COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 24 (1980) 187-213© NORTH-HOLLAND PUBLISHING COMPANY

ANALYSIS OF CERTAIN UNILATERAL PROBLEMS IN VONKARMAN PLATE THEORY BY A PENALTY METHOD-PART 1.

A VARIATIONAL PRINCIPLE WITH PENALTY

K. OHTAKENational Aerospace Laboratory, Tokyo, Japan

],T, ODEN and N. KIKUCHITexas Institute for Computational Mechanics. The University of Texas at AustilL Austin, Texas, US.A,

Received R October 1979Revised manuscript received 15 November 1979

1. Introduction

This is the first of a two-part paper dealing with the analysis of a class of contact problemsin von Karman's large-deflection theory of thin clastic plates. Physically, the problemsconsidered herein involve the large transverse deflections and buckling of clamped and simplysupported plates, the lateral displacement of which is constrained by the presence of a rigid,plane, frictionless foundation parallel to the middle plane of the undeformed plate,

Such problems are very important in structural applications, particularly in analysis anddesign of stiffened panel structures, where lateral supports of thin plates are provided by strutsand beams. Moreover. the theory and methods developed in this work are easily extended to avariety of complex unilateral conditions which are frequently encountered in structural designproblems but which are usually ignored because of their inherent nonlinear character.

Mathematically, the problems considered here are characterized by a system of nonlinearpartial differential equations in the transverse displacement w of points on the middle plane ofthe plate, and the tangential components of displacement lia' a = L 2, subject to a unilateralconstraint of the form w 2: -b, where b is the initial distance between the middle plane and arigid frictionless plane below the plate. The classical theory of von Karman [1910] provides anacceptable model for such phenomena.

To provide focus for this study. two model problems are selected which exhibit the majorfeatures of nonlinear constrained plate problems and which fall naturally within the frame-work of the von Karman theory: (1) the problem of large deflections of clamped plates subject-ed to transverse loads and unilateral constraints and (2) the problem of large deflections andbuckling of simply supported plates subjected to in-plane boundary compressions (referred tohere as a thrusted plate) and unilateral constraints.

Part 1 of this investigation is devoted to the formulation and qualitative analysis ofnonlinear unilateral plate problems of the type described above. After some preliminaries arelisted in the section following this introduction, we develop variational principles for problems

188 K. Ohtake et aL Analysis of problems in von Karman plate theory by a penalty method - Part I

of this type which are characterized by systems of nonlinear variational inequalities. Theseprinciples are developed in section 3 of this paper. Similar principles have been developed byDuvaut and Lions [1974], Naumann [1975]. John and Naumann [1976], Do [19751 and others.However, these variational principles do not necessarily provide a convenient basis forcomputational methods.

Thus, in preparation for a study of finite element approximations and the construction ofnumerical methods for such problems, a penalty formulation is developed in section 4. Thisformulation involves the introduction of a penalty term which effectively adds to the totalpotential energy -(2ET' In (w + b): dx, where E is an arbitrary positive number and (w + b)_ =mine w + b), O}. Thus. this term is identically zero whenever w satisfies the constraint w + b ~ 0 butis positive if w violates the constraint.

Sections 5 and 6 are devoted to the qualitative analysis of the penalized variationalformulation of unilateral problems in nonlinear plate theory. It is shown in section 5 that theoperators appearing in the penalized variational problem can be characterized as pseu-domonotone and that. therefore, a constructive proof of the existence of solutions can beobtained. This fact is known and has been exploited for similar variational problems for vonKarman plates without penalty by Duvaut and Lions [1974] and Naumann [1975]; however,the most widely used techniques for studying nonlinear plate and shell problems is to use thefixed-point theorems of Leray and Schauder [19341 (see e.g. Knightly [1967]. and Bergerr 1967]). On the other hand, such nonconstructive methods are awkward for use in developingan approximation theory and numerical methods, and it is this fact which governed the choiceof the approach used here.

In section 5 we also address the question of the behavior of the penalized solutions (u .. w,)and show that. as E tends to zero, the solution of the penalized problem converges to solutionof the nonlinear plate problem with unilateral constraints. We are also able to show that thequantity E-1(W< + b)_ converges to a quantity -p, as E goes to zero, which represents thecontact pressure developed on the plate due to the unilateral constraint.

In addition, the question of regularity of solutions of the penalized problem is discussed insection 6. We prove regularity theorems for cases in which E > 0 and obtain estimates of thesolutions in terms of E. the initial gap b, and the loads f and Sa. For instance, for the clampedplate with unilateral constraints. we use the results of Agmon, DougHs. and Nirenberg [1959]and John and Naumann [19761 to show that if

f E LP(fl). 2 $ P < 00, then w. E W4·I'(fl) nWk2(fl)

for E > 0 and fl of class C4. Results of this type, including estimates, are derived for simply

supported plates. Finally section 6 is concluded with a brief discussion of the results of Duvautand Lions [1974] on the question of uniqueness of solutions for clamped, transversely loaded,unilaterally supported plates.

Part 2 of this paper is devoted to the finite element approximation and numerical analysis ofunilateral plate problems.

2. Preliminaries

We consider the large transverse deflections of a thin elastic isotropic plate whose middlesurface is an open bounded domain fl in fR2 with a Lipschitzian boundary r. According to

K. Ohtake et aI.. Analysis of problems in von Karozan plate theory by a penalty method -Part 1 189

von Karman's theory, the governing equations arc

D.12w - l(lT o/3(U, W )W,I3).a =/ } ,

IT al3(U' W).13 = 0

where

(2.1)

(2.2)

Here D = E/3/12(1 - VZ

) is the flexural rigidity of the plate, a constant. with E Young'smodulus. 1 the plate thickness, and v Poisson's ratio: .1z is the biharmonic operator. W thetransverse deflection. and IT,,fj the membrane stress tensor given by (2.2) as a function of thein-plane displacement vector U = (Ul' uz) and of w: / is the transverse load per unit area of themiddle surface. The summation convention is used (I ~ a, {3, A, µ. ~ 2), and commas denotedifferentiation with respect to material Cartesian coordinates Xu in the middle plane (UA.IL =c1UA/c1xµ etc.). The elasticities Ea13Aµ are, as usual. given by Ea13Aµ = E(1 + IIfl [O"AO{:lµ +v(1- Vf10afjOAµ].

Among the boundary conditions we wilJ consider are those for a clamped plate,

aww =O=-onrall U = 0 on r, (2.3)

(2.4)

n being the outward normal to r. and those for a simply supported plate with applied in-planetractions Sa on a portion r2 of r (r = F, uF2' r, nrz = cP ).

w=O=M"(w)Onr.u:~onrl ].IT al3(u, W)lI{:l - S" on rz

Here M,,(w) is the normal bending momcnt.

M,,(w) = -D[v.1w + (1- lI)w .a{:lll"lIp 1. (2.5)

.1 being the Laplacian in two dimensions. Our numerical techniques developed in part 2 of thispaper are applicable to arbitrary boundary conditions,

Some writers have criticized the von Karman theory on the grounds that it is derived fromlargely heuristic physical arguments which lack mathematical precision and which ignore thequestion of consistency with the general theory of finite elasticity. However. a rigorousanalysis of the von Karman equations as a first approximation of the equations of elasticity hasbeen provided by Ciarlet and Destuynder [1979] for the case in which the strain energy is aquadratic form in the Green-Saint Venant strains. While the Ciarlet-Destuynder analysis stillrests on assumptions of infinitesimal strains, they nevcrtheless were able to show that the vonKarman equations are, in fact. correct in this casc to terms of first order in a thicknessparameter whenever certain boundary conditions are assumed to hold. The acceptable

190 K. Ohtake l'1 al.. Analysis of problems in ron Kanl/an plate theory by a pel/alty method - Part I

boundary conditions include those of clamped plates but not simply supported plates. In thecase of simply supported plates. the lack of consistency of the von Karman equations with thethree-dimensional theory is manifested only in second-order terms in an asymptotic expansion.and this results in a boundary-layer effect, present only in a strip near the edges, in which thestress distribution deviates somewhat from the linear distribution assumed in the von Karmantheory. Qualitatively. the theory still seems to be adequate for modelling the global behaviorof clamped and simply supported plates. We will base all of our subsequent analysis on thevon Karman theory. It is a rich theory leading to equations with an interesting mathematicalstructure. More importantly, the von Karman theory leads to an acceptable model of a varietyof nonlinear phenomena that cannot be captured by the classical linear theory.

In addition to the boundary conditions needed to define a boundary value problem in thevon Karman theory, we also consider unilateral conditions representing the effect of the platecoming in contact with a frictionless surface upon deflecting out of its plane. Such constraintson the admissible deflections of plates usually assume the form of inequalities or "one-sided"conditions on various kinematical variables and stress resultants. Several such one-sidedconditions for nonlinear plate theory have been investigated by. e.g., Duvaut and Lions [19741,Naumann [1975]. [1977], Do [1975], [19761, John [1977], and others. We will consider physicalsituations such as those in fig. I in which the transverse displacement of a thin plate isconstrained by the presence of a flat, rigid, frictionless plane located a distance b under themiddle plane of the plate for which all admissible transverse displacements must satisfy

W 2: -b.

0'

f

~} ZET2: ;j\=r b

(2.6)

b)

~//I(/~

S 7.~----_:: ~ .2777 77777777//77/777/77/

Fig. 1. Deformation of thin plates with unilateral constraints.

K. Ohtake et al.. Analysis of problems in von Kannan plate theory by a penalty met/rod -Part 1 191

Physically. if W > -b, then the plate does not come in contact with the rigid foundation and noreactive force is developed on the plane. On the other hand. if w = -b at some point(XI, X2) E fl. then the plate is in contact with the plane and there is developed a transversereactive force p on the plate. Thus. p = 0 if w> -b and p ~ 0 if w = -b. or

w + b ~ 0 P 2:0). (2.7)

p(w+b)=Oinfl

The last condition in (2.7) is a form of the complementarity condition of mathematicalprogramming (Abadie [1967], p. 21) in which the reactive force p is interpreted as a Lagrangemultiplier associated with the constraint (2.6). Similar conditions have been studied by Do[1975] in the analysis of unilateral problems for von Karman plates.

Throughout the remainder of part I we will concentrate on the following two modelproblems.

Problem 1. Unilateral problem for a clamped plateGiven f. E "I3Aµ, and b > O. define

p = D.12w - t«(T,.I3(u, w)w.,,),(3 - f. ]mil.

(T ,,(3 (u, w) = Ea(3Aµ ( U A.µ +!w .A w.µ )

Find (u. w) (u = (UI- U2) such that

p2:0, W2:-b,P(W+b)=Oj'mfl

(T ,,(3(u. w),,6 = 0

oWu" = 0 on r, w = ()and -a = 0 on r

11

(2.8)

(2.9)

This system describes the deformation of a clamped plate under a transverse load /. thetransverse displacement w being constrained by the presence of a rigid frictionless planelocated a distance b under the plane X3 = -t/2 of the plate in its reference configuration; p isthe contact pressure,

Problem 2. Unilateral problem for a simply supported plate with an applied thrustGiven f, Sa. b > 0, find (u, w) such that

p2:0. W2:-b,P(W+b)=O).mfl

(T "13(U, w),,6 = 0

Ila = 0 on rJ, (T al3(u, w)1ItJ = S" on r~w = 0 and Mn (w ) = 0 on r = t.ut2

where p and (T ,,(3(u, w) are defined in (2.8).

(2.10)

192 K. Ohtake et aL Analysis of problems in VOII Karmali plate theory by a pellalty method - Part I

This system describes the deformation of a simply supported plate subjected to prescribedin-plane tractions Sa on the boundary r of its middle plane. Again. the transverse deflection isconstrained as in problem I. The "in-plane" boundary conditions are mixed: u is prescribed aszero on an open portion rl of the boundary r of {} and the membrane stress is prescribed bya given "thrust" 5.. on a portion r2 of r, r = f1 U f2.

3. Variational principles

In this section we develop variational principles for the model problems described in theprevious section, Toward this end, we introduce the notations

WnlJ'(il) = Sobolev space of order (m, p), m 2: 0, 1 < P < 00 IZ({}) = {v E W2,2({}): v = ()on r} WnI.P({}) = (W"""(n)y ,

M({}) = (v E WI.2({}): v = 0 a.e. on TI}

(3.1 )

In our definitions of various spaces it is understood that the values of functions on the boundarymay be interpreted in the sense of the trace of v; i.e., .. v = 0 on r" means I'(v) = 0 where l' is thetrace operator. Recall that Wm

•p

({}) is equipped with the norm

Ilvll",,,, = {r '} IDavlp dx}l/P,Jfllaf;t. ..

where dx = dXI dx~. Also, if u = (li" liz) E W""P({}), we write

Let us also define

wl/2.2(rt) = the completion of C(r1) in Wlf2.2(F);

W 1/2.2(r) is complete when equipped with the norm

Similarly, we define

e(F1) = the completion of C~(Ft)in e(F).

(3.2)

(3.3)

Now we can define W-t/2.2(rl) as the dual of W 1f2.2(Ft). All boundary conditions givenhenceforth are understood to be interpreted in the sense of these spaces of traces of functionsdefined on {}.

K. Ohtake el al.. Analysis of problems in von Karman plate theory by II penalty method - Pan 1 193

We begin by introducing the bilinear forms

A: W2.2(12) X W2•2(f1)_fR ]

A(u, v) = D In {L1uL1v - (1- v)[u, v]) dx

and

B: W1.2(f1) x WI.2(f1)_fR )

B(u. v) = t L EatJAµUA.µVa.P dx '

where in (3.4) we use the notation

We also will use the forms

a (u; w, z) = t In (J' aP(U, w)w,,,z./J dx

and

(3.4)

(3,5)

(3,6)

(3.7)

The functional iA (w, w) is the strain energy due to bending in the plate produced by atransverse displacement w(u = 0) and the functional ~B(u, u) is the strain energy due toin-plane deformations of the middle surface of the plate excluding the effects of rotationsw." (w = 0). The elasticities E atJAµ are assumed to be either constants. as in the previoussection, or given functions satisfying

1 $ a, {3, A. µ.. s; 2

max IIE"fIAµIIL'(fl)< M < +00I SU'./J. A. SA s.:! (3.8)

a.e. in f1.

'tIE,.tJ such that EatJ ;: €/Ja' where all = constant> O.

Under these conditions (D = Et)/l2(1- v2) being a positive constant). the following Korn-type

194 K. Ohtake et aL Analysis of problems in VOII Kanllan plate theory by a penalty lIIethod - Part 1

inequalities can be shown to hold (see, e.g. Naumann [19751, [1977]):

Clllwll~.2:::; A(w, w):::; c;llwllt2 Vw E Wk2(fl)

c21Iwll~.z:::;A(w, w)+ c L w2 ds:::; c~lIwll~,2 Vw E W2.2(fl)

c31Iwll~,2:::;A(w. w):::; c~llwllt2 Vw E Z(fl)

and

(3.9)

Vu EM(fl). (3.10)

Here c and the Cj and c;, i = t, 2, 3. 4. are positive constants independent of wand u.If f E L2(fl) and Sa E L2(r) (the restriction to r. being zero) a.e given transverse loads and

in-plane surface tractions, respectively, then the linear functional

F: WI.2(fl) x W2.2(12)_fR l

F(u, w) = r fw dx + t r SaUa dsJo J,)(3.11)

represents the negative of the potential energy of the applied forces on the plate. The totalpotential energy is then

1 I 1I1(u, w) = '2 A(w, w)+ '2 B(u, u) + '2 a(u; w, w)- F(u, w)+ r(u. w), (3.12)

where r(u, w) denotes boundary terms representing the situation in which applied edge forcesand moments may perform work.

A variational statement of the clamped plate prohlem (2.9) can be constructed which in-volves a variational inequality coupled with a variational equality for the displacements (u, w).

THEOREM 3.1. Let (1I, w) he a solution of (2.8). Then

A(w, z - w)+ a(u; w, z - w)~ F(o, z - w)

B(u,v)+b(w;v)=O VvEWl.2(fl)

where

K = {z E w~·2(n): z 2: -b in n}.

Vz E K )(3.13)

(3.14)

Conversely, let (u. w)E Wk2(fl)xK be a solution of (3.13) for fEL2(fl). Then

(i) p = DJ. 2W - t(O'u/J(lI, W )w.u).µ - f ~ 0

in a distributional sense.

K. Ohtake et al.. Analysis of problems in von Karman plate theory by a penalty method - Part I 195

(ii) The following weak form of the complementarity condition holds:

A(w, w + b)+ a(u: w, w + b)- F(o, w + b) = 0,

(iii) <Tap(u, w).p = ()in a distributional sense.Moreover, if wE W4

.2(D) nWk2(D), then p E e(D) and

(3.15)

p(w+b)=O a.e. in D.

Remark 3.1. According to the Sobolev imbedding theorem (see, e.g. Adams [1974]), ifm > nIp, fl bounded in fAn, then W"'J'(fl)4CI(.Q), and this imbedding is compact. There-fore, if z E W2

.2(fl), the condition z 2: -b can be imposed pointwise in fl. 0

Remark 3.2. By"p ~ 0 in a distributional sense:' we infer the following interpretation. Let~+(fl) = {cP E ~(fl): cP 2: O}.Then ~+(fl) is a positive cone of test functions, We then define adistribution p to be non-negative. written p ~ O. whenever (p, cPl!il ~ 0 'tJcP E ~+(fl), where(., .h denotes duality pairing on ~'(fl) x ~(fl). 0

Proof of theorem 3.1. We first note that the constraint conditions (2.9) imply thatp(z - w)~O 'tJz ~ -b. Using this fact and the Green's formula

r ~w~z dx = r Z~2W dx + r (~w aaz - z aa~W) dsIn In Jr n 11

we have

A(w, z - w)+ a(u: w, z - w) = D r (z - W)~2W dx - t r [<Tat!(u, w)w.al.t!(z - w)dxIn In= r [p(z-w)+f(z-w)]dx.In

Recall that (2.9) implies p(z - w) 2: O. Thus, we obtain (3.13).. Equation (3.13h followsimmediately from (2.9h by multiplying it by v"' integrating by parts. and using the definition(3.5) of B(. , .).

Next. let (u, w) be a solution (3.13) and set z = w + </>' </> E ~+(fl) = {y E 0) (,Q): y ~ 0 in fl}.An integration by parts yields

(3.16)

where (., .)", denotes duality pairing on 0J'(fl) x 0)(fl). Note that this result means that thedistribution p = D~ 2W - t(<Tat!w.n)Jl - f is nonnegative in fl (see remark 3.2).

Set z = 2w + b and then z = -b in (3.13)). It is obvious that in both cases z E K. From thepair of inequalities resulting from these choices of z. we conclude that (3.15) holds.

196 K. Ohtake el aI.. Analysis of problems in vOIr KamJall plate theory by a penalty method - Part I

Finally, if wE W-l·2(fl)n W~·2(fl), then p EL2(n) and an integration of (3.15) by parts(using Green's formulas for functions in these spaces) yields

i pew + b) dx = O.11

But we have p 2: 0 and w + b 2: O. Therefore,

p(w+b)=O a.e. in n.Finally, taking v E (0J(nW in (3.13h and integrating by parts leads to

(an/l(u, w)Jl' vn)", = 0,

from which (ii) follows. 0Note that. in the region fl2 C fl where w = -b, we can conclude from (3.16) that

But t/J 2: 0 is arbitrary in 0J+(fl). Therefore, this leads to f ~ 0 in fl2 in a distributional sense,This implies that the given surface load must be negative (downwards) if the plate is to comeinto contact with the foundation.

THEOREM 3.2. Let (u, w) be a solution of (2.10). Then

where

A(w, z - w)+ a(u; w, Z - w)2: F(o. z - w)

B(u.v)+b(w;v)=F(v,O) VvEM(n)

KI = {z EZen): z 2: -b in n}.

Vz EK, 1 (3.17)

(3.18)

Conversely. let (u, w) E M(fl) X KJ be a solution of (3.17) for Sa E e(r), f E e(fl). Thenthe following hold:

P=DL12w-t(.aa/l(u .. w)w.n),/l-f2:0 in0)'(fl) 1

A(w, w + b) + a(u, w, w + b)- F(o, w + b) = 0

auµ(u, w)./l = 0 in 0)'(fl)

(3.19)

Moreover. if fl is Lipschitzian and wE W-l·2(fl) nZ(fl), then p E e(fl), aa/1(U, w) E

K. Ohtake et a/" Analysis of problems in von Karman plate theory by (I penalty method - Part I 197

WI.2(fl). tip E L""(r) and

lTnfJ(u, w)tlp = s'.Mn(w) = () a.e.

p(w+b)=Oa.e.

in L2(r)

onrinfl

(3.20)

Proof. The proof of this result follows steps analogous to those used in the proof oftheorem 3.1 and, for brevity. will be omitted. For complete details, see Ohtake [1979]. 0

Hereafter. we will refer to the variational versions of problems 1 and 2 as PV1 and PV2,respectively,

Since the operators intrinsic in PV1 and PV2 are derivable from the total potential energyand since the constraint sets K and Kt convex, it is a simple matter to show that theminimizers of n on these sets are respective solutions of the variational inequalities charac-terizing PV1 and PV2. We record this fact in the following theorem.

THEOREM 3.3, Let (u, w) E W~I·2(fl) x K be a minimizer of II overthe convex set W k2(fl) xK. where K is defined in (3.14). Then (u, w) satisfies (3.13). Similarly. if (u, w)EM(fl)XK.minimizes n on M(fl)XKh with Kt given in (3.18), then (u, w) satisfies (3.17). 0

4. A penalty formulation

Theorem 3.3 establishes that the minimization of the energy functional n over the convexset W~·2(fl) x K (or M(fl) x Kt) may correspond to the variational inequality formulation ofproblem 1 (or problem 2). Such constrained minimization problems can be approximated byunconstrained minimization problems by the introduction of a penalty functional np• The basicideas are well known. Suppose n is proper, weakly lower semicontinuous. and coercive on anonempty closed convex set Ko of a reflexive Banach space V, and that we wish to minimize IIon Ko. We convert this problem into an unconstrained minimization problem by introducing afunctional np: V -.IR satisfying the following conditions:

(i) np is weakly lower semicontinuous,

(ii) np(v) 2: 0, and np(v) = 0 if and only if v E Ko.

Then, for any € > 0, the penalized functional

IJI.(v) = n(v )+- np(v)€

(4.1)

(4.2)

is weakly lower semicontinuous and coercive on all of V. Thus. for each € > 0, there exists aminimizer u. E V of JI.. Finally, whenever (4.1) holds, the sequence {u.} of minimizersobtained as € -.0 has a subsequence, also denoted by {u.}, which converges weakly in V to aminimizer u E KII of the original functional n:

u. -. u weakly in V as € -.0: inf II(v) = II(u).vEKo

(4.3)

198 K. Ohtake et al., Analysis of problems ill VOII Karmali plate theory by a penalty method - Part I

If both II and IIp are Gateaux differentiable on V, then solutions of this penalizedminimization problem also satisfy the variational equality

I(DII(u.), v) +~DlIp(u.), v) = 0€

'rIv E V, (4.4)

where (, •. ) signifies duality pairing on V' x V.For the unilateral constraint sets K and Kt in PVI and PV2. we can introduce the penalty

functional

where we use the notation

(tj»_ == min(tj>, 0).

We next test IIp to see if it qualifies as a penalty functional.

(4,5)

(4.6)

THEOREM 4.1. Let IIp: K (or K,)--+IR be given by (4.5) where K is defined in (3.14) andKI in (3.18). Then IIp satisfies conditions (4.1). Moreover, IIp is Frechet differentiable and'rIz E W6·2(D) (or Z(D)),

(DIIp(w). z) = In (w + b)_z dx.

Proof. It is sufficient to prove that the real-valued function

is continuous and convex on IR. Note that

(4.7)

Therefore tj>(XI)--+tj>(X2) as Xt--+X2. Next, we decompose tj> as tj>(x) = r/J' p(x) where r/J(a)=1/2 a2 is convex and p(x) = (x + b)_ is increasing. Therefore tj>(x) = t/J . p is convex. It followsthat In tj>(w) dx is weakly lower semicontinuous.

Next, we show that In tj> dx is Frechet differentiable. Consider. for simplicity, the functionr/J(x) = 1/2 x:. Clearly. t/J(y) - t/J(x) - x_(y - x) = 1/2(y_ - X_)2 - x_y+ + x_x+ = 1/2(y_ - x-f- x_y+ 2: 1/2(y_ - X_)2 2: O. Exchanging x and y gives r/J(y) - t/J(x):s -y_(x - y). Hence,

O:s t/J(y) - r/J(x) - x_(y - x):s (x_ - y_)(x - y).

K. Ohtake et al.. Analysis of problems in von Kannall plate theory by a penalty method - Part 1 199

Some algebraic manipulations reveal that

Returning now to 4>(w) = 1/2(w + b):, we define

w(w. Z - w) = I In [4>(z) - 4>(w) - (w + b )_(z - w)] dx Iso that

w(w.z-w)~ r Iz-wI2dx=llz-wll~.2'JII

Hence,

from which it follows that In 4> dx is Frechet differentiable and that

(DJ/p(w).z-w)= r (w+b)_(z-w)dx.JII o

Collecting these results, we now introduce U1Ico1lstrained variational problems for thepenalized energy functional ll.(u, w) = J/(u, w)+ €-IJ/p(W).

Problem PVl. We define the Penalized Variational Problem 1 as follows: Given f E L 2(fl)and € >0. find (u .. w.)E wb·2(f1)x W~.2(f1) such that

A(w.,z)+a(u.: w•. z)+.!. r (w, +b)-z dx =F(o.z)€ In

B(u •. v) + b(w.: v) = 0 'Vv E Wl.2(fl)

'iz E W~"(I1) I.(4,8)

Problem PV2. We define the Penalized Variational Problem 2 as follows: Given f E e(f1),- 2S" EL (r), and € >0. find (u •. w,)EM(il)XZ(fl) such that

A(w •. z)+ a(u,: w., z)+.!. r (w, + b)_z dx = F(o. z)€ JII

B(u .. v) + b(w,: v) = t i.~S"v" dx 'Vv E M(fl)

'iz E Z(11) j.(4.9)

200 K. Ohtake et al .. Analysis of problems in von Karman plate theory by a penalty method - Part J

We easily verify that if (uE. wE)E W~·2(IJ)X Wk2(IJ) is a minimizer of n over W~·2(fl)xW~'~IJ), then (UE, WE) satisfies (4.8). Likewise. if (uE, wE)EM(IJ)xZ(IJ) is a minimizer of II.over M({J) x Z({J), then (U., WE) satisfies (4.9). In the next section, we will prove that solutions(U., w.) of problem PVI (or PV2) converge to a solution (u, w) of the correspondingvariational problem as f ~ 0 in some sense.

5. Existence and convergence of solutions to PVl and PV2

The success of the penalty formulation. of course, depends upon the existence of solutionsof PVl and PV2 for any E > 0 and the behavior of these solutions as E ~O. To address the firstof these issues. we recall a fundamental theorem due to Brezis [1968].

Lemma 5.1. Let W be a reflexive Banach space. K a nonempty closed convex subset of W,and T an operator from W into its dual W' satisfying

(i) T is pseudomonotone (i.e. if {Uj} is a sequence in W converging weakly to U E Wand iflim sup (T(uj). Uj-u)~O, then liminf(T(uj), Uj-v)~(T(u). u-v) VvEW, (.,.) denoting

j-oo j-.oo

duality pairing on W' x W),(ii) T is coercive (i.e. (Tv. v)/lIvllw ~+oo as IIvllw ~oo). Then. for each f E W', there exists at

least one W E K such that

Moreover, if wE int K then the equality holds in (5.1).We also have (see e.g. Oden r 1979a 1):

(T(w), z - w)2:f(z - w) VzEK. (5.1)

o

LEMMA 5.2. If T: W ~ W', W being a reflexive Banach space, and T = A + B where A ismonotone and B is completely continuous (i.e. Uj~U weakly in W imples B(uj)~B(u)strongly in W'). then T is pseudomonotone. 0

It is a fortunate circumstance that. under mild conditions, the operators appearing in ourvariational formulations are pseudomonotone and coercive on the sets K and K, introducedearlier. We will now proceed to prove this assertion. By appropriate choices of spaces, thefollowing results apply to both PVl and PV2. Consider, for example. PV2 for which theunderlying sets are

K, = {y EZ(fl)C W2.2({J): y 2: -b in {J} and

M(IJ) = {v E W1.2({J): v = 0 a.e. on FI}'

Let Til: KI~Z'(IJ) be an operator defined for fixed u EM({J) by

(Tllw, z) = A(w. z)+ a(u; w. z). (5.2)

Here ( ... ) denotes duality pairing on Z/(IJ)x Z({J). For [EL2(IJ). we define the linear

K. Ohtake et aL Analysis of problems ill von Karmali plate theory by a penalty method - Part 1 201

functional (f, z) = F(o, z) = In Iz dx. Finally, we introduce the penalty operator

(5.3)

Then the penalized variational problem PV2 for the thrusted simply supported plate for eachfixed E > 0 assumes the form

Find (u, w)EM(il)xZ(il) such that

1(Tu(w).z)+-(Tp(w),z)=F(o,z) Vz EZ(il)E

B(u,v)+b(w.v)=F(v,O) VvEM(il)

(5.4)

From the definitions (3.8) and (3.10) it is easily verified that constants C(I and C1 exist suchthat

IB(U., v)1 S CJlulldIVII1.2]B(v, v)2: c.llvlli.2

(5.5)

for every u, v EM (il). Moreover, there exist constants C2,C3> 0 such that for every v EM (il)and w EZ(il)

In addition. the trace theorem gives

where

This means that the transformation

v ~-b(v, w)+ t ( Sava dsJr!

"Iv EM(il),

(5.6)

(5.7)

(5.8)

defines a linear continuous functional on W 1.2(il). Therefore, from the Lax-Milgram theorem (seee.g. Showalter [1977] or Oden [1979b, p, 3161) there exists exactly one solution u = u(w) E M(il)for each wE Z(il)(and, of course. each E). We will express the dependence of the solution u on w

202 K. Ohtake et II/" Allalysis of problems in von Karman plate theory by a penalty methocl- Part 1

hy introducing an operator G: W2,2(!2) ~ W 1.2(.0) such that

u = G(w), (5.9)

(5.10)

Naumann [19761 has shown that the solution u = G(w) of (5.4h (with Tp =0) has thefollowing properties:

Ilulll.2$ const(llwllL + a) 1Ilu - u*111.2$ const(llwllJA + Ilw*III.4)llw- w*111.-l

where u* = G(w*). With u = G(w) known for each w, we can eliminate u from (5.3), IfT = T GCw)+ E-1Tp• then T: Z(!2)~Z'(!2) and

(Tw, z) = A(w, z)+ a(G(w): w. z)+ E-1(T,,(w). z),

and (5.4) becomes

(5.11 )

(Tw. z) = <f. z) Vz E Z(!2). (5.12)

The operator T also appears in the penalized clamped plate problem. If we replace thespaces M(n) by W,I?(n) and Z(il) by W[.-2(il), the variational problem PYI can be rewrittenin the form (5.12).

Continuing, let us decompose the operator T in (4.12) into the sum of three operators T.,T2, T3: Wk2(il)~ W-2,2(il) defined by

(T,w, z) = A(w, z). (T2w, z) = a(G(w): w, z), (5.13)

for w, z E W~.2(il), where G is defined in (5,9). We easily verify that T. is linear, coercive andcontinuous, and that therefore T, is monotone, bounded and hemicontinuous. Moreover. inview of (3.9). T. satisfies

Also, T3 is monotone and Lipschitz continuous. Indeed,

(T3 W - T3Z. W - z) = E-I ( [( w + b )_ - (z + b )_](w - z) dxJ!J

2: E-1 L [(w + b)_ - (z + b )_12 dx 2: 0,

and

(T3w-T3z,y)SE-· ([(w+b)_-(z+b)_]ydxJ[)$ E-'llw - zllz.zllyllz.2'

(5.14)

(5.15)

(5,16)

K. Ohtake et al.. Analysis of problems in lion Kannall plate theory by a penalty method - Part I 203

Thus, T. + T3 is coercive, monotone, and continuous.In view of lemma 5.2 we need only prove that T2 is completely continuous.Using Holder's inequality. (3.9) and (5.10), we easily verify that for WI, W2E Wk2(!2),

(5.17)

where IIT2w\ - T2w211* is the norm on W-2.2(Q). Let {wm} be a sequence in W~.2(!2) whichconverges weakly to w, By the Sobolev imbedding theorem, {w",} converges strongly to W inWI.4(!2)::> w~·2(n) since W2.2(.Q) is compact in WI.4(.Q). Thus, IIT2wm - T2wll* ~O as rn ~CXl. Itfollows that T is pseudomonotone on W6·2(.Q). Identical arguments show that T is alsopseudo monotone on Z(.Q).

It remains to be shown that T is coercive. In view of (3.9), (Tlw, w) 2: Clllwll~.2'and in viewof (3.8),

(€afj = ~ (lid.,., + UI3.a + W,,., W.I3))'

L2 = -2r L E"I3Aµ( UA,J.L +~W,AW JJ )G,.,/J(W) dx

= -2[B(u, u)+ b(w, u)] (u = G(w»).

I -For PVl we see that L2 = O. For PV2. -2LI = In S"U" ds and a bound in terms of a of (5.8)can be easily obtained (see proposition 5.2 below).

Finally, turning to T3 we have

Collecting these results, we have

(5,18)

where a = 0 for PVL and a is given by (5.8). Thus, for PVL T: W~·2(.Q)~ W-2.2(.Q) is

pseudomonotone and coercive and for PV2, T: Z(.Q)~Z'(.Q) is pseudomonotone and iscoercive if a is sufficiently small.

In summary, these results and lemma 5.1 lead to the following propositions:

PROPOSITION 5.1, Let (3.8) hold, Then there exists at least one solution (u, w)EW~·2(.Q) x w~.2(n) of problem PVl. 0

204 K. Ohtake e/ aL Analysis of problems in von Karmali plate theory by a penalty me/hod - Part I

PROPOSITION 5.2. Let (3.8) hold. Moreover, let a < 50 for a suitable 50 > 0, where a is definedin (5,8). Then there exists at least one solution (u, w) EM(il) x Z(il) of problem PV2. 0

Having established the existence of solutions to the penalized problems for each € > 0, wenow pass on to the question of the convergence of sequences of solutions as € ~O. Our firstresult in this direction is the following proposition concerning PV1:

PROPOSITION 5.3. Let (3.8) hold. Then there exists a subsequence of solutions {(U., w.)} ofPVl obtained as €~O which converges weakly in Wl/(il)x W~,2(il) to a solution (u, w)EW ,\.2(il) x K of the variational problem VI.

Proof Since the variational problem "8(u,v)+b(w;v)=O \fvEWk2(fl)" has a solutionfor every w. it suffices to show that the sequence {w.} of solutions to (5.12) has a weaklyconvergent subsequence whose limit w is a solution of (3.13). From (5.17).

Thus, IIw.lb.2 is uniformly bounded independent of €, It follows that there exists a weaklyconvergent subsequence, also denoted {w.}, whose limit w E W~·2(fl).

We next show that this limit wE K. Indeed, from the definition of WE as a solution to PV1,

or

I.e.

II(w. + b )-lIo:S cV;.

Taking € ~ 0 yields

L (w + b): dx = O.

c>o. (5,19)

Hence, w EK.Next. we note that for any z E K.

Thus, since Tpz = 0 for Z E K.

K. Ohtake et a/.. Analysis of problems in von Karman plate theory by a penalty method - Part I 205

Since T2 is completely continuous, we have (T2w., z - w.)-+(T2w. z - w) as € -+0.Next note that

limsup(T.w., w. - w)~ lim sup{a(G(w.): w., w - w.)- few - w.)} =0 .• ....0 • ....{I

But since TI is pseudomonotone, this implies that

(T,w, z - w) 2: lim sup(T1w., z - w.).-oCI

2: lim sup{(T2w., w. - z)- (f, w. - z)}.-+CJ

= (T2w, W - z)+(f, z - w).

Hence. w is a solution of the variational inequality (3.13).Similar arguments lead to the proof of the parallel proposition:

o

PROPOSITION 5.4. Let (3.R) hold and a of (4.8) be sufficiently small. Then there exists asubsequence of solutions {(u, w.)} of PV2 obtained as € -+0 which converges weakly inM(il) x Z(il) to a solution (u, w) E M(il) X K. of the variational problem V2. 0

We note that the penalty method also yields a weak form of the contact pressure p.

PROPOSITION 5.5. Under the conditions of proposition 4.3 (or 4.4), there exists asubsequence {w.} of solutions of PVI (or PV2) and an element -p E W-2

.2(il)(or Z'(il)) such that

(l/t:)Tpw. -+ -p weakly in W-2•2(il) (or Z'(il)), where p is the contact force

Proof. Since

we have

where M is a constant independent of € and liz II is the norm of z on Wk2(il) or Z(fnwhichever is applicahle. Hence {€-'Tpw.} is bounded in W-2

.2(il) (or Z'(fl)) and a sub-

sequence exists which converges weakly in this dual space to an element denoted -po Clearly,(-p, z) = (f - T,w - T2w. z). 0

200 K. Ohtake et al.. Analysis of prohlems in von Karman plate theory by a penalty method - Part I

6. Regularity and uniqueness

A complete regularity theory for unilateral problems for von Karman plates is not available.Some regularity results for the unconstrained clamped and simply supported cases have beenobtained by Lions [1969] and John and Naumann [1976], but their analysis dealt with thestress-function formulation of the governing equations. Our objective in this section is todevelop regularity results for the penalized variational problems corresponding to the generaldisplacement formulation of unilateral problems for von Karman plates. Our results pertainonly to the penalized formulation for €. > 0 and do not carry through to the limiting case ofplates with unilateral constraints.

From this point onward. we assume, for convenience. that fl is a smooth open boundeddomain in fA2 of class eo or c2

m for sufficiently large m. However, many of our results holdunder considerably weaker conditions. In preparation for our major regularity results, we willrecord some preliminary ideas and notations.

Let F denote a differential operator with constant coefficients defined on a suitable class offunctions with smooth domain fl C fAn. The operator F* satisfying

i uFv dx = i vF* u dx/} fl

't/u, v E C~(fl)

is the formal adjoint of F For V a linear space of functions such that C;(fl)C V C C"'(fl), wedefine

0= {v E C"(fl): ( uF*v dx = ( vFu dxIn )/}In particular, if

{ - av }VI= vEC"'(fl):v=-=Oonran

't/u E V}. (6.1)

V2 = {v E C"'(D): v = M,,(v) = 0 on nV3 = {v E (C"(il)f u = 0 on r}

Vol = {v E (C"'(D)f: u = 0 on rio E"P.l.µUA.,.np = 0 on r2}

we define

(6.2)

't/u E Vi} i= 1,2

(6.3)

The next lemma is due to Agmon [19591.

't/u E Vi} i = 3. 4

K. Ohtake et aL Allalysis of problems ill VOII Kannan plate theory by a pellalty method - Part I 207

LEMMA 6.1. Let F be a properly elliptic differential operator of order 2/ in the domainIi C fAn. Let the space V be defined as the null space of a system of boundary operatorswhich cover F on r. Next, for, > 1 and nt, an integer such that 0 ~ m ~ 2/, let

Um"(fl) = closure of V in Wm·'(fl).

If

(6.4 )

and

w EL'(fl), s> 1 (6.5)

with 0 defined by (6.1), then

and constants em exist such that

Ilwllm., ~ C,,,(Co+llwllll.,),

Vz E eJ. (6.6)

(6.7)

(6.R)

where 0 ~ nt ~ 2l. 1/, + 1/,' = L 0Lemmas of this type giving estimates up the boundary are due to Agmon [19591 and were

expanded by Schechter [1963]. The general method is discussed in the classic paper of Agmon,Douglis and Nirenberg [1959]. The case in which 111 is a real number was treated by Schechter[1963]. This lemma was a crucial tool in the theory developed by John and Naumann [19761.

We now return to the penalized variational problem for clamped plates with umlateralconstraints:

Find (u, w)E Wk2(fl)x Wk2(fl) such that

A(w. z)+ a(u; w. z)+ €-I L (w + b)-(z) dx = F(o, z)

Vz E W~·2(fl) ~ (6.9)

B(u.v)+b(w,v)=O

(6.10)

Here E > 0 is fixed, f is given in LP(fl), 1 < p < 0:" and b is a positive constant appearing in theunilateral constraint condition.

PROPOSITION 6.1. Let f E LP(fl), 1 < P < 00, let E > 0 be given, and let the conditions ofproposition 4.1 hold with E"fJAI-' constants. Let (u, w) be a solution of problem (6.9) (i.e. PVl).Then

(i) If I < P < 2.

wE W4.p-S(12) n W~·2(fl) ] .

u E W4,p-S(12)n W~·2(12)

208 K. Ohlake et aI.. Analysis of problems in VOII Karman plate theory by a penalty method - Part 1

(ii) If 2 sP < 00,

wE W4.s'(fl) n W~,2(fl) j.u E W4J>(fl) n W b"\a)

Moreover, the following estimates hold:(iii) If 1< P < 2.

IlwI14.P-B S ff] 1lIuIl4.p-(/ s C~[ff

(iv) If 2sp <00,

IIwl14,P s C[lffj3 + Cal + €-'~ltllo.p + €-lllbll"." 1.Ilu114.Ps C;([f]3 + (1 + €-l~ltlloJ> + Ilbllo",)2

(6.11 )

(6.12)

(6.13)

Here 8 is an arbitrary positive real number such that p - () > 1; Cb, C;, Ci are positiveconstants: and

Our proof of this proposition will make use of the following lemmas.

LEMMA 6.1. For 1< r < 2 and u E WI.'(!1), vEL 2(fl). we have

Iluvllo .•s Ilu 11...11 v 110.2.

o (6.14)

(6.15)

Proof. Let lIs + I/g = 1 and qr = 2, for 1 < r < 2. The Sobolev embedding theorem ensuresthat

WI.'(!1)E e,/(2-')(fl).

Using this fact with Holder's inequality, we have

lIuvllo., s {In lul'lvl' dxr'{(I )1/J (i )1/P}I/'

S n lui" dx n Ivl,q dx

= lIullo.2r'(2-rJlvllo.2S Ilullt.rllvllo.2' oLEMMA 6.2. For u E e(!1), 1< r < 2. v E WI.2(!1), and 8 a positive number such that

r - 0 > I,

Iluvll".r-B s cliullo.rllvllI.2where c is a positive constant.

(6.16)

(6.17)

K. O/rtake et al.. Analysis of problems in von Kamwn plate theory by a penalty method - Part I 209

Proof. This result follows easily from arguments analogous to those of the previous lemmaand will be omitted, D

Proof of proposition 6.1. Let (u, w) be a solution of (6.9). We will demonstrate the proof forthe case 1< r < 2. The proof for 2:s r < 00 follows easily from similar arguments. Our proof isbroken down into the following steps:

10 We specify U in lemma 6.1 as U~CCX(fl) of (6.2)). Let 1<r<2. and l/r'+I/r=Ll/r + l/r" = 1/2. Then the condition 1 < r < 2 guarantees that r' > 2, r" > 2. We use C;, i =O. 1,2 •... as positive constants. Then, for v EO). use of the Holder inequality and theSobolev imbedding theorem (W2

•2(fl) ~ WV(fl) for r" > 2) leads to

IL E"/JAµ!l" (VA.,.)./J dxl = I~L En/JA,.(WAW.,.).tlV.a dxlM

:S T Ilwlk,·llwll.dlvllll.'·

:S AJ C)llwll~.lllvllo.,.where M is a constant in (3,8) and C3> n.

Applying lemma 6.1 to (6.17) with 2/ = 2, m = 0 + 2/ = 2, results in

(6.18)

where C4 = MC3/2.20 Since u is in W2.r (fl), the estimate (6.17) holds with v replaced by u. Thus, using the

Korn-type inequality (3.10). we get, for r' > 2,

Csllulln.2I1uIL:s Csllulli.2:S 18(u. u)l:s Ib(w: 11)1

:s M2C31IwIIUlullo.r"

Since W1.2(fl) is continuously embedded in L"(fl) for r' > 2, (6.18) yields

Ilulb., :s C(,llwllt2' (6.19)

Since !lA.,.1l E L'(fl) and W.e. E L"(fl), a simple calculation reveals that

(6.20)

Expanding (6.20) and using lemmas 6.1 and 6.2, we find that for (J a positive number such thatr-(J>l,

(6.21)

with C7 = CM C("

210 K. Ohtake et aL Analysis of problems in von Karman plate theory by a penalty method - Part 1

3° Next we will show that Ilwlb.2S constllf1lu.p for p> I. Recall that for problem PVl,a(G(w); w. w)2: 0, Also, for the penalty term note that (w + b)_w = (w + b): - b(w + b)_ 2: 0,Therefore. from (6,9)., A(w, w)sF(o, w). Hence, in view of (3.9)" constllwll~.2SIlf1lo,pllwl/(J,p"Since W2,2(fl) 4 W',p'(fl) for p' > 1. we can conclude that a constant ell exists such that

'Vp> 1. (6.22)

4° Next we will choose r so that I < r S p. r < 2. According to (6.20) and in view of the factthat (w + b)_ E CO(fl). we must have

We want to apply lemma 6.1 to the bending equation (6.9) •. Identify U of (6.1) with 01 of(6.3). Then for z E 01. we apply Green's formula to A(w. z) of (6.9)1 and obtain, for r'" > 2,1/(r - 0) + l/r'" = I.

IL wL12z dx I s (I/(uatlw,a ),tlll +~ (I/bl/oJ + IltIlo.r) )I/zl/o.r'"s (C.lllwll~.2+ €-'llwllo.r + €-'llblln,r + Ilfllo.r)llzllo.r·'

:5 (C9C~Ilf1I~.p+ C9€-'llfllo.p + €-IIIbllo.p+ 1IfIIu.p)llzllo.r~'

Therefore, from lemma 6.1 (with 0 = 01• 21 = 4, m = 0+21 = 4) and from (6.22) we mayconclude that

(6.23)

where [J] is given in (6.14).5° We now proceed to establish the regularity of u for 1 < p < 2. From (6.23)

We recall that for sufficiently smooth domains if

B(u, v) = L f· v dx

then

u E W'<+2.p(fl) n W (~.2(fl) and Ilulls+2.p:5 cllfl/f.p(see Simader [19721). Thus.

K. Ohtake et at.. Analysis of problems in von Kannan plate theory by a penalty method - Part 1 211

Since

we have

(6.24)

Vz E Wk2(!1) ~. (6.25)

(6.26)

for 1 < r < p. 1 < P < 2. r - 8 > 1. and C2 = CClIl.

This completes the proof for the case 1< p < 2. The proof for 2::s p < 1 follows from similararguments and is omitted for the sake of brevity. See Ohtake [1979] for additional details. 0

In the case of simply supported plates we recall that problem PV2 takes the form

Find (u, w) E M(fl) x Z(fl) such that

A(w,z)+a(u;w,z)+€-l (w+b)_zdx= (fzdxIn JaB(u, v)+ b(w; v) = L.~S"v" dx Vv E Wl,·2(fl)

By following arguments which parallel those used in the proof of proposition 6.1, we canestablish the following parallel regularity theorem for PV2:

PROPOSITION 6.2. Let f E LP(fl), S" E U(T), 1 < P < 00, let € > a be given, and let theconditions of proposition 4.2 hold with E afjAµ. constants. Let (u, w) be a solution of (6.25) (i,e.PV2). Then

(i) If 1< p < 2.

w E W~·p-8(n) nZ(fl) ).

u E W~,p-8(!1)nM(n)

w E W~.p(fl) nZ(fl) Iu E W4./I(fl) nM(fl)

Moreover, the following estimates hold:(iii) If 1 < P < 2.

IlwI14.P-8 ::sun IIlull-l.p-O::S C1IIflf

Ilwll-l,p ::S C:~Dl + C':(1 + €-l)(llfllo.'1 + C2a) + €-'lIbllo.p + Ilfllo.p IIlull~.p ::S C~{[fD3+ (1 + €-l)(llfllo.p + C~a)+ €-tllbllo.,1 + 1lf1111./,Y

(6.27)

(6.28)

(6.29)

212 K. Ohtake et al.. Analysis of problems in von Karman plate theory by a penalty method - Part I

Here 0 is an arbitrary positive real number such that p - 0 > 1; C;;, C{. C'i. and C3 are positiveconstants, a is defined by (5,8). and

o (6.30)

We note that these regularity results do not hold in the limiting case E ~O. The penaltyterm, of course, functions as a regularization of the solution of the constrained problem. Forthe fully constrained case. the contact pressure p may exist only in W-1-'J'(fl) or in somesmaller space. Thus, the general problem of obtaining regularity results for variationalinequalities involving fourth-order operators remains open, even in the case of linear platetheory (e.g. when In D~w~(z - w)dx 2: IJlf(z - w)dx \fz in W~·2(fl) with z 2: -b). A one-dimensional fourth-order case was studied by Stampacchia [1975] who showed that forf E L2(fl). wE W3

.2(fl). If the bending moment is continuous across the contact boundary, as

in the classical small-deflection analysis of Timoshenko and Woinowsky-Kreiger [1959], thenwe could argue that wE W3.5-

1I.2(fl), 0> O. This is. perhaps, the most smoothness one should

expect for these problems.In the case of problem PVl we can generally expect unique solutions for sufficiently small

data. In fact. Duvaut and Lions [19741 discussed the question of uniqueness for certainunilateral von Karman plate problems and. together with Naumann [1975], pointed out thatsuch problems may have a unique solution for small data. In the case of our PenalizedVariational Problem PVl we can easily show that the "smallness" of w depends upon the sizeof IlflloJ" Indeed, if WI and W2 are two distinct solutions of PV1, then it can be shown that

(6.31 )

where C is a positive constant. Thus, WI = W2 if Ilfllo./, is small enough.

Acknowledgment

We gratefully acknowledge the support of the work reported herein by the U.S. Air ForceOffice of Scientific Research under Contract F-49620-78-C-0083.

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P.G. Ciarlct amI P. Dcstuyndcr. A justification of a nonlinear model in plate theory. Comp/lf. Meths. Appl. Mee/I.Eng. 9 (1979) 227-258.

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