Extensional Vibrations of Elastic Plates

1959 ◽  
Vol 26 (4) ◽  
pp. 561-569
Author(s):  
R. D. Mindlin ◽  
M. A. Medick
Keyword(s):  
Three Dimensional ◽  
Infinite Plate ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Wave Numbers ◽  
Complex Wave ◽  
Shear Modes ◽  
Thickness Shear

Abstract A system of approximate, two-dimensional equations of extensional motion of isotropic, elastic plates is derived. The equations take into account the coupling between extensional, symmetric thickness-stretch and symmetric thickness-shear modes and also include two face-shear modes. The spectrum of frequencies for real, imaginary, and complex wave numbers in an infinite plate is explored in detail and compared with the corresponding solution of the three-dimensional equations.

scholarly journals On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

scholarly journals Bechmann’s Number for Harmonic Overtones of Thickness-Shear Vibrations of Rotated Y-Cut Quartz Crystal Plates

Coatings ◽  
2020 ◽  
Vol 10 (7) ◽  
pp. 667
Author(s):  
Han Zhang ◽  
Yumei Chen ◽  
Ji Wang
Keyword(s):  
Experimental Data ◽  
Wave Propagation ◽  
Quartz Crystal ◽  
Three Dimensional ◽  
Approximate Solutions ◽  
Superior Performance ◽  
Shear Modes ◽  
Practical Experiences ◽  
Quartz Crystal Resonators ◽  
Thickness Shear

A procedure based on approximate solutions of three-dimensional equations of wave propagation is utilized for calculating Bechmann’s number for the harmonic overtones of thickness-shear modes in the rotated Y-cut quartz crystal plates. Bechmann’s number is used for the optimization and improvement of electrodes to yield superior performance in the design of quartz crystal resonators. Originally, Bechmann’s number is found through practical experiences, and analytical results were provided afterward to enable optimal design of novel resonator structures. The outcomes in this study are from a simplified theoretical prediction and they are consistent with known empirical results, making it is possible to design optimal quartz crystal resonators for cases without adequate experimental data for a higher frequency and smaller size.

Axially Symmetric Waves in Elastic Rods

1960 ◽  
Vol 27 (1) ◽  
pp. 145-151 ◽  
Author(s):  
R. D. Mindlin ◽  
H. D. McNiven
Keyword(s):  
Cross Section ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Elastic Rods ◽  
Axially Symmetric ◽  
Elastic Rod ◽  
One Dimensional ◽  
Axial Shear ◽  
Wave Numbers ◽  
Complex Wave

A system of approximate, one-dimensional equations is derived for axially symmetric motions of an elastic rod of circular cross section. The equations take into account the coupling between longitudinal, axial shear, and radial modes. The spectrum of frequencies for real, imaginary, and complex wave numbers in an infinite rod is explored in detail and compared with the analogous solution of the three-dimensional equations.

Self-Organization in Three-Dimensional Hydrodynamic Turbulence Self-Organization in Three-Dimensional Hydrodynamic Turbulence

1990 ◽  
Vol 45 (9-10) ◽  
pp. 1059-1073 ◽  
Author(s):  
G. Knorr ◽  
J. P. Lynov ◽  
H. L. Pécseli
Keyword(s):  
Euler Equations ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Self Organization ◽  
Two Dimensional ◽  
Hydrodynamic Turbulence ◽  
Available Energy ◽  
Wave Numbers ◽  
Negative Helicity ◽  
Incompressible Euler

Abstract The three-dimensional incompressible Euler equations are expanded in eigenflows of the curl operator, which represent positive and negative helicity flows in a particularly simple and convenient way. Four different basic types of interactions between eigenflows are found. Two represent an "inverse cascade", the interaction familiar from the two-dimensional Euler equations, in which only modes of the same sign of the helicity interact. The other two interactions mix positive and negative helicity modes. Only these interactions can transport all of the available energy to higher wave numbers. Initial conditions, which lead to the appearance of structures and self-organization, are discussed.

Spectral Properties of Superlattices with Anisotropic Inhomogeneities. The Transition from 3D to 2D Disorder

2012 ◽  
Vol 190 ◽  
pp. 597-600
Author(s):  
V.A. Ignatchenko ◽  
A.V. Pozdnyakov
Keyword(s):  
Density Of States ◽  
Spectral Properties ◽  
Three Dimensional ◽  
Dynamic Susceptibility ◽  
Two Dimensional ◽  
Continuous Transition ◽  
Wave Numbers ◽  
Correlation Properties

Waves in superlattice (SL) contained inhomogeneities with anisotropic correlation properties are considered. The anisotropy of the correlation during the transition from 3D to 2D disorder is characterized by the parameter , where and are the correlation wave numbers along the axis of the SL and in the plane of the its layers, respectively ( and are the correlation radii). Dependencies of both the dynamic susceptibility and density of states at the continuous transition from the isotropic three-dimensional inhomogeneities () to the two-dimensional ones () have been obtained.

Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates

1951 ◽  
Vol 18 (1) ◽  
pp. 31-38 ◽  
Author(s):  
R. D. Mindlin
Keyword(s):  
Uniqueness Theorem ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Dimensional Theory ◽  
Rotatory Inertia ◽  
One Dimensional ◽  
Edge Conditions

Abstract A two-dimensional theory of flexural motions of isotropic, elastic plates is deduced from the three-dimensional equations of elasticity. The theory includes the effects of rotatory inertia and shear in the same manner as Timoshenko’s one-dimensional theory of bars. Velocities of straight-crested waves are computed and found to agree with those obtained from the three-dimensional theory. A uniqueness theorem reveals that three edge conditions are required.

Reflections on the Theory of Elastic Plates

1985 ◽  
Vol 38 (11) ◽  
pp. 1453-1464 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
Shear Deformation ◽  
Fourth Order ◽  
Transverse Shear ◽  
Three Dimensional ◽  
Order Theory ◽  
Sixth Order ◽  
Interior Solution ◽  
Transverse Shear Deformation ◽  
Two Dimensional ◽  
Elastic Plates

We depart from a three-dimensional statement of the problem of small bending of elastic plates, for a survey of approximate two-dimensional theories, beginning with Kirchhoff’s fourth-order formulation. After discussing various variational statements of the three-dimensional problem, we describe the development of two-dimensional sixth-order theories by Bolle´, Hencky, Mindlin, and Reissner which take account of the effect of transverse shear deformation. Additionally, we report on an early analysis by Le´vy, on a direct two-dimensional formulation of sixth-order theory, on constitutive coupling of bending and stretching of laminated plates, on higher than sixth-order theories, and on an asymptotic analysis of sixth-order theory which leads to a fourth-order interior solution contribution with first-order transverse shear deformation effects included, as well as to a sequentially determined second-order edge zone solution contribution.

An exact three-dimensional solution for normal loading of inhomogeneous and laminated anisotropic elastic plates of moderate thickness

1992 ◽  
Vol 437 (1899) ◽  
pp. 199-213 ◽  
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Dimensional Solution ◽  
Important Special Case ◽  
Normal Loading ◽  
Edge Conditions ◽  
Anisotropic Elastic ◽  
Anisotropic Elastic Material ◽  
Edge Displacement

An exact three-dimensional solution is presented for the deformation and stress distribution in an elliptical plate under uniform normal loading of the lateral surfaces, and clamped along its edge. The plate is assumed to be of constant, moderate thickness and composed of anisotropic elastic material which is inhomogeneous in the through-thickness direction but symmetric about the mid-plane. The only material symmetry assumed is that of reflectional symmetry in planes parallel to the mid-plane. A transfer matrix method is used which, without making any further assumptions, gives the exact solution at each point in the plate in terms of the stress and displacement at the mid-plane. The two-dimensional differential equations governing these mid-plane values are found to be the same as those for an equivalent homogeneous plate whose constant elastic moduli are determined by appropriate through-thickness weighted averages of the inhomogeneous moduli. The solution of the two-dimensional problem is known for such a plate when subject to the specified surface and edge conditions, and yields a closed form analytical solution that satisfies all the governing equations and surface conditions of the full three-dimensional elasticity problem, with edge displacement conditions satisfied on the mid-plane. The important special case of an anisotropic laminated plate is given by assuming piecewise constant properties through the thickness.

Theories for Elastic Plates Via Orthogonal Polynomials

1981 ◽  
Vol 48 (4) ◽  
pp. 900-904 ◽  
Author(s):  
S. Krenk
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Displacement Function ◽  
Shear Stresses ◽  
Transverse Displacement ◽  
Normal Strain ◽  
Two Dimensional ◽  
Elastic Plates

A complementary energy functional is used to derive an infinite system of two-dimensional differential equations and appropriate boundary conditions for stresses and displacements in homogeneous anisotropic elastic plates. Stress boundary conditions are imposed on the faces a priori, and this introduces a weight function in the variations of the transverse normal and shear stresses. As a result the coupling between the two-dimensional differential equations is described in terms of a single difference operator. Special attention is given to a truncated system of equations for bending of transversely isotropic plates. This theory has three boundary conditions, like Reissner’s, but includes the effect of transverse normal strain, essentially through a reinterpretation of the transverse displacement function. Full agreement with general integrals to the homogeneous three-dimensional equations is established to within polynomial approximation.

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