A variational free-discontinuity formulation of brittle fracture was given by Francfortand Marigo [1], where the total energy is minimized with respect to the crackgeometry and the displacement field simultaneously. The entire evolution of cracksincluding their initiation and branching is determined by this minimization principlerequiring no further criterion. However, a direct numerical discretization of themodel faces considerable difficulties as the displacement field is discontinuous inthe presence of cracks.

A Continuum Phase Field Model for Fracture

Smoothed finite element method (S-FEM) has attracted lots of attentions in the fields of computational mechanics, especially in solid mechanics and heat transfer problems. In computational fluid dynamics, works on S-FEM were limited to two-dimensional problems. This work aims to extend the S-FEM to three-dimensional (3D) incompressible laminar flows. Wedge element grids and grids with mixed wedge and hexahedral elements are formulated for 3D incompressible laminar flows based on the cell-based S-FEM (CS-FEM). To reduce numerical oscillations, we implemented the streamline-upwind/Petrov-Galerkin method (SUPG) together with the stabilized pressure gradient projection (SPGP). Several examples are presented, including the Beltrami flow, lid-driven cavity flow, backward facing step flow and microchannel flow, to validate and examine the presented method. The results indicate that wedge elements and mixed wedge-hexahedral elements based on the CS-FEM have higher computational efficiency than that of hexahedral elements based on the CS-FEM for the same level of computational accuracy. It is also found that the present CS-FEM performed better than the standard FEM in dealing with pressure stability. The flow characteristics are well captured by the CS-FEM using the mixed wedge-hexahedral elements, and the numerical results are acceptable compared to those of STAR-CCM+.

A cell-based smoothed finite element method (CS-FEM) for three-dimensional incompressible laminar flows using mixed wedge-hexahedral element

On realizing specific failure initiation criteria in the phase field model

The aims of the work were first to obtain the fatigue crack growth and life numerically and empirically and then compare the numerical results and experimental data. Grade 1 titanium plates with both symmetrically and anti-symmetrically waisted cracks were due to tension tests first to receive the load versus displacement diagrams and mechanical properties of strength and stiffness. Next from cyclic tests the load versus cycles curves were constructed and fatigue life obtained. The numerical analysis using finite element method and ANSYS workbench was to yield the numerical results of crack growth and life. From fracture mechanics, the critical crack length was considered as final failure, and then, the modified numerical results were received. The estimated crack growth rates showed that at high load ratios the rates were faster and the lives shorter; however, at low load rations both were reverse. The results in load versus cycles curves from two numerical method were found within acceptable errors with the experimental data. 

Zhimin Xu

Papers

Phase-field model for brittle fracture based on the inner-element edge-based smoothed finite method (IES-FEM)