Finite Element Phase Field Research Articles

Coupled crystal plasticity finite element-phase field model with kinetics-controlled twinning mechanism for hexagonal metals

Hexagonal metals plastically deform through evolution of dislocations and twins. Deformation twinning is unidirectional and results in twin domain associated with crystal reorientation. Growth of a twin domain is accomplished through nucleation and motion of twinning dislocations/ disconnections (TDs) at atomic scale, and described by migration of twin boundaries (TBs) at micro/macro-scales. Corresponding to kinetics-controlled migration mechanisms of twin boundaries, a coupled crystal plasticity finite element-phase field (CPFE-PF) model was developed through implementing microscale twinning model with anisotropic mobilities of twin boundaries. Crystal plasticity finite element model (CPFEM) is used to solve elastic fields and plastic deformation (carried by both dislocation slip and twinning). Phase field (PF) method is adopted to spatially distinguish twin domains from matrix and track the migration of TBs. Migration of TBs is controlled by both anisotropic mobility coefficients that are correlated to experimental measured twinning strain rate, and the twin driving force contributed by strain energy that is calculated from CPFEM, and change of interface energy including chemical energy and gradient energy that is calculated using finite volume method. It is worth mentioning that experimental measured interface energies of TBs and a critical gradient criterion are adopted in PF model to confine the width of TBs. Our CPFE-PF model exhibits the capability of predicting twin growth with experimental observed twin morphologies, and can be further developed to study twinning related microstructural evolution and mechanical behavior for hexagonal metals.

An inverse elastic method of crack identification based on sparse strain sensing sheet

Reliable damage detection over large areas of structures can be achieved by spatially quasi-continuous structural health monitoring enabled by two-dimensional sensing sheets. They contain dense arrays of short-gauge sensors, which increases the probability to have sensors in direct contact with damage (e.g. crack opening) and thus identify (i.e. detect, localize, and quantify) it at an early stage. This approach in damage identification is called direct sensing. Although the sensing sheet is a reliable and low-cost technology, the overall structural health monitoring system that is using it might become complex due to large number of sensors. Hence, intentional reduction in number of sensors might be desirable. In addition, malfunction of sensors can occur in real-life settings, which results in unintentional reduction in the number of functioning sensors. In both cases, reduction in the number of (functioning) sensors may lead to lack of performance of sensing sheet. Therefore, it is important to explore the performance of sparse arrays of sensors, in the cases where sensors are not necessarily in direct contact with damage (indirect sensing). The aim of this research is to create a method for optimizing the design of arrays of sensors, that is, to find the smallest number of sensors while maintaining a satisfactory reliability of crack detection and accuracy of damage localization and quantification. To achieve that goal, we first built a phase field finite element model of cracked structure verified by the analytical model to determine the crack existence (detection), and then we used the algorithm of inverse elastostatic problem combined with phase field finite element model to determine the crack length (quantification) and location (localization) by minimizing the difference between the sensor measurements and the phase field finite element model results. In addition, we experimentally validated the method by means of a reduced-scale laboratory test and assessed the accuracy and reliability of indirect sensing.

Modeling interactions of natural and two-phase fluid-filled fracture propagation in porous media

In this paper, a novel computational framework is introduced for simulation of multiphase flow, geomechanics, and fracture propagation in porous media based on Biot’s model for poroelasticity by focusing on interactions between hydraulic and natural fractures. Since realistic porous media contain many natural fractures, it is important not only to stimulate hydraulic fractures but also to study the interaction between natural and hydraulic fractures. Here, state-of-the-art numerical modeling of natural and hydraulic fractures using a diffusive adaptive finite element phase field approach is employed. The locally mass conservative enriched Galerkin finite element methods (EG) are utilized to model two-phase flow in propagating fractures with relative permeability and capillary pressure. Geomechanics approximated by a continuous Galerkin finite element method is coupled to multiphase flow by applying an iteratively coupled scheme. Numerical examples are presented that demonstrate the effectiveness of this framework for different propagation scenarios by varying the degrees of physics. In addition, the capabilities to perform high-fidelity simulations on complex fracture networks, with randomly joined diffusive natural fractures, are illustrated.

Prediction of pore evolution during selected laser melting solidification by a finite element-phase field model

Porosity control in parts built by additive manufacturing is still a great challenge. In this work, we developed a finite element-phase field model to simulate pore evolution during selected laser melting solidification, and focused on the effects of energy density on the porosity. The model reproduced the solidification phenomena observed experimentally and revealed the quantitative relation between porosity and energy density. It was found that the critical energy density of Ti-6Al-4V alloy for full densification is 45 J mm−3 to 75 J mm−3, which is in good accord with the experimental results.

Evolution and formation mechanism of surface nanorelief associated with martensite twin colony (MTC) in Mn-based alloy

The evolution and formation mechanism of surface nanorelief associated with martensitic twin colony (MTC) in Mn-based alloy are studied by use of the phase-field finite element method. The calculation results show that the maximum height of nanorelief is about 30 nm and the maximum nanorelief angle is about 3.2°. The MTC can form during the microstructure evolution. The nanorelief in the same MTC or in different MTCs with the same orientation may have the similar features, which is different from that in the adjacent MTC. The top and bottom edges of one nanorelief correspond to different martensite twin boundaries (MTBs). A bigger stress and a higher plastic deformation distribute around MTC interface than MTB, and the interfacial migration of MTC is greater than MTB. The grain orientation and external stress directly affect the nanorelief and MTC. The biclinic nanorelief (tent-shape and V-shape) evolves from the monoclinic nanorelief. The surface relief angle of MTC is proposed to classify the nanorelief angles and the sum of relief angels (θ1 and θ2) for two adjacent MTCs satisfies θ1 + θ2 = 90°. For different MTCs with the same orientation, their surface relief angles equal to each other.

An implementation of the phase-field model based on coupled thermomechanical finite element solvers for large-strain twinning, explicit dynamic fracture and the classical Stefan problem

The implementation of a phase-field model in finite elements usually requires significant expertise and involves the development of a user element with additional degrees of freedom. An alternative implementation of the phase-field model within a thermo-mechanical finite element simulation package was presented in (Cho et al 2012 Int. J. Solids Struct. 49 1973–1992), where the phase-field variable is treated as the temperature degree of freedom. However, this approach has only been used for small strain phase-field modelling of martensitic transformations and quasistatic phase-field modelling of fracture. In this work, we present a phase-field finite element implementation via the temperature degree of freedom for several additional cases from the literature: (i) the large-strain phase-field description of deformation twinning presented in (Clayton and Knap 2011 Physica D 240 841–858), (ii) phase-field description of brittle fracture with inertial effects based on the theory from (Molnár and Gravouil 2017, Finite Elem. Anal. Des. 130 27–38) and (Miehe et al 2010 Int. J. Numer. Methods Eng. 83 1273–1311) and (iii) the classical Stefan problem of solidification presented in (Mackenzie and Robertson 2002 J. Comput. Phys. 181 526–544). The last problem involves the temperature and phase-field variables as unknowns.

A generalized phase field multiscale finite element method for brittle fracture

SUMMARYA generalized multiscale finite element method is introduced to address the computationally taxing problem of elastic fracture across scales. Crack propagation is accounted for at the microscale utilizing phase field theory. Both the displacement‐based equilibrium equations and phase field state equations at the microscale are mapped on a coarser scale. The latter is defined by a set of multinode coarse elements, where solution of the governing equations is performed. Mapping is achieved by employing a set of numerically derived multiscale shape functions. A set of representative benchmark tests is used to verify the proposed procedure and assess its performance in terms of accuracy and efficiency compared with the standard phase field finite element implementation.

A Navier‐Stokes‐Korteweg Model for Dynamic Wetting based on the PeTS Equation of State

AbstractDynamic wetting of component surfaces can be investigated by finite element phase field simulations. Often these models use a double‐well potential or the van der Waals equation to define the local part of the free energy density at a point of the computational domain. In order to give the present model a stronger physical background the molecular dynamics based perturbed Lennard‐Jones truncated and shifted (PeTS) equation of state is used instead. This results in phase field liquid‐vapor interfaces that agree with the physical density gradient between the two phases. In order to investigate dynamic scenarios, the phase field description is coupled to the compressible Navier‐Stokes equations. This coupling requires a constitutive equation that complies with the surface tension of the liquid‐vapor interface resulting from the PeTS equation of state and is comparable to the so‐called Korteweg tensor.

Phase-Field Material Point Method for dynamic brittle fracture with isotropic and anisotropic surface energy

A novel phase field material point method is introduced for robust simulation of dynamic fracture in elastic media considering the most general case of anisotropic surface energy. Anisotropy is explicitly introduced through a properly defined crack density functional. The particular case of impact driven fracture is treated by employing a discrete field approach within the material point method setting. In this, the equations of motion and phase field governing equations are solved independently for each discrete field using a predictor–corrector algorithm. Contact at the interface is resolved through frictional contact conditions. The proposed method is verified using analytical predictions. The influence of surface energy anisotropy and loading conditions on the resulting crack paths is assessed through a set of benchmark problems. Comparisons are made with the standard Phase Field Finite Element Method and experimental observations.

Coupled phase field finite element model for crack propagation in elastic polycrystalline microstructures

This paper is aimed at the development of a coupled finite element-phase field modeling framework for simulating fracture processes in anisotropic elastic materials under finite deformation conditions with tension-compression asymmetry of crack phase. A novel Helmholtz free energy density is proposed in terms of a phase field order parameter dependent elastic Green-Lagrange strain tensor. It accommodates anisotropy of the material and unilateral conditions of material damage. The paper also addresses numerical instabilities, which adversely affect computational fracture simulations. Solutions based on viscous stabilization are suggested to effectively overcome the convergence issues of nonlinear FE solvers. Numerical examples are carried out to highlight different aspects of the developed framework. Numerical results with the proposed model are successfully compared with experimental results for validation. The model admits the unilateral conditions of the crack phase and predicts the correct crack kinking in mode II fracture. Special attention is drawn to numerical details affecting the efficiency and accuracy of the phase field models.

Leakage current and polarization domains in ferroelectric nanogenerators for energy harvesting

AbstractEnergy harvesting based on the transformation of mechanical into electrical energy is under investigation. Mechanical deformations from ambient parasitic vibrations serve as energy source for our nanogenerator concept with barium titanate as ferroelectric material. The energy harvesting process is enabled by the designed polarization domain topology. Due to the deformation in the ferroelectric film, the polarization domains reorder resulting alternating charge density within electrodes. Defects and other impurities in the single crystal turn the ferroelectric film into a semiconductor. We account for leakage current mechanisms in our finite element phase field model phenomenologically. Ohm's law (OL) and space‐charge‐limited‐ current (SCLC) are presented by linear or quadratic functions for the current density, respectively. The distance between the electrodes as well as the ferroelectric film thickness have an immense influence on the leakage current density. Further investigations focus on the influence of polarization domain topology on the leakage current.

Crack dynamics and crack tip shielding in a material containing pores analysed by a phase field method

Many naturally occurring materials, such as wood and bone, have intricate porous micro-structures and high stiffness and toughness to density ratios. Here, the influence of pores in a material on crack dynamics in brittle fracture is investigated. A dynamic phase field finite element model is used to study the effects of pores with respect to crack path, crack propagation velocity and energy release rate in a strip specimen geometry with circular pores. Four different ordered pore distributions are considered, as well as randomly distributed pores. The results show that the crack is attracted by the pores; this attraction is stronger when there is more energy available for crack growth. Crack propagation through pores also enables higher crack propagation velocities than are normally seen in strip specimens without pores (i.e. homogeneous material), without a corresponding increase in energy release rate. It is further noticed that as the porosity of an initially solid material increases, the crack tip is increasingly likely to become shielded or arrested, which may be a key to the high relative strength often exhibited by naturally occurring porous materials. We also find that when a pore is of the same size as the characteristic internal length then the pore does not localise damage. Since the characteristic internal length only regularises the damage field and not the strain end kinetic energy distributions, crack dynamics are still affected by small pores.

Dynamic crack propagation in wood fibre composites analysed by high speed photography and a dynamic phase field model

Using an experimental setup, with a high-speed camera to track crack tip velocity, dynamic fracture is studied in wood fibre polylactic acid (PLA) composite and pure PLA. The experiments are analysed quantitatively in terms of the relation between energy release rate and crack tip velocity, and qualitatively in terms of branching occurrence and fracture surface appearance. Branching occurs frequently in PLA specimens but not in wood fibre composite specimens, in spite of high energy release rates. Scanning electron microscopy images of the fracture surfaces show that the fracture surfaces in wood fibre composite materials are rugged and uneven compared to PLA, whose surfaces are smoother.The experimental results are compared to numerical results, obtained using a dynamic phase field finite element model. Simulations correlate well with experiments with respect to the relation between energy release rate and crack tip velocity. For PLA, the simulations also predict branching correctly, but for wood fibre composites, the simulations slightly over-predict the amount of branching and point to a need for further development of fracture models in order to better capture the constitutive behaviour of these heterogeneous materials.

An adaptive variational procedure for the conservative and positivity preserving Allen–Cahn phase-field model

In this paper, we present an adaptive variational procedure for unstructured meshes to capture fluid–fluid interfaces in two-phase flows. The two phases are modeled by the phase-field finite element formulation, which involves the conservative Allen–Cahn equation coupled with the incompressible Navier–Stokes equations. The positivity preserving variational formulation is designed to maintain the bounded and stable solution of the Allen–Cahn equation. For the adaptivity procedure, we consider the residual-based error estimates for the underlying differential equations of the two-phase system. In particular, the adaptive refinement/coarsening is carried out by the newest vertex bisection algorithm by evaluating the residual error indicators based on the error estimates of the Allen–Cahn equation. The coarsening algorithm avoids the storage of the tree data structures for the hierarchical mesh, thus providing the ease of numerical implementation. Furthermore, the proposed nonlinear adaptive partitioned procedure aims at reducing the amount of coarsening while maintaining the convergence properties of the underlying nonlinear coupled differential equations. We investigate the adaptive phase-field finite element scheme through the spinodal decomposition in a complex curved geometry and the volume-conserved interface motion driven by the mean curvature flow for two circles in a square domain. We then assess the accuracy and efficiency of the proposed procedure by modeling the free-surface motion in a sloshing tank. In contrast to the non-adaptive Eulerian grid counterpart, we demonstrate that the mesh adaptivity remarkably reduces the degrees of freedom and the computational cost by nearly half for similar accuracy. The mass loss in the Allen–Cahn equation via adaptivity process is also reduced by nearly three times compared to the non-adaptive mesh. Finally, we apply the adaptive numerical framework to solve the application of a dam-breaking problem with topological changes.

Electric leakage current density in phase field simulations for nanogenerator concepts

AbstractFor the purpose of energy harvesting on the nanoscale with ferroelectric barium titanate (BaTiO3), our nanogenerator concept transforms parasitic mechanical oscillations into usable electric energy. Experimental difficulties occur in sample preparation, e.g., surface roughness, bonding contact, and leakage currents. Latter can be considered in our finite element phase field model, such that the nanogenerator concept can be optimized. Leakage current density is implemented from different phenomenological approaches. First, Ohm's law represents a linear relation between leakage and electric field. Second, the Space‐Charge‐Limited Current (SCLC) relation assumes a quadratic dependency on the electric field. So far, SCLC is suitable for one‐dimensional problems, however, in two or three dimensions it is not found in literature. Therefore, we discuss a reformulation to bring into account that the electrical field E is a vector out of ℝ3 in our model. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Flexoelectricity, strain gradients, and singularities in ferroelectric nanostructures

The effect of flexoelectricity on the formation and evolution of domain structures in ferroelectric materials is developed by integrating strain gradient theory into a finite element phase field model. Length scales associated with elastic strain gradients and the corresponding polarization gradient across a domain wall are integrated into the time-dependent Ginzburg–Landau theory and numerical simulated. Theoretical relations of a shear strain gradient along an electrode/dielectric interface are first solved and verified numerically using the finite element model. A singularity in the strain gradient–induced polarization is shown to occur as the elastic strain gradient length scale approaches a flexoelectric length scale. The theory and finite element modeling is then extended to quantify strain gradient electromechanics near 180° and 90° tetragonal phase domain structures. It is shown that the strain gradient length scale [Formula: see text] strongly influences changes in strain across the domain walls but has a negligible effect on the polarization. Strain gradient effects become negligible when [Formula: see text], where [Formula: see text] is the polarization domain wall length scale.

Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model

This work presents phase field fracture modeling in heterogeneous porous media. We develop robust and efficient numerical algorithms for pressure-driven and fluid-driven settings in which the focus relies on mesh adaptivity in order to save computational cost for large-scale 3D applications. In the fluid-driven framework, we solve for three unknowns pressure, displacements and phase field that are treated with a fixed-stress iteration in which the pressure and the displacement–phase-field system are decoupled. The latter subsystem is solved with a combined Newton approach employing a primal–dual active set method in order to account for crack irreversibility. Numerical examples for pressurized fractures and fluid filled fracture propagation in heterogeneous porous media demonstrate our developments. In particular, mesh refinement allows us to perform systematic studies with respect to the spatial discretization parameter.

A finite element phase field model for relaxor ferroelectrics

AbstractA mechanically coupled phase field model is presented for the domain evolution and mesoscopic response of relaxor ferroelectrics. In the model the spontaneous polarization is treated as order parameter. The model is derived from thermodynamic analysis including the material force theory. Random field theory is adopted to take the disorder of relaxor ferroelectrics into account. Results show that the model is capable of reproducing relaxor features, such as domain miniaturization, small remnant polarization and large piezoelectric response. Dependence of these features on the random field strength is discussed. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modeling dendritic solidification of Al–3%Cu using cellular automaton and phase-field methods

We compared a cellular automaton (CA)–finite element (FE) model and a phase-field (PF)–FE model to simulate equiaxed dendritic growth during the solidification of cubic crystals. The equations of mass and heat transports were solved in the CA–FE model to calculate the temperature field, solute concentration, and the dendritic growth morphology. In the PF–FE model, a PF variable was used to identify solid and liquid phases and another PF variable was considered to determine the evolution of solute concentration. Application to Al–3.0wt.% Cu alloy illustrates the capability of both CA–FE and PF–FE models in modeling multiple arbitrarily-oriented dendrites in growth of cubic crystals. Simulation results from both models showed quantitatively good agreement with the analytical model developed by Lipton–Glicksman–Kurz (LGK) in the tip growth velocity and the tip equilibrium liquid concentration at a given melt undercooling. The dendrite morphology and computational time obtained from the CA–FE model are compared to those of the PF–FE model and the distinct advantages of both methods are discussed.

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.