Phase Field Modeling Of Fracture Research Articles

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.

Performance decay analysis of cementitious composite cladding structure under stochastic aging

Façade cladding structure made of cementitious composite is prone to cracking due to its brittle characteristics, despite being a non-load-bearing member. Served as building exteriors, aging of cementitious composite may further raise the risk of fracture failure under accidental loads, causing both financial loss and safety hazards. To address this issue, a computational approach, fusing together a novel physics-based stochastic aging model and phase field fracture model, is developed to assess the mechanical performance decay of aging cladding structure. In this study, stochastic leaching under inherent material uncertainty is taken as the primary aging concern, and the analysis is assisted with extended support vector regression for substantial efficiency enhancements. The proposed method is carefully validated against two designed benchmark problems and a real-life application. It is demonstrated to be able to predict the time-dependent reliability of cementitious composite cladding structures precisely and efficiently under stochastic leaching-induced aging.

Cohesive Zone Interpretations of Phase-Field Fracture Models

Abstract Unlike micromechanics failure models that have a well-defined crack path, phase-field fracture models are capable of predicting the crack path in arbitrary geometries and dimensions by utilizing a diffuse representation of cracks. However, such models rely on the calibration of a fracture energy (Gc) and a regularization length-scale (lc) parameter, which do not have a strong micromechanical basis. Here, we construct the equivalent crack-tip cohesive zone laws representing a phase-field fracture model, to elucidate the effects of Gc and lc on the fracture resistance and crack growth mechanics under mode I K-field loading. Our results show that the cohesive zone law scales with increasing Gc while maintaining the same functional form. In contrast, increasing lc broadens the process zone and results in a flattened traction-separation profile with a decreased but sustained peak cohesive traction over longer separation distances. While Gc quantitatively captures the fracture initiation toughness, increasing Gc coupled with decreasing lc contributes to a rising fracture resistance curve and a higher steady-state toughness—both these effects cumulate in an evolving cohesive zone law with crack progression. We discuss the relationship between these phase-field parameters and process zone characteristics in the material.

Phase-field finite deformation fracture with an effective energy for regularized crack face contact

Phase-field models are a leading approach for realistic fracture problems. They treat the crack as a second phase and use gradient terms to smear out the crack faces, enabling the use of standard numerical methods for simulations. This regularization causes cracks to occupy a finite volume in the reference, and leads to the inability to appropriately model the closing or contacting -- without healing -- of crack faces. Specifically, the classical idealized crack face tractions are that the shear component is zero, and that the normal component is zero when the crack opens and identical to the intact material when the crack closes. Phase-field fracture models do not replicate this behavior. This work addresses this shortcoming by introducing an effective crack energy density that endows the regularized (finite volume) phase-field crack with the effective properties of an idealized sharp crack. The approach is based on applying the QR (upper triangular) decomposition of the deformation gradient tensor in the basis of the crack, enabling a transparent identification of the crack deformation modes. By then relaxing over those modes that do not cost energy, an effective energy is obtained that has the intact response when the crack faces close and zero energy when the crack faces are open. A highlight of this approach is that it lies completely in the setting of finite deformation, enabling potential application to soft materials and other settings with large deformation or rotations. The model is applied to numerically study representative complex loadings, including (1) cyclic loading on a cavity in a soft solid that shows the growth and closing of cracks in complex stress states; and (2) cyclic shear that shows a complex pattern of crack branching driven by the closure of cracks.

Application of phase-field fracture theories and digital volume correlation to synchrotron X-ray monitored fractures in human trabecular bone: A case study

Fracture processes of trabecular bone have been studied using various approaches over the years. However, reliable methods to analyse fracture at the single trabecula level are limited. In this study, a digital volume correlation (DVC) and a phase-field fracture model are applied and contrasted for human trabecular bone to analyse its failure under global compression at high resolution.A human trabecular bone sample was fractured in situ under synchrotron-based X-ray micro computed tomography (CT). Reconstructed CT data was then used in DVC algorithms to obtain high-resolution displacement fields in the bone at different load steps. A high-resolution specimen-specific structural mesh was discretized from the CT data and used for the phase-field simulation of the fracturing bone.The DVC analysis showed opening mode cracks as well as shear mode cracks. Strains in cracked regions were analysed. The load distribution in the trabecular structure resulted in two completely separated fracture regions in the sample body. A phenomenon that was also captured in the phase-field model. The results encourage us to believe improvements in boundary conditions and material models are worthwhile pursuing. Findings in this study support further development of a phase-field method to analyse fracture in samples with complex morphology, such as trabecular bone, and the capacity of DVC to quantify strains and slowly growing stable fractures during step-wise loading of trabecular bone.

Uncovering the intrinsic deficiencies of phase-field modeling for dynamic fracture

The phase-field fracture (PFF) approach has achieved great triumphs in modeling quasi-static fracture. Nevertheless, its reliability in serving dynamic fractures still leaves something to be desired, such as the prediction of the limiting crack velocity. Using a pre-strained fracture configuration, we discovered a disturbing phenomenon that the crack limiting speed identified by the dynamic PFF model is not related to the specific material, which seriously deviates from the experimental observation. To ascertain the truth, we first ruled out the correlation between the limiting crack velocity on the phase-field characteristic scale and external loading. Afterward, by switching between different crack surface density functions and degradation functions, we reach an astonishing conclusion that the limiting crack velocity predicted by the dynamic PFF model relies on the model itself. This non-physical phenomenon signifies that the commonly used dynamic PFF model has inherent defects. From the perspective of classical linear elastic fracture mechanics (LEFM), the local wave velocity degradation can be responsible for the limiting crack velocity in the phase-field modeling of dynamic fracture.

Research on crack propagation of 3D printed material with complex cracks based on the phase-field fracture model

Research on crack propagation of 3D printed material with complex cracks based on the phase-field fracture model

Variational crack phase-field model for ductile fracture with elastic and plastic damage variables

Based on the study on characteristic features of existing crack phase-field (PF) models for ductile fracture, we propose a new model endowed with elastic and plastic damage variables. In the review part, the emphasis is placed on the plastic driving force and degrading fracture toughness that enable PF models to represent damage evolution in elastoplastic materials. Attention is also paid to the damage evolution tendency in terms of the definition of the yield function. Based on the reviewed features, we originally formulate the proposed model, which separately accommodates two damage variables for elasticity and plasticity. Its constitutive work density consists of elastic, plastic hardening, and damage hardening components. Evolution laws for plasticity and damage are derived by using separate threshold (yield) functions while having similar formats so that those for damage hold variational and thermodynamic consistencies. Also, the proposed model is equipped with thresholds and coefficients to control the amount of damage driving force. Several numerical examples are presented to demonstrate the characteristic features of the proposed model and the ability to mimic the typical fracture behavior of elastoplastic materials.

An adaptive local algorithm for solving the phase-field evolution equation in the phase-field model for fracture

In the phase-field model for fracture, material damage can be characterized by a variable called the crack phase-field. Usually-two sub-problems controlled by the equilibrium equation and the evolution equation, respectively, are needed to be solved in the whole domain. In this paper, an adaptive local algorithm is proposed for the unified phase-field model to solve the evolution equation in the local domain based on the fact that the value of phase-field is non-zero only around the crack. In the initial unified phase-field model, the boundedness and irreversibility conditions are not easy to be satisfied. In this paper, a history field is proposed to convert the inequality into equality in the evolution equation, and the boundedness and irreversibility conditions can be treated easily as in the classical phase-field model. To further obtain the local algorithm, some failure criteria are introduced to identify the damaged area, and the elements in the local domain are updated adaptively in each iteration step. The local algorithm can be implemented easily since the boundary conditions of the local domain are the same as in the non-local algorithm. Numerical examples have shown that the proposed local algorithm can obtain the same results as the non-local algorithm.

Applications of a phase-field fracture model to materials with inclusions

Fracture in materials with inclusions is considered in the computational model of this contribution which allows prediction of crack propagation. The inclusions cause that cracks may appear inside the materials or along matrix-inclusion interfaces. The presented model can treat them both using two internal variables in a fashion of damage mechanics so that fracture is a consequence of the damage. The first variable is defined at the interface represented by a thin degradable adhesive layer so that an appropriate stress-strain relationship can be obtained as in cohesive zone models. The second variable is defined in the structural domains as a phase-field fracture variable which causes elastic properties degradation in a narrow material strip that forms a smeared crack. Both these damaging schemes are expressed in a unique quasi-static energy evolution process. The numerical solution approach is thus rendered from a variational form with a staggered time stepping procedure related to a separation of deformation variables from the damage ones and using non-linear programming algorithms implemented together within an own MATLAB finite element code. The numerical simulations with the model include simplified structural and material elements.

The Ogden and the extended tube model as backbone in describing electroactive polymers: advancements in modelling nonlinear behaviour and fracture.

Hyperelastic constitutive relations form the basis of advanced models for novel materials. Such elastic deformation potentials are the backbone for complex material formulations at elastic and inelastic deformations, especially when embedded into powerful frameworks like generalized standard materials, as well as multiphysical and multiscale formulations. With the focus on electroactive polymers, the article at hand demonstrates the derivation of a variational, rate-dependent electromechanical model for quasi-incompressible polymers and the derivation of an electromechanical model for regularized fracture mechanics by means of the phase-field method. Starting at the prominent Ogden and the extended tube model, some developments from the last decades are revisited and presented via the principle of virtual power, for instance, the established mixed element formulation, nonlinear viscoelasticity and electromechanical coupling. An electromechanically fully coupled representative crack element is used to derive a novel phase-field model for fracture. A key property of the proposed model is the ability to describe the electrical free-space behaviour inside the crack gap, which is demonstrated by adopting three common crack-face conditions. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

A general framework for decomposing the phase field fracture driving force, particularised to a Drucker–Prager failure surface

Due to its computational robustness and versatility, the phase field fracture model has become the preferred tool for predicting a wide range of cracking phenomena. However, in its conventional form, its intrinsic tension–compression symmetry in damage evolution prevents its application to the modelling of compressive failures in brittle and quasi-brittle solids, such as concrete or rock materials. In this work, we present a general methodology for decomposing the phase field fracture driving force, the strain energy density, so as to reproduce asymmetrical tension–compression fracture behaviour. The generalised approach presented is particularised to the case of linear elastic solids and the Drucker–Prager failure criterion. The ability of the presented model to capture the compressive failure of brittle materials is showcased by numerically implementing the resulting strain energy split formulation and addressing four case studies of particular interest. Firstly, insight is gained into the capabilities of the model in predicting friction and dilatancy effects under shear loading. Secondly, virtual direct shear tests are conducted to assess fracture predictions under different pressure levels. Thirdly, a concrete cylinder is subjected to uniaxial and triaxial compression to investigate the influence of confinement. Finally, the localised failure of a soil slope is predicted and the results are compared with other formulations for the strain energy decomposition proposed in the literature. The results provide a good qualitative agreement with experimental observations and demonstrate the capabilities of phase field fracture methods to predict crack nucleation and growth under multi-axial loading in materials exhibiting asymmetric tension–compression fracture behaviour.

Adaptive isogeometric analysis–based phase-field modeling of brittle electromechanical fracture in piezoceramics

The investigation of brittle fracture in piezoceramics under complex electromechanical loading is critical for their durable design and optimal utilization. Phase-field modeling offers a convenient and effective strategy to tackle three-dimensional (3D) fracture problems through the regularization of sharp crack topologies. This paper aims to develop an adaptive phase-field model to study electromechanical fracture in piezoceramics via an isogeometric formulation based on polynomial splines over hierarchical T-meshes (PHT-splines). In particular, we (i) consider the evolution of the crack phase-field within the framework of coupled electromechanical constitutive relationships, (ii) implement PHT-splines to make adaptive refinement computationally efficient and overcome the limitation of nonuniform rational B-splines-based isogeometric formulations, (iii) benchmark our findings with experiments and other numerical studies, and (iv) capture complex crack propagation patterns including deflection and twisting under different complex electromechanical loading conditions in 2D and 3D cracked piezoceramics. The computational efficiency of the implemented phase-field model in cracked piezoceramics is improved through facilitating the adaptive mesh refinement during crack propagation. The proposed scheme lays down the foundation for modelling the diffusive electromechanical fracture in cracked piezoceramics.

Towards validation of crack nucleation criteria from V-notches in quasi-brittle metallic alloys: Energetics or strength?

We discuss crack nucleation observations from a series of experiments that have proven to be particularly challenging for model validation. In particular, we focus attention on crack nucleation from a series of V-notched specimens of an AISI 4340 steel alloy subjected to four-point bending. Details of the specimen geometry, loading, and experimentally-measured forces are provided for V-notches with opening angles of 30, 60, and 90 degrees. Despite the relatively simple geometry and loading, and the fact that the specimens exhibit small scale yielding, model-based simulations that can accurately reproduce the critical forces at crack nucleation across the full set of specimens have proven to be elusive. Along these lines, we present model-based simulation results from finite-element discretizations of phase-field models for fracture. We provide results using both a well-established phase-field model for brittle materials as well as a recently-developed phase-field model for ductile materials. Importantly, both models tie crack nucleation to critical values of energy, and both are shown to be lacking in their accuracy to varying degrees. We also use standard finite-element calculations of a calibrated plasticity model to demonstrate that much greater accuracy in the calculated critical forces can be obtained with the use of a particular strength envelope.

A phase field model for fatigue fracture in piezoelectric solids: A residual controlled staggered scheme

Prediction of the fatigue life of piezoelectric devices and structures is of scientific and engineering significance. In this work, a phase field model for fatigue fracture in piezoelectric materials is developed by means of the Hamilton principle. Two classical phase field fracture models are unified in the present framework. A cumulative history variable is introduced to quantify the degradation of the fracture toughness under cyclic loading. The proposed model accounts for three representative electric boundary conditions on crack faces describing the fatigue fracture behaviors in different physical situations. The residual controlled staggered scheme is extended to address the fatigue fracture problems in the context of electro-elasticity. The present algorithm overcomes the convergence problem for the monolithic approach and shows higher accuracy with lower computational effort in comparison to the single-pass staggered scheme. Systematic numerical simulations are carried out in 2D and 3D cases. The effects of the electric boundary condition, external electric field, and the geometry of the crack tip on the fatigue fracture behaviors of piezoelectric solids have been discussed. The effect of the electric boundary conditions on the fatigue life varies depending on the direction of the external electric field. The present work is beneficial to assess the lifetimes of piezoelectric devices in practical applications.

Three-dimensional phase-field modeling of brittle fracture using an adaptive octree-based scaled boundary finite element approach

In this paper, an adaptive phase-field approach is proposed for three-dimensional fracture modeling in brittle materials. The scaled boundary finite element method (SBFEM) is used to solve the phase-field and elasticity equations in a staggered scheme. This is motivated by the ability of the SBFEM to handle polyhedral elements with hanging nodes straightforwardly. Such elements occur in octree meshes, which facilitate rapid transitions in element size and lend themselves to adaptive refinement. The SBFEM also provides an energy-based error indicator, which follows directly from the semi-analytical solution approach and does not require stress recovery. The error indicator is used to ensure the solution accuracy for both fields and combined with a criterion based on phase-field evolution. The computational cost associated with 3D fracture modeling is further reduced by exploiting the geometric similarity of cells occurring in balanced octree meshes. To validate the proposed method, several classical benchmark problems are solved, resulting in efficient and robust solutions compared to the literature.

Fracture Characterization of Advanced High Strength Steel USS CR980XG3 Using Phase-Field Diffusive Approach in Ductile Solids

In this work, the utilization of a phase-field ductile fracture model in the failure analysis of advanced steels is investigated. The importance of advanced steels is potentially proven, for instance in automotive industry, due to its light weight, which entails the crucial role of fracture analysis in these structures. A third generation advanced high strength USS CR980XG3™️ AHSS material is considered to perform fracture analyses. For this purpose, the necessary data regarding the stress distribution and fracture patterns from digital image correlation tests are utilized for subsequent numerical experimentations.A recent phase-field model of ductile fracture is employed herein for the analysis of crack advance in this class of materials. The significance of the choice of material properties using this model is shown through the analysis of an experimental fracture benchmark called Shear Fracture specimen, through assessment of crack evolution and force diagrams.