Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Guided waves propagation in anisotropic hollow cylinders by Legendre polynomial solution based on state-vector formalism Mingfang Zheng a,b,c , Cunfu He a , Yan Lyu a, ⁎ , Bin Wu a a College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China b School of Environment and Civil Engineering, Dongguan University of Technology, Dongguan 523808, China c State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace Engineering, Xi'an Jiaotong University, Xi'an 710049, China ARTICLEINFO Keywords: Anisotropic hollow cylinder Guided waves State-vector formalism Legendre polynomial expansion ABSTRACT A spectral approach was presented in the computation of dispersion curves for the general anisotropic hollow cylinders. The derivation is based on the hybrid method of the state-vector formalism and Legendre polynomials expansion, which was previously adopted for the anisotropic plates. This method will lead to an eigenvalue/ eigenvector problem for the calculation of wavenumbers and displacement profiles. This hybrid method avoids solving the transcendental dispersion equation. A closed-form solution for the hollow cylinder, involving mul- tiple integral expressions, is demonstrated. A stable scheme for the integration expansion was established by re- expanding the expansion operators from the first round Legendre polynomial expansion versus the displace- ments. Usually, the traditional matrix methods are based on root-finding algorithms, which is difficult to im- plement in anisotropic tubes. In this research, the hybrid approach we proposed provides a reliable mathema- tical solution of wave propagations in an anisotropic hollow cylinder. Applications will be illustrated using isotropic and orthotropic hollow cylinders, in which the isotropic case agrees well with the results by global matrix method. The dispersion curves of orthotropic hollow cylinders, when the out radius set to approximate infinity, are compared to its corresponding anisotropic plate, which is obtained from our previous work. Furthermore, the displacement and stress profiles will be given and analyzed for an orthotropic tube, which has 10 mm thickness with an out radius of 50 mm. 1. Introduction The hollow cylinders are widely used in many engineering appli- cations, such as pipelines of offshore oil platform, pressure pipes, au- tomobile transmission shaft, aerospace structures, etc. Discussions re- garding the vibration and wave propagation characteristics in the hollow cylinder were prosperous in recent decades. Elastic guided waves possess great potentials for nondestructive evaluation [1,2] and structural health monitoring [3,4] of the cylindrical structures due to their ability to detect the inaccessible regions. The propagation beha- viors of guided waves are required for pipeline inspections and eva- luations. It is necessary to determine the dispersion curves for the un- derstanding of wave propagation. The excitation efficiency and mode selection should be the key points in the system design. In addition, the priori knowledge of guided waves can also be used to explain the re- ceived wave signals in order to characterize the defects [3]. However, these investigations on wave propagation characteristics require a comprehensive mathematical model, in which the conventional matrix method is difficult to implement in hollow cylindrical anisotropic structures. The analytical solution of isotropic cylindrical structures has been thoroughly investigated by Mindlin [5],Pao [6,7], Gazis [8,9] and Zemanek [10], Viktorov [11], Achenbach [12], and Auld [13].Butitis more complex or even impossible to obtain analytical solutions for the anisotropic pipes. Dispersion software, developed by the NDE group at the Imperial College, clearly pointed out its limitations: when dealing with cylindrical structures, the material must be isotropic [14]. The polynomial series approach has been presented to solve the wave propagation problem for more than fifty years. Mirsky [15], McNiven and Mengi [16] developed Frobenius power series expansion to solve elastic wave propagation in orthotropic cylinders. The dis- placement components in the circumferential and axial directions are expressed in term of trigonometric functions, while the radial dis- placement field along the thickness is expanded by power series. But the power series are not orthogonal, and will give rise to numerical instability. After that, Maradudin et al. [17] developed the Laguerre orthogonal polynomial approach to solve linear acoustic waves in https://doi.org/10.1016/j.compstruct.2018.09.042 Received 27 February 2018; Received in revised form 21 August 2018; Accepted 17 September 2018 ⁎ Corresponding author. E-mail addresses: zhengmingfang@xjtu.edu.cn (M. Zheng), hecunfu@bjut.edu.cn (C. He), lvyan@bjut.edu.cn (Y. Lyu), wb@bjut.edu.cn (B. Wu). Composite Structures 207 (2019) 645–657 Available online 20 September 2018 0263-8223/ © 2018 Elsevier Ltd. All rights reserved. T