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Characteristic directions of certain equations of plasticity

Published online by Cambridge University Press:  26 February 2010

J. L. Ericksen
J. L. Ericksen
Affiliation:
Applied Mathematics Branch, Mechanics Division, U.S. Naval Research Laboratory, Washington 25, D.C., U.S.A.
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Extract

Various attempts have been made to identify the slip lines or Lüder lines which are observed in solids with surfaces of discontinuity or characteristic surfaces associated with solutions of equations of plasticity. Results such as those obtained in [1], together with the observed fact that such lines occur in a variety of types of experiments, indicate that, for two well-known theories of plasticity, characteristic surfaces fail to exist in situations in which such lines are observed. This can come about in two ways, one being that real characteristic directions do not exist, the other being that they do, but that the characteristic surface elements do not unite to form surfaces. The latter situation seems to arise from the fact that, even in truly three-dimensional problems, the equations considered admit only a finite number of characteristic directions. Results such as those given in [2] indicate that, if real characteristic directions do not always exist, there is some doubt as to whether one can identify such lines with surfaces of discontinuity. Another point to be considered is the ease with which solutions may be obtained. For equations possessing real characteristics, the method of characteristics is a powerful tool to use in solving two-dimensional problems. It is noted in [3], Ch. X, that, in axially symmetric problems, one cannot use this method to obtain solutions of the von Mises equations. In treating such problems, it may be easier to use equations which appear to be more complicated, but which possess real characteristics. These facts suggest that plasticity equations which always possess real characteristic directions are to be preferred to those which do not. Some workers in plasticity appreciate this, as is indicated by remarks made in [4]. However, no one has taken a rather general theory of plasticity, such as the theory of perfectly plastic solids, and attempted to determine which of the equations included in it have this property. We made an unsuccessful attempt to do so for a theory which is roughly equivalent. The purpose of this paper is to present this theory, to indicate the basic mathematical problem involved, and to record a partial solution of it.

Type
Research Article
Information
Mathematika , Volume 3 , Issue 1 , June 1956 , pp. 56 - 62
Copyright
Copyright © University College London 1956

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References

1.Thomas, T. Y., “On the characteristic surfaces of the von Mises plasticity equations”, J. Rational Mech. Anal., 1 (1952), 343–357.Google Scholar
2.Ericksen, J. L., “Singular surfaces in plasticity”, J. Math. Phys., 34, (1955) 74–79.Google Scholar
3.Hill, R., The Mathematical Theory of Plasticity (Oxford, 1950).Google Scholar
4.Prager, W., “Theory of plasticity: a survey of recent achievements II”, The Engineer, 199 (1955), 116–118.Google Scholar
5.Prager, W., Discontinuous fields of plastic stress and flow, Brown University report DA-2598/10 (1954).Google Scholar
6.Rivlin, R. S. and Ericksen, J. L., “Stress-deformation relations for isotropic materials”, J. Rational Mech. Anal., 4 (1955), 323–425.Google Scholar
7.Reiner, M., “A mathematical theory of dilatancy”, American J. Math., 67 (1945), 350–362.Google Scholar
8.Prager, W., “Strain hardening under combined stresses”, J. Appl. Phys., 16 (1945), 837–840.Google Scholar
9.Weitzenböck, R., “Zur Tensoralgebra”, Math. Zeitschrift, 10 (1921), 80–87.CrossRefGoogle Scholar
10.Rivlin, R. S., Further remarks on the stress-deformation relations for isotropic materials, Brown University report DA-3487/1 (1955).Google Scholar
11.Knopp, K., Theory of Functions (Dover, New York, 1945).Google Scholar
12.Geiringer, H., “Some recent results in the theory of an ideal plastic body”, Adv. in Appl. Mech., 3 (1953), 197–294.CrossRefGoogle Scholar
13.Baker, M. and Ericksen, J. L., “Inequalities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids”, J. Wash. Acad. Sci., 44 (1954), 33–35.Google Scholar
14.Ericksen, J. L., “A consequence of inequalities proposed by Baker and Ericksen”, J. Wash. Acad. Sci., 45 (1955), 268.Google Scholar
15.Ericksen, J. L., “Characteristic surfaces of the equations of motion for non-Newtonian fluids”, Z. angew. Math, und Phys., 4 (1953), 260–267.Google Scholar