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<p>We study stochastic homogenisation of free-discontinuity surface functionals defined on piecewise rigid functions which arise in the study of fracture in brittle materials. In particular, under standard assumptions on the density, we show that there exists a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2930_Article_IEq1.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 almost surely and that it can be represented by a surface integral. In addition, the effective density can be characterised via a suitable asymptotic cell formula and is deterministic under an ergodicity assumption. We also show via <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2930_Article_IEq2.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>-convergence that the homogenised functional defined on piecewise rigid functions can be recovered from a Griffith-type model by passing to the limit of vanishing elastic deformations.</p>

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Stochastic homogenisation for functionals defined on asymptotically piecewise rigid functions

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  • 发表日期：

    2025年1月27日

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  • Antonio Flavio Donnarumma,
  • Manuel Friedrich

摘要

We study stochastic homogenisation of free-discontinuity surface functionals defined on piecewise rigid functions which arise in the study of fracture in brittle materials. In particular, under standard assumptions on the density, we show that there exists a \(\Gamma \) Γ -limit almost surely and that it can be represented by a surface integral. In addition, the effective density can be characterised via a suitable asymptotic cell formula and is deterministic under an ergodicity assumption. We also show via \(\Gamma \) Γ -convergence that the homogenised functional defined on piecewise rigid functions can be recovered from a Griffith-type model by passing to the limit of vanishing elastic deformations.