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Modeling the Behavior of Heterogeneous Materials with Non Linear Couplings using “Translated Fields”

Published online by Cambridge University Press:  01 February 2011

S. Berbenni ,
V. Favier  and
M. Berveiller
S. Berbenni
Affiliation:
Lab. Physics and Mechanics of Materials (LPMM), UMR CNRS 7554, ENSAM 4, rue Augustin Fresnel 57078 Metz Cedex 03, France
V. Favier
Affiliation:
Lab. Physics and Mechanics of Materials (LPMM), UMR CNRS 7554, ENSAM 4, rue Augustin Fresnel 57078 Metz Cedex 03, France
M. Berveiller
Affiliation:
Lab. Physics and Mechanics of Materials (LPMM), UMR CNRS 7554, ENSAM 4, rue Augustin Fresnel 57078 Metz Cedex 03, France
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Abstract

The determination of the behavior of heterogeneous materials with complex physical and mechanical couplings constitutes a challenge in the design of new materials and the modeling of their effective properties. In real inhomogeneous materials, the simultaneous presence of elastic mechanisms and non linear inelastic ones (viscoplastic, magnetic, ferroelectric, shape memory effect etc.) leads to a complex non linear coupling between the mechanical fields which is tricky to represent in a simple and efficient way. Hence, for many situations the effective global behavior does not follow the same structure than the local constitutive one. Regarding space-time couplings for instance, a heterogeneous material composed of phases described by Maxwell elements can not be considered as a Maxwellian solid at the macro scale.

In this paper, we introduce a new micro-macro approach based on translated fields in its generalized form to be applied to different coupled phenomena. The local total strain (rate) is composed additively of an elastic strain (rate) and an inelastic one which is no more limited to be “stress free” as considered originally by Kröner. An extended (non conventional) self-consistent model is then proposed starting from the integral equation for a translated strain (rate) field and using the projection operators algebra introduced by Kunin. The chosen translated field is the compatible inelastic strain (rate) of the fictitious inelastic heterogeneous medium submitted to a uniform unknown boundary condition. The self-consistency condition amounts to define analytically these boundary conditions so that a relative simple and compact strain (rate) concentration equation is obtained.

In order to illustrate the method, the case of a non linear elastic-viscoplastic coupling is developed and applied to different classes of composites and polycrystals.

Type
Research Article
Information
MRS Online Proceedings Library (OPL) , Volume 881: Symposium CC – Coupled Nonlinear Phenomena Modeling and Simulation for Smart, Ferroic and Multiferroic Materials , 2005 , CC1.9
Copyright
Copyright © Materials Research Society 2005

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References

1 Hershey, A.V., J. Appl. Mech. 21, 236–240 (1954).Google Scholar
2 Kröner, E., Z. Phys. 151, 504–518 (1958).Google Scholar
3 Kröner, E., J. Mech. Phys. Solids 15, 319–329 (1967).Google Scholar
4 Dederichs, P.H., Zeller, R., Z. Phys. 259, 103–116 (1973).Google Scholar
5 Hill, R., J. Mech. Phys. Solids 13, 89–101 (1965).Google Scholar
6 Hutchinson, J.W., Proc. R. Soc. London A348, 101–127 (1976).Google Scholar
7 Berveiller, M., Zaoui, A., J. Mech. Phys. Solids 26, 325–344 (1979).Google Scholar
8 Molinari, A., Canova, G.R., Ahzi, S., Acta Metall. 35, 2983–2994 (1987).Google Scholar
9 Lebensohn, R.A., Tomé, C.N., Acta Metall. Mater. 41 (9), 2611–2624 (1993).Google Scholar
10 Masson, R., Zaoui, A., J. Mech. Phys. Solids 47, 1543–1568 (1999).Google Scholar
11 Suquet, P., in: Sanchez-Palencia, E., Zaoui, A. (eds.), Homogenization techniques for composite media. Springer, Berlin, pp. 193–278 (1985).Google Scholar
12 Weng, G.J., J. Appl. Mech. 48, 779–784 (1981).Google Scholar
13 Kröner, E., Acta Metall. 9, 155–161 (1961).Google Scholar
14 Budiansky, B., Wu, T.T., in: Proc. 4th U.S. Nat. Congr. Appl. Mech. ASME, pp. 1175–1185 (1962).Google Scholar
15 Li, J., Weng, G.J., J. Mech. Phys. Solids 45, 1069–1083 (1997).Google Scholar
16 Rougier, Y., Stolz, C., Zaoui, A., Acad, C.R.. Sci. Paris 318, 145–151 (1994).Google Scholar
17 Hashin, Z., Int. J. Engng. Sci. 7, 11–36 (1969).Google Scholar
18 Molinari, A., Ahzi, S., Kouddane, R., Mech. Mater. 26, 43–62 (1997).Google Scholar
19 Paquin, A., Sabar, H., Berveiller, M., Arch. Appl. Mech. 69, 14–35 (1999).Google Scholar
20 Sabar, H., Berveiller, M., Favier, V., Berbenni, S., Int. J. Solids Struct. 39, 3257–3276 (2002).Google Scholar
21 Berbenni, S., Favier, V., Lemoine, X., Berveiller, M., Mat. Sci. Engng. A372, 128–136 (2004).Google Scholar
22 Kunin, I.A., Int. J. Engng. Sci. 19 (12), 1551–1561 (1981).Google Scholar
23 Kröner, E., in: Gittus, E., Zarka, E. (eds.), Modeling small deformation of polycrystals. Elsevier, London, pp. 229–292 (1986).Google Scholar
24 Kröner, E., in: Weng, G.J., Taya, M., Abe, M. (eds.), Micromechanics and Inhomogeneity. Springer, Berlin, pp. 197–211 (1989).Google Scholar
25 Eshelby, J.D., Proc. R. Soc. London A241, 376–396 (1957).Google Scholar
26 Taylor, G.I., J. Inst. Metals 61, 307–324 (1938).Google Scholar
27 Reuss, A., Angew, Z.. Math. Mech. 9, 49–58 (1929).Google Scholar