Phase Field Modeling Of Fracture Research Articles

A micromorphic phase-field model for brittle and quasi-brittle fracture

In this manuscript, a robust and variationally consistent technique is proposed for local treatment of the phase-field fracture irreversibility. This technique involves an extension of the phase-field fracture energy functional through a micromorphic approach. Consequently, the phase-field is transformed into a local variable, while a micromorphic variable regularizes the problem. The local nature of the phase-field variable enables an easier implementation of its irreversibility using a pointwise ‘max’ with system level precision. Unlike the popular history variable approach, which also enforces local fracture irreversibility, the micromorphic approach yields a variationally consistent framework. The efficacy of the micromorphic approach in phase-field fracture modelling is demonstrated in this work with numerical experiments on benchmark brittle and quasi-brittle fracture problems in linear elastic media. Furthermore, the extensibility of the micromorphic phase-field fracture model towards multiphysics problems is demonstrated. To that end, a theoretical extension is carried out for modelling hydraulic fracture, and relevant numerical experiments exhibiting crack merging are presented. The source code as well as the data set accompanying this work would be made available on GitHub (https://github.com/ritukeshbharali/falcon).

A modified damage and fracture phase field model considering heterogeneity for rock‐like materials

AbstractDamage and fracture are the most extensive failure modes of rock materials, which may easily induce disaster and instability of engineering structures. This study developed a nonlocal damage fracture phase field model for rocks considering the heterogeneity of rocks. The modified phase field model introduced the heterogeneity of fracture parameters and modified the governing equations. Meanwhile, the free energy was constructed by the elastic strain energy sphere‐bias decomposition and the plastic strain energy. As for the numerical implementation, the three layers finite elements method structure was used in the frame of the finite element method. The ability of the modified phase field model has been illustrated by reproducing the experiment results of rock samples with pre‐existing cracks under compression.

Phase-field modeling of stochastic fracture in heterogeneous quasi-brittle solids

Owing to the random nature of heterogeneity, damage and fracture behavior of quasi-brittle materials exhibits a considerable degree of uncertainty. Computational modeling of stochastic fracture in quasi-brittle materials has become an indispensable tool for analysis and design of engineering structures. To this end, we present in this paper a computational framework to capture probabilistic fracture in heterogeneous quasi-brittle solids by combining the random field theory and the phase-field cohesive zone model (PF-CZM). The spatial variation of the material strength and fracture energy is represented by a cross-correlated bivariate random field generated by the Karhunen–Loève expansion. The recently proposed PF-CZM is employed to simulate the stochastic crack nucleation and propagation in quasi-brittle solids. The objectivity of the Monte-Carlo simulation is achieved by imposing a specific condition on the phase-field length scale parameter and the correlation length of the random field. In particular, upon this condition the width of the fracture process zone (FPZ) is considerably smaller than the correlation length of the random field such that the material inside the FPZ does not exhibit significant spatial variations of mechanical properties. As the fracture energy is intrinsically incorporated in the PF-CZM, it is unnecessary in this case to explicitly consider the FPZ width. The resulting probabilistic PF-CZM together is applied to the Monte-Carlo simulations of fracture in concrete structures of different geometries. It is shown that the stochastic simulation results are insensitive to both the phase-field length scale parameter and the finite element mesh discretization as in the previous deterministic analyses. Enhanced with the specific condition on the involved characteristic lengths, the PF-CZM provides a viable tool for stochastic simulations of damage and fracture in quasi-brittle structures.

On poroelastic strain energy degradation in the variational phase-field models for hydraulic fracture

Though a number of formulations have been proposed for phase-field models for hydraulic fracture in a fully saturated porous medium, the definition of the degraded poroelastic strain energy varies from one model to another. This study explores previously proposed forms of the poroelastic strain energy with diffused fracture and assesses their ability to recover the explicit fracture opening aperture. We then propose a new form of degraded poroelastic strain energy derived from micromechanical analyses. Unlike the previously proposed models, our poroelastic strain energy degradation depends not only on the phase-field variable (damage) but also on the type of strain energy decomposition. Comparisons against closed form solutions suggest that our proposed model can recover crack opening displacement more accurately irrespective of Biot’s coefficient or the pore-pressure distribution. We then verify our model against the plane strain hydraulic fracture propagation, known as the KGD fracture, in the toughness dominated regime. Finally, we demonstrate the model’s ability to handle complex hydraulic fracture interactions with a pre-existing natural fracture.

An isotropic viscoelastic phase field fracture model for flexural loading of freshwater columnar ice

This work presents the development and implementation of a coupled isotropic phase field fracture and viscoelastic computational model to simulate the fracture of freshwater columnar ice undergoing flexural stresses. An existing incremental constitutive model is extended to capture ice behavior up to and just after material fracture occurs. Crack initiation and propagation is modeled through a Griffith’s energy-based phase field formulation that includes contributions from viscous energy dissipation. The variational formulation and implementation of the material model in the open-source platform Dolfinx is discussed. We present 3-point flexural bending experimental data of freshwater columnar ice grown in controlled conditions which are used to inform the model’s mechanical properties. We find that the coupled model captures the creep and fracture behavior of ice specimens well for strain rates on the order of 10−5s−1 at −10 °C. Finally we show that although our model includes viscoplastic effects, freshwater ice under these conditions develops little viscous energy and therefore instantaneous mechanisms dominate its fracture behavior.

Proposal of m-Index for Rating Fracture and Damage Models by Their Ability to Represent a Set of Distinctive Experiments

A recent comparative study revealed that the commonly offered experimental validations of peridynamics and phase-field fracture models have been insufficient because they involved only nondistinctive experiments, i.e., experiments that can be closely fitted by different models that at the same time give very different predictions in important practical applications. The comparisons showed that the peridynamic and phase-field models are incapable of simulating a set of 11 distinctive experiments—experiments that are critical for assessing the accuracy of different models and are representative of fracture behavior of engineering structures. Practical applications would be helped by common adoption of a model index that would compare the predictive capability of various fracture models quantitatively. Proposed here for further discussion is an example of a possible numerical index, the m-Index, which attempts to characterize how well the optimal calibration of model parameters can match the experimental evidence, such as the fracture patterns, measured response curves, size effect, and crack-parallel stress effect. Included are only the distinctive experiments. As an example, the m-Index is here calculated for a set of seven fracture models whose performance was previously compared with 11 distinctive experiments. This previous comparison of seven models is here extended to an eighth model, proposed as a fresh improvement of peridynamics. The choice of distinctive experiments is one of the subjects calling for further discussion. Despite inevitable imperfections, a widely adopted index for appraising new material models would mitigate waste of researchers’ effort and grant funds, as well as the space in scientific journals and conference programs.

Advancements in Phase-Field Modeling for Fracture in Nonlinear Elastic Solids under Finite Deformations

The prediction of failure mechanisms in nonlinear elastic materials holds significant importance in engineering applications. In recent years, the phase-field model has emerged as an effective approach for addressing fracture problems. Compared with other discontinuous fracture methods, the phase-field method allows for the easy simulation of complex fracture paths, including crack initiation, propagation, coalescence, and branching phenomena, through a scalar field known as the phase field. This method offers distinct advantages in tackling complex fracture problems in nonlinear elastic materials and exhibits substantial potential in material design and manufacturing. The current research has indicated that the energy distribution method employed in phase-field approaches significantly influences the simulated results of material fracture, such as crack initiation load, crack propagation path, crack branching, and so forth. This impact is particularly pronounced when simulating the fracture of nonlinear materials under finite deformation. Therefore, this review outlines various strain energy decomposition methods proposed by researchers for phase-field models of fracture in tension–compression symmetric nonlinear elastic materials. Additionally, the energy decomposition model for tension–compression asymmetric nonlinear elastic materials is also presented. Moreover, the fracture behavior of hydrogels is investigated through the application of the phase-field model with energy decomposition. In addition to summarizing the research on these types of nonlinear elastic body fractures, this review presents numerical benchmark examples from relevant studies to assess and validate the accuracy and effectiveness of the methods presented.

Implementation of a phase field damage model with a nonlinear evolution equation in an FFT-based solver

This paper focuses on the numerical implementation of phase-field models of fracture using the Fast Fourier Transform (FFT) based numerical method. Recent studies on phase-field models focus on the discussions of the choice of the value of regularization length proposed to smear the discontinuity of the sharp crack. Some studies argue that it should be considered as a material property because it has a significant impact on the mechanical behavior of a material in some phase-field models, for instance, in the model proposed by Miehe. However, our results in this study for heterogeneous materials have shown that the choice of regularization length not only affects the macroscopic mechanical behavior but also the local crack propagation patterns. As a result, it can be challenging to select an appropriate value that produces both accurate macroscopic responses and local crack patterns for certain phase-field models, such as Miehe’s model. Thus, the phase-field model proposed by Wu, which has been proven to reduce the length sensitivity for homogeneous material, has been successfully implemented in an FFT-based solver with the application of Newton–Krylov algorithm.The length sensitivity of Wu’s phase-field model for heterogeneous materials is also investigated in this paper. Our tests show that Wu’s phase-field model has partial length sensitivity for heterogeneous materials. However, the main reason for this sensitivity is that one phase enters the damage zone of other phase, which is different from Miehe’s model. As a result, a set of criteria for accurately choosing the regularization value has been established for Wu’s model in this work. Meanwhile, it has also been found that Wu’s model may be more suitable than Miehe’s model for brittle failure due to the introduction of an elastic stage.

An elastoplastic phase-field model for dynamic fracture of nickel-based super-alloys

This paper presents a elastoplastic phase-field model for predicting crack nucleation, propagation, and branching, among other things, in elastoplastic solids subjected to dynamic loading. During the entire fracture process of the nickel-based super-alloys, with the accumulation of plastic strain, to some extent the elastic response of the material is reduced and the rate of phase-field evolution is affected, this means that the elastic strength and phase-field evolution of the material will be influenced by the yield surface and the hardening function. Therefore, we construct elastic and plastic energy degradation functions to more accurately reflect the effects of mechanical fields on the damage field, which distinguish between two successive failure mechanisms: (i) elastic strain and (ii) simultaneous elastoplastic localization deformation. In a variational framework, we derive the governing equations and the corresponding weak forms. Then, damage evolution criterion combining elastoplastic and resistance energies and a plastic yielding criterion are derived. From a numerical point of view, we give model parameters’ selection criteria. Finally, the proposed model is applied to some impact scenarios with varying impact velocities or pre-existent cracks. The comparison of numerical and experimental results shows it can be easily used for identifying material gradation profiles that manipulate crack propagation.

Two different phase field models of the explicit material point method for brittle dynamic fracture

This paper proposes two different phase-field models for brittle dynamic fracture using the explicit material point method (MPM). Based on the consistent stiffness matrix, Model I directly solves the linear equations of the activated Euler grid to obtain the nodal phase field; Model II presents a method for calculating the lumped stiffness matrix and solves the phase field by using a residual iterative method. In addition, we developed the phase field MPM (PH-MPM) code based on GPU parallelization. Based on two one-dimensional models, we study the homogeneous and nonhomogeneous performances of the two models and verify their accuracy by comparing them with the analytical solution. We explored the computational accuracy and efficiency of the two models by performing classic two-dimensional numerical experiments. The results show that both models can analyse fractures with high precision. Meanwhile, compared with Model Ⅰ, the computational efficiency of Model Ⅱ has apparent advantages.

Structural-dependent mechanical behaviors of ink-jet printed films: Indentation testing and phase field fracture modeling

The utilization of ink-jet printing presents immense potential in the production of film sensors for gas turbines. This study delves into the mechanical properties of ink-jet printed films with varying thickness, through the indentation tests. In addition, the implementation of an anisotropic phase field fracture model enabled the simulation of the deformation and cracking mechanisms of ink-jet films with different microstructures. Emphasis is placed on the evaluation of the accuracy of indentation tests for films with intricate structures, while considering the effects of thickness and micro-void distributions. This study provides innovative insights into the structural-dependent failure mechanism of ink-jet films, offering a new perspective on the design of films with high mechanical strength.

A phase-field model for thermo-elastic fracture in quasicrystals

In this paper, a thermo-elastic phase-field model for brittle fracture in quasicrystals is developed. The kind of quasicrystals is not preset, so this model is suitable for almost all thermo-elastic quasicrystal solids. The phonon force and thermal load trigger the variation of the phonon elastic energy, and then drive crack initiation and propagation. The thermal conductivity in the cracked region is assumed to be degraded, which means cracks are adiabatic. For the static penny-shaped crack problem, the results based on the present model agree well with the existing analytical solution. For the quasi-static crack problems, both the tensile/shear fracture in a thermal environment and the thermal shock cracking failure can be dealt with by the present model. Several examples of 1D hexagonal and 2D decagonal quasicrystals are performed to evaluate the effects of thermal load and phason elastic field. The simulation results show that the thermal load can influence the peak force and fracture displacement. For the single-edge notched tension test, the effect of the thermal load on the peak force is small, but it on the fracture displacement is obvious. For the single-edge notched shear test, the thermal load has an obvious influence on both the peak force and fracture displacement. For the thermal shock cracking problem, the phason elastic field has a significant effect on the cracking time and crack number. The present model can serve as a powerful tool to simulate various thermo-elastic fracture problems in quasicrystals.

A concise review of small-strain phase-field modeling of ductile fracture

The computational modeling of ductile fracture is a complex problem due to the simultaneous presence of various competing dissipation mechanisms at the microscale resulting in evolving displacement discontinuities at the macroscale. Phase-field modeling of brittle fracture has emerged as one of the more computationally efficient smeared approaches for simulating crack propagation in solids. In recent years, there has been significant attention towards extending this approach to ductile fracture. The first papers on this topic were published approximately a decade ago, and since then, there has been a growing number of new contributions proposing different approaches and applications. The purpose of this work is to offer an extensive bibliography on phase-field modeling of ductile fracture, together with a concise presentation of some of the most widely used formulations, specifically limited to small-strain plasticity under monotonic loading. Few aspects of particular interest are also discussed in detail.

A phase-field framework for brittle fracture in quasi-crystals

Most of quasi-crystals are brittle at room temperature due to their specific cluster structures. Phase-field fracture models have demonstrated a powerful ability to predict brittle crack evolution. This paper develops a phase-field framework for modeling macroscopic brittle fracture in quasi-crystal solids. The kind of quasi-crystals is not preset in the phase-field model; therefore, this model is valid for almost all quasi-crystal solids. The phase-field model for the first time introduces a volume fraction parameter into the fracture toughness to reflect the effect of phason wall. Furthermore, a new numerical implementation approach of phase-field fracture models in Comsol Multiphysics is presented. Several examples of 1D hexagonal and 2D decagonal quasi-crystals are performed. The developed model is validated against the reported results of benchmark problems. Further investigations indicate that the phason field peculiar to quasi-crystals has a certain influence on the crack paths in asymmetric fracture problems and on the force–displacement curves in both symmetric and asymmetric fracture problems. When the fracture toughness is fixed, the increase of the volume fraction parameter results in the decrease of the peak force and the increase of the failure displacement. When the influence of the phason field on the fracture toughness is considered, both the peak force and failure displacement would sharply decrease, which indicates that the degradation of fracture toughness due to the phason field plays a more important role than the phason elastic energy in the fracture of quasi-crystals. Furthermore, the relation between the phonon-phason energy transformation and the elastic constants of QCs has been qualitatively analyzed. The developed phase-field modeling framework provides a guidance for understanding the fracture mechanism of quasi-crystals and assessing the safety of quasi-crystal structures in engineering practice.

An SPH-based FSI framework for phase-field modeling of brittle fracture under extreme hydrodynamic events

An SPH-based FSI framework for phase-field modeling of brittle fracture under extreme hydrodynamic events

Phase-field modeling of thermal cracking in hardening mass concrete

The manuscript is concerned with the phase-field modeling of thermal cracking in massive hardening concrete structures. Owing to the exothermic reactions, known as hydration, a considerable amount of heat is generated during the setting and hardening of concrete. Since concrete has relatively low thermal conductivity and fairly high heat capacity, sharp temperature gradients may occur between the interior and exterior parts of massive concrete structures. This uneven temperature distribution results in a non-uniform thermal expansion that, in turn, may lead to thermal cracking in conjunction with restraining external boundary conditions. Hence, it is of key importance to conduct the crack risk assessment of massive concrete structures at various stages of their construction and thereafter where real-size experiments are often unrealizable. To this end, we develop a multi-field computational approach to simulate thermal cracking in hardening concrete. In particular, we propose a novel chemo-thermo-mechanical model developed within the framework of the theory of reactive porous media coupled with a quasi-brittle phase-field regularized cohesive zone model to explicitly account for the initiation and propagation of thermal cracks. In the model, chemical, thermal, mechanical, and phase-field fracture problems are coupled through the constitutive equations. To our best knowledge, this is the first study where a quasi-brittle phase-field fracture model is used to model thermal cracking in hardening massive concrete through a novel incremental formulation. The potential of the proposed approach is demonstrated through the benchmark thermomechanical problems of a normal concrete member and a roller-compacted concrete dam. In the latter, original multi-field interface elements are devised between the lifts.

On the numerical and mesh-dependent parameters in a computationally enhanced phase-field fracture model coupled with a novel mesh refinement strategy

The phase-field method has been proven as a robust and computationally efficient approach to model the propagation of fractures in brittle solids. However, the performance of this technique in the context of finite element method can be questioned due to restrictions in the mesh structure and the element size to capture the fracture as a diffusive damaged region. This study is dedicated to developing a methodology for finding an appropriate length-scale parameter to model the fracturing process in a way that matches the physical character of failure in materials. The fracture process zone is chosen as the key feature in this study to propose relationships for estimating the length-scale parameter based on the tensile strength and cracking properties, and the robustness of the method is verified using experimental data. To employ the phase-field method in modelling large-scale domains and complex geometries, a novel mesh refinement strategy is developed to increase the computational efficiency based on predicting a corrected tensile strength limit depending on the element size to capture the crack-tip effectively. The proposed mesh refinement strategy reduces the computational effort significantly. Reliability and robustness of the developed relationships are successfully examined by simulating benchmark cases and comparisons with physically measured data.

Higher-order nonlocal operator theory for phase-field modeling of ductile fracture in elasto-plastic materials

In this work, we propose a novel approach based on the higher-order nonlocal operator for the phase-field modeling of ductile fracture in elasto-plastic materials. The present method introduces the total energy function consisting of the elastic, plastic, and fracture terms. The plasticity is coupled with the fracture through the degradation function which is applied to the tensile part of the elastic strain energy. The proposed higher-order nonlocal operator method brings several advantages over the original nonlocal operator method that requires only first-order terms. Moreover, the proposed method does not require the direct computation of kernel function or moment matrix derivatives. Therefore, this approach can improve the computational efficiency and simplify in implementation. The accuracy and effectiveness of the proposed method are demonstrated through various numerical examples, which have the ability to detect complex patterns of ductile fracture, such as crack propagation and plastic localization.

A coupled approach to predict cone-cracks in spherical indentation tests with smooth or rough indenters

Indentation tests are largely exploited in experiments to characterize the mechanical and fracture properties of the materials from the resulting crack patterns. This work proposes an efficient theoretical and computational framework, whose implementation is detailed for 2D axisymmetric and for 3D geometries, to simulate indentation-induced cracking phenomena caused by non-conforming contacts with indenter profiles of an arbitrary shape. The formulation hinges on the coupling of the MPJR (eMbedded Profile for Joint Roughness) interface finite elements which embed the indenter profile to efficiently solve the contact problem between non-planar bodies, and the phase-field model for brittle fracture to simulate crack evolution and nonlocal damage in the substrate. The novel framework is applied to predict cone-crack formation in the case of indentation tests with smooth spherical indenters, with validation against experimental data. Then, the methodology is employed for the very first time in the literature to assess the effect of surface roughness superimposed on the shape of the smooth spherical indenter. In terms of physical insights, numerical predictions quantify the dependencies of the critical load for crack nucleation and the crack radius on the amplitude of roughness in comparison with the behavior of smooth indenters. Again, the consistency with available experimental trends is noticed.

Modelling of Fatigue Microfracture in Porous Sintered Steel Using a Phase-Field Method.

Porosity in sintered materials negatively affects its fatigue properties. In investigating its influence, the application of numerical simulations reduces experimental testing, but they are computationally very expensive. In this work, the application of a relatively simple numerical phase-field (PF) model for fatigue fracture is proposed for estimation of the fatigue life of sintered steels by analysis of microcrack evolution. A model for brittle fracture and a new cycle skipping algorithm are used to reduce computational costs. A multiphase sintered steel, consisting of bainite and ferrite, is examined. Detailed finite element models of the microstructure are generated from high-resolution metallography images. Microstructural elastic material parameters are obtained using instrumented indentation, while fracture model parameters are estimated from experimental S-N curves. Numerical results obtained for monotonous and fatigue fracture are compared with data from experimental measurements. The proposed methodology is able to capture some important fracture phenomena in the considered material, such as the initiation of the first damage in the microstructure, the forming of larger cracks at the macroscopic level, and the total life in a high cycle fatigue regime. However, due to the adopted simplifications, the model is not suitable for predicting accurate and realistic crack patterns of microcracks.