A New Strategy for Treating Frictional Contact in Sheii Structures using Variational Inequalities Nagi El-Abbasi A thesis submitted in confomiity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mechanical and Industrial Engineering University of Toronto 0 Copyright by Nagi El-Abbasi 1999 National Library Bibliotheque nationale du Canada Acquisitions and Acquisitions et Bbliographic Services services bibliographiques 395 WeUington Street 395, nie Wellington OttawaON K1AON4 OttawaON K1A ON4 Canada canada The author has granted a non- L'auteur a accordé une licence non exclusive licence dowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or seil reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/fiim, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celleci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. A New Strategy for Treating Frictional Contact in Shell Structures using Variational Inequalities Nagi Hosni El-Abbasi, Ph.D., 1999 Graduate Department of Mechanical and Industrial Engineering University of Toronto Abstract Contact plays a fundamental role in the deformation behaviour of shell structures. Despite their importance, however, contact effects are usually ignored andor oversimplified in finite element modelling. Existing solution techniques for frictional contact problems involving shell structures suffer from two main deficiencies. Firstly, commonly used shell elements involve basic assumptions, which are not appropriate for contact problems, since they do not: (i) account for variations of displacements and stresses in the transverse direction, and (ii) ailow for double-sided contact. The second deficiency is in the modelling of contact. To the author's knowledge none of the existing techniques are based on the more accurate and mathematically consistent variational inequalities formulation. Typically, the variational formulations are used which employ contact elements. These contact elements are dependent on user-defined parameters that affect the accuracy of solution. In view of the above, three aspects of the problem are accordingly examined. The fint is concerned with the development of a reliable thick shell element, which accounts for the thickness change, the normal stress and strain thiaugh the thickness and accommodates double-sided contact. An assumed naturd strain formulation is used to avoid shear locking, and a new director interpolation scheme is utilised to prevent thickness locking. Large deformations and rotations are accounted for by invoking the appropnate objective stress and strain measures. The second aspect of the work is concemed with the development of variational inequalities formulations for large deformation analysis of fnctional contact in shell structures. The kinematic contact conditions are expressed in terms of the physical contacting surfaces of the shell. Lagrange multipliers are used to ensure that the constraints are accurately satisfied and that the solution is free from user defined parameters. Finally, the numerical predictions are verified experimentally, compared with commercial finite element codes, and with theoretical solutions where available. A number of case studies involving contact, fiction, large deformations and double-sided contact are also exarnined. The results reveal that the new higher order shell element is superior to classical shell elements for thick shell applications, and maintains its high level of accuracy in thin shell problems. Furthemore, the new frictional contact formulation is more accurate than traditional variational techniques. Acknowledgements 1 offer my sincere gratitude to Dr. S.A. Meguid for his expert advice, technical guidance, and his commitment and suppon throughout the course of my research. 1 aiso wish to thank the members of the Engineering Mechanics and Design Laboratory; specifically, Mr. A. Czekanski, Dr. M. Refaat, Mr. J.C. Stranart and Dr. G. Shagal. The financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the Aluminum Company of Amenca (ALCOA) and the University to Toronto is gratefully ac know ledged. Contents Abstract .................................................................... i Acknowledgements ........................................................... üi Contents ................................................................... iv List of Figures .............................................................. viu... List of Tables ................................................................mi .. Notation .................................................................... mi... 1 Introduction and Justification ................................................ 1 1.1 Contact in Shell Structures ............................................. 1 1.2 Justification of the Study ..............................................4 1.3 Aims of the Study ..................................................... 4 1.4 Method of Approach .................................................. 5 1.5 Layout of Thesis ...................................................... 5 2 Literahire Review ........................................................... 8 2.1 Modelling of Shell Structures .......................................... 8 2.1.1 Kirchhoff-Love Type Shell Elements ................................9 2.1.2 Shear-Deformable Shell Elements ................................... 9 2.1.3 Higher Order Thick S hell Elements ................................. 12 2.1.4 Patch Tests ...................................................... 12 2.2 Limitations of Existing Shell Models ................................... 13 2.3 Classical Theories of Contact. ......................................... 14 2.3.1 Hertz Theory of Contact ........................................... 14 2.3.2 Non-Hertzian Contact ............................................. 15 2.4 Techniques Adopted in Modelling Frictional Contact ..................... 15 2.4.1 Variational Approach ............................................. 15 2.4.2 Solution Techniques .............................................. 16 2.4.3 Contact Elemcnts ................................................. 17 2.5 Variational Inequalities Approach ...................................... 18 2.6 Contact in Shell Structures ............................................ 19 2.7 Large Deformation Elastic Analysis .................................... 21 2.7.1 Finite Rotations ..................................................23 3 Development of a New Thick Shell Element .................................. 24 3.1 Existing Thick Shell Elements ......................................... 24 3.2 New Continuum Based Shell Mode1 ................................... -25 3.3 Four-noded Shell Element ............................................ 29 3.4 Thickness Locking ................................................... 30 3.5 Discretization of Shell Element ........................................ 31 3.6 Variational Formulation ............................................. -33 3.6.1 Consistent Loading ............................................... 35 3.7 Numerical Examples ................................................. 37 3.7.1 Patch Tests ..................................................... -38 3.7.2 Flat Cantilever Bearn ............................................. 38 3.7.3 Curved Cantilever Beam ......................................... -40 3.7.4 Pinched Hemisphere .............................................. 42 3.7.5 Pinched Cylinder ................................................. 44 3.7.6 Clamped-Clamped Thick Bearn .................................... 46 3.7.7 Sphencal Shell Under Pressure ..................................... 47 4 Anaiysis of Large Deformation Frictionai Contact in Sheiis using Variational Inequalities ............................................................... 50 4.1 Kinematic Contact Conditions ......................................... 50 4.2 Variational Inequalities for Continuum ................................. 53 4.3 Reduced Variational Inequality ........................................ 54 4.4 Variational Inequdities for Shell Stmctures ............................ -55 4.5 Solution Technique .................................................. 56 4.6 Discretization ....................................................... 57 4.6.1 Contact Constraints .............................................. -57 4.6.2 Friction Terms ................................................... 59 4.6.3 Finite Element Solution ...........................................59 4.7 Verifkation Exarnples ................................................ 60 4.7.1 Three Beam Contact .............................................. 61 4.7.2 Ring Compression ................................................61 4.7.3 Strip Friction Test ................................................ 67 4.7.4 Belt-Pulley Assembly ............................................ -68 4.7.5 Strip Compression Test ..........................................