Showing papers in "Journal of The Mechanics and Physics of Solids in 1970"
TL;DR: In this article, a simplified theoretical model of this behaviour is obtained by extending R. Hill's theory of expanding a cylindrical or spherical cavity in an elastic-plastic material to ensure compatibility between the volume of material displaced by the indenter and that accommodated by elastic expansion.
Abstract: The theory of rigid perfectly-plastic solids predicts indentation pressures, using wedge-shaped or conical indenters, which depend only on the geometry of the indenter and the yield stress of the material. With blunt wedges or with materials having a low ratio of Young's modulus, E, to yield stress, Y, the material displaced by the indenter is accommodated by an approximately radial expansion of the surrounding material. The indentation pressure then falls below the rigid perfectly-plastic value. In these circumstances, measurements of indentation pressure for a variety of indenter geometries are shown to correlate with the single parameter (E/Y) tan β, where β is the angle of inclination of the indenter to the surface at the edge of the indentation. This parameter may be interpreted as the ratio of the strain imposed by the indenter to the yield strain of the material.
A simplified theoretical model of this behaviour is obtained by extending R. Hill's theory of expanding a cylindrical or spherical cavity in an elastic-plastic material to ensure compatibility between the volume of material displaced by the indenter and that accommodated by elastic expansion.
1,137 citations
TL;DR: In this paper, a statistical theory of material strength is proposed, where the strength of the elements is assumed to be a statistic a quantity, and as the material is loaded elements fracture randomly throughout the body causing localized stress concentrations.
Abstract: A Statistical theory of material strength is proposed. Materials are considered to be imperfect heterogeneous continua composed of discrete volume elements whose characteristics are related to material structure and imperfections. The strength of the elements is assumed to be a statistic a quantity, and as the material is loaded elements fracture randomly throughout the body causing localized stress concentrations. The accumulation of these breaks results in overall failure. By relating strength to material structure this theory attempts to bridge the gap between the microscopic and continuum approaches to fracture mechanics. The theory is applied to composite materials reinforced with whiskers and continuous fibers. Comparisons with experimental data show good agreement. Results for whisker-reinforced composites appear to provide a good prediction of strength and an explanation of the disparity between the strength of individual whiskers and the strength of the composites made from them.
288 citations
TL;DR: The particle-velocity histories associated with the compression waves produced by the planar impact of polymethyl methacrylate plates were observed by means of laser interferometry as mentioned in this paper.
Abstract: The particle-velocity histories associated with the compression waves produced by the planar impact of polymethyl methacrylate plates were observed by means of laser interferometry The impact velocity and the thickness of material through which the wave passed were varied from 006 to 064 mm μsec and from 6 to 37 mm respectively Over this range of impact velocities, the observations disclosed a shock followed by a smooth transition to the maximum particle-velocity; the speed and magnitude of the shock varied non-linearly with the impact stress For impact velocities below 03 mm μsec , the wave was steady A steady-wave analysis based on finite linear viscoelasticity theory has been shown to be in good agreement with the experimental observations
74 citations
TL;DR: In this article, an incremental shear stress is applied to a thinwalled tube in a time of about 30 μsec and the speed of propagation of the resulting stress wave is measured.
Abstract: Some of the difficulties inherent in attempts to study the rate dependence of the mechanical properties of solids by means of longitudinal wave-propagation experiments are discussed, and it is concluded that these difficulties can be avoided by studying the propagation of a pure shear pulse applied while the material is being slowly deformed in pure shear. A new apparatus is described, in which an incremental shear stress is applied to a thinwalled tube in a time of about 30 μsec and the speed of propagation of the resulting stress wave is measured. The apparatus is also used to test short tubular specimens in pure shear by the split Hopkinson-bar technique. Results of wave-propagation tests on mild steel, copper and aluminium show that the initial response is essentially elastic, and this is confirmed by results obtained in tests on short specimens of copper and aluminium. From the latter tests, incremental stress-strain curves are derived for a strain rate approaching 100 sec−1, and the rate dependence of the flow stress is compared with that obtained from compression tests at constant strain rate.
60 citations
TL;DR: In this paper, it is suggested that transient creep at high temperatures arises principally as a result of the dispersal of entanglements by the climb mechanism, with a rate constant that depends on stress and temperature in the same way as does the secondary creep rate.
Abstract: It is suggested that transient creep at high temperatures arises principally as a result of the dispersal of entanglements by the climb mechanism. The dispersal of the entanglements is assumed to follow a unimolecular reaction kinetics with a rate constant that depends on stress and temperature in the same way as does the secondary creep rate. The analysis shows that the strain (e) versus time (t) relation can be represented by e=e 0 +e. 3 t+ β−1 K [1- exp (−K e 3 t)] , where e0 is the instantaneous strain on loading, e 3 , the secondary creep rate, K e 3 the rate constant, and β the ratio of initial to secondary creep rate. The experimental creep data on several b.c.c. and f.c.c. metals and alloys correlate quite well with the proposed mechanism. The constants β and K were found to be independent of temperature and stress. The proposed formulation becomes inapplicable for correlating creep data in polycrystals at low stresses because of the significant contribution of grain-boundary sliding to the total creep at these stress levels.
60 citations
TL;DR: In this paper, the incremental boundary-value problem is shown to be amenable to the inequality-constrained minimization of a quadratic functional of the plastic multiplier field, and a weak sufficient condition for uniqueness of solution and overall stability is formulated.
Abstract: E lastoplastic constitutive laws with non-associated flow laws and work-hardening or non-hardening or work-softening behaviour are assumed. The incremental boundary-value problem is shown to be amenable to the inequality-constrained minimization of a quadratic functional of the plastic multiplier field. A weak sufficient condition for uniqueness of solution and overall stability is formulated. Application methods founded on finite-element discretization are indicated for solving threedimensional problems.
54 citations
TL;DR: In this paper, a crack model is introduced in which the non-linear force-deformation relation, just ahead of the crack tip, is taken into account, leading to an integral equation of the Fredholm type, which is solved numerically by approximating it by a finite system of linear equations.
Abstract: A crack model is introduced in which the non-linear force-deformation relation, just ahead of the crack tip, is taken into account. The analysis of the crack model leads to an integral equation of the Fredholm type, which is solved numerically by approximating it by a finite system of linear equations. The distributions of stress and deformation are calculated for different loads increasing from zero to the maximum value of stable equilibrium. The conditions, for which instability is reached, are also determined by investigating the eigenvalues of the system. Comparison of the results with existing theories is made.
48 citations
TL;DR: In this article, the results of the Hertz fracture experiments on two surfaces of float glass and on specimens of polished-plate glass in the condition as-received from the manufacturer and after various etching treatments in hydrofluoric acid were analyzed in terms of the theory of flaw statistics originated by W. Weibull.
Abstract: Hertz fracture experiments were carried out on the two surfaces of specimens of float glass and on specimens of polished-plate glass in the condition as-received from the manufacturer and after various etching treatments in hydrofluoric acid. The results are analysed in terms of the theory of flaw statistics originated by W. Weibull. An acceptable degree of agreement is obtained between the experimental results and values predicted using simple flaw distribution functions which have been determined for each of the five surfaces investigated. Brief consideration is given to the objections which have been raised to the application of flaw statistics analysis to this type of experiment. The present results suggest that these objections are not justified.
47 citations
TL;DR: In this article, the authors extended the self-consistent polycrystalline model of inelastic deformation to include time-dependent behavior and showed that an appreciable amount of the observed dependence of behavior on prior history of combined stressing can be explained by grain interaction effects.
Abstract: The application of the self-consistent polycrystalline model of inelastic deformation has been extended to include time-dependent (i.e. creep) behavior. Comparison of the predictions of a simple power-law micro-constitutive relation in the model with experiments suggests that an appreciable amount of the observed dependence of behavior on prior history of combined stressing can be explained by grain interaction effects. The existence of a flow potential for the self-consistent polycrystalline model is demonstrated, and flow potential surfaces, calculated from a simple power-law micro-constitutive relation, are shown for several cases of combined tension-torsion deformation. To a good approximation these surfaces exhibit a kinematic translation in stress space with prior deformation history.
41 citations
TL;DR: In this paper, the deformation and displacement of a cylindrical element are investigated under boundary conditions similar to those for an element in the interior of a composite, and solutions for the stress and displacement in a typical element are obtained for the typical element.
Abstract: A composite material consisting of an elastic matrix reinforced by parallel elastic fibres may be regarded as composed of approximately circular cylindrical elements, each of which contains a concentric circular fibre. Solutions are obtained for the stress and displacement in a typical element under boundary conditions which approximate those for an element in the interior of a composite. Over the greater part of the length of a cylindrical element the deformation is found to be very close to a uniform extension, but large shear stresses can occur near the ends of the element and these may tend to break the bond between the fibre and the matrix. Numerical results are given in some illustrative cases.
40 citations
TL;DR: In this article, the propagation condition for infinitesimal plane waves superposed on a state of homogeneous strain in isotropic elastic solids is expressed in a simple form in terms of certain instantaneous moduli of elasticity referred to the principal axes of the underlying strain.
Abstract: The propagation condition for infinitesimal plane waves superposed on a state of homogeneous strain in isotropic elastic solids is expressed in a simple form in terms of certain instantaneous moduli of elasticity referred to the principal axes of the underlying strain. The constitutive laws of Hadamard, Green, and harmonic materials are obtained with some economy by using the propagation condition in this form. Under certain restrictions these materials are shown to satisfy the strong ellipticity condition.
TL;DR: In this paper, Biaxial stress tests were performed on thin-wall tubes of polycrystalline 2024-T81 aluminum at temperatures of 150°C and 250°C.
Abstract: Biaxial stress tests were performed on thin-wall tubes of polycrystalline 2024-T81 aluminum at temperatures of 150°C and 250°C. The nominal metallurgical stabilization temperature for this alloy is 190°C. Transient and steady-state creep strain rates exhibited a considerable dependence on load path history. For a prescribed history it is possible to determine unique surfaces of constant creep strain rate. For the zero history, involving a single loading from the origin to a prescribed point in stress space, surfaces of constant steady-state strain rate, at elevated temperature, have the same shape as room temperature yield surfaces of moderate offset. In the temperature and small strain regions considered here, room-temperature yield surfaces were found to be unaffected by elevated temperature deformation. The changes in shape of room-temperature yield surfaces, due to room-temperature plastic deformation caused corresponding changes in the elevated temperature surfaces of constant steady-state creep rate. At a given stress point, an outward local motion of the yield surface resulted in a corresponding outward local motion of the steady-state creep rate surfaces. The experimental determination of surfaces of constant flow potential was also attempted.
TL;DR: In this article, a plane horizontally-polarized shear wave propagates into an undisturbed brittle elastic material containing a void, and the conditions for initiation of crack propagation at an instantaneous velocity upon diffraction of the wave by the void are investigated.
Abstract: A plane horizontally-polarized shear wave propagates into an undisturbed brittle elastic material containing a void. This paper inquires into the conditions for initiation of crack propagation at an instantaneous velocity upon diffraction of the wave by the void. The investigation consists of two parts. In the first, the particle velocity and the shear stresses are determined for diffraction of a transient wave of arbitrary shape by a wedge-shaped void which produces a running crack at the instant the wave front strikes the tip of the wedge. In the second, the balance of rate-of-energy is employed to determine that shape of the incident pulse which is consistent with crack propagation at an instantaneous velocity. It is shown that near the wave front the incident displacement wave must be of the form of a square root of the general argument t−ax−by. For this particular type of incident wave, the instantaneous velocity of crack propagation is computed as a function of the angle of incidence, the “amplitude” of the incident wave, and of the material parameters.
TL;DR: In this paper, a single long elastic fiber is considered which is embedded in and bonded to a plane matrix, and the problem is reduced to the solution of an integral equation; this equation is derived and solved for two shapes of fibres.
Abstract: A new method of analysing stress and strain in the matrix of a fibre-reinforced material within the elastic region is demonstrated. To this end, a single long elastic fibre is considered which is embedded in and bonded to a plane matrix. The problem is reduced to the solution of an integral equation; this equation is derived and solved for two shapes of fibres. The calculated interface shear stress for a rectangular fibre is compared with experiment and good agreement is found. The effect of the fibre shape and the elastic constants on the interface shear stress is discussed. A general curve that can be used to estimate the interface shear stress reasonably far away from the end region is given.
TL;DR: In this paper, a smooth circular inclusion embedded in an infinite plate subjected to uniaxial tension is viewed as a simple model for the effects on a pin-joint.
Abstract: A smooth circular inclusion embedded in an infinite plate subjected to uniaxial tension is viewed as a simple model for the effects on a pin-joint. Concern is with the stress concentrations and also with the regions of separation between inclusion and plate which arise when the applied tension exceeds some critical value. The latter depends on the amount of initial compression of the inclusion and plate. Inclusions and plates of dissimilar isotropic elastic materials and of dissimilar isotropic viscoelastic materials are treated. A method of solution is formulated when the contact between inclusion and plate is perfectly smooth. In the elastic case, the singular integral equation for the contact pressure and applied tension for given contact angle are most readily determined when the four elastic constants satisfy a particular relation, and the computation accuracy is very high. The solution is illustrated for identical materials. In the viscoelastic case, one restriction on the four independent response functions leads to a similar singular integral equation in which time occurs only as a parameter. A further restriction reduces the integral equation to that for an associated elastic problem, and the contact-angle history for prescribed tension history is determined directly from the elastic solution. Again, the solution is illustrated for identical materials with a simple model for the viscoelastic response.
TL;DR: In this paper, the use of a general non-linear yield condition for problems in plane strain of granular materials in which body forces are not negligible and for which computation would be necessary using the linear Coulomb condition is discussed.
Abstract: Attention is drawn to the use of a general non-linear yield condition for problems in plane strain of granular materials in which body forces are not negligible and for which computation would be necessary using the linear Coulomb condition. The theory is extended and applied to the problem of indentation of a semi-infinite mass of material by a flat smooth rigid punch, using a power law yield function.
TL;DR: In this paper, the effect of diameter on work-to-fracture has been determined over a large range of diameters (10 μm to 5 mm) and this also shows general agreement with the above theory.
Abstract: Measurements of absorbed energy in a miniature Charpy test have been made on brittle-fibre ductile-matrix composites (tungsten wires in copper, copper 10 w/o tin and phosphor bronze). The variation of work-to-fracture with fibre volume-fraction has been found and for ductile matrices supports the theory of G.A. Cooper and A. Kelly given in 1967. The effect of fibre diameter on work-to-fracture has been determined over a large range of diameters (10 μm to 5 mm) and this also shows general agreement with the above theory. It is predicted theoretically that a nonhomogeneous arrangement of fibres will lead to a higher work-to-fracture and experimental evidence is provided to support this prediction.
TL;DR: In this paper, the effect of workhardening in large plastic flow and how it relates to nonhardening theory was investigated. And the results indicated that workhardened solutions may reveal strain discontinuities at the limiting case of zero work-hardening.
Abstract: To understand the effect of workhardening in large plastic flow and how it relates to nonhardening theory, an experimental program of wedge flattening has been performed Wedges of 20°, 30°, 45° and 60° semi-angle were prepared from copper with five levels of prestrain and from two aluminum alloys Testing was undertaken with care to insure meaningful quantitative data and to permit geometric similarity to occur Systematic and large effects occurred in the behavior of the specimens as a result of workhardening For instance, it was found that the width of the deformed top could be less than the truncated wedge value, constraint factors could be twice those predicted by nonhardening analysis, and the anvil pressure could decrease with increasing angle whereas nonhardening theory predicts an increase An existing nonhardening slip-line solution which contains strain discontinuities produced by velocity discontinuities was verified by extrapolating the experimentally obtained results to zero workhardening This suggests that workhardening solutions may reveal strain discontinuities at the limiting case of zero workhardening and that perturbation techniques based on nonhardening solutions may be useful in solving workhardening problems
TL;DR: A n-interaction diagram between bending moment and shear force in a beam is not a proper yield surface, and the convexity property of plasticity theory does not necessarily hold as discussed by the authors.
Abstract: A n “interaction diagram” between bending moment and shear force in a beam is not a proper yield surface, and the convexity property of plasticity theory does not necessarily hold. An empirical curve is, however, satisfactory for design in the practical range, although improvement to the theory remains possible for very short beams.
TL;DR: In this paper, the angle of deviation of ray from wave normal for plane elastic waves whose normal lies in a zonal plane of a hexagonal (transversely isotropic) medium (or an equivalent plane) is presented.
Abstract: Expressions for the angle of deviation, Δ, of ray from wave normal for plane elastic waves whose normal lies in a zonal plane of a hexagonal (transversely isotropic) medium (or an equivalent plane) are presented. Although values of Δ have been calculated from general expressions for rays or energy-flux vectors, the present forms have some computational convenience and allow certain general conclusions about the variation of Δ with wave normal to be stated in terms of inequalities involving the elastic stiffnesses. The particular case of cubic crystals is illustrated. The findings have relevance to ultrasonic wave propagation in many elastically anisotropic solids including fibre composites.
TL;DR: In this paper, an energy principle is derived which provides upper bounds on the plastic deformations that occur in an elastic-plastic structure subjected to blast loading, and a safe estimate of the critical permanent deformations in problems for which the complete solution may require lengthy numerical calculation.
Abstract: An energy principle is derived which provides upper bounds on the plastic deformations that occur in an elastic-plastic structure subjected to blast loading. Applications are made to several problems for which complete solutions are known in order to assess the accuracy and the applicability of the technique. Reasonably accurate bounds are obtained with a substantial saving in effort over the complete analysis. The present technique provides a safe estimate of the critical permanent deformations in problems for which the complete solution may require lengthy numerical calculation.
TL;DR: In this paper, a perturbation approach to the analysis of elastic post-buckling and imperfection sensitivity of discrete structural systems is presented, where the response of imperfect systems is related to a dominant postbuckling path of the perfect system.
Abstract: A new perturbation approach to the analysis of elastic post-buckling and imperfection sensitivity of discrete structural systems is presented. This approach by-passes the lowest branching point and obtains estimates of the post-buckling response of the perfect system and the imperfection-sensitivity of imperfect systems by equilibrium projections at constant load from the fundamental equilibrium path, this being possible when the response of imperfect systems is related to a dominant post-buckling path of the perfect system. The new scheme is conceptually simpler than the conventional perturbation approach (which makes projections from the lowest critical equilibrium state of the perfect system), and in addition it overcomes an inherent weakness of the conventional approach which arises when energy coefficients vary rapidly with the load or when there are neighbouring critical equilibrium states on the fundamental equilibrium path close to the one under consideration. It should be particularly effective when the dominant post-buckling path lies close to the fundamental path. A feature of the new approach is that while it can be used as a powerful ad hoc method in regions remote from the lowest critical point it nevertheless retains the essential characteristics of a branching theory able to resolve fine details in the vicinity of a critical equilibrium state. In the presence of a dominant post-buckling path the method can be used in the presence of discrete, simultaneous and near-simultaneous critical points, being particularly useful in the latter case when conventional approaches may experience difficulties. The new approach is applied to the analysis of the imperfection-sensitivity of a buckling model exhibiting an asymmetric point of bifurcation and to the post-buckling analysis of an Euler strut, these studies confirming the promise of the approach.
TL;DR: In this article, a slip-line field solution is proposed for the steady-state plane strain extrusion of rigid/perfectlyplastic material through a frictionless cosine-shaped die.
Abstract: I t is expected that extrusion through dies with regular shapes, for which the divergence of the stream lines is moderate and the velocity field is continuous should yield higher process efficiency than that through conventional conical and square dies. This paper deals with the exact solution for the steady-state extrusion through such a die. The slip-line field solution is proposed for the steady-state plane strain extrusion of rigid/perfectlyplastic material through a frictionless cosine-shaped die. The extrusion pressures are calculated for various different reductions. From numerical calculations the extrusion pressure, P 2k , has been determined as −1.02 In (1 −R), where P is the extrusion pressure, 2k is the yield strength of the strip, and R is the desired reduction of the die, which is nearly equal to the work density for homogeneous compression. The calculated extrusion pressures were compared with those for axisymmetric extrusion and found to be somewhat smaller. The validity of this solution for the range of reductions is given.
TL;DR: In this article, the strengthening and weakening effects of inhomogeneities in an elastic continuum are studied theoretically, and a general criterion for predicting when such interaction tends to be attractive or repulsive is presented.
Abstract: T he strengthening and weakening effects of inhomogeneities in an elastic continuum are studied theoretically. Particular examination is given of the interaction between inhomogeneities, and between an inhomogeneity and a source, such as a dislocation, a point defect, or a loaded crack or cavity. A general criterion for predicting when such interaction tends to be attractive or repulsive is presented.
TL;DR: In this article, it was shown that Robotnov and Shesterikov's criterion is inadequate for the determination of the creep life of a column and that the agreement between theory and experiment is very poor.
Abstract: G.N. Rabotnov and S.A. Shesterikov proposed a criterion for creep buckling based on the stability theory of dynamic systems, but the agreement between theory and experiment is very poor. Here, the dynamic response of a column disturbed by a lateral impulse is investigated, and as a result it is shown that Robotnov and Shesterikov's criterion is inadequate for the determination of the creep life of a column.
TL;DR: In this paper, a multidimensional regressional analysis has been developed which can be used to interpret numerically superplasticity data for the aluminium-copper system determined by D.L. Holt and W.A. Backofen.
Abstract: A multidimensional regressional analysis has been developed which can be used to interpret numerically superplasticity data. The method has been evaluated by applying it to the analysis of the data for the aluminium-copper system determined by D.L. Holt and W.A. Backofen. Possible optimization procedures have been illustrated and practical applications are discussed. It is pointed out that the analytical method should have broader applications than those examined in the present case.
TL;DR: In this article, an iterative approximation for small disturbances passing through, and being scattered by, a deflectional signal of arbitrary amplitude was developed for small extensional signals passing through a non-uniform disturbance, but are carried at a considerably slower speed.
Abstract: This paper illustrates some of the interactions between extensional and deflectional waves in a nonlinearly elastic string. An iterative approximation is developed for small disturbances passing through, and being scattered by, a deflectional signal of arbitrary amplitude. The disturbances produced by purely transverse motions of one end of a semi-infinite string are described. By ensuring that no secular terms arise in the solution, many physical interaction effects are illustrated. For example, it is shown that extensional signals passing through a non-uniform disturbance are not carried along exact characteristics, but are carried at a considerably slower speed, even though the governing equations are non-dispersive.
TL;DR: In this paper, a system of equations governing finite one-dimensional strain waves in a strain-rate sensitive elastic viscoplastic material was extended to include the effects of heating caused by compression and plastic dissipation.
Abstract: In previous work by the present writers, a system of equations was developed governing finite one-dimensional strain waves in a strain-rate sensitive elastic viscoplastic material, and in the present paper these equations are extended to include the effects of heating caused by compression and plastic dissipation. By assuming adiabatic conditions, a non-linear integral equation is established for the temperature increase. This equation is simplified by using the previously-discussed approximation of steady-state propagation of the wave and a nearly exact solution is given for this form of the equation. As a result, it is shown that the temperature can be expressed as a function of the density only in a steady-state wave and is independent of the plastic strain-rate relation used to describe the material. On the basis of the analysis, examples are given showing temperature changes and other features of shock waves in rate-sensitive plastic solids.
TL;DR: In this paper, the finite deformation theory of E.H. Lee was used to calculate the Hugoniot P, V, T states of Armco iron between the pressure limits of the elastic yield point and the polymorphic phase transition at 131 kb.
Abstract: The finite deformation theory of E.H. Lee is used to calculate the Hugoniot P , V , T states of Armco iron between the pressure limits of the elastic yield point and the polymorphic phase transition at 131 kb. The temperature rise at the upper pressure limit is calculated to be 44·0°±1·0° C .