On stability in linear viscoelasticity

MECH. RES. COMM.

ON

STABILITY

F. Bloom Department University

Vol.3, 143-150, 1976.

IN L I N E A R

Pergamon Press.

Printed in USA.

VISCOELASTICITY

of Mathematics and Computer Science, of South Carolina, Columbia, S.C. 29208

(Received 22 December 1975; accepted as ready for print 31 January 1976)

Introduction Various proofs of uniqueness and stability within the theory of isothermal linear viscoelasticity [1,2,33 have depended strongly on the assumption that the relaxation tensor satisfies certain definiteness conditions. Using a logarithmic convexity argument due to Knops & Payne [ 4 ], Bloom [5] has recently derived growth estimates and lower bounds for a class of uniformly bounded solutions to some initial-boundary value problems for isothermal linear viscoelastic bodies; no definiteness assumptions concerning the relaxation tensor of the material are made in this latter work. In this note we wish to indicate how the argument employed in [5] can be used to derive some new stability results for viscoelastic bodies. The Viscoelastic

Stability Problem in a Hilbert

Space Settin$.

Let ~cE 3 be a bounded domain with smooth boundary T > 0 be a real number. of isothermal

In the cylinder ~×(-=,T)

linear viscoelasticity

Z~ and let

the equations

are

~2u ~u k 0(x) ~t--~(~,t) ~ [~ijkl(X,0)~--~l(~,0)] ~ ~xj ~

(i)

t ~ Zu k + ~--~ f_~ ~-~ gijkl(X,t-T)Z-~l(X,T)dT

provided the relaxation assumption

tensor @ijkl(X,t-T)

that lim gijkl(X,t-T)

to (I) we have associated

satisfies

= 0 uniformly

initial and boundary

u(x,t)

= 0,

u(x,O)

~u = f(x), -~-~(x,O) : ~g(x)'~

= 0

(x,t)eZ~×(-~,T)

in x.

the usual In addition

data, viz, (2)

on ~; f,gc~C O (~)

Scientific Communication

143

144

F. BLOOM

and we assume

that the past history

u(x,t)

is prescribed,

u(x,~)

: U(x,T),

We also assume are Lebesgue

measurable

denote

support

Hilbert

under the norms

0(x) and the functions

for each te(-~,~);

in ~ whose

spaces

and

components

<~'~>+

of C0(a)

weH+

Now let L(H+,H_) from,

3

in H.

~w.

(5)

[63 that H+cH algebraically Also we define

H

to be

i i dx)2] 3

G(t)eL2((-~,=);

~ . ~ i 3 [ 3Vk [@(t)v]1 p-7-x-~3--~7. gijkl(~'t)~-~l]' ~

of C~(~)

]

(6)

be the space of bounded

H+ to H_, define

If

under the norm

I/(i 2×l ~ 3

2

to C~(2).

~v. ~ w 3x.l 3x.i dx~'

~ f

with H+ dense

sup Eli viwid

fields

by the inner products

~w.

llzll_

belong

vector

H and If+ to be the completion

then it is easy to show

and topologically,

(4)

the set of t h r e e - d i m e n s i o n a l

~

the completion

finally we make the

(~,t)(2x(-~,~)

induced

z f viwid ~

respectively,

g::~.7(',t)

assumption

Let C0(2) we define

(3)

with ess inf p > O, and ~ijkl(.,t)eC~(2),

9ijkl([ ~t) : gklij(['t)'

with compact

veetor

(x,T)~×(-~,0)

that the density

symmetry

of the displacement

i.e.,

and that ~-~ 9ijk I exists usual

Vol.3, No.3

linear operators

i(H+,H_))

VveH+,

by

te(-~,~),

(7)

]

and set N~ : G(0) Then,

and

K(t-T)

as shown

following

= 38--~ G(t-T)~

in [5], the system

initial-value

problem

(8) (i) - (3) is equivalent

in Hilbert

to the

space:

t ~tt - Nu + f u(0)

= ~f'

u(T)

: U(T),

K(t-T)u(T)dT

= 0, 0 s t < T

(9)

ut(0)~ = g~

(I0)

-~ < T < 0

We are interested (ii) for which

(ii) in solutions

utecl([0,T);~

ueC2([0,T);

H) and utteC([0,T);

H+) of H_).

(9) Note that

Vol. 3, No. 3

STABILITY IN VISCOELASTICITY

NEL(H+,H_)

is symmetric

that K(t),

Kt(t)eL2((-~,~);

U(t)ccl((-~,-);

on H+, by virtue L(H+,H

H+) with lim U(t)

~

145

of (4); we also assume

)) and that : f, lim Ut(t)

t~0 -~

~

= g, limlIU(t)JJ+=0,

t+0 -~

~

t÷-~

~

0 and /

I IU(r)l I+dr < ~.

As we make no use of any definiteness

assumptions

concerning

in general,

non w e l l - p o s e d

results

of Dafermos

Stability

the o p e r a t o r

N, the p r o b l e m

and the existence

(9) - (Ii) is,

and uniqueness

[63 do not apply.

Under P e r t u r b a t i o n s

of the Initial D at a

For any t, 0 s t < T, we define the total energy K(t) = __9l- is the kinetic

K(t) + P(t) where I = - ~
P(t)

Nu(t)>~_ is the potential

II~(t)ll+

H+)l sup

N = {weC2([0,r);

E(t) :

energy and

energy and we set

< N2)

(i2)

[0,T) for some real number N. Theorem define

(Bloom

[5]).

Then we have the following

Let ueN be any solution

F(t;B,t 0) ~ llu(t)lJ 2 + 6(t+t0 )2 where

non-negative

real numbers.

11 11 ,

If K(t)

result:

of (9) - (ii) and 6, t O are ar bi tr ar y

satisfies

~v~H+

(13)

with (i)

< e ~T sup l J f t ( t ) l l

(14)

[0,~)

then F ( t ; B , t

O) s a t i s f i e s

FF" - F '2

-2F(2F(0)

+ 8),

+

JJK(t)JJ +

0 ~ t < T

(i5)

where f(t)

~

[(t)

kI

sup

[0,~)

k 2

sup

JJKt(t)JJ

[0,~) 0

kI

~ ~[TN 4 + (N 2 +

k 2

~ yTN2Z

0

(i)

l lvll

2jJ~JJ+)/ JJu(T)IJ+dT]

llu(T) J+dT

y is the e m b e d d i n g

-< Y l l v l l ,

Fv H+

constant,

i.e.,

as H+cH t o p o l o g i c a l l y ,

(i6)

146

F. B L O O M

Now, [(0)

suppose

s -(k I

that

sup

IIK(t)II

EO,~)

Since,

sup

llKt(t)ll)

(17)

VveH+,

it follows

from

(14) that

s y~llK(0)II

(18)

~

Therefore,

(17) is implied

~ - k sup

IIm(t)ll,

[0,~)

d2 d--~(in F(t))

by the somewhat

(19)

yT

2

we may set 6 = 0 in (15) and reduce

(tl,t 2) where

tion of gensen's

inequality

t2-t t2-tl[

we may assume

F(t)

that

z I lu(t)[ I2 > 0.

6 = t/T.

, tc(tl,t 2)

[4] that, without (21) holds

As H+cH,

for all tc[0,T),

i.e.,

0 < t < T

I l u~( t ) l l #

sup

(21)

any loss of generality,

topologically

1 1 2 ( T ) I I + 2 -<

<

An applica-

the estimate

t2-t I

I ]u(t)l I2 < [] Ifl I]2(I-~)I lu(T)[I 2~

-

(20)

now yields

F(t2)]

It is easy to prove

2

this

t-t 1

s [F(tl)]

II2(T)[i

condition

~ 0, 0 ~ t I < t < t 2 ~ T

on any interval

where

simpler

k ~ k I + zk

~

Given that (19) holds, estimate to

F(t)

sup

I l y l l + ,2

~ II~(o)ll

[0,~)

e(o)

+ k2

[0,~)

(13),

llKt(t)ll

s 0, i.e., that

~

I<%#(o)~>I

our hypothe s e s

F(0)

Vol. 3, No. 3

(22)

as well as algebraically,

~

(23)

N4

[0,T) and, therefore,

there

exists

I lu(t)l I2 s A[max(IIfll 2, We summarize Theorem

I

satisfies

E(o)

~

-~

our results

(13), sup

[0,~)

(14).

ll~(t)ll

(18), which sup

[0,~)

IIK(t)l! ~

of (9) - (ii) where

with

k

the stability [(0)

~ kI +

kT~

- KIT~+k2

K(t)

l_k

estimate

2

(24) for each te[0,T).

s -k for some k > 0.

is satisfied when <

(24)

If yT

that

0 _< t < T

as

-

suppose

A > 0 such that

I Ig112)] 2(I-~),

Let ucN be a solution

then u satisfies Now,

a constant

(13) and

(14) are,

In view of F(0)

s 0 if (25)

V o l . 3, N o . 3

STABILITY

IN VISCOELASTICITY

147

which may be considered as a further restriction on K(t). previous theorem may, therefore, Theorem II satisfies

The

be rephrased as follows:

Let ucN be a solution of (9) - (ii), where K(t) (13),

(14), and suppose that E(0) ~ -k for some k > 0.

If K(t) satisfies

(25) then u satisfies

(24) for each te[0,T).

We now consider the case F(0) > 0.

If K(t) satisfies

(14) then F(0) > 0 is implied by the somewhat

(13),

simpler condition

E(0) > -(kiT Y + k 2) sup IIKt(t)I I _ [0,~) and the estimate FF" - F '2 ~

Now,

(15) may be sharpened to read

-2F2 [2F(0) + i], (t+t0)2 6

te[0,T)

(26)

(26) may be rewritten in the form

d2 r F(t~B't0 2Lln 2+e] a 0, dt (t+t 0 )

0 ~ t < T, e ~ 4F(0)/8

As in [4], we can apply Jensen's

(27)

inequality to (27) to deduce

F(t;B,t 0) < (i + t ) 2 + e F ( 0 ; B , t 0 ) to t

x

F(T;B't0)

T )-2-el

exp~{ln[F(O;B,to)(1 + ~

}

(28)

which, when coupled with the assumption that i lim ~ In{F(T;B,t0)/(T+t0 )2+e} = 0 T~ yields,

(29)

for all t~[0,T),

llu(t)Ii2 ~

~

(t

to

If we now choose

+ i)2+c(

I1 11 2

+ 6t

B,t 0 so that 8t~ ~

)

II II 2,

(30)

we obtain the

stability estimate

II where

(t)ll 2

(t0,T;E)llfll 2,

~(t0;T;c)

Theorem II satisfies

0 _< t

~ 2( T to + I) 2+c

We summarize our results

(31)

as

Let ueN be a solution of (9) - (II), where K(t) (13) and

(14).

Suppose that

E(0) > -(kiT Y + k 2) sup IIKt(t)II E0,~) and that

< T

(32)

148

F. B L O O M

lim

ln{(llu(T)ll

2 + B(T+t 0

)2

)/(T+t

Vol.3,

o

)2+E

} :

0

No.3

(33)

T~ ~

where 6t~

e z 4F(0)/B

s IIfl I2

~(t 0 T;e)

B, t o are n o n - n e g a t i v e

Then u satisfies

~ 2( T

'

(31)

constants

for all te[0,T)

satisfying

where

+ I) 2+~

tO

Finally, K(t)

and

suppose

satisfies

that

(13) and

E(0)

(14),

> -k for some k ~ 0.

the a d d i t i o n a l

Then,

if

restriction

A

sup llKt(t)II > k / ( k l T Y + k 2) [0,~) implies that F(0) > 0 and we have Theorem E(0)

IV

Let u(N be a s o l u t i o n

> -k for some k ~ 0.

(34), and

(34)

(33) obtains,

for

of

(9) - (ii) and suppose

If K(t)

satisfies

B, t

non-negative

2 -< II ll"f"2 , then u satisfies the 0 estimate 6t 0 where

¢(t0,T,s)

A n_n E x a m p l e

E 2( T + i) 2+e and to

i__nnO n e - D i m e n s i o n a l

We c o n s i d e r body w h o s e 9(t)

:

e

-It

relaxation

setting

22u_

~ f u +2

~ E F(0)/B.

u(x,0)

= f(x),

one-dimensional is of the

linear

viscoelastic

form

~2u(x,T)

of m o t i o n

becomes

dT = 0

We also have

(36)

associated

initial

and

= 0, ut(x,0)

te(-~,T) = g(x),

(37) xE[0,1]

(38)

in addition,

u(x,T)

= U(X,T),(X,T)E[0,1]x(-~,0)

For our H i l b e r t

(39)

space H we take the c o m p l e t i o n

set of all r e a l - v a l u e d

C ~ functions



I - / v(x)w(x)dx

y = i in the o n e - d i m e n s i o n a l

case.

of C0([0,1]),

on [0,i] w h i c h

x = 0 and x = i, under the n o r m i n d u c e d by

(2)

(2)

data

= u(l,t)

the

satisfying

(35)

l t e-X(t-~)

u(0,t)

and,

constants

and

(31) for all tc[0,T),

Viscoelasticity

p = I our e q u a t i o n

(x,t)~[0,1]x(-~,T).

boundary

function

(14),

, X > 0

After

for

a homogeneous

(13),

that

vanish

at

V o l . 3, No. 3

while

STABILITY

H+ is the i

~ ~

~ f

+

completion

149

IN VISCOELASTICITY

of C0([0,1])

under

the n o r m

induced

by

8v 8w 8x 8x dx

0

82

-It In view of K(t-T) ~

(35),

G(t) ~

= Xe -1(t-~)

= e

82 8x 2"

82

-so that N -- while 8x 2 ~ 8x 2 Also, I

N ~ ^N = {veC2([0,T); ~

A simple E(o)

(/ i [ S v ( x ' t ) ] 2 d x ) ~ 8x

H+) I sup [0,T)

calculation

(~-{

=

shows

0

< N2 } '

that

+ g (x)]dx

~ 0

(40)

> 0, where

(41)

so that F(0)

= [(0)

+ kll

+ k212

I 18f.

}{TN 2 + (N 2 + 2[/0

k1 =

2

~

t~-~) dx]

i o

1

8u

-

)/ [/ (~-~)2dx]2dr} -~

0

(42)

I

k2 = TN2f0 [/ 1 (~x) 8U.2 dx]TdT -~

Note

that

0

l[K(t)II

one-dimensional the

= le -It and

case

under

IIKt(t)ll

= 12 e -At so that

eonsideration,

(13) and

for the

(14) assume

form

9'(0)

~ -<

In view 9(t)

= e

with

< ~ TI 2

(43)

of

(35), we t h e r e f o r e

-It

i , 0 < I s T

require

that (44)

A

If u~N and

(44) obtains

then

F(t;6,t 0) : / l u 2 ( x , t ) d x + 8(t+t0 )2 0 satisfies (15), with F(0) given by (41) and state

the

following

Proposition

direct

consequence

Let ueN be a solution

of

(45) (42),

of t h e o r e m (36)

and we can IV:

- (39), with

~

0 < I s I/T, and suppose that i lu2 lim ~ in{(/ (x,T)dx + 8 ( T + t 0 ) 2 ) / ( T + t 0 )2+e} T÷~ 0 for

6,t 0 non-negative

constants

satisfying

= 0

(46)

gt 20 <_ /If2(x)dx. 0

150

F.

Then u(x,t) lu2 f 0

BLOOM

Vol.3,

No.3

satisfies

^ 1 2 (x,t)dx ~ ¢(t0,T;e) f f (x)dx 0 A

for all t, 0 -~ t < T, where

¢ : 2(t~ + I)

(47) 2+E

with s ~ 4F(0)/8.

References I.

Edelstein, W. S. & M. E. Gurtin, "Uniqueness Theorems in the Linear Dynamic Theory of Anisotropic Viscoelastic Solids, Arch. Rational Mech. Anal., vol 17, 47 (1964).

2.

Odeh, F. & I. Tadjbakhsh, "Uniqueness in the Linear Theory of Viscoelasticity", Arch. Rational Mech. Anal., vol 18, 144 (1965).

3.

Beevers, C. E., "Some Continuous Dependence Results in the Linear Dynamic Theory of Anisotropic Viscoelasticity", Journal de M6chanique, vol 14, i, (1975).

4.

Knops, R. J. & L. E. Payne, "Growth Estimates for Solutions of Evolutionary Equations in Hilbert Space with Applications in Elastodynamzcs " " , Arch. Rational Mech. Anal., vol 41, 363, (1971).

5.

Bloom, F. "Growth Estimates for Solutions to Initial-Boundary Value Problems in Viscoelasticity" to appear in the J. Math. Anal. Appl.

.

Dafermos, C. M., "An Abstract Volterra Equation with Applications to Linear Viscoelasticity", J. Diff. E~s., vol. 7, 554, (1970).