ACTA MECHANICA SINICA, Vol.19, No.6, December 2003 The Chinese Society of Theoretical and Applied Mechanics Chinese Journal of Mechanics Press, Beijing, China Allerton Press, INC., New York, U.S.A.
ISSN 0567-7718
A FINITE ELEMENT MODEL FOR NUMERICAL SIMULATION OF THERMO-MECHANICAL FRICTIONAL CONTACT PROBLEMS* ZHANG Hongwu ( ~ ) t
HAN Wei ($~ '~)
CHEN Jintao ( N i , . ~ )
DUAN Qinglin ( ~ N ~ )
(State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China) A B S T R A C T : Two kinds of variational principles for numerical simulation of heat transfer and contact analyses are respectively presented. A finite element model for numerical simulation of the thermal contact problems is developed with a pressure dependent heat transfer constitutive model across the contact surface. The numerical algorithm for the finite element analysis of the thermomechanical contact problems is thus developed. Numerical examples are computed and the results demonstrate the validity of the model and algorithm developed. KEY
WORDS:
heat transfer, contact analysis, programming
1 INTRODUCTION Among the different schemes, the iteration method and the mathematical programming method are the most commonly used methods in the finite element method for the thermo-mechanical frictional contact analysis. As compared to the great deal of interest shown in the mechanical contact, little progress is made on the thermo-mechanical frictional contact problems due to the difficulty of the problems. Barber introduced a pressure dependent contact conductance across the contact surface, and illustrated the existence of a solution, which is unique for a sufficiently small contact conductance [x]. Curnier and Taylor presented a finite element model which includes a thermo-mechanical coupling for lubricated contact problems [2]. Johansson and Klarbring presented a model combined with the conventional equations of linear thermoelasticity for the bulk materim of the contacting bodies [3]. Wriggers and Miehe carried out a study for thermoelastic contact problems with the finite element method, and a finite element model for the large deformation process was developed [41. Recently, Pantuso et al. presented a fi-
method, the finite element method
nite element procedure for the analysis of the fully coupled thermo-elasto-plastic response of solids including contact conditions[q. The constraint function method was employed to impose the contact conditions at the Gauss points of the contact surface. A heat transfer equation in the gap region between the contact interfaces of the two contact bodies is introduced in this paper to simulate the heat transfer in the contact gap where the materials are generally air and moisture. The parametric variational principle developed and improved by Zhong et al. and Zhang et al. [6,7] is used for the numerical simulation of the frictional contact problems. To solve the coupled problems, a combined parametric quadratic programming and iteration algorithm is introduced so that the heat transfer and mechanical contact in the thermomechanical frictional contact problem can be solved simultaneously.
2 GOVERNING
EQUATIONS
Consider a pair of two-dimensional bodies /21 and /22(/2 = ~21 + /22) in the thermo-mechanical frictional contact prior to contact. The mechanical
Received 25 June 2002, revised 6 May 2003 * The project supported by the National Key Basic Research Special Foundation (G1999032805), the National Natural Science Foundation of China (50178016, 10225212) and the Foundation for University Key Teacher by the Ministry of Education of China t E-mail:
[email protected]
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boundary S (~) of the bodies consists of disjoint parts Sp(~), S(~~) and S~(~), where the superscript a = 1, 2 denote the two bodies. S (~) indicates force boundary, S (~) the given displacement b o u n d a r y and Sr(~) the possible contact boundary. For the heat transfer analysis, Fi(~) (a = 1, 2, i = 1, 2, 3) indicate three kinds of thermal boundary conditions. Symbol 6* means the initial gap between the two contact bodies, pc = { p t , p n } T is the force vector acting on the contact surface S! ~) . The contact states are classified according to three categories, which are defined by inequalities in the contact local co-ordinate system t-n. du~ ~), du~~) , Pt and p~ are tangential and normal incremental displacements and forces at contact point with respect to the local co-ordinate system, respectively, it is the frictional coefficient. The Coulomb's frictional law is used. We can decompose the contact relative displacement vector into two parts duc = du~ + du~
(1)
where du~ and d u p are the "elastic" and "plastic (slip or separation)" relative displacement vectors, respectively. The relationship between the contact relative displacement vector and the frictional force can be expressed as dpr = D r
= D e ( d u e - du p)
P~ + itP~ < 0
]'2 = - P ~ + itP~ 0, X2 = 0 or X1 = 0, X2 _> 0) and normal directions (X3 -> 0), respectively. From Eqs.(2), (3) and (5) and by introducing the slack variables vk, we obtain the following complementary equations fo + wkduc - mkx
Xkvk = 0
+ vk = 0
Xk,vk _> 0
k = 1,2,3
(4)
g3 -- P~ + c3 Then the contact plastic (slip or separation) incremental relative displacement can be obtained as
(6)
where f~ is the initial value of the yield function at time to and
For the heat transfer analysis, the boundary and initial thermal conditions can be expressed by the following equations T ( x , y, t) = T ( x , y)
~,y 9 r~")
(7)
(~J,j)~
~ , y 9 r2(~)
(s)
(lijTj)ni = -a(T - To~)
~, y 9 r~(~)
(9)
T ( x , y , to) = T o ( x , y )
x , y 9 S?,t = to
= q
(10)
(2)
where D~ is contact penalty matrix. In order to model the frictional phenomenon in thermo-mechanical structures, we employ a plasticitylike dissipative functional approach. To this end, the following slip (yield) functions are used f l =
2003
where T is the temperature, A~i the coefficient of thermal conductivity, p the mass density of the materials, t the time, ni the unit normal vector, q the given heat flux on the boundary, T ~ the given ambient temperature and c~ the given convection coefficient on the boundary. There are still thermal conditions on the thermal contact boundaries. Two kinds of heat transfer roodels across the contact surfaces are considered. First it is assumed t h a t the heat transfer follows the ideal thermal boundary concept, i.e. there is not thermal resistance when the bodies are in sticking or sliding contact states. Thus we have
T2) (x, y, t) =
