacaWGSbaabeaakiaadgeadaWgaaWcbaGaamOAaiaad2gaaeqaaO [email protected]@  - so dA n i = F li 1 lqn Jd v q 0 d w n 0 =J F ki 1 n k 0 d A 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbGaamOBamaaBaaaleaaca WGPbaabeaakiabg2da9iaadAeadaqhaaWcbaGaamiBaiaadMgaaeaa cqGHsislcaaIXaaaaOGaeyicI48aaSbaaSqaaiaadYgacaWGXbGaam OBaaqabaGccaWGkbGaamizaiaadAhadaqhaaWcbaGaamyCaaqaaiaa icdaaaGccaWGKbGaam4DamaaDaaaleaacaWGUbaabaGaaGimaaaaki abg2da9iaadQeacaWGgbWaa0baaSqaaiaadUgacaWGPbaabaGaeyOe I0IaaGymaaaakiaad6gadaqhaaWcbaGaam4AaaqaaiaaicdaaaGcca [email protected]@

 

 

3.8 The Lagrange strain tensor

 

The Lagrange strain tensor is defined as

E= 1 2 ( F T FI)or E ij = 1 2 ( F ki F kj δ ij ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHfbGaeyypa0ZaaSaaaeaacaaIXa aabaGaaGOmaaaacaGGOaGaaCOramaaCaaaleqabaGaamivaaaakiab gwSixlaahAeacqGHsislcaWHjbGaaiykaiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaab+gacaqGYbGaaGPaVlaaykW7caaMc8Ua aGPaVlaadweadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0ZaaS aaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaamOramaaBaaaleaacaWG RbGaamyAaaqabaGccaWGgbWaaSbaaSqaaiaadUgacaWGQbaabeaaki abgkHiTiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaaa [email protected]@

The components of Lagrange strain can also be expressed in terms of the displacement gradient as

E ij = 1 2 ( u i x j + u j x i + u k x j u k x i ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaa daWcaaqaaiabgkGi2kaaykW7caWG1bWaaSbaaSqaaiaadMgaaeqaaa GcbaGaeyOaIyRaaGPaVlaadIhadaWgaaWcbaGaamOAaaqabaaaaOGa ey4kaSYaaSaaaeaacqGHciITcaaMc8UaamyDamaaBaaaleaacaWGQb aabeaaaOqaaiabgkGi2kaaykW7caWG4bWaaSbaaSqaaiaadMgaaeqa aaaakiabgUcaRmaalaaabaGaeyOaIyRaaGPaVlaadwhadaWgaaWcba Gaam4AaaqabaaakeaacqGHciITcaaMc8UaamiEamaaBaaaleaacaWG QbaabeaaaaGcdaWcaaqaaiabgkGi2kaaykW7caWG1bWaaSbaaSqaai aadUgaaeqaaaGcbaGaeyOaIyRaaGPaVlaadIhadaWgaaWcbaGaamyA [email protected]@

 

The Lagrange strain tensor quantifies the changes in length of a material fiber, and angles between pairs of fibers in a deformable solid.  It is used in calculations where large shape changes are expected.

 

To visualize the physical significance of E, suppose we mark out an imaginary tensile specimen with (very short) length l 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYgadaWgaa [email protected]@  on our deforming solid, as shown in the picture. The orientation of the specimen is arbitrary, and is specified by a unit vector m, with components m i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGTbWaaSbaaSqaaiaadMgaaeqaaa [email protected]@ .  Upon deformation, the specimen increases in length to l= l 0 +δl [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGSbGaeyypa0JaamiBamaaBaaale [email protected]@ . Define the strain of the specimen as

ε L ( m i )= l 2 l 0 2 2 l 0 2 = δl l 0 + ( δl ) 2 2 l 0 2 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamitaaqaba GccaGGOaGaamyBamaaBaaaleaacaWGPbaabeaakiaacMcacqGH9aqp daWcaaqaaiaadYgadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGSb WaaSbaaSqaaiaaicdaaeqaaOWaaWbaaSqabeaacaaIYaaaaaGcbaGa aGOmaiaadYgadaWgaaWcbaGaaGimaaqabaGcdaahaaWcbeqaaiaaik daaaaaaOGaeyypa0ZaaSaaaeaacqaH0oazcaWGSbaabaGaamiBamaa BaaaleaacaaIWaaabeaaaaGccqGHRaWkdaWcaaqaamaabmaabaGaeq iTdqMaamiBaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqa [email protected]@

Note that this definition of strain is similar to the definition ε=δl/ l 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzcqGH9aqpcqaH0oazcaWGSb [email protected]@  you are familiar with, but contains an additional term.  The additional term is negligible for small δl [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@ . Given the Lagrange strain components E ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbWaaSbaaSqaaiaadMgacaWGQb [email protected]@ , the strain of the specimen may be computed from

ε L (m)=mEmor    ε L ( m i )= E ij m i m j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamitaaqaba GccaGGOaGaaCyBaiaacMcacqGH9aqpcaWHTbGaeyyXICTaaCyraiab gwSixlaah2gacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaab+gacaqGYbGaaeiiaiaabccacaqGGaGaeqyTdu2a aSbaaSqaaiaadYeaaeqaaOGaaiikaiaad2gadaWgaaWcbaGaamyAaa qabaGccaGGPaGaeyypa0JaamyramaaBaaaleaacaWGPbGaamOAaaqa baGccaWGTbWaaSbaaSqaaiaadMgaaeqaaOGaamyBamaaBaaaleaaca [email protected]@

We proceed to derive this result. Note that

d x i = l 0 m i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamiEamaaBaaaleaacaWGPb aabeaakiabg2da9iaadYgadaWgaaWcbaGaaGimaaqabaGccaWGTbWa [email protected]@

is an infinitesimal vector with length and orientation of our undeformed specimen.  From the preceding section, this vector is stretched and rotated to

d y k =( δ kj + u k x j )d x j =( δ kj + u k x j ) l 0 m j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyEamaaBaaaleaacaWGRb aabeaakiabg2da9maabmaabaGaeqiTdq2aaSbaaSqaaiaadUgacaWG QbaabeaakiabgUcaRmaalaaabaGaeyOaIyRaaGPaVlaadwhadaWgaa WcbaGaam4AaaqabaaakeaacqGHciITcaaMc8UaamiEamaaBaaaleaa caWGQbaabeaaaaaakiaawIcacaGLPaaacaWGKbGaamiEamaaBaaale aacaWGQbaabeaakiabg2da9maabmaabaGaeqiTdq2aaSbaaSqaaiaa dUgacaWGQbaabeaakiabgUcaRmaalaaabaGaeyOaIyRaaGPaVlaadw hadaWgaaWcbaGaam4AaaqabaaakeaacqGHciITcaaMc8UaamiEamaa BaaaleaacaWGQbaabeaaaaaakiaawIcacaGLPaaacaWGSbWaaSbaaS [email protected]@

The length of the deformed specimen is equal to the length of  dy, so we see that

l 2 =d y k d y k =( δ kj + u k x j ) l 0 m j ( δ ki + u k x i ) l 0 m i =( δ ij + u i x j + u j x i + u k x i u k x j ) l 0 2 m j m i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadYgadaahaaWcbeqaaiaaik daaaGccqGH9aqpcaWGKbGaamyEamaaBaaaleaacaWGRbaabeaakiaa dsgacaWG5bWaaSbaaSqaaiaadUgaaeqaaOGaeyypa0ZaaeWaaeaacq aH0oazdaWgaaWcbaGaam4AaiaadQgaaeqaaOGaey4kaSYaaSaaaeaa cqGHciITcaaMc8UaamyDamaaBaaaleaacaWGRbaabeaaaOqaaiabgk Gi2kaaykW7caWG4bWaaSbaaSqaaiaadQgaaeqaaaaaaOGaayjkaiaa wMcaaiaadYgadaWgaaWcbaGaaGimaaqabaGccaWGTbWaaSbaaSqaai aadQgaaeqaaOWaaeWaaeaacqaH0oazdaWgaaWcbaGaam4AaiaadMga aeqaaOGaey4kaSYaaSaaaeaacqGHciITcaaMc8UaamyDamaaBaaale aacaWGRbaabeaaaOqaaiabgkGi2kaaykW7caWG4bWaaSbaaSqaaiaa dMgaaeqaaaaaaOGaayjkaiaawMcaaiaadYgadaWgaaWcbaGaaGimaa qabaGccaWGTbWaaSbaaSqaaiaadMgaaeqaaaGcbaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8Uaeyypa0ZaaeWaaeaacqaH0oazdaWg aaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSYaaSaaaeaacqGHciITca aMc8UaamyDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaaykW7 caWG4bWaaSbaaSqaaiaadQgaaeqaaaaakiabgUcaRmaalaaabaGaey OaIyRaaGPaVlaadwhadaWgaaWcbaGaamOAaaqabaaakeaacqGHciIT caaMc8UaamiEamaaBaaaleaacaWGPbaabeaaaaGccqGHRaWkdaWcaa qaaiabgkGi2kaaykW7caWG1bWaaSbaaSqaaiaadUgaaeqaaaGcbaGa eyOaIyRaaGPaVlaadIhadaWgaaWcbaGaamyAaaqabaaaaOWaaSaaae aacqGHciITcaaMc8UaamyDamaaBaaaleaacaWGRbaabeaaaOqaaiab gkGi2kaaykW7caWG4bWaaSbaaSqaaiaadQgaaeqaaaaaaOGaayjkai aawMcaaiaadYgadaWgaaWcbaGaaGimaaqabaGcdaahaaWcbeqaaiaa ikdaaaGccaWGTbWaaSbaaSqaaiaadQgaaeqaaOGaamyBamaaBaaale [email protected]@

Hence,  the strain for our line element is

ε L ( m i )= l 2 l 0 2 2 l 0 2 = 1 2 ( u i x j + u j x i + u k x j u k x i ) m i m j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamitaaqaba GccaGGOaGaamyBamaaBaaaleaacaWGPbaabeaakiaacMcacqGH9aqp daWcaaqaaiaadYgadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGSb WaaSbaaSqaaiaaicdaaeqaaOWaaWbaaSqabeaacaaIYaaaaaGcbaGa aGOmaiaadYgadaWgaaWcbaGaaGimaaqabaGcdaahaaWcbeqaaiaaik daaaaaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqa amaalaaabaGaeyOaIyRaaGPaVlaadwhadaWgaaWcbaGaamyAaaqaba aakeaacqGHciITcaaMc8UaamiEamaaBaaaleaacaWGQbaabeaaaaGc cqGHRaWkdaWcaaqaaiabgkGi2kaaykW7caWG1bWaaSbaaSqaaiaadQ gaaeqaaaGcbaGaeyOaIyRaaGPaVlaadIhadaWgaaWcbaGaamyAaaqa baaaaOGaey4kaSYaaSaaaeaacqGHciITcaaMc8UaamyDamaaBaaale aacaWGRbaabeaaaOqaaiabgkGi2kaaykW7caWG4bWaaSbaaSqaaiaa dQgaaeqaaaaakmaalaaabaGaeyOaIyRaaGPaVlaadwhadaWgaaWcba Gaam4AaaqabaaakeaacqGHciITcaaMc8UaamiEamaaBaaaleaacaWG PbaabeaaaaaakiaawIcacaGLPaaacaWGTbWaaSbaaSqaaiaadMgaae [email protected]@

giving the results stated.

 

 

 

3.9 The Eulerian strain tensor

 

The Eulerian strain tensor is defined as

E * = 1 2 (I F T F 1 )or E ij * = 1 2 ( δ ij F ki 1 F kj 1 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHfbWaaWbaaSqabeaacaGGQaaaaO Gaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaaCysaiab gkHiTiaahAeadaahaaWcbeqaaiabgkHiTiaadsfaaaGccqGHflY1ca WHgbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaiykaiaaykW7caaM c8UaaGPaVlaab+gacaqGYbGaaGPaVlaaykW7caaMc8UaamyramaaDa aaleaacaWGPbGaamOAaaqaaiaacQcaaaGccqGH9aqpdaWcaaqaaiaa igdaaeaacaaIYaaaaiaacIcacqaH0oazdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyOeI0IaamOramaaDaaaleaacaWGRbGaamyAaaqaaiab gkHiTiaaigdaaaGccaWGgbWaa0baaSqaaiaadUgacaWGQbaabaGaey [email protected]@

Its physical significance is similar to the Lagrange strain tensor, except that it enables you to compute the strain of an infinitesimal line element from its orientation after deformation.

 

Specifically, suppose that n denotes a unit vector parallel to the deformed material fiber, as shown in the picture.  Then

ε E (n)= l 2 l 0 2 2 l 2 =n E * nor    ε E ( n i )= E ij * n i n j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyraaqaba GccaGGOaGaaCOBaiaacMcacqGH9aqpdaWcaaqaaiaadYgadaahaaWc beqaaiaaikdaaaGccqGHsislcaWGSbWaaSbaaSqaaiaaicdaaeqaaO WaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaiaadYgadaahaaWcbeqa aiaaikdaaaaaaOGaeyypa0JaaCOBaiabgwSixlaahweadaahaaWcbe qaaiaacQcaaaGccqGHflY1caWHUbGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caqGVbGaaeOCaiaabccacaqGGa Gaaeiiaiabew7aLnaaBaaaleaacaWGfbaabeaakiaacIcacaWGUbWa aSbaaSqaaiaadMgaaeqaaOGaaiykaiabg2da9iaadweadaqhaaWcba GaamyAaiaadQgaaeaacaGGQaaaaOGaamOBamaaBaaaleaacaWGPbaa [email protected]@

The proof is left as an exercise.

 

 

 

3.10 The Infinitesimal strain tensor

 

The infinitesimal strain tensor is defined as

ε= 1 2 ( u+ ( u ) T )or      ε ij = 1 2 ( u i x j + u j x i ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1oGaeyypa0ZaaSaaaeaacaaIXa aabaGaaGOmaaaadaqadaqaaiaahwhacqGHhis0cqGHRaWkdaqadaqa aiaahwhacqGHhis0aiaawIcacaGLPaaadaahaaWcbeqaaiaadsfaaa aakiaawIcacaGLPaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaab+gacaqGYbGaaeiiaiaabccacaqGGaGaae iiaiaabccacqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyyp a0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGaey OaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIha daWgaaWcbaGaamOAaaqabaaaaOGaey4kaSYaaSaaaeaacqGHciITca WG1bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaa [email protected]@

where u is the displacement vector.  Written out in full

ε ij [ u 1 x 1 1 2 ( u 1 x 2 + u 2 x 1 ) 1 2 ( u 1 x 3 + u 3 x 1 ) 1 2 ( u 2 x 1 + u 1 x 2 ) u 2 x 2 1 2 ( u 2 x 3 + u 3 x 2 ) 1 2 ( u 3 x 1 + u 1 x 3 ) 1 2 ( u 3 x 2 + u 2 x 3 ) u 3 x 3 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlabggMi6kaaykW7caaM c8UaaGPaVpaadmaabaqbaeqabmWaaaqaamaalaaabaGaeyOaIyRaam yDamaaBaaaleaacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWc baGaaGymaaqabaaaaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaada qadaqaamaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaIXaaabeaa aOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOGaey4kaS YaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaikdaaeqaaaGcbaGa eyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaaaakiaawIcacaGLPa aaaeaadaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaWaaSaaaeaa cqGHciITcaWG1bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaam iEamaaBaaaleaacaaIZaaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi 2kaadwhadaWgaaWcbaGaaG4maaqabaaakeaacqGHciITcaWG4bWaaS baaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawMcaaaqaamaalaaabaGa aGymaaqaaiaaikdaaaWaaeWaaeaadaWcaaqaaiabgkGi2kaadwhada WgaaWcbaGaaGOmaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaa igdaaeqaaaaakiabgUcaRmaalaaabaGaeyOaIyRaamyDamaaBaaale aacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqa baaaaaGccaGLOaGaayzkaaaabaWaaSaaaeaacqGHciITcaWG1bWaaS baaSqaaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaI YaaabeaaaaaakeaadaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaaba WaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaikdaaeqaaaGcbaGa eyOaIyRaamiEamaaBaaaleaacaaIZaaabeaaaaGccqGHRaWkdaWcaa qaaiabgkGi2kaadwhadaWgaaWcbaGaaG4maaqabaaakeaacqGHciIT caWG4bWaaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaqaam aalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaadaWcaaqaaiabgkGi 2kaadwhadaWgaaWcbaGaaG4maaqabaaakeaacqGHciITcaWG4bWaaS baaSqaaiaaigdaaeqaaaaakiabgUcaRmaalaaabaGaeyOaIyRaamyD amaaBaaaleaacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcba GaaG4maaqabaaaaaGccaGLOaGaayzkaaaabaWaaSaaaeaacaaIXaaa baGaaGOmaaaadaqadaqaamaalaaabaGaeyOaIyRaamyDamaaBaaale aacaaIZaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqa baaaaOGaey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaik daaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIZaaabeaaaaaa kiaawIcacaGLPaaaaeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWcba GaaG4maaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaiodaaeqa [email protected]@

 

The infinitesimal strain tensor is an approximate deformation measure, which is only valid for small shape changes.  It is more convenient than the Lagrange or Eulerian strain, because it is linear.

 

Specifically, suppose the deformation gradients are small, so that all u i / x j <<1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG1bWaaSbaaSqaaiaadM gaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGc [email protected]@ . Then the Lagrange strain tensor is

E ij = 1 2 ( u i x j + u j x i + u k x j u k x i ) 1 2 ( u i x j + u j x i ) ε ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaa daWcaaqaaiabgkGi2kaaykW7caWG1bWaaSbaaSqaaiaadMgaaeqaaa GcbaGaeyOaIyRaaGPaVlaadIhadaWgaaWcbaGaamOAaaqabaaaaOGa ey4kaSYaaSaaaeaacqGHciITcaaMc8UaamyDamaaBaaaleaacaWGQb aabeaaaOqaaiabgkGi2kaaykW7caWG4bWaaSbaaSqaaiaadMgaaeqa aaaakiabgUcaRmaalaaabaGaeyOaIyRaaGPaVlaadwhadaWgaaWcba Gaam4AaaqabaaakeaacqGHciITcaaMc8UaamiEamaaBaaaleaacaWG QbaabeaaaaGcdaWcaaqaaiabgkGi2kaaykW7caWG1bWaaSbaaSqaai aadUgaaeqaaaGcbaGaeyOaIyRaaGPaVlaadIhadaWgaaWcbaGaamyA aaqabaaaaaGccaGLOaGaayzkaaGaeyisIS7aaSaaaeaacaaIXaaaba GaaGOmaaaadaqadaqaamaalaaabaGaeyOaIyRaaGPaVlaadwhadaWg aaWcbaGaamyAaaqabaaakeaacqGHciITcaaMc8UaamiEamaaBaaale aacaWGQbaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kaaykW7caWG 1bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIyRaaGPaVlaadIhada WgaaWcbaGaamyAaaqabaaaaaGccaGLOaGaayzkaaGaeyisISRaeqyT [email protected]@

so the infinitesimal strain approximates the Lagrange strain.  You can show that it also approximates the Eulerian strain with the same accuracy.

 

Properties of the infinitesimal strain tensor

 

 For small strains, the engineering strain of an infinitesimal fiber aligned with a unit vector m can be estimated as

ε e (m)= l l 0 l 0 ε ij m i m j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyzaaqaba GccaGGOaGaaCyBaiaacMcacqGH9aqpdaWcaaqaaiaadYgacqGHsisl caWGSbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaamiBamaaBaaaleaaca aIWaaabeaaaaGccqGHijYUcqaH1oqzdaWgaaWcbaGaamyAaiaadQga aeqaaOGaamyBamaaBaaaleaacaWGPbaabeaakiaad2gadaWgaaWcba [email protected]@

 Note that

trace( ε ) ε kk = u 1 x 1 + u 2 x 2 + u 3 x 3 = dVd V 0 d V 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqG0bGaaeOCaiaabggacaqGJbGaae yzamaabmaabaGaaCyTdaGaayjkaiaawMcaaiabggMi6kabew7aLnaa BaaaleaacaWGRbGaam4AaaqabaGccqGH9aqpdaWcaaqaaiabgkGi2k aadwhadaWgaaWcbaGaaGymaaqabaaakeaacqGHciITcaWG4bWaaSba aSqaaiaaigdaaeqaaaaakiabgUcaRmaalaaabaGaeyOaIyRaamyDam aaBaaaleaacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGa aGOmaaqabaaaaOGaey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSbaaS qaaiaaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIZaaa beaaaaGccqGH9aqpdaWcaaqaaiaadsgacaWGwbGaeyOeI0Iaamizai aadAfadaWgaaWcbaGaaGimaaqabaaakeaacaWGKbGaamOvamaaBaaa [email protected]@

 (see below for more details)

 The infinitesimal strain tensor is closely related to the strain matrix introduced in elementary strength of materials courses.  For example, the physical significance of the (2 dimensional) strain matrix

[ ε xx γ xy γ yx ε yy ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqaciaaaeaacqaH1o qzdaWgaaWcbaGaamiEaiaadIhaaeqaaaGcbaGaeq4SdC2aaSbaaSqa aiaadIhacaWG5baabeaaaOqaaiabeo7aNnaaBaaaleaacaWG5bGaam iEaaqabaaakeaacqaH1oqzdaWgaaWcbaGaamyEaiaadMhaaeqaaaaa [email protected]@

is illustrated in the figure.

 

To relate this to the infintesimal strain tensor, let { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa [email protected]@  be a Cartesian basis, with e 1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa [email protected]@  parallel to x and e 2 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa [email protected]@  parallel to y as shown.  Let ε ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ [email protected]@  denote the components of the infinitesimal strain tensor in this basis.  Then

ε 11 = ε xx ε 22 = ε yy ε 12 = ε 21 = γ xy /2= γ yx /2 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabew7aLnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaamiEaiaadIha aeqaaaGcbaGaeqyTdu2aaSbaaSqaaiaaikdacaaIYaaabeaakiabg2 da9iabew7aLnaaBaaaleaacaWG5bGaamyEaaqabaaakeaacqaH1oqz daWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaeqyTdu2aaSbaaS qaaiaaikdacaaIXaaabeaakiabg2da9iabeo7aNnaaBaaaleaacaWG 4bGaamyEaaqabaGccaGGVaGaaGOmaiabg2da9iabeo7aNnaaBaaale [email protected]@

 

 

3.11 Engineering shear strains

 

For a general strain tensor (which could be any of E [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@ , E * [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHfbWaaWbaaSqabeaacaGGQaaaaa [email protected]@  or ε [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@ , among others), the diagonal strain components ε 11 , ε 22 , ε 33 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaaGymaiaaig daaeqaaOGaaiilaiabew7aLnaaBaaaleaacaaIYaGaaGOmaaqabaGc [email protected]@  are known as `direct’ strains, while the off diagonal terms  ε 12 = ε 21 ε 13 = ε 31 ε 23 = ε 32 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaaGymaiaaik daaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiaaikdacaaIXaaabeaa kiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabew7aLnaaBa aaleaacaaIXaGaaG4maaqabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGa aG4maiaaigdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7cqaH1oqzdaWgaaWcbaGaaGOmaiaaiodaaeqa [email protected]@  are known as ‘shear strains’

 

The shear strains are sometimes reported as ‘Engineering Shear Strains’ which are related to the formal definition by a factor of 2 i.e.

γ 12 =2 ε 12 γ 13 =2 ε 13 γ 23 =2 ε 23 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHZoWzdaWgaaWcbaGaaGymaiaaik daaeqaaOGaeyypa0JaaGOmaiabew7aLnaaBaaaleaacaaIXaGaaGOm aaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7cqaHZoWzdaWgaaWcbaGaaGymaiaaiodaaeqaaOGa eyypa0JaaGOmaiabew7aLnaaBaaaleaacaaIXaGaaG4maaqabaGcca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8Uaeq4SdC2aaSbaaSqaaiaaikdacaaIZaaabeaakiabg2 [email protected]@

 

This factor of 2 is an endless source of confusion.  Whenever someone reports shear strain to you, be sure to check which definition they are using.  In particular, many commercial finite element codes output engineering shear strains.

 

 

 

 

 

 

3.12 Decomposition of infinitesimal strain into volumetric and deviatoric parts

 

The volumetric infinitesimal strain is defined as trace(ε) ε kk [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqG0bGaaeOCaiaabggacaqGJbGaae yzaiaabIcacaWH1oGaaiykaiaaykW7caaMc8UaeyyyIORaeqyTdu2a [email protected]@

The deviatoric infinitesimal strain is defined as e=ε 1 3 Itrace(ε) e ij = ε ij 1 3 δ ij ε kk [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbGaaeypaiaahw7acqGHsislda WcaaqaaiaaigdaaeaacaaIZaaaaiaahMeacaaMc8UaaeiDaiaabkha caqGHbGaae4yaiaabwgacaqGOaGaaCyTdiaacMcacaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7cqGHHjIUcaaMc8UaaGPaVlaaykW7 caaMc8UaamyzamaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcq aH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOeI0YaaSaaaeaa caaIXaaabaGaaG4maaaacqaH0oazdaWgaaWcbaGaamyAaiaadQgaae [email protected]@

 

The volumetric strain is a measure of volume changes, and for small strains is related to the Jacobian of the deformation gradient by ε kk J1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaam4AaiaadU [email protected]@ .  To see this, recall that

J=det[ 1+ u 1 x 1 u 1 x 2 u 1 x 3 u 2 x 1 1+ u 2 x 2 u 2 x 3 u 3 x 1 u 3 x 2 1+ u 3 x 3 ]( 1+ u 1 x 1 )( 1+ u 2 x 2 )( 1+ u 3 x 3 )1+ u 1 x 1 + u 2 x 2 + u 3 x 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeyypa0Jaciizaiaacwgaca GG0bWaamWaaeaafaqabeWadaaabaGaaGymaiabgUcaRmaalaaabaGa eyOaIyRaamyDamaaBaaaleaacaaIXaaabeaaaOqaaiabgkGi2kaadI hadaWgaaWcbaGaaGymaaqabaaaaaGcbaWaaSaaaeaacqGHciITcaWG 1bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaale aacaaIYaaabeaaaaaakeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWc baGaaGymaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaiodaae qaaaaaaOqaamaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaIYaaa beaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaaaaGcba GaaGymaiabgUcaRmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaI YaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaa GcbaWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaikdaaeqaaaGc baGaeyOaIyRaamiEamaaBaaaleaacaaIZaaabeaaaaaakeaadaWcaa qaaiabgkGi2kaadwhadaWgaaWcbaGaaG4maaqabaaakeaacqGHciIT caWG4bWaaSbaaSqaaiaaigdaaeqaaaaaaOqaamaalaaabaGaeyOaIy RaamyDamaaBaaaleaacaaIZaaabeaaaOqaaiabgkGi2kaadIhadaWg aaWcbaGaaGOmaaqabaaaaaGcbaGaaGymaiabgUcaRmaalaaabaGaey OaIyRaamyDamaaBaaaleaacaaIZaaabeaaaOqaaiabgkGi2kaadIha daWgaaWcbaGaaG4maaqabaaaaaaaaOGaay5waiaaw2faaiabgIKi7o aabmaabaGaaGymaiabgUcaRmaalaaabaGaeyOaIyRaamyDamaaBaaa leaacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaa qabaaaaaGccaGLOaGaayzkaaWaaeWaaeaacaaIXaGaey4kaSYaaSaa aeaacqGHciITcaWG1bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaeyOaIy RaamiEamaaBaaaleaacaaIYaaabeaaaaaakiaawIcacaGLPaaadaqa daqaaiaaigdacqGHRaWkdaWcaaqaaiabgkGi2kaadwhadaWgaaWcba GaaG4maaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaiodaaeqa aaaaaOGaayjkaiaawMcaaiabgIKi7kaaigdacqGHRaWkdaWcaaqaai abgkGi2kaadwhadaWgaaWcbaGaaGymaaqabaaakeaacqGHciITcaWG 4bWaaSbaaSqaaiaaigdaaeqaaaaakiabgUcaRmaalaaabaGaeyOaIy RaamyDamaaBaaaleaacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaWg aaWcbaGaaGOmaaqabaaaaOGaey4kaSYaaSaaaeaacqGHciITcaWG1b WaaSbaaSqaaiaaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaa [email protected]@

The deviatioric strain is a measure of shear deformation (shear deformation involves no volume change).

 

 

 

 

 

3.13 The Infinitesimal rotation tensor

 

The infinitesimal rotation tensor is defined as

w= 1 2 ( u ( u ) T )or     w ij = 1 2 ( u i x j u j x i ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH3bGaeyypa0ZaaSaaaeaacaaIXa aabaGaaGOmaaaadaqadaqaaiaahwhacqGHhis0cqGHsisldaqadaqa aiaahwhacqGHhis0aiaawIcacaGLPaaadaahaaWcbeqaaiaadsfaaa aakiaawIcacaGLPaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caqGVbGaaeOCaiaabccacaqGGaGaaeiiaiaabccacaWG3bWaaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9maalaaabaGaaGymaaqa aiaaikdaaaWaaeWaaeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWcba GaamyAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqa aaaakiabgkHiTmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGQb aabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqabaaaaaGc [email protected]@

Written out as a matrix, the components of w ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEhadaWgaa [email protected]@  are

w ij [ 0 1 2 ( u 1 x 2 u 2 x 1 ) 1 2 ( u 1 x 3 u 3 x 1 ) 1 2 ( u 2 x 1 u 1 x 2 ) 0 1 2 ( u 2 x 3 u 3 x 2 ) 1 2 ( u 3 x 1 u 1 x 3 ) 1 2 ( u 3 x 2 u 2 x 3 ) 0 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG3bWaaSbaaSqaaiaadMgacaWGQb aabeaakiaaykW7caaMc8UaaGPaVlaaykW7cqGHHjIUcaaMc8UaaGPa VlaaykW7daWadaqaauaabeqadmaaaeaacaaIWaaabaWaaSaaaeaaca aIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGaeyOaIyRaamyDamaa BaaaleaacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaG OmaaqabaaaaOGaeyOeI0YaaSaaaeaacqGHciITcaWG1bWaaSbaaSqa aiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabe aaaaaakiaawIcacaGLPaaaaeaadaWcaaqaaiaaigdaaeaacaaIYaaa amaabmaabaWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaigdaae qaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIZaaabeaaaaGccqGH sisldaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaaG4maaqabaaake aacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaaaOGaayjkaiaa wMcaaaqaamaalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaadaWcaa qaaiabgkGi2kaadwhadaWgaaWcbaGaaGOmaaqabaaakeaacqGHciIT caWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabgkHiTmaalaaabaGaey OaIyRaamyDamaaBaaaleaacaaIXaaabeaaaOqaaiabgkGi2kaadIha daWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaaabaGaaGimaa qaamaalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaadaWcaaqaaiab gkGi2kaadwhadaWgaaWcbaGaaGOmaaqabaaakeaacqGHciITcaWG4b WaaSbaaSqaaiaaiodaaeqaaaaakiabgkHiTmaalaaabaGaeyOaIyRa amyDamaaBaaaleaacaaIZaaabeaaaOqaaiabgkGi2kaadIhadaWgaa WcbaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaaabaWaaSaaaeaacaaI XaaabaGaaGOmaaaadaqadaqaamaalaaabaGaeyOaIyRaamyDamaaBa aaleaacaaIZaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGym aaqabaaaaOGaeyOeI0YaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaai aaigdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIZaaabeaa aaaakiaawIcacaGLPaaaaeaadaWcaaqaaiaaigdaaeaacaaIYaaaam aabmaabaWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaiodaaeqa aaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaGccqGHsi sldaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaaGOmaaqabaaakeaa cqGHciITcaWG4bWaaSbaaSqaaiaaiodaaeqaaaaaaOGaayjkaiaawM [email protected]@

Observe that w ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG3bWaaSbaaSqaaiaadMgacaWGQb [email protected]@  is skew symmetric: w ij = w ji [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG3bWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iabgkHiTiaadEhadaWgaaWcbaGaamOAaiaadMga [email protected]@

 

A skew tensor represents a rotation through a small angle.  Specifically, the operation d y i =( δ ij + w ij )d x j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyEamaaBaaaleaacaWGPb aabeaakiabg2da9maabmaabaGaeqiTdq2aaSbaaSqaaiaadMgacaWG QbaabeaakiabgUcaRiaadEhadaWgaaWcbaGaamyAaiaadQgaaeqaaa GccaGLOaGaayzkaaGaamizaiaadIhadaWgaaWcbaGaamOAaaqabaaa [email protected]@  rotates the infinitesimal line element d x j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamiEamaaBaaaleaacaWGQb [email protected]@  through a small angle θ= w ij w ij /2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCcqGH9aqpdaGcaaqaaiaadE hadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaam4DamaaBaaaleaacaWG [email protected]@  about an axis parallel to the unit vector n i = ijk w kj /(2θ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGc caWG3bWaaSbaaSqaaiaadUgacaWGQbaabeaakiaac+cacaGGOaGaaG [email protected]@ .  (A skew tensor also sometimes represents an angular velocity). 

To visualize the significance of  w ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG3bWaaSbaaSqaaiaadMgacaWGQb [email protected]@ , consider the behavior of an imaginary, infinitesimal, tensile specimen embedded in a deforming solid.  The specimen is stretched, and then rotated through an angle ϕ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  about some axis q.  If the displacement gradients are small, then ϕ<<1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcqGH8aapcqGH8aapcaaIXa [email protected]@ .

 

The rotation of the specimen depends on its original orientation, represented by the unit vector m.  One can show (although one would rather not do all the algebra) that w ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG3bWaaSbaaSqaaiaadMgacaWGQb [email protected]@  represents the average rotation, over all possible orientations of m, of material fibers passing through a point.

 

As a final remark, we note that a general deformation can always be decomposed into an infinitesimal strain and rotation

u i x j = 1 2 ( u i x j + u j x i )+ 1 2 ( u i x j u j x i )= ε ij + w ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadwhadaWgaa WcbaGaamyAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQga aeqaaaaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaaeWaae aadaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaaqabaaakeaa cqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaakiabgUcaRmaala aabaGaeyOaIyRaamyDamaaBaaaleaacaWGQbaabeaaaOqaaiabgkGi 2kaadIhadaWgaaWcbaGaamyAaaqabaaaaaGccaGLOaGaayzkaaGaey 4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGa eyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadI hadaWgaaWcbaGaamOAaaqabaaaaOGaeyOeI0YaaSaaaeaacqGHciIT caWG1bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIyRaamiEamaaBa aaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaacqGH9aqpcqaH1oqz daWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIaam4DamaaBaaale [email protected]@

Physically, this sum of ε ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ [email protected]@  and w ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG3bWaaSbaaSqaaiaadMgacaWGQb [email protected]@  can be regarded as representing two successive deformations [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  a small strain, followed by a rotation, in the sense that

d y i =( δ ik + w ik )( δ kj + ε kj )d x j d x i +( ε ij + w ij )d x j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyEamaaBaaaleaacaWGPb aabeaakiabg2da9maabmaabaGaeqiTdq2aaSbaaSqaaiaadMgacaWG RbaabeaakiabgUcaRiaadEhadaWgaaWcbaGaamyAaiaadUgaaeqaaa GccaGLOaGaayzkaaWaaeWaaeaacqaH0oazdaWgaaWcbaGaam4Aaiaa dQgaaeqaaOGaey4kaSIaeqyTdu2aaSbaaSqaaiaadUgacaWGQbaabe aaaOGaayjkaiaawMcaaiaadsgacaWG4bWaaSbaaSqaaiaadQgaaeqa aOGaeyisISRaamizaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHRa Wkdaqadaqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaaqabaGccqGH RaWkcaWG3bWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawM [email protected]@

first stretches the infinitesimal line element, then rotates it.

 

 

 

3.14 Principal values and directions of the infinitesimal strain tensor

The three principal values e i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGLbWaaSbaaSqaaiaadMgaaeqaaa [email protected]@  and directions n (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbWaaWbaaSqabeaacaGGOaGaam [email protected]@  of the infinitesimal strain tensor satisfy

ε n (i) = e i n (i) or   ε kl n l (i) = e i n l (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaaceqaaiaahw7acqGHflY1caWHUbWaaW baaSqabeaacaGGOaGaamyAaiaacMcaaaGccqGH9aqpcaWGLbWaaSba aSqaaiaadMgaaeqaaOGaaCOBamaaCaaaleqabaGaaiikaiaadMgaca GGPaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVdqaaiaab+gacaqGYbGa aeiiaiaabccacqaH1oqzdaWgaaWcbaGaam4AaiaadYgaaeqaaOGaam OBamaaDaaaleaacaWGSbaabaGaaiikaiaadMgacaGGPaaaaOGaeyyp a0JaamyzamaaBaaaleaacaWGPbaabeaakiaad6gadaqhaaWcbaGaam [email protected]@

Clearly, e i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGLbWaaSbaaSqaaiaadMgaaeqaaa [email protected]@  and n (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbWaaWbaaSqabeaacaGGOaGaam [email protected]@  are the eigenvalues and eigenvectors of ε [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@ .  There are three principal strains and three principal directions, which are always mutually perpendicular. 

 

Their significance can be visualized as follows.

1.      Note that the decomposition u i x j = ε ij + w ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadwhadaWgaa WcbaGaamyAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQga aeqaaaaakiabg2da9iabew7aLnaaBaaaleaacaWGPbGaamOAaaqaba [email protected]@  can be visualized as a small strain, followed by a small rigid rotation, as shown in the picture.

2.      The formula ε n (i) = e i n (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1oGaeyyXICTaaCOBamaaCaaale qabaGaaiikaiaadMgacaGGPaaaaOGaeyypa0JaamyzamaaBaaaleaa caWGPbaabeaakiaah6gadaahaaWcbeqaaiaacIcacaWGPbGaaiykaa [email protected]@  indicates that a vector n is mapped to another, parallel vector by the strain.

3.      Thus, if you draw a small cube with its faces perpendicular to n (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbWaaWbaaSqabeaacaGGOaGaam [email protected]@  on the undeformed solid, this cube will be stretched perpendicular to each face, with a fractional increase in length e i =δ l i / l 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGLbWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaeqiTdqMaamiBamaaBaaaleaacaWGPbaabeaakiaac+ca [email protected]@ .  The faces remain perpendicular to n (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbWaaWbaaSqabeaacaGGOaGaam [email protected]@  after deformation.

4.      Finally, w rotates the small cube through a small angle onto its configuration in the deformed solid.

 

 

 

3.15  Strain Equations of Compatibility for infinitesimal strains

 

It is sometimes necessary to invert the relations between strain and displacement [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  that is to say, given the strain field, to compute the displacements. In this section, we outline how this is done, for the special case of infinitesimal deformations.

 

For infinitesimal motions the relation between strain and displacement is

ε ij = 1 2 ( u i x j + u j x i ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqa amaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaOqaai abgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaey4kaSYaaSaa aeaacqGHciITcaWG1bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIy [email protected]@

Given the six strain components ε ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ [email protected]@  (six, since ε ij = ε ji [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiaadQgacaWGPbaabeaa [email protected]@  ) we wish to determine the three displacement components u i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaa [email protected]@ . First, note that you can never completely recover the displacement field that gives rise to a particular strain field.  Any rigid motion produces no strain, so the displacements can only be completely determined if there is some additional information (besides the strain) that will tell you how much the solid has rotated and translated.  However, integrating the strain field can tell you the displacement field to within an arbitrary rigid motion.

 

Second, we need to be sure that the strain-displacement relations can be integrated at all. The strain is a symmetric second order tensor field, but not all symmetric second order tensor fields can be strain fields. The strain-displacement relations amount to a system of six scalar differential equations for the three displacement components ui.

 

To be integrable, the strains must satisfy the compatibility conditions, which may be expressed as

ipm jqn 2 ε mn x p x q =0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHiiIZdaWgaaWcbaGaamyAaiaadc hacaWGTbaabeaakiabgIGiopaaBaaaleaacaWGQbGaamyCaiaad6ga aeqaaOWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaH1o qzdaWgaaWcbaGaamyBaiaad6gaaeqaaaGcbaGaeyOaIyRaamiEamaa BaaaleaacaWGWbaabeaakiabgkGi2kaadIhadaWgaaWcbaGaamyCaa [email protected]@

Or, equivalently

2 ε ij x k x l + 2 ε kl x i x j 2 ε il x j x k 2 ε jk x i x l =0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiabew7aLnaaBaaaleaacaWGPbGaamOAaaqabaaakeaa cqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaeyOaIyRaamiEam aaBaaaleaacaWGSbaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi2oaa CaaaleqabaGaaGOmaaaakiabew7aLnaaBaaaleaacaWGRbGaamiBaa qabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOa IyRaamiEamaaBaaaleaacaWGQbaabeaaaaGccqGHsisldaWcaaqaai abgkGi2oaaCaaaleqabaGaaGOmaaaakiabew7aLnaaBaaaleaacaWG PbGaamiBaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQgaae qaaOGaeyOaIyRaamiEamaaBaaaleaacaWGRbaabeaaaaGccqGHsisl daWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiabew7aLnaaBa aaleaacaWGQbGaam4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqa aiaadMgaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGSbaabeaaaa [email protected]@

Or, once more equivalently

2 ε 11 x 2 2 + 2 ε 22 x 1 2 2 2 ε 12 x 1 x 2 =0 2 ε 11 x 3 2 + 2 ε 33 x 1 2 2 2 ε 13 x 1 x 3 =0 2 ε 22 x 3 2 + 2 ε 33 x 2 2 2 2 ε 23 x 2 x 3 =0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaalaaabaGaeyOaIy7aaWbaaS qabeaacaaIYaaaaOGaeqyTdu2aaSbaaSqaaiaaigdacaaIXaaabeaa aOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaaaaO Gaey4kaSYaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaH 1oqzdaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEam aaDaaaleaacaaIXaaabaGaaGOmaaaaaaGccqGHsislcaaIYaWaaSaa aeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaH1oqzdaWgaaWcba GaaGymaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaI XaaabeaakiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOGaey ypa0JaaGimaaqaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaa aOGaeqyTdu2aaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiabgkGi2k aadIhadaqhaaWcbaGaaG4maaqaaiaaikdaaaaaaOGaey4kaSYaaSaa aeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaH1oqzdaWgaaWcba GaaG4maiaaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaI XaaabaGaaGOmaaaaaaGccqGHsislcaaIYaWaaSaaaeaacqGHciITda ahaaWcbeqaaiaaikdaaaGccqaH1oqzdaWgaaWcbaGaaGymaiaaioda aeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaakiabgk Gi2kaadIhadaWgaaWcbaGaaG4maaqabaaaaOGaeyypa0JaaGimaaqa amaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyTdu2aaS baaSqaaiaaikdacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaqhaaWc baGaaG4maaqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacqGHciITda ahaaWcbeqaaiaaikdaaaGccqaH1oqzdaWgaaWcbaGaaG4maiaaioda aeqaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaIYaaabaGaaGOmaa aaaaGccqGHsislcaaIYaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaa ikdaaaGccqaH1oqzdaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcbaGaey OaIyRaamiEamaaBaaaleaacaaIYaaabeaakiabgkGi2kaadIhadaWg [email protected]@       2 ε 11 x 2 x 3 x 1 ( ε 23 x 1 + ε 31 x 2 + ε 12 x 3 )=0 2 ε 22 x 3 x 1 x 2 ( ε 31 x 2 + ε 12 x 3 + ε 23 x 1 )=0 2 ε 33 x 1 x 2 x 3 ( ε 12 x 3 + ε 23 x 1 + ε 31 x 2 )=0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaalaaabaGaeyOaIy7aaWbaaS qabeaacaaIYaaaaOGaeqyTdu2aaSbaaSqaaiaaigdacaaIXaaabeaa aOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOmaaqaaaaakiabgkGi2k aadIhadaqhaaWcbaGaaG4maaqaaaaaaaGccqGHsisldaWcaaqaaiab gkGi2kaaykW7caaMc8oabaGaeyOaIyRaamiEamaaBaaaleaacaaIXa aabeaaaaGcdaqadaqaaiabgkHiTmaalaaabaGaeyOaIyRaeqyTdu2a aSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiabgkGi2kaadIhadaqhaa WcbaGaaGymaaqaaaaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kabew7a LnaaBaaaleaacaaIZaGaaGymaaqabaaakeaacqGHciITcaWG4bWaa0 baaSqaaiaaikdaaeaaaaaaaOGaey4kaSYaaSaaaeaacqGHciITcqaH 1oqzdaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEam aaDaaaleaacaaIZaaabaaaaaaaaOGaayjkaiaawMcaaiabg2da9iaa icdaaeaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiabew 7aLnaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacqGHciITcaWG4bWa a0baaSqaaiaaiodaaeaaaaGccqGHciITcaWG4bWaa0baaSqaaiaaig daaeaaaaaaaOGaeyOeI0YaaSaaaeaacqGHciITcaaMc8UaaGPaVdqa aiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOWaaeWaaeaacq GHsisldaWcaaqaaiabgkGi2kabew7aLnaaBaaaleaacaaIZaGaaGym aaqabaaakeaacqGHciITcaWG4bWaa0baaSqaaiaaikdaaeaaaaaaaO Gaey4kaSYaaSaaaeaacqGHciITcqaH1oqzdaWgaaWcbaGaaGymaiaa ikdaaeqaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaIZaaabaaaaa aakiabgUcaRmaalaaabaGaeyOaIyRaeqyTdu2aaSbaaSqaaiaaikda caaIZaaabeaaaOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGymaaqaaa aaaaaakiaawIcacaGLPaaacqGH9aqpcaaIWaaabaWaaSaaaeaacqGH ciITdaahaaWcbeqaaiaaikdaaaGccqaH1oqzdaWgaaWcbaGaaG4mai aaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaIXaaabaaa aOGaeyOaIyRaamiEamaaDaaaleaacaaIYaaabaaaaaaakiabgkHiTm aalaaabaGaeyOaIyRaaGPaVlaaykW7aeaacqGHciITcaWG4bWaaSba aSqaaiaaiodaaeqaaaaakmaabmaabaGaeyOeI0YaaSaaaeaacqGHci ITcqaH1oqzdaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaeyOaIyRa amiEamaaDaaaleaacaaIZaaabaaaaaaakiabgUcaRmaalaaabaGaey OaIyRaeqyTdu2aaSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiabgkGi 2kaadIhadaqhaaWcbaGaaGymaaqaaaaaaaGccqGHRaWkdaWcaaqaai abgkGi2kabew7aLnaaBaaaleaacaaIZaGaaGymaaqabaaakeaacqGH ciITcaWG4bWaa0baaSqaaiaaikdaaeaaaaaaaaGccaGLOaGaayzkaa [email protected]@

It is easy to show that all strain fields must satisfy these conditions -  you simply need to substitute for the strains in terms of displacements and show that the appropriate equation is satisfied.  For example,

2 ε 11 x 2 2 + 2 ε 22 x 1 2 2 2 ε 12 x 1 x 2 = 4 u 1 x 1 x 2 2 + 4 u 2 x 2 x 1 2 2 2 x 1 x 2 1 2 ( u 1 x 2 + u 2 x 1 )=0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiabew7aLnaaBaaaleaacaaIXaGaaGymaaqabaaakeaa cqGHciITcaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaaakiabgU caRmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyTdu2a aSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaqhaa WcbaGaaGymaaqaaiaaikdaaaaaaOGaeyOeI0IaaGOmamaalaaabaGa eyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyTdu2aaSbaaSqaaiaaig dacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqa baGccqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiabg2da9m aalaaabaGaeyOaIy7aaWbaaSqabeaacaaI0aaaaOGaamyDamaaBaaa leaacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGymaa qaaaaakiabgkGi2kaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaaa aOGaey4kaSYaaSaaaeaacqGHciITdaahaaWcbeqaaiaaisdaaaGcca WG1bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaDaaa leaacaaIYaaabaaaaOGaeyOaIyRaamiEamaaDaaaleaacaaIXaaaba GaaGOmaaaaaaGccqGHsislcaaIYaWaaSaaaeaacqGHciITdaahaaWc beqaaiaaikdaaaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaae qaaOGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaGcdaWcaaqa aiaaigdaaeaacaaIYaaaamaabmaabaWaaSaaaeaacqGHciITcaWG1b WaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaa caaIYaaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kaadwhadaWgaa WcbaGaaGOmaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaigda [email protected]@

and similarly for the other expressions.

Not that for planar problems for which ε 13 = ε 23 =0 and  d ε ij d x 3 =0, [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIXaGaaG4maa qabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaaGOmaiaaiodaaeqaaOGa eyypa0JaaGimaiaabccacaqGHbGaaeOBaiaabsgacaqGGaWaaSaaae aacaWGKbGaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaaaOqaaiaa dsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaaakiabg2da9iaaicdaca [email protected]@  all of these compatibility equations are satisfied trivially, with the exception of the first: 2 ε 11 x 2 2 + 2 ε 22 x 1 2 2 2 ε 12 x 1 x 2 =0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiabew7aLnaaBaaaleaacaaIXaGaaGymaaqabaaakeaa cqGHciITcaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaaakiabgU caRmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyTdu2a aSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaqhaa WcbaGaaGymaaqaaiaaikdaaaaaaOGaeyOeI0IaaGOmamaalaaabaGa eyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyTdu2aaSbaaSqaaiaaig dacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqa baGccqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiabg2da9i [email protected]@

 

It can be shown that

 (i) If the strains do not satisfy the equations of compatibility, then a displacement vector can not be integrated from the strains.

(ii) If the strains satisfy the compatibility equations, and the solid simply connected (i.e. it contains no holes that go all the way through its thickness), then a displacement vector can be integrated from the strains. 

(iii) If the solid is not simply connected, a displacement vector can be calculated, but it may not be single valued [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  i.e. you may get different solutions depending on how the path of integration encircles the holes.

 

Now, let us return to the question posed at the beginning of this section.  Given the strains, how do we compute the displacements?

2D strain fields

For 2D (plane stress or plane strain) the procedure is quite simple and is best illustrated by working through a specific case

 

As a representative example, we will use the strain field in a 2D (plane stress) cantilever beam with Young’s modulus E and Poisson’s ratio ν [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  loaded at one end by a force P. The beam has a rectangular cross-section with height 2a and out-of-plane width b.  We will show later (Sect 5.2.4) that the strain field in the beam is

ε 11 =2C x 1 x 2 ε 22 =2νC x 1 x 2 ε 12 =( 1+ν )C( a 2 x 2 2 ),C= 3P 4E a 3 b [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0JaaGOmaiaadoeacaWG4bWaaSbaaSqaaiaaigda aeqaaOGaamiEamaaBaaaleaacaaIYaaabeaakiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlabew7aLnaaBaaaleaacaaIYaGaaGOm aaqabaGccqGH9aqpcqGHsislcaaIYaGaeqyVd4Maam4qaiaadIhada WgaaWcbaGaaGymaaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGa aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cq aH1oqzdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0ZaaeWaaeaa caaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaGaam4qamaabmaaba GaamyyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadIhadaqhaaWc baGaaGOmaaqaaiaaikdaaaaakiaawIcacaGLPaaacaGGSaGaaGPaVl aaykW7caaMc8Uaam4qaiabg2da9maalaaabaGaaG4maiaadcfaaeaa caaI0aGaamyraiaadggadaahaaWcbeqaaiaaiodaaaGccaWGIbaaaa [email protected]@

 

We first check that the strain is compatible.  For 2D problems this requires

2 ε 11 x 2 2 + 2 ε 22 x 1 2 2 2 ε 12 x 1 x 2 =0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiabew7aLnaaBaaaleaacaaIXaGaaGymaaqabaaakeaa cqGHciITcaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaaakiabgU caRmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyTdu2a aSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaqhaa WcbaGaaGymaaqaaiaaikdaaaaaaOGaeyOeI0IaaGOmamaalaaabaGa eyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyTdu2aaSbaaSqaaiaaig dacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqa baGccqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiabg2da9i [email protected]@

which is clearly satisfied in this case.

 

For a 2D problem we only need to determine u 1 ( x 1 , x 2 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaa [email protected]@  and u 2 ( x 1 , x 2 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaikdaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaa [email protected]@  such that

u 1 x 1 = ε 11 , u 2 x 2 = ε 22 and  u 1 x 2 + u 2 x 1 =2 ε 12   [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamyDamaaBaaale aacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGymaaqa aaaaaaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaaGymaiaaigdaaeqaaO GaaiilamaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaIYaaabeaa aOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOmaaqaaaaaaaGccqGH9a qpcqaH1oqzdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaaGPaVlaaykW7 caqGHbGaaeOBaiaabsgacaqGGaWaaSaaaeaacqGHciITcaWG1bWaaS baaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaI YaaabaaaaaaakiabgUcaRmaalaaabaGaeyOaIyRaamyDamaaBaaale aacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGymaaqa aaaaaaGccqGH9aqpcaaIYaGaeqyTdu2aaSbaaSqaaiaaigdacaaIYa [email protected]@ .

The first two of these give

ε 11 = u 1 x 1 =2C x 1 x 2 ε 22 = u 2 x 2 =2νC x 1 x 2 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaa igdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaa GccqGH9aqpcaaIYaGaam4qaiaadIhadaWgaaWcbaGaaGymaaqabaGc caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyTdu2aaSbaaSqa aiaaikdacaaIYaaabeaakiabg2da9maalaaabaGaeyOaIyRaamyDam aaBaaaleaacaaIYaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGa aGOmaaqabaaaaOGaeyypa0JaeyOeI0IaaGOmaiabe27aUjaadoeaca WG4bWaaSbaaSqaaiaaigdaaeqaaOGaamiEamaaBaaaleaacaaIYaaa [email protected]@

We can integrate the first equation with respect to x 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaa [email protected]@  and the second equation with respect to x 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaa [email protected]@  to get

u 1 =C x 1 2 x 2 + f 1 ( x 2 ) u 2 =νC x 1 x 2 2 + f 2 ( x 1 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0Jaam4qaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikdaaaGc caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaamOzamaaBaaale aacaaIXaaabeaakiaacIcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGa aiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaadwhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcqGH sislcqaH9oGBcaWGdbGaamiEamaaBaaaleaacaaIXaaabeaakiaadI hadaqhaaWcbaGaaGOmaaqaaiaaikdaaaGccqGHRaWkcaWGMbWaaSba aSqaaiaaikdaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaaGymaaqaba [email protected]@

where f 1 ( x 2 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiaaigdaaeqaaO [email protected]@  and f 2 ( x 1 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiaaikdaaeqaaO [email protected]@  are two functions of x 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaa [email protected]@  and x 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaa [email protected]@ , respectively, which are yet to be determined.  We can find these functions by substituting the formulas for u 1 ( x 1 , x 2 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaa [email protected]@  and u 2 ( x 1 , x 2 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaikdaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaa [email protected]@  into the expression for shear strain

ε 12 = 1 2 ( u 1 x 2 + u 2 x 1 )=( 1+ν )C( a 2 x 2 2 ) 1 2 ( C x 1 2 νC x 2 2 + d f 1 d x 2 + d f 2 d x 1 )=( 1+ν )C( a 2 x 2 2 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaaykW7cqaH1oqzdaWgaaWcba GaaGymaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOm aaaadaqadaqaamaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaIXa aabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOGa ey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaikdaaeqaaa GcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaaaakiaawIca caGLPaaacqGH9aqpdaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawI cacaGLPaaacaWGdbWaaeWaaeaacaWGHbWaaWbaaSqabeaacaaIYaaa aOGaeyOeI0IaamiEamaaDaaaleaacaaIYaaabaGaaGOmaaaaaOGaay jkaiaawMcaaaqaamaalaaabaGaaGymaaqaaiaaikdaaaWaaeWaaeaa caWGdbGaamiEamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgkHiTi abe27aUjaadoeacaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaOGa ey4kaSYaaSaaaeaacaWGKbGaamOzamaaBaaaleaacaaIXaaabeaaaO qaaiaadsgacaWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiabgUcaRmaa laaabaGaamizaiaadAgadaWgaaWcbaGaaGOmaaqabaaakeaacaWGKb GaamiEamaaBaaaleaacaaIXaaabeaaaaaakiaawIcacaGLPaaacqGH 9aqpdaqadaqaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaaca WGdbWaaeWaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Ia amiEamaaDaaaleaacaaIYaaabaGaaGOmaaaaaOGaayjkaiaawMcaaa [email protected]@

We can re-write this as

( d f 2 d x 1 +C x 1 2 )+( d f 1 d x 2 νC x 2 2 2( 1+ν )C( a 2 x 2 2 ) )=0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaamaalaaabaGaamizaiaadA gadaWgaaWcbaGaaGOmaaqabaaakeaacaWGKbGaamiEamaaBaaaleaa caaIXaaabeaaaaGccqGHRaWkcaWGdbGaamiEamaaDaaaleaacaaIXa aabaGaaGOmaaaaaOGaayjkaiaawMcaaiabgUcaRmaabmaabaWaaSaa aeaacaWGKbGaamOzamaaBaaaleaacaaIXaaabeaaaOqaaiaadsgaca WG4bWaaSbaaSqaaiaaikdaaeqaaaaakiabgkHiTiabe27aUjaadoea caWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaOGaeyOeI0IaaGOmam aabmaabaGaaGymaiabgUcaRiabe27aUbGaayjkaiaawMcaaiaadoea daqadaqaaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG4b Waa0baaSqaaiaaikdaaeaacaaIYaaaaaGccaGLOaGaayzkaaaacaGL [email protected]@

The two terms in parentheses are functions of x 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  and x 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ , respectively.  Since the left hand side must vanish for all values of  x 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  and x 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ , this means that

( d f 2 d x 1 +C x 1 2 )=ω ( d f 1 d x 2 νC x 2 2 2( 1+ν )C( a 2 x 2 2 ) )=ω [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaabmaabaWaaSaaaeaacaWGKb GaamOzamaaBaaaleaacaaIYaaabeaaaOqaaiaadsgacaWG4bWaaSba aSqaaiaaigdaaeqaaaaakiabgUcaRiaadoeacaWG4bWaa0baaSqaai aaigdaaeaacaaIYaaaaaGccaGLOaGaayzkaaGaeyypa0JaeqyYdCha baWaaeWaaeaadaWcaaqaaiaadsgacaWGMbWaaSbaaSqaaiaaigdaae qaaaGcbaGaamizaiaadIhadaWgaaWcbaGaaGOmaaqabaaaaOGaeyOe I0IaeqyVd4Maam4qaiaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaa GccqGHsislcaaIYaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGL OaGaayzkaaGaam4qamaabmaabaGaamyyamaaCaaaleqabaGaaGOmaa aakiabgkHiTiaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaaakiaa wIcacaGLPaaaaiaawIcacaGLPaaacqGH9aqpcqGHsislcqaHjpWDaa [email protected]@

where ω [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  is an arbitrary constant.  We can now integrate these expressions to see that

f 1 =( 2(1+ν)C a 2 ω ) x 2 C 3 (2+ν) x 2 3 +c f 2 =ω x 1 C 3 x 1 3 +d [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamOzamaaBaaaleaacaaIXaaabe aakiabg2da9maabmaabaGaaGOmaiaacIcacaaIXaGaey4kaSIaeqyV d4MaaiykaiaadoeacaWGHbWaaWbaaKazbamabeqcbawaaiaaikdaaa GccqGHsislcqaHjpWDaiaawIcacaGLPaaacaWG4bWaaSbaaSqaaiaa ikdaaeqaaOGaeyOeI0YaaSaaaeaacaWGdbaabaGaaG4maaaacaGGOa GaaGOmaiabgUcaRiabe27aUjaacMcacaWG4bWaa0baaSqaaiaaikda aeaacaaIZaaaaOGaey4kaSIaam4yaaqaaiaadAgadaWgaaWcbaGaaG OmaaqabaGccqGH9aqpcqaHjpWDcaWG4bWaaSbaaSqaaiaaigdaaeqa aOGaeyOeI0YaaSaaaeaacaWGdbaabaGaaG4maaaacaWG4bWaa0baaS [email protected]@

where c and d are two more arbitrary constants. Finally, the displacement field follows as

u 1 =C x 1 2 x 2 C 3 (2+ν) x 2 3 +2(1+ν)C a 2 x 2 ω x 2 +c u 2 =νC x 1 x 2 2 C 3 x 1 3 +ω x 1 +d [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadwhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcaWGdbGaamiEamaaDaaaleaacaaIXaaabaGaaGOm aaaakiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHsisldaWcaaqaai aadoeaaeaacaaIZaaaaiaacIcacaaIYaGaey4kaSIaeqyVd4Maaiyk aiaadIhadaqhaaWcbaGaaGOmaaqaaiaaiodaaaGccqGHRaWkcaaIYa GaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaGaam4qaiaadggadaah aaWcbeqaaiaaikdaaaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaey OeI0IaeqyYdCNaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRiaa dogaaeaacaaMc8UaamyDamaaBaaaleaacaaIYaaabeaakiabg2da9i abgkHiTiabe27aUjaadoeacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGa amiEamaaDaaaleaacaaIYaaabaGaaGOmaaaakiabgkHiTmaalaaaba Gaam4qaaqaaiaaiodaaaGaamiEamaaDaaaleaacaaIXaaabaGaaG4m aaaakiabgUcaRiabeM8a3jaadIhadaWgaaWcbaGaaGymaaqabaGccq [email protected]@

The three arbitrary constants ω [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ , c and d can be seen to represent a small rigid rotation through angle ω [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  about the x 3 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa [email protected]@  axis, together with a displacement (c,d) parallel to ( x 1 , x 2 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamiEamaaBaaaleaacaaIXa aabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiykaaaa @[email protected]  axes, respectively.

 

 

3D strain fields

 

For a general, three dimensional field a more formal procedure is required. Since the strains are the derivatives of the displacement field, so you might guess that we compute the displacements by integrating the strains.  This is more or less correct.  The general procedure is outlined below.

 

We first pick a point x 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bWaaSbaaSqaaiaaicdaaeqaaa [email protected]@  in the solid, and arbitrarily say that the displacement at x 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bWaaSbaaSqaaiaaicdaaeqaaa [email protected]@  is zero, and also take the rotation of the solid at  x 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bWaaSbaaSqaaiaaicdaaeqaaa [email protected]@  to be zero. Then, we can compute the displacements at any other point x in the solid, by integrating the strains along any convenient path.  In a simply connected solid, it doesn’t matter what path you pick.

Actually, you don’t exactly integrate the strains [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  instead, you must evaluate the following integral

u i (x)= x0 x U ij (x,ξ)d ξ j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaahIhacaGGPaGaeyypa0Zaa8qCaeaacaWGvbWaaSbaaSqa aiaadMgacaWGQbaabeaakiaacIcacaWH4bGaaiilaiabj67a4jaacM cacaaMc8Uaamizaiabe67a4naaBaaaleaacaWGQbaabeaaaeaacaWH [email protected]@

where

U ij (x,ξ)= ε ij (ξ)+( x k ξ k )[ ε ij (ξ) ξ k ε kj (ξ) ξ i ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbWaaSbaaSqaaiaadMgacaWGQb aabeaakiaacIcacaWH4bGaaiilaiabj67a4jaacMcacqGH9aqpcqaH 1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiikaiabj67a4jaacM cacqGHRaWkcaGGOaGaamiEamaaBaaaleaacaWGRbaabeaakiabgkHi Tiabe67a4naaBaaaleaacaWGRbaabeaakiaacMcadaWadaqaamaala aabaGaeyOaIyRaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaakiaa cIcacqqI+oaEcaGGPaaabaGaeyOaIyRaeqOVdG3aaSbaaSqaaiaadU gaaeqaaaaakiabgkHiTmaalaaabaGaeyOaIyRaeqyTdu2aaSbaaSqa aiaadUgacaWGQbaabeaakiaacIcacqqI+oaEcaGGPaaabaGaeyOaIy RaeqOVdG3aaSbaaSqaaiaadMgaaeqaaaaaaOGaay5waiaaw2faaaaa @[email protected]

Here, x k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaadUgaaeqaaa [email protected]@  are the components of the position vector at the point where we are computing the displacements, and ξ j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH+oaEdaWgaaWcbaGaamOAaaqaba [email protected]@  are the components of the position vector ξ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  of a point somewhere along the path of integration. The fact that the integral is path-independent (in a simply connected solid) is guaranteed by the compatibility condition.  Evaluating this integral in practice can be quite painful, but fortunately almost all cases where we need to integrate strains to get displacement turn out to be two-dimensional.

 

 

 

 

3.16 Cauchy-Green Deformation Tensors

 

There are two Cauchy-Green deformation tensors [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  defined through

 The Right Cauchy Green Deformation Tensor   C= F T F C ij = F ki F kj [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdbGaeyypa0JaaCOramaaCaaale qabaGaamivaaaakiabgwSixlaahAeacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8Uaam4qamaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaWGgbWaaSbaaSqaaiaadUgacaWGPbaabeaakiaa [email protected]@

 The Left Cauchy Green Deformation Tensor     B=F F T B ij = F ik F jk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHcbGaeyypa0JaaCOraiabgwSixl aahAeadaahaaWcbeqaaiaadsfaaaGccaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaamOqamaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaWGgbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaa [email protected]@

They are called `left’ and `right’ tensors because of their relation to the `left’ and ‘right’ stretch tensors defined below.  They can be regarded as quantifying the squared length of infinitesimal fibers in the deformed configuration, by noting that if a material fiber dx= l 0 m [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWH4bGaeyypa0JaamiBamaaBa [email protected]@  in the undeformed solid is stretched and rotated to dy=ln [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWH5bGaeyypa0JaamiBaiaah6 [email protected]@  in the deformed solid, then

l 2 l 0 2 =mCm l 0 2 l 2 =n B 1 n [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadYgadaahaaWcbeqaai aaikdaaaaakeaacaWGSbWaa0baaSqaaiaaicdaaeaacaaIYaaaaaaa kiabg2da9iaah2gacqGHflY1caWHdbGaeyyXICTaaCyBaiaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVpaalaaabaGaamiBamaaDaaaleaacaaIWaaabaGaaGOmaaaaaOqa aiaadYgadaahaaWcbeqaaiaaikdaaaaaaOGaeyypa0JaaCOBaiabgw SixlaahkeadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGHflY1caWH [email protected]@

 

 

 

3.17 Rotation tensor, and Left and Right Stretch Tensors

 

The definitions of these quantities are

 The Right Stretch Tensor U= C 1/2 = ( F T F ) 1/2 U ij = C ij 1/2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHvbGaeyypa0JaaC4qamaaCaaale qabaGaaGymaiaac+cacaaIYaaaaOGaeyypa0ZaaeWaaeaacaWHgbWa aWbaaSqabeaacaWGubaaaOGaeyyXICTaaCOraaGaayjkaiaawMcaam aaCaaaleqabaGaaGymaiaac+cacaaIYaaaaOGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadw fadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0Jaam4qamaaDaaa [email protected]@

 The Left Stretch Tensor   V= B 1/2 V ij = B ij 1/2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHwbGaeyypa0JaaCOqamaaCaaale qabaGaaGymaiaac+cacaaIYaaaaOGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadAfadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaamOqamaaDaaaleaacaWG [email protected]@

 The Rotation Tensor         R=F U 1 = V 1 F R ij = F ik U kj 1 = V ik 1 F kj [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHsbGaeyypa0JaaCOraiabgwSixl aahwfadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGH9aqpcaWHwbWa aWbaaSqabeaacqGHsislcaaIXaaaaOGaeyyXICTaaCOraiaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa dkfadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaamOramaaBa aaleaacaWGPbGaam4AaaqabaGccaWGvbWaa0baaSqaaiaadUgacaWG QbaabaGaeyOeI0IaaGymaaaakiabg2da9iaadAfadaqhaaWcbaGaam yAaiaadUgaaeaacqGHsislcaaIXaaaaOGaamOramaaBaaaleaacaWG [email protected]@

 

To calculate these quantities you need to remember how to calculate the square root of a matrix.  For example, to calculate the square root of C, you must

1.      Calculate the eigenvalues of C [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  we will call these λ n 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBdaqhaaWcbaGaamOBaaqaai [email protected]@ , with n=1,2,3.  Since C and B are both symmetric and positive definite, the eigenvalues λ n 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBdaqhaaWcbaGaamOBaaqaai [email protected]@  are all positive real numbers, and therefore their square roots λ n [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBdaWgaaWcbaGaamOBaaqaba [email protected]@  are also positive real numbers.

2.      Calculate the eigenvectors of C and normalize them so they have unit magnitude.  We will denote the eigenvectors by c (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHJbWaaWbaaSqabeaacaGGOaGaam [email protected]@ .  They must be normalized to satisfy c (n) c (n) =1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHJbWaaWbaaSqabeaacaGGOaGaam OBaiaacMcaaaGccqGHflY1caWHJbWaaWbaaSqabeaacaGGOaGaamOB [email protected]@

3.      Finally, calculate C 1/2 = n=1 3 λ n c (n) c (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdbWaaWbaaSqabeaacaaIXaGaai 4laiaaikdaaaGccqGH9aqpdaaeWaqaaiabeU7aSnaaBaaaleaacaWG UbaabeaakiaahogadaahaaWcbeqaaiaacIcacaWGUbGaaiykaaaaki abgEPielaahogadaahaaWcbeqaaiaacIcacaWGUbGaaiykaaaaaeaa [email protected]@ , where c [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  denotes a dyadic product (See Appendix B). In components, this can be written C ij 1/2 = n=1 3 λ n c i (n) c j (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaa0baaSqaaiaadMgacaWGQb aabaGaaGymaiaac+cacaaIYaaaaOGaeyypa0ZaaabmaeaacqaH7oaB daWgaaWcbaGaamOBaaqabaGccaWGJbWaa0baaSqaaiaadMgaaeaaca GGOaGaamOBaiaacMcaaaGccaWGJbWaa0baaSqaaiaadQgaaeaacaGG OaGaamOBaiaacMcaaaaabaGaamOBaiabg2da9iaaigdaaeaacaaIZa [email protected]@

4.      As an additional bonus, you can quickly compute the inverse square root (which is needed to find R) as

C 1/2 = n=1 3 1 λ n c (n) c (n) or   C ij 1/2 = n=1 3 1 λ n c i (n) c j (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdbWaaWbaaSqabeaacqGHsislca aIXaGaai4laiaaikdaaaGccqGH9aqpdaaeWaqaamaalaaabaGaaGym aaqaaiabeU7aSnaaBaaaleaacaWGUbaabeaaaaGccaWHJbWaaWbaaS qabeaacaGGOaGaamOBaiaacMcaaaGccqGHxkcXcaWHJbWaaWbaaSqa beaacaGGOaGaamOBaiaacMcaaaaabaGaamOBaiabg2da9iaaigdaae aacaaIZaaaniabggHiLdGccaaMc8UaaGPaVlaaykW7caaMc8Uaae4B aiaabkhacaqGGaGaaeiiaiaadoeadaqhaaWcbaGaamyAaiaadQgaae aacqGHsislcaaIXaGaai4laiaaikdaaaGccqGH9aqpdaaeWaqaamaa laaabaGaaGymaaqaaiabeU7aSnaaBaaaleaacaWGUbaabeaaaaGcca WGJbWaa0baaSqaaiaadMgaaeaacaGGOaGaamOBaiaacMcaaaGccaWG JbWaa0baaSqaaiaadQgaaeaacaGGOaGaamOBaiaacMcaaaaabaGaam [email protected]@

 

To see the physical significance of these tensors, observe that

1.      The definition of the rotation tensor shows that

R=F U 1 F=RU R= V 1 FF=VR [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahkfacqGH9aqpcaWHgbGaey yXICTaaCyvamaaCaaaleqabaGaeyOeI0IaaGymaaaakiabgsDiBlaa hAeacqGH9aqpcaWHsbGaeyyXICTaaCyvaaqaaiaahkfacqGH9aqpca WHwbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaeyyXICTaaCOraiaa [email protected]@

2.      The multiplicative decomposition of a constant tensor F=RU [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOuaiabgwSixl [email protected]@  can be regarded as a sequence of two homogeneous deformations [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  U, followed by R.  Similarly, F=VR [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOvaiabgwSixl [email protected]@  is R followed by V.

3.      R is proper orthogonal (it satisfies R R T = R T R=I [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHsbGaeyyXICTaaCOuamaaCaaale qabaGaamivaaaakiabg2da9iaahkfadaahaaWcbeqaaiaadsfaaaGc [email protected]@  and det(R)=1), and therefore represents a rotation.  To see this, note that U is symmetric, and therefore satisfies U T = U 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHvbWaaWbaaSqabeaacqGHsislca WGubaaaOGaeyypa0JaaCyvamaaCaaaleqabaGaeyOeI0IaaGymaaaa [email protected]@ , so that

R T R= ( F U 1 ) T ( F U 1 ) = U T F T F U 1 = U 1 U 2 U 1 =I [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahkfadaahaaWcbeqaaiaads faaaGccaWHsbGaeyypa0ZaaeWaaeaacaWHgbGaeyyXICTaaCyvamaa CaaaleqabaGaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaamaaCaaale qabaGaamivaaaakiabgwSixpaabmaabaGaaCOraiabgwSixlaahwfa daahaaWcbeqaaiabgkHiTiaaigdaaaaakiaawIcacaGLPaaaaeaaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH9aqpca WHvbWaaWbaaSqabeaacqGHsislcaWGubaaaOGaeyyXICTaaCOramaa CaaaleqabaGaamivaaaakiabgwSixlaahAeacqGHflY1caWHvbWaaW baaSqabeaacqGHsislcaaIXaaaaaGcbaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8Uaeyypa0JaaCyvamaaCaaaleqabaGa eyOeI0IaaGymaaaakiabgwSixlaahwfadaahaaWcbeqaaiaaikdaaa GccqGHflY1caWHvbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaeyyp [email protected]@

and det(R)=det(F)det(U-1)=1

  1. U can be expressed in the form

U= λ 1 u (1) u (1) + λ 2 u (2) u (2) + λ 3 u (3) u (3) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCyvaiabg2da9iabeU7aSnaaBaaale aacaaIXaaabeaakiaahwhadaahaaWcbeqaaiaacIcacaaIXaGaaiyk aaaakiabgEPielaahwhadaahaaWcbeqaaiaacIcacaaIXaGaaiykaa aakiabgUcaRiabeU7aSnaaBaaaleaacaaIYaaabeaakiaahwhadaah aaWcbeqaaiaacIcacaaIYaGaaiykaaaakiabgEPielaahwhadaahaa WcbeqaaiaacIcacaaIYaGaaiykaaaakiabgUcaRiabeU7aSnaaBaaa leaacaaIZaaabeaakiaahwhadaahaaWcbeqaaiaacIcacaaIZaGaai ykaaaakiabgEPielaahwhadaahaaWcbeqaaiaacIcacaaIZaGaaiyk [email protected]@

where u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGOaGaam [email protected]@  are the three (mutually perpendicular) eigenvectors of U. (By construction, these are identical to the eigenvectors of C).  If we interpret u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGOaGaam [email protected]@  as basis vectors, we see that U is diagonal in this basis, and so corresponds to stretching parallel to each basis vector, as shown in the figure below.

The decompositions

F=RU [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOuaiabgwSixl [email protected]@  and F=VR [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOvaiabgwSixl [email protected]@

are known as the right and left polar decomposition of F. (The right and left refer to the positions of U and V).  They show that every homogeneous deformation can be decomposed into a stretch followed by a rigid rotation, or equivalently into a rigid rotation followed by a stretch. The decomposition is discussed in more detail in the next section.

 

3.18 Principal stretches

 

The principal stretches can be calculated from any one of the following (they all give the same answer)

  1. The eigenvalues of the right stretch tensor U
  2. The eigenvalues of the left stretch tensor V
  3. The square root of the eigenvalues of the right Cauchy-Green tensor C
  4. The square root of the eigenvalues of the left Cauchy-Green tensor B

The principal stretches are also related to the eigenvalues of the Lagrange and Eulerian strains.  The details are left as an exercise.

 

There are two sets of principal stretch directions, associated with the undeformed and deformed solids.

  1. The principal stretch directions in the undeformed solid are the (normalized) eigenvectors of U or C.  Denote these by u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGOaGaam [email protected]@ .
  2. The principal stretch directions in the deformed solid are the (normalized) eigenvectors of V or B. Denote these by v (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bWaaWbaaSqabeaacaGGOaGaam [email protected]@ .

 

To visualize the physical significance of principal stretches and their directions, note that a deformation can be decomposed as F=RU [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOuaiabgwSixl [email protected]@  into a sequence of a stretch followed by a rotation. 

 

Note also that

  1. The principal directions u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGOaGaam [email protected]@  are mutually perpendicular.  You could draw a little cube on the undeformed solid with faces perpendicular to these directions, as shown above.
  2. The stretch U will stretch the cube by an amount λ i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBdaWgaaWcbaGaamyAaaqaba [email protected]@  parallel to each u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGOaGaam [email protected]@ .  The faces of the stretched cube remain perpendicular to u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGOaGaam [email protected]@ .
  3. The rotation R will rotate the stretched cube so that the directions u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGOaGaam [email protected]@  rotate to line up with v (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bWaaWbaaSqabeaacaGGOaGaam [email protected]@ .
  4. The faces of the deformed cube are perpendicular to v (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bWaaWbaaSqabeaacaGGOaGaam [email protected]@

 

The decomposition F=VR [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOvaiabgwSixl [email protected]@  can be visualized in much the same way.  In this case, the directions u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGOaGaam [email protected]@  are first rotated to coincide with v (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bWaaWbaaSqabeaacaGGOaGaam [email protected]@ .  The cube is then stretched parallel to each v (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bWaaWbaaSqabeaacaGGOaGaam [email protected]@  to produce the same shape change.

 

We could compare the undeformed and deformed cubes by placing them side by side, with the vectors v (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bWaaWbaaSqabeaacaGGOaGaam [email protected]@  and u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGOaGaam [email protected]@  parallel, as shown in the figure. 

 

 

 

3.19 Generalized strain measures

 

The polar decompositions F=RU [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOuaiabgwSixl [email protected]@  and F=VR [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOvaiabgwSixl [email protected]@  provide a way to define additional strain measures.  Let λ i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBdaWgaaWcbaGaamyAaaqaba [email protected]@  denote the principal stretches, and let u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGOaGaam [email protected]@  and v (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bWaaWbaaSqabeaacaGGOaGaam [email protected]@  denote the normalized eigenvectors of U and V.  Then one could define strain tensors through

Lagrangian Nominal strain:                  i=1 3 ( λ i 1) u (i) u (i) Lagrangian Logarithmic strain:             i=1 3 log( λ i ) u (i) u (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaabYeacaqGHbGaae4zaiaabk hacaqGHbGaaeOBaiaabEgacaqGPbGaaeyyaiaab6gacaqGGaGaaeOt aiaab+gacaqGTbGaaeyAaiaab6gacaqGHbGaaeiBaiaabccacaqGZb GaaeiDaiaabkhacaqGHbGaaeyAaiaab6gacaqG6aGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccadaaeWbqa aiaacIcacqaH7oaBdaWgaaWcbaGaamyAaaqabaGccqGHsislcaaIXa GaaiykaiaahwhadaahaaWcbeqaaiaacIcacaWGPbGaaiykaaaakiab gEPielaahwhadaahaaWcbeqaaiaacIcacaWGPbGaaiykaaaaaeaaca qGPbGaaeypaiaabgdaaeaacaqGZaaaniabggHiLdaakeaacaqGmbGa aeyyaiaabEgacaqGYbGaaeyyaiaab6gacaqGNbGaaeyAaiaabggaca qGUbGaaeiiaiaabYeacaqGVbGaae4zaiaabggacaqGYbGaaeyAaiaa bshacaqGObGaaeyBaiaabMgacaqGJbGaaeiiaiaabohacaqG0bGaae OCaiaabggacaqGPbGaaeOBaiaabQdacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cadaaeWbqaaiGacYgacaGGVbGaai4zaiaacIcacqaH7oaBdaWgaaWc baGaamyAaaqabaGccaGGPaGaaCyDamaaCaaaleqabaGaaiikaiaadM gacaGGPaaaaOGaey4LIqSaaCyDamaaCaaaleqabaGaaiikaiaadMga caGGPaaaaaqaaiaabMgacaqG9aGaaeymaaqaaiaabodaa0GaeyyeIu [email protected]@

The correspoinding Eulerian strain measures are

Eulerian Nominal strain:                  i=1 3 ( λ i 1) v (i) v (i) Eulerian Logarithmic strain:             i=1 3 log( λ i ) v (i) v (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaabweacaqG1bGaaeiBaiaabw gacaqGYbGaaeyAaiaabggacaqGUbGaaeiiaiaab6eacaqGVbGaaeyB aiaabMgacaqGUbGaaeyyaiaabYgacaqGGaGaae4CaiaabshacaqGYb GaaeyyaiaabMgacaqGUbGaaeOoaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaWaaabCaeaacaGGOaGaeq4U dW2aaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaaGymaiaacMcacaWH2b WaaWbaaSqabeaacaGGOaGaamyAaiaacMcaaaGccqGHxkcXcaWH2bWa aWbaaSqabeaacaGGOaGaamyAaiaacMcaaaaabaGaaeyAaiaab2daca qGXaaabaGaae4maaqdcqGHris5aaGcbaGaaeyraiaabwhacaqGSbGa aeyzaiaabkhacaqGPbGaaeyyaiaab6gacaqGGaGaaeitaiaab+gaca qGNbGaaeyyaiaabkhacaqGPbGaaeiDaiaabIgacaqGTbGaaeyAaiaa bogacaqGGaGaae4CaiaabshacaqGYbGaaeyyaiaabMgacaqGUbGaae OoaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiamaaqahabaGaciiBaiaac+gaca GGNbGaaiikaiabeU7aSnaaBaaaleaacaWGPbaabeaakiaacMcacaWH 2bWaaWbaaSqabeaacaGGOaGaamyAaiaacMcaaaGccqGHxkcXcaWH2b WaaWbaaSqabeaacaGGOaGaamyAaiaacMcaaaaabaGaaeyAaiaab2da [email protected]@

Another strain measure can be defined as

Green's strain:            E G i=1 3 1 2 ( λ i 2 1) v (i) v (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqGhbGaaeOCaiaabwgacaqGLbGaae OBaiaabEcacaqGZbGaaeiiaiaabohacaqG0bGaaeOCaiaabggacaqG PbGaaeOBaiaabQdacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaahweadaWgaaWcbaGa am4raaqabaGccaqG9aGaaeiiamaaqahabaWaaSaaaeaacaaIXaaaba GaaGOmaaaacaGGOaGaeq4UdW2aa0baaSqaaiaadMgaaeaacaaIYaaa aOGaeyOeI0IaaGymaiaacMcacaWH2bWaaWbaaSqabeaacaGGOaGaam yAaiaacMcaaaGccqGHxkcXcaWH2bWaaWbaaSqabeaacaGGOaGaamyA aiaacMcaaaaabaGaaeyAaiaab2dacaqGXaaabaGaae4maaqdcqGHri [email protected]@

This can be computed directly from the deformation gradient as

E G 1 2 ( F F T I ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHfbWaaSbaaSqaaiaadEeaaeqaaO GaaeypaiaabccadaWcaaqaaiaabgdaaeaacaqGYaaaamaabmaabaGa aCOraiabgwSixlaahAeadaahaaWcbeqaaiaadsfaaaGccqGHsislca [email protected]@

and is very similar to the Lagrangean strain tensor, except that its principal directions are rotated through the rigid rotation R.

 

 

3.20  Measure of rate of deformation - the velocity gradient

 

We now list several measures of the rate of deformation. The velocity gradient is the basic measure of deformation rate, and is defined as

L= y v L ij = v i y j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHmbGaeyypa0Jaey4bIe9aaSbaaS qaaiaahMhaaeqaaOGaaCODaiabggMi6kaadYeadaWgaaWcbaGaamyA aiaadQgaaeqaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG2bWaaSbaaS qaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGQbaa [email protected]@

It quantifies the relative velocities of two material particles at positions y and y+dy in the deformed solid, in the sense that

d v i = v i (y+dy) v i (y)= v i y j d y j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamODamaaBaaaleaacaWGPb aabeaakiabg2da9iaadAhadaWgaaWcbaGaamyAaaqabaGccaGGOaGa aCyEaiabgUcaRiaadsgacaWH5bGaaiykaiabgkHiTiaadAhadaWgaa WcbaGaamyAaaqabaGccaGGOaGaaCyEaiaacMcacqGH9aqpdaWcaaqa aiabgkGi2kaadAhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITca WG5bWaaSbaaSqaaiaadQgaaeqaaaaakiaadsgacaWG5bWaaSbaaSqa [email protected]@

The velocity gradient can be expressed in terms of the deformation gradient and its time derivative as

y v= F ˙ F 1 v i y j = F ˙ ik F kj 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHhis0daWgaaWcbaGaaCyEaaqaba GccaWH2bGaeyypa0JabCOrayaacaGaeyyXICTaaCOramaaCaaaleqa baGaeyOeI0IaaGymaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaSaaaeaacqGH ciITcaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEam aaBaaaleaacaWGQbaabeaaaaGccqGH9aqpceWGgbGbaiaadaWgaaWc baGaamyAaiaadUgaaeqaaOGaamOramaaDaaaleaacaWGRbGaamOAaa [email protected]@

To see this, note that

d v i = d dt d y i = d dt ( F ij d x j )= F ˙ ij d x j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamODamaaBaaaleaacaWGPb aabeaakiabg2da9maalaaabaGaamizaaqaaiaadsgacaWG0baaaiaa dsgacaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaaca WGKbaabaGaamizaiaadshaaaWaaeWaaeaacaWGgbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaadsgacaWG4bWaaSbaaSqaaiaadQgaaeqaaa GccaGLOaGaayzkaaGaeyypa0JabmOrayaacaWaaSbaaSqaaiaadMga [email protected]@

and recall that d y j = F ji d x i d x j = F jk 1 d y k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyEamaaBaaaleaacaWGQb aabeaakiabg2da9iaadAeadaWgaaWcbaGaamOAaiaadMgaaeqaaOGa amizaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHshI3caWGKbGaam iEamaaBaaaleaacaWGQbaabeaakiabg2da9iaadAeadaqhaaWcbaGa amOAaiaadUgaaeaacqGHsislcaaIXaaaaOGaamizaiaadMhadaWgaa [email protected]@ , so that

d v i = F ˙ ij F jk 1 d y k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamODamaaBaaaleaacaWGPb aabeaakiabg2da9iqadAeagaGaamaaBaaaleaacaWGPbGaamOAaaqa baGccaWGgbWaa0baaSqaaiaadQgacaWGRbaabaGaeyOeI0IaaGymaa [email protected]@

 

 

 

3.21 Stretch rate and spin (vorticity) tensors

 

The stretch rate tensor is defined as D=( L+ L T )/2 D ij =( L ij + L ji )/2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHebGaeyypa0ZaaeWaaeaacaWHmb Gaey4kaSIaaCitamaaCaaaleqabaGaamivaaaaaOGaayjkaiaawMca aiaac+cacaaIYaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaadseadaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaeyypa0ZaaeWaaeaacaWGmbWaaSbaaSqaaiaadMgaca WGQbaabeaakiabgUcaRiaadYeadaWgaaWcbaGaamOAaiaadMgaaeqa [email protected]@

The spin tensor or Vorticity tensor is defined as W=( L L T )/2 W ij =( L ij L ji )/2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHxbGaeyypa0ZaaeWaaeaacaWHmb GaeyOeI0IaaCitamaaCaaaleqabaGaamivaaaaaOGaayjkaiaawMca aiaac+cacaaIYaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaadEfadaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaeyypa0ZaaeWaaeaacaWGmbWaaSbaaSqaaiaadMgaca WGQbaabeaakiabgkHiTiaadYeadaWgaaWcbaGaamOAaiaadMgaaeqa [email protected]@

 

A general velocity gradient can be decomposed into the sum of stretch rate and spin, as

L=D+W L ij = D ij + W ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHmbGaeyypa0JaaCiraiabgUcaRi aahEfacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua amitamaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGebWaaS baaSqaaiaadMgacaWGQbaabeaakiabgUcaRiaadEfadaWgaaWcbaGa [email protected]@

The stretch rate quantifies the rate of stretching of material fibers in the deformed solid, in the sense that

1 l dl dt =nDn= n i D ij n j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaaigdaaeaacaWGSbaaam aalaaabaGaamizaiaadYgaaeaacaWGKbGaamiDaaaacqGH9aqpcaWH UbGaeyyXICTaaCiraiabgwSixlaah6gacqGH9aqpcaWGUbWaaSbaaS qaaiaadMgaaeqaaOGaamiramaaBaaaleaacaWGPbGaamOAaaqabaGc [email protected]@

is the rate of stretching of a material fiber with length l and orientation n in the deformed solid.  To see this, let dy=ln [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCyEaiabg2da9iaadYgaca [email protected]@ , so that

d dt dy= dl dt n+l dn dt [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgaaeaacaWGKbGaam iDaaaacaWGKbGaaCyEaiabg2da9maalaaabaGaamizaiaadYgaaeaa caWGKbGaamiDaaaacaWHUbGaey4kaSIaamiBamaalaaabaGaamizai [email protected]@

By definition,

d dt dy= d dt ( Fdx )= F ˙ dx= F ˙ ( F 1 dy )= F ˙ F 1 dy=Ldy=(D+W)ln [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgaaeaacaWGKbGaam iDaaaacaWGKbGaaCyEaiabg2da9maalaaabaGaamizaaqaaiaadsga caWG0baaamaabmaabaGaaCOraiabgwSixlaadsgacaWH4baacaGLOa GaayzkaaGaeyypa0JabCOrayaacaGaeyyXICTaamizaiaahIhacqGH 9aqpceWHgbGbaiaacqGHflY1daqadaqaaiaahAeadaahaaWcbeqaai abgkHiTiaaigdaaaGccaWGKbGaaCyEaaGaayjkaiaawMcaaiabg2da 9iqahAeagaGaaiaahAeadaahaaWcbeqaaiabgkHiTiaaigdaaaGccq GHflY1caWGKbGaaCyEaiabg2da9iaahYeacqGHflY1caWGKbGaaCyE aiabg2da9iaacIcacaWHebGaey4kaSIaaC4vaiaacMcacqGHflY1ca [email protected]@

Hence

(D+W)ln= dl dt n+l dn dt [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaaCiraiabgUcaRiaahEfaca GGPaGaeyyXICTaamiBaiaah6gacqGH9aqpdaWcaaqaaiaadsgacaWG SbaabaGaamizaiaadshaaaGaaCOBaiabgUcaRiaadYgadaWcaaqaai [email protected]@

Finally, take the dot product of both sides with n, note that since n is a unit vector dn/dt [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCOBaiaac+cacaWGKbGaam [email protected]@  must be perpendicular to n and therefore ndn/dt=0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbGaeyyXICTaamizaiaah6gaca [email protected]@ .  Note also that nWn=0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbGaeyyXICTaaC4vaiabgwSixl [email protected]@ , since W is skew-symmetric.  It is easiest to show this using index notation: n i W ij n j = n i ( L ij L ji ) n j /2=0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO Gaam4vamaaBaaaleaacaWGPbGaamOAaaqabaGccaWGUbWaaSbaaSqa aiaadQgaaeqaaOGaeyypa0JaamOBamaaBaaaleaacaWGPbaabeaaki aacIcacaWGmbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTiaa dYeadaWgaaWcbaGaamOAaiaadMgaaeqaaOGaaiykaiaad6gadaWgaa [email protected]@ .  Therefore

n(D+W)ln= dl dt nn+ln dn dt nDln= dl dt [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbGaeyyXICTaaiikaiaahseacq GHRaWkcaWHxbGaaiykaiabgwSixlaadYgacaWHUbGaeyypa0ZaaSaa aeaacaWGKbGaamiBaaqaaiaadsgacaWG0baaaiaah6gacqGHflY1ca WHUbGaey4kaSIaamiBaiaah6gacqGHflY1daWcaaqaaiaadsgacaWH UbaabaGaamizaiaadshaaaGaeyO0H4TaaCOBaiabgwSixlaahseacq GHflY1caWGSbGaaCOBaiabg2da9maalaaabaGaamizaiaadYgaaeaa [email protected]@

 

The spin tensor W can be shown to provide a measure of the average angular velocity of all material fibers passing through a material point. 

 

The vorticity vector is another measure of the angular velocity.  It is defined as

w=curl(v) w i = ijk v k y j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahEhacqGH9aqpcaWGJbGaamyDaiaadk hacaWGSbGaaiikaiaahAhacaGGPaGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaadEhadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcqGHiiIZ daWgaaWcbaGaamyAaiaadQgacaWGRbaabeaakmaalaaabaGaeyOaIy RaamODamaaBaaaleaacaWGRbaabeaaaOqaaiabgkGi2kaadMhadaWg [email protected]@

 

It is related to the spin tensor as

ω=2dual(W) ω i = ijk W jk [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8acqGH9aqpcaaIYaGaamizaiaadw hacaWGHbGaamiBaiaacIcacaWHxbGaaiykaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaeqyYdC3aaSbaaSqaaiaadMgaaeqaaOGa eyypa0JaeyOeI0IaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4Aaa [email protected]@

Where dual (W) denotes the dual vector of the skew tensor W.

 

The vorticity vector has the property that, for any vector g, Wg= 1 2 ω×g W ji g i = 1 2 jki ω k g i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahEfacaWHNbGaeyypa0ZaaSaaaeaaca aIXaaabaGaaGOmaaaacaWHjpGaey41aqRaaC4zaiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caWGxbWaaSbaaSqaaiaadQgacaWGPbaabeaa kiaadEgadaWgaaWcbaGaamyAaaqabaGccqGH9aqpdaWcaaqaaiaaig daaeaacaaIYaaaaiabgIGiopaaBaaaleaacaWGQbGaam4AaiaadMga aeqaaOGaeqyYdC3aaSbaaSqaaiaadUgaaeqaaOGaam4zamaaBaaale [email protected]@ .

 

A motion satisfying W=curl(v)= 0 is said to be irrotational [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqacKqzGfaeaa [email protected]@  such motions are of interest in fluid mechanics.

 

 

 

 

3.22 Spatial (Eulerian) description of acceleration

 

The acceleration of a material particle is, by definition

a= v t | x=const [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpdaabcaqaamaalaaaba GaeyOaIyRaaCODaaqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWc baGaaCiEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabe [email protected]@

 

In fluid mechanics, it is often convenient to use a spatial description of velocity and acceleration [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  that is to say the velocity field is expressed as a function of position y in the deformed solid as v(y,t) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacaGGOaGaaCyEaiaacYcacaWG0b [email protected]@ .   The acceleration of the material particle with instantaneous position y in the deformed solid can be expressed as

a i = v i y k y k t + v i t | y i =const = L ik v k + v i t | y i =const =( D ik + W ik ) v k + v i t | y i =const =( D ik + W ik ) v k + v i t | y i =const [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyyamaaBaaaleaacaWGPbaabe aakiabg2da9maalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGPbaa beaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaam4AaaqabaaaaOWaaS aaaeaacqGHciITcaWG5bWaaSbaaSqaaiaadUgaaeqaaaGcbaGaeyOa IyRaamiDaaaacqGHRaWkdaabcaqaamaalaaabaGaeyOaIyRaamODam aaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadshaaaaacaGLiWoa daWgaaWcbaGaamyEamaaBaaameaacaWGPbaabeaaliabg2da9iaado gacaWGVbGaamOBaiaadohacaWG0baabeaakiabg2da9iaadYeadaWg aaWcbaGaamyAaiaadUgaaeqaaOGaamODamaaBaaaleaacaWGRbaabe aakiabgUcaRmaaeiaabaWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqa aiaadMgaaeqaaaGcbaGaeyOaIyRaamiDaaaaaiaawIa7amaaBaaale aacaWG5bWaaSbaaWqaaiaadMgaaeqaaSGaeyypa0Jaam4yaiaad+ga caWGUbGaam4CaiaadshaaeqaaaGcbaGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlabg2da9maabmaabaGaamiramaaBaaa leaacaWGPbGaam4AaaqabaGccqGHRaWkcaWGxbWaaSbaaSqaaiaadM gacaWGRbaabeaaaOGaayjkaiaawMcaaiaadAhadaWgaaWcbaGaam4A aaqabaGccqGHRaWkdaabcaqaamaalaaabaGaeyOaIyRaamODamaaBa aaleaacaWGPbaabeaaaOqaaiabgkGi2kaadshaaaaacaGLiWoadaWg aaWcbaGaamyEamaaBaaameaacaWGPbaabeaaliabg2da9iaadogaca WGVbGaamOBaiaadohacaWG0baabeaakiabg2da9maabmaabaGaamir amaaBaaaleaacaWGPbGaam4AaaqabaGccqGHRaWkcaWGxbWaaSbaaS qaaiaadMgacaWGRbaabeaaaOGaayjkaiaawMcaaiaadAhadaWgaaWc baGaam4AaaqabaGccqGHRaWkdaabcaqaamaalaaabaGaeyOaIyRaam ODamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadshaaaaacaGL iWoadaWgaaWcbaGaamyEamaaBaaameaacaWGPbaabeaaliabg2da9i [email protected]@

 

 

3.23 Acceleration - spin [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa [email protected]@  vorticity relations

 

In fluid mechanics, equations relating the acceleration to the spatial velocity field are useful.  In particular, it can be shown that

*   a i = v i t | x k =const = v i t | y k =const + 1 2 y i ( v k v k )+2 W ik v k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaabcaqaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWG PbaabeaaaOqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaam iEamaaBaaameaacaWGRbaabeaaliabg2da9iaadogacaWGVbGaamOB aiaadohacaWG0baabeaakiabg2da9maaeiaabaWaaSaaaeaacqGHci ITcaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiDaaaa aiaawIa7amaaBaaaleaacaWG5bWaaSbaaWqaaiaadUgaaeqaaSGaey ypa0Jaam4yaiaad+gacaWGUbGaam4CaiaadshaaeqaaOGaey4kaSYa aSaaaeaacaaIXaaabaGaaGOmaaaadaWcaaqaaiabgkGi2cqaaiabgk Gi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaaiikaiaadAhadaWg aaWcbaGaam4AaaqabaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaai ykaiabgUcaRiaaikdacaWGxbWaaSbaaSqaaiaadMgacaWGRbaabeaa [email protected]@

*  a i = v i t | x k =const = v i t | y k =const + 1 2 y i ( v k v k )+ ijk ω j v k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaabcaqaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWG PbaabeaaaOqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaam iEamaaBaaameaacaWGRbaabeaaliabg2da9iaadogacaWGVbGaamOB aiaadohacaWG0baabeaakiabg2da9maaeiaabaWaaSaaaeaacqGHci ITcaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiDaaaa aiaawIa7amaaBaaaleaacaWG5bWaaSbaaWqaaiaadUgaaeqaaSGaey ypa0Jaam4yaiaad+gacaWGUbGaam4CaiaadshaaeqaaOGaey4kaSYa aSaaaeaacaaIXaaabaGaaGOmaaaadaWcaaqaaiabgkGi2cqaaiabgk Gi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaaiikaiaadAhadaWg aaWcbaGaam4AaaqabaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaai ykaiabgUcaRiabgIGiopaaBaaaleaacaWGPbGaamOAaiaadUgaaeqa aOGaeqyYdC3aaSbaaSqaaiaadQgaaeqaaOGaamODamaaBaaaleaaca [email protected]@

*  ijk a k y j = ω i t | x=const D ij ω j + v k y k ω i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgIGiopaaBaaaleaacaWGPbGaamOAai aadUgaaeqaaOWaaSaaaeaacqGHciITcaWGHbWaaSbaaSqaaiaadUga aeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGQbaabeaaaaGccq GH9aqpdaabcaqaamaalaaabaGaeyOaIyRaeqyYdC3aaSbaaSqaaiaa dMgaaeqaaaGcbaGaeyOaIyRaamiDaaaaaiaawIa7amaaBaaaleaaca WH4bGaeyypa0Jaam4yaiaad+gacaWGUbGaam4CaiaadshaaeqaaOGa eyOeI0IaamiramaaBaaaleaacaWGPbGaamOAaaqabaGccqaHjpWDda WgaaWcbaGaamOAaaqabaGccqGHRaWkdaWcaaqaaiabgkGi2kaadAha daWgaaWcbaGaam4AaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaai [email protected]@

 

Deriving these relations is left as an exercise.

 

 

 

3.24 Rate of change of volume

 

We have seen that

J=det(F) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacqGH9aqpciGGKbGaaiyzaiaacs [email protected]@

quantifies the volume change associated with a deformation, in that

JdV=d V 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacaWGKbGaamOvaiabg2da9iaads [email protected]@

 

In rate form:

dJ dt = dJ d F ij d F ij dt =J F ji 1 F ˙ ij =J L ii =J v i y i =J D ii [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWGkbaabaGaam izaiaadshaaaGaeyypa0ZaaSaaaeaacaWGKbGaamOsaaqaaiaadsga caWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaaaaGcdaWcaaqaaiaads gacaWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOqaaiaadsgacaWG 0baaaiabg2da9iaadQeacaWGgbWaa0baaSqaaiaadQgacaWGPbaaba GaeyOeI0IaaGymaaaakiqadAeagaGaamaaBaaaleaacaWGPbGaamOA aaqabaGccqGH9aqpcaWGkbGaamitamaaBaaaleaacaWGPbGaamyAaa qabaGccqGH9aqpcaWGkbWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqa aiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabe aaaaGccqGH9aqpcaWGkbGaamiramaaBaaaleaacaWGPbGaamyAaaqa [email protected]@ .

 

The trace of D, trace of L or the trace of grad(v)  are therefore measures of rate of change of volume.

 

 

 

 

3.25 Infinitesimal strain rate and rotation rate

 

For small strains the rate of deformation tensor is approximately equal to the infinitesimal strain rate, while the spin can be approximated by the time derivative of the infinitesimal rotation tensor

d dt ε= d dt 1 2 ( u+ ( u ) T )Dor      ε ˙ ij D ij d dt w= d dt 1 2 ( u ( u ) T )Wor      w ˙ ij W ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaalaaabaGaamizaaqaaiaads gacaWG0baaaiaahw7acqGH9aqpdaWcaaqaaiaadsgaaeaacaWGKbGa amiDaaaadaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaGaaCyDai abgEPielabgEGirlabgUcaRmaabmaabaGaaCyDaiabgEPielabgEGi rdGaayjkaiaawMcaamaaCaaaleqabaGaamivaaaaaOGaayjkaiaawM caaiaaykW7caaMc8UaaGPaVlabgIKi7kaahseacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8Uaae4BaiaabkhacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiqbew7aLzaacaWaaSbaaSqaaiaadMga caWGQbaabeaakiabgIKi7kaadseadaWgaaWcbaGaamyAaiaadQgaae qaaaGcbaWaaSaaaeaacaWGKbaabaGaamizaiaadshaaaGaaC4Daiab g2da9maalaaabaGaamizaaqaaiaadsgacaWG0baaamaalaaabaGaaG ymaaqaaiaaikdaaaWaaeWaaeaacaWH1bGaey4LIqSaey4bIeTaeyOe I0YaaeWaaeaacaWH1bGaey4LIqSaey4bIenacaGLOaGaayzkaaWaaW baaSqabeaacaWGubaaaaGccaGLOaGaayzkaaGaeyisISRaaC4vaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caqGVbGaae OCaiaabccacaqGGaGaaeiiaiaabccacaqGGaGabm4DayaacaWaaSba aSqaaiaadMgacaWGQbaabeaakiabgIKi7kaadEfadaWgaaWcbaGaam [email protected]@

The approximation is because the infinitesimal strain and rotation involve derivatives with respect to position in the reference configuration, while the stretch rate and spin tensors are defined in terms of spatial derivatives.   Similarly, you can show that

                                                    d dt u i x j = F ˙ ij = ε ˙ ij + w ˙ ij L ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgaaeaacaWGKbGaam iDaaaadaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaaqabaaa keaacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaakiabg2da9i qadAeagaGaamaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcuaH 1oqzgaGaamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHRaWkceWG3b GbaiaadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyisISRaamitamaa [email protected]@

 

 

 

3.26 Other deformation rate measures

 

The rate of deformation tensor can be related to time derivatives of other strain measures.  For example the time derivative of the Lagrange strain tensor can be shown to be

dE dt = F T DF E ˙ ij = F ki D kl F lj [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHfbaabaGaam izaiaadshaaaGaeyypa0JaaCOramaaCaaaleqabaGaamivaaaakiab gwSixlaahseacqGHflY1caWHgbGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UabmyrayaacaWaaSbaaSqa aiaadMgacaWGQbaabeaakiabg2da9iaadAeadaWgaaWcbaGaam4Aai aadMgaaeqaaOGaamiramaaBaaaleaacaWGRbGaamiBaaqabaGccaWG [email protected]@

Other useful results are

 For a pure rotation R ˙ R T +R R ˙ T =0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHsbGbaiaacqGHflY1caWHsbWaaW baaSqabeaacaWGubaaaOGaey4kaSIaaCOuaiabgwSixlqahkfagaGa [email protected]@ , or equivalently R ˙ R T = ( R ˙ R T ) T [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHsbGbaiaacqGHflY1caWHsbWaaW baaSqabeaacaWGubaaaOGaeyypa0JaeyOeI0YaaeWaaeaaceWHsbGb aiaacqGHflY1caWHsbWaaWbaaSqabeaacaWGubaaaaGccaGLOaGaay [email protected]@ .  To see this, recall that R R T =I [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHsbGaaCOuamaaCaaaleqabaGaam [email protected]@  and evaluate the time derivative. 

 If the deformation gradient is decomposed into a stretch followed by a rotation as F=RU [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOuaiabgwSixl [email protected]@  then D=R( U ˙ U 1 + U 1 U ˙ ) R T /2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHebGaeyypa0JaaCOuaiabgwSixp aabmaabaGabCyvayaacaGaeyyXICTaaCyvamaaCaaaleqabaGaeyOe I0IaaGymaaaakiabgUcaRiaahwfadaahaaWcbeqaaiabgkHiTiaaig daaaGccqGHflY1ceWHvbGbaiaaaiaawIcacaGLPaaacqGHflY1caWH [email protected]@  and W= R ˙ R T +R( U ˙ U 1 U 1 U ˙ ) R T /2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHxbGaeyypa0JabCOuayaacaGaey yXICTaaCOuamaaCaaaleqabaGaamivaaaakiabgUcaRiaahkfacqGH flY1daqadaqaaiqahwfagaGaaiabgwSixlaahwfadaahaaWcbeqaai abgkHiTiaaigdaaaGccqGHsislcaWHvbWaaWbaaSqabeaacqGHsisl caaIXaaaaOGaeyyXICTabCyvayaacaaacaGLOaGaayzkaaGaeyyXIC [email protected]@

 

For small strains the rate of change of  Lagrangian strain E  is approximately equal to the rate of change of infinitesimal strain ε [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ :

dE dt d dt ε E ˙ ij d dt ε ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHfbaabaGaam izaiaadshaaaGaeyisIS7aaSaaaeaacaWGKbaabaGaamizaiaadsha aaGaaCyTdiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlqadweagaGaamaaBaaaleaacaWGPbGaamOAaaqa baGccqGHijYUdaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaacqaH1o [email protected]@

 

3.27 Path lines, streamlines, and vortex lines

 

Path lines, streamlines, and vortex lines are useful concepts in fluid mechanics.

 

* A path line is the curve traced by a material particle as it moves through space.   If the curve is described in parametric form by y i (λ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGcca [email protected]@ , with λ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  a scalar, then the curve satisfies

d y i (t) dt = v i (X,t) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadMhadaWgaaWcba GaamyAaaqabaGccaGGOaGaamiDaiaacMcaaeaacaWGKbGaamiDaaaa cqGH9aqpcaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaahIfaca [email protected]@

 

* A stream line is a curve that is everywhere tangent to the spatial velocity vector.   In general, streamlines may be functions of time.  If y i (λ,t) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGcca [email protected]@  is the parametric representation of the curve, at time t , then y i (λ,t) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGcca [email protected]@  is a member of the family of solutions to the differential equation

d y i (λ,t) dλ = v i ( y(λ),t ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadMhadaWgaaWcba GaamyAaaqabaGccaGGOaGaeq4UdWMaaiilaiaadshacaGGPaaabaGa amizaiabeU7aSbaacqGH9aqpcaWG2bWaaSbaaSqaaiaadMgaaeqaaO WaaeWaaeaacaWH5bGaaiikaiabeU7aSjaacMcacaGGSaGaamiDaaGa [email protected]@

For the particular case of a steady flow, the spatial velocity field is (by definition) independent of time, and therefore the curves are fixed in space.

 

* A vortex line is a curve that is everywhere tangent to the vorticity vector.   These curves satisfy the differential equation

d y i (λ,t) dλ = ω i ( y(λ),t ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadMhadaWgaaWcba GaamyAaaqabaGccaGGOaGaeq4UdWMaaiilaiaadshacaGGPaaabaGa amizaiabeU7aSbaacqGH9aqpcqaHjpWDdaWgaaWcbaGaamyAaaqaba GcdaqadaqaaiaahMhacaGGOaGaeq4UdWMaaiykaiaacYcacaWG0baa [email protected]@

Again, for the special case of a steady flow the vortex lines are independent of time.

 

 

 

3.27 Reynolds Transport Relation

 

The Reynolds transport theorem is a useful way to calculate the rate of change of a quantity inside a volume that deforms with a solid (e.g. the total mass of a volume).  Let ϕ(y,t) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjaacIcacaWH5bGaaiilaiaads [email protected]@  be any scalar valued property of a material particle at position y in the deformed solid.   The Reynolds transport relation states that rate of change of the total value of this property within a volume V of a deformed solid can be calculated as

d dt V ϕdV = V ( ϕ t | x=const +ϕ v i y i ) dV= V ( ϕ t | x=const +ϕ D kk ) dV= V ( ϕ t | y=const ) dV+ S ( ϕ v k n k ) dA [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqy1dyMaamizaiaadAfaaSqaaiaadAfaaeqaniab gUIiYdGccqGH9aqpdaWdrbqaamaabmaabaWaaqGaaeaadaWcaaqaai abgkGi2kabew9aMbqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWc baGaaCiEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabe aakiabgUcaRiabew9aMnaalaaabaGaeyOaIyRaamODamaaBaaaleaa caWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqaba aaaaGccaGLOaGaayzkaaaaleaacaWGwbaabeqdcqGHRiI8aOGaamiz aiaadAfacqGH9aqpdaWdrbqaamaabmaabaWaaqGaaeaadaWcaaqaai abgkGi2kabew9aMbqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWc baGaaCiEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabe aakiabgUcaRiabew9aMjaadseadaWgaaWcbaGaam4AaiaadUgaaeqa aaGccaGLOaGaayzkaaaaleaacaWGwbaabeqdcqGHRiI8aOGaamizai aadAfacqGH9aqpdaWdrbqaamaabmaabaWaaqGaaeaadaWcaaqaaiab gkGi2kabew9aMbqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcba GaaCyEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaa aOGaayjkaiaawMcaaaWcbaGaamOvaaqab0Gaey4kIipakiaadsgaca WGwbGaey4kaSYaa8quaeaadaqadaqaaiabew9aMjaadAhadaWgaaWc baGaam4AaaqabaGccaWGUbWaaSbaaSqaaiaadUgaaeqaaaGccaGLOa [email protected]@

 

Note that the material volume V and surface S convect with the deforming solid [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  they are not control volumes.

 

To see this, note that we can’t take the time derivative inside the integral because the volume changes with time as the solid deforms.   But we can map the integral back to the reference configuration, which is time independent [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  the derivative can then be taken inside the integral.

d dt V ϕdV = d dt V 0 ϕJdV = V 0 ( J ϕ t | x=const +ϕ J t ) dV = V 0 ( ϕ t | x=const +ϕ D kk ) JdV= V ( ϕ t | x=const +ϕ D kk ) dV [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacaWGKbaabaGaamizai aadshaaaWaa8quaeaacqaHvpGzcaWGKbGaamOvaaWcbaGaamOvaaqa b0Gaey4kIipakiabg2da9maalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqy1dyMaamOsaiaadsgacaWGwbaaleaacaWGwbWa aSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aOGaeyypa0Zaa8quae aadaqadaqaamaaeiaabaGaamOsamaalaaabaGaeyOaIyRaeqy1dyga baGaeyOaIyRaamiDaaaaaiaawIa7amaaBaaaleaacaWH4bGaeyypa0 Jaam4yaiaad+gacaWGUbGaam4CaiaadshaaeqaaOGaey4kaSIaeqy1 dy2aaSaaaeaacqGHciITcaWGkbaabaGaeyOaIyRaamiDaaaaaiaawI cacaGLPaaaaSqaaiaadAfadaWgaaadbaGaaGimaaqabaaaleqaniab gUIiYdGccaWGKbGaamOvaaqaaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabg2da9maapefabaWaaeWa aeaadaabcaqaamaalaaabaGaeyOaIyRaeqy1dygabaGaeyOaIyRaam iDaaaaaiaawIa7amaaBaaaleaacaWH4bGaeyypa0Jaam4yaiaad+ga caWGUbGaam4CaiaadshaaeqaaOGaey4kaSIaeqy1dyMaamiramaaBa aaleaacaWGRbGaam4AaaqabaaakiaawIcacaGLPaaaaSqaaiaadAfa daWgaaadbaGaaGimaaqabaaaleqaniabgUIiYdGccaWGkbGaamizai aadAfacqGH9aqpdaWdrbqaamaabmaabaWaaqGaaeaadaWcaaqaaiab gkGi2kabew9aMbqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcba GaaCiEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaa kiabgUcaRiabew9aMjaadseadaWgaaWcbaGaam4AaiaadUgaaeqaaa GccaGLOaGaayzkaaaaleaacaWGwbaabeqdcqGHRiI8aOGaamizaiaa [email protected]@

 

The last result follows by noting that ϕ t | x=const = ϕ t | y=const + ϕ y i v i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaWaaSaaaeaacqGHciITcqaHvp GzaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaahIhacqGH 9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccqGH9aqpda abcaqaamaalaaabaGaeyOaIyRaeqy1dygabaGaeyOaIyRaamiDaaaa aiaawIa7amaaBaaaleaacaWH5bGaeyypa0Jaam4yaiaad+gacaWGUb Gaam4CaiaadshaaeqaaOGaey4kaSYaaSaaaeaacqGHciITcqaHvpGz aeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiaadAhada [email protected]@ .  Then note that ϕ y i v i +ϕ v i y i = (ϕ v i ) y i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeqy1dygabaGaey OaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGccaWG2bWaaSbaaSqa aiaadMgaaeqaaOGaey4kaSIaeqy1dy2aaSaaaeaacqGHciITcaWG2b WaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaa caWGPbaabeaaaaGccqGH9aqpdaWcaaqaaiabgkGi2kaacIcacqaHvp GzcaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaqaaiabgkGi2kaa [email protected]@  and apply the divergence theorem to this term.

 

3.28 Transport Relations for material curves and surfaces

 

Similar transport relations can be derived for material curves and surfaces which convect with a deformable solid or fluid.

 

Let C be a material curve in a deformable solid; and let S be an interior surface with normal vector n.  Let ϕ(y,t) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjaacIcacaWH5bGaaiilaiaads [email protected]@  be any scalar valued property of a material particle at position y in the deformed solid.   Then

  1. d dt C ϕ τ i ds = C ( δ ij ϕ t | x=const +ϕ v i y j ) τ j ds [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqy1dyMaeqiXdq3aaSbaaSqaaiaadMgaaeqaaOGa amizaiaadohaaSqaaiaadoeaaeqaniabgUIiYdGccqGH9aqpdaWdrb qaamaabmaabaWaaqGaaeaacqaH0oazdaWgaaWcbaGaamyAaiaadQga aeqaaOWaaSaaaeaacqGHciITcqaHvpGzaeaacqGHciITcaWG0baaaa GaayjcSdWaaSbaaSqaaiaahIhacqGH9aqpcaWGJbGaam4Baiaad6ga caWGZbGaamiDaaqabaGccqGHRaWkcqaHvpGzdaWcaaqaaiabgkGi2k aadAhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSba aSqaaiaadQgaaeqaaaaaaOGaayjkaiaawMcaaiabes8a0naaBaaale aacaWGQbaabeaaaeaacaWGdbaabeqdcqGHRiI8aOGaamizaiaadoha [email protected]@
  2. d dt S ϕ n i dA = S ( δ ij ϕ t | x=const + δ ij ϕ v k y k ϕ v j y i ) n j dA [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqy1dyMaamOBamaaBaaaleaacaWGPbaabeaakiaa dsgacaWGbbaaleaacaWGtbaabeqdcqGHRiI8aOGaeyypa0Zaa8quae aadaqadaqaamaaeiaabaGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaa beaakmaalaaabaGaeyOaIyRaeqy1dygabaGaeyOaIyRaamiDaaaaai aawIa7amaaBaaaleaacaWH4bGaeyypa0Jaam4yaiaad+gacaWGUbGa am4CaiaadshaaeqaaOGaey4kaSIaeqiTdq2aaSbaaSqaaiaadMgaca WGQbaabeaakiabew9aMnaalaaabaGaeyOaIyRaamODamaaBaaaleaa caWGRbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaam4Aaaqaba aaaOGaeyOeI0Iaeqy1dy2aaSaaaeaacqGHciITcaWG2bWaaSbaaSqa aiaadQgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabe aaaaaakiaawIcacaGLPaaacaWGUbWaaSbaaSqaaiaadQgaaeqaaaqa [email protected]@

 

To show the first result, start by mapping the integral to the reference configuration, then take the time derivative, and map back to the current configuration, as follows

d dt C ϕ τ i ds = d dt C 0 ϕ F ij τ j 0 d s 0 = C 0 ( dϕ dt | x F ij +ϕ d F ij dt ) τ j 0 d s 0 = C ( dϕ dt | x δ ik +ϕ d F ij dt F jk 1 ) τ k ds = C ( δ ij ϕ t | x=const +ϕ v i y j ) τ j ds [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacaWGKbaabaGaamizai aadshaaaWaa8quaeaacqaHvpGzcqaHepaDdaWgaaWcbaGaamyAaaqa baGccaWGKbGaam4CaaWcbaGaam4qaaqab0Gaey4kIipakiabg2da9m aalaaabaGaamizaaqaaiaadsgacaWG0baaamaapefabaGaeqy1dyMa amOramaaBaaaleaacaWGPbGaamOAaaqabaGccqaHepaDdaqhaaWcba GaamOAaaqaaiaaicdaaaGccaWGKbGaam4CamaaBaaaleaacaaIWaaa beaaaeaacaWGdbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aO Gaeyypa0Zaa8quaeaadaqadaqaamaaeiaabaWaaSaaaeaacaWGKbGa eqy1dygabaGaamizaiaadshaaaaacaGLiWoadaWgaaWcbaGaaCiEaa qabaGccaWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgUcaRiab ew9aMnaalaaabaGaamizaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaaGcbaGaamizaiaadshaaaaacaGLOaGaayzkaaGaeqiXdq3aa0ba aSqaaiaadQgaaeaacaaIWaaaaOGaamizaiaadohadaWgaaWcbaGaaG imaaqabaaabaGaam4qamaaBaaameaacaaIWaaabeaaaSqab0Gaey4k IipaaOqaaiabg2da9maapefabaWaaeWaaeaadaabcaqaamaalaaaba Gaamizaiabew9aMbqaaiaadsgacaWG0baaaaGaayjcSdWaaSbaaSqa aiaahIhaaeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGRbaabeaaki abgUcaRiabew9aMnaalaaabaGaamizaiaadAeadaWgaaWcbaGaamyA aiaadQgaaeqaaaGcbaGaamizaiaadshaaaGaamOramaaDaaaleaaca WGQbGaam4AaaqaaiabgkHiTiaaigdaaaaakiaawIcacaGLPaaacqaH epaDdaWgaaWcbaGaam4AaaqabaGccaWGKbGaam4CaaWcbaGaam4qaa qab0Gaey4kIipakiabg2da9maapefabaWaaeWaaeaadaabcaqaaiab es7aKnaaBaaaleaacaWGPbGaamOAaaqabaGcdaWcaaqaaiabgkGi2k abew9aMbqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaaCiE aiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaakiabgU caRiabew9aMnaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGPbaa beaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaaaaGcca GLOaGaayzkaaGaeqiXdq3aaSbaaSqaaiaadQgaaeqaaaqaaiaadoea [email protected]@

To show the second, apply the same process to the surface integral.  The details are left as an exercise…

 

 

3.28 Circulation and the circulation transport relation

The circulation of the velocity field around a closed curve C is defined as

I C = C vτds [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaam4qaaqabaGccq GH9aqpdaWdrbqaaiaahAhacqGHflY1caWHepGaamizaiaadohaaSqa [email protected]@ ,

where τ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  is a unit vector tangent to the curve.  If C is a reducible curve (i.e. if there is a regular, open surface S bounded by C  that lies within the configuration) then Stokes theorem shows that

I C = C vτds = S curl(v)mdA [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaam4qaaqabaGccq GH9aqpdaWdrbqaaiaahAhacqGHflY1caWHepGaamizaiaadohaaSqa aiaadoeaaeqaniabgUIiYdGccqGH9aqpdaWdrbqaaiaadogacaWG1b GaamOCaiaadYgacaGGOaGaaCODaiaacMcacqGHflY1caWHTbGaamiz [email protected]@

The circulation transport relation states that

I c t | x=const = C v t | x=const τds [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaWaaSaaaeaacqGHciITcaWGjb WaaSbaaSqaaiaadogaaeqaaaGcbaGaeyOaIyRaamiDaaaaaiaawIa7 amaaBaaaleaacaWH4bGaeyypa0Jaam4yaiaad+gacaWGUbGaam4Cai aadshaaeqaaOGaeyypa0Zaa8quaeaadaabcaqaamaalaaabaGaeyOa IyRaaCODaaqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaaC iEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaakiab gwSixlaahs8acaWGKbGaam4CaaWcbaGaam4qaaqab0Gaey4kIipaaa [email protected]@

for any material curve (i.e. a curve that convects with material particles within a body).  To see this recall the transport relation for a material curve, and set ϕ= v i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjabg2da9iaadAhadaWgaaWcba [email protected]@

d dt C v i τ i ds = C ( v j t | x=const + v i v i y j ) τ j ds [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaamODamaaBaaaleaacaWGPbaabeaakiabes8a0naa BaaaleaacaWGPbaabeaakiaadsgacaWGZbaaleaacaWGdbaabeqdcq GHRiI8aOGaeyypa0Zaa8quaeaadaqadaqaamaaeiaabaWaaSaaaeaa cqGHciITcaWG2bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIyRaam iDaaaaaiaawIa7amaaBaaaleaacaWH4bGaeyypa0Jaam4yaiaad+ga caWGUbGaam4CaiaadshaaeqaaOGaey4kaSIaamODamaaBaaaleaaca WGPbaabeaakmaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGPbaa beaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaaaaGcca GLOaGaayzkaaGaeqiXdq3aaSbaaSqaaiaadQgaaeqaaaqaaiaadoea [email protected]@

Note that

v i v i y j = 1 2 ( v i v i ) y j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGcda WcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamyAaaqabaaakeaacqGH ciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaaaakiabg2da9maalaaaba GaaGymaaqaaiaaikdaaaWaaSaaaeaacqGHciITcaGGOaGaamODamaa BaaaleaacaWGPbaabeaakiaadAhadaWgaaWcbaGaamyAaaqabaGcca [email protected]@

and hence

C ( v i v i y j ) τ j ds= C d( v i v i ) ds ds=0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaWaaeWaaeaacaWG2bWaaSbaaS qaaiaadMgaaeqaaOWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaa dMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGQbaabeaaaa aakiaawIcacaGLPaaacqaHepaDdaWgaaWcbaGaamOAaaqabaaabaGa am4qaaqab0Gaey4kIipakiaadsgacaWGZbGaeyypa0Zaa8quaeaada WcaaqaaiaadsgacaGGOaGaamODamaaBaaaleaacaWGPbaabeaakiaa dAhadaWgaaWcbaGaamyAaaqabaGccaGGPaaabaGaamizaiaadohaaa aaleaacaWGdbaabeqdcqGHRiI8aOGaamizaiaadohacqGH9aqpcaaI [email protected]@

because C is a closed curve.

 

Kelvin’s circulation theorem is a direct consequence of this result.   The theorem states that if the acceleration is the gradient of a potential, then the circulation around any closed material curve remains constant.   To see this, let

v i t | x=const = ϕ y i I c t | x=const = C ϕ y i τ i ds = C ϕ s ds =0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaqGaaeaadaWcaaqaaiabgkGi2k aadAhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG0baaaaGa ayjcSdWaaSbaaSqaaiaahIhacqGH9aqpcaWGJbGaam4Baiaad6gaca WGZbGaamiDaaqabaGccqGH9aqpdaWcaaqaaiabgkGi2kabew9aMbqa aiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaaGcbaWaaqGaae aadaWcaaqaaiabgkGi2kaadMeadaWgaaWcbaGaam4yaaqabaaakeaa cqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaahIhacqGH9aqpca WGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccqGH9aqpdaWdrbqa amaalaaabaGaeyOaIyRaeqy1dygabaGaeyOaIyRaamyEamaaBaaale aacaWGPbaabeaaaaGccqGHflY1cqaHepaDdaWgaaWcbaGaamyAaaqa baGccaWGKbGaam4CaaWcbaGaam4qaaqab0Gaey4kIipakiabg2da9m aapefabaWaaSaaaeaacqGHciITcqaHvpGzaeaacqGHciITcaWGZbaa aiaadsgacaWGZbaaleaacaWGdbaabeqdcqGHRiI8aOGaeyypa0JaaG [email protected]@