Extremal paths and holonomic constitutive laws in elastoplasticity
Authors:
B. D. Reddy, J. B. Martin and T. B. Griffin
Journal:
Quart. Appl. Math. 45 (1987), 487-502
MSC:
Primary 73E99
DOI:
https://doi.org/10.1090/qam/910456
MathSciNet review:
910456
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Extremal stress paths between any two stresses are investigated for elastic-plastic materials, extending existing results which hold for the case when one of the stress points is at the origin. Assumptions about the differentiability of the various work and complementary work functions are relaxed, and it is shown that the maximum complementary work $\hat U$ is a potential for the strain in the sense that the strain lies in the subdifferential of $\hat U$. In the same way the minimum work $\hat W$ is a potential for stress. Parallel investigations with respect to maximum complementary plastic work and plastic work show that these quantities are potentials for stress and plastic strain increment, respectively. A holonomic constitutive law based on the relations between stress and strain, obtained when the stress history follows an extremal path, is constructed.
- Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753
- Bernard Budiansky, A reassessment of deformation theories of plasticity, J. Appl. Mech. 26 (1959), 259–264. MR 0103653
- M. Z. Cohn, G. Maier, and D. E. Grierson (eds.), Engineering plasticity by mathematical programming, Pergamon International Library of Science, Technology, Engineering and Social Studies, Pergamon Press, Oxford-Elmsford, N.Y., 1979. MR 692946
- Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. MR 0463993
- T. B. Griffin, B. D. Reddy, and J. B. Martin, A numerical study of holonomic approximations to problems in plasticity, Internat. J. Numer. Methods Engrg. 26 (1988), no. 6, 1449–1466. MR 944803, DOI https://doi.org/10.1002/nme.1620260614
Hencky, H., Zur Theorie plastischer Deformationen und der hierdurch in Material herforgerufenen Nachspannungen, Zeit. Ang. Math. Mech. 4, 323 (1924)
Kachanov, L. M., Fundamentals of the Theory of Plasticity, Mir, Moscow, 1974
Maier, G., Complementary plastic work theorems in piecewise-linear elastoplasticity, Internat. J. Solids and Structures 5, 261 (1969)
Martin, J. B., A complementary energy bounding theorem for time-independent materials. In Developments in Theoretical and Applied Mechanics (ed. D. Frederick and E. H. Harris). Pergamon, New York, 1971, p. 517
Martin, J. B., Plasticity: Fundamentals and General Results, MIT Press, Cambridge, Mass., 1975
Nadai, A.L., Plasticity, McGraw-Hill, New York, 1931
Ponter, A. R. S., Convexity conditions and energy theorems for time-independent materials, J. Mech. Phys. Solids 16, 283 (1968)
- A. R. S. Ponter and J. B. Martin, Some extremal properties and energy theorems for inelastic materials and their relationship to the deformation theory of plasticity, J. Mech. Phys. Solids 20 (1972), 281–300. MR 349113, DOI https://doi.org/10.1016/0022-5096%2872%2990024-5
Rockafellar, R. T., Convex Analysis, Princeton Univ. Press, Princeton, 1970
- B. D. Reddy and T. B. Griffin, Variational principles and convergence of finite element approximations of a holonomic elastic-plastic problem, Numer. Math. 52 (1988), no. 1, 101–117. MR 918319, DOI https://doi.org/10.1007/BF01401024
Soechting, J. F. and Lance, R. H., A bounding principle in the theory of work-hardening plasticity, J. Appl. Mech. 36, 228 (1969)
Aubin, J.-P. and Ekeland, I., Applied Nonlinear Analysis, Wiley, New York, 1984
Budiansky, B., A reassessment of deformation theories of plasticity, J. Appl. Mech. 26, 259 (1959)
Cohn, M. Z. and Maier, G. (eds.), Engineering Plasticity by Mathematical Programming (Proc. Nato Adv. Studies Inst., Univ. of Waterloo, Aug. 1977), Pergamon, New York, 1979
Ekeland, I. and Temam, R. Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976
Griffin, T. B., Reddy, B. D., and Martin, J. B., A numerical study of holonomic approximations to problems in plasticity, Internat. J. Numer. Methods Engrg. (to appear)
Hencky, H., Zur Theorie plastischer Deformationen und der hierdurch in Material herforgerufenen Nachspannungen, Zeit. Ang. Math. Mech. 4, 323 (1924)
Kachanov, L. M., Fundamentals of the Theory of Plasticity, Mir, Moscow, 1974
Maier, G., Complementary plastic work theorems in piecewise-linear elastoplasticity, Internat. J. Solids and Structures 5, 261 (1969)
Martin, J. B., A complementary energy bounding theorem for time-independent materials. In Developments in Theoretical and Applied Mechanics (ed. D. Frederick and E. H. Harris). Pergamon, New York, 1971, p. 517
Martin, J. B., Plasticity: Fundamentals and General Results, MIT Press, Cambridge, Mass., 1975
Nadai, A.L., Plasticity, McGraw-Hill, New York, 1931
Ponter, A. R. S., Convexity conditions and energy theorems for time-independent materials, J. Mech. Phys. Solids 16, 283 (1968)
Ponter, A. R. S. and Martin, J. B., Some extremal properties and energy theorems for inelastic materials and their relationship to the deformation theory of plasticity, J. Mech. Phys. Solids 20, 281 (1972)
Rockafellar, R. T., Convex Analysis, Princeton Univ. Press, Princeton, 1970
Reddy, B. D. and Griffin, T. B., Variational principles and convergence of finite element approximations of a holonomic elastic-plastic problem (submitted for publication)
Soechting, J. F. and Lance, R. H., A bounding principle in the theory of work-hardening plasticity, J. Appl. Mech. 36, 228 (1969)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73E99
Retrieve articles in all journals
with MSC:
73E99
Additional Information
Article copyright:
© Copyright 1987
American Mathematical Society