Nonlinear finite elements/Overview of balance laws

Governing equations in the deformed configurationEdit

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 calculusEdit

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 theoremEdit

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 theoremEdit

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 massEdit

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 momentumEdit

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 momentumEdit

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 energyEdit

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 inequalityEdit

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 configurationEdit

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 massEdit

The Lagrangian form of the balance of mass is


Balance of linear momentumEdit

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 momentumEdit

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 energyEdit

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 inequalityEdit

The entropy inequality in Lagrangian form is


In terms of the first P-K stress we have


Conjugate work measuresEdit

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.