Crack Phase Field Research Articles

Investigation on mixed mode I/II crack propagation in nitrate ester plasticized polyether propellant: Experimental and numerical study

The fracture of solid propellant is predominantly attributed to the existence of mixed mode cracks, so it is essential to investigate the the fracture behavior of solid propellant with mixed mode I/II crack. This paper presents fracture characteristics of nitrate ester plasticized polyether (NEPE) propellant under different crack inclination angles (β = 30°–90°). Based on the combination of a drawing machine and a high-speed camera, the mechanical response, crack propagation velocity and crack-path morphology were investigated. The critical equivalent stress intensity factor Keqc was calculated to assess the fracture toughness of the NEPE propellant, and a potential simplified criterion related to the stress intensity factor was proposed. The experimental results demonstrated that the NEPE propellant with mixed mode I/II crack exhibited blunting fracture phenomena during crack propagation, resulting in fluctuating crack propagation velocity. As the crack inclination angle decreases, the fracture toughness of the NEPE propellant increases and then decreases, and the value of Keqc reaches its maximum at β = 45°. Furthermore, numerical studies based on bond-based peridynamic (BBPD) were performed by modeling the crack propagation process of the NEPE propellant, including the crack phase field diagram and the load–displacement curve of the NEPE propellant. The simulation results were then compared with the experiments.

A mixed-element phase field method for the fracture analysis of beams

The prediction of fracture in thin structures is crucial for the safety assessments in various fields of engineering. Modeling methods such as the phase-field method typically assume damage to be constant across the thickness of structural models like beams, plates and shells which, especially in load cases coupling stretching and bending, is an imperfect approximation. In this contribution, fracture behaviors along the thickness direction of Euler–Bernoulli beam considering axial deformation are addressed with a phase-field modeling approach. To achieve this, a mixed-element beam model is introduced, which combines 1D beam elements representing the displacement field with 2D rectangular elements describing a crack phase-field. The nonlinear coupled governing equations are solved using a staggered scheme. The transfer of variables between the coupled fields occurs at the quadrature points which in turn need to have corresponding geometric locations. Numerical examples ranging from the problems of fracture in standalone beams to lattice structures show the applicability of the approach, and detailed comparisons with a standard 2D model confirm its accuracy and efficiency.

Phase-field modeling of fracture for ferromagnetic materials through Maxwell’s equation

Electro-active materials are classified as electrostrictive and piezoelectric materials. They deform under the action of an external electric field. Piezoelectric material, as a special class of active materials, can produce an internal electric field when subjected to mechanical stress or strain. In return, there is the converse piezoelectric response, which expresses the induction of the mechanical deformation in the material when it is subjected to the application of the electric field. This work presents a variational-based computational modeling approach for failure prediction of ferromagnetic materials. In order to solve this problem, a coupling between magnetostriction and mechanics is modeled, then the fracture mechanism in ferromagnetic materials is investigated. Furthermore, the failure mechanics of ferromagnetic materials under the magnetostrictive effects is studied based on a variational phase-field model of fracture. Phase-field fracture is numerically challenging since the energy functional may admit several local minima, imposing the global irreversibility of the fracture field and dependency of regularization parameters related discretization size. Here, the failure behavior of a magnetoelastic solid body is formulated based on the Helmholtz free energy function, in which the strain tensor, the magnetic induction vector, and the crack phase-field are introduced as state variables. This coupled formulation leads to a continuity equation for the magnetic vector potential through well-known Maxwell’s equations. Hence, the energetic crack driving force is governed by the coupled magneto-mechanical effects under the magneto-static state. Several numerical results substantiate our developments.

Numerical analysis of fracture in core-shell particle reinforced composites

The fracture of core-shell particle reinforced composites was simulated using a crack phase field method. A pre-existing crack may grow along the interface between the core and shell (core debonding), may penetrate the shell and the core (trans-core fracture), or may grow in the matrix (brittle fracture in the matrix). Shell fracture behaviours resulting from the competition between the fracture resistance and fracture driving force are crucial to the crack growth mechanisms. It was found that moderate strength and low modulus of the shell favoured trans-core fracture, and low strength and high modulus of the shell favoured core debonding. The effects on the overall mechanical properties were also identified. The composite strength usually benefited from stiffer and tougher components, regardless of the crack growth mechanisms, while the composite toughness was more complex. The toughening effect was highly related to how the crack evolved, as core debonding improved the toughness at the significant cost of strength, and trans-core fracture contributed most to the toughness with a slight reduction in strength compared with the matrix. These findings indicate design strategies for epoxy composites which can achieve a balance of strength and toughness, enabling the design of safer lightweight composites for electric vehicles and transport applications.

Cracking and thermal resistance in concrete: Coupled thermo-mechanics and phase-field modeling

Modeling damage and fracture in concrete accurately is crucial for a comprehensive analysis of structures under concurrent thermal and mechanical actions. This paper presents a novel coupled thermo-mechanical phase field model for concrete at high temperatures. Derived from principles of thermodynamics and unified phase field theory, the model integrates temperature-dependent thermal and mechanical properties, accounting for thermal expansion and transient creep. Mechanical damage, represented by the crack phase field, is augmented by a temperature-dependent thermal damage variable for heat-induced degradation. Addressing the impact of cracks on heat conduction and stress redistributions, the model introduces a novel approach to depict thermal resistance, seamlessly integrating mechanics, heat transfer, and crack propagation. Governing equations for damage evolution are derived, and the numerical implementation is detailed. Numerical illustrations showcase the model's ability to simulate diverse thermo-mechanical cracking patterns without explicit fracture path tracking or assumed criteria. Moreover, the model captures the influence of thermal actions on concrete cracking behavior and mechanical performance. This research is significant for predicting multi-field coupling damage in concrete and structures exposed to elevated temperatures.

Adaptive isogeometric analysis–based phase-field modeling of interfacial fracture in piezoelectric composites

This paper implements an adaptive phase-field model to investigate the interfacial fracture in transversely isotropic piezoelectric composites under different electromechanical loadings using an isogeometric framework based on polynomial splines over hierarchical T-meshes (PHT-splines). The model introduces the interface and crack phase-field parameters to regularize the sharp interface and crack topologies, respectively and the regularized energy terms are incorporated corresponding to crack and interface evading the need for line integration along them. It also modifies the energy functional to scrutinize the interaction between interfacial damage and crack propagation in piezocomposites and thus is capable of capturing the fracture patterns including interfacial debonding, matrix cracking and their interactions under different electric fields. An adaptive mesh refinement scheme using PHT-splines is implemented to efficiently resolve for interface and crack phase-fields and track the complex crack propagation paths without any ad hoc criteria. The efficacy and robustness of the proposed model are demonstrated using a set of examples simulating the electromechanical fracture in piezoelectric composites. Moreover, different fracture mechanisms in composite materials including the crack nucleation, interfacial debonding, matrix cracking, crack propagation and coalescence are found to be precisely captured using the present modeling technique.

Coupled thermo‐viscoplastic fracture model for ductile‐brittle failure of amorphous glassy polymers with phase‐field approach

AbstractAmorphous glassy polymers have an extensive use in the industrial sectors including micro‐electronics, medical industry and aerospace, therefore their design and usage have become a significant task nowadays. The fracture response of these polymers may vary from ductile to brittle depending on several factors such as entanglement density, temperature level and external loading rate. The ductile response is driven by diffuse shear zones exhibiting volume–preserving inelastic deformations while the brittle response is manifested by very small crack‐like defects composed of a sequence of fibrillar bridges separated by micro‐voids, thereby connotating void formation consisting of nucleation and propagation steps. The presents study is focused on the description of shear yielding and crazing phenomenon in terms of their respective evolution equations. In addition, an extension towards the modelling of the fracture is employed via the crack phase–field approach, considering ductile and brittle failure simultaneously. This is provided by the novel failure criterion that features a critical amount of plastic strain and void volume fraction. Since the proposed approach unitedly models the macroscopic crack initiation and propagation for ductile or brittle failure, it is asserted to be more physically grounded compared to present models in the literature. Constitutive formulations for shear yielding, crazing, and void volume fraction are derived with their specific forms starting with the local and conductive component of the dissipation inequality. The performance of model is evaluated after developing the local and global Newton–type update algorithms for the dissipative internal and primary field variables, respectively and it has been analysed by fitting of several experimental data of homogeneous and inhomogeneous tests. The findings reveal the remarkable temperature dependency on the type of failure as well as the interaction between loading rate and temperature change owing to dissipative heating in the solid.

Adaptive fourth-order phase-field modeling of ductile fracture using an isogeometric-meshfree approach

The fourth-order phase-field modeling of ductile fracture in elastic–plastic materials is performed via an adaptive isogeometric-meshfree approach. In the developed phase-field model, the total energy functional consists of the elastic contribution and the dissipated contribution because of fracture and plasticity. The coupling of the plasticity to fracture is implemented by a degradation function that is applied to the elastic energy. The present fourth-order phase-field model is capable of relaxing the mesh size requirements while accurately regularizing sharp cracks. To further enhance the computational efficiency, the isogeometric-meshfree approach is adopted for the numerical implementation of the phase-field model within a staggered computational framework. The developed approach can flexibly implement the C1-continuity of a crack phase field that is required by the fourth-order model. Moreover, an adaptive mesh refinement strategy is developed, which includes the gradient-based refinement indicators and the field transfer operators. Numerical simulations of a series of representative cases show that the developed fourth-order model can accurately and efficiently capture complex ductile fracture patterns including plastic localization, crack initiation, propagation, and merging, which demonstrates the reliability of the adaptive fourth-order phase-field modeling of ductile fracture.

Cohesive zone phase field model for electromechanical fracture in multiphase piezoelectric composites

This paper introduces a coupled electromechanical finite deformation phase field model for crack propagation and interfacial decohesion in multiphase piezoelectric composites with interfaces. The crack phase field model is augmented with cohesive traction-separation laws at the material interfaces, derived from a cohesive potential function. A Gibbs free energy density function is proposed, allowing for the incorporation of the anisotropic elastic stiffness of the piezoelectric material. Numerical simulations exhibiting different failure mechanisms are carried out to demonstrate the efficacy of the model. Effects of external electric field on crack evolution and the competition between penetration and deflection of a crack impinging on an interface are investigated. Limited verification tests are conducted with theoretical results. Finally, the model is used to simulate fracture in nonuniform piezocomposite microstructures. The effect of crack propagation on the evolution of the electric field with different crack face conditions are analyzed. Differences in the electromechanical responses of piezocomposites due to different fiber distributions are also observed.

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.

Numerical analysis of fracture in interpenetrating phase composites based on crack phase field model

A numerical model based on crack phase field analysis is introduced to study the quasi-static fracture process in interpenetrating phase composites (IPCs). Materials were considered elastic solids, and the interface was assumed to be perfectly bonded. Tougher and stiffer tougheners lead to more fracture in the brittle phase, but less fracture in the toughening phase. Thus, the overall fracture performance results from competition between increasing breakage in the brittle phase and declining breakage in the toughening phase. The toughening mechanisms are discussed from both stress-strain and crack topology viewpoints. The toughening phase transfers the load from the crack tip to the whole domain until the maximum stress is reached, and impeded crack growth occurs afterwards. The load transferring and impediment effects made the brittle phase engage in fracture, and several crack propagation patterns were identified for the sacrificial fracture behaviour, namely, crack deflection, crack bridging, crack branching, microcracking and crack blocking. Moreover, fracture in three different microstructures (co-continuous, particle-reinforced, laminar) was compared, and the most effective toughening morphology depends on the tougheners and the loading states. This methodology enables optimum microstructures to be identified to achieve high toughness in aerospace and energy generation applications, increasing safety and reducing weight.

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.

A thermodynamically consistent finite strain phase field approach to ductile fracture considering multi-axial stress states

Phase field models for ductile fracture have gained significant attention in the last two decades due to their ability in implicitly tracking the nucleation and propagation of cracks. However, most crack phase field formulations for elastoplastic solids focus only on the effects of plastic deformation, and do not consider the different multi-axial stress states that may arise in practical designs. In this work, a thermodynamically consistent phase field approach coupled with finite strain plasticity, considering multi-axial stress states is presented. In order to account for the coupling between plasticity and stress states, the Stress-Weighted Ductile Fracture Model (SWDFM) is utilized. The SWDFM represents a criterion for predicting ductile crack initiation under both monotonic and cyclic loadings based on histories of an internal plastic variable, stress triaxiality, and the Lode angle parameter. The excellent performance of the SWDFM for predicting ductile crack initiation motivates for its incorporation into a phase field approach for predicting both crack initiation and propagation through degradation of the fracture toughness. Moreover, based on the second law of thermodynamics, exact requirements are imposed on the rate at which the fracture toughness can evolve. A novel function for degrading the plastic yield surface during the evolution of damage is introduced. This function, in line with experimental observations, leads to an accumulation of plastic deformation in damaged regions of a solid, and avoids numerical instabilities arising from concentrations of large plastic deformations in severely damaged regions. For validating the proposed model, results of computational simulations are compared to data from selected tests considering different multi-axial stress states. Comparisons of the numerical results with data from laboratory experiments demonstrate the capabilities of the proposed framework.

Multi phase-field modeling of anisotropic crack propagation in 3D fiber-reinforced composites based on an adaptive isogeometric meshfree collocation method

An anisotropic multi phase-field model is developed in this paper to investigate the interaction between the crack propagation and interfacial damage in a fiber-reinforced composite. The proposed modeling approach can capture the fracture patterns of composite microstructures through the introduction of the crack phase field and the interface phase field. By considering the free energy related to the interface and crack phase fields, the interfacial debonding, matrix cracking and the interaction of two fracture patterns can thus be simulated. The anisotropic phase-field approach is further adopted to describe the interface interaction associated with a crack. The phase-field equations are solved using the isogeometric-meshfree collocation approach to achieve high computational efficiency. An adaptive h-refinement scheme is incorporated into the phase-field formulations using phase-field variables and their gradients as the error indicators. The proposed method is shown to be effective and robust in both case studies of 2D and 3D fiber-reinforced composite microstructures. Moreover, fracture behaviors including the crack initiation, propagation, coalescence, interfacial debonding, and matrix cracking in composite microstructures are found to be precisely modeled by the proposed approach.

Multilevel global–local techniques for adaptive ductile phase-field fracture

This paper outlines a rigorous variational-based multilevel Global–Local formulation for ductile fracture. Here, a phase-field formulation is used to resolve failure mechanisms by regularizing the sharp crack topology on the local state. The coupling of plasticity to the crack phase-field is realized by a constitutive work density function, which is characterized through a degraded stored elastic energy and the accumulated dissipated energy due to plasticity and damage. Two different Global–Local approaches based on the idea of multiplicative Schwarz’ alternating method are proposed: (i) A global constitutive model with an elastic–plastic behavior is first proposed, while it is enhanced with a single local domain, which, in turn, describes an elastic–plastic fracturing response. (ii) The main objective of the second approach is to introduce an adoption of the Global–Local model toward the multilevel local setting. In (ii), an elastic–plastic global constitutive model is augmented with two distinct local domains; in which, the first local domain behaves as an elastic–plastic material and the second local domain is modeled due to the fractured state. To further reduce the computational cost, predictor–corrector adaptivity within the Global–Local concept is introduced. An adaptive scheme is devised through the evolution of the effective global plastic flow (for only elastic–plastic adaptivity), and through the evolution of the local crack phase-field state (for only fracture adaptivity). Thus, two local domains are dynamically updated during the computation, resulting in a two-way adaptivity procedure. The overall response of the Global–Local approach in terms of accuracy/robustness and efficiency is verified using single-scale problems. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

Phase-field modeling of electromechanical fracture in piezoelectric solids: Analytical results and numerical simulations

The optimal design, reliable manufacture and durable utilization of electromechanical systems demand objective modeling of failure under coupled electric field and mechanical actions. This is still a challenging and largely an open issue despite the large volumes of contributions from the discontinuous approach and the phase-field community for brittle fracture. In this work we propose a length scale insensitive phase-field cohesive zone model (PF-CZM) for electromechanical fracture in linear piezoelectric solids, overcoming all the issues of mesh dependence, crack tracking complexity, length scale sensitivity and crack nucleation inconsistency. More specifically, in the electromechanically coupled phase-field constitutive relations, the evolution law for the crack phase-field is established from the geometric regularization of sharp cracks and the energetic criterion in global form. In order to capture the unsymmetric mechanical and electrical fracture behavior, the mechanical energy release rate contributed from the positive cone of the effective stress in energy norm is employed as the crack driving force, resulting in a hybrid formulation. With a set of phase-field characteristic functions optimal for cohesive fracture, the critical stress upon crack nucleation and the post-peak softening regime are derived in close-form for the first time, and the experimentally observed effects of the electric field on the fracture load are predicted qualitatively. The proposed electromechanical PF-CZM is numerically implemented in the multi-field finite element method and applied to several representative examples. In all cases, as those for purely mechanical problems, the predicted crack patterns and fracture loads are insensitive to the phase-field length scale and the experimental results are well reproduced. In particular, the effect of electric field on the deflection of crack paths can be captured. These merits make the proposed PF-CZM promising in the modeling of electromechanical fracture in piezoelectric solids.

Bayesian inversion for anisotropic hydraulic phase-field fracture

In this work, we employ a Bayesian inversion framework to fluid-filled phase-field fracture. We develop a robust and efficient numerical algorithm for hydraulic phase-field fracture toward transversely isotropic and orthotropy anisotropic fracture. In the fluid-driven coupled problem, three primary fields for pressure, displacements, and the crack phase-field are solved while direction-dependent responses (due to the preferred fiber orientation) are enforced via penalty-like parameters. A new crack driving state function is introduced by avoiding the compressible part of anisotropic energy to be degraded. Next, we use a successful extension of the anisotropic hydraulic phase-field fracture as a departure point for Bayesian inversion to estimate material parameters. To this end, we employ the delayed rejection adaptive Metropolis–Hastings (DRAM) algorithm to identify the parameters. The focus is on uncertainties arising from different variables, including elasticity modulus, Biot’s coefficient, Biot’s modulus, dynamic fluid viscosity and Griffith’s energy release rate in the case of the isotopic hydraulic fracture while in the anisotropic setting, we will have additional penalty-like parameters to be identified. Several numerical examples substantiate our algorithmic developments.

Interaction of anisotropic crack phase field with interface cohesive zone model for fiber reinforced composites

A new phase field model considering the interfacial damage for different configurations of a fiber reinforced composite is proposed and formulated. Crack and non local interface are considered to be diffused. A coupled traction separation law based on a potential function is adopted to represent the behavior of the interface. Anisotropy is introduced into the elastic equilibrium by considering the distinct contributions of fiber and matrix in different modes. The present model captures the predominant failure phenomena in a composite such as matrix failure, delamination by considering the role of fiber orientation, interface fracture properties and configuration of lamina. The proposed formulation is extended to a fiber reinforced composite lamina consisting of two fiber families oriented in different directions. Parametric studies are conducted to understand the effect of anisotropy parameter, length scales, fracture properties of fiber, matrix and interface on crack propagation and mechanical response of the whole system. Numerical examples are performed to validate the proposed model, understand the anisotropic crack growth for unidirectional and woven fiber reinforced composites, study the interaction of anisotropic crack with composite-composite interface and metal-composite interface.

Interaction between interfacial damage and crack propagation in quasi-brittle materials

A thermodynamically consistent phase field formulation for modeling the interactions between interfacial damage and bulk brittle fracture is presented. A regularization scheme is considered for both the interface and the crack phase field. A coupled exponential cohesive zone law is adopted to model the interface which has the contributions of both normal and tangential displacement jump components. A novel nonlocal approach is devised to evaluate the smoothened values of jump at the regularized interface using element-specific geometric information. Such a description of the interface allows a more realistic mechanical response of any composite system in terms of accurately representing failure modes such as matrix/bulk cracking, deflection and or branching of crack at interface, through thickness penetration at interface, and delamination. The possibility of these would depend on modulus mismatch, inclination of the interface, length of the interface and relative fracture toughness of the interface to the bulk. Several numerical examples are solved to validate the performance of the proposed model by peel test, crack interaction with an inclined interface, crack at a bimaterial interface, and stiff-soft interface crack interaction.

Interior-point methods for the phase-field approach to brittle and ductile fracture

The governing equations of the variational approach to brittle and ductile fracture emerge from the minimization of a non-convex energy functional subject to irreversibility constraints. This results in a multifield problem governed by a mechanical balance equation and evolution equations for the internal variables. While the balance equation is subject to kinematic admissibility of the displacement field, the evolution equations for the internal variables are subject to irreversibility conditions, and take the form of variational inequalities, which are typically solved in a relaxed or penalized way that can lead to deviations of the actual solution. This paper presents an interior-point method that allows to rigorously solve the system of variational inequalities. With this method, a sequence of perturbed constraints is considered, which, in the limit, recovers the original constrained problem. As such, no penalty parameters or modifications of the governing equations are involved. The interior-point method is applied in both a staggered and a monolithic scheme for both brittle and ductile fracture models. In order to stabilize the monolithic scheme, a perturbation is applied to the Hessian matrix of the energy functional. The presented algorithms are applied to three benchmark problems and compared to conventional methods, where irreversibility of the crack phase-field is imposed using a history field or an augmented Lagrangian.