<p>We show how to explicitly compute the homogenised curvature energy appearing in the isotropic <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2024_2415_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-limit for flat and for curved initial configuration Cosserat shell models, when a parental three-dimensional minimisation problem on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2024_2415_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> for a Cosserat energy based on the second-order dislocation density tensor <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2024_2415_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="158" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha :=\overline{R} ^T \textrm{Curl}\overline{R} \in \mathbb {R}^{3\times 3}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>:</mo> <mo>=</mo> <msup> <mover> <mi>R</mi> <mo>¯</mo> </mover> <mi>T</mi> </msup> <mtext>Curl</mtext> <mover> <mi>R</mi> <mo>¯</mo> </mover> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2024_2415_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$\overline{R}\in \textrm{SO}(3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>R</mi> <mo>¯</mo> </mover> <mo>∈</mo> <mtext>SO</mtext> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is used.</p>

Explicit formula for the $\Gamma $-convergence homogenised quadratic curvature energy in isotropic Cosserat shell models

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2025年3月13日

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Maryam Mohammadi Saem,
Emilian Bulgariu,
Ionel-Dumitrel Ghiba,
Patrizio Neff

摘要

We show how to explicitly compute the homogenised curvature energy appearing in the isotropic $\Gamma $ $Γ$ -limit for flat and for curved initial configuration Cosserat shell models, when a parental three-dimensional minimisation problem on $\Omega \subset \mathbb {R}^3$ $Ω \subset R^{3}$ for a Cosserat energy based on the second-order dislocation density tensor $\alpha :=\overline{R} ^T \textrm{Curl}\overline{R} \in \mathbb {R}^{3\times 3}$ $α : = {\bar{R}}^{T} Curl \bar{R} \in R^{3 \times 3}$ , $\overline{R}\in \textrm{SO}(3)$ $\bar{R} \in SO (3)$ is used.

Explicit formula for the \(\Gamma \)-convergence homogenised quadratic curvature energy in isotropic Cosserat shell models

摘要