Isotropic Integrity Bases for Vectors and Second-Order Tensors

Part I
  • A. J. M. Spencer
  • R. S. Rivlin

Abstract

In previous papers [2, 3] it has been shown how an arbitrary matrix polynomial in any number of symmetric 3 × 3 matrices may be expressed in a canonical form. From these results an integrity basis under the orthogonal transformation group for an arbitrary number of symmetric 3 × 3 matrices has been derived. This consists of traces of products formed from the matrices which have total degree six or less in the matrices. In deriving these results a number of theorems were obtained which enabled us to express a product formed from any number of 3 × 3 matrices, whether symmetric or non-symmetric, as a sum of products of particular types formed from these matrices, with coefficients which are polynomials in traces of products formed from the matrices.

Keywords

aG111 Titan Tral 

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References

  1. [1]
    Pipkin, A. C., & R. S. Rivlin: The formulation of constitutive equations in continuum physics. I. Arch. Rational Mech. Anal. 4, 129 (1959).CrossRefGoogle Scholar
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    Spencer, A. J. M., & R. S. Rivlin: Further results in the theory of matrix polynomials. Arch. Rational Mech. Anal. 4, 214 (1960).CrossRefGoogle Scholar
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    Spencer, A. J. M., & R. S. Rivlix: The theory of matrix polynomials and its application to the mechanics of isotropic continua. Arch. Rational Mech. Anal. 2, 309 (1959).CrossRefGoogle Scholar
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    Green, A. E., R. S. Rivlin & A. J. M. Spencer: The mechanics of non-linear materials with memory. II. Arch. Rational Mech. Anal. 3, 82 (1959).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • A. J. M. Spencer
    • 1
    • 2
  • R. S. Rivlin
    • 1
    • 2
  1. 1.University of NottinghamUK
  2. 2.Brown UniversityProvidenceUSA

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