Continuum mechanics/Nonlinear elasticity

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

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 \psi\, is the Helmholtz free energy, then the stress-strain behavior for such a material is given by

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

\boldsymbol{S} = 2~\rho_0~\cfrac{\partial \psi}{\partial \boldsymbol{C}}

where \boldsymbol{S} is the 2nd Piola-Kirchhoff stress and \rho_0 is the mass density in the reference configuration.

Isotropic hyperelasticity

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

\begin{align} 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{align}

In other words,

\psi(\boldsymbol{C}) \equiv \psi(I_1, I_2, I_3)

Therefore, from the chain rule,

\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

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

Hence we can also write

\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

\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

\boldsymbol{\sigma} = 2~\rho~\left[a_2~\boldsymbol{\mathit{1}} + a_0~\boldsymbol{B} + a_1~\boldsymbol{B}^2\right]

where \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 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

\begin{align} \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{align}

(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

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

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

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