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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]

To capture the stress corrosion cracking (SCC) in metallic materials, a phase‐field framework considering localized plastic deformation and stress states is established. A new function of critical energy release rate is proposed to describe the degradation of cracking resistance of materials in the SCC process. This proposed framework can reproduce SCC results consistent with experimental observations in C‐ring steel SCC tests, especially capturing the directionally characteristic of the cracks. Furthermore, the influences of localized plastic deformation and stress states on the SCC process are studied. This novel phase‐field framework can capture (1) the influences of coupled electricity, mechanics to the SCC mechanisms; (2) the contribution of plastic strain energy as the driving force for the phase‐field; (3) the potential and preferential crack initiation and propagation morphology within the complex stress and strain domain; and (4) the dependence of SCC rate and propagation direction on stress state and localized plastic deformation. Highlights: A phase‐field framework for SCC prediction of elastoplastic materials is established.The novel framework is implemented in finite element model and validated by tests.The framework can capture SCC morphology within the complex stress and strain domain.Influences of stress states and localized plastic deformation on SCC are clarified. [ABSTRACT FROM AUTHOR]

In vesicle membranes, the minimum of the elastic bending energy determines the equilibrium shape. We can reformulate the energy using a phase‐field function and optimize it using the standard gradient flow approach. The method, in its typical formulation, allows topological changes in the membrane; however, in certain events, like in the simulation of blood cells, maintaining the initial topology of the membrane is crucial. In this study, we add a constraint on the phase‐field method to preserve the topology. We note that even if the phase‐field method is formulated using a hyperbolic tangent function, during a change in topology, the membrane's profile deviates from the tanh function. Owing to this idea, the constraint imposes an additional requirement on the profile to maintain the tanh shape, thus preserving the topology. We perform extensive experiments in two‐dimensional and three‐dimensional scenarios to demonstrate that our method preserves the topology during the simulation. [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]

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]

In the present work, the conservative phase-field lattice Boltzmann model is applied to simulate plunging breakers in shallow waters. The solver is computationally efficient as it is explicit in time stepping and does not require a solution for the pressure Poisson equation. Nevertheless, the solver can simultaneously handle a large Reynolds number ( $ =10^4 $ = 10 4 ) and a high-density ratio of 1000. Simulations are performed by varying initial steepness and dispersion parameters, which influence the nonlinearity degree of wave (or the growth speed of instability) and the oscillatory motion (limited by the bottom) on the breaking characteristics. Detailed analyses of kinematics, air entrainment, vorticity, and energy dissipation are presented. [ABSTRACT FROM AUTHOR]

This work studies crack propagation in particulate nanocomposites using the phase-field method. The crack propagation has been simulated in a wide range of loadings and the critical load for the crack growth has been obtained. Surface tension, as an inelastic stress, is introduced in the model in a thermodynamically consistent way. The effect of surface tension on the crack tip velocity and the crack evolution has been discussed. The finite element method via COMSOL multiphysics software has been utilized to solve the coupled phase-field and elasticity equations. Modeling and prediction of crack propagation for nanocomposites including different nanoparticles and under different loadings are the main purposes of this work. It is found that the kinetics and morphology of the crack propagation depend on the elastic moduli and the surface energy of nanoparticles as well as their longitudinal and angular distances to each other. [ABSTRACT FROM AUTHOR]

This study unveils a numerical paradigm that amalgamates the partially saturated lattice Boltzmann method (PSLBM) with the non‐isothermal quantitative phase‐field (PF) model. This innovative integration equips us with a prognostic tool ready to elucidate the progression and motion of solid‐air dendritic growth in the presence of both natural and forced convection. The PSLBM is employed to compute the flow of the solution and the interaction forces between the fluid and solid dendrites. Concurrently, the PF model is utilized to simulate the formation of solid‐air dendrites. The reliability of calculating of interaction forces between the fluid and solid was confirmed through a numerical case study involving fluid flow around a stationary cylinder. The results indicate that this model is applicable for simulating the growth and evolution of single/multiple solid‐air dendrites under the influence of convection, whether they are stationary or in motion. The promotion of the upstream side dendritic arms and the inhibition of the downstream dendritic arms increase with the intensification of natural convection. As the initial undercooling is raised, the capacity of natural convection to reshape dendritic morphology gradually diminishes. With the enhancement of forced convection intensity, due to alterations in the flow pattern, the downstream dendritic arms do not consistently exhibit growth suppression. The motion of solid‐air dendrites induced by forced convection counteracts the influence of convection, resulting in slightly faster growth of the downstream dendritic arms compared to the upstream arms. Simultaneously, it fosters the formation of secondary dendritic branches in the upstream zone. [ABSTRACT FROM AUTHOR]

Studying Arctic ice formation stays in the focus of research groups over the past decades in the context of ice cover changes, thermal budget and climate agenda in general. Nevertheless, the phenomenon's underlying mechanisms are still not completely understood and described. The main reason for the lack in understanding is the limited experimental access to the field data when it comes to the processes that occur below the ice floe. Thus, there is a need to build competent analogies between the natural (ocean water–ice) and laboratory (here: binary alloy) conditions of the experiment as a step of data preparation for the verification of the mathematical model. In the current paper, the existing qualitative models describing the process of melting and crystallization were expanded and the experimental method was developed copying the layering of the natural ocean water–ice mixture. The experimental set‐up for studying the solidification within the intermediate zone was designed for Al–Cu alloys being the system with appropriate solidus line for creating a sufficient concentration gradient and by that temperature dependent phase fraction under isothermal conditions. The gained experimental data were used for validating a binary phase‐field model for solidification considering moving boundaries. The model includes the description of the free energy of both phases and their respective diffusion coefficients. It allows modeling a binary system at a mesoscopic spatial level by including the concentration‐driven phase transition and resolidification in the two‐phase region. The novel results will help the quantitative understanding of solidification phenomena and are highly‐evaluated from interdisciplinary point of view, including glaciology and geosciences, being ultimately significant for the understanding the global climate change landscape. [ABSTRACT FROM AUTHOR]

Brittle material Mode I fracture may be characterized by the double cleavage drilled compression test. For linear elastic materials, the critical energy release rate, or fracture toughness, can be estimated simply using the linear elastic fracture mechanics. For other types of constitutive behavior, the material parameter has to be determined with numerical fracture modeling. In this paper, we have used two approaches, the phase-field damage model and the cohesive elements, in order to estimate the critical energy release rate of an elastoplastic material. Firstly, we assessed the numerical models and discussed their parameters by comparison of available data from double cleavage drilled compression experimental tests run on a silica glass. Both phase-field damage and cohesive zone models were able to reproduce fracture initiation at the observed macroscopic stress for the linear elastic material. However, the material toughness could not be predicted by the phase-field approach due to the result dependence on the model regularization parameter. Secondly, an elastoplastic methyl methacrylate polymer was submitted to the compression test in our lab. Both models were then extended for elastic-perfectly plastic materials. Crack initiation was obtained at the observed macroscopic strain for similar critical energy release rate ranges for both approaches, providing good confidence in the estimated material toughness. [ABSTRACT FROM AUTHOR]

The state-of-the-art fractographic guidelines for glasses and fine ceramics, ASTM C1678, are only applicable to limited fracture scenarios. This work outlines a new method to analyze fracture scenarios outside ASTM C1678′s scope. The proposed methodology consists of a two-step process: initially, optical images of the fracture surfaces are obtained, and visual analysis is used to highlight the characteristic fractographic features. Next, the selected features are automatically compared to a baseline set of virtual fracture patterns tailored to any relevant fracture scenarios obtained by numerical phase-field simulations. As validation, scenarios with fractures originating at the glass plates' edges for both sharp and chamfered corners were analyzed. Strength estimates within 6% of experimental results were achieved with the proposed method. In the future, this technique could be implemented to carry out fractographic analysis on brittle components of arbitrary shapes and loadings, hence substantially extending the application range and accuracy of fractographic methods. [ABSTRACT FROM AUTHOR]

Hydrogen grain boundary (GB) trapping is widely accepted as the main cause for hydrogen induced intergranular failure. Several studies were conducted to unveil the role of GBs on hydrogen transport; however, a clear understanding is yet to be attained. This is due to the limitations of the state-of-the-art experimental procedures for such highly kinetic processes. In this study, we aim at providing a deeper understanding of hydrogen-GB interactions using full-field representative volume element (RVE). The phase-field method is chosen for generating RVEs, since it is an appropriate numerical tool to represent GBs. A novel fully-kinetic formulation for hydrogen diffusion and GB trapping is presented, which is compatible with the phase-field based RVEs. GB diffusivity (D gb) and trap-binding energy (E gb) were used as parameters to understand the interactions between diffusion and GB trapping. Uptake and permeation simulations were performed with constant and gradient occupancy boundary conditions respectively. In both cases, increasing E gb , increased the hydrogen GB occupancy. The permeation simulations showed that the hydrogen flux along the GBs increased with increasing both, D gb and, surprisingly, E gb. Since trapping increases the hydrogen occupancy along GBs, it also increases the occupancy gradients, resulting in a higher flux. This led to the conclusion that, in the case of an external occupancy gradient, GB trapping and diffusion cooperate, rather than compete, to increase the hydrogen flux. On the other hand, the decisive factor for the retention of hydrogen at the GBs in permeation simulations was D gb rather than E gb. [Display omitted] • A new full-field model for H diffusion and GB trapping is presented. • The model shows the kinetic interactions of GB diffusivity and trapping. • The build-up of H due to trapping increases the permeation flux. • GB diffusion and trapping constructively interact, increasing H flux. [ABSTRACT FROM AUTHOR]

Brittle materials are known for their violent and unpredictable cracking behavior. A behavior which is dictated by a combination of microscopical material defects and the competition between the potential energy of the system and the surface energy of the material. In this study, we present the implementation of a dynamic fracture phase-field model with a new crack driving force into a commercial finite element (FE) solver and examine its behavior using three different tension-compression splits. After validating the implementation, we use the model to investigate its predictive capacity on quasi-statically loaded L-shaped soda-lime glass specimens with varying critical load levels. The dynamic fracture phase-field model predicted similar crack propagation to what was found in the literature for quasi-static and dynamic validation cases. By varying the critical load level for the L-shaped soda-lime glass specimens using the new crack driving force, the model predicted a positive correlation between the initial crack propagation speed and the critical load level, similar to what was seen in the experiments. However, the predicted crack propagation speed decreased quicker than the experimental crack propagation speed. The tension-compression splits had an impact on the predicted crack propagation paths. Overall, the proposed crack driving force used in the dynamic fracture phase-field model seems to capture the relation between critical load and initial crack propagation speed and thus enables crack predictions for specimens of varying strength. [ABSTRACT FROM AUTHOR]

In this paper a phase‐field formulation based on an extended F‐criterion (the normalized strain energy release rate criterion) is proposed to simulate tensile‐compressive‐shear rock fractures. By applying the F‐criterion, the phase‐field crack‐driving energy decomposition is determined by a direction search which maximizes the local fracture dissipation. In compressive‐shear states, the computation is supplemented by an explicitly expressed confinement‐dependent mode‐II fracture energy release rate, and the cracking angle is determined by both the fracture energy and strain states. The hybrid formulation and alternate minimization algorithm are adopted for the numerical examples in this paper. Fractures for rock and rock‐like specimens subjected to compression demonstrate the ability of the present model in capturing tensile‐compressive‐shear rock fracture behaviors. [ABSTRACT FROM AUTHOR]

Fracture of viscoelastic materials is considered to be a complex phenomenon due to their highly rate sensitive behavior. In this context, we are interested in the quasi-static response of a viscoelastic solid subjected to damage. This paper outlines a new incremental variational based approach and its computational implementation to model damage in viscoelastic solids. The variational formalism allows us to embed the local constitutive equations into a global incremental potential, the minimization of which provides the solution to the mechanical problem. Softening damage models in their local form are known to result in spurious mesh-sensitive results, and hence, non-locality (or regularization) has to be introduced to preserve the mathematical relevance of the problem. In the present paper, we consider two different regularization techniques for the viscoelastic damage model: a particular phase-field and a lip-field approach. The model parameters are calibrated to obtain some equivalence between both these approaches. Numerical results are then presented for the bidimensional case and both these approaches compare well. Numerical results also demonstrate the ability of the model to qualitatively represent the typical rate-dependent behaviour of the viscoelastic materials. Besides, the novelty of the present work also lies in the use of lip-field approach for the first time in a viscoelastic context. [ABSTRACT FROM AUTHOR]

We propose a phase-field model of shear fractures using the deviatoric stress decomposition. This choice allows us to use general three-dimensional Mohr–Coulomb's failure function for formulating the relations and evaluating peak and residual stresses. We apply the model to a few benchmark problems of shear fracture and strain localization and report remarkable performance. Our model is able to capture conjugate failure modes under biaxial compression test and for the slope stability problem, a challenging task for most models of geomechanics. [ABSTRACT FROM AUTHOR]

This work studies the evolution of damage in periodic composites with hyperelastic constituents prone to mechanical degradation under sufficient loading. The micromechanical problem is solved for quasistatic far-field loading for plane-strain conditions, using the finite strain high-fidelity general method of cells (FSHFGMC) approach to discretize the conservation equations. Damage is treated as degradation of material cohesion, modeled by a material conservation law with a stress-dependent damage-source (sink) term. The two-way coupled formulation with the internal variable representing damage is reminiscent of the phase-field approach to gradual cracks growth, albeit with a mechanistically derived governing equation, and with important theoretical differences in consequences. The HFGMC approach consists in enforcing equilibrium in each phase (in the cell-average sense) by stress linearization, using instantaneous tangent moduli, and subsequent iterative enforcement of continuity conditions, a formulation arguably natural for composite materials. The inherent stiffness of the underlying differential equations is treated by use of a predictor–corrector scheme. Various examples are solved, including those of porous material developing cracks close to the cavity, for various sizes and shapes of the cavity, damage in a two-phase composite of both periodic and random structure, etc. The proposed methodology is physically tractable and numerically robust and allows various generalizations. [ABSTRACT FROM AUTHOR]

We propose the truncated nonsmooth Newton multigrid method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems with linear time complexity. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without regularization or the introduction of a local history field. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler–Lagrange equations of the spatial problem for any load and initial iterate. On the way, we show several crucial convexity and regularity properties of the models considered here. Numerical comparisons to an operator-splitting algorithm show a considerable speed increase, without loss of robustness. [ABSTRACT FROM AUTHOR]

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]

• A multi-physics thermomechanical phase-field Allen-Cahn formulation is built for studying powder bed fusion processes. • The underlying physics of the phase changes in the powder bed fusion is exposed. • An adaptive mesh refinement strategy is used to capture the melting and solidification frontiers. • Automatic scaling is used to overcome different orders of magnitude of the field variables. • The effect of the phase-field length scale on the temperature, melt pool geometry and stress distributions is explored. The present study investigates the thermomechanical simulation of powder bed fusion (PBF) process, focusing on the characterization of the melt pool. With the aim of comprehensively understanding the complex multi-physics phenomena inherent to the PBF process, our objective is to establish a robust and refined computational framework that is capable of systematic evaluation of the dynamic evolution of the melt pool geometry and temporal characteristics across the various stages of its formation. For this purpose, the Allen-Cahn phase field formulation is integrated with the elastoplastic material model based on J 2 plasticity theory and the model implementation is conducted based on a finite element method (FEM) framework using the Multi-physics Object-Oriented Simulation Environment (MOOSE). As an objective, the model tends to depict the intricate, nonlinear, and non-equilibrium nature of the melt pool, encompassing multi-physics process of the PBF, while assuming powder particles as a homogenized continuum medium and neglecting complex interactions between the particles and their surroundings. This includes the separation of geometry into two domains (i) the powder layer and (ii) the substrate, the melting and solidification of the material, the phase-changing process, and the resulting induced distortion and residual stresses. Additionally, the model incorporates a phase indicator to account for the degree of solidification or consolidation of powder, whose evolution describes diffusion, solidification and shrinkage phenomena. This approach enables the tracking of the movement of the melting front of the metallic material induced by a heat source, effectively dividing the domain into regions of soft and hard solid states with a diffusive interface in between. Furthermore, to enhance the model's fidelity, an adaptive mesh refinement (AMR) strategy is employed to precisely capture important aspects of the phase changing process, such as characteristics of the solidification front and powder melting. To assess the model's performance, it is tested by simulating the processing of a three-dimensional Inconel 625 thin plate, as well as a benchmark example, where the influence of parameters such as temperature and phase change on the mechanical response of the system during the solidification process is elucidated. [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 this paper, we propose a simple and fast shape transformation method. This method is based on the Allen-Cahn (AC) partial differential equation and uses the edge stop function to constrain evolution. We use the operator splitting method to control the equation for splitting, and use the explicit Euler's method to solve the discrete equation. Based on the source shape and the target shape, our phase-field model can perform rapid shape transformation when the appropriate parameters are selected. In order to verify the feasibility of our proposed phase-field model, numerical experiments on shape transformation are carried out in two-dimensional and three-dimensional space. The results show that the established mathematical model can not only perform shape transformation between simple shapes, but also have a good effect on shape transformation between complex shapes. We also propose a numerical solution using the Crank-Nicolson numerical scheme to handle the model we introduced. This numerical scheme utilizes the multigrid method and Gauss-Seidel iteration for solving. The results of the numerical experiments demonstrate the effectiveness of this numerical method in handling shape transformations in three-dimensional space. [ABSTRACT FROM AUTHOR]

Here, we study the multi-component Cahn–Hilliard (multiCH) system and the three-phase fluid system by a novel numerical scheme. The nonlinear terms in the multiCH system are obstructors to establishing numerical schemes. In our proposed method, several scalar auxiliary variables are defined to represent the nonlinear terms. Then, the control equations can be converted into equivalent forms where the nonlinear terms are replaced by the scalar auxiliary variables. Finally, we only need to discretize the equivalence equation, where the second-order backward difference formula (BDF2) and the second-order central difference method are applied in the time and the spatial domain, respectively. The entire calculation process is linear and decoupled. In each time iteration, we apply a fast linear multigrid algorithm to the constant coefficient linear elliptic equation. For the incompressible flow part, we employ a pressure correction method to decouple velocity and pressure during time discretization. We rigorously prove that the numerical solution is unique and shows that the resulting numerical solution satisfies the dissipation of modified energy. The temporal second-order accuracy test, the energy dissipation test, the comparison test with the classical SAV in CPU time, and other various numerical tests show the good performance of our method. • A novel linear and second-order scheme is proposed for N-phase fluids. • The unique solvability and energy stability are proved analytically. • Our proposed scheme works well for various phase-field fluid processes. • The numerical implementation is highly efficient. [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]

In this paper, we propose a phase-field model for electro-mechanical fracture in brittle materials and provide an open-source implementation of the proposed model using a newly developed finite element (FE) toolbox, Gridap in Julia. Here, we have considered electric potential as an additional kinematic descriptor along with the displacement and phase-field variable. Using the virtual power principle, we have derived force balances for the electro-mechanical forces and a force balance associated with the damage. To incorporate the strong electro-mechanical coupling effect into the model, we have considered the Helmholtz free energy as a function of the electric field vector, strain tensor, phase-field variable, and the gradient of the phase-field variable. The proposed model ensures that the constitutive relations of the thermodynamic fluxes are on the satisfaction of the thermodynamic laws and can readily accommodate dissipative energy effects whenever needed. The proposed model provides complex crack paths as a solution to the governing partial differential equations (PDEs), and thus bypasses the need for re-meshing, enrichment of FE shape functions, and an explicit tracking of the crack surfaces. The use of Gridap makes the FE implementation of the proposed model exceedingly compact and user-friendly requiring very low memory usage and providing a high degree of flexibility to the users in defining weak forms of the governing PDEs. We have validated the proposed model and its implementation against a compact tension test on a single edge notched plate and a set of three-point bending tests on a notched beam made of lead zirconium titanate (PZT)-4 piezoelectric ceramics. [ABSTRACT FROM AUTHOR]

We prove second-order optimality conditions for an optimal control problem of tracking-type for a time-discrete regularized phase-field fracture or damage propagation model. The energy minimization problem describing the fracture process contains a penalization term for violation of the irreversibility condition in the fracture growth process, as well as a viscous regularization corresponding to a time-step restriction in a temporal discretization of the problem. In the control problem, the energy minimization problem is replaced by its Euler–Lagrange equations. While the energy minimization functional is convex due to the viscous approximation, the associated Euler–Lagrange equations are of quasilinear type, making the control problem nonconvex. We prove second-order necessary as well as second-order sufficient optimality conditions without two-norm discrepancy. [ABSTRACT FROM AUTHOR]

• Computational framework for the design of Dielectric Elastomer prototypes attaining predefined target configurations. • Two-field Finite Element implementation, with displacements and the scalar electric potential as unknown fields. • Phase-field driven multi-material topology optimisation. • Consideration of arbitrary number of different Dielectric Elastomer materials. • Numerical experiments support the applicability of the proposed approach. This paper presents a novel engineering strategy for the design of Dielectric Elastomer (DE) based actuators, capable of attaining complex electrically induced shape morphing configurations. In this approach, a multilayered DE prototype, interleaved with compliant electrodes spreading across the entire faces of the DE, is considered. Careful combination of several DE materials, characterised by different material properties within each of the multiple layers of the device, is pursued. The resulting layout permits the generation of a heterogenous electric field within the device due to the spatial variation of the material properties within the layers and across them. An in-silico or computational approach has been developed in order to facilitate the design of new prototypes capable of displaying predefined electrically induced target configurations. Key features of this framework are: (i) use of a standard two-field Finite Element implementation of the underlying partial differential equations in reversible nonlinear electromechanics, where the unknown fields ot the resulting discrete problem are displacements and the scalar electric potential; (ii) introduction of a novel phase-field driven multi-material topology optimisation framework allowing for the consideration of several DE materials with different material properties, favouring the development of heterogeneous electric fields within the prototype. This novel multi-material framework permits, for the first time, the consideration of an arbitrary number of different N DE materials, by means of the introduction of N − 1 phase-field functions, evolving independently over the different layers across the thickness of the device through N − 1 Allen-Cahn type evolution equations per layer. A comprehensive series of numerical examples is analysed, with the aim of exploring the capability of the proposed methodology to propose efficient optimal designs. Specifically, the topology optimisation algorithm determines the topology of regions where different DE materials must be conveniently placed in order to attain complex electrically induced configurations. [ABSTRACT FROM AUTHOR]

The efficient simulation of complex fracture processes is still a challenging task. In this contribution, an enriched phase-field method for the simulation of 2D fracture processes is presented. It has the potential to drastically reduce computational cost compared to the classical phase-field method (PFM). The method is based on the combination of a phase-field approach with an ansatz transformation for the simulation of fracture processes and an enrichment technique for the displacement field as it is used in the extended finite element method (XFEM) or generalised finite element method (GFEM). This combination allows for the application of significantly coarser meshes than it is possible in PFM while still obtaining accurate solutions. In contrast to classical XFEM / GFEM, the presented method does not require level set techniques or explicit representations of crack geometries, considerably simplifying the simulation of crack initiation, propagation, and coalescence. The efficiency and accuracy of this new method is shown in 2D simulations. [ABSTRACT FROM AUTHOR]

Investigations of directionally solidifying melt pools during metal additive manufacturing (AM) reveal that the resulting subgrain cellular structures often grow along crystalline orientations different from the temperature gradient direction, some of which are not even along preferred crystallographic directions. It is well-known that dendrite orientation results from the growth competition between the heat flow direction and preferred crystallographic orientation. Specifically, the competition between interfacial anisotropy and process anisotropy (thermal gradient and interface velocity) during directional solidification leads to rich morphological diversity of the resulting dendritic structures, including tilted dendrites and seaweed patterns. The orientation selection mechanisms of such patterns remain unexplored at high velocity in the frame of AM. This study examines the tilted growth of cellular-dendritic arrays as a function of the misorientation angle (θ R) between the thermal gradient and crystal lattice directions and other relevant control parameters. We use a phase-field model to explore dendritic evolution in a binary alloy during high-velocity solidification in two-dimensions. We find marked effects of thermal gradient, growth velocity, alloy composition, and anisotropy parameters on the possible growth directions and the primary arm spacing, constitutional undercooling, microsegregation, and secondary phases that arise during dendritic solidification. Our work provides a detailed yet concise presentation on the tilted growth and morphological transition for a broad range of thermal conditions in the high-velocity regime and the full range of θ R for establishing orientation selection maps. These results reasonably agree with experimental measurements and should have qualitative relevance for controlling the subgrain structure, chemical segregation, and texture randomization in a wide range of commercial dendrite materials relevant to AM. [Display omitted] [ABSTRACT FROM AUTHOR]

Microstructure evolution, which plays a critical role in determining materials properties, is commonly simulated by the high-fidelity but computationally expensive phase-field method. To address this, we approximate microstructure evolution as a time series forecasting problem within the domain of deep learning. Our approach involves implementing a cost-effective surrogate model that accurately predicts the spatiotemporal evolution of microstructures, taking an example of spinodal decomposition in binary and ternary mixtures. Our surrogate model combines a convolutional autoencoder to reduce the dimensional representation of these microstructures with convolutional recurrent neural networks to forecast their temporal evolution. We use different variants of recurrent neural networks to compare their efficacy in developing surrogate models for phase-field predictions. On average, our deep learning framework demonstrates excellent accuracy and speedup relative to the "ground truth" phase-field simulations. We use quantitative measures to demonstrate how surrogate model predictions can effectively replace the phase-field timesteps without compromising accuracy in predicting the long-term evolution trajectory. Additionally, by emulating a transfer learning approach, our framework performs satisfactorily in predicting new microstructures resulting from alloy composition and physics unknown to the model. Therefore, our approach offers a useful data-driven alternative and accelerator to the materials microstructure simulation workflow. [Display omitted] [ABSTRACT FROM AUTHOR]

We present a multiscale modeling framework that integrates density functional theory (DFT) with a phase-field model (PFM) to explore the intricate dynamics of grain growth in nanocrystalline α -Fe single-phase alloy in the presence of chromium (Cr) segregation. Simulated results for equilibrium segregation in stationary grain boundary (GB) agree with the Mclean isotherm, validating our model. Polycrystal simulations featuring nanocrystalline grains at different temperatures reveal that the grain growth kinetics depends on the ratio of Cr diffusivity to intrinsic GB mobility. Without Cr segregation at GB, the relationship between the square of average grain size (d 2) and time (t) demonstrates a linear correlation. With Cr segregation at GB, the d 2 vs. t plot initially follows the same linear growth trajectory as observed without segregation up to a threshold grain size, beyond which it deviates with a decreasing slope. The threshold grain size decreases with increasing temperature from 700K to 900K. Notably, at 1000K, grain growth without and with Cr segregation both follow a linear trajectory, the latter having a smaller slope from the beginning. We develop an analytical formulation based on Cahn's solute drag theory to predict grain growth in the presence of solute segregation at GB and use it to validate our simulation results. • Multiscale modeling approach, integrating phase-field and DFT simulations to explore nanocrystalline grain growth of α -Fe in presence of Cr segregation. • DFT calculation of segregation energy, grain boundary (GB) energy, and GB width are used in phase-field simulation. • Phase-field simulations of bicrystals are used to validate equilibrium GB segregation against McLean's isotherm. • Grain size evolution during grain growth in polycrystal in presence of Cr segregation deviates from linear trend for 700 K to 900 K. • Simulation results for grain growth in polycrystal in presence of Cr segregation is in excellent agreement with the analytical solution. [Display omitted] [ABSTRACT FROM AUTHOR]

In this paper, an adaptive multi-patch isogeometric phase-field model is presented for brittle fracture in isotropic/orthotropic piezoelectric materials. The polynomial splines over hierarchical T-meshes (PHT-splines) functions are employed to represent both geometry and field variables. Nitsche's method provides the continuity of all field variables, including displacement, electric potential, and phase-field at the coupling edge. The local refinement uses a mesh refinement strategy determined by a user-defined threshold for the phase-field variable. An approach that integrates hybrid formulation with staggered solving is utilized for the numerical implementation. The proposed method's accuracy, reliability, and robustness are demonstrated using several fracture simulations, including tensile tests of rectangular and grooved ceramic plates with an edge crack, plate with a single hole, porous plate, and ceramic laminate with different material orientations. • Adaptive multi-patch isogeometric phase-field model for brittle fracture in piezoelectric materials is developed. • Nitsche's method is employed to provide continuity of displacement, electric potential, and phase-field at the coupling edge. • Generalized Miehe decomposition method is extended to the brittle fracture of piezoelectric materials. • Local adaptive refinement is employed to enhance the computational efficiency. [ABSTRACT FROM AUTHOR]

A variational phase field model for dynamic ductile fracture is presented. The model is designed for elasto-viscoplastic materials subjected to rapid deformations in which the effects of heat generation and material softening are dominant. The variational framework allows for the consistent inclusion of plastic dissipation in the heat equation as well as thermal softening. It employs a coalescence function to degrade fracture energy during regimes of high plastic flow. A variationally consistent form of the Johnson–Cook model is developed for use with the framework. Results from various benchmark problems in dynamic ductile fracture are presented to demonstrate capabilities. In particular, the ability of the model to regularize shear band formation and subsequent damage evolution in two- and three-dimensional problems is demonstrated. Importantly, these phenomena are naturally captured through the underlying physics without the need for phenomenological criteria such as stability thresholds for the onset of shear band formation. [ABSTRACT FROM AUTHOR]

Many hydraulic fracturing experiments suggest that low-viscosity fluid tends to generate a complex fracture network, which may be beneficial for geo-energy production. However, the precise impacts of low-viscosity fluid on the fracture nucleation and propagation are unknown. This study focuses on the stress jump at weak interfaces (e.g., grain boundaries or natural fractures) caused by the contrast in Biot's coefficient, which is prevalent in hard rocks, and how failure may develop. We found that fracture nucleation at weak interfaces becomes favorable under certain Biot's coefficient contrast and fluid viscosity. Our numerical simulations of fluid injection through a borehole demonstrate that low-viscosity fluid injection can nucleate isolated fractures at remote interfaces without connecting to the main propagating fracture. These findings imply the necessity to consider Biot's coefficient variations within a rock mass in applications such as hydraulic fracturing or induced seismicity, especially when a diffused pressure gradient persists due to the low-viscosity fluid. • Tensile stress can develop at an interface whose Biot coefficient is lower. • A contrast in Biot coefficient and low fluid viscosity can promote remote fracturing. • Remote fracturing may be engineered to stimulate stiff and tight rock formations. [ABSTRACT FROM AUTHOR]

Phase-field has been effectively applied to many complex problems according to the mesh based method. However, the computational speed of the numerical method based on phase-field still needs improved. In this paper, an improved localized radial basis function collocation method (LRBFCM) based on the adaptive support domain is employed to the phase-field methods. The proposed adaptive support domain can increase the stability of the LRBFCM, and the improved LRBFCM is much more efficient than the traditional finite element method (FEM) in coupling with phase-field methods. The proposed approach is further applied to the single-phase dendrite solidification, two-phase sintering, and three-phase wetting phenomena. We compare the efficiency of the proposed LRBFCM with different numerical methods, which show that the LRBFCM combined with the Fourier spectral method can deal with the three-phase model with more than ten million nodes easily. [ABSTRACT FROM AUTHOR]

Spatiotemporal variation of the thermal gradient in the melt pool inherited from different heat input patterns or other non-equilibrium transient effects during additive manufacturing can significantly affect the resulting subgrain microstructure evolution. To examine the impact of this variation, we approximate the thermal gradient by various isotherm patterns that move with constant velocity following directional solidification. We report the first three-dimensional phase-field simulations to investigate the effects of isotherm patterns on the cellular structures typically observed in solidified melt pools. Results indicate that small variations in the isotherm can considerably impact the microstructural features. We use appropriate statistical characterizations of the solid fraction, solid percolation, and solute partitioning behavior to demonstrate the influence of isotherm patterns on the dendritic structures and semisolid mushy zones. Consistent with experimental observations, we find that non-planar isotherms produce finer cells and reduced microsegregation compared to planar isotherms. Also, we note that a tilt of the isotherm leads to a tilted state of the resulting cellular arrays. Our findings will help in understanding the qualitative aspects of the influence of temperature gradient patterns on the evolution of solidification morphologies, mushy zones, and secondary phases, which are crucial for the macroscopic description of the solidified material. [ABSTRACT FROM AUTHOR]

[Display omitted] • Creep fracture and rafting property of superalloy are studied by phase-field simulation. • Dissolution fracture of rafted γ' phase happens under thermal loading. • Rafting fracture of γ' phase induces the strain jump in Co-based superalloy. • Continuous heating brings the large damage variable and creep strain. The creep fracture and rafting of Co-based superalloys under thermal loading are critical properties for the long time working at high temperature, the crystal plasticity phase-field (CPPF) model is developed by incorporating the creep damage theory. The study of creep in Co-5Al-14V (at.%) superalloy in states of isothermal, continuous heating and overheating is realized. The rafting mechanism and evolution kinetics of γ′–Co 3 (Al, V) phase are analyzed in processes of directional coarsening and dissolution fracture under thermal loading stress, the incomplete P-type rafting morphology with low volume fraction of γ′ phase shows a reduced creep resistance. The creep strain has a steep rise for the rafting fracture and degeneration induced by overheating stress. The overheating followed by cooling triggers the recovery of γ′ rafts and improves the creep resistance. In addition, the dissolution and fracture of γ′ rafts are driven by the chemical potential, while the rafting and re-precipitation of γ′ phase are driven by the elastic potential under the creep stress. At the tertiary creep stage, the damage variable and creep strain increase in the order of isothermal creep, overheating and continuous heating. The studies show the creep fracture and strain properties in stress-diffusion mechanisms by coupling experiment morphology and CPPF simulation, the relationship of creep fracture and rafting properties of superalloys is established. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze a novel class of arbitrarily high-order and unconditionally energy stable schemes for the anisotropic phase-field dendritic crystal growth model, which is a highly nonlinear system that combines the anisotropic Allen–Cahn equation with the thermal equation. The proposed schemes are based on an extrapolated and linearized Runge–Kutta method for an auxiliary variable reformulation of the crystal growth model. A delicate implementation demonstrates that the proposed method can be realized in a very efficient way, requiring only the solution of a coupled linear elliptic system at each time step. We prove that the constructed schemes satisfy the energy dissipation property and demonstrate the convergence order through a consistency error analysis. To the best of our knowledge, this is the first unconditionally stable scheme of arbitrarily high-order for the anisotropic phase-field dendritic crystal growth model. Numerical experiments for two and three dimensional dendritic crystal growth are simulated to verify our theoretical accuracy as well as the efficiency of the proposed method. Furthermore, to emphasize the superiority of the high-order schemes, we present a numerical comparison that directly contrasts its performance with the lower-order schemes. [ABSTRACT FROM AUTHOR]

Inconel 740H (IN740H) has emerged as an important candidate for the advanced ultra-supercritical (AUSC) steam turbines due to its superior microstructural stability and creep resistance under service conditions. In the present work, we have reparameterized a multicomponent Ni-based superalloy (IN740H) into an equivalent ternary Ni-Al-Mo superalloy, based on the partitioning coefficients of the elements, using the thermodynamic database (CALPHAD). We use this thermodynamic description to employ a quantitative phase-field model to assess the long-term stability of γ ′ precipitates in IN740H utilizing GPU-based supercomputing architecture. The assessment helps us to enhance our understanding of the effect of the atomic diffusivity of Mo on coarsening kinetics of the γ ′-precipitates in equivalent ternary Ni-Al-Mo superalloy. Investigation reveals that our phase-field model can accurately predict the experimentally observed coarsening kinetics in IN740H. [Display omitted] • A GPU-accelerated and CALPHAD integrated phase-field model is employed to simulate microstructure evolution in IN740H. • This model accurately captures the phase equilibria and the effect of multicomponent diffusion on long-term coarsening. • The simulation results agree well with the experimentally observed coarsening kinetics in IN740H. • Molybdenum addition significantly decreases the coarsening rate of oredered precipitates. [ABSTRACT FROM AUTHOR]

This study presents a unified formulation of topology optimization with a finite strain nonlocal damage model using the continuous adjoint method. For the primal problem to describe the material response including deterioration, we consider the standard Neo–Hookean constitutive model and incorporate crack phase-field theory for brittle fracture within the finite strain framework. For the optimization problem, the objective function is set to accommodate multiple objectives by weighting each sub-function, and the continuous adjoint method is employed to derive the sensitivity. Thus, both the governing equations for primal and adjoint problems are written as strong forms and hold at any moment and at any location in the continuum body or on its boundary. Accordingly, the proposed formulation is independent of the requirements from numerical implementation, such as element type or discretization method. In addition, the reaction–diffusion equation is used to update the design variable in an optimizing process, by which the continuous distribution of the design variable, as well as material properties, are realized. After the basic performance of the proposed formulation is demonstrated with a simple numerical setup, two-material (matrix and inclusion materials) and single-material (material and null) topology optimizations are presented, and discussions are made. • Topology optimization considering nonlocal damage constraint is presented. • The formulation is developed based on the finite strain framework. • The objective function allows multiple objectives in one optimization. • The continuous adjoint method is used to derive sensitivity. • Both single-material and two-material topology optimizations are demonstrated. [ABSTRACT FROM AUTHOR]

[Display omitted] • Microstructure evolution and characteristics of lamella structure is simulated by three-dimensional phase-field simulation. • The impact mechanism of Al content on lamellar spacing is revealed. • Models of Nucleation and growth of multi-variants γ lamella are investigated. • Evolution kinetics of γ lamella shows distinct larger rate in longitudinal direction. The D0 19 - α 2 / L1 0 - γ alternate lamellae microstructure in TiAl alloys brings the superior mechanical property, the lamellae with multi-variant γ phase are key characteristic of morphology-property causality. The nucleation, growth and coarsening of multi-variants γ are investigated in TiAl alloys by using the three-dimensional phase-field simulation, the γ morphology, lamellar spacing (LS) and the predicted yield strength of alloy are in agreement with the experiment. It is found that the γ variants can nucleate and grow up separately to form the single-variant lamella, or through the symbiotic nucleation and growth to form multi-variant lamella, the growth of γ lamella follows the terrace-ledge-kink mechanism. The growth rate along the longitudinal direction is distinctly larger than that of the thickness direction, resulting in the lamellar structure. A three-stage growth kinetics, nucleation, growth and coarsening of γ lamella in thickness is revealed. As Al content changes, the LS is enlarged with the reduced lamella number. The influence of the LS on the mechanical properties is evaluated by using the Hall-Petch relationship. The results give an insight for the formation and evolution mechanisms of multi-variant γ lamella, and discover the internal relations of lamellar morphology and mechanical properties of TiAl alloys. [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.   • Bigger particles, and higher C-rates and depth of delithiation promote material cracking. • Ni-richer NMC electrodes withstand larger C-rates. • Functionally-graded with composition gradient designs reduce mechanical degradation. [ABSTRACT FROM AUTHOR]

Phase-field modeling has increasingly been used to study microstructural evolution in fission gas bubbles in nuclear fuel to improve understanding of fission gas release. To improve computational efficiency, often only vacancies and gas atoms are included as defect species. In this case, the net effects of vacancy and interstitial production, recombination, and biased sink absorption are included as a net vacancy source, or net vacancy source combined with an effective sink. However, there has been a lack of clarity on what parameter values should be used for these approaches to best match the more complete physical picture that includes interstitials and vacancies. Here, we compare a phase-field model of void growth to analytical models for the source-only and source plus sink approach to gain insight into how the phase-field models can be parameterized effectively. The source-only approach provides greater flexibility to match growth rates determined from the full vacancy-interstitial picture. A strategy was developed for determining the value of the net vacancy source term by comparing to an analytical model that includes vacancy and interstitial production, recombination, and biased sink absorption. This strategy can be used to parameterize phase-field models of fission gas bubble growth. • Net vacancy production can be approximated by vacancy source-only and source+sink approaches. • Quasi-steady state analytical models explain behavior of phase-field models. • Vacancy source-only models have more flexibility to match vacancy-interstitial models. • Fitting source term in source-only model provides a good approximation to full vacancy-interstitial model. [ABSTRACT FROM AUTHOR]

Skyrmion, a local bubble-like topological magnetization structure, can collectively emergent in magnets in a lattice form skyrmion crystal (SkX). SkX has great application potential in functional devices because it can manipulate material properties via coupling with atomic lattices. The lattice defects such as vacancy widely exist in the SkX as well, and they have rich dynamic behaviors and have great implications for the host material. However, although the nature of ideal SkX is well studied, the characteristics of SkX defects are relatively underdeveloped. Here, we deeply studied the structural properties of a vacancy defect in the SkX by a thermodynamic phase-field simulation. We found that the higher external magnetic and temperature fields favor rigid skyrmions (crystal), in which the SkX vacancy is less deformed, while the lower fields favor softer skyrmions where the SkX vacancy structure is considerably deformed. Such unique deformation and stability of the SkX vacancy are mainly the results of the competition between free energies in the view of thermodynamics. Our study demonstrated that the external field-controlled static properties of SkX vacancy highly depend on the quasiparticle nature of skyrmions. This indicates the properties of SkX defects can be controlled by the SkX features under different fields, which should open an avenue for the study and design of smart materials and advanced devices by engineering skyrmion crystals. [Display omitted] [ABSTRACT FROM AUTHOR]

We investigate the morphodynamics of an ice layer over a turbulent stream of warm water using numerical simulations. At low water speeds, characteristic streamwise undulations appear, which can be explained by the Reynolds analogy between heat and momentum transfer. As the water speed increases, these undulations combine with spanwise ripples of a much greater length scale. These ripples are generated by a melting mechanism controlled by the instability originating from the ice–water interactions, and, through a melting/freezing process, they evolve downstream with a migration velocity much slower than the turbulence characteristic velocity. [Display omitted] • First numerical evidence of the formation of ice ripples by means of DNS • Near-wall turbulent structures induce streamwise patterns on the ice surface • At high Reynolds, spanwise ice ripples emerge and overlap to the streamwise patterns • When ripples emerge, the Reynolds analogy between heat and momentum transfer fails [ABSTRACT FROM AUTHOR]

In this article, we propose a thermodynamically consistent phase-field model for thermo-mechanical fracture and provide an open-source implementation of the proposed model using a recently developed finite element toolbox, Gridap in Julia. Here, we have derived the balance equations for the thermo-mechanical fracture by invoking the virtual power principle and determined the constitutive relations for the thermodynamic fluxes based on the satisfaction of the thermodynamic laws. Our proposed formulation provides an equation of temperature evolution that can easily accommodate dissipative effects such as viscous damping. We provide very compact and user-friendly open-source codes for implementing the proposed model using Gridap in Julia that requires very low memory usage and gives a high degree of flexibility to the users in defining weak forms of the governing partial differential equations (PDEs). We have validated the proposed model and its implementation against such standard results available in the literature as crack propagation in the cruciform shape material, single edge notched plate, bi-material beam, and a quenching test. [ABSTRACT FROM AUTHOR]

A unified framework for an impromptu switching between the coupled (i.e., the monolithic), the staggered, and the uncoupled solution strategies for the phase-field based computations of material fracture is presented. As a model problem, a 2d quasi-brittle solid phase-field formulation, which is based on the tension-compression split energy functional, is chosen. In order to fit within the proposed framework, the classical staggered approach is reformulated as a loop of uncoupled monolithic steps that we call modified-staggered. Numerical examples show that an automatic switching between the coupled and the modified-staggered solution methods (when the former fails) may shorten the computational times (compared to the pure staggered approach) for an order of magnitude. [ABSTRACT FROM AUTHOR]

In this article, we develop a reduced basis method for efficiently solving the coupled Stokes/Darcy equations with parametric internal geometry. To accommodate possible changes in topology, we define the Stokes and Darcy domains implicitly via a phase-field indicator function. In our reduced order model, we approximate the parameter-dependent phase-field function with a discrete empirical interpolation method (DEIM) that enables affine decomposition of the associated linear and bilinear forms. In addition, we introduce a modification of DEIM that leads to non-negativity preserving approximations, thus guaranteeing positive-semidefiniteness of the system matrix. We also present a strategy for determining the required number of DEIM modes for a given number of reduced basis functions. We couple reduced basis functions on neighboring patches to enable the efficient simulation of large-scale problems that consist of repetitive subdomains. We apply our reduced basis framework to efficiently solve the inverse problem of characterizing the subsurface damage state of a complete in-situ leach mining site. [ABSTRACT FROM AUTHOR]

In this paper we deal with the inverse problem of determining cavities and inclusions embedded in a linear elastic isotropic medium from boundary displacement's measurements. For, we consider a constrained minimization problem involving a boundary quadratic misfit functional with a regularization term that penalizes the perimeter of the cavity or inclusion to be identified. Then using a phase field approach we derive a robust algorithm for the reconstruction of elastic inclusions and of cavities modelled as inclusions with a very small elasticity tensor. [ABSTRACT FROM AUTHOR]

Based on alumina ceramics, we employ the phase‐field method to study the effects of thermal conductivity, specific heat, density, thermal expansion coefficient, Young's modulus, and fracture toughness on thermal shock cracks. The results show that increasing thermal conductivity and fracture toughness will reduce thermal shock damage. That is, the long crack length becomes shorter, or the crack density becomes smaller. However, increasing the thermal expansion coefficient and Young's modulus will increase thermal shock damage. It is consistent with the previous thermal shock theory. The effect of material parameters on crack propagation speed was also considered. In addition, we carried out a thermal shock test of the zirconia. The results of the phase‐field calculation are the same as the thermal shock results of the zirconia. This paper verifies that the phase‐field method is suitable for simulating thermal shock cracks in other ceramics. [ABSTRACT FROM AUTHOR]