Linear elastic¶

Constitutive theory¶

One can see from the equations in the Linear elastic that as $\firstlame \to \infty$, it is necessary that $\nabla\cdot \bm{u} \longrightarrow 0$ which gives an idea of alternative strategies. One approach is to define an auxiliary variable $p$, and rewrite constitutive equation (77) as

(148)¶\[ \begin{aligned} \bm \sigma(\bm u, p) &= -p \, \bm{I} + 2 \mu \bm \varepsilon, \\ p &= - \firstlame \trace \bm \varepsilon. \end{aligned} \]

Alternatively, we can use the definition of hydrostatic pressure i.e., $p_{\text{hyd}} = - \frac{\trace \bm \sigma}{3}$ and arrive at

(149)¶\[ \begin{aligned} \bm \sigma(\bm u, p_{\text{hyd}}) &= -p_{\text{hyd}} \, \bm{I} + 2 \mu \bm \varepsilon_{\text{dev}}, \\ p_{\text{hyd}} &= -\bulk \trace \bm \varepsilon. \end{aligned} \]

where $\bm \varepsilon_{\text{dev}} = \bm \varepsilon - \frac{1}{3} \trace \bm \varepsilon ~ \bm{I}$ is the deviatoric part of the linear strain tensor and $\bulk$ is the bulk modulus. We present a general constitutive equation as

(150)¶\[ \begin{aligned} \bm \sigma(\bm u, p) &= \left(\bulk_p \trace \bm \varepsilon -p \right) \bm{I} + 2 \mu \bm \varepsilon_{\text{dev}}, \\ p &= -\left(\bulk - \bulk_p \right) \trace \bm \varepsilon. \end{aligned} \]

where

(151)¶\[ \bulk_p = \frac{2 \mu \left(1 + \nu_p \right)}{3 \left(1 - 2 \nu_p \right)}, \]

is the primal portion of the bulk modulus, defined in terms of $\nu_p$ with $-1 \leq \nu_p < \nu$, where $\nu$ is the physical Poisson’s ratio. The standard full-train formulation (148) is obtained using $\nu_p = 0$, and the deviatoric formulation (149) with $\nu_p = -1$.

Strong and weak formulations¶

The boundary-value problem (Strong form) for constitutive equation (150) may be stated as follows: Given body force $\rho \bm g$, Dirichlet boundary $\bar{\bm u}$ and applied traction $\bar{\bm t}$, find the displacement and pressure-like variables $(\bm u, p) \in \mathcal{V} \times \mathcal{Q} $ (here $\mathcal{V} = H^1(\Omega), \mathcal{Q} = L^2(\Omega) $ ), such that:

(152)¶\[ \begin{aligned} -\nabla \cdot \bm \sigma - \rho \bm g &= 0, \qquad \text{in $\Omega$} \\ -\nabla\cdot\bm u - \frac{p}{\bulk - \bulk_p} &= 0, \qquad \text{in $\Omega$} \\ \bm{u} &= \bar{\bm u}, \quad \text{on $\partial \Omega^{D}$} \\ \bm \sigma \cdot \bm n &= \bar{\bm t}. \qquad \text{on $\partial \Omega^{N}$} \end{aligned} \]

with $\bm n$ be the unit normal on the boundary and its weak formulation as:

(153)¶\[ \begin{aligned} \int_{\Omega}{ \nabla \bm{v} \tcolon \bm{\sigma}} \, dv - \int_{\partial \Omega}{\bm{v} \cdot \bar{\bm t}} \, da - \int_{\Omega}{\bm{v} \cdot \rho \bm{g}} \, dv &= 0, \quad \forall \bm v \in \mathcal V \\ \int_{\Omega} q \left( -\nabla\cdot\bm u - \frac{p}{\bulk - \bulk_p} \right) \, dv & = 0. \quad \forall q \in \mathcal Q \end{aligned} \]

Command-line interface¶

To enable the incompressible, mixed linear elastic model, use the model option -model elasticity-mixed and set the material parameters listed in Mixed linear elastic model options. Any parameter without a default option is required.

Table 21 Mixed linear elastic model options¶
Option	Description	Default Value
`-model elasticity-mixed-linear`	Required to enable the mixed linear elastic model.
`-E [real]`	Young’s modulus, $E > 0$
`-nu [real]`	Poisson’s ratio, $\nu \leq 0.5$. Note, this model is only validated for $\nu = 0.5$.
`-nu_primal [real]`	Primal part of the Poisson’s ratio for Jacobian operator, $-1 \leq \nu_p < \nu \leq 0.5$. This parameter may improve solver performance and should not affect the solution.	`-1.0`
`-nu_primal_pc [real]`	Primal part of the Poisson’s ratio for preconditioner operator, $-1 \leq \nu_{pc} < \nu \leq 0.5$. This parameter may improve solver performance and should not affect the solution.	`-nu_primal` value

An example using the mixed linear elastic model can be run via

$ ./bin/ratel-quasistatic -options_file examples/ymls/ex02-quasistatic-elasticity-mixed-linear-face-forces-pcjacobi.yml