Elastic Damage Model Obtained By Topological Method

There are many models for various materials or the different damaging process in the damage mechanics, but the physical defect is all considered in the physical equation in the Euclidean space. The basic idea is to select a damage variable, and then to establish the constitutive equations containing the damage variable and the evolution equations. In solving the constitutive equations containing the damage variable and the evolution equations together with other equations of continuum mechanics constitute the initial-value problems of damage mechanics. However, the damaging mechanism is different greatly for the different materials, Even if the material is identical; the damaging mechanism is also entirely different because of the different condition. This is the reason there are so many damaging theories. More importantly, these damaging theories are only for a certain kind of material and a certain kind of damaging mechanism.The damage problem is incompatible problem in Euclidean space. It's in terms of space whether the problem is non-coordinated or not. The problems are considered in more general space instead of Euclid's space in this paper. The main idea of this paper is to establish correspondence between the defect and some geometric bending space, and to describe the damage of the materials with some tensors in the bending space.The elastic damage is described by topological method in this paper. At first, the quasi-plastic strain tensor reflecting the effect of damage is defined. Then the quasi-plastic damage coefficient tensor is defined. The corresponding expression of the quasi-plastic strain tensor and the quasi-plastic damage coefficient tensor is obtained in other damage theories thirdly. The corresponding relation containing these basic tensors between elastic damage defect and Riemannian space is established by topological method fourthly. Finally, the continuity equation of elastic damage in the Riemannian space is obtained. In this paper, the corresponding expression of the quasi-plastic strain tensor and the quasi-plastic damage coefficient tensor is obtained in other damage theories. Then, the continuity equation of elastic damage in Riemannian space is obtained. So the difficulties that exist in other damaging theories with nonlinear complex equations set solved are avoided successfully. A new method is provided in the researches of the damaging mechanics and in solving nonlinear problem.