Derivatives of fourth order Kronecker power systems with applications in nonlinear elasticity

A natural way to describe systems with polynomial nonlinearities is using the Kronecker product. Particularly, third-order Kronecker power systems can express a wide range of systems from electronic engineering to nonlinear elasticity. But such development (e.g. equations of motions of elastic structures from nonlinear strain energy) requires standard formulation for the derivative of the Kronecker power of vectors with respect to the same vector. Such standard way cannot be found in literature. This paper presents a method to obtain the derivatives of Kronecker powers of vectors with respect to itself up to a power of four and also third-order Kronecker Power systems containing those terms in a concise matrix form. The matrix expression of these systems provides new approach for efficient numerical implementation, organized analysis and linearization. To demonstrate the strength of this method, an example of application for a finite element nonlinear Euler beam is also presented.