Hygrothermal and Imperfection Effects on FG Nanobeam Vibration via a Refined Integral Shear Theory

3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory

Due to the asymmetry of the material gradation, the neutral surface in functionally graded (FG) beams is dislocated from the mid-plane, so mid-plane-based model may lead to inaccurate results in FG material (FGM) beam analysis. However, a little attention is paid to the neutral surface-based vibration in FG beams, especially when the material properties are temperature dependent. This study addresses the analysis of the neutral axis (NA) position effect on fundamental frequency of FG Timoshenko beams with temperature-dependent properties under nonlinear temperature rise. First, a formula for exact position of the neutral surface in the temperature-dependent beams is derived for general nonlinear temperature rise distribution and power law of material grading. Then, the dislocation of the NA from the central one is examined along material grading index and various temperature distributions. Finally, the effect of exact NA position and temperature distributions on fundamental frequency is investigated to reveal when the NA position and nonlinear temperature distribution should be considered in vibration analysis of the beams. The main results obtained in this study are that the exact position of NA is derived in an analytical expression for temperature-dependent FG beams and its effect on fundamental frequency of the beam is examined. Numerical analysis shows that the exact position of the NA has a significant impact on natural frequency and it cannot be ignored in the thermal vibration analysis of FG beams, especially in the case of high-temperature environment.

Exact solutions of steady-state dynamic responses of a laminated composite double-beam system interconnected by a viscoelastic layer in hygrothermal environments

This article seeks to investigate the effect of geometrical uncertainties on the free vibration of Euler–Bernoulli Functionally Graded (FG) beams resting on Winkler-Pasternak elastic foundation. In this scenario, the uncertainties are linked to length and thickness of FG beam using Symmetric Gaussian Fuzzy Number (SGFN). The governing equations of motion regarding free vibration of the uncertain model are derived by combining the Symmetric Gaussian Fuzzy Number with Hamilton's principle and double parametric form of fuzzy numbers. The natural frequencies of the uncertain models are computed using the double parametric form-based Navier's approach for Hinged-Hinged (H–H) boundary condition. The double parametric form-based Hermite-Ritz approach was also used to calculate the natural frequencies of Hinged-Hinged (H–H), Clamped-Hinged (C-H), and Clamped–Clamped (C–C) boundary conditions. Natural frequencies obtained using Navier's method and Hermite-Ritz method are used to validate the results of the uncertain model which exhibit strong agreement. A comprehensive parametric analysis is also conducted with respect to various graphical and tabular results to investigate the fuzziness or spreads of natural frequencies in relation to various uncertain parameters.

In the present study, a nonlocal finite element method (FEM) is proposed to investigate the free vibration of functionally graded (FG) nanobeams resting on two parameters, the Winkler–Pasternak elastic foundation. Using the Eringen’s nonlocal elasticity theory, the Euler–Bernoulli beam model is implemented. The equations of motion are obtained by using Hamilton’s principle. Material properties of the beam vary in the thickness (height) direction based on the power law. The frequencies of functionally graded nanobeam are obtained for simply supported (S-S) boundary conditions with various values of power law exponent, small-scale (nonlocal) parameter, Winkler foundation parameter, and Pasternak foundation parameter. Vibration response of nano-scaled functionally graded beam resting on the Winkler–Pasternak elastic foundation is investigated via the nonlocal finite element method. A comparison of the numerical results of the present study with those from the open literature demonstrates a good agreement. Also, the difference between classical elasticity theory and nonlocal elasticity theory is discussed in this study.

This study explores the free vibration behavior of multi-cracked functionally graded (FG) nanobeams situated on Winkler–Pasternak elastic foundations. Cracks are modeled as rotational springs, effectively dividing the beam into sub-beams. The present model is formulated for metal–ceramic FG nanobeams composed of two isotropic phases with material properties varying continuously across the thickness based on a power-law distribution. The analysis incorporates Euler–Bernoulli beam theory and Eringen’s nonlocal elasticity theory to capture size-dependent effects. A fourth-order differential equation with constant coefficients is formulated, and an analytical approach is utilized to calculate the frequency parameter values. Validation with existing literature in specific cases confirms the accuracy of the obtained results. The effects of various factors, including the transverse gradient index, nonlocal parameter, Winkler and Pasternak foundation parameters, crack severity, and crack locations, on the first four frequency parameters are investigated. Unlike existing studies, the proposed model allows for any number of cracks (multiple cracks), offering a generalized framework for realistic damage scenarios. This is the first study to analytically address the combined effects of multiple cracks with elastic foundations in FG nanobeams. This framework provides a robust tool for analyzing FG nanobeams with cracks under diverse boundary conditions and material gradations. These findings have significant implications for the design and optimization of nanostructures in aerospace, mechanical, and biomedical applications.

Free vibration analysis is carried out on axially inhomogeneous beams resting on Winkler-Pasternak elastic foundation. The material properties of the beam like Young’s modulus, modulus of rigidity and material density are considered to be varying along the length direction following constant, linear and exponential material models. The beam is subjected to different combinations of clamped and simply supported boundary conditions. The formulation is based on Timoshenko beam theory and energy method along with Hamilton’s principle is used to derive the governing equations. The effect of material gradation and the 2 parameters of elastic foundation on the natural frequencies are studied in detail. The present results are validated by comparing them with established ones and satisfactory matching is observed.
 HIGHLIGHTS
 
 Free vibration behavior of axially functionally graded beams in investigated
 Three different material models and three boundary conditions are considered
 Effect of two parameter elastic foundation is considered
 Timoshenko beam theory and Rayleigh Ritz method are employed
 It is found that the stiffness of elastic foundation significantly affects the natural frequency of the beam
 
 GRAPHICAL ABSTRACT

The present study aims to carry out finite element analysis to investigate the natural and whirl frequencies of a functionally graded (FG) rotor-bearing system modeled following the exponential law gradation. The FG shaft is modeled using the Timoshenko beam theory (TBT). The thermal gradation of FG shaft material properties is achieved following the exponential temperature distribution method (ETD). The derivation of element stiffness, mass and gyroscopic matrices has been carried out including the material and thermal gradation effects. FG shaft, consisting of a mixture of Stainless Steel (SS) and Zirconium Dioxide (ZrO2), is investigated, where the volume fraction of metal decreases toward the outer radius and ceramic volume fraction increases. A Python code has been developed to perform the finite element analysis to extract the Eigenvalues (natural frequencies) of the FG rotor bearing subjected to different thermal gradients. The analysis presents the effects of variations in material gradation and thermal gradients on the natural frequencies and whirl frequencies of the FG rotor-bearing system.

An accurate shear deformation theory for vibration and buckling of FGM sandwich plates in hygrothermal environment

In the present work, effect of hygrothermal environment on vibration and buckling behavior of embedded functionally graded elliptical plate under uniform in plane compression is studied. The properties of elliptical plate vary in transverse direction following power law. The functionally graded elliptical plate is considered to be resting on the Winkler–Pasternak elastic foundation. The governing equations are derived using the principle of virtual work and solved by employing the Rayleigh–Ritz method. The algebraic polynomials are employed to satisfy the different boundary constraints. The advantage of the presented mathematical model over the previously reported methods is that it eliminates the constraints regarding edge conditions, and it is simple and computationally fast. The inclusive results depicting the effect of various parameters namely, material property exponents, foundation parameters, aspect ratio on mechanical and thermomechanical buckling, and natural frequency of embedded functionally graded elliptical plate in a hygrothermal environment are reported after the test of convergence and extensive comparisons. The study shows that increase in foundation moduli lead to an increase in natural frequency and buckling parameter. Furthermore, it is noticed that the temperature and moisture concentration remarkably affect the buckling and vibration behavior of functionally graded elliptical plate.

Stiffened functionally graded (FG) folded plates are being widely used these days in aeronautical, automotive, and naval applications which are often exposed to extreme environmental conditions characterized by high temperature and moisture concentrations. However, a numerical model to estimate the dynamic response of such structures in hygrothermal environment is almost non-existent. It is from this perspective that, a MATLAB-based finite element (FE) model is developed to assess the free vibration characteristics of stiffened FG folded panels in hygrothermal environment and is presented for the first time in this article. The governing equation is derived following the first order shear deformation theory including hygro-elastic and thermo-elastic relations. Temperature and moisture concentrations are assumed to vary both linearly and uniformly through the plate’s thickness to numerically simulate hygrothermal environment. Non-linear strains are incorporated in the present FE model to consider the hygrothermal effect. Another FE model is also developed in COMSOL Multiphysics software for these structures and is compared with the MATLAB-based model. Numerical simulations reveal that the free vibration characteristics of stiffened FG folded plates are significantly affected by hygrothermal environment. The present FE models can be effectively utilized by engineers and researchers to analyze and optimize the dynamic behavior of stiffened FG folded plates. The results are also expected to serve as a benchmark for future research works.

This paper addresses the static deformation of simply supported rectangular micro/nano plates made of functionally graded (FG) materials based on the three-dimensional nonlocal elasticity theory of Eringen. The plates are assumed to be simply supported and rested on a Winkler-Pasternak elastic foundation. Elasticity modulus is assumed to obey an exponential law along the thickness direction of the micro/nano plate. Using the Fourier series, a displacement field is defined that satisfies simply supported boundary condition and reduces three elasticity equations to two independent equations. The closed-form bending response is achieved by exerting boundary conditions of the lateral surfaces. Numerical results are presented to investigate the influences of the gradient index of the material properties, nonlocal parameter and stiffness of elastic foundation on the mechanical behavior of the plates.

ABSTRACTThis article presents an investigation on the buckling of functionally graded (FG) truncated conical shells under an axial load resting on elastic foundations within the shear deformation theory (SDT). The governing equations are solved using the Galerkin method, and the closed-form solution of the axial buckling load for FG conical shells on elastic foundations within the SDT is obtained. Various numerical examples are presented and discussed to verify the accuracy of the closed-form solution in predicting dimensionless buckling loads for FG conical shells on the Winkler–Pasternak elastic foundations within the SDT.

This paper aims to analyze dynamic response of axially functionally graded (FG) Timoshenko beams on Winkler–Pasternak elastic foundation. Spring constant and material inhomogeneity are used as research parameters in the dynamic analysis. In this study, ODEs in Laplace domain are calculated by using complementary functions method. The analysis is performed in the Laplace domain, and calculations are converted to the time domain by using Durbin’s algorithm. The effects of material gradient index, translational and rotational spring constants, slenderness ratio, different boundary conditions and impulsive loads on the dynamic response of FG Timoshenko beams are investigated.

The sandwich structures contain three or more layers attached to the core. In the current research, a three-layered sandwich microplate containing functionally graded (FG) porous materials as core and piezoelectric nanocomposite materials as face sheets subjected to electric field resting on Pasternak foundation is chosen as a model to investigate its vibrational behavior. To make the face sheets stiffer, they are reinforced by carbon nanotubes (CNTs) via different distribution patterns which result in changing their properties along the thickness direction. An innovative quasi-3D shear deformation theory with five unknowns, Hamilton’s principle, and modified couple stress theory are hired to gain equations of motion related to the abovementioned microstructure. Eventually, the evaluation of materials’ properties, geometry specifications, foundation moduli, and hygrothermal environment on vibrational behavior of such structures became easier using the presented results of the current study in figure format. As an instance, it is revealed that CNTs’ volume fraction elevation causes mechanical properties improvement, and in the following, natural frequency increment. Besides, considering the hygrothermal environment causes significant effects on the results.