含均匀应力夹杂的平面问题研究

The strength of composite materials with elastic inhomogeneities is closely related to the uniformity of the internal stresses inside the inhomogeneities. For example, when an inhomogeneity achieves a uniform hydrostatic stress field, the interfacial stress on the entire interface becomes a constant eliminating the possibility of local cracking/failure and therefore enhancing the overall strength of the composite material. This project aims to study the plane deformation of elastic inhomogeneities enclosing uniform stresses. For the macroscale, we construct the special shapes of inhomogeneities with uniform stresses in half-plane, wedge-shaped and circular matrices, respectively. For the microscale, we examine the existence of non-circular inhomogeneities with uniform stresses by considering the effects of interface tension and interface elasticity. The main innovations include that we propose for the first time the methodology for designing inhomogeneities with uniform stresses when the matrix is finite or the interface is imperfect, and derive for the first time the analytic solutions for the shape of a single inhomogeneity with uniform stress in a half-plane or wedge-shaped matrix. In summary, the objective of this project is to demonstrate the evolution of the shapes of inhomogeneities ensuring uniform internal stresses in the vicinity of the edge of the matrix or at small length scales, and to promote the theory of elastic inhomogeneities. The results obtained in this project may form the basis for the shape design of particles and fibers in composite materials at both macro- and micro-scales.

夹杂内部应力的均匀性与复合材料的强度密切相关，比如当夹杂具有均匀静水应力时，整个界面上的应力将变为常数，从而避免了界面局部失效的可能性，提高了复合材料的整体强度。本项目通过理论分析、数值模拟和实验验证的手段，研究含均匀应力夹杂的平面问题。对于宏观尺度，分别在半平面基体、楔形基体和圆形基体内，建立具有均匀应力夹杂的形状；对于微观尺度，考虑夹杂与基体之间的界面张力和界面弹性，研究具有均匀应力的非圆形夹杂的存在性。创新点包括：发展针对有限大基体和非理想界面的含均匀应力夹杂的分析方法，推导半平面、楔形域内单个含均匀应力夹杂形状的解析解等。通过本项目的研究，揭示含均匀应力夹杂在基体外边界附近及微纳米尺度下的形状演化特性，促进弹性夹杂理论的发展，为高强度复合材料中颗粒/纤维的形状设计提供理论依据。

夹杂内部应力的均匀性与复合材料的强度密切相关，比如当夹杂具有均匀静水应力时，整个界面上的应力将变为常数，从而避免了界面局部失效的可能性，提高了复合材料的整体强度。本项目对含均匀应力夹杂的几个二维问题进行了系统研究。对于理想界面，研究了双材料结构(包括退化为半平面基体的情形)、楔形基体和圆形基体内具有均匀应力夹杂的形状构建，同时研究了多个具有均匀应力夹杂问题中的静水应力假设；对于非理想界面，研究单个具有均匀应力的非圆形夹杂的存在性。创新点包括：发展了针对有限大基体和非理想界面的含均匀应力夹杂的分析方法，证明了夹杂内部应力的静水特性是建立多个均匀应力夹杂形状的必要条件，探讨了双材料界面、材料自由表面对具有均匀应力夹杂形状的影响。通过本项目的研究，揭示含均匀应力夹杂在材料边界附近及微纳米尺度下的形状演化特性，促进弹性夹杂理论的发展，为先进颗粒/纤维复合材料中的高强度设计提供理论支撑。