Linear elastic

Constitutive theory

For the linear elastic model, the strain energy density is given by

\[ \psi = \frac{\firstlame}{2} (\trace \bm{\varepsilon})^2 + \secondlame \bm{\varepsilon} \tcolon \bm{\varepsilon} . \]

The constitutive law (stress-strain relationship) is therefore given by its gradient,

(77)\[ \bm\sigma(\bm{u}) = \frac{\partial \psi}{\partial \bm{\varepsilon}} = \firstlame (\trace \bm\varepsilon) \bm{I} + 2 \secondlame \bm\varepsilon, \]

where the colon represents a double contraction (over both indices of \(\bm{\varepsilon}\)), \(\bm{\varepsilon}\) is (small/infintesimal) strain tensor defined by

(78)\[ \bm{\varepsilon} = \dfrac{1}{2}\left(\nabla \bm{u} + \nabla \bm{u}^T \right), \]

and the Lamé parameters are given in terms of Young’s modulus \(E\), and Poisson’s ratio \(\nu\) by

\[ \begin{aligned} \firstlame &= \frac{E \nu}{(1 + \nu)(1 - 2 \nu)}, \\ \secondlame &= \frac{E}{2(1 + \nu)}. \end{aligned} \]

The constitutive law (stress-strain relationship) can also be written as

(79)\[ \bm{\sigma} = \mathsf{C} \tcolon \bm{\varepsilon}. \]

For notational convenience, we express the symmetric second order tensors \(\bm \sigma\) and \(\bm \varepsilon\) as vectors of length 6 using the Voigt notation. Hence, the fourth order elasticity tensor \(\mathsf C\) (also known as elastic moduli tensor or material stiffness tensor) can be represented as

(80)\[ \mathsf C = \begin{pmatrix} \firstlame + 2\secondlame & \firstlame & \firstlame & & & \\ \firstlame & \firstlame + 2\secondlame & \firstlame & & & \\ \firstlame & \firstlame & \firstlame + 2\secondlame & & & \\ & & & \secondlame & & \\ & & & & \secondlame & \\ & & & & & \secondlame \end{pmatrix}. \]

Note that the incompressible limit \(\nu \to \frac 1 2\) causes \(\firstlame \to \infty\), and thus \(\mathsf C\) becomes singular.

Strong and weak formulations

In this section we present the governing equations of elasticity in small deformation. In small strain (linear elasticity), the boundary-value problem (Strong form) for constitutive equation (77) 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 variable \(\bm u \in \mathcal{V}\) (here \(\mathcal{V} = H^1(\Omega)\) ), such that:

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

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

(82)\[ \int_{\Omega}{ \nabla \bm{v} \tcolon \bm{\sigma}} \, dv - \int_{\partial \Omega}{\bm{v} \cdot \bar{\bm t}} \, da - \int_{\Omega}{\bm{v} \cdot \rho \bm{g}} \, dv = 0, \quad \forall \bm v \in \mathcal V. \]

Command-line interface

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

Table 17 Linear elastic model options

Option	Description	Default Value
-model elasticity-linear
-E [real]	Young’s modulus, \(E > 0\)
-nu [real]	Poisson’s ratio, \(\nu \leq 0.5\).

An example using the linear elastic model can be run via

$ ./bin/ratel-quasistatic -options_file examples/ymls/ex02-quasistatic-elasticity-linear-mms.yml