Three-Dimensional Axisymmetric Piezothermoelastic Solution of a Transversely Isotropic Piezoelectric Clamped Circular Plate

1998 ◽  
Vol 65 (1) ◽  
pp. 178-183 ◽  
Author(s):  
S. Kapuria ◽  
P. C. Dumir ◽  
S. Segupta
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Dimensional Solution ◽  
Infinite Systems ◽  
Finite Set ◽  
Thickness Parameter

Three-dimensional solution in terms of potential functions is presented for a transversely isotropic piezoelectric clamped circular plate subjected to axisymmetric ther-moelectromechanical load. The boundary conditions are satisfied using Fourier-Bessel expansions yielding two uncoupled infinite systems of algebraic equations for the arbitrary constants. These are solved to any desired degree of accuracy by truncating to finite set of equations. Results are presented to illustrate the effect of the thickness parameter. These would help assess two-dimensional theories of piezoelectric plates.

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.

Theory of laminated elastic plates I. isotropic laminae

1988 ◽  
Vol 324 (1582) ◽  
pp. 565-594 ◽  
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Plane Stress ◽  
Three Dimensional ◽  
Laminated Plates ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Dimensional Solution ◽  
Stress Theory

In this paper (part I) we establish a theory for stretching and bending of laminated elastic plates in which the laminae are different isotropic linearly elastic materials. The theory gives exact solutions of the three-dimensional elasticity equations that satisfy all the interface traction and displacement continuity conditions, with no traction on the lateral surfaces; the only restriction is that edge boundary conditions can be satisfied only in an average manner, rather than point by point. The method, which is based on a generalization of Michell’s exact plane stress theory, yields exact solutions for each lamina. These solutions are generated in a very straightforward manner by solutions of the approximate two-dimensional classical equations of laminate theory and contain sufficient arbitrary constants to enable all the continuity and lateral surface boundary conditions to be satisfied. The values of the constants depend only on the lamina thicknesses and the elastic constants. Thus, for a given laminate and for any boundary-value problem , it is necessary only to solve the appropriate two-dimensional plane problem, and the corresponding exact three-dimensional laminate solution follows by straightforward substitutions. The two-dimensional solution may be derived by any of the available methods, including numerical methods. An important feature of the theory is that it determines the interfacial shearing tractions, as well as the in-plane stress components. The procedure is illustrated by applying the theory to three problems involving stretching and bending of laminated plates containing circular holes.

scholarly journals KOITER ESTIMATE REVISITED

2010 ◽  
Vol 20 (01) ◽  
pp. 1-42 ◽  
Author(s):  
MONIQUE DAUGE ◽  
ERWAN FAOU
Keyword(s):  
Energy Estimate ◽  
Three Dimensional ◽  
Two Dimensional ◽  
A Posteriori ◽  
Modeling Error ◽  
Dimensional Solution ◽  
Manifold With Boundary ◽  
Thickness Parameter ◽  
Posteriori Estimation ◽  
Heuristic Estimate

We prove a general adimensional energy estimate between the solution of the three-dimensional Lamé system on a thin clamped shell and a displacement reconstructed from the solution of the classical two-dimensional Koiter model. This estimate only involves the thickness parameter ε, constants attached to the mid-surface S, the two-dimensional energy of the solution of the Koiter model and "wavelengths" associated with this latter solution. This bound is in the same spirit as Koiter's heuristic estimate in Ref. 26 and can be viewed as an a posteriori estimation of the modeling error by means of the two-dimensional solution. It is general with respect to the geometry of the mid-surface S which is an arbitrary smooth manifold with boundary. Taking boundary layer terms into account, we prove that our estimates are sharp in the cases of plates and elliptic shells.

Separation of the 3D stress state of a loaded plate into two-dimensional tasks: bending and symmetric compression of the plate

2021 ◽  
Vol 103 (3) ◽  
pp. 53-62
Author(s):  
Victor Revenko ◽  
Andrian Revenko
Keyword(s):  
Boundary Conditions ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Partial Solution ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Normal Stresses ◽  
Equilibrium Equations ◽  
Dimensional Boundary ◽  
The Right

The three-dimensional stress-strain state of an isotropic plate loaded on all its surfaces is considered in the article. The initial problem is divided into two ones: symmetrical bending of the plate and a symmetrical compression of the plate, by specified loads. It is shown that the plane problem of the theory of elasticity is a special case of the second task. To solve the second task, the symmetry of normal stresses is used. Boundary conditions on plane surfaces are satisfied and harmonic conditions are obtained for some functions. Expressions of effort were found after integrating three-dimensional stresses that satisfy three equilibrium equations. For a thin plate, a closed system of equations was obtained to determine the harmonic functions. Displacements and stresses in the plate were expressed in two two-dimensional harmonic functions and a partial solution of the Laplace equation with the right-hand side, which is determined by the end loads. Three-dimensional boundary conditions were reduced to two-dimensional ones. The formula was found for experimental determination of the sum of normal stresses via the displacements of the surface of the plate.

Green's function boundary conditions in two-dimensional and three-dimensional atomistic simulations of dislocations

1998 ◽  
Vol 77 (1) ◽  
pp. 231-256 ◽  
Author(s):  
S. Rao ◽  
C. Hernandez ◽  
J. P. Simmons ◽  
T. A. Parthasarathy ◽  
C. Woodward
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Three Dimensional ◽  
Atomistic Simulations ◽  
Two Dimensional

Contact Temperature in Dry Bearings. Three Dimensional Theory and Verification

1981 ◽  
Vol 103 (2) ◽  
pp. 243-251 ◽  
Author(s):  
A. Floquet ◽  
D. Play
Keyword(s):  
Fourier Transform ◽  
Boundary Conditions ◽  
Experimental Verification ◽  
Three Dimensional ◽  
Contact Temperature ◽  
3D Analysis ◽  
New Work ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Infra Red

Boundary conditions were arbitrarily specified in an earlier two dimensional (2D) analysis of contact temperature. In this new work a general three dimensional (3D) Fourier transform solution is obtained from which for specific cases, the boundary conditions can be estimated. Further, experimental verification of 3D analysis was performed using infra-red technique.

Effects of thermal and electric boundary conditions on fracture of three-dimensional thermopiezoelectric media

2017 ◽  
Vol 29 (6) ◽  
pp. 1255-1271 ◽  
Author(s):  
MingHao Zhao ◽  
Yuan Li ◽  
CuiYing Fan
Keyword(s):  
Stress Intensity ◽  
Boundary Conditions ◽  
Stress Intensity Factors ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Boundary Integral ◽  
Displacement Discontinuity ◽  
Green Functions ◽  
Intensity Factors

An arbitrarily shaped planar crack under different thermal and electric boundary conditions on the crack surfaces is studied in three-dimensional transversely isotropic thermopiezoelectric media subjected to thermal–mechanical–electric coupling fields. Using Hankel transformations, Green functions are derived for unit point extended displacement discontinuities in three-dimensional transversely isotropic thermopiezoelectric media, where the extended displacement discontinuities include the conventional displacement discontinuities, electric potential discontinuity, as well as the temperature discontinuity. On the basis of these Green functions, the extended displacement discontinuity boundary integral equations for arbitrarily shaped planar cracks in the isotropic plane of three-dimensional transversely isotropic thermopiezoelectric media are established under different thermal and electric boundary conditions on the crack surfaces, namely, the thermally and electrically impermeable, permeable, and semi-permeable boundary conditions. The singularities of near-crack border fields are analyzed and the extended stress intensity factors are expressed in terms of the extended displacement discontinuities. The effect of different thermal and electric boundary conditions on the extended stress intensity factors is studied via the extended displacement discontinuity boundary element method. Subsequent numerical results of elliptical cracks subjected to combined thermal–mechanical–electric loadings are obtained.

scholarly journals Visualization of CD2 interaction with LFA-3 and determination of the two-dimensional dissociation constant for adhesion receptors in a contact area.

1996 ◽  
Vol 132 (3) ◽  
pp. 465-474 ◽  
Author(s):  
M L Dustin ◽  
L M Ferguson ◽  
P Y Chan ◽  
T A Springer ◽  
D E Golan
Keyword(s):  
Contact Area ◽  
Dissociation Constants ◽  
Three Dimensional ◽  
Lateral Diffusion ◽  
Solution Phase ◽  
Phospholipid Bilayer ◽  
Cell Contact ◽  
Two Dimensional ◽  
Adhesion Receptors ◽  
Dimensional Solution

Many adhesion receptors have high three-dimensional dissociation constants (Kd) for counter-receptors compared to the KdS of receptors for soluble extracellular ligands such as cytokines and hormones. Interaction of the T lymphocyte adhesion receptor CD2 with its counter-receptor, LFA-3, has a high solution-phase Kd (16 microM at 37 degrees C), yet the CD2/LFA-3 interaction serves as an effective adhesion mechanism. We have studied the interaction of CD2 with LFA-3 in the contact area between Jurkat T lymphoblasts and planar phospholipid bilayers containing purified, fluorescently labeled LFA-3. Redistribution and lateral mobility of LFA-3 were measured in contact areas as functions of the initial LFA-3 surface density and of time after contact of the cells with the bilayers. LFA-3 accumulated at sites of contact with a half-time of approximately 15 min, consistent with the previously determined kinetics of adhesion strengthening. The two-dimensional Kd for the CD2/LFA-3 interaction was 21 molecules/microns 2, which is lower than the surface densities of CD2 on T cells and LFA-3 on most target or stimulator cells. Thus, formation of CD2/LFA-3 complexes should be highly favored in physiological interactions. Comparison of the two-dimensional (membrane-bound) and three-dimensional (solution-phase) KdS suggest that cell-cell contact favors CD2/LFA-3 interaction to a greater extent than that predicted by the three-dimensional Kd and the intermembrane distance at the site of contact. LFA-3 molecules in the contact site were capable of lateral diffusion in the plane of the phospholipid bilayer and did not appear to be irreversibly trapped in the contact area, consistent with a rapid off-rate. These data provide insights into the function of low affinity interactions in adhesion.

Study of the Effect of Boundary Conditions on Residual Stresses in Welding Using Element Birth and Element Movement Techniques

2003 ◽  
Vol 125 (4) ◽  
pp. 432-439 ◽  
Author(s):  
Ihab F. Z. Fanous ◽  
Maher Y. A. Younan ◽  
Abdalla S. Wifi
Keyword(s):  
Residual Stresses ◽  
Boundary Conditions ◽  
Three Dimensional ◽  
Welding Process ◽  
Computation Time ◽  
The Other ◽  
Two Dimensional ◽  
The Media ◽  
Three Dimensional Models ◽  
Element Movement

The structure in which the welding process is performed highly affects the residual stresses generated in the welding. This effect is simulated by choosing the appropriate boundary conditions in modeling the welding process. The major parameters of the boundary conditions are the method by which the base metal is being fixed and the amount of heat being applied through the torch. Other parameters may include the coefficients of thermal heat loss from the plate which may simulate the media in which the welding is taking place. In modeling the welding process, two-dimensional forms of approximation were developed in analyzing most of the models of such problem. Three-dimensional models analyzing the welding process were developed in limited applications due to its high computation time and cost. With the development of new finite element tools, namely the element movement technique developed by the authors, full three-dimensional analysis of the welding process is becoming in hand. In the present work, three different boundary conditions shall be modeled comparing their effect on the welding. These boundary conditions shall be applied to two models of the welding process: one using the element birth technique and the other using the element movement technique showing the similarity in their responses verifying the effectiveness of the latter being accomplished in a shorter time.

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