Crack Phase Field Research Articles

A novel FFT-based phase field model for damage and cracking behavior of heterogeneous materials

A novel numerical method is developed for three-dimensional modeling of damage and cracking in heterogeneous rock-like materials. Two key issues are addressed. For the first issue, influences of materials heterogeneities such as pores and inclusions on damage evolution and cracking processes are investigated by a homogenization approach with Fast Fourier Transform technique. For the second issue, the nucleation and propagation of cracks from diffuse damage evolution are formulated in Fourier space and described by a phase-field method. To do this, an efficient numerical procedure is developed for the stress–strain relationships and crack phase field propagation. A new elastic degradation function is proposed in order to describe a large range of cracking processes. A range of heterogeneous materials with different microstructure are generated and performed numerically to study effects of pores and inclusions on the damage evolution and cracking process in heterogeneous materials.

Phase field modeling of fracture in Quasi-Brittle materials using natural neighbor Galerkin method

Recently developed phase field models of fracture require a diffusive crack representation based on an introduction of a crack phase field. We outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles. In this work, the numerical implementation of the phase field model for brittle fracture using natural neighbor Galerkin method is presented. Phase field method diffuses the sharp discontinuity by introducing a regularized crack functional. The notion of regularized crack functional provides a basis for defining suitable convex dissipation functions which govern the evolution of the crack phase field. The use of natural neighbors to dynamically decide the compact support at a nodal point makes it truly mesh free and nonlocal. The method uses smooth non-polynomial type Sibson interpolants, which are C0 at a given node and C∞ everywhere else. This allows to capture the interface region in a phase field model more accurately with only few nodes. For a given accuracy, there is an improvement in the computational time with the use of natural neighbor interpolants. Several benchmark problems were solved to investigate the efficiency of the numerical implementation of phase field model using natural neighbor Galerkin method.

A phase field based discrete fracture model (PFDFM) for fluid flow in fractured porous media

In this study we propose a novel phase field based discrete fracture model (PFDFM) to simulate fluid flow in fractured porous media. The common discrete fracture models represent fractures by sharp topology in an explicit way, regardless of using conforming or non-conforming mesh. Inspired by the definition of crack phase field, the sharp fracture topology is treated as a diffusive one in the solution of fluid flow problems, and the integration of fluid flow equation over fractures can be transformed to the one over the matrix. The algorithm to determine the fracture phase field and finite element discretization have been described in detail. The performance of the proposed method is validated against the classic discrete fracture model on several numerical cases in both two and three dimensions. We further investigate the convergency behavior of the proposed method through sensitivity analysis to mesh resolution and fracture parameters. Numerical results are obtained, which demonstrates that the proposed method is accurate, convergent and quite promising for simulating fluid flow in fractured porous media.

A phase-field model for fracture of unidirectional fiber-reinforced polymer matrix composites

This study presents a crack phase-field approach for anisotropic continua to model, in particular, fracture of fiber-reinforced matrix composites. Starting with the variational formulation of the multi-field problem of fracture in terms of the deformation and the crack phase fields, the governing equations feature the evolution of the anisotropic crack phase-field and the balance of linear momentum, presented for finite and small strains. A recently proposed energy-based anisotropic failure criterion is incorporated into the model with a constitutive threshold function regulating the crack initiation in regard to the matrix and the fibers in a superposed framework. Representative numerical examples are shown for the crack initiation and propagation in unidirectional fiber-reinforced polymer composites under Mode-I, Mode-II and mixed-mode bending. Model parameters are obtained by fitting to sets of experimental data. The associated finite element results are able to capture anisotropic crack initiation and growth in unidirectional fiber-reinforced composite laminates.

An adaptive global–local approach for phase-field modeling of anisotropic brittle fracture

This work addresses an efficient Global–Local approach supplemented with predictor–corrector adaptivity applied to anisotropic phase-field brittle fracture. The phase-field formulation is used to resolve the sharp crack surface topology on the anisotropic/non-uniform local state in the regularized concept. To resolve the crack phase-field by a given single preferred direction, second-order structural tensors are imposed to both the bulk and crack surface density functions Accordingly, a split in tension and compression modes in anisotropic materials is considered. A Global–Local formulation is proposed, in which the full displacement/phase-field problem is solved on a lower (local) scale, while dealing with a purely linear elastic problem on an upper (global) scale. Robin-type boundary conditions are introduced to relax the stiff local response at the global scale and enhancing its stabilization. Another important aspect of this contribution is the development of an adaptive Global–Local approach, where a predictor–corrector scheme is designed in which the local domains are dynamically updated during the computation. To cope with different finite element discretizations at the interface between the two nested scales, a non-matching dual mortar method is formulated. Hence, more regularity is achieved on the interface. Several numerical results substantiate our developments.

Phase-field modeling of porous-ductile fracture in non-linear thermo-elasto-plastic solids

Phase-field methods to regularize sharp interfaces represent a well established technique nowadays. In fracture mechanics, recent works have shown the capability of the method for brittle as well as ductile problems formulated within the fully non-linear regime. In this contribution, we introduce a framework to simulate porous-ductile fracture in isotropic thermo-elasto-plastic solids undergoing large deformations. Therefore, a modified Gurson–Tvergaard–Needleman GTN-type plasticity model is combined with a phase-field fracture approach to account for a temperature-dependent growth of voids on micro-scale followed by crack initiation and propagation on macro-scale. The multi-physical formulation is completed by the incorporation of an energy transfer into the thermal field such that the temperature distribution depends on the evolution of the plastic strain and the crack phase-field. Eventually, this physically comprehensive fracture formulation is validated by experimental data.

An efficient gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture

A reproducing kernel gradient smoothing meshfree formulation is proposed for the fourth-order phase field modeling of brittle fracture. In order to circumvent the complexity and lower efficiency of meshfree gradient computation, a reproducing kernel gradient smoothing formulation is presented with particular reference to quadratic basis functions. Both first- and second-order meshfree smoothed gradients are discussed, in which the first-order smoothed gradients are expressed as a reproducing kernel form and meet the quadratic consistency conditions of Galerkin weak form, and the second-order smoothed gradients are then computed through performing a direct differentiation on the first-order smoothed gradients. It turns out that the resulting second-order smoothed gradients satisfy the standard gradient reproducing conditions of meshfree approximations. Subsequently, the smoothed gradients of meshfree shape functions are employed to discretize the Galerkin weak forms of equilibrium and crack phase field equations, where the numerical integration and smoothed gradient construction are specially emphasized. The tensile–compressive strain decomposition is adopted to prevent compressive crack evolution. Numerical results demonstrate the effectiveness of the proposed gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture.

Phase field modeling of crack growth with double-well potential including surface effects

A thermodynamically consistent phase field approach for fracture including surface stresses is presented. The surface stresses which are distributed inside a finite width layer at the crack surface are introduced as a result of employing some geometrical nonlinearities, i.e., by defining energy terms per unit volume at the current time and evaluating gradient of the order parameter in the current configuration. A double-well barrier term is included in the structure of the Helmholtz free energy which allows one to use the energy terms per unit volume at the current time when introducing the surface stresses in the current approach. Thus, the surface stresses are introduced in a similar way to the interfacial stresses in phase transformations. The differences in the modeling of crack growth with considering the surface stresses and without it are discussed. It is shown how the surface stresses affect the stress fields and consequently the crack nucleation and propagation. The finite element method is utilized to solve the coupled equations of mechanics and crack phase field. It is emphasized that the surface stresses affect the driving force for both the crack nucleation and propagation by disturbing the momentum balance. Thus, a different external loading is required in the presence of the surface stresses.

A three-dimensional fracture pattern diagram of staggered platelet structures

In order to design composites that mimic the remarkable balance of properties such as strength, toughness, and stiffness of staggered platelet structures in nature, it is crucial to understand their load transfer and failure mechanisms. Recently, we proposed an analytical model to predict the stress distribution within staggered platelet structures for a wide range of constituent materials’ moduli and geometric parameters in the elastic response regime. Here, based on the model, a fracture pattern diagram featuring three distinct mechanisms categorized according to the failure sequences of soft tip, soft shear zone, and hard platelet is constructed. The proposed fracture map is capable of capturing the transition of failure mechanisms observed in crack phase field simulations and draws parallels to mechanisms seen in experiments. Our study sheds light on the origin of failure mechanism transitions and enables rational designs of future staggered platelet composites with unprecedented properties.

A finite-strain phase-field approach to ductile failure of frictional materials

This work presents a modeling framework for the ductile failure of frictional materials undergoing large deformations with a focus on soil mechanics. Crack formation and propagation in soil can be modeled in a convenient way by the recently developed continuum phase-field approach to fracture. Within this approach sharp crack discontinuities are regularized. It allows the use of standard discretization methods for crack discontinuities and is able to account for complex crack paths. In the present contribution, we develop a computational modeling framework for the phase-field approach to ductile fracture in frictional materials. It combines a non-associative Drucker–Prager-type elastic-plastic constitutive model with an evolution equation for the crack phase field in terms of an elastic-plastic energy density. An important aspect is the development of an isotropic hardening mechanism that accounts for both friction and cohesion hardening. In order to guarantee a locking- and hourglass-free response, a modified enhanced element formulation, namely the consistent-gradient formulation, is employed as a key feature of the finite-element implementation. The performance of the formulation is demonstrated by means of representative numerical examples that describe soil crack formation rooted in elastic-plastic fracture mechanics.

Modelling progressive failure in multi-phase materials using a phase field method

In this paper, a new phase field method is proposed for modelling of progressive failure in multi-phase materials. Material properties of the interface between inclusion and matrix are regularized by an auxiliary interface phase field. In addition, crack initiation and propagation are simulated by using another crack phase field. Different failure mechanisms such as interface debonding, matrix cracking and the interaction between these two failure mechanisms are modelled in a unified framework. For general application of the framework, an image processing method is employed to identify the individual phases for a given multi-phase material. The proposed method is implemented into the commercial software ABAQUS through a user subroutine UEL (user defined element). The derived method is validated through an example of a single fiber reinforced composite system. Moreover, a procedure for choosing parameters of the proposed phase field model is discussed. Further, the validated method is applied to fracture analysis of a multi-phase concrete structure and complex failure mechanisms within and across the phases are captured.

An iteration scheme for phase field model for cohesive fracture and its implementation in Abaqus

Recently, phase field modelling of fracture has gained much attentions because of its benefits in predicting cracks initiation and propagation without any ad-hoc criterion. In the model cracks are smeared into the surrounding areas by introducing an auxiliary crack phase field which should be solved either by iteration or using sufficiently small loading increment, and this makes the method be computationally demanding. The introduction of other complex models e.g. cohesive zone model results in nonlinear damage sub-problems which may further increase the number of iteration steps. In this study, an improved staggered iteration scheme by which the nonlinear sub-problems are solved by a one-pass procedure is proposed. Such that the computational costs can be reduced and the stability of iteration can be improved. In particular, a unified phase field model which integrates varies cohesive relationships is investigated. The phase field model as well as the iteration scheme are implemented through Abaqus subroutine UEL (user defined element). The present study is validated through comparing both of crack propagation path and load-displacement curve with existing results. Moreover, comparisons on total iteration steps and the elapsed total computational time are made between the proposed iteration scheme and the existing scheme, and the results demonstrate that the proposed iteration scheme is very suitable for the considered problem.

Phase field modeling of crack propagation under combined shear and tensile loading with hybrid formulation

The crack phase field model has been well established and validated for a variety of complex crack propagation patterns within a homogeneous medium under either tensile or shear loading. However, relatively less attention has been paid to crack propagation under combined tensile and shear loading or crack propagation within composite materials made of two constituents with very different elastic moduli. In this work, we compare crack propagation under such circumstances modelled by two representative formulations, anisotropic and hybrid formulations, which have distinct stiffness degradation schemes upon crack propagation. We demonstrate that the hybrid formulation is more adequate for modeling crack propagation problems under combined loading because the residual stiffness of the damaged zone in the anisotropic formulation may lead to spurious crack growth and altered load–displacement response.

Thermodynamically consistent and scale-dependent phase field approach for crack propagation allowing for surface stresses

A thermodynamically-consistent phase field approach for crack propagation which includes the following novel features is presented. (1) Scale dependency was included by relating the length scale to the number of cohesive interatomic planes at the crack tip. Because of this, the developed theory is applicable from the atomistic to the macroscopic scales. (2) The surface stresses (tension) are introduced by employing some geometrical nonlinearities even in small strain theory. They produce multiple contributions to the Ginzburg-Landau equation for crack propagation. (3) Crack propagation in the region with compressive closing stresses is eliminated by employing a stress-state-dependent kinetic coefficient in the Ginzburg-Landau equation. (4) The importance of analysis of the thermodynamic potential in terms of stress-strain curves is shown. The developed theory includes a broad spectrum of the shapes of stress-strain relationships. The finite element method is utilized to solve the complete system of crack phase field and mechanics equations. The effect of the above novel features is analyzed numerically for various model problems.

On the phase field modeling of crack growth and analytical treatment on the parameters

A thermodynamically consistent phase field model for crack propagation is analyzed. The thermodynamic driving force for the crack propagation is derived based on the laws of thermodynamics. The Helmholtz free energy satisfies the thermodynamic equilibrium and instability conditions for the crack propagation. Analytical solutions for the Ginzburg–Landau equation including the surface profile and the estimation of the kinetic coefficient are found. It is shown how kinetic coefficient affects the local stress field. The local critical stress for the crack propagation is calibrated with the theoretical strength which gives the value of the crack surface width. The finite element method is utilized to solve the complete system of crack phase field and mechanics equations. The analytical expressions for the crack surface profile and the crack tip velocity are verified with the numerical solutions. It is shown that under mode I cracking and proper calibration of parameters, phase field models always agree with Griffith’s theory.

Robust numerical implementation of non-standard phase-field damage models for failure in solids

Recently several non-standard phase-field models different from the standard one for brittle fracture (Bourdin et al. 2000, 2008; Miehe et al. 2010a) have been proposed for the modeling of damage and fracture in solids. On the one hand, linear elastic behavior before the onset of damage and even cohesive cracking induced quasi-brittle failure can be considered by such non-standard phase-field models. On the other hand, they present great challenges to the numerical implementation since the damage boundedness is no longer automatically fulfilled. In this work, the numerical implementation of the unified phase-field damage model (Wu 2017, 2018) is addressed, though it also applies to the standard and other non-standard ones. In particular, an iterative alternate minimization (AM) algorithm enhanced with path-following strategies is presented. The phase-field or damage sub-problem is solved by the bound-constrained optimization solver in which the boundedness and irreversibility conditions of the crack phase-field are exactly enforced. Moreover, material softening induced snap-backs typically for localized failure in solids can also be effectively dealt with. The equilibrium paths in terms of the fracture surface and of the indirect displacement (e.g., the crack mouth opening or sliding displacement) are discussed in the context of the AM algorithm. The AM algorithm with the indirect displacement control is advocated, since both prescribed external forces and displacements can be naturally dealt with, which is more versatile than the one with the fracture surface control applicable only for prescribed displacements. Representative examples of benchmark tests show that the AM algorithm enhanced with the indirect displacement control is extremely robust even for large increment sizes. The insensitivity of the unified phase-field damage model to the incorporated length scale and mesh size is also confirmed.

Modelling of environmentally assisted material degradation in the crack phase-field framework

The simulation of crack propagation was conducted with a diffusive crack model in a variational framework. More, the physically sound introduction of mass transport and coupling mechanisms due to environmentally assisted effects could be realised in this framework. The objective consists of the application of the phase-field approach towards the simulation of environmentally assisted material degradation with the advantage of a non-required predefined crack path and a mesh-independent non-local formulation that facilitates the damage evolution with respect to material softening. The sharp crack is regularised by the introduction of a phase-field order parameter leading to a diffusive crack formulation. Besides the equations originating from the linear momentum balance an additional evolution equation for the crack phase-field is introduced. Furthermore, mass transport is simulated by a diffusion equation. The description delivered by the variational phase-field framework is able to simulate crack propagation according to published numerical test cases. Additionally, the calculation of stress intensity factors is possible as well as crack resistance curves that describe stable crack propagation.

Area of lineal-path function for describing the pore microstructures of cement paste and their relations to the mechanical properties simulated from μ-CT microstructures

The pore distribution of a cement paste strongly affects its mechanical behavior such as its stiffness and strength. Porosity is an influential parameter that can be used to identify the complex pore microstructures of cement paste, but it has limitations as a scalar parameter. In this study, the lineal-path function, a low-order probability function, is investigated as a supplement or an alternative parameter for describing the microstructural characteristics of cement paste microstructures. In particular, the area of the lineal-path function is used as a measure of the pore microstructural characteristics, which can be linked with its properties.A relatively new method for simulating crack propagation, the crack phase field model, is used to evaluate the stiffness and tensile strength of cement paste microstructures and the evaluated properties are linked to the proposed characterization parameters. The evaluation is performed on virtual specimens obtained from micro-level computerized tomography (μ-CT) images of real cement paste specimens. The validity of the microstructure-property relations obtained from the proposed characterization parameters and the crack phase field model are confirmed through the statistical analysis of dozens of specimens.It is concluded that the correlation between the area of the lineal-path function and the mechanical properties is very strong. The parameter could potentially be used as a supplementary or an alternative parameter to describe the pore microstructures of cement paste.

A modified Gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling

The modeling of failure in ductile materials must account for complex phenomena at the micro-scale, such as nucleation, growth and coalescence of micro-voids, as well as the final rupture at the macro-scale, as rooted in the work of Gurson (J Eng Mater Technol 99:2–15, 1977). Within a top–down viewpoint, this can be achieved by the combination of a micro-structure-informed elastic–plastic model for a porous medium with a concept for the modeling of macroscopic crack discontinuities. The modeling of macroscopic cracks can be achieved in a convenient way by recently developed continuum phase field approaches to fracture, which are based on the regularization of sharp crack discontinuities, see Miehe et al. (Comput Methods Appl Mech Eng 294:486–522, 2015). This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns. In this work, we develop a new theoretical and computational framework for the phase field modeling of ductile fracture in conventional elastic–plastic solids under finite strain deformation. It combines modified structures of Gurson–Tvergaard–Needelman GTN-type plasticity model outlined in Tvergaard and Needleman (Acta Metall 32:157–169, 1984) and Nahshon and Hutchinson (Eur J Mech A Solids 27:1–17, 2008) with a new evolution equation for the crack phase field. An important aspect of this work is the development of a robust Explicit–Implicit numerical integration scheme for the highly nonlinear rate equations of the enhanced GTN model, resulting with a low computational cost strategy. The performance of the formulation is underlined by means of some representative examples, including the development of the experimentally observed cup–cone failure mechanism.

Phase field modeling of fracture in anisotropic brittle solids

A phase field model of fracture that accounts for anisotropic material behavior at small and large deformations is outlined within this work. Most existing fracture phase field models assume crack evolution within isotropic solids, which is not a meaningful assumption for many natural as well as engineered materials that exhibit orientation-dependent behavior. The incorporation of anisotropy into fracture phase field models is for example necessary to properly describe the typical sawtooth crack patterns in strongly anisotropic materials. In the present contribution, anisotropy is incorporated in fracture phase field models in several ways: (i) Within a pure geometrical approach, the crack surface density function is adopted by a rigorous application of the theory of tensor invariants leading to the definition of structural tensors of second and fourth order. In this work we employ structural tensors to describe transverse isotropy, orthotropy and cubic anisotropy. Latter makes the incorporation of second gradients of the crack phase field necessary, which is treated within the finite element context by a nonconforming Morley triangle. Practically, such a geometric approach manifests itself in the definition of anisotropic effective fracture length scales. (ii) By use of structural tensors, energetic and stress-like failure criteria are modified to account for inherent anisotropies. These failure criteria influence the crack driving force, which enters the crack phase field evolution equation and allows to set up a modular structure. We demonstrate the performance of the proposed anisotropic fracture phase field model by means of representative numerical examples at small and large deformations.