Michael J. Borden

A phase-field model for fracture in piezoelectric ceramics

A phase-field model is presented for modeling the fracture of piezoelectric ceramics. The implementation of several different crack face boundary conditions, including conducting, permeable, and insulating or impermeable, as well as energetically consistent is described. The approach to the latter involves a finite deformation framework for piezoelectricity. In addition, a new function that governs material degradation is proposed to eliminate the presence of high phase-field values in the vicinity of large electric fields. The new function is found to lead to improved brittle material behavior as well. Results are presented that demonstrate the capability of the model to capture complicated phenemona that arise in piezoelectric fracture, including crack retardation, acceleration, and turning.

Conformal Refinement and Coarsening of Unstructured Hexahedral Meshes

This paper describes recently developed procedures for local conformal refinement and coarsening of all-hexahedral unstructured meshes. Both refinement and coarsening procedures take advantage of properties found in the dual or “twist planes” of the mesh. A twist plane manifests itself as a conformal layer or sheet of hex elements within the global mesh. We suggest coarsening techniques that will identify and remove sheets to satisfy local mesh density criteria while not seriously degrading element quality after deletion. A two-dimensional local coarsening algorithm is introduced. We also explain local hexahedral refinement procedures that involve both the placement of new sheets, either between existing hex layers or within an individual layer. Hex elements earmarked for refinement may be defined to be as small as a single node or as large as a major group of existing elements. Combining both refinement and coarsening techniques allows for significant control over the density and quality of the resulting modified mesh.

Isogeometric Failure Analysis

Isogeometric analysis is a versatile tool for failure analysis. On the one hand, the excellent control over the inter-element continuity conditions enables a natural incorporation of continuum constitutive relations that incorporate higher-order strain gradients, as in gradient plasticity or damage. On the other hand, the possibility of enhancing a basis with discontinuities by means of knot insertion makes isogeometric finite elements a suitable candidate for modeling discrete cracks. Both possibilities are described and will be illustrated by examples.

Phase-Field Formulation for Ductile Fracture

Phase-field models have been a topic of much research in recent years. Results have shown that these models are able to produce complex crack patterns in both two and three dimensions. A number of extensions from brittle to ductile materials have been proposed and results are promising. To date, however, these extensions have not accurately represented strains after crack initiation or included important aspects of ductile fracture such as stress triaxiality. This work describes a number of contributions to further develop phase-field models for fracture in ductile materials.