On the averaging of symmetric positive-definite tensors

In this paper we present properly invariant averaging procedures for symmetric positive-definite tensors which are based on different measures of nearness of symmetric positive-definite tensors. These procedures intrinsically account for the positive-definite property of the tensors to be averaged. They are independent of the coordinate system, preserve material symmetries, and more importantly, they are invariant under inversion. The results of these averaging methods are compared with the results of other methods including that proposed by Cowin and Yang (J. of Elasticity 46 (1997) pp. 151-180.) for the case of the elasticity tensor of generalized Hooke's law.