2-d frictionless contact analysis of elastic continua with finite element method

Contact mechanics, as a subdiscipline of elasticity theory, studies the behaviors of the solid bodies which are interacted each other by contacting. So, contact mechanics takes a wide place in engineering disciplines, which are interested to solid bodies. In the field of the structural engineering, some of the contact mechanics applications are structure-soil interactions, bridge supports,structural connection designs and eartquake damping systems. By the nature of the problems, contact mechanics has complex boundary conditions, so analytic methods provide very small range of possibilities. Because of that, numerical methods has been frequently used to solve the problems. The finite element method is one of the featured contact mechanics solution methods. Nowadays, many programs like ANSYS and ABAQUS contains special finite element and solution algorithms for contact problems. While designing with these programs, it is valuable for engineers to have the knowledge of the theory behind in the aspect of interpreting the results. Pioneer analitic solution on contact mechanics was found by Hertz who also proved existence of elecromagnatic wave.After that many famous researchers have glorious work on this topic both analitic and numeric such as Galin, Oden, Wriggers, Bathe etc. In variational aspect first work that explains general boundary conditions on contact surface was done by Signorini. These boundary conditions had similar characteristics between Krush-Kuhn-Tucker conditions which are necessary conditions of general constrainted optimization problems. Fichera who was Signorini's student proved uniqueness of solution of frictionless contact problem by using virtual work principle and this study also was accepted birth of variational inequalities that is one of the main topic on applied mechanics and physics. First topic in this study is fundementals of elasticity, since it creates a foundation for finite element formulation. Differential equilibrium equations which are the most general motion equation in the static situations have been given. Basic relations which is between stains and displacement is introduced. After that, the relations between stress and strain have been revealed by using the generalized Hooke's Loaw in 3D for a lineer homogeneus body. Finally, the formulas have been established between surface tractions which are stress type boundary condition and stress tensor. In the accordiance of some assumptions, 3-dimensional elasticity problem can reduced to 2-dimension. With the scope of thesis, the assumptions of the plane stress and plane stress problem have been mentioned. By these assumptions, strains in the third dimension has been eliminated from the problem and stresses or strains in plane become main unknown of plane problems and other unknowns get zero or constant values. Then, the generalized constitutive equations and the differential equilibrium equations have been reduced to 2-dimension. With explaining differential equation and boundary condition, than virtual work principle is introduced. By multiplying the differential equilibrium equations by the virtual displacements which are assumed as boundary conditions already stisfied and taking the integral on the volume, the work expression of the virtual displacements on the body has been acquired. Stress boundary conditions have been included in the virtual work equation with the assist of divergence theorem. Last form of virtual work equation represent that the work is done by internal forces must be equal to the work is done by extarnal forces. Finally, a two-dimensional representation of virtual work expressions has been obtained for two-dimensional finite element analysis. The finite element formulation is started with the expression of the displacement on the element in terms of nodal displacements through the shape functions which are interpolation functions over domain of element. Because of nodal displacements are indepented of spatial coordinates, derivative of displacement is only related with shape functions. Stresses over element have been obtained by use of constitutive equations and the equations entegrated to the virtual work expression. The finite element equation system has been gained by using arbitrariness of the virtual displacements. Finally, element stiffness matrices and equivalent load vector have been generated in integral form. Isoparametric formulation has been introduced, which is frequently used for two and three-dimensional elements in the finite element method, which enables obtaining of the shape functions systematic way, which simplifies the integral bounds and the use of Gauss-Legendre integration method. Master element for the rectangular element has been represented and the shape functions of it has been acquired. Domain of the master element of rectangular element is square area bounded by $-1$ and $1$ both dimentions. Diffential relationships between global axis and master axis have been by Jacobien transformation. With explanation differential area in terms of master coordinates, stiffness matrix for the rectangular element has been generated in natural gauss numeric integration boundary. System matrix in finite element method is union of all element's stiffness matrix. The assemblage process of system matrix is addition of the stiffness terms of degree of freedom in elements which are connected to same node. This process is described on 2 elements and the pseudo code of this phase is given. Displacement terms in all system can be explained as unknown nodal displacements and known nodal displacements which are boundary conditions. In similar approach external load vector can be sperate two parts whics are known boundary forces and unknown support reactions. Giving primacy unknown degree of freedom in numbering process enable separete system matrix into two part which first part of system mathch unknown displacement against known boundary forces and second part of system accord solved displacement and known displacement to support reaction. By starting to Chapter 3, the basic concepts of the frictionless contact analysis is introduced. Kinematic relationships of two bodies, which are contacting each other, have been revealed and gap function has been mentioned. Gap function explained as distance between arbitrary point in slave surface and its perpendecular projection on master surface using by minimization of the distance phenomena. Then, boundary conditions of frictionless contact analysis have been defined and insertion of this boundary conditions to virtual work equations have been mentioned in integral form of multiplication of contact pressure and gap function. In numerical contact mechanics there are lots of contact enforcement method such as Lagrange Multipliers method, penalty method, augmented Lagrange Multipliers method, direct constraint elimination method etc. Because of Lagrange Multipliers method can be called as semi-analytical way in numeric method, it is choosen for to treatment of contact constraints. For the solution of boundary condition defined problem, Lagrange multiplier method has been introduced. Contact term, which is inserted to the virtual work equation, has been expressed in terms of Lagrange multipliers and variation expressions have been acquired. Finally, the effect of the Lagrange multipliers on the system matrices has been shown and solution algorithm in pseudo code has been given. After the inclusion of the boundary conditions and the virtual work expression into elasticity equations, element formulations about the contact analysis have been mentioned. Geometrical expressions, which are in the scope of this study, has been presented and stiffness matrices have been produced. An elasticity problem, as an example, has been solved by use of different contact elements and results have been compared to ANSYS. Results of the all contact element type are coincidence with Ansys results. Finally, the numerical applications of the formulated contact elements are made. The contact problem of a sylindrical object to a flat object which has analitical solution has been solved with the aid of symmetry that the problem has. Error between numeric solution and analitical solution is obtained as 4\%. Then, a pinned connection detail, which has a wide application in the steel structures, has been solved. The problem contains multibody contact and external loads are imposed system as displacement. Not only normal stress which is direction on load and shear stress but also bending stress are observed on the pin. In the third numerical application, the bending behaviour of two stacked simple beams, which are contacting each other at the begening, has been investigated. Vertical displacements are acquired as nearly half of analitical displacement of one beam resisting the same force. Then, the situation of initial gap between the beams has been handled and results has been shown.