Infinite Thin Plate Research Articles

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

AbstractA dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.

3640 減衰を考慮した非線形弾性材料の一般化された一次元熱弾性解析(S24-1 熱衝撃・熱弾性,S24 数理・数値解析による先端材料評価)

This paper deals with the one-dimensional generalized thermoelasticity in a homogeneous isotropic infinite thin plate exhibiting an inelastic behavior, when a damping effect is taken into account. The plate is initially at zero temperature and subjected to a ramp heating on the top surface. The governing equations for the temperature change, heat flux, particle velocity and axial stress are expressed by a set of first-order partial differential equations. This system of equations has been numerically analyzed by employing the method of characteristics. Finally, the effects of the damping coefficient on time histories of the axial stress and temperature change are illustrated graphically.

A phenomenological model of fatigue crack growth in perfectly plastic infinite plates under completely reversed uniaxial loading

The fatigue failure of a thin infinite center-cracked plate under completely reversed uniaxial loading is considered. A two-stage fatigue crack model including the incubation and crack propagation stages is constructed. The stress distribution in the vicinity of the crack tip is described using the concept of a conventional elastic crack. The crack-tip plastic zone is simulated by a Dugdale thin plastic zone, and the condition for the movement of the failure front is given by criteria of damage mechanics. It is shown that the fatigue crack growth rate in perfectly plastic materials with a plastic zone of constant length is a power-law function of the stress intensity factor range. This relationship is quadratic when the length of the plastic zone is not constant

Analytical and numerical solution of non-stationary state of stress in a thin viscoelastic plate

The response of a thin infinite viscoelastic plate to the transverse non-stationary loading is investigated and the analytical and the numerical solutions of the problem are presented. The analytical solution implies from the theory of thin plates and the so-called Timoshenko–Mindlin correction is used to achieve better compliance with 3D theory of continuum. The resulting relations for deflection, deflection angle and stress components are derived. Then the FE simulation of the problem is performed. Finally, both solutions are compared and their compliance proves the correctness of both the analytical solution and the numerical results.

Diffraction of flexural waves by cracks in orthotropic thin elastic plates. I Formal solution

The problem of flexural wave diffraction by a semi-infinite crack in an infinite orthotropic thin plate is considered. Such models have application to the ultrasonic non-destructive inspection of thin components, such as aeroplane wings. For simplicity, the plate is modelled using Kirchhoff theory, and the crack is chosen to be aligned along one of the principal directions of material orthotropy. For incident plane waves, an exact analytical expression for the scattered field is derived by means of the Wiener–Hopf technique. In this model problem, the Wiener–Hopf kernel is scalar and its factorization is expressed in terms of simple, definite, non-singular contour integrals. A detailed numerical evaluation of the solution will be provided in the second part of this work.

The problem of a circular patch

The problem of reinforcing an infinite thin elastic plate with a circular cut-out by means of a concentric circular patch attached to the plate so that the circumferences of the cut-out and the patch overlap is solved. The stress state in the plate and in the patch, generated by stresses at infinity in the plane of the plate, is investigated. Examples are presented.

Thermal Analysis of Welds

11R7. Thermal Analysis of Welds. - N. Nguyen. WIT Press, Southampton UK. 2004. 334 pp. ISBN 1-85312-951-8. $198.00. Reviewed by C A Rossit (Dept of Eng, Inst of Appl Mech and CONICET, 8000 Bahia Blanca, Argentina).This is a well-written book, which possesses an outstanding organization and clear and didactic figures. The book provides a very complete, up-to-date coverage of known solutions to an important variety of thermal problems related to welding processes. In the reviewer’s opinion the author synthesizes analytical expressions useful for the subject but with such a degree of abbreviation that it is not possible to follow the derivation of the expressions unless one is an expert on the subject. It would have helped if specific references were provided for each formulation. Chapter 1 introduces the characteristics of different heat sources associated with the welding process. Chapter 2 deals with the method of analysis for solving heat conduction differential equation. In Eq. (2.11) page 43 of this chapter one reads “∂2T∂x2+∂2T∂y2+∂2T∂z2=1α∂T∂t+fx,y,z,twhere fx,y,z,t represents the additional heat-generation function in the body…” and it is not clear the reason why the author places fx,y,z,t with positive sign on the right-hand side member, since this is not usual in the classical texts of the subject matter. On the other hand, the additional heat-generation must be divided by “k” in order to be compatible with the rest of the equation. After he divides it by “k” he could call it f1x,y,z,t for instance. Chapters 3 to 7 are concerned with the derivation of the analytical solution for different kinds of heat sources in semi-infinite and infinite bodies, thin and thick plates, and cylinders and spheres. Thus the author deals first with point, line, plane heat sources, and then by superposition principle analysis the bivariate Gaussian heat sources. Finally he focuses on the situations of spherical, single and double ellipsoidal density heat sources. Chapters 8 to 11 are addressed to demonstrate the usefulness of analytical solutions in solving thermal problems of welded components: transient temperature distribution, thermal stresses, residual stresses, and microstructures. The book contains three appendixes related to thermal properties of selected materials, error function characteristics, and documentation for Fortran programs developed by the author, respectively. The book also provides a CD-Rom, with a package called WHEATSIM, which is useful for the researcher in order to simulate welding situations. From a conceptual viewpoint it would have been interesting a short discussion, at least, of the possible importance of coupled thermal-mechanical phenomena and also dynamic effects in the case of very thin structures.

Buckling and vibration of a plate on elastic foundation subjected to in-plane compression and moving loads

The stability and dynamic displacement response of an infinite thin plate resting on a Winkler-type or a two-parameter elastic foundation have been investigated when the system is subjected to in-plane static compressive forces and a distributed load moving with a constant advance velocity. The amplitude of the moving load was assumed to be either constant or of harmonic variation and damping of a linear hysteretic nature for the foundation were considered. Formulations in the transformed field domains of time, space, and moving space were employed. The steady-state response to a moving harmonic load and the response to a moving load of constant amplitude were obtained using a double Fourier transform. Analyses were performed: (i) to investigate the effects of various parameters, such as the load velocity, load frequency, and damping, on the deflected shapes, maximum displacements, and critical values of the velocity, frequency, and in-plane compression, and (ii) to examine how the in-plane compression affects the stability and vibration of the system. Expressions to predict the critical (resonance) velocity, critical frequency, and in-plane buckling force were proposed.

Green’s functions for an anisotropic thin plate with a crack or an anticrack

Based on the octet formalism established by the authors, Green’s functions for an infinite anisotropic elastic thin plate with a Griffith crack or an anticrack (rigid line inclusion) are studied. The plate can be laminated or consist of a continuously inhomogeneous material in the thickness direction. Two systems of problems of the plate under non-self-equilibrated loads are solved. The first is associated with discontinuous in-plane displacements and slopes, in-plane concentrated forces and out-of-plane concentrated moments. The second is associated with a transverse concentrated force. The Green functions for an infinite plate with a crack or an anticrack and the surface Green functions for an infinite plate with a crack are obtained in an exact closed form. The surface Green functions and associated stress resultants at any point throughout the line where the crack is located are presented in a real form. The intensity factor of membrane stress resultants and bending moments are obtained in a real form for the Griffith crack subjected to concentrated loads.

Modelling of Sound Transmission through Lightweight Elements with Stiffeners

The sound reduction index of lightweight panels with stiffeners is calculated using the wave approach applied to an infinite thin plate line connected to periodically spaced beams and excited in flexure by a single or random (diffuse field) incident plane acoustical wave. The reaction forces and moments at the line connections are calculated from the flexural and torsional line impedance of the beams. The finite size of the panel is taken into account using a spatial windowing technique applied to the infinite panel. The case of two stiffened panels with a cavity in between and no structural connections between panels is also considered using the same approach; using SEA, the structural flanking path at the panel boundaries through any receiving structure can also be taken into account. Comparisons to measured results are given and discussed.

Material property estimation in thin plates using focused, synthetic-aperture acoustic beams.

The method developed here exploits the wide angular range of focused acoustic probes and the large synthetic aperture of scanned transducers to permit a rapid and reliable estimation of material properties in thin plates. It is found in several tests with various materials that estimates of elastic behavior using this method agree with contact measurements to within less than 5%. The method utilizes transmission (or reflection) coefficient reconstruction for an infinite thin plate, across a wide range of frequency and wave number, from which elastic property estimates are made. Data collected over a large synthetic acoustic aperture are processed with temporal and spatial Fourier transforms applied to change the acquired data from the coordinate and time domains to the wave number and frequency domains. Extrinsic real-beam effects on the data are accounted for with a complex transducer point analysis. Transmission measurements yield reconstructed data extending to the mode cutoffs, permitting easy and nearly unambiguous estimation of a subset of the elastic stiffnesses. For anisotropic plates, elastic stiffnesses are estimated with an inversion procedure that uses only limited data carefully selected from different portions of the measured scattering coefficient. Estimates are made by reconstructing in a stepwise fashion, based on sensitivity studies, where only one stiffness is estimated from the data at any one time, restricting the optimization to a robust one-dimensional search.

Sound radiation from a thin infinite plate in contact with a layered inhomogeneous fluid

In this paper the sound radiation from a thin infinite plate in contact with a layered inhomogeneous fluid subjected to single point excitation is studied. Sources contributing to the inhomogeneity are discussed, and the fluid is analytically expressed as a layered inhomogeneous field. Using Hankel transform, the equations governing the fluid–structure interaction are solved. The asymptotic form of the radiation pressure at far field is obtained using the method of stationary phase. Numerical examples show that the sound radiation patterns from a plate in contact with an inhomogeneous fluid is much more complicated than that in a homogeneous fluid. Three different radiation patterns (bell, disc, and their combination) were observed in the numerical examples, and the radiation patterns are sensitive to the field parameters. Depending on frequency, the sound pressure at far field is either magnified or reduced compared with that in a homogeneous fluid. And beyond some frequency the sound pressures are almost zero.

Green’s functions for the bending of thin plates under various boundary conditions and applications: a review

Green’s functions of a point dislocation for the bending problem of an infinite thin plate containing an arbitrarily shaped hole under displacement, stress, and mixed boundary conditions and for a bimaterial problem are stated and the review of the related works is given. The point dislocation is defined by the discontinuity of the plate deflection angle. The original formulation for these problems consists of two parts: a principal part containing singular and multi-valued terms, and a complementary part containing only holomorphic terms. The Green’s functions were obtained in closed forms by using the complex stress function method in junction with the rational mapping function. The applications of the Green’s functions are demonstrated in studying the interaction of a hole or inclusion with a line crack in an infinite plate subjected to bending. The Green’s functions can also be used as the kernels for boundary integral representations in boundary element method analysis, where they can notably simplify the procedure of the standard boundary integral equations.

A new method for piezothermoelastic problems of solids of crystal class 6 mm in Cartesian coordinates

Applications of piezoelectric materials for the development of “intelligent” structural and mechanical systems through sensing, actuation, and control have received considerable interest recently. In this document, we present a new analytical method for Piezothermoelastic problems of solids of crystal class 6 mm in Cartesian coordinate system. In the formulation presented, potential functions are introduced to represent the coupled thermal, elastic, and mechanical fields, which satisfy the thermal, mechanical, and electrical equilibrium and prescribed boundary conditions. It is shown that the formulation presented here is simpler and direct than that presented by Ashida et al. [Int. J. Engng. Sci. 31 (3) (1993) 373; J. Thermal Stresses 17 (1994a) 361; ZAMM, Z. Angew. Math. Mech. 74 (2) (1994b) 87; JSME Int. J. 37 (1994c) 334; JSME Int. J. 37 (1994d) 341] and it eliminates the need to discuss special cases. To demonstrate applications of the technique, a thin infinite piezoelectric plate subjected to thermal, electrical, and mechanical loads at a rectangular patch is considered. Numerical calculations for the stresses and the electrical potential are carried out for a cadmium selenide body exhibiting class 6 mm symmetry. Results of these calculations are presented graphically.

Sound radiation of a thin infinite plate in light and heavy fluids subject to multi-point excitation

This paper presents the mathematical expressions for sound radiation from a thin infinite isotropic plate coupled with light and heavy fluids and subject to multi-point excitation. The force at each point is phase dependent. Two typical cases are investigated theoretically; one is the two-point excitation with forces in phase and out of phase (moment). The other one is multi-point excitation with two rows of forces in phase and out of phase. The second one can be used to simulate the sound radiation from a rigid body mounting system with multi-point mounts on a thin infinite plate. It is found that the calculation of total sound pressure in the far field for the above cases cannot be simply considered as the superposition from a single point force excitation as is normally done for the sound pressure calculation from multiple point sound sources. Examples are presented to show the radiation patterns of the above two cases based on the derived equations. The results unveil that, as the plate is coupled with water, the sound radiation is stronger and much more dependent of the excitation frequency, especially as the plate is excited by a moment. It implies that the force excitations at high-order harmonic frequencies may contribute more structure-borne noise than that at basic excitation frequency even though their amplitudes are much smaller.

Dynamic Analysis of a Rigid Body Mounting System with Flexible Foundation Subject to Fluid Loading

This paper presents an investigation of the force transmission from a rigid body mounting system to a flexible foundation with light and heavy fluid loading under the force and moment excitation. The analytical expression has been derived in which the flexible foundation effects are incorporated into a revised system stiffness matrix that is derived from the receptance matrix at mounting points. A typical case with a thin infinite plate as the foundation has been studied with the point and transfer receptances theoretically and numerically analysed in the case of light and heavy fluid loading. The results show that, compared with the rigid foundation, the force transmission is reduced and system natural frequencies are shifted. The detailed analysis demonstrates that the force reduction and frequency shifting are more obvious at low frequencies where the receptance value is significant. The study is also carried out to compare the transfer receptances from different waves in plate as it couples with water with the objective to simplify the calculation of receptance. It is found that, in the low frequency and after a short distance from driving point, the transfer receptance calculation for the heavy fluid loading can be simplified by only accounting the contribution from free wave which may easily be evaluated from the point receptance in air. It implies the plate response under heavy fluid loading could be directly derived from that with light fluid loading.

A Packet of Elastic Plates Joined Along a Periodic System of Segments

A packet of infinite thin elastic plates joined along a periodic system of collinear segments is considered. The elastic properties and thicknesses of the plates can be different. The plates are loaded by tensile forces at infinity. An algorithm for determining the complex potentials governing stresses in the plates is constructed by solving the Riemann matrix boundary‐value problem. The stress‐intensity factors are found and their graphs are given.

On the uniqueness of the problem of acoustic diffraction by an infinite plate with local irregularities

The question concerning the uniqueness of the solution to the problem of the acoustic diffraction by an immersed and isolated thin infinite plate with a finite scatterer is studied. It is shown that, to provide the uniqueness of the solution, the conditions at the scatterer must lead to an energy inequality for a source-free field, which determines the absence of the energy-carrying field components at infinity. A formula that generalizes the Sommerfeld formula is obtained and is used to prove the uniqueness of the solution to the problem of diffraction by a plate immersed in an acoustic medium. For the problem of diffraction of a flexural wave by an irregularity of the plate, the uniqueness theorem is proved only for the case of a fixed or hinged edge. When boundary conditions of a general form are imposed on the scatterer in an isolated plate, the uniqueness of the solution is generally lost, which is also corroborated by an example.

On transient oscillations of plates in moving fluids

In recent years, various groups of researchers have looked at the two-dimensional motions of an undamped infinite thin elastic plate lying under a uniformly moving incompressible inviscid fluid. The plate is driven, usually by a single frequency time-harmonic line-source switched on at a finite time. The system’s behaviour is interesting as it can be shown to be absolutely unstable for flow velocities above a critical value, and below this the long-time solution is convectively unstable (downstream of the source) for a sufficiently low forcing frequency. These results do not appear particularly plausible from a physical point of view, and there is some question regarding the realisation of long-time steady behaviour, and so this article attempts to examine ways in which the model problem can be improved. In particular, the effects of introducing plate thickness and fluid compressibility to the model are studied. This is carried out by comparing the morphology of the original and modified solutions in the complex wavenumber space. It is found that, in the limit of small fluid-to-plate density ratio, the two problems exhibit qualitatively identical behaviour. However, the addition of structural damping is shown herein to lead to a very different solution – the initial boundary value problem is absolutely unstable at all flow velocities. Various other modifications to the original model, including finiteness of the plate, three-dimensional effects and nonlinearity, are discussed and their impact on the long-time response of the system is assessed.

Thermoelastic bending of orthotropic plates acted on by concentrated heat sources with arbitrary heat transfer

The thermoelastic problem is solved for an infinite thin orthotropic plate acted on by internal heat sources. Arbitrary heat transfer is assumed between the plate and its surroundings. Results from a numerical study are presented.