# 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

(107)#\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

(108)#\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 (108): find $$(\diff \bm{u}, \diff p) \in \mathcal{V} \times \mathcal{Q}$$ such that

(109)#\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

(110)#\begin{aligned} \diff \bm{P} &= \diff \bm{F}\, \bm{S} + \bm{F} \diff \bm{S}, \\ \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 (110) is

(111)#\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 (44), we derive split

(112)#$\diff \bm{S} = \underbrace{\frac{\partial \bm{S}_{vol}}{\partial \bm{E}} \tcolon \diff \bm{E}}_{\diff \bm{S}_{vol}} + \underbrace{\frac{\partial \bm{S}_{iso}}{\partial \bm{E}} \tcolon \diff \bm{E}}_{\diff \bm{S}_{iso}} + \underbrace{\frac{\partial \bm{S}_{vol}}{\partial p} \diff p}_{\diff \bm{S}_{p}},$

then,

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

and

(115)#\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

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

Note

If we use single field isochoric model (50) the linearization of the volumetric stress becomes

(116)#$\diff \bm{S}_{vol} = \kappa J^2 (\bm{C}^{-1} \tcolon \diff \bm{E}) \bm{C}^{-1} + \frac{\kappa}{2} (J^2 -1) \diff \bm{C}^{-1}.$

$$\diff \bm{S}_{p} = 0$$ and we only need to solve the first equation of (108).

Deriving $$\diff\bm{S}$$ for isochoric Mooney-Rivlin material

For the Mooney-Rivlin model (58), we derive split

(117)#$\diff \bm{S} = \underbrace{\frac{\partial \bm{S}_{vol}}{\partial \bm{E}} \tcolon \diff \bm{E}}_{\diff \bm{S}_{vol}} + \underbrace{\frac{\partial \bm{S}_{iso}}{\partial \bm{E}} \tcolon \diff \bm{E}}_{\diff \bm{S}_{iso}} + \underbrace{\frac{\partial \bm{S}_{vol}}{\partial p} \diff p}_{\diff \bm{S}_{p}},$

where $$\diff \bm{S}_{p}$$ and $$\diff \bm{S}_{vol}$$ are the same as (114), (113) (or (116) if we use single field stress (75)) and the isochoric part is

(118)#\begin{aligned} \diff \bm{S}_{iso} &= -\frac{4}{3}(\bm{C}^{-1} \tcolon \diff \bm{E}) \left( \mu_1 J^{-2/3} + 4 \mu_2 J^{-4/3} \right) \bm{C}^{-1} \bm{E}_{dev}\\ &- \frac{1}{3} \left( \mu_1 J^{-2/3} + 2 \mu_2 J^{-4/3} \right) \left( 2 \mathbb{{I}_1}(\diff \bm{E})\, \bm{C}^{-1} + \mathbb{{I}_1} \diff \bm{C}^{-1} \right) \\ &- \frac{8}{3}(\bm{C}^{-1} \tcolon \diff \bm{E}) \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{E})\bm{I}_3 - \bm{E} \right) \\ &+2 \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\diff \bm{E})\bm{I}_3 - \diff \bm{E} \right) \\ & + (c_1 + c_2) \bm{C}^{-1} - \frac{4}{3} \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{E}) + 2 \mathbb{{I}_2}(\bm{E})\right) \diff \bm{C}^{-1} \end{aligned}

where

(119)#\begin{aligned} c_1 &= \frac{16}{9} \mu_2 J^{-4/3} (\bm{C}^{-1} \tcolon \diff \bm{E}) \left( \mathbb{{I}_1}(\bm{E}) + 2 \mathbb{{I}_2}(\bm{E}) \right). \\ c_2 &= -\frac{4}{3} \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\diff \bm{E}) + 2 \mathbb{{I}_1}(\bm{E}) \mathbb{{I}_1}(\diff \bm{E}) - 2 \bm{E} \tcolon \diff \bm{E} \right). \end{aligned}
Deriving $$\diff\bm{S}$$ for isochoric Ogden material

Similar to the Neo-Hookean model we have

(120)#$\diff \bm{S} = \underbrace{\frac{\partial \bm{S}_{vol}}{\partial \bm{E}} \tcolon \diff \bm{E}}_{\diff \bm{S}_{vol}} + \underbrace{\frac{\partial \bm{S}_{iso}}{\partial \bm{E}} \tcolon \diff \bm{E}}_{\diff \bm{S}_{iso}} + \underbrace{\frac{\partial \bm{S}_{vol}}{\partial p} \diff p}_{\diff \bm{S}_{p}},$

where $$\diff \bm{S}_{p}$$ and $$\diff \bm{S}_{vol}$$ are similar to Neo-Hookean and Mooney-Rivlin models and the isochoric part is

(121)#\begin{aligned} \diff \bm{S}_{iso} &= \sum_{i=1}^3 \diff S_i^{iso} \hat{\bm{N}_i} \hat{\bm{N}_i}^T + S_i^{iso} \left( \diff \hat{\bm{N}_i} \hat{\bm{N}_i}^T + \hat{\bm{N}_i} \diff \hat{\bm{N}_i}^T\right) \end{aligned}

For example computing $$\diff S_1^{iso}$$ from (71) gives

(122)#\begin{aligned} \diff S_1^{iso} &= -\frac{2\diff \beta_1}{\beta_1^3} \sum_{j=1}^N \frac{m_j}{3} \left[ 2\operatorname{\tt expm1}(\alpha_j \ell_1) - \operatorname{\tt expm1}(\alpha_j \ell_2) - \operatorname{\tt expm1}(\alpha_j \ell_3) \right] J^{-\alpha_j/3} \\ &+ \frac{1}{\beta_1^2} \sum_{j=1}^N \frac{m_j \alpha_j}{3} \left[ 2 \diff \ell_1 \exp(\alpha_j \ell_1) - \diff \ell_2 \exp(\alpha_j \ell_2) - \diff \ell_3 \exp(\alpha_j \ell_3) \right] J^{-\alpha_j/3} \\ &- \frac{1}{\beta_1^2} \sum_{j=1}^N \frac{m_j \alpha_j}{9} \left[ 2 \operatorname{\tt expm1}(\alpha_j \ell_1) - \operatorname{\tt expm1}(\alpha_j \ell_2) - \operatorname{\tt expm1}(\alpha_j \ell_3) \right] J^{-\alpha_j/3} (\bm{C}^{-1} \tcolon \diff \bm{E}) \end{aligned}

To compute $$\diff \beta_i$$ we differentiate $$\bm{C} = \sum_{i=1}^3 \beta_i^2 \hat{\bm{N}_i} \hat{\bm{N}_i}^T$$ as

(123)#$\diff \bm{C} = \sum_{i=1}^3 2 \beta_i \diff \beta_i \hat{\bm{N}_i} \hat{\bm{N}_i}^T + \beta_i^2 \left(\diff \hat{\bm{N}_i} \hat{\bm{N}_i}^T + \hat{\bm{N}_i} \diff \hat{\bm{N}_i}^T \right)$

and used the fact that the eigenvectors are orthonormal i.e., $$\langle \hat{\bm{N}_i}, \hat{\bm{N}_j} \rangle = \delta_{ij}$$ so that $$\langle \hat{\bm{N}_i}, \diff \hat{\bm{N}_i} \rangle = 0$$, we can multiply (123) from the left and right by $$\hat{\bm{N}_i}$$

(124)#$\langle \hat{\bm{N}_i}, \diff \bm{C} \hat{\bm{N}_i} \rangle = 2 \beta_i \diff \beta_i \Rightarrow \diff \beta_i = \frac{1}{\beta_i} \langle \hat{\bm{N}_i}, \diff \bm{E} \hat{\bm{N}_i} \rangle$

By differentiating $$\bm{E} \hat{\bm{N}_i} = \beta_i^E \hat{\bm{N}_i}$$ we will arrive at

$\diff \bm{E} \hat{\bm{N}_i} + \bm{E} \diff \hat{\bm{N}_i}= \diff \beta_i^E \hat{\bm{N}_i} + \beta_i^E \diff \hat{\bm{N}_i}$

taking the inner product of above equation with $$\hat{\bm{N}_j}, \, j\neq i$$ simplifies to

\begin{aligned} &\quad \langle \hat{\bm{N}_j}, \diff \bm{E} \hat{\bm{N}_i} \rangle + \langle \diff \hat{\bm{N}_i}, \bm{E} \hat{\bm{N}_j} \rangle = \beta_i^E \langle \diff \hat{\bm{N}_i}, \hat{\bm{N}_j} \rangle \\ &\quad \langle \hat{\bm{N}_j}, \diff \bm{E} \hat{\bm{N}_i} \rangle + \beta_j^E \langle \diff \hat{\bm{N}_i}, \hat{\bm{N}_j} \rangle = \beta_i^E \langle \diff \hat{\bm{N}_i}, \hat{\bm{N}_j} \rangle \\ &\quad \langle \diff \hat{\bm{N}_i}, \hat{\bm{N}_j} \rangle = \frac{1}{\beta_i^E - \beta_j^E} \langle \diff \bm{E} \hat{\bm{N}_i}, \hat{\bm{N}_j} \rangle \\ &\quad \diff \hat{\bm{N}_i} = \sum_{j \neq i} \frac{1}{\beta_i^E - \beta_j^E} \langle \diff \bm{E} \hat{\bm{N}_i}, \hat{\bm{N}_j} \rangle \hat{\bm{N}_j} \end{aligned}

and finally the linearization of $$\ell_i = \log \beta_i$$ is

$\diff \ell_i = \frac{\diff \beta_i}{\beta_i}.$

which complete $$\diff\bm{S}$$ for Ogden model.

## Current configuration#

Similar to what we have shown in (85), we can write the first equation of (108) in the current configuration (see (86) ) which yields to the Jacobian form

(125)#\begin{aligned} \int_{\Omega_0} \nabla_x \bm{v} \tcolon \left( \diff \bm{\tau} - \bm{\tau} \left( \nabla_x \diff \bm{u} \right)^T \right) 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 the second equation is similar to (109) and we can use (92)

$\diff\bm\tau - \bm\tau\left( \nabla_x \diff\bm u \right)^T = \left( \nabla_x \diff\bm u \right) \bm\tau + \bm F \diff\bm S \bm F^T.$

to compute the linearization of the first equation in current configuration for different material models.

Representation of $$\bm F \diff\bm S \bm F^T$$ for isochoric Neo-Hookean material

Based on the split (112), we can derive

(126)#$\bm{F} \diff \bm{S}_{vol} \bm{F}^{T} = -p J \trace (\diff \bm \epsilon) \bm{I}_3 + 2 p J \diff \bm \epsilon,$
(127)#$\bm{F} \diff \bm{S}_{p} \bm{F}^{T} = -J \bm{I}_3 \diff p,$

where $$\diff \bm \epsilon$$ is defined in (90) and we have used (97) and (98). The isochoric part can be simplified as

(128)#\begin{aligned} \bm{F} \diff \bm{S}_{iso} \bm{F}^{T} &= -\frac{2}{3} \mu J^{-2/3} \left( 2 \trace (\diff \bm \epsilon) \bm{e}_{dev} + \trace(\diff \bm e) \bm{I}_3 - \mathbb{{I}_1} \diff \bm \epsilon \right), \end{aligned}

where $$\mathbb{{I}_1} = \trace(\bm{b})$$.

Note

If we use single field isochoric model the linearization of the volumetric stress becomes

(129)#$\bm{F} \diff \bm{S}_{vol} \bm{F}^{T} = \kappa J^2 \trace (\diff \bm \epsilon) \bm{I}_3 - (J^2 -1) \diff \bm \epsilon.$

$$\diff \bm{S}_{p} = 0$$ and we only need to solve the first equation of (125).

Representation of $$\bm F \diff\bm S \bm F^T$$ for isochoric Mooney-Rivlin material

The $$\bm{F} \diff \bm{S}_{p} \bm{F}^T$$ and $$\bm{F} \diff \bm{S}_{vol} \bm{F}^T$$ in (117) are the same as (127), (126) (or (129) if we use single field formulation ). The isochoric part can be derived as

(130)#\begin{aligned} \bm{F} \diff \bm{S}_{iso} \bm{F}^T &= -\frac{4}{3}(\trace (\diff \bm \epsilon)) \left( \mu_1 J^{-2/3} + 4 \mu_2 J^{-4/3} \right) \bm{e}_{dev}\\ &+ \frac{2}{3} \left( \mu_1 J^{-2/3} + 2 \mu_2 J^{-4/3} \right) \left( \mathbb{{I}_1} \diff \bm \epsilon -\mathbb{{I}_1}(\diff \bm{e})\, \bm{I}_3 \right) \\ &- \frac{8}{3}(\trace (\diff \bm \epsilon)) \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{e})\bm{b} - \bm{b} \bm{e} \right) \\ &+2 \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\diff \bm{e})\bm{b} - \bm{b} \diff \bm \epsilon \bm{b} \right) \\ & + (c_1 + c_2) \bm{I}_3 + \frac{8}{3} \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{e}) + 2 \mathbb{{I}_2}(\bm{e})\right) \diff \bm \epsilon \end{aligned}

where

(131)#\begin{aligned} c_1 &= \frac{16}{9} \mu_2 J^{-4/3} (\trace (\diff \bm \epsilon)) \left( \mathbb{{I}_1}(\bm{e}) + 2 \mathbb{{I}_2}(\bm{e}) \right). \\ c_2 &= -\frac{4}{3} \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\diff \bm{e}) + 2 \mathbb{{I}_1}(\bm{e}) \mathbb{{I}_1}(\diff \bm{e}) - 2 \trace (\bm{b} \bm{e} \diff \bm \epsilon) \right). \end{aligned}

and we have used (100), (101), and

\begin{aligned} \bm{F} \bm{C}^{-1} \bm{E}_{dev} \bm{F}^T &= \bm{F}^{-T} \left( \bm{E} - \frac{1}{3} \mathbb{{I}_1}(\bm{E}) \bm{I}_3 \right) \bm{F}^T = \bm{e}_{dev}. \\ \bm{F} \bm{E} \bm{F}^T &= \frac{1}{2} \left( \bm{F} \bm{C} \bm{F}^T - \bm{F} \bm{F}^T \right) = \bm{b} \bm{e}. \\ \diff \bm{E} &= \frac{1}{2} \left( \bm{F}^T \diff \bm{F} \bm{F}^{-1} \bm{F} + \bm{F}^T \bm{F}^{-T} \diff \bm{F}^T \bm{F} \right) = \bm{F}^T \diff \bm \epsilon \bm{F}. \\ \bm{E} \tcolon \diff \bm{E} &= \trace(\bm{E} \diff \bm{E}) = \trace(\bm{E} \bm{F}^T \diff \epsilon \bm{F}) = \trace (\bm{F} \bm{E} \bm{F}^T \diff \epsilon) = \trace(\bm{b} \bm{e} \diff \bm \epsilon ). \end{aligned}
Representation of $$\bm F \diff\bm S \bm F^T$$ for isochoric Ogden material

Based on the split (120), the $$\bm{F} \diff \bm{S}_{p} \bm{F}^T$$ and $$\bm{F} \diff \bm{S}_{vol} \bm{F}^T$$ are the same as (127), (126) (or (129) if we use single field formulation ) and the isochoric part is

(132)#\begin{aligned} \bm{F} \diff \bm{S}_{iso} \bm{F}^{T} &= \sum_{i=1}^3 \beta^2_i \diff S_i^{iso} \hat{\bm{n}_i} \hat{\bm{n}_i}^T + \beta_i S_i^{iso} \left( \bm{F} \diff \hat{\bm{N}_i} \hat{\bm{n}_i}^T + \hat{\bm{n}_i} \diff \hat{\bm{N}_i}^T \bm{F}^T \right) \\ &= \sum_{i=1}^3 \left(\diff \tau_i^{iso} - 2 \frac{\diff \beta_i}{\beta_i} \tau_i^{iso} \right) \hat{\bm{n}_i} \hat{\bm{n}_i}^T + \tau_i^{iso} \bm{A}_i \end{aligned}

where for example $$\diff \tau_1^{iso}$$ can be written

\begin{aligned} \diff \tau_1^{iso} &= \sum_{j=1}^N \frac{m_j \alpha_j}{3} \left[ 2 \diff \ell_1 \exp(\alpha_j \ell_1) - \diff \ell_2 \exp(\alpha_j \ell_2) - \diff \ell_3 \exp(\alpha_j \ell_3) \right] J^{-\alpha_j/3} \\ &- \sum_{j=1}^N \frac{m_j \alpha_j}{9} \left[ 2 \operatorname{\tt expm1}(\alpha_j \ell_1) - \operatorname{\tt expm1}(\alpha_j \ell_2) - \operatorname{\tt expm1}(\alpha_j \ell_3) \right] J^{-\alpha_j/3} \trace \left(\diff \bm \epsilon \right) \end{aligned}

and

$\bm{A}_i = 2 \frac{\diff \beta_i}{\beta_i} \hat{\bm{n}_i} \hat{\bm{n}_i}^T + \diff \left( \hat{\bm{n}_i} \hat{\bm{n}_i}^T \right) + \left( \left( \nabla_x \diff \bm{u} \right) \hat{\bm{n}_i} \hat{\bm{n}_i}^T + \hat{\bm{n}_i} \hat{\bm{n}_i}^T \left( \nabla_x \diff \bm{u} \right)^T \right)$

where $$\tau_i^{iso}$$ is defined in (74) and we have used $$\bm{F} \hat{\bm{N}_i} = \beta_i \hat{\bm{n}_i}$$ and $$\tau_i^{iso} = \beta_i^2 S_i^{iso}$$. Note that the linearization of principal stretches, $$\beta_i$$, and eigenvectors, $$\hat{\bm{n}_i}$$, are similar to initial configuration but are written in terms of Green-Euler tensor i.e.,

$\diff \beta_i = \frac{1}{\beta_i} \langle \hat{\bm{n}_i}, \diff \bm{e} \hat{\bm{n}_i} \rangle.$
$\quad \diff \hat{\bm{n}_i} = \sum_{j \neq i} \frac{1}{\beta_i^e - \beta_j^e} \langle \diff \bm{e} \hat{\bm{n}_i}, \hat{\bm{n}_j} \rangle \hat{\bm{n}_j}.$