Green's Function for Multilayers with Interfacial Membrane and Flexural Rigidities¹

A three-dimensional Green's function for a material system consisting of anisotropic and linearly elastic planar multilayers with interfacial membrane and flexural rigidities has been derived. The Stroh formalism and two-dimensional Fourier transforms are applied to derive the general solution for each homogeneous layer. The Green's function for the multilayers is then solved by imposing the surface boundary condition, the interfacial displacement continuity condition, and the interfacial traction discontinuity condition. The last condition is given by the membrane and bending equilibrium equations of the interphases modeled as Kirchhoff plates. Numerical results that demonstrate the validity and efficiency of the formulation are presented for the case of a stack of silicon thin films embedded in epoxy.     

Influence of porosity on the flexural and vibration response of gradient plate using nonpolynomial higher-order shear and normal deformation theory

The present article deals with flexural and vibration response of functionally graded plates with porosity. The basic formulation is based on the recently developed non-polynomial higher-order shear and normal deformation theory by the authors’. The present theory contains only four unknowns and also accommodate the thickness stretching effect. The effective material properties at each point are determined by two micromechanics models (Voigt and Mori–Tanaka scheme). The governing equations for FGM plates are derived using variational approach. Results have been obtained by employing a C⁰ continuous isoparametric Lagrangian finite element with eight degrees of freedom per node. Convergence and comparative study with the reported results in the literature, confirm the accuracy and efficiency of the present model and finite element formulation. The influence of the porosity, various boundary conditions, geometrical configuration and micromechanics models on the flexural and vibration behavior of FGM plates is examined.     

Nonlinear flexural analysis of functionally graded plates under different loadings using RBF based meshless method

The nonlinear flexural response of functionally graded plates under different transverse loading is investigated using multiquadric radial basis function (MQRBF) method. The mathematical formulation of the actual physical problem of the plate subjected to mechanical loading is presented utilizing Levinson shear deformation theory and von Karman nonlinear kinematics. These non-linear governing differential equations of equilibrium are linearized using quadratic extrapolation technique. The effect of the volume fraction exponent on the nonlinear flexural response of simply supported and clamped plates under uniformly distributed, sinusoidal, line and point loads is studied.     

A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation

A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation is developed. The displacement field is chosen based on assumptions that the in-plane and transverse displacements consist of bending and shear components, and the shear components of in-plane displacements give rise to the parabolic variation of shear strain through the thickness in such a way that shear stresses vanish on the plate surfaces. Therefore, there is no need to use shear correction factor. Material properties of functionally graded plate are assumed to vary according to power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as Pasternak foundation. Equations of motion are derived using Hamilton’s principle. Closed-form solution of rectangular plates is derived, and the obtained results are compared well with three-dimensional elasticity solutions and third-order shear deformation theory solutions. Finally, the influences of power law index, thickness ratio, foundation parameter, and boundary condition on the natural frequency of plates have been investigated.     

Nonlinear flexural analysis of laminated composite plates using RBF based meshless method

The paper presents the nonlinear flexural response of laminated composite plates. The mathematical formulation of the actual physical problem of the laminated composite plate subjected to mechanical loading is presented utilizing higher order shear deformation theory and von-Karman nonlinear kinematics. These nonlinear governing differential equations of equilibrium are linearized using quadratic extrapolation technique. A meshfree technique based on multiquadric RBFs is used for analysis of the problems. Isotropic, orthotropic and laminated composite plates with immovable simply supported and clamped edges are analyzed.     

Stochastic nonlinear bending response of functionally graded material plate with random system properties in thermal environment

This study deals with the stochastic nonlinear bending response of functionally graded materials (FGMs) plate with uncertain system properties subjected to transverse uniformly distributed load in thermal environments. The system properties such as material properties of each constituent’s material, volume fraction index and transverse load are taken as independent random input variables. The material properties are assumed to be temperature independent (TID) and temperature dependent (TD). The basic formulation is based on higher order shear deformation theory with von-Karman nonlinear strain kinematics using modified C ⁰ continuity. A direct iterative based nonlinear finite element method in conjunction with first-order perturbation technique developed by last two authors for the composite plate is extended for the FGM plate to compute the second order statistics (mean and standard deviation) of the nonlinear bending response of the FGM plates. Effects of TD, TID material properties, aspect ratios, volume fraction index and boundary conditions, uniform temperature and non-uniform temperature distribution on the nonlinear bending are presented in detail through parametric studies. The present outlined approach has been validated with the results available in the literature and independent Monte Carlo simulation.     

Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress

A microstructure-dependent nonlinear theory for axisymmetric bending of circular plates, which accounts for through-thickness power-law variation of a two-constituent material, is developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, temperature-dependent properties, and the von Kármán geometric nonlinearity. Classical and first-order shear deformation theories are considered in the study. The modified couple stress theory contains a material length scale parameter that can capture the size effect in a functionally graded material plate. The theories presented herein can be used to develop analytical solutions of bending, buckling, and free vibration for the linear case and finite-element models for the nonlinear case to determine the effect of the geometric nonlinearity, power-law index, and microstructure-dependent constitutive relations on linear and nonlinear response of axisymmetric analysis of circular plates.     

Time dependent response of low velocity impact induced composite conical shells under multiple delamination

This paper presents the time dependent response of multiple delaminated angle-ply composite pretwisted conical shells subjected to low velocity normal impact. The finite element formulation is based on Mindlin’s theory incorporating rotary inertia and effects of transverse shear deformation. An eight-noded isoparametric plate bending element is employed to satisfy the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. A multipoint constraint algorithm is incorporated which leads to asymmetric stiffness matrices. The modified Hertzian contact law which accounts for permanent indentation is utilized to compute the contact force, and the time dependent equations are solved by Newmark’s time integration algorithm. Parametric studies are conducted with respect to triggering parameters like laminate configuration, location of delamination, angle of twist, velocity of impactor, and impactor’s displacement for centrally impacted shells.     

Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties

As a first endeavor, the thermal buckling and postbuckling analysis of functionally graded (FG) annular plates with material properties graded in the radial direction is presented. The formulation is derived based on the first-order shear deformation theory (FSDT) and the geometrical nonlinearity is modeled using Green’s strain in conjunction with von Karman’s assumptions. The material properties are temperature-dependent and graded according to the power law distribution. It is assumed that the temperature varies along the radial direction. Using the virtual work principle, the pre-buckling and postbuckling equilibrium equations and the related boundary conditions are derived. Differential quadrature method (DQM) as an efficient numerical technique is adopted to solve the governing equations. The presented formulation and the method of solution are validated by performing convergence and comparison studies with available results in the literature. Finally, the effects of volume fraction index, geometrical parameters, mechanical/thermal properties of the constituent materials and boundary conditions on the thermal buckling and postbuckling behavior of the radially graded annular plate are evaluated and discussed.     

Nonlinear analysis of smart functionally graded annular sector plates using cylindrically orthotropic piezoelectric fiber reinforced composite

This work deals with the geometrically nonlinear thermo-electro-elastic analysis of functionally graded (FG) annular sector plates integrated with the annular patches of cylindrically orthotropic piezoelectric fiber reinforced composite (PFRC). The annular patches with an external voltage across their thickness act as the distributed actuators and their performance in controlling the nonlinear flexural deformations of the host FG plates is investigated. The temperature field is assumed to be spatially uniform over the plate surfaces and varied through the thickness of the substrate FG plates. The temperature-dependent material properties of the FG plates are assumed to be graded in the thickness direction of the plates according to a power-law distribution while the Poisson’s ratio is assumed to be a constant over the domain of the substrate plate. A finite element model of the overall smart FG annular sector plate is developed based on the first order shear deformation theory and the Von Karman nonlinear strain–displacement relations. The governing nonlinear finite element equations are derived employing the principle of minimum potential energy and solved using direct iteration method. The numerical results illustrate significant control authority of the cylindrically orthotropic PFRC annular patches for active control of nonlinear deformations of the substrate FG annular sector plates. The numerical results also reveal the best radial and circumferential locations of the annular PFRC patches for effective control. For a specified circumferential stretch of the annular PFRC patches, their minimum radial length is numerically estimated in such a way that the performance of the overall smart FG plate is not affected significantly. The effects of the material properties and the temperature of the host FG plate on the performance of the annular PFRC patches are also discussed.     

Thermoelastic analysis of advanced sandwich plates based on a new quasi-3D hybrid type HSDT with 5 unknowns

This paper presents a thermoelastic bending analysis of functionally graded sandwich plates by using a new quasi-3D hybrid type higher order shear deformation theory (HSDT). The mathematical model contains only 5 unknowns as the first order shear deformation theory (FSDT). The nonlinear term of the temperature field is modeled in such way that can be different from the shape functions of the displacement field. The mechanical properties of functionally graded layers of the plate are assumed to vary in the thickness direction according to a power law distribution. The governing equations for the thermoelastic bending analysis are obtained through the principle of virtual work and solved via Navier-type solution. Results reveal: (a) the good performance of the present generalized formulation; (b) the significant influence of the nonlinear temperature field on the displacements and stresses results. Consequently, discussion on nonlinear temperature field influences should be further considered in the literature.     

考虑横向剪切效应时圆柱型正交各向异性球面扁壳在横向载荷作用下的非线性压屈

本文应用参考文献所介绍的修正迭代法探讨了横向载荷作用下,圆柱型正交各向异性圆底球面扁壳的非线性稳定同题,得出了这一同题的解析解。在求解过程中,本文放弃了经典板壳理论的Kirchhoff假定,从而考虑了横向剪应变对于弯曲变形的影响, 计算结果表明:本文的分析方法和结论是正确的;对于正交各向异性复合材料板壳而言,横向剪切效应是值得注意的。     

A layerwise finite element formulation for vibration and damping analysis of sandwich plate with moderately thick viscoelastic core

Abstract

Most previous studies of viscoelastic sandwich plates were based on the assumption that damping was only resulting from shear deformation in the viscoelastic core. However, extensive and compressive deformations in the viscoelastic core should also be considered especially for sandwich plates with moderately thick viscoelastic core. This paper presents a finite element formulation for vibration and damping analysis of sandwich plates with moderately thick viscoelastic core based on a mixed layerwise theory. The face layers satisfy the Kirchhoff theory while the viscoelastic core meets a general high-order deformation theory. The viscoelastic core is modeled as a quasi-three-dimensional solid where other types of deformation such as longitudinal extension and transverse compression are also considered. To better describe the distribution of the displacement fields, auxiliary points located across the depth of the sandwich plate are introduced. And based on the auxiliary points, the longitudinal and transverse displacements of the viscoelastic core are interpolated independently by Lagrange interpolation functions. Quadrilateral finite elements are developed and dynamic equations are derived by Hamilton’s principle in the variation form. Allowing for the frequency-dependent characteristics of the viscoelastic material, an iterative procedure is introduced to solve the nonlinear eigenvalue problem. The comparison with results in the open literature validates the remarkable accuracy of the present model for sandwich plates with moderately thick viscoelastic core, and demonstrates the importance of the higher-order variation of the transverse displacement along the thickness of the viscoelastic core for the improvement of the analysis accuracy. Moreover, the influence of the thickness and stiffness ratios of the viscoelastic core to the face layers on the damping characteristics of the viscoelastic sandwich plate is discussed.     

Combined effect of shear and large deformation on stress intensity factors

An attempt has been made to study the influence of large deformation on the stress intensity factor in a cracked plate subjected to bending including shear deformation. It is assumed that singular terms for stress resultants and strains in the case of large deformation have the same angular distribution and order of singularity as in the case of a linear problem. With this in view the small deformation singular element has been used at the crack tip region surrounded by large deformation plate bending elements. The finite element analysis, based on total Lagrangian formulation combined with the modified Newton–Raphson technique, has been used to get numerical results. Several examples connected with large deformation of cracked plates subjected to bending are studied. Using the above technique stress intensity factors for linear and non-linear cases have been compared.     

Influence of neutral surface position on the nonlinear stability behavior of functionally graded plates

Nonlinear behavior of functionally graded material (FGM) skew plates under in-plane load is investigated here using a shear deformable finite element method. The material is graded in the thickness direction and a simple power law based on the rule of mixture is used to estimate the effective material properties. The neutral surface position for such FGM plates is determined and the first order shear deformation theory based on exact neutral surface position is employed here. The present model is compared with the conventional mid-surface based formulation, which uses extension-bending coupling matrix to include the noncoincidence of neutral surface with the geometric mid-surface for unsymmetric plates. The nonlinear governing equations are solved through Newton–Raphson technique. The nonlinear behavior of FGM skew plates under compressive and tensile in-plane load are examined considering different system parameters such as constituent gradient index, boundary condition, thickness-to-span ratio and skew angle. An erratum to this article can be found at     

Post-buckling analysis of skew plates subjected to combined in-plane loadings

Post-buckling behavior of laminated composite, sandwich and functionally graded skew plates is analyzed in the present work. The problem formulation is based on higher-order shear deformation theory and von Kármán’s nonlinear kinematics. Linear mapping is used to transform the physical domain into the computational domain. Chebyshev polynomials are used for spatial discretization of governing differential equations and boundary conditions. The nonlinear terms are linearized using quadratic extrapolation technique. The effect of the skew angle on the buckling and post-buckling response of the composite, sandwich and FGM-clamped skew plates is investigated for different combinations of in-plane compressive loadings.     

Analytical solution to the bending of a nonlinearly hardening wide curved bar

An analytical solution to the partially plastic deformation of a nonlinearly hardening wide curved bar is derived. The bar considered has a narrow rectangular cross-section and is under pure bending. The analytical treatment is based on Tresca’s yield criterion, its associated flow rule and a Swift-type nonlinear hardening law. Taking numerical limits, the complete solution is verified in comparison to the linear hardening solution available in the literature.     

An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates

In this paper, an efficient and simple higher order shear and normal deformation theory is presented for functionally graded material (FGM) plates. By dividing the transverse displacement into bending, shear and thickness stretching parts, the number of unknowns and governing equations for the present theory is reduced, significantly facilitating engineering analysis. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with 3-dimensional and quasi-3-dimensional solutions and those predicted by other plate theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of functionally graded plates.     

Thermoelastic bending analysis of functionally graded sandwich plates

The thermoelastic bending analysis of functionally graded ceramic–metal sandwich plates is studied. The governing equations of equilibrium are solved for a functionally graded sandwich plates under the effect of thermal loads. The sandwich plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity, Poisson’s ratio of the faces, and thermal expansion coefficients are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used taking into account the symmetry of the plate and the thickness of each layer. Field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory are derived. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. The influences played by the transverse normal strain, shear deformation, thermal load, plate aspect ratio, side-to-thickness ratio, and volume fraction distribution are studied. Numerical results for deflections and stresses of functionally graded metal–ceramic plates are investigated.     

Deformation and failure of blast-loaded square plates

Numerical results for clamped, thin square steel plates subjected to blast loading are presented. The numerical analysis is based on a finite element formulation, which includes the nonlinear effects of geometry and material as well as strain rate sensitivity. A phenomenological interactive failure criterion comprising bending, tension and transverse shear is proposed to predict the various modes of failure. A node release algorithm is developed to simulate the progression of plate rupture from the boundary. The analysis is continued in the post-failure phase to account for the free flight deformation of the torn plate. The predicted failure modes for a blast-loaded plate are presented and compared with previously published experimental data.