“…Let us consider bold-italicXp and bold-italicxp to be the position vectors of one point p of the mid-surface in the reference configuration C 0 and the current configuration C t , respectively. The position vector of one point q of the shell structure located at distance z from the mid-surface can be defined as (Mellouli et al, 2019a, 2019b, 2019c)where h is the thickness; bold-italicD is the initial shell director, which is perpendicular to the undeformed mid-surface; and bold-italicd1 and bold-italicd2 represent the double director shell vectors in the deformed configuration (see Figure 2). Figure 2.Definition of the coordinate systems.…”
Section: Double Director Shell Formulationmentioning
confidence: 99%
“…The strain–displacement matrices are given at node I in Appendix 1. Details of the strain discretization are mentioned in Mallek et al (2019c).…”
This research makes a first attempt to investigate the dynamic characteristics of functionally graded carbon nanotube–reinforced composite plates and shell structures with surface-bonded piezoelectric layers. A variational formulation is derived based on the linear double director shell theory to ensure realistic parabolic variation of transverse shear strain along the thickness direction. The assumed natural strains method is adopted to enhance the accuracy of the four-node piezoelectric shell element developed in this study. Numerical studies are conducted to validate the efficiency and numerical stability of the proposed model to predict the behavior of piezolaminated composite shell structures. Furthermore, dynamic responses are extended to functionally graded carbon nanotube–reinforced composite shells covered by two active layers. The host structure is reinforced by single-walled carbon nanotubes, which are assumed to be graded through the thickness direction with different types of distributions and embedded in a polymer matrix. The effect of the volume fractions, distribution type, and geometrical parameters of the carbon nanotubes is examined.
“…Let us consider bold-italicXp and bold-italicxp to be the position vectors of one point p of the mid-surface in the reference configuration C 0 and the current configuration C t , respectively. The position vector of one point q of the shell structure located at distance z from the mid-surface can be defined as (Mellouli et al, 2019a, 2019b, 2019c)where h is the thickness; bold-italicD is the initial shell director, which is perpendicular to the undeformed mid-surface; and bold-italicd1 and bold-italicd2 represent the double director shell vectors in the deformed configuration (see Figure 2). Figure 2.Definition of the coordinate systems.…”
Section: Double Director Shell Formulationmentioning
confidence: 99%
“…The strain–displacement matrices are given at node I in Appendix 1. Details of the strain discretization are mentioned in Mallek et al (2019c).…”
This research makes a first attempt to investigate the dynamic characteristics of functionally graded carbon nanotube–reinforced composite plates and shell structures with surface-bonded piezoelectric layers. A variational formulation is derived based on the linear double director shell theory to ensure realistic parabolic variation of transverse shear strain along the thickness direction. The assumed natural strains method is adopted to enhance the accuracy of the four-node piezoelectric shell element developed in this study. Numerical studies are conducted to validate the efficiency and numerical stability of the proposed model to predict the behavior of piezolaminated composite shell structures. Furthermore, dynamic responses are extended to functionally graded carbon nanotube–reinforced composite shells covered by two active layers. The host structure is reinforced by single-walled carbon nanotubes, which are assumed to be graded through the thickness direction with different types of distributions and embedded in a polymer matrix. The effect of the volume fractions, distribution type, and geometrical parameters of the carbon nanotubes is examined.
“…Table 4 The parameters of (10, 10) armchair SWCNT obtained from molecular dynamics [32]. It is worth noting that, the shear modulus 13 G is taken equal to 12 G whereas 23 G is taken equal to 12 1.2G [32].…”
Section: Table3mentioning
confidence: 99%
“…The governing equations were extracted by the Hamilton method, and for solving them, the researchers used an analytical approach. Mallek et al [13] analyzed the electromechanical behavior of composite shell structures with embedded piezoelectric layers. For solving the equations, they used the 3D-shell model based on discrete double directors shell elements.…”
This paper presents free vibration of the double-curved shells and panels with piezoelectric layers in a thermal environment. Vibration characteristics of elliptical, spherical, cycloidal, and toro circular shells of revolution are studied in detail. These structures are made of a based carbon nanotubes (CNTs) core and piezoelectric layers at the upper and lower surfaces. It is supposed that temperature changes linearly through the thickness direction. Reissner-Mindlin and the first order shear deformation (FSDT) theories are implemented to derive the governing equations of the considered structures. The distribution of nanotubes is assumed to be linear along the thickness direction. For solving the equation, the General Differential Quadrature (GDQ) method is implemented to investigate the dynamic behavior of the structures. Finally, the effects of the boundary conditions, the thickness of piezoelectric layers, the functional distribution of CNTs, thermal environment and kinds of the circuit (opened-circuit and closedcircuit) are analyzed. Eigenvalue system is solved to obtain natural frequencies. It is delineated that the obtained fundamental frequency by the closed -circuit is smaller than those obtained by the opened-circuit. Another interesting result is that the natural frequency is decreased by increasing temperature.
“…Indeed, shear correction coefficients can be simply acquired for linear isotropic material (5/6), but their determination is more complex for composites, especially laminated structures (Hajlaoui et al, 2015). Hence, HSDT theories were proposed to overcome the shear coefficient limitations (Wali et al, 2015; Zghal et al, 2018, Mallek et al, 2018). This class of kinematic theories has been recently equipped with piezoelectric capability.…”
This article investigates geometrically nonlinear and linear analysis of multilayered shells with integrated piezoelectric materials. An efficient nonlinear shell element is developed to solve piezoelastic response of laminated structure with embedded piezoelectric actuators and sensors. A modified first-order shear deformation theory is introduced in the present method to remove the shear correction factor and improve the accuracy of transverse shear stresses. The electric potential is assumed to be a linear function through the thickness of each active sub-layer. Several numerical tests for different piezolaminated geometries are conducted to highlight the reliability and efficiency of the proposed implementation in linear and geometrically nonlinear finite element analysis.
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