本构关系对连续介质损伤力学基本方程适定性的影响——一维弹性损伤情况

 许多工程材料当变形超过某一极限时由于损伤都会出现应变软化行为。首先从非局部理论出发,推导了应变梯度损伤本构方程;然后利用一阶拟线性偏微分方程组的特征理论,在一维弹性损伤情况下分析了两种不同的本构模型,即 Kachanov损伤本构方程与应变梯度损伤本构方程,对连续介质损伤力学基本方程适定性的影响。结果表明,当损伤发展时,与 Kachanov损伤本构模型相关的连续介质损伤力学的基本方程是不适定的;而与应变梯度损伤本构模型相关的,则无论损伤是否发展,其基本方程始终是适定的。这说明在连续介质损伤力学的本构方程中必须考虑材料内部微结构的尺度效应。

INFLUENCE OF THE CONSTITUTIVE EQUATIONS ON THE WELL-POSEDNESS OF FUNDAMENTAL EQUATIONS IN CONTINUUM DAMAGE MECHANICS--THE CASE OF ONE DIMENSION ELASTIC DAMAGE

Many engineering materials show so called strain softening behavior when deformed beyond a certain limit. How to describe this phenomenon appropriately is the subject of this paper. Firstly, a gradient dependent damage constitutive equation is derived by the nonlocal approach. And then the influence of two of the damage models, the Kachanov and the gradient dependent constitutive equations, on the well posed properties of the fundamental equations in continuum damage mechanics is studied according to the characteristic method of the quasi linear partial differential equations under the case of one dimension elastic damage. The conclusions are accomplished that the differential equations of continuum damage mechanics dependent on the former are ill posed but the latter are well posed when the defects in materials grow with loading. These results show that the scale effects of microstructures should be accounted in the damage constitutive equations of materials.