Showing papers in "Journal of The Mechanics and Physics of Solids in 1996"

A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites

Abstract: A variational formulation is employed to derive a micromechanics-based, explicit nonlocal constitutive equation relating the ensemble averages of stress and strain for a class of random linear elastic composite materials. For two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), we show that the leading-order correction to a macroscopically homogeneous constitutive equation involves a term proportional to the second gradient of the ensemble average of strain. This nonlocal constitutive equation is derived in explicit closed form for isotropic material in the one case in which there exists a well-founded physical and mathematical basis for describing the material's statistics: a matrix reinforced (or weakened) by a random dispersion of nonoverlapping identical spheres. By assessing, when the applied loading is spatially-varying, the magnitude of the nonlocal term in this constitutive equation compared to the portion of the equation that relates ensemble average stresses and strains through a constant “overall” modulus tensor, we derive quantitative estimates for the minimum representative volume element (RVE) size, defined here as that over which the usual macroscopically homogeneous “effective modulus” constitutive models for composites can be expected to apply. Remarkably, for a maximum error of 5% of the constant “overall” modulus term, we show that the minimum RVE size is at most twice the reinforcement diameter for any reinforcement concentration level, for several sets of matrix and reinforcement moduli characterizing large classes of important structural materials. Such estimates seem essential for determining the minimum structural component size that can be treated by macroscopically homogeneous composite material constitutive representations, and also for the development of a fundamentally-based macroscopic fracture mechanics theory for composites. Finally, we relate our nonlocal constitutive equation explicitly to the ensemble average strain energy, and show how it is consistent with the stationary energy principle.

Stress and grain growth in thin films

Abstract: The mechanical properties of polycrystalline thin films with thickness of 1 μm or less depend strongly on the grain geometry, the grain size, and the way in which the crystallographic orientations of the grains are distributed. Grain growth during film formation or during post-deposition annealing can play a dominant role in defining these microstructural characteristics, and therefore, the mechanical properties of films. Stress can suppress or promote grain growth. In the latter case, stress promotes texture evolution during grain growth. Grain growth can serve as a stress relief mechanism in both elastically isotropic and anisotropic materials, and can also promote plastic yielding.

The fractal crushing of granular materials

Abstract: A study has been made of the micro mechanical origins of the irrecoverable compression of aggregates which comprise brittle grains. The terms “yielding” and “plastic hardening” are used in the discipline of soil mechanics to describe the post-elastic behaviour of granular media. These “plastic” phenomena are here related to the successive splitting of grains. Grains are taken to split probabilistically; the likelihood increasing with applied (macroscopic) stress; but reducing with any increase in the co-ordination number and with any reduction in particle size. When the effect of the co-ordination number dominates; a simple numerical model confirms published findings that a fractal distribution of particle sizes evolves from the compression of an aggregate of uniform grains. Taking the production of new surface area from the particle size distributions produced by the numerical model; a work equation is used to deduce the plastic compression of voids; for one-dimensional compression of the aggregate. This too is shown to be in agreement with experimental data; and in particular confirms the linearity of plots of voids ratio versus the logarithm of stress. The gradient of these plots is for the first time related to fundamental material parameters.

A computational procedure for rate-independent crystal plasticity

Abstract: In the rate-independent theory of crystal elasto-plasticity there have been three long-standing problems. The first is to determine which slip systems are active, and the second is to determine the increments of shear on the active slip systems. Third, because of the typical multiplicity of slip systems in ductile crystals, the selection of slip systems required to produce an arbitrary deformation increment is not necessarily unique. The purpose of this paper is to present a robust calculation scheme which determines a unique set of active slip systems and the corresponding shear increments in a rate-independent theory. We show by comparing the predictions from our computational procedure for the rate-independent theory against corresponding predictions from a procedure for a similar but rate-dependent theory (with a low value of the rate sensitivity parameter) that the results from the two procedures are essentially indistinguishable.

Exact second-order estimates for the effective mechanical properties of nonlinear composite materials

Abstract: Motivated by previous small-contrast perturbation estimates, this paper proposes a new method for estimating the effective behavior of nonlinear composite materials with arbitrary phase contrast. The key idea is to write down a second-order Taylor expansion for the phase potentials, about appropriately defined phase average strains. The resulting estimates, which are exact to second order in the contrast, involve the “tangent” modulus tensors of the nonlinear phase potentials, and reduce the problem for the nonlinear composite to a linear problem for an anisotropic thermoelastic composite. Making use of a well-known result by Levin for two-phase thermoelastic composites, together with estimates of the Hashin-Shtrikman type for linear elastic composites, explicit results are generated for two-phase nonlinear composites with statistically isotropic particulate microstructures. Like the earlier small-contrast asymptotic results, the new estimates are found to depend on the determinant of the strain, but unlike the small-contrast results that diverge for shear loading conditions in the nonhardening limit, the new estimates remain bounded and reduce to the classical lower bound in this limiting case. The general method is applied to composites with power-law constitutive behavior and the results are compared with available bounds and numerical estimates, as well as with other nonlinear homogenization procedures. For the cases considered, the new estimates are found to satisfy the restrictions imposed by the bounds, to improve on the predictions of prior homogenization procedures and to be in excellent agreement with the results of the numerical simulations.

Some basic fracture mechanics concepts in functionally graded materials

Abstract: In this paper, the crack-tip fields in a general nonhomogeneous material are summarized. The fracture toughness and R-curve of functionally graded materials (FGMs) are studied based on the crack-bridging concept and a rule of mixtures. It is shown that the fracture toughness is significantly increased when a crack grows from the ceramic-rich region into the metal-rich region in an alumina-nickel FGM. By applying the concept of the toughening mechanism to the study of the strength behavior of FGMs, it is found that the residual strength of the alumina-nickel FGM with an edge crack on the ceramic side is quite notch insensitive.

Dynamically propagating shear bands in impact-loaded prenotched plates—I. Experimental investigations of temperature signatures and propagation speed

Abstract: The initiation and propagation of shear bands are investigated by subjecting prenotched plates to asymmetric impact loading (dynamic mode-II). The materials studied are C-300 (a maraging steel) and Ti-6Al-4V. A shear band emanates from the notch tip and propagates rapidly in a direction nearly parallel to the direction of impact. When the impact velocity is higher than a critical value, the shear band propagates throughout the specimen. The shear band arrests inside the specimen when the impact velocity is below this critical value. In the latter case and for the C-300 steel, a crack initiates and propagates from the tip of the arrested shear band at an angle to the direction of shear band propagation. Microscopic examinations of the shear band and crack surfaces reveal a ductile mode of shear failure inside the shear band and an opening mode of failure for the crack. The coexistence of shear banding and fracture events in the same specimen signifies a transition in the modes of failure for this material under the conditions described. For Ti-6Al-4V, the only mode of failure observed is shear banding. While the transition is induced by changes in loading conditions, the different behaviors of these two materials suggest it is also related to material properties. The experimental investigation focuses on both the thermal and the mechanical aspects of the propagation of shear bands. Real time temperature histories along lines intersecting and perpendicular to and along the shear band path are recorded by means of a high speed infrared detector system. Experiments show that the peak temperatures inside the propagating shear bands increase with impact velocity. The highest temperature measured is in excess of 1400 °C or approximately 90% of the melting point of the C-300 steel. For Ti-6Al-4V, the peak temperatures are approximately 450 °C. In the mechanical part of the study, high speed photography is used to record the initiation and propagation of shear bands. Recorded images of propagating shear bands at different impact velocities provide histories of the speed of shear band propagation for the C-300 steel. A strong dependence of shear band speed on the impact velocity is found. The highest speed observed for the C-300 steel is approximately 1200 ms−1 or 40% of its shear wave speed.

On the toughness of ductile adhesive joints

Abstract: Crack propagation along one of the interfaces between a thin ductile adhesive layer and the elastic substrates it joins is considered. The layer is taken as being elastic-plastic, and the fracture process of the interface is modeled by a traction-separation law, characterized by the peak separation stress 6 and the work of separation per unit area To. Crack growth resistance curves for mode I loading of the adhesive joint are computed, with emphasis on steady-state toughness, as a function of three extrinsic effects : layer thickness, layer-substrate modulus mismatch, and initial residual stress in the layer. Conditions under which separation first occurs well ahead of the initial crack tip are discussed. 1. SPECIFICATION OF THE MODEL This paper continues the study begun by Tvergaard and Hutchinson (1994) in which an embedded fracture zone model is applied to the mode I fracture of an adhesive joint comprised of a thin elastic-plastic metal layer joining two elastic substrates. The present work employs the model to investigate the influence on joint toughness of both the elastic mismatch between the layer and the substrates and the residual stress in the layer. As in the earlier study, the thickness of the ductile layer is another extrinsic variable which comes into play. The approach adopted was first introduced by Needleman (1987) to study particle debonding in metal matrices and subsequently by Tvergaard and Hutchinson (1992, 1993) to model crack growth resistance in homogeneous solids and along interfaces. A traction-separation law simulating the fracture process is embedded within an elastic-plastic continuum as a boundary condition along the line extending ahead of the crack. In the case of an interface joining dissimilar materials, the separation law necessarily involves both the normal and shear tractions and the two associated relative displacements of the surfaces across the interface.

A general macroscopic description of the thermomechanical behavior of shape memory alloys

Abstract: The paper presents a macroscopic description that allows the simulation of the global thermomechanical behavior of shape memory alloys (SMA). Use is made of the thermodynamics of irreversible processes framework. Two internal variables are taken into account: the volume fraction of self-accommodating (pure thermal effect) and oriented (stress-induced) product phase. A specific free energy, valid in the total range of phase transition, is defined with particular attention paid to the interaction term. A study of the thermodynamic absolute equilibrium during phase transition proves its instability, and hence explains the hysteretic behavior of SMA. The kinetic equations for the internal variables are written in such a general way that the model could comply with the second law of thermodynamics. The postulate of five yield functions (each of them being related to one process) permits the phase transition criteria to be defined and the kinetic equations related to each process through consistency equations to be derived. The parameters of the model have been identified for three particular SMA, and the simulated results show good agreement with experiments.

Finite thermoelastic fracture criterion with application to laminate cracking analysis

Abstract: A stress energy criterion for the development of a new finite size crack surface under temperature and load input, in the presence of thermal residual stresses, is established on the basis of an energy release formulation. It is shown that if a fixed specific crack opening surface energy γ exists, an upper bound on the critical energy release can be established on the basis of the thermoelastic principle of minimum complementary energy. On the basis of a variational formulation of thermoelastic laminate analysis a relation is established which permits construction of thermoelastic solutions for cracked laminates by a simple replacement in the corresponding isothermal solution. The results described above are applied to establish relations between crack density and standard deviation of crack interdistances of crack distribution in a cross-ply laminate layer, and load/temperature inputs of the laminate. It is shown that for certain ranges of crack densities, one small to medium and one large, the standard deviation is not needed. The physical and experimental significance of the results obtained is discussed.

Dynamically propagating shear bands in impact-loaded prenotched plates—II. Numerical simulations

Abstract: The experimental observations of dynamic failure in the form of propagating shear bands and of the transition in failure mode presented in Part I of this investigation is analyzed. Finite element simulations are carried out for the initiation and propagation of shear-dominated failure in prenotched plates subjected to asymmetric impact loading. Coupled thermomechanical simulations are carried out under the assumption of plane strain. The simulations take into account finite deformations, inertia, heat conduction, thermal softening, strain hardening and strain-rate hardening. The propagation of shear bands is assumed to be governed by a critical plastic strain criterion. The results demonstrate a strong dependence of band propagation speed on impact velocity, in accordance with experimental observations. The calculations reveal an active plastic zone in front of the tip of the propagating shear bands. The size of this zone and the level of the shear stresses inside it do not change significantly with the impact velocity or the speed of shear band propagation. Shear stresses are uniform inside this zone except near the band tip where higher rates of strain prevail. The shear band behind the propagating tip exhibits highly localized deformations and intense heating. Temperature rises are relatively small in the active plastic zone compared with those inside the well-developed shear band behind the propagating tip. The calculations also show shear band speeds and temperature rises that are in good agreement with experimental observations. Computed temperature fields confirm the experimental observation that dissipation continues behind the propagating shear band tip. In addition, the numerical results capture the arrest of the shear band. The arrested shear band is first subjected to reverse shear. Subsequently, the arrested band is subjected to mixed-mode loading which eventually leads to tensile failure at an angle about 30 ° to the band.

Small and large deformation of thick and thin-film multi-layers: Effects of layer geometry, plasticity and compositional gradients

Abstract: The thermomechanical response of multi-layered materials subjected to small and large deformation during temperature excursions is examined in this paper. General bilayer and trilayer plates with comparable layer thicknesses, as well as the limiting cases of thin films on thicker substrates with and without compositionally graded interfaces are examined, all within the context of the classical Kirchoff theory for thin plates. Closed-form analytical formulations for small elastic deformation are presented whereby explicit expressions for stress/curvature relations are obtained for any general bilayer or graded trilayer with isotropic elastic properties, but anisotropic strains. The effects of the variation of Poisson ratio through the thickness of layered and compositionally graded materials on the evolution of multiple curvatures are analyzed. New theoretical results are presented on the effects of layer geometry, plastic flow and compositional gradation on large deformation (small strains and small rotations) in bilayer and trilayer systems comprising thick or thin-film layers. It is shown that the small deformation theory predictions for the generalized plane strain state provide an upper bound for curvature evolution among all the cases considered. By recourse to analytical methods and three-dimensional finite element modeling involving shell elements, particular attention is devoted to the occurrence of bifurcation in the solution for curvature evolution and the associated geometry changes in the thermoelastoplastic response of layered materials during thermal excursions. The model systems chosen for analyses include NiAl2O3 layers with a sharp or compositionally graded interfaces, AlSi thin-film bilayers and a compositionally graded interlayer sandwiched between layers in In0.12Ga0.88As and GaAs for applications in microelectronics and optoelectronics, and a carbon/ epoxy laminated composite.

Eshelby's inclusion problem for polygons and polyhedra

Abstract: An algorithmic closed-form solution is derived for Eshelby's problem for polygonal and polyhedral inclusions. Illustrative calculations are presented for two- and three-dimensional problems. Also it is proven that polyhedra with constant Eshelby's tensor do not exist.

Crack tip fields in strain gradient plasticity

Abstract: Solutions are presented for mode I and mode II crack tip fields for plane strain deformations of an elastic-plastic solid whose constitutive behavior depends on both strains and strain gradients. The constitutive law is the simplest generalization of the J2 deformation theory of plasticity to include strain gradient effects. Only one new constitutive parameter enters, a length parameter characterizing the scale over which gradient effects become important. The formulation is cast within the framework of coupled stress theory. Crack tip solutions are obtained which display the transition from the HRR fields, governing behavior in an intermediate region with the plastic zone, to the dominant fields at the tip. The dominant fields are obtained in closed form, and finite element methods have been used to produce the solution over the entire field. Some of the difficulties encountered in arriving at an accurate numerical scheme are detailed. Implications of the solutions for fracture are discussed, as are avenues for further research.

The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate

Abstract: The debonding problem for a composite layer that consists of a homogeneous substrate and a non-homogeneous coating is considered. It is assumed that the problem is one of plane strain or generalized plane stress and the elastic medium contains a crack along the interface. It is further assumed that the thermomechanical properties of the medium are continuous functions of the thickness coordinate with discontinuous derivatives and the kink line of the property distributions corresponds to the “interface”. The mixed-mode crack problem is formulated for arbitrary crack surface tractions and sample results are given for uniform normal and shear tractions. The main variables in the problem are two dimensionless length parameters and a nonhmogeneity constant. Calculated results consist of primarily the stress intensity factors and the strain energy release rate and are partly intended to provide benchmark solutions for further numerical studies.

The role of strain gradients in the grain size effect for polycrystals

Abstract: The role of grain size on the overall behaviour of polycrystals is investigated by using a strain gradient constitutive law for each slip system for a reference single crystal. Variational principles of Hashin-Shtrikman type are formulated for the case where the strain energy density is a convex function of both strain and strain gradient. The variational principles are specialized to polycrystals with a general multi-slip strain gradient constitutive law. An extension of the Hashin-Shtrikman bounding methodology to general strain gradient composites is discussed in detail and then applied to derive bounds for arbitrary linear strain gradient composites or polycrystals. This is achieved by an extensive study of kernel operators related to the Green's function for a general “strain-gradient” linear isotropic incompressible comparison medium. As a simple illustrative example, upper and lower bounds are computed for linear face-centred cubic polycrystals: a size effect is noted whereby smaller grains are stiffer than large grains. The relation between the assumed form of the constitutive law for each slip system and the overall response is explored.

Cracking of thin films bonded to elastic-plastic substrates

Abstract: Thin bonded films have many applications In information storage and processing systems, for example, conducting, semiconducting and insulating films are used in integrated circuits, and thin magnetic films are used in disk storage systems In many cases, thin bonded films are in a state of residual tension, which can lead to film cracking Because cracking can alter desired film properties, methods for predicting it are needed The geometry considered in this work is one in which cracks or flaws oriented normal to the film-substrate interface propagate (or “channel”) across the film It is assumed that the film is brittle and the substrate is ductile Plane strain fracture analyses are used to investigate the channel cracking of elastic thin films in residual tension in the presence of yielding in the substrate material Although crack channeling induces yielding in the substrate, channel crack extension in the brittle film occurs under small scale yielding conditions The case of an elastic film bonded to an elastic substrate has been considered in earlier work, and is used as the basis for the current study A numerical model is used to extend the results from the fully elastic problem so that plastic yielding of the substrate is allowed Results are presented for an elastic-perfectly plastic substrate and for substrates exhibiting strain hardening A simple shear lag model of the problem without hardening in the substrate is discussed, which gives reasonable predictions for the dependence of dimensionless fracture quantities on the normalized loading over a wide range of material mismatches In addition, a method is presented by which shear lag modeling can be extended to cases in which the substrate exhibits strain hardening

Fracture mechanics for the design of ceramic multilayer actuators

Abstract: In a multilayer actuator, each internal electrode terminates an edge inside the active ceramic. Around the edge, the nonuniform electric field generates an incompatible strain field, which, in its turn, generates stresses and may cause the ceramic to crack. The industry has been exploring alternative electrode configurations to alleviate the stress concentration. The effort has been empirical and benefited little from numerical simulations. An inherent difficulty is that the actuator ceramics have nonlinear electro-mechanical interactions, of which no unified mathematical description is now available. In this paper, we develop a crack nucleation model that includes essential features of this nonlinearity. The model applies to both paraelectrics and ferroelectrics. Attention is focused on situations where the small-scale saturation conditions prevail. That is, the driving voltage is low enough so that the bulk of the ceramics is linearly dielectric, except for a cylinder of a small radius around the electrode edge. Inside the cylinder, large strains result from electrostriction or polar rotation. We identify a parameter group that determines the cracking condition; details in the material description only affect a dimensionless coefficient. Everything else being fixed, a critical layer thickness exists, below which a multilayer actuator will not crack around its internal electrode edges. Merits and limitations of the small-scale saturation model are discussed. We analyze this model analytically for a paraelectric with perfect polarization saturation, and estimate the value of the dimensionless coefficient in the model.

The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models

Abstract: Of interest here is the scale size effect on the stability of finitely strained, rate-independent solids with periodic microstructures. Using a multiple scales asymptotic technique, we express the critical load at the onset of the first instability and the corresponding eigenmode in terms of the scale size parameter e. The zeroth order e terms in these expansions depend on the standard (first order gradient) macroscopic moduli tensor, while all the higher order e terms require the determination of higher order gradient macroscopic moduli. These macroscopic moduli, which are calculated by solving appropriate boundary value problems on the unit cell, relate the macroscopic (unit cell average) stress rate increment to the macroscopic displacement rate gradients. The proposed general theory is subsequently applied to the investigation of the failure surfaces in periodic solids of infinite extent. For these solids one can define in macroscopic strain space a microscopic (local) failure surface, which corresponds to the onset of the first bulking-type instability in the solid, and a macroscopic (global) failure surface, which corresponds to the onset of the first long wavelength instability in the solid. The determination of the macrofailure surface is considerably easier than the determination of the microfailure surface, for it requires the calculation of the standard macroscopic moduli tensor. In addition, the regions where the two surfaces coincide is of significant practical interest, for a macroscopic localized mode of deformation (e.g. in the form of a shear band or a kink band) appears in the post-bifurcation regime. The prediction of these coincidence zones is based on a necessary criterion that depends on the higher order gradient macroscopic moduli. A detailed example is given for the case of layered composites, in view of the possibility of obtaining closed form expressions for all the required macroscopic moduli and in view of the existence of an analytical solution to the microscopic failure problem. Two applications are presented, one for a foam rubber composite and another for a graphite-epoxy composite whose properties have been determined experimentally. Following the verification of the above mentioned necessary criterion for the coincidence of the micro- and macrofailure surfaces in the two examples, the presentation is concluded by a discussion and suggestions for further work.

Inelastic microstructural failure mechanisms in crystalline materials with high angle grain boundaries

Abstract: Microstructurally-induced failure mechanisms in crystalline materials with coincident site-lattice (CSL) high angle grain boundaries (GBs) have been investigated. A multiple-slip rate-dependent crystalline constitutive formulation that is coupled to the evolution of mobile and immobile dislocation densities and specialized computational schemes have been developed to obtain a detailed understanding of the interrelated physical mechanisms that result in material failure. A transmission scalar has also been introduced to investigate slip-rate transmission, blockage and incompatibility at the GB. The combined effects of high angle GB misorientation, mobile and immobile dislocation densities, strain hardening, geometrical softening, localized plastic strains, and slip-rate transmission and blockage on failure evolution in face centered cubic (f.c.c.) crystalline materials have been studied. Results from the present study are consistent with experimental observations that single dislocation pile-ups result in a transgranular failure mode for the ∑9 CSL GB, and that symmetric double dislocation pile-ups result in an intergranular failure mode for the ∑17b CSL GB.

A theory of local limiting speed in dynamic fracture

Abstract: A nonlinear continuum analysis is developed to show that stable, steady-state crack motion is limited not only by the macroscopic Rayleigh wave speed as asserted by the established theory of dynamic fracture, but also by a local wave speed governed by the elastic response near the crack tip. The local limiting speed ensures that a subsonic steady-state field can be established in highly nonlinear material regions prior to rupture. A two-dimensional triangular lattice with nearest-neighbor interatomic bonding is studied as a model nonlinear elastic solid that is isotropic under infinitesimal strains, but becomes anisotropic and nonlinear when the lattice is heavily stretched. The local limiting speed is determined by considering the most critical state of deformation on the verge of bond rupture. If the critical state is assumed to be under equibiaxial stressing, the local limiting speed is found to be v 1 = c s σ max μ , where cs is the macroscopic shear wave speed, μ is the shear modulus and σmax is the equibiaxial cohesive strength of the solid (i.e. the maximum equibiaxial tensile stress that a flawless solid can stand without spontaneous rupture). The generality of this result is discussed by relaxing the restrictions in the model problem. It is also shown that lattice dispersion in front of a crack tip can further reduce the speed of bond-breaking stress waves with wavelength on the order of a few atomic spacings. This study lends further support for a viewpoint previously discussed by the author that high speed dynamic fracture involves a competition between a high inertia local crack-tip field and the surrounding low inertia apparent crack field. Motivated by recent molecular dynamics simulations of crack propagation in a 6–12 Lenard-Jones lattice, a variational principle for steady-state deformation is used together with a conjugate gradient minimization algorithm to compute atomistic responses near the tip of a crack moving with constant speed in a similar Lenard-Jones lattice. The computation is performed over a block which moves with the crack and is subjected along the boundary to a low inertia displacement field based on existing solutions for cracks moving in linear elastic solids. The critical velocity at the onset of local crack branching is found to be 0.30cs, in almost exact agreement with the earlier molecular dynamics study. In this case, the local limiting speed is calculated to be v1 = 0.37cs, which is 20% larger than the observed value. This difference can be attributed to the effects of local lattice dispersion. The results are fully supportive of the notion that global-local inertia competition is a key to understanding dynamic fracture instabilities.

Ductile crack growth−III. Transition to cleavage fracture incorporating statistics

Abstract: The fracture resistance of ferritic steels in the ductile/brittle transition regime is controlled by the competition between ductile tearing and cleavage fracture. Under typical conditions, a crack initiates and grows by ductile tearing but ultimate failure occurs by catastrophic cleavage fracture. In this study the tearing process is simulated using void-containing cell elements embedded within a conventional elastic-plastic continuum; details of the cell model are discussed in Parts I and II of this article. Weakest link statistics is incorporated into the cell element model and this new model is employed to predict the onset of unstable cleavage fracture. Our approach differs from previous analyses in several important ways. The elastic-plastic field computed for crack growth by ductile tearing is fully integrated with a weakest link cleavage fracture model. The model also accounts for the competition between the nucleation of voids from carbide inclusions and the unstable cracking of inclusions precipitating catastrophic cleavage fracture. This model leads immediately to a natural definition of the Weibull stress measure pertinent to cleavage fracture. The model is not restricted by the extent of plastic deformation and ductile tearing. Two effects are associated with ductile crack growth: the cumulative sampling volume is increased and the crack tip constraint is altered. Both effects have important roles which are treated within the present cleavage fracture model. Load-displacement behavior, ductile tearing resistance and transition to cleavage fracture are investigated for several different test geometries and a range of microstructural parameters. It is found that certain variations in microstructure can result in pronounced effects on the cleavage fracture toughness though they have no effect on the ductile tearing resistance preceding cleavage. Rate effects on ductile tearing and transition to cleavage fracture are also discussed. The model predicts trends in ductile/brittle transition that are consistent with available experimental data.

Stress concentrations around multiple fiber breaks in an elastic matrix with local yielding or debonding using quadratic influence superposition

Abstract: A new computational technique, called the quadratic influence superposition (QIS) technique, is developed to study the stresses around arbitrary arrays of fiber breaks in a unidirectional composite loaded in simple tension, and consisting of elastic fibers in a matrix, which is either elastic-perfectly plastic or which can debond at the interface leaving residual friction. The method involves extending a recently developed break influence superposition (BIS) technique, where to model the behavior of damaged (yielded or debonded) matrix elements, we use special compensating shear stress profiles and develop the corresponding influence functions. The QIS technique appears to be at least an order of magnitude more efficient than other numerical schemes as the computation time is tied mainly to the amount of damage, and it is more accurate than a simpler version of this technique developed earlier. In illustrative examples, the method determines the Mode I fiber and matrix stress distributions around a “center crack” consisting of up to 31 contiguous fiber breaks. Incremental treatment is needed to establish the extent of the inelastic regions and the results, which achieve excellent agreement with exact shear lag analyses, clearly show that QIS calculated these correctly. Results show that the extent of the matrix damage region increases approximately linearly with applied load and nonlinearly with the number of breaks. The stress concentrations and overload profiles along nearby unbroken fibers are altered as compared to the fully elastic case with magnitudes reduced but length scales increased.

Dynamic crack propagation in piezoelectric materials—Part I. Electrode solution

Abstract: An analysis is performed for the transient response of a semi-infinite, anti-plane crack propagating in a hexagonal piezoelectric medium. The mixed boundary value problem is solved by transform methods together with the Wiener-Hopf and Cagniard-de Hoop techniques. As a special case, a closed form solution is obtained for constant speed crack propagation under external anti-plane shear loading with the conducting electrode type of electric boundary condition imposed on the crack surface (a second type of boundary condition is considered in Part II of this work). In purely elastic, transversely isotropic elastic solids, there is no antiplane mode surface wave. However, for certain orientations of piezoelectric materials, a surface wave will occur—the BleusteindashGulyaev wave. Since surface wave speeds strongly influence crack propagation, the nature of antiplane dynamic fracture in piezoelectric materials is fundamentally different from that in purely elastic solids, exhibiting many features only associated with the indashplane modes in the elastic case. For a general distribution of crack face tractions, the dynamic stress intensity factor and the dynamic electric displacement intensity factor are derived and discussed in detail for the electrode case. As for inplane elastodynamic fracture, the stress intensity factor and energy release rate go to zero as the crack propagation velocity approaches the surface wave speed. However, the electric displacement intensity does not vanish.

Configurational forces and the basic laws for crack propagation

Abstract: This paper develops a framework for dynamical fracture, concentrating on the derivation of basic field equations that describe the motion of the crack tip in two space-dimensions. The theory is based on the notion of configurational forces in conjunction with a mechanical version of the second law.

Morphologically representative pattern-based bounding in elasticity

Abstract: A general theory for the homogenization of heterogeneous linear elastic materials that relies on the concept of “morphologically representative pattern” is given. It allows the derivation of rigorous bounds for the effective behaviour of the Voigt-Reuss-type, which apply to any distribution of patterns, or of the Hashin-Shtrikman-type, which are restricted to materials whose pattern distributions are isotropic. Particular anisotropic distributions of patterns can also be considered: Hashin-Shtrikman-type bounds for anisotropic media are then generated. The resolution of the homogenization problem leads to a complex composite inclusion problem with no analytical solution in the general case. Here it is solved by a numerical procedure based on the finite element method. As an example of possible application, this procedure is used to derive new bounds for matrix-inclusion composites with cubic symmetry as well as for transversely isotropic materials.

Some elementary connections between curvature and mismatch strain in compositionally graded thin films

Abstract: A thin film structure with through-the-thickness variation of properties and/or mismatch strain is considered. The relationship of the overall curvature of the film to the variation of properties and mismatch strain is reviewed. It is shown that, if the material properties are known, the mismatch strain distribution in the film can be expressed in terms of the dependence of curvature on film thickness. In addition, the case of film growth under conditions for which the mismatch strain of deposited material depends on the local strain conditions of the growth surface is considered. By means of an illustration, it is shown that the final state of strain within a free film following growth depends on the constraint conditions that were imposed on the film during its growth.

Dynamic crack propagation in piezoelectric materials—Part II. Vacuum solution

Abstract: In Part I of this work, antiplane dynamic crack propagation in piezoelectric materials was studied under the condition that crack surfaces behaved as though covered with a conducting electrode. Piezoelectric surface wave phenomena were clearly seen to be critical to the behavior of the moving crack. Closed form results were obtained for stress and electric displacement intensities at the crack tip in the subsonic crack speed range; the major result is that the energy release rate vanishes as the crack speed approaches the surface (Bleustein-Gulyaev) wave speed. In this paper, an alternative assumption is made that between the growing crack surfaces there is a permeable vacuum free space, in which the electrostatic potential is nonzero. By coupling the piezoelectric equations of the solid phase with the electric charge equation in the vacuum region, a closed form solution is again obtained. In contrast to the electrode case of Part I, this case allows both applied charge and applied traction loading. In addition, the work of Part I is extended to examine piezoelectric crack propagation over the full velocity range of subsonic, transonic and supersonic speeds. Several aspects of the results are explored. The energy release rate in this case does not go to zero when the crack propagating velocity approaches the surface wave speed, even if there is only applied traction loading. When the crack exceeds the Bleustein-Gulyaev wave speed, the character of the crack-tip singularities of the physical fields depends on both speed regime and type of loading. At the other extreme, the quasi-static limit of the dynamic solution furnishes a set of new static solutions. The general permeability assumptions made here allow for fully coupled conditions that are ruled out by the a priori interfacial assumptions made in previously published solutions.

Progressive damage in discrete models and consequences on continuum modelling

Abstract: In phenomenological damage models, damage is very often understood as a degradation of the elastic stiffness of the material. The unrealistic features of damage localisation as a result of strain softening are usually circumvented by the introduction of an internal length in the continuum description that scales the localisation process and controls the size of the damage/strain localisation zone. This paper focuses on the motivation for introducing such an internal length. It is based on the analysis of failure in lattice models presenting an initial disorder in strength. The simple model introduced here and studied numerically, should be amenable to a continuum description for large enough lattice sizes. In the lattice modelling, an internal length appears as a correlation length owing to spatial redistributions and interactions during the failure process. The variations of this length with the system size are studied numerically and theoretically with an original model based on percolation theory, which accounts for the spatial interactions. The analysis shows that the internal length increases with the damage of the system, and finally reaches a finite (lattice size independent) value at the peak load. A full statistical analysis of the local stress is provided and discussed in the formalism of multifractals, so as to extract the salient scaling features of the model.

Modeling of hydrogen transport and elastically accommodated hydride formation near a crack tip

Abstract: Transient hydrogen diffusion and elastically accommodated hydride formation coupled with material elastic deformation are studied in a hydride forming system. The constitutive behavior of the material is modeled as isotropically linear elastic and account is taken of the effect of the dilatational strain induced by the solute hydrogen and formed hydride. The concept of terminal solid solubility of hydrogen as affected by stress is described and the mode of hydrogen diffusion through the two-phase material (matrix + hydride) is discussed. Probabilistic precipitation of hydride is modeled in the neighborhood of a stationary sharp crack tip under mode I plane strain loading, fixed hydrogen concentration on the crack surfaces and the outer boundary, and a uniform initial hydrogen concentration below the stress-free terminal solid solubility. A full transient finite element analysis allows for numerical monitoring of the development and expansion of the hydride zone. Information about the shape, size and density of the hydride in the hydride zone is obtained. The mechanistic effects of the solute hydrogen and hydride formation on the stress intensity at the crack tip are analyzed and their consequence on the fracture toughness resistance of the material is discussed.