Normal Deformation Theory Research Articles

Static and Stability Characteristics of Geometrically Imperfect FGM Plates Resting on Pasternak Elastic Foundation with Microstructural Defect

The static and stability characteristics of geometrically imperfect functionally graded material (FGM) plate with a microstructural defect (porosity) resting on Pasternak elastic foundation are investigated. The formulations are based on hybrid higher-order shear and normal deformation theory. A new mathematical expression is presented to accomplish the effective material properties of the material with porosity inclusion. A generic function is employed to model various modes of geometric imperfection. The equations of motion are derived using variational principle. Convergence and comparison studies with reported results ensure the reliability and accuracy of the present solution. Influence of geometric imperfection, porosity, foundation parameters, and boundary constraints on the static and stability behavior of FGM plate is examined. The results reflect that the geometric imperfection and porosity have a significant influence on the structural response of FGM plate.

Size dependent bending analysis of two directional functionally graded microbeams via a quasi-3D theory and finite element method

Abstract This paper presents the flexural behaviour of two directional functionally graded (2D-FG) microbeams subjected to uniformly distributed load with various boundary conditions. A four-unknown shear and normal deformation theory or quasi-3D one is employed based on the modified couple stress theory, Ritz method and finite element formulation. The material properties are assumed to vary through the thickness and longitudinal axis and follow the power-law distribution. Firstly, the static deformations of conventional FG microbeams are investigated to verify the developed finite element code. For the convergence studies, a simply supported FG microbeam is considered by employing various number of elements in the problem domain, aspect ratios, material length scale parameters and gradient indexes. The verification of the developed code is established and then extensive studies are performed for various boundary conditions. Secondly, since there is no reported data regarding to the analysis of 2D-FG microbeams, verification studies are performed for 2D-FG beams with different aspect ratios and gradient indexes. The effects of the normal and shear deformations as well as and material length scale parameters on the flexural behaviour of the 2D-FG microbeams are investigated. Finally, some new results for deflections of conventional FG and 2D-FG microbeams for various boundary conditions are introduced for the first time and can be used as reference for future studies.

Imperfection sensitivity of the post-buckling characteristics of functionally gradient plates using higher-order shear and normal deformation theory

The aim of the present paper is to investigate the post-buckling responses of geometrically imperfect gradient plate using hybrid deformation plate theory (HHDT). This theory is accountable for realistic transverse shear distribution along with thickness stretching effect. The geometric imperfection is incorporated using various imperfection functions in the transverse direction only. Parametric studies have been carried out to present new results using finite element method with C0 continuous element. The consequences of geometric nonlinearity, various geometric imperfection, and geometric configuration on the Post-buckling characteristics of FGM plate is investigated.

Bending Analysis of Two Directional Functionally Graded Beams Using A Four-Unknown Shear and Normal Deformation Theory

The bending behaviour of two-directional functionally graded beams (FGBs) subjected to various sets of boundary conditions is investigated by using a shear and normal deformation theory and the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. A simply supported conventional FGB problem is studied to validate the developed code. The comparison studies are performed along with the analytical solutions and the results from previous studies. The numerical calculations in terms of maximum dimensionless transverse deflections, dimensionless axial and transverse shear stresses are performed for various gradation exponents, aspect ratios (L/h) and sets of boundary conditions. The effects of the gradation exponents on the accuracy and the robustness of the SSPH method are also investigated for the two directional functionally graded beams which are having clamped-free boundary condition..

Nonlocal Free Vibration Analysis of FG-Porous Shear and Normal Deformable Sandwich Nanoplate with Piezoelectric Face Sheets Resting on Silica Aerogel Foundation

The present work is devoted to the free vibration analysis of sandwich nanoplate with functionally graded porous core and piezoelectric face sheets. The rectangular elastic sandwich nanoplate is resting on silica aerogel foundation based on Vlasov’s model foundation. Material properties of the core layer such as Young’s modulus, shear modulus and density are supposed to vary along the thickness direction by means of uniform and nonuniform function. The governing equations of motion are derived from Hamilton’s principle using sinusoidal shear and normal deformation theory. An iterative technique is presented to solve seven governing equations of motion. After all, the natural frequency verification with corresponding studies is surveyed. Eventually, the numerical results are scrutinized for plate aspect ratio, thickness ratio, nonlocal parameter, porosity index, Young’s modulus and height of silica aerogel foundation. The numerical results indicate that the Young’s moduli and height of silica aerogel foundation, porosity index of core layer, thickness ratio and nonlocal parameter have significant effects on free vibration response of the sandwich nanoplate.

Free vibration analysis of a three-layered microbeam based on strain gradient theory and three-unknown shear and normal deformation theory

Free vibration analysis of a three-layered microbeam including an elastic micro-core and two piezo-magnetic face-sheets resting on Pasternak's foundation are studied in this paper. Strain gradient theory is used for size-dependent modeling of microbeam. In addition, three-unknown shear and normal deformations theory is employed for description of displacement field. Hamilton's principle is used for derivation of the governing equations of motion in electro-magneto-mechanical loads. Three micro-length-scale parameters based on strain gradient theory are employed for prediction of vibrational characteristics of structure in micro-scale. The results show that increase of three micro-length-scale parameters leads to significant increase of three natural frequencies especially for increase of second micro-length-scale parameter. This result is according to this fact that stiffness of a micro-scale structure is increased with increase of micro-length-scale parameters.

Analytical investigation of bending response of FGM plate using a new quasi 3D shear deformation theory Effect of the micromechanical models

In this paper, a new refined quasi-three-dimensional (3D) shear deformation theory for the bending analysis of functionally graded plate is presented. The number of unknown functions involved in this theory is only four against five or more in the case of the other shear and normal deformation theories. Due to its quasi-3D nature, the stretching effect is taken into account in the formulation of governing equations. In addition, the effect of different micromechanical models on the bending response of these plates is studied. Various micromechanical models are used to evaluate the mechanical characteristics of the FG plates whose properties vary continuously across the thickness according to a simple power law. The present theory accounts for both shear deformation and thickness stretching effects by a parabolic variation of displacements across the thickness, and the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The problem is solved for a plate simply supported on its edges and the Navier solution is used. The results of the present method are compared with others from the literature where a good agreement has been found. A detailed parametric study is presented to show the effect of different micromechanical models on the flexural response of a simply supported FG plates.

Influence of Porosity on the Flexural and Free Vibration Responses of Functionally Graded Plates in Thermal Environment

This paper examines the influence of porosities on the flexural and free vibration response of functionally graded material (FGM) plates based on the authors’ recently developed non-polynomial higher-order shear and normal deformation theory. The theory accommodates the nonlinear variation in the in-plane and transverse displacements in the thickness coordinates. It also contains the hyperbolic shear strain shape function in the displacement field with only four unknowns. A new mathematical model has also been proposed to incorporate the effects of porosity in the FGM plate. Various numerical examples have been solved to ascertain the accuracy, efficiency, and applicability of the present formulation. The effects of porosity, volume fraction index, plate thickness, aspect ratio, boundary conditions and temperature have been discussed in details. The obtained results can be treated as a benchmark for future studies.

Bending analysis of advanced composite plates using a new quasi 3D plate theory

In this paper, a refined higher-order shear deformation theory including the stretching effect is developed for the analysis of bending analysis of the simply supported functionally graded (FG) sandwich plates resting on elastic foundation. This theory has only five unknowns, which is even less than the other shear and normal deformation theories. The theory presented is variationally consistent, without the shear correction factor. The present one has a new displacement field which introduces undetermined integral variables. Equations of motion are obtained by utilizing the Hamilton's principles and solved via Navier's procedure. The convergence and the validation of the proposed theoretical numerical model are performed to demonstrate the efficacy of the model.

A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates

This paper presents a new shear deformation theory including the stretching effect for free vibration of the simply supported functionally graded plates. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the functionally graded plate without using shear correction factors. This theory has only five unknowns, which is even less than the other shear and normal deformation theories. The present one has a new displacement field which introduces undetermined integral variables. The equation of motion of the vibrated structure obtained via the classical Hamilton’s principle and solved using Navier’s steps. The validation of the proposed theoretical model is performed to demonstrate the efficacy of the model. It can be concluded that the present theory is not only accurate but also simple in predicting the natural frequencies of functionally graded plates with stretching effect.

Thermoelastic analysis of laminated and functionally graded sandwich cylindrical shells with two refined higher order models

ABSTRACTAnalytical solutions for laminated and functionally graded sandwich open cylindrical shells under mechanical and thermal loads are presented using a refined higher order shear and normal deformation theory. Temperature variation through thickness is assumed as thickness coordinate polynomial. Present study also extends the classical thickness criteria with more reliable extension to moderately thick shells. Navier solution method is used to solve system of equations derived using principle of minimum potential energy for all edges diaphragm supported. Two kinds of sandwich panels with core or face sheets made of thickness graded material are studied. Several examples are numerically evaluated to establish the accuracy of present models.

Elastostatics of laminated and functionally graded sandwich cylindrical shells with two refined higher order models

Elastostatics of laminated and functionally graded (FG) sandwich open cylindrical shells is studied using a refined higher order shear and normal deformation theory. Displacement based approach with twelve middle surface displacement parameters representing bending and membrane response of cylindrical shell is considered. An extended thickness criterion, (h/R)2≪1, is used to make the present theory more reliable over large range of thickness ratios of shell. Basic equations are obtained using minimum potential energy principle and solved subsequently with Navier solution method for cylindrical shells with all edges diaphragm supported. Two kinds of FG sandwich panels are considered having FG layer with Voigt’s rule of mixture following power law gradation of volume fraction. Results show an excellent accuracy with the three dimensional results available in literature for laminated shells. Also, superiority of present formulation over other higher order theories is shown by carrying out a number of comparative studies.

Stacking sequence optimization for maximizing the first failure initiation load followed by progressive failure analysis until the ultimate load

For laminated plates deformed statically with transverse normal tractions applied on a major surface, we use honeybees inspired Nest-Site Selection (NeSS) optimization algorithm, a third order shear and normal deformable plate theory (TSNDT), and the finite element method to find an optimum stacking sequence that maximizes the first failure load according to the Tsai-Wu criterion. An optimum lay-up of fibers in different layers depends upon the starting estimate and is not unique. In the TSNDT, we express each displacement component at a point asa complete polynomial of degree three in the thickness coordinate. Subsequent to finding an optimum stacking sequence, we perform progressive failure analysis to determine the ultimate load by degrading elasticities of the material at an integration point where the failure criterion has just been satisfied. Without incrementing the load, we repeat the analysis to find if the failure at one integration point has led to failure at other integration points. When the failure ceases to propagate, we increase the load and repeat the analysis. We take the degradation of material properties to be permanent. At the ultimate load deflections of one or more points become extremely large with a minute increment in the applied load. We also find the ultimate load for different load distributions on a major surface, boundary conditions and plate geometries. It is found that for a clamped plate, the ultimate load can be 40% more than the first ply failure load. However, for a simply supported plate, the ultimate load essentially equals the first failure load. The stacking sequence for the optimal first failure load need not have the maximum ultimate load.

Free vibration of functionally graded open cylindrical shells based on several refined higher order displacement models

Free vibration analysis of functionally graded (FG) open cylindrical shells is presented here using various refined higher order theories. Present study undertakes the displacement based approach including higher order shear and normal deformation theory (HOSNT) along with first order shear deformation theory (FOST) and higher order shear deformation theory (HSDT) models. Difficulty of obtaining three dimensional (3D) solutions and errors associated with classical shell theory (CST) necessitates the requirement of higher order models. Present study takes into account moderately thick shells unlike CST, by considering square of ratio of thickness to radius of shell less than unity, instead of the classical assumption of considering ratio of thickness to radius less than unity. Here Navier method of solution with double trigonometric functions for displacement terms is used to analytically reduce the given set of partial differential equations (PDEs) to an eigenvalue problem. Results are computed using MATLAB and comparison between various higher order models is discussed based on the consideration of middle surface displacement parameters. Present results should establish benchmark solutions for free vibration analysis of isotropic/orthotropic FG cylindrical panels. Functionally graded material properties are graded according to power law variation in thickness direction. Various shell solutions, based on other theories and 3D solutions available in the literature are compiled along with present solutions.

Investigation of electric field effect on size-dependent bending analysis of functionally graded porous shear and normal deformable sandwich nanoplate on silica Aerogel foundation

In the present research electro-mechanical bending behavior of sandwich nanoplate with functionally graded porous core and piezoelectric face sheets is carried out. Vlasov’s model foundation is utilized to model the silica Aerogel foundation. Two functions are considered for nonuniform variation of material properties of the core layer along the thickness direction such as Young’s modulus, shear modulus, and density. The governing equations are deduced from Hamilton’s principle based on sinusoidal shear and normal deformation theory. In order to solve seven governing equations, an iterative technique is accomplished. After all, deflection and stresses are verified with corresponding literatures. Eventually, the numerical results reveal that applied voltage, plate aspect ratio, thickness ratio, nonlocal parameter, porosity index, Young’s modulus, and height of silica Aerogel foundation have substantial effects on the electro-mechanical bending response of functionally graded porous sandwich nanoplate.

Free vibration of axially loaded composite beams using a four-unknown shear and normal deformation theory

This paper presents free vibration of composite beams under axial load using a four-unknown shear and normal deformation theory. The constitutive equation is reduced from the 3D stress-strain relations of orthotropic lamina. The governing differential equations of motion are derived using the Hamilton’s principle. A two-node C1 beam element is developed by using a mixed interpolation with linear and Hermite-cubic polynomials for unknown variables. Numerical results are computed and compared with those available in the literature and commercial finite element software (ANSYS and ABAQUS). The comparison study illustrates the effects of normal strain, lay-ups and Poisson’s ratio on the natural frequencies and load-frequency curves of composite beams.

Flexural analysis of laminated composite and sandwich beams using a four-unknown shear and normal deformation theory

This paper presents flexural analysis of composite and sandwich beams using a quasi-3D theory, which considers simultaneously three effects such as normal and shear deformation as well as anisotropy coupling. The axial and transverse displacements are assumed to be cubic and parabolic variation through the beam depth. In order to solve problem, two-node C1 beam elements with six degrees of freedom per node are developed. Numerical examples are carried out and the results are compared with those available in literature to validate the accuracy of the present theory. The effects of fibre angle, lay-up and span-to-height ratio on displacements and stresses are studied. Some new results, which can be useful for future references, are also given.

Nonlinear flexural and vibration response of geometrically imperfect gradient plates using hyperbolic higher-order shear and normal deformation theory

Abstract This paper investigates the sensitivity of nonlinear flexural and vibration response of shear deformable functionally graded material (FGM) plates to the initial geometrical imperfections. The formulations are based on recently developed non-polynomial higher-order shear and normal deformation theory by authors. The theory accounts for nonlinear variation in the in-plane and transverse displacement through the thickness. It also accommodates thickness stretching effect without employing shear correction factor. The novelty of this theory is that it contains only four unknowns, unlike several other shear deformation theories which contain five or more unknowns. The equations of motion are derived using variational principle. The initial geometric imperfections have been incorporated using generic imperfection function. Material properties of the FGM plates are assumed to be varying continuously in the thickness direction according to a simple power law and exponential law. Convergence and comparison studies have been carried out to establish the authenticity and reliability of the solution. The numerical results are highlighted with various geometric imperfections and system parameters.

Large amplitude free flexural vibration analysis of finite element modeled FGM plates using new hyperbolic shear and normal deformation theory

In this paper, displacement based new hyperbolic higher-order shear and normal deformation theory (HHSNDT) is introduced for the geometrically nonlinear vibrations response of FGM plates. The proposed theory accounts for nonlinear in-plane and transverse displacement through the plate thickness. Unlike any other theory, the number of unknown functions involved in the present theory is only four, as against five or higher in the case of other well-known shear deformation theories. It also accounts the stretching effects across the thickness and does not require any shear correction factor. The fundamental equations of FGM plate are obtained using variational principle and von Karman theory is employed for large transverse deflection. Voigt and Mori–Tanaka model is used with the conjunction of exponential law and power law to estimate the graded material properties. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. A comprehensive numerical study is carried out based on the present theory to examine the influence of the homogenization techniques, geometrical parameter, amplitude ratio and boundary conditions on the vibration response of the graded plate.

Free vibration and flexural response of functionally graded plates resting on Winkler–Pasternak elastic foundations using nonpolynomial higher-order shear and normal deformation theory

ABSTRACTIn the present work, the flexural and vibration response of a functionally graded plate resting on Pasternak elastic foundation is analyzed using a recently developed nonpolynomial higher-order shear and normal deformation theory by the authors. The novelty of this theory is that it contains only four unknowns and also accommodates the thickness stretching effect. Two kinds of micromechanics models, namely, the Voigt and Mori–Tanaka models, are considered. Material properties of the functionally graded plates are assumed to vary continuously in the thickness direction according to either a simple power law or an exponential law. Finite element formulation is done using C° continuous Lagrangian quadrilateral nine-noded elements with eight degrees of freedom per node. The equations of motion are derived using a variational approach. Convergence and comparison studies are carried out to establish the authenticity and reliability of the solutions. The effect of various boundary conditions, geometric conditions, micromechanics models, and foundation parameters on the flexural and vibration response of the functionally graded plate are investigated.