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A Note on Rivlin’s Identities and Their Extension

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Abstract

An analytic method is developed for the derivation of a series of tensor identities that include Rivlin’s identities as a special case. The derivation is based on taking the derivatives of the Cayley–Hamilton equation. The identities generated involve multiple tensors on the n-dimensional vector space.

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References

  1. R.S. Rivlin, Further remarks on the stress-deformation relations for isotropic materials. J. Rational Mech. Anal. 4 (1955) 681–702.

    MathSciNet  Google Scholar 

  2. A. Hoger and D.E. Carlson, On the derivative of the square root of a tensor and Guo’s rate theorems. J. Elasticity 14 (1984) 329–336.

    MATH  MathSciNet  Google Scholar 

  3. M. Scheidler, The tensor equation AX+XA=Φ(A,H), with applications to kinematics of continua. J. Elasticity 36 (1994) 117–153.

    MATH  MathSciNet  Google Scholar 

  4. R.S. Rivlin and J.L. Ericksen, Stress-deformation relations for isotropic materials. J. Rational Mech. Anal. 4 (1955) 323–425.

    MathSciNet  Google Scholar 

  5. J.S. Lew, The generalized Cayley–Hamilton theorem in n dimensions. Zeitschrift für Angewandte Mathematik und Physik 17 (1966) 650–653.

    MATH  ADS  MathSciNet  Google Scholar 

  6. R.S. Rivlin and G.F. Smith, On identities for 3×3 matrices. Rendiconti di Matematica Serie VI 8 (1975) 345–353.

    MATH  MathSciNet  Google Scholar 

  7. F.G. Gantmakher, The Theory of Matrices. Chelsea, New York (1959).

    Google Scholar 

  8. C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics. Springer, Berlin (1992).

    MATH  Google Scholar 

  9. D.E. Carlson and A. Hoger, On the derivatives of the principal invariants of a second-order tensor. J. Elasticity 16 (1986) 221–224.

    MATH  MathSciNet  Google Scholar 

  10. Z.-H. Guo, Derivatives of the principal invariants of a 2nd-order tensor. J. Elasticity 22 (1989) 185–191.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Jaric, On the gradients of the principal invariants of a second order tensor. J. Elasticity 44 (1996) 285–287.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Houston, Houston, TX, 77204, U.S.A.

    Guansuo Dui & Yi-Chao Chen

Authors
  1. Guansuo Dui
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  2. Yi-Chao Chen
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Mathematics Subject Classifications (2000)

15A24, 74A99.

Guansuo Dui: Present address: Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing, 100044, P. R. China.

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Dui, G., Chen, YC. A Note on Rivlin’s Identities and Their Extension. J Elasticity 76, 107–112 (2004). https://doi.org/10.1007/s10659-004-5903-1

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  • DOI: https://doi.org/10.1007/s10659-004-5903-1

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