fahmi_dry_cyc_impr.eps Acta Polytechnica Vol. 52 No. 2/2012 Modeling of Concrete Creep Based on Microprestress-solidification Theory P. Havlásek, M. Jirásek Abstract Creep of concrete is strongly affected by the evolution of pore humidity and temperature, which in turn depend on the environmental conditions and on the size and shape of the concrete member. Current codes of practice take that into account only approximately, in a very simplified way. A more realistic description can be achieved by advanced models, such as model B3 and its improved version that uses the concept of microprestress. The value of microprestress is influenced by the evolution of pore humidity and temperature. In this paper, values of parameters used by the microprestress-solidification theory (MPS) are recommended and their influence on the creep compliance function is evaluated and checked against experimental data from the literature. Certain deficiencies of MPS are pointed out, and a modified version of MPS is proposed. Keywords: creep, concrete, compliance function, Kelvin chain, solidification, microprestress, finite elements. 1 Introduction In contrast to metals, concrete exhibits creep already at room temperature. This phenomenon results in a gradual but considerable increase in deformation at sustained loads, and needs to be taken into account in the design and analysis of concrete structures. The present paper examines an advanced concrete creep model, which extends the original B3 model [1] and uses the concepts of solidification [5, 6] and micro- prestress [3, 4, 2]. The main objective of the paper is to clarify the role of non-traditional model param- eters and provide hints on their identification. The creep tests performed by Kommendant, Polivka, and Pirtz [8], Nasser and Neville [9], and Fahmi, Polivka and Bresler [7] are used as a source of experimental data, which are compared with the results of numer- ical simulations. References [8] and [9] were focused mainly on creep of sealed concrete specimens sub- jected to elevated but constant temperatures. Refer- ence [7] studied creep under variable temperature for both sealed and drying specimens. The same refer- ences were used in [2] to demonstrate the functional- ity of the Microprestress-Solidification Theory, which is the constitutive model described in Section 2. All numerical computations have been performed using the finite element package OOFEM [10–12] developed mainly at the CTU in Prague by Bořek Patzák. 2 Description of the material model The complete constitutive model for creep and shrinkage of concrete can be represented by the rhe- ological scheme shown in Figure 1. It consists of (i) a non-aging elastic spring, representing instantaneous elastic deformation, (ii) a solidifying Kelvin chain, representing short-term creep, (iii) an aging dashpot with viscosity dependent on the microprestress, S, representing long-term creep, (iv) a shrinkage unit, representing volume changes due to drying, and (v) a unit representing thermal expansion. All these units are connected in series, and thus the total strain is the sum of the individual contributions, while the stress transmitted by all units is the same. Attention is fo- cused here on the mechanical strain, composed of the first three contributions to the total strain, which are stress-dependent. In the experiments, shrinkage and thermal strains were measured separately on load- σσ ε E0 E1 η1 E2 η2 EM ηM SS S S CS ksh αT εa εv εf εsh εT solidification Fig. 1: Rheological scheme of the complete hygro-thermo-mechanical model 34 Acta Polytechnica Vol. 52 No. 2/2012 free specimens and subtracted from the strain of the loaded specimen under the same environmental con- ditions. It should be noted that even after subtrac- tion of shrinkage and thermal strain, the evolution of mechanical strain is affected by humidity and tem- perature. The reference case is so-called basic creep, i.e. creep in sealed conditions and at room temper- ature. Dry concrete creeps less than wet concrete, but the process of drying accelerates creep. Elevated temperature leads to faster cement hydration and thus to faster reduction of compliance due to aging, but it also accelerates the viscous processes that are at the origin of creep and the process of micropre- stress relaxation. Solidification theory [5] reflects the process of con- crete aging due to cement hydration, which leads to the deposition of new layers of solidified hydration products (mainly calcium-silicate-hydrate gels, C-S- H). It is assumed that the creep of C-S-H is described by non-aging viscoelasticity, and aging is caused by the growth in volume of the solidified material, which leads to a special structure of the compliance func- tion, reflected by model B3. According to this model, basic creep is described by a compliance function of the form Jb(t, t′) = q1 + q2 ∫ t t′ ns−m s − t′ + (s − t′)1−n ds+ (1) q3 ln [ 1 + (t − t′)n ] + q4 ln t t′ , t ≥ t′, where t is the current time (measured as the age of the concrete, expressed in days), t′ is the age at load application, n = 0.1, m = 0.5, and q1, q2, q3 and q4 are parameters determined by fitting of experi- mental results or estimated from concrete compo- sition and strength using empirical formulae. The first (constant) term corresponds to the compliance q1 = 1/E0 of the elastic spring in Figure 1, the sec- ond and third terms to the solidifying viscoelastic material (in numerical simulations approximated by a solidifying Kelvin chain), and the fourth term to an aging viscous dashpot with viscosity ηf (t) = t/q4. Microprestress-solidification theory is an exten- sion of the above model to variable humidity and temperature. It replaces the explicit dependence of viscosity ηf on time by its dependence on the so- called microprestress, S, which is governed by a sep- arate evolution equation. The microprestress is un- derstood as the stress in the microstructure generated due to large localized volume changes during the hy- dration process. It builds up at very early stages of microstructure formation and then is gradually re- duced by relaxation processes. The microprestress is considered to be much bigger than any stress acting on the macroscopic level, and therefore it is not influ- enced by the macroscopic stress. Additional micro- prestress is generated by changes in internal relative humidity and temperature. This is described by the non-linear differential equation dS dt + ψS(T, h)c0S2 = k1 ∣∣∣∣d(T lnh) dt ∣∣∣∣ (2) in which T denotes the absolute temperature, h is the relative pore humidity (partial pressure of water vapor divided by the saturation pressure), c0 and k1 are constant parameters, and ψS is a variable fac- tor that reflects the acceleration of microprestress re- laxation at higher temperature and its deceleration at lower humidity (compared to the standard condi- tions). Owing to the presence of the absolute value operator on the right-hand side of (2), additional mi- croprestress is generated by both drying and wetting, and by both heating and cooling, as suggested in [2]. The dependence of factor ψS on temperature and humidity is assumed in the form ψS(T, h) = exp [ QS R ( 1 T0 − 1 T )] · (3)[ αS + (1 − αS)h2 ] where QS is the activation energy, R is the Boltz- mann constant, T0 is the reference temperature (room temperature) in absolute scale and αS is a pa- rameter. The default parameter values recommended in [2] are QS/R = 3000 K and αS ≈ 0.1. As discussed in [3], high microprestress facilitates sliding in the microstructure and thus accelerates creep. Therefore, the viscosity of the dashpot that represents long-term viscous flow is assumed to be inversely proportional to the microprestress. This viscosity acts as a proportionality factor between the flow rate and the stress. The model is thus described by the equations σ = ηf dεf dt (4) ηf = 1 cS (5) with a constant parameter c, which is not indepen- dent and can be linked to the already introduced pa- rameters. It suffices to impose the requirement that, under standard conditions (T = T0 and h = 1) and constant stress, the evolution of flow strain should be logarithmic and should exactly correspond to the last term of the compliance function (1) of model B3. A simple comparison reveals that c = c0q4. At the same time, we obtain the appropriate initial condi- tion for microprestress, which must supplement dif- ferential equation (2). The initial condition reads S0 = 1/(c0t0), where t0 is a suitably selected time that precedes the onset of drying and temperature variations. As already mentioned, parameters q1, q2, q3 and q4 are related to basic creep and can be predicted 35 Acta Polytechnica Vol. 52 No. 2/2012 from the composition of the concrete mixture and its average 28-day compressive strength using empirical formulae [1]. The part of the compliance function that contains q2 and q3 is related to viscoelastic ef- fects in the solidifying part of the model. In numer- ical simulations, this part of compliance is approx- imated by Dirichlet series corresponding to a solid- ifying Kelvin chain. The stiffnesses and viscosities of individual Kelvin units can be conveniently deter- mined from the continuous retardation spectrum of the non-aging compliance function that describes the solidifying constituent. The flow term representing long-term creep is handled separately. For general applications with variable environmental conditions, it is necessary to determine parameters c0 and k1 that appear in the microprestress evolution equation (2) and indirectly affect the flow viscosity. The model response is also influenced by parameters QS/R and αS , for which default values have been recommended. As will be shown later, a better agreement with ex- perimental results can be obtained if the default val- ues are adjusted. Also, the assumption that changes of T lnh contribute to the build-up of microprestress independently of their sign will be shown to be too simplistic. 3 Numerical simulations In this section, experimental data are compared to results obtained with MPS theory, which reduces to the standard B3 model in the special case of basic creep. All examples concerning drying and thermally induced creep have been run as a staggered problem, with the heat and moisture transport analyses pre- ceding the mechanical analysis. The available experi- mental data contained the mechanical strains (due to elasticity and creep), with the thermal and shrinkage strains subtracted. 3.1 Experiments of Kommendant, Polivka and Pirtz (1976) At the time of writing, the original report was not available to the authors of the present paper; there- fore the experimental data as well as the recom- mended basic creep parameters (q1 = 20.0, q2 = 70.0, q3 = 5.6 and q4 = 7.0, all in 10−6/MPa) were taken from [2]. Under constant uniaxial load and constant temperature, it is assumed that there are similar con- ditions in the whole specimen. This allowed all com- putations to be carried out on just one finite element. Figure 2 shows the experimental (points) and cal- culated (curves) compliance functions for two dif- ferent ages at loading and three different levels of temperature. For the younger age at loading, t′ = 28 days, the computed curves provide an excellent fit of the measured data for temperatures 23 ◦C and 43 ◦C. For the highest temperature, T = 71 ◦C, the compliance is somewhat overpredicted for load dura- tions from 1 week to 2 years. For the higher age at loading, t′ = 90 days, the measured data are under- predicted for all temperatures, but starting from load durations of 10 days all the creep rates are predicted very well. The reduced accuracy can be attributed to the general tendency of the B3 model to overem- phasize the effect of aging. 0 20 40 60 80 100 120 0.1 1 10 100 1000 co m pl ia nc e J [1 0-6 /M P a] duration of loading t-t’ [days] ex. data T=23°C T=23°C ex. data T=43°C T=43°C ex. data T=71°C T=71°C 0 20 40 60 80 100 120 0.1 1 10 100 1000 co m pl ia nc e J [1 0-6 /M P a] duration of loading t-t’ [days] ex. data T=23°C T=23°C ex. data T=43°C T=43°C ex. data T=71°C T=71°C Fig. 2: Experimental data (Kommendant, Polivka and Pirtz) and computed compliance functions for age at load- ing t′ = 28 days (top) and t′ = 90 days (bottom) In this example, the present results agree with those presented in the original work [2], which veri- fies the correct implementation. The calculated data are independent of parameters c0 and k1. One can obtain exactly the same curves as T = 23 ◦C in Fi- gure 2 just by substituting parameters q1–q4, age at loading t′ and the duration of loading t − t′ into the full version of the B3 model; see equation 1. The original B3 model contains a simple extension to basic creep at constant elevated temperatures; see section 1.7.2 in [1]. The actual age at loading and the load duration are replaced by the equivalent age 36 Acta Polytechnica Vol. 52 No. 2/2012 and the equivalent load duration, which evolve faster at elevated temperatures. The calculated compliance functions for default values of activation energies, as- sumed water content w = 200 kg/m3 and average 28-day compressive strength f̄c = 35 MPa are shown in Figure 3. For loading at age t′ = 28 days, the ini- tial compliance is overestimated and for the highest temperature 71 ◦C the rate of creep for longer load- ing durations is too low. The compliance functions for loading at age t′ = 90 days fit the experimental data nicely except for the highest temperature. In all cases the rate of creep is captured better by MPS theory. 0 20 40 60 80 100 120 0.1 1 10 100 1000 co m pl ia nc e J [1 0-6 /M P a] duration of loading t-t’ [days] ex. data T=23°C B3 T=23°C ex. data T=43°C B3 T=43°C ex. data T=71°C B3 T=71°C 0 20 40 60 80 100 120 0.1 1 10 100 1000 co m pl ia nc e J [1 0-6 /M P a] duration of loading t-t’ [days] ex. data T=23°C B3 T=23°C ex. data T=43°C B3 T=43°C ex. data T=71°C B3 T=71°C Fig. 3: Experimental data (Kommendant, Polivka and Pirtz) and computed compliance functions for age at load- ing t′ = 28 days (top) and t′ = 90 days (bottom) using the original model B3 3.2 Experiments of Nasser and Neville (1965) Nasser and Neville studied the creep of cylindrical concrete specimens subjected to three different lev- els of temperature. In their experiments, all speci- mens were sealed in water-tight jackets and placed in a water bath in order to guarantee constant tem- perature. At the age of 14 days the specimens were loaded to 35%, 60% or 69% of the average compres- sive strength at the time of loading; unfortunately, just the lowest load level is in the range in which concrete creep can be considered as linear. Paper [9] does not contain enough information to allow the pa- rameters of MPS theory to be predicted, but the values q1 = 15, q2 = 80, q3 = 24 and q4 = 5 (all in 10−6/MPa) published in [2] again provide good agreement at room temperature, see the first graph in Figure 4. 0 20 40 60 80 100 1 10 100 1000 co m pl ia nc e J [1 0-6 /M P a] duration of loading t-t’ [days] ex. data T=21°C standard 0 20 40 60 80 100 120 1 10 100 1000 co m pl ia nc e J [1 0-6 /M P a] duration of loading t-t’ [days] ex. data T=71°C standard modified 0 20 40 60 80 100 120 1 10 100 1000 co m pl ia nc e J [1 0-6 /M P a] duration of loading t-t’ [days] ex. data T=96°C standard modified Fig. 4: Experimental data (Nasser and Neville) and com- pliance functions for temperatures 21 ◦C, 71 ◦C and 96 ◦C 37 Acta Polytechnica Vol. 52 No. 2/2012 0 20 40 60 80 100 120 1 10 100 1000 co m pl ia nc e J [1 0-6 /M P a] duration of loading t-t’ [days] ex. data T=21°C B3 T=21°C ex. data T=71°C B3 T=71°C ex. data T=96°C B3 T=96°C Fig. 5: Experimental data (Nasser and Neville) and com- pliance functions obtained with the original model B3 for temperatures 21 ◦C, 71 ◦C and 96 ◦C For the higher temperature, T = 71 ◦C, the agree- ment is good up to 20 days at loading, but afterwards the computed rate of creep is too low. A remedy can be sought in modifying the activation energy. Re- duction of QS/R from the default value 3 000 K to the adjusted value of 2 200 K leads to an excellent fit; see the curve labeled in Figure 4 as modified. Unfor- tunately, the prediction for the highest temperature (T = 96 ◦C) is improved only partially. Changes in activation energy have no influence on the results when the temperature is close to the room temperature. Before loading, the specimens had been subjected to an environment at the given temperature, which accelerated the hydration pro- cesses in concrete, i.e. the maturity of concrete. In other words, the higher the temperature, the lower the initial compliance. On the other hand, for longer periods of loading the higher temperature accelerates the rate of bond breakages, which accelerates creep. This justifies the shape of the obtained curve for the medium temperature, which is different from the one published in [2], where the initial compliance for this temperature was higher than for the room tempera- ture. The compliance functions obtained with the B3 model are shown for all tested temperatures in Fi- gure 5. Again, default values were used for the activa- tion energies, assumed water content w = 200 kg/m3 and compressive strength f̄c = 35 MPa. Experimen- tal data for the room temperature and for the high- est temperature are captured nicely, but the compli- ance function for T = 71◦C is overestimated (until 100 days of load duration), and the final rate of creep seems to be too low. 3.3 Experiments of Fahmi, Polivka and Bresler (1972) In these experiments, all specimens had the shape of a hollow cylinder with inner diameter 12.7 cm, outer diameter 15.24 cm and height 101.6 cm. The weight ratio of the components of the concrete mix- ture was water : cement : aggregates= 0.58 : 1 : 2. From this we can estimate that the concrete mixture contained approximately 520 kg of cement per cu- bic meter. The average 21-day compressive strength was 40.3 MPa. Using CEB-FIP recommendations, the 28-day strength can be estimated as 42.2 MPa. The experiment was performed for four different his- tories of loading, temperature and relative humidity. The loading programs of the first two specimens are summarized in Tables 1 and 2, the other two loading programs with cyclic thermal loading are specified in Tables 3 and 4. Table 1: Testing program of the sealed specimen with one temperature cycle (Data set #1) time duration T RH σ [day] [◦C] [%] [MPa] 21 23 100 0 37 23 98 −6.27 26 47 98 −6.27 82 60 98 −6.27 10 23 98 −6.27 25 23 98 0 Table 2: Testing program of the drying specimen with one temperature cycle (Data set #2) time duration T RH σ [day] [◦C] [%] [MPa] 18 23 100 0 14 23 50 0 37 23 50 −6.27 108 60 50 −6.27 10 23 50 −6.27 25 23 50 0 Table 3: Testing program of the sealed specimen sub- jected to several temperature cycles (Data set #3). As- terisks denote a section which is repeated 4× time duration T RH σ [day] [◦C] [%] [MPa] 21 23 100 0 35 23 98 −6.27 9 40 98 −6.27 5 60 98 −6.27 14 23 98 −6.27 7∗ 60 98 −6.27 7∗ 23 98 −6.27 7 60 98 −6.27 12 23 98 −6.27 40 23 98 0 38 Acta Polytechnica Vol. 52 No. 2/2012 Table 4: Testing program of the drying specimen sub- jected to several temperature cycles (Data set #4). As- terisks denote a section which is repeated 4× time duration T RH σ [day] [◦C] [%] [MPa] 18 23 100 0 14 23 50 0 33 23 50 −6.27 15 60 50 −6.27 14 23 50 −6.27 7∗ 60 50 −6.27 7∗ 23 50 −6.27 7 60 50 −6.27 13 23 50 −6.27 14 23 50 0 The four parameters of the B3 model describing the basic creep, q1, q2, q3 and q4, were determined from the composition of the concrete mixture and from the compressive strength using empirical for- mulae according to [1]. The result of this prediction exceeded expectations; only minor adjustments were necessary to get the optimal fit (see the first part of the strain evolution in Figure 6). The following values were used: q1 = 19.5, q2 = 160, q3 = 5.25 and q4 = 12.5 (all in 10 −6/MPa). They differ signifi- cantly from the values recommended in [2], q1 = 25, q2 = 100, q3 = 1.5 and q4 = 6, which do not provide satisfactory agreement with experimental data. MPS theory uses three additional parameters, c0, k1 and c, but parameter c should be equal to c0q4. It has been found that the remaining parameters c0 and k1 are not independent. What matters for creep is only their product. For different combinations of c0 and k1 giving the same product, the evolution of microprestress is different but the evolution of creep strain is exactly the same. Since microprestress is not directly measurable, c0 and k1 cannot (and need not) be determined separately. In practical computations, k1 can be set to a fixed value (eg. 1 MPa/K), and c0 can be varied until the best fit with experimen- tal data is obtained; in all the following figures c0 is specified in MPa−1day−1. All other parameters were used according to standard recommendations. A really good fit of the first experimental data set (98% relative humidity, i.e., h = 0.98) was ob- tained for c0 = 0.235 MPa −1day−1; see Figure 6. The agreement is satisfactory except for the last interval, which corresponds to unloading. It is worth noting that the thermally induced part of creep accounts for more than a half of the total creep (compare the ex- perimental data with the solid curve labeled basic in Figure 6). Unfortunately, with default values of the other parameters, the same value of c0 could not be used to fit experimental data set number 2, because it would have led to overestimation of the creep (see the dashed curve in Figure 7). In the first loading in- terval of 37 days, creep takes place at room tempera- ture and the best agreement would be obtained with parameter c0 set to 0.940 MPa−1day−1; see the dash- dotted curve in Figure 7. However, at the later stage when the temperature rises to 60◦C, the creep would be grossly overestimated. A reasonable agreement during this stage of loading is obtained with c0 re- duced to 0.067 MPa−1day−1 (solid curve in Figure 7), but then the creep is underestimated in the first in- terval in Figure 6 left. Raising parameter αS from its recommended value 0.1 to 0.3 (short-dashed curve in Figure 7 right) has approximately the same effect as decreasing c0 from 0.235 to 0.067 MPa−1day−1. Parameter αS controls the effect of reduced humid- ity on the rate of microprestress relaxation, and its modification has no effect on the response of sealed specimens. 0 200 400 600 800 1000 1200 1400 1600 1800 0 50 100 150 200 m ec ha ni ca l s tr ai n [1 0-6 ] age of concrete [day] experimental data basic c0 = 0.067 c0 = 0.235 c0 = 0.671 Fig. 6: Mechanical strain evolution for sealed specimens, with relative pore humidity assumed to be 98%, loaded by compressive stress 6.27 MPa at time t′ = 21 days 0 200 400 600 800 1000 1200 1400 1600 1800 0 50 100 150 200 m ec ha ni ca l s tr ai n [1 0-6 ] age of concrete [day] experimental data c0 = 0.067 c0 = 0.235 c0 = 0.235 modified c0 = 0.940 Fig. 7: Mechanical strain evolution for drying specimens at 50% relative environmental humidity, loaded by com- pressive stress 6.27 MPa at time t′ = 32 days 39 Acta Polytechnica Vol. 52 No. 2/2012 0 500 1000 1500 2000 2500 3000 0 20 40 60 80 100 120 140 160 180 m ec ha ni ca l s tr ai n [1 0-6 ] age of concrete [day] experimental data c0 = 0.235 basic Fig. 8: Mechanical strain evolution for sealed speci- men, loaded by compressive stress 6.27 MPa at time t′ = 21 days and subjected to cyclic variations of tem- perature 0 1000 2000 3000 4000 5000 6000 0 20 40 60 80 100 120 140 160 180 m ec ha ni ca l s tr ai n [1 0-6 ] age of concrete [day] experimental data c0 = 0.235 Fig. 9: Mechanical strain evolution for drying speci- mens, loaded by compressive stress 6.27 MPa at time t′ = 32 days and subjected to cyclic variations of tem- perature For the last two testing programs described in Ta- bles 3 and 4, the agreement between the experimental and computed data is reasonable only until the end of the second heating cycle (solid curves in Figure 8 and Figure 9). For data set 3, the final predicted compliance exceeds the measured value almost twice (Figure 8), for data set 4 almost five times (Figure 9). In order to obtain a better agreement, parameter c0 would have to be reduced, but this would result in an underestimation of the creep in the first two testing programs. The experimental data show that the tem- perature cycles significantly increase the creep only in the first cycle; during subsequent thermal cycling their effect on creep diminishes. Therefore it could be beneficial to enhance the material model by adding internal memory, which would improve the behavior under cyclic thermal loading, while the response to sustained loading would remain unchanged. Another deficiency of the model is illustrated by the graphs in Figure 10. They refer to the first set of experiments. As documented by the solid curve in Figure 6, a good fit was obtained by setting pa- rameter c0 = 0.235 MPa−1day−1, assuming that the relative pore humidity is 98%. The pores are ini- tially completely filled with water; however, even if the specimen is perfectly sealed, the relative humidity decreases slightly due to the water deficiency caused by the hydration reaction. This phenomenon is re- ferred to as self-desiccation. 0 200 400 600 800 1000 1200 1400 1600 1800 0 50 100 150 200 m ec ha ni ca l s tr ai n [1 0-6 ] age of concrete [day] experimental data RH = 95% RH = 96% RH = 97% RH = 98% RH = 99% RH = 100% basic Fig. 10: Mechanical strain evolution for sealed speci- mens, loaded by compressive stress 6.27 MPa from age 21 days, with the assumed relative humidity of the pores varied from 95% to 100%. Parameters of MPS theory: k1 = 1 MPa/K, c0 = 0.235 MPa −1 day−1 The problem is that the exact value of pore hu- midity in a sealed specimen and its evolution in time are difficult to determine. In simple engineering cal- culations, a constant value of 98% is often used. Un- fortunately, the response of the model is quite sensi- tive to this choice, and the creep curves obtained with other assumed values of pore humidity in the range from 95% to 100% would be different; see Figure 10. The source this strong sensitivity is the assumption that the instantaneously generated microprestress is proportional to the absolute value of the change of T ln(h); see the right-hand side of (2). Rewriting (2) as dS dt + ψS(T, h)c0S 2 = k1 ∣∣∣∣lnh dT dt + T h dh dt ∣∣∣∣ (6) we can see that at (almost) constant humidity close to 100%, the right-hand side is proportional to the magnitude of the temperature rate, with proportion- ality factor k1| ln(h)| ≈ k1(1−h). For instance, if the assumed humidity is changed from 99% to 98%, this proportionality factor is doubled. 40 Acta Polytechnica Vol. 52 No. 2/2012 0 200 400 600 800 1000 1200 1400 0 50 100 150 200 m ec ha ni ca l s tr ai n [1 0-6 ] age of concrete [day] experimental data original MPS improved MPS Fig. 11: Mechanical strain evolution for sealed speci- mens loaded by compressive stress 6.27 MPa at time t′ = 21 days 0 500 1000 1500 2000 0 50 100 150 200 m ec ha ni ca l s tr ai n [1 0-6 ] age of concrete [day] experimental data original MPS κT = -ln(0.98) κT adjusted improved Fig. 12: Mechanical strain evolution for drying speci- mens loaded by compressive stress 6.27 MPa at time t′ = 32 days 4 Improved material model and its validation As a simple remedy to overcome these problems, the microprestress relaxation equation (2) is replaced by dS dt + ψS(T, h)c0S2 = k1 ∣∣∣∣Th dhdt − κT kT (T ) dT dt ∣∣∣∣ (7) with kT (T ) = e −cT (Tmax−T ) (8) in which κT and cT are new parameters and Tmax is the maximum reached temperature. With κT = 0.02, the creep curves in Figure 10 plotted for different assumed pore humidities would be almost identical with the solid curve, which nicely fits the experimen- tal results. The introduction of a new parameter pro- vides more flexibility, which is needed to improve the fit of the second testing program in Figure 7, with combined effects of drying and temperature varia- tion. For sealed specimens and monotonous thermal loading, only the product c0k1κT matters, and so the good fit in Figure 7 could be obtained with different combinations of κT and c0. The results are shown in Figures 11 and 12 for sus- tained thermal loading (data sets 1 and 2) and in Fig- ures 13 and 14 for cyclic thermal loading (data sets 3 and 4). Default values of parameters αS , αR, αE and activation energies are used. In these plots, data se- ries labeled original MPS show results obtained with standard MPS. The data series κT = − ln(0.98) were obtained with c0 = 0.235 MPa−1day−1, k1 = 1 MPa/K, κT = 0.020 203 and cT = 0. The data series κT adjusted correspond to parameters c0 = 0.235 MPa−1day−1, k1 = 4 MPa/K, κT = 0.005 051 and cT = 0. Note that in the case of constant relative humidity (Fig- ures 11 and 13) these series coincide with the data series original MPS. 0 200 400 600 800 1000 1200 1400 0 20 40 60 80 100 120 140 160 180 m ec ha ni ca l s tr ai n [1 0-6 ] age of concrete [day] experimental data original MPS improved Fig. 13: Mechanical strain evolution for sealed speci- men, loaded by compressive stress 6.27 MPa at time t′ = 21 days and subjected to cyclic variations of tem- perature 0 500 1000 1500 2000 2500 0 50 100 150 200 m ec ha ni ca l s tr ai n [1 0-6 ] age of concrete [day] experimental data original MPS κT = -ln(0.98) κT adjusted improved Fig. 14: Mechanical strain evolution for drying speci- mens, loaded by compressive stress 6.27 MPa at time t′ = 32 days and subjected to cyclic variations of tem- perature 41 Acta Polytechnica Vol. 52 No. 2/2012 The best agreement with experimental data is ob- tained with c0 = 0.235 MPa−1day−1, k1 = 4 MPa/K, κT = 0.005 051 and cT = 0.3 K−1; these series are la- beled improved. In Figure 11, only a small change can be observed compared to data series original MPS; these differences arise when the temperature ceases to be monotonous. For the sealed specimen (Figure 11), this change is detrimental, but looking at Figures 13 and 14, this deterioration is negligible compared to the substantial improvement in the case of cyclic thermal loading. 5 Conclusions The material model based on MPS theory has been successfully implemented into the OOFEM finite el- ement package, and has been used in simulations of concrete creep at variable temperature and humidity. MPS theory performs well for standard sustained levels of temperature and load levels within the linear range of creep, provided that the activation energy is properly adjusted. For higher sustained tempera- tures (above 70 ◦C) the experimental data are repro- duced with somewhat lower accuracy. For sealed specimens subjected to variable tem- perature, the results predicted by MPS theory are very sensitive to the assumed value of relative pore humidity (which is slightly below 100% due to self- desiccation). In order to overcome this deficiency, a modified version of the model has been proposed and successfully validated. Excessive sensitivity to the specific choice of relative humidity has been elimi- nated. Also, it has become easier to calibrate the model because thermal and moisture effects on creep are partially separated. The original MPS theory grossly overestimates creep when the specimen is subjected to cyclic tem- perature. A new variable kT has been introduced in order to reduce the influence of subsequent thermal cycles on creep. This modification does not affect creep tests in which the evolution of temperature is monotonous. Acknowledgement Financial support for this work was provided by projects 103/09/H078 and P105/10/2400 of the Czech Science Foundation. 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