Nonlinear finite elements/Overview of balance laws

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Governing equations in the deformed configuration[edit]

The governing equations for a continuum (apart from the kinematic relations and the constitutive laws) are

  1. Balance of mass
  2. Balance of linear momentum
  3. Balance of angular momentum
  4. Balance of energy
  5. Entropy inequality

Two important results from calculus[edit]

There are two important results from calculus that are useful when we use or derive these governing equations. These are

  1. The Gauss divergence theorem.
  2. 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[edit]

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 written as

where the subscripts and represents the values on the two sides of the jump discontinuity with normal .

Reynold's transport theorem[edit]

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 . Then

To refresh your memory, recall that the material time derivative is given by

Conservation of mass[edit]

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[edit]

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[edit]

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[edit]

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 us zero.

For purely mechanical problems, and . So we can write

This shows that and are conjugate in power.

Entropy inequality[edit]

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 temperature.

In differential form the Clausius-Duhem inequality can be written as

In terms of the specific internal energy, the entropy inequality can be expressed as

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[edit]

The Lagrangian form of the governing equations can be obtained using the relations between the various measures in the deformed and reference configurations.

Balance of mass[edit]

The Lagrangian form of the balance of mass is

Balance of linear momentum[edit]

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[edit]

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[edit]

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

Entropy inequality[edit]

The entropy inequality in Lagrangian form is

In terms of the first P-K stress we have

Conjugate work measures[edit]

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

  1. The Cauchy stress () and the rate of deformation ().
  2. The nominal stress () and the rate of the deformation gradient ().
  3. 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.