Phase Field Modeling of Crack Growth with Viscoplasticity

The fracture of viscoplastic materials is a complex process due to its time-dependent and plastic responses. Numerical simulation for fractures plays a significant role in crack prediction and failure analysis. In recent years, the phase field model has become a competitive approach to predict crack growth and has been extended to inelastic materials, such as elasto-plastic, viscoelastic and viscoplastic materials, etc. However, the contribution of inelastic energy to crack growth is seldom studied. For this reason, we implement the fracture phase field model coupled with a viscoplastic constitutive in a finite element framework, in which the elastic energy and inelastic energy are used as crack driving forces. The implicit algorithm for a viscoplastic constitutive is presented; this procedure is suitable for other viscoplastic constitutive relations. The strain rate effect, creep effect, stress relaxation effect and cyclic loading responses are tested using a single-element model with different inelastic energy contributions. A titanium alloy plate specimen and a stainless-steel plate specimen under tension are studied and compared with the experimental observations in the existing literature. The results show that the above typical damage phenomenon and fracture process can be well reproduced. The inelastic energy significantly accelerates the evolution of the phase field of viscoplastic materials. For cyclic loadings, the acceleration effect for low frequency is more significant than for high frequency. The influence of the weight factor of inelastic energy fi on the force-displacement curve mainly occurs after reaching the maximum force point. With the increase of fi, the force drops faster in the force-displacement curve. The inelastic energy has a slight effect on the crack growth paths.