Prolate Inclusions Research Articles

Concentration tensors preserving elastic symmetry of multiphase composites

The micromechanics of composites with multiple phases of different shapes embedded into a matrix phase, typically requires symmetrization strategies for their homogenized elasticity tensors, in particular so if the popular Mori–Tanaka estimate is employed. We here explore the implications of such symmetrization techniques, on the concentrations tensors, i.e. on the relations between macroscopic strains imposed onto a representative volume element of microelastic matter, and the microscopic phase strains developing across the materials’ microstructure. Thereby, we adopt the important idea of Mori and Tanaka to approximate the phase strains by the homogeneous strains inside an Eshelbian inhomogeneity embedded into an infinite matrix, together with the phase strains fulfilling the strain average rule; while we refrain from the identification of the strain in the matrix phase as the auxiliary strain imposed remotely at the infinite matrix of Eshelby’s matrix-inhomogeneity problem. Instead of this identification, we allow for a general multilinear relation between auxiliary strains and RVE-related macroscopic strains, and we express the homogenized stiffness as (i) a function of the conversion tensor quantifying the aforementioned multilinear relation, and (ii) as the symmetrized Mori–Tanaka estimate. In this way, the conversion tensor and all phase concentration tensors can be determined in a way which allows the overall elastic stiffness to remain symmetric. This is illustrated by means of two benchmark examples concerning matrix–inclusion composites hosting both spherical and prolate inclusions, with the matrices being isotropic and transversely isotropic, respectively. These examples benefit from analytical expressions for the Hill tensor governing the underlying matrix-inhomogeneity problems.

Elastic Percolation in Nanocomposites with Impenetrable Ellipsoidal Inclusion (Comprehensive Study of Geometry and Interphase Thickness)

In this paper, effects of percolation, shape, and interphase thickness of inclusions have been studied on the effective elastic modulus of polymer nanocomposites. Inclusion distribution within the RVE has been prescribed using the Monte Carlo method, and existence of a relationship between percolation and effective Young’s modulus has been investigated. Further studies were carried out to determine the effect of increased spherical, oblate, and prolate particles in conjunction with different interphase thicknesses. Results suggest that the elastic modulus increases with the interphase thickness, and the maximum strength is associated with prolate inclusions. It has been concluded that percolation has a significant effect on the strength of polymer nanocomposites, and that it results in a spike in the value of Young’s modulus.

New orientation effect in piezo-active 1–3-type composites

This paper demonstrates the influence of the mutual orientation of the poling direction of single-crystal and ceramic components on the properties of 1–0–3 composites containing two ferroelectric components and a piezo-passive polymer. Effective properties to be considered are hydrostatic piezoelectric performance, anisotropy of squared figures of merit and electromechanical coupling factors. We demonstrate that the elastic and piezoelectric anisotropy of the 0–3 connectivity ferroelectric ceramic/polymer matrix with prolate inclusions leads to large hydrostatic piezoelectric coefficients (dh∗ and gh∗) and squared figure of merit (dh∗gh∗) in a 1–0–3 0.67Pb(Mg1/3Nb2/3)O3–0.33PbTiO3 single crystal/(Pb1–xCax)TiO3 ceramic/araldite composite with x = 0.20–0.25. In this type of composite values of maxgh∗ ∼102 mV m/N and max(dh∗gh∗) ∼10−11 Pa−1 are achieved at specific volume-fraction and rotation-angle ranges due to a new orientation effect in the presence of a highly anisotropic 0–3 matrix. Such composites are of interest for sensor, actuator and energy-harvesting applications.

Investigation of thermal conductivity enhancement in bakelite–graphite particulate filled polymeric composite

This paper presents an investigation of thermal conductivity enhancement of a particulate filled polymeric composite. Different samples of bakelite–graphite composite material were prepared by compression molding technique. Conductive filler (graphite) percentage varied from 30% to 55% by volume. Thermal conductivities of composite samples were measured by using P.A Hilton Heat Conduction Unit H-940. ASTM standard E1225-99 was followed for numerical calculation. Gradual increase in thermal conductivity has been observed, with increase in conductive filler percentage in composite sample. Since many theoretical/empirical models are available in the literature for the estimation of effective thermal conductivity of filled composite systems. Therefore, the experimental results of thermal conductivity were plotted along with the available theoretical models. It was found that the Mori–Tanaka model for prolate inclusions and the Hamilton–Crosser model yield the closest approximation to the measured thermal conductivity of filled composite system where the particle shape morphology is important.

Analysis of the Piezoelectric Performance of Modern 0–3-Type Composites Based on Relaxor-Ferroelectric Single Crystals

Effective piezoelectric coefficients e 33*, d 33*, g 33*, and h 33*, squared figures of merit d 33*g 33* and dh *gh *, and other related parameters of novel 0–3 and 0–3–0 composites based on single crystals of relaxor-ferroelectric solid solutions of (1−x)Pb(Mg1/3Nb2/3)O3−xPbTiO3 are studied using the effective field method, effective medium method and finite element method. The volume-fraction dependences of a number of parameters and figures of merit are determined using the above methods and compared for composites with aligned prolate spheroidal inclusions. Four parameter ratios that determine interconnections between the effective electromechanical constants of the 0–3-type composites are first introduced. These parameter ratios are used to interpret the longitudinal dielectric and piezoelectric response of composite structures with prolate inclusions that exhibit high piezoelectric activity. The role of piezoelectric anisotropy and activity of the composite components in forming the longitudinal piezoelectric effect is analysed. Specific novel composite architectures with high piezoelectric sensitivity (longitudinal and hydrostatic) are identified and discussed.

POROUS SILICON DIELECTRIC FUNCTION MODELING FROM EFFECTIVE MEDIUM THEORIES

The spectral variation of the effective dielectric constant in porous silicon thin films is modeled using effective medium theories. The pores in the material are supposed to be spheroid prolate inclusions in a homogeneous matrix. We employ modified Maxwell–Garnett and Bruggeman expressions obtained from previous work. Fitting to experimental reflectance spectra and the refractive index at the visible spectral range is satisfactory. Best results in both models are obtained using the spheroidal inclusions of 0.9 eccentricity. The best adjusted porosity values for two samples with different parameters of preparation are 47% with Bruggeman's expression and 67% with Maxwell–Garnett's expression, which are very close to the experimental ones of 44% and 67% respectively, obtained through gravimetric methods. Our model is useful for determining and for checking the influence of the preparation parameters on the optical and structural properties.

The Effect of Debonding Angle on the Reduction of Effective Moduli of Particle and Fiber-Reinforced Composites

In an effort to uncover the effect of interfacial partial debonding on the reduction of composite stiffness, a reduced moduli approach is proposed for the fictitious inclusions which are used to replace the original partially debonded inclusions. The fictitious inclusions are now perfectly bonded to the matrix and any micromechanical theory can be called upon to estimate the moduli of the composite. Using the volume of the inclusion directly beneath the interfacial cracks under the considered loading mode as a measure of damage, a set of anisotropic damage parameters is established in terms of the debonding angle, providing the reduced moduli for the fictitious inclusions. Specific considerations include debonding on the top and bottom of spheres and prolate inclusions, debonding on the lateral surface of spheres and oblate inclusions, and debonding on the top and bottom of circular fibers and elliptic cylinders. The reductions of the five transversely isotropic moduli for the partially debonded particle composites and the nine orthotropic moduli for the partially debonded fiber composites are examined as the debonding angle increases. The theory is also compared with some finite element results, and it suggests that the concept proposed to estimate the reduced moduli of the fictitious inclusions is a viable one.

Finite element modelling of 3-3 piezocomposites

This paper demonstrates the influence of the mutual orientation of the poling direction of single-crystal and ceramic components on the properties of 1–0–3 composites containing two ferroelectric components and a piezo-passive polymer. Effective properties to be considered are hydrostatic piezoelectric performance, anisotropy of squared figures of merit and electromechanical coupling factors. We demonstrate that the elastic and piezoelectric anisotropy of the 0–3 connectivity ferroelectric ceramic/polymer matrix with prolate inclusions leads to large hydrostatic piezoelectric coefficients (dh∗ and gh∗) and squared figure of merit (dh∗gh∗) in a 1–0–3 0.67Pb(Mg1/3Nb2/3)O3–0.33PbTiO3 single crystal/(Pb1–xCax)TiO3 ceramic/araldite composite with x = 0.20–0.25. In this type of composite values of maxgh∗ ∼102 mV m/N and max(dh∗gh∗) ∼10−11 Pa−1 are achieved at specific volume-fraction and rotation-angle ranges due to a new orientation effect in the presence of a highly anisotropic 0–3 matrix. Such composites are of interest for sensor, actuator and energy-harvesting applications.

Influence of thermal residual stresses on the composite macroscopic behavior

The influence of the thermal residual stress on the deformation behavior of a composite has been analyzed with a new micromechanical method. The method is based on secant moduli approximation and a new homogenized effective stress to characterize the plastic state of the matrix. It is found that the generated thermal residual stresses after cooling and their influence on the subsequent deformation behavior depends significantly on the aspect ratio of the inclusions. With prolate inclusions, the presence of thermal residual stresses generate a higher compressive hardening curves of the composite, but it is reversed with oblate inclusions. For particle reinforced composite, thermal residual stresses induce a tensile hardening curve higher than the compressive one and this is in agreement with experimental observations.

Transversely isotropic moduli of two partially debonded composites

The effective transversely isotropic moduli of two hybrid composites containing both partially debonded and perfectly bonded spheroidal inclusions are derived. In this derivation a fictitious, transversely isotropic inclusion is introduced to replace the isotropic, partially debonded one so that Eshelby's solution of a perfectly bonded inclusion could be used. [Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. LondonA241, 376–396]. Two types of debonding configuration are considered: the first type occurs on the top and bottom of the oblate inclusions and the second one exists on the lateral surface of the prolate inclusions. Albeit approximate, the method is simple and capable of providing explicit results for the five independent moduli in terms of the volume concentrations and aspect ratio of the partially-debonded and perfectly-bonded inclusions. The results are given for the spherical and various inclusion shapes. It is shown that, with spherical inclusions, the longitudinal Young's modulus E11 and axial shear modulus μ12 in type 1, and the transverse Young's modulus E22, plane-strain bulk modulus k23, and the axial and transverse shear moduli in type 2, can all be greatly affected by partial debonding. Examination on the influence of inclusion shape indicates that disc-shaped inclusions in the first type and prolate ones is the second type lead to stronger moduli reduction than spheres.

Plasticity of isotropic composites with randomly oriented and packeted inclusions

Abstract Based on an energy approach, the nonlinear elastoplastic behavior of a two-phase, isotropic composite with two kinds of inclusion morphologies are determined as a function of inclusion shape at the low concentration range. Both types of morphology involve the three-dimensional randomly oriented spheroidal inclusions, but one is homogeneously dispersed, resulting in an ordinary two-phase composite, and the other possesses a packeted structure in a form similar to a polycrystalline arrangement. The overall elastoplastic response of these two kinds of composite is found to be strongly dependent upon the inclusion shape and their morphological arrangement. Disc-shaped inclusions generally give a superior reinforcing effect, but when the inclusions become very stiff or totally rigid, needle-shaped inclusions tend to be more effective. In line with the known elastic behavior, the overall elastoplastic response of an ordinary two-phase composite is markedly stiffer than that of the packeted composite at a given inclusion shape and concentration. The theory is finally compared with the finite-element calculations of a particle-reinforced composite and of a packeted one with oblate and prolate inclusions, and also with some experimental data for a silicon-carbide/aluminium system. In both cases, reasonable agreement is found in the comparison.

The Elastoplastic Behavior of a Class of Two-Phase Composites Containing Rigid Inclusions

The theoretical principle for the determination of the overall elastoplastic behavior for three types of two-phase composites reinforced with rigid spheroidal inclusions is established. In the first type the inclusions are aligned unidirectionally (1-D) whereas in the second type they are randomly oriented on a plane (2-D), both leading to transversely isotropic composites. The third type of composite involves a 3-D random orientation leading to a globally isotropic material. The theory is based on the energy approach recently proposed by Qiu and Weng (1992) and, while intended only for a modest concentration of inclusions, it can cover a wide range of inclusion shapes, from discs to spheres and all the way to needles (or fibers). It is shown, among others, that the axial response of the 1-D composite is the strongest when reinforced with fibers whereas discs provide the strongest reinforcement under the transverse loading. The outcome is reversed for the 2-D composite. As a consequence of a uniform boundary displacement, the 3-D isotropic composite reinforced with the extreme shapes of rigid needles (aspect ratio → ∞) or discs (aspect ratio → 0) turns out to be also ideally rigid with the entire external stress being carried by the rigid inclusions. For other inclusion shapes, the elastoplastic behavior with prolate inclusions are almost always stiffer as compared to those with oblate shapes with a reciprocal aspect ratio. The asymptotic moduli of the two-phase composites containing the two extreme inclusion shapes are given in terms of the disc and needle density parameters, and the overall elastoplastic behavior for the three different microgeometries are also given as a function of these parameters.

Constitutive models for ductile solids reinforced by rigid spheroidal inclusions

We study the effective constitutive response of composite materials made of rigid spheroidal inclusions dispersed in a ductile matrix phase. Given a general convex potential characterizing the plastic “in the context of J 2-deformation theory” behavior of the isotropic matrix, we derive expressions for the corresponding effective potentials of the rigidly reinforced composites, under general loading conditions. The derivation of the effective potentials for the nonlinear composites is based on a variational procedure developed recently by Ponte Castaneda (1991a, J. Mech. Phys. Solids 39, 45–71). We consider two classes of composites. In the first class, the spheroidal inclusions are aligned, resulting in overall transversely isotropic symmetry for the composite. In the second class, the inclusions are randomly oriented, and thus the composite is macroscopically isotropic. The effective response of composites with aligned inclusions depends on both the orientation of the loading relative to the inclusions and on the inclusion concentration and shape. Comparing the strengthening effects of rigid oblate and prolate spheroids, we find that prolate spheroids give rise to stiffer effective response under axisymmetric “relative to the axis of transverse isotropy” loading, while oblate spheroids provide greater reinforcement for materials loaded in transverse shear. On the other hand, nearly spherical “slightly prolaterd spheroids are most effective in strengthening the composite under longitudinal shear. Thus, the optimal shape for strengthening composites with aligned inclusions depends strongly on the loading mode. Alternatively, the properties of composites with randomly oriented spheroidal inclusions, being isotropic, depend only on the concentration and shape of the inclusions. We find that both oblate and prolate inclusions lead to significant strengthening for this class of composites.

Relaxation kinetics and relaxed stresses caused by interface diffusion around spheroidal inclusions

The kinetics of relaxation by interface diffusion around spheroidal inclusions is discussed using the Eshelby method for ellipsodal inclusions and a free energy approach. The relaxation time for the process is derived in the case where the direction of an applied uniaxial stress is parallel to the rotation axis of symmetry of the speroid. The aspect-ratio dependence of the relaxanion time is derived. For prolate inclusions with large aspect ratios, the relaxation time increases proportionally to the aspect ratio. On the other hand, for oblate inclusions with small aspect ratios, the relaxation time becines constant as the aspect ratio approaches zero. The constant value of the relaxation time depends on the elastic moduli of the inclusion but independent of the elastic moduli of the matrix. The hydrostatic stress in the inclusion after the relaxation and its aspect-ratio dependence are also discussed. Although the relaxation time for a rod-like inclusion is longer compared with that for a disk-like inclusion, the relaxed stress in the former is much smaller than that in the latter.

Effect of inclusion shape on the stiffness of nonlinear two-phase composites

constitutive relations are established for the axisymmetric deformation of an incompressible power-law matrix containing aligned, rigid spheroidal inclusions. The range of inclusion shapes considered is from thin disks with aspect ratios of 100: 1 to whiskers with aspect ratios of 50: 1. Results are presented for several matrix hardening exponents between n = 1 (linearly elastic) and n = 10. It is found that at a fixed matrix hardening exponent and a fixed volume fraction of inclusions, slender prolate inclusions give rise to a greater increase in stiffness than any other inclusion shape with the same or smaller aspect ratio. It is also found that the stiffening which results from a fixed volume fraction of inclusions is a strong function of matrix nonlinearity; the stiffness of the composite relative to the matrix material is much greater for a highly nonlinear matrix than for a linear matrix.

Anomalous tensoelectric effects in gallium arsenide tunnel diodes

Anomalous tensoelectric phenomena induced in a tunnel p-n junction by a concentrated load and by hydrostatic compression were studied. The anomalous tensoelectric effects are caused by the action of concentrators of mechanical stresses in the vicinity of the p-n junction, giving rise to local microplastic strain. Under the conditions of hydrostatic compression prolate inclusions ∿100–200 a long play the role of concentrators. Analysis of irreversible changes in the current-voltage characteristics of tunnel p-n junctions made it possible to separate the energy levels of the defects produced with plastic strain of gallium arsenide.

Ordered structures ink random grain boundaries

The effective transversely isotropic moduli of two hybrid composites containing both partially debonded and perfectly bonded spheroidal inclusions are derived. In this derivation a fictitious, transversely isotropic inclusion is introduced to replace the isotropic, partially debonded one so that Eshelby's solution of a perfectly bonded inclusion could be used. [Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. LondonA241, 376–396]. Two types of debonding configuration are considered: the first type occurs on the top and bottom of the oblate inclusions and the second one exists on the lateral surface of the prolate inclusions. Albeit approximate, the method is simple and capable of providing explicit results for the five independent moduli in terms of the volume concentrations and aspect ratio of the partially-debonded and perfectly-bonded inclusions. The results are given for the spherical and various inclusion shapes. It is shown that, with spherical inclusions, the longitudinal Young's modulus E11 and axial shear modulus μ12 in type 1, and the transverse Young's modulus E22, plane-strain bulk modulus k23, and the axial and transverse shear moduli in type 2, can all be greatly affected by partial debonding. Examination on the influence of inclusion shape indicates that disc-shaped inclusions in the first type and prolate ones is the second type lead to stronger moduli reduction than spheres.

Electromagnetic prospecting by Turam cable induction versus conduction effects

This paper discusses two ways by which a conductor buried in the ground can give a secondary field in Turam prospection. First, if Turam cable is grounded at both ends, and if the ore body is long and parallel to the cable, the secondary field is produced according to the Ampère law: it decreases like 1 R and its vertical component vanishes over the apex. Much more often, and in all cases where the Turam transmitter consists of an insulated loop, the signal cornes from induced currents: the magnetic field decreases like 1 R 2 and its vertical component is maximum over the apex. However, this maximum is shifted when the cable lies too near the ore body and when the ground surface is not horizontal. Digital and analog computations allowed us to establish ten anomaly charts for various shapes of prolate inclusions.