Nonlinear finite elements/Rate form of hyperelastic laws

Rate equationsEdit

Let us now derive rate equations for a hyperelastic material.

First elasticity tensorEdit

We start off with the relation

 

Then the material time derivative of   is given by

First elasticity tensor

 

where the fourth order tensor   is call the first elasticity tensor. This tensor has major symmetries but not minor symmetries. In index notation with respect to an orthonormal basis

 

Proof:

We have

 

Using the product rule, we have

 

Therefore,

 

Second elasticity tensorEdit

Similarly, if we start off with the relation

 

the material time derivative of   can be expressed as

Second elasticity tensor

 

where the fourth order tensor   is called the material elasticity tensor or the second elasticity tensor. Since this tensor relates symmetric second order tensors it has minor symmetries. It also has major symmetries because the two partial derivatives are with the same quantity and an interchange does not change things. In index notation with respect to an orthonormal basis

 

Proof:

We have

 

Again using the product rule, we have

 

Therefore,

 

Relation between first and second elasticity tensorsEdit

The first and second elasticity tensors are related by

 

Proof:

Recall that the first and second Piola-Kirchhoff stresses are related by

 

Taking the material time derivative of both sides gives

 

Using the expression for   above, we get

 

Now

 

Therefore,

 

Now

 

That means

 

which gives us

 

In index notation,

 

Therefore,

 

Fourth elasticity tensorEdit

Now we will compute the spatial elasticity tensor for the rate constitutive equation for a hyperelastic material. This tensor relates an objective rate of stress (Cauchy or Kirchhoff) to the rate of deformation tensor. We can show that

Fourth elasticity tensor for the Kirchhoff stress

 

where

 

The fourth order tensor   is called the spatial elasticity tensor or the fourth elasticity tensor. Clearly,   cannot be derived from the store energy function   because of the dependence on the deformation gradient.

Proof:

Recall that the Lie derivative of the Kirchhoff stress is defined as

 

We have found that

 

We also know from Continuum_mechanics/Time_derivatives_and_rates#Time_derivative_of_strain that

 

where   is the spatial rate of deformation tensor. Therefore,

 

In index notation,

 

or,

 

where

 

Alternatively, we may define   in terms of the Cauchy stress  , in which case the constitutive relation is written as

Fourth elasticity tensor for the Cauchy stress

 

where

 

The proof of this relation between the spatial and material elasticity tensors is very similar to that for the rate of Kirchhoff stress. Many authors define this quantity   as the spatial elasticity tensor. Note the factor of  . This form of the spatial elasticity tensor is crucial for some of the calculations that follow.

Relation between first and fourth elasticity tensorsEdit

The first and fourth elasticity tensors are related by

 

In the above equation   is the elasticity tensor that relates the rate of Kirchhoff stress to the rate of deformation.

Instead, if we use the Cauchy stress and the spatial elasticity tensor   that relates the Cauchy stress to the rate of deformation), the above relation becomes

 

Proof:

Recall that

 

Therefore,

 

Also recall that

 

Therefore, using index notation,

 

Now,

 

In index notation

 

Using this we get

 

or,

 

Now,

 

Therefore,

 

Also,

 

So we have

 

Note:

The fourth order tensor

 

which depends on the symmetry of   is called the third elasticity tensor, i.e.,

 

Therefore, the relation between the first and third elasticity tensors is

 

or,

 

In index notation

 

Therefore,

 

An isotropic spatial elasticity tensor?Edit

An isotropic spatial elasticity tensor cannot be derived from a stored energy function if the constitutive relation is of the form

 

where

 

Since a significant number of finite element codes use such a constitutive equation, (also called the equation of a hypoelastic material of grade 0) it is worth examining why such a model is incompatible with elasticity.

Start with a constant and isotropic material elasticity tensorEdit

Let us start of with an isotropic elastic material model in the reference configuration. The simplest such model is the St. Venant-Kirchhoff hyperelastic model

 

where   is the second Piola-Kirchhoff stress,   is the Lagrangian Green strain, and   are material constants. We can show that this equation can be derived from a stored energy function.

Taking the material time derivative of this equation, we get

 

Now,

 

where   is the second (material) elasticity tensor.

Therefore,

 

which implies that

 

In index notation,

 

Now, from the relations between the second elasticity tensor and the fourth (spatial) elasticity tensor, we have

 

Therefore, in this case,

 

or,

 

where  . So we see that the spatial elasticity tensor   cannot be a constant tensor unless  .

Alternatively, if we define

 

we get

 

Start with a constant and isotropic spatial elasticity tensorEdit

Let us now look at the situation where we start off with a constant and isotropic spatial elasticity tensor, i.e.,

 

In index notation,

 

Since

 

multiplying both sides by   we have,

 

Therefore, substituting in the expression for a constant and isotropic  , we have

 

or,

 

or,

 

Since   which gives us  , we can write

 

Alternatively, if we define

 

we get

 

and therefore,

 

Hypoelastic material of grade 0Edit

A hypoelastic material of grade zero is one for which the stress-strain relation in rate form can be expressed as

 

where   is constant. When the material is isotropic we have

 

We want to show that hypoelastic material models of grade 0 cannot be derived from a stored energy function. To do that, recall that

 

and

 

For a material elasticity tensor   to be derivable from a stored energy function it has to satisfy the Bernstein integrability conditions. We have

 

Also, due to the interchangeability of derivatives,

 

Therefore,

 

These integrability conditions have to be satisfied by any material elasticity tensor.

At this stage we will use the relation

 

If we plug this into the integrability condition we will see that

 

If we multiply both sides by   we are left with

 

This is an unphysical situation and hence shows that a hypoelastic material of grade zero requires that   for it to be derivable from a stored energy function.

Proof:

Let us simplify the notation by writing  . Then,

 

Then,

 

and

 

As this stage we use the identities (see Nonlinear finite elements/Kinematics#Some_useful_results for proofs)

 

and

 

Therefore we have

 

and

 

Equating the two, we see that the terms that cancel out are

 

and

 

Therefore,

 

implies that

 

or,

 

In other words,

 

Now, if we multiply both sides by   we get

 

or,

 

Next, multiplying both sides by   gives

 

or,

 

Finally, multiplying both sides by   gives

 

Therefore,