Derivative of a function of a nonsymmetric second-order tensor

Abstract

Abstract.Exact explicit expressions are obtained for an isotropic tensor-valued function of a nonsymmetric second-order tensor, and its derivative, without resort to eigenvector calculations.These are then used to derive explicit expressions for the material time derivative of the general strain measures in terms of the deformation rate tensor. Introduction.Isotropic tensor-valued functions of symmetric second-order tensors are used to express various strain measures in the kinematics of finite deformation. These strain measures are often expressed in a spectral form with respect to the principal triad of the right stretch tensor (left stretch tensor, if the strain measure is Eulerian)\ Hill [5]. Using the Hamilton-Cayley theorem, a strain measure of this kind can be represented in the coordinate-invariant form. Ting [14] obtains closed-form expressions for the general isotropic functions of a symmetric second-order tensor. Among the various strain measures, logarithmic strain is considered to have certain advantages in constitutive modeling; Hill [6]. However, the relation between the rate of stretch (or deformation rate) and the material time derivative of the logarithmic strain is very complicated. Hill [6] obtains this relation in component form with respect to the principal triad of the right stretch tensor. Later, Hill [7] obtains a relation between the deformation rate and

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

“…[13] can be achieved by replacing f (λ i , I 1 , I 3 ) and D in Eqs. 44) and (47) will reduce to the results obtained by Balendran and Nemat-Nasser [5] .…”

“…In the present paper, only the case of three real distinct eigenvalues is considered. The results for the case of two complex eigenvalues can be obtained by similar procedures proposed by Balendran and NematNasser [5] . Then, A can be written as [5] …”

“…where E i , i = 1, 2, 3 corresponding to the eigenvalues λ i , are the eigenprojections of A, and take the following coordinate-invariant forms [5] :…”

“…In 1996, Balendran and Nemat-Nasser [5] gave the exact explicit expressions for polynomial nonsymmetric isotropic tensor functions and their derivatives without resort to eigenvector calculations. De Souza Neto [6] directly calculated the derivative of tensor power series of a nonsymmetric tensor.…”

“…This subclass of tensor functions is more general than those solved by existing methods [5][6][7][8][9][10][11] ; the basis-free expressions for the derivatives of those tensor functions are obtained by solving a tensor equation [13] . In the present paper, another scalar function besides the one used in Ref.…”