M-eigenvalues of the fourth-order partially symmetric tensors play an important role in the analysis of nonlinear elastic materials. In this paper, M-identity tensor is introduced to establish some new eigenvalue localization sets for fourth-order partially symmetric tensor. It is revealed that the new eigenvalue localization sets are tighter than some existing results. Numerical examples demonstrate the effectiveness of the results obtained. As applications, some bound estimations for the M-spectral radius and some checkable sufficient conditions for the positive definiteness of the fourth-order partially symmetric tensor are obtained based on the new eigenvalue localization sets.
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