CONTINUITY EQUATION IN RIEMANNIAN SPACE EXPRESSED WITH DAMAGE VARIABLE IN EUCLID’S SPACE

As a kind of defect, the damage of materials is usually described with continuous damaging variable in Euclid’s Space in which a virtual configuration is inducted. But it is insufficient only in Euclid’s Space to describe more complicated defects and to solve the incompatibility caused by damage. In addition, there is no uniform model for various materials or the different damaging courses of the same material. With the fourth-order damaging tensor in Euclid’s Space, third-order quasi-plastic strain tensor, fourth-order extra-matter tensor of damaged plasmodium were defined in the natural state in Riemannian Space, and the continuity equation expressed by the extra-matter tensor was presented in Riemannian Space. Consequently, the corresponding relation between damage defects and Riemannian Space was established, and Bianchi identical equation was used to express the incompatibility of the damaged plasmodium. So the deterioration of material caused by damage can be discussed in a curving space. Finally, an example about anisotropic damage was presented.