Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/77700
Title: Positive definiteness of paired symmetric tensors and elasticity tensors
Authors: Huang, ZH
Qi, L 
Keywords: Elasticity tensor
M-eigenvalue
Paired symmetric tensor
Positive definiteness of tensor
Semidefinite relaxation
Issue Date: 2018
Publisher: North-Holland
Source: Journal of computational and applied mathematics, 2018, v. 338, p. 22-43 How to cite?
Journal: Journal of computational and applied mathematics 
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.
URI: http://hdl.handle.net/10397/77700
ISSN: 0377-0427
EISSN: 1879-1778
DOI: 10.1016/j.cam.2018.01.025
Appears in Collections:Journal/Magazine Article

Access
View full-text via PolyU eLinks SFX Query
Show full item record

Google ScholarTM

Check

Altmetric


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.