https://www.crm.sns.it/media/course/3060/miehe+hofacker+welschinger10.pdf

A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that Γ-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.

Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations

A new approach for modelling discrete cracks in meshfree methods is described. In this method, the crack can be arbitrarily oriented, but its growth is represented discretely by activation of crack surfaces at individual particles, so no representation of the crack's topology is needed. The crack is modelled by a local enrichment of the test and trial functions with a sign function (a variant of the Heaviside step function), so that the discontinuities are along the direction of the crack. The discontinuity consists of cylindrical planes centred at the particles in three dimensions, lines centred at the particles in two dimensions. The model is applied to several 2D problems and compared to experimental data. Copyright © 2004 John Wiley & Sons, Ltd.

Cracking particles: A simplified meshfree method for arbitrary evolving cracks

A phase-field description of dynamic brittle fracture

An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non-smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi-field problems. Copyright © 2010 John Wiley & Sons, Ltd.

The extended/generalized finite element method: An overview of the method and its applications

SUMMARY A methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial dierential equation loses hyperbolicity. The approach is limited to rate-independent materials, so that the transition occurs on a set of measure zero. The discrete discontinuity is treated by the extendednite element method (XFEM) whereby arbitrary discontinuities can be incorporated in the model without remeshing. Loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model. A new method was developed for the case when the discontinuity ends within an element; it facilitates the modelling of crack tips that occur within an element in a dynamic setting. The method is applied to several dynamic crack growth problems including the branching of cracks. Copyright ? 2003 John Wiley & Sons, Ltd.

Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment

A new vector level set method for modelling propagating cracks in the element-free Galerkin (EFG) method is presented. With this approach only nodal data are used to describe the crack; no geometrical entity is introduced for the crack trajectory, and no partial differential equations need to be solved to update the level sets. The nodal description is updated as the crack propagates by geometric equations. The advantages of this approach, here introduced and analysed for the two-dimensional case, are particularly promising in three-dimensional applications, where the geometrical description and evolution of an arbitrary crack surface in a complex solid is very awkward. In addition, new methods for crack approximations in EFG are introduced, using a jump function accounting for the displacement discontinuity along the crack faces and the Westergard's solution enrichment near the crack tip. These enrichments, being extrinsic, can be limited only to the nodes surrounding the crack and are naturally coupled to the level set crack representation. Copyright © 2002 John Wiley & Sons, Ltd.

A vector level set method and new discontinuity approximations for crack growth by EFG

A method for modelling arbitrary growth of dynamic cracks without remeshing is presented. The method is based on a local partition of unity. It is combined with level sets, so that the discontinuities can be represented entirely in terms of nodal data. This leads to a simple method with clean data structures that can easily be incorporated in general purpose software. Results for a mixed-mode dynamic fracture problem are given to demonstrate the method.

/pdf/the-extended-finite-element-method-for-dynamic-fractures-4z6nu40f9o.pdf

The extended finite element method for dynamic fractures

A new method for modeling discontinuities, such as cracks, in the element free Galerkin method is presented. A jump function is used for the displacement discontinuity along the crack faces and the Westergard’s solution enrichment near the crack tip. These enrichments, being extrinsic, can be limited only to the nodes surrounding the crack. The method is coupled to a new vector level set method [1] so with this approach only nodal data are used to describe the crack, no geometrical entity is introduced for the crack trajectory, and no partial differential equations need be solved to update the level sets.

https://link.springer.com/content/pdf/10.1007%2F978-3-642-56103-0_3.pdf

New Methods for Discontinuity and Crack Modeling in EFG

Meshfree methods with discontinuous radial basis functions and their numerical implementation for elastic problems are presented We study the following radial basis functions: the multiquadratic (MQ), the Gaussian basis functions and the thin-plate basis functions These radial basis functions are combined with step function enrichments directly or with enriched Shepard functions The formulation is coupled with level set methods and requires no explicit representation of the discontinuity Numerical results show the robustness of the method, both in accuracy and convergence

Jingxiao Xu

Papers

Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment

A vector level set method and new discontinuity approximations for crack growth by EFG

The extended finite element method for dynamic fractures

New Methods for Discontinuity and Crack Modeling in EFG

Discontinuous Radial Basis Function Approximations for Meshfree Methods