4 Model Problems in Heat Transfer and Solid Mechanics
Heat conduction and solid mechanics are used throughout the course because they expose the same finite element machinery through different physics. Larson and Bengzon treat both within a common finite element framework, while Arnold derives the heat equation directly from balance and Fourier conduction (Larson and Bengzon 2010; Arnold 2011).
4.1 Heat conduction
For temperature \(T(\bx,t)\), conservation of energy and Fourier’s law lead to
\[ \rho c\,\dot T - \divg(k\grad T)=q. \]
The steady problem is obtained by setting \(\dot T=0\).
4.2 Linear elasticity
For displacement \(\bu\), the infinitesimal strain tensor is
\[ \beps(\bu) = \frac12\left(\grad\bu+(\grad\bu)^T\right). \]
The quasistatic balance equation is
\[ -\divg\bsig=\bfm, \]
with constitutive relation \(\bsig=\mathbb{C}:\beps(\bu)\) for linear elasticity.
4.3 Hyperelasticity
For finite deformation, the deformation gradient is
\[ \bF=\mathbf{I}+\Grad\bu, \]
and a hyperelastic material is described through a stored-energy density \(W(\bF)\). The nonlinear variational problem will be revisited in the nonlinear finite element module.
4.4 Common variational structure
Despite their different physical meanings, the problems above can be organized through a residual statement
\[ R(u;v)=0 \qquad\forall v\in V. \]
This common structure is the reason the same finite element software framework can treat heat transfer, linear elasticity, and nonlinear solid mechanics.