Analytic Tensor Functions

Abstract

In the previous chapter we discussed isotropic and anisotropic tensor functions and their general representations. Of particular interest in continuum mechanics are isotropic tensor-valued functions of one arbitrary (not necessarily symmetric) tensor. For example, the exponential function of the velocity gradient or other non-symmetric strain rates is very suitable for the formulation of evolution equations in large strain anisotropic plasticity. In this section we focus on a special class of isotropic tensor-valued functions referred here to as analytic tensor functions. In order to specify this class of functions we first deal with the general question how an isotropic tensor-valued function can be defined.

Exercises

7.1.

Let $\mathbf {R}\left( \omega \right)$ be a proper orthogonal tensor describing a rotation about some axis $\mathbf {e}\in \mathbb {E}^3$ by the angle $\omega$. Prove that $\mathbf {R}^a\left( \omega \right) =\mathbf {R}\left( a\omega \right)$ for any real number a.

7.2.

Specify the right stretch tensor U (7.5)$_1$ for simple shear utilizing the results of Exercise 4.1.

7.3.

Prove the properties of analytic tensor functions (7.21).

7.4.

Represent the tangen moduli for the Ogden material (6.12) in the case of simple shear by means of (7.49)–(7.50) and by using the result of Exercises 4.14 and 6.10.

7.5.

Prove representation (7.54) for eigenprojections of diagonalizable second-order tensors.

7.6.

Calculate eigenprojections and their derivatives for the tensor A (Exercise 4.15) using representations (7.81)–(7.85).

7.7.

Calculate by means of the closed-form solution $\exp \left( \mathbf {A}\right)$ and $\exp \left( \mathbf {A}\right) ,_\mathbf {A}$, where the tensor A is defined in Exercise 4.15. Compare the results for $\exp \left( \mathbf {A}\right)$ with those of Exercise 4.16.

7.8.

Compute $\exp \left( \mathbf {A}\right)$ and $\exp \left( \mathbf {A}\right) ,_\mathbf {A}$ by means of the recurrent procedure with the precision parameter $\varepsilon =1\cdot 10^{-6}$, where the tensor A is defined in Exercise 4.15. Compare the results with those of Exercise 7.7.