4. Kinetics

 

 

 

 

Our next objective is to outline the mathematical formulas that describe internal and external forces acting on a solid.  Just as there are many different strain measures, there are several different definitions of internal force.  We shall see that internal forces can be described as a second order tensor, which must be symmetric.  Thus, internal forces can always be quantified by a set of six numbers, and the various different definitions are all equivalent.

 

4.1 Surface traction and internal body force

 

Forces can be applied to a solid body in two ways. 

(i) A force can be applied to its boundary: examples include fluid pressure, wind loading, or forces arising from contact with another solid. 

(ii) The solid can be subjected to body forces, which act on the interior of the solid.  Examples include gravitational loading, or electromagnetic forces.

 

 

 The surface traction vector t at a point on the surface represents the force acting on the surface per unit area of the deformed solid.

 

Formally, let dA be an element of area on a surface.  Suppose that dA is subjected to a force dP [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@ .  Then

t= lim dA0 dP dA [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH0bGaeyypa0ZaaCbeaeaaciGGSb GaaiyAaiaac2gaaSqaaiaadsgacaWGbbGaeyOKH4QaaGimaaqabaGc [email protected]@

The resultant force acting on any portion S of the surface of the deformed solid is

P= S tdA [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHqbGaeyypa0Zaa8quaeaacaWH0b [email protected]@

Surface traction, like `true stress,’ should be thought of as acting on the deformed solid.

 Normal and shear tractions

The traction vector is often resolved into components acting normal and tangential to a surface, as shown in the picture. 

 

The normal component is referred to as the normal traction, and the tangential component is known as the shear traction.

 

Formally, let n denote a unit vector normal to the surface.  Then

t n =( tn )n t t =t t n [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH0bWaaSbaaSqaaiaah6gaaeqaaO Gaeyypa0ZaaeWaaeaacaWH0bGaeyyXICTaaCOBaaGaayjkaiaawMca aiaah6gacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaCiDamaaBaaaleaacaWH0baabeaakiabg2da9iaahshacq [email protected]@

 

 

 The body force vector denotes the external force acting on the interior of a solid, per unit mass.

 

Formally, let dV denote an infinitesimal volume element within the deformed solid, and let ρ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  denote the mass density (mass per unit deformed volume).  Suppose that the element is subjected to a force dP [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@ .  Then

b= 1 ρ lim dV0 dP dV [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHIbGaeyypa0ZaaSaaaeaacaaIXa aabaGaeqyWdihaamaaxababaGaciiBaiaacMgacaGGTbaaleaacaWG KbGaamOvaiabgkziUkaaicdaaeqaaOWaaSaaaeaacaWGKbGaaCiuaa [email protected]@

The resultant body force acting on any volume V  within the deformed solid is

P= V ρbdV [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHqbGaeyypa0Zaa8quaeaacqaHbp GCcaaMc8UaaCOyaiaaykW7caWGKbGaamOvaaWcbaGaamOvaaqab0Ga [email protected]@

 

4.2 Traction acting on planes within a solid

 

Every plane in the interior of a solid is subjected to a distribution of traction.  To see this, consider a loaded, solid, body in static equilibrium.  Imagine cutting the solid in two.  The two parts of the solid must each be in static equilibrium.  This is possible only if forces act on the planes that were created by the cut.

 

 

 

 

 The internal traction vector T(n) represents the force per unit area acting on a section of the deformed body across a plane with outer normal vector n.

 

Formally, let dA be an element of area in the interior of the solid, with normal n.  Suppose that the material on the underside of dA is subjected to a force d P (n) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCiuamaaCaaaleqabaGaai [email protected]@  across the plane dA.  Then

T(n)= lim dA0 d P (n) dA [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaaiikaiaah6gacaGGPaGaey ypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaadsgacaWGbbGa eyOKH4QaaGimaaqabaGcdaWcaaqaaiaadsgacaWHqbWaaWbaaSqabe [email protected]@

Note that internal traction is the force per unit area of the deformed solid, like `true stress’

 

 

 

 The resultant force acting on any internal volume V with boundary surface A within a deformed solid is

P= A T(n)dA+ V ρbdV [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHqbGaeyypa0Zaa8quaeaacaWHub Gaaiikaiaah6gacaGGPaGaamizaiaadgeacqGHRaWkaSqaaiaadgea aeqaniabgUIiYdGcdaWdrbqaaiabeg8aYjaaykW7caWHIbGaaGPaVl [email protected]@

The first term is the resultant force acting on the internal surface A, the second term is the resultant body force acting on the interior V.

 

 

 

 

 

 Newton’s third law (every action has an equal and opposite reaction) requires that

T(n)=T(n) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaaiikaiabgkHiTiaah6gaca [email protected]@

To see this, note that the forces acting on planes separating two adjacent volume elements in a solid must be equal and opposite.

 

 

 

 Traction acting on different planes passing through the same point are related, in order to satisfy Newton’s second law (F=ma).

 

Let { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  be a Cartesian basis.  Let T i ( e 1 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO [email protected]@ , T i ( e 2 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO [email protected]@ , T i ( e 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO [email protected]@  denote the components of traction acting on planes with normal vectors in the e 1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa [email protected]@ , e 2 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa [email protected]@ , and e 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa [email protected]@  directions, respectively.  Then, the traction components T i (n) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO [email protected]@  acting on a surface with normal n are given by

T i (n)= T i ( e 1 ) n 1 + T i ( e 2 ) n 2 + T i ( e 3 ) n 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO Gaaiikaiaah6gacaGGPaGaeyypa0JaamivamaaBaaaleaacaWGPbaa beaakiaacIcacaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiaad6 gadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaa dMgaaeqaaOGaaiikaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGPa GaamOBamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadsfadaWgaaWc baGaamyAaaqabaGccaGGOaGaaCyzamaaBaaaleaacaaIZaaabeaaki [email protected]@

where n i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaa [email protected]@  are the components of n.

 

To see this, consider the forces acting on the infinitessimal tetrahedron shown in the figure.  The base and sides of the tetrahedron have normals in the e 2 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaWHLbWaaSbaaSqaaiaaik [email protected]@ , e 1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaWHLbWaaSbaaSqaaiaaig [email protected]@  and e 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaWHLbWaaSbaaSqaaiaaio [email protected]@  directions.  The fourth face has normal n.  Suppose the volume of the tetrahedron is dV, and let d A 1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyqamaaBaaaleaacaaIXa [email protected]@ d A 2 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyqamaaBaaaleaacaaIYa [email protected]@ d A 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyqamaaBaaaleaacaaIZa [email protected]@ d A n [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyqamaaBaaaleaacaWGUb [email protected]@  denote the areas of the faces.  Assume that the material within the tetrahedron has mass density ρ [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  and is subjected to a body force b. Let a denote the acceleration of the center of mass of the tetrahedron. Then, F=ma for the tetrahedron requires that

T(n)d A (n) +T( e 1 )d A 1 +T( e 2 )d A 2 +T( e 3 )d A 3 +ρbdV=ρdVa [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaaiikaiaah6gacaGGPaGaam izaiaadgeadaahaaWcbeqaaiaacIcacaWGUbGaaiykaaaakiabgUca RiaahsfacaGGOaGaeyOeI0IaaCyzamaaBaaaleaacaaIXaaabeaaki aacMcacaWGKbGaamyqamaaBaaaleaacaaIXaaabeaakiabgUcaRiaa hsfacaGGOaGaeyOeI0IaaCyzamaaBaaaleaacaaIYaaabeaakiaacM cacaWGKbGaamyqamaaBaaaleaacaaIYaaabeaakiabgUcaRiaahsfa caGGOaGaeyOeI0IaaCyzamaaBaaaleaacaaIZaaabeaakiaacMcaca WGKbGaamyqamaaBaaaleaacaaIZaaabeaakiabgUcaRiabeg8aYjaa hkgacaWGKbGaamOvaiabg2da9iabeg8aYjaadsgacaWGwbGaaCyyaa [email protected]@

Recall that T( e i )=T( e i ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaaiikaiabgkHiTiaahwgada WgaaWcbaGaamyAaaqabaGccaGGPaGaeyypa0JaeyOeI0IaaCivaiaa [email protected]@  and divide through by d A n [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyqamaaBaaaleaacaWGUb [email protected]@ :

T(n)T( e 1 ) d A 1 d A (n) T( e 2 ) d A 2 d A (n) T( e 3 ) d A 3 d A (n) +ρb dV d A (n) =ρ dV d A (n) a [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaaiikaiaah6gacaGGPaGaey OeI0IaaCivaiaacIcacaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaaiyk amaalaaabaGaamizaiaadgeadaWgaaWcbaGaaGymaaqabaaakeaaca WGKbGaamyqamaaCaaaleqabaGaaiikaiaad6gacaGGPaaaaaaakiab gkHiTiaahsfacaGGOaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacM cadaWcaaqaaiaadsgacaWGbbWaaSbaaSqaaiaaikdaaeqaaaGcbaGa amizaiaadgeadaahaaWcbeqaaiaacIcacaWGUbGaaiykaaaaaaGccq GHsislcaWHubGaaiikaiaahwgadaWgaaWcbaGaaG4maaqabaGccaGG PaWaaSaaaeaacaWGKbGaamyqamaaBaaaleaacaaIZaaabeaaaOqaai aadsgacaWGbbWaaWbaaSqabeaacaGGOaGaamOBaiaacMcaaaaaaOGa ey4kaSIaeqyWdiNaaCOyamaalaaabaGaamizaiaadAfaaeaacaWGKb GaamyqamaaCaaaleqabaGaaiikaiaad6gacaGGPaaaaaaakiabg2da 9iabeg8aYnaalaaabaGaamizaiaadAfaaeaacaWGKbGaamyqamaaCa [email protected]@

Finally, let d A n 0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyqamaaBaaaleaacaWGUb [email protected]@ .  We can show that

d A 1 d A (n) = n 1 d A 2 d A (n) = n 2 d A 3 d A (n) = n 3 lim d A (n) 0 dV d A (n) =0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWGbbWaaSbaaS qaaiaaigdaaeqaaaGcbaGaamizaiaadgeadaahaaWcbeqaaiaacIca caWGUbGaaiykaaaaaaGccqGH9aqpcaWGUbWaaSbaaSqaaiaaigdaae qaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVpaalaaabaGaamizaiaadgeadaWgaaWcbaGaaG OmaaqabaaakeaacaWGKbGaamyqamaaCaaaleqabaGaaiikaiaad6ga caGGPaaaaaaakiabg2da9iaad6gadaWgaaWcbaGaaGOmaaqabaGcca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7daWcaaqaaiaadsgacaWGbbWaaSbaaSqaaiaaiodaaeqaaaGcba GaamizaiaadgeadaahaaWcbeqaaiaacIcacaWGUbGaaiykaaaaaaGc cqGH9aqpcaWGUbWaaSbaaSqaaiaaiodaaeqaaOGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaCbeaeaa ciGGSbGaaiyAaiaac2gaaSqaaiaadsgacaWGbbWaaWbaaWqabeaaca GGOaGaamOBaiaacMcaaaWccqGHsgIRcaaIWaaabeaakmaalaaabaGa amizaiaadAfaaeaacaWGKbGaamyqamaaCaaaleqabaGaaiikaiaad6 [email protected]@

so

T(n)=T( e 1 ) n 1 T( e 2 ) n 2 T( e 3 ) n 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaaiikaiaah6gacaGGPaGaey ypa0JaaCivaiaacIcacaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaaiyk aiaad6gadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWHubGaaiikai aahwgadaWgaaWcbaGaaGOmaaqabaGccaGGPaGaamOBamaaBaaaleaa caaIYaaabeaakiabgkHiTiaahsfacaGGOaGaaCyzamaaBaaaleaaca [email protected]@

or, using index notation

T i (n)= T i ( e 1 ) n 1 + T i ( e 2 ) n 2 + T i ( e 3 ) n 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO Gaaiikaiaah6gacaGGPaGaeyypa0JaamivamaaBaaaleaacaWGPbaa beaakiaacIcacaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiaad6 gadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaa dMgaaeqaaOGaaiikaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGPa GaamOBamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadsfadaWgaaWc baGaamyAaaqabaGccaGGOaGaaCyzamaaBaaaleaacaaIZaaabeaaki [email protected]@

The significance of this result is that the tractions acting on planes with normals in the e 1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa [email protected]@ , e 2 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa [email protected]@ , and e 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa [email protected]@  directions completely characterize the internal forces that act at a point.  Given these tractions, we can deduce the tractions acting on any other plane.  This leads directly to the definition of the Cauchy stress tensor in the next section.

 

 

 

 

4.3 The Cauchy (true) stress tensor

 

Consider a solid which deforms under external loading. Let { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  be a Cartesian basis.  Let T i ( e 1 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO [email protected]@ , T i ( e 2 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO [email protected]@ , T i ( e 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO [email protected]@  denote the components of traction acting on planes with normals in the e 1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa [email protected]@ , e 2 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa [email protected]@ , and e 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa [email protected]@  directions, respectively, as outlined in the preceding section 

 

 Define the components of the Cauchy stress tensor σ ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa [email protected]@  by

σ ij = T j ( e i ) { σ 11 = T 1 ( e 1 ) σ 12 = T 2 ( e 1 ) σ 13 = T 3 ( e 1 ) σ 21 = T 1 ( e 2 ) σ 22 = T 2 ( e 2 ) σ 23 = T 3 ( e 2 ) σ 31 = T 1 ( e 3 ) σ 32 = T 2 ( e 3 ) σ 33 = T 3 ( e 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaaceqaaiabeo8aZnaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpcaWGubWaaSbaaSqaaiaadQgaaeqaaOGa aiikaiaahwgadaWgaaWcbaGaamyAaaqabaGccaGGPaaabaGaeyyyIO 7aaiqaaqaabeqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGc cqGH9aqpcaWGubWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaahwgada WgaaWcbaGaaGymaaqabaGccaGGPaGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdp WCdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaamivamaaBaaa leaacaaIYaaabeaakiaacIcacaWHLbWaaSbaaSqaaiaaigdaaeqaaO GaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaaca aIXaGaaG4maaqabaGccqGH9aqpcaWGubWaaSbaaSqaaiaaiodaaeqa aOGaaiikaiaahwgadaWgaaWcbaGaaGymaaqabaGccaGGPaaabaGaeq 4Wdm3aaSbaaSqaaiaaikdacaaIXaaabeaakiabg2da9iaadsfadaWg aaWcbaGaaGymaaqabaGccaGGOaGaaCyzamaaBaaaleaacaaIYaaabe aakiaacMcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIYa GaaGOmaaqabaGccqGH9aqpcaWGubWaaSbaaSqaaiaaikdaaeqaaOGa aiikaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGPaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaaikdacaaIZaaabeaaki abg2da9iaadsfadaWgaaWcbaGaaG4maaqabaGccaGGOaGaaCyzamaa BaaaleaacaaIYaaabeaakiaacMcaaeaacqaHdpWCdaWgaaWcbaGaaG 4maiaaigdaaeqaaOGaeyypa0JaamivamaaBaaaleaacaaIXaaabeaa kiaacIcacaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaaiodacaaIYaaabeaakiabg2 da9iaadsfadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaaCyzamaaBaaa leaacaaIZaaabeaakiaacMcacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH dpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaOGaeyypa0JaamivamaaBa aaleaacaaIZaaabeaakiaacIcacaWHLbWaaSbaaSqaaiaaiodaaeqa [email protected]@

Then, the traction  T i (n) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO [email protected]@  acting on any plane with normal n follows as

T(n)=nσor      T i (n)= n j σ ji [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaaiikaiaah6gacaGGPaGaey ypa0JaaCOBaiabgwSixlaaho8acaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8Uaae4BaiaabkhacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa dsfadaWgaaWcbaGaamyAaaqabaGccaGGOaGaaCOBaiaacMcacqGH9a qpcaWGUbWaaSbaaSqaaiaadQgaaeqaaOGaeq4Wdm3aaSbaaSqaaiaa [email protected]@

 

To see this, recall the last result from the preceding section

T i (n)= T i ( e 1 ) n 1 + T i ( e 2 ) n 2 + T i ( e 3 ) n 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO Gaaiikaiaah6gacaGGPaGaeyypa0JaamivamaaBaaaleaacaWGPbaa beaakiaacIcacaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiaad6 gadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaa dMgaaeqaaOGaaiikaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGPa GaamOBamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadsfadaWgaaWc baGaamyAaaqabaGccaGGOaGaaCyzamaaBaaaleaacaaIZaaabeaaki [email protected]@

and substitute for T i ( e j ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO [email protected]@  in terms of the components of the Cauchy stress tensor

T i (n)= σ 1i n 1 + σ 2i n 2 + σ 3i n 3 = n j σ ji [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaO Gaaiikaiaah6gacaGGPaGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaigda caWGPbaabeaakiaad6gadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcq aHdpWCdaWgaaWcbaGaaGOmaiaadMgaaeqaaOGaamOBamaaBaaaleaa caaIYaaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaaIZaGaamyAaa qabaGccaWGUbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaamOBamaa BaaaleaacaWGQbaabeaakiabeo8aZnaaBaaaleaacaWGQbGaamyAaa [email protected]@

 

The Cauchy stress tensor completely characterizes the internal forces acting in a deformed solid.  The physical significance of the components of the stress tensor is illustrated in the figure: σ ji [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOAaiaadM [email protected]@  represents the ith component of traction acting on a plane with normal in the e j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaadQgaaeqaaa [email protected]@  direction.

 

Note the Cauchy stress represents force per unit area of the deformed solid.  In elementary strength of materials courses it is called `true stress,’ for this reason.

 

HEALTH WARNING: Some texts define stress as the transpose of the definition used here, so that T(n)=σnor      T i (n)= σ ij n j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaaiikaiaah6gacaGGPaGaey ypa0JaaC4WdiabgwSixlaah6gacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8Uaae4BaiaabkhacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa dsfadaWgaaWcbaGaamyAaaqabaGccaGGOaGaaCOBaiaacMcacqGH9a qpcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamOBamaaBaaa [email protected]@ .  In this case the first index for each stress component denotes the direction of traction, while the second denotes the normal to the plane.  We will see later that Cauchy stress is always symmetric, so there is no confusion if you use the wrong definition.  But some stress measures are not symmetric (see below) and in this case you need to be careful to check which convention the author has chosen.

 

 

4.4 Other stress measures [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa [email protected]@  Kirchhoff, Nominal and Material stress tensors

 

Cauchy stress σ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ [email protected]@  (the actual force per unit area acting on an actual, deformed solid) is the most physical measure of internal force.  Other definitions of stress often appear in constitutive equations, however. 

 

The other stress measures regard forces as acting on the undeformed solid.  Consequently, to define them we must know not only what the deformed solid looks like, but also what it looked like before deformation.  The deformation is described by a displacement vector u(x) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  and the associated deformation gradient

F=I+u F ij = δ ij + u i x j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCysaiabgUcaRi abgEGirlaahwhacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGgbWaaSbaaS qaaiaadMgacaWGQbaabeaakiabg2da9iabes7aKnaaBaaaleaacaWG PbGaamOAaaqabaGccqGHRaWkdaWcaaqaaiabgkGi2kaadwhadaWgaa WcbaGaamyAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQga [email protected]@

as outlined in Section 2.1. In addition, let J=det(F) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeyypa0Jaciizaiaacwgaca [email protected]@

 

We then define the following stress measures

 

  Kirchhoff stress  τ=Jσ τ ij =J σ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaCiXdOGaeyypa0JaamOsaKaaal aaho8acaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabes8a0PWaaSba aSqaaiaadMgacaWGQbaabeaakiabg2da9iaadQeacqaHdpWCdaWgaa [email protected]@

 Nominal (First Piola-Kirchhoff) stress   S=J F 1 σ S ij =J F ik 1 σ kj [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaC4uaiabg2da9iaadQeacaWHgbWaaW baaSqabeaacqGHsislcaaIXaaaaOGaeyyXICDcaaSaaC4WdiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4uaO WaaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadQeacaWGgbWa a0baaSqaaiaadMgacaWGRbaabaGaeyOeI0IaaGymaaaakiabeo8aZn [email protected]@

 Material (Second Piola-Kirchhoff) stress   Σ=J F 1 σ F T Σ ij =J F ik 1 σ kl F jl 1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaC4Odiabg2da9iaadQeacaWHgbWaaW baaSqabeaacqGHsislcaaIXaaaaOGaeyyXICDcaaSaaC4WdiabgwSi xRGaaCOramaaCaaaleqabaGaeyOeI0IaamivaaaakiaaykW7caaMc8 UaaGPaVlaaykW7cqqHJoWudaWgaaWcbaGaamyAaiaadQgaaeqaaOGa eyypa0JaamOsaiaadAeadaqhaaWcbaGaamyAaiaadUgaaeaacqGHsi slcaaIXaaaaOGaeq4Wdm3aaSbaaSqaaiaadUgacaWGSbaabeaakiaa [email protected]@

 

The inverse relations are also useful [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  the one for Kirchhoff stress is obvious [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  the others are

σ= 1 J FS σ ij = 1 J F ik S kj [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaC4Wdiabg2da9OWaaSaaaeaaca aIXaaabaGaamOsaaaacaWHgbGaeyyXICTaaC4uaKaaalaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7kiabeo8aZnaaBaaale aacaWGPbGaamOAaaqabaGccqGH9aqpjaaWcaaMc8UcdaWcaaqaaiaa igdaaeaacaWGkbaaaKaaalaadAeakmaaBaaaleaacaWGPbGaam4Aaa [email protected]@                σ= 1 J FΣ F T σ ij = 1 J F ik Σ kl F jl [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaC4Wdiabg2da9OWaaSaaaeaaca aIXaaabaGaamOsaaaacaWHgbGaeyyXICTaaC4OdKaaalabgwSixRGa aCOramaaCaaaleqabaGaamivaaaakiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaadMga caWGQbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaadQeaaaGaam OramaaDaaaleaacaWGPbGaam4Aaaqaaaaakiabfo6atnaaBaaaleaa caWGRbGaamiBaaqabaGccaWGgbWaa0baaSqaaiaadQgacaWGSbaaba [email protected]@

 

The Kirchoff stress has no obvious physical significance. 

 

The nominal stress tensor can be regarded as the internal force per unit undeformed area acting within a solid, as follows

1.      Visualize an element of area dA in the deformed solid, with normal n, which is subjected to a force d P (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHqbWaaWbaaSqabeaacaGGOa [email protected]@  by the internal traction in the solid;

2.      Suppose that the element of area dA has started out as an element of area d A 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaaicdaae [email protected]@  with normal n 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabe[email protected]@  in the undeformed solid, as shown in the figure;

3.      Then, the force d P (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHqbWaaWbaaSqabeaacaGGOa [email protected]@  is related to the nominal stress by d P j (n) =d A 0 n i 0 S ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGqbWaa0baaSqaaiaadQgaae aacaGGOaGaaCOBaiaacMcaaaGccqGH9aqpcaWGKbGaamyqamaaBaaa leaacaaIWaaabeaakiaad6gadaqhaaWcbaGaamyAaaqaaiaaicdaaa [email protected]@

 

To see this, note that one can show that

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]@

Recall that the Cauchy stress is defined so that

d P i (n) =dA n j σ ji [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGqbWaa0baaSqaaiaadMgaae aacaGGOaGaaCOBaiaacMcaaaGccqGH9aqpcaWGKbGaamyqaiaad6ga daWgaaWcbaGaamOAaaqabaGccqaHdpWCdaWgaaWcbaGaamOAaiaadM [email protected]@

 

Substituting for dA n j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbGaamOBamaaBaaaleaaca [email protected]@  and rearranging shows that

d P i (n) =Jd A 0 n k 0 ( F kj 1 σ ji )=d A 0 n k 0 S ki [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGqbWaa0baaSqaaiaadMgaae aacaGGOaGaaCOBaiaacMcaaaGccqGH9aqpcaWGkbGaamizaiaadgea daWgaaWcbaGaaGimaaqabaGccaWGUbWaa0baaSqaaiaadUgaaeaaca aIWaaaaOWaaeWaaeaacaWGgbWaa0baaSqaaiaadUgacaWGQbaabaGa eyOeI0IaaGymaaaakiabeo8aZnaaBaaaleaacaWGQbGaamyAaaqaba aakiaawIcacaGLPaaacqGH9aqpcaWGKbGaamyqamaaBaaaleaacaaI Waaabeaakiaad6gadaqhaaWcbaGaam4AaaqaaiaaicdaaaGccaWGtb [email protected]@

 

The material stress tensor can also be visualized as force per unit undeformed area, except that the forces are regarded as acting within the undeformed solid, rather than on the deformed solid.  Specifically

1.      The infinitesimal force d P (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHqbWaaWbaaSqabeaacaGGOa [email protected]@  is assumed to behave like an infinitesimal material fiber in the solid, in the sense that it is stretched and rotated just like an small vector dx in the solid

2.      This means that we can define a (fictitious) force in the reference configuration d P (n0) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHqbWaaWbaaSqabeaacaGGOa [email protected]@  that is related to d P (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHqbWaaWbaaSqabeaacaGGOa [email protected]@  by Fd P (n0) =d P (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAeacqGHflY1caWGKbGaaCiuamaaCa aaleqabaGaaiikaiaah6gacaaIWaGaaiykaaaakiabg2da9iaadsga [email protected]@  or F ij d P j (n0) =d P i (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaamizaiaadcfadaWgaaWcbaGaamOAaaqabaGcdaahaaWcbeqa aiaacIcacaWHUbGaaGimaiaacMcaaaGccqGH9aqpcaWGKbGaamiuam aaBaaaleaacaWGPbaabeaakmaaCaaaleqabaGaaiikaiaah6gacaGG [email protected]@ .

3.      This fictitious force is related to material stress by   d P i (n0) =d A 0 n j 0 Σ ji [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGqbWaa0baaSqaaiaadMgaae aacaGGOaGaaCOBaiaaicdacaGGPaaaaOGaeyypa0Jaamizaiaadgea daWgaaWcbaGaaGimaaqabaGccaWGUbWaa0baaSqaaiaadQgaaeaaca [email protected]@

 

To see this, substitute into the expression relating d P (n) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHqbWaaWbaaSqabeaacaGGOa [email protected]@  to nominal stress to see that

F ik d P k (n0) =d A 0 n j 0 S ji [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadUgaae qaaOGaamizaiaadcfadaqhaaWcbaGaam4AaaqaaiaacIcacaWHUbGa aGimaiaacMcaaaGccqGH9aqpcaWGKbGaamyqamaaBaaaleaacaaIWa aabeaakiaad6gadaqhaaWcbaGaamOAaaqaaiaaicdaaaGccaWGtbWa [email protected]@

Finally multiply through by F li 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaqhaaWcbaGaamiBaiaadMgaae [email protected]@ , note F li 1 F ik = δ lk [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaqhaaWcbaGaamiBaiaadMgaae aacqGHsislcaaIXaaaaOGaamOramaaBaaaleaacaWGPbGaam4Aaaqa [email protected]@ , and rearrange to see that

d P l (n0) =d A 0 n j 0 S ji F li 1 =d A 0 n j 0 Σ jl [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGqbWaa0baaSqaaiaadYgaae aacaGGOaGaaCOBaiaaicdacaGGPaaaaOGaeyypa0Jaamizaiaadgea daWgaaWcbaGaaGimaaqabaGccaWGUbWaa0baaSqaaiaadQgaaeaaca aIWaaaaOGaam4uamaaBaaaleaacaWGQbGaamyAaaqabaGccaWGgbWa a0baaSqaaiaadYgacaWGPbaabaGaeyOeI0IaaGymaaaakiabg2da9i aadsgacaWGbbWaaSbaaSqaaiaaicdaaeqaaOGaamOBamaaDaaaleaa caWGQbaabaGaaGimaaaakiabfo6atnaaBaaaleaacaWGQbGaamiBaa [email protected]@

where we have noted that Σ jl = S ji F li 1 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfo6atnaaBaaaleaacaWGQbGaamiBaa qabaGccqGH9aqpcaWGtbWaaSbaaSqaaiaadQgacaWGPbaabeaakiaa [email protected]@

 

In practice, it is best not to try to attach too much physical significance to these stress measures.  Cauchy stress is the best physical measure of internal force [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  it is the force per unit area acting inside the deformed solid.  The other stress measures are best regarded as generalized forces (in the sense of Lagrangian mechanics), which are work-conjugate to particular strain measures.  This means that the stress measure multiplied by the time derivative of the strain measure tells you the rate of work done by the forces.  When setting up any mechanics problem, we always work with conjugate measures of motion and forces.

 

Specifically, we shall show later that the rate of work W ˙ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  done by stresses acting on a small material element with volume d V 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGwbWaaSbaaSqaaiaaicdaae [email protected]@  in the undeformed solid (and volume dV [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@  in the deformed solid) can be computed as

W ˙ = D ij σ ij dV= D ij τ ji d V 0 = F ˙ ij S ji d V 0 = E ˙ ij Σ ji d V 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadEfagaGaaiabg2da9iaadseadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeq4Wdm3aaSbaaSqaaiaadMgacaWG QbaabeaakiaadsgacaWGwbGaeyypa0JaamiramaaBaaaleaacaWGPb GaamOAaaqabaGccqaHepaDdaWgaaWcbaGaamOAaiaadMgaaeqaaOGa amizaiaadAfadaWgaaWcbaGaaGimaaqabaGccqGH9aqpceWGgbGbai aadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaam4uamaaBaaaleaacaWG QbGaamyAaaqabaGccaWGKbGaamOvamaaBaaaleaacaaIWaaabeaaki abg2da9iqadweagaGaamaaBaaaleaacaWGPbGaamOAaaqabaGccqqH JoWudaWgaaWcbaGaamOAaiaadMgaaeqaaOGaamizaiaadAfadaWgaa [email protected]@

where D ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaWgaaWcbaGaamyAaiaadQgaae [email protected]@  is the stretch rate tensor, F ˙ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadAeagaGaamaaBaaaleaacaWGPbGaam [email protected]@  is the rate of change of deformation gradient, and E ˙ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadweagaGaamaaBaaaleaacaWGPbGaam [email protected]@  is the rate of change of Lagrange strain tensor.  Note that Cauchy stress (and also Kirchhoff stress) is not conjugate to any convenient strain measure [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  this is the main reason that nominal and material stresses need to be defined.  The nominal stress is conjugate to the deformation gradient, while the material stress is conjugate to the Lagrange strain tensor. 

 

 

 

 

 

4.5 Stress measures for infinitesimal deformations

 

For a problem involving infinitesimal deformation (where shape changes are characterized by the infinitesimal strain tensor and rotation tensor) all the stress measures defined in the preceding section are approximately equal.

σ ij τ ij S ij Σ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGHijYUcqaHepaDdaWgaaWcbaGaamyAaiaadQgaaeqaaOGa eyisISRaam4uamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHijYUcq [email protected]@

To see this, write the deformation gradient as F ij = δ ij + u i / x j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiab gUcaRiabgkGi2kaadwhadaWgaaWcbaGaamyAaaqabaGccaGGVaGaey [email protected]@ ; recall that J=det(F)1+ u k / x k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacqGH9aqpciGGKbGaaiyzaiaacs hacaGGOaGaaCOraiaacMcacqGHijYUcaaIXaGaey4kaSIaeyOaIyRa amyDamaaBaaaleaacaWGRbaabeaakiaac+cacqGHciITcaWG4bWaaS [email protected]@ , and finally assume that for infinitesimal motions u i / x j <<1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaa qabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaakiab [email protected]@ .  Substituting into the formulas relating Cauchy stress, Nominal stress and Material stress, we see that

σ ij = 1 J F ik S kj 1 1+ u p / x p ( δ ip + u i x p ) S pj = S pj +... S pj [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGkbaaaiaadAeadaWg aaWcbaGaamyAaiaadUgaaeqaaOGaam4uamaaBaaaleaacaWGRbGaam OAaaqabaGccqGHijYUdaWcaaqaaiaaigdaaeaacaaIXaGaey4kaSIa eyOaIyRaamyDamaaBaaaleaacaWGWbaabeaakiaac+cacqGHciITca WG4bWaaSbaaSqaaiaadchaaeqaaaaakmaabmaabaGaeqiTdq2aaSba aSqaaiaadMgacaWGWbaabeaakiabgUcaRmaalaaabaGaeyOaIyRaam yDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWc baGaamiCaaqabaaaaaGccaGLOaGaayzkaaGaam4uamaaBaaaleaaca WGWbGaamOAaaqabaGccqGH9aqpcaWGtbWaaSbaaSqaaiaadchacaWG QbaabeaakiabgUcaRiaac6cacaGGUaGaaiOlaiabgIKi7kaadofada [email protected]@

The same procedure will show that material stress and Cauchy stress are approximately equal, to within a term of order u i / x j <<1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaa qabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaakiab [email protected]@

 

 

4.6 Principal Stresses and directions

 

For any stress measure, the principal stresses σ i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbaabeaaaa [email protected]@  and their directions n (i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah6gadaahaaWcbeqaaiaacIcacaWGPb [email protected]@ , with i=1..3 are defined such that

n (i) σ= σ i n (i) or     n j (i) σ jk = σ i n k (i) (no sum on i) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah6gadaahaaWcbeqaaiaacIcacaWGPb GaaiykaaaakiabgwSixNaaGkaaho8akiabg2da9iabeo8aZnaaBaaa leaacaWGPbaabeaakiaah6gadaahaaWcbeqaaiaacIcacaWGPbGaai ykaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caqGVbGaaeOCaiaabccacaqGGaGaaeiiaiaabccacaWGUbWaa0baaS qaaiaadQgaaeaacaGGOaGaamyAaiaacMcaaaGccqaHdpWCdaWgaaWc baGaamOAaiaadUgaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadM gaaeqaaOGaamOBamaaDaaaleaacaWGRbaabaGaaiikaiaadMgacaGG PaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aacIcacaqGUbGaae4BaiaabccacaqGZbGaaeyDaiaab2gacaqGGaGa [email protected]@

Clearly,

  1. The principal stresses are the (left) eigenvalues of the stress tensor
  2. The principal stress directions are the (left) eigenvectors of the stress tensor

 

The term `left’ eigenvector and eigenvalue indicates that the vector multiplies the tensor on the left. We will see later that Cauchy stress and material stress are both symmetric.  For a symmetric tensor the left and right eigenvalues and vectors are the same.

 

Note that the eigenvectors of a symmetric tensor are orthogonal.  Consequently, the principal Cauchy or material stresses can be visualized as tractions acting normal to the faces of a cube. The principal directions specify the orientation of this special cube.

 

One can also show that if σ 1 > σ 2 > σ 3 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaO GaeyOpa4Jaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaeyOpa4Jaeq4W [email protected]@ , then σ 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaa [email protected]@  is the largest normal traction acting on any plane passing through the point of interest, while σ 3 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaiodaaeqaaa [email protected]@  is the lowest.  This is helpful in defining damage criteria for brittle materials, which fail when the stress acting normal to a material plane reaches a critical magnitude.

In the same vein, the largest shear stress can be shown to act on the plane with unit normal vector m shear =( m 1 + m 3 )/ 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaae4CaiaabIgaca qGLbGaaeyyaiaabkhaaeqaaOGaeyypa0JaeyOeI0Iaaiikaiaah2ga daWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWHTbWaaSbaaSqaaiaaio [email protected]@  (at 45o to the m 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCyBamaaBaaaleaacaaIXaaabeaaaa [email protected]@  and m 3 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCyBamaaBaaaleaacaaIZaaabeaaaa [email protected]@  axes), and its magnitude is τ max = 1 2 ( σ 1 σ 3 ) [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiXdq3aaSbaaSqaaiaab2gacaqGHb GaaeiEaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaa cIcacqaHdpWCdaWgaaWcbaGaaGymaaqabaGccqGHsislcqaHdpWCda [email protected]@ .  This observation is useful for defining yield criteria for metal polycrystals, which begin to deform plastically when the shear stress acting on a material plane reaches a critical value.

 

4.7 Hydrostatic and Deviatoric Stress; von Mises effective stress

 

Given the Cauchy stress tensor σ [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa [email protected]@ , the following may be defined:

 

The Hydrostatic stress is defined as σ h =trace(σ)/3 σ kk /3 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGObaabeaaki abg2da9iaabshacaqGYbGaaeyyaiaabogacaqGLbGaaeikaiaaho8a caGGPaGaai4laiaaiodacqGHHjIUcqaHdpWCdaWgaaWcbaGaam4Aai [email protected]@

The Deviatoric stress tensor is defined as σ ij = σ ij σ h δ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaafaWaaSbaaSqaaiaadMgaca WGQbaabeaakiabg2da9iabeo8aZnaaBaaaleaacaWGPbGaamOAaaqa baGccqGHsislcqaHdpWCdaWgaaWcbaGaamiAaaqabaGccqaH0oazda [email protected]@

The Von-Mises effective stress is defined as σ e = 3 2 σ ij σ ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGLbaabeaaki abg2da9maakaaabaWaaSaaaeaacaaIZaaabaGaaGOmaaaacuaHdpWC gaqbamaaBaaaleaacaWGPbGaamOAaaqabaGccuaHdpWCgaqbamaaBa [email protected]@

 

The hydrostatic stress is a measure of the pressure exerted by a state of stress.  Pressure acts so as to change the volume of a material element.

 

The deviatoric stress is a measure of the shearing exerted by a state of stress. Shear stress tends to distort a solid, without changing its volume.

 

The Von-Mises effective stress can be regarded as a uniaxial equivalent of a multi-axial stress state.  It is used in many failure or yield criteria.  Thus, if a material is known to fail in a uniaxial tensile test (with σ 11 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa [email protected]@  the only nonzero stress component) when σ 11 = σ crit [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaam4yaiaadkhacaWGPbGa [email protected]@ , it will fail when σ e = σ crit [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGLbaabeaaki abg2da9iabeo8aZnaaBaaaleaacaWGJbGaamOCaiaadMgacaWG0baa [email protected]@  under multi-axial loading (with several σ ij 0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa [email protected]@  )

 

The hydrostatic stress and von Mises stress can also be expressed in terms of principal stresses as

σ h =( σ 1 + σ 2 + σ 3 )/3 σ e = 1 2 { ( σ 1 σ 2 ) 2 + ( σ 1 σ 3 ) 2 + ( σ 2 σ 3 ) 2 } [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aaSbaaSqaaiaadIgaae qaaOGaeyypa0ZaaeWaaeaacqaHdpWCdaWgaaWcbaGaaGymaaqabaGc cqGHRaWkcqaHdpWCdaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcqaHdp WCdaWgaaWcbaGaaG4maaqabaaakiaawIcacaGLPaaacaGGVaGaaG4m aaqaaiabeo8aZnaaBaaaleaacaWGLbaabeaakiabg2da9maakaaaba WaaSaaaeaacaaIXaaabaGaaGOmaaaadaGadaqaamaabmaabaGaeq4W dm3aaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaai aaikdaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa ey4kaSYaaeWaaeaacqaHdpWCdaWgaaWcbaGaaGymaaqabaGccqGHsi slcqaHdpWCdaWgaaWcbaGaaG4maaqabaaakiaawIcacaGLPaaadaah aaWcbeqaaiaaikdaaaGccqGHRaWkdaqadaqaaiabeo8aZnaaBaaale aacaaIYaaabeaakiabgkHiTiabeo8aZnaaBaaaleaacaaIZaaabeaa aOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5Eaiaaw2 [email protected]@

 

The hydrostatic and von Mises stresses are invariants of the stress tensor [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@  they have the same value regardless of the basis chosen to define the stress components.

 

4.8 Stresses near an external surface or edge [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa [email protected]@  boundary conditions on stresses

 

Note that at an external surface at which tractions are prescribed, some components of stress are known.  Specifically, let n denote a unit vector normal to the surface, and let t denote the traction (force per unit area) acting on the surface.  Then the Cauchy stress at the surface must satisfy

n i σ ij = t j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO Gaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadsha [email protected]@

 

For example, suppose that a surface with normal in the e 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa [email protected]@  direction is subjected to no loading.  Then (noting that n i = δ i2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO [email protected]@  ) it follows that σ 2i =0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaadM [email protected]@ . In addition, two of the principal stress directions must be parallel to the surface; the third (with zero stress) must be perpendicular to the surface.

 

The stress state at an edge is even simpler.  Suppose that surfaces with normals in the e 2 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa [email protected]@  and e 1 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa [email protected]@  are traction free.  Then σ 1i = σ 2i =0 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaadM gaaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaikdacaWGPbaabeaa [email protected]@ , so that 6 stress components are known to be zero.