** **

** **

** **

**5. Conservation Laws for Continua**

In
this section, we generalize Newton’s laws of motion (conservation of linear and
angular momentum); mass conservation; and the laws of thermodynamics for a
continuum.

**5.1 Mass Conservation**

The
total mass of any subregion within a deformable solid must be conserved. We can write express this condition as a
constraint in several different ways:

In integral form:

Or, (using Reynolds
transport relation) we can write a *local*
mass conservation equation

Alternatively, in spatial
form

**5.2 Linear
momentum balance in terms of Cauchy stress**

Let denote the Cauchy stress distribution within a
deformed solid. Assume that the solid is
subjected to a body force **,
**and let and denote the displacement, velocity and
acceleration of a material particle at position * * in
the deformed solid.

**F**=m**a**) can be expressed
as

Written out in full

Note that the derivative is taken with respect to position in
the actual, deformed solid. For the special (but rather common) case of a solid
in static equilibrium in the absence of body forces

**Derivation:** Recall that the resultant force
acting on an arbitrary volume of material *V*
within a solid is

where
**T(**n**)** is the internal traction acting on the surface *A *with normal **n** that bounds *V*.

The linear momentum of the
volume *V* is

where
**v** is the velocity vector of a
material particle in the deformed solid. Express **T** in terms of and set

Apply
the divergence theorem to convert the first integral into a volume integral,
and note that the Reynolds transport equation implies that

so

Since this must hold for
every volume of material within a solid, it follows that

as stated.

We can
also write this in spatial form by recalling that

so that

**5.3 Angular momentum balance in terms
of Cauchy stress**

Conservation
of angular momentum for a continuum requires that the Cauchy stress satisfy

i.e. the
stress tensor must be symmetric.

**Derivation:** write down the equation for balance
of angular momentum for the region *V *within
the deformed solid

Here, the left hand side is the resultant moment (about the
origin) exerted by tractions and body forces acting on a general region within
a solid. The right hand side is the
total angular momentum of the solid about the origin.

We can
write the same expression using index notation

Express **T** in terms of and re-write the first integral as a volume
integral using the divergence theorem

We may
also show that

Substitute
the last two results into the angular momentum balance equation to see that

The
integral on the right hand side of this expression is zero, because the
stresses must satisfy the linear momentum balance equation. Since this holds for any volume *V*, we conclude that

which is the result we
wanted.

**5.4 Equations of motion in terms of
other stress measures**

In terms
of nominal and material stress the balance of linear momentum is

Note that
the derivatives are taken with respect to position in the *undeformed *solid.

The
angular momentum balance equation is

To derive
these results, you can start with the integral form of the linear momentum
balance in terms of Cauchy stress

Recall
that area elements in the deformed and undeformed solids are related through

In
addition, volume elements are related by . We can use these results to re-write the
integrals as integrals over a volume in the *undeformed
solid* as

Finally,
recall that and that to see that

Apply the
divergence theorem to the first term and rearrange

Once
again, since this must hold for any material volume, we conclude that

The linear
momentum balance equation in terms of material stress follows directly, by
substituting into this equation for in terms of

The angular momentum
balance equation can be derived simply by substituting into the momentum
balance equation in terms of Cauchy stress

**5.5 Work done by Cauchy stresses**

Consider a solid with mass density in its initial configuration, and density in the deformed solid. Let denote the Cauchy stress distribution within
the solid. Assume that the solid is
subjected to a body force ** **(per unit mass)**, **and let and denote the displacement, velocity and
acceleration of a material particle at position * * in
the deformed solid. In addition, let

denote the
stretch rate in the solid.

The **rate of work done
by Cauchy stresses per unit deformed volume** is then . This energy is either dissipated as heat or
stored as internal energy in the solid, depending on the material behavior.

We
shall show that the rate of work done by internal forces acting on any
sub-volume *V* bounded by a surface *A *in the deformed solid can be
calculated from

Here,
the two terms on the left hand side represent the rate of work done by
tractions and body forces acting on the solid (work done = force x velocity). The
first term on the right-hand side can be interpreted as the work done by Cauchy
stresses; the second term is the rate of change of kinetic energy.

**Derivation: **Substitute
for in terms of Cauchy stress to see that

Now, apply the divergence
theorem to the first term on the right hand side

Evaluate
the derivative and collect together the terms involving body force and stress
divergence

Recall
the equation of motion

and note
that since the stress is symmetric

to see
that

Finally,
note that

Finally, substitution leads to

as
required.

**5.6 Rate of mechanical work in terms
of other stress measures**

The **rate
of work done per unit undeformed volume by Kirchhoff stress **is

The **rate
of work done per unit undeformed volume by Nominal stress **is

The **rate
of work done per unit undeformed volume by Material stress **is

This shows
that nominal stress and deformation gradient are *work conjugate*, as are material stress and Lagrange strain.

In addition, the rate of work done on a volume of the undeformed solid can be expressed as

**Derivations: **The proof of the first result (and the stress power of
Kirchhoff stress) is straightforward and is left as an exercise. To show the second result, note that and to re-write the integrals over the undeformed
solid; then and apply the divergence theorem to see that

Evaluate the derivative,
recall that and use the equation of motion

to see that

Finally,
note that and re-write the second integral as a kinetic
energy term as before to obtain the required result.

The
third result follows by straightforward algebraic manipulations note that by definition

Since is symmetric it follows that

**5.7 Rate of mechanical work for
infinitesimal deformations**

For infintesimal motions all stress measures are equal; and
all strain rate measures can be approximated by the infinitesimal strain tensor
. The rate of work done by stresses per unit
volume of either deformed or undeformed solid (the difference is neglected) can
be expressed as ,
and the work done on a volume of the solid is

**5.8 The principle of Virtual Work**

The principle of virtual
work forms the basis for the finite element method in the mechanics of solids
and so will be discussed in detail in this section.

Suppose
that a deformable solid is subjected to loading that induces a displacement
field ,
and a velocity field . The loading consists of a prescribed
displacement on part of the boundary (denoted by ), together with a traction **t** (which may be zero in places) applied
to the rest of the boundary (denoted by ). The
loading induces a Cauchy stress . The stress field satisfies the angular
momentum balance equation .

The
principle of virtual work is a different way of re-writing partial differential
equation for linear moment balance

in an
equivalent *integral* form, which is
much better suited for computer solution.

To
express the principle, we define a **kinematically
admissible virtual velocity field **,
satisfying on . You can visualize this field as a small
change in the velocity of the solid, if you like, but it is really just an
arbitrary differentiable vector field.
The term `kinematically admissible’ is just a complicated way of saying
that the field is continuous, differentiable, and satisfies on - that is to say, if you perturb the velocity by
,
the boundary conditions on displacement are still satisfied.

In
addition, we define an associated *virtual
velocity gradient, and virtual stretch rate* as

The
principal of virtual work may be stated in two ways.

**First version of the principle of
virtual work**

The
first is not very interesting, but we will state it anyway. Suppose that the Cauchy stress satisfies:

1. The boundary condition on

2. The linear momentum balance equation

Then the *virtual
work equation*

is satisfied for *all* *virtual
velocity fields*.

**Proof: ** Observe that since the Cauchy stress is
symmetric

Next, note that

Finally,
substituting the latter identity into the virtual work equation, applying the
divergence theorem, using the linear momentum balance equation and boundary
conditions on and we obtain the required result.

**Second version of the principle of virtual work**

The converse of this
statement is much more interesting and useful.
Suppose that satisfies the virtual work equation

*for all* virtual velocity fields . Then the stress field must satisfy

3. The boundary condition on

4. The linear momentum balance equation

The
significance of this result is that it gives us an alternative way to solve for
a stress field that satisfies the linear momentum balance equation, *which avoids having to differentiate the
stress*. It is not easy to
differentiate functions accurately in the computer, but it is easy to integrate
them. The virtual work statement is the
starting point for any finite element solution involving deformable solids.

**Proof: **Follow
the same preliminary steps as before, i..e.

and substitute into the
virtual work equation

Apply the divergence
theorem to the first term in the first integral, and recall that on ,
we see that

Since this must hold for
all virtual velocity fields we could choose

where
is an arbitrary function that is positive
everywhere inside the solid, but is equal to zero on . For this choice, the virtual work equation
reduces to

and since the integrand is
positive everywhere the only way the equation can be satisfied is if

Given
this, we can next choose a virtual velocity field that satisfies

on . For this choice (and noting that the volume
integral is zero) the virtual work equation reduces to

Again, the integrand is
positive everywhere (it is a perfect square) and so can vanish only if

as stated.

**5.9 The Virtual Work equation in
terms of other stress measures.**

It is often convenient to
implement the virtual work equation in a finite element code using different
stress measures.

To do so, we define

1.
The actual
deformation gradient in the solid

2.
The virtual rate
of change of deformation gradient

3.
The virtual rate
of change of Lagrange strain

In addition, we define (in
the usual way)

1. Kirchhoff
stress** **

2.
Nominal (First
Piola-Kirchhoff) stress** **

3.
Material (Second Piola-Kirchhoff) stress** **

In terms of these
quantities, the virtual work equation may be expressed as

Note
that all the volume integrals are now taken over the *undeformed solid* this is convenient for computer applications,
because the shape of the undeformed solid is known. The area integral is evaluated over the *deformed* solid, unfortunately. It can be expressed as an equivalent integral
over the undeformed solid, but the result is messy and will be deferred until
we actually need to do it.

**5.10 The Virtual Work equation for
infinitesimal deformations.**

For infintesimal motions, the Cauchy, Nominal, and Material
stress tensors are equal; and the virtual stretch rate can be replaced by the
virtual infinitesimal strain rate

There is no need to distinguish between the volume or surface
area of the deformed and undeformed solid.
The virtual work equation can thus be expressed as

for all kinematically
admissible velocity fields.

As
a special case, this expression can be applied to a quasi-static state with .
Then, for a stress state satisfying the static equilibrium equation and boundary conditions on ,
the virtual work equation reduces to

In which * are kinematically admissible displacements components
** on **S*_{2})
and .

Conversely,
if the stress state satisfies for every set of kinematically admissible
virtual displacements, then the stress state satisfies the static equilibrium equation and boundary conditions on .

**5.10 The
first and second laws of thermodynamics for continua**

Consider a sub-region *V* of a deformed solid with surface *A*.
Define:

The heat flux vector **q **, which is defined so that is the heat flux crossing an internal surface with
area *dA *and normal

**n**in the deformed solid;

The heat supply *q* , defined so that *dQ=* *qdV* is the heat supplied from an
external source into a volume element d*V*
in the deformed solid;

The net heat flux into the solid

The net
rate of mechanical work done on the solid

The
total kinetic energy

The
total internal energy where is the specific internal energy (internal
energy per unit mass)

The
total entropy ,
where is the specific entropy (entropy per unit
mass)

The
temperature of the solid .

The net
external entropy supplied to the volume

The specific free energy

The
total free energy

The ** first law of thermodynamics**
then requires that

for any volume *V*.

This condition can also be
expressed as

To see this,

- recall that

- the divergence theorem gives

- Therefore

- Note also that

where
is the mass density per unit reference
volume. Finally

This
must hold for all *V*, giving the
required result.

The ** second law of thermodynamics **specifies
that the net entropy production within

*V*must be non-negative, i.e.

This can also be expressed as

(this condition is know as
the **Clausius-Duhem **inequality).

To see this, simply
substitute the definitions and use the divergence theorem.

The first and second laws can
be combined to yield the *free energy imbalance*

This can also be expressed as

To see the first result,

- note that

- Use the first law to see that

where
we have noted that temperature is always positive. This yields the solution.

The second result follows by
integrating the local form and using the stress-power work expression.

**5.11 Conservation laws for a control
volume**

To model solids, it is usually convenient to write the
equations of motion for a volume that moves with the solid. When modeling fluids, it is often preferable
to consider a fixed spatial volume, through which the fluid moves with
time. To this end,

We
consider a fixed region in space *R*,
bounded by a surface *B*.

Material
flows through the region with velocity field **v**(**y**,*t*).

The solid has mass density per unit deformed
volume (in the spatial configuration) ;
and is subjected to a body force **b **per
unit mass.

A heat flux **q** flows through the solid; while an external source injects heat
flux *Q* per unit deformed volume.

The
conservation laws can be expressed in terms of integrals over the fixed spatial
region (which does not move with the solid) as follows

**Mass Conservation: **

**Linear Momentum Balance **

**Angular Momentum Balance**

**Mechanical Power Balance **

**First law of thermodynamics**

** **

**Second law of thermodynamics**

These results can all be
derived from the conservation laws for a material volume, using a similar
approach. Consider the mass conservation
equation as a special case. Start with
the local form

Integrate over a fixed
spatial volume

Note the *R* is independent of time, and use the divergence theorem

As a second example, for the
linear momentum balance equation, start with the local form

Note that, using mass
conservation

Therefore

Integrating this expression
over the fixed control volume and using the divergence theorem gives the stated
answer.

A similar approach can be
used to obtain the remaining results details are left as an exercise.

**5.12 Transformation of kinematic and
kinetic variables under changes of reference frame**

Since physical laws must be constructed so as to be
independent of the choice of reference frame, the behavior of kinematic and
kinetic variables, the field equations, and constitutive equations under a
change of reference frame is of interest. The concept of a reference frame, and the
various relations involved in changing reference frames, are both rather
obscure concepts. There are several
reasons for this:

1. One source of confusion arises
because Newtonian mechanics relies on the concept of an *inertial* frame, and Newton’s law **F**=m**a** only holds in this
frame. The statement ‘the laws of
physics are independent of reference frame’ does not mean that **F**=m**a**
in all reference frames it means that all observers must describe Newton’s
laws with respect to the same inertial frame, and must do so in a consistent
manner. Frame indifference is not the
theory of relativity…

2. A second source of confusion stems
from the use of a *reference configuration*
to quantify shape changes of a solid.
We nearly always use the undeformed solid as reference, which gives the
impression that the reference configuration, like the deformed configuration,
is associated with the inertial frame.
In fact, the reference configuration is completely arbitrary, and even
though all observers might choose the same initial configuration of a solid as
reference, *they will all assume that the
reference configuration is fixed*.
The reference configuration is *not*
attached to the inertial frame.
Moreover, since the reference configuration is arbitrary, two observers
could choose different reference configurations, and still devise equations
that describe the same physical process.
Of course the exact form of the governing equations will change with the
choice of reference configuration. There
are no restrictions governing transformation of reference configuraiton between
observers, beyond the fact that two reference configurations must be related by
an invertible 1:1 mapping.

To make the concept of a change in reference frame in classical
continuum mechanics precise, we first introduce the intertial frame. As in all preceding discussions, we assume
that the inertial frame is a three-dimensional Euclidean space, and let denote a point in the inertial frame. We then define Newtonian measures of velocity
and acceleration vectors in the usual way as

This inertial frame could be viewed by an observer who
rotates and translates with respect to the inertial frame. To this observer, all physical quantities
associated with the inertial frame would appear to be translated and rotated in
the opposite sense. We describe this
apparent transformation of space with respect to the observer as a rigid rotation
and translation thus, the position vector of a point seen by
the observer is related to position in the inertial frame by

where is an arbitrary fixed point in the inertial
frame, is an arbitrary vector, and is a proper orthogonal tensor, representing a
rigid rotation. It is convenient to
introduce the spin associated with the relative rotation of the inertial frame
and the observer’s frame

We will denote quantities in the observer’s reference frame with a star superscript; -for example mass density, body force, Cauchy stress ; those without superscripts will be assumed to be defined in the inertial frame.

All observable physical
quantities must transform in particular ways under a change of observer. Specifically:

Scalar quantities, such as density or
temperature are *invariant* they have the same value in all reference
frames.

Quantities such as body force, a line element
in the deformed solid; the normal to a surface; the velocity vector, the
acceleration vector, and so on, are physical vectors defined in the inertial
reference frame. They can be regarded
as connecting two points in the inertial frame, and must transform with the
line connecting these two points under a change of reference frame. Thus, a normal vector to a deformed surface,
body force, velocity, acceleration vectors must transform as

Vectors that transform in this way are said to be **frame indifferent**, or **objective**. Note (1) frame indifference does not mean
that vectors are invariant quite the opposite, in fact it means that all observers must describe the
same physical quantity; (2) vector quantities we make frequent use of in solid
mechanics need not necessarily be frame indifferent. For example, the normal to the reference
configuration of a solid would *not* be
frame indifferent; nor would a material fiber within the reference
configuration.

Tensor quantities that map a frame indifferent
vector onto another frame independent vector are similarly said to be **frame indifferent, **or **objective. **Examples include the stretch
rate tensor (which specifies the relative velocity of two ends of an
infinitesimal material fiber in the spatial configuration); or Cauchy stress
(which maps the normal to a surface in the spatial configuration to the
physical traction vector. A frame
indifferent tensor must transform as

Again, not all tensors are frame
indifferent. The deformation gradient;
the spin tensor; the Lagrange strain tensor are *not* frame indifferent.

Frame indifference can also be looked at as follows. Let be an inertial basis. Under a change of observer, the basis
vectors transform as (relative to the observer, they appear to
rotate with the observed frame. It is
important to note that are time dependent). Now, we can compute components in either
basis

*The components of a
vector with respect to a basis that rotates with an inertial frame is
independent of the observer*. This is one interpretation of
what we mean by a physical process being independent of the observer.

We now examine how several kinematic and kinetic variables
commonly used in continuum mechanics transform under a change of observer. To describe deformations, a reference
configuration must be selected. A
material particle in the reference configuration is identified by a time
independent vector **X** in *reference space*. The choice of reference space is arbitrary;
and there is no reason why different observers will necessarily adopt the same
reference space. Discussions are
greatly simplified, however, if we choose to assume that all observers use the
same space for the reference configuration (behavior under a change of
reference configuration can be treated separately).

**The
deformation mapping** transforms as

**The
deformation gradient **transforms as

**The left
Cauchy Green strain Lagrange strain, the
right stretch tensor are invariant**

**The
right Cauchy Green strain, Eulerian strain, left stretch tensor are frame indifferent**

**The
velocity gradient and spin tensor **transform as

** **

**The
velocity and acceleration vectors **transform as

(the
additional terms in the acceleration can be interpreted as the centripetal and
coriolis accelerations)

**The
Cauchy stress is frame indifferent ** (you can see this from the formal definition,
or use the fact that the virtual power must be invariant under a frame change)

**The material
stress is frame invriant **

**The
nominal stress transforms as ** (note that this transformation rule will
differ if the nominal stress is defined as the transpose of the measure used
here…)

By
this time you are probably asking yourself why anyone could possibly care about
all this. This is a fair question issues of frame indifference arise rather
rarely in practice. They do come up, however, when we define *constitutive equations* for a material which must relate deformation measures to
internal force measures. A new
constitutive equation is a new physical law and it is important to make sure that the new
law behaves correctly under a change of observer. Most modern constitutive equations try to
describe the underlying microscopic processes that govern its response, and if
this is done properly, the law will be frame indifferent. But some constitutive laws are just
curve-fits some mathematical relationship between a
deformation measure and a force measure and not all possible relationships will
transform correctly.

Problems arise most commonly
in trying to develop *rate forms* of
constitutive equations, which are intended to relate some measure of strain
rate to stress rate. This is because, even
if a vector or tensor is itself frame indifferent, its time derivative is
generally not. For example, although
position vector satisfies and is frame indifferent, this does not mean
that . Similarly, for the rate of change of Cauchy
stress is not frame indifferent, because

In fact,
only quantities that are *invariant*
under a change of observer can be differentiated safely with respect to time their time derivatives remain invariant.

This means that if
constitutive equations are expressed in rate form for example something that looks at first
glance like the rate form of an elastic constitutive equation

(here **C** is a fourth order constant tensor) the constitutive equation will *not* be frame indifferent.

There
are various fixes for this the constitutive law can be written in terms
of invariant quantities (eg by relating the rate of change of material stress
to Lagrange strain rate); they can be derived from physical principles, in
which case frame indifferent measures usually emerge naturally from the
treatment; or frame indifferent measures of time derivatives can be specially
constructed.

As
a specific example, one way to construct a frame indifferent stress rate is to use
the rate of change of stress components with respect to a basis that rotates
with the solid (this is what an observer rotating with the material would
actually see). This sounds easy we just choose some basis vectors with each basis vector parallel to a
particular material fiber. But this
doesn’t quite work, because of course the basis vectors won’t generally remain
orthogonal under an arbitrary deformation.
So rather than attach to particular material fibers, we simply
suppose that they rotate with the *average*
angular velocity of all material fibers passing through a particular
point. This means that

where **W** is the spin tensor. Now,
the time derivative of stress can be written as

Here,
the first term can be interpreted as the stress rate seen by an observer
rotating with the embedded basis; the second and third are the rates of change
of stress arising from the rotation of the material. The first term is of particular interest, and
is called the ** Jaumann stress rate. **It
is

**defined as**

It is easy to show that is frame indifferent. Many constitutive equations assume that
material stretch rate is proportional to this special stress rate. For example, we could write

Provided
that is a frame indifferent fourth-order tensor,
this constitutive equation would be frame indifferent.

This
raises another question, of course.
What does it mean for a fourth-order tensor to be frame
indifferent? Hopefully you can answer
this question for yourself!