# Isochoric Hyperelasticity#

Ratel solves the momentum balance equations using unstructured high-order finite/spectral element spatial discretizations.

## Initial configuration#

In the total Lagrangian approach for the incompressible Neo-Hookean hyperelasticity problem, the discrete equations are formulated with respect to the initial configuration. In this formulation, we solve for displacement and pressure $$\bm{u} \left( \bm{X} \right), p$$ in the reference frame $$\bm{X}$$. The notation for elasticity at finite strain is inspired by [Hol00] to distinguish between the current and initial configurations. As explained in the Continuum Mechanics section, we denote by capital letters the reference frame and by small letters the current one.

### Weak form#

We multiply the following strong form

(57)#\begin{aligned} \nabla_X \cdot \bm{P} + \rho_0 \bm{g} &= 0 \quad \text{in} \quad \Omega_{0} \\ -\frac{1}{2 J} (J^2 -1) - \frac{p}{\kappa}&= 0 \quad \text{in} \quad \Omega \end{aligned}

by test functions $$(\bm{v}, q)$$ and integrate by parts to obtain the weak form for finite-strain incompressible hyperelasticity: find $$(\bm{u}, p) \in \mathcal{V} \times \mathcal{Q} \subset H^1 \left( \Omega_0 \right) \times L^2 \left( \Omega_0 \right)$$ such that

(58)#\begin{aligned} \int_{\Omega_0}{\nabla_X \bm{v} \tcolon \bm{P}} \, dV - \int_{\Omega_0}{\bm{v} \cdot \rho_0 \bm{g}} \, dV - \int_{\partial \Omega_0}{\bm{v} \cdot \left( \bm{P} \cdot \hat{\bm{N}} \right)} \, dS &= 0, \quad \forall \bm{v} \in \mathcal{V}, \\ \int_{\Omega_0} q \cdot \left( -\frac{1}{2 J} (J^2 -1) - \frac{p}{\kappa} \right) J \, dV &= 0, \quad \forall q\in \mathcal{Q}, \end{aligned}

where $$\bm{P} \cdot \hat{\bm{N}}|_{\partial\Omega}$$ is replaced by any prescribed force/traction boundary condition written in terms of the initial configuration. This equation contains material/constitutive nonlinearities in defining $$\bm{S}(\bm{E})$$, as well as geometric nonlinearities through $$\bm{P} = \bm{F}\, \bm{S}$$, $$\bm{E}(\bm{F})$$, and the body force $$\bm{g}$$, which must be pulled back from the current configuration to the initial configuration.

### Newton linearization#

To derive a Newton linearization, we need the Jacobian form of the (58): find $$(\diff \bm{u}, \diff p) \in \mathcal{V} \times \mathcal{Q}$$ such that

\begin{aligned} \int_{\Omega_0} \nabla_X \bm{v} \tcolon \diff \bm{P} dV &= \text{rhs}, \quad \forall \bm{v} \in \mathcal{V}, \\ \int_{\Omega_0} q \cdot \left( \diff K J + K \diff J \right)dV &= 0, \quad \forall q \in \mathcal{Q} \end{aligned}

where $$K = -\frac{1}{2 J} (J^2 -1) - \frac{p}{\kappa}$$ and

(59)#\begin{aligned} \diff \bm{P} &= \frac{\partial \bm{P}}{\partial \bm{E}} \tcolon \diff \bm{E} + \frac{\partial \bm{P}}{\partial p} \diff p = \diff \bm{F}\, \bm{S} + \bm{F} \diff \bm{S} - J F^{-T} \diff p, \\ \diff K J + K \diff J &= J \frac{\partial K}{\partial \bm{E}} \tcolon \diff \bm{E} + J \frac{\partial K}{\partial p} \diff p + K \frac{\partial J}{\partial \bm{E}} \tcolon \diff \bm{E} \end{aligned}

with $$\diff \bm{F} = \nabla_X\diff \bm{u}$$ and

$\diff \bm{E} = \frac{\partial \bm{E}}{\partial \bm{F}} \tcolon \diff \bm{F} = \frac{1}{2} \left(\diff \bm{F}^T \bm{F} + \bm{F}^T \diff \bm{F} \right).$

The linearization of the second equation of (59) is

(60)#\begin{aligned} \diff K J + K \diff J &= -\frac{1}{2} (J^2 + 1) (\bm{C}^{-1} \tcolon \diff \bm{E}) - J \frac{\diff p}{\kappa} + K J (\bm{C}^{-1} \tcolon \diff \bm{E}), \\ &= -\left( J^2 + J \frac{p}{\kappa} \right) \bm{C}^{-1} \tcolon \diff \bm{E} - J \frac{\diff p}{\kappa} \end{aligned}

The linearization of the second Piola-Kirchhoff stress tensor, $$\diff \bm{S}$$, depends upon the material model.

Deriving $$\diff\bm{S}$$ for isochoric Neo-Hookean material

For the Neo-Hookean model (25), we derive split

$\diff \bm{S} = \underbrace{\frac{\partial \bm{S}_{vol}}{\partial \bm{E}} \tcolon \diff \bm{E}}_{\diff \bm{S}_{vol}} + \underbrace{\frac{\partial \bm{S}_{iso}}{\partial \bm{E}} \tcolon \diff \bm{E}}_{\diff \bm{S}_{iso}},$

then,

(61)#$\diff \bm{S}_{vol} = -p J (\bm{C}^{-1} \tcolon \diff \bm{E}) \bm{C}^{-1} - p J \diff \bm{C}^{-1},$

and

(62)#\begin{aligned} \diff \bm{S}_{iso} &= -\frac{2}{3}\mu J^{-2/3} (\bm{C}^{-1} \tcolon \diff \bm{E}) \left(\bm{I}_3 - \frac{1}{3} \mathbb{{I}_1} \bm{C}^{-1} \right) \\ &- \frac{1}{3} \mu J^{-2/3} \left( 2 \operatorname{trace} \diff \bm{E} \, \bm{C}^{-1} + \mathbb{{I}_1} \diff \bm{C}^{-1} \right), \\ &= -\frac{4}{3}\mu J^{-2/3} (\bm{C}^{-1} \tcolon \diff \bm{E}) \bm{C}^{-1} \bm{E}_{dev} - \frac{1}{3} \mu J^{-2/3} \left( 2 \operatorname{trace} \diff \bm{E} \, \bm{C}^{-1} + \mathbb{{I}_1} \diff \bm{C}^{-1} \right) \end{aligned}

where

$\diff \bm{C}^{-1} = \frac{\partial \bm{C}^{-1}}{\partial \bm{E}} \tcolon \diff \bm{E} = -2 \bm{C}^{-1} \diff \bm{E} \, \bm{C}^{-1} .$