damage-induced nonassociated inelastic flow in rock salt

Pergamon

International Journal of Plasticity, Vol. 10, No. 6, pp. 623-642, 1994 Copyright © 1994 Elsevier Science Ltd Printed in the USA. All rights reserved

0749-6419/94 $6.00 + .00

0749-6419(94)E0017-9

D A M A G E - I N D U C E D N O N A S S O C I A T E D I N E L A S T I C F L O W

IN R O C K S A L T

K.S. CHAN,* N.S. BRODSKY,'~ A.F. FOSSUM,'~ S.R. BODNER,*§ and D.E. MUNSON~

*Southwest Research Institute, iRE/SPEC Inc., and ~tSandia National Laboratories¶

Abstract-The multimechanism deformation coupled fracture model recently developed by CrlAN et al. [1992], for describing time-dependent, pressure-sensitive inelastic flow and damage evolution in crystalline solids was evaluated against triaxial creep experiments on rock salt. Guided by experimental observations, the kinetic equation and the flow law for damage-induced inelastic flow in the model were modified to account for the development of damage and inelastic dilatation in the transient creep regime. The revised model was then utilized to obtain the creep response and damage evolution in rock salt as a function of confining pressure and stress difference. Comparison between model calculation and experiment revealed that damage-induced inelastic flow is nonassociated, dilatational, and contributes significantly to the macroscopic strain rate observed in rock salt deformed at low confining pressures. The inelastic strain rate and volumetric strain due to damage decrease with increasing confining pressures, and all are suppressed at sufficiently high confining pressures.

I. INTRODUCTION

Natural salt deposits are considered desirable host rocks for permanent disposal of radioactive waste because the creep characteristics of natural salt allow closure of the disposal room, leading to eventual encapsulation of the radioactive waste and minimizing the possibility of leakage and contamination of the environment. A potential failure mode in the salt deposits is tertiary creep, which can result in time-dependent fracture in bedded natural salt deposits (MuNsON et al. [1989]). Recent concern over the potential for the development of creep-induced damage zones in the shafts and rooms excavated from bedded salt deposits has led to the need for constitutive models that are capable of describing the time-, temperature-, and pressure-dependent inelastic flow behavior of rock salt in the presence of creep-induced microcracks or cavities. This need has gen- erated considerable interest in developing reliable, t ime-dependent constitutive equations for describing the creep response of rock salt under triaxial compression (e.g., SENSENY [1983]; LANGER [1984]; WALLNER [1984]; MUNSON & DAWSON [1984]; AUBERTIN, GILL, & LADANYI [1991]).

Most of the constitutive equations for salt, however, do not treat creep-induced damage that usually appears in the form of microcracks (BRACE, PAIn~DI~O, a SCHO~ [1966]; CRUDEN [1974]). In triaxial compress ion of britt le, elastic solids, microcracks tend to form and propagate in the max imum compressive stress direction (BRACE & BO~,mOLAKIS [1963]; BRACE, PAULDINO, & SCHOLZ [1966]; HORn & NEMAT-NASSER [1985]; ASHBY &

§Permanent address: Technion, Dept. of Mech. Eng., Haifa, Israel. ¶A U.S. DOE facility.

623

624 K.S. CHAN et al.

HALLAM [1986]). Creep-induced microcracks in rock salt are expected to form and behave in a somewhat similar manner. The sliding and opening of creep-induced microcracks under triaxial compression creep can lead to an increase in the inelastic strain rate in the transient creep regime (CRUDEN [1974]), inelastic dilatancy (BRACE, PAULDINC, & SCHOLZ [1966]), and tertiary creep (KACHANOV [1958]; CRUDEN [1974]). This coupling of creep deformation and damage on the creep response of rock salt has been treated only recently in constitutive models (CRISTESCU ~ HUNSCrm [1992]; CHAN, BODNER, FOSSUM & MUNSON [1992]; AUBERTIN, SOAOULA, & GIrL [1993]) that incorporate both creep and damage processes for describing the time-dependent inelastic response of rock salt, and for modeling of rupture of rock salt (NAm & SINOH [1973]).

The multimechanism deformation coupled fracture (MDCF) constitutive model by CHAN et al. [1992], is an extension of the multimechanism deformation model (M-D) proposed by MUNSON and DAWSON [1984] to include creep-induced damage in the form of microcracks and cavities. The extended model couples both creep and damage mechanisms for describing time-dependent, pressure-sensitive inelastic flow in rock salt under nonhydrostatic triaxial compression. In particular, both damage and dislocation flow processes contribute to the overall inelastic strain rate. As in the M-D model, the creep rate that originates from dislocation mechanisms is essentially pressure-insensitive and incompressible. The damage-induced inelastic strain rate, on the other hand, is pressure- sensitive and dilatational and is considered to arise from the opening of microcracks present in the material. A quantitative measure of damage is described in terms of the continuum damage variable, co, (KACHANOV [1958]), while the development of damage with inelastic deformation is provided through an appropriate evolution equation. The MDCF model allows prediction of the complete creep curve, including previously un- modeled tertiary creep, for rock salt subjected to nonhydrostatic triaxial compression. Extensive evaluation of the MDCF model, however, has not been possible because of the lack of creep data obtained under low confining pressures and for durations suffi- cient to reach the tertiary creep regime.

Two assumptions were made in the original development of the MDCF model. The first is that the development of creep-induced damage leads to the onset of tertiary creep and that creep-induced damage is not expected to be present, at least not in significant amounts, in the primary or secondary creep regimes, as observed in most metals and alloys. The second assumption is that the flow law is associated and derivable from a single flow potential. The objective of this article is to present results of an investigation that evaluates these two aspects of the MDCF model by conducting triaxial creep experiments for rock salt and by comparing experimental results with model calculations.

The article is divided into four parts. The first part describes the MDCF model and modification of the kinetic equation and the flow law for damage-induced inelastic flow. Highlights of the experimental results of Waste Isolation Pilot Plant (WlPP) salt under creep constitute the second part of the article. In the third part, the associativity of damage-induced inelastic flow in WIPP salt is examined. In classical plasticity, the term "associativity" is used to indicate that the direction of the plastic strain rate associated with a stress point on the yield surface is normal to the yield surface. Although the concept of a yield surface is not used in the current formulation, effective work-conjugate stress measures that can be considered as flow potentials for dislocation slip and damage- induced inelastic deformation are used. As a consequence, the concept of associativity is still meaningful. In this article, the term a s s o c i a t i v i t y means the same effective work- conjugate stress measure that is used in the scalar kinetic equation is also used to derive

Inelastic flow in rock salt 625

the flow law so that normality applies when one considers the effective work-conjugate stress measure as being the flow potential. By the same token, the term "non-associativity" means that the effective work-conjugate stress measure used in the scalar kinetic equation is different from the one used in deriving the flow law, so that normality does not apply, leading to the implication that the work-conjugate stress measure in the scalar kinetic equation does not represent the flow potential. In the fourth part, the revised model is used to illustrate the suppression of the creep response and damage evolution in rock salt by confining pressure. Both the experimental and theoretical results are to demonstrate that damage-induced inelastic flow in rock salt is present in all three stages of the creep curve and contributes to a significant part of the macroscopic strain rate observed in rock salt crept at low confining pressures. Furthermore, damage-induced flow leads to inelastic dilatation. A nonassociated flow formulation is required to represent these deformation characteristics of damage-induced flow in rock salt.

I!. COUPLED CREEP-DAMAGE CONSTITUTIVE MODEL

The MDCF model is a constitutive model that provides a continuum description of the creep response and the associated damage evolution in crystalline solids, such as rock salt. Both dislocation motion and creep-induced damage are assumed to contribute directly to the macroscopic inelastic strain rate. The generalized average kinetic equation for the coupled creep and damage-induced inelastic flow is given as (FossoM et al. [1988]; CHAN et al. [1992])

ao~q aO;'q ~1 __ t~Oij ~ceq -'1- aO------~'j ~eq, (1)

• c and "~ are work-conjugate where ~ / i s the inelastic strain rate tensor; OeqC, Oeq, eeq, ee q equivalent stress measure and equivalent inelastic strain rates for the dislocation and damage mechanisms, respectively. Elastic strains resulting from stiffness change due to creep damage are expected to be small when compared to the inelastic strains accom- panied by creep; consequently, damage-induced elastic strains are ignored in this article.

The effects of creep damage on inelastic flow are modeled in two ways in the MDCF model. The softening effect associated with the reduction of effective load-bearing area due to damage is modeled using the continuum damage mechanics approach. The Kachanov damage variable, ~, is used as a scalar measure of the damage level with the development of damage during deformation represented by an evolution equation (KAcrIA~OV [1958]). Furthermore, damage also directly contributes to the inelastic strain rate through the opening of microcracks and microvoids (BRACE ~ BO~,mOLAKIS [1963]; WA~SrI & BRACE [1966]; BRACE, PAULDING, & SCHOLZ [1966]; HUTCHINSON [1983]; COSTIN [1983]; HORII & NEMAT-NAssER [1985]; HANSEN & FOSSUM [1986]; ASHBY & HALLAM [1986]; KEMEDY [1991]). This is modeled in the MDCF model in terms of a kinetic equation for damage-induced inelastic flow that is additional to and independent of the kinetic equation for dislocation flow mechanisms. Details of the work-conjugate stress measures, kinetic equations, the flow law, and the evolution equation for damage in the MDCF model are summarized as follows. Note that compression is taken to be posi- tive in this article.


II. 1. Work-conjugate equivalent stress measures

The work-conjugate equivalent stress measure, 0"~q, for dislocation-induced flow is the Tresca stress,

C 0"eq 10"1 -- 031, (2)

where o~ and 0 3 are the maximum and minimum principal stresses, respectively. The Tresca stress measure is preferred over von Mises' because experimental measurements of flow surface and inelastic strain rate vectors are in better agreement with the former formulation (MuNsON et al. [1989]).

The work-conjugate equivalent stress measure for damage-induced flow is assumed to consist of three terms as represented by (CHAN et al. [1992])

11 _ Ol Ix6 o~ = IOl - 0.31 - X2XTsgn(I, - trl) 3xTs~-(-I[ - 0.,) - x 1 0 . 3 H ( - ° 3 ) ' (3)

where I~ is the first invariant of Cauchy stress, the xis are material constants, sgn( ) represents the signum function, and H ( ) denotes the Heaviside step function. The first term on the right-hand side of eqn (3) represents the driving force for shear-induced damage, which manifests as slip-induced microcracks with "wing-tips" or grain boundary cracks whose opening and sliding lead to irreversible inelastic strain in addition to that originating from dislocation mechanisms. The third term represents the opening of microcracks by the maximum tensile stress, a3. The second term in eqn (3), which is in the form of f ( I~ - al) , represents the suppression of the opening and sliding of microcracks by a confining pressure.

II.2. Kinetic equation for dislocation f low

The kinetic equation representing the creep rate, ~q, due to dislocation mechanisms is based on the M-D model. In this formulation, the inelastic strain rate is given by (MtINSON & DAWSON [1984])

k~q =F~s (4)

where F is the transient function representing transient creep behavior, and ks, the overall steady-state strain rate, is the sum of individual steady-state strain rates, ksi, for three independent dislocation mechanisms. The steady-state strain rates of the individual mechanisms, each of which is taken to be thermally activated, are (MvNso~q [1979]; MUNSON & DAWSON, [1984])

~' = A ' e - O ' / R r ( ~ ( l a - o ~ ) f ' (5)

ks~ A2e_Q2/RT( O )n2 = (6) U(1 - co)


I( ° ~s3 = IHI(B, e - e ' /Rr + B2e-Qz/Rr) sinh q (1 - oJ) Iz

oo) (7)

where the As and Bs are constants; Qis are activation energies; Tis the absolute temperature; R is the universal gas constant; o is the generalized stress, which is taken as the stress difference and given by eqn (1); # is the shear modulus; nis are the stress exponents; q is the stress constant; and a0 is the stress limit of the dislocation slip mechanism. The dislocation climb mechanism, designated by subscript 1, dominates at low stress and high temperature. The undefined mechanism, designated by subscript 2, dominates at low stress and temperature, and the glide mechanism, denoted by subscript 3, controls at high stress for all temperatures. The 1 - c0 term in eqns (5-7) represents the reduction in the effective loading-bearing area due to the presence of damage.

The transient function, F, (MuNsON, Fosstru, & SENSENY [1989]) is

F = 1, ~'= eL (8)

which is composed of a work-hardening branch, an equilibrium branch, and a recovery branch. In eqn (8), A and 6 represent the work-hardening and recovery parameters, respectively; ~" is the hardening variable; and e~ is the transient strain limit. Only isotropic hardening is considered. The temperature and stress dependence of the transient strain limit is represented by (MtrNSON, FOSSUM, & SENSENY [1989]),

K cT{ o )m c , = o~ ~ , i , ( 1 - o ~ ) ' (9)

where Ko, c, and m are constants. The evolution rate, ~:, of the isotropic hardening variable ~" is governed by

[ = ( F - 1)(k~), (10)

which diminishes to zero when the steady-state condition is achieved.

11.3. Kinetic equation for damage-induced f low

The kinetic equation for damage-induced inelastic flow originally proposed in the MDCF model was (CnAN et al. [1992])

628 K.S. CHAN el al.

where Cl, c2, and n 3 are material constants. Eqn (11) shows a linear relation between Ce% and ~0 aside from the modifying factor (1 - co) on the stress. The form of eqn (11) was selected based on theoretical results of HUTCHINSON [1983] that indicate that the inelastic strain rate due to microcracks depends linearly on the microcrack density for a dilute concentration of constrained microcracks. Because of the linear relation, damage growth leads immediately to tertiary creep.

Two discrepancies exist between eqn (11) and the creep damage process observed in WIPP salt: (1) creep damage accumulates in the transient creep region, but does not lead to tertiary creep immediately, and (2) damage-induced inelastic strain rate exhibits a transient behavior similar to dislocation-induced creep. To account for the damage behavior observed in WIPP salt, eqn (11) has been modified and is given by the expression as follows (FossvM et al. [1993]):

keq = F°~k~, (12)

where

F~° = Fexp[ c4(° - c5) (13)

is the transient function for damage-induced inelastic flow, and

~'2 = cl w°eC3'~ [sinh ( CZOe% H ( °~q) ) ] - w)tz (14)

is the kinetic equation for damage-induced flow during steady-state creep, with

C 1 = c o ( B l e ( - Q , / R 1 ) + B2eC-Q2/RT~) , (15)

and Co, c2, c3, c4, c5, and n3 are material constants; Wo is the initial value of the damage variable, w. The Bs and Qs are constants in dislocation glide mechanisms. This particular form of kinetic equation allows ke~q to exhibit a transient behavior by virtue of the transient function, F ~, which is related to the transient function for creep, F, through eqn (13). The form of ks is formulated such that it remains fairly constant for small values of w so that tertiary creep is not prematurely activated. A potential draw- back of eqn (12) is that ~e% is sensitive to Wo. On the other hand, eqn (14) draws atten- tion to the importance of the initial damage on the inelastic flow behavior of WIPP salt.

Damage development is described by the damage evolution equation (BODNER [1985]; BODNER & CHAN [1986]; CHAN et al. [1992])

= - - w In [~reqH(rleq)] - h(w,T, I1), X5

(16)

where x3, x4, and x5 are material constants, and h (w, T, I~ ) is the damage healing function whose form remains to be determined. The damage healing term h (~0, T, I1 ), is expected to depend on the damage variable, w, temperature, T, and the first invariant, I1, of Cauchy stress. Recent work (BRODSKY [1993]) showed that healing of damage in WIPP salt was rapid and the kinetics could be described in terms of an Arrhenius equa-


tion. Incorporation of this recent result into the healing term in eqn (16) is currently underway. The healing term, however, is taken to be zero in this article.

For creep under a constant stress and without initial damage (O~o = 0) and healing, eqn (16) may be integrated with respect to time to obtain the current value for o~, leading to (CHAN et al. [1992])

= exp - [°~qH(°~'q)] x3t , (17)

which has the feature that damage development is very small for small values of [ a~q H ( a ~ ) ] x3 t, but increases rapidly when [ a~'q H ( a e ) ] x3 t becomes larger and exceeds a certain value. Thus, eqn (17) provides a feature that is equivalent to the use of a critical stress-time criterion for damage development. According to this formulation, a thresh- old stress of a~'q below which creep rupture would never occur, is not predicted. In- stead, creep failure would always occur when a~q > 0, but the time to rupture might be extremely long. Creep rupture is predicted not to occur only for a~q _< 0, a condition which depends on the stress difference and the confining pressure, as defined in eqn (3).

I1.4. Flow law

In the MDCF model, inelastic flow was originally assumed to be associated for both dislocation and damage mechanisms. The associated flow law was obtained by taking the stress derivative of the work-conjugate equivalent stress measures shown in eqns (2) and (3) and substituting the results into eqn (1), leading to

~ij • (CeCq -I- ~e~q) [blSij + b2tij] - k e ~ q [ b 4 ( ~ i j - mij) + bsnij], (18)

where

sij =ai i - ½ o~, 6ij (19)

is the stress deviator, 6ij is the Kronecker delta, and

tii = si, s k j - 2 j260 (20)

is the deviator of the square of the deviatoric stress (PRAGER [1945]). The coefficients in eqn (18) are given by,

co s (2~ ) bl = 4~2cos(3 ¢ ) (21)

sin b2 = J2 cos(31t') (22)

x2 6[ lx6-, b 4 = T 3 s g n ( I i - - O l ) (23)

b5 = xl H ( - a 3 ) , (24)

630 K . S . CHAN et al.

where q~ is the Lode angle, J2 is the second invariant of the deviatoric stress, mij = dol /doi j , and nij = das/doij are given in an earlier article (CHAN et al. [1992]).

Instead of associativity, it is also possible that damage-induced inelastic flow is nonassociated. To examine this possibility, a different work-conjugate equivalent stress measure, O~'q, is assumed and used in conjunction with eqn (1) to obtain the flow law for damage-induced inelastic flow. The proposed function for aeq is

* X 2 .268 Oeq ---- I ° , - - ~31 - - - 3 - - [ l , - - o , ] - - x , o 3 H ( - ~ 3 ) , ( 2 5 )

which leads to the same flow law as described in eqn (18), but with

b4 - - X 2 X8 3 ' (26)

where x8 is a material constant, while h i , b2, and b5 are given in eqns (21), (22), and (24). Note that eqn (25) is modified from eqn (3) by taking x6 -- 1 and adding an additional constant, xs. For both associated and nonassociated flow formulation, the volumetric strain rate, tkk, can be obtained f rom eqn (18) and is given by

kkk = --2 b 4 keq, (27)

since skx = t~k = 0, 6kk = 3, and mkk = 1. The main difference between the associated and nonassociated formulat ions is that the values for b4, which are given by eqns (23) and (26), respectively, are not the same.

111. T R I A X I A L C R E E P E X P E R I M E N T S

Complete experimental creep curves, including tertiary creep, of clean salt f rom the natural bedded salt formations at the W I P P site in southeastern New Mexico are avail- able for comparison with theoretical calculations. Even though the data base is not extensive, these curves can be analyzed through the MDCF model for axial, lateral, and volumetric strains. In these tests, conventional triaxial compression creep experiments were conducted at a constant stress difference of 25 MPa with various values of confining pressure (P), ranging f rom 0.5 to 15 MPa. The load up times ranged f rom 1 to 16 min. The testing technique, specimen characterization, and preliminary evaluation of the axial strain have already been given by FossuM et al. [1993]. These test results will be summarized as background for use here in a more complete analysis based on the modified MDCF model.

The axial, lateral, and volumetric strains were measured as a function of time of creep. Figure 1 shows the axial and volumetric strains observed in W I P P salt tested at a stress difference of 25 MPa for confining pressures of 1 and 15 MPa. Similar creep curves for the same stress difference but for confining pressures of 2 and 3.5 M P a are shown in Fig. 2. The influence of the confining pressure on the creep response is evident. As shown in Fig. 1, deformat ion occurred more readily at a confining pressure of 1 MPa than at 15 MPa. Tertiary creep and volumetric strain were evident at 1 M P a pressure, but not at 15 MPa pressure. Another important observation in Fig. 1 is that creep dam-


z

15.0

9.0

3.0

- 3 . 0

- 9 . 0

Axial Strain (P = 1 MPa)

Axial Strain (P = 15 MPa)

(P = 15 MPa)

WIPP Salt (~1 - cr 3 = 25 MPa 25"C

Volumetric Strain (P = 1 iPa ) - - MDCF Model

....... Experiment

- 1 5 . 0 , ! , t , I

0.0 5 . 0 1 0 . 0 1 5 . 0

TIME: (DAY)

Fig. 1. Experimental creep data of WIPP salt tested at cr 1 - 03 = 25 MPa under a confining pressure of 1 or 15 MPa and comparison with model calculations.

WIPP Salt 25.0 al - a3 = 25 MPa

25"C Axial Strain Axial Strain 20.0 MDCF Model /r~. (PJ 2MPa) ~.. (P = 3.5 MPa)

....... i.iiii i. ii.....iii i !~.!~.~..." ' ' ~ io.o

5.0

Volumetric Strain (P = 3.5 MPa)

0.0 - -

~ i ~ ~ . . , ~ : ~ . ~ , , , , ~ Volumetric Strain (P=2 iPa) - 5 . 0 , I , I ~ ' ¢ ' ; I ~ I

0.0 10.0 20.0 30.0 40.0

(DAY) Fig. 2. Experimental creep data of W I P P salt tested at al - 03 = 25 MPa under a confining pressure of 2 or 3.5 MPa and comparison with model calculations.


age, as measured by the volumetric strain, developed early in the primary creep regime and accumulated throughout all three stages (primary, secondary, and tertiary) of creep in W I P P salt tested at 1 MPa. Similar observations were made on the results shown in Fig. 2.

The value of the minimum strain rate, /:rain, observed in W I P P salt crept at a stress difference of 25 MPa was not a constant, but decreased with increasing confining pressures, as shown in Fig. 3. Specifically, the minimum strain rate was reduced significantly by a small increase in the confining pressure at low pressure levels (0.5-3.5 MPa). At higher confining pressures (3.5-15 MPa), the minimum strain rates were essentially independent of the confining pressure, Fig. 3. Similar results have also been obtained for a different rock salt (HuNSCHE & SCHULZE [1993]). The decrease in the minimum strain rate also corresponded to a decrease in the volumetric strain with increasing confining pressure, Fig. 4. The experimental evidence suggested that creep damage contributed to a higher deformation rate in W I P P salt at low confining pressures. In addition, the damage-induced inelastic flow was reduced when creep damage was suppressed by a higher confining pressure. At a sufficiently high confining pressure, creep damage was totally suppressed and the minimum strain rate corresponded to the creep rate due to dislocation motion only, whose value is essentially independent of the confining pressure. Thus, the apparent strong influence of pressure on the triaxial creep response of W I P P salt at low confining pressure appears to be a manifestation of the suppression of damage-induced inelastic flow by a confining pressure.

WlPP Salt 25,C

10 -e

o~ f ~ r } Experimental Data

tu-

10.?

N

10-a 0.0 3.0 6.0 9.0 12.0 15.0

CONRNING PRESSURE, MPa

Fig. 3. Experimental and calculated minimum strain rates as a function of a confining pressure for creep of WIPP salt at o~ - (73 ~ 25 MPa. The pressure dependence at low confining pressure is due to contribution of damage to the macroscopic strain rate. Influence of confining pressure on the minimum strain rate is drasti- cally reduced at high confining pressures (after FOSStJM et al . , [1993]).


g ,<

E 3 0 >

1.0

0.0

-t .O

- 2 . 0

- 3 . 0

o

o

e~ = 0

- 4 . 0 , I , I 0.0 5.0 15.0

WIPP Salt ~1- Ga = 25 MPa 25'C o Experiment

MDCF Model (~=~)

I

1 0 . 0

CONRNING PRESSURE, MPa

Fig. 4. Measured and calculated volumetric strains for creep of W l P P salt tested at al - o3 = 25 MPa under various confining pressures. The volumetric strains were obtained at the time when the minimum strain (creep) rate was observed. Calculation is based on the nonassociated flow formulation.

IV. NONASSOCIATIVITY OF DAMAGE-INDUCED INELASTIC FLOW

The question concerning the associativity of damage-induced flow was examined by comparing model calculations against experimental data of axial, lateral, and volumetric strains for creep of W I P P salt at low confining pressures. Material constants for W l P P salt are summarized in Table 1. Model constants related to dislocation mechanisms (M-D model) were determined previously using creep data obtained at a confining pressure of 15 MPa (MuNsoN et al. [1989]). The same set of material constants was used and is shown at the left column in Table 1. Additional material constants related to damage-induced flow and damage evolution are shown in the right column of Table 1. They were determined by fitting the model to the creep curves of W l P P salt at low confining pressures, (e.g., 1 MPa) or by assuming values thought to be physi- cally realistic (e.g., O~o = 1 x 10-4). The goodness of individual values of material constants was judged by the reproducibility of experimental results by the model.

Note that in Table 1, two values of x6 are reported; x6 = 1 for P < 1.5 MPa, and x6 = 0.65 for P > 1.5 MPa. The pressure dependence of damage-induced inelastic flow was nonlinear for W l P P salt; hence, x6 = 0.65. However, the b4 term in the associated law (eqn 23) becomes singular at zero confining pressure ( P = 0) when x6 < 1. As a result, x6 was taken to be 1 for P < 1.5 MPa, and x6 = 0.65 for P > 1.5 MPa. This restriction on x6 does not apply in the b4 term for the nonassociated flow law (eqn 26). However, to make a one-to-one comparison, the same set of x6 values was used for both flow laws. For the nonassociated flow law, x8 = 0.1. All other constants were identical for both formulations.

Fig. 5 shows comparison of calculated and measured creep curves for W l P P salt tested at 25°C subjected to tr~ - a3 = 25 MPa and P = 1 MPa, while Fig. 6 presents results for the lateral and volumetric strains. Model calculations were performed based on the


Table 1. Material constants f o r W I P P (c l ean ) salt

E la s t i c P r o p e r t i e s M - D M o d e l C o n s t a n t s D a m a g e M o d e l C o n s t a n t s

/~ 12.4 G P a E 31 .0 G P a ~, 0 .25

A~ 8 .386 E 2 2 sec 1 .k-? = 10,0 QI 1.045 X l0 t X3 = 5.5

n I 5 .5 X 4 = 3 .0 BI 6 .086 E6 sec i x5 = 1.0 × 1013 ( M P a ) 5 5-sec

; % = 1 fo r P < 1 . 5 M P a x(, = 0 .65 fo r P > 1.5 M P a x7 = I M P a .v~ = 0. I

A 2 9 .672 E l 2 sec ~ co = 7 .0 x 107 Q2 4 .18 × 104 c 2 = 100.0

n2 5 .0 c 3 = 10.0 B2 3 .034 E-2 s e c - I c4 = 6 .0

c5 = 25 M P a

o o 20 .57 M P a n3 = 3 q 5 .335 E3

R 8 .3143 J / m o l ° K c% = 1 × 10 -4

m 3 .0 K o 6 .275 E5 c 0 . 0 0 9 1 9 8 K

- - 1 7 . 3 7 - - 7 . 7 3 8

6 0 .58

associated flow law. Despite the good agreement obtained for the axial strain (Fig. 5), large discrepancies were observed for the lateral and volumetric strains (Fig. 6). For this case, b4 = 2.5 and ekk = 5~ffq according to eqn (27). Since the axial strain rate (~) and the creep rate (~q) are accurately modeled, the damage-induced inelastic strain rate, ~;0q, must be correct since k = ~Cq "t- I~e~°q. Consequently, the overpredictions of the lateral and volumetric strains by the associated flow formulation indicated that the b4 value was too large, meaning that the damage-induced inelastic flow in WIPP salt was nonassociated. This notion was supported by the results shown in Figs. 5 and 7, which shows that calculated axial (Fig. 5) lateral and volumetric strains (Fig. 7) based on the nonassociated flow formulation are in good agreement with experimental data. For this case, b 4 -= 0.33 and kkk = 0.67 ~e~q according to eqn (27). From these results, it is clear that nonassociativity in WIPP salt arises from damage-induced inelastic flow only.

V. PRESSURE D E P E N D E N C E OF D A M A G E - I N D U C E D INELASTIC FLOW

Comparison of model calculations and experimental results for axial and volumetric strains is shown in Fig. 1 for confining pressures of 1 and 15 MPa, with corresponding values for the damage variable, co, shown in Fig. 8. When comparing calculational results based on average parameter values, it should be noted that individual creep curves may show an experimental scatter of about a factor of 2.5 for identical test conditions. Fig- ures 1 and 8 illustrate that damage accumulation at 1 MPa leads to (1) a higher deformation rate (damage-induced inelastic strain rate) in the transient creep regime, (2) an inelastic dilatational strain in the transient creep regime, and (3) tertiary creep. At 15 MPa confining pressure, creep damage was suppressed as co = 0, tertiary creep was


25.0 WIPP Salt al-a3 = 25 MPa P = 1 MPa MDCF Model

20.0 25.C - - ~

;5.o Experiment - ~ ~

~ 10.0 0

5,0 ~

0 . 0 - - I ~ I , I , I , I

0.0 1.0 2.0 5.0 4.0 5.0

TIME (DAY) Fig. 5. Compar i son of calculated and experimental creep curves for W I P P salt tested at a stress difference o f 25 MPa and 1 MPa confining pressure.

absent, and the corresponding volumetric strain was nil. Thus, the inelastic flow behavior of WIPP salt is dilatational at P -- 1 MPa due to damage accumulation, but is incompressible at P = 15 MPa when damage is totally suppressed. Comparison of model calculations and experimental results for confining pressures of 2 and 3.5 MPa is shown

0 . 0 i

-10.0

- 2 0 . 0

_z ty ~ - 3 o . o

- 4 0 . 0

- 5 0 . 0

!- Experimental Volumetric Strain ~t~. .~_A", ," , ,A A " A Z - , A ~N~. ~ u O 0 0 0 0 0 0 0 ~ 0 & /~ /~ A A &8. \ \ o ooooooo

~ '- Experimental Lateral Strain

WlPP Salt . . . . ~l~31M~5a MPa ~ /- Lateral Strain 25"C - - MDCF Model

0.0 1.0 2.0 3.0 4.0

TIME (DAY)

Fig. 6. Compar ison of calculated and experimental lateral and volumetric strains for W I P P salt crept at a stress difference o f 25 M P a and 1 MPa confining pressure. Calculation is based on the associated flow law.

636 K . S . CHAy et al.

v z

ev

0 , 0 •

Volumetric Strain _, .o

_ . o

- 6 . 0

WlPP Salt -8 .o c l - o 3 = 25 MPa

P = 1 MPa 0 \ 25'C

- 1 0 . 0 lines: MDCF Model o symbols: Experiment

- 1 2 . 0 , I , I , I ,

0 . 0 1.0 2 . 0 3 . 0

TIME (DAY)

I

4.0

Fig. 7. Comparison of calculated and experimental lateral and volumetric strains for WIPP salt crept at a stress difference of 25 MPa and I MPa confining pressure. Calculation is based on the nonassociated flow law.

in Fig. 2. At these intermediate pressure levels, di latat ional flow was present, but its magnitude was reduced. The corresponding results of the damage variable for these cases are also presented in Fig. 8. The shape o f the ~0-time curves for low confining pressures (1 and 2 MPa) is similar to those presented earlier by AUBERTIN et al. [1993].

The various inelastic strain rate componen t s that consti tute the calculated inelastic strain rate, k, for creep o f W I P P salt at 1 M P a pressure are shown in Fig. 9 as a func-

5 130

h i r 3

C3

0.3

0.2

0 .1

o.o

WlPP Salt 0.1 - 0"3 = 25 MPa 25"C MDCF Model

P = 1 MPa P = 2 MPa .--~ x~7. /~ j

~ P = 3.5 and 15 MPa

- -0 .1 i I , I i I

0 . 0 1 0 . 0 2 0 . 0 3 0 . 0

TIME (DAY)

Fig. 8. Values of the d a m a g e var iable , ~, co r r e spond ing to the creep curves shown in Figs. 1 and 2.


10 s

t .}

td

r- lff 7 t.O

WlPP Salt o ~ 3 = 25 MPa P = 1 MPa 25'C MDCF Model

• " ¢ " ( 0

1 0 ~ , I , l . l . i , I T l l | i I i l ' l ' h l i l l | • I • l , h l . i l l l l . I , t , h i , l l l l |

10 ~ 10 .3 10 .2 10 -1 10 0

DAMAGE VARIABLE

Fig. 9. Comparison of the creep rate (~q) and damaged-induced inelastic strain rate (~q) to the macroscopic strain rate (~) showing ee~ is a significant portion of the macroscopic strain rate.

tion of the damage variable, c0. In Fig. 9, the curve labeled as ~e~q was calculated via eqn (12), with ~' given by eqn (14); the former represents the inelastic strain rate induced by creep damage. The creep rate due to dislocation motion is shown as eeq,'C while

is the sum of ~e~q and ~q. The value of ~' is essentially constant at low values of c0, but increases rapidly when ¢0 exceeds 0.02.

Three important observations were made of the results shown in Fig. 9. First, ~q was a significant part of ~. Second, ~q was greater than ~q over the whole range of damage values (or equivalently, the time of creep) examined. Third, the initial decreases of e, ~eq, and ~q at low values of the damage variable, ¢o, were due to reduction in the value of the transient function, F, with time of creep (or damage), which is presented in Fig. 10. The value for the transient function, F, at the minimum strain rate was =4, compared to 1 for steady state creep. Based on these results, it is evident that damage- induced inelastic flow dominates, or contributes significantly to, the deformation response of WIPP salt at low confining pressures, and this domination begins in the primary creep region. As damage accumulates, tertiary creep is initiated without the presence of a true steady-state creep, as evidenced by the relatively large value for F when the minimum value of ~ was observed (see Fig. 10). In the presence of damage, the minimum strain rate does not correspond to true steady state creep, but is formed as the consequence of termination of transient creep by the onset of tertiary creep due to damage.

The minimum strain rates corresponding to the "secondary" regime of the calculated and experimental creep curves are presented in Fig. 3, which show that for both cases, the minimum creep rate increases rapidly with decreasing confining pressures for pressures less than approximately 5 MPa. This increase in the minimum strain rate is due to the contribution of damage-induced inelastic flow. At pressures above 5 MPa, the


u_

Z o_ I,-- ¢0 Z

LL I.-- Z LLI 0") Z <~ cl- I-"

102

101

WIPP Salt ~1-~ = 25 MPa P = 1MPa 25"C

10 0 , i . t . l . = . = = , , ! . , . i . J . l . = t l l | . i . i . l . i . J , i l l . J . i . , . , . i ] 1 1 1

10 .4 10 .3 10 .2 10 -1 10 0

DAMAGE VARIABLE

Fig. 10. Values o f the transient function, F, as a funct ion of the damage variable.

calculated minimum strain (creep) rate is dependent weakly on the confining pressure. This weak dependence, which arises from the dependence of diffusion on confining pressure, was modeled using the relation (FRosT & ASrIBY [1982])

k = ~ 0 e x p [ - ( P - Po)V*/kT], (28)

where P0 and k0 are the reference pressure (15 MPa) and the creep rate at P0, respectively; V* is the activation volume for diffusion, and k is the Boltzmann's constant. For rock salt, V* = 8.53 x 10 -29 m 3 (FRosT 8, AShBy [1982]). Since dislocation climb is generally considered to be controlled by diffusion, eqn (28) was applied to the two climb mechanisms in the M-D model, but not to the dislocation glide mechanism. For the pressure levels examined, the effect of pressure on the activation volume is negligi- ble and smaller than the experimental scatter. However, this pressure effect is included in the model calculations for the sake of completeness.

Comparison of the calculated and measured inelastic volumetric strains for various confining pressures is presented in Fig. 4. Calculated based on the nonassociated flow formulation, the volumetric strain values reported in Fig. 4 represent the volume strains at the time when the creep strain rate reaches its minimum value. The calculated inelastic volumetric strain is largest at P = 0.5 MPa, and it decreases with increasing pressures. At confining pressures above -- 5 MPa, the inelastic volumetric strain is zero due to suppression of damage by confining pressure. The calculated results are in fair agreement with experimental data, Fig. 4. Thus, pressure-sensitive flow in WIPP salt at low confining pressures is the consequence of creep damage contributing directly to the macroscopic inelastic strain. Suppression of creep damage by a high confining pressure eliminates damage-induced flow and inelastic dilatation.


VI. DISCUSSION

Important findings of this investigation on WIPP salt include (1) the presence of significant damage-induced inelastic flow in the transient creep regime, (2) the suppression of damage-induced inelastic flow and dilatation by a confining pressure, and (3) the nonassociativity of the flow formulation required for describing the deformation characteristics of the damage-induced inelastic flow. A possible origin of the pressure- sensitive, nonassociated damage-induced inelastic flow in WlPP salt is shown schemat- ically in Fig. 11, which shows the postulated creep damage processes at high and low confining pressures. As indicated in an earlier paper (CHAN et al. [1992]), creep damage in WIPP salt is envisioned to occur in the form of microcracks that exist either within grains or at grain boundaries. At high confining pressures (Fig. 11 a), these microcracks are closed and they could not propagate either by opening or sliding due to high normal compressive stresses and frictional stresses acting on the crack surfaces. As a consequence, damage-induced inelastic flow is nil (~e~q = 0), and the resulting inelastic deformation, arising entirely from dislocation flow mechanisms, is plastically incompressible and independent of confining pressure. On the other hand, the microcracks can propagate in the shear mode during transient creep at low confining pressure, Fig. 1 lb (HoRn & NEMAT-NASSER [1985]; ASHBY & HALLAM [1986]; KEMEDY [1991]). Slid- ing or shearing of these microcracks can account for the deviatoric component of the damage-induced inelastic strain rate, ~e~. Furthermore, some of the microcracks that slide and propagate by shear may develop wing-tips that are aligned parallel to the maximum compressive stress axis, Ol. Opening of these wing-tip cracks would account for the dilatational component of the /~e~ (BRACE, PAULDING, & SCHOLZ [1966]; COSTIN [1983]; HORn & NEMAT-NASSER [1985]; AstraY & HALLAM [1986]; KEMEDY [1991]). Appar- ently, there is no one-to-one correspondence between shearing of microcracks and opening of wing cracks so that damage-induced inelastic flow is nonassociated. One possible

O" 2 el" (~3

/

/

/

~1 / Clicr°SocedraWck ng'Tip

/ L.- Non-Sliding / ~ Mierocrack

, 4._ O~ or ~ 3 02 or 0"3 ._.= /

f

1' O" t

(a) High Confining Pressure

(~t Sliding Microcrack ,~ / / ~ With Open Wing-'lips

/ ~ Sliding Microcrack

ol

(b) Low or No Confining Pressure

Fig. 11. Damage mechanisms envisioned during creep deformation of rock salt under triaxial compression: (a) closure of microcracks by a high confining pressure; and (b) generation and opening of microcracks at low or no confining pressure. Sliding or shearing of microcracks leads to the deviatoric component, while opening of wing-tip microcracks lead to the dilatational component of the damage-induced inelastic strain

"oJ r a t e , F.eq.

640 K.S . CHAN et al.

explanation is that only a fraction of microcracks that shear develop wing-tips, and that the remaining sliding or shearing microcracks do not develop wing-tip cracks, but propagate along their original paths with plastic dissipation by dislocation slip at crack tips. This rationale is supported by the smaller b4 value for nonassociated flow (b4 = 0.33 for nonassociated flow and b4 = 2.5 for associated flow). Since sliding of the shear microcracks and opening of the wing-tip cracks are opposed by a confining pressure, damage-induced inelastic flow, ~e~q should be pressure dependent and should be totally suppressed at a sufficiently high confining pressure, as observed in Figs. 3 and 4.

As indicated in an earlier paper (CHAN et al. [1992]), the work-conjugate equivalent stress measure defined in eqn (3) and its modified form in eqn (25), which can be considered as the flow potentials for damage-induced inelastic flow, were developed with the creep damage processes described in Fig. 11 in mind. Since the wing-tip cracks are always aligned parallel to a~, the relevant stresses opposing opening of the wing-tip cracks are a 2 and a3 for triaxial compression. The form of eqn (3) leads to anisotropy in damage-induced inelastic flow for creep under triaxial compression and in tension, which is consistent with experimental observations. Note that both eqns (3) and (25) are similar to the equivalent stress function that contain the -/2, 11, and tensile 03 proposed by LECKIE and HAYHURST [1974,1977] for creep damage development.

A possible limitation of the proposed model is in the use of a scalar damage variable, co, for representing creep damage. The scalar damage variable, which has been chosen for its simplicity, is valid for representing creep damage due to shearing microcracks that are distributed uniformly in the salt. It is also valid for representing creep damage due to formation of wing-tip cracks, providing that loading is proportional. Further evaluation of the model in other loading conditions is required before a full assessment of the scalar damage approach can be made.

VII. CONCLUSIONS

1. Damage-induced inelastic strain rate, which constitutes a significant portion of the macroscopic strain rate, commences in the transient regime and accumulates throughout the three regimes of creep in WIPP salt tested at low confining pressures.

2. The direct contribution of creep damage to the macroscopic inelastic strain rate may be modeled in terms of a kinetic equation that relates the damage-induced inelastic strain rate to the damage variable and the work-conjugate equivalent stress

measure. 3. Damage-induced inelastic flow in WIPP salt is dilatational and pressure-dependent.

A nonassociated flow formulation, with two slightly different work-conjugate equivalent stress measures in the kinetic equation and the flow law, is required to represent the deformation characteristics of damage-induced flow.

4. Pressure-dependent creep in WIPP salt at low confining pressures is the consequence of damage contributing directly to the macroscopic inelastic strain rate. Suppression of creep damage by a high confining pressure leads to elimination of damage-induced inelastic flow, plastic dilatation, and eventually to pressure- independent flow characteristics of the dislocation mechanisms.

5. The MDCF model has been modified to account for damage accumulation in the transient creep regime and for nonassociativity of the inelastic flow due to dam-


age. The modified model is capable of representing the creep response of WIPP salt both at low and high confining pressures.

Acknowledgements--The clerical assistance of Ms. Julie McCombs of Southwest Research Institute is appreciated. This work was supported by the U.S. Department of Energy (DOE), Contract No. DE-AC04- 76DP00789.

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Southwest Research Institute P.O. Drawer 28510 San Antonio, TX 78228-0510, USA

RE/SPEC Inc. Rapid City, SD 57709, USA

Sandia National Laboratories Albuquerque, NM 87185, USA

( Received in f inal revised Jbrm 10 January 1994)

damage-induced nonassociated inelastic flow in rock salt

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