A statistically homogeneous random bar with the bond-based peridynamic properties of constituents is considered for a static case. For both statistically homogeneous composites and homogeneous remote loading, the effective properties of both the peridynamic composites and locally elastic ones are described by constant tensor of the local effective moduli. However, even for locally elastic composites subjected to inhomogeneous loading, the effective deformations are described by a nonlocal (either the differential or integral) operator. The current paper is dedicated to the estimation of effective deformations of1D statistically homogeneous peridynamic composite bar for the prescribed self equilibrated body-forces. A subsequent simplification is described by the dilute approximation method assuming coincidence of the effective field with the applied field when interaction of inclusions is negligible. The method is based on estimation of a perturbator introduced by one inclusion that is, in fact, the solutions of the basic problem for one inclusion inside the infinite peristatic matrix subjected to the body-force. The statistical averages of both the displacements and stresses are estimated by summation of these perturbators for all possible locations of inclusions.

中文翻译：

---

受非均质力作用的周缘随机结构杆的有效变形

对于静态情况，考虑具有成分的基于键的蠕动特性的统计均质随机条。对于统计上均质的复合材料和均质的远程荷载，围动态复合材料和局部弹性复合材料的有效特性都由局部有效模量的恒定张量来描述。但是，即使对于承受不均匀载荷的局部弹性复合材料，有效变形也由非局部（微分或积分）算子描述。当前论文致力于针对规定的自平衡体力估算一维统计上均匀的绕动复合杆的有效变形。当夹杂物的相互作用可忽略不计时，假定有效场与外加场重合，则用稀疏近似法描述随后的简化。该方法是基于对一个包含物引入的扰动的估计，实际上是对无限大的静力学矩阵中一个包含物的基本问题的解决方案，该基本存在的问题受到了身体的作用力。通过对包含物所有可能位置的这些扰动器求和，可以估算出位移和应力的统计平均值。