Nonlocal nonlinear analysis of functionally graded plates using natural neighbour Galerkin method

Nonlocal nonlinear analysis of functionally graded plates using third-order shear deformation theory

Functionally graded materials (FGM) are an advanced class of engineering composites constituting of two or more distinct phase materials described by continuous and smooth varying composition of material properties in the required direction. In this work, the effect of the material homogenization scheme on the flexural response of a thin to moderately thick FGM plate is studied. The plate is subjected to different loading and boundary conditions. The formulation is developed based on the first-order shear deformation theory. The mechanical properties are assumed to vary continuously through the thickness of the plate and obey a power-law distribution of the volume fraction of the constituents. The variation of volume fraction through the thickness is computed using two different homogenization techniques, namely rule of mixtures and Mori–Tanaka scheme. Comparative studies have been carried out to demonstrate the efficiency of the present formulation. The results obtained from the two techniques have been compared with the analytical solutions available in the literature. In addition to the above a parametric study bringing out the effect of boundary conditions, loads, and power-law index has also been presented.

A nonlocal higher-order shear deformation approach for nonline ar static analysis of magneto-electro-elastic sandwich Micro/Nano-plates with FG-CNT core in hygrothermal environment

ABSTRACTWhere deflections of thick plates are concerned, sufficient studies have been performed on circular and rectangular plates using first-order shear deformation theory (FSDT). Less attention, however, has been placed on the use of higher-order theories and plates of other shapes. This article proposes a deflection model for a simply supported and under uniformly loaded equilateral triangular plate that fulfils the third-order shear deformation theory (TSDT). Comparison of maximum plate deflections using this exact TSDT model with the FSDT and the simplified TSDT models reveals that the former exhibits better correlation agreement with the exact TSDT. Using the plate deflection from the exact TSDT, a refined shear correction factor, which is a function of Poisson's ratio, relative thickness, and location on plate, is extracted for the FSDT model. Results suggest that the usual shear correction model of 5/6 in Mindlin plates is highly accurate when dealing with moderately thick triangular plates made from negative Poisson's ratio materials, while the prescription of the refined shear correction factor obtained herein is advised for very thick triangular plates made from large Poisson's ratio materials. The results avail an FSDT deflection model for very thick triangular plates with the accuracy of exact TSDT.

For moderately thick plates, the use of First order Shear Deformation Theory (FSDT) with a constant shear correction factor of 5/6 is sufficient to take into account the plate deflection arising from transverse shear deformation. For very thick plates, the use of Third order Shear Deformation Theory (TSDT) is preferred as it allows the shear strain distribution to be varied through the plate thickness. Therefore no correction factor is required in TSDT, unlike FSDT. Due to the complexity involved in TSDT, this paper obtains a more accurate shear correction factor for use in FSDT of very thick simply supported and uniformly loaded isosceles right triangular plates based on the TSDT. By matching the maximum deflections for this plate according to FSDT and TSDT, a variable shear correction factor is obtained. Results show that the shear correction factor for the simplified TSDT, i.e. 14/17, is least accurate. The commonly adopted shear correction factor of 5/6 in FSDT is valid only for very thin or highly auxetic plates. This paper provides a variable shear correction for FSDT deflection that matches the plate deflection by TSDT. This variable shear correction factor allows designers to justify the use of a commonly adopted shear correction factor of 5/6 even for very thick plates as long as the Poisson’s ratio of the plate material is sufficiently negative.

In the present work, flexural analysis of a thin to moderately thick FGM plate subjected to transverse loads have been studied using finite element method. The formulation is developed based on the First order Shear Deformation Theory (FSDT). The mechanical properties are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents. FGM’s are typically heterogeneous in nature and a homogenization scheme is generally adopted for the analysis. To ascertain the effect of homogenization schemes on the material properties, the variation of volume fraction through the thickness have been computed using two different material homogenization techniques; namely the Rule of Mixtures and Mori-Tanaka scheme. In calculating the effective material properties through the thickness numerical integration have been used to evaluate the integrals which is easier and faster when compared to symbolic integration. A detailed discussion comparing the results from both the homogenization schemes have been presented. In addition to that a detailed parametric study bringing out the effect of boundary conditions, loading intensities and volume fraction index have been presented. Convergence tests and comparison studies have been carried out to demonstrate the efficiency of the present formulation.

The present study investigates the free vibration behavior of rotating beams made of functionally graded materials (FGMs) with a tapered geometry. The material properties of the beams are characterized by an exponential distribution model. The stiffness and mass matrices of the beams are derived using the principle of virtual energy. These matrices are then evaluated using three different beam theories: Bernoulli–Euler (BE) or Classical Beam Theory (CBT), Timoshenko (T) or First-order Shear Deformation Theory (FSDT), and Reddy (R) or Third-order Shear Deformation Theory (TSDT). Additionally, the study incorporates uncertainties in the model parameters, including rotational velocity, beam material properties, and material distribution. The mean-centered second-order perturbation method is employed to account for the randomness of these properties. To ensure the robustness and accuracy of the probabilistic framework, numerical examples are presented, and the results are compared with those obtained through the Monte Carlo simulation technique. The investigation explores the impact of critical parameters, including material distribution, taper ratios, aspect ratio, hub radius, and rotational speed, on the natural frequencies of the beams is explored within the scope of this investigation. The outcomes are compared not only with previously published research findings but also with the results of 3-Dimensional Finite Element (3D-FE) simulations conducted using ANSYS to validate the model’s effectiveness. The comparisons demonstrate a strong agreement across all evaluations. Specifically, it is observed that for thick beams, the results obtained from FSDT and TSDT exhibit a greater agreement with the 3D-FE simulations compared to CBT. It is shown that the coefficient of variation (C.O.V.) of first mode eigenvalue of TSDT, FSDT and CBT are approximately identical for random rotational velocity and discernible deviations are noted in CBT compared to FSDT and TSDT in the case of random material properties. The findings suggest that TSDT outperforms FSDT by eliminating the need for a shear correction coefficient, thereby establishing its superiority in accurately predicting the natural frequencies of rotating, tapered beams composed of FGMs.

Review and comparison of thin and thick FGM plate theories using a unified buckling formulation

This article develops the modified couple stress theory to study the free vibration of bi-directional functionally graded microplates subjected to multidimensional temperature distribution. Third-order shear deformation and classical theories of plates are adapted for free vibration analysis of thick and thin microplates, respectively. Employing the third-order shear deformation theory, both normal and shear deformations are considered without the need for shear correction factor. Material of the bi-directional functionally graded microplate is graded smoothly through the length and thickness of the microplate. Gradient of the material is assumed to obey from the power law in terms of the volume fraction of the constituents. Assuming the uniform and nonuniform temperature distributions, the effect of thermal environment on dynamic behavior of the microplate is discussed in detail. Applying the Ritz method, the displacement field is expanded by admissible functions which satisfy the essential boundary conditions, and Hamilton principle is employed to determine the natural frequencies of the microplate. Developed model has been applied to determine the natural frequencies in problems of thin/thick, one-directional/bi-directional functionally graded, and homogeneous/nonhomogeneous microplates. Effects of parameters such as the thermal environment, power law indexes [Formula: see text] and length scale parameter on free vibration of these problems are studied in detail. The results show that higher values of length scale parameter and temperature rise decrease the natural frequency of the bi-directional functionally graded microplate. According to results obtained by classical and third-order shear deformation theories, the third-order shear deformation theory is proposed for vibration analysis of microplates with thickness-to-length ratio less than five.

In this work, different shear deformation theories are used for the first time to investigate the effect of shear deformation on the dynamic response of laminated plates and track slab on a Pasternak foundation under moving load. The Pasternak formulation was used to simulate the interaction between the slab and the elastic foundation. Using Hamilton’s principle, the governing equations are derived based on the third-order shear deformation theory (TSDT) in the higher-order shear deformation theory (HSDT). Additionally, governing equations based on the first-order shear deformation theory (FSDT) and the classical laminate plate theory (CLPT) are also derived. The analytical solution of the classical plate theory (CPT) is derived as an example for isotropic thin plates and used as a benchmark solution to verify the accuracy of the governing equations based on TSDT, FSDT, and CLPT. Then, the effect of the thickness-to-width ratio on the dynamic response of the laminated slab is investigated, along with the effects of the Pasternak foundation coefficient, the vertical load movement speed, and load eccentricity on the dynamic response of the laminated plate and the track slab. The effect of uncertainty in the modulus of elasticity on the analytical results is also examined. The results indicate that considering shear deformation is essential in the dynamic analysis of CRTSII plate ballast slabs, but higher-order shear deformation theory should be avoided due to its computational cost and complexity. This study provides a benchmark solution for further work.

Studies of vibration of plates have matured and are a well-established branch of research in structural dynamics. They have a vast range of applications in engineering and technology. But not much work can be found on vibration analysis of Functionally Graded Materials (FGMs) as compared to isotropic and composite plates and shells. FGMs are those in which the volume fraction of the two or more constituent materials is varied, as a power-law distribution, continuously as a function of position along certain dimension(s) of the structure From the perspective of finite element method (FEM) studies of FGM, Praveen and Reddy [3], studied the static and dynamic responses of functionally graded (FG) ceramic-metal plate accounting for the transverse shear deformation, rotary inertia and moderately large rotations in the Von-Karman sense, in which the effect of an imposed temperature field on the response of the FG plate was discussed in detail. Ng et al. [4] dealt with the parametric resonance of FG rectangular plates under harmonic in-plane loading. Ferreira and Batra [5] provided a global collocation method for natural frequencies of FG plates by a meshless method with first order shear deformation theory (FSDT). Woo et al. [6] presented an analytical solution for the nonlinear free vibration behavior of FGM plates, where the fundamental equations were obtained using the Von-Karman theory for large transverse deflection, and the solution was based in terms of mixed Fourier series. Zhao et al. [7] studied the free vibration analysis of metal and ceramic FG plates using the element-free kp-Ritz method. The FSDT was employed to account for the transverse shear strain and rotary inertia, mesh-free kernel particle functions were used to approximate the two-dimensional displacement fields and the eigen-equation was obtained by applying the Ritz procedure to the energy functional of the system. Batra and Jin [8] used the FSDT coupled with the FEM to study the free vibrations of an FG anisotropic rectangular plate with various edge conditions. Also, Batra and Aimmanee [9] studied a higher order shear and normal deformable plate theory by FEM. Many studies conducted on FGMs are related to the analysis of free vibration by applying FSDT (see Other forms of shear deformation theory, such as the third order-shear deformation theory (TSDT) that accounts for the transverse effects, have been considered. Cheng and Batra [13] www.intechopen.

In this paper, spectral finite elements (SFEs) are developed for wave propagation analysis of isotropic curved beams using three different beam models: (1) the refined third-order shear deformation theory (TOT), (2) the first-order shear deformation theory (FSDT), and (3) the classical shell theory (CST). The formulation is validated by comparing the results for the wavenumber dispersion relations and natural frequencies with the published results based on the FSDT. The numerical study reveals that even for a very thin curved beam with radius-to-thickness ratio of 1000, the wavenumbers predicted by the CST at high frequencies show significant deviation from those of the shear deformable theories, FSDT and TOT. The FSDT results for the wavenumber of the flexural displacement mode differ significantly from the TOT results at high frequencies even for thin beams. The deviation increases and occurs at lower frequencies with the decrease in the radius-to-thickness ratio. The results for wave propagation response show that the CST yields highly erroneous response for flexural mode wave propagation even for thin beams and at a relatively low frequency of 20 kHz. The FSDT results too differ by unacceptably high margin from the TOT results for flexural wave response of thin beams at frequencies greater than 100 kHz, which are typically used for structural health monitoring (SHM) applications. For thick beams, FSDT results for the tangential wave response also show large deviation from the TOT results.

The paper presents a numerical assessment of the performance of the Refined Zigzag Theory to the analysis of bending (deflection and stress distributions) and free vibration of functionally graded material plates, monolayer and sandwich, under a set of different boundary conditions. The numerical assessment is performed comparing results from Refined Zigzag Theory using Ritz method with those from 3D, quasi-3D, and 2D theories and finite element method. In the framework of 2D theories, equivalent single-layer theories of different orders (sinusoidal, hyperbolic, inverse-hyperbolic, third-order shear deformation theory, first-order shear deformation theory, and classical plate theory) have been used to investigate deformation, stresses, and free vibration and compared with results from the Refined Zigzag Theory. After validating the convergence characteristics and the numerical accuracy of the developed approach using orthogonal admissible functions, a detailed parametric numerical investigation is carried out. Bending under transverse pressure and free vibration of functionally graded material square and rectangular plates of a different aspect ratio under various combinations of geometry (core-to-face sheet thickness ratio and plate to thickness ratio), boundary conditions and law of variation of volume fraction constituent in the thickness direction (power-law functionally graded material, exponential law functionally graded material, and sigmoidal-law functionally graded material) is studied. Monolayer and sandwich plates with homogeneous core and functionally graded face sheets are considered for the assessment. It is concluded that the Refined Zigzag Theory generally predicts the global (deflection and frequencies) and local (displacement and stress distributions) response of functionally graded material sandwich plates, more accurately than first-order shear deformation theory and third-order shear deformation theory, while retaining its simplicity.

Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method

A simplified numerical procedure has been developed to study the hygrothermal bending behaviour of fibre reinforced plastics bridge deck. The bridge deck has been modelled as a combination of laminated plates and closely spaced box-shaped stiffeners using the finite element method. The eight noded isoparametric plate bending element with seven degrees of freedom per node is used to formulate the plate element. The box-shaped stiffeners are modelled using three noded isoparametric beam element having same degrees of freedom per node as the plate element. A computer code has been developed in MATLAB 2013. The stress distributions throughout the thickness of the deck plate are determined considering third order shear deformation theory (TSDT) as well as first order shear deformation theory (FSDT). A comparative study of FSDT and TSDT under thermal and moisture load is presented here. The temperature distribution over the surface of the deck plate is considered here in two ways, namely, uniform temperature distribution and sinusoidal temperature distribution over the surface of the deck. The elevated moisture concentration has been considered to be uniform throughout the surface of the bridge deck.