First-order Shear Deformation Theory Research Articles

The vibration behaviour-based meshfree moving Kriging method of functionally graded sandwich plates placed on Kerr foundation

ABSTRACT This paper presents a pioneering application of the meshfree moving Kriging (MK) method in conjunction with improved first-order shear deformation theory (i-FSDT) to study the vibration behaviour of functionally graded sandwich (FGSW) plates with a hard-core sandwiched between two FG skin layers placed on a Kerr foundation (KF). One significant advantage of MK interpolation is its Kronecker delta property, which allows for direct enforcement of boundary conditions (BCs). Unlike the original MK method, this approach eliminates the need to pre-define the correlation parameter, which might affect approximation accuracy. The governing equation was obtained using Hamilton’s concept. The Matlab code-based Newmark-β method is used to solve the motion equation of FGSW plates. The performance and accuracy of the proposed formula are verified through comparative examples. Then, the effects of geometrical dimensions, material properties, and BCs on the vibration behaviour of FGSW plates are examined in detail.

A Mathematical Approach to the Buckling Problem of Axially Loaded Laminated Nanocomposite Cylindrical Shells in Various Environments

In this study, the solution of the buckling problem of axially loaded laminated cylindrical shells consisting of functionally graded (FG) nanocomposites in elastic and thermal environments is presented within extended first-order shear deformation theory (FOST) for the first time. The effective material properties and thermal expansion coefficients of nanocomposites in the layers are computed using the extended rule of mixture method and molecular dynamics simulation techniques. The governing relations and equations for laminated cylindrical shells consisting of FG nanocomposites on the two-parameter elastic foundation and in thermal environments are mathematically modeled and solved to find the expression for the axial buckling load. The numerical results of the current analytical approach agree well with the existing literature results obtained using a different methodology. Finally, some new results and interpretations are provided by investigating the influences of different parameters such as elastic foundations, thermal environments, FG nanocomposite models, shear stress, and stacking sequences on the axial buckling load.

Analysis and optimization for buckling behavior of functionally graded face layers and porous core sandwich plates resting on pasternak elastic foundation

In this paper, the first-order shear deformation theory (FSDT) is employed to derive analytical solutions for the buckling analysis of functionally graded material—porous—functionally graded material (FGM-porous-FGM) sandwich plate resting on Pasternak elastic foundation. The top and bottom layers of the plate consist of functionally graded materials that vary according to the exponential law along the thickness direction, while the core layer is composed of a porous material. The governing equations are derived by using the principle of minimum total potential energy. Based on the Navier’s solution, the displacements are obtained to satisfy the boundary conditions. The results are then compared with existing published literature showing the accuracy of the proposed solutions. Moreover, parametric studies investigate the effects of material parameters, geometric dimensions, foundation coefficients and face-core-face thickness ratios on the critical buckling load of the rectangular FGM-porous-FGM sandwich plate. In the case of uniaxial loading, the critical bucking load is almost twice for the case of biaxial loading, no matter the value of volume fraction index. For all the cases of face-core-face thickness ratio, the biaxial critical buckling load linearly depends on the porosity coefficient. The porosity coefficient is crucial when and only when the volume of the core is almost eight times bigger than the volume of the face layer. Optimization studies are then presented in order to: (i) maximize the critical buckling load and (ii) optimize the input variables corresponding to a given value of the critical buckling load. The proposed study aims to provide valuable insights into the design and application of FGM-porous-FGM sandwich plates in various engineering fields, contributing to the development of more resilient and efficient structural components.

Connection between geometric dimension and dynamic response of bistable composite laminate with variable curvatures

Geometric dimension significantly affects the stable configuration and thereby the nonlinear dynamic response of bistable composite laminate. Previous studies have not focused on the minimum dynamic snap-through excitation of stable configuration transformation. This paper aims to address this gap by taking into account the effect of curvature on the bistable composite laminate with different geometric dimensions. The room-temperature equilibrium stable states of unsymmetric composite plates are obtained by cooling down from the cured temperature with variable curvatures. The governing equations are established by the first-order shear deformation theory, von Karman type nonlinear strain-displacement relation, and Rayleigh-Ritz method. The single- and double-well vibration mechanism are explored by solving the nonlinear equations via fourth-order Runge-Kutta method, and are visualized by bifurcation diagram, phase portrait, time history, Poincare maps and amplitude spectrum. The evolution of the dynamic response under the foundation excitation is estimated, and the effect of geometric dimension on the intra- and inter-well oscillation behaviors are illustrated in detail via theoretical model. Numerical results with finite element method are also obtained qualitatively consistent with the theoretical results. In view of the efficiency of the theoretical model, various geometric dimensions are considered to elucidate the universality of the vibration evolution. The nonlinear phenomenon that emerged in the dynamic response of composite structures provides a conceptual design of energy harvester and adaptive structure.

Elementary-level intrusive coupling of machine learning for efficient mechanical analysis of variable stiffness composite laminates: a spatially-adaptive fidelity-sensitive computational framework

AbstractMechanical analysis of the complex configurations of composite laminates can be computationally prohibitive based on accurate higher-order theories, especially when the analyses involve multiple realizations corresponding to different sets of input parameters such as uncertainty quantification, optimization, reliability and sensitivity analysis. Efficient lower-order theories should not be adopted in such situations since the error accumulates with multiple realizations, leading to poor outcomes. We propose an elementary-level coupling of machine learning for efficient, yet accurate mechanical analysis of laminated composites based on finite element simulations coupled with gaussian process regression. The generic parameter space of material properties, mesh size, number of layers, and ply angle in composite laminates are accounted for forming an efficient mapping with the augmentation of lower-order theory-based elementary-level structural matrices. The computationally efficient machine learning models predict the difference in the elements of the stiffness matrix for higher-order zigzag theory (HOZT) and first-order shear deformation theory (FSDT) at the first stage. Based on such machine learning-based difference mapping, we augment the elementary stiffness matrices obtained using FSDT efficiently to the equivalent of HOZT theory without any additional computational expenses (referred to here as augmented FSDT, or aFSDT). However, it is not necessary to augment all the elements in the analysis domain which might otherwise lead to unnecessary computational expenses and loss in accuracy. To achieve an optimal level of computational efficiency and accuracy, we further propose spatially-adaptive fidelity-sensitive coupling of machine learning, only for the elements within the analysis domain where it is necessary to adopt higher-order theories. The selective augmentation strategy essentially brings in a scope of integrating physics-based insights of critical stress resultant distribution into the algorithm based on best theory diagram. Subsequently, the global structural matrices are computed exploiting such adaptive criteria containing a mixed set of elements formed using FSDT and aFSDT, which leads to an accuracy equivalent to HOZT in the mechanical analysis of composite laminates almost at the computational expense of FSDT. The proposed spatially-adaptive fidelity-sensitive scheme ensures optimal performance in terms of computational efficiency by augmenting selective elements while minimizing the loss of accuracy due to the involvement of surrogates. Detailed numerical results are presented for static, dynamic and stability characterization of composite laminates including the demonstration for variable stiffness composite configurations based on the efficient machine learning-assisted elementary-level intrusive computational framework, wherein the notion of engineering judgement is introduced concerning the trade-off between computational efficiency and required level of accuracy.

Stability and bifurcation analysis of simply supported rectangular nanoplate embedded on visco-Pasternak foundation based on first-order shear deformation theory

Vibrations of simply supported nanoplates embedded on visco-Pasternak foundation and stressed by in-plane periodic forces are investigated. The governing size-dependent equations employ the first-order shear deformation theory, nonlinear von Kármán strains, and the nonlocal elasticity theory. Reducing the governing system is carried out by the Bubnov-Galerkin method, which is based on two-mode model, thus the resulting system contains two ordinary differential equations with size- and time-dependent coefficients. Analysing the linearized system, the stability of the nanoplate structure is studied depending on the foundation parameters, force parameters and small-scale parameter. Nonlinear dynamics approaches are employed to determine the nature of vibration after the loss of stability. Time history, Poincare section, Lyapunov exponent are analysed for chosen values of excitation parameters, visco-Pasternak foundation parameters to indicate small-scale effects and nonlinear effects. The novelty of the work consists of the combined application of the nonlocal theory, first-order shear deformation theory, Floquet theory, methods of nonlinear dynamics and obtaining new results by a numerical experiment.

Spectral method for bending analysis of rectangular functionally graded plates

This study investigates the bending behavior of functionally graded rectangular plates using the spectral collocation method to offer precise modeling. We apply the First-order Shear Deformation Theory (FSDT) to accurately capture the effects of shear deformation in our plate behavior models. The effectiveness of the proposed method is demonstrated by a comparison of the results obtained with those of literature.

Nonlinear Thermomechanical Low-Velocity Impact Behaviors of Geometrically Imperfect GRC Beams.

This paper studies the thermomechanical low-velocity impact behaviors of geometrically imperfect nanoplatelet-reinforced composite (GRC) beams considering the von Kármán nonlinear geometric relationship. The graphene nanoplatelets (GPLs) are assumed to have a functionally graded (FG) distribution in the matrix beam along its thickness, following the X-pattern. The Halpin-Tsai model and the rule of mixture are employed to predict the effective Young modulus and other material properties. Dividing the impact process into two stages, the corresponding impact forces are calculated using the modified nonlinear Hertz contact law. The nonlinear governing equations are obtained by introducing the von Kármán nonlinear displacement-strain relationship into the first-order shear deformation theory and dispersed via the differential quadrature (DQ) method. Combining the governing equation of the impactor's motion, they are further parametrically solved by the Newmark-β method associated with the Newton-Raphson iterative process. The influence of different types of geometrical imperfections on the nonlinear thermomechanical low-velocity impact behaviors of GRC beams with varying weight fractions of GPLs, subjected to different initial impact velocities, are studied in detail.

Computational analysis of free-corner-induced failure of thick angle-ply composite laminates established upon a global-local model

The complex three-dimensional (3D) state of interlaminar stresses in the vicinity of a free corner in composite laminates brings about delamination, matrix cracking, and overall failure. This study focuses on devising a straightforward and efficient computational framework for analyzing the free-corner-induced failures in mechanically and thermally-loaded thick angle-ply composite laminates based on an innovative global-local finite element model. In so doing, the global region is dealt with by the first-order shear deformation theory, and the local sub-model region near the free corner is scrutinized using the 3D layer-wise theory. Distinctive risk parameters associated with the meso-level damage modes, including ply failure, delamination, and transverse matrix cracking are well-defined under the corresponding criteria. The precision of the model is ensured through verifications in which satisfactory alignment between the current outcomes and those of other available works in the literature substantiates the efficacy of the proposed approach. Furthermore, the impacts of the configuration parameters such as fiber orientation and thickness of plies upon the risk parameters are fully examined. It is found that, generally, by increasing the angle of fibers between two adjacent layers and diminishing the ply thickness, the composite laminate becomes more critical. The matrix cracking mode is dominated regarding the delamination and is in close agreement with the ply failure analysis. The proposed computational infrastructure, occasioning time and financial savings is valuable and enlightening for designing and manufacturing composite structures.

Stability Analysis of Curved Beams Based on First-Order Shear Deformation Theory and Moving Least-Squares Approximation

Based on the first-order shear deformation theory (FSDT) and moving least-squares approximation (MLS), a new meshfree method that considers the effects of geometric nonlinearity and the pre- and post-buckling behaviors of curved beams is proposed. An incremental equilibrium equation is established with the Updated Lagrangian (UL) formulation under the von Karman deflection theory. The proposed method is applied to several numerical examples, and the results are compared with those from previous studies to demonstrate its convergence and accuracy. The pre- and post-buckling behaviors of the curved beam with different parameters, such as vector span ratios, bending forms, inclusion angles, boundary conditions, slenderness ratios, and axial shear stiffness ratios, are also investigated. The effects of the parameters on the buckling response are demonstrated. The proposed method can be extended to the study of double nonlinearities of curved beams in the future. This extension will provide a more scientific reference basis for the structural selection of curved girder structures in practical engineering.

Large amplitude vibration of rotating graphene reinforced titanium alloy dovetailed blades with general boundary conditions

Engine blades are generally installed on disks via dovetail joints, which can alleviate stress concentrations at the blade root and prevent heat transfer. As a result, the connection stiffness is strongly affected by the clearance between the blade and disk as well as the rotation speed of the blade. This paper first proposes a functionally graded graphene-reinforced titanium alloy trapezoid plate with elastic boundary conditions to investigate the large amplitude vibration of rotating dovetailed blades. The elastic boundaries are modelled by applying a series of artificial springs, and the related admissible displacement functions are constructed by orthogonal polynomials through the Gram–Schmidt process. The governing equations of the functionally graded graphene platelet reinforced composite (FG-GPLRC) trapezoid plate are derived by the first-order shear deformation theory (FSDT) and Lagrange equation. The numerical results of the large amplitude vibration of the plate are obtained by the weighted residual method combined with a direct iterative algorithm and verified through analytical solutions of the harmonic balance method. The effects of the stiffness of the constraint springs, rotation speed and structural parameters on the linear and nonlinear free vibrations of the FG-GPLRC trapezoid plate are thoroughly studied.

Isogeometric Analysis of Functionally Graded Plates using High-Order Theories

Functionally Graded Materials are composites, usually of ceramic and metal, in which the volume fraction of their constituents varies smoothly along an interest direction. These materials were initially proposed as a solution for thermal barriers development, but are currently used in different applications. The composition variation results in a gradual change in their properties, enhancing the performance of structures subjected to thermal and mechanical loads, but making structural analysis more complex. Plates, on the other hand, are three-dimensional flat structures in which the thickness is much smaller than their other two dimensions. Different theories have been proposed for structural analysis of plates. These theories can be grouped based on the considerations adopted for the behavior of transverse shear deformations. In this regard, there are the Kirchoff-Love Theory (also known as the Classical Plate Theory - CPT), which disregards the effect of these deformations, the Reissner-Mindlin Theory (also known as the First-Order Shear Deformation Theory - FSDT), which considers shear deformations constant throughout the plate thickness, and Higher-Order Shear Deformation Theories (HSDT’s), which consider a nonlinear variation of shear deformations through different approaches. Among these theories, the most robust and accurate ones are evidently the HSDT’s. However, it is important to clarify that they require the displacement field to have a C1 continuity, which is complex for isoparametric finite elements. Thus, Isogeometric Analysis emerges as a more viable alternative by employing high continuity elements based on Non-Uniform Rational B-Splines (NURBS) functions. Therefore, this paper presents a NURBS-based isogeometric formulation for buckling and free vibration analyses of functionally graded plates using a HSDTs. A series of tests were conducted to assess the accuracy of this formulation considering examples available in the literature with different geometries, boundary conditions, and materials. The obtained results present excellent agreement with the reference solutions for thin and thick plates.

Semi-analytical modeling and vibration analysis of joined FGP-GPLRC thin-walled cylindrical-spherical shells with arbitrary boundary conditions

In this paper, the vibration characteristics of the joined thin-walled cylindrical-spherical shells with graphene platelet (GPL) reinforcement in metal foams are investigated based on a relatively novel Gegenbauer-Ritz method. The material properties of the coupled shells are also obtained using the Halpin-Tsai micromechanical modeling. The equations for the mechanical modeling of the coupled shell are given by applying the first-order shear deformation theory (FSDT) and the energy method. In addition, the spring stiffness penalty method is employed to obtain arbitrary boundary conditions at both ends of the shell. The final solution for the coupled shell is obtained by the Gegenbauer-Ritz method. Finally, the results of this paper are compared with the literature results and those obtained by the finite element method (FEM) to prove the accuracy of the approach. On this basis, the effects of arbitrary boundary conditions, GPL distribution modes, porosity distribution patterns, structural openings, and geometrical parameters on the vibration characteristics of the joined cylindrical-spherical shells are discussed.

Elastic local buckling of double-skinned UHPC composite plates considering interfacial shear slip

This study explores the elastic local buckling behavior of double-skinned ultra-high-performance concrete (DS-UHPC) composite plates, focusing on the effects of interfacial shear slip. A theoretical model, based on the first-order shear deformation theory for sandwich plates, was developed to predict the elastic local buckling behavior of these composite plates. An analytical solution for the elastic buckling coefficient under uniform compression with a simply supported boundary condition was derived. The results reveal that interfacial shear stiffness and the proportion of cross-sectional steel content significantly affect the elastic buckling coefficient. Specifically, lower interfacial shear stiffness and higher steel content proportion result in a decreased buckling coefficient, while increasing interfacial shear stiffness reduces the sensitivity of the buckling coefficient to variations in steel content proportion. Finite element models were established and validated against previous studies on the buckling performance of single-layer steel-concrete composite plates and classical solutions of buckling coefficients without interfacial shear slip. Semi-analytical solutions for the elastic buckling coefficient under various boundary conditions were also developed through a comprehensive set of finite element results. Building on these findings, a design procedure for DS-UHPC composite plates is proposed, emphasizing the optimization of interfacial connectors to minimize shear slip effects on local buckling. This procedure ensures the prevention of local buckling in both the external steel plates and the entire composite plate, thereby enhancing structural stability.

Nonlinear analysis of functionally gradient magneto-electro-elastic porous cylindrical shells considering thermal effects

This paper presents a static analysis of large deflection behavior in functionally graded magneto-electro-elastic porous (FG-MEEP) cylindrical shells using the geometric fully nonlinear finite element (FE) method. The governing equations are derived based on the first-order shear deformation theory (FOSD) and the strain-displacement relationship considering large rotations. A nonlinear dynamic model is developed employing the Hamiltonian principle. Four gradient models of temperature variation along the thickness are constructed, such as uniform, linear, sinusoidal and heat conduction. Moreover, the porosity effects of FG-U, FG-V, FG-O and FG-X distributions are taken into account. The accuracy and efficiency of the proposed model are verified by comparing with the results of existing literature and commercial software. The subsequent extensive research is conducted to investigate the influence of various parameters on the mechanical response of the FG-MEEP structure. It is concluded that adopting the large rotation theory (LRT) can yield more precise outcomes for structures experiencing significant deformations under thermal environments, providing valuable insights for future research on FG-MEEP structures.

Free vibrational mode shapes and frequencies of sandwich folded plates standing on folded foundation and structured by auxetic honeycomb core and FG-GPLRC coating layers

In this article, for the first time, the free vibration and modal shapes of sandwich folded plate with honeycomb core are studied. The current article investigates the free vibrational mode shapes and frequencies of sandwich folded plate made of auxetic honeycomb core and graphene platelet-reinforced functionally graded composite (FG-GPLRC) coating layers which standing on the normal-shear folded supports. The structures are supported by normal-shear folded supports. The negative Poisson’s ratio exhibited by honeycomb materials results in distinct physical and mechanical properties, profoundly impacting the behavior of structures constructed from this material. While structures with basic shapes like rectangular, circular, and spherical configurations find limited applications across various industries, the versatility of folded structures makes them highly suitable for a wide range of industrial applications, given their more complex geometries. Dynamic equations are formulated within the framework of the first-order shear deformation theory (FSDT) for each flat section of the folded plate. The continuous equality conditions for displacements, rotations, and stresses at the junction edges of the two flat sections are rigorously satisfied. To effectively solve these equations, a robust numerical technique known as the generalized differential quadrature method (GDQM) is employed specifically tailored for the 2D analysis of folded plates. Furthermore, the study investigates the influence of various parameters, including material properties and geometry, on the frequencies and mode shapes of the folded structures, shedding light on the intricate interplay between these factors in shaping the dynamic response of the structures.

Free vibration analysis of a functionally graded porous nanoplate in a hygrothermal environment resting on an elastic foundation

This research investigates the free vibrational behavior of a functionally graded porous (FGP) nanoplate resting on an elastic Pasternak foundation in a hygrothermal environment. The nanoplate is modeled based on the nonlocal strain gradient theory (NSGT) and considering several plate theories including the CPT (classical plate theory), the FSDT (first-order shear deformation theory), and the TSDT (third-order shear deformation theory). Several patterns are investigated for the dispersion of pores, and the surface effects are incorporated to enhance the precision of the model. The governing equations and boundary conditions are derived via Hamilton's principle and an exact solution is provided via the Navier method. The impacts of several parameters on the natural frequencies are inspected such as length scale and nonlocal parameters, surface effects, porosity parameter, hygrothermal environment, and coefficients of the foundation. The results show that the impact of the porosity parameter on the natural frequencies of nanoplates is significantly dependent on the porosity distribution pattern. It is discovered that by increasing the porosity parameter from 0 to 0.6, the relative changes of natural frequencies vary from a decrease of 30 % to an increase of 6 %.

Static, free vibration, and buckling analysis of functionally graded plates using the dual mesh control domain method

In this paper, the Dual Mesh Control Domain Method (DMCDM) put forward by Reddy is applied to solve linear static, free vibration, and buckling problems of functionally graded plates modeled using the First-Order Shear Deformation Theory (FSDT). The material properties are assumed to vary continuously through the thickness of the plate according to a power-law. Formulations are presented for linear triangular (3-noded) and bilinear quadrilateral (4-noded) primal elements of arbitrary shape. The influence of the power-law exponents, length to thickness ratio, boundary conditions, and plate skewness on the numerical solution is systematically analyzed. Additionally, the numerical solutions using the DMCDM are compared against those from the Finite Element Method (FEM) to demonstrate the robustness of the DMCDM as a strong competitor to well-established numerical techniques such as the FEM.

Static and Transient Response Analysis of Arbitrary Laminated Composite and Sandwich Shells by the FSDT Meshless Method

This paper, for the first time, presents a novel meshless method based on the first-order shear deformation theory (FSDT) and moving-least squares (MLS) approximation to carry out static and transient response analysis on arbitrary laminated and sandwich shell structures. The novelty of the method lies in the implementation of MLS in the curvilinear shell formulation, and the derivation of meshless governing equations on the static and transient response of arbitrary laminated and sandwich shells according to the principle of minimum potential energy and Hamilton’s principle, respectively. The concept of a Convected coordinate system is adopted to deal with arbitrary curvilinear surfaces. Both the position and displacement vectors of shells are approximated by MLS, which is conceptually the same procedure as that in the isoparametric finite element method (FEM). The numerical integration of the stiffness matrices is conducted by the simple Gaussian integral in the Convected coordinate system. Because the moving-least squares (MLS) approximation does not satisfy the Kronecker delta condition, the full transformation method is used to modify the meshless governing equations. Numerical examples are calculated to investigate the static and transient response analysis of laminated and sandwich structures with different geometrical shell shapes, including square plates, spherical shells, doubly-curved shells, and cylindrical shells. The numerical tests show that the proposed method is reliable, and robust and produces accurate results compared to the analytical and numerical solutions taken from the previous literature.

New method for predicting the free vibration characteristics of composite laminates based on generalized cell and spectral‐Tchebychev methods

AbstractThis paper presents a new method for analyzing the free vibration characteristics of composite laminates under arbitrary boundary conditions by integrating the generalized method of cells (GMC) and the Two‐Dimensional Spectral‐Tchebychev (2D‐ST) method. First, a meso‐macro correlation matrix is constructed based on the GMC, and the macroscopic performance parameters of the composite laminates are predicted. Then, virtual boundary springs are introduced to simulate the arbitrary boundary conditions, and the energy equation of the laminate is derived based on the first‐order shear deformation theory (FSDT) and Hamilton's principle. Furthermore, an approximate displacement function of the laminate is derived by using the 2D‐ST Tchebychev polynomials, and the Gauss‐Lobatto points are selected for meshless discretization of the structure to obtain the discretized characteristic equation, which is solved to obtain the free vibration frequency of the laminate. Based on Matlab programming, the free vibration frequencies of the laminate under different conditions are calculated, and the convergence, accuracy and computational efficiency of the calculation method are discussed, and the effects of boundary conditions, stacking angle, geometrical parameters, and material parameters on the free vibration frequencies of the laminate are analyzed. The numerical results show that the method has the advantages of fast convergence, high accuracy and high efficiency, and the method can be convenient and fast for parametric studies.Highlights The free vibration characteristics of composite laminates are studied. Combining generalized cell method with spectral‐Tchebychev method. The method has the characteristics of fast convergence, high accuracy and high efficiency. A parametric study is conducted on the free vibration of laminated panels.