Quastistatic

If we neglect the inertial term (\(\rho \frac{\partial \vec{v}}{\partial t} \approx \vec{0}\)), then time dependence only arises from history-dependent constitutive equations and boundary conditions. Our solution vector is the displacement vector and the elasticity equation reduces to

(131)\[\begin{gather} % Solution \vec{s}^T = \left( \vec{u} \quad \ p \right)^T, \\ % Elasticity \vec{f}(t) + \boldsymbol{\nabla} \cdot \left(\boldsymbol{\sigma}^\mathit{dev}(\vec{u}) - p\boldsymbol{I}\right) = \vec{0} \text{ in }\Omega, \\ % Pressure \vec{\nabla} \cdot \vec{u} + \frac{p}{K} = 0 \text{ in }\Omega, \\ % Neumann \boldsymbol{\sigma} \cdot \vec{n} = \vec{\tau} \text{ on }\Gamma_\tau, \\ % Dirichlet \vec{u} = \vec{u}_0 \text{ on }\Gamma_u, \\ p = p_0 \text{ on }\Gamma_p. \end{gather}\]

Because we will use implicit time stepping, we place all of the terms in the elasticity equation on the LHS. Using trial functions \({\vec{\psi}_\mathit{trial}^{u}}\) and \({\psi_\mathit{trial}^{p}}\) and incorporating the Neumann boundary conditions, we write the weak form as

(132)\[\begin{gather} % Displacement \int_\Omega {\vec{\psi}_\mathit{trial}^{u}} \cdot \vec{f}(t) + \nabla {\vec{\psi}_\mathit{trial}^{u}} : \left(-\boldsymbol{\sigma}^\mathit{dev}(\vec{u}) + p\boldsymbol{I} \right)\, d\Omega + \int_{\Gamma_\tau} {\vec{\psi}_\mathit{trial}^{u}} \cdot \vec{\tau}(t) \, d\Gamma, = 0 \\ % Pressure \int_\Omega {\psi_\mathit{trial}^{p}} \cdot \left(\vec{\nabla} \cdot \vec{u} + \frac{p}{K} \right) \, d\Omega = 0. \end{gather}\]

Residual Pointwise Functions

Identifying \(F(t,s,\dot{s})\), we have

(133)\[\begin{split}\begin{gathered} F^u(t,s,\dot{s}) = \int_\Omega {\vec{\psi}_\mathit{trial}^{u}} \cdot{\color{blue} \underbrace{\color{black}\vec{f}(t)}_{\color{blue}{f_0^u}}} + \nabla {\vec{\psi}_\mathit{trial}^{u}} :{\color{blue} \underbrace{\color{black}\left(-\boldsymbol{\sigma}^\mathit{dev}(\vec{u}) + p\boldsymbol{I}\right)}_{\color{blue}{f_1^u}}} \, d\Omega + \int_{\Gamma_\tau} {\vec{\psi}_\mathit{trial}^{u}} \cdot {\color{blue}\underbrace{\color{black}\vec{\tau}(t)}_{\color{blue}{f_0^u}}} \, d\Gamma, \\ % \end{gathered}\end{split}\]

(134)\[\begin{gathered} F^p(t,s,\dot{s}) = \int_\Omega {\psi_\mathit{trial}^{p}} \cdot {\color{blue}\underbrace{\color{black}\left(\vec{\nabla} \cdot \vec{u} + \frac{p}{K} \right)}_{\color{blue}{f_0^p}}} \, d\Omega. \end{gathered}\]

Jacobians Pointwise Functions

With two fields we have four Jacobian pointwise functions for the LHS:

(135)\[\begin{align} % JF uu J_F^{uu} &= \frac{\partial F^u}{\partial u} + s_\mathit{tshift} \frac{\partial F^u}{\partial \dot{u}} = \int_\Omega \nabla {\vec{\psi}_\mathit{trial}^{u}} : \frac{\partial}{\partial u}(-\boldsymbol{\sigma}^\mathit{dev}) \, d\Omega = \int_\Omega {\psi_\mathit{trial}^{u}}_{i,k} \, {\color{blue} \underbrace{\color{black}\left(-C^\mathit{dev}_{ikjl}\right)}_{\color{blue}{J_{f3}^{uu}}}} \, {\psi_\mathit{basis}^{u}}_{j,l}\, d\Omega \\ % JF up J_F^{up} &= \frac{\partial F^u}{\partial p} + s_\mathit{tshift} \frac{\partial F^u}{\partial \dot{p}} = \int_\Omega \nabla{\vec{\psi}_\mathit{trial}^{u}} : \boldsymbol{I} {\psi_\mathit{basis}^{p}} \, d\Omega = \int_\Omega {\psi_\mathit{trial}^{u}}_{i,k} {\color{blue} \underbrace{\color{black}\delta_{ik}}_{\color{blue}{J_{f2}^{up}}}} \, {\psi_\mathit{basis}^{p}} \, d\Omega \\ % JF pu J_F^{pu} &= \frac{\partial F^p}{\partial u} + s_\mathit{tshift} \frac{\partial F^p}{\partial \dot{u}} = \int_\Omega {\psi_\mathit{trial}^{p}} \left(\vec{\nabla} \cdot {\vec{\psi}_\mathit{basis}^{u}}\right) \, d\Omega = \int_\Omega {\psi_\mathit{trial}^{p}} {\color{blue}\underbrace{\color{black}\delta_{jl}}_{\color{blue}{J_{f1}^{pu}}}} {\psi_\mathit{basis}^{u}}_{j,l} \, d\Omega\\ % JF pp J_F^{pp} &= \frac{\partial F^p}{\partial p} + s_\mathit{tshift} \frac{\partial F^p}{\partial \dot{p}} = \int_\Omega {\psi_\mathit{trial}^{p}}{\color{blue}\underbrace{\color{black}\frac{1} {K}}_{\color{blue}{J_{f0}^{pp}}}} {\psi_\mathit{basis}^{p}} \, d\Omega \end{align}\]