Internal friction: Some considerations of the frequency response of rocks and other metallic and non-metallic materials

Int. J. Rock Mech. Mining Sci. Vol. 1, pp. 231-254. Pergamon Press 1964. Printed in Great Britain.

I N T E R N A L FRICTION: SOME CONSIDERATIONS OF THE F R E Q U E N C Y RESPONSE OF ROCKS A N D OTHER METALLIC A N D NON-METALLIC MATERIALS PETER B. ATTEWELLand DAVID BRENTNALL Postgraduate School in Mining, University of Sheffield Abstract--Although several workers, including the authors, have shown that the attenuation exponent in rocks tends to increase linearly with frequency, such behaviour is not general for all materials: metals, in particular, project several internal friction peaks over a spectrum of frequencies, each peak being ascribed to a certain physical process operating in the material. This article reviews in a simple manner some basic concepts behind the frequency response of various materials since a general knowledge of the behaviour of solids under constraint is an essential prerequisite to a more fundamental investigation of a particular material. All strains are considered recoverable, the vibrating stress levels being low and the behaviour of the material being described by its anelastic properties. The review takes no consideration of the high stress regime within which the logarithmic decrement has been found to increase strongly due usually to plastic losses. 1. I N T R O D U C T I O N IDEALLY elastic bodies do not show any dependence upon the rate of deformation, the necessary stress being determined by the extent of the deformation and not by the time processes. The internal stresses set up in a body exactly balance the externally applied stresses, this balance being maintained until the original stress and strain-free state is allowed to return. Under these conditions, Hooke's law of stress-strain proportionality holds. When a real material is stressed, elastic energy of deformation is stored in the body. Removal of the constraint allows the recovery processes to operate but some fraction of the elastic energy changes irreversibly into heat energy: solids which behave in this manner are said to be viscoelastic, their associated energy losses being ascribed to what is known as 'internal friction'. In a viscoelastic body during the period of applied deformation, energy absorbed by the viscous elements retards the elastic strain; similarly, energy is dissipated when the elastic phase relaxes as the strain is recovered. Internal friction is therefore responsible for the difference in energy between the work expended in deformation and the work recovered in relaxation. Two types of mechanical loss comprise internal friction. There are those losses associated with the long-term stress-strain non-linearity (so-called 'hysteresis'): in this case, the energy loss per cycle is constant and the logarithmic decrement (to be considered later) is independent of frequency. GEMANTand JACKSON[14] found for many materials that the logarithmic decrement was constant over a wide frequency band. Secondly, there are those losses associated with velocity gradients--rate of straining effects---especially marked in long chain polymers. Of these, a Maxwellian loss is irrecoverable while with the Voigt type of deformation, the strain can approach a constant value asymptotically with time and recover slowly with time. Internal friction--or damping--comprises such physical processes as plastic slip or flow, grain boundary friction, dislocation movements, and scattering at the boundaries of the 231

232

P E T E R B. A T T E W E L L A N D D A V I D B R E N T N A L L

constituent grains and voids. An experimental study of damping in materials offers :tn insight into their physical structure in addition to verifying or disproving any hypotheses t:~l~ the anelastic behaviour of polycrystalline materials and polymers. An understanding of damping capacity is important in mechanical design; such as in the development of anti-vibration mountings for sensitive and delicate machinery or in problems involving mechanical fatigue and internal resonance. For example, dangerous vibrations can develop in machinery when the exciting frequencies and the natural frequencies of fundamental components in the machine coincide. By arranging for the critical machine frequencies to correspond to that frequency at which the logarithmic decrement 'peaks', the sharper damping at resonance effectively stabilizes the machine during a particularly difficult period. A study of the frequency response of rubber and polymeric materials generally is important in the context of shock absorbers and anti-vibration mountings. Rubber is a unique material in that it is essentially incompressible (Poisson's ratio approx. 0.5) and its damping properties will be shown to be dependent upon frequency and temperature. Finally, in the more immediate context of rock mechanics, there is some evidence for supposing that shock waves are attenuated to an extent dependent upon the relationship of their frequency composition to the rock structure. Analogies have been drawn between rocks and electrical filter networks in the sense that the former tend to pass a particular band of frequencies more easily than they pass those frequencies outside the band. This behaviour is of practical significance when explosive blasting or seismic prospecting is to be conducted economically and efficiently, and it was with this particular problem in mind together with the realization that no single material can be studied in isolation that the present work was undertaken. 2. E X P E R I M E N T A L ASSESSMENT OF I N T E R N A L F R I C T I O N The mechanical behaviour of many glasses, rubbers and high polymers has been described by a linear viscoelastic law in which the stress-strain relationships can be written down as a linear differential equation which simply involves stress, strain and their time derivatives. Considering single components of stress and strain, the viscoelastic law is usually expressed n as P~r = Q Ewhere P and Q are linear operators of the form oS,an D n, D being the time derivative ~/~t. In order to define completely the mechanical behaviour of a linear viscoelastic solid, derivatives of all orders are required but it is usual to adopt only a limited number of terms in the expansion and ignore higher orders. Internal friction can be defined by the ratio A 14//W where W represents the total elastic energy stored in the stressed body and A W is the energy lost when the body is unloaded. The Q-factor, quantitatively defining the internal friction, is related to the work lost per cycle in a vibration test by the equation:

The Q-factor for a particular material is a function of frequency, stress level and temperature; values range widely from about 106 for a quartz resonator to between 10 and 105 for metals, for polymers to between 1 and 10, and between 10 and 103 for rocks. DAVIES [9] quotes Q-factors for water at 4°C of about 3 × 10a at a frequency of 17 Mc/sec and for air of about 104 at 6 kc/sec.

INTERNAL FRICTION: SOME CONSIDERATIONS OF THE FREQUENCY RESPONSE

233

Internal friction by free vibration techniques When energy is dissipated by internal friction, the strain is caused to lag behind the applied stress. Calling this phase angle/3 it can be seen that if ER and ~z denote the elastic strain in phase with the stress and the non-elastic strain component at right angles to the stress then: tan/3=-

~R

_/3.

(2)

Referring to ZENER [43] we have that:

AW

/3 =

Q-I =

(2zr)-I W"

(3)

Now, by allowing the oscillations of a freely vibrating system to decay naturally in amplitude, the logarithmic decrement 8 can be defined as the natural logarithm of the ratio of two successive excursions on opposite sides of equilibrium, or:

An 8 = In An+ 1"

(4)

If the internal friction is not dependent upon the amplitude of vibration, the graph of amplitude against frequency on log-linear paper will be of straight line form. On the other hand, if the damping is a function of vibration amplitude then the slope of the resultant curve will define the log. dec. for a given amplitude. The log. dec. is related to the phase angle/3 by the expression: 8 =/3=

(5)

and from equations (1), (3), and (5), all the characteristic relationships can be drawn.

Internal friction by forced vibration It is often inconvenient to attempt to set a low Q material into free vibration. More often, a condition of forced vibration is set up in which the test specimen is driven at a constant amplitude; under these conditions, the fractional decrease in vibrational energy per cycle gives a measure of the internal friction, the response of the system being a maximum at the resonant frequency. The energy of vibration is proportional to the amplitude squared, the logarithmic decrement being given by:

AW

8 = 0.5 ~ .

(6)

Considering the shape of the curve of amplitude response against sinusoidal exciting force, it is clear that the area under the curve decreases as Q increases and vice versa. If f r denotes the resonance frequency of the specimen and if 3fis the change in frequency of the sinusoidal driving force necessary to change the amplitude from 1/~/2 times its maximum on one side offr to 1/~/2 times its maximum on the other side then: •r(pass band)

,r3f

fr

fr"

_

(7)

77"

Now, since Q = ~, then:

Q-1 = ~ .

(8)

234

PETER B. ATTEWELL AND DAVID BRENTNALL

The technique used by the present authors in their laboratory (the apparatus being shown schematically in Fig. 1) was initiated by QUIMBY [33] and developed to its present form by TERRY [36]. The frequency of the driving signal is calibrated by matching it to an accurate signal derived from a Muirhead decade oscillator via the stationary Lissajous figure described on the oscilloscope screen. •

,

~

I

LlssoJous f i g u r e ~

DecoOe

I

I °scill,a'Or I

~] 1~-3 pin supporting chuck / I nickel or Permendur -- ' I/2 in. dia. polarized rod Detecting coil IOmH

Amplifier

~

Volve

voltmeter

I Rock specimen

FIG. 1. Composite oscillator.

The alternating magnetic field in the driving coil induces forced mechanical vibrations in the composite rod as a result of the direct magnetostrictive effect in the transducer. The resulting voltage on the detecting coil is due to the inverse magnetostrictive effect and is proportional to the velocity amplitude of the mechanical vibrations. From the readings of the valve voltmeter corresponding to the different frequencies about a natural resonance frequency, the mechanical resonance curve for the composite rod can be deduced. Using different lengths of nickel rod as transducers, and working on the higher odd harmonics, the frequency range was extended from approximately 7 to 100 kc/sec. The lower frequency limit was determined by the extreme length of specimen while the higher frequencies were limited by the eddy current losses which effectively impaired the transducer efficiency. For a composite rod of two components [37]: Qe=

QT ~-MI~:1 - - Q~')

1 + M2~

Ol

(9)

where sufficesT, l and 2 refer to the composite rod, transducer, and specimen respectively, and M denotes the mass. By measunng the mechanical resonance curve for the composite rod and transducer singly, the value for Qz can be derived. O n the assumption that a rock

INTERNAL FRICTION: SOME CONSIDERATIONS OF THE FREQUENCY RESPONSE

235

can be reasonably represented by a Burgers model [36] the attenuation in the solid can be represented by: sin h -1 sin a =

1

-

(10)

where l denotes length and n defines the number of harmonics. A simpler, approximate expression was derived by BORN [7] viz: nT/-

a -- 2QI,

(11)

Some results derived by the authors using the resonance technique are given later in the paper Internal friction through wave attenuation techniques

The fundamental aspects of the propagation of viscoelastic waves have been considered fairly recently by KOLSKY[22] and are not repeated here. Internal friction, usually expressed as the phase difference between stress and strain at a particular frequency, is a direct function of a complex elastic modulus analogous to a complex impedance in electrical network theory. When low amplitude, undispersed, sinusoidal waves are propagated through a timedependent solid, the decay can usually be taken as exponential with distance, the expression in its simplest form being: g(x) ~- go exp (--ax)

(12)

where ct is the attenuation constant, usually expressed in terms of units per unit distance or units per wavelength, a can be related to Q through the expression: eL = ¢rfc-la -1

(13)

wherefis the fundamental frequency of the wave and c is its velocity. If Q is high valued, very long specimens would be required in any experimental investigation in order to reproduce significant amplitude differences. For this reason, the fundamental wave propagation technique is applied invariably to low Q materials such as plastics through the longitudinal mode of vibration within the audio frequency range. Experiments on thin strips, fibres or filaments of material have been performed by BALLOUand SMITH [3] polymers; WITTE et al. [40] polymers; NOLLE [27, 28, 29] rubbers; HILLIER [16] and HILLIERand KOLSKY[17] polythene, neoprene, nylon. Short rise-time, high amplitude pulses can be propagated by means of swinging hammers, high velocity bullets, or explosive detonators and provided that the free-surface 'jumps' due to repeated pulse reflections can be continuously recorded then relatively small specimens will yield attenuation data over long distances of travel. KOLSKY [22], for example, by detonating small charges of lead azide at the ends of polystyrene, polythene, and perspex rods was able to record the changing shape of the stress pulse as it repeatedly travelled over distances that were a multiple of the rod length. Examples of the type of oscillogram projected by Kolsky's capacitance gauges are given in Fig. 2 together with the typical attenuation effect resulting from a differentiation of the oscillograms. It can be seen that as the step corners on the osciUogram traces become progressively rounded, so the pulse

236

P E T E R B. A T T E W E L L

AND DAVID

BRENTNALL

length increases and the reflections gradually overlap. Ultimately, the particle movement becomes continuous indicating that the centre of gravity of the rod is moving at constant velocity. Similar experiments have been conducted by one of the authors (P.A.) on ~cdimentary and igneous rocks: in this case however pulses from explosive detonators of different intensity were monitored at an initial steel bar before being passed through a low fricticm grease joint into rods of the selected rocks.

FO

A

8

6~

4 d

2~ 400

300

200 Time,

IO0 /~ sec

FIG. 2(C). Showing the change in particle velocity after pulses have travelled through different lengths of polythene. A, 30 cm. B, 60 cm. C, 90 cm. After KOLSKV [22].

Provided that certain assumptions can be made concerning the frequency/attenuation characteristics of the material, then a mathematical analysis of the changes in pulse shape becomes feasible both for direct detonation on the study material and also for the more general problem of pulse transmission from elastic to viscoelastic media. The usual technique, which is not set down here, is to represent some function, which defines the pulse at its origin, as a Fourier integral and then to replace the expression for some distance of pulse travel by a Fourier series, choosing a frame of reference which moves with the pulse. In the experiments just described, low amplitude attenuation/frequency characteristics can be used to predict the attenuated shape of high amplitude pulses propagated through linear viscoelastic solids. Quite apart from any obvious discrepancies due to &function approximation and lateral inertia dispersion, there may well be differences in the predicted and actual pulse configuration due simply to this amplitude effect. Attenuation values derived from direct readings of the pulses may well differ substantially from the low amplitude values. Nevertheless, low amplitude pulse tests have a very useful rrle to play in dynamic analysis: electromechanical techniques of the type used by the present authors were first developed by HUGHES et al. [18] to measure longitudinal and shear wave velocities in rocks and subsequently adapted by AUBERGERand RINEHART[2] to record attenuation. The authors use a pulse generator of suitable output impedance to supply a 20V, 10 ~sec pulse to barium titanate disks resonating in the longitudinal mode (Fig. 3). Rock specimens sandwiched between two identical disks receive a mechanical pulse which is transmitted into

INTERNAL FRICTION: SOME CONSIDERATIONS OF THE FREQUENCY RESPONSE

237

and converted by the second ceramic to an electrical pulse available for display on an oscilloscope screen. The length of ceramic immediately determines the frequency of excitation while the attenuation can be calculated from the strength of signal projected by specimens of different thickness (Fig. 4). Frequencies between 104 kc/sec and 5 Mc/sec have been covered in this manner by the authors, the attenuation c~being expressed in decibels per unit length, viz: A2 20 logl0 A1 (14) 5--

/1-12 where A1 and A2 denote the amplitudes of the first arrival from specimens of thickness/1 and

/2. Pulse generator

4

Rock specimen

'scope

©

Y, D

¥2

f i

FIG. 3. Piezoelectricexcitationof rocks.

JAMIESONand HOSKINS[20] have shown that by using Pyrex wedges of calculated shape, waves initially generated in the longitudinal mode can be transformed into the shear mode and then after passing through the test specimen can be converted back in the longitudinal sense for ease of measurement. Shear waves have received rather less experimental attention than longitudinal waves but with the advent of this comparatively simple and inexpensive technique the present situation is likely to be remedied in the future. 3. ANELASTICITY AND RHEOLOGICAL MODELS Rheological models perform a useful function in acting as the first step in the development of rational thought as to the inherent causes and processes of plastic deformation in solids. Nevertheless, the-theory that attributes plastic deformation to progressive slip along preferred planes within the individual crystals is infinitely more complex than the classical Hookean-Newtonian rheological concepts. Reference should be made to the extensive literature for detailed descriptions of mechanical models in which springs deform according to Hooke's law and viscous dashpots obey Newton's law of viscosity. Both Maxwell and Kelvin-Voigt elements contain first-order terms and therefore describe the dependence upon the rate of loading. The Maxwell model allows for no delayed recovery while the Kelvin-Voigt model describes no purely elastic

238

PETER B. ATTEWELL AND DAVID BRENTNALL

strain. The standard linear solid (another spring in series with a Kelvin-Voigt model, Fig. 5) --originally suggested by BOLTZ~tANN [6J--although limited, is more representative of deformation processes in solids since it incorporates the elements of instantaneous response, delayed response, and flow. F. . . . . .

~

.

.

.

.

.

.

.

.

.

.

.

FiG. 5. S t a n d a r d linear solid rheological model.

The standard linear solid can be interpreted by relating the elastic modulus under high frequency conditions to the modulus when the relaxation process is allowed to terminate. The former can be termed the dynamic or unrelaxed modulus and the latter, the static or relaxed modulus. By equating stress and its first derivative with respect to time to the strain and strain rate we have that: a l cr -~- a2o" ~--- bl~

+ b2e.

(16)

Now letting r, and % be the times of relaxation of stress for constant strain and of strain for constant stress, and denoting the relaxed and unrelaxed moduli by ER and Eu respectively, it can be shown that: ER Eu

~-, %

(17)

the general equation for the solid being written in the form:

+ T,~ = ER(, + ~-~).

(18)

NowIcK [31] introduces a dimensionless proportionality constant AM called the relaxation strength which relates the 'equilibrium non-elastic strain' to the perfectly elastic strain which obeys Hooke's law. It provides a measure of the total relaxation, viz: Ev ER (EuER) ~ . -

AM-

-

(19)

Now, the single relaxation time, ~-, associated with the standard linear solid is equal to half the sum of the two particular relaxation times [10] and the internal friction can be expressed by the equation: Q-1 = A i oJr ]

~ ~ i-~--~;~;2}

where oJ = 27rfi

(20)

INTERNAL FRICTION: SOME CONSIDERATIONS OF THE FREQUENCY RESPONSE

239

Thus, when the angular frequency of vibration is equal to the reciprocal of the relaxation time of the process causing the relaxation, the internal friction will reach a maximum value. On either side of this peak, for higher and lower frequencies, Q-1 tends to zero. If a further Newtonian dashpot is inserted in series with the standard linear solid model, the resulting Burgers model--widely used by Terry and his collaborators to describe the mechanical response of coal--allows for a degree of permanent strain. A Boltzmann superposition treatment considers the mechanical behaviour of the specimen to be a function of its entire previous strain history: that is, the total deformation can be considered as comprising a number of independent incremental deformations, the actual behaviour of the specimen being calculated by the simple addition of the effects that would occur when the deformations t o o k place singly. Taking strain as the independent variable, according to the Boltzmann model: I_ d~ . , = e, + ~(t - t') ~-, ctt. (21) oO

E can be taken as the relaxed modulus, • is the strain at some time t and t' must satisfy the condition that t >~ t' >/0. The memory function ~(t -- t') implies that a small change in strain 8 ~ produces a change of stress ~(o)8~ instantaneously and ~(t -- t')8~ after a time (t -- t'). This theory, then, embodies the concept of a linear superposition of effects. The stress-strain curves for different materials subjected to high rates of loading have been found to conform to equation (21) with a memory function of the form:

• (t--t')=Aexp{--t~

t'}

(22)

where A is a non-dimensionless constant. Again this implies (as for equation (20)) that these materials will exhibit an internal friction peak at a frequency equal to r-1.

Relaxation time In many materials, and particularly in metals, it is found that the relaxation time is an exponential function of temperature, viz:

AH

1-= ~'0 exp ~

(23)

where AH is the activation energy, R is the gas constant and T is the absolute temperature. Since Q-1 is dependent only upon the parameter cot the internal friction can be observed as a variable with r by maintaining o~ constant and changing the temperature. The curve of Q-1 against T -1 will again show a maximum at oJ~ equals unity. In body-centred cubic metals, relaxation peaks are caused by the diffusion of interstitial atoms to the minimum energy positions in the stress fields of the mobile dislocations. Now, curves of Q-1 against T -1 derived at frequencies o,1 and o,2 are linearly displaced (AT-1) but describe the same relaxation time. Since ~o1"r1(0_1> = co~'2(o_x) we have that:

AH

coI exp R-T1 ---- oJ2exp

AH RTg."

(24)

Thus,

In oo_._2 tol

AH = R A-~:i. Q

(25)

240

P E T E R B. A T T E W E L L A N D D A V I D B R E N T N A L L

For a given relaxation time, the diffusion coefficient for the interstitial atoms is given [10] by: d - - -a°2 36r

(26)

d = do exp (A-~T).

(27)

ao being the interatomic spacing. d varies with temperature as,

Relaxation spectrum The anelastic properties of many materials subjected to stress levels insufficient to cause a permanent set are generally independent of stress amplitude and the number of strain cycles but are highly dependent upon both frequency and temperature. If Q-1 is plotted against the frequency of vibration, metals in particular project curves comprising a number of internal friction peaks, each peak being associated with a particular physical mechanism and operating in a different region of the frequency spectrum. Figure 6 shows diagrammatically a typical relaxation spectrum and indicates the several mechanical processes associated with the losses throughout the observed frequency range. Each peak can be described by its own relaxation time and it is possible, by the insertion of suitable constants, to express each peak in terms of equation (18). Temperature at 0"8 c/s

400

300

200% (scoleB)

Constant frequency but decreasing temperature Oq Pairs of solute Twin f ' ~ % . S c o l e El I / ~ atoms , f - - ~ \ boundaries Polycrystalline aluminum ,0-2~ ' [ \ (~ grain boundary effect

$ o I0-

4

o = 10_5

~/I ~

I t I Grain I boundaries l

l I

I I ~) "Hypothetical" material (Dashed line}

~ ' , , ~ ~ o= Scae A [ ~' ~ ~ r~'~.~o I 'I = Intershhol I <-solute I (;3 Itl at°Ts I Tronsver~ / thermql currents / / ! Intercryst / \ / \ 1

,0-2-,__,

lO-4q~ ID I0 -s'

Peek values iO-6 tar various materials and I ] I i I I\~/' I \t-" ~ mechanisms IO~4 IO-12 lO io iO-e 10-6 10-4 iO-.Z 1.0 I02 I04 I0e (scale A) Constant temperature but decr.,easing frequency ..-......Ib

FIG. 6. Internal friction peaks for a 'hypothetical' and several real metals and the associated microstructural mechanisms : (A)Cu-Mn alloy (movement of twin interfaces); (B) German silver (thermal currents in transverse vibrations); (C) alpha brass (stress induced by preferential orientation of axis joining pairs of solute atoms); and (D) polycrystalline brass (intercrystalline thermal currents). After LAZANand GOODMAN,Shock and Vibration Handbook, 1961, by courtesy of McGraw-Hill Book Company, inc. One point concerning relaxation times is important. If several relaxation mechanisms each described by its unique ~- are operating at the same time, it does not follow that the total Q-1 is the simple superposition of the individual values of internal friction due to the relaxation mechanisms acting separately. If the sources of relaxation are mutually disturbed, relaxation coupling is said to have developed and the extent of the resultant relaxation can greatly exceed that due to simple superposition [31]. Nowick [30] detected such coupling as a consequence of the irregular arrangement of interfaces in a metal (see next section).

INTERNAL FRICTION" SOMECONSIDERATIONSOF THE FREQUENCY RESPONSE

241

4. SOME SOURCES OF I N T E R N A L FRICTION

Grain boundary relaxation Stress relaxation can occur along the boundaries of contiguous grains. These boundaries act in a viscous manner creating an angle of loss which increases uniformly with temperature (temperature dependence of viscosity)--ZwIKKER [45]. Such relaxation usually results in a large, broad internal friction peak [10]. Kf~ [21] proved the significance of grain boundaries by discovering a broad internal friction peak in the region of 300°C in polycrystalline aluminimum but observed no peak in aluminium single crystals (Fig. 7(a)). He further found that the modulus (proportional to frequency squared) at different temperatures showed a sharp drop for the polycrystalline specimen but not for the single crystal specimen (Fig. 7(b)). This substantiates the previous contention that at high temperatures, the grain boundaries tend to deform viscously. O'IC 0"09 0.0~

Polycrystaltine

vO 0.07

~luminum

~- 0.06

o.o~ g H

0.04 0.05

0'02

"Single crystal" 0.01

aluminum×,....--.×

0

IO0

200

300

Temperature of measurement,

400 °C

500

FIG. 7(a). Variation of internal friction with temperature in polycrystalline and 'single crystal' aluminium (frequency of vibration = 0'8 c/sec at room temperature). After IO2 [21].

0.70 0.65 o "= o

0.60

:) •46

0-55

g ii

" ~ x ~

0"50

/

"Single crystal" aluminum

Polycrystalline

0.45 0-40 0"350

I

I00

I

200

I

300

Temperature of measurement~

I ~ 400

500

°C

FIG. 7(b). Variation of modulus (p) with temperature for polycrystalline and single crystal aluminium. After Kfl [21].

242

PETER B. ATTEWELL

AND DAVID BRENTNALL

If an interface is subjected to shear stress, the latter undergoes a process of relaxation until the stress reaches zero. The stress redistribution causes concentrations along the edges of the interface and in the polycrystalline case causes a superposition of the relaxations of the individual interfaces. Nowick showed the relaxation time for a single interface to be proportional to the linear dimension of the interface but since there can be a random distribution of contact areas when crystals are assembled in aggregate form the internal friction peak is distended to accommodate the different relaxation times so explaining the broad band of this peak. If the distribution of the contact areas of the interfaces is narrow (such as would occur for example in a welt-graded material comprising an agglomeration of near-spheroidal particles) then a number of the individual relaxation times would be approximately equal, but instead of causing a simple superposition of relaxation times, slip along adjacent interfaces would induce larger shear stress across their neighbours (coupling effects) to produce magnified viscous losses. A much higher--but narrower--friction peak would result. Before leaving the question of grain boundary effects, we must mention that fatigue can lead to intercrystalline corrosion--a loosening of neighbouring grains--and thus to the development of a large number of minute cracks which will increase the damping considerably [13]. Under these circumstances, the amplitude and breadth of the friction peak is closely dependent upon the amplitude of the vibration force which will determine the frictional forces across the walls of the hair cracks.

Movement of low energy twin interfaces WORRELL [41, 42] noted an internal friction peak for a C u - M n alloy at about 0°C (Fig. 8). Since the lattice is effectively intact across such an interface the friction mechanism cannot be related to that described for the grain boundary case.

50 i

25 J

Tempero)ure, 0 I

°C -25 [

-50

0'007

0"006

:o .

g

0.005

0,004

o 0"003

O, 0 0 2

0"00 I

3'0

9 l 3-5

I 4'0

4-5

IO00/T

FIG. 8. Typical variation of internal friction of Cu-Mn alloy with temperature. This internal friction is supposedly due to movement of twin interfaces (after Worrell). Reference, ZENER[43].

INTERNAL

FRICTION"

SOME

CONSIDERATIONS

OF THE

FREQUENCY

RESPONSE

243

Preferential ordering of interstitial atoms (Fig. 9) The effect of interstitial atoms of carbon and nitrogen upon the internal friction in a-iron was first explained by SNOEK [35] who showed that an internal friction peak no longer occurred when all traces of carbon and nitrogen were removed from the iron. Without considering here in detail the mechanisms whereby the internal friction peak develops, it is known that under a stress system which shows rapid reversals, the atoms which occupy interstitial positions in the atomic lattice are in a state of continuous motion tending either towards, or away from, a preferred orientation. 650 I

Temperature, °C 625 500 475 4.50 47'5 4 0 0 375 350 I t f t ] ~ I I

325 t

1200 I100 I0O0

900 r

o - observations

800 f

at 620 c//s

too x To 60o

oi

500

3OO 2OO I00 0

12

t

I 13

I

I 14

I

I 15

I

I 16

t

17

IO00/T

FIG. 9. Internal friction due to stress induced by preferential orientation of axes joining pairs of solute atoms. Reference, ZENER[43]. In the case of glass, FITZGERALD [12] interpreted internal friction peaks in terms of the diffusion of sodium and potassium ions to preferential positions in the atomic lattice.

Internal friction due to thermoelastic effects Mechanical strains are always accompanied by small changes in temperature (positive or negative depending upon the mode of strain), the non-uniform strain distribution which accompanies sinusoidal reversals producing temperature gradients within the material. The flow of heat into and out of an element of the material produces non-elastic strain. High frequency processes are usually adiabatic and since there is insufficient time during a half cycle for heat flow to occur, the energy loss is small. On the other hand, at sufficiently low frequencies the specimen is maintained in temperature equilibrium, the processes are isothermal and the energy loss is again minimal. At intermediate frequencies, mechanical energy is less easily converted into heat energy and the internal friction losses are at a maximum.

244

PETER B. ATTEWELLAND DAVID BRENTNALL

If a very thin strip of material is vibrated in the flexural mode, for each half cycle of vibration the convex side is expanded (cooled) at the same time as the concave side is compressed (heated). At intermediate frequencies interrupted heat flow should take place leading to an internal friction peak. BENNEWLTZand R6TGER [4] performed experiments upon ferrous and non-ferrous metals and glass (Fig. 10) and showed that the experimental conditions were 0.6 0-004

0-003 O

0

1000

~ Si

350

/

I00

tO Frequency,

t'O

c/s

FIG. 10. Internal friction as measured by Bennewitz and Ri3tger. Calculated maximum values indicated by vertical lines. Reference, ZENER[43]. very nearly describable in terms of a standard linear solid having a single relaxation time r given by d,~ r .... (28) rrZD where d is the thickness across the heat path and D is the thermal diffusion constant equal to S/Cvp; where S is the thermal conductivity and Cv is the specific heat per unit volume. The two German workers re-performed their experiments on silver to decide the characteristic friction peak shown in Fig. 11. In a polycrystalline solid, there is an internal friction loss associated with the elastic anisotropy of a particular grain together with the random orientation of the grains comprising the solid. The transition from isothermal to adiabatic behaviour is therefore a function of frequency, grain size, and the thermal diffusion coefficient. If these three variables are combined in dimensionless form (.(d2/D)--where d is a linear dimension of the grain size--RANDELL et al. [34] showed that for all frequencies and all grain sizes, the graph Q-1 peaked at a value of (fd2/D) equal to unity (Fig. 12). It has been shown that for high values of the dimensionless combination of parameteIs, Q-1 is almost linear with 1/~/f (Fig. 13). In the isothermal case (small grain size and low frequencies) the heat flow between grains is virtually independent of frequency and from this we can deduce that the internal friction is proportional to the frequency (Fig. 14). The thermal diffusivity of rocks varies from about 0.005 cm2/sec for a limestone to 0.031 cm2/sec for a quartzite (International Critical Tables). These values are naturally low

INTERNAL FRICTION: SOME CONSIDERATIONS OF THE FREQUENCY RESPONSE

245

2.0 :~+ ,'5

0"5

O

+

I00l I

{

I

r

?

v:

I

t0J i l l /

I

I

1

Illl

I'0

c/sec

FIG. 11. Check on the theory of damping of transverse vibrations by thermal currents. Example of German silver (after Bennewitz and R6tger). Reference, ZENER [43].

vd2/D I0

0.01 I

0.1 I

[.0 I

9

o

~

I0 i

I00 J

~

78

+/

I000 I

IO,O00 [

A AT 6000 o AT 12,000 ~6,000 ~

6 x 5 O ~_

÷

4

3

I

0

0.1

I

I

I'0 vVZd,

FIG.

I

r

I

I Illll

I0

r

I00

cm/sed/e

12. Internal friction in polycrystalline brass (after Randell, Rose and Zener). Reference, ZENER[43].

when compared with the thermal diffusivity of metals (for example, brass has a quoted figure of 0.38 cm2/sec--ZENER [43], Table 11, p. 90) and, in addition, rocks are relatively poorly graded and often polyminerallic. We would expect, therefore, little thermoelastic effect in rocks, any dissipation that might be piesent being impressed as a 'broad band' loss covering an extensive fiequency range and being absorbed by complementary friction mechanisms.

246

PETER B. A T T E W E L L A N D D A V I D B R E N T N A L L Harmonic

654 I

3

2

I

I

l

14

/

. I0

I 1

/

/xj

:,= 8 o E o~

T~ 6 4

2

0

4 6 I0 12 [I/(frequency)t~=j xl03 in [c/s] '/2

14

16

FIG. 13. I/v'fdependence of internal friction in the adiabatic case (after Zener and Randell). Reference, ZENER [43].

Harmonic

2.~ 2<

I.E

i-c 0"E 0-E 0.4 0"2

0

6

12

18

Frequency,

24 kc/s

30

36

FIO. ]4. Internal friction/frequency relationship in isothermal case (after Zener and Randell). Reference, ZENZR [43].

INTERNAL FRICTION: SOME CONSIDERATIONS OF THE FREQUENCY RESPONSE

247

5. FREQUENCY RESPONSE OF SOME NON-METALLIC MATERIALS At low stresses, rubbers and plastics follow a linear viscoelastic law. The stress limits of linearity are of the same order of magnitude for both rubbers and plastics but the limits within which the strains are linear become smaller in plastics. If the viscoelastic behaviour is linear the shear modulus can be expressed in terms of its real and imaginary pa~ts, the former representing or describing the stiffness of the material and the latter describing its damping capacity. Since the dynamic behaviour of these materials is strongly temperature and frequency dependent [1] one would expect to find a critical frequency band or bands within which the internal friction becomes more pronounced. At high frequencies, the material behaves stiffly (its 'storage modulus' is high while its 'loss modulus'--imaginary part of the expression--is low): this is the so-called 'glassy' region in which the 'molecular curling and uncurling' is too slow to follow the stress (Fig. 15). At low frequencies or high temperature both the real and imaginary parts of the Low frequency or high temp. =

Rubbery

High frequency or low temp. Glassy ,~

Tronsition

Loss foctor ,, r

/GI

_

\

\ log scole Increasing

frequency at constant

temperature

or

Decreasing temperature at constant frequency

FIG. 15. Dependence of storage modulus (G1), loss modulus (GO, resultant modulus (Gr) and the loss factor '1 on temperature and frequency for plastics and rubbers. After Lazan and Goodman, Shock and Vibration Handbook, 1961, by courtesy of McGraw-Hill Book Company Inc.

expression are small: this results in a 'rubbery' region where the molecular disturbances are in phase with the stress and so cause minimum energy loss. At intermediate temperatures and frequencies within the transition region where the loss modulus tends to maximum, the molecular processes grow out-of-phase with the cyclic stress and the resultant lagging cyclic strain causes a dissipation of damping energy. Figures 16, 17, 18 show the attenuation peaks and confirm the observation that for increasing frequency the internal friction peaks develop at correspondingly higher temperatures: such an effect is, however, significant only when a sufficiently high frequency range is investigated. In terms of rheological models, KOLSKY[22] using the data on high polymers derived by LETrrERSICI-I [25] demonstrated that both the Maxwell and Voigt models were inadequate in describing the frequency response of these materials but that a standard linear solid gave reasonable agreement over about a decade of frequency.

248

PETER B. ATTEWELL AND DAVID BRENTNALL

It is found that the most widely applicable general rule is that the attenuation exponent (expressed in decibels per unit distance or decibels per wavelength) is proportional to some power of the frequency over a substantial frequency range. WEGELand WALTHER[39] found, for metals and glasses, the attenuation exponent to be proportional to fn where 1-54 > n > 0-87. QUIMBY [33] for similar materials found the attenuation exponent to be ,~50

"/

'~Cb c:

o

\

c

5C' L

- 4,7)

- 2 ()

{

<

4 0

;3 0

8 <:

°C

Temp,

FI~. 16. Bulk wave attenuation against temperature at a frequency of 10 Mclsec for Hevea (©), GR-S (x), and butyl (A) rubbers (after IVEY,MROWCAand GUTH[19]). 450

i'OMc/~

i/.

.¢=

"~" 3oc 400kc/

/

E

/

// /]

44kc.~-/~--/_./

'V,/

/

/"

],

~,

//', \ // "\ \

jill

i

'

F\ ~\

"'~\ \\

/ / / x.)(~ /

\i

\

--

\

N \

!

/%:k/

~ ~5c

//! ,,/.-" /

//I .'7

"'-. J

/3Mc

400ki

-40

-20

0 Temperoture,

\'.

-~

;

I

20

°C

40

60

80

FIG. 17. Attenuation per cm (solid curves) and attenuation per wavelength (dashed curves, arbitrary scale) for butyl at different temperatures (after IVEY, MROWCAand GtrrH [19]). proportional to the square of the frequency. On the other hand, MASON and MCSKIM1N [26] found that for metals the attenuation of both longitudinal and torsional waves was proportional to frequency at the lower frequencies and to the fourth power of the frequency at higher frequencies. At much higher frequencies, they discovered a grain size effect in that diffusion processes were operative independent of frequency when the grains were of size > 3A.

INTERNAL FRICTION: SOME CONSIDERATIONS OF THE FREQUENCY RESPONSE

249

In their vibration tests on various types of wood, KROGER and ROHLOFF [24] and HEARMON [15] showed that the internal friction was very dependent both upon the orientation of specimen with respect to the wood grain and also upon porosity. They found the internal friction to be amplitude dependent but to be independent of frequency in the range l0 c/sec to 10 kc/sec. Frequency effects in rocks have, of course, a particular significance in seismic prospecting where the frequency composition of a pulse will determine its behaviour in rocks of different mineral content and grain size. BIRCH and BANCROFT [5], subjecting a granite cylinder 8 ft long by 9 in. dia. to frequencies in the range 140-1600 c/see in the longitudinal, torsional and flexural modes, found the attenuation to be roughly proportional to frequency. BORN

E

g o

~ x ~ - - x

2t'c

x

t kc

-I0

I0

50 Temperature,

50

70

90

°C

FIG. 18. Attenuation peaks with temperature in butyl (A) and GR-S ( × ) gum stocks (after WITTE, MROWCA and GtrrH [40]). [7] using bars of shale, limestone and sandstone in lengths up to 6 ft and subjected to frequencies in the range 102-104 c/see again found the attenuation to be proportional to frequency. Saturation with water, however, caused the attenuation to be equal to the sum of a term proportional to frequency and a term proportional to f L BRUCKSHAW and MAHANTA [8] subjecting different rocks to lateral vibrations in the range 40-120 c/see similarly found the attenuation exponent to vary linearly with frequency although at lower frequencies in the range 40-70 c/see they found the increase to be more rapid. Yet again, both EWING and PRESS[11] and PESELNICKand ZIETZ [32] confirm the attenuation-frequency proportionality, the former in the very low frequency range--0.003-0.007 c/sec--and the latter in the megacycle range (Fig. 19). The work of KRISHNAMURTHIand BALAKRISHNA[23] is particularly interesting in that they relate their results to scattering and grain size effects. It is shown that attenuation in rocks is generally higher than in amorphous substances of negligible grain size or in metals and can be ascribed to the relatively inhomogeneous nature of rocks. Attenuation was shown to increase with frequency to some limit (Fig. 20) but that for limestone (fine-grained, homogeneous, close-packed) the attenuation was slight but linear with frequency (absence of scattering). On the other hand, attenuation in shale was shown to increase rapidly with

250

P E T E R B. A T T E W E L L

!

~.a0

'

AND DAVID

'

I

l

A

2

4

6

BRENTNALL

I

_~ _ _ _.L 8

IC_a~te_ _ _~ 12 14

I0

Frequency, Me/see

16

18

FIG. 19. Dilatational wave absorption against frequency for several limestones (after PESELNICK ZIETZ[32]). S-l, Solenhofen limestone. I-l, fairly uniform limestone. H-l, non-uniform limestone.

and

30 I. Deccanfrap

25 E

2O

2. Dolerite 3. Limestone 4. Shale 5. Marble

4

"5 ~ 15

g

.Q

I0

t I

I

1

i

]

1

2

3

4

5

6

7

Frequency,

Mc//s

FIG. 20. Attenuation against frequency for several rocks (after KRISHNAMURTHIand BALAKRISHNA[23]). frequency (similar grain size but loose packing) while the larger grain sizes in dolerite and marble caused the attenuation to become independent of frequency. The general conclusion was that for low porosity rocks the higher acoustical absorption was probably due to the increased number of grain boundaries per unit volume. Some of the results derived by the present authors using the resonance technique are presented in Figs. 21, 22, 23. The attenuation exponent appears to vary linearly with

I N T E R N A L F R I C T I O N : SOME C O N S I D E R A T I O N S O F T H E F R E Q U E N C Y RESPONSE

251

120

I00

80

,

/

6O

/ /

X C op 20

x ~n /

/

/< 0

r

i

40

2O

I

60

Frequency,

I

80

I00

kc/s

FIo. 21. Frequency response of Darley Dale sandstone: line of best fit calculated from equation (10) (points × ) and equation (11) (points 0).

60

5C

4C

°

/

o

<

2C

/

/ 0

20

I

40 Frequency,

I

60

I

80

I

I00

kc/s

FIG. 22. Frequency response of calcareous sandstone: line of best fit calculated from equation (10) (points × ) and equation (ll) (points O).

252

P E T E R B. A T T E W E L L

AND DAVID

BRENTNALL

frequency over a range of approximately 5-100 kc/sec (using higher harmonics)and therefore generally conforms with the results of the previous workers. Attenuation is inevitably due to scattering at grain and pore boundaries, as evidenced by the change in slope of the graphs with the change in the degree of porosity exhibited by the rocks. Although the curves of Peselnick and Zietz (Fig. 19) and of the present authors (Figs. 21, 22, 23) can be extrapolated linearly back to the origin, the very low frequency ranges have, in fact, received very little experimental study. Frequencies of between 2 and 100 c/sec are very important seismically and the work of USHER [38] suggests that AW/ W in representative rocks increases gradually from this lower limit but becomes constant before 40 c/sec. This

25

%

20

1,¢

7c 15

g g <

I0

0

20

40 Frequency,

60

80

I00

kc/s

FI~. 23. Frequency response of Pennant sandstone: line of best fit calculated from equation (10) (points x ) and equation (11) (points ©). work conforms with the results derived by the present authors but fixes more exactly the slight frequency dependence in the range 2-40 c/sec, a feature masked by the extrapolation to zero in Figs. 19, 21, 22, 23. 6. CONCLUSION AUBERGER and RINEHART [2] sampling over the 100 kc/sec to 2 Mc/sec range in rocks showed that attenuation peaks occurred at harmonic frequencies which could be related to the predominant grain size of the rocks. In other words, when the length of the propagated pulse approaches the grain size of the rock, the constituent crystals are set into resonance causing local increases in the attenuation/frequency curves. The effect of such higher harmonics might still be detectable in the megacycle range but any 'peaking' would certainly be less sharp. We might, however, in this range expect some internal friction effects associated

INTERNAL FRICTION: SOME CONSIDERATIONSOF THE FREQUENCY RESPONSE

253

with intracrystalline m e c h a n i s m s although at low v i b r a t i o n amplitudes these might r e m a i n undetected. Scattered friction peaks in the lower frequency range of approximately 5 100 kc/sec w o u l d n o t usually develop at r o o m temperature in view of the large pulse length to grain size ratio. The very low frequency range covering a few cycles per m i n u t e to a b o u t 1 kc/sec is rather less a m e n a b l e to experimental t r e a t m e n t a n d accurate recording b u t material b e h a v i o u r at such frequencies can be effectively reproduced by subjecting specimens to higher temperatures; in other words, b y i n d u c i n g conditions more favourable to plastic deformation. If we are prepared to regard rock as a material--albeit a relatively complicated m a t e r i a l - a n d assess its b e h a v i o u r in v i b r a t i o n by a d o p t i n g the same criteria as for more h o m o g e n e o u s materials, then we may, as m i n i n g engineers, be able more effectively to c o n t r o l the perf o r m a n c e of our explosives u n d e r g r o u n d . Acknowledgements This work forms part of the rock mechanics research programme currently in progress at Sheffield under the general direction of Dr. A. Roberts, Director of the Postgraduate School in Mining of the University of Sheffield.

REFERENCES 1. ALFREYT. Mechanical behaviour of high polymers, Interscience, N.Y. (1948). 2. AUBERGERM. and RaNEHARTJ. S. J. Geophys. Res. 66, 191 (1961). 3. BALLOUJ. W. and SMXTHJ. C. J. AppL Phys. 20, 493 (1949). 4. BENNEWrrzK. and R6TGER,H. Phys. Zeitschr. 37, 578 (1936). 5. BmCH F. and BANCROFTD. Bull. Seis. Soc. Amer. 28, 243-254 (1938). 6. BOLTZMANNL. Pogg. Ann. Erg. 7, 624 (1876). 7. BORNW. J. Geophysics, 6, 132-148 (1941). 8. BRUCKSEIAWJ. McG. and MAHA~rrAP. C. Petroleum, 17, 14--18 (1954). 9. DAVmS R. M. 'Internal Friction of Solids' Some recent developments in rheology, British Rheologists Club (1950). 10. DIETERG. E. Mechanical Metallurgy, p. 227. McGraw-Hill, N.Y. (1961). 11. EWING M. and PRESSF. Bull. Seismol. Soc. Amer. 44, 127-147, 471-479 (1954). 12. FITZGERALDJ. V. J. Amer. ceram. Soc. 34, 314 (1951). 13. GEMANTA. Frictional Phenomena, p. 352. Chemical Publ. Co., N.Y. (1950). 14. GEMANTA. and JACKSONW. Phil. Mag. 23, 960 (1937). 15. HEARMONR. F. S. The elastic and plastic properties of natural wood; Mechanical properties of wood and paper. (Editor: R. Meredith) North Holland Publ. Co., Amsterdam (1953). 16. HILLIERK. W. Proc. Phys. Soc. 62B, 701 (1949). 17. HILLIERK. W. and KOLSKYH. Proc. Phys. Soc. 62B, 111 (1949). 18. HOGnESD. S., PONDROMW. L. and MIMSR. L. Phys. Rev. 75, 1552-1556 (1949). 19. IVEYD. G., MROWCAB. A. and GtrrH E. J. Appl. Phys. 20, 486-492 (1949). 20. JAMIESONJ. C. and HOSKINSH. Geophysics, 28, 87 (1963). 21. K~ T. S. Phys. Rev. 71, 533 (1947); 72, 41 (1947). 22. KOLSKYH. Viscoelastic waves, Proc. Int. Symp. on Stress Wave Propagation in Materials, (Editor: N. Davids), Interscience, N.Y., pp. 59-89 (1960). 23. KRISHNAMURTHIM. and BALAKR1SHNAS. Geophysics, 22, 268-274 (1957). 24. KROGERF. and ROHLOFE,E. The internal friction of wood, Z. Phys. 110, 58 (1938). 25. LETHERSlCHW. Brit. J. Appl. Phys. 1, 294 (1950). 26. MASONW. P. and MCSKIMINH. J. J. Acoust. Soc. Amer. 19, 464-473 (1947). 27. NOLLEA. W. J. acoust. Soc. Amer. 19, 194 (1947). 28. NOLLEA. W. J. AppL Phys. 19, 753 (1948). 29. NOLLEA. W. J. Polymer Sci. 5, 1 (1949). 30. NOWlCKA. S. or. Appl. Phys. 22, 925 (1951). 31. NOWlCKA.. S. Internal friction in metals: Progress in metal physics, Vol. 4, 15. Pergamon Press, London (1953). 32. PESELr,aCK L. and ZIETZI. Geophysics, 24, 285-296 (1959). 33. QUIMBYS. L. Phys. Rev. 25, 558-573 (1925). 34. RANDELLR. H., ROSEF. C. and ZENERC. Phys. Rev. 56, 343 (1939).

254

PETER B. ATTEWELL AND DAVID BRENTNALL

35. SNOEKJ. Physica, 6, 591 (1939). 36. TERRY N. B. 'The Elastic Properties of Coal VI--Some measurements of internal damping and some considerations of viscoelastic bchaviour' N.C.B.M.R.E., Report No. 2080 (1957). 37. TERRY N. B. Brit. J. Appl. Phys. 8, 270 (1959). 38. USHER M. J. Elastic behaviour of rocks at low frequencies, Ph.D. Thesis, Imperial College (1961). 39. WEGEL R. L. and WALTnER H. Internal dissipation in solids for small cyclic strains, Physics, 6, 141-157 (1935). 40. Wia-rE R. S., MROWCAB. A., and GUTH E. J. Appl. Phys. 20, 481 (1949). 41. WORRELLF. T. J. Appl. Phys. 19, 929 (1948). 42. WORRELLF. T. J. Appl. Phys. 22, 1257 (1951). 43. ZENER C. Elasticity and anelasticity of metals, p. 60. University of Chicago Press, Chicago (1948). 44. ZENER C. and RANDELLR. H. Trans. A.LM.E. 137, 41 (1940). 45. ZWIKKERC. Physicalproperties of solid materials, p. 116. Pergamon Press, London (1954).