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.
nonsymmetric tensor derivative of tensor function scalar function fourth-order tensor
Chinese Library Classification
2000 Mathematics Subject Classification
This is a preview of subscription content, log in to check access.
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
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
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
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
Ogden R. Non-linear elastic deformations[M]. Chichester: Ellis Horwood, 1984.Google Scholar
Huang Z P. Foundations of continuum mechanics[M]. Beijing: Higher Education Press, 2003 (in Chinese).Google Scholar