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A general theory for nonlocal softening plasticity of integral-type. (English) Zbl 1388.74018

Summary: This paper deals with a comparison of several models, proposed in the literature, of softening plasticity with internal variables regularized by nonlocal averaging of integral type.
The analyzed models of softening plasticity are divided in two groups. The former complies with models based on the principle of nonlocal maximum plastic dissipation; the latter is given by models formulated ad hoc without any recourse to thermodynamic requirements.
In general, these models are not easy to compare. Actually, the formulations based on the principle of nonlocal maximum plastic dissipation appears to be quite different in terms of the free energy and of the yield function. On the other side, it is not trivial to check the thermodynamic admissibility of nonlocal models ad hoc formulated.
To highlight the fundamental properties of these models in order to exploit the similarities and the differences between them, a general treatment of softening plasticity with internal variables is preliminarily proposed in the framework supplied by convex analysis and by the generalized standard material.
It is shown that this framework provides the suitable tools to perform a theoretical analysis of such nonlocal problems.The proposed nonlocal formulation of softening plasticity combines in a quite general way the effect of local and nonlocal internal variables in the expressions of the free energy and of the elastic domain.
As a consequence, several models of softening plasticity based either on thermodynamic considerations or on ad hoc formulations can be derived from the proposed model. The maximum dissipation principle is provided for each analyzed model.

MSC:

74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
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[1] Acharya, A.; Bassani, J. L.: Incompatibility and crystal plasticity, Journal of the mechanics and physics of solids 48, 1565-1595 (2000) · Zbl 0963.74010
[2] Aifantis, E. C.: The physics of plastic deformation, International journal of plasticity 3, 211-247 (1987) · Zbl 0616.73106 · doi:10.1016/0749-6419(87)90021-0
[3] Bassani, J. L.; Needleman, A.; Van Der Giessen, E.: Plastic flow in a composite: a comparison of nonlocal continuum and discrete dislocation predictions, International journal of solids and structures 38, 833-853 (2001) · Zbl 1004.74006 · doi:10.1016/S0020-7683(00)00059-7
[4] Bažant, Z. P.; Lin, F. -B.: Nonlocal yield-limit degradation, International journal for numerical methods in engineering 26, 1805-1823 (1988) · Zbl 0661.73041 · doi:10.1002/nme.1620260809
[5] Borino, G., Failla, B., 2000. Thermodynamic consistent plasticity models with local and nonlocal internal variables. In: CD-Rom Proceedings of European Congress an Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, September 11 – 14.
[6] Borino, G.; Polizzotto, C.: Comments on nonlocal bar revisited by christer nilsson, International journal of solids and structures 36, 3085-3091 (1999) · Zbl 1056.74536
[7] Borino, G.; Fuschi, P.; Polizzotto, C.: A thermodynamic approach to nonlocal plasticity and related variational principles, Journal of applied mechanics, ASME 66, 952-963 (1999)
[8] Cosserat, E.; Cosserat, F.: Théorie des corps déformables, (1909) · JFM 40.0862.02
[9] De Borst, R.: Some recent issues in computational failure mechanics, International journal for numerical methods in engineering 52, 63-95 (2001)
[10] Dillon, O. W.; Kratochvil, J.: A strain gradient theory of plasticity, International journal of solids and structures 6, 1513-1533 (1970) · Zbl 0262.73036 · doi:10.1016/0020-7683(70)90061-2
[11] Edelen, D. G. B.; Laws, N.: On the thermodynamics of systems with nonlocality, Archive for rational mechanics and analysis 43, 24-35 (1971) · Zbl 0225.73004
[12] Ericksen, J. L.; Truesdell, C.: Exact theory of stress and strain in rods and shells, Archive for rational mechanics and analysis 1, 295-323 (1958) · Zbl 0081.39303 · doi:10.1007/BF00298012
[13] Eringen, A. C.: On nonlocal plasticity, International journal of engineering science 19, 1461-1474 (1981) · Zbl 0474.73028 · doi:10.1016/0020-7225(81)90072-0
[14] Eringen, A. C.: Theories of nonlocal plasticity, International journal of engineering science 21, 741-751 (1983) · Zbl 0519.73024 · doi:10.1016/0020-7225(83)90058-7
[15] Eringen, E. C.; Speziale, C. G.; Kim, B. S.: Crack-tip problem in nonlocal elasticity, Journal of the mechanics and physics of solids 25, 339-355 (1977) · Zbl 0375.73083 · doi:10.1016/0022-5096(77)90002-3
[16] Fleck, N. A.; Hutchinson, J. W.: A reformulation of strain gradient plasticity, Journal of the mechanics and physics of solids 49, 2245-2271 (2001) · Zbl 1033.74006 · doi:10.1016/S0022-5096(01)00049-7
[17] Halphen, B.; Nguyen, Q. S.: Sur LES matériaux standards géné ralisés, Journal de mécanique 14, 39-63 (1975) · Zbl 0308.73017
[18] Hiriart-Urruty, J. B.; Lemarechal, C.: Convex analysis and minimization algorithms I – II, (1993)
[19] Houlsby, G.T., 1996. Derivation of incremental stress – strain response for plasticity models based on thermodynamic functions. In: Fleck, N.A., Cocks, A.C.F. (Eds.), Proceedings of the International Union of Theoretical and Applied Mechanics (IUTAM). Symposium on Mechanics of Granular and Porous Materials, Cambridge, July 15 – 17, 1996. Kluwer Academic Publishers, pp. 161 – 172.
[20] Houlsby, G. T.; Puzrin, A. M.: A thermomechanical framework for constitutive models for rate-independent dissipative materials, International journal of plasticity 16, 1017-1047 (2000) · Zbl 0958.74011 · doi:10.1016/S0749-6419(99)00073-X
[21] Jirásek, M.; Rolshoven, S.: Comparison of integral-type nonlocal plasticity models for strain-softening materials, International journal of engineering science 41, 1553-1602 (2003) · Zbl 1211.74039 · doi:10.1016/S0020-7225(03)00027-2
[22] Kroner, E.: Elasticity theory of materials with long range cohesive forces, International journal of solids and structures 3, 731-742 (1967) · Zbl 0163.19402 · doi:10.1016/0020-7683(67)90049-2
[23] Lasry, D.; Belytschko, T.: Localization limiters and transient problems, International journal of solids and structures 24, 581-597 (1988) · Zbl 0636.73021 · doi:10.1016/0020-7683(88)90059-5
[24] Lemaite, J.; Chaboche, J. L.: Mechanics of solids materials, (1994)
[25] Lubliner, J.: Plasticity theory, (1990) · Zbl 0745.73006
[26] De Sciarra, F. Marotti: Nonlocal and gradient plasticity, International journal of solids and structures 41, 7329-7349 (2004) · Zbl 1076.74013
[27] Needleman, A.: Material rate dependence and mesh sensitivity in localization problems, Computer methods in applied mechanics and engineering 67, 69-86 (1988) · Zbl 0618.73054 · doi:10.1016/0045-7825(88)90069-2
[28] Nilsson, C.: Nonlocal strain softening bar revisited, International journal of solids and structures 34, 4399-4419 (1997) · Zbl 0942.74591 · doi:10.1016/S0020-7683(97)00019-X
[29] Nilsson, C.: Author’s closure, International journal of solids and structures 36, 3093-3100 (1999)
[30] Pijaudier-Cabot, G.; Bažant, Z. P.: Nonlocal damage theory, Journal of engineering mechanics, ASCE 113, 1512-1533 (1987)
[31] Planas, J.; Elices, M.; Guinea, G. V.: Cohesive cracks versus nonlocal models: closing the gap, International journal of fracture 63, 173-187 (1993)
[32] Polizzotto, C.; Borino, G.; Fuschi, P.: A thermodynamic consistent formulation of nonlocal and gradient plasticity, Mechanics research communications 25, 75-82 (1998) · Zbl 0923.73014 · doi:10.1016/S0093-6413(98)00009-3
[33] Rajagopal, K. R.; Srinivasa, A. R.: Mechanics of the inelastic behaviour of material – part 2, inelastic response, International journal of plasticity 14, 969-995 (1998) · Zbl 0978.74014 · doi:10.1016/S0749-6419(98)00041-2
[34] Reddy, B. D.; Martin, J. B.: Algorithms for the solution of internal variable problems in plasticity, Computer methods in applied mechanics and engineering 93, 253-273 (1991) · Zbl 0744.73023 · doi:10.1016/0045-7825(91)90154-X
[35] Romano, G.: New results in subdifferential calculus with applications to convex optimization, Applied mathematics and optimization 32, 213-234 (1995) · Zbl 0828.49015 · doi:10.1007/BF01187900
[36] Romano, G.; Rosati, L.; De Sciarra, F. Marotti: An internal variable theory of inelastic behaviour derived from the uniaxial rigid-perfectly plastic law, International journal of engineering science 31, 1105-1120 (1993) · Zbl 0781.73022 · doi:10.1016/0020-7225(93)90085-9
[37] Salençon, J.: Calcul à la rupture et analyse limite, (1992)
[38] Simo, J. C.: A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part I – continuum formulation, Computer methods in applied mechanics and engineering 66, 199-219 (1988) · Zbl 0611.73057 · doi:10.1016/0045-7825(88)90076-X
[39] Sluys, L. J.; De Borst, R.: Wave propagation and localization in rate dependent crack medium, model formulation and one dimensional examples, International journal of solids and structures 29, 2945-2958 (1992)
[40] Strömber, L.; Ristinmaa, M.: FE-formulation of a nonlocal plasticity theory, Computer methods in applied mechanics and engineering 136, 127-144 (1996) · Zbl 0918.73118 · doi:10.1016/0045-7825(96)00997-8
[41] Svedberg, T.; Runesson, K.: Thermodynamically consistent nonlocal and gradient formulations of plasticity, In nonlocal aspects in solid mechanics Proceedings of EUROMECH colloquium 378, 32-37 (1998) · Zbl 0948.74054
[42] Vermeer, P.A., Brinkgreve, R.B.J. 1994. A new effective non-local strain measure for softening plasticity. In: Chambon, R., Desrues, J., Vardoulakis I. (Eds.), Localization and Bifurcation Theory for Soils and Rocks, A.A. Balkema, Rotterdam, Aussois, France, September 6 – 9, 1993, pp. 89 – 100.
[43] Ziegler, H., 1983. An Introduction to Thermomechanics, second edition. North Holland, Amsterdam. · Zbl 0531.73080
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