An elastoplastic multi-level damage model for ductile matrix composites considering evolutionary weakened interface

An elastoplastic multi-level damage model considering evolutionary weakened interface is developed in this work to predict the effective elastoplastic behavior and multi-level damage evolution in particle reinforced ductile matrix composites (PRDMCs). The elastoplastic multi-level damage model is micromechanically derived on the basis of the ensemble-volume averaging procedure and the first-order effects of eigenstrains. 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 particles having mildly or severely weakened interface, and a multi-level damage model [Lee, H.K., Pyo, S.H., in press. Multi-level modeling of effective elastic behavior and progressive weakened interface in particulate composites. Composites Science and Technology] in accordance with the Weibull’s probabilistic function is employed to describe the sequential, progressive weakened interface in the composites. Numerical examples corresponding to uniaxial, biaxial and triaxial tension loadings are solved to illustrate the potential of the proposed micromechanical framework. A series of parametric analysis are carried out to investigate the influence of model parameters on the progression of weakened interface in the composites. Furthermore, the present prediction is compared with available experimental data in the literature to verify the proposed elastoplastic multi-level damage model.     

An elastoplastic damage model for metal matrix composites considering progressive imperfect interface under transverse loading

An elastoplastic damage model considering progressive imperfect interface is proposed to predict the effective elastoplastic behavior and multi-level damage progression in fiber-reinforced metal matrix composites (FRMMCs) under transverse loading. The modified Eshelby’s tensor for a cylindrical inclusion with slightly weakened interface is adopted to model fibers having mild or severe imperfect interfaces [Lee, H.K., Pyo, S.H., 2009. A 3D-damage model for fiber-reinforced brittle composites with microcracks and imperfect interfaces. J. Eng. Mech. ASCE. doi:10.1061/(ASCE)EM.1943-7889.0000039]. An elastoplastic model is derived micromechanically on the basis of the ensemble-volume averaging procedure and the first-order effects of eigenstrains. A multi-level damage model [Lee, H.K., Pyo, S.H., 2008a. Multi-level modeling of effective elastic behavior and progressive weakened interface in particulate composites. Compos. Sci. Technol. 68, 387–397] in accordance with the Weibull’s probabilistic function is then incorporated into the elastoplastic multi-level damage model to describe the sequential, progressive imperfect interface in the composites. Numerical examples corresponding to uniaxial and biaxial transverse tensile loadings are solved to illustrate the potential of the proposed micromechanical framework. A series of parametric analysis are carried out to investigate the influence of model parameters on the progression of imperfect interface in the composites. Furthermore, a comparison between the present prediction and experimental data in the literature is made to assess the capability of the proposed micromechanical framework.     

On the overall elastic moduli of composites with spherical coated fillers

A micromechanical approach is presented to estimate the overall linear elastic moduli of three phase composites consisting of two phase coated spherical particles randomly dispersed in a homogeneous isotropic matrix. The theoretical method is based on Eshelby’s equivalent inclusion method and its recent extension by Shodja and Sarvestani [J. Appl. Mech. 68 (2001) 3] to evaluate the local field variables in case of double (multi) inhomogeneities. Using Tanaka–Mori theorem [J. Elasticity 2 (1972) 199] and a decomposition of Green’s function integral equation, the pair-wise average phase values of stress and strain in two interacting coated particles are estimated. Following Ju and Chen [Acta Mech. 103 (1994) 103; Acta Mech. 103 (1994) 123] the ensemble phase volume average of stress and strain fields can be evaluated within a representative volume element containing a finite number of coated particles. Comparisons with classical bounds are presented to illustrate the accuracy of the proposed method.     

On micromechanical modeling of particulate composites with inclusions of various shapes

The effective elastic properties of statistically homogeneous two-phase particulate composites are considered. Several first-order micromechanical models are re-written in terms of the inclusion compliance contribution tensor (H-tensor). This tensor is a convenient tool to evaluate contribution of arbitrarily shaped inclusions and cavities to the overall composite properties.For any inclusion shape, the procedure starts with calculation of the H-tensor for a single inclusion. The non-interaction approximation is obtained by direct summation. More advanced micromechanical schemes are derived by substituting the non-interaction inclusion compliance contribution tensor into the formulae provided in the paper. The proposed procedure is illustrated by considering several two-dimensional and three-dimensional examples.     

THE VALIDITY OF THE MODIFIED MORI-TANAKA METHOD FOR COMPOSITES WITH SLIGHTLY WEAKENED INTERFACE

This paper presents a direct Mori-Tanaka approach to calculate the effective moduli ofparticle-reinforced composites and fiber-reinforced composites with spring like imperfect interfaces.Bya comparison between these results and those obtained from the approximate Mori-Tanaka method de-veloped by Qu for composites with slightly weakened interface,the validity of the Qu's method is re-vealed.     

Micromechanics-based constitutive modeling for unidirectional laminated composites

A micromechanics-based constitutive model is developed to predict the effective mechanical behavior of unidirectional laminated composites. A newly developed Eshelby’s tensor for an infinite circular cylindrical inclusion [Cheng, Z.Q., Batra, R.C., 1999. Exact Eshelby tensor for a dynamic circular cylindrical inclusion. J. Appl. Mech. 66, 563–565] is adopted to model the unidirectional fibers and is incorporated into the micromechanical framework. The progressive loss of strength resulting from the partial fiber debonding and the nucleation of microcracks is incorporated into the constitutive model. To validate the proposed model, the predicted effective stiffness of transversely isotropic composites under far field loading conditions is compared with analytical solutions. The constitutive model incorporating the damage models is then implemented into a finite element code to numerically characterize the elastic behavior of laminated composites. Finally, the present predictions on the stress–strain behavior of laminated composite plate containing an open hole is compared with experimental data to verify the predictive capability of the model.     

A continuum micromechanical theory of overall plasticity for particulate composites including particle size effect

Classical continuum micromechanics cannot predict the particle size dependence of the overall plasticity for composite materials, a simple analytical micromechanical method is proposed in this paper to investigate this size dependence. The matrix material is idealized as a micropolar continuum, an average equivalent inclusion method is advanced and the Mori–Tanaka's method is extended to a micropolar medium to evaluate the effective elastic modulus tensor. The overall plasticity of composites is predicted by a new secant moduli method based on the second order moment of strain and torsion of the matrix in a framework of micropolar theory. The computed results show that the size dependence is more pronounced when the particle's size approaches to the matrix characteristic length, and for large particle sizes, the prediction coincides with that predicted by classical micromechanical models. The method is analytical in nature, and it can capture the particle size dependence on the overall plastic behavior for particulate composites, and the prediction agrees well with the experimental results presented in literature. The proposed model can be considered as a natural extension of the widely used secant moduli method from a heterogeneous Cauchy medium to a micropolar composite.     

Micromechanical modeling of packing and size effects in particulate composites

This paper is devoted to the introduction of packing and size effects in micromechanical predictions of the overall elastic moduli of particulate composite materials. Whereas micromechanical models derived from the classical ‘point approach’ are known to be unable to model such effects, it is shown that the so-called ‘morphologically representative pattern-based approach’ (MRP-based approach) offers new means of taking some geometrical parameters into account such as the mean distance between nearest-neighbor particles or their size, so as to predict the dependence of the overall moduli on these parameters, at least in a relative way. Moreover, when internal lengths, such as the thickness of interphase shells of coated particles, are introduced, absolute size effects can be predicted as well. Illustrative applications are reported in view of comparisons between such new treatments and the predictions of some classical models which are shown to coincide with the ones derived from MRP-based models in definite limiting cases only.     

Elastoplastic modeling of circular fiber-reinforced ductile matrix composites considering a finite RVE

A micromechanical elastoplastic damage model considering a finite RVE is proposed to predict the overall elastoplastic damage behavior of circular fiber-reinforced ductile (matrix) composites. The constitutive damage model proposed in our preceding work (Kim and Lee, 2009) considering a finite Eshelby’s tensor (Li et al., 2005, Wang et al., 2005) is extended to accommodate the elastoplastic behavior of the composites. On the basis of the exterior-point Eshelby’s tensor for circular inclusions and the ensemble-averaged effective yield criterion, a micromechanical framework for predicting the effective elastoplastic damage behavior of ductile composites is derived. A series of numerical simulations are carried out to illustrate stress–strain response of the proposed micromechanical framework and to examine the influence of a Weibull parameter on the elastoplastic behavior of the composites. Furthermore, comparisons between the present predictions and experimental data available in the literature are made to further assess the predictive capability of the proposed model.     

Modeling the macroscopic behavior of two-phase nonlinear composites by infinite-rank laminates

A new approach is proposed for estimating the macroscopic behavior of two-phase nonlinear composites with random, particulate microstructures. The central idea is to model composites by sequentially laminated constructions of infinite rank whose macroscopic behavior can be determined exactly. The resulting estimates incorporate microstructural information up to the two-point correlation functions, and require the solution to a Hamilton–Jacobi equation with the inclusion concentration and the macroscopic fields playing the role of ‘time’ and ‘spatial’ variables, respectively. Because they are realizable, by construction, these estimates are guaranteed to be convex, to satisfy all pertinent bounds, to exhibit no duality gap, and to be exact to second order in the heterogeneity contrast. Sample results are provided for two- and three-dimensional power-law composites, and are compared with other homogenization estimates, as well as with numerical simulations available from the literature. The estimates are found to give physically sensible predictions for all the cases considered, even for extreme values of the nonlinearity and heterogeneity contrast. Interestingly, in the case of isotropic porous materials under hydrostatic loadings, the estimates agree exactly with standard Gurson-type models for viscoplastic porous media.     

Modeling of anisotropic polydispersed composites by progressive micromechanical approach

A progressive micromechanical method is presented in order to predict the elastic constants of polydispersed composites including multi-directional or randomly oriented reinforcement particles. Heterogeneities of various types are introduced into the matrices in a gradual manner. At each step, the Mori-Tanaka method is used to obtain the stiffness tensor of the intermediate medium used as a matrix of the following step. The proposed method is capable of introducing any kind of heterogeneities based on their dimensions, orientations, mechanical properties, and volume fractions to the matrix. Furthermore, suitable probability density functions can be defined for physical and structural parameters of the composite, including the level of the filler-matrix interfacial bonding, the aspect ratio, and the orientation of reinforcement particles. The efficiency of the iterative approach and the convergence of the solution are studied by computing the stiffness tensors of unidirectional and bidirectional particulate composites. The results of the present study are also compared with the literature data for a randomly oriented particulate composite.     

The asymptotic homogenization elasticity tensor properties for composites with material discontinuities

The classical asymptotic homogenization approach for linear elastic composites with discontinuous material properties is considered as a starting point. The sharp length scale separation between the fine periodic structure and the whole material formally leads to anisotropic elastic-type balance equations on the coarse scale, where the arising fourth rank operator is to be computed solving single periodic cell problems on the fine scale. After revisiting the derivation of the problem, which here explicitly points out how the discontinuity in the individual constituents’ elastic coefficients translates into stress jump interface conditions for the cell problems, we prove that the gradient of the cell problem solution is minor symmetric and that its cell average is zero. This property holds for perfect interfaces only (i.e., when the elastic displacement is continuous across the composite’s interface) and can be used to assess the accuracy of the computed numerical solutions. These facts are further exploited, together with the individual constituents’ elastic coefficients and the specific form of the cell problems, to prove a theorem that characterizes the fourth rank operator appearing in the coarse-scale elastic-type balance equations as a composite material effective elasticity tensor. We both recover known facts, such as minor and major symmetries and positive definiteness, and establish new facts concerning the Voigt and Reuss bounds. The latter are shown for the first time without assuming any equivalence between coarse and fine-scale energies (Hill’s condition), which, in contrast to the case of representative volume elements, does not identically hold in the context of asymptotic homogenization. We conclude with instructive three-dimensional numerical simulations of a soft elastic matrix with an embedded cubic stiffer inclusion to show the profile of the physically relevant elastic moduli (Young’s and shear moduli) and Poisson’s ratio at increasing (up to 100 %) inclusion’s volume fraction, thus providing a proxy for the design of artificial elastic composites.     

Vibration of two bonded composites: effects of the interface and distinct periodic structures

Small elastic vibrations of two particulate composites that are caused by a non-plane time-harmonic wave are investigated. Effects of the adhesive interface and distinct periodic structures on the transmission and reflection of acoustic waves are rigorously analyzed. A two-scale asymptotic expansion with interfacial correctors is introduced to account for the macro- and micromechanical effects on wave propagation. An efficient algorithm is developed for computing first and second order corrections for the coefficients that depend on the composites microstructure and the interfacial constraint.     

Discrete modeling of crack bridging by a discontinuous platelet with a controlled interface

Crack bridging by discontinuous fibers can make brittle materials tougher by transferring stresses from the crack tip to elsewhere in the matrix material. One important aspect of crack bridging is the nature of the interface between the fibers and the matrix material. In this paper, a two-dimensional numerical model of bridging a Mode I loaded crack by linear elastic discontinuous platelets is developed for two different types of interfaces. The first type is a perfectly bonded interface. The second type is an imperfect interface described as a stick–slip interface. A shear-lag model to predict platelet pullout is developed in detail to verify the numerical implementation of the stick–slip interface. An example of a crack tip bridged by a platelet is examined for both interfaces. The perfectly bonded interface will reduce the Stress Intensity Factor (SIF) of the crack greatly but introduces new stress concentrations at the platelet ends. The stick–slip interface can be tailored to also reduce the SIF while not introducing new stress concentrations.     

Modelling of mechanical behaviour of damageable particulate composites

A simple discrete structural model has been developed [V.V. Moshev, L.A. Golotina, Non-linear mechanical models for particulate elastometric composites III. Macrocrack initiation from randomly scattered microdamages, Ukrainian Polymer Journal 4(1-2) (1995) 70–84.] for particular composites; it is capable of describing the entire life-cycle of the material from its virgin state through microdamage accumulation to macrofailure. The model geometry corresponds to chain-like cross-sections tied in series where each cross-section represents two clamps interconnected in parallel by a number of elastic links. In extension, some links become ruptured in a random manner reflecting microdamage accumulation which tends to enhance longitudinal stiffness nonuniformity. A stage is reached, when the specimen looses its longitudinal elastic stability. Sudden rupture occurs at the most vulnerable location. The first version of the model was developed for a set of material parameters that were chosen arbitrarily. The phenomena of damage accumulation and macrocrack formation were well represented qualitatively. Examination of random structures of particulate composites and application of a physical discretization approach [V.V. Moshev, O.C. Garishin, Physical discretization approach to evaluation of elastic moduli of highly filled granular composites, Int. J. Solids and Structures 30 (17) (1993) 2347–2355.], however, have suggested the possibility to a further refinement of the model with more realistic structural features. The objective of this work is to provide such a refinement.     

Remarks on formulating an adhesion problem using Euler’s elastica (draft)

Three formulations for the problem of an elastica adhering to a rigid surface are discussed and compared. These include stationary principles, the surface integral of Eshelby’s energy-momentum tensor, and the material (configurational) force balance. The configuration at static equilibrium is predicted in closed form for a pair of structures that arise in nano- and microscale applications.     

Application of results on Eshelby tensor to the determination of effective poroelastic properties of anisotropic rocks-like composites

The present work is devoted to the determination of the macroscopic poroelastic properties of anisotropic elastic porous materials saturated by a fluid under pressure. It makes use of the theoretical results provided by Withers [Withers, P.J., 1989. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Philosophical Magazine A 59 (4), 759–781.] for the problem of an ellipsoidal inclusion embedded in a transversely isotropic elastic medium. The particular case of a spherical inclusion is very important for rock-like composites such as argillite and shales. The implementation of these results in a micromechanical theory of poroelasticity allows to quantify the effects of the solid matrix anisotropy and of pore space on the effective poromechanical properties. Closed form expressions of Biot tensor and of Biot modulus are presented as well as numerical applications for anisotropic shales.     

Effective moduli for micropolar composite with interface effect

Size-dependence is well observed for metal matrix composites, however the classical micromechanical model fails to describe this phenomenon. There are two different ways to consider this size-dependency: the first approach is to include the nonlocal effect by idealizing the matrix material as a high order continuum (e.g., micropolar or strain gradient); the second is to take into account the interface effect. In this work, we combine these two approaches together by introducing the interface effect into a micropolar micromechanical model. The interface constitutive relations and the generalized Young–Laplace equation for micropolar material model are firstly presented. Then they are incorporated into the micropolar micromechanical model to predict the effective bulk and shear moduli of a fiber-reinforced composite. Two intrinsic length scales appear: one is related to the microstructure of the matrix material, the other comes from the interface effect. The size-dependent effective moduli due to the nonlocal effect and interface effect can be synchronized or desynchronized for nanosize fibers, depending on the nature of the interface. For the relatively large fiber size, the size-dependence is dominated by the nonlocal effect. As expected, when the fiber size tends to infinity, classical result can be recovered.     

Effective elastic properties of solids with defects of irregular shapes

Pores and defects in real materials often have very irregular shapes. Thus, micromechanical modeling based on the analytical solutions of elasticity becomes inapplicable. The objective of this paper is to present a computational procedure to calculate the contribution of the irregularly shaped defects into the effective moduli of two-dimensional elastic solids. In this procedure, the cavity compliance tensor is constructed numerically for an individual defect, and then used in the elastic potential-based approach to predict the effective moduli of porous solids. Two computational methods are used in this paper to calculate the components of a cavity compliance tensor: finite element analysis and numerical conformal mapping. Application of this procedure to the regular hole shapes produces results that are in good correspondence with analytical predictions.     

Electromechanical response of (3–0, 3–1) particulate,fibrous, and porous piezoelectric composites with anisotropic constituents: A model based on the homogenization method

An analytical framework based on the homogenization method has been developed to predict the effective electromechanical properties of periodic, particulate and porous, piezoelectric composites with anisotropic constituents. Expressions are provided for the effective moduli tensors of n-phase composites based on the respective strain and electric field concentration tensors. By taking into account the shape and distribution of the inclusion and by invoking a simple numerical procedure, solutions for the electromechanical properties of a general anisotropic inclusion in an anisotropic matrix are obtained. While analytical forms are provided for predicting the electroelastic moduli of composites with spherical and cylindrical inclusions, numerical evaluation of integrals over the composite microstructure is required in order to obtain the corresponding expressions for a general ellipsoidal particle in a piezoelectric matrix. The electroelastic moduli of piezoelectric composites predicted by the analytical model developed in the present study demonstrate excellent agreement with results obtained from three-dimensional finite-element models for several piezoelectric systems that exhibit varying degrees of elastic anisotropy.