Trigonometric Shear Deformation Theory Research Articles

Effect of bi-directional material gradation on thermo-mechanical bending response of metal-ceramic FGM sandwich plates using inverse trigonometric shear deformation theory

PurposeThe purpose of this study is to investigate the bending analysis of metal (Ti-6Al-4V)-ceramic (ZrO2) functionally graded material (FGM) sandwich plate with material property gradation along length and thickness direction under thermo-mechanical loading using inverse trigonometric shear deformation theory (ITSDT). FGM sandwich plate with a ceramic core and continuous variation of material properties has been modelled using Voigt’s micro-mechanical model following the power law distribution method. The impact of bi-directional gradation of material properties over the bending response of FGM plate under thermo-mechanical loading has been investigated in this work.Design/methodology/approachIn this study, gradation of material properties for FGM plates is considered along length and thickness directions using Voigt’s micromechanical model following the power law distribution method. This type of FGM is called bi-directional FGMs (BDFGM). Mechanical and thermal properties of BDFGM sandwich plates are considered temperature-dependent in the present study. ITSDT is a non-polynomial shear deformation theory which requires a smaller number of field variables for modelling of displacement function in comparison to poly-nominal shear deformation theories which lead to a reduction in the complexity of the problem. In the present study, ITSDT has been utilized to obtain the governing equations for thermo-mechanical bending of simply supported uni-directional FGM (UDFGM) and BDFGM sandwich plates. Analytical solution for bending analysis of rectangular UDFGM and BDFGM sandwich plates has been carried out using Hamilton’s principle.FindingsThe bending response of the BDFGM sandwich plate under thermo-mechanical loading has been analysed and discussed. The present study shows that centre deflection, normal stress and shear stress are significantly influenced by temperature-dependent material properties, bi-directional gradation exponents along length and thickness directions, geometrical parameters, sandwich plate layer thickness, etc. The present investigation also reveals that bi-directional FGM sandwich plates can be designed to obtain thermo-mechanical bending response with an appropriate selection of gradation exponents along length and thickness direction. Non-dimensional centre deflection of BDFGM sandwich plates decreases with increasing gradation exponents in length and thickness directions. However, the non-dimensional centre deflection of BDFGM sandwich plates increases with increasing temperature differences.Originality/valueFor the first time, the FGM sandwich plate with the bi-directional gradation of material properties has been considered to investigate the bending response under thermo-mechanical loading. In the literature, various polynomial shear deformation theories like first-order shear deformation theory (FSDT), third-order shear deformation theory (TSDT) and higher-order shear deformation theory (HSDT) have been utilized to obtain the governing equation for bending response under thermo-mechanical loading; however, non-polynomial shear deformation theory like ITSDT has been used for the first time to obtain the governing equation to investigate the bending response of BDFGM. The impact of bi-directional gradation and temperature-dependent material properties over centre deflection, normal stress and shear stress has been analysed and discussed.

On aeroelastic flutter assessment of thick nanoshell convening two-phase fluid flow using mathematical simulation and deep-neural networks

It is well known that flow-induced vibration occurs often in many different sectors. Dynamic fluid forces are produced by the flow, and these forces cause the system to become excited. One of the most prominent flows in nature and industrial domains, such as industrial facilities, is the multiphase flow which may contribute to creating noise and vibration. As one specific example of a multiphase flow, a two-phase flow relies heavily on flow arrangement for its characteristics. Furthermore, despite the most recent developments and advancements, flow-induced vibration is still considered an open topic. So, in this work, for the first time, frequency analysis of a thick nanoshell convening two-phase fluid flow subjected to aerodynamic pressure is presented. Via nonlocal strain gradient theory (NSGT), Hamilton's principle, trigonometric shear deformation theory, the reappraisal of the fluid-structure interaction relation, and non-conservative aerodynamic pressure formulation, the governing equations, and various boundary conditions of the system are presented. After that, using the generalized differential quadrature method (GDQM) the obtained governing equations are solved, numerically. For more verification of the results, the results are compared with the outcomes of the artificial neural network (ANN) model. For this purpose, the obtained results using mathematical modeling are presented to generate the training dataset and test dataset of the neural network in batches. Finally, some suggestions for improving the stability of the thick nanoshell convening two-phase fluid flow subjected to aerodynamic pressure are presented for future handbooks.

Size-Dependent Frequency Analysis of Higher-Order Microplates with FGP Core and Polymeric CNTRC Faces Considering Piezoelectricity

The present study examines a microplate with a porous structure and two nanocomposite piezoelectric layers. All the layers’ properties are graded functionally, bonded to each other, and supported by an elastic foundation that can withstand both normal and shear loads. Additionally, carbon nanotubes are used to increase the electro-mechanical performance of the piezoelectric patches, which are exposed to an externally applied electric voltage. Using a higher-order trigonometric shear deformation theory and von Karman’s assumptions, the kinematic relations are demonstrated. The governing motion equations are derived using Hamilton’s principle and variational technique, and the modified couple stress theory is employed to take the scale effect into account. An analytical method based on Fourier series functions is used to solve the differential motion equations, and the impact of diverse factors such as porosity percentage, pore distribution patterns, carbon nanotubes distribution patterns, and other key parameters on the normalized frequencies of the model is analyzed after verifying the accuracy of the results. The findings of this research may aid in the development and production of smart structures and devices with increased efficiency.

Thermal Stress Analysis of Composite Laminates using Trigonometric Shear Deformation Theory and Finite Element Method

Laminated composite materials offer a versatile design approach for achieving the desired levels of stiffness and strength by selecting specific lamination schemes. The Trigonometric Shear Deformation Theory (TrSDT) effectively addresses the appropriate distribution of transverse shear strains throughout the plate thickness while maintaining stress-free boundary conditions on the plate's top surfaces. Consequently, there is no need for a shear correction factor. In this research paper, we use the Trigonometric Shear Deformation Theory (TrSDT) that takes into account the influence of transverse shear deformation. The in-plane displacement field incorporates a sinusoidal function with respect to the thickness coordinate to accommodate the effects of shear deformation. Theories that involve trigonometric functions based on the thickness coordinate in the displacement fields are collectively referred to as Trigonometric Shear Deformation Theories (TrSDTs). In the present study, we conduct a thermal stress analysis of Laminated Composite Plates using the TrSDT. This theory eliminates the need for shear correction factors and provides a more accurate distribution of interlaminar stresses compared to other methods like CPT and FOST. We assess deflection and stress at various locations and for different aspect ratios under thermal loads using TrSDT. Stress evaluations are carried out analytically, and the results are validated by comparing them with existing findings from the literature. To further verify our findings, we model a composite laminate under thermal loads using the commercial Finite Element Method tool ABAQUS, and our results are validated against those obtained with the TrSDT for plates with simply supported boundary conditions.

Numerical investigation of sandwich plate in bending by a new inverse shear deformation theory based on finite element analysis

Employing innovative kinematic function, a new inverse trigonometric shear deformation theory (nITSDT), variationally suitable, is developed to acquire pertinent data on the bending of sandwich plates subjected to transverse loads with variable aspect ratios(S). This nITSDT eliminates the need for shear correction factors since the transverse shear stress is directly determined by constitutive relations on the two extreme faces of sandwich plate satisfying the shear stress free surface circumstances. The governing equations and boundary conditions of the nITSDT are obtained by applying the dynamic version of the virtual work principle. For sandwich plates with simplly supports, solution is given by MATLAB code using Finite Element (FE) based on nITSDT. The findings of displacements and stresses are supported by those of more comprehensive theories, and the exact solution serves to highlight the viability of the proposed theory.

Magneto-electro-thermo-elastic frequency response of functionally graded saturated porous annular plates via trigonometric shear deformation theory

Magneto-electro-thermo-elastic frequency response of functionally graded saturated porous annular plates via trigonometric shear deformation theory

Modeling the Thermoelastic Bending of Ferric Oxide (Fe2O3) Nanoparticles-Enhanced RC Slabs.

Nanoparticles, by virtue of their amorphous nature and high specific surface area, exhibit ideal pozzolanic activity which leads to the formation of additional C-S-H gel by reacting with calcium hydroxide, resulting in a denser matrix. The proportions of ferric oxide (Fe2O3), silicon dioxide (SiO2), and aluminum oxide (Al2O3) in the clay, which interact chemically with the calcium oxide (CaO) during the clinkering reactions, influence the final properties of the cement and, therefore, of the concrete. Through the phases of this article, a refined trigonometric shear deformation theory (RTSDT), taking into account transverse shear deformation effects, is presented for the thermoelastic bending analysis of concrete slabs reinforced with ferric oxide (Fe2O3) nanoparticles. Thermoelastic properties are generated using Eshelby's model in order to determine the equivalent Young's modulus and thermal expansion of the nano-reinforced concrete slab. For an extended use of this study, the concrete plate is subjected to various mechanical and thermal loads. The governing equations of equilibrium are obtained using the principle of virtual work and solved using Navier's technique for simply supported plates. Numerical results are presented considering the effect of different variations such as volume percent of Fe2O3 nanoparticles, mechanical loads, thermal loads, and geometrical parameters on the thermoelastic bending of the plate. According to the results, the transverse displacement of concrete slabs subjected to mechanical loading and containing 30% nano-Fe2O3 was almost 45% lower than that of a slab without reinforcement, while the transverse displacement under thermal loadings increased by 10%.

Eigen value analysis of composite hollow shafts using modified EMBT formulation considering the shear deformation along the thickness direction

Composite hollow shafts are used in power transmission applications due to their high specific stiffness and high specific strength. The dynamic characteristics of these shafts are important for transmission applications. Dynamic modelling of these shafts is generally carried out using Equivalent Modulus Beam Theory (EMBT) and Layerwise Beam Theory (LBT) formulations. The EMBT formulation is modified by considering stacking sequence, shear normal coupling, bending twisting coupling and bending stretching coupling. It is observed that modified EMBT formulation is underestimating the shafts stiffness at lower length/mean diameter (l/dm) ratios. In the present work, a new formulation is developed by adding shear deformation along the thickness direction to the existing modified EMBT formulation. The variation of shear deformation along the thickness direction is found using different shear deformation theories, i.e., first-order shear deformation theory (FSDBT), parabolic shear deformation theory (PSDBT), trigonometric shear deformation theory (TSDBT), and hyperbolic shear deformation theory (HSDBT). The analysis is performed at l/dm ratios of 5, 10, 15, 20, 25, 30, 35, and 40 for carbon/epoxy composites, E-glass/epoxy composites, and boron/epoxy composite shafts. The results show that new formulation has improved the bending natural frequency of the composite shafts for l/dm < 15 in comparison with modified EMBT. The effect of new formulation is more significant for the second and third bending modes of natural frequencies.

Static analysis of an exponentially varying functionally graded beam using trigonometric shear deformation theory

The present study investigates the bending behavior of a functionally graded beam under transverse loads (sinusoidal and uniform) using trigonometric shear deformation theory (TSDT). Boundary value problem are derive using principle of virtual work. The Navier-type closed-form solution is obtained under simply supported boundary conditions. The material properties across the thickness directions vary according to the exponential law of distribution. The present theory does not need a problem-dependent shear correction factor since it accurately accounts for the distribution of transverse shear stress across the beam thickness. Results from higher-order shear and normal deformation theory (HOSNT), exact elasticity theory, and other theories are compared with the current approach to ensure its accuracy and efficacy. The results of present theory is in excellent agreement with the exact elasticity theory.

A new trigonometric shear deformation theory for free vibration of graphene reinforced metal–matrix nanocomposite plate submerged in fluid medium

A novel trigonometric shear deformation plate theory (TSDPT) is proposed for the vibrational characteristics of the functionally graded graphene-reinforced metal matrix nanocomposite (FG-GRMMNC) plate immersed in the fluid medium. Combining trigonometric and inverse trigonometric functions, a new shape function is presented to determine the distribution of the transverse shear strains and stresses without using any transverse shear correction factors. Using a combination of the Halpin–Tsai micromechanical model and the rule of mixtures, the properties of three distributions of GRMMNC are investigated. Aluminum serves as the matrix material, and graphene nanoplatelets are utilized as the reinforcing material. Due to the ideal and incompressible fluid medium, the influence of hydrostatic pressures and free surface waves on structures is disregarded. The governing equations are derived using the Hamilton’s principle, taking into account the influences of three plate–fluid interaction models. The Galerkin’s method is then utilized to determine the natural frequencies of the FG-GRMMNC plate with simply supported boundary conditions. To confirm the validity of the proposed theory, the obtained outcomes are compared with those found in the published works. On the vibrational characteristics of the nanocomposite plate, influences of material properties, distribution patterns, geometrical parameters, and fluid environments are also investigated in depth.

Comparative studies between Semi-analytical and shear deformation theories for functionally graded beam under bending

In this paper, the bending response of functionally graded (FG) beam using three different approaches. viz. semi-analytical, higher-order shear and normal deformation theory (HOSNT), and trigonometric shear deformation theory (TSDT) under transverse loading conditions has been investigated. The boundary value problem (BVP) of the HOSNT and TSDT is derived using principle of virtual work whereas semi-analytical approach consist of formation of two-point BVP along the beam depth. All three models satisfy zero shear stress conditions at the top and bottom surfaces of the beam. The material properties of the FG beam vary across the thickness direction according to the power-law distribution. Aluminium-zirconia and aluminium-alumina are two types of compositions used to create FG beams. For comparison purpose, the authors have generated results for semi-analytical and HOSNT, being not available in the published literature. The numerical results are compared with previously published solutions to assess the correctness and effectiveness of the three models.

Thermoelastic bending analysis of laminated composite shells using a trigonometric shear and normal deformation theory

In the present study, a trigonometric shear and normal deformation theory using trigonometric functions in the displacement field is developed and employed for the static analysis of laminated composite spherical shells subjected to sinusoidal mechanical/thermal loads with simply supported boundary conditions. In the present theory, the effect of both transverse shear and normal strain is considered. The present theory satisfies the zero transverse shear stress conditions at the top and the bottom surfaces of the shell. By using the principle of virtual work, the governing equations and boundary conditions are derived. Navier’s solution procedure is employed to solve the governing differential equations and satisfying the boundary conditions. For validation of the present theory, obtained results are compared with higher-order theories and 3D elasticity solutions wherever possible. Thermal stresses in laminated plates are deduced from the present shell theory and compared with the exact thermoelastic solutions. Thermal stresses presented for the laminated composite spherical shells can serve as a benchmark for future work.

Thermal buckling behavior of power and sigmoid functionally graded material sandwich plates using nonpolynomial shear deformation theories

PurposeThe purpose of this article is to carry out the thermal buckling analysis of power and sigmoid functionally graded material Sandwich plate (P-FGM and S-FGM) under uniform, linear, nonlinear and sinusoidal temperature rise.Design/methodology/approachThermal buckling of FGM Sandwich plates namely, FGM face with ceramic core (Type-A) and homogeneous face layers with FGM core (Type-B), incorporated with nonpolynomial shear deformation theories are considered for an analytical solution in this investigation. Effective material properties and thermal expansion coefficients of FGM Sandwich plates are evaluated based on Voigt's micromechanical model considering power and sigmoid law. The governing equilibrium and stability equations for the thermal buckling analysis are derived based on sinusoidal shear deformation theory (SSDT) and inverse trigonometric shear deformation theory (ITSDT) along with Von Karman nonlinearity. Analytical solutions for thermal buckling are carried out using the principle of minimum potential energy and Navier's solution technique.FindingsCritical buckling temperature of P-FGM and S-FGM Sandwich plates Type-A and B under uniform, linear, non-linear, and sinusoidal temperature rise are obtained and analyzed based on SSDT and ITSDT. Influence of power law, sigmoid law, span to thickness ratio, aspect ratio, volume fraction index, different types of thermal loadings and Sandwich plate types over critical buckling temperature are investigated. An analytical method of solution for thermal buckling of power and sigmoid FGM Sandwich plates with efficient shear deformation theories has been successfully analyzed and validated.Originality/valueThe temperature distribution across FGM plate under a high thermal environment may be uniform, linear, nonlinear, etc. In practice, temperature variation is an unpredictable phenomenon; therefore, it is essential to have a temperature distribution model which can address a sinusoidal temperature variation too. In the present work, a new sinusoidal temperature rise is proposed to describe the effect of sinusoidal temperature variation over critical buckling temperature for P-FGM and S-FGM Sandwich plates. For the first time, the FGM Sandwich plate is modeled using the sigmoid function to investigate the thermal buckling behavior under the uniform, linear, nonlinear and sinusoidal temperature rise. Nonpolynomial shear deformation theories are utilized to obtain the equilibrium and stability equations for thermal buckling analysis of P-FGM and S-FGM Sandwich plates.

A novel trigonometric shear deformation theory for the buckling and free vibration analysis of functionally graded plates

In the current work, a four-node rectangular finite element with assumed natural shear strains, based on the exact neutral surface position and new higher-order shear deformation theory is employed to perform mechanical buckling and free vibration analysis of functionally graded plates. The present theory has only five unknowns and satisfies zero transverse shear stress at the top and bottom surfaces of the plate, therefore a shear correction factor is not necessary. The accuracy of the proposed formulation is evaluated by comparing the obtained results with those available in the literature.

Numerical Analysis of Thick Isotropic and Transversely Isotropic Plates in Bending using FE Based New Inverse Shear Deformation Theory

In this work, using new inverse trigonometric kinematic displacement function, the bending solution of simply supported isotropic and transversely isotopic thin, moderately thin and thick square plates with aspect ratio variations is given. The paper introduces a new inverse trigonometric shear deformation theory (nITSDT) for the bi-directional bending study, which is variationally compatible. The transverse shear stress can be obtained directly from the constitutive relationships on the top and bottom surfaces of the plate that satisfy the shear stress free surface conditions, so the theory does not need a factor for shear correction. The dynamic version of the virtual work principle is used to obtain the governing equations and boundary conditions of the theory. The Finite Element (FE) solution has been developed using MATLAB code based on the present theory for simply supported laminated composite plates. In order to illustrate the efficiency of the proposed theory, the results of displacements and stresses are compared with those of other refined theories and exact solution. The findings obtained from the use of the theory are found to agree well with the precise results of elasticity.

Thermal Buckling of Laminated Composite Plates Using a Simple Four Variable Plate Theory

In this study, the thermal buckling behavior of composite laminate plates cross-ply and angle-ply all edged simply supported subjected to a uniform temperature field is investigated, using a simple trigonometric shear deformation theory. Four unknown variables are involved in the theory, and satisfied the zero traction boundary condition on the surface without using shear correction factors, Hamilton's principle is used to derive equations of motion depending on a Simple Four Variable Plate Theory for cross-ply and angle-ply, and then solved through Navier's double trigonometric sequence, to obtain critical buckling temperature for laminated composite plates. Effect of changing some design parameters such as, orthotropy ratio (E1/E2), aspect ratio (a/b), thickness ratio (a/h), thermal expansion coefficient ratio (α2/α1), are investigated, which have the same behavior and good agreement when compared with previously published results with maximum discrepancy (0.5%).

Free vibration and buckling analysis of FGM plates using inverse trigonometric shear deformation theory

PurposeThe purpose of this paper is to carry out free vibration and buckling analysis of functionally graded material (FGM) plate.Design/methodology/approachEquilibrium and stability equations of FGM rectangular plate under different boundary conditions are derived using finite element method-based inverse trigonometric shear deformation theory (ITSDT). Eight-noded rectangular plate element with seven degrees of freedom at each node is used for the present analysis. The power-law distribution method has been considered for the continuously graded variation in composition of the ceramic and metal phases across the thickness of a functionally graded plate.FindingsThe finite element formulation incorporated with ITSDT and provisions of the constitutive model of FGM plate has been implemented in a numerical code to obtain the natural frequency and critical buckling load under uniaxial and biaxial compressive load. The influence of material gradation, volume fraction index, span to thickness ratio and boundary constraints over free vibration and buckling response has been studied.Originality/valueDevelopment and validation of finite element methodology using ITSDT to predict the structural response of the FGM plates under different loading, geometric and boundary conditions.

Active Control of Thermo-mechanical Buckling of Composite Laminated Plates Using Piezoelectric Actuators

This paper is concerned with the active control of thermo-mechanical buckling of composite laminated plates using piezoelectric facesheets as actuators. The four-variable trigonometric shear deformation theory and Hamilton’s principle are applied to formulate the governing equation of structural system. The temperature feedback control strategy is proposed to conduct the active control of thermal-mechanical buckling. The simulation results show that the thermo-mechanical buckling of composite laminated plates can be effectively controlled by the presented control method. With a specific control gain, the critical mechanical buckling load can remain constant at different temperatures. The effects of geometric parameters, fiber angle, stacking sequence, position of piezoelectric layer and boundary conditions on the active control of thermo-mechanical buckling are also investigated.

Optimization of thermal buckling control for composite laminates with PFRC actuators using trigonometric shear deformation theory

This work aims to explore optimal design and control of thermal buckling for composite laminates with piezoelectric fiber reinforced composite (PFRC) actuators. Four-variable trigonometric shear deformation theory without introduction of a shear correction factor is used in the structural modeling. The high accuracy of the present approach is examined by analyzing the critical buckling temperature and natural frequency compared with references results. A temperature feedback control approach is proposed in designing the controller. Taking fiber angle as the optimization design variable, layerwise optimization approach (LOA) is applied to maximize critical buckling temperature of piezolaminated plate and a set of stacking sequences are obtained. The resulting analysis demonstrates that the proposed control approach can clearly increase the critical buckling temperature of piezo-laminated plates. The optimization process developed is useful in devising stacking sequences for the piezo-laminated structures in thermal environments.

Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory

In the present paper, a new trigonometric two-variable shear deformation beam nonlocal strain gradient theory is developed and applied to investigate the combined effects of nonlocal stress and strain gradient on the bending, buckling and free vibration analysis of nanobeams. The model introduces a nonlocal stress field parameter and a length scale parameter to capture the size effect. The governing equations derived are solved employing finite element method using a 3-nodes beam element, developed for this purpose. The predictive capability of the proposed model is shown through illustrative examples for bending, buckling and free vibration of nanobeams. Comparisons with other higher-order shear deformation beam theory are also performed to validate its numerical implementation and assess its accuracy within the nonlocal context.