Homogenisation in finite elasticity for composites with a high contrast in the vicinity of rigid-body motion
Homogenisation in finite elasticity for composites with a high contrast in the vicinity of rigid-body motion
复制标题
刚体运动附近具有高对比度的复合材料的有限弹性均匀化
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
S. Neukamm
中科院分区：
文献类型：
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作者：
M. Cherdantsev;K. Cherednichenko;S. Neukamm
我们提出了一个多尺度渐近框架，用于分析有限应变设置中具有高对比度的周期性两种材料复合材料的宏观行为。我们的推导始于复合材料的几何非线性描述，该复合材料由刚性材料基体和柔软的周期性分布的夹杂物组成，其中组件之间的对比度与周期 $varepsilon 耦合。我们假设变形场是复合材料边界某些部分的刚体运动，并考虑各种加载方案，这些加载方案由附加参数 $lambda_varepsilon$ 控制。我们表明，对于特定方案，整个材料的弹性位移很小，为 $varepsilon o0$并且刚性材料中的变形梯度位于边界条件指定的刚体运动附近。这允许在均质化极限下用刚性组件的线性化版本替换非线性材料定律。软组件上变形梯度的行为取决于 $lambda_varepsilon.$ 的渐进性。特别是，在 $lambda_varepsilonsim1$ 的情况下，它们显示为一阶，并且对均质能量的相关贡献通过在适当重新缩放的梯度场上定义的原始存储能量密度的拟凸包络来表示。作为主要结果，我们本着 $Gamma$ 收敛的精神推导了一个极限能量泛函，并建立了一个精确的两尺度展开来最小化序列。
We present a multiscale asymptotic framework for the analysis of the macroscopic behaviour of periodic two-material composites with high contrast in a finite-strain setting. Our derivation starts with the geometrically nonlinear description of a composite consisting of a stiff material matrix and soft, periodically distributed inclusions, where the contrast between the components is coupled with the period $varepsilon.$ We assume that the deformation field is a rigid-body motion on some part of the boundary of the composite, and consider various loading regimes, which are controlled by an additional parameter $lambda_varepsilon.$ We show that for particular regimes the elastic displacements are small throughout the material as $varepsilon o0$ and that the deformation gradients in the stiff material are situated in the vicinity of the rigid-body motion specified by the boundary condition. This allows to replace, in the homogenisation limit, the nonlinear material law of the stiff component by its linearised version. The behaviour of the deformation gradients on the soft component depends on the asymptotics of $lambda_varepsilon.$ In particular, in the case $lambda_varepsilonsim1$ they are shown to be of order one, and the related contribution to the homogenised energy is expressed via the quasiconvex envelope of the original stored-energy density defined on suitably rescaled gradient fields. As a main result we derive, in the spirit of $Gamma$-convergence, a limit energy functional and establish a precise two-scale expansion for minimising sequences.