Efficient Numerical Modeling of Hertzian Line Contact for Material With Inhomogeneities

2012 ◽  
Author(s):  
Xiaoqing Jin ◽  
Zhanjiang Wang ◽  
Qinghua Zhou ◽  
Leon M. Keer ◽  
Qian Wang
Keyword(s):  
Half Plane ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
General Purpose ◽  
Form Solution ◽  
Inclusion Problem ◽  
Computational Domain ◽  
Line Contact ◽  
Equivalent Inclusion Method ◽  
Parametric Studies

The present work proposes an efficient and general-purpose numerical approach for handling two-dimensional inhomogeneities in an elastic half plane. The inhomogeneities can be of any shape, at any location, with arbitrary material properties (which can also be non-homogeneous). To perform the numerical analysis, we first derive an explicit closed-form solution for a rectangular inclusion with uniform eigenstrain components, where the inclusion is aligned with the surface of the half plane. In view of the equivalent inclusion method, an inhomogeneity problem can be converted to a corresponding inclusion problem. In order to determine the distribution of the equivalent eigenstrain, the computational domain is meshed into rectangular elements whose resultant contributions can be efficiently computed using an efficient algorithm based on fast Fourier transform (FFT). In principle, there is no specific limitation on the type of the external load, although our major concern is the contact analysis. Parametric studies are performed and typical results highlighting the deviation of the current solution from the classical Hertzian line contact theory are presented.

Diffraction by a half-plane perpendicular to the distinguished axis of a general gyrotropic medium

1977 ◽  
Vol 55 (4) ◽  
pp. 305-324 ◽  
Author(s):  
S. Przeździecki ◽  
R. A. Hurd
Keyword(s):  
Half Plane ◽  
Fourier Transforms ◽  
Diffraction Problem ◽  
Plane Waves ◽  
Closed Form Solution ◽  
Basic Feature ◽  
Form Solution ◽  
Electromagnetic Plane Wave ◽  
Edge Diffraction ◽  
Special Cases

An exact, closed-form solution is found for the following half-plane diffraction problem: (I) The medium surrounding the half-plane is both electrically and magnetically gyrotropic. (II) The scattering half-plane is perfectly conducting and oriented perpendicular to the distinguished axis of the medium. (III) The incident electromagnetic plane wave propagates in a direction normal to the edge of the half-plane.The formulation of the problem leads to a coupled pair of Wiener–Hopf equations. These had previously been thought insoluble by quadratures, but yield to a newly discovered technique : the Wiener–Hopf–Hilbert method. A basic feature of the problem is its two-mode character i.e. plane waves of both modes are necessary for the spectral representation of the solution. The coupling of these modes is purely due to edge diffraction, there being no reflection coupling. The solution obtained is simple in that the Fourier transforms of the field components are just algebraic functions. Properties of the solution are investigated in some special cases.

Scattering by hard and soft parallel half-planes

1981 ◽  
Vol 59 (12) ◽  
pp. 1879-1885 ◽  
Author(s):  
R. A. Hurd ◽  
E. Lüneburg
Keyword(s):  
Plane Wave ◽  
Closed Form ◽  
Half Plane ◽  
Diffraction Problem ◽  
Closed Form Solution ◽  
Normal Derivative ◽  
Form Solution ◽  
The Other ◽  
Total Field

We consider the diffraction of a scalar plane wave by two parallel half-planes. On one half-plane the total field vanishes whilst on the other its normal derivative is zero. This is a new canonical diffraction problem and we give an exact closed-form solution to it. The problem has applications to the design of acoustic barriers.

The Sliding With Coulomb Friction of a Rigid Indentor Over a Power-Law Inhomogeneous Linearly Viscoelastic Half-Plane

1984 ◽  
Vol 51 (2) ◽  
pp. 289-293 ◽  
Author(s):  
J. R. Walton
Keyword(s):  
Shear Modulus ◽  
Half Plane ◽  
Power Law ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Plane Boundary ◽  
Physical Parameters ◽  
Title Problem ◽  
Rigid Indentor

In a previous paper, the title problem was solved for a homogeneous power-law linearly viscoelastic half-plane. Such material has a constant Poisson’s ratio and a shear modulus with a power-law dependence on time. In this paper, the shear modulus is assumed also to have a power-law dependence on depth from the half-plane boundary. As in the earlier paper, only a quasi-static analysis is presented, that is, the enertial terms in the equations of motion are not retained and the indentor is assumed to slide with constant speed. The resulting boundary value problem is reduced to a generalized Abel integral equation. A simple closed-form solution is obtained from which all relevant physical parameters are easily computed.

A Closed-Form Solution for the Eshelby Tensor and the Elastic Field Outside an Elliptic Cylindrical Inclusion

2011 ◽  
Vol 78 (3) ◽  
Author(s):  
Xiaoqing Jin ◽  
Leon M. Keer ◽  
Qian Wang
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Elastic Field ◽  
Form Solution ◽  
Eshelby Tensor ◽  
Cylindrical Inclusion ◽  
Closed Form Expression ◽  
Equivalent Inclusion Method ◽  
Explicit Analytical Solution ◽  
Present Solution

From the analytical formulation developed by Ju and Sun [1999, “A Novel Formulation for the Exterior-Point Eshelby’s Tensor of an Ellipsoidal Inclusion,” ASME Trans. J. Appl. Mech., 66, pp. 570–574], it is seen that the exterior point Eshelby tensor for an ellipsoid inclusion possesses a minor symmetry. The solution to an elliptic cylindrical inclusion may be obtained as a special case of Ju and Sun’s solution. It is noted that the closed-form expression for the exterior-point Eshelby tensor by Kim and Lee [2010, “Closed Form Solution of the Exterior-Point Eshelby Tensor for an Elliptic Cylindrical Inclusion,” ASME Trans. J. Appl. Mech., 77, p. 024503] violates the minor symmetry. Due to the importance of the solution in micromechanics-based analysis and plane-elasticity-related problems, in this work, the explicit analytical solution is rederived. Furthermore, the exterior-point Eshelby tensor is used to derive the explicit closed-form solution for the elastic field outside the inclusion, as well as to quantify the elastic field discontinuity across the interface. A benchmark problem is used to demonstrate a valuable application of the present solution in implementing the equivalent inclusion method.

A Symbolic-Numerical Approach for the Synthesis of Planar Rigid-Body Guidance

1994 ◽  
Author(s):  
Yong Chen ◽  
Jinyuan Fu
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Fourth Order ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
Form Solution ◽  
System Of Nonlinear Equations ◽  
Single Variable ◽  
All Solutions ◽  
Order Algebraic

Abstract Based on the pole method and the computer symbolic manipulating technique, a fourth order algebraic equation with single variable has been derived for the closed form solution of planar rigid-body guidance. The newly developed method eliminates the requirement of sloving system of nonlinear equations. All solutions of the problem can be obtained directly and easily without the need of initial values. A numerical example is given demonstrating the efficiency and advantages of the method.

The effect of a nonlinear Mohr–Coulomb criterion on borehole stresses and damage-zone estimate

1994 ◽  
Vol 31 (1) ◽  
pp. 104-109 ◽  
Author(s):  
Yarlong Wang
Keyword(s):  
Oil Sands ◽  
Damage Zone ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
Solution Procedure ◽  
Form Solution ◽  
Far Field ◽  
Circular Opening ◽  
Oil Sand ◽  
Coulomb Criterion

To investigate the possible error introduced by the assumption of a linear Mohr–Coulomb criterion on the stress distribution near a circular opening, a numerical approach is used to calculate both the damage-zone radius and the stresses near a circular opening. A general solution procedure for the stress calculation under a uniform far-field loading is presented and a closed-form solution for the near-opening stresses is given for a cohesionless medium. Based on the numerical results calculated, it is concluded that the linear assumption is acceptable for a strong rock with an unconfined compressive strength that is of similar magnitude to the far-field stress. However, the tangential stress calculated with this linear assumption may be overestimated, but the damage zone may be underestimated in weaker rock or soil such as an oil sand formation. Key words : Mohr–Coulomb criterion, circular opening, nonlinearity, oil sands, Gauss–Legendre method.

Closed-Form Solution of the Frictional Sliding Contact Problem for an Orthotropic Elastic Half-Plane Indented by a Wedge-Shaped Punch

2014 ◽  
Vol 618 ◽  
pp. 203-225 ◽  
Author(s):  
Aysegul Kucuksucu ◽  
Mehmet A. Guler ◽  
Ahmet Avci
Keyword(s):  
Contact Problem ◽  
Half Plane ◽  
Mixed Boundary ◽  
Closed Form Solution ◽  
Contact Problems ◽  
Form Solution ◽  
Orthotropic Materials ◽  
Principal Axes ◽  
The Coefficient Of Friction ◽  
Processing Techniques

In this paper, the frictional contact problem of a homogeneous orthotropic material in contact with a wedge-shaped punch is considered. Materials can behave anisotropically depending on the nature of the processing techniques; hence it is necessary to develop an efficient method to solve the contact problems for orthotropic materials. The aim of this work is to develop a solution method for the contact mechanics problems arising from a rigid wedge-shaped punch sliding over a homogeneous orthotropic half-plane. In the formulation of the plane contact problem, it is assumed that the principal axes of orthotropy are parallel and perpendicular to the contact. Four independent engineering constants , , , are replaced by a stiffness parameter, , a stiffness ratio, a shear parameter, , and an effective Poisson’s ratio, . The corresponding mixed boundary problem is reduced to a singular integral equation using Fourier transform and solved analytically. In the parametric analysis, the effects of the material orthotropy parameters and the coefficient of friction on the contact stress distributions are investigated.

Moving Loads on an Elastic Half-Plane With Hysteretic Damping

2001 ◽  
Vol 68 (6) ◽  
pp. 915-922 ◽  
Author(s):  
A. Verruijt ◽  
C. Cornejo Co´rdova
Keyword(s):  
Wave Velocity ◽  
Half Plane ◽  
High Speed ◽  
Closed Form Solution ◽  
Point Load ◽  
Form Solution ◽  
Moving Loads ◽  
Hysteretic Damping ◽  
Loading And Unloading ◽  
Dynamic Amplification

A closed-form solution is presented for the problem of a moving point load on an elastic half-plane with hysteretic damping. The problem has been studied in order to investigate the dynamic amplification of stresses and displacements if the velocity of the load approaches the Rayleigh wave velocity or the shear wave velocity in the elastic medium. This is relevant for the construction of high speed railway lines on relatively soft soils. Hysteretic damping is introduced as pseudo-viscous damping, assuming that the damping in a full cycle of loading and unloading is independent of the frequency.

The Frictional Contact Problem of a Rigid Stamp Sliding over a Graded Medium

2016 ◽  
Vol 681 ◽  
pp. 155-174 ◽  
Author(s):  
M.A. Guler ◽  
M. Ozturk ◽  
A. Kucuksucu
Keyword(s):  
Contact Problem ◽  
Half Plane ◽  
Frictional Contact ◽  
Fourier Transforms ◽  
Closed Form Solution ◽  
Form Solution ◽  
Linear Elastic ◽  
Rigid Stamp ◽  
Sharp Edges ◽  
Parameter Coefficient

In this study, the contact problem for a graded elastic half-plane in frictional contact with a rigid stamp is considered. The plane contact problem is assumed to be linear elastic and the Poisson's ratio is assumed to be constant. Analytical formulation of the study includes Fourier transforms of the governing equations and boundary conditions. The resulting integral equation is solved numerically. Contact pressure, in-plane stress and the stress intensity factor at the sharp edges of the contact are evaluated and demonstrated for various stamp profiles. The results are compared with a closed form solution for homogeneous isotropic half-plane indented by rigid stamps. The effects of the nonhomogeneity parameter, coefficient of friction and stamp profiles on the contact and in-plane stresses are analyzed in detail.

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