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Reconstruction of a fully anisotropic elasticity tensor from knowledge of displacement fields. (English) Zbl 1329.35344

The paper studies the reconstruction of the fully anisotropic elasticity tensor \(C\) when finitely-many displacement field solutions are known for the linear elasticity problem \(-\nabla \cdot (C:(\nabla u+(\nabla u)^{T}))=0\) in a bounded and smooth domain \(X\subset \mathbb{R}^{3}\) with a prescribed boundary condition \(u=g\) on \(\partial X\). The authors assume that there exist six solutions \(u^{1},\dots ,u^{6}\) to the above elasticity problem which form a basis of \(S_{3}(\mathbb{R})\) at every \(x\in \Omega \subset X\) and there exist \(N\) additional solutions \(u^{6+1},\dots ,u^{6+N}\) which build a family of \(3N\) matrices whose expressions are explicit in terms of \( \{\epsilon ^{(j)}\), \(\partial _{\alpha }\epsilon ^{(j)}\), \(1\leq \alpha \leq 3\), \(1\leq j\leq 6+N\}\) and which span a hyperplane of \(S_{6}(\mathbb{R})\) at every \(x\in \Omega \). This leads to a reconstruction procedure through the decomposition of every \(S_{6}(\mathbb{R})\)-valued function \(C\) as the product of a scalar function \(\tau \) times a normalized anisotropic structure \(\widetilde{C}\) such that the determinant of the Voigt counterpart of \(\widetilde{C}\) has determinant 1. Once \(\widetilde{C}\) is reconstructed, \(\tau \) can be recovered from an equation satisfied by \(\nabla \log \tau \).
The authors first prove that if these two hypotheses hold for two families of displacement fields \(\left\{ u^{(j)}\right\} _{j=1}^{6+N}\) and \(\left\{ u^{\prime (j)}\right\} _{j=1}^{6+N}\) corresponding to the elasticity tensors \(C\) and \(C^{\prime }\), then \(C\) and \(C^{\prime }\) each can be uniquely reconstructed over \(\Omega \) from the knowledge of their corresponding solutions, with the following stability estimate for every integer \(p\geq 0\) : \[ \left\| C-C^{\prime }\right\| _{W^{p,\infty }(\Omega )}+\left\| \mathrm{div}C-\mathrm{div}C^{\prime }\right\| _{W^{p,\infty }(\Omega )}\leq K\sum_{j=1}^{N+6}\left\| \epsilon ^{(j)}-\epsilon ^{\prime (j)}\right\| _{W^{p,\infty }(\Omega )}. \] The main result states that if \(C \) is \(C^{3}\)-close to constant or if \(C\) is smooth (at least of class \(C^{3}\)) and satisfies the Runge approximation property, there exists a nonempty open set of smooth enough boundary conditions generating displacement fields characterizing \(C\) uniquely and stably in the sense of the above result. For the proof, the authors justify the properties of the reconstruction algorithm for which they build explicit polynomial expressions.

MSC:

35R30 Inverse problems for PDEs
35J47 Second-order elliptic systems
35Q74 PDEs in connection with mechanics of deformable solids
35C11 Polynomial solutions to PDEs
74G75 Inverse problems in equilibrium solid mechanics
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