任意形状热夹杂位移场的三角形单元离散算法

平面夹杂模型在纤维增强型复合材料中有广泛应用.复合材料内部通常含有不规则形状夹杂,而夹杂物的存在能严重影响材料的机械力学性能,往往导致应力集中及裂纹萌生等失效先兆.先前关于多边形夹杂的研究大多数关注受均匀本征应变下的应力/应变解,而对位移的分析较少. 基于格林函数方法和围道积分,本文给出了平面热夹杂边界线单元的封闭解析解,可方便应用于受任意分布本征应变的任意形状平面热夹杂位移场的数值计算.当夹杂受均匀本征应变时, 只需将该夹杂边界进行一维离散,因而本文方法可直接得出受均匀分布热本征应变的任意多边形夹杂位移场的封闭解析解.当夹杂区域存在非均匀分布本征应变时,可将该区域划分为足够小的三角形单元进行数值计算. 众所周知,应力应变场在多边形夹杂顶点处具有奇异性,容易导致数值计算上的处理困难及相应的数值稳定性问题; 然而本文工作表明,在多边形顶点处位移场是连续有界的, 因而数值稳定性较好.本文算法可以便捷高效地通过计算机编程实现. 文中给出的验证算例,均体现了本文离散方法的高精度、以及计算编程的鲁棒性. 

A TRIANGULAR ELEMENT DISCRETIZATION FOR COMPUTING DISPALCEMENT OF AN ARBITRARILY SHAPED THERMAL INCLUSION1)

Inclusion models have been widely used to explore the micromechanical properties of fiber-reinforced composite materials. The composite materials usually contain irregularly shaped inclusions, which can severely affect the mechanical properties of the materials. Abundant research has demonstrated that the stress localizations as well as the sites of crack initiation are predominantly detected in the neighborhood of nonmetallic inclusions. Previous studies on polygonal inclusion mainly focused on the stress/strain solutions under uniform eigenstrains, while the analyses on displacement are limited. Based on the method of Green's function and contour integral, this work presents a closed-form solution for a line element along the boundary of a two-dimensional thermal inclusion. The proposed method of solution is effective for determining the displacement of an arbitrarily shaped inclusion subjected to any distributed dilatational eigenstrain. In the case of uniform eigenstrain, only the boundary of the inclusion needs to be discretized into line elements; therefore, the proposed method analytically yields the closed-form solution for the displacement of an arbitrary polygonal inclusion subjected to uniform thermal eigenstrain. When the eigenstrain is non-uniformly distributed in the inclusion, the resulting displacements may be evaluated by discretizing the thermal inclusion into a system of triangular elements. It is known that the stress and strain fields exhibit singularities at the vertices of a polygonal inclusion. Such singularity issue can be intractable in numerical evaluations of the stresses/strains in the vicinity of the vertices, leading to a commonly seen yet tricky phenomenon of numerical instability. In contrast, the present work shows that the displacement is continuous and bounded at the corners of the polygon. Other than the merit of numerical discretization, the derived closed-form solutions may be conveniently programmed on a personal computer, while the corresponding algorithm seems to be straightforward, facilitating a high accurate and expeditious evaluation of the displacements. Benchmark examples demonstrate the computational efficiency and numerical robustness of the proposed method.