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Vol. 58 Zavarise, G., Wriggers, P. (Eds.) Trends in Computational Contact Mechanics 354 p. 2011 [978-3-642-22166-8] Vol. 57 Stephan, E., Wriggers, P. Modelling, Simulation and Software Concepts for Scientific-Technological Problems 251 p. 2011 [978-3-642-20489-0] Vol. 54: Sanchez-Palencia, E., Millet, O., Béchet, F. Singular Problems in Shell Theory 265 p. 2010 [978-3-642-13814-0] Vol. 53: Litewka, P. Finite Element Analysis of Beam-to-Beam Contact 159 p. 2010 [978-3-642-12939-1] Vol. 52: Pilipchuk, V.N. Nonlinear Dynamics: Between Linear and Impact Limits 364 p. 2010 [978-3-642-12798-4] Vol. 51: Besdo, D., Heimann, B., Klüppel, M., Kröger, M., Wriggers, P., Nackenhorst, U. Elastomere Friction 249 p. 2010 [978-3-642-10656-9] Vol. 50: Ganghoffer, J.-F., Pastrone, F. (Eds.) Mechanics of Microstructured Solids 2 102 p. 2010 [978-3-642-05170-8] Vol. 49: Hazra, S.B. Large-Scale PDE-Constrained Optimization in Applications 224 p. 2010 [978-3-642-01501-4] Vol. 48: Su, Z.; Ye, L. Identification of Damage Using Lamb Waves 346 p. 2009 [978-1-84882-783-7] Vol. 47: Studer, C. Numerics of Unilateral Contacts and Friction 191 p. 2009 [978-3-642-01099-6] Vol. 46: Ganghoffer, J.-F., Pastrone, F. (Eds.) Mechanics of Microstructured Solids 136 p. 2009 [978-3-642-00910-5] Vol. 45: Shevchuk, I.V. Convective Heat and Mass Transfer in Rotating Disk Systems 300 p. 2009 [978-3-642-00717-0]

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Trends in Computational Contact Mechanics

Giorgio Zavarise, Peter Wriggers (Eds.)

123

Prof. Giorgio Zavarise University of Salento Department of Innovation Engineering Via per Monteroni - Edificio "La Stecca" 73100 Lecce Italy

Prof. Dr.-Ing. habil. Peter Wriggers Leibniz Universitaet Hannover Institut fuer Kontinuumsmechanik Appelstr. 11 30167 Hannover E-mail: [email protected] http://www.ikm.uni-hannover.de

ISBN: 978-3-642-22166-8 e-ISBN: 978-3-642-22167-5

DOI 10.1007/ 978-3-642-22167-5

Lecture Notes in Applied and Computational Mechanics ISSN 1613-7736

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Preface

Contact mechanics is a science that has a great impact on everyday life and is presentin many different fields. These include civil, mechanical and environmental engin-eering, but also medicine since locomotion as well as functional joints do not workwithout friction. In one application friction is needed – like traction of car tyres –and in another application friction produces wear and costs – like in bearings. Thusit is of the utmost interest to have reliable and efficient methods and associated ana-lysis tools that can be applied to a vast range of contact problems.

Using the power of today’s computers many complex contact problems can besolved with numerical simulation tools. Despite the progress that has been reachedwith respect to the implementation of contact algorithms in commercial codes, vividresearch is still going on in the area of contact mechanics. Thus, within the lastyears, computational contact mechanics has been a topic of intense research. Theaim of the development is to devise robust solution schemes and new discretizationtechniques, which can be applied to different problem classes in engineering andscience.

These are wide-ranging and include computational aspects of discretization tech-niques using finite and boundary element methods. Special solution algorithmsfor single- and multi-processor computing environments are of great interest forefficient solutions. Furthermore, multi-scale approaches have been applied suc-cessfully to contact problems and multi-field formulations were used for thermo-mechanical or electro-thermo-mechanical applications involving contact. Discreteelement models include always contact and pose a challenge for the numerical treat-ment due to the high number of particles. Finally, problems like rolling wheels andtyres need special contact formulations and special algorithmic approaches.

Technical applications incorporate different interface problems. Examples arefailure processes in heterogeneous materials, textile and laminated composites, in-teraction between road and tyres, hip implants or artificial knee joints as well asspraying of particles on surfaces and impact analysis of cars.

The present book summarizes work in the area of computational contactmechanics that was presented at the 1st International Conference on Computational

Preface

Contact Mechanics in Lecce, Italy. The authors discuss different theoreticalmethodologies, algorithms for the solution of contact problems and apply these todifferent engineering problems.

Hannover and Lecce, April 2011P. Wriggers and G. Zavarise

VI

Table of Contents

Contact Modelling in Entangled Fibrous Materials 1Damien Durville

3D Contact Smoothing Method Based on Quasi-C1 Interpolation 23Maha Hachani and Lionel Fourment

On a Geometrically Exact Theory for Contact Interactions 41Alexander Konyukhov and Karl Schweizerhof

Finite Deformation Contact Based on a 3D Dual Mortar and Semi-SmoothNewton Approach 57Alexander Popp, Michael W. Gee and Wolfgang A. Wall

The Contact Patch Test for Linear Contact Pressure Distributions in 2DFrictionless Contact 79G. Zavarise and L. De Lorenzis

Finite Deformation Thermomechanical Contact Homogenization Framework 101Ilker Temizer and Peter Wriggers

Analysis of Granular Chute Flow Based on a Particle Model IncludingUncertainties 121F. Fleissner, T. Haag, M. Hanss and P. Eberhard

Soft Soil Contact Modeling Technique for Multi-Body System Simulation 135Rainer Krenn and Andreas Gibbesch

A Semi-Explicit Modified Mass Method for Dynamic Frictionless ContactProblems 157David Doyen, Alexandre Ern and Serge Piperno

Table of Contents

An Explicit Asynchronous Contact Algorithm for Elastic-Rigid BodyInteraction 169Raymond A. Ryckman and Adrian J. Lew

Dynamics of a Soft Contractile Body on a Hard Support 193A. Tatone, A. Di Egidio and A. Contento

Two-Level Block Preconditioners for Contact Problems 211C. Janna, M. Ferronato and G. Gambolati

A Local Contact Detection Technique for Very Large Contact andSelf-Contact Problems: Sequential and Parallel Implementations 227V.A. Yastrebov, G. Cailletaud and F. Feyel

Cauchy and Cosserat Equivalent Continua for the Multiscale Analysis ofPeriodic Masonry Walls 253Daniela Addessi and Elio Sacco

Coupled Friction and Roughness Surface Effects in Shallow SphericalNanoindentation 269P. Berke and T.J. Massart

Application of the Strain Rate Intensity Factor to Modeling MaterialBehavior in the Vicinity of Frictional Interfaces 291Elena Lyamina and Sergei Alexandrov

Unilateral Problems for Laminates: A Variational Formulation withConstraints in Dual Spaces 321Franco Maceri and Giuseppe Vairo

Contact Modelling in Structural Simulation – Approaches, Problemsand Chances 339Rolf Steinbuch

VIII

Contact Modelling in Entangled FibrousMaterials

Damien Durville

Abstract An approach to model contact-friction interactions between beams withinassemblies of fibers is presented in this paper in order to simulate the mechanicalbehaviour of entangled structures at the scale of individual fibers using the finiteelement method. The determination of contact elements associating pairs of mater-ial particles is based on the detection of proximity zones between beams and on theconstruction of intermediate geometries approximating the actual contact zone, andallowing to consider contact along zones of non-zero lengths. The penalty methodfor contact is improved by adjusting the penalty parameter for each contact zone,thus stabilizing contact algorithms and allowing to handle high numbers of contactelements. Applications to samples of textile materials involving few hundreds offibers are presented to demonstrate the abilities of the method. The presented ex-amples are related to the simulation of woven fabrics – computation of the initialconfiguration and application of test loadings – and the identification of the trans-verse mechanical behaviour of a twisted textile yarn.

1 Introduction

Entangled fibers are involved in different types of structures and materials, rangingfrom biological tissues to technical textiles used as reinforcements in composites.The characteristic features of the nonlinear mechanical behaviour of media consti-tuted by fibers rely mostly on contact-friction interactions developed between in-dividual fibers. The finite element simulation can be usefully employed to betterunderstand elementary mechanisms ruling the global behaviour of such structuresand identifying their mechanical properties. The approach presented in this paperis aimed to this purpose. Its goal is to consider samples of reduced size of fibrous

Damien DurvilleLMSSMat, Ecole Centrale Paris/CNRS UMR8579, Grande Voie des Vignes,92290 Chatenay-Malabry, France; e-mail: [email protected]

G. Zavarise & P. Wriggers (Eds.): Trends in Computational Contact Mechanics, LNACM 58, pp. 1–22.springerlink.com c© Springer-Verlag Berlin Heidelberg 2011

D. Durville

materials, taking into account all individual fibers contained in these samples, andmodeling contact-friction interactions between them. The mechanical problem isset in the form of determining the equilibrium of an assembly of fibers, submittedto large displacements and finite strains. A kinematically enriched beam model isused to represent fibers, and assuming quasistatic loadings, an implicit scheme isemployed to solve the problem.

The issue of modelling contact between beams, at the core of our approach, hasbeen addressed in various ways in the literature, in a finite element framework. Usualtechniques to determine contact between deformable surfaces seem hardly applic-able to the case of beams. These techniques, based on master/slave strategies, checkcontact at nodes on one surface, associating to each node a target on the oppositesurface, frequently using the normal direction to the first surface. Because crossingsbetween beams can occur anywhere with respect to the finite element discretization,checking contact at nodes is often not sufficient in the case of beams. To overcomethis difficulty, some authors [4–6] use a minimum distance criterion to characterizethe location of contact. This approach has the advantage to determine accuratelythe location of contacts between beams and to provide a symmetrical treatment ofinteracting beams, and is particularly suitable for situations where the contact limitsto only one point within each contact zone. However, in cases where contact is to beconsidered as a continuous phenomenon along a contact zone of non-zero length,the notion of minimum distance loses its relevance.

In order to handle the various contact configurations that can be encountered de-pending on the angle formed between interacting beams, we propose another way ofdetermining contact, based on the determination of proximity zones between beamsand on the construction of intermediate geometries in each proximity zone. Theintermediate geometry, whose role is to approximate the geometry of the actualunknown contact zone, is used as a support for the contact discretization. Contactelements are generated between pairs of material particles on the surfaces of in-teracting beams that are predicted to enter into contact at some discrete locationsdefined on the intermediate geometry. This way contact is determined consideringsymmetrically both interacting beams with respect to the intermediate geometry,and a discretization size for contact can be defined. Another advantage of the intro-duction of proximity zones is to reduce the cost of the contact search by splittingit into two different tasks: the first one at a global level consisting only of associ-ating coarsely pairs of close parts of beams, and the second one performed moreaccurately at the local level to associate particles for contact elements.

The high density of contacts involved in entangled structures is a challenge forthe convergence of nonlinear algorithms dedicated to contact friction. In order tostabilize contact algorithms, two main improvements to the penalty method are im-plemented: a quadratic regularization for small penetrations to smooth transitionsbetween contacting and non-contacting status, and a local adjustment of the penaltyparameter for each contact zone to control the maximum penetration. The combin-ation of these two ingredients is essential to make algorithms converge for casesinvolving up to nearly 100 000 contact elements.

2

Contact Modelling in Entangled Fibrous Materials

The paper is organized as follows. Section 2 summarizes the way the global prob-lem is set, and describes the 3D beam model used to represent fibers, based on threevector fields (nine degrees of freedom) according to Antman’s theory with two un-constrained directors. Section 3 presents the approach to determine contact betweenbeams based on the construction of proximity zones and intermediate geometries.Section 4 is dedicated to the mechanical models for contact-friction interactions,and to some algorithmic aspects. Three applications are then presented in Section 5to demonstrate the capabilities of the approach. The first presented application is atest of sliding between two orthogonal beams to illustrate the taking into account ofa moving frictional contact. The second one deals with the simulation of samplesof woven fabrics, made of 480 fibers, from the determination of the unknown initialconfiguration of such structures, until the characterization by biaxial and shear load-ing tests. The third example refers to the identification of the transversal behaviourof a twisted textile yarn constituted of 250 fibers, and crushed between moving rigidtools.

2 Mechanical Equilibrium of an Assembly of Entangled Fibers

2.1 Principle of Virtual Work

In order to simulate the mechanical behaviour of an entangled structure, we con-sider it as an assembly of N fibers submitted to various loadings and undergoinglarge displacements, and we characterize the global displacement solution u by thefollowing principle of virtual work, using a full-Lagrangian formulation:

Find u kinematically admissible, such that ∀v kinematically admissibleN

∑I=1

(∫Ω I

0

Tr

(s(u)

DEDu

·v)

dωωω +N

∑J=I

W IJcf (u,v)

)=

N

∑I=1

W Iext(v). (1)

In the above expression, the internal work for each fiber is expressed as functionof the second Piola–Kirchhoff stress tensor s and the Green–Lagrange strain tensorE; W IJ

cf is the virtual work of contact-friction interactions between fibers I and J,and W I

ext is the virtual of external loads applied to the fiber I. While the expressionof contact-friction interactions is the main subject of the paper, the beam modelemployed to represent the behaviour of each fiber is now briefly described.

2.2 3D Beam Model

The model used to account for the mechanical response of fibers describes the kin-ematics of each cross-section by the means of three kinematical vectors, following

3

D. Durville

Fig. 1 Kinematical beam model.

Antman’s theory [1]. According to this model, the position x of any particle of thebeam, identified by its coordinates (ξ1,ξ2,ξ3) in the reference material configura-tion is assumed to be expressed as follows:

x(ξξξ ) = x0(ξ3)+ ξ1g1(ξ3)+ ξ2g2(ξ3). (2)

In this equation, ξ1 and ξ2 stand for the transverse coordinates of the particle in thecross-section, and ξ3 for its curvilinear abscissa along the beam axis (see Figure 1).The three kinematical vectors involved in the model, defined on the centroidal lineof the beam, are the position of the center of the cross-section, x0(ξ3), and twodirectors of the cross-section, g1(ξ3) and g2(ξ3). Accordingly, the displacement ofany particle ξ of the beam is expressed as

u(ξξξ ) = u0(ξ3)+ ξ1h1(ξ3)+ ξ2h2(ξ3). (3)

where u0(ξ3) is the displacement of the center of the cross-section, and h1(ξ3)and h2(ξ3) are the variations of the cross-section directors. The variations of cross-section directors are unconstrained, and both the norms of these directors and the re-lative angle between them may vary. According to this first order kinematical model,cross-sections are assumed to remain plane but may deform depending on the vari-ations of cross-sections directors. For instance, intially circular cross-sections cantake can any elliptical shape depending on the applied mechanical loading. In par-

4


ticular, the Poisson effect – contraction of the cross-section generated by an axialstretch – is naturally considered by this model.

The use of three kinematical vector fields (nine degrees of freedom) to describethe kinematics of each cross-section allows the derivation of a full 3D strain tensor,including in particular planar deformations of cross-sections. Standard constitutivelaws can thus be employed to express 3D stresses in the beam as function of strains.Cases of specific behaviours, such as for example reduced bending or torsional stiff-nesses for some particular fibers, can be considered either by using an orthotropicconstitutive law, or by changing artificially the values of the moments of inertia usedin the calculation of stresses.

The 3D beam model is implemented under a nonlinear finite strain framework.

3 Geometrical Handling of Contacts within an Assembly ofFibers

The case of entangled structures presents a specific context for the contact detectioncharacterized by a high number of interacting fibers, and by large relative displace-ments between fibers making the arrangement of fibers continuously changing.

The goal of the contact detection is to predict locations where contact is likely tooccur, and to provide with pairs of entities between which non-penetrating condi-tions can be formulated. Because of the high number of fibers possibly considered,the global algorithm for detecting contact must be fast.

3.1 Continuous Geometrical Approach of Contact between Fibers

3.1.1 Considerations of Various Contact Configurations

Diversified situations of contacts depending on the angle between fibers and on theextension of the contact zone can be encountered in a general assembly of fibers.When fibers cross each other with an angle close to 90, the contact is almost point-wise. Yet for fibers forming locally a small angle, contact can be viewed as a con-tinuous phenomenon along a contact surface that can be assimilated to a line ofgiven length. In cases where two fibers are almost parallel or where a fiber is woundaround another, the contact may be continuous all along the fibers.

For the pairing task in the contact detection, that consists in associating parts ofthe structure that are likely to come into contact, we seek a unique strategy ableto handle the various contact configurations that can be encountered in a generalassembly of fibers. The minimum distance criterion, employed in most of methodsdealing with contact between beams to determine contact points, is effective in caseof crossing between fibers, where the contact zone can be reduced to a point. How-ever, if contact is continuous along a zone of non-zero length, the distance between

5

D. Durville

fibers is almost constant in this region; the search of a minimum distance loses itsrelevance and another way of associating points in contact is required.

Various strategies are available for the consideration of contact between deform-able surfaces. To couple contact entities, most of them take points on one of thesurfaces, and determine corresponding target points on the opposite surface, usingfor example the normal direction to the first surface for this search. Such strategies,that demonstrate efficiency for deformable surfaces, seem nevertheless hard to ad-apt to the case of contact between deformable beams. Besides the fact it provides anon-symmetrical treatment of both interacting beams, the use of normal directions(planes orthogonal to the centerline of beams) raises difficulties in regions with highcurvatures.

3.1.2 Pairing of Portions of Fibers

In order to preserve the continuous aspect of contact along a zone, instead of coup-ling directly points through a minimum distance criterion, we suggest to associatefirst pairs of portions of fibers that are close to each other. By this way, we try toconsider simultaneously the geometry of both fibers in contact in order to betterapproximate the geometry of the actual contact zone. To this end, we proceed asfollows. First we determine proximity zones between fibers in the whole assembly,defining a proximity zone as a pair of portions of fibers that are stated to be close toeach other. Next, for each proximity zone, we determine an intermediate geometry,defined as the average between the two line segments constituting the proximityzone. This intermediate geometry can be viewed as an approximation of the actualcontact zone. Instead of defining contact on one beam with respect to the other,contact is now defined on this intermediate geometry with respect to both beams.Because it depends on both geometries, normal directions determined from this in-termediate geometry are better suited to the search of contact than those determinedonly from one of both beams.

3.2 Proximity Zones

As the number of contacts in entangled structures may be high, the determinationof proximity zones meets a first need of reducing the cost of the contact search byoperating first a coarse localization of contact. The purpose of this task is to delimitpairs of intervals on beam centerlines that are estimated to be close to each other. Itis performed considering any possible pair of beams in the assembly. For a pair ofbeams (I,J), test points generated only to evaluate the distance between beams, aredistributed on the first beam I according to a discretization size lzp, and the closestdistance to the other beam is calculated for each of these test points. The k-th testpoint on the first beam being identified by its curvilinear abscissa sI

k = klzp, wesearch the curvilinear abscissa of the closest point on the centerline opposite beam,

6


Fig. 2 Determination of proximity zones.

denoted sJk∗ (Figure 2). Considering the centerline of beams in their discretized form

by finite elements, the closest point is either an orthogonal projection on a finiteelement or a node.

Giving a proximity criterion ∆prox, we define the k-th proximity zone betweenbeams I and J, denoted ZIJ

k , in the following way:

ZIJk =

([sI

k1,sI

k2], [sJ

k1

∗,sJ

k2

∗]), such that

∀k ∈ [k1,k2],dist(x(sIk),x(sJ

k∗)) ≤ ∆prox, (4)

where dist(·, ·) stands for the distance between two points.The process of determination of proximity zones provides a set of pairs of inter-

vals.

3.3 Intermediate Geometries

For each proximity zone, we define an intermediate geometry as the average of thetwo line segments delimited on the centerlines of beams. The position of a pointon the intermediate geometry related to the zone ZIJ

k , and identified by its relativecurvilinear abscissa ζ is calculated as

xIJint(ζ ) =

12

(x(sI

k1+ ζ (sI

k2− sI

k1))+ x

(sJ

k1

∗ + ζ (sJk2

∗ − sJk1

∗)))

, ζ ∈ [0,1], (5)

and the tangent vector to the intermediate geometry tIJint(ζ ) is obtained straightfor-

ward by derivating this expression.The intermediate geometry can be viewed as a first approximation of the geo-

metry of the unknown actual contact zone. It will be used as a referential with re-spect to which contact between the two beams will be analyzed.

7

D. Durville

3.4 Discretization by Contact Elements

Owing to its symmetrical position with respect to the two interacting beams, theintermediate geometry affords a proper location from which to consider contact.Despite the contact zone is geometrically considered as continuous, we chose toapproach contact along this zone by the means of discrete elements. These contactelements are defined at discrete locations on the intermediate geometry, and are con-stituted by the two material particles located on the surfaces of both beams that canbe predicted to enter into contact at these discrete locations. Taking the intermediategeometry as referential to consider contact allows us to formulate the contact detec-tion in the form of the question: Which particles on interacting beams are likely tocome into contact at a given point of the intermediate geometry?

Contact elements are generated according to the following procedure. First, wedetermine the discrete locations on the intermediate geometry where the contact isto be checked. The number of contact elements to be generated is calculated de-pending on the length of the intermediate geometry and on a given discretizationsize. This discretization size is chosen according to the smaller finite element lengthon both beams, denoted hmin, and to the polynomial degree of shape functions. Ifquadratic shape functions are used, the discretization size is fixed so as to get twocontact elements per finite element, in order to be consistent with the number ofconstraints these finite elements can support. Denoting Lint the length of the inter-mediate geometry, the number of contact elements Nc is calculated as

Nc =[

2Lint

hmin

], (6)

where [·] stands for the floor function, and the relative abscissa ζk of contact ele-ments on the intermediate geometry are calculated as

ζk =k−1

Nc −1, k = 1, . . . ,Nc. (7)

Then, the determination of the pair of particles likely to come into contact at eachpoint xIJ

int(ζk) on the intermediate geometry is performed in two steps. First, the twobeam cross-sections candidate to contact at the relative abscissa ζk on the interme-diate geometry, are selected at the intersections between the orthogonal plane tothe intermediate geometry and the centerlines of both beams (see Figure 3). Thecurvilinear abscissae sI

k and sJk of these cross-sections are characterized by

(xI(sI

k)−xIJint(ζk), t

IJint(ζk)

)= 0,(

xJ(sJk)−xIJ

int(ζk), tIJint(ζk)

)= 0. (8)

In a second step, the two particles candidate to contact constituting the con-tact element are positionned on the outline of these cross-sections. Each of theseparticles, identified by their coordinates ξξξ I

k and ξξξ Jk in the reference configuration, is

8


Fig. 3 Intermediate geometry.

Fig. 4 Positions of particles candidate to contact on the outline of selected cross-sections.

placed at the intersection between the projection onto the cross-section of the direc-tion between the centers of cross-sections and the outline of this cross-section (seeFigure 4).

The global procedure associates to each contact test location identified by itsrelative abscissa ζc on the intermediate geometry, a contact element, denoted Ec(ζc),and defined as the pair of material particles predicted to enter into contact at thislocation :

Ec(ζc) = (ξξξ Ik,ξξξ

Jk). (9)

9

D. Durville

3.5 Kinematical Contact Conditions

Non-penetration conditions are commonly expressed for each contact element bydefining a gap function between contact particles and prescribing this function to re-main positive. A normal direction according to which the distance between particlesis measured, is required to evaluate this gap function. The role of this normal direc-tion is predominant since it determines the direction according to which the contactis considered between the two beams. This normal direction is evaluated in differ-ents ways depending on the angle θ between the two interacting beams. If the angleis greater than a given criterion θcross, reflecting a situation of crossing between thetwo beams, the normal direction is taken as the normalized vector product betweenthe tangent vectors to both beams. When the angle is smaller than a criterion θparall,indicating that beams are nearly parallel, the normal direction is taken as the direc-tion between the centers of the cross-sections candidate to contact. For intermediateangles, the normal direction is calculated as a linear combination between the lasttwo expressions. This can be summarized as follows:

if |θ | > θcross, N(ζk) =tI(ξ I

k 3)× tJ(ξ Jk 3)

‖tI(ξ Ik 3

)× tJ(ξ Jk 3

)‖ ,

if |θ | < θparall, N(ζk) =xI(ξ I

k 3)−xJ(ξ Jk 3)

‖xI(ξ Ik 3

)−xJ(ξ Jk 3

)‖ ,

if θparall ≤ |θ | ≤ θcross,

N(ζk) =θcross −|θ |

θcross −θparall·

xI(ξ Ik 3)−xJ(ξ J

k 3)‖xI(ξ I

k 3)−xJ(ξ J

k 3)‖

+|θ |−θparall

θcross −θparall·

tI(ξ Ik 3)× tJ(ξ J

k 3)‖tI(ξ I

k 3)× tJ(ξ J

k 3)‖ .

(10)

The kinematical condition to be fulfilled at each contact element is expressed bymeans of the gap function denoted gN as follows:

gN(ζk) =(xI(ξξξ I

k)−xJ(ξξξ Jk),N(ζk)

)≥ 0. (11)

4 Mechanical Models for Contact and Friction

4.1 Quadratic Regularization of the Penalty Method for Contact

Dealing with an entangled structure with a high number of contacts, the conver-gence for the contact problem is practically unreachable using the standard penaltymethod. The high number of contacts, the softness of fibers and the connections

10


between them favor unstable situations where oscillations of contact in one regionhave repercussions in neighboughring regions. For these reasons, contact needs tobe stabilized through proper adaptations of the penalty method.

Using a quadratic regularization of the penalty is a first way to stabilize the con-tact algorithm. Such a regularization consists in employing a quadratic function forvery small gaps, below a given regularization threshold preg, and to express the nor-mal reaction RN as function of the gap:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

if gN > 0, RN = 0

if − preg ≤ gn ≤ 0, RN =kN

2pregg2

N ,

if gn < −preg, RN = −kNgN − kN

2preg.

(12)

The quadratic regularization appears to be particularly effective for contact ele-ments holding very low contact forces, and whose contact status may easily changefrom one iteration to another. In this case, the quadratic regularization of the pen-alty smooths the change of contact stiffness between contacting and non-contactingstatus. However, in order to be really effective, a small, but significant, proportionof contact elements has to be concerned by this regularization, and so needs to havepenetrations lower than the threshold preg.

4.2 Local Adjustment of the Penalty Parameter

The need to control the penetration to ensure the effectiveness of the penalty regu-larization leads to the second improvement of the penalty, by adjusting locally thepenalty coefficient in order to limit the maximum penetration. Because resultantcontact forces can be very different from one contact zone to another, and can alsovary widely, penetrations of very different orders would be expected if a uniqueand constant penalty coefficient was used for all contact zones. To avoid such cir-cumstances, the penalty coefficient is adjusted for each proximity zone so as themaximum penetration within this zone is equal to a given maximum allowed penet-ration, denoted pmax. Penalty parameters are iteratively adjusted during the solutionfor each loading step. At the i-th iteration of this process, the penalty parameter ki

cis adjusted for each proximity zone in function of the previous parameter ki−1

c in thefollowing way:

kic =

gN,max

pmaxki−1

c , (13)

where gN,max is the maximum penetration measured on the proximity zone. Similartechniques to adjust the penalty parameter have already been suggested [2]. Onedifference here is that the parameter is not adjusted at the level of each contactelement, but more globally for a set of contact elements contained in a proximityzone.

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4.3 Regularized Coulomb’s Law for Friction

As far as tangential reactions at contact elements are concerned, a regularized Cou-lomb’s law accounting for a small reversible relative displacement before the grosssliding occurs is formulated in an incremental way. Knowing the reversible tangen-tial displacement at the previous step gn−1

T,rev, and the increment of relative tangentialdisplacement at the current step ∆gn

T , a trial reversible relative tangential displace-ment gn,tr

T,revfor the current step is computed as follows:

gn,trT,rev = gn−1

T,rev + ∆gnT . (14)

The current reversible tangential displacement is evaluated according to

if ‖gn,trT,rev‖ ≤ gT,max, gn

T,rev = gn,trT,rev, (15)

else gnT,rev = gT,max

gn,trT,rev

‖gn,trT,rev

‖ , (16)

where gT,max is the maximum allowed reversible tangential displacement.

4.3.1 Transfer of History Variables Related to the Friction Model

The reversible part of the friction law requires a transmission of the history variablegn

T,rev from one step to the next. However, the fact that contact elements have nocontinuity in time raises difficulties regarding this transfer. Since this informationcannot be attached to contact elements, whose constituting particles are constantlychanging, the vector of reversible tangential displacement is stored for each particleof both beams at the end of each loop on the determination of contact. For the nextiteration on contact determination (either within the same step or at the begining ofthe next step), for any generated contact element, the value of the vector of revers-ible tangential displacement is first interpolated for each contact particle from thevalues stored at the previous iteration on the beam holding the particle, and then in-terpolated for the contact element between the vectors determined at both particles.

4.4 Algorithmic Aspects

The global problem is solved using an implicit method, within a quasi-static frame-work. It involves nonlinearities of various kinds that need different algorithms to besolved, depending essentially on whether the concerned nonlinear quantities can beeasily linearized or not.

Consistent with our approach of the contact problem, the global problem canbe divided into two nested nonlinear subproblems, the first one dealing with the

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statement of linearized kinematical contact conditions (Eq. 11), and the second onededicated to solving the mechanical problem satisfying these conditions. For the lat-ter problem, under fixed unilateral contact conditions, most nonlinear quantities canbe differentiated, and a Newton–Raphson type algorithm can be employed to solvesimultaneously the nonlinearities related to the contact status, the sliding status, andthe nonlinear terms involved in the internal virtual work of beams.

A linearization of the first level problem seems very hard to obtain since the twononlinear handled entities, namely the contact elements and the normal directionsfor contact, are defined through geometrical constructions, and not directly by dif-ferentiable equations. Newton-like algorithms can therefore not be employed. Toaccount for the nonlinear character of this problem, iterations on the determinationof the two entities are simply made using two nested fixed point algorithms. Thisleads to the following algorithm, made of three nested loops, to solve the problemfor each loading step:

First level loop – fixed point algorithm on the determination of contact elements

Second level loop – fixed point algorithm on the determination of normal dir-ections for contactThird level loop – Newton–Raphson algorithm to solve

• contact status,• sliding status,• finite strains nonlinearities.

Although the convergence of the two first level fixed point iterations is hard to prove,from our experience, three iterations for each of these two loops is generally suffi-cient to obtain a good convergence on the global solution for not too large loadingincrements.

5 Applications

5.1 Test of Alternate Sliding between Two Beams

A test of sliding between beams is performed to illustrate the effectiveness of thetransfer of history variables related to the friction model to follow sliding phenom-ena. Two 4 millimeter long crossing beams, with a radius of 0.1 mm, and a Youngmodulus equal to 4000 MPa, are considered. A force of 0.1 N is applied at bothends of one of the beam, while the other beam is maintained vertically at ends (Fig-ure 5). An alternate translation of 0.75 mm in their longitudinal direction, dividedin 25 increments, is applied at both ends of each beams. A friction coefficient of0.2 is considered, with a maximum allowed reversible tangential displacement of0.5 micrometers. The maximum penetration allowed for each proximity zone, usedto adjust the penalty ocefficient for contact, is equal to 5×10−4 mm, that is to say0.5% of the radius of the beam.

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Fig. 5 Description of the cyclic alternate loading applied to the beams.

Fig. 6 Deformed configurations at end of each alternate motion.

After a first translation, the full cycle of alternate displacements is repeated twice.For each alternate motion, there is first a sticking phase at contact between beams,before the gross sliding occurs. Each beam is discretized with 10 quadratic finiteelements, and during the sliding phase, the location of contact goes accross twosuccessive elements on each beam (Figure 6). The total number of iterations perloading step is between 13 and 23, depending on the step.

The resultant horizontal force applied to each of the beams is plotted in Figure 7.The curves for the two successive cycles exactly superimpose. They exhibit clearly asticking and a sliding phase. The slight slope in the sliding part is presumably due tothe curvature of the bent beams, which makes the orientation of the tangential planeat contact vary as the location of contact along the beams changes, while the forceis measured in the horizontal plane. Small irregularities on the curve are related tothe sliding from one finite element to the next, and may be due to the discontinuityin curvature at the junction between these two elements. However, the ability ofthe model to reproduce the transition between the sticking and the sliding statusdemonstrates the effectiveness of the transfer of history variables between contactelements which are discontinuous in time.

5.2 Modeling of Woven Fabrics

When woven textiles are employed as reinforcements to manufacture textile com-posites, their mechanical properties have to be known in particular to assess theirformability. Indentifying local mechanisms responsible for the global response of

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Displacement at beam’s end (mm)

Hor

izon

tal f

orce

(N

)

two full cycles

Fig. 7 Horizontal interaction force between the two sliding beams.

Fig. 8 Starting configuration for the calculation of the initial geometry.

such structures is also important to better understand these materials. The simula-tion at the scale of individual fibers can be very helpful in this context. To have agood representation of the media constituting these materials, a reasonable numberof fibers has to be considered for each yarn.

5.2.1 Calculation of the Initial Geometry

As the layout of fibers within a woven fabric cannot be determined a priori, theinitial geometry of the structure to study has to be computed by trying to reproducethe way yarns are intertwined together through the weaving process. As the processitself is not easy to simulate, the way we use to obtain this initial geometry is to make

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Table 1 Characteristic features of the woven sample model.

Material characteristics

Fiber diameter 0.052 mmYoung Modulus 73 000 MPaPoisson ratio 0.3

Model characteristics

Coefficient of reduction of bending stiffness 0.12Friction coefficient 0.1Maximum allowed penetration 2×10−4 mmNumber of fibers 480Number of finite elements per fiber 40Total number of DOFs 349 920Number of contact elements ≈ 90 000

yarns move gradually above or below each other at their crossings, depending on thechosen weaving pattern. To do this, starting from an initial configuration where allyarns lay on the same plane and interpenetrate each other (Figure 8), the directionof contact between fibers belonging to different yarns is temporarily chosen verticaland oriented according to the stacking order defined at each crossing by the weavingpattern. This way fibers of different yarns are gradually moved until there is no morepenetration between yarns. Once this step is achieved, classical contact conditionsare considered, and the equibrium of the fabrics is found, just applying small tensileloads at ends of yarns.

This procedure is illustrated on an example involving ten yarns, each of themconstituted by 48 fibers. Starting from the same initial configuration, two weavingpatterns – a plain weave and a twill weave – are applied. Main features of the modelare summarized in Table 1. Fifteen steps are necessary to compute the initial con-figuration, with in average 30 total Newton iterations per step.

The final shapes of both weaves are shown in Figures 9 and 10. Cuts of thesamples at some steps of this initial procedure (Figures 11 and 12) show the re-arrangement of fibers within yarns.

5.2.2 Application of Test Loadings

Once the initial configuration has been computed, biaxial tensile tests and shearloading tests that are commonly performed to characterize woven fabrics, can besimulated through the application of appropriate boundary conditions.

Biaxial tensile tests, prescribing identical elongations in warp and weft direc-tions, are simulated using 20 loading steps, with in average 45 Newton iterationsper step. These tests exhibit a nonlinearity at the start of the loading curve (Fig-ure 13). This nonlinearity is likely related to the possible compression of yarnscross-sections at crossings, allowed by the free spaces existing between fibers. This

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Fig. 9 Computed initial configuration for the plain weave.

Fig. 10 Computed initial configuration for the twill weave.

nonlinearity is stronger for the plain weave than for the twill weave, probably dueto the fact that fibers are initially more undulated in the plain weave than in the twillweave.

A cyclic shear loading test is performed on the plain weave sample, by applyingalternate opposite displacements to edges of the sample in the weft direction, in43 loading steps. Approximately 25 Newton iterations are needed in average foreach step. This test demonstrates an hysteretic behaviour due to friction interactionsbetween fibers (Figure 15).

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D. Durville

Fig. 11 Cuts of the plain weave sample during the computation of the initial configuration.

Fig. 12 Cuts of the twill weave sample during the computation of the initial configuration.

5.3 Identification of the Transverse Mechanical Behaviour of aTwisted Textile Yarn

A model is developed to study locally the effects of the transverse compressionundergone by yarns at their crossings in woven fabrics, in order to better understandthe influence of various parameters such as the tensile force, the torsion, and eventhe disorder in the initial layout of fibers, on their transverse behaviour.

So as to simulate a test experiment consisting in crushing a tightened twistedyarn (Figure 16) by means of two moving rigid tools, a finite element model (char-acteristics in Table 2) representing a yarn made of 250 fibers is considered. Westart with a configuration where all fibers are parallel and in the form of a com-pact arrangement. A disorder is first introduced in this initial layout applying smallrandom perturbations to the straight trajectories of fibers. As these random perturb-

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0 0.005 0.01 0.0150

50

100

150

200

250

300

Axial strain

Axi

al fo

rce

(N)

Twill weave

Plain weave

Fig. 13 Biaxial loading curves for the plain and twill weave samples.

Fig. 14 Plain weave sample under a shear loading.

ations cause interpenetration between fibers, a first stage of simulation is needed tofind a new equilibrium configuration fulfiling contact conditions between fibers, andan equilibrated disordered configuration is obtained. A tensile force is then appliedand the yarn is twisted until a given torsion.

To simulate the transverse compression, contact conditions between fibers andtwo moving rigid tools are considered. The transverse compression is applied in150 steps, approximately 30 Newton iterations being necessary to solve each step.Evolutions of the central cross-section of yarn are represented in Figure 17, showingsome interesting features. Fibers are more tightened at the periphery of the yarn than

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−20 −15 −10 −5 0 5 10 15 20−4

−3

−2

−1

0

1

2

3

4

Shear angle (degrees)

She

ar fo

rce

(N)

Fig. 15 Shear loading curve for the plain weave sample.

Fig. 16 Transverse compression of a twisted yarn between two moving rigid tools.

at the center, due to the elongation induced by the deflection of outside fibers. Theseoutside fibers produce a confinement effect in relation with the global twisting, in-creasing the density of fibers at the periphery of the yarn. The global deformationinduces large relative displacements between fibers that may be compared to flowsof granular materials.

The global loading curve (Figure 18) shows some irreguralities that could berelated to slidings between groups of fibers corresponding to possible alignementsbetween fibers appearing at some steps of the loading. This kind of curve has simil-arities with behaviours observed on generalized entangled media such as wools [3].

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Table 2 Characteristic features of the twisted yarn model.

Material characteristics

Fiber diameter 0.02 mmYoung Modulus 73 000 MPaPoisson ratio 0.3

Model characteristics

Coefficient of reduction of bending stiffness 0Friction coefficient 0.05Maximum allowed penetration 10−4 mmNumber of finite elements per fiber 15Number of fibers 250Number of DOFs 69 750Number of contact elements ≈ 17 000

Fig. 17 Evolution of the yarn median cross-section during the transverse compression.

6 Conclusion

The geometrical approach to the detection of contact between beams based on thedetermination of intermediate geometries at the level of proximity zones allows toconsider contact for beams arranged according to various layouts. The high numbers

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Lecture Notes in Applied and Computational Mechanics · 2016. 2. 9. · Lecture Notes in Applied and Computational Mechanics Edited by F. Pfeiffer and P. Wriggers Further volumes