SOME APPLICATIONS OF THE FRACTAL PARAMETRIC-HOMOGENEOUS FUNCTIONS

Fractals ◽  
1994 ◽  
Vol 02 (02) ◽  
pp. 311-314 ◽  
Author(s):  
FEODOR M. BORODICH
Keyword(s):  
Fractal Dimension ◽  
Half Space ◽  
Solid Mechanics ◽  
Contact Problems ◽  
Field Equations ◽  
Arbitrary Degree ◽  
Homogeneous Functions ◽  
Hertz Problem

The purpose of this paper is to present some recent results concerning applications of so-called parametric-homogeneous (ph) functions to some problems of solid mechanics. The ph-functions have often fractal graphs and can be nowhere differentiable. An example of ph-function is the Weierstrass-Mandelbrot one. We introduce some ways of construction of the ph-function of both arbitrary degree and fractal dimension. We consider the discrete Hertz problem of contact between a punch, whose shape is described by a positive ph-function, and a deformable half-space. General expressions for changes of all functions, giving the solutions of the contact problems for a fractal punch, are derived exactly, without solving the field equations.

Bleustein-Gulyaev Waves in an Inhomogeneous Layered Piezoelectric Half-Space

2006 ◽  
Vol 306-308 ◽  
pp. 1223-1228
Author(s):  
Fei Peng ◽  
Hua Rui Liu
Keyword(s):  
Fourier Transform ◽  
Phase Velocity ◽  
Half Space ◽  
Electric Potential ◽  
Piezoelectric Layer ◽  
Field Equations ◽  
Transform Method ◽  
Coupled Field ◽  
Coupled Field Equations ◽  
The Fourier Transform

The propagation of Bleustein-Gulyaev (BG) waves in an inhomogeneous layered piezoelectric half-space is investigated in this paper. Application of the Fourier transform method and by solving the electromechanically coupled field equations, solutions to the mechanical displacement and electric potential are obtained for the piezoelectric layer and substrate, respectively. The phase velocity equations for BG waves are obtained for the surface electrically shorted case. When the layer and the substrate are homogenous, the dispersion equations are in agreement with the corresponding results. Numerical calculations are performed for the case that the layer and the substrate are identical LiNbO3 except that they are polarized in opposite directions. Effects of the inhomogeneities induced by either the layer or substrate are discussed in detail.

Steady-State Motion of a Line Mechanical/Heat Source Over a Half-Space: A Thermoelastodynamic Solution

1997 ◽  
Vol 64 (3) ◽  
pp. 562-567 ◽  
Author(s):  
L. M. Brock ◽  
H. G. Georgiadis
Keyword(s):  
Steady State ◽  
Heat Source ◽  
Half Space ◽  
Constant Velocity ◽  
Characteristic Length ◽  
Surface Displacement ◽  
Contact Problems ◽  
Governing Equations ◽  
Asymptotic Expressions ◽  
Thermoelastic Characteristic

An asymptotic solution within the bounds of steady-state coupled thermoelastodynamic theory is given for the surface displacement and temperature due to a line mechanical/heat source that moves at a constant velocity over the surface of a half-space. This problem is of basic interest in the fields of contact mechanics and tribology, and an exact formulation is considered. The results may serve as a Green’s function for more general thermoelastodynamic contact problems. The problem may also be viewed as a generalization of the classical Cole-Huth problem and the associated Georgiadis-Barber correction. Asymptotic expressions are obtained by means of the two-sided Laplace transform, and by performing the inversions exactly. The range of validity of these expressions is actually quite broad, because of the small value of the thermoelastic characteristic length appearing in the governing equations.

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