Hertzian fracture at unloading

Hertzian fracture through indentation of flat float glass specimens by steel balls has been examined experimentally. Initiation of cone cracks has been observed and failure loads together with contact and fracture radii determined at monotonically increasing load but also during unloading phases. Contact of dissimilar elastic solids under decreasing load may cause crack inception triggered by finite interface friction and accordingly the coefficient of friction was determined by two different methods. In order to make relevant predictions of experimental findings, a robust computational procedure has been developed to determine global and local field values in particular at unloading at finite friction. It was found that at continued loading it is possible to specify in advance how the contact domain divides into invariant regions of stick and slip. The maximum tensile stress was found to occur at the free surface just outside the contact contour, the relative distance depending on the different elastic compliance properties and the coefficient of friction. In contrast, at unloading invariance properties are lost and stick/slip regions proved to be severely history dependant and in particular with an opposed frictional shear stress at the contact boundary region. This causes an increase of the maximum tensile stress at the contour under progressive unloading. Predictions of loads to cause crack initiation during full cycles were made based on a critical stress fracture criterion and proved to be favourable as compared to the experimental results.     

Effect of thickness and boundary conditions on the behavior of viscoelastic layers in sliding contact with wavy profiles

In this work, the sliding contact of viscoelastic layers of finite thickness on rigid sinusoidal substrates is investigated within the framework of Green's functions approach. The periodic Green's functions are determined by means of a novel formalism, which can be applied, in general, to either 2D and 3D viscoelastic periodic contacts, regardless of the contact geometry and boundary conditions.Specifically, two different configurations are considered here: a free layer with a uniform pressure applied on the top, and a layer rigidly confined on the upper boundary. It is shown that the thickness affects the contact behavior differently, depending on the boundary conditions. In particular, the confined layer exhibits increasing contact stiffness when the thickness is reduced, leading to higher loads for complete contact to occur. The free layer, instead, becomes more and more compliant as thickness is reduced.We find that, in partial contact, the layer thickness and the boundary conditions significantly affect the frictional behavior. In fact, at low contact penetrations, the confined layer shows higher friction coefficients compared to the free layer case; whereas, the scenario is reversed at large contact penetrations. Furthermore, for confined layers, the sliding speed related to the friction coefficient peak is shifted as the contact penetration increases. However, once full contact is established, the friction coefficient shows a unique behavior regardless of the layer thickness and boundary conditions.     

Stress field at a sliding frictional contact: Experiments and calculations

A MEMS-based sensing device is used to measure the normal and tangential stress fields at the base of a rough elastomer film in contact with a smooth glass cylinder in steady sliding. This geometry allows for a direct comparison between the stress profiles measured along the sliding direction and the predictions of an original exact bidimensional model of friction. The latter assumes Amontons’ friction law, which implies that in steady sliding the interfacial tangential stress is equal to the normal stress times a pressure-independent dynamic friction coefficient μ_d, but makes no further assumption on the normal stress field. Discrepancy between the measured and calculated profiles is less than 14% over the range of loads explored. Comparison with a test model, based on the classical assumption that the normal stress field is unchanged upon tangential loading, shows that the exact model better reproduces the experimental profiles at high loads. However, significant deviations remain that are not accounted for by either calculations. In that regard, the relevance of two other assumptions made in the calculations, namely (i) the smoothness of the interface and (ii) the pressure-independence of μ_d is briefly discussed.     

Adhesion and friction of an elastic half-space in contact with a slightly wavy rigid surface

The paper deals with the estimation of the pressure distribution, the shape of contact and the friction force at the interface of a flat soft elastic solid moving on a rigid half-space with a slightly wavy surface. In this case an unsymmetrical contact is considered and justified with the adhesion hysteresis. For soft solids as rubber and polymers the friction originates mainly from two different contributions: the internal friction due to the viscoelastic properties of the bulk and the adhesive processes at the interface of the two solids. In the paper the authors focus on the latter contribution to friction. It is known, indeed, that for soft solids, as rubber, the adhesion hysteresis is, at least qualitatively, related to friction: the larger the adhesion hysteresis the larger the friction. Several mechanisms may govern the adhesion hysteresis, such as the interdigitation process between the polymer chains, the local small-scale viscoelasticity or the local elastic instabilities. In the paper the authors propose a model to link, from the continuum mechanics point of view, the friction to the adhesion hysteresis. A simple one-length scale roughness model is considered having a sinusoidal profile. For partial contact conditions the detached zone is taken to be a mode I propagating crack. Due to the adhesion hysteresis, the crack is affected by two different values of the strain energy release rate at the advancing and receding edges respectively. As a result, an unsymmetrical contact and a friction force arise. Additionally, the stability of the equilibrium configurations is discussed and the adherence force for jumping out of contact and the critical load for snapping into full contact are estimated.     

Comprehensive bounded asymptotic solutions for incomplete contacts in partial slip

Solutions for the traction distributions and corresponding sub-surface state of stress adjacent to the edge of an incomplete contact suffering partial slip are found. The effects of frictional shakedown and a synchronously varying in-plane tension on the solution are found in closed form. The value of the asymptote, and its characterisation by just three independent parameters is illustrated by applying it to the finite problem of a rigid, tilted punch pressed onto a half-plane, and suffering partial slip induced by the application of in-plane tension.     

Contact with friction of a rigid cylinder with an elastic half-space

An exact solution to the problem of indentation with friction of a rigid cylinder into an elastic half-space is presented. The corresponding boundary-value problem is formulated in planar bipolar coordinates, and reduced to a singular integral equation with respect to the unknown normal stress in the slip zones. An exact analytical solution of this equation is constructed using the Wiener-Hopf technique, which allowed for a detailed analysis of the contact stresses, strain, displacement, and relative slip zone sizes. Also, a simple analytical solution is furnished in the limiting case of full stick between the cylinder and half-space.     

Frictional energy dissipation in materials containing cracks

Kachanov's simplified model of microcrack interaction is applied to an investigation of the behaviour of a cracked body under predominantly compressive periodic loading, so that the cracks experience periods of closure and slip, with frictional dissipation. The model is shown to be equivalent to a discrete elastic frictional system with each crack representing one node. Theorems and algorithms from such systems are applied to determine the conditions under which the system shakes down to a state with no slip and hence no energy dissipation in friction. For conditions not too far beyond the shakedown state, the dissipation is significantly affected by the initial conditions, but with larger oscillating loads, it becomes a unique and increasing function of load amplitude. The effect of crack interaction is assessed by comparison with an uncoupled model, for which the dissipation is obtained as a summation of closed form expressions over the crack population. For small numbers of cracks, the results are significantly dependent on the randomly chosen crack locations and sizes, but with larger populations, a statistically significant decrease in dissipation is observed with increasing interaction terms.     

Mechanics of adhesive contact on a power-law graded elastic half-space

We consider adhesive contact between a rigid sphere of radius R and a graded elastic half-space with Young's modulus varying with depth according to a power law E=E₀(z/c₀)^k (0<k<1) while Poisson's ratio ν remaining a constant. Closed-form analytical solutions are established for the critical force, the critical radius of contact area and the critical interfacial stress at pull-off. We highlight that the pull-off force has a simple solution of P_cr=−(k+3)πRΔγ/2 where Δγ is the work of adhesion and make further discussions with respect to three interesting limits: the classical JKR solution when k=0, the Gibson solid when k→1 and ν=0.5, and the strength limit in which the interfacial stress reaches the theoretical strength of adhesion at pull-off.     

The role of friction on sharp indentation

Frictional effects on sharp indentation of strain hardening solids are examined in this paper. The results of finite element simulations in a wide range of solids allow us to derive two simplified equations, accounting for the influence of the friction coefficient on hardness. Comparisons between the simulations and instrumented micro-indentation experiments are undertaken to ensure the validity of the former to metallic materials. Quantitative estimates of the role of friction on the development of pileup and sinking-in around the contact boundary are also given in the paper. These results provide a physical insight into the plastic flow features of distinctly different solids brought into contact with sharp indenters. Overall, the investigation shows that the amount of pileup can be used to set the range of validity of the two hardness equations indicated above. Friction has the largest influence on the contact response of solids exhibiting considerable piling-up effects (whose parameter , see text for details), whereas materials developing moderate pileup or sinking-in are less sensitive to friction. Finally, a methodology is devised to assess the influence of the friction coefficient on mechanical properties extracted through indentation experiments.     

Matched asymptotic expansions for twisted elastic knots: A self-contact problem with non-trivial contact topology

We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist, and held at both endpoints at infinity. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically nonlinear. The problem is formulated as a nonlinear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. We obtain a family of equilibrium solutions depending on a single loading parameter (proportional to applied twisting moment divided by square root of pulling force), which are asymptotically valid in the limit of a loose knot, ε→0. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium.     

Statistical model of nearly complete elastic rough surface contact

In the area of homogeneous, isotropic, linear elastic rough surface normal contact, many classic statistical models have been developed which are only valid in the early contact when real area of contact is infinitesimally small, e.g., the Greenwood–Williamson (GW) model. In this article, newly developed statistical models, built under the framework of the (i) GW, (ii) Nayak–Bush and (iii) Greenwood’s simplified elliptic models, extend the range of application of the classic statistical models to the case of nearly complete contact. Nearly complete contact is the stage when the ratio of the real area of contact to the nominal contact area approaches unity. At nearly complete contact, the non-contact area consists of a finite number of the non-contact regions (over a finite nominal contact area). Each non-contact region is treated as a mode-I “crack”. The area of each non-contact region and the corresponding trapped volume within each non-contact region are determined by the analytical solutions in the linear elastic fracture mechanics, respectively. For a certain average contact pressure, not only can the real area of contact be determined by the newly developed statistical models, but also the average interfacial gap. Rough surface is restricted to the geometrically-isotropic surface, i.e., the corresponding statistical parameters are independent of the direction of measurement. Relations between the average contact pressure, non-contact area and average interfacial gap for different combinations of statistical parameters are compared between newly developed statistical models. The relations between non-contact area and average contact pressure predicted by the current models are also compared with that by Persson’s theory of contact. The analogies between the classic statistical models and the newly developed models are also explored.     

Subsonic semi-infinite crack with a finite friction zone in a bimaterial

Propagation of a semi-infinite crack along the interface between an elastic half-plane and a rigid half-plane is analyzed. The crack advances at constant subsonic speed. It is assumed that, ahead of the crack, there is a finite segment where the conditions of Coulomb friction law are satisfied. The contact zone of unknown a priori length propagates with the same speed as the crack. The problem reduces to a vector Riemann–Hilbert problem with a piece-wise constant matrix coefficient discontinuous at three points, 0, 1, and ∞. The problem is solved exactly in terms of Kummer's solutions of the associated hypergeometric differential equation. Numerical results are reported for the length of the contact friction zone, the stress singularity factor, the normal displacement u₂, and the dynamic energy release rate G. It is found that in the case of frictionless contact for both the sub-Rayleigh and super-Rayleigh regimes, G is positive and the stress intensity factor K_II does not vanish. In the sub-Rayleigh case, the normal displacement is positive everywhere in the opening zone. In the super-Rayleigh regime, there is a small neighborhood of the ending point of the open zone where the normal displacement is negative.     

Unloading an elastic-plastic contact of rough surfaces

A statistical model for the unloading of elastic-plastic contact of rough surfaces is presented for a single load-unload cycle. The hystereses of load-separation and load real contact area behavior are analyzed for a wide range of surface roughness and loading conditions. The residual topography of the unloaded rough surfaces is also analyzed and the new distribution functions of asperity heights and summit radii of curvature along with a corresponding GW residual plasticity index are presented. A new modified plasticity index (MPI) is suggested which considers the energy dissipation due to unrecovered plastic deformations. This MPI varies from zero for purely elastic contacts to unity for purely plastic contacts and hence, can better define the level of plasticity of contacting rough surfaces compared to the original GW plasticity index.     

Steady sliding frictional contact problem for a 2d elastic half-space with a discontinuous friction coefficient and related stress singularities

The steady sliding frictional contact problem between a moving rigid indentor of arbitrary shape and an isotropic homogeneous elastic half-space in plane strain is extensively analysed. The case where the friction coefficient is a step function (with respect to the space variable), that is, where there are jumps in the friction coefficient, is considered. The problem is put under the form of a variational inequality which is proved to always have a solution which, in addition, is unique in some cases. The solutions exhibit different kinds of universal singularities that are explicitly given. In particular, it is shown that the nature of the universal stress singularity at a jump of the friction coefficient is different depending on the sign of the jump.     

Applicability of the Hertz contact theory to rail-wheel contact problems

Summary Rail-wheel contact problems have been analyzed by applying three-dimensional finite element models. Based on these models, the applicability of the Hertz contact theory (HCT) to rail-wheel contact problems is verified in the present paper. Beside a standard rail, also a crane rail and a switching component are considered in the verification. In the case of a contact between the standard rail UIC60 and the standard wheel UICORE, different transverse contact positions are analyzed. Numerically calculated distributions of the contact pressure for different types of rails with respect to different initial contact positions agree with the results from the HCT only if either the contact zone does not spread into a region of changed surface curvature or if plastification does not occur. Finally, the convective part of the dissipation power due to friction is calculated, which cannot be provided directly by the HCT. Received 5 August 1998; accepted for publication 6 May 1999     

On the sliding instabilities at rough surfaces

In this paper, the onset of sliding between two elastic half-spaces in contact, subjected to a tangential force, is studied within the framework of critical phenomena. First, it is shown that the contact domain between two rough surfaces is a lacunar set and that the distribution of contact stresses is multifractal. By applying an increasing tangential force, under constant normal load, the so-called regime of partial-slip comes into play. However, the continuous and smooth transition to full sliding, predicted by the classical Cattaneo-Mindlin theory, is not confirmed by the experiments, which show marked frictional instabilities. A numerical multi-scale procedure is proposed, taking into account the redistribution of stress, consequent to partial-slip, among the contact areas at all scales. It is shown that the lacunarity of the contact domain delays the onset of instability, when compared to compact Euclidean domains. Independently of the assumptions made for the frictional behaviour at the scale of the asperities (Coulomb friction for meso-scale asperities, adhesion for micro-scales), renormalization permits the critical value of the tangential force which provides the instability to be found. Moreover, the multifractal analysis of the domains where the shear resistance is activated captures the size-scale effects on the friction coefficient, currently evidenced by the experiments.     

Self-sustained slip pulses of finite size between dissimilar materials

The problem of a steady-state slip pulse of finite size between dissimilar materials is studied. It is shown that for a Coulomb friction law, there is a continuous set of possible solutions for any slip propagation velocity and any slip length. These solutions are, however, nonphysical because they show a singular behaviour of the slip velocity at one extremity of the pulse, which implies a crack-like behaviour. In order to regularize these solutions, we introduce a modified friction law due to Prakash and Clifton (Experimental Techniques in the Dynamics of Deformable Solids, Vol. AMD-165, pp. 33-48; J. Tribol. 120 (1998) 97), which consists in introducing in the Coulomb friction law a relaxation time for the response of the shear stress to a sudden variation of the normal stress. Then, we show that even for a slip velocity-dependent characteristic time, the degeneracy of the solutions is not suppressed and a physical pulse is not selected. This result shows the absence of steady state self-healing pulses within the modified friction law and is consistent with recent finite-difference calculations (J. Geophys. Res. 107 (2002) 10).     

Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments

This paper presents a modified regularized formulation of the Ambrosio-Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8), 1319-1342]. We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches.     

On oblique contact of creeping solids

Oblique indentation of power-law creeping solids by a rigid die is analysed in three dimensions with perfectly plastic behaviour emerging as an asymptotic case. Indenter profiles are prescribed to be axisymmetric for simplicity but not by necessity. Invariance and generality is aimed at, as the problem is governed by only four essential parameters, i.e. the die profile, p, the indentation angle, γ, the power-law exponent, n, and the coefficient of friction, μ. The solution strategy is based on a self-similar transformation resulting in a reduced problem corresponding to flat die indentation of complete contact. The reduced auxiliary problem, being independent of loading, history and time, was solved by a three-dimensional finite element analysis characterized by high accuracy. Subsequently, cumulative superposition was used to resolve the original problem and global and invariant relations between force, depth and contact area were determined. Detailed results are given for the location and shape of the contact region and stick/slip contours as well as for local states of surface stresses and deformation at flat and spherical indenters. Due to the asymmetry prevailing, it was found that in the spherical case, contact contours proved to be oval and shifted, although with normal and tangential forces only weakly coupled. Finite friction as compared to full adhesion proved to have only a minor effect on global relations. The framework laid down may be applied to the contact of structural assemblies subjected especially to elevated temperatures and also to various issues such as compaction of powder aggregates, flattening of rough surfaces and plastic impact.     

Non-slipping adhesive contact between mismatched elastic spheres: A model of adhesion mediated deformation sensor

A generalized JKR model is established for non-slipping adhesive contact between two dissimilar elastic spheres subjected to a pair of pulling forces and a mismatch strain. We discuss the full elastic solution to the problem as well as the so-called non-oscillatory solution in which tension and shear tractions along the contact interface is decoupled from each other. The model indicates that the mismatch strain has significant effect on the contact area and the pull-off process. Under a finite pulling force, a pair of adhering spheres is predicted to break apart spontaneously at a critical mismatch strain. This study suggests an adhesion mediated deformation sensing mechanism by which cells and molecules can detect mechanical signals in the environment via adhesive interactions.