Applied Mathematics and Mechanics

, Volume 28, Issue 9, pp 1249–1257 | Cite as

Basis-free expressions for derivatives of a subclass of nonsymmetric isotropic tensor functions

  • Wang Zhi-qiao  (王志乔)
  • Dui Guan-suo  (兑关锁)Email author


The present paper generalizes the method for solving the derivatives of symmetric isotropic tensor-valued functions proposed by Dui and Chen (2004) to a subclass of nonsymmetric tensor functions satisfying the commutative condition. This subclass of tensor functions is more general than those investigated by the existing methods. In the case of three distinct eigenvalues, the commutativity makes it possible to introduce two scalar functions, which will be used to construct the general nonsymmetric tensor functions and their derivatives. In the cases of repeated eigenvalues, the results are acquired by taking limits.

Key words

nonsymmetric tensor derivative of tensor function scalar function fourth-order tensor 

Chinese Library Classification

O331 O183.2 

2000 Mathematics Subject Classification

74A20 74C15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Dui G S, Chen Y C. Basis-free representations for the stress rate of isotopic materials[J]. Int J Solids Struct, 2004, 41:4845–4860.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Miehe C. Exponential map algorithm for stress updates in anisotropic multiplicative elastoplasticity for single crystals[J]. Int J Num Meth Eng, 1996, 39:3367–3390.zbMATHCrossRefGoogle Scholar
  3. [3]
    Sansour C, Kollmann F G. Large viscoplastic deformations of shells. Theory and finite element formulation[J]. Comput Mech, 1998, 21:512–525.zbMATHCrossRefGoogle Scholar
  4. [4]
    Steinmann P, Stein E. On the numerical treatment and analysis of finite deformation ductile single crystal plasticity[J]. Comput Methods Appl Meth Engrg, 1996, 129:235–254.zbMATHCrossRefGoogle Scholar
  5. [5]
    Balendran B, Nemat-Nasser S. Derivative of a function of a nonsymmetric second-order tensor[J]. Quart Appl Math, 1996, 54:583–600.zbMATHMathSciNetGoogle Scholar
  6. [6]
    De Souza Neto E A. The exact derivative of the exponential of a nonsymmetric tensor[J]. Comput Methods Appl Meth Engrg, 2001, 190:2377–2383.zbMATHCrossRefGoogle Scholar
  7. [7]
    Itskov M, Aksel N. A closed-form representation for the derivative of non-symmetric tensor power series[J]. Int J Solids Struct, 2002, 39:5963–5978.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Itskov M. Application of the Dunford-Taylor integral to isotropic tensor functions and their derivatives[J]. Proc R Soc London A, 2003, 459:1449–1457.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Itskov M. Computation of the exponential and other isotropic tensor functions and their derivatives[J]. Comput Methods Appl Meth Engrg, 2003, 192:3985–3999.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Lu J. Exact expansions of arbitrary tensor functions F(A) and their derivatives[J]. Int J Solids Struct, 2004, 41:337–349.zbMATHCrossRefGoogle Scholar
  11. [11]
    Dui G S. Discussion on ‘Exact expansions of arbitrary tensor functions F(A) and their derivatives’[J]. Int J Solids Struct, 2005, 42:4514–4515.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    Dui G S, Ren Q, Shen Z. Time rates of Hill’s strain tensors[J]. J Elast, 1999, 54:129–140.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Wang Z Q, Dui G S. On the derivatives of a subclass of tensor functions of a nonsymmetric tensor[J]. Int J Solids Struct, 2007 (in Press).Google Scholar
  14. [14]
    Del Piero G. Some properties of the set of 4th-order tensors, with application to elasticity[J]. J Elast, 1979, 9:245–261.zbMATHCrossRefGoogle Scholar
  15. [15]
    Kintzel O, Basar Y. Fourth-order tensors—tensor differentiation with applications to continuum mechanics. Part I: classical tensor analysis[J]. ZAMM, 2006, 86:291–311.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Dui G S, Wang Z D, Jin M. Derivatives on the isotropic tensor functions[J]. Science in China Ser G, 2006, 49:321–334.zbMATHCrossRefGoogle Scholar
  17. [17]
    Ogden R. Non-linear elastic deformations[M]. Chichester: Ellis Horwood, 1984.Google Scholar
  18. [18]
    Huang Z P. Foundations of continuum mechanics[M]. Beijing: Higher Education Press, 2003 (in Chinese).Google Scholar

Copyright information

© Editorial Committee of Appl. Math. Mech. 2007

Authors and Affiliations

  • Wang Zhi-qiao  (王志乔)
    • 1
  • Dui Guan-suo  (兑关锁)
    • 1
    Email author
  1. 1.Institute of Engineering MechanicsBeijing Jiaotong UniversityBeijingP. R. China

Personalised recommendations