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A method for the numerical solution of contact problems
MECH. RES. C O M M .
A METHOD
Vol.3, 307-312, 1976.
FOR THE NUMERICAL
SOLUTION
Pergamon Press.
OF CONTACT
Printed in USA.
PROBLEMS *
E. Betz, New South Wales, Australia M. Levinson Department of Civil Engineering and Engineering Mechanics, McMaster University, Hamilton, Ontario, Canada (Received 23 March 1976; accepted as ready for print 2 April 1976)
Introduction ~he numerical solution of cont_;~t~problems in solid mechanics poses certain difficulties since the force pressing the EDdies together, the cont;~t surface, and the relative ap[Eoach of the bodies cannot be specified arbitrarily. In fact, one may specify only one of these quantities and must obtain the others as part of the solution. It may happen that the force pressing the bodies ~ogether, as in the case of paper mill rolls [i], or the relative approach (interference), as in the extrusion of a plastic sheet bet~sen two sensibly rigid rolls [2], is known. In both these cases a numerical solution m y be achieved by assuming a contact surface which is then adjusted in same iterative fashion if the solution so fo%md does not s a ~ f y the requirements of the ~oblem. Our present concern, however, is to d e ~ l within a specific context only how one solves a contact ~oblem once the contact surface is specified. Previous papers, e.g. [2,3], which have cons~ered this type of problem have not detailed how this is ~ be dnne and so other in~sstigatnrs of similar ~oblems must start ab initio. ~be purpose of the present paper is to bring a typical ~ocedure for the numerical solution of such c0mtact problems to the attention of those engineers ccmcerned with the/r solution. In additicm, suggestions for ~ l i n g with problems more cumPlicated than the one comsidered below will also be given. ~he specific context in ~%ich we shall present our procedure for solving contact problems numerically is that of the indentation by a rigid cylinder of a relatively soft layer bonded ~o another rigid cylinder. ~ situation arises in various industrial settings, e.g. the pulp and paper industry. Here the two cylinders are of ~ a r a b l e diameter and the deformable layer is
* ~ work ~ s par~ally supported by the Nat/trial Re~---~archCouncil of Canada through PRAI Grant No. P-7312. Scientific Communication - abbreviated
307
308
E. BE~gZ and M. LEVINSON
Vol. 3, No. 4
relatively thin. Other industrial applications may involve quite different geometries, e.g. a small ~ h b e r cylinder mounted on a slender shaft which drives a large, virtlm] ly rigid cylinder by friction. The work presented in this paper is applicable to the entire range of possible geometries although our immediate focus of interest is the geometry associated with paper mill rolls. Such a system is shown in Figure 1 where sane pertinent notation is also introduced. Tne thickness of the rubberlike layer is somewhat exaggerated in the sketch.
Ro+R-uo
FIG. 1 ~tation
configuration of paper mill rolls.
Problem formulation The present work is based on the asmmptions that the rubberlike layer may be treated as a linearly elastic material and that inertia and friction forces may be neglected.
~hese asstm~tions are not fundamental to the procedure pre-
sented and nonlinear, inelastic, dynamic contact problems may be treated in like fashion; the choice of sample problem is for the sake of simplicity.
The
present work, done by tb~ finite element method, will be compared with an analytic (series) solution of the same problem found previously [4]. More realistic, viscoelastic studies of paper mill roll cover problems must be done numerically [i].
We also assume that our problem is one of plane strain because
of the large length to diameter ratio of paper mill rolls. The present work is done in terms of constant strain triangular elements and furthe/mDre we assume that only that part of the covering layer in the vicinity of the contact surface need be s ~ e d .
This ass~pt/on can be justi-
fied, a posteriori, by numerical ccmparisons and it is a practically convenient assumption because it reduces crmputational effort.
Since O-y, in
Figure 2, is an axis of symmetry for the quasistatic problem being st~ied we note that only one half of the region of interest need be considered.
Vol.3, No. 4
NUMERICAL SOLUTION OF CONTACT PROBELMS
309
y Y
Bond surface Ro\
,
This surface constrained in x direction p~cal triangular element i
~
x
o
urtace x
e-
o
FTG. 3
FIG. 2 Finite element model.
Diagr~,matic representatic~ of displacements at contact surface.
Iteration procedure
The inner surface of the rubber covering layer is bonded to a rigid cylindrical core so that nodal points on the bond surface have zero displacement.
On
the axis of symmetry O-y, nodal displacements in the x direction must vanish but there is no constraint in the radial d~r~ction.
On the basis of a numeri-
cal experiment it was found, if the length of the strip is several times the contact length, that boundary conditions at the far end of the strip are immaterial; this end surface is unconstrained in our calculations. where the contact b e ~
On r = Ro,
the two cylinders occurs, ~e choose a given contact
arc and then enforce appropriate b o t a n y
conditions by use of the following
iterative p~ocedure which causes an arbitrarily chosen set of no~a] load vectors in the radial direction to converge nunerically to a set of load vectors which simultaneously satisfy the equilibrium equations for the system and deform the rubber layer to the profile of the indenter; this is shown in figure 3 above.
In a~ition, by forcing the load to vanish at the end points of the
contact surface we insure that the problem solved is a contact problem and not a punch problem, this point is discussed in [4]. We also obtain, as part of the solution, the relative approach of the cylinders corresponding to the chosen contact surface and the resultant force pressing the cylinders together. ~he computation is started with an arbitrarily chosen set of load vectors which, in general, will lead to displacements not satisfying the geometric conditions. Then the first iteration on the chosen set of load vectors is made by consid-
310
E. BETZ and M. LEVINSON
Vol.3, No.4
ering an arc of radius R, that of the indenter, whose center c is on the O-y axis and such that a best fit is established to the mean of the displacements derived from the initial c o n ~ t i o n . A (~)j = where
Aj
Tne position of c is found from
B.
-/2n+ -~2
(i)
n 7 [{ (~) 2 2 i=l - (xi + 6xi) }%
=
(2)
- (Yi + 6Yi)]
and Bj = Aj when i = n+l.
~xi and ~Yi are displacements as shown in Figure 3.
j is the iteration n ~ ,
n is the total ntm~er of surface nodal points along
the (positive) x axis, and i ranges from 1 to n. The location of this mean arc centre (Oc---)jof radius R is o-,t~ared with the locations on the O-y axis of the centres of arcs, of radius R passing through the load points P eji
=
z
to give error coefficients e0. defined as ]z
(~) ji - (~) j
(3)
where (Oc---)jiis the arc centre correspoDding to the ith load point.
Then the
loads are adjusted for the next iteration acoording to the equation
(Fi) j+ 1
=
(Fi) j
+
(Fi) j eji (~Yl) j
(4)
For the first iteration, now being discussed, j = i. ~e
procedure is repeated as many times as necessary m t i l every eji is suffi-
ciently small.
In the present work this criterion was chosen to be
(5)
eji <- 0.01 (~yl)j where i ranges from 1 to n+l.
A check as to whether we have studied a contact problem or a punch problem is made at Pn+2"
In order to insure that it is indeed a contact problem we re-
quire that en+ 2 for the last iteration should always be less than or ec91~] to zero; this implies separation, or at least non-penetration, of the two bodies beyond the specified contact surface,
we have been able, by judicious choice
of initial load vectors, to set a limit of i0 on the iteration c~unter j. fact, our convergence criteria has been satisfied by many fewer iterations when the initial loads are chosen by an experienced person.
Sample numerical results we now present results for bond surface stresses, in Figure 4 below, which
In
Vol • 3, No. 4
N U M E R I C A L S O L U T I O N OF CONTACT PROBLEMS
31 1
are compared with some results gi~m% by Harm and ievinson [4] who sttzlied
the present problem a n a l y t i c a l l y . S
=
Ri/R o
=
For the case considered,
139/144,
c o m p a r i s o n o f t h e bond s u r f a c e
-% = 1 , and
eo = 0 . 0 5
s h e a r and n o r m a l s t r e s s e s
reveals
that
the
a n a l y t i c and numerical r e s u l t s are i n good agreement; these stresses are the ones of greatest interest. ~he displacement on the surface, at and beyond the contact zone, disfDrted for clarity in Figure 5, shows that our iteration procedure has indeed produced a solution of the roll contact pi-oblem in which separation between the cylinders occurs, as it must, when 8 > 8o.
0.1
S = 139/144 eo =
-o r analyt~al ",~ numer~al '~ Tr8 analytkml . . . . . . . ~,~, numerical.................
,r-
)LI.I
0
I
I
i
0
15
i ,-
2E
8/8o -1
0/0o
Indenter
~1
~--ENst~ ~yer
FIG. 4
FIG.5
Ccmparison of bond surface stress distributions.
Displacm~nt at and beyond contact surface of rolls.
Discussion
We have shown how one may use the finite element method to deal with contact ~oblems of a fairly simple nature, i.e. where one of the bodies is sensibly rigid and %t~re the systau is s ~ i c cular).
(the shapes hn~ived need not be cir-
With minor n~dification our method can be extended to a) the case
where both contacting bodies are deformable and b) the case where there is asymmetry no matter whether it is due to geumetry, surface frictional forces, or any other cause.
In the first case we would proceed exactly as in the
present work except for the following obvious change.
It %r~id, in that case,
be necessary to enforce displao~ent crmpatibility between both bodies at
31 2
E. BETZ and M. LEVINSON
Vol. 3, No. 4
contact nodes instead of making one body conform to a pre-assigned deformation pattern (the shape of the rigid body) on the contact surface. The second case, that of an asymmetrical system,
for whatever reason,
quires a more subtle revision of our method than does the first one.
reInstead
of now requiring that the contact surface be specified precisely it is allowed to "float" somewhat, i.e. we choose a set of load vectors at nodal contact points such that the end point loads no longer are required to vanish.
We
then proceed as before and consider the problem solved when the displacements at loaded nodes meet the required geometric conditions and the displacements at nodes beyond the loads satisfy a condition of separation or, at least, non-interpenetration of the contacting bodies.
The contact area is deter-
mined, approximately, by extrapolating the pressure distribution until it vanishes.
~is
should not trouble one if we recall that the nt~nerical solu-
tion of contact problems in terms of a specified surface of contact is usually only a matter of convenience; this point was discussed in the first paragraph of the present paper.
~fer~c~
i.
Levinson, M., R.C. Batra, E. Betz and H.T. Hahn, "Paper Mill Rolls From Elasticity to Tneamoviscoelasticity", Proceedings of the Second Symposium on Applications of Solid Mechanics, McMaster University, 1974
.
Lynch, F.de S., "A Finite Elem~nt Method of Viscoelastic Stress Analysis with Application to P~lling Oontact Problems", Int.J. Num.Meth.Eng i, 379-394 (1969).
o
4.
Engel, P.A., "Rolling and Impact on a Linearly Viscoelastic Slab", Int. J.Num.Meth.Eng. 5, 465-479 (1973). Hahn, H.T. and M. Levinson, "Indentation of an Elastic Layer(s) Bonded to a Rigid Cylinder - I. Quasistatic Case Without Friction", Int. J. Mech. Sci., 16, 489-502 (1974).
Abbreviated Paper - For further information, please contact the authors.
A METHOD
Vol.3, 307-312, 1976.
FOR THE NUMERICAL
SOLUTION
Pergamon Press.
OF CONTACT
Printed in USA.
PROBLEMS *
E. Betz, New South Wales, Australia M. Levinson Department of Civil Engineering and Engineering Mechanics, McMaster University, Hamilton, Ontario, Canada (Received 23 March 1976; accepted as ready for print 2 April 1976)
Introduction ~he numerical solution of cont_;~t~problems in solid mechanics poses certain difficulties since the force pressing the EDdies together, the cont;~t surface, and the relative ap[Eoach of the bodies cannot be specified arbitrarily. In fact, one may specify only one of these quantities and must obtain the others as part of the solution. It may happen that the force pressing the bodies ~ogether, as in the case of paper mill rolls [i], or the relative approach (interference), as in the extrusion of a plastic sheet bet~sen two sensibly rigid rolls [2], is known. In both these cases a numerical solution m y be achieved by assuming a contact surface which is then adjusted in same iterative fashion if the solution so fo%md does not s a ~ f y the requirements of the ~oblem. Our present concern, however, is to d e ~ l within a specific context only how one solves a contact ~oblem once the contact surface is specified. Previous papers, e.g. [2,3], which have cons~ered this type of problem have not detailed how this is ~ be dnne and so other in~sstigatnrs of similar ~oblems must start ab initio. ~be purpose of the present paper is to bring a typical ~ocedure for the numerical solution of such c0mtact problems to the attention of those engineers ccmcerned with the/r solution. In additicm, suggestions for ~ l i n g with problems more cumPlicated than the one comsidered below will also be given. ~he specific context in ~%ich we shall present our procedure for solving contact problems numerically is that of the indentation by a rigid cylinder of a relatively soft layer bonded ~o another rigid cylinder. ~ situation arises in various industrial settings, e.g. the pulp and paper industry. Here the two cylinders are of ~ a r a b l e diameter and the deformable layer is
* ~ work ~ s par~ally supported by the Nat/trial Re~---~archCouncil of Canada through PRAI Grant No. P-7312. Scientific Communication - abbreviated
307
308
E. BE~gZ and M. LEVINSON
Vol. 3, No. 4
relatively thin. Other industrial applications may involve quite different geometries, e.g. a small ~ h b e r cylinder mounted on a slender shaft which drives a large, virtlm] ly rigid cylinder by friction. The work presented in this paper is applicable to the entire range of possible geometries although our immediate focus of interest is the geometry associated with paper mill rolls. Such a system is shown in Figure 1 where sane pertinent notation is also introduced. Tne thickness of the rubberlike layer is somewhat exaggerated in the sketch.
Ro+R-uo
FIG. 1 ~tation
configuration of paper mill rolls.
Problem formulation The present work is based on the asmmptions that the rubberlike layer may be treated as a linearly elastic material and that inertia and friction forces may be neglected.
~hese asstm~tions are not fundamental to the procedure pre-
sented and nonlinear, inelastic, dynamic contact problems may be treated in like fashion; the choice of sample problem is for the sake of simplicity.
The
present work, done by tb~ finite element method, will be compared with an analytic (series) solution of the same problem found previously [4]. More realistic, viscoelastic studies of paper mill roll cover problems must be done numerically [i].
We also assume that our problem is one of plane strain because
of the large length to diameter ratio of paper mill rolls. The present work is done in terms of constant strain triangular elements and furthe/mDre we assume that only that part of the covering layer in the vicinity of the contact surface need be s ~ e d .
This ass~pt/on can be justi-
fied, a posteriori, by numerical ccmparisons and it is a practically convenient assumption because it reduces crmputational effort.
Since O-y, in
Figure 2, is an axis of symmetry for the quasistatic problem being st~ied we note that only one half of the region of interest need be considered.
Vol.3, No. 4
NUMERICAL SOLUTION OF CONTACT PROBELMS
309
y Y
Bond surface Ro\
,
This surface constrained in x direction p~cal triangular element i
~
x
o
urtace x
e-
o
FTG. 3
FIG. 2 Finite element model.
Diagr~,matic representatic~ of displacements at contact surface.
Iteration procedure
The inner surface of the rubber covering layer is bonded to a rigid cylindrical core so that nodal points on the bond surface have zero displacement.
On
the axis of symmetry O-y, nodal displacements in the x direction must vanish but there is no constraint in the radial d~r~ction.
On the basis of a numeri-
cal experiment it was found, if the length of the strip is several times the contact length, that boundary conditions at the far end of the strip are immaterial; this end surface is unconstrained in our calculations. where the contact b e ~
On r = Ro,
the two cylinders occurs, ~e choose a given contact
arc and then enforce appropriate b o t a n y
conditions by use of the following
iterative p~ocedure which causes an arbitrarily chosen set of no~a] load vectors in the radial direction to converge nunerically to a set of load vectors which simultaneously satisfy the equilibrium equations for the system and deform the rubber layer to the profile of the indenter; this is shown in figure 3 above.
In a~ition, by forcing the load to vanish at the end points of the
contact surface we insure that the problem solved is a contact problem and not a punch problem, this point is discussed in [4]. We also obtain, as part of the solution, the relative approach of the cylinders corresponding to the chosen contact surface and the resultant force pressing the cylinders together. ~he computation is started with an arbitrarily chosen set of load vectors which, in general, will lead to displacements not satisfying the geometric conditions. Then the first iteration on the chosen set of load vectors is made by consid-
310
E. BETZ and M. LEVINSON
Vol.3, No.4
ering an arc of radius R, that of the indenter, whose center c is on the O-y axis and such that a best fit is established to the mean of the displacements derived from the initial c o n ~ t i o n . A (~)j = where
Aj
Tne position of c is found from
B.
-/2n+ -~2
(i)
n 7 [{ (~) 2 2 i=l - (xi + 6xi) }%
=
(2)
- (Yi + 6Yi)]
and Bj = Aj when i = n+l.
~xi and ~Yi are displacements as shown in Figure 3.
j is the iteration n ~ ,
n is the total ntm~er of surface nodal points along
the (positive) x axis, and i ranges from 1 to n. The location of this mean arc centre (Oc---)jof radius R is o-,t~ared with the locations on the O-y axis of the centres of arcs, of radius R passing through the load points P eji
=
z
to give error coefficients e0. defined as ]z
(~) ji - (~) j
(3)
where (Oc---)jiis the arc centre correspoDding to the ith load point.
Then the
loads are adjusted for the next iteration acoording to the equation
(Fi) j+ 1
=
(Fi) j
+
(Fi) j eji (~Yl) j
(4)
For the first iteration, now being discussed, j = i. ~e
procedure is repeated as many times as necessary m t i l every eji is suffi-
ciently small.
In the present work this criterion was chosen to be
(5)
eji <- 0.01 (~yl)j where i ranges from 1 to n+l.
A check as to whether we have studied a contact problem or a punch problem is made at Pn+2"
In order to insure that it is indeed a contact problem we re-
quire that en+ 2 for the last iteration should always be less than or ec91~] to zero; this implies separation, or at least non-penetration, of the two bodies beyond the specified contact surface,
we have been able, by judicious choice
of initial load vectors, to set a limit of i0 on the iteration c~unter j. fact, our convergence criteria has been satisfied by many fewer iterations when the initial loads are chosen by an experienced person.
Sample numerical results we now present results for bond surface stresses, in Figure 4 below, which
In
Vol • 3, No. 4
N U M E R I C A L S O L U T I O N OF CONTACT PROBLEMS
31 1
are compared with some results gi~m% by Harm and ievinson [4] who sttzlied
the present problem a n a l y t i c a l l y . S
=
Ri/R o
=
For the case considered,
139/144,
c o m p a r i s o n o f t h e bond s u r f a c e
-% = 1 , and
eo = 0 . 0 5
s h e a r and n o r m a l s t r e s s e s
reveals
that
the
a n a l y t i c and numerical r e s u l t s are i n good agreement; these stresses are the ones of greatest interest. ~he displacement on the surface, at and beyond the contact zone, disfDrted for clarity in Figure 5, shows that our iteration procedure has indeed produced a solution of the roll contact pi-oblem in which separation between the cylinders occurs, as it must, when 8 > 8o.
0.1
S = 139/144 eo =
-o r analyt~al ",~ numer~al '~ Tr8 analytkml . . . . . . . ~,~, numerical.................
,r-
)LI.I
0
I
I
i
0
15
i ,-
2E
8/8o -1
0/0o
Indenter
~1
~--ENst~ ~yer
FIG. 4
FIG.5
Ccmparison of bond surface stress distributions.
Displacm~nt at and beyond contact surface of rolls.
Discussion
We have shown how one may use the finite element method to deal with contact ~oblems of a fairly simple nature, i.e. where one of the bodies is sensibly rigid and %t~re the systau is s ~ i c cular).
(the shapes hn~ived need not be cir-
With minor n~dification our method can be extended to a) the case
where both contacting bodies are deformable and b) the case where there is asymmetry no matter whether it is due to geumetry, surface frictional forces, or any other cause.
In the first case we would proceed exactly as in the
present work except for the following obvious change.
It %r~id, in that case,
be necessary to enforce displao~ent crmpatibility between both bodies at
31 2
E. BETZ and M. LEVINSON
Vol. 3, No. 4
contact nodes instead of making one body conform to a pre-assigned deformation pattern (the shape of the rigid body) on the contact surface. The second case, that of an asymmetrical system,
for whatever reason,
quires a more subtle revision of our method than does the first one.
reInstead
of now requiring that the contact surface be specified precisely it is allowed to "float" somewhat, i.e. we choose a set of load vectors at nodal contact points such that the end point loads no longer are required to vanish.
We
then proceed as before and consider the problem solved when the displacements at loaded nodes meet the required geometric conditions and the displacements at nodes beyond the loads satisfy a condition of separation or, at least, non-interpenetration of the contacting bodies.
The contact area is deter-
mined, approximately, by extrapolating the pressure distribution until it vanishes.
~is
should not trouble one if we recall that the nt~nerical solu-
tion of contact problems in terms of a specified surface of contact is usually only a matter of convenience; this point was discussed in the first paragraph of the present paper.
~fer~c~
i.
Levinson, M., R.C. Batra, E. Betz and H.T. Hahn, "Paper Mill Rolls From Elasticity to Tneamoviscoelasticity", Proceedings of the Second Symposium on Applications of Solid Mechanics, McMaster University, 1974
.
Lynch, F.de S., "A Finite Elem~nt Method of Viscoelastic Stress Analysis with Application to P~lling Oontact Problems", Int.J. Num.Meth.Eng i, 379-394 (1969).
o
4.
Engel, P.A., "Rolling and Impact on a Linearly Viscoelastic Slab", Int. J.Num.Meth.Eng. 5, 465-479 (1973). Hahn, H.T. and M. Levinson, "Indentation of an Elastic Layer(s) Bonded to a Rigid Cylinder - I. Quasistatic Case Without Friction", Int. J. Mech. Sci., 16, 489-502 (1974).
Abbreviated Paper - For further information, please contact the authors.