Measure-valued structured deformations (Q6616428)

The authors consider a bounded domain \(\Omega \subset \mathbb{R}^{N}\) and the energy \(E:SBV(\Omega ;\mathbb{R}^{d})\rightarrow \lbrack 0,+\infty )\) written as: \(E(u;\Omega )=\int_{\Omega }W(\nabla u)dx+\int_{\Omega \cap S_{u}}\psi ([u],\nu _{u})d\mathcal{H}^{N-1}(x)\), where \(u\) is the displacement field, \(S_{u}\) the (possibly empty) discontinuity set of \(u\), \( [u]\) the jump of \(u\) along the discontinuity, \(\nu _{u}\) the normal to the discontinuity, \(W:\mathbb{R}^{d\times N}\rightarrow \lbrack 0,+\infty )\) the bulk energy density and \(\psi :\mathbb{R}^{d}\times \mathbb{S} ^{N-1}\rightarrow \lbrack 0,+\infty )\) the interfacial energy density, both being continuous functions and satisfying suitable structural assumptions. The authors replace \(u\) and its gradient \(\nabla u\) by a triple \((\kappa ,g,G)\), where \(\kappa =S_{u}\), \(g:\Omega \setminus \kappa \rightarrow \mathbb{R}^{d}\) is the piecewise differentiable field that represents the macroscopic deformation and \(G:\Omega \setminus \kappa \rightarrow \mathbb{R} ^{d\times N}\) is the piecewise continuous matrix-valued field that captures the contribution at the macroscopic level of smooth submacroscopic changes. They define the space \(mSD(\Omega ;\mathbb{R}^{d}\times \mathbb{R}^{d\times N})=BV(\Omega ;\mathbb{R}^{d})\times \mathcal{M}(\Omega ;\mathbb{R}^{d\times N})\), and the energy \(I:mSD\rightarrow \lbrack 0,+\infty )\) through relaxation as: \(I(g,G;\Omega )=inf\{liminf_{n\rightarrow \infty }E(u_{n};\Omega ):\{u_{n}\}\in \mathcal{R}(g,G;\Omega )\}\), where, for every open set \(U\subset \Omega \), \(\mathcal{R}(g,G;U)=\{\{u_{n}\}\subset SBV(U; \mathbb{R}^{d}):u_{n}\overset{\ast }{\rightharpoonup }g\mid _{U}\) in \(BV(U; \mathbb{R}^{d})\) and \(\nabla u_{n}\overset{\ast }{\rightharpoonup }G\mid _{U} \) in \(\mathcal{M}(U;\mathbb{R}^{d\times N})\}\) is the set of admissible sequences. The first main result proves a representation theorem for this energy: for all \((g,G)\in mSD(\Omega ;\mathbb{R}^{d}\times \mathbb{R} ^{d\times N})\), \(I(g,G;\Omega )=J(g,G;\Omega )=\int_{\Omega }H(\nabla g,G^{a})dx+\int_{\Omega \cap S_{g}}h^{j}([g],\frac{dG_{g}^{j}}{d(\mathcal{H} ^{N-1}\llcorner S_{g})},\nu _{g})d\mathcal{H}^{N-1}(x)+\int_{\Omega }h^{c}( \frac{dD^{c}g}{d\left\vert D^{c}g\right\vert },\frac{dG_{g}^{c}}{d\left\vert D^{c}g\right\vert })d\left\vert D^{c}g\right\vert (x)+\int_{\Omega }h^{c}(0, \frac{dG_{g}^{s}}{d\left\vert G_{g}^{s}\right\vert })d\left\vert G_{g}^{s}\right\vert (x)\), where \(H:\mathbb{R}^{d\times N}\times \mathbb{R} ^{d\times N}\rightarrow \lbrack 0,+\infty )\), \(h^{j}:\mathbb{R}^{d}\times \mathbb{R}^{d\times N}\times \mathbb{S}^{N-1}\rightarrow \lbrack 0,+\infty )\) , and \(h^{c}:\mathbb{R}^{d\times N}\times \mathbb{R}^{d\times N}\rightarrow \lbrack 0,+\infty )\) are suitable bulk, surface, and Cantor-type relaxed energy densities, \(Dg=D^{a}g+D^{s}g=D^{a}g+D^{j}g+D^{c}g=\nabla g\mathcal{L} ^{N}+]g]\otimes \nu _{g}\mathcal{H}^{N-1}\llcorner S_{g}+D^{c}g\), and \( G=G^{a}+G^{s}=G^{a}+G_{g}^{j}+G_{g}^{c}+G_{g}^{s}\), where \(G^{a}\ll \mathcal{ L}^{N}\), \(dG_{g}^{j}=\frac{dG}{d\left\vert D^{j}g\right\vert }d\left\vert D^{j}g\right\vert \), \(dG_{g}^{c}=\frac{dG}{d\left\vert D^{c}g\right\vert } d\left\vert D^{c}g\right\vert \), \(G_{g}^{s}=G-G^{a}-G_{g}^{j}-G_{g}^{c}\). The authors give the expressions of the functions \(H\), \(h^{j}\), and \(h^{c}\) through minimization problems involving \(E\) or \(E^{\infty }\) defined as: \( E^{\infty }(u;\Omega )=\int_{\Omega }W^{\infty }(\nabla u)dx+\int_{\Omega \cap S_{u}}\psi ([u],\nu _{u})d\mathcal{H}^{N-1}(x)\), where \(W^{\infty }(A)=limsup_{t\rightarrow +\infty }W(tA)/t\), on special classes. For the proof, the authors first exhibit links between the functions \(H\), \(h^{j}\), and \(h^{c}\). They prove a sequential characterization of the relaxed energy densities \(H\), \(h^{j}\), and \(h^{c}\). They use and improve properties of the functional spaces they work with. The second main result proves a quite similar representation for the energy, but in the case of trace constraints of Dirichlet type on a piece of the boundary. The paper ends with some further discussions and examples.