A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate

A new hyperbolic shear deformation theory taking into account transverse shear deformation effects is presented for the buckling and free vibration analysis of thick functionally graded sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded sandwich plates are obtained using the Navier solution. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories with more number of unknown functions.     

FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations

In this paper, the generalized differential quadrature (GDQ) method is applied to study the dynamic behavior of functionally graded materials (FGMs) and laminated doubly curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner-Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Firstly, the differential quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner-Mindlin and the Toorani-Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A two-constituent functionally graded lamina consists of ceramic and metal those are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of shell and panel structures considered.     

Dynamic contour error compensation in micro/nano machining of hardened steel by applying elliptical vibration sculpturing method

In this study, a novel dynamic contour error compensation technique has been proposed for the elliptical vibration cutting process achieved through the ultra-precision amplitude control. The influence of the contour error, triggered due to the inertial vibrations of the friction-less feed drive system, on the machining accuracy deterioration has been experimentally investigated. In order to reduce the contour error, a compensation method utilizing a real-time amplitude control in the elliptical vibration cutting process has been applied. In the proposed method, the dynamic motion error along the depth of cut direction is detected by utilizing the precise linear encoders installed on the feed drive system. The motion error in real-time is subsequently converted into cancelling amplitude command for the vibration control system of the ultrasonic vibrator, thus, guaranteeing that the envelope of the vibration amplitudes auto-tracks the dynamic reference position of the motion axis in the depth of cut direction. Due to this, a constant nominal depth of cut can be obtained even though the inertial vibrations disturb the feed drive control during machining. A series of experimental investigations have been conducted in order to analyze the machining performance by employing the proposed method. The maximum machining error is observed to significantly decrease from 0.6 to 0.04 μm by applying the proposed compensation method. Finally, the micro dimple array with a structural height from about 200 to 600 nm could be accurately fabricated with a maximum machining error of 36.8 nm, which verified the feasibility of the proposed amplitude control compensation method.     

Natural frequencies of composite plates with random material properties using higher-order shear deformation theory

Composites are known to display a considerable amount of scatter in their material properties due to large number of parameters associated with the manufacturing and fabrication processes. In the present work, the material properties have been taken as random variables for accurate prediction of the system behavior. Higher order shear theory including rotatory inertia effects has been accounted for in the system dynamic equations. A first order perturbation technique has been employed to obtain the solution of the governing equations. An approach has been outlined for obtaining closed form expressions for the variances of eigen solutions. The effects of side to thickness ratio and variation in standard deviation of the material properties have been investigated for cross-ply symmetric and anti-symmetric laminates. The mean and standard deviations of the first five natural frequencies have been worked out for laminated rectangular plates with all edges simply supported. The higher order shear deformation theory results have been validated with Monte Carlo simulation results and compared with the results based on classical laminate and first order shear deformation theories.     

Non-linear beam vibration problems and simplifications in finite element models

When finite element formulations are used to study the non-linear vibration problems, some simplifications, which are not consistent with the governing variational principles, are commonly employed. Three such simplifications are critically reviewed here, through beam finite element models. The first one, equivalent/ quasi-linearisation technique is shown to have a reduced non-linear stiffness. The second, where in neglect of in plane displacements takes place, is seen to register an excessive non-linear stiffness. Thirdly, when both these simplifications are introduced together, they produce results closer to those of variationally correct ones,rather fortuitously. The objective of this paper is to highlight the necessity of formulating this class of problems in a variationally correct and consistent manner. Numerical computations are performed systematically, using two different beam finite element models for various commonly studied boundary conditions and suitable conclusions are drawn.     

Macrotexture influence on vibrational mechanisms of the tyre–road noise of an asphalt rubber pavement

The road surface is one of the most important factors that have influence on the current traffic noise. Usually, for dense surfaces, impacts of the tyre on the pavement generate vibrations which are the dominant mechanisms in the tyre–road noise. In this study, the effect of muffling these vibrations, by the incorporation of crumb rubber (CR) from wasted tyres into asphalt pavements, has been evaluated acoustically. Close proximity measurements have been carried out to register the sound emission generated in the contact zone between a reference tyre and an experimental asphalt pavement with CR. The analysis of the measurements indicates that the incorporation of CR as well as the air voids content has less influence than the macrotexture of the road surface on the acoustical behaviour of this experimental asphalt pavement.     

Local and global instability and vibrations of a slender system consisting of two coaxial elements

The stability and free vibrations of a geometrically non-linear system built of a pipe and a rod are discussed in the paper. The pipe and the rod were connected between the mountings by an elastic element and the whole system was subjected to Euler’s load by external force whose direction is unchanging. The boundary problem (static and free vibrations) was formulated and solved, and numerical computations connected to the bifurcation load and natural frequency were carried out. The boundary problem was formulated using Hamilton’s principle and the straightforward expansion method. The calculations were carried out for different characteristic parameters of the considered system, such as rigidity, the placement of the elastic element, pre-stressing of the system as well as the flexural rigidity asymmetry factor of a column.     

A unified computational methodology for dynamic thermoelasticity with multiple subdomains under the GSSSS framework involving differential algebraic equation systems

Abstract

A novel and general computational methodology for thermal stress problems with multiple subdomains is presented under the unified generalized single-step single-solve (GSSSS) framework for first- and second-order differential algebraic equations. It enables arbitrary number of subdomains and the coupling of different but compatible time-stepping algorithms ensuring second-order time accuracy in all differential and algebraic variables. The framework permits implicit/explicit coupling and subcycling; however, only selected coupling of algorithms in different subdomains is focused upon. Numerical examples encompassing transient heat conduction with quasi-static thermal stresses, and thermally-induced vibrations are illustrated.