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<p>In this paper we consider the fourth-order elliptic equation describing the Kirchhoff-Love model for pure bending of a thin solid symmetric plate under a transverse load. We derive Hashin-Shtrikman bounds on the complementary energy, and calculate these bounds explicitly for mixtures of two isotropic materials in dimension <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>d</mi> <mo>=</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$d=2$</EquationSource> </InlineEquation>. Moreover, we give bounds on the bulk and shear moduli of an isotropic two-phase composite material, and show that these bounds are optimal, i.e.&#xa0;that they are attainable by finite-rank sequential laminates. These results pave the way for numerous applications of the homogenization theory in optimal design problems for stationary elastic plates, for example the development of an optimality criteria method for two-dimensional compliance minimization problems.</p>

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Explicit Bounds on the Energy and Moduli of a Composite Elastic Plate

  • 期刊论文

  • 发表日期：

    2026年6月2日

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  • Krešimir Burazin,
  • Jelena Jankov Pavlović

摘要

In this paper we consider the fourth-order elliptic equation describing the Kirchhoff-Love model for pure bending of a thin solid symmetric plate under a transverse load. We derive Hashin-Shtrikman bounds on the complementary energy, and calculate these bounds explicitly for mixtures of two isotropic materials in dimension d = 2 $d=2$ . Moreover, we give bounds on the bulk and shear moduli of an isotropic two-phase composite material, and show that these bounds are optimal, i.e. that they are attainable by finite-rank sequential laminates. These results pave the way for numerous applications of the homogenization theory in optimal design problems for stationary elastic plates, for example the development of an optimality criteria method for two-dimensional compliance minimization problems.