The effects of roughness on the area of contact and on the elastostatic friction FEM simulation of micro-scale rough contact and real world applications Doctoral Dissertation submitted to the Faculty of Informatics of the Università della Svizzera Italiana in partial fulfillment of the requirements for the degree of Doctor of Philosophy presented by Alessandro Pietro Rigazzi under the supervision of Prof. Dr. Rolf Krause October 2014 Dissertation Committee Prof. Dr. Illia Horenko Università della Svizzera Italiana, Switzerland Prof. Dr. Igor Pivkin Università della Svizzera Italiana, Switzerland Prof. Dr. Marco Paggi Institute for Advanced Studies Lucca, Italy Prof. Dr. Friedemann Schuricht Technische Universität Dresden, Germany Dissertation accepted on 22 October 2014 Prof. Dr. Rolf Krause Research Advisor Università della Svizzera Italiana, Switzerland Prof. Dr. Stephan Wolf PhD Program Director i I certify that except where due acknowledgement has been given, the work pre- sented in this thesis is that of the author alone; the work has not been submitted previously, in whole or in part, to qualify for any other academic award; and the con- tent of the thesis is the result of work which has been carried out since the official commencement date of the approved research program. Alessandro Pietro Rigazzi Lugano, 22 October 2014 ii To my late father, for supporting me in learning numbers before letters. iii iv If numbers aren’t beautiful, I don’t know what is. Paul Erdos˝ v vi Abstract Roughness is everywhere. Every object, every surface we touch or look at, is rough. Even when it looks smooth and flat, if analyzed at the proper length scale, it will reveal roughness. Thus, macro-scale, micro-scale, and even nano-scale roughness exist. What is even more fascinating, is that most of the rough structures, which can be observed at a given length scale, repeat themselves at smaller length scales, as in a fractal. The first implication of the rough nature of surfaces is that what we perceive as a full, solid, and smooth contact area, is in reality a collection of fragmented microscopical contact patches, composed of single contact points. Given its intrinsic complexity, the modeling of the real area of contact has been the subject of a huge amount of studies, which yielded different and contrasting results. As it is easy to imagine, the real area of contact is crucial for many real-world applications, such as the prediction of wear and fretting, charge and heat conduction, and frictional effects. Let alone think of how an acting load is normally believed to be uniformly distributed over the contact surface, and how variable and uneven it must, in reality, appear at microscopic length scales. The importance of roughness, together with our knowledge in parallel computing and fast solution methods, are the premises of the current work. In this study, we an- alyze rough contact between realistic surfaces, and we resolve it numerically at micro- scale, to understand its meso- and macro-scale effects. We do this by means of the Finite Element Method, in combination with an optimal multi-grid strategy and a spa- tial decomposition to perform the computations on highly parallel super-computers. We concentrate on the type of surfaces for which it is believed that molecular and chemical effects can be neglected. We simulate the contact between an elastic cube and diverse rigid rough surfaces, under different loading conditions, and we derive empirical laws which describe the influence of well known roughness parameters on important features such as contact evolution and static friction production. We also define bounds on the uncertainty of our measurements, to make clear the level up to which our predictions have to be considered reliable and applicable. Literature on roughness is densely populated by models, approaches, and theoret- ical predictions about the evolution of the real area of contact. An exhaustive com- parison of our results with such corpus of works, articles, books, and theses would be vii viii infeasible. We therefore compare our results on the real area of contact to the pre- dictions of two widely accepted theories (one by B. N. J. Persson, the other by A. W. Bush, R. D. Gibson, and T. R. Thomas), which have often proved to be interpretable as asymptotical bounds, for systems at low pressures. For large pressures, and conse- quent large areas of contact, we also compare our results to the newly developed and semi-empirical theory by Yastrebov, Molinari, and Anciaux. Finally, we test our method on the real world problem of tyre-asphalt interactions on wet roads, comparing the results obtained by our method to data from other studies, collected on real highways and runways, and to a theoretical model, which is close, in the assumptions, to our numerical experiments. Acknowledgements I wish to thank everyone who supported me during this study: Rolf Krause for be- ing both an academic and a research advisor, providing me with knowledge, motiva- tion, method, and scientific spirit, Illia Horenko and Igor Pivkin for their suggestions throughout the development of this work, Friedemann Schuricht for his lectures and his remarks on elasticity and contact problems, and Marco Paggi for his useful ad- vices about theory, methods, and literature in contact mechanics. I also acknowledge the collaboration of a non-disclosed industrial partner, which provided me with useful insights, based on its long-time experience in this field. I thank my family and my girlfriend for continuous support, and for coping with me going through the ups and downs every research brings. Thanks to my best friends, the five of them, who kept me laughing, and helped me seeing everything in the right perspective. Thanks to my musical family, for keeping my brain at the right tempo. I also would like to thank Marco Favino for fighting back to back –literally– with me during these years, helping me when I felt stuck, blocked, or inconclusive in this work. Finally, I thank Gioacchino Noris, who was there during my bachelor and master years, and I had the luck to find again on my side, before the final effort which lead to this thesis. ix x Contents Contents xi List of Figures xv List of Tables xvii 1 Introduction: constitutive elements 1 1.1 Roughness and rough contact . .1 1.2 Framework of numerical experiments . .1 1.3 Brief introduction to friction . .2 1.3.1 Historical remarks and basic concepts . .3 1.3.2 The nature of friction . .5 1.3.3 Anomalous or not fully understood effects of roughness on friction6 1.4 Outline of the present work . .7 2 Fundamentals in elasticity and contact theory 9 2.1 Elasticity models . .9 2.1.1 Kinematics . 10 2.1.2 Balance laws . 12 2.2 Contact problems in elasticity . 16 2.2.1 Signorini problem . 17 2.3 Hertzian contact . 17 2.3.1 Geometry of contact region . 18 2.3.2 Point-load, contact area, and pressure distribution . 19 2.3.3 A remark on validity . 22 3 State-of-the-art models in rough contact theory 25 3.1 Nayak’s characterization of a random process . 25 3.1.1 Characterization of an isotropic random process . 26 3.1.2 Moments of the Power Spectral Density (PSD) . 27 3.1.3 Surface statistics . 28 3.2 Realistic self-affine rough surfaces . 36 xi xii Contents 3.2.1 Self-affinity and randomness of real surfaces . 37 3.2.2 Shape of the PSD . 38 3.2.3 Experimental determination of C(q) .................. 40 3.2.4 Limitations of the description . 42 3.3 Bush-Gibson-Thomas’s model . 42 3.3.1 Asperity cap modeling . 42 3.3.2 Probability distribution of asperity density . 43 3.3.3 Bearing area . 45 3.3.4 Precision of bearing area fraction Ae(t) ................ 47 3.3.5 Real area of contact through Hertzian contact . 47 3.4 Persson’s theory . 50 3.4.1 Constitutive elements . 51 3.4.2 Probability distributions P(σ, ζ) and P(ζ) .............. 51 3.4.3 Comparison with the BGT model . 57 3.4.4 Extension to kinetic friction computation for elastomers . 58 3.5 Yastrebov-Anciaux-Molinari’s theory . 58 3.5.1 Constitutive elements . 58 3.5.2 Derivation of the contact law . 59 3.5.3 Comparison with BGT and Persson’s models . 60 3.6 Theories of rough contact, concluding remarks . 61 4 Implementation 63 4.1 Rough surfaces generation . 63 4.1.1 Algorithm . 64 4.1.2 Parallelization in frequency and time . 66 4.1.3 Quality of the generated surfaces . 67 4.2 Iterative Signorini problem . 70 4.2.1 Generic algorithm . 70 4.2.2 Boundary conditions . 73 4.2.3 Distance function . 74 4.3 The Finite Element Method . 76 4.3.1 Galerkin approximation . 76 4.3.2 Finite element discretization . 77 4.3.3 Solution method and discretization . 78 4.3.4 Software performance . 80 5 Numerical studies of microscopic rough contact 85 5.1 Datasets . 85 5.1.1 Rough surfaces characterization . 86 5.1.2 Characterization of the elastic cube . 89 5.2 Normal load: validation against state-of-the-art models . 89 5.2.1 Normal load strategy . 90 xiii Contents 5.2.2 Summary of the results . 90 5.2.3 Contact evolution at low pressures . 95 5.2.4 Contact evolution at medium and large pressures . 99 5.2.5 Differences from other studies . 102 5.2.6 Periodic boundary conditions . 103 5.2.7 Conclusive remarks . 103 5.2.8 Plots of polynomial coefficients for different numerical settings . 104 5.3 Computation of elastostatic friction by means of shear tests .