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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-limits as the thickness <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(h\ll 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>≪</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and the size of the material microstructure <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\epsilon \ll 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>≪</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> vanish. In the regime <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\epsilon \ll h\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>≪</mo> <mi>h</mi> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(h\ll \epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>≪</mo> <mi>ϵ</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月2日

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  • Klaus Böhnlein,
  • Lucas Bouck,
  • Stefan Neukamm,
  • David Padilla-Garza,
  • Kai Richter

摘要

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 \(\Gamma \) Γ -limits as the thickness \(h\ll 1\) h ≪ 1 and the size of the material microstructure \(\epsilon \ll 1\) ϵ ≪ 1 vanish. In the regime \(\epsilon \ll h\) ϵ ≪ h 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 \(h\ll \epsilon \) h ≪ ϵ .