Second-order Shear Deformation Theory Research Articles

Second–order models for the bending analysis of thin and moderately thick circular cylindrical shells

The suitability of employing the second-order shear deformation theory to static bending problems of thin and moderately thick isotropic circular cylindrical shells was investigated. Two variant forms of the polynomial second-order displacement models were considered. Both models account for quadratic expansions of the surface displacements along the shell thickness, although the second model (SSODM) was augmented by the initial curvature term. The equilibrium equations were derived by use of the principle of virtual work. Navier analytical solutions were obtained under simply supported boundary conditions. The results of the displacements and stresses revealed that the theory formulated on the SSODM provides a good depiction of the bending response of thin and moderately thick shells and are in close agreement with those of the first and higher-order shear deformation theories (FSDT; HSDT). The ability of the theory formulated on the first model (FSODM) to predict adequate values of displacements and stresses in thin shells was found to be significantly affected by changes in length to radius of curvature (l/a) ratios.

Nonlinear mechanics of sandwich plates: Layerwise third-order thickness and shear deformation theory

The nonlinear mechanics of sandwich plates is studied using a layerwise third-order thickness and shear deformation theory. In particular, the two face sheets are modelled using a second-order shear deformation theory (well-justified in case of a zero-shear stress only at one outer surface) and the core is modelled with a third-order thickness and transverse shear deformation theory. After introducing continuity of displacements at the interfaces between the core and the face sheets, 16 independent kinematic parameters are retained. An independent parameter is kept to describe the thickness deformation, which allows for introduction of the corresponding boundary condition; this represents a significant novel contribution. Geometric nonlinearity in all the kinematic parameters is introduced. Numerical results are presented for deflection of a sandwich plate under pressure. A thin core was considered, since it is numerically more critical, and a wide range of core stiffnesses was studied in order to verify the validity of the proposed model till the limit case of two plates joined by a core of negligible stiffness (i.e., two independent plates). Numerical results are compared to those obtained by a commercial finite element (FE) program with three-dimensional solid elements till the limit when the FE code fails for large deformations of the extremely weak core.

Wave dispersion characteristics of graphene reinforced nanocomposite curved viscoelastic panels

This paper tries to investigate the wave phenomenon in nanoshell made of graphene nanoplatelets (GNPs) reinforced nanocomposites. The second-order shear deformation theory in curvilinear coordinate is utilized to develop the doubly-curved shell as a continuous structure and the general nonlocal strain gradient theory is adopted to calculate nonlocality and strain gradient size-dependency. The effective material properties are estimated through the Halpin–Tsai micromechanical model and a modified rule of mixture. A virtual work of Hamilton statement is employed to obtain the governing motion equations and then a harmonic solution based on an analytical procedure is developed to find the wave response. Afterwards, a parametric study is performed to analyze the effects of the linear spring and damper coefficients, small-scale parameters, different curves, radius to thickness ratio, GNP's weight fraction as well as its number of layers on the phase velocity response of the GNPs reinforced doubly curved nanoshell with various shape panels resting on a visco-Winkler medium.

Stress analysis of the five layer circular sandwich plate subjected to uniform distributed load by layerwise theory along with second order shear deformation theory

ABSTRACT Layerwise theory (LT) along with the second order shear deformation theory (SSDT) is used to determine the stress distribution in a simply supported circular sandwich plate subjected to a uniformly distributed load. Two adhesive layers are used to adhere the core to their neighbouring functionally graded face sheets. According to the results, the finite element analysis (FEA) findings give almost the same estimations on planar stresses compared to first order shear deformation theory (FSDT) as well as SSDT. Additionally, the out-of-plane shear stresses obtained by FSDT, are slightly different from those of FEA. The differences are decreased by using SSDT.

Wave propagation of FG polymer composite nanoplates reinforced with GNPs

This study examines the wave propagation of the functionally graded polymer composite (FG-PC) nanoplates reinforced with graphene nanoplatelets (GNPs) resting on elastic foundations in the framework of the nonlocal strain gradient theory incorporating both stiffness hardening and softening mechanisms of nanostructures. To this end, the material properties are based on the Halpin-Tsai model, and the expressions for the classical and higher-order stresses and strains are consistently derived employing the second-order shear deformation theory. The equations of motion are then consistently derived using Hamilton\'s principle of variation. These governing equations are solved with the help of Trial function method. Extensive numerical discussions are conducted for wave propagation of the nanoplates and the influences of different parameters, such as the nonlocal parameter, strain gradient parameter, weight fraction of GNPs, uniform and non-uniform distributions of GNPs, elastic foundation parameters as well as wave number.

A comprehensive analytical study on functionally graded carbon nanotube-reinforced composite plates

This paper presents a model which is effective and simple based upon Second Order Shear Deformation Theory (SSDT) to comprehensive study of Functionally Graded Carbon Nanotube Reinforced Composite (FG-CNTRC) plates including size effects for the first time. Also it is the first time that the size-dependent static, stability and dynamic response of FG-CNTRC plates considering Winkler–Pasternak elastic foundation effects are investigated. It is assumed that the material properties of FG-CNTRC are varied through the thickness direction using four different distributions of carbon nanotubes (CNTs). In order to include the size effects in our modeling, the nonlocal elasticity theory presented by Eringen is utilized. The influences of different parameters such as volume fraction of carbon nanotubes, nonlocality and elastic foundation effects on the response of FG-CNTRC plates are studied. Numerical results prove high accuracy and reliability of the present method in comparison with other available numerical or analytical methods.

Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory

The effective elastic-piezoelectric properties of nanostructures have been shown to be strongly size-dependent. In this paper, a nonlocal second-order shear deformation formulation is presented to study the size-dependent thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core. Temperature is considered as uniform and nonlinear distributions across plate’s thickness direction. Based on the developed nonlocal second-order shear deformation theory, the size-dependent equations of motion are derived. The nonlocal thermal buckling responses of simply supported nanoplates are solved via Navier method. The reliability of present approach is verified by comparing the existing results provided in the open literature. The influences of nonlocal parameter, gradient index, electric voltage, and Winkler–Pasternak parameters on the thermal buckling characteristics of functionally graded nanoplates are examined.

Temperature-dependent flexural wave propagation in nanoplate-type porous heterogenous material subjected to in-plane magnetic field

ABSTRACTThis study is focused on the wave propagation analysis of nanoplate made of temperature-dependent porous functionally graded (FG) materials rested on Winkler–Pasternak foundation under in-plane magnetic field. The material properties of FG nanoplate are supposed to vary through the thickness direction and described by power-law rule, in which the porosity distribution is considered as an even pattern. Hamilton’s principle is utilized to derive the governing equations on basis of second-order shear deformation theory in conjunction with nonlocal strain gradient theory. The influence of small-length parameters, thermal distribution, magnetic field, material composition, porosity, and Winkler–Pasternak foundation on wave dispersion is explored.

Wave dispersion of mounted graphene with initial stress

This paper is focused on the initial stress affected wave dispersion in mounted graphene. To capture the size-dependent and shear deformation effects, the second-order shear deformation theory (SSDT) and nonlocal strain gradient elasticity theory are used to model the graphene. A size-dependent shear deformation plate model is established on the basis of the orthotropic constitutive equations, the nonlocal strain gradient theory as well as the SSDT. Analytical solutions are developed for the wave frequency and phase velocity. The influences of small-scale parameters, initial stress and elastic medium on wave propagation behaviors of the graphene sheets are explored. It is found that the developed model reasonably interprets the softening effects of flexural frequencies and phase velocities. Unlike the classical (scaling-free) model, the developed size-dependent model shows a reasonably good agreement with the experimental frequencies and phase velocities. The wave frequencies and phase velocities of single layer graphene sheets (SLGS) will decrease with increasing compression load, and can increase by increasing tension load. The developed size-dependent 2D continuum model is hopeful to provide a possible theoretical approach to explore the wave behaviors of Graphene-like 2D materials.

Wave propagation in functionally graded nanocomposites reinforced with carbon nanotubes based on second-order shear deformation theory

ABSTRACTIn the present article, as a first endeavor, the wave propagation in functionally graded nanocomposites reinforced with carbon nanotubes is investigated on the basis of second-order shear deformation theory. Four different types of functionally graded nanocomposites are presented. An analytical method is used to find the circular frequencies and phase velocities. To show the accuracy of the present methodology, our results for the free vibration are compared with the results of functionally graded plates available in the literature. The influences of different parameters are also investigated on the circular frequencies and phase velocities.

Using high-order shear deformation theory in the analysis of Lamb’s waves propagation in materials reinforced with two families of fibers

The paper analyses the problem of elastic waves propagation through the material reinforced by two families of extensible fibers using high-order shear deformation theories. The material reinforced with two families of fibers can be seen as a material made of an arbitrary number of plies reinforced with one family of fibers, which are at angles \(\phi \) and \(-\phi \), in relation to the lines of symmetry, so that it can be considered to be a composite laminate. Elasticity constants are derived using strain energy and the theory of invariants. The second-order shear deformation theory was used for creating dynamic equations of motion. Using analytical methods, dispersion curves and diagrams of phase velocities in the polar coordinate system were obtained. The emergence of quasi-modes, in the forms of quasi- bending qA, quasi-extensional qS and quasi-horizontally polarized modes qSH, was shown. The numerical part was done in MATLAB software using combinations of symbolic and numeric values.

Interface fracture in orthotropic composite plates using second-order shear deformation theory

The second-order shear deformation theory is used in this study to calculate the stresses and the energy release rates in orthotropic composite plates. A novel double-plate system is utilized with the imposition of the proper kinematic constraints in the interface plane of a double-plate system. The governing equations of the system were derived and as a demonstrative example a simply-supported plate subjected to a point force was analyzed. Using Lévy plate formulation, the plate problem was solved by a state-space model, incorporating four different regions. The distribution of the stress resultants and the interlaminar stresses in the uncracked part were also determined. Moreover, the distributions of the mode-II and mode-III energy release rates along the crack front were calculated by the J-integral. The 3D finite element model of the plate was created providing reference data for the analytical model. The results show reasonably good agreement between the analytical and numerical results. Also, the present model eliminates the physical inconsistency of previous models and reveals that under mixed-mode II/III condition, the energy release rate is not contributed by the bending, twisting moments and shear forces at all.

Free vibration analysis of solar functionally graded plates with temperature-dependent material properties using second order shear deformation theory

Second-order shear deformation theory (SSDT) is employed to analyze vibration of temperature-dependent solar functionally graded plates (SFGP’s). Power law material properties and linear steady-state thermal loads are assumed to be graded along the thickness. Two different types of SFGP’s such as ZrO2/Ti-6Al-4V and Si3N4/SUS304 are considered. Uniform, linear, nonlinear, heat-flux and sinusoidal thermal conditions are imposed at the upper and lower surface for simply supported SFGPs. The energy method is applied to derive equilibrium equations, and solution is based on Fourier series that satisfy the boundary conditions (Navier’s method). Non-dimensional results are compared for temperature-dependent and temperature-independent SFGP’s and validated with known results in the literature. Numerical results indicate the effect of material composition, plate geometry, and temperature fields on the vibration characteristics and mode shapes. The results obtained using the SSDT are very close to results from other shear deformation theories.

Thermal Free Vibration Analysis of Temperature-Dependent Functionally Graded Plates Using Second Order Shear Deformation

This research has been conducted to approach second-order shear deformation theory (SSDT) to analysis vibration characteristics of Functionally Graded Plates (FGP’s). Material properties in FGP's were assumed to be temperature dependent and graded along the thickness using a simple power law distribution in term of the volume fractions of the constituents. FGP was subjected to a linear and nonlinear temperature rise. The energy method was chosen to derive the equilibrium equations. The solution was based on the Fourier series that satisfy the simply supported boundary condition (Navier's method). Numerical results indicated the effect of material composition, plate geometry, and temperature fields on the vibration characteristics and mode shapes. The results revealed that, the temperature field and volume fraction distribution had significant effect on the vibration of FGPs. It was observed the second order theory was very close to the other shear deformation theorem as reported in the literature.

Natural Frequency of F.G. Rectangular Plate by Shear Deformation Theory

Natural frequency of functionally graded (F.G.) rectangular plate is carried out by using second-order shear deformation theory (SSDT). The material properties of functionally graded rectangular plates, except the Poisson's ratio, are assumed to vary continuously through the thickness of the plate in accordance with the exponential law distribution. The equations of motion are obtained by energy method. Numerical results for functionally graded plates are given in dimensionless graphical forms and the effects of material properties on natural frequency are determined.

Second-Order Shear Deformation Theory to Analyze Stress Distribution for Solar Functionally Graded Plates#

Second-order shear deformation theory (SSDT) is applied to evaluate the displacement and stress fields of a solar functionally graded plate (SFGP) due to mechanical loadings. The material properties are graded by a simple power law between Full-ceramic and Full-metal at upper and lower surfaces, respectively. Navier's method is applied to find analytical results for derived equations by using energy method in case of simply supported boundary conditions. The effects of the material grading index of the plate on the stresses and displacements are investigated. It is revealed that the longitudinal stresses in the functionally graded (FG) plates lie between full-metal and full-ceramic plates. It is found that the neutral axes for SFGP move to upper surface and not at the mid-surface as predicted in the homogeneous plates. The SSDT has computed acceptable results for in-plane stresses and displacement fields when compared with the existing literatures.

Axisymmetric Stress Analysis of a Thick Conical Shell with Varying Thickness under Nonuniform Internal Pressure

A mathematical approach based on the perturbation theory has been used for axisymmetric stress analysis of a thick conical shell with varying thickness under nonuniform internal pressure. The equilibrium equations have been derived using the energy principle and considering the second-order shear deformation theory (SSDT), which includes shear deformation effects. This system of ordinary differential equations with variable coefficients has been solved analytically using the matched asymptotic expansion method of the perturbation theory. A comparison of the results with the finite-element method and the first-order shear deformation theory shows that the SSDT can predict the displacements and stresses of the shell for a wide range of thicknesses as well with less calculations than other analytical methods such as the Frobenius series method.

Free Vibration of Laminated Composite Circular Arches by Dynamic Stiffness Analysis

The dynamic stiffness method for studying the vibration characteristics of the laminated composite shallow circular arches is introduced in this article. The exact dynamic stiffness matrix is formulated from the analytical solutions of the governing differential equations of motion of the laminated circular arches, based on second-order shear deformation theory. The application of the present method is illustrated by miscellaneous numerical examples, in which the effects of material orthotropy, the lamination scheme, the curvature ratio, length-to-thickness ratio, and boundary condition upon the natural frequencies of the composite arches are investigated. Compared to those solutions available in the literature, the numerical results demonstrate the accuracy and effectiveness of the present method.

Free vibrations of laminated composite plates using second-order shear deformation theory

A complete set of linear equations of the second-order theory of laminated composite plates are obtained. A generalized Levy type solution in conjunction with the state space concept is used to analyze the free vibration behavior of cross-ply and antisymmetric angle-ply laminated plates. Exact fundamental frequencies of cross-ply plate strips are obtained for arbitrary boundary conditions. The exact analytical solutions are obtained for thick and moderately thick plates as well as for thin plates and plate strips. It is shown that the results of the second-order theory are very close to the results of the first-order and third-order theories reported in the literature, and different from those of the classical Kirchhoff’s theory for thick laminates.