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• Computers & Svucrvrrs Vol. 27. No. 4. PP. 4E-466. 1987 0045.7949187 13.00 + 0.00 Printed in Great Britain. Pcrgamon Journals Ltd.

FINITE ELEMENT SOLUTIONS OF TWO-DIMENSIONAL CONTACT PROBLEMS BASED ON A CONSISTENT

MIXED FORMULATION

T. Y. CHANG, A. F. SALEEB and S. C. SHYU

Department of Civil Engineering, University of Akron, Akron, OH 44325, U.S.A.

Abstract-A consistent mixed finite element method for solving two-dimensional contact problems is presented. Derivations of stiffness equations for contact elements are made from a perturbed Lagrangian variational principle. For a contact element, both the displacement and pressure fields are independently assumed. In order to achieve a consisent formulation, thus avoiding any numerical instability, the pressure function is assumed in such a way that all non-contact modes in deformations must be excluded. Stiffness equations for four-noded and six-noded contact elements are given. Four numerical examples are included lo demonstrate the methodology.

1. INTRODUCTION Conditions involving contact between two deform- able solids, or between a solid and another medium, constitute an important class of problems in engineer- ing applications. For example, contact of a standing or rolling pneumatic tire with the ground, design of mechanical components such as gears and bearings, and modeling of interfacial effect in metal extrusions, etc. are typical problems of practical interest. The object of a contact analysis is to determine the contact area and pressure distribution at the interface between two bodies. This class of problems is gener- ally known to be strongly non-linear, and hence difficult to analyze since the contact areas are not known a priori and they are changing in size and shape as the load is being applied.

In the early years, the analysis of stresses and deformations developed during the contact of two or more elastic bodies was obtained via the classical elasticity method (l-31. Solutions of such problems are restricted to bodies of simple geometries, e.g. cylinders or spheres. With the advent of finite element method, the ability to obtain engineering solutions for contact problems has obviously been widened. Using constant strain elements, some frictionless contact problems based on the potential energy the- orem were solved by an indirect approach  in which the contact conditions were treated as subsidiary equa- tions. Extension of this method to contact problems with friction was given in [5l in that an irreversible stick-slip process was modeled. Application of a similar technique to study the effects of clearance, load and friction on turbine blade fastenings can be found in [6,7J. An incremental solution procedure for solving contact problems with various friction con- ditions was presented in . Another noteworthy approach is the formulation of flexibility matrices for elements along the contact boundary [91.

Although the finite element method can be applied to solving contact problems in many different ways, three major approaches may be identified on the basis of variational formulations. These are: (i) Lagrange multiplier method; (ii) penalty method; and (iii) mixed (or hybrid) method. These three approaches, though different in concept, are closely related. Some discussions on the merits and drawbacks of these approaches are given as follows.

In the Lagrange multiplier method, the constrained conditions for a contact problem are satisfied by introducing Lagrange parameters in the variational statement [IO-151. In this approach, both the nodal displacements and Lagrange parameters are treated as indpendent variables. Applications of the Lagrange multiplier method are many. For example, a numer- ical procedure to study the impact-contact process of axisymmetric elastic bodies with geometric non- linearity was given in [l 11. Extension of the method to the contact analysis of thin elastic shells was shown in . A solution procedure for the analysis of planar and axisymmetric contact problems was presented in , in which the total potential of the contact forces was included in the variational equa- tion to enforce the contact condition. A two-level contact algorithm was employed in , wherein a linear solution was obtained first, then the nonlinear solution was found by a Newton iteration procedure. In [l5], the Lagrange multiplier method was applied to elastodynamic contact problems with friction. In all the above work, through the use of Lagrange multipliers, corresponding contact elements can be derived in a rather straightforward manner. One major drawback of this approach, however, is that the size of the resulting system of equations will increase due to the use of independent Lagrange variables. Moreover, the associated stiffness matrix is indefinite and contains zero diagonal terms which

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• require special handling procedure in the solution of lincdr equations.

In the penalty method, the contact pressure is assumed to be proportional to the amount of pene- trations by introducing a pointwise penalty param- eter. With this assumption, one can obtain the final stiffness equations which do not contain additional variables as the Lagrange multiplcr method. Penalty formulation of contact problems with Coulombs friction was previously introduced in [ 16) in the form of variational inequalities. Mathematical formulation of the penalty method for contact problems, discus- sion of existence and uniqueness of solutions, when Coulomb friction law is utilized, also have been given, e.g. in [l7-211. Numerical applications of this method can be found in [22-251. Note, however, that in the penalty method, the contact presssure is as- sumed to be proportional to the point-wise pene- trations. That is, the approximating functions for the pressure in a contact element are of the same order as the displacement functions. This in turn may lead to violation of the well-known local Babuska-Breui stability condition . Hence, the corresponding pressure distribution in the contact region will gener- ally be oscillatory. In order to control such a numer- ical instability phenomenon, a reduced integration scheme must be utilized[27-291. Aside from the aforementioned inconsistency, one drawback of this method is that the accuracy of the approximate solution depends strongly on the penalty number employed.

As an alternative, a mixed (or hybrid) finite element method may be employed for solving contact problems. Application of this method to contact problems is not new. For instance, a two-dimensional contact element was given in (301, in which both the displacement and stress fields were independently approximated. In (31,321, a hybrid contact element was derived from a modified complementary energy principle by introducing the continuity on the contact surface as a constraint condition and treating the contact forces as additional field variables. More recently, a mixed method based on a perturbed Lagrangian formulation was presented in , in which piece-wise constant pressure was assumed for a linear displacement contact element. However, pres- sure distributions for a more general class of elements was not mentioned. It is also noted that the key to the success of a mixed finite element formulation for contact analysis is the appropriate choice of pressure functions in conjuction with the assumed displace- ment polynomials. Unfortunately, no guidelines can be found in the literature for making such a choice.

In this paper, a consistent mixed formulation of two-dimensional contact elements is presented. The formulation is based on a perturbed Lagrangian variational principle in which both the displacement and pressure functions in a contact element are independently assumed. In our approximations, the displacements are assumed to be C-continuous

whereas the pressure is C--continuous. By imposing the stationarity condition of the potential functional with respect to the presssure parameters, it is possible to eliminate the pressure variables on the element level. Hence, the resulting element stiffness equations involve only the nodal displacements as global unknowns. More importantly, specific guidelines are given on the proper choice of pressure distributions so that both numerical instability and element over- constrained response can be avoided. For the sake of clarity, this paper is limited to frictionless contact of linearly elastic bodies with small deformations. Extension of the present work to a more general class of problems, e.g. by including the effects of frictions and large deformations, should be apparent. Four numerical examples are included to illustrate the application of the proposed mixed elements.

2. FINITE ELEMENT FORMlJLATlONS

This paper is limited to frictionless contact of elastic bodies with small deformations. For the solu- tion of nonlinear finite element equations, the con- ventional incremental loading procedure is employed.

In this section, the kinematic relations to describe two bodies in contact and an incremental form of the variational equation are given. Additionally, the finite element equations for derivation of mixed contact elements are briefly outlined.

2.1. Kinematics

Consider two bodies, designated as A and B (shown in Fig. l), each of which occupies a domain

Fig. I. Contact of two bodies.

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