New exact results for the effective electric, elastic, piezoelectric and other properties of composite ellipsoid assemblages

In this paper we present a unified treatment of composite ellipsoid assemblages in the setting of uncoupled phenomena like conductivity and elasticity and coupled phenomena like thermoelectricity and piezomagnetoelectricity. The building block of this microgeometry is a confocal ellipsoidal particle consisting of a (possibly void) core and a coating. All space is filled up with such units which have different sizes but possess the same aspect ratios. The confocal ellipsoids may have the same orientation in space or may be randomly oriented. The resulting microgeometry simulates two-phase composites in which the reinforcing components are short fibers or elongated particles. Our main interest is in obtaining information of an exact nature on the effective moduli of this microgeometry whose effective tensor symmetry structure depends on the packing mode of the coated ellipsoids. This information will sometimes be complete like the full effective thermoelectric tensor of an assemblage which contains aligned ellipsoids in which the coating is isotropic and the core is arbitrarily anisotropic. In the majority of the cases however the maximum achievable exact information will be only partial and will appear in the form of certain exact relations between the effective moduli of the microgeometry. These exact relations are obtained from exact solutions for the fields in the microstructure for a certain set of loading conditions. In all the considered cases an isotropic coating can be combined with a fully arbitrary core. This covers the most important physical case of anisotropic fibers in an isotropic matrix. Allowing anisotropy in the coating requires the fulfillment of certain constraint conditions between its moduli. Even though in this case the presence of such constraint conditions may render the anisotropic coating material hypothetical, the value of the derived solutions remains since they still provide benchmark comparisons for approximate and numerical treatments. The remarkable feature of the general analysis which covers all treated uncoupled and coupled phenomena is that it is developed solely on the basis of potential solutions of the conduction problem in the same microgeometry.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.     

A variational multiscale constitutive model for nanocrystalline materials

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior phase and a grain boundary effected zone (GBAZ). A rate-independent isotropic porous plasticity model is employed to describe the GBAZ, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The constitutive models of both phases are formulated in a small strain framework and extended to finite deformation by use of logarithmic and exponential mappings. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization where a log-normal grain size distribution is assumed. It is shown that the proposed model is able to capture the inverse Hall-Petch effect, i.e., loss of strength with grain size refinement. Finally, the predictive capability of the model is validated against experimental results on nanocrystalline copper and nickel.     

Modal analysis of piezoelectric bodies with voids. II. Finite element simulation

The paper is concerned with the eigenvalue problems for piezoelectric bodies with voids in contact with massive rigid plane punches and coved by the system of open-circuited and short-circuited electrodes. The linear theory of piezoelectric materials with voids for porosity change properties according to Cowin–Nunziato model is used. The generalized statements for eigenvalue problem are obtained in the extended and reduced forms. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. The orthogonality relations for eigenvectors are obtained in different forms. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical, electric and “porous” boundary conditions and the moduli of piezoelectric solid with voids change, are established.     

Molecular dynamics simulation of defects production in the vicinity of voids

We report on the results of computer simulation of point defect production near voids in crystalline Cu at primary knock-on atom (PKA) kinetic energies ranging from 5 to 1000?eV. The PKA energy dependence of numbers of created defects are revealed. The threshold energy for a stable vacancy formation is found to be much smaller than that for an interstitial atom, which results in a biased formation of vacancies in the void proximity in the whole investigated range of PKA energies. Dissolution of small voids by subthreshold irradiation is simulated. The impact of considered radiation effects on kinetics of radiation damage is discussed.     

Plasticity size effects in voided crystals

The shear and equi-biaxial straining responses of periodic voided single crystals are analysed using discrete dislocation plasticity and a continuum strain gradient crystal plasticity theory. In the discrete dislocation formulation, the dislocations are all of edge character and are modelled as line singularities in an elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and annihilation are incorporated through a set of constitutive rules. Over the range of length scales investigated, both the discrete dislocation and strain gradient plasticity formulations predict a negligible size effect under shear loading. By contrast, under equi-biaxial loading both plasticity formulations predict a strong size dependence with the flow strength approximately scaling inversely with the void spacing. Excellent agreement is obtained between predictions of the two formulations for all crystal types and void volume fractions considered when the material length scale in the non-local plasticity model is chosen to be (about 10 times the slip plane spacing in the discrete dislocation models).     

Effective pressure-sensitive elastoplastic behavior of particle-reinforced composites and porous media under isotropic loading

The composite under investigation consists of an elastoplastic matrix reinforced by elastic particles or weakened by pores. The material forming the matrix is pressure-sensitive. The Drucker–Prager yield criterion and a one-parameter non-associated flow rule are employed to formulate the yield behavior of the matrix. The objective of this work is to estimate the effective elastoplastic behavior of the composite under isotropic tensile and compressive loadings. To achieve this objective, the composite sphere assemblage model of Hashin [Z. Hashin, The elastic moduli of heterogeneous materials, ASME J. Appl. Mech. 29 (1962) 143–150] is used. Exact solutions are thus derived as estimations for the effective secant and tangent bulk moduli of the composite. The effects of the loading modes and phase properties on the effective elastoplastic behavior of the composite are analytically and numerically evaluated.     

A constitutive model for plastically anisotropic solids with non-spherical voids

Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.     

Constitutive models for power-law viscous solids containing spherical voids

In the analysis of non-linear porous solids, it is commonplace to employ a spherical unit cell owing to the simplicity it affords. The macroscopic constitutive response of the material is then predicted based upon either uniform traction or linear displacement/velocity boundary conditions applied to the outer surface of the cell. In this investigation, we carry out a careful computational analysis of the effect of these two types of boundary conditions on the macroscopic response of the (idealized) porous solid and in particular, we explore the sensitivity of the predicted response to the macroscopic stress, void volume fraction and material non-linearity. The numerical results are then used as a basis for establishing an approximate constitutive model that is expressed in a compact, explicit form. The study is carried out in the context of an incompressible, isotropic power-law viscous matrix material, and the computational analysis is focused on axisymmetric deformation of the unit cell. While the macroscopic strain-rate potential is found to exhibit a dependence on the third invariant of the macroscopic stress deviator, this dependence is slight (particularly for the linear displacement/velocity boundary condition) and, toward developing an approximate strain-rate potential applicable to general macroscopic stress states, a simple averaging scheme is employed to suppress the role of this quantity. Guided by the numerical results as well as by various previously proposed constitutive relations, an approximate generalized elliptic form for the macroscopic strain-rate potential is then proposed. The constitutive potential which is ultimately developed involves a fairly simple dependence upon the void volume fraction and the properties of the matrix material, yet it gives rise to predictions that agree well with the detailed unit cell calculations over the full range of properties and macroscopic stress states considered.     

Measurement, modelling and simulation of defects in as-grown Czochralski silicon

Defects in as-grown and heat-treated 150 and 200 mm Czochralski silicon crystals are investigated for different crystal pulling conditions and thermal treatments. First results are presented using noncontact carrier recombination imaging for detection of electrically active defects. The defect densities and sizes are compared with the results of computer modelling, suggesting that the observed defects in as-grown material are most likely large voids, possibly partially filled with silicon oxide. In contrast, the defects observed after prolonged heat treatments are silicon oxide precipitates with a density which is several orders of magnitude larger. The voids nucleate at temperatures above 1100°C by a homogeneous nucleation process and grow further to the observed size during further cooling of the crystal. They are responsible for the midfield breakdown of 10–50 nm gate oxides.