×

zbMATH — the first resource for mathematics

An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals. (English) Zbl 0984.74068
Summary: Variational approaches for nonlinear elasticity show that Hill’s incremental formulation for the prediction of the overall behaviour of heterogeneous materials yields estimates which are too stiff and may even violate rigorous bounds. This paper aims at proposing an alternative ‘affine’ formulation, based on a linear thermoelastic comparison medium, which could yield softer estimates. It is first described for nonlinear elasticity and specified by making use of Hashin-Shtrikman estimates for the linear comparison composite; the associated affine self-consistent predictions are satisfactorily compared with incremental and tangent ones for power-law creeping polycrystals. Comparison is then made with the second-order procedure, and some limitations of the affine method are pointed out; explicit comparisons between different procedures are performed for isotropic, two-phase materials. Finally, the affine formulation is extended to history-dependent behaviours; application to the self-consistent modelling of the elastoplastic behaviour of polycrystals shows that it offers an improved alternative to Hill’s incremental formulation.

MSC:
74Q15 Effective constitutive equations in solid mechanics
74E30 Composite and mixture properties
74A40 Random materials and composite materials
74Q20 Bounds on effective properties in solid mechanics
74E15 Crystalline structure
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berveiller, M.; Zaoui, A., An extension of the self-consistent scheme to plastically-flowing polycrystals, J. mech. phys. solids, 26, 325-344, (1979) · Zbl 0395.73033
[2] Buryachenko, V., The overall elastoplastic behavior of multiphase materials with isotropic components, Acta mechanica, 119, 93-117, (1996) · Zbl 0877.73040
[3] Gilormini, P., A critical evaluation of various nonlinear extensions of the self-consistent model, (), 67-74
[4] Hill, R., Continuum micro-mechanics of elastoplastic polycrystals, J. mech. phys. solids, 13, 89-101, (1965) · Zbl 0127.15302
[5] Hu, G., A method of plasticity for general aligned spheroidal void of fiber-reinforced composites, Int. J. plasticity, 12, 439-449, (1996) · Zbl 0884.73035
[6] Hutchinson, J.W., Elastic – plastic behaviour of polycrystalline metals and composites, Proc. R. soc. lond., A319, 247-272, (1970)
[7] Hutchinson, J.W., Bounds and self-consistent estimates for creep of polycrystalline materials, Proc. R. soc. lond., A348, 101-127, (1976) · Zbl 0319.73059
[8] Kröner, E., Zur plastischen verformung des vielkristalls, Acta metall., 9, 155-161, (1961)
[9] Lebensohn, R.; Tomé, C.N., A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys, Acta metall. mater., 41, 2611-2624, (1993)
[10] Levin, V.M., Thermal expansion coefficients of heterogeneous materials, Mekh. tverd. tela, 2, 8, 38-94, (1967)
[11] Masson, R.; Zaoui, A., Self-consistent estimates for the rate-dependent elastoplastic behaviour of polycrystalline materials, J. mech. phys. solids, 47, 1543-1568, (1999) · Zbl 0976.74010
[12] Molinari, A.; Canova, G.R.; Ahzi, S., A self-consistent approach of the large deformation polycrystal viscoplasticity, Acta metall., 35, 12, 2983-2994, (1987)
[13] Nebozhyn, M.V., Ponte Castañeda, P., 2000. The second-order procedure: corrected results for isotropic, two-phase composites. J. Mech. Phys. Solids, submitted
[14] Ponte Castañeda, P., The effective mechanical properties of nonlinear isotropic composites, J. mech. phys. solids, 39, 1, 45-71, (1991) · Zbl 0734.73052
[15] Ponte Castañeda, P., Exact second-order estimates for the effective mechanical properties of nonlinear composite materials, J. mech. phys. solids, 44, 6, 827-862, (1996) · Zbl 1054.74708
[16] Ponte Castañeda, P.; Suquet, P., Nonlinear composites, Adv. appl. mech., 34, 171-302, (1998) · Zbl 0889.73049
[17] Qiu, Y.P.; Weng, G.J., A theory of plasticity for porous materials and particle-reinforced composites, J. appl. mech., 59, 261-268, (1992) · Zbl 0825.73037
[18] Rougier, Y.; Stolz, C.; Zaoui, A., Self-consistent modelling of elastic-viscoplastic polycrystals, C. R. acad. sci. Paris, Série II, 318, 145-151, (1994) · Zbl 0787.73031
[19] Suquet, P., Overall potentials and extremal surfaces of power law or ideally plastic materials, J. mech. phys. solids, 41, 981-1002, (1993) · Zbl 0773.73063
[20] Suquet, P., Overall properties of nonlinear composites: a modified secant moduli theory and its link with ponte castañeda’s nonlinear variational procedure, C. R. acad. sci. Paris, Série II, 320, b, 563-571, (1995) · Zbl 0830.73046
[21] Suquet, P., Effective properties of nonlinear composites. continuum micromechanics, (), 197-264 · Zbl 0883.73051
[22] Suquet, P.; Ponte Castañeda, P., Small-contrast perturbation expansions for the effective properties of nonlinear composites, C. R. acad. sci. Paris, Série II, 317, 1515-1522, (1993) · Zbl 0844.73052
[23] Talbot, D.R.S.; Willis, J.R., Variational principles for inhomogeneous nonlinear media, IMA J. appl. math., 35, 39-54, (1985) · Zbl 0588.73025
[24] Tandon, G.P.; Weng, G.J., A theory of particle-reinforced plasticity, J. appl. mech., 55, 1, 126-135, (1988)
[25] Taylor, G.I., Plastic strain in metals, J. inst. metals, 62, 307-324, (1938)
[26] Willis, J.R., Bounds and self-consistent estimates for the overall moduli of anisotropic composites, J. mech. phys. solids, 25, 185-202, (1977) · Zbl 0363.73014
[27] Willis, J.R., Variational and related methods for the overall properties of composites, Adv. appl. mech., 21, 1-78, (1981) · Zbl 0476.73053
[28] Willis, J.R., The overall response of composite materials, J. appl. mech., 50, 1202-1209, (1983) · Zbl 0539.73003
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.