Thick Elastic Plate Research Articles

Vibration and Stability of Thick Plates on Elastic Foundations

Natural frequencies and buckling stresses of a thick isotropic plate on two-parameter elastic foundations are analyzed by taking into account the effect of shear deformation, thickness change, and rotatory inertia. Using the method of power series expansion of the displacement components, a set of fundamental dynamic equations of a two-dimensional, higher-order theory for thick rectangular plates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are used to solve the eigenvalue problems of a simply supported thick elastic plate. To assure the accuracy of the present theory, convergence properties of the minimum natural frequency and the buckling stress are examined in detail. The distribution of modal transverse stresses are obtained by integrating the three-dimensional equations of motion in the thickness direction. The present approximate theories can accurately predict the natural frequencies and buckling stresses of thick plates on elastic foundations as compared with Mindlin plate theory and classical plate theory.

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.

A DYNAMIC MODEL FOR THICK ELASTIC PLATES

In this paper the general equations of motion for thick elastic plates with arbitrary shape are derived by using a variational principle. In addition to the influences of the bending, the transverse shear deformation and the rotatory inertia, the proposed theory also contains the effects of the transverse normal stress and the membrane forces. The equations presented in this paper can be reduced to those deduced by E. Reissner and R. D. Mindlin. Some numerical results are compared with those obtained from the Reissner–Mindlin plate theory and the classical plate theory. It is found that for thick plates there exists a dense region of frequencies and the position of frequencies is shifted, so that the influence of the transverse normal stress must be considered.

Equations for thick elastic plates with partially electroded piezoelectric actuators and higher order electric fields

A system of two-dimensional equations for the flexural motion of an elastic plate with both electroded and unelectroded portions of piezoelectric actuators attached to the top and bottom surfaces are obtained. These equations include the extension and the bending effects of the piezoelectric actuators and the effects of rotatory inertia and shear deformations of the elastic plate. They are valid when the elastic plate is not thin and the actuators are not thin compared to the elastic plate. The equations also include the quadratic variation of the electric potential along the thickness of the actuators. It is shown that this quadratic electric potential variation is directly coupled to the flexure of the actuators and its effect is of the same order as the bending stiffness of the actuators which is usually small.

Reciprocal theorem method for the forced vibration of elastic thick rectangular plates

In this paper, reciprocal theorem method (RTM) is generalized to solve the problems for the forced vibration of thick rectangular plates based on the Reissner's theory. The paper derives the dynamic basis solution of thick rectangular plates; and the exact analytical solution of the steady-state responses of thick rectangular plates with three clamped edges and one free edge under harmonic uniformly distributed disturbing forces is found by RTM. It is regarded as a simple, convenient and general method for calculating the steady-state responses of forced vibration of thick rectangular plates.

Coupled instabilities for a thick elastic plate under thrust

For a thick incompressible hyperelastic plate under biaxial thrust, flexural and constitutive (homogeneous) instabilities are considered. In the present study their interaction (coupling) is discussed, when the critical conditions coincide for both kinds of instabilities. With the help of branching theory, especially the Fredholm Alternative theorem, the interactive post-critical equilibrium configurations are defined.

On an Axisymmetric Bending of Annular Thick Plates. 2nd Report. Annular Thick Plates with Clamped Edge.

This study is concerned with the axisymmetric bending of uniform, isotropic, elastic thick plates. In particular, we analyze an annular thick plates a portion of which is clamped between two smooth rigid plane surfaces. Using Reissner's theory of plates, expressions for the stresses in the region constrained within the clamp are obtained and appropriate continuity conditions are satisfied at the edge of that region. The numerical results for the case of axisymmetric bending of an annular thick plate taking into consideration the clamped edge condition are summarized as technological data.

Buckling instabilities of thick elastic plates subjected to in-plane stresses

Buckling instabilities of a simply supported thick elastic plate subjected to in-plane stresses are studied. Taking into account the effects of shear deformations and thickness changes, buckling loads and buckling displacement modes of thick plates are obtained. Buckling of thick plates can occur in the (x α, x 3)-planes involving either in-plane or out-of-plane mode shapes with arbitrary integer numbers r and s of wavelengths. The same asymptotic buckling load appears at higher displacement modes with wrinkling instabilities for the in-plane and out-of-plane problems. Based on the power series expansion of displacement components, a set of fundamental equations of a two-dimensional higher-order plate theory is derived through the principle of virtual displacements. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a thick plate. The convergence properties of the buckling loads for the in-plane and out-of-plane problems of a simply supported square plate are examined in detail. It is noticed that the present approximate theories can predict the buckling loads of an extremely thick plate more accurately compared to other refined theories and classical plate theory.

Quaternary evolution of the Corinth Rift and its implications for the Late Cenozoic evolution of the Aegean

SUMMARY We present geological and morphological observations at different scales to constrain rates of faulting and the distribution of deformation in the seismically active Aegean region. We focus first on the 130 km long Corinth Rift, an asymmetric graben where a flight of terraces of marine origin are uplifted. We show that the edges of the terraces lie in the footwall of the normal fault bounding the Corinth Rift and correspond to sea-level highstands of laic Pleistocene age. Using a detailed analysis of aerial and SPOT imagery supported by field observations, we have mapped 10 terrace platforms and strandlines ranging in elevation from 10 to 400 m over distances of 2 to 20 km from the fault. The elevation of the terraces' inner edges was estimated at 172 sites with an error of ±5m. This data set contains a precise description of the uplift and flexure of 10 different palaeohorizontal lines with respect to the present sea level. To date the deformation, we correlate the Corinth terraces with late Pleistocene oxygen-isotope stages of high sea-level stands and with global sea-level fluctuations. Using a thick elastic plate model consistent with our current understanding of the earthquake cycle and a boundary-element technique we reproduce the geometry of the shorelines to constrain both mechanical parameters and the slip on the fault. We show that the seismogenic layer behaves over the long term as if its elastic modulus were reduced by a factor of about 1000. All the terraces are fitted for fault slip increasing in proportion to terrace age, and the component of regional uplift is found to be less than 0.3 mm yr−1. The best fits give a slip rate of 11±3 mm yr−1 on the main rift-bounding fault over the last 350 kyr. Other geological and morphologic information allows us to estimate the total age of the main fault (∼1 Ma) and to examine the mechanical evolution of the Corinth Rift. The minimum observed sediment thickness in the Gulf places an extreme check on the results of the modelling and a lower bound on slip rate of 6–7 mm yr−1 (40 per cent less than estimated with modelling). Even this slip rate is nearly 10 times higher than for comparable features in most of the Aegean and elsewhere in the world. At a larger scale, the spacing and asymmetry of the rift systems in the Aegean suggest strain localization in the upper mantle, with slow extension starting 15 Myr ago or earlier. The more recent (1 Myr), rapid phase of rifting in Corinth partly reactivated this earlier phase of extension. The younger faulting in Corinth appears to result from its present location in the inhomogeneous stress field (process zone) of the south-westward propagating tip of the southern branch of the North Anatolian Fault. We extend these relations to propose a mechanical model for the Late Cenozoic evolution of the Aegean. As the Arabia/Europe collision progressed in eastern Turkey it caused Anatolia to move to the west and the North Anatolian Fault to propagate into the Aegean, where the early slow extension started to be modified about 5 Ma ago. The process of propagation dramatically increased the activity of some but not all of the earlier rifts. The model we present is compatible with tectonic observations, as well as with the seismicity, the palaeomagnetic rotations and the displacement field now observed with GPS and SLR.

Second order displacement fields of a thick elastic plate under thrust-the incompressible case

Branching of a thick incompressible plate of hyperelastic material under uniaxial thrust is studied. The acceptable barreling second order displacement field is described, corresponding to the first order lower flexure mode. An application, clarifying the theory, is presented for a Mooney-Rivlin material

THERMAL SHOCK UPON THE SURFACE OF A PLATE RESTING ON A THERMOELASTIC ISOTROPIC HALF-SPACE

The problem of the nonstationary vibrations of an infinite elastic uniform thick plate resting on a thermoelastic isotropic half-space is solved. The equations of the plate motion take into consideration the rotary inertia and transverse shear deformations, and the half-space equations take the thermal flow lag into account. Both welded and smooth contacts between layer and half-space are considered. Nonstationary vibrations are excited by the instantaneous change in the ambient temperature above the plate to some magnitude that then does not change with time or by the instantaneous change in the plate temperature. The plate nonstationary vibrations, in their turn, give rise to three types of plane surface of strong discontinuity propagating in the thermoelastic half-space. The solution behind the wave fronts up to the contact boundary is constructed by virtue of ray series. Unknown functions entering into the ray series coefficients and into the equations of the plate motion are determined from the bound...

Free vibration and stability of thick elastic plates subjected to in-plane forces

Natural frequencies and buckling loads of a simply supported thick plate subjected to in-plane initial tensile and/or compressive forces are analysed. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order plate theory is derived through Hamilton's principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a thick plate. In order to assure the accuracy of the present theory, convergence properties of the minimum natural frequency and the buckling load for the out-of-plane problem of a simply supported square plate are examined in detail. It is noticed that the present approximate theories can predict the frequencies and buckling loads of an extremely thick plate more accurately compared to other refined theories and classical plate theory.

Late Cenozoic flexural deformation of the middle U.S. Atlantic passive margin

Despite the century‐long recognition of regional epeirogeny along the middle Atlantic passive margin, relatively few studies have focused on understanding postrift uplift mechanisms. Here, we demonstrate that epeirogenic uplift of the central Appalachian Piedmont and subsidence of the Salisbury Embayment represent first‐order, flexural isostatic processes driven by continental denudation and offshore deposition. Our results show that regional epeirogenic processes, present on all Atlantic‐type passive margins, are best resolved by specific stratigraphic and geomorphic relationships, rather than topography. A simple one‐dimensional geodynamic model, constrained by well‐dated Baltimore Canyon trough, Coastal Plain, and lower Susquehanna River (Piedmont) stratigraphy, simulates flexural deformation of the U.S. Atlantic margin. The model represents the passive margin lithosphere as a uniformly thick elastic plate, without horizontal compressive stresses, that deforms flexurally under the stress of strike‐averaged, vertically applied line loads. Model results illustrate a complex interaction among margin stratigraphy and geomorphology, the isostatic response to denudational and depositional processes, and the modulating influence of exogenic forces such as eustasy. The current elevation, with respect to modern sea level, of fluvial terraces and correlative Coastal Plain deposits or unconformities is successfully predicted through the synthesis of paleotopography, eustatic change, and margin flexure. Results suggest that the middle U.S. Atlantic margin landward of East Coast Magnetic Anomaly is underlain by lithosphere with an average elastic thickness of 40 km (flexural rigidity, D = 4 × 1023 N m), the margin experiences an average, long‐term denudation rate of approximately 10 m/m.y., and the Piedmont has been flexurally upwarped between 35 and 130 meters in the last 15 m.y. Long‐term isostatic continental uplift resulting from denudation and basin subsidence resulting from sediment loading are accommodated primarily by a convex‐up flexural hinge, physiographically represented by the Fall Zone. Our results elucidate an inherent danger in using topography alone to constrain late‐stage passive margin deformation mechanisms. Only through careful synthesis of field stratigraphic and geomorphic elements such as fluvial terraces, Coastal Plain deposits, and offshore stratigraphy can age control be extended from the offshore depositional setting to the erosionally dominated continent. This study demonstrates that despite a relatively subdued topography, the middle U.S. Atlantic margin experiences progressive flexural isostatic deformation similar to that proposed for high‐relief margins characterized by great escarpments. Thus margin topographic diversity remains a function of other factors, such as lithospheric composition and/or structure, supracrustal stratigraphy and structure, degree of drainage integration, drainage divide migration, and climate.

Buckling of a pervasively faulted lithosphere

We investigated the buckling response of a faulted elastic plate under horizontal compression using the finite element technique to better understand the effect of faults on the elastic behavior of a plate. We studied the effect of changes in fault spacing, depth and dip on the effective Young's modulus, buckling stress and wavelength. Our model consists of a thick elastic plate whose entire upper surface is cut by evenly spaced faults. We impose either an initial sinusoidal deformation with a fixed wavelength or a random deformation to the grid. A fault is represented as a free surface with no resolved shear stress and is allowed to slip in a specified direction using the method of ‘slippery nodes’. With the assumption of free slip on the faults, our model results represent an end member case in which the buckling wavelength and buckling stress are minimized by the presence of the faults.

The application of a two-dimensional higher-order theory for the analysis of a thick elastic plate

The application of approximate equations of a two-dimensional theory for the analysis of stress and displacement distributions of a thick elastic plate with fixed boundaries is presented. By using the method of power series expansion of displacement components, a set of fundamental equations of a two-dimensional higher-order plate theory is derived through the principle of virtual work. In order to assure the accuracy of the present theory, the governing equations of several sets of truncated approximate theories are solved numerically by the finite difference method. Higher-order finite difference operators with error terms of o( h 4) are applied in order to validate the numerical accuracy of results. The convergence properties of the present numerical solutions have been shown to be accurate for the expanded displacement components with respect not only to the number of finite difference mesh intervals but also to the order of approximate theories. Stress components are determined by satisfying the equilibrium equations of a three-dimensional elastic continuum and the stress boundary conditions on the upper and lower surfaces of a plate. A direct comparison is also made with the numerical results of the integral method obtained by using the solutions of a three-dimensional elastic continuum.

An equation of motion for a thick viscoelastic plate

In this paper an equation of motion is presented for a general thick viscoelastic plate, including the effects of shear deformation, extrusion deformation and rotatory inertia. This equation is the generalization of equations of motion for the corresponding thick elastic plate, and it can be degenerated into several types of equations for various special cases.

A model of thick plate deformation and its application to the isostatic movements due to surface, subsurface and internal loadings

SUMMARY A model of thick plate deformation is presented without any restriction on stress or strain conditions in the plate. An analytical solution is given for the vertical displacement of a thick elastic plate. The present method is exact and applicable to 3-D plate deformation problems associated with surface, subsurface and internal loadings. The conventional thin plate theory, which ignores the variation of vertical strain of a plate, may not be appropriate to be used as a basic model to study deformation problems of the continental lithosphere, since it is shown that the actual surface isostatic adjustment due to surface loading could reach an amount twice as much as predicted by thin plate theory, even on a relatively thin plate with 30 km thickness and Young’s modulus E = lo00 kbar and Poisson’s ratio Y = 0.25. The isostatic adjustment due to surface load depends on the stress condition at the lower boundary of the elastic lithosphere. If there is an active mantle upwelling or downwelling force acting on the boundary, then the isostatic adjustment, depending on the magnitude of the mantle force, is either greatly hindered or amplified to a degree that is beyond the normal prediction of isostatic movements. In addition, the viscous effect of mantle material on the isostatic movements is shown to be negligible on geological time-scales if the mantle viscosity is below ld4poise. It is also shown that the Airy isostasy of the total sediment load in a surface depression with small wavelength could be achieved only if a considerable period of geological time is allowed for the stress relaxation of viscoelastic or viscous material in the lithosphere. The rheological property of the continental lithosphere has a significant effect on basin subsidence caused by sediment loading alone, while pure elastic lithosphere restricts surface basin development because of the strong basement support throughout the evolution history.

Green's functions for torsional body force problems of a dissimilar elastic thick plate.

Green's functions are shown for axisymmetric torsional body force problems of a dissimilar elastic thick plate. Green's functions are defined as solutions to the problem of a dissimilar elastic thick plate subjected to a torsional body force acting on a circle in the interior of the thick plate. As a special case for the Green's functions shown, we can obtain Green's functions for torsional body force problems of an elastic half-space, an elastic thick plate, a dissimilar elastic solid, an elastic layer on an elastic half-space, among others. A general method of solution is presented for torsional body force problems of a dissimilar elastic thick plate. The Green's functions shown here are obtained by applying the general method of solution. As an example of the application of Green's functions, we consider the torsion of a hemispherical rigid body embedded in the dissimilar thick plate. Numerical results for the problem are shown.

Buckling instabilities in a thick elastic three-ply composite plate under thrust

The buckling of a three-ply sandwich composite plate is studied in the context of finite deformation incompressible nonlinear elasticity. As is the case of a noncomposite plate, buckled configurations involving either barrelling or flexure are possible. However, the ordering of the failure thrusts for certain composite configurations is more complicated than the ordering that occurs in the noncomposite case. In particular, under suitable conditions, the thrusts at which barrelling and flexural buckling modes occur can be interlaced. Furthermore, transitions in the ordering of the different failure modes may occur as the aspect ratio of the plate is varied.

Coupling between the ocean and an ice shelf

The possibility of long-period ocean waves coupling to an ice shelf is investigated. A thick elastic plate model is used for the ice shelf with comparisons made to the simpler thin-plate model. The strain set up on the ice shelf by a normally incident single frequency ocean wave is calculated by completely solving the equations governing the velocity potential for such a system. In the absence of measurements on an ice shelf, existing measurements of long-period strain on an ice tongue are used to estimate the required incident amplitude in the open water to induce the observed oscillations. It is found that the height of seas required indicates that ocean wave driving is a plausible forcing mechanism for observed oscillations.