Phase-field Modeling Approach Research Articles

A phase-field modeling approach of fracture propagation in poroelastic media

This paper proposes a phase field model for fracture in poroelastic media. The porous medium is modeled based on the classical Biot poroelasticity theory and the fracture behavior is controlled by the phase field model. Moreover, the fracture propagation is driven by the elastic energy where the phase field is used as an interpolation function to transit fluid property from the intact medium to the fully broken one. We use a segregated (staggered) scheme and implement our approach in Comsol Multiphysics. The proposed model is verified by a single-phase solid subjected to tension and a 2D specimen subjected to an increasing internal pressure. We also compare our results with analytical solutions. Finally, we show 2D and 3D examples of internal fluid injection to illustrate the capability of the proposed approach.

Phase field modeling of sintering: Role of grain orientation and anisotropic properties

The influence of anisotropic properties of powder particles on microstructural evolution during solid-state sintering processes is analyzed. Two types of anisotropy studied in the current work are direction-dependent interface diffusion anisotropy, and grain orientation dependent grain boundary energy anisotropy. A phase field modeling approach is utilized to assess how the individual anisotropic characteristics influence morphological changes during sintering. In addition, a novel approach for updating grain orientation after rigid-body rotation of particles during powder compaction accompanying the sintering process is developed. It is observed that conventional isotropic microstructural analysis over-simplifies the material behavior and demonstrates faster microstructural evolution. Direction-dependent diffusion produces gradual shape change of the particles by mass transfer from high curvature region to low curvature region along the particle surface, and delays the onset of the grain growth process. Depending on the grain boundary misorientation and inclination, anisotropic grain boundary energy may induce faster or slower grain boundary migration rate. Orientation update during the process plays a critical role in the consolidation kinetics, as well as in the final microstructural configuration. Variation in grain orientation and its evolution during the sintering process can produce different grain morphologies for the same initial condition.

Solid-State Dewetting of Gold Aggregates/Islands on TiO2 Nanorod Structures Grown by Oblique Angle Deposition.

A composite film made of a stable gold nanoparticle (NP) array with well-controlled separation and size atop a TiO2 nanorod film was fabricated via the oblique angle deposition (OAD) technique. The fabrication of the NP array is based on controlled, Rayleigh-instability-induced, solid-state dewetting of as-deposited gold aggregates on the TiO2 nanorods. It was found that the initial spacing between as-deposited gold aggregates along the vapor flux direction should be greater than the TiO2 interrod spacing created by 80° OAD to control dewetting and produce NP arrays. A numerical investigation of the process was conducted using a phase-field modeling approach. Simulation results showed that coalescence between neighboring gold aggregates is likely to have caused the uncontrolled dewetting in the 80° deposition, and this could be circumvented if the initial spacing between gold aggregates is larger than a critical value smin. We also found that TiO2 nanorod tips affect dewetting dynamics differently than planar TiO2. The topology of the tips can induce contact line pinning and an increase in the contact angle along the vapor flux direction to the supported gold aggregates. These two effects are beneficial for the fabrication of monodisperse NPs based on Rayleigh-instability-governed self-assembly of materials, as they help to circumvent the undesired coalescence and facilitate the instability growth on the supported material. The findings uncover the application potential of OAD as a new method to fabricate structured films as template substrates to mediate dewetting. The reported composite films would have uses in optical coatings and photocatalytic systems, taking advantage of their ability to combine plasmonic nanostructures within a nanostructured dielectric film.

Abstract A11: Calibration, selection and validation of tumor growth models

Abstract As a result of the highly heterogeneous tumor microenvironment and a series of complex phenomena that occur at different scales, several modeling approaches have emerged. The tumor growth models can be categorized by the behaviors they aim to capture, the hallmarks of cancer that are being considered, and the temporal and biological scales represented. Moreover, these models can also be differentiated according to the mathematical framework that is being employed, being either discrete models, continuum models, or hybrid models. Therefore, with this large set of possible models, and with each model choice having many unknown parameters, it is important to develop an adaptive model selection and validation framework in order to choose the best model for the desired application. In this work, we employ the Occam Plausibility Algorithm (OPAL) framework [1] for model selection and validation of a large body of tumor growth models. We select several continuum tumor growth models, half of them using the phase-field modeling approach, derived from the models developed in [2], and the other half being reaction-diffusion type models derived from [3]. The OPAL framework brings together model calibration, determination of sensitivities of outputs to parameter variances, and calculation of model plausibilities for model selection. Firstly, we identify the model classes and compute the sensitivity of the quantity of interest (QoI), here the tumor volume as a function of time. The models with parameters that do not influence the QoI are eliminated. The next step consists in dividing the models into Occam categories according to their number of parameters, where the models with fewer parameters are assembled to category 1. Following the principle of Occam's Razor, one of main goals of OPAL is to select the simplest model among the several models that leads to the same prediction. The models from the first category are calibrated using Bayes' rule for a specific calibration scenario. The next step is to compute the plausibilities for the models in this category and choose the most plausible model. Finally, with the most plausible model known, we solve the problem in the validation scenario. The model is valid depending on the accuracy with which the model predictions of the QoI agree with experimental data. If the model is not valid, we move to the next category; otherwise, we select this model as being the simplest and the most plausible valid model. We demonstrate that the OPAL is able to assist in the selection of the best tumor growth model from a defined set of models. The model selected is valid for particular scenarios. In this work, we consider avascular tumor growth models to reproduce the data obtained in a murine model of glioma growth. The brain data is generated with diffusion weighted magnetic resonance imaging over a period of ten days. The methodologies and results confirm that the OPAL strategy provides a powerful framework for model selection, parameters estimation, and model validation in the presence of uncertainties.

A phase-field study on the oxidation behavior of Ni considering heat conduction

Phase-field modeling approach has been used to study the oxidation behavior of pure Ni when considering heat conduction. In this calculation, the dependence of the coefficient of the Cahn–Hilliard equation Lc on the temperature T was considered. To this end, high-temperature oxidation experiments and phase-field modeling for pure Ni were performed in air under atmospheric pressure at 600, 700, and $$800\,^{\circ }\hbox {C}$$ . The oxidation rate was measured by thermogravimetry and Lc at these temperatures was determined via interactive algorithm. With the Lc $${-}T$$ relationship constructed, oxidation behavior of Ni when considering heat conduction was investigated. The influence of temperature boundaries on the oxidation degree, oxide film thickness, and specific weight gain were discussed. The phase-field modeling approach proposed in this study will give some highlights of the oxidation resistance analysis and cooling measures design of thermal protection materials.

Investigation of wing crack formation with a combined phase‐field and experimental approach

AbstractFractures that propagate off of weak slip planes are known as wing cracks and often play important roles in both tectonic deformation and fluid flow across reservoir seals. Previous numerical models have produced the basic kinematics of wing crack openings but generally have not been able to capture fracture geometries seen in nature. Here we present both a phase‐field modeling approach and a physical experiment using gelatin for a wing crack formation. By treating the fracture surfaces as diffusive zones instead of as discontinuities, the phase‐field model does not require consideration of unpredictable rock properties or stress inhomogeneities around crack tips. It is shown by benchmarking the models with physical experiments that the numerical assumptions in the phase‐field approach do not affect the final model predictions of wing crack nucleation and growth. With this study, we demonstrate that it is feasible to implement the formation of wing cracks in large scale phase‐field reservoir models.

A phase-field modeling approach of hydraulic fracture in saturated porous media

Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.

Phase-field modeling of brittle and ductile fracture in shells with isogeometric NURBS-based solid-shell elements

In this paper, we investigate fracture in shells with a phase-field modeling approach. The shell model is based on solid-shell kinematics with small rotations and displacements and is discretized using quadratic Non-Uniform Rational B-Spline basis functions. Membrane and shear locking is alleviated through the Assumed Natural Strain approach. The solid-shell formulation is combined with a brittle phase-field model for elastic materials, as well as with a ductile fracture model for elasto-plastic materials exhibiting J2 plasticity with isotropic hardening. Several examples demonstrate the ability of the proposed framework to capture crack initiation, propagation, merging and branching phenomena as well as crack bulging effects in shells under different states of loading.

Consistent simulation of droplet evaporation based on the phase-field multiphase lattice Boltzmann method.

In the present article, we extend and generalize our previous article [H. Safari, M. H. Rahimian, and M. Krafczyk, Phys. Rev. E 88, 013304 (2013)] to include the gradient of the vapor concentration at the liquid-vapor interface as the driving force for vaporization allowing the evaporation from the phase interface to work for arbitrary temperatures. The lattice Boltzmann phase-field multiphase modeling approach with a suitable source term, accounting for the effect of the phase change on the velocity field, is used to solve the two-phase flow field. The modified convective Cahn-Hilliard equation is employed to reconstruct the dynamics of the interface topology. The coupling between the vapor concentration and temperature field at the interface is modeled by the well-known Clausius-Clapeyron correlation. Numerous validation tests including one-dimensional and two-dimensional cases are carried out to demonstrate the consistency of the presented model. Results show that the model is able to predict the flow features around and inside an evaporating droplet quantitatively in quiescent as well as convective environments.

Nonlocal damage model using the phase field method: Theory and applications

A new nonlocal, gradient based damage model is proposed for isotropic elastic damage using the phase field method in order to show the evolution of damage in brittle materials. The general framework of the phase field model (PFM) is discussed and the order parameter is related to the damage variable in continuum damage mechanics (CDM). The time dependent Ginzburg–Landau equation which is also termed the Allen–Cahn equation is used to describe the damage evolution process. Specific length scale which addresses the interface region in which the process of changing undamaged solid to fully damaged material (microcracks) occurs is defined in order to capture the effect of the damaged localization zone. A new implicit damage variable is proposed through the phase field theory. Details of the different aspects and regularization capabilities are illustrated by means of numerical examples and the validity and usefulness of the phase field modeling approach is demonstrated.

Plastic deformation induced microstructure evolution through gradient enhanced crystal plasticity based on a non-convex Helmholtz energy

A gradient crystal plasticity model in the framework of continuum thermodynamics and rate variational formulation is presented for the description of plastic deformation patterning in a system with non-convex energetic hardening. The paper focuses on the extension of the 1D deformation patterning analysis of Yalcinkaya et al. (2011) to 2D for monotonic loading histories. Solution algorithm is based on the simultaneous solution of displacement and plastic slip fields, which have been considered as primary variables. The incorporation of non-convexity in the plastic slip potential in the Landau–Devonshire form makes the framework dual to phase field modeling approaches with a strong coupling between the deformation and the plastic slip fields. In the phase field modeling approaches the coupling is rather weak, i.e. fields do not have to be coupled as in the current approach based on the decomposition of the total strain. The numerical examples illustrate the intrinsic formation of (laminate type) microstructures and their evolution under mechanical loading together with the macroscopic hardening–softening–stress plateau response. The effect of different number of slip systems, loading rates and boundary conditions are investigated in detail.

Plastic slip patterning through rate dependent non‐convex gradient‐enhanced plasticity

AbstractThe paper presents a non‐convex rate dependent strain gradient plasticity framework for the description of plastic slip patterning in metal crystals. The non‐convexity is treated as an intrinsic property of the free energy of the material. Departing from explicit expressions for the free energy, the non‐convex strain gradient crystal plasticity model is derived in a thermodynamically consistent manner, including the accompanying slip law. For the numerical solution of the problem, the displacement and the plastic slip fields are considered as primary variables. These fields are determined on a global level by solving simultaneously the linear momentum balance and the resulting slip evolution equation. The slip law differs from classical ones in the sense that it naturally includes a contribution from the non‐convex free energy term, which enables patterning of the deformation field. The formulation of the computational framework is partially dual to a Ginzburg Landau type of phase field modeling approach. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An effective continuum approach for modeling non-equilibrium structural evolution of protein nanofiber networks

We quantify the formation and evolution of protein nanofibers using a new phase field modeling framework and compare the results to transmission electron microscopy measurements (TEM) and time-dependent growth measurements given in the literature. The modeling framework employs a set of effective continuum equations combined with underlying nanoscale forces and chemical potential relations governing protein nanofiber formation in solution. Calculations based on the theoretical framework are implemented numerically using a nonlinear finite element phase field modeling approach that couples homogenized protein molecular structure via a vector order parameter with chemical potential relations that describe interactions between the nanofibers and the surrounding solution. Homogenized, anisotropic molecular and chemical flux relations are found to be critical in obtaining nanofiber growth from seed particles or a random monomer bath. In addition, the model predicts both sigmoidal and first-order growth kinetics for protein nanofibers for unseeded and seeded models, respectively. These simulations include quantitative predictions on time scales of typical protein self-assembly behavior which qualitatively match TEM measurements of the RADA16-I protein and growth rate measurements for amyloid nanofibers from the literature. For comparisons with experiments, the numerical model performs multiple nanofiber protein evolution simulations with a characteristic length scale of ∼2.4 nm and characteristic time scale of ∼9.1 h. These results provide a new modeling tool that couples underlying monomer structure with self-assembling nanofiber behavior that is compatible with various external loadings and chemical environments.

Deformation patterning driven by rate dependent non-convex strain gradient plasticity

A rate dependent strain gradient plasticity framework for the description of plastic slip patterning in a system with non-convex energetic hardening is presented. Both the displacement and the plastic slip fields are considered as primary variables. These fields are determined on a global level by solving simultaneously the linear momentum balance and the slip evolution equation which is postulated in a thermodynamically consistent manner. The slip law differs from classical ones in the sense that it includes a non-convex free energy term, which enables patterning of the deformation field. The formulation of the computational framework is at least partially dual to a Ginzburg–Landau type of phase field modeling approach. The essential difference resides in the fact that a strong coupling exists between the deformation and the evolution of the plastic slip, whereas in the phase field type models the governing fields are only weakly coupled. The derivations and implementations are done in a transparent 1D setting, which allows for a thorough mechanistic understanding, not excluding its extension to multidimensional cases.

Assigning physical significance to the diffuse interface between terraces in phase-field modeling of steps on crystal surfaces: modeling step–step interaction

We apply a modified phase-field modeling approach to the analysis of steps on a crystalline surface. Specifically, we are interested in capturing phenomena associated with the interaction between steps. To this end, we assign a physical significance to the form of the interfacial region between terraces (i.e. steps), inherent in the phase field approach, by tuning the multi-well potential to produce long-range interaction energies varying as 1/l2, where l is half the distance between steps. Resultant repulsive interactions between adjacent steps of the same sign are shown to affect step-flow kinetics in a manner consistent with curvature-driven interfacial relaxation. This phenomenon is further demonstrated to cause dislocation-driven (spiral) crystal growth kinetics to deviate, for large supersaturation and Burgers vector values, from the classical quadratic growth law. Attractive interactions between adjacent steps of opposite sign, also resulting from the finite interfacial width, are briefly explored particularly with respect to their possible impact on two-dimensional nucleation.

Simulation of crack propagation in fiber-reinforced bulk metallic glasses

Based on a phase-field model for deformation in bulk metallic glasses (BMGs), shear band formation and crack propagation in the fiber-reinforced BMG are investigated. Ideal unbroken fibers embedded in the BMG matrix are found to significantly influence the shear banding and crack propagation in the matrix. The crack propagation affected by fibers’ length and orientation is quantitatively characterized and is described by micromechanics models for composite materials. Furthermore, fractures in some practical fiber-reinforced BMG composites such as tungsten-reinforced Zr-based BMG are simulated. The relation between the enhanced fracture toughness and the mechanical properties of fiber reinforcements is determined. Different fracture modes of BMG-matrix composites are identified from the systematic simulation studies, which are found to be consistent with experiments. The simulation results suggest that the phase-field modeling approach could be a useful tool to assist the fabrication and design of BMG composites with high fracture toughness and ductility.

A new approach to modeling liquid crystal elastomers using phase field methods

A phase field modeling framework is developed to quantify domain structure evolution in nematic phase liquid crystal elastomers. A hyperelastic energy function is combined with liquid crystal energy relations to formulate a constitutive model for liquid crystal elastomers that undergo thermo-mechanical loads and finite deformation. A set of balance laws and constitutive relations are defined which lead to coupling behavior when finite deformation is introduced within the energy description. The theoretical framework is implemented numerically using a non-linear finite element phase field modeling approach which couples deformation of the elastomer network with microscopic liquid crystal domain structure evolution. A comparison of monodomain and polydomain behavior is analyzed to illustrate spontaneous deformation and polydomain evolution during heating and mechanical stretching. Many of the essential constitutive relations governing these materials are obtained without the use of explicit phenomenological coupling between the liquid crystals and the host elastomer.

A continuum thermodynamics formulation for micro-magneto-mechanics with applications to ferromagnetic shape memory alloys

A continuum thermodynamics formulation for micromagnetics coupled with mechanics is devised to model the evolution of magnetic domain and martensite twin structures in ferromagnetic shape memory alloys. The theory falls into the class of phase-field or diffuse-interface modeling approaches. In addition to the standard mechanical and magnetic balance laws, two sets of micro-forces and their associated balance laws are postulated; one set for the magnetization order parameter and one set for the martensite order parameter. Next, the second law of thermodynamics is analyzed to identify the appropriate material constitutive relationships. The proposed formulation does not constrain the magnitude of the magnetization to be constant, allowing for spontaneous magnetization changes associated with strain and temperature. The equations governing the evolution of the magnetization are shown to reduce to the commonly accepted Landau–Lifshitz–Gilbert equations for the case where the magnetization magnitude is constant. Furthermore, the analysis demonstrates that under certain limiting conditions, the equations governing the evolution of the martensite-free strain are shown to be equivalent to a hyperelastic strain gradient theory. Finally, numerical solutions are presented to investigate the fundamental interactions between the magnetic domain wall and the martensite twin boundary in ferromagnetic shape memory alloys. These calculations determine under what conditions the magnetic domain wall and the martensite twin boundary can be dissociated, resulting in a limit to the actuating strength of the material.

Continuum thermodynamics of ferroelectric domain evolution: Theory, finite element implementation, and application to domain wall pinning

A continuum thermodynamics framework is devised to model the evolution of ferroelectric domain structures. The theory falls into the class of phase-field or diffuse-interface modeling approaches. Here a set of micro-forces and governing balance laws are postulated and applied within the second law of thermodynamics to identify the appropriate material constitutive relationships. The approach is shown to yield the commonly accepted Ginzburg–Landau equation for the evolution of the polarization order parameter. Within the theory a form for the free energy is postulated that can be applied to fit the general elastic, piezoelectric and dielectric properties of a ferroelectric material near its spontaneously polarized state. Thereafter, a principle of virtual work is specified for the theory and is implemented to devise a finite element formulation. The theory and numerical methods are used to investigate the fields near straight 180° and 90° domain walls and to determine the electromechanical pinning strength of an array of line charges on 180° and 90° domain walls.