• Advances in nano research
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• v.16 no.3
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• pp.289-301
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• 2024

In this work, an analytical model employing a new higher-order shear deformation beam theory is utilized to investigate the bending behavior of axially randomly oriented functionally graded carbon nanotubes reinforced composite nanobeams. A modified continuum nonlocal strain gradient theory is employed to incorporate both microstructural effects and geometric nano-scale length scales. The extended rule of mixture, along with molecular dynamics simulations, is used to assess the equivalent mechanical properties of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) beams. Carbon nanotube reinforcements are randomly distributed axially along the length of the beam. The equilibrium equations, accompanied by nonclassical boundary conditions, are formulated, and Navier's procedure is used to solve the resulting differential equation, yielding the response of the nanobeam under various mechanical loadings, including uniform, linear, and sinusoidal loads. Numerical analysis is conducted to examine the influence of inhomogeneity parameters, geometric parameters, types of loading, as well as nonlocal and length scale parameters on the deflections and stresses of axially functionally graded carbon nanotubes reinforced composite (AFG CNTRC) nanobeams. The results indicate that, in contrast to the nonlocal parameter, the beam stiffness is increased by both the CNTs volume fraction and the length-scale parameter. The presented model is applicable for designing and analyzing microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS) constructed from carbon nanotubes reinforced composite nanobeams.

• Steel and Composite Structures
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• v.28 no.1
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• pp.13-24
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• 2018

A size-dependent novel hyperbolic shear deformation theory of simply supported functionally graded beams is presented in the frame work of the non-local strain gradient theory, in which the stress accounts for only the nonlocal strain gradients stress field. The thickness stretching effect (${\varepsilon}_z{\neq}0$) is also considered here. Elastic coefficients and length scale parameter are assumed to vary in the thickness direction of functionally graded beams according to power-law form. The governing equations are derived using the Hamilton principle. The closed-form solutions for exact critical buckling loads of nonlocal strain gradient functionally graded beams are obtained using Navier's method. The derived results are compared with those of strain gradient theory.

• Steel and Composite Structures
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• v.33 no.3
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• pp.403-432
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• 2019

Axially functionally graded (AFG) beams are a new class of composite structures that have continuous variations in material and/or geometrical parameters along the axial direction. In this study, the exact analytical solutions for the free vibration of AFG and uniform beams with general elastic supports are obtained by using Euler-Bernoulli beam theory. The elastic supports are modeled with linear rotational and lateral translational springs. Moreover, the material and/or geometrical properties of the AFG beams are assumed to vary continuously and together along the length of the beam according to the power-law forms. Accordingly, the accuracy, efficiency and capability of the proposed formulations are demonstrated by comparing the responses of the numerical examples with the available solutions. In the following, the effects of the elastic end restraints and AFG parameters, namely, gradient index and gradient coefficient, on the values of the first three natural frequencies of the AFG and uniform beams are investigated comprehensively. The analytical solutions are presented in tabular and graphical forms and can be used as the benchmark solutions. Furthermore, the results presented herein can be utilized for design of inhomogeneous beams with various supporting conditions.

• Steel and Composite Structures
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• v.18 no.1
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• pp.1-28
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• 2015

In this paper, the dynamic response of functionally gradient steel (FGS) composite cylindrical panels in steady-state thermal environments subjected to impulsive loads is investigated for the first time. FGSs composed of graded ferritic and austenitic regions together with bainite and martensite intermediate layers are analyzed. Thermo-mechanical material properties of FGS composites are predicted according to the microhardness profile of FGS composites and approximated with appropriate functions. Based on the three-dimensional theory of thermo-elasticity, the governing equations of motionare derived in spatial and time domains. These equations are solved using the hybrid Fourier series expansion-Galerkin finite element method-Newmark approach for simply supported boundary conditions. The present solution is then applied to the thermo-elastic dynamic analysis of cylindrical panels with three different arrangements of material compositions of FGSs including ${\alpha}{\beta}{\gamma}M{\gamma}$, ${\alpha}{\beta}{\gamma}{\beta}{\alpha}$ and ${\gamma}{\beta}{\alpha}{\beta}{\gamma}$ composites. Benchmark results on the displacement and stress time-histories of FGS cylindrical panels in thermal environments under various pulse loads are presented and discussed in detail. Due to the absence of similar results in the specialized literature, this paper is likely to fill a gap in the state of the art of this problem, and provide pertinent results that are instrumental in the design of FGS structures under time-dependent mechanical loadings.

• Composites Research
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• v.16 no.5
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• pp.21-27
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• 2003

Using the theory of linear piezoelectricity, the dynamic response of a central crack in a functionally graded piezoelectric ceramic under anti-plane shear impact is analyzed. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. By using the Laplace and Fourier transform, the problem is reduced to two pairs of dual integral equations and then into Fredholm integral equations of the second kind. Numerical values on the dynamic stress intensity factors are presented to show the dependence of the gradient of material properties and electric loading.

• Steel and Composite Structures
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• v.43 no.6
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• pp.797-808
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• 2022

This paper presents an investigation on the free vibration characteristics of functionally graded nanocomposite double-beams reinforced by single-walled carbon nanotubes (SWCNTs). The double-beams coupled by an interlayer spring, resting on the elastic foundation with a linear layer and shear layer, and is simply supported in thermal environments. The SWCNTs gradient distributed in the thickness direction of the beam forms different reinforcement patterns. The materials properties of the functionally graded carbon nanotube-reinforced composites (FG-CNTRC) are estimated by rule of mixture. The first order shear deformation theory and Euler-Lagrange variational principle are employed to derive the motion equations incorporating the thermal effects. The vibration characteristics under several patterns of reinforcement are presented and discussed. We conducted a series of studies aimed at revealing the effects of the spring stiffness, environment temperature, thickness ratios and carbon nanotube volume fraction on the nature frequency.

• Steel and Composite Structures
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• v.23 no.3
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• pp.263-271
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• 2017

A theoretical study was carried-out of mode II delamination fracture behavior of the End Loaded Split (ELS) functionally graded beam configuration with considering the material non-linearity. The mechanical response of ELS was modeled analytically by using a power-law stress-strain relation. It was assumed that the material is functionally graded transversally to the beam. The non-linear fracture was investigated by using the J-integral approach. Equations were derived for the crack arm curvature and zero axes coordinate that are needed for the J-integral solution. The analysis developed is valid for a delamination crack located arbitrary along the beam height. The J-integral solution was verified by analyzing the strain energy release rate with considering material non-linearity. The effects of material gradient, non-linear material behavior and crack location on the fracture were evaluated. The solution derived is suitable for parametric analyses of non-linear fracture. The results obtained can be used for optimization of functionally graded beams with respect to their mode II fracture performance. Also, such simplified analytical models contribute for the understanding of delamination fracture in functionally graded beams exhibiting material non-linearity.

• Steel and Composite Structures
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• v.43 no.1
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• pp.31-54
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• 2022

This paper is concerned with the buckling behavior of functionally graded graphene reinforced porous nanocomposite beams based on the finite element method (FEM) using two variables trigonometric shear deformation theory. Both Young's modulus and material density of the FGP beam element are simultaneously considered as grading through the thickness of the beam. The finite element approach is developed using a nonlocal strain gradient theory. The governing equations derived here are solved introducing a 3-nodes beam element, and then the critical buckling load is calculated with different porosity distributions and GPL dispersion patterns. After a convergence and validation study to verify the accuracy of the present model, a comprehensive parametric study is carried out, with a particular focus on the effects of weight fraction, distribution pattern of GPL reinforcements on the Buckling behavior of the nanocomposite beam. The effects of various structural parameters such as the dispersion patterns for the graphene and porosity, thickness ratio, boundary conditions, and nonlocal and strain gradient parameters are brought out. The results indicate that porosity distribution and GPL pattern have significant effects on the response of the nanocomposite beams, and the results allows to identify the most effective way to achieve improved buckling behavior of the porous nanocomposite beam.

• Steel and Composite Structures
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• v.45 no.3
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• pp.331-348
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• 2022

The main goal of this article is to develop the finite element formulation based on the nonlocal strain gradient and the refined higher-order deformation theory employing a new function f(z) to investigate the static bending and free vibration of functionally graded porous (FGP) nanobeams. The proposed model considers the simultaneous effects of two parameters: nonlocal and strain gradient coefficients. The nanobeam is made by FGP material that exists in un-even and logarithmic-uneven distribution. The governing equation of the nanobeam is established based on Hamilton's principle. The authors use a 2-node beam element, each node with 8 degrees of freedom (DOFs) approximated by the C¹ and C² continuous Hermit functions to obtain the elemental stiffness matrix and mass matrix. The accuracy of the proposed model is tested by comparison with the results of reputable published works. From here, the influences of the parameters: nonlocal elasticity, strain gradient, porosity, and boundary conditions are studied.

• Smart Structures and Systems
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• v.19 no.3
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• pp.309-322
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• 2017

In this paper, the buckling, and free vibration analysis of tapered functionally graded carbon nanotube reinforced composite (FG-CNTRC) micro Reddy beam under longitudinal magnetic field using finite element method (FEM) is investigated. It is noted that the material properties of matrix is considered as Poly methyl methacrylate (PMMA). Using Hamilton's principle, the governing equations of motion are derived by applying a modified strain gradient theory and the rule of mixture approach for micro-composite beam. Micro-composite beam are subjected to longitudinal magnetic field. Then, using the FEM, the critical buckling load, and natural frequency of micro-composite Reddy beam is solved. Also, the influences of various parameters including ${\alpha}$ and ${\beta}$ (the constant coefficients to control the thickness), three material length scale parameters, aspect ratio, different boundary conditions, and various distributions of CNT such as uniform distribution (UD), unsymmetrical functionally graded distribution of CNT (USFG) and symmetrically linear distribution of CNT (SFG) on the critical buckling load and non-dimensional natural frequency are obtained. It can be seen that the non-dimensional natural frequency and critical buckling load decreases with increasing of ${\beta}$ for UD, USFG and SFG micro-composite beam and vice versa for ${\alpha}$. Also, it is shown that at the specified value of ${\alpha}$ and ${\beta}$, the dimensionless natural frequency and critical buckling load for SGT beam is more than for the other state. Moreover, it can be observed from the results that employing magnetic field in longitudinal direction of the micro-composite beam increases the natural frequency and critical buckling load. On the other hands, by increasing the imposed magnetic field significantly increases the stability of the system that can behave as an actuator.