A closed-form representation for the derivative of non-symmetric tensor power serie

In the present paper a closed-form representation for the derivative of non-symmetric tensor power series is proposed. Particular attention is focused on the special case of repeated eigenvalues. In this case, a non-symmetric tensor can possess no spectral decomposition (in diagonal form) such that the well-known solutions in terms of eigenprojections as well as basis-free representations for isotropic functions of symmetric tensor arguments cannot be used. Thus, our representation seems to be the only possibility to calculate the derivative of non-symmetric tensor power series in a closed form. Finally, this closed formula is illustrated by an example being of special importance in large strain anisotropic elasto-plasticity. As such, we consider the exponential function of the velocity gradient under simple shear. Right in this loading case the velocity gradient has a triple defective eigenvalue excluding the application of any other solutions based on the spectral decomposition