A numerical convergence study regarding homogenization assumptions in phase field modeling

From a mathematical point of view, phase field theory can be understood as a smooth approximation of an underlying sharp interface problem. However, the smooth phase field approximation is not uniquely defined. Different phase field approximations are known to converge to the same sharp interface problem in the limiting case—if the thickness of the diffuse interface converges to zero. In this respect and focusing on numerics, a question that naturally arises is as follows: What are the convergence rates of the different phase field models? The paper deals precisely with this question for a certain family of phase field models. The focus is on an Allen–Cahn‐type phase field model coupled to continuum mechanics. This model is rewritten into a unified variational phase field framework that covers different homogenization assumptions in the diffuse interfaces: Voigt/Taylor, Reuss/Sachs and more sound homogenization approaches falling into the range of rank‐one convexification. It is shown by means of numerical experiments that the underlying phase field model—that is, the homogenization assumption in the diffuse interface—indeed influences the convergence rate. Copyright © 2017 John Wiley & Sons, Ltd.     

Modeling impact‐induced damage and debonding using level sets in a sharp interface Eulerian framework

This work presents a level‐set–based sharp interface technique to simulate the evolution of damage in ductile materials under high velocity impact conditions. The level‐set method is adopted to track all interfaces including damage zones within the materials. Two types of damage are considered, ie, the creation of spall zones due to damage accumulation in homogeneous ductile materials and interfacial debonding in heterogeneous materials. Spall is simulated using continuum damage models and a level‐set–based crack generation and evolution algorithm. Three continuum damage models are tested for metal targets subjected to flyer impact; the results from the current code (SCIMITAR3D) are compared with the two widely used computer codes EPIC and CTH, and to experimental data; it is found that the computer codes are in good agreement among each other, but agreement of all methods with experimental data is not uniform. At material interfaces, damage is handled using a cohesive zone model and evolving level sets to create void spaces because of material separation due to debonding. Finally, ductile damage combined with debonding is simulated in an Al‐Ni laminate impacted by a projectile. The results demonstrate the ability of the present approach to simulate various types of damage in materials with heterogeneities and inclusions.     

Modeling of non-equilibrium solute diffusion upon rapid solidification: effective mobility versus kinetic energy

Abstract

The effective mobility approach is compared with the kinetic energy approach in terms of sharp interface modeling and phase-field modelling of non-equilibrium solute diffusion upon rapid solidification of binary alloys. The two approaches are equivalent for modelling of long range solute diffusion in bulk phases, but only the effective mobility approach can introduce the non-equilibrium solute diffusion effect to short range solute diffusion at a sharp interface or within a diffuse interface. Addition of the kinetic energy terms results in an unreasonable non-bilinear expression of the flux and thermodynamic driving force in the free energy production of interface migration or phase field propagation, whereas the effective mobility approach allows the thermodynamic extremal principle workable.     

Coupled heat conduction and thermal stress analyses in particulate composites

This study introduces two micromechanical modeling approaches to analyze spatial variations of temperatures, stresses and displacements in particulate composites during transient heat conduction. In the first approach, a simple micromechanical model based on a first order homogenization scheme is adopted to obtain effective mechanical and thermal properties, i.e., coefficient of linear thermal expansion, thermal conductivity, and elastic constants, of a particulate composite. These effective properties are evaluated at each material (integration) point in three dimensional (3D) finite element (FE) models that represent homogenized composite media. The second approach treats a heterogeneous composite explicitly. Heterogeneous composites that consist of solid spherical particles randomly distributed in homogeneous matrix are generated using 3D continuum elements in an FE framework. For each volume fraction (VF) of particles, the FE models of heterogeneous composites with different particle sizes and arrangements are generated such that these models represent realistic volume elements “cut out” from a particulate composite. An extended definition of a RVE for heterogeneous composite is introduced, i.e., the number of heterogeneities in a fixed volume that yield the same expected effective response for the quantity of interest when subjected to similar loading and boundary conditions. Thermal and mechanical properties of both particle and matrix constituents are temperature dependent. The effects of particle distributions and sizes on the variations of temperature, stress and displacement fields are examined. The predictions of field variables from the homogenized micromechanical model are compared with those of the heterogeneous composites. Both displacement and temperature fields are found to be in good agreement. The micromechanical model that provides homogenized responses gives average values of the field variables. Thus, it cannot capture the discontinuities of the thermal stresses at the particle-matrix interface regions and local variations of the field variables within particle and matrix regions.     

Theoretical foundations for integrating sound in interactive interfaces: identifying temporal and spatial information conveyance principles

This paper proposes theoretical foundations for conveying temporal (i.e. relating to time) and spatial (i.e. relating to space) information using auditory cues in interactive systems. Three theoretical models are developed to aid the design of auditory interfaces, including an audio integration model that outlines an end-to-end process for adding sounds to interactive interfaces, a temporal audio model that provides a framework for when to integrate these sounds to meet certain performance objectives and a spatial audio model that provides a framework for adding spatialisation cues to interface sounds. The models presented in this paper, which are each coupled with a set of design guidelines theorised from the literature, put forward a structured process for integrating sounds in interactive interfaces.     

Reinvestigation of the tensile strength and fracture property of Ni(1 1 1)/α-Al2O3(0 0 0 1) interfaces by first-principle calculations

First-principle calculations are performed to reinvestigate the mechanical tensile property and failure characteristic of Ni/Al₂O₃ interfaces, in order to clear the inconsistence existed in the literatures. Four types of interface models without initial lateral stresses are used, i.e., Al-terminated O-site, O-terminated Al-site, Al-terminated Al-site and Al-terminated H-site models. Two kinds of tensile methods, viz., uniaxial extension and uniaxial tension, are adopted to check the mechanical responses of these interface models. It is found that the results under uniaxial extension are generally consistent with those under uniaxial tension, including the overall shapes of stress–strain curves and the values of tensile strengths. Moreover, the initial lateral stresses have an apparent influence on the mechanical properties of the interfaces during the loading process, such as tensile strength, fracture strain and the work of separation. Our simulation results also clarified that, under tensile loading, the most stable O-terminated Al-site interface model tends to fracture in a brittle way along the sublayer between in-plane Ni–Ni atomic bonds, while all of the Al-terminated interface models will fail in a ductile fracture manner with relatively lower stress levels, breaking along the interlayer between the Ni(1) and Al(1) layers.     

Hemispherical and diffuse reflectance and transmittance of a slightly inhomogeneous film bounded by rough, unparallel interfaces

The optical reflectance and transmittance of an ideal thin film are calculated in a well-known way. As far as a non-ideal thin film is concerned - i.e., a slightly inhomogeneous thin film bounded by rough, unparallel interfaces - three categories of spectral coefficients can be defined, i.e.: specular reflectance and direct transmittance (light intensity flux along the optical axis), hemispherical reflectance and transmittance (light intensity flux integrated over the solid half angle π), and diffuse reflectance and transmittance (light intensity flux scattered around the optical axis) coefficients. In this paper a model recently introduced for the specular and direct coefficients is generalized to calculate also the hemispherical and diffuse coefficients of a non-ideal film.     

An adaptive fixed mesh method for modelling and simulating of unsteady two‐phase flows with sharp physical interfaces

We present in this paper a new computational method for simulation of two‐phase flow problems with moving boundaries and sharp physical interfaces. An adaptive interface‐capturing technique (ICT) of the Eulerian type is developed for capturing the motion of the interfaces (free surfaces) in an unsteady flow state. The adaptive method is mainly based on the relative boundary conditions of the zero pressure head, at which the interface is corresponding to a free surface boundary. The definition of the free surface boundary condition is used as a marker for identifying the position of the interface (free surface) in the two‐phase flow problems. An initial‐value‐problem (IVP) partial differential equation (PDE) is derived from the dynamic conditions of the interface, and it is designed to govern the motion of the interface in time. In this adaptive technique, the Navier–Stokes equations written for two incompressible fluids together with the IVP are solved numerically over the flow domain. An adaptive mass conservation algorithm is constructed to govern the continuum of the fluid. The finite element method (FEM) is used for the spatial discretization and a fully coupled implicit time integration method is applied for the advancement in time. FE‐stabilization techniques are added to the standard formulation of the discretization, which possess good stability and accuracy properties for the numerical solution. The adaptive technique is tested in simulation of some numerical examples. With the test problems presented here, we demonstrated that the adaptive technique is a simple tool for modelling and computation of complex motion of sharp physical interfaces in convection–advection‐dominated flow problems. We also demonstrated that the IVP and the evolution of the interface function are coupled explicitly and implicitly to the system of the computed unknowns in the flow domain. Copyright © 2005 John Wiley & Sons, Ltd.     

Micromechanical modeling of interface damage of metal matrix composites subjected to transverse loading

A three-dimensional finite element micromechanical model was developed to study effects of thermal residual stress, fiber coating and interface bonding on the transverse behavior of a unidirectional SiC/Ti–6Al–4V metal matrix composite (MMC). The presented model includes three phases, i.e. the fiber, coating and matrix, and two distinct interfaces, one between the fiber and coating and the other between coating and matrix. The model can be employed to investigate effects of various bonding levels of the interfaces on the initiation of damage during transverse loading of the composite system. Two different failure criteria, which are combinations of normal and shear stresses across the interfaces, were used to predict the failure of the fiber/coating (f/c) and coating/matrix (c/m) interfaces. Any interface fails as soon as the stress level reaches the interfacial strength. It was shown that in comparison with other interface models the predicted stress–strain curve for damaged interface demonstrates good agreement with experimental results.     

Crystallography-based prediction of plastic anisotropy of polycrystalline materials

The on-line prediction of metal sheet formability requires that both material characterization (texture identification) and yield loci predetermination be done in very shor time intervals. Of two applicable approaches, i.e., continuum mechanics and crystallography-based methods, only the latter are suitable for this purpose. Several models of plasticity of a polycrystalline material were reviewed, and their applicability to the prediction of plastic anisotropy of face-centered cubic (FCC) metals was evaluated. A tailored set of cold-rolled copper alloy samples was designed and manufactured, representing the wide spectrum of textures and cold work levels typical for the sheet metal industry. The texture was quantitatively described in the form of the orientation distribution functions derived by the inversion of four incomplete pole figures. The Taylor-Bishop-Hill model was applied in order to calculate the planar variation of the plastic strain ratio. The continuum mechanics of textured polycrystals approach was also used for the prediction of the plastic strain-rate ratio for the same set of deformed materials. The theoretical predictions were compared with the plastic strain ratios measured in tensile tests using strain gauges. The applicability of the models for prediction of the plastic anisotropy of FCC metals was discussed in view of the operating deformation mechanisms and other factors such as strain hardening sensitivity and grain size/shape effects.     

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase‐field approximations of sharp domains. Leveraging the properties of the phase‐field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase‐field solutions of the Allen‐Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase‐field in the diffuse boundary region and a uniform mesh for the representation of the physics‐based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty‐type methods. In the context of imaging‐based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, ie, the interface width of the phase‐field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body.     

On the continuum modeling of materials with kinematic structure

Summary In this work, a phenomenological field-based approach to the formulation of models for structured materials or continua is presented. The corresponding results are in particular relevant to that class of such materials for which thedistribution of (micro)structure at each material point and the evolution of this distribution may influence the behavior of the material as a whole (e.g., in nematic liquid crystals with variable orientation). Essential to the underlying approach is the assumption or idealization that the structured continuum in question is characterized kinematically by additional degrees of freedom in comparison to standard continua. On this basis, the kinematics and balance relations for the structure continuum are formulated with respect to a (generalized) kinematic space via direct generalization of standard kinematics and balance relations. In particular, the formation of the latter for the structured continuum is based on the corresponding (total) energy balance. Indeed, analogous to the standard case, the assumed Euclidean frame-indifference of this balance, together with the transformation properties of the fields appearing in it, determine the forms of the remaining balance relations for the structured continuum. With these general results in hand, account is next taken of the fact that the degrees of freedom of the standard continuum constitute a subset of those of the structured continuum. As already established in previous work, this fact can be represented in a mathematically-precise fashion as a fibre bundle, with base space the standard kinematic space, i.e., three-dimensional Euclidean point space, and total space the kinematic space for the structured continuum. In this context, the kinematic space for the structure itself is represented by the typical fibre of the fibre bundle. Among other results, one obtains on this basis a split of the momentum balance for the structured continuum into momentum balances for the standard continuum and for the structure. Further, the fibre bundle representation induces naturally forms of all fields and balance relations with respect to standard kinematic space, i.e., forms averaged over the degrees of freedom of the structure. In the last part of the work, the results of the current approach are applied to the special cases of rigid-rod- and rigid-body-like structure, and compared with previous work.     

Anti-plane transient fracture analysis of the functionally gradient elastic bi-material weak/infinitesimal-discontinuous interface

The mechanical model was established for the Dirac-type anti-plane transient fracture problem of the weak-discontinuous interface between two FGMs half-planes. Integral transform was adopted to derive Cauchy singular integral equation and Erdogan’s allocation method was used to calculate transient stress intensity factors numerically. The numerical solutions of the weak-discontinuous case were contrasted with those of the infinitesimal-discontinuous one. Two possible effective methods to diminish the peak values of transient stress intensity factors are discussed. One is to reduce the weak-discontinuity of the interface, i.e., to make the ratio of the two non-homogeneity parameters be close to 1.0 and to avoid the case that the signs of the two non-homogeneity parameters are different. Another is to make a compromise between the weak-discontinuity and the all-continuity, i.e., to make FGMs interface infinitesimal-discontinuous. Simple method was suggested for the realization of the infinitesimal-discontinuity of FGMs interface. From the strong-discontinuous interface to the weak- discontinuous one, and then to the infinitesimal-discontinuous one, this is a law and trend of the development of composite interfaces. To design and manufacture infinitesimal-discontinuous interfaces may be a brand-new effective approach to enhance the reliability of composite structures, and the first rank infinitesimal-discontinuity is enough to improve the mechanical performances of composites notably.     

The interfacial characteristic of SiCp/AZ91 magnesium matrix composites fabricated by stir casting

The particle/matrix interfaces in SiCp/AZ91 composite fabricated by stir casting were investigated using transmission electron microscope (TEM) equipped with ultra-thin window energy dispersive X-ray analysis (EDAX) system. Chemical reactions indeed occurred at the interfaces. According to EDAX results, the interfacial reaction products are considered to contain Al₄C₃, MgO, and Mg₂Si phases. The interfaces can be classified into three types (interfaces I, II, and III) according to morphological features of the interfaces: (1) for interface I, interfacial reaction products were in direct contact with the surface of SiCp; (2) for interface II, interfacial reaction products were not in direct contact with the surface of SiCp; (3) for interface III, interfacial reaction products were not observed at the interfaces, i.e., interface III was simply formed by the two surfaces of SiCp and matrix. Mg₁₇Al₁₂ and Al₈Mn₅ precipitate phases heterogeneously nucleated at the particle/matrix interfaces.     

Microstructural effects of processing in the plastic-bonded explosive Composition A-3

Multiscale model development for composites such as plastic-bonded explosives (PBXs) is critical for assessing structural integrity and behavior and for developing new materials. For models to be successful they must reproduce bulk material responses while incorporating important microstructural features. These features arise at the mesoscale during processing, and while they have commensurate effects on composite material response they are not easily identified or characterized. Here, we study Composition A-3 as a representative PBX, chosen both for relevance to the field and as a tractable formulation to model. Composition A-3 is formulated by mixing micron-scale explosive crystals with an emulsified polyethylene solution, then breaking the emulsion to form small crystal/polyethylene agglomerates suitable for pressing and machining. Key aspects of this formulation process that may potentially affect the microstructure were identified. Specifically, the emulsification chemicals in a model system were found to partially dissolve or degrade the explosive and create a diffuse interface between the crystal and binder. The diffuse interfaces, along with some of the chemicals that remain in the composite after manufacture, create a heterogeneous multicomponent system that likely influences adhesion, void formation, and crack formation. The observed interfaces may be difficult to model. These results are compared with previous interfacial studies in other PBX materials, and the necessity of including such data in mesoscale models is discussed.     

Micromechanics based ply level material degradation model for unidirectional composites

The damaged response of a composite lamina depends on various mechanisms that take place at the microlevel, i.e., at the level of the fiber and matrix. The present work focuses on developing a ply level continuum damage model for point-wise stiffness degradation through simplified representation of the microlevel damage. A three dimensional micromechanical analysis of a single cell representative volume element is carried out for various volume fraction, and levels of damage. The model brings out the coupled effect of damage on the effective point-wise ply level stiffness. Further, the numerical results are employed to develop a functional continuum representation of stiffness degradation as a function of the damage parameters and fiber volume fraction perturbations. The micromechanics model is consistent with experimentally observed stiffness degradation, i.e., a strong influence of fiber breakage and fiber matrix debond, and a weak influence of normal cracking of matrix. The proposed model can be considered as an improved version of the widely accepted diffused (meso) damage models, i.e., DML. The study also gives a generalized and consistent definition for the free energy, which can be used for modeling growth of damage.