Continuum mechanics/Nonlinear elasticity: Wikis

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Study guide

Up to date as of January 14, 2010

From Wikiversity

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 $\psi\,$ 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, ρ 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 ρ0 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 λ and μ are material constants that have to be determined by experiments. Such a linear relation is physically possible only for small strains.