The stress on an elastic half-space due to sectionally smooth-ended punch

The axisymmetric contact problem for an elastic half-space and a rigid punch is considered using integral transform methods. The end of the punch is sectionally smooth and there is incomplete penetration. The normal stress under the punch is calculated and found to have an elliptic integral type of singularity.     

A moving punch on an infinite viscoelastic layer

Summary The problem of a rigid punch pressed against and moved on the surface of an elastic or viscoelastic layer is studied. It is shown that the governing equations reduce to the same integral equation for the elastic contact problem. Two particular motions of the punch are considered. In the first case the punch moves at a constant speed along a straight line on the surface of a viscoelastic layer. In the second case the punch moves at a constant speed along a circular path. Finally, the special case of a punch moving on a layer of a standard linear viscoelastic solid is studied. The equation is identical to a punch of modified shape pressed on an elastic layer.The work presented here was supported by the National Science Foundation under Grant GK 35163 with the University of Illinois.With 1 figure     

Contact problem for an orthotropic half-space

Numerical and analytical solutions of the 3D contact problem of elasticity on the penetration of a rigid punch into an orthotropic half-space are obtained disregarding the friction forces.A numericalmethod ofHammerstein-type nonlinear boundary integral equations was used in the case of unknown contact region, which permits determining the contact region and the pressure in this region. The exact solution of the contact problem for a punch shaped as an elliptic paraboloid was used to debug the program of the numerical method. The structure of the exact solution of the problem of indentation of an elliptic punch with polynomial base was determined. The computations were performed for various materials in the case of the penetration of an elliptic or conical punch.     

三维运动接触问题的广义Галин定理

本文给出了求解一类相当普遍的三维运动接触问题的分析解法,并且把仅仅对静态接触问题成立的 定理推广到动力学情形,作了严格证明。作为例子,对接触区为椭圆的情形给出了积分形式的解,并且作了数值计算,由这些结果可以看出运动压体速度的效应。     

三维运动接触问题的广义Галин定理

本文给出了求解一类相当普遍的三维运动接触问题的分析解法,并且把仅仅对静态接触问题成立的 定理推广到动力学情形,作了严格证明。作为例子,对接触区为椭圆的情形给出了积分形式的解,并且作了数值计算,由这些结果可以看出运动压体速度的效应。     

Indentation of a Rigid Body into an Elastic Plate

The problem of a punch shaped like an elliptic paraboloid pressed into an elastic plate is studied under the assumption that the contact region is small. The action of the punch on the plate is modeled by point forces and moments. The method of joined asymptotic expansions is used to formulate the problem of one–sided contact for the internal asymptotic representation; the problem is solved with the use of the results obtained by L. A. Galin. The coordinates of the center of the elliptic contact region, its dimensions, and the angle of rotation are determined. The moments which ensure translational indentation of the punch are calculated and an equation that relates displacements of the punch to the force acting on it is given.     

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.     

Determination of optimal shape of a moving punch with friction taken into account

The optimization problem for the contact interaction between a rigid punch and an elasticmediumis considered. It is assumed that that the punch is under the action of some prescribed forces and momenta and moves along a surface bounding a half-space filled with an elastic medium. It is also assumed that themotion is quasistatic and the friction forces arising in the region of contact are taken into account. The punch shape is considered as the desired design variable, and the integral functional characterizing the discrepancy between the pressure distribution in the region of contact that corresponds to the optimized shape of the punch and a given goal distribution of pressure is taken as the minimizing criterion. The optimal shape can be determined efficiently by solving the following two problems: first, to obtain the optimal pressure distribution and then to solve a boundary value problemfor the elastic half-space under the action of the obtained normal pressure and friction forces. By way of example, the optimal shape is analytically determined for a punch of rectangular shape in horizontal projection.     

On a conical punch pressing into an elastic layer

A conical punch presses into an elastic layer resting on a rigid foundation. Precision numerical results for the radius of the circle of contact, the force required, and the displacement of the center of the punch are presented.     

Oscillation of a Punch Moving on the Free Surface of an Elastic Half Space

The contact problem concerning oscillation of a circular rigid punch, moving uniformly at sub-Rayleigh speed along the surface of an elastic half space, is investigated using a three-dimensional formulation. Slow motion of the punch is considered, which implies that the characteristic time for the external loading is much larger than the time interval necessary for shear waves to propagate across the punch. An asymptotic solution for the vertical oscillation of the punch is given. It is shown that the vertical displacement of the punch can approximately be described by the equation of dynamics for a system of one degree of freedom with viscous friction. The dependence of the coefficients for effective viscosity and stiffness, occurring in this equation, on the speed of the punch and Poisson ratio of the half space, is investigated. The solution for the non-stationary problem concerning a suddenly applied moving point load is also obtained, correcting and extending the result known so far. Mathematics Subject Classifications (2000) 74H10, 74J05, 74M15.     

Uniform indentation of the elastic half-space by a rigid rectangular punch

The problem considered is that of a rigid flat-ended punch with rectangular contact area pressed into a linear elastic half-space to a uniform depth. Both the lubricated and adhesive cases are treated. The problem reduces to solving an integral equation (or equations) for the contact stresses. These stresses have a singular nature which is dealt with explicitly by a singularity-incorporating finite-element method. Values for the stiffness of the lubricated punch and the adhesive punch are determined: the effect of adhesion on the stiffness is found to be small, producing an increase of the order of 3%.     

The application of asymptotic solutions to characterising the process zone in almost complete frictionless contacts

A method is developed for characterising the nature of the plastic zone which develops along the boundary of any notionally complete frictionless contact but where, in practice, there is some small rounding. The approach consists of an outer asymptote, the solution for a semi-infinite square ended rigid punch, whose validity sets the upper limit to the load, and a nested inner asymptote, the solution for a semi-infinite rounded punch, which sets the lower limit to the load. The technique is applied, as an example, to a circular punch, and explicit values of the load given to ensure that the singular field characterises the local stress field to within a given degree of accuracy.     

Exact solution of a flat smooth punch on a piezoelectric half plane

This paper derives an exact solution for a flat smooth punch applied on a piezoelectric half plane. The piezoelectric solid occupies the lower half plane, and a flat rigid punch is applied on it. As the permittivity of the air (environment) is far less than that of the piezoelectric material, the electric induction of air may be neglected. The permittivity of the punch is also far less than that of the piezoelectric material and consequently the normal component of the electric displacement vanishes at the contact boundary. The exact solution is obtained by eigen-function representation and analytic continuation. The distribution of pressure under the punch has been found. The electric field along the surface of the lower half plane is extracted in a closed form.     

New,real fundamental solutions to the transient thermal contact problem in a piezoelectric strip under the coupling actions of a rigid punch and a convective heat supply

A thermo-electro-mechanical contact analysis has been performed for a finite piezoelectric strip, which is subjected to the joint actions of a rigid, flat punch and a transient convective heat supply. The Laplace transform and Fourier sine and cosine transforms were applied in solving the governing equations. A detailed analysis of the characteristic roots of the corresponding characteristic equation was made. Real fundamental solutions were derived, which can readily lead to real solutions to the thermo-electro-mechanical quantities. A Cauchy-type singular integral equation was obtained for the stated problem and then solved numerically. Closed form solutions of a special case were obtained. To obtain the accurate solution in the time domain, an effective numerical inversion algorithm of the Laplace transform was applied. Detailed analyses were performed to reveal the variation law of temperature, contact stress beneath the punch, stress intensity factor at the punch edge and strain with time. Parametric studies were performed to discover the effects of the layer thickness on the distribution of temperature, contact stress beneath the punch and stress intensity factor at the punch edge.     

Inclined circular flat punch on a transversely isotropic piezoelectric half-space

Summary Utilizing the general solution of transversely isotropic piezoelectricity, the paper analyzes the problem of an inclined rigid circular flat punch indenting a transversely isotropic piezoelectric half-space. The potential theory method is employed and generalized to take into account the effect of the electric field in piezoelectric materials. Assuming that the punch is maintained at a constant electric potential, exact expressions for the elastoelectric field are derived in terms of elementary functions. It is noted that the solution corresponding to a flat circular punch centrally loaded by a concentrated force can be obtained as a special case. Received 15 December 1998; accepted for publication 9 March 1999     

Effect of intermediate principal stress on flat-ended punch problems

The intermediate principal stress has certain effects on the yield strength of metallic materials under complex stress states. The flat-ended punch problem is a classical and fundamental problem in plasticity theory and mechanical engineering in which the metal beneath a flat-ended punch is under complex stress states. Using the finite difference codes, fast Lagrangian analysis of continua and Unified Strength Theory, the effect of the intermediate principal stress on the flat-ended punch problem is analyzed in this paper. First, the limit pressures of strip and circular punches pressed into an elastoplastic and homogeneous metallic medium are calculated by the two-dimensional finite difference method. The problems of square and rectangular punches are analyzed by the three-dimensional finite difference method. Finally, the effect of the intermediate principal stress on flat-ended punch problems with different punch geometries is analyzed.     

Axisymmetric bonded punch problem: A complete solution

Summary Explicit expressions are derived for the field of stresses and displacements around an axisymmetric punch bonded to a transversely isotropic elastic half-space. The method is based on the new results in potential theory obtained by the author earlier. A flat centrally loaded circular punch is considered as an example. Specific computations were performed in order to compare the elastic field in the vicinity of a bonded punch with similar parameters for a smooth punch.

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Eine vollständige Lösung des Stempelproblems bei Axialsymmetrie und Haften

Übersieht Für die Spannungs- und Verschiebungsfelder eines transversal-isotropn elastischen Halbraums, der unter Haftbedingung von einem axialsymmetrischen Stempel belastet ist, werden explizite Formeln hergeleitet. Die dazu benutzte Methode beruht auf neueren Ergebnissen des Autors in der Potentialtheorie. Als Beispiel wird ein zentrisch belasteter, flacher Kreisstempel behandelt. Als numerische Ergebnisse werden die Verschiebungsfelder in der Umgebung eines haftenden und glatten Stempels bei gleichen Parameterbedingungen zum Vergleich vorgestellt.

     

基于厚度法确定小冲杆试验能量转变温度

以A106管材钢为研究对象,进行了小冲杆断裂试验,获得断后试样的最小厚度;在对试样断裂后的变形分析中,建立了试样厚度减薄率与温度的关系,规避了低温试验中载荷和位移测量不准确导致小冲杆能量转变温度不确定的问题。结果表明:基于厚度法可以准确确定小冲杆断裂试验的能量转变温度;小冲杆断裂试验后,试样最薄厚度的减小率与温度曲线关系呈现S型,有上平台、转变区、下平台三个典型区域;根据厚度减薄率与温度曲线关系可以得到小冲杆的能量转变温度。采用厚度法得到的小冲杆断裂试验的转变温度与能量法获得的结果基本一致。     

Plastic deformation of a wedge by a sliding punch

We present a self-similar solution of the problem of deformation of an ideally plastic wedge by a sliding punch with regard to contact friction; such a solution generalizes the well-known solutions of the problem of wedge penetration into a plastic half-space and of compression of an ideally plastic wedge by a plane punch. The problem is of interest for modeling the processes of plastic deformation of rough surfaces of metal pieces by a rigid tool.     

Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch

This paper studies the contact vibration problem of an elastic half-space coated with functionally graded materials (FGMs) subject to a rigid spherical punch. A static force superimposing a dynamic time-harmonic force acts on the rigid spherical punch. Firstly, we give the static contact problem of FGMs by a least-square fitting approach. Next, the dynamic contact pressure is solved by employing the perturbation method. Lastly, the dynamic contact stiffness with different dynamic contact displacement conditions is derived for the FGM coated half-space. The effects of the gradient index, coating thickness, internal friction, and punch radius on the dynamic contact stiffness factor are discussed in detail.