Exponential Finite Elements for a Phase Field Fracture Model

Sharp interface material models can be related to phase field models by introducing an order parameter, whose value is assigned to the different phases of a material. The elastic material law is coupled to the evolution equation of the order parameter and cracking is addressed as a phase transition problem instead of a moving boundary value problem. A regularization parameter ϵ controls the width of the diffuse cracks represented by the order parameter and the underlying sharp interface model can be recovered from the phase field model by the limit ϵ → 0. However, in numerical simulations using standard finite elements with linear shape functions, the minimum value of ϵ is restricted by the grid size and therefore the discretization of the crack field requires extensive mesh refinement for small values of ϵ. In this work, we construct special 2d shape functions which take into account the exponential character of the crack field and its dependence on the parameter ϵ. Especially in simulations with small values of ϵ and a rather coarse mesh, the elements with exponential shape functions perform significantly better than standard linear elements. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On an Energetic Interpretation of a Phase Field Model for Fracture

In the pioneering work by Griffith, it is assumed that a crack propagates, if this is energetically favorable. However, this original formulation requires a pre-existing initial crack. In order to bypass this deficiency of classical Griffith theory, Francfort and Marigo advocate a global variational criterion, where the total energy is minimized with respect to any admissible displacement field and crack set. Bourdin's regularized approximation of this variational formulation makes use of a continuous scalar field to indicate cracks. Based on this regularization a phase field fracture model is formulated. The crack field is assumed to follow a Ginzburg-Landau type evolution equation, and cracking is addressed as a phase transition problem. The coupled problem of mechanical balance equations and the evolution equation is solved using the finite element method combined with an implicit time integration scheme. The numerical solution naturally yields the crack evolution including crack propagation, kinking, branching and initiation without any additional criteria. In this work we study the driving mechanisms behind the crack evolution in the phase field fracture model and compare to the purely energetic considerations of the underlying variational formulation. (© 2011 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 Approach for Dynamic Fracture

In phase field fracture models cracks are indicated by the value of a scalar field variable which interpolates smoothly between broken and undamaged material. The evolution equation for this crack field is coupled to the mechanical field equations in order to model the mutual interaction between the crack evolution and mechanical quantities. In finite element simulations of crack growth at comparatively slow loading velocities, a quasi-static phase field model yields reasonable results. However, the simulation of fast loading or the nucleation of new cracks challenges the limits of such a formulation. Here, the quasi-static phase field model predicts brutal crack extension with an artificially high crack speed. In this work, we analyze to which extend a dynamic formulation of the mechanical part of the phase field model can overcome this paradox created by the quasi-static formulation. In finite element simulations, the impact of the dynamic effects is studied, and differences between the crack propagation behavior of the quasi-static model and the dynamic formulation are highlighted. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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

The 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 Approximation of Dynamic Brittle Fracture

The numerical assessment of fracture has gained importance in fields like the safety analysis of technical structures or the hydraulic fracturing process. The modelling technique discussed in this work is the phase field method which introduces an additional scalar field. The smooth phase field distinguishes broken from undamaged material and thus describes cracks in a continuum. The model consists of two coupled partial differential equations - the equation of motion including the constitutive behaviour of the material and a phase field evolution equation. The crack growth follows implicitly from the solution of this system of PDEs. The numerical solution with finite elements can be accelerated with an algorithm that performs computationally extensive tasks on a graphic processing unit (GPU). A numerical example illustrates the capability of the model to reproduce realistic features of dynamic brittle fracture. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heat Conduction in a Phase Field Model for Martensitic Transformation

We study the martensitic transformation with a phase field model, where we consider the Bain transformation path in a small strain setting. For the order parameter, interpolating between an austenitic parent phase and martensitic phases, we use a Ginzburg-Landau evolution equation, assuming a constant mobility. In [1], a temperature dependent separation potential is introduced. We use this potential to extend the model in [2], by considering a transient temperature field, where the temperature is introduced as an additional degree of freedom. This leads to a coupling of both the evolution equation of the order parameter and the mechanical field equations (in terms of thermal expansion) with the heat equation. The model is implemented in FEAP as a 4-node element with bi-linear shape functions. Numerical examples are given to illustrate the influence of the temperature on the evolution of the martensitic phase. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Model for Martensitic Transformations

The martensitic transformation is described using a phase field model which is in mathematical terms the regularization of a sharp interface approach. In this work, up to two martensitic orientation variants are considered. The evolution of microstructure is assumed to follow a time dependent Ginzburg-Landau equation. The coupled problem of the mechanical balance equation and the evolution equations is solved using finite elements and an implicit time integration scheme. In this work, the global energy evolution during the martensitic transformation and the influence of external loads on the formation of the different martensitic phases are studied in 2d. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Determination of a Parameter h(t) for a Phase Field Model

A phase field model with an unknown parameter h(t) is considered. The existence, uniqueness and continuous dependence upon the data of the solution (u,φ, h) are demonstrated.     

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)     

守恒相场模型弱解的Blow—up

本文讨论一类守恒相场模型弱解的性态,证明当ａ２ｐ－１＜０及初始数据充分大时解在有限时刻Ｂｌｏｗ－ｕｐ     

Isogeometric Analysis of a Phase Field Model for Darcy Flows with Discontinuous Data

The authors consider a phase field model for Darcy flows with discontinuous data in porous media; specifically,they adopt the Hele-Shaw-Cahn-Hillard equations of[Lee,Lowengrub,Goodman,Physics of Fluids,2002] to model flows in the Hele-Shaw cell through a phase field formulation which incorporates discontinuities of physical data,namely density and viscosity,across interfaces. For the spatial approximation of the problem,the authors use NURBS—based isogeometric analysis in the framework of the Galerkin method,a computational framework which is particularly advantageous for the solution of high order partial differential equations and phase field problems which exhibit sharp but smooth interfaces. In this paper,the authors verify through numerical tests the sharp interface limit of the phase field model which in fact leads to an internal discontinuity interface problem; finally,they show the efficiency of isogeometric analysis for the numerical approximation of the model by solving a benchmark problem,the so-called"rising bubble" problem.     

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)     

Simulation of Surface Wetting by Droplets Using a Phase Field Model

In the proposed phase field model a continuous order parameter indicates the phase distribution (liquid/gas). An energy density functional which is dependent on the surface tensions and defined by three contributions yields the total energy of the system. An equilibrium state is then computed by minimizing this energy of the system using an evolution equation. Details of the algorithmic implementation are discussed by illustrative examples. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of Size Effects in Ferroelectric Materials using a Phase Field Model

An electro-mechanically coupled phase field model for domain evolution in ferroelectric materials is presented. The inner length scale introduced by the model gives rise to size effects, especially in the context of the poling behavior of polycrystals. Such size effects are investigated by 2D numerical simulations for barium titanate polycrystals. Ferroelectric hysteresis curves and coercive fields are calculated for two different transition conditions for the order parameter at the grain boundaries. The results show that there is a significant size effect for the investigated polycrystal systems. (© 2015 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)     

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)     

Fracture Model for a Thin Layer of a Fibrous Composite

A model for a flat isolated layer of a unidirectional fibrous composite with a regular structure is constructed to investigate the possible variants of its failure development. An integrodifferential equation for determining the forces in fibers is obtained. Primary attention is focused on examining the failure process after the rupture of one fiber. This causes a drastic redistribution of stresses, which can lead to a failure of adjacent fibers owing to the increased load on them, to an interfacial shear fracture, and to the matrix cracking. It is shown that the development of layer failure is determined by the strength of fibers, the crack resistance of the matrix in axial tension and transverse shear, and also by the adhesion strength of the matrix-fiber interface. The sufficient conditions of applicability of the brittle fracture model are formulated.