Showing papers in "Journal of The Mechanics and Physics of Solids in 1993"

A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials

Abstract: Aconstitutive model is proposed for the deformation of rubber materials which is shown to represent successfully the response of these materials in uniaxial extension, biaxial extension, uniaxial compression, plane strain compression and pure shear. The developed constitutive relation is based on an eight chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network. The eight chain model accurately captures the cooperative nature of network deformation while requiring only two material parameters, an initial modulus and a limiting chain extensibility. Since these two parameters are mechanistically linked to the physics of molecular chain orientation involved in the deformation of rubber, the proposed model represents a simple and accurate constitutive model of rubber deformation. The chain extension in this network model reduces to a function of the root-mean-square of the principal applied stretches as a result of effectively sampling eight orientations of principal stretch space. The results of the proposed eight chain model as well as those of several prominent models are compared with experimental data of Treloar (1944, Trans. Faraday Soc. 40, 59) illustrating the superiority, simplicity and predictive ability of the proposed model. Additionally, a new set of experiments which captures the state of deformation dependence of rubber is described and conducted on three rubber materials. The eight chain model is found to model and predict accurately the behavior of the three tested materials further confirming its superiority and effectiveness over earlier models.

A phenomenological theory for strain gradient effects in plasticity

Abstract: A Strain Gradient Theory of plasticity is introduced, based on the notion of statistically stored and geometrically necessary dislocations. The strain gradient theory fits within the general framework of couple stress theory and involves a single material length scale l. Minimum principles are developed for both deformation and flow theory versions of the theory which in the limit of vanishing l, reduce to their conventional counterparts: J2 deformation and J2 flow theory. The strain gradient theory is used to calculate the size effect associated with macroscopic strengthening due to a dilute concentration of bonded rigid particles; similarly, predictions are given for the effect of void size upon the macroscopibic softening due to a dilute concentration of voids. Constitutive potentials are derived for this purpose.

Compressive failure of fibre composites

Abstract: A review of experimental data and elementary theoretical formulas for compressive failure of polymer matrix fibre composites indicates that the dominant failure mode is by plastic kinking. Initial local fibre misalignment plays a central role in the plastic kinking process. Theoretical analyses and numerical results for compressive kinking are presented, encompassing effects of strain-hardening, kink inclination, and applied shear stress. The assumption of rigid fibres is assessed critically, and the legitimacy of its use for polymer matrix composites is established.

The influence of plasticity on mixed mode interface toughness

Abstract: Calculations are reported for the mixed mode toughness of an interface joining an elastic-plastic solid to a solid which does not yield plastically. A potential function of the components of the crack face displacements is used to generate the tractions along the interface where the fracture processes causing separation occur. The two main parameters characterizing this potential are the work of separation per unit area and a peak normal stress. This description of the interface separation process is embedded within the continuum description as a boundary condition on the interface linking the adjoining solids. Small-scale yielding in plane strain is considered with the remote field specified by the magnitude and phase of the mixed mode stress intensity factors. Crack growth resistance curves are computed for a range of the most important nondimensional material parameters and for various combinations of remote mixed mode loading. Particular emphasis is placed on the ratio of the steady-state interface toughness to the “intrinsic” work of separation as it depends on plastic yielding and on the combination of modes 1 and 2. Plasticity enhances the interface toughness for all modes of loading, but substantially more so in the presence of a significant mode 2 component of loading than in near-mode 1 conditions.

A general anisotropic yield criterion using bounds and a transformation weighting tensor

Abstract: A gspeneral Expression for the yield surface of polycrystalline materials is developed. The proposed yield surface can describe both isotropic and anisotropic materials. The isotropic surface can be reduced to either the Tresca or von Mises surface if appropriate, or can be used to capture the yield behavior of materials (e.g. aluminum) which do not fall into either category. Anisotropy can be described by introducing a set of irreducible tensorial state variables. The introduced linear transformation is capable of describing different anisotropic states, including the most general anisotropy (triclinic) as opposed to existing criteria which describe only orthotropic materials. Also, it can successfully describe the variation of the plastic strain ratio (R-ratio), where polycrystalline plasticity models usually fail. A method for obtaining the material constants using only uniaxial test data is described and utilized for the special case of orthotropic anisotropy. Finally, the use of tensorial state variables together with the introduced mathematical formulation make the proposed yield function a very convenient tool for numerical implementation in finite element analysis.

Approximate models for ductile metals containing non-spherical voids—Case of axisymmetric prolate ellipsoidal cavities

Abstract: T he aim of this paper is to extend the classical Gurson analysis of a hollow rigid ideal-plastic sphere loaded axisymmetrically to an ellipsoidal volume containing a confocal ellipsoidal cavity, in order to define approximate models for ductile metals containing non-spherical voids. Only axisymmetric prolate cavities are considered here. The analysis makes an essential use of an “expansion” velocity field satisfying conditions of homogeneous boundary strain rate on every ellipsoid confocal with the cavity. A two-field estimate of the overall yield criterion is presented and shown to be reducible, with a few approximations, to a Gurson-like criterion depending on the “shape parameter” of the cavity. The accuracy of this estimate is assessed through comparison with some results derived from a numerical minimization procedure. The two-field approach is also used to derive an approximate evolution equation for the shape parameter ; comparison with some finite element simulations reveals a reasonable qualitative agreement, and suggests a slight modification of the theoretical formula which leads to acceptable quantitative agreement. The application of these results to materials containing axisymmetric prolate ellipsoidal cavities with parallel or random orientations is finally discussed.

On Improved Network Models for Rubber Elasticity and Their Applications to Orientation Hardening in Glassy Polymers

Abstract: T hree-dimensional molecular network theories are studied which use a non-Gaussian statistical mechanics model for the large strain extension of molecules. Invoking an affine deformation assumption, the evolution of the network — consisting of a large number of molecular chains per unit volume, which are initially randomly oriented in space — is shown to be governed by a balance equation in orientation space. Eulerian and Lagrangian type formulations of these balance equations are given, and the closed-form analytical solution for the so-called Chain Orientation Distribution Function is derived. This full network model is then used to describe the large strain inelastic behaviour of rubber-like materials. Detailed comparisons with experimental results and with two approximate models, namely the classical three-chain model and a very recently proposed eight-chain model, are provided for different types of deformation and rubbers. Finally, the network model is applied to describe the orientational hardening in amorphous glassy polymers, and confronted with experimental data for polycarbonate. The inherent physical limitations of the network theory for both applications are discussed.

Micromechanics modelling for the constitutive behavior of polycrystalline shape memory alloys. I: Derivation of general relations

Abstract: A MICROMECHANICS constitutive model has been proposed in this paper to describe the pseudoelastic and shape memory behavior of polycrystalline shape memory alloys under various temperatures The derivation of the model is based on the thermodynamics, micromechanics and microstructural physical mechanism analysis of the material during deformation and it is shown that the inelastic deformation of the material in the mechanical and/or thermal loading processes is associated with some temperature, stress state and loading history dependent yielding surfaces which microscopically correspond to the forward and reverse transformation (or reorientation) processes, respectively

Overall potentials and extremal surfaces of power law or ideally plastic composites

Abstract: A method is proposed for bounding the overall properties of a class of composite materials in terms of the properties of the individual phases and of their arrangement. It applies to power law materials and, as a special case, to rigid ideally plastic materials. A link between the overall potential of a nonlinear composite and the overall energy of a fictitious linear composite is presented with no assumptions on the arrangement of the phases. With this method, any upper bound available for linear materials can easily be transposed to nonlinear materials. A new characterizing of the external surface of ideally plastic composites is given. The possible applications of these bounds are illustrated in a study on two-phase isotropic composites and the predictions of the bounds are compared with Finite Element cell calculations.

A continuum model of a thermoelastic solid capable of undergoing phase transitions

Abstract: We construct explicitly a Helmholtz free energy, a kinetic relation and a nucleation criterion for a one-dimensional thermoelastic solid, capable of undergoing either mechanically- or thermally-induced phase transitions. We study the hysteretic macroscopic response predicted by this model in the case of quasistatic processes involving stress cycling at constant temperature, thermal cycling at constant stress, or a combination of mechanical and thermal loading that gives rise to the shape-memory effect. These predictions are compared qualitatively with experimental results.

Models for breakdown-resistant dielectric and ferroelectric ceramics

Abstract: Models for dielectric breakdown are proposed and analysed, with emphasis on concepts leading to breakdown-resistant materials. The Griffith energy balance is extended to cracks under combined electrical and mechanical loading, and to conductive tubular channels. Breakdown strength for a perfect crystal is estimated by an analogue of the Frenkel model. In a crystal subjected to an electric field the equilibrium displacement of the electron clouds is described by a curve with periodicity of the lattice constant. A theory of breakdown-resistant laminates is proposed on the basis of charge relocation, facilitated by breakdown of the weak layers and the interfaces. A process by which a conducting path grows like a crack in ferroelectric ceramics is discussed, followed by an outline of fields around conducting cracks in piezoelectric ceramics.

Micromechanics modelling for the constitutive behavior of polycrystalline shape memory alloys. II: Study of the individual phenomena

Abstract: T he constitutive relation for various phenomena of SMA ( superelasticity, rubber-like elasticity, ferroelasticity, elastic anomaly, shape memory effect ) is studied in detail and compared with the available experimental data. It is shown that the micromechanical model developed in Part I can satisfactorily describe the main peculiarities of the macroscopic thermomechanical constitutive behavior in the course of uniaxial mchanical and/or thermal loadings and that the existing phenomenological models are special cases of the proposed theory under proportional loading conditions. Some theoretical predictions and discussions for complex loading paths are also given which are yet subject to experimental verification.

Micromechanical modeling of large plastic deformation and texture evolution in semi-crystalline polymers

Abstract: A micromechanically-based composite model is proposed to study large plastic deformation and texture evolution in semi-crystalline polymers. The microstructure of many semi-crystalline polymers consists of co-existing crystalline and amorphous phases locally associated with each other in a fine plate-like morphological structure. An aggregate of two-phase composite inclusions is used to model these materials. Special consideration is given to molecular chain inextensibility within the crystalline phase. The introduction of a back stress tensor in the constitutive model of the amorphous phase accounts for hardening due to deformation-induced molecular alignment. Interface compatibility and traction equilibrium are enforced within each composite inclusion. A Sachs-like model and two newly-developed self-consistent-like hybrid models are proposed to relate volume-average deformation and stress within the two-phase composite inclusion to the remote (macroscopic) fields. Applications of these composite models arc made to predict stress strain behavior and texture evolution in initially isolropic high density polyethylene (HOPE) under different modes of straining.

Theoretical investigation of phenomena caused by heterogeneous surface tension in solids

Abstract: From the analysis of the mechanical equilibrium of an interface between two different media (the generalized Laplace equation) it follows that, in addition to the discontinuity of the normal stress, there exists a discontinuity of the tangential stress across this interface due to the surface tension gradient. Using the solution of the corresponding problem for a system “elastic solid-elastic surface” the generalization of the Cassie equation for the contact angle on the heterogeneous surface is obtained and the level of stress arising from the interaction of surface-active melt with metal is estimated.

Optimal design of biaxial tensile cruciform specimens

Abstract: F or experimental investigations concerning the mechanical behaviour under biaxial stress states of rolled sheet metals, mostly cruciform flat specimens are used. By means of empirical methods, different specimen geometries have been proposed in the literature. In order to evaluate the suitability of a specimen design, a mathematically well defined criterion is developed, based on the standard deviations of the values of the stresses in the test section. Applied to the finite element method, the criterion is employed to realize the shape optimization of biaxial cruciform specimens for isotropic elastic materials. Furthermore, the performance of the obtained optimized specimen design is investigated in the case of off-axes tests on anisotropic materials. Therefore, for the first time, an original testing device, consisting of hinged fixtures with knife edges at each arm of the specimen, is applied to the biaxial test. The obtained results indicate the decisive superiority of the optimized specimens for the proper performance on isotropic materials, as well as the paramount importance of the proposed off-axes testing technique for biaxial tests on anisotropic materials.

Surface roughening and branching instabilities in dynamic fracture

Abstract: W hy are most experimentally observed terminal fracture speeds only around half of the theoretically predicted value, i.e. the Rayleigh wave speed c R ? A wavy-crack model to be discussed in this paper gives a short answer to the above question: at high speeds, cracks tend to propagate along a wavy fracture path so that the apparent crack velocity can be maintained at 0.5 c R in order to maximize the time-rate of energy being absorbed into the fracture process. The wavy-crack model is motivated by experimental observations that rapidly moving cracks develop roughened fracture surfaces. The essence of that model is to separate the microscopic crack-tip motion with local velocity v c from the macroscopically observable crack motion with apparent velocity v a . From the macroscopic point of view, the energy going into the fracture process per unit time per unit length of the crack front is approximately Γ a = (1- v a / c R ) v a G * a , where G * a denotes the quasi-static energy release rate. By propagating along a wavy path, a crack is able to maintain its apparent velocity v a at 0.5 c R to maximize Γ a , while it may locally propagate at a significantly higher velocity in accordance with the local energy balance between the crack driving force and the material resistance to fracture. Much theoretical investigation on the wavy-crack problem and applications remains to be done in future work. For present purposes, a perturbation analysis is used to gain some preliminary insights, particularly on issues regarding the stability of a mode I fracture path during dynamic crack propagation. The perturbation results are supportive of the notion that wavy fracture paths become favorable at high crack speeds, that the apparent crack which moves only at around 0.5 c R favors a mode I path and tends to suppress a branching tendency, and that the local crack tip motion with significantly higher velocity promotes crack branching. Discussions on various aspects of dynamic fracture indicate that the wavy-crack model is capable of explaining important discrepancies currently existing between theory and experiments. In particular, analyses indicate that the basic mechanism of dynamic crack branching is somewhat like a thermally activated kinetic process. The fracture energy supplied from the applied loads acts as the driving force, the high inertia, branch-promoting local crack tip field acts as a nucleation source of microbranches and the relatively low inertia, branch-suppressing apparent crack sets an energetic barrier for macroscopic branching. This energy barrier is controlled by the macroscopic non-singular T a stress, in that it may be increased by a more compressive T a and decreased by a less compressive T a .

Instability of a biaxially stressed thin film on a substrate due to material diffusion over its free surface

Abstract: The configuration of an elastically strained thin film bonded to a relatively thick elastic substrate over a plane interface is considered. The free energy of the system is taken to be the surface energy of the free surface, which is initially flat, and the elastic strain energy. It is assumed that the film material can change the shape of its free surface by means of mass diffusion along the surface, and that this mass transport occurs coherently. As a result of this diffusion process, the free energy of the system changes due to a change in surface shape and due to a change in the elastic energy. If the change in free energy as the surface shape departs from planar is positive, then this change will tend to occur spontaneously and the flat shape is unstable. The stability of a biaxially stressed thin film due to material diffusion over its surface is considered here under both two- and three-dimensional conditions. The stability condition is derived in the form of a difference between two positive definite quantities, one associated with surface energy and the other associated with strain energy, and the sign of this difference depends on the relative stiffnesses of the materials, the thickness of the film, the surface energy of the film material, and the initial elastic stress in the film. It is demonstrated that the configuration with a flat free surface is unstable under sinusoidal perturbations in the shape of the surface for any combination of parameters, provided that the wavelength of the perturbation is larger than some critical value. Numerical results are presented for the critical value as a function of film thickness for several values of the ratio of elastic stiffnesses of the materials.

Elastic moduli of a composite of hollow spheres in a matrix

Abstract: L ow density foams deform primarily by bending of the cell edges. Their mechanical properties could be improved by increasing the flexural rigidity of the wall. The introduction of a high volume fraction of thin walled hollow spheres into a foamed matrix produces a composite cellular solid with a sandwich cell wall: the walls of the spheres act as the faces of the sandwich while the foamed matrix acts as the core. As a first step to producing cellular solids with sandwich cell walls, in this paper we describe the elastic moduli of a composite cellular solid made by introducing thin walled hollow spheres into a solid matrix.

A theoretical and experimental investigation of buckling induced delamination growth

Abstract: A theoretical and experimental investigation of a laminated plate with an embedded, initially circular delamination, loaded in uniaxial compression is described. The main issue concerns the combined postbuckling and crack growth behaviour in general situations at arbitrary crack contours and applied to fibre-reinforced carbon-epoxy composite plates. In a numerical procedure a standard FE-code is used to solve the structural postbuckling problem and extended to account also for contact effects. To predict interlaminar crack growth an automatic finite element procedure was employed for the purpose. Delamination growth in laminates was detected by the aid of acoustic emission. After successive crack increments, the evolution of the delamination front was mapped using an ultrasonic C-scan technique and also for comparison by X-ray. Predicted and observed results are compared as regards postbuckling behaviour and in particular the magnitude and shapes of delamination growth fronts.

Highly transient elastodynamic crack growth in a bimaterial interface : higher order asymptotic analysis and optical experiments

Abstract: A higher Order asymptotic analysis of the transient deformation field surrounding the tip of a crack running dynamically along a bimaterial interface is presented. An asymptotic methodology is used to reduce the problem to one of the Riemann-Hilbert type. Its solution furnishes displacement potentials which are used to evaluate explicitly the near-tip transient stress field. Crack-tip fields corresponding to crack speeds up to the lower of the two shear wave speeds are investigated. An experimental study of dynamic crack growth in PMMA steel interfaces using the optical method of CGS and high speed photography, is also described. Transonic terminal speeds (up to 1.4cPMMAS) and initial accelerations ( $ 108 ms2) are reported and discussed. Transient effects are found to be severe and more important than in homogeneous dynamic fracture. For subsonic crack growth, these experiments arc used to demonstrate the necessity of employing a fully transient expression in the analysis of optical data to predict accurately the complex dynamic stress intensity factor history.

Coupled estimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite

Abstract: We improve the classical Hashin-Shtrikman and Walpole estimates of the effective properties for an isotropic mixture assembled from two isotropic elastic materials. The planar elasticity problem is considered. Unlike the prior estimates which bound the bulk and shear moduli independently, our estimates are coupled and more restrictive. The set of the bulk modulus-shear modulus pairs turns out to be bounded in the plane of the values of these moduli by two straight lines (the Hashin-Shtrikman or Walpole bulk modulus estimates) and also by two fractional linear curves. To obtain the new estimates we use the translation method, which provides a general approach to both Hashin-Shtrikman (well-ordered materials) and Walpole (badly-ordered materials) cases; the method also provides the estimates for anisotropic mixtures.

Steady-state flow in classical elastoplasticity: Applications to repeated rolling and sliding contact

Abstract: T he sthady-state assumption is used to analyse mechanical problems involving moving loads. A general formulation in the case of elastoplasticity is given. A uniqueness theorem is proved. An efficient and reliable algorithm for the calculation of stresses and strains either for an arbitrary number of loading passes or directly for the stabilized state is presented. Two numerical procedures are proposed: (i) the pass-by-pass stationary method (PPSM) which rigorously treats the case of a single load pass; (ii) the direct stationary method (DSM) going straight to the steady state in the case of a repeatedly moving load. Although the scope of application of these methods is wide, numerical results obtained in the case of rolling and sliding contact only are presented and compared with already published results.

Coupled stress-diffusion: Case II

Abstract: I n this paper , a macroscopic continuum formulation is developed for predicting Case II diffusion in polymers. This formulation is based on micromechanical considerations in that the free energy density for the solvent-polymer system is calculated using information from the microscopic structure of the system. Formulation here means a complete set of governing balance equations and constitutive laws that may be used to solve initial boundary value problems. The analysis begins with the development of the proper statements of momentum and mass balance for a mixture of a solvent and a polymer capable of undergoing a solvent induced rubber-glass transition. Subsequently, the expression for the local reduced entropy inequality is derived and used to determine the form of the constitutive relations in terms of the free energy function for the mixture. An extension of the Flory-Huggins free energy of mixing to the transient case is then presented. The result of this formulation is a complete set of coupled field equations. An example initial boundary value problem is then solved to demonstrate that the qualitative features of Case II diffusion behavior are replicated.

The tough to brittle transition in brittle matrix composites

Abstract: A THEoRY is presented to predict the ultimate tensile strength of brittle matrix composites as a function of underlying material parameters, and specifically to investigate the origin of the tough to brittle transition often observed in these materials as the fiber-matrix interfacial sliding resistance tau is increased. The theory relaxes the usual assumption of global load sharing of the load transfer from broken to unbroken fibers in the composite [CURTIN, W. A., J. Am. Ceram. Soc. 74, 2837 (1991)] by taking the load to be equally distributed among only N(f) fibers around a broken fiber (local load sharing). The composite is then modeled as a collection of independent fiber bundles with Nf fibers per bundle, and composite failure occurs when the weakest bundle fails. Composite strength is thus controlled by the strength distribution of size-N(f) bundles, which is calculated here by analytical and simulation techniques. As N(f) --textgreater infinity the global load sharing results for composite strengths are regained, but significant composite strength degradation is predicted for bundle sizes N(f) less-than-or-equal-to 100. An ansatz relating N(f) to material parameters is then proposed and calculations of the strengths of C-Nicalon composites agree well with experiment. Model calculations on a Nicalon-LAS glass composite show that local load sharing effects lead to a tough to brittle transition between 100 and 200 MPa, much lower than predicted by the global load sharing theory although still larger than found experimentally.

Higher-order analysis of crack tip fields in elastic power-law hardening materials

Abstract: A HIGHER-ORDER asymptotic analysis of a stationary crack in an elastic power-law hardening material has been carried out for plane strain, Mode 1. The extent to which elasticity affects the near-tip fields is determined by the strain hardening exponent n. Five terms in the asymptotic series for the stresses have been derived for n = 3. However, only three amplitudes can be independently prescribed. These are K1, K2 and K5 corresponding to amplitudes of the first-, second- and fifth-order terms. Four terms in the asymptotic series have been obtained for n = 5, 7 and 10; in these cases, the independent amplitudes are K1, K2 and K4. It is found that appropriate choices of K2 and K4 can reproduce near-tip fields representative of a broad range of crack tip constraints in moderate and low hardening materials. Indeed, fields characterized by distinctly different stress triaxiality levels (established by finite element analysis) have been matched by the asymptotic series. The zone of dominance of the asymptotic series extends over distances of about 10 crack openings ahead of the crack tip encompassing length scales that are microstructurally significant. Furthermore, the higher-order terms collectively describe a spatially uniform hydrostatic stress field (of adjustable magnitude) ahead of the crack. Our results lend support to a suggestion that J and a measure of near-tip stress triaxiality can describe the full range of near-tip states.

Model studies of fibre breakage and debonding in a metal reinforced by short fibres

Abstract: F or a metal reinforced by short, brittle fibres the development of damage by fibre cracking or decohesion is studied, taking into account the interaction between these two types of damage. The metal matrix composite, containing a periodic array of aligned fibres, is represented in terms of a cell model, and solutions for the stress and strain fields are determined numerically. Failure at the interface is modelled in terms of a cohesive zone model that accounts for decohesion by normal separation as well as by tangential separation, whereas fibre fracture is simply represented by a critical value of the average tensile stress on a cross-section. The effect of various material parameters and of the macroscopic stress state on the two types of damage is investigated, together with the subsequent mode of void growth.

Piezoelectric properties of multiphase fibrous composites: Some theoretical results

Abstract: A number of exact results are established for overall moduli of a piezoelectric composite medium consisting of many perfectly-bonded transversely isotropic phases of cylindrical shape and arbitrary transverse geometry. It is shown that for three-phase media of this type three universal relationships, which are independent of geometry at given volume fractions, connect six of these effective physical constants. When the phases have equal transverse rigidities in shear, exact values of certain overall moduli can be derived for multiphase systems. The explicit formulae depend solely on the concentrations and phase moduli and are unaffected by the transverse geometry of the inclusions. Specifically, seven out of a total of 10 overall moduli of a transversely isotropic composite can be found. The remaining three constants p, e15 and k11 are shown to obey an exact relation, which also applies to other physical phenomena, such as magnetoelectric and thermoelectric effects. The result is a generalization of the relations found by H ill [J. Mech. Phys. Solids12, 199 (1964)] for purely elastic media and by Mendelson[J. Appl. Phys.46, 917 (1975)] for the purely dielectric problem.

An analysis of texture and plastic spin for planar polycrystals

Abstract: A major source of induced anisotropy in metals undergoing large strain is the preferential reorientation of single crystals. We present a macroscopic description of this textural anisotropy for an idealized planar aggregate of single crystals with two slip systems. We derive an analytical expression for the plastic spin associated with crystallographic slip and use it to obtain an equation of evolution for the single crystal orientation. The single microstructural parameter that appears in this equation is defined in terms of the slip system geometry. We introduce a continuous distribution function to describe orientation of crystals in an aggregate and obtain analytical solutions to the conservation equation governing its evolution. Such solutions are either monotonic or periodic, depending upon the relative magnitudes of stretching, spin and the microstructural parameter. Using an orientation average, we determine the average plastic spin in terms of the microstructural parameter and a second rank tensor related to the anisotropy in the orientation distribution. Finally, for constant velocity gradients, we show that the eigenvectors of this tensor rotate with half the difference between the macroscopic and average plastic spins.

Multi-fracture of ceramic composites

Abstract: This work presents a mechanistic model for the multi-fracture process of uniaxially reinforced fibrous ceramic composites under monotonically increasing tension parallel to the fiber direction. The model employs an energy criterion to account for the progression of matrix cracks, bridged by intact fibers, and Weibull failure statistics to relate the failure of the fibers. Consideration is given to the interactions between the foregoing failure processes as well as to the effects of various material parameters on the response of the composite.

Quasi-static crack advance under a range of constraints—Steady-state fields based on a characteristic length

Abstract: A numerical investigation of a crack growing under steady-state, quasi-static conditions has been performed within the framework of a boundary layer formulation whereby the remote loading is fully specified by the first two terms in Williams' expansion, characterized by k 1 and T . Mode I, plane strain crack tip fields have been obtained for strain-hardening and non-hardening materials over a wide range of K 1 and T combinations. A length scale for the boundary layer problem is ( K 1 /σ 0 ) 2 , where σ 0 is the material's yield stress in tension. Rescaling physical coordinates by ( K 1 /σ 0 ) 2 results in a family of self-similar solutions parameterized by T /σ 0 . Moreover, these fields can be arranged into a one-parameter near-tip field based on a characteristic length L g , which scales with the smallest dimension of the plastic zone. Specifically, the numerically determined fields collapse into a single near-tip distribution when physical coordinates are rescaled by L g . Thus loading and crack geometry enter into the description of the near-tip field only through L g , which therefore scales the intensity of the near-tip fields. Consequently, a one-parameter crack growth criterion is rigorously valid for steady crack growth under well-contained yielding, when the oneparameter field dominates over microstructurally significant size scales, i.e. any postulated local fracture criterion can be expressed as the requirement that L g attains a critical value L gc . The latter provides a single, unified criterion to assess quantitatively loading and crack geometry effects on fracture toughness.