The repulsion property in nonlinear elasticity and a numerical scheme to circumvent it (Q6611843)

Let \(\Omega \subset \mathbb{R}^{m}\), \(m=2\) or \(m=3\), be a bounded domain occupied by a nonlinearly elastic body in its reference configuration. A deformation of the body is a map \(u:\Omega \rightarrow \mathbb{R}^{m}\), \(u\in W^{1,1}(\Omega)\), that is one-to-one almost everywhere and satisfies the condition \(det\nabla u(x)>0\) for a.e. \(x\in \Omega\). The energy stored under such a deformation is given by \(E(u)=\int_{\Omega}W(\nabla u(x))dx\), where \(W:M_{+}^{m\times m}\rightarrow \lbrack 0,\infty)\) is the stored energy density of the material and \(M_{+}^{m\times m}\) denotes the set of real \(m\times m\) matrices with positive determinant. \N\NThe authors consider stored energy functions of the form \(W(F)=\widetilde{W}(F)+h(detF)\) for \(F\in M_{+}^{m\times m}\), where \(\widetilde{W}\geq 0\) is \(W^{1,p}\) -quasiconvex and satisfies \(k_{1}\left\Vert F\right\Vert ^{p}\leq \widetilde{ W}(F)\leq k_{2}[\left\Vert F\right\Vert ^{p}+1]\) for \(F\in M_{+}^{m\times m}\), \(p\in (m-1,m)\), for some positive constants \(k_{1}\), \(k_{2}\), and \(h(\cdot)\) being a \(C^{2}(0,\infty)\) convex function such that \(h(\delta)\rightarrow \infty\) as \(\delta \rightarrow 0^{+}\), and \(\frac{h(\delta)}{\delta} \rightarrow \infty\) as \(\delta \rightarrow \infty\). The displacement is prescribed on the boundary \(u(x)=u^{h}(x)=Ax\) on \(\partial \Omega\), where \(A\in M_{+}^{m\times m}\) is fixed. Assuming that \(\Omega \subset \subset \Omega ^{e}\), where \(\Omega ^{e}\) is a bounded, open, and connected set with smooth boundary, the authors define the extension \(u_{e}\) of \(u\) by \(Ax\) on \(\Omega ^{e}\setminus \Omega\), for \(p>m^{2}/(m+1)\) the well-defined distribution \(Det\nabla u(\phi)=-\int_{\Omega}\frac{1}{m}([adj\nabla u]u)\cdot \nabla \phi dx\), \(\forall \phi \in C_{0}^{\infty}(\Omega)\), and the set of admissible deformations as: \(\mathcal{A}_{x_{0}}=\{u\in W^{1,p}(\Omega):u\mid_{\partial \Omega}=u^{h}\), \(u_{e}\) satisfies (INV) on \(\Omega\), \(det\nabla u>0\) a.e., \(Det\nabla u=(det\backslash \nabla u) \mathcal{L}^{m}+\alpha_{u}\delta_{x_{0}}\}\), where \(\alpha_{u}\geq 0\) is a scalar depending on the map \(u\) that represents the volume of the cavity, \(\delta_{x_{0}}\) denotes the Dirac measure with support at \(x_{0}\) that represents the position of the cavity, and (INV) is a condition proposed by \textit{S. Müller} and \textit{S. J. Spector} in [Arch. Ration. Mech. Anal. 131, No. 1, 1--66 (1995; Zbl 0836.73025)] in \(\Omega ^{e}\). \N\NThe authors start proving that taking \(p\in (m-1,m)\), a symmetric matrix \(A\in M_{+}^{m\times m}\), a deformation \(u_{0}\in \mathcal{A}_{x_{0}}\) with finite energy and \(\alpha_{u_{0}}>0\), and a sequence \((u_{n})\subset \mathcal{A}_{x_{0}}\) satisfying \(\alpha_{u_{n}}=0\) for all \(n\) and \(u_{n}\rightharpoonup_{n\rightarrow \infty}u_{0}\) in \(W^{1,p}(\Omega)\), then \(E(u_{n})\rightarrow_{n\rightarrow \infty}\infty\). To circumvent this repulsion property, they introduce a penalized energy functional: \(I_{\varepsilon}^{\tau}(u,v)=\int_{\Omega}[\widetilde{W}(\nabla u(x))+h(det\nabla u(x)-v(x))]dx+\int_{\Omega}[\frac{\varepsilon ^{\alpha}}{ \alpha}\left\Vert \nabla v(x)\right\Vert ^{\alpha}+\frac{1}{q\varepsilon ^{q}}\phi_{\tau}(v(x))]dx\), where \(\tau >0\), \(\alpha >1\), \(\frac{1}{\alpha}+\frac{1}{q}=1\), \((u,v)\in \mathcal{U}=\{(u,v)\in W^{1,p}(\Omega)\times W^{1,\alpha}(\Omega):u\mid_{\partial \Omega}=u^{h}\), \(u_{e}\) satisfies (INV) in \(\Omega\), \(det\nabla u>v\geq 0\) a.e., \(Det\nabla u=(det\nabla u) \mathcal{L}^{m}\), \(v\mid_{\partial \Omega}=0\}\), and \(\phi_{\tau}: \mathbb{R}\rightarrow \lbrack 0,\infty)\) is a continuous function, positive in \((0,\tau)\), and vanishing in \(\mathbb{R}\setminus (0,\tau)\). They prove that this penalized energy functional has always a minimizer \((u_{\varepsilon}^{\tau},v_{\varepsilon}^{\tau})\in \mathcal{U}\). If \(A\) is such that \(h^{\prime}(detA)\leq 0\), this global minimizer is \((u^{h},0)\). For any sequence \(\varepsilon_{j}\rightarrow 0\), the sequences \(\{u_{\varepsilon_{j}}^{\tau}\}\) and \(\{v_{\varepsilon_{j}}^{\tau}\}\) have subsequences which converge weakly to \(u^{\tau}\) in \(W^{1,p}(\Omega)\) and weakly\(^{\ast}\) in \(\mathcal{M}(\Omega)\) to a nonnegative Radon measure \(\nu ^{\tau}\). Moreover, \(u^{\tau}\mid_{\partial \Omega}=u^{h}\), \(u_{e}^{\tau}\) satisfies (INV) in \(\Omega\), and \(Det\nabla u^{\tau}=(det\nabla u^{\tau})\mathcal{L}^{m}+\nu_{s}^{\tau}\), where \(det\nabla u^{\tau}\in L^{1}(\Omega)\) with \(det\nabla u^{\tau}>0\) a.e. in \(\Omega\) and \(\nu_{s}^{\tau}\) is the singular part of \(\nu ^{\tau}\) with respect to Lebesgue measure. \N\NThe main result proves that if \(\{\tau_{k}\}\) and \(\{\varepsilon_{r}\}\) are sequences such that \(\tau_{k}\rightarrow \infty\) and \(\varepsilon_{r}\rightarrow 0^{+}\), and \((u_{k,r},v_{k,r})\) is a minimizer of \(I_{\varepsilon_{r}}^{\tau_{k}}\) over \(\mathcal{U}\), there exist subsequences, such that \(u_{k}\rightharpoonup_{k\rightarrow \infty}u^{\ast}\) in \(W^{1,p}(\Omega)\) and \(v_{k}\rightharpoonup_{k\rightarrow \infty}\nu ^{\ast}\) in \(\mathcal{M}(\Omega)\). Moreover, \(\underline{lim}_{k\rightarrow \infty}I_{\varepsilon_{k}}^{\tau_{k}}(u_{k},v_{k})\geq \int_{\Omega}W(\nabla u^{\ast}(x))dx+c\), for some positive constant \(c\). In the final sections of their paper, the authors specialize their results in the radial case and \(m=3\): they prove that the minimizers of the penalized functional satisfy Euler-Lagrange equations and they present results of numerical simulations obtained using MATLAB.