Almansi-Hamel model

The Almansi-Hamel elastic constitutive model.

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Almansi-Hamel strain measure is given by $\mathbf{e}=\frac{1}{2}(1-{\mathbf{B}}^{-1})$.

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{2\mu }{J}{\mathbf{e}}^{\prime }+\frac{\kappa }{J}\mathrm{tr}(\mathbf{e})1$$

Cauchy tangent stiffness

$${\mathcal{T}}_{ijkL}(\mathbf{F})=\frac{\mu }{J}\left[B_{jk}^{-1}{F}_{iL}^{-T}+B_{ik}^{-1}{F}_{jL}^{-T}-\frac{2}{3}{\delta }_{ij}{B}_{km}^{-1}{F}_{mL}^{-T}-2e_{ij}^{\prime }{F}_{kL}^{-T}\right]+\frac{\kappa }{J}\left[{\delta }_{ij}{B}_{km}^{-1}{F}_{mL}^{-T}-\mathrm{tr}(\mathbf{e}){\delta }_{ij}{F}_{kL}^{-T}\right]$$