Classical Kirchhoff Theory Research Articles

Asymptotic Study of Plate Bending for a Strongly Orthotropic Materia

The technique of asymptotic averaging has been developed for three-dimensional partial differential equations with rapidly oscillating coefficients. For example, for the equations of elasticity theory. Then it was modified and applied to thin bodies in the form of plates (homogeneous or inhomogeneous, with even front surfaces or not), described by the three-dimensional theory of elasticity. In these cases, asymptotic solutions were constructed with respect to one small parameter, which is usually the ratio of the plate thickness to the characteristic plan size. The averaging technique in this case also reduces the dimension of the problem, i.e. reduces the three-dimensional boundary value problem to some two-dimensional one. In this article, we substantiate the application of the method to a problem with two small parameters in the case of a homogeneous thin, highly orthotropic plate bent by a surface load without taking into account body forces. The second small parameter is the ratio of the transverse elastic moduli to the moduli in the plan of the plate. It is shown that strong orthotropy is equivalent to an increase in the thickness of the equivalent plate. A procedure for obtaining the stress distribution over the plate thickness is described for three approximations. The first approximation gives the classical Kirchhoff theory, also called the Kirchhoff-Love theory, and the third approximation coincides with the Ambartsumian theory and allows one to find transverse shear and normal stresses. The consideration of cylindrical bending makes it possible to find solutions in the framework of classical plate theories in the form of formulas, as well as three approximations of the asymptotic theory, which simplifies comparison. Examples are considered when the averaged orthotropic moduli are taken for a single-layer fibrous composite.

Quasi-Static Solution of a Problem of Thermal Shock Acting on a Plate

In most research works related to stress analysis of thin structures subjected to temperature change it is assumed that variability of temperature in time is limited. In classical theory of thermoelastic thin structures the case of strong dynamical effects is excluded from consideration. Under this assumption, from three-dimensional equations of thermoelasticity the equations of thermoelastic plates and shells of Kirchhoff-type are derived. In this paper we focus on the analysis of stress-strain state of a plate in strongly dynamical temperature field, for which the classical Kirchhoff theory is not suitable. The analysis is carried out by a mathematical method without making any hypotheses. A numerical example is considered.

THE REFINED THEORY OF ELASTIC PLATES AS ASYMPTOTIC APPROACH OF 3D PROBLEM

The refined theory of elastic thin and thick plates is constructed by the asymptotic method for reducing three-dimensional (3D) equations of linear elasticity to two-dimensional ones without the use of any assumptions. The resulting refined theory is much more complicated than the known classical Kirchhoff theory: the required values of the refined theory vary in thickness of the plate by more complex laws; the system of partial differential equations of the refined theory has a higher order than the system of equations of the classical theory. A comparison of the obtained theory with the popular refined theory of Timoshenko and E. Reissner, taking into account the transverse shear deformation is made. It is shown that the inclusion only of the transverse shear deformation is insufficient. In addition to the transverse shear deformation, many additional terms having the same order as the transverse shear deformation must be taken into account.

A modified Kirchhoff plate theory for analyzing thermo-mechanical static and buckling responses of functionally graded material plates

In this paper, we present a modified Kirchhoff theory for analyzing thermo-mechanical static and buckling responses of functionally graded material (FGM) isotropic and sandwich plates. In comparison to the classical Kirchhoff theory, the proposed method is taken account into the shear deformation effects, then it can be applied for both moderately thick and thin plates. On the other hand, comparing to the first order shear deformation theory (FSDT) or Reissner-Mindlin plate theory, the number of independent unknowns reduces one variable and retains four degrees of freedom (Dofs) per node. The shear locking phenomenon is also negligible due to extending from the classical Kirchhoff theory. Furthermore, a new normalization of a quartic form is used to build the shape functions in the radial point interpolation method (RPIM). The novel imposing essential boundary conditions related to slopes of deflections has been proposed, then the enforcement of these gradient deflections becomes easily and directly as the same in finite element analysis. The obtained numerical results for the thermo-mechanical static and buckling problems from the proposed method are stable and well accurate prediction as comparing to the exact solutions and other published analyses.

The multidimensional stationary-phase method for an asymptotic estimate of edge effects in the multiple Kirchhoff diffraction by curved surfaces

An analytical approach is proposed to study the contribution of edge effects in the multiple high-frequency diffraction, according to guidelines of classical Kirchhoff theory in (scalar) wave propagation. We start from a suitable asymptotic analysis of the Kirchhoff diffraction integral, here set up in a generalized (iterated) form to describe the multiple reflections from an arbitrary sequence of curved reflecting (smooth) surfaces. The explicit formula obtained for a concrete example of double reflection is compared with the results from direct numerical simulation.

A model for sound propagation between two adsorbing microporous plates.

A model describing the sound propagation between two infinite adsorbing plates is proposed in order to investigate the extension to the audible sound range of the Frequency Response method applied to the measurement of diffusion in micropores. The model relates adsorption parameters (i.e., diffusivity and equilibrium constant) to an acoustic quantity (i.e., propagation constant). The equations describing sound propagation in the presence of adsorbing boundaries are obtained on the basis of the classical Kirchhoff theory [(1868). Ann. Phys. (Leipzig) 134, 177-193]. The solution is derived using the Low Reduced Frequency Approximation method [Tijdeman, (1975). J. Sound Vib. 39, 1-33].

Ultimate Carrying Capacity of Elastic-Plastic Plates on a Pasternak Foundation

In the present work, the problem of an infinite elastic perfectly plastic plate under axisymmetrical loading conditions resting on a bilateral Pasternak elastic foundation is considered. The plate is assumed thin, thus making it possible to neglect the shear deformation according to the classical Kirchhoff theory. Yielding is governed by the Johansen's yield criterion with associative flow rule. A uniformly distributed load is applied on a circular area on the top of the plate. As the load is increased, a circular elastic-plastic region spreads out starting from the center of the loaded area, whereas the outer unbounded region behaves elastically. Depending on the size of the loaded area, a further increase of the load may originate two or three different elastic-plastic regions, corresponding to different yield loci. A closed form solution of the governing equations for each region is found for a special value of the ratio between Pasternak soil moduli. The performed analysis allows us to estimate the elastic-plastic behavior of the plate up to the onset of collapse, here defined by the formation of a plastic mechanism within the plate. The corresponding collapse load and the sizes of the elastic-plastic regions are thus found by imposing the boundary and continuity conditions between the different regions. The influence of the soil moduli, plate bending stiffness, and size of the loaded area on the ultimate bearing capacity of the plate is then investigated in detail.

Flexural vibrations of clamped-free rhombic plates with corner stress singularities. Part II: comparison of solutions

An ample review of the published literature of skew (rhombic) plate vibrations in a companion Part I paper serves as a background that motivates the need for accurate solutions incorporating stress singularity-based methodologies for analyzing the titled problem. Such an accurate method is presented in this Part II paper. The prime focus here is that the vibration analysis explicitly considers the bending stress singularities that occur in the two opposite, clamped-free corners with obtuse angles of the rhombic plates. The strength of these singularities increases significantly, as the obtuse angles at the clamped-free corners exceeds 95o. A single-field energy-based Ritz procedure is employed with the dynamical energies derived from classical Kirchhoff thin-plate theory. The normal displacement of the rhombic plate is approximated as a hybrid series of (i) admissible and mathematically complete algebraic polynomials, and (ii) corner functions which account for both the kinematic boundary conditions and the bending stress singularities at the obtuse clamped-free corners. It is surmised from extensive convergence studies that the corner functions accelerate the convergence of upper bounds on the exact solutions, and that these functions are required if accurate solutions are to be obtained for highly skewed plates (including very thin ones incorporating shear deformable plate theories). Accurate non-dimensional frequencies and normalized contours of the vibratory transverse displacement are presented for rhombic plates having a large enough skew angle of 750 (i.e., obtuse angle of 1650), so that a significant influence of clamped-free corner stress singularities may be understood. Given the double symmetry axis of the rhombus domain, accurate solutions for a number of isosceles and right triangular plates (depending on the symmetry axis invoked) with various combinations of clamped, free, and sliding edges are also available from the frequency and mode shape data presented. Frequency data obtained from the present analysis are compared with nearly six decades of previously published data obtained using alternative theoretical plate analyses (including shear deformable ones) and classic experimental bench tests, clearly answering the inquiry of whether the stress singularities at the obtuse clamped-free corners of rhombic plates must be explicitly taken into consideration to obtain accurate three to four significant digit upper bounds on exact clamped-free skew (rhombic) plate vibration solutions reported in this work.

Modeling of consistent second‐order plate theories for anisotropic materials

AbstractBy Fourier‐series expansion in thickness direction of the plate with respect to a basis of scaled Legendre polynomials, several equivalent (and therefore exact) two‐dimensional formulations of the three‐dimensional boundary‐value problem of linear elasticity in weak formulation for a plate with constant thickness are derived. These formulations are sets of countably many PDEs, which are power series in the squared plate parameter. For the special case of a homogeneous monoclinic material, we obtain an approximative plate theory in finitely many PDEs and unknown variables by the truncation approach of the uniform‐approximation technique. The PDE system is reduced to a scalar PDE expressed in the mid‐plane displacement. The resulting second‐order theory, considered as a first‐order theory, is equivalent to the classical Kirchhoff theory for the special case of an isotropic material and equivalent to Huber's classical theory for an actual monoclinic material. However it remains shear‐rigid as a second‐order theory. Therefore, it is modified by an a‐priori assumption to a theory for monoclinic materials, that presumes the former equivalences, considered as a first‐order theory, but is in addition equivalent to Kienzler's theory as a second‐order theory for the special case of isotropy, which implies further equivalences to established shear‐deformable theories, especially the Reissner‐Mindlin theory and Zhilin's plate theory. The presented new second‐order plate theory for monoclinic materials is finally a system of two coupled PDEs of differentiation order six in two variables.

Analysis of the secondary anisotropic creep of layered metal-composite plates with account of their weakened resistance to the transverse shear 2. Model of deformation

On the basis of the second variant of Timoshenko theory, the problem of deformation of layered metal-composite plates of regular structure is formulated with account of their weakened resistance to the transverse shear under the conditions of secondary creep. The mechanical behavior of annular plates is investigated. It is shown that the application of the classical Kirchhoff theory to the calculation of the secondary creep of relatively thick layered plates in bending leads to the prediction of an unjustifiably understated compliance of thin-walled metal-composite designs. The calculations performed showed the existence of significant rates of transverse shear strains in such plates, whose account is obligatory and which are the main reason for the accumulation of creep strains.

Localized bending vibrations of piezoelectric plates

This paper investigates the existence and propagation of electro-elastic bending waves localized at the free edge of a piezoelectric plate. The problem is considered within the framework of the high-order refined plate theory introduced by Ambartsumian. The condition for existence of a localized bending wave is obtained, and the dispersion equation solved with respect to a dimensionless frequency. It is shown that the piezoelectric effect can increase the attenuation coefficient for a localized wave by up to 70% compared with that for a purely elastic plate, thus significantly decreasing the depth of penetration. The problem is also solved within the classical Kirchhoff theory. A comparison of results is carried out between two theories.

Optimization of a functionally graded circular plate with inner rigid thin obstacles. I. Continuous problems

AbstractOptimal control problems are considered for a functionally graded circular plate with inner rigid obstacles. Axisymmetric bending and stretching of the plate is studied using the classical Kirchhoff theory. The plate material is assumed to vary according to a power‐law distribution in terms of the volume fractions of the constituents. Four optimal design problems are considered for the elastic circular plate. The state problem is represented by a variational inequality with a monotone operator and the design variables (i.e., the thickness and the exponent of the power‐law) influence both the coefficients and the set of admissible state functions. We prove the existence of a solution to the above‐mentioned optimal design problems.

Flexural waves in structured elastic plates: Mindlin versus bi-harmonic models

The paper presents an analytical approach to modelling of Bloch–Floquet waves in structured Mindlin plates. The emphasis is given to a comparative analysis of two simplified plate models: the classical Kirchhoff theory and the Mindlin theory for dynamic response of periodic structures. It is shown that in the case of a doubly periodic array of cavities with clamped boundaries, the structure develops a low-frequency band gap in its dispersion diagram. In the framework of the Kirchhoff model, this band gap persists, even when the radius of the cavities tends to zero. A clear difference is found between the predictions of Kirchhoff and Mindlin theories. In Mindlin theory, the lowest band goes down toω= 0 as the radius of the cavities tends to zero, which is linked with the contrasting behaviour of the corresponding Green functions.

Symmetry Reduced Dynamics of Charged Molecular Strands

The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonlocal when the screened Coulomb interactions, or Lennard–Jones potentials between pairs of charges, are included. The nonlocal dynamics is derived in the convective representation of continuum motion by using modified Euler–Poincaré and Hamilton–Pontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry group of the system. This approach uses the process of affine Euler–Poincaré reduction initially developed for complex fluids. On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant formulation in terms of Lagrange–Poincaré equations of field theory. In another revealing perspective, the convective representation of the nonlocal equations of molecular strand motion is transformed into quaternionic form.

An Exact Derivation from Three-Dimensions of the Linear Equations for the Bending of an Elastically Monoclinic Plate and Associated Three-Dimensional Solutions

The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are subject to a special set of body and surface loads that leave the analogous plate loads arbitrary.

Diffraction of stochastic electromagnetic fields by a hole in a thin film with real optical properties

The classical Kirchhoff theory of diffraction is extended to the case of real optical properties of a screen and its finite thickness. A spectral power density of diffracted electromagnetic fields by a hole in a thin film with real optical properties was calculated. The problem was solved by use of the vector Green theorems and related Green function of the boundary value problem. A spectral and spatial selectivity of the considered system was demonstrated. Diffracted patterns were calculated for the coherent and incoherent incident fields in case of holes array in a screen of perfect conductivity.

Statics and dynamics of simply supported polygonal Reissner-Mindlin plates by analogy

Free and forced vibrations of moderately thick, transversely isotropic plates loaded by lateral forces and hydrostatic (isotropic) in-plane forces are analyzed in the frequency domain. Influences of shear, rotatory inertia, transverse normal stress and of a two-parameter Pasternak foundation are taken into account. First-order shear-deformation theories of the Reissner–Mindlin type are considered. These theories are written in a unifying manner using tracers to account for the various influencing parameters. In the case of a general polygonal shape of the plate and hard-hinged support conditions, the Reissner-Mindlin deflections are shown to coincide with the results of the classical Kirchhoff theory of thin plates. The background Kirchhoff plate, which has effective (frequency-dependent) stiffness and mass, is loaded by effective lateral and in-plane forces and by imposed fictitious “thermal” curvatures. These deflections are further split into deflections of linear elastic prestressed membranes with effective stiffness, mass and load. This analogy for the deflections is confirmed by utilizing D'Alembert's dynamic principle in the formulation of Lagrange, which yields an integral equation. Furthermore, the analogy is extended in order to include shear forces and bending moments. It is shown that in the static case, with no in-plane prestress taken into account, the stress resultants for certain groups of Reissner-type shear-deformable plates are identical with those resulting from the Kirchhoff theory of the background. Finally, results taken from the literature for simply supported rectangular and polygonal Mindlin plates are yielded and verified by analogy in a quick and simple manner.

Analytical solution of spatial elastica and its application to kinking problem

An analytical solution is presented for the spatially large deformation of a thin elastic rod (spatial elastica) which is naturally straight and uniform with equal principal stiffnesses and is subjected to terminal loads. The elastica can suffer not only flexure and torsion as in the classical Kirchhoff theory, but also extension and shear. The present solution is expressed in integral form and described in terms of only four parameters. This solution clears the difficulty with the polar singularity in the use of Euler angles. Hence, the numerical analysis is possible for various boundary value problems with no limitation.In this paper we study the post-buckling behavior of an elastica under the terminal twist and uniaxial end-shortening, and give a theoretical explanation to commonly observed phenomena such as secondary bifurcation, formation of a kink, snap-through behavior. The contact problem is analyzed in the case where the elastica contacts with itself and forms a kink. These results are available for other analysis, e.g., based on finite element approximations.

Plate bending problems using the nonsingular boundary element formulation and C1-continuous elements

The numerical implementation of an advanced boundary element formulation for the solution of plate bending problems within the classical Kirchhoff theory is presented. A direct and nonsingular formulation is used which is applicable to planar plates of arbitrary shape with arbitrary boundary conditions and subjected to static or harmonically varying loads. Owing to the nonsingular formulation, the regular integration quadratures give sufficiently accurate results, but the use of C 1-continuous elements appears to be necessary. The Overhauser elements are employed on smooth parts of the boundary contour, while ‘corner’ elements are used to represent discontinuities (such as corners) in geometry. Several numerical solutions are presented and compared to existing exact solutions.

Comparison between shear deformable and Kirchhoff theories for bending, buckling and vibration of antisymmetric angle-ply laminated plates

A generalized Lévy-type solution in conjunction with the state space concept is developed for the bending, buckling and vibration of antisymmetric angle-ply laminated plates. The exact solutions are applicable to rectangular plates with two opposite edges simply supported and the remaining ones subjected to a combination of clamped, simply supported and free boundary conditions. The solutions are obtained for the classical Kirchhoff theory and the numerical results are compared with their counterparts using the first order transverse shear deformation theory. The comparisons show that the results obtained within the classical laminated theory can be significantly inaccurate.