弹性损伤理论的几何-拓扑描述

GEOMETRICAL TOPOLOGY OF ELASTIC DAMAGE THEORY

材料内部微观几何缺陷通常是作为物理非线性问题在本构方程中考虑。针对连续介质弹性损伤理论作几何拓扑,采用非完整标架方法把材料内部微观几何缺陷转化为材料空间的弯曲,并体现在基本几何法则中。首先由连续损伤变量定义拟塑性张量,给出这些基本张量所满足的连续性方程和基本几何法则。由此建立了弹性损伤缺陷与Riemann流形的对应关系,将物理非线性问题转化为物理线性和材料所在空间的弯曲之和。最后讨论了二维情况下,各向同性晶格材料受各向异性损伤的算例。

The microscopic geometrical defects of materials are usually taken into account in the constitutive equation as a physical nonlinear problem. In this paper, the geometrical topology of elastic damage theory is given and the microscopic geometrical defects of materials are translated into the bending of the space, which is reflected in the geometrical equations. At first, this paper defines some quasi-plastic tensors with continuous damage tensor, which satisfy the continuity equations and the geometric laws. As a result, the corresponding relation between elastic damage defects and Riemann Space is established, and the physical nonlinear problem is converted to a physical linear problem together with a bending of space. Finally, an example of anisotropic damage of isotropic materials in two-dimensions is discussed.