Showing papers in "Journal of The Mechanics and Physics of Solids in 1984"
TL;DR: In this paper, a set of elastic-plastic constitutive relations that account for the nucleation and growth of microvoids is analyzed numerically, based on the set of constitutive relation for axisymmetric and plane strain notched tensile specimens.
Abstract: Ductile fracture in axisymmetric and plane strain notched tensile specimens is analyzed numerically, based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of microvoids. Final material failure by void coalescence is incorporated into the constitutive model via the dependence of the yield function on the void volume fraction. In the analyses the material has no voids initially; but as the voids nucleate and grow, the resultant dilatancy and pressure sensitivity of the macroscopic plastic flow influence the solution significantly. Considering both a blunt notch geometry and a sharp notch geometry in the computations permits a study of the relative roles of high strain and high triaxiality on failure. Comparison is made with published experimental results for notched tensile specimens of high-strength steels. All axisymmetric specimens analyzed fail at the center of the notched section, whereas failure initiation at the surface is found in plane strain specimens with sharp notches, in agreement with the experiments. The results for different specimens are used to investigate the circumstances under which fracture initiation can be represented by a single failure locus in a plot of stress triaxiality vs effective plastic strain.
702 citations
TL;DR: In this article, the authors consider a single degree of freedom elastic system undergoing frictional slip, where the system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speedV 0.
Abstract: We consider quasistatic motion and stability of a single degree of freedom elastic system undergoing frictional slip. The system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speed V 0 We adopt rate and state dependent frictional constitutive relations for the slider which describe approximately experimental results of Dieterich and Ruina over a range of slip speeds V . In the simplest relation the friction stress depends additively on a term A In V and a state variable θ; the state variable θ evolves, with a characteristic slip distance, to the value − B In V , where the constants A, B are assumed to satisfy B > A > 0. Limited results are presented based on a similar friction law using two state variables. Linearized stability analysis predicts constant slip rate motion at V 0 to change from stable to unstable with a decrease in the spring stiffness k below a critical value k cr . At neutral stability oscillations in slip rate are predicted. A nonlinear analysis of slip motions given here uses the Hopf bifurcation technique, direct determination of phase plane trajectories, Liapunov methods and numerical integration of the equations of motion. Small but finite amplitude limit cycles exist for one value of k , if one state variable is used. With two state variables oscillations exist for a small range of k which undergo period doubling and then lead to apparently chaotic motions as k is decreased. Perturbations from steady sliding are imposed by step changes in the imposed load point motion. Three cases are considered: (1) the load point speed V 0 is suddenly increased; (2) the load point is stopped for some time and then moved again at a constant rate; and (3) the load point displacement suddenly jumps and then stops. In all cases, for all values of k :, sufficiently large perturbations lead to instability. Primary conclusions are: (1) ‘stick-slip’ instability is possible in systems for which steady sliding is stable, and (2) physical manifestation of quasistatic oscillations is sensitive to material properties, stiffness, and the nature and magnitude of load perturbations.
489 citations
TL;DR: In this article, an axisymmetric model for cavity growth in polycrystalline metal is presented, in which a cavitating facet is represented as disk-shaped, and the model dimensions are taken to represent spacings between neighbouring cavitating facets.
Abstract: Creep rupture in a polycrystalline metal at a high temperature, by cavity growth on a number of grain boundary facets, is studied numerically. An axisymmetric model problem is analysed, in which a cavitating facet is represented as disk-shaped, and the model dimensions are taken to represent spacings between neighbouring cavitating facets. For the grains both power law creep and elastic deformations are taken into account, and the description of cavity growth is based on an approximate expression that incorporates the coupled influence of grain boundary diffusion and power law creep. The cases considered include creep-constrained cavity growth at low stresses, where the voids link up to form grain boundary cracks at relatively small overall strains, as well as the power law creep dominated behaviour at higher stress levels, where rupture occurs at large overall strains. The numerical results are compared with results based on various simplified analyses.
139 citations
TL;DR: In this article, the authors considered the shear wave propagation in a half-space of a nonlinear material, where the surface of the halfspace is subjected to a time dependent but spatially uniform tangential velocity.
Abstract: O ne-dimensional shear wave propagation in a half-space of a nonlinear material is considered. The surface of the half-space is subjected to a time dependent but spatially uniform tangential velocity. The half-space material exhibits strain hardening, thermal softening and strain rate sensitivity of the flow stress. For this system, a well-defined band of intense shear deformation can develop adjacent to the loaded surface, even though the material has no imperfections or other natural length scale. Representative particle velocity and strain profiles, which have been obtained numerically, are described for several different models.
122 citations
TL;DR: In this article, the dynamic crack propagation experiments have been performed using wedge loaded double cantilever beam specimens of an austenitized, quenched and tempered 4340 steel.
Abstract: Dynamic crack propagation experiments have been performed using wedge loaded double cantilever beam specimens of an austenitized, quenched and tempered 4340 steel. Measurements of the dynamic stress intensity factor have been made by means of the optical method of caustics. The interpretation of experimental data, obtained from the shadow spot patterns photographed with a Cranz-Schardin high speed camera, is based on an elastodynamic analysis. The instantaneous value of the dynamic stress intensity factor KdI is obtained as a function of crack tip velocity. Finally, the interaction of reflected shear and Rayleigh waves with the moving crack tip stress field is considered.
117 citations
TL;DR: In this paper, the effective elasticity tensor of a composite is defined to be the four-tensor C which relates the average stress to the average strain, and an integral equation for this traction is derived for simple, body-centered and face-centered cubic lattices.
Abstract: The effective elasticity tensor of a composite is defined to be the four-tensor C which relates the average stress to the average strain. We determine it for an array of rigid spheres centered on the points of a periodic lattice in a homogeneous isotropic elastic medium. We first express C in terms of the traction exerted on a single sphere by the medium, and then derive an integral equation for this traction. We solve this equation numerically for simple, body-centered and face-centered cubic lattices with inclusion concentrations up to 90% of the close-packing concentration. For lattices with cubic symmetry the effective elasticity tensor involves just three parameters, which we compute from the solution for the traction. We obtain approximate asymptotic formulas for low concentrations which agree well with the numerical results. We also derive asymptotic results for C at high inclusion concentrations for arbitrary lattice geometries. We find them to be in good agreement with the numerical results for cubic lattices. For low and moderate concentrations the approximate results of Nemat - Nasser et al., also agree well with the numerical results for cubic lattices.
102 citations
TL;DR: In this paper, the deformation-dependent part of the internal energy is considered as a symmetric function of the principal stretches rather than depending only on their product, as in the original development.
Abstract: In the first part of the paper, consisting of Sections 2 and 3, physical grounds are adduced for weakening the concept of strictly entropic elasticity as applied to elastomeric materials. The resulting notion of modified entropic elasticity is shown to provide an alternative formulation of a model of rubberlike thermoelasticity proposed by C hadwick (1974). In the second part (Sections 4 and 5) Chadwick's model is generalized by allowing the deformation-dependent part of the internal energy to be a symmetric function of the principal stretches rather than depending only on their product, as in the original development. Consistently with the molecular theory of polymer networks, and with experimental findings, the extended model predicts that the deviatoric stress is not entirely entropic in origin, but arises in part from changes of internal energy. The energetic fraction of the retractive force in an extended cylinder is calculated and discussed in some detail and the paper concludes with a correlation of theoretical results with measurements on specimens of a lightly cross-linked natural rubber.
82 citations
TL;DR: In this article, a new method for deriving rigorous bounds on the effective elastic constants of a composite material is presented and used to derive a number of known as well as some new bounds.
Abstract: A new method for deriving rigorous bounds on the effective elastic constants of a composite material is presented and used to derive a number of known as well as some new bounds. The new approach is based on a presentation of those constants as a sum of simple poles. The locations and strengths of the poles are treated as variational parameters, while different kinds of available information are translated into constraints on these parameters. Our new results include an extension of the range of validity of the Hashin-Shtrikman bounds to the case of composites made of isotropic materials but with an arbitrary microgeometry. We also use information on the effective elastic constants of one composite in order to obtain improved bounds on the effective elastic constants of another composite with the same or a similar microgeometry.
78 citations
Abstract: The hardening model proposed by Z. Mroz based on the uniaxial fatigue behavior of many metals is adopted to derive an incremental constitutive equation for general three-dimensional problems. This constitutive law is then employed in the analysis of metal forming problems to assess the influence of loading cycles, of the types involved in standard forming processes, on the ultimate formability of sheet metals. The predicted forming limit curves differ quantitatively from results obtained via an isotropie hardening model and differ qualitatively from those obtained via a kinematic model. Also investigated are the effects of such loading cycles on material response to simple tensile loading, which is often used to characterize a material. Significant differences between the present model and the other two models considered are observed in such characterizers of simple tensile behavior as the stress-strain curve, the anisotropy parameter and the uniform elongation. These differences suggest a rather simple experiment to identify the proper material model to be used in analyses of problems which involve loading cycles. Comparisons with some experimental results reveal that the employment of an anisotropic hardening model, such as the generalized Mroz model derived herein, is indeed crucial in accurately predicting material response to complicated loading histories.
76 citations
TL;DR: In this article, a model is proposed for an elastic-plastic solid with an excess of voids in a disk-shaped cluster embedded in a uniform background distribution, which is used to study the effect of a void cluster on plastic flow localization.
Abstract: I nfinite band calculations indicate that the process of flow localization in voided solids is highly sensitive to non-uniformity in void distribution. In this paper, a model is proposed for an elastic-plastic solid with an excess of voids in a disk-shaped cluster embedded in a uniform background distribution. The model is used to study the effect of a void cluster on plastic flow localization. Substantial reductions in ductility due to nonuniformity only occur for quite large clusters when the triaxiality of the overall stresses is low, as in uniaxial tension. At higher stress triaxialities, a small cluster can be severely deleterious.
69 citations
TL;DR: An exact relation between the thermal expansion coefficient and the bulk modulus of statistically isotropic polycrystalline aggregates composed of crystals of hexagonal, tetragonal or trigonal symmetry was developed in this article.
Abstract: An exact relation is developed between the thermal expansion coefficient and the bulk modulus of statistically isotropic polycrystalline aggregates composed of crystals of hexagonal, tetragonal or trigonal symmetry. This relation is exploited to derive simple close bounds for the thermal expansion coefficient in terms of single crystal properties. Comparison of bounds to experimentally obtained expansion coefficients shows fair to very good agreement.
TL;DR: In this paper, a complete solution for crack growth in the plane of the crack was obtained by matching the relevant stress components and particle velocities to the dominant terms of appropriate elastic fields at the elastic-plastic boundary.
Abstract: M ode -I crack growth under conditions of generalized plane stress has been investigated. It has been assumed that near the plane of the crack in the loading zone, the simple stress components corresponding to a central fan field maintain validity up to the elastic-plastic boundary. By the use of expansions of the particle velocities in the coordinate y , and by matching of the relevant stress components and particle velocities to the dominant terms of appropriate elastic fields at the elastic-plastic boundary, a complete solution has been obtained for e y in the plane of the crack. The solution applies from the propagating crack tip up to the moving elastic-plastic boundary. The strain fields for a self-similar crack nucleating at a point and for steady-state propagation of a crack have been considered as special cases.
TL;DR: In this article, an extensive theoretical investigation of f.c. crystals under [110] loading in the channel die compression test is presented, and two lattice orientations known from experiment to be stable relative to the channel axes, through large deformations, are investigated for each of four hardening laws.
Abstract: An extensive theoretical investigation of f.c.c. crystals under [110] loading in the channel die compression test is presented. Two lattice orientations known from experiment to be stable relative to the channel axes, through large deformations, are investigated for each of four hardening laws. These are Taylor's classical isotropic hardening rule, a 2-parameter empirical rule from the metallurgical literature, the “simple theory” of anisotropic latent hardening( Havner and Shalaby , Proc. R. Soc. A 358 ,47 (1977)), and a modification of the simple theory proposed by pfirce et al., Acta Met. 30 , 1087 (1982). Predictions of active systems, equal multiple-slip and consequent lattice stability, finite shape change, and lateral constraint stress are the same for all theories, corresponding to minimum rate of plastic work, and are in general agreement with experiments on copper crystals by Wonsiewicz and Chin , Met. Trans. 1 , 2715 (1970) and Wonsiewicz et al. , Met. Trans. 2 , 2093 (1971). The predictions of latent hardening differ among the theories, however, depending upon whether there is relative rotation of material and lattice. The potential significance of experimental studies of latent hardening in these particular stable lattice orientations is emphasized.
TL;DR: In this article, an austenitic AISI Type 304 stainless steel, a ferritic A533B pressure vessel steel and a Ti-7Al-2Cb-1Ta alloy were tested using a servocontrolled MTS axial-torsion testing machine.
Abstract: An austenitic AISI Type 304 stainless steel, a ferritic A533B pressure vessel steel and a Ti-7Al-2Cb-1Ta alloy were tested using a servocontrolled MTS axial-torsion testing machine. Tests involved changes in strain rate between 10−8 and 10−3 s−1 and intermittent creep periods of less than 1200 s duration. The tests show that inelastic work is not a suitable repository for modeling strain (work)-hardening and the Bauschinger effect is found to be rate dependent. Upon an increase in stress level, creep rate can decrease. This anomaly can be reproduced by a theory of viscoplasticity based on overstress previously proposed by the first author and his co-workers.
TL;DR: In this article, the steady state crack propagation problems of elastic-plastic materials in Mode I, plane strain under small scale yielding conditions were investigated with the aid of the finite element method.
Abstract: Steady state crack propagation problems of elastic-plastic materials in Mode I, plane strain under small scale yielding conditions were investigated with the aid of the finite element method. The elastic-perfectly plastic solution shows that elastic unloading wedges subtended by the crack tip in the plastic wake region do exist and that the stress state around the crack tip is similar to the modified Prandtl fan solution. To demonstrate the effects of a vertex on the yield surface, the small strain version of a phenomenological J2, corner theory of plasticity (Christoffersen, J. and Hutchinson, J. W. J. Mech. Phys. Solids, 27, 465 C 1979) with a power law stress strain relation was used to govern the strain hardening of the material. The results are compared with the conventional J2 incremental plasticity solution. To take account of Bauschinger like effects caused by the stress history near the crack tip, a simple kinematic hardening rule with a bilinear stress strain relation was also studied. The results are again compared with the smooth yield surface isotropic hardening solution for the same stress strain curve. There appears to be more potential for steady state crack growth in the conventional J2 incremental plasticity material than in the other two plasticity laws considered here if a crack opening displacement fracture criterion is used. However, a fracture criterion dependent on both stress and strain could lead to a contrary prediction.
TL;DR: In this article, the effective linear elastic behavior of random media subjected to inhomogeneous mean fields is studied and the authors show that the effective elastic moduli show dispersion, i.e., they depend on the wave vector of the mean field.
Abstract: The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e∗ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lame parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.
TL;DR: In this paper, the impact of a long rod onto a large target using an Eulerian finite difference scheme was studied, and the impact velocity of 1.5 km s −1 was chosen to be low enough for metal strength to be an important parameter characterising the impact.
Abstract: The normal impact of a long rod onto a large target is studied using an Eulerian finite difference scheme. The impact velocity of 1.5 km s −1 is chosen to be low enough for metal strength to be an important parameter characterising the impact. It is also sufficiently high for the rod to flow as a jet, which is consumed as it penetrates the target. The first numerical study neglects the material strength representation, so that the flow is inviscid. On impact, the flat face of the rod strikes the flat face of the target, and one dimensional analysis is used to check numerical predictions for the initial impact pressure and velocity. A steady state penetration is quickly achieved, at a velocity which is in agreement with theoretical predictions of jet flow. In the second numerical study, an elastic-perfectly plastic representation of material strength is included within the calculation. It is then found that the rod has to travel several rod diameters into the target before the penetration velocity falls from the one dimensional impact value to a steady state value. This implies that the resistance to flow increases with the depth into the target, and consequently the penetration achieved by a rod will be dependent on its diameter, as well as its length.
TL;DR: In this article, a n asymptotic solution is given for Mode II dynamic fields in the neighborhood of the tip of a steadily advancing crack in an incompressible elastic solid (plane strain).
Abstract: A n asymptotic solution is given for Mode II dynamic fields in the neighborhood of the tip of a steadily advancing crack in an incompressible elastic—perfectly-plastic solid (plane strain). It is shown that, like for Modes I and III (G ao and N emat -N asser , 1983), the complete dynamic solution for Mode II predicts a logarithmic singularity for the strain field, but unlike for those modes which involve no elastic unloading, the pure Mode II solution includes two elastic sectors next to the stress-free crack surfaces. This is in contradiction to the quasi-static solution which predicts a small central plastic zone, followed by two large elastic zones, and then two very small plastic zones adjacent to the stress-free crack faces. The stress field for the complete dynamic solution varies throughout the entire crack tip neighborhood, admitting finite jumps at two shock fronts within the central plastic sector. This dynamic stress field is consistent with that of the stationary crack solution, and indeed reduces to it as the crack growth speed becomes zero.
TL;DR: In this article, a complete study of the phenomena of incubation, initiation and propagation of cracks in thin plates, when they are subjected to a compressive pulse, which is subsequently reflected from the free boundaries of the plate and changed to complicated wave-trains, was undertaken.
Abstract: When a stress wave (tensile or compressive) impinges on a crack existing in an elastic medium, reflection, refraction and diffraction-phenomena take place. A result of diffraction is the loading of the crack. While compressive stress-waves do not create any stress concentration at the tip of an existing crack, tensile stress-waves develop stresses at the tip which may cause a propagation of the crack. If the tensile pulse is weak the crack may propagate by steps under the action only of successive tensile stress-pulses, whereas intermediate compressive-stress pulses do not have any influence. A complete study of the phenomena of incubation, initiation and propagation of cracks in thin plates, when they are subjected to a compressive pulse, which is subsequently reflected from the free boundaries of the plate and changed to complicated wave-trains, was undertaken in this paper, based on the method of caustics. Interesting results that were derived from this experimental study are presented.
TL;DR: In this paper, the authors present closed-form solutions to interactive buckling models, which exhibit a form of Bifurcation involving looping equilibrium paths, identified recently from topological considerations, and are put forward as evidence of its typicality.
Abstract: E xact , closed-form, solutions to a recent interactive buckling model proposed by B udiansky and H utchinson (1979) are presented in full for the first time. These exhibit a form of Bifurcation involving looping equilibrium paths, identified recently from topological considerations, and are put forward as evidence of its typicality. A second, related, typical form is also identified in the response of the model. The predictions of two different asymptotic approaches, the first from traditional structural mechanics and the second making use of the mathematical concept of determinacy, are compared with the exact solutions.
TL;DR: In this article, the S elf Consistent Scheme approximation method is applied to evaluate the thermal expansion coefficient of statistically isotropic polycrystalline aggregates, in terms of single crystal thermoelastic properties and polycrystal elastic properties.
Abstract: T he S elf Consistent Scheme approximation method is applied to evaluate the thermal expansion coefficient of statistically isotropic polycrystalline aggregates, in terms of single crystal thermoelastic properties and polycrystal elastic properties. The case of orthorhombic crystals is considered in detail.
TL;DR: In this article, a simple model was proposed for the interpretation of the non-circular form of the Rayleigh wavefronts emitted by a fast running crack in a plate.
Abstract: A simple model was proposed for the interpretation of the non-circular form of the Rayleigh wavefronts emitted by a fast running crack in a plate. The surface deformation around the crack tip, due to the high stress concentration there, propagated as a surface wave after fracture of this zone. On the other hand, the moving singularity of the crack tip created a dynamic stress field of varying intensity with time all over the specimen. This dynamic stress field resulted in a significant change of the mechanical properties of a strain-rate dependent material and therefore it influenced the velocity of propagation of fracture-Rayleigh wavefronts. An analysis of this varying dynamic strain field explained the non-circular form of Rayleigh waves, accompanying the propagating crack. For the experimental evaluation of the K1-factor the method of dynamic caustics was used in conjunction with the high-speed photography technique.
TL;DR: In this article, a simple procedure is used to determine the effective pipe length associated with the instability of circumferential crack growth in a piping system. But it is assumed that the pipe-ends remain fixed, i.e. they are built-in, throughout this operation.
Abstract: A simple procedure is currently used to determine the effective pipe length associated with the instability of circumferential crack growth in a piping system. This procedure involves a separation of the complete piping system into two elastic parts at the cracked cross-section, the application of equal and opposite moments M to the cut faces, and the equation of the effective pipe length with El∥φ∥ M where φ is the rotational discontinuity generated at this section, E is Young's modulus and I is the second moment of area of the pipe at the cracked section. It is presumed that the pipe-ends remain fixed, i.e. they are built-in, throughout this operation. This paper shows that this procedure refers to the stability of a crack in a piping system which is subject to either a fixed displacement or a fixed rotation at a built-in end. The viability of the simple procedure is therefore underscored by the present paper's analysis.
TL;DR: In this paper, a simple perturbation procedure was proposed to determine the first order change of eigenmodes and critical loads at bifurcation buckling of slender structures when cavities or reinforcements are introduced.
Abstract: It is shown that a simple method may be applied in order to determine the first order change of eigenmodes and critical loads at bifurcation buckling of slender structures when cavities or reinforcements are introduced. The perturbation procedure developed is applicable whenever a characteristic diameter of the structural geometry perturbation is one order less than the wave length of the unperturbed eigenmode and necessary computations involve at most the solution of two linear boundary value problems. Situations when the procedure may be further simplified are discussed and statements are made regarding requisite conditions for the buckling load definitely to increase or decrease due to geometry perturbations. Application of the method is illustrated by means of four cases of engineering interest, which involve perforated and reinforced beams and plates with holes and cracks.
TL;DR: In this article, the inclined strip yield model with continuous distributions of infinitesimal dislocations was utilized to express the crack tip plasticity in this model and the fatigue crack tip blunting process was approximated by sequential activation of two slip lines under loading, and crack closure during unloading was taken into account.
Abstract: A computational model was developed to numerically analyse fatigue striations. The inclined strip yield model with continuous distributions of infinitesimal dislocations was utilized to express the crack tip plasticity in this model. The fatigue crack tip blunting process was approximated by sequential activation of two slip lines under loading, and crack closure during unloading was taken into account. The plastic zone at a growing fatigue crack tip at the maximum load was independent of the crack growth up to ten cycles while the reversed plastic zone decreased in a size to one twentieth of that at the maximum load as the crack grew. The ratio of these plastic zone sizes and also the crack tip opening displacement were quite different from the simple prediction by J.R. Rice for a stationary crack. The computed striation spacings were compared with the observed ones and moderate agreement between them obtained.
TL;DR: In this paper, the authors examined the case where a pipe of length L is built-in at one end and the other end is subject to an imposed displacement or rotation, and concluded that the instability criterion is the same irrespective of whether a displacement and rotation is imposed at a builtin end.
Abstract: The paper examines the case where a pipe of length L is built-in at one end and the other end is subject to an imposed displacement or rotation. The criterion for instability of growth of a circumferential through-wall crack is shown to depend on the pipe-end boundary conditions as well as the pipe geometry, crack size and crack location. The worst possible case is that where there is only a force, but no moment, at the pipe-end. However, this is probably an artificial situation which is unlikely to arise in practice. A pipe is more likely to be built-in at both ends, and for this situation, it is concluded that the instability criterion is the same irrespective of whether a displacement or rotation is imposed at a built-in end.
TL;DR: In this article, the effect of applied stress on internal variable bifurcation was analyzed and connections were made between the internal (microscopic) picture and the thermo-elastic (macroscopic) view of the phenomenon.
Abstract: P hase changes involving bifurcation of an internal variable are discussed. It is shown that one particular type of such bifurcation is likely to be preferred above the others, and we present an analysis of the effect of applied stress on that type of bifurcation. Connections are made between the internal (microscopic) picture and the thermoelastic (macroscopic) picture of the phenomenon.