Second-order model of linear elastic orthotropic plates

MechanicsResearchCommunications,Vol.27, No. 6, pp. 659-668, 2000 Copyright© 2000ElsevierScienceLtd Printedin the USA. Allrightsreserved 0093-6413/00/S-seefrontmatter

Pergamon

PIh S0093-6413(00)00146-4

SECOND-ORDER

MODEL OF LINEAR ELASTIC ORTHOTROPIC

PLATES

A. Stawianowska and J.J. Telega Institute of Fundamental Technological Research, Polish Academy of Sciences, Swi~tokrzyska 21, 00-049 Warsaw, Poland e-mail :[email protected]; [email protected]

(Received 16 February 2000; acceptedfor 24 August 2000) Introduction There are two ways of deriving equations of structures such like plates and shells. The first one, more traditional, consists in making a priori some kinematical or]and stress assumptions, cf. [7]. The second approach exploits the fact that such thin structures have one dimension small in comparison with the remaining two ones. In this case we have at our disposal in a natural manner a small parameter -the thickness. In the second case the starting point is a three-dimensional problem posed on a domain characterized by the small parameter. The two-dimensional model of plate is obtained by passing with this parameter to zero, cf.[1, 3, 4, 9, 11]. In this manner one derives models which are described by the term (U ° , a °) of the asymptotic expansion. The problem is to satisfy the boundary conditions by, in our case, the term ( U 2 , a 2) appearing in the asymptotic expansion. This term may be called a corrector, similarly to the homogenization theory, cf. [8]. Until now two approaches have been proposed to deal with higher-order terms. For instance, for clamped plates the second-order term U 2 does not satisfy the condition U 2 = 0 on the part of the boundary where the plate is clamped. To overcome this difficulty Destuynder [6] studied the boundary layer, cf. also Dauge and Gruais [5]. An alternative approach was proposed by Raoult [10]. To deal with the second-order term of the asymptotic expansion, she appropriately defined boundary conditions, cf. our formula (16). Incorporating the second-order term, the resulting plate model is more exact. The asymptotic analysis performed by Destuynder [6] and Raoult [10] is confined to plates made of isotropic linear elastic materials. The aim of this contribution is to construct a second-order model of orthotropic plates. Obviously, here the expression "second-order model" means that such a model includes the second-order term (U 2, a 2) of the asymptotic expansions (20), (21). In the case of isotropic material our model reduces to the model derived by Raoult [10]. The study of convergence is here confined to the formulation of two theorems. In a forthcoming paper we shall propose a second-order dynamic model of orthotropic elastic plates. 659

660

A. S L A W I A N O W S K A and J.J. TELEGA

Basic equations Let /2 C Td2 be a bounded, sufficiently regular domain, being the mid-plane of the plate. In its underformed state the plate occupies the region ~c = ~ x [ - e , el. Here e > 0 is "small". We set: F = 0)/2, Foe = F × ( - e , e), F~. = /2 × {-i-e}. The plate is made of a linear elastic and homogeneous orthotropic material. The basic equations are given by:

-0)j caij :

efi,

(1)

c~j = c, jk~ ~k~(~u),

(2)

~j(cv) = ~(0~ cvj + 0j cv~),

(3)

cv = 0

on r : ,

Cai3 = + egi

(4)

on r ~ .

(5)

Here C~r, cU denote the stress tensor and the displacement vector, respectively. Throughout this paper Latin indices take values 1,2, 3 and Greek indices run over 1, 2. The summation convention is consequently used, unless otherwise stated. In the case of orthotropy the elasticity tensor C = (Cijkl) satisfies :

Ciijj # O,

Cklkl # 0

for k # 1

(6)

(no summation over repeated indices). The remaining coefficients Cijkt vanish. The variational formulation formally equivalent to Eqs.(1)-(5) takes the form of the Hellinger-Reissner principle :

find ( ca, cU) E S c x V c such that

Vr ~ S c

vv

vo

oB(

ca( ca, r ) + cB(r, cU) = 0,

v)--

- [ of, v, dx- [ cg, v, t3~

whereV c = {re

H I ( B e ) 3 1 V = O on

(7)

(8)

r~:

Feo} and S ~ = L ' ( B c,F;). Here [ F ~ d e n o t e s t h e s p a c e o f

symmetric 3 × 3 matrices ; moreover V~

~e,

VeH~(/3~) ~

CB(~,V)=-/~,j(V)d~,

(9)

B~ V (o, "r) E Z c × ~e where

A = C -1.

CA(a, "r) = / A i j k t awl rij dx,

(10)

Under usual symmetry and coercivity assumptions [3], problem (7), (8) is

uniquely solvable. To prove this assertion one can use Brezzi's theorem [2].

C h a n g e o f variables and scalings To use the method of asymptotic expansions it is convenient to work with the fixed domain B = /'2 × ( - 1 , 1), cf. [3, 4, 7, 8, 10]. To this end we first consider our plate as belonging to

SECOND-ORDER ORTHOTROPIC PLATES

66l

a family of plates occupying in their underformed state the regions ~ x [-e, e]. Those plates are clamped on their lateral surfaces F~ = F x I-e, e] and subject to body forces ~f and surface forces *g. The variational formulation is given by Eqs. (7), (8), where e is to be replaced by e. Next we introduce two transformations: i: L2(B ~)-~ L2(t~), V ¢•

a.e. X • ~ ,

L2(8), k:

L 2(r~. U F_~ ) - - , L 2 ( r + t3 F - ) ,

V ¢ • L 2 (B), H e r e F + = ~ x {1},

j(dp)(Xl,X2, Z3) = (~(.TI,X2,eX3);

a.e. (x,~) • 1"2, k(¢)(xbx2,'4-1) = tb(xl,x2,+e).

F - = 1"2 x {-1}. We also set F 1 = F × [-1, 1]. Then we have

a~Z = j(~0"¢,~),

0"~3 = e-1J(~0"~3),

u~. = j ( ~ v . ) ,

°'~3 = e-2J(~0"33),

(11)

(12)

v~ = ej(~u3).

Furthermore, we assume that

j(tfa) = 1°,

Problem

j(~f3) = eft,

k(e9,~) = eg °,

k('93) = e2g;.

(13)

P~

This problem is posed on the e-independent domain B. It is formulated as follows: find ( a e, U e) q ~ x X such that

v1" • x VV•X

A*(~, ", 1") + B'(1", u ~) = 0,

(14)

B~(a ", V ) = - f f~ Vi d x - / g° V~ dI ".

(15)

s

r:~

Here L' = L2(B, ff:~)and X = X12 x X3, where 1

X12 = {V E Hi(B)2

I f V~ dx3 -1 1

1

= O,

f za V,~ n,~ dx3 = 0:on F },

-1

(16)

X3 = ( V • HI(B) I f ( 1 - x~)Va dz3 = 0 on F}. -1

Substituting (11) and (12) into Eq.(14) we conclude

A~(tr, 1") : Ao(a, 1") + e2A2(a, 1") + E4A4(o", 1"),

(17)

where

Ao(cr, r) = (Ac,~.,6 a.~6, r,~),

A4(o', 1") -- (A3333 0"33, 7"33),

A2( tr, 1") : (AaB3a 0"33, Ta/~) + 2(Aa363 0"35, T3ct) + (A33-:,6 0"~5, T33).

Here ( . , .) denotes the scalar product in L2(B) ;

for instance :

(0"~e,

dx. B

(18) (19)

662

A. SLAWIANOWSKA and J.J. TELEGA

The decomposition (17) suggests an asymptotic expansion of the form, ef. [10],

0,¢

0,o + C20,2 + 640,4 .jr,..

(20)

U ~ = U ° + c2U 2 + c4U 4 + ...

(21)

:

It is well known that the second-order term {a 2, U 2} cannot be such that U 2 belongs to V. In other words U 2 cannot satisfy the boundary condition on/'o1. To remedy this difficulty Destuynder [6] examined the boundary layer, cf. aJso [5]. A different approach was proposed by Raoult I10], who introduced the space X for displacements, here defined by (16). P r o b l e m s ( po) a n d ( po f ) Identifying the terms linked with •°in ( P ' ) we get the following problem: find {0,0, U o} • ~ x X such that

Vr • .,U,, (P°)

I VV

Ao(0, °, "r) + B ( r , U °) = O,

e X,

B(0, °,V)

=

(22)

F(V).

(23)

Furthermore, we introduce the space VI
vuL = { u

I~(u)

= 0,

%~(u) = 0),

(24)

~33 = 0}.

(25)

and the following space of stresses s=

{~• 21~3=0,

The last space can be identified with L2(13, IF~.); moreover [3] yuL = { v I u3 = w • Ho~(a),

go -- ~ - x3 0~w,

~ • Ho'(a)}.

Now we are in a position to introduce the following plane stress problem: find {a °, U °} • 5' x VKL such that

V-r E S,

Ao(o"°,'r) + B ( r , U °) = 0,

(26)

(P°f) v v • vuL,

B(0, o, v )

=

F(v).

(27)

It is well-known that if {0,°, U °} is a solution of (po), then { { ~ o , 0, 0}, U °} (where U ° • VKL) solves problem (pof), [10]. For 0,0 • S we get: 733(U °) = - C ~ 3 C~33 "Y~z(U°); hence

= D~.

~

~

(u°),

(28)

where

Dct/3AU = Cet~AU -- Coq333 C33133 CA,u33.

(29)

Problem (pof) splits into two problems: (i) the plane elasticity problem u ° • Hol(a) 2,

[2 D ~ ,

-r~u(u°),-r~z(v)] = F(-v~,, 0),

VveHo(1 s2),~

(30)

SECOND-ORDER ORTHOTROPIC PLATES

663

(ii) the orthotropic plate problem w ° e H~o(~),

2

3IDylls, O~w°,Oo~w] = F(xa Oow,-w),

o

w ° - u3,

Vw E H~o(~2).

(31)

Here [., .] denotes the scalar product in L2(j2) or the duality pairing between H-I(J2) 2 and Hol(~2)2. Remark 1. For materials inhomogeneous in the transverse direction problems (30) and (31) become coupled, cf. [7, 8]. In the case of isotropy Eqs.(30) and (31) reduce to, cf. [10],

[ g u °, v ] : r ( - v ,

0),

2 E 3 1 - v 2 [ AwL Aw ] = F(x3 0 ~ w , - w ) , respectively. Here A denotes the laplacian. Moreover we have ~o

E 1 - .~ [(1 - ~) %z(u °) + ~ ~ . ( u °) ~o,1,

=

where E, u denote the Young modulus and Poisson ratio, respectively. o o , Let us come back to problem (P°). Our aim is to find a solution {{%Z, o'33 We assume that the loading functional F is given by F(V)

= - ( f~, V~ )

[

-

o.o 33}, U ° } ~

X X X.

(32)

g~ V~ dr,

F+ u F -

where fo E L2(B), gO E L2( F + U F - ) . The equations to be satisfied by 0°3 E L2(B) 2 and 0"~3 E L2(B) are respectively given: v v e x~,

(o°~3, o3vo) = - (o~, o.y~) + (f$, yo) +

[

go y~ dr,

(33)

g~ v3 dr.

(34)

F+ u F -

v v3 ~ x3,

(a~3, O~v3) = - (a%, O~v3) + (I~, v3) +

/ F+~F-

By using Eq.(28) and Lemma 3 of Raoult [10] we get =

J-I

o,

_

z;

where b c,~ ° = - x 3 Dao~u OAuw°. In the case of isotropy Eq.(35) reduces to, cf. [10], o ~ 2 E o o'~3 = - (1 - X3)l--~v2OaAw . The following relation was derived by Ra~ult [10] 1

/(f~

+ 0oa°3)dx3 + g~+ + g~- = 0,

664

A. SLAWIANOWSKA and J.J. TELEGA

where g~+ = g~ It÷,

g~- = g~ IF- • Hence for the orthotropic plate we get 1

(36) -I

In the case of isotropy the last equation reduces to 1

2 E A2 w o f o+ o31---u 2 = f~ dx3 + g3 + g3 • -1

To determine a~3 we apply Lemma 3 of Raoult [10] once again. Finally we get x3

+ /1

-1

+ .+o_.o_

+ ( g ~ + - g ~ - ) ( 3 x 3 - x 3) 4

(37)

-1

We observe that the stress 0"33 does not depend on material properties.

Problems (p2) and (p2f) Identifying the terms linked with e2 in (pc) we get the following problem: find (a ~, U 2) E 27 × X such that VT 6 S , Ao(o "~, r) + B ('r, U 2) = - A ~ ( a °, r), (38) V V e X,

B ( a 2, V) = 0.

(39)

Similarly to previous section we first consider a truncated problem ( p 2 f ) : find (0 2, U 2) E S x X (pZf)

/

Vl"e Z,

t

v v ~ VKL,

Ao(a 2,r) + B ( r , U 2) =

B(,, 2, V )

=

-A2 (a °, r),

(40)

O.

(41)

From now on we assume that the loading functional is given by F o ( v ) = _ (fo, V~) -

/

g~ V3 dF,

(42)

F+ u F -

where j o e H I ( ~ ) , f~ E L2(B), gO3e L~(F + t.J r - ) . Now taking successively in Eq.(40) : v = (raZ , 0, 0), v = (0, ra3, 0) and r = (0, 0, r33) we get a.~62 = D,~-v67,~0(U 2) - Aa~33 D,~/~-~6o'33,° =

w 2 + A33cq~ D¢~/~76 [7,~6(u°) x 3 -

u~2

-

x3 0~

w 2

-

A.363(z3

~ x 2 3 0 7 6 w°J,

-

~1 z 33) D 6 ~

(43) w 2 =- u~,

~

(44)

w°

(45)

SECOND-ORDER ORTHOTROPIC PLATES

665

The last relation yields

, . ~ ( u 2)

=

lx'3) D6vw,, &Y.~v w° 3

Aa363(x3 -

7 a 0 ( U 2) -- X30qa~ W2 -

1 - ~Aa3¢¢ D~c.y6 [ 0 ~ %s(u °) x] -

~x~ O.a-y~ w°].

In the case of isotropy we have

U2~ = u~ - z3 cg,~w2 + 1 -"/ 2 !2~ 1

- 1 ----7 =

O.Aw °[z3+l

- --

1-v

0 ~ . ~ ( u o) v

~ x] ( 5 -

z~ 7..(u °) +

1---~ 2

1)],

z~ Aw °.

Let us pass to the derivation of the strong equation satisfied by w 2. Substituting (43)-(44) into (41) and integrating the obtained equation twice by parts we get D,~Z;~. 0,~xu

D~zxu A33~i De¢.~ oq,~Ox,,-~w °

=

8 -'i-~D,~xu Ao3#3 D~s,, O.y6,,~xu w °

1

(46)

D~,/3az AaB33 0,xz f x3 0"~3 dx3.

-

-1

In the case of isotropy the last equation simplifies to, of. [10] 1

2

E

5 -1- ---- ' -v~

A2w2 = _

2E

3(1 - /22) (1 - /2)

8 + v A3wO + 10

~

v

/~

f

o

*'L'3 0.33 dx3.

-1

Substituting Eqs.(36) and (37) into Eq.(46) we get 1

-1 - [

3

D~,B,~ Acta33 +

~3

-1

(47)

1

s ~ a,,~o~Do,,.,,lO.,,(g~++g~_+ffgd~)"

~

= ,2

-1

In the case of isotropy we obtain 1

2- E~/x2w2 1 - v3 2

z3

1

1-vv /X x3 f~d~3+ 10~-~,)/X[ /~dx3+g~++9~-]. -1

-1

-1

As we have mentioned earlier, the functions u~,2 w 2 and c9,,w2 do not vanish on F = 0~. Therefore it is interesting to know what boundary conditions are satisfied by these functions. function U32 belongs to X3 therefore 1

/(i - :X) ui ~ -!

= o on F.

Since the

666

A. S L A W I A N O W S K A and J.J. TELEGA

Substituting Eq.(44) into the last relation we finally obtain W2 -~- ~I

D c,,OXu A c ~ 3 3 O.~UW° on F.

(48)

Similarly, since U~EX12 therefore we have 1

/ U~ dx3

0

F.

Oil

-1

Substituting Eq.(45) into the last relation we get

1

u~ = ~ A33;~u D~u~5 0~%6(u °)

on F.

(49)

Finally, the second condition defining the space X12, now applied to U~, yields 1

X3 U2

nct

dx3 = 0

on

F.

-1

Hence

O,~w2 = - 8

~__L2A.a~a D,,~xu O,~ OAuw° + l o~

Aa3.~ D~,~u O,~ Oxuw°

on F.

(50)

,

In the case of isotropy the boundary conditions (48), (49), (50) assume simpler forms, cf. [10]

w2 O.w2

_

u 10 (1

_

-

8+v

lO (1

-

u) .)

Aw o

on F,

O. A w °

onF,

L,'

u~

6 ( 1 - . - - ~ 0o %u(u °) on /'.

-

Convergence From the mathematical viewpoint the method of asymptotic expansions is a formal one. Hence the need for rigorous proofs of convergence of the sequences {~}~>0 and {Ut}~>0. The convergence of the zeroth order is standard, cf. [3], [8]. We observe that convergence theorems formulated in [3,10] pertain to homogeneous and isotropic plates. T h e o r e m 1. For each F ° E X* one has U ~ -~ U ° o'~B --+ o ' o a3 ,

in Hi(B) 3 strongly as

e --* 0,

(51)

¢a~3

in L2(B),

(52)

--+ O,

E2a~3

0

where (or°, U °) is the solution of problem (pof), 0. More precisely, one usually proves only the weak convergence. The strong convergence is then proved by using just this weak convergence and 1

1

the property that 1" --* A ( r , T)~ is a norm on Z' equivalent to the usual norm. Moreover we use the form (17) of A t, cf. [9]. Here A(T, ~-) = AI(~-, r ) . With a similar problem of strong convergence we have to deal with the convergence of the second order. We pass to the formulation of this result.

S E C O N D - O R D E R ORTHOTROPIC PLATES Theorem

667

2. Under the assumption (42) the existence of a solution for problem ( p 2 f ) implies

that f1i ( U ~ - U °) --* U2 1 a~ o fi( ~ _ ~e)~ ~o,,2

1

in Hi(B) 3 strongly as

c

~(a~3 - o~3)° - , o,

E --* 0,

( ~ 3 - ~ 3 ) -* 0 in L2(t~)

(53)

as ~ -~ 0, c.

(54)

The proof will be given in a separate paper.

Interpretation Let us now examine the second-order plate model, where the transverse displacement is approximated by

r3 = (v~ + ? u~) I=~=o = w ° + ? ~ 2 .

(55)

Taking into account (55) in Eq.(36) we get 2

1

D~Bx ~

O~x.

r3 = f f~ a~z + g~+ + g~-. -1

From Eqs.(48) and (50) we obtain the boundary conditions satisfied by r3 and Onr3 on F: r3

=

21

E ~

A33a/3 D ~ u

0nr 3 = E ]-6 A33a# Dc.~;~u On O~,,w ° - s 2

O~w ° ~

on F,

Aa3a3 D,~,~,u On Ox,w °

on F.

a=l,2

We recall that w ° E Ho2(12), i.e.:

w° = 0

and

OnW° = 0

on F. Now let us consider the real

plate of thickness 2e, the volume density ~p and subject to the surface force eg3 on F~.. We assume that ep=pe2, e9+ = g+e 3. According to (12) eU3 satisfies : eU3 ( z l , x 2 , e z 3 )

= e -1 U~ ( z b x 2 , x 3 ) .

For sufficiently small e we have: e-2(~U3 - Z~) (zl, z2, 0) ~- z32 (Zl, z2), Then, after (55), er 3 =

z~(za, x2) = Z~(zx, x~, 0).

Z~ -I- e 2 Z32.

(56)

Consequently, (56) and (36) yield 2 "~ DaBxu OaB~u er3 = -- 2e ep +

g3+ "

(57)

The boundary conditions are 1 er3 = ~ e2A33aB Do~xu O~uz~,

0n

%

1 8 3 = -~ e2A33aa Daa~, On Oxuz~ - .tu e2 E

on F,

Aa3a3 D,~a.~6 On O.~sz~,

(58) on F.

(59)

a=l,2

Obviously, z~ is the solution to

2 e~

o =

2e "p + °g+~,

z~ e z h a ) .

(60)

668

A. SLAWIANOWSKA and J.J. TELEGA

Concluding remarks The analysis performed clearly exhibits that the second-order terms improve thin plate models. It seems that such models cannot be obtained by standard engineering methods. The second-order terms play the role similar to correctors in the theory of homogenization, cf. [8 ]. They improve the convergence as E tends to zero. Indeed, by taking into account these terms the weak convergence becomes the strong convergence thus allowing also for an approximation o f derivatives of displacements and stresses. In a forthcoming paper we shall propose a second-order dynamic model of orthotropic plates.

References [1] G. Anzellotti, S. Baldo, D. Percivale, Dimension reduction in variational problems, asymptotic development in/'-convergence and thin structures in elasticity, Asymptotic Anal.,9 (1994) 61-100. [2] F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers, P~v. Fran§alse Automat. Informat. Recherche Operationnelle,

S~r. Rouge, Anal. Num~r., R-2 (1974) 129-151. [3] P.G. Ciarlet, Mathematical elasticity,vol.ll:Theory o/plates, North-Holland, Amsterdam 1997. [4] P.G. Ciarlet, P. Destuynder, A justificationof the two-dimensional linear plate model, J. mdc., 18 (1979) 315-344. [5] M. Dauge, I. Gruals, Asymptotics of arbitrary order for a thin elastic clamped plate, I. Optimal error estimates, Asymptotic Anal., 13 (1996) 167-197; II. Analysis of the boundary layer term, ibid., 16 (1998) 99-124. [6] P.G. Destuynder, Sur une justification des modules de plaques et de coques par les mdthodes asymptotiques, Thtse, Universitd Pierre et Marie Curie, Paris 1980. [7] T. Lewifiski, Effective models of composite periodic plates - I. Asymptotic solution, Int. J. Solids Struct., 27 (1991) 1155-1172; II. Simplifications due to symmetries, ibid., 1173-1184; Ill. Two-dimensional approaches, ibid., 1185-1203. [8] T. Lewifiski, J.J. Telega, Plates laminates and shells: asymptotic analysis and homogenization, World Scientific, Singapore, in press, [9] A. Raoult, Contributions a l'dtude des modules d'dvolution des plaques et ~ l'approximation d'dquation d'evolution lindalres du second ordre par les mdthodes multipas., Th~se 3~me cycle, Universitd Pierre et Marie, Paris 1980. [10] A. Raoult, Construction d'un module d'dvolution de plaques avec terrne d'inertie de rotation, Annali Mat. pura appl., 139 (1985) 361-400. [11] A. Slawianowska, J.J. Telega, Asymptotic analysis of anisotropic nonlinear elastic membranes, Bull. Pol. Acad. Sci., Tech. Sci., 47 (1999) 115-126.