Abstract
The formulation of a backward difference algorithm based on an internal variable description for piecewise linear yield surfaces is presented. Attention is restricted to an associated flow rule and isotropic material behaviour. The Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules are considered in detail. The algorithm has the advantages of being fully linked to the governing principles and avoiding the inherent problems associated with corners on the yield surface. It is used to identify return paths in stress space for the Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules. These return paths provide a basis against which heuristically developed algorithms can be compared.
Sommario
Il lavoro presenta la formulazione di un particolare algoritmo di differenza all'indietro basato su una variabile interna descrittiva di una superficic di snervamento lincare a tratti. L'interesse del lavoro e'ristretto allo studio della legge di evoluzione ed ad un comportamento isotropo del materiale. Vongono csaminate in dettaglio le superfici di snervamento di Tresca e di Mohr-Coulomb per comnportamento perfettemente plastico ed incrudimento lineare. L'algoritmo presenita il vantaggio di essere completamente integrato nella formulazione ed evita quei problemi connessi con una descrizione della superficie di snervamento spigolosa. Esso e' usato per identificare, nello spazio delle tensioni, il cammino di ritorno per superfici di snervamento alla Tresca e Mohr-Coulomb che descrivono leggi di comportamento perfettament plastico ed incrudimento lineare. Questi cammini di ritorno costituiscono una base di confronto con algoritmi sviluppati su basi piu' euristiche.
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CarterP. and MartinJ. B., ‘Work bounding functions for plastic materials’, Journal of Applied Mechanics, 43 (1976) 434–438.
CrisfieldM. A., ‘Plasticity computations using the Mohr-Coulomb yield criterions’, Engineering Computations, 4 (1987) 300–308.
Crisfield, M.A., ‘Consistent schemes for plasticity computation with the Newton-Raphson method’, Proc. 1st Int. Conf. Computational Plasticity: Models, Software and Applications (eds D. R. J. Owen, E. Hinton and E. Onate), Part 1, Pineridge Press, 1987, pp. 133–159.
DeBorstR., ‘Integration of plasticity equations for singular yield functions’, Computers and Structures, 26 (1987) 823–829.
EveR. A., ReddyB. D. and RockafellarR. T., ‘An internal variable theory of elastoplasticity based on the maximum plastic work inequality’, Quarterly Journal of Applied Mathematics, 48 (1990) 59–83.
KriegR. D. and KriegD. B., ‘Accuracies of numerical solution methods for the elastic-perfectly plastic model’, Journal of Pressure Vessel Technology, ASME, 99 (1977) 510–515.
MaierG., ‘Quadratic programming and theory of elastic-perfectly plastic structures’, Meccanica, 3 (1968) 265–273.
MartinJ. B. and NappiA., ‘An internal variable formulation of perfectly plastic and linear kinematic and isotropic hardening relations with a Von Mises yield condition’, European Journal of Mechanics, A-Solids, 9 (1990) 107–131.
MartinJ. B., ReddyB. D., GriffinT. B. and BirdW. W., ‘Applications of mathematical programming concepts to incremental elastic-plastic analysis’, Engineering Structures, 9 (1987) 171–176.
MartinJ. B., ‘An internal variable approach to the formulation of finite element problems in plasticity’, Physical Nonlinearities in Structural Analysis (eds J.Holt and J.Lemaitre), Springer-Verlag, Berlin, 1981, pp. 165–176.
MarquesJ. M. M. C., ‘Stress computations in elastoplasticity’, Engineering Computations, 1 (1984) 42–51.
OrtizM. and MartinJ. B., ‘Symmetry-preserving return mapping algorithms and incrementally extremal paths: a unification of concepts’, International Journal for Numerical Methods in Engineering, 28 (1989) 1839–1853.
OrtizM. and PopovE. P., ‘Accuracy and stability of integration algorithms for elastoplastic constitutive relations’, International Journal for Numerical Methods in Engineering, 21 (1985) 1561–1576.
OwenD. R. J. and HintonE., Finite Elements in Plasticity — Theory and Practice, Pineridge Press, Swansea, 1980.
Pankaj and Bićanić, N., ‘On multivector stress returns in Mohr-Coulomb plasticity’, Computational Plasticity: Models, Software and Applications (eds D. R. J. Owen, E. Hinton and E. Onate), Part 1, Pineridge Press, 1989, pp. 421–436.
PramonoE. and WilliamK., ‘Implicit integration of composite yield surfaces with corners’, Engineering Computations, 6 (1989) 186–197.
RiceJ. R., ‘Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity’, Journal of Mechanics and Physics of Solids, 19 (1971) 433–455.
RiceJ. R. and TraceyD. M., ‘Computational fracture mechanics’, Numerical and Computer Methods in Structural Mechanics (eds S. J.Fenves et al), Academic Press, New York, 1973, pp. 585–623.
SimoJ. C., KennedyJ. G. and GovindjeeS., ‘Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms’, International Journal for Numerical Methods in Engineering, 26 (1988) 2161–2185.
WilkinsM. L., ‘Calculation of elastic-plastic flow’, Methods of Computational Physics (eds B.Alder et al.), Academic Press, New York, 1964.
ZienkiewiczO. C., The Finite Element Method, McGraw-Hill, London, 1977.
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Rencontré, L.J., Bird, W.W. & Martin, J.B. Internal variable formulation of a backward difference corrector algorithm for piecewise linear yield surfaces. Meccanica 27, 13–24 (1992). https://doi.org/10.1007/BF00452999
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DOI: https://doi.org/10.1007/BF00452999