Showing papers in "Journal of The Mechanics and Physics of Solids in 1976"

On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states

Abstract: The strain required to initiate ductile failure in three low-alloy, quenched and tempered steels has been determined in multi-axial stress-states. The ductility was found to depend markedly both on the orientation of the stress system with respect to the rolling direction and on the tri-axiality of the stress-state. In some cases, ductile failure occurred at plastic strains which were only a few times the yield strain. Metallographic studies have been used to compare the size, shape and orientation of the holes which cause failure initiation with the isotropic continuum analysis of F.A. Mc Clintock (1968). The application of ductile-fracture models to directional steels is discussed with particular reference to the effects of directionality and stress-state on the condition for flow localization to occur between holes.

Effect of thickness inhomogeneities in internally pressurized elastic-plastic spherical shells

Abstract: The behaviour of elastic-plastic spherical shells under internal pressure is investigated numerically for thickness-to-radius ratios ranging from cases of thin shells to very thick shells. The shells under consideration are made of strain-hardening elastic-plastic material with a smooth yield-surface. Attention is restricted to axisymmetric deformations, and results are presented for initial thickness inhomogeneities in various axisymmetric shapes. For smooth thickness-variations in the shape of the critical bifurcation mode, the reduction in maximum pressure is studied together with the distribution of deformations in the final collapse mode. Also, the possibility of flow localization due to more localized, initially thin regions on a spherical shell is investigated.

Volume changes associated with the deformation of rubber-like solids

Abstract: V olume changes accompanying the deformation of rubber-like solids are analysed on the basis of isotropic elasticity theory. In particular, a simple, but general, result relating the volume and the stretch in simple tension is obtained. This is achieved by the introduction of certain modified principal stretches which allow the dependence of the strain energy on the isochoric and the dilatational parts of the deformation to be considered separately. The fact that volume changes in rubber-like solids are typically of order 0.01 % is used to linearize the stress-deformation relations in the dilatation. This enables the dilatation to be given explicitly as a function of the stretch in simple tension. Specific results are obtained for certain classes of constitutive law and good agreement with the experimental data for simple tension is demonstrated. Results for equi-biaxial tension and pure shear are also given in anticipation of further experimental data becoming available. The need for volume-change data for a wide variety of types of strains is emphasized.

The self-force on a planar dislocation loop in an anisotropic linear-elastic medium

Abstract: A proper prescription for the in-plane self-force on each element of a plane dislocation loop is developed by computing the first variation of the loop self-energy during an arbitrary virtual planar change in the loop configuration. The appropriate self-energy is defined to be the strain energy exterior to a tube of radius e surrounding the loop. The expression derived for the self-force on a loop element ds depends on the local curvature at ds , on certain elastic data for an infinite straight dislocation tangent to the loop at ds , and only weakly on the ‘cut-off’ radius e. The theory of stress fields of dislocations in anisotropic media is sufficiently advanced to permit easy numerical evaluation of the self-force expression. The analysis further reveals that the singular behavior of the self-stresses in the plane of the loop near an element ds is that of an infinite straight dislocation tangent to ds plus a curvature-dependent logarithmic singularity which is proportional to the line-tension of this tangent dislocation.

Bifurcation phenomena in the plane compression test

Abstract: A basic , compression, bifurcation problem is studied by methods similar to those used by R. Hill and J. W. Hutchinson (1975) for the corresponding tension problem. Bifurcations from a state of homogeneous in-plane compression loading are investigated for a rectangular block of incompressible material constrained to undergo plane deformations. The sides of the block are tractionfree, and it is loaded compressively by a uniform, shear-free, relative displacement of its ends. For a wide class of incrementally-linear time-independent materials only two instantaneous moduli enter into the analysis. Diffuse modes of both symmetric and antisymmetric bifurcation are examined in the elliptic regime of the governing equations, and the possibility of localized modes is considered both inside and outside this regime. Lowest bifurcation stresses are computed for essentially the entire range of possible combinations of material properties and geometry, and these are compared with results obtained by Hill and Hutchinson for the tension problem. The limiting case of large thickness (the semi-infinite block) is considered, confirming the results of M. A. Biot (1965).

The propagation of surface waves in bodies with material boundaries

Abstract: A study is made of the propagation of Love and Rayleigh waves in elastic bodies whose boundaries support surface stresses, such as crystals cleaved in vacuo or bodies upon which are deposited extremely thin films. The surfaces of such bodies are herein modelled as two-dimensional elastic continua that adhere without slipping to their substrates and in which is located residual stress. The corresponding boundary condition gives rise to behaviour which is contrasted with that occurring in the absence of surface stress. The results are also related to previous studies of surface waves in thin strata superposed on elastic half-spaces, from which the present theory differs by taking into account residual stress. The behaviour is shown to be sensitive to the relative values of the residual stress, elastic moduli, and density of the surface. Attention is drawn to the relevance of the study to those signal-processing devices which utilize surface waves in thin films.

Yielding on inclined planes at the tip of a crack loaded in uniform tension

Abstract: Plane-strain yielding from a crack in an infinite elastic body is represented here by a distribution of edge dislocations on two planes inclined at angles ±ga to the crack plane, and the equilibrium condition is solved numerically. Approximate analytical expressions are obtained for the plastic-zone length, the crack opening displacement, and the J -integral, as functions of the applied stress and α. A comparison with a co-planar model of the plastic zone gives very similar results for α ≈ 65°. It is shown that fracture criteria based either on a critical crack opening displacement (COD) or on a critical value of J are always different, and the use of the former may lead to critical defect-sizes which are twice as large as those given by the latter. Furthermore, COD appears not to be a well-defined material property. The critical J criterion gives a fracture stress which is α-dependent : this may be responsible for deviations towards results of linear elastic fracture mechanics when post-yield fracture mechanics is used to analyse extensive yielding. The changes in the stress field of the crack due to the existence of the plastic zone are also discussed.

The elastic field caused by a general ellipsoidal inclusion and the application to martensite formation

Abstract: The elastic field throughout an ellipsoidal inclusion in an indefinitely-extended anisotropic material is investigated when an eigenstrain (a stress-free transformation strain) is periodically distributed throughout the inclusion. This is an extention of the results obtained by J.D. Eshelby (1961) for uniform eigenstrains and by R.J. Asaro and D.M. Barnett (1975) for polynomial eigenstrains. The solution is applied to the evaluation of elastic strain energies of a disc-shaped martensite with alternating twins and of a spherical precipitate with a banded structure. The significant amount of the elastic strain energies explains the necessity of the supercooling of austenite steel for the martensitic transformation to occur.

Plastic yielding on inclined slip-planes at a crack tip

Abstract: Plastic yield at crack tips on singular slip-planes, inclined to the crack plane, has been studied under plane-strain conditions for combined tension, hydrostatic stress, and in-plane shear. The singular integral equation, which represents the equilibrium condition of edge dislocations on the slip-planes, is transformed into a Fredholm integral equation in order to avoid difficulties that occur with its numerical solution. Results are presented for the slip-band length, the plastic crack-tip opening displacement, stress fields, and crack-opening contours. A series expansion of the results obtained numerically confirms approximate analytical expressions given by J.R. Rice (1974), up to the third-order in the applied stresses. The results of finite element methods agree with values of the crack-tip opening displacement obtained here to within 10 per cent. Ahead of the crack tip, the principal tensile stresses exceed the principal shear stresses by a factor of 10, approximately.

Yielding from a crack with finite root-radius loaded in uniform tension

Abstract: For the situation of plane strain, yielding from an embedded crack with finite root-radius and loaded in uniform tension has been studied. The plastic zone was represented by a coplanar array of edge dislocations and the equilibrium equations were solved numerically. Approximate analytical formulae for the dependence of the plastic-zone size and of the crack opening displacement on the ratio of the applied and friction stresses were constructed. A comparison of the results with those for the situation of anti-plane strain has been made and application of these results to the study of crack initiation from a blunt notch and of fracture toughness discussed.

Fracture trajectories in a biaxially stressed plate

Abstract: A specially designed rig has been used to examine the phenomena of crack growth in centrallynotched plates of polymethyl methacrylate under biaxial stress. Attention has been directed particularly towards investigating the phenomenon of path instability, viz. the deviation of a crack from its expected path with increasing transverse stress. Cracks grown under a biaxial stress system, whose transverse component, acting parallel to the crack, exceeds the normal component, do so in an S-shaped curve centred on the original, straight notch. Stress intensity factors for such a crack are estimated by superposition of related known solutions. On the assumption that crack extension takes place with opening displacements only at the tip, that is, the crack tip stress field remains symmetrical, the observed dependence of path geometry on the degree of stress biaxiality is explained.

Optimal design of partially discretized grillages

Abstract: Optimal layout of the beams of a grillage has so far only been discussed for grillage-like continua consisting of dense arrangements of beams of infinitesimal cross-sections. The present paper is concerned with grillages consisting of discrete main beams and dense arrangements of secondary beams of infinitesimal cross-sections, the typical secondary beam being simply supported on two neighbouring main beams and normal to them. A condition for the optimal spacing of the main beams is established. The proposed method is illustrated by examples in which first the optimal location of a prescribed number of main beams is determined and then the optimal number of main beams is derived in dependence of the coefficients associated with the specific cost function of the beams.

Elastic-plastic analysis of the yield process in mild steel

Abstract: A theoretical analysis of the particular yield-zone pattern in mild steel is presented. The constitutive relation adopted utilizes work of W.G. Johnston and J.J. Gilman (1959) and G.T. H ahn (1962) in conjunction with a newly-proposed criterion for the production of the initial, mobile dislocations. Numerical analysis was undertaken for the solution of an initial-value problem using the continuously distributed dislocations model. The patterns obtained for twisted bars of circular and square cross-sections closely resemble the well-known strain figures. It was also found that (i) the particular patterns come from the negative-slope characteristic of the stress-strain relation, (ii) the yield drop takes place when a small yield-zone, already formed around a defect at a lower nominal stress, begins to spread, and (iii) a scale-effect due to a characteristic length introduced in the proposed criterion exists. Besides, two experiments were conducted which gave support to the constitutive equation and the newly-proposed criterion.

Necking of pressurized spherical membranes

Abstract: T he necking of spherical membranes subject to a prescribed increase in enclosed volume is investigated. Attention is restricted to axisymmetric deformations. The materials considered are incompressible, isotropic, time-independent and incrementally linear. A complete set of axisymmetric bifurcation modes is considered and a simple relation is found to govern the critical stress for bifurcation into a given mode. The limiting critical stress and the corresponding mode for short wavelengths are investigated and related to the results obtained from an independent local-necking analysis. Two perturbation methods are employed to study the growth of initial imperfections: one is valid for arbitrary modes, but restricted to small deviations from sphericity, and the other is valid only for the local-necking mode, but is not restricted to small deviations. The effect of path-dependent material behavior on the onset of local necking is explored. Path-dependent material behavior is found to encourage the preferential growth of short wavelength imperfections. Path-independent materials are shown to exhibit significant sensitivity to initial imperfections in the localized-necking mode, although this sensitivity is far less than for a path-dependent material. When account is taken of initial material-property inhomogeneities as well as initial thickness imperfections, it seems that no definite conclusion can be drawn concerning the appropriateness or inappropriateness of an explanation of the onset of localized necking based on a smooth yield-surface plasticity theory and assuming the presence of such initial inhomogeneities.

Deformation field of a misfitting inclusion

Abstract: J.D. Eshelby (1957, 1959) has calculated the deformation field associated with an ellipsoidal inclusion in a state of homogeneous strain within an infinite matrix. Since most real precipitates occur with facets, the strain within such an inclusion is not uniform. Thus, plate precipitates of θ ′ in Al-Cu and η in Al-Au have coherent broad faces with mismatches of 1.34 and 4.95 % respect- ively and semicoherent or disordered interfaces at the edges with residual mismatches of about −4.3 and −1.00% normal to the broad faces. The deformation field in the matrix around such precipitates has been calculated using Kelvin's (1848) result for the stress field due to a point force. The calculations show the existence of high stresses near the edges of the precipitates where they have an appreciable misfit. Unlike the case of an ellipsoidal inclusion, the stress fields of these precipitates have dilatational components which can affect the diffusion of solute atoms to them and, thus, the kinetics of interface migration. The behavior of alloys containing these precipitates indicates that the moduli of the precipitates are somewhat greater than those of the matrices. The present calculations, based on the assumption that the two moduli are the same, underestimate the actual deformation field in the matrix. In real systems, therefore, the effects of the deformation field on misfit dislocation nucleation and kinetics of interface migration are likely to be somewhat greater in general.

Shock-wave evolution in a chemically reacting solid

Abstract: The evolutionary behavior of shock waves in the chemically-reacting solid (explosive) PBX-9404 has been observed using laser interferometry. These experimental results, along with the shockamplitude relation for reacting, elastic materials, permit the determination of the energy release rate at the shock front as a function of compression. The results show that shock-wave evolution in PBX-9404 is dominated at low amplitudes by mechanical dissipation effects and at high amplitudes by the exothermic decomposition reaction at the front.

The fracture of a glass half-space by projectile impact

Abstract: The brittle fracture of glass blocks subjected to intense impulsive loading is studied. A series of projectile-impact experiments was made and the fracture-zone growth was analysed as a function of time. It was found that the crack structure was essentially independent of the type of glass used or of the presence of embedded bubbles and visible defects. A marked effect was noted, however, when pre-existing micro-flaws were removed from the impact surface by acid-etching several of the target specimens prior to use. The experimental results are compared with two-dimensional axi-symmetric finite-difference calculations in which elastic-plastic constitutive models are contrasted with an intuitive representation of brittle fracture in which local failure occurs whenever any principal stress component exceeds in tension some small prescribed value. It is shown that, whichever model is employed, the development of the fracture zone is strongly influenced by a feed-back effect from the interaction of the elastic solution in the virgin material with signals propagating forward from the existing fracture-zone.

Matched asymptotic expansions in nonlinear fracture mechanics—I. longitudinal shear of an elastic perfectly-plastic specimen

Abstract: A method of analysis based upon matched asymptotic expansions is proposed for a cracked specimen which is subjected to longitudinal shear (mode III) loading. This gives the small-scale yielding estimate of linear fracture mechanics as a first approximation, and provides systematic refinements which take account of the nonlinear interaction between the elastic and the plastic regions. Explicit solutions can be generated for any specimen which is amenable to a linear elastic analysis. Fracture parameters, such as crack opening displacement and the J integral, are expressed as power series in the ratio of applied stress to yield stress, and three terms are given explicitly. These are defined from linear elastic solutions alone. The edge-cracked strip and cracking from a semi-circular notch are studied as examples. Comparison with an exact solution for the former geometry suggests that the three-term expansions give useful results up to 75 % of limit load. The latter example is new and shows the effect of a notch on a crack at loads beyond the normal range of validity of linear elastic fracture mechanics.

Matched asymptotic expansions in nonlinear fracture mechanics—II. Longitudinal shear of an elastic work-hardening plastic specimen

Abstract: Earlier analysis given by T.M. E dmunds and J.R. W illis (1976) is extended to deal with cracks in elastic work-hardening plastic specimens subjected to longitudinal shear loads. Solutions are expressed in terms of a set of parameters that are determined from linear elastic solutions alone. It is proved, for any specimen geometry and any loading symmetric about the plane of the crack, that a ‘plastic-zone correction’, obtained by solving a linear elastic problem for a crack which is a length r y longer than the actual crack, provides a two-term asymptotic expansion for the J -integral, if r y is defined suitably in terms of the linear elastic stress concentration factor and the initial slope of the work-hardening curve. The general method is applied in detail to a strip of finite width containing an edge crack, for which the effect of the work-hardening on the maximum extent of the plastic zone and on the J -integral is summarized graphically.

Shock waves in diatomic chains—II. nonlinear analysis

Abstract: Solutions are presented for the response of semi-infinite diatomic chains to a step jump in velocity. Integral transform theory and contour integration are used to express the solutions as definite integrals for the acoustic and optical branch contributions to the response. The contribution of an end mode is indicated for the case of particles that are unequally spaced within a unit cell. Asymptotic approximations are obtained for the contribution of the optical and acoustic branches to the wave solution when the wave propagates far into the lattice. Asymptotic estimates are obtained, also, for the discontinuous speeds at which the head of the pulse travels in a general diatomic chain. Shock profiles for the special case of a simple diatomic chain and for a general diatomic chain are discussed, and upper bounds are obtained for the maximum shock responses that are possible in such chains with the given shock condition.

Estimates of the rupture life of structural components subjected to proportional cyclic loading

Abstract: The process of creep in metals is associated with physical mechanisms which cause internal damage. This damage weakens the material; so that for a given stress level, the strain-rate increases with time. As a result of this behaviour, stress redistribution occurs which can greatly influence the ruptuie life of structural components. By use of the appropriate constitutive relations which model the tertiary portion of the creep curve, it is possible to estimate the life of a structural component by use of finite element methods. Unfortunately, the procedure is demanding on computer time; and, as a result, considerable attention has been given to the possibility of the use of bounding techniques which ease the computing problem and which are particularly useful at the early stage of design. Techniques have already been developed for bounding the rupture life of structural components subjected to constant loading and these are found to be useful. In this paper, bounding procedures are developed for structural components subjected to proportional cyclic loading. The results may be expressed in terms of a representative rupture stress, so that structural-component performance can readily be related to material behaviour.

Interfacial de-bonding caused by a screw dislocation pile-up at a circular cylindrical rigid inclusion

Abstract: An array of continuously-distributed screw dislocations piled up against a circular cylindrical rigid inclusion is analyzed by the complex-variable method. Both uniformly applied shearing load at infinity and internal friction stress opposing the movement of dislocations are taken into account. The pile-up tip is away from the matrix-inclusion interface, its distance from the interface being determined by the condition that the stresses should be finite everywhere in the solid. Stress distributions on the interface are determined, and de-bonding of the interface, namely the formation of initial voids or cracks, is discussed. Stress and displacement near the tip of these initial voids are then analyzed. This analysis is combined with the virtual work argument of A.A. Griffith (1920) to yield a criterion for the initial voids to grow along the interface. The critical void-growth load is expressed by the sum of two terms, one proportional to the friction stress and the other inversely proportional to the square-root of the inclusion radius.

Thermal-energy relaxation in shock-compressed solids

Abstract: I n order to examine the significance of thermal-energy relaxation in shock-compressed solids, a continuum theory is formulated following along lines similar to those for energy relaxation in gases. The model predicts the features that are well known for molecular gases, namely, a two-wave structure and a high temperature, precursor shock. For a given particle velocity, the latter propagates faster than an ordinary shock. This permits the theory to be tested by the examination of the experimental shock velocity versus particle velocity relation for solids. An analysis of six substances (Cu, Al, Pb, Au, NaCl, α-Fe) seems to reveal a need for the re-examination of all the experimental data, and it is suggested that the standard techniques used in order to assess the Hugoniot data may have overlooked the relaxation effect in some substances, such as NaCl, Au and Pb.