# Continuum mechanics/Nonlinear elasticity

There are two types models of nonlinear elastic behavior that are in common use. These are :

• Hyperelasticity
• Hypoelasticity

## Hyperelasticity

Hyperelastic materials are truly elastic in the sense that if a load is applied to such a material and then removed, the material returns to its original shape without any dissipation of energy in the process. In other word, a hyperelastic material stores energy during loading and releases exactly the same amount of energy during unloading. There is no path dependence.

If ${\displaystyle \psi \,}$ is the Helmholtz free energy, then the stress-strain behavior for such a material is given by

${\displaystyle {\boldsymbol {\sigma }}=\rho ~{\boldsymbol {F}}\bullet {\cfrac {\partial \psi }{\partial {\boldsymbol {E}}}}\bullet {\boldsymbol {F}}^{T}=2~\rho ~{\boldsymbol {F}}\bullet {\cfrac {\partial \psi }{\partial {\boldsymbol {C}}}}\bullet {\boldsymbol {F}}^{T}}$

where ${\displaystyle {\boldsymbol {\sigma }}}$ is the Cauchy stress, ${\displaystyle \rho }$ is the current mass density, ${\displaystyle {\boldsymbol {F}}}$ is the deformation gradient, ${\displaystyle {\boldsymbol {E}}}$ is the Lagrangian Green strain tensor, and ${\displaystyle {\boldsymbol {C}}}$ is the left Cauchy-Green deformation tensor.

We can use the relationship between the Cauchy stress and the 2nd Piola-Kirchhoff stress to obtain an alternative relation between stress and strain.

${\displaystyle {\boldsymbol {S}}=2~\rho _{0}~{\cfrac {\partial \psi }{\partial {\boldsymbol {C}}}}}$

where ${\displaystyle {\boldsymbol {S}}}$ is the 2nd Piola-Kirchhoff stress and ${\displaystyle \rho _{0}}$ is the mass density in the reference configuration.

### Isotropic hyperelasticity

For isotropic materials, the free energy must be an isotropic function of ${\displaystyle {\boldsymbol {C}}}$. This also mean that the free energy must depend only on the principal invariants of ${\displaystyle {\boldsymbol {C}}}$ which are

{\displaystyle {\begin{aligned}I_{\boldsymbol {C}}=I_{1}&={\text{tr}}(\mathbf {C} )=C_{ii}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}\\II_{\boldsymbol {C}}=I_{2}&={\tfrac {1}{2}}\left[{\text{tr}}(\mathbf {C} ^{2})-({\text{tr}}~\mathbf {C} )^{2}\right]={\tfrac {1}{2}}\left[C_{ik}C_{ki}-C_{jj}^{2}\right]=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\\III_{\boldsymbol {C}}=I_{3}&=\det(\mathbf {C} )=\lambda _{1}^{2}\lambda _{2}^{2}\lambda _{3}^{2}\end{aligned}}}

In other words,

${\displaystyle \psi ({\boldsymbol {C}})\equiv \psi (I_{1},I_{2},I_{3})}$

Therefore, from the chain rule,

${\displaystyle {\cfrac {\partial \psi }{\partial {\boldsymbol {C}}}}={\cfrac {\partial \psi }{\partial I_{1}}}~{\cfrac {\partial I_{1}}{\partial {\boldsymbol {C}}}}+{\cfrac {\partial \psi }{\partial I_{2}}}~{\cfrac {\partial I_{2}}{\partial {\boldsymbol {C}}}}+{\cfrac {\partial \psi }{\partial I_{3}}}~{\cfrac {\partial I_{3}}{\partial {\boldsymbol {C}}}}=a_{0}~{\boldsymbol {\mathit {1}}}+a_{1}~{\boldsymbol {C}}+a_{2}~{\boldsymbol {C}}^{-1}}$

From the Cayley-Hamilton theorem we can show that

${\displaystyle {\boldsymbol {C}}^{-1}\equiv f({\boldsymbol {C}}^{2},{\boldsymbol {C}},{\boldsymbol {\mathit {1}}})}$

Hence we can also write

${\displaystyle {\cfrac {\partial \psi }{\partial {\boldsymbol {C}}}}=b_{0}~{\boldsymbol {\mathit {1}}}+b_{1}~{\boldsymbol {C}}+b_{2}~{\boldsymbol {C}}^{2}}$

The stress-strain relation can then be written as

${\displaystyle {\boldsymbol {S}}=2~\rho _{0}~\left[b_{0}~{\boldsymbol {\mathit {1}}}+b_{1}~{\boldsymbol {C}}+b_{2}~{\boldsymbol {C}}^{2}\right]}$

A similar relation can be obtained for the Cauchy stress which has the form

${\displaystyle {\boldsymbol {\sigma }}=2~\rho ~\left[a_{2}~{\boldsymbol {\mathit {1}}}+a_{0}~{\boldsymbol {B}}+a_{1}~{\boldsymbol {B}}^{2}\right]}$

where ${\displaystyle {\boldsymbol {B}}}$ is the right Cauchy-Green deformation tensor.

#### Cauchy stress in terms of invariants

For w:isotropic hyperelastic materials, the Cauchy stress can be expressed in terms of the invariants of the left Cauchy-Green deformation tensor (or right Cauchy-Green deformation tensor). If the w:strain energy density function is ${\displaystyle W({\boldsymbol {F}})={\hat {W}}(I_{1},I_{2},I_{3})={\bar {W}}({\bar {I}}_{1},{\bar {I}}_{2},J)={\tilde {W}}(\lambda _{1},\lambda _{2},\lambda _{3})}$, then

{\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}&={\cfrac {2}{\sqrt {I_{3}}}}\left[\left({\cfrac {\partial {\hat {W}}}{\partial I_{1}}}+I_{1}~{\cfrac {\partial {\hat {W}}}{\partial I_{2}}}\right){\boldsymbol {B}}-{\cfrac {\partial {\hat {W}}}{\partial I_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]+2{\sqrt {I_{3}}}~{\cfrac {\partial {\hat {W}}}{\partial I_{3}}}~{\boldsymbol {\mathit {1}}}\\&={\cfrac {2}{J}}\left[{\cfrac {1}{J^{2/3}}}\left({\cfrac {\partial {\bar {W}}}{\partial {\bar {I}}_{1}}}+{\bar {I}}_{1}~{\cfrac {\partial {\bar {W}}}{\partial {\bar {I}}_{2}}}\right){\boldsymbol {B}}-{\cfrac {1}{3}}\left({\bar {I}}_{1}~{\cfrac {\partial {\bar {W}}}{\partial {\bar {I}}_{1}}}+2~{\bar {I}}_{2}~{\cfrac {\partial {\bar {W}}}{\partial {\bar {I}}_{2}}}\right){\boldsymbol {\mathit {1}}}-\right.\\&\qquad \qquad \qquad \left.{\cfrac {1}{J^{4/3}}}~{\cfrac {\partial {\bar {W}}}{\partial {\bar {I}}_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]+{\cfrac {\partial {\bar {W}}}{\partial J}}~{\boldsymbol {\mathit {1}}}\\&={\cfrac {\lambda _{1}}{\lambda _{1}\lambda _{2}\lambda _{3}}}~{\cfrac {\partial {\tilde {W}}}{\partial \lambda _{1}}}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\cfrac {\lambda _{2}}{\lambda _{1}\lambda _{2}\lambda _{3}}}~{\cfrac {\partial {\tilde {W}}}{\partial \lambda _{2}}}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+{\cfrac {\lambda _{3}}{\lambda _{1}\lambda _{2}\lambda _{3}}}~{\cfrac {\partial {\tilde {W}}}{\partial \lambda _{3}}}~\mathbf {n} _{3}\otimes \mathbf {n} _{3}\end{aligned}}}

(See the page on the left Cauchy-Green deformation tensor for the definitions of these symbols).

### Saint-Venant–Kirchhoff material

The simplest constitutive relationship that satisfies the requirements of hyperelasticity is the Saint-Venant–Kirchhoff material, which has a response function of the form

${\displaystyle {\boldsymbol {S}}=\lambda ~{\text{tr}}({\boldsymbol {E}})~{\boldsymbol {\mathit {1}}}+2~\mu ~{\boldsymbol {E}},}$

where ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are material constants that have to be determined by experiments. Such a linear relation is physically possible only for small strains.