George J. Weng

Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions

Abstract Based on Mori and Tanakas concept of “average stress” in the matrix and Eshelbys solutions of an ellipsoidal inclusion, an approximate theory is established to derive the stress and strain state of constituent phases, stress concentrations at the interface, and the elastic energy and overall moduli of the composite. Both “stress-free” strain (polarization strain) and “strain-free” stress (polarization stress) are employed in these derivations under the traction- and displacement-prescribed conditions. The theory was developed first for a general multiphase, anisotropic composite with arbitrarily oriented anisotropic inclusions; explicit results are then given for a suspension of uniformly distributed, multiphase isotropic spheres in an isotropic matrix. Numerical results for stress concentrations in the spherical inclusions and at the interface are given for a 2-phase composite. Further, it is shown that the derived moduli are related to the Hashin-Shtrikman bounds and that, when the shear moduli are equal, the overall bulk modulus of a 2-phase composite reduces to Hills exact solution. As compared with experimental data, the theory also provides reasonably accurate estimates for the Youngs modulus of some 2- and 3-phase composites.

A Theory of Particle-Reinforced Plasticity

A simple, albeit approximate, theory is developed to determine the elastoplastic behavior of particle-reinforced materials. The elastic, spherical particles are uniformly dispersed in the ductile, work-hardening matrix. The method proposed combines Mori-Tanaka’s concept of average stress in elasticity and Hill’s discovery of a decreasing constraint power of the matrix in polycrystal plasticity. Under a monotonic, proportional loading the latter was characterized, approximately, by the secant moduli of the matrix. The theory is established for both traction and displacement-prescribed boundary conditions, under which, the average stress and strain of the constituents and the effective secant moduli of the composite are explicitly given in terms of the secant moduli of the matrix and the volume fraction of particles. In particular, the yield stress and work-hardening modulus of the composite are shown to be inversely proportional to the deviatoric part of average stress concentration factors of the matrix, and therefore will increase (or decrease) with increasing hard (or soft) particle concentration. It is also found that, even if the matrix is plastically incompressible, the composite as a whole is not. Comparison between the theory and the experiment for a silica/epoxy system shows a reasonable agreement. The theory is also compared with a recently developed one by Arsenault and Taya; while both give the same initial yield stress for the composite, the work-hardening modulus predicted by their theory is found to be higher.

The theoretical connection between Mori-Tanaka's theory and the Hashin-Shtrikman-Walpole bounds

Abstract Mori-Tanakas theory with the general anisotropic constituents has been recast into a new form and it is shown that this form bears an identical structure to that developed by Walpole for the bounds. The equivalent polarization stress and strain in the former theory are exactly those chosen by Hashin-Shtrikman and Walpole and the average stress and strain of the matrix phase are equal to the image stress and strain imposed on the approximate fields by Walpole to meet the required boundary conditions. The consequence is that the effective moduli of the composite containing either aligned or randomly-oriented, identically shaped ellipsoidal inclusions always have the same expressions as those of the H-S-W bounds, only with the latters comparison material identified as the matrix phase and Eshelbys tensor interpreted according to the appropriate inclusion shape. This connection allows one to draw a line of important conclusions regarding the predictions of the M-T theory, and it also points to the conditions where this theory can always be applied safely without ever violating the bounds and where such an application might be less reliable.

On the application of Mori-Tanaka's theory involving transversely isotropic spheroidal inclusions

Abstract Based on the new structure recently established by Weng for the Mori-Tanaka theory, the effective elastic moduli of three types of composite containing transversely isotropic spheroidal inclusions are explicitly derived. For a multiphase composites with aligned, identically shaped inclusions, the derived moduli are believed to be generally reliable, where the three extreme cases involving circular fibers, spheres, and thin discs all lie on or within the respective Hashin-Shtrikman-Walpole bounds. For a multiphase aligned composite whose inclusion phases differ in shape, the M-T moduli tensor can lose its diagonal symmetry, which, for a hybrid composite containing fibers and another aligned spheroids, is found to be severest when the spheroids take the shape of thin discs, and tends to decrease as their aspect ratio increases. When the transversely isotropic spheroidal inclusions are randomly oriented in an isotropic matrix, the M-T moduli with spherical inclusions are shown to always lie on or within the isotropic Hashin-Shtrikman-Walpole bounds. Such a desired property however is not always assured with other inclusion shapes, where the needle and disc-like inclusions may cause the M-T moduli and Walpoles self-consistent estimates to lie outside the H-S-W bounds.

The overall elastoplastic stress-strain relations of dual-phase metals

Abstract T wo simple , albeit approximate, theories are developed to estimate the stress-strain relations of dual-phase metals of the inclusion-matrix type, where both phases are capable of undergoing plastic flow. The first one is based upon Hills recognition of a weakening constraint power in a plastically deforming matrix, whereas the second one is based on Kroners elastic constraint in the treatment of the single inclusion-matrix interaction. The inclusion-inclusion interaction at finite concentration is accounted for by the Mori-Tanaka method in both cases. Consistent with the known elastic behavior, the first theory discloses that the geometrical arrangement of the constituents has a significant influence on the overall elastoplastic response. When the harder phase takes the position of the matrix the composite is far Stiffer than that when it takes the position of inclusions. The strong elastic constraint associated with the second theory tends to provide an upper-bound type of estimate regardless of whether the matrix is the harder phase or the softer, and, therefore, it is suggested that this theory be used only for the class of composites whose matrix is the harder phase. Both theories are finally applied to predict the stress-strain relations of dual-phase stainless steels, and the results are found to be in satisfactory agreement with the test data.

The connections between the double-inclusion model and the Ponte Castaneda–Willis, Mori–Tanaka, and Kuster–Toksoz models

Abstract In this paper, it is shown that the double-inclusion model (Hori, M., Nemat-Nasser, S., 1993. Double-inclusion model and overall moduli of multi-phase composites. Mech. Mater. 14, 189–206) carries more theoretical connections with other micromechanical models than what is presently realized. In the past, only connections with the Mori–Tanaka (MT) model (Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571–574) and the self-consistent model (Hill, R., 1965. A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222; Budiansky, B., 1965. On the elastic moduli of some heterogeneous material. J. Mech. Phys. Solids, 13, 223–227) for aligned inclusions have been established. By choosing the shape and the relative orientation of the inclusion and the matrix judiciously, the double-inclusion model can produce results for a two-phase composite containing randomly oriented ellipsoidal inclusions for the Ponte Castaneda–Willis (PCW) model (Ponte Castaneda, P., Willis, J.R., 1995. The effect of spatial distribution on the effective behavior of composite materials and cracked media. J. Mech. Phys. Solids 43, 1919–1951), MT model, and Kuster–Toksoz (KT) model (Kuster, G.T., Toksoz, M.N., 1974. Velocity and attenuation of seismic waves in two-phase media: I Theoretical formulation. Geophysics, 39, 587–606). These connections have also shed some light into the possible microgeometries for the MT and KT models. The microstructure for the PCW model is already known, and it is now established that the outer shape and orientation of the double inclusion is exactly the spatial distribution ellipsoid of the PCW model. The result also proves that the KT model, widely used in the geophysics community, actually provides a result that is identical to the PCW model and, thus, has a well-defined microstructure that was previously said to be non-existent.

On Eshelby's inclusion problem in a three-phase spherically concentric solid, and a modification of Mori-Tanaka's method

Abstract The elastic field in the three-phase, spherically concentric solid due to a stress-free transformation strain in the inclusion is obtained. This analysis considers both homogeneous and a polynomial-type, nonlinear transformations. In the former case, the strain field in the inclusion is found to be also homogeneous under a hydrostatic transformation, but becomes heterogeneous under a deviatoric one. The mean fields are uncoupled between a hydrostatic and a deviatoric case, and even between one shear and the other. The elastic energies of the solid under these transformations are also derived. In light of this new development, a modification is suggested for the Mori-Tanaka method. The resulting effective shear modulus of a two-phase composite lies between the Hashin-Shtrikman bounds, and the predicted Youngs modulus also compares well with the experimental data.

Tunneling resistance and its effect on the electrical conductivity of carbon nanotube nanocomposites

In this paper, we examined the effect of electron tunneling upon the electrical conductivity of carbon nanotube (CNT) polymer nanocomposites. A CNT percolating network model was developed to account for the random distribution of the CNT network using Monte Carlo simulations, where the tunneling resistance between CNTs was established based on the electron transport theory. Our work shows several novel features that result from this tunneling resistance: (i) direct contact resistance is the result of one-dimensional electron ballistic tunneling between two adjacent CNTs, (ii) the nanoscale CNT-CNT contact resistance should be represented by the Landauer-Buttiker (L-B) formula, which accounts for both tunneling and direct contact resistances, and (iii) the difference in contact resistance between single-walled CNTs (SWCNTs) and multi-walled CNTs (MWCNTs) can be modeled by the channel number in the L-B model. The model predictions reveal that the contact resistance due to electron tunneling effects in nanoc...

Influence of polarization orientation on the effective properties of piezoelectric composites

The effective properties of 0–3- and 1–3-type piezoelectric composites of lead zirconate titanate and vinylidene fluoride–trifluoroethylene with different polarization status in both phases are calculated using an effective-medium theory. The effects of volume fraction and polarization orientation on the effective behavior are presented in detail. The theory gives results in reasonable agreement with recent experimental ones. The theoretical predictions demonstrate the interesting behavior of the composites and provide a general guideline for optimizing the microstructural scale of the composites for piezoelectric transducers and pyroelectric sensors.

Transversely isotropic moduli of two partially debonded composites

Abstract 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 Eshelbys 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. London A241 , 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 Youngs modulus E 11 and axial shear modulus μ 12 in type 1, and the transverse Youngs modulus E 22 , plane-strain bulk modulus k 23 , 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.