Conformal contact between layered foundations and punches

We study the contact interaction between rigid punches and viscoelastic foundations with thin coatings for the cases in which the punch and coating surfaces are conformal (mutually repeating). Such problems can arise, for example, when the punch immerses into a solidificating coating before its complete solidification; as a result, the surface takes the shape of the punch base. Examples of such coatings can be a layer of glue, concrete at its young age, many polymeric materials. We consider plane contact problems for inhomogeneous aging viscoelastic basements in the case of their conformal contact with rigid punches. We present the statements of the problems and derive their basic mixed integral equation. The solution of this equation is constructed by using the generalized projection method. We present numerical computations of model problems, including the problem in which the shape of the punch base is described by a rapidly oscillating function.     

Indentation of punches into a piezoceramic body: two-dimensional contact problem of electroelasticity

The paper establishes the relationship between the solutions of the static contact problems of elasticity (no friction) for an isotropic half-plane and problems of electroelasticity for a transversely isotropic piezoelectric half-plane with the boundary perpendicular to the polarization axis. This allows finding the contact characteristics in the electroelastic case from the known elastic solution, without the need to solve the electroelastic problem. The contact problems of electroelasticity for different types of wedge-shaped punches (flat punch with rounded one or two edges, half-parabolic punch, and a periodic system of punches) are solved as examples Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 55–70, November 2008.     

On Indentation of a System of Irregularly Shaped Rigid Punches into a Coated Foundation

We consider the contact problem of interaction between a coated viscoelastic foundation and a system of rigid punches in the case where the punch shape is described by rapidly varying functions. A system of integral equations is derived, and possible versions of the statement of the problem are given. The analytic solution of the problem is constructed for one of the versions.     

A unified treatment of axisymmetric adhesive contact problems using the harmonic potential function method

A unified treatment of axisymmetric adhesive contact problems is provided using the harmonic potential function method for axisymmetric elasticity problems advanced by Green, Keer, Barber and others. The harmonic function adopted in the current analysis is the one that was introduced by Jin et al. (2008) to solve an external crack problem. It is demonstrated that the harmonic potential function method offers a simpler and more consistent way to treat non-adhesive and adhesive contact problems. By using this method and the principle of superposition, a general solution is derived for the adhesive contact problem involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile. This solution provides analytical expressions for all non-zero displacement and stress components on the contact surface, unlike existing ones. In addition, the newly derived solution is able to link existing solutions/models for axisymmetric non-adhesive and adhesive contact problems and to reveal the connections and differences among these solutions/models individually obtained using different methods at various times. Specifically, it is shown that Sneddon’s solution for the axisymmetric punch problem, Boussinesq’s solution for the flat-ended cylindrical punch problem, the Hertz solution for the spherical punch problem, the JKR model, the DMT model, the M-D model, and the M-D-n model can all be explicitly recovered by the current general solution.     

On indentation of a pair of rigid punches connected by an elastic beam into an elastic half-plane with regard to the friction and adhesion forces in the contact region

The contact problem of indentation of a pair of rigid punches with plane bases connected by an elastic beam into the boundary of an elastic half-plane is considered under the conditions of plane strain state. The external load is generated by lumped forces applied to the punches and a uniformly distributed normal load acting on the beam.It is assumed that the contact between the punch and the elastic half-plane can be described by L. A. Galin’s statement, i.e., it is assumed that the adhesion acts in the interior part of each of the contact regions and the tangential stresses obeying the Coulomb law act on their boundaries.With the symmetry taken into account, the problem is stated only for a single punch, and solving this problem is reduced to a system of four singular integral equations for the tangential and normal stresses in the adhesion region and the contact pressure in the sliding zones. The solution of the constitutive system together with three conditions of equilibrium of the system of punches connected by a beam is constructed by direct numerical integration by the method of mechanical quadratures.As a result of the numerical analysis, the contact stress distribution functions were constructed and the values of the sliding zones and the punch rotation angle were determined for various values of the geometric, elastic, and force characteristics.     

Development of a solution method for frictional contact problems with complex loading

In this work, solution methods for frictional contact problems are extended to the case of moving punches and to the external loading history-dependent system states. To solve the frictional contact problems in the contact area, an iterative method is developed and implemented. Solutions of two-dimensional problems are constructed using the boundary element method. Numerical analysis is aimed at the quantitative study of effects such as the interaction of contact pressure and friction forces, estimates of the friction force differences due to the differences in the choice of local basis for the calculation of normal pressure and friction forces, and evaluation of the effects of complex loading (rotation of the rigid punch after its preliminary penetration into the solid). We find that, for the same definition of the friction force, different initial approximations lead to the same solution. At the same time, the friction forces defined either as projections onto the common tangent plane or as projections onto the plane tangent to the punch can differ quite substantially. Similar conclusions are derived for the solutions corresponding to single or multiple loading steps. The work relies on the variational principle for the solution of contact problems and numerical algorithms developed for the problems with one-sided constraints. The variational principle was first applied by Signorini [1] to the determination of the stress-strain state in a linearly deformed body in a rigid smooth shell. The modern view of the problem and its generalizations to the frictional problems and some other problems involving unilateral constraints in given in the monograph [2]. Finite difference and finite element methods in application to the problems with unilateral constraints are described in [3]. Analytical solution methods are developed in the monographs [4–6].     

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.     

Axi-symmetric bodies of equal material in contact under torsion or shift

Summary Two axi-symmetric bodies are pressed together, so that their axes of symmetry coincide with the contact normal and the normal force is held constant. A small torque about the contact normal or a small tangential force is applied. For bodies of equal material, the normal and tangential stress states are uncoupled, and can solved separately. The surfaces of the bodies are thought as a superposition of infinitesimal rigid flat-ended punches. Consequently, the normal stress distribution can be calculated as a summation of differential flat punch solutions. A formula results, which is identical with the solution of Green and Collins. After application of a torque an annular sliding area forms at the border of the contact area. For reasons of symmetry, the common displacement of the inner stick area must be a rigid body rotation. Similarly to the normal problem, the solution can be thought as a superposition of rigid punch rotations. The tangential solution can be derived analogically, in form of a superposition of rigid punch displacements. The present method also solves the problem of simultanous normal and torsional or tangential loading with complete adhesion. As an example, Steuermann's problem for polynomial surfaces of the formA _2nr²ⁿis solved. The solutions for constant normal forces can be used as basic functions for loading histories with varying normal and tangential forces.     

Galin’s problem for a spatial elastic wedge

We study a three-dimensional contact problem on the indentation of an elliptic punch into a face of a linearly elastic wedge. The wedge is characterized by two parameters of elasticity and its edge is subjected to the action of an additional concentrated force. The other face wedge is free from stresses. The problem is reduced to an integral equation for the contact pressure. An asymptotic solution of this equation is obtained which is effective for a given contact region fairly remote from the edge. Calculations are performed that allow one to evaluate the effect of a force applied outside the contact region on the contact pressure distribution. The problem under study is a generalization of L. A. Galin’s problem on a force applied outside a circular punch on an elastic half-space [1, 2]. In a special case of a wedge with an opening angle of 180° and zero contact ellipse eccentricity, the obtained asymptotic relation coincides with the expansion of Galin’s exact solution in a series. Problems of indentation of an elliptic punch into a spatial wedge with the face not loaded outside the contact region have been studied previously. For example, the paper [3] dealt with the case of a known contact region (asymptotic method) and the paper [4] considered the case of an unknown contact region (numerical method). The solution of Galin’s problem allowed the authors of [2] to reduce the contact problem on the interaction of several punches applied to a half-space to a system of Fredholm integral equations of the second kind (Andreikin-Panasyuk method). A topical direction in contact mechanics is the model of discrete contact as well as related problems on the interaction of several punches [2, 5–8]. The interaction of several punches applied to a face of a wedge can be treated in a similar manner and an asymptotic solution can be obtained for the case where a concentrated force is applied at an arbitrary point of this face beyond the contact region rather than on the edge.     

On Brinell and Boussinesq indentation of creeping solids

As an alternative to traditional tensile testing of materials subjected to creep, indentation testing is examined. Axisymmetric punches of shapes defined by smooth homogeneous functions are analysed in general at power law behaviour both from a theoretical and a computational point of view. It is first shown that by correspondence to nonlinear elasticity and self-similarity the problem to determine time-dependent properties admits reduction to a stationary one. Specifically it is proved that the creep rate problem posed depends only on the resulting contact area but not on specific punch profiles. As a consequence the relation between indentation depth and contact area is history independent. So interpreted, the solution for a flat circular cylinder (Boussinesq) is not only of intrinsic interest but serves as a reference solution to generate results for various punch profiles. This is conveniently carried out by cumulative superposition and in particular ball indentation (Brinell) is analysed in depth. A carefully designed finite element procedure based on a mixed variational principle is used to provide a variety of explicit results of high accuracy pertaining to stress and deformation fields. Universal relations for hardness at creep are proposed for Boussinesq and Brinell indentation in analogy with the celebrated formula by Tabor for indentation of strain-hardening plastic materials. Quantitative comparison is made with a diversity of experimental data attained by earlier writers and the relative merits of indentation strategies are discussed.     

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.     

New analytical and numerical results for two-dimensional contact profiles

Recently, a generalized Coulomb law for elastic bodies in contact has been developed by the author, which assumes that the tangential traction is the difference of the slip stress of the contact and the stick area, whereby each stick area corresponds to a smaller contact area. It holds for multiple contact regions also. Several applications for elastic half planes, half spaces, thin and thick layers and impact problems have been published. For plane contact of equal bodies with friction, it provides exact solutions, and the interior stress field can be expressed with analytical results in closed form. In this article, a singular superposition of flat punch solutions is outlined, in which the punches are aligned with an edge of the contact area. It is shown that this superposition satisfies Coulomb's inequalities directly, and new results for the Muskhelishvili potentials of several profiles are presented. It is illustrated how problems of singularity and multi-valuedness of complex functions can be solved in closed form, and the Chebyshev approximation used by earlier authors can be avoided. For comparison, some previous solutions for symmetric profiles are appended. Some results for the interior stress field, the pressure, the frictional traction and the surface displacements are compared with FEM solutions of an equivalent problem. The small differences between both methods show characteristic features of the FEM model and the theoretical assumptions, and are shortly explained. Further, this example can be used as benchmark test for FEM and BEM programs.     

Contact problems for a circular plate with sliding fixation at the end face

Two mixed elasticity problems of punch indentation into a circular plate placed without clearance in a rigid cylindrical holder with smooth walls are considered. In the first problem, the plate lies without friction on a rigid base, and in the second problem, the plate is rigidly fixed to the base. The problems are solved by a method that was developed for bodies of finite dimensions and is based on the properties of closed systems of orthogonal functions. Each of the problems is reduced to two integral equations, namely, a Volterra integral equation of the first kind for the contact pressure function and a Fredholm integral equation of the first kind for the derivatives of the displacement of the plate upper surface outside the punch. The displacement function is sought as the sum of a trigonometric series and a power function with a root singularity. After truncation, the obtained illposed system of linear algebraic equation has a stable solution. A method for solving Volterra integral equations is given. The contact pressure distribution function and the dimensionless indentation force are determined. Examples of calculation of the plate interaction with the plane punch are given. Contact problems were earlier studied for a rectangle and a circular plate with a stress-free end both without taking account of their fixation [1, 2] and with regard for their fixation [3, 4]. The solution method described here was used to study the interaction of elastic hollow cylinder of finite length with a rigid bandage and a rigid insert [5, 6]. Other papers dealing with contact problems for bodies of finite dimensions, in particular, for a circular plate, should also be mentioned. In these papers, the problems under study were solved by the method of homogeneous solutions [7, 8] and by the method of coupled series-equations [9].     

Axisymmetric contact between an annular rough punch and a surface nonuniform foundation

The axisymmetric contact problem of interaction between a two-layer foundation and a rigid annular punch is considered under the assumption that the surface nonuniformity of the upper layer and the shape of the punch base are described by rapidly varying functions. The integral equation of the problem containing two rapidly varying functions is derived, and two versions of the problem are considered. Their solutions were first constructed by the generalized projection method. As an illustration, the model problem is analyzed numerically to demonstrate the high efficiency of the method.     

Adhesion between elastic cylinders based on the double-Hertz model

A cohesive zone model for two-dimensional adhesive contact between elastic cylinders is developed by extending the double-Hertz model of Greenwood and Johnson (1998). In this model, the adhesive force within the cohesive zone is described by the difference between two Hertzian pressure distributions of different contact widths. Closed-form analytical solutions are obtained for the interfacial traction, deformation field and the equilibrium relation among applied load, contact half-width and the size of cohesive zone. Based on these results, a complete transition between the JKR and the Hertz type contact models is captured by defining a dimensionless transition parameter μ, which governs the range of applicability of different models. The proposed model and the corresponding analytical results can serve as an alternative cohesive zone solution to the two-dimensional adhesive cylindrical contact.     

Ⅲ型界面裂纹面受变载荷Pxmtn作用下的自相似解

通过复变函数论的方法，对Ⅲ型界面裂纹表面受变载荷$Px^mt^n$作用下的动态扩 展问题进行了研究. 采用自相似函数的方法可以获得解析解的一般表达式. 应用 该法可以很容易地将所讨论的问题转化为Riemann-Hilbert问题, 然后应 用Muskhelishvili方法就可以较简单地得到问题的闭合解. 利用这些解 并采用叠加原理，就可以求得任意复杂问题的解.     

The dynamic indentation of an elastic half-space by a rigid punch

The two-dimensional contact problem between a rigid die and an elastic half-space is considered. A numerical method of solution is proposed which involves an iterative process which is continued until the correct solution is obtained according to certain criteria. The method is general enough and can handle punches of arbitrary shape as well as time-dependent indentation velocities. The treatment is unified for subsonic, transonic and supersonic indentations. The numerical procedure is checked with analytical results which are known in several special cases and good agreement is obtained. Results are presented for the smooth as well as frictional indentation by a wedge-shaped die and for a smooth parabolic punch.     

Indentation of a Punch with a Fine-Grained Base into an Elastic Foundation

The linear contact problem for a system of small punches located periodically on a part of the boundary of an elastic foundation is studied. An averaged contact problem is derived using the Marchenko–Khruslov averaging theory. An asymptotic formula is obtained for the translational capacity of a smooth punch with a fine-grained flat base.     

有摩擦有间隙的三维接触问题有限元法及应用

介绍了一种求解三维弹性体间有摩擦、有间隙的接触问题有限元解法。文中推导出四种接触状态的接触条件。该方法通过赫兹问题验证，有限元解与精确解吻合较好。应用本方法对汽轮机汽缸法兰连接进行了应力计算，获得了法兰结合面上压应力分布规律并得出了一些重要结论。