The Explicit Determination of the \(p\)-th Root of an Arbitrary Second-Order Tensor and Its Derivative

In nonlinear elasticity, the square root of a tensor arises in the polar decomposition of the deformation gradient, and in many other applications in other areas as well. In this work, given a positive integer \(p\), we derive an explicit expression for the principal \(p\)-th root of a real-valued second-order tensor, which is not necessarily diagonalizable, whose eigenvalues do not lie on the closed negative real axis, but which is otherwise arbitrary, for any underlying space dimension \(n\). We also present a method for the explicit evaluation of the derivative of the \(p\)-th root of a tensor.