Abstract
The M-eigenvalue of elasticity M-tensors play important roles in nonlinear elastic material analysis. In this paper, we establish an upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors without irreducible conditions, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.
Highlights
Tensor eigenvalue problems play an important role in numerical multilinear algebra [1,2,3,4,5,6,7], and they have a wide range in medical resonance [8], imaging spectral hypergraph theory [9], automatical control [10,11,12,13]
A fourth-order partially symmetric tensor is useful in nonlinear elastic material analysis [3,16,17,18,19,20,21,22,23,24,25,26,27]
Ostrosablin [16] first constructed a complete system of eigentensors for the fourth rank tensor of elastic modulus, and Nikabadze [18] generalized these results and constructed a full system of eigentensors for a tensor of any even rank, as well as a complete system of eigentensor-columns for a tensor-block matrix of any even rank [22,23]
Summary
Tensor eigenvalue problems play an important role in numerical multilinear algebra [1,2,3,4,5,6,7], and they have a wide range in medical resonance [8], imaging spectral hypergraph theory [9], automatical control [10,11,12,13]. A fourth-order real tensor A = ( aijkl ) is called a partially symmetric tensor, denoted by. A fourth-order partially symmetric tensor with n = 2 or 3, called the elasticity tensor, can be used in the two/three-dimensional field equations for a homogeneous compressible nonlinearly elastic material for static problems without body forces [27]. Based on structural properties of elasticity M-tensors, He et al [26] proposed some bounds for the minimum M-eigenvalue under irreducible conditions. Irreducibility is a i,j∈ N,i 6= j relatively strict condition for elasticity M-tensor Inspired by these observations, we want to present sharp bounds for the minimum M-eigenvalue of elasticity M-tensors by describing eigenvectors precisely without irreducible conditions, which improve existing results in [26]. Numerical examples are proposed to verify the efficiency of the obtained results
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