Micromechanical analysis of friction anisotropy in rough elastic contacts

Computational contact homogenization approach is applied to study friction anisotropy resulting from asperity interaction in elastic contacts. Contact of rough surfaces with anisotropic roughness is considered with asperity contact at the micro scale being governed by the isotropic Coulomb friction model. Application of a micro-to-macro scale transition scheme yields a macroscopic friction model with orientation- and pressure-dependent macroscopic friction coefficient. The macroscopic slip rule is found to exhibit a weak non-associativity in the tangential plane, although the slip rule at the microscale is associated in the tangential plane. Counterintuitive effects are observed for compressible materials, in particular, for auxetic materials.     

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.     

A numerical analysis of partial slip problems under Hertzian contacts

In this study the tangential partial slip problems in Hertzian contact regions are treated by a numerical technique. The tangential loading may include tangential forces in the contact plane and a twisting moment normal to the contact plane. The Coulomb’s law of friction and the property that the direction of friction must oppose the relative motion lead to nonlinear equations. The Newton-Raphson method is utilized to solve these nonlinear equations. Numerical results for tangential tractions and sizes of stick and slip zones may be determined, and they agree with existing analytical results for circular contacts.     

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.     

Transition from elastohydrodynamic to mixed lubrication in highly loaded squeeze contacts

We analyze the highly loaded strongly non-stationary squeeze process of an oil film sandwiched between an elastic spherical ball and a rigid rough substrate. We show that the coupling between the elastic properties of the contacting solids, the oil rheology, the surface roughness and the applied load determines a wide range of lubrication conditions from fully elastohydrodynamic to mixed and even boundary lubrication. In particular we find that increasing (decreasing) the surface roughness (the applied normal load) speeds up the squeeze process, anticipates and shrinks the time interval during which the transition to mixed lubrication conditions occurs. On the contrary, the initial separation between the approaching bodies only marginally affects the transition time. We also observe that, in mixed lubrication conditions, the highest asperity-asperity contact pressure occurs in the annular region where the separation between solids takes its minimum value. One then concludes that surface damage and wear should nucleate in the outer region of the contact.     

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.     

A slightly corrected Greenwood and Williamson model predicts asymptotic linearity between contact area and load

The author presents a very simple and slightly corrected version of the well-known Greenwood and Williamson (GW) model which closely follows the predictions of more complicated and computationally expensive theory such as the Bush, Gibson and Thomas (BGT) theory [Bush, A.W., Gibson, R.D., Thomas, T.R., 1975. Wear 35, 87]. This new model (which I call GW modified in the following) still treats the asperities of the rough surface as spherical cups, but, this time, the curvature of spheres is not constant and instead depends on the asperity height. The GW modified theory is, in particular, able to predict, in the limiting case of large separations, the same asymptotic linearity between contact area and load as in the BGT theory. This, in turn, proves that the BGT asymptotic linear area-load relation is not a consequence of having taken into account that the contact between the asperities is actually elliptic and, therefore, of having included the spread of asperity curvature at a given height (which incidentally makes the treatment very complicated), but a consequence of having included only the influence of asperity heights on the curvature distribution of the summits. I also give a simple explanation for why the GW modified model and the BGT theory follow exactly the same asymptotic behavior. Indeed, I show that the surface summits can be treated as perfectly spherical cups (all those at the same height having the same radius of curvature) as their height is increased to very large values. In fact, in the asymptotic limit of large separation, the mean curvature of summits is shown to increase proportionally to the summit height, whereas the difference of the two principal curvatures approaches a finite constant value. The consequence of this is that the Hertzian contact between the approaching elastic (initially flat) half-space and the summits exactly resemble that between an elastic half-space and a sphere.     

The adhesive contact of viscoelastic spheres

We have formulated the restricted self-consistent model for the adhesive contact of linear viscoelastic spheres. This model is a generalization of both the Ting (J. Appl. Mech. 33 (1966) 845) approach to the viscoelastic contact of adhesionless spheres and the restricted self-consistent model for adhesive axisymmetric bodies. We also show how the model can be used in practice by giving a few examples of numerical solutions.     

Bio-inspired mechanics of reversible adhesion: Orientation-dependent adhesion strength for non-slipping adhesive contact with transversely isotropic elastic materials

Geckos and many insects have evolved elastically anisotropic adhesive tissues with hierarchical structures that allow these animals not only to adhere robustly to rough surfaces but also to detach easily upon movement. In order to improve our understanding of the role of elastic anisotropy in reversible adhesion, here we extend the classical JKR model of adhesive contact mechanics to anisotropic materials. In particular, we consider the plane strain problem of a rigid cylinder in non-slipping adhesive contact with a transversely isotropic elastic half space with the axis of symmetry oriented at an angle inclined to the surface. The cylinder is then subjected to an arbitrarily oriented pulling force. The critical force and contact width at pull-off are calculated as a function of the pulling angle. The analysis shows that elastic anisotropy leads to an orientation-dependent adhesion strength which can vary strongly with the direction of pulling. This study may suggest possible mechanisms by which reversible adhesion devices can be designed for engineering applications.     

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.     

The role of binder mobility in spontaneous adhesive contact and implications for cell adhesion

The standard view of mechanical adhesive contact is as a competition between a reduction in free energy when surfaces with bonding potential come into contact and an increase in free energy due to elastic deformation that is required to make these surfaces conform. An equilibrium state is defined by an incremental balance between these effects, akin to the Griffith crack growth criterion. In the case of adhesion of biological cells, the molecules that tend to form surface-to-surface bonds are confined to the cell wall but they are mobile within the wall, adding a new phenomenon of direct relevance to adhesive contact. In this article, the process of adhesive contact of an initially curved elastic plate to a flat surface is studied for the case in which the binders that account for adhesion are able to migrate within the plate. This is done by including entropic free energy of the binder distribution in the total free energy of the system. By adopting a constitutive assumption that binders migrate at a speed proportional to the local gradient in chemical potential, the transient growth of an adhesion zone due to binder transport is analyzed. For the case of a plate of very large extent, the problem can be solved in closed form, whereas numerical methods are invoked for the case of a plate of limited extent. Results are presented on the rate of growth of an adhesion zone in terms of system parameters, on the evolution of the distribution of binders and, in the case of a plate of limited extent, on the long-term limiting size of the adhesion zone.     

On the relationship between ultrasonic and micromechanical properties of contacting rough surfaces

In this work, it is suggested that a unique set of the interfacial stiffness constants, K_N and K_T, is sufficient to characterize the macroscopic elastic response of an interface between two rough contacting surfaces regardless of the direction of incidence of the ultrasonic wave. It is also shown that by combining ultrasonic spectroscopy with the theoretical procedures developed for a single imperfect interface, the stiffness constants of a double interface can be successfully recovered. The values of the stiffness constants determined from ultrasonic measurements are related to the micromechanical interaction and topography of the contacting surfaces using a micromechanical model of two rough surfaces in contact.     

A Molecular Dynamics Model for Single Adhesive Contact

The normal adhesive contact between a pair of asperities is performed using molecular dynamics. To simplify the problem, the equivalent contact problem of sphere–plane interaction is solved. Displacements in the contact zone are very small compared to the asperity size, therefore, the computational model is focused on the neighborhood of the contact area. The adhesion between the asperity and the plane is calculated as a sum of interactions between atoms of the asperity and the plane. A computational experiment of pull-on and pull-off is carried out to study the influence of the adhesion on the formation of the contact forces and deformations. The numerical results are compared with theoretical predictions.     

A model of cell motility leading to biphasic dependence of transport speed on adhesive strength

A rolling model for cell motility is proposed here where the movement of cell is treated as a result of the continuous release and growth of adhesions at the trailing and leading edge of the cell, respectively. The appearance of actin polymerization is key in this model as it breaks the symmetry of adhesion characteristics. The cell speed predicted here is in the correct range and exhibits a biphasic relationship with the cell-substrate adhesive strength which is consistent with experimental observations. We will show that this biphasic dependence of cell speed on adhesivity is due to the interplay between the energy dissipation associated with cell movement and the thermal fluctuations of actin filaments necessary for polymerization. Our results also suggest that the mobility of adhesion molecules is not only unnecessary but may actually limit cell motility.     

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.     

Contact of rough surfaces: A simple model for elasticity, conductivity and cross-property connections

Two rough plates with multiple contacts of diverse microgeometries are pressed against one another. Two types of contacts can be distinguished: Hertzian ones and “welded” areas. We find that for circular contacts (1) the two types produce the same effect on the incremental stiffness of the interface and on the effective conductivity across it if their contact areas are the same; (2) for both contact types, the compliance and the conductivity are controlled by the same microstructural parameter, where S_k is the kth contact area; (3) the explicit cross-property connection is established that gives the (incremental) stiffness in terms of the conductivity; it holds for an arbitrary mixture of Hertzian and welded contacts and does not require any knowledge of their microgeometries. Whereas the two types of contacts produce the same effect on the incremental stiffness, the Hertzian contacts cause non-linearities (the incremental stiffness increases with loading). The non-linearity is controlled by the microstructural parameter that is sensitive to contrasts of curvatures of the contacting parts. Model predictions are generally in good agreement with experimental data on conductivities of rough metal surfaces and stiffnesses of rough rock surfaces.     

大气呼吸模式激光推进的机理分析及数值模拟

就大气呼吸模式的推进机理作了较为系统的分析和研究,导出了考虑反射面几何约束的冲量及冲量耦合系数的近似解析表达式。在数值模拟方面,采用前期点爆炸自模拟解和后期高分辨率PPM格式相结合的新方法,计算了固定的抛物面反射镜聚焦入射平行激光击穿空气形成的高温等离子体流场及其对反射面产生的推动作用。结果表明,推进效应(飞行器获得的推力、总冲量、冲量耦合系数等)与反射面的几何参数和入射激光强度密切相关。83J的激光单脉冲能量算例得到的冲量耦合系数为246Ns/J,略高于同种工况下W.O.Schall等实验得到的耦合系数。     

Self-similar problems of elastic contact for non-convex punches

Self-similar problems of contact for non-convex punches are considered. The non-convexity of the punch shapes introduces differences from the traditional self-similar contact problems when punch profiles are convex and their shapes are described by homogeneous functions. First, three-dimensional Hertz type contact problems are considered for non-convex punches whose shapes are described by parametric-homogeneous functions. Examples of such functions are numerous including both fractal Weierstrass type functions and smooth log-periodic sine functions. It is shown that the region of contact in the problems is discrete and the solutions obey a non-classical self-similar law. Then the solution to a particular case of the contact problem for an isotropic linear elastic half-space when the surface roughness is described by a log-periodic function, is studied numerically, i.e. the contact problem for rough punches is studied as a Hertz type contact problem without employing additional assumptions of the multi-asperity approach. To obtain the solution, the method of non-linear boundary integral equations is developed. The problem is solved only on the fundamental domain for the parameter of self-similarity because solutions for other values of the parameter can be obtained by renormalization of this solution. It is shown that the problem has some features of chaotic systems, namely the global character of the solution is independent of fine distinctions between parametric-homogeneous functions describing roughness, while the stress field of the problem is sensitive to small perturbations of the punch shape.