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On the constitutive modeling of transient creep in polycrystalline ice: Reply to the comments of M. Aubertin
Cold Regions Science and Technology, 20 ( 1992 ) 315-319 Elsevier Science Publishers B.V., Amsterdam
315
Discussion
On the constitutive modeling of transient creep in polycrystalline ice*: Reply to the comments of M. Aubertin M a o S. WLIa a n d S. S h y a m Sunder b aDepartment of Engineering Mechanics, University of Nebraska-Lincoln, 219 Bancroft, Lincoln, NE 68508, USA bDepartment of Civil Engineering, Massachusetts Institute of Technology, Room 1-346, Cambridge, MA 02139, USA (Received 8 June 1991; accepted 13 June 1991 )
We appreciate Michel Aubertin's ( 1992 ) interest in our paper (Shyam Sunder and Wu, 1990) and would like to respond to his c o m m e n t s on the following: 1. Ratio of initial creep rate to steady-state creep rate. 2. Non-recoverability of transient strain. 3. Static and dynamic recovery. 4. Incubation period in stress-dip tests. 5. Strain rate decomposition. We will show that the discusser's comments on items ( 1 ), (2) and (5) in relation to our transient creep model are incorrect. The physical basis of his interpretations of item (3) is unclear. The comments on item (4) are well taken and reinforce what we have discussed in our paper but can be debated in the absence of conclusive experimental data for ice. The conclusions that the discusser draws regarding the ability of our model to describe stress relaxation p h e n o m e n a and internal structure evolution after a period of zero creep rate are also incorrect. These five items are discussed in the following. Ratio of creep rates. The discusser took issue with the relationship proposed by Mellor and Cole (1983) which was derived on the basis of experimental data for polycrystalline ice:
*Cold Reg. Sci. Technol., 18: 267-294.
~0 r=__=C
(1)
Es
where r denotes the ratio between the initial creep rate and the steady-state creep rate and C is a constant. The initial creep rate refers to the strain rate measured after imposition of a stress j u m p Aa,, while the steady-state creep rate denotes the strain rate attained under the constant stress a' = a0 + Aa~ and ao equal to zero. It is noteworthy that the proportionality between the initial strain rate and the steady-state strain rate has also been observed for other materials (see the references contained in Takeuchi and Argon, 1976). The discusser, however, emphasized that "the initial creep rate is largely dependent on the entire mechanical history of the material". He apparently established the inadequacy of eq. (1) by showing that the ratio after a second stress j u m p is a nonlinear function of the initial and final states of stress a' and a":
~o ( a " ] m-n
(2)
is-\a'./ where ~0 is the initial strain rate measured after the imposition of the second stress j u m p and m, n are the constant-structure and the standard power-law exponents, respectively. Note that the discusser uses the term "initial strain rate" in a general sense to mean the strain rate measured immediately follow-
316
M.S. WU AND S. SHYAM SUNDER
ing a stress j u m p which is imposed at steady state reached under the previous stress state, while the same term is used in eq. ( 1 ) to mean the strain rate measured after the very first stress jump. Our model equation actually agrees with eq. (2). Equation ( 1 ), which is a specialization of a more general relation in our model, represents the ratio of initial and steady-state strain rates, where the initial state refers to a relaxed state prior to the application of any external load. Under this condition, the ratio r is given by: r=l+
Boo = C
(3)
where Bo denotes the initial value of the non-dimensional drag stress B which characterizes the structure offering isotropic resistance to deformation. The back stress R in our model equals zero initially because a non-zero back stress would imply that the material is initially anisotropic. I f an additional load is applied at (or before) steady state so that the new stress level is a, then the ratio r is given by: r=l+
(4)
which shows a nonlinear dependence on the current values of the stress and internal variables. Note that eq. (4) reduces to eq. (3) when R = 0. Equation (4) is interpreted as follows. First, the initial strain rate is a function o f the current values of the internal structural variables and the variable (the applied stress) associated with the observable variable (the strain). The current values of the internal variables are averaged quantities which are produced by the entire past history of deformation, and the instantaneous response of the material is governed by its current state. These concepts constitute the very essence of internal variable modeling. Second, the steady-state strain rate as described by the conventional power law represents a state of saturation characterized by a stress-independent constant. Thus the ratio r between the two strain rates is still a function of the current values of the internal and associated variables. We note that eq. (2) proposed by the discusser cannot describe the ratio r properly when a creep
stress is first applied to a specimen because a' is zero. The internal variable interpretation suffers from no such inconsistency and is superior to the discusser's interpretation ofeq. (2) which reads "... r is dependent on the initial and final states of stress in a non-linear fashion ...". In a broader perspective, a material state should be described by structural variables and not by the stress because two specimens can have exactly the same current stress but a very different history of loading and hence structure. Non-recoverability of transient strain. The discusser quotes Budd and Jacka (1989) as providing experimental evidence that "at least a portion of the transient creep strain of ice is non-recoverable". Budd and Jacka, however, meant that before the m i n i m u m creep rate is reached primary creep is composed of a recoverable anelastic component and a non-recoverable component, as described by the Andrade power relation. In rate form, this relation is:
~p=K+ 3flt-2/3
(5)
where ~p is the total primary creep rate, K and fl are stress-dependent parameters, and t is the time. On the right side of eq. ( 5 ), the first t e r m K in fact describes the rate of non-recoverable primary creep, while the second term describes the rate of the recoverable component. In the model of Shyam Sunder and Wu (1990), the total creep rate before reaching the m i n i m u m creep rate is composed similarly of the rate of a recoverable component, called transient creep rate, and the rate of a non-recoverable component which is described by the conventional power law for viscous creep. This power law describes the m i n i m u m creep rate when steady state is reached, assuming no tertiary creep takes over. Consequently, our model does contain a non-recoverable creep component during primary creep that is identical in form to the first term of Budd and Jacka's equation ( K i n eq. 5). Both Duval et al. ( 1991 ) and Shyam Sunder and Wu (1990) used the term transient creep strain (e~) to mean the strain obtained by subtracting from the total strain (E) the elastic strain (~e) and the nonrecoverable strain (ev) computed from the viscous power law. By performing unloading tests, Duval et
ON THE CONSTITUTIVE MODELING OF TRANSIENT CREEP IN POLYCRYSTALLINE ICE: REPLY TO AUBERTIN'S COMMENTS
al. (1991 ) found that transient creep does have a non-recoverable component. The loading period of their creep tests is approximately three hours, while the unloading period is approximately seven hours. They concluded that the total non-recoverable creep is composed of this transient component in addition to that described by the viscous power law. Although their results should be taken into consideration, it should be noted that the non-recoverable transient component becomes less important at typical strain rates of engineering interest, i.e., moderate to high strain rates ( > 10- s s- ~). This is also noted by Duval et al. ( 1991 ). On the issue of mathematical modeling, we note that a yield surface or static recovery is not necessary for modeling non-recoverable transient creep. It is possible to formulate equations in which the structural variables do not return to their original values or would take an infinitely long time to do so. On the other hand, we do agree with the discusser's remarks that the introduction of an isotropic internal stress AY does not preclude the incorporation of a normalizing parameter D for the "active" stress ( Ia - R I - A Y ) that generates nonlinear response. That D is interpreted as an additional isotropic hardening variable with an evolution equation of its own rather than just a normalizing parameter should however be treated with caution. Although D may characterize long-range isotropic hardening, there is as yet no physical justification for D and AY to enter the rate equation in different ways. In this connection it is noteworthy that viscoplastic theories for metals are generally of three types: those with no yield surface (AY= 0) but with evolving R and D, those with a yield surface and a constant D, and those with D = A Y (see the paper by Freed and Walker, 1990). Examples of category one models include those of Shyam Sunder and Wu (1990) and Lowe and Miller (1986), while the model of Chaboche (1989) belongs to the second category. Aubertin's model for rock salt (Aubertin et al., 1990) incorporates not only the yield surface but also evolution for all three internal variables R, AY and D, and does not belong to any of the three categories above. It is interesting to compare their model with that of Lowe and Miller (1986). In the former, short- and long-range isotropic hardening is
317
characterized by two variables A Y and D which enter the creep rate equation in different ways. In the latter, isotropic resistance due to short- and longrange barriers is characterized by two parameters D~ and D 2 which are added to give the composite normalizing parameter D. No isotropic internal stress is used. Static and dynamic recovery. Increased thermal activity at high homologous temperatures would indeed favor the operation of diffusion mechanisms. The diffusion coefficient of ice, however, is three orders of magnitude smaller than the typical values for metals (Goodman et al., 1981 ). This small value suggests that static recovery during ice deformation, which is dependent on time, is less significant in the intermediate- to high-rate loading of ice. A more significant recovery mechanism during ice deformation is the climb of basal dislocations by diffusion with the mechanical stress as the driving "force". Because it is promoted by a mechanical stress (or strain), this represents dynamic rather than static recovery. Also, the cross-slip of dislocations does not appear to be a plausible mechanism of dynamic recovery in ice because ice has a very low stacking-fault energy. In any case, static recovery can be added to our model as noted by the discusser, but it will only be necessary for modeling slow-rate phenomena ( < 10- 5 s- ~). Incubation period. We have discussed on pages 288-291 of our previous paper the phenomenon of the incubation period as observed in stress-dip tests. The discusser is concerned that internal variables in our model would stop evolving because the creep rate is zero during this period. We have however taken the view that whether the creep rate during this period is exactly zero can be debated (see our previous paper). To the best of our knowledge, data on the stress-dip test of ice are very limited. Furthermore, this issue has also been a matter of contention in the metals literature (see the remarks on p. 1557, Takeuchi and Argon, 1976), even though far more comprehensive and detailed data are available on the stress-dip test of metals. As far as our model is concerned, an instantaneous zero creep rate due to the balance of forward creep and reverse transient creep can be predicted during the incubation period. Moreover, the discusser is incorrect in saying that out model is incapable
318
M.S.WUANDS. SHYAMSUNDER
of predicting a new steady state if the creep rate is zero for a finite period of time. This misunderstanding arises from the fact that the time rate of the internal variables in Shyam Sunder and Wu (1990) is expressed in the form of a function o ~ ( 1 - 1 / F ) multiplied by the total creep rate, i.e., o~( 1 - 1/F)~cr, where c~ is a parameter and F i s the ratio of the creep rate ~cr to the minimum creep rate ~v. This ratio is the variable r in the case of constant-stress creep discussed previously. When the creep rate is zero, 1 / F would be infinite so that the time rate of the internal variables is not zero; the time rate is in fact proportional to the transient strain rate since ( 1 - 1/F) ~cr = ~t. Consequently, our model does not have the severe restrictions that the discusser claims. We would also like to point out that the creep rate is not zero in a stress relaxation test in which the total strain is suddenly frozen after a period of straining. Rather, the total strain rate is zero and the creep rate is given by:
~cr= -~e= - 6 / E
strain, time and strain rate are independent of the loading parameters, viz., stress and temperature. The results of this non-dimensionalization have been compared to the data of Jacka (1984) as presented by Ashby and Duval (1985) (see Shyam Sunder and Wu, 1989). Finally, we would like to remark that existing creep models for ice clearly require further development to adequately describe cyclic tests and other non-monotonic tests involving complex stress histories. An example is the experimental observation by Cole (1990) that the fully-reversed cyclic stressstrain curve of ice exhibits a rather sharp decrease in slope when the applied stress (tension or compression) exceeds certain values. Cole suggested that this feature might represent a dislocation break-away process. Based on the history of the development of models for metals, it is expected that new and more complex evolution equations will have to be proposed on the basis of extensive experimental studies.
(6)
where 1 / E is the compliance of the specimen. This implies that the creep strain increases while the elastic strain decreases, i.e., the elastic strain energy is dissipated. Recovery may not participate in this process and the stress would relax to the internal stress in the material. Creep occurs at constant structure under such a condition. If recovery does occur, then the stress can relax to a magnitude below that of the internal stress attained previously. In our model, internal variables will evolve under such a test, resulting in recovery and stress relaxation. This is true even if the total strain is frozen after it has become zero, because the subsequent but brief constant-structure stress relaxation will make ~t non-zero and re-initiate recovery. Strain rate decomposition. The strain rate decomposition used in our model does not yield inconsistencies in the stress and temperature dependence of transient and steady-state flow. In both components, the temperature dependence is described by Arrhenius' Law and the strain rate is proportional to the third power of the driving "force". In fact, it is precisely because of this same dependence that our model can be non-dimensionalized so that the relationships among the normalized variables of
References Ashby, M.F. and Duval, P., 1985. The creep of polycrystalline ice. Cold Reg. Sci. Technol., I 1: 285-300. Aubertin, M., 1991. Discussion: On the constitutive modeling of transient creep in polycrystallineice. Cold Reg. Sci. Technol., 20 (1992) 225-227. Aubertin, M., Gill, D.E. and Ladanyi, B., 1990. An internal variable model for the creep of rocksalt. Rock Mech. Rock Eng., 24:81-97. Budd, W.F. and Jacka, T.H., 1989. A review of ice rheology for ice sheet modeling. Cold Reg. Sci. Technol., 16: 107144. Chaboche, J.L., 1989. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int. J. Plasticity, 5: 247302. Cole, D.M., 1990. Reversed direct-stress testing of ice: initial experimental results and analysis. Cold Reg. Sci. Technol., 18: 303-321. Duval, P., Kalifa, P. and Lestringant, R., 1991. Visco-elasticity and the compressive failure of polycrystalline ice. Proceedings of the Sixth International Cold Regions Engineering Specialty Conference, ASCE, February 26-28, W. Lebanon, NH, pp. 494-503. Freed, A.D. and Walker, K.P., 1990. Steady-state and transient Zener parameters in viscoplasticity: drag strength versus yield strength. Proceedings of the Eleventh US National Congress of Applied Mechanics, Tucson, AZ, May 1990, pp. $328-$337.
ON THE CONSTITUTIVE MODELING OF TRANSIENT CREEP IN POLYCRYSTALLINE ICE: REPLY TO AUBERTIN'S COMMENTS
Goodman, J.D., Frost, H.J. and Ashby, M.F., 1981. The plasticity of polycrystalline ice. Philos. Mag. A, 43: 665-695. Jacka, T.H., 1984. The time and strain required for the development of minimum strain rates in ice. Cold Reg. Sci. Technol., 8:261-268. Lowe, T.C. and Miller, A.K., 1986. Modeling internal stresses in the nonelastic deformation of metals. J. Eng. Mater. Technol., 108: 365-373. Mellor, M. and Cole, D.M., 1983. Stress/strain/time relations for ice under uniaxial compression. Cold Reg. Sci. Teehnol., 6: 207-230.
319
Shyam Sunder, S. and Wu, M.S., 1989. A differential flow model for polycrystalline ice. Cold Reg. Sci. Technol., 16: 45-62. Shyam Sunder, S. and Wu, M.S., 1990. On the constitutive modeling of transient creep in polycrystalline ice. Cold Reg. Sci. Technol., 18: 267-294. Takeuchi, S. and Argon, A.S., 1976. Steady-state creep of single-phase crystalline matter at high temperature. J. Mater. Sci., 11: 1542-1566.
315
Discussion
On the constitutive modeling of transient creep in polycrystalline ice*: Reply to the comments of M. Aubertin M a o S. WLIa a n d S. S h y a m Sunder b aDepartment of Engineering Mechanics, University of Nebraska-Lincoln, 219 Bancroft, Lincoln, NE 68508, USA bDepartment of Civil Engineering, Massachusetts Institute of Technology, Room 1-346, Cambridge, MA 02139, USA (Received 8 June 1991; accepted 13 June 1991 )
We appreciate Michel Aubertin's ( 1992 ) interest in our paper (Shyam Sunder and Wu, 1990) and would like to respond to his c o m m e n t s on the following: 1. Ratio of initial creep rate to steady-state creep rate. 2. Non-recoverability of transient strain. 3. Static and dynamic recovery. 4. Incubation period in stress-dip tests. 5. Strain rate decomposition. We will show that the discusser's comments on items ( 1 ), (2) and (5) in relation to our transient creep model are incorrect. The physical basis of his interpretations of item (3) is unclear. The comments on item (4) are well taken and reinforce what we have discussed in our paper but can be debated in the absence of conclusive experimental data for ice. The conclusions that the discusser draws regarding the ability of our model to describe stress relaxation p h e n o m e n a and internal structure evolution after a period of zero creep rate are also incorrect. These five items are discussed in the following. Ratio of creep rates. The discusser took issue with the relationship proposed by Mellor and Cole (1983) which was derived on the basis of experimental data for polycrystalline ice:
*Cold Reg. Sci. Technol., 18: 267-294.
~0 r=__=C
(1)
Es
where r denotes the ratio between the initial creep rate and the steady-state creep rate and C is a constant. The initial creep rate refers to the strain rate measured after imposition of a stress j u m p Aa,, while the steady-state creep rate denotes the strain rate attained under the constant stress a' = a0 + Aa~ and ao equal to zero. It is noteworthy that the proportionality between the initial strain rate and the steady-state strain rate has also been observed for other materials (see the references contained in Takeuchi and Argon, 1976). The discusser, however, emphasized that "the initial creep rate is largely dependent on the entire mechanical history of the material". He apparently established the inadequacy of eq. (1) by showing that the ratio after a second stress j u m p is a nonlinear function of the initial and final states of stress a' and a":
~o ( a " ] m-n
(2)
is-\a'./ where ~0 is the initial strain rate measured after the imposition of the second stress j u m p and m, n are the constant-structure and the standard power-law exponents, respectively. Note that the discusser uses the term "initial strain rate" in a general sense to mean the strain rate measured immediately follow-
316
M.S. WU AND S. SHYAM SUNDER
ing a stress j u m p which is imposed at steady state reached under the previous stress state, while the same term is used in eq. ( 1 ) to mean the strain rate measured after the very first stress jump. Our model equation actually agrees with eq. (2). Equation ( 1 ), which is a specialization of a more general relation in our model, represents the ratio of initial and steady-state strain rates, where the initial state refers to a relaxed state prior to the application of any external load. Under this condition, the ratio r is given by: r=l+
Boo = C
(3)
where Bo denotes the initial value of the non-dimensional drag stress B which characterizes the structure offering isotropic resistance to deformation. The back stress R in our model equals zero initially because a non-zero back stress would imply that the material is initially anisotropic. I f an additional load is applied at (or before) steady state so that the new stress level is a, then the ratio r is given by: r=l+
(4)
which shows a nonlinear dependence on the current values of the stress and internal variables. Note that eq. (4) reduces to eq. (3) when R = 0. Equation (4) is interpreted as follows. First, the initial strain rate is a function o f the current values of the internal structural variables and the variable (the applied stress) associated with the observable variable (the strain). The current values of the internal variables are averaged quantities which are produced by the entire past history of deformation, and the instantaneous response of the material is governed by its current state. These concepts constitute the very essence of internal variable modeling. Second, the steady-state strain rate as described by the conventional power law represents a state of saturation characterized by a stress-independent constant. Thus the ratio r between the two strain rates is still a function of the current values of the internal and associated variables. We note that eq. (2) proposed by the discusser cannot describe the ratio r properly when a creep
stress is first applied to a specimen because a' is zero. The internal variable interpretation suffers from no such inconsistency and is superior to the discusser's interpretation ofeq. (2) which reads "... r is dependent on the initial and final states of stress in a non-linear fashion ...". In a broader perspective, a material state should be described by structural variables and not by the stress because two specimens can have exactly the same current stress but a very different history of loading and hence structure. Non-recoverability of transient strain. The discusser quotes Budd and Jacka (1989) as providing experimental evidence that "at least a portion of the transient creep strain of ice is non-recoverable". Budd and Jacka, however, meant that before the m i n i m u m creep rate is reached primary creep is composed of a recoverable anelastic component and a non-recoverable component, as described by the Andrade power relation. In rate form, this relation is:
~p=K+ 3flt-2/3
(5)
where ~p is the total primary creep rate, K and fl are stress-dependent parameters, and t is the time. On the right side of eq. ( 5 ), the first t e r m K in fact describes the rate of non-recoverable primary creep, while the second term describes the rate of the recoverable component. In the model of Shyam Sunder and Wu (1990), the total creep rate before reaching the m i n i m u m creep rate is composed similarly of the rate of a recoverable component, called transient creep rate, and the rate of a non-recoverable component which is described by the conventional power law for viscous creep. This power law describes the m i n i m u m creep rate when steady state is reached, assuming no tertiary creep takes over. Consequently, our model does contain a non-recoverable creep component during primary creep that is identical in form to the first term of Budd and Jacka's equation ( K i n eq. 5). Both Duval et al. ( 1991 ) and Shyam Sunder and Wu (1990) used the term transient creep strain (e~) to mean the strain obtained by subtracting from the total strain (E) the elastic strain (~e) and the nonrecoverable strain (ev) computed from the viscous power law. By performing unloading tests, Duval et
ON THE CONSTITUTIVE MODELING OF TRANSIENT CREEP IN POLYCRYSTALLINE ICE: REPLY TO AUBERTIN'S COMMENTS
al. (1991 ) found that transient creep does have a non-recoverable component. The loading period of their creep tests is approximately three hours, while the unloading period is approximately seven hours. They concluded that the total non-recoverable creep is composed of this transient component in addition to that described by the viscous power law. Although their results should be taken into consideration, it should be noted that the non-recoverable transient component becomes less important at typical strain rates of engineering interest, i.e., moderate to high strain rates ( > 10- s s- ~). This is also noted by Duval et al. ( 1991 ). On the issue of mathematical modeling, we note that a yield surface or static recovery is not necessary for modeling non-recoverable transient creep. It is possible to formulate equations in which the structural variables do not return to their original values or would take an infinitely long time to do so. On the other hand, we do agree with the discusser's remarks that the introduction of an isotropic internal stress AY does not preclude the incorporation of a normalizing parameter D for the "active" stress ( Ia - R I - A Y ) that generates nonlinear response. That D is interpreted as an additional isotropic hardening variable with an evolution equation of its own rather than just a normalizing parameter should however be treated with caution. Although D may characterize long-range isotropic hardening, there is as yet no physical justification for D and AY to enter the rate equation in different ways. In this connection it is noteworthy that viscoplastic theories for metals are generally of three types: those with no yield surface (AY= 0) but with evolving R and D, those with a yield surface and a constant D, and those with D = A Y (see the paper by Freed and Walker, 1990). Examples of category one models include those of Shyam Sunder and Wu (1990) and Lowe and Miller (1986), while the model of Chaboche (1989) belongs to the second category. Aubertin's model for rock salt (Aubertin et al., 1990) incorporates not only the yield surface but also evolution for all three internal variables R, AY and D, and does not belong to any of the three categories above. It is interesting to compare their model with that of Lowe and Miller (1986). In the former, short- and long-range isotropic hardening is
317
characterized by two variables A Y and D which enter the creep rate equation in different ways. In the latter, isotropic resistance due to short- and longrange barriers is characterized by two parameters D~ and D 2 which are added to give the composite normalizing parameter D. No isotropic internal stress is used. Static and dynamic recovery. Increased thermal activity at high homologous temperatures would indeed favor the operation of diffusion mechanisms. The diffusion coefficient of ice, however, is three orders of magnitude smaller than the typical values for metals (Goodman et al., 1981 ). This small value suggests that static recovery during ice deformation, which is dependent on time, is less significant in the intermediate- to high-rate loading of ice. A more significant recovery mechanism during ice deformation is the climb of basal dislocations by diffusion with the mechanical stress as the driving "force". Because it is promoted by a mechanical stress (or strain), this represents dynamic rather than static recovery. Also, the cross-slip of dislocations does not appear to be a plausible mechanism of dynamic recovery in ice because ice has a very low stacking-fault energy. In any case, static recovery can be added to our model as noted by the discusser, but it will only be necessary for modeling slow-rate phenomena ( < 10- 5 s- ~). Incubation period. We have discussed on pages 288-291 of our previous paper the phenomenon of the incubation period as observed in stress-dip tests. The discusser is concerned that internal variables in our model would stop evolving because the creep rate is zero during this period. We have however taken the view that whether the creep rate during this period is exactly zero can be debated (see our previous paper). To the best of our knowledge, data on the stress-dip test of ice are very limited. Furthermore, this issue has also been a matter of contention in the metals literature (see the remarks on p. 1557, Takeuchi and Argon, 1976), even though far more comprehensive and detailed data are available on the stress-dip test of metals. As far as our model is concerned, an instantaneous zero creep rate due to the balance of forward creep and reverse transient creep can be predicted during the incubation period. Moreover, the discusser is incorrect in saying that out model is incapable
318
M.S.WUANDS. SHYAMSUNDER
of predicting a new steady state if the creep rate is zero for a finite period of time. This misunderstanding arises from the fact that the time rate of the internal variables in Shyam Sunder and Wu (1990) is expressed in the form of a function o ~ ( 1 - 1 / F ) multiplied by the total creep rate, i.e., o~( 1 - 1/F)~cr, where c~ is a parameter and F i s the ratio of the creep rate ~cr to the minimum creep rate ~v. This ratio is the variable r in the case of constant-stress creep discussed previously. When the creep rate is zero, 1 / F would be infinite so that the time rate of the internal variables is not zero; the time rate is in fact proportional to the transient strain rate since ( 1 - 1/F) ~cr = ~t. Consequently, our model does not have the severe restrictions that the discusser claims. We would also like to point out that the creep rate is not zero in a stress relaxation test in which the total strain is suddenly frozen after a period of straining. Rather, the total strain rate is zero and the creep rate is given by:
~cr= -~e= - 6 / E
strain, time and strain rate are independent of the loading parameters, viz., stress and temperature. The results of this non-dimensionalization have been compared to the data of Jacka (1984) as presented by Ashby and Duval (1985) (see Shyam Sunder and Wu, 1989). Finally, we would like to remark that existing creep models for ice clearly require further development to adequately describe cyclic tests and other non-monotonic tests involving complex stress histories. An example is the experimental observation by Cole (1990) that the fully-reversed cyclic stressstrain curve of ice exhibits a rather sharp decrease in slope when the applied stress (tension or compression) exceeds certain values. Cole suggested that this feature might represent a dislocation break-away process. Based on the history of the development of models for metals, it is expected that new and more complex evolution equations will have to be proposed on the basis of extensive experimental studies.
(6)
where 1 / E is the compliance of the specimen. This implies that the creep strain increases while the elastic strain decreases, i.e., the elastic strain energy is dissipated. Recovery may not participate in this process and the stress would relax to the internal stress in the material. Creep occurs at constant structure under such a condition. If recovery does occur, then the stress can relax to a magnitude below that of the internal stress attained previously. In our model, internal variables will evolve under such a test, resulting in recovery and stress relaxation. This is true even if the total strain is frozen after it has become zero, because the subsequent but brief constant-structure stress relaxation will make ~t non-zero and re-initiate recovery. Strain rate decomposition. The strain rate decomposition used in our model does not yield inconsistencies in the stress and temperature dependence of transient and steady-state flow. In both components, the temperature dependence is described by Arrhenius' Law and the strain rate is proportional to the third power of the driving "force". In fact, it is precisely because of this same dependence that our model can be non-dimensionalized so that the relationships among the normalized variables of
References Ashby, M.F. and Duval, P., 1985. The creep of polycrystalline ice. Cold Reg. Sci. Technol., I 1: 285-300. Aubertin, M., 1991. Discussion: On the constitutive modeling of transient creep in polycrystallineice. Cold Reg. Sci. Technol., 20 (1992) 225-227. Aubertin, M., Gill, D.E. and Ladanyi, B., 1990. An internal variable model for the creep of rocksalt. Rock Mech. Rock Eng., 24:81-97. Budd, W.F. and Jacka, T.H., 1989. A review of ice rheology for ice sheet modeling. Cold Reg. Sci. Technol., 16: 107144. Chaboche, J.L., 1989. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int. J. Plasticity, 5: 247302. Cole, D.M., 1990. Reversed direct-stress testing of ice: initial experimental results and analysis. Cold Reg. Sci. Technol., 18: 303-321. Duval, P., Kalifa, P. and Lestringant, R., 1991. Visco-elasticity and the compressive failure of polycrystalline ice. Proceedings of the Sixth International Cold Regions Engineering Specialty Conference, ASCE, February 26-28, W. Lebanon, NH, pp. 494-503. Freed, A.D. and Walker, K.P., 1990. Steady-state and transient Zener parameters in viscoplasticity: drag strength versus yield strength. Proceedings of the Eleventh US National Congress of Applied Mechanics, Tucson, AZ, May 1990, pp. $328-$337.
ON THE CONSTITUTIVE MODELING OF TRANSIENT CREEP IN POLYCRYSTALLINE ICE: REPLY TO AUBERTIN'S COMMENTS
Goodman, J.D., Frost, H.J. and Ashby, M.F., 1981. The plasticity of polycrystalline ice. Philos. Mag. A, 43: 665-695. Jacka, T.H., 1984. The time and strain required for the development of minimum strain rates in ice. Cold Reg. Sci. Technol., 8:261-268. Lowe, T.C. and Miller, A.K., 1986. Modeling internal stresses in the nonelastic deformation of metals. J. Eng. Mater. Technol., 108: 365-373. Mellor, M. and Cole, D.M., 1983. Stress/strain/time relations for ice under uniaxial compression. Cold Reg. Sci. Teehnol., 6: 207-230.
319
Shyam Sunder, S. and Wu, M.S., 1989. A differential flow model for polycrystalline ice. Cold Reg. Sci. Technol., 16: 45-62. Shyam Sunder, S. and Wu, M.S., 1990. On the constitutive modeling of transient creep in polycrystalline ice. Cold Reg. Sci. Technol., 18: 267-294. Takeuchi, S. and Argon, A.S., 1976. Steady-state creep of single-phase crystalline matter at high temperature. J. Mater. Sci., 11: 1542-1566.