double-power 指数算法徐变模型

CEMENT and CONCRETE RESEARCH. Vol. 14, pp. 793-806, 1984. Printed in the USA. 0008-8846/84 $3.00+00. Copyright (c) 1984 Pergamon Press, Ltd. DOUBLE-POWER LOGARITHMIC LAW FOR CONCRETE CREEP Zden~k P. Ba~ant I and Jenn-Chuan Chern 2 Center for Concrete and Geomaterials The Technological Institute, Northwestern University Evanston, Illinois 60201, U.S.A. (Communicated by F.H. Wittmann) (Received Feb. 17, 1984) ABSTRACT An improvement of the double-power law for creep at constant temperature and moisture content is proposed. Comparisons with available test data indicate that the final slopes of long-term creep curves, as indicated by the double-power law, are pre- dominantly on the high side. This is remedied by introducing a transition to a straight line in the logarithmic scale of load duration. The strain at the transition as well as the slope of the straight line are the same for all ages at loading. The strain and the slope at the transition point are continuous, while the curvature is discontinuous. ~e new law is also found to significantly limit the occurrence of divergence of the creep curves and of negative values at the ends of the relaxation curves calculated by the superposition principle. Extensive statistical comparisons with test data from the literature justify the proposed law. Introduction Although the double-power law [2,3] provides a relatively good descrip- tion of the existing test data and the creep of concrete at constant tempera- ture and water content, called the basic creep, one can detect some deviations which seem to be systematic rather than random. In particular, the final slope of the curves of strain versus the logarithm of load duration appears to be somewhat too steep when long-term tests are considered. This study exam- ines whether this can be remedied by introducing a transition to a logarithmic law for long times. Review of Double-Power Law The basic creep of concrete may be relatively well described by the double-power law [1,2,3,7]: 1 ¢i (t'-m + a)(t - t')n (i) J(t,t') = ~-0 + EO Iprofessor of Civil Engineering and Center Director 2Graduate Research Assistant 793 794 Vol. 14, No. 6 ZoP. Bazant and J.-C. Chern in which J(t,t') is the compliance function (also called the creep function), which represents the strain at age t caused by a unit uniaxial constant stress acting since age t'; E 0 is the asymptotic modulus, which may be visu- alized as the left-hand side asymptotic value of the curve of J(t,t') versus log(t -t'), and n, m, ~, and ~i are material parameters. Their typical values are n = 1/8, m = 1/3, and if t and t' are in days then ~ = 0.05 and ~i = 3 to 6. Also, E 0 = 1.5E28 where E28 is the conventional elastic modulus at age 28 days. Since (t -t') n = exp[n in(t- t')], the plots of J(t,t') versus log(t -t ') at constant t' have the shape of exponentials. Eq. 1 has a remarkably broad range of applicability. It yields accep- table values for ages at loading from about 1 day to many years, and for load durations from 1 second to several decades. It also yields acceptable compli- ance values for rapidly (dynamically) applied loads, and the dynamic modulus is approximately obtained from Eq. 1 as the value of i / J (t '+ A,t') for ~ =10 -7 day, whereas the conventional (static) elastic modulus is obtained as the value of i/J(t' +&,t') for A = 0.i day [24]. Since four parameters, namely E0, ~i, m and ~ are required to describe the age dependence of the elastic modulus we see that only one additional parameter, n, is needed to describe all creep. Extensive statistical regression analyses of practically all test data available in the literature revealed that the double-power law exhibits, on the whole, smaller errors than other formulas for concrete creep proposed before [4,5,6,1,2,3,7]. The power function of load duration t - t ' , involved in Eq. i, was first introduced by Straub [8] and Shank [25]. Wittmann et al. [12] gave supporting arguments for the power function based on the activation energy theory, and Cinlar, Ba~ant and Osman [26] gave other supporting argu- ments based on a certain reasonable hypothesis for the stochastic nature of the physical mechanism of creep. Others, e.g. Branson [9,10], introduced a power function of age t' It has often been commented that a power function of t- t' predicts too much creep for longer durations, exceeding 1 month. These critical comments were, however, incorrect since they resulted from using the power function to describe only that part of the creep strain that is in addition to the con- ventional short-time strain, approximately the strain for load duration 0.i day [2]. With this approach, the horizontal asymptotic value in Fig. 1 is obtained too high, and in order to fit the test data for medium load dura- tions (i day to 30 days) one needs to introduce a relatively high curvature by using a high value for exponent n, about n = 1/3. This then inevitably causes the power curve to shoot above the data points for longer load dura- tions. Recently it has been discovered [2], however, that the applicabil ity range becomes much broader if the power function is used to describe the entire creep strain including that which occurs for very short load durations (from 10 -6 second to 0.i day). This represents a fundamental difference from the earlier use of the power function, and invalidates the aforementioned critical comments. Since the left-hand asymptote (given by I/E 0) is much lower than considered in the classical studies (Fig. i), exponent n required to fit the creep data between 1 and 30 day durations is much smaller, roughly 1/8. This then causes that the power curve has a much smaller positive cur- vature in the log- time plot, and thus does not overshoot the test data for long creep durations. (There remains, however, an overshoot for very long creep durations, and that is what we try to improve here.) Note also that another advantage of including the entire creep strain in the power law is that I/E 0 can be considered age-independent, while the ear- lier approach in which only the creep strain after approximately 0.i day was Vol. 14, No. 6 795 CREEP, DOUBLE-POWER LAW, TRANSITION 1 1 ~ t,'~B • _ ~_" . . . . ,_-?,_ . . . . . 1-,. 11 I I J ( t , t ' ) , ,/~ / / ~ ,~__ ( t - t ) - - . .~ , ,~o~ ~.~ .,~,~'~ ,~. , - .~ / / / / ~~.t ,~ , ~. / ~ , / ~u~ /~/ /~ / • ,, . . . . . . . . . . . . I,o~,~.,~o:~I2 :~ . ~ ~,~ ~ _ ~o lday Io~ (t-t') ~ lo~(t-t') FIG. i. Creep Curves in Actual and Log FIG. 2. (a) Theoretical Recovery Time Scales (a = true elastic deforma- Curve Obtained by Principle of Super- tion, b =true creep, a '= conventional position, Showing No Divergence; elastic deformation, b' = conventional (b) Typical Curves of Doub±e-Power creep). Logarithmic Law. described by the power function, required considering an age-dependent elastic modulus E(t'). Although the double-power law is intended to describe only the basic creep it may be used as an approximation for creep in drying environment provided the cross section is relatively massive, with a thickness over about 30 cm. For such cross sections, the average creep deformation in the cross section is closer to that of a sealed cylinder than to that of a standard six-inch cylinder exposed to a drying environment. Proposed Formulation When the compliance values J(t,t') are plotted against log(t -t') for various constant values of h at constant t', it is found that for longer durations (over several years) the double-power law yields in most cases pre- dictions that are somewhat on the high side. Especially, the final slope ap- pears to be in most cases higher than the measured one (Fig. i). A remedy can be achieved by introducing a transition to straight lines in the log-time scale at a certain creep duration 0 L. This may be accomplished by the follow- ing formulas l ~ (t '-m + ~)(t- t') n for t - t ' ~ ~L (double-power law) J(t,t') = ~0 + E0 n~L in t - t ' 1 + @L = E 0 -~-~ + E----~--- for t - t' ~ 0 L (log-law) (2) 796 Vol. 14, No. 6 Z.P. Bazant and J.-C. Chern " ~L ~I/n in which 8 L = / ~ - • (3) ~(t '-m + ~)y According to these formulas, which may be called the Double-Power Loga- rithmic Law, the slope at the transition times 8 L (in log-time scale) is the same for all ages t' at loading, and the value of J(t,t') at which the transi- tion occurs is also the same. So, the ranges of validity of the double-power law and of the logarithmic law are separated by a horizontal line (see Fig. 2b); for longer t' the transition occurs at longer creep duration. For very high ages at loading, the double-power law is valid throughout the entire lifetime. Diversence of Creep Curves If the principle of superposition is used, as a crude approximation, to calculate the creep recovery curves from the creep curves for various ages at loading, the recovery curves can be either monotonic, with a monotonic decay to an asymptotic value, or nonmonotonic, with a minimum followed by a mono- tonic rise up to a certain asymptotic value. Nonmonotonic recovery curves are thermodynamically impossible for a non-aging material, however, they are thermodynamically admissible in case of aging [11,30]. With some exceptions, which might be due to statistical scatter or the effect of drying, most test data show a monotonic decay [11,12,31]. Thus, although recovery cannot in fact be accurately predicted using the linear principle of superposition [12], it seems preferable to use compliance functions that cannot yield nonmonotonic recovery curves upon superposition. This condition is verified when, for the same age t, the creep curve for a higher age at loading t' has a higher slope, i.e. ~[~J(t,t')/~t]/~t' ~ 0 or [ii] $2j(t~t') ~ 0. (4) 8t~t' It may be readily verified that the logarithmic law in Eq. 2 never vio- lates the inequality in Eq. 4 (Fig. 2a). On the other hand, the double-power law (Eq. 2) violates this inequality beginning with a certain creep duration. It is possible to always satisfy with Eq. 2 the inequality in Eq. 4 if the following condition is verified ~ [i - n]n ~L ~l~mf ~ (5) max in which [ m-n _n+l ] f = Max t' n (i + at 'm ) --fi-- . (6) max Comparison With Test Data Same as the double-power law, Eq. 2 should be applied only to creep at constant water content, called the basic creep. Only under such conditions the creep represents a constitutive property of the material. The creep observed on drying specimens is not a constitutive property but an average property of the specimen as a whole, since the drying causes in the specimen a highly nonuniform distribution of water content and of stress, produces microcracking, and leads to great differences in creep at various points. ~ empirical description of the mean creep of drying specimens requires, there- fore, much more complicated formulas. Eq. 2 has six material parameters, E0, n, m, a, 90, @L, which have to be Vol. 14, No. 6 797 CREEP, DOUBLE-POWER LAW, TRANSITION determined from test data. Similarly to previous works [2,3], this may be accomplished by minimizing the sum of squared deviations A of Eq. 2 from the given data. When all six material parameters are considered as unknown, the optimization problem is a nonlinear one. The optimum fits have been obtained by Marquardt-Levenberg algorithm, for which an efficient library subroutine exists. The test data used in optimization have been extracted from a com- puterized data bank set up at Northwestern University. Since most experimen- talists did not take their readings at times uniformly spaced in the loga- rithmic time scale, the raw data as reported are biased in that some readings are crowded at certain times and others are too sparse. For this reason, the test data from the literature have been smoothed visually by hand. At the same time, the hand smoothing approximately achieves elimination of the meas- urement error, which needs to be done since structures do not feel this error. The hand-smoothed curves were characterized by data points placed at regular intervals in the log(t- t') scale, using two points per decade. The deviations of Eq. 2 from test data have been characterized by the coefficient of variation defined [3,7, 31] as: N n n - 1 ±( i 1 I =- (7) ' - i j ~ ' j n lj i=l i 3 in which Jij (i= l,...,n) are the characteristic points of the data set number j ~laced at regular spacing in log-time scale) on the creep curves reported in the data set; n = number of all data points on all curves within the data set, A i j=vert ica l deviations of Eq. 2 from these data points, ] j= mean ordinate of all data points fro~ the data set number j, ~j = coefficien~ of variation for data set number j, ~ = the overall coefficient of variation for all data sets combined (j = i .... ,N). Fig. 3 shows the optimum fits of the data sets reported in Refs. 15, 16, 17, 18, 19, 20, 21 and 23. For comparison, Fig. 3 shows the optimal fits previously achieved with the double-power law [2,3]. The coefficients of variation are summarized in Table la. The overall coefficient of variation - - for the fits by the double-power law is m = 5.45% [2], and for the present - - Eq. 2 we achieve ~ = 3.9%. We see that some improvement is achieved by Eq. 2. More importantly, we may note that the final slopes achieved with Eq. 2 are better. This is important for extrapolation to longer times. Therefore, we determine for all creep curves the final slope, which we draw graphically as illustrated in Fig. 3 by the dashed lines, and we compare these experi- mental slopes with the value of ~J(t,t ')/Slog(t- t') according to Eq. 2 for t~e last sampling time of each curve. The combined coefficient of variation - - for the deviations from the measured final slopes for all data tests, ~f, is found to be 34% for the optimum fits by the double-power law, and 29% for Eq. 2; see Fig. 3 and Table lb. Thus, we see that the present formulation achieves an appreciable improvement in the representation of the final slopes of the creep curves, and therefore also in the extrapolation to longer times. The foregoing fits of test data (Fig. 3) have been obtained without the nondivergence restriction (Eqs. 4 - 6). If this restriction is imposed in data fitting, the optimum fits must obviously get worse. Such fits are shown in Fig. 4, and the corresponding coefficient of variation for all data sets - - - - is found to be ~ = 5.54%, while for the double-power law it is ~ = 5.45%. 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