Geometry update driven by material forces for simulation of brittle crack growth in functionally graded materials

Abstract

SUMMARYFunctionally graded materials (FGMs) are advanced materials that possess continuously graded properties, such that the growth of cracks is strongly dependent on the gradation of the material. In this work a thermodynamic consistent framework for crack propagation in FGMs is presented, by applying a dissipation inequality to a time-dependent migrating control volume. The direction of crack growth is obtained in terms of material forces as a result of the principle of maximum dissipation. In the numerical implementation a staggered algorithm-deformation update for fixed geometry followed by geometry update for fixed deformation-is employed within each time increment. The geometry update is a result of the incremental crack propagation, which is driven by material forces. The corresponding mesh is generated by combining Delaunay triangulation with local mesh refinement. Furthermore a Newton algorithm is proposed, taking into account mesh transfer of displacements for crack propagation in incremental elasticity. In two numerical examples brittle crack propagation in FGMs is investigated for various directions of strength gradation within the structures.

Finite strain fracture of plates and shells with configurational forces and edge rotations

“…An introduction to configurational forces and their relation to the J ‐integral are provided in Morton Gurtin's book . Finite element applications have been performed by Steinmann and coworkers and more recently by Mahnken . We use a slightly different notation, but the result is the same as in this book.…”

“…Figure contains a representation of force and moment at the crack tip and surrounding elements, some contributing, as detailed in Mahnken , for the total value at the tip. The procedure was adopted because polynomial‐based low‐order elements produce insufficiently accurate stresses.…”

“…This is ideal for estimating the direction. However, for brittle materials, because the implementations make use of normalized quantities (see the discussion by Mahnken ), the alternative is the virtual crack closure technique by Krueger . Furthermore, because it is based on the crack tip opening displacement, it is known to have high predictive capabilities for large strain plasticity .…”

Goal‐oriented adaptive refinement for phase field modeling with finite elements

“…To obtain a proper error estimator, one has to first compute the complete state solution for the whole time interval and then solve for the dual variable backward in time. However, as a main practical drawback, this strategy would require the storage of the dual solutions for all time steps. In order to allow individual spatial discretizations at each time step for the time dependent coupled problem, field variables are transformed between different meshes, for example, . In Step 2, the number of maximum nodes is limited to maxkn in order to restrict the overall computation time. In case, that this comes in conflict with the global restrictions , a message is given by the algorithm. Several possibilities can be envisage for spatial refinement, for example, .…”

“…In order to allow individual spatial discretizations at each time step for the time dependent coupled problem, field variables are transformed between different meshes, for example, .…”