填充液体夹杂物的弹性体的均匀化：小变形极限
Homogenization of elastomers filled with liquid inclusions: The small-deformation limit
摘要：
本文推导了描述含液体夹杂弹性体在准静态小变形下宏观力学响应的均匀化方程。推导进行了周期性微结构的材料通过两个尺度的渐近分析。重点是非耗散情况下，当弹性体是弹性固体，构成夹杂物的液体是弹性流体，分离固体弹性体与液体夹杂物的界面是具有初始表面张力的弹性界面，并且夹杂物最初是$n$ -球形（$n=2,3$）形状。值得注意的是，尽管由于界面处的初始表面张力而在夹杂物内存在局部残余应力，但这种填充的弹性体的宏观响应证明是不含残余应力的线性弹性固体的宏观响应，因此其简单地以有效弹性模量$\bar{\textbf{L}}$为特征。此外，尽管本体和界面中的局部弹性模量不具有较小的对称性（由于界面处存在残余应力和初始表面张力），但所得的有效弹性模量$\bar{\textbf{L}}$确实具有常规线弹性固体（即，$\bar{L}_{ijkl}=\bar{L}_{jikl}=\bar{L}_{ijlk}$）的标准较小对称性。作为第一个应用程序，数值计算结果和分析的各向同性悬浮液的不可压缩液体$2$ -球形夹杂物的单分散尺寸嵌入在各向同性不可压缩弹性体的有效弹性模量。
英文摘要：
This paper presents the derivation of the homogenized equations that describe the macroscopic mechanical response of elastomers filled with liquid inclusions in the setting of small quasistatic deformations. The derivation is carried out for materials with periodic microstructure by means of a two-scale asymptotic analysis. The focus is on the non-dissipative case when the elastomer is an elastic solid, the liquid making up the inclusions is an elastic fluid, the interfaces separating the solid elastomer from the liquid inclusions are elastic interfaces featuring an initial surface tension, and the inclusions are initially $n$-spherical ($n=2,3$) in shape. Remarkably, in spite of the presence of local residual stresses within the inclusions due to an initial surface tension at the interfaces, the macroscopic response of such filled elastomers turns out to be that of a linear elastic solid that is free of residual stresses and hence one that is simply characterized by an effective modulus of elasticity $\bar{\textbf{L}}$. What is more, in spite of the fact that the local moduli of elasticity in the bulk and the interfaces do not possess minor symmetries (due to the presence of residual stresses and the initial surface tension at the interfaces), the resulting effective modulus of elasticity $\bar{\textbf{L}}$ does possess the standard minor symmetries of a conventional linear elastic solid, that is, $\bar{L}_{ijkl}=\bar{L}_{jikl}=\bar{L}_{ijlk}$. As a first application, numerical results are worked out and analyzed for the effective modulus of elasticity of isotropic suspensions of incompressible liquid $2$-spherical inclusions of monodisperse size embedded in an isotropic incompressible elastomer.
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