Showing papers in "Journal of The Mechanics and Physics of Solids in 1969"
TL;DR: In this article, a variational principle is established to characterize the flow field in an elastically rigid and incompressible plastic material containing an internal void or voids, and an approximate Rayleigh-Ritz procedure is developed and applied to the enlargement of an isolated spherical void in a nonhardening material.
Abstract: The fracture of ductile solids has frequently been observed to result from the large growth and coalescence of microscopic voids, a process enhanced by the superposition of hydrostatic tensile stresses on a plastic deformation field. The ductile growth of voids is treated here as a problem in continuum plasticity. First, a variational principle is established to characterize the flow field in an elastically rigid and incompressible plastic material containing an internal void or voids, and subjected to a remotely uniform stress and strain rate field. Then an approximate Rayleigh-Ritz procedure is developed and applied to the enlargement of an isolated spherical void in a nonhardening material. Growth is studied in some detail for the case of a remote tensile extension field with superposed hydrostatic stresses. The volume changing contribution to void growth is found to overwhelm the shape changing part when the mean remote normal stress is large, so that growth is essentially spherical. Further, it is found that for any remote strain rate field, the void enlargement rate is amplified over the remote strain rate by a factor rising exponentially with the ratio of mean normal stress to yield stress. Some related results are discussed, including the long cylindrical void considered by F.A. McClintock (1968, J. appl. Mech . 35 , 363), and an approximate relation is given to describe growth of a spherical void in a general remote field. The results suggest a rapidly decreasing fracture ductility with increasing hydrostatic tension.
4,156 citations
TL;DR: In this article, the overall elastic moduli of some solid composite materials are evaluated, first by bounding them precisely, and secondly by a self-consistent estimate, at both random and aligned orientations, and at arbitrary volume concentration.
Abstract: The overall elastic moduli of some solid composite materials are evaluated, first by bounding them precisely, and secondly by a ‘self-consistent’ estimate. Transversely isotropic inclusions of ‘needle’ and ‘disc’ shapes are particularly considered, at both random and aligned orientations, and at arbitrary volume concentration.
464 citations
TL;DR: In this article, the authors extended the theory of long rod penetration to take account of the deformation of a soft rod against a rigid target and the penetration of a rigid projectile into a soft target.
Abstract: T he theory of long rod penetration as given in a previous paper by the author is extended to take account of the deformation of a soft rod against a rigid target and the penetration of a rigid projectile into a soft target. It is shown that it is theoretically possible to have a decrease in depth of penetration with increasing impact velocity, and a method for deducing the average radius of the hole is given. The theory is compared with experimental results.
201 citations
TL;DR: In this paper, an expression for the dynamic elastic field of a crack when one of its tips moves arbitrarily in the plane of the crack, starting from rest, was found with the help of a theorem of Bateman's.
Abstract: With the help of a theorem of Bateman's an expression is found for the dynamic elastic field of a crack when one of its tips moves arbitrarily in the plane of the crack, starting from rest. The linear isotropic theory of elasticity is used, and only states of anti-plane strain (mode III deformation) are considered. The crack is initially of finite length and subject to any static anti-plane loading. The solution obtained becomes inaccurate in regions into which disturbances reflected at the other tip have penetrated. The error is estimated for some special cases. The results are used to discuss the equation of motion of a crack tip.
191 citations
TL;DR: In this paper, the effective complex shear modulus of two-phase media of the composite sphere model was derived for the case of spherical voids and perfectly rigid spherical inclusions embedded in a matrix media.
Abstract: Some of the methods of analysis used to obtain the effective mechanical properties of heterogeneous elastic materials are reviewed with respect to the possibility of establishing extensions to heterogeneous viscoelasticity. In certain cases, a simple transition from heterogeneous elasticity results to the more general viscoelasticity forms can be effected. However, in some other cases, where this cannot be done, a new procedure is derived for establishing bounds upon the effective viscoelastic mechanical properties. This new procedure involves the development and application of two viscoelastic minimum theorems. Rigorous, closed form analytical expressions are found for upper and lower bounds of the effective complex shear modulus in the cases of spherical voids and perfectly rigid spherical inclusions embedded in a matrix media in accordance with the composite sphere model. For this same geometric model, but composed of two independently specified viscoelastic phases, rather than one medium with voids or rigid inclusions, an approximate formula is derived for the effective complex shear modulus. This formula along with the exact expression for the effective complex bulk modulus, which is available, provides a complete formulation of the effective properties for heterogeneous viscoelastic media of the composite sphere model type. The minimum theorems are also used to obtain some bounds information for the effective properties of two phase media with no geometric restrictions upon the interface between phases.
165 citations
TL;DR: In this paper, the external stress required to hold a screw dislocation in equilibrium in a two-phase material formed by joining two dissimilar elastic half-planes is determined using the Peierls model of a dislocation.
Abstract: The external stress required to hold a screw dislocation in equilibrium in a two-phase material formed by joining two dissimilar elastic half-planes is determined using the Peierls model of a dislocation. The displacement field on the slip plane is also obtained when the dislocation is at the interface. The solution to this problem that appears in the literature is based on a singular dislocation model and is valid only when the dislocation is at some distance away from the interface, since the expression for the necessary external stress for equilibrium of the dislocation becomes infinite when the dislocation is at the interface. By taking into consideration the atomic forces on the slip plane, the present work yields a finite value of external stress needed to cause penetration of the barrier and reduces to the known solution for a singular dislocation when the distance of the dislocation from the interface is large.
70 citations
TL;DR: In this article, it was shown that stresses and strains in an infinite tensile sheet with a hole consisting of an arbitrary number of branches of different lengths and directions emerging from a common origin can be determined by means of methods due to N. Muskhelishvili.
Abstract: It is shown that stresses and strains in an infinite tensile sheet with a hole consisting of an arbitrary number of branches of different lengths and directions emerging from a common origin can be determined by means of methods due to N. Muskhelishvili. The complex stress functions are calculated and the stress-intensity factors at the tips of the branches are studied. Numerical results are given for branched crack contours comprising a straight main crack with one sidebranch of varying length and direction. Also, symmetric forking with arbitrary forking angle is studied. By using a quasi-static point of view necessary conditions for forking are derived.
60 citations
TL;DR: In this paper, the mechanics of the build-up of large local plastic strains at the free surface of a polycrystal under fatigue loading is shown, and the increase of local plastic strain in the slices with cycles of loading is calculated by applying the analogy between plastic strain and external force.
Abstract: The mechanics of the build-up of large local plastic strains at the free surface of a polycrystal under fatigue loading is shown. Two closely located thin slices in a most-favourably-oriented crystal located at a free surface of the polycrystal are assumed to have an initial resolved shear stress of a few psi of opposite sign. Under cyclic loading, the two thin slices slide in opposite directions, one during the forward loading and the other during the reversed loading. Based on the dependency of slip on the resolved shear stress, the increase of local plastic strain in the slices with cycles of loading is calculated by applying the analogy between plastic strain and external force. The calculated local plastic shear strain in both slices (one positive and the other negative) reaches 100 per cent at the free surface in a few hundred cycles. These large plastic shear strains clearly cause the start of an extrusion or intrusion and the nucleation of a fatigue crack.
56 citations
TL;DR: In this article, it was shown that the usual definition of a kinematically admissible velocity field is unnecessarily restrictive for the validity of the upper bound inequality of limit analysis, which can be used to derive overestimates of quantities of interest in metal-forming processes which have not been considered hitherto.
Abstract: The usual definition of a kinematically admissible velocity field is unnecessarily restrictive for the validity of the upper bound inequality of limit analysis. In consequence, it is shown that this inequality can be used to derive overestimates of quantities of interest in metal-forming processes which have not been considered hitherto. In particular, upper bounds can be obtained for the total load in the presence of Coulomb friction on the tool/workpiece interface. In illustration, the technique is applied to problems of compression and extrusion.
52 citations
TL;DR: In this article, the authors examined the buckling of imperfection sensitive elastic structures from a statistical point of view and showed that the statistics of the critical load distribution are highly dependent on both the mean and the dispersion about the mean of the imperfections.
Abstract: The buckling of imperfection-sensitive elastic structures is examined from a statistical point of view. Given the relation between the imperfection and the buckling load, an account is given of the dependence of the statistical parameters of the critical load distribution upon the parameters of the imperfection distribution. It is shown that the statistics of the critical load distribution are highly dependent on both the mean and the dispersion about the mean of the imperfections. The question of the probability of failure at a specified nominal load, and its dependence on the degree of uncertainty of both load and imperfection, is also analysed. Numerical results are obtained on the basis of a normal imperfection distribution for two classes of structures, namely symmetric and asymmetric structures.
51 citations
TL;DR: In this article, an incremental analysis of the hydrostatic bulging of a circular sheet clamped at the periphery is performed on the basis of both an incremental theory and the corresponding total strain theory of plasticity, where the material of the sheet is assumed to have strainhardening capacity and to be anisotropic in the thickness direction.
Abstract: The hydrostatic bulging of a circular sheet clamped at the periphery is analysed on the basis of both an incremental theory and the corresponding total strain theory of plasticity. The material of the sheet is assumed to have strain-hardening capacity and to be anisotropic in the thickness direction. It is found that the incremental theory predicts that as the polar strain increases the pressure reaches a maximum and then decreases, whereas the total strain theory gives unsatisfactory results. Incremental calculations have been made for the pressure, the polar strain and the polar height at the maximum pressure (instability pressure) as functions of assumed strain-hardening characteristics and the degree of anisotropy in the thickness direction of the sheet.
TL;DR: In this paper, some fairly general thermodynamic constitutive equations are derived for an elastic-plastic body, which are then reduced to fairly simple forms by making plausible physical assumptions.
Abstract: Some fairly general thermodynamic constitutive equations are derived for an elastic-plastic body, which are then reduced to fairly simple forms by making plausible physical assumptions. The reduced equations retain sufficient flexibility to facilitate fitting to experimental results but are also sufficiently simple to offer the prospect of solving problems. An illustrative example is given to support this latter statement.
TL;DR: In this article, the elastic interaction between an edge dislocation and a circular inclusion with a slipping interface is considered, and the behaviour of the dislocation can be classified in terms of two parameters formed from the elastic constants.
Abstract: The elastic interaction between an edge dislocation and a circular inclusion with a slipping interface is considered. The behaviour of the dislocation can be classified in terms of two parameters formed from the elastic constants. For certain values of these parameters, the interaction is not restricted to either simple attraction or repulsion, and the dislocation has equilibrium positions that correspond to saddle points in the interaction energy.
TL;DR: In this paper, a theoretical analysis is made of the dimensional changes, stresses and plastic flow accompanying an allotropic transformation in one component of a fiber composite, taking into account the simultaneous stress relaxation which occurs.
Abstract: A theoretical analysis is made of the dimensional changes, stresses and plastic flow accompanying an allotropic transformation in one component of a fibre composite, taking into account the simultaneous stress relaxation which occurs. The theory predicts the transformation dimensional changes observed experimentally under various temperature cycling conditions and provides an estimate of the stresses. Extending the theory to include the effect of a variation in temperature, several categories of coefficients of thermal expansion are obtained, none of which correspond to the rule of mixtures. The predicted transition from one coefficient of expansion to another during continuous cooling is confirmed experimentally and the resulting dimensional changes verified.
TL;DR: In this article, a method of bounding the shakedown domain for shells is proposed, starting from W.T. Koiter's inadaptation theorem and using the consequences following from the associated flow rule for piece-wise linear yield surfaces.
Abstract: A method of bounding the shakedown domain for shells is proposed. Starting from W.T. Koiter's ‘inadaptation’ theorem and using the consequences following from the associated flow rule for piece-wise linear yield surfaces a procedure is developed to bound from above the shakedown loading programme for a shell. For multi-parameter loadings a time independent interaction surface bounding the shakedown domain from above is obtained whenever the elastic and the limit analysis solutions for the individual loads are known. An example of shakedown interaction curve for a two-parameter loading of a cylindrical shell is given.
TL;DR: In this article, the authors re-examine the original problem in the light of the modern theory of optimal plastic design and apply general optimality criteria to full circular as well as annular plates.
Abstract: Hopkins and Prager (1955) used an intuitive approach based on the concept of competing yield mechanisms to discuss plastic minimum-weight design of a circular plate with piecewise constant cross section. Since this paper was written, the theory of optimal plastic design has progressed considerably, but subsequent papers on optimal plate design were exclusively concerned with plates of continuously varying cross section. In view of the lesser manufacturing cost of plates with piecewise constant cross section, the present paper re-examines the original problem in the light of the modern theory of optimal plastic design. General optimality criteria are established and applied to full circular as well as annular plates. The relation of the designs obtained for annular plates to the singular designs investigated by Mpegarefs (1966, 1967) is discussed.
TL;DR: In this article, upper and lower bounds for both real and imaginary parts of the complex rigidity and bulk moduli of the system were derived for both elastic and viscous moduli.
Abstract: If a system , consisting of firmly bonded isotropic linearly viscoelastic phases, behaves macroscopically as a homogeneous isotropic material under oscillatory deformation, upper and lower bounds can be set to both the real and imaginary parts of the complex rigidity and bulk moduli of the system. These reduce to the Voigt and Reuss bounds on the elastic moduli when the phases are purely elastic, and to the corresponding bounds on the shear viscosity and bulk viscosity when the phases are purely viscous.
TL;DR: In this paper, three special forms of material functions are introduced into the Green-Rivlin constitutive equations of nonlinear viscoelasticity, and two of these reduce the third-order equations to single integral forms, rationally and experimentally derived by others.
Abstract: Three special forms of material functions are introduced into the Green-Rivlin constitutive equations of nonlinear viscoelasticity. Two of these reduce the third-order equations to single integral forms, rationally and experimentally derived by others. Published creep and stressrelaxation data are analysed using both the proposed and previously published constitutive equations, particular attention being given to multi-step data. Specific results show the limits of applicability of the third-order constitutive relations for the materials considered.
TL;DR: In this paper, a rational explanation can be given of the seemingly complicated manner in which the fatigue properties of annealed low-carbon steel depend on the number of grains in a specimen cross-section, the polycrystal grain size and the yield point behaviour.
Abstract: I t is shown, by comparison of fatigue and yield strength results previously reported for several materials, that a rational explanation can be given of the seemingly complicated manner in which the fatigue properties of annealed low-carbon steel depend on the number of grains in a specimen cross-section, the polycrystal grain size and the yield point behaviour This dependence varies according to whether the fatigue stress is greater than, equal to, or less than the yield stress of the same material
TL;DR: In this article, it was shown that friction at the interface between the rolls and a thin strip leads to an appreciable region in the centre of the arc of contact where the strip does not slip relative to the rolls.
Abstract: The elastic stresses in a thin strip as it passes between rolls have been studied with a view to predicting the onset of plastic reduction. It has been shown that the load to initiate plastic reduction of hard materials increases very rapidly for thin gauges but, how ever thin the strip, some reduction is shown to be theoretically possible. The stresses in the strip which control the onset of plastic flow are critically dependent upon the frictional conditions at the interface between the rolls and the strip. It is shown that friction at the interface between the rolls and a thin strip leads to an appreciable region in the centre of the arc of contact where the strip does not slip relative to the rolls. Under these conditions plastic reduction occurs at entry and at exit to the roll bite but does not occur in the central region of no relative motion.
TL;DR: In this article, the propagation of finite amplitude waves with polar symmetry through an isotropic elastic-plastic work-hardening material is studied, and the required number of interface conditions to determine the interface path and wave motion in two regions is obtained for each type.
Abstract: Equations governing the propagation of finite amplitude waves with polar symmetry through an isotropic elastic-plastic work-hardening material are formulated. Formal characteristics solutions for elastically and plastically deforming regions are given, and the respective validity conditions obtained. Matching conditions across a moving interface between elastic and plastic regions at which the normal stress is continuous are determined from the balance laws and constitutive response. Analogous to the situation for waves in uni-axial strain, six types of interaction can occur corresponding to interface propagation speeds in the six ranges limited by the elastic and plastic wave propagation speeds. Given the initial motion, the required number of interface conditions to determine the interface path and wave motion in the two regions is obtained for each type. The role of vanishing plastic work rate is shown precisely. Linearized equations for infinitesimal deformation and a parabolic work-hardening law are recovered, and give rise to displacement potentials which satisfy a non-homogeneous wave equation in both regions. Global solutions can then be formally expressed in terms of wave functions.
TL;DR: In this article, a general perturbation theory for the branching analysis of perfect and imperfect discrete conservative structural systems is presented without resorting to a scheme of diagonalization, and the absence of such a scheme distinguishes the present development from an earlier study by the author.
Abstract: A general , perturbation theory for the branching analysis of perfect and imperfect discrete conservative structural systems is presented. Such systems are best analysed without resort to a scheme of diagonalization, and the absence of such a scheme distinguishes the present development from an earlier study by the author. The tensor notation and the system of sliding axes employed in that study are however of considerable analytical value and are therefore retained. The theory is presented for both a general and a specialized class of system and some general features of the perturbation scheme are established. For imperfect systems the concept of a spiralling eigenvector is introduced to yield the equations of imperfection-sensitivity explicitly in terms of the post-buckling derivatives of the perfect system and in a form that can be directly employed in numerical analysis.
TL;DR: In this paper, a nonsteady analytical solution for plane strain necking of V-notched, rigid/perfectly-plastic tensile bars was obtained by E.H. Lee, where the plastic zone does not remain geometrically similar, velocity discontinuity is absent, and plastic flow results in a convex shape.
Abstract: An analytical nonsteady solution for plane strain necking of V-notched, rigid/perfectly-plastic tensile bars was obtained by E.H. Lee in 1952. In his solution the plastic zone remains geometrically similar, a velocity discontinuity occurs on the rigid-plastic boundary, and when carried to the point of separation, the plastic flow results in a wedge-shaped neck. In this paper a new analytical solution is given for the same problem. However, the plastic zone does not remain geometrically similar, velocity discontinuities are absent, and plastic flow results in a convexshaped neck. Comparisons with measurements of neck profiles obtained under conditions approximating plane strain indicate that the new solution is a good representation of the physical situation n metals. Thus, it appears to be useful for estimating basic stress-strain relations from tensile tests after necking, and for estimating the stresses and strains required for fracture from notched bar tests.
TL;DR: Chadwick and Morland as mentioned in this paper presented a closed-form solution for a uniform pressure applied instantaneously and smoothly released on the surface of a spherical cavity in an infinite elastic-plastic medium.
Abstract: T his paper presents a closed-form solution for a uniform pressure applied instantaneously and smoothly released on the surface of a spherical cavity in an infinite elastic-plastic medium. The analysis is for a linearized theory, including plastic work-hardening, and complements the solution for the starting problem (P. Chadwick and L.W. Morland) when the applied pressure is maintained. The assumed wave-pattern is an attenuating elastic loading discontinuity front followed in turn by an expanding region of continuous elastic loading, an attenuating plastic loading discontinuity front, and a continuous elastic unloading region. After a finite time the plastic discontinuity is totally annulled, leaving purely elastic disturbances. An explicit solution is obtained by prescribing the attenuation of the plastic loading front and determining the required pressure release on the boundary over a short initial time; the subsequent release is prescribed. A variety of cases is investigated to determine the effects of varying the plastic front attenuation, the pressure amplitude, the final pressure release, and the difference between perfectly-plastic and work-hardening materials. Estimates of the maximum distance the plastic discontinuity front travels before annulment are deduced, and shown to be significantly lower than corresponding distances in the starting problem. Conditions for the onset of reverse yield are noted. A typical solution is illustrated.
TL;DR: In this article, a method of postbuckling analysis is described which is related to some work of W.T. K oiter. Differences in starting data and objectives between this method and the author's recent general theory of equilibrium behavior near critical points are pointed out.
Abstract: A method of post-buckling analysis is described which is related to some work of W.T. K oiter . Differences in starting data and objectives between this method and the author's recent general theory of equilibrium behaviour near critical points are pointed out. Some remarks are made about convergence difficulties which can arise when critical points are close together.
TL;DR: In this paper, a generic instant during a process of deformation of a body of isotropic rigid/plastic material with a regular yield surface is considered, at which the state of stress is spherically symmetric.
Abstract: A generic instant during a process of deformation of a body of isotropic rigid/plastic material with a regular yield surface is considered, at which the state of stress is spherically symmetric. The equations defining the direction of strain-rate in the material are solved to find the associated velocity fields, and a possible bifurcation is indicated. A uniqueness criterion given by R. H ill in (1957) is applied, using these fields, to the problem of a spherical shell expanded by internal pressure. Two resulting inequalities sufficient for uniqueness are expressed in terms of the distributions of hardening and uniaxial yield stress at the generic instant. The possibility exists that in certain circumstances bifurcation may take place before the pressure has reached an analytic maximum.
TL;DR: In this article, a modification to the Rayleigh-Ritz method is described for three-dimensional linear elasticity and, by way of illustration, for the problem of plate bending.
Abstract: Details are given of a modification to the Rayleigh-Ritz method which, by extending the field of definition of the coordinate functions, improves the convergence in predicting the values of stress concentrations in elastic continua. The extension to the field of definition requires that the coordinate functions are not only complete in energy but also allows early derivatives of the displacement to be continuous. The procedure is described for three-dimensional linear elasticity and, by way of illustration, for the problem of plate bending.
TL;DR: In this article, a method of determining the subsequent yield condition is proposed on the basis of theoretical and experimental studies of the yield surfaces in the deviatoric plane, and the yield surface as calculated by this method are compared with the results obtained by the biaxial testing of brass plate.
Abstract: A method of determining the subsequent yield condition is proposed on the basis of theoretical and experimental studies of the yield surfaces in the deviatoric plane. The yield surfaces as calculated by this method are compared with the results obtained by the biaxial testing of brass plate. A fairly good agreement is found between the two values thus obtained. The normality of the plastic strain vector with regard to the subsequent yield surface is also confirmed.
TL;DR: In this article, a general energy principle applicable to a wide range of both time dependent and time independent materials is particularized to a range of constitutive relations which describe the creep and plastic behaviour of metals.
Abstract: A general energy principle, applicable to a wide range of both time dependent and time independent materials is particularized to a range of constitutive relations which describe the creep and plastic behaviour of metals. When applied to the study of the deformation of a body subject to dead loading, bounds are obtained on the total deformation of the body. The techniques provides a useful direct means of deriving energy theorems for a wide range of constitutive relationships.
TL;DR: In this article, an energy theorem for a class of time dependent materials was derived, which provides a generalization of a theorem due to J.B. Martin for time independent materials.
Abstract: An energy theorem is derived for a class of time dependent materials, which provides a generalization of a theorem due to J.B. Martin for time independent materials. Convexity conditions are derived for minimum work and maximum complementary work functions, and the theorem is related to the complementary energy theorem of elasticity. Linear visco-elastic and non-linear Maxwell materials are treated.