Plane contact problem for an elastic rectangular punch and a half-plane with initial stresses

International Applied Mechanics -     

Frictionless indentation by an elastic punch: a dynamic Hertzian contact problem

Frictionless indentation of an elastic half-plane by a relatively blunt, symmetric elastic punch at an ar: bitrary speed is analyzed by treating the more general problem of frictionless Hertzian contact between elastic solids. As in the quasi-static problem, the analysis assumes that the solid surface contours are approximately flat. In addition, the contact strip expands at a constant rate and the imposed rigid body motions and surface contours are represented by polynomial curves. Homogeneous function techniques allow analytic solutions to the basic mathematical problem. As an example, the general results are then applied to the uniformly accelerating parabolic punch on a half-plane.     

A misfitting elastic inclusion in an infinite plane containing a crack

The problem discussed in this paper is that of a misfitting circular inclusion in an infinite elastic medium which contains a straight crack. The crack is stress free. The stresses develop in the elastic medium because of the misfit. The point force method is used to solve the problem. The problem reduces to finding two sets of complex potential functions: {(z), (z)}: One for the infinite medium and the other for the misfitting inclusion. The solution has been obtained in closed form. Graphs are drawn for stress intensity at the crack tip and also for normal, shear and hoop stresses at the common interface of medium and misfitting inclusion.     

Uniform motion of a plane punch on the boundary of an elastic half-plane

The dynamic contact problem of a plane punch motion on the boundary of an elastic half-plane is considered. The punch velocity is constant and does not exceed the Rayleigh wave velocity. The moving punch deforms the elastic half-plane penetrating into it so that the punch base remains parallel to itself at all times. The contact problem is reduced to solving a two-dimensional integral equation for the contact stresses whose two-dimensional kernel depends on the difference of arguments in each variable. A special approximation to the kernel is used to obtain effective solutions of the integral equation. All basic characteristics of the problem including the force of the punch elastic action on the elastic half-plane and the moment stabilizing the punch in the horizontal position in the process of penetration are obtained. A similar problem was considered in [1] and earlier in the “mode of steady-state motions” in [2, 3] and in other publications.     

Identification of a plane crack in an elastic body by invariant integrals

We consider the problem of plane crack identification in an elastic body from the results of static tests. We show that the crack plane, its volume under homogeneous normal loading, and the coordinates of the central point are uniquely determined from the results of three static tests by uniaxial tension in three mutually perpendicular directions. We obtain explicit formulas for these crack characteristics in terms of the corresponding invariant integrals, which can be calculated if the stresses and displacements are measured on the external boundary of the body in the experiments mentioned above. These formulas are exact for the problem about a crack in an infinite medium. If the elastic body boundedness is taken into account and it is assumed that the crack characteristic dimensions are small compared with the distance from the crack to the body boundary, then the obtained formulas can be considered as approximate ones.     

An approach to the contact problem for a circular punch on an elastic foundation

Rektorys’ approach is used in implementing the Ritz method to solve the contact problem for a circular punch on an elastic foundation of general form __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 65–71, April 2008.     

The penny-shaped crack at a bonded plane with localized elastic non-homogeneity

This paper examines the axisymmetric problem pertaining to a penny-shaped crack which is located at the bonded plane of two similar elastic halfspace regions which exhibit localized axial variations in the linear elastic shear modulus, which has the form G(z)=G₁+G₂e^±ζz. The equations of elasticity governing this type of non-homogeneity are solved by employing a Hankel transform technique. The resulting mixed boundary value problem associated with the penny-shaped crack is reduced to a Fredholm integral equation of the second kind which is solved in a numerical fashion to generate the crack opening mode stress intensity factor at the tip.     

含圆形嵌体弹性平面中径向裂纹问题的超奇异积分方程方法

根据含圆形嵌体平面问题在极坐标下的弹性力学基本解,使用Betti互换定理,在有限部积分意义下将问题归结为两个以裂纹岸位移间断为基本未知量、对于Ⅰ型和Ⅱ型问题相互独立的超奇异积分方程,对含圆形嵌体弹性平面中的径向裂纹问题进行了研究.根据有限部积分原理,建立了问题的数值算法.计算结果表明,嵌体半径、裂纹位置及材料剪切弹性模量等都对裂纹应力强度因子具有较为明显的影响.     

Sliding frictional contact between a rigid punch and a laterally graded elastic medium

Analytical and computational methods are developed for contact mechanics analysis of functionally graded materials (FGMs) that possess elastic gradation in the lateral direction. In the analytical formulation, the problem of a laterally graded half-plane in sliding frictional contact with a rigid punch of an arbitrary profile is considered. The governing partial differential equations and the boundary conditions of the problem are satisfied through the use of Fourier transformation. The problem is then reduced to a singular integral equation of the second kind which is solved numerically by using an expansion–collocation technique. Computational studies of the sliding contact problems of laterally graded materials are conducted by means of the finite element method. In the finite element analyses, the laterally graded half-plane is discretized by quadratic finite elements for which the material parameters are specified at the centroids. Flat and triangular punch profiles are considered in the parametric analyses. The comparisons of the results generated by the analytical technique to those computed by the finite element method demonstrate the high level of accuracy attained by both methods. The presented numerical results illustrate the influences of the lateral nonhomogeneity and the coefficient of friction on the contact stresses.     

Rigid punch on an elastic semispace with a depth-varying poisson ratio

International Applied Mechanics -     

Scattering of plane, elastic waves by a plane crack of finite width

The diffraction of time-harmonic, vertically polarized, plane elastic waves by a crack of finite width is investigated with the aid of the integral-equation method. Using the integral representation for the particle displacement of the scattered field together with the constitutive equation, it is shown that the resulting integral equations uncouple for this kind of obstacle. In them, the amount by which the components of particle displacement jump across the crack occur as unknown quantities. The integral equations are solved numerically. Normalized power scattering characteristics and scattering cross-sections are computed.The research reported in this paper has been supported by the Netherlands organization for the advancement of pure research (Z.W.O.).     

Stress intensity factor in a contact problem for a plane crack under an antiplane shear wave

The frictional contact interaction of the finite edges of a plane crack under the action of a normally incident harmonic shear wave that produces antiplane deformation is studied. The influence of the forces of contact interaction on the stress intensity factor is analyzed Published in Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 115–119, September 2007.