Abstract

In this paper the authors develop the governing equations for a finite element model of micro-cracking based on a novel approach, eschewing differential equations and continuum mechanics. Instead of first stating the continuum balance laws and constitutive relations followed by discretization, the body is discretized first and then the equations of equilibrium are directly stated for the discretized body. It is shown that, as a result, the balance laws and constitutive relations can be entirely stated in terms of edge forces and lengths rather than strains. Furthermore, by ensuring that microcracks always propagate along the dual mesh which represents the possible fracture microplanes in the element, the need for creating additional nodes, gap elements or cohesive zones is avoided. Finally, the notion of the survival probability of a fracture microplane is introduced and the transition probability evolution is described by using probabilistic notions from population models. Thus the resulting governing equations can be solved by a conventional elastic predictor, followed by a nonlocal fracture corrector, making this convenient to augment conventional elements with fracture abilities.

摘要

在本文中，作者基于一种新方法开发了微裂纹有限元模型的控制方程，避开了微分方程和连续介质力学。不是先陈述连续平衡律和本构关系再离散化，而是先离散化物体，然后直接陈述平衡方程为离散体。结果表明，平衡定律和本构关系可以完全用边缘力和长度而不是应变来表示。此外，通过确保微裂纹始终沿代表单元中可能的断裂微平面的双网格传播，避免了创建额外节点、间隙单元或粘性区域的需要。最后，引入了断裂微平面的生存概率概念，并利用种群模型中的概率概念描述了过渡概率演化。因此，得到的控制方程可以通过传统的弹性预测器求解，然后是非局部断裂校正器，从而方便地增加具有断裂能力的传统元件。