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

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


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


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


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