Phase field model of brittle fracture based on polynomial degradation function

基于多项式型退化函数的脆性断裂相场模型

¹ School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China
² School of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China
³ Xi'an Modern Control Technology Research Institute, Xi'an 710065, China

Received: 14 December 2021

Abstract

In order to make the phase field model better describe the brittle fracture process of materials, a polynomial degradation function is given in this paper, and the elastic limit stress and strain corresponding to the phase field model, the critical phase field value, and the critical stress and strain formulas are deduced so as to quantitatively describe the fracture change process of materials. From the formula of the critical phase field value, it can be found that the value decreases with the increasing of number of functions continuously, which indicates that the damage degree of the model decreases gradually when it is damaged. The influence of the degradation function on the fracture process of the material is analyzed by a numerical example. When the function order is small, the phase field model can keep the linear elastic response process well. With the increasing of number of functions, the decreasing trend of the function increases at the beginning, which will aggravate the damage trend inside the material and cause the material to damage earlierly, indicating that the change trend of the degradation function will affect the failure process of the material to a certain extent.

摘要

为了使相场模型更好地描述材料脆性断裂过程, 给出了一个多项式型退化函数, 并推导了相场模型对应的弹性极限应力与应变、临界相场值, 以及临界应力与应变公式, 定量地描述材料的断裂变化过程。从临界相场值公式可以发现, 随着函数次数的增大, 临界相场值不断减小, 这表明模型在破坏时的损伤程度逐渐减小。通过算例分析了退化函数对材料断裂过程的影响, 在函数次数较小时, 可以使相场模型很好地保持线弹性响应过程。随着次数的增大, 函数在初始阶段下降趋势增大, 会使材料内部的损伤趋势加剧, 引起材料更早地发生破坏, 表明退化函数的变化趋势在一定程度上也会影响材料的失效变化过程。