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This work proposes a novel approach for coupling non‐isothermal fluid dynamics with fracture mechanics to capture thermal effects within fluid‐filled fractures accurately. This method addresses critical aspects of calculating fracture width in enhanced geothermal systems, where the temperature effects of fractures are crucial. The proposed algorithm features an iterative coupling between an interface‐capturing phase‐field fracture method and interface‐tracking thermo‐fluid‐structure interaction using arbitrary Lagrangian–Eulerian coordinates. We use a phase‐field approach to represent fractures and reconstruct the geometry to frame a thermo‐fluid‐structure interaction problem, resulting in pressure and temperature fields that drive fracture propagation. We developed a novel phase‐field interface model accounting for thermal effects, enabling the coupling of quantities specific to the fluid‐filled fracture with the phase‐field model through the interface between the fracture and the intact solid domain. We provide several numerical examples to demonstrate the capabilities of the proposed algorithm. In particular, we analyze mesh convergence of our phase‐field interface model, investigate the effects of temperature on crack width and volume in a static regime, and highlight the method's potential for modeling slowly propagating fractures. [ABSTRACT FROM AUTHOR]

Significant interest and developments in phase-field fracture modeling have emerged. It approximates fracture continuously using a length-scale parameter, accurately simulating fracture initiation and crack propagation in brittle materials like Polymethyl Methacrylate (PMMA) with nonlinear elasticity. This study proposes a phase-field approach for simulating PMMA fracture behavior using ABAQUS with a nonlinear elastic material model (UMAT) and a three-layered element composition (UEL). The staggered approach sequentially solves displacements and phase-field variables. Numerical results are validated against experiments considering various geometries, stress concentrators, and pre-cracks. The computational approach accurately predicts fracture initiation and crack growth in complex patterns and diverse loading conditions. [ABSTRACT FROM AUTHOR]

Understanding the relation between the microstructure and the material's effective behavior is an important aspect in inverse computational materials engineering. Especially in the context of localized, inelastic phenomena like plasticity and crack growth, the microstructure morphology plays a crucial role. Due to the stochastic nature of heterogeneous media, a statistical analysis over multiple simulations is necessary, since even with the same material, the simulated crack paths and effective crack lengths are highly dependent on the specific locations of microstructural features. A relevant factor that limits this type of investigation is the high cost of real microstructure data. This work presents a digital workflow for exploring the fracture properties of materials. Therein, the required statistical analyses are facilitated by an algorithm that reconstructs multiple realization of a material structure given a single example. The reconstructed structures are discretized with a regular non-conforming mesh with a diffuse interface and crack representation. Crack phase-field simulations are conducted in order to analyze the effective response. An in-depth introduction to the required methods is given together with a statistical evaluation of the conducted numerical experiments. It is concluded that the statistical variation of the effective material behavior overshadows morphological trends in the presented case. This confirms the relevance and utility of complementing simulation-based workflows with microstructure reconstruction and statistical analyses. [ABSTRACT FROM AUTHOR]

Phase-field models of brittle fracture are typically endowed with a decomposition of the elastic strain energy density in order to realistically describe fracture under multi-axial stress states. In this contribution, we identify the essential requirements for this decomposition to correctly describe both nucleation and propagation of cracks. Discussing the evolution of the elastic domains in the strain and stress spaces as damage evolves, we highlight the links between the nucleation and propagation conditions and the modulation of the elastic energy with the phase-field variable. In light of the identified requirements, we review some of the existing energy decompositions, showcasing their merits and limitations, and conclude that none of them is able to fulfil all requirements. As a partial remedy to this outcome, we propose a new energy decomposition, denoted as star-convex model, which involves a minimal modification of the volumetric-deviatoric decomposition. Predictions of the star-convex model are compared with those of the existing models with different numerical tests encompassing both nucleation and propagation. [ABSTRACT FROM AUTHOR]

In the realm of fracture mechanics and phase-field modeling, the utilization of exponential finite element (EFE) shape functions has shown promise in accurately predicting fracture responses, particularly in scenarios with intricate crack propagation paths. However, the computational complexity associated with EFE shape functions, including higher integration schemes and orientation requirements, prompts the need for optimization. This study delves into the critical aspect of integration point density and aims to determine the ideal balance between accuracy and computational efficiency in the context of EFE shape functions. The research focuses on functionally graded materials, which inherently exhibit spatial variations in solution quantities due to their graded material system. Through a systematic investigation, the study aims to identify the optimal number of integration points required to maximize the computational advantages offered by EFE shape functions while minimizing the associated computational resources. By conducting a comprehensive analysis across various loading scenarios, such as tension and shear, this study intends to establish a quantitative relationship between integration point density and prediction accuracy for fracture responses. [ABSTRACT FROM AUTHOR]

Lithium Nickel Manganese Cobalt Oxides (NMC) are one of the most used cathode materials in lithium-ion batteries, and they will become more relevant in the following years due to their potential in electric vehicles. Unfortunately, this material experiences microcracking during the battery operation due to the volume variations, which is detrimental to the battery performance and limits the lifetime of the electrodes. Thus, understanding mechanical degradation is fundamental for the development of advanced batteries with improved capacity and limited degradation. In this work, we propose a chemo-mechanical model, including a stochastic phase-field fracture approach, to design structurally stable NMC electrodes. We include the degradation in the mechanical and chemical contributions. The heterogeneous NMC microstructure is considered by representing the material's tensile strength with a Weibull distribution function, which allows to represent complex and non-deterministic crack patterns.We use our model to provide a comprehensive analysis of mechanical degradation in NMC111 electrodes, including the effect of particle size, C-rate, and depth of charge and discharge. Then, we analyze the influence of the electrode composition (namely, Ni content) on the structural integrity. We use this information to provide design guides for functionally-graded electrodes with high capacity and limited degradation.

Fracture of materials with rate-dependent mechanical behaviour, e.g. polymers, is a highly complex process. For an adequate modelling, the coupling between rate-dependent stiffness, dissipative mechanisms present in the bulk material and crack driving force has to be accounted for in an appropriate manner. In addition, the resistance against crack propagation can depend on rate of deformation. In this contribution, an energetic phase-field model of rate-dependent fracture at finite deformation is presented. For the deformation of the bulk material, a formulation of finite viscoelasticity is adopted with strain energy densities of Ogden type assumed. The unified formulation allows to study different expressions for the fracture driving force. Furthermore, a possibly rate-dependent toughness is incorporated. The model is calibrated using experimental results from the literature for an elastomer and predictions are qualitatively and quantitatively validated against experimental data. Predictive capabilities of the model are studied for monotonic loads as well as creep fracture. Symmetrical and asymmetrical crack patterns are discussed and the influence of a dissipative fracture driving force contribution is analysed. It is shown that, different from ductile fracture of metals, such a driving force is not required for an adequate simulation of experimentally observable crack paths and is not favourable for the description of failure in viscoelastic rubbery polymers. Furthermore, the influence of a rate-dependent toughness is discussed by means of a numerical study. From a phenomenological point of view, it is demonstrated that rate-dependency of resistance against crack propagation can be an essential ingredient for the model when specific effects such as rate-dependent brittle-to-ductile transitions shall be described. [ABSTRACT FROM AUTHOR]

Variational phase-field models of fracture are widely used to simulate nucleation and propagation of cracks in brittle materials. They are based on the approximation of the solutions of a free-discontinuity fracture energy by two smooth function: a displacement and a damage field. Their numerical implementation is typically based on the discretization of both fields by nodal P 1 Lagrange finite elements. In this article, we propose a nonconforming approximation by discontinuous elements for the displacement and nonconforming elements, whose gradient is more isotropic, for the damage. The handling of the nonconformity is derived from that of heterogeneous diffusion problems. We illustrate the robustness and versatility of the proposed method through series of examples. [ABSTRACT FROM AUTHOR]

In the recent decade, there has been a growing interest in using the phase-field approach to model fracture processes in various materials. Conventional phase-field implementations can simulate fracture processes using bi-linear finite element (LFE) shape functions but at the expense of a very fine mesh. In contrast, exponential finite element (EFE) shape functions can predict sharp gradients in solution variables with coarse meshes due to their exponential nature. A potential advantage lies in reducing the number of elements in the problem without losing accuracy in the solution. However, EFE shape functions do not yield a good approximation unless they are oriented relative to the expected crack propagation path. This study uses an approximate analysis using LFE shape functions to orient the EFE shape functions before the computations. Computational advantages are reported in terms of accuracy in predicted load responses and the computational times incurred. [ABSTRACT FROM AUTHOR]

The paper proposes a novel staggered phase-field framework for modelling brittle and ductile fractures in monotonic and cyclic loading regimes. The algorithm consists of two mesh layers (displacement and phase field) and a single special-purpose, user-defined finite element, which controls global convergence of the coupled problem and passing of the solution variables between mesh layers. The proposed algorithm is implemented into FE software ABAQUS. For the problem of high cyclic fatigue, a cycle-skipping scheme is also introduced. The proposed methodology is verified on the usual benchmark examples. Small-strain theory is applied, but it has been demonstrated that extension to large strains is straightforward using only the ABAQUS built-in option. The efficiency and stability of the proposed framework was proven by comparison of computational time and the number of iterations per increment in the RCTRL scheme. [ABSTRACT FROM AUTHOR]

This work proposes a thermodynamically consistent phase-field model for anisotropic brittle material under the hypotheses of plane stress, small deformation and constant temperature. The model is derived from the principle of virtual power, the first and second laws of thermodynamics in the form of the Clausius-Duhem inequality. The degradation effect on the material behavior is given by a fourth-order degradation tensor introduced as an internal variable that evolves according to the current strain state rather than the conventional scalar degradation function of phase-field models. Therefore, local anisotropy can be induced, changing the material mechanical behavior differently in all directions organically. The proposed degradation tensor is defined in the global coordinate system and therefore is sensitive to any change in the principal directions of the strain and stress states. To demonstrate the model’s capability of representing damage in isotropic and transversely isotropic materials, some benchmark examples were carried out and the evolution of the damage components was analyzed.

The paper proposed a novel framework for efficient simulation of crack propagation in brittle materials. In the present work, the phase field represents the sharp crack surface with a diffuse fracture zone and captures the crack path implicitly. The partial differential equations of the phase field models are solved with physics informed neural networks (PINN) by minimizing the variational energy. We introduce to the PINN-based phase field model the degradation function that decouples the phase-field and physical length scales, whereby reducing the mesh density for resolving diffuse fracture zones. The numerical results demonstrate the accuracy and efficiency of the proposed algorithm.

Phase-field models of brittle fracture can be regarded as gradient damage models including an intrinsic internal length. This length determines the stability threshold of solutions with homogeneous damage and thus the strength of the material, and is often tuned to retrieve the experimental strength in uniaxial tensile tests. In this paper, we focus on multiaxial stress states and show that the available energy decompositions, introduced to avoid crack interpenetration and to allow for unsymmetric fracture behavior in tension and compression, lead to multiaxial strength surfaces of different but fixed shapes. Thus, once the length scale is tailored to recover the experimental tensile strength, it is not possible to match the experimental compressive or shear strength. We propose a new energy decomposition that enables the straightforward calibration of a multi-axial failure surface of the Drucker-Prager type. The new decomposition, which hinges upon the theory of structured deformations, encompasses the volumetric-deviatoric and the no-tension models as special cases. Preserving the variational structure of the model, it includes an additional free parameter that can be calibrated based on the experimental ratio of the compressive to the tensile strength (or, if possible, of the shear to the tensile strength), as successfully demonstrated on two data sets taken from the literature. [ABSTRACT FROM AUTHOR]

The paper proposes a novel staggered phase-field framework for modelling brittle and ductile fractures in monotonic and cyclic loading regimes. The algorithm consists of two mesh layers (displacement and phase field) and a single special-purpose, user-defined finite element, which controls global convergence of the coupled problem and passing of the solution variables between mesh layers. The proposed algorithm is implemented into FE software ABAQUS. For the problem of high cyclic fatigue, a cycle-skipping scheme is also introduced. The proposed methodology is verified on the usual benchmark examples. Small-strain theory is applied, but it has been demonstrated that extension to large strains is straightforward using only the ABAQUS built-in option. The efficiency and stability of the proposed framework was proven by comparison of computational time and the number of iterations per increment in the RCTRL scheme.

• MEX process using PLA is studied for the development of bone surrogates. • Experimental and numerical analyses of real and artificial bones. • Influence of process parameters on mechanical behaviour. • Numerical modelling through homogenisation technique and phase-field approach. The limited availability of human bone samples for investigation leads to the demand for alternatives. Bone surrogates are crucial in promoting research on the intricate mechanics of osseous tissue. However, solutions are restricted to commercial brands, which frequently fail to faithfully replicate the mechanical response of bone, or oversimplified customised simulants designed for a specific application. The manufacturing and assessment of reliable bone surrogates made of polylactic acid via material extrusion-based additive manufacturing are presented in this work. An experimental and numerical study with 3D-printed dog-bone and prismatic specimens was carried out to characterise the polymeric feedstock and analyse the influence of process parameters under three-point bending and quasi-static conditions. Besides, three porcine rib samples were considered as a reference for the development of the artificial bones. Bone surrogates were manufactured from the 3D-scanned real bone geometries. In order to reproduce the trabecular and cortical bone, a lattice structure for the infill and a compact shell surrounding the core were employed. Infill density and shell thickness were evaluated through different printing configurations. Additionally, a computational analysis based on the phase-field approach was conducted to simulate the experimental tests and predict fracture. The modelling considered homogenisation of the infill material. Outcomes demonstrated the potential of the presented methodology. Maximum force and flexural stiffness were compared to real bone properties to find the optimal printing configuration, replicating the flexural mechanical behaviour of bone tissue. Certain configurations accurately reproduce the studied properties. Regarding the numerical model, strength and stiffness prediction was validated with experimental results. The presented methodology enables the manufacturing of artificial bones with accurate geometries and tailored mechanical properties. Furthermore, the described modelling strategy offers a powerful tool for designing bone surrogates. [ABSTRACT FROM AUTHOR]

The phase-field approach has proven to be a powerful tool for the prediction of crack phenomena. When it is applied to inelastic materials, it is crucial to adequately account for the coupling between dissipative mechanisms present in the bulk and fracture. In this contribution, we propose a unified phase-field model for fracture of viscoelastic materials. The formulation is characterized by the pseudo-energy functional which consists of free energy and dissipation due to fracture. The free energy includes a contribution which is related to viscous dissipation that plays an essential role in coupling the phase-field and the viscous internal variables. The governing equations for the phase-field and the viscous internal variables are deduced in a consistent thermodynamic manner from the pseudo-energy functional. The resulting model establishes a two-way coupling between crack phase-field and relaxation mechanisms, i.e. viscous internal variables explicitly enter the evolution of phase-field and vice versa. Depending on the specific choice of the model parameters, it has flexibility in capturing the possible coupled responses, and the approaches of recently published formulations are obtained as limiting cases. By means of a numerical study of monotonically increasing load, creep and relaxation phenomena, rate-dependency of failure in viscoelastic materials is analysed and modelling assumptions of the present formulation are discussed. [ABSTRACT FROM AUTHOR]

The mechanical behaviour of many materials, including polymers or natural materials, significantly depends on the rate of deformation. As a consequence, a rate-dependent ductile-to-brittle fracture transition may be observed. For toffee-like caramel, this effect is particularly pronounced. At room temperature, this confectionery may be extensively deformed at low strain rates, while it can behave highly brittle when the rate of deformation is raised. Likewise, the material behaviour does significantly depend on temperature, and even a slight cooling may cause a significant embrittlement. In this work, a thorough experimental investigation of the rate-dependent deformation and fracture behaviour is presented. In addition, the influence of temperature on the material response is studied. The experimental results form the basis for a phase-field modelling of fracture. In order to derive the governing equations of the model, an incremental variational principle is introduced. By means of the validated model, an analysis of the experimentally observed ductile-to-brittle fracture transition is performed. In particular, the coupling between the highly dissipative deformation behaviour of the bulk material and the rate-dependent fracture resistance is discussed. • Strain rate-dependant transition from ductile behaviour to brittle fracture. • Experimental investigation of the deformation and fracture of toffee. • Fracture phase-field modelling based on an incremental variational principle. • Rate-dependent fracture resistance. • Equivalence of an increase in strain rate and a decrease in temperature. [ABSTRACT FROM AUTHOR]

Summary: The phase‐field approach is a promising technique for the realistic simulation of brittle fracture processes, both in quasi‐static and transient analysis. Considering fast loading, experimental evidence indicates a strong relationship between the rate of strain and the material's resistance against fracture, which can be considered by a dynamic increase factor for the strength of the material. The paper at hand presents a novel approach within the framework of phase‐field models for brittle fracture. A rate‐dependent fracture toughness is formulated as a function of the rate of crack driving strain components, which results in higher strength for faster loading. Beside the increased amount of energy necessary to evolve a crack at a high strain rate loading situation, the model incorporates quasi‐viscous stress‐type quantities that are not directly related to the formation of the crack and exist only in the phase‐field transition zone between broken and sound material. The governing strong form equations for a transient simulation are derived and the relevant information for an implementation of the model into a finite element code is outlined in detail. The performance of the model is demonstrated for static and dynamic benchmark simulations and for a comparison to experimental findings. [ABSTRACT FROM AUTHOR]

Within this work, we develop a phase-field description for simulating fractures in nearly incompressible materials. It is well-known that low-order approximations generally lead to volume-locking behaviors. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor–Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on three numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson's ratio approximating the incompressible limit, are presented. [ABSTRACT FROM AUTHOR]

We propose a physics and chemistry-based constitutive framework to predict the stress and brittle failure responses of thermo-chemically aged elastomers. High-temperature aging in the presence of oxygen causes chemical reactions inducing significant changes in the elastomer macromolecular network. Experimental studies agree that the increase in effective chemical crosslinks is dominant, leading to increased material stiffness and ultimately causing brittle failure of aged elastomers. The main novelty of this work relies on directly connecting the evolution of the effective crosslink density to the mechanical properties which control stress and failure response of thermo-chemically aged elastomers. We equip the Arruda–Boyce hyperelastic constitutive equations to predict the stress–strain response until failure with phase-field to capture the induced brittle failure via a strain-based criterion for fracture. Four material properties associated with the stress and failure responses evolve to capture the detrimental effects of thermo-chemical aging. The evolution of the four material properties is characterized by the change in the effective crosslink density, obtained based on chemical characterization tests, for any given aging temperature and duration. The established constitutive framework is first solved analytically for the case of uniaxial tension in a homogeneous bar to highlight the interconnection between the four material properties. Then, the framework is numerically implemented within a finite element (FE) context via a user-element subroutine (UEL) in the commercial FE software Abaqus. The framework is validated versus a set of experimental results available in the literature. The comparison confirms that the proposed constitutive framework can accurately predict the stress–strain and failure responses of thermo-chemically aged elastomers. Further numerical examples are provided to demonstrate the effects of evolving material properties on the response of specimens containing pre-existing cracks exploiting the capabilities of the phase-field approach. [Display omitted] • A constitutive framework is developed for thermo-chemically aged elastomers. • Four material properties capture the aging detrimental effects on stress and failure. • The material properties evolve by changes in the macromolecular network of elastomers. • There are no additional fitting constants that do not carry physical meaning. [ABSTRACT FROM AUTHOR]

Fatigue crack growth is decisive for the design of thin-walled structures such as fuselage shells of air planes. The cold rolling process, used to produce the aluminium sheets this structure is made of, leads to anisotropic mechanical properties. In this contribution, we simulate the fatigue crack growth with a phase-field model due to its superior ability to model arbitrary crack paths. A fatigue variable based on the Local Strain Approach describes the progressive weakening of the crack resistance. Anisotropy regarding the fracture toughness is included through a structural tensor in the crack surface density. The model is parameterised for an aluminium AA2024-T351 sheet material. Validation with a set of experiments shows that the fitted model can reproduce key characteristics of a growing fatigue crack, including crack path direction and growth rate, considering the rolling direction. • Development of the first phase-field model for fatigue fracture in anisotropic media. • Acquisition of set of anisotropic fracture and cyclic parameters for a AA2024-T351. • Parametrisation of the model for anisotropic aluminium sheet material. • Reproduction of experimentally observed effects in crack path and crack growth rate. [ABSTRACT FROM AUTHOR]

Abstract A continuum-based model for the surfactant-driven fracture of closely-packed particle rafts is extended to examine the influence of micro-scale density variability. The model treats the particle monolayer as an elastic sheet endowed with a critical fracture energy that can be reduced through interaction with a flowing surfactant. In addition to the displacement of the monolayer, the model employs a surfactant damage field that serves as both an indicator function for the surfactant concentration as well as the damage to the monolayer. Spatial variability in the particle packing is incorporated in the model through a continuum mapping approach. The formulation gives rise to a coupled system of nonlinear partial differential equations with an irreversibility constraint. The evolution equations are recast in variational form and discretized with an adaptive finite element method. Simulations are provided to demonstrate convergence of the model, illustrate the sensitivity of the fracture process to variations in the initial packing fraction field, and make comparisons with experimental observations. The results indicate that crack bifurcations can occur in regions with spatially uniform packing as well as spatially variable packing, suggesting that both the macro-scale mechanics and the random aspects of the packing contribute to these instabilities. The model is also used to predict the response of these systems to multiple injection sources and obstacles in the domains. Finally, the model is extended to non-planar surfaces as a means to study systems in which confinement and jamming can only occur due to multiple injection sites. [ABSTRACT FROM AUTHOR]

A phase-field theory of brittle fracture in elastoplastic solids hosting mobile interstitial solute species is developed in this paper. The theory, which is formulated within the framework of modern continuum mechanics, provides a systematic way to describe the interplay between solute migration and solid deformation and fracture. A specialization of the theory, which accounts for both solute-induced deformation and solute-assisted fracture as well as for their mutual effects on solute migration, is selected for numerical studies. Toward this end, a numerical model based on the finite-element method for spatial discretization and a splitting scheme with sub-stepping for the time integration is proposed. The model is applied to the study of hydrogen-assisted crack propagation of high-strength steel specimens under sustained loads. The solutions obtained are compared with numerical and experimental results reported in the literature. It is shown that the proposed model has the capability to capture important features presented in the studied phenomenon. [ABSTRACT FROM AUTHOR]

Over the past few decades, the phase-field method for fracture has seen widespread appeal due to the many benefits associated with its ability to regularize a sharp crack geometry. Along the way, several different models for including the effects of pressure loads on the crack faces have been developed. This work investigates the performance of these models and compares them to a relatively new formulation for incorporating crack-face pressure loads. It is shown how the new formulation can be obtained either by modifying the trial space in the traditional variational principle or by postulating a new functional that is dependent on the rates of the primary variables. The key differences between the new formulation and existing models for pressurized cracks in a phase-field setting are highlighted. Model-based simulations developed with discretized versions of the new formulation and existing models are then used to illustrate the advantages and differences. In order to analyze the results, a domain form of the J-integral is developed for diffuse cracks subjected to pressure loads. Results are presented for a one-dimensional cohesive crack, steady crack growth, and crack nucleation from a pressurized enclosure. • Formulations for modeling pressurized cracks in phase-field for fracture models are examined and compared. • A derivation for a new formulation is presented, illustrating differences with existing models. • Development of a domain form of the J-integral for regularized cracks with pressurized faces. [ABSTRACT FROM AUTHOR]

This study introduces a novel implementation of exponential finite element (EFE) shape functions within the phase field model for predicting fracture responses in functionally graded materials (FGMs). The proposed approach utilizes an effective fracture toughness concept to analyze load–displacement responses and crack paths in various examples of FGMs. To optimize computational efficiency, a mixed scheme combining both linear finite element (LFE) and EFE shape functions is employed. Specifically, only the critical elements are corrected using EFE shape functions. Comparative analysis of simulation results against converged outcomes demonstrates the superiority of the EFE scheme, even when employing coarser meshes. The EFE approach accurately predicts load–displacement responses and crack paths for the evaluated examples. The proposed implementation offers a reliable and efficient tool for studying fracture behavior in FGMs. Despite the higher integration schemes and orientation requirements associated with EFE shape functions, the additional computational effort is found to be negligible. Implementation aspects include a staggered iteration scheme, a hybrid tension–compression splitting scheme, and automatic orientation of EFE shape functions. [Display omitted] • Exponential finite element shape functions (EFE) used with the phase field model. • EFE predicted more accurately the load–displacement responses and crack paths. • Notable computational advantages compared to linear finite element shape functions. • The computational resources required are only marginally higher and is negligible. [ABSTRACT FROM AUTHOR]

The phase-field method has gained much attention as a novel method to simulate fracture due to its straightforward way allowing to cover crack initiation and propagation without additional conditions. More recently, it has also been applied to fatigue fracture due to cyclic loading. This publication gives an overview of the main phase-field fatigue models published to date. For the first time, we present all models in a unified variational framework for best comparability. Subsequently, the models are compared regarding their most important features. It becomes apparent that they can be classified in mainly two categories according to the way fatigue is implemented in the model — that is as a gradual degradation of the fracture toughness or as an additional term in the crack driving force. We aim to provide a helpful guide for choosing the appropriate model for different applications and for developing existing models further. • Presentation of most existing phase-field fatigue models in common framework for the first time. • Categorisation of the models in mainly two classes according to their structure. • Numerical comparison of the two model classes. • Discussion of incorporation of plasticity and acceleration techniques. [Display omitted] [ABSTRACT FROM AUTHOR]

• The unified peridynamic correspondence principle is proposed on the basis of the energy compensation method. • A new peridynamic deformation gradient, shape matrix and force state are derived and redefined. • The nonlocal phase field damage constitutive model is developed in the context of peridynamics. • The linearization method of the peridynamics-based phase-field model (PD-PFM) is first presented. • The PD-PFM not only solves the zero-energy mode of the conventional peridynamic correspondence principle, but also provides the rational criterion for the breakage of the peridynamic bond. The peridynamic correspondence model is a promising and attractive candidate for the modeling of localized failure in solids, in that it can incorporate many local classical damage constitutive models through introducing a nonlocal averaged deformation gradient. However, the zero-energy mode and the limited bond breaking criterion greatly restrict the potential applications of this correspondence model. To address these two problems, a unified nonlocal peridynamics-based phase-field damage theory is proposed within the framework of thermodynamics. Firstly, the unified correspondence principle for the displacement field is developed on the basis of the energy compensation method in such a way that a new peridynamic deformation gradient, shape matrix and force state are derived and redefined. And then, the proposed principle is applied to derive the nonlocal phase-field damage constitutive model that the general nonlocal phase-field flux, phase-field flow state and phase-field internal force are defined. Moreover, we propose a mixed variational derivative method to obtain the coupled equilibrium governing equation and present the general linearization approach of the proposed peridynamics-based phase-field model (PD-PFM), where the double states for the coupled displacement and phase fields are derived in detail. It will be found that PD-PFM can not only resolve the zero-energy mode existed in the conventional peridynamic correspondence model, but also provide a rational criterion for the breakage of the peridynamic bond. Some representative numerical examples including mixed mode fracture of single-material media and interface fracture of ceramic coating systems are presented for the validation of PD-PFM. The satisfactory results show both quantitative and qualitative agreement with the available experiment. [ABSTRACT FROM AUTHOR]

The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns but is computationally demanding. The model is a strongly nonlinear non-convex variational inequality that demands the resolution of small length scales. The current standard algorithm for its solution, alternate minimization, is robust but converges slowly and demands the solution of large, ill-conditioned linear subproblems. In this paper, we propose several advances in the numerical solution of this model that improve its computational efficiency. We reformulate alternate minimization as a nonlinear Gauss-Seidel iteration and employ over-relaxation to accelerate its convergence; we compose this accelerated alternate minimization with Newton's method, to further reduce the time to solution, and we formulate efficient preconditioners for the solution of the linear subproblems arising in both alternate minimization and in Newton's method. We investigate the improvements in efficiency on several examples from the literature; the new solver is five to six times faster on a majority of the test cases considered. © 2016 The Authors International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd. [ABSTRACT FROM AUTHOR]

The mechanical behaviors of polymer composites exhibit strong time-dependent nonlinearities, and the tensile and compressive strengths can be vastly different. Such a strength differential effect poses a great challenge in fracture modeling using existing phase-field models. To address this problem, this study develops a new viscoelastic fracture phase-field model, in which the strain tensor spectral decomposition is proposed to decompose the elastic energy of the equilibrium and non-equilibrium branches into tensile and compressive components. The non-negativity of elastic energy and the thermodynamic consistency are theoretically verified. Various case studies, spanning from material uniaxial tension–compression and compression under isostatic pressure to fracture processes of realistic structural components, are used to validate the proposed method. Results of the proposed method are compared with those of existing methods and the actual testing data of polymer composites to demonstrate the effectiveness of the method. It is shown that the proposed method can accurately model not only the strong strength differential effect but also the isostatic pressure dependence of viscoelastic materials. Furthermore, the proposed method offers a more precise prediction of the fracture path and can deal with the loading rate effect. [ABSTRACT FROM AUTHOR]

The 3D printing or material extrusion-based additive manufacturing has evolved from a promising fabrication technology to a mature method that can be integrated into numerous applications. However, this technique involves a large number of variables that significantly affect the resulting structure. In addition, this dependence hinders the development of numerical models to estimate the mechanical behaviour of 3D-printed components with different printing configurations. Hence, the phase-field approach is presented to predict crack propagation through a relatively simple energy balance minimisation problem. Nevertheless, this computational method requires specific parameters to be determined. Therefore, an experimental methodology based on tensile tests is proposed to mechanically characterise the material and analytically define the necessary fracture parameters, including strength and critical energy release rate, from the experimental results. Unnotched and notched specimens, fabricated via material extrusion using a sustainable thermoplastic, are studied under different configurations to analyse fracture mechanisms while addressing strategies to minimise printing defects. Additionally, an open-source numerical predictive tool by means of the phase-field fracture modelling is developed, along with the assessment of the essential length scale parameter. The combination of experimental and numerical studies validates the proposed methodology, and also demonstrates the ease of reproducing for further case studies.

We present a novel computational framework to simulate the electromechanical response of self-sensing carbon nanotube (CNT)-based composites experiencing fracture. The computational framework combines electrical-deformation-fracture finite element modelling with a mixed micromechanics formulation. The latter is used to estimate the constitutive properties of CNT-based composites, including the elastic tensor, fracture energy, electrical conductivity, and linear piezoresistive coefficients. These properties are inputted into a coupled electro-structural finite element model, which simulates the evolution of cracks based upon phase-field fracture. The coupled physical problem is solved in a monolithic manner, exploiting the robustness and efficiency of a quasi-Newton algorithm. 2D and 3D boundary value problems are simulated to illustrate the potential of the modelling framework in assessing the influence of defects on the electromechanical response of meso- and macro-scale smart structures. Case studies aim at shedding light into the interplay between fracture and the electromechanical material response and include parametric analyses, validation against experiments and the simulation of complex cracking conditions (multiple defects, crack merging). The presented numerical results showcase the efficiency and robustness of the computational framework, as well as its ability to model a large variety of structural configurations and damage patterns. The deformation-electrical-fracture finite element code developed is made freely available to download. • A phase field model for electro-mechanical fracture of CNT-based composites is presented. • Mean field homogenisation is combined with deformation-electrical-fracture modelling. • First phase field fracture modelling investigation on piezoresistive materials. • A BFGS-based monolithic solution scheme is used for robustness and efficiency. • 2D/3D case studies reveal the interplay between strains, electric fields and damage. [ABSTRACT FROM AUTHOR]

Phase-field models have been a topic of much research in recent years. Results have shown that these models are able to produce complex crack patterns in both two and three dimensions. A number of extensions from brittle to ductile materials have been proposed and results are promising. To date, however, these extensions have not accurately represented strains after crack initiation or included important aspects of ductile fracture such as stress triaxiality. This work introduces a number of contributions to further develop phase-field models for fracture in ductile materials. These contributions include: a cubic degradation function that provides a stress–strain response prior to crack initiation that more closely approximates linear elastic behavior, a derivation of the governing equations in terms of a general energy potential from balance laws that describe the kinematics of both the body and phase-field, introduction of a yield surface degradation function that provides a mechanism for plastic softening and corrects the non-physical elastic deformations after crack initiation, a proposed mechanism for including a measure of stress triaxiality as a driving force for crack initiation and propagation, and a correction to an error in the configuration update of an elastoplastic return-mapping algorithm for J 2 flow theory. We also present a heuristic time stepping scheme that facilitates computations that require a relatively long load time prior to crack initiation. A number of numerical results will be presented that demonstrate the effects of these contributions. [ABSTRACT FROM AUTHOR]

Phase-field modeling of fracture phenomena in solids is a very promising approach which has gained popularity within the last decade. However, within the finite element framework, already a two-dimensional quasi-static phase-field formulation is computationally quite demanding, mainly for the following reasons: (i) the need to resolve the small length scale inherent to the diffusive crack approximation calls for extremely fine meshes, at least locally in the crack phase-field transition zone, (ii) due to non-convexity of the related free-energy functional, a robust, but slowly converging staggered solution scheme based on algorithmic decoupling is typically used. In this contribution we tackle problem (ii) and propose a faster and equally accurate approach for quasi-static phase-field computing of (brittle) fracture using a monolithic solution scheme which is accompanied by a novel line search procedure to overcome the iterative convergence issues of non-convex minimization. We present a detailed critical evaluation of the approach and its comparison with the staggered scheme in terms of computational cost, accuracy and robustness. [ABSTRACT FROM AUTHOR]

Martensitic steels form a material class with a versatile range of properties that can be selected by varying the processing chain. In order to study and design the desired processing with the minimal experimental effort, modeling tools are required. In this work, a full processing cycle from quenching over tempering to mechanical testing is simulated with a single modeling framework that combines the features of the phase-field method and a coupled chemo-mechanical approach. In order to perform the mechanical testing, the mechanical part is extended to the large deformations case and coupled to crystal plasticity and a linear damage model. The quenching process is governed by the austenite-martensite transformation. In the tempering step, carbon segregation to the grain boundaries and the resulting cementite formation occur. During mechanical testing, the obtained material sample undergoes a large deformation that leads to local failure. The initial formation of the damage zones is observed to happen next to the carbides, while the final damage morphology follows the martensite microstructure. This multi-scale approach can be applied to design optimal microstructures dependent on processing and materials composition. [ABSTRACT FROM AUTHOR]

Despite many noteworthy contributions, the computational modeling of arbitrary fracture configurations and in particular, the quantification of size effect in quasi-brittle solids, still remains a challenging issue. In this work we proposed a model scaling approach for fracture and size effect simulations in solids. It employs only the geometric transformation relations and thus applies to any material model for fracture. Those popular models, i.e., the cohesive zone model, the smeared crack model and the phase-field cohesive zone model, are considered. With the geometric scaling factor intrinsically incorporated in the formulation, the resulting scaled models are able to predict the energetic size effect of a series of geometrically similar structures by parametric numerical simulations of the reference one. The 1-D analytical results of all the three scaled models are coincident with those from the underlying unscaled counterparts, validating the proposed model scaling approach. As it is independent of the mesh discretization and insensitive to the length scale, the scaled phase-field cohesive zone model (PF-CZM) is adopted for application to several representative benchmark tests of concrete and ice with mode-I and mixed I+II failure modes. The numerical results demonstrate that, owing to the strength-based crack nucleation criterion, the energy-based crack propagation criterion and the variational principle based path chooser, as well as the geometric scaling factor, are all incorporated into a standalone framework, the scaled PF-CZM is able to predict both the type II size effect law of pre-cracked quasi-brittle structures and the type I size effect law of intact ones. Moreover, those laborious manual manipulations in the numerical modeling of a large size range of geometrically similar structures, e.g., spatial discretization, enforcement of loading and boundary conditions, postprocessing of output data, etc., need to be done only once on the reference structure. These advantages make the scaled PF-CZM rather promising in the quantification of size effect in quasi-brittle structures. [ABSTRACT FROM AUTHOR]

Abstract: Phase-field models based on the variational formulation for brittle fracture have recently been gaining popularity. These models have proven capable of accurately and robustly predicting complex crack behavior in both two and three dimensions. In this work we propose a fourth-order model for the phase-field approximation of the variational formulation for brittle fracture. We derive the thermodynamically consistent governing equations for the fourth-order phase-field model by way of a variational principle based on energy balance assumptions. The resulting model leads to higher regularity in the exact phase-field solution, which can be exploited by the smooth spline function spaces utilized in isogeometric analysis. This increased regularity improves the convergence rate of the numerical solution and opens the door to higher-order convergence rates for fracture problems. We present an analysis of our proposed theory and numerical examples that support this claim. We also demonstrate the robustness of the model in capturing complex three-dimensional crack behavior. [Copyright &y& Elsevier]