Coulomb Friction Law Research Articles

A Stiction Oscillator with Canards: On Piecewise Smooth Nonuniqueness and Its Resolution by Regularization Using Geometric Singular Perturbation Theory

In mechanics, one often describes microscopic processes such as those leading to friction between relative interfaces using macroscopic variables (relative velocity, temperature, etc.) in order to avoid models of intangible complexity. As a consequence, such macroscopic models are frequently nonsmooth, a prominent example being the Coulomb law of friction. In many cases, these models are perfectly adequate for engineering purposes. Formally, however, since the Fundamental Theorem of Existence and Uniqueness does not apply to these situations, one generally expects that these models possess forward nonuniqueness of solutions. Consequently, numerical computations of such systems might possibly unknowingly discard certain solutions. In this paper, we try to shed further light on this issue by studying solutions of a simple friction oscillator subject to stiction friction. The stiction law is a simple nonsmooth model of friction that is a modification of Coulomb based on the fundamental observation that the dynamic friction force, when the mass is in motion, is smaller than the static friction force during stick. The resulting piecewise smooth vector field of this discontinuous model does not follow the classical Filippov convention, and the concept of a Filippov solution cannot be used. Furthermore, some Carathéodory solutions, i.e., absolutely continuous solutions satisfying the differential equation in a weaker sense, are nonphysical. Therefore, we introduce the concept of stiction solutions. These are the Carathéodory solutions that are physically relevant, i.e., the ones that follow the stiction law. However, we find that some of the stiction solutions are forward nonunique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further understanding of the nonunique dynamics, we then introduce a general regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. Using geometric singular perturbation theory, we identify a repelling slow manifold that separates the forward slipping from the forward sticking solutions, leading to high sensitivity to the initial conditions. On this slow manifold we find canard trajectories that have the physical interpretation of delaying the slip onset. Most interestingly, we find that these new solutions do not correspond to stiction solutions in the piecewise-smooth limit, and are therefore seemingly nonphysical, yet they are robust and appear generically in the class of regularizations we consider. Finally, we show that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, which are otherwise disconnected.

Ultimate strength evaluation of multi-stage cold forming technique for manufacture of thin-walled vessels

The multi-stage process of cold forging of thin-walled steel vessels was evaluated taking into account technological heredity. The quality of the product is evaluated by the degree of deviation of its stress- strain state from the ultimate states of the forming limit diagram. The process is calculated based on the model of large plastic deformations of the anisotropic shell, which takes into account the dynamics and contact interactions of the shell with the tool. The model was numerically implemented in the LS-DYNA®package. Numerical simulation was performed using the simulation tools of the package library, such as the models of plastic flow of an anisotropic sheet with the power law isotropic strain hardening, associated with the Barlat criterion Yld 2000-2d, the Peng-Landel potential of nonlinear elastic behavior of a polyurethane die, and the Coulomb friction law of contact interaction of the product with the tool. The model constants for low-carbon sheet steel DC04EK 0.7 mm and polyurethane SKU-PFL were determined from the experimental data. The forming limit curve was plotted using as the basic data the distortions of the coordinate grid near the zones of strain localization and failure of the vessel during the technological process without intermediate annealing and the results of failure tests under uniaxial tension. The features of the strain paths are studied at thе control points of the vessel at each stage of the technological process, including the sequence of drawing, bulging and reducing operations. The calculated strain paths were verified by the experiment, in which pressing equipment was used as the test facility. It was found that the operation of work piece bulging after its rapid drawing leads to the limit state and therefore requires a preliminary recovery of the plasticity resource by annealing. The obtained results demonstrate the advantages of forming the relief of the vessel by smaller degrees of bulging and greater degrees of reducing for eliminating the limit states and intermediate annealing.

A new method for distortion calculations in additive manufacturing: Contact analysis between a workpiece and clamps

For the first time, contact theory is incorporated into the deformation calculation of additive manufacturing. Compared with the traditional boundary constraints of pure rigidity or elasticity, the model in this study considers the contact between a workpiece and clamps. The Coulomb friction law is utilized to solve the contact problems. A penalty function method is used to solve the Coulomb friction problems, and the tangential and normal behaviors of the contact parts are analyzed in detail. For both rigid and contact constraints, the deformation trend is consistent with the measured results; however, the deformation distribution of the contact constraints is closer to the real distribution. The maximum deformation that is obtained with the rigid constraints is only 60.6% of the actual measured value, compared to 90.4% with the contact constraints. The mean square error of multiple points between the results by rigid constraints and the measurement is 0.23, while it is 0.14 with contact constrains. Therefore, the incorporation of contact theory can improve the accuracy of deformation simulations for additive manufacturing. Relative motion analysis of the contact part between the workpiece and the clamp during electron beam freeform fabrication demonstrates that the dominant role of the transverse deformation of the substrate decreases with the increase in the number of deposition layers, and warping deformation gradually increases to become the main component of the contact force.

EXPERIMENTAL STUDY ON STEADY DYNAMIC FRICTION OF MWCNTs MIXED LUBRICANTS

This paper investigates the steady sliding behavior of multiwalled carbon nanotubes (MWCNTs) mixed lubricants at the metal–metal interface in direct shear tests. Slide-free-slide (SFS) experiments were performed to understand dynamic frictional stress of the sliding surfaces above the critical velocity. The experimental observations show that the dynamic stress decreases with increase in sliding velocity, but the same increases with normal stress. Further, in the case of change in concentration of the MWCNTs in the lubricant, the dynamic stress increases initially and then decreases to a minimum value and then further increases with addition of more nanoparticles in the lubricant. Dynamic stress and corresponding critical velocity are found to be minimum about 1.6% (wt./vol.) of MWCNTs concentration. Magnitude of cohesion as well as the coefficient of friction were also determined experimentally with the Coulomb friction law. The friction results are discussed in terms of the scaling laws in three regimes, namely viscous, rolling and sliding. These laws are justified on the basis of the mixed lubrication regime of Stribeck curve. Surface morphology of the test specimens before and after the experiment was also examined using SEM and EDS tests. No evidence of surface damage owing to motion of the nanoparticles was observed.

Frictional contact analysis of sliding joints with clearances between flexible beams and rigid holes in flexible multibody systems

Joints with clearances in flexible multibody systems have been investigated for many years. However, the previous work is mainly concerned with the lower pair joints with small clearances. The aim of this paper is to present a formulation for the sliding joint with clearance between a flexible beam and a rigid hole, with special focus on the frictional contact. We thoroughly discuss a contact detection method for a flexible beam with circular cross-section and a rigid hole with rectangular cross-section. We employ the layer-wise contact detection concept and propose an efficient and robust approach to address the two-dimensional ellipse–rectangle contact detection problem within the rigid hole cross-section. After the potential contact points on the beam surface and on the hole cross-section are determined, a beam–rigid hole frictional contact element is proposed. The measure of tangential interaction is defined in the current configuration, and the changes of both contact points are taken into consideration. The constitutive relationship for the tangential contact is modeled by the Coulomb friction law with a penalty regularization, and the real tangential force vector in the current step is determined via a trial step and a subsequent return-mapping scheme. In addition, the discontinuity problem across the beam element boundaries in the case of large sliding is circumvented by a simple and effective transformation. Finally, two numerical examples are presented to verify the proposed contact detection method and the beam–rigid hole frictional contact element, and to demonstrate the influence of the frictional contact on the dynamic performance of flexible multibody systems.

Pressure and sliding velocity dependent surface asperity based friction model: Application to springback simulation

For accurate predictions of the formability and springback in the finite element sheet forming simulations, significant advances have been achieved in constitutive modeling that captures complex deformations occurring in the sheet forming process. However, friction behavior has been still modeled and implemented as a simple way though it is one of critical factors for accurate numerical simulations. For instance, sheet metal forming simulations often employ the Coulomb friction law with a constant friction coefficient, but it is known to be a function of other factors such as contact pressure, sliding rate and other surface quality. In this study, the microscopic surface asperity based friction model originally proposed by Westeneng [1], as one of microscale level friction models, was revisited by using additional contact assumptions. In particular, the model added the strain rate effect to the existing contact pressure dependent model in order to include the effect of both contact pressure and sliding velocity during sheet forming process. The calculated contact friction coefficient was compared to that by the measured for dual-phase 780 steel sheet under different pressure and sliding velocity. Moreover, the predicted friction model was employed in the simulation of U-draw bending springback by using a user friction subroutine, which was evaluated by the comparison of experimentally measured springback profile

A modal based reduction technique for wide loose interfaces and application to a turbine stator

Aim of this research is the prediction of the non-linear forced response of a stator bladed disk for aeronautical applications in the presence of friction contacts at the hook joints (casing – vane segment contact). Due to the large extension of the contact interfaces the use of classic node-to-node contact elements based on the Coulomb friction law becomes cumbersome. Although such approach appears convenient for problems involving small contact regions, it is computationally expensive when large contact interfaces with refined meshes are present between components. In this research, a novel modal interface reduction method is combined with a layer of Jenkins contact elements in order to efficiently predict the effects of friction on the non-linear forced response of the stator bladed disk. The wide rail/hook contact regions are therefore suitable for the application of the presented technique. The goodness of the proposed methodology is quantified both in terms of accuracy and time costs savings on the calculation of non-linear forced responses by exploiting cyclic symmetry hypotheses and the Harmonic Balance Method.

A 2D explicit numerical scheme–based pore pressure cohesive zone model for simulating hydraulic fracture propagation in naturally fractured formation

AbstractIn this work, a 2D pore pressure cohesive zone model is presented to simulate the hydraulic fracture propagation in naturally fractured formation, in which the fracturing process is governed by a bilinear cohesive zone model equipped with Coulomb's friction law and the fluid flow within the hydraulic fracture is described by the lubrication equation. In this model, fluid leak‐off into rock matrix is ignored and a fully explicit temporal integration scheme is adopted to overcome the convergence issue of conventional implicit scheme (eg, Newton‐Raphson method). The advantage of the proposed model is that it does not require any special crossing criterion to determine the interaction behavior when the hydraulic fracture hits the natural fracture. Implementation of the model was described in detail, and then, the model was verified with well‐known analytical solution of KGD problem and criteria of hydraulic fracture crossing natural fracture. The capability of the proposed model to capture the interaction behavior between hydraulic fracture and natural fracture was demonstrated by the good agreement of the modeling result and analytical solution. Several numerical cases were performed to investigate the impact of key factors on fracture network evolution during hydraulic fracturing treatment. Results show that fracture network is not only governed by the spatial distribution of natural fracture, but also greatly affected by the initial hydraulic aperture of natural fracture and in situ stress contrast.

Constraint conditional finite element method for off-axial interfacial sliding of fiber reinforced composite

The main toughening mechanism of Ceramics Matrix Composite (CMC) is the fiber/matrix interfacial debonding and sliding. To simulate of the contact problem (such as interfacial sliding), penalty method etc have been used in general finite element method. But, these method need the repeated calculations, thus the CPU time is long and the accuracy depends on the number of repetitions. On the other hand, in actual CMCs, the fibers are not necessarily oriented along the loading direction, and the fiber diameter also fluctuates along the axis. Therefore, in this study, Constraint Conditional Finite Element Method (CC-FEM) was formulated to analyze the “off-axis interfacial sliding” with high precision and in a short time without repeated calculations. In CC-FEM, the equality of nodal displacements at the interface and the equilibrium of contact forces are assumed as constraint conditions, in which Coulomb's friction law and equivalent friction coefficient were taken into account. In addition, the validity was verified especially for “off-axis interfacial sliding” by comparing with general-purpose finite element software ANSYS. In the both cases of on- and off-axial interfacial sliding, the resultant stress distributions of the fiber and matrix agreed well with those of ANSYS. As compared to the case of on-axial interfacial sliding, the matrix stress recovered more steeply because of the higher equivalent frictional coefficient.

Modeling of partial dome collapse of La Soufri\xe8re of Guadeloupe volcano: implications for hazard assessment and monitoring

Over the past 9,150 years, at least 9 flank collapses have been identified in the history of La Soufrière of Guadeloupe volcano. On account of the volcano’s current unrest, the possibility of such a flank collapse should not be dismissed in assessing hazards for future eruptive magmatic as well as non-magmatic scenarios. We combine morphological and geophysical data to identify seven unstable structures (volumes ranging from 1 × 106 m3 to 100 × 106 m3), including one that has a volume compatible with the last recorded flank collapse in 1530 CE. We model their dynamics and emplacement with the SHALTOP numerical model and a simple Coulomb friction law. The best-fit friction coefficient to reproduce the 1530 CE event is tan(7°) = 0.13, suggesting the transformation of the debris avalanche into a debris flow, which is confirmed by the texture of mapped deposits. Various friction angles are tested to investigate less water-rich and less mobile avalanches. The most densely populated areas of Saint-Claude and Basse-Terre, and an area of Gourbeyre south of the Palmiste ridge, are primarily exposed in the case of the more voluminous and mobile flank collapse scenarios considered. However, topography has a prominent role in controlling flow dynamics, with barrier effects and multiple channels. Classical mobility indicators, such as the Heim’s ratio, are thus not adequate for a comprehensive hazard analysis.

Modification of friction for straightforward implementation of friction law

Friction phenomena exist in almost every mechanical device. Due to its complicated nature and influence on the system performance, extensive dynamic simulations are often required in the early system design stage. In this work, a novel approach for eliminating the numerical discontinuity in the classical Coulomb law and its extension is developed. Specifically, the method improves the computation process instead of modifying the Coulomb friction model. The estimated error of this procedure is derived under a simple and idealized model with an externally applied sinusoidal force. Two application examples of a single-body system are used to verify the proposed method, namely, 1-DOF mass-spring system with a moving belt and static ground. Results indicate that the proposed approach reveals the characteristics of the classical Coulomb friction law and its development and eliminates oscillations in the simulated friction force. Furthermore, a 3-DOF multi-body system is simulated to investigate the difference between the LuGre model and the proposed approach.

Frictional Characteristics of Metal Friction Pairs and the Amontons and Coulomb Friction Laws

Based on the interpretation of the laws of friction of Amontons and Coulomb as equations of linear regression of the friction force on the normal pressure force corresponding to the statistical processing of the experimental results, the values of the friction coefficients for six metal pairs are calculated. The principal difference between the values of the friction coefficient calculated in this way and the values calculated by simply dividing the friction force by the normal force is emphasized. Quantitatively, the difference can be more than twofold.

Modelling and dynamic analysis of an anti-stall tool in a drilling system including spatial friction

This paper investigates the effects of a down-hole anti-stall tool (AST) in deviated wells on the drilling performance of a rotary drilling system. Deviated wells typically induce frictional contact between the drill-string and the borehole, which affects the drill-string dynamics. In order to study the influence of such frictional effects on the effectiveness of the AST in improving the rate-of-penetration and drilling efficiency, a model-based approach is proposed. A dynamic model with coupled axial and torsional dynamics of a drilling system including the down-hole tool in an inclined well is constructed. Furthermore, the frictional contact between the drill-string and the borehole is modelled by a set-valued spatial Coulomb friction law affecting both the axial and torsional dynamics. These dynamics are described by state-dependent delay differential inclusions. Numerical analysis of this model shows that the rate-of-penetration and drilling efficiency increases by inclusion of the AST, both in the case with and without spatial Coulomb friction. Furthermore, a parametric design study of the AST in different inclined drilling scenarios is performed. This study reveals a design for the AST, which gives optimal drilling efficiency, robustly over a broad range of inclined drilling scenarios.

Contact Problem for a Rigid Punch and an Elastic Half Space as an Inverse Problem

We solve a contact problem of indentation of a punch into an elastic half space with regard for the friction and in the presence of the zones of adhesion, sliding, and separation. The applied approach is based on the statement of the problem in the form of the inverse problem in which the Coulomb law of friction is used as an additional condition in the regions with friction. In the formulation of the inverse problem, we take into account the presence of the zones of adhesion whose sizes are unknown. The correctness of the solution of the inverse problem is analyzed. The proposed approach, in combination with the procedure of discretization, enables us to determine the zones of microsliding alternating with the zones of adhesion and separation.

Modeling and analysis of the nonlinear indentation problems of functionally graded elastic layered solids

In tribological and thermo-electro-mechanical applications with an appropriate gradation of properties, functionally graded materials provide superior performance for damage and wear resistance of contact systems and thermal and mechanical responses for other mechanical devices. This paper presents a nonlinear mixed variational-based computational model for the analysis of nonlinear plane indentation problems of elastic functionally graded layered elastic solids. The functionally graded elastic layer is perfectly bonded to the elastic substrate and is normally loaded by a cylindrical rigid punch. In addition to nonlinearity inherent to the contact problem, the developed model accounts for the geometric nonlinearity in the sense of larger displacements and rotations, but smaller strains. The modulus of elasticity as well as Poisson's ratio of the graded layer are varying along the thickness of the layer according to both exponential and power laws. On the contrary, in the conventional homogeneous finite element formulation the isoparametric graded finite element formulation is adopted on the level of Gaussian integration points to realize the gradation in material properties. The friction at the contact interface is modeled using Coulomb's friction law. Moreover, the inequality contact constraints are exactly satisfied throughout the contact interface by employing the Lagrange multiplier method, where the indentation force and the displacement fields are treated as independent variables. The obtained results of contact pressure, tangential contact stress, stress distribution, and indentation force show a significant influence of the material distribution, gradation law, gradient index, and thickness of the functionally graded layer.

Regularized Coulomb Friction Laws for Ice Sheet Sliding: Application to Pine Island Glacier, Antarctica.

The choice of the best basal friction law to use in ice‐sheet models remains a source of uncertainty in projections of sea level. The parameters in commonly used friction laws can produce a broad range of behavior and are poorly constrained. Here we use a time series of elevation and speed data to examine the simulated transient response of Pine Island Glacier, Antarctica, to a loss of basal traction as its grounding line retreats. We evaluate a variety of friction laws, which produces a diversity of responses, to determine which best reproduces the observed speedup when forced with the observed thinning. Forms of the commonly used power law friction provide much larger model‐data disagreement than less commonly used regularized Coulomb friction in which cavitation effects yield an upper bound on basal friction. Thus, adoption of such friction laws could substantially improve the fidelity of large‐scale simulations to determine future sea level.

Modeling tangential contact of lap joints considering surface topography based on Iwan model

Abstract A contact model to capture the tangential responses of lap joints is proposed. Based on the micro-contact theory and the Coulomb friction law, the tangential contact of the contacting asperities undergone certain normal load is simplified to Jenkins element in the parallel-series Iwan model, then a continuous function possessing an explicit physical significance is developed to describe the yield force distribution of the contacting asperities on the joint interface. In order to validate the proposed model, the predicted results are compared with the published experimental results. A definition and a criterion of slip ratio are introduced to quantifiably evaluate the contact states of lap joints. The influencing factors of the distribution function and the slip ratio were analyzed.

Ping-pong ball cannon: Why do barrel and balls fly in the same direction?

An impressive ping-pong ball cannon can be made by placing a bottle of liquid nitrogen at the bottom of a container and quickly covering it with, say, 1500 ping-pong balls. The liquid turns rapidly into a gas whose mounting pressure explodes the bottle, sending a swarm of balls upward out of the container. Surprisingly, the container also moves upward. This is a counterintuitive effect because the balance of forces, that is, Newton's third law does not seem to allow the container to move upwards. We explain the effect as a consequence of granular jamming in combination with Coulomb's friction law.

Modeling of fault slip during hydraulic stimulation in a naturally fractured medium

Assessing the intensity of potential seismicity induced by fault slip is a mandatory task in the development of enhanced geothermal systems. In this paper numerical simulations are performed to investigate the propagation of hydraulic fractures and the slip behavior of existing faults due to fluid injection. The cohesive zone model is used in combination with finite cohesive elements to model the hydraulic fractures and the existing faults while the fault shear strength is assumed to follow the Coulomb friction law. Our focus is on the role of the friction conditions on the fault slip behavior. The simulation results show that faults with small friction coefficients tend to slip at low slip rates while faults with a higher friction coefficient tend to slip at rates that are higher than the unstable slip rate threshold. It is also demonstrated that under specific frictional conditions, the sequential stimulation mechanism of permeability enhancement is possible. The results suggest that a good characterization of the fault frictional conditions is required to successfully predict the fault slip pattern and that lowering the fault friction coefficient could potentially reduce the fault slip rate.

A reformulation of the nonsmooth approaches of M. Frémond and J.J. Moreau for rigid body dynamics

This paper deals with the nonsmooth dynamics of a rigid bodies system. The proposed theory is inspired by the formalism of J.J. Moreau and that of M. Frémond and relies on the notion of percussion which is the integral of the contact force during the duration of the collision. Contrary to classical discrete element models, it is here assumed that percussions can be expressed as a function of only the velocity before the impact. This assumption is checked for the usual mechanical constitutive laws for collisions derived from a pseudopotential of dissipation or the Coulomb friction law. Motion equations are then reformulated taking into account simultaneous collisions of solids. A mathematical study of the new model is presented: the existence and uniqueness of the solution are discussed according to the regularity of both the forces (Lebesgue‐density occurring during the regular evolution of the system) and the percussions (Dirac‐density describing the collision). In the light of the principles of thermodynamics, a condition on the internal percussion assuring that the collision is thermodynamically admissible, is established. Finally, an application of this new model to the motion of a system of rigid disks, including simultaneous collisions is presented.