Arbitrary Load Distribution on a Layered Half Space

2000 ◽  
Vol 122 (4) ◽  
pp. 672-681 ◽  
Author(s):  
N. Schwarzer
Keyword(s):  
Half Space ◽  
Load Distribution ◽  
Elastic Field ◽  
Stress Fields ◽  
Method Of Images ◽  
Elementary Functions ◽  
Homogeneous Case ◽  
Arbitrary Load ◽  
Homogeneous Half Space ◽  
Layered Half Space

This paper develops a method which allows one to calculate the complete elastic field (stress field and displacements) of layered materials of transverse and complete isotropy under given load conditions. It is assumed that the layered body consists of an infinite half-space and various infinite planes which are all ideally bonded to each other. Thus, the interfaces are parallel to the surface of the resulting “coated half space.” The approach is based on the method of images in classical electrostatics. The final solution for an arbitrary load problem can be presented as a series of potential functions, where corresponding functions may be interpreted as “image loads” the analogous to “image charges.” The solution for the elastic field for any arbitrary stress distribution on the surface of the coated half space can be obtained in a relatively straightforward manner by using the method described here as long as the corresponding solution for the homogeneous half space is known. Further, if this solution of the homogeneous case may be expressed in terms of elementary functions, then the solution for the coated half space is elementary, too. Explicit formulas for the stress fields for some particular examples are given. [S0742-4787(00)01204-2]

Quasi-Static Strain and Stress Fields due to a Moment Tensor in Elastic–Viscoelastic Layered Half-Space

2014 ◽  
Vol 171 (8) ◽  
pp. 1669-1693 ◽  
Author(s):  
Akinori Hashima ◽  
Yukitoshi Fukahata ◽  
Chihiro Hashimoto ◽  
Mitsuhiro Matsu’ura
Keyword(s):  
Half Space ◽  
Moment Tensor ◽  
Stress Fields ◽  
Static Strain ◽  
Layered Half Space

The Elastic Field Resulting From Elliptical Hertzian Contact of Transversely Isotropic Bodies: Closed-Form Solutions for Normal and Shear Loading

1997 ◽  
Vol 64 (3) ◽  
pp. 457-465 ◽  
Author(s):  
M. T. Hanson ◽  
I. W. Puja
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Elastic Field ◽  
Transversely Isotropic ◽  
Normal Pressure ◽  
Shear Loading ◽  
Potential Functions ◽  
Hertzian Contact ◽  
Elementary Functions ◽  
Shear Traction

This analysis presents the elastic field in a half-space caused by an ellipsoidal variation of normal traction on the surface. Coulomb friction is assumed and thus the shear traction on the surface is taken as a friction coefficient multiplied by the normal pressure. Hence the shear traction is also of an ellipsoidal variation. The half-space is transversely isotropic, where the planes of isotropy are parallel to the surface. A potential function method is used where the elastic field is written in three harmonic functions. The known point force potential functions are utilized to find the solution for ellipsoidal loading by quadrature. The integrals for the derivatives of the potential functions resulting from ellipsoidal loading are evaluated in terms of elementary functions and incomplete elliptic integrals of the first and second kinds. The elastic field is given in closed-form expressions for both normal and shear loading.

The Elastic Field in a Half-Space With a Circular Cylindrical Inclusion

1996 ◽  
Vol 63 (4) ◽  
pp. 925-932 ◽  
Author(s):  
L. Z. Wu ◽  
S. Y. Du
Keyword(s):  
Half Space ◽  
Elastic Field ◽  
Elastic Half Space ◽  
Elliptic Integrals ◽  
Cylindrical Inclusion ◽  
Stress Fields ◽  
Function Technique ◽  
Explicit Solutions ◽  
Elastic Fields ◽  
Complete Elliptic Integrals

The problem of a circular cylindrical inclusion with uniform eigenstrain in an elastic half-space is studied by using the Green’s function technique. Explicit solutions are obtained for the displacement and stress fields. It is shown that the present elastic fields can be expressed as functions of the complete elliptic integrals of the first, second, and third kind. Finally, numerical results are shown for the displacement and stress fields.

Elastic Displacement and Stress Fields Induced by a Dislocation of Polygonal Shape in an Anisotropic Elastic Half-Space

2012 ◽  
Vol 79 (2) ◽  
Author(s):  
H. J. Chu ◽  
E. Pan ◽  
J. Wang ◽  
I. J. Beyerlein
Keyword(s):  
Half Space ◽  
Dislocation Loop ◽  
Hydrostatic Stress ◽  
Jump Condition ◽  
Elastic Half Space ◽  
Biaxial Stress ◽  
Stress Fields ◽  
Elastic Displacement ◽  
Homogeneous Half Space ◽  
Anisotropic Elastic

The elastic displacement and stress fields due to a polygonal dislocation within an anisotropic homogeneous half-space are studied in this paper. Simple line integrals from 0 to π for the elastic fields are derived by applying the point-force Green’s functions in the corresponding half-space. Notably, the geometry of the polygonal dislocation is included entirely in the integrand easing integration for any arbitrarily shaped dislocation. We apply the proposed method to a hexagonal shaped dislocation loop with Burgers vector along [1¯ 1 0] lying on the crystallographic (1 1 1) slip plane within a half-space of a copper crystal. It is demonstrated numerically that the displacement jump condition on the dislocation loop surface and the traction-free condition on the surface of the half-space are both satisfied. On the free surface of the half-space, it is shown that the distributions of the hydrostatic stress (σ11 + σ22)/2 and pseudohydrostatic displacement (u1 + u2)/2 are both anti-symmetric, while the biaxial stress (σ11 − σ22)/2 and pseudobiaxial displacement (u1 − u2)/2 are both symmetric.

Controlled source tensor magnetotellurics

Geophysics ◽  
1991 ◽  
Vol 56 (9) ◽  
pp. 1456-1461 ◽  
Author(s):  
Xiaobo Li ◽  
Laust B. Pedersen
Keyword(s):  
Half Space ◽  
Transfer Functions ◽  
Unit Vector ◽  
Impedance Tensor ◽  
Wave Excitation ◽  
Controlled Source ◽  
Practice Analysis ◽  
Homogeneous Half Space ◽  
Dipole Sources ◽  
Layered Half Space

Impedance tensor and tipper vectors, known to connect the electromagnetic surface components for plane‐wave excitation, are shown to be uniquely defined for horizontal electric or horizontal magnetic dipole sources. Two independent source polarizations are needed for their estimation in practice. Analysis of impedance tensors and tipper vectors for a layered half‐space shows that the impedance tensor can be antidiagonalized by rotating the measurement system so that one of the measurement directions coincides with the direction to the transmitter dipole. The tipper vector points towards the transmitter dipole. In the static limit, all transfer functions are real, and simple analytic results for a homogeneous half‐space show that impedance elements are proportional to the inverse of the product of conductivity and distance between source and receiver, while the tipper vector is a unit vector pointing towards the transmitter dipole.

scholarly journals Seismic Rayleigh waves on an exponentially graded, orthotropic half-space

2006 ◽  
Vol 463 (2078) ◽  
pp. 495-502 ◽  
Author(s):  
Michel Destrade
Keyword(s):  
Half Space ◽  
Rayleigh Waves ◽  
Seismic Waves ◽  
Symmetry Axis ◽  
Mass Density ◽  
Homogeneous Case ◽  
Marked Contrast ◽  
Homogeneous Half Space ◽  
Shear Horizontal ◽  
Exponentially Graded

Efforts at modelling the propagation of seismic waves in half-spaces with continuously varying properties have mostly been focused on shear-horizontal waves. Here, a sagittally polarized (Rayleigh type) wave travels along a symmetry axis (and is attenuated along another) of an orthotropic material with stiffnesses and mass density varying in the same exponential manner with depth. In contrast to what could be expected at first sight, the analysis is very similar to that of the homogeneous half-space, with the main and capital difference that the Rayleigh wave is now dispersive. The results are illustrated numerically for (i) an orthotropic half-space typical of horizontally layered and vertically fractured shales and (ii) for an isotropic half-space made of silica. In both the examples, the wave travels at a slower speed and penetrates deeper than in the homogeneous case. In the second example, the inhomogeneity can force the wave amplitude to oscillate as well as decay with depth, in marked contrast with the homogeneous isotropic general case.

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