A remark on a boundary contact problem in linear elasticity

manuscripta mathematica

m a n u s c r i p t a math. 63, 455 - 468 (1989)

9 Springer-VerlagI989

A REMARK ON A BOUNDARY CONTACT PROBZEM

IN LINEAR ELASTICITY Rainer Schumann

In C 8 3 the author used a boundary contact problem to define an operator ac~ing on the boundary of a given domain. Here we show that this operator is a pseudodifferentlal opera,or and compute its principal symbol.

Introduction, and statement of the result

1. ,

.

,

,,,

t

,,,H,

,,,

,, ,,,,

.

,

,

In order to p~ove C I+~- regularity for the solution of Signorini's problem in linear, n-dlmenslonal elasticity the author proposed to use a boundary contact problem (see ~8~ ). This p~ocedure made it possible to reduce Signorini's problem to a variational inequality for a scalar function. Let us state the problem. We suppose that an elastic body occupies abounded domain i ~ . c ~ n (in~2) with smooth boundary'S4 . The linear elastic behaviour of the body is described by the elastic coefficients a ~ (1) symmetry: a ~

= a~

= a~

455

= a~

satisfying

(i,j,~,p=1,...,n),

SCHUMANN

with ~ioc = ~ i (i,~ =1,...,n; s11mmation convention applied,

1 '12

=

i,r =1

I '" 12i )"

Here we explicitely compute the symbol of the pseudodlfferential operator given in Definition 1.1. later on if the~ material is homogeneous and isotropic. In this case we have (1.1)

aii ii=2/~+X;

lj aij

=~

,

ajii=/=' j

From the symmetry property we ge~

fori+

aij._ =/~ f o r

j. i $ J. In

a~j~ = O. Here ~,/a > 0 ara the

the remaining cases we have so-called Lam6 coefficients.

Using summation convention the equilibrium configuration of the body is described by

(i=l,...,n; ~.,.:= Q/~ =~ ).

(1.2) - 4 (a~'~ ~ ub= fi which we abbreviate by

Au = f. To supplement (1.2) by

boundary conditions we define the stress tensor O"i responding to the displacement vector .~2 (u) := aija~--~ u j . Let ~ r

~L.

u: G

-~n by

denote the outer unit normal

Then the boundary stress vector T(d4e)u together

with its normal and tangential components T ~ u given by 9 ~vu

cor-

(T~d@)u) i =

=(T(~)u.,)~

if;

and Ttu is

n~ = aij O/b uJn~ ,

~ a/~ uOng ni)~ , = ta " ij

T tU = ~ ( ~ ) U - T4V~. With these preliminaries we can define a pseudodifferential operator now. Definition 1.1. Suppose we are given ~ 6 ~6G~(~,~

Co~ ( ~ A ~ ) "

Let

~I) be the unique solution (of. Prop. 1.5.,

456

SOHUMANN

below) cf the boundary contact problem A~=

(1.3)

0

in~'~,

-~'~' : ~', ~t ~ P f , : : -(m(,,,)~).,*,.

We se~

:

Remark 1.2. let us assume ~ : ~ + ,,

,

~.

o on

and that an |

subset

u

~i~. belongs to the (x I,. . .,Xn_ 1)-hyperplane of ~n.Then em r the boundary conditions of (i.3) read as follows

~

=

~,, ~ ( ~

l~rthermore

=

~(~

:

...

:

~

6n_

1

(~

:

o.

P~ = -~(~).

Pre~osltion 1.3.

(1) The boundary contact problem (1.3)

,

has a unique solution provided that

(1.4)

~

where ~ : =

= {07 f u l u = a + Bx

with a e ~ n

B a skew-sym-

met~ric c;ons~ant, real (n~n) matrix; u.~v= 0 on ~ _ ~ } . (li) Assuming Ca~

(1.4) the operator

P: Cf(~.C~.)

is we11-deflned; it is a pseudodifferential

e~erator of order I: PELI(~i'Z). Its principal symbol on ~ i s ~I(P) (x',

~,)

fbr (x,,o)~r, Proof.

= ~/,(/,.+~)(~,.+z~-ll~"l

r ' E m n-1 -

fol

.

Per (i). The geometrical condition (.I.4) implies

the coerciveness of the quadratic functional correspon. ding to A (cf. Ne~as and Hlava~ek ~ 6 ] , p. 101; Schumann

~r8], ~ o p . 3.8.).

457

SCHUMANN

For (ii). Th~s is the contents of Section 2.

2. i

On elliptic boundary value problems ii

2.1.

ii

ii

i

li,.

ill

General remarks~.

ii

We shall need some aspects of the

general theory of elliptic BVP's. De~ailed expositions emphasizing different aspects can be found in Ds

E2],

Vol.8| Grubb [3]; H~rmander E4], Vol. 3| Rempel and Schulse ~7]. Suppose _CA is the domain from S e c t i ~ I. Per functions u: ~

~

>C N (N ~ N) we define the trace operators

c ' ~ ( ~ , c ~)

~ c'~(~e.,r

by ~u-= (1/1)~

a~u/,~?J

-- i ~ ~'IU/~4~J (J=1,2,...) where ~2 denotes the inner unit normal t.o ~.~ . Le~

A: 0 ~ ( ~ , r N) ~ > C ~ , ~

linear elliptic (matrix)differential

N)

denote a

operator of degree m

with C ~-coefficients which we suppose to be defined on Rn, for simplicity. For any function u E C~ ( ~ , C N) we denote by u ~ its extension by zero outside ~ ,

Thus u ~ is a (vec-

tor) distribution on ]{n. The difference

A(u ~

- (Au) e is

a distribution with support in ~ - ~ tha5 depends only on A

and on ~'u:=( ,roU, se~

A(u ~

for N A on P

(2.1)

, ~_lU)T; for any ~ e " '

- (Au) ~ =: N A ~ U .

(~,C N) we

To get an explicit expression

we assume that A is given by

A(x,Dx) = ~

Aj(x,Dx,)Dnj

where x=(x',x~), Dj = Dxj I = (I/i)~j, Dx, = (DI,...,Dn_I) , D x = (DI,...,Dn). The (matrix) differential operators Aj

458

SCHUMANN

are supposed to have degree not greater than (m-j) and to contain only derivatives with respect to x'. Then on the multiple layer NAY can be represented by

(2.2)

N&V = {

~-k-1

~

l=0

Al+k+l(X''0;Dx'~ vl e D(k)~(x n)

wi~h v = ( V o , . . . , V m _ l ) g C ' ( ~

m-1 , i ~1= 8N) where 8(x n) de-

notes the ~-distribution with respect to x n. Furthermore the so-called Calder6n operator is defined by

(2.3)

Or:=

~'/(QNAV)i~] m-1

where Q is; a~proper parametrix of A. C maps C ~ ( ~ , i ~ l ~e

itself. The components of

eN)

C = (Cjl)are pseudediffe-

z e n t i ~ Qperate=H:OjlELJ-l()A'A,

oN). Of. Dieudonn6 ~2],

Vol. 8, Oh. 23.48; H8rmander [4], Ch. 20.1.

2.2.

The boundaz'y contact problem.

The differential ope-

rator from (.1.2)i'can be written in the form (2.1) with m=2, N=n:

AU~ = AOt~ + AIDlu §

A

2 2Dnu

with n-1 ~ , # =1

n-I AI = ( ~

A2

=

ann

( ij ),

a ~

~n + eg---, ~ ai~ Dec ) I ~i,j_~n.

459

SCHUMANN

Thus we have on I NA~U: y ~A1r0u~(Xn)

+ x2Gu~#(~ n) + A2roU|

} .

We denote the principal symbol matrix of A by a:

a Cx, ~ ) = ( ~,#~--i aiJ

'

Analogously we set: al(x',0; ~') = p~inclpal symbol

'

'

6~2_l(A1)(x ',0; ~i)i for the

matrices of the operators A 1 (l=0, I ,2)~,,

(x;o)e ~, ~ I ~ n-l- [ 0 ]

The Calfler6n operator' is a. matrix

pseudediffe ern~ial operator from C ~ ( ~ l

; ~N~N)

into i~-

self: C = (Cjl) (j,l=0,1). The principal symbols Cjl(X',~) 99-- 6"j_I (Cjl )(x', ~' ) en ~ las where

are given by the following formu-

a -1(x',O; ~l, ~n ) is the inverse matrix ef

a(x',O; }", ~'n): %0(x" ~') :

j" P+~')

a-1 (x, ,o; f', In ) [a 1 (x',o; ~") +

Ool(Z',~') = Olo(:X', ~') :

P+(~') p ~( ~' )

a2(x',O,

CI~'n 9 r')~'n} 2~----F"

a-l(x',o; ~',[n ) a2(x',o, r') 2=d~ni a-1 (=',0; ~', ~n) {~l(x',O; ~') V n ,

,

2~i 011(x',

) =

f

a-l(x',o; [',~n) a2(x',o; (') ~n

a ~n

Here we assume t h a t ( x ' , O ) e P , ~'+ 0 and tha~ ~+(~1) is a

460

SCHUMANN

positively oriented contour in the upper complex half-plane

~n

E ~ I Ira ~n > 0 }

enclosing the zeros

~n of

det a ( x ' , O ; ~ ' , ~ n) = 0 such that Im ~n>O (cf. Dieudonn$ 62], Vol. 8, Oh. 23.48/49; H~rmander Er

cendi~lan of (1.3) is written in the form

The b o u n d a ~

B~u = ~

wi~

we have

Bru

and

B( I ) -_

~u-= (~ou, 6u) ~ and B:= (B(O),B(1)).

o

o

i:

a

n

@o

i

@ee

n

...

abbreviates summation from ~=I

We dene~te the rows of B~ by b i fine a boundary operator n-1 n-1 b~.~ ( ~=I ~ ~ , . . . , ~=I 7-

aln

n

an-l,2

~

to ~ = n - 1 . Next we de-

~ ~ an1'''''ann

We remark that b1 2 T :=

~bn is related to: the boundary stress operator by

461

n

an_l, n

(i=O,...,n-1).

,

1

o)

...

eee

n

an-l, I where ~-i

,

o

@el

D

On P

(~,O,...~O):~Z~

=

(

Vol. 3, Ch.20.1).

~"

SCHUMANN

~ u , = - i ~(@~iu c u D ,

and P ~ =

-i bn~"u sn F i Z u is

the selutlen ef (I .3)i corresponding to the boundary da%-um ~. Since the boundary value problem (1.3) is elliptic (cf. subsection 2.3., toe) we may conclude that the "augmente@" epera~or "A:

c,O (/~,C n)

>

u,

c,~C.~.,r

~(

,r

Au

N

,

N

Bru

),

~a

has a right parametrix Q, il.e. ~ Q = I + E where R is a teN gula~izing operant. It: ~s easy to see that Q can be chosen

a~

(,2.+) ~(f,g)

= ~fQ(e ~ + Qt+xU(g - ~::rCQ(:~)]l.~ )~]a~

fo~ f ~ C ~(~,gn)i,

g6C~(~/~

,r

'

where U is a proper

righ~ ira~ametrix of the operator BC (in the sense ef Deuglis and

Nirenberg) : (BC) U = I + R I, R I regularizing

(cf. Dieudonu6 ~2],Vol. 8, Ch.23.50; H~rmander ~4~,Vol.3, Ch.20.I.). The existence, uniqueness and regularity results for the elliptic BVP (1.3) (cf. Agmen, Deuglis and Nirenberg [1~,Ch. 10) ~ p l y

the existence of a linear continuous ope-

rator G,

such that

AG(f,g) = f in /I and B ~ G ( f , g )

= g on ~/i f~r

all (f, g) ~ C ~ ( ~ , cn) ~ O ~ ( B / I , ~n) 9 From the equations G~u and

= u

A Q = I @ R

for all u ~ C ~ ( ~ , ~ we get.

G(f,g) = ~(f,g) - G~(f,g)

for all (f,g) ~ C "~ ( ~ , c n ) > ~

C ~ ( ~ / l , ~ n) and

~(~,g)

n)

= ~'(f,g)-

~'~(:~,g).

462

SCHUMANN

(Of. the reasoning in Dieudonn$ ~2], Vol. 7, Oh. 23.25.) Putting f:=O in (2.4) and using the definition of the Calder6n operator (2.3) we conclude

F~(o,g)

oug

=

-

~(o,g).

This shows that P is a pseudedifferential operator which has the following representation on r :

(2.5) P ~ : - i

bn rG(O,(W,G,...,O) T)

= - i bnCU( ~ , 0 , . . . { 0 ) T + ib n rG~(O, ( ~ , O , . . . , O ) T ) . Looking at the orders of the pseudodifferential operators O and U we easily get P ~ Ll(~_O.).

2.3.

Oomputation of 6~I(P).

New we use the concrete form

(I .I) of the elastic coefficients to calculate the principal symbol of P thus finishing the proof of Prop. 1.3,(ii). We have

a 2 = diag ( ~ , / , 2 ~ + ~ ) r O

0

...

0

0

...

here, and

I

0

/~D 1

...

0

0'

. e e

0

0

B=

b~

: ::

e e e

o . e

0

0

(~])l

, ~

(b (~

eeo

eee

e , e

... /~Dn_ I

D2

,""

,

o

I

e e e

el

e|

0

... /~

o,

. . . , o, ( ~ , + Z ) )

0

bn(') )

For Zam6's system the inverse matrix

F =

the principal symbol a is given by

j2

-

4

463

' s,1 =

,i )

SCHUMANN

I ,ran _ fo} ,

1 = Krenecke=' s symbol (of. Kecs and

Teodorescu [ 5], 0h.4.3.1.2., e.g.). Let us define a matrix

w=

(wJ,z) (j,z = ~,...,n)

C2.6)

for

wJ,l(~ , ):=

by

~"~'nF~ e

I~ ~( ~')'

d~n

~',~,,)

2~i

,i (

t~ O, ~§ O, ~;~ ~n-1 where ~ +( ~t)~ is the contou~ used

in subsection 2.2. te compute the cemponenSs of th~ Calder6n operator. From (2.6) we conclude that W,ls C @@ ( ~+,C n) satisfies the following system of ODE's: a(x',O; ~',Dt) w l(~',t) = 0 for t>O,

~'@ O, (x,,o) E r . Here we set for

1 = l,...,n

w,1 = ( w1,1'''''wn,1)T" Now we determine the functions wJ,l explicitely. We define

/6:= ( ~ + ~ ) [ 4 / ~ ( ~ + A ) ~

-I,

V := (3/~ + l ) [ 4/~ (2/, + 1 ) ] -1 anr remark t h a t ~2-k~ = E 2 ( 2 / ~ ) ] theorem gives

(2/~.! -1 _/~ =~2 . The residue~

-1 and

W(1',t ) = i -I e - l u

l~'l- I~'1~-i~'i ~ ~1~'13

-

.

II'1 ~

"'" ...

I~'13

1I'1"

2/~1I'1

1~,1~

lrl -~K

~,,

I~'1

II'1 9 eee

oee

"d

k

11'1

1t'1

464

... iri§

/

r.Q

I1

0 II 0

o

~i~. ~!,~

o

~1~ ~1~ I

I ~'1~

~1~ ~!~

"~

9

.o " e~ T-

,, A

9:

~

~i~

~

~m'~_

i

o

".

.:

i~l~

I!

4~

O

|

t

o

9

9

:

~1~ ~ '

o

C~

t"~

:

4,

,I

~

o

:

m

LO ~D

SCHIE~ANN

~e

D~w(~',o~J

S'p=.(b n)

(.2.8)

(

~ ~, i~ . 2.(2/,. 9 z):. I ~1 ' 2 C2/,...~) I fl "'"

112 ~.

Le~ c denote the principal symbol matrix of the Cal~er6n eperater: c = (Ckl). We set im c(x', ~l) for the image of the pro:Jecter c(x', ~l): c2n____~c2n. Then we ha~e (.2.9)

~(W(~l'O) )~ im e(x',~1)~= ~ D t -~i~;~i

~er (~',o)EP, ~'~mn-1 -~0~

Col. D•

~cn

1

~23, Vol. 8,

Thm. 23.49.5; H~rmander [4], Vel. 3, Thm. 20.I.3.)J. Fzem (2.7) and (2.9) we conclude that the restriction ef ~pr(B) to

im c(x',~')>: ~pr(B)]im c ( x ' , ~ ' ) : im c(x',~')

>Cn

is bijective what corresponds to the elliptlclty ef (1.3). Instead te compute the right: pa~ametrix U of BC fully we prefe~ ta calculate the symbol efl P in a mere direct way. We propose to find ~-* 6 ~2n such that

(2.1o)

~pr(~) c(~',I'~# = e~:= (~,O,..,O)T~ n

The bijectlvity of % r ( B ) a unique -* ~ Cn such that

er r(~) Therefore ~ :=

on

(

gives the existence of

~tw(~"~

',o) /

(w(u

D~w(,~,,o)./~

(.2.5) implies

(2.11)

im c

6"p=(P) (=', ~')

466

fulfills

= e~ (2.1o). Then

SOHUMANN

= - i 6"pr(bn)(X',~') o(x', ~') O-p~,(U)(x', 1I) eI

(wc ,o) ) twi ;;i It remains to determine the numerical value of ~ (2,7) we get ~ n = iv-lJ~~

. From

and ~j = - ( 2 / a + ; L ) - I v - ~

Ej

for J = I,...,n-I. Then (2.8) and (2.11) give

e'pr(~')(~',~')

= (2/,.+R.)-12/,(;~./.,)I~"i

p , ~, ~ ~n-1 _ [ o ] .

:~or (T',O)~ Q.e.D.

REFERENOES

[1]

AGMON, S., A, DOUGLIS and L. NIRENBERG: Estimates near the bmundaz~ fer selutions of elliptic partial ~ifferential equa~iens satisfying general boundary cenditlens (II). Oemm. Pu~e Appl. Math. 17 (1964), 35-92 DIEUDONNE, Jo: GrundzGge der medernen Analysis. Vels. I - 8. Vieweg Braunschweig und VEB Deutsche~ u der Wiss., Berlin 1975 - 1983

[)2

GRUBS, G.: Functiena] calculus ef pseudedifferentlal boundary preblems. Birkh~uset, Basel 1986

[4"J

HORMANDER, L.:: The analysis ef llnearpartial differential epe~ate~s. Vol. I - 4 . Springer-Verlag, Nerlin 1983 - 1985

E53

KECS, W., and P, TEODORESOU: An introduction to the theory of distributions with technical applications (Russian). Mir, Mescew 1978

Z6]

NECVAS, J., and I. HLAYAOEK: Mathematical theory of

467

S01~IqN

elastic and elasteplastic bodies. SNTL Pragu~and Elsevier, Amsterdam 1981

[7]

REMPEL, S., and B.-W. SOHULZE: Index theary ef elliptic beundary preblems. Akademie-Verlag, Berlin 1982

[8]

SCHUMANN, R.: Regulanity fer Signerini's pzoblem in linear elasticity. Preprint No. 50, SFB 256, 1988

Ralner Schumann Karl-Marx-Universit~t Sektien Mathematik DDR 7010 Leipzig

(Received August 26, 1988)

468