In phase-field models the damage evolution problem is considered as a minimisation problem of a Griffith-like energy functional, governed by the principles of irreversibility , stability and energy balance. Herein, we consider phase-field models characterised by different degradation and energy dissipation functions. With a proper choice for the characteristic functions in phase-field models, it is possible to reproduce cohesive fracture in a one-dimensional setting. We consider a one-dimensional bar with stress softening , which exhibits homogeneous deformations provided that the length of the bar length is below a state-dependent critical value. Otherwise, the bar will lose stability and show a localised response. It appears that the phase-field method can partially reproduce the response of a cohesive zone model , for instance the traction–separation law, but not all aspects of the model, like the dissipated energy. For a one-dimensional problem, the crack nucleation load varies smoothly from that predicted by a strength criterion to that of a toughness criterion for different lengths of the bar. We have compared the one-dimensional results with the numerical solutions in a two-dimensional setting, which yielded a very good agreement.

Phase-field modelling of cohesive fracture

Phase-field models for brittle fracture in anisotropic materials result in a fourth-order partial differential equation for the damage evolution. This necessitates a $$\mathcal {C}^1$$
 continuity of the basis functions. Here, Powell-Sabin B-splines, which are based on triangles, are used for the approximation of the field variables as well as for the the description of the geometry. The use of triangles makes adaptive mesh refinement and discrete crack insertion straightforward. Bezier extraction is used to cast the B-splines in a standard finite element format. A procedure to impose Dirichlet boundary condition is provided for these elements. The versatility and accuracy of the approach are assessed in two case studies, featuring crack kinking and zig-zag crack propagation. It is also shown that the adaptive refinement well captures the evolution of the phase field.

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The use of Powell-Sabin B-Splines in a higher-order phase-field model for crack kinking

An IGA based nonlocal gradient-enhanced damage model for failure analysis of cortical bone

Geometrically nonlinear extended isogeometric analysis for cohesive fracture with applications to delamination in composites

X-IGALME: Isogeometric analysis extended with local maximum entropy for fracture analysis

The objective of this study is to present an extended isogeometric formulation for cohesive fracture. The approach exploits the higher order interelement continuity property of nonuniform rational B‐splines (NURBS), in particular the higher accuracy that results for the stress prediction, which yields an improved estimate for the direction of crack propagation compared to customary Lagrangian interpolations. Shifting is used to ensure compatibility with the surrounding discretization, where, different from extended finite element methods, the affected elements stretch over several rows perpendicular to the crack path. To avoid fine meshes around the crack tip in case of cohesive fracture, a blending function is used in the extension direction of the crack path. To comply with standard finite element data structures, Bezier extraction is used. The absence of the Kronecker‐delta property in the higher order interpolations of isogeometric analysis impedes the enrichment scheme and compatibility enforcement. These issues are studied comprehensively at the hand of several examples, while crack propagation analyses show the viability of the approach.

/pdf/extended-isogeometric-analysis-for-cohesive-fracture-2nrdudzcn5.pdf

Farshid Fathi

Papers

Extended isogeometric analysis for cohesive fracture

Geometrically nonlinear extended isogeometric analysis for cohesive fracture with applications to delamination in composites

X-IGALME: Isogeometric analysis extended with local maximum entropy for fracture analysis