## 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

### Citations:

Zbl 1127.35070; Zbl 0893.73006
Full Text:

### References:

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