  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
demonstrate that our method is effective.
