Showing papers in "Journal of The Mechanics and Physics of Solids in 1974"

Limitations to the small scale yielding approximation for crack tip plasticity

Abstract: Recent finite-element results by S G Larsson and A J Carlsson suggest a limited range of validity to the 'small scale yielding approximation,' whereby small crack tip plastic zones are correlated in terms of the elastic stress intensity factor It is shown with the help of a model for plane strain yielding that their results may be explained by considering the non-singular stress, acting parallel to the crack at its tip, which accompanies the inverse square-root elastic singularity Further implications of the non-singular stress term for crack tip deformations and fracturing are examined It is suggested that its effect on crack tip parameters, such as the opening displacement and J-integral, is less pronounced than its effect on the yield zone size

A crack along the interface of a circular inclusion embedded in an infinite solid

Abstract: The two-dimensional problem of an arc shaped crack lying along the interface of a circular elastic inclusion embedded in an infinite matrix with different elastic constants is considered. Based on the complex variable method of Muskhelishvili, closed-form solutions for the stresses and the displacements around the crack are obtained when general biaxial loads are applied at infinity. These solutions are then combined with A.A. Griffith's virtual work argument to give a criterion of crack extension, namely the de-bonding of the interface. The critical applied loads are expressed explicitly in terms of a function of the inclusion radius and the central angle subtended by the crack arc. In the case of simple tension the critical load is inversely proportional to the square-root of the inclusion radius. By analyzing the variation of the cleavage stress near the crack tip, the deviation of the crack into the matrix is discussed. The case of uniaxial tension is worked out in detail.

Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension

Abstract: The bifurcation problem governing the onset of axisymmetric necking in a circular cylindrical specimen in uniaxial tension is analysed. The specimen is made of an incompressible elastic/plastic material. One end is subject to a prescribed uniform axial displacement relative to the other and both ends are shear free. The true stress at bifurcation is greater than the stress at which the maximum load is attained by an amount which depends on (a) the radius to length ratio of the specimen, (b) the ratio of the elastic shear modulus to the tangent modulus, and (c) the derivative of the tangent modulus with respect to the stress. Bifurcation takes place immediately following attainment of the maximum load when the specimen is sufficiently slender.

On the temperature rise at the tip of a fast running crack

Abstract: T he high energy concentration at the tip of a running crack leads to irreversible deformations, and a great amount of the deformation energy is set free as heat. Assuming that this moving heat source is of circular shape, the temperature distribution around the crack tip has been calculated. The temperatures are dependent on the radius of the heat source and the crack velocity. Some examples for the material glass are given. The very high temperatures computed lead to the supposition that the observed light emission during fast fracture is of thermal origin.

Elastic waves in a fiber-reinforced composite

Abstract: T he propagation of time-harmonic elastic waves in a fiber-reinforced composite is studied. The circular fibers are assumed to be parallel to each other and randomly distributed with a statistically uniform distribution. The direction of propagation and the associated particle motion are considered to be normal to the fibers. It is shown that the average waves in the composite separate into compressional and shear types. General formulae for the complex wave number giving the phase velocity and the damping are obtained. It is shown that these formulae lead to the Hashin-Rosen expressions for the transverse bulk modulus and the lower bound for the transverse rigidity, if the correlation in the positions of the fibers can be ignored. The correlation terms, for exponential correlation, are shown to have a significant effect on the damping property of the composite, especially at high frequencies and concentrations.

On the overall moduli of non-linear elastic composite materials

Abstract: F rom the work of R. Hill on constitutive macro-variables it is known that for an inhomogeneous elastic solid under finite strain an overall elastic constitutive law may be defined. In particular, the volume average of the strain energy of the solid is a function only of the volume-averaged deformation gradient. In view of the importance of this result it is re-derived in this paper as a prelude to a discussion of composite materials. A composite material consisting of a dilute suspension of initially spherical inclusions embedded in a matrix of different material is considered. For second-order isotropic elasticity theory an expression for the overall bulk modulus of the composite material is obtained in terms of the moduli of the constituents. When the inclusions are vacuous a known result for the bulk modulus of porous materials is recovered. In certain situations the strengthening/ weakening effects of the inclusions are less pronounced in the second-order theory than in the linear theory.

Crack propagation in an elastic solid subjected to general loading— IV. Obliquely incident stress pulse

Abstract: T he stress intensity factors of a half-plane crack extending nonuniformly in an isotropic elastic solid subjected to stress wave loading are considered. A plane stress pulse is obliquely incident on the crack, and the wavefront strikes the crack at some initial time. At some arbitrary later time, the crack begins to extend at a nonuniform rate. It is found that the mode I and mode II stress intensity factors each have the form of the product of a universal function of instantaneous cracktip speed with the stress intensity factor for an equivalent stationary crack. An energy-rate balance fracture criterion is applied to obtain an equation of motion for the crack tip and to determine the actual delay time between the arrival of the incident wave and the onset of fracture as a function of angle of incidence of the loading wave.

The yield-point loads of singly-notched pin-loaded tensile strips

Abstract: P lane stress and plane strain yield-point loads have been calculated for single-edge notched strips when tensile loading is applied through pins. These yield-point loads are lower than those for fixed-grip loading, and the percentage drop is larger in plane stress than in plane strain. The calculations were required for assessing various creep and fatigue crack propagation tests. Excellent agreement was found between experimental yield-point loads and the plane stress calculations based on the von Mises yield criterion. However, no such agreement was found when further tests were carried out on doubleedged notched specimens. Possible reasons for this discrepancy are discussed. The plane strain calculations may also be used for the ‘double cantilever bend’ specimen by a simple substitution.

Dynamic non-shakedown theorem for elastic perfectly-plastic continua

Abstract: Sufficient and necessary conditions of a kinematic nature are established for alternating plasticity or incremental collapse (i.e. inadaptation or non-shakedown) of elastic perfectly-plastic media subjected to given histories of loads and thermal strains, in the presence of significant inertia and viscous damping forces. The classical second shakedown theorem, due to W.T. Koiter, is thus extended to the dynamic range.

A plastic flow rule at a yield vertex

Abstract: A yield vertex having three distinct facets at the uniaxial stress point is considered. An associated plastic flow rule is constructed using a new 3 × 3 coupled hardening matrix. This has the property that the incipient shear modulus is less than the elastic value. The approach is novel and distinct from previous work having that conclusion. In particular, all the relevant incipient moduli governing fully active in-plane loading can be fitted, if desired, to those values which J 2 -deformation theory would require. To that extent the proposed incremental theory therefore legitimizes the use of the latter moduli, for example in certain ‘paradoxical’ buckling problems. When the moduli of the incremental theory are so chosen, the domain of stress-rate vectors enforcing loading is a calculable pyramid which contracts from the Mises half-space as the stress increases beyond yield. This domain becomes wider, at a given stress, as the initial smooth curvature of the stress-strain curve is imagined to become sharper.

Crack propagation in wedged double cantilevered beam specimens

Abstract: A closed form analytical solution of crack propagation in double cantilevered beam specimens opened at a constant rate has been found. Hamilton's principle for non-conservative systems was applied to describe the crack motion, under the assumption of a Bernoulli-Euler beam. The criterion of crack propagation is a critical bending moment at the crack tip. The calculations of beam motion take into account wave effects in the Bernoulli-Euler theory of elastic beams. The beam shape during the crack motion is found with a similarity transformation and expressed by Fresnel integrals. The boundary conditions satisfied are the fixed ones of zero bending moment and constant beam opening rate at the load end of the specimen and the moving ones of zero deflection and zero slope of the deflected beam at the tip of the moving crack. The fracture represents a moving critical bending moment. The analytical results show that the specific fracture surface energy is a unique function of the ratio of the crack length squared to the time subsequent to loading and this is computed from the recorded time-dependence of the crack length.

Finite element treatment of a uniformly moving elastic-plastic crack tip

Abstract: Summary A uniformly growing elastic-plastic crack tip is investigated by the aid of a finite element mesh that is moving with the crack tip through the material. At the boundary of the mesh singular elastic conditions are applied, characterized by the stress-intensity factor K. The macroscopic differences between the static and the steadily growing plastic zone at a crack tip are evaluated as functions of the work-hardening properties of the material, and the results are used to discuss the degree of applicability of static crack solutions to moving crack situations. The J-integral may be modified to yield an expression for the irreversible energy released to the region at the crack tip under steady-state conditions. The division of this energy into one part absorbed by plastic dissipation and another part transferred to the non-continuous fracture zone at the actual crack tip is investigated. The finite element representation uses incremental loading and successive relaxation of crack tip nodal forces to model the situation, together with a material description that includes isotropic work-hardening.

Toughness in obliquely-stressed fibrous composites

Abstract: A n earlier treatment of toughness for continuous, uniform, fibre reinforced materials given by M.R. Piggott (1970) is extended to the case where the stress is not parallel to the fibres. Experiments on pairs of fibres crossing cracks obliquely are used to reveal the effect on fibre strength of fibre flexure at the crack. The theory indicates that, so long as splitting parallel to the fibres does not occur, the fracture surface energy γ φ for a material stressed at an angle φ to the fibres is given with sufficient accuracy for brittle fibres by the approximate formula γ φ = γ o (1−2.4 A tan φ), where γ o is the surface energy for fracture normal to the fibre direction, and A is a non-dimensional parameter depending on the force exerted by the matrix on the fibres, and involving, in particular, the ratio of matrix flow stress to the fibre ultimate tensile strength. For ductile fibres, the work of fracture increases with the angle φ at a rate depending on fibre breaking stress. The form of fracture surface and the onset of splitting are also discussed.

The theory of a rigid circular disc ground anchor buried in an elastic soil either with adhesion or without adhesion

Abstract: This paper presents a classical elastostatic analysis of the following situation. A rigid circular disc of radius a is buried in an elastic soil at a depth h below a stress-free surface. The disc is subject to a normal force T resulting in uniform normal displacement of the disc of amount α. Two problems are solved. In the first, the elastic soil is assumed to adhere to the underside of the disc and a solution is obtained by perturbation methods for a h . For the second, the material on the underside of the disc is assumed to have broken away; here, an exact solution is found for the limiting case a h → 0 . The analysis is pertinent to the recently innovated civil engineering technique which utilizes ground anchors to support the retaining walls of excavations.

Plastic flow and dislocation structure at small strains in OFHC copper deformed in tension, torsion and combined tension-torsion

Abstract: OFHC copper specimens of 39 μm grain size were deformed to small strains (up to 8%) in tension, torsion and combined tension-torsion at 300 K and the resulting dislocation structures, distributions and densities were determined using transmission electron microscopy. Employing the von Mises yield criterion and the plastic-work hypothesis good agreement was obtained for the three testing conditions for (i) equivalent stress \ gs vs equivalent strain \ g3p curves, (ii) the dislocation structure, distribution and density ρ as a function of \ g3p, and (iii) \ gs as a function of ρ 1 2 . Furthermore, upon comparing the \ gs vs ρ 1 2 curve for polycrystalline copper with the τRSS vs ρ 1 2 curve for single crystals, an average Taylor factor M= (σ/τRSS) of approximately 3.2 was obtained, which is in good accord with that predicted theoretically for FCC metals. Almost equally good correlations for the stressstrain curves and for the dislocation density were obtained on the basis of maximum shear stress τmax and maximum shear strain γpmax as on the basis of \ gs and \ g3P. Therefore, the present results do not permit a positive decision on the question whether the dislocation density correlates better with \ gs and \ g3P or with τmax and γPmax. A single test in which the direction of straining in torsion was reversed yielded a density and distribution of dislocations (and a corresponding value of \ gs) equivalent to those that developed at a smaller strain in unidirectional straining.

Non-linear viscoelastic behaviour of polymers: An implicit equation approach

Abstract: I t is proposed that a comprehensive description of the non-linear viscoelastic behaviour of polymers can be most readily obtained using an implicit form of the constitutive relation. Extensive creep, constant strain-rate and stress relaxation measurements on oriented polyethylene terephthalate and creep measurements on isotropic polypropylene and polymethylmethacrylate have tested the usefulness of the proposed implicit constitutive equation. First, this equation has been shown to describe accurately the stress dependence of the relaxed and unrelaxed creep response. Secondly, it enables the strain dependence of the stress relaxation response to be accurately predicted from the observed creep behaviour. Finally, making simplifying assumptions regarding the form of the two response functions involved, good first-order predictions were obtained for the creep curves at any stress, the stress relaxation behaviour at any strain and the constant strain-rate behaviour at any strain-rate using only six parameters.

Dynamics of ideal fibre-reinforced rigid-plastic beams

Abstract: A mechanism is proposed by which discontinuities in slope can propagate along an ideal fibr-ereinforced beam which is inextensible in the direction of its axis. The equations of motion of the beam are formulated, including the dynamical conditions which must be satisfied at the discontinuity. Constitutive equations for a rigid-plastic fibre-reinforced beam are established, and it is shown that slope discontinuities may propagate in a strain-hardening material, but are stationary in a perfectlyplastic beam. The theory is illustrated by its application to the problem of a beam moving in a direction normal to its axis brought to rest by striking a rigid stop at its mid-point. It is shown that in the subsequent motion slope discontinuities travel outwards from the centre of the beam. A complete explicit solution is obtained for the case of a beam with linear strain-hardening.

Plastic waves in a rate sensitive material—I. Waves of uniaxial stress†

Abstract: Summary Experiments have been carried out in which longitudinal plastic waves have been propagated in thin-walled tubes of alpha-titanium. Strain-time profiles recorded in these experiments show evidence of (i) stress levels considerably above quasi-static values at the same strain, (ii) decay of the amplitude of the elastic precursor, and (iii) variation with distance of propagation of the speed at which a given level of strain propagates. These features of the strain-time profiles are interpreted as indicating that a strain-rate dependent theory is necessary to describe the observed wave phenomena. Numerical solutions based on such a theory agree reasonably well with experimental results. For these solutions a plastic strain-rate function is used which agrees with data from quasi-static and split Hopkinson bar tests on the same tubes.

Initiation of spherical elastic-plastic boundaries due to loading at a spherical cavity☆

Abstract: The spherical waves in an elastic-plastic, isotropically work-hardening medium generated by radial stress uniformly applied at a spherical cavity r=r0 are studied (r denoting radial distance). The radial stress and its time derivative at the cavity may be discontinuous at time t=t0. If the applied radial stress is continuous while its time derivative is not, the discontinuity at (r0, t0) propagates into r > r0 along the characteristics and/or the elastic-plastic boundaries. If the applied radial stress itself is discontinuous, the discontinuity may propagate into r > r0 in the form of a shock wave, or a centered simple-wave, or a combination of both. In any case, the solutions in the neighborhood of (r0, t0) are obtained for all possible combinations of discontinuous loadings applied at r=r0. This is a systematic study on the nature of the solution near (r0, t0) where the applied load is discontinuous. Solutions for special materials, such as linearly work-hardening or ideally-plastic ones, and for special applied loadings at the cavity obtained by other workers, in which the nature of the solutions near (r0, t0) are assumed a priori rather than determined, are compared with the results obtained here. Some of the solutions are found to be in error because of incorrect a priori assumptions.

The yield-point loads of symmetrically-notched metal strips

Abstract: T heoretical plastic yield-point loads are calculated for symmetrically-notched metal strips in plane stress, assuming that the material obeys the von Mises yield criterion. Deep-notch solutions due to R. Hill are extended to cover all notch depths. The numerical results are presented in simple empirical formulae. The purpose of the work is to provide a way of discriminating between metals that obey the Tresca or the von Mises yield criteria.

Interaction between the effects of local and overall imperfections on the buckling of elastic columns

Abstract: A. van der Neut has studied the buckling of elastic columns made from thin-walled members, with particular reference to the effects of imperfections. In this paper, a graphical method is used to extend his work and to make an exhaustive study of the combined effects of both ‘local’ and ‘overall’ imperfections. The resulting picture is remarkably simple, and the effects of imperfections are well described by the celebrated Perry formula in conjunction with a single imperfection parameter compounding simply the local and overall imperfections. Experiments on small-scale rubber model columns substantiate the main results of the theoretical analysis.

The tensile failure modes of metal-matrix composite materials

Abstract: The strengths of individual boron fibers extracted from various as-received and thermally fatigued aluminum alloy matrix materials were measured. The results are described in terms of a Weibull distribution, and strengths of composites fabricated from these fibers are calculated in terms of lower and upper bounds. Tests conducted on composite specimens indicated that strengths approaching the upper bounds can be achieved in composites fabricated by normal diffusion bonding techniques. Cyclic temperature changes effectively reduced the strength values toward the lower bounds. It was concluded that this effect resulted from the degradation of the strength of the fiber-matrix bond.

Plastic waves in a rate sensitive material—II. Waves of combined stress†

Abstract: Summary Experiments are reported in which combined longitudinal and torsional plastic waves are generated in thin-walled tubes of alpha-titanium by subjecting pre-torqued tubes to longitudinal impact. Longitudinal and torsional strain-time profiles are recorded at several stations along the specimen. These strain-time profiles exhibit features which cannot be explained within the framework of a strain-rate independent theory. The latter theory requires the wave generated under the loading employed to consist of, successively, a fast simple wave, an intermediate constant state region, a slow simple wave, and a final constant-state region. The experiments show no evidence of an intermediate constant-state region; furthermore, a final constant-state region is not observed even though the latest times of observation are greater than the time at which, according to a strain-rate independent theory, a constant strain-rate region is expected. Straightforward generalization of the rate-dependent theory to allow for combined stress states leads to a theory which predicts the observed strain-time profiles with good accuracy.

Thermodynamic systems for solids and liquids

Abstract: The basic mathematical consequences which follow from the laws of thermodynamics are explored in a systematic new treatment for establishing fluid thermodynamic systems for a number of materials under conditions where they behave as fluids. For metals and many other solids, but less so for liquids, the specific heat at constant volume cν is sensibly constant at all temperatures above room temperature. The partial differential equation of thermodynamics which expresses the constancy of cν is easily solved, the solution involving two arbitrary functions of the specific volume. Various approaches are presented to illustrate how one may choose these functions to accord with experimental observations over large thermodynamic ranges, and so produce practical thermodynamic systems. Two complementary thermodynamic systems are presented, which embody significant experimental results; the first is based on linear shock velocity/particle velocity (U, u) relations; the second on the limiting value of the specific heat ratio c p c v at high temperatures. They are complementary in the sense that the first has analytically complicated non-Hugoniot properties which differ insignificantly from the comparatively simple non-Hugoniot properties of the second; whilst the second has analytically complicated Hugoniot properties which differ insignificantly from the simple Hugoniot properties of the first. Together, then, the two systems may be combined to give comprehensive practical simplicity. The main interest in these thermodynamic systems lies in the study of the behaviour of liquids, metals and other solids in shock transitions and under extreme conditions such as occur in high velocity impact or in explosion phenomena; but they are also of importance in stress-wave analysis, where complete thermodynamic systems are required in order to derive stress-strain relationships which do not neglect the effects of temperature changes.

Effect of the secondary slip system on early fatigue damage

Abstract: A face-centered cubic polycrystal under cyclic loading is considered. It is shown that slip in the primary slip system causes the resolved shear stress in a second slip system to increase to the critical value and then this second slip system slides. Slip in this second slip system greatly increases the rate of the local plastic strain build-up in the primary slip system. Hence the occurrence of the second active slip system may greatly increase the rate of extrusion and intrusion commonly observed in face-centered cubic metals.

Elastic-plastic deformations of thickwalled circular cylinders under transient heating and combined axial tension and torsion loads

Abstract: The method of successive elastic solutions has been used to predict the stress-strain history of a thick-walled circular cylinder when it is subjected to rapid heating on its outer surface together with various combinations of axial tension and torsion. The extent of the zones of plasticity is calculated for both a constant yield stress and for a yield stress which reduces with increase of temperature.

Dislocation loops in hexagonal crystals

Abstract: Summary In this paper, E Kroner's representation of the Green's function is first used to derive closed-form solutions for the components of the displacement and stress fields created by an arbitrary dislocation loop situated in a basal plane of a hexagonal crystal The interaction energy of two dislocation loops in basal planes is then calculated Previously, only certain stress components were known at points coplanar with the loop

Stress and strain distributions beyond general yield in the Charpy V-notched specimen

Abstract: T he visioplasticity method has been used to obtain the stress and strain distributions during planestrain bending in a standard Charpy V-notched specimen of low carbon steel. The specimen was slit longitudinally through the notch with a square grid inscribed on the interface. The instantaneous grid distortion was obtained during a three-point incremental bending test. A computer program was developed for the calculation of the instantaneous stress and strain distributions using the experimentally determined displacement field and the true stress-strain curve of the steel.

Some theoretical aspects of hydrostatic extrusion and allied processes

Abstract: A lthough the mechanical principles of various extrusion processes are well understood in outline, very little is known about the stresses in the billet and product outside the working zone. Yet this is crucial to the optimal use of fluid pressure in hydrostatic forming. As a preliminary to a thorough investigation, some perspective is provided here by delimiting the range of pressure under which abnormal modes of flow occur. These are modes in which the billet either bulges or thins ahead of the die, or the product swells behind it. The analysis is based on standard slip-line theory and on overloading criteria at the stress singularities at entry and exit. Data are also presented on the distribution of stress along the boundaries of the working zone under arbitrary loading systems.

Failure analysis of unidirectional fiber composites under combined axial tension and shear

Abstract: A method of analysis is proposed for investigating the mechanics of failure of unidirectional, fiber composite materials subjected to axial tension and shear. The mechanisms of failure are assumed to result from the interaction of the applied shear stress and local matrix shear stress concentrations which arise as a result of the scattered fiber breaks that occur throughout the material. Two modes of failure are identified. One is associated with the unstable growth of shear failure regions in the matrix. The other is primarily a tensile failure mode which is influenced by the applied shear. The analysis predicts that a variety of shear-tensile stress failure surfaces are possible, depending on material properties. The results suggest that radical changes in the shape of failure surfaces can occur as the result of environmental effects. This has significant implications for reliability.