A self-consistent approach to dynamic effective properties of a composite reinforced by distributed spherical particles

Summary A self-consistent approach to dynamic effective properties of a composite reinforced by randomly distributed spherical inclusions is studied. The coherent plane waves propagating through the particle-reinforced composite are of attenuation nature. It implies that there is an analogy between the particle-reinforced composite and the effective medium with complex-valued elastic constants from the viewpoints of wave propagation. A composite sphere consisting of the inclusion, the matrix and the interphase between them is assumed embedded in the effective medium. The effective wavenumbers of the coherent plane waves propagating through the particle-reinforced composite are obtained by the dynamic self-consistent conditions which require that the forward scattering amplitudes of such a composite sphere embedded in the effective medium are equal to zero. The dynamic effective properties (effective phase velocity, effective attenuation and effective elastic constants) obtained by the present dynamic self-consistent approach for SiC-Al composites are compared numerically with that obtained by the effective field approach at various volume concentrations. It is found that there is a good agreement between the two approaches at a relatively low frequency and low volume concentration but the numerical results deviate from each other at a relatively high frequency and high volume concentration.     

Ultrasonic measurements of microelectronic resins

The elastic moduli, measured with the ultrasonic technique, of commercial silica filled epoxy resins used in the electronic circuits are reported. Measurements of velocity propagation and attenuation were carried out in large temperature and frequency ranges. Predictions of the theoretical models were compared with the experimental values. Explicit expressions of the elastic moduli were derived as functions of filler content and the properties of the matrix and the fillers. The influences of frequency and temperature on the elastic moduli and attenuation are discussed.     

Propagation of low-frequency elastic waves in double-porositymedia containing viscoelastic component in the pores

A self-consistent scheme named the effective field method (EFM) is applied for the calculation of the velocities and quality factors of elastic waves propagating in double-porosity media. A double-porosity medium is considered to be a heterogeneous material composed of a matrix with primary pores and inclusions that are represent by flat (crack-like) secondary pores. The prediction of the effective viscoelastic moduli consists of two steps. First, we calculate the effective viscoelastic properties of the matrix with the primary small-scale pores (matrix homogenization). Then, the porous matrix is treated as a homogeneous isotropic host where the large-scale secondary pores are embedded. Spatial distribution of inclusions in the medium is taken into account via a special two-point correlation function. The results of the calculation of the viscoelastic properties of double-porosity media containing isotropic fields of crack-like inclusions and double-porosity media with some non-isotropic spatial distributions of crack-like inclusions are presented.     

Eshelby tensors for a spherical inclusion in microelongated elastic fields

Eshelby tensors are found for a spherical inclusion in a microelongated elastic field. Here, a special micromorphic model is introduced to describe the damaged material which defines the damage as the formation and the growth of microcracks and microvoids occurred in the material at the microstructural level. To determine the new material coefficients of the model, an analogy is established between the damaged body and the composite materials and then Mori–Tanaka homogenization technique is considered to obtain overall material moduli. Following this idea, the determination of the Eshelby tensors which establish the relation between the strains of the matrix material and of the inclusion becomes the first task. Introducing the concept of eigenstrain and microeigenstrain, the general constitutive theory is given for a homogeneous isotropic centrosymmetric microelongated media with defects. Then by the use of Green’s functions, micro and macro elastic fields are presented for the case of spherical inclusions embedded in an infinite microelongated material. Thus, the Eshelby tensors are obtained for a microelongated elastic field with a spherical inclusion and it is also shown that the classical Eshelby tensors can be obtained as a limit case of the microelongation.     

Dynamic mechanical response of elastic spherical inclusions to impulsive acoustic radiation force excitation

Acoustic radiation force impulse imaging has been used clinically to study the dynamic response of lesions relative to their background material to focused, impulsive acoustic radiation force excitations through the generation of dynamic displacement field images. Dynamic displacement data are typically displayed as a set of parametric images, including displacement immediately after excitation, maximum displacement, time to peak displacement, and recovery time from peak displacement. To date, however, no definitive trends have been established between these parametric images and the tissues' mechanical properties. This work demonstrates that displacement magnitude, time to peak displacement, and recovery time are all inversely related to the Young's modulus in homogeneous elastic media. Experimentally, pulse repetition frequency during displacement tracking limits stiffness resolution using the time to peak displacement parameter. The excitation pulse duration also impacts the time to peak parameter, with longer pulses reducing the inertial effects present during impulsive excitations. Material density affects tissue dynamics, but is not expected to play a significant role in biological tissues. The presence of an elastic spherical inclusion in the imaged medium significantly alters the tissue dynamics in response to impulsive, focused acoustic radiation force excitations. Times to peak displacement for excitations within and outside an elastic inclusion are still indicative of local material stiffness; however, recovery times are altered due to the reflection and transmission of shear waves at the inclusion boundaries. These shear wave interactions cause stiffer inclusions to appear to be displaced longer than the more compliant background material. The magnitude of shear waves reflected at elastic lesion boundaries is dependent on the stiffness contrast between the inclusion and the background material, and the stiffness and size of the inclusion dictate when shear wave reflections within the lesion will interfere with one another. Jitter and bias associated with the ultrasonic displacement tracking also impact the estimation of a tissue's dynamic response to acoustic radiation force excitation.     

Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures

The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions centered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix-vector products with such matrices may be calculated by the fast fourier transform technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are performed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly distributed inclusions allows predicting the composite effective elastic constants with sufficient accuracy.     

Dynamic mechanical and ultrasonic properties of polyurea

Dynamic mechanical analysis (DMA) and ultrasonic measurements were carried out to study the temperature and frequency dependences of viscoelastic properties of polyurea. Master curves of Young’s storage and loss moduli were developed from the DMA data. Relaxation spectra were subsequently calculated by means of two approximate models, and the apparent activation energy of molecular rearrangements was also determined based on the temperature dependence of the time-temperature shift factor. Velocity and attenuation of longitudinal and shear ultrasonic waves in polyurea were measured in the 0.5-2 MHz frequency range between −60 and 30 °C temperatures. The complex longitudinal and shear moduli were computed from these measurements. Combining these results provided an estimate of the complex bulk and Young’s moduli at high frequencies. The results of the DMA and temperature and frequency shifted ultrasonic measurements are compared and similarities and deviations are discussed.     

On application of classical Eshelby approach to calculating effective elastic moduli of dispersed composites

The problem of finding effective elastic moduli of media with spheroid inclusions in case of small concentration of these inclusions is addressed. A number of particular solutions, both known and new, were obtained as limit transitions and asymptotical expansion of the general solution, based on Eshelby’s approach. A special attention was paid to determining the ranges of applicability of the obtained asymptotical solutions. It was shown that for spheroid inclusions the areas of applicability of the asymptotic solutions are determined by two parameters: the ratio of elastic moduli of the inclusion and the matrix and aspect ratio of the inclusions.     

Dispersion relations and modes of wave propagation in inclusion-reinforced composite plates

This paper is intended to examine the effect of inclusion shapes, inclusion contents, inclusion elastic constants, and plate thickness on the dispersion relations and modes of wave propagation in inclusion-reinforced composite plates. The shape of inclusion is modeled as spheroid that enables the composite reinforcement geometrical configurations ranging from sphere to short and continuous fiber. Mori–Tanaka mean-field theory is used to predict the effective elastic moduli of the composite plate explicitly. The effective elastic moduli are able to elucidate the effect of inclusion’s shape, stiffness, and volume fraction on the composite’s anisotropic elastic behavior. The resulting moduli are then used to determine the dispersion relations and the modal patterns of Lamb waves using the dynamic stiffness matrix method. The types (symmetric or antisymmetric) of Lamb waves in an isotropic plate can be classified according to the wave motions are symmetrical or antisymmetric about the midplane of the plate. Classifying the wave type in an anisotropic plate is not as simple as that in an isotropic plate, and has not received proper attention in the literature. The wave types and orders are identified by analyzing the dispersion curves and inspecting the calculated modal patterns, and the results indicate that the Lamb waves in an orthotropic composite plate can also be classified as either symmetric or antisymmetric waves. It is also found that the inclusion contents, aspect ratios and plate thickness affect propagation velocities, higher-order mode cutoff frequencies, and modal patterns. Propagation speed is generally increased with the aspect ratio, e.g., using longer fibers generally results in a higher propagation speed.     

Analysis on the variances of material and structural properties based on random field theory

Based on the random field theory (RFT) and the stochastic finite element method (SFEM), the variances of the mechanical properties of materials and structures are studied. Manufacturing processes can easily lead to the spatial variations of the load and the material properties such as moduli and density. Characterizing the elastic moduli, load and density with one-dimensional random fields, the analytical solutions for the coefficient of variations (COVs) of effective material moduli, displacement and natural frequencies of beams are obtained. Then, with the fiber and matrix properties, volume fraction modeled by two-dimensional random fields and the fiber angle as a single random variable, a Monte Carlo simulation (MCS) is performed to generate the variances of effective modulus of fiber-reinforced composite laminar plate. Compared with the previous numerical conclusions, the present results reveal that the variances of effective material properties and structural displacement are greatly dependent on both the random fields and the sizes of structures in theory.     

High-spatial-resolution sub-surface imaging using a laser-based acoustic microscopy technique

Scanning acoustic microscopy techniques operating at frequencies in the gigahertz range are suitable for the elastic characterization and interior imaging of solid media with micrometer-scale spatial resolution. Acoustic wave propagation at these frequencies is strongly limited by energy losses, particularly from attenuation in the coupling media used to transmit ultrasound to a specimen, leading to a decrease in the depth in a specimen that can be interrogated. In this work, a laser-based acoustic microscopy technique is presented that uses a pulsed laser source for the generation of broadband acoustic waves and an optical interferometer for detection. The use of a 900-ps microchip pulsed laser facilitates the generation of acoustic waves with frequencies extending up to 1 GHz which allows for the resolution of micrometer-scale features in a specimen. Furthermore, the combination of optical generation and detection approaches eliminates the use of an ultrasonic coupling medium, and allows for elastic characterization and interior imaging at penetration depths on the order of several hundred micrometers. Experimental results illustrating the use of the laser-based acoustic microscopy technique for imaging micrometer-scale subsurface geometrical features in a 70-μm-thick single-crystal silicon wafer with a (100) orientation are presented.     

Effective moduli of hyperelastic porous media at large deformation

Summary.  Among the parameters affecting the overall material properties of porous media, the most significant involve the micromechanical morphology, the matrix material behavior and the applied load range. Considering a unit cell for the porous medium, several approaches of the material response are developed, which yield the effective properties of the medium. Numerical results are presented and compared with experimental or analytical data available in literature. Proposed formulations impose several material characterizations ranging from linear elastic to incompressible hyperelastic. In the case of nonlinear materials, a special formulation has been developed permitting prediction of the porous material moduli. This formulation considers a special nonlinear form for the strain energy function under specific loading conditions. The proposed method yields simple formulas approximating the effective moduli of porous media, which are useful for design purposes. Received August 3, 2001; revised August 14, 2002 Published online: January 16, 2003 Acknowledgements The first of the authors is grateful to his mentor Dr. Paul J. Blatz for his encouragement all these years for continuous research on the nonlinear theories of hyperelastic materials.     

Dispersion of elastic waves in periodically inhomogeneous media

Propagation of time-harmonic elastic waves through periodically inhomogeneous media is considered. The material inhomogeneity exists in a single direction along which the elastic waves propagate. Within the period of the linear elastic and isotropic medium, the density and elastic modulus vary either in a continuous or a discontinuous manner. The continuous variations are approximated by staircase functions so that the generic problem at hand is the propagation of elastic waves in a medium whose finite period consists of an arbitrary number of different homogeneous layers. A dynamic elasticity formulation is followed and the exact phase velocity is derived explicitly as a solution in closed form in terms of frequency and layer properties. Numerical examples are then presented for several inhomogeneous structures.     

Local Electroelastic Field and Effective Electroelastic Moduli of Piezoelectric Nanocomposites with Interface Effect

Due to the large ratio of surface area to volume in nanoscale objects, the property of surfaces and interfaces likely becomes a prominent factor in controlling the behavior of nano-heterogeneous materials. In this work, based on the Gurtin-Murdoch surface/interface elastic theory, a distinct expression is derived for embedded nano-inclusion in an infinite piezoelectric matrix coupled with interface effect. For the problem of a spherical inclusion in transversely isotropic piezoelectric medium, we reach a conclusion that the elastic and electric field are uniform when eigen-strain and eigen-electric field imposed on the inclusion are uniform even in the presence of the interface influence. The electroelastic fields in the inclusion are related to both interface electroelastic parameters and the radius of the inclusion. Then overall properties of the composites are estimated by using the dilute distribution model. Numerical results reveal that the effective electroelastic moduli are function of the interface parameters and the size of the nano-inhomogeneities.     

Nonlinear wave scattering at the flexible interface of a granular dimer chain

Uncompressed granular dimer chains composed of repetitive pairs of heavy-light spherical, linearly elastic beads exhibit interesting intrinsic responses. The dynamics of these highly discontinuous nonlinear media is governed by the mass ratio scaling the mass disparity of each heavy-light pair of beads. In particular, it has been theoretically and experimentally shown that they support countable infinities of anti-resonances at a discrete set of mass ratios leading to solitary pulses propagating through the dimers with no attenuation or distortion. Conversely, they support countable infinities of resonances at a different discrete set of mass ratios, leading to substantial and rapid attenuation of propagating pulses due to energy scattering from low-to-high frequencies and wavenumbers by means of radiating traveling waves. In this work we computationally study nonlinear scattering of impeding pulses at the interface of an impulsively excited dimer chain with a dispersive elastic boundary, namely, a finite linear string resting on an elastic foundation. We develop a computational algorithm which, through iteration and interpolation at successive time steps, accurately computes (and ensures convergence of) the highly discontinuous contact forces and displacements at the flexible interface of the granular medium. This enables accurate computation of wave transmission, reflection, localization or multi-scale nonlinear scattering at the flexible interface for varying mass ratios of the dimer and the interface parameters. We show that, depending on the mass ratio of the dimer and the stiffness of the elastic foundation, the nonlinear scattering at the flexible interface may lead to significant reduction of the maximum contact force at the interface, and, thus, drastically affect the transmitted and reflected energy at the flexible boundary. In fact, an inverse relation between the stiffness of the elastic foundation and the residual energy transferred from the dimer chain to the flexible boundary is found. Moreover, for sufficiently small mass ratios of the dimer chain transient breathers are realized close to the interface in the form of localized “fast” oscillations of light granules of the dimer that entrap shock energy and then release in a slow time scale back to the chain and the flexible boundary. This work paves the way for studying highly discontinuous and nonlinear scattering phenomena at interfaces of granular media with flexible continua.     

Elastic Constants of Particle and Fiber Reinforced Metal Matrix Composites

Abstract

A model has been developed to predict the elastic moduli in composites reinforced with both particles and fibers. In the model the matrix material and the particles, which are assumed to be homogeneously distributed, form an effective matrix. The characteristics of this effective matrix is calculated using a theory formulated by Ledbetter and Datta. The effective matrix is then considered to be reinforced with fibers lying in one plane but randomly oriented in that plane. The effect of the 2-dimensionally random orientation of the fibers on the elastic moduli of the composites is determined in two steps. First the composite cylinders model by Hashin and Rosen for an aligned fiber system is employed, and then a geometric averaging procedure suggested by Christensen and Waals is performed. Using this model, the Young's and shear moduli were calculated for three samples with different aluminum matrices and volume fractions of particles (9, 13, and 17%) but the same fiber content (6%). The same elastic moduli were also determined using ultrasonic velocity measurements. The agreement between calculated and measured elastic moduli is found to be very good. Also, the elastic anisotropics between directions of the fiber rich plane and that normal to the plane could be predicted by the model.     

Elastic properties of rubber particles in toughened PMMA: ultrasonic and micromechanical evaluation

Ultrasonic measurements and micromechanical models are used to evaluate elastic properties of rubber particles dispersed in toughened polymers. Ultrasonic phase velocities and attenuation spectra of rubber-toughened poly(methyl methacrylate) (PMMA) with different rubber particle fractions are measured for longitudinal as well as transverse waves. The ultrasonic properties of rubber-toughened PMMA are found to depend markedly on the rubber particle fraction. The bulk and shear moduli determined from the measured velocities are in turn used to estimate those moduli of the particles based on existing micromechanics models, namely the three-phase model and the Hashin–Shtrikman upper and lower bounds. The bulk modulus of the particle estimated by the three-phase model is found to be in close agreement with the result of previous investigators. Implications of the Hashin–Shtrikman bounds for the particle moduli are also examined.     

Dynamic Effective Properties of Particle-Reinforced Composites with Viscoelastic Matrix

The frequency-dependent dynamic effective properties of the particle-reinforced composites with the viscoelastic matrix are studied. Several equations to predict the effective wavenumber of the coherent plane waves propagating through particle-reinforced composites are discussed and the equation given by Gubernatis, J.E., [‘Effects of microstructure on speed and attenuation of elastic waves in porous materials’, Wave Motion, 6, 1984, 579–589] based on the independent scattering approximation is used in this paper. The effective phase velocity, the effective attenuation and the effective elastic moduli are evaluated. Numerical calculations are carried out for two kinds of composites, namely, Lead-Epoxy and Glass-Epoxy and the numerical results show that the frequency-dependent dynamic effective properties are related to both the multiple scattering effects among the distributed particles and the viscous dissipative effects of the viscoelastic matrix. However, these effects in the composites with distributed heavy particles (lead) and light particles (glass) are of evidently different features.     

Amplitude-dependent attenuation of compressive waves in curved granular crystals constrained by elastic guides

We study the wave propagation in a curved chain of spherical particles constrained by elastic guides under the axial impact of a falling mass. We characterize the force transmission properties of the chain by varying the striker’s mass and the chain’s curvature. Experimental tests demonstrate amplitude-dependent attenuation of compressive waves propagating through the curved chain. In particular, we observe that the curved systems present an improved transmission of small dynamic disturbances relative to that of strong excitations, resulting from the close interplay between the granular particles and the softer elastic medium. We also find that the transmission of the compressive waves through the chains is dependent on the initial curvature imposed to the system. Numerical simulations, based on an approach that combines discrete element and finite element methods, corroborate the experimental results. The findings suggest that hybrid structures composed of granular particles and linear elastic media can be employed as new passive acoustic filtering materials that selectively transmit or mitigate excitations in a desired range of pressure amplitudes.     

Elastic constants of particle and fiber reinforced metal matrix composites

A model has been developed to predict the elastic moduli in composites reinforced with both particles and fibers. In the model the matrix material and the particles, which are assumed to be homogeneously distributed, form an effective matrix. The characteristics of this effective matrix is calculated using a theory formulated by Ledbetter and Datta. The effective matrix is then considered to be reinforced with fibers lying in one plane but randomly oriented in that plane. The effect of the 2-dimensionally random orientation of the fibers on the elastic moduli of the composites is determined in two steps. First the composite cylinders model by Hashin and Rosen for an aligned fiber system is employed, and then a geometric averaging procedure suggested by Christensen and Waals is performed. Using this model, the Young's and shear moduli were calculated for three samples with different aluminum matrices and volume fractions of particles (9, 13, and 17%) but the same fiber content (6%). The same elastic moduli were also determined using ultrasonic velocity measurements. The agreement between calculated and measured elastic moduli is found to be very good. Also, the elastic anisotropies between directions of the fiber rich plane and that normal to the plane could be predicted by the model.This article is dedicated to Professor Dr. Paul Höller on the occasion of his 65th birthday.