A normal crack in an elastic half-space with stress-free surface

1993 ◽  
Vol 16 (8) ◽  
pp. 563-579 ◽  
Author(s):  
P. A. Martin ◽  
L. Päivärinta ◽  
S. Rempel
Keyword(s):  
Free Surface ◽  
Half Space ◽  
Elastic Half Space

scholarly journals Transient Excitation of an Elastic Half Space by a Point Load Traveling on the Surface

1969 ◽  
Vol 36 (3) ◽  
pp. 505-515 ◽  
Author(s):  
D. C. Gakenheimer ◽  
J. Miklowitz
Keyword(s):  
Exact Solution ◽  
Free Surface ◽  
Steady State ◽  
Wave Front ◽  
Half Space ◽  
Displacement Field ◽  
Elastic Half Space ◽  
Point Load ◽  
Limit Case ◽  
Transient Waves

The propagation of transient waves in a homogeneous, isotropic, linearly elastic half space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are derived for the interior of the half space and for all load speeds. Wave-front expansions are obtained from the exact solution, in addition to results pertaining to the steady-state displacement field. The limit case of zero load speed is considered, yielding new results for Lamb’s point load problem.

scholarly journals Dynamic Stress and Displacement in an Elastic Half-Space with a Cylindrical Cavity

2011 ◽  
Vol 18 (6) ◽  
pp. 827-838 ◽  
Author(s):  
İ. Coşkun ◽  
H. Engin ◽  
A. Özmutlu
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Half Space ◽  
Cylindrical Cavity ◽  
Circular Cross Section ◽  
Elastic Half Space ◽  
Least Square ◽  
Polar Coordinates ◽  
Wave Number ◽  
Forcing Function

The dynamic response of an elastic half-space with a cylindrical cavity in a circular cross-section is analyzed. The cavity is assumed to be infinitely long, lying parallel to the plane-free surface of the medium at a finite depth and subjected to a uniformly distributed harmonic pressure at the inner surface. The problem considered is one of plain strain, in which it is assumed that the geometry and material properties of the medium and the forcing function are constant along the axis of the cavity. The equations of motion are reduced to two wave equations in polar coordinates with the use of Helmholtz potentials. The method of wave function expansion is used to construct the displacement fields in terms of the potentials. The boundary conditions at the surface of the cavity are satisfied exactly, and they are satisfied approximately at the free surface of the half-space. Thus, the unknown coefficients in the expansions are obtained from the treatment of boundary conditions using a collocation least-square scheme. Numerical results, which are presented in the figures, show that the wave number (i.e., the frequency) and depth of the cavity significantly affect the displacement and stress.

MATHEMATICAL ANALYSIS OF ELASTIC SURFACE WAVES IN TOPOGRAPHIC WAVEGUIDES

1999 ◽  
Vol 09 (05) ◽  
pp. 755-798 ◽  
Author(s):  
A. S. BONNET-BEN DHIA ◽  
J. DUTERTE ◽  
P. JOLY
Keyword(s):  
Free Surface ◽  
Half Space ◽  
Guided Waves ◽  
Transverse Energy ◽  
Eigenvalue Problems ◽  
Low Frequency ◽  
Elastic Half Space ◽  
High Frequencies ◽  
Guided Mode ◽  
Adjoint Operators

We present here a theoretical study of the guided waves in an isotropic homogeneous elastic half-space whose free surface has been deformed. The deformation is supposed to be invariant in the propagation direction and localized in the transverse ones. We show that finding guided waves amounts to solving a family of 2-D eigenvalue problems set in the cross-section of the propagation medium. Then using the min-max principle for non-compact self-adjoint operators, we prove the existence of guided waves for some particular geometries of the free surface. These waves have a smaller speed than that of the Rayleigh wave in the perfect half-space and a finite transverse energy. Moreover, we prove that the existence results are valid for arbitrary high frequencies in the presence of singularities of the free boundary. Finally, we prove that no guided mode can exist at low frequency, except maybe the fundamental one.

scholarly journals Approximate formula for the H/V ratio of Rayleigh waves in incompressible orthotropic half-spaces coated by a thin elastic layer

2017 ◽  
Vol 39 (4) ◽  
pp. 365-374
Author(s):  
Pham Chi Vinh ◽  
Tran Thanh Tuan ◽  
Le Thi Hue
Keyword(s):  
Free Surface ◽  
Rayleigh Wave ◽  
Half Space ◽  
Rayleigh Waves ◽  
Elastic Layer ◽  
Approximate Formula ◽  
Elastic Half Space ◽  
Displacement Amplitude ◽  
Vertical Displacements ◽  
Thin Elastic Layer

This paper is concerned with the propagation of Rayleigh waves in an incompressible orthotropic elastic half-space coated with a thin incompressible orthotropic elastic layer. The main purpose of the paper is to establish an approximate formula for the Rayleigh wave H/V ratio (the ratio between the amplitudes of the horizontal and vertical displacements of Rayleigh waves at the traction-free surface of the layer). First, the relations between the traction amplitude vector and the displacement amplitude vector of Rayleigh waves at two sides of the interface between the layer and the half-space are created using the Stroh formalism and the effective boundary condition method. Then, an approximate formula for the Rayleigh wave H/V ratio of third-order in terms of dimensionless thickness of the layer has been derived by using these relations along with the Taylor expansion of the displacement amplitude vector of the thin layer at its traction-free surface. It is shown numerically that the obtained formula is a good approximate one. It can be used for extracting mechanical properties of thin films from measured values of the  Rayleigh wave H/V ratio.

scholarly journals Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects

2016 ◽  
Vol 472 (2186) ◽  
pp. 20150800 ◽  
Author(s):  
R. Chebakov ◽  
J. Kaplunov ◽  
G. A. Rogerson
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Boundary Conditions ◽  
Half Space ◽  
Elastic Half Space ◽  
Interior Solution ◽  
Homogeneous Half Space ◽  
Classical Boundary ◽  
Non Local ◽  
Rayleigh Wave Speed

The dynamic response of a homogeneous half-space, with a traction-free surface, is considered within the framework of non-local elasticity. The focus is on the dominant effect of the boundary layer on overall behaviour. A typical wavelength is assumed to considerably exceed the associated internal lengthscale. The leading-order long-wave approximation is shown to coincide formally with the ‘local’ problem for a half-space with a vertical inhomogeneity localized near the surface. Subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface. The order of the correction is greater than the order of the better-known correction to the governing differential equations. The refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface. As an illustration, the effect of non-local elastic phenomena on the Rayleigh wave speed is investigated.

Surface Instability and Splitting in Compressed Brittle Elastic Solids Containing Crack Arrays

1982 ◽  
Vol 49 (4) ◽  
pp. 761-767 ◽  
Author(s):  
L. M. Keer ◽  
S. Nemat-Nasser ◽  
A. Oranratnachai
Keyword(s):  
Free Surface ◽  
Half Space ◽  
Special Class ◽  
Compressive Force ◽  
Surface Damage ◽  
Elastic Half Space ◽  
Surface Instability ◽  
Elastic Solids ◽  
Longitudinal Splitting ◽  
Brittle Solids

Surface instability in brittle solids may occur at relatively small values of the inplane compressive force if the solid contains shallow cracks parallel to its free surface. The instability may produce surface damage by spallation. Similarly, the buckling load of a longitudinally compressed strip that contains an array of central cracks is affected to a great extent by the size and the relative spacing of these cracks. The instability in this case may result in longitudinal splitting of the strip. To illustrate these phenomena, the compression of an elastic half space and a layer, each containing an array of coplanar equally spaced cracks, is studied for a special class of hypoelastic materials, and the corresponding weakening due to cracks is analytically estimated.

Oscillation of a Punch Moving on the Free Surface of an Elastic Half Space

2004 ◽  
Vol 75 (3) ◽  
pp. 247-265 ◽  
Author(s):  
S. N. Gavrilov ◽  
G. C. Herman
Keyword(s):  
Free Surface ◽  
Half Space ◽  
Elastic Half Space
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