Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces

A micromechanical framework is proposed to predict the effective elastic behavior and weakened interface evolution of particulate composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [Qu, J., 1993a. Eshelby tensor for an elastic inclusion with slightly weakened interfaces. Journal of Applied Mechanics 60 (4), 1048–1050; Qu, J., 1993b. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mechanics of Materials 14, 269–281] is adopted to model spherical particles having imperfect interfaces in the composites and is incorporated into the micromechanical framework. Based on the Eshelby’s micromechanics, the effective elastic moduli of three-phase particulate composites are derived. A damage model is subsequently considered in accordance with the Weibull’s probabilistic function to characterize the varying probability of evolution of weakened interface between the inclusion and the matrix. The proposed micromechanical elastic damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress–strain responses. Comparisons between the present predictions with other numerical and analytical predictions and available experimental data are conducted to assess the potential of the present framework.     

On two models of arbitrarily curved three-dimensional thin interphases in elasticity

In boundary value problems involving thin interphases, it is often desirable to have a model of an interphase which makes possible to solve for the fields in the adjacent media without having to solve for the fields in it. This is usually achieved in the literature by replacing the interphase by a geometrical surface with appropriately designed “imperfect interface” conditions on it. In the present study, carried out in the setting of elasticity, another option is explored: the geometry of the interphase is left intact, and conditions are devised for the displacements and tractions pertaining to the media adjacent to the interphase and evaluated at both sides of it such that they will simulate the presence of the interphase. Those conditions do not involve the fields within the interphase, yet they depend on its material properties and on those of the adjacent media as well, and make possible to solve for the fields in the adjacent media without having to solve for the fields in the interphase. The formulation is given in a parallel orthogonal curvilinear coordinate system suitable for the modeling arbitrarily curved three-dimensional interphases of constant thickness. Both types of the above described interphase models are tested in the setting of a coated infinite fiber embedded in a matrix which is subjected to an anti-plane shear loading and an in-plane transverse shear loading at infinity, and their predictions are compared with the exact solutions for the fields in the three-phase configuration consisting of the interphase and its adjacent media. The model in which the interphase geometry is left intact is observed to perform generally better than the one in which the interphase is replaced by an interface.     

A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media

An arbitrarily curved three-dimensional anisotropic thin interphase between two anisotropic solids is considered. The purpose of this study is to model this interphase as a surface between its two neighbouring media by means of appropriately devised interface conditions on it. The analysis is carried out in the setting of unsteady heat conduction and dynamic elasticity, and makes use of the simple idea of a Taylor expansion of the relevant fields in thin regions. It consists of a generalization of a previous study by Bövik [1994. On the modelling of thin interface layers in elastic and acoustic scattering problems. Q. J. Mech. Appl. Math. 47, 17-42] which was confined to the isotropic setting. The remarkable feature of the presently derived anisotropic interface model is that formally it has a more compact form than that of Bövik's isotropic version. This is achieved by a judicious choice of surface differential operators which have been used in the derivation, and makes possible to show that several previously known classical interface models are recovered as special cases of the one obtained in this study, once suitable assumptions are made on the magnitude of the conductivity and elasticity tensors of the interphase.     

Effective properties of elastic periodic composite media with fibers

A series solution to obtain the effective properties of some elastic composites media having periodically located heterogeneities is described. The method uses the classical expansion along Neuman series of the solution of the periodic elasticity problem in Fourier space, based on the Green's tensor, and exact expressions of factors depending on the shape of the inclusions. Some properties of convergence of the solution are presented, more specifically concerning the elasticity tensor of the reference medium, showing that the convergence occurs even for empty fibers. The solution is extended for rigid inclusions. A comparison is made with previous exact solutions for a fiber composite made of cylindrical fibers with circular cross-sections and with previous estimates. Different examples are presented for new situations concerning the study of fiber composites: composites with elliptic cross-sections and multi-phase fibrous 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.     

Influences of imperfect interfaces on effective properties of multiferroic composites with coated inclusion

In order to predict the effective properties of multiferroic composite materials, the effective material constants of multiferroic composites with the coated inclusion and imperfect interface are investigated. Based on the generalized self-consistent theory, the closed-form solutions of the effective material constants are derived. For the composites with piezomagnetic inclusion, piezoelectric coating and polymer matrix, numerical calculations are performed to present the influences of the imperfect interface cooperating with the coating on the effective material constants. From the results, it can be observed that the effective constants can be enhanced by the coating but reduced by the imperfect interface. Moreover, the coating has the shielding effects on the imperfect interface for the composite structures with its higher filling ratio.     

Dispersion effects of extensional waves in pre-stressed imperfectly bonded incompressible elastic layered composites

The effect of an imperfect interface, on time-harmonic extensional wave propagation in a pre-stressed symmetric layered composite is considered. The bimaterial composite consists of incompressible isotropic elastic materials. The shear spring type resistance model employed to simulate the imperfect interface can accommodate the extreme cases of perfect bonding and a fully slipping interface. The dispersion relation obtained by formulating the incremental boundary-value problem and the use of the propagator matrix technique, is analyzed at the low and high wavenumber limits. For the perfectly bonded and imperfect interface cases in the low wavenumber region, only the fundamental mode has a finite phase speed, while other higher modes have an infinite phase speed when the dimensionless wavenumber approaches zero. However, for the fully slipping interface in the low wavenumber region, both the fundamental mode and the next lowest mode have finite phase speeds. In the high wavenumber region, when the dimensionless wavenumber tends to infinity, the phase speeds of the fundamental mode and the higher modes depend on the phase speeds of the surface and interfacial waves and on the limiting phase speed of the composite. An expression to determine the cut-off frequencies is obtained from the dispersion relation. Numerical examples of dispersion curves are presented, where when the material has to be prescribed either Mooney–Rivlin material or Varga material is assumed. The effect of the imperfect interface is clearly evident in the numerical results.     

Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities

This work aims at estimating the size-dependent effective elastic moduli of particulate composites in which both the interfacial displacement and traction discontinuities occur. To this end, the interfacial discontinuity relations derived from the replacement of a thin uniform interphase layer between two dissimilar materials by an imperfect interface are reformulated so as to considerably simplify the characteristic expressions of a general elastic imperfect model which is adopted in the present work and include the widely used Gurtin–Murdoch and spring-layer interface models as particular cases. The elastic fields in an infinite body made of a matrix containing an imperfectly bonded spherical particle and subjected to arbitrary remote uniform strain boundary conditions are then provided in an exact, coordinate-free and compact way. With the aid of these results, the elastic properties of a perfectly bonded spherical particle energetically equivalent to an imperfectly bonded one in an infinite matrix are determined. The estimates for the effective bulk and shear moduli of isotropic particulate composites are finally obtained by using the generalized self-consistent scheme and discussed through numerical examples.     

An improved model of thin cylindrical piezoelectric layers between isotropic elastic media

Radially polarized piezoelectric fibers are now considered to be used in smart composites. The piezoelectric layers in these fibers are generally anisotropic in the transverse direction and therefore difficult to analyze when dynamic loads are involved. The present paper presents a theoretical study of the dynamic behavior of radially polarized cylindrical piezoelectric layers between isotropic elastic media. A new interphase model is developed to provide simple yet accurate evaluation of the dynamic response of such anisotropic elastic layers. Unlike the traditional interface-spring model, the current interphase model satisfies the equations of motion of the layers and can provide reliable prediction of the stress and displacement. A comparison between the developed model, the interface-spring model and the finite element analysis is conducted. The results clearly show the advantages of the current model over the traditional interface-spring model in simulating anisotropic layers. Numerical examples based on this interphase model for different interphase thicknesses, loading frequencies and material combinations are presented to evaluate the dynamic behavior of multilayered elastic media.     

Critical states in the interaction of discontinuous waves with interfaces between elastic media

The ray method is used to develop a procedure for studying the kinematics and dynamics of the interaction of discontinuous waves with interfaces between elastic media. The critical states of such interactions are analyzed. For the cases of a free surface and two surfaces contacting with and without sliding, it is established that the kinematic and dynamic conditions for critical states to occur are satisfied simultaneously Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 79–90, February 2009.     

Morphological stability of epitaxial thin elastic films by van der Waals force

Summary  The morphological stability of epitaxial thin elastic films on a substrate by van der Waals force is discussed. It is found that only van der Waals force with negative Hamaker constant tends to stabilize the film, and the lower bound for the Hamaker constant is also obtained for the stability of thin film. The critical value of the undulation wavelength is found to be a function of both film thickness and external stress. The charateristic time-scale for surface mass diffusion scales to the fourth power to the wavelength of the perturbation. Received 4 December 2000; accepted for publication 31 July 2001     

Influence of interfaces on effective properties of nanomaterials with stochastically distributed spherical inclusions

The method of conditional moments is generalized to include evaluation of the effective elastic properties of particulate nanomaterials and to investigate the size effect in those materials. Determining the effective constants necessitates finding a stochastically averaged solution to the fundamental equations of linear elasticity coupled with surface/interface conditions (Gurtin–Murdoch model). To obtain such a solution the system of governing stochastic differential equations is first transformed to an equivalent system of stochastic integral equations. Using statistical averaging, the boundary-value problem is then converted to an infinite system of linear algebraic equations. A two-point approximation is considered and the stress fluctuations within the inclusions are neglected in order to obtain a finite system of algebraic equations in terms of component-average strains. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix and randomly distributed spherical inhomogeneities. As a numerical example a nanoporous material is investigated assuming a model in which the interface effects influence only the bulk modulus of the material. In that model the resulting shear modulus is the same as for the material without surface effects. Dependence of the bulk moduli on the radius of nanopores and on the pore volume fraction is analyzed. The results are compared to, and discussed in the context of other theoretical predictions.     

Scattering of harmonic elastic waves by a plane interface crack with linear adhesive tips in a layered half space

In this paper, the scattering of elastic waves by an interface crack with linear adhesive tips in a layered half space is considered. By use of integral transform and integral equation methods, the singular integral equations of this problem are derived, which are transformed into a set of algebraic equations by means of contour integration and Chebyshev polynomials expanding technique. The numerical results of the adhesive region and stress amplitudes are given in this paper.     

Second-harmonic generation of two-dimensional elastic wave propagation in an infinite layered structure with nonlinear spring-type interfaces

The two-dimensional elastic wave propagation in an infinite layered structure with nonlinear interlayer interfaces is analyzed theoretically to investigate the second-harmonic generation due to interfacial nonlinearity. The structure consists of identical isotropic linear elastic layers that are bonded to each other by spring-type interfaces possessing identical linear normal and shear stiffnesses but different quadratic nonlinearity parameters. Explicit analytical expressions are derived for the second-harmonic amplitudes when a single monochromatic Bloch mode propagates in the structure in arbitrary directions by applying the transfer-matrix approach and the Bloch theorem to the governing equations linearized by a perturbation method. The second-harmonic generation by a single nonlinear interface and by multiple consecutive nonlinear interfaces are shown to be profoundly influenced by the band structure of the layered structure, the fundamental Bloch wave mode, and its propagation direction. In particular, the second harmonics generated at multiple consecutive interfaces are found to grow cumulatively with the propagation distance when the phase matching occurs between the Bloch modes at the fundamental and double frequencies.     

On the elastic properties of model suspensions as investigated by creep recovery measurements in shear

 The elastic properties of model suspensions with spherical monodisperse hydrophilic glass spheres that were dispersed in a Newtonian liquid were determined in creep and creep recovery measurements in shear with a magnetic bearing torsional creep rheometer. The creep and creep recovery measurements were performed depending on the applied level of shear stresses ranging from 0.19 Pa to 200 Pa. Since the recoverable creep compliances of the chosen suspending medium (i.e. a low molecular weight polyisobutylene) were far below the lower limit of the resolution of the creep rheometer it can be considered to behave as purely viscous. By applying a large shear stress in the creep tests the investigated suspensions with a volume fraction of Φ _t =0.35 behave as Newtonian liquids, too. For these suspensions no significant recoverable creep compliances could be detected, as well. In contrast to the Newtonian state of suspensions at high shear stresses, where a shear induced ordering of the particles can be expected, a non-Newtonian behaviour arises by applying a very low shear stress in the creep test. In this state large recoverable creep compliances were detected for the suspensions. The magnitude of the recoverable creep compliances of the suspensions exceeded the largest creep compliances of polymer melts that are reported in the literature by more than two decades. From the results obtained by creep recovery measurements with a magnetic bearing torsional creep rheometer it can clearly be concluded that the particle structure present in the chosen model suspension gives rise to a pronounced elasticity. Received: 21 November 2000 Accepted: 12 July 2001     

Numerical method and models for anti-plane strain of a system with a thin elastic wedge

The paper presents a general method to find asymptotics for a (multi-)wedge system containing a thin wedge. It employs separation of the symmetric and anti-symmetric parts of the boundary displacements and tractions of the wedge. The method is applicable when the angle of the thin wedge turns to zero. A physical interpretation of the derived equations is obtained by using power expansions of non-polynomial functions, which appear after the Mellin transform. We establish that the first term in the expansion of the symmetric part corresponds to shear, while the first term of the anti-symmetric part describes deflection of the wedge axis. Numerical experiments, performed by using a code developed on the basis of the theory, show that using only the first terms of the expansions insignificantly influence accuracy: the approximate results coincide with the exact values of roots to the third significant digit even for the wedge angle of 30°.     

Uniqueness for the solutions of elastic thin plates and shallow shells as well as their contact problem with half space

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｇｏｖｅｒｎｉｎｇｅｑｕａｔｉｏｎｓｆｏｒａｎｉｎｖｅｒｔｅｄｅｌａｓｔｉｃｔｈｉｎｓｈａｌｌｏｗｓｈｅｌｌｏｎｅｌａｓｔｉｃｈａｆｓｐａｃｅｉｎｓｔａｔｉｃｂｅｎｄｉｎｇａｒｅ[1]4-Ｅｉｈｉ2ｋｗ=02=2/ｘ2 2/ｙ2,2ｋ=ｋｙ(2/ｘ2) ｋｘ(2/ｙ2),(1)Ｄｉ4ｗ 2ｋ=ｑ-ｐ,(2)ｗ(ｘ,ｙ)=λｅ 2Ｇｅ4πＧｅ(λｅ Ｇｅ)∫∫ｓｐ(ξ,η)ｄξｄη(ｘ-ξ)2 (ｙ-η)2,(3)ｗｈｅｒｅｉｎＥｑｓ.(1),(2)ａｎｄ(3),ｗ(ｘ,ｙ)ｉｓｔｈｅｄｅｆｌｅｃｔｉｏｎｏｆｔｈｅｓｈｅｌｌ,(ｘ,ｙ)ｉｓｔｈｅｍ…