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A numerical analysis of a class of contact problems with friction in elastostatics
COMPUTER METHODS IN APPLIED MECHANICS NORTH-HOLLAND PUBLISHING COMPANY
A NUMERICAL
AND ENGINEERING
34 (1982) 821-845
ANALYSIS OF A CLASS OF CONTACT WITH FRICTION IN ELASTOSTATICS
L.T. CAMPOS, The Texas Institute for Computational
J.T. ODEN
Mechanics,
PROBLEMS
and N. KIKUCHI
The University of Texas at Austin, Austin, U.S.A.
TX 78712,
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 twodimensional 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 problems. 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 problems, 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 00457825/82/0000-0000/$02.75
@ 1982 North-Holland
L.T. Campos et al., Numerical
822
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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. Nevertheless, 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 complications 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 following system of equations and inequalities, Cij(U),j
+fi
uj = 0
=
0
Vii(U)
T
on TD,
=
CT,,n,= ti
a, = 0 .
aT=O
if u,
a”
Us = 0
if (all<
a, < 0, 3 A L 0 u T = -hul
EijklUk,l
are characterized
by the
in 0 ,
on rF,
Y~(cF~[. if u, = s.
such that
if laTl = v,la,l
.
in this study
,
I
on rC. .
(1.1)
L.T. Campos et al., Numerical analysis 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. vector; u = u(x), where x = (x1, x2, . . . , xN) is a - U = (Ul, 4.. . ) uN) is the displacement 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,. 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 boundary r; a, is the normal stress on the boundary
-fiare components
Un
=
Un(U) =
Uij(U)ninj
and UTi are the components OTT= g’ri(u) =
fTij(U)nj
=
Eijk&.Jtinj
to the
,
of the stress vector tangent to r, -
a,(u)ni
.
n = normal displacement of particles on the boundary r. -Un=U' - 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)
x24
L.T.
Campos
et al..
Numerical
analysis
of contact problems
with friction
in elastostatics
Here,
v = {u = (II,,
u2, . . .,
uN) E
(H’(fi))”
1y(v) = 0 a.e. on
r,} .
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 this work we employ notations and conventions component of 2, on Tc. Here and throughout 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 displacement 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 p resents 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
IIvII1= (I n
(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). Consequently, 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,
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
ui = 0
on r,,
lwl
~UT[= g+3
ti = CT&
on r~,
eIEl=g
onTc,
orire, h 10
82.5
(1.6)
such that UT= -haT
on rC.
We can then replace i(tl, 0) of (1.3) by (1.7) and f E V’ becomes
(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 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
L.T. Campos et al., Numerical
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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 energy functional, existence, uniqueness,
and a perturbed
variational
principle
2.1. The 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) = $a(v, v)- f(v)+j(v)
(2.1)
represents the potential energy associated with the static friction problem for the Coulomb law (g prescribed), where the same notation used in describing the variational principle (1.9) was used. The usefulness of considering (2.1) will be clear in what follows. 2.2. Existence and uniqueness We now summarize the major results concerning existence and uniqueness of solutions obtained by Duvaut and Lions [7], for the simplified problem where g is prescribed. The most direct approach to the issue of existence of solutions to (1.9) is to call upon and variational inequalities (see, e.g.. the following theorem from convex analysis P31). THEOREM of the form sequentially possesses at inequality
2.1. Let V be a reflexive Banach space and I : V + E a proper, coercive, functional I = F + + where F is convex and Gateaux-differentiable and, therefore. weakly lower semicontinuous on V, and C$ is convex and lower semicontinuous. Then I least one minimizer on V. Moreover, any minimizer u of I satisfies the variational
(DF(u),u-u)++(v)-4(u)d
(2.2)
VU’S V
where DF(u) is the Gateaux differential of F at u and ( . , . ) denotes duality pairing on V' X V. In the present F(v)=
application ia(v, v)-f(v),
of this theorem,
I is given by (2.1) and one may take
4(v)=i(v).
Under conditions (1.4) F is strictly convex and Gateaux verified that the friction functional i is convex and continuous
(2.3)
differentiable on V. It is easily on V. Thus, all of the conditions
L.T. Carnpos et al., Numerical analysis 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 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)1q(r)
v r E I? ,
where R is the finite-dimensional space of infinitesimal R 3 a(r, r) = 0). Under the stronger condition,
(2.4) rigid motions
of the body (r E
(2.5)
iw - If( 2 cll& If t-E R (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 uniqueness theorem.
existence
and
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 - u2 E 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)
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L.T. Campos et al., Numerical
analysis
of contact problems with friction in elastostatics
for given I > 0, wherein VT, of course, is the tangential W e easily verify vector v on rC (y(q)E H”‘(r,)). C#&(u+ ev) E C’[O, l] f or any U,V E V. Our approximation
component of the trace y(v) of the that v+&(v) is convex and that of j is then
id4 = I,,g4eW ds. The following
result can be proved
(2.7) by a straightforward
calculation.
LEMMA 2.1. The functional j E : 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 @j,(u),
v) =
That je provides LEMMA
2.2.
I,,g$ &(U + eV)ds.
hrl
an approximation
If g E L”(T,),
of j is established
(2.8)
in the next lemma.
g 2 0 a.e., then, for all v E V, (2.9)
[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
On sets of nonzero +_ V T-
0 VT
calculation
measure
gives
on Tc-, we define
iflvTIS&. if lvT1 > E
.
Then, [j(v) - jE(v)l 5 I,, g] Iv+] - lv$l+ E/21 ds +,I,,. gl gds+g
IvTI+ t1i2&)i”T171 ds
g ds,
from which (2.9) follows. Having functional,
approximated
j by jF, we can
now
introduce
the
regularized
potential
energy
L.T. Campos et al., Numerical
I,:
v-+/x;
analysis of contact problems with friction in elastostatics
IE(0)=~a(u,v)-f(u)+j,(v).
829
(2.10)
This functional is strictly convex, coercive for meas r, > 0, and Gateaux differentiable of V. Thus, we have the following theorem.
on all
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 THEOREM
2.3. Let
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. 2.4. Let u be the solution of (1.9) and u, the solution of (2.11) for fixed E > 0. Then, there exists a constant C independent of E such that THEOREM
(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) - WjE(uE),u, -
u>.
Since jE is convex and Ghteaux differentiable, &(u) - jE(u,) 2
U-
UE)
iE(uE)l
+ IL(u)
(Qi(U=),
Thus, using (1.4) we have
m0llu- u,llT~Ii
-
-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,,.
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 of regular finite-element approximations of Sobolev spaces (see [4]): If functions forming the basis of V,, contain complete polynomials of degree rk(and V,,, where Pk(fl) is the space of polynomials of degree k) and if we are @-zY(R))~ n V, m > 0, then there exists a constant C, independent of 0 and h, and u,, E V,, such that p = min(k + 1 - s, m - s),
property the shape (Pk(0))” C given u E an element
s = 0, 1 .
(3.1)
In addition, mild restrictions on R and the spaces V,,, we can expect estimates to hold on traces of functions on a0 and to hold for negative s (see [2]),
of the type (3.1)
(3.2) with p possibly a negative real number. Having constructed such finite dimensional spaces, problem (1.9) takes the form: Find u,, E V, such that
4% % - wI)+j(Vh)-j(Uhpf(uh Likewise, the approximation such that, for given E > 0,
of the regularized
a(G, vh)+ (Dj&E), Uh)= f(Q)
v Uh
- Uh)
the finite
E
problem
v Vh (5 Vh .
element
approximation
v,. (2.11) consists
of
(3.3) of seeking
U; E V,,
(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 we may conclude that the results concerning existence and dimensional approximation, 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 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 interpolation estimates (3.1) and independent of h such that
fl V be the solution of (1.9) and uh the solution of (3.3). Let (3.2) hold and let k I 1. Then there exists a constant
831
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics jlu - UJ, 5 Ch .
PROOF.
(3.6)
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,,)
.
Noting that a(&,, t),, - u,) = a(&, z),,- u + u - u,,) ,
@(u, 2, - u) = a(u, 2, - 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(& -f(vh
-
z)h
-
u)
U)+j(D)--(U)+j(2)h)-j(Uh).
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
(3.7)
v,.
From (1.4) we have mCJl/ua(u
-
uhjj:
uh,
u
s
-
a(u
vh)
-
s
uh,
M(lu
u
-
-
(3.8)
uh) , uhlll
[(u
-
vhlll
(3.9)
.
Using (3.1) the second term on the right-hand side gives for any uh, a(%
oh
-
u)
s
[(u(lO
(Iu
-
vh[(O
5
cIIIu112
II”
-
VhllO
g
c21jull%2
*
(3.10)
On the other hand, using (3.1) again, we obtain f(u -
uh)
s
llfll0
IIu
-
vh/CI
s
cllfllIJh*11412
(3.11)
.
Next we make use of (3.2) together with the trace theorem to obtain,
5 ~~11~l1~c?.r,~*11~dl~/*.r~ 5 Gll~ll1/2.~~~*11~02,R .
We have thus estimated Young’s inequality, ubs(1/4c)u*+&b*
all terms in (3.7). Introducing
Qu,bEIW,QE70,
(3.8H3.12)
(3.12) into (3.7) and applying
x32
L. T. Cumpos
et ul.. Numerical
analysis
of contact problems
to the term M/u - ~~11,//u - z+,//, and simplifying,
we obtain
with frictim
in elastostatics
the estimate
(m,,- F)[lU- u,,llf-=(SC34llull;+ C,llz&+ qflloII4 + C~llglllir.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 of (3.12),
on g we have, instead
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 approximation problem (3.4), which we state in the following.
THEOREM
final
error
estimate
for
the
.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.
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 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.
L.T. Campos et al., Numerical
The steps in Step 1. First, The idea here without friction pressures. For Oden, Kikuchi, friction. In this
analysis of contact problems with friction in elastostatics
833
the algorithm for the analysis of the general case (ii) are listed as follows: a finite element approximation of the problem without friction is obtained. is to compute, as a first approximation, normal contact pressures produced to be used later as data for a problem with friction but with prescribed normal this purpose we employ the reduced-integration, exterior penalty method of and Song [18] and Kikuchi and Oden [13] for contact problems without 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 I-C
ds = J[(uZ - s)+z),,n] ,
(4.2)
where J( ) represents a numerical quadrature rule and u “hLis 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 Note that this problem is nonlinear: The function is, therefore, unknown. However, any of several problem without difficulty. In the present study scheme with projection described in, for example, Step 2. Having calculated ui’ for a specified normal contact pressure by setting
uh
E vh .
(4.3)
(u;k - s)+ is the positive part of (u;A - s) and iteration schemes can be used to solve this we employ the successive over-relaxation Kikuchi and Oden [13]. ,sl and h, we calculate nodal values of the
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 (TEL for 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
x34
LT. Cumpos et al., Numerical analysis of contact problems with friction in eiastostatics
(4.5) and define
(4.6) for any Us, 2)hE V,, where &(vh) is defined Then a correction UE of 11;’ for frictional
in a similar way as in (2.2). 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.
L.T. Campus et al., Numerical analysis of contact problems with friction in elastostatics
r
E =
Pressure = 3OO/unit length
l,OOO/(length)'
v = 0.3 vF
835
length
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 100
i
80 -
60 -
1
(b) Deformed
2
5
6
7
a
sliding part
Configuration stick +
Fig. 2. Computed
4
3
part
4-i
results: (a) frictional stresses along surface CD and (b) the deformed
finite element
mesh.
X36
L. T. Campos et al.. Numerical
analysis
of contact problems with friction in elastostutics
Al
f
Fig. 3. Finite element
model
of an elastic slab resting
on a Winkler
foundation.
(b)
Fig. 4. Computed with friction.
displacements
for an elastic
slab resting
on a Winkler
foundation:
(a) without
friction
and (b)
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
El Frictional Stress 0 Contact Pressure
Fig. 5. Computed
frictional stress and contact pressure along surface AB.
838
L.T. Campos
et al., Numerical
analysis
of contact problems
with friction in elastostatics
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
839
250
Contact pressure:
200 -
Classical Solution (no friction) o
Numerical Solution (no friction)
q
Numerical Solution (with friction)
150 Stress due to friction force:
A
Numerical Solution
100
50
0
Fig. 8. Comparison between solution along surface AB.
distribution
of computed
pressure
and stress
due to friction
and a classical
Hertz
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
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’.
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
l-7
E
= 1000
v = 0.3 “F
= 0.3
1
i’
------I
I2 Fig. 10. Indentation
of an elastic body by a flat annular
Applied
rigid stamp:
undeformed
Force
1
I---
Fig. 11. Computed
deformed
D
configuration.
configuration.
841
842
L.T. Campos
et al., Numerical
analysis
of contact problems
with friction in elastostatics
'-0.25
no friction with friction
I
Fig. 12. Computed
displacements
I
and comparison
Numerical Solution 0
no friction
.
with friction
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 the axial stress and a, the tangential stress which becomes contact boundary. a,, represents 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
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
843
(Ri
--- no friction -
with friction
Numerical Solution
Fig.13. Computed
stresses and comparison
frictional effects the two numerical studied the indentation of an elastic the numerical solutions with and adhesion, studied by Shibuya et al. From these results we concluded in the study of this Signorini’s type
0
no friction
.
with friction
with analytic solutions;
from [22].
solutions were then compared. In the second place, we half-space by a flat annular rigid punch and compared both without friction with an analytic result for the case of [22]. the usefulness and accuracy of the numerical scheme used of problems with friction. Therefore, a detailed analysis of
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 uniqueness 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.
References [l] R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975). [2] I. Babuska and A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method. in: A.K. Aziz. ed., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Academic Press, New York. 1072) pp. l-Xc). [3] H. Brezis. Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces d’Hilhert (North-Holland, New York, 1973). [4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam. lY78). Seminaires de Mathematiques SupCrieur\ [S] P.G. Ciarlet, Numerical analysis of the finite element method. (Presses de L’UniversitC de Montreal, Montreal, 1976). [6] G. Duvaut, Problemes mathematiques de la mecanique-Equilibre d’un solide Clastique avec contact unilateral de frottement de Coulomb. C.R. Acad. Sci.. Paris. No. 290. Serie A-263, 265. 1980. [7] G. Duvaut and J.L. Lions. Les Inequations en Mecanique et en Physique (Dunod, Paris. 1072). [8] I. Ekland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam. 1976). [9] R. Glowinski, J.L. Lions and R. Tremolieres, Analyse Numerique des Inequations Variationelles, Vols. I. 7. (Dunod, Paris, 1976). [lo] J.J. Kalker, The computation of three-dimensional rolling contack with dry friction. Internat. J. Numer. Meths. Engrg. 14 (1979) 12951307. Ph.D. dissertation. The University of [ 1 l] N. Kikuchi, Analysis of contact problems using variational inequalities. Texas at Austin, 1977.
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics [12] N. Kikuchi, [13] [14] [15] [16] [17] [18]
[19] [20] [21] [22]
[23] [24]
845
A minimization problem for non-differentiable functionals arising in a class of frictional contact problems, Ninth Southwestern Graduate Research Conference in Applied Mechanics, 1978. N. Kikuchi and J.T. Oden, Contact problems in elasticity, TICOM Rept. 79-8, The University of Texas at Austin, 1979. J.L. Lions and G. Duvaut, Un probleme d’elasticite avec Frottement, J. Mecanique 3 (1971) 409-420. J.J. Moreau, Proximite et dualite dans un espace Hilbertien, Bull. Sot. Math. France 93 (1965) 273-299. J.T. Oden, Applied Functional Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1979). J.T. Oden, RIP methods for Stokesian flows, TICOM Rept. 80-11, The University of Texas at Austin, 1980. Oden J.T., N. Kikuchi and Y.J. Song, Reduced integration and exterior penalty methods for finite element approximation of contact problems in incompressible elasticity, TICOM Rept. 80-2., The University of Texas at Austin, 1980. J.T. Oden and J.N. Reddy, An Introduction to the Mathematical Theory of Finite Elements (Wiley, New York, 1976). P.D. Panagiotopoulos, On the unilateral contact problem of structures with a non-quadratic strain energy density, Internat. J. Solids and Structures 13 (1977) 253-361. M.J. Sewell, On dual approximation principles and optimization in continuum mechanics, Philos. Trans. Roy London Ser. A 265 (1969) 319-351. T. Shibuya, T. Koizumi and I. Nakahara, An elastic contact problem for a half-space idented by a flat annular rigid stamp in the presence of adhesion, Proc. 25th Japan National Congress for Applied Mechanics, Vol. 25, pp. 497-504. Y.J. Song, J.T. Oden and N. Kikuchi, Discrete LBB conditions for RIP-finite element methods, TICOM Rept. 80-7, The University of Texas at Austin, 1980. J.R. Turner, The frictional unloading problem on a linear elastic half-space, J. Inst. Math. Appl. 24 (1979) 439-469.
A NUMERICAL
AND ENGINEERING
34 (1982) 821-845
ANALYSIS OF A CLASS OF CONTACT WITH FRICTION IN ELASTOSTATICS
L.T. CAMPOS, The Texas Institute for Computational
J.T. ODEN
Mechanics,
PROBLEMS
and N. KIKUCHI
The University of Texas at Austin, Austin, U.S.A.
TX 78712,
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 twodimensional 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 problems. 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 problems, 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 00457825/82/0000-0000/$02.75
@ 1982 North-Holland
L.T. Campos et al., Numerical
822
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. Nevertheless, 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 complications 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 following system of equations and inequalities, Cij(U),j
+fi
uj = 0
=
0
Vii(U)
T
on TD,
=
CT,,n,= ti
a, = 0 .
aT=O
if u,
a”
Us = 0
if (all<
a, < 0, 3 A L 0 u T = -hul
EijklUk,l
are characterized
by the
in 0 ,
on rF,
Y~(cF~[. if u, = s.
such that
if laTl = v,la,l
.
in this study
,
I
on rC. .
(1.1)
L.T. Campos et al., Numerical analysis 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. vector; u = u(x), where x = (x1, x2, . . . , xN) is a - U = (Ul, 4.. . ) uN) is the displacement 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,. 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 boundary r; a, is the normal stress on the boundary
-fiare components
Un
=
Un(U) =
Uij(U)ninj
and UTi are the components OTT= g’ri(u) =
fTij(U)nj
=
Eijk&.Jtinj
to the
,
of the stress vector tangent to r, -
a,(u)ni
.
n = normal displacement of particles on the boundary r. -Un=U' - 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)
x24
L.T.
Campos
et al..
Numerical
analysis
of contact problems
with friction
in elastostatics
Here,
v = {u = (II,,
u2, . . .,
uN) E
(H’(fi))”
1y(v) = 0 a.e. on
r,} .
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 this work we employ notations and conventions component of 2, on Tc. Here and throughout 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 displacement 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 p resents 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
IIvII1= (I n
(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). Consequently, 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,
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
ui = 0
on r,,
lwl
~UT[= g+3
ti = CT&
on r~,
eIEl=g
onTc,
orire, h 10
82.5
(1.6)
such that UT= -haT
on rC.
We can then replace i(tl, 0) of (1.3) by (1.7) and f E V’ becomes
(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 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
L.T. Campos et al., Numerical
826
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 energy functional, existence, uniqueness,
and a perturbed
variational
principle
2.1. The 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) = $a(v, v)- f(v)+j(v)
(2.1)
represents the potential energy associated with the static friction problem for the Coulomb law (g prescribed), where the same notation used in describing the variational principle (1.9) was used. The usefulness of considering (2.1) will be clear in what follows. 2.2. Existence and uniqueness We now summarize the major results concerning existence and uniqueness of solutions obtained by Duvaut and Lions [7], for the simplified problem where g is prescribed. The most direct approach to the issue of existence of solutions to (1.9) is to call upon and variational inequalities (see, e.g.. the following theorem from convex analysis P31). THEOREM of the form sequentially possesses at inequality
2.1. Let V be a reflexive Banach space and I : V + E a proper, coercive, functional I = F + + where F is convex and Gateaux-differentiable and, therefore. weakly lower semicontinuous on V, and C$ is convex and lower semicontinuous. Then I least one minimizer on V. Moreover, any minimizer u of I satisfies the variational
(DF(u),u-u)++(v)-4(u)d
(2.2)
VU’S V
where DF(u) is the Gateaux differential of F at u and ( . , . ) denotes duality pairing on V' X V. In the present F(v)=
application ia(v, v)-f(v),
of this theorem,
I is given by (2.1) and one may take
4(v)=i(v).
Under conditions (1.4) F is strictly convex and Gateaux verified that the friction functional i is convex and continuous
(2.3)
differentiable on V. It is easily on V. Thus, all of the conditions
L.T. Carnpos et al., Numerical analysis 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 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)1q(r)
v r E I? ,
where R is the finite-dimensional space of infinitesimal R 3 a(r, r) = 0). Under the stronger condition,
(2.4) rigid motions
of the body (r E
(2.5)
iw - If( 2 cll& If t-E R (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 uniqueness theorem.
existence
and
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 - u2 E 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)
828
L.T. Campos et al., Numerical
analysis
of contact problems with friction in elastostatics
for given I > 0, wherein VT, of course, is the tangential W e easily verify vector v on rC (y(q)E H”‘(r,)). C#&(u+ ev) E C’[O, l] f or any U,V E V. Our approximation
component of the trace y(v) of the that v+&(v) is convex and that of j is then
id4 = I,,g4eW ds. The following
result can be proved
(2.7) by a straightforward
calculation.
LEMMA 2.1. The functional j E : 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 @j,(u),
v) =
That je provides LEMMA
2.2.
I,,g$ &(U + eV)ds.
hrl
an approximation
If g E L”(T,),
of j is established
(2.8)
in the next lemma.
g 2 0 a.e., then, for all v E V, (2.9)
[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
On sets of nonzero +_ V T-
0 VT
calculation
measure
gives
on Tc-, we define
iflvTIS&. if lvT1 > E
.
Then, [j(v) - jE(v)l 5 I,, g] Iv+] - lv$l+ E/21 ds +,I,,. gl gds+g
IvTI+ t1i2&)i”T171 ds
g ds,
from which (2.9) follows. Having functional,
approximated
j by jF, we can
now
introduce
the
regularized
potential
energy
L.T. Campos et al., Numerical
I,:
v-+/x;
analysis of contact problems with friction in elastostatics
IE(0)=~a(u,v)-f(u)+j,(v).
829
(2.10)
This functional is strictly convex, coercive for meas r, > 0, and Gateaux differentiable of V. Thus, we have the following theorem.
on all
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 THEOREM
2.3. Let
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. 2.4. Let u be the solution of (1.9) and u, the solution of (2.11) for fixed E > 0. Then, there exists a constant C independent of E such that THEOREM
(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) - WjE(uE),u, -
u>.
Since jE is convex and Ghteaux differentiable, &(u) - jE(u,) 2
U-
UE)
iE(uE)l
+ IL(u)
(Qi(U=),
Thus, using (1.4) we have
m0llu- u,llT~Ii
-
-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,,.
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 of regular finite-element approximations of Sobolev spaces (see [4]): If functions forming the basis of V,, contain complete polynomials of degree rk(and V,,, where Pk(fl) is the space of polynomials of degree k) and if we are @-zY(R))~ n V, m > 0, then there exists a constant C, independent of 0 and h, and u,, E V,, such that p = min(k + 1 - s, m - s),
property the shape (Pk(0))” C given u E an element
s = 0, 1 .
(3.1)
In addition, mild restrictions on R and the spaces V,,, we can expect estimates to hold on traces of functions on a0 and to hold for negative s (see [2]),
of the type (3.1)
(3.2) with p possibly a negative real number. Having constructed such finite dimensional spaces, problem (1.9) takes the form: Find u,, E V, such that
4% % - wI)+j(Vh)-j(Uhpf(uh Likewise, the approximation such that, for given E > 0,
of the regularized
a(G, vh)+ (Dj&E), Uh)= f(Q)
v Uh
- Uh)
the finite
E
problem
v Vh (5 Vh .
element
approximation
v,. (2.11) consists
of
(3.3) of seeking
U; E V,,
(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 we may conclude that the results concerning existence and dimensional approximation, 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 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 interpolation estimates (3.1) and independent of h such that
fl V be the solution of (1.9) and uh the solution of (3.3). Let (3.2) hold and let k I 1. Then there exists a constant
831
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics jlu - UJ, 5 Ch .
PROOF.
(3.6)
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,,)
.
Noting that a(&,, t),, - u,) = a(&, z),,- u + u - u,,) ,
@(u, 2, - u) = a(u, 2, - 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(& -f(vh
-
z)h
-
u)
U)+j(D)--(U)+j(2)h)-j(Uh).
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
(3.7)
v,.
From (1.4) we have mCJl/ua(u
-
uhjj:
uh,
u
s
-
a(u
vh)
-
s
uh,
M(lu
u
-
-
(3.8)
uh) , uhlll
[(u
-
vhlll
(3.9)
.
Using (3.1) the second term on the right-hand side gives for any uh, a(%
oh
-
u)
s
[(u(lO
(Iu
-
vh[(O
5
cIIIu112
II”
-
VhllO
g
c21jull%2
*
(3.10)
On the other hand, using (3.1) again, we obtain f(u -
uh)
s
llfll0
IIu
-
vh/CI
s
cllfllIJh*11412
(3.11)
.
Next we make use of (3.2) together with the trace theorem to obtain,
5 ~~11~l1~c?.r,~*11~dl~/*.r~ 5 Gll~ll1/2.~~~*11~02,R .
We have thus estimated Young’s inequality, ubs(1/4c)u*+&b*
all terms in (3.7). Introducing
Qu,bEIW,QE70,
(3.8H3.12)
(3.12) into (3.7) and applying
x32
L. T. Cumpos
et ul.. Numerical
analysis
of contact problems
to the term M/u - ~~11,//u - z+,//, and simplifying,
we obtain
with frictim
in elastostatics
the estimate
(m,,- F)[lU- u,,llf-=(SC34llull;+ C,llz&+ qflloII4 + C~llglllir.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 of (3.12),
on g we have, instead
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 approximation problem (3.4), which we state in the following.
THEOREM
final
error
estimate
for
the
.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.
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 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.
L.T. Campos et al., Numerical
The steps in Step 1. First, The idea here without friction pressures. For Oden, Kikuchi, friction. In this
analysis of contact problems with friction in elastostatics
833
the algorithm for the analysis of the general case (ii) are listed as follows: a finite element approximation of the problem without friction is obtained. is to compute, as a first approximation, normal contact pressures produced to be used later as data for a problem with friction but with prescribed normal this purpose we employ the reduced-integration, exterior penalty method of and Song [18] and Kikuchi and Oden [13] for contact problems without 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 I-C
ds = J[(uZ - s)+z),,n] ,
(4.2)
where J( ) represents a numerical quadrature rule and u “hLis 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 Note that this problem is nonlinear: The function is, therefore, unknown. However, any of several problem without difficulty. In the present study scheme with projection described in, for example, Step 2. Having calculated ui’ for a specified normal contact pressure by setting
uh
E vh .
(4.3)
(u;k - s)+ is the positive part of (u;A - s) and iteration schemes can be used to solve this we employ the successive over-relaxation Kikuchi and Oden [13]. ,sl and h, we calculate nodal values of the
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 (TEL for 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
x34
LT. Cumpos et al., Numerical analysis of contact problems with friction in eiastostatics
(4.5) and define
(4.6) for any Us, 2)hE V,, where &(vh) is defined Then a correction UE of 11;’ for frictional
in a similar way as in (2.2). 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.
L.T. Campus et al., Numerical analysis of contact problems with friction in elastostatics
r
E =
Pressure = 3OO/unit length
l,OOO/(length)'
v = 0.3 vF
835
length
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 100
i
80 -
60 -
1
(b) Deformed
2
5
6
7
a
sliding part
Configuration stick +
Fig. 2. Computed
4
3
part
4-i
results: (a) frictional stresses along surface CD and (b) the deformed
finite element
mesh.
X36
L. T. Campos et al.. Numerical
analysis
of contact problems with friction in elastostutics
Al
f
Fig. 3. Finite element
model
of an elastic slab resting
on a Winkler
foundation.
(b)
Fig. 4. Computed with friction.
displacements
for an elastic
slab resting
on a Winkler
foundation:
(a) without
friction
and (b)
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
El Frictional Stress 0 Contact Pressure
Fig. 5. Computed
frictional stress and contact pressure along surface AB.
838
L.T. Campos
et al., Numerical
analysis
of contact problems
with friction in elastostatics
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
839
250
Contact pressure:
200 -
Classical Solution (no friction) o
Numerical Solution (no friction)
q
Numerical Solution (with friction)
150 Stress due to friction force:
A
Numerical Solution
100
50
0
Fig. 8. Comparison between solution along surface AB.
distribution
of computed
pressure
and stress
due to friction
and a classical
Hertz
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
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’.
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
l-7
E
= 1000
v = 0.3 “F
= 0.3
1
i’
------I
I2 Fig. 10. Indentation
of an elastic body by a flat annular
Applied
rigid stamp:
undeformed
Force
1
I---
Fig. 11. Computed
deformed
D
configuration.
configuration.
841
842
L.T. Campos
et al., Numerical
analysis
of contact problems
with friction in elastostatics
'-0.25
no friction with friction
I
Fig. 12. Computed
displacements
I
and comparison
Numerical Solution 0
no friction
.
with friction
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 the axial stress and a, the tangential stress which becomes contact boundary. a,, represents 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
L.T. Campos et al., Numerical analysis of contact problems with friction in elastostatics
843
(Ri
--- no friction -
with friction
Numerical Solution
Fig.13. Computed
stresses and comparison
frictional effects the two numerical studied the indentation of an elastic the numerical solutions with and adhesion, studied by Shibuya et al. From these results we concluded in the study of this Signorini’s type
0
no friction
.
with friction
with analytic solutions;
from [22].
solutions were then compared. In the second place, we half-space by a flat annular rigid punch and compared both without friction with an analytic result for the case of [22]. the usefulness and accuracy of the numerical scheme used of problems with friction. Therefore, a detailed analysis of
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 uniqueness 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.
References [l] R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975). [2] I. Babuska and A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method. in: A.K. Aziz. ed., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Academic Press, New York. 1072) pp. l-Xc). [3] H. Brezis. Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces d’Hilhert (North-Holland, New York, 1973). [4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam. lY78). Seminaires de Mathematiques SupCrieur\ [S] P.G. Ciarlet, Numerical analysis of the finite element method. (Presses de L’UniversitC de Montreal, Montreal, 1976). [6] G. Duvaut, Problemes mathematiques de la mecanique-Equilibre d’un solide Clastique avec contact unilateral de frottement de Coulomb. C.R. Acad. Sci.. Paris. No. 290. Serie A-263, 265. 1980. [7] G. Duvaut and J.L. Lions. Les Inequations en Mecanique et en Physique (Dunod, Paris. 1072). [8] I. Ekland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam. 1976). [9] R. Glowinski, J.L. Lions and R. Tremolieres, Analyse Numerique des Inequations Variationelles, Vols. I. 7. (Dunod, Paris, 1976). [lo] J.J. Kalker, The computation of three-dimensional rolling contack with dry friction. Internat. J. Numer. Meths. Engrg. 14 (1979) 12951307. Ph.D. dissertation. The University of [ 1 l] N. Kikuchi, Analysis of contact problems using variational inequalities. Texas at Austin, 1977.
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