Dynamical Response of an Elastic Half-Space to Tangential Surface Loadings

1960 ◽  
Vol 27 (3) ◽  
pp. 559-567 ◽  
Author(s):  
Chi-Chang Chao
Keyword(s):  
Half Space ◽  
Tangential Force ◽  
Elastic Half Space ◽  
The Body ◽  
Plane Boundary ◽  
Tangential Displacement ◽  
Reciprocal Relation ◽  
Dynamical Response ◽  
Static Case ◽  
Vertical Force

Exact and closed-form solutions are obtained for both tangential and vertical surface displacements of a homogeneous isotropic elastic half-space due to the application, at a point on the surface, of a concentrated force tangential to the plane boundary and varying with time as the Heaviside unit function. Similar expressions for the displacements in the interior of the body along a line directly below the applied force are derived. Solutions are obtained by a semi-inverse method with the aid of Laplace and Hankel transforms. The reciprocal relation in the static case, between tangential displacement due to a vertical force and vertical displacement due to a tangential force, is preserved in the dynamic case.

scholarly journals Influence of tangential displacement on the adhesion strength of a contact between a parabolic profile and an elastic half-space

2017 ◽  
Vol 4 (8) ◽  
pp. 161010 ◽  
Author(s):  
Valentin L. Popov ◽  
Iakov A. Lyashenko ◽  
Alexander E. Filippov
Keyword(s):  
Half Space ◽  
Adhesion Strength ◽  
Adhesion Force ◽  
Tangential Force ◽  
Elastic Half Space ◽  
Tangential Displacement ◽  
Bodies Of Revolution ◽  
Parabolic Profile ◽  
Rotationally Symmetric ◽  
Adhesive Contacts

The adhesion strength of a contact between a rotationally symmetric indenter and an elastic half-space is analysed analytically and numerically using an extension of the method of dimensionality reduction for superimposed normal/tangential adhesive contacts. In particular, the dependence of the critical adhesion force on the simultaneously applied tangential force is obtained and the relevant dimensionless parameters of the problem are identified. The fracture criterion used coincides with that suggested by Johnson. In this paper, it is used to develop a method that is applicable straightforwardly to adhesive contacts of arbitrary bodies of revolution with compact contact area.

Dynamic response of an elastic half-space to time-dependent surface tractions over an embedded spherical cavity. III (With comments on a paper by R. D. Gregory)

1969 ◽  
Vol 309 (1498) ◽  
pp. 313-329 ◽  
Keyword(s):  
Dynamic Response ◽  
Half Space ◽  
Rayleigh Waves ◽  
Spherical Cavity ◽  
Elastic Half Space ◽  
Time Dependent ◽  
Plane Boundary ◽  
Surface Motion ◽  
Harmonic Solution ◽  
Time Harmonic

Discussion of the problem of an elastic half-space with spherical cavity is continued in respect of Rayleigh waves on the plane boundary. Displacements in the initial and first group of higher order Rayleigh waves are derived by using the time-harmonic solution developed in part I of this series with attention confined to the case of time-harmonic normal stress at the cavity. These are employed to find also the response to an exponential shock at the cavity and graphs are presented showing the surface motion due to the initial Rayleigh waves. Finally, in an appendix to the paper, some comments are given on a recent paper by R. D. Gregory on the problem of the half-space with cavity.

On Elastic Line Contact

1972 ◽  
Vol 39 (4) ◽  
pp. 1125-1132 ◽  
Author(s):  
J. J. Kalker
Keyword(s):  
Half Space ◽  
Surface Displacement ◽  
Elastic Half Space ◽  
Characteristic Dimension ◽  
The Body ◽  
Contact Theory ◽  
Total Force ◽  
Asymptotic Results ◽  
Elastic Line ◽  
Hertz Problem

Two-dimensional elastic half-space contact theory suffers from the defect that the surface displacement with respect to infinity becomes infinitely large when the total force carried by the half space is different from zero. Several authors removed this defect by altering the geometry so that the depth of the elastic body becomes finite. In the present paper another approach is chosen by considering contact areas which are many times as long as they are wide, but which still are small as compared with a characteristic dimension of the body which is approximated by the half space. As one of the examples, the Hertz problem is considered, and the asymptotic results are compared with the exact theory. It is found that errors of 10–15 percent are found when the contact ellipse is twice as long as wide, and errors of 2–3 percent are encountered when the ratio of the axes is five.

The azimuthal and polar radiation patterns obtained from a horizontal stress applied at the surface of an elastic half space

1962 ◽  
Vol 52 (1) ◽  
pp. 27-36
Author(s):  
J. T. Cherry
Keyword(s):  
Surface Wave ◽  
Half Space ◽  
Rayleigh Waves ◽  
Poisson’S Ratio ◽  
Horizontal Stress ◽  
Elastic Half Space ◽  
The Body ◽  
Body Waves ◽  
Poisson's Ratio ◽  
A Value

Abstract The body waves and surface waves radiating from a horizontal stress applied at the free surface of an elastic half space are obtained. The SV wave suffers a phase shift of π at 45 degrees from the vertical. Also, a surface wave that is SH in character but travels with the Rayleigh velocity is shown to exist. This surface wave attenuates as r−3/2. For a value of Poisson's ratio of 0.25 or 0.33, the amplitude of the Rayleigh waves from a horizontal source should be smaller than the amplitude of the Rayleigh waves from a vertical source. The ratio of vertical to horizontal amplitude for the Rayleigh waves from the horizontal source is the same as the corresponding ratio for the vertical source for all values of Poisson's ratio.

Fourier integral representation of the Green function for an anisotropic elastic half-space

1993 ◽  
Vol 443 (1918) ◽  
pp. 367-389 ◽  
Keyword(s):  
Green Function ◽  
Integral Representation ◽  
Half Space ◽  
Isotropic Material ◽  
Fourier Integral ◽  
Elastic Half Space ◽  
Material Symmetry ◽  
Vertical Force ◽  
Anisotropic Elastic ◽  
The Green Function

This paper describes the development of a Fourier integral representation of the Green function for an anisotropic elastic half-space. The representation for an isotropic material is integrated in closed form and shown to reduce to Mindlin’s solution. An application of the anisotropic representation is made to deduce the exact displacement caused by a two-dimensional periodic vertical force distribu­tion applied to the interior of a half-space with cubic material symmetry.

Self similar solutions to adhesive contact problems with incremental loading

1968 ◽  
Vol 305 (1480) ◽  
pp. 55-80 ◽  
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Half Space ◽  
Boundary Value ◽  
Contact Problems ◽  
Adhesive Contact ◽  
Elastic Half Space ◽  
The Body ◽  
Similar Solutions ◽  
Dimensional Argument

The boundary-value problem for axisymmetric distortion of an elastic half space by a rigid indentor is formulated. A dimensional argument is used to infer the form of the distribution of radial displacement within the contact circle in terms of the shape of the body, assuming the load to be applied progressively, with interfacial friction sufficient to prevent any slip taking place between the indentor and the half space. This obviates the need for solving a preliminary integral equation for the boundary conditions, as proposed by Goodman (1962) and Mossakovski (1963). The resulting boundary-value problem is cast in the form of an integral equation of Wiener-Hopf type, which has been solved in a separate paper (Spence 1968, referred to as II). The solution is used to calculate stresses, displacements and contact radii for adhesive indentation by (i) a flat faced cylinder, (ii) an almost flat conical indentor and (iii) a sphere. The results are compared with those for frictionless indentation, for a range of values of Poisson’s ratio (iv). Adhesive indentation of a half space by a sphere of radius R rolling with angular velocity ω and linear velocity V (excluding dynamical effects) is also treated, and a value found for the creep 1 ( V / R ω in the absence of torsional or tractive forces.

Reflection of plane waves from the boundary of an initially stressed incompressible half-space

2017 ◽  
Vol 24 (2) ◽  
pp. 406-433 ◽  
Author(s):  
M Shams
Keyword(s):  
Initial Stress ◽  
Half Space ◽  
Plane Waves ◽  
Strain Energy Function ◽  
Prototype Strain ◽  
Incompressible Material ◽  
Elastic Half Space ◽  
Theory Of Elasticity ◽  
Plane Boundary ◽  
Infinitesimal Deformations

In this paper, nonlinear theory of elasticity is used to study the effect of initial stress on plane waves in an incompressible material. For this problem, the initial stress is not associated with a finite elastic deformation and the material is assumed to be isotropic in the absence of the initial stress. The theory of superposition of infinitesimal deformations on finite deformation is applied to a problem of plane incremental motions in an initially stressed incompressible homogeneous elastic half-space. The general formulation of the problem is presented first and then specialized using a prototype strain energy function. Homogeneous plane waves are considered and the analysis is carried out for incompressible materials in both the deformed and the undeformed reference configurations. In addition to this, problems for the reflection of small amplitude homogeneous waves from the plane boundary of an initially stressed half-space are also considered and graphical results are included, which show the effect of initial stress on reflection. It is noted that the reflection coefficients in this case behave in a similar fashion when the initial stress is a pre-stress.

scholarly journals A STUDY ON GROSS SLIP AND FRETTING WEAR OF CONTACTS INVOLVING A POWER-LAW GRADED ELASTIC HALF-SPACE

2019 ◽  
Vol 17 (1) ◽  
pp. 47 ◽  
Author(s):  
Markus Hess
Keyword(s):  
Half Space ◽  
Power Law ◽  
Fatigue Cracks ◽  
Tangential Force ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Fretting Wear ◽  
Partial Slip ◽  
Graded Materials ◽  
Closed Form Solutions

For the steady wear state of two contact problems involving power-law graded materials, closed-form solutions are derived in terms of pressure distribution and limiting shapes of profile. Both gross slip of an initially flat-ended cylindrical punch on a power-law graded half-space and the load-controlled fretting wear under partial slip of an initially parabolic indenter are studied. In the case of gross slip at fixed penetration depth there exists a certain exponent of elastic inhomogeneity, for which the effective volume change takes its maximum value. To minimize wear due to fretting under partial slip, an amplitude dependent design of the material gradient is necessary. For large amplitudes of the tangential force a gradient ranged from a soft surface to a hard ground is beneficial, small amplitudes require a reverse gradient characterized by a hard surface and a soft ground. However, the choice of the material gradient also has a decisive influence on the strength of stress singularities at the contact edge and thus the initiation of fretting fatigue cracks, which is why it is discussed in more detail.

Sign in / Sign up

Export Citation Format

Share Document