Transverse Shear Deformation Research Articles

Stall nonlinear aeroelastic effects and active control of thin-walled composite blade with piezoelectric patches embedded

Stall nonlinear flutter behavior and active control of anisotropic composite blade have been investigated based on piezoelectric actuation. The blade body comprises an elliptical composite host structure and a lightweight honeycomb skin structure. The composite blade with elliptical section is modeled as a thin-walled beam structure with circumferential asymmetric stiffness (CAS) configuration, exhibiting transverse shear deformation, warping restraint effect, and vertical bending, with structural tailoring implemented. The stall-induced flutter suppression and nonlinear aeroelastic effects of linearized dynamic response characteristics of blade incorporating stall nonlinear aerodynamic model are investigated. The nonlinear aeroelastic coupled equations are reduced to ordinary equations by the Galerkin method, with the aerodynamic force decomposed by strip theory. Active feedback control based on boundary moment control strategy is developed to enhance the vibrational behavior and dynamic response to aerodynamic excitation and stabilize structures that might be damaged in the absence of control. Various active feedback control schemes tuned by optimal proportional-derivative (PD) controller are implemented. The excellent control effect is not only confirmed by the time response with its vibration amplitude greatly suppressed but also confirmed by the frequency change and comparison. The research provides a way for rare study of amplitude suppression of stall nonlinear aeroelastic effect of thin-walled composite blade with CAS configuration and piezoelectric patches embedded.

Specified models in the problems of the deformation of multilayer plates on a rigid foundations

The analysis and estimation of the stress-strain state (SSS) of multilayered plates on a rigid foundation under the action of stationary transverse loading is an urgent task. As its includes the calculations of strength and deformability of various homogeneous and multilayer coatings. This is the calculation of pavement on relatively rigid bridge structures, or on a non-deformable underlying layer, protective multilayer coatings of flat elements of structures of greater rigidity than roofing, parts and more. Combining materials with isotropic and transversely isotropic physical characteristics into a multilayer package allows you to create multifunctional structures. The SSS of such structures, due to their structural heterogeneity and relatively low transverse stiffness of the individual layers, is significantly associated with the influence of transverse shear deformations and transverse compression deformations. Therefore, the problem of refined modeling of SSS plates, which would take into account these types of deformations, is urgent. Based on the decomposition of the SSS plate into flexural and unflexuralcomponents, it is proposed to optimize the design scheme of deformation of a rectangular multilayer plate on a rigid foundation. The essence of optimization is to consider such a calculation scheme of the plate, in which the SSS of the plate would be fully described by only one component, namely the unflexural component of SSS. To do this, instead of the actual design of the multilayer plate, which is deformed without separation from the foundation, it is proposed to consider the design scheme of the plate, which is formed by symmetrical completion relative to the contact surface of the plate with the foundation. In this case, the plate will be bilaterally symmetrically loaded relative to the middle surface of the plate, and the thickness of the plate will double. SSS plate will be unflexural, which greatly simplifies its modeling. For unflexural SSS, a two-dimensional, high-degree iterative approximation, but three-dimensional models of deformation of multilayer rectangular plate on a rigid foundation with isotropic and transverse-isotropic layers are constructed in an elastic formulation.That models takes full account deformationof transverseshearand transverse compression at transverse loading of a plate. By solving the test problems of deformation of isotropic and transversely isotropic plates on a rigid foundation with sliding and rigid contact with the foundation, and comparing the solutions with the exact three-dimensional solutions of these problems obtained by known methods, the accuracy of the proposed refined models is estimated.The limits of admissible parameters of elastic characteristics of transversely isotropic plates for application of the offered models are established.

Porosity-dependent free vibration and transient responses of functionally graded composite plates employing higher order thickness stretching model

Tailoring the mechanical properties in hybrid materials, like functionally graded materials (FGMs), has become more pragmatic with the recent progress in material manufacturing and design. The porosities in FGMs, which develop during the manufacturing process, may become detrimental to armored plates or bulletproof designs. On the other hand, gradient-porosity distributions may have broad advantages in the design of lightweight and variable-stiffness aircraft components. The present study investigates the free vibration and transient responses of porosity-gradient FGM plates. A higher order theory, which considers the transverse shear and normal deformation, warping of the transverse cross-section, and higher order rotary inertia, is utilized for the first time to model the deformation of the porous FGM plate. The advantages of this model are (i) higher order kinematic terms for membrane and bending deformations due to the coupled membrane-bending behavior of FGMs in addition to the thickness-stretching effects, and (ii) the flexibility of applying external loads at different points along the thickness direction of the FGM plate. An analytical solution technique, popularly known as Navier’s approach, is adopted for the spatial solutions, and Newmark’s average acceleration method is utilized for the temporal solutions. The influence of the geometrical and porosity parameters on the structure’s vibration response is studied. Natural frequencies are affected significantly with uniform porosity distribution, while graded porosities can tweak vibration response in a small range but with better control. In the absence of the elasticity solutions, the present results may be considered the possible benchmark solutions for comparison of the other two-dimensional (2D) kinematic models.

Behavior of anchor cable bolts with c-shaped tube and cable bolts in shear test

The gradual development of coal mining in deep areas has brought new challenges to the surrounding rock support system. It is imperative to design surrounding rock support elements that exhibit strong tensile and shear resistance and are suitable for pretension load. In this paper, an anchor cable bolt with a c-shaped tube (ACC) is proposed, and in order to study the shear performance of the ACC, the mechanical properties of the ACC and ordinary cable bolts with various diameters and pretension loads are tested. In addition, a comparative analysis is performed to investigate the effect of diameter and pretension load on the shear strength of the ACC and cable bolts. The results show that compared with ordinary cable bolts, the ACC exhibits better shear resistance, where its ability to withstand transverse shear load and shear deformation increased by more than 26.8% and 7.0%, respectively. The diameter and pretension load of the bolt significantly affect its shear performance. The larger the diameter of the bolt, the greater the peak shear load. By increasing the pretension load, the shear load and shear displacement at the peak shear load decrease. The failure of the ACC and ordinary cable bolt is mainly due to a combination of shear and tensile failures. In addition, the action mechanism of the ACC is preliminarily explored, and it provides some references for clarifying the mechanical properties of the ACC.

Bending and Free Vibration Analysis of Functionally Graded Sandwich Pile on Elastic Foundation Using Rayleigh-Ritz Method

The present study aims to carry out a parametric investigation on a pile made of different materials, and subjected to static and dynamic loading resulting from the weight of the structure and the shear force acting on the head of the pile. It is worth specifying that the pile is anchored in the ground on a Winkler-Pasternak elastic foundation that is made up of five different layers. In this study, the equilibrium equations were treated using the high-order theory with a new shear deformation shape function of a cylindrical pile made of an isotropic material and a sandwich functionally graded material.The results obtained with a numerical study based on the minimization of energies by the Rayleigh-Ritz method was carried out in order to highlight the influence of the geometric ratio, the volume fraction of the functionally graded material, and the type of loading on the vibration frequencies and the admissible stresses in order to determine the most appropriate material for the pile so that extreme stresses can be absorbed. The precision and the applicability of the method with new solution of the transverse shear deformation are demonstrated through this study with modeling of the layered property of the soil, were then compared with those reported in previous work available in the literature. It turned out that these results are in good agreement with each other, for the different materials of pile and for all boundary conditions considered. The results of pile with sandwich functionally graded material, it is revealed that the role of this material has an influence on the type of loading, especially, in the case statique of a pile subjected to a compound bending weight and the shear effect, in case dynamique the pile with vibration flexible or rigid .

Steady-state and transient vibration analysis of a thermally loaded power law-based functionally graded rotor system with induced porosities

The dynamic analysis of a power law-based functionally graded (FG) rotor-bearing system with induced porosities for non-uniform porosity distributions has been performed using the finite element method. A novel porous rotor element considering the effects of transverse shear deformation, gyroscopic moments, translational and rotational inertial has been developed for non-uniform porosity distributions based on the Timoshenko beam theory. Material properties are graded based on the power law along the radial direction of the FG shaft. The nonlinear temperature distribution law has been considered in conjunction with the power law to vary the temperature across the cross-section of the FG shaft. Material is graded in such a way that the stainless steel is comprised at the core of the FG shaft and the outer periphery of the FG shaft is made of Zirconia. The steady-state and transient time and frequency responses of a thermally loaded porous FG rotor-bearing system have been computed for both types of uneven porosity distribution using the Houbolt time marching technique and Fast Fourier Transform, respectively. The free vibration analysis has also been performed on the porous FG rotor-bearing system for various parameters such as power law index, the volume fraction of porosity and porosity distribution under different thermal gradients. The reduction of critical speed with increased amplitude is observed from the steady-state and transient frequency responses with an increase in volume fraction of porosity and temperature gradients

A novel modeling method for the mechanical behavior of 3D woven fabrics considering yarn distortion

Three-dimensional woven fabrics feature complex meso-architectures, such as yarn torsion and cross-section variation. This paper aims to developing a novel numerical modeling method to simulate the mechanical behaviors of reinforcements considering realistic yarn geometries. The initial as-woven unit cell is first simulated by the digital element analysis. A reconstruction algorithm is then used to generate mesoscale high-fidelity solid geometric models from digital chains. Fiber directional orientation within torsional yarns of the high-fidelity model is described by a digital-chain tracing method and analyzed in detail. Furthermore, based on the high-fidelity model, a hyper-elastic constitutive model is employed for the numerical analysis of the mechanical properties of reinforcements. The finite element analysis (FEA) of transverse compression, tensile deformation and in-plane shearing deformation combined with in-situ Micro-CT experiments are carried out to verify the method. The proposed method agrees well with experimental tests. Defining fiber orientations of torsional yarns using the digital-chain tracing method leads to more accurate simulation results. The developed numerical method can be extended for the composites design and structure optimization due to its high-fidelity simulation capabilities.

Optimization of anti-bending deformation of aerogel sandwich thermal protection structure

The failure of the thermal protection structure is very important to the safety of the aircraft structure. In order to accurately predict the flexural behavior of the aerogel sandwich thermal protection structure, first of all, for the sandwich structure in the form of statically indeterminate beam with variable cross-section, the beam theory considering the transverse shear deformation of the core layer was deduced. Secondly, by comparing the deflections, the forms of deflection curve and the maximum strains of the upper panel, the accuracy of the theoretical prediction was analyzed. Then, according to the requirements of heat protection and integrity of the thermal protection structure, the optimization analyses of the parameters of the core layer, such as the thickness, the density and the elastic modulus, were carried out. The results showed that the beam theory considering the transverse shear deformation of the core layer had better prediction accuracy. The maximum deflection error was the smallest, which was -17%, and the form of the deflection curve was the most reasonable. Within the range of the physical parameters of the core layer studied, the theoretical basis and guiding direction were given for the regulation of the performance of the aerogel core layer.

Post-buckling behavior of rectangular multilayer FG-GPLRC plate with initial geometric defects subjected to non-uniform in-plane compression loads in thermal environment

A semi-analytical method is established to investigate post-buckling behavior of functionally graded graphene platelets reinforced composite (FG-GPLRC) plate with initial geometric defects under non-uniform in-plane compression loads. Buckling governing equations are derived base on a new unified framework of transverse shear deformation plate theory, and the reliability and accuracy of proposed method are validated. Parametric studies are systematically performed for the first time to elaborate the effects of GPLs, structural characteristics, initial geometric defects, loading cases and thermal environment on post-buckling behavior of FG-GPLRC plate. This article can provide theoretical guidance for the design and optimization of FG-GPLRC plate.

Porosity-dependent buckling analysis of elastically supported FGM sandwich plate via new tangent HSDT: A meshfree approach

In this paper, new tangent shape function-based higher-order transverse shear deformation theory (NTHSDT) is proposed to compute the buckling behavior of the elastically supported functionally graded material (FGM) sandwich plates under porous medium. The proposed theory is found to be variationally consistent and fulfills the zero traction boundary conditions on the bottom and top layer without a shear correction factor. The material properties are presumed to be graded in the thickness direction as characterized by a modified power law distribution in terms of volume fraction of constituents. The governing equations are derived using Hamilton’s Principle. A strong form of solution discretizes the governing equations by employing a thin plate spline radial basis function-based collocation (TSRBFC) method. The proposed theory is efficient, reliable, and is in close agreement with the results in the literature. Comparison studies show that the NTHSDT is more accurate than other plate theories and is simple in analyzing buckling behavior. A parametric study is done to examine the effects of grading index, porosity index, sandwich schemes, aspect ratio, side-to-length thickness ratio and foundation stiffness.

Nonlinear forced vibration analysis of laminated composite doubly-curved shells enriched by nanocomposites incorporating foundation and thermal effects

Assessment of the geometrically nonlinear forced vibration of the doubly-curved laminated composite shallow shell panels enriched by nanocomposites have been carried out in this paper. The proposed shells incorporate some layers composed of fibres and nanocomposite-reinforced resin, resting on the two types Winkler and Pasternak elastic foundations. Since the thin and moderately thick shells are investigated, the first-order transverse shear deformation theory (FSDT) is employed for extending the governing equations. The achieved system of differential equations incorporating the geometrical nonlinear terms is solved by applying the principles of the Galerkin method. This approach yields a closed-form semi-analytical solution for motion differential equation of the shell which is solved by fourth-order Runge-Kutta method. The consequence of the present approach is a strong and robust tool without any time-consuming computational process. The applicability of the proposed method is verified by considering some benchmark problems from the literature and comparing the relevant results with those are achieved in this study. Finally, by solving several examples, the effects of various factors on the forced-vibration of proposed shells are evaluated.

Sound transmission characteristics of a composite sandwich plate using multi-layer first-order zigzag theory

This study investigates sound transmission through a composite sandwich plate with a soft core. Vibro-acoustic governing equations based on a new simple multi-layer first-order zigzag theory (MFZT) are presented. The transverse shear deformations of the face layers and core are described by different shear deformation theories, satisfying the interface’s continuity condition. First-order shear deformation theory (FSDT) is used for the top and bottom layer face, while linear functions are used to model the transverse and in-plane displacements of the soft core, respectively. The vibro-acoustic governing equations are derived based on Hamilton’s principle. The suggested model can accurately predict the vibration and sound transmission loss (STL) of the soft core composite sandwich plate under simply supported. The accuracy and convergence of the present model and numerical solutions are validated versus available studies. On that basis, the vibration and STL of composite sandwich plates with soft cores are studied, and the influence of parameter changes (face layer ply angles, incident elevation angles, face layer/core thickness and density) on the STL is discussed. The results show that the material properties can significantly affect the sound transmission loss and peak/valley frequencies of a composite sandwich plate with a soft core.

A kinematically consistent second-order computational homogenisation framework for thick shell models

This paper presents a kinematically consistent second-order computational homogenisation scheme for shear deformable shells. The proposed framework can accurately evaluate the membrane, bending, and transverse shear components of the shell resultants and tangent operators, whilst showing no size dependency on the fine scale model. To date, a proper extension of second-order homogenisation to a thick shell model, such as the 5-parameter formulation, remains non-trivial due to the difficulties in projecting the macroscopic transverse shear strains to the fine scale whilst satisfying the stress boundary conditions on the top and bottom faces. To overcome this, the paper proposes a novel volumetric constraint on the fluctuation moment field, that is used in conjunction with a set of constraints obtained through an orthogonality condition. The result is a consistent downscaling procedure that can properly downscale all the macroscopic shell strains. More notably, a pure transverse shear deformation can be achieved, thus producing a uniform parabolic shear stress distribution in homogeneous materials. The key equations for upscaling are obtained through the modified Hill–Mandel condition. In particular, the shell tangent operators are derived in closed form as a function of the Taylor upper bound and softening terms coming from the fluctuation matrices. The proposed framework is general and can incorporate full geometric and material nonlinearities across all the length scales of interest. Through a series of benchmarks, it is demonstrated that the constitutive tangents computed through the present framework correspond well with analytical solutions. Finally, excellent agreements in model responses, including through-thickness stress distributions are found between the multi-scale (FE2) and the full-scale models in the numerical benchmarks, featuring the nonlinear loading of thin and thick heterogeneous panels.

Parametric resonance of shear deformable nanotubes: A novel higher-order model incorporating nonlinearity from both curvature and inertia

The research on nonlinear parametric resonance is of significance in the application of nanotubes in nanoelectromechanical systems. To capture the nonlinear behaviors of the nanotube in an accurate way, a size-dependent nonlinear model fully considering the transverse shear deformation, the nonlinearity from curvature and inertia is proposed for the first time. Specifically, the size-dependency is modeled by the nonlocal strain gradient theory coupled with surface effect. By introducing the accurate curvature, Zhang Fu's higher-order beam theory is developed to establish the refined displacement field. Besides, the axial nonlinear inertial force caused by transverse vibration is derived by introducing the hypothesis of “non-extensible beam”. The two-step perturbation-incremental harmonic balance method (TSP-IHBM) is developed to obtain both the stability boundary and bifurcation diagram. The accuracy is confirmed through convergence analysis and the validation by Runge Kutta method. Results provide four bifurcation topologies with different stiffness parameters and reveal the relationships between the stability boundary and the bifurcation diagram. It is found that the inertia nonlinearity may change the bifurcation diagram from stiffness hardening characteristics to softening characteristics. Also, parameter analysis provides the influence law of different size-dependent effects on the stability boundary and bifurcation diagram.

ДОСЛІДЖЕННЯ ДЕФОРМУВАННЯ БАГАТОШАРОВОЇ ТРАНСВЕРСАЛЬНО-ІЗОТРОПНОЇ ПЛИТИ НА ЖОРСТКІЙ ОСНОВІ ЗА БЕЗЗГИНОВОЮ УТОЧНЕНОЮ КОНТИНУАЛЬНОЮ МОДЕЛЛЮ

The high-precision estimation of the stress-strain state (SSS) of multilayered plates on a rigid foundation under the action of stationary transverse loading is an urgent task. As its includes the calculations of strength and deformability of various homogeneous and multilayer coatings. This is the calculation of road surface on relatively rigid bridge structures, or on a non-deformable underlying layer or calculation protective multilayer coatings of flat structural elements of greater rigidity than coatings, etc. The combining of materials with isotropic and transversal-isotropic physical characteristics into a multilayer package allows creating of the multifunctional designs. The SSS of such structures due to their structural heterogeneity and the relatively low transverse stiffness of the individual layers is significantly associated with the effect of transverse shear deformations and transverse compression deformations. Therefore, the problem of refined modelling of SSS of plates, which takes into account these types of deformations, is an urgent one. Based on the decomposition the SSS of plate into the flexural and unflexural components, it is proposed to optimize the design diagram of deformation a rectangular multilayer plate on a rigid foundation. The essence of optimization is to consider such a design diagram of the plate, in which the SSS of plate would be fully described by only one component, namely the unflexural component of SSS. To do this, instead of the actual design of the multilayer plate, which is deformed without separation from the foundation, it is suggested to consider the design diagram of the plate, which is formed by supplementing it with a symmetric one about the contact surface of the foundation. In this case, the plate will be symmetrically loaded with respect to the middle surface of the plate, and the thickness of the plate will double. The SSS of plate will be unflexural, which greatly simplifies its modeling. For unflexural SSS, a twodimensional and high-degree iterative approximation but three-dimensional by the nature reflection of SSS, model of deformation of multilayer rectangular plates on a rigid foundation with isotropic and transverse-isotropic layers is constructed in an elastic formulation. This model takes full account deformations of transverse shear and of transverse compression at transverse loading of a plate. Calculations of homogeneous and two-layer transverse-isotropic plates on a rigid foundation under the action evenly distributed and localized transverse loads on the surface of a plate are performed by the finite difference method.

Geometrically nonlinear electro-thermo-static analysis of piezoelectric/CNT/GPL/fibre/polymer sandwich panels with double curvature resting on elastic foundation

Evaluation of the electro-thermo-static behaviour of the shell panels of double curvature using an analytical method is carried out in this paper. The proposed panels are made of a hybrid composite, and considered to be rested on the Winkler-Pasternak elastic foundation. The hybrid composite shell contains two thin piezoelectric sheets at its bottom and top surfaces, and some plies made of carbon fibres and polymeric resin which is reinforced by the randomly oriented nanocomposites. The employed nanocomposites for some layers are carbon nanotubes (CNTs) and for some other are graphene nanoplatelets (GPLs). The governing nonlinear system of differential equations of orthotropic doubly-curved shell panels are developed based on the first-order transverse shear deformation theory (FSDT) which incorporates moderately thick and also thin shells. The Gauss Quadrature method is applied to obtain the forces and moments due to the thermo-electrical loads, and the Galerkin method is employed to solve the developed system of equations which leads to a closed-form load-deformation relationship (equilibrium path of the shell). The novelty of the present study can be expressed in the solution and employed hybrid composite materials. First, by employing the proposed closed-form solution no time consuming computations are required. On the other hand, the effects of the piezoelectric and two different nanocomposite materials on the thermo-electro-mechanical nonlinear static behaviour of the hybrid composite shells can be evaluated. The accuracy of the present method is validated by comparing the achieved results for the some benchmark problems with those are existent in the literature. The impacts of various electro-thermo-mechanical and geometrical characteristics of the proposed shells on their equilibrium paths are also considered by carrying out an extended parametric study.

A new electro-mechanical finite formulation for functionally graded graphene reinforced composite laminated thick plates with piezoelectric actuator

Graphene is regarded as one of the most attractive reinforcements for composite materials attributing to the extraordinary physical characteristics. However, existing models in the literatures might meet severe challenges for the interlaminar stress prediction of functionally graded graphene reinforced composite (FG-GRC) laminated thick plate integrated with piezoelectric fiber reinforced composite (PFRC) actuator under electro-mechanical loadings. If transverse shear deformations cannot be described accurately, the mechanical performance of FG-GRC laminated plate with PFRC actuator will be significantly impacted by the electro-mechanical coupling effect and the sudden change of material characteristics at the interfaces. Thereby, a new electro-mechanical coupled plate model with only seven independent displacement variables is to be proposed in this paper. Employing the Reissner mixed variational theorem (RMVT), the precision of the transverse shear stresses considering the electro-mechanical coupling effect can be improved. Moreover, the second-order derivatives of in-plane displacement parameters have been removed from the transverse shear stress components, which can greatly simplify the finite element formulation. Thus, based on the proposed electro-mechanical coupled plate model, a simple C0-type finite element formulation is developed for interlaminar shear stress analysis of FG-GRC laminated thick plate with a PFRC actuator. The 3D elasticity solutions and the results obtained from other models are used to assess the performance of the proposed finite element formulation. Additionally, comprehensive parametric studies are achieved for the influences of graphene volume fraction, distribution pattern, electro-mechanical loading, boundary conditions, lamination scheme and geometrical parameters of the plate on the deformations and stresses of the FG-GRC laminated plate with a PFRC actuator.

Angular Displacement Control for Timoshenko Beam by Optimized Traveling Wave Method

The vibration of flexible structures in spacecraft, such as large space deployable reflectors, solar panels and large antenna structure, has a great impact on the normal operation of spacecraft. Accurate vibration control is necessary, and the control of angular displacement is a difficulty of accurate control. In the traditional control method, the mode space control has a good effect on suppressing low-order modes, but there is control overflow. The effect of traveling wave control on low-order modes is worse than the former, but it has the characteristics of broadband control. It can better control high-order modes and reduce control overflow. In view of the advantages and disadvantages of the two control methods, based on Timoshenko beam theory, this paper uses vector mode function to analyze the modal of spacecraft cantilever beam structure, establishes the system dynamic equation, and puts forward an optimized traveling wave control method. As a numerical example, three strategies of independent mode space control, traditional traveling wave control and optimized traveling wave control are used to control the active vibration of beam angle. By comparing the numerical results of the three methods, it can be seen that the optimal control method proposed in this paper not only effectively suppresses the vibration, but also improves the robustness of the system, reflecting good control performance. An innovation of this paper is that the Timoshenko beam model is adopted, which considers the influence of transverse shear deformation and moment of inertia on displacement and improves the accuracy of calculation, which is important for spacecraft accessory structures with high requirements for angle control. Another innovation is that the optimized traveling wave control method is exquisite in mathematical processing and has good results in global and local vibration control, which is not available in other methods.

New HSDT for free vibration analysis of elastically supported porous bidirectional functionally graded sandwich plate using collocation method

The present paper deals with the free vibration response of elastically supported porous bidirectional functionally graded soft-core sandwich rectangular plate with proposed new tangent shape function-based higher-order transverse shear deformation theory. The theory is accomplished to uphold the continuity of transverse shear stresses and zero shear stress conditions on both the extreme surfaces of a plate. The bidirectional gradation varies through the thickness ( z-axis) and length ( x-axis) directions. The effective material properties are calculated via the modified power-law. The governing differential equations (GDEs) of motion are acquired via the Hamilton principle. The inverse multi-quadric radial basis function-based Collocation method is implemented to discretize the GDEs obtained by strong form formulation. The current theory and method’s accuracy and efficacy are validated by comparing the present results with existing results in the literature. Studies show that the results of the proposed theory are in close agreement with 3D approach. Numerical results are obtained with a different scheme of sandwich layers for free vibration analysis. Effects of grading index, porosity fraction, two parameters foundation, different scheme, and the span to thickness ratio on free vibration response have been discussed. New results for free vibration response in porous media are presented with different skin materials.

Mathematical Modeling and Analytical Solution of Thermoelastic Stability Problem of Functionally Graded Nanocomposite Cylinders within Different Theories

Revolutionary advances in technology have led to the use of functionally graded nanocomposite structural elements that operate at high temperatures and whose properties depend on position, such as cylindrical shells designed as load-bearing elements. These advances in technology require new mathematical modeling and updated numerical calculations to be performed using improved theories at design time to reliably apply such elements. The main goal of this study is to model, mathematically and within an analytical solution, the thermoelastic stability problem of composite cylinders reinforced by carbon nanotubes (CNTs) under a uniform thermal loading within the shear deformation theory (ST). The influence of transverse shear deformations is considered when forming the fundamental relations of CNT-patterned cylindrical shells and the basic partial differential equations (PDEs) are derived within the modified Donnell-type shell theory. The PDEs are solved by the Galerkin method, and the formula is found for the eigenvalue (critical temperature) of the functionally graded nanocomposite cylindrical shells. The influences of CNT patterns, volume fraction, and geometric parameters on the critical temperature within the ST are estimated by comparing the results within classical theory (CT).