Showing papers in "Journal of The Mechanics and Physics of Solids in 2000"

Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces

Abstract: Some materials may naturally form discontinuities such as cracks as a result of deformation. As an aid to the modeling of such materials, a new framework for the basic equations of continuum mechanics, called the "peridynamic" formulation, is proposed. The propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived. Material stability and its connection with wave propagation is investigated. It is demonstrated by an example that the reformulated approach permits the solution of fracture problems using the same equations either on or off the crack surface or crack tip. This is an advantage for modeling problems in which the location of a crack is not known in advance.

Numerical experiments in revisited brittle fracture

Abstract: The numerical implementation of the model of brittle fracture developed in Francfort and Marigo (1998. J. Mech. Phys. Solids 46 (8), 1319–1342) is presented. Various computational methods based on variational approximations of the original functional are proposed. They are tested on several antiplanar and planar examples that are beyond the reach of the classical computational tools of fracture mechanics.

Isotropic constitutive models for metallic foams

Abstract: The yield behaviour of two aluminium alloy foams (Alporas and Duocel) has been investigated for a range of axisymmetric compressive stress states. The initial yield surface has been measured, and the evolution of the yield surface has been explored for uniaxial and hydrostatic stress paths. It is found that the hydrostatic yield strength is of similar magnitude to the uniaxial yield strength. The yield surfaces are of quadratic shape in the stress space of mean stress versus effective stress, and evolve without corner formation. Two phenomenological isotropic constitutive models for the plastic behaviour are proposed. The first is based on a geometrically self-similar yield surface while the second is more complex and allows for a change in shape of the yield surface due to differential hardening along the hydrostatic and deviatoric axes. Good agreement is observed between the experimentally measured stress versus strain responses and the predictions of the models.

An extended model for void growth and coalescence

Abstract: A model for the axisymmetric growth and coalescence of small internal voids in elastoplastic solids is proposed and assessed using void cell computations. Two contributions existing in the literature have been integrated into the enhanced model. The first is the model of Gologanu-Leblond-Devaux, extending the Gurson model to void shape effects. The second is the approach of Thomason for the onset of void coalescence. Each of these has been extended heuristically to account for strain hardening. In addition, a micromechanically-based simple constitutive model for the void coalescence stage is proposed to supplement the criterion for the onset of coalescence. The fully enhanced Gurson model depends on the flow properties of the material and the dimensional ratios of the void-cell representative volume element. Phenomenological parameters such as critical porosities are not employed in the enhanced model. It incorporates the effect of void shape, relative void spacing, strain hardening, and porosity. The effect of the relative void spacing on void coalescence, which has not yet been carefully addressed in the literature. has received special attention. Using cell model computations, accurate predictions through final fracture have been obtained for a wide range of porosity, void spacing, initial void shape, strain hardening, and stress triaxiality. These predictions have been used to assess the enhanced model. (C) 2000 Elsevier Science Ltd. All rights reserved.

Mechanism-based strain gradient plasticity—II. Analysis

Abstract: A mechanism-based theory of strain gradient (MSG) plasticity has been proposed in Part I of this paper. The theory is based on a multiscale framework linking the microscale notion of statistically stored and geometrically necessary dislocations to the mesoscale notion of plastic strain and strain gradient. This theory is motivated by our recent analysis of indentation experiments which strongly suggest a linear dependence of the square of plastic flow stress on strain gradient. Such a linear dependence is consistent with the Taylor plastic work hardening model relating the flow stress to dislocation density. This part of this paper provides a detailed analysis of the new theory, including equilibrium equations and boundary conditions, constitutive equations for the mechanism-based strain gradient plasticity, and kinematic relations among strains, strain gradients and displacements. The theory is used to investigate several phenomena that are influenced by plastic strain gradients. In bending of thin beams and torsion of thin wires, mechanism-based strain gradient plasticity gives a significant increase in scaled bending moment and scaled torque due to strain gradient effects. For the growth of microvoids and cavitation instabilities, however, it is found that strain gradients have little effect on micron-sized voids, but submicron-sized voids can have a larger resistance against void growth. Finally, it is shown from the study of bimaterials in shear that the mesoscale cell size has little effect on global physical quantities (e.g. applied stresses), but may affect the local deformation field significantly.

On the plasticity of single crystals: free energy, microforces, plastic-strain gradients

Abstract: This study develops a general theory of crystalline plasticity based on classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on plastic strain-gradients. The microforce balances are shown to be equivalent to yield conditions for the individual slip systems, conditions that account for variations in free energy due to slip. When this energy is the sum of an elastic strain energy and a defect energy quadratic in the plastic-strain gradients, the resulting theory has a form identical to classical crystalline plasticity except that the yield conditions contain an additional term involving the Laplacian of the plastic strain. The field equations consist of a system of PDEs that represent the nonlocal yield conditions coupled to the classical PDE that represents the standard force balance. These are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. A viscoplastic regularization of the basic equations that obviates the need to determine the active slip systems is developed. As a second aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. As an application of the theory, the special case of single slip is discussed. Specific solutions are presented: one a single shear band connecting constant slip-states; one a periodic array of shear bands.

A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals

Abstract: The energy balance equation for elastoplastic solids includes heat source terms that govern the conversion of some of the plastic work into heat. The remainder contributes to the stored energy of cold work due to the creation of crystal defects. This paper is concerned with the fraction β of the rate of plastic work converted into heating. We examine the status of the common assumption that β is a constant with regard to the thermodynamic foundations of thermoplasticity and experiments. A general internal-variable theory is introduced and restricted to abide by the second law of thermodynamics. Experimentally motivated assumptions reduce this theory to a special model of classical thermoplasticity. The only part of the internal energy not determined from the isothermal response is the stored energy of cold work, a function only of the internal variables. We show that this function can be inferred from stress and temperature data from a single adiabatic straining experiment. Experimental data from dynamic Kolsky-bar tests at various strain rates yield a unique stored energy function. Its knowledge is crucial for the determination of the thermomechanical response in non-isothermal processes. Such a prediction agrees well with results from dynamic tests at different rates. In these experiments, β is found to depend strongly on both strain and strain rate for various engineering materials. The model is successful in predicting this dependence. Requiring β to be constant is thus an approximation of dubious validity.

Lattice incompatibility and a gradient theory of crystal plasticity

Abstract: In the finite-deformation, continuum theory of crystal plasticity, the lattice is assumed to distort only elastically, while generally the elastic deformation itself is not compatible with a single-valued displacement field. Lattice incompatibility is shown to be characterized by a certain skew-symmetry property of the gradient of the elastic deformation field, and this measure can play a natural role in a nonlocal, gradient-type theory of crystal plasticity. A simple constitutive proposal is discussed where incompatibility only enters the instantaneous hardening relations, and thus the incremental moduli, which preserves the classical structure of the incremental boundary value problem.

Experimental analysis of deformation mechanisms in a closed-cell aluminum alloy foam

Abstract: The evolution of plastic deformation in a cellular Al alloy upon axial compression is monitored through a digital image correlation procedure. Three stages in the deformation response have been identified. The first involves localized plastic straining at cell nodes. It occurs uniformly and leads to a nominal loading modulus appreciably lower than the stiffness. The second comprises discrete bands of concentrated strain containing cell membranes that experience plastic buckling, elastically constrained by surrounding cells. In this phase, as the loading increases, previously formed bands harden, giving rise to new bands in neighboring regions. The localized bands exhibit a long-range correlation with neighboring bands separated by 3–4 cells along the loading direction. This length scale characterizes the continuum limit. Thirdly, coincident with a stress peak, σo, one of the bands exhibits complete plastic collapse. As the strain increases, this process repeats, subject to small stress oscillations around σo.

Crack patterns in thin films

Abstract: A two-dimensional model of a film bonded to an elastic substrate is proposed for simulating crack propagation paths in thin elastic films. Specific examples are presented for films subject to equi-biaxial residual tensile stress. Single and multiple crack geometries are considered with a view to elucidating some of the crack patterns which are observed to develop. Tendencies for propagating cracks to remain straight or curve are explored as a consequence of crack interaction. The existence of spiral paths is demonstrated.

An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals

Abstract: Variational approaches for nonlinear elasticity show that Hill’s incremental formulation for the prediction of the overall behaviour of heterogeneous materials yields estimates which are too stiff and may even violate rigorous bounds. This paper aims at proposing an alternative ‘affine’ formulation, based on a linear thermoelastic comparison medium, which could yield softer estimates. It is first described for nonlinear elasticity and specified by making use of Hashin–Shtrikman estimates for the linear comparison composite; the associated affine self-consistent predictions are satisfactorily compared with incremental and tangent ones for power-law creeping polycrystals. Comparison is then made with the second-order procedure (Ponte Castaneda, P., 1996. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids, 44 (6), 827–862) and some limitations of the affine method are pointed out; explicit comparisons between different procedures are performed for isotropic, two-phase materials. Finally, the affine formulation is extended to history-dependent behaviours; application to the self-consistent modelling of the elastoplastic behaviour of polycrystals shows that it offers an improved alternative to Hill’s incremental formulation.

Gradient-dependent deformation of two-phase single crystals

Abstract: In this work, a gradient- and rate-dependent crystallographic formulation is proposed to investigate the macroscopic behaviour of two-phase single crystals. The slip-system-based constitutive formulation relies on strain-gradient concepts to account for the additional strengthening mechanism associated with the deformation gradients within a single crystal with a high volume fraction of dispersed inclusions. The resulting total slip resistance in each active system is assumed to be due to a mixed population of forest obstacles arising from both statistically stored and geometrically necessary dislocations. The non-local theory is implemented numerically into the finite element method and used to investigate the effect of the relevant microstructural (i.e., size and volume fraction of precipitated inclusions) and deformation-gradient-related length scales on the macroscopic behaviour of a typical nickel-based superalloy single crystal. An analytical framework to link the strain-gradient effects at the microscopic level with the macroscopic behaviour of an equivalent homogeneous single crystal is also proposed.

Superimposed finite elastic–viscoelastic–plastoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation

Abstract: The paper presents a phenomenological material model for a superimposed elastic–viscoelastic–plastoelastic stress response with damage at large strains and considers details of its numerical implementation. The formulation is suitable for the simulation of carbon-black filled rubbers in monotonic and cyclic deformation processes under isothermal conditions. The underlying key approach is an experimentally motivated a priori decomposition of the local stress response into three constitutive branches which act in parallel: a rubber–elastic ground–stress response, a rate-dependent viscoelastic overstress response and a rate-independent plastoelastic overstress response. The damage is assumed to act isotropically on all three branches. These three branches are represented in a completely analogous format within separate eigenvalue spaces, where we apply a recently proposed compact setting of finite inelasticity based on developing reference metric tensors. On the numerical side, we propose a time integration scheme which exploits intrinsically the modular structure of the proposed constitutive model. This is achieved on the basis of a convenient operator split of the local evolution system, which we decouple into a stress evolution problem and a parameter evolution problem. The constitutive functions involved in the proposed model are specified for a particular filled rubber on the basis of a parameter identification process. The paper concludes with some numerical examples which demonstrate the overall response of the proposed model by means of a representative set of numerical examples.

A new class of extremal composites

Abstract: The paper presents a new class of two-phase isotropic composites with extremal bulk modulus. The new class consists of micro geometrics for which exact solutions can be proven and their bulk moduli are shown to coincide with the Hashin–Shtrikman bounds. The results hold for two and three dimensions and for both well- and non-well-ordered isotropic constituent phases. The new class of composites constitutes an alternative to the three previously known extremal composite classes: finite rank laminates, composite sphere assemblages and Vigdergauz microstructures. An isotropic honeycomb-like hexagonal microstructure belonging to the new class of composites has maximum bulk modulus and lower shear modulus than any previously known composite. Inspiration for the new composite class comes from a numerical topology design procedure which solves the inverse homogenization problem of distributing two isotropic material phases in a periodic isotropic material structure such that the effective properties are extremized.

Multiphase composites with extremal bulk modulus

Abstract: This paper is devoted to the analytical and numerical study of isotropic elastic composites made of three or more isotropic phases. The ranges of their effective bulk and shear moduli are restricted by the Hashin–Shtrikman–Walpole (HSW) bounds. For two-phase composites, these bounds are attainable, that is, there exist composites with extreme bulk and shear moduli. For multiphase composites, they may or may not be attainable depending on phase moduli and volume fractions. Sufficient conditions of attainability of the bounds and various previously known and new types of optimal composites are described. Most of our new results are related to the two-dimensional problem. A numerical topology optimization procedure that solves the inverse homogenization problem is adopted and used to look for two-dimensional three-phase composites with a maximal effective bulk modulus. For the combination of parameters where the HSW bound is known to be attainable, new microstructures are found numerically that possess bulk moduli close to the bound. Moreover, new types of microstructures with bulk moduli close to the bound are found numerically for the situations where the aforementioned attainability conditions are not met. Based on the numerical results, several new types of structures that possess extremal bulk modulus are suggested and studied analytically. The bulk moduli of the new structures are either equal to the HSW bound or higher than the bulk modulus of any other known composite with the same phase moduli and volume fractions. It is proved that the HSW bound is attainable in a much wider range than it was previously believed. Results are readily applied to two-dimensional three-phase isotropic conducting composites with extremal conductivity. They can also be used to study transversely isotropic three-dimensional three-phase composites with cylindrical inclusions of arbitrary cross-sections (plane strain problem) or transversely isotropic thin plates (plane stress or bending of plates problems).

Grain-size effect in viscoplastic polycrystals at moderate strains

Abstract: This work deals with the prediction of grain-size dependent hardening in FCC and BCC polycrystalline metals at moderately high strains (2–30%). The model considers 3–D, polycrystalline aggregates of purely viscoplastic crystals, and simulates quasi-static deformation histories with a hybrid finite element method implemented for parallel computation. The hardening response of the individual crystals is considered to be isotropic, but modified to include a physically motivated measure of lattice incompatibility which is supposed to model, in the continuum setting, the resistance to plastic flow provided by lattice defects. The length-scale in constitutive response that is required on dimensional grounds appears naturally from physical considerations. The grain-size effect in FCC polycrystals and development of Stage IV hardening in a BCC material are examined. Though the grain-size does not enter explicitly into the constitutive model, an inverse relationship between the macroscopic flow stress and grain-size is predicted, in agreement with experimental results for deformation of FCC polycrystals having grain-sizes below 100 microns and at strains beyond the initial yield (>2%). The development of lattice incompatibility is further shown to predict a transition to Stage IV (linear) hardening upon saturation of Stage III (parabolic) hardening.

On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media

Abstract: Higher order (so-called strain gradient) homogenised equations are rigorously derived for an infinitely extended periodic elastic medium with the periodicity cell of a small size e, in the presence of a fixed body force f, via a combination of variational and asymptotic techniques. The coefficients of these equations are explicitly related to solutions of higher order unit cell problems. The related higher order homogenised solutions are shown to be best possible in a certain variational sense, and it is shown that these solutions are close to the actual solutions up to higher orders in e. We derive a rigorous full asymptotic expansion for the energy I e, f and also show that its higher order terms are determined by the higher order homogenised solutions. The resulting variational construction generates higher order effective constitutive relations which are in agreement with those proposed by phenomenological strain gradient theories.

Substrate curvature due to thin film mismatch strain in the nonlinear deformation range

Abstract: The physical system considered is a thin film bonded to the surface of an initially flat circular substrate, in the case when a residual stress exists due to an incompatible mismatch strain in the film. The magnitude of the mismatch strain is often inferred from a measurement of the curvature it induces in the substrate. This discussion is focused on the limit of the linear range of the relationship between the mismatch strain and the substrate curvature, on the degree to which the substrate curvature becomes spatially nonuniform in the range of geometrically nonlinear deformation, and on the bifurcation of deformation mode from axial symmetry to asymmetry with increasing mismatch strain. Results are obtained on the basis of both simple models and more detailed finite element simulations.

Mechanics of a discrete chain with bi-stable elements

Abstract: It has become common to model materials supporting several crystallographic phases as elastic continua with non (quasi) convex energy. This peculiar property of the energy originates from the multi-stability of the system at the microlevel associated with the possibility of several energetically equivalent arrangements of atoms in crystal lattices. In this paper we study the simplest prototypical discrete system—a one-dimensional chain with a finite number of bi-stable elastic elements. Our main assumption is that the energy of a single spring has two convex wells separated by a spinodal region where the energy is concave. We neglect the interaction beyond nearest neighbors and explore in some detail a complicated energy landscape for this mechanical system. In particular we show that under generic loading the chain possesses a large number of metastable configurations which may contain up to one (snap) spring in the unstable (spinodal) state. As the loading parameters vary, the system undergoes a number of bifurcations and we show that the type of a bifurcation may depend crucially on the details of the concave (spinodal) part of the energy function. In special cases we obtain explicit formulas for the local and global minima and provide a quantitative description of the possible quasi-static evolution paths and of the associated hysteresis.

A theory of subgrain dislocation structures

Abstract: We develop a micromechanical theory of dislocation structures and finite deformation single crystal plasticity based on the direct generation of deformation microstructures and the computation of the attendant effective behavior. Specifically, we aim at describing the lamellar dislocation structures which develop at large strains under monotonic loading. These microstructures are regarded as instances of sequential lamination and treated accordingly. The present approach is based on the explicit construction of microstructures by recursive lamination and their subsequent equilibration in order to relax the incremental constitutive description of the material. The microstructures are permitted to evolve in complexity and fineness with increasing macroscopic deformation. The dislocation structures are deduced from the plastic deformation gradient field by recourse to Kroner's formula for the dislocation density tensor. The theory is rendered nonlocal by the consideration of the self-energy of the dislocations. Selected examples demonstrate the ability of the theory to generate complex microstructures, determine the softening effect which those microstructures have on the effective behavior of the crystal, and account for the dependence of the effective behavior on the size of the crystalline sample, or size effect. In this last regard, the theory predicts the effective behavior of the crystal to stiffen with decreasing sample size, in keeping with experiment. In contrast to strain-gradient theories of plasticity, the size effect occurs for nominally uniform macroscopic deformations. Also in contrast to strain-gradient theories, the dimensions of the microstructure depend sensitively on the loading geometry, the extent of macroscopic deformation and the size of the sample.

Cohesive zone modeling of crack nucleation at bimaterial corners

Abstract: The prediction of crack nucleation from bimaterial corners that have the same corner angle (and therefore singularity) via stress intensity factors is fairly well established. The objective of this work was to examine crack nucleation from a range of corner angles and see if a more general crack nucleation criterion could be formulated. A series of experiments was conducted using an aluminum-epoxy bimaterial specimen loaded under 4-point bending. In addition to the usual measurements of load and an associated displacement, the displacements near the corner were measured using moire interferometry. Numerical analyses were first conducted assuming a rigid interface. However, the resulting displacements differed from the measured ones, especially near the corner and along the interface. The interface was then modeled as a separate constitutive entity by incorporating a cohesive zone model in the numerical analysis. Following calibration via an interface crack configuration (zero corner angle), the cohesive zone model yielded displacements that were in good agreement with the measured values for all the other corner angles that were considered. The predicted failure loads were also in good agreement with the experimental results. Thus the consistent nucleation criterion was that the area under the traction-separation curve and its maximum traction (the dominant cohesive zone model parameters) remain the same. The numerical solutions indicated that the plastic deformation in the epoxy was small and that failure was predominantly in opening mode. In addition, the critical vectorial crack opening displacement and mode-mix were independent of the corner angle. Finally, a simple design parameter was proposed for predicting the failure load of a bimaterial specimen with an arbitrary corner angle, based on the failure load of a bimaterial specimen with an interface crack.

On the continuum formulation of higher gradient plasticity for single and polycrystals

Abstract: This paper develops a geometrically linear formulation of higher gradient plasticity of single and polycrystalline material based on the continuum theory of dislocations and incompatibilities As a result, a phenomenological but physically motivated description of hardening is obtained, which incorporates for single crystals second order spatial derivatives of the plastic deformation gradient and for polycrystals fourth order spatial derivatives of the plastic strains into the yield condition Moreover, these modifications mimic the characteristic structure of kinematic hardening, whereby the backstress obeys a nonlocal evolution law For the one-dimensional example of an infinite shear layer the relation between the characteristic length l and the width w of a localized elasto–plastic shear band is examined in detail for both cases

An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity

Abstract: A novel constitutive formulation is developed for finitely deforming hyperelastic materials that exhibit isotropic behavior with respect to a reference configuration The strain energy per unit reference volume, W, is defined in terms of three natural strain invariants, K1–3, which respectively specify the amount-of-dilatation, the magnitude-of-distortion, and the mode-of-distortion Distortion is that part of the deformation that does not dilate Moreover, pure dilatation (K2=0), pure shear (K3=0), uniaxial extension (K3=1), and uniaxial contraction (K3=−1) are tests which hold a strain invariant constant Through an analysis of previously published data, it is shown for rubber that this new approach allows W to be easily determined with improved accuracy Albeit useful for large and small strains, distinct advantage is shown for moderate strains (eg 2–25%) Central to this work is the orthogonal nature of the invariant basis If η represents natural strain, then {K1,K2,K3} are such that the tensorial contraction of (∂Ki/∂η) with (∂Kj/∂η) vanishes when i≠j This result, in turn, allows the Cauchy stress t to be expressed as the sum of three response terms that are mutually orthogonal In particular (summation implied) t=Ai∂W/∂Ki, where the ∂W/∂Ki are scalar response functions and the Ai are kinematic tensors that are mutually orthogonal

Micromechanical simulation of the failure of fiber reinforced composites

Abstract: The strength of unidirectionally reinforced fiber composites is simulated using the three dimensional shear lag model of Landis, C. M., McGlockton, M. A. and McMeeking, R. M. (1999) (An improved shear lag model for broken fibers in composites. J. Comp. Mat. 33, 667–680) and Weibull fiber statistics. The governing differential equations for the fiber displacements and stresses are solved exactly for any configuration of breaks using an influence superposition technique. The model predicts the tensile strength of well bonded, elastic fiber/matrix systems with fibers arranged in a square array. Length and strength scalings are used which are relevant for elastic, local load sharing composites. Several hundred Monte Carlo simulations were executed to determine the statistical strength distributions of the composite for three values of the fiber Weibull modulus, m =5, 10 and 20. Stress–strain behavior and the evolution of fiber damage are studied. Bundle sizes of 10×10, 15×15, 20×20, 25×25, 30×30 and 35×35 fibers of various lengths are investigated to determine the dependence of strength on the composite size. The validity of weakest link statistics for composite strength is examined as well.

Modeling of the competition between shear yielding and crazing in glassy polymers

Abstract: Fracture in amorphous glassy polymers involves two mechanisms of localized deformations: shear yielding and crazing. We here investigate the competition between these two mechanisms and its consequence on the material's fracture toughness. The mechanical response of the homogeneous glassy polymer is described by a constitutive law that accounts for its characteristic softening upon yielding and the subsequent progressive orientational strain hardening. The small scale yielding, boundary layer approach is adopted to model the local finite-deformation process in front of a mode I crack. The concept of cohesive surfaces is used to represent crazes and the traction-separation law incorporates craze initiation, widening and breakdown leading to the creation of a microcrack. Depending on the craze initiation sensitivity of the material, crazing nucleates at the crack tip during the elastic regime or ahead of the crack. As the crazes extend, plasticity develops until an unstable crack propagation takes place when craze fibrils start to break down. Thus, the critical width of a craze appears to be a key feature in the toughness of glassy polymers. Moreover, the opening rate of the craze governs the competition between shear yielding and brittle failure by crazing.

Atomistic simulations on the tensile debonding of an aluminum-silicon interface

Abstract: In this paper we present Modified Embedded Atom Method (MEAM) simulations of the deformation and fracture characteristics of an incoherent interface between pure FCC aluminum and diamond cubic silicon. As a first approximation, the study only considers the normal tensile separation of a [100] interface with the principal crystallographic axis of the aluminum and the silicon aligned. The MEAM results show that the relaxed interface possesses a rippled structure, instead of a planar atomic interface, and such ripples act as local stress concentrators and initiation sites for interfacial failure. The stress–strain (traction–displacement) response of aluminum and silicon blocks attached at an interface depends on the distance from the interface that the boundary conditions are applied, i.e. the size of the atomic blocks, and the location of the measured opening displacement. Point vacancy defects near the interface are found to decrease the maximum normal tensile stress that the interface can support at a rate almost linearly proportional to the number fraction of the dispersed defects. A crack-like vacancy defect in the bulk aluminum or silicon must reach an area fraction (projected to the surface normal to the tensile axis) of about 50 or 30%, respectively, in order to shift the failure from the interface to the bulk materials. It is further demonstrated that the present results are consistent with continuum-based traction separation laws, provided that the opening displacement is measured near the physical boundary of the deforming cohesive zone (±10 A from the boundary of the Al–Si interface). As the opening displacements are measured farther from the interface, the traction–displacement response approaches that of classical linear elastic fracture mechanics.

A discrete dislocation analysis of mode I crack growth

Abstract: Small scale yielding around a plane strain mode I crack is analyzed using discrete dislocation dynamics. The dislocations are all of edge character, and are modeled as line singularities in an elastic material. At each stage of loading, superposition is used to represent the solution in terms of solutions for edge dislocations in a half-space and a complementary solution that enforces the boundary conditions. The latter is non-singular and obtained from a finite element solution. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation are incorporated into the formulation through a set of constitutive rules. A relation between the opening traction and the displacement jump across a cohesive surface ahead of the initial crack tip is also specified, so that crack growth emerges naturally from the boundary value problem solution. Material parameters representative of aluminum are employed. For a low density of dislocation sources, crack growth takes place in a brittle manner; for a low density of obstacles, the crack blunts continuously and does not grow. In the intermediate regime, the average near-tip stress fields are in qualitative accord with those predicted by classical continuum crystal plasticity, but with the local stress concentrations from discrete dislocations leading to opening stresses of the magnitude of the cohesive strength. The crack growth history is strongly affected by the dislocation activity in the vicinity of the growing crack tip.

The influence of imperfections on the nucleation and propagation of buckling driven delaminations

Abstract: The influence of prototypical imperfections on the nucleation and propagation stages of delamination of compressed thin films has been analyzed Energy release rates for separations that develop from imperfections have been calculated These demonstrate two characteristic quantities: a peak that governs nucleation and a minimum that controls propagation and failure These quantities lead to two separate criteria that both need to be satisfied to cause failure They involve a critical film thickness for nucleation and a critical imperfection wavelength for buckling Implications for the avoidance of failure are discussed

Adhesive contact of elastic–plastic spheres

Abstract: We examine the process of decohesion of two adhering elastic–plastic spheres following mutual indentation beyond their elastic limit. In the first instance, it is assumed that during unloading to the point of pull-off, the deformation is predominantly elastic. First, elastic–plastic loading is modelled by a fine grid finite element analysis revealing that, for elastic–perfectly plastic materials, the contact pressure at the end of loading p o is approximately uniform. Next, the contact size and pressure distribution during elastic unloading from a uniform pressure, in the absence of adhesion, is found by the method of rigid punch decomposition. The pressure distribution approaches that of Hertz as the contact size approaches zero. The distribution of adhesive traction has been found in two ways. First, using the singular traction distribution of linear elastic fracture mechanics, and second, using a step cohesive law, whereby a constant adhesive stress σ o acts between surfaces separated by less than δ o , while the surfaces separated by more than δ o , are traction-free. The cohesive zone solution depends on two non-dimensional parameters, while the asymptotic singular solution depends on one non-dimensional parameter only. The limit where the singular model provides a good approximation for the more accurate cohesive zone solution is defined. Finally, a decohesion map is deduced, which divides the parameter space into the regions where decohesion process is governed by different physical mechanisms. The deduced mechanisms are compared with the existing experimental data.

A theory of stress-softening in incompressible isotropic materials

Abstract: A general theory of isotropic stress-softening in incompressible isotropic materials is developed. The principal idea is that a stress-softening material is an inelastic material that has selective memory of only the maximum previous deformation to which it is subjected. This memory dependence is incorporated within general material response functions that are monotone decreasing functions of a stress-softening variable, which is a monotone increasing function of the maximum previous strain experienced by the material. A loading criterion is introduced to identify when the material is loaded along its virgin deformation path where the maximum previous strain is its current value, and to identify when it is unloaded to deform subsequently as an ideal isotropic elastic material in both elastic loading and unloading, so long as the maximum previous strain is not exceeded. The effect of loading from a configuration of maximum previous strain is to further stress-soften the material. Results demonstrating the effects of stress-softening are obtained for general isotropic stress-softening materials in simple uniaxial extension and in simple shear. A simplified analytical model together with a special softening function are introduced to illustrate some general results and to provide specific analytical and graphical examples. Both general and model-specific analytical results obtained for simple uniaxial extension are shown to be consistent with the overall ideal phenomenological behavior exhibited in experiments by others on stress-softening in simple tension and compression. Similar but totally new results for simple shear are derived, and their relation to effects in simple tension are discussed. It is demonstrated that the larger effect of softening occurs in the simple uniaxial extension, the effect in even a gross equivalent simple shear being small. All results are obtained from general three-dimensional constitutive equations.