Indentation of an Elastic Layer by an Array of Punches Moving With Steady Velocity

1971 ◽  
Vol 38 (1) ◽  
pp. 92-98 ◽  
Author(s):  
L. M. Keer ◽  
C. Sve
Keyword(s):  
Integral Equation ◽  
Asymptotic Solution ◽  
Elastic Layer ◽  
Wave Speed ◽  
Plane Elasticity ◽  
Contact Length ◽  
Dual Series Equations ◽  
Infinite Array ◽  
Rayleigh Wave Speed ◽  
Steady Velocity

A solution is presented for the problem of an elastic layer that is indented by an infinite array of punches moving with a steady velocity below the Rayleigh wave speed. The layer is assumed to be loaded symmetrically about its midplane. The equations of plane elasticity are used to develop dual series equations which are reduced to a single Fredholm integral equation. An asymptotic solution to the integral equation is developed for the case when the ratio of contact length to punch spacing is small, and it is compared with a numerical solution. Numerical calculations for the effective stiffness of the layer and the stresses along the midplane are included.

APPLICATION OF DUAL SERIES EQUATIONS TO WAVY CONTACT BETWEEN PIEZOELECTRIC MATERIALS AND AN ELASTIC SOLID

2014 ◽  
Vol 06 (04) ◽  
pp. 1450046 ◽  
Author(s):  
YUE-TING ZHOU ◽  
ZHENG ZHONG
Keyword(s):  
Integral Equation ◽  
Piezoelectric Materials ◽  
Elastic Solid ◽  
Contact Length ◽  
Elementary Functions ◽  
Isotropic Solid ◽  
Full Contact ◽  
Dual Series Equations ◽  
Loading Case ◽  
The Relationship

In this paper, the wavy contact between piezoelectric materials and an isotropic solid is considered. The Papkovich–Neuber potentials for the isotropic solid and three harmonic functions for piezoelectric materials are also presented. The stated problem is reduced to a pair of dual series equations and then recast as an integral equation of the Abel type. Employing the product relation for trigonometric functions and the Mehler integral yields an exact solution of the reduced Abel type integral equation. The relationship between contact length and the level of loading, and the distribution of the surface normal stress are given in terms of elementary functions. The derived results agree well with the previous ones for the purely elastic solid. It is found that a critical loading exists for the disturbance. For limiting cases, such as the low level of loading case and full contact case, corresponding contact behaviors are presented. Numerical analyses are done to reveal the influence of the level of loading on the contact behaviors.

Impedance of Strip-Traveling Waves on an Elastic Half Space: Asymptotic Solution

1974 ◽  
Vol 41 (2) ◽  
pp. 412-416
Author(s):  
S. H. Crandall ◽  
A. K. Nigam
Keyword(s):  
Asymptotic Solution ◽  
Rayleigh Wave ◽  
Half Space ◽  
Traveling Wave ◽  
Load Distribution ◽  
Wave Speed ◽  
Elastic Half Space ◽  
Surface Load ◽  
Shear Wave Speed ◽  
Rayleigh Wave Speed

The dynamic normal-load distribution across a strip that is required to maintain a plane progressive wave along its length is studied for the case where the strip is of infinite length and lies on the surface of a homogeneous isotropic elastic half space. This configuration is proposed as a preliminary idealized model for analyzing the dynamic interaction between soils and flexible foundations. The surface load distribution across the strip and the motion of the strip are related by a pair of dual integral equations. An asymptotic solution is obtained for the limiting case of small wavelength. The nature of this solution depends importantly on the propagation velocity of the strip-traveling wave in comparison with the Rayleigh wave speed, the shear wave speed and the dilatational wave speed. When the strip-traveling wave propagates faster than the Rayleigh wave speed, a pattern of trailing Rayleigh waves is shed from the strip. The limiting amplitude of the trailing waves is provided by the asymptotic solution.

A semianalytical and finite-element solution to the unbonded contact between a frictionless layer and an FGM-coated half-plane

2017 ◽  
Vol 24 (2) ◽  
pp. 448-464 ◽  
Author(s):  
Jie Yan ◽  
Changwen Mi ◽  
Zhixin Liu
Keyword(s):  
Integral Equation ◽  
Finite Element ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Singular Integral ◽  
Material Properties ◽  
Elastic Layer ◽  
Coated Substrate ◽  
Receding Contact ◽  
Contact Size

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

scholarly journals Computation of Energy Release Rates for a Nearly Circular Crack

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Nik Mohd Asri Nik Long ◽  
Lee Feng Koo ◽  
Zainidin K. Eshkuvatov
Keyword(s):  
Integral Equation ◽  
Energy Release Rate ◽  
Energy Release ◽  
Release Rate ◽  
Linear Equations ◽  
Circular Crack ◽  
Plane Elasticity ◽  
Hypersingular Integral Equation ◽  
Hypersingular Integral ◽  
Energy Release Rates

This paper deals with a nearly circular crack, in the plane elasticity. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over a considered domain, and it is then transformed into a similar equation over a circular region, , using conformal mapping. Appropriate collocation points are chosen on the region to reduce the hypersingular integral equation into a system of linear equations with unknown coefficients, which will later be used in the determination of energy release rate. Numerical results for energy release rate are compared with the existing asymptotic solution and are displayed graphically.

scholarly journals Dual-series equations formulation for static deformation of plates with a partial internal line support

2007 ◽  
Vol 34 (3) ◽  
pp. 221-248 ◽  
Author(s):  
Yos Sompornjaroensuk ◽  
Kraiwood Kiattikomol
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Hankel Transform ◽  
Internal Line ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Distributed Load ◽  
Dual Series Equations ◽  
Finite Hankel Transform ◽  
Standard Techniques

The paper deals with the application of dual-series equations to the problem of rectangular plates having at least two parallel simply supported edges and a partial internal line support located at the centre where the length of internal line support can be varied symmetrically, loaded with a uniformly distributed load. By choosing the proper finite Hankel transform, the dual-series equations can be reduced to the form of a Fredholm integral equation which can be solved conveniently by using standard techniques. The solutions of integral equation and the deformations for each case of the plates are given and discussed in details.

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