Existence of solutions to obstacle problems for linear and nonlinear elastic plates

Mathl. Comput. Modelling Vol. 28, No. 4-8, pp. 55-66, 1998 @ 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0895-7177198 $19.00 + 0.00

PII:SO8957177(98)00108-3

Existence of Solutions to Obstacle Problems for Linear and Nonlinear Elastic Plates W. R. BIELSKI Polish Academy of Sciences, Institute of Geophysics ul. Ksiecia Janusza 64, 01-452 Warsaw, Poland J. J. TELEGA Polish Academy of Sciences Institute of Fundamental Technological Research ul Swigtokrzyska 21, 00-049 Warsaw, Poland

Abstract-obstacle contact problems for geometrically linear and nonlinear elastic plates were studied, provided that friction is negligible. Three linear and two nonlinear plate models were investigated. Unilateral contact with a rigid obstacle occurs through one of the faces of a plate. Already for linear models such contact problems lead, in general, to nonconvex minimization problems. Existence theorems formulated for plate models were considered. @ 1998 Elsevier Science Ltd. All rights reserved. Keywords-obstacle

problem, Kirchhoff plates, Moderately thick plates, Nonconvexity.

1. INTRODUCTION Obstacle

contact

conditions

Methods

of variational

for thin plates are usually set up on the mid-plane

inequalities

are usually employed to solve such unilateral

of a plate, cf. [I-S].

contact

problems.

For linear Kirchhoff plates, the stretching and bending problems are then uncoupled. If a real situation is described precisely, then these problems become coupled, even after linearization of the unilateral contact condition (2.11), see Remark 2.4 and [4]. The objective of this contribution is to solve the existence problems for rigorously formulated unilateral contact of a plate with a smooth, rigid obstacle. Five plate models are investigated: three linear and two geometrically nonlinear ones. Two linear and one nonlinear of these models describe

moderately

thick plates:

Reissner’s

model [5], linear and nonlinear

refined models

[6-91.

The unilateral contact of lower faces of plates is taken into account rigorously. General existence results for nonconvex minimization problems were formulated by Baiocchi et al. [lo]. We shall Existence results for geometrically apply them to the geometrically linear obstacle problems. nonlinear plates have been obtained by using Cea’s theorem [11,12].

PART I. GEOMETRICALLY LINEAR 2. THIN

PLATES

PLATES

In this section, we shall be concerned with a unilateral contact problem of a Kirchhoff plate with a rigid obstacle provided that the contact occurs through the lower face of the plate. An The authors were partially supported by the State Committee No. 3 P404 013 06.

for Scientific Research (Poland)

through Grant

Typeset by AA,.&QX 55

56

W.

R. BIELSKI AND J. J. TELEGA

extension to the case where both the lower and upper faces may come into contact with rigid obstacles is straightforward. Let 0 C P2 be a sufficiently smooth domain and l? = dR its boundary.

Throughout this

paper, R will always denote the mid-plane of the undeformed plate. The plate occupies the region fz x (-h, h) c B3. The boundary I’ is decomposed into two parts: Fe and I?1 such that

r=fa~~l,ronrl

= 0.

Let v = (vi), zli = ui(Za, z) be a displacement vector of a point (CC&, z) E Cl x (-h, h), a = 1,2; i = 1,2,3.

Throughout this paper, Greek (Latin) indices take values 1,2 (1,2,3) and the

summation convention applied to repeated indices. The axis z is directed downwards. Assuming the classical Kirchhoff-Love kinematical hypothesis, we write (1,13,14]

Here u = (u,)

stands for the in-plane displacement vector whilst w denotes the transverse

displacement or deflection. By Cijkl, we denote the elasticity tensor of the material of the plate. We assume that the plane z = 0 is the plane of the material symmetry; hence, ~,4~s = ~333~= 0. Fora thin elastic plate, the constitutive relationships take the form UcY@=

&3X&Xp(U),

fl,3

=

2’&3X3~X3b’h

033

=

0,

(2.2)

As usual. where gij are components of the stress tensor, and Core.+ = QV,~ - ~~433~33~~~~~~~. the strain-displacement relations are given by

(Z+$j) Sij(V) = U(Q)

The elasticity tensor

Cijkl Satisfies

we assume that cijkl E Lm(R

x

the

=

2

USUFLlSyIUIIM?t~~

conditions: Cijkl = Cjikl = cklij; moreover,

(-h, h)) and

3x1 > 0,

cijkr(Xtn)&jEkl

2 Xl&j&j,

(H)

for each E E Mi and a.e. (L,) E s2 x (-h, h), w here &II:is the space of symmetric 3 x 3 matrices. Let N = (Nao) and M = (Map) denote the membrane force tensor and moment tensor. respectively, defined by

s s h

N,s =

~a4

-h

dz,

h

-h

Taking account of (2.1) in (2.3), we obtain

(2.3)

ma0 dz.

(2.4)

where (2.5) h

A c@Xp

=

/ -h

c a,ab dz,

(2.6)

Existence

of Solutions to Obstacle Problems

57

The hypothesis (H) implies that there exists a constant X > 0 such that

A aRXp(~v)X~PXXp L ~XcxRXuR7 B aRX/&)XaRXXp 1

(2.7)

~XaRXaR,

for a.e., x = (z,) E 52 and each x = (~~0) E Ml:. Obviously, Asp,+ similarly for B&Dx~.

= Aaa~p = ~~~~~ and

For Reissner’s plates, in Section 3, we also need the following elastic moduli:

s h

&R =

~3~33

(2.8)

dz,

-h

which satisfy the coercivity condition

for each < E W2 and a.e., x E R. In the absence of the obstacle, the equilibrium equations of the plate are given by N aL?$ + Pa = 03

W+3a: + P = 0,

in a.

(2.10)

We assume that the distributed loading (pa,p) is such that p, E L2(Q) and p E L2(0). Suppose that a continuous function f : f21 + W, z = f(x), fl c 01, determines a rigid obstacle. The unilateral condition is specified by (cf. also [4]) (2.11)

w(x) + h I f(x, + us(x) - hw,,(x)).

We recall that the z-axis is directed downwards. Thus, the lower face of the plate may come into contact with the rigid obstacle. We introduce the set K = {(u, w) E H1(R)2 x H2(Sl) 1 (2.11) is satisfied for x E Q} . We observe that if K is nonempty then in general it is a nonconvex set. provided that f is a concave function. In this section, two types of boundary conditions are dealt with: (ii; ;I;;

(2.12)

K is a convex set

~~;~~;O~onIol

where mess (IO) > 0 and n stands for the outer unit normal vector to I’. Other types of boundary conditions can also be considered. We set V =

(u,w) E H’(52)2 x H2(n) ) u = 0, w = g

VI = H’(fi)2

4u,

v>=

b(w, t) =

s n

sR

(2.13)

x H,f(st),

(2.14)

-hp~,Jx)wtu)~~~tv) dx,

(2.15)

%.r~Jx)~ap(~)~(t)

(2.16)

dz,

where u, v E H’(a)2 and t, w E H2(Cl). For Case (i), the functional of the external loading is assumed in the form

L(u, w) = n(~aua + pu)) dx + s

/(l-1

rauo + qw - IL22

>

dl?,

(2.17)

58

W. R. BIELSKIAND J. J. TELEGA

where T,, q, A? E L2(I’,).

In the second case, we take (2.18)

and then T, E L2(I'). The functionals of the total potential energy are given by

J(u7w) = Jl(U, w) =

&I, u) + ;qw. w) - L(u, w),

$(u,u) + gw, w) -

(2.19)

Ll(U, w).

(2.20)

The function J is coercive and weakly lower semicontinuous on V, cf. [l-3]. Let us denote by 77, the space of rigid body motions in the case of boundary conditions (ii). Those motions are readily obtained from the equation a(u, u) + b(w, 20) = 0, (2.21) from which we deduce

73= {P = (Pa)I Pl = a1

+

bz2, pz =

a2

-

bz1),

(2.22)

where a,.b E W. Now, we are in position to formulate the first (nonconvex) minimization problem. PROBLEM

(P) . Find inf{

J(u, w) 1 (u, w) E K f~ V}.

We tacitly assume that K II V # 0. We observe that on account of the unilateral condition (2.11) the in-plane and transverse displacements are interconnected. Consequently, Problem (P) cannot be decomposed into membrane and plate problems. We recall that if the obstacle condition is imposed on the mid-plane of the plate, then both problems are independent and only the bending problem is of a unilateral type, cf. [1,3,15]. The strong form of the above minimization problem is now formulated provided that its solution is sufficiently regular. PROBLEM

(Pd).

Find (ti(x),C(x))

and R(x),

a(x),

R(x)

(x E a)

such that

on rl, g(x,

ii,tZ) 2 0,

R(x) 5 0,

R(x)g(x,ii,tZ)

= 0,

in R,

where s parametrizes Pr and (7,) denotes the unit vector tangent to Pi, cf. (1,151. Obviously, R denotes the reaction of the obstacle and g(x, ii, 271) = f (z, + i.&(x) - hlil,a(x)) - C(x) - h.

Problem (Pa) can be derived by applying Rockafellar’s theory of duality [16) to Problem (P). Prior to proving the first existence theorem, we are formulating the following lemma.

Existence of Solutions to Obstacle Problems LEMMA 2.1.

Let f : R + W be a continuous function.

59

The set K defined by (2.12) is weakly

closed. Suppose that {u*, UJ~},EN c K is a sequence converging weakly to (u,w) E H1(R)2 x I-12(n). The Rellich-Kondrachov lemma yields u” --) u in L2(fl)2 strongly and wn + w, w$ + w,~ in L2(Cl) strongly as n -+ 00. Hence, at least for a subsequence, still denoted by {u”, w~}~~N, we have PROOF.

u”(x) --+ u(x) a.e. x E 0,

wn(x) + w(x),

Vwn(x) + VW(X) 8.e. x E R,

as n + co. Passing to the limit in the inequality w*(x) + h I f(z,

+ u:(x) - hw>(x)),

which holds at least almost everywhere in R, we obtain w(x) + h I f(z,

+ u,(x) - hw,o(x)),

Thus (u, w) E K. The first existence result is now formulated as follows. THEOREM 2.2. Problem (P) possesses at least one solution (i&C) E K fl V. PROOF. Taking account of Lemma 2.1, the indicator function ~mv(u,

if (u,w) E KrlV,

0,

w) =

1 +oo,

otherwise

is weakly lower semicontinuous on H’ (fl)2 x H2(0). inf {(J +

IKN)(

u,w)l

(U,W)E

The case where H'(R)2

x H2(iq}

= +oo,

being precluded, by applying the existence results due to Baiocchi et al. [lo], the existence of (fi, G) solving Problem (P) readily follows. I REMARK 2.3. The next minimization problems means finding inf{(Jr

+

IK~v~)(u,w)

I (u,w)E

Pl)

H'(fl)2 x H2(Q}.

The functional J1 + IK~v, is weakly lower semicontinuous on H’(fl)2 x H2(n) and coercive on [IYI’(R)~]/R x @(S-I). Th us, Problem (PI) possesses a solution (f&C) E K n VI provided that, cf. [lo] Ll(P,O)

of each p E R I-IKz, Kz

=

n

J

P&l fh +

flI0, Jr~crPa

where = {u E H’(fl)2

1 ~{%a}n~N

c k

3{‘ha}n~N

c H’(flj2,

with q,, + 00, u,, + u weakly and (ue + q,,u,,,O) E K,Vn E

W},

for any us E H’(R)? REMARK 2.4. The linearization of the r.h.s. of (2.11) yields, cf. [4] af (x) w(x) + h I f(x) + ,,aIua(x)

- hw,a(x)l = f(x) •t

af (xl az:v,(x, a

h),

W. Ft.BIELSKI

60 provided

that

AND

J.J.TELEGA

f is of class Cl; we recall that x = (2,).

af

(4 -yjy&-(X,

Hence,

h) + w(x) I f(x) - h.

(2.23)

a

The inward

unit normal

u = (Vi) =

vector to the obstacle

am --,-4) 8x1

(

Now, (2.23) may be written

= f(x)

Kr = {(u,w)

ax2

01:

G(X, h)b(x) + w(x) - h(x) I 0,

E H’(R)2

x H2(S2) ( (2.24) is satisfied

The convex minimization

that

(2.24)

in 0))

(2.25)

problem

1(u,w) E KI n V},

Pl)

because the functional J is strictly convex. Moreover, this problem the variational inequality, see [1,3]. Find (ti, G) E Ki such that

~(%U)--~+b(~,w-*) We observe

in Sl,

the set of constraints

inf {J(u,w) is uniquely solvable, equivalent to solving

(J-cqgj7pJ)-1’z.

af(x)

- h. We introduce

which is now convex.

is given by

in the form, cf. [4]

vivi - fi(x) L 0 where fi(x)

surface

even after linearization,

V(u,w)

>L(u,-ii,w_*), the membrane

and bending

E

problems

Kl.

is

Pv.1.)

are still coupled.

REMARK 2.5. Of practical interest is the case of a rigid punch pressed a certain distance, say hc 2 0, into the elastic plate. To treat such a case, we assume that for u = 0, w = 0, and he = 0 we have h 5 f(~~), x E R. For hc > 0, inequality (2.11) is to be replaced by

w(x) + h 6 ho + f(x, + u,(x) - hw,,(x)). 3. REISSNER’S In a simple model of moderately replaces

(2.1)i

PLATE

thick plates accounting

(2.26)

MODEL for transverse

shear deformations,

one

by, cf. [5,8]

%(X, 2) = %&g + %%&4,

(x, 2) E R x [-h, h].

Here cpo! (cr = 1,2) stand for the rotations of the plate tion (2.1)~ remains unchanged. The strain measures are

transverse

cross-sections.

For &(w, cp) = 0, Kirchhoff’s model studied in the previous section is recovered. Let us denote by T = (57,)the transverse shear force vector. The constitutive are given by (2.4)i

and

(3.1) Assump

relationships

Existenceof Solutionsto ObstacleProblems

61

The first equilibrium equation is still given by (2.10) 1, whilst the remaining equations are M ~O,D- Ta + ma = 0,

(3.4)

Ta,,+p=O,

provided that the obstacle is absent. We pass to the formulation of the boundary conditions. We assume them in the following form: u = 0,

cp=o,

w=o,

on

(3.5)

r0,

where meas (I’c) > 0. We set Vz = {(u,w,cp)

e H1(R)2 x Hi(R)

x H’(R)2

1u = 0, cp = 0, w = 0 on

If a continuous function f : 521 + W, R c $21, z = f(x)

Fs} .

(3.6)

defines the obstacle, then the condition

of noninterpenetration is given by w(x) + h 5 f(%

+ u,(x)

x E R.

+ k,(x)),

(3.7)

Consequently, the set of kinematically admissible displacements is defined by Ks = {(u,uJ,c~) E Vs 1 (3.7) is satisfied in Q}.

(3.3)

The functional of the total potential energy is expressed by

+%3&(w

(PPB(W,

cp)l

dx

-

Lz(u,

w, cp>,

where J52(u,

‘WY9)

=

(PUG + pw + m&a) s l-l

&r +

J r1

(r&X + qw + fi&,)

dr.

(3.10)

HereP,, P, ma E L2(Q) and ro, q, A& E L2(l?l) are prescribed functions. Now we are in position to formulate the following. PROBLEM (Ps).

Find inf{J2(u,w,cp)

I (ww,cp)

E

K2].

In general, this problem is also nonconvex. The next existence result is formulated as follows. THEOREM 3.1. Problem (Pz) possesses st least one minimizer (6, J, @) E Ks. PROOF. Reasoning similarly as in the proof of Lemma 2.1, we conclude that Ks is a weakly closed set in VZ. By virtue of (2.9),(3.5) and knowing that mess (Fc) > 0, we easily infer that the functional Js is coercive on V.. Obviously, it is also weakly lower semicontinuous on H1(R)2 x Hi(Q) x JYJ~(~)~.The existence of (ti, 5, @) E K2 solving (P2) follows by applying the results of the paper [lo].

4. REFINED

I

MODEL

OF MODERATELY

THICK

PLATES

Reissner’s plate model is not always satisfactory, and hence, the need for refined models, cf. [S-g]. One of such models consists in replacing (3.1) by a more accurate approximation specified by %X(x, 2) = &Y(x) + z G(X)

- ; (i)?

(%X(x) + w..(x))]

9

(4.1)

W. R. BIELSKIAND J. J. TELEGA

62

still preserving (2.1)s. The displacement field (u, ut) have the same meaning as previously; similarly (Pi stands for the components of the rotation vector of the plate transverse cross-sections. Now, the deformation measures are given by (2.5) and (3.2), i.e., &u)

= u(a,B)r

PcrP(yl) = ‘P(a,P)1

&B(W) = --2o,cQ3, The constitutive

(4.2)

d,(w, PO)= 9% + w&K.

equations take the form

Nero= Map = (4.3)

mcxp = T, =

and in the absence of the obstacle, the equilibrium equations are given by Nap,4 + P, = 0,

in 0,

M ao,pa+ Tcw+ P = 0,

in SE,

M a4,4 - T, + m, = 0,

in R.

(4.4)

The distributed loadings p,, p, and m, are prescribed. We assume that p,,p, m, E L2(i2). The boundary conditions imposed on u and cp are the same as in the previous section. Further, w E H2(n) and w = $$ = 0 on Fo, mess (I’o) > 0. We define the space V-s=

{

(u, 20,cp) E H’(R)2 x @(Cl) x H’(R)2 1 u = 0 = cp, w = g

The set of kinematically

= 0 on FO . >

(4.5)

, x E a}.

(4.6)

admissible displacement fields is given by

K3 = {(II, w, ‘p) E V3 I w(x) + h .5 f (za + ‘IL,(X) - ;rp,(x)

- fw,&))

This set is weakly closed in Vs, provided that the function f is continuous. The functional of the total potential energy has the form .73(u,w,cp) = A

2 J{i-l

A a~~,~a&)44

-%P(W))

(Pdcp)

+ Baoxcc P~CP)PX&) [

- W)]

+ fb~a(w7

+ i&+&4

cpPdw7 P)}

dx

(4.7)

Each element of the boundary loading, i.e., r,,q, Z&, and fi is of class L2(I’r). Now, we are in position to formulate the obstacle problem for the refined model of moderately thick plates. PROBLEM

(Pa).

Find inf(Js(u,

w, ‘P) I (u, w, cp>E &I.

As previously, this problem is in general nonconvex. We formulate the following. THEOREM

4.1. The functional J3 attains minimum on the set Ks.

PROOF. Since meas (Fo) > 0, therefore the functional 5s is coercive on Vs. Being convex and finite, it is weakly lower semicontinuous. The set Ks is weakly closed. The existence of a minimizer follows, cf. [lo]. I

Existence of Solutionsto ObstacleProblems

PART II. GEOMETRICALLY NONLINEAR 5.

63

PLATES

VON KPiRMAN PLATES

A lucid account of the theory of thin nonlinear plates is included in the book by Fung [17]. This model is still based on Kirchhoff-Love kinematical hypothesis, thus (2.1) is still valid. Now the strain measures are

&+7(w)=

-w&3,

(5.1)

where, as previously, eclo(u) = uca,~). We note that only the first strain measure is nonlinear. In the c8se of small displacements w,~w,~ M 0 and the linear model of Section 2 is recovered. The constitutive equations have the form

ho = Aa~wap(u, w), MY/3= &l3xpQ&4.

(5.2)

We recall that N, M are the membrane forces tensor and bending moments tensor, respectively. In the absence of the obstacle, the equilibrium equations are given by (2.10)1 and

MYABa+ vk3w,P),a+ P

=

0,

in R.

(5.3)

We impose the following boundary conditions: u = 0,

aw’

w=dn= Hence, an appropriate

0

on

r0,

mess l?0 > 0, (5.4)

on r.

space for displacements is cf. [12] v4

=

{(u,

w)

E H’(fq2

x H(f(R)

1u = 0 on r0

(5.5)

The functional of the total potential energy is given by

(5.6)

The nonlinear strain measure (5.1)1 renders the functional J4 nonconvex on H1(fi)2 x Hz(n) and on V4. This functional is weakly lower semicontinuous and bounded from below [12]. For the obstacle problem, the set of kinematically admissible fields is specified by K4 = {(u,w) E V4 1 (2.11) is satisfied for x E a}.

(5.7)

We assume that K4 # 0. We can formulate the obstacle contact problem. PROBLEM (P4).

Find inf{ J4(u, w) I h w>E fh).

To prove the existence theorem, we shall need the following result, cf. [ll]. Let V be a refkxive Banach space and J : V --) R a weakly lower-semicontinuous If U c V is a bounded and weakly closed set, then there exists at least one mhimizer

THEOREM 5.1.

functional. ofJinU.

W. R.

64

BIELSKI AND J. J. TELECA

The existence of a solution to Problem (P4) is ensured from the following. THEOREM

5.2. The functional 54 has at least one minimizer on the set K4.

PROOF. The set defined by U = {(ww)

E v, I J4(u,w) 5 0)

(5.8)

is bounded and weakly closed. The boundedness of the set U is a consequence of growth condition of the functional J4, see (5.6); detailed proof is provided in [7,12,18]. Obviously, it is nonempty because (0,O) E U. As previously, the set K4 is weakly closed. Consequently, the set Ku=UnK4#0

(5.9)

is bounded and weakly closed. By applying Theorem 5.1, we conclude that J4 attains its minimum value on Ku. I REMARK 5.3. Let a minimizer (ti,3) solve Problem (P4), then J4(CG) I J(u,w), for each (u,‘w) E K4. Since (0,O) E K4, therefore

and definition (5.8) becomes clear. REMARK 5.4. Various static and dynamic unilateral contact problems for von Kkmh plates were studied by Duvaut and Lions [19,20], provided that convex unilateral conditions are defined directly on the mid-plane 0. The method of variational inequalities is then a convenient tool, cf. also [3]. Nonconvex unilateral conditions, still defined on the mid-plane were examined by Panagiotopoulos [2] by applying the so-called hemivariational inequalities.

6. REFINED

NONLINEAR THICK

MODEL PLATES

OF MODERATELY

In Section 4, we have studied the obstacle contact problem for the relined linear model of moderately thick plates. The same contact problem will now be studied for a nonlinear model, where the displacement vector (TJ~)is approximated by

(6.1) 2)3(x, 2) = w(x), and the strain measures are given by

The constitutive

relationships are specified by (4.3)2,3,4 and Nap = A+~ea~(u,

w).

(6.3)

Similarly, as in von K&rmbn plate model, only the strain measure e,o(u, w) is nonlinear. In the absence of the obstacle, the equilibrium equations are given by N,P,P + P, = 0,

in Q,

Mcrp,~a + (&Bw,P),~ + G,, + P = 0, !mu4,~ - T, + mu = 0,

in Q, in Sl.

(6.4)

Existence

of Solutions to Obstacle

Problems

65

Mathematical study of the nonlinear model, just introduced, was performed by us in [7,18]; however, only classical, bilateral boundary conditions were imposed. Neither the obstacle problem has been dealt with. Similarly as in [7,18], we impose the following kinematic boundary conditions: u = 0,

cp = 0,

on

meas (PO) > 0,

r0,

W=dnfi=O

(6.5)

7 0nP .

Consequently, the space to which kinematically admissible fields belong is V5 = {(u, 20,cp) E K’(fi)2 x H:(n)

x H’(s2)2 ] u = cp = 0 on PO}.

(fw

The functional of the total potential energy is given by, cf. [7,18] 1 Js(u,

w, ‘p)

=

-

2

11n

A a~~pw(u,

w)exJu,

w)

where p,,p, m, E L2(sZ) and T,, n;ih. E L2(l?i) are prescribed. For the obstacle contact problem, the set of kinematically given by KS

=

{(u,

w, cp) E v5

I w(x)

+ h 5

f

(za

+ u,(x)

-

admissible displacement

$P&)

-

t+,.(x))

, x E

fields is

O}.

(6.8)

We are now formulating the last obstacle problem. PROBLEM

(P5). Find

inf{Js(u,w,cp)

I (u,w, ‘P) E KS).

The existence of a solution to this minimization problem is ensured by the following. THEOREM

6.1. Problem

(Ps) h as at least one minimizer

(ti, 6, $5) E KS.

PROOF. Adapting the proof of Lemma 2.1, we readily conclude that the set K5 is weakly closed

in VS. In our papers [7,18], we have proved that the set defined by Ul

=

{fu,

w,cp)

E vs

I Js(u,w(P)

IO)

VW

is nonempty, bounded and weakly closed. Hence, the set KS nUi is also nonempty, bounded and weakly closed. The functional Js being weakly lower semicontinuous, the existence of a minimizer (ti, 6, cp) follows by applying Theorem 5.1. I

7. FINAL

REMARKS

The study of obstacle contact problems for geometrically linear plates exhibits that a rigorous setting leads, in general, to nonconvex minimization problems, provided that friction is neglected. Similar situation arises for geometrically linear beams and shells. Problems with nonassociated friction laws cannot be formulated in the form of extremum principles, cf. [l-4,19,20], in contrast with the case when the normal stresses are prescribed, cf. (1,211. Necessary conditions for the minimization problems studied were out of scope of the present paper. Towards this end, one can use the method of hemivariational inequalities [2,22]. The strong formulation has been dealt with for the linear Kirchhoff plate model only. We suggested that the form can be derived by applying the theory of duality expounded in [16]. In a similar manner, one can find strong forms for the remaining models examined in the paper.

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