Stick, slip, and opening of wavy frictional faults: A numerical approach in two dimensions

A simple numerical approach for calculating theq-dependence of the scattering intensity in small-angle X-ray or neutron scattering (SAXS/SANS) is discussed. For a user-defined scattering density on a lattice, the scattering intensityI(q) (qis the modulus of the scattering vector) is calculated by three-dimensional (or two-dimensional) numerical Fourier transformation and spherical summation inqspace, with a simple smoothing algorithm. An exact and simple correction for continuous rather than discrete (lattice-point) scattering density is described. Applications to relatively densely packed particles in solids (e.g.nanocomposites) are shown, where correlation effects make single-particle (pure form-factor) calculations invalid. The algorithm can be applied to particles of any shape that can be defined on the chosen cubic lattice and with any size distribution, while those features pose difficulties to a traditional treatment in terms of form and structure factors. For particles of identical but potentially complex shapes, numerical calculation of the form factor is described. Long parallel rods and platelets of various cross-section shapes are particularly convenient to treat, since the calculation is reduced to two dimensions. The method is used to demonstrate that the scattering intensity from `randomly' parallel-packed long cylinders is not described by simple 1/qand 1/q4power laws, but at cylinder volume fractions of more than ∼25% includes a correlation peak. The simulations highlight that the traditional evaluation of the peak position overestimates the cylinder thickness by a factor of ∼1.5. It is also shown that a mix of various relatively densely packed long boards can produceI(q) ≃ 1/q, usually observed for rod-shaped particles, without a correlation peak.

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A numerical approach for the Riesz space-fractional Fisher' equation in two-dimensions

International Journal of Computer Mathematics ◽

10.1080/00207160.2015.1105363 ◽

2015 ◽

Vol 94 (2) ◽

pp. 296-315 ◽

Cited By ~ 19

Author(s):

Xiaogang Zhu ◽

Yufeng Nie ◽

Jungang Wang ◽

Zhanbin Yuan

Keyword(s):

Numerical Approach ◽

Two Dimensions ◽

Riesz Space ◽

Fisher Equation

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A new methodology to simulate subglacial deformation of water saturated granular material

The Cryosphere Discussions ◽

10.5194/tcd-9-3617-2015 ◽

2015 ◽

Vol 9 (4) ◽

pp. 3617-3660 ◽

Cited By ~ 2

Author(s):

A. Damsgaard ◽

D. L. Egholm ◽

J. A. Piotrowski ◽

S. Tulaczyk ◽

N. K. Larsen ◽

...

Keyword(s):

Mechanical Response ◽

Fluid Pressure ◽

Fluid Phase ◽

Large Degree ◽

Numerical Approach ◽

Unconsolidated Sediments ◽

Close Monitoring ◽

Stick Slip ◽

Rate Dependent ◽

Water Saturated

Abstract. The dynamics of glaciers are to a large degree governed by processes operating at the ice–bed interface, and one of the primary mechanisms of glacier flow over soft unconsolidated sediments is subglacial deformation. However, it has proven difficult to constrain the mechanical response of subglacial sediment to the shear stress of an overriding glacier. In this study, we present a new methodology designed to simulate subglacial deformation using a coupled numerical model for computational experiments on grain-fluid mixtures. The granular phase is simulated on a per-grain basis by the discrete element method. The pore water is modeled as a compressible Newtonian fluid without inertia. The numerical approach allows close monitoring of the internal behavior under a range of conditions. The rheology of a water-saturated granular bed may include both plastic and rate-dependent dilatant hardening or weakening components, depending on the rate of deformation, the material state, clay mineral content, and the hydrological properties of the material. The influence of the fluid phase is negligible when relatively permeable sediment is deformed. However, by reducing the local permeability, fast deformation can cause variations in the pore-fluid pressure. The pressure variations weaken or strengthen the granular phase, and in turn influence the distribution of shear strain with depth. In permeable sediments the strain distribution is governed by the grain-size distribution and effective normal stress and is typically on the order of tens of centimeters. Significant dilatant strengthening in impermeable sediments causes deformation to focus at the hydrologically more stable ice–bed interface, and results in a very shallow cm-to-mm deformational depth. The amount of strengthening felt by the glacier depends on the hydraulic conductivity at the ice–bed interface. Grain-fluid feedbacks can cause complex material properties that vary over time, and which may be of importance for glacier stick-slip behavior.

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Transitions and singularities during slip motion of rigid bodies

European Journal of Applied Mathematics ◽

10.1017/s0956792518000062 ◽

2018 ◽

Vol 29 (5) ◽

pp. 778-804 ◽

Cited By ~ 1

Author(s):

P. L. VÁRKONYI

Keyword(s):

Coulomb Friction ◽

Rigid Bodies ◽

Two Dimensions ◽

Multiple Point ◽

Point Contacts ◽

Stick Slip ◽

Unilateral Contacts ◽

Rigid Rod ◽

Body Theory ◽

Slip Motion

The dynamics of moving solids with unilateral contacts are often modelled by assuming rigidity, point contacts, and Coulomb friction. The canonical example of a rigid rod with one endpoint slipping in two dimensions along a fixed surface (sometimes referred to as Painlevé rod) has been investigated thoroughly by many authors. The generic transitions of that system include three classical transitions (slip-stick, slip reversal, and liftoff) as well as a singularity called dynamic jamming, i.e., convergence to a codimension 2 manifold in state space, where rigid body theory breaks down. The goal of this paper is to identify similar singularities arising in systems with multiple point contacts, and in a broader setting to make initial steps towards a comprehensive list of generic transitions from slip motion to other types of dynamics. We show that – in addition to the classical transitions – dynamic jamming remains a generic phenomenon. We also find new forms of singularity and solution indeterminacy, as well as generic routes from sliding to self-excited microscopic or macroscopic oscillations.

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On the ability of a Darcy-scale model to capture wormhole formation during the dissolution of a porous medium

Journal of Fluid Mechanics ◽

10.1017/s0022112002007735 ◽

2002 ◽

Vol 457 ◽

pp. 213-254 ◽

Cited By ~ 208

Author(s):

F. GOLFIER ◽

C. ZARCONE ◽

B. BAZIN ◽

R. LENORMAND ◽

D. LASSEUX ◽

...

Keyword(s):

Porous Medium ◽

Coupled System ◽

Propagation Rate ◽

Numerical Approach ◽

Injection Rate ◽

Two Dimensions ◽

Scale Model ◽

Flow Parameters ◽

Volume Method ◽

Macroscopic Equations

Dissolution of a porous medium creates, under certain conditions, some highly conductive channels called wormholes. The mechanism of propagation is an unstable phenomenon depending on the microscopic properties at the pore scale and is controlled by the injection rate. The aim of this work is to test the ability of a Darcy-scale model to describe the different dissolution regimes and to characterize the influence of the flow parameters on the wormhole development. The numerical approach is validated by model experiments reflecting dissolution processes occurring during acid injection in limestone. Flow and transport macroscopic equations are written under the assumption of local mass non-equilibrium. The coupled system of equations is solved numerically in two dimensions using a finite volume method. Results are discussed in terms of wormhole propagation rate and pore volume injected.

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Statistics of Sliding on Periodic and Atomically Flat Surfaces

Frontiers in Mechanical Engineering ◽

10.3389/fmech.2021.742684 ◽

2021 ◽

Vol 7 ◽

Author(s):

Maja Srbulovic ◽

Konstantinos Gkagkas ◽

Carsten Gachot ◽

András Vernes

Keyword(s):

Equations Of Motion ◽

Two Dimensions ◽

Analytical Models ◽

Fourier Series Expansion ◽

Stick Slip ◽

Time Resolved ◽

Flat Surfaces ◽

Atomic Force ◽

Reaction Forces ◽

Atomically Flat

Among the so-called analytical models of friction, the most popular and widely used one, the Prandtl-Tomlinson model in one and two dimensions is considered here to numerically describe the sliding of the tip within an atomic force microscope over a periodic and atomically flat surface. Because in these PT-models, the Newtonian equations of motion for the AFM-tip are Langevin-type coupled stochastic differential equations the resulting friction and reaction forces must be statistically correctly determined and interpreted. For this, it is firstly shown that the friction and reaction forces as averages of the time-resolved ones over the sliding part, are normally (Gaussian) distributed. Then based on this, an efficient numerical scheme is developed and implemented to accurately estimate the means and standard deviations of friction and reaction forces without performing too many repetitions for the same sliding experiments. The used corrugation potential is the simplest one obtained from the Fourier series expansion of the two-dimensional (2D) periodic potential, e.g., for an fcc(111) surface, which permits sliding on both commensurate and incommensurate paths. In this manner, it is proven that the PT-models predict both frictional regimes, namely the structural superlubricity and stick-slip along (in)commensurate sliding paths, if the ratio of mean corrugation and elastic energies is properly set.

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Boundary Pointwise Control for Diffusion Hopfield Neural Network

International Journal of Nanotechnology and Molecular Computation ◽

10.4018/jnmc.2010010102 ◽

2010 ◽

Vol 2 (1) ◽

pp. 13-29 ◽

Cited By ~ 1

Author(s):

Quan-Fang Wang

Keyword(s):

Neural Network ◽

Finite Element Method ◽

Optimal Control ◽

Hilbert Space ◽

Theoretical Study ◽

Control Process ◽

Hopfield Neural Network ◽

Numerical Approach ◽

Two Dimensions ◽

Control Solution

For a close to practical neural network in biology field, in this paper the author address the diffusion Hopfield neural network (HNN) with boundary pointwise control. In the framework of variational method at Hilbert space, the theoretical study finds and characterizes the boundary optimal control solution. Furthermore, with the numerical approach consist of finite element method (FEM) and conjugate gradient method (CGM), computational demonstration is performed for three neurons in two dimensions case. This approach adequately interpreted the effectiveness and feasibility of the control process in a realistic sense.

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Fretting Contact Analysis on Three-Dimensional Elastic Layered Half Space

Journal of Tribology ◽

10.1115/1.4004104 ◽

2011 ◽

Vol 133 (3) ◽

Cited By ~ 24

Author(s):

Zhan-jiang Wang ◽

Wen-zhong Wang ◽

Fan-ming Meng ◽

Jia-xu Wang

Keyword(s):

Computing Time ◽

Three Dimensional ◽

Numerical Approach ◽

Stick Slip ◽

Coated Materials ◽

History Of ◽

Unit Stress ◽

Analytical Relations ◽

Areas Of Interest ◽

Or History

A numerical approach for solving the fretting contact on coated or layered materials, with consideration of loading history, is presented in the paper. The fretting problem was solved by using a semi-analytical method (SAM), in which analytical relations between a unit stress and corresponding displacements or stresses were obtained through the use of the Papkovich–Neuber potentials. Conjugate gradient method (CGM) and fast Fourier transform (FFT) technique were employed to increase the solution speed. The algorithm was very effective since the meshes applied to the positions were just in the contact areas of interest, which saves the computing time. The fretting contact of coated materials was studied and the effects of stick-slip behaviors were analyzed. Results show that the coupled effects between the shear tractions and the pressure make the contact behaviors quite different with the solutions from same materials. The solutions depend on the path or history of the loading process when the ball is under dynamic loads, and the contact behaviors rely on the degree of dissimilarity of material properties.

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Analysis of a Rocking and Walking Punch—Part II: General Numerical Solution Using Quadratic Programming

Journal of Applied Mechanics ◽

10.1115/1.1641062 ◽

2004 ◽

Vol 71 (2) ◽

pp. 234-239 ◽

Cited By ~ 3

Author(s):

D. Nowell

Keyword(s):

Steady State ◽

Quadratic Programming ◽

Piecewise Linear ◽

Central Zone ◽

Linear Representation ◽

Numerical Approach ◽

Stick Slip ◽

Slip Regime ◽

Special Cases ◽

Traction Distribution

This paper pursues a numerical approach to the solution of the contact problem of a rigid punch on an incompressible half-plane, subjected to a shearing force together with a normal force which may be offset from the centerline of the punch. A piecewise-linear representation of the shear tractions is employed and quadratic programming techniques are used to solve the problem. This method enables an arbitrary load history to be followed. Results are presented which show excellent agreement with other solutions to the special cases of monotonic and steady-state cyclic loading. It is shown that the traction distribution reaches a steady-state cycle after only a few cycles of loading. The existence of an interesting stick-slip regime, where a central zone of slip is bordered by two stick regions is highlighted.

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Optimization and Design of a Railway Wheel Profile Based on Interval Uncertainty to Reduce Circular Wear

Mathematical Problems in Engineering ◽

10.1155/2020/9579510 ◽

2020 ◽

Vol 2020 ◽

pp. 1-10

Author(s):

Yongjie Lu ◽

Yun Yang ◽

Jianxi Wang ◽

Bowen Zhu

Keyword(s):

Sampling Technique ◽

Numerical Approach ◽

Interval Uncertainty ◽

Wheel Wear ◽

Stick Slip ◽

Mechanics Model ◽

Design Cycle ◽

Speed Fluctuation ◽

Slip Area ◽

Wheel Profile

Wheel tread wear is a form of wheel damage that can seriously affect the performance of freight vehicles. A new numerical approach to optimizing wheel profiles can reduce circular wear on the LM wheel in the design cycle. This approach considers the influence of different line conditions and speed fluctuation on wheel wear, along with the performance of the wheel and the rail as the materials wear. In this approach, a nonlinear numerical optimization model for the wheel tread profile is built through a backpropagation (BP) neural network method. The multipoint Kik–Piotrowski (KP) contact mechanics model is applied to calculate the wheel/rail normal force, tangential creep force, the stick-slip area, and the size and shape of the contact patch. The optimal profile is obtained through the genetic algorithm (GA) method. In order to better reflect the random characteristics of wheel/rail matching and interval uncertainty, a random sampling technique is used to generate a random data sample at typical operating speeds.

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