Shear Deformation Theory Research Articles

Thermomechanical Nonlinear Vibration of Axially Loaded Functionally Graded Material Cylindrical Panels Including Porosity

For the first time, the combined influences of axial compressive load, porosity, geometric imperfection, elastic edge restraint, and elevated temperature on the nonlinear free vibration of functionally graded material (FGM) cylindrical panels are investigated in this paper. Unlike previous studies, the structural model considered in this paper includes practical situations and conditions arising in the design process and application of FGM cylindrical panels. The properties of material constituents are assumed to be temperature-dependent, and the effective properties of porous FGM are determined using a modified rule of mixture. Governing equations in terms of deflection and stress function are established based on first-order shear deformation theory, taking into account von Kármán–Donnell nonlinearity and initial geometric imperfection. Analytical solutions are assumed to satisfy simply supported boundary conditions, and the Galerkin method is applied to derive a time differential equation containing both quadratic and cubic nonlinear terms. This differential equation is numerically integrated employing the fourth-order Runge–Kutta scheme to determine the frequencies of nonlinear free vibration. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies of porous FGM cylindrical panels. The study reveals that natural frequencies are strongly decreased and that frequency nonlinearity is more significant due to axial compressive loads.

Nonlinear Analysis of the Multi-Layered Nanoplates

This text investigates the bending/buckling behavior of multi-layer asymmetric/symmetric annular and circular graphene plates through the application of the nonlocal strain gradient model. Additionally, the static analysis of multi-sector nanoplates is addressed. By considering the van der Waals interactions among the layers, the higher-order shear deformation theory (HSDT), and the nonlocal strain gradient theory, the equilibrium equations are formulated in terms of generalized displacements and rotations. The mathematical nonlinear equations are solved utilizing either the semi-analytical polynomial method (SAPM) and the differential quadrature method (DQM). Also, the available references are used to validate the results. Investigations are conducted to examine the effect of small-scale factors, the van der Waals interaction value among the layers, boundary conditions, and geometric factors.

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.

Closed-Form Solution for Scale-Dependent Frequencies of Shear Deformable Cellular Microplates Coated by Piezoelectric Polymeric Skins Consisting of FG Distributed CNTs

In this work, a rectangular cellular microplate is taken into consideration which is embedded between two functionally graded carbon nanotube-reinforced composite layers that have the piezoelectric ability. Different patterns for the carbon nanotube dispersion are considered. Moreover, an external electrical voltage is applied to them. The displacements are described based on a higher-order shear deformation theory which is called hyperbolic theory and the size influence is captured via the modified couple stress formulations. The governing motion equations are then derived using the variational technique and Hamilton’s principle. Then, for simply supported edge conditions, a closed-form solution is provided. Next, it turns to validate the results in simpler states, and after that, the effect of the most prominent parameters on the results is investigated. It is observed that by increasing the external applied voltage to the face sheets, the frequencies are reduced. Also, the natural frequencies have a tendency to decrease as the dimensionless material length scale parameter increases. This study’s outcomes may help design and manufacture micro-/nano-electro systems, sensors and actuators, small-scale devices, and other engineering structures.

A unified Jacobi-Ritz-spectral BEM for vibro-acoustic behavior of spherical shell

In order to solve the vibration and acoustic characteristics of spherical shell in light fluid environment, based on Jacobi-Ritz-spectral BEM, a unified analysis formula for acoustic vibration of spherical shell under arbitrary boundary conditions is established. Based on the First-order shear deformation theory (FSDT) and domain decomposition method (DDM), the theoretical model of spherical shell structure is established. The improved Jacobi polynomial is innovatively used to construct the displacement tolerance function of the spherical shell. Based on the spectral Kirchhoff-Helmholtz integral formula, the theoretical model of the acoustic fluid outside the spherical shell is established. The special form of Jacobi polynomial Chebyshev polynomial is used to describe the excitation sound pressure on the spherical shell segment, so as to ensure that the generalized coordinates of the structure can be perfectly matched with the acoustic boundary element nodes. In addition, the integration along the meridian of the shell is used to control the movement of the fluid, which simplifies the surface integration. The CHIEF method is used to solve the problem of non-unique solution of acoustic variables of rotary structure. Compared with the published literature, numerical simulation results and experimental results, the proposed method has higher calculation accuracy. In addition, based on this method, the influence of boundary conditions, geometric dimensions and other factors on the acoustic and vibration characteristics of spherical shells is discussed, which accumulates data for analyzing the acoustic and vibration behavior of spherical shells.

Analysis of FGM-type porous sandwich plates subjected to free vibration using Student Order Theory

This research adopts a high-order shear deformation theory to explore the free vibration characteristics of square porous sandwich plates constructed from functionally graded materials (FGMs). This theory effectively addresses shear deformation effects without necessitating a shear correction factor. The material properties of the FGMs change continuously throughout the plate's thickness, following a power-law distribution (P-FGM) based on the volume fractions of the component materials. The outer layers are metal, while a uniform ceramic layer constitutes the core. The governing equations are derived from the principle of virtual displacements. This study examines three different porosity distributions. By utilizing Hamilton's principle, the equations of motion were formulated. The numerical analysis demonstrates how the distribution of materials, geometry, and porosity affect the free vibration responses of FGM sandwich plates. The validity of the present theory is studied by comparing certain results obtained with other results published in the literature.

Analysis of Vibration Characteristics for Rotating Braided Fiber-Reinforced Composite Annular Plates with Perforations

In the current study, a comprehensive numerical model for analyzing the vibrational characteristics of braided fiber-reinforced composite (BFRC) rotating annular plate with perforations under diverse boundary constraints was introduced. This model employs the differential quadrature finite element method (DQFEM), which was developed based on the first-order shear deformation theory (FSDT) and coordinate transformation approach. The BFRC material, specifically a two-dimensional biaxial orthogonal fabric, was utilized to fabricate the annular plate with two distinct types of holes: circular and sector-shaped. The model’s convergence, accuracy, numerical stability, and reliability were confirmed through comparative assessments utilizing data from the literature, from ABAQUS software, and from experimental findings. The analysis focuses on studying the influences of structural properties, material parameters, and boundary restraints on the frequencies of vibration for BFRC rotating annular plates with holes. This theoretical model helps provide scientific basis and technical guidance for the stability and lightweight design of rotating annular plates, such as rotor structures in aircraft engines.

Localized Stochastic Dynamic Solution for Laminated Coupled Open Conical-Cylindrical Cabin System Based on the Condensation Method

This paper proposed a technical solution for quickly and accurately solving the stochastic dynamic characteristics of laminated coupled open conical-cylindrical cabin system. On the basic of first-order shear deformation theory (FSDT), the two-dimensional spectral Chebyshev method was introduced to obtain a unified dynamic matrix that included laminated conical and cylindrical panels. The middle part of the cabin was selected as the target substructure, while the precise dynamic condensation theory was used as a tool to establish a local dynamic analysis model of the cabin system based on comprehensive consideration of complex boundary conditions, coupling conditions, random load excitation and other factors. From a numerical analysis perspective, the local-level dynamic responses obtained from the model were validated to match well with the global-level results from the finite element method. Based on this numerical validation, a dynamic parametric analysis scheme was proposed, focusing on the local dynamic characteristics of the target substructure. This scheme analyzed the impact of the structural parameters of the middle part of cabin on the dynamic behaviors of the cabin system, providing technical guidance for the design optimization of the cabin system in engineering applications.

Free Vibration and Stability Analyses of Functionally Graded Plates Resting on Elastic Foundations Based on 2D and Quasi-3D Shear Deformation Theories Using the Finite Strip Method

Free Vibration and Stability Analyses of Functionally Graded Plates Resting on Elastic Foundations Based on 2D and Quasi-3D Shear Deformation Theories Using the Finite Strip Method

Nonlinear combined resonance of magneto-electro-elastic plates

The existing articles on the nonlinear resonances of magneto-electro-elastic (MEE) structures are mainly devoted to the parametric resonance and primary resonance, neglecting the coupling effect between parametric and forced excitations. To reveal this issue, the coupled longitudinal and transverse excitations are considered to investigate the combined resonance of MEE plates, in which the simply supported ends is considered. The expressions of magnetic-, electric- and displacement- fields are determined using the Maxwell's equation and first order shear deformation theory (FSDT). Applying the Galerkin method, the nonlinear ordinary differential equations are derived. The combined resonance problem of MEE plates with simply supported ends is solved by developing the method of varying amplitude (MVA). And the resonance trajectory is depicted by amplitude-frequency diagram in the follow-up analysis, in which the complicated dynamical phenomena with jump, hysteresis and multi-stable solution can be observed. To reveal the dynamic mechanism, comprehensive numerical analyses including electric potential, magnetic potential, excitations, temperature variation and other factors are conducted, the bifurcation and chaotic dynamic behaviors are also analyzed. The results elucidate that these parameters under discussion play significant roles.

Dynamic of composite nanobeams resting on an elastic substrate with variable stiffness

This study introduces a novel approach by combining finite simulation and an enhanced shear deformation theory to analyze the dynamic behavior of multi-layer composite nanobeams supported by an elastic foundation. The calculation formulae are derived from nonlocal theory in order to account for the impact of size effect. An intriguing aspect of this research is the presence of intricate curved profiles in the two material layers of the beam. The elastic foundation exhibits varying stiffness throughout the beam's length, accounting for many imperfections in the beam and including the drag parameter of the material. These elements contribute to the steady evolution of the issue model towards more realistic structures. Nevertheless, the computation gets intricate and impedes the precise approach. However, our work has successfully resolved this difficult and significant difficulty using the two-node element. This research demonstrates the temporal evolution of the dynamic reactions of the point with the greatest displacement and velocity, as dictated by the law of change. Simultaneously, it also presents visual representations of every point on the beam as it undergoes temporal variations. These conclusions possess both scientific and practical value, establishing a foundation for real-world implementations.

Electromagnetic frequency prediction pattern for electrical circuited shell based on FSTT theory using electromagnetic wave prediction method

This research focuses on electromagnetic analysis of electrical circuited shell to predict possible failure using wave prediction method. The equation is established considering axial and lateral hydrostatic pressure based on first order shear deformation theory of shell motion using the wave propagation approach and classic Flügge shell equations. For validation of accuracy of this research method, a comparison carried out with experimental data. With this method the effects of circuit parameters as well as different t power-law exponent on the frequencies are investigated. The results showed fantastic agreement between the present method and experimental ones.

Solving for the Thermal Vibration Behaviors of Functionally Graded Stepped Conical Curved Plates with Point Connection Conditions

This paper proposed a degree-of-freedom reduction model for the rapid and unified solution of the modal and steady-state response behavior of functionally graded stepped conical curved plates with point connections under thermal environments. Within the framework of the first-order shear deformation theory, the two-dimensional spectral Chebyshev method and the fixed interface modal synthesis method were combined to derive the degree-of-freedom reduction model for functionally graded conical curved plates under thermal environments. Each reduction model was assembled considering external load excitations. Virtual springs were used to equivalently handle the coupling interface forces and point connection interface forces, further establishing a thermal vibration response analysis model for functionally graded stepped conical curved plates with point connections. In numerical examples, this degree-of-freedom reduction model was compared with experimental results and overall finite element results, validating the accuracy of the model and the advantages of rapid solution. Additionally, this degree-of-freedom reduction model was applied to quickly analyze the effects of material parameters and point connection parameters on the thermal steady-state response of the structure, providing a theoretical basis for the environmental adaptability design, verification, and evaluation of such structures.

Large-amplitude nonlinear vibrations characteristics of bi-directional functionally graded plate having microstructure defects in hygro-thermal environment

The present article investigates the nonlinear hygro-thermal vibration characteristics of bi-directional functionally graded plates (BFGP) with microstructural defects. The distribution of material properties is anticipated to vary gradually along both thickness and length directions according to the modified power-law distribution. A novel expression has been developed to solve for the nonlinear temperature variation along the thickness of the BFGP. The micro-structural defects in terms of porosity have been incorporated in the formulation using a newly developed porosity model for a BFGP. A refined non-polynomial trigonometric higher-order shear deformation theory (HSDT) with a von Karman sense of geometric nonlinearity has been employed for the vibration response of BFGP. The finite element formulation has been carried out utilizing the C0 continuous Lagrangian quadrilateral element with nine nodes and eight degrees of freedom per node. The equations of motion have been derived using a variational approach. Various validation and comparative studies have been conducted to substantiate the current formulation's authenticity. The effect of the geometric nonlinearity, boundary constraints, volume fraction index along thickness (n) and volume fraction index along length (q), porosity index, temperature, and moisture concentration difference on the vibration frequency ratios (VFR) has been covered in depth. The VFR is more sensitive to the volume fraction indices (VFI) ' n′ than 'q'. Incorporating porosity decreases the VFR of BFGP, but it increases at higher amplitude ratios when BFGP becomes more porous. The VFR of BFGP exhibits nonlinear behavior with increasing moisture concentration and also varies non-linearly with the rise in amplitude ratio due to the hygro-thermal environment.

3D micromechanics assisted finite element approach for bending analysis of straight/wavy CNT-reinforced multi-scale hybrid composite plate

This study examines the micromechanical and structural behavior of straight and wavy carbon nanotube (CNT) reinforced multi-scale fiber-reinforced hybrid composite (MFRHC) plates. An important feature of MFRHC is the incorporation of nano-scale CNTs into two-phase carbon fiber-reinforced composites (CFRC), enhancing their micromechanical and macroscopic properties, particularly the bending and stress behaviors. In CNT fiber-reinforced composites, the fiber waviness can arise from manufacturing imperfections which demands in-depth analysis of MFRHC plates reinforced with both straight or wavy CNTs to characterize their relative influences. In this work, the CNTs are assumed to be continuous and distributed uniformly within the CFRC, and the waviness is considered to be sinusoidal spanning both longitudinal and transverse planes. A mathematical formulation is devised, based on a three-dimensional mechanics of materials (MOM) model, to analyze the micromechanical behavior of the hybrid composite and the model’s accuracy are compared with the multi-inclusion Mori-Tanaka method and the Voigt/Reuss rule of mixtures. The MOM model indicates that the waviness pattern of the CNTs significantly influences the effective stiffness properties of MFRHC in comparison to that of the straight CNTs. Following this, a finite element model is constructed to investigate the bending deflection and stress properties of a simply-supported symmetric cross-ply (0/90/0) MFRHC laminate under sinusoidal transverse loading, utilizing first-order shear deformation theory (FSDT). The results suggest that the bending deflection decreases for the MFRHC laminate when the waviness factor ( Ψ ) is ≤ 0.05 , but increases when Ψ > 0.05 . Finally, the bending stress behavior is examined for both straight and wavy CNT-reinforced MFRHC laminates considering their geometrical parameters ( a / h , z / h ), and the CNTs’ waviness factor ( Ψ ), respectively.

Dynamic instability and nonlinear response analysis of nanocomposite sandwich arches with viscoelastic cores

This paper presents a comprehensive study of the nonlinear dynamic behavior and snap-through phenomena in sandwich arch structures with viscoelastic cores and carbon nanotube-reinforced nanocomposite face sheets. Subjected to uniform time-dependent pressure shocks, these arches exhibit complex snap-through behavior critical for practical engineering applications. Utilizing third-order shear deformation theory, the study accurately captures nonlinear behaviors. The viscoelastic core, modeled with the Kelvin-Voigt law, enhances damping and reduces vibration amplitudes. Numerical solutions are obtained using a Chebyshev-based Ritz method, Newmark integration, and Newton-Raphson method. The Budiansky-Ruth criterion evaluates dynamic buckling loads. Key findings include significant instability near buckling loads, increased buckling loads and vibration damping due to viscoelastic effects, reduced buckling loads with foam cores, improved performance with CNTs, and more pronounced CNT effects with greater deflections. Additional conclusions highlight the sensitivity of dynamic snap-through to geometric parameters and the superior accuracy of the proposed approach compared to traditional models. This research advances the understanding and design strategies for nonlinear sandwich arch structures, enhancing predictive capabilities in complex structural systems.

Vibration analysis using the theory of exponential shear deformation for laminated plates

In this research, the free vibration response of laminated composite plates is studied using a refined exponential shear strain theory. The most interesting feature of this theory is that it allows exponential distributions of transverse shear strains, and verifies zero-shear boundary conditions on the plate surfaces without the use of shear correction factors. Some stiffness constants are written as a series. The number of independent unknowns in the present theory is four, compared with five in other shear deformation theories. The equations of motion are obtained from Hamilton's principle and Navier's method is used to determine the exact solution for antisymmetric cross-laminated plates. The numerical results found in the present analysis for free vibration are presented and compared with those available in the literature. The proposed theory is not only accurate, but also effective in predicting the fundamental frequencies of laminated composite plates.

Modal analysis of plates resting on elastic foundation based on the first-order shear deformation theory and finite element method

This paper investigates the free vibration characteristics of plate structures supported by a Pasternak elastic foundation, utilizing the first-order shear deformation theory (FSDT). FSDT simplifies the plate theory by considering only first-order shear deformation, enhancing formulation simplicity. Additionally, employing plate theory reduces computational complexity, as 2D models entail fewer degrees of freedom compared to their 3D counterparts. The finite element method (FEM) with 8-node quadrilateral element is employed for computational analysis, implemented using MATLAB. First, a comparison is made with some existing data to show the accuracy and reliability of the research. Numerical examples are then presented of the influence of the effects of thickness variation, foundation parameters and boundary conditions on frequency are investigated. The results show that the method converges very fast and reliability when compared to other research findings. The results of the research can be applied to many different engineering applications related to plates resting on elastic foundation.

Nonlinear dynamic characteristics of smart FG-GPLRC sandwich varying thickness truncated conical shell with internal resonance for first three order modes

This paper examines the 1:1:1 internal resonant nonlinear dynamic characteristic of the simply supported varying thickness functionally graded graphene platelets reinforced composite (FG-GPLRC) smart truncated sandwich conical shell subject to the combined effects of transverse load and in-plane force. The truncated smart sandwich conical shell is composed of an FG-GPLRC varying thickness core and two magneto-electro-elastic face layers, whose material properties and constitutive relations are individually identified by the rule of mixture, improved Halpin-Tsai approach and generalized Hooke's law. Utilizing the first-order shear deformation theory (FSDT), von Karman's geometrical nonlinearity, Hamilton's principle and Galerkin technique, the 3DOF dimensionless nonlinear dynamic formulations for the truncated smart FG-GPLRC conical shell are established. The multiple-scale technique is applied to developing the averaged equations for the truncated smart FG-GPLRC conical shell under combined resonance. The frequency-response and force-response curves, Poincare maps, phase portraits, time history diagrams, bifurcation and maximum Lyapunov exponent diagrams can be portrayed by the nonlinear equation solver and Runge-Kutta approach. The effects of the damping and tuning parameters, transverse and in-plane forces on the 1:1:1 internal resonant nonlinear dynamic characteristic of truncated smart varying thickness FG-GPLRC sandwich conical shell are examined.