A method was developed for contact problems which may be either frictional or frictionless and may involve extensive sliding between deformable bodies. It was based on an assumed stress hybrid approach and on an incremental variational principle for which the Euler's equations of the functional include the equilibrium and compatibility conditions at the contact surface. The tractions at an assumed contact surface were introduced as Lagrangian multipliers in the formulation. It was concluded from the results of several example solutions that the extensive sliding contact between deformable bodies can be solved by the present method

Kubomura, K.

Pian, T. H. H.

English

NASA Technical Reports Server

SOLUTIONS OF CONTACT PROBLEMS BY THE ASSUMED STRESS HYBRID MODEL 
Kenji Kubomura 
System Development Corporation 
Theodore H .H . Pian 
Department of Aeronautics and Astronautics 
Massachusetts Institute of Technology 
SUMMARY 
A method has been developed for contact problems which may be  either  frictional or 
frictionless and may involve extensive sliding between deformable bodies. It i s  based on an 
assumed stress hybrid approach and on an  incremental variational  principle for  which  the 
Euler's equations of the functional  include  the  equilibrium and compatibility conditions at 
the contact surface. The tractions at an assumed contact surface are introduced as Lagrang- 
ian  multipliers i n  the formulation. It has been concluded from the results of several example 
solutions that  the  extensive sliding  contact between deformable bodies can be solved by  the 
present  method. 
INTRODUCTION 
The finite element method has been applied by many authors for solving  solid mechan- 
ics problems which involve undetermined contact surfaces. They include the relatively 
simple Hertz  contact problem for which there i s  no sliding between  contact surfaces and the 
small displacement model can be used [ l ,  21. They also include problems involving relative 
sliding  either  with  friction or i n  frictionless conditions [31. Existing solutions are based 
largely  on the  conventional assumed displacement finite element model. 
The present paper i s  based on an assumed stress approach and  on  an  incremental vari- 
ational  principle for which  the Euler's equations of the functional  include  the  equilibrium 
and compatibility conditions at the contact surface. An assumed contact surface is  inserted 
between bodies in  contact and i s  divided  into elements. Contact tractions are independently 
assumed i n  terms of unknown values of such nodes of the contact elements. Thus, a finite 
element equation  includes  nodal displacements and nodal contact  tractions as unknown. 
This paper i s  to present the  variational  principle and the corresponding finite element imple- 
mentation  for this problem. 
21 1 
https://ntrs.nasa.gov/search.jsp?R=19800024263 2020-03-21T15:57:04+00:00Z
SYMBOLS 
Coordinates before  deformation 
Displacements defined  within  an  element 
Displacements defined  along element boundaries 
Tractions 
Contact  tractions 
Coefficient  of  friction 
The whole boundary of the  body 
The contact surface 
The portion  of  the boundary  where  displacements are  prescribed 
The portion  of  the boundary  where  loads are  applied. 
Volume 
Strains 
Stresses 
Quantity  tangential to the  contact surface 
Quantity normal to the contact surface 
Matrix 
Tensor 
Prescribed quantity 
Quantity  pertinent to  body A 
Quantity  pertinent to body B 
Quantity  pertinent  to bodies A and B 
Quantity  pertinent  to element N 
incremental auantitv 
21 2 
I 
GENERAL INCREMENTAL ASSUMED STRESS FORMULATION FOR CONTACT PROBLEM 
The requirements for contact (the conditions  of  contact)  are as follows: 
( 1 )  At the point of contact between two bodies tractions exerted 
on each other are  the same in  magnitude and are opposite i n  
directions. 
(2) The normal tractions are compressive and the tangential 
tractions  counteract relative movement of the bodies. 
(3) There should be no gap and no penetration of material points 
at  the  place  of  contact. 
A B 
Consider two bodies A and B shown in  Fig. 1 with volumes V and V , and boundary 
B A B 
surfaces, a@ and 2V which are composed of portions, S,,- and Sf, and SA and S 
These two bodies share a contact surface, S through which they interact. The previously 
B 
U U '  
C 
mentioned conditions of  contact i n  incremental form are 
+  AT^) + (TB + AT B ) = 0 
n  n 
(TB  +  AT^ ) B 
S G -  + ( T ~  n +  AT^) n 
21 3 
and the signs in  Eqs. 3 and 4 are chosen such that  tangential  tractions on this surface act  to 
restrain  the relative movement of contacting points. 
A f inite element method which i s  based on  a variational  principle  with  relaxed con- 
tinuity requirement at interelement boundaries i s  defined as a hybrid model [4]. Boland and 
Pian [5] have applied  an incremental assumed stress hybrid method for  large  deflection 
analyses of thin elastic structures. The functional IT that has been derived in the refer- 
ence [5]i based on the Updated Lagrangian coordinate system i s  used as ;base for   der iv ing 
the functional for the present problem. 
me 
The conditions of no gap and no overlapping on t,he plac%of  contact are $traduced 
into the functional by means of Lagrangian multipliers, T + A T  and 7 + A T  . The 
functional K becomes Pmc, i.e. n  n S S 
mc 
AT. ( AU. - AU. dS -A -40 
I I 
A +B 
s C N  s C N  
21 4 
The Euler's equations of T are C 
rnc 
On the contact surface S , in addition to Eqs. 5 and 6 
cN 
(TB n + ATB) n + (7 n + AT n ) = o  
Introducing frictional constraint on 7 + A? and 'i +z such that 
S S n n' 
cu 
T + A T  < + , , T  + A T  
S S - [n  -n) 
and rearranging Eqs. 11, 12,  13, 14 and 15, results i n  Eqs. 1, 2, 3, and 4. Thus, 
it has been proved that the conditions of contact are Eulers's  equations of the functional 7i' C 
21 5 
These Euler  equations are the strain displacement relations, the mechanical boundary 
condition, the stress equilibrium requirements along  the  interelement boundaries  and the 
conditions of contact. Since they are only satisfied in an average sense within an increment 
they cannot be expected to satisfy these conditions i n  the usual sense. It is, therefore, 
necessary to consider a compatibility check, a stress equilibrium check,  and a  contact check. 
It i s  seen that Eq. 7 already has al l  of these built-in checks. The 5 t h  and 6-th 
integral terms in the  functional correspond to the equilibrium check and 7-th term to the 
compatibility check. Also, the compatibility and equilibrium checks of the contact surface 
(contact check) are easily identified, the equilibrium check being  the 8-th integral term 
and the compatibility check, the 9-th term. 
FINITE ELEMENT IMPLEMENTATION 
Since the aim of the present  work i s  to solve contact problems by the finite element 
method, expressions arising from nonlinearities,  not due to contact, are excluded from 
equations. A technique for solving these equations, with only contact nonlinearities, wi l l  
be discussed here. 
Neglecting  nonlinearities not due to contact, the assumed stress hybrid  functional, 
77 takes the form 
mc 
3 
cN 
The stresses AS are expressed in  terms of  a  finite number of stress parameters, A P  and 
the element boundary displacements interpolated in terms of the nodal displacements, CJ, 
and AZ. Also, coordinates are interpolated in terms of their nodal values. As a solid 
continuum i s  subdivided into elements, the  contact surface is  also discretized into  finite 
number of elements referred to here as ''contact elements" with "contact nodes. 'I The con- 
tact traction J + AI i s  interpolated in terms of i t s  nodal values, t. 
Y 
+ 
N 
Thus, interpolatbns  otthem are: A z  = P a ~ 2 ,  A: = 5 A:, lJ = q, 
A = ACJ and 1 + A$ = M i. Substituting these interpolations into Eq. lz, 
H 
v 
Y 
and defining the following matrices 
R0 L T -  T d S ,  R: 
N N = \ s h j T * [ ( g A  + ) -  (GB + gB)]dS 
A 
-E N IC. 
c N  
FA = \ M - L  d S ,  F L  = M - L B d S  
4N " - *  
A 
C C 
JcN JcN 
the functional, becomes 
mcN 
21 7 
in  which S i  and 5; result from  stress equilibrium checks on S ,  and on the contact 
surface S , and RC , from the init ial mjsmatch  checks on S . N 
n 
0 
cN N cN 
Equation 17 contains three unknown vectors, the incremental stress parameters AZ, the 
incremental displacements AI, and the contact stresses . Whereas, the vector are 
independent on the element level, a q  and are not. Thus, eliminating AZ from Eq. 17, 
n' becomes w 
mc 
Summing  up over a l l  elements and !aking the  va r ia t i ons  of t h e  f u n c t i o n a l  w i t h  
respec t   to  a q  and t ,  resulk in 
' W  r u  
where 
K hr = GjN* T tiN - 1  - S N ,  Q = x(,aN + Q i  - Ro ) u -E N * N  N 
&C 
K =  
Equation 19 represents the total assembled finite element matrix equation. 
21 8 
I 
TECHNIQUE OF ITERATIVE SOLUTION 
Once  the  contacting bodies are  adequately constrained such that  the inverse of the 
global stiffness matrix, 5-1 i n  Eq. 19 can be calculated, it can be used throughout the 
iteration procedure. Thus, i n  locating the contact surface only K needs to be recomputed 
i n  each  iteration. The global stiffness matrix 5 remains constant % uring this process.  Even 
in  the case of  material and/or large deflection nonlinearities, it i s  possible to use a modified 
Newton-Raphson method; hence the  global stiffness matrix 51 may remain constant during 
this process. 
For a two-dimensional problem, before each iteration,  the  contact surface i s  a  line, 
f ixed  in the coordinate system, but  not  to the contacting bodies. Such line i s  assumed 
known i n  order to perform the necessary integrals. But if, before the iteration, one has 
assumed the location  of  the  contact surface and the positions on i t  that  the  contacting nodes 
of  the bodies wi l l  occupy, then it would appear that  the  displacement  increment can be 
specified. Th is  i s  not the case. Instead, the problem i s  solved for the displacement incre- 
ments and i f  the contact surface found therefrom i s  not  coincident  with  the one presupposed, 
a new contact surface location i s  calculated, and then  an iteration can be  followed. 
It has been found that the length  of the  contact  element  which  yields best results i s  
the same as the  length of the contacting side of boundary elements of the bodies. In order 
to facilitate programming, the nodes of the contact  element  are chosen to be coincident 
with those of one of  the  contacting bodies. As a  result  of  variations i n  load, the place  of 
contact changes: thus a vital part  of the  solution i s  to establish a procedure for calculating 
this change. A tr ial and error scheme i s  employed because i t  i s  virtually impossible to form- 
ulate a variationalprincipleincluding unknown variables  which locate  the surface of contact. 
The overall strategy for solving the contact problem i s  discussed here. First, an incre- 
ment in  the external load or prescribed displacement i s  applied. Second, a contact surface 
i s  assumed together with  the points on it through which nodes of the bodies contact  each 
other. Also, the types of  contact  (sliding or non-sliding) at each of the above-mentioned 
points areassumed .For the initial  calculation  of the first load increment, the above assump- 
tions  are made simply by  inspection,  and  for  the first iteration  after each new load increment, 
the converged solution of the previous load step i s  used. Third, a l l  the necessary matrices 
are calculated and assembled. At the i-th iteration of the N-th load step, incremental 
displacement, AU and contact  ractions '(Tk + A T  ) are solved from a finite element 
matrix equation, Fourth, knowing  the total displacement U at the end of,the previous k 
loading step N-1, the total displacements U + 'A U on the boundary, and contact 
tractions (Tk + A? ) are checked to determine i f  they satisfy the conditions of contact. 
If they do not satisfy these conditions, the location  of the assumed contact 
surface i s  modifi'ed and the procedure repeated until they do. Next, a 
convergence test i s  made, and i f  the  solution i s  not convergent, the location  of  the 
I c 
k N-1 
N-1 
k k 
I 
k 
21 9 
contact surface i s  further  modified and  the solution procedure  repeated. 
To determine i f  the  solution satisfies the  conditions of contact,  the following 
assurances are made: 
( 1 )  That nodes of either body beyond the last contacting nodes from the 
previous iteration have  not  penetrated the other  body. 
(2) That tractions a t  the contacting nodes are compressive. These normal 
tractions  can  be  calculated  by  three  different methods; (a) from the 
stress coefficients, (b) from the equivalent nodal forces, and (c) from 
the contact tractions, $ + A.. Here the last method was used. 
(3) That the relationship between normal and tangential contact tractions, 
i s  satisfied. Depending on which of the above checks, i f  any, i s  violated, one of the 
following procedures i s  employed to modify  the assumed location  of the  contact surface. 
(a) If ( 1 )  i s  violated, the contact surface may be extended to include the 
points at  which  penetration has occurred. 
(b) If (2) i s  violated, the contact surface i s  reduced by ekcluding nodes 
at  which  the  tractions  are  tensile. 
(c) If (3) i s  violated, sliding i s  allowed to occur. 
After  the  conditions  of  contact  are satisfied, a  test  for  convergence can be made by 
calculating  the  following  quantity: 
where AU i s  the displacement at the k-th degree of freedom. If R i s  less than a precribed 
quantity, say 0.01, the solution i s  considered as converged. k 
EXAMPLE SOLUTIONS 
The finite element model and solution scheme are  applied  to problems of contact 
between a disk and a semi-infinite half-plane. The overall mesh pattern, the location of 
220 
the prescribed displacement and relevant dimensions are shown in Fig. 2-a, with the area 
immediately surrounding the contact surface shown in great detail in  Fig. 2-b. The semi- 
infinite half-plane has been modeled  by a finite one with  overall dimensions  much larger 
than those of the disk. The basic element used i s  four-node quadrilateral element derived 
by assuming  seven P -parameters and I inear displacement distribution  along each  edge. 
Five-node and six-node elements are also introduced in  transition regions between coarser 
and finer meshes. Contact tractions  along each contact element are approximated by linear 
interpolations. 
Non-Sliding Contact 
Problems are solved for the case with both  applied loads  and prescribed displacements 
at the top of the disk. In the problems, the ratio  of Young's moduli are varied over a range 
from 1 to and slightly different mesh patterns near the contact surface are used to 
accornrnoda te'. node-to-node and node-to-internode contacts. Loads or displacements are 
applied by hethree increments until the length of contact surface becomes about 2.4 mm, 
For each increment, the converged solutions are reached with three or four iterations. For 
these solutions, the best results for contact tractions are obtained when calculated from 
equivalent nodal forces and are compared excellently  with the Hertz  solution i n   a l l  cases. 
Frictionless Contact With Extensive SI iding 
The half disk and the semi-infinite  half-plane are  also  used to demonstrate the capa- 
bi l i ty of this forrnulat ion to solve extensive sliding contact problems.  Since the contact 
between the two bodies i s  frictionless, the solution i s  independent of the path; thus a Hertz 
solution i s  again available for comparison. Solutions are obtained for prescribed displace- 
ments at the top of the disk by eight increments. Stress distributions on the p l a n e  of con- 
tact, for two prescribed displacements, are plotted in Fig. 3 where zero position represents 
the point of initial contact. It i s  seen that the solution agrees almost exactly  with  that  of 
the Hertz solution. It i s  noted that the center of symmetry of the stress distribution moves 
to the left  as the half-disk slides in that  direction. 
Frictional Contact With Extensive SI iding 
The same problem i s  again solved here with  friction between the disk and the half- 
plane as an added consideration. No solution to this problem, analytic or otherwise, can 
be found; thus, results arrived  at here wil l  be justified by comparison with the results of the 
previous section and  by  showing that they satisfy the conditions of contact. 
The normal tractions at the contacting nodes between the disk and the half-plane for 
every 4th displacement increment are shown in Fig. 3 and are compared with those of the 
frictionless case. Because of  friction, i t can be seen that  the displacement of the  contact 
surface is  retarded. That the normal tractions of the plane and disk are equal i n  magnitude 
and opposite in sign i s  also evident i n  the figure. This implies that the normal tractions 
satisfy a condition of contact. A condition of sliding  contact requires the ratio between 
normal  and tangential components of tractions to be constant  and equal to the coefficient  of 
221 
fr iction . They were approximately verified for all contacting nodes. Finally, the con- 
tacting surfaces of  the disk and  the  plane  are shown in  Fig. 4 along  with  the  locations  of the 
.nodes obtained in previous solutions. It can be seen that  friction retards the movement of 
the  contact surface, and as in the  previous  solution,  through averaging  over  the  entire con- 
tact surface, the  contact  condition  of no separation or penetration i s  satisfied. 
C.0NCLUS IO NS 
An  incremental  variational  principle and a corresponding finite element 
formulation have been made for contact problems based on an assumed stress 
hybrid method. An iterative scheme for the solution has been developed. 
Successful applications  of the present method  for plane  elasticity problems 
have been demonstrated for 
(a) Non-sliding problems with node-to-node contact 
and with node-to-internode  contact, 
(b) Frictionless contact with extensive sliding, and 
(c) Frictional contact with extensive sliding. 
The present method should be extended to problems involving  material 
and/or geometrical  nonlinearities in  addition  to  contact  nonlinearity. 
REFERENCES 
Tsuta, T. and Yamaji, S . ,  "Finite Element Analysis of Contact Problem", 
Theory and Practice in Finite Element Structure Analysis, University of 
Tokyo Press, 177-  194 ( 1973). 
Wilson, E.A. and Parsons, B., "Finite Element Analysis of Elastic Contact 
Problems Using Differential Displacements", Int. J. Num. Meth. Engng., 
Vol . 2, 387-395 (1970). 
Chan, S. H. and Tuba, I .S., "A Finite Element Method for Contact Problems 
of Solid Bodies", Int. J. Mech. Sci., Vol. 13, 615-639, (1971). 
Pian, T. H .H . , "Finite Element Methods by  Variational  Principle  with Relaxed 
Continuity Requiremenfll, Variational Methods i n  Engineering, Edited by 
C.A. Brebbia and H. Tottenham, Southhampton University Press, England, 
., 
September, Session 1 1 1 ,  pp. 3.1-3.24, (1972). 
Boland, P. L. and Pian, T.H .H ., "Large Deflection Analysis of Thin Elastic 
Structures by the Assumed Stress Hybrid  Finite Element Method", Computers 
and Structures, Vol. 7, pp. 1-12, (1977). 
L 743.6 mm 
Figure  2.- Mesh pa t te rn .  
223 
T R A C T 1  ON 
(hg/mm) 0- TRACTTON ON HALF- PLANE 
600 I (NEGATT E, u= 0.2 1 
a" TRACTTON ON VTSK (p= 
0 TRACTTON (p= 0 . 0 )  
HERTZ SOLUTTON 
EIGHT  TNCREMENTS 
DX= - 0 . 4 ,  VY= -0  
-2 .0  - 7  . o  0 1 .0  (mm) X 
Figure 3.- Normal t rac t ion  on contact surface.  
0.2)  
. 3  
A PLANE (WZTff FRZCTZON) mml X 
v I I . B 1 I 
-0.6 -0.5 - 0 . 4  -0.3 -0.2 - 0 . 1  
Figure 4.- Location of nodes. 
224 


Solutions of contact problems by the assumed stress hybrid model

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