Keywords: elastoplastic constitutive equation, plastic flow rule, non-associate, nose.

The phenomenological elastoplastic constitutive equation based on a plastic flow rule requires both plastic potential function and yield function. If the plastic potential function different from the yield function is adopted, the model falls within the framework of the non-associated plasticity. In particular for materials with frictional characteristics like sand, rock, and concrete, the yield surface and the plastic potential are often chosen to be different to be able to simulate well the deformation behaviour, whilst the elastoplastic stiffness matrix of the model becomes non-symmetric [1,2,3].

Past esearch elaborates the constitutive model under the non-proportional loading behaviour and the response envelopes of the stress rate-stretching in the stress rate-stretching space has been considered clarifying the mechanical properties of the models [4,5,6,7,8]. It has been pointed out that the response envelopes of the models adopting a non-associate flow rule exhibit a peculiar nose, i.e. non-convexity of the response envelope, to some particular directions of stretching [5,9,10], whilst some noses are observed for the models adopting an associated flow rule [4]. The existence of a nose in the stress rate space means that the material in an elastoplastic state behaves stiffer than in an elastic state to certain stretching directions.

Although its occurrence in the stress rate response envelope has been miss-understandably explained as the result of the adoption of the non-associated flow rule, the authors [8,11] explained its occurrence as the natural property of the model adopting the pressure dependent yield function.

On the other hand, it has been clarified that the elastoplastic constitutive model adopting the non-associated plastic flow does not fulfil the condition for the plastic relaxation of the second work rate or stiffness moduli, and looses positive definiteness of the elastoplastic matrix in the hardening regime, i.e. the so-called Drucker's postulate on stability [1,3].

This article focuses on not only the nose observed in the stress rate response envelope but also on the stability condition in a sense of Drucker's postulate [1], the fundamental features on the mechanical properties of the models adopting the associated and the non-associated flow rule are examined for the pressure dependent yielding and plastic flow materials, in both hardening and softening states under the triaxial condition.

1

Drucker, D.C., "A definition of a stable inelastic material", J. Appl. Mech. Trans. ASME, Vol. 26, pp. 101-106, 1959.

2

Runesson, K. and Mroz, Z., "A note on nonassociated plasticity flow rules", Int. J. Plasticity, Vol. 5, pp. 639-658, 1989. doi:10.1016/0749-6419(89)90005-3

3

Raniecki, B. and Bruhns, O.T., "Bounds to bifurcation stresses in solids with non-associated flow law at finite strain", J. Mech. Phys. Solids, 29, pp. 153-172, 1981. doi:10.1016/0022-5096(81)90021-1

4

Gudehus G., "A comparison of some constitutive laws for soils under radially symmetric loading and unloading", Proc. 3rd Int. Conf. Numer. Meth. Geomech. Aachen, Vol. 4, pp. 1309-1323, 1979.

5

Vermeer, P.A., "A five-constant model unifying well-established concepts", Results of Int. Workshop on Constitutive Relations for Soils, Grenoble, pp. 175-197, 1982.

6

Royis, P. and Doanh, T., "Theoretical analysis of strain response envelops using incrementally non-linear constitutive equations", Int. J. Numer. Anal. Meth. Geomech. Vol. 22, pp. 97-132, 1998. doi:10.1002/(SICI)1096-9853(199802)22:2<97::AID-NAG911>3.0.CO;2-Z

7

Chambon, R., Caillerie, D., Desrues, J. and Crochepeyre, S., "A comparison of incremental behaviour of elastoplastic and CloE models", Int. J. Numer. Anal. Meth. Geomech. Vol. 235, pp. 295-316, 1999. doi:10.1002/(SICI)1096-9853(19990410)23:4<295::AID-NAG970>3.3.CO;2-I

8

Tsutsumi, S., Hashiguchi, K., Okayasu, Y., Saitoh, K. and Sugimoto, M., "Mechanical response of subloading surface model with tangential plasticity", J. Appl. Mech., JSCE, Vol.4, pp. 375-382, 2001.

9

Molenkamp F. and Ommen A.V., "Peculiarity of non-associativity in plasticity of soil mechanics", Int. J. Numer. Anal. Meth. Geomech. Vol. 11, pp. 659-661, 1987. doi:10.1002/nag.1610110611

10

Hashiguchi K., "Inexpediency of the non-associated flow rule", Int. J. Numer. Anal. Meth. Geomech. Vol. 15, pp. 753-756, 1991. doi:10.1002/nag.1610151005

11

Tsutsumi, S. and Hashiguchi, K., "General non-proportional loading behavior of soils", Int. J. Plasticity, Vol. 21(10), pp. 1941-1969, 2005. doi:10.1016/j.ijplas.2005.01.001

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