I Giorgio

F Dell'isola

D J Steigmann

Axisymmetric deformations of a 2nd grade elastic cylinder

Keywords: Woven fabrics, elastic surface theory, strain gradients, elastic cylinder

published or not. The documents may come

Axisymmetric deformations of a 2nd grade elastic cylinder

I. Giorgio, F Dell'Isola, D J Steigmann

Introduction

Our purpose in this short communication is to illustrate the equilibrium response of 2nd grade elastic cylinders in axial tension and compression. The cylinder is regarded as a surface composed of intersecting fibers that offer elastic resistance to stretching, flexure and twist.

Conventional plate and shell theories account for elastic resistance of the surface to strain, twist and normal bending. In the present model, discussed comprehensively in [START_REF] Giorgio | Pattern formation in the threedimensional deformations of fibered sheets[END_REF], a non-standard elastic resistance to geodesic bending is also taken into account (see also [START_REF] Misra | Isola, Pantographic metamaterials show atypical Poynting effect reversal[END_REF]). This latter mode of bending is the source of a strain-gradient effect, which is not present in the constitutive equations of conventional theories of plates and shells but which is nevertheless non-negligible in fiber sheets. The general topic is the subject of extensive current research on pantographic sheets. We refer to [START_REF] Dell'isola | Pantographic metamaterials: an example of mathematically driven design and of its technological challenges[END_REF] for a comprehensive review and bibliography.

We assume the surface to be in a natural, relaxed state in a plane configuration wherein the fibers are orthogonal, straight and untwisted. The surface is then mapped to a right circular cylinder capped by rigid rings at its ends, and subsequently extended or compressed in the presence of various boundary conditions. The predicted deformations are obtained using the commercial software package COMSOL Multiphysics T M . This code furnishes a particularly convenient platform for our purpose. In particular, it requires as input only the explicit expression for the strain-energy function.

The program then constructs an associated weak form of the relevant equilibrium equations together with its finite-element implementation. Applications of the finite element method specific to second-gradient or micromorphic elasticity [START_REF] Barchiesi | Mechanical metamaterials: a state of the art[END_REF][START_REF] Kim | Gradient elasticity theory for fiber composites with fibers resistant to extension and flexure[END_REF][START_REF] Altenbach | On generalized Cosserat-type theories of plates and shells: a short review and bibliography[END_REF][START_REF] Khakalo | Modelling size-dependent bending, buckling and vibrations of 2d triangular lattices by strain gradient elasticity models: applications to sandwich beams and auxetics[END_REF][START_REF] Abali | Three-dimensional elastic deformation of functionally graded isotropic plates under point loading[END_REF][START_REF] Khludnev | Delaminated thin elastic inclusions inside elastic bodies[END_REF] are discussed in [START_REF] Maurin | Isogeometric analysis for nonlinear planar pantographic lattice: discrete and continuum models[END_REF][START_REF] Greco | A reconstructed local B formulation for isogeometric Kirchhoff-Love shells[END_REF][START_REF] Greco | An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods[END_REF][START_REF] Balobanov | Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity[END_REF][START_REF] Cazzani | Constitutive models for strongly curved beams in the frame of isogeometric analysis[END_REF][START_REF] Niiranen | Variational formulations, model comparisons and numerical methods for Euler-Bernoulli microand nano-beam models[END_REF][START_REF] Khakalo | Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software[END_REF].

In Section 2 we recall the formulation introduced in [START_REF] Giorgio | Pattern formation in the threedimensional deformations of fibered sheets[END_REF] for the strain-energy function together with the underlying kinematical framework. This is reduced, in Section 3, to a one-dimensional model for axisymmetric deformations, with the axial coordinate along the reference cylinder playing the role of the single space dimension. Section 4 concludes with a presentation and discussion of numerical examples.

Strain-energy function, axisymmetry

Our objective is to determine the position field r(u 1 , u 2 ) of the surface in equilibrium, where u α are the Cartesian coordinates of a material point on a plane.

These are taken to be u 1 = z and u 2 = R θ, where z ∈ [-L, L] is the axial coordinate on the right circular cylinder, of radius R, onto which the plane is mapped, and θ ∈ [-π, π] is the azimuthal angle on the cylinder.

Thus the plane configuration of the sheet is a rectangle of length 2L and width 2πR.

The model is based on the strain-energy function [START_REF] Giorgio | Pattern formation in the threedimensional deformations of fibered sheets[END_REF][START_REF] Giorgio | Buckling modes in pantographic lattices[END_REF]

W =w( L , M , J) + 1 2 A L |L α L β r ,αβ | 2 + 1 2 A M |M α M β r ,αβ | 2 + 1 2 A Γ |L α M β r ,αβ | 2 , (1)
where A L , A M , A Γ are material constants, L α and M α (α = 1, 2) are the Cartesian components of the unit tangent vectors to the orthogonal fibers on the plane; and

L =E αβ L α L β , M = E αβ M α M β , J = |L α M β r ,α × r ,β | (2)
are the extensional fiber strains and the areal dilation, where

E αβ = 1 2 (r ,α • r ,β -δ αβ ), (3)
in which δ αβ is the Kronecker delta, is the Lagrange strain. Here and elsewhere, r ,α = ∂r/∂u α and r ,αβ =

∂ 2 r/∂u α ∂u β .
The second and third terms in Eq. ( 1) attribute energy to the fiber stretches and their gradients in the directions of the fibers, and to the normal and geodesic bending of the fibers. The fourth term accounts for twist of the fibers as the surface deforms, and for the cross derivatives of the fiber stretches in directions orthogonal to the fibers. The latter derivatives induce stretch along parallel fibers in a small neighborhood of a material point. We refer the interested reader to [START_REF] Giorgio | Pattern formation in the threedimensional deformations of fibered sheets[END_REF][START_REF] Giorgio | Buckling modes in pantographic lattices[END_REF] for a full discussion of the physical significance of the various terms represented in Eq. ( 1).

For the strain-dependent term in Eq. ( 1) we assume [START_REF] Giorgio | Pattern formation in the threedimensional deformations of fibered sheets[END_REF][START_REF] Giorgio | Buckling modes in pantographic lattices[END_REF]

w( L , M , J) = 1 2 (E L 2 L + E M 2 M ) -G(ln J + 1 -J), (4)
where E L,M and G are positive constants. This energy does not include a term proportional to L M and therefore does not exhibit a Poisson effect with respect to the fiber axes. The term involving J penalizes fiber collapse (J → 0) by imposing unbounded growth of the energy, whereas the remaining terms are appropriate for smallto-moderate fiber strains.

In this work we assume the fibers to be straight on the reference plane and oriented at ±45 • to the edges of the rectangle, with

L 1 = L 2 = √ 2 2 and M 1 = -M 2 = - √ 2 2 . (5)
We seek axisymmetric deformations that map the cylinder to the surface of revolution described by

r = r(z)e r (θ) + ξ(z)k, (6)
where r and ξ are the radial and axial coordinates of the circle z = const. of material points on the cylinder, k is a unit vector aligned with the axis of the cylinder and e r (θ) is the radial unit vector in the cross section of the cylinder at azimuth θ.

One-dimensional formulation

A straightforward calculation furnishes

r ,1 = r e r + ξ k, r ,2 = r R e θ (7)
and

r ,11 = r e r + ξ k, r ,22 = - r R 2 e r , r ,12 = r ,21 = r R e θ , (8)
where (•) = d(•)/dz and e θ = k × e r is the azimuthal unit vector in a cross-sectional plane.

From these we obtain the fiber strains

L = M = 1 2 (E 11 + E 22 ), (9)
where

E 11 = 1 2 (r ) 2 + (ξ ) 2 -1 and E 22 = 1 2 r R 2 -1 , (10)
whereas E 12 = 0, and the surface dilation

J = r R (r ) 2 + (ξ ) 2 1/2 , (11)
together with

|L α L β r ,αβ | 2 = |M α M β r ,αβ | 2 = 1 4 r - r R 2 2 + 4 r R 2 + (ξ ) 2 (12)
and

|L α M β r ,αβ | 2 = 1 4 r + r R 2 2 + (ξ ) 2 . (13)
Substitution into Eq. ( 1) yields a strain-energy function of the form W = W (r, r , r , ξ , ξ ). ( 14)

Examples

All examples pertain to a cylinder of radius R = 1 cm and length L = 10 cm. We assume the function ξ(z)

to
ξ(L + h) -ξ(L) = hξ (L) + o(h), (15)
whereas the rigidity of the collar implies that ξ(L + h)ξ(L) = h. Our boundary condition provides the leadingorder approximation to this condition for small h. Similar reasoning justifies the boundary condition r (L) = 0.

This ensures that r(L + h) = r(L) at leading order.

Figures 1 and2 depict the deformed meridians in equilibrium; i.e., plots of normalized r vs. ξ, corresponding to imposed axial extension or compression.

These exhibit a local effect associated with the boundary conditions, while in the interior of the sheet the radius approaches a nearly uniform value. This stands in contrast to the classical necking response of a pure membrane model in tension, in which the membrane radius decreases monotonically as the symmetry plane z = 0 is approached. However, pure membrane theory is not appropriate in the present circumstances as it does not account for the higher-gradient effects associated with fiber bending and twisting resistance. For small compressive axial displacement, we predict a reduction in radius relative to its reference value. The radius is seen to increase as the axial displacement is increased. Presumably this transition to a deformation mode in which the radius is larger and the fiber curvatures are smaller serves to lower the overall energy and may be associated with post-buckling response. This conclusion is not definitive, however, as we have not conducted an analysis of potential bifurcation.

A perspective view of the compressed cylinder, showing the deformed fiber trajectories, is provided in Fig-

7 NFigure 1 :Figure 2 :

 712 Figure 1: Tension test with fully clamped boundary conditions.Each color refers to a different imposed axial displacement.

ure 3 .Figure 3 :Figure 4 :Figure 5 :

 3345 Figure 3: Perspective view of the compressed cylinder. Lengths are normalized with respect to L.

Figures 4 and 5

 5 Figures 4 and 5 depict the deformed surface under tension and compression with zero double force. The response is very similar to that of the fully clamped cylinder, but minor quantitative differences are observable.In general the double force is a vector µ[START_REF] Steigmann | Equilibrium of elastic lattice shells[END_REF] which is given, in the present circumstances, by

Acknowledgments: The work of DJS is supported by the US National Science Foundation through grant number CMMI-1538228.


Imen Lassoued
email: lassoued.imen@yahoo.fr

Ezzeddine Zagrouba
email: ezzeddine.zagrouba@fms.rnu.tn

Youssef Chahir
email: chahir@unicaen.fr

A new approach of action recognition based on Motion Stable Shape (MSS) features

Action recognition is actually considered as one of the most challenging areas in computer vision domain. In this paper, we propose a new approach based on utilization of motion boundaries to generate Motion Stable Shape (MSS) features to describe human actions in videos. In fact, we have considered actions as a set of human poses. Temporal evolution of each human pose is modeled by a set of new MSS feature's. Motion stable shapes of considered poses are defined by specific regions located at the borders of movements. Our modelisation is composed of different steps. First, a volume of optical flow frames highlighting the principal motions in poses is substracted. Then, motion boundaries are computed from the previous optical flow frames. Finally, maximally Stable Extremal Regions (MSER) are applied to motion boundaries frames in order to obtain MSS features. To predict classes of different human actions, the MSS features are combined with a standard bag-of-words representation. To prove the efficiency of our developed model, we have performed a set of experiments on four datasets: Weizmann, KTH, UFC and Hollywood. Obtained experimental results show that the proposed approach significantly outperforms state-ofthe-art methods.

I. INTRODUCTION

Human action recognition is an important component of video analysis with potential applications in video indexing, surveillance, gesture recognition and analysis of sport events. In fact, the state-of-the-art methods related to action recognition can be divided into two categories: global and local representation. Indeed, global representation methods [START_REF] Moshe | Actions as space time shapes[END_REF][START_REF] Efros | Recognizing action at a distance[END_REF] are generally restricted to some specific video actions because they rely on exact human localization or silhouette extraction. Therefore, many difficulties should be treated such as camera motion, dynamic and cluttered backgrounds, lighting changes, etc. Some other works focus on local features representation [START_REF] Marcin | Human focused action localization in video[END_REF][START_REF] Klaser | Will person detection help bag-of-features action recognition[END_REF][START_REF] Sun | Hierarchical spatiotemporal context modeling for action recognition[END_REF]. The local features methods allow to recognize a rich set of actions ranging from simple periodic motion (running, waving) to interactions (shaking hands, kissing), even in difficult realistic conditions [START_REF] Marcin | A spatio-temporal descriptor based on 3d-gradients[END_REF][START_REF] Bregonzio | Recognising action as clouds of spacetime interest points[END_REF][START_REF] Yeffet | Local trinary patterns for human action recognition[END_REF][START_REF] Taylor | Convolutional learning of spatio-temporal features[END_REF]. The principal contribution of this work is a development of a novel visual representation of human actions based on motion boundaries of the optical flow. The temporal evolution of different parts of human body is modeled with a set of particular regions of motion boundaries named motion stable shape (MSS). Given a human action sequence, the optical flow between any two consecutive frames is extracted. Their motion boundaries are derived and normalized to gray-scale images. Then, salient regions are detected using a Maximally Stable Extremal Regions (MSER) [START_REF] Matas | Robust wide baseline stereo from maximally stable extremal regions[END_REF] feature detector. For each MSS, a local descriptor is computed. These descriptors are used as visual words that contain local optical flow and boundaries motions information. The remaining of this paper is structured as follows: In Section 2, an overview of existing methods for action classification and recognition are presented. In Section 3, we have introduced our new developed method based on MSS features for representing different actions in videos. In Section 4, different experiments have been performed on several actions datasets and results have been presented. Conclusion and ideas for further work are summarised in Section 5.

II. STATE OF THE ART

In the literature, two main categories for action representation can be distinguished. The first one is global representation, witch aims to Utilize knowledge of the human location in the video and therefore learn a pattern of actions that capture the characteristics and overall body movements without any idea of the body parts. The second category is local representation where is based entirely on the descriptors of the local areas in a video, without any prior knowledge about the human positioning nor its members.

A. Global action representation

Global methods are based on the structure and dynamics of the whole body to represent human actions. Global models represent an action using descriptors of appearance and movements of either the whole body or the region of interest surrounding an actor. These representations generally depend on the silhouette extraction or structured grid that represents the area covered by the person performing an action. Global models are widely applicable because they do not rely on the identification and monitoring of various body parts. To characterize general motion and appearance of actions, many information derived from silhouettes can be used. In this case, global dynamics of body are supposed to be discriminative enough to recognize human actions. In [START_REF] Aaron | The recognition of human movement using temporal Templates[END_REF], silhouette information are used to represent actions. In this context, Motion Energy Image (MEI) and Motion History Image (MHI) are introduced to extract temporal information from video sequences. In the binary MEI, silhouettes are extracted from a single view and the difference between consecutive frames are aggregated. The MEI indicates therefore where motion occurs. At the same time, MHIs are used in combination with MEIs to weight regions that occurred more recently in time. Another way to model actions consists to use Spacetime Shapes [START_REF] Moshe | Actions as space time shapes[END_REF]. A space-time shape encodes both, spatial and dynamic information of a given human action. More precisely, the spatial information describe location and orientation of torso and limbs while dynamic information represent global motion of body and limbs. Note that human silhouettes are, in general, computed using background substraction techniques. Gorelick et al. [START_REF] Moshe | Actions as space time shapes[END_REF] propose to compute local shape properties from the solution of Poisson equation. Those properties include local saliency, action dynamics, shape structure and orientation. A sliding temporal window is then used to extract space-time chunks of 10 frames length with an overlap of 5 frames. After that, those chunks are described by a highdimensional feature vector and are matched against spacetime shapes of test-sequences. The authors employ a nearestneighbor classifier to vote for the associated class. In almost all cases, global action representation is based on a combination between the optical flow with the shape information. Moreover, Efros et al. [START_REF] Efros | Recognizing action at a distance[END_REF] propose a method to recognize human actions in low-resolution videos. In the beginning, human-centered tracks are obtained from sports footage. As a second step, motion information are encoded by blurred optical flow. The horizontal and vertical optical flow as well as positive and negative components yielding four different motion channels are separated. To classify a human action, the test sequence is aligned to a labeled data set of actions. Their method has shown promising results on different sport video datasets such as: ballet, tennis, and soccer video sequences. The drawback of this approach is that they only consider full actions of completely visible people in simple scenarios (i.e., no occlusion and simple backgrounds). Furthermore, Jhuang et al. [START_REF] Hueihan | A biologically inspired system for action recognition[END_REF] propose a biologically-inspired system for action recognition. Their approach is based on extension of the static object recognition method proposed by Serre et al. [START_REF] Lior | Robust object recognition with cortex-like mechanisms[END_REF] in the spatial-temporal domain. The original form features are replaced by motion-direction ones obtained from: gradientbased information, optical flow and space-time oriented filters. Lassoued et al. [START_REF] Lassoued | Video Action Classification: A New Approach combining Spatio-teporal Krawtchouk Moments and Laplacian Eiginmaps[END_REF] represent actions by 3D silhouettes and describe it by different types of spatio-temporal moments. Multi-class SVM is then used to classify different actions. Schindler et al. [START_REF] Luc | Action snippets: How many frames does human action recognition require[END_REF] propose a method that combines both motion and appearance information to characterize human actions. For each frame, appearance information are extracted from the responses of Gabor filters. Motion information are extracted from optical flow filters. Finally, a linear multi-class SVM estimates the final class label for a given test sequence.

A different way to model human actions was proposed by Ali et al. [START_REF] Matas | Robust wide baseline stereo from maximally stable extremal regions[END_REF]. In their approach, they use concepts from the theory of chaotic systems to model and analyze non-linear dynamics of human actions. Klaser et al. [START_REF] Marcin | A spatio-temporal descriptor based on 3d-gradients[END_REF] and Laptev et al. [START_REF] Ivan | Retrieving actions in movies[END_REF] proposed two promising global approaches for action localization in realistic video. Both methods propose an initial filtering to identify possible action localizations and to reduce the computational complexity. To avoid an exhaustive spatio-temporal search for localizing actions, Laptev et al. [START_REF] Ivan | Retrieving actions in movies[END_REF] use a human key-pose detector trained on keyframes. In a second step, actions are generated and represented as cuboids with different temporal extents and aligned to the detected keyframes. The cuboid region is represented by a set of appearance (histograms of oriented spatial gradients) and motion (histograms of optical flow) features which are learned in an AdaBoost classification scheme. These previous features can be organized in different spatial and temporal layouts within the cuboid search window. Klaser et al. [START_REF] Marcin | Human focused action localization in video[END_REF] propose a generic pre-filtering approach to detect and track humans in video sequences. Action localization is done with a temporal sliding window classifier on the human tracks. For the desription of actions, the authors introduce a spatiotemporal extension of histograms of oriented gradients (HOG) [START_REF] Navneet | Histograms of oriented gradients for human detection[END_REF], which that extracts appearance and motion information. Jiang et al. [START_REF] Jones | Relevance feedback for real-world human action retrieval[END_REF] focus on general information of the event, and use it to locate human activity or human action.

B. Local action representation

The local spatio-temporal characteristics describe form and movement for local video area. they offer a relatively independent representation of events with respect to spatio-temporal scales as well as the confusion of the background with the different movement in the scene. These characteristics are generally extracted directly from the video which avoids possible failures of other pretreatment methods such as segmentation or human movement detection. The literature propose many approaches for local spatio-temporal features extraction in videos. Laptev et al. [START_REF] Laptev | Learning realistic human actions from movies[END_REF] extend the Harris corner detector to 3D domain to determine the space-time interest points corresponding to local regions characterized by significant spatial and temporal changes. Dollar et al. [START_REF] Dollr | Behavior recognition via sparse spatio-temporal features[END_REF] descriptors are based on normalized brightness, gradient and optical flow information. Ones et al. [START_REF] Jiang | High-level event recognition in unconstrained videos[END_REF] use detector proposed by Dollar et al. [START_REF] Dollr | Behavior recognition via sparse spatio-temporal features[END_REF] to detect interest points and use k-means to cluster them. The novelty is that it integrated the relevance feedback mechanism using SVM-ABRS for action classification. Bregonzio et al. [START_REF] Bregonzio | Recognising action as clouds of spacetime interest points[END_REF] have extended this approach with 2D Gabor filters of different orientations. Hessian et al. [START_REF] Willems | An efficient dense and scaleinvariant spatio-temporal interest point detector[END_REF] have made a spatio-temporal detector based on the determinant of Hessian matrix. Wong and Cipolla et al. [START_REF] Wong | Extracting spatiotemporal interest points using global information[END_REF] have added global information to the interest point detection by applying non-negative matrix factorization (NNMF) on the entire video sequence. The locations extracted by all these approaches are sparse and detect salient motion patterns. To describe spatio-temporal points, Schldt et al. [START_REF] Lassoued | Video Action Classification: A New Approach combining Spatio-teporal Krawtchouk Moments and Laplacian Eiginmaps[END_REF] use higher order derivatives (local jets). Scovanner et al. [START_REF] Scovanner | A 3-dimensional SIFT descriptor and its application to action recognition[END_REF] extend the popular SIFT descriptor to the spatio-temporal domain. Willems et al. [START_REF] Willems | An efficient dense and scaleinvariant spatio-temporal interest point detector[END_REF] generalize the image SURF (Speeded-Up Robust Features) descriptor to the video domain by computing weighted sums of uniformly sampled responses of spatio-temporal Haar wavelets. Yeffet et al. [START_REF] Marcin | A spatio-temporal descriptor based on 3d-gradients[END_REF] propose Local Trinary Patterns for videos as extension of Local Binary Patterns (LBP). Klazer et al. [START_REF] Klaser | Will person detection help bag-of-features action recognition[END_REF] combine histograms of oriented gradients (HOG) and histograms of optical flow (HOF). Spatio-temporal interest points encode video information at a given location in space and tim. On the contrary, trajectories tracks spatial point over time and captures motion information. Messing et al. [START_REF] Messing | Activity recognition using the velocity histories of tracked keypoints[END_REF] extract features trajectories by tracking Harris3D interest points. Trajectories are represented by a sequence of log-polar quantized velocities and used it for action classification. Matikainen et al. [START_REF] Matikainen | Trajectons: Action recognition through the motion analysis of tracked features[END_REF] extract trajectories using a standard KLT tracker, cluster the trajectories and compute an affine transformation matrix for each cluster center. The elements of the matrix are then used to represent the trajectories. Sun et al. [START_REF] Sun | Hierarchical spatiotemporal context modeling for action recognition[END_REF] compute trajectories by matching SIFT descriptors between two consecutive frames. They impose a unique match constraint among the descriptors and discarded matches that are too far. Actions are described with intra-and inter-trajectory statistics. Sun et al. [START_REF] Mu | Activity recognition using dense longduration trajectories[END_REF] combine both KLT tracker and SIFT descriptor matching to extract long-duration trajectories. To assure a dense coverage with trajectories, random points are sampled for tracking within the region of existing trajectories. Spatiotemporal statistics of the trajectories are then used to discriminate different actions. Raptis and al. [START_REF] Raptis | Tracklet descriptors for action modeling and video analysis[END_REF] track feature points in regions of interest. They compute tracket descriptors as concatenation of HOG and HOF descriptors along the trajectories. Jain and al. [START_REF] Jain | Better exploiting motion for better action recognition[END_REF] decompos visual motion into dominant and residual motions, both in the extraction of the space-time trajectories and for the computation of descriptors, significantly improves action recognition algorithms. they design a new motion descriptor, the DCS descriptor, based on differential motion scalar quantities, divergence, curl and shear features. It captures additional information on the local motion patterns enhancing results. Jain and al apply the recent VLAD coding technique proposed in image retrieval provides a substantial improvement for action recognition. Xin and al [START_REF] Xin | ARCH: Adaptive recurrent-convolutional hybrid networks for long-term action recognition[END_REF] propose a learning framework using static, dynamic and sequential mixed features to solve different fundamental problems such as spatial domain variation and temporal domain polytrope. They utilise a cognitive-based data reduction method and a hybrid network up on networks architecture and extract human action representations for spatial and temporal interferences and adaptive to variations in both action speed and duration.

Methods of video actions representation are divided into two classes. Global action representation that are particularly effective when used to recognize the videos aligned spatially and temporally. These methods are not robust to occlusions (eg truncated actors), significant perspective changes, and changes in duration as they focus on the overall structure. The main idea of the proposed method is to use motion boundaries surfaces to represent human action. More precisely, the temporal evolution of different parts of human body is modeled by a set of particular regions named Motion Stable Shape(MSS). These regions are localized on the boundaries surfaces. Figure 1 summarises the different steps of our developed method. Our method is composed of different steps. The first step consists to estimate the optical flow between every two consecutive frames in the human action sequence.

Thereafter, the motion boundaries are derived and identified from the optical flow fields. In the third step, MSER regions are detected from the motions boundaries surfaces to obtain MSS sets. The final step is dedicated to classification of different image sequences using MSSs descriptors combined to bag-of-words model.

A. Motion boundaries field extraction

Motion boundaries of different image are obtained by computing spatial derivatives of each optical flow by applying the gradient function. This processing consists in eliminating the stable motions of camera and conserving the motion boundaries and changes in the optical flow fields. In fact, motion boundary is more discriminant for action classification than optical flow fields. The boundary motion function 'Bound' for each frame I is defined as follows:

Bound(⌦) = div[u, v] = d(u) dx + d(v) dy (III.1)
Where ⌦ is the optical flow variation for each frame I. u and v are respectively the horizontal and vertical components of optical flow at position (x,y). Indeed, surface boundaries are an excellent and simple source of visual motion information. As known, optical flow became discontinuous due to independent motion of different objects located in videos. As a result, we get motion boundaries between adjacent image regions having different velocities. These motion boundaries provide information about position and orientation of surface boundaries in the scene. Moreover, analysis of occlusion or disocclusion of pixels at motion boundaries can provide information about relative depth ordering of neighboring surfaces. Information about surface boundaries and depth ordering can be useful for different tasks like navigation, video compression and object recognition. The optical flow is calculated using the proposed TV-L1 variational method [START_REF] Zach | A duality based approach for realtime tvl1 optical flow[END_REF]. TV-L1 is a very efficient algorithm. In fact, variational methods are among the most popular and successful approaches for computing optical flow between two frames [START_REF] Zach | A duality based approach for realtime tvl1 optical flow[END_REF]. Among the reasons of popularity of the TV-L1 method are: a very appealing properties of the two terms in the energy formulation of the problem, the robust L1 norm in terms of data fidelity and the total variation (TV) regularization that smoothes the flow while preserving strong discontinuities. Specifically in [START_REF] Zach | A duality based approach for realtime tvl1 optical flow[END_REF], a very clean and efficient algorithm for calculating TV-L1 optical flows between gray scale images is provided. This algorithm can maintain discontinuities in the flow field and affords greater robustness against illumination changes, occlusion and noise.

B. MSS implementation

In this step, the MSS is detected from motion boundary fields using the maximally stable extremal regions (MSER) detector [START_REF] Matas | Robust wide baseline stereo from maximally stable extremal regions[END_REF]. This former extracts a set of stable connected regions from a gray scale image. These regions are defined by an extremal property of intensity function in the region and on its outer boundary. The set of MSERs is closed under continuous geometric transformations. Particularly, MSER has been widely used in image matching and object recognition, and present a better recognition performance [START_REF] Scovanner | A 3-dimensional SIFT descriptor and its application to action recognition[END_REF] in several applications. MSER depends on the threshold of the image ⇠ , if we give them some threshold value the pixels below that threshold value are white and all those above or equal are black.it also depends on the Threshold of maximum area variation between extremal regions ⇢. ⇢ is a threshold when an extremal region is maximally stable. Typical values of ⇢ range from 0.1 to 1.0.

Most of the detected MSS are localized on the boundary of silhouettes where motions are more frequent. Figure2 shows the different steps of MSS features implementation.

Given a set of MSSs, for each video and I i (i = 1 . . . N) is the set of frame from , the representative of ' Rep( )' is defined as follows

Rep( ) = i=1..N [ Ii2 feat(I i ) (III.2)
Where

feat(I i ) = j=1....s [ j MSS ij = MSER(Bound(⌦)) (III.3)
where 0 s 0 is the number of detected MSER regions in the image I i .

Below, we present the MSER algorithm used to detect a set of MSS features using motion boundaries. We describe each feature MSS ij by the discrete polynomial Krawtchouk moments defined by yap et al. [START_REF] Yap | Image Analysis by Krawtchouk Moments[END_REF]. Krawtchouk moments have the interesting property of extracting the local characteristics of the image.they can recognize non-frontal images although it is trained up with frontal images. In fact, a good recognition rate is obtained using these moments when images are corrupted by noise. Based on Krawtchouk weighted polynomials defined by yap et al. [START_REF] Yap | Image Analysis by Krawtchouk Moments[END_REF], The Krawtchouk moments of the order (n + m) for MSS ij is defines as :

&v nm = Nx X x=0 Ny X y=0 K n,m MSS ij (x, y) (III.4)
Where MSS ij (x, y) is the function intensity and Kn is the polynomial orthonormal Krawtchouk 2D proposed by Yap et al. [START_REF] Yap | Image Analysis by Krawtchouk Moments[END_REF] defined as:

K n,m = Kn (x; p x , N x ) Km (y; p y , N y ) (III.5)

C. Bag of MSS features

As mentioned above, each input video is described by a set of (MSSs) descriptors vectors. Based on these MSSs, the image sequences are classified using a bag-of-words model [START_REF] Nistr | Scalable Recognition with a Vocabulary Tree[END_REF]. This model aims to represent videos by the occurrence of features or descriptors named visual words. Thus, videos are identified and compared using histograms of the visual word occurrences. Using this technique, various levels of abstraction are obtained after building a tree structure by repetitive clustering descriptors. We consider a class the one with the most similar visual word in each level.

The proposed approach aims to group descriptors of the detected MSS features into important visual words containing local optical flow and boundary motions. In particular, signatures are constructed by counting the number of visual word (MSS descriptors) in the video sequence. Figure 3 illustrates a diagram of video representation and classification with MSS features descriptors. Finally, a machine learning algorithm to train and classify visual word signatures is applied.

IV. EXPERIMENTAL RESULTS

The experimentation process is composed of six steps. The first one consists in estimating the optical flow in image sequences. After that, the motion boundary for each frame is computed. In the following step, we apply the MSER detector to obtain Motion Stable Shape (MSS)witch depends on the complexity of surfaces. After computing, each videos have generated a number of MSS features between 500 and 700. Fourth, the MSS feature are described using Krawtchouk moments [START_REF] Lassoued | Video Action Classification: A New Approach combining Spatio-teporal Krawtchouk Moments and Laplacian Eiginmaps[END_REF]. Fifth, we use hierarchical k-means to cluster descriptors into 278 visual words. We use these visual words signatures to classify acion sequences. Sixth, random ferns process is trained with 150 trees each one contains 1024 leaf nodes witch are used to make the last classification. Moments order is chosen after a series of experiments to study its influence on the classification result. Several tests of classification were performed on different datasets for several orders. Figure 4 shows the variation of average classification rate for different Krawtchouk moment orders. Curvesof figure 4 show clearly that the best classification rate in different datasets has been obtained for the order 60.

To asses objectively the proposed work, a set of experiment was performed in different datasets KTH, WEIZMAN, UCF sports and Hollywood (Figure5). KTH dataset [START_REF] Schuldt | Recognizing human actions: A local SVM approach[END_REF] consists in six human action classes: walking, jogging, running, boxing, waving and clapping. The Weizmann actions dataset [START_REF] Efros | Recognizing action at a distance[END_REF] is composed of nine different types of action classes: bending downwards, running, walking, skipping, jumpingjack, jumping forward, jumping in place, galloping sideways, waving with two hands, and waving with one hand. For these two datasets, video backgrounds are homogeneous and static which simplifies computing. The choice of KTH and WEIZMAN datasets is explained by the fact that they are the most used in similar works. Furthermore, Hollywood [START_REF] Yeffet | Local trinary patterns for human action recognition[END_REF] and UCF sports [START_REF] Rodriguez | Action mach: A spatio-temporal maximum average correlation height lter for action recognition[END_REF] datasets are caracterized by higher action complexity. We remind that Hollywood dataset videos has been collected from 69 Hollywood movies and representing 12 action classes(answering the phone, driving car, eating, fighting, getting out of car, hand shaking, hugging, kissing, running, sitting down, sitting up and standing up). Training and test sequences come from different movies. Table 1 shows the average classification rate for weizmann dataset compared to other work using different descriptors in bag-of-words classifier. We note that, the average classification rate of our proposed method is higher than other works witch prove its good effencicy. Table 1 illustrates an average recognition results. Indeed, the use of MSERs regions placed on the surface of boundary motion volumes improves the results by 3%. This demonstrates the advantages of analyzing stable regions of motion boundary. In table I, four different feature representations and descriptor combinations taken from the literature are compared with our developped method. For the two first features : gray-scale image data and optical flow magnitude field, both are described by 3D SIFT. Concerning the binary volumes feature, they are described by a 3D shape context. Finally, our method uses MSS feature and described by krawtchouk moments. The best result is achieved by our MSS features with a classification rate of 98.7%. This highlighted the benefit of using flow volumes and stable motion boundaries as feature representation. This proves that boundary surfaces of motions contain enough information to properly classify actions in videos.

Table II shows experimental result of our developed method using bag-of-word classifier and comparison with state-ofart works using other different classifiers for different video actions datasets. This comparison proves that using MSS features with bag-of-words classifier improves results in the tested datasets. Inded, we have obtained a classification rate arround 98% for the two action datasets Weizmanna and KTH and this result is considered among the best compared to other methods sited in the table2. Nevertheless, classification rate of our method is becoming lower for UCF and Holluwood dataset(86% in UCF and 58% in Holluwood). But these last results remain higher than those obtained by other methods shown in table2. We can explain the decrease in classification rate UCF and Holluwood datasets by the fact that they contain more complex actions besides background mobility. we note that, our results are globally similar to those presented by XIN et al. [START_REF] Xin | ARCH: Adaptive recurrent-convolutional hybrid networks for long-term action recognition[END_REF] and Jain et al [START_REF] Jain | Better exploiting motion for better action recognition[END_REF].

In order to study computation complexity, we have investigated theoretical and practical complexity of our method. We have computed MSS temporal feature and their Krawtchouk moments for each video.

The theorical complexity Cx can be written as [START_REF] Bregonzio | Recognising action as clouds of spacetime interest points[END_REF] 72.8 Raptis et al. [START_REF] Raptis | Tracklet descriptors for action modeling and video analysis[END_REF] 96.67 Matikainen et al. [START_REF] Matikainen | Trajectons: Action recognition through the motion analysis of tracked features[END_REF] 82.6 Klaeser et al. [START_REF] Klaser | Will person detection help bag-of-features action recognition[END_REF] 84.3 Our method 98.7 KTH Kovashka et al. [START_REF] Kovashka | Learning a hierarchy of discriminative space-time neighborhood features for human action recognition[END_REF] 94.5 Youan et al. [START_REF] Yuan | Discriminative video pattern search for efficient action detection[END_REF] 93.7 Le at al. [START_REF] Le | Learning hierarchical invariant spatio-temporal features for action recognition with independent subspace analysis[END_REF] 93.9 Gilbert et al. [START_REF] Illingworth | Action recognition using mined hierarchical compound features[END_REF] 94.5 xin et al. [START_REF] Xin | ARCH: Adaptive recurrent-convolutional hybrid networks for long-term action recognition[END_REF] 95.2 Our method 98.3

UCF sports

Wang et suter [START_REF] Wang | Recognizing Human Activities from Silhouettes: Motion Subspace and Factorial Discriminative Graphical Model[END_REF] 85.6 Klaeser et al. [START_REF] Klaser | Will person detection help bag-of-features action recognition[END_REF] 86.7 Kovashka et al. [START_REF] Kovashka | Learning a hierarchy of discriminative space-time neighborhood features for human action recognition[END_REF] 82.27 Le et al.

86.5 Our method 86.8

Hollywood

Wang et suter [START_REF] Wang | Recognizing Human Activities from Silhouettes: Motion Subspace and Factorial Discriminative Graphical Model[END_REF] 47.7 Taylor et al. [START_REF] Taylor | Convolutional learning of spatio-temporal features[END_REF] 46.6 Guilbert et al. [START_REF] Illingworth | Action recognition using mined hierarchical compound features[END_REF] 50.9 Li ey al [START_REF] Le | Learning hierarchical invariant spatio-temporal features for action recognition with independent subspace analysis[END_REF] 53.3 xin et al. [START_REF] Xin | ARCH: Adaptive recurrent-convolutional hybrid networks for long-term action recognition[END_REF] 63.1 Jain et al. [START_REF] Jain | Better exploiting motion for better action recognition[END_REF] 62.5 proposed method 53 a bag-of-words model to classify different actions for image sequences in datasets KTH, Wezmainn, UFC and Hollywwod.

The experimental results show that this method captures effeciently action information in videos and show good performances compared to state-of-the-art approaches for action classification. Future work will focus on the combination of our MSS features with other features based on appearance and spatial destribution to further improve recognition performance.

Figure 1 .

 1 Figure 1. Flowchart of the proposed method

Figure 2 .

 2 Figure 2. MSS feature of human motion

Algorithm 1 :

 1 MSS implementation Require: Bound(⌦) Ensure: setof M SS ⇠ = find( Threshold intensity) ER=Extremal Regions(⇠) ⇢ = find(T hreshold max area variation) setof M SS = Maximally Stable Extremal Regions(⇢,ER )

Figure 3 .

 3 Figure 3. Illustration for video representation and classification with MSS features descriptors

Figure 4 .

 4 Figure 4. Average of classification rate vs different Krawtchouk moments orders

Figure 5 .

 5 Figure 5. Examples of video actions from datasets: (a) UCF sports , (b)KTH, (c)hollywood and (d) Weizmann

Cx=

  Nbr ⇤ (O(2 ⇤ nlog(n) + n 2 + nb m (ns ⇤ log(log(ns))))) (IV.1) Where (nbr) is the number of frame considered in video,(n) is the number of pixels in each frame, (nb m ) is the number of MSS feature in each frame and ns is a dimensionality of each MSS.Nine training video clips from the KTH dataset were used To analyze the computatio complexity of MSS extraction. The nine previous video clips have resolution of 160 ⇤ 120 pixels and approximately 1000 frames. The computation time is devided as follows : 35% of stable boundary motion computation, 30% of the optical flow computing and the rest of the time is consumed equally between moments descriptors and motion boundary computation.V. CONCLUSION This paper introduces a new approach based on MotionStable Shape (MSS) for video action recognition. To obtain MSS feature, each video is transformed to a set of motion boundaries volume. MSERs regions are then detected in computed motion boundaries. We have used MSS signatures with

Table I ACTION

 I RECOGNITION PERFORMANCE COMPARISON ON THE WEIZMANN

	DATASET FOR DIFFERENT COMBINATIONS OF FEATURES
	Methods	Average classification rate (%)
	Gray+ 3D SIFT	90.22
	Flow+ 3D sift	93.11
	Binary MSV+ 3D shape context	96.67
	Proposed method	98.7

Table II COMPARISON

 II OF THE MSS DESCRIPTOR TO THE-STATE-OF-THE-ART, AS REPORTED IN THE CITED PUBLICATIONS IN DIFFERENT DATASETS

	Dataset	Methods	Average classification rate (%)
		Laptev et al. [13]	68.4
		Bregonzio et al.
	Weizmann

978-1-5090-4320-0/16/$31.00 ©2016 IEEE


Karina A Valenzuela Aracena

Jesica G Benito

Luc Oger

Irene Ippolito

Rodolfo O Uñac

Ana M Vidales
email: avidales@unsl.edu.ar

Frequency-amplitude behavior for incipient movement of grains under vibration

Keywords: vibration, incipient movement, forced oscillator, 3D DEM simulation

This paper focus on the onset of the movement of particles placed on a horizontal rough surface subjected to a vertical sinusoidal vibration. We track the problem through experiments, theory and numerical simulations. The frequencies needed to put particles in movement decays exponentially with the input amplitude of the oscillations. This behavior is explained through a simple approach where a forced damped harmonic oscillator is used with a spring constant representing all the interactions between the particle and the surface.

The numerical results compare well with the experimental ones, proving that the forces included in the numerical calculations are suitable to account for the main aspects of the problem, even though that the complexity of the surface is not fully accounted for.

Describing the way in which the frequency varies with amplitude could be relevant when technological applications like, for instance, surface cleaning take place.

Introduction

The displacement of grains in industrial processes frequently implies external energy supply through agitation in conveyor belts, truck transport, drilling operations, among many others. As a result, the manipulation of granular matter always carries secondary effects, desired or not. Segregation, crushing, jamming, flow fluctuations, particle resuspension and saltation are just some examples. Much scientific attention is dedicated to the solution of the problems related to granular materials behavior, but many questions, especially those related to the microscopic interaction between grains and surfaces, are not yet solved.

It has been proven that the initiation of movement of particles deposited on a surface is crucial to predict their possible further resuspension by air flow turbulence [START_REF] Henry | Progress in particle resuspension from rough surfaces by turbulent flows[END_REF]; [START_REF] Reeks | Kinetic models for particle resuspension in turbulent flows: theory and measurement[END_REF][START_REF] Valenzuela Aracena | Experiments and numerical modeling for the movement and resuspension of grains[END_REF]). The concept of "fluid threshold" (or the "aerodynamic threshold") is already introduced by Bagnold many years ago [START_REF] Bagnold | The Physics of Blown Sand and Desert Dunes[END_REF]; [START_REF] Chepil | Dynamics of wind erosion: I. Nature of movement of soil by wind[END_REF]; [START_REF] Zingg | Wind tunnel studies of the movement of sedimentary material[END_REF]). On the other hand, initial destabilization of sand grains by the wind is important to answer if transport of sandy matter in desert geography can be expected [START_REF] Oger | Model of the saltation transport by Discrete Element Method coupled with wind interaction[END_REF]).

In the same way, vibration, as a perturbation of particle's equilibrium, is relevant in many theoretical and applied problems, like, for instance, the removal of small particles from surfaces in engineering applications [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF]). Likewise, in granular matter manipulation stages, the mechanical vibration of grains, either horizontal or vertical, is also an important external perturbation.

Naturally, if the vertical vibration can compensate the gravitational acceleration, jumps are observed as typical movements for the grains. Studies of the fluidization of granular beds subjected to vibrations are commonly found in the literature [START_REF] Eshuis | Phase diagram of vertically shaken granular matter[END_REF], [START_REF] Renard | Vertically shaken column of spheres. Onset of fluidization[END_REF], [START_REF] Mawatari | Effect of particle diameter on fluidization under vibration[END_REF]), where the critical intensity of the external excitations to fluidize the grains is investigated. Moreover, even jamming of a drained granular medium can be observed depending on the time needed for rearrangements of the grains [START_REF] D′anna | Vibration-induced jamming transition in granular media[END_REF]). In recent years, the formation of patterns on the surface of vibrated layers of grains has been extensively studied. Depending on the whole set of parameters (amplitude and frequency of the excitation, shape and size of grains and container, and so on) different kinds of standing waves can be observed, with varied textures [START_REF] Melo | Transition to parametric wave patterns in a vertically oscillated granular layer[END_REF]; [START_REF] Melo | Hexagons, kinks, and disorder in oscillated granular layers[END_REF]; [START_REF] Metcalf | Standing wave patterns in shallow beds of vibrated granular material[END_REF]). On the other hand, basic studies concerning the excitation of small spheres deposited on a horizontal platform has been conducted in the past [START_REF] Ripperger | Measurement of adhesion forces in air with the vibration method[END_REF]; [START_REF] Tippayawong | Analysis of microparticle resuspension in turbulent flows with horizontally vibrating surface[END_REF]).

The first step to describe a system consisting of a particle sitting on a vibrating surface can be performed considering a mechanical approach as the one presented in [START_REF] Dybwad | A sensitive new method for the determination of adhesive bonding between a particle and a substrate[END_REF].

In that work, the author studies the problem of a particle of mass m deposited on a vibrated surface (quartz resonator) as two coupled springs. The particle is linked to the surface through an elastic force with constant k. The quartz resonator has a mass M and an elastic constant K, and its resonance frequency only depends on its elastic properties. The frequency of the coupled system (particle -quartz resonator) results to depend on the elastic force constant, k, of the particle. The behavior of that frequency is found to contradict the mass loading expectation, i.e., that the frequency of an oscillating system will decreases when its mass increases. An experimental estimation of the value of the elastic constant shows that the actual area of contact of the two touching solids (particle and surface) is very much less than their geometrical area, i.e., than the area of contact that could be determined through a direct calculation, given that it may be lowered by the presence of roughness on the surfaces [START_REF] Dybwad | A sensitive new method for the determination of adhesive bonding between a particle and a substrate[END_REF]).

As a second step in the description of the perturbation of a particle deposited on a surface, several linear and nonlinear oscillation models are introduced and analyzed in [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF] to demonstrate whether particle removal is possible for soft and hard particles on smooth and rough surfaces. One of the conclusions in that paper is that the natural frequency may be reduced by up to two orders of magnitude in case that the distance between two contacts at which the particle touches the surface is much smaller than the particle radius. Besides, they emphasize the idea that behavior of particles is like if they were linked to the surface by springs [START_REF] Dabros | Motion of latex spheres tethered to a surface[END_REF]; [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF]).

Experiments using aerosol particles deposited on a vibrating glass surface demonstrate that the wall vibration plays two opposite roles, i.e., to increase the separation force between particles and the wall, and to increase the particle deposition rate due to active capture of aerosol particles [START_REF] Theerachaisupakij | Effects of wall vibration on particle deposition and reentrainment in aerosol flow[END_REF]). In the same way, analytical models show that frequency of particle-surface interaction is found to significantly influence the removal rate of micro-particles from a vibrating surface [START_REF] Tippayawong | Analysis of microparticle resuspension in turbulent flows with horizontally vibrating surface[END_REF]). Besides, the description of the re-entrainment process implies the analysis of adhesion and capillary forces between the particle and the surface, the inertia of the particle when accelerated and its contact interaction with the particles belonging to the surface, especially in cases where roughness is of the order of the size of the beads deposited on it.

For that reason, the vibration of a surface as an indirect method for the measurement of adhesion forces between a particle and a wall is used in [START_REF] Hein | Analysis of adhesion forces between particles and wall based on the vibration method[END_REF], by correlating particle re-entrainment events with the acting acceleration.

To conclude, we may say that the dynamic involved in the initiation of a particle movement on a surface subjected to an external perturbation is not simple, and the inclusion of all the parameters playing a role is not straightforward [START_REF] Mullins | Effect of geometry on particle adhesion[END_REF]), even when only a threshold velocity for movement initiation is looked for [START_REF] Soepyan | Threshold velocity to initiate particle motion in horizontal and near-horizontal conduits[END_REF]). As a result, the conditions under which the incipient movement of grains subjected to vibration is expected are still an open-question topic, even for particles of the millimeter sizes.

The aim of the present study is to predict the critical conditions for a particle to move when it rests on a vibrating surface, and to evaluate the physical parameters involved in this process. To pursue this objective we perform series of experiments over rough surfaces (built with micrometer glass beads) where different particle sizes (within the millimeter range) are deposited for vibration. We also develop a theoretical approach that describes the main aspects of the experimental behavior found here. The idea behind is to find the critical frequency-amplitude pairs needed to initiate the movement of the deposited grains for different bead sizes and materials and to prove that the system behaves like a forced damped oscillator with a weak elastic constant (compared to that of the particle's material)

given by the force interaction between particle and surface. In this sense we follow the former ideas developed in the works by [START_REF] Dybwad | A sensitive new method for the determination of adhesive bonding between a particle and a substrate[END_REF] and [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF].

To reinforce our goal, we also perform some numerical tests using a Discrete Element Method with the appropriate interactions between the grains and the surface, to model the experimental conditions [START_REF] Herrmann | Modeling granular media with the computer[END_REF]; [START_REF] Luding | Collisions & contacts between two particles[END_REF]). Indeed, we simulate a set of experimental results where a good agreement is found when we include the presence of elastic forces due to the bead material and also those due to adhesion and capillary effects.

Experiments

Experimental setup and materials

The device for the experiments consists of the sinusoidal vertical vibration of a horizontal rough surface on which beads are placed in order to study the onset of their movement. The rough surface is built by gluing glass beads on a flat circular plate with a diameter of 6 cm.

We work with two different rough surfaces: one, made with 250 µm glass beads and the other one of 500 µm glass beads, with respective size dispersions of 44 µm and 85 µm. A Scanning Electron Microscope (SEM) image is shown in Figure 1 to illustrate the surface topography created by the glued beads for the case of 250 µm. It is easy to see in some regions, 2 or 3 layers of beads due to the manufacturing method of the surface. This is also observed in the 500 µm surface.

Concerning the free particles deposited on the surface, we use glass beads whose mean diameters are: (0.93 ± 0.02) mm, (1.98 ± 0.04) mm, (3.26 ± 0.09) mm and (3.98 ± 0.03) mm; and stainless steel ball bearings of 2, 3 and 4 mm.

The plate is fixed to the drive arm of a mechanical oscillator and can vibrate with a selected frequency given by a wave generator connected to it. The mechanical oscillator has a drive arm that allows fixing on it the plate with the rough surface to be vibrated. The amplitude of oscillation of this arm is proportional to the signal (measured in volts) of the wave generator connected to the mechanical oscillator. The linear correspondence between voltage and amplitude is verified and calibrated carefully. To this end we make use of two different techniques. The first one consists in the tracing, with the help of a fine fiber pen, of the trajectory of the oscillating arm-plate system. The double check is achieved by recording the movement of the arm-plate system with a video camera Casio EXILIM EX-ZR400.

We also verify the dependence of the amplitude given by the generator on the different frequency ranges and that the sinusoidal shape of the signal does not change with time. For the range involved in the present experiments, constant amplitude can be provided, independently of the working frequency.

Another important point is to know if the critical frequency depends on the rate of increment during the experimental run. Indeed, we detect a dependence on the frequency rate, so a controlled ramp is provided with a fixed rate equal to 0.63 rad/s 2 in all the experiments.

In the present experiments we use sinusoidal waves with a given well controlled amplitude, A, and frequency, ω. Figure 2 gives a sketch of the experimental set up. The values selected and measured for the amplitudes used in the experiments are: 0.73, 1. 28, 1.91, 2.28, 2.58, 2.87, 3.12, 3.47, 3.88 and 4.03 mm, with an error of ± 0.06 mm. The error in the different frequencies provided by the generator is of the order of 1 rad/s. Initially, ten spheres with a given diameter are deposited randomly over the surface. They are carefully separated to prevent possible contacts or hits among them or with the lateral edges which are transparent thin walls stuck around the disk to avoid the flow out of the free particles.

The desired amplitude is selected and fixed in the wave generator (see (1) in Figure 2) and the platform is subjected to an oscillation starting from a frequency of 0.10 Hz (approximately 0.63 rad/s). Next, the frequency is increased in fixed steps at a rate of 0.10 Hz/s.

With the help of a magnifying glass, we check for the incipient movement of particles and record the critical frequency, ω c , at which at least half of the deposited spheres starts to move from their original positions. Movement is considered when a particle has a displacement of the order of its radius. This procedure is also recorded by the digital video camera referred above, at 140 fps, to complete the determination of ω c (see ( 5) and ( 6) in Figure 2). Once the critical frequency is reached, the experimental run ends, and the oscillator is turned off.

A new value for the amplitude of oscillation is chosen and the spheres are redistributed over the surface. The frequency is set to 0.10 Hz and the excitation is started again with the same increasing rate. The critical frequency for the new amplitude value is registered in the same way as explained before.

To avoid important humidity effects, experiments are always performed with a relative ambient humidity between 35% and 55%. This allows having the same capillary interaction between the spheres and the surface [START_REF] Kohonena | On capillary bridges in wet granular materials[END_REF]; [START_REF] Rabinovich | Capillary forces between surfaces with nanoscale roughness[END_REF]).

Finally, for a given surface and a given deposited sphere size and material, a set of 200 different positions on the surface are inspected in order to average the results shown in the next section.

Results and discussion

In our experiments, we observe that positions of the beads which initiate their motion are randomly distributed on the surface. In Figure 3 we show the results obtained for the critical frequencies, ω c , needed to put into incipient movement at least 50% of the glass particles deposited on the vibrating surface as the amplitude of the excitation is increased.

The error bars for the amplitude are indicated and those for the statistical error in the determination of the critical frequency are of the order of the symbol size. Part (a) of the figure corresponds to the 250 µm surface and part (b) to 500 µm. The different ratios between the size of the free spheres and the size of the beads belonging to the surface are indicated in each case.

On the other hand, our results demonstrate to be practically independent of the size of the beads used to build the surface. This implies that the roughness of the surface created by the glued spheres is such that the characteristic length of it is not dictated by the size of the constituting beads, but rather by the geometrical arrangement of them with the glue. In other words, the topography of the surface created by the glued grains masks the influence of their own sizes.

According to the definition of ω c , the set of points divides the upper region corresponding to the existence of movement of at least 50% of thespheres from the lower part with absence of movement or with incipient movement of less than 50% of the particles.

The results for both surfaces and all ratios are quite similar. They can be described by an exponential decay in all cases, independently of the size ratio. So, as the amplitude of the oscillation increases, the critical frequency needed to initiate the particle's movement decreases.

In order to know whether the behavior is still the same for a different material, we perform a series of experiments with stainless steel spheres deposited on the same two rough surfaces. By following the same procedure explained earlier, we obtain the results shown in Figure 4. Comparing the two parts of the figure, it is clear that the behavior for both surfaces is practically the same, with a slight displacement of the data to higher frequencies for the smaller spheres. As indicated by the curves, the behavior can also be fitted by an exponential decay. It is necessary to note that experiments with the smaller 1mm stainless steel beads are not performed to avoid static charge effects.

According to the results shown so far, a clear dependence on the excitation amplitude is experimented by the critical frequency needed for a given particle to start its motion. In the scheme that the system could behave like a lightly damped forced oscillator, it is expected that one natural resonance frequency will be present. Nevertheless, the exponential decay of the critical frequency as the vibration amplitude increases is very solid and, as we will see in the next section, the simple scheme of a forced oscillator has to be rethought to explain our present results [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF]).

Theoretical approach

Model of the results

In this section, we explain, from a theoretical point of view, the possible physical basis leading to the experimental results presented above. Using a basic approach, we analyze the problem of a spherical particle on top of a vibrating bed. This system can be represented as a forced damped harmonic oscillator like the one sketched in Figure 5.

As presented in the introduction, we assume that the key point is to consider that our system behaves like a harmonically excited oscillator with a spring representing the effective interactions between particle and surface. These interactions are due to the intermolecular forces and capillary forces, which lead us to assume that their effect is like an elastic bond ("spring") between the particle and the surface, with stiffness constant k [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF]).

In this way, the stiffness constant represents the effective bond seen by the excited particles. Here, it is important to clarify that the attractive part of the effective interaction force in our model could be attributed to the presence of van der Waals and capillary effects. Indeed, these forces are still important when compared with gravitational ones even in the case of particles with few millimeters size. Both forces increase with the radius of the particle and it has been calculated that their value are not negligible respect to the weight [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF], [START_REF] Zhu | Discrete particle simulation of particulate systems: Theoretical developments[END_REF]). For capillary forces, although the relative humidity values at which experiments are performed is not so high, it is enough to generate capillary bridges among the particles and the surface that have to be considered in the problem [START_REF] Herminghaus | Dynamics of wet granular matter[END_REF]). On the other hand, were these forces not present, the sole inertial force due to the external excitation would yield to a constant critical acceleration, which is not the case, as will be shown below.

Like it is clearly discussed in [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF], the oscillatory motion of a particle on a surface may be caused by the mechanical vibrations of it and nonlinear effects can be present. The fact that the oscillation of a rough surface can cause a complex vibration movement of the particle deposited on it makes it necessary to consider a nonlinear behavior. In the case of a forced oscillator with a nonlinear restitution force [START_REF] Pain | The physics of vibrations and waves[END_REF]) it is calculated that, depending on the sign of the nonlinear term, the effective spring constant decreases with the deformation of the spring, which is propose in our model below.

It is very difficult to measure the elastic properties of the bond between particles and surface but, our experimental results show (as plotted further below) that a non linear behavior is present. Given the complexity of resolving the equations of motion for such a nonlinear behavior [START_REF] Wang | The role of damping in phase imaging in tapping mode atomic force microscopy[END_REF]), we propose an approximation by considering a simple linear harmonic damped oscillator where we introduce a correction on the dependence of the natural frequency on the oscillation amplitude [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF]).

We assume an external harmonic force exerted on the particle through the spring by the vibrating bed with amplitude A and frequency ω/2π. Using standard oscillation analysis and with the help of the sketch in Figure 5, let figure out which is the amplitude that the particle will see when external excitation is turned on.

To follow the theoretical model proposed here, one only has to remember the main features of a forced damped harmonic oscillator. Thus, the amplitude seen by the particle can be calculated following the standard resolution as:

2 2 2 2 0 0 * ) ( ) ( / ) ( γω ω ω ω + - = m F A (1)
In equation ( 1), ω 0 and γ are the natural frequency of the system and the damping constant of the elastic bond between the particle and the surface, respectively. As known, ω 0 will depend on the stiffness of the bond.

In the present case, experiments are done by varying the frequency of the external excitation, so the amplitude of the force, F 0 , is not a constant but varies with Aω 2 , where A is the input amplitude of the oscillator, fixed at each experimental run. Thus, as the run goes on, the amplitude seen by the sphere is:

2 2 2 2 0 2 * ) ( ) ( ) ( γω ω ω ω ω + - = A A (2)
Using equation ( 2), the acceleration undergone by the particle can be calculated as 𝐴𝐴 * (𝜔𝜔)𝜔𝜔 2 . When a different value for the amplitude A is selected (a new experimental run begins by increasing the frequency) new series of A * (ω) values are seen by the particle as the experiment goes on and, as a consequence, the velocity and the acceleration suffered by the particle change.

To illustrate our point we apply the theoretical model to two of the experiments performed above. We consider glass spheres of 1mm deposited on the 250 µm and on the 500 µm glass beads surfaces. Figure 6 shows the forces undergone by the particle deposited on the oscillating surface for each value of the amplitude and calculated by using 𝐹𝐹 * = 𝑚𝑚𝐴𝐴 * (ω)ω 2 . Note that in our present model, the curves will be the same for both types of surfaces. The mass of the sphere is 1.053 x 10 -6 kg, by taking 2500 kg/m 3 as the glass density. For a better visualization of the values we plot the results in a semi-log scale. The meaning of the horizontal lines (very close one to the other) is explained below.The ten curves correspond to the different amplitudes of excitation used in the experiments and they are enumerated in the legend. Besides, a damping constant of 200 s -1 is considered. This value will be (as seen below) of the order of the natural frequencies entering in equation 2, thus modeling a lightly damped harmonic oscillator. In our checks, the model shows to be almost insensitive to the value of this constant starting at 100 s -1 and up to one order of magnitude. Only the range of oscillation frequencies involved in our experiments is displayed in the plots.

Taking into account that some authors find that the stiffness of a spherical particle is not constant but increases with particle deformation [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF]), it is not unreasonable to think that, in our present scenario, k will depend on the amplitude of the oscillations. It is worth noting that it is very difficult to quantify the elastic properties of the bond between particle and surface. For that reason some assumptions and estimations are done for the values of ω 0 used in the figures.

In our present case, it is expected that particle deformation will be larger as the applied acceleration increases and, thus, according to [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF], the stiffness k will increases. Experimentally, the critical acceleration decreases as the input amplitude A increases (see below). Consequently, k decreases for increasing values of the input A. Given that the mass of the particles is a constant, a decrease in k means a decrease in the natural frequency ω 0 of the system. Taken into account that an exponential decay is evident among the critical frequency-amplitude pairs found in our experiments, we propose, as a reasonable trend, an exponential decay of the natural frequency with the amplitude A. This idea is also in coincidence with the expected decaying behavior of vibrating frequencies as the restoring force becomes non-linear [START_REF] Kovacic | Forced vibrations of oscillators with a purely nonlinear power-form restoring force[END_REF]).

Considering the range of values obtained for the experimental critical frequencies and their relation with the different input amplitudes, we propose that the natural frequencies ω 0 obey the decaying behavior with A displayed in Figure 7. It is worthy to say that these are the values for ω 0 which are used to plot the curves in Figure 6. Now, consider the force necessary to overcome the interaction forces holding a particle attached to the surface (spring breaking). The horizontal lines in Figure 6 represent this force, assuming that its value is constant for a given particle size. Although we do not know this precise value, we can say that its magnitude is of the order of the nano-Newtons for micrometric particles on flat surfaces [START_REF] Ripperger | Measurement of adhesion forces in air with the vibration method[END_REF]). Again, taking into account the range of values for ω c found in our experiments, we are only able to estimate detaching forces of 267 nN for 250 µm and 259 nN for 500 µm surfaces. This rough estimation seems reasonable if one thinks that, in our present case, we deal with 1mm particles on surfaces where the attaching forces are expected to decrease because of the presence of roughness. We will discuss these values in the next section below.

The successive intersections of the horizontal lines with the different curves in Figure 6 give the critical frequencies at which incipient movement of the particles is expected at a

given amplitude of excitation. The values obtained at the intersections are plotted, respectively, in each part of Figure 8, along with the corresponding experimental results evaluated from the data in Figure 3(a) and (b). The agreement is quite good.

In the same way, taking the different values for ω c obtained from the intersections, one can calculate the maximum critical acceleration, 𝑎𝑎 𝑐𝑐 = 𝐴𝐴𝜔𝜔 𝑐𝑐 2 , at which the vibrating plate is subjected when 50% of particle movement is starting as a function of the amplitude. The results are respectively presented for both surfaces in Figure 9, where the comparison with experimental data is also shown. Here again, the agreement is good and we can appreciate the above mentioned decreasing trend of ω 0 as the amplitude increases.

Discussion

The theoretical approach developed here allows the plotting of the force seen by the moving beads as the external frequency increases, for different amplitudes. Given that in all the cases the moving particle is the same, the force calculus for both rough surfaces yields In this way, one could conclude that the interaction of the beads with both surfaces is quite similar and this is likely due to the way that the surfaces are constructed. The high degree of disorder present in the space distribution of the glued beads on the surface is more important that their size, thus, the moving particles interact in a similar way with both surfaces.

It is interesting to mention that in experiments for ideally rigid particles on a rigid plane which is vibrated sinusoidally, the so called Froude criterion is suitable to characterize critical detachment frequencies when only inertial forces are present. As expressed above, adhesion forces have to be taken into account. For that reason, the experimental data cannot be represented by that criterion and one would have to modify that criterion to embody both the elastic and adhesion properties of the particle-plate system [START_REF] Eshuis | Phase diagram of vertically shaken granular matter[END_REF][START_REF] Renard | Vertically shaken column of spheres. Onset of fluidization[END_REF]).

The model is able to describe quite well the experimental dependence of the critical frequencies on the input amplitude. Only the idea of a harmonic oscillator resonating at the natural frequency corresponding to the particle material is not enough, in coincidence with the ideas proposed by [START_REF] Dabros | Motion of latex spheres tethered to a surface[END_REF] and by [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF]. Although the exponential decrease of the critical frequency ω c with the amplitude A is not explicitly derived from the theoretical calculus, it can be appropriately deduced by the present model as soon as an appropriate fitting of the spring stiffness ω 0 is implemented in the model.

Finally, the decrease of the acceleration as A increases is an indication that a non-linear behavior of the natural frequency of the system could be happening. Indeed, the assumption for the dependence of 𝜔𝜔 0 on the amplitude, as displayed by Figure 7, is necessary to explain, through this simplified model, the results obtained experimentally.

Simulation Model

In this Section we present the outline of our simulation model to show which are the main forces contributing to a good description of the problem and validating that the inclusion of elastic, adhesion and capillary forces are enough to explain, from a numerical approach, the experimental behavior found.

Numerical Setup

We use a classical Discrete Element Method (DEM) model to represent the dynamical interaction between the moving particles and the rough surface. The present model follows the two-dimensional formulation of [START_REF] Savage | Disorder, diffusion, and structure formation in granular flows[END_REF] and the three-dimensional extension used by [START_REF] Oger | How disorder can diminish avalanche risks: effect of size distribution[END_REF]. In order to make the reading clearer all the equations are regrouped in the Appendix.

The particles are modeled as spheres using a "soft-particle" approach (equations A1toA3),

where each sphere can have multiple contacts that can persist for extended durations (typically during 50 time steps). Normal and tangential forces can develop at the contact between two spheres.When only compressive forces are allowed, the simulations represent dry, non-cohesive particle assemblies. The presence of tensile forces at the contacts models cohesive particles. In this case, short range interactions can occur up to the breakage of the adhesive forces or the breaking of a liquid bridge. No long range interactions are present in this model. In this way, bonds between contacting spheres can be introduced at the initial stage of the packing or during the oscillation process, thus mimicking an assembly of particles with tensile strengths.

In our model, a tensile (negative) value is possible up to a limiting tensile strength defined for the bond, after which the bond breaks. These bonds can arise, as expected, due to the presence of van der Waals and/or capillary forces (see equations A4 and A5). When no cohesive forces are present, a contact takes place only when two spheres overlap, and the normal and tangential forces increase as the centers of the particles approach each other.

The tangential force has also a critical tensile strength below which it is modeled as a viscoelastic one and, above it, the bond breaks and the tangential force follows the Mohr Coulomb law.

In order to simulate local contact effects and non-perfect spherical particles, a rolling resistance parameter has been introduced recently [START_REF] Ai | Assessement of rolling resistance models in discrete element simulations[END_REF]). This parameter represents the effects of the shape of the particle on its rolling ability and the possible resistance due to plastic deformation around the contact and/or possible viscous hysteresis.

We consider this parameter in our model and its value is taken between the range 0 (no effect) to 0.4, which is the maximum value that the slope angle can attain when the rolling resistance torque equals the one produced by gravity acting on the particle.

In order to simulate the experimental set up, we first create a rough surface of glued spheres with diameter 500 µm. We randomly deposit up to three layers of these spheres on the surface. In all cases, they are sitting on a stable position under gravity with one contact in the case of the first layer, or up to three contacts for the two upper layers. Thereafter, all the spheres are assumed as fixed in their positions and motionless respect to the vibrating plate.

They only interact with the upper moving spheres. This glued layer simplifies the global interaction calculation, as only sphere contacts are present in the simulation, i.e., no flat wall-sphere interactions are possible.

Once the rough surface is ready, 200 spheres with 1 mm diameter are randomly deposited on it to consider a large number of different configurations. To earn computing time, we prevent the particles to see each other, i.e., no billiard balls effects are present during all the simulation process. After the moving spheres attained equilibrium positions, the surface constituted by the glued spheres is put under vertical oscillation at a given amplitude and frequency by moving the center of these spheres. Due to the interaction forces, this process will consequently generate the oscillation of all the spheres. It is worthy to mention here that this oscillation will limit the choice of the time step for simulations. Indeed, besides the classical soft DEM rule of using the length of the time steps for the numerical integration as 1/50 of the duration of the collisions, we have to take into account that the displacement provoking the oscillation of the glued spheres will generate an overlap with the upper spheres that has also to follow that soft DEM rule.

Since increasing the frequency step by step like in the experiments would consume a very long computer time, we run the program a few times for each frequency-amplitude pair and look for spheres in movement after a given time. Thus, we can easily get the 50% of moving spheres threshold with a good statistic. The set of input parameters used in the simulation are given in Table 1.

Results and Discussion

In Figure 10 We understand that the main reason for explaining the difference in behavior between simulations and experiments has to do with the roughness of the surface. In simulations, the number of layers and the mean distance between spheres in these upper layers do not have the topological complexity of the surface generated through real glued spheres. In this way, the ease of the moving beads to overcome the barrier to jump from a hole is greater for the case of simulations than for experiments. This is evident especially for lower amplitudes where the critical frequencies observed in simulations are respectively lower. We also try other ratios and confirm this behavior. This will be the subject of further research.

Conclusion

In this work, we aim to determine the critical moment when particles deposited on a vibrating surface are put into movement. Although the experimental set up is rather simple, the explanation of the exponential decaying relation found for the critical frequencyamplitude pairs is not straightforward.

As discussed in previous works [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF]), our experiments

show that the frequencies are quite lower than those related to the elastic properties of the particles material (6x10 6 rad/s or higher). These frequencies follow an exponential decay with the amplitude of the sinusoidal external excitation.

Following the idea that particles behave like a forced damped oscillator, we develop a simple resonant spring model where the key point stands on the assumption that the stiffness of the vibrating spring encompasses all the moving particle interactions with the surface and it results to be quite lower than the one associated to glass stiffness alone.

Besides, it is necessary to assume that the natural frequency of that spring depends on the particle deformation and, as a consequence, on the acceleration amplitude of the vibrations. This is in line with other authors' arguments, where the stiffness constants of oscillators are analyzed as a function of the external excitation for both linear and non-linear restoring forces [START_REF] Ziskind | Particle behavior on surfaces subjected to external excitations[END_REF]; [START_REF] Kovacic | Forced vibrations of oscillators with a purely nonlinear power-form restoring force[END_REF]). The idea of a harmonic oscillator resonating at the natural frequency corresponding to the particle material, alone, is not enough. In this way, the theoretical model proposed here describes the experimental behavior quite closely, explaining the exponential decay obtained for the critical frequencies, although it is not explicitly derived from the theoretical calculus. .28, 1.91, 2.28, 2.58, 2.87, 3.12, 3.47, 3.88, 4.

  the same curves shown in Figure6. On the other hand, the cohesion force (adhesion plus eventual capillary bridges) experimented by the particles when they try to detach from the surface (spring breaking) is different. The two values proposed (267 nN and 259 nN) for each case are very close. Such equal values are evidencing the similar behavior found whencomparing Figures 3 (a) and (b). They are deduced by drawing horizontal lines in Figure6that take into account the experimental range of values within which ω c varies with A. In other words, by taking into account the frequency range, the intersections of the horizontal line with each amplitude curve give the correct values of the experimental critical frequencies. We have to say that these force values in Figure6are just rough estimations obtained from the model. They represent the force needed to destabilize the particle and are inside the wide range of adhesion forces estimations found in the literature[START_REF] Zhu | Discrete particle simulation of particulate systems: Theoretical developments[END_REF],[START_REF] Ripperger | Measurement of adhesion forces in air with the vibration method[END_REF],[START_REF] Jones | Adhesion Forces between Glass and Silicon Surfaces in Air Studied by AFM: Effects of Relative Humidity, Particle Size, Roughness, and Surface Treatment[END_REF]). Evidently, if one could measure those force values, the inverse way could be performed to predict the critical frequencyamplitude ranges originally measured in our experiments.

  we show the simulation results for the critical pair frequency-amplitude for the case of 500 µm surface and glass beads with diameter 1 mm. The corresponding experimental and theoretical results are shown for comparison. The exponential decay is recovered again in simulations and the agreement with experiments is reasonably good. The lines in the figure indicate fitting exponential curves.

Figure 1 :

 1 Figure 1: SEM image from the surface of 250 µm glass beads.

Figure 2 :

 2 Figure 2: Sketch of the experimental set up used to measure the critical amplitude-

Figure 3 :

 3 Figure 3: Critical frequency as a function of the vibration amplitude for glass beads on (a)

Figure 4 :

 4 Figure 4: Critical frequency as a function of the vibration amplitude for stainless steel

Figure 5 :

 5 Figure 5: Sketch of the forced damped harmonic oscillator representing our model system.

Figure 6 :

 6 Figure 6: Calculated forces undergone by the particle deposited on the oscillating surface

  03 millimeters. Note the semi-log scale. The horizontal lines correspond to the detaching forces for each surface: 250 µm (upper) and 500 µm (lower).

Figure 7 :

 7 Figure 7: Exponential decay assumption for the dependence of the natural frequency, ω 0 ,

Figure 8 :

 8 Figure 8:Comparison between the experimental and theoretical values for the critical

Figure 9 :

 9 Figure 9: Comparison between the experimental and theoretical values for the maximum

Figure 10 :

 10 Figure 10: Comparison with simulation results for critical frequency-amplitude pairs for

Table Table 1: Input parameters for the 500 µm surface and 1 mm moving glass beads at 70% of relative humidity

 Table

	Parameter	Value
	Density for glass	2500 kg/m 3
	Friction coefficient, µ i	0.5
	Restitution coefficient	0.8
	Stiffness, K n Stiffness, K t	10 6 N/m 1.6x10 6 N/m
	Dashpot constant,b n	0.11kg/s
	Dashpot constant,b t	0.4b n
	Rolling resistance	0.3
	Surface tension, γ sl Hamaker coefficient, A H	72 mN/m 6.5 x 10 -20 J
	Figure captions

Acknowledgements

Authors want to acknowledge financial support from Universidad Nacional de San Luis through PROICO 310114, from CONICET through PIP 353, from MYNCyT-ECOS Sud, Argentine-France, project A15E03 and LIA CONICET-CNRS Physique et Mécanique de Fluides.

Contrary to the theoretical model, where the elastic constant can be thought of as the resultant of all the interactions of the particle with the surface, in the simulation model, the lack of a real topological description of the surface roughness is responsible that the critical frequencies result lower than expected, especially for lower amplitudes where space barriers are more difficult for the moving particles to overcome. In this sense, a deeper study of the critical frequency-amplitude pairs for incipient movement of particles on a highly controlled monolayer of glued spheres is in course.

Appendix

Spring-dashpot model for particle contact

The i th particle is characterized by its radius r i , the position of its center (x i ,y i ) and the angular rotation θ i around its center. Interparticle forces exist only when two spheres overlap or when the cohesive forces keep them in close contact, and the normal and tangential contact forces increase as the centers of the particles approach each other.

The normal force F n at the contact is modeled as viscoelastic. It consists of an elastic (a linear spring) and a viscous damping (a linear dashpot) contributions, described as follows:

where K n is the spring constant for normal forces, δ is the relative normal displacement between the centers of the two particles in contact, σ is the distance between their two centers, r i and r j are the two radii of the particles, v n is the relative normal velocity and b n is the dashpot constant for normal forces.

The force in the tangential direction is also modeled as a viscoelastic one where a linear spring and a linear dashpot are used as follows:

where K t is the spring constant for tangential forces, δ t is the relative lateral displacement during all the duration of the contact, v t is the relative tangential velocity, and b t is the dashpot constant for tangential forces. F t is also limited to a maximum value which is chosen according to a Coulomb friction law where slipping can occur:

where µ i is the coefficient of friction. The tangential force acts in a direction opposite to that of the relative tangential velocity v t .

The definition of the two spring constants and the two dashpot terms can be directly related to the classical mechanical parameters describing a material, i.e., Young's modulus, Poisson ratio and restitution coefficient (see, for example, [START_REF] Mishra | On the determination of contact parameters for realistic DEM simulations of ball mills[END_REF]).

In the present model, we can introduce bonds between spheres that are in contact depending of the surrounding water content. The decrease of the cohesive force (F cap ) during the pulling apart process of two adjacent spheres is modeled using the equation proposed by [START_REF] Charlaix | Adhesion forces between wetted solid surfaces[END_REF]:

where cos(θ), defined from the wetting angle, is around 0.84 for water on glass, γ sl is the liquid surface tension and R is the effective radius of the two interacting particles.

Finally, the van der Waals force between particles, F VW , is modeled through an always attractive force as [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF]):

where A H is the Hamaker coefficient, R is, as in (A4), the effective radius and D is the distance between the surfaces.