Displacement Potential Function Research Articles

3-D elasticity buckling solution for simply supported thick rectangular plates using displacement potential functions

This paper concentrates on the elastic buckling of thick plates. The shear strains in thick structures such as thick plates are important and thus, cannot be neglected. On the other hand, shear strains in the analysis of thick plates leads to complicity of governing equations. In this paper, with the use of displacement potential functions, the governing equations are simplified to two partial differential equations for the potential functions, one of which is second order and the other is forth order. These PDEs are established for a rectangular isotropic body, so that they are applicable to any arbitrary thickness of plate with no limitation on its thickness ratio. By solving the governing differential equations using separation of variables method and satisfying the exact boundary conditions, an analytical solution is obtained for linear elastic buckling of simply supported rectangular thick plates, subjected to in-plane either uniaxial or biaxial static loads, one of which could also be tensile force. Then critical buckling load is expressed in terms of non-dimensional buckling factor. The results of this paper are compared with other analytical and numerical works for thin and moderately thick plates, and also with numerical works for thick plates, which proves an excellent agreement between the results of this paper and others.

Elastic Waves in Generalized Thermo-Piezoelectric Transversely Isotropic Circular Bar Immersed in Fluid

AbstractIn this paper, a mathematical model is developed to study the wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of circular cross-sections immersed in inviscid fluid. The present study is based on the use of the three-dimensional theory of elasticity. Three displacement potential functions are introduced to uncouple the equations of motion and the heat and electric conductions. The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied based on Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity. The frequency equations of the coupled system consisting of cylinder and fluid are developed under the assumption of perfect-slip boundary conditions at the fluid-solid interfaces, which are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 material bar immersed in fluid. The computed non-dimensional frequencies are compared with Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity for longitudinal and flexural modes of vibrations. The dispersion curves are drawn for longitudinal and flexural modes of vibrations. Moreover, the dispersion of specific loss and damping factors are also analyzed for longitudinal and flexural modes of vibrations.

Analytical solution of the asymmetric transient wave in a transversely isotropic half-space due to both buried and surface impulses

With the aid of a complete set of two scalar potential functions, the problem of transient wave propagation in transversely isotropic half-space, subjected to time dependent tractions applied on a finite patch at an arbitrary depth below the free surface of the half-space is investigated. With the use of the displacement–potential function relationships in a cylindrical coordinate system, the coupled equations of motion are uncoupled; resulting in two separate partial differential equations one of which is second order and the other is fourth order. These two partial differential equations are solved with the aid of both Fourier series expansion and joint Hankel–Laplace integral transforms. The solutions are also investigated in details for tractions varying with time as Heaviside step function, which may be used as a kernel in any integral based method for more complicated elastodynamic initial-boundary value problems. Moreover, some displacement Green׳s functions are numerically evaluated for a synthetic transversely isotropic material to graphically demonstrate the transient motion of the free surface of the half-space.

Coupled thermoelastic analysis of a multi-layered hollow cylinder based on the C–T theory and its application on functionally graded materials

The coupled thermoelastic behavior of a multi-layered or functionally graded hollow cylinder under generally transient thermal and mechanical loadings is investigated in this paper. The generalized Chandrasekharaiah–Tzou (C–T) thermoelasticity theory, based on the dual-phase-lag hyperbolic heat conduction model, is used to describe the nature of finitely thermal and elastic wave speeds. The Kapitza thermal interface model indicates that the temperature can jump across the interface but the heat flux in the radial direction is continuous. With the Laplace transform method eliminating the time-dependence of the coupled partial differential equations, the problem is theoretically solved by introducing a displacement potential function. A numerical Laplace inversion technique is then employed to transform the results into the time domain. Finally, the effect of heat conduction models, inertial and coupling effects, loading types, interfacial imperfection, Biot number, and functionally graded profiles on the thermal and elastic results is analyzed. Also, the physically unrealistic discontinuity in displacement is observed when the cylinder is subjected to a Dirac delta-type thermal loading.

유체에 부분적으로 둘러싸인 사각형 용기의 접수진동

This paper deals with a theoretical method to calculate natural frequencies of a fixed–free rectangular tank partially in contact with an outer water gap. Orthogonal polynomials satisfying the boundary conditions of the tank are used as admissible functions in the Rayleigh–Ritz method. A quarter model of the liquid-coupled system is constructed and it is simplified to a line supported flat plate in contact with the liquid. The liquid displacement potential functions satisfying the Laplace equation and water boundary conditions are derived, and the finite Fourier transform is accomplished in conjunction with the compatibility requirement along the contacting interfaces between the tank and water. An eigenvalue problem is derived so that the natural frequencies of the wet rectangular tank can be extracted. The predictions from the proposed analytical method show good agreement with the finite element analysis results.

Analytical technique for electroelastic field in piezoelectric bodies belonging to point group D ∞

To elucidate the electroelastic field in woods, which are piezoelectric bodies belonging to point group D∞, we construct an analytical technique for general solutions to electroelastic problems in these bodies. First, the constitutive equations are derived considering the microstructures and their combined behaviors. Then, the displacement and electric field are expressed in terms of two types of displacement potential functions and the electric potential function, and their governing equations are obtained using the fundamental equations for the electroelastic field. As a result, the electroelastic field quantities are found to be expressed in terms of four functions, namely two elastic displacement potential functions and two piezoelastic displacement potential functions, each of which satisfies a Laplace equation with respect to the appropriately transformed spatial coordinates. As an application of the technique, the electroelastic field in a semi-infinite body subjected to a prescribed electric potential on its surface is analyzed, and the numerical results are illustrated. This novel technique serves to investigate the electroelastic field inside wooden materials.

Analysis of rectangular piles subjected to lateral loads in nonhomogeneous soil using a Winkler model approach

SummaryAn analytical approach using a Winkler model based on two lateral soil displacement components in a three‐dimensional soil is investigated to provide analytical solutions of horizontal response of a rectangular pile subjected to lateral loads in nonhomogeneous soil. The two lateral displacement components of a soil surrounding the rectangular pile are represented by the Fourier series of displacement potential functions in the elastic three‐dimensional analysis. The lateral stiffness coefficient of the rectangular pile shaft in nonhomogeneous soil is derived from the rocking stiffness coefficient taking into account rocking rotation of a rigid pile shaft. The relationship between horizontal displacement, rotation, moment, and shear force for the rectangular pile subjected to horizontal loads in nonhomogeneous soil is obtainable in the form of the recurrence equation. The formulation of lateral displacement and rotation for a rectangular pile subjected to lateral loads on the pile base in nonhomogeneous soil is proposed by taking into account Mindlin's equation and the equivalent thickness for soil layers in the equivalent elastic method. The difference of lateral behavior between square and circular piles subjected to lateral loads is insignificant. The effect of aspect ratio of the rectangular pile on the lateral behavior is great for the lower stiffness ratio between pile and soil and the larger length–equivalent diameter ratio. The effect of the value of Poisson's ratio of soil on lateral stiffness coefficient is relatively small except Poisson's ratio close to 0.5. The comparison of the results calculated by the current method for a rectangular pile subjected to lateral loads in nonhomogeneous soil has shown good agreement with those obtained from the analytical methods and the finite element method. Copyright © 2015 John Wiley & Sons, Ltd.

Surface Effect on Diffractions of Elastic Waves and Stress Concentration near a Cluster of Cylindrical Nanoholes Arranged as Quadrate Shape

We consider the multiple scattering of elastic waves (P-wave and SV-wave) by a cluster of nanosized cylindrical holes arranged as quadrate shape. When the radius of the holes shrinks to nanometers, the surface elasticity theory is adopted in analysis. Using the displacement potential method and wave functions expansion method, we obtain that the multiple scattering fields induced by incident P- and SV-waves around the holes are derived. The dynamic stress concentration around the holes is calculated to illustrate the effect of surface effects on the multiple scattering of P- and SV-waves.

Natural vibrations of thick circular plate based on the modified Mindlin theory

Outline of the modified Mindlin theory is presented in which the Mindlin mathematical model with three differential equations of motion for total deflection and rotations is decomposed into a single equation for pure bending vibrations with transverse shear and rotary inertia effects and two differential equations for in-plane shear vibrations. The governing equations are transformed from orthogonal to polar coordinate system for the purpose of circular plate vibration analysis. The fourth order differential equation of flexural vibrations is split further into two second order equations of Bessel type. Also, the in-plane shear differential equations are transformed to Bessel equation by introducing displacement potential functions. The exact values of natural frequencies are listed and compared with FEM results.

Stress wave analysis of thermo-piezoelectric solid bar of polygonal cross-sections immersed in fluid

Purpose – The purpose of this paper is to study the problem of wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of polygonal (triangle, square, pentagon and hexagon) cross-section immersed in fluid is using Fourier expansion collocation method, with in the frame work of linearized, three-dimensional theory of thermo-piezoelectricity. Design/methodology/approach – A mathematical model is developed to study the wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of polygonal cross-sections immersed in fluid is studied using the three-dimensional theory of elasticity. Three displacement potential functions are introduced, to uncouple the equations of motion and the heat and electric conductions. The frequency equations are obtained for longitudinal and flexural (symmetric and antisymmetric) modes of vibration and are studied numerically for triangular, square, pentagonal and hexagonal cross-sectional bar immersed in fluid. Since the boundary is irregular in shape; it is difficult to satisfy the boundary conditions along the curved surface of the polygonal bar directly. Hence, the Fourier expansion collocation method is applied along the boundary to satisfy the boundary conditions. The roots of the frequency equations are obtained by using the secant method, applicable for complex roots. Findings – From the literature survey, it is clear that the free vibration of an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of polygonal cross-sectional bar immersed in fluid have not been analyzed by any of the researchers, also the previous investigations in the vibration problems of transversely isotropic thermo-piezoelectric solid bar of circular cross-sections only. So, in this paper, the wave propagation in thermo-piezoelectric cylindrical bar of polygonal cross-sections immersed in fluid are studied using the Fourier expansion collocation method. The computed non-dimensional frequencies are plotted in the form of dispersion curves and its characteristics are discussed, also a comparison is made between non-dimensional wave numbers for longitudinal and flexural modes piezoelectric, thermo-piezoelectric and thermo-piezoelectric polygonal cross-sectional bars immersed in fluid. Research limitations/implications – Wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of polygonal cross-sectional bar immersed in fluid have not been analyzed by any of the researchers, also the previous investigations in the vibration problems of transversely isotropic thermo-piezoelectric solid bar of circular cross-sections only. So, in this paper, the wave propagation in thermo-piezoelectric cylindrical bar of polygonal cross-sections immersed in fluid are studied using the Fourier expansion collocation method. The computed non-dimensional frequencies are plotted in the form of dispersion curves and its characteristics are discussed, also a comparison is made between non-dimensional wave numbers for longitudinal and flexural modes of piezoelectric, thermo-piezoelectric and thermo-piezoelectric polygonal cross-sectional bars immersed in fluid. Originality/value – The researchers have discussed the wave propagation in thermo-piezoelectric circular cylinders using three-dimensional theory of thermo-piezoelectricity, but, the researchers did not analyzed the wave propagation in an arbitrary/polygonal cross-sectional bar immersed in fluid. So, the author has studied the free vibration analysis of thermo-piezoelectric polygonal (triangle, square, pentagon and hexagon) cross-sectional bar immersed in fluid using three-dimensional theory elasticity. The problem may be extended to any kinds of cross-sections by using the proper geometrical relations.

Elastocapillary deformations on partially-wetting substrates: rival contact-line models.

A partially-wetting liquid can deform the underlying elastic substrate upon which it rests. This situation requires the development of theoretical models to describe the wetting forces imparted by the drop onto the solid substrate, particularly those at the contact-line. We construct a general solution using a displacement potential function for the elastic deformations within a finite elastic substrate associated with these wetting forces, and compare the results for several different contact-line models. Our work incorporates internal contributions to the surface stress from both liquid/solid Σls and Σsg solid/gas solid surface tensions (surface stress), which results in a non-standard boundary-value problem that we solve using a dual integral equation. We compare our results to relevant experiments and conclude that the generalization of solid surface tension Σls ≠ Σsg is an essential feature in any model of partial-wetting. The comparisons also allow us to systematically eliminate some proposed contact-line models.

Wave propagation in a transversely isotropic magneto-electro-elastic solid bar immersed in an inviscid fluid

Abstract Wave propagation in a transversely isotropic magneto-electro-elastic solid bar immersed in an inviscid fluid is discussed within the frame work of linearized three dimensional theory of elasticity. Three displacement potential functions are introduced to uncouple the equations of motion, electric and magnetic induction. The frequency equations that include the interaction between the solid bar and fluid are obtained by the perfect slip boundary conditions using the Bessel functions. The numerical calculations are carried out for the non-dimensional frequency, phase velocity and attenuation coefficient by fixing wave number and are plotted as the dispersion curves. The results reveal that the proposed method is very effective and simple and can be applied to other bar of different cross section by using proper geometric relation.

Exact thermal elasticity solution for non-compliant pipe-in-pipe systems

In this study, thermo-elastic differential equations for the displacement potential function are solved to obtain the functional axial deflections for the segments between the bulkheads of a pipe-in-pipe system. The functional axial deflections, the axial displacements from the bulkhead forces and the axial strain induced by the seabed friction are used to calculate the shear force exerted on each bulkhead of the pipe-in-pipe system via an iterative approach. Both pipe-in-pipe systems with equally and unequally spaced bulkheads are discussed.

TORSIONAL ANALYSIS OF PIEZOELECTRIC HOLLOW BARS

Analytical formulations are presented for Saint-Venant's torsion of orthotropic piezoelectric hollow members. It is assumed that the cross-section consists of straight and curved segments with the same thickness. Governing equations are based on Prandtl's stress and electric displacement potential functions. Some examples are solved using the presented formulations and verified by the finite element method solutions. The results show the accuracy of the present method for torsional analysis of thin- to moderately thick-walled hollow bars.

Solution of a flat elliptical crack in an electrostrictive solid

Based on the assumption that the elastic strain of electrostrictive materials is a higher-order small quantity, this paper studies the 3D problem of an infinite electrostrictive solid with a flat elliptical crack which is electrically permeable. According to existing solutions of similar problems in pure elastic materials, with the displacement function method, we first derived explicit expression for displacement potential function and obtained stress field near the crack and open displacement of crack surface. Then, the general solution for the stress intensity factor was derived, and the corresponding solutions were also presented for a penny-shaped crack and a permeable line-crack as two special cases of the present problem. Finally, numerical results were given to discuss the effect of environment at infinity and electric field inside the crack on the stress-intensity factors.

Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane

The normal indentation of a rigid circular disk into the surface of a transversely isotropic half-space reinforced by a buried inextensible thin film is addressed. By virtue of a displacement potential function and the Hankel transform, the governing equations of this axisymmetric mixed boundary value problem are represented as a dual integral equation, which is subsequently reduced to a Fredholm integral equation of the second kind. Two important results of the contact stress distribution beneath the disk region as well as the equivalent stiffness of the system are expressed in terms of the solution of the Fredholm integral equation. When the membrane is located on the surface or at the remote boundary, exact closed-form solutions are presented. For the limiting case of an isotropic half-space the results are verified with those available in the literature. As a special case, the elastic fields of a reinforced transversely isotropic half-space under the action of surface axisymmetric patch loads are also given. The effects of anisotropy, embedment depth of the membrane, and material incompressibility on both the contact stress and the normal stiffness factor are depicted in some plots.

Two Temperature Magneto-Thermoelasticity with Initial Stress: State Space Formulation

Magneto-thermoelastic interactions in an initially stressed isotropic homogeneous elastic half-space with two temperatures are studied using mathematical methods under the purview of the L-S model of linear theory of generalized thermoelasticity. The formalism deals with the state space approach with the purpose of counteracting the difficulties of handling the displacement potential functions. Of specific concern here is the propagation of waves owing to ramp type increase in temperature and load. The medium is considered to be permeated by a uniform magnetic field. The expressions for different field parameters such as displacement, temperature, strain, and stress in the physical domain are obtained by applying a numerical inversion technique. Results of some earlier workers have been deduced from the present formulation. Numerical work is also performed for a suitable material with the aim of illustrating the results.

Generalized Thermo Elastic Waves in a Rotating Plate Subjected to Ramp Type Increase in Boundary Temperature

The generalized thermo elastic waves in a rotating plate subjected to ramp type increase in boundary temperature is studied based on Lord-Shulman (LS) and Green-Lindsay (GL) generalized two dimensional theory of thermo elasticity. Two displacement potential functions are introduced to uncouple the equations of motion and heat conduction. The frequency equations are obtained by the traction free and ramp type thermal boundary conditions using Bessel function solutions. The numerical calculations are carried out for the material Zinc. The computed non-dimensional frequency and attenuation coefficient are plotted as dispersion curves.

Wave propagation in a generalized thermo elastic circular plate immersed in fluid

In this paper, the wave propagation in generalized thermo elastic plate immersed in fluid is studied based on the Lord-Shulman (LS) and Green-Lindsay (GL) generalized two dimensional theory of thermo elasticity. Two displacement potential functions are introduced to uncouple the equations of motion. The frequency equations that include the interaction between the plate and fluid are obtained by the perfectslip boundary conditions using the Bessel function solutions. The numerical calculations are carried out for the material Zinc and the computed non-dimensional frequency, phase velocity and attenuation coefficient are plotted as the dispersion curves for the plate with thermally insulated and isothermal boundaries. The wave characteristics are found to be more stable and realistic in the presence of thermal relaxation times and the fluid interaction.

Axisymmetric circular indentation of a half-space reinforced by a buried elastic thin film

The analytical treatment of an axisymmetric rigid punch indentation of an isotropic half-space reinforced by a buried extensible thin film is addressed. With the aid of appropriate displacement potential functions, Hankel transforms, and some mathematical techniques, the mixed boundary value problem under consideration is reduced to a Fredholm integral equation of the second kind. The most interesting results of the problem, including the equivalent normal stiffness of the system and the contact stress distribution beneath the rigid punch, are expressed in terms of the solution of the obtained Fredholm integral equation. Some limiting cases corresponding to inextensible and extremely extensible thin films, a surface-stiffened half-space, axisymmetric surface loading, and infinite embedment are studied and the available results in the literature are used for verification purposes. Some dimensionless plots are provided to show the effects of extensibility of the membrane on the system stiffness. It is observed that neglecting the elastic behavior of thin films and utilizing the simplified inextensible assumption for their modeling can lead to unrealistic predictions. It is shown that the axisymmetric problems concerning the elastic half-spaces including surface effects can be equivalently modeled by a surface-reinforced elastic medium. Taking the surface effects into account, the problem of an isotropic half-space under the action of a surface patch load is treated by employing the present approach.