Infinite Thin Plate Research Articles

The micromechanics theory of classical plates: a congruous estimate of overall elastic stiffness

A micromechanics theory is set forth for classical, or Love–Kirchhoff plate. A generalized eigenstrain theory, an eigen-curvature formulation, is proposed, which can be viewed as the analogue, or counterpart of the eigenstrain formulation in linear elasticity. This thin plate version of micromechanics is capable of dealing with heterogeneous inclusions, or inhomogeneities, whose size is comparable with the thickness of thin plates, under these circumstances, the continuum micromechanics theory is no longer applicable. The paper consists of three parts. In the first part, the solution of the elliptical inclusion in an infinite thin plate is revised. In the second part, several variational inequalities of the Love–Kirchhoff plate are derived, including a Hashin–Shtrikman/Talbot–Willis type principle. In the third part, as an application, exact variational estimates are given to bound the effective elastic stiffness, and a self-consistent scheme is also discussed. The newly derived bounds are congruous with Love–Kirchhoff plate theory, i.e. they are genetic to the governing equations of Love–Kirchhoff plate. They may serve as the alternatives together with the Hashin–Shtrikman bounds in linear elastostatics in the design process of composite plates.

Bending of a thin plate containing a rigid inclusion and a crack

The bending of a thin infinite plate with a line crack and an arbitrarily shaped rigid inclusion is analyzed. The superposition principle is used to reduce the original formulation to two subsidiary problems. A distribution of dislocation is assumed along the crack line. The solution is obtained in an integral form by using the Green function of a point dislocation. The stress functions for both subsidiary problems are obtained by employing the rational mapping function technique. The stress intensity factors are obtained in terms of the dislocation density function. Numerical results are demonstrated for the plate containing a square rigid inclusion and a line crack.

Two unequal cracks with coalesced plastic zones – the generalised Dugdale model approach

The problem of two collinear hairline unequal cracks weakening an infinite thin elastic plate is considered. The faces of the cracks open in mode I type deformation on account of the uniform constant uniaxial tension applied at infinity. The tension is increased to the limit such that the plastic zones developed at two interior tips of the unequal cracks get coalesced. Each of the plastic zones developed is closed by distributing a normal cohesive stress distribution over its rim. The problem reduces to dual Hilbert problems, the solution of which gives a set of non-linear simultaneous equations to determine the length of the plastic zones. Expressions are derived to obtain crack opening displacement at the tips of the cracks. An illustrative numerical example is considered to study the variation of load ratio required for the closure of the prescribed plastic zone.

Study of stress and strain fields in an infinite plate exposed to hydrogen diffusion

Strain and stress fields, which accompany the diffusion flux through an infinite thin plate, are studied. A complex tension is realized in a plate despite an admission, and is developed in the course of time. An essential inhomogeneity and instationarity of deformation results in the appearance of additional elastic-viscous stresses depending on a strain velocity. The shear components of elastic-viscous stresses have large values. These values are able to change the present structure.

INVESTIGATION OF VIBRATION POWER TRANSMISSION OVER A RECTANGULAR EXCITATION AREA USING EFFECTIVE POINT MOBILITY

This paper investigates the vibration response and power transmission for a thin infinite plate which is excited over a rectangular area by a uniform conphase force distribution. The use of the effective point mobility concept allows the distribution of power flow and vibration over the contact region to be determined. The distribution is found to depend on the physical and geometrical properties of the plate and the Helmholtz number (based on the width of the contact region). This demonstrates that the surface mobility of the contact region decreases with increasing frequency of excitation and increasing contact area. The effect of the shape of the contact region on the surface mobility is highlighted by comparing the results for a strip, a circular region and a rectangular region with various aspect ratios for a fixed contact area. Experimental measurements confirm the nature of the relationship between power flow and Helmholtz number for an infinite plate.

Displacement and flexural stresses of a loaded elastic plate on a viscous liquid

The nonstationary deformation of an infinite thin elastic plate that is symmetrically loaded and floats on a highly viscous liquid is characterized by two similarity parameters and the shape of the load. The first similarity parameter is proportional to the rigidity of the plate and inversely proportional to the fourth power of the radial extent of the load. The second is proportional to the radial extent of the load raised to three halves and inversely proportional to the viscosity of the liquid. The equations describing the deformation of the plate and the motion of the liquid are solved analytically and applied to the rebound of Fennoscandia since the Weichselian glaciation. The corresponding interpretation of uplift data of the lithosphere in Scandinavia implies that the ice cap did not cover the southern part of Scandinavia. Moreover, the calculated flexural stresses of the lithosphere in Sweden are small in those regions where no weak earthquakes occur.

Stability of flexural-gravity waves and quadratic interactions

The processes leading to strong distortion of the amplitude of a periodic flexural-gravity wave propagating in a fluid layer of finite depth beneath an infinite thin elastic plate are investigated. It is shown that three-wave resonance with noise harmonics of the flexural-gravity waves may lead to instability of a wave with the wave vector kw, where |kw|<kmin. Depending on |kw|, either two copropagating noise harmonics or two noise harmonics whose wave vectors make a nonzero angle with the vector kw may be most strongly amplified during the initial instants of time.

The general fundamental solution of the sixth-order Reissner and Mindlin plate bending models revisited

The sixth-order differential equation system of the Reissner and Mindlin plate bending models describe mathematically the plate problem, where lines normal to the mid-plane before deformation remain straight and inextensible in the deformed configuration, but not necessarily normal to the reference plane anymore. The non-normality condition is due to the consideration of transverse shearing strains, disregarded in the classical bi-harmonic Kirchhoff plate model. As the plate thickness is reduced and/or the transverse shear modulus is increased, the deformed configuration is less dependent of the transverse shear strains and, in the limit as the plate thickness approaches zero and/or the transverse shear modulus approaches ∞, the transverse shear strain effects vanish and the problem is exactly that described by the classical plate model. In this paper, we investigate the fundamental solutions of both the fourth-order Kirchhoff and the sixth-order Reissner and Mindlin plate models. We consider a transversely isotropic material and show that the fundamental solution of the bi-harmonic problem can be obtained directly from the general fundamental solution of the sixth-order plate problem, in the limit as the plate thickness approaches zero and/or the transverse shear modulus approaches ∞. This solution is in agreement with the analytical solution of an infinite thin clamped circular plate submitted to a unitary concentrated load acting at its center.

Solutions without preacceleration to the one-dimensional Lorentz-Dirac equation

An approach to the problem of violation of causality in classical electrodynamics is proposed. This approach is based on the construction of exact analytic solutions without preacceleration to the Lorentz-Dirac equation in an electrostatic field that vanishes identically outside a certain region. Exact solutions are given for the potential well and the linear potential wall, cases for which Plass [Rev. Mod. Phys. 33, 37 (1961)] already found the corresponding preaccelerative solutions. In addition, an exact solution which differs from the one found by Plass is given for a thin infinite charged plate. Finally, an exact solution is constructed for a special electrostatic field. All these nonpreaccelerative solutions have a jump in the acceleration at points where the electrostatic field has a jump; this implies that they cannot be obtained as solutions to the usual integro-differential equation associated to the Lorentz-Dirac equation.

Coupling and attenuation of waves in plates by randomly distributed attached impedances

The average response of an infinite thin plate with statistically homogeneous attached random impedances is examined. The added impedances, which represent typical heterogeneities that might occur on complex shells, provide light coupling between the extensional, shear, and flexural waves. The mean plate response is formulated in terms of the Dyson equation which is solved within the assumptions of the first-order smoothing approximation, or Keller approximation, valid when the heterogeneities are weak. Scattering attenuations are derived for each propagation mode. It is shown that the attenuation of one wave type due to coupling to another is proportional to the modal density of the other wave type. Thus the attenuation of extensional and shear waves is predominantly due to mode conversion into flexural waves and is proportional to the large modal density of flexural waves. The flexural degrees of freedom serve as a sink for the energy of the membrane modes and constitute for them an effective fuzzy structure. The specific case of delta-correlated springs is considered for purposes of illustration.

On elastic joints containing interfacial cracks

We study quasistatic deformations of a massive elastic body bonded to an inifinite elastic layer of finite thickness whose other surface is glued to a half space. The adhesive is modeled as a nonlinear elastic material. It is assumed that interfacial cracks are present at the body/adhesive interface and at the interface between the elastic layer and adhesive material closer to the half space. Variational formulations of the problems when the material of the half space is modeled as isotropic linear elastic or as rigid are given and are shown to be equivalent to the corresponding boundary-value problems. Analytical solutions of two example problems are given; one involves a long rigid cylinder bonded to a rigid half space and the other a thin infinite linear plate bonded to a rigid half space. The results depicted graphically include the dependence of the crack closure length on the applied loads and the dependence of the gradient of the potential energy on the crack length for the first problem, and distributions of the normal stress and normal displacement on the crack surface for the second problem.

Conservation laws and group-invariant solutions of the von Kármán equations

The present paper is concerned with the classical von Kármán equations for large deflection of thin elastic plates both in 2 (space) and in 2 + 1 (space-time) dimensions. All infinitesimal divergence symmetries of two action functionals whose Euler-Lagrange equations coincide with the von Kármán equations are established and the corresponding conservation laws obtainable through Noether's theorem are determined. Wave phenomena in infinite thin elastic plates are studied using group-invariant solutions to the time-dependent von Kármán equations, the derived conservation laws being taken into consideration.

Development of plastic strips and fracture in the vicinity of two unequal cracks in a plate

We studied the process of development of plastic strains and fracture in tension of a thin infinite plate with two collinear unequal cracks. An ideally elastoplastic material of the plate satisfies the Tresca condition. We investigated the propagation of narrow plastic strips on the continuation of the cracks by using the well-known method for the simulation of such strips by segments with jumps of displacements satisfying the plasticity condition. We also present a general algorithm for the solution of the problem under consideration. Within the framework of this algorithm, the results for two equal cracks are obtained as a special case. In the case of unequal cracks, we computed the value of loading whose application results in the joining of strips lying between the cracks and deduced equations for the determination of the length of these strips and crack tip opening displacements both at the time of joining of the strips and after this event. The numerical analysis of results demonstrates that the largest strips appear (prior to their joining and opening) between the cracks in the vicinities of the tips of the longer crack. According to the strain criterion, the processes of fracture and joining of the cracks are initiated just at the indicated sites. It is shown that, under a load of 0.6 σt, at the ends of cracks, one observes the initiation of secondary strips following the primary ones and making an angle of 59° with them (just as in the case of a single crack). We also indicate the principal shortcomings of the solution of the problem suggested by Theocaris [Eng. Fract. Mech,18, No. 3, 545–560 (1981)].

Average response of an infinite plate on a random elastic foundation

The average response of infinite thin plates with attached random impedances is examined. The added impedance, which represents typical heterogeneities that might occur on complex shells, provides light coupling between the three propagation modes. The problem is formulated in terms of the Dyson equation which governs the mean plate response. It is solved within the assumptions of the Keller (smoothing) approximation which is valid when the heterogeneities are weak. Scattering attenuations are derived for each propagation mode. The specific case of delta-correlated springs provides a simple intuitive result for a statistically homogeneous plate. The attenuation of one wave type due to coupling to another is shown to be proportional to the modal density of the other wave type. Thus the attenuation of extensional and shear waves is predominantly due to mode conversion into flexural waves and is proportional to the large modal density of flexural waves. The flexural degrees of freedom serve as a sink for the energy of the membrane modes and constitute for them an effective fuzzy structure. A similar result is expected for the more complicated submerged shell case. A discussion of diffuse energy transport on such a plate will also be presented. [Work supported by ONR.]

Response of an infinite plate on a honeycomb foundation to a rigid cylindrical impactor

Summary The response of a thin infinite plate on a honeycomb foundation to impact by a rigid cylindrical projectile is investigated by continuum mechanics. The influence of the honeycomb mass accumulating at the front plate is included in the analysis. The plate is assumed to be rigid-perfectly-plastic and to obey the Hodge yield condition and associated flow rule. The honeycomb is modeled as a continuum with discontinuous density. The effects of finite deformation and membrane forces, in addition to transverse shear forces and bending moments are considered in the present formulation. Excellent correlation of the predicted results with corresponding experimental data is observed. The effect of the crushed honeycomb mass is shown to be negligible.

Effect of a gradient heating on the stress-strain state of a plate

The solution of a thermoelasticity problem is constructed and the effect of a preliminary cooling and heating by cylindrical heat sources of different intensities on the thermoelastic state of an infinite thin plate is investigated.

Elastohydrodynamic response of an ice sheet to forced sub-surface uplift

T he objective of this study is to investigate the dynamic response of an ice sheet under transient subsurface uplift. The ice sheet is modeled as an infinite thin plate undergoing small deflections that is pushed vertically upwards by a rigid flat-ended cylinder ; the water underneath the ice sheet is modeled using incompressible potential flow theory. Ultimately, the deflection of the ice sheet is expressed in terms of an integral of the unknown indentation force. For a given motion of the cylinder, the corresponding indentation force must be determined by solving a Volterra integral equation of the first kind. After some numerical experimentation, it proved advantageous to transform the governing equation into an integro-difierential equation. Once the indentation force is obtained, the other physical quantities of interest — the temporal and spatial distributions of the displacement, moment, shear, the buoyant force, the inertial force of the water underneath the ice sheet and the inertial force of the ice sheet — can be evaluated numerically. The results reveal that the inclusion of the hydrodynamic reaction is of major importance.

A finite element alternating approach to the bending of thin plates containing mixed mode cracks

A finite element alternating procedure developed to deal with the cracked plate in symmetric bending is extended to analyse the more realistic case with single or multiple mixed mode cracks. This modified procedure involves the new derivation of the analytical solution for an infinite thin plate containing a crack subjected to arbitrary anti-symmetric equivalent shear force on crack surfaces. Due to the natural limitation of Kirchhoff assumptions for a thin plate in bending, the appropriate treatment of constant twisting moments on the crack surfaces is devised. The interaction effect among cracks on the calculation of bending (symmetric) and shear (anti-symmetric) stress intensity factors is also discussed. Several numerical examples are solved to demonstrate the validity and efficiency of the approach.

Elastic interaction of two thin infinite plates joined along segments of a straight line

Elastic interaction of two thin infinite plates joined along segments of a straight line

Transient sub-surface uplift of a floating ice sheet

In this paper, the transient response of an ice sheet subjected to sudden uplift is investigated. The ice sheet is modeled as an infinite thin plate undergoing small deflections that is being pushed vertically upwards by a rigid flatended cylinder. The water underneath the ice sheet is modeled using incompressible potential flow theory. For a given motion of the cylinder, the current problem is reduced to a Volterra integral equation of the first kind for the indentation force. It is solved by transforming the governing equation into an integro-differential equation. For the physical quantities of interest, comparisons are made with related experiments.