On the formulation and iterative solution of small strain plasticity problems
Author:
Kerry S. Havner
Journal:
Quart. Appl. Math. 23 (1966), 323-335
DOI:
https://doi.org/10.1090/qam/99938
MathSciNet review:
QAM99938
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Abstract |
References |
Additional Information
Abstract: This paper is concerned with a general method of formulation and iterative solution of small displacement plasticity problems, using the Hencky-Nadai hardening law as mathematical model for the material behavior. Beginning with a minimum energy principle for small thermal-mechanical strains under simple external loading, quasi-linear partial differential equations are formulated and a method of iteration by successive solutions is proposed. A finite-difference discretization of the equations (in two dimensions) is obtained through minimization of the total potential energy function, leading to positive definite symmetric matrices for general boundary configurations.
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D. C. Drucker, Some Implications of Work Hardening and Ideal Plasticity, Quart. Appl. Math. 7, 411–418 (1950)
D. C. Drucker, A More Fundamental Approach to Plastic Stress-Strain Relations, Proc. First U. S. Natl. Cong. Appl. Mech., 1951, pp. 487-491
D. C. Drucker, A Definition of Stable Inelastic Material, J. Appl. Mech. 26, 101-106 (1959)
D. C. Drucker, Plasticity, Proc. First Symp. Naval Struct. Mech., Pergamon Press, New York, 407-455 (1960)
V. D. Kliushnikov, On Plasticity Laws for Work-Hardening Materials, J. Appl. Math. Mech. (PMM) 22, 129-160 (1958)
P. M. Naghdi, Stress-Strain Relations in Plasticity and Thermo-plasticity, Proc. Second Symp. Naval Struct. Mech., Pergamon Press, New York, 121-169 (1960)
J. L. Sanders, Jr., Plastic Stress-Strain Relations Based on Linear Loading Functions, Proc. Second U. S. Natl. Cong. Appl. Mech., 455-460 (1954)
B. Budiansky, A Reassessment of Deformation Theories of Plasticity, J. Appl. Mech. 26, 259-264 (1959)
V. D. Kliushnikov, On a Possible Manner of Establishing the Plasticity Relations, J. Appl. Math. Mech. (PMM) 23, 405-418 (1959)
V. D. Kliushnikov, New Concepts in Plasticity and Deformation Theory, J. Appl. Math. Mech. (PMM) 23, 1030-1042 (1959)
B. Budiansky and O. L. Mangasarian, Plastic Stress Concentration at a Circular Hole in an Infinite Sheet Subjected to Equal Biaxial Tension, J. Appl. Mech. 27, 59-64 (1960)
O. L. Mangasarian, Stresses in the Plastic Range Around a Normally Loaded Circular Hole in an Infinite Sheet, J. Appl. Mech. 27, 65-73 (1960)
H. J. Greenberg, W. S. Dorn, E. H. Wetherell, A Comparison of Flow and Deformation Theories in Plastic Torsion of a Square Cylinder, Proc. Second Symp. Naval Struct. Mech., Pergamon Press, New York, 279-296 (1960)
H. J. Greenberg, On the Variational Principles of Plasticity, Rep. A11-S4, Grad. Div. Appl. Math., Brown Univ. (1949)
D. C. Drucker, Variational Principles in the Mathematical Theory of Plasticity, Proc. Symposia Appl. Math. 8, McGraw-Hill, New York, 7-22 (1958)
L. M. Kachanov, Variational Principles for Elastic-Plastic Solids, Prikl. Mat. Mekh. 6, 187-196 (1942)
A. A. Ilyushin, Some Problems in the Theory of Plastic Deformations, Prikl. Mat. Mekh. 7, 245-272 (1943)
A. Mendelson and S. S. Manson, Practical Solution of Plastic Deformation Problems in Elastic-Plastic Range, NASA TR R-28 (1959)
R. Courant, K. Friedricks, H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math Annalen 100, 32-74 (1928)
G. E. Forsythe and W. R. Wasow, Finite-Difference Methods for Partial Differential Equations, John Wiley, New York, Chap. 3 (1960)
S. D. Conte, K. L. Miller, C. B. Sensenig, The Numerical Solution of Axisymmetric Problems in Elasticity, Proc. Fifth Symp. Ballistic Missile and Space Technology, Academic Press, New York,. 4, 173-202 (1960)
R. Foye, The Elastic Characteristics of Variable Thickness Cantilevered Plates, Rep. NA60H-573,. North American Aviation, Inc., Columbus Div. (1961)
K. S. Havner, Solution of Two Variable Thermal and Centrifugal Stress Problems in Cartesian or Cylindrical Geometry, AiResearch Manufac. Co. Engr. Rept. AM-5505-R, Phoenix, July 1963
K. S. Havner, The Finite-Difference Solution of Two Variable Thermal and Mechanical Deformation Problems, J. Spacecraft and Rockets 2, 542–549 (1965)
D. Greenspan, On the Numerical Solution of Problems Allowing Mixed Boundary Conditions, J. Franklin Inst. 277, 11–30 (1964)
W. Prager, On Isotropic Materials with Continuous Transition from Elastic to Plastic State, Proc. Fifth Int. Cong. Appl. Mech., 234-237 (1939)
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Article copyright:
© Copyright 1966
American Mathematical Society
