Two analytical models for the study of periodic fibrous elastic composite with different unit cells

In this work, the effective elastic moduli of two-phase fibrous periodic composites are obtained by means of the Asymptotic Homogenization Method (AHM) and eigenfunction expansion-variational method (EEVM), for different types of parallelogram cells. The constituents exhibit transversely isotropic properties. A doubly periodic parallelogram array of cylindrical inclusions under longitudinal shear is considered. The behavior of the shear elastic coefficient for different geometry arrays of the cell related to the angle of the fibers is studied. Some numerical examples and comparisons with other theoretical results demonstrate that both methods (AHM and EEVM) are efficients for the analysis of composites with presence of rhombic cell. The effect of the configuration of the cells on the shear effective property is observed.     

Dielectric modeling of multiphase composites

In this paper effective medium theory based on Clausius–Mossoti relation was used to predict dielectric properties of multiphase composite system. The composite consisted of E-glass fibers (plain weave S-glass, J.B. Martin) embedded in bisphenol A diglycidylether epoxy matrix (D.E.R. 324, D.O.W. chemicals) with hollow ceramic spherical inclusions (SF 14, P.Q. Corp.) at different volume fractions. In many engineering applications, materials with designed dielectric properties are often sought. An important question in engineering design of a composite material is how the overall properties of the composite depend on those of the individual constituents. Mixing the epoxy resin with hollow ceramic inclusions can effectively reduce dielectric constant of the resin, which is often desirable, rendering composites as good candidates for many applications, especially in telecommunications. Compensation for degradation of mechanical properties of matrix (due to inclusion insertion) is obtained by embedding fibers into the mixture. Measurements of dielectric constant and loss tangent for this multiphase composite system were conducted in the region between 0.1 and 100 kHz range using DEA 290 (T.A. Instruments) dielectric analyzer and experimental results are presented. Good correlation between analytical model and experimental results was observed throughout all frequency range of investigation.     

Micromechanics modeling of viscoelastic properties of hybrid composites with shunted and arbitrarily oriented piezoelectric inclusions

A comprehensive micromechanics model is developed to estimate the effective viscoelastic properties of hybrid composites containing polymer matrix, conductive inclusions and shunted piezoelectric inclusions. The model is derived using the viscoelastic correspondence principle in conjunction with the Mori-Tanaka approach and the orientation averaging scheme. Three dimensional complex moduli are explicitly presented for hybrid composites with any orientation distribution. The model is first validated by comparison with available experimental results. Then, the loss factors are examined for hybrid composites with inclusions of various volume fractions and of shapes ranging from thin disks to long fibers. It is seen that hybrid composites with randomly oriented inclusions exhibit shear loss factors which are not possible with monolithic piezoelectric plate. Furthermore, the numerical results indicate that composites with long spheroid inclusions provide the best damping performance. The results recommend that aligned inclusion composites are good for alleviating longitudinal oscillations. If oscillation energy needs to be dissipated in all directions and for all modes, three dimensional random composites should be used. It is also observed that spherical inclusion composites cannot improve shear damping irrespective of the orientation and the volume fraction. In general, to achieve a pronounced damping piezoelectric inclusions that lie in aspect ratio range 0.1?α?2 should be avoided.     

Micromechanical modeling of imperfect textile composites

After fabrication the carbon-carbon (C/C) plain weave textile composites often show a certain degree of geometrical or material disorder including yarn waviness and misalignment or nesting of individual fiber tows together with intrinsic material porosity observed at all relevant scales. A brief survey of recently developed approaches for estimating overall elastic stiffnesses or thermal conductivities of such composite systems is presented in this paper. Depending on the source and type of available geometrical data the homogenization scheme usually relies either on finite element (FEM) simulations performed for a suitable Periodic Unit Cell (PUC) or employs one of the popular averaging techniques such as the Mori-Tanaka (MT) method. While existing applications of both methodologies are encouraging, there still exists a number of steps to be completed in the course of the future research.     

Influence of parallelogram cells in the axial behaviour of fibrous composite

Effective longitudinal shear moduli closed-form analytical expressions of two-phase fibrous periodic composites are obtained by means of the asymptotic homogenization method (AHM) for a parallelogram array of circular cylinders. This work is an extension of previous reported results, where elastic, piezoelectric and magneto-electro-elastic composites for square and hexagonal arrays with perfect contact were considered. The constituents exhibit transversely isotropic properties. A doubly period-parallelogram array of cylindrical inclusions under longitudinal shear is studied. The behaviour of the anisotropic shear elastic coefficients is studied for several cell geometry arrays. Numerical examples and comparisons with other theoretical results demonstrate that the present model is efficient for the analysis of composites in which the periodic cell is rectangular, rhombic or a parallelogram. The effect of the arrangement of the cells on the shear effective property is discussed. The present method can provide benchmark results for other numerical and approximate methods.     

Micromechanical analysis of effective properties of magneto-electro-thermo-elastic multilayer composites

An exact matrix method, originally proposed for evaluating effective elastic constants of generally anisotropic multilayer composites, is further developed for a micromechanical analysis of multilayers with various coupled physical effects including piezoelectricity, piezomagnetism, thermoelasticity (in consideration of entropy), and the Biot’s poroelasticity. The results for a BaTiO₃-CoFe₂O₄ magneto-electro-thermo-elastic (METE) multilayer coincide with those calculated using other micromechanical models based on the Mori-Tanaka method and the asymptotic homogenization method. It is shown that the present method can efficiently handle the most general type of multilayers with an arbitrary number of general anisotropic layers. Analytical expressions for effective material properties of a transversely isotropic METE multilayer composite are derived, from which those for functionally graded METE multilayers can be directly obtained. The effects of crystallographic orientations and volume fractions of constituting layers on the magnetoelectric coefficients are investigated for BaTiO₃-CoFe₂O₄ and LiNbO₃-CoFe₂O₄ multilayer composites. It is thus demonstrated that the present model can be used for the layout/material optimization of these METE multilayers to obtain a maximum product property such as the magnetoelectric, pyroelectric, and pyromagnetic coefficients. It is also shown that the same method can be used to predict the effective properties of poroelastic multilayers.     

The development of compression damage zones in fibrous composites

Recent experimental work (Narayanan S, Schadler LS. Mechanisms of kink band formation in graphite/epoxy compsites: a micromechanical experimental study. Comp Sci Technol 1999; 59:2201-13) suggests that kink bands in unidirectional continuous carbon fiber reinforced polymer composites initiate from damage zones formed under axial compressive loads. A damage zone consists of a cluster of locally crushed fibers and broken fibers, that are often fractured at an angle, θ > 0°, normal to the fiber axis. Typically, under compressive loads, fiber breaks in damage zones form roughly along a plane at an angle φ, normal to the fiber axis. These damage zones produce stress concentrations which can lead to instabilities in the nearby fiber and matrix and initiate microbuckling and kink bands. This paper extends a micromechanical influence function technique based on earlier shear lag fiber composite models. Our modified technique calculates the fiber axial and matrix shear stress concentrations due to multiple angled and crushed fibers in arbitrary configurations. Modeling reveals that angled or ‘shear’ breaks (θ > 0°) can lead to higher shear stress concentrations in the matrix than transverse breaks (θ=0°). Also we find that the damage zone is more likely to form at an angle φ, which is greater than that of its individual fiber breaks, θ. When φ is slightly greater than θ, the shear stress in the surrounding matrix regions within the damage zone achieves a maximum, potentially weakening the matrix and interface and consequently leading to kink band formation. Monte Carlo simulations incorporating this stress analysis predict that the initiation and propagation of crushed and angled breaks progress roughly along an angle, φ ≈ 17° in a linear elastic system. When possible, our model results are compared to strain measurements of fiber composites under compression obtained by Narayanan and Schadler using micro-Raman spectroscopy (MRS).     

Effective properties of composites with embedded piezoelectric fibres

The present work deals with the modelling of 1–3 and 0–3 composites made of piezoceramic fibres embedded in a soft non-piezoelectric matrix. We especially focus on the longitudinal and transversal effective piezoelectric constants as a function of several micromechanical parameters such as the fibre volume fraction or the aspect ratio. Finite-element results, based on the concept of a cell model, are compared with results of analytical approaches. This study is restricted to linear piezoelectricity and quasistatic cases.     

The effective properties of smart composites with linear coupling behaviors

The analytical expressions for effective thermal, mechanical, electrical, and magnetic moduli of biphasic smart composites with the linear coupling behavior are developed by employing Mori–Tanaka theory and Dunn model, combined with the equivalent inclusion method for an ellipsoidal inclusion embedded in a transversely isotropic medium. These expressions reflected directly the contributions of each phase and that of the coupling taken place between any combinations of thermal, mechanical, electric, and magnetic behavior and provide a physical insight into how a composites consisting of inclusions perform. For a fibrous reinforced smart composite, the explicit expressions for the sum and product properties are derived and are helpful in understanding the coupling behaviors. The effective properties and the coupling effects as a function of microstructure are discussed theoretically and numerically for piezocomposite and piezomagnetic composite. The results are compared with that of the rule of mixture.     

Micromechanical analysis for thermoviscoplastic behavior of unidirectional fibrous composites

A three-dimensional micromechanics formulation for fiber-reinforced composites containing viscoplastic matrix materials is presented. The micromechanics model is based on the relaxation of the coupling effect between the normal and shear stresses. Three variations of Bodner's theory of viscoplasticity are used to predict the thermoviscoplastic behavior of unidirectional metal-matrix composites; first, the original isotropic-hardening model of Bodner and Partom; second, Ramaswamy's extension of the theory through the inclusion of a back stress; and third, Stouffer and Bodner's extension of the theory through a special form of directional hardening. Comparisons with numerical solutions and experimental data of other researchers are made to demonstrate the accuracy of the model. Micromechanical analyses of metal-matrix composites under both in-phase and out-of-phase thermomechanical fatigue-loading conditions are also presented for comparison with experiments and previous models.     

Lifetime distributions for unidirectional fibrous composites under creep-rupture loading

Monte-Carlo simulations and theoretical modeling are used to study the statistical failure modes and associated lifetime distribution of unidirectional 2D and 3D fiber-matrix composites under constant load. Within the composite the fibers weaken over time and break randomly, and the matrix undergoes linear viscoelastic creep in shear. The statistics of fiber failure are governed by the breakdown model of Coleman (1958a), which embodies a Weibull hazard functional of fiber load history imparting power-law sensitivity to fiber load with exponent , and Weibull lifetime characteristics with shape parameter . The matrix has a power-law creep compliance in shear with exponent . Fiber load redistribution at breaks is calculated using a shear-lag mechanics model, which is much more realistic than idealized rules based on equal, global or local load-sharing. The present study is concerned only with the `avalanche' failure regime discussed by Curtin and Scher (1997) which occurs for sufficiently large , and whereby the composite lifetime distribution follows weakest-link scaling. The present Monte-Carlo failure simulations reveal two distinct failure modes within the avalanche regime: For larger , where fiber failure is very sensitive to load level, the weakest link volume fails in a `brittle' manner by the gradual growth of a cluster of mostly contiguous fiber breaks, which then abruptly transitions into a catastrophic crack. For smaller , where this load sensitivity is much less, the weakest link volume shows `tough' behavior, i.e., distributed damage in terms of random fiber failures until the failure of a critical volume and its catastrophic extension. The transition from brittle to tough failure mode for each within the avalanche regime is gradual and depends on the matrix creep exponent and Weibull exponent . Also, as increases above zero the sensitivity of median composite lifetime to load level increasingly deviates from power-law scaling known to occur in the elastic matrix case, =0. By probabilistic modeling of the dominant failure modes in each regime we obtain distribution forms and various scalings for damage growth, and for carefully chosen sets of parameter values we analytically extend simulation results on small composites (limited by current computer power) to more realistic sizes. Our analytical strength distributions are applicable for >2 in 2D, and 4 in 3D. The 2D bound coincides with the avalanche-percolation threshold derived by Curtin and Scher (1997) using entirely different arguments.     

Explicit modeling the progressive interface damage in fibrous composite: Analytical vs. numerical approach

Two micromechanical, representative unit cell type models of fiber reinforced composite (FRC) are applied to simulate explicitly onset and accumulation of scattered local damage in the form of interface debonding. The first model is based on the analytical, multipole expansion type solution of the multiple inclusion problem by means of complex potentials. The second, finite element model of FRC is based on the cohesive zone model of interface. Simulation of progressive debonding in FRC using the many-fiber models of composite has been performed. The advantageous features and applicability areas of both models are discussed. It has been shown that the developed models provide detailed analysis of the progressive debonding phenomena including the interface crack cluster formation, overall stiffness reduction and induced anisotropy of the effective elastic moduli of composite.     

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.     

Micromechanical modeling of fracture initiation in 7050 aluminum

Mechanical testing and finite element calculations have been carried out to characterize the fracture initiation behavior of the high-strength aluminum alloy 7050-T7451. Results show that fracture initiation is well predicted for two specimen types of differing constraint using the stress-modified, critical plastic strain micromechanical model. The relation between stress triaxiality and critical plastic strain was found from a series of notched tensile specimens. Data from these tests are interpreted using both companion finite element modeling and common, semi-empirical relations, and these two approaches are compared. Multiple, interrupted tests of standard, highly constrained single edge notched bend specimens are used to obtain the J–R curve in 7050 for small amounts of tearing to experimentally identify initiation. Companion modeling and the stress-modified, critical plastic strain relation are used to find the length scale for fracture, l^*, needed for initiation predictions. The calibrated stress-modified, critical plastic strain relation and length scale are then used to predict fracture initiation of a low-constraint specimen. The prediction is within 5% of the experimental measurements. Finally, various aspects of the procedure followed in the present work are compared to previous efforts using similar approaches.     

Meshfree modeling and homogenization of 3D orthogonal woven composites

In this paper, evaluation of 3D orthogonal woven fabric composite elastic moduli is achieved by applying meshfree methods on the micromechanical model of the woven composites. A new, realistic and smooth fabric unit cell model of 3D orthogonal woven composite is presented. As an alternative to finite element method, meshfree methods show a notable advantage, which is the simplicity in meshing while modeling the matrix and different yarns. Radial basis function and moving kriging interpolation are used for the shape function constructions. The Galerkin method is employed in formulating the discretized system equations. The numerical results are compared with the finite element and the experimental results.     

Micromechanical deformations in PP/lignocellulosic filler composites: Effect of matrix properties

Polypropylene composites were prepared from three different PP matrices, a homopolymer, a random and a heterophase copolymer, and corn cob to study the effect of matrix characteristics on deformation and failure. The components were homogenized in an internal mixer and compression molded to 1 mm thick plates. Mechanical properties were characterized by tensile testing, while micromechanical deformations by acoustic emission measurements and fractography. The results proved that the dominating micromechanical deformation process may change with matrix properties. Yield stress determined from the stress vs. strain traces may cover widely differing processes. Debonding is the dominating process when the adhesion of the components is poor, while matrix yielding and/or filler fracture dominate when adhesion is improved by the introduction of a functionalized polymer. The dominating deformation mechanism is determined by component properties and adhesion. Interfacial adhesion, matrix yield stress and the inherent strength of the reinforcement can be limiting factors in the improvement of composite strength. The properties of polymer composites reinforced with lignocellulosic fillers are determined by micromechanical deformation processes, but they are independent of the mechanism of these processes.     

Three-dimensional analysis of load transfer micro-mechanisms in fibre/matrix composites

This study gives a detailed analysis of load distributions around fibre breaks in a composite. In contrast to other studies reported in the literature, the analysis considers different configurations of composite damage from the failure of a few to the failure of many fibres. The model considers three types of matrix behaviours (elastic, elastic–plastic and viscoelastic) with or without debonding at the broken fibre/matrix interface. In this way, the usual limitations of the finite element approach are overcome so as to take into account the number and interactions of broken fibres whilst maintaining an evaluation of the various fields (stresses in particular).     

A comparative study of the load transfer mechanisms of the carbon nanotube reinforced polymer composites with interfacial crystallization

With the help of shear-lag theory, load transfer analysis is performed on the carbon nanotube reinforced polymer composites with interfacial crystallization of different morphologies, including transcrystallinity layer (TCL) and nanohybrid shish-kebab (NHSK) structures. By comparison, we find that the TCL structures can ease the burden of the CNT while the NHSK structures can lead to a fluctuating distribution of the axial stress in the CNT. Both structures can improve the effective elastic modulus of the composites, though the effect of the TCL structures is more pronounced. Besides, the enhancement of the load transfer efficiency of the composites is also observed, the study of the interfacial stress on different kinds of interfaces shows that the reinforcing effect of the TCL structures is sensitive to both the CNT/crystalline polymer interface and crystalline polymer/amorphous polymer interface, while the major decisive factor for the NHSK structures is confined to be the CNT/crystalline polymer interface because of the interlocking effect. Based on these features, some suggestions are given for tailoring the high-performance carbon nanotube reinforced polymer composites.     

Modeling damage and load redistribution in composites under tension–tension fatigue loading

A fatigue model developed for composite laminates and based on the cycle-by-cycle probability of failure has been modified to account for damage creation and evolution and its effect on cycles to failure. The residual strength of different parts of the laminate is determined during cyclic loading and damage such as matrix cracking is quantified along with its effect on load redistribution and cycles to failure of different parts of the laminate. The model does not require any curve fitting or experimentally measured data other than basic material static strength values and their associated experimental scatter. The model is applied to uni-directional and cross-ply laminates. A stress-based approach using energy minimization and calculus of variations is used. The model predictions range from fair to excellent.