Phase Field Modeling of Brittle and Ductile Fracture

The phase field modeling of brittle fracture was a topic of intense research in the last few years and is now well-established. We refer to the work [1-3], where a thermodynamically consistent framework was developed. The main advantage is that the phase-field-type diffusive crack approach is a smooth continuum formulation which avoids the modeling of discontinuities and can be implemented in a straightforward manner by multi-field finite element methods. Therefore complex crack patterns including branching can be resolved easily. In this paper, we extend the recently outlined phase field model of brittle crack propagation [1-3] towards the analysis of ductile fracture in elastic-plastic solids. In particular, we propose a formulation that is able to predict the brittle-to-ductile failure mode transition under dynamic loading that was first observed in experiments by Kalthoff and Winkler [4]. To this end, we outline a new thermodynamically consistent framework for phase field models of crack propagation in ductile elastic-plastic solids under dynamic loading, develop an incremental variational principle and consider its robust numerical implementation by a multi-field finite element method. The performance of the proposed phase field formulation of fracture is demonstrated by means of the numerical simulation of the classical Kalthoff-Winkler experiment that shows the dynamic failure mode transition. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Model for Ductile to Brittle Failure Mode Transition

The 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)     

Ductile failure with gradient plasticity coupled to the phase-field fracture at finite strains

Most metals fail in a ductile fashion, i.e, fracture is preceded by significant plastic deformation. The modeling of failure in ductile metals must account for complex phenomena at micro-scale, such as nucleation, growth and coalescence of micro-voids. In this work, we start with von-Mises plasticity model without considering void generation. The modeling of macroscopic cracks can be achieved in a convenient way by the continuum phase field approaches to fracture, which are based on the regularization of sharp crack discontinuities [1]. This avoids the use of complex discretization methods for crack discontinuities and can account for complex crack patterns. The key aspect of this work is the extension of the energetic and the stress-based phase field driving force function in brittle fracture to account for a coupled elasto-plastic response in line with our recent work [3]. We develop a new theoretical and computational framework for the phase field modeling of ductile fracture in elastic-plastic solids. To account for large strains, the constitutive model is constructed in the logarithmic strain space, which simplify the model equations and results in a formulation similar to small strains. We demonstrate the modeling capabilities and algorithmic performance of the proposed formulation by representative simulations of ductile failure mechanisms in metals. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Towards Phase Field Modeling of Ductile Fracture in Gradient-Extended Elastic-Plastic Solids

The modeling of failure in ductile metals must account for complex phenomena at a micro-scale as well as the final rupture at the macro-scale. Within a top-down viewpoint, this can be achieved by the combination of a micro-structure-informed elastic-plastic model with a concept for the modeling of macroscopic crack discontinuities. In this context, it is important to account for material length scales and thermo-mechanical coupling effects due to dissipative heating. This can be achieved by the construction of non-standard, gradient-enhanced models of plasticity with a full embedding into continuum thermodynamics [1,2]. The modeling of macroscopic cracks can be achieved in a convenient way by recently developed continuum phase field approaches to fracture based on regularized crack discontinuities. This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns within a pure continuum formulation. Moreover, the phase field modeling of fracture is related to gradient theories of continuum damage mechanics, and fits nicely the structure of constitutive models for gradient plasticity. The main focus of this work is the extensions to gradient thermoplasticity and phase field formulation of ductile fracture, conceptually in line with the work [3]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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

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 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 Phase-Field Model of Ductile Fracture at Finite Strains

This work outlines a variational-based framework for the phase field modeling of ductile fracture in elastic-plastic solids at large strains. The phase field approach regularizes sharp crack discontinuities within a pure continuum setting by a specific gradient damage model with geometric features rooted in fracture mechanics. Based on the recent works [1, 2], the phase field model of ductile fracture is linked to a formulation of gradient plasticity at finite strains in order to ensure the crack to evolve inside the plastic zones. The thermodynamic formulation is based on the definition of a constitutive work density function including the stored elastic energy and the dissipated work due to plasticity and fracture. The proposed canonical theory is shown to be governed by a rate-type minimization principle, which determines the coupled multi-field evolution problem. Another aspect is the regularization towards a micromorphic gradient plasticity-damage setting which enhances the robustness of the finite element formulation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase Field Modeling of Fracture in Plates and Shells

The 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 fracture

The variational formulation of brittle fracture as formulated for example by Francfort and Marigo in [1], where the total energy is minimized with respect to any admissible crack set and displacement field, allows the identification of crack paths, branching of preexisting cracks and even crack initiation without additional criteria. For its numerical treatment a continuous approximation of the model in the sense of Γ-convergence has been presented by Bourdin in [2]. In the regularized Francfort–Marigo model cracks are represented by an additional field variable (secondary variable) s∈[0,1] which is 0 if the material is cracked and 1 if it is undamaged. In this work, we reinterpret the crack variable as a phase field order parameter and address cracking as a phase transition problem. The crack growth is governed by the evolution equation of the order parameter which resembles the Ginzburg–Landau equation. The numerical treatment is done by finite elements combined with an implicit Euler scheme for the time integration. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of a Phase Field Model for Elasto-plastic Fracture

Phase field modelling of brittle fracture is very well understood today. However, the attempts of investigation of elasto-plastic fracture by the phase field approach are limited. This contribution deals with the investigation of a phase field model for elasto-plastic fracture. Based on a free energy density comprising elastic, fracture and plastic contributions, the model describes an extension of the linear elastic model towards von Mises plasticity. In this work it is analyzed numerically to which extend analytical findings concerning the interpretation of the model parameters in 1D are transferable to 2D scenarios. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A projection method for phase field models in micromagnetics

Magnetic materials have been finding increasingly wider areas of application in industry and therefore, as indicated by the reviews [1], [2] and [3], there is an increased interest in the efficient modeling of such materials that have an inherent coupling between the magnetic and mechanical characteristics. A particular challenge in the modeling of such materials is the algorithmic preservation of the geometric constraint on the magnetization field, that remains constant in magnitude [4]. In earlier works, [5] and [6], we presented a phase field model within a geometrically exact incremental variational framework where the geometric property of the magnetization director is exactly preserved pointwise by nonlinear rotational updates at the nodes. In the current work however, we present an alternative approach that involves an operator split along with a projection step for the magnetization vector. This method provides significant advantages in terms of speed and ease of implementation at the cost of the maximum time step size used. The current work therefore presents comparative study of the the two methods. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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

The 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)     

Phase-Field Modeling of Hydraulic Fracture

Hydraulically driven fracture has gained more and more research activity in the last few years, especially due to the growing interest of the petroleum industry. Key challenge for a powerful simulation of this scenario is an effective modeling and numerical implementation of the behavior of the solid skeleton and the fluid phase, the mechanical coupling between the two phases as well as the incorporation of the fracture process. Existing models for hydraulic fracturing can be found for example in [1], where the crack path is predetermined, or in [2] who use a phase field fracture model in an elastic framework, however without incorporating the fluid flow. In this work we propose a new compact model structure for the Biot-type fluid transport in porous media at finite strains based on only two constitutive functions, that is the free energy function ψ and a dissipation potential ϕ that includes the incorporation of an additional Poiseuille-type fluid flow in cracks. This formulation is coupled to a phase field approach for fracture and is fully variational in nature, as shown in [3]. In contrast to formulations with a sharp-crack discontinuity, the proposed regularized approach has the main advantage of a straight-forward modeling of complex crack patterns including branching. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed Variational Principles and Robust Finite Element Design of Gradient Plasticity at Finite Strains

The modeling of size effects in elastic-plastic solids, such as the width of shear bands or the grain size dependence in polycrystals, must be based on non-standard theories which incorporate length-scales. This is achieved by models of strain gradient plasticity, incorporating spatial gradients of selected micro-structural fields which describe the evolving dissipative mechanisms. The key aspect of this work is to provide a rigorous incremental variational formulation and mixed finite element design of additive finite gradient plasticity in the logarithmic strain space. We start from a mixed saddle point principle for metric-type plasticity, which is specified for the important model problem of isochoric plasticity with gradient-extended hardening/softening response. To this end, we propose a novel finite element design of the coupled problem incorporating a local-global solution strategy of short- and long-range fields. This includes several new aspects, such as extended Q1P0-type and MINI-type finite elements for gradient plasticity [4]. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermoviscoplasticity of Glassy Polymers in the Logarithmic Strain Space Based on the Free Volume Theory

In this contribution a new constitutive model of finite thermo-visco-plastic behavior of amorphous glassy polymers and details of its numerical implementation are outlined. In contrast to existing kinematical approaches to finite plasticity of glassy polymers, the formulation applies a plastic metric theory based on an additive split of Lagrangian Hencky-type strains into elastic and plastic parts [1, 3]. The characteristic strain hardening of the model is derived from a polymer network model, the thermo-visco-plastic flow rule in the logarithmic strain space uses structures of the free volume flow theory [4]. The integration of this micromechanically motivated approach in a three-dimensional computational model is the key novel aspect of this work. An important aspect of this work is the model validation based on experimental findings, whereas the excellent performance of the proposed formulation is demonstrated by means of numerical examples. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled Finite Thermoviscoplasticity of Glassy Polymers

The contribution is concerned with the constitutive modeling of the temperature and thermomechanical coupling effects on the rate–dependent finite plastic behavior of glassy polymers. In contrast to the existing kinematic approaches to the finite plasticity of glassy polymers, we propose a distinct kinematic framework constructed in the logarithmic strain space [5]. The logarithmic framework is extremely attractive due to the fact that it allows a very efficient algorithmic treatment of finite plasticity akin to the geometrically linear theory. The evolution law of plastic strains is adopted from Argon's double kink theory [1, 3]. Temperature–induced softening is incorporated by thermal disassociation of the secondary bonds in the polymer network [6, 2]. In the FE analysis of the coupled BVPs, the staggered scheme is employed [7]. The proposed formulation is validated by simulating various experimental data. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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

The 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)     

Plastic deformation of rough surfaces

Friction forces are only transferred within the the real area of contact A_real, which is usually smaller than the apparent area of contact A_o. The maximum friction stress τ_fric is therefore determined by the shear limit τ_max in the area of real contact and the fraction of the real area of contact (τ_fric = τ_max (A_real/A_o)). For rough surfaces the size of A_real is governed o by the plastic deformation of the surface roughness. We present a fully elasto-plastic halfspace contact formulation based on the work of Jacq et al. [1]. Linear elastic-plastic material behavior is modeled based on v. Mises plasticity with isotropic hardening. The algorithm gives the residual stress as well as the full plastic deformation field due to a frictionless normal contact. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonfocusing Instabilities in Coupled, Integrable Nonlinear Schrödinger pdes

nonfocusing instabilities that exist independently of the well-known modulational instability of the focusing NLS equation. The focusing versus defocusing behavior of scalar NLS fields is a well-known model for the corresponding behavior of pulse transmission in optical fibers in the anomalous (focusing) versus normal (defocusing) dispersion regime [19], [20]. For fibers with birefringence (induced by an asymmetry in the cross section), the scalar NLS fields for two orthogonal polarization modes couple nonlinearly [26]. Experiments by Rothenberg [32], [33] have demonstrated a new type of modulational instability in a birefringent normal dispersion fiber, and he proposes this cross-phase coupling instability as a mechanism for the generation of ultrafast, terahertz optical oscillations. In this paper the nonfocusing plane wave instability in an integrable coupled nonlinear Schr?dinger (CNLS) partial differential equation system is contrasted with the focusing instability from two perspectives: traditional linearized stability analysis and integrable methods based on periodic inverse spectral theory. The latter approach is a crucial first step toward a nonlinear , nonlocal understanding of this new optical instability analogous to that developed for the focusing modulational instability of the sine-Gordon equations by Ercolani, Forest, and McLaughlin [13], [14], [15], [17] and the scalar NLS equation by Tracy, Chen, and Lee [36], [37], Forest and Lee [18], and McLaughlin, Li, and Overman [23], [24]. Received February 9, 1999; accepted June 28, 1999     

Global existence in rate-independent infinitesimal strain gradient plasticity

We propose a model of infinitesimal strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. The model is a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the non-symmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied. Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: ṗ.τ = 0 on the microscopically hard boundary Γ_D ⊂ ∂Ω and [Curl p ].τ = 0 on the microscopically free boundary ∂Ω\Γ_D, where τ are the tangential vectors at the boundary ∂Ω. Moreover, a weak reformulation of the infinitesimal model allows for a global in-time solution of the corresponding rate-independent initial boundary value problem. The method of choice are a formulation as a quasi-variational inequality with symmetric and coercive bilinear form. Use is made of new Hilbert-space suitable for dislocation density dependent plasticity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)