# CXXVIII. A theoretical derivation of the plastic properties of a polycrystalline face-centred metal

01 Nov 1951-Philosophical Magazine Series 1 (Taylor & Francis Group)-Vol. 42, Iss: 334, pp 1298-1307

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.

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01 May 1952TL;DR: The connection between the elastic behavior of an aggregate and a single crystal is considered in this article, with special reference to the theories of Voigt, Reuss, and Huber and Schmid.

Abstract: The connection between the elastic behaviour of an aggregate and a single crystal is considered, with special reference to the theories of Voigt, Reuss, and Huber and Schmid. The elastic limit under various stress systems is also considered, in particular, it is shown that the tensile elastic limit of a face-centred aggregate cannot exceed two-thirds of the stress at which pronounced plastic distortion occurs.

7,944 citations

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2,385 citations

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TL;DR: In this paper, a review of continuum-based variational formulations for describing the elastic-plastic deformation of anisotropic heterogeneous crystalline matter is presented and compared with experiments.

Abstract: This article reviews continuum-based variational formulations for describing the elastic–plastic deformation of anisotropic heterogeneous crystalline matter. These approaches, commonly referred to as crystal plasticity finite-element models, are important both for basic microstructure-based mechanical predictions as well as for engineering design and performance simulations involving anisotropic media. Besides the discussion of the constitutive laws, kinematics, homogenization schemes and multiscale approaches behind these methods, we also present some examples, including, in particular, comparisons of the predictions with experiments. The applications stem from such diverse fields as orientation stability, microbeam bending, single-crystal and bicrystal deformation, nanoindentation, recrystallization, multiphase steel (TRIP) deformation, and damage prediction for the microscopic and mesoscopic scales and multiscale predictions of rolling textures, cup drawing, Lankfort ( r ) values and stamping simulations for the macroscopic scale.

1,573 citations

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Brown University

^{1}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.

Abstract: A new rate dependent constitutive model is developed for polycrystals subjected to arbitrarily large strains. The model is used to predict deformation textures and large-strain strain hardening behavior following various stress-strain histories for single phase f.c.c. aggregates that deform by crystallographic slip. Examples involving uniaxial and plane strain tension and compression are presented which illustrate how texture influences polycrystalline strain hardening, in particular these examples demonstrate both textural strengthening and softening effects. Input to the model includes the description of single crystal strain hardening and latent hardening along with strain rate sensitivity, all properties described on the individual slip system level. The constitutive formulation used for the individual grains is essentially that developed by Peirce et al . [6, Acta metall . 31, 1951 (1983)] to solve rate dependent boundary value problems for finitely deformed single crystals. Inclusion of rate dependence is shown to overcome the long standing problem of nonuniqueness in the choice of active slip systems which is inherent in the rate independent theory. Because the slipping rates on all slip systems within each grain are unique in the rate dependent theory, the lattice rotations and thus the textures that develop are unique. In addition, the model makes it possible to study how strain rate sensitivity on the slip system, and single grain, levels is manifested in polycrystalline strain rate sensitivity. The model is also used to predict “constant offset plastic strain yield surfaces” for materials that are nearly rate insensitive—these calculations describe the development of rounded “yield surface vertices” and the resulting softening of material stiffness to a change in loading path that vertices imply. For our rate dependent solid this reduction in stiffness occurs after small but finite loading increments. Finally the model is used to carry out an imperfection-based sheet necking analysis both for isotropic and strongly textured sheets. The results show that larger strain hardening rates, and strain rate sensitivity, on the slip system level both increase the failure strains, as expected, but also demonstrate a strong influence of texture on localized necking.

1,557 citations

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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.

1,243 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

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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

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TL;DR: The plasticity of metals has been the subject of many recent papers as discussed by the authors, but, owing to the complexity of the subject, there is but little agreement between different researches and attempts to extract simple generalisations from the very complex phenomena have been made chiefly in two directions: engineers have used test bars of certain specially simple form, such as uniform round bars which they have subjected to twisting or tension, and they have found the effect on their test of varying physical conditions.

Abstract: The plasticity of metals has been the subject of many recent papers but, owing to the complexity of the subject, there is but little agreement between different researches. Attempts to extract simple generalisations from the very complex phenomena have been made chiefly in two directions (1) Engineers have used test bars of certain specially simple form, such as uniform round bars which they have subjected to twisting or tension, and they have found the effect on their test of varying physical conditions. (2) Mathematicians have assumed an ideal plastic material and have given it properties which may or may not be possessed by some real material. They have then analysed the distributions of stress and strain when this ideal material is subjected to given external forces or distortions.

315 citations

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TL;DR: The observed strain-ratio relationship is used in conjunction with the assumption of maximum work during a given strain to calculate the criterion of yield and is found that this is very close to, but not identical with, the Mises-Heneky criterion.

Abstract: The assumption that the work done during a small plastic strain is a maximum as the yield-stress criterion is varied is shown to give rise to a connexion between the yield-stress and the strain-ratio relationship The strain-ratio relationship is that which exists between the ratios of principal stress differences and the ratios of the corresponding strain differences It is common to assume that this relationship is one of simple proportionality Experiments, however, show that this assumption is not true in metals The observed strain-ratio relationship is used in conjunction with the assumption of maximum work during a given strain to calculate the criterion of yield It is found that this is very close to, but not identical with, the Mises-Hencky criterion

16 citations