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Homogenization and global responses of inhomogeneous spherical nonlinear elastic shells. (English) Zbl 1094.74046

Summary: Homogenization of radially inhomogeneous spherical nonlinear elastic shells subject to internal pressure is studied. The equivalent homogeneous material is defined in such a way that it gives rise to exactly the same global response to the pressure load as that of the inhomogeneous shell. For a shell with general strain-energy function and inhomogeneity, the strain-energy function of the equivalent homogeneous material is determined explicitly. The resulting formula is used to study layered composite shells. The equivalent homogeneous material for an infinitely fine layered composite shell is examined, and is found to give not only the same global response, but also the same average stress field as the composite shell does.

MSC:

74Q15 Effective constitutive equations in solid mechanics
74K25 Shells
74E30 Composite and mixture properties
74B20 Nonlinear elasticity
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[1] R.M. Christensen, Mechanics of Composite Materials. Wiley Interscience (1979). · Zbl 0393.46049
[2] J.D. Eshelby, Elastic inclusions and inhomogeneities. In: I.N. Sneddon and R. Hill (eds.), Progress in Solid Mechanics, Vol. 2. North-Holland, Amsterdam (1961) pp. 89–140. · Zbl 0097.17602
[3] G. Franfort and F. Murat, Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal. 94 (1986) 307–334. · Zbl 0604.73013
[4] Z. Hashin, Analysis of composite materials – A survey. J. Appl. Mech. 50 (1983) 481–505. · Zbl 0542.73092
[5] Z. Hashin, Large isotropic elastic deformation of composites and porous media. Int. J. Solids Struct. 21 (1985) 711–720. · Zbl 0575.73051
[6] R. Hill, Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids 11 (1963) 357–372. · Zbl 0114.15804
[7] R. Hill, The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15 (1967) 79–95.
[8] R. Hill, On constitutive macro-variables for heterogeneous solids at finite strain. Proc. R. Soc. Lond. A 326 (1972) 131–147. · Zbl 0229.73004
[9] R.V. Kohn, Recent progress in the mathematical modelling of composite materials. In: G. Sih et al. (eds.), Composite Material Response: Constitutive Relations and Damage Mechanisms. Elsevier (1988) pp. 155–176.
[10] L. Lahellec, F. Mazerolle and J.C. Michel, Second-order estimate of the macroscopic behavior of periodic hyperelastic composites: Theory and experimental validation. J. Mech. Phys. Solids 52 (2004) 27–49. · Zbl 1074.74048
[11] G.W. Milton, The theory of composites. Cambridge University Press (2002). · Zbl 0993.74002
[12] G.W. Milton and R.V. Kohn, Variational bounds of the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36 (1988) 597–629. · Zbl 0672.73012
[13] S. Müller, Homogenization of nonconvex integral functions and cellular elastic materials. Arch. Rational Mech. Anal. 99 (1987) 189–211.
[14] S. Nemat-Nasser and M. Hori, Micromechanics: Overall Properties of Heterogeneous Solids. Elsevier, Amsterdam (1993). · Zbl 0924.73006
[15] R.W. Ogden, Extremum principles in non-linear elasticity and their application to composites – I Theory. Int. J. Solids Struct. 14 (1978) 265–282. · Zbl 0384.73022
[16] R.W. Ogden, Non-linear elastic deformations. Dover Publications (1997). · Zbl 0938.74014
[17] P. Ponte Castañeda, The overall constitutive behaviour of nonlinear elastic composites. Proc. R. Soc. Lind. A 422 (1989) 147–171. · Zbl 0673.73005
[18] P. Ponte Castañeda, Nonlinear composite materials: Effective constitutive behavior and microstructure evolution. In: P. Suquet (ed.), Continuum Micromechanics, CISM Courses and Lecture Notes No. 377. Springer (1997) pp. 131–195. · Zbl 0883.73050
[19] P. Ponte Castañeda and P. Suquet, Nonlinear composites. Adv. Appl. Mech. 34 (1998) 171–302. · Zbl 0889.73049
[20] P. Ponte Castañeda and E. Tiberio, A second-order homogenization method in finite elasticity and applications to black-filled elastomers. J. Mech. Phys. Solids 48 (2000) 1389–1411. · Zbl 0984.74070
[21] P. Ponte Castañeda and J.R. Willis, On the overall properties of nonlinearly viscous composites. Proc. R. Soc. Lond. A 416 (1988) 217–244. · Zbl 0635.73006
[22] M. Sahimi, Heterogeneous materials I, linear transport and optical properties. Springer (2003a). · Zbl 1028.74001
[23] M. Sahimi, Heterogeneous materials II, nonlinear and breakdown properties and atomistic modeling. Springer (2003b). · Zbl 1028.74002
[24] U. Saravanan and K.R. Rajagopal, On the role of inhomogeneities in the deformation of elastic bodies. Math. Mech. Solids 8 (2003) 349–376. · Zbl 1059.74015
[25] U. Saravanan and K.R. Rajagopal, A comparison of the response of isotropic inhomogeneous elastic cylindrical and spherical shells and their homogenized counterparts. J. Elast. (2004), in press.
[26] U. Saravanan and K.R. Rajagopal, Inflation, extension, torsion and shearing of an inhomogeneous compressible right circular annular cylinder. Math. Mech. Solids (2005), in press. · Zbl 1134.74014
[27] P. Suquet, Effective properties of nonlinear composites. In: P. Suquet (ed.), Continuum Micromechanics, CISM Courses and Lecture Notes No. 377. Springer (1997) pp. 197–264. · Zbl 0883.73051
[28] D.R.S. Talbot and J.R. Willis, Variational principles for inhomogeneous non-linear media. J. Appl. Math. 35 (1985) 39–54. · Zbl 0588.73025
[29] J.R. Willis, Variational and related methods for the overall properties of composite materials. In: C.-S. Yih (ed.), Advances in Applied Mechanics, Vol. 21 (1981) 2–78. · Zbl 0476.73053
[30] J.R. Willis, Elasticity theory of composites. In: H.G. Hopkins and M.J. Sewell (eds.), Mechanics of Solids. Pergamon, Oxford (1982) pp. 653–686. · Zbl 0508.73052
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