Nonlinear finite elements/Objectivity of constitutive relations

A detailed description of thermoelastic materials can be found in Continuum mechanics/Thermoelasticity. In this discussion we will avoid the complications induced by including the temperature.

In the material configuration, a hyperelastic material satisfies two requirements:

1. a stored energy function (${\displaystyle W}$) exists for the material.
2. the stored energy function depends locally only on the deformation gradient.

Given these requirements, if ${\displaystyle {\boldsymbol {N}}={\boldsymbol {P}}^{T}}$ is the nominal stress (${\displaystyle {\boldsymbol {P}}}$ is the first Piola-Kirchhoff stress tensor), then

${\displaystyle {\boldsymbol {N}}^{T}(\mathbf {X} ,t)={\boldsymbol {P}}(\mathbf {X} ,t)=\rho _{0}~{\frac {\partial W}{\partial {\boldsymbol {F}}}}\left[\mathbf {X} ,{\boldsymbol {F}}(\mathbf {X} ,t)\right]}$

The stored energy function ${\displaystyle W(\mathbf {X} ,{\boldsymbol {F}})}$ is said to be objective or frame indifferent if

${\displaystyle W(\mathbf {X} ,{\boldsymbol {Q}}\cdot {\boldsymbol {F}})=W(\mathbf {X} ,{\boldsymbol {F}})}$

where ${\displaystyle {\boldsymbol {Q}}}$ is an orthogonal tensor with ${\displaystyle {\boldsymbol {S}}\cdot {\boldsymbol {Q}}^{T}={\boldsymbol {\mathit {1}}}}$.

This objectivity condition can be achieved only if (in the material configuration)

${\displaystyle W(\mathbf {X} ,{\boldsymbol {F}})={\hat {W}}(\mathbf {X} ,{\boldsymbol {C}})~;~~{\boldsymbol {C}}={\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}}$

since ${\displaystyle {\boldsymbol {C}}_{r}={\boldsymbol {F}}^{T}\cdot {\boldsymbol {Q}}^{T}\cdot {\boldsymbol {Q}}\cdot {\boldsymbol {F}}={\boldsymbol {C}}}$.

We can show that

Proof:

The stress strain relation for a hyperelastic material is

${\displaystyle {\boldsymbol {P}}=\rho _{0}~{\frac {\partial W}{\partial {\boldsymbol {F}}}}\left[\mathbf {X} ,{\boldsymbol {F}}(\mathbf {X} ,t)\right]}$

The chain rule then implies that

${\displaystyle {\boldsymbol {P}}:{\boldsymbol {T}}=\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}:\left({\frac {\partial {\boldsymbol {C}}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}\right)}$

for any second order tensor ${\displaystyle {\boldsymbol {T}}}$.

Now, using the product rule of differentiation,

${\displaystyle {\frac {\partial {\boldsymbol {C}}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}={\frac {\partial }{\partial {\boldsymbol {F}}}}\left({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}\right):{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}^{T}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot \left({\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}\right)}$

or,

${\displaystyle {\frac {\partial {\boldsymbol {C}}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}=\left({\boldsymbol {\mathsf {I}}}^{T}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot \left({\boldsymbol {\mathsf {I}}}:{\boldsymbol {T}}\right)={\boldsymbol {T}}^{T}\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot {\boldsymbol {T}}}$

where ${\displaystyle {\boldsymbol {\mathsf {I}}}}$ is the fourth order identity tensor. Therefore,

${\displaystyle {\boldsymbol {P}}:{\boldsymbol {T}}=\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}:\left({\boldsymbol {T}}^{T}\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot {\boldsymbol {T}}\right)}$

Using the identity

${\displaystyle {\boldsymbol {A}}:({\boldsymbol {B}}^{T}\cdot {\boldsymbol {C}})=({\boldsymbol {C}}\cdot {\boldsymbol {A}}^{T}):{\boldsymbol {B}}=({\boldsymbol {B}}\cdot {\boldsymbol {A}}):{\boldsymbol {C}}}$

we have

${\displaystyle {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}:({\boldsymbol {T}}^{T}\cdot {\boldsymbol {F}})=\left[{\boldsymbol {F}}\cdot \left({\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\right)^{T}\right]:{\boldsymbol {T}}\quad {\text{and}}\quad {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}:({\boldsymbol {F}}^{T}\cdot {\boldsymbol {T}})=\left[{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\right]:{\boldsymbol {T}}}$

Therefore, invoking the arbitrariness of ${\displaystyle {\boldsymbol {T}}}$, we have

${\displaystyle {\boldsymbol {P}}=\rho _{0}~{\boldsymbol {F}}\cdot \left({\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\right)^{T}+\rho _{0}~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}}$

Since ${\displaystyle {\boldsymbol {C}}={\boldsymbol {C}}^{T}}$ we have

${\displaystyle \left({\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\right)^{T}={\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}}$

which implies that

${\displaystyle {\boldsymbol {P}}=2~\rho _{0}~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\quad {\text{or}}\quad {\boldsymbol {N}}=2~\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}}$

Recall the relations between the 2nd Piola-Kirchhoff stress tensor and the first Piola-Kirchhoff stress tensor (and the nominal stress tensor)

${\displaystyle {\boldsymbol {S}}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {P}}}$

Therefore, we have

${\displaystyle {\boldsymbol {S}}=2~\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}}$

Also from the relation between the Cauchy stress and the 2nd Piola-Kirchhoff stress tensor

${\displaystyle {\boldsymbol {\sigma }}=J^{-1}~{\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}={\cfrac {\rho }{\rho _{0}}}~{\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}}$

we have

${\displaystyle {\boldsymbol {\sigma }}=2~\rho ~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}\quad {\text{or}}\quad {\boldsymbol {\tau }}=2~\rho _{0}{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}}$

We may also express these relations in terms of the Lagrangian Green strain

${\displaystyle {\boldsymbol {E}}={\frac {1}{2}}({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}-{\boldsymbol {\mathit {1}}})={\frac {1}{2}}({\boldsymbol {C}}-{\boldsymbol {\mathit {1}}})}$

Then we have

${\displaystyle {\frac {\partial W}{\partial {\boldsymbol {C}}}}={\frac {\partial W}{\partial {\boldsymbol {E}}}}:{\frac {\partial {\boldsymbol {E}}}{\partial {\boldsymbol {C}}}}={\frac {1}{2}}~{\frac {\partial W}{\partial {\boldsymbol {E}}}}}$

Hence, we can write

The stored energy function ${\displaystyle W}$ is objective if and only if the Cauchy stress tensor is symmetric, i.e., if the balance of angular momentum holds. Show this.

1. ↑ C. Truesdell and W. Noll, 1992, The Nonlinear Field Theories of Mechanics:2nd ed., Springer-Verlag, Berlin