Stress Corrosion Theory of Crack Propagation With Applications … Fracture/Final... · 2018. 10....

VOL. 15, NO. 1 REVIEWS OF GEOPHYSICS AND SPACE PHYSICS FEBRUARY 1977

Stress Corrosion Theory of Crack Propagation With Applications to Geophysics ORSON L. ANDERSON AND PRISCILLA C. GREW

Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90024

The theory of stress corrosion for slow crack propagation is reviewed in the light of classical Griffith theory of fracture. Experimental data for stress corrosion cracking for glasses, ceramics, and metals are reviewed. We suggest that stress corrosion cracking plays an important role in the intrusion of magmas and in the transport of magmas upward through the lithosphere. It is shown that the effect of decreasing temperature (at progressively shallower levels along the geotherm) would be to decrease the crack velocity by several orders of magnitude if other factors were equal. We also propose that stress corrosion may be an important process in time-dependent earthquake phenomena such as premonitory behavior and earthquake aftershocks. We suggest that slow cracking in the earth is not seismically detectable but may nevertheless precede the terminal (catastrophic) phase of the fracture that is discerned as an earthquake. The seismically quiet periods before some earthquakes and the seismically quiet regions beneath some volcanoes may in fact be regimes of slow crack propagation. Slow crack propagation in a lithospheric plate may provide access routes for magmas which give rise to prominent linear volcanic chains.

CONTENTS

Introduction .......................................................................................... 77 Griffith theory and fast crack propagation ..................................... 78

Ene.rgy balance concept of Griffith .............................................. 78 Mott extension of Griffith concept: the kinetic energy term... 78 Relations between crack length and propagation velocity ....... 78 Terminal velocity ............................................................................. 79 Velocity of fault rupture in earthquakes ..................................... 79

Stress corrosion cracking: theory and experiment ......................... 80 Stress corrosion cracking: a time-dependent fracture process in

metallic and nonmetallic materials ........................................... 80 Continuum formulations in fracture mechanics ......................... 81 Theory of stress corrosion cracking ............................................. 82 Experimental data on stress corrosion cracking ........................ 90

Acoustic emission and slow crack propagation ............................. 94 Geophysical and geological applications for the theory of slow

crack propagation by stress corrosion ..................................... 96 Earthquake phenomena .................................................................. 96 Applications of stress corrosion theory to magmatic intrusion 97 Stress corrosion and geothermal reservoirs ................................ 101

Conclusion ............................................................................................ 101 Notation ................................................................................................ 101 References ............................................................................................. 102

I. INTRODUCTION

The purpose of this paper is to review some aspects of the theory of slow crack propagation by stress corrosion and to apply these considerations to several geophysical and geological processes.

The fracture process most familiar to us is irreversible catastrophic failure, in which a crack propagates with a velocity of the order of 10 • m/s. Many materials, however, under appropriate conditions undergo an earlier phase of slow crack propagation that precedes catastrophic fracture. The earlier phase consists of the subcritical slow propagation of a small flaw before it reaches the critical size for spontaneous catastrophic growth. The time of slow propagation, at velocities less than 10 -: m/s, can be very long in comparison with the time for the latter phase of rapid crack growth. The rate of subcritical slow crack growth in many materials is highly sensitive to factors such as temperature and the chemistry of the crack environment.

Stress corrosion cracking is environmentally induced subcritical crack growth, usually under applied or residual tensile stress, 'although it may also occur owing to a shear stress

Copyright ̧ 1977 by the American Geophysical Union.

[Tetelman and McEvily, 1967, p. 425]. A basic principle of fracture mechanics is that rapid crack propagation occurs when the stress intensity near the tip of a crack equals or exceeds a critical value. Stress corrosion cracking occurs in the presence of an aggressive chemical environment at stress intensities above a threshold value but below the critical stress intensity factor necessary for fast crack propagation. The chemical constituents of the environment reduce the energy requirements for crack growth by reacting with the cracking material at or near the crack tip.

Stress corrosion cracking has been most extensively studied in metals because of the need to prevent failure far below yield stress in high-strength alloys such as those used in the aerospace industry. Recently, stress corrosion has also been experimentally studied in many nonmetallic materials, such as glasses, ceramics, sapphire [Wiederhorn, 1968], vitreous carbon [Nadeau, 1974], portland cement [Nadeau et al., 1974], and quartz [Scholz, 1972]. The phenomenon called static fatigue in glass, ceramics, and quartz is caused by stress corrosion [Charles and Hillig, 1962]. Little experimental information on the existence or characteristics of stress corrosion

cracking in rocks is presently available, although time-dependent phenomena associated with earthquakes have been attributed to static fatigue [e.g., Scholz, 1972]. While a great deal of caution must be used in suggesting that stress corrosion phenomena in coarse-grained rocks should be analogous to stress corrosion in glass, metals, and very fine grained ceramics, we believe that our understanding of the process of slow crack growth in rocks can be greatly increased by applications of stress corrosion research on these other materials, just as the understanding of fracture processes in polycrystalline ceramics has been increased by results of metallurgical studies and work on single crystals and glass.

Theories of crack propagation have been developed at the continuum and the atomic levels. The continuum approach relies on continuum mechanics and classical thermodynamics, while the particle or atomic approach uses statistical mechanics. The critically important region of a propagating crack is the crack tip, where extension of the crack occurs. In the particle model the propagation of the crack is represented by the breaking of bonds between atoms at the crack tip. This separation of atoms, however, is caused by forces transmitted from distances that are very great in comparison to the bond lengths in the solid. These forces are more conveniently repre-

Paper number 6R0573. 77

78 ANDERSON AND GREW: STRESS CORROSION THEORY OF CRACK PROPAGATION

sented by continuum models. The kinetics of stress corrosion are determined by processes on the atomic scale at the crack tip. The kinetics of slow crack propagation, as studied in the laboratory, are measured in terms of stress intensity factors from a continuum standpoint.

We first review the energy balance concept of Griffith [ 1920], which, by inclusion of kinetic energy [Mort, 1948], leads to expressions for the velocity of crack propagation in a brittle material [Roberts and Wells, 1954; Anderson, 1959; Berry, 1960; Dulaney and Brace, 1960; Erdogan, 1968]. We then review theories of slow crack growth at subcritical stress intensities. Following a discussion of experimental studies of parameters influencing environmentally assisted slow crack growth in various materials, we suggest several roles that stress corrosion cracking may play in geological processes.

II. GRIFFITH THEORY AND FAST CRACK PROPAGATION

A. Energy Balance Concept of Griffith Griffith [1920] formulated the first energy balance theory of

crack propagation for the fracture of ideally brittle materials. He reasoned that a crack existing in a perfectly elastic body under applied tension is in equilibrium, that is, on the verge of extension, if the energy of the system is at a minimum. The basic notion of the concept is the first law of thermodynamics. The total energy of the system U is the sum [Lawn and Wil- shaw, 1975]

U= (--WL + Us)+ Us (1)

where the mechanical energy terms (in parentheses) are WL, the work done by loads that displace the outer boundary of the body, and Us, the stored potential elastic strain energy in the body. The term Us is surface energy, which in the case of an ideal brittle solid (with no other form of energy dissipation) is identified with the thermodynamic true surface energy of the solid being cracked. Griffith used as a model system an elliptical sharp crack extending through the thickness of a thin plate. Surface free energy is assumed to be the only form of energy dissipation. The crack has a length 2a; the plate is homogeneous, isotropic, and perfectly elastic. The two-dimensional crack has a total surface area of 4a per unit thickness (width) of the plate. There is equilibrium if for a virtual increment in crack length the mechanical energy just balances the surface energy. For the crack to advance, energy is needed to create fresh fractured surfaces, and elastic energy is released by the advance of the crack.

Since a crack of length 2a spreads symmetrically and two new surfaces are created, the two-dimensional crack has a surface area of 2a per unit thickness of the plate (or width of the crack), so the energy involved in forming the surface is

Us = 4a7 (2)

where 7 is the surface energy. In the ideal case, 7 is the true surface energy, that is, the work done in creating new surface area by breaking atomic bonds. In real materials the work of fracture (called the fracture surface energy) is orders of magnitude greater than the true surface energy of the solid.

For the case of plane strain (a thick plate) the total system energy per unit width of crack (unit thickness of the plate) is

U = [-•r(1 - ?)a•'a•'/E] + 4a7 (3)

where v is Poisson's ratio, E is Young's modulus, and a is a uniform applied tensional stress. The substitution of E for the quantity El(1 - ?) yields the expression appropriate for plane stress (a thin plate).

If we apply the equilibrium condition

dU/da - 0 (4) the familiar Griffith fracture criterion for constant load and plane strain conditions is obtained:

a = [2E7Dra(1 - ?')]•/•' (5)

This equation is a necessary condition for elastic crack propagation in that it expresses the requirement that the overall free energy of the system must be reduced for spontaneous propagation.

B. Mott Extension of Griffith Concept.' The Kinetic Energy Term

Once a crack has begun to propagate, a volume element of material near the sides of the crack must be displaced per- pendicular to the crack plane. The rate at which this material can move limits the velocity of crack propagation. Mott [1948] showed on dimensional grounds that the kinetic energy (per unit width of crack) U•c associated with the motion of the material at the crack boundary should be

U•c = •k'dt)2a2tr2(1 -- F2)2/E2 (6)

where k' is a numerical constant depending on v, d is the density of the material, and v is the crack velocity.

The kinetic energy term U•c is included in the expression for the total system energy:

u = + + u,, (7)

Thus as the crack propagates, part of the mechanical energy (applied load plus strain energy stored in the elastic medium) is used to create fresh surfaces and part is required to displace material near the tip of the crack.

C. Relations Between Crack Length and Propagation Velocity

An expression for the kinetic energy in terms of crack length can be obtained with (3) and (5) by considering the case of constant force loading in which 'the applied tension a,. is raised (quasistatically) to just beyond the Griffith equilibrium level, and is thereafter held constant as the crack extends from its initial length 2a0' [Lawn and Wilshaw, 1975, p. 94]:

UK -- •ra •'a•'(1 -- ?') 1 -- E

(8)

By equating (6) and (8) the crack velocity is expressed by

v = k' d(1- v•') ' 1 - -- (9) Thus the crack asymptotically approaches a terminal velocity at large a much greater than a0. The terminal velocity vr is

vr = [2•rE/k'd(1 - ?')]•/•'

or since the velocity of longitudinal sound waves V,• is

(10)

VL = (E/d) •/:, (11 )

the terminal velocity may be written

vr = VL[2•r/k'(1 - v")] x/z (12)

Roberts and Wells [1954], Anderson [1959], and Charles

ANDERSON AND GREW: STRESS CORROSION THEORY OF CRACK PROPAGATION 79

[1961, p. 31] used the approximating assumption that

(13)

at large a, which is valid when the velocity of the crack is very close to the terminal velocity. They applied the Griffith criterion (5) at t = 0, where t is time, and found the following equation relating velocity and crack length:

v= k'd(1 - p") 1- (14)

The terminal velocity expected from this equation is the same as that implied by (9). Anderson [1959] reported that integration of (14) gives the following relationship between time and crack length:

t=-- a 1 - -- +aotanh 1- • (15) /)T

Berry[1960], Dulaney and Brace [1960], and Erdogan [1968] consider (14) to be in error because of assumption (13). Berry [1960] derived equations of motion for crack growth, using the device of adjusting a dimensionless parameter to approach the Griffith condition for fracture. 'The crack velocity-crack length relations are complex functions which have resisted integration' [Berry, 1960, p. 216], and the attempted integration of (9) yields a result in which tl•e constant of integration is infinite. The explanation for this feature is that the Griffith condition represents an '[unstable] equilibrium, and that the sample must be subjected to a stress infinitesimally greater than the critical Griffith value before the crack begins to increase in length' [Berry, 1960, p. 198].

Erdogan [1968] followed Berry's [1960] line of reasoning and assumed that the applied load is slightly greater than the critical load corresponding to the Griffith criterion (5). He and Berry obtained similar equations for the relation between time and crack length. Erdogan's [1968, p. 515, equation 36] equation is

t = a-2-ø a--a-- 1 --- (2n 2- 1 + 2n •' /)T ao ao

1/2 a 1/ In a--a-- 1 + --- (2n •'- 1 -n •'ln(2-2n •' ao ao

(16) where

ac = ' 1 - -- (19) a a

assuming that ac = 0 when v = 0 at a = ao. The form of the dependence of crack velocity as a function of crack length for the case of constant tensile stress is shown in Figure 1.

Erdogan [1968, p. 557] relates crack velocity to the energy balance concept:

In a fracturing solid, the crack will propagate along the surface offering the least thermodynamic resistance, and the velocity of propagation at the crack periphery will depend on the difference between (a) the rate at which the work is done on the solid by the external loads and (b) the sum of the rates of stored recoverable energy, kinetic energy, and dissipative energy.

Considering the theoretical terminal velocities for crack propagation in various physical situations, Erdogan [1968, p. 558] concludes that

if the crack can be maintained to propagate along a plane [without branching], the Rayleigh wave velocity in plane and axially symmetric problems and the shear wave velocity in antiplane problems form the upper limits for the respective fracture propagation velocities ... for propagation beyond these velocities, energy has to be generated rather than dissipated at the crack periphery.

E. Velocity of Fault Rupture in Earthquakes

Rupture velocities associated with earthquakes are about 3 km/s, that is, close to the shear wave velocity Vs in crustal rocks such as granite [Johnson and Scholz, 1976, p. 883]. For the great Chilean earthquake of May 22, 1960, for example, Press et al. [1961] found that rupture velocity on a fault length of 1000-1200 km (the zone of aftershocks) was about 3-3.5 km/s.

In a mathematical analysis, Burridge [1973] considered theoretically admissible speeds for shear cracks. While he agrees with Erdogan [1968] that Vs is the only admissible speed for antiplane strain, he suggests that a crack propagating in three dimensions (in which the crack edge is a combination of plane and antiplane strain) 'may spread on a circle expanding with the S-wave speed or possibly the P-wave speed' [Burridge, 1973, p. 455].

Apparent rupture velocities higher than Vs have been measured in stick-slip experiments on rocks [Johnson and Scholz, 1976, p. 881], but these velocities 'could be artificially high,' since the 'time of rupture arrival may have been measured in a direction oblique to the direction of rupture propagation.'

a• = na n < 1 (17)

and a• is the critical load, slightly greater than a. For the Griffith criterion (5), n = 1.

D. Terminal Velocity Roberts and Wells [1954] evaluated k' in (10). For a material

with v = 0.25 they found

(2•r/k') •/•'= 0.38 (18)

According to the considerations of the Griffith theory outlined above and from (12) and (18) it is evident that once the critical condition (5) has been satisfied, the crack accelerates very rapidly to a terminal velocity about equal to 40% of the velocity of longitudinal sound waves in the material. Berry [1960] derived the following expression from (9) for the acceleration a• of the crack:

0.8

0.6

ß 0.4

0.2 n = 1

.... n<l

1 0 5 10 15

a/a o Fig. 1. Theoretical crack velocity-crack length relations for con-

stant tensile loading (adapted from Berry [1960, p. 198, Figure 2] and Lawn and Wilshaw [1975, p. 97, Figure 5.4]).


III. STRESS CORROSION CRACKING: THEORY AND EXPERIMENT

A. Stress Corrosion Cracking: A Time-Dependent Fracture Process in Metallic and Nonmetallic Materials

Many materials exhibit time-dependent fracture, that is, when they are subjected to a static load, they fail after some time interval that depends inversely on the magnitude of the load. Griffith theory, as outlined in section II, fails to account for this phenomenon, because it predicts 'that a stress only infinitesimally less than the critical value [from (5)] would be sustained indefinitely' [Berry, 1972, p. 75].

One type of time-dependent fracture is caused by stress corrosion. There is no one definition of stress corrosion cracking that is universally agreed upon; in the most general terms it is slow (or 'stable') crack growth resulting from the combined effects of corrosion and stress. Some workers have restricted the definition to static stress or more specifically to static tensile stress [Brown et al., 1972, p. 88]. For purposes of the present paper a useful definition of stress corrosion cracking is 'environment-induced subcritical crack growth under sustained stress' [Speidel, 1971a, p. 289], in which the term subcritical refers to conditions such that the stress intensity is less than the critical stress intensity factor for unstable crack propagation under plane strain loading (see below).

At present, there is no general theory of stress corrosion cracking that predicts from first principles the experimental observations on alloys and nonmetallic materials [Speidel and Hyatt, 1972, p. 282]. The basic premise of several theories for time-dependent fracture is that the 'failure process is characterized by an activation energy [such as that for the rupture of atomic bonds], and the height of the energy barrier is reduced by some function of the applied stress' [Berry, 1972, p. 77].

According to the definition used in the present paper, stress corrosion cracking may be distinguished from other forms of slow crack growth and from phenomena that are due to corrosion alone [Brown et al., 1972; Speidel, 1971a]. The definition excludes types of purely mechanical failure (independent of environment) such as fatigue (due to intermittent loading) and crack growth due to creep in the absence of corrosion. It excludes purely chemical phenomena such as intergranular corrosion. Corrosion fatigue is excluded because it involves an intermittent rather than a sustained stress. Some workers [e.g., Speidel, 1971a, p. 290] include liquid metal embrittlement and hydrogen embrittlement (both observed in alloys) within this definition of stress corrosion cracking.

Stress corrosion cracking is a term that has been used mainly in the metallurgical literature to describe the behavior of alloys in adverse environments. In glasses and ceramics the phenomenon called static fatigue (or delayed failure) is attributed to stress corrosion [e.g., Charles and Hillig, 1962]. Glass, for example, is relatively strong when it supports a load for a short time, but it is relatively weak if it is loaded for a longer time in the presence of water. A common explanation of static fatigue is that it results 'from a stress-dependent chemical reaction... which causes a preexisting flaw to grow to the critical dimensions for spontaneous crack propagation' [Ritter and Sherburne, 1971, p. 601 ].

Stress corrosion in metals can have dramatic effects: 'It seems a ridiculous result that the amount of chloride in drink- ing water can cause the cracking of a thick stainless steel section in few hours' [Staehle, 1969, p. 4]. Stress corrosion cracking was first recognized as a practical problem in metals during the second half of the nineteenth century, when cold- drawn brass cartridge cases cracked in India during the rainy

season. This cracking was caused by residual stresses in cartridge brass 'exposed to moist air containing ammonia vapor' [Tetelman and McEvily, 1967, p. 425]. Other examples of stress corrosion cracking followed in (1) brass condenser tubing used in the developing electric power generation industry, (2) riv- eted boiler steel of late nineteenth century steam locomotives, (3) stainless steels used in the paper, chemical, and petroleum industries of the 1930's, (4) aluminum alloys for the rigid airship Zeppelin, (5) magnesium alloys for aircraft in the 1930's, (6) titanium alloys for aerospace applications in the 1950's, and (7) high-strength alloys for the Saturn V launch vehicle and the lunar landing module of the NASA Apollo program [Speidel, 1971a; Brown, 1972a, b; Speidel and Hyatt, 1972, p. 305]. 'This experience confirmed the growing supposition that SCC [stress corrosion cracking] is a general phenomenon to be expected in all alloy systems' [Brown, 1972b, p. 5].

In spite of extensive research on stress corrosion in metals, 'it is still a relatively little understood phenomenon about which there is considerable ambiguity and argument' [Speidel, 1971a, p. 1]. Stress corrosion in metals, 'one of the truly insidious phenomena of metallurgical pathology' [Staehle, 1969, p. 3], is of obvious social importance to prevent the loss of vehicles in the air or at sea and to assure 'predictable long lives of alloy components in environments in which failure within a period of 20 years would result in a serious economic disaster,' for example, in nuclear power plants [Scully, 1971, p. 1].

Much experimental work has been done on stress corrosion (and static fatigue) in glasses and ceramics (reviewed by Wiederhorn [1966, 1974] and Wachtman [1974]). As in metal- lurgy this work has to some extent been stimulated by the need for practical applications. Delayed failure in glass was first described by Grenet in 1899, who, according to Wiederhorn [1966, p. 294],

observed that glass could statically support a given weight for a time and then would fail... this fact was appreciated by the French champagne makers of the time who never used the same bottle twice, although bottles were expensive, thus saving many a good bottle of wine from destruction.

More recent studies of slow crack growth in glass have con- cerned applications for space shuttle windows [Wiederhorn et al., 1974a].

Little is known about stress corrosion in rocks. Charles [1959, pp. 241-242] conducted some preliminary static fatigue experiments on granite and several silicate minerals. He concluded that 'granite is subject to considerable fatigue' and that it showed 'slow crack growth under water vapor' particularly in a 'saturated steam atmosphere' at 240øC. Charles [1959, p. 244] found experimentally that albite, spodumene, and horn- blende 'show a sensitivity to atmospheric fatigue,' as do single crystals of quartz and sapphire.

Scholz [1968a, b] suggested that static fatigue due to stress corrosion could account for creep in brittle rock at low temperatures. Scholz and Martin [1971, p. 474] postulated that this low-temperature creep (which they described as being dilatant, i.e., producing inelastic volume increases accompanied by microfracturing events) 'can be accounted for by static fatigue within individual grains, although some of the strain could be produced by the slow crack growth leading to static fatigue.'

Scholz [1972, p. 2113] reaffirmed his postulate that static fatigue is the mechanism of brittle creep in rocks and further suggested that 'static fatigue on a large scale may be the mechanism of certain similar time-dependent phenomena observed for earthquakes' and that 'triggering of earthquakes by water injection ... may involve intrinsic weakening by stress


corrosion as well as strictly mechanical pore pressure effects, as had been previously assumed.' Rice [1975, p. 1535] agreed with Scholz that time-dependent earthquake phenomena could involve 'macroscopic rock creep due to the growth of microfissures by stress corrosion cracking in the surrounding groundwater.' Anderson and Perkins [1974a] and Perkins and Anderson [1974] also considered the possible role that stress corrosion cracking might play in earthquake mechanisms and in magmatic intrusion.

Wu and Thomsen [1975, p. 173] also postulated 'stress-aided corrosion' in experiments of microfracturing in Westerly granite and suggested that these experimental studies have applications for improving mine safety, namely, that pillars in deep mines could be 'strengthened effectively ... by isolation from water.'

Westwood [1974] and Westwood et al. [1974] have reviewed another process of environment-sensitive fracture in rock, that of the dramatic increased speeds in drilling rates achievable by manipulation of the chemical environment during rotary dia- mond drilling. If n-alcohols and cationic flotation agents are added to the drilling medium, rates of drill penetration in granite can be increased manyfold. This phenomenon is attributed to 'adsorption-induced changes in near-surface dislocation mobility,' although 'attempts to reveal the extent of dislocation motion around indentations ... have been so far unsuccessful' [ Westwood et al., p. 111 ]. Freiman [1974, 1975a, b] has shown that slow crack propagation velocities in glass immersed in straight chain alcohols or alkanes depend on the water content in the alcohols or on the chain length in the alcohols and alkanes; he states that the crack growth in both alcohols and alkanes 'is apparently controlled by the same stress corrosion-diffusion mechanism' [Freiman, 1975a].

Grew [1976] has proposed that stress corrosion cracking may play an important role in mineralization, particularly in the formation of porphyry copper deposits. B. Continuum Formulations in Fracture Mechanics

Stress concentration versus Griffith energy balance. In consideration of stress corrosion cracking, attention is focused on processes at or near the crack tip. A second condition for crack propagation, in addition to the Griffith energy condition (5), is that local conditions at the atomic level at the tip of the crack must permit the breaking of atomic bonds. Inglis [1913] showed that local stresses near a notch tip could 'rise to a level several times that of the applied stress' [Lawn and Wilshaw, 1975, p. 2].

The elastic stress concentration factor for an 'infinitely' sharp crack is [Tetelman and McEvily, 1967, p. 48]

K,• = 2(a/p )•/• = O'max/O' (20)

where K• is the stress concentration factor, 0 is the radius of curvature at the tip of the crack (the 'root radius'), and amax is the maximum tensile stress at the tip of the crack. This concen- trated stress may be set equal to the theoretical stress required to break bonds at the crack tip in order to solve for the applied force needed for fracture.

As Wachtman [1974, p. 509] has noted, it has been debated whether the Griffith energy condition and the condition of stress concentration at the crack tip are independent or equivalent criteria for fracture. In fracture mechanics, crack propagation is generally treated in terms of a critical stress intensity factor near the crack tip rather than by the Griffith energy concept [Wachtman, 1974; Evans, 1974a, b]. Although the first law of thermodynamics must be satisfied during crack growth, as in the Griffith theory, it is not clear how the internal energy

of the system changes during crack propagation. As Evans [1974b, p. 20] states,

it is evident that the surface energy ... must increase and that there is an energy change due to plastic work at the crack tip... but it is not known whether energy is dissipated as heat, acoustic energy, etc. which should be included in the total energy balance. Therefore, it is only possible to express the fracture condition in terms of an empirical critical 'fracture surface energy' ... which may contain certain unknown energy contributions.

One of the unknown contributions may be from the thermal energy of the solid, as was suggested by Poncelet [1965]. In fact, Poncelet challenges the basic concept of Griffith as given by (1) and (7) as being incomplete because it amounts to the assumption of isothermal behavior of the freshly cleaved surface. Poncelet [1965, p. 11] argues that

it is... erroneous to assume that all of the surface energy of the newly formed surfaces is produced isothermally from the strain energies stored in the body; at the beginning of a fracture, a considerable proportion of it is provided from the thermal energy of the body.

This is the statistical mechanical viewpoint. The breaking of bonds which are not reformed supplies the heat of sublimation of the solid (per broken bond) to the surface and kinetic energy terms. Poncelet [1965, p. 53] argues that the 'surface energy of bodies is a free energy, not a potential energy, as stated by Griffith' and that what is missing in the Griffith hypothesis is the S dT thermodynamic term (where S is entropy). He argue,s that newly formed surfaces are always cooler than the original body and hence dT is negative for a positive increment da in crack length. The inclusion of the S dT term, in Poncelet's view, eliminates the requirement for a preexisting small flaw (implicit in the Griffith theory) for crack propagation.

Stress intensity factor. The stress intensity factor [Irwin, 1958; Lawn and Wilshaw, 1975, pp. 52-66] describing the intensity of the local stress field for an infinitely sharp elastic crack is

Kx = a(a•ra) a/2 (21)

where a is a dimensionless parameter depending on the specimen and crack geometry and I refers to mode I fracture, which is defined as the opening mode with separation of crack walls under tensile stress. (Some workers believe that stress corrosion cracking only occurs in mode I fracture.)

Theoretically, the critical stress intensity factor is proportional to the strength of the material and may be used 'as an engineering parameter to evaluate the relative strength of different materials' [Wiederhorn, 1969a, p. 101]. For fast fracture, as is described in section II, there exists a critical stress intensity factor Kxc above which rapid unstable crack propagation occurs. Stress corrosion cracking is slow crack growth at stress intensities less than K•c, with velocities typically less than 10 -• m/s [Evans, 1974b, p. 20]. Much of the lifetime of a stress corrosion specimen is therefore spent in the slow growth region [Speidel and Hyatt, 1972, p. 197].

Crack extension force and fracture surface energy. In fracture mechanics the Griffith energy concept has been extended to account for the behavior of real materials. The theory incorporates a critical parameter analogous to the Griffith surface energy that characterizes the resistance to fracture in a real material and provides a means for incorporating non- linear dissipative.terms in the potential energy term.

The crack extension force (the mechanical strain energy release rate per unit width of the crack front) is defined as [Lawn and Wilshaw, 1975, p. 50]


G = -d(-W,• + Ue)/da (22)

For plane strain (a thick plate) the fracture mechanics expression for the variation of mechanical energy in the crack system (in its simplest form) is [Tetelman and McEvily, 1967, p. 53]

G = Kx2(1 - v2)/E (23)

If a dissipative component is considered in the fracture process, the fracture surface energy may be defined as [Lawn and Wilshaw, 1975, p. 79]

-2r = -dUs/da (24)

At Griffith equilibrium 'the so-called Irwin-Orowan general- ization of the Griffith criterion' [Lawn and Wilshaw, 1975, p. 79] is

= 2r (25)

The crack extends if G > 2F.

If there is no dissipative component in the work of creating new fracture surface,

F = 'y (26)

where 'y is the reversible surface energy of an ideal brittle solid. Cohesive zone models. More sophisticated treatments of

nonlinearity near the crack tip are outgrowths of the cohesive zone model of Barenblatt [1962] as developed by Willis [1967], Rice [1968a, b, 1974], and Smith [1974]. According to the cohesive zone model the crack system can be divided into two zones: (1) a 'relatively small, inner zone immediately surrounding the crack tip' in which 'the non-linear, separation processes operate' and (2) the surrounding outer zone in which the linear elastic material 'serves the [unction of a medium for

transmitting the applied forces to the inner regions' [Lawn, 1975, p. 470]. Barenblatt [1962] argued that the nonlinear crack tip zone must extend over several atomic spacings along the crack interface, and he proposed that cohesive forces in this zone hold the two sides of the crack together. The region of the cohesive zone is thus very small in comparison to the crack length. The work of separating the material at the crack tip is expressed in terms of a 'modulus of cohesion,' which is related to an intrinsic surface tension.

Willis [1967] showed that the Barenblatt cohesive zone model is essentially identical to Griffith theory. Rice [1968a] further confirmed this equivalence and developed a path-independent J integral to approximate the strain concentration near the crack tip (assuming that nonlinearity is confined to a small region immediately surrounding the crack tip). Rice [1968a, p. 381] showed that the integral J is equivalent to the crack extension force G under appropriate assumptions.

Smith [1974, p. 221] has pointed out that in brittle crystalline solids 'the cohesive zone model gives the zone R to be of the same order as the distance b0 between atoms,' so that a limitation is placed on this model, since 'the model is essentially of the continuum type and stresses should not vary markedly over atomic dimensions.' Therefore Smith [1974, p. 221] suggests that it would be more appropriate to 'conduct calculations using models in which a finite region in the vicinity of a crack tip contains discrete atoms that are subject to an appropriate interaction law.'

The logical extension of the modulus of cohesion is the consideration of cohesive forces between atoms. One approach to crack propagation at the atomic level is that offered by Chang [1969, p. 309], who considered crack propagation in

terms of an 'atom cohesion parameter' instead of the Griffith surface energy. The relationship between Chang's [1969, p. 309] atomistic approach and the Griffith continuum theory is that

the atom cohesion energy parameter ... is directly calculable from the interatomic potential, the fracture toughness of a given material according to the atomic model is then directly proportional to the product of the atom cohesion energy parameter (per unit interatomic advance of the crack front) and the distance between the subcrack and the main crack.

The subcrack in Chang's model is a nucleation step that forms some distance from the main crack and then joins it as the crack propagates.

C. Theory of Stress Corrosion Cracking Stress corrosion cracking and the Griffith energy balance

concept. The Griffith theory may be extended to account for stress corrosion cracking if for example, it is postulated that 'surface active materials in the environment ... reduce the

fracture surface energy to a critical value' [Berry, 1972, p. 76]. Thus part of the energy required to form a new surface is supplied by a chemical reaction [Wiederhorn, 1969a, p. 103]. In such a model the rate of chemical reaction or the rate of

migration of chemical agents to and from the crack tip confers the time dependence observed in the fracture process.

The usefulness of introducing F is that the crack can be considered in terms of Griffith theory, that is, the crack is in thermodynamic or 'Griffith' equilibrium when 'for a virtual increment in area, the work done by applied forces transmitted to the [crack] tip just balances the increase' in the fracture surface energy, which corresponds 'to a reduced, interfacial value for the solid in contact with a chemical environment'

[Lawn, 1975, p. 470]. Speidel [1971a] and Speidel and Hyatt [1972] have reviewed

several theories developed to account for stress corrosion cracking in metals. The hypotheses may be divided into two groups, depending on the postulated site of reaction of the damaging species that causes a reduction in the energy requirements for crack growth. The two groups are ( 1 ) those hypotheses that invoke damaging species reacting at the crack tip and (2) those that invoke damaging species reacting in the interior of the material being cracked. The hypotheses that have generally been put forward to explain stress corrosion in ceramics and glass belong to the first group.

The overall process at the crack tip may consist of several steps according to Speidel and Hyatt [1972, p. 283]:

(1) transport of reactants to the surface, (2) adsorption of the reactants, (3) reaction on the surface, (4) desorption of the products, and (5) transport of the liberated products from the surface into the bulk environment.

It is not clear which is the rate-limiting step or whether any one step is rate limiting in most cases.

General experimental observations. Since many theories for stress corrosion cracking have been empirically derived to explain experimental data, it is appropriate in this section to consider first some of the general experimental observations of stress corrosion cracking before reviewing the major theories.

There are three general methods for presenting experimental data for precracked stress corrosion specimens [Blackburn et al., 1972, pp. 275-277]. Figure 2 is an example of a static fatigue diagram in which the breaking strength is plotted versus the logarithm of the time to failure. Alternatively, the ordinate may be plotted in terms of the stress intensity factor K• or the ratio Kic/gi. In Figure 2 the nearly horizontal


IS i [ • PORCEL•~ ,.

• O PORCELAIN

- o

=11 ........

8

-I O I 2 3 4 Lo9 time (sec.}

Fig. 2. Static fatigue curves for porcelain. Experimental points are from Baker and Pres•n [1946]; curves are those calculated from (31) when appropriate constants are selected. Reproduced from Stuart and Anderson [1953, p. 421, Figure 2].

portions of the curves for long times to failure suggest the existence of a static fatigue limit, or a value of the stress intensity factor below which stress corrosion cracking does not occur.

Figure 3 is a far more commonly used graph for displaying stress corrosion cracking data. The logarithm of the stress corrosion crack velocity is plotted against the stress intensity factor. Laboratory measurements on many metallic and nonmetallic materials yield the general curve shape shown in Fig- ure 3. Following Wiederhorn [1967] the three main regions of the curve are labeled I, II, and III. Experiments on alloys have shown that 'there are also transitions between the regions' [Speidel and Hyatt, 1972].

In region I there is a linear relation between the logarithm of the crack velocity and the crack tip stress intensity. One interpretation of region I is that the crack propagation is 'controlled by the kinetics of a chemical reaction between the solid and the environment that fills the crack' [Speidel and Hyatt, 1972. p. 286]. In the plateau region, region II, the crack velocity is independent of stress intensity. One explanation for the existence of the plateau is that mass transport (e.g., diffusion) of corrosive reactants to the tip of the crack is the rate-limiting factor. In this region, which corresponds to higher crack velocities than does region I, the crack cannot propagate faster than the maximum velocity at which the damaging species can move to the crack tip.

Region III corresponds to very high fractions of the critical stress intensity Kxc. Wiederhorn [1967, p. 412] stated that 'no satisfactory explanation has been found for crack propagation in region Ill.' It has been suggested that the kinetics of fracture in this region represent 'a mixture of corrosive and mechanical failure' [Evans, 1972, p. 1137; Hodkinson and Naareau, 1975, p. 855]. This region thus shows only slight departures from fast fracture as discussed in section II. It will not be discussed further here.

Not all of the three regions are observed during slow crack growth in a given material in a specific environment. For example, graphite does not show regions I and II [Hodkinson and Nadeau, 1975, p. 885], and region II is absent for commercial aluminum alloys [Speidel, 1971a, p. 301]. Region III is missing for polycrystalline alumina in air with 50% relative humidity [Evans, 1972, p. 1143].

Since it is very difficult to prove the existence of zero crack growth under given experimental conditions, there is some doubt about the existence of a threshold stress intensity factor for stress corrosion cracking Kxscc in some materials. Such a threshold appears not to exist in aluminum alloys [Brown, 1972b, p. 9], although 'the kinetics [of stress corrosion crack propagation] may simply decrease continuously to ever smaller but finite values as K is decreased.'

In the literature on subcritical crack growth and static fatigue it is often assumed that there is a static fatigue limit, or that there is a threshold stress intensity factor Kxscc (Figure 3) below which stress corrosion cracking does not take place. Shand [1954a, b] has reviewed the indirect experimental evidence, which supports the belief that there exists a static fatigue limit for glass. ß

The existence of the K•scc threshold is difficult to prove experimentally, and it has been said that 'minimum [crack] growth rates depend primarily on the patience of the observer' [Speidel and Hyatt, 1972, p. 144]. Crack propagation in alloys has been measured at velocities as low as 10 -• m/s, or a fraction of a centimeter per year [e.g., Speidel, 1971a, p. 300]. Such values take years to measure, and it is unlikely that laboratory measurements can be made at much slower rates. Speidel and Hyatt [1972, pp. 145-146] have proposed that for practical purposes the K•scc determined for engineering be arbitrarily defined as

stress intensity at which the stress-corrosion crack-tip velocity reaches 10 -8 cm/sec, or about 1.4 X 10 -5 in./hr. This is the lowest crack-tip velocity that can be conveniently measured within 30 days. It also happens to correspond to the crack growth rate that one would observe if everywhere along the crack front [in an aluminum alloy] one atomic bond would fracture every second.

A third type of plot for static fatigue data is that of the universal fatigue curve [Mould and Southwick, 1959] in which the ratio at/a•c is plotted versus the ratio xP/XPo.s, where at is strength, a•c is the stress needed for fracture in an inert environment, xp is time to failure, and XPo.5 is time to failure at 50%

KiSCC

!

Kic

STRESS INTENSITY

Fig. 3. Generalized representation of effect of the stress intensity factor on kinetics of stress corrosion cracking. The ordinate scale has been adjusted; the origin represents zero crack growth rate. Some systems show an apparent zero growth rate at a certain limiting value of K denoted K•scc [from Brown, 1972a, p. 10, Figure 3; Evans, 1972, p. 1137, Figure 1!.


of the stress O'Ic [Evans, 1972, p. 1143]. Evans noted that the time to failure at constant stress can be evaluated from an integration of the K-v diagram'

f O-1½. ,I, = •/a (27)

Figures 4 and 5 show the relationship between these two graphical means of displa.ying the stress corrosion data. 'The rapidly varying region at small •/•o.5 is due to region II of the K-v diagram and the less stress sensitive region at larger •/•o.5 is due to region I' [Evans, 1972, p. 1143].

In alloys, stress corrosion cracking is considered to be 'a nucleation and growth process' [Brown, 1972b, p. 4]. In the absence of a preexisting flaw, there is an initial nucleation step of corrosion pit formation in which the primary function of pitting is to 'permit establishment of a local environment conducive to cracking.' For alloy steels in seawater, for example, the 'rate of growth of stress corrosion cracking is faster than it is for pitting by a factor of about 11Y, and fast fracture [described in section II] propagates at about 10 •ø times faster than stress corrosion cracking' [Brown, 1972b, p. 4].

In the present paper we consider only the case of precracked material. In this section we discuss theories to explain the kinetics of stress corrosion cracking in regions I and II (Figure 3). Region III may be treated largely in terms of our discussion in section II. It should be noted that in the examples discussed in the present section the crack is assumed to be immersed in a fluid environment, but no fluid was assumed to be in the crack in the discussion in section II.

i0-1

10-2

10-3

10-4

io -5

•0-6

10 '7

10-8

10-9

Io-•O

i i I I i

{ Kic I

ol ø ß'"'e"e'e I

i ß I I

o o! • I I

/o ' i i ß I I •

? , I

3 4 5 C 7

o AIR 50% I RE•'ATIVE HUMIDITY, 25øC I

ß TOLUENE, Z5øC I

I I

o/o I

KI(Nm'3/2 x I0 '6) Fig. 4. Crack velocity and stress intensity factor data for poly-

crystalline alumina in moist air with 50% relative humidity and in Toluene. Reproduced from Et)ans [1972, p. 1143, Figure 7].

1.0 ' 1 i

O. 9 ••'•• 0.7

00.6

b' 0.5

0.4

0.3

0.2

. DATA FROM BURKE ET AL - •CALCULATED FOR TESTS IN

AIR, 50% HUMIDITY - ---CALCULATED FOR TESTS IN

TOLUENE _

0.1 I i I I I I 1 I0 '•ø !0 'e I0 '6 I0 '4 I0 'a I I0 z 10 4

Fig. 5. Times to failure for polycrystalline alumina, calculated by an integration of the K-t) diagram (Figure 4) and plotted as a universal fatigue curve. Data points shown were obtained with a similar system by Burke et al. [1971]. Reproduced from Et)ans [1972, p. 1143, Figure 81.

The terminal crack velocity in the theory of fast fracture is controlled by the velocity of sound in the solid being cracked. In stress corrosion cracking the limiting crack velocity is controlled by diffusion and viscous processes in the fluid that determine its transfer to the tip of the crack. If the crack extends by an increment, further rupture cannot take place until the fluid in the crack has moved by the same increment and reaction products have moved in the opposite direction.

Empirical equations for stress corrosion crack velocity. Crack growth in region I is strongly dependent on stress intensity. The K-v curve in region I fits an empirical equation of the form [Speidel, 1971a; Speidel and Hyatt, 1972]

v = v, exp c, gi (28)

in which v• and el are experimentally determined constants. Laboratory measurements 'are not accurate enough to exclude definitely a Ki 2 dependence on the logarithm of the crack velocity in region I' [Speidel, 1971a, p. 300].

Stress corrosion crack velocities in region I, measured at various temperatures, fit the following empirical equation [ Wiederhorn and Bolz, 1970, p. 545]:

v = Vo exp (-E* + bKx)/RT (29)

where v0, b, and E* (the apparent activation energy) are experimentally determined constants. Crack growth is i'ndependent of stress intensity in region If; in the plateau region the dependence of crack velocity on temperature can be approximated by [Speidel and Hyatt, 1972, p. 195]

v = v2 exp (-E*/RT) (30)

where v• is another experimentally determined constant. In general, region I is characterized by a strong dependence

of crack velocity on stress intensity and by a high apparent activation energy. Region II kinetics are independent of stress and have a low apparent activation energy, which might be explained by a mass transport process in which the plateau velocity is proportional to the concentration of damaging species at the crack tip.


An advantage of the K-v plot over the time-to-failure test for static fatigue is that it becomes possible to distinguish between kinetics of region I and those of region II. A time-to-failure test for static fatigue integrates, according to Speidel and Hyatt [1972, pp. 196-197],

over regions I and lI of the K-v curve. It would be expected, therefore, that TTF (time-to-failure) tests would yield apparent activation energies somewhere between 3.8'and 27 kcal/mole (for aluminum alloys), but biased toward the larger value because most of the lifetime of a smooth stress-corrosion specimen is probably spent in the slow-growth region. Historically, activation energies measured with TTF tests indeed follow that description.

Stuart and Anderson equation fnom reaction rate theory. Stuart and Anderson [1953] considered the problem of static fatigue in glass. They derived an expression for crack velocity in terms of the net rate of breaking chem'ie, al bonds dN/dt, where N is the number of bonds. They assumed that the forward rate of fracturing bonds is determined by the energy of the applied load and by f*, the average free energy of activation per particle for the fracture process, while the backward rate of repairing bonds is determined by the energy of the applied load and by f* minus fs, where f8 is the effective average surface energy per particle (an interfacial energy between the glass and the atmosphere). Stuart and Anderson proceeded from the energy balance concept of Griffith, assuming that a crack is not propagated until the elastic strain energy exceeds the surface energy. Since in the continuum approach, 'strain energy' means the potential energy of strain per unit volume and 'surface energy' means the potential energy of surface formation per unit area, it was necessary for Stuart and Anderson [1953, p. 419] to consider 'the dimensions of strain energy per particle and surface energy per particle as... equal and therefore different from those found for strain energy and surface energy, respectively, in continuum physics.' It should also be noted that the Stuart and Anderson formulation differs from the Griffith expressions by a numerical factor in the surface energy term, since the surface energy is expressed per particle rather than for a crack of length 2a. The Stuart and Anderson formulation was developed for application to glasses rather than to crystalline solids.

The Stuart and Anderson equation is

I ( dN = (-AF* + F•/2) exp dt exp RT 2NkT 2

- exp 2 NkT + (31)

where k is Boltzmann's constant, T is absolute temperature, h is Planck's constant, 3, is the average distance that particles must separate to go from the initial or unfractured state to the fractured state, AF* is the average free energy of activation per mole of particles in the fracture process, F• is the effective average surface energy per mole of particles, R is the gas constant, and c is a stress concentration factor (the constant of proportionality assumed to exist between the average macroscopic stress a and the stress near the fracturing bond, so that ca is the actual or real stress near the fracturing bond).

The justification for the inclusion of the term k T/h comes from the theory of reaction rates and in this case is assumed to arise from inclusion of the rates of flow over the energy barrier in the forward and reverse directions [Stuart and Anderson, 1953, p. 419]. In more recent formulations this type of rate theory has been applied to crack propagation in lattice trapping models (see below).

The Stuart and Anderson equation can be rewritten in a simplified form in terms of potential energy (PE) and surface energy (SE):

X dN XNkt (-AF* +(SE/2))2sinh( P•-•_SE ) v= d•- h exp RT ..... (32)

The strain energy is distributed equally in the forward and reverse rates, the hyperbolic sine law thus resulting.

Stuart and Anderson [1953] showed that through integration, asymptotic expansions, and choice of appropriate empirical constants, (31) can be reformulated as a general equation that can be made to fit experimental data for static fatigue (Figure 2). In particular, they considered two special cases: (1) large stress and short breaking time (e.g., less than 1 s in' pyr/:x'glass) ani:l (2) small stress and very long breaking time. The resulting equations are given below. For strain energy per particle that is large in comparison to the surface energy per particle (short breaking time),

(caX 2•7•)_1n caX lnt =- 2Nok T 2Nok T

kT (-AF* F• ) (33) - In -•- exp R-•-- + 2-•--• For the' case in which strain energy per particle is small in comparison to the surface energy (very long breaking time),

2Nok T/ 2NokT 2kT - In 4kT

kT (- AF* F• ) (34) - In -•-exp R T + 2-•-• In (34), strain energy is small, and t • m as strain energy

approaches the surface energy. This equation therefore predicts a static fatigue limit.

It should be noted that in the Stuart and Anderson expressions the potential energy term is proportional to the first power of a rather than to the second power, as will be found in certain other theoretical treatments of slow crack growth. For purposes of comparison to other equations for static fatigue, (33) for short times can be simplified, by rearranging terms, to

lnt-• -A• - B•a - lna (35)

where A• and B• are constant positive parameters evaluated by separate experiments.

Other theoretical treatments of static fatigue. Another equation to fit data for static fatigue was proposed by Elliott [1958, p. 225]. His equation, rearranged for purposes of comparison, is

ln t= (A2/a 2) - B2 (36)

where A• and B• are constants. Elliott reasoned that the slow growth rate of a Griffith crack in a corrosive atmosphere is controlled by diffusion of the corrosive agent through 'fissured corrosion products' to the root of the crack. H e then combined the Griffith criterion for fracture (5) with an equation for the growth law of oxide films to obtain an equation equivalent to (36). It should be noted that (36) represents integration of an equation of the general form

v = Vo exp (-PE) (37)

in which the potential energy term depends on the second power of stress instead of the first power, as it did in the Stuart and Anderson formulation (35).


Charles [1958] developed an equation for static fatigue in rods of soda-lime glass. He postulated that the static fatigue was caused by a corrosive mechanism in which water from the atmosphere reacts with cations (such as Na +) in the glass and thus causes hydrolysis and an increased concentration of hydroxyl ions. He assumed that the flaw growth process had a simple Arrhenius dependence on temperature. Charles [1958, p. 1557] assumed that the corrosion rate conforms 'to an arbitrary power function of stress,' so that the penetration velocity of the crack is

v = k"(a.,) '• + k ø (38)

where a,• is the tensile stress at the crack tip, k ø is the corrosion rate of the material under zero stress, and k" is a constant. Charles [1958, p. 1560] determined experimentally that n = 16 for delayed failure in soda-lime glass. The Charles [1958] equation may be expressed for purposes of comparison with other static fatigue equations as

In t -• -As - Ba in a (39)

where As and Bs are constants, Bs corresponding to n. Poncelet [1948] earlier obtained an expression for static fatigue equivalent to Charles's (39). Poncelet assumed flaw genesis under stress rather than a preexisting flaw, which is required by the Griffith theory.

Mould and Southwick [1959, p. 590] reviewed the Stuart and Anderson, Elliott, and Charles equations for static fatigue. It should be noted that there is a misprint on page 590 of the Mould and Southwick article in that the Stuart and Anderson

equation is erroneously shown to contain the term b/a instead of ba. The equation appears correctly elsewhere in the Mould and Southwick paper, but the misprint has subsequently been copied in the literature [e.g., Schmitz and Metcalfe, 1966, p. 1].

MouM and Southwick [1959, p. 590] showed that of all the previous equations for static fatigue, Stuart and Anderson's (31) best fit the universal fatigue curve for abraded glass (Fig- ure 6). They noted that the complete expression of the Stuart and Anderson equation was necessary to fit the curve because the value of (33) is an asymptotic expansion for large stress and short time. They criticized the Stuart and Anderson theory

because as the time to failure becomes indefinitely short, the stress increases indefinitely according to (35), while real material has an instantaneous strength. Mould and Southwick preferred the Charles equation to fit the experimental data over a greater range of times to failure because of its behavior at vanishingly small elapsed time.

In summary, the problems inherent in the various theories of static fatigue discussed by Mould and Southwick [1959] include the following questions: (1) whether the stress is finite at vanishingly short times (the instantaneous strength), (2) whether there is some stress limit below which the solid will

never fail even at indefinitely long times (the existence of the static fatigue limit), and (3) what the value of the exponent for stress in the potential energy terms is. Mould and Southwick [1959, p. 591] suggested that a complete theory for static fatigue should include treatment of (1) 'the stress-dependent interaction between the glass and an attacking medium' and (2) 'the availability of the medium at the tip of the crack.' They observed that the Charles (38) only considers 'the dependence [of the interaction] on the stress,' while the Elliott equation only considers the 'availability of the medium,' i.e., the rate controlled by the diffusion of the corrosive agent through fissured corrosion products.

The Charles and Hillig equation. Charles and Hillig [1962] developed a theory of static fatigue in brittle amorphous solids (glass) which was later extended by Hillig and Charles [1965] to brittle crystalline solids. The Charles and Hillig equation was derived by assuming that (1) the reaction rate is not controlled by the transport of reactants to the reaction site, (2) the reaction between the solid and its environment is stress dependent, and (3) the activation energy is derived from an Arrhenius plot of the corrosion rate versus temperature and is a function of the tensile stress [Charles and Hillig, 1962, p. 514]. The geometry of a preexisting crack (namely, its root radius) is assumed to be altered by the corrosive reaction; sharpening of the crack finally results in rupture.

The Charles and Hillig [1962, p. 515] equation is

v' = v0 exp (-E '• + aV '• - '¾VM/p)/RT (40)

where v' is the local velocity of corrosion normal to the inter-

1.4

12

1.0

0.8

0.4

(12

4 -3 -2 -I 0 I 2 .3 4 5

LOG •o (t,/, t, o.5)

Fig. 6. Comparison of universal fatigue curve with various theoretical treatments. Experimental points with vertical error bars were measured on abraded glass by Mould and Southwick [1959, p. 588]. Curve A is the theory of Taylor [1947], curve B is (35) [Stuart and Anderson, 1953], curve C is (36) [Elliott, 1958], and curve D is (39) [Charles, 1958]. Reproduced from MouM and Southwick [1959, p. 590, Figure 8].

ANDERSON AND GREW; STRESS CORROSION THEORY OF CRACK PROPAGATION 87

crack interface; Vo is the preexponential kinetic factor; E s is the zero-stress Arrhenius energy (the stress-free activation energy), including terms expressing the chemical potential difference and energy barrier for reacting species; •, is the interfacial surface energy between the glass and the reaction products; V• is the molar volume of the glass; p is the radius of curvature of the crack tip; -•,V•/p is the molar free energy contribution associated with the curvature of the surface; o. is the crack tip tensile stress; and V s is the activation volume.

The term V s, V s = t9 AE s / t9 o. (41 )

is called an 'activation volume' by 'analogy to terms ... observed for the effects of hydrostatic pressure on various kinetic processes' [Hillig and Charles, 1965, p. 4]. It is assumed to be a negative quantity for tension.

In reviewing several theories of static fatigue, Wiederhorn [1966, pp. 303-304] observed that the Charles and Hillig equation 'explains the existence of a static fatigue limit' and ac- counts for the dependence of static fatigue on temperature and the environment.

The Wiederhorn and Bolz equation. Wiederhorn and Bolz [1970] expressed the Charles and Hillig equation in terms of the fracture mechanics parameter Ki, assuming that for a two- dimensional Griffith crack

o. = 2K•/Orp )•/2 (42) and

E* = E s + •V•/p (43) so that Charles and Hillig's (40) can be modified to [Wieder- horn and Bolz, 1970, p. 548, equation 9]

v = v0 exp [-E* + 2VSK•/Orp)•/2]/Rr (44) 'At stresses greater than the fatigue limit, crack sharpening occurs, and p decreases to a small value limited by the structure of the glass' [ Wiederhorn and Bolz, 1970, p. 548]. Equation (44) is identical to empirical (29) if

V • = (b/2)Orp) •/• (45)

is very similar in form to the Stuart and Anderson equation in the special case of (47).

Equations (29) and (44) are now widely used in discussions of region I kinetics of stress corrosion cracking. The predic- tions of the theory in (44) are in good agreement with experimental data for region I. Equation (44) predicts that 'the slope of region I in the V-K curve [Figure 3] should decrease as the radius of the crack tip increases,' and experimental observations in metals have indirectly confirmed this: there is a 'reduced stress dependence of the crack tip velocity in alloys of reduced yield strength because of the correspondingly enhanced plastic relaxation' [Speidel and Hyatt, 1972, pp. 286-287].

Slow crack growth in vacuum. As Wachtman [1974, p. 510] has noted, not all subcritical crack growth in glass is due to stress corrosion in aggressive environments. Subcritical crack growth has been experimentally studied in vacuum (<10 -4 tort) by Wiederhorn et al. [1974b], who, citing the Stuart and Anderson [1953] theory, explained this slow crack growth in terms of thermally activated processes such as those used to explain the time-dependent strength of brittle materials. It should be noted that this effect was predicted by Poncelet's [1948, 1965] theory. If the bond rupture is considered to occur only at the crack tip, the stress at the crack tip can assist in overcoming the activation barrier for crack growth, and the resulting equation for crack growth from reaction rate theory is

v = v0 exp (-AGS/RT) (48)

where AG s is the change in free energy of a bond going from the initial to the activated state for fracture. Wiederhorn et al. [1974b] indicate that this change in free energy can be expressed in terms of an activation enthalpy (usually referred to as the activation energy) AH s, an activation entropy AS s, and an activation volume A V s, so that

AG s = AH s + PAV s - TAS s (49)

where P is the pressure at the crack tip, which can be related to the applied stress in the case of an elliptical crack by

Equations (29) and (44) for region I of the K-v plot (see Figure 3) both involve a dependence of the logarithm of the crack velocity on a linear, rather than a quadratic, term in stress. Creager and Paris [1967] showed theoretically by calculation of the elastic stress field for blunt cracks that for acceleration of stress corrosion cracking to occur, the reaction rate should depend on stress more strongly than by a one-half power law.

We note that the Stuart and Anderson equation in the case where the surface energy term is so small that it may be neglected is similar to the Wiederhorn and Bolz equation. If we start with (32), because sinh x -• e x for large x (small surface energy),

Nkr -AF* + (SE/2) 2 exp PE - SE (46) v = • exp RT 2RT or

2XNkT v- h exp (-AF* + PE/2)/RT (47)

In the original form of the Stuart and Anderson (31), PE has the form co.XA/Nk T, where co' is analogous to the K• term and XA has the dimensions of volume. Thus although Wiederhorn and Bolz's (44) is derived from quite different assumptions, it

P = -t K,/(•rp)•/• (50)

so that with the entropy term included in the preexponential v0 the equation is

v = v0 exp [-AH s + {K•VS/Orp)•/•]/RT (51)

This equation, which is similar to (44), fits the experimental data on a number of glasses exhibiting subcritical growth in vacuum at temperatures from 25 ø to 400øC. An example of data from Wiederhorn et al. [1974b] is shown in Figure 7.

Wiederhorn et al. [1974b] discounted alkali ion diffusion as the cause of the slow crack growth, because the activation energy for crack motion (60-175 kcal/mol) is far less than that for sodium diffusion in glass (15-35 kcal/mol). Viscous processes were also discounted except for temperatures above 400øC, at which they can account for the observed cessation of crack growth. In addition, these authors noted that glass fractures in a brittle manner at temperatures nearly as high as the glass transition temperature.

Theory of chemical controls on stress corrosion cracking. Wiederhorn [1972] derived an equation similar to Charles and Hillig's (44) to account for the dependence of crack kinetics in region I on the concentration of hydroxyl ions [OH-]. 'It is believed that hydroxyl ions attack the glass network, causing


10'5

10'6

10. 7

10-8

I I

352øC

300øC 200oc 107øC

147øC 24øC

I I 0.5 0.6

STRESS INTENSITY FACTOR, MN/m '3/2 Fig. 7. Effect of temperature on subcritical crack propagation in

61% lead glass in vacuum (<10 -5 torr). Modified from Wiederhorn et al. [1974b, p. 337, Figure 2].

siloxane bond cleavage. This supposition ... is supported by the susceptibility of glass to corrosion by solutions with high pH' [ Wiederhorn, 1972, p. 81 ].

The rate of chemical reaction can be related to [OH-] and to the effective concentration of siloxane bonds [C] at the crack tip by [Wiederhorn, 1972, p. 84, equation 4]

v* = k._• [OH_],•,[C ] exp (- AG•/RT) (52)

where n' is the order of the rate-limiting chemical reaction with respect to [OH-] and AG * is the change in free energy as the chemical reaction goes from the initial to the final state. This equation is found by letting the free energy include the energy due to a chemical reaction of order n', so that

AG = AG • + n'RT In [OH-] (53)

Such a procedure is allowed by any statistical mechanical theory involving thermal activation, such as the Stuart and Anderson [ 1953] theory.

The free energy of activation AG * is

AG • = AE • -TAS • + PAV '• + VM3'/P (54)

where P is the hydrostatic pressure at the crack tip. For a state of plane stress,

AG • = AE • -- TAS '• - (aVe/3) + (VM'y/p) (55)

and if the temperature dependence of the hydroxyl ion concentration is

[OH-] = A' exp (-AH/RT) (56)

where A' depends on the concentration of the reactants and AH is the enthalpy of reaction, and if the crack velocity v is proportional to the rate of reaction, the crack velocity is [Wiederhorn, 1972, p. 84, equation 8]

v = B[OH-] '• exp [-AE • + (aAV•/3)

-- (VM'•/O)]/RT (57)

Martin's [1972, p. 1414] experimental data on the rate of slow crack growth in quartz showed that the rate depends on the partial pressure of water in the atmosphere surrounding the crack. His experimental results were for the most part in agreement with the general equation developed by Wiederhorn [1969b]:

v = vo•P.2o'•exp(-E • + aV* -'yVM/p)/RT (58)

which is the same as Charles and Hillig's (40)' with the introduction of a constant • and P, the partial pressure of water.

An analogous expression for static fatigue in quartz presented by Scholz [1972, p. 2111, equation 3] is

(t) = toC.2o-" exp [(u - K'a)/RT] (59)

where (t) is the mean time to failure; to, a, u, and K' are material constants; and C.2o is the 'water concentration.' Scholz [1972, pp. 2107-2108] notes that 'to compare vapor-solid reactions only, C.2o can be replaced by P.•o, the partial pressure of H,O.'

The crack radius of curt•ature. In Wiederhorn and Bolz's (44) the second term in the exponent includes p, the radius of curvature of the crack. We now examine how the rate of

change of p compares with the rate of change of crack length a. Charles and H#lig [1962, p. 522] considered this problem, and they concluded that 'if a flaw proceeds to failure by stress corrosion, a small change [increase] in flaw length is accompanied by a large reduction in flaw tip radius.'

The conclusion of Charles and Hillig quoted above does not, however, follow from Stuart and Anderson's [1953] (31). Balance of the values for PE and SE at the crack tip does not require a special relationship between p and crack length. In fact, p was assumed to be constant in the derivation from the Stuart and Anderson theory in (47), which is similar to Wie- derhorn and Bolz's (44). While assumption of constant p may be questioned theoretically, it is clear from many experiments [e.g., Wiederhorn, 1967; Speidel and Hyatt, 1972; Wachtman, 1974] that the crack velocity in the stress corrosion range depends primarily on stress intensity, temperature, and the chemistry of the environment. It is not necessary to include p as a variable in the empirical description of the experimental results. Unfortunately, the crack tip radius cannot be measured directly during a crack propagation test [Speidel and Hyatt, 1972, p. 287; Martin and Durham, 1975, p. 4843].

According to the theory of Charles and Hillig [1962], changes in the crack tip radius, such as sharpening, occur during the corrosion process and alter the rate of crack extension. In experimental studies of the mechanism of slow crack growth in quartz, however, Martin and Durham [1975, p. 4843] found that the radius might indeed remain approximately constant during crack propagation. They found that

each macroscopic crack consists of a series of nearly equally spaced microscopic cleavage cracks inclined to the axis of greatest compression. As each minute cleavage crack lengthens... we are continually initiating new microcracks, each presumably with the same radius of curvature, as the macroscopic crack propagates.

They concluded that the controlling mechanism was instead


the partial pressure of water at the crack tip; the water at the tip 'reduces the stress necessary to rupture bonds by approximately an order of magnitude below that for a silicate with no water available' [Martin and Durham, 1975, p. 4843].

Mass transport kinetics theory of stress corrosion cracking in region II. The apparent activation energy for region II of the K-t) curve (Figure 3), found by conducting experiments at various temperatures but holding other variables constant, is much lower than that for region I. For example, in titanium alloys the activation energy of region I is about 28 kcal/mol, while that of region II is about 5 kcal/mol [Beck, 1971, p. 65]. It has been suggested that the rate-limiting process for region 1 is a chemical reaction, while that for region II is mass transport of the reacting species to the tip of the crack.

Since 'no one really knows what happens at a crack tip because it is inaccessible to direct experimental observation at the atomic scale' [Beck, 1971, p. 78], there is some dis- agreement on the exact model to be used for the mass transport kinetics. The interested reader is referred to the papers by Bbck and Grens [1969] and Beck [1971] for proposed mathematical models of these kinetics. The models for region II behavior of titanium and aluminum alloys have been reviewed by Speidel and Hyatt [1972, pp. 297-298].

Briefly, in the mass transport kinetics models it is assumed that the velocity of crack propagation is limited by the transport of halide ions to the crack tip. Both the diffusivity and the characteristics of electrolytic migration determine the halide ion mass flux to the crack tip. The activation energy of 4-5 kcal/mol measured for region II 'is well within the range predictable for the activation of a process limited by ionic transport' [Speidel and Hyatt, 1972, p. 298]. The diffusivity of halide ions is inversely proportional to the viscosity of the solution, and indeed, as the mass transport kinetics model predicts, the stress corrosion crack velocity in region II varies inversely with the viscosity of the solution filling the crack.

Diffusion as a rate-limiting process in slow crack growth. Robertson [1966] derived an equation to account for rates of crack propagation in liquid metal embrittlement, and he suggested that his model might be applicable, with modifi- cations, to stress corrosion in a liquid environment. In the model a crack 'is assumed to propagate by solution of the solid in the liquid under the influence of an applied stress, with volume diffusion of the dissolved solute through the liquid' as the rate-limiting step [Robertson, 1966, p. 1478]. The Rob- ertson [1966, p. 1479] equation is

1)( 2aa2 _ CøDI22'Y (• • 1) (60) t•= kT where Co is the equilibrium concentration of solute in the liquid in the presence of an unstressed surface, D is the diffusion coefficient of the solute in the liquid, and •2 is the atomic volume of the solid. The term -y is the solid-liquid interfacial energy.

The reader interested in additional diffusion models for slow

crack growth is referred to Dutton [1974] for equations treat- ing propagation rates controlled by (1) grain boundary diffusion (a creep rupture model), (2) lattice diffusion (crack growth by absorption of vacancies at the crack tip from the surrounding lattice by a process of bulk diffusion), (3) surface diffusion (rate of removal of atoms from the crack tip followed by their deposition on the flat surface of the crack by surface diffusion), and (4) vapor phase transport (evaporation of atoms at the crack tip followed by their condensation on a stress-free flat surface of the crack).

Wiederhorn [1974, p. 626] commented that although alkali ion diffusion in glass falls in approximately the same activation energy range [as that determined by fitting experimental data to an equation such as (29)], a diffusion mechanism for fracture is not likely since slow crack growth is observed in glasses such as silica that contain low concentrations of alkali ions (• 1 ppm for silica).

However, it may be that diffusion processes are important in crack propagation in regions I and II, in which they may be involved as the diffusion of reaction species in the fluid within the crack rather than as diffusion within the solid.

A tomistic models of stress corrosion cracking. 'The lack of a satisfactory theoretical basis for describing crack-tip events at the atomic or molecular level' has been a great obstacle to our understanding of crack propagation, for crack growth is 'fun- damentally a non-linear problem' because it involves the rupture of bonds [Lawn, 1975, p. 469]. Poncelet [1948, 1965] and Stuart and Anderson [1953] approached the problem of static fatigue from the atomistic point of view by considering that the probability of bonds being broken during crack growth is determined by Maxwell-Boltzmann statistics. Their approach was purely statistical. Recently, other approaches have been used to relate macroscopic parameters of fracture such as G and K• to processes at the atomic level.

A feature essential to theories of thermally activated crack growth is a 'periodically varying energy barrier' [Lawn, 1975, p. 469]. The total surface energy, for example, 'is determined explicitly by the non-linear separation processes which operate within the non-linear zone, so that 1` must be expected to oscillate with atomic periodicity' [Lawn, 1975, p. 472]. The crack propagates according to the condition that G exceeds 21' 'where the energy barrier for forward fluctuations [e.g., kink motion] falls below that for backward fluctuations' [Lawn and Wilshaw, 1975, p. 176].

Thomson and Fuller [1974] have developed a qualitative model of slow crack growth in terms of processes at the atomic level at the crack tip. This model applies to brittle materials in which cracks at the local level are atomically sharp cleavage cracks. Using the lattice trapping model of Thomson et al. [1971] and Hsieh and Thomson [1973], they 'translate' the Charles and Hillig [1962] theory 'into a discrete atomic lan- guage' [Thomson and Fuller, 1974, p. 289].

The crack is considered to open 'by the evaporation of atoms from the surfaces of the crack and from the crack tip' [Thomson and Fuller, 1974, p. 289]. In the model the evaporation occurs at kink sites on the surface. The equation for the growth of the crack is

AFgrowth = E_• - 2U, - (.Oao•?a/E) < 0 (61)

where AFgrowth is the free energy of crack growth, Eo is a normal bond energy, U, is the energy by which a bond is stretched by stress at the crack tip, ao is the external stress, a is the half-length of the crack, E is Young's modulus, and I is one interplanar atomic spacing across the crack plane, that is, the lattice parameter. The onset of slow crack growth becomes thermodynamically favorable when AFgrowth -- 0. Equating this expression to zero (to obtain the stress at the crack tip in the thermodynamic equilibrium state), Thomson and Fuller find

O'eq = 0.84a,• (62)

where aeq is the stress at the crack tip in the thermodynamic equilibrium state and a,• is the maximum stress which a bond is capable of exercising between two atoms. Thus the Thomson


IO-4

iO -5 ß

&& ß &

15OO Nm- 2 I

x t 300 I I

-;..

• 10 -6

10 -7

10-8 O IO

30

2'4

"- G, Jm-2 Fig. 8. Comparison between theory (solid lines) and experiment

(data points) for (10T2) fracture of sapphire in water vapor at room temperature. Mechanical energy release rates G were calculated from Ki by (23). Partial pressures of water for the various experiments are indicated. Reproduced from Lawn [1975, p. 479, Figure 6].

and Fuller model predicts a range of stress intensity factors, from 0.84Kxc to Kic (between the thermodynamic equilibrium state and the dynamic equilibrium state), over which slow crack growth can occur. Thomson and Fuller [1974, p. 293] state that their theory is equivalent to that of Charles and Hillig. The term E0, for example, can be replaced by the heat of solution for the case in which the entire crack tip is in the presence of a fluid.

Using the lattice trapping model originated by Thomson and his co-workers, Lawn [1975] developed an atomistic model to account for slow crack growth (fracture on the (1012) cleavage) in sapphire in the presence of water vapor (Figure 8). Lawn assumed that a modification of the surface energy term could account for chemically enhanced crack growth. It is assumed that [Lawn and Wilshaw, 1975, p. 178]

r/A + B --, B* (63)

where rt molecules of environmental species A interact with crack tip bonds B to cause passage over an activation barrier into a ruptured state B*. For an extension ba of a crack of unit width in which NA is the surface density of crack plane bonds, the number of bonds broken during extension of the crack is

fin = NA(Sa (64) The total energy change is the sum of mechanical and chemical energy terms [Lawn, 1975, p. 476]:

dU/da = -G + (#•, - /as - r]#A)NA (65) where the # terms are chemical potentials. Lawn and Wilshaw [1975, pp. 178-179] use an extension of the parameter F to express the energy of the material-environment interface:

2r': - #B)NA (66)

Then in accordance with the energy balance concept, dU/da = -G + 2F' - rt#aNa (67)

Lawn [1975, p. 476] writes the following expression for reaction-limited crack velocity:

v,. = V(T)(paO/pa•) "•' exp (-Uo*/kT) exp (G/2NAkT) (68) where V(T) is a 'slowly varying factor,' pa ø is the gas pressure at the crack mouth, pa • is the gas pressure at some reference state, and U0* is a collection of 'various uncertain energy constants.'

Lawn [1975, p. 478] also presents an equation to account for the kinetics of migration of the reacting species in the environment along the narrow crack interface to the tip of the crack. The equation for the transport-limited crack velocity is

Vt = t•aypAø/[rINA(27rmk T) •/•'] (69)

where • is an 'attenuation factor associated with the increasing incidence of retarding, diffuse molecule/wall collisions as the gas approaches the crack tip' (and a function of G, although it is regarded as an empirical constant over any limited range of G) [Lawn, 1975, pp. 477-478], a s is the lattice spacing along the y axis representing the 'reaction cross section' per unit width of crack front presented to the impinging gas molecules by the crack tip bonds, and rn is the molecular mass of the gaseous species.

When these two equations for the overall process controlled by both transport and chemical reaction are combined [Lawn, 1975, p. 478],

(v/v ) + (v/vy : 1 (70) At low values of G (or K), v.is much smaller than vt, and the crack velocity approaches the value for the reaction-limited case vr given by (68). At high G (or K), vr is much larger than vt, so that the crack velocity approaches the transport-limited case vt (69).

Figure 8 shows the comparison between the curves con- structed from (68), (69), and (70) by the insertion of selected empirical constants and the experimental data for the stress corrosion cracking of sapphire.

Conclusion. At present the theory of stress corrosion cracking is still in a very active stage of development. The equations developed by Charles and Hillig and by Wiederhorn and Bolz appear to explain satisfactorily the experimental observations of region I behavior. They must be supple- mented, however, with considerations of mass transport kinetics to explain plateau region II. There is considerable dis- agreement about what role diffusion might play in the kinetics of stress corrosion cracking.

Recently, there have been some advances in the modeling of slow crack propagation at the atomic level, such as the theories developed by Thomson, Lawn, and others. In these theories and in the theory of slow crack propagation in vacuum, approaches similar to those originally taken by Stuart and An- derson have been followed. The theory of slow crack growth is still largely empirical, however, because little is really known about the actual processes that take place at or near the crack tip.

D. Experimental Data on Stress Corrosion Cracking In this section we review some of the experimental data that

have been accumulated on the variables which influence stress corrosion cracking, such as temperature, viscosity, pressure,


THE DOUBLE-TORSION SPECIMEN

UPPER SUR FACE

CRACK FRONT UNCRACKED MATERIAL

LOWER SURFACE

Fig. 9. Double-torsion specimen configuration used in studies of stress corrosion cracking. Modified from Et•ans [1974b, p. 25, Figure 7, p. 32, Figure 11].

and humidity. We also discuss some of the textural characteristics of stress corrosion cracking, such as intergranular versus transgranular fracture, crack branching, and effects of preferred orientation.

Extensive experimental studies have been made of stress corrosion cracking in metals, glasses, and ceramics. The materials tested have included (1) noncrystalline solids such as glasses, (2) homogeneous monomineralic polycrystalline ag- gregates such as aluminh, (3) single crystals of minerals such as sapphire [Wiederhorn, 1968; Bansal and Heuer, 1974] and quartz [Scholz, 1968a, b, 1972; Martin, 1972; Martin and Dur- ham, 1975], (4) heterogeneous ceramics such as porcelain, and (5) metallic alloys such as steel. For reference a typical experimental sample configuration is shown in Figure 9.

Experiments by Charles [1959] and Scholz [1972] on quartz and granite suggest that stress corrosion cracking may be an important process in the fracture of rock-forming silicates. It should be emphasized, however, that sufficient experimental work has not yet been done on stress corrosion cracking in geological materials to elucidate the special features of slow crack growth in rocks and minerals.

It is known that cracks in rocks are commonly intergranular rather than transgranular. Therefore experimental data for glasses and single crystals may not be strictly applicable to slow crack propagation in rocks. Stress corrosion cracking is intergranular under some conditions in metallic alloys, but there are obvious problems in extrapolating experimental data for ductile alloys to brittle geological materials.

The effect of grain size on stress corrosion cracking in rocks is as yet undetermined. It has not been established that experimental data for ceramic materials with very small grain sizes (e.g., 20 um) can correctly be applied to coarse-grained rocks (e.g., grain sizes of up to 1 cm).

We believe that on the basis of preliminary experimental data for minerals and rocks the general theory of stress corrosion can be applied to subcritical crack propagation in geological materials. In order to consider certain applications for geophysics we wish to focus on several of the important variables known to influence stress corrosion cracking in other materials.

Effect of temperature. Experiments on metals, ceramics, and minerals have shown that crack propagation by stress corrosion is enhanced by increased temperature if other parameters, such as the stress intensity factor and the chemical composition and pressure of the fluid environment in the crack, are held constant (Figure 10). These results are consistent with the theoretical model of stress corrosion as a thermally activated process. A temperature dependence for stress corrosion cracking is noted for both region I and region II; as was noted above, the apparent activation energy for region I is higher than that for region I I.

Increase of crack velocity with temperature at a constant stress intensity factor has been well documented in such materials as soda-lime-silicate glass in water [Wiederhorn and Bolz, 1970], high lead glass in water [Wiederhorn and Johnson, 1973a], HS 130 type silicon carbide in air [Evans, 1974a], pore-


10-4 '• ,••:x:•oo'o'o• 105øC E

Oc ' • 21 øC

•0 • 3ø0 • oc -11

• 10'6

•O 10. 8

•C e I-- 3 rn Aqueous KI Solution

u• Potential-450 mV vs EH2/H + pH •6

, I, , I, ,, ,I 0 10 2o 3o

STRESS INTENSITY (MN/m 3/2) Fig. 10. Effect of temperature on stress corrosion cracking in an

aluminum alloy. Regions I and II are well defined [from Speidel, 1971a, p. 313, Figure 19].

free polycrystalline alumina [Evans, 1974a], and quartz in an atmosphere.with variable humidity [Scholz, 1968a, b, 1972; Martin, 1972].

The effect of high temperatures on crack velocity is considerable. For example, in alumina [Evans, 1974a] at constant Kx, increasing the temperature by 100øC at 1200øC increases the crack velocity at the same stress intensity factor by 6 orders of magnitude. Wiederhorn and Bolz's (44) for region I fits data for temperatures up to about 0.7 of the solidus of alumina, and it apparently might even hold for higher temperatures (Figure !l). Creep experiments are often performed in this temperature range, and therefore it is possible that experiments will ultimately show that the Wiederhorn and Bolz equation applies well into the creep range.

The question of whether stress relaxation resulting from creep would diminish the stress concentration at the tip of the crack depends on the distribution of stress around the crack and the model for the energy balance. Hillig and Charles [ 1965] showed that as long as the plastic zone ahead of the crack, defined by the yield point, is further from the crack tip than the radius of curvature of the crack, there will be little relaxation effect on the stress concentration at the tip of the crack. We might expect therefore that stress relaxation would have a large effect on fast fracture but not much effect on slow cracking by stress corrosion. According to this argument, stress corrosion cracking might exist well into the creep range.

In terms of reaction rate theory it may be very difficult to distinguish between an equation for creep and an equation for crack velocity. They both depend upon the idea that a chemical bond is broken and the probability that a certain number of bonds are broken and reformed in a unit time. It is likely that the activation energy will be about the same for both processes. The activation entropies and the activation volumes

will be quite different for the two processes, but these quan- tities elude experimental investigations. Activation energies for slow crack propagation and for creep are similar in the case of polycrystalline alumina (Table 1).

Effect of viscosity of the fluid environment. We are not aware of systematic studies of dependence of stress corrosion rates on viscosity of the fluid in cracks propagating through ceramic materials. However, experiments have been done on stress corrosion of titanium alloys in H Cl-water-glycerine solutions of variable viscosity [Blackburn et al., 1973, p. 149] and on aluminum alloys (Figure 12) [Speidel, 1971a]. In titanium alloys the velocity of crack propagation at a constant stress intensity factor and constant temperature was found to decrease linearly with increasing viscosity over 4 orders of magnitude of velocity and viscosity. As is evident in Figure 12, changing the viscosity of the fluid appears to affect the kinetics of only region II, not region I [Speidel, 197 la, p. 320; Speidel and Hyatt, 1972, p. 199]. The result is consistent with the hypothesis that crack propagation velocity in region II depends on mass transport kinetics. Since the viscosity of the fluid controls the ease with which the fluid can penetrate to the crack tips, we propose that the same relation should hold for slow cracking in rocks. Other things being equal (e.g., chemical corrosivity of the fluid and temperature), the lower the viscosity of the fluid in the crack, the greater the ease of stress corrosion. Elsewhere [Anderson and Perkins, 1974b, 1975; Perkins and Anderson, 1974] and in this paper we also explore the mechanical importance of viscosity as a limiting variable determining whether cracks filled with magma can maintain a sufficient propagation rate to reach the upper crust or surface of the earth.

Effect of humidity and of pH of the environment. Increased humidity increases the rate of stress corrosion crack growth by shifting region II to higher velocities and by shifting region I to lower stress intensities (Figure 13). Martin [1972] and Martin and Durham [1975] report that the rate of growth of stress corrosion cracks in single crystals of quartz increases as either the temperature or the partial pressure of water in the test environment is incre•tsed.

Wiederhorn and Johnson [1973a, b] studied the effect of electrolyte pH on crack propagation in glasses. The slope of region I of the K-v diagram was found to be dependent upon pH. Silica glass, for example, shows a steeper slope in more acidic solutions, so that for comparable values of the stress intensity factor the velocity of crack propagation is higher in more basic environments. From (44), Wiederhorn and Johnson [1973b] showed that the slope is proportional to the activation volume of the chemical reaction controlling the fracture and inversely proportional to the radius of curvature at the crack tip. Since the slope depends on pH, the radius of curvature or the activation volume must also depend on pH, although the reasons for this dependence and the precise nature of it are not known.

Effect of pressure. In considering the effect of pressure we must be careful to distinguish three effects. There are the effects of pressure upon (1) the stress tensor of the solid being cracked, (2) the mechanical properties of the fluid in the crack, and (3) the particular chemical reactions taking place between the fluid in the crack and the solid crack tip.

Wiederhorn and Johnson [1971, p. 585] studied the static fatigue of glasses immersed in water at atmospheric pressures and at pressures from 6 to 7 k bar. They found that high pressure (at least for these particular glasses in this pressure range) 'has no effect on static fatigue properties,' since identi-


10'3- 10-1

10-2 10 '2

•.. 10 '4

• 10'5 10-6

10'7

1300øC ß O O

:_1 ,ooc.ooooc [ I'

,i

ß

I 600oc

ß CRACK GROWTH RATE

O ACOUSTIC EMISSION RATE

o-8 , I I I I ..... I 2.0 2.4 2.8 3.2 3.6 4,0

K I (MN m '3/2) Fig. i i. Effect of temperature on crack velocity and acoustic emission rate in pore-free polycrystalline alumina. Modified

from Evans [i 974a, p. 39, Figure 8].

cal data for static fatigue at these different pressures were obtained. Correspondingly, there should be little effect on stress corrosion crack velocity. Gerberich [1974] suggested on theoretical grounds that increased externally applied hydrostatic pressure should increase the threshold stress intensity K•scc in high-strength steel and titanium alloys immersed in aqueous environments. From this point of view, one would expect, contrary to the results of Wiederhorn and Johnson, that high pressure would affect the stress corrosion crack velocity. Gerberich's arguments were based on the assumption that the material being cracked was an alloy subject to plastic flow. One could speculate on the basis of these two results that the pressure effect on stress corrosion might depend on the degree to which the material is nonelastic.

Although experimental results are not available for ceramics or rock materials, we can make the following general state- ments on the basis of preliminary evidence from work on metals:

1. If raising the hydrostatic pressure dramatically increases the viscosity of the fluid in the crack, the rate of crack propagation should be decreased in region II, in which mass transport kinetics are dominant.

2. The effect of pressure on the chemical corrosion reactions at the crack tip will depend on the particular chemical equilibria or disequilibria between the fluid and the material being cracked. If the particular reactions are enhanced by increased pressure, the chemical response to increased pressure should dominate any deceleration caused by pressure-induced higher viscosity.

High hydrostatic pressure must help to keep the fluid in the tip of the crack, maintaining physical proximity between fluid and solid and facilitating chemical reaction in that region. Similarly, we would expect an enhancement of crack velocity with pressure from any fluid when the chemical potential of the fluid and the reactants at the tip of the crack are influenced by pressure.

Whether increased pressure will increase or decrease crack velocity depends upon the competition between numerous effects. Pressure could well increase velocity in one given situa- tion and decrease it in another.

Stress corrosion and preferred orientation. In applying the theory of stress corrosion to geological crack propagation, not only must we scale up from the fine grain sizes of laboratory- tested ceramics to coarser-grained silicates in rocks, but we also must consider that many rocks, especially those thought to be from lower crustal and mantle source regions, have a strong preferred orientation of their constituent mineral grains.

TABLE 1. Activation Energy for Al•.Oa Ceramics at High Temperature

Activation Energy, Phenomenon kcal/mol Source

Self-diffusion of oxygen 140 Oishi and Kingely [1960] Creep 145 Cannon [ 1971 ] Crack propagation • 130 A.G. Evans (personal

communication, 1975)


z o

o

o

10-4

10-6

10-8 [ lo-lO

o

Alloy 7079-T651 2.5 cm Thick Plate

2 Molar KI Solution (H20+Glycerol) _ Temperature 23 øC

Potential-450 mV vs EH2/H+ VISCOSITY •7 (CENTIPOISE)

o-o-o--oo 7.2

o.o-ocx:x• 24

c• 417

o

10 20 30

STRESS INTENSITY (MN/m 3/2) Fig. 12. Effect of viscosity on stress corrosion cracking in an

aluminum alloy immersed in water-glycerol mixtures [from Speidel, 197 l a, p. 318, Figure 24].

Stress corrosion cracking rates are affected not only by environment but also by the microstructure of the material being cracked. For example, in high-strength aluminum alloys there is a 'large reduction of crack growth rate when the crack orientation is changed with respect to the preferred orientation of the grain boundaries' [Speidel and Hyatt, 1972, p. 162]. Crack velocities in region II are 4 orders of magnitude faster in one orientation than in another [Speidel and Hyatt, 1972, p. 155, Figure 28]. The direction of fastest crack propagation is in the direction of the lineation.

In titanium alloys, according to Blackburn et al. [1973, p. 128],

specimens selected so that the general cracking plane is aligned parallel to the basal planes [of strongly oriented alloy] exhibit the most severe susceptibility . . . [however] in highly susceptible alloys, the influence of texture becomes less pronounced and... [stress corrosion] will occur irrespective of specimen orientation.

Although experimental results for metals are probably not directly applicable to the case of rock materials, we conclude from these preliminary studies that the fabric of rocks would be a variable influencing the kinetics of stress corrosion cracking in certain directions. Strongly foliated rocks would be unlikely to behave isotropically unless they were cracking in a particularly corrosive environment, which would dominate the effects of preferred orientation.

Topology of stress corrosion cracks. In some alloys, charac- teristic fracture features are observed in specimens that failed under stress corrosion cracking. We cite in this section some of the fractographic features noted in titanium alloys. In the future it might be possible to correlate fractographic features in rocks with various regimes of stress corrosion cracking.

Blackburn et al. [1972, p. 337] note that titanium alloys that have undergone cracking in the different regions of the K-v diagram (Figure 3) show

a wide variety of fracture t9Pologies ... depending on alloy composition, microstructural factors, environment, and stress level . . . in most cases crack growth occurs in region I by intergranular separation... in region II by transgranular cleav, age ... and in the supercritical region (Kxc) by microvoid coalescence... between regions I and II, mixed intergranular and transgranular fracture is observed ... between region II and unstable (fast) fracture, mixed transgranular cleavage and dim- pled fracture is observed.

The correlation of transgranular cracking at high stress intensities and intergranular cracking at low stress intensities 'is a fairly common one' and has also been observed in steels [Speidel, 1971b, p. 347].

In aluminum alloys, stress corrosion cracking 'typically occurs along grain boundaries whereas cracking associated with mechanical fractures resulting from fatigue, creep rupture, tensile overload, etc., generally is transgranular' [Brown et al., 1972, p. 87]. Stress corrosion cracking in nickel alloys is also predominantly intergranular [Boyd and Berry, 1972]. Inter- granular cracks are common in rocks. An intergranular geometry would be consistent with an origin by stress corrosion if an analogy may be drawn between the behavior of rocks and that of alloys. Transgranular cracks in rocks would be consistent with an origin by fast fracture. Because rocks are brittle and metals are ductile, however, the confirmation of this analogy must await experimental studies of slow crack propagation in rock materials.

Crack branching. Two types'of crack branching of stress corrosion cracks have been recognized in alloys: (1)'microbranching, in which the crack front splits into several local cracks with separation distances of the order of a grain diame- ter' [Speidel, 1971b, p. 345] and (2) 'macrobranching, in which the crack separates into two or more macroscopic components that tend to diverge.' Macrobranching in alloys appears to be confined to region II of the K-t) diagram.

Conditions for macrobranching in alloys appear to be that (1) it is confined to region II, (2) the fracture path must be relatively isotropic, (3) 'the stress corrosion crack should not follow exclusively microstructural features such as, grain boundaries or cleavage planes with a strong preferred orientation' [Speidel, 1971b, p. 345]. Microbranching is observed in region I as well as in region II.

Speidel [1971b] has suggested that the reason that macrobranching is confined to region II is that the propagation of two cracks (the two branches) requires twice the strain energy that would be needed for the propagation of one crack alone. In order for one crack not to outrun the other, 'both branches should propagate with about the same speed . .. it is only in this region [II] that minor differences in crack length (and stress intensity) do not result in major differences in crack growth rate' [Speidel, 1971b, p. 352].

IV. ACOUSTIC EMISSION AND SLOW CRACK PROPAGATION

In materials science the term acoustic emission is used to denote 'a transient elastic wave generated by the rapid release of energy within a material' [Green and Anderson, 1972, p. 336]. Acoustic emission is synonymous with the term micro- seism, which is more commonly used in North American rock mechanics literature. Earthquake studies in the field are nor- mally carried out at lower frequency ranges (less than 11Y Hz) than are acoustic emission laboratory studies (greater than 10: Hz in geologic materials and 10•-11Y Hz in metals [Hardy, 1972]).

Although in geological materials the origin of acoustic emission is not well understood, acoustic emission in ceramic mate-


z

10'4

10'5

10'6

10-7

30%

H20(I )

ee• e

ß o II

lO%

o øß ß

o; 1.0%

ß

•e•e ß 0.20,4

0.017%

I

4.0 5.0 6.0 7.0 8.0

STRESS INTENSITY FACTOR, K I, N/m 3/2 x 10 5 Fig. 13. Effect of relative humidity on crack velocity in soda-lime-silica glass in a nitrogen atmosphere. Only half the

experimental points used to determine the line for region Ill are shown [after Wiederhorn, 1967, p. 409, Figure 3].

rials has been correlated with the propagation of fractures through (transgranular) and between (intergranular) mineral grains. In addition, dislocation motion, deformation twinning, and cleavage are all sources of elastic energy release that can be monitored acoustically [Noone and Mehan, 1974, p. 227]. Slowly propagating cracks in both polycrystalline and poly- phase ceramics 'emit energy pulses that are detected as acoustic emissions' [Lange, 1974, p. l t]. Graham and Alers [1974] measured the number of acoustic emissions produced as a function of crack length during slow crack growth in polycrystalline ceramics. Using photographs of fracture surfaces taken with the scanning electron microscope, they were able to deduce the number of acoustic emission events per grain fractured in alumina, Lucalox (commercial alumina), magnesia, and spinel. They observed 'a one-to-one correlation between the area swept out by the advancing crack front for each emission burst and the area of a single grain ... for large grained spinel the correlation with grain size failed and a better correlation was indicated with the spacing between pores in the material' [Graham and Alers, t974, p. t87].

Acoustic activity is greatest in ceramics which are either coarse grained (particularly if they are highly anisotropic), multiphased, inhomogeneous, or preflawed [Noone and Me- han, 1974, p. 227]. All these characteristics are typical of many

rock materials. Acoustic emission has been correlated with crack propagation in single crystals and glass as well. Scholz [1972] showed that the rate of microfracturing (i.e., the number of microseisms per unit time) in single crystals of quartz is proportional to the crack velocity. This is consistent with the results of Evans and Linzer [1973], who found that the acoustic emission rate of porcelain in the water is directly proportional to the crack velocity.

Byerlee and Peselnick [1970, p. 83] made compression studies of specimens of plate glass in which open slits were precut at an angle of 30 ø to the direction of loading. No elastic shocks were observed for samples of glass which were not precut. In. the samples which contained the slits, no 'seismic' activity was observed until a load of 570 kg, at which 'a crack suddenly propagated for a distance of about 1 mm from the end of the slit... This sudden crack growth was accompanied by a burst of "seismic" energy which lasted for about t0 msec.' Byerlee and Peselnick [1970, p. 84] observed that 'no elastic shocks were detected during slow crack growth or during crack closure on release. of the load.'

Of particular interest for geophysical applications are the experiments during which for sufficiently slow crack growth under stress corrosion, there are no acoustic emissions. Fig- ures 11 and 14 show that in polycrystalline alumina and porce-


106

105 z

,,, 104

Z

o • 103

z 02 o 1

C--. 10 o

1

10-1

10-2

10-7

10-8

ß CRACK VELOCITY

O ACOUSTIC EMISSION RATE

ß , O ,oO

O

! !

I I

o

,/

10'9 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

K I (MN m '3/2) Fig. 14. Crack growth rate and acoustic emission rate in porcelain as a function of stress intensity factor [from Evans and

Linzer, 1973, p. 576, Figure 5].

lain no acoustic emissions are detected for crack propagation rates of the order of centimeters to kilometers per year, even though crack growth is proceeding. Concerning their experiments on glass, Byedee and Peselnick [1970, p. 85] reported that 'the results described here show that sudden crack growth does give rise to elastic shocks, but they also show that cracks can grow stably without generating high-frequency "seismic" energy that could be detected by the instruments we used.' At present, however, the question of whether rocks undergo slow crack growth without acoustic emission has not yet been satisfactorily answered by experiments.

V. GEOPHYSICAL AND GEOLOGICAL APPLICATIONS FOR THE THEORY OF SLOW CRACK PROPAGATION

BY STRESS CORROSION

A. Earthquake Phenomena Scholz [1968a] suggested that stress corrosion in slow crack

propagation may be a controlling process in time-dependent geological processes, in particular, in earthquake aftershocks. He pointed out that the time sequence in aftershocks depends upon a time-dependent strength, which he identified as static fatigue, and he suggested that the physical mechanism controlling aftershock sequences is represented by the form of the static fatigue expression given by Scholz [1972] and reproduced here as (59). He suggested that 'the mechanism of the

static fatigue process in such systems is thought to be weakening by stress corrosion. The primary reactions which weaken silicates are hydration reactions.' He further pointed out the time delay law 'depends on the mechanism and is the same for earthquakes and microfractures (in the laboratory).' In other words, he implies that there is no problem with scaling his experimental results to fracture in the earth.

Martin [1972] pointed out that dilation associated with an earthquake has the effect of changing the partial pressure of water and therefore affects static fatigue, as given in (58).

Booker [1974] suggested that porous media effects might explain the distribution of earthquake foreshocks, earthquake aftershocks, and delayed creep. He argued that a fracture that redistributes shear stress in a porous medium pro- duces local fluid pore pressure changes. For a simple crack model in which a fracture surface is slowly reloaded after an initial drop in shear stress, the shear stress of the material adjacent to the fracture varies with time. The frequency of aftershocks associated with reloading decays as time -x. While such porous media effects may explain the observed sequences, we believe that stress corrosion may also be taking place and that it may contribute to the time-dependent behavior.

The effect of a major earthquake might be to redistribute the groundwater in the vicinity of the fault, so that some cracks that were dry before the earthquake might be immersed in water. This is reflected in changes in water levels of wells and


springs which take place after earthquakes. In this case the static fatigue limit of rocks would be changed. Thus one can suppose that the distribution of activation energies is altered in the vicinity of the fault at constant stress. This is equivalent, with respect to static fatigue, to the assumption of a change in the distribution of breaking stresses at constant activation energy subsequent to an earthquake.

Many characteristics of fault activity attributed to creep or fluid diffusion may be due to stress corrosion cracking at very slow velocities. A number of cases are well documented in which portions of a fault are seismically quiet prior to an earthquake on the same segment. This quiescence does not mean that crack propagation by stress corrosion is not in progress. In laboratory experiments, absence of acoustic emission is observed at low stress intensity factors in region I (see Figures 11 and 14). In region I, stress corrosion (reaction rate controlled) behavior is dominant. At low values of the stress intensity factor no acoustic emissions are observed for very low velocities of crack propagation. These very low velocities furthermore are of the same order of magnitude as some geological processes (centimeters to meters per year). It is possible that slow crack propagation, unaccompanied by acoustic emission but controlled by chemical reactions between fluid and rocks being cracked, may take place on seismically quiet portions of fault zones moving with rates of the order of a few centimeters per year.

Changes in the composition of the fluid environment or changes in temperature could speed up or slow down the crack propagation at constant values of the stress intensity factor. Thus even if the crack is propagating in response to a steady stress, the rate of propagation could change (perhaps to acoustically detectable velocities) if new fluid were injected or if fluids were removed. The velocity of crack propagation could also change if there were a change in the stress intensity factor due to regional stress variation or if the propagating crack crossed a boundary into material of different chemical composition.

It may well be that the mechanical effects of fluid migration which change rock pore pressure and effective stress will dominate fault behavior and swamp stress corrosion effects. We suggest that stress corrosion cracking controls the induction time of subcritical crack growth (equivalent to region I of the stress intensity-velocity curves) but that at higher values of the stress intensity factor and higher crack propagation velocities, fluid transport becomes the dominant factor. Then at the critical stress intensity, fast cracking progresses in the catastrophic fashion typical of earthquake faulting. Injecting fluid into an inactive fault could result in substantially increasing the crack velocity by changing the kinetics of chemical reactions at the tip of the crack. Certain corrosive liquids and water weakening could reduce the breaking strength of the chemical bonds at the crack tip. The importance of fluids in the control of fault behavior has been dramatically demon- strated at the Rangely oil field by Raleigh [1972]. Much of the effect is certainly due to changes in pore pressure in the rocks and is explainable in terms of effective stress. However, time- dependent friction and fracture properties may be significant, and environmentally assisted subcritical crack growth may affect the rate of response. Rice [1975, p. 1535], considering precursors to shallow-focus earthquakes, offered a similar sug- gestion:

Time effects could also enter in the form of macroscopic rock creep, due to the growth of microfissures by stress corrosion cracking in the surrounding groundwater [Scholz, 1968]. Such crack growth under sustained shear stress would lead to overall

dilatancy, and hence induced pore fluid suctions would... play a role in strengthening the rock or slowing the rate of creep toward failure provided that the time scale is short in comparison with diffusion times.

Martin and Durham [1975, p. 4843] also hold a similar view titat 'water, apart from its mechanical effect [such as in hydrofracturing], plays an important role in upper crustal deformation and failure. It reduces the stress at crack tips necessary to rupture bonds by approximately an order of magnitude below that for a silicate with no water available .... 'They go on to propose that the chemical effect of water may suggest a 'mechanism whereby deformation may proceed withotit requiring a continuous augmentation of the stress field. Such time-dependent dilatancy may be particularly important in earthquake regions.'

According to the dilatancy-diffusion hypothesis [e.g., Nur, 1972; Aggarwal et al., 1973; Scholz et al., 1973; Whitcomb et al., 1973; Hanks, 1974], the following events precede an earthquake: (1) accumulation of tectonic strain and increasing effective stress, (2) production of dilatancy at a rate faster than pore water can flow into volumes created by newly formed cracks, (3) undersaturation of rocks and a decrease in pore pressure as crack volume exceeds fluid volume and the fracture strength of rock increases (dilatancy hardening), (4) return to saturated state by fluid flow, and (5) rising pore pressure, triggering an earthquake 'just as it does through fluid injection' [Scholz et al., 1973, p. 804]. Increased radon content in well water prior to earthquakes has been attributed to increased fresh surface area in cracks and to an increase in the

flow rate of pore water [Scholz et al., 1973, p. 807]. Since the dilatancy-diffusion hypothesis invokes variable

crack propagation rates in the presence of variable humidity, it is important that slow crack growth by stress corrosion, known to be dependent upon these parameters, be in- corporated into the model.

B. Applications of Stress Corrosion Theory to Magmatic Intrusion

Introduction. The discussion of igneous intrusion in this paper chiefly concerns oceanic basaltic suites and nonorogenic intrusions and does not treat syntectonic igneous intrusions of orogenic zones.

Whenever igneous rocks at the surface of the earth are interpreted to reflect thermal anomalies at depth in the earth's interior, assumptions are necessarily made about the physical mechanisms by which magmas are transported upward through the earth's lithosphere. Some workers have considered these mechanisms and have used earthquake studies to deduce the nature of upward migration of magma, particularly in the Hawaiian volcanoes [Fiske and Jackson, 1972; Koyanagi and Endo, 1971; Jackson and Shaw, 1975; Shaw and Swanson, .1970]. Howe•ver, the majority of geological and geophysical studies have emphasized surface observations of exposed volcanic and plutonic rocks, petrologic inferences of depth of magma generation, and theoretical considerations of flow within the asthenosphere. Less attention has been given to the physics of magma transport between the source area and the upper portion of the crust. A better understanding is needed of the physical mechanisms by which magma is transported through the lithosphere. Otherwise, inferences from surface geology of the distribution of thermal anomalies in time and space may not be valid. The lithosphere may well act as a filter, obscuring our observations of thermal anomalies in the asthenosphere.

For example, a one-to-one correspondence is often assumed


between hot spots in the asthenosphere and volcanic centers on the earth's surface. Yet it is very likely that many injected magmas which initially intrude the lithosphere are arrested and never reach the upper crust or surface [Marsh and Carmi- chael, 1974]. They are probably inhibited from upward progress because the propagation of fluid-filled cracks is so sensitive to slight variations in a number of thermal and mechanical variables. Consideration of such constraints on magmatic transport is essential to the interpretation of periodic eruptive activity, because periodicity could be due to reactivation of and changes in velocity of crack propagation rather than to renewed heating.

Popular models for the transport of magma through the lithosphere include (1) propagation through cracks at the base of the lithosphere in tension [e.g., Isacks et al., 1968], (2) upwelling of magmas in the form of diapirs or irregular masses of magma passing through the lithosphere [e.g., Green, 1971, p. 58, Figure 7], (3) abyssal wedges feeding large intrusions [ Wager and Brown, 1967, p. 205, Figure 125], and (4) pencillike or tubelike cylindrical necks transporting magmas from down- going slabs to chains of volcanoes at plate margins [Marsh and Carmichael, 1974, Figure 10].

There are more complex several-stage models which involve propagation by crack, diapir, or pipe to an intermediate cham- ber in the lithosphere where the magma differentiates before further periodic eruption. One-stage models are generally thought to be necessary to account for deep mantle xenoliths transported rapidly to the surface in explosive eruptions of kimberlites and alkalic magmas, but two-step processes have been invoked to explain xenolith populations that contain both mantle rocks and shallow-level cumulates [Jackson, 1968].

In our treatment of magmatic intrusion we shall use a model of intrusion that considers cracks filled with magma propagating upward through the lithosphere. We will apply the stress corrosion theory of slow crack propagation to this model.

While we believe that an important type of magma transport is through fluid-filled cracks, we recognize that there are many other modes of magmatic intrusion not involving cracks, such as those which result in the formation of diapirs and batho- !iths. We consider only the crack type in this paper.

Magmas rising through the lithosphere.' some constraints. Experiments on ceramics and other materials and the theory of stress corrosion imply that only a small fraction of magmas generated at depth rise to crustal levels because the propagation of fluid-filled cracks is constrained by many factors. These factors include (1) supply of fluid to the tip of the crack, necessitating relatively low viscosity and possibly the presence of a volatile phase, (2) chemical reactions between the fluid in the crack and the enclosing rocks, (3) maintenance of a state of stress appropriate for crack propagation, (4) crack size, and (5) temperature. The crucial problem in magma transport from the asthenosphere to the near surface is the maintenance of conditions such that the magma in the crack does not completely solidify. An especially important con- straint appears to be the need for a continuous supply of fluid to the crack tip during crack propagation.

Roberts [1970] and Weenman [1971] have shown that the behavior of a crack filled with a fluid such as a magma will be controlled by the stress field around the crack, even if the hydrostatic pressure is large in comparison to the magnitude of the stress field. A crack will become unstable and propagate if fluid is injected into the crack and if there is tension in the solid immediately in front of the crack. Weertman also showed

that if there is an accumulation of melt at the base of a lithospheric plate, if the density of the melt is less than that of the plate, and if there is tension in the plate, then the conditions are sufficient for nucleation of a crack. Once the crack is

nucleated in the base of the plate in tension, the crack will extend upward, providing the liquid follows in the tip of the extension. A symbiotic relationship exists between the crack and the liquid filling it, for if the liquid does not follow into the tip of the crack, then hydrostatic stress overwhelms the tension field. The crack cannot go faster than the fluid (magma and its volatiles) can flow in the channel provided by the crack, and the speed of the fluid is limited by its own viscosity.

Since the supply of fluid to the tip of the crack is important to crack propagation, the role of volatiles must be crucial. A volatile phase at the tip of the crack at lithostatic pressure will allow the crack to accelerate to high speeds, since the viscosity of a gas phase is many orders of magnitude smaller than that of magmas. A low-viscosity precursor of the magma in the tip of the crack has been previously suggested by Currie and Ferguson [1970] to account for the geometry of lamprophyre dikes in the Canadian shield.

The chemistry of intruding fluid relative to material in the crack tip will be a rate-determining factor due to the domi- nance of stress corrosion at low velocities of crack propagation. At these low velocities, behavior of the crack is expected to be that of region I of the stress intensity-velocity curves. The lower lithosphere is probably a region of low stress intensity factors, because rocks at depth in the earth's interior are not strong enough to support large stresses, and creep processes are dominant. As the crack propagates upward, however, the stress intensity factor may increase, the velocity of propagation thus increasing.

If the crack is too narrow or the velocity of the crack is too small, then the magma solidifies. Thus the ascent velocity of the magma (or the crack velocity) is a controlling feature of magma transport. To have a sufficiently rapid ascent velocity, there must be a continuous recharge of liquid at the lithosphere boundary injecting new melt into the crack. Maalqle [1973] showed that if the velocity of alkali basalt magma is about 1 cm/s, then the center of the conduit remains liquid, provided the conduit is at least 2.5 m wide and there is a sufficiently large reservoir to supply the magma in the crack. In cracks that are too narrow (such as 1 cm in width) the magma would solidify, and crack propagation would stop [McBirney and Williams, 1969, pp. 180-182].

Perhaps the most surprising result of the recent work on slow crack growth is the temperature effect on crack velocity (Figures 10, 11, and 15). As the crack rises through the lithosphere the temperature decreases, and this effect alone, if other effects were equal, would tend to decrease the crack velocity greatly (by several orders of magnitude). If the magma is to reach the surface, other mechanisms have to compensate for the decreased temperature. The temperature at the top of a fracture containing magma would depend on such factors as the heat loss to the wall rocks, heat generation from the latent heat of crystallization, and perhaps heat of viscous dissipation. These factors may compensate for the decrease in temperature as the earth's surface is approached. It could be that stress corrosion cracking has no role in determining the rate of intrusion. Rather, mechanical factors due to the viscosity of the magma and the presence or absence of volatiles may control crack propagation. Physical constraints on upward magma migration appear to be so great that there must be much more molten material at depth than ever reaches the


lOO

lO

1

o 1 -, lO-

o lO '2

o: 0-3

10-4 --

10-5 --

10-6 2.0

T/"I' m = 0.68

/ MAGMATIC ASCENT VELOCITY Maal•e, 1973

0.63

LOWER LIMIT IN ACOUSTIC DETECTION

I METER/YEAR

0.55

0.38

I I I I I 2.8 3.2 3.6 4.0

K I STRESS INTENSITY FACTOR (MN m '3/2) 4.4

Fig. 15. Data from Figure I l expressed as fractions of the melting temperature. T is the temperature of experiment, and T,• is the melting temperature. For a given Ki an increase in temperature shows a great increase in crack propagation velocity. Acoustic emissions are not observed for the lowest crack growth rates.

upper levels of the crust. Only in those few instances where the conditions for crack propagation are appropriate will the magma rise. Considerations of crack behavior lead us to reaf- firm the importance of the mechanical and tectonic characteristics of the lithosphere in controlling the location and nature of igneous provinces.

Magma as the corrosive environment in a slowly propagating crack. We have seen that the velocity of crack propagation at a constant stress intensity factor is accelerated by the stress corrosion process under the following conditions in metal alloys, glass, and polycrystalline ceramics: (1) increased temperature, (2) changes in the pH of the fluid (also depending on the composition of the glass), and (3) decreased viscosity. Increased pressure might enhance stress corrosion by increasing the stress intensity factor and by enhancing the chemical corrosion reactions. Alternatively, however, it might increase the value of Kxscc, as was suggested by Gerberich [1974].

of the area of freshly created surface (opened by cracking) to the volume of solution in the region of the crack tip. If fluid is not continually supplied to the crack tip, stress corrosion effects are diminished.

M agmas have several characteristics which would enhance stress corrosion in crack propagation by analogy with the experimental results discussed above. Magma is intruded at an elevated temperature, and the reaction of the magma with the country rock and the corrosive activity of associated gases should speed crack propagation. In most cases, magmas intrude rocks with which they are not in chemical equilibrium. Assimilation of and reaction with host rock and con-

tamination of ascending magmas provide evidence for this chemical interaction. Examples are known of dikes and sills which show little or no chemical interaction with the host

rock. Nevertheless, chemical reactions may occur in the crack tip, and the resulting corrosion may be sufficient to corn-

Increased pressure also should have the same enhancing effect pensate for the decreasing temperature of the surroundings as by maintaining fluid in the tip of the crack, where reaction can the magma ascends through the lithosphere. take place. Scully [1971] has pointed out that a very important A variety of acids have been found in natural magmas. 'At ratio controlling stress corrosion cracking in metals is the ratio magmatic temperature (>600øC) and low pressures, HCI and


HF are very weakly dissociated' [Carmichael et al., 1974, p. 314];' however, according to Carmichael et al. [1974, p. 315],

a cooling fumarolic gas rich in HCI will become strongly acid, and a Kilauea steam condensate of 2 N HCI is a good example of this .... The increasingly corrosive capacity of a cooling fumarolic gas will undoubtedly change its composition as it migrates through rocks on its way to the surface.

Such corrosive fluids may reach sufficiently high concentrations in the crack tip, especially at shallow depths in the earth, to enhance crack propagation.

In order to understand the chemical corrosion associated with magmatic intrusion into cracked host rocks, answers to a number of questions listed below are needed (these questions are similar to those suggested by Pourbaix [1971, p. 23]):

1. What is the composition of the corroding solution? (This is not a simple question, because experiments have shown that the composition of fluid in the tip of the crack may be radically different from the bulk composition of the fluid environment.)

2. What is the initial chemical composition of the pore fluid in the rock and of the fluid inclusions and pores in minerals of the rock? This is particularly important for consideration of domains in which interstitial melts are postulated to exist.

3. What are the chemical constituents and structural (e.g., preferred orientation) characteristics of the rock being intruded?

4. What are the effects on crack propagation rates of grain boundaries in the rock?

5. What are the corrosion products deposited inside the crack cavity? In metals, reaction zones form, and there are oxides deposited behind the advancing cracks. Superficially, they appear to resemble fillings deposited in hydrothermal veins of rocks.

Pourbaix [1971] pointed out that in order to understand stress corrosion in metals it is necessary to know, for the internal cracks, the thermodynamic equilibria for all the solid phases in the presence of the relevant solutions and theoretical regions of stability and instability. In addition, it would be necessary to know the kinetics of corrosion inside the cavities.

Answers to these questions will require extensive information from the disciplines of igneous and metamorphic petrology. It is beyond the scope of this paper to consider them in any detail. It should be possible, however, to develop some qualitative sense of the reactions between magmas, particularly basaltic ones, and intruded host rock.

Slowly propagating cracks and linear volcanic chains. Daly [1933, p. 245] wrote that the alignment of volcanic islands such as 'Hawaii, Samoa, the Society Islands, and other groups, is usually explained by assuming through-going fissures in the crust.' More recently, it has been popular to assign the origin of 'assembly line' volcanic activity (such as that in Hawaii and the Yellowstone region) to the overriding of asthenospheric hot spots by lithospheric plates. It has been shown that the direction and the rate of plate movements deduced from the assumption of fixed hot spots are inconsistent with plate motion measured by other methods [Jackson, 1976]. Con- sequently, it has been necessary to propose that the hot spots are not fixed but migrate. This hypothesis, moreover, fails to explain the nonlinear rate of propagation in some chains [Jackson, 1976].

The 'propagating fracture' hypothesis has been one of several (including 'plume' and 'hot spot' theories) advanced to explain the linearity and age relationships of the Hawaiian

volcanoes (see summary by Dalrymple et al. [1973]). Since the time sequence of progresson within the volcanic chains (such as the Leeward Islands) implies progression at rates of as much as 24 cm/yr [Jackson, 1976], we would like to reexam- ine, in light of the theory of slow crack propagation, the notion that these volcanic progressions may represent periodic intrusions along a propagating fracture which allows magma to rise from the mantle.

Slowly propagating cracks in the lithosphere, accompanied by magmatic activity, have been suggested by several other authors, such as McDougall [1971], Turcotte and Oxburgh [1973], Turcotte [1974], Oxburgh and Turcotte [1974], and /tnguita and Hernan [1975]. McDougall [1971] proposed a model for linear volcanic chains in which magmas are tapped from the asthenosphere by a spreading lithospheric fracture analogous to those proposed by Turcotte and Oxburgh./tnguita and Hernan [1975] have postulated tensional fractures propagated from the African continent toward the Atlantic Ocean to explain the age sequences observed in volcanoes of the Canary Islands. They specify that 'the fracturing process must be repeated in time and propagate with an irregular velocity' [/tnguita and Hernan, 1975, p. 18]. Apparent velocities of the fracture propagation deduced from the radiometric ages for purposes of comparison with plate motions range from 0.9 to 26.7 cm/yr [/tnguita and Hernan, 1975, p. 13].

In laboratory experiments on polycrystalline ceramics, cracks have been measured with propagation rates as slow as a few centimeters per year, even at temperatures which represent a large fraction of the melting temperatures of the materials. Stress corrosion is active at this rate of propagation. Linear sequences of volcanoes thus might form from a slowly propagating fluid-filled crack undergoing stress corrosion. The cracking would not necessarily generate acoustic signals, particularly at the slowest rates of propagation. Seismic events do appear to be correlated with movement of magma in Kilauea volcano [e.g., Koyanagi and Endo, 1971; Koyanagi et al., 1972].

In this model it is not necessary to generate a consistently monotonic sequence of linear volcanic progression; magmas might rise repeatedly through older weakened previously cracked portions of the lithosphere along the same fracture trend, in addition to rising along ihe advancing portion of the crack [Vogt, 1974]. Propagating fractures would not need to be collinear with plate motion or proceed at the same velocity as the plates.

In experiments on ceramics, cracks are observed to grow very slowly, speed up, and bifurcate. Slowly propagating cracks in the lithosphere might explain variations in the apparent velocity of transgressive volcanism and abrupt offsets in the geometry of a volcanic chain (by en echelon or bifurcating cracks).

We visualize a slowly propagating crack in the Pacific plate, the rate and direction being controlled by the moments exerted on the boundaries of the plate. The crack is continuous, but volcanoes are rare. Magma upwells to the surface along the crack only in those few spots where all the conditions for magma intrusion to the surface are satisfied (cf. the 'curtain of fire' effect of focusing eruptions along certain loci in fissures [Jackson and Shaw, 1975]). In this model, magma is available in many areas at the base of the crack. The apparent rate of propagation of a linear chain is controlled by the bending moments in the plate and only indirectly by the velocity of the plate.

One of the configurations for experimen,tal monitoring of very slow crack propagation is a close geometrical analog of


our visualization of a cracking lithospheric plate (Figure 9). Fractures do not necessarily have to propagate from an edge, although it is experimentally convenient to initiate them there. It is also to be noted that fractures can be healed. A fracture

that began at the edge of a plate might eventually be healed at the edge but remain active in the interior of the plate.

C. Stress Corrosion and Geothermal Reservoirs

The Los Alamos Scientific Laboratory is working on the development of a hot dry rock system to extract geothermal energy. In this system, water would be circulated in hot rocks at depth in a large crack produced by hydrofracturing. Demarest [1976] has considered the possibility that slow crack propagation by stress corrosion might destroy a dry hot rock geothermal well.

Using presently available stress corrosion data for ceramics and glass, Demarest found that rocks in the proposed geothermal reservoir would undergo slow cracking that would se- riously limit the operation of the reservoir. However, by an alternate calculation using in situ hydrofracturing measurements and a different theoretical treatment, he found that the stress which can be safely contained in a geothermal reservoir is 'only slightly lower than the stress which causes hydrofracturing' [Demarest, 1976, p. 3].

Since considerable uncertainties remain in the prediction of stress corrosion cracking thresholds for the geothermal system, there is a need for experimental data on slow cracking in rocks in a hydrothermal environment before the conflicting conclusions can be satisfactorily resolved.

IV. CONCLUSION

There is a need for research on the role that stress corrosion

may play in geological and geophysical processes. In this paper we have made some preliminary suggestions of possible applications of the theory of stress corrosion cracking to geological problems.

There is at present a serious question whether stress corrosion is a significant process in the fracture of rocks. Many of the cracks in rocks are intergranular, rather than transgranular, so that the experimental data for slow cracking in glass and single crystals may have little application to understanding the behavior of cracks in the immediate vicinity of grain boundaries between phases of different composition. Stress corrosion in many materials, however, is intergranular [Brown et al., 1972], and therefore the suggested mode of slow cracking in rocks may be correct. The effect of grain size on cracks is uncertain, so the research results on fine-grained ceramics may not be applicable to rocks with coarser grain sizes. Very little is known at present about the effect of pressure on stress corrosion, and therefore the very existence of stress corrosion at depth in the earth is in question. Experi- ments by Wiederhorn and Johnson [1971], however, showed that 'high pressure [6-7 kbar] has no effect on static fatigue properties of glass,' that is, the stress corrosion cracking rates are the same in glass under atmospheric and higher pressures. It is not known whether the same would be true for rocks.

At present there is a lack of experimental information on the nature of slow crack growth in rocks and on the question of whether slow crack growth in rocks is accompanied by acoustic emission. However, there appear to be a number of geological phenomena, some in corrosive environments, in which slow crack propagation is postulated to take place. The intent of this paper is to encourage a new look at these geological processes. The wide variety of materials in which stress corro-

sion cracking has been documented suggests that rocks are not exceptionally immune to this fracture process.

NOTATION

a half-length of crack. ac acceleration of crack. a0 initial crack half-length. ay lattice spacing along y axis. A an environmental species.

A' constant. A•,A:,Aa constants.

b constant.

B crack tip bonds. B•, B:, Ba constants.

B* ruptured state. c stress concentration factor.

c• constant. Co equilibrium concentration of solute in liquid in

the presence of an unstressed surface. d density. D diffusion coefficient of solute in liquid. E Young's modulus.

Eo normal bond energy. E* apparent activation energy. E * zero-stress Arrhenius energy (stress-free activation

energy). fs effective average surface energy per particle (in-

terfacial energy between glass and atmosl•here). f* average free energy of activation per particle for

the fracture process. AFgrowt h free energy of crack growth.

AF* average free energy of activation per mole of particles in the fracture process.

Fs effective average surface energy per mole of particles.

G crack extension force (mechanical stress energy release rate per unit width of crack front).

AG '• change in free energy. h Planck's constant.

AH enthalpy of reaction. AH • activation enthalpy.

k Boltzmann's constant. k', k" constants.

k ø corrosion rate of material under zero stress. k,• stress concentration facttor. K stress intensity factor.

K, stress intensity factor for mode I fracture. K,c critical stress intensity factor.

Kxscc threshold stress intensity factor. I one interplanar atomic spacing across the crack

plane. m molecular mass of gaseous species. n constant.

n' order of rate-limiting chemical reaction. N number of chemical bonds.

NA surface density of crack plane bonds. [OH-] concentration of hydroxyl ions.

PA ø gas pressure at crack mouth. PA • gas pressure at some reference state.

P pressure. PE potential energy.

R gas constant. S entropy.

3S • activation entropy.


SE surface energy. (t) mean time to failure.

t time. to constant. T absolute temperature. u constant.

U total energy of the system. UK kinetic energy. Us stored potential elastic strain energy.

Uo* collection of energy terms. Us surface energy. U, energy by which bond is stretched by stress at

crack tip. v crack velocity.

v' local velocity of corrosion normal to crack interface.

Vo preexponential kinetic factor. v•, v: constants.

t•r reaction-limited crack velocity. t•t transport-limited crack velocity. t•r terminal velocity.

V(T) slowly varying preexponential factor. V • activation volume.

VL velocity of longitudinal sound waves. VM molar volume of glass. Vs shear wave velocity.

WL work done by loads that displace the outer boundary of a body.

a constant.

• constant. 'r reversible surface energy of ideal brittle solid. F fracture surface energy.

F' energy of material-environment interface. r• molecules of environmental species A. K attenuation factor.

K' constant.

X average distance that particles must separate to go from initial, or unfractured, state to fractured state.

• chemical potential. v Poisson's ratio. p radius of curvature of crack (root radius).

amax maximum tensile stress at crack tip. a,, tensile stress at crack tip. ar•' maximum stress bond can exercise between two

atoms.

a stress.

ac critical load strength. aic stress needed for fracture in inert environment. a r strength.

aeq stress at crack tip in thermodynamic equilibrium state.

•b time to failure at 50% of the stress a•c. •2 atomic volume of solid.

Acknowledgments. We thank R. Lee Aamodt, Don L. Anderson, Edward S. Grew, Everett D. Jackson, Christopher H. Scholz, Gene Simmons, and Johannes Weertman for comments on an early draft. One of us (P.C.G.) thanks A. S. Teteiman for a useful course on fracture. Detailed comments by two anonymous reviewers of a previous version of this paper are gratefully acknowledged. However, the authors accept full responsibility for the conclusions (and any errors) in this paper. This research was supported by NSF grant DES75-04869 and U.S. Atomic Energy Commission contract AT(04-3)-34. Pub- lication 1442 of the Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California.

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(Received June 10, 1975; accepted July 14, 1976.)

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