Symmetrical frictionless indentation over a uniformly expanding contact region—I. Basic analysis

The study of two-dimensional wave propagation in a half-space due to indentation by a rigid smooth indentor of a general shape at a non-uniform velocity is approached by assuming that the indentor shape and displacement history can be represented by polynomial curves in, respectively, a spatial variable and the time. As a first step symmetric indentation over a contact region expanding at a constant sub-Rayleigh wave speed is considered. Since superposition will yield more general forms attention is confined to polynomials homogeneous of degree $\text{n ? 1}$ with n + 1 terms and arbitrary coefficients. By homogeneous function techniques all the field variables in the half-space for any $\text{n}$ are obtained as single integrals. Conditions for the existence of singularities in the stresses and particle velocities are examined and some general results with bearing on the indentation problem are discussed.     

The frictionless rolling contact of a rigid circular cylinder on a semi-infinite granular material

The resulting flow and deformation of a semi-infinite granular material under a rolling, smooth rigid circular cylinder is investigated using a perturbation method. Based on the double-shearing theory of granular flow, complete stress and velocity fields, resistance to rolling and the permanent displacement of surface particles are determined to first order; when the internal friction angle is zero, the solutions reduce to those obtained in the corresponding analysis for Tresca or von-Mises materials. The solution scheme and the double-shearing model for granular flow both find their origins in the work of A.J.M. Spencer.     

Frictionless indentation by a rigid wedge: The effect of tangential displacements in the contact zone

Recent dynamic analyses of frictionless rigid indentation on linearly elastic half-planes assume that the tangential displacement of half-plane points in contact with the indentor is negligible. This assumption is used as a justification for denning the contact zone size, although not known in advance, on the undeformed half-plane, and for uncoupling the tangential and normal displacements in the contact zone. This article examines the validity of this assumption for the symmetric wedge with a constant indentation speed and a contact zone which, as denned on the undeformed half-plane, extends at a constant subcritical rate. Both the coupled and uncoupled solutions are obtained analytically and compared. The difference between the half-plane size defined on the undeformed half-plane and its size on the deformed half-plane may actually be greater for the uncoupled solution. Moreover, the coupled solution does not exhibit the wedge apex stress singularities found for the uncoupled solution.     

Shape optimization in frictionless contact problems

A finite element approach for shape optimization in two-dimensional (2-D) frictionless contact problems is presented in this work. The goal is to find the shape that gives a constant distribution of stresses along the contact boundary. The whole formulation, including mathematical model for the unilateral problem, sensitivity analysis and geometry definition is treated in a continuous form, independently of the discretization in finite elements. Shape optimization is performed by direct modification of geometry through B-spline curves and an automatic mesh generator is used at each new configuration to provide the finite element input data for numerical analysis and sensitivity computations. Using augmented-Lagrangian techniques (to solve the contact problem) and an interior-point mathematical-programming algorithm (for shape optimization), we obtain several results reported at the end of the article.     

Enhanced multiple-point beam-to-beam frictionless contact finite element

In this paper a new contact finite element for beams with circular cross-sections is presented. The element is an enhancement of the previously formulated point-wise beam-to-beam contact finite elements to be used in cases when beams get in contact at very acute angles. In such situation, if beam deformations in the vicinity of the contact zone are taken into account, the contact is not point-wise but it extends to a certain area. To cover such a case in a more realistic way, two additional pairs of contact points are introduced to accompany the original single pair of contact points. The central pair is determined using the orthogonality conditions for the beam axes and the positions of two extra points are defined on one beam axis by a shift of the local co-ordinate. This shift depends on beams geometry and the current angle between tangent vectors at the central contact point. The appropriate kinematic variables for normal contact together with their finite element approximation are derived. Basing on the weak form for normal contact and its linearisation, the tangent stiffness matrix and the residual vector are derived. The new element is tested using author’s computer programs and comparisons with the point-wise contact elements are made.     

A direct automated procedure for frictionless contact problems

The solution of elastostatic bodies in frictionless contact is obtained by an automated direct method which exploits the theory of linear elasticity and circumvents the need for the inclusion of artificial interface elements, mathematical programming techniques or computation of contact pressure. The method is simple and economical to use and can be easily appended to existing numerical schemes such as the finite element method. The formulation and numerical algorithm are presented for body combinations which are independent of relative tangential displacements along the contact surface. The method is illustrated through an elementary example amenable to hand calculation. Numerical results for more realistic problems are given and compared to known solutions. It is concluded that the method provides a powerful means for both the analysis and design of contacting bodies when used in conjunction with a finite element computer program.     

Quasistatic indentation of a rubber-covered roll by a rigid roll

The nonlinear elastic problem involving the indentation of a slightly compressible rubber-like layer bonded to a rigid cylinder and indented by another rigid cylinder is analysed by the finite element method. Both the geometric and material nonlinearities are accounted for. The finite element formulation of the problem is based upon a variational principle recently proposed by Cescotto and Fonder, and is valid for both slightly compressible and incompressible materials. The results computed and presented graphically include the shape of the indented surface, the pressure distribution over the contact surface, and the stress distribution at the bond surface. For the same contact width, the results for the compressible material are found to differ significantly from those for the case when the rubber-like layer is assumed to be incompressible.     

The elasto-plastic indentation of a half-space by a rigid sphere

Using the finite element method, a detailed study of the deformations and stresses produced in an elastic-perfectly plastic half-space indented by a rigid sphere was done. The analysis covers the transition region from the maximum elastic contact load to a state where this load has been increased one hundredfold. Experimental results available in the literature are in good agreement with the analysis. In solving repeatedly the large number of linear equations involved in the solution of the problem, it was found profitable, in order to save computer time, to modify the direct elimination method. This technique is described in some details in the paper.     

Elastic-plastic analysis of frictionless contact at interfacial crack tips

The asymptotic elastic behavior of an interfacial crack occurring between two dissimilar isotropic media is reviewed. Distinct solutions, based on differing assumptions regarding crack-face boundary conditions, can be generated. The assumption of traction-free faces generally leads to oscillatory singular asymptotic fields which mathematically cause crack-face interpenetration, an inconsistency which can be alleviated by alternatively assuming asymptotic frictionless contact. For predominant tensile loading, the elastically-calculated ratio of contact length to crack size is typically very small, but may become appreciable when shear loading is applied. In either case, the singular crack-tip stresses cannot be sustained in materials capable of limited plastic flow, and small scale yielding (SSY) should be considered. In an extension of previous work [11], we identify conditions for SSY within surrounding dominant elastic regions of both traction-free and frictionless contact types. For the latter case, approximate closed form expressions for the plastic zone size and shape are obtained as the locus of points where the elastically-calculated Mises stress equals the tensile yield strength, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS% baaSqaaiaadMhacaWGZbaabeaaaaa!39D2!\[\sigma _{ys} \]. The maximum extent of this plastic zone is approximately % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadU% eadaqhaaWcbaGaaeysaiaabMeaaeaacaWGJbaaaOGaai4laiaaikda% cqaHdpWCdaqhaaWcbaGaamyEaiaadohaaeaacaaIYaaaaaaa!4042!\[3K_{{\text{II}}}^c /2\sigma _{ys}^2 \], where K _II ^c is the closed crack-tip bimaterial stress intensity factor. Precise SSY numerical calculations for an elastic/perfectly-plastic material atop a rigid substrate indicate that the asymptotic stress field in the plastically-deforming material is composed of two fan regions and two constant state regions. Within the plastic zone, the interfacial and crack-face tractions asymptotically reach constant values. Compressive crack-face tractions persist even when contained inelastic crack-tip deformation is included.     

Dynamic contact analysis technique for rapidly sliding elastic bodies with node-to-segment contact and differentiated constraints

For a stabilized Newmark time integration of dynamic contact problems of the rapidly sliding bodies, considering the equality and inequality contact constraints and a high-speed contact point motion sliding on the deforming contact surface, the velocity and acceleration contact constraints are derived. Also, to suppress the numerical oscillations accompanied by the node-to-segment contact of the finite element models, a pseudo-node-to-node contact technique is suggested with the linear shape function elements having the almost equal segment lengths on the contact surface. The numerical simulations are performed with a high-speed punch moving on the beam and the high-speed rotating disks to check the stability and accuracy of the solution.     

压痕诱发GaAs塑性变形区的层错结构

用电子显微镜研究了压痕诱发砷化镓单晶中的塑性变形．结果表明,在压痕周围除了产生玫瑰型分布的位错组态和二次对称的孪晶结构外,还形成了层错．层错有两种类型：一种由两个具有不同Burgers矢量的不全位错组成,另一种则由具有相同Burgers矢量的不全位错组成．     

A formulation for frictionless contact problems using a weak form introduced by Nitsche

In this paper a finite element formulation for frictionless contact problems with non-matching meshes in the contact interface is presented. It is based on a non-standard variational formulation due to Nitsche and leads to a matrix formulation in the primary variables. The method modifies the unconstrained functional by adding extra terms and a stabilization which is related to the classical penalty method. These new terms are characterized by the presence of contact forces that are computed from the stresses in the continuum elements. They can be seen as a sort of Lagrangian-type contributions. Due to the computation of the contact forces from the continuum elements, some additional degrees-of-freedom are involved in the stiffness matrix parts related to contact. These degrees-of-freedom are associated with nodes not belonging to the contact surfaces.     

A novel finite element formulation for frictionless contact problems

This article advocates a new methodology for the finite element solution of contact problems involving bodies that may undergo finite motions and deformations. The analysis is based on a decomposition of the two-body contact problem into two simultaneous sub-problems, and results naturally in geometrically unbiased discretization of the contacting surfaces. A proposed two-dimensional contact element is specifically designed to unconditionally allow for exact transmission of constant normal traction through interacting surfaces.     

High-speed sliding indentation of ceramics: thermal effects

A diamond indentor is attached to the periphery of an aluminium wheel and slid against ferrite and sapphire surfaces at high speed. A high-frequency response infrared sensor is used to measure the average diamond tip temperature and a piezoelectric dynamometer is used to measure the normal and tangential forces acting on the indentor. The analytical procedure estimates the heat flux at the contact as the product of the shear traction and the tangential velocity. The heat flux is used in conjunction with a Green's function to calculate the interface temperature and the tangential surface stress. The calculated and measured grain tip temperatures are compared and the relative magnitudes of the mechanical and thermal surface stresses are calculated. Finally, the importance of thermal effects on the wear and surface finishing of ceramics is discussed.     

Axisymmetric contact problem for a frictionless elastic layer indented by an elastic cylinder

This paper is concerned with the elastostatic contact problem of a semi-infinite cylinder compressed against a layer lying on a rigid foundation. It is assumed that all the contacting surfaces are frictionless and that only compressive normal tractions can be transmitted through the interfaces. Upon loading the contact along the layer-foundation interface shrinks to a circular area whose radius is unknown. The analysis leads to a system of singular integral equations of the second kind. The integral equations are solved numerically and the contact pressures, extent of the contact area between the layer and the foundation, and the stress intensity factor round the edge of the cylinder are calculated for various material pairs.     

A sliding superconducting contact

We have found that solder coated spring wire brushes can make a sliding superconducting contact with a rotating superconducting commutator. With a relative velocity of 0.5 cm s^?1 between brushes and rotor, we observe critical currents of up to 0.1 A. Below the critical current we detect no resistance above 10^?7 Ω.     

The dynamic indentation of an elastic-viscoplastic work-hardening slab by a rigid punch

A unified theory for elastic-viscoplastic work-hardening materials, which requires neither a yield criterion nor loading or unloading conditions, is implemented to solve two-dimensional dynamic problems. Specifically, the theory is applied to the dynamic indentation by a rigid indenter of a slab made of an elastic-viscoplastic material. The contact area between the indenter and the slab at any time is not known in advance but should be determined from the process of the solution. An iterative numerical procedure is proposed by which the complete solution is determined from the dynamic elastic-viscoplastic equations, the moving mixed boundary conditions, the requirement that the contact normal stresses are compressive and that no interpenetration occurs outside the contact area. The method is applied to the indentation problem of a viscoplastic slab by a long rigid circular cylinder, and by a wedge-shaped die. Comparisons with the corresponding perfectly elastic problems are given.     

A minimum principle for frictionless elastic contact with application to non-Hertzian half-space contact problems

Summary A variational principle governing the frictionless contact between two elastic bodies is established, which is valid both for linear and for non-linear elasticity. In the case of linear elasticity it appears to lead to an infinite dimensional convex quadratic programming problem. It is applied to the half-space geometry in linear elasticity and it is established that non-Hertzian normal half-space contact problems are physically meaningful.A Hertzian and a non-Hertzian normal contact problem are investigated numerically, to which end the principle is discretised on a triangular network. In the case of the Hertz problem it is found that the exact relationships between penetration, maximum pressure, and total normal force are well satisfied. The form of the contact area is given only crudely, unless the discretisation network is considerably refined. It appeared that such a refinement is only necessary close to the edge, in which case passable results will be obtained.