OF SALT DURING STRESS
MODEL OF TRANSIENT
Experimental Programs Division National Laboratories, Albuquerque,
School of Mechanical Cornell University,
and Aerospace Engineering Ithaca, NY 14853
An empirical model is developed that predicts accurately the transient response of salt creep to incremental and decremental changes in stress and temperature. Even though the model is empirical, it is derived from a firm theoretical framework for the micromechanical deformation processes of both steady-state and transient creep. In the model, the transient creep functions modify the steady-state creep behavior, which in turn is based upon creep mechanisms and the deformation-mechanism map. The model is applied successfully to experimental stress-drop
results obtained for salt.
Over the last few years an intensified national research and development program has been directed toward permanent disposal
of radioactive waste. Prime candidates for geologic disposal
are naturally occurring bodies of rock salt because, with time, the creep of salt encapsulates the waste. In order to predict the consequences of creep on encapsulation, as well as the short-term structural stability of the repository, a constitutive model is required for salt creep in the appropriate stress and temperature range. Because the assurance of public safety
requires satisfactory results from numerical calculations of design and performance, there is considerable incentive to provide sophisticated models for these calculations. A primary obstacle in formulating a creep model is the difficulty describing creep
*This work supported by the U.S. Department of Energy.
ISSUES IN ROCK MECHANICS
during decrements in stress (unloading). In the repository unloading occurs because of stress redistribution resulting convergence of the storage rooms and cooling of the waste.
Creep of pure materials, such as salt, is a very complex process that has not yet fully yielded to theoretical analysis. Concerted efforts to understand the micromechanical aspects have provided a deformation mechanism framework for modeling certain
parts of the creep process. In particular, models of micromechanical mechanisms, such as those proposed by Weertman (1968), are especially useful in understanding steady-state creep. Steady-state creep is an equilibrium condition between workhardening and recovery where the structure (internal dislocation and defect array) is constant. The individual mechanisms that control steady-state creep. have been largely identified. These mechanisms and their temperature and stress regimes are commonly represented by deformation-mechanism maps (Munson,
stress changes and for primary creep is shown schematically in Figure 1. The theoretical analysis of transient creep is much less developed than for steady-state creep. Description of the transient response involves not only transient creep during the first stage of creep, but also transients that occur because of changes in applied stress and temperature. This complex response to changes in loading has proven difficult to model. Several empirical models have been advanced, with varying success, to simulate the transient response. Early empirical models that consisted of shifting segments, either in time or strain, of individual creep curves (at the new conditions of stress) to the time and strain condition of the initial creep curve (just before the stress change) are inadequate in decremental loading. More recently, the transient response has been treated according to
first-order kinetics where the model causes a characteristic
decay between the steady-state creep rate at the new stress and temperature and the creep rate at the time and strain of the stress change (Webster, et al., 1969 and Herrmann, et al., 1980). Unless special restrictions are enforced, this model can predict negative creep (strain recovery). A model given by Krieg (1980) that evokes workhardening and recovery is typical of another group of quite complicated models based on plasticity-creep
Perhaps the best current approach to the description of transient creep is based on micromechanical concepts. Dislocation mechanics is used to develop mechanism models which explain certain aspects of transient behavior. They have concentrated principally on transients observed during decremental stress changes and the concurrent changes in internal structure and backstress. Typical of these models are the anelastic backflow
WORK HARDENING RECOVERY MODEL
INCREMENTAL LOADI NG ?/(1 )
?s(1) ?s(2) > >
?(1) > ?(2)>?(3)
models of Gibeling and Nix (1981). Such models are too detailed for our purposes; however, the experimental observations are instructive and form the foundation for the empirical model developed here.
In salt, internal substructures and clusters of dislocations (called structure) have been observed as a consequence of the creep process. These structures give rise to an internal backstress that opposes the applied stress. At any given applied stress, a structure develops that is in equilibrium. This equilibrium structure is observed experimentally as a steady-state creep rate. If we begin with a specimen that has less internal structure than the equilibrium structure, a potential exists that drives the structure toward equilibrium. By the same token, if we begin with a specimen that has more internal structure than the equilibrium structure, a potential exists that drives the structure toward equilibrium. Although the potential function is not known, schematically it must appear as in Figure 2.
The minimum in the potential function is the condition. This state has some unique properties. equilibrium or steady-state strain rate is a function
equilibrium First, the of tempera-
(?s = f(T,?))
as demonstrated experimentally.
ISSUES ROCK MECHANICS IN
the tional total
equilibrium tion only
a functhe ap,
STRUCTURE DENSITY -?,?
is expressed experimental
rate rate vs vs time strain or
empirically quite ated experimentally
well known because the relationship is for primary creep in each standard
test. This work-hardening branch relationship starting from the creep is indicated in Figure 3.
of the strain rate vs strain virgin condition for primary
The strain rate vs strain relationship for the recovery branch is not well known because the relationship is not experimentally accessible using a standard creep test. This results from not being able to exceed the structure density at equilibrium in a standard creep test. Because of nonunique initial states for the recovery branch, a family of curves are obtained that start from the initial state and end at the equilibrium state. Two such curves are shown in Figure 3.
In the difficult form shown to treat. in Figure This is 3, the recovery family of curves overcome by reframing the curves.
manipulations for transient
an aborigine By introducing
the beginning) internal state
parameter, ? , we can reduce the recovery family to a single curve
that joins the workhardening curve at ? ?.
increases in workhardening, but decreases
The state parameter
recovery, always However, transient in
WORK HARDENING RECOVERY MODEL
I?WORKHARD ? STATE (VIRGIN)
? ,? (EQUILIBRIUm1)
INITIAL STATE OVERHARDENED
strain always increases aborigine construction the state parameter is
in either workhardening of the transient creep shown in Figure 4.
or recovery. curve in terms
In developing the model, we follow and extend the earlier results of Munson and Dawson (1979). Steady-state creep of salt is the result of micromechanical mechanisms, as specified through the deformation-mechanism map, acting additively to produce the
deformation (Munson, 1979). Thus, for n mechanisms, the
i s = i=l ?s.(T,o, s) ?. 1
where T is temperature, factor. The creep rate rate through
? effective is simply
and S a structure the steady-state
( = F( s
ISSUES ROCKMECHANICS IN
i i i i
EXACT LINEAR APPROXlIVIATION
STATE PARAMETER RATIO (? / ?t )
Construction of the Transient Creep Curve. strain rate. The linear approxima-
where F is the normalized
shownin Figure 4, to the exact function for F gives
1 --E t
where ? and 6 are
' ? > ?t
and recovery parameters,
respectively, and ? t
evolutionary equation for ? is
the transient strain
? : sign(6t (6t) . _f)
WORK HARDENING RECOVERY MODEL
[ I I I I I I I
c?(1) > o(2)
"- /?t ?-? ?t -. (2) (1)
or applied stress (either incremental
or decremental) takes
strain transient be-
?s 8'00xlO-? '? = /s
I I i
]--I I i I I i
strain curves of Figure 4 representing the previous temperature
and stress condition
STATE PARAMETER (?)
? ? .......... ......
(D = O.?8 ?Pa
accommodated by progressing from one aboriginal curve to
another. This strat-
?(1) = O. ?pa 96
100 200 300 400 500
changes with The important
processes, and the
ration of both
Experimental data are comparatively rare for stress-drop or decremental stress-change tests that include complete recovery to a new steady-state rate. The experimental results reported by Robinson et al (1974) of one complete test of this type are shown in Figure 5b. Under the initial stress of 0.96 MPa, the
reached a steady-state
observed. Upon recovery,
of 2.8 x 10-4/s.
the stress drops to 0.48 MPa, a new creep rate of 4.7 x 10-7/s
a new steady-
state rate of 8.0 x 10-6/s.
ISSUES IN ROCK MECHANICS
produces response to the
Application of the appropriate branch of the model the predicted recovery transient and steady-state creep shown in Figure 5. Comparison of the model calculation experimental data is quite accurate for this experiment.
Recent theoretical advances have begun to explain the underlying micromechanical processes that control transient creep response. These recent advances, when combined with earlier analyses of the micromechanical processes leading to steady-state creep, suggest a framework for the development of an empirical creep model for salt that treats workhardening, equilibrium, and recovery during creeping deformation in response to stress loadings and unloadings. Application of the model to a high-temperature decremental stress-change test that underwent complete recovery after the stress change showed very
Gibeling, J.C., and Nix, W.D., 1981, "Observations Backflow Following Stress Reductions During Metals," Acta ?et., 29, p. 1769.
of an Elastic Creep of Pure
Herrmann, W., Wawersik, W.R., and Lauson, H.S., 1980, "A Model for Transient Creep of Southeastern New Mexico Rock Salt," SAND80-2172, Sandia National Laboratories, Albuquerque, NM.
Krieg, R.D., Halite,"
1980, "A SAND80-1195, NM.
Unified Creep-Plasticity Sandia National
Model for Laboratories,
D.E., 1979, "Preliminary (with Application to
Deformation-Mechanism WIPP)," SAND79-0076,
Map for Sandia
Robinson, S.L., Energy and
Burke, P.M., and Sherby, O.D., 1974, Subgrain Size--Creep Rate Relations
Phil. Mag., 29, p. 432.
"Activation in Sodium
Webster, G.A., Cox, A.P.D., and Dorn, J.E., Between Transient and Steady-State
1969, "A Relationship Creep at Elevated
Theory of Steady-State