Investigation of the Age-Dependent Constitutive Relations of Mortar
Huang Hsing Pan 1 and George J. Weng 2
Abstract: The stress-strain relation of cement binder is age-dependent and so is the behavior of mortar. In this study, we conducted a concurrent experimental and theoretical investigation into the effect of material age on the properties of both cement binder and mortar.
First the stress-strain curves of the binder were measured over a wide range of aging time from 7 days to 18 months. Two types of binder —one with the cement paste alone and the other with the fly ash/cement combination—were chosen in the tests. The test results were simulated with a modified, age-dependent Burgers rheological model. Then measured were the stress-strain curves of the mortar that contains the same types of cement at three levels of aggregate volume concentrations: c 1 ¼ 29, 38, and 49%. A micromechanics-based composite model making use of the secant-moduli was then introduced to predict the age-dependent behavior of mortar at various aging times and aggregate concentrations. Direct comparison between the measured data and the composite predictions is found to be in close agreement. It suggests that the proposed micromechanics model could be a viable approach to the estimate of age-dependent constitutive relations of mortar. DOI: 10.1061/(ASCE)EM.1943-7889.0000323. © 2012 American Society of Civil Engineers.
CE Database subject headings: Stress strain relations; Cement; Aggregates; Mortars; Micromechanics.
Author keywords: Age-dependent stress-strain relations; Cement paste; Aggregates; Mortar; Micromechanics.
Introduction
It is widely recognized that an accurate constitutive description for the behavior of cement paste and concrete (or mortar) is critical to their safe and long-life operation. For this reason many constitutive models have been proposed in the past (Carreira and Chu 1985;
Tsai 1988; Mander and Priestley 1998; Alexander and Milne 1995; Yip 1998; Du et al. 2010). Most of these models are phenom- enological. Although useful in their own right, the influence of aggregate concentration on the concrete (or mortar) behavior cannot be explicitly accounted for in these approaches. To address the influence of aggregate concentrations, some micromechanics approaches have been introduced recently by Kuo et al. (2008), Scheiner and Hellmich (2009), Grassl and Pearce (2010), and Pan and Weng (2010). These studies have focused on the modulus of elasticity and aspect-ratio dependence of the stress-strain curves of mortar, creep and relaxation of concrete, influence of tempera- ture on the thermal mismatch between aggregates and cement paste, and strain-rate sensitivity of mortar, respectively. The microme- chanics approach has proven to be particularly useful to the predic- tion of concrete or mortar properties as the aggregate concentration changes.
In addition to the dependence of aggregate concentration, the properties of concrete (or mortar) are also known to be sensitive to the environment, such as temperature, moisture, and aging time.
To address these issues, a significant effort has been directed. In particular, Di Luzio (2009) have adopted a Maxwell chain to model the behavior of uncracked concrete and used it to study the time- dependent cracking process; Zhang et al. (2009) have examined the moisture distribution in early age concrete; and Chen et al.
(2010) have measured the triaxial compressive strength in water environments.
The focus in this study is on the influence of material age on the stress-strain relations of both the cement paste and mortar. The novel feature of this work is that it covers a rather wide range of aging time, from 7 days to one and a half years (18 months).
It will also include a wide range of aggregate volume concentration, from 0 (cement paste itself) to 49%. The experimental tests will also include the peak stress and peak strain at each aging time and aggregate concentration. Most studies on the material age often focus on the early age properties, typically up to 28 days. For in- stance. in the creep test programs of Laplante (1993) and Atrushi (2003) [as cited by Scheiner and Hellmich (2009) in their model verifications], the material ages were 20 h, 27 h, 3 days, 7 days, and 28 days in the former and 1, 2, 3, 4, 6, and 8 days in the latter.
In the study of cement hydration and the hydration effect on creep, shrinkage, and cracking by Schutter (2004), Mabrouk et al. (2004), and Pane and Hansen (2008), the material ages were from 2 to 800 h (approximately 1 month). If parts of the cement paste are replaced by pozzolanic materials (silica fume, slag, or fly ash), a longer aging time is required (typically 56 or 90 days) because of the late pozzolanic reaction. It is quite possible that the aging time of 28 (or 56 or 90) days may not have fully reached the steady-state without (or with) the pozzolanic replacement. Seen some limited tests covering the early to long-term age of hardened concrete have been seen, but the focus has been either on the modu- lus of elasticity or on the failure strength (Katz 1996; Al-Khaiat and Fattuhi 2001; Lopez et al. 2007), none on the stress-strain relations.
Because the stress-strain relation is a vital component of the con- stitutive laws, we believe that there is an urgent need to provide such data. It is with this belief that this work has been carried out.
1 Professor, Dept. of Civil Engineering, Kaohsiung Univ. of Applied Sciences, Kaohsiung 807, Taiwan. E-mail: pam@cc.kuas.edu.tw
2 Professor, Dept. of Mechanical and Aerospace Engineering, Rutgers Univ., New Brunswick, NJ 08903 (corresponding author). E-mail:
weng@jove.rutgers.edu
Note. This manuscript was submitted on November 24, 2010; approved on August 4, 2011; published online on August 6, 2011. Discussion period open until August 1, 2012; separate discussions must be submitted for individual papers. This paper is part of the Journal of Engineering Me- chanics, Vol. 138, No. 3, March 1, 2012. ©ASCE, ISSN 0733-9399/2012/
3-297 –306/$25.00.
In the following, experimental data is presented on the stress- strain relations of cement pastes, and then the data of mortar using the same pastes. To translate the observed behaviors into useful constitutive models, a composite model will also be devised that treats the cement paste as the matrix and the sand aggregates as spheroidal inclusions. The measured age-dependent data of the cement paste will be taken as the input and used to predict the age-dependent stress-strain relations of the mortar. The predicted results will be compared with the test data.
Experimental Investigation
Two types of cement binder and six mortars at three aggregate con- centrations, each with nine material ages —7 days, 14 days, 28 days, 2 months, 3 months, 6 months, 9 months, 12 months and 18 months, have been tested. The cement binders used in mortar were the same as those tested individually, so that the properties of cement binder obtained in the first program could be used in the developed composite model for a direct comparison with the measured stress-strain curves of mortar obtained in the second program.
The experimental setups are similar to those used in recent studies (Kuo et al. 2008; Pan and Weng 2010).
Cement is Portland type I. The first type of binder is the 100%
cement paste with a 0.45 water-to-cement ratio (w/c), and the sec- ond one is a fly ash/cement binder with a 0.41 water-to-binder ratio (w/b), where fly ash/cement binder consists of 85% cement and 15% fly ash by weight, approximately. Fly ash was supplied by Hsinta thermal power plant (Taiwan) and is categorized as F-type.
These two types are to be referred to as the C-group and FC-group, standing for cement and fly ash/cement, respectively. Mortar was made from these two types of binder with three aggregate volume concentrations, c 1 ¼ 0:29, 0.38, and 0.49 (aggregates are referred to as phase 1 in the composite model, and the binder as phase 0, the matrix phase). The aggregate is the sand consisting of 99% quartz in accordance with C109 Ottawa standard sand. The particle size of the sand is approximately 0 :7 ∼ 1:0 mm with a specific gravity of 2.65, and the shape is similar to a prolate inclusion with an average aspect-ratio (length-to-diameter ratio), α ¼ 1:13. A typical optical image is shown in Fig. 1. The elastic bulk and shear moduli of the sand are κ 1 ¼ 19:46 GPa and μ 1 ¼ 18:44 GPa, respectively (Baalbaki et al. 1991). Table 1 is the mixture proportion of the C-group cementitious materials, and Table 2 is for the FC-group, in which c 1 is the volume fraction of aggregate. In both Tables, the materials at c 1 ¼ 0, denoted as C0 and FC0, represent the associ- ated binder itself without aggregates.
Table 1. Mixture Proportions (in kg) of Cement Paste Binder Composite (the C-Group)
Material c 1 Water Cement Sand
C0 0 586 1302 —
C29 29% 415 921 775
C38 38% 362 804 1014
C49 49% 298 623 1301
Note: c 1 is the volume fraction of aggregates.
Fig. 1. Optical image of the aggregates
Table 2. Mixture Proportions (in kg) of Fly Ash/Cement Binder Composite (the FC-Group)
Material c 1 Water Cement Fly ash Sand
FC0 0 546 1133 200 —
FC29 29% 384 796 140 878
FC38 38% 334 693 123 1029
FC49 49% 275 570 101 1316
Note: c 1 is the volume fraction of aggregates.
Fig. 2. (a) an installation of extensometers for longitudinal and lateral
strains; (b) a combination of shear and cleavage failure for C0 materials
Cylindrical specimens were prepared using steel molds with the size of ϕ100 × 200 mm. The mold of the specimen was removed at 24 h of age after casting. All specimens were subjected to moist curing until the moment of testing. Six specimens of each material were tested, and a total of 864 specimens in Tables 1 and 2 were measured for the stress-strain relations. Each specimen was loaded under uniaxial compression by an MTS machine (a close-loop servohydraulic testing machine) with a constant strain-rate ε ¼ 1 × 10 _ 5 =s, until the specimen failed. Used were two exten- someters located in the half height of a specimen with a 4 mm range to measure the longitudinal and lateral strains. Fig. 2(a) displays the installation of extensometers on the specimen parallel and perpendicular to the direction of applied load to record the longitudinal and lateral strains. Together with the applied stress, these data allowed plotting of the stress-strain curves and determi- nation of the initial modulus of elasticity and Poisson ’s ratio of the binder and mortar at each material age.
Under a constant strain-rate, the specimens of cement binder and mortar fail with rapid brittle failure. The failure pattern of the specimen belongs to a combination of shearing and cleavage shown in Fig. 2(b). The specimen passing through the peak stress is very venerable to fast fragmentation, and it is hard to record the postpeak stress-strain data. Although the applied strain could be strictly controlled by a very small strain-rate on the test specimen to obtain the few postpeak data if the applied stress is near to and beyond the peak stress (Wee et al. 1996), the recorded descending stress-strain curve is very short because of the brittle nature of cementitious materials. Moreover, the nature of stress-strain rela- tion of the descending portion is not the same as that of the ascend- ing one because of possible cracking and different strain rates that set in. The descending part of the stress-strain curve becomes significant only when the mortar materials contain some fibers
in practical applications (Katz 1996; Nataraja et al. 1999). For these reasons, we only focus on the ascending branch of the stress-strain curve. We have also measured the maximum stress of the stress- strain curve (peak stress) and the strain corresponding to the maxi- mum stress (peak strain); these data are given in Tables 3 and 4.
The measured initial Young ’s modulus and Poisson’s ratio are given in Tables 5 and 6. We shall use the modified Burgers model as de- picted in Fig. 3 to represent the nonlinear viscoelastic behavior of the age-dependent cement paste.
The measured stress-strain curves of the C- and FC-group cement pastes at various aging times are shown in Figs. 4 and 5, respectively. The variations of peak stress and peak strain for both cement paste and mortar are given in Figs. 6 – 9, and the variations of initial Young ’s modulus are plotted in Figs. 10 and 11. Finally, the stress-strain curves of the mortar as the aggregate concentra- tion increases from 0.29 to 0.38, and to 0.49, are depicted in Figs. 12 – 17. In Figs. 4, 5, and 12 – 17, it was observed that the stress-strain curves appear to reach a plateau before failure at the age of 7 days, and no plateau is found at 18 months. This is because cement hydration and pozzolanic reaction in cementitious binder have gradually developed with aging time according to the Table 3. Peak Stress and Corresponding Strain for Cement Paste Binder Composite (the C-Group)
c 1 0% 29% 38% 49%
Material age stress (MPa) strain (10 3 ) stress (MPa) strain (10 3 ) stress (MPa) strain (10 3 ) stress (MPa) strain (10 3 )
7 days 42.31 6.81 43.42 4.54 43.97 4.33 44.05 4.28
14 days 44.85 6.66 46.07 4.44 47.68 4.21 48.57 4.13
28 days 48.23 6.22 49.80 4.22 52.25 4.15 52.90 4.09
2 months 51.74 5.93 54.01 4.12 55.92 3.98 56.41 3.94
3 months 53.32 5.51 56.14 4.10 58.15 3.95 58.74 3.88
6 months 54.45 5.48 57.04 3.98 59.07 3.78 60.01 3.79
9 months 54.89 5.30 57.78 3.81 59.85 3.70 61.14 3.66
12 months 55.25 5.20 58.34 3.77 60.57 3.63 61.43 3.62
18 months 55.30 5.13 58.80 3.77 61.29 3.63 61.93 3.61
Table 4. Peak Stress and Corresponding Strain for Fly Ash/Cement Binder Composite (the Fc-Group)
c 1 0% 29% 38% 49%
Material age stress (MPa) strain (10 3 ) stress (MPa) strain (10 3 ) stress (MPa) strain (10 3 ) stress (MPa) strain (10 3 )
7 days 45.91 6.43 47.51 4.40 49.03 4.26 49.21 4.06
14 days 48.97 6.21 50.79 4.22 52.61 4.13 53.70 4.01
28 days 52.22 5.95 54.42 4.08 56.81 4.04 57.35 3.91
2 months 55.79 5.64 57.90 4.01 60.34 3.92 61.40 3.82
3 months 57.15 5.51 60.05 3.93 62.69 3.85 63.86 33.74
6 months 58.26 5.26 61.10 3.88 63.55 3.72 64.75 3.59
9 months 58.45 5.12 61.59 3.83 64.13 3.69 65.24 3.51
12 months 58.85 4.96 62.14 3.79 64.52 3.60 65.92 3.49
18 months 59.02 4.86 62.41 3.77 65.20 3.53 66.46 3.45
Fig. 3. A modified Burgers model for the age-dependent cement-based
binder
Table 5. Young ’s Modulus (in Gpa) and Poisson’s Ratio for Cement Paste Binder Composite (the C-Group)
c 1 0% 29% 38% 49%
Material age E ν E ν E ν E ν
7 days 10.70 0.174 16.02 0.165 17.77 0.163 20.27 0.161
14 days 11.28 0.174 16.80 0.165 18.63 0.163 21.09 0.161
28 days 12.47 0.173 17.80 0.165 19.48 0.163 22.03 0.161
2 months 13.22 0.173 18.63 0.165 20.34 0.163 23.27 0.160
3 months 14.17 0.173 19.04 0.164 21.20 0.163 23.77 0.160
6 months 15.00 0.173 19.85 0.164 21.89 0.163 24.61 0.160
9 months 15.58 0.172 20.63 0.164 22.58 0.162 24.98 0.160
12 months 16.16 0.172 21.12 0.164 22.89 0.162 25.42 0.159
18 months 16.61 0.172 21.36 0.164 23.19 0.162 25.72 0.159
Table 6. Young ’s Modulus (in Gpa) and Poisson’s Ratio for Fly Ash/Cement Binder Composite (the FC-Group)
c 1 0% 29% 38% 49%
Material age E ν E ν E ν E ν
7 days 12.05 0.174 17.35 0.164 19.47 0.162 22.06 0.159
14 days 12.48 0.174 18.20 0.164 20.22 0.162 22.57 0.159
28 days 13.59 0.174 19.24 0.164 21.12 0.162 23.57 0.158
2 months 15.09 0.174 20.64 0.164 22.34 0.162 24.71 0.158
3 months 16.12 0.173 21.09 0.163 23.01 0.161 25.35 0.158
6 months 16.66 0.173 21.82 0.163 23.70 0.161 26.08 0.158
9 months 17.18 0.173 22.51 0.163 24.04 0.160 26.47 0.157
12 months 17.52 0.173 22.71 0.162 24.16 0.160 26.76 0.157
18 months 17.93 0.172 22.83 0.162 24.57 0.160 26.99 0.157
strain (10
-3)
0 1 2 3 4 5 6 7
0 10 20 30 40 50 60 70 80
simulated curve experimental result
7 days 14 days 28 days 2 months 3 months 6 months 9 months 12 months 18 months W/C=0.45
c1=0
stress (MPa)
Fig. 4. Simulations and experiments for cement paste binders (c 1 ¼ 0)
strain (10
-3)
0 1 2 3 4 5 6 7
0 10 20 30 40 50 60 70 80
simulated curve experimental result
7 days 14 days 28 days 2 months 3 months 6 months 9 months 12 months
18 months W/B=0.41
c1=0