Mixed elasticity (incompressible)

For the incompressible and nearly incompressible single-phase materials where the motions are constrained by spatial conditions, it is most beneficial to split the deformation locally into a so-called volumetric (dilational) part and an isochoric (distortional) part. In order to be able to define strain energy functions that are separable into volumetric and isochoric parts, we decompose the deformation gradient in a multiplicative sense by

(140)\[ \bm{F} = (J^{1/3} \bm{I}) \bar{\bm{F}} = J^{1/3} \bar{\bm{F}} \]

where \(J^{1/3} \bm{I}\) describes the purely volumetric deformation and \(\bar{\bm{F}}\) captures the isochoric or volume-preserving deformation since \(\operatorname{det} \bar{\bm{F}} = \operatorname{det} J^{-1/3} \bm{F} = 1\) . Similar decomposition can be obtained for other tensor such as the right Cauchy-Green tensor \(\bm{C}\) as

(141)\[ \bar{\bm{C}} = \bar{\bm{F}}^T \bar{\bm{F}} = J^{-2/3} \bm{C} \]

where we call \(\bar{\bm{F}}\) and \(\bar{\bm{C}}\) the modified deformation gradient and modified right Cauchy-Green tensor, respectively. The modified principal stretches \(\bar{\lambda}_i\) and modified invariants \(\mathbb{\bar{I}}_i\) are

(142)\[ \bar{\lambda}_i = J^{-1/3} \lambda_i, \, i=1,2,3 \]
(143)\[ \begin{aligned} \mathbb{\bar{I}}_1 &= J^{-2/3}\mathbb{I}_1, & \mathbb{\bar{I}}_2 &= J^{-4/3}\mathbb{I}_2, & \mathbb{\bar{I}}_3 &= \left(J^{-2/3}\right)^3 \mathbb{I}_3 = 1. \end{aligned} \]

For the strain energy in terms of modified invariants \(\psi \left( \bar{\bm{C}} \right) = \psi \left( \mathbb{\bar{I}}_1, \mathbb{\bar{I}}_2, \mathbb{\bar{I}}_3 \right)\) we can derive the stress relations as

(144)\[ \bm{S} = 2 \frac{\partial \psi}{\partial \bm{C}} = 2 \sum_{i=1}^3 \frac{\partial \psi}{\partial \mathbb{\bar{I}}_i} \frac{\partial \mathbb{\bar{I}}_i}{\partial \bm{C}} \]

where the derivative of modified invariants are

(145)\[ \begin{aligned} \frac{\partial \mathbb{\bar{I}}_1}{\partial \bm{C}} &= \frac{\partial (J^{-2/3}\mathbb{I}_1)}{\partial \bm{C}} = J^{-2/3} \left( \bm{I} -\frac{1}{3} \mathbb{I}_1 \bm{C}^{-1}\right), \\ \frac{\partial \mathbb{\bar{I}}_2}{\partial \bm{C}} &= \frac{\partial (J^{-4/3}\mathbb{I}_2)}{\partial \bm{C}} = J^{-4/3} \left( \mathbb{I}_1 \bm{I} - \bm{C} - \frac{2}{3} \mathbb{I}_2 \bm{C}^{-1}\right), \\ \frac{\partial \mathbb{\bar{I}}_3}{\partial \bm{C}} &= 0, \end{aligned} \]

where we have used

(146)\[ \begin{aligned} \frac{\partial J}{\partial \bm{C}} &= \frac{1}{2} J \bm{C}^{-1}, & \frac{\partial J^{-2/3}}{\partial \bm{C}} &= -\frac{1}{3} J^{-2/3} \bm{C}^{-1}. \end{aligned} \]

Note that the derivatives above can be written with respect to \(\bm{E}\) by use of chain rule and \(\bm{C} = 2\bm{E} + \bm{I}\) as

(147)\[ \frac{\partial }{\partial \bm{E}} = \frac{\partial }{\partial \bm{C}} \frac{\partial C}{\partial \bm{E}} = 2 \frac{\partial }{\partial \bm{C}}. \]

The following sections showcase the constitutive models for incompressible materials and their numerical implementation in Ratel.