Reddy's Third-order Shear Deformation Research Articles

A meshfree formulation for size-dependent thermal buckling and post-buckling behaviour of porous microplates on elastic foundation subjected to localized heating

The semi-analytical framework is developed for in-plane thermal stresses within the microplates under localized heating. Localized heating is modelled using Fourier Series expansions over the complete domain of the plate. These stresses within the plate are estimated after solving the in-plane thermoelasticity problem using Airy's stress approach. Utilizing these stresses, present work also studies the buckling and post-buckling behaviour of a porous metal foam microplate resting on Winkler and Pasternak elastic foundations. The plate is modelled using Reddy's third-order shear deformation theory (TSDT) and von Kármán geometric nonlinearity, and the size-dependent effects are included using the modified strain gradient theory (MSGT). Galerkin's weighted residual method converts the partial differential equations (PDEs) into algebraic equations. The post-buckling equilibrium path is obtained using the modified Newton-Raphson method. The mode shapes of the microplate for the various configurations of the plate with different boundary conditions due to localized thermal loading are plotted. Also, these mode shapes are compared with the mode shapes of the microplate due to in-plane mechanical loading. The parametric effect of porosity, elastic foundation parameters, thickness of plate, size of plate, boundary conditions and loading concentrations on the buckling and post-buckling behaviour of the plate is studied.

NURBS-based isogeometric formulation for linear and nonlinear buckling analysis of laminated composite plates using constrained and unconstrained TSDTs

Two isogeometric plate models employing Reddy's third-order shear deformation theory (TSDT) and unconstrained third-order shear deformation theory (UTSDT) are presented and compared for linear and nonlinear buckling analysis of laminated composite plates with and without imperfection and subjected to different inplane loads. Cubic non-uniform rational B-spline (NURBS) basis functions that easily satisfy C1 continuity of the IGA-TSDT model are employed. The total Lagrangian approach in conjunction with the principle of virtual work is used to derive the governing equations. The primary and secondary solutions are traced using a tangent based arc-length method with a simple branch switching technique. The performance of the models is evaluated by validation and comparison with solutions obtained using ANSYS, Navier method (for linear analysis only), and those in the literature. The buckling response is significantly affected by pre-buckling boundary conditions and strain-displacement relations. The nonlinear buckling approach, among other approaches, is observed to be the most accurate methodology for an arbitrarily laminated composite plate. Further, IGA-UTSDT with nine DOF gives marginal improvement over IGA-TSDT with five DOF at the cost of computation. The IGA-TSDT is observed to be superior to FEM-TSDT in terms of computation demand and performance.

Nonlinear poro-thermo-forced vibration in curved sandwich magneto-electro-elastic shells under hygrothermal environment

This research employs a multiple scales perturbation approach to evaluate the nonlinear wave propagation behaviors of a doubly curved sandwich composite piezoelectric shell with a flexible core in a hygrothermal environment. Stress and strain calculations for the flexible core and face sheets are carried out using Reddy's third-order shear deformation theory (TSDT) and third-order polynomial theory, respectively. The study explores the synergistic effects of a multilayered shell, flexible core, and magneto-rheological layer (MR) in revealing the nonlinearity of both in-plane and vertical moment within the core. The Halpin–Tsai model is employed to derive the properties of polymer/carbon nanotube/fiber (PCF) and polymer/graphene platelet/fiber (PGF) three-phase composite shells. The governing equations for the multiscale shell are derived using Hamilton's formulation. The research investigates temperature variations, diverse distribution patterns, curvature ratios, and magnetic fields through numerical analysis, presenting the results graphically and prior research has demonstrated the accuracy of these methods. Notably, these factors exert significant influence on the frequency-amplitude curves of the smart structure.

A systematic methodology for port-Hamiltonian modeling of multidimensional flexible linear mechanical systems

This article introduces a novel systematic methodology for modeling a class of multidimensional linear mechanical systems that directly allows to obtain their infinite-dimensional port-Hamiltonian representation. While the approach is tailored to systems governed by specific kinematic assumptions, it encompasses a wide range of models found in current literature, including ℓ-dimensional elasticity models (where ℓ = 1, 2, 3), vibrating strings, torsion in circular bars, classical beam and plate models, among others. The methodology involves formulating the displacement field using primary generalized coordinates via a linear algebraic relation. The non-zero components of the strain tensor are then calculated and expressed using secondary generalized coordinates, enabling the characterization of the skew-adjoint differential operator associated with the port-Hamiltonian representation. By applying Hamilton's principle and employing a specially developed integration by parts formula for the considered class of differential operators, the port-Hamiltonian model is directly obtained, along with the definition of boundary inputs and outputs. To illustrate the methodology, the plate modeling process based on Reddy's third-order shear deformation theory is presented as an example. To the best of our knowledge, this is the first time that a port-Hamiltonian representation of this system is presented in the literature.

Nonlinear dynamic analysis of aircraft CFRP sandwich wings under explosive blast loading: Introducing SVM-DNN algorithm to predict dynamical information

The aircraft wings are a crucial component, particularly during mechanical shock excitation. Thus, for the first time, an inventive adaptable model to simulate the nonlinear dynamics of the wings under explosive blast loading at the moment of the collision is described in this article. It is crucial to increase the structure's stability because of explosive blast loading. The mentioned structure is made of two composite face sheets and a carbon fiber-reinforced polymer (CFRP). Functionally graded (FG) graphene origami (GOri)-enabled auxetic metallic metamaterial (GOEAM) is introduced to improve the stability of this kind of applicable structure, especially in air maneuvers. The analysis is conducted utilizing Reddy's third-order shear deformation of Reddy theory (TSDRT), incorporating Hamilton's principle and Von Kármán nonlinear geometric assumptions. The governing and boundary equations are accurately computed at the domain and boundary edges of the doubly curved structure. The coupled mesh-free radial point interpolation method (MRPIM) and Newton-Raphson method are applied for numerically simulating the current doubly curved structure by employing Newmark's temporal integration method with a constant-average acceleration approach. The present findings are cross-checked against the machine learning method and open-source results from the literature. Based on physical data, machine learning integrates supervised machine learning to predict the nonlinear vibration of the system. In this case, nonlinear transient bending properties are found using hybrid machine learning methods and solutions. The findings demonstrate the significance of GOEAM for nonlinear vibrations of the composite system. The current study's conclusions could be used as a benchmark and useful recommendations for future structural design techniques with enhanced features.

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°.

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.

Static Buckling and Free Vibration Analysis of Aligned CNTs Reinforced Composite Plates

This work introduces an analysis of the nonlinear buckling and free vibration behavior of polymer plates reinforced with aligned carbon nanotubes using Reddy's third-order shear deformation plate theory and incorporating Theodore von Kármán's geometric nonlinearity. The polymer plates were enhanced with single-walled carbon nanotubes assumed to exhibit either uniform distribution or functionally graded distribution across the thickness. The equations of motion were established through Hamilton’s principle and then solved by the Galerkin method and Airy’s stress function for the composite plates with fully simply supported edges. The investigation focused on assessing the effects of carbon nanotube distribution, volume fraction, and geometrical parameters on the buckling load and fundamental frequency parameters of composite plates through numerical results.

Dynamic Behavior Modeling of Natural-Rubber/Polybutadiene-Rubber-Based Hybrid Magnetorheological Elastomer Sandwich Composite Structures.

This study investigates the dynamic characteristics of natural rubber (NR)/polybutadiene rubber (PBR)-based hybrid magnetorheological elastomer (MRE) sandwich composite beams through numerical simulations and finite element analysis, employing Reddy's third-order shear deformation theory. Four distinct hybrid MRE sandwich configurations were examined. The validity of finite element simulations was confirmed by comparing them with results from magnetorheological (MR)-fluid-based composites. Further, parametric analysis explored the influence of magnetic field intensity, boundary conditions, ply orientation, and core thickness on beam vibration responses. The results reveal a notable 10.4% enhancement in natural frequencies in SC4-based beams under a 600 mT magnetic field with clamped-free boundary conditions, attributed to the increased PBR content in MR elastomer cores. However, higher magnetic field intensities result in slight frequency decrements due to filler particle agglomeration. Additionally, augmenting magnetic field intensity and magnetorheological content under clamped-free conditions improves the loss factor by from 66% to 136%, presenting promising prospects for advanced applications. This research contributes to a comprehensive understanding of dynamic behavior and performance enhancement in hybrid MRE sandwich composites, with significant implications for engineering applications. Furthermore, this investigation provides valuable insights into the intricate interplay between magnetic field effects, composite architecture, and vibration response.

Thermal vibration analysis of FG-GPLRC doubly curved shells partially resting on Kerr foundation based on higher-order shear deformation theory

In this paper, we present a novel analytical approach for investigating the influence of non-homogeneous Kerr elastic foundations on the nonlinear dynamical characteristics of functionally graded graphene platelets reinforced composite (FG-GPLRC) doubly curved shells. A new aspect of this study is the investigation of various distributions of the Kerr foundation, either centrally within or along the edges of the nanocomposite shells, a model not previously explored. Three distinct distributions of graphene nanoplatelets reinforcement are examined, utilizing the rule of mixtures and the Halpin-Tsai micromechanical model. The theoretical framework is based on Reddy's third-order shear deformation theory. The equations of motion, formulated as partial differential equations, are efficiently resolved through the application of Galerkin's method. The effectiveness of this approach resides in its utilization of integration to tackle the distribution issue of the elastic foundation, all without the need for deploying intricate algorithms. The validity of our proposed method is established through meticulous comparisons with findings from prior literature. Furthermore, this paper delves into the effect of material properties, thermal conditions, boundary conditions, geometrical parameters, and four configurations of the shells on the vibrational responses. Of notable significance is our thorough investigation of the impact of variations in the distributions, stiffness, and geometrical parameters of the elastic foundation on natural frequencies and the nonlinear transient response of these shells, all of which hold great importance in structural design. We present the obtained results systematically in various formats, including tables, 2D and 3D graphs, and contour plots, providing a comprehensive overview of the research outcomes. These findings have profound implications for the field of engineering, contributing to advancements in novel structural designs and their diverse applications.

Stability of higher-order lattice composite cylindrical shell reinforced with graphene platelets by means of a Chebyshev collocation-based semi-analytical approach

In this study, we present a novel analysis approach for lattice composite cylindrical shells reinforced with Graphene Platelets (GPL) nanoparticles. Our primary contribution lies in the investigation of these advanced structures, incorporating nanocomposite reinforcement, orthotropic inhomogeneity, and semi-analytical methods. The lattice composite comprises an anisogrid lattice laminated shell, reinforced with functionally graded GPL. We model this structure using a global continuous orthotropic deep shell approach, integrating the Halpin-Tsai and rule of mixtures homogenization strategies to estimate equivalent mechanical properties. We derive theoretical formulations utilizing Reddy's third-order shear deformation theory and nonlinear Sanders' kinematic assumptions, tailored for deep thick shells. Nonlinear equilibrium equations are obtained using Hamilton's principle and Hooke's constitutive law, leading to linearized bifurcation equations through adjacent-equilibrium and membrane pre-buckling analysis. Our stability analysis employs a semi-analytical method combining trigonometric expansion and Chebyshev collocation functions. Validation through parametric examples demonstrates the accuracy and efficiency of our approach, unveiling insights into the impact of lattice composite and geometric parameters on the stability response of these innovative structures.

Nonlinear frequency analysis of porous Bi directional functionally graded beams utilizing reddy shear deformation theory

The nonlinear frequency response of bi-directional functionally graded porous beams experienced range of various end conditions is investigated in this work. The end conditions which are simply supported, clamped-simply supported, clamped-clamped, and clamped-free are taken by using the Von Karman geometric nonlinearity, Green's tensor and Reddy third-order shear deformation theory. A generalized differential quadrature technique (GDQM) accompanied by direct numerical iterance approach is proposed to solve equations. The findings are presented to aid in future research into the effects of various gradient indices, vibration amplitude ratios, porosity coefficients, shear and elastic substrate parameters, boundary conditions, and vibration frequencies on the bi-directional functionally graded beams. The outcomes of this research have practical applications and can be utilized to enhance the design of bi-directional beams. The results are also highly useful in anticipating and identifying potential causes of failure in these beams.

Nonlinear transient response of sandwich beams with functionally graded porous core under moving load

Nonlinear transient response of sandwich beams possessing functionally graded porous core under the action of moving load was examined in this study. With this regard, Reddy's third-order shear deformation theory and the geometrical nonlinearity of von Kármán assumption were utilized to construct the governing equation system. The set of governing equations was solved by the Gram-Schmidt-Ritz method in conjunction with iterative procedures based on the time-integration of Newmark for the convergence in time and geometrical domains. According to the methodology, we received accurately stable results of nonlinear free and forced vibration of the sandwich beams. Several important parameters such as slenderness ratio, layer thickness's ratio, porous coefficient, types of porous distribution, etc. that have a significant impact on nonlinear deflection of the beams were taken into account. Based on numerical experiments, it can be disclosed that the sandwich beams carrying porous distribution in form of functionally graded materials are much better than the beams with uniform porous distribution in terms of strength and stiffness. The porous coefficient also plays an important role in changing the ability of loading resistance.

Nonlinear Finite Element Model for Bending Analysis of Functionally-Graded Porous Circular/Annular Micro-Plates under Thermomechanical Loads Using Quasi-3D Reddy Third-Order Plate Theory.

A nonlinear finite element model for axisymmetric bending of micro circular/annular plates under thermal and mechanical loading was developed using quasi-3D Reddy third-order shear deformation theory. The developed finite element model accounts for a variation of material constituents utilizing a power-law distribution of a two-constituent material, three different porosity distributions through plate thickness, and geometrical nonlinearity. The modified couple stress theory was utilized to account for the strain gradient effects using a single material length scale parameter. Three different types of porosity distributions that have the same overall volume fraction but different enhanced areas were considered as a form of cosine functions. The effects of the material and porosity distribution, microstructure-dependency, the geometric nonlinearity, and various boundary conditions on the bending response of functionally-graded porous axisymmetric microplates under thermomechanical loads were studied using the developed nonlinear finite element model.

Nonlinear vibrations of multiscale composite beams on a nonlinear softening foundation

This paper describes a new approach to evaluate the instability of beams on a softening elastic foundation. The basis of this approach is solving a nonlinear frequency problem. The beam is made of multiscale composite materials based on carbon nanotubes (CNTs) and long fibers. Homogenization methods are utilized in a sequential process to obtain the effective material properties of these composites. Furthermore, the thermal environment's effect on the beam's response is examined. Reddy's third-order shear deformation theory approximates the induced displacements in the composite beam. Also, the kinetic and strain energies of the beam are calculated based on the von Kármán nonlinear strains and linear thermoelasticity theories. A recently reported Lagrangian nodal weak formulation (LNWF) method is implemented to obtain the linear and nonlinear stiffness and mass matrices governing the beam's vibrations. The modified coefficients for nonlinear stiffness matrices are obtained using the residual Galerkin method across time. Finally, the nonlinear natural frequencies are calculated by employing the iterative displacement control strategy. In the results section, in addition to examining various parameters on nonlinear free vibrations, the effect of these parameters on the displacement required for the loss of stability of the beam placed on the softening foundation is also examined.

Supersonic flutter and free vibration features of functionally graded material nanobeams incorporating surface effects

In the present work, a numerical investigation is provided for free vibrational and aeroelastic stability features of nanobeams made of Functionally Graded Material (FGM) resting on an elastic Pasternak foundation. The FGM nanobeam's mechanical properties are considered to differ based on power-law and exponential functions in the thickness direction. The effect of size is modeled in terms of the Gurtin-Murdoch surface elasticity theory and the Nonlocal Strain Gradient Theory (NSGT) is employed for incorporating the surface effects. Mathematical modeling of the FGM nanobeam is oriented by three various beam theories including Reddy's third-order Shear Deformation Beam Theory (TSDBT), Timoshenko beam theory (TBT), and Euler-Bernoulli Beam Theory (EBT), moreover, the aerodynamic pressure is determined in terms of the linear supersonic piston theory. It is concluded that when the surface effects are incorporated, the width-to-length ratio of the FGM nanobeam plays a sensible role in determining the natural frequencies; but it has no sensible effect on the flutter boundaries. Numerical results reveal that the importance of the surface effects increases as the values of thickness-to-length and width-to-length ratios decrease.

Analytical and meshless DQM approaches to free vibration analysis of symmetric FGM porous nanobeams with piezoelectric effect

The current study is an first attempt to develop a vibrational-based piezoelectrically actuated smart nanoactuator model based on assumptions of Reddy third-order shear deformation beam model together with nonlocal strain gradient theory. The considered model is a sandwich structure consisting of piezoelectric layers under an external electric field, and symmetric to the midplane functionally graded (FG) porous core. Expressed by displacements equations of motion are derived based on a dynamic version of Hamilton's principle. Eigenvalue problem of free vibration of simply supported nanobeam is solved analytically by using properties of Fourier series, and numerically via meshless Differential Quadrature Method (DQM). The study includes a convergence study for the numerical approach, as well detailed discussion on obtained results involving effects of nonlocal parameters, external mechanical and electrical loads, material gradation coupled with diverse porosity distribution, and influence of external environment on dynamic behavior of the nanostructure. The present paper, in view of included optimization study, may positively affect nanodevices reliability and extend its operation by avoiding resonance phenomenon.

Experimental verification of multi-fibre hybridization influence on dynamic deflection and stress values of curved composite panel

The finite-element time-dependent deflection and stress responses of the shallow composite panels subjected to variable mechanical loadings (uniformly distributed load and sinusoidally distributed load) are reported in this article. The study reveals the influence of the advanced fibre hybridization (glass–carbon–kevlar) on the dynamic responses and establishes the accuracy of the numerical responses by comparing them with the experimental values. The numerical steady-state deflections of the hybrid composite structure are evaluated using a generic mathematical model derived through Reddy's third-order shear deformation in conjunction with the finite-element technique and Newmark's time integration scheme. The experimentally evaluated mechanical properties of the fabricated composites are utilized in the computations. The validity of the computed solutions is ascertained with in-house experimental (transient deflection) results. Besides, the numerical model is extended to assess the parametric dependence (aspect ratio, thickness ratio, curvature ratio, geometry, hybrid schemes and support conditions) of the dynamic deflection/stress responses of hybrid composite shallow shell panels subjected to variable mechanical loading types.

Linear and nonlinear static bending of sandwich beams with functionally graded porous core under different distributed loads

In this investigation, linear and nonlinear bending analyses of sandwich beams with functionally graded cores are determined under different types of distributed loads. These sandwich beams are composed of two isotropic faces and a porous core with different gradients of internal pores. The governing formulation used to describe the beam’s linear and nonlinear behavior is constructed from Reddy's third-order shear deformation theory and nonlinear strain–displacement relations of von Kármán. The Gram-Schmidt orthogonalization procedure is adopted to generate numerically stable functions for the displacement field to solve the beam problems with various boundary conditions. Then, the Ritz method is utilized to find out linear and nonlinear bending results in conjunction with the iterative technique. The accuracy of our solutions is validated, and our numerical results agree well with some cases available in the literature. New results of the sandwich beams based on several effects of porosity coefficient, slenderness ratio, loading types, porous distributions of the core, etc., are presented in graphical and tabular forms, serving as a benchmark solution for future studies.