Isotropic Elastic Plate Research Articles

ANALYTICAL DETERMINATION OF STRESS INTENSITY FACTOR FOR PLATE BENDING PROBLEMS

International Journal of Computational Engineering ScienceVol. 04, No. 01, pp. 109-119 (2003) No AccessANALYTICAL DETERMINATION OF STRESS INTENSITY FACTOR FOR PLATE BENDING PROBLEMSLALITHA CHATTOPADHYAYLALITHA CHATTOPADHYAYAdvanced Research Group, Structures Division, National Aerospace Laboratories, P.O. Box 1779, Kodhihalli, Bangalore 560 017, India Search for more papers by this author https://doi.org/10.1142/S1465876303000776Cited by:3 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractThe problem of estimating the bending stress distribution in the neighborhood of a crack located on a single line in an isotropic elastic plate of constant thickness is examined. The plate containing cracks is infinite and its boundary is subjected to bending moment or twisting moment. Using classical plate theory and integral transform techniques, the general formulae for the bending moment, twisting moment and bending stress distribution in an elastic plate acted upon by moment forces and containing cracks located on a single line are derived. The solution is obtained in detail, for the case in which there is a single crack in an infinite plate and the bending stress intensity factor is determined in a closed form and the results are compared with those obtained from finite element analysis. Close agreements between the analytical solution, numerical solution and reference solution were found in all cases.Keywords:Stress intensity factorplatecrackbendingintegral transform References M. L. Williams, ASME J. Appl. Mech. 28, 78 (1961). Crossref, Google ScholarG. C. Sih, P. C. Paris and F. Erdogan, ASME J. Appl. Mech. 29, 306 (1962). Crossref, Google ScholarJ. K. Knowles and N. M. Wang, J. Math. Phys. 39, 223 (1960). Crossref, Google ScholarR. S. Alwar and K. N. Ramachandran, Eng. Fracture Mech. 17, 323 (1983), DOI: 10.1016/0013-7944(83)90083-8. Crossref, Google ScholarE. Reissner, Quart. Appl. Math. 5, 55 (1947). Crossref, Google ScholarM. J. Vizet al., Int. J. Fracture 72, 21 (1995). Crossref, Google ScholarA. Zucchini, C.-Y. Hui and A. T. Zehnder, Int. J. Fracture 104, 387 (2000), DOI: 10.1023/A:1007699314793. Crossref, Google Scholar S. P. Timoshenko , Theory of Plates and Shells ( McGraw Hill Book Company , Auckland , 1959 ) . Google Scholar I. N. Sneddon , Use of Integral Transforms ( McGraw Hill Book Company , New York , 1972 ) . Google ScholarE. F. Rybicki and M. F. Kanninen, Eng. Fracture Mech. 9, 931 (1977), DOI: 10.1016/0013-7944(77)90013-3. Crossref, Google Scholar Y. Murakami , Stress Intensity Factors Handbook 2 ( Pergamon Press , Oxford , 1987 ) . Google Scholar FiguresReferencesRelatedDetailsCited By 3Two-Dimensional Crack and Contact Problems – Transform MethodArabinda Roy and Rasajit Kumar Bera6 Dec 2019EXACT SOLUTION FOR BENDING OF AN ELASTIC PLATE CONTAINING A CRACK AND SUBJECTED TO A CONCENTRATED MOMENTLALITHA CHATTOPADHYAY, S. SRIDHARA MURTHY and S. VISWANATH20 November 2011 | International Journal of Computational Methods, Vol. 04, No. 02Analytical Solution for an Orthotropic Elastic Plate Containing CracksLalitha Chattopadhyay1 Aug 2005 | International Journal of Fracture, Vol. 134, No. 3-4 Recommended Vol. 04, No. 01 Metrics History Received 28 August 2002 Accepted 24 January 2003 KeywordsStress intensity factorplatecrackbendingintegral transformPDF download

Plate Bending Problem for a Finite Doubly Connected Domain with a Partially Unknown Boundary

The problem of bending of an isotropic elastic plate is solved for a finite doubly connected domain whose external boundary is a convex polygon and internal boundary is a smooth closed contour. The bending problem is reduced to the analytical solution of the Riemann—Hilbert problem for a ring. The deflection of the median surface and the shape of the plate's internal boundary are found

Selected topics in the history of the two-dimensional biharmonic problem

This review article gives a historical overview of some topics related to the classical 2D biharmonic problem. This problem arises in many physical studies concerning bending of clamped thin elastic isotropic plates, equilibrium of an elastic body under conditions of plane strain or plane stress, or creeping flow of a viscous incompressible fluid. The object of this paper is both to elucidate some interesting points related to the history of the problem and to give an overview of some analytical approaches to its solution. This review article contains 758 references.

Analysis of the dispersion relation for an incompressible transversely isotropic elastic plate

The dispersion relation associated with harmonic wave propagation in an incompressible, transversely isotropic elastic plate is derived. Such a material is characterized by only three material constants, contrasting with five in the corresponding compressible case. Motivated by a numerical investigation, asymptotic expansions, giving phase speed and frequency as functions of wave number, are derived in both the long and short wave regimes. These approximations, which owing to the constitutive simplifications are readily available, are shown to provide excellent agreement with the corresponding numerical solution. It is envisaged that the detailed investigation carried out in this paper will aid numerical inversion of the transform solutions often used in impact problems. Additionally, the asymptotic investigation provides the necessary basis for future studies to derive asymptotically approximate models to describe long and short wave motion.

The Exact Theory of Plate Bending

A new approach is introduced for the analysis and calculation of homogeneous, isotropic elastic plates of constant thickness under arbitrary bending loads. This theory can be called “exact” because it leads to exact values of the generalized 2D quantities. Moreover, contrary to classical plate theories, it is not limited to relatively thin plates.

A problem of the bending of a plate for a doubly-connected domain bounded by polygons

The problem of the bending of an isotropic elastic plate, bounded by two convex polygons is considered. It is assumed that the internal boundary of the plate is simply supported and normal bending moments act on each section of the external contour in such a way that the angle of rotation of the middle surface of the plate is a piecewise-constant function. With respect to the complex potentials, which express the bendings of the middle surface (Goursat's formula), the problem is reduced to a Riemann-Hilbert boundary-value problem for a circular ring, the solution of which is constructed in analytic form. Estimates are given of the behaviour of these potentials in the neighbourhood of the corner points.

Use of the two channels method to determine the guided waves damping coefficients of an asymmetrically fluid-loaded plate: Theory and experiment

Exact and approximate formalisms describing the interactions of acoustic plane waves with an elastic isotropic plate immersed between two different fluids are presented. This is an extention of the Fiorito, Madigosky, and Berall (FMU) theory. Resonant approximations are given for the reflection and transmission coefficients, which assume light fluids loading. These approximations show that the resonance widths are the sum of two independent partial widths, each of them being related to one fluid and to the plate physical properties. In addition, experimental results are presented for a steel plate; the damping coefficients of the propagating leaky Lamb waves along the loaded plate are measured in two different loading situations. In the first situation the plate is embedded in water (water loading on both sides), while in the second situation it is loaded with water on one side and acetone on the other side. The experimental damping coefficient of the acetone-steel-acetone structure is deduced therefrom. The damping coefficient obtained in each case is then compared to the theoretical half resonance width, and the experimental results are found to be in good agreement with the theoretical predictions.

On the Equivalence of Dispersion Relations Resulting from Rayleigh-Lamb Frequency Equation and the Operator Plate Model

The Rayleigh-Lamb frequency equations for the free vibrations of an infinite isotropic elastic plate are expanded into the infinite power series and reduced to the polynomial frequency and velocity dispersion relations. The latter are compared to those of the operator plate model developed in [Losin, N. A., 1997, “Asymptotics of Flexural Waves in Isotropic Elastic Plates,” ASME J. Appl. Mech., 64, No. 2, pp. 336–342; Losin, N. A., 1998, “Asymptotics of Extensional Waves in Isotropic Elastic Plates,” ASME J. Appl. Mech., 65, No. 4, pp. 1042–1047] for both symmetric and antisymmetric vibrations. As a result of comparative analysis, the equivalence of the corresponding dispersion polynomials is established. The frequency spectra, generated by Rayleigh-Lamb equations, are illustrated graphically and briefly discussed with reference to those published in [Potter, D. S., and Leedham, C. D., 1967, “Normalized Numerical Solution for Rayleigh’s Frequency Equation,” J. Acoust. Soc. Am., 41, No. 1, pp. 148–153].

𝕊-matrix theory applied to acoustic scattering by asymmetrically fluid-loaded elastic isotropic plates

Exact and approximate formalisms describing the interactions of acoustic plane waves with an elastic isotropic plate immersed between two different fluids (asymmetrically fluid-loaded plate) are presented. This constitutes an extension of the Fiorito, Madigosky, and Überall (FMU) [R. Fiorito et al., J. Acoust. Soc. Am. 66, 181 (1979)] theory, refined later by Freedman [A. Freedman, J. Sound Vib. 82, 181 (1982); ibid. 82, 197 (1982)]. The method, based upon the multichannel resonant scattering theory derived from quantum physics [A. Bohm, Quantum Mechanics, Foundations, and Applications (Springer, New York, 1993)], consists of two parts. First, a 2×2 scattering 𝕊-matrix in which the diagonal elements are the two reflection coefficients and the off-diagonal elements are the two transmission coefficients, is built. Second, in order to compare our results with the FMU theory, resonant approximations are given for these coefficients, assuming light fluids when compared to the plate. The approximated coefficients show that the resonance widths are the sum of two independent partial widths, each of them being related to one fluid and to the plate physical properties. Of importance in this extension is the fact that the eigenvalues of the matrix, which reveal all the resonant features of the immersed plate, allow the separation between antisymmetrical and symmetrical modes, contrary to the reflection and transmission coefficients. The eigenvalues also allow the analysis of resonances despite the overlapping phenomenon. For the computations, the exact coefficients and eigenvalues, rather than their approximate forms, are used. This allows us to check the validity of the exact part of the theory under any fluid loading and not especially for light fluid loading.

Control of transient thermoelastic displacement of a two-layered composite plate constructed of isotropic elastic and piezoelectric layers due to nonuniform heating

In this study, the theoretical analysis of the control of transient thermoelastic displacement is developed for a two-layered composite rectangular plate constructed of an isotropic elastic and a piezoelectric layer due to nonuniform heat supply. The transient three-dimensional temperature in a two-layered composite rectangular plate is analyzed by the methods of Laplace and finite cosine transformations. Exact solutions for isotropic elastic and piezoelectric plates of crystal class mm2 are used in the theoretical analysis. A three-dimensional transient piezothermoelastic solution is developed for a simple-supported combined plate. The analysis yields an appropriate electric potential which when applied to the piezoelectric plate, suppresses the induced thermoelastic displacement in the thickness direction at the midpoint on the free surface of the isotropic plate. As an example, numerical calculations are carried out for an isotropic rectangular plate made of steel, bonded to a piezoelectric plate of cadmium selenide. Some numerical results are shown for temperature change, displacement and stress in transient state, when the transient thermoelastic displacement are controlled.

Three-dimensional effects for through-thickness cylindrical inclusions in an elastic plate

A quasi-three-dimensional theoretical closed-form solution for a cylindrical inclusion embedded in an isotropic elastic plate of finite thickness is obtained using the Kane-Mindlin theory. The present solution gives an insight into the transition from plane stress to plane strain. It is shown that strong three-dimensional effects exist near the inclusion even in very thin plates. Moreover, it is shown that two-dimensional plane stress and plane strain solutions are not always the two limits for a three-dimensional problem. The effects of plate thickness and inclusion radius, material constants of the inclusion and the matrix, as well as interference fitting on the stress concentration factor are discussed. The three-dimensional affected zone around the inclusion can be determined from the present solution. The variation of the lateral contraction and shear stresses is discussed. The comparison between the finite element results and the present results shows good agreement.

Large amplitude free vibration of sandwich parabolic plates—Revisited

Two simplified methods were developed, one by Mazumdar [1] and another by Banerjee [2] for the study of large amplitude transverse vibration of thin homogeneous isotropic elastic plates. The present method is a modification of the above two approaches for the analysis of large amplitude transverse vibration of thin isotropic, homogeneous sandwich elastic plates of arbitrary shapes. As an illustration of the method, an interesting example of the vibration of a sandwich parabolic plate with an isotropic core is examined. Graphs corresponding to numerical results are drawn and compared with available results.

Effects of Pre-Stress on Impact Waves in an Incompressible Elastic Plate

We study the effects of pre-stress on the propagation of impact waves resulting from a sudden line load applied on the surface of an infinite isotropic incompressible elastic plate placed on a smooth rigid foundation. The pre-stress is assumed to be a uni-axial compression or an all-round pressure (tension). Our interest is in the behaviour of the impact waves when the pre-stress approaches any of the critical values for buckling instability. The problem is solved using Laplace and Fourier transforms. The Laplace inversion is carried out analytically and the Fourier inversion is evaluated both numerically and with the aid of the method of stationary phase.

On the theory of wave propagation in anisotropic plates

A theoretical framework is developed which describes wave propagation in infinite homogeneous elastic plates of unrestricted anisotropy. The approach exploits the propagator matrix which is the exponential of the fundamental elasticity matrix underlying Stroh9s formalism of anisotropic elastodynamics. The matrices of plate impedance and admittance are introduced, and their analytical properties are established, which appear fruitful for computing and analysing the plate wave spectra. On this basis, the dispersion equation can be cast into the form of a real equation involving the monotonic function, whose zeros and poles are the wave velocities in a given plate subjected to different boundary conditions (traction–free or clamped faces). It is proved that three fundamental wave branches exist for any orientation of wave propagation in an anisotropic plate with traction–free faces, and that those branches are missing if one or both faces are clamped. The intrinsic symmetry of the wave motion in an arbitrary plate is revealed. The general formalism is applied to elaborate the long–wavelength low–frequency approximation for a (thin) free plate of unrestricted anisotropy. The frequency–dispersive wave velocities, displacements and tractions at the onset of the fundamental wave branches are derived explicitly. The conditions for the extreme velocity values and some other useful universal connections are determined for an arbitrary thin plate, and exemplified for specific anisotropy cases.

Transfer and Green functions based on modal analysis for Lamb waves generation

This work presents an easy way to deduce the tensorial transfer and Green functions for Lamb waves generated in isotropic elastic plates. These functions could be applied to obtain the response of each propagating mode in the ensemble of excited modes arising from any sort of pulsed excitation (wedge transducers, lasers, etc.). The transfer function is based on modal analysis development. Not only is it easy to manipulate but also allows the avoidance of laborious calculations for each kind of Lamb waves source. Theoretical predictions are compared with those of Viktorov [I. A. Viktorov, Rayleigh and Lamb Waves (Plenum, New York, 1967)] and with experimental measurements of Lamb waves generated by the wedge-transducer method.

Stress singularity analysis in space junctions of thin plates

The stress singularity in space junctions of thin linearly elastic isotropic plate elements with zero bending rigidities is investigated. The problem for an intersection of infinite wedge-shaped elements is considered first and the solution for each element, being in the plane stress state, is represented in terms of holomorphic functions (Kolosov–Muskhelishvili complex potentials) in some weighted Hardy-type classes. After application of the Mellin transform with respect to radius, the problem is reduced to a system of linear algebraic equations. By use of the residue calculus during the inverse Mellin transform, the stress asymptotics at the wedge apex is obtained. Then the asymptotic representation is extended to intersections of finite plate elements. Some numerical results are presented for a dependence of stress singularity powers on the junction geometry and on membrane rigidities of plate elements.

Harmonic generation by Rayleigh–Lamb waves in isotropic elastic plates

Harmonic generation by Rayleigh–Lamb waves of finite amplitude in homogeneous, isotropic, stress-free elastic plates is investigated theoretically. A bifrequency primary wave field is considered, in which the two waves propagate in single yet arbitrary Rayleigh–Lamb modes. Solutions for the second-harmonic and difference-frequency components are obtained via modal decomposition and use of reciprocity relations. Two conditions are required for generation and resonant amplification of these spectral components, power flow, and phase matching. Second-harmonic generation is considered first. Although phase matching is possible between a primary wave in the first symmetric mode and its second harmonic in the second asymmetric mode, the power flow is found to be zero. However, with one primary wave in each of the first symmetric and asymmetric modes, both phase matching and power flow may be achieved with the difference-frequency wave generated in the first asymmetric mode. In particular, resonant parametric amplification of the difference-frequency wave can be achieved under conditions where dispersion prevents efficient coupling from occurring between the primary waves and other frequency components. Explicit analytic solutions are presented and discussed. [Work supported by the Brazilian Ministry of Science and Technology, and the National Science Foundation.]

Three Dimensional Axisymmetric Piezothermoelastic Solution of Clamped Circular Elastic Plate Bonded to Piezoceramic Plate

A three dimensional (3D) solution in terms of potential functions is presented for a finite clamped circular isotropic elastic plate bonded to a piezoceramic plate subjected to axisymmetric thermoelectromechanical load. The boundary and interface conditions are satisfied using Fourier-Bessel expansions yielding an infinite system of algebraic equations for the arbitrary constants. These are solved to any desired degree of accuracy by truncating to a finite set of equations. Results are presented to illustrate the effect of the thickness parameter. The present solution would help to ascertain the accuracy and range of validity of two dimensional (2D) theories of piezoelectric hybrid plates.

Asymptotics of Extensional Waves in Isotropic Elastic Plates

The long and short-wave asymptotics of order O(η6),η=kh, for free extensional vibrations of an infinite isotropic elastic plate are studied. The asymptotic model for flexural and extensional wave motion applicable for both long and short-wave approximations and for any materials is developed. The velocity and frequency dispersion relations for extensional waves are derived in analytical form from the system of three-dimensional dynamic equations of linear elasticity. All dispersion equations and the group velocity formula are presented as explicit functions in material parameter γ=cs2/cL2 (the ratio of the velocities squared of the flexural and extensional waves) without any correction factors as in the Reissner-Mindlin theory. Variations of the velocity and frequency spectra depending on Poisson’s ratio ν are illustrated graphically. The results are discussed and compared to those obtained and summarized by Mindlin (1960), Mindlin and Medick (1959), Tolstoy and Usdin (1953, 1957), Achenbach (1973), and Graff (1991).

La théorie exacte de la flexion des plaques

A new approach is introduced for the analysis and computation of homogeneous, isotropic elastic plates subjected to any bending loads. This approach is based upon a partitioning, which is independent of thickness, of the 3D reference solution into both a part with large variation length and localized effects, whose generalized stress-displacement values are zero. The part with large variation length can easily be computed from data as well as from generalized quantities that represent ‘averages’ in the thickness. These generalized quantities coincide with those obtained from the 3D reference solution; they constitute the solution to an ‘exact’ theory of plate bending.