Governing equations in the deformed configuration
The governing equations for a continuum (apart from the kinematic relations
and the constitutive laws) are
- Balance of mass
- Balance of linear momentum
- Balance of angular momentum
- Balance of energy
- Entropy inequality
Two important results from calculus
There are two important results from calculus that are useful when we
use or derive these governing equations. These are
- The Gauss divergence theorem.
- The Reynolds transport theorem
These are useful enough to bear repeating at this point in the context of
second order tensors.
The Gauss divergence theorem
The divergence theorem relates volume integrals to surface integrals.
Let be a body and let be its boundary with outward unit normal
. Let be a second order tensor valued function of . Then,
if is differentiable at least once (i.e., ) then
It is often of interest to consider the situation where is not
continuously differentiable, i.e., when there are jumps in within the
body. Let represent the set of surfaces internal to the
body where there are jumps in . In that case, the divergence theorem is
where the subscripts and represents the values on the two sides of
the jump discontinuity with normal .
Reynold's transport theorem
The transport theorem shows you how to
calculate the material time derivative of an integral. It is a
generalization of the Leibniz formula.
Let be a body in its current configuration and let be
its surface. Also, let .
If is a scalar valued function of and then
If is a vector valued function of and .
To refresh your memory, recall that the material time derivative is given by
Conservation of mass
For the situation where a body does not gain or lose mass, the balance of mass
is written as
Sometimes, this equation is also written in conservative form as
If the material is incompressible then the density does not change with
time and we get
For Lagrangian descriptions we can show that
where is the initial density.
Conservation of linear momentum
The balance of linear momentum is essentially Newton's second applied to
continua. Newton's second law can be written as
where the linear momentum is given by
and the total force is given by
where is the density, is the spatial velocity, is
the body force and is the surface traction. Therefore, the balance
of linear momentum can be written as
Now using the transport theorem, we have
From the conservation of mass, the second term on the right hand side is zero
and we are left with
Therefore, the balance of linear momentum can be written as
Using Cauchy's theorem () and the divergence theorem we
can show that the balance of linear momentum can be written as
In index notation,
Conservation of angular momentum
We also have to make sure that the moments are balanced. This requirement takes
the form of the conservation of angular momentum and can be written as
We can show that this equation reduces down to the requirement that the
Cauchy stress is symmetric, i.e.,
Conservation of energy
The balance of energy can be expressed as
where is the internal energy per unit mass, is the heat flux vector
and is the heat source per unit volume.
We may also write this equation as
where , the rate of deformation tensor, is the symmetric part of the
velocity gradient. We can do this because the contraction of the skew
symmetric part of the velocity gradient with the symmetric Cauchy stress gives
For purely mechanical problems, and . So we can write
This shows that and are conjugate in power.
The entropy inequality is useful in determining which forms of the constitutive
equations are admissible. This inequality is also called the dissipation
inequality. In its Clausius-Duhem form, the inequality may written as
where is the specific entropy (entropy per unit mass) and is the
In differential form the Clausius-Duhem inequality can be written as
In terms of the specific internal energy, the entropy inequality can be
If we define the Helmholtz free energy (the energy that is available to do
mechanical work) as
we can also write,
Governing equations in the reference configuration
The Lagrangian form of the governing equations can be obtained using the
relations between the various measures in the deformed and reference
Balance of mass
The Lagrangian form of the balance of mass is
Balance of linear momentum
The Lagrangian form of the balance of linear momentum is
where is the nominal stress and indicates that the
derivatives are with respect to . In terms of the first Piola-Kirchhoff
stress we can write
In index notation,
Balance of angular momentum
The balance of angular momentum in Lagrangian form is
In terms of the first Piola-Kirchhoff stress
In terms of the second Piola-Kirchhoff stress
Balance of energy
In the material frame, the balance of energy takes the form
where is the heat flux per unit reference area.
In terms of the first Piola-Kirchhoff stress tensor we have
The entropy inequality in Lagrangian form is
In terms of the first P-K stress we have
Conjugate work measures
Whatever measures we choose to use to represent stress and strain (or a rate
of strain), their product should give us a measure of the work done (or the
power spent). This measure should not depend on the chosen measures.
Therefore, the correct combination of stress and strain should be
work conjugate or power conjugate. Three commonly used power
conjugate stress and rate of strain measures are
- The Cauchy stress () and the rate of deformation ().
- The nominal stress () and the rate of the deformation gradient ().
- The second P-K stress () and the rate of the Green strain ().
We can show that, in the absence of heat fluxes and sources,
Many more work/power conjugate measures can be found in the literature.