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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

Matrix Product Symmetric Matrice Orthogonal Group Matrix Polynomial Outer Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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
  2. [2]
    Spencer, A. J. M., & R. S. Rivlin: Further results in the theory of matrix polynomials. Arch. Rational Mech. Anal. 4, 214 (1960).CrossRefGoogle Scholar
  3. [3]
    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
  4. [4]
    Rivlin, R. S.: Further remarks on the theory of matrix polynomials. J. Rational Mech. Anal. 4, 681 (1955).Google Scholar
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    Rivlin, R. S.: The formulation of constitutive equations in continuum physics. II. Arch. Rational Mech. Anal. 4, 262 (1960).CrossRefGoogle Scholar
  6. [6]
    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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