Applications of conservation integral to indentation with a rigid punch

When a substrate of the brittle material is indented by a relatively rigid, square-ended punch, a singular stress field and K-dominant region will arise at the near-surface adjacent to the punch corners. The singularity of this stress field is identical with the mode I crack. The stress intensity factor in this case represents the intensification of the stress fields induced by indentation. In present article, a new method to determine the stress intensity factors of the indentation problems is developed based on the conservation law. The physical meanings of the proposed method and some numerical analysis have been investigated.     

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.     

The indentation of an elastic layer by a rigid stamp under conditions of complete adhesion

Summary In this paper the indentation of an elastic layer by a rigid stamp is treated under conditions of complete adhesion beneath the stamp, where the ratio of the half-width of the contact region and the thickness of the layer is assumed to be small. The cases of a flat stamp and a polynomial one are considered successively and two applications are treated.

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Die Einprägung einer elastischen Schicht von einem starren Stempel bei vollständiger Befestingung under conditions of complete adhesion

Zusammenfassung Die vorliegende Arbeit befaßt sich mit dem Eindringen eines starren Stempels in eine elastische Schicht für den Fall, daß die Unterseite des Stempels auf der Schicht befestigt ist.Dabei wird angenommen, daß die halbe Breite des Berührungsgebietes klein ist gegenüber der Dicke der Schicht.Zwei Sonderfälle werden näher untersucht: der flache Stempel und der polynomförmige Stempel. Zwei Anwendungen werden behandelt.

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With 6 Figures     

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 indentation of a half-space of hexagonal elastic material by a circular punch of arbitrary end-profile

Summary The problem is considered of the indentation by a smooth rigid punch of a half-space composed of linear elastic material of hexagonal symmetry whose plane boundary is parallel to the basal planes. The case is considered in which the area of contact between the punch and the half-space is circular, the end of the punch with is in contact with the half-space having an arbitrary profile. An integral equation is formulated and solved for the boundary value of the normal displacement in the half-space, and an expression is derived for the distribution of pressure under the punch.     

Thermal and slip mechanism effects in rapid indentation by a flat punch

Summary The rapid indentation of a fully-coupled thermoelastic body by a flat, smooth rigid punch is studied in a 2D dynamic analysis. A semi-infinite punch width is assumed to be a valid short-time model, and a zone of slip mechanisms is assumed to form under the punch edge, and to relax the stresses there. The mathematical problem is reduced by superposition and transform methods to a Wiener-Hopf equation which, despite the dependence of poles and branch points on the time transform parameter, can be solved exactly. Imposition of the stress relaxation property of the zone then removes the singularity in punch edge stresses, thereby giving a relation for the contact region temperature change. Because a thermoelastic characteristic length of 0(10^–4)m is the scale factor, this transform can be inverted for so-called long times without invalidating the short-time nature of the model. The inversion shows that, despite the absence of actual plasticity and contact friction, nominal but not negligible temperature increases can occur in the contact region.     

The problem of a rigid punch on a non-homogeneous elastic half space

The elastostatic problem of a rigid punch on an elastic half space is considered. The medium is assumed to exhibit a non-homogeneity varying with depth. Using the Fourier Transform Technique, the mixed boundary value problem is reduced to a singular integral equation which is solved numerically. The effect of non-homogeneity on the stress distribution under the punch and on the stress singularity is studied. The influence of Poisson's ratio on the results is also considered.     

Dynamic rigid indentation induced by sliding frictionless contact

Dynamic rigid indentation induced by sliding frictionless contact is studied by considering a rigid smooth half-wedge which simultaneously translates tangentially and normally at constant speeds with respect to the surface of an elastic half-space. A more general problem is treated by homogeneous function techniques and the solutions are then applied. Among other results, it is found that if no tangential speed is imposed, the half-wedge must approach a flat punch configuration and a maximum normal force must be applied. The normal half-space surface displacements are also plotted.     

Dynamic response of a slab of elastic-viscoplastic material that exhibits induced plastic anisotropy

Various 2-dimensional problems of the dynamic loading of a slab are solved for a material characterization that is elastic-viscoplastic and exhibits anisotropic work-hardening. The governing constitutive equations are based on a unified formulation which requires neither a yield criterion nor loading or unloading conditions. They include multi-dimensional anisotropic effects induced by the plastic deformation history. The theory also considers plastic compressibility which depends on the extent of the anisotropy. A numerical procedure for solving the equations is developed which incorporates the history dependent anisotropic hardening effects. Cases considered are the dynamic penetration of a slab by a rigid cylindrical indenter, and a distributed force rapidly applied over part of the slab surface. Both conditions of fixed and free rear surfaces of the slab are examined. A uniaxial problem is also considered in which different bases for the anisotropic hardening law are examined.     

On the indentation of an inhomogeneous anisotropic elastic material by multiple straight rigid punches

A generalised plane strain problem concerning the indentation of an inhomogeneous anisotropic elastic material by mutiple straight rigid punches is considered. The problem is reduced to a boundary integral equation with the stresses over the contact regions being represented in terms of Chebyschev polynomials. The boundary integral equation is solved numerically for some particular antiplane contact problems involving one or two contact regions and the stress intensity factors at the ends of the contact regions are calculated. The effect of anisotropy and inhomogeneity on the stress intensity factors is examined through the illustrative examples. The analysis is relevant for a class of geomechanics problems involving inhomogeneous materials.     

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.     

Stress intensity factors in a cracked infinite elastic wedge loaded by a rigid punch

The paper considers splitting a plane elastic wedge-shaped solid through the application of a rigid punch. It is assumed that the coefficient of friction on the contact area is constant, the problem has a plane of symmetry with respect to loading and geometry, and the crack lies in the plane of symmetry. The problem is formulated in terms of a system of integral equations with the contact stress and the derivative of the crack surface displacement as the unknown functions. The solution is obtained for an internal crack and for an edge crack. The results include primarily the stress intensity factors at the crack tips, and the measure of the stress singularity at the wedge apex, and at the end points of the contact area.     

Sliding and indentation by a rigid half-wedge with friction and displacement coupling effects

Dynamic rigid indentation induced by sliding contact in the presence of Coulomb friction is studied by considering a rigid half-wedge which simultaneously translates tangentially and normally at constant speeds with respect to the surface of an elastic half-plane. The tangential displacements of half-plane points in the contact zone are not a priori assumed to be negligible, and are thus coupled with the normal displacements in imposing the contact zone boundary conditions. The contact zone itself, as defined on the undeformed half-plane, extends at a constant rate. A complete solution to the problem shows that the coupling effects may be small for low indentation speeds and relatively flat half-wedges. The coupling effect decreases the singularity order at the half-wedge apex, while the friction increases it. Indeed, the two effects may cancel out in this and other aspects of the solution. It is also found that apex particle velocity singularities occur only due to the tangential motion of the half-wedge.     

Stress state of a half-plane with cracks under rigid punch action

A contact problem in elasticity theory for an isotropic half-plane with a set of curvilinear cracks, into which a rigid punch with the foundation of convex shape is indented, is considered. Coulomb friction is assumed to exist between the punch and the half-plane, while the crack faces are under conditions of either stick or smooth contact on contact parts. On the basis of integral representation for Kolosov-Muskhelishvili complex potentials by derivatives of displacement discontinuities along the crack contours and pressure under the punch, the problem is reduced to a system of complex Cauchy type singular integral equations of first and second kind. An algorithm is proposed to find solution of these equations by the method of mechanical quadratures using an iterative procedure. Two numerical examples are presented.     

Fundamental solution of a circular rigid punch problem for a half plane

A fundamental solution (a Green's function) of a circular rigid punch problem is derived. The punch on a half plane contacts smoothly at both ends of the contact region and a concentrated force or a point dislocation apply at an arbitrary point. Two cases where the punch is kept vertically and the punch is inclined are considered. Complex stress functions and a mapping function are used and a closed form solution is derived. Some results of stress distribution, contact length and resultant moment on the contact region are shown in figures.     

Piezoelectric indentation of a flat circular punch accompanied by frictional sliding and applications to scanning probe microscopy

Indentation of a piezoelectric half-space by a flat circular indenter accompanied by frictional sliding is considered. Full-field electroelastic solutions in elementary functions are obtained. The solution is based on the correspondence principle between elastic and piezoelectric problems. Stiffness relations between applied load and resulting displacement are given in elementary functions. In conjunction with the conical and spherical solutions, given previously by Makagon et al. [A. Makagon, M. Kachanov, S.V. Kalinin, E. Karapetian, Indentation and frictional sliding of spherical and conical punches into piezoelectric half-space, Physical Review B 76 (2007) 064115 (14)], this work completes the set of limiting cases of tip geometries utilized in lateral force microscopy (LFM) technology. Implications for quantitative interpretation of scanning probe microscopy (SPM) data and tribological data are analyzed.