Let us now derive rate equations for a hyperelastic material.
We start off with the relation
Then the material time derivative of is given by
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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,
Similarly, if we start off with the relation
the material time derivative of can be expressed as
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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,
The first and second elasticity tensors are related by
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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,
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
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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
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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.
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
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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 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 tensor
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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 tensor
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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,
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,