Phase-Field Modeling and Computation of Crack Propagation and Fracture

摘要

The purpose of this project has been to develop continuum phase- field models in concert with numerical methods for their solution to study dynamic brittle fracture. In contrast to discrete descriptions of fracture, phase-field descriptions do not require numerical tracking of discontinuities in the displacement field. This greatly reduces implementation complexity. During this project we have studied the basic formulations of the phase-field fracture theory, leading to second order partial differential equations (PDEs), along with the effect of adding higher-order gradients to the standard phase- field theory, leading to fourth and higher order PDEs. We have derived the thermodynamically consistent governing equations for the phase-field models by way of both balance law approaches and variational principles based on energy balance assumptions. We have completed studies on the implementation of second and fourth order phase-field methods for fracture brittle elastic materials as well as for fracture in brittle piezoelectric materials. We found that the fourth order phase-field model leads to higher regularity in the exact phase- field solution, which is exploited by the smooth function spaces utilized in isogeometric analysis. This increased regularity improves the convergence rate of the numerical solution and opens the door to higher-order convergence rates for fracture problems.