Phase Field Modeling

This article is devoted to extension of the recently developed enhanced local damage model for failure prediction in bi-material structures. Compared to non-local models, the enhanced local model offers lower computational cost while the inherent mesh-dependency issue is treated. By defining equivalent strain based on the bi-energy norm concept and Mazars's criterion, which considers both tensile and compressive strain components, the model aligns with the behavior of quasi-brittle materials. The state of material point is indicated by a damage parameter, ranging from 0 to 1, to represent the evolution from being fully intact to complete failure. An efficient staggered scheme is introduced, in which the equilibrium equation and the update of damage parameter are decoupled. The proposed model is validated with a series of three-point bending experimental tests on PMMA/Al6061 specimens reported by Lee and Krishnaswamy (2000). Good agreement is observed between the proposed model and experimental data, as well as numerical results from other authors, in crack path prediction.

Dikes supply magma to most volcanic eruptions. Understanding how propagating dikes may, or may not, reach the surface is thus one of the fundamental tasks for volcanology. Many, perhaps most, dike segments injected from magma sources do not reach the surface to feed volcanic eruptions. Instead, the dike segments become arrested (stop their propagation), commonly at or close to contacts between mechanically dissimilar layers/units, at various crustal depths. This means that many and perhaps most volcanic unrest periods with dike injections do not result in eruptions. There are several conditions that make dike arrest likely, but the main one is layering where the layers have contrasting mechanical properties. Such layering means that local stresses are heterogeneous and anisotropic and, therefore, in some layers unfavourable for dike propagation – hence the dike arrest. Here I show that once a dike has formed, however, its very existence tends to make the local stress field along the dike homogeneous (with invariable orientation of principal stresses) and favourable (with dike-parallel orientation of the maximum compressive principal stress) for later dike injections. This means that subsequence dikes may use an earlier dike as a path, either along the margin or the centre of the earlier dike, thereby generating a multiple dike. Because earlier feeder-dikes form potential paths for later-injected dikes to the surface, many volcanic eruptions are fed by multiple dikes. Examples include recent eruptions in the volcanoes Etna (Italy) and Kilauea (Hawaii), and the Icelandic volcanoes Krafla, Hekla,  Fagradalsfjall, and the Sundhnukur crater row. Thus, multiple dikes favour dike propagation to the surface; thereby making dike-fed eruptions easier.

Oscillation marks are ripples formed on the surface of continuously cast material. They may cause cracking and decrease the yield of the process since some material must be grinded away to avoid crack growth. A study of break-out shells, a fullscale water model study and full-scale experiments in four different plants have been performed to analyse the formation of oscillation marks. The hypothesis initiating the studies was that there is an optimal oscillation frequency. Material cast at the optimal frequency will have smaller oscillation marks and fewer cracks and, maybe most important, all marks are of the same character and depth. The optimal oscillation frequency is determined by its relation to casting velocity and interfacial tension between metal and protective medium, e.g. slag:

Isolated metallic lithium (“dead” lithium) formation is one of the key challenges to enabling the practical application of lithium-metal anodes. Here, we identify a charge-discharge asymmetry mechanism through linear stability analysis and phase-field modeling and propose a simple symmetry parameter to quantify the difficulty of forming isolated metallic lithium. Our results suggest that this symmetry parameter is highly correlated with the experimentally measured Coulombic efficiency. The symmetry parameter provides a mechanistic understanding of the interconnected relationship between the battery’s performance and various factors, like material, geometry, and loading parameters. Our results show that completely suppressing dendrites is unnecessary as long as they are controllable and do not form isolated metallic lithium. Our analysis provides insights into mitigating isolated metallic lithium and guidelines for lithium-metal battery design.

Modeling microstructure evolution in electrochemical systems is vital for understanding the mechanism of various electrochemical processes. In this work, we propose a general phase field framework that is fully variational and thus guarantees that the energy decreases upon evolution in an isothermal system. The bulk and interface free energies are decoupled using a grand potential formulation to enhance numerical efficiency. The variational definition of the overpotential is used, and the reaction kinetics is incorporated into the evolution equation for the phase field to correctly capture capillary effects and eliminate additional model parameter calibrations. A higher-order kinetic correction is derived to accurately reproduce general reaction models such as the Butler-Volmer, Marcus, and Marcus-Hush-Chidsey models. Electrostatic potentials in the electrode and the electrolyte are considered separately as independent variables, providing additional freedom to capture the interfacial potential jump. To handle realistic materials and processing parameters for practical applications, a driving force extension method is used to enhance the grid size by three orders of magnitude. Finally, we comprehensively verify our phase field model using classical electrochemical theory.

We present a novel computational framework to assess the structural integrity of welds. In the first stage of the simulation framework, local fractions of microstructural constituents within weld regions are predicted based on steel composition and welding parameters. The resulting phase fraction maps are used to define heterogeneous properties that are subsequently employed in structural integrity assessments using an elastoplastic phase field fracture model. The framework is particularized to predicting failure in hydrogen pipelines, demonstrating its potential to assess the feasibility of repurposing existing pipeline infrastructure to transport hydrogen. First, the process model is validated against experimental microhardness maps for vintage and modern pipeline welds. Additionally, the influence of welding conditions on hardness and residual stresses is investigated, demonstrating that variations in heat input, filler material composition, and weld bead order can significantly affect the properties within the weld region. Coupled hydrogen diffusion-fracture simulations are then conducted to determine the critical pressure at which hydrogen transport pipelines will fail. To this end, the model is enriched with a microstructure-sensitive description of hydrogen transport and hydrogen-dependent fracture resistance. The analysis of an X52 pipeline reveals that even 2 mm defects in a hard heat-affected zone can drastically reduce the critical failure pressure.

We present TDS Simulator, a new software tool aimed at modelling thermal desorption spectroscopy (TDS) experiments. TDS is a widely used technique for quantifying key characteristics of hydrogen-material interactions, such as diffusivity and trapping. However, interpreting the output of TDS experiments is non-trivial and requires appropriate post-processing tools. This work introduces the first software tool capable of simulating TDS curves for arbitrary choices of material parameters and hydrogen trap characteristics, using the primary hydrogen diffusion and trapping models (Oriani, McNabb-Foster). Moreover, TDS Simulator contains a specific functionality for loading experimental TDS data and conducting the inverse calibration of a selected transport model, providing automatic estimates of the density and binding energy of each hydrogen trap type in the material. In its first version, TDS Simulator is provided as a MATLAB App, which is made freely available to the community and provides a simple graphical user interface (GUI) to make use of TDS Simulator straightforward. As reported in the present manuscript, the outputs of TDS Simulator have been extensively validated against literature data. Demonstrations of automatic determination of trap characteristics from experimental data through the optimization tool are also provided. The present work enables an efficient and straightforward characterization of hydrogen-material characteristics relevant to multiple applications, from nuclear fusion to the development of hydrogen-compatible materials for the hydrogen economy. TDS Simulator can be downloaded from https://mechmat.web.ox.ac.uk/codes.

Linear elastic fracture mechanics (LEFM) models have been used to estimate crevasse depths in glaciers and to represent iceberg calving in ice sheet models. However, existing LEFM models assume glacier ice to be homogeneous and utilize the mechanical properties of fully consolidated ice. Using depth-invariant properties is not realistic as the process of compaction from unconsolidated snow to firn to glacial ice is dependent on several environmental factors, typically leading to a lower density and Young's modulus in upper surface strata. New analytical solutions for longitudinal-stress profiles are derived using depth-varying properties based on borehole data from the Ronne Ice Shelf and are used in an LEFM model to determine the maximum penetration depths of an isolated crevasse in grounded glaciers and floating ice shelves. These maximum crevasse depths are compared to those obtained for homogeneous glacial ice, showing the importance of including the effect of the upper unconsolidated firn layers on crevasse propagation. The largest reductions in the penetration depth ratio were observed for shallow grounded glaciers, with variations in Young's modulus being more influential than firn density (maximum differences in crevasse depth of 46 % and 20 %, respectively), whereas firn density changes resulted in an increase in penetration depth for thinner floating ice shelves (95 %–188 % difference in crevasse depth between constant and depth-varying properties). Thus, our study shows that the firn layer can increase the vulnerability of ice shelves to fracture and calving, highlighting the importance of considering depth-dependent firn layer material properties in LEFM models for estimating crevasse penetration depths and predicting rift propagation.

The triaxial test is used in a laboratory to investigate the behaviour of geotechnical materials (e.g. clays and sands). The difficulty in measuring some properties of granular media such as energy changes throughout the test have motivated the current numerical simulations of this test. This paper presents a description of a series of triaxial tests using LIGGGHTS open source Discrete Element Modelling software in a study of how energy is dissipated in granular media. The simulated triaxial tests are being carried out on cube shaped samples with six mesh walls enclosing the particles. Three of the walls (i.e. bottom, left and back walls) are fixed in position while the other three walls are allowed to move. Energy dissipation will be investigated by tracing changes of the energy terms at various time steps and applying the principle of energy conservation. The relationship between confining pressure, particle size distribution, friction coefficient and the voids ratio with energy d...

Rapid solidification experiments on thin film aluminum samples reveal the presence of lattice orientation gradients within crystallizing grains. To study this phenomenon, a single-component phase-field crystal (PFC) model that captures the properties of solid, liquid, and vapor phases is proposed to model pure aluminium quantitatively. A coarse-grained amplitude representation of this model is used to simulate solidification in samples approaching micrometer scales. The simulations reproduce the experimentally observed orientation gradients within crystallizing grains when grown at experimentally relevant rapid quenches. We propose a causal connection between defect formation and orientation gradients.

Reel-laying rigid pipelines induce cyclic deformation that affect the section. Among other, excessive ovality may jeopardize pipe strength during installation & in-service. Previous works shown that conventional material stress-strain relation is no longer suitable for pipeline reel-lay simulations. This paper introduces an advanced model for material behaviour. The evolution of the material behavior and its variations on the cyclic loads paths is detailed. The model uses stressstrain relations and yield surface evolution from the first load curve, with Lüders plateau leading to stabilized cycle with material property variations in cycles. The model has been implemented in finite element software. This model calibration and validation are discussed in this paper. Simulation accuracy is improved compared to published results.

A methodology to predict failure of composite materials is proposed. It is based on local shakedown analysis on the microscopic level of the composite and the use of homogenisation technique to determine the influence of each component (matrix, fiber and interface) on the macroscopic response of such composite.

The phase field method has gathered significant attention in the past decade due to its versatile applications in engineering contexts, including fatigue crack propagation modeling. Particularly, the phase field cohesive zone method (PF-CZM) has emerged as a promising approach for modeling fracture behavior in quasi-brittle materials, such as concrete. The present contribution expands the applicability of the PF-CZM to include the modeling of fatigue-induced crack propagation. This study critically examines the validity of the extended PF-CZM approach by evaluating its performance across various fatigue behaviours, encompassing hysteretic behavior, S-N curves, fatigue creep curves, and the Paris law. The experimental investigations and validation span a diverse spectrum of loading scenarios, encompassing pre-and post-peak cyclic loading, as well as low-and high-cyclic fatigue loading. The validation process incorporates 2D and 3D boundary value problems, considering mode I and mixed-modes fatigue crack propagation. The results obtained from this study show a wide range of validity, underscoring the remarkable potential of the proposed PF-CZM approach to accurately capture the propagation of fatigue cracks in concrete-like materials. Furthermore, the paper outlines recommendations to improve the predictive capabilities of the model concerning key fatigue characteristics.

The presence of hydrogen traps within a metallic alloy influences the rate of hydrogen diffusion. The electro-permeation (EP) test can be used to assess this: the permeation of hydrogen through a thin metallic sheet is measured by suitable control of hydrogen concentration on the front face and by recording the flux of hydrogen that exits the rear face. Additional insight is achieved by the more sophisticated three stage EP test: the concentration of free lattice hydrogen on the front face is set to an initial level, is then dropped to a lower intermediate value and is then restored to the initial level. The flux of hydrogen exiting the rear face is measured in all three stages of the test. In the present study, a transient analysis is performed of hydrogen permeation in a three stage EP test, assuming that lattice diffusion is accompanied by trapping and de-trapping. The sensitivity of the three stage EP response to the depth and density of hydrogen traps is quantified. A significant difference in permeation response can exist between the first and third stages of the EP test when the alloy contains a high number density of deep traps.

A nonlinear phase-field model is developed to simulate corrosion damage. The motion of the electrode−electrolyte interface follows the usual kinetic rate theory for chemical reactions based on the Butler−Volmer equation. The model links the surface polarization variation associated with the charging kinetics of an electric double layer (EDL) to the mesoscale transport. The effects of the EDL are integrated as a boundary condition on the solution potential equation. The boundary condition controls the magnitude of the solution potential at the electrode−electrolyte interface. The ion concentration field outside the EDL is obtained by solving the electro−diffusion equation and Ohm's law for the solution potential. The model is validated against the classic benchmark pencil electrode test. The framework developed reproduces experimental measurements of both pit kinetics and transient current density response. The model enables more accurate information on corrosion damage, current density, and environmental response in terms of the distribution of electric potential and charged species. The sensitivity analysis for different properties of the EDL is performed to investigate their role in the electrochemical response of the system. Simulation results show that the properties of the EDL significantly influence the transport of ionic species in the electrolyte.

Magnesium alloys exhibit great potential for use as implants that gradually and entirely dissolve in the human body after healing. Rapid corrosion in body fluids and the susceptibility to pitting corrosion and stress corrosion cracking are the main reasons limiting the widespread use of Mg alloys as a biodegradable material. The concurrence of mechanical loading and a corrosive environment dramatically accelerates the corrosion rate and promotes crack propagation, leading to the sudden failure of implants. A phase-field model is developed to simulate the corrosion of bioabsorbable metals in biological fluids. The model incorporates both Mg dissolution and the transport of Mg ions in solution. The framework captures pitting corrosion and incorporates the role of mechanical fields in enhancing the corrosion of biodegradable metals. The model is validated against in vitro corrosion data on Mg alloys immersed in simulated body fluid. The potential of the model to capture mechano-chemical effects is demonstrated by considering bioabsorbable coronary stents subjected to mechanical loading. The results show that pitting severely compromises the structural integrity of the stent and the application of mechanical loading initiates a pit-to-crack transition and crack propagations, leading to premature fracture after a short time in solution. This work extends phase-field modeling to biomaterial degradation and provides a novel mechanistic tool for assessing the service life of bioabsorbable medical devices.

Scale-free model for the coupled evolution of discrete dislocation bands and multivariant martensitic microstructure is developed. In contrast to previous phase field models, which are limited to nanoscale specimens, this model allows treating nucleation and evolution of martensite at evolving dislocation pileups, twin tips, and shear bands in a sample of an arbitrary size. Model is applied for finite element (FE) simulations of plastic strain-induced phase transformations (PTs) in a polycrystalline sample under compression and shear. Solution explains the one to two orders of magnitude reduction in PT pressure by plastic shear; existence of incompletely transformed stationary state and optimal shear strain for strain-induced synthesis of high pressure phases.

In the race to reduce global CO2 emissions and achieve net-zero, chemomechanics must play a critical role in the technological development of current and next-generation batteries to improve their energy storage capabilities and their lifetime. Many degradation processes arise through mechanics via the development of diffusion-induced stress and volumetric strains within the various constituent materials in a battery. From particle cracking in lithium-ion batteries to lithium dendrite-based fracture of solid electrolytes in solid-state batteries, it is clear that significant barriers exist in the development of these energy storage systems, where chemomechanics plays a central part. To accelerate technological and scientific advances in this area, multi-scale and highly coupled multiphysics modelling must be carried out that includes mechanics-based phenomena. In this perspective article, we provide an introduction to chemomechanical modelling, the various physical problems that it addresses, and the issues that need to be resolved in order to expand its use within the field of battery technology.

Selective laser melting (SLM) is a promising manufacturing technique where the part design, from performance and properties process control and alloying, can be accelerated with integrated computational materials engineering (ICME). This paper demonstrates a process-structure-properties-performance modeling framework for SLM. For powder-bed scale melt pool modeling, we present a diffuse-interface multiphase computational fluid dynamics model which couples Navier–Stokes, Cahn–Hilliard, and heat-transfer equations. A computationally efficient large-scale heat-transfer model is used to describe the temperature evolution in larger volumes. Phase field modeling is used to demonstrate how epitaxial growth of Ti-6-4 can be interrupted with inoculants to obtain an equiaxed polycrystalline structure. These structures are enriched with a synthetic lath martensite substructure, and their micromechanical response are investigated with a crystal plasticity model. The fatigue performance of these...

Steel structures are designed to operate in an elastic domain, but sometimes plastic strains induce damage and fracture. Besides experimental investigation, a phase-field damage model (PFDM) emerged as a cutting-edge simulation technique for predicting damage evolution. In this paper, a von Mises metal plasticity model is modified and a coupling with PFDM is improved to simulate ductile behavior of metallic materials with or without constant stress plateau after yielding occurs. The proposed improvements are: (1) new coupling variable activated after the critical equivalent plastic strain is reached; (2) two-stage yield function consisting of perfect plasticity and extended Simo-type hardening functions. The uniaxial tension tests are conducted for verification purposes and identifying the material parameters. The staggered iterative scheme, multiplicative decomposition of the deformation gradient, and logarithmic natural strain measure are employed for the implementation into finit...

This article overviews a new, recent success of phase-field modeling: its application to predicting the evolution of the corrosion front and the associated structural integrity challenges. Despite its important implications for society, predicting corrosion damage has been an elusive goal for scientists and engineers. The application of phase-field modeling to corrosion not only enables tracking the electrolyte-metal interface, but also provides an avenue to explicitly simulate the underlying mesoscale physical processes. This lays the groundwork for developing the first generation of mechanistic corrosion models, which can capture key phenomena such as film rupture and repassivation, the transition from activationto diffusion-controlled corrosion, interactions with mechanical fields, microstructural and electrochemical effects, intergranular corrosion, material biodegradation, and the interplay with other environmentally assisted damage phenomena such as hydrogen embrittlement.

A quantitative phase-field model is developed for the investigation of crevice corrosion of iron in salt water. Six types of ionic species and some associated chemical reactions have been considered. In addition to the transient distributions of ion concentrations and electric potential in the electrolyte, some physical and chemical properties related to corrosion, such as overpotential, pH value and corrosion rate, under different metal potentials are studied. Benchmarking of the phase-field model against a sharp interface model is conducted. The corrosion rates predicted by the models are in the same order of magnitudes with experimental results.

Controlling volume fractions of nanoparticles in a matrix can have a substantial influence on composite performance. This paper presents a topology optimization algorithm that designs nanocomposite structures for objectives pertaining to stiffness and strain sensing. Local effective properties are obtained by controlling local volume fractions of carbon nanotubes (CNTs) in an epoxy matrix, which are assumed to be well dispersed and randomly oriented. The method is applied to the optimization of a plate with a hole structure. Several different allowable CNT volume fraction constraints are examined, and the results show a tradeoff in preferred CNT distributions for the two objectives. It is hypothesized that the electrode location plays an important role in the strain sensing performance, and a surrogate model is developed to incorporate the electrode boundary as a set of additional design variables. It is shown that optimizing the topology and boundary electrode location together lea...

Combined experiments and computational modelling are used to increase understanding of the suitability of the Single-Edge Notch Tension (SENT) test for assessing hydrogen embrittlement susceptibility. The SENT tests were designed to provide the mode I threshold stress intensity factor (th) for hydrogen-assisted cracking of a C110 steel in two corrosive environments. These were accompanied by hydrogen permeation experiments to relate the environments to the absorbed hydrogen concentrations. A coupled phase-field-based deformation-diffusion-fracture model is then employed to simulate the SENT tests, predicting th in good agreement with the experimental results and providing insights into the hydrogen absorption-diffusion-cracking interactions. The suitability of SENT testing and its optimal characteristics (e.g., test duration) are discussed in terms of the various simultaneous active time-dependent phenomena, triaxiality dependencies, and regimes of hydrogen embrittlement susceptibility.

Amoeboid cells exhibit a highly dynamic motion that can be directed by external chemical signals, through the process of chemotaxis. Here, we propose a three-dimensional model for chemotactic motion of amoeboid cells. We account for the interactions between the extracellular substances, the membrane-bound proteins, and the cytosolic components involved in the signaling pathway that originates cell motility. We show two- and three-dimensional simulations of cell migration on planar substrates, flat surfaces with obstacles, and fibrous networks. The results show that our model reproduces the main features of chemotactic amoeboid motion. Our simulations unveil a complicated interplay between the geometry of the cell's environment and the chemoattractant dynamics that tightly regulates cell motion. The model opens new opportunities to simulate the interactions between extra- and intra-cellular compounds mediated by the matrix geometry.

We present a new mechanistic, phase field-based formulation for predicting hydrogen embrittlement. The multi-physics model developed incorporates, for the first time, a Taylor-based dislocation model to resolve the mechanics of crack tip deformation. This enables capturing the role of dislocation hard

We present a new mechanistic, phase field-based formulation for predicting hydrogen embrittlement. The multi-physics model developed incorporates, for the first time, a Taylor-based dislocation model to resolve the mechanics of crack tip deformation. This enables capturing the role of dislocation hard

An electro-chemo-mechanical phase-field formulation is developed to simulate pitting and stress corrosion in polycrystalline materials. The formulation incorporates dependencies of mechanical properties and corrosion potential on crystallographic orientation. The model considers the formation and charging dynamics of an electric double layer through a new general boundary condition for the solution potential. The potential of the model is demonstrated by simulating corrosion in polycrystalline materials with various grain morphology distributions. The results show that incorporating the underlying microstructure yields more extensive defects, faster defect kinetics, and irregular pit and crack shapes relative to a microstructurally-insensitive homogeneous material scenario.

In this work, we describe our contribution to the Purdue-SANDIA-LLNL Damage Mechanics Challenge. The phase field fracture model is adopted to blindly estimate the failure characteristics of the challenge test, an unconventional three-point bending experiment on an additively manufactured rock resembling a type of gypsum. The model is formulated in a variationally consistent fashion, incorporating a volumetric–deviatoric strain energy decomposition, and the numerical implementation adopts a monolithic unconditionally stable solution scheme. Our focus is on providing an efficient and simple yet rigorous approach capable of delivering accurate predictions based solely on physical parameters. Model inputs are Young’s modulus
, Poisson’s ratio
, toughness
and strength
(as determined by the choice of phase field length scale
). We show that a single mode I three-point bending test is sufficient to calibrate the model, and that the calibrated model can then reliably predict the force versus displacement responses, crack paths and surface crack morphologies of more intricate three-point bending experiments that are inherently mixed-mode. Importantly, our peak load, crack trajectory and crack surface morphology predictions for the challenge test, submitted before the experimental data was released, show a remarkable agreement with experiments. The characteristics of the challenge, and how changes in these can impact the predictive abilities of phase field fracture models, are also discussed.

A control system or computer aided quality control system methodology is presented to determine surface defects in the continuous casting of steel slabs. When no defects are present, slabs can be either hot charged or direct rolled , thereby increasing the throughput of the system and decreasing energy consumption .

Particle-in-Cell (PIC) methods employ particle representations of unknown fields, but also employ continuum fields for other parts of the problem. Thus projection between particle and continuum bases is required. Moreover, we often need to enforce conservation constraints on this projection. We derive a mechanism for enforcement based on weak equality, and implement it in the PETSc libraries. Scalability is demonstrated to more than 1B particles.

The aim of this study is to investigate multi-dimensional vector variational problems considering data uncertainty in each of the objective functional and constraints. We establish the robust necessary and sufficient efficiency conditions such that any robust feasible solution could be the robust weakly efficient solution for the problems under consideration. Emphatically, we present robust efficiency conditions for multi-dimensional scalar, vector, and vector fractional variational problems by using the notion of a convex functional.

The effect of cold rolling on the microstructure evolution and mechanical properties of a cold rolled Fe-0.3C-17Mn-1.5Al TWIP steel was studied. The plate samples were cold rolled with reductions of 20, 40, 60 and 80%. The structural changes were associated with the development of deformation twinning and shear bands. The average spacing between twin boundaries in the transverse section of the rolled plates decreased from ~190 to 36 nm with an increase in the rolling reduction from 20 to 40%. Upon further rolling to 80% reduction the twin spacing remained at about 30 nm. The cold rolling resulted in significant increase in strength as revealed by tensile tests at an ambient temperature. The offset yield stress approached 1440 MPa, and the ultimate tensile strength increased to 1630 MPa after rolling reduction of 80%. Such significant strengthening was attributed to the development of specific structure consisting of deformation nanotwins with high dislocation density.

In this work we develop a fully implicit Hybrid High-Order algorithm for the Cahn-Hilliard problem in mixed form. The space discretization hinges on local reconstruction operators from hybrid polynomial unknowns at elements and faces. The proposed method has several advantageous features: (i) It supports fairly general meshes possibly containing polyhedral elements and nonmatching interfaces; (ii) it allows arbitrary approximation orders; (iii) it has a moderate computational cost thanks to the possibility of locally eliminating element-based unknowns by static condensation. We perform a detailed stability and convergence study, proving optimal convergence rates in energy-like norms. Numerical validation is also provided using some of the most common tests in the literature.

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We present a theoretical and computational model for the behavior of a porous solid undergoing two interdependent processes, the finite deformation of a solid and species migration through the solid, which are distinct in bulk and on surface. Nonlinear theories allow us to systematically study porous solids in a wide range of applications, such as drug delivery, biomaterial design, fundamental study of biomechanics and mechanobiology, and the design of sensors and actuators. As we aim to understand the physical phenomena at a smaller length scale towards comprehending the fundamental biological processes and the miniaturization of devices, the surface effect becomes more pertinent. Although existing methodologies provide the necessary tools to study coupled bulk effects for deformation and diffusion; however, very little is known about fully coupled bulk and surface poroelasticity at finite strain. Here we develop a thermodynamically consistent formulation for multiphysics processes of surface and bulk poroelasticity, specialized for soft hydrated solids, along with a corresponding finite element implementation. Our multiphysical approach captures the interplay between competing processes of finite deformation and species diffusion through the bulk and surface, and provides invaluable insight when surface effects are important.

Numerical simulations and parametric studies of notched rectangular specimens subjected to dynamic tensile loads were performed. The simulations were based on two-dimensional finite element analysis to predict the brittle fracture path using the phase-field approach. The parametric studies investigated the influence of geometric parameters and the loading speed on crack path propagation. An empirical model based on the sparse proper generalized decomposition learning technique was created to predict the crack path. This model provides a quick prediction of the global behavior of the crack path circumventing the CPU cost of the full finite element method simulation.

In this study, the finite element method was used to predict the brittle fracture path using the phase-field approach. The staggered algorithm was adopted owing to its proved robustness. In order to reduce the computational time, an optimization method of matrix assembly was proposed. Then, a comparative study was performed between the isotropic and the hybrid formulations. Numerical tests were computed to compare these formulations. Because of the lack of experimental validation, tensile tests were achieved in order to validate the phase-field approach. A good agreement between the experimental and numerical results in terms of crack pattern was illustrated for the hybrid formulation.

Hydrogen-assisted cracking (HAC) behavior of peak-aged, as-received material of aerospacegrade, and over-aged, based on the API 6ACRA-718 standard, microstructural variants of Ni alloy 718 was studied using slow strain rate test (SSRT) under in-situ hydrogen (H) environment. The effect of cooling rate post solution annealing (SA) treatment on HAC susceptibility was also examined. The influence of H testing protocols with and without H pre-charging on HAC susceptibility was evaluated. The link between microstructural features and HAC fracture features was established using advanced electron microscopy. The results suggest that the microstructural variant of alloy 718, with sufficient initial ductility and clean grain boundaries, may be suitable for service applications in H environments. The over-aged variant obtained by air cooling post SA and followed by aging at 774°C for 6 hours exhibited better HAC resistance due to the presence of d phase sporadically decorated along the GBs. The slower cooling rates post SA increased HAC susceptibility. The yield strength of the over-aged variants aged at 802°C for 8 hours, regardless of SA cooling rates, was lower than the minimum requirement expected by the API standard, and the HAC susceptibility was severe exhibiting intergranular fracture due to continuous decoration of d phase along the GBs. The peak-aged variant demonstrated severe HAC susceptibility showing transgranular fracture. HAC susceptibility of all variants was greater for SSRT of non-H pre-charged specimens under in-situ H environment at 20°C than at 80°C. H pre-charging enhanced the HAC susceptibility further when tested at 20°C and the role of microstructure was largely suppressed. HAC susceptibility was manifested as both transgranular and intergranular fracture modes depending on the slip localization activity and H availability.

A phase-field model of alloy solidification is coupled to a new heat transfer finite element model of the laser powder deposition process. The robustness and accuracy of the coupled model is validated by studying spacing evolution under the directional solidification conditions in laser powder deposition of Ti-Nb alloys. Experimental Ti-Nb samples reveal the microstructure on a longitudinal section with significant change in the size of the dendrites across the sample. Quantitative phase-field simulations of directional solidification under local steady-state conditions extracted from the results of the finite element thermal model confirmed this behavior. Specifically, the phase-field simulations agree with the results of the analytical model of Hunt in predicting a minimum spacing value, which is due to the mutual effects of the increasing temperature gradient and decreasing solidification velocity towards the bottom of the microstructure. This work demonstrates the potential of coupling the phase-field method to complex heat transfer conditions necessary to simulate topologically complex microstructure morphologies present in laser powder deposition and other industrially relevant casting conditions.

The phase field method has gathered significant attention in the past decade due to its versatile applications in engineering contexts, including fatigue crack propagation modeling. Particularly, the phase field cohesive zone method (PF-CZM) has emerged as a promising approach for modeling fracture behavior in quasi-brittle materials, such as concrete. The present contribution expands the applicability of the PF-CZM to include the modeling of fatigue-induced crack propagation. This study critically examines the validity of the extended PF-CZM approach by evaluating its performance across various fatigue behaviors, encompassing hysteretic behavior, S-N curves, fatigue creep curves, and the Paris law. The experimental investigations and validation span a diverse spectrum of loading scenarios, encompassing pre- and post-peak cyclic loading, as well as low- and high-cyclic fatigue loading. The validation process incorporates 2D and 3D boundary value problems, considering mode I and mixed-modes fatigue crack propagation. The results obtained from this study show a wide range of validity, underscoring the remarkable potential of the proposed PF-CZM approach to accurately capture the propagation of fatigue cracks in concrete-like materials. Furthermore, the paper outlines recommendations to improve the predictive capabilities of the model concerning key fatigue characteristics.

Diffuse interface descriptions offer many advantages for the modeling of microstructure evolution. However, the numerical representation of moving diffuse interfaces on discrete numerical grids involves spurious grid friction, which limits the overall performance of the model in many respects. Interestingly, this intricate and detrimental effect can be overcome in finite difference (FD) and fast Fourier transformation (FFT)-based implementations by employing the so-called sharp phase-field method (SPFM). The key idea is to restore the discretization-induced broken translational invariance (TI) in the discrete phase-field equation by using analytic properties of the equilibrium interface profile. We prove that this method can indeed eliminate spurious grid friction in the three-dimensional space. Focusing on homogeneous driving forces, we quantitatively evaluate the impact of spurious grid friction on the overall operational performance of different phase-field models. We show that t...

The phase-field method provides a powerful framework for microstructure evolution modeling in complex systems, as often required within the framework of integrated computational materials engineering. However, spurious grid friction, pinning and grid anisotropy seriously limit the resolution efficiency and accuracy of these models. The energetic resolution limit is determined by the maximum dimensionless driving force at which reasonable model operation is still ensured. This limit turns out to be on the order of 1 for conventional phase-field models. In 1D, grid friction and pinning can be eliminated by a global restoration of Translational Invariance (TI) in the discretized phase-field equation. This is called the sharp phase-field method, which allows to choose substantially coarser numerical resolutions of the diffuse interface without the appearance of pinning. In 3D, global TI restricts the beneficial properties to a few specific interface orientations. We propose an accurate ...

The effectiveness of the mechanism of precipitation strengthening in metallic alloys depends on the shapes of the precipitates. Two different material systems are considered: tetragonal γ′′ precipitates in Ni-based alloys and tetragonal θ′ precipitates in Al-Cu-alloys. The shape formation and evolution of the tetragonally misfitting precipitates was investigated by means of experiments and phase-field simulations. We employed the method of invariant moments for the consistent shape quantification of precipitates obtained from the simulation as well as those obtained from the experiment. Two well-defined shape-quantities are proposed: (i) a generalized measure for the particles aspect ratio and (ii) the normalized λ2, as a measure for shape deviations from an ideal ellipse of the given aspect ratio. Considering the size dependence of the aspect ratio of γ′′ precipitates, we find good agreement between the simulation results and the experiment. Further, the precipitates’ in-plane shap...

We develop a continuum phase-field model for the simulation of diffusion limited solid-solid phase transformations during lithium insertion in LiFePO 4-nano-particles. The solid-solid phase boundary between the LiFePO 4 (LFP)-phase and the FePO 4 (FP)-phase is modeled as a diffuse interface of finite width. The model-description explicitly resolves a single LiFePO 4-particle, which is embedded in an elastically soft electrolyte-phase. Furthermore, we explicitly include anisotropic (orthorhombic) and inhomogeneous elastic effects, resulting from the coherency strain, as well as anisotropic (1D) Li-diffusion inside the nano-particle. In contrast to other related research work, we employ an Allen-Cahn-type phase-field approach for the diffuse interface modeling of the solid-solid phase boundary. The model contains an extra non-conserved order parameter field to distinguish the two different phases. The evolution of this order parameter field is controlled by an extra kinetic parameter independent from the Li-diffusion. Further, the effect of the nano-particle's size on the kinetics of FP to LFP phase transformations is investigated by means of both model. Both models predict a substantial increase in the steady state transformation velocity as the particle-size decreases down to dimensions that are comparable with the width of the interface between the FP and the LFP-phase. However, the extra kinetic parameter of the Allen-Cahn-type description may be used to reduce the strength of the velocity-increase with the decreasing particle size. Further, we consider the influence of anisotropic and inhomogeneous elasticity on the lithiation-kinetics within a rectangularly shaped LiFePO 4-particle embedded in an elastically soft electrolyte. Finally, the simulation of equilibrium shapes of LiFePO 4-particles is discussed. Within a respective feasibility study, we demonstrate that also the simulation of strongly anisotropic particles with aspect ratios up to 1/5 is possible.

In this work, the phase-field approach to fracture is extended to model fatigue failure in high- and low-cycle regime. The fracture energy degradation due to the repeated externally applied loads is introduced as a function of a local energy accumulation variable, which takes the structural loading history into account. To this end, a novel definition of the energy accumulation variable is proposed, allowing the fracture analysis at monotonic loading without the interference of the fatigue extension, thus making the framework generalised. Moreover, this definition includes the mean load influence of implicitly. The elastoplastic material model with the combined nonlinear isotropic and nonlinear kinematic hardening is introduced to account for cyclic plasticity. The ability of the proposed phenomenological approach to naturally recover main features of fatigue, including Paris law and Wöhler curve under different load ratios is presented through numerical examples and compared with e...