scholarly journals Sliding Contact Between a Cylindrical Punch and a Graded Half-Plane With an Arbitrary Gradient Direction

2015 ◽  
Vol 82 (4) ◽  
Author(s):  
Chen Peijian ◽  
Chen Shaohua ◽  
Peng Juan
Keyword(s):  
Half Plane ◽  
Integral Transformation ◽  
Contact Region ◽  
Fourier Integral ◽  
Graded Materials ◽  
Special Cases ◽  
Cylindrical Punch ◽  
The Moment ◽  
Gradient Variation ◽  
Stable Sliding

Contact behavior of a rigid cylindrical punch sliding on an elastically graded half-plane with shear modulus gradient variation in an arbitrary direction is investigated. The governing partial differential equations and the boundary conditions are achieved with the help of Fourier integral transformation. As a result, the present problem is reduced to a singular integral equation of the second kind, which can be solved numerically. Furthermore, the presently general model can be well degraded to special cases of a lateral gradient half-plane and a homogeneous one. Normal stress in the contact region is predicted with different material parameters, which is usually used to estimate the possibility of surface crack initiation. The moment that is needed to ensure stable sliding of the cylindrical punch on the contact surface is further predicted. The result in the present paper should be helpful for the design of novel graded materials with surfaces of strong abrasion resistance.

scholarly journals Consideration of wear in plane contact of rectangular punch and elastic half-plane

2019 ◽  
pp. 138-141
Author(s):  
V. M. Onyshkevych ◽  
G. T. Sulym
Keyword(s):  
Half Plane ◽  
Integral Transformation ◽  
Initial Approximation ◽  
Vertical Displacement ◽  
Fourier Integral ◽  
Mean Value ◽  
Theory Of Elasticity ◽  
Contact Stresses ◽  
Dual Integral Equations ◽  
Plane Contact

The plane contact problem on wear of elastic half-plane by a rigid punch has been considered. The punch moves with constant velocity. Arising thermal effects are neglected because the problem is investigated in stationary statement. In this case the crumpling of the nonhomogeneities of the surfaces and abrasion of half-plane take place. Out of the punch the surface of half-plane is free of load. The solution for problem of theory of elasticity is constructed by means of Fourier integral transformation. Contact stresses are found in Fourier series which coefficients satisfy the dual integral equations. It leads to the system of nonlinear algebraical equations for unknown coefficients by a method of collocations. This system is reduced to linear system in the partial most interesting cases for computing of maximum and minimum wear. The iterative scheme is considered for investigation of other nonlinear cases, for initial approximation the mean value of boundary cases is used. The evolutions of contact stresses, wear and abrasion in the time are given. For both last cases increase or invariable of vertical displacement correspondently is obtained. In the boundary cases coincidence of results with known is obtained.

Dynamical (harmonic) interface stress field in the half-plane covered by the prestretched layer under a strip load

2005 ◽  
Vol 40 (3) ◽  
pp. 225-234 ◽  
Author(s):  
S D Akbarov ◽  
C Guler
Keyword(s):  
Stress Distribution ◽  
Half Plane ◽  
Integral Transformation ◽  
Three Dimensional ◽  
Engineering Application ◽  
Fourier Integral ◽  
Dynamical Problem ◽  
Harmonic Load ◽  
Normal Stress Distribution ◽  
Application Fields

Within the framework of the piecewise homogeneous body model, by employing the three-dimensional linearized theory of elastic waves in initially stressed bodies the dynamical problem of the stress distribution in a half-plane covered with a prestretched layer is investigated. It is assumed that the free face plane of the covered layer is subjected to a uniformly distributed harmonic load acting on a strip extending to infinity in the x3 direction, which is perpendicular to the x1-x2 plane and is of width 2a in the x1 direction. The plane-strain state in the x1-x2 plane is analysed. The corresponding boundary-value problems are investigated by employing the exponential Fourier integral transformation. The numerical results regarding the interface normal stress distribution are presented. The influences of the problem parameters and pre-stretching of the covered layer on this distribution are analysed. Practical engineering application fields of the results are suggested.

scholarly journals Exact Solutions of Brinkman Type Fluid Between Side Walls Over an Infinite Plate

2018 ◽  
Author(s):  
Saeed Islam ◽  
Muhammad Asif ◽  
Samiul Haq
Keyword(s):  
Integral Transformation ◽  
Infinite Plate ◽  
Fourier Integral ◽  
Bottom Plate ◽  
Physical Parameters ◽  
Shear Stresses ◽  
Special Cases ◽  
Depth Analysis ◽  
Side Walls ◽  
Oscillating Shear Stress

In this paper Brinkman type fluid over an infinite plate between side walls is being investigated. The flow is generated by oscillating shear stress of the bottom plate and the solutions are obtained by using Fourier integral transformation. The obtained results are presented in steady and transient states for both sin and cos shear stresses. The general solutions are reduced to some special cases corresponding, namely to the Brinkman type fluid over an infinite plate and flow of a Newtonian viscous fluid. Graphical illustrations are carried out to have in depth analysis of the involved physical parameters

The Influence of Grain Size and Grain Size Distribution on Sliding Frictional Contact in Laterally Graded Materials

2012 ◽  
Vol 157-158 ◽  
pp. 964-969 ◽  
Author(s):  
Romik Khajehtourian ◽  
Saeed Adibnazari ◽  
Samaneh Tashi
Keyword(s):  
Grain Size ◽  
Half Plane ◽  
Frictional Contact ◽  
Crack Nucleation ◽  
Functionally Graded ◽  
Scale Model ◽  
Effective Parameters ◽  
Graded Materials ◽  
Macro Scale ◽  
Comparison Of The Results

The sliding frictional contact problem for a laterally graded half-plane has been considered. Two finite element (FE) models, in macro and micro scales have been developed to investigate the effective parameters in contact mechanics of laterally graded materials loaded by flat and triangular rigid stamps. In macro scale model, the laterally graded half-plane is discretized by piecewise homogeneous layers for which the material properties are specified at the centroids by Mori-Tanaka method. In micro scale model, functionally graded material (FGM) structure has been modeled as ideal solid quadrant particles which are spatially distributed in a homogeneous matrix. Boundary conditions and loading is the same in both models. The microstructure has modeled as rearrangement and sizes changing of particles are possible to provide the possibility of crack nucleation investigation in non-singular regions. Analyses and comparison of the results showed that micro and macro scale results are in very good agreement. Also, increasing the grains aspect ratio and using optimum distribution of grains decrease stress distribution roughness on the surface. Therefore, the possibility of surface cracking far from stamp’s edges decreased.

On the Interaction at Anti-Flat Deformation of Stress Concentrators of the Type of Cracks and Stringers with Regard to the Layer Manufactured from Miscellaneous Materials

2019 ◽  
Vol 828 ◽  
pp. 81-88
Author(s):  
Nune Grigoryan ◽  
Mher Mkrtchyan
Keyword(s):  
Integral Transform ◽  
Composite Layer ◽  
Singular Integral Equations ◽  
Original System ◽  
Fourier Integral ◽  
Quadrature Formulas ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Tangential Forces

In this paper, we consider the problem of determining the basic characteristics of the stress state of a composite in the form of a piecewise homogeneous elastic layer reinforced along its extreme edges by stringers of finite lengths and containing a collinear system of an arbitrary number of cracks at the junction line of heterogeneous materials. It is assumed that stringers along their longitudinal edges are loaded with tangential forces, and along their vertical edges - with horizontal concentrated forces. In addition, the cracks are laden with distributed tangential forces of different intensities. The case is also considered when the lower edge of the composite layer is free from the stringer and rigidly clamped. It is believed that under the action of these loads, the composite layer in the direction of one of the coordinate axes is in conditions of anti-flat deformation (longitudinal shift). Using the Fourier integral transform, the solution of the problem is reduced to solving a system of singular integral equations (SIE) of three equations. The solution of this system is obtained by a well-known numerical-analytical method for solving the SIE using Gauss quadrature formulas by the use of the Chebyshev nodes. As a result, the solution of the original system of SIE is reduced to the solution of the system of systems of linear algebraic equations (SLAE). Various special cases are considered, when the defining SIE and the SLAE of the task are greatly simplified, which will make it possible to carry out a detailed numerical analysis and identify patterns of change in the characteristics of the tasks.

scholarly journals PARABOLIC KAZHDAN–LUSZTIG BASIS, SCHUBERT CLASSES, AND EQUIVARIANT ORIENTED COHOMOLOGY

2019 ◽  
Vol 19 (6) ◽  
pp. 1889-1929
Author(s):  
Cristian Lenart ◽  
Kirill Zainoulline ◽  
Changlong Zhong
Keyword(s):  
Cohomology Ring ◽  
Flag Varieties ◽  
Special Cases ◽  
Oriented Cohomology ◽  
Schubert Classes ◽  
New Interpretation ◽  
The Moment ◽  
The Right ◽  
Relationship Of ◽  
The Relationship

We study the equivariant oriented cohomology ring $\mathtt{h}_{T}(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in $\mathtt{h}_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory $\mathfrak{h}$ corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in $\mathfrak{h}_{T}(G/P)$. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.

Thermoplastic Instability for the Transient Contact Problem of Two Sliding Half-Planes

1986 ◽  
Vol 53 (3) ◽  
pp. 565-572 ◽  
Author(s):  
A. Azarkhin ◽  
J. R. Barber
Keyword(s):  
Steady State ◽  
Pressure Distribution ◽  
Half Plane ◽  
Initial Pressure ◽  
Approximate Solutions ◽  
Steady State Solution ◽  
Fourier Integral ◽  
Numerical Results ◽  
Initial Size ◽  
Transient Contact

We study the time dependent problem of a nonconducting half-plane sliding on the surface of a conductor with heat generation at the interface due to friction. The conducting half-plane is slightly rounded to give a Hertzian initial pressure distribution. Relationships are established for temperature and thermoelastic displacements due to a heat input of cosine type through the surface, and then these are used to obtain the solution in the form of a double Fourier integral. Numerical results show that, if the ratio of the initial size of the area of contact to that in the steady state is less than some critical value, the area of contact and the pressure distribution change smoothly toward the steady state solution. Otherwise the area of contact goes through bifurcation. The bifurcation accelerates the process. Numerical results are compared with previous approximate solutions.

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.

Boussinesq's problem for a flat-ended cylinder

1946 ◽  
Vol 42 (1) ◽  
pp. 29-39 ◽  
Author(s):  
Ian N. Sneddon
Keyword(s):  
Free Surface ◽  
Rigid Body ◽  
Elastic Medium ◽  
Normal Displacement ◽  
Full Account ◽  
Elastic Stresses ◽  
Special Cases ◽  
Cylindrical Punch ◽  
Distribution Of Stress ◽  
Boussinesq’S Problem

1. In a recent paper(1) expressions were found for the elastic stresses produced in a semi-infinite elastic medium when its boundary is deformed by the pressure against it of a perfectly rigid body. In deriving the solution of this problem—the ‘Boussinesq’ problem—it was assumed that the normal displacement of a point within the area of contact between the elastic medium and the rigid body is prescribed and that the distribution of pressure over that area is determined subsequently. The solutions for the special cases in which the free surface was indented by a cone, a sphere and a flat-ended cylindrical punch were derived, but no attempt was made to give a full account of the distribution of stress in the interior of the medium in any of these cases.

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