Diffusive Crack Modeling Research Articles

A Review on phase-field modeling of fracture

In cases with complicated crack topologies, the computational modeling of failure processes in materials owing to fracture based on sharp crack discontinuities fails. Diffusive crack modeling based on the insertion of a crack phase-field can overcome this. The phase-field model (PFM) portrays the fracture geometry in a diffusive manner, with no abrupt discontinuities. Unlike discrete fracture descriptions, phase-field descriptions do not need numerical monitoring of discontinuities in the displacement field. This considerably decreases the complexity of implementation. These qualities enable PFM to describe fracture propagation more successfully than numerical approaches based on the discrete crack model, especially for complicated crack patterns. These models have also demonstrated the ability to forecast fracture initiation and propagation in two and three dimensions without the need for any ad hoc criteria. The phase-field model, among numerous options, is promising in the computer modeling of fracture in solids due to its ability to cope with complicated crack patterns such as branching, merging, and even fragmentation. A brief history of the application of the phase-field model in predicting solid fracture has been attempted. An effort has been made to keep the conversation focused on recent research findings on the subject. Finally, some key findings and recommendations for future research areas in this field are discussed.

Hydraulic fracture in poro-hydro-elastic media

The prediction of fluid-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing, the so-called “fracking”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developing cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media that provides a rigorous approach to a diffusive crack modeling based on the introduction of a regularized crack surface. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. It shows that the quasi-static problem of elastically deforming, fluid-saturated porous media at fracture is related to a minimization principle for the evolution problem. The existence of this minimization principle for the coupled problem is advantageous with regard to a new unconstrained stable finite element design, while previous space discretizations of the saddle point principles are constrained by the LBB condition. This proposed formulation includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length.

Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media

The prediction of fluid- and moisture-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing, the so-called “fracking”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developed cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media. It provides a rigorous geometric approach to a diffusive crack modeling based on the introduction of a constitutive balance equation for a regularized crack surface and its modular linkage to a Darcy–Biot-type bulk response of hydro-poro-elasticity. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. A modular concept is outlined for linking of the diffusive crack modeling with the hydro-poro-elastic response of the porous bulk material. This includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length. This proposed modular model structure is exploited in the numerical implementation 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 hydro-poro-elastic bulk response. We demonstrate the performance of the phase field formulation of fracture for a spectrum of model problems of hydraulic fracture. A slight modification of the framework allows the simulation of drying-caused crack patterns in partially saturated capillar-porous media.

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.

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.

Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic–plastic solids

This work presents a generalization of recently developed continuum phase field models from brittle to ductile fracture coupled with thermo-plasticity at finite strains. It uses a geometric approach to the diffusive crack modeling based on the introduction of a balance equation for a regularized crack surface and its modular linkage to a multi-physics bulk response developed in the first part of this work. This evolution equation is governed by a constitutive crack driving force. In this work, we supplement the energetic and stress-based forces for brittle fracture by additional forces for ductile fracture. These are related to state variables associated with the inelastic response, such as the amount of plastic strain and the void volume fraction in metals, or the amount of craze strains in glassy polymers. To this end, we define driving forces based on elastic and plastic work densities, and barrier functions related to critical values of these inelastic state variables. The proposed thermodynamically consistent framework of ductile phase field fracture is embedded into a formulation of gradient thermo-plasticity, that is able to account for material length scales such as the width of shear bands. It is applied to two constitutive model problems. The first is designed for the analysis of brittle-to-ductile failure mode transition in the dynamic failure analysis of metals. The second is constructed for a quasi-static analysis of crazing-induced fracture in glassy polymers. A spectrum of simulations demonstrates that the use of barrier-type crack driving forces in the phase field modeling of fracture, governed by accumulated plastic strains in metals or crazing strains in polymers, provide results in very good agreement with experiments.

Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids

Abstract This work presents a generalization of recently developed continuum phase field models for brittle fracture towards fully coupled thermo-mechanical and multi-physics problems at large strains. It outlines a rigorous geometric approach to the diffusive crack modeling based on the introduction of a balance of regularized crack surface, governed by a crack phase field. The regularized crack surface functional is based on a crack surface density function, that describes the macroscopic crack surface in the bulk material per unit of the reference volume. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies. The formulation proposed is essentially a gradient damage theory, however, equipped with critical ingredients rooted in fracture mechanics. A modular concept is outlined for the linking of the diffusive crack modeling with complex multi-field response of the bulk material, where focus is put on the model problem of finite thermo-elasticity. This concerns a generalization of crack driving forces from the energetic definitions towards stress-based criteria, the constitutive modeling of heat conduction across cracks and convective heat exchanges at crack faces based on additional constitutive functions. This is achieved by approximating surface load integrals of the sharp crack approach by distinct volume integrals. We demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples.

Variational Phase Field Modeling of Fracture Caused by Fluid Transport in Poroelastic Solids

AbstractThe numerical modeling of failure mechanisms due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by recently developed diffusive crack modeling concepts, which are based on the introduction of a crack phase field. Such an approach is conceptually in line with gradient‐extended continuum damage models which include internal length scales. In this paper, we extend our recently outlined mechanical framework [1–3] towards the phase field modeling of fracture in the coupled problem of fluid transport in deforming porous media. Here, extremely complex crack patterns may occur due to drying or hydraulic induced fracture, the so called fracking. We develop new variational potentials for Biot‐type fluid transport in porous media at finite deformations coupled with phase field fracture. It is shown, that this complex coupled multi‐field problem is related to an intrinsic mixed variational principle for the evolution problem. This principle determines the rates of deformation, fracture phase field and fluid content along with the fluid potential. We develop a robust computational implementation of the coupled problem based on the potentials mentioned above and demonstrate its performance by the numerical simulation of complex fracture patterns. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase field modeling of complex crack patterns in multi‐physics problems

AbstractRecently developed continuum phase field models for brittle fracture show excellent modeling capability in situations with complex crack topologies including branching in the small and large strain applications. This work presents a generalization towards fully coupled multi‐physics problems at large strains. A modular concept is outlined for the linking of the diffusive crack modeling with complex multi field material response, where the focus is put on the model problem of finite thermo‐elasticity. This concerns a generalization of crack driving forces from the energetic definitions towards stress‐based criteria, the constitutive modeling of degradation of non‐mechanical fluxes on generated crack faces. Particular assumptions are made on the generation of convective heat exchanges approximating surface load integrals of the sharp crack approach by distinct volume integrals. The coupling effect is also shown in generation of cracks due to thermally induced stress states. We finally demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase field modeling of fracture in rubbery polymers. Part I: Finite elasticity coupled with brittle failure

This work presents a new phase field model for rate-independent crack propagation in rubbery polymers at large strains and considers details of its numerical implementation. The approach accounts for micro-mechanically based features of both the elastic bulk response as well as the crack toughness of idealized polymer networks. The proposed diffusive crack modeling based on the introduction of a crack phase field overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies. The crack phase field governs a crack density function, which describes the macroscopic crack surface in the polymer per unit of the reference volume. It provides the basis for the constitutive modeling of a degrading free energy storage and a crack threshold function with a Griffith-type critical energy release rate, that governs the crack propagation in the polymer. Both the energy storage as well as the critical energy release due to fracture can be related to classical statistical network theories of polymers. The proposed framework of diffusive fracture in polymers is formulated in terms of a rate-type variational principle that determines the evolution of the coupled primary variable fields, i.e. the deformation field and the crack phase field. On the computational side, we outline a staggered solution procedure based on a one-pass operator split of the coupled equations, that successively updates in a typical time step the crack phase field and the displacement field. Such a solution algorithm is extremely robust, easy to implement and ideally suited for engineering problems. We finally demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples.

Phase Field Modeling of Fracture in Plates and Shells

AbstractThe numerical modeling of failure mechanisms in plates and shells due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. In this paper, we extend ideas recently outlined in [1, 2] towards the phase field modeling of fracture in dimension‐reduced continua with application to Kirchhoff plates and shells. The introduction of history fields, containing the maximum reference energy obtained in history, provides a very transparent representation of the coupled balance equations and allows the construction of an extremely robust operator split technique. The performance of the proposed models is demonstrated by means of representative numerical examples. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Phase Field Model for Ductile to Brittle Failure Mode Transition

AbstractThe computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in dynamic problems with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. We outline a conceptual framework for phase field models of crack propagation in brittle elastic and ductile elastic‐plastic solids under dynamic loading and investigate the ductile to brittle failure mode transition observed in the experiment performed by Kalthoff and Winkeler [3]. We develop incremental variational principles and consider their numerical implementations by multi‐field finite element methods. To this end, we define energy storage and dissipation functions for the plastic flow including the fracture phase field. The introduction of local history fields that drive the evolution of the crack phase field inspires the construction of robust operator split schemes. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation

The modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations of complex crack topologies including branching. This drawback can be overcome by a diffusive crack modeling based on the introduction of a crack phase field as proposed in Miehe et al. (Comput Methods Appl Mech Eng 19:2765–2778, 2010a; Int J Numer Meth Eng 83:1273–1311, 2010b), Hofacker and Miehe (Int J Numer Meth Eng, 2012). In this work, we summarize basic ingredients of a thermodynamically consistent, variational-based model of diffusive crack propagation under quasi-static and dynamic conditions. It is shown that all coupled field equations, in particular the balance of momentum and the gradient-type evolution equation for the crack phase field, follow as the Euler equations of a mixed rate-type variational principle that includes the fracture driving force as the mixed field variable. This principle makes the proposed formulation extremely compact and provides a perfect basis for the finite element implementation. We then introduce a local history field that contains a maximum energetic crack source obtained in the deformation history. It drives the evolution of the crack phase field. This allows for the construction of an extremely robust operator split scheme that updates in a typical time step 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.

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)

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)

Configurational‐Force‐Based Adaptive FE Solver for a Phase Field Model of Fracture

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 diffusive crack modeling, based on the introduction of a crack phase field as outlined in [1, 2]. Following these formulations, we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles and, as an extension to [1, 2], consider their numerical implementations by an efficient h‐adaptive finite element method. A key problem of the phase field formulation is the mesh density, which is required for the resolution of the diffusive crack patterns. To this end, we embed the computational framework into an adaptive mesh refinement strategy that resolves the fracture process zones. We construct a configurational‐force‐based framework for h‐adaptive finite element discretizations of the gradient‐type diffusive fracture model. We develop a staggered computational scheme for the solution of the coupled balances in physical and material space. The balance in the material space is then used to set up indicators for the quality of the finite element mesh and accounts for a subsequent h‐type mesh refinement. The capability of the proposed method is demonstrated by means of a numerical example. (© 2010 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.