Theory

Linear Elasticity

Description

The motion of a single continuous medium is governed by a system of partial differential equations that are mathematical expressions of the balance law for the mass, linear momentum, angular momentum, and total energy. These so-called governing equations can be written in the Eulerian form (i.e. in the current configuration of the continuous body), or in the Lagrangian form (i.e. in the reference configuration of the continuous body).

The governing equations written in the Lagrangian setting are especially convenient for the study of the deformation of elastic solids. These equations are generally nonlinear. The sources of the nonlinearity are:

  • transformation between the reference and current configuration,

  • response of the material of interest (specified by a constitutive equation).

The current configuration of the given continuous body is in many situations almost identical to the reference one (in a sense of “small” deformation). Therefore, in such situations, it makes perfect sense to work with an approximate linear version of the governing equations.

The basic stationary linear elasticity model reads:

\begin{alignat}{2} - \operatorname{div} \mathcal \tsigma(\vu) &= \vf &&\quad \textrm{in}\ \Omega, \\ \vu &= \vu_D &&\quad \textrm{on}\ \Gamma_D, \\ \vu \cdot \vn &= 0 &&\quad \textrm{on}\ \Gamma_S, \\ \tsigma(\vu) \cdot \vn &= \vg &&\quad \textrm{on}\ \Gamma_N, \\ \tsigma(\vu) \cdot \vn &= 0 &&\quad \textrm{on}\ \Gamma_0 = \partial \Omega \setminus (\Gamma_D \cup \Gamma_N \cup \Gamma_S), \end{alignat}

where

  • \(\vu\) is the displacement vector,

  • \(\tsigma\) is the Cauchy stress tensor for isotropic elastic solids given by the constitutive relation

    \begin{align} \tsigma(\vu) &= 2 \mu \teps(\vu) + \lambda \tr \left( \teps(\vu) \right) \tI, \end{align}

    also known as the Hooke law,

  • \(\lambda\) and \(\mu\) are referred to as Lamé parameters,

  • \(\teps(\vu)\) is the linearised strain, i.e. \(\teps(\vu) = \operatorname{sym}(\nabla \vu)\),

  • \(\tI\) is the identity tensor,

  • \(\Omega\) is the computational domain,

  • \(\vu_D\) is the Dirichlet boundary condition on \(\Gamma_D\) (boundary clamping),

  • \(\vg\) is the Neumann boundary condition on \(\Gamma_N\) (boundary forcing),

  • \(\vf\) is the term representing sources and sinks in \(\Omega\) (volume forcing).