cleavage and creep fracture of rock salt

$: Cleavage and creep fracture of rock salt$
Pergamon Arfa mater. Vol. 44, No. 9, pp. 3553-3565, 1996

Copyright ((2 1996 Acta Metallurgica Inc. Published by Elsevier Science Ltd

PI1 S1359-6454(96)00004-3 Printed m Great Britain. All rights reserved 1359-6454196 $15.00 + 0.00

CLEAVAGE AND CREEP FRACTURE OF ROCK SALT-f

K. S. CHAN’, D. E. MUNSONZ, S. R. BODNER’” and A. F. FOSSUM4$ ‘Southwest Research Institute, San Antonio, TX 78238; ‘Sandia National Laboratoriesg, Albuquerque,

NM 87185, U.S.A.; 3Technion, Department of Mechanical Engineering, Haifa, Israel; and 4RE/SPEC Inc., Rapid City. SD 57709, U.S.A.

(Received 26 July 1995; in revised ,form 12 December 1995)

Abstract-The dominant failure mechanism in rock salt at ambient temperature is either cleavage or creep fracture. Since the transition of creep fracture to cleavage in a compressive stress field is not well understood, failure of rock salt by cleavage and creep fracture is analyzed in this paper to elucidate the effect of stress state on the competition between these two fracture mechanisms. For cleavage fracture. a shear crack is assumed to cause the formation and growth of a symmetric pair of wing cracks in a predominantly compressive stress field. The conditions for wing-crack instability are derived and presented as the cleavage fracture boundary in the fracture mechanism map. Using an existing creep fracture model, stress conditions for the onset of creep fracture and isochronous failure curves of specified times-to-rupture are calculated and incorporated into the fracture mechanism map. The regimes of dominance by cleavage and creep fracture are established and compared with experimental data. The result indicates that unstable propagation of cleavage cracks occurs only in the presence of tensile stress. The onset of creep fracture is promoted by a tensile stress, but can be totally suppressed by a high confining pressure. Transition of creep fracture to cleavage occurs when critical conditions of stress difference and tensile stress for crack instability are exceeded. Copyright mc 1996 Acta Metallurgica Inc.

1. INTRODUCTION

Natural, bedded salt formations are being considered as host rocks for the permanent disposal of nuclear waste. It is presumed that long-term creep of salt over time would result in complete encapsulation and isolation of the waste material from the environment. However, complete waste isolation can be maintained only when there is an effective sealing system in the shafts that connect to the underground repository. The presence of damage in the form of microcracks in salt can increase the permeability and therefore the potential for fluid flow around the sealing system. The development and healing of damage or microcracks in salt are thus important factors to be considered in the design and performance assessment of seals.

In underground rooms and shafts, the microcracks may be introduced during excavation or formed as a consequence of creep under low confinement pressures, since the stresses encountered in the repository are predominantly compressive. However, low-level tensile stresses might be encountered at certain locations in the salt structure [l]. Thus, a possibility exists that cleavage cracks may be induced or creep-initiated microcracks may propagate by

tWork supported by U.S. Department of Energy (DOE), Contract No. DE-AC04-94AL85000.

ICurrently at Sandia National Laboratories, Albuquerque, NM 87185, U.S.A.

$A DOE Facility.

cleavage under the action of low-level tensile stresses. Unstable propagation of cleavage cracks in salt can affect adversely the effectiveness of the sealing system and the structural integrity of the repository.

Studies of creep response of rock salt under triaxial compression have revealed that the creep damage process is pressure dependent and dilatational [2-51. Both the onset of tertiary creep and creep rupture can be delayed or suppressed by the presence of a high confining pressure [3-S]. Physically, creep damage in rock salt occurs in the form of microcracks that exist either within grains or at grain boundaries [l, 6, 71. During creep at low confining pressures, some of these microcracks that slide and propagate by shear may develop wing tips that are aligned parallel to the maximum compressive stress axis, CT, [I, 6, 71. Wing cracks may also be nucleated on the tips of dislocation pileups, which act like shear cracks, by the Stroh process [8]. Opening of these wing cracks would lead to dilatational flow [4, 51. However, at high confining pressures, these microcracks may be closed due to high normal compressive and frictional stresses acting on the crack surfaces and thus could not propagate either by opening or sliding. As a consequence, inelastic flow due to microcracks is absent and the resulting inelastic deformation, arising from dislocation flow mechanisms [9], is isochoric and independent of confining pressure [4, 51. A set of constitutive equations for describing the inelastic flow and damage evolution due to creep and creep- induced microcracks in rock salt from the Waste

3553

3554 CHAN et al.: FRACTURE OF ROCK SALT

Isolation Pilot Plant (WIPP) site was formulated by the authors and reported in earlier publications [3-S, 10, 111.

The occurrence of cleavage in salt subjected to tension at ambient temperature is well documented [12-141. Nucleation of cleavage cracks in salt by dislocation mechanisms was summarized by Strotzki and Haasen [14]. A fracture mechanism map con- taining these three cleavage regions was constructed in a plot of normalized stress and homologous temperature for rock salt by Gandhi and Ashby [13]. In a recent investigation [15], cleavage fracture in WIPP salt was studied by axial compression of thin disks on edge and the fracture process was documented using high-speed video photography. The indirect tension experiments indicated that cleavage occurred at the center of the thin salt disk where tensile stresses were induced as the result of axial compression. The cleavage crack propagated as wing cracks in the maximum compressive stress (axial) direction. The cleavage fracture strength of WIPP salt was determined to be about 2 MPa (1.96 f 0.18 MPa). At tensile stresses below the cleavage fracture strength, WIPP salt was found to exhibit primary, secondary and tertiary creep behaviors.

The initiation and propagation of wing cracks in an elastic solid subjected to compression have been analyzed by a number of investigators [16-211. The wing crack results of Nemat-Nasser and Horii [16-181 indicated that brittle and ductile modes of fracture are possible. A brittle-ductile diagram depicting the transition of brittle-to-ductile fracture in a normalized stress plot of KIc/zy& versus a,/a, was proposed [18], where KIc is the fracture toughness, ry is the yield stress in shear, lu is the half length of the shear crack, and cr3 and 0, are the minimum and maximum principal stresses, respectively. The calculated failure diagram contains a brittle region, a ductile region and a transition region [18]. In another study [21], a wing-crack model was developed and used to construct a fracture mechanism map for rock salt, which depicts failure regimes in the principal stress space that include (1) elastic fracture by wing-crack initiation; (2) wing- crack propagation and coalescence; (3) plastic yielding; and (4) creep deformation. Creep damage and fracture were not considered in these studies, however.

The objective of this paper is to present results of an investigation that examined the conditions for the onset of cleavage and creep fracture in rock salt. In the first part of this paper, a wing-crack analysis is performed to identify the critical stress conditions for the onset of unstable cleavage fracture. In the second part, an existing constitutive model that describes coupled creep and damage in rock salt is used to develop the failure curve for the onset of creep fracture as well as isochronous failure curves for specified times-to-rupture. Based on these results,

representations of the fracture mechanism map for depicting cleavage and creep fracture in rock salt are then constructed. The calculated fracture mechanism map is compared with experimental data and previous calculations in the literature.

2. UNSTABLE WING-CRACK FRACTURE

Several recent investigations have considered the initiation and propagation of wing cracks from a shear crack in a compressive stress field [1621]. A rigorous formulation of this crack problem has been given by Nemat-Nasser and Horii in terms of singular integral equations, with the solutions obtained by numerical means [l&18]. Closed-form, but approximate estimates of the stress intensity factors at the tips of the wing cracks have also been provided by Horii and Nemat-Nasser [18], and by Ashby and Hallam [20]. The approach of these investigators will be followed here to obtain the stress intensity factors for both short and long wing cracks. The approximate stress intensity factor solution for the wing cracks will then be used to establish the critical stress conditions for the onset of unstable propagation of wing cracks by cleavage.

Figure 1 shows a schematic of the crack problem considered. A shear crack or dislocation pileup of length 21, is subjected to a predominantly compressive stress field of principal stresses cr,, gZ and u?, with compression taken to be positive. Specifically, cr, is the maximum principal stress in the positive (compressive) direction, g3 is the minimum principal stress and cr2 is the intermediate principal stress. As the stress increases, sliding along the surfaces of the shear crack (or a dislocation pile-up) causes the formation of a pair of symmetric wing cracks at the tips of the shear crack, as shown in Fig. 1. Wing cracks developed at the tips of a shear crack are generally curved [18], Fig. l(a). For simplicity, the wing cracks are assumed to be straight and aligned parallel to the 0, direction, as shown in Fig. l(b). The length of the wing cracks is I, and the angle between the shear crack and the cr, axis is ti.

Analyses of the stress intensity factor for both short and long wing cracks in an infinite plate are considered. The result for the short wing crack is obtained first, which is subsequently to be used in the long wing crack analysis as the limiting condition when the length of wing crack becomes small. Short wing cracks are defined as L < Lo, where L = l/lo, and L, is the initial value of L where the short and long wing crack solutions are matched. The value of Lo is to be evaluated later. For short wing cracks, the mode I stress intensity factor, K,, at the tip of the wing cracks arises from two contributors: (I) the normal tensile stress associated with the crack-tip field of the shear crack, and (2) the external normal stress, g?. By considering the wing cracks as infinitesimal kinks, the contribution to K, due to

CHAN et al.: FRACTURE OF ROCK SALT 3555

the shear crack has been obtained based on the results of Cotterell and Rice [22] as

K, = 2 k,, JI

(1)

where the mode II stress intensity factor of the shear crack, k,,, is given by

ks = (5, - $G)&, with

z, = f(g, - az)sin 2$

(4

(2)

(3)

t t t 01

(b)

-

Fig. 1. Geometries of short and long wing cracks at the tips of a shear crack of length 210: (a) curved short wing cracks, (b) straight short wing cracks, and (c) long wing cracks

subjected to a point force, F.

and

g1 + 03 (T =~+~cos2* n 2 (4)

and p’ is the coefficient of friction; r” and a, are the shear and normal stresses acting on the shear crack, respectively. The contribution of the compressive stress, gjr normal to the wing cracks to the stress intensity factor is [23]

K, = -CT+ (5)

which is added to equation (1) to give

as the expression for the total stress intensity factor due to the shear crack and the lateral stress, crl.

For long wing cracks, a straight crack of length 21 is considered. The loading due to the shear crack is represented as a point force, F, as shown in Fig. l(c). The mode I stress intensity, K,, at the tip of the long wing crack is then computed based on the point force, F, acting at the center of the crack [24] and the confinement stress, cr3, acting over the length of the crack, leading to

K = 2(7, - ~‘a,)& sin $ 1 (7)

where

is obtained by matching the long crack solution with the short crack result at L = Lo.

The growth of the wing crack was assumed to occur when K, = kc. Applying this criterion to equation (7) and rearranging the terms leads to

which can be used to examine the growth of the wing cracks from an initial length of Lo to a large L value (e.g. L > 10).

Figure 2 shows the calculated results of stress differences as a function of the normalized crack length, L = l/lo, at four different confining pressures, P, for the axisymmetric stress state of c, I> gz > gi, with o2 = cri = P. For these calculations, lo = 1 mm, kc = 0.6 MPa 6, and $ = 45 and 70.5” in Figs 2(a) and (b), respectively. These crack angles have been chosen because the experimentally observed crack angle for wing cracks in brittle solids is 70.5” at short crack lengths and 45” at longer crack lengths [18]. The Klc value is the average of experimentally determined values for WIPP salt [25]. The results indicated that the growth of the

CHAN et al.: FRACTURE OF ROCK SALT

WIPP salt, 2sc

I,=tmm

’ .‘c’,‘,““l lo’ ld

NORMALIZED CRACK LENGTH, L

(a)

PzSMPa

P I-2.5 MP.3

P..5MPa WIPP Salt. 25’c K, = 0.6 MPa&i I.=lmm lq = 70.5

10” x 10” lo’ ld

NORMAUZED CRACK LENGTH. L

@I

Fig. 2. Stress difference required for extending wing cracks from an initial normalized crack length of L = I for various confining pressures. Tensile stresses are indicated by negative values of the confining pressure, P. The crack angle, Y, is 45” in (a) and 70.5” in (b), p’ = 0. 10 = 1 mm.

wing crack is stable and requires increasing stress difference values in both the unconfined and con- fined conditions (P > 0). A maximum in the stress difference versus normalized crack length curve is observed only for loading involving a tensile (TV stress (P < 0). This result indicates that unstable wing crack extension occurs only in the presence of a tensile stress acting normal to the wing cracks.

Relations between the stress difference, tensile stress and the normalized crack length at the onset of unstable crack growth are derived from equation (7) by imposing the condition that

dK -_=O a1 (10)

at crack instability. This leads to

-P Jz sin * sin 2* GI - fl3 x(L + K)’ 5 (11)

for the case of frictionless sliding (II’ = 0). This particular value of p’ has been chosen because shear along a slip plane leading to nucleation by dislocation pileup is most likely in salt in the dislocation creep regime. Equation (11) is then substituted into equation (9) to give

n(L + K)’ 5k~

O’ - O3 = (2L + K)& sin * sin 2* (12)

and

p=_-K!%!!L (2L + lc,JZ

(13)

as the expressions for the critical stress difference and tensile stress at the onset of unstable cleavage fracture.

Previously, Nemat-Nasser and Horii [ 16-181 have presented a wing-crack analysis in which the wing cracks were modeled in terms of continuous dislocations and singular integral equations. The approach allowed short wing cracks to be modeled as curved cracks. Since the wing-crack model reported here has been developed based on the assumption of straight cracks, it is therefore an approximate solution only, but has the advantage that explicit expressions of the critical stresses could be obtained. An approximate wing-crack solution has also been reported earlier by Ashby and Hallam [20]. Their solution differs from the present one in that it includes a microcrack coalescence term, which is not necessary for the problem considered here. The accuracy of the present wing-crack model is compared with the results of these previous investigators in Fig. 3, which presents results of normalized stress, cr,,,/?&/Ktc, as a function of normalized crack length, L, which is the ratio of l/C, for various stress ratio of 1 = ~,/a,. The comparison indicates that the present wing-crack model is fairly accurate for L > 0.5. The agreement is best for i, < 0, for which al is tensile. The result indicates that equations (12) and (13) are fairly accurate for L > Lo, where Lo = 0.5.

3. CREEP FRACTURE

Inelastic flow in rock salt can occur by dislocation, microfracture and damage healing mechanisms. Creep due to three different dislocation mechanisms is considered in the constitutive equations formulated by Munson and Dawson [lo], which have been referred to as the Multimechanism Deformation (M-D) model. The M-D constitutive equations have been extended to include continuum, isotropic damage as a fully coupled variable that enhances the stress influence by reduction of the effective area and also contributes directly to the inelastic strain rate


[l 1, 4, 51. The total inelastic strain rate equation thereby becomes pressure-dependent since the sub- sidiary equations include the effect of pressure to suppress damage development, i.e. the opening of microcracks. The extended model, referred to as the Multimechanism Deformation Coupled Fracture (MDCF) model, has been applied successfully for representing the creep and fracture response of WIPP salt subjected to triaxial compression [3-S].

In the MDCF formulation [4, 5, 111, the total strain rate, @:,, for a solid deformed under isothermal

o,(=P) - Approximate SohAm

%(=P) ---- Nemat-Nasrer 8 l-hi (1962)

._ lo-’ 100 lo’ ld


(4

w l--F-t - Approximate Solution 0$=P) ,Y o,(=P) ---- ArhbyhHalhm(1996)

. . loo I ‘~‘-” ’ “‘.‘.‘,“8” m.lLLLikJ

16’ 100 lo’ ld


@)

Fig. 3. Normalized critical stress versus normalized crack length for cleavage fracture by the growth of wing cracks. The approximate solution obtained in this study is compared against (a) solution of Nemat-Nasser and Horri [ 161, who used continuum dislocations and singular integral equations to represent the wing cracks; and (b) approximate

solution of Ashby and Hallam [20].

conditions is given as the sum of the elastic strain rate, i;, and the inelastic strain rate, il. The latter is decomposed into contributions from the various deformation mechanisms and is described by the generalized kinetic equation given by Ref. [5],

(14)

where u&, 02, 02, i&, i; and i: are power-conjugate equivalent stress measures and equivalent inelastic strain rates for the dislocation creep, shear damage and tensile damage mechanisms, respectively. It is necessary to distinguish between shear and tensile damages by separate terms because the kinetics of damage in salt is significantly different under shear and tensile loads. In equation (14), the power- conjugate equivalent stress measure plays the role of a flow potential for each of the deformation mechanisms, and the derivative with stress gives the flow direction. The inelastic strain rate measures for individual mechanisms were formulated as a scalar function of the appropriate stress measure, temperature and internal variables that represent the current states of deformation and damage. A complete description of the formulation of the equivalent stress and inelastic strain rate measures in the dislocation creep term [lo], and the shear damage and tensile damage terms are described elsewhere [5]. Evaluation of the magnitudes of the strain rate contributions for individual deformation mechanisms from experimental data is also presented in those papers [3, 4, lo]. In this paper, the failure loci of cleavage fracture and creep damage are calculated for rock salt. For these calculations, only expressions for the conjugate equivalent stress measures and the damage evolution equations are required. Because of this, expressions for the inelastic strain rate measures are not presented here, but can be found in earlier publications [5, lo].

The conjugate equivalent stress measure for creep, (r&, is formulated based on the stress difference as given by

where (T, and oi are the maximum and minimum principal stresses. The Tresca equivalent stress measure is preferred over that of von Mises because experimental measurements of the flow surface and inelastic strain rate vector are in better agreement with the former formulation [26].

The conjugate equivalent stress measure, a$, for shear damage-induced flow is assumed to consist of two terms as represented by [4, 111,

x L 1, - Cl 3x7 sgn(Z, - 0,) 1 Ih (15)

where 1, is the first invariant of Cauchy stress; the x,s are material constants, sgn() is the Signum


function, and H() denotes the Heaviside step function. The first term represents shear-induced creep damage, which manifests as slip-induced microcracks. Some of these microcracks develop “wing tips” that generate irreversible inelastic strains that add to those originating from dislocation mechanisms. The second term in equation (15), which is of the form off(l, - a,), models phenomenologi- tally the effects of stress state, including triaxial compression or extension, on the sliding of microcracks and the opening of wing-tip microcracks. Specifically, 0, is the maximum principal stress in the positive (compressive) direction, q3 is the minimum principal stress and cr? is the intermediate principal stress. The term (II - cr,) reduces to o2 + (r3, so the effective pressure of confinement would be (cr? + a3)/2. This would correspond to the experimental “confining pressure” in most cases of interest.

The conjugate equivalent stress measure, cr$ for tensile damage-induced flow is taken to be [5],

0;; = -x,cr3H(-(51) (16)

which is intended to represent the opening of microcracks by a tensile stress, cr3. Here, H() is the Heaviside step function. Since the conjugate stress measures for damage depend on the individual principal stress components, an anisotropic influence is present in damage development even though damage is treated as a scalar quantity. Taken together with the basic equation (14) equation, (15) and (16) also control the direction of inelastic straining due to damage and would, in general, indicate different values in the various stress directions. Nonisotropy of inelastic straining is therefore introduced within an overall isotropic format.

Damage development is modeled using the Kachanov damage variable [27] with the damage evolution equation given by

where a; and a; are power-conjugate stress measures; ,xlr, x4, x5, x, (with i = s or t for shear or tensile damage, respectively) are material constants; to is a reference time. For creep under constant stress, integration of equation (17) with time to the failure condition tr, wf, leads to

and

with

for 0 < z < 1. Similar expressions have also been derived for 0 < 1 /z < 1.

For a given loading path signified by a prescribed value of z, the parameters on the right-hand-side of equations (18) and (19) are material constants, except for the time-to-rupture, tf and the critical value of the damage variable, w,, at creep rupture. Previous work has shown that creep rupture of rock salt can be considered to occur at a critical value of wF = 0.15 [.5]. Based on a critical wr value as the creep fracture criterion, equations (18) and (19) give the failure stress for a specified time-to-rupture, and can be used to generate isochronous failure curves [28] in an appropriate stress space when used in conjunction with equations (15) and (16).

For creep under triaxial compression, the tensile damage term, a:, in equation (20) is zero, leading to z = 0. Setting z = 0 in equation (18) results in

as the expression relating the power-conjugate equivalent stress measure and time-to-rupture for triaxial compression. The boundary of stresses below which creep failure would not occur was calculated using equation (21) by setting tr = co, leading to

a$ = 0 (22)

as the condition for the absence of creep damage. The corresponding volumetric strain rate, ill, is also zero for a$ d 0. The implication is that & = 0 and isochoric (nondilatant) flow occurs at stresses below the creep fracture boundary defined by a:; = 0.

4. COMPARISON OF MODEL AND EXPERIMENT

4.1. Cleavage fracture

The cleavage fracture strength of rock salt from the Waste Isolation Pilot Plant (WIPP) site has been measured [15] by indirect tension tests using a Brazilian test. There are additional fracture strength test data from Hunsche [29] for salt from the ASSE mine. Some information about cleavage fracture strength of salt subjected to tension is also available in the literature [12, 141. These experimental results have been used to evaluate the wing-crack model. The wing-crack model requires input of the initial length of the wing crack. For comparison with experimental data, the value of lo was obtained by matching the wing-crack model with the tensile strength of WIPP salt obtained by indirect tensile tests conducted under moderate loading rates. The value of 1, deduced by this method was 7 mm, which is half of the arain diameter of WIPP salt. A comnarison of the


0.0 10.0 20.0 30.0 40.0 STRESS DIFFERENCE, MPa

50.0

Fig. 4. Comparison of calculated and measured values of tensile strength of rock salt as a function of stress difference. Experimental data are from Brodsky [15], Hunsche [29],

Skrotzki and Haasen [I41 and Stokes [12].

wing-crack model with the full set of test data is shown in Fig. 4. The other test data tends to confirm the trend indicated by the model. The calculated curve was based on La = 0.5, lo = 7 mm, $ = 45’ and I& = 0.6 MPa 6 for WIPP salt [25].

The calculated curve shows that the critical tensile stress decreases with increasing stress difference. For uniaxial tension where lr~, - c3 1 = ox, the fracture mechanism changes from wing crack to Griffith crack fracture [30]. Figure 4 shows that cleavage fracture changes from the wing crack to the Griffith crack process at lr~, - 0~1 = crj = 3.38 MPa. This transition point was computed by first obtaining the value of the normalized crack length, L, at which the condition of 1 CJ, - o3 / = o3 was met in the wing-crack solution. The calculation indicated that L = 0.672 and 10, - oil = g3 at 3.33 MPa. The critical tensile stress for Griffith fracture was then calculated using

where the half length, a, of the Griffith crack is taken to be the sum of the length of the wing crack (Ll,,, since L = I/&,) and the projected crack length (lo cos I./I) of the shear crack on the plane normal to cr), leading to

a = lo(cos $ + L). (24)

Using equations (23) and (24), the calculated critical stress difference and tensile stress for fracture of the Griffith crack were both 3.38 MPa, which is in good agreement with the wing-crack results. Comparison of model calculation with independent experimental data in Fig. 4 indicates that the tensile strength decreases with increasing stress difference. The WIPP salt experimental data from the Brazilian indirect tensile tests are also plotted in Fig. 4. The curve marked “wing crack” should go through the average of cleavage fracture at a given stress difference value. Although not exactly in accordance with the best fit to the experimental results, the calculated curve is an

acceptable and possible approximation to the data. The discrepancy between the calculated curve and experimental data is probably because the different rock salts do not exhibit identical cleavage fracture strength at a given stress difference.

Theoretical failure loci for cleavage fracture of rock salt in the principal stress plane have also been calculated using the wing-crack model. The results are compared with experimental data of rock salt in Fig. 5, which shows the fracture loci in the -cr, vs -crZ stress space in which tension is plotted as positive and compression as negative. The tensile strength of rock salt is about 2-4.6 MPa when (T? = 0, as shown in Fig. 5. The experimental data from Hunsche indicate that the tensile strength of ASSE salt decreases with increasing values of compression stress in the intermediate stress, g2, direction. At large compression stresses (aZ > 30 MPa), the tensile strength of ASSE strength decreases to very low values (<0.5 MPa). The dependence of tensile strength is qualitatively reproduced by the wing crack calculation shown in Fig. 5. The initial length of the wing crack. 2&, used in this calculation was taken to be 14 mm, which was the grain size, d, for WIPP salt. The grain size for ASSE salt was not reported [29]. The grain size of the salt in the studies compiled by Skrotzki and Haasen [14] was in the 0.7-6 mm range, while it was between 0.3 and 2 mm for the salt studied by Stokes [12]. These investigators reported tensile strength as a function of grain size. Results were extrapolated to obtain the tensile fracture strength for d = 14 mm, shown in Fig. 5. The extrapolated and actual values differ only by small amounts. The agreement between model calculation and experimental data is reasonable.

In Fig. 5, the experimentally determined tensile

25~ c

-15

I

Fig. 5. Comparison of calculated failure loci of rock salt against experimental data from Brodsky [15], Hunsche [29],

Skrotzki and Haasen [I41 and Stokes [12].


Table I. Damage model constants for WIPP salt

%I =6 x: = 9 .Yis = 5.5 X3r = 40 14 = 3 I< = 231.0 MPa x, = 15.15 MPa X6 = 0.75 x: = I MPa fil = I s

ui = 0.15

fracture strength of rock salt appears to be independent of the intermediate principal stress, 02. This is because only the portion of the experimental data at low -02 values is presented in Fig. 5 in order to show a detailed view of the fracture loci in the biaxial tension stress space. When a complete set of experimental data with -oZ ranging from 0 to -40 MPa is included, the experimental tensile strength of rock salt definitely decreases with increasing compressive stress, cr:. This dependence of tensile fracture strength on the intermediate principal stress, oZ, can be readily inferred from the results shown in Fig. 4.

4.2. Creep fracture

An isochronous failure curve is a stress locus of points with identical times of rupture. Stress states that lie inside an isochronous failure curve for a particular time-to-rupture, t,, would have a time- of-rupture greater than t,, while those outside the isochronous failure curve would have a time-to- rupture less than t,. Isochronous failure curves can be obtained from the MDCF model through equations (18t(20) once the particular time-to-rupture, tr, is specified and the critical damage value of ur = 0.15 is invoked.

The isochronous failure curves for WIPP salt subjected to indirect tension have been calculated using equations (18) and (19), the appropriate equations for the conjugate stress measures and the model constants for WIPP clean salt, which are presented in Table 1. These material constants were determined elsewhere by fitting the model to experimental creep curves [5]. The calculated isochronous failure curves, for the times-to-rupture of 1 h, 1 day and 1 year, are rectangular stress contours that exhibit the characteristic of decreasing times-to-rupture with increasing stress levels. The isochronous curves are plotted together with the computed cleavage fracture boundary and compared against cleavage fracture and tensile creep data of WIPP salt in Fig. 6. The computed cleavage fracture boundary represents the stress locus at which cleavage fracture would occur for a shear crack of length 21, = 14 mm. The superposition of the computed isochronous failure curves and the cleavage fracture boundary in Fig. 6 allow one to delineate the failure regimes of creep rupture and cleavage

fracture. At stresses below the cleavage fracture boundary, cleavage fracture would not occur and failure would occur by creep damage. At stresses at or above the cleavage fracture boundary, cleavage fracture would intervene and replace creep rupture as the dominant failure mechanism. Excellent agreement was obtained between model calculations and experimental data in that the change in fracture mode observed experimentally occurs (within experimental scatter) almost exactly where the theoretical transition boundary occurs. Both the experimental data and model calculations indicate that at tensile stresses below the cleavage fracture boundary, failure of salt would occur by creep rupture, whereas at the boundary, failure is by cleavage fracture. Existing creep cracks with lengths of the order of the grain size (14 mm) can trigger cleavage when the combination of tensile stress and stress difference exceeds the cleavage fracture boundary, shown as the solid line in Fig. 6. The solid line was calculated based on the average value of the cleavage fracture strength of WIPP salt, which was 1.96 + 0.18 MPa. As a result, some of the cleavage fracture data points in Fig. 6 fall below the solid line, giving the appearance that cleavage fracture occurs below the cleavage fracture boundary. In reality, a distribution of flaw size exists in salt and the cleavage fracture boundary would be represented more appropriately by a scatter band, which may be incorporated into the current calculation (solid line) if the flaw distribution in WIPP salt is known and a probabilistic treatment of cleavage fracture is employed.

For creep under triaxial compression, the boundary of stresses below which creep failure would not occur was calculated using equation (22) and the material constants for WIPP salt listed in Table 1. The calculated creep failure boundary predicted from

!,=lhr

___________________. I WIPP salt 25°C

30.0

20.0

10.0

i - Cleavage Fracture Boundary Wing-Crack Model

lswhronous Faikr Curves MDCF Mcdel

Experimental Data Cleavage Fracture creep Rupture

,Griilith Crack ; :: I

I I I I

1.0 2.0 3.0 4.0

TENSILE STRESS, MPa

Fig. 6. Calculated cleavage fracture boundary and isochronous failure curves in the stress space of stress difference and tensile stress compared with experimental data of WIPP salt obtained under cleavage fracture and creep conditions. The cleavage fracture boundary is

= 7 mm, Lo = 0.5, +!I = 45” and KIc = 0.6 MPa for cleavage fracture in salt.


o.o~‘- h h Ib I ’ 0.0 5.0 10.0 15.0 20.0 2.5.0

CONFINING PRESSURE, o,(MPa)

Fig. 7. Comparison of calculated creep failure boundary against creep data (Van Sambeek et al. [33], Senseny [32], Fossum ef al. [34], Wawersik and Hannum [31]) and quasi-static fracture data (Wawersik and Hannum [31]) of

WIPP salt.

the material constants obtained from the original creep curves is compared with experimental creep

[31-341 and quasi-static fracture [31] data of WIPP salt in Fig. 7. Dilatational flow with damage is predicted for stresses above the failure boundary, while isochoric creep without damage is predicted for stresses below the failure boundary. With the exception of a few data points, the calculated failure boundary agrees with the experimental results. The discrepancy between the calculated curve and the experimental data might be due to the presence of small amounts of clay particles, which act as damage initiation sites, in otherwise “clean” salt. On the other hand, there are no test points for which isochoric creep was observed above the calculated boundary curve.

4.3. Fracture mechanism map

Theoretical fracture mechanism maps have been calculated using the MDCF and the wing-crack models. The cleavage fracture boundaries have been calculated based on the wing-crack model and the Griffith fracture criterion [30]. The failure boundaries of the creep damage mechanisms are isochronous failure curves and have been calculated using the MDCF model. The isochronous curve with a failure time of infinity is used to define the boundary between regions where creep occurs with and without damage. Isochronous curves of a time-to-rupture of 1 h are used to depict failure regimes where creep crack growth is expected to dominate due to the short failure time. The regions between tr = oc, and t,- = 1 h depict regimes where failure by creep (or plastic flow) occurs with concurrent microcrack damage. On this basis, two forms of fracture mechanism map have been constructed: (1) in the stress space

of stress difference, 0, - 03, and confining pressure, c3 (= c2 = P), and (2) in the stress space of - 0, and -cr3 (= -02).

The fracture mechanism map in Fig. 8, which is constructed in the stress space of stress difference, 01 - g?, and confining pressure, crl (=gz = P), contains seven different failure regimes: (1) Region A, where isochoric creep occurs without rupture; (2) Region C, where dilatational flow (creep or plasticity) occurs with microcrack damage; (3) Region D, where fracture occurs by wing-crack extension; (4) Region E, where creep failure occurs by a mixture of tensile and shear damage in the forms of microcracks; (5) Region F, where the dominant failure mechanism is tensile creep crack growth; (6) Region G, where cleavage fracture by the extension of wing cracks dominates; and (7) Region H, where cleavage or brittle inter-granular fracture (BIF) by extension of the Griffith crack dominates. At the extremely low stress region of C, creep rupture could occur by damage with little or no dilatation. The damage behaviors in Regions G and D are similar at high stress differences. The portion of the boundary separating these two regions might not be distinguish- able in practice and was represented in terms of a dashed line in Fig. 8. Comparison of the calculated fracture mechanism map with experimental data of WIPP salt and ASSE salt [29] is presented in Fig. 9, which shows that the calculated cleavage and creep fracture boundaries agree in general with both sets of experimental data.

The calculated fracture mechanism map for the stress space of -0, and -03 is presented in Fig. 10. In this case, only the creep fracture and cleavage fracture boundaries are shown. The dashed/dotted line represents the stress loci within which isochoric creep without damage occurs. The solid line in Fig. IO represents the cleavage fracture loci outside which unstable cleavage fracture occurs. The region between the dashed/dotted line and the solid line depicts stress states where dilatational creep with

I

-5.0 0.0 5.0 10.0 15.0 20.0

0, VW

Fig. 8. A calculated fracture mechanism map for rock salt in the stress difference, 0, ~ g3, and confining pressure,

u,(rr, = 0~ = P), stress space.


-5.0 0.0

Fig. 9. Calculated fracture mechanism map compared against experimental data of WIPP salt (Van Sambeek et al. [33], Senseny [32], Fossum rt al. [34], Wawersik and

Hannum [31]) and ASSE salt (Hunsche [29]).

damage and creep fracture would occur after an unspecified time of creep. The isochronous failure curves for 1 h, 1 day and 1 year, which lie between creep fracture and cleavage fracture boundaries, are shown in Fig. 11. The isochronous curves terminate at the cleavage fracture boundaries indicating the dominant fracture mechanism changes from creep rupture to cleavage fracture. The calculated fracture mechanism map is compared with experimental data of WIPP and ASSE salt in Fig. 12, which shows good agreement between models and experiment.

The fracture mechanism map for rock salt plotted in the normalized stress and homologous temperature space was updated and is shown in Fig. 13. Figure 13 is a modification of the map reported previously by

Calculated Failure Boundaries - Cleavage Boundary

Wing-Crack Model -.-Creep Fracture Boundaty

MDCF Model

10.0 - a,=0

,?I = 03

-50.0 - ?.‘.

I Fracture

. 25°C if=” 1. I * I. I. I I. I -50.0 -30.0 -10.0 10.0

- Us, MPa

Fig. 10. Calculated fracture mechanism map for rock salt in the oi and ~,(a, = ~2 = P) stress space.

Calculated Failure Boundaries -Cle,avogs Boundary, WlngGmck Model -.-Creep ,=m*m. MOCF Model

-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0

- og, MPa

Fig. 11. Calculated creep fracture region with calculated isochronous failure curves,

Gandhi and Ashby [ 131. Most of the fracture regions are identical; minor changes to cleavage 1, 2 and 3 have been made after incorporation of additional experimental data. Cleavage 1, 2 and 3 are accompanied with little, small (Z 1%) and large (Z 10%) amounts of plastic strains, respectively. The cleavage 2 boundary was calculated based on the Griffith fracture criterion using 2a = 14 mm, where the crack length, 2~7, was assumed to equal the grain size, 14 mm, of WIPP salt. The cleavage fracture boundaries corresponding to 2a = 0.4 m and 2a = 2 m are also shown. Cleavage fracture will not occur at stresses below a given boundary unless a

Calculated Failure Boundaries

-Cleavage Boundary, Wing-Crack Model -.- Cnep F,acturs. MDCF Model

Triaxlal Compresslon Data Cleavage Fracture Data 0 Incompressible Creep A Indirect Tension . DIlatIonal Creep 0 lndlmct Tsnsion . QuasCStatic Fracture 0 TensIon

x Tension

Creep without

-50.0 -30.0 -10.0

- D3, MPa 10.0

Fig. 12. Comparison of calculated fracture mechanism map against experimental data of rock salt. The cleavage fracture data are from Brodsky [15], Hunsche [29], Skrotzki and Hassen [14] and Stokes [12]. The triaxial creep data are from Van Sambeek et al. [33], Senseny [32], Fossum et al. [34] and Wawersik and Hannum [31]. The quasi-static triaxial

comparison data are from Wawersik and Hannum [31].


Rock WI

~ynamlc Fmctum

. Hunsche 0 BndSky

Clewago 21E.I.F.2 ?? stokes

Cleavage YS.I.F.3 0 stokss 0 Skmlzkl and Haas

Tmnsgmnular creep Fractum (MIcrovoids)

Homologous Temperature, T/Tm

Fig. 13. Fracture mechanism map for rock salt in the normalized stress versus homologous temperature plot obtained by updating the Gandhi and Ashby version [13] with recent experimental data and by adding the calculated Cleavage 1 boundaries for several values of crack length, 2~. The experimental data are from Brodsky [15], Skrotzki and

Haasen [14], Hunsche [29] and Stokes [12].

crack longer than the critical crack length exists for that stress in the salt. Although the stress conditions of the repository are largely compressive, and therefore involve the normalized compressive stress of the fracture mechanism map companion to Fig. 13, there are small regions around underground openings that contain small tensile stress components. These possible tensile components are believed to be contained within the shaded area shown in Fig. 13. As is apparent, a relatively large existing initial, critical crack length is necessary for a cleavage crack to propagate in the available tensile stress fields of the WIPP facility. Within the shaded region, in the absence of a critical crack, creep induced trans- and inter-granular microcracks can develop; however, the time to rupture for these conditions is extremely long.

5. DISCUSSION

Previously, Ashby et al. [35] has reported a fracture mechanism map for rock salt by plotting the Mises equivalent stress versus the confining pressure, P, for axisymmetric triaxial compression. In this fracture mechanism map, failure mechanisms such as yielding, necking, cleavage and plastic failure are depicted without a creep fracture regime. In a separate paper, Hallam and Ashby [21] reported a failure map for rock salt in the 0, and o3 stress space. The failure mechanisms depicted in this failure map included

yielding, creep deformation, wing-crack initiation, and wing-crack propagation and coalescence. In- itiation of cracks by creep and creep fracture has not been considered by these authors. In this work, we are concerned mainly with cleavage and creep fracture in the comparatively low confining pressure regimes. Because of permeability considerations, whether or not dilatational inelastic flow accompa- nies creep and creep fracture is important and, consequently, is represented in the fracture mechanism map given in this work. Since different types of failure are considered, the present fracture mechanism map representations are somewhat different from the ones presented by Ashby et al. [35] and by Hallam and Ashby [21].

Despite the difference in emphasis, there are some similarities in the failure plots presented in this investigation and those of Hallam and Ashby [21]. As a result, a comparison between these two results is warranted. According to Hallam and Ashby, initiation of wing cracks by the elastic stress field of the shear crack, i.e. fracture initiation, occurs at 120,211

O’ = &[(l - A)(1 +:;I’2 - (1 + J”)$] (25)

where

and the corresponding critical stress for fracture propagation is given by [20, 211

O’ = &[{ 1 - 3. - p((Z 1”) - 0.43X) (27)

{0.23L + l/(&l + L)“))]

which has been developed by Ashby and Hallam [20] based on the consideration of stable propagation and coalescence of wing cracks. Using p’ = 0, lo = 7 mm, and a critical normalized crack length, L, of 1.5, the failure curves for fracture initiation and fracture propagation were calculated using equations (25)-(27) and the results compared with those from this study in Fig. 14. As shown in Fig. 14, the failure curve for elastic wing-crack initiation lies in the center of the fracture mechanism map where incompressible creep without damage occurs. Except in the tensile stress regime, the creep fracture boundaries lie outside the elastic fracture initiation boundaries. This finding suggests that creep deformation relaxes the elastic stress of the shear crack and delays the development of wing cracks in a creeping matrix. The failure curves for fracture propagation lie between the creep fracture boundary and the unstable cleavage fracture boundaries. The calculated fracture propagation curve is in agreement with the quasistatic fracture data of WIPP salt [31]. The shape of the fracture propagation curve is similar to that of the


isochronous curve for 1 day, as shown in Fig. 14. Thus, there is agreement between the present results and those of Hallam and Ashby in the sense that failure by microcrack damage dominates in this regime.

Figures 10 and 12 are also consistent with the brittleductile diagram introduced by Horii and Nemat-Nasser for rock fracture [18]. The creep deformation regime where compressible creep without damage occurs corresponds to the ductile regime, while the cleavage fracture regime is obviously the brittle region. Accordingly, the creep fracture regime would be the transition region. Within the creep fracture regime, the time-to-rupture varies with individual stress states. Because of this, isochronous failure curves are required to distinguish the propensity for creep failure at individual stress states in this region.

A more complete picture of the fracture process in rock salt emerges as the result of this study. From Fig. 14, it is clear that the elastic stress field associated with a shear crack or a dislocation pileup is sufficient to introduce cleavage wing cracks at low levels of stress difference in rock salt if creep deformation by dislocation mechanisms does not occur at ambient temperature. Because of creep, the onset of wing- crack formation is delayed and occurs at higher levels of stress difference than that calculated based on the elastic solution. Additionally, the wing cracks may also be closed by a confining pressure. As a result, the stress difference at the onset of creep fracture increases with increasing confining pressure, Fig. 9. Since the propagation of a wing crack is always stable in the presence of a confining pressure (Fig. 2),

Present (nvestigation . Umsl-Stailc Fracture Data -Cleavage Boundary Hallam and Ashby

Wing-Crack Model ----- Fracture lnltlatlon - ‘-Creep Fracture Boundary --- Fracture Propagation

MDCF Model -x-lsochronous Curve (t, z 1 day)

10.0 - 0, = 0 q 3 D3

I. I ( I. I. I. t, 1, I

-50.0 -30.3 -10.0 10.0

- as, MPa

Fig. 14. Comparison of calculated boundaries for cleavage fracture and creep fracture, and isochronous curves for a rupture time of one day against fracture initiation and propagation boundaries calculated based on the model of Hallam and Ashby [21] and quasi-static triaxial compression fracture data of WIPP salt from Wawersik and Hannum

[311.

the relevant damage processes in the creep fracture regime are the initiation, stable propagation and coalescence of wing cracks for triaxial compression. In the presence of a small tensile stress, creep-induced wing cracks can propagate as unstable cleavage cracks if the combination of stress difference and tensile stress exceeds that required for crack stability. The tensile stress required to trigger unstable cleavage fracture decreases with increasing stress difference. Because of this, both stress difference and tensile stress levels are important in considering possible failure by cleavage in a salt structure. Comparison of theory and experiment indicate that the critical stress difference and tensile stress are well described by equations (12) and (13).

The current model is formulated in terms of the principal stresses. For a generalized stress state, a transformation of the stress tensor into the principal stress planes must be performed. Once on the principal stress planes, the transformed stress tensor can be compared with the failure boundaries to determine whether or not a particular fracture condition has been exceeded. This approach has been applied successfully for predicting creep failure of a salt structure via finite-element structural analyses

1361.

6. SUMMARY

Failure of rock salt by cleavage and creep fracture has been analyzed to elucidate the effect of stress state on the competition between these two fracture mechanisms. The regimes of dominance by cleavage and creep fracture have been established, presented in the form of a fracture mechanism map, and compared against experimental data. The results indicate that unstable growth of cleavage cracks occurs only in the presence of a tensile stress acting normal to the wing crack, and the critical tensile stress for triggering cleavage fracture decreases with increasing stress difference. The creep fracture boundary depends on the stress difference and the pressure. Stresses below the creep fracture boundary would undergo isochoric creep without damage or failure, while stresses at the cleavage fracture boundary would fail by cleavage. Stresses that fall between the creep and cleavage fracture boundaries would fail by accumulation of creep or plastic damage in the form of wing cracks. These wing cracks might propagate as cleavage cracks if they are subsequently subjected to stresses corresponding to the cleavage fracture boundary. The calculated fracture boundaries are in good agreement with experimental data on rock salt.

Acknowledgements-This work was supported by U.S. Department of Energy (DOE), Contract No. DE-AC04- 94AL85000. The authors thank Dr Nancy Brodsky, formerly with REjSPEC and currently at Sandia, for the use of previously unpublished data of WIPP salt. The clerical assistance of MS Julie A. McCombs and MS Patty A. Soriano and the editorial assistance of MS Deborah J.


Stowitts. all of Southwest Research Institute, are 21. appreciated.

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