Consistent Phase Field Research Articles

A consistent phase field model for hydraulic fracture propagation in poroelastic media

We present a novel phase field method for modeling hydraulic fracture propagation in poroelastic media. In this approach, a new phase field evolution equation is derived to account for damage dependent poro-elastic parameters (Biot’s coefficient, Biot’s modulus and porosity). The fluid flow obeys Darcy’s seepage law in the entire domain including the damage zone, where the rock permeability is assumed to be anisotropic, following the maximum principal strain. The fully coupled problem is solved by a staggered scheme in which the mechanical equilibrium and fluid flow equations are linearized and solved using a Newton–Raphson(NR) method. Several numerical results are presented to investigate the effectiveness of the proposed formulation. First, stability and convergence of the method are verified on a set of benchmark problems considering different time steps and mesh sizes. Second, it is shown that if the poroelastic parameters are kept constant and do not change with the phase field parameter, i.e. reducing to standard phase field approaches in the literature, the model will tend to underestimate the fracture length and overestimate the pore pressure. Finally, we study the interaction of a propagating hydraulic fracture in porous media with inclined natural fractures, and simulate the hydraulic fracture propagation with different perforation phase angle.

Phase field approach to mode-I fracture by introducing an eigen strain tensor: General theory

A thermodynamically consistent phase field theory for fracture is developed. An alternative approach to describe the reduction in the slope of the stress–strain curve and the reduction of the stresses to zero during damage is utilized by introducing transformation strain instead of the degradation function. The main requirements for the thermodynamic potential are presented and used to present the general energy terms. The analysis of the equilibrium solution is formulated which describes the desired stress–strain curves and for which the infinitesimal damage produces an infinitesimal change in the equilibrium elastic modulus. Analytical analysis of the equations is performed to calibrate phase field parameters for a general case. The explicit form of the Ginzburg-Landau equation is derived based on which the evolution of the order parameter is obtained. The importance of the analysis of the thermodynamic potential in terms of stress–strain curves is shown. The developed theory includes a broad spectrum of the shapes of stress–strain relationships.

A non-isothermal thermodynamically consistent phase field model for damage, fracture and fatigue evolutions in elasto-plastic materials

This paper presents a general thermodynamically consistent non-isothermal phase field framework to model effects of damage, fracture and fatigue evolutions in elasto-plastic materials under the hypothesis of small strains. The proposed methodology is obtained from the principle of virtual power, energy balance and second law of thermodynamics in the form of a generalized Clausius–Duhem inequality for entropy. Aspects of the energy degradation functions are thoroughly investigated for an isotropic elasto-plastic material with viscous dissipation and constant specific heat. A new combined degradation function is proposed to degrade both elastic and plastic energy densities as an alternative to the classical degradation functions. Our conceptual framework leads to thermodynamically consistent models that may include contributions not usually considered in the literature, as temperature and inertia effects as well as time-rate dependent processes. The governing nonlinear transient equations obtained are solved adopting a semi-implicit time integration scheme combined with the classical Newton–Raphson iterative procedure. Results for an I-shaped specimen made of 7075-T7351 aluminum alloy under different conditions are presented. The proposed model is able to reproduce qualitative and quantitatively the ductile fracture and the fatigue phenomena. In particular for fatigue, both the S–N experimental data and the Paris crack growth curve over cycles are recovered. In addition, the cycle jump strategy reduces four times the CPU time for fatigue simulations.

Matrix-precipitate interface-induced martensitic transformation within nanoscale phase field approach: Effect of energy and dimensionless interface width

Martensitic transformation induced by the matrix-precipitate interface (or other internal surfaces) for single and two martensitic variants is studied using a thermodynamically consistent multiphase phase field approach. Three order parameters are considered; two of them describe the austenite (A) ↔ martensite (M) and variant Mi ↔variant Mj transformations in a matrix, and the third one describes the finite width matrix - non-transforming precipitate interface. The energy of the matrix-precipitate interface reduces during A→M phase transformation from the value for energy of A-precipitate interface, γA, to value for energy of M-precipitate interface, γM, due to its dependence on the order parameter related to the austenite↔martensite transformation. Such an interface increases the temperature for barrierless martensite nucleation well above the critical temperature for A→M transformation. The nucleation temperatures strongly depend on the ratio Δ¯ of the widths of the matrix-precipitates interface and A−M interface. New “phase diagram” for transformation temperatures between austenite, martensite, and premartensite versus Δ¯ has been presented for neglected mechanics for two cases when magnitude of Δγ=γM−γA is larger than the energy of the A−M interface (0.2 N/m). For Δγ=−0.5 N/m, below a critical width ratio Δ¯*, a layer of premartensite appears jump-like within the matrix-precipitate interface and progresses with reducing temperature, until it loses its stability and jump-like transforms to complete martensite in the entire matrix. However, for Δ¯≥Δ¯*, the entire matrix transforms to martensite without any premartensite. For Δγ=−0.3 N/m, no premartensite appears and the A matrix completely transforms into M at lower temperatures than the case with Δγ=−0.5 N/m. The combined effect of the energy of the matrix-precipitate interface, Δ¯, precipitation-induced misfit strains, and applied displacements on the boundary of the sample on nucleation of martensite and complex microstructure evolution in the systems with a single and two martensitic variant(s) is studied. Obtained results are important for controlling cyclic martensitic transformations in shape memory and elastocaloric alloys and designing alloys with desired characteristics of martensitic transformations.

Controlling Sliding Droplets with Optimal Contact Angle Distributions and a Phase Field Model

AbstractWe consider the optimal control of a droplet on a solid by means of the static contact angle between the contact line and the solid. The droplet is described by a thermodynamically consistent phase field model from [Abels et al., Math. Mod. Meth. Appl. Sc., 22(3), 2012] together with boundary data for the moving contact line from [Qian et al., J. Fluid Mech., 564, 2006]. We state an energy stable time discrete scheme for the forward problem based on known results, and pose an optimal control problem with tracking type objective.

A second order fully-discrete linear energy stable scheme for a binary compressible viscous fluid model

We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flows, derived from the generalized Onsager Principle. The hydrodynamic model possesses not only the variational structure in its constitutive equation, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model using the energy quadratization method into an equivalent form and then discretize the reformulated model to obtain a semidiscrete partial differential equation system using the Crank-Nicolson method in time. The semi-discrete numerical scheme preserves the mass conservation and energy dissipation law in time. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using 2nd order finite difference methods, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and numerical examples on phase separation due to spinodal decomposition in binary polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications, respectively.

A linear energy and entropy-production-rate preserving scheme for thermodynamically consistent crystal growth models

We present a linear, second order, energy and entropy-production-rate preserving scheme for a thermodynamically consistent phase field model for dentritic crystal growth, combining an energy quadratization strategy with the finite element method. The scheme can be decomposed into a series of Poisson equations for efficient numerical implementations. Numerical tests are carried out to verify the accuracy of the scheme and simulations are conducted to demonstrate the effectiveness of the scheme on benchmark examples.

Phase field parameters for battery compounds from first-principles calculations

In this work, we present and apply schemes to determine from first-principles calculations the relevant effective parameters used in phase field theory simulations of battery compounds. In particular, we derive that a consistent free energy density can be obtained by mean-field sampling, which is especially suited for materials with different configurations on a lattice, such as alloys or Li intercalation batteries. In addition, it is demonstrated that mean-field sampling can be performed very effectively with the use of special quasi random structures and that experimentally determined free energy density parameters for ${\mathrm{LiFePO}}_{4}$ are reproduced by density functional theory calculations. The additional computation of interfacial and strain energy parameters allows us to present a consistent phase field parametrization of ${\mathrm{Li}}_{2}{\mathrm{FeSiO}}_{4}$ without relying on experimental data.

Phase field modeling of crack growth with double-well potential including surface effects

A thermodynamically consistent phase field approach for fracture including surface stresses is presented. The surface stresses which are distributed inside a finite width layer at the crack surface are introduced as a result of employing some geometrical nonlinearities, i.e., by defining energy terms per unit volume at the current time and evaluating gradient of the order parameter in the current configuration. A double-well barrier term is included in the structure of the Helmholtz free energy which allows one to use the energy terms per unit volume at the current time when introducing the surface stresses in the current approach. Thus, the surface stresses are introduced in a similar way to the interfacial stresses in phase transformations. The differences in the modeling of crack growth with considering the surface stresses and without it are discussed. It is shown how the surface stresses affect the stress fields and consequently the crack nucleation and propagation. The finite element method is utilized to solve the coupled equations of mechanics and crack phase field. It is emphasized that the surface stresses affect the driving force for both the crack nucleation and propagation by disturbing the momentum balance. Thus, a different external loading is required in the presence of the surface stresses.

Energy and entropy preserving numerical approximations of thermodynamically consistent crystal growth models

We present a numerical scheme that preserves the total energy and the entropy production rate, termed the energy and entropy production rate preserving scheme, for a general class of thermodynamically consistent phase field models for dendritic crystal growth derived from the first and second law of thermodynamics. The scheme is second order in time, linear and energy and entropy production rate preserving for any time steps. The scheme is first discretized in time aided by the energy quadratization (EQ) method and then in space using compact finite difference methods. The linear system resulting from the scheme is shown to be uniquely solvable at both the semi-discrete and the fully discrete level. Mesh refinement tests are performed to show the second-order time convergence rate in the scheme. Several numerical examples of dendritic crystal growth are provided to demonstrate the accuracy and efficiency of the scheme. The effects of various model parameters on growth patterns of the crystal are further investigated in details with the numerical solver. The approach to developing the energy and entropy production rate-preserving numerical scheme proposed in this study is so general that it can be applied to a wide range of thermodynamically consistent models not limited to phase field models.

Simulation and experimental studies on microstructure evolution of resolidified dendritic TiCx in laser direct deposited Ti-TiC composite

This paper describes the formation process of resolidified dendritic TiCx in laser direct deposited (LDD) Ti-40 vol%TiC composite via a combination of a thermodynamic consistent phase field (PF) model and an LDD model. The LDD model simulated the LDD process of this composite and provided the temporal and spatial temperature fields in the molten pool for the PF model, which predicted the solidification process of TiCx dendrites at different zones in the molten pool. The improved PF model introduced a grain index to link different crystallographic orientations for polycrystalline growth, which improved the computational efficiency by solving one PF equation instead of multiple phase variables. A two-sublattice model was utilized to establish the thermodynamic database of TiCx. The predicted temperature history, deposition geometry as well as the morphology of polycrystalline TiCx dendrites matched the experimental observation. As the temperature decreases, microsegregation occurred to the distribution of carbon over the dendrite where dendrite tips had a higher carbon concentration. Due to the increased undercooling, TiCx dendrites grew at an increased speed from top to bottom of the molten pool. Within 0.007 s, the dendrites were fully developed and inhibited further growth of each other.

Phase-field model of pitting corrosion kinetics in metallic materials

Pitting corrosion is one of the most destructive forms of corrosion that can lead to catastrophic failure of structures. This study presents a thermodynamically consistent phase field model for the quantitative prediction of the pitting corrosion kinetics in metallic materials. An order parameter is introduced to represent the local physical state of the metal within a metal-electrolyte system. The free energy of the system is described in terms of its metal ion concentration and the order parameter. Both the ion transport in the electrolyte and the electrochemical reactions at the electrolyte/metal interface are explicitly taken into consideration. The temporal evolution of ion concentration profile and the order parameter field is driven by the reduction in the total free energy of the system and is obtained by numerically solving the governing equations. A calibration study is performed to couple the kinetic interface parameter with the corrosion current density to obtain a direct relationship between overpotential and the kinetic interface parameter. The phase field model is validated against the experimental results, and several examples are presented for applications of the phase-field model to understand the corrosion behavior of closely located pits, stressed material, ceramic particles-reinforced steel, and their crystallographic orientation dependence.

Thermodynamically consistent and scale-dependent phase field approach for crack propagation allowing for surface stresses

A thermodynamically-consistent phase field approach for crack propagation which includes the following novel features is presented. (1) Scale dependency was included by relating the length scale to the number of cohesive interatomic planes at the crack tip. Because of this, the developed theory is applicable from the atomistic to the macroscopic scales. (2) The surface stresses (tension) are introduced by employing some geometrical nonlinearities even in small strain theory. They produce multiple contributions to the Ginzburg-Landau equation for crack propagation. (3) Crack propagation in the region with compressive closing stresses is eliminated by employing a stress-state-dependent kinetic coefficient in the Ginzburg-Landau equation. (4) The importance of analysis of the thermodynamic potential in terms of stress-strain curves is shown. The developed theory includes a broad spectrum of the shapes of stress-strain relationships. The finite element method is utilized to solve the complete system of crack phase field and mechanics equations. The effect of the above novel features is analyzed numerically for various model problems.

On the phase field modeling of crack growth and analytical treatment on the parameters

A thermodynamically consistent phase field model for crack propagation is analyzed. The thermodynamic driving force for the crack propagation is derived based on the laws of thermodynamics. The Helmholtz free energy satisfies the thermodynamic equilibrium and instability conditions for the crack propagation. Analytical solutions for the Ginzburg–Landau equation including the surface profile and the estimation of the kinetic coefficient are found. It is shown how kinetic coefficient affects the local stress field. The local critical stress for the crack propagation is calibrated with the theoretical strength which gives the value of the crack surface width. The finite element method is utilized to solve the complete system of crack phase field and mechanics equations. The analytical expressions for the crack surface profile and the crack tip velocity are verified with the numerical solutions. It is shown that under mode I cracking and proper calibration of parameters, phase field models always agree with Griffith’s theory.

On hydrodynamic phase field models for binary fluid mixtures

Two classes of thermodynamically consistent hydrodynamic phase field models have been developed for binary fluid mixtures of incompressible viscous fluids of possibly different densities and viscosities. One is quasi-incompressible, while the other is incompressible. For the same binary fluid mixture of two incompressible viscous fluid components, which one is more appropriate? To answer this question, we conduct a comparative study in this paper. First, we visit their derivation, conservation and energy dissipation properties and show that the quasi-incompressible model conserves both mass and linear momentum, while the incompressible one does not. We then show that the quasi-incompressible model is sensitive to the density deviation of the fluid components, while the incompressible model is not in a linear stability analysis. Second, we conduct a numerical investigation on coarsening or coalescent dynamics of protuberances using the two models. We find that they can predict quite different transient dynamics depending on the initial conditions and the density difference although they predict essentially the same quasi-steady results in some cases. This study thus cast a doubt on the applicability of the incompressible model to describe dynamics of binary mixtures of two incompressible viscous fluids especially when the two fluid components have a large density deviation.

Effects of gravity on the thermo-hydrodynamics of moving contact lines

In this paper, we demonstrate the effects of gravity on the interfacial thermo-hydrodynamics as modulated by the patterned wettability gradients placed on the surfaces of a narrow fluidic channel. We investigate the dynamics of contact line motion of two-component incompressible immiscible liquid mixtures under the framework of a thermodynamically consistent phase field model. We validate our model with the experimental results available in the literature in the purview of thermocapillary-actuated microscale transport. We show that the gravity-induced forces in the presence of the thermocapillarity effect play a unique role on the interfacial dynamics at small scales, leading to a uniform movement of the interface in the channel, and offer a greater degree of controllability in the filling/wetting rate in the capillary. We show that the change in patch width, placed on the walls of the channel, leads to a change in the filling dynamics into the capillary. Also, we investigate the rate of different dissipations during the movement of the contact line along the channel and show that the kinetic energy modulated dissipation provides maximum energy to the motion of the contact line.

Modeling elasto-viscoplasticity in a consistent phase field framework

Existing continuum level phase field plasticity theories seek to solve plastic strain by minimizing the shear strain energy. However, rigorously speaking, for thermodynamic consistency it is required to minimize the total strain energy unless there is proof that hydrostatic strain energy is independent of plastic strain which is unfortunately absent. In this work, we extend the phase-field microelasticity theory of Khachaturyan et al. by minimizing the total elastic energy with constraint of incompressibility of plastic strain. We show that the flow rules derived from the Ginzburg-Landau type kinetic equation can be in line with Odqvist's law for viscoplasticity and Prandtl-Reuss theory. Free surfaces (external surfaces or internal cracks/voids) are treated in the model. Deformation caused by a misfitting spherical precipitate in an elasto-plastic matrix is studied by large-scale three-dimensional simulations in four different regimes in terms of the matrix: (a) elasto-perfectly-plastic, (b) elastoplastic with linear hardening, (c) elastoplastic with power-law hardening, and (d) elasto-perfectly-plastic with a free surface. The results are compared with analytical/numerical solutions of Lee et al. for (a-c) and analytical solution derived in this work for (d). In addition, the J integral of a fixed crack is calculated in the phase-field model and discussed in the context of fracture mechanics.

Wall-bounded multiphase flows of N immiscible incompressible fluids: Consistency and contact-angle boundary condition

We present an effective method for simulating wall-bounded multiphase flows consisting of N (N⩾2) immiscible incompressible fluids with different densities, viscosities and pairwise surface tensions. The N-phase physical formulation is based on a modified thermodynamically consistent phase field model that is more general than in a previous work, and it is developed by considering the reduction consistency if some of the fluid components were absent from the system. We propose an N-phase contact-angle boundary condition that is reduction consistent between N phases and M phases (2⩽M⩽N−1). We also present a numerical algorithm for solving the N-phase governing equations together with the contact-angle boundary conditions developed herein. Extensive numerical experiments are presented for several flow problems involving multiple fluid components and solid-wall boundaries to investigate the wettability effects with multiple types of contact angles. In particular, we compare simulation results with the de Gennes theory for the contact-angle effects on the liquid drop spreading on wall surfaces, and demonstrate that our method produces physically accurate results.

A non-isothermal thermodynamically consistent phase field framework for structural damage and fatigue

We present a general thermodynamically consistent non-isothermal non-local framework for the evolution of damage, fatigue and fracture in materials under the hypothesis of small deformation. The approach is based on the principle of virtual power (PVP), the balance of energy and the second law of thermodynamics in the form of the generalized Clausius–Duhem inequality for the entropy. In addition to the usual physical fields, the model uses the phase field approach to describe the evolution of both damage and fatigue. The kinematic descriptor (phase field) for damage is considered a continuous dynamical variable whose evolution equation is obtained by the PVP. The kinematic descriptor (another phase field) for fatigue is a continuous internal variable whose evolution equation is considered as a constitutive relation to be determined in a thermodynamically consistent way. The behavior of particular material classes can be specified by their corresponding free-energy potentials (which gives the reversible parts of the involved thermodynamic forces) and their associated pseudo-potentials of dissipation (which gives the irreversible parts of the involved thermodynamic forces). To exemplify our general framework, we present the case of an isotropic linear elastic material with viscous dissipation and constant specific heat. The corresponding case of irreversible damage is also presented by using penalization. The considered damage and fatigue phase field approach is a framework from which other methods in the literature may be recovered. The model is approximated by the nodal high-order finite element method with explicit fourth-order Runge–Kutta time integration. Results for one-dimensional examples are presented and conclusions are addressed.

Simulation of Environmentally-Assisted Material Degradation by a Thermodynamically Consistent Phase-Field Model

Environmentally-assisted material degradation involves mass transport and mechanical processes interacting in the material. A well-known example is hydrogen-induced stress-corrosion cracking. One major challenge within this scope is the quantification of the coupling mechanisms in question. The computational modeling of environmentally-assisted cracks is the key objective ofthis investigation and realised within the theory of gradient-extended dissipative continua with length-scales. The modeling of sharp crack discontinuities is replaced by a diffusive crack model based onthe introduction of a crack phase-field to maintain the evolution of complex crack topologies. Withina thermodynamical framework allowing for mechanical and mass transport processes the crack phase-field is capable to model crack initiation and propagation bythe finite element method. As complexcrack situations such as crack initiation, curvilinear crack patterns and crack branching are usuallyhard to realise with sharp crack models, they can be assessedwithout the requirement of a predefinedcrack path within this method. The numerical modeling of a showcase demonstrates a crack initiationas well as a crack propagation situation with respect to the determination of stress-intensity factors; acrack deviation situation with a curvilinear crack path is modeled by the introduction of a geometricalperturbation and a locally enhanced species concentration