Positive definiteness of paired symmetric tensors and elasticity tensors

Abstract

In this paper, we consider higher order paired symmetric tensors and strongly paired symmetric tensors. Elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors. A (strongly) paired symmetric tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite. Positive definiteness of elasticity and higher order elasticity tensors is strong ellipticity in solid mechanics, which plays an important role in nonlinear elasticity theory. We mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional (strongly) paired symmetric tensors. We first show that the concerned (strongly) paired symmetric tensor is positive definite if and only if its smallest M -eigenvalue is positive. Second, we propose several necessary and sufficient conditions under which the concerned (strongly) paired symmetric tensor is positive definite. Third, we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional (strongly) paired symmetric tensor can be written as a sum of squares of polynomials, and further, propose several necessary and/or sufficient conditions to judge whether the concerned (strongly) paired symmetric tensors are positive definite or not. Fourth, by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest M -eigenvalue of a fourth order three dimensional (strongly) paired symmetric tensor, by which we can check positive definiteness of the concerned tensor. The preliminary numerical results confirm our theoretical findings.

On the M-eigenvalue estimation of fourth-order partially symmetric tensors

“…Generally speaking, the study on high order tensors have attracted much attention of researchers, which made tensor analysis an important tool in theoretical physics, continuum mechanics and many other areas of science and engineering [2,3,4,5,6,21,24,28,27,22,12]. Particulary, Wang et al [25] established Zeigenvalue inclusion theorems, and the upper bounds for the largest Z-eigenvalue of a weakly symmetric nonnegative tensor was obtained.…”

M-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors

“…Note that f A (x, y) is positive definite if and only if Meigenvalues of A are positive [7]. Hence, effective algorithms for finding M-eigenvalue and the corresponding eigenvector have been implemented [8][9][10][11][12][13][14][15][16]. Due to the complexity of the tensor eigenvalue problem [17,18], it is difficult to compute all M-eigenvalues.…”

Multilinear Time Invariant System Theory

“…Similarly, these notions proved to be useful in numerical solutions of master equations associated with Markov processes on extremely large state spaces [12]. Even-order paired tensors were originally proposed by Huang and Qi [13] in the context of elasticity tensors in solid mechanics. It turns out that compared to even-order non-paired tensors, the even-order paired tensors are easier for bookkeeping and can be conveniently manipulated using tensor algebra for MLTI systems, see Section 3.…”