Phase Field Modeling Of Fracture Research Articles

A mesoscale modelling approach coupling SBFEM, continuous damage phase-field model and discrete cohesive crack model for concrete fracture

This study develops an innovative numerical approach for simulating complex mesoscale fracture in concrete. In this approach, the concrete meso-structures are generated using a random aggregate generation and packing algorithm. Each aggregate is modelled by a single scaled boundary finite element method (SBFEM) based polygon with the boundary discretized only. The damage and fracture in the mortar is simulated by the continuous damage phase-field regularized cohesive zone model (PF-CZM), and the aggregate-mortar interfaces are modelled by zero-thickness cohesive interface elements (CIEs) with nonlinear softening separation-traction laws. This new approach thus takes full advantages of different methods, including the semi-analytical accuracy and high flexibility in mesh generation and transition of SBFEM, the mesh and length-scale independence of PF-CZM, and the ease-of-use of CIEs in modelling discrete interfacial fracture. These advantages are demonstrated by successful simulations of a few 2D and 3D benchmark examples in mode-I and mixed-mode fracture.

Statistical effects of pore features on mechanical properties and fracture behaviors of heterogeneous random porous materials by phase-field modeling

Heterogeneous materials with randomly distributed pores are ubiquitous, such as sintered silver nanoparticles, concrete materials, 3D printed polymers, and natural bones. Recent experimental investigations have revealed that porosity and also pore-related geometries (size, number, shape, distribution and alignment) have significant impacts on the mechanical behavior of random porous materials. However, existing studies focus on the porosity effect while ignoring other pore features such as pore size and pore shape. Our research is dedicated to a computational framework for generating isotropic/anisotropic random porous materials using Gaussian random fields with stochastic pore size and shape factor and addressing the mechanical properties and behavior of brittle fractures using a fracture phase-field model with a preferred degradation function. Sintered silver nanoparticles with typical randomly distributed pores, as representative porous materials, are chosen for their promising applications in emerging fields such as power electronics and wearable devices. In order to emphasize the effect of pore size and shape, 420 random samples with a fixed porosity were generated to discuss the stress–strain response during fracture and to establish statistical relationships between pore feature distributions and mechanical properties such as Young's modulus, UTS, and average historical energy. Our findings suggest that the statical attributes of the pore sizes and shape factors significantly affect the material performance related to the mechanical properties and fracture behavior, which could give a better understanding of the random porous materials and guide reliability-based material design optimization.

Explicit phase-field material point method with the convected particle domain interpolation for impact/contact fracture in elastoplastic geomaterials

A novel explicit phase-field material point method with the convected particle domain interpolation (ePF-CPDI) is proposed for solving large deformation dynamic impact/contact fracture problems in elastoplastic geomaterials. In this method, we derive an explicit rate-dependent phase-field fracture model relying on the microforce balance law and the second law of thermodynamics. A coupled explicit phase-field plasticity model is then developed to describe dynamic elastoplastic fracture responses of geomaterials. Here the explicit time integration strategy and the staggered solution scheme are utilized in this work to solve the coupled-field governing equations based on the material point method. To eliminate the numerical noises caused by material points crossing cell boundaries, the convected particle domain interpolation technique is adopted to improve the computational accuracy in large deformation simulation. Furthermore, the proposed approach combining with the particle to particle contact algorithm is extended to deal with more complicated high-velocity impact and multi-body contact elastoplastic fracture problems. Numerical experiments are employed to validate the high accuracy and excellent capability of the proposed method and discuss the influences of main parameters on the phase-field fracture modeling.

Phase-field modeling of crack growth and mitigation in solid oxide cells

Fracture and crack growth is one of the main degradation mechanisms in solid oxide cells (SOCs). However, the modeling of crack growth in SOCs is challenging due to their complex microstructures and possible plasticity development within the Ni particles in Ni-based SOC electrodes. In this study, a phase-field fracture model is developed, which incorporates the SOC microstructures and phase-dependent material properties, including yield strength, fracture toughness in the bulk and at the interphase boundaries. The model is employed to study crack initiation and growth under thermal and redox cycling on the hydrogen electrode side of SOCs. The simulation results demonstrate that under thermal cycling, work-zone cracking dominates in electrolyte-supported SOCs with cracks initiated at the triple-phase boundaries, while only minor mechanical degradation occurs in hydrogen-electrode-supported SOCs after hundreds of thermal cycles. Under redox cycling, through-cracking of yttria-stabilized zirconia (YSZ) in the hydrogen electrode and electrolyte layers dominates. The simulation results suggest several crack-mitigation strategies, including decreasing the porosity in the hydrogen electrode support layer and synchronizing thermal strain to balance oxidation strain.

Extension of the spatially adaptive phase-field model to various forms of fracture

The phase field approach has proved to be efficient and has received ample attention amongst the available techniques to model fracture. However, high computational cost still imposes substantial difficulties in the phase-field simulation of fractures. This contribution is based on a recently proposed variational approach for spatial adaptivity in a phase-field model of fracture. The main idea is to consider the regularisation length ϵ as a space-dependent variable in the argument of the energy functional. We extend this now by implementing a strain energy split to ensure that only the tensile energy drives the crack propagation. The displacement, phase field, and optimal regularisation length are then determined locally by minimisation of the modified energy functional. Subsequently, the computed optimal regularisation length is used to refine the mesh size locally. The resultant solution procedure is implemented in the finite element library FEniCS. Numerical investigations on selected examples of different fracture modes demonstrate that the spatially adaptive phase field model has a comparable convergence rate, but a subjacent energy convergence curve resulting in significant computational savings. Moreover, it also computes the peak force more accurately illustrating its potential for usage in practical applications.

A Review on phase-field modeling of fracture

In cases with complicated crack topologies, the computational modeling of failure processes in materials owing to fracture based on sharp crack discontinuities fails. Diffusive crack modeling based on the insertion of a crack phase-field can overcome this. The phase-field model (PFM) portrays the fracture geometry in a diffusive manner, with no abrupt discontinuities. Unlike discrete fracture descriptions, phase-field descriptions do not need numerical monitoring of discontinuities in the displacement field. This considerably decreases the complexity of implementation. These qualities enable PFM to describe fracture propagation more successfully than numerical approaches based on the discrete crack model, especially for complicated crack patterns. These models have also demonstrated the ability to forecast fracture initiation and propagation in two and three dimensions without the need for any ad hoc criteria. The phase-field model, among numerous options, is promising in the computer modeling of fracture in solids due to its ability to cope with complicated crack patterns such as branching, merging, and even fragmentation. A brief history of the application of the phase-field model in predicting solid fracture has been attempted. An effort has been made to keep the conversation focused on recent research findings on the subject. Finally, some key findings and recommendations for future research areas in this field are discussed.

A comparative assessment of different adaptive spatial refinement strategies in phase-field fracture models for brittle fracture

For the smeared approximation of a discrete crack, phase-field fracture simulations of brittle materials require suitable finite element meshes in regions where crack propagation is expected to get an accurate resolution of the phase-field function. The straightforward option is to pre-refine the mesh in regions of the expected crack paths. However, this could lead to very computationally intensive simulations due to the high number of elements. Alternatively, adaptive spatial refinement of the finite element mesh is utilized based on appropriate error indicators to obtain the required accuracy in the areas of crack propagation. Different error indicators can be used: the most common one for phase-field fracture simulations is the threshold-based approach, in which elements are refined depending on the value of the phase-field function. Alternatively, the Kelly error indicator can be used as a criterion for spatial adaptivity. It considers the jumps in the gradients of the phase-field function between the elements. We additionally introduce here an error indicator based on configurational forces, that depend on the Eshelby stress tensor. For mode I loading in linear elastic fracture mechanics, the configurational forces have a close connection to the J-Integral and the critical fracture energy Gc, respectively. Therefore, a suitable norm of the configurational forces is introduced as an error indicator here. These three error indicators are introduced and compared to each other in terms of accuracy and efficiency by means of numerical examples for crack growth in the single edge notched shear test.

Effect of calcium leaching on the fracture properties of concrete

In this work, acceleration leaching and three-point bending (TPB) tests were performed on single-notched TPB beam specimens to explore the evolution of the fracture properties of concrete subjected to calcium leaching. The load-crack mouth opening displacement (P-CMOD) curves obtained for concrete at various leaching times were adopted to analytically calculate parameters that can reflect the fracture properties of concrete, such as the crack extension resistance (KR), fracture toughness (KIcini and KIcun), elastic modulus (E), tensile strength (ft), and fracture energy (Gf). Furthermore, a phase-field fracture model for concrete that considers calcium leaching effects is established by incorporating the time-dependent damage variables E, ft, and Gf. The results reveal that the KR curves of the leached concrete increase with crack propagation but lower and flatten with more leaching time. The fracture parametersKIcini, KIcunE, ft, and Gf rapidly decrease with leaching time in the first 60 days. However, with more leaching time, the impact of calcium leaching on the concrete fracture properties gradually diminishes. The variation in fracture parameters of leached concrete can be described by a series of exponential functions. The residual value fracture parameters of the fully leached concrete are estimated to be approximately 60–80 % of those of sound concrete. The established fracture model reproduces the P-CMOD curves of concrete at each leaching time, which verifies the rationality of the model. The numerical simulation reveals that crack propagate faster in leached concrete than in sound concrete, and leached concrete is more prone to cracking.

Fast staggered schemes for the phase-field model of brittle fracture based on the fixed-stress concept

Phase-field models are promising to tackle various fracture problems where a diffusive crack is introduced and modeled using the phase variable. Owing to the non-convexity of the energy functional, the derived partial differential equations are usually solved in a staggered manner. However, this method suffers from a low convergence rate, and a large number of staggered iterations are needed, especially at the fracture nucleation and propagation. In this study, we propose novel staggered schemes, which are inspired by the fixed-stress split scheme in poromechanics. By fixing certain invariants of the stress when solving the damage evolution, the displacement increment is expressed in terms of the increment of the phase variable. The relation between these two increments enables a prediction of the displacement and the active energy based on the increment of the phase variable. Hence, the maximum number of staggered iterations is reduced, and the computational efficiency is improved. We present three fast staggered schemes by fixing the first invariant, second invariant, or both invariants of the stress, denoted by S1, S2, and S3 schemes. The performance of the schemes is then verified by comparing with the standard staggered scheme through three benchmark examples, i.e., tensile, shear, and L-shape panel tests. The results exhibit that the force–displacement relations and the crack patterns computed using the fast schemes are consistent with the ones based on the standard staggered scheme. Moreover, the proposed S1 and S3 schemes can largely reduce the maximum number of staggered iterations and total cpu time in all benchmark tests. The S2 scheme performs comparably except in the L-shape panel test, where the underlying assumption is violated in the region close to the crack.

An extended hydrostatic–deviatoric strain energy density decomposition for phase-field fracture theories

The interest in using phase-field theories to numerically analyze fracture has sky-rocketed in the last years. However, in phase-field fracture models are splits, or decompositions, of the strain energy density vital to avoid interpenetration of crack surfaces and to select physically trustworthy crack paths. The most popular decomposition strategies use either a spectral decomposition or a hydrostatic–deviatoric decomposition. Both decompositions have significant disadvantages; the most important is that none of them can handle mixed-mode load scenarios in compression. To circumvent these problems, a generalized decomposition method is derived that unifies some features of the hydrostatic–deviatoric and spectral decompositions, enhanced with a classical Mohr–Coulomb failure criterion. The derived decomposition scheme has the potential to judge whether or not a compressive deformation field will assist in the crack driving process in brittle materials. The enhanced decomposition is scrutinized in numerical models and revealing biaxially loaded crack experiments in global compression. Simulations using the decomposition scheme capture the experiments in a remarkable way: complex crack patterns are reproduced, as well as critical loads. The enhanced decomposition strategy hence provides mechanistic insight into fracture processes in brittle materials subject to mixed-mode loads.

A thermo-mechanical phase-field fracture model: Application to hot cracking simulations in additive manufacturing

Thermal fracture is prevalent in many engineering problems and is one of the most devastating defects in metal additive manufacturing. Due to the interactive underlying physics involved, the computational simulation of such a process is challenging. In this work, we propose a thermo-mechanical phase-field fracture model, which is based on a thermodynamically consistent derivation. The influence of different coupling terms such as damage-informed thermomechanics and heat conduction and temperature-dependent fracture properties, as well as different phase-field fracture formulations, are discussed. The model is numerically implemented with the finite element method. Finally, the model is applied to simulate the hot cracking in additive manufacturing. Thereby not only the thermal strain but also the solidification shrinkage is considered. As for the thermal profile, both analytical temperature solution and numerical thermal field around the melting pool are taken into account. Based on the latter approach, the influence of different process parameters is further studied. The study reveals that the solidification shrinkage strain takes a dominant role in the formation of the circumferential crack, while the temperature gradient is mostly responsible for the central crack. Process parameter study demonstrates further that a higher laser power and slower scanning speed are favorable for keyhole mode hot cracking while a lower laser power and quicker scanning speed tend to form the conduction mode cracking. The numerical predictions of the hot cracking patterns are in good agreement with similar experimental observations, showing the capability of the model for further studies.

Combined diffused material interface and hybrid phase-field model for brittle fracture in heterogeneous composites

In this article, we propose a novel approach for modeling brittle fracture in heterogeneous composites using a combined diffused material interface method and a hybrid phase-field model (PFM). We have used the diffused material interface method to derive analytical expressions for the regularized material properties across the material interfaces that are treated as an input to the PFM for the simulation of brittle fractures in composites. The use of diffused material interface method regularizes the material properties in such a way that one can avoid the discontinuity in stress across the material interface and thus bypass the requirement of explicit tracking of material interfaces for finite element implementation of the hybrid PFM. We have implemented the proposed combined model using a recently developed open-source finite element toolbox, Gridap in Julia. We have considered tests on a single circular fiber inclusion under tension, a bi-material rectangular plate under tension, single square inclusion under tensile loading, a notched composite beam with a central notch under symmetric three-point bending, and a rectangular plate with multiple fibers under tensile loading to show how the proposed combined model can be used to simulate brittle fracture in composites.

Nonlinear field-split preconditioners for solving monolithic phase-field models of brittle fracture

One of the state-of-the-art strategies for predicting crack propagation, nucleation, and interaction is the phase-field approach. Despite its reliability and robustness, the phase-field approach suffers from burdensome computational cost, caused by the non-convexity of the underlying energy functional and a large number of unknowns required to resolve the damage gradients. In this work, we propose to solve such nonlinear systems in a monolithic manner using the Schwarz preconditioned inexact Newton (SPIN) method. The proposed SPIN method leverages the field split approach and minimizes the energy functional separately with respect to displacement and the phase-field, in an additive and multiplicative manner. In contrast to the standard alternate minimization, the result of this decoupled minimization process is used to construct a preconditioner for a coupled linear system, arising at each Newton’s iteration. The overall performance and the convergence properties of the proposed additive and multiplicative SPIN methods are investigated by means of several numerical examples. A comparison with widely-used alternate minimization is also performed showing a significant reduction in terms of execution time. Moreover, we also demonstrate that this reduction grows even further with increasing problem size.

Multiscale phase-field approach for investigation of anisotropic fracture properties of architected materials

We propose a computational approach to determining the direction-dependent fracture toughness of architected materials. Using the principles of multiscale mechanics and phase-field modelling of fracture, our proposition enables straightforward determination of the macroscopic toughness of the homogenized material. The most important feature of our computational scheme is the loading arrangement which ensures the crack propagation along a macroscopic straight line in the same direction as the initial notch, irrespective of the material’s orientation. Forcing the crack propagation in a specific direction, we extract the toughness of the homogenized material in that direction by utilizing the equivalence of dissipation in micro and macro scales. Diffused representation of crack, both in micro and macro scales, through a phase-field formalism has been adopted to simulate crack propagation and establish the necessary expression for quantifying direction-dependent macroscopic toughness. We verify the capability of the proposed approach by computing the fracture toughness of a brittle layered media and comparing the result with that available in the literature. We subsequently apply the algorithm to investigate the fracture behaviour of brittle architected lattice materials of two types– a material with an array of circular holes and triangular lattice. The nature of fracture anisotropy and the effect of relative density on fracture behaviour of these materials, as derived through the present technique, are highlighted.

A mode-adjustable phase-field model for brittle fracture by regulating distortional crack driving energy

A mode-adjustable phase-field model is proposed by introducing a weight parameter to regulate the relative contribution of the volumetric and distortional crack driving energy. Altering the weight parameter results in the change of cracking mode. The weight parameter can be considered as a material parameter and determined through the comparison of experimental and simulated results. The proposed phase-field model is employed to simulate the fracture experiments for polymethylmethacrylate and Al 7075-T651 aluminum. Applying the experimentally determined parameter, the mode-adjustable phase-field model provides better simulations than the classical phase-field model for both materials. Afterwards, a shear test of a single edge notched plate is simulated and the results show that for the volumetric-deviatoric split method, as the weight parameter increases, the failure pattern during crack propagation exhibits a transition from the mode-I dominant failure to the mode-II dominant failure gradually. On the contrary, a mode-I dominant cracking always occurs with the initial crack deflection angle of about 70.5°, regardless of the value of weight parameter for the spectral decomposition split method. To summarize, the proposed mode-adjustable phase-field model can reflect the difference of crack driving mechanism, which is helpful for capturing and understanding the fracture characteristics of diverse materials.

Modeling fracture propagation in poro-elastic media combining phase-field and discrete fracture models

We present a novel model for fluid-driven fracture propagation in poro-elastic media. Our approach combines ideas from dimensionally reduced discrete fracture models with diffuse phase-field models. The main advantage of this combined approach is that the fracture geometry is always represented explicitly, while the propagation remains geometrically flexible. We prove that our model is thermodynamically consistent. In order to solve our model numerically, we propose a mixed-dimensional discontinuous Galerkin scheme with a computational grid fully conforming to the fractures. As the fracture propagates, the diffuse phase-field acts as indicator to identify new fracture facets to be added to the discrete fracture network. Numerical experiments demonstrate that our approach reproduces classical scenarios for fracturing porous media.

A graded interphase enhanced phase-field approach for modeling fracture in polymer composites

The superiority of nanosized filler particles in improving the elastic and fracture behavior of polymers, in comparison to microsized inclusions, has been substantiated by several experimental observations. Accurate modeling of crack propagation in such heterogeneous materials, however, involves resolution of complex crack topologies while taking into account the coalescence and branching of multiple cracks. Such complexity renders the traditional sharp crack modeling approaches, such as those based on the idea of partition of unity, to be of limited suitability, especially in 3D, since these involve explicit tracking of the evolving crack surfaces. Consequently, phase-field fracture approaches have come up as an attractive alternative to the traditional sharp crack models, especially when complex crack topologies need to be handled. Furthermore, standard two-phase continuum models, owing to their lack of the necessary length scale, are unable to capture the previously mentioned smaller is stronger size effect. In order to remedy this shortcoming, in context of finite strain hyperelasticity, a graded interphase based enhancement of the standard first-order continuum model was introduced in our previous work. The present contribution extends the idea of graded interphases to the phase-field fracture approach. Herein, an interphase region around filler particles, as observed experimentally, having continuously varying or graded material properties is considered. Fracture behavior of the composite material can be controlled by means of the degree of grading employed in determining the elastic and fracture properties within the interphase region. An optimal combination of the graded interphase parameters yields a tougher macroscopic response, as observed in case of tougher interphases, in comparison to the one obtained with a standard phase-field fracture model, The appropriateness of the introduced technique for modeling a wide spectrum of experimentally observed fracture behaviors, depending upon the degree of adhesion between the filler and the matrix phases, is depicted by means of extensive numerical experimentation.

A mixed mode phase-field model of ductile fracture

We present the first mixed mode phase-field model of ductile fracture. The contribution of crack opening and shearing deformations to the propagation of a crack is expressed by introducing two phase fields. Constitutive relations are then introduced to couple and distinguish these phase fields. Special attention is given to the maximum shear stress and its effect on the development of fractures. The proposed model is validated by tensile testing experiments found in the literature on Al 2024 T-351. Model predictions of the crack path and load-displacement response are compared to a plane-strain tension experiment, a round bar tension experiment and a notched round bar tension experiment. The model is shown to accurately capture the slant and cup-cone crack paths as well as force-displacement curves.

A phase field solution for modelling hyperelastic material and hydrogel fracture in ABAQUS

The phase field fracture model is attracting significant interest. To model fracture in hyperelastic material and hydrogel, we have implemented a robust two- and three-dimensional phase field method in the commercial finite element code ABAQUS/Standard. The method is based on the rate-independent variational principle of diffuse fracture and also exploits the analogy between the phase field evolution law and the heat transfer equation, enabling the use of Abaqus’ in-built features and sparing the need for defining user elements. The framework is shown to accommodate both staggered and monolithic solution schemes. The approach can properly simulate the fracture for both hyperelastic material and hydrogel under different boundary conditions. Several examples are provided to demonstrate the robustness of the method. The provided source codes and the tutorials make it easy for practicing engineers and scientists to model crack propagation in hyperelastic and gel materials.

Phase-field modeling of brittle fracture in heterogeneous bars

We investigate phase-field modeling of brittle fracture in a one-dimensional bar featuring a continuous variation of elastic and/or fracture properties along its axis. Our main goal is to quantitatively assess how the heterogeneity in elastic and fracture material properties influences the observed behavior of the bar, as obtained from the phase-field modeling approach. The results clarify how the elastic limit stress, the peak stress and the fracture toughness of the heterogeneous bar relate to those of the reference homogeneous bar, and what are the parameters affecting these relationships. Overall, the effect of heterogeneity is shown to be strictly tied to the non-local nature of the phase-field regularization. Finally, we show that this non-locality may amend the ill-posedness of the sharp-crack problem in heterogeneous bars where multiple points compete as fracture locations.