Time-dependent boundary integral equations for multiply connected plates

The existence of distributional solutions is discussed for theinitial-boundary value problems associated with the motion ofa thin, elastic, multiply connected plate, and for the boundaryequations arising from integral representations of such solutions.     

The problem of the reinforcement of a plate with a cutout by a two-dimensional patch

The problem of the reinforcement of an elastic plate with a cutout by means of a two-dimensional patch, which completely covers the cutout and is rigidly fixed to the plate along its boundary, is considered. The cutout and the patch can be of arbitrary shape. The problem is reduced to a system of three singular integral equations using of special integral representations which describe the stress state in the plate and in the patch. The unique solvability of the system is proved. Examples are presented.     

Stress state near loaded holes in anisotropic plates

Special representations of the solution are constructed and solving integral equations of the problem of the elastic equilibrium of a finite anisotropic plate weakened by an elliptical hole or rectilinear crack are derived. The absence of the unknown function for the boundary of the internal hole (crack) makes it possible to propose an effective algorithm for the problem's numeric solution. The results of calculations, which illustrate the effect of the external boundary and material anisotropy on the stress distribution near loaded holes of different sizes, are presented. Direct comparison with the finite-element method indicates that the proposed algorithm significantly lowers the amount of input data, the computer time, and the required volume of memory with comparable accuracy.Novosibirsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 45–51, 1990.     

Quasi-three-dimensional theory for solution of dynamic thermoelastic problem of laminated composite plates

An uncoupled dynamic thermoelastic problem for laminated composite plates has been considered. The hypotheses used take into account the nonlinear distribution of temperature and displacements over the thickness of a laminated plate. On the basis of these hypotheses a quasi-three-dimensional (layerwise) theory is constructed that makes it possible to investigate the internal thermal and stress-strain states, as well as the edge effects of the boundary layer type for laminated plates. Systems of the heat conduction and motion equations are derived using the variational method. The order of the equations depends on the number of layers and terms in expansions of temperature and displacements of each layer. An analytical solution of the dynamic thermoelastic problem is presented for a cross-ply laminated rectangular plate with simply supported edges. The reliability of the results is confirmed by a comparison with the known exact solutions. The results based on the proposed theory can be used for verifying various two-dimensional plate theories when solving the dynamic thermoelastic problems for laminated composite plates.     

Elastoplastic stability of glass-reinforced plastic plates with a central metal layer

The problem of the stability of a three-layer plate with a central plastic layer of metal sandwiched between elastic glass-reinforced plastic outer layers is considered. The presence of a metal layer restrains the development of creep strains in the glass-reinforced plastic and makes it possible to neglect the viscous strain components. The general equations of the problem are obtained, and the approximate Il'yushin formulation [1] is considered. An example is presented for a rectangular plate in pure shear. It is shown that the elastic anisotropic layers play the part of a load-relieving system for the central plastic layer [3], which results in an increase in the over-all critical load for the layered plate.Kalinin Polytechnic Institute. Translated from Mekhanika Polimerov, No. 5, pp. 909–915, September–October, 1969.     

Integral equations of the first kind in the theory of oscillating plates

A modified matrix of fundamental solutions is used to derive and solve first-kind integral equations for the problem of high-frequency harmonic oscillations of an infinite elastic plate with a hole when Dirichlet or Neumann conditions are prescribed on the boundary curve.     

Boundary integral equations for bending of thermoelastic plates with transmission boundary conditions

The solution of an initial‐boundary value problem for bending of a piecewise‐homogeneous thermoelastic plate with transverse shear deformation is represented as various combinations of single‐layer and double‐layer time‐dependent potentials. The unique solvability of the boundary integral equations generated by these representations is proved in spaces of distributions. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamic stresses in a compound body with circular crack under sliding contact on an interface

We have considered the three-dimensional problem of harmonic loading of a circular crack in an elastic composite consisting of two dissimilar half-spaces under sliding contact on the surface of their bonding. The defect is situated in one of the half-spaces perpendicularly to the interface of materials. Using the representations of solutions in the form of Helmholtz potentials, we have reduced the problem to a boundary integral equation for the function of dynamic defect opening. Based on the numerical solution of this equation, we have obtained the frequency dependences of mode I stress intensity factor near the crack for different relations between the elastic moduli of components of the composite and the depths of crack location with respect to the interface.     

弹性半空间地基上四边自由矩形板稳态振动的解析解

采用双重Fourier变换,分析得到弹性半空间地基受竖向稳态荷载作用下的积分变换解．与四边自由矩形板的振动解析解相结合,得出弹性半空间地基上四边自由矩形板稳态振动的解析解．还给出算例及参数影响分析．     

Thin elastic and periodic plates

This paper is devoted to the study of a three dimensional model of elastic periodic plate when the thickness e of the plate and the size ω of the periods are small. In the three studied limits (e → 0 then ω → 0), (ω → 0 then e → 0) and lately (e and ω → 0 together) the three dimensional equation of elasticity are approached by the two dimensional general equations of a linear anisotropic plate, the stretching and bending being coupled. This study points out the importance of the ratio of the two small parameters, indeed the moduli occuring in the two dimensional equations are different in the three limits. In each case a convergence proof is carried out.     

The contact problem when there are friction forces in the contact area for a three-component cylindrical base

The plane contact problem of elasticity theory on the interaction when there are friction forces in the contact area of an absolutely rigid cylinder (punch) with an internal surface of a cylindrical base, consisting of two circular cylindrical layers rigidly connected to one another and with an elastic space, is considered. The layers and space have different elastic constants. A vertical force and a counterclockwise torque, act on the punch, and the punch – base system is in a state of limiting equilibrium,. An exact integral equation of the first kind with a kernel represented in an explicit analytical form, is obtained for the first time for this problem using analytical calculation programs. The main properties of the kernel of the integral equation are investigated, and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of polynomials in products of the powers of the moduli of the displacement of the layers and the half-space. A solution of the integral equation is constructed by the direct collocation method, which enables the solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the interconnection between the punch displacement and the forces and torques acting on it are calculated as a function of the geometrical and mechanical parameters of the layers and the space. The results of the calculations in special cases are compared with previously known results.     

Bending of plates made of oriented glass-reinforced plastics

A plate made of oriented glass-reinforced plastic (GRP) is treated as a regular medium with numerous anisotropic layers. The problem of bending is solved in the elastic formulation of the problem without the use of the hypothesis of undeformed normals and a preliminary specification of the law of distribution of shearing stresses in transverse planes. The important effect of shear strains on the deflections and stresses in plates of oriented GRP is illustrated in the example of a simply supported square plate with transverse isotropy. An estimate is given of the errors introduced by arbitrary specification of the law of distribution of shearing stresses.Mekhanika Polimerov, Vol. 1, No. 3, pp. 129–136, 1965     

A theory of dislocations and disclinations in elastic plates

The problem of the stress state of a thin elastic plate, containing dislocations and disclinations, is considered using Kirchhoff's theory. The problem of the equilibrium of a multiply connected plate with Volterra dislocations with specified characteristics is formulated. The problem of the flexure of an annular slab resulting from a screw dislocation and a twisting disclination is solved. The solutions of problems of concentrated (isolated) dislocations and disclinations in an unbounded plate as well as the dipoles of dislocations and disinclinations are found. It is shown that a screw dislocation in a thin plate is equivalent to the superposition of two orthogonal dipoles of torsional disclinations. By taking the limit from a discrete set of defects to their continuous distribution, a theory of thin plates with distributed dislocations and disclinations is constructed. Solutions of problems of the flexure of circular and elliptic plates with continuously distributed disclinations are obtained. An analogy is established between the problem of the flexure of a plate with defects and the plane problem of the theory of elasticity with mass forces, and also between a plane problem with dislocations and disclinations and the problem of the flexure of a plate with specified distributed loads.     

Boundary element bending analysis of heated elastic plates

Generally, the boundary integral equations for nonlinear and inhomogeneous problems need to be formulated in terms of boundary and domain integrals. The present paper declares the possibility of the transformation of the domain integral term into the boundary integral for the thermal bending analysis of a thin elastic plate,provided that the temperature can be expressed by a linear variation over the thickness.     

On the plane deformation of fibre-reinforced viscoelastic plates

The realization method of elastic solutions is used to solve the problem of bending of a viscoelastic plate reinforced unidirectionally by elastic fibres. Numerical computations are carried out for three kinds of external load. The plane deformation of this plate is discussed.     

The breaking of a non-homogeneous fiber embedded in an infinite non-homogeneous medium

The torsion of an infinite non-homogeneous elastic cylindrical fiber, containing a penny-shaped crack embedded in an infinite non-homogeneous elastic material is considered. The cylinder and elastic medium have different shear moduli. Using integral transformation techniques the solution of the problem is reduced to the solution of dual integral equations. Later on the solution of the dual integral equations is transformed into the solution of a Fredholm integral equation of the second kind, which is solved numerically. Closed form expressions are obtained for the stress intensity factor and numerical values for the stress intensity factors are graphed to demonstrate the effect of non-homogeneity of the fiber and infinite medium. In the end the stress singularity is obtained when the crack touches the infinite non-homogeneous medium (matrix).     

Damping of sound and vibration by flow nonlinearity in the apertures of a perforated elastic screen

An analysis is made of the influence of flow nonlinearity inthe apertures of a perforated elastic plate on the damping ofsound and flexural vibrations. Fluid is forced through the perforationsby the pressure differential established across the plate bythe incident disturbance. The Reynolds number is assumed tobe sufficiently large that separation occurs, and the reciprocatingaperture flows form ‘jets’ on alternate sides ofthe plate. The growth of these jets is modelled by means ofa nonlinear equation proposed by Cummings (1986). This equationis solved simultaneously with a generalized bending wave equationderived by the author (Howe, 1995a) which governs motions ofa perforated elastic plate whose lengths scales are large comparedto the aperture spacing. It is shown that significant attenuationsof large amplitude acoustic waves can occur except when thefrequency is so small that the plate is acoustically transparent.Bending waves are also damped provided the amplitude of theplate surface velocity is not too large and the frequency issmall enough to ensure the formation of substantial jets inthe apertures. Numerical results are given for large amplitudesound waves incident on a perforated screen in air, and forbending waves propagating over aluminium and steel screens immersedin either air or water.     

A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points

We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of these modes in the presence of multiple localized defects for a wide range of two-dimensional geometries. The defects result in eigenfunctions with a weak singularity that is resolved by decomposing the solution as a superposition of Green’s functions plus a smooth regular part. This method is applied to a variety of regular and irregular domains and two key phenomena are observed. First, careful placement of clamping points can entirely eliminate particular eigenvalues and suggests a strategy for manipulating the vibrational characteristics of rigid bodies so that undesirable frequencies are removed. Second, clamping of the plate can result in partitioning of the domain so that vibrational modes are largely confined to certain spatial regions. This numerical method gives a precision tool for tuning the vibrational characteristics of thin elastic plates.     

Asymptotic of the Solution of the Contact Problem for a Thin Elastic Plate and a Viscoelastic Layer

The contact problem for a thin elastic rigid plate described by the elasticity equations and a viscoelastic layer is solved. The ratio of the thicknesses of the plate and the layer is a small parameter, while the ratio of the Young’s moduli of the layer and the plate is proportional to the cube of this parameter. The asymptotic expansion of the solution is constructed. A theorem on the estimate of the error of asymptotic approximation is formulated. Such problem appears in geophysics, in modeling of the Earth crust–magma interaction.     

On the optimal design of plates for a given deflection

The optimum design problem of an elastic plate for a given deflection is considered. The design variable is chosen to be the thickness of the plate. Using the principle of stationary mutual potential energy first introduced by Shield and Prager, a sufficient optimality condition (which makes the volume stationary) is derived for plates under bending caused by general loading conditions. As an example, the optimum profile of a simply supported circular plate under a given rotationally symmetric loading is obtained.