General Imperfect Interface Research Articles

General imperfect interface model for spherical–circular inclusion composites

General imperfect interface model for spherical–circular inclusion composites

An over-penalized weak Galerkin method for parabolic interface problems with time-dependent coefficients

Based on the idea of classical discontinuous Galerkin and weak Galerkin finite element methods, we introduce an over-penalized term in the new scheme as a part of stabilization for solving parabolic interface problems. From the double-valued functions defined on interior edges of elements, it is natural to generate jumps of the over-penalized term. An over-penalized weak Galerkin (OPWG) finite element method can be applied very well to interface problems with general imperfect interface. Importantly, the diffusion coefficients of the interface problems depend on both temporal and spacial variables, not only limited in space as usual. With the use of (Pk(K),Pk−1(e),[Pk−1(K)]d) elements, semi-discrete and fully discrete schemes with backward Euler approximation in time are presented, and then the semi-discrete one is analyzed to be unconditionally stable. To analyze error estimates of semi-discrete and fully discrete schemes directly by error equation, we can just have optimal convergence order in energy norm. By virtue of the introduction of an elliptic projection operator, optimal error estimates of those schemes in L2 norm can be proved. Numerical examples are given to validate the efficiency and optimal convergence orders of the new schemes.

Ellipsoidal inhomogeneity with anisotropic incoherent interface. Multipole series solution and application to micromechanics

The ellipsoidal inhomogeneity with transversely isotropic incoherent interface is considered in the conductivity and elasticity context. The general imperfect interface is modeled by the first order approximation of thin transversely isotropic interphase layer. This model expands the Gurtin et al. (1998) theory of curved deformable interfaces in solids with a nanometer-scale microstructure to the incoherent interfaces between the dissimilar elastic materials and provides a certain insight into the interface moduli. The rigorous analytical solution to the conductivity and elasticity problems has been obtained by the multipole expansion method in terms of ellipsoidal solid harmonics. An accurate fulfillment of the imperfect interface conditions reduces the model boundary value problem to the linear algebraic system for multipole strengths. These results apply equally to the inhomogeneities with anisotropic interphases and nano level incoherent interfaces. The obtained solutions are valid for the non-uniform far loading and are readily incorporated in the many-particle (finite cluster or representative unit cell) model of heterogeneous solid with anisotropic incoherent interface. The tensors of effective conductivity and elastic stiffness of composite with ellipsoidal inhomogeneities are evaluated using Maxwell homogenization scheme. The obtained accurate numerical data indicate that the effective properties of composite may vary widely due to shape of inhomogeneities and the interface anisotropy ratio. Taking the incoherency of interface into account may increase reliability of predicting the behavior of nanostructured solids.

Identification of elastic properties of interphase and interface in graphene-polymer nanocomposites by atomistic simulations

This article tackles the problem of identification of elastic continuum model by atomistic simulations for graphene polymer nanocomposite. The Atomistic Local IdentificAtion of Stiffness method, so-called ALIAS method, is developed to estimate the local stiffness tensor at all points of polymer graphene laminate nanocomposite. Results suggest that the graphene can be modeled at continuum scale by a general imperfect interface with zero thickness. Moreover, the identification procedure reveals the existence of interphase on either side of the graphene with a thickness of 1 nm, which is one and a half times stiffer than the polymer bulk matrix. The identified continuum model is used to study the effective elastic properties of nanocomposites with sandwich microstructure. This study at continuum scale reveals a softening effect due the very low stiffness of slip along graphene plane. The softening due to the interfaces is preponderant in relation to the interphase stiffening. Finally, the continuum model also suggests that the wrinkling of graphene increases the stiffness of nanocomposites.

Representative unit cell model of elastic spherical particle composite with interphase and/or general imperfect interface

We consider the representative unit cell (RUC) model of the elastic spherical particle composite with (a) interphase layer and (b) general imperfect interface in the form proposed by Bövik (1994). The rigorous analytical solution to both RUC model problems is obtained by the multipole expansion method in a unified manner. Fulfillment of the matrix-to-inhomogeneity elastic contact conditions reduces the boundary value problem to the linear algebraic system for multipole strengths and provides a highly efficient algorithm for numerical study. The proposed approach applies to all physically consistent models of imperfect interface including the nonlinear ones. In so doing, use of the vector spherical harmonics makes the solving procedure simple and straightforward. Comparison of the accurate numerical data obtained from the interphase and interface models confirms correctness of the Bövik model and illustrates its applicability limits in the multi-particle environment. The developed RUC model enables consideration of composite with the spherical particles of diverse size, properties, interphases and interface bonding, with an adequate account of interaction between the inhomogeneities. Comparison with the laboratory test data for syntactic foam shows that a careful modeling of microstructure is a prerequisite for reliable prediction of the effective elastic properties of composite.

The effective thermal conductivity of composites with interfaces oscillating in two directions around a curved surface

In many situations of practical or/and theoretical interest, the assumption that the interfaces between constituent phases of a composite are smooth is no longer appropriate, and the consideration of rough interfaces at microscopic scale is necessary. However, in micromechanics, when the interfaces between the constituent phases of composites become rough, all classical well-known micromechanical schemes resorting to Eshelby's formalism cannot be applicable and the problem of determining the effective properties of composites become largely open. The present work aims to determine the effective thermal conductivity of a composite in which the interfaces between its constituent phases are perfectly bonded but oscillate quickly around a curved surface and along two directions. To achieve this objective, a two-scale homogenization method is proposed. In the first-scale homogenization, or microscopic-to-mesoscopic upscaling, the interfacial zone in which the interface oscillates is homogenized as an equivalent interphase by applying an asymptotic analysis. The thermal properties of the equivalent interphase can generally be determined by using a numerical approach based on the fast Fourier transform (FFT) method. In particular case where the equivalent interphase is very thin, this interphase is then replaced with a general imperfect interface situated at its middle surface. By applying the equivalent inclusion method, every inclusion with imperfect interface is further substituted by an equivalent inclusion with perfect interface. In the second scale homogenization, or mesoscopic-to-macroscopic upscaling, due to the fact that the interfaces are perfect, the effective thermal conductivity can be analytically obtained by using some well-known classical micromechanical schemes. To illustrate the two-scale homogenization method proposed in this work, the case of a layered

Modeling and simulating the effects of a general imperfect interface in fibrous piezoelectric composites

Interfacial behavior within fibrous piezoelectric composites (PZCs) may dramatically degrade the overall electromechanical performance and simultaneously vary the effective coupling moduli of these intelligent materials. In the present paper, we first recapitulate a physics-based general interfacial relation, which is rigorously derived from an ideal three-phase configuration and thus owns explicit physical meaning, to characterize this behavior between two different piezoelectric media. Afterwards, the explicit strong and weak governing equations of fibrous PZC boundary value problems (BVPs) are constructed, and a computational approach is then developed with the help of the extended finite element method (XFEM) to account for the discontinuities within both the primary (i.e., the electric potential and displacement fields) and the secondary fields (i.e., the normal electric displacement and normal traction fields). To achieve a benchmark problem with analytical exact solution, a simplified general interfacial relation is introduced, and a fibrous PZC BVP involving a cylindrical simplified general interface is designated and analytically solved. Hereafter, the convergence performance and validity of the elaborated computational approach are tested in detail. Eventually, discussions are further made on the influence of material composition and inhomogeneity shapes on the electroelastic coupling behavior of PZCs, and a few concluding remarks are drawn.

An asymptotic derivation of a general imperfect interface law for linear multiphysics composites

The paper is concerned with the derivation of a general imperfect interface law in a linear multiphysics framework for a composite, constituted by two solids, separated by a thin adhesive layer. The analysis is performed by means of the asymptotic expansions technique. After defining a small parameter ε, which will tend to zero, associated with the thickness and the constitutive coefficients of the intermediate layer, we characterize three different limit models and their associated limit problems: the soft interface model, in which the constitutive coefficients depend linearly on ε; the hard interface model, in which the constitutive properties are independent of ε; the rigid interface model, in which they depend on 1ε. The asymptotic expansion method is reviewed by taking into account the effect of higher order terms and by defining a general multiphysics interface law which comprises the above aforementioned models.

The microscopic conduction fields in the multi-coated sphere composites under the imposed macroscopic gradient and flux fields

Explicit analytical expressions of the microscopic conduction (gradient or flux) fields in the d-dimensional (\(d=2,3\)) multi-coated sphere assemblages under the imposed macroscopic gradient and flux fields are presented. Limiting procedures are developed to derive the results for most important specific composites, which include the inclusion composites with highly conducting imperfect interface, lowly conducting (with Kapitza resistance) imperfect interface, general imperfect interface, and those with anisotropic coatings. When the volume proportion of the outermost shells of the assemblages approaches 1, the simplified results for the dilute suspensions of the complex spherically symmetric inclusions in a major matrix are deduced.

Elastic fields and effective stiffness tensor of spheroidal particle composite with imperfect interface

An analytical approach to the elasticity homogenization problem for a spheroidal particle composite with general imperfect interface has been developed. The considered model takes into account the volume content and elastic moduli of constituents, shape, size and orientation of inhomogeneities and interface stiffness. The multipole expansion method has been applied to obtain an accurate solution to the problem for a single spheroidal inhomogeneity with imperfect interface. This solution constitutes a mathematical background of the Maxwell homogenization scheme. The formula for effective stiffness is derived by equating the induced dipole moment of equivalent inclusion to the total dipole moment of individual inhomogeneities. Numerical study shows a significant combined effect of interface stiffness and particle size, shape and orientation on the stress field and macroscopic stiffness of composite.

Effective elastic moduli of fiber-reinforced composites with interfacial displacement and stress jumps

The generalized self-consistent method (GSCM) is applied to obtain the closed-form estimates for the 5 elastic moduli of a transversely isotropic composite consisting of an isotropic matrix reinforced by unidirectional isotropic fibers of circular cross-section. The interfaces between the matrix and the fibers in this composite are taken to be imperfect and characterized by the general elastic isotropic model which includes as particular cases the widely used elastic spring-layer and membrane-type imperfect interface models. The displacement, strain and stress fields in an infinite homogeneous medium containing a composite cylinder with a general imperfect interface and subjected to each of 4 elementary remote uniform loadings are specified and used in deriving the estimates for the effective elastic moduli. These results extend and bridge the special relevant results reported in the literature for fiber-reinforced composites with imperfect interfaces characterized by the spring-layer or membrane-type imperfect interface model. Numerical examples are also given to illustrate some results.

Homogenization of fibrous piezoelectric composites with general imperfect interfaces under anti-plane mechanical and in-plane electrical loadings

The present paper aims mainly to estimate the size-dependent effective properties of fibrous piezoelectric composites with general imperfect interfaces. The interface model used states that the displacement, traction, electric potential, and normal electric displacement all suffer jumps across an interface. In addition, it can degenerate into the well-known special ones by employing appropriate high-contrast interfacial parameters. To achieve our objective, an auxiliary inhomogeneity problem of a circular fiber embedded in an infinite cylindrical reference phase via general imperfect interface under anti-plane mechanical and in-plane electrical boundary conditions is analytically solved. This solution allows us to apply the well-known micromechanical schemes such as the dilute, Mori–Tanaka to obtain the closed-form expressions for the size-dependent overall properties of composites under consideration. Some numerical examples are provided for illustrating the features of the obtained general results.

Closed-form estimates for the effective conductivity of isotropic composites with spherical particles and general imperfect interfaces

The present paper is concerned with estimating the effective conductivity of a composite consisting of spherical particles dispersed inside a matrix in a statistically homogeneous and isotropic way. The interface between every spherical particle and the matrix is thermally imperfect and described by the general linear isotropic imperfect model resulting from the replacement of an interphase of weak thickness by an interface of zero thickness. This general model includes as extreme particular cases Kapitza’s (or lowly conducting) thermal resistance model and the highly conducting thermal imperfect interface model. The fundamental solution is derived for the problem of a spherical particle embedded, via a general imperfect interface, in an infinite matrix undergoing a remote uniform intensity boundary loading. With the help of this fundamental solution, closed-form estimates for the size-dependent effective conductivity of the composite are deduced by using the dilute distribution, Mori–Tanaka, self-consistent and generalized self-consistent schemes. These results, incorporating as particular ones all the relevant estimates reported in the literature, are discussed and compared through numerical examples.

The strong and weak forms of a general imperfect interface model for linear coupled multifield phenomena

Imperfect interfaces are present in a variety of practical situations in mechanics and physics. In particular, they play an essential role in determination of mechanical and physical properties of composite materials. In a recent paper by Gu and He (2011), a general imperfect interface model for linear coupled multifield phenomena is rigorously derived which includes as special ones the widely used linear imperfect interface models proposed on the basis of some phenomenological considerations. The present paper aims at the establishment of the weak form for the boundary value problem of composites with general imperfect interfaces, which is the first key step towards numerically implementing the imperfect interface effects within the framework of finite element. To achieve this objective, firstly, the general imperfect interface model is reformulated in such a way that a particularly simple and compact strong form is obtained for it. Then, its weak form is derived in the general context of coupled multifield phenomena and expressed in a unified way for various linear mechanical and physical phenomena. Finally, the general results are particularized to transport phenomena, elasticity and piezoelectricity, together with detailed discussions on extreme situations in each case.

Thermomechanics of solids with general imperfect coherent interfaces

The objective of this contribution is to develop a thermodynamically consistent theory for general imperfect coherent interfaces in view of their thermomechanical behavior and to establish a unified computational framework to model all classes of such interfaces using the finite element method. Conventionally, imperfect interfaces with respect to their thermal behavior are often restricted to being either highly conducting (HC) or lowly conducting (LC) also known as Kapitza. The interface model here is general imperfect in the sense that it allows for a jump of the temperature as well as for a jump of the normal heat flux across the interface. Clearly, in extreme cases, the current model simplifies to HC and LC interfaces. A new characteristic of the general imperfect interface is that the interface temperature is an independent degree of freedom and, in general, is not a function of only temperatures across the interface. The interface temperature, however, must be computed using a new interface material parameter, i.e., the sensitivity. It is shown that according to the second law, the interface temperature may not necessarily be the average of (or even between) the temperatures across the interface. In particular, even if the temperature jump at the interface vanishes, the interface temperature may be different from the temperatures across the interface. This finding allows for a better, and somewhat novel, understanding of HC interfaces. That is, a HC interface implies, but is not implied by, the vanishing temperature jump across the interface. The problem is formulated such that all types of interfaces are derived from a general imperfect interface model, and therefore, we establish a unified finite element framework to model all classes of interfaces for general transient problems. Full details of the novel numerical scheme are provided. Key features of the problem are then elucidated via a series of three-dimensional numerical examples. Finally, we recall since the influence of interfaces on the overall response of a body increases as the scale of the problem decreases, this contribution has certain applications to nano-composites and also thermal interface materials.

General imperfect interfaces

The objective of this contribution is to develop a thermodynamically consistent theory for general imperfect interfaces and to establish a unified computational framework to model all classes of such interfaces using the finite element method. The interface is termed general imperfect in the sense that it allows for a jump of the temperature as well as for a jump of the normal heat flux across the interface. Conventionally, imperfect interfaces with respect to their thermal behavior are restricted to being either highly-conducting (HC) or lowly-conducting (LC) also known as Kapitza. For a HC interface the temperature is continuous across the interface while the jump of the normal heat flux is admissible. On the contrary, a LC interface does not allow for a jump of the heat flux across the interface but it does allow for a temperature jump. The temperature jump of a LC interface is frequently assumed to be proportional to the average heat flux across the interface. In this contribution we prove that this common assumption is indeed an appropriate condition to (sufficiently and not necessarily) satisfy the second law of thermodynamics.While HC and LC interfaces are generally accepted and well established today, the general imperfect interfaces remain poorly understood. Here we propose a thermodynamically consistent theory of general imperfect interfaces and we show that the dissipative structure of the interface suggests firstly to classify such interfaces as semi-dissipative (SD) and fully-dissipative (FD). Secondly, for a FD interface the interface temperature shall be considered as an independent degree of freedom and a new (constitutive) equation is obtained to calculate the interface temperature using a new interface material parameter i.e. the sensitivity. Furthermore, we show how all types of interfaces are derived from a FD general imperfect interface model. This finding allows us to establish a unified finite element framework to model all classes of interfaces. Full details of the novel numerical scheme are provided. Key features of general imperfect interfaces are then elucidated via a series of three-dimensional numerical examples. In particular, we show that according to the second law the interface temperature may not necessarily be the average of (or even between) the temperatures across the interface. Finally, we recall since the influence of interfaces on the overall response of a body increases as the scale of the problem decreases, this contribution has certain applications to nano-composites.

Interfacial discontinuity relations for coupled multifield phenomena and their application to the modeling of thin interphases as imperfect interfaces

Interfacial continuity and discontinuity relations are needed in dealing with a variety of mechanical and physical phenomena in heterogeneous media. The present work consists of two parts. In the first part concerned with perfect interfaces, two orthogonal projection operators reflecting the interfacial continuity and discontinuity of the field variables of coupled mechanical and physical phenomena are introduced and some coordinate-free interfacial relations involving the surface decomposition of a generic linear constitutive law are deduced. In the second part dedicated to the derivation of a general imperfect interface model for coupled multifield phenomena by applying Taylor's expansion to a 3D curved thin interphase perfectly bonded to its two neighboring phases, the interfacial operators and relations given in the first part are used directly so as to render the derivation more direct and to write the final interfacial jump relations characterizing the model in a unified and compact way. The general imperfect interface model obtained in the present work includes as special cases all the relevant ones reported in the literature.

Coordinate-free derivation and weak formulation of a general imperfect interface model for thermal conduction in composites

In a great number of situations of practical interest, the interfaces between the constituent phases of a composite turn out to be imperfect. In the context of thermal conduction, an interface is said to be imperfect if the requirement that both the temperature and the normal heat flux be continuous across the interface is not satisfied. A powerful method based on mathematical asymptotic analysis has been proposed and developed in the literature by several authors for the derivation of linear imperfect interface models of thermal conduction. This method consists in replacing an interphase of small uniform thickness between two-phases by an imperfect interface of null thickness characterized by the temperature and normal heat flux jump relations deduced by carrying out an appropriate asymptotic analysis. The objective of the present work is threefold. Firstly, it aims to explicitly show and emphasize the key role played by Hadamard’s relation in the method. Secondly, it has the purpose of using a coordinate-free differential geometry theory and Hadamard’s relation to render the method coordinate-free. Thirdly and most importantly, the present work gives a weak formulation for the problem concerning the steady thermal conduction in a composite with the interfaces described by the general temperature and normal heat flux jump relations derived. This weak formulation is a key step toward solving the problem by the extended finite element method (XFEM) presented in a companion paper.

EFFECT OF IMPERFECT INTERFACE ON THERMAL STRESSES-ASSISTED MATRIX CRACKING IN FIBER COMPOSITES

In this article we study the effects of imperfect bonding on stress intensity factors (SIFs) calculated at a radial matrix crack in a fiber (inclusion) composite subjected to thermal loading. We use analytic continuation to extend existing series methods to obtain series representations of deformation and stress fields in both the inclusion and the surrounding matrix in the presence of the crack. The interaction between the crack and the inclusion is demonstrated numerically for different elastic materials, geometries, and varying degrees of bonding (represented by imperfect interface parameters) at the interface. The results show that, generally, SIF decreases monotonically with the interface imperfection at both nearby and distant crack tips. The sliding and nonslipping interfaces have a relatively small effect on the reduction of thermal stresses-induced SIF as compared to a general imperfect interface. Therefore, the pronounced effect of the imperfect fiber-matrix bond on the thermal stresses-assisted SIF at the crack tips should not be ignored.

Elastic properties of particulate composites with imperfect interface

An effective analytical approach is developed for the problem of particulate composites containing spherical inclusion with imperfect interface between the matrix and spherical inclusions. In this paper, a general interface model for a variety of interfacial defects has been presented, in which both displacement discontinuity across the interface and the elastic moduli varing with radius outside of the inclusion are considered. The imperfect interface conditions are appropriate in the case of thin coatings on the inclusion. Furthermore, in the case of thin elastic interphase, the displacement field and the stress field in the inclusion and matrix are exactly solved for the boundary problem of hydrostatic compression of an infinite spherical symmetrical body by Frobenius series, and the expression of the normal interface parameter, Dr, is derived. In addition, it has been proved that two previous results derived in some literatures by considering the interface to be a thin interphase with displacement jump or with some variance in its moduli can be reverted from the present formula, respectively. Numerical results are given to demonstrate the significance of the general imperfect interface effects.