Numerical–analytical model for transient dynamics of elastic-plastic plate under eccentric low-velocity impact

The aim of this study is to present an efficient model for the analysis of complicated nonlinear transient dynamics of an elastic-plastic plate subjected to a transversely eccentric low-velocity impact. A mixed numerical–analytical model is presented to predict the transient dynamic behaviours consisting of either plate impact responses or wave propagations induced by the impact in a plate with an arbitrary shape and support. This hybrid approach has been validated by comparison with results of laboratory tests performed on an elastic-perfectly plastic narrow plate eccentrically struck by an elastic sphere, and results of a three-dimensional finite element (FE) analysis for an elastic-perfectly plastic simply-supported rectangular plate eccentrically struck by an elastic sphere. The advantages of this hybrid approach are in the simplification of local contact force formulation, computational efficiency over the FE model, and convenient application to parametric study for eccentric impact behaviour. The hybrid approach can provide accurate predictions of the plate impact responses and plate wave propagations.     

Temperature stresses in an annular anisotropic plate with temperature-dependent heat transfer coefficient

The nonsteady temperature stresses are determined for an annular plate, an infinite plate with a circular opening, and a circular plate, whose coefficients of heat transfer from the lateral surfaces are functions of temperature.Physico-Mechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. Translated from Mekhanika Polimerov, No. 5, pp. 949–950, September–October, 1971.     

Concentrated force solutions for an inhomogeneous thick elastic plate

The technique developed in refs. [1-5] is applied to the problem of a concentrated line force acting in the interior of an infinite plate. The plate is of arbitrary thickness, is isotropic, but is inhomogeneous in that the elastic moduli are any specified functions, not necessarily continuous, of the through-thickness coordinate. The mechanical properties of the plate are not necessarily symmetric about the mid-surface. The solution is based on the classical solution for a concentrated force in a thin elastic plate. This classical solution is extended to give exact closed form solutions for the displacement and stress in the thick inhomogeneous plate. For a plate that is not symmetric an in-plane force gives rise to bending as well as stretching deformations. Higher order force singularities are also considered, as is the problem of a concentrated force on the boundary of a semi-infinite symmetric plate.     

Théorèmes d'existence pour un modèle membranaire « proprement invariantde plaque non linéairement élastique

We establish the “local” existence of an injective solution to the nonlinear, “properly invariant”, membrane plate model, stated in [1] and [2], successively for the clamped plate submitted to forces parallel to its plane and for the plate submitted to a boundary condition of place of “extended” state.     

Vibration of an elastic plate under action of an incompressible fluid in case of N = 0 approximation of I. Vekua's hierarchical models

This article deals with the study of the interaction between an elastic plate with constant thickness and an incompressible fluid when the plate is approximated by the N?=?0 order term of Vekua's hierarchical model.     

A Study on the Propagation of Plane Stress Waves across the Thickness of a Plate by the Method of Analytic Continuation in Time

The interaction of plane tension/compression waves propagating within a plate perpendicularly to its surface is considered. The analytic solution is obtained by a modified method of characteristics for the one-dimensional wave equation used in problems on an impact of a rigid body on the surface of a plate. The displacements, velocities, and stresses in the plate are determined by the edge disturbance caused by the initial velocity and the stationary force field of masses of the striker and the plate. The method of analytic continuation in time put forward allows a stress analysis for an arbitrary time interval by using finite expressions. Contrary to a stress analysis in the frequency domain, which is commonly used in harmonic expansion of disturbances, the approach advanced allows one to analyze the solution in the case of discontinuous first derivatives of displacements without calculating jumps in summing series. A generalized closed-form solution is obtained for stresses in an arbitrary cycle n(t), which is determined by the multiplicity of the time of wave travel across the double thickness of the plate. A method of recurrent solution based on calculating the convolution of repeated integrals of the initial form of disturbance at t = 0 is elaborated. The procedure can be used for evaluating the maximum stress and the contact time in a plane impact on the surface of a plate.     

The flow past a cylinder in a rotating frame. II. A flat plate

Summary In a rapidly rotating frame, the slow flow past a flat plate is examined by a Wiener-Hopf method and it is shown that the infinite velocity in the potential flow at the edges of the plate can be made finite by the action of viscosity in the Stewartson layer. It is shown that, for small angles of incidence, there are small eddies at the two edges of the plate, but that as the angle is decreased, they disappear, and the dividing streamline meets the plate at the two edges only.

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Zusammenfassung In einem schnell rotierenden Koordinatensystem wird die langsame Strömung an einer ebenen Platte durch das Wiener-Hopf Verfahren behandelt. Es wird gezeigt, dass die unendliche Geschwindigkeit in einer Potentialströmung an den Kanten der Platte durch die Wirkung der Zähigkeit innerhalb der Stewartson-Schichten endlich gemacht werden kann. Ferner sieht man, dass bei kleinen Anstellwinkeln kleine Wirbel an den zwei Kanten der Platte entstehen die aber verschwinden, sobald der Winkel gegen Null geht; in diesem Fall trifft die Trennungsstromlinie die Platte an den zwei Kanten.

     

Flexure of thin inhomogeneous isotropic plates

The problem of the flexure of an inhomogeneous plate with a hole may be reduced, by the small-parameter method, to the solution of a series of problems for a homogeneous plate with a hole. These problems are solved using classical complex-potential methods. For an infinite plate with a circular hole, the results of numerical calculations are given; from these results, it follows that inhomogeneity of the material has a considerable influence on the stress.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 43–49, 1987.     

Transient analysis of a potential pulse in a piezoelectric (622) crystal plate

Transient analysis of the wave propagation due to an electric potential pulse concentrated at a point on the boundary of an infinite piezoelectric plate of the crystal class (622) resting on an infinite elastic medium is considered. The electric potential is studied for an exponential pulse and step pulse. Numerical results are obtained for theβ-quartz plate that rests on aluminium half-space.     

Hydrodynamic analogy for direct shear force and bending moment evaluation in a plate

A brand new interpretation of the plate bending equations is given using hydrodynamic analogy. It permits one to determine directly the shear forces and bending moments of a plate without the need of finding deflections. In engineering design of a plate it is more important to know shear forces and bending moments than the deflections. The existing numerical methods of solution of plate problems consist in determining deflections; then shear forces and bending moments are obtained by differentiating the deflection three and two times which produces great loss of accuracy. The hydrodynamic analogy method has the advantage over other numerical methods because the shear forces and bending moments are obtained directly, without the need of finding deflections and because they are obtained with better accuracy. The hydrodynamic analogy can be applied to a plate of arbitrary shape, with arbitrary boundary conditions under an arbitrary loading.     

Free vibrations of a cantilevered plate with a polymer coating

The article considers some of the special characteristics of the free vibrations of a steel cantilevered plate, with the formation of a polymer coating on the plate. It compares the results of an experimental investigation of the effect of internal stresses in the coating on the vibrations and on the static bending of the plate. It demonstrates the possibility of determining the internal stresses in the coating from the change in the frequency of the natural vibrations of a plate with a coating.Institute for the Mechanics of Metal-Polymer Systems, Academy of Sciences of the Belorussian SSR, Gomel'. Translated from Mehanika Polimerov, No. 3, pp. 541–544, May–June, 1970.     

Hydromagnetic flow near an accelerated plate with suction

The two-dimensional unsteady flow of a conducting viscous incompressible fluid past, an infinite flat plate with uniform suction, is considered in the presence of a uniform magnetic field. For a constant time, it is shown that for a given Hartmann numbera, as the cross Reynolds number β (corresponding to the suction velocity of the plate) increases, the velocity at any point of the fluid decreases and the skin friction at the plate increases. The results also hold good for a given β, asa increases if the magnetic lines of force are fixed relative to the fluid and are just opposite for the magnetic lines of force fixed relative to the plate.     

Deformation of a composite plate on an elastic foundation by local loads

Bending of an elastic annular composite plate with a light filler lying on an elastic foundation is considered. The plate is subjected to local loads. To describe the kinematics of the package, asymmetric across its thickness, the hypotheses of broken normal is accepted. The reaction of foundation is described based on the Winkler model. A system of equilibrium equations is constructed, and its exact solution in displacements is found. Numerical solutions for a metal-polymer sandwich plate are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 109–120, January–February, 2007.     

Stress and fracture analysis in delaminated orthotropic composite plates using third-order shear deformation theory

The third-order shear deformable plate theory is applied in this work to calculate the stresses and energy release rates in delaminated orthotropic composite plates with straight crack front. The delaminated parts are modeled by the general third-order plate theory, while a double-plate model with interface constraint is developed for the uncracked portion of the plate. The governing equations of the uncracked part are formulated by considering the equilibrium and the displacement continuity along the interface. As an example, a simply-supported delaminated orthotropic plate subjected to a point force is solved adopting Lévy plate formulation and the state-space approach. The mode-II and mode-III energy release rate distributions along the crack front were calculated by the J-integral. To verify the analytical results the 3D finite element model of the plate was constructed and the energy release rates were calculated by the virtual crack-closure technique. A previous second-order plate theory solution was also utilized in the course of the comparison. The results indicate a good agreement between analysis and numerical computation and that third-order theory is better in some cases than the second-order approximation.     

Plate models with state-dependent damping coefficient and their quasi-static limits

We study long-time dynamics of a class of plate models with a state-dependent damping coefficient and their quasi-static limits. We first present the problem in abstract form and then prove the existence of finite-dimensional global attractors and their upper semicontinuity in the quasi-static limit, i.e., in the case when the mass density of plate tends to zero. Our proofs involve a recently developed method based on “compensated” compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, von Karman and Berger plate models with different types of boundary conditions and damping coefficients. Our results can be also applied to the nonlinear wave equations in an arbitrary dimension.     

Scattering of S0 Lamb mode in plate with multiple damage

A study of scattering properties of S₀ mode Lamb wave in an infinite plate with multiple damage is presented. Plate theory and wave function expansion method are used to derive the analytical solutions for the scattering wave field in plate with a single damage, and by using the addition theorems of Bessel functions, interference phenomena between scattering wave fields from different damage is investigated. Measurements agree well between theoretical results and FE simulation study of plate with two damage and validity of the model is confirmed. Numerical results of scattering displacement field in plate with two and three damage are graphically presented and discussed. An assessment of effects of damage geometric properties on the scattering properties is made.     

Wave propagation and band gaps in homogenized phononic plates — modelling by spectral decomposition

The paper deals with analysis of the elastic waves in the Reissner-Mindlin type of plates formed by strongly heterogeneous structures. The homogenized plate model involves frequency-dependent mass coefficients associated with the plate cross-section rotations and the plate deflections. Intervals of frequencies called band gaps exist for which these coefficients constituting a mass matrix can be negative, so that certain wave modes cannot propagate. A spectral decomposition based method is proposed which is suitable to compute the plate response for an external loading by periodic forces with frequencies in range of the band gaps. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The construction of exact solutions in the floating-plate problem

It is shown that it is possible to construct, by an inverse method, exact solutions of the problem of the flexural-gravitational oscillations of a floating elastic plate. The results obtained are used to check the accuracy of numerical solutions of the problem. It is shown that the numerical algorithm given in Ref. [Khabakhpasheva TI. The plane problem of an elastic floating plate. In Continuum Dynamics. Inst. Gidrodinamiki SO Ross Akad Nauk 2000;16:166–9.], predicts, with high accuracy, the values of the amplitudes of the oscillations of the plate and the distributions of the bending moments and hydrodynamic pressure for a wide frequency range.     

Thermal post-buckling analysis of square plates resting on elastic foundation: A simple closed-form solutions

Thermal post-buckling paths of homogeneous, isotropic, square plate configurations resting on elastic foundation (Winkler type) subjected to biaxial compressive thermal loads are expressed as simple closed-form solutions by using the Rayleigh–Ritz method based on coupled displacement fields. Geometric non-linearity of von-Karman type is considered. The in-plane displacement field variations used in the formulation of Rayleigh–Ritz method are derived by using the governing in-plane static differential equations of the plate which in turn simplifies the difficulty of assuming an in-plane displacement field variations of the square plate. Accuracy and robustness of the proposed closed-form solutions are demonstrated by using the non-linear finite element formulation results which are obtained from an equilibrium path approach.     

Flow of a generalized Maxwell fluid induced by a constantly accelerating plate between two side walls

The unsteady flow of a viscoelastic fluid with the fractional Maxwell model, induced by a constantly accelerating plate between two side walls perpendicular to the plate, is investigated by means of the integral transforms. Exact solutions for the velocity field are presented under integral and series forms in terms of the derivatives of generalized Mittag–Leffler functions. The corresponding solutions for Maxwell fluids are obtained as limiting cases for β → 1. In the absence of the side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate.