Upper bounds for the homogenization problem in nonlinear elasticity: the incompressible case

Abstract We consider periodic homogenization of hyperelastic models incorporating incompressible behavior via the constraint $$\det (\nabla u)=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>det</mml:mo> <mml:mo>(</mml:mo> <mml:mi>∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We show that the ’usual’ homogenized integral functional $$\int W_{\textrm{hom}}(\nabla u)\,dx$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∫</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mtext>hom</mml:mtext> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> , where $$W_{\textrm{hom}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mtext>hom</mml:mtext> </mml:msub> </mml:math> is the standard multicell-formula of non-convex homogenization restricted to volume preserving deformations, yields an upper bound for the $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> -limit as the scale of periodicity tends to zero.