Checkable Criteria for the M-Positive Definiteness of Fourth-Order Partially Symmetric Tensors
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Checkable Criteria for the M-Positive Definiteness of Fourth-Order Partially Symmetric Tensors Suhua Li1 · Yaotang Li1 Received: 28 July 2019 / Revised: 24 October 2019 / Accepted: 28 November 2019 © Iranian Mathematical Society 2019
Abstract Based on the matrix unfolding technique of a tensor, three easily checkable sufficient conditions for the M-positive definiteness of fourth-order partially symmetric tensors are given. Numerical examples show that the proposed results are efficient. Keywords Partially symmetric tensors · M-positive definiteness · Unfolding matrix Mathematics Subject Classification 15A69 · 15A18 · 65F15
1 Introduction The equilibrium equations [1,2] ci1 i2 i3 i4 (1 + ∇u)u i3 ,i4 i2 = 0
(1.1)
are of great importance in the theory of elasticity [3], where u i (X )(i = 1, 2, 3) is the displacement field (X is the coordinate of a material point in the reference configuration), ci1 i2 i3 i4 is the component of elastic modulus tensor C = (ci1 i2 i3 i4 ) ∈ R3×3×3×3 and has the following property: ci1 i2 i3 i4 = ci2 i1 i3 i4 = ci1 i2 i4 i3 = ci3 i4 i1 i2 , ∀i 1 , i 2 , i 3 , i 4 ∈ 3 = {1, 2, 3}.
Communicated by Abbas Salemi.
B
Yaotang Li [email protected] Suhua Li [email protected]
1
School of Mathematics and Statistics, Yunnan University, Kunming 650091, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
and Eq. (1.1) is strongly elliptic if and only if Cx yx y =
3
ci1 i2 i3 i4 xi1 yi2 xi3 yi4 > 0
(1.2)
i 1 ,i 2 ,i 3 ,i 4 =1
holds for all unit vector x ∈ R3 and y ∈ R3 . For common usage, (1.2) is called the strong ellipticity condition. In the past several decades, considerable effort has been made to seek the sufficient or necessary criteria for the strong ellipticity condition such as literatures [4–9]. However, easily verifiable criteria are few because Cx yx y in (1.2) can be equivalently written as Ax yx y, where A = (ai1 i2 i3 i4 ) ∈ R3×3×3×3 is a partially symmetric tensor, that is, ai1 i2 i3 i4 = ai3 i2 i1 i4 = ai1 i4 i3 i2 = ai3 i4 i1 i2 and ai1 i2 i3 i4 = 41 (ci1 i2 i3 i4 + ci3 i2 i1 i4 + ci1 i4 i3 i2 + ci3 i4 i1 i2 ). In 2009, Qi et al. [10] presented that the strong ellipticity condition holds if and only if the partially symmetric tensor A is M-positive definite which is defined as follows. Without loss of generality, in this paper, we consider a more general partially symmetric tensor, that is, A = (ai1 i2 i3 i4 ) ∈ Rm×n×m×n with ai1 i2 i3 i4 = ai3 i2 i1 i4 = ai1 i4 i3 i2 = ai3 i4 i1 i2 , ∀i 1 , i 3 ∈ m = {1, 2, . . . , m}, ∀i 2 , i 4 ∈ n.
Definition 1.1 [10] A partially symmetric tensor A = (ai1 i2 i3 i4 ) ∈ Rm×n×m×n is called an M-positive definite tensor, if Ax yx y =
m
n
ai1 i2 i3 i4 xi1 yi2 xi3 yi4 > 0
(1.3)
i 1 ,i 3 =1 i 2 ,i 4 =1
holds for any unit vectors x = (xi ) ∈ Rm and y = (yi ) ∈ Rn . Qi et al. [10] proved that a fourth-order real partially symmetric tensor is M-positive definite if and only if its smallest M-eigenvalue is positive, whereas the computation for the M-eigenvalues of a partially symmetric tensor
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