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

|
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

|
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

|
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,
![{\displaystyle {\dot {\boldsymbol {P}}}={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {S}}+{\frac {1}{2}}~{\boldsymbol {F}}\cdot [{\boldsymbol {\mathsf {C}}}:({\dot {{\boldsymbol {F}}^{T}}}\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot {\dot {\boldsymbol {F}}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7e7642194e2cb36e9189b29faf4e48481eb5a8)
Now

That means

which gives us
![{\displaystyle {\dot {\boldsymbol {P}}}={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {S}}+{\boldsymbol {F}}\cdot [{\boldsymbol {\mathsf {C}}}:({\boldsymbol {F}}^{T}\cdot {\dot {\boldsymbol {F}}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cca27be2a2e0ca359bfc1b4ea31764d8c1adfaf)
In index notation,
![{\displaystyle {\begin{aligned}{\dot {P}}_{iJ}&={\dot {F}}_{iL}~S_{LJ}+F_{iN}~[{\mathsf {C}}_{NJML}~F_{kM}~{\dot {F}}_{kL}]\\&=S_{LJ}~\delta _{ik}{\dot {F}}_{kL}+{\mathsf {C}}_{NJML}~F_{iN}~F_{kM}~{\dot {F}}_{kL}\\&=(\delta _{ik}~S_{JL}+F_{iN}~{\mathsf {C}}_{NJML}~F_{kM})~{\dot {F}}_{kL}&={\mathsf {A}}_{iJkL}~{\dot {F}}_{kL}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a103d12d4a73d852a3020eebf5400dbacd6d3317)
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
![{\displaystyle {{\mathcal {L}}_{\varphi }[{\boldsymbol {\tau }}]={\boldsymbol {\mathsf {c}}}:{\boldsymbol {d}}\equiv {\boldsymbol {\mathsf {c}}}_{\tau }:{\boldsymbol {d}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f20077b866873bb65174e6ed6ec7c038cb328db)
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
![{\displaystyle {\mathcal {L}}_{\varphi }[{\boldsymbol {\tau }}]={\overset {\circ }{\boldsymbol {\tau }}}={\boldsymbol {F}}\cdot {\dot {\boldsymbol {S}}}\cdot {\boldsymbol {F}}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7830721feba4b36f1b2815adbf4a7afff18e57)
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,
![{\displaystyle {\mathcal {L}}_{\varphi }[{\boldsymbol {\tau }}]={\boldsymbol {F}}\cdot [{\boldsymbol {\mathsf {C}}}:({\boldsymbol {F}}^{T}\cdot {\boldsymbol {d}}\cdot {\boldsymbol {F}})]\cdot {\boldsymbol {F}}^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1707d29e5c65cc893df54ada749eb03a43d918)
In index notation,
![{\displaystyle ({\mathcal {L}}_{\varphi }[{\boldsymbol {\tau }}])_{ij}=F_{iK}~{\mathsf {C}}_{KLMN}~F_{kM}~d_{kl}~F_{lN}~F_{jL}=F_{iK}~F_{jL}~F_{kM}~F_{lN}~{\mathsf {C}}_{KLMN}~d_{kl}={\mathsf {c}}_{ijkl}~d_{kl}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/576a65d7984237c1c86d3c7f6407ed3a0521abcc)
or,
![{\displaystyle {\mathcal {L}}_{\varphi }[{\boldsymbol {\tau }}]={\boldsymbol {\mathsf {c}}}:{\boldsymbol {d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/780488ef65da6432d54470f7aebb9cfb92544d14)
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
![{\displaystyle {{\mathcal {L}}_{\varphi }[{\boldsymbol {\sigma }}]={\boldsymbol {\mathsf {c}}}:{\boldsymbol {d}}=J^{-1}~{\boldsymbol {\mathsf {c}}}_{\tau }:{\boldsymbol {d}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e45fe264516e4e913acec99a3bd147caa9eec8)
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.
The first and fourth elasticity tensors are related by
![{\displaystyle {{\mathsf {A}}_{iJkL}=F_{Jj}^{-1}[{\mathsf {c}}_{ijkl}+\tau _{jl}~\delta _{ik}]~F_{Ll}^{-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7054e14f46d9964ad51c65186e8fd99efbc74a)
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
![{\displaystyle {{\mathsf {A}}_{iJkL}=J~F_{Jj}^{-1}[{\mathsf {c}}_{ijkl}+\sigma _{jl}~\delta _{ik}]~F_{Ll}^{-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63f447ef51466f55ac4e15d6af21965a24e03ee9)
|
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,
![{\displaystyle {\boldsymbol {\mathsf {A}}}:{\dot {\boldsymbol {F}}}=[{\boldsymbol {\mathsf {a}}}:({\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1})]\cdot {\boldsymbol {F}}^{-T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed0f9c564ea2b6142a5ea6139e11f16c07843b72)
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
![{\displaystyle {\mathcal {L}}_{\varphi }[{\boldsymbol {\tau }}]={\boldsymbol {\mathsf {c}}}:{\boldsymbol {d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/780488ef65da6432d54470f7aebb9cfb92544d14)
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
[edit | edit source]
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,
![{\displaystyle {\mathsf {c}}_{ijkl}=F_{iI}~F_{jJ}~F_{kK}~F_{lL}~\left[\lambda ~\delta _{IJ}~\delta _{KL}+\mu ~(\delta _{IK}~\delta _{JL}+\delta _{JK}~\delta _{IL})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a1725efdb37603cb3b3facfb1b7af68c0a2728e)
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
[edit | edit source]
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
![{\displaystyle {\mathsf {C}}_{ABCD}=F_{Ai}^{-1}~F_{Bj}^{-1}~F_{Ck}^{-1}~F_{Dl}^{-1}~\left[\lambda ~\delta _{ij}~\delta _{kl}+\mu (\delta _{ik}~\delta _{jl}+\delta _{jk}~\delta _{il})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f94be158a0583341fde687eab08fdef14fc168a)
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
![{\displaystyle {\mathcal {L}}_{\varphi }[{\boldsymbol {\sigma }}]={\boldsymbol {\mathsf {c}}}:{\boldsymbol {d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5511973380a46aedb36ef8adcd201750127281b8)
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,
![{\displaystyle {\begin{aligned}{\frac {\partial {\mathsf {C}}_{IJKL}}{\partial C_{MN}}}=&\lambda ~{\frac {\partial J}{\partial C_{MN}}}~A_{IJ}~A_{KL}+\\&J~\lambda ~\left[{\frac {\partial A_{IJ}}{\partial C_{MN}}}~A_{KL}+A_{IJ}~{\frac {\partial A_{KL}}{\partial C_{MN}}}\right]+\\&\mu ~{\frac {\partial J}{\partial C_{MN}}}~(A_{IK}~A_{JL}+A_{JK}~A_{IL})+\\&J~\mu ~\left[{\frac {\partial A_{IK}}{\partial C_{MN}}}~A_{JL}+A_{IK}~{\frac {\partial A_{JL}}{\partial C_{MN}}}+{\frac {\partial A_{JK}}{\partial C_{MN}}}~A_{IL}+A_{JK}~{\frac {\partial A_{IL}}{\partial C_{MN}}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4eca63d9933a883b7105b08671e25ceb6d148b)
and
![{\displaystyle {\begin{aligned}{\frac {\partial {\mathsf {C}}_{IJMN}}{\partial C_{KL}}}=&\lambda ~{\frac {\partial J}{\partial C_{KL}}}~A_{IJ}~A_{MN}+\\&J~\lambda ~\left[{\frac {\partial A_{IJ}}{\partial C_{KL}}}~A_{MN}+A_{IJ}~{\frac {\partial A_{MN}}{\partial C_{KL}}}\right]+\\&\mu ~{\frac {\partial J}{\partial C_{KL}}}~(A_{IM}~A_{JN}+A_{JM}~A_{IN})+\\&J~\mu ~\left[{\frac {\partial A_{IM}}{\partial C_{KL}}}~A_{JN}+A_{IM}~{\frac {\partial A_{JN}}{\partial C_{KL}}}+{\frac {\partial A_{JM}}{\partial C_{KL}}}~A_{IN}+A_{JM}~{\frac {\partial A_{IN}}{\partial C_{KL}}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3771d7d4aa411ca4aa197192f80f09ee7f163b0a)
As this stage we use the identities (see Nonlinear finite elements/Kinematics#Some_useful_results for proofs)

and

Therefore we have
![{\displaystyle {\begin{aligned}{\frac {\partial {\mathsf {C}}_{IJKL}}{\partial C_{MN}}}=&{\cfrac {J~\lambda }{2}}~A_{MN}~A_{IJ}~A_{KL}-\\&{\cfrac {J~\lambda }{2}}~\left[(A_{IM}~A_{JN}+A_{JM}~A_{IN})~A_{KL}+(A_{KM}~A_{LN}+A_{LM}~A_{KN})~A_{IJ}\right]+\\&{\cfrac {J~\mu }{2}}~A_{MN}~(A_{IK}~A_{JL}+A_{JK}~A_{IL})-\\&{\cfrac {J~\mu }{2}}~\left[(A_{IM}~A_{KN}+A_{KM}~A_{IN})~A_{JL}+(A_{JM}~A_{LN}+A_{LM}~A_{JN})~A_{IK}\right.+\\&\qquad \quad \left.(A_{JM}~A_{KN}+A_{KM}~A_{JN})~A_{IL}+(A_{IM}~A_{LN}+A_{LM}~A_{IN})A_{JK}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e018193884e46f0b6f8297ac9758c7509828b5b3)
and
![{\displaystyle {\begin{aligned}{\frac {\partial {\mathsf {C}}_{IJMN}}{\partial C_{KL}}}=&{\cfrac {J~\lambda }{2}}~A_{KL}~A_{IJ}~A_{MN}-\\&{\cfrac {J~\lambda }{2}}~\left[(A_{IK}~A_{JL}+A_{JK}~A_{IL})~A_{MN}+(A_{KM}~A_{LN}+A_{LM}~A_{KN})A_{IJ}\right]+\\&{\cfrac {J~\mu }{2}}~A_{KL}~(A_{IM}~A_{JN}+A_{JM}~A_{IN})-\\&{\cfrac {J~\mu }{2}}~\left[(A_{IK}~A_{LM}+A_{KM}~A_{IL})~A_{JN}+(A_{JK}~A_{LN}+A_{JL}~A_{KN})~A_{IM}+\right.\\&\qquad \qquad \left.(A_{JK}~A_{LM}+A_{JL}~A_{KM})~A_{IN}+(A_{IK}~A_{LN}+A_{IL}~A_{KN})~A_{JM}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38ee8653002b6d8159b07ad24cf55cce02e1a5f0)
Equating the two, we see that the terms that cancel out are

and
![{\displaystyle {\begin{aligned}{\cfrac {J~\mu }{2}}~&\left[(A_{IM}~A_{KN}+A_{KM}~A_{IN})~A_{JL}+(A_{JM}~A_{LN}+A_{LM}~A_{JN})~A_{IK}\right.+\\&\left.(A_{JM}~A_{KN}+A_{KM}~A_{JN})~A_{IL}+(A_{IM}~A_{LN}+A_{LM}~A_{IN})A_{JK}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99d6efdbcb96b4ea5df6a8c869749b1f5ff4ab25)
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,
