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Relaxation behavior and modeling
International Journal of Plasticity 17 (2001) 1419±1436 www.elsevier.com/locate/ijplas
Relaxation behavior and modeling Erhard Krempl * Mechanics of Materials Laboratory, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA Received in final revised form 23 October 2000
Abstract Load relaxation tests deliver several orders of magnitude of inelastic strain rate data while elastic strains are converted into inelastic strains [see Lemaitre and Chaboche, 1994. (Mechanics of Solid Materials, Oxford University Press, Cambridge p. 264)]. Hart used this test for providing information on the inelastic deformation behavior for modeling purposes. Characteristic relaxation curves were obtained with ductile metals and alloys at room and high temperature showing a scaling relation derived from Hart's theory. Subsequent testing with servo-controlled testing machines and strain measurement on the gage length showed that an increase of prior strain rate also increased the average relaxation rate. For relaxation tests starting in the ¯ow stress region, the relaxation curves can be independent of the stress and strain at the start of the relaxation tests. For the modeling of these newly found relaxation behaviors and other phenomena the viscoplasticity theory based on overstress (VBO) has been introduced. It is shown that VBO admits a long-term (asymptotic) solution that can be used with sucient accuracy for the ¯ow stress region of the stress±strain diagram. The longterm solution predicts the observed relaxation behaviors and that the relaxation curves coincide when shifted along the stress axis. This behavior is observed for the recently obtained data and is con®rmed by two sets of the Hart-type data when they are plotted according to the new method. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Creep; A. Stress relaxation; B. Constitutive behaviour; B. Elastic-viscoplastic material; State variable model
1. Relaxation testing The load relaxation test to determine material properties for modeling was initially introduced by Hart and Solomon (1972) after a critical analysis of the * Tel.: +1-518-276-6985; fax: +1-518-276-2623. E-mail address: [email protected] 0749-6419/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0749-6419(00)00092-9
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phenomenology of plastic deformation. The relaxation test provides a convenient means of obtaining a span of strain rates while the developing inelastic strain is comparatively small and equal to the elastic strain. Curves of constant hardness are obtained from relaxation tests. Multiple hardness curves are determined at a sequence of nominally constant strain levels and subsequent reloading. Hart and Solomon (1972) show that starting from the origin there are three regions in a loading event: the initial elastic region, the transition to fully established inelastic ¯ow and the ¯ow stress region where the tangent modulus is small compared to the elastic modulus. They recommend that all relaxation tests originate in the last region, see their Fig. 2 where load vs. time is plotted and C designates the desirable region. On re-loading the ¯ow stress characteristic of the strain rate must be reached before a new relaxation test is to be started. If this condition is not ful®lled hardness curves can cross, see Krempl (1988) for a discussion of some ®ndings reported in the literature. Crosshead displacement was controlled and load was the dependent variable. The test data were analyzed and corrected to ®nd the inelastic strain rate and the true stress from the crosshead displacement and from the load. The results are a series of hardness curves plotting log stress vs. log strain rate as shown in Fig. 1 taken from Yamada and Li (1973). It is seen that all curves start at a strain rate of 10 4 1/s and end at 10 8 1/s (the strain rate numbers are approximate). There is also a straight line with an arrow pointing towards the lower left that marks a scaling relation proposed by Hart. ``The scaling relation can be used to test the existence of a mechanical equation of state. If the scaling relation exists, one will be able to : : superpose by translation log; log" any one of the log log" curves onto any of the others of the same material in such a way that the overlapping segments of each curve match within experimental error. A master curve can, therefore, be constructed . . . to satisfy the condition of one parameter family of curves.'' Yamada and Li (1973), p. 2134. An equation of state can be de®ned as an expression involving n (usually three or four) variables such that the remaining variable is uniquely determined once n-1 variables are known. Hart and others who adopted his method published many other test results that show the same characteristics of the relaxation curves, called hardness curves. They all start at around 10 4 1/s and extend to 10 9 1/s at the most. A shift is always possible provided the tests were not performed in the transient region. A master curve is constructed and this master curve extends to much smaller strain rates than the original curves. Although the in¯uence of strain rate on the stress±strain diagrams was explored, see Wire et al. (1976), pp. 678 and 679, the in¯uence of prior strain rate on the relaxation behavior was apparently never investigated. 2. Relaxation testing using servo control Servo-controlled testing and strain measurement on the gage length using a clipon extensometer became available after Hart published his ideas. Now, they are
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Fig. 1. A series of room temperature hardness curves for Type 304 stainless steel, from Yamada and Li (1973). N1±N4 are the consecutive runs with one specimen. After completing the relaxation test at N1 the specimen was reloaded to N2 for a new relaxation test. A total of four such relaxation tests were performed with one specimen.
nearly standard methods of testing. The strain rate used in loading up to the relaxation event can be controlled easily and accurately. Further, strain-control using a clip-on extensometer on the gage length with feedback represents the best available ``hard testing machine'' needed for relaxation testing. Setting the strain rate equal to zero simulates the relaxation test. For the servo-control to operate a
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Fig. 2. The same hardness curves as in Fig. 1. This time, the curves are re-plotted in Fig. 2a using a linear stress scale. A vertical shift yields one master curve, see Fig. 2b.
dierence signal between command and feedback is needed which represents a deviation form the ``ideal'' hard testing machine. But that deviation is negligible for a well-tuned system. Krempl (1979) reported room temperature relaxation test results obtained with a servo-controlled testing machine. In the transient free region and for a ®xed time interval the relaxation drop for annealed type 304 stainless steel was shown to be independent of the stress and strain at the beginning of the relaxation test. It increased with prior strain rate in a nonlinear fashion. The testing was extended to other materials, see Krempl and Nakamura (1998) for details. These observations pertain to the regions where plastic ¯ow is fully established, the ¯ow stress has been reached and the tangent modulus is much less
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than the elastic modulus. This region is referred to as ``transient free region'' by Hart, see above. In this region the following properties were found.
Box 1. Relaxation properties found 1. For a given strain rate and for a given relaxation time the stress drop can be independent of the stress and strain at which relaxation starts. Examples of this behavior have been found for alloys that work harden (Positive tangent modulus, example Type 304 Stainless Steel.), show perfect plastic ¯ow (A zero tangent modulus, example Ti-alloy.) and exhibit work softening (Negative tangent modulus, example modi®ed 9 Cr 1Mo steel at 538 C). In the instances where the relaxation curves have been measured it was shown that not only the relaxation drops but also the stress vs. time curves can be identical, see Bordonaro and Krempl (1992) and Majors and Krempl (1994). 2. The stress drops in a constant time interval depend on prior strain rate. They are nonlinearly related to the prior strain rate. A tenfold increase of the strain rate increases the stress drop by much less than a factor of ten. 3. At the end of relaxation periods of constant duration the test associated with the fastest (slowest) prior strain rate can exhibit the smallest (largest) stress magnitude. 4. Once straining is resumed at the end of the relaxation periods the ¯ow stress characteristic of the strain rate is reached quickly (transition strains as low as 0.01% can be observed.). The stress±strain curve after relaxation follows the curve with the same constant strain rate without relaxation periods. The relaxation periods are forgotten. In other words, no strain rate history eect has been observed.
Property 1 above should be considered an ideal property that not all metals and alloys exhibit. There are some examples where there is a dependence of the stress drop on the stress or strain from which relaxation starts, see Bordonaro and Krempl (1992) for Nylon 66 and Krempl and Nakamura (1998) for metals and alloys. There will be a discussion of this aspect below. 3. Relation to the experimental results obtained with Hart's method On ®rst glance there does not seem to be an overlap between the ®ndings of the Hart school of thought and the present observations. This is true for items 2 and 3 since we have found no reference where the strain rate was changed. There are, however, commonalties for items 1 and 4.
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Although not speci®cally mentioned, the experiments performed according to Hart's methodology are taking advantage of the fact that after a relaxation event the stress±strain curve is reached quickly again, see item 4. Since transients are to be avoided a new hardness run can only be started after the original ¯ow stress has been reached. Item 1 relates to the relaxation curve, speci®cally the stress drop and the stress vs. time curves. Such curves are not reported by Hart, rather the results are given as log : log vs: log " curves. The strain rate axis can be interpreted as a measure of time since the strain rate varies in a consistent but nonlinear manner with an increase of time. A series of statements pertaining to the viscoplasticity theory based on overstress (VBO) are given below which demonstrate that property 1 is the consequence of the overstress dependence of the inelastic strain rate and the attainment of an asymptotic solution which holds when the transient free region is reached. The pure overstress version of VBO requires that the relaxation curves (stress vs. time) obtained in the ¯ow stress, transient free region are congruent. It is of interest to see whether the hardness curves share this property. To explore this aspect, the data are re-plotted on a linear stress scale leaving the nonlinear time represented by the strain rates. These curves should overlap after a shift along the stress axis. Figs. 2a and b and 3a and b show the results for stainless steel and niobium, respectively. Figs. 2a and 3a display the individual hardness curves that were shifted to obtain one ``master curve'' in Figs. 2b and 3b, respectively. It is seen that the shifts are possible with a minimum of error. The data obtained by the Hart method can be reinterpreted and con®rm the results obtained recently, see property 1 above. Other data, see Yamada and Li (1973, 1974), Hart et al. (1975) and Wire et al. (1976), could be analyzed as well. 4. Modeling the relaxation behavior The purpose of this section is to demonstrate how VBO can qualitatively model the observed relaxation behavior. To this end we are reproducing the VBO theory ®rst. Then we investigate the long-term solutions and their in¯uence on the modeling of relaxation tests. 4.1. Small strain, isotropic VBO for variable, low homologous temperature 4.1.1. The ¯ow law On the assumption of volume preserving inelastic deformation the ¯ow law can be written as d 1 3 s : : : e eel ein s F dt E 2
g
1
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Fig. 3. The room temperature hardness curves of high purity niobium shown in Fig. 1 of Yamada and Lee (1974) are re-plotted in Fig. 3a using a linear stress scale. A vertical shift such that the curves coincide at the lowest left point yields one master curve, see Fig. 3b. Note, that Run 105 shows the largest deviation. Since it is the test with the smallest strain the transient free region may not have been reached at this strain.
q 3 g g is the overstress invariant with s the stress deviator 2 tr : q : : is the and g the equilibrium stress deviator. 23 tr ein ein F Ek eective inelastic strain rate and F [ ] is the positive, increasing ¯ow function with the dimension of 1/time and F[0]=0. A frequently used ¯ow function is the power law. The positive, decreasing viscosity function, with k0 6 0 and the dimension of time can be considered a variable relaxation time. Modi®ed power laws and single and double exponential functions have been used here. The elastic constants E and where
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n, as well as other constants, can be functions of temperature. The [ ] following a : symbol denotes ``function of.'' The deviatoric strain rate is e and the superscripts el in and denote elastic and inelastic parts, respectively. The symbol t designates time. The viscosity and the ¯ow function are usually functions of =D where D is a positive quantity with the dimension of stress. It can be considered a state variable with a growth law and then it is called the drag stress. Unless otherwise mentioned D will be a constant in this paper. The deviatoric formulation has to be augmented by the volumetric elastic relation : d 1-2 : tr 3 T 2 tr" dt E where is the coecient of thermal expansion, T is the absolute temperature and a superposed dot designates dierentiation with respect to time. " and are the strain and stress tensor, respectively. The time derivatives in Eqs. (1) and (2) ensure the path independence of the elastic deformation for variable temperature and temperature dependent material properties, see Lee and Krempl (1991). For constant temperature the expressions simplify to the stress rates. It is seen from Eq. (1) that in the case of rest, when all time rates are zero, s g=0. The equilibrium stress g is therefore the stress that can be sustained at rest. When the applied stress is zero at rest the equilibrium stress must also be zero. On application of a stress the equilibrium stress increases but has to be dierent from the applied stress so that inelastic deformation can develop. When not at rest, g represents the stress that must be overcome to generate inelastic deformation. It can be thought of as a measure of the defect structure of a metal. The growth law of the equilibrium stress to be given needs to be hysteretic so that loading/unloading behavior can be modeled. The evolution has to be path-dependent to represent plastic deformation properly. The terms ``back stress'' and ``eective stress'' are used in the literature and their meanings are close to the equilibrium stress and overstress, respectively. Subtle differences must be kept in mind. The back stress in plasticity is identi®ed with kinematic hardening and is responsible for modeling the Bauschinger eect. The equilibrium stress, however, is not the repository for modeling of the Bauschinger eect. The kinematic stress of VBO has that function, see below. The equilibrium stress cannot be the internal stress, which by equilibrium considerations must be equal to the applied stress. The designation internal stress for the equilibrium stress is wrong and is not acceptable. There is no question that internal stresses exist in a specimen for which a macroscopically uniform stress distribution is assumed. The internal stress distribution is non-uniform. However, when integrated over the specimen cross-section it must be equal to the applied stress. The equilibrium stress is a state variable that, like all true state variables, cannot be measured and controlled. But the presence of such a state variable can be inferred from transient tests and a comparison of the observations with the predictions of a model, see Krempl (1995, 1996). These tests have established the advantage of the equilibrium stress in modeling transient phenomena. Despite the usefulness
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of the equilibrium stress in explaining unusual phenomena, it cannot be said that the equilibrium stress is the only quantity that has such capabilities. But no other has been found as of now. 4.1.2. Growth laws for the state variables The growth law for the equilibrium stress consists of elastic and inelastic hardening terms, a dynamic recovery term and a term that involves the kinematic stress which is needed so that positive, negative or zero slope can be modeled at the maximum strain of interest. The growth law for the equilibrium stress is @ : : s g g f : : g s Ts 1 f 3 @T E E k k A E The ®rst term on the right hand side is only for variable temperature and ensures the path independence of the g during initial deformation. The positive, decreasing shape function is bounded by 1 > E > EEt . It controls the transition from initial quasi-linear behavior to fully established inelastic ¯ow. The ®rst two terms in the parenthesis following E are the elastic and inelastic hardening terms, the term with the negative sign is the dynamic recovery term. This growth law can also be written in terms :of the ¯ow function F. The last term multiplied by the kinematic stress deviator f is needed so that an appropriate long-term asymptotic solution can be developed, see below. The growth law for the kinematic stress is given in the Prager hardening form by : 2 : E^ t s g f E^ t ein 3 E k
4
E^ t is the tangent modulus at the maximum inelastic strain of interest that can be positive, zero or negative (Et is the same quantity referred to the total strain. The two quantities are related by E^ t Et = 1 Et =E and dier very little for most engineering materials.) The isotropic stress A is given the simple growth law : A Ac Af
: A
5
where Ac and Af are constants with no dimension and the dimension of stress, respectively. The ®rst controls the speed with which the ®nal value Af is reached. A more complicated growth law for the isotropic stress is needed when complex cyclic behavior is to be modeled, see Ruggles and Krempl (1990) and Choi and Krempl (1991). These equations constitute the VBO model for small strain and metals and alloys (the inelastic deformations are volume preserving). They are coupled, nonlinear sti dierential equations and for their solution initial conditions must be known. The growth laws for the state variables are homogeneous of degree one in the rates and represent rate-independent behavior. This property is not obvious for Eq. (3) when it is written in the stress form. Using the ¯ow law Eq. (3) can be rewritten as
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@ : g @T
: : eel 2: ein Ts E E 1 3
: g
f A
E 1
2 ^ : in Et e E 3
6
where the homogeneity of degree one in the rates is apparent. It will be demonstrated below that the dependence of the shape function on the overstress invariant introduces a rate dependence during the transition from the initial quasi elastic region to fully established plastic ¯ow. 4.1.3. Properties in very slow and very fast loading The responses to very slow and very fast loading at constant temperature are of interest. To obtain the slow response the uniaxial version of Eq. (1) is tranformed to an integral equation with 32 s11 and 32 g11 G. For loading with a constant strain rate it is advantageous to change the integration variable from time to strain to yield " 1 " 1 dG 1 " 1 ds ds d E G 0 G0 exp exp 0 k d" k 0 7 If the strain rate is reduced to smaller and smaller values the right hand side tends to zero and we have lim G 0 so that the stress is equal to the equilibrium stress ! 0
for very slow loading. Since the growth law for g is essentially rate-independent a nonzero equilibrium stress exists and the growth law represents a solid. This property and the fact that G at rest suggests the name equilibrium stress. To obtain the response for in®nitely fast loading the uniaxial version of the ¯ow law is used again after application of the chain rule we get 1
d=d" 1 G E Ek
8
Taking the limit for very fast strain rates we obtain lim
! 1
1
d=d" E
0. The
fast response is, therefore, elastic. The slow and the fast responses bracket the responses for all other loading rates. Fig. 4 gives a sketch of these results. 4.1.4. Fully established inelastic ¯ow and asymptotic solutions of VBO In demonstrating that VBO has a long-term solution that applies to the region of established ¯ow stress isothermal conditions are considered. At all times the identity s s
g g
f f
9
holds. It shows that the stress is composed of the overstress s g, the dierence between the equilibrium stress and the kinematic stress g f and the contribution of the kinematic stress. The kinematic stress can be used to set the slope at the maximum strain of interest. It can be positive (strain hardening), zero or negative (strain softening).
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Fig. 4. Schematic showing the fast and the slow monotonic response of VBO.
This writing also suggests investigating the stress dierences. It will be shown that the two stress dierences have a long-term solution whereas the stress need not. The ¯ow law can, after some straight forward calculations be written as the integral equation t t 3 1 E : : d e g s g s0 g0 exp t0 2 1 k t0 1 t 3 1 ds d 10 exp 2 1 k Similar manipulations of the growth law of the equilibrium stress, Eq. (6), for isothermal conditions, results in g
f g0
f0 exp
: ! t : s g s d k A t0
t t0
: f exp
t
: ! ds d A 11
The subscript zero denotes initial conditions. For the isotropic stress we have t : Ac d 12 A Af A0 Af exp t0
The arguments of the exponential functions are positive. If time grows without bounds the initial conditions of the overstress and of g f vanish since the exponential function goes to zero. The ®nal value Af of the isotropic stress is also independent of the initial condition. Only the kinematic stress can exhibit a non-vanishing in¯uence of the initial condition. It could be used to model tension/compression
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asymmetry. The asymptotic value of the isotropic stress is Af . The long-term solutions for the overstress and for g f in Eqs. (10) and (11), respectively, are found following Ho (1998). Both equations are dierentiated with respect to time. For large times these equations in indeterminate expressions for the rates which must be resolved. : result : Then s g 0 and : : n :o s g f
13
where {} denotes asymptotic value. This result allows the statement that f sets the slope of the stress-strain curve at the maximum strain of interest. This result shows that ultimately the stress, equilibrium stress and kinematic stress will have the same rate and the distance between any two of the three curves will be constant. The results shown in Eq. (13) are used in evaluating the asymptotic solution of Eq. (10)
s
g =kf
: g 2=3 Ee
: : g 1 2=3E ein
14
It is seen that the overstress is rate dependent and that this dependence is nonlinear and is determined by the viscosity function k[ ]. This equation can be expressed in terms of the ¯ow function ns
go
Ff
g
2 : in e 3
15
Again the nonlinear rate sensitivity of the overstress is evident. The arguments of the viscosity and the ¯ow functions are not explicitly written to allow for a pure overstress dependence, for a dependence on the drag stress or for a dependence on both. For the asymptotic solution to hold it is necessary for the drag stress to have reached the ®nal value. The constant overstress that pertains to the stationary solution is determined by the strain rate and can have any value. The long-term solution of Eq. (11) is s g g f Af 16 The rate-independent contribution to the stress is given by this expression since Af and the expression are rate independent. The directions of the rate-dependent and of the rate-independent contributions to the stress are the same. Both point in the direction of the overstress. Their magnitudes can be easily q n : odetermined to be Af 3 with 2 tr g f g f from Eq. (16) and Ff g Ekff g g from Eq. (15). A geometrical interpretation of the asymptotic solution is given by Krempl (1996).
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4.2. Application to the modeling of relaxation for pure overstress dependence We assume that the viscosity function or the ¯ow function depends only on the overstress. When the asymptotic region is reached the overstress will be constant. The relaxation tests are performed after the transient free region has been reached. The overstress does not change as the stress±strain curve develops to higher and higher strains. The ¯ow law for relaxation is 1: 3 s g : : s F 17 0 eel ein E 2 This equation can be rewritten as : s
3E F 2 1
s
g
3 s g 2 1 k
18
It is seen that the relaxation rate depends only on the overstress. So all relaxation tests with identical prior strain rate and performed in the transient free region have initially the same overstress and the stress rate is initially the same. The growth laws for the other state variables depend only on the overstress and on g f. These quantities do not depend on stress or strain directly. It follows that further values of the relaxation rate are also independent of the strain or stress at which the relaxation starts. The evolution of the relaxation curves is the same for any point in the transient free region as long as the prior strain rate is the same. Box 2. Modeling of relaxation properties 1. VBO can model the described behaviors. In real experiments the tangent modulus is not strictly constant as required but changes somewhat. It is still assumed that the asymptotic solution is applicable. 2. Since the overstress increases nonlinearly with the strain rate the relaxation rate will, therefore, increase with an increase in prior strain rate. These prediction are found in experiments, see property 2 given previously. 3. Property 3 is also predicted by VBO as demonstrated in Krempl and Nakamura (1998) using the data by Yaguchi and Takahashi (1999). 4. The modeling of the quick transition from the relaxation test to the ¯ow stress characteristic of the strain rate mentioned under property 4 requires that the in¯uence of initial conditions vanishes quickly. Once the ¯ow stress is reached in a simulation any in¯uence of the initial conditions has vanished. The above analyses show that the in¯uence of the initial conditions of the overstress and of g f will vanish. No permanent in¯uence of the relaxation holds is predicted by VBO. Figs. 6±8 of Majors and Krempl (1994) demonstrate that VBO is capable of modeling repeated relaxation tests with loading intervals between them.
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5. Discussion 5.1. Relaxation behavior The above discussion has demonstrated that in the ``transient free region'' there is a characteristic relaxation behavior that was initially explored by Hart and coworkers. However, the in¯uence of a change in the prior strain rate had not been explored and this topic has increased the challenges to modeling. New experiments with servo-controlled testing machines explored the in¯uence of prior strain rates on relaxation behavior. The ``transient free region'' corresponds to the concept of fully established ¯ow stress used elsewhere in the literature. VBO models this region through a long-time asymptotic solution that is very well applicable at small strains. It is shown that the existence of the asymptotic solution requires the overstress to be constant. This fact explains without any diculty that the relaxation behavior can be independent of the stress and strain at the beginning of the relaxation test. If either the drag stress or the isotropic stress is not constant with all the other conditions unchanged then the relaxation behavior may depend on the stress and strain at the beginning of the relaxation test. It is seen from Eqs. (10) and (11) that the functions which appear as arguments of the negative exponential functions are dierent for the overstress and for g f. In theory it is possible for the overstress to reach the asymptotic limit while the rate independent contribution to the stress is still evolving or vice versa. [The expressions, which appear in the exponentials, Eqs. (10) and (11) are dierent. The asymptotic limit can, therefore, be reached with dierent speeds.] This dierence in the approach to the long-term solution, and the existence of rate-dependent and rate-independent contributions to the stress is a basic characteristic of VBO, which brings ¯exibility to the modeling. 5.2. Creep experiments The question arises whether the creep behavior can have the same regularity as the relaxation behavior. If the `transient free region' is considered, the overstress is constant for a given prior strain rate. The ¯ow law for creep is obtained from Eq. (1) 3 s0 : : e ein F 2
g 0
3 s0 g 2 Ek
19
where the subscript 0 indicates that the stress is constant. It is seen that for all tests with the same initial strain rate the overstress and, therefore, the initial creep rate is the same. To obtain equal creep curves for dierent initial strains it is necessary for the equilibrium stress to evolve in the same way in the strain intervals that are covered by the creep tests. This is only true for a linear evolution of the equilibrium
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stress. It is known from the properties of the VBO model that for the asymptotic solution the stress±strain diagram and the equilibrium stress±strain diagram have equal slopes, see Eq. (13). If we assume that this model is a good qualitative picture of real behavior, then the equilibrium stress±strain diagram has the same tangent modulus as the real stress±strain diagram. The real stress±strain diagram is seldom linear and neither is the equilibrium stress±strain curve. It can therefore not be expected that the same creep curves be obtained for dierent initial strains. Given the nonlinear dependence of strain rate on overstress a small variation in the slope of the equilibrium stress can have a signi®cant in¯uence on the measured creep rate. So creep tests are very sensitive to variation in the tangent modulus. Generally creep curves that are independent of the initial strain should not be expected. Note that there is a fundamental dierence in the creep and the relaxation test. The evolution of the equilibrium curve with time is small and occurs at a constant total strain in relaxation. Considerable strain can develop for the creep test and the evolution of the equilibrium stress plays a signi®cant role. 5.3. High homologous temperature behavior For high homologous temperature behavior the hardening due to inelastic deformation and the recovery due to diusion are in competition. To account for this behavior, a static recovery term is introduced in the growth law for the equilibrium stress taking the Baily-Orowan format as a model, see Majors and Krempl (1994) and Tachibana and Krempl (1995, 1997, 1998). This new term is not of degree one in the rates so that time matters in the evolution of the equilibrium stress. In addition a softening of the isotropic stress is needed to account for cyclic softening and tertiary creep. Under these conditions the long-term asymptotic solution may not exist. The methodology may still be useful but the interpretation of the results must be done with care. 5.4. A physically based overstress model The concept of the eective stress and the equilibrium (back) stress has been used in many materials science-oriented, physically based papers. An example is the approach of Gupta and Li (1970) who have applied their model to relaxation. They : start with the Orowan equation for the inelastic strain rate " p bv ( is a geometric factor, is the density of mobile dislocations, b is their Burgers vector magnitude and is their average velocity) and use the Johnston±Gilman relation B i m (B is the average velocity at unit eective stress, m* is the dislocation : : velocity-stress exponent and i is the internal stress). During relaxation "p "e 1 d E dt (E* is the combined elastic modulus of the testing specimen, and d=dt is the relaxation rate). The three formulae can be combined to yield d dt
K 0
i m
20
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where K' is the combination of the other constants. Gupta and Li integrate this expression for constant K 0 , constant i and constant m* to yield
i K t a
n
21
with n 1= m 1, K=(K 0 (m* 1))-n, t is time, presumably zero at the start of relaxation and a is an integration constant. It is seen that Eq. (20) is akin to Eq. (18) if F[ ] is interpreted as a power function. The physical approach and the experiment based approach come to qualitatively the same expressions. There are, however, substantial dierences. VBO cannot yield Eq. (21) since in VBO the equilibrium stress (equivalent to the internal stress i ) is not constant in a relaxation test. Also the approach by Gupta and Li does not give the constitutive equation, equivalent to the set of coupled nonlinear equations, see Eqs. (1)±(5), that enable the simulation of any kind of test. For VBO relaxation is a special type of test that can be obtained using the special conditions of the test, i.e. strain rate is equal to zero. It leads to a set of dierential equation that can be solved. Other tests are the creep test and stress and displacement control in cyclic loading to name just a few. The Gupta and Li approach is totally dedicated to relaxation and the approach cannot be used to predict creep. Inserting the conditions for creep, i.e. stress rate is equal to zero in VBO results in Eq. (19) and the evolution equations for the state variables. For the Gupta and Li approach using the condition of zero stress rate equal to zero in Eq. (20) results in i which is not describing any creep curve. In the development of VBO monotonic loading at dierent rates (stress or strain control), creep and relaxation as well as cyclic loading were considered. These preconditions and the experimental base enable the application of VBO to any kind of test. The Gupta and Li theory, despite its physical base, applies only to relaxation, it fails to give sensible results in creep. It follows that a mechanistic base alone does not guarantee adequate results. Mechanistic approaches are usually developed to parameterize responses to a given input (example creep curves give the response for the input constant stress; relaxation yields the response for constant strain). But these responses do not constitute a material model that can be applied to any loading. These theories apply only to the test condition for which they had been developed. A constitutive equation or material model should be competent to model the responses for the tests discussed here. VBO has that capability. Using their method Gupta and Li (1970) were able to determine the internal stress i as a function of strain. Their Figs. 14±16 and 18 show that the eective stress equivalent to the overstress tends to vary little as the strain increases. The results obtained from various relaxation tests at dierent strain levels are plotted as stress vs. strain and resemble very much the behavior of VBO depicted schematically in Fig. 1 of Krempl (1996). Gupta and Li (1970) assume that the equilibrium (internal) stress is constant during relaxation but changes during tensile straining. This is not predicted by VBO where the equilibrium stress changes during relaxation and during tensile loading The changing equilibrium stress is a major factor in predicting the endpoints of the relaxation tests, see Krempl and Nakamura (1998).
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6. Conclusions The relaxation tests initially advocated by Hart have been extended to dierent test conditions and to using servo-controlled testing and strain measurement on the gage length. New test conditions include use of dierent strain rates preceding the relaxation. The relaxation rate depends nonlinearly on the strain rate preceding the relaxation test. These and other properties recently measured in relaxation tests are predicted qualitatively by the viscoplasticity theory based on overstress (VBO). The overstress (eective stress) dependence and the long-term asymptotic solution of VBO are the basis for these predictions. It is also shown that the hardness curves measured by Hart and collaborators can be re-interpreted using the overstress model. A discussion of the physically based approach proposed by Gupta and Li (1970) shows that it is not capable of predicting realistic creep behavior. It is argued that the response function, which generalizes the observations in a particular type of test, i. e. relaxation, is, despite its physical basis, not a suitable constitutive equation. A constitutive equation must be synthesized from the responses to monotonic loading using dierent rates, to cyclic loading and to creep and relaxation test conditions. Acknowledgements This research was supported by the Department of Energy Grant DE-FG0296ER14603. D. Alexander prepared Figs. 2 and 3.
References Bordonaro, C.M., Krempl, E., 1992. Eect of strain rate on the deformation and relaxation behavior of 6/ 6 Nylon at room temperature. Polymer Eng. and Science 32, 1066±1072. Choi, S.H., Krempl, E., 1991. Viscoplasticity theory based on overstress: the modeling of ratcheting and cyclic hardening of AISI type 304 stainless steel. Nuclear Engineering and Design 133, 401±410. Gupta, I., Li, J.C.M., 1970. Stress relaxation, internal stress, and work hardening in some BCC metals and alloys. Metallurgical Transactions 1, 2323±2330. Hart, E.W., Solomon, H.D., 1972. Load relaxation studies of polycrystalline high purity Aluminum. Acta Metallurgica 21, 295±307. Hart, E.W., Li, C.-Y., Yamada, H., Wire, G.L, 1975. Phenomenological theory: a guide to constitutive relations and fundamental deformation properties. In: Argon, A.S. (Ed.) Constitutive Equations in Plasticity, The MIT Press, pp. 149±197. Ho, K., 1998. Application of the Viscoplasticity Theory Based on overstress to the Modeling of Strain Aging of Metals and to the Modeling of Solid Polymers, Speci®cally to Nylon 66. PhD dissertation, Rensselaer Polytechnic Institute, Troy, NY, August. Krempl, E., 1979. An experimental study of room-temperature rate sensitivity, creep and relaxation of AISI Type 304 Stainless Steel. J. Mech. Phys. Solids 27, 363±365. Krempl, E., 1988. Phenomenological modeling of viscoplasticity. Revue Phys. Appl. 23, 331±338. Krempl, E., 1995. The overstress dependence of inelastic rate of deformation inferred from transient tests. Materials Science Research International 1, 3±10. Krempl, E., 1996. A small strain viscoplasticity theory based on overstress. In: Krausz, A., Krausz, K. (Eds.), Uni®ed Constitutive Laws of Plastic Deformation. Academic Press, pp. 281±318. Krempl, E., Nakamura, T., 1998. The in¯uence of the equilibrium stress growth law formulation on the modeling of recently observed relaxation behaviors. JSME International Journal Series A 41, 103±111.
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Lee, K.D., Krempl, E., 1991. Uniaxial thermo-mechanical loading. Numerical experiments using the thermal viscoplasticity theory based on overstress European. J. Mech., A/Solids 10, 173±192. Lemaitre, J., Chaboche, J.-L., 1994. Mechanics of Solid Materials. Oxford University Press, Cambridge. Majors, P.S., Krempl, E., 1994. The isotropic viscoplasticity theory based on overstress applied to the modeling of modi®ed 9 Cr-1 Mo steel at 538 C. Materials Science and Engineering A186, 23±34. Ruggles, M.B., Krempl, E., 1990. The interaction of cyclic hardening and ratcheting for AISI Type 304 stainless steel at room temperature-II. Modeling with the viscoplasticity theory based on overstress. Journal of the Mechanics and Physics of Solids 38, 587±597. Tachibana, Y., Krempl, E., 1994. Modeling of high homologous temperature deformation behavior using the viscoplasticity theory based on overstress (VBO): Part I-Creep and tensile behavior. J. Eng. Mat. Technology 117, 456±461. Tachibana, Y., Krempl, E., 1997. Modeling of high homologous temperature deformation behavior using the viscoplasticity theory based on overstress (VBO): Part II-Characteristics of the VBO model. J. Eng. Mat. Technology 119, 1±6. Tachibana, Y., Krempl, E., 1998. Modeling of high homologous temperature deformation behavior using the viscoplasticity theory based on overstress (VBO): Part III-A simpli®ed model. J. Eng. Mat. Technology 120, 193±196. Wire, G.L., Ellis, F.V., Li, C.-Y., 1976. Work hardening and mechanical equation of state in some metals in monotonic loading. Acta Metallurgica 26, 677±685. Yaguchi, M., Takahashi, Y., 1999. Uni®ed constitutive model for modi®ed 9Cr-1Mo steel incorporating dynamic strain aging eect. JSME International Journal Series A. 42, 1±10. Yamada, H., Che-Yu, Li., 1993. Stress relaxation and mechanical equation of state in austenitic stainless steels. Metallurgical Transactions 4, 2133±2136. Yamada, H., Che-Yu, Li., 1974. Stress relaxation and mechanical equation of state in B.C.C. metals in monotonic loading. Acta Metallurgica 22, 249±253.
Relaxation behavior and modeling Erhard Krempl * Mechanics of Materials Laboratory, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA Received in final revised form 23 October 2000
Abstract Load relaxation tests deliver several orders of magnitude of inelastic strain rate data while elastic strains are converted into inelastic strains [see Lemaitre and Chaboche, 1994. (Mechanics of Solid Materials, Oxford University Press, Cambridge p. 264)]. Hart used this test for providing information on the inelastic deformation behavior for modeling purposes. Characteristic relaxation curves were obtained with ductile metals and alloys at room and high temperature showing a scaling relation derived from Hart's theory. Subsequent testing with servo-controlled testing machines and strain measurement on the gage length showed that an increase of prior strain rate also increased the average relaxation rate. For relaxation tests starting in the ¯ow stress region, the relaxation curves can be independent of the stress and strain at the start of the relaxation tests. For the modeling of these newly found relaxation behaviors and other phenomena the viscoplasticity theory based on overstress (VBO) has been introduced. It is shown that VBO admits a long-term (asymptotic) solution that can be used with sucient accuracy for the ¯ow stress region of the stress±strain diagram. The longterm solution predicts the observed relaxation behaviors and that the relaxation curves coincide when shifted along the stress axis. This behavior is observed for the recently obtained data and is con®rmed by two sets of the Hart-type data when they are plotted according to the new method. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Creep; A. Stress relaxation; B. Constitutive behaviour; B. Elastic-viscoplastic material; State variable model
1. Relaxation testing The load relaxation test to determine material properties for modeling was initially introduced by Hart and Solomon (1972) after a critical analysis of the * Tel.: +1-518-276-6985; fax: +1-518-276-2623. E-mail address: [email protected] 0749-6419/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0749-6419(00)00092-9
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phenomenology of plastic deformation. The relaxation test provides a convenient means of obtaining a span of strain rates while the developing inelastic strain is comparatively small and equal to the elastic strain. Curves of constant hardness are obtained from relaxation tests. Multiple hardness curves are determined at a sequence of nominally constant strain levels and subsequent reloading. Hart and Solomon (1972) show that starting from the origin there are three regions in a loading event: the initial elastic region, the transition to fully established inelastic ¯ow and the ¯ow stress region where the tangent modulus is small compared to the elastic modulus. They recommend that all relaxation tests originate in the last region, see their Fig. 2 where load vs. time is plotted and C designates the desirable region. On re-loading the ¯ow stress characteristic of the strain rate must be reached before a new relaxation test is to be started. If this condition is not ful®lled hardness curves can cross, see Krempl (1988) for a discussion of some ®ndings reported in the literature. Crosshead displacement was controlled and load was the dependent variable. The test data were analyzed and corrected to ®nd the inelastic strain rate and the true stress from the crosshead displacement and from the load. The results are a series of hardness curves plotting log stress vs. log strain rate as shown in Fig. 1 taken from Yamada and Li (1973). It is seen that all curves start at a strain rate of 10 4 1/s and end at 10 8 1/s (the strain rate numbers are approximate). There is also a straight line with an arrow pointing towards the lower left that marks a scaling relation proposed by Hart. ``The scaling relation can be used to test the existence of a mechanical equation of state. If the scaling relation exists, one will be able to : : superpose by translation log; log" any one of the log log" curves onto any of the others of the same material in such a way that the overlapping segments of each curve match within experimental error. A master curve can, therefore, be constructed . . . to satisfy the condition of one parameter family of curves.'' Yamada and Li (1973), p. 2134. An equation of state can be de®ned as an expression involving n (usually three or four) variables such that the remaining variable is uniquely determined once n-1 variables are known. Hart and others who adopted his method published many other test results that show the same characteristics of the relaxation curves, called hardness curves. They all start at around 10 4 1/s and extend to 10 9 1/s at the most. A shift is always possible provided the tests were not performed in the transient region. A master curve is constructed and this master curve extends to much smaller strain rates than the original curves. Although the in¯uence of strain rate on the stress±strain diagrams was explored, see Wire et al. (1976), pp. 678 and 679, the in¯uence of prior strain rate on the relaxation behavior was apparently never investigated. 2. Relaxation testing using servo control Servo-controlled testing and strain measurement on the gage length using a clipon extensometer became available after Hart published his ideas. Now, they are
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Fig. 1. A series of room temperature hardness curves for Type 304 stainless steel, from Yamada and Li (1973). N1±N4 are the consecutive runs with one specimen. After completing the relaxation test at N1 the specimen was reloaded to N2 for a new relaxation test. A total of four such relaxation tests were performed with one specimen.
nearly standard methods of testing. The strain rate used in loading up to the relaxation event can be controlled easily and accurately. Further, strain-control using a clip-on extensometer on the gage length with feedback represents the best available ``hard testing machine'' needed for relaxation testing. Setting the strain rate equal to zero simulates the relaxation test. For the servo-control to operate a
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Fig. 2. The same hardness curves as in Fig. 1. This time, the curves are re-plotted in Fig. 2a using a linear stress scale. A vertical shift yields one master curve, see Fig. 2b.
dierence signal between command and feedback is needed which represents a deviation form the ``ideal'' hard testing machine. But that deviation is negligible for a well-tuned system. Krempl (1979) reported room temperature relaxation test results obtained with a servo-controlled testing machine. In the transient free region and for a ®xed time interval the relaxation drop for annealed type 304 stainless steel was shown to be independent of the stress and strain at the beginning of the relaxation test. It increased with prior strain rate in a nonlinear fashion. The testing was extended to other materials, see Krempl and Nakamura (1998) for details. These observations pertain to the regions where plastic ¯ow is fully established, the ¯ow stress has been reached and the tangent modulus is much less
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than the elastic modulus. This region is referred to as ``transient free region'' by Hart, see above. In this region the following properties were found.
Box 1. Relaxation properties found 1. For a given strain rate and for a given relaxation time the stress drop can be independent of the stress and strain at which relaxation starts. Examples of this behavior have been found for alloys that work harden (Positive tangent modulus, example Type 304 Stainless Steel.), show perfect plastic ¯ow (A zero tangent modulus, example Ti-alloy.) and exhibit work softening (Negative tangent modulus, example modi®ed 9 Cr 1Mo steel at 538 C). In the instances where the relaxation curves have been measured it was shown that not only the relaxation drops but also the stress vs. time curves can be identical, see Bordonaro and Krempl (1992) and Majors and Krempl (1994). 2. The stress drops in a constant time interval depend on prior strain rate. They are nonlinearly related to the prior strain rate. A tenfold increase of the strain rate increases the stress drop by much less than a factor of ten. 3. At the end of relaxation periods of constant duration the test associated with the fastest (slowest) prior strain rate can exhibit the smallest (largest) stress magnitude. 4. Once straining is resumed at the end of the relaxation periods the ¯ow stress characteristic of the strain rate is reached quickly (transition strains as low as 0.01% can be observed.). The stress±strain curve after relaxation follows the curve with the same constant strain rate without relaxation periods. The relaxation periods are forgotten. In other words, no strain rate history eect has been observed.
Property 1 above should be considered an ideal property that not all metals and alloys exhibit. There are some examples where there is a dependence of the stress drop on the stress or strain from which relaxation starts, see Bordonaro and Krempl (1992) for Nylon 66 and Krempl and Nakamura (1998) for metals and alloys. There will be a discussion of this aspect below. 3. Relation to the experimental results obtained with Hart's method On ®rst glance there does not seem to be an overlap between the ®ndings of the Hart school of thought and the present observations. This is true for items 2 and 3 since we have found no reference where the strain rate was changed. There are, however, commonalties for items 1 and 4.
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Although not speci®cally mentioned, the experiments performed according to Hart's methodology are taking advantage of the fact that after a relaxation event the stress±strain curve is reached quickly again, see item 4. Since transients are to be avoided a new hardness run can only be started after the original ¯ow stress has been reached. Item 1 relates to the relaxation curve, speci®cally the stress drop and the stress vs. time curves. Such curves are not reported by Hart, rather the results are given as log : log vs: log " curves. The strain rate axis can be interpreted as a measure of time since the strain rate varies in a consistent but nonlinear manner with an increase of time. A series of statements pertaining to the viscoplasticity theory based on overstress (VBO) are given below which demonstrate that property 1 is the consequence of the overstress dependence of the inelastic strain rate and the attainment of an asymptotic solution which holds when the transient free region is reached. The pure overstress version of VBO requires that the relaxation curves (stress vs. time) obtained in the ¯ow stress, transient free region are congruent. It is of interest to see whether the hardness curves share this property. To explore this aspect, the data are re-plotted on a linear stress scale leaving the nonlinear time represented by the strain rates. These curves should overlap after a shift along the stress axis. Figs. 2a and b and 3a and b show the results for stainless steel and niobium, respectively. Figs. 2a and 3a display the individual hardness curves that were shifted to obtain one ``master curve'' in Figs. 2b and 3b, respectively. It is seen that the shifts are possible with a minimum of error. The data obtained by the Hart method can be reinterpreted and con®rm the results obtained recently, see property 1 above. Other data, see Yamada and Li (1973, 1974), Hart et al. (1975) and Wire et al. (1976), could be analyzed as well. 4. Modeling the relaxation behavior The purpose of this section is to demonstrate how VBO can qualitatively model the observed relaxation behavior. To this end we are reproducing the VBO theory ®rst. Then we investigate the long-term solutions and their in¯uence on the modeling of relaxation tests. 4.1. Small strain, isotropic VBO for variable, low homologous temperature 4.1.1. The ¯ow law On the assumption of volume preserving inelastic deformation the ¯ow law can be written as d 1 3 s : : : e eel ein s F dt E 2
g
1
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1425
Fig. 3. The room temperature hardness curves of high purity niobium shown in Fig. 1 of Yamada and Lee (1974) are re-plotted in Fig. 3a using a linear stress scale. A vertical shift such that the curves coincide at the lowest left point yields one master curve, see Fig. 3b. Note, that Run 105 shows the largest deviation. Since it is the test with the smallest strain the transient free region may not have been reached at this strain.
q 3 g g is the overstress invariant with s the stress deviator 2 tr : q : : is the and g the equilibrium stress deviator. 23 tr ein ein F Ek eective inelastic strain rate and F [ ] is the positive, increasing ¯ow function with the dimension of 1/time and F[0]=0. A frequently used ¯ow function is the power law. The positive, decreasing viscosity function, with k0 6 0 and the dimension of time can be considered a variable relaxation time. Modi®ed power laws and single and double exponential functions have been used here. The elastic constants E and where
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n, as well as other constants, can be functions of temperature. The [ ] following a : symbol denotes ``function of.'' The deviatoric strain rate is e and the superscripts el in and denote elastic and inelastic parts, respectively. The symbol t designates time. The viscosity and the ¯ow function are usually functions of =D where D is a positive quantity with the dimension of stress. It can be considered a state variable with a growth law and then it is called the drag stress. Unless otherwise mentioned D will be a constant in this paper. The deviatoric formulation has to be augmented by the volumetric elastic relation : d 1-2 : tr 3 T 2 tr" dt E where is the coecient of thermal expansion, T is the absolute temperature and a superposed dot designates dierentiation with respect to time. " and are the strain and stress tensor, respectively. The time derivatives in Eqs. (1) and (2) ensure the path independence of the elastic deformation for variable temperature and temperature dependent material properties, see Lee and Krempl (1991). For constant temperature the expressions simplify to the stress rates. It is seen from Eq. (1) that in the case of rest, when all time rates are zero, s g=0. The equilibrium stress g is therefore the stress that can be sustained at rest. When the applied stress is zero at rest the equilibrium stress must also be zero. On application of a stress the equilibrium stress increases but has to be dierent from the applied stress so that inelastic deformation can develop. When not at rest, g represents the stress that must be overcome to generate inelastic deformation. It can be thought of as a measure of the defect structure of a metal. The growth law of the equilibrium stress to be given needs to be hysteretic so that loading/unloading behavior can be modeled. The evolution has to be path-dependent to represent plastic deformation properly. The terms ``back stress'' and ``eective stress'' are used in the literature and their meanings are close to the equilibrium stress and overstress, respectively. Subtle differences must be kept in mind. The back stress in plasticity is identi®ed with kinematic hardening and is responsible for modeling the Bauschinger eect. The equilibrium stress, however, is not the repository for modeling of the Bauschinger eect. The kinematic stress of VBO has that function, see below. The equilibrium stress cannot be the internal stress, which by equilibrium considerations must be equal to the applied stress. The designation internal stress for the equilibrium stress is wrong and is not acceptable. There is no question that internal stresses exist in a specimen for which a macroscopically uniform stress distribution is assumed. The internal stress distribution is non-uniform. However, when integrated over the specimen cross-section it must be equal to the applied stress. The equilibrium stress is a state variable that, like all true state variables, cannot be measured and controlled. But the presence of such a state variable can be inferred from transient tests and a comparison of the observations with the predictions of a model, see Krempl (1995, 1996). These tests have established the advantage of the equilibrium stress in modeling transient phenomena. Despite the usefulness
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of the equilibrium stress in explaining unusual phenomena, it cannot be said that the equilibrium stress is the only quantity that has such capabilities. But no other has been found as of now. 4.1.2. Growth laws for the state variables The growth law for the equilibrium stress consists of elastic and inelastic hardening terms, a dynamic recovery term and a term that involves the kinematic stress which is needed so that positive, negative or zero slope can be modeled at the maximum strain of interest. The growth law for the equilibrium stress is @ : : s g g f : : g s Ts 1 f 3 @T E E k k A E The ®rst term on the right hand side is only for variable temperature and ensures the path independence of the g during initial deformation. The positive, decreasing shape function is bounded by 1 > E > EEt . It controls the transition from initial quasi-linear behavior to fully established inelastic ¯ow. The ®rst two terms in the parenthesis following E are the elastic and inelastic hardening terms, the term with the negative sign is the dynamic recovery term. This growth law can also be written in terms :of the ¯ow function F. The last term multiplied by the kinematic stress deviator f is needed so that an appropriate long-term asymptotic solution can be developed, see below. The growth law for the kinematic stress is given in the Prager hardening form by : 2 : E^ t s g f E^ t ein 3 E k
4
E^ t is the tangent modulus at the maximum inelastic strain of interest that can be positive, zero or negative (Et is the same quantity referred to the total strain. The two quantities are related by E^ t Et = 1 Et =E and dier very little for most engineering materials.) The isotropic stress A is given the simple growth law : A Ac Af
: A
5
where Ac and Af are constants with no dimension and the dimension of stress, respectively. The ®rst controls the speed with which the ®nal value Af is reached. A more complicated growth law for the isotropic stress is needed when complex cyclic behavior is to be modeled, see Ruggles and Krempl (1990) and Choi and Krempl (1991). These equations constitute the VBO model for small strain and metals and alloys (the inelastic deformations are volume preserving). They are coupled, nonlinear sti dierential equations and for their solution initial conditions must be known. The growth laws for the state variables are homogeneous of degree one in the rates and represent rate-independent behavior. This property is not obvious for Eq. (3) when it is written in the stress form. Using the ¯ow law Eq. (3) can be rewritten as
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@ : g @T
: : eel 2: ein Ts E E 1 3
: g
f A
E 1
2 ^ : in Et e E 3
6
where the homogeneity of degree one in the rates is apparent. It will be demonstrated below that the dependence of the shape function on the overstress invariant introduces a rate dependence during the transition from the initial quasi elastic region to fully established plastic ¯ow. 4.1.3. Properties in very slow and very fast loading The responses to very slow and very fast loading at constant temperature are of interest. To obtain the slow response the uniaxial version of Eq. (1) is tranformed to an integral equation with 32 s11 and 32 g11 G. For loading with a constant strain rate it is advantageous to change the integration variable from time to strain to yield " 1 " 1 dG 1 " 1 ds ds d E G 0 G0 exp exp 0 k d" k 0 7 If the strain rate is reduced to smaller and smaller values the right hand side tends to zero and we have lim G 0 so that the stress is equal to the equilibrium stress ! 0
for very slow loading. Since the growth law for g is essentially rate-independent a nonzero equilibrium stress exists and the growth law represents a solid. This property and the fact that G at rest suggests the name equilibrium stress. To obtain the response for in®nitely fast loading the uniaxial version of the ¯ow law is used again after application of the chain rule we get 1
d=d" 1 G E Ek
8
Taking the limit for very fast strain rates we obtain lim
! 1
1
d=d" E
0. The
fast response is, therefore, elastic. The slow and the fast responses bracket the responses for all other loading rates. Fig. 4 gives a sketch of these results. 4.1.4. Fully established inelastic ¯ow and asymptotic solutions of VBO In demonstrating that VBO has a long-term solution that applies to the region of established ¯ow stress isothermal conditions are considered. At all times the identity s s
g g
f f
9
holds. It shows that the stress is composed of the overstress s g, the dierence between the equilibrium stress and the kinematic stress g f and the contribution of the kinematic stress. The kinematic stress can be used to set the slope at the maximum strain of interest. It can be positive (strain hardening), zero or negative (strain softening).
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Fig. 4. Schematic showing the fast and the slow monotonic response of VBO.
This writing also suggests investigating the stress dierences. It will be shown that the two stress dierences have a long-term solution whereas the stress need not. The ¯ow law can, after some straight forward calculations be written as the integral equation t t 3 1 E : : d e g s g s0 g0 exp t0 2 1 k t0 1 t 3 1 ds d 10 exp 2 1 k Similar manipulations of the growth law of the equilibrium stress, Eq. (6), for isothermal conditions, results in g
f g0
f0 exp
: ! t : s g s d k A t0
t t0
: f exp
t
: ! ds d A 11
The subscript zero denotes initial conditions. For the isotropic stress we have t : Ac d 12 A Af A0 Af exp t0
The arguments of the exponential functions are positive. If time grows without bounds the initial conditions of the overstress and of g f vanish since the exponential function goes to zero. The ®nal value Af of the isotropic stress is also independent of the initial condition. Only the kinematic stress can exhibit a non-vanishing in¯uence of the initial condition. It could be used to model tension/compression
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asymmetry. The asymptotic value of the isotropic stress is Af . The long-term solutions for the overstress and for g f in Eqs. (10) and (11), respectively, are found following Ho (1998). Both equations are dierentiated with respect to time. For large times these equations in indeterminate expressions for the rates which must be resolved. : result : Then s g 0 and : : n :o s g f
13
where {} denotes asymptotic value. This result allows the statement that f sets the slope of the stress-strain curve at the maximum strain of interest. This result shows that ultimately the stress, equilibrium stress and kinematic stress will have the same rate and the distance between any two of the three curves will be constant. The results shown in Eq. (13) are used in evaluating the asymptotic solution of Eq. (10)
s
g =kf
: g 2=3 Ee
: : g 1 2=3E ein
14
It is seen that the overstress is rate dependent and that this dependence is nonlinear and is determined by the viscosity function k[ ]. This equation can be expressed in terms of the ¯ow function ns
go
Ff
g
2 : in e 3
15
Again the nonlinear rate sensitivity of the overstress is evident. The arguments of the viscosity and the ¯ow functions are not explicitly written to allow for a pure overstress dependence, for a dependence on the drag stress or for a dependence on both. For the asymptotic solution to hold it is necessary for the drag stress to have reached the ®nal value. The constant overstress that pertains to the stationary solution is determined by the strain rate and can have any value. The long-term solution of Eq. (11) is s g g f Af 16 The rate-independent contribution to the stress is given by this expression since Af and the expression are rate independent. The directions of the rate-dependent and of the rate-independent contributions to the stress are the same. Both point in the direction of the overstress. Their magnitudes can be easily q n : odetermined to be Af 3 with 2 tr g f g f from Eq. (16) and Ff g Ekff g g from Eq. (15). A geometrical interpretation of the asymptotic solution is given by Krempl (1996).
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4.2. Application to the modeling of relaxation for pure overstress dependence We assume that the viscosity function or the ¯ow function depends only on the overstress. When the asymptotic region is reached the overstress will be constant. The relaxation tests are performed after the transient free region has been reached. The overstress does not change as the stress±strain curve develops to higher and higher strains. The ¯ow law for relaxation is 1: 3 s g : : s F 17 0 eel ein E 2 This equation can be rewritten as : s
3E F 2 1
s
g
3 s g 2 1 k
18
It is seen that the relaxation rate depends only on the overstress. So all relaxation tests with identical prior strain rate and performed in the transient free region have initially the same overstress and the stress rate is initially the same. The growth laws for the other state variables depend only on the overstress and on g f. These quantities do not depend on stress or strain directly. It follows that further values of the relaxation rate are also independent of the strain or stress at which the relaxation starts. The evolution of the relaxation curves is the same for any point in the transient free region as long as the prior strain rate is the same. Box 2. Modeling of relaxation properties 1. VBO can model the described behaviors. In real experiments the tangent modulus is not strictly constant as required but changes somewhat. It is still assumed that the asymptotic solution is applicable. 2. Since the overstress increases nonlinearly with the strain rate the relaxation rate will, therefore, increase with an increase in prior strain rate. These prediction are found in experiments, see property 2 given previously. 3. Property 3 is also predicted by VBO as demonstrated in Krempl and Nakamura (1998) using the data by Yaguchi and Takahashi (1999). 4. The modeling of the quick transition from the relaxation test to the ¯ow stress characteristic of the strain rate mentioned under property 4 requires that the in¯uence of initial conditions vanishes quickly. Once the ¯ow stress is reached in a simulation any in¯uence of the initial conditions has vanished. The above analyses show that the in¯uence of the initial conditions of the overstress and of g f will vanish. No permanent in¯uence of the relaxation holds is predicted by VBO. Figs. 6±8 of Majors and Krempl (1994) demonstrate that VBO is capable of modeling repeated relaxation tests with loading intervals between them.
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5. Discussion 5.1. Relaxation behavior The above discussion has demonstrated that in the ``transient free region'' there is a characteristic relaxation behavior that was initially explored by Hart and coworkers. However, the in¯uence of a change in the prior strain rate had not been explored and this topic has increased the challenges to modeling. New experiments with servo-controlled testing machines explored the in¯uence of prior strain rates on relaxation behavior. The ``transient free region'' corresponds to the concept of fully established ¯ow stress used elsewhere in the literature. VBO models this region through a long-time asymptotic solution that is very well applicable at small strains. It is shown that the existence of the asymptotic solution requires the overstress to be constant. This fact explains without any diculty that the relaxation behavior can be independent of the stress and strain at the beginning of the relaxation test. If either the drag stress or the isotropic stress is not constant with all the other conditions unchanged then the relaxation behavior may depend on the stress and strain at the beginning of the relaxation test. It is seen from Eqs. (10) and (11) that the functions which appear as arguments of the negative exponential functions are dierent for the overstress and for g f. In theory it is possible for the overstress to reach the asymptotic limit while the rate independent contribution to the stress is still evolving or vice versa. [The expressions, which appear in the exponentials, Eqs. (10) and (11) are dierent. The asymptotic limit can, therefore, be reached with dierent speeds.] This dierence in the approach to the long-term solution, and the existence of rate-dependent and rate-independent contributions to the stress is a basic characteristic of VBO, which brings ¯exibility to the modeling. 5.2. Creep experiments The question arises whether the creep behavior can have the same regularity as the relaxation behavior. If the `transient free region' is considered, the overstress is constant for a given prior strain rate. The ¯ow law for creep is obtained from Eq. (1) 3 s0 : : e ein F 2
g 0
3 s0 g 2 Ek
19
where the subscript 0 indicates that the stress is constant. It is seen that for all tests with the same initial strain rate the overstress and, therefore, the initial creep rate is the same. To obtain equal creep curves for dierent initial strains it is necessary for the equilibrium stress to evolve in the same way in the strain intervals that are covered by the creep tests. This is only true for a linear evolution of the equilibrium
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stress. It is known from the properties of the VBO model that for the asymptotic solution the stress±strain diagram and the equilibrium stress±strain diagram have equal slopes, see Eq. (13). If we assume that this model is a good qualitative picture of real behavior, then the equilibrium stress±strain diagram has the same tangent modulus as the real stress±strain diagram. The real stress±strain diagram is seldom linear and neither is the equilibrium stress±strain curve. It can therefore not be expected that the same creep curves be obtained for dierent initial strains. Given the nonlinear dependence of strain rate on overstress a small variation in the slope of the equilibrium stress can have a signi®cant in¯uence on the measured creep rate. So creep tests are very sensitive to variation in the tangent modulus. Generally creep curves that are independent of the initial strain should not be expected. Note that there is a fundamental dierence in the creep and the relaxation test. The evolution of the equilibrium curve with time is small and occurs at a constant total strain in relaxation. Considerable strain can develop for the creep test and the evolution of the equilibrium stress plays a signi®cant role. 5.3. High homologous temperature behavior For high homologous temperature behavior the hardening due to inelastic deformation and the recovery due to diusion are in competition. To account for this behavior, a static recovery term is introduced in the growth law for the equilibrium stress taking the Baily-Orowan format as a model, see Majors and Krempl (1994) and Tachibana and Krempl (1995, 1997, 1998). This new term is not of degree one in the rates so that time matters in the evolution of the equilibrium stress. In addition a softening of the isotropic stress is needed to account for cyclic softening and tertiary creep. Under these conditions the long-term asymptotic solution may not exist. The methodology may still be useful but the interpretation of the results must be done with care. 5.4. A physically based overstress model The concept of the eective stress and the equilibrium (back) stress has been used in many materials science-oriented, physically based papers. An example is the approach of Gupta and Li (1970) who have applied their model to relaxation. They : start with the Orowan equation for the inelastic strain rate " p bv ( is a geometric factor, is the density of mobile dislocations, b is their Burgers vector magnitude and is their average velocity) and use the Johnston±Gilman relation B i m (B is the average velocity at unit eective stress, m* is the dislocation : : velocity-stress exponent and i is the internal stress). During relaxation "p "e 1 d E dt (E* is the combined elastic modulus of the testing specimen, and d=dt is the relaxation rate). The three formulae can be combined to yield d dt
K 0
i m
20
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where K' is the combination of the other constants. Gupta and Li integrate this expression for constant K 0 , constant i and constant m* to yield
i K t a
n
21
with n 1= m 1, K=(K 0 (m* 1))-n, t is time, presumably zero at the start of relaxation and a is an integration constant. It is seen that Eq. (20) is akin to Eq. (18) if F[ ] is interpreted as a power function. The physical approach and the experiment based approach come to qualitatively the same expressions. There are, however, substantial dierences. VBO cannot yield Eq. (21) since in VBO the equilibrium stress (equivalent to the internal stress i ) is not constant in a relaxation test. Also the approach by Gupta and Li does not give the constitutive equation, equivalent to the set of coupled nonlinear equations, see Eqs. (1)±(5), that enable the simulation of any kind of test. For VBO relaxation is a special type of test that can be obtained using the special conditions of the test, i.e. strain rate is equal to zero. It leads to a set of dierential equation that can be solved. Other tests are the creep test and stress and displacement control in cyclic loading to name just a few. The Gupta and Li approach is totally dedicated to relaxation and the approach cannot be used to predict creep. Inserting the conditions for creep, i.e. stress rate is equal to zero in VBO results in Eq. (19) and the evolution equations for the state variables. For the Gupta and Li approach using the condition of zero stress rate equal to zero in Eq. (20) results in i which is not describing any creep curve. In the development of VBO monotonic loading at dierent rates (stress or strain control), creep and relaxation as well as cyclic loading were considered. These preconditions and the experimental base enable the application of VBO to any kind of test. The Gupta and Li theory, despite its physical base, applies only to relaxation, it fails to give sensible results in creep. It follows that a mechanistic base alone does not guarantee adequate results. Mechanistic approaches are usually developed to parameterize responses to a given input (example creep curves give the response for the input constant stress; relaxation yields the response for constant strain). But these responses do not constitute a material model that can be applied to any loading. These theories apply only to the test condition for which they had been developed. A constitutive equation or material model should be competent to model the responses for the tests discussed here. VBO has that capability. Using their method Gupta and Li (1970) were able to determine the internal stress i as a function of strain. Their Figs. 14±16 and 18 show that the eective stress equivalent to the overstress tends to vary little as the strain increases. The results obtained from various relaxation tests at dierent strain levels are plotted as stress vs. strain and resemble very much the behavior of VBO depicted schematically in Fig. 1 of Krempl (1996). Gupta and Li (1970) assume that the equilibrium (internal) stress is constant during relaxation but changes during tensile straining. This is not predicted by VBO where the equilibrium stress changes during relaxation and during tensile loading The changing equilibrium stress is a major factor in predicting the endpoints of the relaxation tests, see Krempl and Nakamura (1998).
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6. Conclusions The relaxation tests initially advocated by Hart have been extended to dierent test conditions and to using servo-controlled testing and strain measurement on the gage length. New test conditions include use of dierent strain rates preceding the relaxation. The relaxation rate depends nonlinearly on the strain rate preceding the relaxation test. These and other properties recently measured in relaxation tests are predicted qualitatively by the viscoplasticity theory based on overstress (VBO). The overstress (eective stress) dependence and the long-term asymptotic solution of VBO are the basis for these predictions. It is also shown that the hardness curves measured by Hart and collaborators can be re-interpreted using the overstress model. A discussion of the physically based approach proposed by Gupta and Li (1970) shows that it is not capable of predicting realistic creep behavior. It is argued that the response function, which generalizes the observations in a particular type of test, i. e. relaxation, is, despite its physical basis, not a suitable constitutive equation. A constitutive equation must be synthesized from the responses to monotonic loading using dierent rates, to cyclic loading and to creep and relaxation test conditions. Acknowledgements This research was supported by the Department of Energy Grant DE-FG0296ER14603. D. Alexander prepared Figs. 2 and 3.
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