Mechanics of deformable solids

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• ConceptPolyconvexity

  A function of an $m \times n$ matrix is polyconvex if it is a convex function of the matrix together with all its minors. For $3 \times 3$ matrices the minors are the entries, the entries of the cofactor matrix and the determinant. Polyconvexity implies Morrey quasiconvexity, and it is much easier to check. It allows energies that blow up as the determinant tends to $0$, so it is the standard hypothesis on stored energies in nonlinear elasticity.

• TheoremBall's existence theorem in nonlinear elasticity

  Let $\Omega \subseteq \R^3$ be a bounded open connected set with Lipschitz boundary, and let $\Gamma_0 \subseteq \partial \Omega$ have positive surface measure. Let $\mathbb{M}^3_+$ be the set of $3 \times 3$ real matrices with positive determinant, and let $W \colon \Omega \times \mathbb{M}^3_+ \to \R$ be a stored energy function such that:

• ConceptRank-one convexity

  A function on $m \times n$ matrices is rank-one convex if it is convex along every line in a rank-one direction $a \otimes b$. It is the weakest of the convexity notions used in the vectorial calculus of variations. Polyconvexity implies quasiconvexity, which implies rank-one convexity, and each is implied by ordinary convexity. For $C^2$ integrands it is the Legendre–Hadamard condition at every matrix.

• ProblemSignorini problem

  Let $\Omega \subseteq \R^3$ be a bounded open set with Lipschitz boundary, the reference shape of a linearly elastic body. Split the boundary into disjoint parts $\Gamma_D$, $\Gamma_N$ and $\Gamma_C$, with $\Gamma_D$ of positive surface measure, and let $n$ be the outward unit normal. For a displacement $u \colon \Omega \to \R^3$ let $\varepsilon(u) = \tfrac12 (\nabla u + \nabla u^T)$ be the strain and $\sigma(u) = \mathcal{C} \varepsilon(u)$ the stress. Here $\mathcal{C}$ is a bounded elasticity tensor with the usual symmetries that is uniformly positive definite on symmetric matrices. Let $f$ be a body force and $g$ a surface force on $\Gamma_N$. Write $u_n = u \cdot n$, $\sigma_n = (\sigma(u) n) \cdot n$ and $\sigma_T = \sigma(u) n - \sigma_n n$. Find $u$ such that:

• ConceptHomogenization

  Homogenization is the study of partial differential equations whose coefficients oscillate on a small scale $\varepsilon$, in the limit $\varepsilon \to 0$. The solutions converge to the solution of a homogenized equation with effective, slowly varying or constant coefficients. For periodic media the effective coefficients are computed from a cell problem on one period. It explains, for example, the effective conductivity of a composite material made of a fine mixture of two materials.

• ConceptRayleigh–Ritz method

  The Rayleigh–Ritz method approximates the eigenvalues of a self-adjoint operator by restricting its Rayleigh quotient to a finite-dimensional space of trial vectors. This reduces the problem to a small matrix eigenvalue problem. By the min-max theorem, each approximate eigenvalue is an upper bound for the corresponding true one, and the bounds improve as the trial space grows. The method underlies the finite element computation of vibration modes and many iterative eigenvalue solvers.