Non-polynomial Higher-order Shear Deformation Theory Research Articles

Dynamic response of honeycomb-FGS shells subjected to the dynamic loading using non-polynomial higher-order IGA

The main goal of this study is to use higher-order isogeometric analysis (IGA) to study the dynamic response of sandwich shells with an auxetic honeycomb core and two different functionally graded materials (FGM) skin layers (namely honeycomb-FGS shells) subjected to dynamic loading. Touratier's non-polynomial higher-order shear deformation theory (HSDT) is used due to its simplicity and performance. The governing equation is derived from Hamilton's principle. After verifying the present approach, the effect of input parameters on the dynamic response of honeycomb-FGS shells is carried out in detail.

Thermoelastic free vibration of rotating tapered porous functionally graded conical shell based on non-polynomial higher-order shear deformation theory

This study uses a non-polynomial higher-order shear deformation theory to calculate the fundamental frequency of a rotating cantilevered porous functionally graded (FG) conical shell with variable thickness. Eight noded isoparametric shell elements having seven degrees of freedom per node are used to discretize the shell. Utilizing a simple power law, the temperature-dependent material characteristics of the FG conical shell are determined. The one-dimensional Fourier heat conduction equation is used to obtain the nonlinear temperature distribution. Lagrange’s equation is used to derive the dynamic equation of motion. Finally, a parametric study is presented.

Vibration response of functionally graded material sandwich plates with elliptical cutouts and geometric imperfections under the mixed boundary conditions

Abstract The present article investigates the effect of elliptical cutouts and geometric imperfections on the vibrational response of functionally graded material (FGM) sandwich plates. Generalised governing equations for the sandwich FGM (SFGM) plate are derived based on non-polynomial higher-order shear deformation theory. Geometric discontinuities have been incorporated as elliptical cutouts in the plates, and the various geometric imperfections are modelled using the generic function. The mathematical modelling has been carried out using the C0 continuity isoparametric finite element formulation by considering four-noded elements with seven degrees of freedoms per node. Convergence and validation studies have been performed to demonstrate the efficiency and accuracy of the present methodology. The influence of volume fraction index, geometric imperfections, and elliptical cutouts on the vibrational frequency of SFGM plates have been analysed under the mixed boundary conditions.

Finite element analysis of smart composite plate structures coupled with piezoelectric materials: Investigation of static and vibration responses

This research presents a generalized-energy-based finite element modeling of smart composite plates with distributed piezoelectric materials for the static and vibration responses under electrical and mechanical loading. The zigzag kinematics in conjunction with a non-polynomial higher-order shear deformation theory is used to derive the model. The plate theory accounts for the non-linear variations of the transverse shear strains across the thickness of the plates and also accommodates the inter-laminar continuity effects of transverse shear stresses at the interfaces. Eight noded isoparametric serendipity elements are used to discretize the physical domain. The solutions in the time domain are obtained from the discretized system of ordinary differential equations with Newmark’s time integration scheme. Several numerical examples are solved for different types of smart composite plates with various piezoelectric materials, boundary conditions, static and dynamic loadings. The classical negative feedback controller is used for deriving the suppressed and controlled vibration responses of the smart composite plates with distributed actuators and sensors. A comparison of the present FE solutions with the elasticity and other analytical and numerical solutions shows good agreement, thereby ensures the applicability of the present finite element model for studying the coupled electromechanical problems of piezoelectricity.

Nonlinear free vibration analysis of laminated composite plates and shell panels using non-polynomial higher-order shear deformation theory

In the present work, nonlinear free vibration analysis of composite laminated plates and shallow cylindrical/spherical/hyperboloid shell panels considering geometric nonlinearity is carried out using non-polynomial inverse trigonometric higher-order shear deformation theory with seven degrees of freedo (DOFs). The present theory assumes parabolic variation of out-of-plane stresses and satisfies traction-free boundary conditions on the top and bottom surfaces of the composite as a priori. A nonlinear finite element model is developed and applied to obtain discretized nonlinear equations. The geometric nonlinearity in sense of Green-Lagrange considering von-Kármán assumptions is incorporated in formulation. An eight noded efficient C 0 continuous isoparametric rectangular finite element is implemented in the present nonlinear analysis. The efficacy and accuracy of the present theory and finite element model is validated with the available literature results. For the analysis, various types of plates and shell panels with different material properties, lamination schemes, thickness ratios, aspect ratios, modular ratios, curvature ratios, and boundary conditions are considered. The effects of these different geometric and material properties on nonlinear frequency ratios at various amplitude ratios are examined in detail.

Stress analysis of smart composite plate structures

In the present study, the electro-mechanical responses of smart laminated composite plates with piezoelectric materials are derived using a two-dimensional (2 D) displacement-based non-polynomial higher-order shear deformation theory. The kinematics of the mathematical model incorporates the deformation of laminates which account for the effects of transverse shear deformation and a non-linear variation of the in-plane displacements using inverse sine hyperbolic function of the thickness coordinate. The equilibrium equations are obtained using the minimization of energy principle known as the principle of minimum potential energy (PMPE) which is also based on a variational approach and the solutions are obtained using Navier’s solution technique for diaphragm supported smart laminated composite plates. The responses obtained in the form of deflection and stresses are compared with three dimensional (3 D) solutions and also with different polynomial and non-polynomial based higher-order theories in the literature. The transverse shear stresses are obtained using 3 D equilibrium equations of elasticity to enhance the accuracy of the present results. Various examples are numerically solved to establish the efficiency of the present model.

Exact Solution for Free Vibration and Buckling of Sandwich S-FGM Plates on Pasternak Elastic Foundation with Various Boundary Conditions

In the present study, free vibration and buckling characteristics of a sandwich functionally graded material (FGM) plate resting on the Pasternak elastic foundation have been investigated. The formulation is based on non-polynomial higher-order shear deformation theory with inverse hyperbolic shape function. A new modified sigmoid law is presented to compute the effective material properties of sandwich FGM plate. The governing equilibrium equations have been derived using Hamilton’s principle. Non-dimensional frequencies and critical buckling loads are evaluated by considering different boundary conditions based on admissible functions satisfying the desired primary and secondary variables. Comprehensive parametric studies have been performed to analyze the influence of geometric configuration, volume fraction exponent, elastic medium parameter, and non-dimensional load parameter on the non-dimensional frequency and critical buckling load. These parametric studies have been done for various boundary conditions and different configurations of the sandwich plate. The computed results can be used as a benchmark for future comparison of sandwich S-FGM plates.

Nonlinear Vibration Analysis of Sigmoid Functionally Graded Sandwich Plate with Ceramic-FGM-Metal Layers

In the present study, free and forced nonlinear vibration characteristics of a sandwich functionally graded material (FGM) plate resting on Pasternak elastic foundation have been investigated. The formulation is based on non-polynomial higher-order shear deformation theory with inverse hyperbolic shape function. A new modified sigmoid law is presented to compute the effective material properties of sandwich FGM plate. The governing equilibrium equations have been formulated using Hamilton’s principle and the governing nonlinear coupled ordinary differential equations are derived using stress function method in conjunction with Galerkin approach. The resulting equations are then solved using fourth order Runge–Kutta time integration scheme for simply supported plate with immovable edges. Comprehensive parametric studies have been performed to analyze the influence of geometric configuration, volume fraction exponent, and elastic medium parameter on nonlinear time series analysis for different configurations of sandwich plate. A nonlinear complex behaviour of the plate is studied using time displacement response, phase-plane plot and Poincare map. For Pasternak foundation, system shows multi-loop periodic nature while it losses its periodicity and shows the quasi-periodic nature for Winkler foundation. In addition, it has been observed that the system shows quasi-periodic route to chaotic nature for thick plate to thin plate.

Refined and generalized hybrid type quasi-3D shear deformation theory for the bending analysis of functionally graded shells

The closed-form solution of a generalized hybrid type quasi-3D higher order shear deformation theory (HSDT) for the bending analysis of functionally graded shells is presented. From the generalized quasi-3D HSDT (which involves the shear strain functions “f(ζ)” and “g(ζ)” and therefore their parameters to be selected “m” and “n”, respectively), infinite six unknowns' hybrid shear deformation theories with thickness stretching effect included, can be derived and solved in a closed-from. The generalized governing equations are also “m” and “n” parameter dependent. Navier-type closed-form solution is obtained for functionally graded shells subjected to transverse load for simply supported boundary conditions. Numerical results of new optimized hybrid type quasi-3D HSDTs are compared with the first order shear deformation theory (FSDT), and other quasi-3D HSDTs. The key conclusions that emerge from the present numerical results suggest that: (a) all non-polynomial HSDTs should be optimized in order to improve the accuracy of those theories; (b) the optimization procedure in all the cases is, in general, beneficial in terms of accuracy of the non-polynomial hybrid type quasi-3D HSDT; (c) it is possible to gain accuracy by keeping the unknowns constant; (d) there is not unique quasi-3D HSDT which performs well in any particular example problems, i.e. there exists a problem dependency matter.

Static response of advanced composite plates by a new non-polynomial higher-order shear deformation theory

Abstract The static responses of functionally graded plates are investigated by using an accurate recently developed higher order shear deformation theory (HSDT), which is also compared with other HSDTs available in the literature. A practical generalized numerical code for solving the plate governing equations, which can include the shear strain shape functions of existing HSDTs, is utilized. The plate governing equations and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FG plates subjected to transverse bi-sinusoidal and distributed loads for simply supported boundary conditions. For the generality of the present HSDT, a continuous isoparametric Lagrangian finite element with 7° of freedom per node are also presented. Results are provided for thick to thin FG plates and for different volume fraction distributions. The accuracy of the present code is verified by comparing it with various HSDTs available in the literature. Results show good agreement between the HSDTs for normal and transversal displacements, normal stresses and in-plane shear stresses. However, opposite occurs for transverse shear stresses. It is because the shear stress results are sensible to the shear strain shape functions used in the formulation of displacement field of a particular HSDT having five unknowns.