Euclid空间中损伤变量表示的Riemann空间中的连续性方程

损伤作为一种缺陷,宏观上通常是在Euclid空间中通过虚拟构形的方式,以连续分布的损伤变量加以描述。但如果描述更复杂的缺陷,处理变形非协调性问题,仅停留在Euclid空间中是不够的。同时,针对工程中不同材料以及同种材料的不同损伤机制,目前尚未建立一个统一的损伤模型。根据Euclid空间中的四阶损伤变量张量,定义了处于自然状态中的损伤变形体在Riemann空间中的三阶拟塑性张量、四阶异物张量,并用其描述损伤缺陷。并给出Riemann空间中异物张量所满足的连续性方程。从而建立了损伤缺陷与Riemann空间的对应关系,以Riemann空间中Bianchi恒等式刻划损伤变形体的非协调性。使得可以在Riemann这样一个弯曲空间中讨论损伤所引起的材料力学性能的劣化。最后给出一个各向异性损伤的算例。

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.