Faculty of Engineering
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Publication Embargo Continuum models incorporating surface energy for static and dynamic response of nanoscale beams(IEEE, 2009-10-09) Liu, Chang; Rajapakse, R. K. N. DNanoscale beams are commonly found in nanomechanical and nanoelectromechanical systems (NEMS) and other nanotechnology-based devices. Surface energy has a significant effect on nanoscale structures and is associated with their size-dependent behavior. In this paper, a general mechanistic model based on the Gurtin-Murdoch continuum theory accounting for surface energy effects is presented to analyze thick and thin nanoscale beams with an arbitrary cross section. The main contributions of this paper are a set of closed-form analytical solutions for the static response of thin and thick beams under different loading (point and uniformly distributed) and boundary conditions (simply-supported, cantilevered, and clamped ends), as well as the solution of the free vibration characteristics of such beams. Selected numerical results are presented for aluminum and silicon beams to demonstrate their salient response features. It is shown that classical beam theory is not accurate in situations where the surface residual stress and/or surface elastic constants are relatively large. An intrinsic length scale for beams is identified that depends on beam surface properties and cross-sectional shape. The present work provides a convenient set of analytical tools for researchers working on NEMS design and fabrication to understand the static and dynamic behavior of nanoscale beams including their size-dependent behavior and the effects of common boundary conditions.Publication Open Access Vertical vibrations of a rigid disk embedded in a poroelastic medium(John Wiley & Sons, Ltd., 1999-12-25) Zeng, X; Rajapakse, R. K. N. Dhis paper considers the steady-state vertical vibrations of a rigid circular disk embedded at a finite depth below the free surface of a poroelastic medium. Biot's elastodynamic theory for porous media is used in the analysis. General solutions for axisymmetric poroelastic fields are obtained by using Hankel integral transforms. Analytical solutions for influence functions corresponding to four types of buried axisymmetric excitations are derived. The embedded disk problem is fomulated in terms of a set of coupled integral equations for unknown traction and pore pressure jumps across the disk. The kernel functions of the integral equations are the influence functions corresponding to buried vertical, radial and pore pressure ring loads. The system of integral equations is solved numerically by discretizing the disk into several concentric annular rings. Selected numerical solutions for displacements, vertical stress and pore pressure due to a buried fully flexible disk (uniform pressure) are also presented. The vertical compliances of a rigid disk are examined for different depths of embedment, poroelastic materials and hydraulic boundary conditions. Solutions for traction and pore pressure jumps are also examined. The present results are useful in the study of dynamic response of embedded foundations and anchors in poroelastic soils.Publication Open Access Fracture analysis of magnetoelectroelastic solids by using path independent integrals(Kluwer Academic Publishers, 2005-02) Tian, W. Y; Rajapakse, R. K. N. DA solution scheme based on the fundamental solution for a generalized edge dislocation in an infinite magnetoelectroelastic solid is presented to analyze problems involving single, multiple and slowly growing impermeable cracks. The fundamental solution for a generalized dislocation is obtained by extending the complex potential function formulation used for anisotropic elasticity. The solution for a continuously distributed dislocation is derived by integrating the solution for an edge dislocation. The problem of a system of cracks subjected to remote mechanical, electric and magnetic loading is formulated in terms of set of singular integral equations by applying the principle of superposition and the solution for a continuously distributed dislocation. The singular integral equation system is solved by using a numerical integration technique based on Chebyshev polynomials. The J i and M-integrals for single crack and multi-cracks problems are derived and their dependence on the coordinate system is investigated. Selected numerical results for the M-integral, total energy release rate and mechanical energy release rate are presented for single, double and multiple crack problems. The case of a slowly growing crack interacting with a stationary crack is also considered. It is found that M-integral presents a reliable and physically acceptable measure for assessment of fracture behaviour and damage of magnetoelectroelastic materials.Publication Embargo Suctions, stresses and strengths in unsaturated sand–bentonite(Elsevier, 2002-05-01) Tang, Gary X; Graham, James; Blatz, James; Gray, Malcolm; Rajapakse, R. K. N. DThis paper presents suction and strength characteristics for a dense, compacted, unsaturated sand–bentonite mixture under a variety of preparation and stressing conditions. Suctions were determined experimentally using thermocouple psychrometers and the filter paper method. They are shown to be related to water contents, saturation, dry densities and osmotic agents. The influence of the initial suctions on strength was evaluated using quick undrained triaxial compression tests, here called ‘constant-mass’ tests. Examination of the suction response to applied external stresses was carried out using stress-controlled triaxial tests along selected stress paths. These tests measured suctions in tests where the mean stress p and deviator stress q were changed systematically to give a series of constant values of Δq/Δp. The results showed that suction decreased as mean stress loading increased. Suction changes appear to be produced only by the mean stress component of the stress tensor and not by the shear stress component. In this compacted sand–bentonite material, for the pressure range at which the tests were performed (p′≤3 MPa), changes in suction produced by changes in mean stress are largely reversible.Publication Embargo Joint slip in steel electric transmission towers(Elsevier, 2003-05-01) Ungkurapinan, N; Rajapakse, R. K. N. D; Chandrakeerthy, SR De S; Yue, S.BJoint slip is the relative displacement of a bolted joint under shear. It is greater in transmission towers as bolt diameters are small, members joined are thin, bearing type joints with a lower clamping force are used, and coefficient of friction of galvanized faying surfaces is low. This study on behaviour of such joints, incorporated 36 joint tests, generated joint slip data and developed mathematical expressions to describe slip and load-deformation behaviour. As currently used construction clearance was found satisfactory, joint slip cannot be eliminated. Hence, incorporation of the reported joint slip data or mathematical expressions in tower analysis software will refine their results.Publication Embargo On the longitudinal harmonic motion of an elastic bar embedded in an elastic half-space(Pergamon, 1987-01-01) Shah, A.H; Rajapakse, R. K. N. DThe present study is concerned with the motion of a long cylindrical elastic bar which is partially embedded in a homogeneous elastic half-space and subjected to a harmonic axial load. Initially Green's functions corresponding to axisymmetric harmonic ring loads are derived and presented explicitly. It is found that the direct extension of elastostatic solution schemes to solve elastodynamic problems may lead to erroneous solutions due to the inability of these algorithms to properly account for inertia effects. Some discrepancies in existing solutions with respect to the inertia component of the bar are shown. An efficient solution scheme, based on Lagrange's equation of motion and a discretization technique, is presented to solve the title problem. Numerical results are presented to illustrate the influence of bar flexibility, mass density, geometry, and frequency of excitation on the axial impedance of the system.Publication Embargo Exact stiffness method for quasi-statics of a multi-layered poroelastic medium(Pergamon, 1995-06-01) Senjuntichai, T; Rajapakse, R. K. N. DA method is presented to study the three-dimensional quasi-static response of a multi-layered poroelastic half-space with compressible constituents. The system under consideration consists of N layers of different thickness and material properties overlying a homogeneous half-space. Fourier expansion, Laplace transforms and Hankel transforms with respect to the circumferential, time and radial coordinates, respectively, are used in the formulation. Laplace-Hankel transforms of displacements and pore pressure at layer interfaces are considered as the basic unknowns. Exact stiffness matrices describing the relationship between generalized displacement and force vectors of a finite layer and a half-space are derived explicitly in the transform space. The global stiffness matrix of a layered system is assembled by considering the continuity of tractions and fluid flow at layer interfaces. The time histories of displacements, stresses and pore pressure are obtained by solving the stiffness equation system for discrete values of Laplace and Hankel transform parameters, and using numerical quadrature schemes for Laplace and Hankel transform inversions. Selected numerical results for different layered systems are presented to portray the influence of layering and poroelastic material properties. The advantage of the present method is that for an N-layered system, it yields a numerically stable symmetric stiffness matrix of order 4N × 4N when compared to the unsymmetric and numerically unstable coefficient matrix of order 8N × 8N associated with the conventional method based on the determination of layer arbitrary coefficients.Publication Embargo Analytical solutions for a surface-loaded isotropic elastic layer with surface energy effects(Pergamon, 2009-11-01) Rajapakse, R. K. N. D; Zhao, XJConsideration of surface (interface) energy effects on the elastic field of a solid material has applications in several modern problems in solid mechanics. The Gurtin–Murdoch continuum model [M.E. Gurtin, A.I. Murdoch, Arch. Ration. Mech. Anal. 57 (1975) 291–323; M.E. Gurtin, J. Weissmuller, F. Larché, Philos. Mag. A 78 (1998) 1093–1109] accounting for surface energy effects is applied to analyze the elastic field of an isotropic elastic layer bonded to a rigid base. The surface properties are characterized by the residual surface tension and surface Lame constants. The general solutions of the bulk medium expressed in terms of Fourier integral transforms and Hankel integral transforms are used to formulate the two-dimensional and axisymmetric three-dimensional problems, respectively. The generalized Young–Laplace equation for a surface yields a set of non-classical boundary conditions for the current class of problems. An explicit analytical solution is presented for the elastic field of a layer. The layer solution is specialized to obtain closed-form solutions for semi-infinite domains. Selected numerical results are presented to show the influence of surface elastic constants and layer thickness on stresses and displacements.Publication Embargo Self-heat generation in piezoelectric stack actuators used in fuel injectors(IOP Publishing, 2009-03-18) Senousy, MS; Mumford, D; Gadala, MS; Rajapakse, R. K. N. DMultilayer piezoelectric actuators are used in fuel injectors due to their quick response, high efficiency, accuracy, low power consumption, and excellent repeatability. Experimental results for soft lead zirconate titanate (PZT) stack actuators have shown that a significant amount of heat is generated when they are driven under high frequency and/or high electric-field magnitudes, both of which occur in fuel injectors. Self-heat generation in these actuators, mainly caused by losses, can significantly affect their reliability and piezoelectric properties, and may also limit their application. Other studies have demonstrated that at large unipolar electric-field magnitudes, displacement–electric-field loss (displacement hysteresis) shows a direct relation with polarization–electric-field loss (dielectric hysteresis). In this paper, a simplified analytical self-heating model is presented. The model directly relates self-heating in multilayer piezoelectric actuators to displacement–electric-field loss (displacement hysteresis). The model developed is based on the first law of thermodynamics, and accounts for different parameters such as geometry, magnitude and frequency of applied electric field, duty cycle percentage, fuel type, and environmental properties. The model shows reasonable agreement with experimental results at low and high electric-field magnitudes.Publication Embargo On a plane crack in piezoelectric solids(Pergamon, 2001-10-01) Rajapakse, R. K. N. D; Xu, X-LA new analytical solution for a piezoelectric plane with an elliptical void is derived by removing the commonly held assumptions that the void boundary is impermeable and a void axis is perpendicular to the poling direction. The approach of Lekhnitskii's complex potential functions is used in the derivation. Applicability of the common practice of reducing a void solution to a crack solution is examined. It is shown that a recently reported solution for exact electric boundary conditions is actually the well known solution for a permeable crack. A unified formulation for plane cracks containing air or vacuum is then developed to account for different electric boundary conditions. Crack closure is taken into consideration in the analysis. The influence of electric boundary conditions and crack orientation on fracture parameters is discussed.