Keywords: large strain, plasticity, damage, effective stress.

If the plasticity without damage is well modelised in large strains [1], the plasticity in the presence of damage has not received a satisfactory framework. Moreover, many studies are restricted to small strains [2]. We recall that a general approach consists in the exclusive use of the Thermodynamics of Irreversible Processes [3,4]. But a more specific approach is based on the concept of effective stress, that introduces a fictitious elastoplastic material in order to compare the elasto-plastic damaged real material and this virtual undamaged one. The main aim of this paper is to give a coherent framework and to point out some limitations of this mechanical approach, by defining with accuracy notions such accompanying body, elastic strain correlation and effective stress.

Our first step. We define a referential body , accompanying the real body , as a virtual body used for mathematical supports only. This virtual body may possess some microstructure and so does not satisfy to Classical Continuum hypotheses, particularly the symmetry of the Cauchy stress tensor and the compatibility conditions. Furthermore, its definition may introduce some internal parameters unaware of its companion real body . Since we are concerned by local constitutive laws, it is sufficient to compare the local maps, and , by , [5]. However this new parameter cannot be an internal variable, since is related to both reversible and irreversible mechanical processes.

Our second step. To overcome this difficulty, the material elements are elastically unloaded [6] by , the respective elastic maps. So, the maps and are concerned by irreversible effects only. A comparison between the "intermediate configurations" and can be made by , as for and . A priori, can be viewed as an internal parameter connected with some irreversible microscopic changes only. Now since , we can write the decompositions of the deformation gradients and the velocity field as

also proposed by an other method [7]. Finally, as is a plastic term only, it is assumed that the internal parameter describes some permanent (under zero stress) geometric changes due to damaging effects [8].

Our third step. A comparison between the stress levels requires a rule of comparison between the elastic strains and , defined by an elastic strain correlation . Then the elimination of strains between the elastic laws and gives a relation between stresses noted . Reciprocally, if , and are known, we can obtain the real elastic law as

( is the Cauchy stress tensor). The fictitious stress is said the "effective stress" associate to the elastic strain correlation . The functions are often obtained by inverse methods and depend on internal damage parameters.

Our fourth step. If the functions are known, the transfer of the elastic range, i.e. loading surface and elastic law, is generally achieved. Concerning the evolution laws of plastic parameters, it turns out that the evolution laws depend on all parameters of the virtual body and some of them may be irrelevant of the actual problems. This presence is a theoretical limitation on the use of the method.

1

R. Souchet, "Leçons sur les grandes déformations", Cépaduès-Editions, Toulouse, France, 2001.

2

J. Lemaître, R. Desmorat, M. Sauzay, "Anisotropic damage law of evolution", Eur. J. Mech., A/Solids, 19, pp. 187-208, 2000. doi:10.1016/S0997-7538(00)00161-3

3

N.R. Hansen, H.L. Schreyer, "A thermodynamically consistent framework for theories of elastoplasticity coupled with damage", Int. J. Solids Structures, Vol.31, No 3, pp ; 359-389, 1994. doi:10.1016/0020-7683(94)90112-0

4

P. Longère, A. Dragon, H. Trumel, T. de Resseguier, X. Deprince, E. Petitpas, "Modèle de comportement dynamique d'un matériau en présence de l'endommagement par cisaillement adiabatique", J. Phys. IV France 12, pp. 327-334, 2002. doi:10.1051/jp4:20020509

5

R. Souchet, "Concerning the fictitious continuum in Damage Mechanics", Int. J. Engng Sciences, 41, (2003), 1975-1988. doi:10.1016/S0020-7225(03)00136-8

6

E.H. Lee, "Elastic-plastic deformation at finite strains", J. Applied Mechanics, 36, pp. 1-6, 1968.

7

G.Z. Voyiadjis, T. Park, "The kinematics of damage for finite-strain elasto-plastic solids", Int. J. Engng Sciences, 37, pp. 803-830, 1999. doi:10.1016/S0020-7225(98)00100-1

8

R. Souchet, "Strain correlation and effective stress in Damage Mechanics", AIMETA 2003, Proceedings of the 16th Congress of Italian Association of Theoretical and Applied Mechanics, Ferrara.

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