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1. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 34 (1982) 821-845
NORTH-HOLLAND PUBLISHING COMPANY
A NUMERICAL ANALYSIS OF A CLASS OF CONTACT PROBLEMS
WITH FRICTION IN ELASTOSTATICS
L.T. CAMPOS, J.T. ODEN and N. KIKUCHI
The Texas Institute for Computational Mechanics, The University of Texas at Austin, Austin, TX 78712,
U.S.A.
1. Introduction
1.1. Introductory comments
This work describes a numerical analysis of a class of contact problems in elastostatics by
finite element methods. Specifically, a new numerical scheme is developed for the analysis of
contact problems with Coulomb friction, in which normal boundary tractions are prescribed.
Error estimates are derived and applications of the method to several representative two-
dimensional boundary-value problems are described. In addition, an algorithm is presented
with which the methods developed here can be used to solve general friction problems in
which the normal tractions are not known in advance.
The general problem of equilibrium of elastic bodies in contact with rough rigid foundations
on which frictional forces are developed remains one of the most difficult problems in solid
mechanics. The issue of existence of solutions to such problems in cases in which Coulomb’s
friction law is assumed to hold is still open, a fact which has prompted some investigators to
question the validity of this law for general elastostatics problems. Moreover, when solutions
do exist, it is rare that uniqueness can be proved and, in fact, non-unique solutions are
common. Inherent in the friction problem is the free-surface problem of identifying a priori
the unknown contact surface. In addition, the presence of friction leads to non-conservative
forces which give rise to non-differentiable forms in variational formulations of these prob-
lems.
It is clear that the only hope for overcoming this formidable list of complications is to
employ numerical methods. Toward constructing a general numerical scheme for such prob-
lems, a variational principle governing a class of contact problems with friction is considered
herein which involves a variational inequality defined on a set of admissible displacements
which satisfy the unilateral contact condition. Formulations of such problems as variational
inequalities were originally investigated by Duvaut and Lions [7], but they were unable to
prove that solutions exist to such problems except in special cases in which the normal contact
pressure urn is prescribed.
The physical problems under study here involve the determination of equilibrium
configurations of a linearly elastic body subjected to external forces and initially at a distance s
from a rigid or deformable foundation. Upon contact of the body and the foundation,
frictional forces are developed according to Coulomb’s law. The types of problems considered
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2. 822 L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
also include rigid punch problems in which a rigid ‘punch’ or ‘stamp’ is indented in an elastic
body.
In variational-inequality formulations of such problems, the location of the free boundary
(contact area) becomes an intrinsic part of the solution and no special devices are needed to
locate it. To minimize difficulties due to the non-differentiability of the friction terms in the
virtual work equations, a regularized form (a smooth perturbation) of the work term due to
friction is constructed in the present study. This term depends upon a perturbation parameter
E > 0; as E tends to zero, the correct non-differentiable friction term is recovered. This
perturbed variational inequality is then used as a basis for the construction of finite element
approximations of the problem.
Since the problem is highly nonlinear, the discrete problem involves systems of nonlinear
inequalities. These are solved in the present study by an iterative procedure which is described
in detail in Section 4.
Since no existence theory is available for the general elastostatics problem with friction,
there does not exist a framework for developing a complete approximation theory. Neverthe-
less, a rather complete analysis of certain very special cases is available, and approximations of
these cases are studied herein. Specifically, if the normal contact pressure is prescribed on the
boundary of the body, then the contact surface is known, and the only significant com-
plications which remain are due to the friction effects on these surfaces. As noted earlier, this
particular subclass of problems was studied by Duvaut and Lions [7] and an existence theory is
available. In the present study, a priori error estimates for finite element approximations of
this class of problems are derived.
It is noted that several other authors have attempted to solve elastostatics problems with
friction by finite element methods. Among them we distinguish Kalker [IO] who presented a
computer code for three-dimensional, steady-state, rolling contact problems with dry friction,
based on the variational principle of Duvaut and Lions. Also noteworthy are the works of
Sewell [21] and Panagiotopoulos [20] who derived a variational formulation for equations
involving self-adjoint operators subject to unilateral constraints. In the paper of Turner [24],
whose work was also based on the variational principle mentioned above, the problem of
contact between a rigid circular cylindrical indentor and an isotropic homogeneous linearly
elastic half-space was studied for the cases of frictionless contact, adhesive contact, and
frictional loading for which analytical solutions are available. Turner also considered a
problem of frictional unloading for which no analytical solutions were available.
1.2. Notation and statement of the problem
The general class of contact problems considered in this study are characterized by the
a, = 0 . aT=O if u,<s,
a” <o, Us = 0 if (all< Y~(cF~[
.
a, < 0, 3 A L 0 such that
uT = -hul if laTl = v,la,l ,
following system of equations and inequalities, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG
Cij(U),j +fi = 0 T Vii(U) = EijklUk,l . in 0 ,
uj = 0 on TD, CT,,n,= ti on rF,
if u, = s. Ion rC. .
(1.1)

3. L.T. Campos et al., Numerical analysis zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF
of contact problems with friction in elastostatics 823
This system describes the classical statement of Signorini’s problem in elastostatics for the
case of a foundation developing frictional forces upon contact which obey Coulomb’s law.
Similar systems are obtained for the rigid-punch problem and for two-body contact problems.
In (1.1) the following conventions and notations are assumed:
- 0 is the elastic body in a bounded open domain in RN with Lipschitz boundary r =
r-D u i;, u i;,.
- rD(rF) are portions of r on which the displacements (tractions) are prescribed.
- rc is the (candidate) contact surface on which the body may come in contact with the
foundation upon the application of loads; it is assumed throughout that rc flrD = 0.
- U = (Ul, 4.. . ) uN) is the displacement vector; u = u(x), where x = (x1, x2, . . . , xN) is a
point in 0.
- aij are components of the stress tensor; its value at a displacement u is
def
aij(U> = J3jklUk.l *
Here and throughout this work index notation and the summation convention are used;
commas denote differentiation with respect to xi; uk,{= &J8xl.
- Eijkl
are the elasticities of the material of which the body is composed. These are given
functions of x assumed to satisfy the following conditions:
max IIGjkllb 5 M,
izG,j.k,lsN
Eijkl = Ejikl = Efjfk = Ekfij 7
for every symmetric tensor A,.
-fi
are components of body force, assumed to be given as functions in L2($2).
- ti are components of surface traction, assumed to be given as functions in L”(f,).
- uiinj = unni + UTi; Iti are the components of a unit vector outward and normal to the
boundary r; a, is the normal stress on the boundary
and UTi
-Un=U'
Un = Un(U) = Uij(U)ninj = Eijk&.Jtinj ,
are the components of the stress vector tangent to r,
OTT
= g’ri(u) = fTij(U)nj - a,(u)ni .
n = normal displacement of particles on the boundary r.
- s is the normalized initial gap between the body 0 and the foundation prior to the
application of loads.
- r+ is the coefficient of friction, assumed to be a given strictly positive constant.
Duvaut and Lions [7] derived the following variational principle characterizing problem
(1.1).
Find a displacement field u in a subset K of the space V of admissible
displacements satisfying the variational inequality,
U(U,V-u)+j(u,zJ)-j(U,U)Lf(V-u) VVEK.
(1.2)

4. x24 L
.T
. Campos e t al.. Nume ric al analysis o f c o ntac t proble ms with fric tio n in e lasto static s
Here,
v = {u=(II,, u2, . . ., uN) E (H’(fi))” 1y(v) = 0 a.e. on r,} . zyxwvutsrqponmlkjihgfedcbaZ
K = {u E V (y(o,)ni - s 5 0 on H”‘(Tc)},
(1.3)
wherein y is the trace operator mapping H’(o) onto H”*(r) and UT is the tangential
component of 2, on Tc. Here and throughout this work we employ notations and conventions
commonly used in the study of partial differential equations in Sobolev spaces (see, e.g.,
[l]) and in the study of contact problems by variational methods (see, in particular. [ 131
for more details).
Eq. (1.2) is merely a statement of the principle of virtual work for an elastic body restrained
by frictional forces. The strain energy of the body corresponding to an admissible displace-
ment 2, is $a(~, u). Thus, a(u, 21 - U) is the work produced by aij(U) through strains caused by
the (virtual) displacement D - u. The linear form f represents the work done by the external
forces and j( * , . ) re presents the work done by the frictional forces. The total virtual work
must be such that (1.2) holds rather than an equality because of the presence of the unilateral
contact constraint, u, - s I 0 on r,. The actual contact surface depends upon the solution u
and is not known in advance.
Note that, in view of the assumptions on the elasticities Eijk,,the bilinear form a(. , . ) is
continuous and V-elliptic; i.e.
where I]- II1 is the norm on V given by zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB
II
v 1
II= (In
(I 3)
Likewise, the work done by the external forces defines a continuous linear functional f E V’.
The normal stress component on the boundary, U”(U), is a function of the solution u and
therefore an unknown. As pointed out earlier, the only existence theory available is that of
Duvaut and Lions for the special case of a prescribed normal contact pressure. In this case.
F, = o-n is assumed to be given on all of r,, i.e., V&~(U)] = g, g given in L”(Tc). Con-
sequently, the contact surface Tc is known in advance and u, is not prescribed on rc. With
these gross simplifications, the boundary conditions in (1.1) reduce to,

5. L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics 82.5
ui = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
0 on r,, ti = CT& on r~, eIEl=g onTc,
lwl<gjuT=O orire, (1.6)
~UT[
= g+3 h 10 such that UT= -haT on rC.
We can then replace i(tl, 0) of (1.3) by
and f E V’ becomes
(1.7)
(l-8)
It is assumed, as before, that fE (L2(0))N and F, is the prescribed normal contact pressure on
1 ‘c.
The variational problem (1.2) now reduces to the following problem:
Find a displacement field u E zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF
V, such that
a(u,v - U)+j(V)-_(U)Lf(V - u), v 2,E v.
(1.9)
1.3. Scope of study
Following this introduction, special attention will be given to the case in which the normal
contact pressure is prescribed, (1.9). In Section 2, a perturbed variational problem is derived
for this case which features a regularization of the work done by friction forces.
In Section 3, a finite element approximation to the class of problems described by the first
part of (1.1) together with the boundary condition (1.6) and definitions (1.7) and (1.8), or
equivalently by (1.9), is considered. In addition, an a priori error estimate is derived for the
case of linear finite element approximations.
We then introduce in Section 4 an algorithm for solving (1.9) by finite element methods. As
a by-product we will be able to obtain a solution to (1.2), where g is not prescribed, by
considering a sequence of approximate solutions of problems of the type (1.9).
Following this analysis, the numerical study of several problems is considered. Among these
is the problem of the deflection of an elastic beam subject to prescribed loads on a surface on
which frictional forces may develop. This problem falls into the category of one in which
normal boundary tractions are prescribed (as, e.g. in (1.9)). Then the deformation of an elastic
beam resting on a Winkler foundation is studied and solutions are compared with those of the
non-frictional case. An interesting feature of this example is that in the case in which friction is
present, plane sections normal to the beam axis prior to deformation do not remain plane after
deformation, thus violating the classical hypothesis used in the analysis of this type of problem
for non-frictional loading.
Finally, two rigid punch problems are analyzed. They involve the indentation of an elastic
half-space by a rigid sphere and a flat annular rigid punch, respectively. For the spherical

6. 826 L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
punch the numerical solution of the non-frictional case is compared to Hertz’s analytical
solution and to the numerical solution of the frictional case. For the flat annular punch
problem, the numerical results for frictional and non-frictional indentations are compared with
analytical solutions obtained for the case of adhesion by Shibuya et al. [‘22].
Finally, in Section 5 of this work, a summary and some conclusions about the problems
analyzed are presented. Also, possible areas for future study are discussed and some possible
generalizations of the present work are pointed out.
2. The zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
energy functional, existence, uniqueness, and a perturbed variational principle
2.1. The zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
energy functional
For friction problems there exists no potential energy in the usual sense, due to the
non-conservative character of the friction forces. However, it is possible to consider that the
proper, convex, weakly lower semicontinuous functional I(u) : V+ [w
,such that
I(v) = $a(v, v)- f(v)+j(v) (2.1)
represents the potential energy associated with the static friction problem for the Coulomb law
(g prescribed), where the same notation used in describing the variational principle (1.9) was
used. The usefulness of considering (2.1) will be clear in what follows.
2.2. Existence and uniqueness
We now summarize the major results concerning existence and uniqueness of solutions
obtained by Duvaut and Lions [7], for the simplified problem where g is prescribed.
The most direct approach to the issue of existence of solutions to (1.9) is to call upon
the following theorem from convex analysis and variational inequalities (see, e.g.. zyxwvuts
P31).
THEOREM 2.1. Let V be a reflexive Banach space and I : V + E a proper, coercive, functional
of the form I = F + + where F is convex and Gateaux-differentiable and, therefore. weakly
sequentially lower semicontinuous on V, and C$ is convex and lower semicontinuous. Then I
possesses at least one minimizer on V. Moreover, any minimizer u of I satisfies the variational
inequality
(DF(u),u-u)++(v)-4(u)d VU’S V (2.2)
where DF(u) is the Gateaux differential of F at u and ( . , . ) denotes duality pairing on V'X V.
In the present application of this theorem, I is given by (2.1) and one may take zyxwvutsrqponmlkjihg
F(v)= ia(v, v)-f(v), 4(v)=i(v). (2.3)
Under conditions (1.4) F is strictly convex and Gateaux differentiable on V. It is easily
verified that the friction functional i is convex and continuous on V. Thus, all of the conditions

7. L.T. Carnpos et al., Numerical analysis zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF
of contact problems with friction in elastostatics 827
of Theorem 2.1 are satisfied by the functional I of (2.1) except coerciveness, and this depends
upon the boundary conditions.
For the case in which meas r, > 0, it is not difficult to show that I is coercive on zyxwvutsrqponmlk
V. In this
case, the strict convexity of F makes it possible to establish that a unique minimizer of I
exists. If r, = 0, then a compatibility condition on the data is needed. Duvaut and Lions [7, p.
1421 show that a necessary condition for the existence of solutions to (1.9) in this case is that
If(r)1
q(r) v r E I? , (2.4)
where R is the finite-dimensional space of infinitesimal rigid motions of the body (r E
R 3 a(r, r) = 0). Under the stronger condition, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH
iw - If( 2 c ll& If t-ER (2.5)
(where ]]$J = Jo r - r dx), they are able to show that I is coercive on V.
In the light of Theorem 2.1, these observations lead to the following existence and
uniqueness theorem.
THEOREM 2.2. Let measID > 0. Then there exists a unique minimizer u of the energy
functional I of (2.1). Moreover, u is also the solution of the variational inequality (1.9).
If ID = 0 and if (2.5) holds, then there exists at least one minimizer of the energy functional I,
and each such minimizer is a solution of the variational inequality (1.9).
In the case r,, = 0, the solution is unique to within an arbitrary rigid motion in R; i.e., if u1
and u2 are distinct solutions of (1.9) then u1- u2E R.
2.3. A perturbed variational principle
The previous results give us conditions under which a solution to the static contact problem
with Coulomb’s friction law (g prescribed), is also a solution to variational inequality (1.9). This
solution is also a unique minimizer of the functional I over V if meas r,, > 0.
One important objective in developing variational principles for friction problems is to
provide a basis for the construction of finite element approximations. However, the direct
approximation of (1.9) by finite elements through the minimization of (2.1) will lead to a
discrete system which does not lend itself to the most popular methods for solving nonlinear
variational inequalities owing to the fact that the functional i : V+ W is nondifferentiable.
To overcome this difficulty, we consider a G-differentiable perturbation is(*) of i(s), which is
a function of a positive real parameter E, that approximates i(a) arbitrarily closely as E is
allowed to approach zero. Similar regularizations have been used to study nondifferentiable
functionals in nonlinear operator theory and optimization (see, e.g. [3,15]) and as a basis for
the numerical solution of variational inequalities (see [9]). We shall choose a regularization of i
which is quite different from that employed by Glowinski et al. [9].
Specifically, we shall introduce the function & : V-t L’(I,) defined by
(2.6)

8. 828 L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
for given I > zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
0, wherein VT, of course, is the tangential component of the trace y(v) of the
vector v on rC (y(q)E H”‘(r,)). W e easily verify that v+&(v) is convex and that
C#&(u+ ev) E C’[O, l] for any U,VE zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
V. Our approximation of j is then
id4 = I,,g4eWds. (2.7)
The following result can be proved by a straightforward calculation.
LEMMA 2.1. The zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
functional jE: V+ R defined by (2.7) is convex, weakly sequentially lower
semicontinuous and, in fact, GBteaux differentiable on all of Vfor all E > 0. Indeed, the G&eaux
derivative of jE is given by zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
@j,(u), zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
v) = hrl I,,g$ &(U + eV)ds.
That je provides an approximation of j is established in the next lemma.
LEMMA 2.2. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
If g E L”(T,), g 2 0 a.e., then, for all v E V,
[j(v) - j6(v)l 5 (211gl10.ffi,k
meas ~C)F
where j is given by (1.7) and jc by (2.7).
PROOF. A direct calculation gives
(2.8)
(2.9)
On sets of nonzero measure on Tc-, we define zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF
V
+_ 0 iflvTIS&.
T-
VT if lvT1 > E .
Then,
[j(v) - jE(v)l 5 I,, g] Iv+] - lv$l+ E/21ds +,I,,. gl zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM
IvT
I
+ t1i2&)
i”T
171
ds
gds+g g ds,
from which (2.9) follows.
Having approximated
functional,
j by jF, we can now introduce the regularized potential energy

9. L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics 829
I,: v-+/x; IE(0)=~a(u,v)-f(u)+j,(v). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO
(2.10)
This functional is strictly convex, coercive for meas r, > 0, and Gateaux differentiable on all
of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
V. Thus, we have the following theorem.
THEOREM 2.3. Let meas r,, > 0. Then, for each E > 0, there exists a unique minimizer u, E V
of the perturbed energy functional I, of (2.10). Moreover, u, is characterized by the variational
equality,
a(u,, v) + (DjE(uE),
v) = f(u) V 2, E V. (2.11)
Clearly, if rD = $3,a similar result holds on V/R provided the external forces satisfy the
compatibility condition described earlier with j now replaced by je.
Next we arrive at the question of whether or not solutions of (2.11) converge to the solution
of (1.9) as E tends to zero. Our next theorem not only establishes that the answer to this
question is affirmative, but it also provides an estimate of the rate of convergence.
THEOREM 2.4. Let u be the solution of (1.9) and u, the solution of (2.11) for zyxwvutsrqponmlkjihgfedcba
fixed E > 0. Then,
there exists a constant C independent of E such that
(2.12)
PROOF. Setting v = u, in (1.9) u = u, - u in (2.11) and subtracting yields,
a(u, - u,24, - u) 2 j(u,) - j(u) - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG
WjE(uE),
u, - u>
.
Since jE is convex and Ghteaux differentiable,
&(u) - jE(u,) 2 (Qi(U=), U - UE)
Thus, using (1.4) we have
m0llu
- u,llT~Ii - iE(uE)l + IL(u) -i(u)1 .
The assertion now follows from Lemma 2.2.
3. Finite element approximations
3.1. An approximation of the set V
We now consider the question of constructing finite element approximations to the problem
of minimizing the energy functional I over the set of all admissible displacement fields V.
Towards this end, we assume that conventional conforming finite element methods are used
to construct a family {V,}, 0 5 h I 1, of finite dimensional subspaces of V C (H’(a))‘“, where h
is a mesh parameter, typically the largest diameter of an element in one of a sequence of
quasi-uniform refinements of meshes approximating fi in such a way that fi = a,,.

10. 830 L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
We shall assume that this family of subspaces {Vh} has the standard interpolation property
of regular finite-element approximations of Sobolev spaces (see [4]): If the shape
functions forming the basis of V,, contain complete polynomials of degree rk(and (Pk(0))” C
V,,, where Pk(fl) is the space of polynomials of degree k) and if we are given u E
@-zY(R))~ n V, m > 0, then there exists a constant C, independent of 0 and h, and an element
u,, E V,, such that
p = min(k + 1 - s, m - s), s = 0, 1 . (3.1)
In addition, mild restrictions on R and the spaces V,,, we can expect estimates of the type (3.1)
to hold on traces of functions on a0 and to hold for negative s (see [2]),
(3.2)
with p possibly a negative real number.
Having constructed such finite dimensional spaces, the finite element approximation of
problem (1.9) takes the form: Find u,, E V, such that
4% %- wI)+j(Vh)-j(Uhpf(uh - Uh) v Uh E v,. (3.3)
Likewise, the approximation of the regularized problem (2.11) consists of seeking U; E V,,
such that, for given E > 0,
a(G, vh)+ (Dj&E), Uh)= f(Q) vVh (5 Vh. (3.4)
Observing that (3.3) and (3.4) are formally identical to (1.9) and to (2.11) and that all the
operator properties of those forms are carried from the continuous problem to its finite
dimensional approximation, we may conclude that the results concerning existence and
uniqueness of solutions to the finite dimensional problem follow with identical conclusions as
in the case of the continuous problem.
3.2. Error estimates for a finite element approximation
We shall now address the question of the convergence of the solution of (3.4) to the solution
of (1.9) as E and h tend to zero. Towards this end, we first note that by following steps
identical to those used in the proof of Theorem 2.4 we can show that
where C is independent of h and of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
E and uh and u ; are the solutions of (3.3) and (3.4)
respectively. The remaining step in the establishment of a final estimate is given in the
following result.
THEOREM 3.1. Let u E (H2(0))N fl V be the solution of (1.9) and uh the solution of (3.3). Let
interpolation estimates (3.1) and (3.2) hold and let k I 1. Then there exists a constant
independent of h such that

11. PROOF.
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
jlu - UJ, 5 Ch .
Adding (1.9) and (3.3) we obtain for all 2)E V and vh E vh,
a(u, 0 - u) + a(u,,, uh - uh) + j(v) -j(u) + j(%) - j(uh) >f(u - u) + f(% - u,,) .
831
(3.6)
Noting that
@(u,2,- u) = a(u, 2,- u,, + u,, - u), a(&,, t),,- u,) = a(&, z),,- u + u - u,,) ,
and adding and subtracting u(u, u - 2)h)we obtain
a(u - uh, u - u,,) 5 a(u - &,, u - vh) + a(& 0 - uh) - f(u - u,,) + a(& z)h - u)
-f(vh - U)+j(D)--(U)+j(2)h)-j(Uh). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM
Since vh C V, we can set 2,= &, and are left with
a(u-uh,u-Uh)Ia(U-Uh,U--h)+u(U,Dh-U)-f(2)h-U)+j(~h)-j(U)
tlVhE v,.
From (1.4) we have
(3.7)
mCJl/u
- uhjj: s a(u - uh, u - uh) ,
a(u - uh, u - vh) s M(lu - uhlll [(u - vhlll .
Using (3.1) the second term on the right-hand side gives for any uh,
(3.8)
(3.9)
a(% oh - u) s [(u(lO (Iu - vh[(O 5 cIIIu112 II” - VhllO g c21jull%2 *
On the other hand, using (3.1) again, we obtain
(3.10)
f(u - uh) s llfll0 IIu - vh/CI s cllfllIJh*11412 .
Next we make use of (3.2) together with the trace theorem to obtain,
(3.11)
5 ~~11~l1~c?.r,~*11~dl~/*.r~
5 Gll~ll1/2.~~~*11~02,R
. (3.12)
We have thus estimated all terms in (3.7). Introducing (3.8H3.12) into (3.7) and applying
Young’s inequality,
ubs(1/4c)u*+&b* Qu,bEIW,QE70,

12. x32 L. T. Cumpos et ul.. Numerical analysis of contact problems with frictim in elastostatics
to the term M/u - ~~11,
//u - z+,//,and simplifying, we obtain the estimate zyxwvutsrqponmlkjihgfedcbaZYX
(m,,
- F)[lU
- u,,llf
-=(SC34llull;+ C,llz&
+ q fllo II4 + C~llg lllir.lill~ll~)~’
*
which. upon choosing F < nr,,, yields the desired estimate.
We remark that if we do not identify (aTh - uT) as a linear functiolial on g we have, instead
of (3.12),
where we used (3.2).
We immediately see that for this case
II24
- UJ, = O(h”‘y .
Use of (3.51, (3.6) and the triangle inequality gives the final error estimate for the
approximation problem (3.4), which we state in the following.
THEOREM .3.Z. Under the conditions of Theorems 2.4 and 3. I we have
where u und ui ure the solutions of (1.9)und (3.4). respectively. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR
4. A numerical solution
4.1. Algorithm
We shall now describe a numerical method for the analysis of static contact problems
involving Coulomb’s law of friction. We shall also demonstrate the application of this method
to Signorini’s problem with friction, i.e., the problem of a linearly elastic zyxwvutsrqponmlkjihgfedcba
body in contact with
a rigid foundation on which friction forces are developed. Then g is no longer assumed to be
given, but is determined through the function z+c~(u) of the unknown solution u.
We, thus, distinguish two different types of conditions on Tc-:
(i) The normal stress distribution is prescribed on rTC.
(ii) The normal stress distribution is not known a priori on rC
Our primary concern here is with case (i). Our analysis of this case is straightforward: We
employ a standard conforming finite element analysis of the regularized problem discussed in
Section 3. This involves adding to the usual stiffness matrix a penalty term, solving the
resulting linear system, and considering the behavior of the approximation as I tends to zero.
However, by employing an existing code developed for the analysis of contact problems
without friction (see [I 1, 13, 17, 18,231) an iterative scheme can be developed which is
applicable to certain classes of problems falling into category (ii) above.

13. L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics 833
The steps in the algorithm for the analysis of the general case (ii) are listed as follows:
Step 1. First, a finite element approximation of the problem without friction is obtained.
The idea here is to compute, as a first approximation, normal contact pressures produced
without friction to be used later as data for a problem with friction but with prescribed normal
pressures. For this purpose we employ the reduced-integration, exterior penalty method of
Oden, Kikuchi, and Song [18] and Kikuchi and Oden [13] for contact problems without
friction. In this method, one seeks minima of the penalized energy functional,
MD) = b(v, u) -f(u) + & I, (v, - s): ds (4.1)
(F, a penalty parameter > 0), except that in the discrete approximations, the integral Jrc is
replaced by a suitable quadrature rule J which may be of an order lower than that necessary
to evaluate this integral exactly.
For the finite-dimensional problem, the contribution to the virtual work due to the penalty
terms is approximated according to
I (UEhL
- s)+v,m
ds = J[(uZ - s)+z),,n]
,
I-C
(4.2)
where J( ) represents a numerical quadrature rule and u “hL
is the (reduced-integration-penalty)
finite element approximation of the displacement field. Specifically, u;‘E V,, C V is the
solution to the discrete problem
u(G’, Vh)+ (lIEl)J[(Ui~- S)+nhn]
= f(uh) tl uh E vh . (4.3)
Note that this problem is nonlinear: The function (u;k - s)+ is the positive part of (u;A - s) and
is, therefore, unknown. However, any of several iteration schemes can be used to solve this
problem without difficulty. In the present study we employ the successive over-relaxation
scheme with projection described in, for example, Kikuchi and Oden [13].
Step 2. Having calculated ui’ for a specified ,sl and h, we calculate nodal values of the
normal contact pressure by setting
where &j are the quadrature points used in J. In the present study, J is taken to be Simpson’s
rule. Then the normal contact pressure u i:, is a continuous, piecewise quadratic polynomial on
the contact surface Tc. Of course, if cl > 0, the unilateral condition Uhn- s IO will not be
satisfied exactly on this surface.
Step 3. Having calculated the approximate displacement u i’ and normal contact pressure
(TELfor the frictionless case, we now use these results to compute data for a problem of the
type described in Section 3 with Coulomb friction but with prescribed normal pressures. We
shall use, for this purpose, the perturbed variational formulation described earlier. Thus, we
next select E > 0 and, using the available function u:‘, identify ‘stick’ or ‘slidding’ conditions
on r,. We set

14. x34 LT. Cumpos et al., Numerical analysis of contact problems with friction in eiastostatics
and define
(4.5)
(4.6)
for any Us, 2)hE zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
V,, where &(vh) is defined in a similar way as in (2.2).
Then a correction UE of 11;’ for frictional effects is computed by solving the linear problem
We remark that if the ‘slidding condition’ (/u&l > &) holds at a node, the contribution to (4.7)
from (Dic,,(ufi), v,,) is represented by the addition of terms to the load vector of the system; on
the other hand, contributions of these frictional terms for lug$l I F are added to the stiffness
matrix of the problem. (The first iteration is performed under a stick condition.)
Step 4. Having obtained the solution u “hof (4.7) we can now calculate tangential stresses
CT~;=
on r,. We return to the problem without friction described in Step 1 and, treating a;lT as
data, resolve the contact problem for the case in which tangential nodal forces agT(&j) are
applied on the contact surface. This leads to new iterates u~‘(*‘,a$” of the displacement field
and the normal contact pressure. We then repeat Step 3 using these corrected solutions and
obtain a second (corrected) approximation of CT:,-.
We continue this process until successive solutions do not differ by a preassigned tolerance.
We remark that while the above procedure has been shown to be convergent for case (i). it
is by no means general nor may it necessarily be convergent in all situations involving case (ii).
It is possible that when tangential forces are applied in Signorini’s problem in the second
iteration, no contact takes place and the process may terminate or diverge. Also. there are no
features of the process that would allow for the detection of multiple solutions. Nevertheless, the
process has proved to be convergent and very effective for a wide class of friction problems and it
appears to work especially well in problems in which adhesion prevails over significant portions of
the contact surface.
4.2. Numerical examples
We are now in a position to study some numerical examples for both types ((i) and (ii)) of
situations on rC.
The first example, corresponding to a situation where case (i) is applicable, consists of a slab
of dimensions 8 x I units, of unit thickness, made of an elastic material with a Young’s
modulus E = 1000 units of force per unit area and a Poisson’s ratio v = 0.3.
The slab is simply supported on one part of its boundary (r,) and subjected to a uniform
pressure p = 0, = F, acting through a frictional surface (TC), with a friction coefficient VF= 0.3.
and also subjected to a prescribed compressive force on one of its ends (fF) as illustrated in
Fig. 1.
The analysis was done using 16 nine-node isoparametric quadratic elements, and for this
case a value of F of lo-’ was employed.

15. L.T. Campus et al., Numerical analysis zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH
of contact problems with friction in elastostatics 835
r
Pressure = 3OO/unit length
E = l,OOO/(length)'
v = 0.3 length
vF
on CD=0.3
Fig. 1. An elastic slab with an applied normal pressure on a side CD on which Coulomb’s law of friction holds.
Convergence was obtained after four iterations. The computed deformed configuration and
frictional stresses are shown in Fig. 2. For this case on Tc we have a, = 300 stress units and
vF= 0.3 . Consequently, g = 90 stress units and we notice that the stick and sliding parts of Tc
are easily identified according to the absolute value of the frictional stress (TTbeing less than g
and equal to g, respectively, which is Coulomb’s law of friction.
(a)
Frictional Stress zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
100
i
80 -
60 -
(b)
1 2 3 4 5 6 7 a
Deformed Configuration sliding
part
+
stick part zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
4-i
Fig. 2. Computed results: (a) frictional stresses along surface CD and (b) the deformed finite element mesh.

16. X36 L. T. Campos et al.. Numerical analysis of contact problems with friction in elastostutics
A-
l
f
Fig. 3. Finite element model of an elastic slab resting on a Winkler foundation.
(b)
Fig. 4. Computed displacements for an elastic slab resting on a Winkler foundation: (a) without friction and (b)
with friction.

17. LT. Campos et al., Numerical analysis of contact problems with friction in elastostatics 837
As a second example, we compute the deformation of an elastic slab on a Winkler
foundation on which frictional forces are developed. We also solve this problem for the
frictionless case. The mesh of 40 biIinear elements is shown in Fig. 3 together with the data for
the problem.
Computed deformed shapes and contact stresses are shown in Figs. 4 and 5. To emphasize
the effects of friction, a coefficient of friction of unity was used. From Fig. 5 we easily identify
stick and shdding conditions on the contact part of the boundary according to 1~1~1
<lo,,/ or
[uT/ = iani because now z+ = 1. In Fig. 4, we observe that in the case without friction not only
is the contact area smaller and the deformation larger, but also, when friction is present plane
sections of the slab do not remain plane during the deformation.
Two final examples are cited which also fall into category (ii) mentioned previously: Contact
problems with friction in which the normal stress on the contact surface is unknown in
advance. Both are rigid punch problems with friction. Specifically, they involve the indentation
of elastic half-spaces by rigid punches (spherical in the case of the third example, and a flat
annular punch in the case of the last example). Both problems have an axisymmetric
geometry, a half-section of which is shown in Figs. 6 and 10, with the corresponding computed
deformed configurations shown in Figs. 7 and 11, respectively. In both cases, biquadratic,
axisymmetric isoparametric elements were used (40 for the first case and 50 for the second).
El Frictional Stress
0 Contact Pressure
Fig. 5. Computed frictional stress and contact pressure along surface AB.

18. 838 L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics

19. L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics 839
250
200
150
100
50
0
Contact pressure:
- Classical Solution (no friction)
o Numerical Solution (no friction)
q Numerical Solution (with friction)
Stress due to friction force:
A Numerical Solution
Fig. 8. Comparison between distribution of computed pressure and stress due to friction and a classical Hertz
solution along surface AB.
For each case, calculations were performed with a penalty parameter cl of lo-” and a
regularization parameter E of lo-’ for the spherical punch and 10F4 for the annular punch,
respectively.
Convergence was obtained on the spherical punch problem with ten complete iterations and
for the flat annular punch with five complete iterations.
In the case of the spherical punch, the classical Hertz solution is known for the frictionless
case. Computed contact pressures for this problem, with and without friction, are compared
with the Hertz solution in Fig. 8. Notice that the presence of friction does produce an increase
in contact pressure of around six percent, and a decrease in the contact surface in such a way
that the total area under the normal pressure curves is constant, for it is a measure of the total
applied force which was the same for both the frictionless and frictional cases. The resolution
of the tangential friction stress is more difficult and required a refinement of the mesh near the

20. 840 L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
separation point between the portion of the contact area in full adhesion with the punch and
that where slidding takes place (see Fig. 8). The computed frictional stress is shown in Fig. 8 and
the computed radial and axial stress components, with and without friction, are shown in Fig. 9
compared with the classical Hertz solution. According to Coulomb’s law of friction slidding
takes place where laTl = Y&~~. Consequently, for this problem, slidding will begin at a point E
(Fig. 8) of the contact boundary where the value laTj is 0.6(a,J. Therefore, in a similar way as
in the first example of Section 4.2, all points on the part EF of the contact boundary AF are in
a slidding condition and the part GF of the tangential friction stress is equal to 0.6 times the
corresponding part of the normal pressure distribution curve (Fig. 8).
Similar calculations were performed for the annular punch for problems with and without
friction and these are compared with the analytical solution of Shibuya. Koizumi, and
Nakahara [22] for the case of full adhesion in Figs. 12 and 13. The computed deformed
configuration is shown in Fig. II. In Fig. 12 the radial (~1,) and axial (w,) components of the
displacement field for the analytic and numerical solutions for both the frictional and
frictionless cases are compared. ABCD represents the relative axial displacement with respect
Fig. 9. Comparison between stress distributions along AC’.

21. L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics 841
l-7
E =
1000
v = 0.3
“F
= 0.3 zyxwvutsrqponmlkjihgfedcbaZYXWVU
i’
I2 ------I
1
Fig. 10. Indentation of an elastic body by a flat annular rigid stamp: undeformed configuration.
Applied Force
I--- 1 D
Fig. 11. Computed deformed configuration.

22. 842 L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
'-0.25
no friction
with friction
I I Numerical Solution
0 no friction
. with friction
Fig. 12. Computed displacements and comparison with analytic solutions; from [22].
to the indentation (E”) of the upper surface of the elastic body, also designated in the same
way in the previous figures. We remark that for the frictional case U, = 0 on that part of the
possible contact boundary (Tc) defined by 0.5 I Ri < R,s 1.0, where Ri and R, designate
respectively the interior and exterior radius of the annular punch. We are therefore in the
presence of a case of adhesion. Fig. 13 represents the comparison between the computed
stresses and the analytic solution for the same problem, on the contact part BC of the possible
contact boundary. a,, represents the axial stress and a, the tangential stress which becomes
very large near the edges of the punch due to the presence of adhesion.
5. Summary, conclusions and future study
In this work a numerical method for the analysis of a class of contact problems in elasticity,
with Coulomb’s friction law, using finite element methods, was presented. This method was
developed based on the study of the static contact problem with friction, done by Duvaut and
Lions [7], and on the study of Signorini’s type of contact problems in elasticity, presented by
Kikuchi [l I] and Kikuchi and Oden [13].
Although the theoretical results apply only to the situation in which the contact pressure is
known, the numerical scheme is capable of handling some Signorini-type problems with
Coulomb’s friction, for which no existence theory is available.
For this type of problem, two main cases were considered. In the first, we studied the
indentation of an elastic half-space by a rigid spherical punch, in the absence of friction forces,
and we compared the numerical solution to the classical Hertz solution. Considering the

23. L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics 843
(Ri
--- no friction
- with friction
Numerical Solution
0 no friction
. with friction
Fig.13. Computed stresses and comparison with analytic solutions; from [22].
frictional effects the two numerical solutions were then compared. In the second place, we
studied the indentation of an elastic half-space by a flat annular rigid punch and compared both
the numerical solutions with and without friction with an analytic result for the case of
adhesion, studied by Shibuya et al. [22].
From these results we concluded the usefulness and accuracy of the numerical scheme used
in the study of this Signorini’s type of problems with friction. Therefore, a detailed analysis of

24. 844 L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
such problems, both analytically and numerically, will constitute an immediate and interesting
subject for future study. Namely, the complete study of the boundary value problem arising on
the formulation of Signorini’s problem with friction constitutes an interesting and very general
problem both in the mechanics and mathematical points of view. This is due to the fact that
the existence of solutions to (1.2) and (1.3) is still an open question because P,,(U) is unknown
on r, and although we may consider u E (IIZ’(R))~ which will imply cr, E H”‘(T,.) it is then
impossible to give meaning to ]a,l. As this term arose directly from the virtual work due to
friction forces and is therefore a direct consequence of the friction law used, it seems to be
necessary to generalize or to consider laws of friction other than of the Coulomb type.
Recently, based on the physical fact that a,, represents the quotient of a force by an area.
Duvaut [6] proposed a non-local friction law to overcome this difficulty in Signorini’s problem
with friction, by considering a regularization a: of a,,, that is, by considering a mapping from
H-“*(rc) into L”(T,) preserving positiveness. it was possible to prove existence and unique-
ness of solutions of this new problem.
Another important aspect for future research is the study of the dynamic analogues to the
problems mentioned in this work, for Coulomb’s friction should be formulated in terms of
velocities instead of displacements. and only a quasi-static assumption on the problems studied
allowed the present simplification.
Acknowledgment
The support of the National Science Foundation under contract NSF ENG 7547846 and the
U.S. Air Force Office of Scientific Research under contract F-49620-78-C-0083 is gratefully
acknowledged.
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of contact problems with friction in elastostatics 845
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