Nonlinear variational inequalities in elastostatics

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Inr. J. E~/IR Jcr Vol 26. No. III. pp. 1041-10.51. I9XX PrInted in Circat Bnla~n. All npht\ reacrvcd

NONLINEAR

VARIATIONAL INEQUALITIES ELASTOSTATICS MUHAMMAD

Mathematics

Department.

Punjab

ASLAM University.

IN

NOOR

New Campus,

Lahore,

Pakistan

Abstract-Variational inequalities characterizing a class of nonlinear contact problems with Coulomb friction are considered. In a variational inequality formulation, the presence of friction leads to nonconservative nondifferentiable forms, which presents a main difficulty to apply the finite element method effectively. It is well known that the finite element error estimate is of order h’ ’ in the energy norm. In this paper. we use the smooth perturbation of the variational inequality to obtain the finite element error estimate. which is of order h in the energy norm. In fact, our estimates improve all of the previously known estimates for elliptic variational inequalities.

1.

INTRODUCTION

Variational inequalities theory as developed mainly by the Italian and French schools in the early 1960s and thereafter, constituted a significant extension of the variational principles and led to new and numerous applications in diverse fields of mathematical and engineering sciences. The variety of problems to which variational inequalities techniques may be applied is impressive and amply representative for the richness of the field. One of the charms of variational inequalities is that a large number of seemingly unrelated free and moving boundary value problems can be studied in a general and unified framework [see l-3 and references therein]. The general problem of equilibrium of elastic bodies on which frictional forces are developed is one of the most difficult and interesting problems from mathematical and solid mechanics points of view. Inherent in the friction problem is the free surface (contact area) problem of identifying a priori the unknown contact surface. However, it has been shown by Kikuchi and Oden [4] that the contact problems in elastostatic with friction can be characterized by a class of variational inequalities. In a variational inequality formulation, the location of the free boundary becomes an intrinsic part of the solution and no special devices are needed to locate it. In addition, the presence of friction leads to non-conservative forces which give arise to non-differentiable forms in a variational inequality formulation of these problems. In most cases, the issue of the existence of solutions to such problems is an open problem. Some special cases have been considered by Demkowicz and Oden [5] and Noor [6,7]. It should be remarked that the formulation of such problems as variational inequalities was originally studied by Duvaut and Lions [8] and they proved the unique existence of solutions of such problems for some special cases. In recent years, numerical techniques including finite element methods are being applied to overcome these difficulties. The main difficulty arises due to the non-differentiable terms in a The direct approximation of variational inequalities variational inequality formulation. containing non-differentiable terms by standard finite elements yields a discrete system. In this case, it has been shown by Pires and Oden [9], Glowinski et al. [lo] and Noor [ll] that the rate of convergence of the error estimate is of order h “* in the energy norm. The Lagrangian multiplier method is also being used to obtain the mixed finite element approximation of the variational inequalities. This approach involves the approximation of two independent fields and may be computationally inefficient. Another approach is to consider an F-differentiable perturbation of non-differentiable functions. This technique has been used quite effectively by Campos et al. [12] and Noor [13] to obtain the error estimates for finite element approximation of variational inequalities coupled with the potential energy functional, which are of order h in the energy norm. In this paper, using the perturbation (regularization) technique, we obtain the general error estimates for a class of nonlinear variational inequalities characterizing nonlinear contact problems in elastostatics with friction. Our results are quite general and

M. A. NOOH

1044

represent a significant improvement of ail the previously obtained error estimates tot variational inequalities of elliptic type that can be found in the literature. Special cases. which can be obtained from our main results, are also discussed. Following this introductory section, the system of differential equations and boundar! conditions for the nonlinear contact problem with friction are stated and characterized by a class of variational inequalities in terms of an unknown equilibrium displacement held. Existence of a unique solution is discussed in Section 3. A perturbed variational principle is introduced in Section 4. It has been shown that the non-differentiable form can be approximated closely by the perturbed differentiable form. In Section 5, we derive our main results.

2. FORMULATION

OF A VARIATIONAL

INEQUALITY

We here consider a genera1 class of nonlinear contact problems in elastostatics with friction. which are characterized by the following system of equations and inequalities: -aij(u).j

ajj(u)

=f;(“)~

u;=O

on

oijklUk,/

=

t,

=

in

Ejjkluk,,

Q

ID on

u.N-ss0,

rF

a(u)(u

G(U) d 0,

. N -s)

le(u)l < v~(cv(u))

implies

/c+(u)1 = pLs(a,(u))

implies there exists

a number

A2 0

such that

= 0

u-r = 0 uT = -Aa,

on I(.

(2.lj

1

The system (2.1) describes the Signorini problem in elastostatics for a case of foundation developing frictional forces upon contact which obey Coulomb’s law. Here we use the following notations and conventions, which are commonly used in the study of partial differential equations and contact problems [3,4]. 52 is the elastic body in a bounded open domain DBwith Lipschitz boundary r = r, U r, U rc. T,(T,) are portions of the boundary I’ on which the displacement (tractions) are prescribed. To is the contact surface on which the body may come in contact with the foundation upon the applications of loads; it is assumed throughout that r, fl rn = $, u = (u, , u2, . . , uN) is the displacement vector, u = U(X), where x = (xi, x2, . . . , x,) is a point in 52. aij(u) are components of the stress tensor; its value at a displacement u is o,(u) = E,,k$k,,, where Eijkl are the elasticities of material of which the body is composed. These are given functions of x satisfying the usual ellipticity (coercivity) and symmetry conditions. H is the space of admissible displacements v, that is H = {v E H’(Q); v = 0 a.e. on I,}, where H1(Q) is the usual Sobolev space of classes of functions with &-partial derivatives of order 1. The space H is a Hilbert space and is equipped with the energy norm

llv II2= 1

“i.j”i.j

dX

R

Here the commas denote partial differentiation: 6%

V

E---F

‘,I

dXj

.

We also use the notation

Ilvllb=/--v .vds W = H”‘* (r,) is the space of normal traces of the admissible displacements v on the contact surface Tc. If 12denotes the unit outward normal to I,-, u E H and y(u) is the trace of u on I(.,

Nonlinear

variational

inequalities

1045

in elastostatics

then y(u) . n E W. The space W is equipped with the norm

M is the constraint

set corresponding to the unilateral condition y(u) . n d s, that is where s is given in W and represents the initial gap between the body Q and the rigid foundation. In fact, M is a closed convex subset of H. M = {v E H; y(v).

n G s in W},

a(u, V) =

In

EijkPk,lui,j do

= the virtual work produced by the stresses. f is a continuous nonlinear differentiable external forces with values given by

functional

(2.2)

on H representing

virtual work of the

(2.3) Here J is the body force depending upon the displacement vector; ti is the tractions on rF and ds is the element surface area measure. We also assume fi E L(IF). S: W’+ L2(rc) is a smoothing linear operator from the dual space W’ representing a transformation of the normal stress uN(u) in W’ into a &(I,-). We assume that S is endowed with the property that S(a,(u)) Furthermore, it has been shown in [14] that

prescribed surface that & E L,(Q) and of W into L2(Ic) mollified stress in 20 a.e. on Ic.

(2.4)

II~M4)llL2(rc) 4 c1 Ilull

where Ci is a constant. p is the coefficient of friction of the contact surface. We assume that p E L,(I,), p > p. 2 0 a.e. on rc. We also assume that nonlinear given function f(u) defined by (2.3) satisfies the following: f(u) is antimonotone, that is

I (f(u) -f(u))(u

(2.5)

- VI dJi=20

R

and

(2.6) where r = r(t) is a non-decreasing function for t E R, l> 0. We note that the function f(u) differentiable and, (f’(u), V) = Jnf(u)u d_x (see [15]), from which it follows that f’(u) antimonotone and

b(u, u) =

I PS(dU)) Id =c

ds

is is

(2.7)

where S is a linear positive-valued compact operator from W’ into L&c). For the prescribed normal contact pressure, F, = u‘, is assumed to be given on all of Tc, that is p(a,(u)) = g, g is given in L,(Tc). Thus the contact surface Tc is known in advance and u, is not prescribed on r c. With these modifications, the problem (2.1) becomes:

Ui =O oijkluk,[

P

Q

rD

on =

fi

on

IFnl= g,

IuT( =g

ujj(u) = Eijk,uk,! in

=f;(“),

-“ij(u).j

rF

IuIT]
implies

implies

3A > 0:

uT =0 uT = --Au, I

on

rc

(2.8)

104(1

M.

tZ

NOOK

The problem (2.8) is introduced and studied by Noor [ 131, where using the perturbanon technique, he has shown that the error estimate for the finite element approximation is of order h in the energy norm. We also replace b(u, u) andf(u) defined by (3.7) and (7.3) respectively by j(u)=

J

* I:

bJ-I.l ds

(2.9)

1-c

and (2. 10) where F, is the prescribed normal contact pressure on Ic. With these preliminaries now established, using the technique of Oden and Pires [14] and Kikuchi and Oden [4], one can show that the nonlinear problem (2.1) can be characterized by a class of highly nonlinear variational inequalities, which is the motivation of our next result. Theorem 2.1 Let u E (H2(Q))N nM variational inequality

be a solution

of problem

(2.1),

a(u, 21- u) + b(u, u) - b(u, u) B (f’(U),

then

u - u),

u is also

a solution

VUEM

of the

(2.11)

and the converse is also true, in the distributional sense. Here f’(u) is the FrCchet differential of the nonlinear differentiable functional f(u) defined by (2.3). Remark 2.1. The variational inequality (2.11) characterizes the Signorini problem in elastostatics with friction. The inequality (2.11) 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 21 is 4 a(v, u). Thus a(u, in- u) is the work produced by the stresses through strains caused by the virtual displacement v - u. The functional f depending on the displacement respresents the work done by the external forces and the form b(. , .) is the work done by the frictional forces.

3. EXISTENCE Since the bilinear

form a(u, v) is symmetric,

RESULT so it is useful

to consider

the functional

defined

by Z[v] =

;u(v,v) + b(v,

v) -f(v)

(3.1)

associated with (2.1), which is known as the potential functional associated with the statistics friction problem for the Coulomb law. Here the bilinear form a(u, v) defined by (2.2) is positive and continuous. Furthermore, it has been shown in [14] that the form b(u, V) defined by (2.7) satisfies the following properties. (i) b( u , v ) 1s 1’mear in the first argument. (ii) b(u, v) is bounded, that is, there exists a constant y > 0 such that Ib(u, v)l d YllUll

IIVII~

VU?

UEH

(iii) For every u, 21, w E H, I@(& v> - @(u, w)l c (u, 21- w> and @(u, v f w) =Z@(u, u) + $(u, (iv) b(u, v) is convex in the second argument. It can be easily shown that the minimum of Z[v] defined

w)

by (3.1) can be characterized

by the

Nonlinear

variational

inequalities

1047

in elaatostatics

variational inequality (2.11). Noor [7] has studied the problem of the unique solution u E A4 of a more general variational inequality of the type a(u, I’ - U) + b(u, U) - b(u, u) 2 (A(u),

II - u),

where A is a nonlinear operator from M into H’ such that A(u) is a special case of (3.2), when A(u) =f’(u).

4. PERTURBED

VARIATIONAL

existence

Vu E M, E H’.

Clearly

of a

(3.2) inequality

(2.11)

PRINCIPLE

From (2.7), we can see that the form b(u, v) is non-differentiable in the second argument, and this presents a main difficulty in using the standard finite element techniques. To apply the finite element methods effectively, one usually considers an F-differentiable perturbation of the non-differentiable form. This type of perturbation, known as the regularization (smooth) technique. has been used to study non-differentiable functions arising in nonlinear operator theory and optimization. Noor [13] and Campos ef (II. [12] used this regularization technique as a basis to obtain the error estimates for the finite element approximation of some classes of variational inequalities. In this paper, we extend these ideas to derive the error estimates for the finite element approximation of the solution of the nonlinear variational inequality (2.1 I). We want to approximate problem (3.1) by a family of regularized problems, which lead to the solution of a variational inequality (2.11). We approximate the frictional form which is non-differentiable in the second variable, by a family of b(. ( .):H x H-R, functionals h,.(. , .), convex and differentiable in the second argument: h, (. , .): H x H + R:

b,(. , .) =

I

PLs(UN(U))&(~f) ds

(4.1)

l-c.

Here the functional

@)F: H+

L,(T,)

is defined

by

(4.2)

for given E > 0, where uT. is the tangential component of the trace y(u) of the vector u on lYc. Following the technique of Campos et al. [12] and using (2.4). we can show that the form 6(. , . ) can be approximated by b, (. , ). Lemma 4.1 If p E L,(T,.),

p b 0 a.e.,

then Qv E H,

IN4 1,)- b(~~,2J)lc (2 ll~ll~.~.~~ mWrcy) Il4lJ~ We now define

the regularized

(perturbed)

potential

energy

functional

(4.3) as

I,. : H + R I,[v]

= +(u,

v) + h,(u, u) -f(u)

(4.4)

The functional I,[v] defined by (4.4) is clearly strictly convex, coercive and F-differentiable on all of H. In this case, one can easily show that uF E H may be characterized by the variational equations a(~~, V) + (b:(u’, Here

the

partial

derivative

u’), v) = (f“(u’),

of b,(u, u) relative

to the

u), Vu E H second

argument

(4.5)

at (u, w) in the

M. A. NOOK

104x

direction of u is given by

Now the question is, does the solution of (4.5) converges to the exact solution u E M of (2.11) as E+O? The answer is yes, and this is the main motivation of our next result, and also provides us an estimate for the rate of convergence. Theorem 4.1 Let u E M be the solution of (2.11) and let 8” be the solution of (4.5), for fixed E > 0. If f’(u) is antimonotone, then I/u - rQ: s c IiUlll (4.7) where C is a constant independent of E. Proof. Taking v = uE in (2.11) and n = uE - u in (4.5) and adding these inequalities, obtain a(r4&- u, uE - U) Gb(u, uE)- b(u, u) - (b’(u”, u), U&--U) + (f’(l.c) -f’(u),

UE- EL>

6 b(u, uE) - b(u, u) - (b’(d, by the antimonotonicity second argument, so

of f’(u).

we

uE), uE - u>

Since bE(u, v) is convex and Frechet

b,(u, u) - bp(u, u’) 3 (b:(u”, tic), u’-

(4.8)

differentiable

a}

in the (4.9)

From (4.8), (4.9) and the coercivity of a(u, v), we have Ib(u, u’) - b,(u, u&)1+ jbE(u, u) - b(u, u)]

LYI@‘- u+

from which, it follows that, by invoking Lemma 4.1, DE - ull: s C II~l/i which is the required result.

5. FINITE

ELEMENT

ERROR

ESTIMATES

We now consider finite element approximation of the variational inequality (2.11) coupled with the energy functional (4.4). Following standard finite element techniques, we construct a partition of &?into a mesh of finite elements over which the displacements are approximated by piecewise polynomials. This defines a finite dimensional subspace S, of H. By constructing a sequence of regular refinements of the mesh, we generate a family {S,}, h > 0, of subspaces of H. It is well known [15] that the subspaces S, exhibit the following asymptotic interpolation properties. If u E (H’(S2))N, r 2 0, then there exists a constant C > 0 such that s = min(3, r - 1)

(5.1)

Let I’: denote the boundary of the mesh that approximates Ic and Z, denote the set of all nodal points e on I$. We assume that E:, c I’: t-l I o. As an approximation of the constraint set M, we consider

Mh = {CbhE C”(E)

: $h

=

Y (Vh)

* N,

vh

E sh,

cph(4

d

W%

Ve E Z,}

(5.2)

where S,, is the .&(Io) projection of s on the space W, of normal traces of functions in S,. Thus, in our discrete model of the friction problem (2.11), we impose the contact conditions only at the nodal points on I’$. Clearly, in general, A.&St n/r.

Nonlinear

variationa!

The finite element approximation such that

inequalities

of problem (2.11) takes the following form: find M,,E Mh

a(%, Uh - Uh) + b(&, %) - b(&> u/l) ?= (f’(u,), Similarly, the approximation that, for given E < 0,

1049

in elastostatics

of the regularized

a(ul;, VI) + (K(&

Vu, E Mh

u/l - &jr

(5.3)

problem (4.5) consists of finding uh E S,, such

G)> v/l) = (f’(G),

Vu, E s,

VA

(5.4)

It is evident that relations (5.3) and (5.4), and all the properties of those forms are carried from the continuous problems to its finite-dimensional approximate problems. We may conclude that all results concerning existence and uniqueness of solutions to the finite dimensional problems follow with similar conclusions. Using essentially the same technique as in Theorem 4.1, we have the following estimate for the finite dimensional case: /I% - Gil: =ZCe I/%l//l

(5.5)

where C is a constant independent of h and E, and &, and ui are the solutions of (5.3) and (5.4) respectively. In order to derive the general estimate for the error, we need the following results, which can be proved using the technique of Noor [16], Janovsky and Whiteman [17] and Westbrook 1181. Lemma 5. I

of h such that

There exist constants C, and C, independent

ll~(~~~)ll~*~*~ ss c1

(5.4)

and

ll~hlll =sG

(5.7)

From (5.7) and (5.5), we have the following estimate: II% - ~~ll1 =GG fi

(5.8)

where C3 is a constant independent of h and E. We now state and derive the general error estimates for u - uh. Theorem 5.2

Let u urn and uh E Mh be the solutions of (2.11) and (5.3) reSpiXtiVt?ly. coercive continuous bilinear form and b(u, v) satisfy the properties antimonotone, then for all v E M and uh E n/fht (i) Mh + M.

Let

a(u,

(i)-(iii).

v) be a If f’ is

a(lL - &, u - uh) s a(u - &,, u - uh) + a(u - &, u - vh) + (f’(u) a(& +

uh

-f’(vh), -

zfh

u)

+

-v)

+

(f’(a),

u-uh)

uh-t,)-+-‘i(uh,2/--uh)

cf’(u),

b(u - Z.Q,U - i&h)+ b(u, 2.~ -

V> +

b(u,,

U - uh)

(5.9)

(ii) M,, c M, a(u - U,,, td- &) s a(f.t - t%h,td- vh) + a(& vh - u) + (f’(&),

u - vh)

+ b(u - uh, U - tdh) + b(uh, U - vh)

(5.10)

Proof. (i) Mh # M. Since u E M and uh E Mh are solutions of (2.11) and (5.3) respectively, by adding (2.11) and (5.3), we have t(v EM, vh E Mh, a(& u) + ~(%r

uh)

c

(f’(u)> +

v b(u,

u)

-

u> -

b(u,

+

(f’(Uh), v)

vh +

b(wsr

ud

uh) -

b(u,,

vd

so

Subtracting a(u, uh) + a(~,, u) from both sides, using (,i)-~iii) and rearranging terms, we get a(u

-&,,

[A -u/,)

au(,.d

-

uh,

+

(f’(u)

+

(~‘(~~),

-

a(uh,

+ b(u,

7! -

l.,,,) + u(u

-f’(&*), 4,

uh -

Z!h -

-

u,, -

-

7!> -

71) f 71) $

Id/,, 11 -

7f> +

a(74

b(u

-

b(Uh,

(f’(u),

2.4 &,

u -

7?h) N -

l)h

2’4,)

14 -

tih)

(5.11)

7jh)

which is the required (5.9). (ii) Mh c 44. Setting 1) = uh in (5.11), we obtain a(u - Uh, U - u/J G ‘$u -

uh,

7.4 -

uh)

+

(f’(u)

-f’(%),

uh -

&)

a -

uh)

which can be written as a(u - uh, u - uh) s a(u - u,,, ld - v,,) + a(& t,h - u) + (f’(u), +

tf’(u>

@(u

-

u uh,

u -

-f’(Uh)t

uh-u)+b(u-Uh,U--h)

-f’(Uh)?

+ b(Uh, s

U - &z) + (f’(u)

u,,) 7&)

+ ff(u,

7,h -

a)

+

{f’(uh),

7.i -

tfh)

+b(u-ff,,u-uh)+b(u,,u-vh)

by using the antimonotonicity off’. Remark 5.1. We also note that if f is a linear continuous functional, then (f’(u), u) -f, V) a well known fact. In this case, the result of Theorem 5.1 is exactly the same as given in (91. Theorem 5.2

Let u E (H2(Q))N fi M be the solution of (2.11) and uh be the solution of (5.3). If f E L,(Q), the angle condition is satisfied; and Mh c M, then (5.12)

~~~-~h~~,=O(h)+C where C is any generic constant. Proof Now, in view of the relation (5.10) can be written in the form

(f’(u),

u) = jnf(u)u

dr, the general error estimate

a(u-u,,u-u,)~ff(u-Uh,U--h)+a(u,v,-u)+

iP

f(&&--h)dX

+ b(u - uh, 2.4- llh) + b(uh, U - tfh) By using coercivity and continuity of a(~, V) and the boundedness

ubsEa2f-

b2 4E’

Vu,btsR

and

E>O,

of b(u, u), we obtain

10s I

Nonlinear variational inequalities in elastostatics

by using the Sobolev embedding theorem; see Ciarlet [ 151 and Lemma 5.1, where Cs are generic constants independent of h and u. Let I,, be the operator of &-interpolation. Then, since 52 c RN, we have H2 c C”(Q), and u E Hz, implying I,u E M,, (see Ciarlet [15] and Glowinski ef al. [lo]). Taking v,, = I,+ in the inequality (5.13), we have IIU-u,ll:=I

IIU-bz41:+

c3 IIU

-4UIlL*(n)+

c2 Ilull

IIU

-bPIlL,(n)+ Cd

(5.14)

Thus from (5.1) and (5.14), it follows that Ilu-u,xll1=W)+C

Now using (5.8), (5.12) and the triangle inequality approximate problem (5.3), which we state as: Theorem

gives the final error estimate

for the

5 3

Under the assumptions of Theorems

5.2 and 5.1, we have

Ilu-uhlll=O(h+v’i+C)

where u and uh are the solutions of (2.11) and (5.4), respectively.

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[9) (lo] [ll] [12] [13] [14] [15] [16] [17]

[18]

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17 December

1987)