Interaction between rubber material and mold during demolding

Von der Fakultät für Maschinenbauder Gottfried Wilhelm Leibniz Universität Hannover

zur Erlangung des akademischen GradesDoktor-Ingenieur

genehmigte Dissertationvon

Dipl.-Ing. Jan-Hendrik Dobberstein

geboren am 10.06.1983 in Hannover

2014

1. Referent: Prof. Dr.-Ing. habil. Dr. h.c. mult. Peter Wriggers2. Referent: Prof. Dr. -Ing. Matthias Kröger

Tag der Promotion: 01.09.2014

Herausgeber:Prof. Dr.-Ing. habil. Dr. h.c. mult. Peter Wriggers

Verwaltung:Institut für KontinuumsmechanikGottfried Wilhelm Leibniz Universität HannoverAppelstraße 1130167 Hannover

Tel: +49 511 762 3220Fax: +49 511 762 5496Web: www.ikm.uni-hannover.de

© Dipl.-Ing. Jan-Hendrik DobbersteinInstitut für KontinuumsmechanikGottfried Wilhelm Leibniz Universität HannoverAppelstraße 1130167 Hannover

Alle Rechte, insbesondere das der Übersetzung in fremde Sprachen, vorbehalten. Ohne Genehmigung des Autors ist es nicht gestattet, dieses Heft ganz oder teilweise auf photomechanischem, elektronischem oder sonstigem Wege zu vervielfältigen.

ISBN 978-3-941302-11-2

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Abstract

This thesis is concerned with the numerical simulation of the demolding process inthe context of tire production. The demand for such a model arose from increasingproblems during the manufacturing process in terms of an undesirable adhesion betweenrubber material and mold. The simulation aims to increase the understanding of thedemolding process and furthermore to provide a tool to determine specific influencingfactors.Since the rubber material is of major importance for the separation process, first anappropriate material model is developed. Beside the typical properties of elastomers,such as the nonlinear material behavior and the viscoelasticity, also the incompletevulcanization at the time of the demolding is considered.To describe the interactions between rubber and mold, an interface model is imple-mented within the finite element method (FEM). This model includes, alongside withthe adhesion, also the classical interactions due to contact. The choice of the modelwas made on the basis of a practically accessible identification of the parameters. Ini-tially the model is implemented with the node to segment discretization strategy usingthe penalty method. In addition, for the first time, an adhesion model is implementedusing the mortar method, where the additional constraints, stemming from contact andadhesion, are treated with the augmented Lagrangian method.All adhesion models, available in the literature, are based on interface laws, at whichthe adhesion force between two points decreases with the distance of these points. Thissoftening behavior can lead to instabilities at the numerical solution. These instabilitiesare more likely to occur, the steeper the force decline is, which is also observed in thesimulations of this work. By means of a parameter variation it is, however, shown, thatrealistic simulation results are solely obtained with unstable adhesion curves. Thus astabilization of the calculations was necessary. For this an arclength method with anadaptive constraint equation is used. Alternatively a stabilization is also achieved,when the inertia effects are considered, and the associated dynamic problem is solved.Finally some numerical examples are presented. This includes the inspection of somequalitative effects at the adhesive contact of elastomers. In addition simulation resultsfor the demolding of a tread block are shown, where the influence of different factorson the demolding force is analyzed.

Keywords: Contact Mechanics, Finite Element Method, Adhesion, Mortar Method,Demolding Process

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Zusammenfassung

Gegenstand dieser Arbeit ist die numerische Simulation des Formentnahmeprozessesim Rahmen der Reifenherstellung. Der Bedarf für solch ein Simulationsmodell ist mitzunehmenden Problemen währen des Produktionsprozesses, in Form eines unerwün-schten Anhaftens des Gummimaterials an der Vulkanisationsform, entstanden. Dabeihat die Simulation das Ziel das Verständnis des Formentnahmeprozesses zu erhöhensowie ein Werkzeug für die gezielte Bestimmung von Einflussparametern bereitzustel-len.Da dem Gummimaterial eine entscheidende Bedeutung für den Ablöseprozess zukommt,wird zunächst ein geeignetes Materialmodell erarbeitet, welches neben den typischenEigenschaften von Elastomeren wie dem nichtlinearen Materialverhalten und der Viskoe-lastizität auch die zum Zeitpunkt der Formentnahme vorliegende, unvollständige Vulka-nisation berücksichtigt.Um die Wechselwirkungen zwischen der Form und dem Reifenmaterial zu beschreiben,wird ein Grenzflächenmodell, welches neben der Adhäsion auch die klassischen Kontakt-Wechselwirkungen einschließt, im Rahmen der Finite Elemente Methode (FEM) im-plementiert. Die Auswahl des Modells wurde auf Basis einer mit vertretbarem experi-mentellen Aufwand durchführbaren Parameterbestimmung getroffen. Das Modell wirdzunächst im Rahmen der node to segment Strategie mit der penalty-Methode imple-mentiert. Weiterhin wird zum ersten mal ein Adhäsionsmodell auf Basis der MortarMethode entwickelt. Dabei werden die zusätzlichen Zwangsbedingungen aus Kontaktund Adhäsion mit dem Augmented Lagrange Verfahren behandelt.Alle in der Literatur verfügbaren Adhäsionsmodelle basieren auf Grenzflächengeset-zen, bei denen die Adhäsionskraft wischen zwei Punkten mit deren Abstand abnimmt.Solch ein Entfestigungseffekt kann mitunter zu Instabilitäten bei der numerischen Lö-sung führen. Dies ist umso ausgeprägter, je steiler der Kraftabfall ist, was auch inden Simulationen zu dieser Arbeit beobachtet werden konnte. Anhand einer Param-etervariation wird gezeigt, dass realistische Simulationsergebnisse ausschließlich mitinstabilen Adhäsionsverläufen erzielt werden, weshalb eine Stabilisierung der Berech-nungen erforderlich ist. Hierzu wird zunächst ein Bogenlängenverfahren mit einer adap-tiven Zwangsbedingung verwendet. Alternativ wird eine Stabilisierung auch durch dieBerücksichtigung der Trägheitseffekte und die damit verbundene Lösung des dyna-mischen Problems erreicht.Abschließend werden einige numerische Ergebnisse vorgestellt. Dies umfasst zunächstdie Betrachtung einiger qualitativer Effekte beim adhäsiven Kontakt von Elastomeren.Weiterhin werden Berechnungen des Formentnahmeprozesses eines Profilklotzes präsen-tiert. Dabei wird der Einfluss unterschiedlicher Parameter auf die Entformungskräfteuntersucht.

Schlagworte: Kontaktmechanik, Finite Elemente Methode, Adhäsion, Mortar Me-thode, Formentnahmeprozesses

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Acknowledgements

This thesis is the result of my research work during the years 2009 to 2014 at theInstitute of Continuum Mechanics (IKM) at the Leibniz Universität Hannover underguidance of Prof. Dr.-Ing. habil. Dr. h.c. mult. Peter Wriggers. The project was sup-ported by the German Research Foundation (DFG).

First of all I would like to thank my supervisor Prof. Peter Wriggers for his constantsupport and his trust. From his guidance I benefited not only concerning my technicalexpertise but also personally. He gave me the freedom to follow own ideas and providedhelpful advice, whenever needed. I am also grateful, that I had the opportunity toresponsibly work with the project partners and also that Prof. Wriggers encouragedme to present my work on international conferences. Both were valuable experiencesto me.

I would also like to thank my second referee Prof. Dr.-Ing. Matthias Kröger for hisearnest interest in my work, as well as the fruitful discussions and the good collabora-tion we had within the joint project. Furthermore I thank Prof. Dr.-Ing. Bernd-ArnoBehrens for chairing the examination committee.

My sincere thanks also go to my colleagues at the IKM for their helpfulness and thepleasant working atmosphere. Especially my office mate Nasim Hajibeik supported andencouraged me in difficult phases of my work. Special thanks are also due to Dr.-Ing.Christian Weißenfels for his patient help with contact mechanics, to Dr.-Ing. StefanLöhnert for his generous help with FEAP and to Dr. Roger A. Sauer for introducingme to the field of computational mechanics. I would also like to acknowledge the workof the technical staff and the office staff.

Last but not least I thank my family for their unconditional support. My parents havealways believed in me and enabled me to attend university. Above all I would like tothank my wife Baukje for her patience and support and all the sacrifices she made.Without her, this work would not have been possible. I am also grateful to my childrenWiebke and Gesa, who always showed me, that there are other important things inlife.

Hannover, September 2014 Jan-Hendrik Dobberstein

vi

Contents

1 Introduction 11.1 Background and state of the art . . . . . . . . . . . . . . . . . . . . . . 21.2 Structure of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background 52.1 Continuum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Balance laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . . 112.1.4 Weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Constraints and constitutive equations . . . . . . . . . . . . . . 152.2.3 Contact virtual work and treatment of contact constraints . . . 16

2.3 Finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Discretization of space . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Discretization of time . . . . . . . . . . . . . . . . . . . . . . . . 232.3.3 Solution algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Contact finite elements 273.1 Node to segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Discretization, projection and contact search . . . . . . . . . . . 273.1.2 Kinematical quantities and constitutive equations . . . . . . . . 293.1.3 Contribution to the weak form . . . . . . . . . . . . . . . . . . . 303.1.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.5 Residual vector and tangent matrix . . . . . . . . . . . . . . . . 323.1.6 Special cases at the contact search . . . . . . . . . . . . . . . . 34

3.2 Mortar method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.2 Integration domain for slave-master pairing . . . . . . . . . . . . 383.2.3 Integration points and Jacobian . . . . . . . . . . . . . . . . . . 413.2.4 Kinematical quantities . . . . . . . . . . . . . . . . . . . . . . . 423.2.5 Contribution to the weak form . . . . . . . . . . . . . . . . . . . 423.2.6 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.7 Residual vector and tangent matrix . . . . . . . . . . . . . . . . 50

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viii

3.2.8 Treatment of corners and edges . . . . . . . . . . . . . . . . . . 51

4 Material model 534.1 Characteristic properties of rubber . . . . . . . . . . . . . . . . . . . . 534.2 Vulcanization process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Linear viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.4 3D continuous material model . . . . . . . . . . . . . . . . . . . . . . . 584.5 Parameter identification . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.5.1 Static parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5.2 Viscoelastic parameters . . . . . . . . . . . . . . . . . . . . . . . 61

4.6 Belt plies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Adhesion 675.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.1.1 Intermolecular forces . . . . . . . . . . . . . . . . . . . . . . . . 675.1.2 Analytical models and experimental observations . . . . . . . . 685.1.3 Numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2 Node to segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2.1 Adhesion gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2.2 Penalty approach and adhesion states . . . . . . . . . . . . . . . 755.2.3 Contribution to the weak form and linearization . . . . . . . . . 765.2.4 Residual vector and tangent matrix . . . . . . . . . . . . . . . . 785.2.5 Patchtest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2.6 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Mortar method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.1 Normal interaction . . . . . . . . . . . . . . . . . . . . . . . . . 835.3.2 Tangential interaction . . . . . . . . . . . . . . . . . . . . . . . 845.3.3 Contribution to the weak form and linearization . . . . . . . . . 855.3.4 Patchtest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6 Solution algorithms 896.1 Modified arclength method . . . . . . . . . . . . . . . . . . . . . . . . . 896.2 Dynamic calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7 Numerical results 957.1 Qualitative Observations . . . . . . . . . . . . . . . . . . . . . . . . . . 957.2 Influence of the adhesion parameters . . . . . . . . . . . . . . . . . . . 977.3 Demolding tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.4 Mortar method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8 Conclusion 111

A Mathematical bases 115A.1 Convective coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 115A.2 Cross product in matrix form . . . . . . . . . . . . . . . . . . . . . . . 115

CONTENTS ix

B NTS contact element 117B.1 Linearization of the kinematical quantities . . . . . . . . . . . . . . . . 117B.2 Linearization of the tangential stress . . . . . . . . . . . . . . . . . . . 118B.3 Vectors for the residual and the tangent matrix . . . . . . . . . . . . . 119

Bibliography 120

Curriculum vitae 130

x CONTENTS

List of Symbols

Operators and symbols

˙(..) Material time derivativediv Divergence operatorgrad Gradient operator∆(..) Linearization of a quantityδ(..) Variation of a quantityΩ (..) Cross product matrix

Continuum mechanics

a AccelerationB0 Reference configuration of a bodyBt Current configuration of a body∂B Boundary of a body∂Bt Boundary of a body, current configuration∂tB Boundary with prescribed traction∂uB Boundary with prescribed displacementb Left Cauchy-Green tensorb̄ Specific body forceC Right Cauchy-Green tensord Symmetric part of the spatial velocity gradientE Total energy of a systemE1, E2, E3 Base vectors of the reference configuratione1, e2, e3 Base vectors of the current configurationF Deformation gradientH Displacement gradienth Entropy fluxI Linear momentumJ Determinant of the deformation gradientK Kinetic energyL Angular momentuml Spatial velocity gradientm Massn Surface normal

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xii CONTENTS

P Mechanical powerQ Thermal powerqi Internal variable

Q̃ Entropy transferqn Heat fluxq Cauchy heat fluxr Heat sourcer̃ Entropy sourcer Lever arm at the balance of angular momentumS Entropys Specific entropyS Second Piola-Kirchhoff stress tensort Surface loadU Inner energyu Specific inner energyu Displacementu0 Initial displacementV , v Volume at the reference and the current configurationv Velocityv0 Initial velocityW Strain energy functionX Position vector at the reference configurationx Position vector at the current configurationη Test functionΘ Absolute temperatureρ0, ρ Density of the reference and the current configurationσ Cauchy stress tensorϕ Mapping between current and reference configurationΨ Helmholtz free energy function

Contact mechanics

(..)1 Quantity of the slave surface(..)2 Quantity of the master surface(..)tr Trial quantity, used within the return mapping algorithma1, a2 Covariant base vectors∂cBt Contact surfacecN , cT Penalty parametersCN , CT Nonlinear complementary functionsfs Slip criterionG Weak form without contact contributionGc Contact contribution to the weak formGcu Virtual work of the contact forcesGcl Weak form of the enforcement of the contact constraints

CONTENTS xiii

gN Normal gapgT Tangential sliding distancegeT Elastic part of the relative tangential movementgsT Slip part of the relative tangential movement∆gT Tangential displacement incrementtN Normal contact pressuretT Tangential contact stress vectorγ̇ Slip rateλN Lagrange multiplier describing the normal contact pressure

λ̂N Augmented Lagrange quantity for the normal contact pressureλT Vector of Lagrange multipliers for the tangential contact stress

λ̂T Augmented Lagrange vector for the tangential contact stressµ Coefficient of frictionξ1, ξ2 Convective coordinatesΠcN , Πc T Normal and a tangential contact potential

Finite element method

(..)h Discretized quantity(..)k Quantity of the last iteration step(..)k+1 Quantity of the current iteration step(..)gp Quantity at a Gauss point(..)n Quantity of the last time step(..)n+1 Quantity of the current time stepJ Mapping between reference and initial configurationj Mapping between reference and current configurationKT Tangent matrix of the quasi static problemK∗T Tangent matrix of the dynamic problemM Mass matrixne Number of finite elementsngp Number of Gauss pointsNI Shape function of node IP External load vectorR Residual force vector∆t Time incrementuI Displacement of node Iwgp Weighting factor at Gauss pointXI Position vector of node I in reference configurationxI Position vector of node I in current configurationβ, γ Parameters of the Newmark methodδIJ Kronecker symbolξ Local coordinates inside the finite elementξJ Coordinates at the position of node JΩe Domain of one finite element

xiv CONTENTS

Ω� Reference configuration of the finite elementΩ0 Initial configuration of the finite element

Node to segment contact element

¯(..) Quantity evaluated at the projection point coordinates(..),α Differentiation with respect to the convective coordinate ξ

α

nm Number of nodes on a master surface elementas Area surrounding the slave nodeāαβ Contravariant metric coefficientsāαβ Covariant metric coefficientsb̄αβ Components of the curvature tensorDα Vector for the node to segment contact elementE, Eα Vectors for the node to segment contact elementH̄αβ, H̄

αβ Auxiliary matrix: co- and contravariant componentsN , Nβ , Nαβ Vectors for the node to segment contact element

T β, T αβ , T̂ αβ Vectors for the node to segment contact elementξ̄α Coordinates of the projection point at the NTS contact element∆tξαn+1 Increments of the projection point coordinates

Mortar contact element

¯(..) Averaged quantityBj Auxiliary matrix for the linearization of the Jacobian determinantBl Auxiliary matrix for the linearizationBn Auxiliary matrix for the linearization of the normal vectorBt α Auxiliary matrix for the linearization of the normalized base vectorsBiαg Auxiliary matrix for the linearization of the integration point coordi-

natesDd Auxiliary matrix for the linearization of the triangle pointsd1, d2, di Auxiliary vectors for the Cyrus-Beck algorithmEint Auxiliary matrix for the linearization of intersection pointsEi Auxiliary matrix for the linearization of the projected nodesEiI Auxiliary matrix for the linearization of projected slave and master

nodesEpc Auxiliary matrix for the linearization of the centroiddetj Jacobian determinant of the triangleMA Shape function for the Lagrange multiplier at slave node Anpkl Number of points of the overlapping polygonn1glob Number of slave nodes on the slave surfacengp Number of Gauss points belonging to a slave noden10 Normal vector at the center of the slave elementnA Normal vector of the slave surface at slave node AP1 Auxiliary matrix for the linearization of points on the slave surface

CONTENTS xv

P1α Auxiliary matrix for the linearization of the averaged base vectorsWg Weighting factor at a Gauss pointx10 Position vector of the center point of the slave elementxig, x

ig,α Integration points and base vectors at these integration points on the

master and the slave surfacex̄int Position vector of the intersection pointx̄dJ Position vector of triangle point J

x̄jI Position vector of node I of surface j projected onto the axillary planex̄pc Position vector of the centroid of the overlapping polygonx̄pJ Position vector of polygon point Jλ Vector of Lagrange multipliersξgp Coordinates of the triangle Gauss points on the auxiliary planeξig Coordinates of the Gauss points on the slave and the master surface

Material model

be Left Cauchy-Green Tensor of the elastic deformation partCi Right Cauchy-Green tensor of the inelastic deformation partE∗ Complex modulusE

′

Storage modulusE

′′

Loss modulusE∞, Ej, ηj Parameters of the linear viscoelastic material modelf90 t90-factorFe Elastic part of the deformation gradientFi Inelastic part of the deformation gradientI1, I2 First and second invariant of the right Cauchy-Green tensorJ Determinant of the deformation gradientqi Generalized parameter to be determinedS Torque at the rheometer testS ′ Storage momentS ′′ Loss momentT Temperaturet90 Optimum cure timetref Equivalent curing time at reference temperatureǫ Strain of the Wiechert modelǫ̂ Amplitude of the harmonic excitationǫej Elastic strain of Maxwell element jǫij Inelastic strain of Maxwell element jΛ, µ1, µ2 Parameters of the Mooney-Rivlin material modelλ Stretchν Poisson ratioσ̂ Amplitude of the stressσEQ Equilibrium stress (finite linear viscoelastic material)

σjNEQ Stress in Maxwell element j (finite linear viscoelastic material)

xvi CONTENTS

τj Relaxation time of Maxwell element j (τj = ηj/Ej)ψ Phase differenceω Angular frequency

Adhesion

f̂aA Extended slip criterion including friction and adhesionGa Adhesion contribution to the weak formgd Decohesion gapga Adhesion gap vectorr Distance of two particless Tension limitsN Tension limit in the normal directionsT Tension limit in the tangential directionta Adhesive stressta Adhesion tension vectorxa Memorized adhesion pointα Maximum adhesion stressβT Parameter coupling the tangential with the normal tension limitγ Penalty parameter for the adhesionδ Rupture gap

λ̂aNA Augmented adhesion stress in the normal direction at slave node Aξaβ Components of the tangential adhesion gapξp Coordinates of the remembered adhesion point

Chapter 1

Introduction

A key interest of tire manufacturers is to improve the tires grip properties in order toenhance safety and driving comfort. This is achieved with a more and more complextread design as well as the use of compounds with better adhesive properties. Howeverdesired the advanced grip properties are in later use, they cause major challenges inproduction.One sub-step at the production process of a tire is the vulcanization of a green tire underpressure inside a mold. At the subsequent extraction phase, called demolding, differentproblems arise. Due to the adhesion on the one hand the mold can be damaged, or theforce necessary to open the mold can exceed the capabilities of the machinery, bothcausing production downtimes and resulting for the former case also in costly repairs.On the other hand the tire can be damaged, leading to safety risks, if this remainsundetected.A prevalent method to avoid these troubles, is, to apply a release agent onto the moldat regular intervals. This approach, however, causes a pollution of the environmentand complicates the production process. Another possibility is a permanent coating ofthe molds. However there is little research on coatings, which minimize the adhesionbetween rubber and metal and, at the same time, withstand the high temperaturesand pressures for a large number of demolding cycles.To systematically develop solution strategies to the problems, the aim is, to obtain a ba-sic understanding of the demolding process. This is pursued by practical experiments aswell as numerical models. Experiments have the drawback, that they are extensive andtheir interpretation can be difficult, since they only provide certain, measurable vari-ables. Numerical simulations allow an inspection of all calculated quantities, therebyenabling a deeper insight into the regarded process and thus creating a better under-standing. This can not only be used to reduce the experimental effort, leading to anadvantage in time, but also new ideas for products or the optimization of processesmight be found. However, despite these advantages, it should be noted, that the sim-ulations can not entirely replace experiments, since they are based on simplifications.Instead experiment and simulation complement one another.In view of this within this work a model to simulate the demolding of not completelyvulcanized tire tread blocks will be developed. For this the complex contact with fric-tion and adhesion, the high temperatures and different states of vulcanization inside the

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2 CHAPTER 1. INTRODUCTION

material as well as the large deformations during the extraction of elaborate patternsare of particular challenge.

1.1 Background and state of the art

In the technical literature only a few contributions treating the demolding of rubbercan be found, and to the authors knowledge there are no research results published onits simulation. Only for the molding of rubber some publications exist. In Dupaix andCash [2009] e.g. the focus is laid on the description of the material, and the moldingprocess is simulated using a two dimensional contact model excluding friction andadhesion.A simulation model for the demolding of tires requires, beside a suitable description ofthe rubber material, a model for contact with friction and adhesion. In order, not to berestricted to certain geometries, in this work a computational model is developed, forwhich, due to the complex geometry of the tread, a simplification to two dimensions isnot admissible.For macroscopic mechanical problems usually the Finite Element Method (FEM) isapplied. An overview on the treatment of computational contact problems with theFEM is e.g. given in Laursen [2002] and Wriggers [2006]. For contact problems onthe nano-scale molecular dynamic simulation methods (MD) are prevalent, see e.g.Frenkel and Smit [2002] for an introduction. Compared to FEM models they havethe advantage, that adhesion is automatically included and chemical reactions at thecontact surface can be described. However with current computing facilities it is notpossible to calculate problems on a larger scale. Therefore for the simulation of thedemolding process the FEM is applied.FEM models including contact of rubber can e.g. be found in Hofstetter et al. [2006]and in Wriggers and Reinelt [2009] for the calculation of friction between rubber andrough, rigid surfaces. In Ziefle and Nackenhorst [2008] a model to study the behavior ofa rolling tire is presented and the contribution of Lee et al. [2006] treats the calculationof the contact of seals.To study the decohesion within a composite material, i.e. the debonding between fibersand matrix material, cohesive zone models have been developed. In this context thework of Xu and Needleman [1994] should be named. Cohesive zone models are usuallybased on interface laws, describing the tensile interactions at the interface between twomaterials. Compressive forces, resulting from the impenetrability between the contactpartners, are not addressed.In Sauer and Li [2007] an FEM model considering contact and adhesion is presented.The model is derived from the interaction between the particles of the contacting bodies,and is therefore suitable to study adhesion on the nano- and on the micro-scale. Modelsfor contact and adhesion on the macro-scale have been developed in Raous et al. [1999]and Talon and Curnier [2003], both including an irreversible decrease of the adhesionstrength. In the contribution of Raous et al. [1999] a possible rate dependency of theadhesive effects is considered, while in Talon and Curnier [2003] adhesion is treated asa constraint and decohesion is modeled with a linear constitutive equation.

1.2. STRUCTURE OF THIS WORK 3

Concerning the description of the material, it has to be noted that rubber is a complexmaterial, which possesses various characteristic properties, differing from those of otherengineering materials such as e.g. metals. Depending on, which property is to be de-scribed, different models have been developed. The nonlinear elastic response for staticdeformations is described by the models of Mooney [1940], Rivlin [1948] and Ogden[1982]. In Heinrich and Kaliske [1997] a model, motivated from the molecular structureis presented. Inelastic effects are considered e.g. in the models of Simo [1987] and Reeseand Govindjee [1998a]. The Mullins effect, which marks a softening of the material af-ter first loading, is e.g. considered in the models of Ernst and Septanika [1999], Besdoand Ihlemann [2003] and more recently with a micro-mechanically motivated approachin De Tommasi et al. [2006]. A numerical model for the Payne effect, which denotes adecrease in dynamic stiffness with an increasing strain amplitude, can e.g. be found inLion and Kardelky [2004]. Models considering the effects of temperature are e.g. givenin Holzapfel and Simo [1996], Lion [1997] and Reese and Govindjee [1998b].

At the demolding the separation between tire and mold does not occur simultaneouslythroughout the interface. Instead a local detachment is initiated at a small region,which then, comparable to a crack, propagates across the contact surface. Since thisprocess occurs within a few seconds, it is important to describe the viscoelasitcity ofthe rubber correctly. In this context also the state of curing has to be considered. Asimulation model for the curing of rubber can e.g. be found in André [2001] and Andréand Wriggers [2005].

1.2 Structure of this work

Subsequent to this introduction in chapter 2 the fundamental theory necessary to un-derstand the mathematical description and the modeling in this work is presented. Thisconsists of a brief summary of solid continuum mechanics, treating the description ofthe behavior of bodies under loading, and the theory of contact mechanics, which marksan extension of the classical continuum mechanics to contact problems. As a tool toobtain a numerical solution of the resulting equations at the end of this section theFEM is introduced.

In chapter 3 the treatment of contact problems with the FEM is specified. In thiscontext two contact elements, based on different discretization techniques, are describedin detail. The first is the well known three dimensional node to segment contactelement for contact problems with large deformations. The second is the more recentlydeveloped three dimensional mortar contact element.

Chapter 4 treats the description of the material. Some characteristic properties ofrubber are briefly described and the employed material model is specified. Also theidentification of the material parameters and the homogenization of a belt ply, whichis used to fix rubber samples at experimental tests, are addressed.

Adhesion is addressed in chapter 5. After a short summary of the background in thissection different computational models coupling contact and adhesion are discussed,and the contact elements of chapter 3 are extended to also include adhesive interactions.

4 CHAPTER 1. INTRODUCTION

All available adhesion models are based on interface laws, at which the adhesion forcedecreases with an increase of a certain gap quantity. This softening behavior can leadto instabilities at the numerical solution, which were also observed at the simulation ofthe demolding process. Therefore in chapter 6 two methods to stabilize the calculationsare regarded. These are an arclength-type solution and a dynamic calculation of themechanical problem.The numerical results are presented in chapter 7. First two qualitative simulations,illustrating some characteristic effects at the adhesive contact of rubber, are presented.Subsequent, using the stabilization methods of chapter 6, for a simple demolding testthe influence of the parameters of the adhesion model on the separation process isregarded. This is followed by simulation results of more complex demolding tests, forwhich different parameters, such as the number of sipes, the material and the fixing ofthe rubber sample, are varied. At the end of this chapter some results for the mortaradhesion-contact model are presented.Finally in chapter 8 the main results of this work are summarized and open points arediscussed. Based on this a perspective on future works is given.

Chapter 2

Background

In this chapter the mathematical bases for the modeling and simulation of mechanicalproblems are presented. The chapter starts with a recapitulation of solid continuummechanics in section 2.1, followed by an extension of the considerations to contactmechanics in section 2.2. The solution of the resulting equations is obtained numericallywith the finite element method (FEM), which is described in section 2.3.

2.1 Continuum mechanics

Solid continuum mechanics is concerned with the behavior of bodies under mechanical,thermal or other loading. To describe and predict this behavior, mathematical equa-tions are used, which can be divided into three different categories. The motion anddeformation of a body are described by kinematical equations. The stresses and otherfield variables inside a continuum body are calculated from the balance laws, whichare fundamental equations obtained from physical observations and valid for any ma-terial. The link between both sets of equations is achieved by the constitutive laws,which provide the relation between deformation and stress. This section gives onlyan overview on the most important equations and those necessary to understand themathematical models in this work. For more information on the subject of continuummechanics the interested reader is referred to the literature (Ogden [1984], Mardsenand Hughes [1994], Chadwick [1999] Holzapfel [2000], Haupt [2002], Truesdell and Noll[2004], Altenbach [2012]).

2.1.1 Kinematics

A body B is conceived as a continuous set of particles or material points P ∈ B. Astime t evolves, this body occupies different regions in Euclidean space E3, referred toas configurations of B. Since cracks, penetrations and singularities within the body arenot allowed, each configuration can be described by a bijective mapping X : B → E3,which positions each material point into E3, see figure 2.1. In order to describe themotion and deformation of such a body a reference configuration B0 is defined, whichcan e.g. be the configuration at the initial time t0. The position vector X of a particle

5

6 CHAPTER 2. BACKGROUND

X x

P

B

B0 Bt

X0 Xt

ϕ

E1, e1

E2, e2

E3, e3

Figure 2.1. Motion and deformation of a continuum body.

P in the reference configuration may be given with the mapping X0 by X = X0 (P ).In the current configuration Bt it reads x = Xt (P ), which can be expressed in termsof X, since

x = Xt (P ) = Xt(X−10 (X)

)= ϕ (X) . (2.1)

The mapping of position vectors from the current to the reference configuration can befound by inversion of (2.1) leading to

X = ϕ−1 (x) . (2.2)

Regarding (2.1) at different points in time, the motion of a particle is described. Sucha formulation of the field variables with respect to the reference coordinates X iscalled Lagrangian or material description, and is mainly used in solid mechanics. Aformulation with respect to the current coordinates x, as in (2.2), is called Eulerian orspatial description. In this case the changes of the field variables at a fixed position inspace are regarded. This concept is employed in fluid mechanics.Introducing a Cartesian coordinate system the position vectors of material points inthe reference- and the current configuration can be written as

X = XjEj and x = xiei , (2.3)

where Ej and ei are the base vectors of the reference and the current configuration.With this the displacement field, defined as

u = x−X (2.4)

can be calculated. As a measure for the deformation, the deformation gradient

F =∂x

∂X=

∂xi∂Xj

ei ⊗ Ej (2.5)

2.1. CONTINUUM MECHANICS 7

is introduced. It is a linear operator, which maps undeformed line elements dX todeformed ones dx via

dx = FdX . (2.6)

With (2.4) the deformation gradient can also be stated as

F = 1+∂u

∂X= 1+H , (2.7)

with H being the displacement gradient. The mapping of (2.6) is supposed to beinvertible, what means that the inverse of the deformation gradient F−1 exists.With the deformation gradient several strain measures can be defined. Motivated fromthe calculation (of the square) of the change in length from the reference to the currentconfiguration, the right Cauchy-Green tensor C is defined by

C = FT · F , dx · dx = dXC dX . (2.8)

Similarly the left Cauchy-Green tensor b is introduced as

b = F · FT , (2.9)

which appears when (dX)2 is expressed in terms of (dx)2. The tensors C and b aredefined in reference and current coordinates, respectively.The conservation laws introduced in the next section require an integration over thevolume of the body. Therefore the transformation of an infinitesimal volume elementof the reference configuration dV , defined by the infinitesimal vectors dX, dY and dZ,to the current configuration is regarded. It is calculated by

dv = det[dx dy dz

]= det

[FdX FdY FdZ

]= J dV , (2.10)

where J is the determinant of the deformation gradient, also known as the Jacobian.As F is invertible and volume elements can not be negative,

J = detF > 0 (2.11)

has to hold.As in this work rubber materials, which show a rate dependent mechanical behavior,are investigated and also dynamic calculations are performed, some time derivativeswill be specified in the following. The velocity and the acceleration of a particle in thereference configuration are calculated by

v =∂x

∂t= ẋ = u̇ and a = v̇ = ẍ = ü . (2.12)

Besides, with the material time derivative of the deformation gradient, given by

Ḟ =∂ẋ

∂X, (2.13)

and the spatial derivative of the velocity

l =∂ẋ

∂x=∂ẋ

∂X

∂X

∂x= ḞF−1 , (2.14)

the material time derivative of the Jacobian follows to

J̇ = J div v . (2.15)

8 CHAPTER 2. BACKGROUND

2.1.2 Balance laws

The balance laws represent fundamental equations, which have to be fulfilled at eachinstant of time. They describe the conservation of the physical quantities mass, linearmomentum, angular momentum and energy, as well as the increase of entropy, and areindependent of the material of the continuum body.

Conservation of mass

In a continuum body the mass m can be either calculated in the current or in thereference configuration by

m =

∫

Bt

ρ dv =

∫

B0

ρ0 dV , (2.16)

from which with (2.10) the relation between the density of the reference and the currentconfiguration can be determined to

ρ0 = Jρ . (2.17)

If a closed system is regarded, no change in mass is allowed, leading to the equation

dm

dt= 0 =

∫

B0

ρ̇0 dV =

∫

B0

(

ρ̇J + ρJ̇)

dV . (2.18)

which can be stated in local or strong form as

ρ̇0 = 0 and ρ̇+ ρ div v = 0 , (2.19)

where the expression inside the second integral has been simplified using (2.15). Equa-tion (2.19) is called strong form, since it is fulfilled in every point, whereas (2.18) isonly satisfied in an averaged sense over the integration domain.

Balance of linear momentum

The linear momentum inside a continuum (or an arbitrary subset of a continuum) B isgiven by

I =

∫

Bt

ρv dv . (2.20)

Its material time derivative has to be equal to the external forces acting on B

d

dtI = Fext . (2.21)

These external forces can originate from surface tensions t, acting on parts of theboundary ∂B of the continuum, and from volume forces ρb̄, which can e.g. result fromgravity, so that the right hand side of (2.21) can be specified to

d

dt

∫

Bt

ρv dv =

∫

Bt

ρb̄ dv +

∫

∂Bt

t da . (2.22)

2.1. CONTINUUM MECHANICS 9

Using the Cauchy theorem the traction vector t can be written as

t = σ · n , (2.23)

with n being the surface normal and σ the second order stress tensor, which uniquelydescribes the state of stress at a material point. Thus also using the divergence theoremthe surface integral in (2.22) can be replaced by a volume integral

∫

∂Bt

t da =

∫

∂Bt

σ · n da =

∫

Bt

divσ dv . (2.24)

Inserting (2.24) into (2.22) and arguing that the equation is satisfied, if the integrandsare zero, the local balance of linear momentum is obtained to

divσ + ρb̄ = ρv̇ . (2.25)

When solving (2.25) for σ the stress at each point of the continuum can be calculated.

Balance of angular momentum

The angular momentum in a continuum relative to a reference point x0 can be stated,with the distance vector r = x− x0, as

L =

∫

Bt

ρ r× v dv . (2.26)

Similar to (2.21) its material time derivative has to balance the external torque Mext,which can be stated with the volume forces ρb̄ and the surface tensions t as

d

dtL =

∫

Bt

r× ρb̄ dv +

∫

∂Bt

r× t da . (2.27)

Again applying the Cauchy and the divergence theorem the surface integral is convertedto a volume integral, leading to

∫

Bt

ρ (ṙ× v + r× v̇) dv =

∫

Bt

(r× ρb̄+ r× divσ + grad r× σ

)dv . (2.28)

With (2.25) and the fact, that ṙ× v = 0 since ṙ = ẋ = v, it follows that

∫

Bt

grad r× σ dv = 0 , (2.29)

which, written in index notation, reveals the symmetry of the Cauchy stress tensor

σ = σT . (2.30)

10 CHAPTER 2. BACKGROUND

Thermodynamics

In this work rubber has to be modeled, which shows a significant energy dissipation ata mechanical loading. Therefore, beside the mechanical balance laws, also continuumthermodynamics will be viewed.To begin with the energy transfer inside a system is regarded. The total energy Econsists of the inner energy U and the kinetic energy K

E = U +K , (2.31)

where U follows with the specific inner energy u to

U =

∫

Bt

ρ u dv (2.32)

and K can be calculated from

K =

∫

Bt

1

2ρv2 dv . (2.33)

The change in energy in the system has to be equal to the sum of the mechanical andthe thermal power P and Q

d

dtE = P +Q , (2.34)

with P being the mechanical power, which stems from the external loads

P =

∫

∂Bt

t · v da +

∫

Bt

ρb̄ · v dv . (2.35)

The thermal power Q is calculated with the heat flux qn over the boundary and theheat source r inside the continuum from

Q =

∫

∂Bt

qn da +

∫

Bt

ρ r dv . (2.36)

Similar to (2.23) the heat flux can be replaced by qn = −q · n, where q is the Cauchyheat flux and n the surface normal. With

∫

∂Bt

t · v da =

∫

Bt

(divσ · v + σ : gradv) dv (2.37)

and the balance of linear momentum (2.25) the strong form of the balance of energy(2.34) can be stated as

ρ u̇ = σ : d+ ρ r − div q , (2.38)

which is known as the first law of thermodynamics. In (2.38) d = 1/2(l+ lT

)is the

symmetric part of the spatial velocity gradient l introduced in (2.14).

2.1. CONTINUUM MECHANICS 11

The first law of thermodynamics states, that in a system energy can not be lost, butis instead converted between different forms of energy. It provides no information onthe direction of the energy conversion and when this conversion takes place. To answerthese questions the entropy

S =

∫

Bt

ρ s dv . (2.39)

is introduced, which can be regarded as a measure of the disorder inside a system. Overits boundaries and from sources entropy can be transfered to the system. According tothe second law of thermodynamics the entropy has to increase, what means that theentropy transfer

Q̃ = −

∫

∂Bt

h · n da +

∫

Bt

ρ r̃ dv (2.40)

has to be less than the change in entropy, leading to the inequality relation

d

dtS − Q̃ ≥ 0 . (2.41)

For most processes the entropy flux h and the entropy source r̃ of (2.40) can be relatedto the heat flux q and the heat source r, respectively by

h =q

Θand r̃ =

r

Θ, (2.42)

with Θ being the absolute temperature. Inserting this into the second law of thermody-namics (2.41) and eliminating the heat source r with the first law (2.38), the ClausiusPlanck inequality is derived, which reads

σ : d− ρ u̇+ ρΘ ṡ ≥ 0 . (2.43)

For an isothermal process (2.43) can with the Helmholtz free energy function

Ψ = ρ (u−Θs) (2.44)

be written asσ : d− Ψ̇ ≥ 0 , (2.45)

which is known as the internal dissipation inequality.

2.1.3 Constitutive equations

The balance laws are fundamental equations and, just as the kinematical description,independent of the material of the continuum. The characteristics of the materials areaccounted for by the constitutive equations. The aim of the constitutive theory is tofind a mathematical relation, which describes the response of a material element to agiven input (Haupt [2002]). Constitutive equations can e.g. describe the relationshipbetween deformation and stress, where the deformation or the stress could be the input

12 CHAPTER 2. BACKGROUND

and the other quantity arises from this input, or e.g. between heat and temperature.These relations are necessary to solve the balance equations. For the deduction ofconsistent constitutive equations several principles exist, of which in the following themost important are briefly described.

The principle of determinism demands, that the current state of stress σ in a materialonly depends on the current deformation and its history, and not on any future events.

The principle of local action states, that the material response at a particle P is onlyinfluenced by the infinitesimal neighborhood of this particle.

The principle of frame indifference requires, that the material behavior is indepen-dent of the observer and its motion.

There exist also other principles, which are not addressed in this work. Altogetherthe formulation of consistent constitutive equations is a complex field, and for a morecomprehensive view on this topic the reader is refered to e.g. Truesdell and Noll [2004],Haupt [2002] and Altenbach [2012].The rubber regarded in this work exhibits inelastic effects. Its stress can be expressedas a function of the current deformation and a set of internal variables qk as

σ = f(C, q1, ..., qn) . (2.46)

The current values of the internal variables are determined implicitly from the defor-mation process and the initial values of the internal variables qk0. Following Haupt[2002] this is expressed with the functional f by

qk(t) = ft0≤τ≤t

[C (τ) , q10, ..., qn0] , (2.47)

where the notation ft0≤τ≤t

denotes a dependency on the deformation history between

starting time t0 and current time t.The elastic response of a rubber material can be modeled using a hyperelastic materiallaw, which means that the stress can be calculated with a scalar function W from

S = 2∂W

∂C, (2.48)

where S = J F−1σF−T is the second Piola-Kirchhoff stress tensor. The actual strainenergy function W and the evolution equations for the internal variables used in thiswork can be found in chapter 4.

2.1.4 Weak form

Beside the balance of linear momentum the solution of a mechanical problem has tofulfill some boundary conditions. In this context the boundary of the continuum bodycan be split into a part ∂uB, where the displacements u are prescribed and the remaining

2.2. CONTACT 13

part ∂tB, on which surface tensions t̄ can act. Together with (2.25) the set of equationsis given by

divσ + ρ(v̇ − b̄

)= 0 in Bt

u = ū on ∂uBt (2.49)

t = σ · n = t̄ on ∂tBt .

If the inertia forces have to be considered for the solution of a problem, additionallyinitial conditions for the displacement and its velocity of the form

u (t = 0) = u0 and u̇ (t = 0) = v0 in Bt (2.50)

are required. Altogether equations (2.49) and (2.50) form an initial boundary valueproblem (IBVP), which in general can not be solved analytically.A numerical solution can be obtained with the Finite Element Method (FEM), whichrequires the IBVP to be formulated in a variational form. To this end the local balanceof linear momentum is multiplied with a test function η, which satisfies

η = 0 on ∂uBt . (2.51)

Subsequent integration over the domain Bt yields, after integration by parts, applicationof the divergence theorem and consideration of (2.51), the weak form of the IBVP as

−

∫

Bt

σ : gradη dv +

∫

Bt

ρ(v̇− b̄

)· η dv +

∫

∂tBt

t̄ · η da = 0 . (2.52)

The test function η can be regarded as a variation of the displacement u, which iswhy (2.52) also corresponds to the principle of virtual work. It includes the boundarycondition for the surface tensions, which is now fulfilled in a weak sense.

2.2 Contact

The simulation of contact between deformable bodies needs further considerations.When bodies, of which at least one is deformable, approach, the area of contact, whichforms between the bodies, and the forces, acting at this contact surface, are unknown.Yet (2.52) only captures deformations due to preexisting loads acting on known sur-faces, and therefore the theory has to be extended to include also contact.For this purpose, starting with the works of Wilson and Parsons [1970] and Chan andTuba [1971], the field of computational contact mechanics has evolved. Introductionsinto this subject can be found in the books by Kikuchi and Oden [1988], Laursen [2002]and Wriggers [2006].

2.2.1 Kinematics

In the following the contact between two deformable bodies, as sketched in figure 2.2,is regarded. In this context, going back to Hallquist [1979], one body B2 is denoted as

14 CHAPTER 2. BACKGROUND

X̄1

X̄2

B10

B20

B1t

B2tϕ

∂cBt

na

Figure 2.2. Contact of two bodies.

x

ξ1

ξ2n

a1

a2

Figure 2.3. Parametrization of a surface with convective coordinates.

the master and the other as the slave body B1. In general the bodies are not allowedto penetrate each other and thus, when they approach, a contact zone ∂cBt develops,at which interactions between the bodies take place.To describe these interactions the surfaces of the master and the slave body areparametrized each by two convective coordinates ξ1 and ξ2, see figure 2.3. With thisparametrization coordinate systems fixed at the material points on each surface are in-troduced. From differential geometry it is known, that their base vectors are calculatedwith the position vector x of the regarded particle from

aα =∂x

∂ξα= x,α with α = 1, 2 , (2.53)

see e.g. Klingbeil [1989]. The third base vector is the normalized surface normal ncalculated by

n =a1 × a2‖a1 × a2‖

. (2.54)

To decide whether contact occurs, two facing surface particles x̄1 and x̄2 are regarded.The particles are in contact, if the normal gap function

gN =(x̄2 − x̄1

)· nc (2.55)

equals zero, at which nc denotes the contact normal, which is specified in Chapter 3.The facing points are determined by projections from one surface onto the other. Since

2.2. CONTACT 15

the projection routine depends on the numerical implementation, it is not described atthis point but can be found for two different implementations also in Chapter 3.If friction or adhesion has to be considered, the two bodies can interact in a directiontangential to the interface. These interactions are usually modeled by constituive equa-tions, which need a description of the tangential movement between both bodies. Therelative tangential velocity can be defined by

ġT = (1− nc ⊗ nc) ·(v̄2 − v̄1

), (2.56)

from which the total sliding distance between time t0 and t1 can be computed from

gT =

t1∫

t0

‖ġT‖ dt . (2.57)

2.2.2 Constraints and constitutive equations

With the normal gap gN the different states of contact can be distinguished. If thebodies are not allowed to penetrate each other, then gN ≥ 0 has to hold for all particleson the contact surface ∂cB. This non penetration condition only generates stresses tNnormal to the contact surfaces. If gN > 0 there is a gap between both particles andthus, if adhesion is not considered, the contact pressure tN is equal to zero. On theother hand if the bodies are in contact, then tN is less than zero. These circumstancesare combined in the Karush-Kuhn-Tucker conditions

gN ≥ 0 , tN ≤ 0 , gN tN = 0 . (2.58)

In the literature sometimes also the persistency condition is mentioned (see e.g. Agelet deSaracibar [1997]), which states, that if the bodies stay in contact, then

ġN = 0 . (2.59)

As mentioned the interaction in tangential direction is described by a constitutive equa-tion, which is usually based on measurements. To model friction effects, the classicallaw of Coulomb is applied in this work.Up to a certain tangential stress ‖tT‖ < µ|tN |, where µ is the models parameter, thebodies stick to each other, meaning that no relative movement of the two contactingsurfaces occurs

ġT = 0 . (2.60)

When the limit µ|tN | is reached, a relative tangential movement of the surfaces calledsliding occurs. In this case the tangential stress

tT = −µ|tN |ġT

‖ġT‖. (2.61)

acts opposite to the direction of movement. To distinguish between stick and slip theslip criterion

fs = ‖tT‖ − µ|tN | ≤ 0 (2.62)

16 CHAPTER 2. BACKGROUND

is introduced, where slip occurs if fs = 0. In this case the relative tangential movementof the surfaces is computed from the evolution equation

ġT = γ̇tT

‖tT‖, (2.63)

in which γ is a plastic parameter and γ̇ ≥ 0 has to hold. Similar to (2.58) the tangentialcontact constraints can also be stated as

γ̇ ≥ 0 , fs ≤ 0 , γ̇ fs = 0 . (2.64)

2.2.3 Contact virtual work and treatment of contact constraints

To model a mechanical system, in which contact occurs, the governing equation (2.52)has to be extended to also include interactions due to contact. To begin with theboundary value problem for two contacting bodies is regarded for the case, that at thecontact zone no tangential interactions occur. With the unknown contact surfaces ∂cB

it

on both bodies (i = 1, 2) and the unknown contact stresses

tic = tiNnc , i = 1, 2 (2.65)

it reads

2∑

γ=1

∫

Bγt

σγ : gradηγ dv −

∫

Bγt

ρ(v̇γ − b̄γ

)· ηγ dv −

∫

∂tBγt

t̄γ · ηγ da

−

∫

∂cB1t

t1c · η1c da−

∫

∂cB2t

t2c · η2c da = 0 ,

(2.66)

where ηic are the variations of the displacements of the particles in contact. The firstthree integral terms arise from the virtual work of both bodies and are in the followingdenoted by Gγ, whereas only the remaining terms compose the contribution Gc ofthe contact constraints to the weak form. Since the contact surfaces on both bodiesmust be equal and the contact stresses have to be equal in magnitude and opposite indirection

t1c = −t2c = tc , (2.67)

Gc can be simplified to

Gc =

∫

∂cBt

tc ·(η2c − η

1c

)da . (2.68)

This can, with (2.65) and the variation δgN of the normal gap, be expressed as

Gc =

∫

∂cBt

tN δgN da , (2.69)

2.2. CONTACT 17

since δn · (x̄2 − x̄1) = 0, which will be shown in chapter 3. Generally the presence ofcontact interactions leads to an extension of the IBVP by a contact contribution Gc

2∑

γ=1

Gγ +Gc = 0 . (2.70)

Lagrange multiplier method

For the enforcement of the contact constraints, different methods, which lead to dif-ferent contact contribution Gc exist. One method is the Lagrange multiplier method,which will be described only for the adhesion and frictionless case, since it is not appliedin this work, but is the basis for the augmented Lagrange multiplier method.Provided that the contact surfaces are known, the non penetration condition (2.55)leads to additional equations in the equation system. These equations can be includedby introducing additional unknowns, the so called Lagrange multipliers λN . The con-tact contribution for the Lagrange multiplier method reads

Gc =

∫

∂cBt

(λN δgN + δλN gN) da , (2.71)

where δ(..) denotes the variation of a quantity. Comparing the first term in (2.71) with(2.69) it can be noticed that the multiplier λN corresponds to the contact stress tN . Thesecond term arises from the fact, that the multipliers are independent unknowns andrepresents a weak formulation of the constraint equation (for details, see e.g. Wriggers[2006]).

Penalty method

The introduction of additional degrees of freedom (DOF) leads to an increase in systemsize and thus for later simulations in an increase in computation time. This can beavoided by a regularization of the contact constraints with the penalty method. Sincethis method is used for the simulations in this work, in the following also tangentialinteractions due to friction are regarded.Within the penalty method the contact stresses are assumed to be dependent on thegap, which is sometimes referred to as the primal variable

tN = cNgN tT =

−cT (gT − gsT ) for stick

−µ |tN |ġT‖ġT‖

for slip ,(2.72)

where cN > 0 and cT > 0 are the so called penalty parameters and gsT is the slip (or

plastic) part of the relative tangential movement. This assumption leads to the contactcontribution

Gc =

∫

∂cBt

(tN δgN − tT · δgT ) da . (2.73)

18 CHAPTER 2. BACKGROUND

The regularization of the contact constraints leads to a violation of the non penetrationand the stick condition ((2.55) and (2.60)). The deviation from the exact solutiondepends on the values of the penalty parameters. For cN → ∞ and cT → ∞ thesolution converges to the exact solution of the Lagrange multiplier method. However,too large penalty parameters lead to an ill-conditioning of the system of equations, sothat the parameters have to be chosen with care. Instead of (2.72) also other relationsbetween gap and stress stemming e.g. from constitutive equations can be used (see e.g.Zavarise et al. [1992] and Bandeira et al. [2004]).

Augmented Lagrange multiplier method

The augmented Lagrange multiplier method was introduced in the mathematical liter-ature in the context of optimization with constraints by Hestenes [1969] and extendedto inequality constraints by Rockafellar [1976]. An early implementation for contactproblems can be found in Wriggers et al. [1985]. Since the augmented method is alsoapplied in this work, it will be described in detail in this section.Generally two different approaches exist. In the mixed primal dual formulation theLagrange multipliers, which appear in the method, are treated as fully independent(denoted as dual) variables, whereas in the so called Uzawa algorithm they are handledas known quantities, which are eliminated from the system of equations and updatedafter each time step. The method can be viewed as a combination of the penalty and theLagrange multiplier method. It was found to be more stable than the penalty method(Alart and Curnier [1991]) and fulfills the non penetration and the stick conditionexactly.Within the method the augmented stresses

λ̂N = λN + cN gN and λ̂T = λT − cT ∆gT (2.74)

are defined, where cN and cT are regularization parameters, λN and λT the Lagrangemultipliers in normal and tangential direction and ∆gT = ġT ∆t is the tangentialdisplacement increment.A mixed primal dual approach for frictional contact problems was first formulated byAlart and Curnier [1991] and extended to large deformations in Pietrzak and Curnier[1999]. There the contact contribution Gc is calculated from the variation of a normaland a tangential contact potential, ΠcN and Πc T , respectively, by

Gc =

∫

∂cBt

(

δλN∂Πc N∂λN

+ δgN∂Πc N∂gN

+ δgT∂Πc T∂gT

+ δλT∂Πc T∂λT

)

da . (2.75)

The normal potential ΠcN is determined directly from Rockafellar [1976] to

ΠcN = −1

2cN|λN |

2 +1

2cN[min (|λN |+ cN gN , 0)]

2 , (2.76)

and the tangential potential reads

Πc T = λT ·∆gT +cT2‖∆gT‖

2 −1

2cT

[

max(

‖λT + cT ∆gT‖ −max(

−µ λ̂N , 0)

, 0)]2

.

(2.77)

2.2. CONTACT 19

After evaluation of the min and max functions and subsequent variation with respectto the variables, the contact contribution Gc is calculated to

Gc = Gcu +Gcl , (2.78)

where

Gcu =

∫

∂cBt

(

λ̂N δgN + λ̂T · δgT

)

da (2.79)

is independent of the state of contact and can be regarded as the virtual work of thecontact forces. Gcl represents a weak formulation of the constraints and reads

Gcl =

∫

∂cBt

(

δλN1

cNλN + δλT ·

1

cTλT

)

da if λ̂N > 0

∫

∂cBt

(δλN gN + δλT ·∆gT ) da if λ̂N ≤ 0 , ‖λ̂T‖ < µ |λ̂N |

∫

∂cBt

(

δλN gN + δλT ·1

cT

[

λT − µ λ̂Nλ̂T

‖λ̂T‖

])

da if λ̂N ≤ 0 , ‖λ̂T‖ ≥ µ |λ̂N | ,

(2.80)where the augmented stresses determine the contact state: λ̂N > 0 corresponds to aninactive contact, ‖λ̂T‖ < µ |λ̂N | to stick and ‖λ̂T‖ ≥ µ |λ̂N | accordingly to the case ofslip.Another approach to deal with the contact constraints, which arrives at almost thesame equations, is pursued in Hüeber and Wohlmuth [2005] and Hüeber et al. [2008].Starting from the virtual work the contact problem for each particle, which can possiblycome into contact, can be stated as

λNp δgNp + λTp · δgTp = 0

subject to gNp ≥ 0 , λNp ≤ 0 , gNp λNp = 0

and γ̇p ≥ 0 , fsp ≤ 0 , γ̇p fsp = 0 .

(2.81)

The inequalities can be treated with an active set strategy, see Luenberger [1984],which, however, requires an additional loop in the iterative solution scheme. In orderto avoid this, two nonlinear complementary functions (NCF) CN and CT are defined,with which the inequalities in (2.81) can be replaced by

δλNpCNp = 0

δλTp ·CTp = 0 .(2.82)

The functions CN and CT are given in Hintermüller et al. [2003] and Hüeber et al. [2008](also see Popp et al. [2009] and Gitterle et al. [2010]). In accordance with Weißenfels[2013] they are in this work divided by the respective regularization parameter andhence read

CN =1

cN

[

λN −min(

0, λ̂N

)]

CT =1

cT

[

max(

µ λ̂N , ‖λT‖)

λT − µ min(

0, λ̂N

)

λ̂T

]

.(2.83)

20 CHAPTER 2. BACKGROUND

The formulation of the constraint equations in terms of (2.82) allows for a solution ofthe contact problem with the semi-smooth Newton method, where the contact searchand the determination of stick and slip are treated within the same iterative schemeas the nonlinearities of the continua. When the min and max functions in (2.83) areevaluated this approach results in the same equations as the one based on the potentials,except that (2.79) is changed to

Gcu =

∫

∂cBt

(λN δgN + λT · δgT ) da . (2.84)

In the following, when the augmented Lagrange multiplier method is addressed, atreatment of the contact constraints with (2.80) and (2.84) is meant.Beside the described methods also other approaches to treat the contact constraints(e.g. Nitsche’s method) exist. For a more comprehensive overview the reader is referredto e.g. Wriggers [2006].

2.3 Finite elements

An analytical solution of the IBVP in the form of (2.52), or (2.70) for contact prob-lems, can only be obtained for special cases. However, in the mathematical and theengineering literature several techniques to obtain approximate numerical solutions ofsuch partial differential equations (PDEs) exist. The most common is the finite ele-ment method (FEM), which is based on a Rayleigh Ritz approximation of the soughtquantities and Galerkins method of weighted residuals. After its first appearance inCourant [1943] it has been successfully applied and extended to a variety of differentapplication areas. For a comprehensive overview the reader is referred to the textbooksof Zienkiewicz et al. [2005], Zienkiewicz and Taylor [2005] and e.g. Wriggers [2001].

2.3.1 Discretization of space

In the following the solution of the mechanical problem of (2.52) at one instant in timeis regarded. The discretization of the contact contributions in (2.70) will be regardedin chapter 3. Within the FEM the continuous domain B is approximated by a set ofne non overlapping subdomains Ωe

B ≈ Bh =

ne⋃

e=1

Ωe , (2.85)

(see Wriggers [2001]) which is illustrated in figure 2.4. This approach also leads toan approximation of the boundaries ∂B of the domain with the discrete boundaries∂Bh. The subdomains Ωe are denoted as finite elements. On each element a numberof nodes is defined, which in figure 2.4 can e.g. be the corners of the elements. Usuallythe elements are of simple geometry. Triangles and squares are used to discretize twodimensional domains, whereas in three dimensional problems mainly tetrahedrons andhexaedrons are employed.

2.3. FINITE ELEMENTS 21

B

Ωe

∂B ∂Bh

Figure 2.4. Discretization of a two dimensional body B.

Following the Rayleigh Ritz approach the solution field u of the PDE is approximatedby a linear combination of different test functions. Within the FEM these test functionsare defined only locally on each element with so called shape functions NI by

u ≈ uh =n∑

I=1

NI (ξ)uI , (2.86)

where uI are the (unknown) displacements at the nodes of the finite element and ξ arelocal coordinates describing the position inside the element. In order that the integralsof (2.52) can be evaluated, the shape functions have to be Cm continuous functions,when the highest derivative of the variables appearing in the weak form is of orderm + 1. Furthermore a standard shape function NI has to be equal to one at node Iand zero at all other nodes, and the sum of all shape functions has to be always equalto one

NI (ξJ) = δIJ ,n∑

I=1

NI (ξ) = 1 , (2.87)

where ξJ are the coordinates at the position of node J .

In this work the isoparametric concept is applied, which means that the geometry isapproximated by the same shape functions as the displacements

x ≈ xh =

n∑

I=1

NI (ξ)xI . (2.88)

Since the shape functions are defined on a reference element, this approximation can beinterpreted as a mapping of the element geometry onto a reference configuration Ω�,see figure 2.5. Ω0 denotes the initial configuration, which can also be mapped on thisnew reference configuration. With this isoparametric mapping it is simple to computethe gradients appearing in the weak form. Also the integration over the domain canbe readily accomplished in the reference configuration.

22 CHAPTER 2. BACKGROUND

ξ

ξ

ξ

ηη

η

F

Ω0

Ωt

Ω�

jJ

Figure 2.5. Isoparametric mapping.

The mappings between the different configurations are calculated to

j =∂x

∂ξ=

n∑

I=1

xI ⊗∂NI∂ξ

,

J =∂X

∂ξ=

n∑

I=1

XI ⊗∂NI∂ξ

.

(2.89)

With this the gradient of the discretized displacement field, e.g. , can be calculated to

graduh =∂uh

∂x=

n∑

I=1

uI ⊗∂NI∂x

=

n∑

I=1

uI ⊗ j−T ∂NI

∂ξ. (2.90)

The integrals over the elements and their boundaries are evaluated numerically usingGaussian quadrature. At this the integration of a function f is replaced by the sum ofthe values of the function at different Gauss points gp multiplied with weighting factorswgp, giving

∫

Ωe

f (ξ) dΩe =

∫

Ω�

f (ξ) det j dΩ� ≈

ngp∑

g=1

f(ξgp)det jgp wgp . (2.91)

Approximating the other field variables η and v̇ with the same shape functions as thedisplacements, an application of the FEM approximation to the governing equationsleads to a transformation of the PDE of (2.52) into a system of nonlinear equations, inwhich the nodal (displacement) values have to be determined. This can be written inmatrix form as

ηT [Mv̇ +R(u)−P] = 0 , (2.92)

where u and η are vectors containing the nodal displacements and their variations, Mis the mass matrix, P a vector containing the external loads and R is the residual force

2.3. FINITE ELEMENTS 23

vector, which results from the internal forces. Since the test function η is arbitrary,(2.92) is only fulfilled, if the expression inside the brackets becomes zero.

2.3.2 Discretization of time

If the motion or deformation of a mechanical system is regarded as it evolves over time,a problem continuous in time results. Regarding this problem only at discrete pointsin time, it can be transferred into sequence of a limited number of problems. To thisend the time is discretized with an adjustable time increment ∆t, which leads to therecursion formula

tn+1 = tn +∆t (2.93)

for the current time tn+1. In (2.92) the accelerations v̇ and the displacements u are noindependent variables. Thus if inertia effects have to be considered in the solution of aproblem, a suitable integration scheme has to be applied, which reflects the dependencybetween both quantities. To this end the method proposed by Newmark [1959] isapplied in this work. The current nodal velocities and displacements, vn+1 and un+1,are calculated with their values from the last time step, denoted by a subscript (..)n,and the previous and the current accelerations a = v̇ by

vn+1 = vn +∆t [(1− γ) an + γ an+1] ,

un+1 = un +∆tvn + (∆t)2

[(1

2− β

)

an + β an+1

]

.(2.94)

The parameters β and γ are restricted by

0 ≤ β ≤1

2, 0 ≤ γ ≤ 1 . (2.95)

The Newmark method is called an implicit time integration scheme, which means thatthe variables to be integrated are determined not only from kinematical quantities,known from the last time step, but also from unknown quantities at the new time step.If the parameters β and γ are chosen to be zero and 0.5, respectively, the dependency onthe unknown accelerations an+1 drops, which results in an explicit integration scheme.Inserting relation (2.94) into (2.92) the system of equations can be solved either for thedisplacements or the accelerations.

2.3.3 Solution algorithms

Newton-Raphson method

The nonlinear equation system resulting from (2.92) can be written as

G(u) = Ma(u) +R(u)−P = 0 . (2.96)

Generally its solution is obtained iteratively using the Newton-Raphson method. Theiterative procedure is derived from a Taylor expansion of (2.96) to the first order at aknown deformation state uk, which reads

G(uk+1

)= G

(uk)+∂G(u)

∂u

∣∣∣∣u = uk

·(uk+1 − uk

)= 0 . (2.97)

24 CHAPTER 2. BACKGROUND

The new displacement field uk+1 is thus calculated to

uk+1 = uk − (K∗T )−1G

(uk), (2.98)

where

K∗T =∂G(u)

∂u

∣∣∣∣u = uk

(2.99)

denotes the Jacobi or tangent matrix. At each time step the iteration with (2.98) isrepeated until the solution converges. The converged solution un from the previoustime step is then chosen as the initial value u1n+1 at the new time step. Anotherpossibility to solve (2.96), especially in the case of high nonlinearities, would be to usearc-length type methods. This approach will be applied and explained in chapter 6.

Implicit solution

In the case of small accelerations it is sometimes sufficient to solve the quasi staticproblem, which results when in (2.96) the term containing the acceleration is neglected.If dynamic effects have to be considered, the equation system is usually solved for thenodal displacements. To this end the Newmark equation (2.94)2 is rewritten as follows

an+1 =1

β (∆t)2(un+1 − un)−

1

β∆tvn −

(1

2β− 1

)

an . (2.100)

Since in (2.96) no velocity dependent terms appear, here only the accelerations areregarded. Inserting (2.100) into (2.96) the tangent matrix for the Newton iteration canbe stated as

K∗T =1

β (∆t)2M+KT , (2.101)

where

KT =∂R(u)

∂u

∣∣∣∣u = uk

(2.102)

is the tangent matrix of the quasi static problem.

Explicit solution

As already mentioned an explicit solution scheme results, when the Newmark parameterβ is chosen to be zero. In this case (2.94)2 reads

un+1 = un +∆tvn + (∆t)2 1

2an , (2.103)

which, inserted into (2.96), leads to the following equation to determine the currentaccelerations:

an+1 = M−1

[

R

(

un +∆tvn + (∆t)2 1

2an

)

−P

]

. (2.104)

2.3. FINITE ELEMENTS 25

With the solution of this equation the unknown displacements un+1 can in turn becalculated from (2.103). Compared to (2.101) the explicit solution via (2.104) is verysimple, as the (sometimes very complex) tangent matrix KT is not required, and onlythe mass matrix M has to be inverted. However this solution scheme is only condi-tionally stable and requires a very small time step size ∆t. An estimation for the timestep size for nonlinear problems can be found in Belytschko et al. [1976].

Algorithms for contact

In the following the discretization Gc h of the contact contribution in (2.70) is assumedto be given. Depending on the applied strategy the contact constraints are imposedon nodes or elements. To calculate the contact contributions Gc h the contact surfaces∂cB

h need to be known ((compare e.g. (2.73) and (2.78))), which is usually not thecase. A simple method to deal with this issue is an active set strategy, see Luenberger[1984], which will be used in this work, when the penalty method is applied. Withinthe penalty method nodes (or elements) are in contact if

gN ≤ 0 . (2.105)

At the beginning of each iterative solution procedure a set of nodes (or elements),denoted as the active set IA, is assumed to be in contact. By checking (2.105) for allpossible candidates, nodes (or elements) can be determined which have to be added toor removed from IA. Depending on the implementation this update can be performedin each iteration, or in a outer loop after convergence with a particular active set hasbeen obtained.Inequality (2.62), describing the tangential contact conditions, is treated with the re-turn mapping algorithm, which was first applied to contact problems in Wriggers [1987].Within the penalty method the relative tangential movement gT is split into an elasticand an inelastic or slip part

gT = geT + g

sT , (2.106)

where the elastic part geT stems from the regularization of the stick condition within thepenalty method. Under the assumption, that stick occurs, meaning that the currenttangential movement is elastic, a trial tangential stress

ttrT n+1 = −cT(gT n+1 − g

sT n

)(2.107)

is calculated, which is used to evaluate the slip criterion (2.62)

f trs n+1 = ‖ttrT n+1‖ − µ |tN | . (2.108)

If f trs n+1 ≤ 0, then the node or element sticks. In this case the gsT does not change from

the last time step and the actual tangential stress tT n+1 is equal to ttrT n+1. Otherwise,

if f trs n+1 > 0, sliding occurs, and the tangential stress has to be projected onto the slipsurface which gives

tT n+1 = µ |tN |ttrT n+1

‖ttrT n+1‖gsT n+1 = g

sT n +

1

cTf trs n+1

ttrT n+1‖ttrT n+1‖

. (2.109)

26 CHAPTER 2. BACKGROUND

Within the augmented Lagrangian method active contact nodes (or elements) and thedifferentiation between sticking and sliding are determined with the augmented stressesas follows:

active contact if λ̂N ≤ 0 , inactive contact if λ̂N > 0 ,

stick if ‖λ̂T ‖ < µ λ̂N , slip if ‖λ̂T‖ ≥ µ λ̂N .(2.110)

Since the augmentation leads to a regularization of the contact constraints, the updateof the active set and of the tangential state can be performed within the same Newton-Raphson loop, the material and geometrical nonlinearities are treated in.

Chapter 3

Contact finite elements

This chapter addresses the discretization of the contact contributions in the weak form.To enforce the contact constraints several techniques have been developed in the lit-erature. In the simplest case, referred to as a node-to-node formulation, they aredirectly established between the nodes of the contacting surfaces, see e.g. Francavillaand Zienkiewicz [1975]. Since this concept needs matching meshes at the interfaces, itis restricted to small deformations.A more general approach are the node to segment (NTS) formulations, where the nodesof one of the surfaces are in contact with the elements of the other. Since this conceptis employed in this work, its implementation within a three dimensional finite elementframework and a treatment of the constraints with the penalty method is described inthe next section.Recently the mortar method has become very popular. This method constitutes anenforcement of the contact constraints between the elements of both contacting sur-faces, and thus represents a segment-to-segment formulation. It is also applied in thiswork, yet with an enforcement of the contact constraints using the augmented Lagrangemultiplier method, and is therefore described in section 3.2.

3.1 Node to segment

Early implementations of 2D NTS strategies can be found in Chan and Tuba [1971] orHallquist [1979]. Extensions to 3D contact problems were developed in Parisch [1989]and, for the case of friction, in Peric and Owen [1992]. A fully consistent linearizationfor 3D frictional contact problems can be found in Laursen and Simo [1993]. Theformulation applied in this work is based on this contribution and can also be found ine.g. Wriggers [2006] or Weißenfels [2013].

3.1.1 Discretization, projection and contact search

Having assigned one surface as the master and the other as the slave surface, as intro-duced in section 2.2, the contact constraints are enforced between the nodes x1 of theslave surface and the elements of the master surface ∂cB

2. The continua are in this work

27

28 CHAPTER 3. CONTACT FINITE ELEMENTS

slave surface

master surface

x1

x2 (ξ1, ξ2)

x21

x22

x23

x24

a22

a21

n2

ξ1

ξ2

Figure 3.1. 3D NTS contact element.

discretized with hexahedrons leading to four sided elements on the surfaces. Using theisoparametric concept, the geometry and the displacement field of these elements areinterpolated with the bi-linear shape functions

NI =1

4

(1 + ξ1I ξ

1) (

1 + ξ2I ξ2), (3.1)

where ξ1 ∈ [−1, 1] and ξ2 ∈ [−1, 1] are the convective coordinates and the index (..)Idenotes the value at node I. With this the position of a point on the master surface iscalculated by

x2 =

nm∑

I=1

NI(ξ1, ξ2

)x2I , (3.2)

where nm denotes the number and x2I the coordinates of the nodes of the master surface

element, see figure 3.1. Recalling (2.53) the base vectors of the master surface followfrom

a2α =nm∑

I=1

NI,α(ξ1, ξ2

)x2I , (3.3)

where here and in the following small Greek letters have values of 1 or 2 and a differ-entiation with respect to ξα is denoted by (..),α.To determine contact, a function

d = ‖x1 − x2(ξ1, ξ2

)‖ (3.4)

describing the distance between a slave node x1 and a point x2 on a facing masterelement is introduced. The closest point to x1 is determined by minimizing d withrespect to the convective coordinates (ξ1, ξ2), which yields

[x1 − x2

(ξ1, ξ2

)]· x2,α

(ξ1, ξ2

)= 0 , (3.5)

3.1. NODE TO SEGMENT 29

where x2,α corresponds to the base vector a2α of the master element. With this in mind

(3.5) denotes a projection of the slave node onto the master element, along the normalof the master surface at the projection point. To solve this nonlinear equation for theconvective coordinates a Newton-Raphson scheme is applied. Neglecting higher orderterms, the Taylor expansion of (3.5) reads

[x1 − x2 k

]· x2 k,α +

(−a2 kα a

2 kβ +

[x1 − x2 k

]x2 k,α β

) (ξβ k+1 − ξβ k

)= 0 , (3.6)

where the superscript (..)k denotes an evaluation at the coordinates ξβ k, calculatedin the last iteration. Solving (3.6) for the new coordinates ξβ k+1, yields the updateformula for the solution. As initial values for the iteration, ξ1 0 = ξ2 0 = 0 are chosen,which corresponds to a projection point at the center of the slave element.Within the NTS strategy for each slave node x1I the facing master element has to beidentified. To this end the master node x2J , closest to x

1I , is determined. x

1I is then

projected onto all elements neighboring x2J . The element, which yields valid values forξ1 and ξ2 is then selected as the facing element. If more than one projection yieldspoints inside the corresponding element, a special treatment is necessary, which will bediscussed in section 3.1.6.

3.1.2 Kinematical quantities and constitutive equations

In the following the coordinate values, which solve (3.5), are denoted by ξ̄α. Alsoall quantities, which have to be evaluated at these coordinates, are marked with anoverbar. Having determined the closest point, the normal gap function is defined withthe normal of the master surface as

gN = n̄2[x1 − x̄2

]. (3.7)

A multiplication with the normal vector yields the relation

gN n̄2 = x1 − x̄2 , (3.8)

which will be of use in subsequent calculations. The increment of the relative tangentialmovement between the previous and the current time step (tn and tn+1, respectively),is calculated from the change of the convective coordinates of the projection point via

∆gT n+1 =(ξ̄αn+1 − ξ̄

αn

)ā2α . (3.9)

The contact constraints are treated with the penalty method, so that the contactstresses are directly related to these kinematical quantities. The normal contact pres-sure tN is calculated with the penalty parameter cN from (2.72) to

tN = cN gN . (3.10)

To determine the correct tangential stress, the return mapping algorithm of section2.3.3 is applied. The covariant components of the trial tangential stress are computedfrom

ttrTα = −cT

(

∆tξβn+1 + ξe βn

)

āαβ n+1 , (3.11)

30 CHAPTER 3. CONTACT FINITE ELEMENTS

where ξeβn are the components of the elastic gap at the previous time step,

∆tξβn+1 = ξ̄βn+1 − ξ̄

βn (3.12)

are the increments of the projection point coordinates and

āαβ = ā2α · ā

2β (3.13)

is the metric tensor. At the initiation of contact at a node the elastic gap is zero. Withthe norm of the trail tangential stress

‖ttrT ‖ =√

ttrTα ttrTβ ā

αβ (3.14)

the trial slip criterionf tr = ‖ttrT ‖ − µ |tN | (3.15)

can be evaluated. Depending on its value, the actual tangential stress and the compo-nents of the new elastic gap result in

tTα =

ttrTα if ftr < 0

µ |tN |ttrTα‖ttrT ‖

if f tr ≥ 0(3.16)

and

ξe βn+1 =

ξe βn +∆tξβn+1 if f

tr < 0

1

cTtTα ā

αβ if f tr ≥ 0 .(3.17)

3.1.3 Contribution to the weak form

The contribution Gc of the contact constraints to the weak form is stated, for the case,that the penalty method is applied, in (2.73). The tangential part can also be writtenin component form as

tT · δgT = tTαδξ̄α . (3.18)

The integration over the contact surface is carried out on the slave surface. With thearea as surrounding each slave node, G

c can be approximated by

Gc =

∫

∂cBt

(tN δgN − tT · δgT ) da ≈

n1glob∑

s=1

(tN δgN − tTα · δξ̄

α)as , (3.19)

where the contributions of all n1glob slave nodes are summarized.To calculate as, the areas of the elements adjacent to the current slave node have tobe determined. With the same local node numbering for the slave surface as for themaster surface, shown in figure 3.1, on each slave element e the edge vectors

de1 = x12 − x

11 , d

e2 = x

14 − x

11 , d

e3 = x

12 − x

13 and d

e4 = x

14 − x

13 (3.20)

3.1. NODE TO SEGMENT 31

are defined. With this the area surrounding a slave node is calculated to

as =1

n1

n1ad∑

e=1

1

2‖de1 × d

e2‖ ‖d

e3 × d

e4‖ , (3.21)

where n1 is the number of nodes on the slave elements and n1ad the number of adjacentelements.

3.1.4 Linearization

Since the weak form will be solved with a Newton-Raphson scheme (see section 2.3.3),the contact contribution (3.19) has to be linearized, giving

∆Gc =

n1glob∑

s=1

(∆tN δgN + tN ∆δgN −∆tTα · δξ̄

α − tTα ·∆δξ̄α)as , (3.22)

where ∆(..) stands for the linearization of a quantity. The variation δgN of the normalgap is calculated