Multiple Diffractions of Elastic Waves by a Rigid Rectangular Foundation: Plane-Strain Model

1976 ◽  
Vol 43 (2) ◽  
pp. 291-294 ◽  
Author(s):  
M. Dravinski ◽  
S. A. Thau
Keyword(s):  
Plane Strain ◽  
Half Space ◽  
Elastic Waves ◽  
Equation Of Motion ◽  
Elastic Half Space ◽  
Strain Type ◽  
Time Required ◽  
Strain Model ◽  
Rectangular Foundation ◽  
Plane Strain Model

A rigid rectangular foundation embedded in an elastic half space moves in a direction perpendicular to the surface of the half space, Fig. 1. The model under consideration is of the plane-strain type. By application of the Laplace, Fourier, and Kontorovich-Lebedev (K-L) transforms, the equation of motion for the foundation is derived. The transient response of the foundation is exact during the period of time required for a longitudinal wave to traverse the base of the foundation twice. Thus the process of multiple diffractions at the corners of the foundation is taken into account.

Multiple Diffractions of Elastic Shear Waves by a Rigid Rectangular Foundation Embedded in an Elastic Half Space

1976 ◽  
Vol 43 (2) ◽  
pp. 295-299 ◽  
Author(s):  
M. Dravinski ◽  
S. A. Thau
Keyword(s):  
Half Space ◽  
Equation Of Motion ◽  
Elastic Half Space ◽  
Embedment Depth ◽  
Sh Wave ◽  
Strain Type ◽  
Time Required ◽  
Elastic Shear ◽  
Base Width ◽  
Rectangular Foundation

A rigid rectangular foundation, embedded in an elastic half space, is subjected to a plane, transient, horizontally polarized shear (SH) wave. Embedment depth of the foundation and the angle of the incidence of the plane wave are assumed to be arbitrary. The problem considered is of the antiplane-strain type. The Laplace and Kontorovich-Lebedev transforms are employed to derive the equation of motion for the foundation during the period of time required for an SH-wave to traverse the base width of the obstacle twice. Therefore this solution includes the process of multiple diffractions at the corners of the foundation.

Multiple Region Contact Solutions for a Flat Indenter on a Layered Elastic Half Space: Plane-Strain Case

1989 ◽  
Vol 56 (2) ◽  
pp. 251-262 ◽  
Author(s):  
T. W. Shield ◽  
D. B. Bogy
Keyword(s):  
Plane Strain ◽  
Half Space ◽  
Contact Region ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Geometrical Parameters ◽  
Restricted Range ◽  
Complete Contact ◽  
The One ◽  
Layered Half Space

The plane-strain problem of a smooth, flat rigid indenter contacting a layered elastic half space is examined. It is mathematically formulated using integral transforms to derive a singular integral equation for the contact pressure, which is solved by expansion in orthogonal polynomials. The solution predicts complete contact between the indenter and the surface of the layered half space only for a restricted range of the material and geometrical parameters. Outside of this range, solutions exist with two or three contact regions. The parameter space divisions between the one, two, or three contact region solutions depend on the material and geometrical parameters and they are found for both the one and two layer cases. As the modulus of the substrate decreases to zero, the two contact region solution predicts the expected result that contact occurs only at the corners of the indenter. The three contact region solution provides an explanation for the nonuniform approach to the half space solution as the layer thickness vanishes.

A New Embedded-Zone Method on Computation of the Transverse Elastic Properties of Fiber-Reinforced Composites

2005 ◽  
Author(s):  
Xiaochun Wang
Keyword(s):  
Plane Strain ◽  
Elastic Properties ◽  
Fiber Reinforced Composites ◽  
Reinforced Composites ◽  
Fiber Reinforced ◽  
Zone Method ◽  
Two Dimension ◽  
The Matrix ◽  
Strain Model ◽  
Plane Strain Model

There are many methods on computation of transverse elastic properties of unidirectional fiber-reinforced composites when using the finite element method, such as three-dimension model, two-dimension plane strain model, unit cell model, etc[1]. But unit cell models could be used only when the fibers are arrayed regularly. The computations of three- and two-dimension plane strain models are tremendous when many fine fibers are spread randomly in the matrix so that the properties of block of composite must be computed. The paper proposes a new embedded-zone method to compute the transverse elastic properties for a block of fiber-reinforced composites containing a great amount of fibers embedded in the matrix stochastically while using very little computational work compared with three- and two-dimension plane strain model. The transverse elastic modulus and shear modulus of unidirectional fiber-reinforced composites are computed.

Thermal Stresses in Layered Electrical Assemblies Bonded With Solder

1991 ◽  
Vol 113 (4) ◽  
pp. 350-354 ◽  
Author(s):  
H. S. Morgan
Keyword(s):  
Residual Stresses ◽  
Plane Strain ◽  
3D Model ◽  
Thermal Stresses ◽  
Three Dimensional ◽  
Creep Models ◽  
Strain Model ◽  
Maximum Stresses ◽  
Generalized Plane Strain ◽  
Plane Strain Model

Thermal stresses in a layered electrical assembly joined with solder are computed with plane strain, generalized plane strain, and three-dimensional (3D) finite element models to assess the accuracy of the two-dimensional (2D) modeling assumptions. Cases in which the solder is treated as an elastic and as a creeping material are considered. Comparison of the various solutions shows that, away from the corners, the generalized plane strain model produces residual stresses that are identical to those computed with the 3D model. Although the generalized plane strain model cannot capture corner stresses, the maximum stresses computed with this 2D model are, for the mesh discretization used, within 12 percent of the corner stresses computed with the 3D model when the solder is modeled elastically and within 5 percent when the solder is modeled as a creeping material. Plane strain is not a valid assumption for predicting thermal stresses, especially when creep of the solder is modeled. The effect of cooling rate on the residual stresses computed with creep models is illustrated.

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