Keywords: non-associated, plasticity, numerical implementation, bipotential, return mapping, ABAQUS.

One the key features of the developments in non-linear finite elements relates to numerical implementation of constitutive laws and thus several theories were developed. The "return mapping" [4] and the "bipotential" [2] methods are one of those, associated respectively to two different classes of materials: the general standard materials (GSM) for the return mapping and the implicit standard materials (ISM) for the bipotential.

The originality of this paper is the complete comparison of the numerical implementation of those both methods in the case of non associated flow rules in plasticity. The particular case of the Armstrong-Frederick's hardening law is considered [1].

In plasticity, the class of general standard material admits a normal dissipation law, associated to plastic potential F. In the framework of associated flow rules, the plastic potential F is known and is equal to the yield criteria f (von Mises,Tresca for example). However, within the framework of non-associated flow rules, the plastic potential F is not equivalent to the yield criteria f, and a certain shape of the plastic potential is postulated.

Thus, a new class of material was introduced by de Saxcé [3], the implicit standard materials, to distort the problem of the postulate in the case of non-associated theories in the GSM. The aim of this approach is to connect by an implicit normality law the plastic strain rate and the stresses. On the one hand, this class corresponds to GSM in the associated flow rule case. On the other hand, in the non-associated case, this class admits a bipotential b, based on a generalization of the Fenchel's inequality. It is a function depending on both stress and plastic strain rate. The unique knowledge of this function b allows simultaneously to define the yield locus and the flow rule although they are not associated.

The first step concerns the determination of the plastic increment. This determination by the bipotential method implies to start from the incremental form of this one associated with a stationary condition. This method results in the same implicit equation as the one obtained with the radial return technique. The second step concerns the determination of the consistent tangent operator obtained by a simple derivation of the stress tensor increment in relation to the strain increment for the return mapping method. In the case of a bipotential, this results in an implicit function and to the use of the implicit function theorem. The shapes of these two consistent tangent operators are different with additionnal terms in the case of the bipotential method.

The numerical implementation is realised in ABAQUS/Standard 6.5 by the means of a UMat subroutine. The practical simple tension-compression case provides similar results in terms of axial stress versus axial strain for both methods, in accordance with the theory. Other loadings have now to be studied in order to improve both methods.

1

P. J. Armstrong and C. O. Frederick, "A mathematical representation of the multiaxial baushinger effect", Note RD/B/N731, Berkeley Nuclear Laboratories, 1966.

2

G. de Saxcé, "Une généralisation de l'inégalité de Fenchel et ses applications aux lois constitutives", C.R. Ac. des Sciences - série II, vol. 314, pp. 125-129, 1992.

3

G. de Saxcé and Z. Q. Feng, "The bipotential method : a constructive approach to design the complete contact law with friction and improved numerical algorithm", Math. Comput. Modelling, vol. 28 (4-8), pp. 225-245, 1998. doi:10.1016/S0895-7177(98)00119-8

4

J. C. Simo and T. J. R. Hughes, "Computational inelasticity", Springer-Verlag, New-York, 1998.

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