Crack Topology Research Articles

A phase‐field model for chemo‐mechanical induced fracture in lithium‐ion battery electrode particles

SummaryCapacity fade in conventional Li‐ion battery systems due to chemo‐mechanical degradation during charge–discharge cycles is the bottleneck in high‐performance battery design. Stresses generated by diffusion‐mechanical coupling in Li‐ion intercalation and deintercalation cycles, accompanied by swelling and shrinking at finite strains, cause micro‐cracks, which finally disturb the electrical conductivity and isolate the electrode particles. This leads to battery capacity fade. As a first attempt towards a reliable description of this complex phenomenon, we propose a novel finite strain theory for chemo‐elasticity coupled with phase‐field modeling of fracture, which regularizes a sharp crack topology. We apply a rigorous geometric approach to the diffusive crack modeling based on the introduction of a global evolution equation of regularized crack surface, governed by the crack phase field. The irreversible evolution of the crack phase field is modeled through a novel critical stress‐based growth function. A modular concept is outlined for linking of the diffusive crack modeling to the complex chemo‐elastic material response of the bulk material. Here, we incorporate standard as well as gradient‐extended Cahn–Hilliard‐type diffusion for the Li‐ions, where the latter accounts for a possible phase segregation. From the viewpoint of the methodology, the separation of modules for the crack evolution and the bulk response provides a highly attractive and transparent structure of the multi‐physics problem. This structure is exploited on the numerical side by constructing a robust finite element method, based on an algorithmic decoupling of updates for the crack phase field and the state variables of the chemo‐mechanical bulk response. We demonstrate the performance of the proposed coupled multi‐field formulation by an analysis of representative boundary value problems. Copyright © 2015 John Wiley & Sons, Ltd.

Rupture in soft biological tissues modeled by a phase‐field method

AbstractWe present an application of the phase‐field method of fracture to the simulation of artery rupture at large strains. To achieve this, the crack driving force function associated with the evolution of the crack phase‐field is modified to account for the inherent anisotropy of the soft biological tissues. The phase‐field methods present a promising and innovative approach to the thermodynamically consistent modeling of fracture. A key advantage lies in the prediction of the complex crack topologies where the cohesive zone approaches to fracture are known to suffer. A regularized crack surface functional is introduced that Γ‐converges to a sharp crack topology for vanishing length scale parameter. The evaluation of the phase‐field follows the minimization of this crack surface functional. The phase‐field variable can be treated as a geometric quantity whose evolution is coupled to the anisotropic bulk response in a modular format in terms of a crack driving state function. A stress‐based anisotropic failure criterion is introduced whose maximum value from the deformation history drives the irreversible crack phase‐field. The formulation is verified by the finite element based simulation of a real arterial cross‐section undergoing rupture in a two‐dimensional setting when subjected to inflation pressure. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A phase-field model for fracture in biological tissues.

This work presents a recently developed phase-field model of fracture equipped with anisotropic crack driving force to model failure phenomena in soft biological tissues at finite deformations. The phase-field models present a promising and innovative approach to thermodynamically consistent modeling of fracture, applicable to both rate-dependent or rate-independent brittle and ductile failure modes. One key advantage of diffusive crack modeling lies in predicting the complex crack topologies where methods with sharp crack discontinuities are known to suffer. The starting point is the derivation of a regularized crack surface functional that [Formula: see text]-converges to a sharp crack topology for vanishing length-scale parameter. A constitutive balance equation of this functional governs the crack phase-field evolution in a modular format in terms of a crack driving state function. This allows flexibility to introduce alternative stress-based failure criteria, e.g., isotropic or anisotropic, whose maximum value from the deformation history drives the irreversible crack phase field. The resulting multi-field problem is solved by a robust operator split scheme that successively updates a history field, the crack phase field and finally the displacement field in a typical time step. For the representative numerical simulations, a hyperelastic anisotropic free energy, typical to incompressible soft biological tissues, is used which degrades with evolving phase field as a result of coupled constitutive setup. A quantitative comparison with experimental data is provided for verification of the proposed methodology.

An efficient approach to the modeling of compressive transverse cracking in composite laminates

A wedge-shaped failure mechanism occurs when a composite ply in a laminate is loaded under transverse compression. Therefore failure models developed for tension are not straightforwardly applicable to compression. For models that represent matrix cracks as discontinuities, the inclined crack topology complicates algorithms considerably. In this paper, a method is introduced to approximate the behavior of inclined cracks while avoiding these complications. Rather than adapting the crack topology, a transformation is applied in the evaluation of the cohesive law. The method is applied on predefined interface elements in a 2D model, as well as on cohesive segments in the extended finite element method in a 3D model. It is found that the compressive failure of composite laminates including the wedge effect can be approximated accurately with this method using a vertical crack plane. In 3D, a variable fracture plane angle is taken into account with the method, which means that the whole range of failure modes, from tensile through shear-dominated to compressive failure can be covered in a single approach.

Abaqus implementation of phase-field model for brittle fracture

A phase-field model for brittle fracture is implemented in the commercial finite element software Abaqus by means of UEL and UMAT subroutines. The phase-field method considerably reduces the implementation complexity for fracture problems as it removes the need for numerical tracking of discontinuities in the displacement field that are characteristic of discrete crack methods. This is accomplished by replacing the sharp discontinuities with a scalar damage phase-field representing the diffuse crack topology wherein the amount of diffusion is controlled by a regularization parameter. The nonlinear coupled system consisting of the linear momentum equation and a diffusion-type equation governing the phase-field evolution is solved simultaneously via a Newton–Raphson approach. The implemented crack propagation model does not require predefined paths for crack growth or user-defined surfaces to simulate crack debonding. Post-processing of simulation results is performed via an additional subroutine implemented in the visualization module.

Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method

We present a new approach based on local partition of unity extended meshfree Galerkin method for modeling quasi-static crack growth in two-dimensional (2D) elastic solids. The approach utilizing the local partition of unity as a priori knowledge on the solutions of the boundary value problems that can be added into the approximation spaces of the numerical solutions. It thus allows for extending the standard basis functions by enriching the asymptotic near crack-tip fields to accurately capture the singularities at crack-tips, and using a jump step function for the displacement discontinuity along the crack-faces. The radial point interpolation method is used here for generating the shape functions. The representation of the crack topology is treated by the aid of the vector level set technique, which handles only the nodal data to describe the crack. We employ the domain-form of the interaction integral in conjunction with the asymptotic near crack-tip field to extract the fracture parameters, while crack growth is controlled by utilizing the maximum circumferential stress criterion for the determination of its propagating direction. The proposed method is accurate and efficient in modeling crack growths, which is demonstrated by several numerical examples with mixed-mode crack propagation and complex configurations.

Stable Cracking Particles Method Based on Stabilized Nodal Integration and Updated Lagrangian Kernel

A stable cracking particles method (CPM) based on updated Lagrangian kernels is proposed. The idea of CPM is to model the crack topology by a set of cracked particles. Hence no representation of the crack surface is needed making the method useful for problems involving complex fracture patterns as they occur in dynamics and under fast loading conditions. For computational efficiency, nodal integration is exploited in the present paper. In order to avoid instabilities, a scheme is presented to stabilized the integration. Moreover, a set of simple cracking rules are proposed in order to prevent numerical fracture. The method is applied to two benchmark problems and shows good accuracy.

Coupled chemomechanics and phase field modeling of failure in electrode materials of Li‐ion batteries

AbstractWe propose a canonical finite strain theory for diffusion‐mechanics coupling for the intercalation induced stress generation in Li‐ion electrode particles. The intrinsic coupling arises from both mechanical pressure gradient‐induced diffusion of Li‐ion particles and diffusion induced swelling/shrinkage leading to mechanical stresses. In addition, we extend the finite strain theory for diffusion‐mechanics coupling to chemomechanical fracture of electrode particles by introducing a nonlocal crack phase field which replaces a sharp crack topology with a smooth diffuse interpolation between the intact and broken states of the material. We employ a semi‐implicit Galerkin‐type finite element method for the solution of resulting set of differential equations. In addition to the mechanical, chemical and crack phase field, we introduce the pressure as an independent field variable in order to reduce the smoothness requirements on the interpolation functions. We illustrate characteristic features of the proposed model by means of representative initial‐boundary value problems. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A SIMPLIFIED MESHLESS METHODS FOR BRITTLE FRACTURE OF CONCRETE

A simplified meshless methods for brittle fracture and nonlinear material is presented. In this method, the crack is modeled by a set of discrete crack segments crossing the entire domain of influence of the meshless shape functions. The key advantage of this method is its simplicity since no representation of the crack topology is needed. A nonlocal stress tensor around the crack tip is used as fracture criterion. A neo-Hooke material in the bulk material is used and a cohesive zone model is employed once discrete cracks occur. We also present consistent linearization of the cohesive zone model. The method is applied to fracture modeling in concrete that is accompanied by excessive cracking and therefore methods that represent the crack path have major drawbacks. We demonstrate the accuracy of the proposed method for complex problems involving mode-I and mixed mode failure.

Computational studies on fragmentation of brittle materials

We study dynamic fragmentation of metallic structures using the cracking particles method with obscuration zone. The cracking particles method is an efficient meshfree method for modeling dynamic fracture. Fracture is modeled by a set of crack segments. The cracking particles method does not require the representation of the crack topology. To avoid numerical fracture observed in discrete approaches, we employ obscuration zones proposed by Mott in his analytical work of fragmentation theory. We also study the influence of initial imperfections with different stochastic input parameters on the results.

Modeling ductile fracture using a simplified meshfree method

A new meshfree method for modeling ductile fracture in metallic materials is presented. The approach combines the Gurson–Tveergaard–Needleman (GTN) model and the meshfree cracking particles method (CPM). The GTN model is employed to model the ductile constitutive behavior in the bulk material while the discontinuous CPM is employed at post-localization. In the CPM, fracture is modeled by a set of crack segments that cross the entire domain of influence of the meshfree shape functions. This facilitates the method as no crack topology and associated crack propagation algorithms need to be implemented. A seamless transition from a continuum to a discontinuum is provided by the loss-of-material-stability criterion. The efficiency and accuracy of the method is shown by two problems.

A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns

SUMMARYThe numerical modeling of dynamic failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies and demands the formulation of additional branching criteria. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. Following our recent works on quasi‐static modeling of phase‐field‐type brittle fracture, we propose in this paper a computational framework for diffusive fracture for dynamic problems that allows the simulation of complex evolving crack topologies. It is based on the introduction of a local history field that contains a maximum reference energy obtained in the deformation history, which may be considered as a measure of the maximum tensile strain in the history. This local variable drives the evolution of the crack phase field. Its introduction provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a very robust algorithmic treatment for elastodynamic problems of diffusive fracture. Here, we extend the recently proposed operator split scheme from quasi‐static to dynamic problems. In a typical time step, it successively updates the history field, the crack phase field, and finally the displacement field. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples, which show the evolution of complex crack patterns under dynamic loading. Copyright © 2012 John Wiley & Sons, Ltd.

Phase Field Modeling of Crack Propagation at Large Strains with Application to Rubbery Polymers

AbstractThe computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities exhibits drawbacks in situations with complex crack topologies. This drawback is overcome by diffusive crack modeling based on the introduction of a fracture phase field characterizing via an auxiliary variable the crack topology. In the following we extend recent advances in phase‐field‐type fracture based on operator split techniques, suggested in Miehe et al. [1], to the modeling of crack propagation in geometrically large deforming solids e.g. rubber‐like materials. An extremely robust algorithmic treatment based on an operator split scheme is introduced consisting of three steps. Updating i) a local history‐field containing the maximum reference energy, ii) the fracture phase field, and iii) the displacement field. We demonstrate the performance of proposed phase field formulation for largely deforming solids by means of a representative numerical example. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Phase Field Model for Three‐Dimensional Dynamic Fracture and its Efficient Numerical Implementation

AbstractThe numerical modeling of dynamic failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies and demands the formulation of additional branching criteria. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. We focus on the extension of a recently developed phase field model for fracture from the quasi‐static setting towards the dynamic setting. It is obtained by taking into account inertial terms and associated dynamic integrators. The introduction of a history field, containing a maximum fracture‐driving energy, provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of an extremely robust operator split technique. In a subsequent step, the proposed model is extended to three dimensional problems. The dynamic treatment opens the door to the analysis of complex fracture phenomena like multiple crack branching and crack arrest. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Thermodynamically­Consistent Phase Field Model for Electromechanical Diffusive Crack Propagation

AbstractThe computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies and for crack initiation in bodies free of defects. This drawback can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. In [1,2], we developed a phase‐field‐model of brittle fracture, which is conceptually in line with gradient‐type continuum damage models. Its global response is governed by an incremental variational principle, whose Euler equations describe the balance of linear momentum and the evolution of the crack topology. In this paper, the proposed model is extended to a thermodynamically consistent four‐field setting of coupled electromechanical fracture, based on a degrading mixed energy‐enthalpy functional and an over‐force‐type dissipation functional. These functionals are expressed in terms of the displacement, the electric potential, the fracture phase and its dual driving‐force fields. Evaluation of the necessary condition yields, in addition to the above mentioned Euler equations, the Gauss‐law of electrostatics. The capability of the proposed framewok for the description of electromechanically coupled crack propagation is demonstrated by means of a representative numerical example. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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

AbstractThe 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.

A phase field model of electromechanical fracture

Structural reliability analyses of piezoelectric solids need the modeling of failure under coupled electromechanical actions. However, the numerical simulation of failure due to fracture based on sharp crack discontinuities may suffer 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 work, we develop a framework of diffusive fracture in piezoelectric solids. We start our investigation with the definition of a crack surface functional of the phase field that Γ -converges for vanishing length-scale parameter to a sharp crack topology. This functional provides the basis for the definition of suitable dissipation functions which govern the evolution of the crack phase field. Based on experimental results available in the literature, we suggest a non-associative dissipative framework where the fracture phase field is driven by the mechanical part of the coupled electromechanical driving force. This accounts for a hierarchical view that considers (i) the decrease of stiffness due to mechanical rupture as the primary action that is followed by (ii) the decrease of electric permittivity due to the generated free space. The proposed definition of mechanical and electrical parts of the fracture driving force follows in a natural format from a kinematic assumption, that decomposes the total strains and the total electric field into energy–enthalpy-producing parts and fracture parts, respectively. Such an approach allows the insertion of well-known anisotropic piezoelectric storage functions without change. We end up with a three-field-problem that couples the displacement with the electric potential and the fracture phase field. The latter is governed by a micro-balance equation, which appears in a very transparent form in terms of a history field containing a maximum fracture source obtained in the time history of the electromechanical process. This representation allows the construction of a very robust algorithmic treatment based on a operator split scheme, which successively updates in a typical time step the history field, the crack phase field and finally the two piezoelectric fields. The proposed model is considered to be the canonically simple scheme for the simulation of diffusive electromechanical crack propagation in solids. We demonstrate its modeling capacity by means of representative numerical examples.

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. Following our recent work [C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically-consistent phase field models of fracture: Variational principles and multi-field fe implementations, International Journal for Numerical Methods in Engineering DOI:10.1002/nme.2861] on phase-field-type fracture, we propose in this paper a new variational framework for rate-independent diffusive fracture that bases on the introduction of a local history field. It contains a maximum reference energy obtained in the deformation history, which may be considered as a measure for the maximum tensile strain obtained in history. It is shown that this local variable drives the evolution of the crack phase field. The introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a new algorithmic treatment of diffusive fracture. Here, we propose an extremely robust operator split scheme that successively updates in a typical time step the history field, the crack phase field and finally the displacement field. A regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added. The proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples.

Cracking particles: a simplified meshfree method for arbitrary evolving cracks

AbstractA 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.

Assessment of the fatigue crack closure phenomenon in damage-tolerant aluminium alloy by in-situ high-resolution synchrotron X-ray microtomography

Synchrotron X-ray microtomography has been utilized for the in-situ observation of steady-state plane-strain fatigue crack growth. A high-resolution experimental configuration and phase contrast imaging technique have enabled the reconstruction of crack images with an isotropic voxel with a 0.7 µm edge. The details of a crack are readily observed, together with evidence of the incidence and mechanical influence of closure. After preliminary investigations of the achievable accuracy and reproducibility, a variety of measurement methods are used to quantify crack-opening displacement (COD) and closure from the tomography data. Utilization of the physical displacements of microstructural features is proposed to obtain detailed COD data, and its feasibility is confirmed. Loss of fracture surface contact occurs gradually up to the maximum load. This is significantly different from tendencies reported where a single definable opening level is essentially assumed to exist. The closure behaviour is found to be attributable mainly to pronounced generation of mode III displacement which may be caused by local crack topology. Many small points of closure still remain near the crack tip, suggesting that the near-tip contact induces crack growth resistance. The effects of overloading are also discussed.