Linear models of dissipation in anelastic solids

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where the causal functions h(t), h(t) represents the response of the system to a unit impulse ~(t) of ... unit step of strain, the stress-strain relations become t. (2.4a).

RIVISTA ]:)EL NUOVO CIM:ENTO

VOL.

l,

N. 2

Aprile-Giugno

1971

Linear Models of Dissipation in Anelastic Solids. M. CA~'VTO a n d F . MAINA~DI I s t i i u t o di F i s i c a dell' Universith - Bologna l s t i t u t o di Geodesia dell' Universith - B o l o g n a

(rieevuto l ' l Aprile 1971)

161 162 162 164 167 169 170 174 174 176 179 179 184 187 193 196

l.

-

1. Introduction. 2. The linear theory of viscoelasticity. 2"1. The stress-strMn relations. 22. The material functions. 2'3. The distribution functions and the operator equation. 2"4. The specific dissipation function. 2"5. The mechanical models. 3. The propagation of viscoelastic waves. 3"1. Statement of the problem. 3"2. The wave-front velocity: absorption and dispersion. 4. The derivative of real order in linear viscoelasticity. 4"1. The operator equation of order v. 4"2. The complex index of refraction. 4"3. Dissipation and experimental checks. APPEnDIx A: The derivative of complex order. APPENDIX B: The Mittag-LclTler function.

Introduction.

I t is w e l l k n o w n t h a t a c t u a l solids s h o w d e v i a t i o n s f r o m p e r f e c t e l a s t i c b e h a v i o u r e v e n a t s m a l l s t r e s s l e v e l s a n d , t h e r e f o r e , a r e r e f e r r e d t o as (~a n e l a s t i c ~>. I n l i n e a r e l a s t i c t h e o r y , t h e c o m p o n e n t s of s t r e s s a r e w r i t t e n as l i n e a r c o m b i n a t i o n s of t h e c o m p o n e n t s of s t r M n ( g e n e r a l i z e d t t o o k e ' s law), so t h a t t h e s t r a i n a t ~ p o i n t i n s t a n t l y a d j u s t s t o t h e s t r e s s . B e c a u s e of t h i s i n s t a n t a n e o u s r e s p o n s e , a p u l s e or s i n e - w a v e t r a v e l l i n g t h r o u g h a p e r f e c t l y e l a s t i c s o l i d w o u l d h a v e a c o n s t a n t e n e r g y , a n d a n i s o l a t e d p e r f e c t l y e l a s t i c s o l i d once s e t i n t o v i b r a t i o n w o u l d c o n t i n u e to v i b r a t e i n d e f i n i t e l y . L a b o r a t o r y s t u d i e s of p r o p a g a t i o n of s t r e s s w a v e s i n solids d e m o s t r a t e t h a t t r a n s m i s s i o n of e l a s t i c e n e r g y is n o t p e r f e c t , b u t is a c c o m p a n i e d b y d i s s i p a t i o n of e n e r g y , t h e m e c h a n i c M e n e r g y b e i n g t r a n s f o r m e d i n t o h e a t . I1 - Rivista deZ Nuovo Cimen~o.

161

162

M. CA~VTO and F. MAINARD1

The dissipation results f r o m imperfections of elasticity within the b o d y ; the causes are due to various physical mechanisms. Presently, there is no satisfactory theory of dissipation in solids a n d m o r e e x p e r i m e n t a l d a t a are required. KOLSKY [1] a n d Z ~ E ~ [2, 3] h a v e studied several physical m e c h a n i s m s of dissipation in metals; recently JACKS0~ a n d A~DERSON [4] h a v e reviewed those responsible for the a t t e n u a t i o n of seismic waves in the E a r t h . I n m a n y of these m e c h a n i s m s , one recognizes t h a t analytically the deviation f r o m perfectly elastic b c h a v i o u r implies the i n t r o d u c t i o n of t i m e effects of linear k i n d into the stress-strain relations. This is the subject of the ((linear viscoelasticity ~, based on the well-known B o l t z m a n n superposition principle. The m e c h a n i c a l b e h a v i o u r of a solid is indeed a function of its entire previous history, n a m e l y the m a t e r i a l b e h a v e s as if it h a d a m e m o r y . This p o i n t of view was developed b y VOLTERRA at the beginning of 1900 as a n h e r e d i t a r y linear t h e o r y of elasticity, t h a n k s to his integral equations. The m a t h e m a t i c a l basis of this t h e o r y was later carefully studied b y BE~CVE~UTi [5]. The handling of the m a t h e m a t i c a l aspects of the linear viscoelasticity has been greatly simplified b y Glcoss [6] b y the i n t r o d u c t i o n of the integral-transf o r m technique. The m8in purpose of the p r e s e n t t h e o r y is to provide linear m e m o r y models which are in a g r e e m e n t with the e x p e r i m e n t a l d a t a on the dissipation in solids, when anelasticity is essentially due to physical m e c h a n i s m s of a h e r e d i t a r y kind. W i t h this in mind, we review in our present work the general features of the linear theory of viscoelasticity a n d t h e n introduce some m e m o r y models to be checked with some e x p e r i m e n t a l d a t a concerning E a r t h a n d metals. F o r this purpose, we use the well-known p a r a m e t e r Q-1 as a m e a s u r e of dissipation. I n m o s t cases, the p a r a m e t e r is a function of the driving f r e q u e n c y and of the t e m p e r a t u r e ; we will detail the dependence on frequency neglecting t h a t on t e m p e r a t m ' e .

2. - The linear theory o f viscoelasticity.

2"1. T h e s t r e s s - s t r a i n r e l a t i o n s . - Viscoelasticity is a p r o p e r t y possessed b y bodies which, when deformed, exhibit b o t h viscous a n d elastic b e h a v i o u r t h r o u g h s i m u l t a n e o u s dissipation a n d storage of mechanical energy. According to H ~ T E ~ [7], a viscoelastic b o d y is specified b y the existence of a functional equation of state connecting stress (0), strain (s), t i m e (t) a n d t e m p e r a t u r e (T): (2.1)

F(~, ~, t, T) = 0 .

(Here, for simplicity, we are restricting the discussion to unidimensional problems.)

LINEAR

]~IODELS

OF

DISSIP&TION

IN

ANELASTIC

SOLIDS

l ~

At sufficiently small (theoretically infinitesimal) strains a n d at c o n s t a n t t e m p e r a t u r e , the b e h a v i o u r of a viscoelastic b o d y is well described b y the linear theory of viscoleasticity (*). According to this theory, the b o d y m a y be considered as a linear s y s t e m with the stress (or strain) as the excitation function (input) a n d the strain (or stress) as the response function (output). F u r t h e r m o r e , two f u n d a m e n t a l h y p o t h e s e s are required: t i m e i n v a r i a n c e a n d causality; the f o r m e r m e a n s t h a t a t i m e shift in the i n p u t results in a n equal shift in the output, the l a t t e r t h a t the o u t p u t for a n y i n s t a n t tl depends on the values of the i n p u t only for

t~tl. Then~ the m o s t general stress-strain relations m a y be w r i t t e n in the following forms: t

(2 o2a) -co t

(2.2b)

g(t) : th(t - - ~)s(7) d ~ , -¢o

where the causal functions h(t), h(t) represents the response of the s y s t e m to a u n i t impulse ~(t) of stress a n d strain, respectively. I t is usual to t a k e the s y s t e m at rest for t ~ 0, so t h a t the eqs. (2.2) reduce to t

s(t) = l h ( t - - ~) a(r) 4 7 ,

(2.3a)

0t

(2.3b)

~(t) ----th(t-- 7) s(7) d r , 0-

where the lower limits of i n t e g r a t i o n are w r i t t e n as 0 to account for the possibility of impulse functions centred at t ~ 0 (McDo~ALD-BRAC~A~ [8]). I n s t e a d of the impulse response we m a y utilize the step responses. I f we denote b y c(t) the response to a u n i t step of stress, a n d m(t) the response to a u n i t step of strain, the stress-strain relations become t

(2.4a)

s(t) = c(0 +) a(t) -{-fc'(t - - T)(~(7) dT: , 0

m(0+)e(t) t

(2.4b)

a(t) =

+fin'(t--v)e(7)dv, 0-

(') According to ZEN]~ [3] the linear viscoelastic behaviour may be a manifestation of the e~istcnce of certain hidden parameters, such as the same temperature.

164

~.

CAPUTO ~ I i ~ F. M A I N A R D I

because we have (2.5a)

h(t) = e(O+) O(t) 4- c'(t),

(2.5b)

h(t) = m(O+) (~(t) 4- m'(t) .

The functions c(t), re(t) are usually referred to as the creep compliance and the relaxation modulus respectively, or, simply, as the material functions. B y introducing the technique of the Laplace transforms, eqs. (2.3), (2.4) are replaced b y v e r y simple algebraic relations connecting the transformed values of stress and strain. I n the (( Laplace representation )) we obtain (in an obvious notation) (2.6a)

E(p) = H(p) S(p) = pC(p)S(p),

(2.6b)

S ( p ) ----H ( p ) E ( p ) -= p M ( p ) E ( p )

.

The coexistence between eqs. (2.6) yields the following condition of compatibility: (2.7)

H ( p ) l ~ ( p ) =- 1 .

Following McDoNALD and BRACIt~AN [8], we call H, H the system functions, a n d C, M the network functions. The r e q u i r e m e n t of physical realizability produces some i m p o r t a n t conditions on these functions. If G(p) denotes a n y one of them, we point out its f u n d a m e n t a l properties: i) G(p) is analytic in the finite, nonzero p a r t of the right-half plane (analyticity property). ii) If p = x + i y and G(iy) ~- lira G(p), W=0

G(iy) -----G * ( - - i y ) (crossing p r o p e r t y ) ,

where the asterisk denotes the complex conjugate. This means t h a t the real p a r t of G(iy) is an even function of y, while the imaginary p a r t is an odd one. iii) The K r o n i g - K r a m e r relations hold between the real p a r t and imagin a r y p a r t of G(iy). 2"2. The material ]unctions. - This is the time to consider the form of the material functions. I n general they are determined b y two components, one constant and the other time dependent, so t h a t we m a y write

(2.8a)

c(t) = [co + ~p(t)]l(t),

(2.Sb)

re(t) = [m= + ~ ( t ) ] l ( t ) ,

L I N E A R MODELS OF DISSIPATION IN ANELASTIC SOLIDS

165

w h e r e l ( t ) d e n o t e s t h e H e a v i s i d e s t e p f u n c t i o n . T h e f u n c t i o n s yJ, v~ a r e posit i v e for t > 0; t h e f o r m e r , c a l l e d t h e c r e e p f u n c t i o n , is a m o n o t o n i c i n c r e a s i n g f u n c t i o n , v a n i s h i n g for t ~--0; t h e l a t t e r , c a l l e d t h e r e l a x a t i o n f u n c t i o n , is a m o n o t o n i c d e c r e a s i n g one, v a n i s h i n g as t - > oo. T h e ( p o s i t i v e ) c o n s t a n t s co, m ~ r e p r e s e n t , t h e r e f o r e , t h e f i n i t e l i m i t s of t h e m a t e r i a l f u n c t i o n s ; t h e f o r m e r of v(t) as t - + 0 +, t h e l a t t e r of re(t) as t--> + oo. T h e y a r e called, r e s p e c t i v e l y , i n s t a n t a n e o u s (or glass) c o m p l i a n c e a n d e q u i l i b r i u m (or s t a t i c ) m o d u l u s . B e c a u s e of t h e c o m p a t i b i l i t y c o n d i t i o n (2.7) a n d of t h e i n i t i a l a n d f i n a l v a l u e t h e o r e m s for L a p l a c e t r a n s f o r m s , i t is e a s y to r e c o g n i z e t h e f o l l o w i n g conditions:

(2.9a)

co

= O,

(2.9b)

m ~ = O,

~(0)

c~,

=

~v(c~) = c0, t > 0 with the following b o u n d a r y conditions: (3.1a)

r(x, t)]~= o

= ro(t) ,

(3.1b)

r(x, t ) L ~

= O.

The response variable r(x, t) m a y be either the displacement the stress a(x, t), or the strain s(x, t). The problem consists of solving the equation of m o t i o n (3.2)

8 z(x, t) 8~ ax = ~ ~ u(x, t)

u(x, t), or

L I N E A R MODELS OF DISSIPATION IN i N E L A S T I C SOLIDS

175

with the aid of the strain-displacement relation

(3.3)

e(x, t) = ~ u(x, t)

and the stress-strain relation, under the b o u n d a r y conditions (3.1) and the initial conditions (3Aa)

r(x, t)[t_ o

(3.4b)

~ r ( x , t)[,-o = 0 .

= 0,

8

The stress-strain relation is the equation describing the mechanical properties of the rod; it is most conveniently t a k e n in the oo ,

(4.7a)

co

~__tt(oo) : a/b ,

(4.7b)

m~ --~/t(0) : m .

I n this case eq. (4.4) gives the stress-strain relation for a viscoelastic body of the first type, with the same terminal characteristics as the S.L.S., b u t exhibiting continuous spectra of retardation-relaxation times.

180

:~t, CAI:'UTO~nd F. MAINARDI I f we t a k e

(4.8)

w(t) -

0< v~l,

F ( 1 - - v) '

with F denoting the g a m m a - f u n c t i o n , the d e r i v a t i v e of order 1 is s u b s t i t u t e d by t h a t of real order v (Appendix A). B y this choice we obtain the stress-strain relation i n t r o d u c e d b y us recently (CAPvTO a n d MAINAICDI [19])

E q u a t i o n (4.9) will be referred to as the o p e r a t o r e q u a t i o n of order v (0 1 t h e d i s t r i b u t i o n s b e c o m e s h a r p e r a n d s h a r p e r u n t i l for ~ = 1 t h e y r e d u c e t o d e l t a - f u n c tions. T h e d i s t r i b u t i o n s (4.16), (4.17) h a v e a l r e a d y b e e n c a l c u l a t e d b y G R o s s [9] i n 1947 w h e n , i n t h e a t t e m p t t o e l i m i n a t e t h e f a u l t s w h i c h a p o w e r l a w s h o w s f o r t h e c r e e p f u n c t i o n , h e p r o p o s e d t h e M i t t a g - L e f f l e r f u n c t i o n as a g e n e r a l e m p i r i c a l l a w for b o t h t h e c r e e p a n d r e l a x a t i o n f u n c t i o n s . W e a r e n o w deriving a similar result by introducing a memory mechanism into the stress-strain r e l a t i o n s b y m e a n s of a c o n v o l u t i o n b e t w e e n t h e f i r s t - o r d e r d e r i v a t i v e a n d a k e r n e l w h i c h is a p o w e r of t h e t i m e . T h i s o p e r a t o r is r e a d i l y e x p r e s s e d b y m e a n s of a d e r i v a t i v e of r e a l o r d e r .

1~

M. CAPUTO ~bnd F. MAINARDI

To c o m p l e t e t h e a n a l y s i s of cq. (4.9), we m u s t consider t h e l i m i t i n g cases o c c u r r i n g w h e n a . b . m = 0. T h e n we o b t a i n a generalization of the N e w t o n , V o i g t a n d Maxwell models s t u d i e d in Subsect. 2"5. I t is easy to show t h a t this is c a r r i e d o u t a c c o r d i n g to t h e following s u b s t i t u t i o n s in the m a t e r i a l functions: (4.18a)

t ~ t ' l F ( i + ~),

(4.18b)

6(t) ~ t - ' l F ( 1 - - ~ )

(4.19a)

exp [ - - tlr] ~ E~[-- (tlr)~] ,

(4.19b)

exp [-- tlF] --~ E , [ - - (tlf)"].

;

T h e s i t u a t i o n is s u m m a r i z e d i n T a b l e V I . TABLE VI. - Models derived ]ron~ the operator equation o] order v [1 nu a(@~/~ct')]a(t)= = [m-F b(~'t~tv)]e(t). ( X ~:0.) Coefficients m a

b

Type

Creep compliance c(t)

Relaxation modulus re(t)

×

0

0

I

%

moo

0

0

X

IV

clt~tF(1 + v)

mit-~lF(1 - - v)

X

0

x

III

7.{1 -- Ev [-- (tifF]}

m ~ ~- mlt-~/F(1 - - v )

0

x

X

II

co + cl t~lF(1 + ~,)

2E~ [-- (tl~Y]

x

x

x

I

%+ Z(1--E~[--(tlrF] }

m~÷2E~[--(t/eF]

4"2. T h e c o m p l e x i n d e x o] re/raction. - As is i n d i c a t e d in Subsect. 3"2, t h e a b s o r p t i o n a n d t h e dispersion of a m e c h a n i c a l w a v e in a linear viscoelastic m e d i u m are ruled b y t h e c o m p l e x i n d e x of r e f r a c t i o n . F o r t h e m o d e l b a s e d on eq. (4.9), we o b t a i n , b e c a u s e of (3.16), (4.7a) a n d (4.10),

(4.20)

LP~J

n(p)

where for c o n v e n i e n c e we p u t (4.21a)

CZ ~ - m / b

(4.21b)

~

= 1/~.

B e c a u s e of t h e c o n d i t i o n (4.2), (4.22)

~' ~

fl- ~ l . O~

'

185

L I N ] ~ A R MOD]~LS O F D I S S I P A T I O N I N A N E L A S T I C S O L I D S

I n the case v = i (S.L.S.), ~, fi are, respectively, the reciprocal of the retardation and relaxation ~imes (see (2.37)). B y p u t t i n g p = ~=i(o in (4.20) we obtain the required complex index of refraction: by eqs. (3.19), (3.21), (3.22), we m a y give the absorption coefficient, the phase velocity a n d the specific-dissipation function as functions of the driving frequency ~o, for any fixed values of ~, ft. I n his analysis of the S.L.S., MEn)AV [20] takes various values for the ratio y for a fixed value of fl = 103 s -~. According to him a good value is ~ ~1.5.

f 10 2

1)

~

10°

-2

7n-2_

,

50 0

.f

1000

,

,

1500

2000

w (rc~d/s)

Fig. 8. - Absorption coefficient over a wide frequency range for some values of ~ for ~=103s-L ~ = 1 . 5 : 1) ~ = 1 , 2) ~ 0 . 7 5 , 3) ~ : 0 . 5 0 , 4) ~=0.25. As an example, we take for g the values 108 s-', and for ? the values 1.1 and 1.5. The effect of the variation of these parameters on the absorption and the dispersion is shown in Fig. 8, 9, for some values of v. Particular a t t e n t i o n is to be devoted to the dispersion rules b y the real p a r t of the index of refraction.

186

M. CAPUTO and F. M&IN~.RDI

~.00

a)

0.95

090

0

b)

8) r

0.8110 o

i

10I

i

102

i

103

i

16

105

~a (rod/s)

Fig. 9. - Dispersion curves o v e r a wide f r e q u e n c y range for some values of v, for ~=103s-v: a) ~ 1 . 1 , b) ~ = 1 . 5 ; 1) v = l , 2) v = 0 . 7 5 , 3) v = 0 . 5 0 , 4) v = 0 . 2 5 , 5) v = 1, 6) v = 0.75, 7) v = 0.50, 8) v-----0.25.

The well-known travelling

formulae

of

dispersion

theory,

valid

for

any

kind

of

waves, are

i)

V=--

c

(e = w a v e - f r o n t

velocity),

~r

ii)

1 dv

1 dnr

vd---~=-

n~ d w '

v w dm - ---- 1 + - - - -

iii)

u

We get the following fundamental

(4.23)

(u -----g r o u p v e l o c i t y ) .

nr do.~

r e s u l t s f o r a n y co:

a)

v ~v ( a n o m a l o u s d i s p e r s i o n ) ,

of ~o,

LIN:EAI~ M O D E L S

OF D I S S I P A T I O N I N A N E L A S T I C S O L I D S

187

because one verifies t h a t , for our model, ii)

(4.24)

n,(co) ~>1 , dn, ~g co of eq. (4.25) ( g e n e r a l i z e d V o i g t m o d e l ) . T h e e x c e l l e n t fit was o b t a i n e d b y CAP~,TO [2¢, 25], b y t h e l e a s t - s q u a r e s m e t h o d , b y u s i n g t h e approximate formula (4.31)

q - l ( o ) ) = ~- o)~ sin O,:n/2) .

T h e v a l u e s of t h e p a r a m e t e r s

are listed in Table IX.

TABL]~ IX. - 1)arameter values ]or the data measured by ZEMANEOK and RUDNICK. LongitudinM vibrations

~ (s-~)

v

Aluminium

2.34.10 s

0.15

E x p e r i m e n t a l d a t a on Q-I a r e also a v a i l a b l e for t h e e l a s t i c w a v e s i n t h e Earth's interior. I n F i g . 17 we r e p o r t t h e d a t a for R a y l e i g h w a v e s ( d a s h e d line) a n d s p h e r o i d a l o s c i l l a t i o n s ( d o t - d a s h e d line), a n d i n F i g . 18 t h e d a t a for L o v e w a v e s ( d a s h e d line) a n d t o r s i o n a l o s c i l l a t i o n s ( d o t - d a s h e d l i n e ) (see CAP~JTO [24]).

. . . . . . . . . . . . . . .

%

. 1/

0

i

5

i

I

10 15 ,,a .lO-3(Hz)

i

20

I

25

Fig. 17. - Specific dissipation in the E a r t h (Rayleigh waves and spheroidal oscillations): comparison between theoretical and experimental curves.

T h e s e d a t a h a v e also b e e n f i t t e d w i t h (4.25), as s h o w n i n F i g . 17, 18, w i t h t h e c o n t i n u o u s l i n e s . M o r e p r e c i s e l y , w e h a v e u s e d t h e p a r t i c u l a r f o r m u l a e (4.31) a n d (4.27); t h e v a l u e s for t h e r e l a t i v e p a r a m e t e r s a r e l i s t e d i n T a b l e X . TABLE X . - Parameter values /or the data pertienent to the Earth. Rayleigh waves and spheroidal oscillations (eq. (4.31))

1 / a = 3 . 7 '10 -~ v = 0.15 = 4.37.10 -3

Love waves and torsional oscillations (eq. (4.27))

=

v

1.16.10

= 0.60

1

LINEAR

MODELS

OF

DISSIPATION

IN

ANELASTIC

193

SOLIDS

lO

!

I i

75

t i

%

i !

i

J

i

O5

! i i J

i

25

5

0 w -I0 3(Hz)

15

20

Fig. 18. - Specific dissipation in the Earth (Love waves and torsional oscillations):

comparison between theoretical and experimentM curves.

One can notice t h a t the a g r e e m e n t b e t w e e n the theoretical curves a n d the observed values is satisfactory except for v e r y low frequencies; this d i s c r e p a n c y is e x p e c t e d a n d is in the right direction because the low-frequency waves sample deep E a r t h m a t e r i a l which is in v e r y different physical conditions.

APPENDIX

A

The derivative of complex

order.

To introduce the notion of d e r i v a t i v e of complex order we shall essentially follow GEL'FA~'D a n d SmLOV [26]. C a u c h y ' s well-known f o r m u l a $ Sn-i

v I TI

(A.1) 0

o

0

0 t

-- ( n - - l ) !

(r)(t--z)"-~dv

(n =

1, 2, ...) ,

0

reduces t h e calculation of t h e n-fold p r i m i t i v e of a function 13 - Rivlsta del N u o v o Cimenfo.

](t)

defined for

194

M. CAPUTO and r. MAINAR])I

t > 0 to a single integral. (A.2)

This f o r m u l a m a y also be w r i t t e n in t h e f o r m ]=(t) = / ( t ) ,

(n--l)! '

where for t < 0 b o t h ](t) a n d t =-~ are replaced b y zero a n d , denotes t h e convolution b e t w e e n ordinary functions. I t would seem quite n a t u r a l to generalize this f o r m u l a to t h e case of a complex index ,~ (Re~ > 0) and a r b i t r a r y generalized function ] c o n c e n t r a t e d on t h e half-line t ~ 0. Then, one defines t h e p r i m i t i v e of order ~ of f as t h e convolution (in a generalized sense)

(A.3)

i(t) = l ( t ) *

( R e 2 > 0) ,

"+

where t~+4 = {t~-~ ~

t > 0,

O,

t- 0 ) .

The v a l i d i t y of this equation m a y be e x t e n d e d to l~e ~ ~ 0, q~¢ is t h e ordinary function t+'-l/P(x ') a n d t h e convolution a p p e a r i n g in t h e last m e m b e r is a n o r d i n a r y one. Then we m a y write t

(A.14)

do

f i'°%) dr

}(t) - r ( x ' ) J ( t - ~)~-~"

(0 < x ' < l ) .

0

Putting (A.15) we obtain t h e a l t e r n a t i v e form, used b y CAPUT0 [25], t

(A.16)

d~

1

~ / ( t ) -- V ( i : - ~ ) J

r ](-'(v)

(T~--~), dr

(o < , < n .

W e recall t h a t , b y hypothesis, eqs. (A.14), (A.15) hold in t h e open intervals (n = 1, 2, ...). )low we are going to see if t h e s e equations are also valid in t h e e x t r e m a of t h e intervals.

n--l 0), attention.

F o r this p u r p o s e we

197

L I N E A R MODELS OF DISSIPATION IN ANELASTIC SOLIDS

i) the expansion theorem, ii) Bromwich's inversion formula. According to m e t h o d i), then,

F(p) m a y be expanded in negative powers of p; 1

co

F(p) ----~ .~o(--1)" (q)'"•

(B.3)

Antitransforming t e r m b y term, we obtain

(qt)'. l(t) ----.=o ~ ( - 1)" F(vn d- 1)" =

(B.4)

In order to use m e t h o d ii), we consider the limit of t h e contour in Fig. 19, as R--> ~ . The limit of t h e vertical line (AB) is t h e Bromwich p a t h (Br), while t h a t of t h e loop, t a k e n counterclockwise (CFEDC), is t h e H a n k e l p a t h (Ha). Imp

- _ -fl ~B

/////"~ //

//

/

o ;

Rep

I I I

i\ \\ \ \\ \

).

I

y

II I

\%

Fig. 19. - Integration contour for the Mittag-Leffier function. We obtain (B.5)

l(t)=~2~i exp[pt]F(p)dp= ~

Br

xp[pt]p,+q d p =

Ha _

1

~e

xp

2~i

[z] -

z~-i -

z" + (qt)"

Ha

where an obvious change of variable has been carried out in the last one.

dz

198

M. CAPUTO a I l d F. MAINARDI

E q u a t i o n s (B.4), (B.5) give, r e s p e c t i v e l y , t h e series r e p r e s e n t a t i o n a n d t h e i n t e g r a l r e p r e s e n t a t i o n of t h e Mittag-Leffler f u n c t i o n of order v a n d arg u m e n t - - (qt) ~ (see EI~DEL¥I [27]). S u c h a f u n c t i o n , i n t r o d u c e d b y LEFFLER in 1903, is u s u a l l y d e n o t e d b y E~; it is 1V~ITTAGasddddddd g e n e r a l i z a t i o n of t h e expon e n t i a l f u n c t i o n w i t h w h i c h it is i d e n t i c a l w h e n t h e order v is 1. W e inc i d e n t a l l y p o i n t o u t a p r o p e r t y of E , w i t h r e g a r d t o t h e d e r i v a t i v e of t h e s a m e order v, w h i c h easily follows f r o m (B.4) a n d (A.12): dr

(B.6)

h7~ E,[ ± (qt) ~] = ± q,E~[ ± (qt),],

w i t h q > O, O< ~,