Since the pioneering work of Eshelby on a single ellipsoidal inclusion embedded in an infinite space, much attention has been devoted in the literature to the question of the uniformity of the stress field inside inclusions surrounded by an elastic matrix. Over the last decade or so, researchers have established the existence of multiple (interacting) inclusions enclosing uniform internal stress distributions when embedded in an infinite elastic matrix subjected to a uniform far-field loading and identified a variety of shapes of such inclusions. In the design of multiple inclusions with uniform internal stresses, it is customary to assume that the uniform stress field inside each inclusion (each with different shear modulus distinct from that of the matrix) is hydrostatic. In this paper, we examine whether this assumption is actually necessary to ensure the required existence of multiple inclusions enclosing uniform stresses. By establishing several theorems in the theory of functions of a complex variable, we prove rigorously that for any collection of multiple inclusions enclosing uniform stresses in an infinite isotropic plane subjected to uniform remote in-plane loading, the internal uniform stress field must indeed be hydrostatic if the corresponding inclusion’s shear modulus is distinct from that of the matrix.

自从埃舍尔比（Eshelby）对嵌入无限空间中的单个椭圆形夹杂物的开创性工作以来，文献已将大量注意力集中在由弹性矩阵包围的夹杂物内部应力场的均匀性问题上。在过去的十年左右的时间里，研究人员已经建立了多个（相互作用的）夹杂物，它们包裹在均匀的远场载荷作用下的无限弹性基体中时，它们具有均匀的内部应力分布，并确定了各种形状的此类夹杂物。在设计具有相同内部应力的多个夹杂物时，通常假设每个夹杂物内部的均匀应力场（每个具有不同于基体的剪切模量）是流体静力学的。在本文中，我们检查了此假设是否确实必要，以确保包含均匀应力的多个夹杂物的存在。通过在复变函数的理论中建立几个定理，我们可以严格证明，对于在无限均匀各向同性平面内承受均一远程面内载荷的，包含均一应力的多个包含物的任何集合，内部均一应力场的确必须是静液压的如果相应的夹杂物的剪切模量与基质的剪切模量不同。