zbMATH — the first resource for mathematics

Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms. (English) Zbl 0661.73058
Rate-independent plasticity and viscoplasticity in which the boundary of the elastic domain is defined by an arbitrary number of yield surfaces intersecting in a non-smooth fashion are considered in detail. It is shown that the standard Kuhn-Tucker optimality conditions lead to the only computationally useful characterization of plastic loading. On the computational side, an unconditionally convergent return mapping algorithm is developed which places no restrictions (aside from convexity) on the functional forms of the yield condition, flow rule and hardening law. The proposed general purpose procedure is amenable to exact linearization leading to a closed-form expression of the so-called consistent (algorithmic) tangent moduli. For viscoplasticity, a closed- form algorithm is developed based on the rate-independent solution. The methodology is applied to structural elements in which the elastic domain possesses a non-smooth boundary. Numerical simulations are presented that illustrate the excellent performance of the algorithm.

74S30 Other numerical methods in solid mechanics (MSC2010)
74C99 Plastic materials, materials of stress-rate and internal-variable type
74P99 Optimization problems in solid mechanics
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
65K05 Numerical mathematical programming methods
Full Text: DOI
[1] Cormeau, Int. j. numer. methods eng. 9 pp 109– (1975)
[2] and , Numercal Methods for Unconstrained Optimization, Prentice-Hall, Englewood Cliffs, N.J., 1983.
[3] and , Les Inequations en Mechanique et en Physique, Dunod, Paris, 1972.
[4] ’NIKE 2D: An implicit, finite deformation, finite element code for analyzing the static and dynamic response of two-dimensional solids’, Lawrence Livermore National Laboratory, Report UCRL-52678, University of California, Livermore, 1984.
[5] Halphen, J. Mecanique 14 pp 39– (1975)
[6] Hughes, Comp. Struct. 8 pp 169– (1978)
[7] Johnson, J. Math. Pures Appl. 55 pp 431– (1976)
[8] Johnson, Numer. Math. 26 pp 79– (1976)
[9] Johnson, J. Math. Anal. Appl 62 pp 325– (1978)
[10] Koiter, Quart. Appl. Math. 11 pp 350– (1953)
[11] Koiter, Prog. Solid Mech. 6 (1960)
[12] Linear and Nonlinear Programming, Addison-Wesley, Menlo Park, CA 1984.
[13] ’A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes’, Meccanica 54-66 (1970). · Zbl 0197.23303
[14] and , Engineering Plasticity and by Mathematical Programming, Pergamon Press, New York, 1979.
[15] Mandel, Int. J. Solids Struct. 1 pp 273– (1965)
[16] ’Problems in plasticity and their finite element approximation’, Ph.D. Thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 1978.
[17] Matthies, Int. j. numer. methods eng. 14 pp 1613– (1979)
[18] ’Application of convex analysis to the treatment of elastoplastic systems’, in and (eds.), Applications of Methods of Functional Analysis to Problems in Mechanics, Springer-Verlag, Berlin, 1976.
[19] Moreau, J. Diff. Eqn. 26 pp 347– (1977)
[20] ’Stress-strain relations in plasticity and thermoplasticity’, in Proc. 2nd. Symp. on Naval Struct. Mechanics, Pergamon Press, London, 1960.
[21] Naghdi, Int. J. Eng. Sci. 13 pp 785– (1975)
[22] Neal, J. Appl. Mech. 23 (1961)
[23] Nguyen, Int. j. numer. methods eng. 11 pp 817– (1977)
[24] Ortiz, Int. j. numer. methods eng. 21 pp 1561– (1985)
[25] Ortiz, Int. j. numer. methods eng. 23 pp 353– (1986)
[26] ’Thermodynamic theory of viscoplasticity’, in Advances in Applied Mechanics., Vol. 11, Academic Press, New York 1971.
[27] Pinsky, Comp. Methods Appl Mech. Eng. 40 pp 137– (1983)
[28] Pshenichny, Numerical Methods in External Problems MIR (1978)
[29] Simo, Comp. Methods Appl. Mech. Eng. 40 pp 301– (1984)
[30] Simo, J. Applied Mechanics · Zbl 0761.73016
[31] and , ’General return mapping algorithms for rate independent plasticity’, in Desai (ed.), Constitutive Equations for Engineering Materials, 1987.
[32] and , Elastoplasticity and Viscoplasticity. Computational Aspects, in press.
[33] and , ’Complementary mixed finite element formulations of elastoplasticity’, Comp. Meth. Appl. Mech. Engng. (submitted). · Zbl 0687.73064
[34] Simo, Comp. Methods Appl. Mech. Eng. 49 pp 221– (1985)
[35] Simo, Comp. Methods Appl. Mech. Eng. 48 pp 101– (1985)
[36] Simo, Int. j. numer. methods eng. 22 pp 649– (1986)
[37] Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, Massachusetts, 1986. · Zbl 0618.00015
[38] ’Sur les √®equations de la plasticite: existence et regularite des solutions’, J. Mechanique, 3-39 (1981).
[39] ’Calculation of elastic-plastic flow’, in et al. (eds.), Methods of Computational Physics 3, Academic Press, New York 1964.
[40] Zienkiewicz, Int. j. numer. methods eng. 8 pp 821– (1974)
[41] The Finite Element Method, 3rd edn, McGraw-Hill, London, 1977.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.