a constitutive model for inelastic flow and damage evolution in solids under triaxial compression

Mechanics of Materials 14 (1992) 1-14 1 Elsevier

A constitutive model for inelastic flow and damage evolution in solids under triaxial compression

K.S. Chan a, S.R. Bodner a,l, A.F. Fossum b and D.E. Munson c

" Southwest Research Institute, San Antonio, TX, USA b RE/SPEC Inc., Rapid City, SD 57709, USA c Sandia National Laboratories, Albuquerque, NM 87185, USA

Received 1 November 1991; revised version received 6 March 1992

A constitutive model for describing time-dependent, pressure-sensitive inelastic flow and damage evolution in crystalline solids under non-hydrostatic compression has been developed on the basis that the relevant damage and dislocation flow processes both contribute to the overall inelastic strain rate. A damage-based kinetic equation is first formulated using the work-conjugate approach and the continuum damage concept. That relation is then added to the dislocation-based kinetic equation of a multi-mechanism deformation (M-D) model to obtain the macroscopic inelastic strain rate. The proposed kinetic relation for the overall inelastic strain rate is shown to be derivable from a flow potential. The kinetic equation indicates plastic dilatancy under triaxial compression when the damage term is activated, and leads to plastic incompressibility when inelastic straining is primarily provided by dislocation flow mechanisms. The dependence of creep rate and plastic dilatancy on confining pressure shown by model calculations for rock salt is in accordance with experimental observations.

I. Introduction

During recent years, there has been consider- able interest in developing reliable, t ime-dependent constitutive equat ions for describing the creep response of rock salt under triaxial compression (e.g., Auber t in et al., 1991; Munson and Dawson, 1984; Senseny, 1983; Langer , 1984). The interest is motivated in par t by the possibility that natural salt deposits can be used as a repository for nuclear waste disposal. Natura l salt deposits are considered desirable host rocks for permanent disposal of radioactive waste because the creep characterist ics of natural salt allow closure of the disposal room, leading to eventual encap- sulation of the radioactive waste minimizing the

Correspondence to: Dr. K.S. Chan, Principal Engineer, South- west Research Institute, 62220 Culebra Road, P.O. Drawer 28510, San Antonio, TX 78228-0510, USA. 1 Permanent address: Technion, Department of Mechanical

Engineering, Haifa, Israel.

possibility of leakage and contaminat ion to the environment . A potent ial failure mode in the salt deposit is tertiary creep, which can result in t ime-dependent interface separat ion in bedded natural salt deposits. To meet safety and regula- tory requirements , the closure times and repository condit ions upon encapsulat ion, as well as possible times to failure and interface separation, of bedded natural salt deposits must be predicted with a high degree of certainty for anticipated storage condit ions (stress, tempera ture , etc.). These goals can be accomplished only if both creep deformat ion and fracture characteristics of the natural salt deposit are accurately predicted. The constitutive model for creep deformat ion in salt are reasonably well-developed; however, there remains a need for a reliable, t ime-dependent , damage-based constitutive model for rock salt.

One of the creep models recently developed for rock salt is the mult i -mechanism deformat ion model ( M - D ) proposed by Munson and Dawson (1984). Based on micromechanist ic concepts de-

0167-6636/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

2 K.S. Chan et a L / Inelastic flow

rived from the deformation mechanism map for salt (Munson, 1979), the M-D model describes the macroscopic relationship between steady-state creep rate and stress in terms of three specific deformation mechanisms in salt: (1) dislocation glide at high shear stress levels (stress > 10 3/z, where /x is the shear modulus); (2) dislocation climb at high temperatures (homologous temper- a ture>0.5) and low shear stress (10-5/z to 10-3/x); and (3) an undefined, but empirically characterized, mechanism in the range of 0.28 to 0.48 homologous temperature. These mechanisms are assumed to occur as additive processes so that their contributions to the steady creep rate are summed in a simple manner. Transient creep is incorporated through the use of a transient function composed of a work hardening branch, an equilibrium branch, and a recovery branch for an internal variable whose development is governed by a separate evolution equation (Munson et al., 1989).

Beside dislocation motion, another possible contribution to the inelastic strain rate during compression of salt is damage in the form of distributed microcracks and cavities generated by the stress state. The fracture mechanism in rock salt under low confining pressure and extensional stresses is the formation of microcracks (Hansen and Fossum, 1986; Horii and Nemat-Nasser, 1985; Walsh and Brace, 1966). The opening of microcracks during extensional loading leads to volumetric strains and plastic dilatation. Because both microcracking and plastic dilatation decrease with increasing confining pressure, it implies that inelastic deformation due to damage is pressure- dependent. Figure 1 shows that the experimentally determined inelastic deformation rate and fracture of rock salt depends on both the stress difference and the confining pressure (Wallner, 1984), which is not described by the M-D creep model or any other dislocation-based constitutive model. However, a possible method for improving the predictive capability of the M-D creep model is to incorporate an expression for describing the contributions of cavities and microcracks to the macroscopic inelastic strain rate. The objective of this article is to extend the M-D model so that inelastic strain rate resulting from the presence of

10 .3

10 "4i

"fO ~

t.6

,.r- 10 "~

10 ~

zO 10 "~

10-~

10-1l

10-1

10 20 30 40 60 80 100

01 " (~3 MPa

Fig. 1. Creep failure data of rock salt for various confining pressures, P, where the lines are fit to experimental data or loading trajectories (Wallner, 1984).

cavities or microcracks in a crystalline solid such as rock salt under compressive loading can be described in terms of a continuum formulation.

One approach, which originated from the work of Kachanov (1958), is to incorporate damage through the use of a continuum damage variable, ~o, in a set of constitutive equations for describing the degradation effects associated with the presence of grain boundary cavities in a creeping solid. In the context of Kachanov's work, the damage variable can be viewed as a measure of the reduction in the net load-carrying area as the consequence of the presence of microcracks and microvoids in the material (Leckie and Hayhurst, 1974, 1977; Murakami and Ohno, 1981; Mu- rakami, 1983). An alternative approach is that the damage term is a softening variable comparable, but with opposite influence, to the hardening term in the constitutive equations (Bodner, 1981, 1985; Bodner and Chan, 1986). In these ap- proaches, the damage term is treated as a modifying factor for either the stress or the hardening variable but not as a direct contributor to the macroscopic inelastic strain rate.

A somewhat different approach by Hutchinson (1983) combines secondary power-law creep of

K.S. Chan et al. / Inelastic flow 3

the matrix with an additional strain rate term due to the presence of penny-shaped cracks in the material matrixi The total inelastic strain rate is that of the matrix plus the contribution, which has both dilatational and deviatoric components, of a distribution of non-interacting microcracks. Hutchinson's treatment, however, does not consider the overall softening effect of the microcracks in the sense of continuum damage on the response of the matrix, nor is damage growth coupled to the damage term. Rodin and Parks (1986) generalized Hutchinson's approach in a formal manner, extended the formulation to large microcrack densities, and suggested that the growth of microcracks could be treated by meth- ods of time-dependent fracture mechanics. In a recent paper, Bassani and Hawk (1990) examined creep crack growth using an approach that coupled the previous investigations (Hutchinson, 1983; Rodin and Parks, 1986) with the concepts of Kachanov. In that manner, the effect of damage on the matrix response is considered, and the formulation includes a dilatational contribution to the inelastic strain rate and pressure dependence of the strain rate.

Work by Hutchinson (1983), Rodin and Parks (1986), and Bassani and Hawk (1990) also has established that the opening of microcracks in damaged materials can contribute directly to the overall deviatoric strain rate as well as to non-re- coverable dilatation. However, in these treat- ments, the controlling factors and manifestations for that inelastic strain rate could be different from those associated with dislocation flow. The approach described in this article is to consider the macroscopic inelastic strain rate to be due to both dislocation flow and damage in the form of microcracks and microvoids. The present formulation considers damage to contribute to inelastic strain rate in two ways: (1) by reducing the load- carrying-capability through a reduction in area; and (2) by direct contribution due to opening of microcracks and voids. The first will be modeled using the Kachanov approach, while the latter will be modeled by developing a kinetic relation for relating damage-induced inelastic strain rate to stress and the damage variable. This damage- based kinetic equation is then added to the dislo-

cation-based flow equation of the M-D model to obtain the overall inelastic strain rate. In certain respects, this summation of inelastic strain rates arising from separate mechanisms is similar to the formulations of Hutchinson and of the later investigators, but the details are different. The current procedure requires determination of a stress measure that is the work conjugate to the kinetic relation for the inelastic strain rate due to damage, and an evolution equation describing the rate of growth of damage. An important feature of the resulting constitutive equations is that damage can develop under a triaxial compressive stress state, as experimental evidence appears to require.

2. Damage model development

The convention used in this article is that compression is taken to be positive and tension negative. One of the desired characteristics in the extension of the M-D model is that the modifica- tion must lead to pressure-dependent flow and plastic dilatancy. A convenient approach that is thermodynamically admissible and consistent with the current M-D formulation is the use of the 3-D generalization of the continuum average kinetic relation,

~O'eq n ~P = O0"eql "P "~ -4- "P (1) ~O-ij Eeql . . . ~O.ij Eeqn

by Fossum et al. (1988), where g~ is the plastic strain rate tensor; EPqn and treq. are the equivalent plastic strain rate, i v and the equivalent eq, stress measure, treq , for the nth deformation mechanism, respectively. Using this approach and considering small strains, the damage-induced strain rate tensor, ~i"}, is derived from the equivalent inelastic strain rate measure due to damage (i.e., microcracking and cavitation) and the stress-derivative of the work-conjugate stress measure, O'eq , according to the relation

OO-eq = o, ,j ( 2 )

4 K.S. Chan et al. / Inelastic f low

which is then added to the inelastic strain rate tensor due to dislocation-based mechanisms in the M - D model. Although the formulation is presented in terms of small strains, this is by no means a necessary restriction.

The first step in applying Eq. (2) is to identify the relevant work-conjugate stress measure. Pre- viously, Hunsche and Albrecht (1990) have examined the failure strength of rock salt under triaxial compression. Their result indicated that the shear strength of rock salt increased nonlinearly with confining pressure, P, in a manner as shown in Fig. 2. This dependence of failure strength on confining pressure can be understood on the basis that rock salt contains crack-like defects whose surfaces, under the action of deviatoric stress, would separate and produce crack opening dis- placements as illustrated in Fig. 3a (Walsh and Brace, 1966; Horii and Nemat-Nasser, 1985). The amount of crack surface opening is, however, suppressed by a confining pressure that tends to cause crack closure in a manner suggested in Fig. 3b (Walsh and Brace, 1966). The opening and closure of the microcracks or other forms of damage should lead to corresponding changes in the inelastic straining response. Based on this interpretation of the data shown in Fig. 2, the work conjugate stress measure, 0-~q, is taken to be

to 0"eq I0-1 - - 0"3 I

- x 2 x 7 sgn(l l - 0"a) ( 3X 7

-- x l o ' 3 H ( - 0 - 3 )

I1 - 0"1 I x6

) (3)

30

el"

¢ 20 <:

e~

0 8 0

/ f I I I I

10 20 30 4.0 50

PRESSURE, M P a

Fig. 2. Failure strength of rock salt for 25°C (Hunsche and Albrecht, 1990).

f ope~micmcrack closed microcrac 1

.. / t / /

- - - \ 0 - - \ I

I \ / ix # *

At zero pressure Unaer confining pm,~ure (a) (bl

Fig. 3. Damage mechanisms envisioned during creep deformation of rock salt: (a) generation and opening of microcracks; (b) closure of microcracks by a confining pressure.

with

Im - 0"31 = 2 cos qt J~2, (4)

where qt is the Lode angle, 11 is the first stress invariant, and J2 is the second invariant of the deviatoric stress. The first and third terms in Eq. (3) are intended to represent, respectively, the opening of crack-like defects due to shear and the principal stress in tension, 0"3, while the second term, which is of the form F ( I 1 - 0"1), repre- sents the suppression of the opening of microcracks by a confining pressure. In Eq. (3), x 1, x 2, x 6 and x 7 are material constants that can be evaluated from experimental data. Taking the stress derivative of Eq. (3) leads one to

~i.t °~ = ~eq[ b l S i j "q- b 2 t i j - b 4 ( ~ i j - m i j ) - b s m i j ] ,

(5)

where

1 S i j = O'ij - - ~ 0 - k k 6 i j (6)

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

2 ti) = S ikSk j -- 5 J 2 6 i j (7)

is the deviator of the square of the deviatoric stress (Prager, 1945), and

m i s = 6ik6kS, (no summation), (8)


for the stress state of triaxial compression along the principal axes considered here. The expression of mij for generalized stress states is presented in the Appendix. The coefficients in Eq. (5) are defined by

cos(2~) (9) bl = J~2 c°s(3gt) '

v;3 sin bE = J--7 cos(3~) ' (10)

( t x6- X2X 6 I1 -- 0-1 b4= - - ~ 3 sgn(-~l--_0-1 ) (11)

b5 = XlH( - 0-3), (12)

where the Lode angle, ~ , is given by (Prager, 1945; Fossum et al., 1988)

2 j3/-----2 - - f f<~<~-f f

(13)

due to symmetry, and J3 is the third invariant of the deviatoric stress. Note that in Eq. (5) b I and b 2 are loading parameters which depend on the stress state, while b 4 and b 5 depend on both the stress state and the material parameters that de- fine the work conjugate stress measure.

The second step in applying Eq. (2) involves developing a relation for ~eq in terms of oJ and 0-eq- Very little experimental data are currently available for this purpose. However, the results in Fig. 1 indicate that the creep rates at high values of stress difference may be described either by a power-law or by a hyperbolic sine function. The results of Hutchinson (1983) suggest that the inelastic strain rate due to microcracks depends linearly on the microcrack density. Thus, a linear relation between ~eq and oJ may be used aside from the modifying factor ( 1 - ~o) on the stress. To allow for flexibility for correlation with experimental data, a kinetic equation for 4eq, which is linear with damage but nonlinear with modified stress, is proposed (Garofalo, 1963),

ie~ = ClO)H(0-~ ) sinh ~-i ~ -~-- ~

where Cl, c2 and n 3 are material constants: tz is the shear modulus; H is the Heaviside function with 0-eq as the argument. As will be discussed later, characteristics of Eq. (14) include the exis- tence of a flow potential and its similarity to a power-law relation at low stress levels (Garofalo, 1963). Combining Eqs. (5) and (14) leads to

[ n3 ~i~j=Cl 0) sinh " ( 1 7 ~ - - ~

X [blS i j Jr- b 2 t t j - b4(~i j - m i j ) - bsmi,] (15)

for the inelastic strain rate tensor due to damage. In the context of continuum damage mechan-

ics, Eq. (15) can be viewed as a mathematical description of the inelastic strain rate resulting from the opening of microcracks and microvoids subjected to an applied stress. Another equation, i.e., an evolution equation, is also required to describe the change in the microcrack and mi- crovoid densities as damage accumulates in the creeping solid. A phenomenological description based on a power law for the growth of creep damage was first proposed by Kachanov (1958) for the uniaxial stress case, which was subse- quently generalized for multiaxial stress conditions by Leckie and Hayhurst (1977). An alternative to the power-law relation is the exponential damage growth law proposed by Bodner (1981). These phenomenological damage growth laws are somewhat different from those based on cavity growth models (Cocks and Ashby, 1982; Cocks and Leckie, 1987). Since detailed information about the mechanisms of damage in rock salt under confining pressure has not been established fully, it is necessary to consider the growth of damage in terms of a phenomenological approach. Following previous work (Bodner, 1981; Bodner and Chan, 1986), the damage evolution equation, tb, is taken as

, ,

t o t o X 3 X [0-eqn(O'eq)] -h(o.), T, /1), (16)

6 KS. Chan et al. / Inelasticflow

where x 3, x4, and x 5 are material constants, and h(w, T, I l) is the damage healing function whose exact form remains to be determined. The damage healing term, h(w, T, I1), is expected to depend on the damage variable, w, temperature, T, and confining pressure. Starting with an initial value, w 0 (e.g., ~o 0 = 1 × 10-10), Eq. (16) may be integrated with respect to time to obtain the current value for ~o. For the case of constant stress, w0 = 0, and neglecting damage healing in- tegration of Eq. (16) with respect to time of creep, t, leads to (Bodner, 1981)

[( x )x] w = e x p - [ ,o ~o x3 , °eq H(°'eq ) ] t (17)

which has the features expected for physical damage development.

3. Incorporation of the damage model into the multi-mechanism deformation model

The incorporation of the creep damage model into the M - D creep equations is relatively simple since dislocation motion and the opening of microcracks and microvoids are considered as separate, additive mechanisms. Using the approach of Eq. (1) proposed by Fossum et al. (1988), the inelastic strain rates induced by dislocation flow, Eeq,'C and by damage mechanisms, ~eq, are added, leading to

Ei, = (~eCq + EWq)[blSij + b2tij ]

- e e q [ b 4 ( • i j - m i j ) + b 5 m i j ] (18)

for the total macroscopic inelastic strain rate tensor.

Since Skk = tkk = 0, the volumetric strain rate, - ~ k , derived from Eq. (18) is

--ekk = 2b4eeq (19)

for triaxial compression, and

--~kk = (2b4 + bs)~eq (20)

for stress states involving a tensile tr 1.

According to Eqs. (3), (14), (15), and (18), plastic dilatation occurs under either tension or compression at low confining pressures, but is totally suppressed in compression when the confining pressure is large, which is in qualitative agreement with experimental observations. Also, Eqs. (19) and (14) indicate zero volume change for hydrostatic compression and dilatation for hydrostatic tension.

When b 5 = 0, Eq. (18) leads to equal dilational components in the % and ~r 3 principal axes, but none in the ~r I direction under triaxial compression. This feature of Eq. (18) is consistent with experimental observations that compression along the % axis leads to microcracks aligned parallel along the stress axis; opening of those microcracks should result in higher dilatational components in the % and ~3 axes. When x ~ > 0 , anisotropic dilatation is also obtained from Eq. (18) for stress states involving a tensile cr 3 since b5 4: 0.

The expression for g~q can be obtained from the IV' -D model. In this formulation, the inelastic strain rate is given by (Munson and Dawson, 1984)

EeCq = F ~ , (21)

where F is the transient function representing transient creep behavior and ~s is the steady-state strain rate. Considered as the overall contribution of three independent dislocation mechanisms act- ing as additive processes, the steady-state strain rates from individual slip mechanisms are summed to give

~s = L~s, , (22) i=1

where ~si is the steady-state strain rate of the ith mechanisms. The steady-state strain rates of the individual mechanisms, each of which is taken to be thermally activated, are (Munson and Dawson, 1984)

( )nl ~s~ =A1 e-O'/m" , (23)

KS. Chan et aL / lnelastic flow 7

e oJ( ) ,24,

~s3 = I n l ( n l e -QI /RT + B 2 e - a 2 / R r )

× sinh( q (0 .~ 0.°) ), (25,

where the As and Bs are constants; Qi s are activation energies; T is the absolute temperature; R is the universal gas constant; 0. is the generalized stress which is taken as the stress difference on the basis of the Tresca criterion; [ H q is the Heaviside step function, with an argument of 0. - 0.0; nis are the stress exponents; q is the stress constant; 0.0 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 transient function, F, (Munson et al., 1989) is

exp A 1 - e--~- '

F = 1, ~ '=¢*, (26)

exp - 6 1 - E--~- , (>~E*,

which is composed of a work-hardening branch, an equilibrium branch, and a recovery branch. In Eq. (26), A and 6 represent the work-hardening and recovery parameters, respectively, and e* is the transient strain limit. The temperature dependence of the transient strain limit is represented by (Munson et al., 1989),

where Ko, c and m are constants. The evolution rate, ~', of the internal variable ~ is governed by

~= ( F - 1)(g~), (28)

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

Extending the M - D model to creep-damaged materials also requires that the reduction in net load-carrying area in the material be taken into account. This is achieved by invoking the Kachanov approach (Kachanov, 1958); all the stress terms in the M - D model (i.e., Eqs. (23), (24), (25), and (27)) are divided by the ( 1 - o~) term. Modeling damage effects therefore involves: (1) the (1 - t o ) term for representing the reduction in load-carrying area; and (2) the ~eq term for modeling the inelastic strain rate contribution due to the opening of microcracks in the damaged material. Additionally, the relation c I = c0(B 1 e -Q1/RT + B 2 e-Qe/RT) , where c o is a material constant, is assumed so that the curves represented by Eqs. (25) and (14) would intersect in stress space. The interaction point then repre- sents the stress where the transition from dislocation glide to damage-induced flow occurs. Such a transition corresponds to moving from a point in the deformation mechanism map (Ashby, 1983; Munson, 1979) to an equivalent position in the fracture mechanism map (Gandhi and Ashby, 1979; Ashby, 1983). For salt, the fracture mechanisms under tension include cleavage, brittle interface fracture, transgranular and intergranular fracture. Some, if not all, of these fracture processes are expected to remain operative under triaxial compression, but their regions of domi- nance in the fracture mechanism map are expected to differ from those in tension. It is also possible that cleavage and brittle interface fracture might be totally suppressed by a confining pressure. To the authors' knowledge, a fracture mechanism map for salt under triaxial compression is not available in the literature.

4. Model calculations

To illustrate the response characteristics of the new formulation, creep rates were calculated as a function of stress difference for various confining pressures. The constants used in these calculations are shown in Table 1. Figure 4 shows the result for a triaxial stress state of 0.1, o'2 = 0.3, a

8 K.S. Chan et al. / Inelastic flow

T a b l e 1

M a t e r i a l c o n s t a n t s fo r c l ean sal t

M - D m o d e l D a m a g e m o d e l E las t i c

c o n s t a n t s c o n s t a n t s p r o p e r t i e s

A x 8 .386 E 2 2 s -1 xl = 0.0 /x 12.4 G P a

Q t 1 . 0 4 5 × 1 0 5 J / m o l x 2 = 3 . 6 E 3 1 . 0 G P a

n 1 5.5 x 3 = 1.0 v 0.25

B I 6 . 0 8 6 E 6 s 1 x 4 = 3 . 0 x 5 = 3.0 × 1 0 - 7

M P a s

A 2 9 .672 E l 2 s - 1 x 6 = 0.85

Q2 4 . 1 8 × 104 J / m o l x 7 = 1 M P a

n 2 5.0 C O = 1 × 10 22 S !

B 2 3 .034 E-2 s - 1 c2 = 5.335 × 103

o'(1 20.57 M P a n 3 = 5

q 5.335 E3

R 8.3143

J / m o l °K

rn 3.0

K 0 6.275 E5

c 0 .009198 K 1

a - 17.37

/3 - 7.738

0.58

confining pressure, P, of 1 MPa, and for a damage value of 0.1 (to -- 0.1). The important result in Fig. 4 is that the creep rate, ~]1, is essentially equal to the dislocation-based strain rate term, ~q, at low values of stress difference but is dominated by the damage-induced strain rate term,

10.' i i .

I0-~ ~11

10"' ! ~ 10. I

~ 10.'

g 10.' to - 0.1

lO-' J --~ 1 M P a P r e s s u r e

10"* i = , i , i , h i i I I

10."1d lO'

STRESS OIFTERENCE, M P o

Fig. 4. C a l c u l a t e d s t e a d y s ta te c r e e p r a t e s as a f u n c t i o n o f s t ress d i f f e r ence for c l ean sal t u n d e r a c o n f i n i n g p r e s s u r e o f 1

M P a .

10-=

10 .3

10" '

10. =

10"*

~ I0"

1 0 " Ca) = 0 . 1

lO- '° - ' ~ ( - 'Eaa) = " 1 M P a P r e s s u r e

10"" , : ' , , , , , L , , , , , i 10' 10 ~

STRE'~3S OIFFERENCE, MPo

Fig. 5. C a l c u l a t e d s t eady s ta te s t r a in r a t e c o m p o n e n t s a n d

v o l u m e t r i c s t ra in r a t e fo r c lean sal t u n d e r a c o n f i n i n g pres-

su r e o f 1 MPa .

~eq, at higher values of stress difference. The corresponding strain rate components are shown in Fig. 5. In Fig. 5, positive strain rates are compressive, while negative strain rates are tensile. The volumetric strain rate is given as - ~ in Fig. 5, which is essentially zero in the regime where ~Cq dominates and is positive in the regime where the ~eq term dominates; positive values of - ~ k indicate plastic dilatation.

Figure 6 shows the effects of confining pressure on the calculated creep rates for various values of stress difference, and the corresponding volumetric strain rates are shown in Figure 7.

C o n f i n i n g P r e s s u r e

10-= 0 5 1 0 2 0 M P a ,0-. / / 10"

.~, 10 .=

~ I0-*

~ lO 10.'

10.'

10.'*

10." ld 10'

STRESS DIFFERENCE, MPo

Fig. 6. Calculated steady state creep rates as a funct ion o f stress di f ference for various levels o f conf ining pressure showing a reduct ion o f creep rate with an increasing level o f conf ining pressure.


10-'

10- ~

10" .

~< 10'' 0¢ Z

10-' E m 10-'

• ~, I0:" ' I ~ 10'' >

10""

10-" id

Confining Pressure, MPa o 5 lO

o) - 0.1

I f / i i i I I I

ld STRESS DIFFERENCE, MPo

Fig. 7. Calculated volumetric strain rates as a function of stress difference for various levels of confining pressure showing a reduction of the volumetric strain rate with an increasing level of confining pressure.

Basically, confining pressure delays both the development of damage and the rapid increase in the creep rate due to increasing values of stress difference. For these cases, x 6 was set to 0.85. As a result, the calculated pressure dependence of creep rates shown in Fig. 6 is slightly nonlinear, while that observed experimentally in rock salt (Wallner, 1984) in Fig. 1 appears to be more nonlinear.

Another means of presenting the results for plastic dilatancy is to compute the ratio of volumetric strain rate to the axial strain rate, ~kk/~ll. Figure 8 shows the calculated values of ~k/~11 for increasing values of confining pressure. For these calculations, x 6 was also set to 0.85. As shown in Fig. 8, the Ekk//Ell ratio is zero at low value of stress difference, but becomes negative when the damage-induced term becomes significant. For compressive ~1~, a negative value of ~kk//Ell indicates plastic dilatation. As the confining pressure is increased, the development of damage and plastic dilatation is delayed to occur at higher values of stress difference in non-linear fashion. The result suggests that a possible means for evaluating the constants in the creep damage equation is to measure the ratio of Ekk//Ell a s a function of stress difference and confining pressure.

1.0

Confining Pressure

o.o 20

. ' ~ - 1 . 0

10

- 2 . 0

01-0.1

- 3 . 0 i i , I , I , I , I i I

0 .0 10.0 20 .0 30 .0 40 .0 SO.O 60 .0

STRESS DIFFERENCE, MPo

Fig. 8. Calculated values of ~kk//Ell showing the onset of plastic dilatancy, delayed by an increasing level of confining pressure.

Figure 9 shows creep curves for compression at zero value of confining pressure for various stress difference levels. In this exercise damage develops according to the evolution equation, Eq. (16), commencing from a small initial value (10-1°). At low levels of stress difference (10-18 MPa), the calculated creep curves exhibit classi- cal primary, secondary, and tertiary creep regimes. The transition from secondary (steady-state) creep to tertiary creep is relatively gradual and influenced by the (1 - to) term in the creep equations. At moderate levels of stress difference (21 and 25 MPa), the three regimes of primary, secondary, and tertiary creep are still discernable. The tran-

0 .20

Pressure - 0 ,3o !

oJs i 25 21

_z < I - - C'3 0 .10 " ' 0 - ,t. H : 01 - o3 (MPa) rr :

0.05 ~ /18 .-'15 0

0 .00 = , I , I , I 0 , 0 S,O 10.0 IS.O 20 .0 25 .0 30 .0 35 .0 40 .0

TIME, 10 s sec

Fig. 9. Calculated creep curves for various levels of stress difference under no confining pressure.

10 K.S. Chan et aL / Inelastic flow

sition from secondary to tertiary creep, however, is rather abrupt. The abrupt change is due to the activation of the ieq term which physically corresponds to the onset of creep damage in the form of opening of microcracks. At high levels of stress difference ( < 30 MPa), tertiary creep dominates and individual creep regimes are no longer distin- guishable. This behavior is somewhat similar to the presence of large flaws in a creeping solid subjected to an applied stress. The corresponding s t ress- rupture life plot, shown in Fig. 10, clearly shows two distinct regimes corresponding to low stress and high stress behaviors. Tertiary creep at low stress is influenced mostly by the ( 1 - w) term. The creep rupture life at low stress difference was therefore computed as the time to reach a damage value of 0.1, which is a relatively large value. Tertiary creep at high stress differences is controlled by the ~eq term, and a small value of damage leads to a large increase in the inelastic strain rate. As a result, the damage variable never reached the value of 0.1. For these cases, the time to rupture was taken as the time where the creep strain began to increase rapidly to reach a critical value of 0.2. Thus, the low stress behavior is similar to creep cavitation with formation of cavities at grain boundaries in metals and ceramics, while the high stress behavior is similar to creep fracture dominated by microcracks.

O~ - 0.1 1<' r(1. )term]%

L dominates j \\ \ \ ~= 0.2

"~ ~ • lOI ~ t LdominatesJ

Pressure - 0 "~o

1 ( ~ ' . . . . . . , a , , . , . , ~ . a . , . , . ~ . a . , . , . ~ , , a . . . . . . . .d . . . . . . . . . . a . i . . . . . . . . a . , . , . , . = , , d

10 e Id 10' ld 10 = ld 10 ~ 10 ~ 10 s ~m (sEc)

Fig. 10. C a l c u l a t e d s t r e s s - r u p t u r e life p lo t s h o w i n g low s t ress

a n d high s t ress reg imes . T h e low s t ress r e g i m e is o b t a i n e d us ing o~ = 0.1 as the r u p t u r e c r i t e r ion , whi le the h igh s t ress

r e g i m e is b a s e d o n a 0.2 c r e e p s t ra in , E c, c r i t e r ion .

10 "2 I

- 104

104

- ~/ 0 0 P = OMPa Z 104

/ & • P = 5.0 MPa / [] • P= 10.0MPa

/ <3 • P = 20.0 MPa

- - M-D Model ] L__ failure - - creep

! I i I 10 20 30 40 60 80 100

0"1 - G3, MPa

Fig. 11. C o m p a r i s o n o f c a l c u l a t e d a n d m e a s u r e d c r e e p ra tes

fo r rock sal t u n d e r va r ious levels o f con f in ing p re s su re , P .

T h e e x p e r i m e n t a l d a t a a re f r o m W a l l n e r (1984), a n d the l ines

a r e m o d e l ca lcu la t ions .

Application of the proposed model to the experimental data of Wallner (1984) is shown in Fig. 11. The material constants shown in Table 1 are for the Waste Isolation Pilot Plant (WIPP) salt in southwestern New Mexico. Procedures for determining material constants in the M - D creep equations have been presented earlier (Munson et al., 1989). The experimental data of Wallner (1984), on the other hand, are for rock salt from the ASSE II mine in Germany. As a result, re-evaluation of material constants in the M - D model was necessary. It was found that most of the model constants for WIPP salt could be used for ASSE salt. The model constants for ASSE salt which are different from those shown in Table 1 are as follows: ~r 0 = 4 5 MPa, x 2=12 , x 6 = 0.65, c o = 5 × 10-12 s - 1, and n 3 = 3. Note that the minimum principal stress, ~r 3, term was not included in the ~req equation (Eq. (3)) for these calculations since H ( - o " 3) - 0 . The material constants in the damage model were obtained by fitting the M - D model to the experimental creep data in the tertiary regime. Figure 11 shows that the pressure dependence of creep rate ob-

K.S. Chan et aL / Inelastic flow 11

served in ASSE salt is well represented by the proposed model.

5. Discussion

Although the concept of a flow potential has not been used in the formulation, the flow law given by Eq. (15) is derivable from the flow potential, q~o~, represented by the relation:

CLO9(1 -- t o ) / x I 4 '° = , (29)

£2

where

I = f(sinh X) "3 d X (30)

and

X = (1 -~o) /x ' (31)

which leads to Eq. (15) upon differentiation by crij. The integral shown in Eq. (30) can be represented in the form of a finite series when n 3 is an integer greater than or equal to unity (Dwight, 1961). Comparing the various forms of the sinh function with the power law in Fig. 12 reveals that X" and (sinh X) n are essentially identical

10' ~ 4 ld n-3 2 1

10"

10-2 z Q 10-s ~ 1: Z n 10-' ~ 2: sinh Z / 3: sinh_(z n) 10 -= / 4: sinh nZ

/ 10-1 . i . . . . . . . . . . . I . i . n m n .n . I i aa l . I . . . . . . . . I l l i

10-' io" IO' ld Z

Fig. 12. Comparison of the functional behavior of the power- law with the hyperbolic sine function shows sinhnx can be approximated by various forms of X n for X ~< 0.6.

when the normalized stress parameter, X, is less than 0.6 (Garofalo, 1963). This result suggests that at low stresses (X ~< 0.6), the flow potential can be approximated by

CltO(1 -- to)p.,x(n3 +1) q~o, = (32)

c 2 ( n 3 + 1)

and the kinetic equation then becomes

(33)

Because of these relations, the inelastic strain rates due to damage that are presented in Figs. 4, 5, and 7 exhibit the appearance of a power-law dependence on stress difference even though the kinetic relation given by Eq. (14) was used in all of the calculations.

In terms of a yield condition, Eq. (3) is the Tresca hexagonal prism in the three-dimensional principal stress space where x~ = 0. The size of the hexagon, however, increases non-linearly with increasing confining pressure. The rate of increase is dictated by b4, whose value depends on material constants including x2, x6, and x 7. When x 6 is less than unity, b 4 exhibits a singularity at P = 0. To alleviate this problem, x 6 is set to be unity when 2 P is less than or equal to 3x 7. Under this circumstance, Eq. (3) is linear and it gives a dilatational rate that is independent of confining pressure for 2 P ~< 3x 7. When x 1 > 0, Eq. (3) is similar to the equivalent stress function that contains the J2, 11, and tensile ~r 3 proposed by Leckie and Hayhurst (1977) for damage development. The difference between the present approach and that of Leckie and Hayhurst (1977) is that the cr~ term is used both in the flow law and the damage evolution equation, while it is used only in the damage evolution equation in the Leckie and Hayhurst approach.

The inelastic strain rate tensor due to dislocation flow mechanisms, ~ , in the M - D model (1989) is also derivable from a flow potential, q~c. For this formulation,

3

4~c= E q~, (34) i=1

12 K.S. Chan et aL / Inelastic flow

where

(1--w)txAie-Q'/Rr( o" )ni+l

~c= (ni+ 1) ~(1-~o1 (35)

for i = 1 and 2, and

q ~ c - ( 1 - ° J ) ~ I H ' ( ~ Bi e 3 q i = 1 Qi/RT)

/x(1 - ~ (36)

for i = 3, are the flow potentials for individual dislocation flow mechanisms. Combining Eqs. (29) and (34) leads to

q~ = q~o~ + q~c (37)

as the flow potential which gives the overall inelastic strain rates for a solid exhibiting dislocation creep and damage-induced inelastic deformation.

The characteristics of the present model have both similarities and differences to those of Hutchinson (1983), Rodin and Parks (1986), and Bassani and Hawk (1990). The similarities are that: (1) damage leads to both deviatoric and dilatational inelastic strain rate components; (2) plastic incompressibility is preserved when the damage-induced inelastic strain rate term is suppressed or not activated; and (3) plastic dilatancy increases with the damage variable or the microcrack density parameter. The volumetric strain rate increases linearly with the damage variable in the present model and in Hutchinson's formulation (1983). In contrast, non-linear relationships between ~kk and the damage variable have been obtained in Bassani and Hawk's work (1990). A significant difference between the current model and those of the other investigators is that the present formulation gives plastic dilatation for both tensile and non-hydrostatic compressive loads, while the others do so only for tensile loading.

In terms of creep rupture life calculation, the present model leads to two different types of response characteristics depending on the ap-

plied stress level. At low stress levels where ~eq is not activated, the damage term, ~o, accumulates and causes an increase in the creep rate mainly through the reduction in the load-carrying area, i.e., the (1 - w ) term. This behavior is consistent with creep behavior under constrained cavitation in which growth of cavities is controlled by creep in the matrix (Dyson, 1976). At high stress levels, where the ~q term is activated, the present model gives response characteristics that are typical of cracked materials.

6. Summary

A constitutive model for describing time-dependent, pressure-sensitive creep damage response of rock salt has been developed. The formulation is based on the continuum damage mechanics concept and on the assumption that creep due to the growth of distributed microcracks and microvoids is an independent mechanism that contributes to the inelastic strain rate. The proposed damage-based kinetic equation is then combined with the dislocation-based kinetic equation in the M-D creep model to obtain the macroscopic inelastic strain rate. It is demon- strated that the proposed kinetic relation for the overall inelastic strain rate is derivable from a flow potential. Furthermore, the kinetic relation gives plastic dilatancy under triaxial compression when the damage term dominates, but leads to plastic incompressibility when inelastic flow is due primarily to dislocation flow mechanisms. The dependence of creep rate and plastic dilatancy on confining pressure shown by model calculations is in accordance with experimental observations.

Acknowledgement

The computing assistance by Ms. L.K. Tweedy and the clerical assistance by Ms. J. McCombs, Southwest Research Institute, are acknowledged. The work was supported by U.S. Department of Energy (DOE), Contract No. DE-AC04-76D P00789.

K.S. Chan et aL / Inelastic flow 13

References

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Fossum, A.F., G.D. CaUahan, L.L. Van Sambeek and P.E. Senseny, (1988), Key Questions in Rock Mechanics, in: P.A. Cundall, R.L. Sterling and A.M. Starfield, eds., Proc. 29th Symposium, Balkema, Rotterdam, p. 35.

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Hunsche, U. and H. Albrecht (1990), Results of true triaxial strength tests on rock salt, Eng. Fract. Mech. 35, 867.

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A p p e n d i x - P r i n c i p a l s t r e s s d e r i v a t i v e

T h e p r i n c i p a l s t r e s s d e v i a t o r s a r e d e t e r m i n e d

f r o m t h e c u b i c e q u a t i o n ( P r a g e r , 1945),

s3 - Jz s - J3 = O, ( A 1 )

w h e r e s is t h e d e v i a t o r i c s t r e s s t e n s o r by l e t t i n g

s = r s in 0 ( A 2 )

14 K.S. Chan et al. / Inelastic flow

and using the trigonometric identity,

3 1 sin 3 0 - 3 s i n 0 + 3 s i n 3 0 = 0 . (A3)

Substituting Eq. (A2) into Eq. (A1) leads to

2 r = _~_j~/2 (A4)

and

3~ J3 sin 30 2 j3/2, (A5)

where Eq. (A3) is invoked. The three possible values of sin 0 give the three principal stresses from the cyclic nature of sin(30 + 2nrr/3), with n = 0, 1, 2, and - ~ r / 6 ~< 0 ~< "rr/6. The first principal stress, o-], is given by

2 1 ( A 6 ) cr I = _~_j~/2 s i n ( O + ~-~) + ~o'kk,

which is differentiated with respective to cr~j to obtain

leading to

sin(O + 2rr /3) 1 Si j mij = 3~iJ -~- ~/3J:

c o s ( 0 + 2 " r r / 3 ) ( s i n 3 0 1 )

- - COS 3 0 ~--~2--2 Sij -- ~ tij

(A8)

for a generalized stress state ~rij with the Lode angle, 0 = ~ , and

mij = 1, for i= j = 1, and

mij = 0, otherwise, (A9)

in the principal stress space.

d o v 1

mi~ = do.,. t , (A7)

a constitutive model for inelastic flow and damage evolution in solids under triaxial compression

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