Commutativity and Non-commutativity of Limits in the Nonlinear Bending Theory for Prestrained Microheterogeneous Plates

Abstract In this paper we study the derivation of nonlinear bending models for prestrained elastic plates from three-dimensional nonlinear elasticity via homogenization and dimension reduction. We compare effective models obtained by either simultaneously or consecutively passing to the $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> -limits as the thickness $$h\ll 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and the size of the material microstructure $$\epsilon \ll 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> vanish. In the regime $$\epsilon \ll h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>≪</mml:mo> <mml:mi>h</mml:mi> </mml:mrow> </mml:math> we show that the consecutive and simultaneous limit are equivalent, and also analyze the rate of convergence. In contrast, we observe that there are several different limit models in the case $$h\ll \epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>≪</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:math> .