Showing papers in "Journal of The Mechanics and Physics of Solids in 1971"
TL;DR: In this article, the theoretical foundations of constitutive relations at finite strain for a class of solids exhibiting inelasticity as a consequence of specific structural rearrangements, on the microscale, of constituent elements of material.
Abstract: This paper is a study of the theoretical foundations of constitutive relations at finite strain for a class of solids exhibiting inelasticity as a consequence of specific structural rearrangements, on the microscale, of constituent elements of material. Metals deforming plastically through dislocation motion are of this class and form the primary application of the theory. The development is in terms of a general internal-variable thermodynamic formalism for description of the microstructural rearrangements, and it is shown how metal plasticity may be so characterized. The principal result is in the normality structure which is shown to arise in macroscopic constitutive laws when each of the local microstructural rearrangements proceeds at a rate governed by its associated thermodynamic force. This provides a theoretical framework for time-dependent inelastic behavior in terms of a “flow potential”, and reduces to statements on normality of strain increments to yield surfaces in the time-independent case. Conventional characterizations of the stress-state dependence of metallic slip are noted to be in accord with this concept of associated forces governing rates, so that the resulting normality structure may be considered directly applicable to metal plasticity.
1,573 citations
TL;DR: In this article, an elastic-plastic stress analysis for a notched bar using the method of finite elements was carried out, and it was shown that the maximum stress does not occur at the elastic interface.
Abstract: An elastic-plastic stress analysis, which allows for linear work-hardening, has been carried out for a notched bar using the method of finite elements. Good agreement has been found with available experimental data and confirmation obtained of the idea that the strain at the notch surface is nearly constant over a large fraction of the root. However, some interesting differences exist between the present results and those apparently predicted from slip-line field theory. First, there is the surprising result that the maximum stress does not occur at the elastic-plastic interface. Secondly, in considering the stress intensification (the ratio, R , of the maximum stress to the yield stress), it is shown that slip-line field theory apparently over-estimates R at low loads but slightly under-estimates it near to general yield. The results for the stress intensification are used to re-analyse existing experimental data in order to find the temperature variation of the local fracture stress: this is found to be nearly constant over a range of 250°C.
351 citations
TL;DR: In this article, a two-dimensional analysis of the stress field around a crack on the plane interface between two bonded dissimilar anisotropic elastic half-spaces is presented, which is then combined with the usual local form of the Griffith virtual work argument to give an explicit fracture criterion which involves a suitably defined stress concentration vector and the specific surface energy of the bonded surfaces.
Abstract: This paper contains a two-dimensional analysis of the stress field around a crack on the plane interface between two bonded dissimilar anisotropic elastic half-spaces. This analysis is then combined with the usual local form of the Griffith virtual work argument to give an explicit fracture criterion which involves a suitably defined ‘stress concentration vector’ and the specific surface energy of the bonded surfaces. This criterion has a simple structure and reduces to the conventional form of Irwin when the two half-spaces are isotropic and identical. The analysis is then extended to cracks moving uniformly and a local fracture criterion with the same structure as the static criterion is derived by an energy balance argument. The criterion is specialized to isotropic half-spaces for illustration, when it predicts that the speed of a crack on an interface between such media will be limited by a speed Vc which is slightly greater than the smaller of the two Rayleigh wave speeds. A by-product of the analysis is an expression for the displacement field of an arbitrary interfacial dislocation, either stationary or moving uniformly.
307 citations
TL;DR: In this article, a non-linear fracture mechanics theory is formulated, in which the critical value of the Rice integral is found by experiment and then the crack starts to grow when it reaches a critical value.
Abstract: Processes taking place in a certain region near a crack-tip play a dominant part for the initial growth of the crack. This region, called the end-region , is generally very small, and its state at the critical load is a material property. Outside the end-region, the material can be treated as a continuum, in which therefore d σ ij d e ij ⩾0. The state of the end-region is described by the value of the Rice integral J , which is a monotonic function of the load. When J reaches a critical value J c , the crack starts to grow. In the case of unstable growth, the criterion J = J c is also one of fracture. J may be determined very accurately, even in cases of large-scale yielding, by finite-element methods, without detailed knowledge of the situation near the crack-tip. The critical value J c may then be found by experiment. Thus, a non-linear fracture mechanics theory can be formulated.
233 citations
TL;DR: In this article, the authors investigated the rate sensitivity of two annealed face-centred cubic metals, aluminium and copper, at elevated temperatures to determine their rate sensitivity, and the results indicate that hot compression is a thermally-activated process.
Abstract: T he mechanical behaviour of two annealed face-centred cubic metals, aluminium and copper, at elevated temperatures has been investigated experimentally to determine their rate sensitivity. The results indicate that hot compression is a thermally-activated process. The experimental value of the activation energy for aluminium obtained from the present work is almost similar to that for creep and self-diffusion. This indicates that hot dynamic compression may be a diffusion-controlled thermally-activated process. However, the experimental activation energy of 74 kcal/mole obtained from the present work for copper is higher than that observed both for creep and self-diffusion. Experimental evidence (D. Hardwick and W.J. McG. Tegart in 1961) suggests that recrystallization is the rate-controlling process in dynamic deformation, while in creep it is usually recovery in the form of sub-grain formation. For aluminium specimens the dislocation density decreases with increasing temperature; for increasing strain rate the dislocation density remains almost constant, while the size of dislocation cells diminishes. If the dislocation density did not change substantially, one should conclude (as did L. Taborský in 1969) that the rise in flow stress caused by high strain rate is due to the reduction of the size of dislocation cell structure and by the increased rate of movement of the moving individual dislocations.
115 citations
TL;DR: In this paper, a crack with steady growth under anti-plane shear was found to have a primary plastic zone included in an angle of ±19.7° ahead of the crack tip, and two very thin secondary (reverse) plastic zones along the crack flanks.
Abstract: F or a crack with steady growth under anti-plane shear, analysis shows a primary plastic zone included in an angle of ±19.7° ahead of the crack tip, and two very thin secondary (reverse) plastic zones along the crack flanks, each included in an angle of 0.37°. Numerical solutions give the shape of the plastic zones which determine the active and residual plastic strains, and give the crack tip displacement, which is approximately 0.07 of that for monotonic loading without growth. The length of the primary plastic zone is almost the same as that without growth, but the thickness is about 3/5 as great. Coupled with ductile fracture criteria, the present results predict initially stable crack growth, whereas analyses based on the simplification of yielding on just one plane predict unstable fracture immediately following initiation.
107 citations
TL;DR: In this article, a nonlinear viscoelastic model with temperature-dependent Young's modulus and viscosity is proposed for the mechanical behavior of metals at high temperatures or low strain-rates.
Abstract: A nonlinear viscoelastic model with temperature-dependent Young's modulus and viscosity is proposed for the mechanical behavior of metals at high temperatures or low strain-rates. The model is then used in the formulation of the one-dimensional solidification problem for a rectangular mold subject to uniform but nonsteady surface temperature and pressure. Although the resulting equations are highly nonlinear, solutions are obtained for the special case of slow cooling using a linearizing perturbation procedure. In particular, analytical expressions are obtained for the stresses in the solidifying skin and for the time required for an air-gap to form between the skin and the mold. For the case of a constant rate of surface-temperature drop, the stresses are found to be uniform through the skin and to remain compressive up to the time of air-gap formation. Furthermore, no air-gap is possible if the pressure exceeds a critical value. For a case of increasing rate of surfacetemperature drop, the stresses in the outer surface of the ingot are found to become tensile prior to air-gap formation.
56 citations
TL;DR: In this paper, the failure mechanism of diametrically-compressed glass cylinders was examined and it was shown that fracture always initiates at the contact surface, and not within the cylinder as predicted by conventional theory.
Abstract: This paper considers the failure mechanism of diametrically-compressed glass cylinders. Examination of failed specimens shows that fracture always initiates at the contact surface, and not within the cylinder as predicted by conventional theory. A tentative explanation for this phenomenon is given in terms of the three-dimensional stress system in the contact region and also of the frictional effects at the interface; in this connection, an analytical solution is given to the problem of adhesive contact between normally-loaded dissimilar elastic cylinders. The results have some relevance to the application of the well-known indirect tensile test for materials such as concrete and rock, and also to the general design of roller bearings.
44 citations
TL;DR: In this paper, the problem of necking in metal specimens under uniaxial tension is investigated by means of the mathematical theory of bifurcation and uniqueness due to R. Hill, with particular reference to the maximum load criterion normally used by engineers.
Abstract: T he problem of necking in metal specimens under uniaxial tension is investigated by means of the mathematical theory of bifurcation and uniqueness due to R. Hill, with particular reference to the maximum load criterion normally used by engineers. The difficulties encountered with rigid/plastic materials are demonstrated. It is shown that, for appropriate boundary-value problems, uniqueness certainly holds for specimens of elastic/plastic material until the load reaches an analytic maximum. An exact necking solution of simple form is exhibited for a circular cylinder of incompressible elastic/plastic material. The stress at which this mode is initiated, although of the order of the elastic (shear) rigidity in magnitude, at least serves to bound the critical stress at necking from above.
43 citations
TL;DR: In this paper, a uniformly moving crack, preceded by a thin plastic zone, in which the yield stress is linearly dependent on the strain rate, is considered, in both plane and anti-plane strain.
Abstract: A uniformly moving crack, preceded by a thin plastic zone, in which the yield stress is linearly dependent on the strain rate, is considered, in both plane and anti-plane strain. This model is used to calculate the variation of the stress intensity factor with crack velocity in the limit of small-scale yielding and the unsteady motion of a crack with a finite plastic zone in anti-plane strain. It is shown that this latter motion differs only slightly from what would be obtained if the yielding were small scale and that if the parameters are chosen appropriately the model can explain the observed relativelyslow velocities of crack propagation.
40 citations
TL;DR: In this article, a theory of plastic behavior of fiber-reinforced composite materials is presented and yield criteria are developed for deciding when the material may flow plastically and also the particular mode under which failure occurs.
Abstract: A theory of plastic behavior of fiber-reinforced composite materials is presented. The theory allows for several failure modes of the composite material, including plastic failure of the matrix material alone and the combined ductile failure of the fibers and the matrix. The latter mode may be readily modified to allow for brittle behavior of the fiber material. Yield criteria are developed for deciding when the material may flow plastically and also the particular mode under which failure occurs. The plastic strain rates associated with each of the failure modes are specified. In view of technological applications, special attention is given to the case of plane biaxial stress of a thin composite sheet. The failure criteria in this case reduce to the usual Tresca criterion in the limiting case of an isotropic material, and to the well known results of A. K elly and G.J. D avies (1965) in the case of uniaxial stress. These results are well substantiated by experimental evidence.
TL;DR: In this article, a metal-matrix composite containing ductile-metallic fibers is described and the critical stress intensity for crack propagation between fibers is a decreasing function of volume fraction.
Abstract: F racture mechanics of a metal-matrix composite containing ductile-metallic fibers is described. Experimental verification of the proposed descriptions has been established with an aluminum-base composite containing uni-directional stainless-steel fibers. Theoretical description of the plastic energy dissipation shows that crack propagation across fibers is very difficult because of the high energy density represented by the fiber. The fiber contribution is an increasing function of volume fraction which results in the critical stress-intensity factor increasing with fiber content. On the other hand, crack propagation between fibers is very easy because the inter-fiber spacing limits the plastic energy dissipation in the matrix. The critical stress intensity for crack propagation between fibers is a decreasing function of volume fraction.
TL;DR: In this article, the Griffith-Irwin-Orowan theory is applied to the fracture of unidirectional composites, and lower bounds on the fracture propagation stress are derived.
Abstract: A n analysis is presented of the mechanics of failure of notched unidirectional composites subjected to tensile load in the fiber direction. The applicability of the Griffith-Irwin-Orowan theory to the fracture of composites is discussed. The effects of fiber debonding, matrix and fiber plasticity, and scatter in fiber strength are considered. Statistical lower bounds on the fracture propagation stress are developed.
TL;DR: In this paper, a simple mechanistic model for time-dependent spallation was developed and correlated with experimental data, including effects of plastic flow and strain-rate sensitivity, and experimental data for two aluminum alloys were consistent to within about 30 percent of the critical stress predictions for the model.
Abstract: A simple mechanistic model for time-dependent spallation is developed and correlated with experimental data. Included in the model are effects of plastic flow and strain-rate sensitivity. Experimental data for two aluminum alloys are consistent to within about 30 per cent of the critical stress predictions for the model.
TL;DR: An experimental method of obtaining flow fields for plane-strain extrusion using printed grids (0.002 in. square) is described, and a slip-line field is constructed by calculating the directions of maximum shear strain-rate from the measured velocity gradients.
Abstract: An experimental method of obtaining flow fields for plane-strain extrusion using printed grids (0.002 in. square) is described, and a slip-line field is constructed by calculating the directions of maximum shear strain-rate from the measured velocity gradients. It is shown that account must be taken in the stress equilibrium equations of variations in flow stress caused by strain-hardening in order to satisfy internal stress consistency and equilibrium of forces.
TL;DR: In this paper, the authors describe creep, recovery and load superposition experiments on an oriented polypropylene monofilament in the temperature range 28-60 C. Although the non-linear viscoelastic behavior exhibited nonlinear behaviour, the results for different temperatures were found to be related by simple time/temperature equivalence.
Abstract: This paper describes creep, recovery and load superposition experiments on an oriented polypropylene monofilament in the temperature range 28–60 C. Although the monofilament exhibited non-linear behaviour, the results for different temperatures were found to be related by simple time/temperature equivalence. The multiple integral representation for the non-linear viscoelastic behaviour is considered, and it is found that the experimental results can be represented, with reasonable accuracy, by first-, second- and third-order stress terms. It is shown that each of the kernels in this representation is shifted by the same factor with change in temperature.
TL;DR: In this article, a new representation for the stress field produced by a semi-infinite accelerating crack in a longitudinal shear field is presented, based on the solution of a diffraction problem given by F.G. Friedlander, and the relationship of the Griffith-Irwin energy balance to the Dugdale-Barenblatt model of small-scale yielding is examined.
Abstract: A new representation , based on the solution of a diffraction problem given by F.G. Friedlander, is presented for the stress field produced by a semi-infinite accelerating crack in a longitudinal shear field. This is employed to give an explicit expression for the field at any point generated by an incident plane stress wave and also to obtain asymptotic expressions for the components of strain and velocity near the crack tip when the crack is loaded in any prescribed manner. The relationship of the Griffith-Irwin energy balance to the Dugdale-Barenblatt model of small-scale yielding is next examined, and it is concluded that the Griffith-Irwin energy balance is acceptable for all but the most rapid loading conditions, provided the yielding is small-scale. Finally, some effects of large-scale yielding are investigated by adopting the Dugdale model, with a plastic zone of finite size, the motion of a crack subjected to a step load being investigated in detail. An important simplifying approximation, which is adequate in this case and likely also to be so in most practical contexts, is noted, which very drastically reduces the complexity of the mathematical analysis. It is concluded that a crack with a finite plastic zone accelerates less rapidly than one for which the yielding is small-scale, although the accelerations predicted are still unrealistically high. It is expected that this defect will be remedied by the future inclusion of rate-dependence into the model.
TL;DR: In this article, a general velocity field, satisfying constitutive requirements at a generic instant, is constructed and employed to investigate the possibility of loss of steady-state deformation in closed-end cylinders under internal and external pressure.
Abstract: Rigid -plastic strain-hardening flow in closed-end cylinders under internal and external pressure is examined from the view-point of obtaining criteria for uniqueness and stability of finite deformation. A general velocity field, satisfying constitutive requirements at a generic instant, is constructed and employed to investigate the possibility of loss of steady-state deformation. Earlier stability bounds, for the case of internal pressure, derived under the assumption that instability occurs in a continuation of the steady-state mode, may then be verified, although with some restrictions as it is shown that, for certain combinations of cylinder geometry and material hardening properties, non-unique deformation might precede a stability loss. For the case of external pressure, a general criterion is established from which the accuracy of the classical tangent-modulus buckling load may be confirmed in the thin-shell approximation.
TL;DR: In this paper, induced birefringence theory is used to account for memory and nonlinearity effects in dielectric properties dependence on deformation history, and memory is used in the context of isotropic materials with memory.
Abstract: Isotropic materials with memory, discussing induced birefringence theory to account for memory and nonlinearity effects in dielectric properties dependence on deformation history
TL;DR: In this paper, the first set of experimental data, reported here, indicates that nonlinear constitutive relations which are adequate and relatively simple may be attainable, and nonlinear interactions between loads may be significant over longer time-intervals than direct responses to loads.
Abstract: A critical evaluation of some previous attempts to characterize nonlinear mechanical behaviour of plastics reveals that they are open to misinterpretation and have a range of validity which is difficult to define. From this study a programme of experiments is motivated and the first set of experimental data, reported here, indicates that nonlinear constitutive relations which are adequate and relatively simple may be attainable. In addition, the data show that nonlinear interactions between loads may be significant over longer time-intervals than direct responses to loads; this is relevant to the definition of what constitutes a virgin material.
TL;DR: In this article, principal and slip-line methods of numerical analysis are presented for plane and axially symmetric deformations of rigid/plastic media obeying generalized Tresca/von Mises constitutive equations, and solutions for the particular problem of combined extension and expansion of a thick-walled circular cylinder under axial force and internal pressure are discussed.
Abstract: Some new principal- and slip-line methods of numerical analysis are presented for plane and axiallysymmetric deformations of rigid/plastic media. These methods are applicable to the solution of problems of incompressible isotropic, or incompressible anisotropic, or (special) compressible isotropic media obeying generalized Tresca/von Mises constitutive equations. The essential idea in the numerical methods considered is an artifice of the reduction of elliptic (and similar) problems of systems of partial differential equations to hyperbolic ones by means of a procedure based upon the initial estimation of one of the field quantities followed by iterations to determine all field quantities. Solutions for the particular problem of combined extension and expansion of a thick-walled circular cylinder under axial force and internal pressure are discussed.
TL;DR: In this paper, a n upper bound analysis is used to examine dead-zone formation in rigid/plastic axially-symmetric converging flow and a quasi-stable internal shear is found to occur under certain conditions.
Abstract: A n upper-bound analysis is used to examine dead-zone formation in rigid/plastic axially-symmetric converging flow. A quasi-stable internal shear is found to occur under certain conditions. Bounds on the geometry for quasi-stable behavior and bounds on the required power are presented. The analysis agrees favorably with experiments.
TL;DR: In this paper, the unsteady motion of a finite crack preceded by thin plastic zones is calculated and the yield stress in the plastic zones was linearly dependent on the strain rate.
Abstract: The unsteady motion of a finite crack preceded by thin plastic zones is calculated. The yield stress in the plastic zones is linearly dependent on the strain rate. It is shown that the small-scale yielding approximation can be used to predict the motion of a crack, even if there is a considerable plastic flow. The effect of the increased yield stress at high strain-rates on the crack speed is assessed.
TL;DR: In this paper, measurements of thermoelastic stresses were made as a function of heating rate, and good agreement was found with the predictions of the thermo-elastic wave theory.
Abstract: U sing pulses of monochromatic MeV electrons from a linear accelerator to heat copper and tantalum samples, measurements of thermoelastic stresses were made as a function of heating rate. The results were compared with the predictions of thermoelastic wave theory, and good agreement was found.
TL;DR: In this article, Griffiths and D.R. Owen showed that the stress maximum lies some way before the elastic-plastic interface, inside the plastic zone, and that the calculated stresses conflict with plastic slip-line field theory.
Abstract: Recent calculations by J.R. Griffiths and D.R.J. Owen (1971) on the growth of the elastic-plastic stresses for the plane strain bending of a V-notched bar reveal an interesting phenomenon : the stress maximum lies some way before the elastic-plastic interface, inside the plastic zone. Later calculations have confirmed this effect, for both work-hardening and perfectly-plastic von Mises and Tresca materials. At low applied loads the calculated stresses conflict with plastic slip-line field theory. This result is important, because it means that notch stresses before general yield cannot readily be deduced by etching up plastically-yielded zones. This paper explains the conflict analytically.
TL;DR: In this paper, the elastic field generated in two bonded isotropic half-planes containing either a circular or a rectangular inclusion, each having the same elastic properties as those of the surrounding half-plane, was analyzed based on the Papkovich-Neuber stress function approach.
Abstract: This paper is concerned with the elastic field generated in two bonded isotropic half-planes containing either a circular or a rectangular inclusion, each having the same elastic properties as those of the surrounding half-plane. The circular inclusion undergoes a transformation which in the absence of the surrounding material would be an arbitrary uniform stress-free strain, while that imposed on the rectangular inclusion corresponds to a pure dilatation. The analysis is based upon the Papkovich-Neuber stress function approach. It is found that the elastic components in the composite plane can be derived from the corresponding components in the homogeneous plane by a set of relations, which are the same for both inclusion shapes, provided that each transformation corresponds to a stress-free dilatation.
TL;DR: In this paper, Drucker's postulate for stability was further developed to reach a uniform treatment of perfectly-plastic and viscoplastic materials, and a single inequality was derived which provided a relation between permanent deflection and response time in a given boundary-value problem.
Abstract: P revious impulsive-loading theorems due to J.B. Martin are reconsidered and further developed to reach a uniform treatment of perfectly-plastic and viscoplastic materials. Starting from the extended form of D.C. Drucker's postulate for stability, and using as a reference state a complete solution of an auxiliary quasi-static problem, a single inequality is derived which provides a relation between permanent deflection and response-time in a given boundary-value problem. It is shown that bounds on the one quantity can be found whenever the other is known. Separate bounds on deflections and response-time can be obtained only in the limiting case of perfectly-plastic material. The theory is illustrated by the example of a clamped circular plate.
TL;DR: In this paper, the deformation of zirconium notched bend specimens has been simulated by a two-dimensional elastic-plastic finite-element program, and small variations of the crack geometry studied.
Abstract: In earlier work, A.R. Luxmoore and P.J. Wyatt described measurements of the deformation of zirconium notched bend specimens, using the Moire effect. This deformation has been simulated by a two-dimensional elastic-plastic finite-element program, and small variations of the crack geometry studied. The numerical and experimental results were compared by studying (a) the crack opening displacement, (b) the crack opening profile, and (c) the size and shape of the plastic zones.
TL;DR: In this article, the influence of strain-hardening and grain size on the early stage of fatigue damage of polycrystal subjected to fluctuating stress is considered, and it is shown that the increase of the strainhardening rate and/or the decrease of the grain size decreases the rate of early fatigue damage.
Abstract: E arly fatigue damage is assumed to correspond to the build-up of local plastic shear strain. The influence of strain-hardening and grain size on the early stage of fatigue damage of a polycrystal subjected to fluctuating stress is considered. The calculations are based on a micromechanics theory proposed recently by the authors. It is shown that the increase of the strain-hardening rate and/or the decrease of the grain size decreases the rate of early fatigue damage. In order to produce 100 per cent local plastic shear strain for a given number of loading cycles, the range of the alternating stress decreases with increasing amounts of mean stress. This result is shown to be rather insensitive to the rate of strain-hardening and is found to lie between the values predicted by Gerber's parabolic law and by the modified Goodman linear law for fatigue failure.
TL;DR: A second-order theory of plane plastic flow is developed in this article, where the second order stresses are hyperbolic and the characteristics are the slip-lines of the classical theory, and simple sets of equations are found which determine the complete second order solution of any problem for which the classical slip-line field analysis is available.
Abstract: A second-order theory of plane plastic flow is developed. The equations for the second-order stresses are hyperbolic and the characteristics are the slip-lines of the classical theory. The characteristics for the second-order velocity field are also the slip-lines; and in both cases, simple sets of equations are found which determine the complete second-order solution of any problem for which the classical slip-line field analysis is available. Situations in which the second-order terms are likely to be important are discussed, and a punch indentation problem is solved as an illustrative example.