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)     

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)     

Thermo-mechanical modeling of turbine blades taking into account structural defects

In turbine blades of aero-engines typical defects are cracks due to high mechanical and thermal loads. The extended finite element method (XFEM) is used for simulations of fracture mechanics problems with cracks. Discontinuities in the displacement and temperature field are allowed and the crack opening displacement and crack tip stress field are reproduced accurately. Since crack closure and non-physical penetration of the crack surfaces may occur under certain load conditions, it becomes necessary to enforce the non-penetration condition for crack surfaces. This contact formulation is assumed to be frictionless. The node-to-segment approach proposed in [3] is extended to ten-node tetrahedral elements with quadratic shape functions. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Adaptive Finite Element Approach for Brittle Fracture

The extension of the finite element method to take discrete fracture and failure modes into account is a current field of research. In recent times, first results in terms of cohesive element formulations have been introduced into commercial applications. Such element formulations are able to cover the discrete behaviour of interfaces between different materials or the mechanical processes of thin layers. These approaches are not suitable for simulations with unknown crack paths in homogeneous materials, due to the initial elastic phase of the material formulation and the necessity to define potential crack paths a priori. The presented strategy starts with an unextended model and modifies the structure during the computations in terms of an adaptive procedure. The idea is to generate additional elements, based on the cohesive element formulation, to approximate arbitrary crack paths. For this purpose, a failure criterion is introduced. For nodes where the limiting value is reached, cohesive elements are introduced between the volume element boundaries of accordingly facets and corresponding nodes are duplicated. Necessary modifications for this application on system level as well as the element and the material formulation are introduced. By means of some numerical examples, the functionality of the presented procedure is demonstrated. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A strong discontinuity based adaptive refinement approach for the modeling of crack branching

In the current work, the physical phenomena of dynamic fracture of brittle materials involving crack growth, acceleration and consequent branching is simulated. The numerical modeling is based on the approach where the failure in the form of cracks or shear bands is modeled by a jump in the displacement field, the so called ‘strong discontinuity’. The finite element method is employed with this strong discontinuity approach where each finite element is capable of developing a strong discontinuity locally embedded into it. The focus in this work is on branching phenomena which is modeled by an adaptive refinement method by solving a new sub-boundary value problem represented by a finite element at the growing crack tip. The sub-boundary value problem is subjected to a certain kinematic constraint on the boundary in the form of a linear deformation constraint. An accurate resolution of the state of material at the branching crack tip is achieved which results in realistic dynamic fracture simulations. A comparison of resulting numerical simulations is provided with the experiment of dynamic fracture from the literature. (© 2011 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)     

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)     

Numerical and experimental investigations of phase segragation effects in binary brazing solders

Solder materials occupy many of fields for technical application (e.g. solder joints in automotive control units or in microelectronic packages). They are required to provide electrical and mechanical connections between different components. Due to their wide range of applications solder alloys are subject to a great variety of microstructural changes such as phase separation and coarsening processes. The micromorphological variations influence strength and life expectation of solder materials, in particular, in very small components such as solder joints in microelectronic packages. In order to analyze the microstructural evolution with a diffusion theory of heterogeneous solid mixtures we employ an extended Cahn-Hilliard phase field model. The diffusion equation under consideration constitutes a partial differential equation involving spatial derivatives of fourth order. Thus, the variational formulation of the problem requires approximation functions which are piecewise smooth and globally C1-continuous. In our contribution we fulfil the continuity requirement by means of rational B-spline finite element basis functions. To illustrate the versatility of this approach numerical simulations of phase decomposition and coarsening controlled by diffusion and by mechanical loading are discussed and compared with experimental results. (© 2010 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)     

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)     

Phase field simulation of thermomechanical fracture

In Francfort and Marigo's variational free-discontinuity formulation of brittle fracture [1] cracking is regarded as an energy minimization process, where the total energy is minimized with respect to any admissible crack set and displacement field. No additional criterion is needed to determine crack paths, branching of cracks and crack initiations. However, a direct discretization of the model is faced with significant technical problems, as it involves minimizations in a set of possibly discontinuous functions. A regularized version of the model has been introduced by Bourdin [2] and based on this, we use the concept of a continuum phase field model to simulate cracking processes. Cracks are indicated by the order parameter of the phase field model and cracking can be regarded as a phase transition problem. Additionally, introducing the heat equation into the model, it is capable to also take account of crack propagation due to thermal stresses. In the numerical implementation, crack parameter as well as temperature are treated as additional degrees of freedom and the coupled field equations are solved using the finite element method together with an implicit time integration scheme. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

爆炸和冲击载荷下金属材料及结构的动态失效仿真

通过数值模拟研究爆炸冲击载荷下金属材料和结构的动态失效规律,对表征爆炸冲击毁伤效应及设计新型抗冲击结构有重要意义.强动载下金属材料失效涉及材料大变形、热力耦合、材料状态变化等多个复杂物理过程,给数值仿真带来了极大挑战,其中包括裂纹、剪切带等复杂失效模式的几何描述、动态失效准则的确定、塑性与损伤耦合演化的描述等问题.针对这些挑战性问题,基于能量变分建立描述金属动态失效过程的热弹塑性相场理论和计算模型,实现了断裂与剪切带失效模式的统一描述,并提出了其显式有限元高效求解策略.进一步将该模型应用于爆炸冲击载荷下金属脆韧失效模式转变、绝热剪切带(ASBs)自组织及冲击波作用下薄壁圆盘失效形式转变三个典型金属动态失效问题,验证了理论模型的准确性及计算模型的稳健性.该工作为后续开展基于仿真的爆炸冲击毁伤评估及防护结构设计研究奠定了基础.     

Fatigue crack growth simulations of homogeneous and bi-material interfacial cracks using element free Galerkin method

In this paper, quasi-static fatigue crack growth simulations of homogeneous and bi-material interfacial cracks have been performed using element free Galerkin method (EFGM) under mechanical as well as thermo-elastic load. The thermo-elastic fracture problem is decoupled into thermal and elastic problems. The temperature distribution obtained by solving heat conduction equation is used as input in the elastic problem to get the displacement and stress fields. Discontinuities in the temperature and displacement fields are captured by extrinsic partition of unity enrichment technique. The values of stress intensity factors have been extracted from the EFGM solution by domain based interaction integral approach. The standard Paris fatigue crack growth law has been implemented for the life estimation of various model problems. The results obtained by EFGM under mechanical and thermo-elastic loads were compared with those obtained by FEM using remeshing approach.     

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

We 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)     

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)     

A conductive crack and a remote electrode at the interface between two piezoelectric materials

An interaction of a tunnel conductive crack and a distant strip electrode situated at the interface between two piezoelectric semi-infinite spaces is studied. The bimaterial is subject by an in-plane electrical field parallel to the interface and by an anti-plane mechanical loading. Using the presentations of electromechanical quantities at the interface via sectionally-analytic functions the problem is reduced to a combined Dirichlet-Riemann boundary value problem. Solution of this problem is found in an analytical form excepting some one-dimensional integrals calculations. Closed form expressions for the stress, the electric field and their intensity factors, as well as for the crack faces displacement jump are derived. On the base of these presentations the energy release rate is also found. The obtained solution is compared with simple particular case of a single crack without electrode and the excellent agreement is found out. An auxiliary plane problem for open and closed cracks between two isotropic materials is also considered. The mathematical model of this problem is identical to the above one, therefore, the obtained solution is used for this model. It is compared with finite element solution of a similar problem and good agreement is found out.     

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)     

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)     

带复合型摩擦裂纹的圆盘动态断裂实验有限元分析

为验证考虑裂纹面接触和动态荷载时,中心裂纹巴西圆盘（CCBD）试件用于分离式Hopkinson压杆（SHPB）系统中测量脆性材料复合型动态断裂韧度的可行性,以及研究裂纹面接触对动态断裂韧度实验结果的影响.通过有限元法建立SHPB CCBD三维有限元模型,计算了不同加载条件下CCBD试件的动态应力强度因子（DSIF）.结果表明:在实验中,将考虑裂纹面接触的应力强度因子(SIF)准静态公式推广为动态公式,需要判定断裂时间是否达到应力平衡的时间条件;压剪复合型加载时,裂纹面接触导致裂纹面应力变化,会对Ⅱ型裂纹的DSIF产生显著影响,不考虑裂纹面接触的影响将会导致Ⅱ型DSIF的测试值偏大.