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Stability analysis of the mixed variable method and its application in wave reflection and transmission in multilayered anisotropic structures

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Abstract

This paper focuses on the reflection and transmission (R/T) problem of elastic waves in multilayered anisotropic structures. Stability analysis of the mixed variable method (MVM) for computing the R/T coefficients of elastic waves in multilayered anisotropic structures is presented. For this purpose, a detailed comparison of the MVM with the other two widely used methods, namely the transfer matrix method (TMM) and the stiffness matrix method (SMM), is made. Although the TMM, the SMM and the MVM are mathematically equivalent, they are quite different in numerical stability. The theoretical analysis shows that the MVM is unconditionally stable for arbitrary wavenumber–thickness products, whereas the TMM and the SMM may become unstable for large or small wavenumber–thickness products, respectively. This conclusion is numerically verified by various examples. Finally, the R/T coefficients of elastic waves in generally anisotropic multilayered structures bounded by two semi-infinite spaces are calculated using the MVM for a quasi-longitudinal or quasi-transverse wave incidence, and the effects of incident angles and wavenumber–thickness products on the R/T coefficients are discussed in detail through an example.

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Acknowledgements

The authors are grateful for the support of the Natural Science Foundation of China (Nos. 11572076 and 91748203).

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  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian, 116023, People’s Republic of China

    Yanhui Zhang & Qiang Gao

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  1. Yanhui Zhang
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Zhang, Y., Gao, Q. Stability analysis of the mixed variable method and its application in wave reflection and transmission in multilayered anisotropic structures. Arch Appl Mech 90, 127–146 (2020). https://doi.org/10.1007/s00419-019-01601-5

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