二维非均质材料应力场的数值化计算方法

等效夹杂方法是求解含杂质材料弹性应力场的一种有效方法，但是其解析求解只适用于椭球/椭圆类杂质问题。本文提出一种基于等效夹杂方法的数值化计算方法，介绍了其基本理论，并引入共轭梯度法求解该方法的一致性条件线性方程组。该方法通过计算区域的数值离散，能够实现对二维任意形状杂质弹性场的求解。将该方法得到的结果与解析解进行比较，验证了该方法的有效性。讨论了数值化等效夹杂方法在效率以及收敛性上的表现。通过对比证明，利用共轭梯度法实现该方法，能在保持精度的同时，相较于高斯消元法具有较大的效率优势。最后通过半椭圆杂质和氧化锆/氧化铝共挤复合材料算例验证了该方法处理任意形状杂质的能力。

A numerical calculation method for stress field of 2D inhomogeneous materials

Equivalent inclusion method is a convenient tool for modeling the stress field of materials embedded with inhomogeneities. However, its analytical applications are limited to elliptical shaped inhomogeneity cases. In this work, a new powerful and versatile numerical extension to the original equivalent inclusion method, called numerical equivalent inclusion method, was proposed. The fundamental theory of the numerical equivalent inclusion method was introduced, and an implementation method, conjugate gradient method, was presented to solve the linear equation group of the equivalency condition of the new method. The method can be applied to 2D inhomogeneity problems with arbitrary shape through a handy numerical discretization. Benchmark comparisons with the analytical results for an elliptical inhomogeneity model illustrated the accuracy of the proposed solution method. The efficiency and convergence of the numerical equivalent inclusion method were also discussed in detail, the results show that the proposed method implemented by conjugate gradient method has significant advantages in efficiency compared with that by Gaussian elimination method, and can maintain the accuracy of the results as well. A half-elliptical inhomogeneity model and a kind of zirconia/alumina coextruded composites were utilized to demonstrate the capability of the new method on solving arbitrarily shaped inhomogeneity problems.