On discontinuous plastic states, with special reference to localized necking in thin sheets
TL;DR: In this article, the authors investigated the permitted discontinuities of stress, velocity, and surface slope in a plastic-rigid sheet deformed in its plane, and the necessary restrictions on the stress-state and rate of workhardening were obtained for any yield function and plastic potential.
Abstract: Permissible discontinuities of stress, velocity, and surface slope are investigated in a plastic-rigid sheet deformed in its plane. One such discontinuity of velocity is shown to be the mathematical idealization of localized necking; the necessary restrictions on the stress-state and rate of workhardening are obtained for any yield function and plastic potential. The results are illustrated by an examination of the modes of necking in notched tension strips. The constraint factors at the yield point are obtained for notches with wedge-shaped or circular roots.
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TL;DR: In this paper, the authors investigated the hypothesis that localization of deformation into a shear band may be considered a result of an instability in the constitutive description of homogeneous deformation.
Abstract: T his paper investigates the hypothesis that localization of deformation into a shear band may be considered a result of an instability in the constitutive description of homogeneous deformation. General conditions for a bifurcation, corresponding to the localization of deformation into a planar band, are derived. Although the analysis is general and applications to other localization phenomena are noted, the constitutive relations which are examined in application of the criterion for localization are intended to model the behavior of brittle rock masses under compressive principal stresses. These relations are strongly pressure-sensitive since inelasticity results from frictional sliding on an array of fissures; the resulting inelastic response is dilatant, owing to uplift in sliding at asperities and to local tensile cracking from fissure tips. The appropriate constitutive descriptions involve non-normality of plastic strain increments to the yield hyper-surface. Also, it is argued that the subsequent yield surfaces will develop a vertex-like structure. Both of these features are shown to be destabilizing and to strongly influence the resulting predictions for localization by comparison to predictions based on classical plasticity idealizations, involving normality and smooth yield surfaces. These results seem widely applicable to discussions of the inception of rupture as a constitutive instability.
2,411 citations
TL;DR: In this article, a rate dependent constitutive model is developed for polycrystals subjected to arbitrarily large strains, and the model is used to predict deformation textures and large-strain strain hardening behavior following various stressstrain histories for single phase f.c. aggregates that deform by crystallographic slip.
1,557 citations
TL;DR: Using a simplified constitutive model of a pointed vertex on subsequent yield loci, the onset of localized necking under biaxial stretching has been predicted and this result supports the hypothesis of vertex-formation on the yield locus under continued plastic flow.
Abstract: B y using a simplified constitutive model of a pointed vertex on subsequent yield loci, namely, such that the equations of deformation-theory of rigid-plastic solids apply for fully-active stress increments, the onset of localized necking under biaxial stretching has been predicted. The predictions agree reasonably well with reported experimental observations. Since localized necking under biaxial stretching of a uniform and homogeneous sheet is impossible when flow theories of plasticity with smooth yield-loci are used, this result supports the hypothesis of vertex-formation on the yield locus under continued plastic flow. The implications of this conclusion with respect to the study of the inception of ductile fracture in solids, viewed as a material instability, may be far-reaching. Still, explanations based on a smooth yield-locus but small initial inhomogeneities cannot be ruled out, and both initial imperfections and yield-vertex effects may contribute in general to localization instabilities.
816 citations
Cites background from "On discontinuous plastic states, wi..."
...The limit-curve predicted by HILL (1952) for a isotropic, rigid-plastic, Mises material is 8; = N I +p (- 1 < p I O), which localized necking is power-law work-hardening, (42) where the angle $ between the maximuni principal strain direction and the normal n is given by $ = arctan (,I -p), \ or 11:…...
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...…sheet is deformed by biaxial tension so that both principal strain increments in the plane of the sheet are positive (i.e. deJd&, = p > 0), there is no line of zero extension in the plane, and the description of localized necking given by HILL (1952) does not predict the onset of localized necking....
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TL;DR: In this article, the dominant singularity governing the plastic behavior at a crack tip is analyzed for conditions of plane stress and plane strain for cracks in both far tensile and far shear fields.
Abstract: Further details of the stress and strain fields associated with the dominant singularity governing the plastic behaviour at a crack tip are presented for conditions of plane stress and plane strain for cracks in both far tensile and far shear fields. Results are obtained for a power hardening material. Limiting cases for non-hardening materials are shown to correspond to certain perfect plasticity solutions.
700 citations
TL;DR: In this paper, a finite element approach is proposed for the solution of the continuum elastic-plastic problems by means of a plastic stress-strain matrix which is derivable by inverting the Prandtl-Reuss equations in plasticity theory.
606 citations
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01 Jan 1950
TL;DR: In this paper, the solution of two-dimensional non-steady motion problems in two dimensions is studied. But the solution is not a solution to the problem in three dimensions.
Abstract: 1. Introduction 2. Foundations of the thoery 3. General theorems 4. The solution of plastic-elastic problems I 5. The solution of plastic-elastic problems II 6. Plane plastic strain and the theory of the slip-line field 7. Two-dimensional problems of steady motion 8. Non-steady motion problems of steady motion 9. Non-steady motion problems in two dimensions II 10. Axial symmetry 11. Miscellaneous topics 12. Platic anisotropy
7,810 citations
TL;DR: In this paper, a general relationship between stress and plastic strain in polycrystalline aggregate is derived for any metal in which individual crystals deform by slipping over preferred planes under a critical shear stress.
Abstract: Summary A general relationship between stress and plastic strain in a polycrystalline aggregate is derived for any metal in which individual crystals deform by slipping over preferred planes under a critical shear stress. Full account is taken of the non-uniform distortion due to mutual constraints between the grains of an aggregate. It is shown that a plastic potential exists which is identical with the yield function. Upper and lower bounds are obtained for an approximate calculation of this function for any applied system of combined stresses.
1,251 citations
TL;DR: In this paper, it was shown that the work-hardness of an isotropic aggregate of face-centred cubic crystals is a function only of the total plastic work if the grains hardened equally.
Abstract: Summary In continuation of a previous paper (Bishop and Hill 1951) it is conjectured that the work done in plastically deforming a polycrystal is approximately equal to that which would be done if the grains were free to deform equally. In conjunction with the principle of maximum plastic work, this enables the yield function of an aggregate to be calculated. This is done for an isotropic aggregate of face-centred cubic crystals, following a determination of the stresses needed to produce multi-slip. The theoretical yield criterion lies between those of Tresca and von Mises, in good agreement with observaton for copper and aluminum. It is shown further that the work-hardening of an aggregate would be a function only of the total plastic work if the grains hardened equally ; the departure from this functional relation is expressed explicitly in terms of the non-uniform hardening.
702 citations
TL;DR: In this paper, the safe loads for a Prandtl-Reuss material subject to surface tractions or displacements which increase in ratio are extended to any perfectly plastic material and any history of loading.
Abstract: Earlier results [1,2]2 on safe loads for a Prandtl-Reuss material subject to surface tractions or displacements which increase in ratio are here extended to any perfectly plastic material and any history of loading.
636 citations