Third-order Shear Deformation Theory Research Articles

Analysis of Laminated Composite Porous Plate under Sinusoidal Load with Various Boundary Conditions

Bending analysis was carried out for a laminated composite porous plate due to sinusoidal loading with various boundary conditions using improved third-order theory. Zero transverse shear stress provided a free surface at the top and bottom of the plate. Also, the authors developed a finite element formulation based on improved third-order shear deformation theory. To circumvent the C1 continuity requirement associated with improved third-order shear deformation theory, a C0 FE formulation was developed by replacing the out-of-plane derivatives with independent field variables. An in-house FORTRAN code was developed for the bending analysis of the laminated porous plate considering a 2D finite element model. The complete thickness of the plate was covered with different porosity patterns. The impacts of various modulus ratios, boundary conditions, thickness ratios, fiber orientation angles, and material parameters were examined for laminated porous plates. There was an 18.8% reduction in deflection in the case of the square plate as compared to rectangular plates, with a porosity value of 0.1, a thickness ratio of 10, and an orientation angle of 0°/90°/0°. According to the current research, adding porosities causes a relatively greater change in deflection rather than stress, thereby aiding in the development of a lightweight structure.

Static and dynamic response of pyramidal lattice sandwich plate with composite face sheets reinforced by graphene platelets

For the pursuit of lightweight structures with excellent mechanical performance, sandwich structures with lightweight cores and strengthened face sheets have become popular in the advanced engineering fields in recent years. In this paper, the nonlinear static and dynamic responses of the sandwich plate constructed with the aluminum pyramidal lattice core and functionally graded graphene platelets reinforced composite (FG-GPLRC) face layers are analyzed. The partial differential equations of the pyramidal lattice sandwich plate considering the geometric nonlinearity are achieved based on the Reddy’s third order shear deformation theory (TSDT), which are solved by the combination of finite element model, Raphson’s method and Newmark numerical integration method. Four different types of the impact loads including the step loading, triangular loading, half-sine loading and exponential loading with the same peak value and duration time are investigated. The influences of the thickness-to-truss radium ratio and the angle of the truss core, as well as the weight fraction and distribution patterns of graphene platelets (GPLs) over the face sheets under different impact loads and different boundary conditions are analyzed. According to the numerical results, it is concluded that the sandwich plate with higher weight fraction of GPLs, lower thickness-to-truss radium ratio, and higher angle of the truss core can exhibit better performance in both static and dynamic loading conditions. The FG-X distribution of GPLs presents the best dynamic performance as compared to the other GPL distributions. The combination of lightweight truss core with the advanced GPLRC face sheets is instructive for potential applications and further research.

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.

The effect of nonsimultaneous impact of three impactors with spherical tip on a beam

Purpose The purpose of this paper is to investigate the nonsimultaneous impact of three impactors with spherical tip on the response of a low-velocity impact on a beam. Design/methodology/approach In this research, the third-order shear deformation theory of the beam with hyperbolic shear-strain function is used. Hamilton’s principle is applied to derive the motion equations. To simulate nonsimultaneous impacts, by using the Hertz nonlinear contact law, the contact of the impactors with different times is simulated. Comparisons with other articles are carried out in the one impactor form. Findings In the parametric study, the histories of the contact force and displacement of the beam are investigated in the presence of only one impactor in the center of the beam and also in the presence of three impactors, one in the center of the beam and the other two around the first impactor with a delay. One of the important and noteworthy points is that the presence of two impactors with a delay causes the maximum contact force and contact time to decrease and the maximum displacement of the beam center to increase. Originality/value The original point of this paper is what is the difference between the impact response of one projectile and three nonsimultaneous projectiles on the beam.

Size-dependent nonlinear vibration analysis of nonlocal magneto-electro-elastic sandwich nanobeam based on Reddy’s third-order shear deformation theory

Size-dependent nonlinear vibration analysis of nonlocal magneto-electro-elastic sandwich nanobeam based on Reddy’s third-order shear deformation theory

Thermal buckling analysis for hybrid and composite laminated plate by using new displacement function

Abstract The analysis of a critical buckling temperature is dependent on the new displacement function field provided in the literature and uses third-order shear deformation theory. The value of the parameter “m,” which determines the new displacement function, closed with the three-dimensional (3D) elasticity results. Thermal critical buckling is analyzed for thick and thin laminated plates by using the Navier solution for symmetric and anti-symmetric cross-ply simply supported boundary conditions. Several design parameters are considered such as extension thermal coefficient ratio (α 1⁄α 2), number of schemes, thickness ratio (a/h), aspect ratio (a/b), and modular ratio (E 1/E 2) when analyzing the dynamic behavior of the critical buckling temperature for different materials, including composite (glass/epoxy) and hybrid (glass/carbon/epoxy), under uniformly distributed temperature load. The accuracy for theoretical results by using Matlab R2019b was checked with other results for different researchers and gave good agreement. Increasing orthotropic ratio and aspect ratio resulted in increasing critical buckling temperature in M1 than M2, whereas increasing thickness ratio, thermal coefficient expansion, and number of layers resulted in decrease in critical buckling temperature in M2 than M1.

Surface crack effect on frequency and vibration mode switching of solar-powered aerospace structures

In the engineering design of modern wing panels and their optimal control system, the first step is to find and predict their dynamic characteristics. These characteristics are dependent on material properties, panel geometry, and boundary conditions. In addition, possible fracture existence, imperfections and environmental conditions may unpredictably change the frequency and modes shape of the panel and then the parameters of a control system. In the present paper, the effect of surface fracture depth and orientation on frequency and vibration mode switching of perovskite solar cells-based aircraft wing panels is investigated for the first time. The PSC (perovskite solar cell) panel is made of a polymer substrate reinforced with even-distributed graphene platelets (GPLs) and thin solar layers to enhance the overall electromechanical performance of the whole structure. Equations of motion of the panel are obtained by the Hamilton approach based on the properties of Reddy's third-order shear deformation theory and linear components of the Green-Lagrange strain tensor. The novel aspect is a line-spring model of surface crack by Cartmell et al. adapted to the model of the PSC panel under consideration. The discretization of the established equations of motion for the cracked panel is carried out by the element-free IMLS-Ritz method with mesh-free features. Validation of the obtained solutions is carried out by comparisons with results from previous studies for cracked laminated plates.First, material and geometric properties effect on vibration characteristics of the PSC structure was investigated. Next, partial surface crack depth and orientation angle are examined to show changes in frequency and vibration modes shape of the panel. It is shown, that there is a mode swapping between the second and third modes when transitioning from an intact panel to a cracked one. As the crack length increases, higher-order mode variations become more noticeable, potentially leading to their swapping or the excitation of new ones. Additionally, the findings demonstrate that the surface crack orientation angle significantly impacts the higher-order modes of the panel. When the orientation angle is 90°, the first six modes align with those of the intact plate. Mode shapes and frequency shifts are significant for crack orientation angles between 0° and 90°.

A compressible layerwise third-order shear deformation theory with transverse shear stress continuity for laminated sandwich plates

Sandwich structures with a soft core typically exhibit multiple deformation modes such as bending, twisting and compression when subjected to transverse loads. Compressive behavior of soft core significantly affects the structural response, which is usually neglected in conventional plate theories. Additionally, conventional plate theories do not take into account transverse shear stress continuity. In order to address this concern and precisely characterize the characteristics of arbitrary multilayer laminated structures, a novel electromechanical coupled finite element model is developed. This model is based on a compressible layerwise third-order shear deformation theory with a soft core. Notably, the proposed model theory requires only twelve degrees of freedom for each quadratic element node. The accuracy of the model is validated first. Afterwards, several numerical investigations are carried out for piezolaminated sandwich structures with different width-to-thickness ratios, reinforcement angle and skew angle. The results demonstrate the applicability of the current model in predicting the behavior of laminated sandwich plates.

Advanced frequency of thick FGM cylindrical shells with fully homogeneous equation

The effects of shear deformation theory and improved shear correction factors on the advanced computation of frequencies by using the fully homogeneous equation for thick functionally graded material (FGM) circular cylindrical shells are studied. It is quite reasonable to consider the extra advanced effect of third-order shear deformation theory (TSDT) of displacements on the varied shear correction coefficient. The values of advanced nonlinear shear correction coefficient are usually functions of the nonlinear coefficient term in TSDT, power-law exponent parameter, and environment temperature. The main achievements in the nonlinear case of displacements and with the varied value of shear correction coefficients, the non-dimensional fundamental frequencies are estimated, investigated, and compared with the values in the linear case. This paper aims to study the fundamental frequencies of very thick FGM cylindrical shells under the advanced effects of shear deformation theory and improved shear correction factors.

Novel Kuhn-Tucker conditions for vibration analysis in a functionally graded porous beam using the R-program

Functionally graded materials provide a flexible and individualized strategy for material design, allowing for optimization of properties and performance for particular purposes. The investigation considers the effects of simply supported (SS), clamped-clamped (CC) and clamped-free (CF) configurations. The study examines the vibration characteristics of bi-directional functionally graded porous beams (BDFGPB) using the third-order shear deformation theory, considering both even and uneven porosity conditions. The Hamilton method is used to derive equilibrium equations for beams, which are then solved using the Kuhn-Tucker technique and R-program. The BDFGPB's validity was verified by comparing it with open literature, revealing deviations of 3.19%, 1.25%, and 2.15% in non-dimensional natural frequency for SS, CC and CF boundary conditions. Furthermore, as the porosity index increases, the dimensionless natural frequency decreases, reducing beam stiffness and rigidity. This study demonstrates that porosity plays a critical role in the design of modern structures, as its ratio greatly impacts their performance and responsiveness.

Geometrical and material optimization of the functionally graded doubly-curved shell by metaheuristic optimization algorithms

This paper presents the simultaneous geometrical and material optimization of functionally graded doubly-curved shells under static load and free vibration. The weak form of governing equations is derived from third-order shear and normal deformation theory. Bending and free vibration analyses were conducted using the finite element method. The functionally graded material (FGM) is composed of metal and ceramic, and Voigt’s rule of mixtures is applied to capture the characteristics of FGMs throughout the shell thickness direction. In this regard, two multi-objective optimization algorithms (i.e., multi-objective particle swarm optimization and multi-objective cuckoo search optimization algorithms are employed. The objective functions combine minimizing the maximum normalized displacement and mass and maximizing the first dimensionless frequency. The results show that where minimizing the mass of the FG shell structure was considered an objective function, the value of the power index is equal to zero for almost all studied cases.

Multimodal Resonance Response of Incompressible Hyperelastic Moderately Thick Cylindrical Shells

Based on the modified third-order shear deformation theory, the harmonic balance method, and the pseudo-arclength continuation method with two-point prediction, the nonlinear forced vibration response of incompressible hyperelastic moderately thick cylindrical shells subjected to a concentrated harmonic load at mid-span and simply supported boundary conditions at both ends is investigated. The algorithmic procedure for solving steady-state periodic solutions of strongly nonlinear systems of differential equations is presented. The structural response characteristics of shells under different excitation amplitudes and structural parameters are analyzed. The numerical results indicate that the aspect ratio of moderately thick hyperelastic cylindrical shells has a significant effect on the natural frequency ratio. Different frequency ratios lead to varying nonlinear mode coupling effects. The coupling effects among modes result in complex nonlinear behavior in the vibration response of each mode, leading to abundant multi-valued phenomena in the response curve.

Free vibration and static analysis of sandwich composite plate with auxetic core and GPLRC facing sheets in hygrothermal environment

The main objective of this study is to investigate the vibration and static analysis of a sandwich composite plate in the hygrothermal environment. The plate consists of three layers such as an Aluminum auxetic core and Graphene platelet-reinforced composite (GPLRC) facing sheets. Based on the Halpin-Tsai theory and Gibson's model, the macro mechanical properties of facing sheets and auxetic core are derived. Via Reddy's third-order shear deformation theory (TSDT), the equations of motion of the sandwich plate are obtained and solved by both Navier and the generalized differential quadrature method (GDQM). The current formulation is validated by comparing the numerical results with those that are reported in the literature. Finally, the influence of different parameters such as the inclined angle of auxetic cells, thickness to the length, and core to total thickness on the natural frequencies, deflection and stresses of the sandwich plate, is examined. Even though some studies have been conducted to investigate the bending or vibration behavior of sandwich structures with negative Poisson's ratio (NPR) cores, the bending and vibration behavior of such structures in the hygrothermal environment is remained unexplored. Additionally, the analysis of stress component variations along the thickness of the plate was a novel feature that has not been addressed in prior studies on the similar structure.

The efficient data-driven solution to estimate the nonlinear bending of sandwich doubly curved panel subjected to transient loading

This paper investigates nonlinear transient bending characteristics of sandwich doubly curved panels under transient loading which is extensively used in aerospace structures. Due to the lack of a dataset for the nonlinear transient bending of the sandwich doubly curved panel for the data-driven solution, mathematical modeling is used. The sandwich doubly curved panel is made of two graphene nanoplatelets (GPLs) reinforced face sheets, and a polymer core. In the mathematical modeling section, the effective Young's modulus is obtained via modified Halpin-Tsai methodology, and the role of the mixture is used to estimate other material properties of the composite face sheets. The presented structure is analyzed using Reddy's third-order shear deformation theory (RTSDT), which incorporates von Kármán nonlinear geometric assumptions and Hamilton's principle. The governing equations and associated boundary conditions are derived accurately by imposing shear stress-free conditions on the upper and lower surfaces of the panel. The meshfree radial point interpolation method (RPIM) with interpolation functions is utilized to determine the unknown degrees of freedom to numerically simulate the mid-plane of the sandwich doubly-curved panel. The Newton-Raphson approach is used to solve the nonlinear ordinary differential using the constant-average acceleration strategy from Newmark's time integration method. After obtaining the dataset, the data is collected, and using an efficient data-driven approach, the results are tested, validated, and trained. In future studies, the presented efficient data-driven approach can be used with low computational cost to simulate nonlinear transient bending characteristics of sandwich doubly curved panels under transient loading for related aerospace industries.

Optimizing fiber paths of tow-steered laminated composites for parametric stability using isogeometric analysis and genetic algorithm

An isogeometric formulation is presented for fiber path optimization of tow-steered composite laminates (TSCL) with minimal parametric instabilities. Here, the fiber path functions of a TSCL is expressed smoothly by Cp−1 continuous non-uniform rational B-splines (NURBS) for engineering shapes, providing accurate field solutions. In this way, the motion equations of a TSCL plate is developed based on the third-order shear deformation theory (TSDT) and the Bolotin’s approximation is used to determine the parametric instability region. Primarily, the accuracy and efficiency of presented framework is measured for a TSCL example with respect to the existing solutions. Then, the optimal fiber paths of TSCL plates are searched using the genetic algorithm (GA) for various geometries, force component combinations, and dynamic-to-static force ratios. Results reveal that the optimal fiber paths are not necessarily symmetric even for regular domains, demonstrating the non-uniform coupling of bending and twisting stiffness in TSCL plates. The optimal design of TSCL fiber paths necessitates the consistency of local stiffness distribution and the resultant bending–twisting mode shape, specially when the dynamic force is high with respect to the static one. Geometry, force component combination, and dynamic-to-static force ratio impose case-specific fiber paths for optimal TSCL plates, requiring both minimal deformation and instability opening.

Advanced Thick FGM Plates-Cylindrical Shells in Linear Unsteady Supersonic Flow

Introduction: Thick functionally graded (FG) plates-cylindrical shells with third-order shear deformation theory (TSDT) of displacement models under thermal vibration in linear unsteady supersonic flow are investigated by using the generalized differential quadrature (GDQ) method. Method: The nonlinear coefficient term of TSDT models and the advanced nonlinear shear correction coefficient expression are used to study the thermal vibration of thick FG plates and cylindrical shells in linear unsteady supersonic flow. At the intersection of FG plates-cylindrical shells, the displacements, stress resultants, and moment resultants could be reasonably assumed in the continuity condition for the differential volume in linear unsteady supersonic flow, respectively. Result: The dynamic equilibrium differential equations of thick FG plates and cylindrical shells in linear unsteady supersonic flow, respectively in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads consist of the linear supersonic aerodynamic pressure and inertia terms can be presented. Two parametric effects including environment temperature and functionally graded material (FGM) power law index on the thermal stress and center displacement of thick FG plates-cylindrical shells in linear unsteady supersonic flow are investigated. Conclusion: The linear unsteady supersonic flow over the outer surface of thick FG platescylindrical shells provides the additional aerodynamic effect on the values of displacements and stresses.

A novel uncertainty quantification method for determining deformations and reliabilities of stochastic laminated composite plates with geometric nonlinearity

Stochastic finite element method (SFEM) for uncertainty quantification is widely applied for analysis of structures with intrinsic randomness. For determining the geometrically nonlinear deformations of laminated composite plates with random fields, the existing intrusive SFEMs have the limitations of low applicability, insufficient accuracy, or low efficiency. To this end, this paper proposes a novel and efficient non-intrusive SFEM incorporating direct probability integral method to achieve probability density functions (PDFs) of stochastic responses and reliabilities of laminated composite plates with geometric nonlinearity. Firstly, the von Kármán strain-displacement relation based on the third-order shear deformation theory is employed to model the geometric nonlinearity of laminated plates. The random field is discretized via Karhunen–Loève expansion. Secondly, the probability density integral equation (PDIE) is derived from the new perspective of probability conservation. The proposed non-intrusive SFEM decouples the equilibrium equation and PDIE to compute the response PDFs and reliabilities of uncertain laminated composite plates in a unified way. Moreover, the criterion which can judge the applicability of geometrically nonlinear theory is suggested for performing uncertainty quantification. Finally, comparisons of the results in terms of Monte Carlo simulation and the literature demonstrate the high accuracy and efficiency of the proposed method. For stochastic laminated plates, the response statistical moments vary nonlinearly with linear increase of load amplitude due to geometric nonlinearity, the deflection variability increases and structural reliability decreases with the increase in variability and correlation length of random field, and the stacking angle significantly affects the stochastic nonlinear deflections and reliabilities.

Effects of functionally graded graphene reinforcements on nonlinear post-local buckling and axial stiffness of laminated channel section struts

The role of local buckling on the behavior of slender members under compression has received considerable attention in structural engineering. For thin-walled sections, in particular, there is a noticeable decrease in the axial compressive stiffness, resulting in a substantial reduction in their load-bearing capacity due to the occurrence of local buckling. The principal purpose of this article is to explore the potential improvements in the postbuckling characteristics of polymeric composite channel section struts subjected to a progressive end-shortening by employing multi-layer graphene sheets reinforcements. The solution methodology incorporates the von Karman geometrical nonlinearity and is based on the layerwise third-order shear deformation theory (LW-TSDT). To verify the accuracy of the results obtained based on LW-TSDT and to evaluate its computational efficiency, a three-dimensional (3D) finite element model is also developed using ABAQUS for the comparative analysis. A thorough examination of nonlinear instability is conducted on composite laminated channel section struts, featuring distinctive graphene distribution patterns through the thickness directions of the flanges and webs to identify the most effective material distribution with the objective of a significant increase in critical end-shortening and axial compressive stiffness. The influence of the geometrical parameters on the critical end-shortening, postbuckling equilibrium paths, and load-bearing capacity of functionally graded graphene reinforced composite (FG-GRC) laminated channel section struts are elicited. The conducted parametric analyses emphasize that altering the distribution patterns of graphene reinforcement across the flanges and web can enhance the critical end-shortening and load-bearing capacity by 80% and 25%, respectively.

Advanced Frequency Study of Thick FGM Cylindrical Shells by Using TSDT and Nonlinear Shear

For the advanced frequency study of thick functionally graded material (FGM) circular cylindrical shells, it is interesting to consider the extra effects of nonlinear coefficient term in third-order shear deformation theory (TSDT) of displacements on the calculation of varied shear correction coefficient. The formulation for the advanced nonlinear shear correction coefficient are based on the energy equivalence principle. The values of nonlinear shear correction coefficient are usually functions of nonlinear coefficient term of TSDT, power-law exponent parameter and environment temperature. The free vibration frequencies of thick FGM circular cylindrical shells are investigated with the simply homogeneous equation by considering that simultaneous effects of the TSDT, the nonlinear shear correction coefficient of transverse shear force and the two direction of mode shapes. The novelty is more important and reasonable for the thick FGM circular cylindrical shells, especially for the ratio of length to thickness is five by considering the effects of advanced nonlinear shear correction coefficient and nonlinear coefficient term of TSDT on the advanced calculation of fundamental first natural frequencies.

Nonlinear dynamic response of FG-GPLRC beams induced by two successive moving loads

In this study, the nonlinear dynamic analysis of functionally graded graphene nanoplatelet reinforced composite (FG-GPLRC) beams is examined. The material properties can be estimated by using the modified Halpin-Tsai model and rule of mixture. Based on the third-order shear deformation theory and the von Kármán assumption, the equations of motion is derived and solved by using Jacobi-Ritz method incorporating with iteration procedure. Several effects such as number of layers, weight fraction of graphene nanoplatelets, material distributions, beam geometry, velocity of moving loads, distance between the loads are considered. For nonlinear forced vibration, the results indicate a considerable dissimilarity in the nonlinear dynamic deflection of the beams under moving loads when compared to linear analysis. The incorporation of additional GPLs into the beams yields a significant improvement in the strength of the beam, resulting in lower deflection. The new findings on nonlinear response of the beams excited by one and two moving loads are presented.