Litewka Finite Element Analysis of Beam-to-Beam Contact

1. Auflage 2010
ISBN: 978-3-642-12940-7
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 175 Seiten

Phenomena occurring during a contact of two bodies are encountered in everyday life. In reality almost every type of motion is related to frictional contact between a moving body and a ground. Moreover, modeling of simple and more complex processes as nailing, cutting, vacuum pressing, movement of machines and their elements, rolling or, finally, a numerical simulation of car crash tests, requires taking contact into account. Therefore, its analysis has been a subject of many research efforts for a long time now. However, it is author's opinion that there are relatively few efforts related to contact between structural elements, like beams, plates or shells. The purpose of this work is to fill this gap. It concerns the beam-to-beam contact as a specific case of the 3D solids contact. A numerical formulation of frictional contact for beams with two shapes of cross-section is derived. Further, a couple of effective methods for modeling of smooth curves representing beam axes are presented. A part of the book is also devoted to analyze some aspects of thermo-electro-mechanical coupling in contact of thermal and electric conductors. Analyses in every chapter are illustrated with numerical examples showing the performance of derived contact finite elements.

Autoren/Hrsg.

Litewka, Przemyslaw

Inhaltsverzeichnis

1;Title Page;2
2;Preface;7
3;Contents;9
4;Introduction;12
4.1;From the Ancient Egypt to the Computer Era;12
4.2;Frictionless Contact between Solids;14
4.3;Methods of Introduction of Contact Constraints;16
4.4;The Finite Element Method in Contact Analysis;18
4.5;Friction Constraints;20
5;Frictionless Beam-to-Beam Contact;23
5.1;Assumptions;23
5.2;Penetration Function;25
5.3;Contact Search;29
5.4;Weak Form and Kinematic Variables for Contact;34
5.5;Discretisation of Kinematic Variables;37
5.6;Residual Vector and Tangent Stiffness Matrix;39
5.7;Numerical Examples;41
5.7.1;Introduction;41
5.7.2;Example 1;42
5.7.3;Example 2;44
5.7.4;Example 3;45
6;Friction in Beam-to-Beam Contact;48
6.1;Friction Model;48
6.2;Kinematic Variables for Friction;52
6.3;Weak Form Components due to Friction;56
6.4;Discretization of Friction Terms Present in Weak Form;58
6.5;Residual Vector and Tangent Stiffness Matrix for Friction;61
6.6;Numerical Examples;66
6.6.1;Introduction;66
6.6.2;Example 1;67
6.6.3;Example 2;70
6.6.4;Example 3;72
6.6.5;Example 4;75
6.6.6;Example 5;77
7;Contact between Smoothed Beams;79
7.1;General Remarks on Smoothing of Contact Facets;79
7.2;Smoothing of 3D Curves;80
7.2.1;General Remarks;80
7.2.2;Inscribed Curve Algorithm;80
7.2.3;Node-Preserving Algorithm;84
7.3;Discretisation and Smooth Beam Contact Finite Elements;87
7.3.1;Inscribed Curve Elements;87
7.3.2;Node-Preserving Elements;89
7.4;Numerical Examples;91
7.4.1;Introduction;91
7.4.2;Example 1;92
7.4.3;Example 2;94
7.4.4;Example 3;96
7.4.5;Example 4;99
7.4.6;Example 5;101
7.4.7;Example 6;103
8;Electric Contact;106
8.1;Introduction;106
8.2;Electro-mechanical Variables for Contact;106
8.3;Weak Formulation of Electro-mechanical Contact;110
8.4;Beam Finite Element for the Electric Current Flow;112
8.5;Discretisation and Electro-mechanical Contact Finite Element;113
8.6;Numerical Examples;116
8.6.1;Introduction;116
8.6.2;Example 1;117
8.6.3;Example 2;118
8.6.4;Example 3;120
8.6.5;Example 4;123
8.6.6;Example 5;124
9;Thermo-mechanical Coupling;127
9.1;Introduction;127
9.2;Thermo-mechanical Beam Finite Element;128
9.3;Variables for Thermo-mechanical Contact;131
9.4;Weak Form for Thermo-mechanical Contact;132
9.5;Discretisation and Thermo-mechanical Contact Beam Finite Element;133
9.6;Numerical Examples;137
9.6.1;Introduction;137
9.6.2;Example 1;137
9.6.3;Example 2;140
10;Summary and Outlook;141
11;Appendix 1 Matrices D and E for Beams with Rectangular Cross-Sections;143
11.1;Components of Matrix D;143
11.2;Components of Matrix E;146
12;Appendix 2 Derivation of Variables $\Delta\delta\xi_{mn}$ and $\Delta\delta\xi_{sn}$;147
13;Appendix 3 Matrices G, H and M in Smoothing Procedures;152
13.1;Components of Matrix G;152
13.2;Components of Matrix H;155
13.3;Components of Matrix M;157
14;Bibliography;159

Leseproben

"Chapter 6 Thermo-mechanical Coupling (p. 121-122)

6.1 Introduction

Analysis of contact with inclusion of coupling between mechanical and thermal fields is a complicated problem because the mutual influences between displacements or strains and temperature are manifested in many different ways. The aspects involved include: heat flow through a real contact area resulting from roughness of contacting surfaces, heat flow through a gas between the bodies, heat flow through radiation; frictional heating; dependence of material properties like elasticity moduli, friction coefficient or heat conduction coefficient on temperature, etc.

The more detailed description of various issues related to the thermo-mechanical coupling in contact can be found in the monographs by Wriggers (2002) and Laursen (2002). One can find there numerous references to the papers and other monographs devoted to the problem of the heat conduction in contact. This phenomenon requires a precise description of geometry of bodies surfaces in the micro scale and a development of a thermo-mechanical physical law for the contact.

To this end statistical methods can be used (Cooper et al. 1969 and Song and Yovanowic 1987). The numerical solution to this problem was a subject of the papers by Zavarise (1991), Zavarise et al. (1992) as well as Wriggers and Zavarise (1993b). Another problem is related to the frictional heating due to the contact. This topic was considered, for instance, in the papers by Wriggers and Miehe (1992) or by Zavarise et al. (1995, 2005).

Numerical treatment of the thermo-mechanical problem generally depends on the type of the heat flow – steady state, independent of time, or transient one with a variation in time. In the former case the problem is relatively simple and monolithic methods can be effectively used, where both types of unknowns, displacements and temperature, are calculated simultaneously. In the more complicated transient state case, staggered methods are preferred, where the problem is solved iteratively with temperature kept constant and solving for displacements in one iteration and vice versa in the subsequent one.

To these methods with respect to the thermo-mechanical contact the papers by Wriggers and Miehe (1992) and by Agelet de Saracibar (1998) were devoted. In this chapter a formulation of the beam-to-beam contact with a thermomechanical coupling in a limited form (Boso et al. 2006) is presented. From the previously mentioned issues of the coupling the heat flow through contact with an assumption of ideally smooth surfaces is taken into account.

The heat transfer through the gas and the radiation are neglected. In the physical model of the beams material only the linear thermal expansion is included. All the physical parameters are treated as independent of time. The analysis presented herein can therefore be considered only as an introduction to a very complicated problem of contact with the coupled fields of displacements and temperature. It can be expanded further, if one takes into account the electric contact discussed in Chapter 5, too. In such a case some additional manifestations of the coupling emerge. They include heat production due to the electric current flow and dependence of electric material and contact properties on temperature."