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On the Commutability of Homogenization and Linearization in Finite Elasticity

# On the Commutability of Homogenization and Linearization in Finite Elasticity,10.1007/s00205-011-0438-7,Archive for Rational Mechanics and Analysis,St

On the Commutability of Homogenization and Linearization in Finite Elasticity
We consider a family of non-convex integral functionals $$\frac{1}{h^2}\int_\Omega W(x/\varepsilon,{\rm Id}+h\nabla g(x))\,\,{\rm d}x,\quad g\in W^{1,p}({\Omega};{\mathbb R}^n)$$ where W is a Carathéodory function periodic in its first variable, and non-degenerate in its second. We prove under suitable conditions that the Γ-limits corresponding to linearization (h → 0) and homogenization ($${\varepsilon\rightarrow 0}$$) commute, provided W is minimal at the identity and admits a quadratic Taylor expansion at the identity. Moreover, we show that the homogenized integrand, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the second variation of W.
Journal: Archive for Rational Mechanics and Analysis - ARCH RATION MECH ANAL , vol. 201, no. 2, pp. 465-500, 2011
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Published in 2009.