It has been known for some time that constitutive models for the large-deformation response of porous elastomers can develop ‘macroscopic’ instabilities as a consequence of loss of strong ellipticity. Indeed, constitutive models obtained by homogenization methods for porous elastomers with periodic and random microstructures can lose strong ellipticity under appropriate loading conditions. For periodic microstructures, it has been shown theoretically, and verified experimentally and numerically, that ‘microscopic’ instabilities consisting in solutions that are periodic on ensembles of unit cells tend to occur before the long-wavelength ‘macroscopic’ instabilities that are captured by the loss of strong ellipticity of the homogenized response. But what about the response of porous elastomers with random microstructures? In this case, microscopic instabilities can be excluded, and the question then arises as to what happens after the onset of a macroscopic instability. Building on earlier work for reinforced elastomers, it is shown here that porous elastomers can undergo twinning after the onset of a macroscopic instability. For this purpose, use is made of a generalized Maxwell-type construction arising from the theory of relaxation and applied to linear comparison variational homogenization estimates for a certain class of two-dimensional porous elastomers consisting of aligned cylindrical pores in a rubber matrix subjected to plane strain loadings. It is shown that such porous elastomers recover Legendre–Hadamard stability by twinning after the onset of a macroscopic instability. Interestingly, the porous elastomers behave like elastic fluids in the twinned region, losing their ability to support shear stresses.

中文翻译：

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多孔弹性体中的孪晶


一段时间以来，人们已经知道，由于失去强椭圆度，多孔弹性体大变形响应的本构模型可能会产生“宏观”不稳定性。事实上，通过均质化方法获得的具有周期性和随机微观结构的多孔弹性体的本构模型在适当的载荷条件下可能会失去强椭圆度。对于周期性微观结构，已经从理论上证明，并通过实验和数值验证，由晶胞系综上的周期性解组成的“微观”不稳定性往往发生在长波长“宏观”不稳定性之前，这些不稳定性是由同质化响应的强椭圆性损失所捕获的。但是具有随机微观结构的多孔弹性体的响应呢？在这种情况下，可以排除微观的不稳定性，那么问题就出现了，即宏观不稳定性开始后会发生什么。在增强弹性体的早期工作的基础上，本文表明，多孔弹性体在宏观不稳定开始后可以发生孪晶。为此，使用了源自松弛理论的广义麦克斯韦型结构，并应用于某一类二维多孔弹性体的线性比较变分均质估计，该弹性体由橡胶基体中对齐的圆柱形孔组成，承受平面应变载荷。结果表明，这种多孔弹性体在宏观不稳定性开始后通过孪生恢复勒让德-哈达玛稳定性。有趣的是，多孔弹性体在孪晶区域中的行为类似于弹性流体，失去了支撑剪切应力的能力。