A direct proof of uniqueness of square-root of a positive semi-definite tensor (Q735291)

This paper gives a direct proof of the uniqueness of the square-root of a positive semi-definite tensor without resorting to the notions of eigenvalues, eigenvectors and spectral decomposition. By studying the kinematics of continuum undergoing finite deformation, it is necessary to make polar decomposition for the second-order deformation gradient tensor. The uniqueness of this procedure is a critical element for development of theories of continuum mechanics, and rests on the uniqueness of the square-root of a second-order (symmetric) positive-definite tensor. For some new materials, such as bio-materials and elastic-perfectly plastic materials, it is necessary to use the positive semi-definite tensors to describe their mechanical and physical behavior. The paper has two motivations, namely: (i) in order to utilize the individual properties, it is important to understand the basic properties, in particular the uniqueness of square-root of a positive-definite tensor, and (ii) in order to extend the polar decomposition theorem to meet both the normal and non-normal deformation gradients, it is necessary to study the uniqueness of the square-root of positive semi-definite tensors. First, the authors prove two lemmas concerning the properties of a second-order positive semi-definite tensor. Then, they present a direct proof of the uniqueness of the square-root of the tensor. The proof does not depend on the finite dimensions of a vector space. By simultaneously changing the considered tensors to second-order negative semi-square ones, the proof will hold and the description of the square-root theorem remains unchanged.