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Invariant Tensor-to-Matrix Mappings for Evaluation of Tensorial Expressions

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Abstract

A class of mappings is presented, parameterized by a variable η, that operates on tensorial expressions to yield equivalent matrical expressions which are then easily evaluated, either numerically or symbolically, using standard matrix operations. The tensorial expressions considered involve scalar, second- and fourth- order tensors, double contractions, inversion and transposition. Also addressed is coordinate transformation and eigenanalysis of fourth- order tensors. The class of mappings considered is invariant, meaning that for a given η the corresponding mapping depends only on the order of the tensor upon which it acts and not, for example, on its physical interpretation (e.g., stress vs. strain, or stiffness vs. compliance). As a result the proposed mappings avoid ad hoc definitions like that of engineering shear strain (i.e., γij:=2∈ij for i ≠ j) which is inconsistent with an invariant mapping. Two convenient choices for the parameter η are presented.

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References

  1. R. Hill, Continuum micro-mechanics of elastoplastic polycrystals. Journal of the Mechanics and Phyisics of Solids 13(1965) 89-101.

    Article  MATH  Google Scholar 

  2. W. Voigt, Lehrbuch der Kristallphysik. Teubner, Leipzig (1928).

    MATH  Google Scholar 

  3. R.F.S. Hearmon, An Introduction to Applied Anisotropic Elasticity. Oxford University Press (1961).

  4. S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco (1963).

    MATH  Google Scholar 

  5. J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices. Clarendon Press, Oxford (1985).

    Google Scholar 

  6. L.J. Walpole, Fourth-rank tensors of the thirty-two crystal classes: Multiplication tables. Proceedings of the Royal Society, London A391(1984) 149-179.

    MathSciNet  ADS  Google Scholar 

  7. M.M. Mehrabadi and S.C. Cowin, Eigentensors of linear anisotropic elastic materials. Quarterly Journal of Mechanics and Applied Mathematics 43(1) (1990) 15-41.

    MATH  MathSciNet  Google Scholar 

  8. T.T. Wu, The effect of inclusion shape on the elastic moduli of a two-phase material. International Journal of Solids and Structures 2(1966) 1-8.

    Article  Google Scholar 

  9. Y. Benveniste, A new approach to the application of Mori-Tanaka's theory in composite materials. Mechanics of Materials 6(1987) 147-157.

    Article  Google Scholar 

  10. J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Physical Society, London A241(1957) 276-303.

    MathSciNet  Google Scholar 

  11. T. Mura, Micromechanics of Defects in Solids. Kluwer Academic Publishers, Dordrecht, second revised edition, (1987).

    Google Scholar 

  12. M. Ferrari, and N. Marzari, AMori-Tanaka theory for textured, short-fiber composites application. In: D. Hui and T.J. Kozik (eds), Composite Material Technology 1989, volume PD-Vol. 24, pages 111-116, New York, 1989, ASME. Also, see Journal of Energy Resources Technology 114(1992) 101-104.

  13. J. Rychlewski, On Hooke's law. Prikl. Matem. Mekhan., 48(3) (1984) 420-435. Translation: Journal of Applied Mathematics and Mechanics 48(3) (1984) 303-314.

    MathSciNet  Google Scholar 

  14. W.L. Bond, The mathematics of the physical properties of crystals. The Bell System Technical Journal 22(1) (1943) 1-72.

    MathSciNet  Google Scholar 

  15. A.N. Norris, Effective moduli of layered media - A new look at an old problem. In: G.J. Weng, M. Taya, and H. Abé (eds), Micromechanics and Inhomogeneity. The Toshio Mura 65th Anniversary Volume, pages 321- 339. Springer-Verlag, New York (1990).

    Google Scholar 

  16. R.C. Reuter Jr, Concise property transformation relations for an anisotropic lamina. Journal of Composite Materials 5(1971) 270-272.

    Google Scholar 

  17. T.C.T. Ting, Invariants of anisotropic elastic constants. Quarterly Journal of Mechanics and Applied Mathematics 40(3) (1987) 431-448.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of California, Berkeley, CA, 94720-1710, U.S.A

    J.C. Nadeau & M. Ferrari

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  1. J.C. Nadeau
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  2. M. Ferrari
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Nadeau, J., Ferrari, M. Invariant Tensor-to-Matrix Mappings for Evaluation of Tensorial Expressions. Journal of Elasticity 52, 43–61 (1998). https://doi.org/10.1023/A:1007539929374

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