Infinite Friction Research Articles

Micro-rheology of a particle in a nonlinear bath: Stochastic Prandtl–Tomlinson model

The motion of Brownian particles in nonlinear baths, such as, e.g., viscoelastic fluids, is of great interest. We theoretically study a simple model for such a bath, where two particles are coupled via a sinusoidal potential. This model, which is an extension of the famous Prandtl–Tomlinson model, has been found to reproduce some aspects of recent experiments, such as shear-thinning and position oscillations [R. Jain et al., “Two step micro-rheological behavior in a viscoelastic fluid,” J. Chem. Phys. 154, 184904 (2021)]. Analyzing this model in detail, we show that the predicted behavior of position oscillations agrees qualitatively with experimentally observed trends; (i) oscillations appear only in a certain regime of velocity and trap stiffness of the confining potential, and (ii), the amplitude and frequency of oscillations increase with driving velocity, the latter in a linear fashion. Increasing the potential barrier height of the model yields a rupture transition as a function of driving velocity, where the system abruptly changes from a mildly driven state to a strongly driven state. The frequency of oscillations scales as (v0−v0*)1/2 near the rupture velocity v0*, found for infinite trap stiffness. Investigating the (micro-)viscosity for different parameter ranges, we note that position oscillations leave their signature by an additional (mild) plateau in the flow curves, suggesting that oscillations influence the micro-viscosity. For a time-modulated driving, the mean friction force of the driven particle shows a pronounced resonance behavior, i.e., it changes strongly as a function of driving frequency. The model has two known limits: For infinite trap stiffness, it can be mapped to diffusion in a tilted periodic potential. For infinite bath friction, the original Prandtl–Tomlinson model is recovered. We find that the flow curve of the model (roughly) crosses over between these two limiting cases.

Universality of jammed frictional packing

Understanding disordered particle packings is of great significance from both theoretical and engineering perspectives. Establishing a quantitative relationship between nonspherical particle shape and disordered packing properties is generally challenging, due to the complex geometry and topology. Here we resolve this issue by numerically investigating disordered jammed packings of various frictional congruent nonspherical particles, including superellipsoids and polyhedra, over a wide range of friction coefficients. We discover several universal packing characteristics across different particle shapes and frictions. In the infinite friction limit, the coordination numbers for all shapes approach the identical lower bound for jamming. The resulting ``random loose packing'' (RLP) state possesses minimal structural correlations, with the packing fraction as a simple monotonic decreasing function of the orientation-averaged excluded volume for different particle shapes. Packings with finite friction can then be understood via a perturbative approach based on RLP. The nature of RLP can be illuminated by the percolation transition of contacting particle network during the quasistatic densification process. Moreover, the large-scale density fluctuations for all jammed frictional packings are also strongly suppressed, broadening the previous claim of hyperuniformity for the frictionless ones.

Implicit Regularization and Momentum Algorithms in Nonlinearly Parameterized Adaptive Control and Prediction.

Stable concurrent learning and control of dynamical systems is the subject of adaptive control. Despite being an established field with many practical applications and a rich theory, much of the development in adaptive control for nonlinear systems revolves around a few key algorithms. By exploiting strong connections between classical adaptive nonlinear control techniques and recent progress in optimization and machine learning, we show that there exists considerable untapped potential in algorithm development for both adaptive nonlinear control and adaptive dynamics prediction. We begin by introducing first-order adaptation laws inspired by natural gradient descent and mirror descent. We prove that when there are multiple dynamics consistent with the data, these non-Euclidean adaptation laws implicitly regularize the learned model. Local geometry imposed during learning thus may be used to select parameter vectors-out of the many that will achieve perfect tracking or prediction-for desired properties such as sparsity. We apply this result to regularized dynamics predictor and observer design, and as concrete examples, we consider Hamiltonian systems, Lagrangian systems, and recurrent neural networks. We subsequently develop a variational formalism based on the Bregman Lagrangian. We show that its Euler Lagrange equations lead to natural gradient and mirror descent-like adaptation laws with momentum, and we recover their first-order analogues in the infinite friction limit. We illustrate our analyses with simulations demonstrating our theoretical results.

Jamming with Tunable Roughness.

We introduce a new model to study the effect of surface roughness on the jamming transition. By performing numerical simulations, we show that for a smooth surface, the jamming transition density and the contact number at the transition point both increase upon increasing asphericity, as for ellipsoids and spherocylinders. Conversely, for a rough surface, both quantities decrease, in quantitative agreement with the behavior of frictional particles. Furthermore, in the limit corresponding to the Coulomb friction law, the model satisfies a generalized isostaticity criterion proposed in previous studies. We introduce a counting argument that justifies this criterion and interprets it geometrically. Finally, we propose a simple theory to predict the contact number at finite friction from the knowledge of the force distribution in the infinite friction limit.

Force networks and jamming in shear-deformed sphere packings.

The emergence of rigidity upon changes of temperature, density, or applied stresses in disordered assemblies of particles is of interest in a wide range of soft matter, from glass formers, gels, foams, and granular matter. Shear jamming of frictional grains presents an interesting special case wherein the application of shear stress leads to rigidity rather than its loss. The formation of self-organized structures that resist shear deformation offers an appealing geometric picture of shear jamming, which nevertheless is incompletely developed, and not well integrated with ideas concerning rigidity in frictionless systems. Exploiting the observation that athermally sheared sphere assemblies develop structural features necessary for shear jamming even in the absence of friction [H. A. Vinutha and S. Sastry, Nature Physics 12, 578 (2016)1745-247310.1038/nphys3658], we analyze conditions for jamming in such assemblies computationally. Solving force and torque balance conditions for their contact geometry, we show, and validate with frictional simulations, that the mean contact number Z equals D+1 (for spatial dimension D=2,3) at jamming for both finite and infinite friction, above the "random loose packing" limit density, at variance with previous analyses of frictional jamming. We show that the shear jamming threshold satisfies the marginal stability condition recently proposed for jamming in frictionless systems. Along lines explored in studying covalent glasses, we perform rigidity percolation analysis for D=2 and find that rigidity percolation precedes shear jamming, which, however, coincides with the percolation of over-constrained regions, leading to the identification of a regime analogous to the intermediate phase observed in covalent glasses. Together, these results provide a geometric description of shear jamming that relate closely with analyses of jamming, rigidity, and the glass transition in frictionless systems, and thus help develop a unified description of jamming phenomenology in diverse disordered matter.

Precision analysis and dynamic stability in the numerical solution of the two-dimensional wheel/rail tangential contact problem

ABSTRACTIn this paper the two-dimensional contact problem is analysed through different mesh topologies and strategies for approaching equations, namely; the collocation method, Galerkin, and the polynomial approach. The two-dimensional asymptotic problem (linear theory) associated with very small creepage (or infinite friction coefficient) is taken as a reference in order to analyse the numerical methods, and its solution is tackled in three different ways, namely steady-state problem, dynamic stability problem, and non-steady state problem in the frequency domain. In addition, two elastic displacements derivatives calculation methods are explored: analytic and finite differences. The results of this work establish the calculation conditions that are necessary to guarantee dynamic stability and the absence of numerical singularities, as well as the parameters for using the method that allows for maximum precision at the minimum computational cost to be reached.

Bathymetric Influence on the Coastal Sea Level Response to Ocean Gyres at Western Boundaries

AbstractIt is our aim with this paper to investigate how the presence of a continental shelf and slope alters the relationship between interior ocean dynamics and western boundary (coastal) sea level. The assumption of a flat-bottomed basin with vertical sidewall at the coast is shown to hide the role that depth plays in the net force acting on the coast. A linear β-plane theory is then developed describing the transmission of sea level over variable depth bathymetry as analogous to the steady advection–diffusion of a thermal fluid. The parameter , relating the friction parameter r to the bathymetry depth H and width , is found to determine the contribution of interior sea level to coastal sea level, with small giving maximum penetration and large maximum insulation. In the small (infinite friction) limit the frictional boundary layer extends far offshore, and coastal sea level tends toward the vertical sidewall solution. Adding simple stratification produces exactly the same result but with reduced effective depth and hence enhanced penetration. Penetration can be further enhanced by permitting weakly nonlinear variations of thermocline depth. Wider and shallower shelves relative to the overall scales are also shown to maximize penetration for realistic values of . The theory implies that resolution of bathymetry and representation of friction can have a large impact on simulated coastal sea level, calling into question the ability of coarse-resolution models to accurately represent processes determining the dynamic coastal sea level.

A note on the crack analogue fretting fatigue model with varying normal load

The crack analogue model was developed to interpret various experimental observations of damage and cracking in fretting fatigue. This method assumes infinite friction at the interface and defines the oscillatory stress-intensity factor at the contact edge when the tangential load cyclically varies while the normal force is constant. However, practical engineering systems are subject to periodic loading in both the normal and shear directions, so that the contact area is not constant any more. Recently, Ciavarella and Berto suggested a crude extension to the crack analogue model in order to include the case of varying normal load, which is still not immediate to use since the singularities move in space and have no equivalent to fatigue from a crack. In this paper, we shall investigate the validity of the proposed model. For this, we shall establish an exact solution for a full stick contact problem with harmonic loading in normal and tangential directions. Here, this solution shows that there is a moving singularity at the edge of the contact area as unloading proceeds. The magnitude of the moving singularity depends on the tangential force difference between unloading and loading curve at constant normal force and the instantaneous value of the contact semi-width. Also, this solution shows that there is a logarithmic singularity which does not move.

Assessing the stability of unreinforced masonry arches and vaults: a comparison of analytical and numerical strategies

ABSTRACTDespite being accepted as a robust assessment of masonry stability, thrust line analysis (TLA) relies on assumptions that can lead to a conservative assessment of stability. This article aims to quantify the extent of these limitations through a comparison of TLA with discrete element modeling (DEM). Two studies are provided. The first study compares TLA with DEM (using fixed input parameters) in assessing the stability of unreinforced masonry arches, semi-circular barrel vaults, multi-ring arches, and groin vaults. The tests demonstrate the types of sliding failures overlooked by the safe theorem due to its assumption of infinite friction. Following these validations, the comparisons between 2D structures and 3D counterparts also give insight into the efficacy of the slicing method. The second study examines the effect of DEM input parameters on the DEM-predicted stability of the considered geometries. While material parameters had limited effect on the determination of stability, for each typology, joint friction angle had a unique impact on stability. These trends are graphically presented and demonstrate how t/R ratios alone are not sufficient to unequivocally confirm stability of the considered vaults. Overall, this research informs the extent of safety for using the geometry-based analysis tool, TLA, for analyzing masonry structures.

Effect of stereotomy on the lower bound value of minimum thickness of semi-circular masonry arches

ABSTRACTConstructing structural systems with the minimal required cross section of its members was a strong motivation for progress whole along the history of building. This article investigates the effect of stereotomy on the minimum thickness value of a semi-circular arch made of masonry: a material with negligible tensile strength. The arch is modeled with its center line to which the loading is assigned according to arch length. Here, instead of some scalar parameters such as the rupture angle and the location of the middle hinge on the intrados, the geometry of the stereotomy is associated with a continuous function. Stereotomy-related constraints are introduced to keep the results feasible from an engineering point of view, and it is also demonstrated that they are essential for a well-posed constrained optimization problem. The necessary condition for a non-vanishing lower bound of minimum thickness values is derived analytically considering the assumptions of limit state analysis, infinite friction, and a no-tension material model. The lower bound thickness to radius ratio (t/R) is found to be t/R = 0.0819. A numerical method is introduced to demonstrate the existence of a valid stereotomy at the lower bound. Multiple admissible stereotomy patterns are presented at various minimum thickness values (higher than the lower bound) to demonstrate that suitable stereotomy for a fixed (t/R) ratio is far not unique—in general even the location of the middle hinge and the rupture angle might vary.

Ambiguous collision outcomes and sliding with infinite friction in models of legged systems

This work explores the collision process at foot contact in models of legged robots. In particular, we highlight that for legged systems the widely used assumption of no sliding of the contact points can yield inconsistent outcomes. For certain contact configurations and system parameters, neither the assumption of lift-off of the trailing foot nor the assumption that it stays fixed on the ground yield valid solutions. The foot will slide, even if an infinite coefficient of friction is assumed. In a related effect, we present configurations in which the solution of the collision process is ambiguous. These behaviors are examined for the models of a minimalistic biped, a passive dynamic walker, and the five-link bipedal robot RAMone. The paper provides background for these non-intuitive behaviors and investigates the influence of system parameters onto the collision behavior of a passive dynamic walker. By studying bipedal models that range from minimalistic to physically accurate, this paper establishes a bridge between the theoretical study of unilateral contact in multibody dynamic simulations and the application of these simulation methods in the modeling of ground contact in actual legged systems.

Static Analysis of Cross Vaults: The Case of the Cathedral of Casertavecchia

In the present paper, on assuming that the material has infinite friction and no cohesion,i.e. it is No-Tension in the sense of Heyman, we study the equilibrium of cross vaults of unequal rise, such as those of some Romanesque and Gothic churches. In particular the case study of the lateral cross vaults of the Cathedral of the old town of Caserta is analyzed. The scope of the paper is not to study in detail and quantitatively any particular vault, rather we give a compressive equilibrium solution (compatible with the unilateral material assumption) for the general case of cross vaults, constructed on arches of unequal rise, and loaded by a uniform load.For the purpose of the structural analysis, the vault is modelled as a thin shell made of Rigid No-Tension material. The webs of the vault that we consider, are supported on two crossing ribs. To obtain the axial force inside the ribs, equilibrated and singular stress fields (that is stress fields concentrated on curves and balanced with the loads) are constructed.An estimate of the thrust forces transmitted by the vault to the peripheral arches and walls is also obtained, so that the stability of these structures can be checked.

Prediction of longitudinal modulus of aligned discontinuous fiber-reinforced composites using boundary element method

AbstractIn this study, the boundary element method is presented for the prediction of longitudinal modulus of aligned discontinuous fiber-reinforced composites. The details of the boundary element formulation model covering infinite friction (stick) contact conditions are given. Both fiber and matrix materials are assumed to display linear elastic material behavior. The formulation is applied to boron/epoxy discontinuous fiber-reinforced composites. The computed results show a very good agreement with the modified Cox model and the finite element analysis with experimental data.

Analysis of frictional contact problems for functionally graded materials using BEM

In this paper, a quadratic boundary element formulation for continuously non-homogeneous, isotropic and linear elastic functionally graded material contact problems with friction is presented. To evaluate domain related integrals, the radial integration method (RIM) based on the use of the approximating the normalized displacements in the domain integrals by a series of prescribed radial basis functions (RBF), leading a meshless scheme, is employed. An exponential variation with spatial coordinates is assumed for Young's modulus of the functionally graded materials (FGM), while Poisson's ratio is assumed to be constant. Under the contact conditions, including infinite friction, frictionless and Coulomb friction, different systems of equations for each body in contact are united. Numerical examples including non-confirming contact are given.

Dynamic Finite Element Analysis for Contact Stress of Dynamic Consolidation

Based on Abaqus/explicit dynamics finite element program, an ax symmetrical numerical model, the infinite fringe condition and friction contact condition were built, and then the surface contact stress condition of the dynamic consolidation was studied. The time-load properties of dynamic consolidation, the spread law of contact pressure for rammer bottom and the friction influence to contact stress between the hammer and foundation were gained. The results indicate that the dynamic consolidation load can be simplified to triangular load with the weight of the hammer itself; the contact stress distribution between the hammer and the foundation is not uniform; and frictionless contact hypothesis can led errors to the simulated result.

Computing transverse elasticity modulus of composites using boundary integral equation method

Abstract In this study, the boundary integral equation (BIE) method is presented for the prediction of the transverse elasticity modulus of composites. The details of the BIE model covering infinite friction (stick) contact conditions are given. Both fiber and matrix materials are assumed to display linear elastic material behavior. The formulation is applied to four composite samples, e-glass/epoxy, s-glass/epoxy, boron/epoxy, and T300/epoxy. It is shown that the computed results are in very good agreement with the Tsai-Hahn equation.

Tensile response of passivated films with climb-assisted dislocation glide

The tensile response of single crystal films passivated on two sides is analysed using climb enabled discrete dislocation plasticity. Plastic deformation is modelled through the motion of edge dislocations in an elastic solid with a lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation incorporated through a set of constitutive rules. The dislocation motion in the films is by glide-only or by climb-assisted glide whereas in the surface passivation layers dislocation motion occurs by glide-only and penalized by a friction stress. For realistic values of the friction stress, the size dependence of the flow strength of the oxidised films was mainly a geometrical effect resulting from the fact that the ratio of the oxide layer thickness to film thickness increases with decreasing film thickness. However, if the passivation layer was modelled as impenetrable, i.e. an infinite friction stress, the plastic hardening rate of the films increases with decreasing film thickness even for geometrically self-similar specimens. This size dependence is an intrinsic material size effect that occurs because the dislocation pile-up lengths become on the order of the film thickness. Counter-intuitively, the films have a higher flow strength when dislocation motion is driven by climb-assisted glide compared to the case when dislocation motion is glide-only. This occurs because dislocation climb breaks up the dislocation pile-ups that aid dislocations to penetrate the passivation layers. The results also show that the Bauschinger effect in passivated thin films is stronger when dislocation motion is climb-assisted compared to films wherein dislocation motion is by glide-only.

Influence of friction and adhesion on the onset of plasticity during normal loading of spherical contacts

Increasing contact loading causes early transformation from elastic to elastic–plastic deformations in many conventional systems as well as micro/nano-electro-mechanical systems. The load required for yielding and the location of the onset of plasticity is critical in the robustness of systems with contacts. For frictionless (such as fully-lubricated) contacts, inception of plastic yielding occurs beneath the contact surface. However, frictional slip (contact shear) and adhesion push the inception of plastic yielding toward the contact surface. The influence of elastic mismatch, shear tractions and adhesive normal tractions on the subsurface stress field is studied analytically by superposition of the Hertzian stress field and the stress field created by the shear and additional (due to adhesion) normal tractions. Specifically, three contact conditions have been studied in this work: (i) frictionless, (ii) finite friction, and (iii) infinite friction (full stick). Also, a finite-element model is developed to verify certain assumptions in the analytical solution for the contact with finite friction. The results obtained are applied to two sets of in situ nanoindentation experiments to explain the change in the yielding behavior of submicrometer polycrystalline aluminum grains.

Unjamming in Dry Granular Matter: Second-Order Phase Transition between Fragile Solid and Dry Fluid (Bearing) by Intermittency.

Dry granular matter, with infinite tangential friction, is modeled as a connected graph ofgrains linked by purely repulsive contacts. The degrees of freedom of a grain are non-slip rotationon, and disconnection from another. The material stability under shear (jamming) is ensured by oddcircuits of grains in contact that prevent the grains from rolling on each other. A dense granularmaterial with high stiffness-to-load ratio has two possible states: fragile solid, blocked by odd circuits,and dry fluid or bearing, in the absence of odd circuits, that flows under shear by creation and glide ofa pair of dislocations as in plasticity of continuous media. In this paper, we introduce the notions of blob, a region of the material containing only even circuits, and of critical contact that closes an oddcircuit. The granular material is then represented, at low energies and critical applied shear, as a chainof blobs connected by critical contacts. The transition between dry fluid and fragile solid occurs byintermittency.

Using Mindlin theory to model friction-dependent shear modulus in granular media

An explicit expression for the effective shear modulus of a random packing of identical spheres is derived as a function of Mindlin’s tangential stiffness with interparticle contact friction. The motivation behind the approach is to incorporate the effect of intergrain friction to predict velocities in unconsolidated sands. The Mindlin friction term, allowing partial slip across the contact area between pairs of spheres, can be viewed as a parameter accounting for the growing macroscopic intergrain friction in sands as burial progresses. Hence, both moduli and velocities will gradually increase as the compressional- to shear-wave velocity ratio [Formula: see text] or Poisson’s ratio [Formula: see text] decreases. An estimate of effective elastic constants in particular shear modulus can be obtained for a spherical grain pack with an arbitrary frictional behavior ranging between two special contact boundary conditions representing infinite friction and zero friction. The proposedmodel predicts a nonlinear transition between the two special grain-contact conditions when compared to previously published linear relationships. Comparison of elastic properties, i.e., dynamic shear-modulus predictions assuming zero contact friction with experimental data on loose glass bead and sand samples undergoing hydrostatic compression, appears to match reasonably well at low confining stress (less than [Formula: see text]) but deviates gradually as stress increases. It is advocated that the increasing effective internal frictional resistance of the experimental core samples control both the frictional attenuation mechanism in loose grain packs under low confining stress for strain amplitudes typical of seismic waves (less than [Formula: see text]) and the higher stress-velocity sensitivity. Circumstantial evidence of this is found in publications describing both laboratory attenuation analysis and consolidation experiments on granular materials with different degrees of competence or static shear strength.