TY - JOUR
T1 - Elasticity M-tensors and the strong ellipticity condition
AU - Ding, Weiyang
AU - Liu, Jinjie
AU - Qi, Liqun
AU - Yan, Hong
PY - 2020/5/15
Y1 - 2020/5/15
N2 - In this paper, we establish two sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition, the elasticity M-tensor. The first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Besides, the elasticity M-tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity M-tensor satisfies the strong ellipticity condition by employing a Perron-Frobenius-type theorem for M-spectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity M-tensors are also established.
AB - In this paper, we establish two sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition, the elasticity M-tensor. The first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Besides, the elasticity M-tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity M-tensor satisfies the strong ellipticity condition by employing a Perron-Frobenius-type theorem for M-spectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity M-tensors are also established.
KW - Alternating projection
KW - Elasticity tensor
KW - M-positive definite
KW - M-tensor
KW - Nonnegative tensor
KW - S-positive definite
KW - Strong ellipticity
UR - http://www.scopus.com/inward/record.url?scp=85077755599&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85077755599&origin=recordpage
U2 - 10.1016/j.amc.2019.124982
DO - 10.1016/j.amc.2019.124982
M3 - 21_Publication in refereed journal
VL - 373
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
SN - 0096-3003
M1 - 124982
ER -