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Elastic constants and their admissible values for incompressible and slightly compressible anisotropic materials. (English) Zbl 1027.74003
Summary: Constitutive relations for incompressible (slightly compressible) anisotropic materials cannot (or could hardly) be obtained through the inversion of generalized Hooke’s law since the corresponding compliance tensor becomes singular (ill-conditioned) in this case. This is due to the fact that the incompressibility (slight compressibility) condition imposes some additional constraints on elastic constants. The problem requires a special procedure discussed in the present paper. The idea of this procedure is based on the spectral decomposition of compliance tensor, but leads to a closed formula for the elasticity tensor without explicit using the eigenvalue problem solution. The condition of nonnegative (positive) definiteness of material tensors restricts the elastic constants to belong to an admissible value domain. For orthotropic and transversely isotropic incompressible as well as isotropically compressible materials, the corresponding domains are illustrated graphically.

74B05 Classical linear elasticity
74E10 Anisotropy in solid mechanics
Full Text: DOI
[1] Spencer, A. J. M.: Constitutive theory for strongly anisotropic solids. In: Continuum theory of the methanics of fibre-reinforced composites. (Spencer, A. J. M., ed.), Wien New York: Springer 1984
[2] Chadwick, P.: Wave propagation in incompressible transversely isotropic elastic media I. Homogeneous plane waves. Proc. R. Ir. Acad.93A (2), 231-253 (1993). · Zbl 0788.73024
[3] Loredo, A., Kl?cker, H.: Generalized inverse of the compliance tensor, and behaviour of incompressible anisotropic materials-application to damage. Mech. Res. Commun.24, 371-376 (1997). · Zbl 0900.73051 · doi:10.1016/S0093-6413(97)00038-4
[4] Theocaris, P. S., Sokolis, D. P.: Spectral decomposition of the compliance fourth-rank tensor for orthotropic materials. Arch. Appl. Mech.70, 289-306 (2000). · Zbl 0993.74013 · doi:10.1007/s004199900066
[5] R?ter, M., Stein, E.: Analysis, finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity. Comp. Meth. Appl. Mech. Engg.190, 519-541 (2000). · Zbl 0969.74067 · doi:10.1016/S0045-7825(99)00286-8
[6] Lempriere, B. M.: Poisson’s ratio in orthotropic materials. AIAA Journal,6, 2226-2227 (1968). · Zbl 0313.76009 · doi:10.2514/3.4974
[7] Jones, E. M.: Mechanics of composite materials. New York: McGraw-Hill 1975.
[8] Itskov, M.: On the theory of fourth-order tensors and their application in computational mechanics. Comp. Meth. Appl. Mech. Engg.189 (2), 419-438 (2000). · Zbl 0980.74006 · doi:10.1016/S0045-7825(99)00472-7
[9] Mehrabadi, M. M., Cowin, S. C.: Eigentensors of linear anisotropic elastic materials. Quart. J. Mech. Appl. Math.43, 15-41 (1990). · Zbl 0698.73002 · doi:10.1093/qjmam/43.1.15
[10] Thomson, W. K. (Lord Kelvin): Elements of a mathematical theory of elasticity. Phil., Trans. R. Soc.166, 481-498 (1856). · doi:10.1098/rstl.1856.0022
[11] Pipkin, A. C.: Constraints in linearly elastic materials. J. Elast.6 (2), 179-193 (1976). · Zbl 0351.73010 · doi:10.1007/BF00041785
[12] Bertram, A.: An introduction of internal constraints in a natural way. ZAMM60, T100-T101 (1980).
[13] Campbell, S. L., Meyer, C. D.: Generalized inverses of linear transformations. New York: Dover 1979. · Zbl 0417.15002
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