Rend. Its. Acc Lincet s 9. v. 297-I05 (199I)

Geofisica. - -

Threedimensional

rheology.

N o t a (*) del Socio MICHELE CaPUTO.

ABs'rm~c'r.- - Some theoretical studies appeared in the rheological literature have been treated without discussion or consideration of the pertinent laboratory experiments and results. We stress here that the uniaxiaI experiments, in general, are not onedimensional and obtain, compare and discuss the creep and relaxation functions obtained from onedimensional and uniaxial experiments data showing that those of the former experiments are more complicated that those of the latter. The interpretation of physical phenomena neglecting the distinction between the two types of experiments gives therefore unreliable results. The importance of the distinction is obvious in the retrieval of the rheological properties of the Earth from geodetic, and seismological studies, as results from previous studies. KEy woe,Ds: Threedimensional; Rheology; Creep; Relaxation. PvL4.SSUNTO. Reoloua trtdimensionale NeUa letteratura delle ricerche di reologia sono apparsi alcum studi cbe non discutono o considerano i pertinenti risultati di laboratorio. In questa Nora si sottolinea che gli esperimenti di Iaboratorio cosiddetti uniassiali, in generale, non sono unidimensionali e per questo si ottengono, confrontano e discutono le funzioni di materiale ottenute da esperimenti puramente unidimensionali ed uniassiali e si mostra che le funzioni di materiale degli esperimenti uniassiali hanno forma piil complicata di quelle degli esperimenti unidimensionali. Pertanto le interpretazioni dei fenomcni fisici the non tengano conto della distinzione ffa unidimensionalit~ ed uniassialit~ portano a risuItati inattendibili. L'importanza di questa distinzione ~ evidente nella determinazione delle propriera reologiche della terra con studi geodeticl e sismologici, in accordo con precedenti studi. -

-

INTRODUCTION T h e i n t e r p r e t a t i o n of the data o b s e r v e d in e x p e r i m e n t s aimed at the study of the theology of materials is of great i m p o r t a n c e a n d m u s t be carried out with great care a n d rigor in o r d e r to o b t a i n consistent a n d realistic results. A m o n g the most c o m m o n data used are those resulting from e x p e r i m e n t s in which a cylindrical sample is subject to a c o n f i n i n g pressure applied to the lateral b o r d e r a n d to a c o n s t a n t pressures or a c o n s t a n t strain or a strain increasing at a c o n s t a n t rate a p p l i e d to the top and b o t t o m of the sample. D u e to the n a t u r e a n d the g e o m e t r y of the e x p e r i m e n t s the theory, is generally d e v e l o p e d a s s u m i n g u n i d i m e n s i o n a l i t y n e g l e c t i n g that the lateral c o n f i n i n g pressure acting o n the sample actually makes the e x p e r i m e n t b i d i m e n s i o n a l . T h e laboratory e x p e r i m e n t s d e s c r i b e d above are p r o p e r l y called uniaxial a n d n o t u n i d i m e n s i o n a l . T h e theory is usually d e v e l o p e d c o n s i d e r i n g the so called creep f u n c t i o n c(t), w h i c h is the strain history observed in the s u p p o s e d l y linear field, in the d i r e c t i o n of the a p p l i e d stress in rigorously o n e d i m e n s i o n a l c o n s t a n t stress e x p e r i m e n t s , or the

(*) Presentata nella seduta del l0 novembre 1990.

98

M.

CAPUTO

relaxation function m(t) which is the stress history in the direction of the applied strain in rigorously onedimensional constant strain experiments. The corresponding onedimensional laboratory experiments would be obtained by imposing lateral zero strain on the cylindrical sample. The two functions, the relaxation and the creep, are not independent, their Laplace Transform (LT), C(p) and M(p), where p is the Laplace variable are related by (1)

p2 C(p)M(p)

= 1.

The knowledge of the creep and relaxation of a given medium allows to perform ideal experiments to obtain immediately the behaviour of the medium with generic time dependent boundary, conditions at the top and bottom of the onedimensional sample which could be a slab limited by parallel planes and extending to infinity. The boundary condition at the top and at the bottom of this slab, defined by the function of time, is the same for all the points of the boundary. In this case the principal strain parallel to the surfaces of the slab is nil and, although the stress components parallel to the surface of the slab are not nil, one may rigorously assume to be in a onedimensional geometric condition because the displacement of the points occurs only in one direction, that normal to the surface of the slab. The same circumstance occurs in the case of pure elasticity; one single elastic parameter, in the geometric configuration described above, describes the behaviour of the medium. In a comparison with elastic problems we recall that when the rigorous onedimensional geometry is not respected, to describe the elastic properties of the medium two parameters are needed. That is in a static uniaxial Iaboratou experiment with the cylindrical sample the Poisson ratio plays an important role in the discussion of the data. In pure elasticity a simplifying assumption, sometimes not too far from reality, is to consider the two Lamd parameters equal, which in turn makes the Poisson ratio equal to 0.25. This allows to discuss elastic problems with a single elastic parameter. The same simplifying assumption may be tentatively made in rheology too, although it seems natural to describe the rheology of compression and expansion at the boundaries with two different independent functions of time (Caputo, 1986); as in elasticity, for the same purpose, one uses two distinct indipendent parameters. However in the interpretation of the laboratory data resulting from the experiments described above one must take into account the fact that they are not onedimensional but bidimensional; which was not always done in some previous interpretation of laboratory data (i.e. Caputo, 1986). In fact many authors interpreted laboratory, data, obtained from uniaxial laboratory experiments, using onedimensional theoretical equations and specifying the confining pressure of each set of data (e.g. Caputo 1986; Carter et al., 1981; Gangi, 1983; Handin eta/., 1986; Webster et al., 1970); the results of the interpretation, that is the theoretical curves fit to the data, are naturally valid for the specified confining pressure but may not be used for other confining pressures; this limitation would not exist if the

99

THREEDIMENSIONAL Rt IEOLOGY

data had been interpreted with a threedimensional theoretical model asit will be shown here. C O M P A R I S O N W I T H T H E RESULTS P R E V I O U S L Y O B T A I N E D

We shall see in this paragraph the difference between the results of the interpretation of the uniaxial laboratory experimental data by means of a onedimensional theory. The theory in general assumes that the stress strain relations of rheology are, using Laplace Transform LT, T,; + ~(T,i - 8,, Trs

(2)

= 8,j(;t + 2pN/3) E,, + 2(,~ + pN) E,j

where the capital letters T,. E,j, N(p), H(p), indicate the LT of the stress -,j, the strain 8,~, h(t) and n(t), which are the functions defining the rheology of the medium, ), and ,x are the Lamd parameters. When 822 = 833 , T22 = r3t eq. (2) may be written

IE,, = (TH - T3~)(,=+ pH)/3pH(a + pN) - (TH + 2T~3)/3(35, + 2,.0,

(3)

/ JEll + 2E~3 = (Tn + 2T33)/(3), + 2~)

The second of eq. (3) simply means that the changes of volume occur according to the laws of pure elasticity. If limpN=. 0 and IimpH = 0 we obtain from (3) (4)

Tll = T .

Elt + 2 E . = 3T11/(3.~, + 2,~)

or

(5)

8.(oo) = - 8,.(~)/2 + .-~/2(3.;. + 2,,.)

rl,(~) = r33(~)

which imply that assymptotically the stress is that of a pure elastic fluid. We obtain also ,~=(m) = 83~(~) = - 8 . ( ~ ) / 2 or the time variation of the dilatation is assymptotically nil. If N = 0 one obtains a rheology described by a single function (Caputo, 1986) which may be used as a first order approximation in the threedimensional interpretation of the laboratory data resulting from uniaxial experiments. By interpreting the data of uniaxial experiments on polycrystalline halite at constant strain rate and constant confining pressure Gangi (1983) and Caputo (1983) assumed that the constant strain rate laboratory data on polycrystalline halite are adequately fitted by the following function defined by the four parameters a, b, ~, ,8 (6)

a(l - e -=t) + b(l - e-.~t).

We shall assume here that eq. (6) is valid also in the linear range inspite of the fact that the laboratory tests, which indicate its validity, probably exceeded the linear range. We should also note that there is evidence that ifi the linear range, the data fit well a simpler function, which contains only 3 arbitrary parameters instead of 4 as (6),

100

M.

CAPUTO

obtained theoretically assuming that the stress strain relations of polycrystalline halite, as well as for many anelastic materials, may be realistically expressed by means of derivatives of fractional order. In fact Bagley and Torvik (1983; 1986) and Torvik and Bagley (1984) proved that the stress strain relations with time derivatives of real order fit exactly the rheological properties of 146 different materials, over frequency ranges which in some instances covered 7 decades, including coming glasses, polymers, nitrile rubber and glassy enamels. Of the 149 materials tested only 3 fit the stress strain relations defined with derivatives of real order with discrepances of at most 20% (Bagley, 1989, personal communication). Introducing the function (6) in (2) and interpreting the data on polycrystalline halite as coming from a onedimensional, and not uniaxial, experiment and therefore assuming that there is zero lateral strain 822 = 0 g3~ = O, one obtains (7)

H~(p)=2~[(~za+eqb)p+~,8(a+b)]/3[8(p+=)(p+eq)(2+2,a)+ - (~a +~qb)p + =~(a + b)].

If one instead interprets rigorously the data as resulting from a uniaxial experiment and considers that the lateral strain 8~.~and 8~ may be not nil and are actually unknown one obtains, substituting the LT of (6) in (3) and eliminating E22 = E~ (8)

H=(32+2~),a[p27c+((~+,8)v~-~a-gb)p+~(v~-a-b)]/ /[-

3),,~cp ~ + {3(), + ,~)(a~ + b~) - 3,~(3~ + 2,~) 8, - 3),~'~(~ + ,~)} pe +

- (3~(3)~ + 2~)(~ +,8) 8 + 3~,~0-~ - 3 ~ ( ~ + ,,~)(a + b)} p - 3 , ~ ( 3 ~ , + 2,,~) ~11

where it is assumed that the experiment was made with confining pressure r~. and applied strain 8~ = 8~t. With the knowledge of H one may then proceed to obtain theoretically the creep and relaxation functions in uniaxial experiments and also in zero lateral strain experiments. THE

CREEP CURVE OF P O L Y C R Y S T A L L I N E H A L I T E

The creep in a uniaxial experiment with confining pressure ":22- . 3 3 - ":~ and applied pressure r11 = ~, is obtained substituting these values and H(p) (given by (8) and not by (7)) in (3) with N = 0. One obtains (9)

Eu=(~l-~3){-3~c~.p3+[(),+2~)(a~+bcfl)-3~cZ(~z+~q)-3,,.*8(3"*,+2.'~)]p2+

+ [3~(a + b)(2 +,~) - 3:'c=~)~ - 3,u8[3), + 2,,~)(~ +,8)] p - 3,~8(3), + 2,,,) ~3}/ /3p2(32 + 2>),,~{vcp2 + [~c(x + 3 ) - a= - b~]p + ~c=~ - =~(a + b)} + + ( ~ - e3)/3fxp + (r + 2~3)/3p(32 + 2~).

The creep is (10)

8~ = A + Bt + Ce +p'' + De +pJ + ( ~ - ~3)/3,~ + (~'~+ 2v3)/3(35~ + 2f~)

i01

THRt~EDIMENSIONAL RHEOLOGY

where the expressions of A, B, C, D, are obtained rewriting (9) as follows El: = (aop 3 + a l p 2 + a 2 p + a 3 ) / p 2 ( p - P l ) ( P - P2) + ('~1 - ":~),/3:.~p +

+ (v~ + 2%)/3p(3)` - 2y.) = A / p + B/p 2 + C/(p - p~) + D / ( p - P2) + + ('~1 - ~)/3,.*p + ( ~ + 2~)/3p(3). + 2~.),

a0 = - [3~).(el - r~)]/[3,~(3). + 2>) ~ ] , a3 = [ - 3~8 ~3(3). + 2,~)(:~ - ~ ) ] / [ 3 , ~ ( 3 2

+ 2,,~)],

a~ = [[(2 + 2 ~ ) ( a a + / , 8 ) - 3 ~ ) , ( = + 3 ) - 38(3), + 2,.)](~-~ - r

/[3(3), + 2t-d,~*~.] (~1)

a2 = [[3~fl(a + b)(2 + ,,.~) - 3~=fl), - 3,=8(3). + 2,~)(~ + 3)](~~ - ~'~)]/

/[3(3), + 2,~t),~vc] C+D=ao-A, Cp2 + Dp~ = B - A(p~ + P2) + a l , B = a3/p~p2 = - 8(~ - r~)/(a + b),

A = (a2 4-(p~ + / 9 2 ) a 3 / p ~ p 2 ) / p ~ p 2 = (~ - r

=

+ b)(), +,=) - 3~'c)`~3 -

3=g(~ + 3)/3)` + 2.,~) +

+ 3(~a +,~b) ~8(3), + 2,,-0/(a + b)}/3~,~(a + b).,x(3>, + 2,.~)

with p,, and P2 solutions of the equation (12)

rc(P + =)(P +fl) - (a= + beq)p - ~ ( a + b) = O .

The steady state creep rate is B = terms of a and b, it is

8~3(r

it may be obtained also in

81,(~) = limopeE,~ = (~'~- ~3)8/(a + b + ~c)-

If one were to use ht(t) instead of h(t) one would obtain El! = 71 ~[p2 -Jr-(~ q-3) P 4- ~3]/P2[ (~a + ,Sb) p + ~q(a + b)] - ~-~/(). + 2f~)p, 811 = - ~1/(). + 2>) + {(~b + ~qa)/~eq(a + b) e + t/(a + b) + (13)

- [ab(~ _ q)2 exp [ - ~3(a + b) t/(~a + 3b)]]/=3(~a +,Sb)(a e + b2)} ~-~8,

8 u ( ~ ) = ~18/(a + b).

The difference between h(t) and hi(t) is obvious. If one seeks the n u m b e r of activation mechanisms (or activation processes), operating in the medium, in the creep function, as it may seem natural, due to the theoretical simplicity of the experiments avoiding discontinuities in the applied forces, the h,(t) would suggest one mechanism while h(t) would suggest two mechanisms. Obviously, within the limits of the assumption N(p) = 0, the correct interpretation would be that deriving from h(t).

102

M. CAPUTO THE

RELA2KATION

FUNCTION

OF

POLYCRYSTALLINE

HALITE

The LT of the relaxation function associated to the value of H(p) is readily found assuming a confining pressure ":22- - . = v3 and a strain 811 = 8 and substituting in (2). W e find for the LT of the relaxation function (14)

Tn = (3). + 2~)(3pH(,,.G + ).~3/(32 + 2~)) + ,~3)/ /3(;~ + ,~)(Hp + ,~(3), + 2~)/3 (2 + ,~)) p.

Since l'm~opH(p) --O, as check, it is seen that "q1(00)--limpTH = ~3 which shows that assymptotical.ly the stress in the sample reaches the value of the confining pressure, or the body, assymptotically, behaves as a fluid. The lateral strain 8_,_~ is obtained substituting T 2 2 - "33 = r3, 811 = 8 and (14) in (2) one finds (15)

E22 = (2~z(~-~ - pTH)I3(2 + 2:~)pH + (~3 - ),8)/(). + 2,.,))/p

and 822(O0) ~"

li[npE. =

3:/2(3),

+ 2,,~) -

8/2

which again indicates that, assymptotically, the body behaves as a fluid. This shows an interesting aspect of the relaxation experiment with respect to the creep and constant strain rate experiments, in fact it allows to study the theology without entering the field of non linearity since the applied strain is fixed and limited a priori. In practice however the most reliable experiment is that with constant strain rate because it avoids the time and the effects implied by the applied pressure :'L (or strain 8l) in order to reach the constant value fixed a priori.

THE

CREEP

\grlTH

ZERO

LATERAL

STIr\IN

The creep in a theoretical unidimensional experiment mav be found assuming zero lateral strain, that is 822 = 833 = 0 and an applied pressure r~L = ~'~; substituting these values and H(t) in (2) one finds (16)

C(p) = E11 =

~1(, U.

-}-pH)/(2 + 2,~)p(pH + ,,~(2 + 2,,~/3)/(). + 2,;))

The assymptotic creep rate is nil and there is no stready state. The assymptotic value of 811 is finite when l i m p H = 0 8u(00) = limpEl~ = ~'1/()- + 2~/3) while in the threedimensional case given by (10) it is not.

103

THREEDIMENSIONAL RHEOLOGY

THE

RELAXATION

FUNCTION

WITH

ZERO

LATERAL

STRAIN

The relaxation function in a unidimensional experiments is found assuming, as before, zero lateral strain, that is 833 = 8= = 0, and a constant applied strain 8ll = 8. Substituting these values and H(p) in (2) one finds Iru(Cr = 8(2 + 2~/3);

(17)

I

I.M~, =

"ql(~) 811(~176 =8~1,

Tn = 8(2 + 2,~,)(pH + ,u(). + 2>/3)/(). + 28t))/p(,a + p H ) ,

which implies that, assymptotically, one experiment reproduces the other. In other words in a constant strain experiment with applied strain 81~ = ~'~/(), + 2~/3) (which is the assymptotic value of the creep experiment with :H = rl), the final value of the stress is again :q. Substituting H from (8) in (16) and (17) we may see that the creep and relaxation functions with zero lateral strain are actually more complicated than those (10) and (14) obtained with constant confining pressure. It is verified that (16) and (17) satisfy eq. (1). The creep and relaxation functions (16) and (17) of the unidimensional experiments with zero lateral strain are those commonly considered in the theoretical discussion of theology in one dimension. THE

MATERIAL

FUNCTION

H

IN T H E

CASE OF

GIL,\NITE

Granite is a very important rock in Earth sciences and has been widely studied in laboratory, experiments. Carter eta[. (1981), in uniaxial creep experiments at different temperatures, found thai the creep is represented by the following function (18)

811 = eo + eo t + er(1 - e -'/=)

where according to. Carter et al. (1981) e0 represents the steady state strain rate, er represents the amplitude of the exponential transient and e0 is the sum of the elastic and anelastic response during loading. In this case using formulae (3) with N = 0 and assuming the confining pressure -;3 = r22 = rc and the applied pressure rll = ~,-+ ~'1, we find, after taking the LT of (18) (19)

EH = (eo + er)/p + eo/p 2 - er/(p + i / r ) = = 8~/3[~p + (.sl + 3~-c)/3(3Z + 2/.L)p + ,,'~/3peH

which gives (2O)

H = 38~(p + 1 / : ) / ( ( A 1 -

er)p 2 + (A1/r + eo)P + eo/=)

where A~-- ((eo + er),,~(3), + 2,~) - 8~0, & '-0 - ,,-~vc)/,,.~(3)~& 2,,-0.

104

M.C.APUTO

The material function (20) of granite is therefore simpler than that (8) of polycrystalline halite, and gives also a simpler relaxation function. The creep with confining pressures ~22 - - ' 3 3 - - ~'3 and r , = 81 is (~1- ~)/3fz + (~, + 2~3)/3(32 + 2,~) + (~ - 0"~)/3(d + B t + (Cexp ( - b t / a ) ) / a ) , a = 81;

b = 811v;

A = (d - ae/b)/b;

c = A~ - er; B = e/b;

d = A,I r +

~0;

e = ~0/v

C = c - a(d - ae/b)/b

the steady state creep rate is e0(~l-~3)/381.

CONCLUSIONS

It has been seen that in the interpretation of creep and relaxation laboratory experiments, in order to obtain the correct creep and relaxation functions one must properly take into account the confining pressure. Since the natural theological phenomena have always two or three dimensions, the distinction between uniaxial and unidimensional experiments has relevant effects; for instance in the case of polycrystalline halite the correct creep function contains two exponentials instead of one as it would result neglecting the confining pressure. The rheology of two and threedimensional natural phenomena is also effected by the wavenumber dependence of the most commonly accepted theological models. In the ideal equation in which the forces acting on the anelastic medium are properly convolved in the time domain with the material function of the medium to give the observable stress or strain, both the material function and the acting forces are unknown. As the frequency dependence of the material function requires observations for a long time (Carter et al., 1981) or at many frequencies (Bagley and Torvik, 1983; 1986) in order to allow its determination with good resolution in the time domain, the wavenumber dependence of the rheology complicates further the solution of the above mentioned equation because it requires observations of the natural phenomena in a great number of points in order to obtain a good resolution in the wavenumber domain. A good example of a study of a natural phenomenon in which this effect is relevant is the postglacial rebound of the Earth's crust, where it is seen (K6rnig and M/.iller, 1989) that the data discussed in the current literature are not sufficient to obtain reliable results.

REFERENCES

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THREEDIMENSIONAL RHEOLOGY

105

CaPuTo M., 1966. Lmear models of dzsszpation whose Q is almost frequency mdependent. Annali di Geofisica, 21(4): 383-393. Cavu'ro M., I979. A model for the fattgue melasttc matertals with frequency independent Q J. Acoust. Soc. Am., 66: 176-179. Caeu'ro M., 1983. Determination of creep, fattgue and activation energyfrom constant strain rate experiments. Tectonophysics, 91: 157-164. Caeu'ro M., 1986. Linear and non linear inverse rheologies of rocks. Tectonophysics, 122" 53-71. CAPUTO M., 1989a. The rheology of anelasttc medta studied by means of the observatzon of the splitting of its etgenfrequencies. J. Acoust. Soc. Am., 86, 5: 1984-1987. Cavu'ro M., 1989b. Rheology and Geodesy. In: F. SacEr-.oo'n~, F. Sal-as6 (eds.), Proceedings of the Symposium on Mathemattcal Geodesy (Pisa), Milano: 167-184. CaPuTo M., 1989c. Rheologies quasi wavenumber in&pendent an a sphere and splitting of the spectral lines. Atti Acc. Lincei Rend. fis., 8, 82: 507-526. Ca.vu'ro M., 1990. The spktttng of the free oscillations of the Earth caused by the rheology. Rend. Fis. Acc. Lincei, s. 9, v. 1: 119-125. CapUTO M., M,~NaRDI F., 1971. Linear models of&sstpation m anelastic solids. Pdvista del Nuovo Cimento, 1, 2: 161-198. Cag'rER N., ANDEmSOND. A., t-Lu-~SENF. D., Km-v'qzR. L., I981. Creep and creep rupture ofgramtte rock. In: Mechantcal behavtour of crustal rock. Geophys. Mon., 24, Amer. Geophys. Un. G~,~GI A., 1983. Transient and steady-state deformation of synthetic rocksalt. Tectonophysics, 91: 137-156. HaNDIN J., RtaSS~-LJ. E., C,u~R N. L., 1986. Experimental deformatzon of rocksalt, in Mineral and Rock Deformatton Laboratory Studies, the Patterson volume Geophys. Monog., 36, Amer. Geophys. Un. K6RNIG M., Mt)t.r.ER G., 1989. Rheological models and interpretatzon ofpostglaczal uplift. Geophys. J. R. Ast. Soc., 13: 529-539. TORWK P. J., BaGLEY R. L., 1984. On the appearence of the fractional derwative zn the behawour of real material. J. of Applied Mechanics, 51(2): 294-298. WEBSTERG. A., Cox A. P. D., 1970. A relationshtp between transient and steady creep at elevated temperature. In: R. W. CaHN (ed.), PhysicalMetallurgy, 2 "a rev. ed., North-Holland Pub. co., Amsterdam: 983-1010. Dipartimento di Fisica Universit~l degli Studi di Roma <~La Sapienza,, Piazzale A. Moro, 5 - 00185 RoMa