Abstract
The explicit computation of the effective elasticity tensor of the material produced by laminating two homogeneous elastic media is used to show that, in 2-dimensional and 3-dimensional linear elasticity, for any isotropic material a whose elasticity tensor is strongly elliptic, but not semipositive definite, we can select very strongly elliptic materials, so that through laminations between these with material a, we can create a nonstrongly elliptic media, whose existence contradicts properties concerning the propagation of elastic waves.
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References
J. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. I, R.J. Knops (ed.), Pitman Res. Notes Math. 17(1977) 187-241.
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag (1989).
G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag (1976).
G. Francfort and G. Milton, Sets of conductivity and elasticity tensors stable under lamination. Comm. Pure Appl. Math. 47(1994) 257-279.
G. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal. 94(1986) 307-334.
G. Geymonat, S. Müller and N. Triantafyllidis, Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Rational Mech. Anal. 122(1993) 231-290.
M. Gurtin, The linear theory of elasticity. Handbuch der Physik Vol. VIa/2. Springer-Verlag (1972).
S. Gutiérrez, Laminations in Linearized Elasticity and a Lusin Type Theorem for Sobolev Spaces. PhD thesis, Carnegie Mellon University (1997).
H. Le Dret, An example of H1-unboundedness of solutions to strongly elliptic systems of partial differential equations in a laminated geometry. Proc. Roy. Soc. Edinburgh A 105(1987) 77-82.
W.H. McConnell, On modeling the aggregate response of laminated thermoelastic media. J. Elasticity 7(1) (1977) 67-78.
J.O. Smith and O.M. Sidebottom, Elementary Mechanics of Deformable Bodies. The MacMillan Company (1969).
L. Tartar, Cours Peccot, Collège de France (1977).
L. Tartar, Homogenization, Compensated Compactness and H-Measures. CBMS-NSF Conference, Santa Cruz, June 1993. Lecture Notes in preparation.
L. Tartar, Homogenization and Hyperbolicity. Special issue of the Ann. Scuola Norm. Sup. Pisa Cl. Sci. in memory of Ennio de Giorgi (1997).
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Gutiérrez, S. Laminations in Linearized Elasticity: The Isotropic Non-very Strongly Elliptic Case. Journal of Elasticity 53, 215–256 (1998). https://doi.org/10.1023/A:1007670013167
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DOI: https://doi.org/10.1023/A:1007670013167