zbMATH — the first resource for mathematics

Self-consistent estimates for the rate-dependent elastoplastic behaviour of polycrystalline materials. (English) Zbl 0976.74010
The paper deals with a new Hill-type formulation for rate-dependent elasto-plasticity. Recalling Kröner’s vs. Hill’s versions of nonlinear self-consistent model, the authors conclude that for the rate-dependent elasto-plasticity Kröner’s approach is inadequate and that Hill’s concept would be more appropriate in the rate-dependent case. Essential features of Hill’s approach to the self-consistent modelling of nonlinear heterogeneous materials are the reduction of nonlinear homogenization procedure to a quasi-linear one through an adequate linearization process, and the derivation of self-consistent concentration equations for the solution of Eshelby-type problems. An alternative linearization method at the local level is adopted as a consistent affine formulation, in the specific context of rate-dependent elasto-plasticity. At each point of the representative volume element of a multiphase material and at any time instant, the authors derive the inelastic strain rate from a potential depending on internal parameters which are described by evolution equations. According to the proposed affine approach, for sufficiently large time, the authors derive the linearized constitutive equations for the total strain rate and for the rate of internal parameters. Using the solution of these differential equations for internal variables, simpler linearized constitutive equations are derived within the framework of rate-dependent elasto-plasticity. At any time a linear viscoplastic homogenization problem, in the presence of eigenstrains, has to be solved. By use of the correspondence principle, i.e. the Laplace-Carson transform, the problem is reduced to an elastic one. The authors put into evidence the following difficulties: a) due to the affine approach, the whole nonlinear problem has an implicit nature; b) due to the viscoelastic coupling, local and global responses need to be known not only at a fixed time, but also at any previous time. The authors undertake the following solution steps: first, they discretize the whole time interval; second, they elaborate an iterative scheme to simultaneously determine the associated unknown variables (stresses and internal parameters derived from the affine model) defined at intermediate times, in such a way that the variables satisfy the concentration and constitutive equations, for some given macroscopic loading path, at any intermediate time. The predictions of the proposed theory for typical mechanical response of rate-dependent elasto-plastic face-centred cubic polycrystals are analyzed and compared with the corresponding predictions from Taylor’s and Kröner-Weng’s model.

74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
74E15 Crystalline structure
Full Text: DOI
[1] Berveiller, M., Zaoui, A., 1979. An extension of the self-consistent scheme to plastically flowingpolycrystals. J. Mech. Phys. Solids 26, 325-344. · Zbl 0395.73033
[2] Beurthey, S., 1997. Modélisations du comportement dalliages de polymères. Ph.D. thesis, ÉcolePolytechnique, France.
[3] Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion and relatedproblems. Proc. Roy. Soc. London A241, 376-396. · Zbl 0079.39606
[4] Gilormini, P., 1995. Insuffisance de lextension classique du modèle autocohérent au comportementnon linéaire. C.R. Acad. Sci. Paris 320 II, 115-122.
[5] Harren, S.V., 1991. The finite deformation of rate dependent polycrystals I and II. J. Mech. Phys.Solids 39, 345-383. · Zbl 0734.73035
[6] Hershey, A.V., 1954. The elasticity of an isotropic aggregate of anisotropic cubic crystals. A.S.M.E.J. Appl. Mech. 21, 236-240. · Zbl 0059.17604
[7] Hill, R., 1965. Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 13,89-101. · Zbl 0127.15302
[8] Hutchinson, J.W., 1970. Elastic – plastic behaviour of polycrystalline metals and composites. Proc.Roy. Soc. London A319, 247-272.
[9] Hutchinson, J.W., 1976. Bounds and self-consistent estimates for creep of polycrystalline metals.Proc. Roy. Soc. London A348, 101-127. · Zbl 0319.73059
[10] Iwakuma, T., Nemat-Nasser, S., 1984. Finite elastic – plastic deformation of polycrystalline metals.Proc. Roy. Soc. London A394, 87-119. · Zbl 0549.73029
[11] Kröner, E., 1958. Berechnung der elastischen Konstaten des Vielkristalls aus den Konstaten desEinkristalls. Z. Phys. 151, 504-518.
[12] Kröner, E., 1961. Zur plastischen Verformung des Vielkristalls. Acta. Metall. Mater. 9, 155-161.
[13] Laws, N., McLaughlin, R., 1978. Self-consistent estimates for the viscoelastic creep compliance ofcomposite materials. Proc. Roy. Soc. London A359, 251-273.
[14] Lebensohn, R., Tome, C.N., 1993. A self-consistent anisotropic approach for the simulation ofplastic deformation and texture development of polycrystals: application to zirconium alloys. Acta.Metall. Mater. 41, 2611-2624.
[15] Lipinski, P., Krier, J., Berveiller, M., 1990. Élastoplasticité des métaux en grandes déformations:comportement global et évolution de la structure interne. Rev. Phys. Appl. 25, 361-388.
[16] Masson, R., 1998. Estimations non linéaires du comportement global de matériaux hétérogènesen formulation affine. Ph.D. thesis, École Polytechnique, France.
[17] Masson, R., Zaoui, A., 1997. From rate-dependent to rate independent self-consistent modeling ofelastoplastic multiphase materials. In: Khan, A.S. (Ed.) , Physics and Mechanics of Finite Plasticand Viscoplastic Deformation. Neat Press, Maryland, U.S.A., pp. 209-210.
[18] Molinari, A., Canova, G.R., Ahzi, S., 1987. A self-consistent approach of the large deformationpolycrystals viscoplasticity. Acta. Metall. Mater. 35, 2983-2994.
[19] Navidi, P., Rougier, Y., Zaoui, A., 1996. Self-consistent modelling of elastic – viscoplastic multiphasematerials. In: Pineau, A., Zaoui, A. (Eds.) , Micromechanics of Plasticity and Damage of MultiphaseMaterials. Kluwer Academic, Dordrecht, pp. 123-130.
[20] Nemat-Nasser, S., Obata, M., 1986. Rate-dependent finite elastoplastic deformation of polycrystals.Proc. Roy. Soc. London A497, 343-375.
[21] Ponte Castañeda, P., 1991. The effective mechanical properties of nonlinear composite materials.J. Mech. Phys. Solids 39, 45-71.
[22] Ponte Castañeda, P., 1996. Exact second order estimates for the effective mechanical properties ofnonlinear composite materials. J. Mech. Phys. Solids 44, 827-862. · Zbl 1054.74708
[23] Rougier, Y., Stolz, C., Zaoui, A., 1994. Self-consistent modelling of elastic – viscoplasticpolycrystals. C.R. Acad. Sci. Paris 381 II, 145-151. · Zbl 0787.73031
[24] Suquet, P., 1995. Overall properties of nonlinear composites: a modified secant moduli theory andits link with Ponte Castañedas nonlinear variational procedure. C.R. Acad. Sci. Paris 320 II,563-571. · Zbl 0830.73046
[25] Taylor, G.I., 1938. Plastic strain in metals. J. Inst. Metals 62, 307-324.
[26] Turner, P.A., Tome, C.N., 1993. Self-consistent modeling of viscoelastic polycrystals: applicationto irradiation creep and growth. J. Mech. Phys. Solids 41, 1191-1211. · Zbl 0775.73103
[27] Weng, G.J., 1981. Self-consistent determination of time-dependent behaviour of metals. J. Appl.Mech. 48, 41-46. · Zbl 0473.73043
[28] Weng, G.J., 1993. A self-consistent relation for the time-dependent creep of polycrystals. Int. J.Plast. 9, 181-198. · Zbl 0783.73025
[29] Zaoui, A., 1997. Structural morphology and constitutive behaviour of micro-heterogeneousmaterials. In: Suquet, P. (Ed.) , Continuum Micromechanics, CISM Lecture Notes. Springer Verlag,Berlin, pp. 291-347. · Zbl 0882.73046
[30] Zaoui, A., Masson, R., 1998. Micromechanics-based modeling of plastic polycrystals: an affineformulation. In: NSF-IMM Symposium on Micromechanical Modeling of Industrial Materials.Seattle, July 1998, to appear.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.