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Homogenization of two-dimensional elasticity problems with very stiff coefficients. (English) Zbl 1134.35010

The purpose of the paper is to describe the asymptotic behaviour of general 2D elastic problems written as \(\text{div}(A^{\varepsilon }e(u^{\varepsilon }))=f\), posed in a smooth domain \(\Omega \) and with homogeneous Dirichlet boundary conditions on \(\partial \Omega \). The fourth-order tensor \( A^{\varepsilon }\) satisfies the stiff conditions \(A^{\varepsilon }(x)\xi :\xi \geq \alpha \| \xi \| ^{2}\) ; \((A^{\varepsilon }(x))^{-1}\xi :\xi \geq (\beta _{\varepsilon }(x))^{-1}\| \xi \| ^{2}\), for every \(\xi \in \mathbb{R}_{S}^{2\times 2}\) and with \( \alpha >0\).
The authors distinguish between the non periodic and the periodic cases. In the non periodic case, the main convergence result asserts that if \((\beta _{\varepsilon })_{\varepsilon }\) weakly converges to some \(\beta \in L^{\infty }(\Omega )\) in the sense of measures, then \( (u_{\varepsilon })_{\varepsilon }\) converges to \(u\) in the weak sense of \(H_{0}^{1}(\Omega ; \mathbb{R}^{2})\) and \((A^{\varepsilon }e(u_{\varepsilon }))_{\varepsilon }\) converges to \( A^{\ast }e(u)\) in the weak sense of \(\mathcal{M}(\Omega ;\mathbb{R} _{S}^{2\times 2})\) for some fourth-order tensor which satisfies the above conditions with \(\beta \) instead of \(\beta _{\varepsilon }\).
In the periodic case, that is assuming that \(A^{\varepsilon }\) and \(\beta _{\varepsilon }\) are respectively associated to some tensor \(A_{\natural }^{\varepsilon }\) or \(\beta _{\varepsilon }^{\natural }\) defined in the unit cell \(Y\), the authors prove a similar result as above, now assuming that \( \lim_{\varepsilon \to 0}\varepsilon ^{2}\int_{Y}\beta _{\varepsilon }^{\natural }(y)\,dy=0\). For the proof of the convergence result in the non periodic case, the authors use some div-curl results presented in [M. Briane and J. Casado-Díaz, Commun. Partial Differ. Equations 32, No. 6, 935–969 (2007; Zbl 1127.35070)]. The convergence result in the periodic case is proved introducing the solution of some cell problem.
The paper ends with some counterexample, the authors proving some convergence result for a 2D composite material for which the \(\Gamma \)-limit of the energies involves some second-order partial derivative of the displacement field. This result is inspired from that of C. Pideri and P. Seppecher in the 3D case [Contin. Mech. Thermodyn. 9, No. 5, 241–257 (1997; Zbl 0893.73006)].

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
74Q05 Homogenization in equilibrium problems of solid mechanics
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