Showing papers in "Journal of The Mechanics and Physics of Solids in 1991"

Family of crack-tip fields characterized by a triaxiality parameter—I. Structure of fields

Abstract: Central to the J-based fracture mechanics approach is the existence of a HRR near-tip field which dominates the actual field over size scales comparable to those over which the micro-separation processes are active. There is now general agreement that the applicability of the J-approach is limited to so-called high-constraint crack geometries. We review the J-annulus concept and then develop the idea of a J-Q annulus. Within the J-Q annulus, the full range of high- and low-triaxiality fields are shown to be members of a family of solutions parameterized by Q when distances are measured in terms of J σ 0 , where σ0 is the yield stress. The stress distribution and the maximum stress depend on Q alone while J sets the size scale over which large stresses and strains develop. Full-field solutions show that the Q-family of fields exists near the crack tip of different crack geometries at large-scale yielding. The Q-family provides a framework for quantifying the evolution of constraint as plastic flow progresses from small-scale yielding to fully yielded conditions, and the limiting (steady-state) constraint when it exist. The Q value of a crack geometry can be used to rank its constraint, thus giving a precise meaning to the term crack-tip constraints, a term which is widely used in the fracture literature but has heretofore been unquantified. A two-parameter fracture mechanics approach for tensile mode crack tip states in which the fracture toughness and the resistance curve depend on Q, i. JC(Q) and JR(Δa, Q), is proposed.

The effective mechanical properties of nonlinear isotropic composites

Abstract: A new variational structure is proposed that yields a prescription for the effective energy potentials of nonlinear composites in terms of the corresponding energy potentials for linear composites with the same microstructural distributions. The prescription can be used to obtain bounds and estimates for the effective mechanical properties of nonlinear composites from any bounds and estimates that may be available for the effective properties of linear composites. The main advantages of the procedure are the simplicity of its implementation, the generality of its applications and the strength of its results. The general prescription is applied to three special nonlinear composites : a porous material, a two-phase incompressible composite and a rigidly reinforced material. The results are compared with previously available results for the special case of power-law constitutive behavior.

A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating Mullins' effect

Abstract: The problem of Mullins' effect in carbon black-filled rubbers is treated from a micro-mechanical viewpoint and a complete continuum constitutive model is developed. To begin with, a first-order accurate free energy function is derived for the composite in terms of the free energy densities of the constituents. Second, an exact relation between averaged macroscopic nonlinear strain measures and averaged nonlinear matrix material strain measures is derived under the assumption of affinely rotating particles and a C0 continuous motion. Third, the notion of strain-induced matrix-particle debonding is incorporated into the free energy density for the material by exploiting ideas from statistical mechanics. The accuracy of the resulting constitutive model is then demonstrated via comparisons to published experimental data.

A micromechanical model of tension-softening and bridging toughening of short random fiber reinforced brittle matrix composites

Abstract: A micromechanical model has been formulated for the post-cracking behavior of a brittle matrix composite reinforced with randomly distributed short fibers. This model incorporates the mechanics of pull-out of fibers which are inclined at an angle to the matrix crack plane and which undergo slip-weakening or slip-hardening during the pull-out process. In addition, the random location and orientation of fibers are accounted for. Comparisons of model predictions of post-cracking tension-softening behavior with experimental data appear to support the validity of the model. The model is used to examine the effects of fiber length, snubbing friction coefficient and interfacial bond behavior on composite post-cracking tensile properties. The scaling of the bridging fracture toughening with material parameters is discussed.

The effect of non-singular stresses on crack-tip constraint

Abstract: The effect of the T-stress on the small-scale yielding field of a crack in plane strain conditions has been examined using modified boundary layer formulations. The numerically calculated stresses at the crack tip are represented by slip line fields for small-strain theory. Positive T-stresses cause plasticity to envelop the crack tip and exhibit a Prandtl field, corresponding to the limiting solution of the HRR field for a nonhardening material. Moderate compressive T-stresses reduce the direct stresses within the plastic zone by decreasing the hydrostatic stress by T. This causes a loss of J-dominance, and a stress distribution represented by an incomplete Prandtl field.

Thermoelastic properties of particulate composites with imperfect interface

Abstract: Effective elastic moduli and thermal expansion coefficient of spherical particle composites with imperfect interfaces are evaluated on the basis of the composite spheres assemblage and generalized self-consistent scheme models. Imperfect interface is defined in terms of interface displacement discontinuities which are linearly related to interface tractions in terms of spring constant parameters. In the case of presence of interphase these parameters are evaluated in terms of interphase characteristics.

J-Dominance of short cracks in tension and bending

Abstract: Edge-Cracked bars with a W ratios less than 0.3 in bending and 0.5 in tension are shown to lose Jdominance. The loss of single parameter characterization is associated with the development of plasticity to the cracked face. Deeper cracks, for which plasticity develops through the ligament without spreading to the cracked face maintain J-dominance into large scale plasticity. The loss of J-dominance is attributed to compressive T-stresses, while geometries which exhibit tensile T-stresses retain J-dominance in accord with modified boundary layer formulations. The solutions for all the geometries can be characterized by J and Tinto large-scale plasticity.

Cavitation instabilities in elastic-plastic solids

Abstract: A cavitation instability occurs when an isolated void in an infinite, remotely stressed elastic-plastic solid grows without bound under no change of remote stress or strain. The cavitation instability can be thought of as a process in which elastic energy stored in the remote field drives the plastic expansion of the void. The paper begins with a brief review of cavitation under spherically symmetric stress states and then goes on to consider the problem for cavitation states under general axisymmetric stressing. It is found that the criterion for cavitation under multiaxial axisymmetric stressing depends on the attainment of a critical value of the mean stress, to a reasonably good approximation. A set of recent experiments is discussed in which cavitation instabilities appear to have occurred. The last section of the paper reviews available theoretical results for the dilatation rates of isolated voids. The most commonly used formulae underestimate the dilatation rate under stress states with moderate to high triaxiality.

On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media

Abstract: T he existence of diagonal symmetry in estimates of overall stiffness tensors of heterogeneous media is examined for several micromechanical models. The dilute approximation gives symmetric estimates for all matrix-based multiphase media. The Mori-Tanaka and the self-consistent methods do so for all two-phase systems, but only for those multiphase systems where the dispersed inclusions have a similar shape and alignment. However, the differential schemes associated with the self-consistent method can predict diagonally symmetric overall stiffness and compliance for multiphase systems of arbitrary phase geometry. A related question is raised about the equivalence of two possible approaches to evaluation of the overall thermal stress and strain tensors. A direct estimate follows from each of the above models, whereas L evin 's results [ Mechanics of Solids 2 , 58 (1967)] permit an indirect evaluation in terms of the estimated overall mechanical properties or concentration factors and phase thermoelastic moduli. These two results are shown to coincide for those systems and models which return diagonally symmetric estimates of the overall stiffness. Finally, model predictions of the overall elastic symmetry of composite media are discussed with regard to the spatial distribution of the phases.

On methods for bounding the overall properties of nonlinear composites

Abstract: A new method for bounding the overall properties of nonlinear composites, proposed byPonte Castaneda (J. Mech. Phys. Solids 39, 45, 1991), is compared with an older prescription based on a generalization to nonlinear behaviour of the Hashin-Shtrikman procedure. It is demonstrated that the two methods generate precisely the same information, and hence that differences noted by Ponte Castaneda arise from comparing optimal bounds obtained from the new procedure with sub-optimal bounds obtained from the original one. The relative advantages of either procedure are discussed.

A three-dimensional analysis of crack trapping and bridging by tough particles

Abstract: The toughness of a brittle material may be substantially improved by adding small quantities of tough particles to the solid. Three mechanisms may be responsible. Firstly, the front of a crack propagating through the solid can be trapped by the particles, causing it to bow out between them. Secondly, the particles may remain intact in the wake of the crack, thereby pinning its faces and reducing the crack tip stress intensity factors. Finally, the toughness may be enhanced by frictional energy dissipation as particles are pulled out in the wake of the crack. This paper estimates the improvement in toughness that might be expected due to these mechanisms, by means of a three-dimensional model. The analysis considers a semi-infinite crack propagating through a brittle matrix material, which contains a regular distribution of tough particles. Particles in the wake of the crack are modelled by finding an appropriate distribution of point forces that pin the crack faces; and the effect of the crack bowing between obstacles is included by means of an incremental perturbation method based on work byRice [J. Appl. Mech.56, 619 (1985)]. The calculation predicts the shape of the crack as it propagates through the solid; the resulting R-curve behaviour; and the length of the bridged zone in the wake of the crack.

On the temperature distribution at the vicinity of dynamically propagating cracks in 4340 steel

Abstract: T he heat generated due to plastic deformation at the tip of a dynamically propagating crack in a metal causes a large local temperature increase at the crack tip which is expected to affect the selection of failure modes during dynamic fracture and to thus influence the fracture toughness of the material. The distribution of temperature at the tips of dynamically propagating cracks in two heat treatments of AISI 4340 carbon steel was investigated experimentally using an array of eight high speed indium antimonide, infrared detectors. Experiments were performed on wedge loaded, compact tension specimens with initially blunted cracks, producing crack speeds ranging from 1900 to 730 m/s. The measurements provide the spatial distribution of temperature increase near the crack tip on the specimen surface. Temperature increases were as high as 465° C over ambient and the region of intense heating (greater than 100° C temperature rise) covered approximately one third of the active plastic zone on the specimen surface. The observed temperature increase profiles clearly show the three-dimensional nature of the fracture process near the specimen surface and provide valuable information regarding the dynamic formation of shear lips and their role in the dissipation of energy during dynamic crack growth. Preliminary temperature measurements performed on side-grooved specimens are also reported.

Determination of higher-order terms in asymptotic elastoplastic crack tip solutions

Abstract: A METHODOLOGY for the calculation of higher-order terms in asymptotic elastoplastic crack tip solutions is developed. The J2-deformation plasticity theory with power law hardening is used to describe the constitutive behavior of the continuum. A two-term expansion of the solution in the near crack tip region is developed. Plane stress and plane strain solutions for a crack in a homogeneous material as well as for a crack lying along the interface between a rigid substrate and an elastoplastic medium are obtained. For the case of a plane strain crack in a homogeneous material, it is shown that, when the hardening capacity of the material is small, the effects of elasticity enter the asymptotic solution to third order or higher, when there is substantial hardening, however, elastic effects enter the solution to second order and the magnitude of the second term in the expansion of the solution is controlled by the J-integral. THE CHARACTERIZATION of the stress and deformation fields in the region near the tip of a crack is essential for the development of sound fracture criteria. HUTCHINSON (1968) and RICE and ROSENGREN (1968) developed the elastoplastic asymptotic solution for the near-tip stresses in a homogeneous material (known as the HRR solution) and showed that the magnitude of the dominant term in the expansion of the solution is determined by the J-integral (RICE, 1968). If the region of dominance of the leading-order term in the expansion of the solution is sufficiently larger than the region over which the fracture micro-mechanisms take place, then the J-integral can be used as the fracture parameter. If the region of J-dominance, however, is smaller than the fracture process zone, then two or more parameters may enter the fracture criterion. LI and WANG 0986) suggested the use of a parameter k2, which is the magnitude of the second term in the near-tip stress plastic solution, as the second parameter to be used together with the J-integral in the fracture criterion. BETEGrN and HANCOCK (1991) used a modified boundary layer formulation of the small-scale yielding problem, in which the boundary conditions are defined in terms of the mode I stress intensity factor Kt and the constant stress term T that enters the near-tip expansion of the elastic solution (LARSSON and CARLSSON, 1973; RICE, 1974), and

Ductile failure of a constrained metal foil

Abstract: A metal foil bonded between stiff ceramic blocks may fail in a variety of ways, including de-adhesion of interfaces, cracking in the ceramics and ductile rupture of the metal. If the interface bond is strong enough to allow the foil to undergo substantial plastic deformation dimples are usually present on fracture surfaces and the nominal fracture energy is enhanced. Ductile fracture mechanisms responsible for such morphology include (i) growth of near-tip voids nucleated at second-phase particles and or interface pores, (ii) cavitation and (iii) interfacial debonding at the site of maximum stress which develops at distances of several foil thicknesses ahead of the crack tip. For a crack in a low to moderately hardening bulk metal, it is known that the maximum mean stress which develops at a distance of several crack openings ahead of the tip does not exceed about three times the yield stress. In contrast, the maximum mean stress that develops at several foil thicknesses ahead of the crack tip in a constrained metal foil can increase continuously with the applied load. Mean stress and interfacial traction of about four to six times the yield of the metal foil can trigger cavitation and/or interfacial debonding. The mechanical fields which bear on the competition between failure mechanisms are obtained by a large deformation finite element analysis. Effort is made to formulate predictive criteria indicating, for a given material system, which one of the several mechanisms operates and the relevant parameters that govern the nominal fracture work. The shielding of the crack tip in the context of ductile adhesive joints, due to the non-proportional deformation in a region of the order of the foil thickness, is also discussed.

Stress concentration at slightly undulating surfaces

Abstract: T his paper presents a first-order perturbation analysis of the stress concentration effects caused by slightly undulating surfaces. The perturbation approach that we use treats the undulating surfaces as being perturbed from a reference state in which the surface is perfectly flat. The magnitude of the perturbation is assumed to be sufficiently small compared to other length scales of the bulk material so that a half-plane model can be used for simplification. First-order-accurate perturbation solutions have been derived for the stress distribution along a sinusoidally wavy surface and for the attenuation of the stress concentration away from the undulating surface. The interactions among different surface perturbation waves are investigated by comparing the result of stress concentration factor at the trough of a single wave perturbation along an otherwise flat surface to that for periodically wavy surface. We also examine some of the 3-D effects by using the perturbation algorithm to calculate the stress concentration at undulating surfaces of elastic half-spaces. In all cases, it is found that wavy surfaces can magnify the bulk stress easily by a factor of 2 or 3 when the surface profile does not deviate substantially from flatness. This stress concentration effect is significant especially for already highly stressed heteroepitaxial semiconductor thin films, suggesting that the surface morphology of the film surfaces can play an important role in nucleating dislocations and crack-like surface flaws before the bulk stress reaches a critical level.

A micromechanics constitutive model of transformation plasticity with shear and dilatation effect

Abstract: B ased on micromechanics, thermodynamics and microscale t → m transformation mechanism considerations a micromechanics constitutive model which takes into account both the dilatation and shear effects of the transformation is proposed to describe the plastic, pseudoelastic and shape memory behaviors of structural ceramics during transformation under different temperatures. In the derivation, a constitutive element (representative material sample) was used which contains many of the transformed m-ZrO2 grains or precipitates as the second phase inclusions embedded in an elastic matrix. Under some basic assumptions, analytic expressions for the Helmholtz and complementary free energy of the constitutive element are derived in a self-consistent manner by using the Mori-Tanaka method which takes into account the interaction between the transformed inclusions. The derived free energy is a function of externally applied macroscopic stress (or strain), temperature, volume fraction of transformed phase and the averaged stressfree transformation strain (eigenstrain) of all the transformed inclusions in the constitutive element, the latter two quantities being considered to be the internal variables describing the micro-structural rearrangement in the constitutive element. In the framework of the Hill-Rice internal variable constitutive theory, the transformation yield function and incremental stress strain relations, in analogy to the theory of metal plasticity, for proportional and non-proportional loading histories are derived, respectively. The theoretical predictions are compared with the available experimental data of Mg-PSZ and Ce-TZP polycrystalline toughening ceramics.

A dynamic indentation technique for the characterization of the high strain rate plastic flow behaviour of ductile metals and alloys

Abstract: The objective of the paper is to describe a dynamic indentation (DI) technique suitable for the determination of the high strain rate flow behaviour of ductile metals and alloys and illustrate its use by characterizing the high strain rate flow behaviour of iron and OFHC copper. The DI technique is first described in detail and the dynamic hardness-strain data of iron and copper obtained using the technique is presented. It is also demonstrated that it is a suitable technique for characterizing the high strain rate flow behaviour as long as certain validity conditions are met. It is shown that these validity conditions are fully met in the case of copper and at low strain levels in iron. The reliability of the DI technique is finally demonstrated by comparing the present data with the literature data on similar materials and finally a critique of the DI technique is provided.

Approximations for the strength distribution and size effect in an idealized lattice model of material breakdown

Abstract: Various random network models have been developed recently to explain certain features of fracture development in materials, including the character of ‘cracks’, the form of the strength distribution, the size effect, and the connection to percolation theory. Applications include fibrous composites, random fuse networks, superconducting networks, dielectrics and elastic lattices. Because of extreme analytical difficulties researchers have relied on Monte Carlo simulation to validate various scaling hypotheses and approximations. Since only small network sizes are presently accessible, features which eventually emerge at the largest scale may not be uncovered. To shed light on this issue we consider a simple, idealized model where elements have strength zero or one with probabilities α or 1 − α, respectively. The load of a failed element is redistributed equally onto the nearest surviving neighbors, and open boundary conditions are considered for simplicity of calculation. Various exact, asymptotic and numerical results are obtained including a careful evaluation of any errors. Features of the results are in conflict with some of those in the literature for more complex stress redistribution situations. For α close to one, the mean strength of the network is dominated by small-scale bm boundary effects which may persist up to relatively large network sizes (1000 × 1000) before large-scale effects ultimately dominate.

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

Abstract: 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.

Acceleration waves, flutter instabilities and stationary discontinuities in inelastic porous media

Abstract: coupled constitutive equations for a saturated porous medium are developed in the frame of the theories of mixtures : the fluid constituent is elastic and the solid skeleton behaves as a rate-independent elastic-plastic solid. Then the existence of real acceleration wave-speeds is considered : actually the analysis centers on the modes in which these wave-speeds cease to be real. An explicit criterion indicating the critical value of the plastic modulus at the onset of a stationary discontinuity (one wave-speed is zero) is derived in both cases where the fluid and solid constituents are compressible and where they are not. Furthermore, it is shown that in some circumstances, some wave-speeds cease to be real in the very early stages of the inelastic deformation process due to the incipience of a flutter instability (two wave-speeds are complex conjugate). When it is not excluded, this mode of loss of hyperbolicity of the dynamic equilibrium equations usually precedes the onset of a stationary discontinuity and may occur right at the inception of plastic loading, that is for an infinitely large plastic modulus. Flutter instability is excluded when the plastic behavior of the solid skeleton is associative and its existence depends strongly on the relative positions of the shear and longitudinal elastic wave-speeds. It is not likely to occur if the shear wave-speed is the smallest elastic wave-speed and then a stationary discontinuity prevails.

Shear wave scattering from a debonded fibre

Abstract: Scattering of an antiplane shear wave from a single fibre partially bonded to a matrix is considered. The region of debonding is modeled as an interface crack with non-contacting faces; the crack opening displacement is represented by Chebyshev polynomials, and a system of equations is derived for the unknown coefficients. The solution is valid for arbitrary values of K 1 a, where K 1 is the incident wavenumber and a the fibre radius; and the semi-angle δ subtended by the crack may vary from zero to π. The general solution simplifies in two limiting situations: (i) if δ is small, and k 1 aδ is also small, then the wavelength greatly exceeds the crack length and the crack is effectively subject to a static loading determined by the dynamic field around the perfectly bonded fibre. Explicit expressions can be obtained for the COD and the scattered field in this case; (ii) when the crack semi-angle is finite, but k 1 a is small, then both the field in the fibre and the crack loading are quasistatic. The dependence of the scattered field on δ is particularly simple; however, the quasistatic theory breaks down at surprisingly low values of k1 a when the fibre becomes almost completely separated from the matrix. The narrow neck joining the matrix and fibre permits the fibre to oscillate at very low frequencies and causes a strong resonance in the scattering cross-section. The general and approximate results for the single fibre are used to estimate the attentuation of a wave propagating through a composite with many fibres and the possibility of quantitative detection of debonding using ultrasound attenuation measurements is discussed.

The formation of filamentary voids in solids

Abstract: We show that solutions which create spherical voids are unstable relative to the formation of long, thin voids that point in the radial direction, for a large class of nonlinearly elastic materials. We compare sufficient conditions for the formation of such “filamentary voids” to criteria for crazing in glassy polymers.

Polycrystalline configurations that maximize electrical resistivity

Abstract: A lower bound on the effective conductivity tensor of polycrystalline aggregates formed from a single basic crystal of conductivity σ was recently established by Avellaneda. Cherkaev, Lurie and Milton. The bound holds for any basic crystal, but for isotropic aggregates of a uniaxial crystal, the bound is achieved by a sphere assemblage model of Schulgasser. This left open the question of attainability of the bound when the crystal is not uniaxial. The present work establishes that the bound is always attained by a rather large class of polycrystalline materials. These polycrystalline materials, with maximal electrical resistivity, are constructed by sequential lamination of the basic crystal and rotations of itself on widely separated length scales. The analysis is facilitated by introducing a tensor s = 0(0I + σ)−1 where 0 > 0 is chosen so that Tr s = 1. This tensor s is related to the electric field in the optimal polycrystalline configurations.

Interface cracks in anisotropic dissimilar materials : An analytic solution

Abstract: Based on linear elastic fracture mechanics, analytic solutions are given for displacement and stress fields of in-plane deformation of two anisotropic half-planes, forming a composite bimaterial, with an interface crack, assuming strictly two-dimensional problems; this requires suitable orientation of the material symmetry axes to ensure decoupling of the anti-plane fracture mode from the in-plane ones. It is shown that the field equations are fully characterized in terms of four dimensionless parameters, and these parameters are expressed in terms of the twelve involved elastic constants, six for each half-plane. Analytic solutions are given for two models: (1) the fully open-crack model, involving oscillatory square-root singularities at crack tips ; and (2) the Comninou model which allows possible small contact zones close to the crack tips. Analytic expressions are obtained for the crack opening displacement, the size of the contact zone, the total force transmitted across the contact zone, and the stress field. The results are discussed and related to those for isotropic bimaterials, given by Gautesen and Dundurs.

Localized Buckling in Long Axially-Loaded Cylindrical-Shells

Abstract: D ouble-scale perturbation analysis of a long elastic cylindrical shell under axial compression reveals a second-order non-linear differential equation for the buckle-pattern amplitude in slow-space. Numerical solution then suggests that the most easily-triggered failure mode is localized along the length. The method is extended to include mode interaction, giving three coupled second-order non-linear differential equations in slow-space. Localized solutions are again found, by combining features of the Lagrangian function with a systematic numerical search procedure. The predicted extent of the localization, about one-and-a-half axial wavelengths when fully developed, compares well with published experiments on long cylinders. Moreover, in contrast to the associated periodic solutions, “square” waves at the minimum critical load are denied ; the predominant waveform turns out to be long axially, again as seen experimentally.

An elastoplastic analysis of the interface crack with contact zones

Abstract: AN ELASTOPLASTIC solution for the interface crack with contact zones is presented in this paper. The elastic asymptotic solution is discussed first and approximate expressions for the Comninou stress intensity factors of a Griffith interface crack are obtained. It is shown that the elastic asymptotic solutions based on the assumption of a closed crack tip predict material interpenetration : material elements near the crack tip are “turned inside out” and cross the bimaterial interface. The problem of a crack along the interface of an elastoplastic material and a rigid substrate is analyzed in detail. Any contact between the crack surface and the substrate is assumed to be frictionless. An elastoplastic asymptotic solution for a closed crack tip under plane strain conditions is presented. It is shown that the asymptotic solution is separable in r and 0, where (r, 0) are polar coordinates at the crack tip. A boundary layer approach is used to study the near tip elastoplastic fields under small-scale yielding conditions. The finite element method is used for the solution of the small-scale yieldmg problem and the validity of the developed plastic asymptotic solution is verified numerically. Large-scale yielding solutions for a Griffith interface crack with contact zones are also presented.

Thermorheologically complex behavior of multi-phase viscoelastic materials

Abstract: T he dynamic correspondence principle of viscoelasticity is used to study the nature of time-temperature behavior of multi-phase composites by means of finite element computation. The composite considered contains viscoelastic inclusions embedded in a viscoelastic matrix. Each phase of the composite is considered to be thermorheologically simple, but the resulting mechanical properties of the composite are thermorheologically complex. The deviation of the composite moduli from thermorheologically simple behavior of the matrix material is shown to occur at frequencies and temperatures where the glass-to-rubber transition of the included phases are reached. Properties of a styrene-butadiene-styrene (SBS) block copolymer are investigated based on the individual phase properties of polystyrene and polybutadiene. To achieve congruence of the results with experimental data, it is necessary to consider a transition phase of properties “intermediate” to those of polystyrene and polybutadiene. Using accurate physical information on the individual phase properties and on the interphase region, it is possible to predict properties of multiphase composites. Although detailed a priori knowledge of such an interphase is usually lacking, it is shown that the computational procedure presented here together with an extended range of test frequencies will aid in estimating the properties of the phase in question.

Effect of Plastic Spin on Localization Predictions for a Porous Ductile Material

Abstract: Various constitutive frameworks for macroscopic large strain elastoplasticity have recently identified the plastic spin as one of the key concepts in the description of anisotropic hardening. These theories involve a particular corotational stress rate that differs from the Jaumann stress rate by terms involving the plastic spin. This stress rate is introduced into a recently proposed material model for combined isotropic and kinematic hardening of a porous ductile solid. The plastic spin is taken to be governed by the simplest possible constitutive law, involving only one additional material parameter. An analysis of so-called unconstrained shearing is used to illustrate the effect of plastic spin on the stress response at large strains and finite material rotations. The effect of the plastic spin, via the corotational stress rate, on the predictions of strain localization in shear bands is studied in terms of simple model analyses. Results for various deformation and void nucleation conditions are discussed. The plastic spin is found to have a significant influence on the onset of localization even though the material rotations are still rather small at that instant. Also the progressive evolution of the shear band after localization until ductile fracture occurs in a void-sheet is strongly affected.

The finite deformation of rate-dependent polycrystals—I. A self-consistent framework

Abstract: A self-consistent model is developed for the finite deformation of rate-dependent polycrystals. The formulation presented rigorously extends H utchinson 's (Proc. R. Soc. (Land.)A319, 247, 1970) implementation of H ill 's (J. Mech. Phys. Solids13, 89, 1965a) theory to include finite deformations and material strain rate sensitivity. The present framework is similar to the recent formulation of N emat -N asser and O bata (Proc. R. Soc. (Lond.)A407, 343, 1986), except that the averaging operations are approached from a different viewpoint. Useful formulae for the numerical implementation of the theory are also presented.

The finite deformation of rate-dependent polycrystals—II: A comparison of the self-consistent and Taylor methods

Abstract: A self-consistence formulation presented in a previous paper (H arren , J. Mech. Phys. Solids39, 345, 1991) is implemented numerically. Aggregates composed of face centered cubic crystals which deform by rate sensitive crystallographic slip are considered. Self-consistent predictions of overall stress strain response and texture development are presented for polycrystals finitely deformed under plane strain compression and uniaxial tension. These predictions are compared to concomitant results obtained from the generalized Taylor model of A saro and N eedleman (Acta Metall. 33, 923, 1985). Discussions concerning the numerical implementation are also included.