Investigation of bending behavior for laminated composite magneto-electro-elastic cylindrical shells subjected to mechanical or electric/magnetic loads

Abstract

Due to the multidisciplinary nature for the solution of magneto-electro-elastic (MEE) shell structures, developing a novel and accurate computational model is both essential and necessary for the practical engineering. The scaled boundary finite element method (SBFEM) is a semi-analytical technique in which only the surfaces or boundaries of the computational domain need to be discretized, while an analytical formulation can be derived in the radial direction of the surrounding area. These advanced features enable the spatial dimension to be reduced by one, while the accuracy of the proposed algorithm is maintained. In this paper, a novel semi-analytical numerical model based on the SBFEM is developed for the bending analysis of the laminated MEE cylindrical shells under the mechanical or electric/magnetic potential loads. According to the three-dimensional (3D) magneto-electro-elasticity theory, the magneto-electro-mechanical coupling equations and the associated boundary conditions in terms of the mechanical displacement as well as the electrical and magnetical potentials are derived in the scaled boundary coordinate system using the weighted-residual method. The analytical expressions for the generalized displacement and internal nodal force fields are determined by applying the state-space method and have been solved by means of the precise integration technique (PIT). Comparisons between the present numerical results for limiting conditions and solutions available in the published work have been carried out to demonstrate the convergence and accuracy of this approach. At the same time, by utilizing the proposed mechanics, the influences of the aspect ratio and stacking configuration on the through-thickness bending behaviors of the laminated MEE cylindrical shells are studied in detail.

Introduction

The application of the magneto-electro-elastic (MEE) shells has increased in many practical engineering such as aerospace, electronic and automotive due to their advantages in stiffness/strength-to-weight ratio reinforce and advanced mechanical–electrical–magnetical coupling properties. Therefore, solutions of the MEE shell problems are not only essential for engineering, but also the important part of related projects like electromagnetism and mechanics of the composite materials. Analysis of the MEE shell problems often requires that the solution algorithms are accurate, robust and small in computer cost. An alternative approach to fulfill these standards for the considered problems is the scaled boundary finite element method (SBFEM) [1], [2], [3], [4], which applies the principle of the virtual work or the weighted residual method to transform the partial differential equations into the ordinary differential equations. Similar to the boundary element method (BEM), this approach proposes that discretizations are confined only to the surfaces or boundaries of the computational domain, reducing the problem dimension by one and making mesh generation much simpler. Furthermore, the accuracy of numerical results can be guaranteed due to the analyticity of the SBFEM in the radial direction of the scaled boundary coordinate system. Some of researches using the SBFEM for numerical simulations include papers in [5], [6], [7], [8], [9] for soil–structure interaction, in [10], [11], [12], [13] for fracture analysis, in [14], [15] for sensitivity analysis and in [16], [17], [18] for coupled fluid–solid analysis, and so on. Most recently, the elastic static and dynamic analyses for beam and plate/shell structures have also been proposed in [19], [20], [21], [22], [23], [24]. In the literature, only one line or one surface of beam and plate/shell structures need to be discretized using the SBFEM. At the same time, due to the formulations being exclusively based on the 3D theory without introducing the kinematics assumption of beam or plate/shell theory, the SBFEM is applicable to solve the bending simulations for both thick and thin shell and plate without suffering from the shear locking problems [21], [22], [23], [24]. Furthermore, the computational domains with curved boundaries can be better represented by using the higher order spectral elements in the SBFEM [21], [22].

Since the MEE shell structures play a very important role in engineering application, a considerable number of analytical approaches and numerical models have been developed by many scholars and engineers. Wang and Zhong [25] derived an exact formulation for the piezoelectric or piezomagnetic laminated cylindrical shell under thermo-mechanical loading based on the power series and Fourier series expansion approaches. The mechanical, electric and magnetic behaviors of the system were measured by solving the algebraic equations and the influence of the stacking sequences on the bending responses is​ discussed in detail. By means of the asymptotic approach, Wu and Tsai [26] treated the bending problems of doubly-curved functionally graded and laminated MEE shells, and the accuracy and convergence rate were demonstrated by comparing with exact solutions. Results concluded that the kinematics field in coupled classical shell theory was not appropriate for the solutions of functionally graded and laminated MEE shells, especially when the structures were subjected to the electric and magnetic loads. Based on the 3D magneto-electro-elasticity theory, Wu et al. [27], [28], [29], [30] investigated the bending and vibration performances of the functionally graded MEE shell structures by use of the asymptotic approach. Considering a power-law exponent distribution in the direction of thickness for the functionally graded material, they investigated the influences of piezoelectric and piezomagnetic effects on the response behaviors of the system. Numerical estimations shown that, the distributions of the elastic and electric displacements as well as the magnetic induction along the thickness were higher-degree polynomials for the moderately thick shells. Wang et al. [31], [32] presented an analytical model to study the bending and free vibration characteristics of the MEE cylindrical panels in the frame of the Hamiltonian system. The natural frequencies and bending results were computed at various boundary conditions, and it was shown that the existence of the magneto-electro-elastic and piezoelectric couplings had a significant influence on the elastic displacements, while the effects of that on the stresses were less significant. Albarody et al. [33], [34] developed an analytical model on the basis of first-order shear deformation theory to compute the bending and free vibration responses of the multi-layered MEE shell structures. Solutions for the center deflection and natural frequency were presented with or without the consideration of thermal loads. Chen et al. [35] applied the propagation matrix method to the bending analysis of the laminated MEE hollow spheres and calculated the elastic deformation as well as the distribution characteristics of electric and magnetic potentials. Results illustrated that the stresses in the laminated MEE shell increased due to the existence of the magnetostrictive layer, while the piezoelectric layer had an opposite influence on the corresponding stresses, and the effects of stacking sequence on the radial stresses of the MEE system were very small. Wu et al. [36] carried out an analytical model to examine bending performances of the laminated MEE shell consisting of a viscoelastic interlayer. In their work, the mechanical, electric and magnetic behaviors under the radial load at the external surface have been investigated, and the elastic displacement, stress, electric and magnetic fields were computed. The solutions presented that the electric displacement, shear stress and magnetic induction near the interlayer decreased with increasing interlayer thickness while the circumferential normal stress increased as the interlayer thickness increased. In the aspect of numerical approach, Rao et al. [37] explored the nonlinear bending characteristics of the MEE laminated plates and shells under mechanical forces or magnetic loads by means of the finite element approach. The deformation responses and electromagnetic behaviors were presented and the effects of the nonlinear strain–displacement relations on responses of the system were discussed. For a composite laminated plate or doubly-curved shell structure, Kattimani [38] investigated the nonlinear bending and vibration behaviors using the finite element method (FEM), the results of deflection and frequency were calculated and compared with that evaluated by the Reissner–Mindlin theory. The numerical results shown that the non-dimensional central deflections from the reference solutions were in good agreement with the FEM measurements, and the effects of the curvature aspect ratio, thickness aspect ratio and boundary conditions on the nonlinear behaviors of the MEE doubly curved shells were significant.

As is well known from the literature mentioned above, the analytical solutions can not only give more scientific insight into the mechanical, electric and magnetic response characteristics for the bending study of the MEE shell structures, but also be easily used for numerical implementation [25]. However, for previous analytical formulations concerning the magneto-electro-elastic coupling, complex mathematical relations must be solved, which is not at all easy to understand for engineers, as presented in [27], [28], [31]. Especially for the more meaningful condition of laminated composite MEE shell structures, the mathematical formulations will become more involved by considering physical quantities of all the MEE shell, such as in the literature [35], [36]. Furthermore, in addition to complex governing equations, variable boundary conditions and complex computational domains are also the difficulties for simulating these problems by using the analytical approach, which makes it unsuitable to deal with mixed boundary condition problems. The numerical simulations for the bending problems of the MEE shell have been dominated by the finite element approach [37], [38]. Although the FEM has been successfully used to solve those problems, the accuracy of the numerical results relies critically on the quality of meshes. Thus, appropriate mesh generation is one of the difficulties for estimation of magneto-electro-elastic characteristics. At the same time, another problem in the application of FEM for the solutions of the MEE shell structures is to avoid “shear and membrane locking” [37]. Special measures such as reduced integration and selective integration must be taken to get rid of such obstacles arising from thickness becoming thinner.

From the literature mentioned above, it is observed that most of works focused attention on the analytical approach and finite element approach in studying the bending problems for laminated MEE shells, while almost very few scholars attempt to investigate the considered problem with other techniques. Nevertheless, simulating the bending problems for laminated MEE shells using the SBFEM has not been found in the literature. Considering the abilities to against shear locking and eliminate costly mesh generation for the SBFEM, the main purpose of this paper is to present applications of the SBFEM along with the precise integration technique (PIT) [39] for the solution of bending problems of the laminated composite MEE cylindrical shells. The MEE cylindrical shell structures are subject to mechanical forces or electric/magnetic potential loading, and the structural parameter analyses are discussed in detail. The remaining sections are organized as follows. In Section 2, the mechanical model for the laminated MEE cylindrical shell problems is presented. In Section 3, the SBFEM formulations for each individual layer are proposed. Solution procedure for the whole system is derived in Section 4. In Section 5, numerical examples are calculated to verify the effectiveness of the SBFEM for the considered problems and the structural parameter analyses are also presented. Concludes with some discussions are drawn in Section 6.