On uniqueness in the theory of plasticity

Author:
D. C. Drucker

Journal:
Quart. Appl. Math. **14** (1956), 35-42

MSC:
Primary 73.2X

DOI:
https://doi.org/10.1090/qam/77386

MathSciNet review:
77386

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Abstract: The fundamental definitions of work-hardening and perfect plasticity have far reaching implications with respect to uniqueness of solution for elastic-plastic bodies. Satisfaction of the basic postulate, that in a cycle work cannot be extracted from the material and the system of forces acting upon it, guarantees an existing solution to be stable but not necessarily unique. Uniqueness follows for the usual linear relation between the increments or rates of stress and strain and also for combinations of such linear forms. Conversely, lack of uniqueness results for an elastic-perfectly plastic body when, for example, the maximum shearing stress criterion of yield is employed with the Mises flow rule.

**[1]**P. Hodge and W. Prager,*A variational principle for plastic materials with strain-hardening*, J. Math. Physics**27**(1948), 1–10. MR**0025914****[2]**R. Hill,*The Mathematical Theory of Plasticity*, Oxford, at the Clarendon Press, 1950. MR**0037721****[3]**B. Budiansky,*Fundamental theorems and consequences of the slip theory of plasticity*, Ph.D. thesis, Brown University, Providence, R. I., 1950**[4]**W. T. Koiter,*Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with a singular yield surface*, Quart. Appl. Math.**11**(1953), 350–354. MR**0059769**, https://doi.org/10.1090/S0033-569X-1953-59769-7**[5]**D. C. Drucker,*A more fundamental approach to plastic stress-strain relations*, Proceedings of the First U. S. National Congress of Applied Mechanics, Chicago, 1951, The American Society of Mechanical Engineers, New York, N. Y., 1952, pp. 487–491. MR**0054502****[6]**D. C. Drucker,*Some implications of work hardening and ideal plasticity*, Quart. Appl. Math.**7**(1950), 411–418. MR**0034210**, https://doi.org/10.1090/S0033-569X-1950-34210-6**[7]**S. B. Batdorf and Bernard Budiansky,*A mathematical theory of plasticity based on the concept of slip*, Tech. Notes Nat. Adv. Comm. Aeronaut.,**1949**(1949), no. 1871, 33. MR**0029681****[8]**J. F. W. Bishop and R. Hill,*A theory of the plastic distortion of a polycrystalline aggregate under combined stresses*, Philos. Mag. (7)**42**(1951), 414–427. MR**0041691****[9]**W. Prager,*General theory of limit design*, Proc. 8th Internatl. Congr. Theoret. Appl. Mech. Istanbul (1952)**[10]**J. L. Sanders,*Plastic stress-strain relations based on infinitely many plane loading surfaces*, Proc. 2nd U. S. Natl. Congr. Appl. Mech., Ann Arbor, Mich., 1954, pp. 455-460**[11]**William Prager and Philip G. Hodge Jr.,*Theory of perfectly plastic solids*, John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951. MR**0051118****[12]**D. C. Drucker,*The significance of the criterion for additional plastic deformation of metals*, J. Colloid Sci.**4**, pp. 299-311 (1949)

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DOI:
https://doi.org/10.1090/qam/77386

Article copyright:
© Copyright 1956
American Mathematical Society