Compressive strength of cement paste as a function of loading rate: Experiments and engineering mechanics analysis

Cement and Concrete Research 58 (2014) 186–200

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Compressive strength of cement paste as a function of loading rate: Experiments and engineering mechanics analysis Ilja Fischer a, Bernhard Pichler a,⁎, Erhardt Lach b, Christina Terner b, Elodie Barraud b, Fabienne Britz b a b

Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien), Karlsplatz 13/202, A-1040 Vienna, Austria French-German Research Institute of Saint-Louis (ISL), 5 rue du Général Cassagnou, F-68300 Saint-Louis, France

a r t i c l e

i n f o

Article history: Received 7 May 2013 Accepted 14 January 2014 Keywords: Compressive strength (C) Creep (C) Cement Paste (D) Mortar (E) Modeling (E)

a b s t r a c t Uniaxial compressive strength of cement paste increases with increasing loading rate. Two experimental campaigns are described: 2-day-old cement paste samples are tested by using quasi-static stress rates. 6-monthold and oven-dried cement paste samples are tested by using quasi-static and high-dynamic strain rates. As for the analysis of the former tests, we develop a nonlinear viscoelastic–brittle model. It explains our test data very well, and it suggests that strength decreases with decreasing loading rate, because the duration of the test increases and this provides the possibility for creep-related damage mechanisms to reduce the strength of the material. As for the analysis of the latter tests, we develop an elasto-brittle model considering crack propagation, at the Rayleigh wave speed, in loading direction. The model is free from fitting parameters and explains our measurements very well, indicating that high-dynamic strength increase is a structural effect. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction The uniaxial compressive strength of cementitious materials increases with increasing loading rate, see, e.g. [1–10]. This strengthening effect is (i) moderate in the regime of quasi-static loading rates which can be applied with classical laboratory equipment, and the effect is (ii) significant in the regime of high-dynamic loading rates which are typically applied with Split Hopkinson Pressure Bars. This complex behavior is typically explained by viscoelasticity of the bulk material, by loading-rate dependent aspects of crack propagation, and by inertia forces, see, e.g., [1,6,11–13]. However, detailed, direct, and accurate measurements of force fields, deformation fields, and acceleration fields in the entire volume of a tested material sample, with a high resolution both in space and time, is a very challenging and yet unsolved task, in particular during a high-dynamic compression test. Herein, we combine experimental investigations with theoretical modeling, with the aims (i) to further increase the available insight into the mechanisms governing the strength of cementitious materials and (ii) to provide new arguments to an ongoing scientific discussion of a still unsolved problem. Because cement paste is the binder of cementitious materials, insight into the strength dependency of cement paste on the loading rate will help to understand the corresponding

⁎ Corresponding author. E-mail address: [email protected] (B. Pichler). 0008-8846/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cemconres.2014.01.013

behavior of mortars and concretes. This is the motivation to devote our main focus to cement pastes. Our experiments comprise quasistatic tests as well as high-dynamic tests on Portland cement pastes with different composition in terms of initial water-to-cement mass ratios amounting to w/c = 0.43 and to w/c = 0.60, respectively. The samples with w/c = 0.43 are tested 2 days after their production. The ones with w/c = 0.60 are tested 6 months after their production, and they are oven-dried right before carrying out the compression experiments. As regards modeling, moderate strength increase in the quasi-static testing regime is explained by a visco-elastic model combined with an ultimate strain criterion. Thereby, we establish a conceptual link between (i) the affinity concept for nonlinear creep by Ruiz et al. [14] and (ii) microcracking observed with an acoustic emission technique during creep tests by Rossi et al. [15], indicating progressive damage evolution during the loading process. The significant strength increase in the high-dynamic testing regime, in turn, is modeled by envisioning that the duration of a compression test is composed of (i) the period of time required to reach a load level under which the first macrocrack nucleates, and (ii) the period of time required for the first crack to propagate through the entire specimen. Macrocrack nucleation is modeled by an elastic limit criterion. As for macrocrack propagation, we envision mode I crack extension in the direction of axial loading, at a speed being equal to the Rayleigh wave speed [16]. The Rayleigh wave speed is estimated by the speed of regular shear waves [17]. The ultimate load is calculated as the product of the loading rate and the test duration. The satisfactory model performance on the cement paste level, provides the motivation to apply the model also to mortar, based on test results from Grote et al. [2].

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List of symbols A d E fc fc,qs fc,ini h i j k ℓ0 m DIF F F˙ J J˙1d Nae sprop t t0 t0,i tcreep ti tref tnucl tprop tult vprop vT w/c γ ΔF Δℓ Δt Δσ ε ε˙ εnlc εnucl εult εult,qs η ηi κ μ ν ρ σ σ˙ σult σult,qs σi τ

area of cross section of specimen diameter of cylindrical specimen Young's modulus uniaxial compressive strength quasi-static value of fc initial value of fc height of cylindrical specimen increment number increment number acoustic emission intensity measurement length for quasi-static strain determination mass of specimen dynamic strength increase factor compressive force rate of F linear creep compliance rate of creep compliance, 1 day after the start of a creep test accumulated number of acoustic events length of crack propagation time variable time instant of sudden loading in a creep test t0 in creep test considered in the i-th increment characteristic time of creep time instant at which the i-th increment starts reference value of t test duration required to arrive at nucleation of the first macrocrack time required for a crack to propagate through the entire specimen duration of a compression test up to the peak load speed of crack propagation speed of shear wave initial water-to-cement mass ratio power law exponent increment of compressive force unloading displacement increment time increment stress increment axial compressive strain rate of ε (“strain rate”) accumulated nonlinear creep strain ε at which the first macrocrack nucleates value of ε at the peak load quasi-static value of εult affinity parameter for nonlinear creep value of η in the i-th increment power law parameter shear modulus Poisson's ratio mass density uniaxial compressive stress rate of σ (“stress rate”) ultimate value of σ quasi-static value of σult value of σ in the i-th increment power law exponent

The paper is structured as follows: Our two experimental campaigns are described in Section 2. In Section 3, we develop the related models, and we check the performance of the model for high-dynamic

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strengthening on the levels of cement paste and mortar. Our results are discussed in Section 4, before we conclude in Section 5. Throughout the entire manuscript, a positive sign in mathematical expressions relates to compression. 2. Experiments In the sequel, we describe two series of destructive uniaxial compression experiments on cement pastes. On the one hand, we investigate the behavior of highly creep-active cement pastes, by testing 2-day-old samples, see Subsection 2.1. On the other hand, we crush 6-month-old samples which (immediately before testing) are oven-dried, in order to obtain samples with a minimum water content, such that creep is reduced to the possible minimum, see Subsection 2.2. 2.1. Quasi-static experiments on 2-day-old cement pastes with w/c = 0.43 A commercial CEM I 42.5N cement and Viennese tap water are used to produce cylindrical cement paste specimens, with a diameter of 29.3 ± 0.4 mm, with a height of 60.3 ± 0.8 mm, and with an initial water-to-cement mass ratio w/c amounting to 0.43, see Table 1. The formworks containing the samples are stored, immediately after their production, in a climate chamber, at 95% relative humidity and 25 °C. 24 h after production, the specimens are taken out of the formworks and both circular end faces are cut away with a metal saw in order to minimize inhomogeneities which might result from production and/or storage. In order to obtain plan-parallel loading surfaces, the specimens are very carefully ground on a belt sander while being laterally supported by a several centimeter thick metal block, exhibiting a cylindrical drilling with a diameter which is less then 0.5 mm larger than one of the specimens. This way, the grinding plane is in very satisfactory approximation perpendicular to the samples' axis. During the second day after production, the specimens are kept in a lime-saturated solution, at a temperature of 25 °C. Storage in lime-saturated solution keeps possible hygral gradients within the samples at the possible minimum. Immediately before testing, each specimen is taken out of the lime-saturated solution, the surface is slightly dried with a towel in order to avoid that water drops would continue dripping off the specimen, then the diameter, height and mass are measured, followed by integrating the samples into the test setup and performing the test. These handling steps are done very quickly, such that the surface of the specimens are still wet during the tests. Surface drying, namely, would result in non-homogeneous drying shrinkage deformations, and the latter induce the risk of microcracking which would alter the material properties of the specimen, see, e.g. [18,19]. A uniaxial compression test requires loading by means of uniform normal tractions, raising the need for a central load application and for a counter measure against friction in the interfaces between specimen and load plates, see Chapter 5 of [20] for an interesting review of classical methods and an innovative flexible support device. While further details regarding the effect of friction on strength and failure mode of cementitious materials can be found in Appendix A, we here proceed with describing the setup chosen for our test campaigns. The experiments are carried out exactly 48 h after production with a uniaxial electro-mechanical universal testing machine of the type Walter & Bai LFM 150. During testing, the specimens are part of the following serial arrangement: load plate – washer – aluminum cylinder – rubber layer – two Teflon layers – specimen – two Teflon layers – rubber layer – aluminum cylinder – washer – load plate, see Fig. 1. The washers are used to ensure a central load application. The adjacent aluminum cylinders widen the stress trajectories from the diameter of the washer, on the one end, to the diameter of the sample, on the other end, such that the central elements of the arrangement are loaded by a homogeneous stress distribution. The rubber layers even out inevitable inaccuracies

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Table 1 Uniaxial compressive strength values of cylindrical cement paste specimens with w/c = 0.43, tested 2 days after their production. Specimen height h[mm]

Specimen diameter d[mm]

Specimen mass m[g]

Mass density ρ[kg/m3]

Force rate F˙ ½kN=s

Maximum force F[kN]

59.70 60.14 61.70 59.54 60.08 60.34 60.58 61.11 60.76 61.85 61.68 60.09 59.49 60.42 59.62 60.22 60.31 60.80 60.65 61.34 60.47 60.63 60.10 60.09 60.06 59.68 60.56 60.96 60.96 61.00 60.48 59.77 61.30 59.05 58.76 60.90 59.75 60.62 58.99 60.46 60.38 61.21 60.33 58.43 59.64 60.84 60.48 58.71 60.38 60.34

29.65 29.40 29.35 28.95 28.85 29.00 29.55 29.50 29.65 28.85 29.65 29.55 29.40 28.85 28.90 28.85 29.00 28.75 29.00 28.90 29.70 29.50 29.75 28.85 28.93 29.55 29.48 29.30 29.60 29.40 29.35 29.80 29.65 28.95 28.85 28.85 29.60 28.75 29.65 29.60 29.55 29.58 28.88 29.45 30.00 28.80 29.00 28.84 29.65 29.55

77.39 77.08 79.52 73.48 74.60 74.73 79.41 79.87 79.34 77.68 80.56 77.60 78.62 75.54 74.03 77.48 75.87 75.93 76.5 76.96 77.65 78.22 80.70 74.63 74.78 77.51 78.47 76.49 79.52 79.85 78.34 71.42 74.37 72.82 72.62 76.43 77.98 75.5 77.18 79.62 79.4 80.21 75.9 79.15 77.26 74.68 74.75 72.54 79.42 78.42

1877 1888 1905 1875 1900 1875 1911 1912 1892 1921 1892 1883 1948 1913 1893 1968 1905 1924 1910 1913 1854 1888 1932 1900 1895 1894 1899 1862 1896 1929 1915 1713 1758 1874 1891 1921 1897 1919 1896 1914 1918 1908 1921 1989 1833 1885 1872 1892 1905 1895

1.060·10−2 1.060·10−2 1.060·10−2 1.060·10−2 1.060·10−2 1.060·10−2 1.060·10−2 1.060·10−2 1.060·10−2 1.060·10−2 1.060·10−1 1.060·10−1 1.060·10−1 1.060·10−1 1.060·10−1 1.060·10−1 1.060·10−1 1.060·10−1 1.060·10−1 1.060·10−1 1.060·10+0 1.060·10+0 1.060·10+0 1.060·10+0 1.060·10+0 1.060·10+0 1.060·10+0 1.060·10+0 1.060·10+0 1.060·10+0 1.060·10+1 1.060·10+1 1.060·10+1 1.060·10+1 1.060·10+1 1.060·10+1 1.060·10+1 1.060·10+1 1.060·10+1 1.060·10+1 1.060·10+1 5.306·10+1 5.306·10+1 5.306·10+1 5.306·10+1 5.306·10+1 5.306·10+1 5.306·10+1 5.306·10+1 5.306·10+1

10.88 10.28 12.29 9.84 12.24 10.64 12.19 11.08 12.18 13.30 13.65 11.46 11.76 12.87 10.87 13.20 13.58 14.21 14.34 14.04 12.69 14.48 14.51 15.10 13.79 14.48 13.84 15.26 15.95 15.95 12.77 16.55 16.54 13.91 14.97 15.90 16.41 14.07 15.12 15.93 15.85 17.79 17.50 13.67 15.73 14.46 15.42 14.87 16.42 16.72

regarding the desired co-planarity of the two opposite specimen surfaces. The two pairs of Teflon layers effectively reduce the transfer of friction-induced shear stresses into the specimen. During the compression tests, specific force rates are prescribed, resulting in ˙ 1:54  10−2 MPa/s specimen-related stress rates ranging from σ¼ ˙ 8:15  101 MPa/s, see Table 1. The 50 measured strength to σ¼ values range from 14.95 MPa to 26.72 MPa and – on average – the measured strength values increase with increasing loading rate, see Fig. 2. Notably, the observed failure mode of the specimens is axial splitting, see Fig. 3. This failure model underlines that the teflon sheets very effectively reduce friction such that the remaining small shear stresses do not influence the failure mode anymore, for more details see Appendix A. Young's modulus of the tested material is determined by using cement paste cylinders with a height of 200.3 ± 1.5 mm and a diameter of 70.5 ± 0.6 mm, see Table 2. The samples are produced, processed, and

Stress rate σ˙ ½MPa=s 1.535 ⋅ 10−2 1.562 ⋅ 10−2 1.567 ⋅ 10−2 1.610 ⋅ 10−2 1.622 ⋅ 10−2 1.605 ⋅ 10−2 1.546 ⋅ 10−2 1.551 ⋅ 10−2 1.536 ⋅ 10−2 1.622 ⋅ 10−2 1.536 ⋅ 10−1 1.546 ⋅ 10−1 1.562 ⋅ 10−1 1.622 ⋅ 10−1 1.616 ⋅ 10−1 1.622 ⋅ 10−1 1.605 ⋅ 10−1 1.633 ⋅ 10−1 1.605 ⋅ 10−1 1.616 ⋅ 10−1 1.530 ⋅ 10+0 1.551 ⋅ 10+0 1.525 ⋅ 10+0 1.622 ⋅ 10+0 1.613 ⋅ 10+0 1.546 ⋅ 10+0 1.554 ⋅ 10+0 1.573 ⋅ 10+0 1.541 ⋅ 10+0 1.562 ⋅ 10+0 1.567 ⋅ 10+1 1.520 ⋅ 10+1 1.536 ⋅ 10+1 1.611 ⋅ 10+1 1.622 ⋅ 10+1 1.622 ⋅ 10+1 1.540 ⋅ 10+1 1.633 ⋅ 10+1 1.536 ⋅ 10+1 1.540 ⋅ 10+1 1.546 ⋅ 10+1 7.724 ⋅ 10+1 8.103 ⋅ 10+1 7.789·10+1 7.507·10+1 8.147·10+1 8.035∙10+1 8.124·10+1 7.686·10+1 7.737·10+1

Compressive strength fc[MPa] 15.76 15.14 18.17 14.95 18.73 16.11 17.78 16.21 17.64 20.35 19.77 16.71 17.33 19.69 16.57 20.19 20.56 21.89 21.72 21.41 18.32 21.19 20.88 23.10 20.99 21.11 20.28 22.65 23.18 23.50 18.88 23.73 23.96 21.13 22.90 24.33 23.85 21.68 21.91 23.15 23.12 25.89 26.72 20.07 22.25 22.20 23.35 22.77 23.79 24.38

stored as described above. Also the experimental setup contains the same elements as before. In addition, two metal rings are fixed to the specimen in a typical distance of ℓ0 = 160 mm, and the loadinginduced change of this distance is measured by five Linear Variable Differential Transformers (LVDTs) which are evenly distributed along the perimeter of the specimen, see Fig. 4. Four different specimens are subjected to cyclic, non-destructive, uniaxial compression tests, between 3 kN and 13.86 kN. The larger load level corresponds to specimenrelated compressive normal stresses amounting to 3.6 MPa, and this is smaller than 25% of the actual compressive strength of the specimens, see Table 1. The two described load levels are kept constant for 20 s. Loading/unloading is performed with a force rate of ± 8 kN/s, such that the characteristic time of unloading is small compared to the characteristic time of creep. Measurements from the third unloading cycle are used to determine Young's modulus, i.e. the individual readings of the five LVDTs are averaged, in order to

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Loading plate Aluminium cylinder Washer 3 cm 3 mm Rubber 2 layers of teflon 3 cm Cylindrical cement paste specimen

6 cm

2 layers of teflon Fig. 3. Typical axial splitting fracture pattern of 2-day-old cement paste specimen with w/c = 0.43, tested under quasi-static conditions.

3 mm Rubber

3 cm Aluminium cylinder

Washer

25

Cylindrical cement paste specimens are produced by using a commercial CEM I 52.5 R cement and Viennese tap water, with the initial water-to-cement mass ratio w/c amounting to 0.60. As for quasi-static testing, length and diameter of the samples amounted to 66.3 ± 7.3 mm and to 29.25 ± 0.04 mm, respectively; and as for the highdynamic testing to 6.63 ± 0.06 mm and 10.11 ± 0.04 mm, respectively; see also Tables 3 and 4. The smaller specimens are produced by using a mold in the form of a plastic board (i) which contains bore holes exhibiting the desired diameter and (ii) which is resting on a glass plate in order to ensure flat bottom surfaces of the specimens, perpendicular to their axis. Slightly more cement paste volume than necessary is poured into the holes. After that, another glass plate is used as a cover. It is removed a few days after production, such that the top surface of the samples could be ground with a fine emery paper, in order to obtain plan-parallel surfaces. The surfaces of the larger specimens are ground using the belt sander, as described before. All specimens are allowed to cure for 6 months under ambient conditions typical for laboratories for macroscopic material testing. Right before testing, they are stored in an oven for 20 h at 75 °C, in order to eliminate most of the free water in the pores. After this treatment, the specimens are allowed to cool down to room temperature. Quasi-static tests are performed again on the conventional uniaxial electro-mechanical universal testing machine of the type Walter & Bai LFM 150, by putting them directly between the two load plates, and by using a strain rate amounting to 2·10−5 s−1. The results from the two tests imply that the quasi-static uniaxial compressive strength of the tested material amounts to

20

f c;qs ¼ 48:15  0:93 MPa

15

see also Table 3 and Fig. 5. The high-dynamic tests are performed on a Split Hopkinson Pressure Bar [21], with strain rates ranging from 2·10+2 s−1 to 2.9·10+3 s−1. The results of the twelve tests exhibit more scatter than observed in quasistatic testing,1 see Table 4. Still, the tests indicate that the compressive strength increases – in the tested strain rate interval – from the level of the quasi-static strength, see Eq. (3), up to 160 MPa, see Fig. 5. The high-dynamic tests destroy the specimens into many small fragments. Inspection of their fracture planes shows that, also in the high-dynamic

Loading plate Fig. 1. Experimental setup for quasi-static testing of 2-day-old cement paste specimens with w/c = 0.43.

determine the unloading displacement increment Δℓ. Young's moduli are determined as

E¼

σ ΔF=A ¼ ε Δℓ=ℓ0

ð1Þ

where ΔF = 10.86 kN and A denote the unloading force increment and the nominal cross-sectional area, respectively. Test results imply that the Young's modulus of the tested material amounts to E ¼ 10:80  0:88 GPa

ð2Þ

stress rate: σ [MPa/s] 10−2

uniaxial compressive strength fc [MPa]

2.2. Quasi-static and high-dynamic experiments on 6-month-old, ovendried cement pastes with w/c = 0.60

10−1

100

101

102

10 5 0 −2

−1

0

1

. log. stress rate: log (σ × 1s / MPa) [–]

2

Fig. 2. Measured uniaxial compressive strength values of 2-day-old cement paste samples with w/c = 0.43, as function of the stress rate, for raw data see Table 1.

1

ð3Þ

This will be (at least partly) explained by the corresponding model, see Subsection 3.4.

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Table 2 Young's modulus values of cylindrical cement paste specimens with w/c = 0.43, tested 2 days after their production. Specimen height h[mm]

Specimen diameter d[mm]

Specimen mass m[g]

Mass density ρ[kg/m3]

Young's modulus E[GPa]

201 202 199 199

70.0 70.0 71.2 70.9

1485 1488 1489 1490

1920 1914 1879 1896

10.83 10.66 9.78 11.92

tests, cracks were propagating (in mode I) in the direction of uniaxial compressive loading (“axial splitting”), for an extended discussion on this property, see Appendix A. Finally, we note that the quasi-static 28-day compressive strength was, unfortunately, not measured, but a recently validated continuum micromechanics model provides a quite detailed insight into the compressive strength evolution of the tested material, as a function of hydration degree, see [22]. This model describes (i) that the final quasi-static strength of cement paste with an initial water-to-cement mass ratio w/c = 0.60 amounts to 55.4 MPa and (ii) that quasi-static strength values amounting to 47.50 MPa and to 48.81 MPa, measured 6 months after specimen production, refer to expected hydration degrees between 91 and 93%. 3. Modeling In the sequel, we describe engineering mechanics models serving as the basis for the analysis of our two experimental campaigns. 3.1. Creep of cementitious materials: fundamentals Test results obtained with 2-day-old cement paste will be analyzed based on a viscoelastic–brittle model (Subsections 3.2 and 3.3). We here establish a conceptual link between the affinity concept for

nonlinear creep, pushed forward by Ruiz et al. [14], and more recent microcracking observations by Rossi et al. [15]. This is followed by a brief description of a model for the linear creep compliance of young cement pastes, taken from Tamtsia et al. [23]. Before we present related details, we provide a brief introduction into the creep phenomenon. Creep is the tendency of materials to deform progressively under the influence of constant stress. The macroscopic phenomenon of creep is likely linked to water transport phenomena at very small scales of observation. The expression “basic creep” refers to transport of water exclusively taking place inside the material, i.e. no water enters or leaves the material sample. In order to observe basic creep, the surface of the material sample has to be sealed, or the test has to be finished in a time which is significantly shorter than the time required for significant amounts of water to enter or leave the sample. Creep of cementitious materials depends – in a non-trivial fashion – on the load level, i.e. on the ratio between the applied loading and the quasi-static strength fc,qs of the material, see, e.g. [24,14]. In more detail, considering creep tests carried out at different load levels smaller than 40%, the observed creep strains exhibit qualitatively very similar temporal evolutions, and creep strain magnitudes are – at any time instant – proportional to the applied loading. This was the motivation to introduce the expression “linear creep”. Considering tests carried out at different load levels larger than 40%, in turn, the observed creep strains exhibit still qualitatively similar temporal evolutions,2 but creep strains increase – at any time instant – overlinearly with increasing load level. This was the motivation to introduce the expression “nonlinear creep”. Linear and nonlinear creep is typically modeled by the phenomenological affinity concept by Ruiz et al. [14], i.e. the rate of creep strains follows as ∂εðt Þ dJ ðt−t 0 Þ ¼η σ dt ∂t

ð4Þ

In Eq. (4), ε is the axial strain, σ is the applied compressive stress, J is a linear creep compliance function, t is the time variable, and t0 is the time instant of sudden loading. η denotes a dimensionless affinity parameter, defined as [24,14]

η¼1þ2

σ f c;qs

!4 ð5Þ

see also Fig. 6. For load levels σ/fc,qs smaller than 40%, η is – in very good approximation – equal to 1. Specifying Eq. (4) for η = 1 results in a classical linear creep law. For load levels larger than 40%. Eq. (5) describes that η increases overlinearly with increasing load level. This way, η amplifies, in Eq. (4), the linear creep response, in order to account for nonlinear creep. The described affinity concept for nonlinear creep represents the current state-of-the-art. As such, it was recently implemented into a micromechanics-based model for creep of sprayed concrete, and this model was successfully applied to the load-level assessment of tunnels driven according to the New Austrian Tunneling Method [25–27].

Fig. 4. Experimental setup for determination of Young's modulus of cement paste samples with w/c = 0.43, tested 2 days after their production.

2 This holds in the so-called primary creep regime. A discussion of failure under sustained loading is beyond the scope of this paper.

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Table 3 Uniaxial compressive strength values of cylindrical cement paste specimens with w/c = 0.60, tested in oven-dried condition, 6 months after their production, with a quasi-static strain rate amounting to ε˙ ¼ 2:0  10−5 s−1 . Specimen height h[mm]

Specimen diameter d[mm]

Specimen mass m[g]

Mass density ρ[kg/m3]

Compressive strength fc[MPa]

71.50 61.11

29.28 29.22

76.94 65.09

1598 1588

47.50 48.81

Knowledge on the microstructural material processes responsible for nonlinear creep is even nowadays rather limited. Recently, however, Rossi et al. [15] carried out interesting acoustic emission measurements parallel to creep tests in the nonlinear creep regime. They found that the number of measurable acoustic events is – in very good approximation – proportional to the total creep strain: h σi Nae ðt Þ ¼ k εðt Þ− E

ð6Þ

In Eq. (6), Nae(t) is the accumulated number of acoustic events measured up to time t, ε(t) is the total strain answer (including an elastic contribution as well as creep), and σ/E the elastic strain contribution. The proportionality parameter k exhibits the physical dimension “acoustic events per unit creep strain”, and it increases with increasing level of loading, see Table 5 for data reported in [15]. Here, we envision that microcracking observed by Rossi et al. [15] is related to the nonlinear creep studied by Ruiz et al. [14]. In order to motivate this modeling idea, we approximate the dependence of k on the load level by the following power law (see also Fig. 7a) k¼κ

σ

!12:03

f c;qs

with κ ¼ 1:5519  10

8

acoustic events unit creep strain

ð7Þ

Using this reliable power law to predict the intensity of acoustic emissions down to load levels of 20%, it follows that significant acoustic emissions are expected for load levels larger than 40%, see Fig. 7b. In other words, the observations of Rossi et al. [15] together with the power law Eq. (7) suggest that acoustic emissions in the linear creep regime (σ/fc,qs b 40%) are negligible compared to the acoustic emissions in the nonlinear creep regime (σ/fc,qs ≥ 40%). This provides us with the sought conceptual link between k and η: both the acoustic emission intensity k and the nonlinear creep-related affinity parameter η begin to increase significantly at a load level of approximately 40%, compare also Figs. 7b and 6. In mathematical terms, the link between k according to Eq. (7) and η according to Eq. (5) reads as k ¼ κ

  η−1 3:01 2

ð8Þ

Eq. (8) establishes a conceptual link between acoustic events measured

by Rossi at al. [15] and the affinity parameter η by Ruiz et al. [14]. From this link, we conclude that the affinity concept accounts – in a phenomenological rather than in an etiologic fashion – for progressive material damage encountered in nonlinear creep tests. The phenomenological nature of the model manifests in the fact that consideration of damage is very much focused on the creep compliance, but leaves the elastic properties of the material and its strength unaffected. Micromechanicsbased models establishing quantitative links between creep-induced damage, on the one hand, and mechanical properties such as elastic stiffness, creep compliance, and strength, on the other hand, would be required to improve the situation. However, such more elaborate models are, unfortunately, not yet available, such that the phenomenological affinity concept for nonlinear creep remains the current state-of-the-art. As such, we check – in Subsections 3.2 and 3.3 – whether or not it is able to explain the results of our quasi-static testing series on 2-dayold cement paste samples with w/c = 0.43. Thereby, we will interpret the accumulated nonlinear creep strain as a damage variable. As for the reanalysis of our experiments on 2-day-old cement pastes, we are left with finding a suitable linear creep compliance function J, see Eq. (4). Notably, Tamtsia et al. [23] performed linear creep tests on cement paste samples (with initial water-to-cement mass ratio amounting to 0.35 to 0.5, respectively), by applying uniaxial compressive loading 18, 24, and 30 h after their production. They measured the short-term linear strain answers ε(t), and adjusted best-fit power laws to the creep strain rates, resulting in the following expression for the rate of the creep compliance dJðt−t 0 Þ t−t 0 ¼ J˙1d t ref dt

!γ with t ref ¼ 1 day

ð9Þ

where J˙1d and the power-law exponent γ denote model constants. J˙1d can be interpreted as the rate of creep compliance, 1 day after the start of a creep test. Tamtsia et al. [23] found that their measured creep behavior could be reliably described with values of J˙1d and γ taken from the following intervals −6 J˙1d ∈½11:2; 79:7  10 =ðMPa dayÞ

γ∈½−0:640; −0:858;

ð10Þ

see Figs. 7 and 8 of [23].

Table 4 Uniaxial compressive strength values of cylindrical cement paste specimens with w/c = 0.60, tested in oven-dried condition, 6 months after their production, under high-dynamic uniaxial compression.   Compressive strength fc[MPa] Strain rate ε˙ s−1 Specimen height h[mm] Specimen diameter d[mm] Specimen mass m[g] Mass density ρ[kg/m3] 6.57 6.66 6.61 6.57 6.57 6.60 6.64 6.62 6.64 6.63 6.80 6.59

10.09 10.09 10.03 10.12 10.12 10.12 10.08 10.10 10.10 10.19 10.14 10.12

0.85 0.96 0.81 0.85 0.85 0.85 0.86 0.85 0.86 0.85 0.87 0.84

1616 1610 1559 1602 1613 1598 1625 1598 1608 1567 1588 1576

7.0·102 2.0·102 5.0·102 5.0·102 5.0·102 5.0·103 1.9·103 2.1·103 2.1·103 2.1·103 2.1·103 2.9·103

74.15 42.01 74.41 48.60 65.40 114.26 132.73 164.39 156.36 133.73 152.64 143.78

192

I. Fischer et al. / Cement and Concrete Research 58 (2014) 186–200

strain rate: ε [1/s] dynamic strength increase factor (DIF): fc /fc,qs [–]

10−5

10−4

10−3

10−2

10−1

100

101

102

103

104

quasi-static tests high-dynamic tests

3.5 3 2.5 2 1.5

Table 5 Experimental results from uniaxial creep experiments on concrete, taken from Rossi et al. [15]. Load level σ/fc,qs [−]

Number of acoustic events per unit creep strain k [−]

0.54 0.59 0.73 0.75 0.80

0.97·105 2.67·105 2.66·106 6.24·106 1.07·107

1 0.5 0 −5

−4

−3

−2

−1

0

1

2

3

Given a specific stress rate σ˙ of interest, the time increment Δt follows from

4

. logarithmic strain rate: log ( ε × 1s) [–]

Δt ¼ Fig. 5. Measured dynamic strength increase factor (DIF) of 6-month-old, oven-dried cement paste samples with w/c = 0.6, as function of the strain rate; ordinate values are normalized with respect to mean quasi-static strength fc,qs = 48.15 MPa, for raw data see Tables 3 and 4.

3.2. Nonlinear creep model for monotonous load increase at constant stress rate Herein, we model the strain evolution during a compression test with a monotonous load increase, at a constant stress rate, right up to the strength of the material. We account for nonlinear creep based on the aforedescribed affinity concept. The increasing accumulated nonlinear creep strain is taken as a measure for progressive damage of the material. As for the related solution concept, we note that the well-known Boltzmann superposition principle is applicable to linear creep problems only, see, e.g. [28], i.e. related convolution integrals are not applicable when it comes to consideration of nonlinear creep. As a remedy, and motivated by the fact that the used creep compliance rate (9) was identified from experiments with constant load, we approximate the linear loading history in a stepwise fashion, see Fig. 8. Our modeling concept consists of solving a series of elementary constant-load creep problems, for which closed-form solutions are still in reach. Elementary solutions are put together by means of transition conditions ensuring that the damage-related nonlinear creep strain remains constant during instantaneous (and hence elastic) load increments. The continuous version of this solution concept is, unfortunately, out of reach, but it can be very reliably approximated by setting the stress increment Δσ equal to a value which is as small as one thousandth of the quasi-static strength f c;qs Δσ ¼ 1000

ð11Þ

3

amplification factor for linear creep η [–]

experiments 2.5

σ fc,qs

1+2

4

Δσ σ˙

The i-th increment starts at time ti = t1 + (i − 1) Δt and ends at time ti+1 = t1 + i Δt  t i btbt iþ1

1 0.4

0.6

t i ¼ t 1 þ ði−1Þ Δt ∀ i ¼ 1; 2; … t iþ1 ¼ t 1 þ i Δt

ð13Þ

The applied stress acting during the i-th increment amounts to i Δσ, i.e. σ ¼ σ i ¼ i Δσ

∀ i ¼ 1; 2; …

ð14Þ

Since the imposed stress is – in the described approximation – constant during every time increment, the creep strain rate given in Eq. (9) represents a reasonable choice for every increment. Accordingly, we specify Eq. (4) for the creep compliance (9) and for the stress level of the i-th increment: ∂εðt Þ ¼ ηi σ i J˙1d ∂t

t−t 0;i t ref

!γ ∀ i ¼ 1; 2; …

ð15Þ

with ηi as the affinity parameter of the i-th increment, following from specification of Eq. (5) for σ = σi ηi ¼ 1 þ 2

σi f c;qs

!4 i ¼ 1; 2; …

ð16Þ

When it comes to modeling the strain evolution ε(t) of a stepwisely approximated compression test, we note that the superposition principle does not hold because of the nonlinearities introduced by the affinity parameter for nonlinear creep, see Eq. (16). Therefore, we integrate the incrementally defined strain rate (15) progressively in time, and we consider the following two transition conditions for the transition from the end of one increment to the subsequent one:

Δε ¼

1.5

0.2

with

1. The strain history ε(t) is discontinuous at all increment boundaries, because the loading is increased in instantaneous stress increments, and these stress jumps result in elastic strain jumps. The latter amount to

2

0

ð12Þ

0.8

1

load level σ / fc,qs [–] Fig. 6. Nonlinear creep as a function of load level: amplification factor for linear creep η, after Ruiz et al. [14].

Δσ E

ð17Þ

2. Nonlinear creep induces damage at the microscale of the material. In accordance with Rossi et al.'s observation, the damage variable is set equal to the accumulated nonlinear creep strain, denoted as εnlc(t). Since instantaneous elastic loading does not result in nonlinear creep-induced microdamage, the damage variable has to be continuous at all increment boundaries.

I. Fischer et al. / Cement and Concrete Research 58 (2014) 186–200

(a) overview

(b) detail

x 106

3

σ fc,qs

κ

8

experiments

τ

6 4 2

x 105

experiments

10

acoustic events per unit creep strain [–]

acoustic events per unit creep strain [–]

12

193

r 2 =96.9%

0 0.4

0.5

0.6

0.7

κ

2

τ

1

0 0.2

0.8

σ fc,qs

0.3

load level σ / fc,qs [–]

0.4

0.5

0.6

load level σ / fc,qs [–]

Fig. 7. Acoustic events per unit creep strain after Rossi et al. [15], see circles, and best-fit power law with constants κ = 1.552 ⋅ 108 acoustic events per unit creep strain, and τ = 12.03.

In more detail, the strain evolution in the first increment is simply equal to the strain history of a fictitious creep test under constant loading σ1 = Δσ, starting at t 0;1 ¼ t 1

ð18Þ

see also Fig. 9(a), (b), (d), and (e). This strain history follows from timeintegration of Eq. (15) and from considering the instantaneous elastic strain increment (17) as t ref t−t 0;1 σ ε ðt Þ ¼ 1 þ η1 σ 1 J˙1d E γþ1 t ref

"

!γþ1 ð19Þ

t 1 btbt 2

Next, we quantify the value of the damage variable at the end of the first increment. To this end, the total creep strain – see the last term on the right-hand side of Eq. (19) – is considered to consist of a linear creep contribution and a nonlinear creep contribution

ε ðt Þ ¼

According to the line of argumentation leading to Eq. (8), this quantity is a measure for the number of acoustically measurable microcracks up to time instant t2. As for quantifying the evolution of the creep strains in the second increment, we envision a second fictitious creep test, with loading σ2, started at time instant t0,2, such that the accumulated nonlinear creep strains reach, at time t2, the value of εnlc(t2) given in Eq. (21), see Fig. 9(c) and (f). In more detail, by setting

!γþ1 t ref t−t 0;1 σ1 þ σ 1 J˙1d E γþ1 t ref |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð20Þ

t 0;2 ¼ t 2 −t ref

# ðγ þ 1Þ ε ðt Þ  nlc 2 η2 −1 σ 2 J˙1d t ref

1 γþ1

ð22Þ

we ensure that both considered fictitious creep tests are characterized, at time t2, by the same amount of damage. In the second fictitious creep test, namely, the accumulated nonlinear creep strain at t = t2 reads by analogy to Eq. (21) as 

t ref t 2 −t 0;2 εnlc ðt 2 Þ ¼ η2 −1 σ 2 J˙1d γþ1 t ref

!γþ1 ð23Þ

Linear creep contribution

!γþ1  t ref t−t 0;1 þ η1 −1 σ 1 J˙1d γþ1 t ref |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Nonlinear creep contribution; ε nlc ðt Þ

for an illustration of the nonlinear creep contribution see the shaded part of Fig. 9(e). The accumulated nonlinear creep strain at the end of the first increment (t = t2) follows from Eq. (20) as t ref t 2 −t 0;1 εnlc ðt 2 Þ ¼ η1 −1 σ 1 J˙1d γþ1 t ref

stress history σ (t)



!γþ1 ð21Þ

ε ðt Þ ¼

σ4 σ3 Δσ

σ2 σ1

t2

t3

!γþ1 t ref t 2 −t 0;1 σ2 þ η1 σ 1 J˙1d E γþ1 t ref " !γþ1 !γþ1 # t ref t−t 0;2 t 2 −t 0;2 ; t 2 btbt 3 þη2 σ 2 J˙1d − γþ1 t ref t ref

ð24Þ

As for the following load increments, the described modeling approach can be generalized in a straightforward fashion. At the end of each increment (i = 2, 3,…), the accumulated nonlinear creep strains are quantified as

Δt

t1

and specifying Eq. (23) for t0,2 from Eq. (21) is equal to εnlc(t2) given in Eq. (22), compare also Fig. 9(e) and (f). After having found the suitable value of t0,2, the second fictitious creep test delivers, in the time interval t2 ≤ t ≤ t3, a strain increase which is representative for the strain increase during the second increment of our stepwise loading history, compare Fig. 9(d) and (f). In addition, we consider that the elastic strain jump at the beginning of the second increment is equal to the one given in Eq. (17), such that the total elastic strain amounts to σ2/E, see Fig. 9(d). This way, the strain evolution reads, in the second increment, as

t4

t5

Fig. 8. Approximation of monotonously increasing load history in a stepwise fashion.

ε nlc ðt i Þ ¼

i−1

 X t ref η j −1 σ j J˙1d γ þ1 j¼1

"

t jþ1 −t 0; j t ref

!γþ1 −

t j −t 0; j t ref

!γþ1 # ð25Þ

I. Fischer et al. / Cement and Concrete Research 58 (2014) 186–200

t2

(a)

Δσ /E

Δσ /E

t1

t2

(d)

fictitious creep test 2

σ1

(b)

t 1 = t 0, 1

↓ ↑

ε nlc (t 2 )

σ 1 /E

t 1 = t 0, 1

(e)

t2

stress history σ (t)

σ 1 =Δσ

stress history σ (t)

σ 2 =2Δ σ

t1 strain history ε (t )

fictitious creep test 1

strain history ε (t )

stress history σ (t)

analyzed stress history

σ2

(c)

t 0, 2 strain history ε (t )

194

ε nlc (t 2 )

↓ ↑

σ2

E

t 0,2 t 2

(f)

Fig. 9. Schematic representation of nonlinear creep model for stepwise increase of compressive stresses: the accumulated nonlinear creep strain (representing the damage variable, see shaded domains) is continuously increasing: (a) analyzed stress history, (b) and (c) stress histories of fictitious creep tests 1 and 2, considered in the first and in the second increment, respectively; (d) strain history related to the analyzed stress history, (e) and (f) strain histories related to the fictitious creep tests 1 and 2.

1 γþ1

Once the starting time t0,i is quantified, the strain evolution in the i-th increment reads as " !γþ1 !γþ1 # i−1 t ref t jþ1 −t 0; j t j −t 0; j σi X ˙ εðt Þ ¼ η j σ j J1d − þ E γþ1 t ref t ref j¼1 " !γþ1 !γþ1 # t ref t−t 0;i t i −t 0;i − ; t i b t b t iþ1 þηi σ i J˙1d γþ1 t ref t ref

ð27Þ 3.3. Nonlinear creep model for quasi-static strength of cement pastes In order to obtain a strength model, we follow the CEB-FIB model code [29] and introduce an ultimate strain criterion, i.e. failure is envisioned if ε(t) according to Eq. (27) reaches the ultimate strain εult,qs, whereby the latter quantity represents a material constant, describing a deformation state at which macrocrack propagation associated with failure of the specimen is observed. The model-predicted compressive strength fc follows as the stress level of the very time increment, in which the strain evolution ε(t) according to Eq. (27) reaches the ultimate strain, i.e. f c ¼ σi

⇔ εðt ult Þ ¼ ε ult;qs

t i ≤t ult bt iþ1

ð28Þ

where tult denotes the time instant at which ε(t) reaches εult,qs. Next, we identify the model parameters involved in Eqs. (11)–(28), in order to fit model outputs to our measurement data, i.e. we identify optimal values for the yet unknown quantities J˙1d , γ, fc,qs, and εult,qs. In order to define search intervals and in order to come up with a highly constraint optimization problem, we introduce the following side conditions • Both J˙1d and γ have to be chosen from the experimentally observed intervals given in Eq. (10). • 2-day-old cement paste will exhibit a larger deformation capacity than mature concrete. Therefore, the ultimate strain has to be larger

than 2.2 ⋅ 10− 3 which is the value typically used for mature concrete [29]; −3

ε ult;qs N2:2  10

:

ð29Þ

• fc,qs in Eq. (16) denotes the “quasi-static” compressive strength, and the latter is commonly defined to be related to a strain rate amounting to 1 ⋅ 10− 5 s− 1. Therefore, the model-predicted compressive strength f c ðε˙ ¼ 1  10−5 s−1 Þ has to be equal to the underlying choice of the “quasi-static” strength fc,qs, i.e. ˙ E  1  10−5 s−1 ¼ 0:108 MPa=s ⇒ σ¼

!

f c ¼ f c;qs

ð30Þ

see the square symbol in Fig. 10. The optimization problem is solved iteratively, by covering the described parameter space with search intervals of iteratively decreasing size. This leads to the following optimal parameters −6 J˙1d ¼ 19:5  10 =ðMPa dayÞ γ ¼ −0:710 −3 f c;qs ¼ 19:2MPa εult;qs ¼ 2:33  10

ð31Þ

stress rate: σ [MPa /s] 10−2

uniaxial compressive strength: fc [MPa]

In order to ensure, that the value of this damage variable is continuous at time ti, the starting time of the fictitious creep test considered during the subsequent increment is set equal to: " # εnlc ðt i Þ ðγ þ 1Þ  ð26Þ t 0;i ¼ t i −t ref ηi −1 σ i J˙1d t ref

10−1

100

101

102

103

104

30

fc,ini 20 measurements fitted model asymptotic strength fc = fc,qs

10

0 −2

−1

0

1

2

3

4

. log. stress rate: log ( σ × 1s / MPa) [–] Fig. 10. Measurement data from Fig. 2 and best-fit model outputs, i.e. strength values obtained from model (11)–(28), evaluated for optimal parameters (31).

I. Fischer et al. / Cement and Concrete Research 58 (2014) 186–200

195

uniaxial compressive stress σ (t) [MPa]

25

σ˙ = 1 · 10+4 σ˙ = 1 · 10+3 σ˙ = 1 · 10+2 σ˙ = 1 · 10+1 σ˙ = 1 · 10±0 σ˙ = 1 · 10−1 σ˙ = 1 · 10−2

20 15 10 5 0

0

0.5

1

1.5

axial strain ε (t) [–]

2

MPa/s, MPa/s, MPa/s, MPa/s, MPa/s, MPa/s, MPa/s,

t ult t ult t ult t ult t ult t ult t ult

= 2.47 · 10−3 s = 2.43 · 10−2 s = 2.36 · 10−1 s = 2.26 · 10±0 s = 2.11 · 10+1 s = 1.91 · 10+2 s = 1.66 · 10+3 s

2.5

x10−3

Fig. 11. Stress–strain diagrams obtained with nonlinear creep model (11)–(28) and the input parameters (31).

The fitted model is able to explain the measured strength values reliably, see Fig. 10. Model-predicted stress–strain diagrams exhibit nonlinearities which increase with decreasing stress rate, see Fig. 11. Also, model-predicted test durations agree well with our experimental observations. Combining our experiments with the phenomenological affinity concept for nonlinear creep, and interpreting the accumulated nonlinear creep strain as a damage variable allows for the following reinterpretation of the loading-rate sensitivity of the compressive strength of creep-active cement paste, in the regime of quasi-static loading rates. Experimentally accessible strength values decrease with decreasing loading rate, because the test duration increases and this provides the possibility for nonlinear creep-related damage mechanisms to reduce the initial strength of the material. This implies that the “quasi static” compressive strength fc,qs is smaller than the initial strength fc,ini of the material, which is equal to the asymptotic strength level in Fig. 10. In mathematical terms, the initial strength is accessible under consideration of purely elastic material behavior (without viscous effects) f c;ini ¼ E εult;qs :

ð32Þ

Experimentally, the initial strength of a material is only accessible, if the loading is applied (i) fast enough as to prevent the evolution of significant creep-associated damage, and (ii) slow enough as to prevent high-dynamic strengthening effects, studied next. 3.4. Elasto-brittle model for high-dynamic strength increase of dry cementitious materials Next, we model the behavior of the 6-month-old and dry cement paste, produced with an initial water-to-cement mass ratio w/c = 0.6. We consider exclusively linear elastic behavior without viscous creep. This way, even “quasi-static” loading rates do not result in nonlinear creep strains, such that cement paste remains practically undamaged, and this preserves the initial strength of the material, i.e. herein we consider f c;qs ¼ f c;ini ¼ E εult;qs :

ð33Þ

Assumption (33) is justified, since the dry specimens did not exhibit any strength increase in the range from the quasi-static loading up to a loading rate of ε˙ ¼ 200 s−1 . Modeling the strength increase observed at even larger loading rates is the focus of the following developments. The essential modeling idea is to envision that the total test duration consists of (i) the period of time required to increase the loading up to a level at which the first macrocrack nucleates, as well as (ii) the period of time required for the first crack to propagate through the entire

specimen.3 The ultimate load will be calculated as the product of the loading rate and the test duration. We start with quantification of the two mentioned periods of time in which the total test duration may be subdivided. The period of time required to increase the load to a level at which the first macrocrack nucleates (denoted as tnucl) is quantified first. We consider that crack propagation starts, once a specific compressive strain is reached. This nucleation strain εnucl is considered to be a material constant which is, in particular, independent of the loading rate. Therefore, εnucl is equal to the strain εult,qs, reached at the peak load of a quasi-static test, and under consideration of Eq. (33) it follows that ε nucl ¼ εult;qs ¼

f c;ini E

ð34Þ

Considering a given specific strain rate ε˙ the loading time required to reach the crack nucleation strain Eq. (34) reads as t nucl ¼

εnucl ε˙

ð35Þ

When it comes to quantification of the time which is required for the first crack to split the material sample, the speed of crack propagation and the length of the specimen in the direction of crack propagation are of interest. In the latter context, we reiterate that inspection of specimen pieces after the high-dynamic testing indicates that the cracks were traveling in the direction of axial loading, similar to the situation observed under quasi-static testing, see Fig. 3. This opening mode of crack propagation is standardly referred to as mode I crack propagation. If the first macrocrack nucleates in the middle of the specimen, each crack tip has to propagate along one half of the cylinder height. However, if it nucleates at a specimen's surface at which the material sample is in contact with the adjacent load plate, the crack has to propagate along the entire cylinder height. Putting together these two bounding cases, the interval of possible macrocrack propagation lengths is given as sprop ¼ ½0:5; 1:0h

ð36Þ

As for the speed of crack propagation νprop, we note that cementitious materials are naturally micro-heterogeneous, i.e. they intrinsically contain a very large number of interfaces joining different phases constituting the materials' microstructure. Weak interfaces are prime candidates for crack propagation paths. Molecular dynamics simulations of mode I crack propagation along such weak interfaces show that the crack is quickly accelerated to the Rayleigh wave speed and limited by

3

In quasi-static testing, the latter period of time is negligible with respect to the former.

196

I. Fischer et al. / Cement and Concrete Research 58 (2014) 186–200

this speed, see Fig. 1 of [16]. Rayleigh waves, in turn, travel only slightly slower than regular shear waves [17,16], such that we introduce shear wave speed νT as a simple estimate of the speed of crack propagation. This estimate is taken as reliable enough for an engineering mechanics model which aims at being as simple as possible, but only as complex as necessary. In this line of reasoning, we estimate the speed of mode I crack propagation simply as vprop

rffiffiffi μ ¼ vT ¼ ρ

ð37Þ

where μ and ρ denote the shear modulus and the mass density of the material, respectively. Finally, the time required for the first macrocrack to split the material sample follows from combining Eqs. (36) and (37) as t prop ¼

sprop vprop

ð38Þ

ð39Þ

The ultimate strain reached during this total test duration follows as ˙ ult : ε ult ¼ εt

ð40Þ

Our elasto-brittle modeling approach suggests that the ultimate stress reached in a compression test is equal to the ultimate strain times the Young's modulus: f c ¼ σ ult ¼ E εult

Specimen

Mass

Quasi-static

Young's

Shear

height h[mm]

density ρ[kg/m3]

strength fc,qs [MPa]

modulus E[GPa]

modulus μ[GPa]

6.625

1597

48.15

14.24

5.533

determined elastic properties of 6-months-old moist-cured Portland cement pastes based on resonance frequency tests: w=c ¼ 0:623 : w=c ¼ 0:566 :

E ¼ 13:38 GPa E ¼ 15:51 GPa

μ ¼ 5:171 GPa μ ¼ 6:068 GPa

ð44Þ

Linear interpolation delivers expected values for the initial water-tocement mass ratio of the samples involved in our testing campaign w=c ¼ 0:600 : E ¼ 14:24 GPa μ ¼ 5:533 GPa

The total period of time which is required to break the specimen into pieces is the sum of the period of time required to reach a load level at which the first crack nucleates, tnucl, see Eq. (35), and the period of time required for the first macrocrack to propagate through the specimen, tprop, see Eq. (38), i.e. t ult ¼ t nucl þ t prop

Table 6 Properties of 6-month-old, oven-dried cement paste samples taken from Table 3 as well as from Eqs. (3) and (45).

ð41Þ

ð45Þ

see also Table 6. Model-predicted dynamic strength increase factors follow from specification of Eq. (43) for the input values listed in Table 6 and for strain rates ranging from 1 ⋅ 10−5 s− 1 to 1 ⋅ 10+4 s−1. Model predictions agree qualitatively very well with our independent DIF measurements, see Fig. 12. In more detail, model-predictions based on a crack propagation length sprop = 0.5 h underestimate the measurements by (on average) 24.84%, and based on a crack propagation length sprop = 1.0 h, they overestimate the measurements by (on average) 5.96%. The model yields a vanishing mean prediction error for sprop ¼ 0:90 h:

ð46Þ

Notably, this value lies between the two limiting cases 0.5 h b sprop b 1.0 h, see Eq. (36) and Fig. 12. The related quadratic correlation coefficient amounts to 2

r ¼ 78:8%; The dynamic strength increase factor is standardly equal to the com pressive strength reached under a specific loading rate, f c ε˙ , divided by the quasi-static strength fc,qs:  f c ε˙ f c;qs

ð42Þ

Combining Eqs. (34)–(42), the model-predicted dynamic strength increase factor is a function of the strain rate ε,˙ of the material properties Young's modulus E, quasi-static compressive strength fc,qs, mass density ρ, and shear modulus μ, as well as of the sample height h: rffiffiffi ˙ ρ εEh DIF ¼ 1 þ ½0:5; 1:0  f c;qs μ

and this is very satisfactory given the significant scatter of the experimental data which is typical for high-dynamic testing. The satisfactory performance of the proposed elasto-brittle model on the level of cement paste motivates us to strive for extended hypothesis testing, also on the level of mortar. In this context, we consider compression tests on mortars, carried out by Grote et al. [2], see Table 7 for the specimen height and material properties. Next, we reformulate

strain rate: ε [1/s] −5

10

ð43Þ

Notably, this expression for the dynamic strength increase factor does not involve any fitting parameters. 3.5. Hypothesis testing on the levels of cement paste and mortar In order to check the predictive capabilities of Eq. (43), we start on the cement paste level, considering results of our own testing campaign, see Subsection 2.2. As for the input values required for the evaluation of the right-hand side of Eq. (43), we note that mean specimen mass density ρ = 1597 ± 20 kg/m3 and mean cylinder height h = 6.625 ± 0.063 mm follow from the data given in Table 4. The mean quasistatic compressive strength is given in Eq. (3), see also Table 6. Unfortunately, the elastic properties of the samples were not determined. As a remedy, we estimate expected values for Young's modulus E and shear modulus μ from elasticity data by Helmuth and Turk [30], who

dynamic strength increase factor (DIF): fc /fc,qs [–]

DIF ¼

ð47Þ

10−4

10−3

10−2

10−1 100 101

102

103 104

test results sprop = h / 2 sprop = h

3

2

1

0 −5

−4

−3

−2

−1

0

1

2

3

4

. logarithmic strain rate: log ( ε × 1s) [–] Fig. 12. Measured dynamic strength increase factors of cement paste from Fig. 5 and corresponding model predictions obtained from evaluating Eq. (43) for specimen properties listed in Table 5 (sprop = crack propagation length; h = specimen height).

I. Fischer et al. / Cement and Concrete Research 58 (2014) 186–200 Table 7 Properties of mortar samples tested by Grote et al. [2]. Strain at onset of

Specimen

Mass

Young's

Poisson's

crack nucleation εult,qs [−]

height h[mm]

density ρ[kg/m3]

modulus E[GPa]

ratio ν[−]

4.10−3

15.24

2100

20

0.2

Eq. (43) based on (34) as well as on the standard elasticity relation between shear modulus μ, Young's modulus E, and Poisson's ratio ν, in form of µ = E/[2(1 + ν)]: DIF ¼ 1 þ ½0:5; 1:0 

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ νÞρ E ε ult;qs ˙ εh

ð48Þ

Model-predicted dynamic strength increase factors are obtained from evaluating Eq. (48) for the properties listed in Table 7 and for strain rates ranging from 1.10−5 s−1 to 1.10+4 s−1. Model predictions agree qualitatively very well with the DIF measurements reported in [2], see Fig. 13. In more detail, model-predictions based on a crack propagation length sprop = 0.5 h underestimate the measurements by (on average) 11.34%, based on a crack propagation length sprop = 1.0 h, they overestimate the measurements by (on average) 23.75%, and the model yields a vanishing mean prediction error for sprop ¼ 0:66h:

ð49Þ

Notably, this value lies between the two limiting cases 0.5 h b sprop b 1.0 h, see Eq. (36) and Fig. 13. The quadratic correlation coefficient amounts to 2

r ¼ 75:6%:

ð50Þ

Comparison of Eq. (50) with Eq. (47) might be an indicator that the scatter in our high-dynamic test series on cement paste was slightly smaller than the scatter observed in the mortar test campaign performed by Grote et al. [2]. 4. Discussion Our experiments and the related models suggest that the dependence of strength of cement paste is governed by the characteristic periods of time associated with three different processes: nonlinear creep,

strain rate: ε [1/s] dynamic strength increase factor (DIF): fc /fc,qs [–]

10−5

10−4

10−3

10−2 10−1 100

101

102

103

104

3

1

−4

−3

−2

−1

• We here define the characteristic time of nonlinear creep, tcreep, as the period of time in which creep-associated damage significantly reduces the initial strength of the material during a compression test right up to the failure of the material. In other words, tcreep quantifies the minimum duration of a compression test, which is required such that the finally measured strength is influenced by damage accumulated during the loading process. From our compression test, we conclude that for 2-day-old cement paste, tcreep is on the order of magnitude of 1 s, while it is on the order of magnitude of several minutes for ovendried and 6-month-old samples. • The characteristic time of loading is equal to the period of time which is required to reach a load level under which the first macrocrack nucleates. Considering elastic behavior and combining Eqs. (34), (35), ˙ E ε, ˙ we obtain and σ¼ t nucl ¼

f c;qs f c;qs ¼ E ε˙ σ˙

ð51Þ

Eq. (51) underlines that tnucl decreases with increasing loading rate and it decreases with decreasing strength of the material. • The characteristic time for crack propagation, tprop, is equal to the period of time which is required for a macrocrack to propagate through the entire sample. It is obtained from combining Eqs. (37) and (38) as rffiffiffi ρ ð52Þ t prop ¼ sprop μ Eq. (52) underlines that tprop decreases with increasing stiffness of the material, and it increases with increasing mass density as well as with increasing size of the specimen measured in the crack propagation direction. In quasi-static testing, the loading time up to macrocrack nucleation typically ranges from some seconds to one minute, and this is significantly larger than the characteristic time of crack propagation. In other words, crack propagation is a quasi-instantaneous process, such a quasi-static test is finished once the first macrocrack starts to propagate. Staying with quasi-static loading rates, we now compare the characteristic times of loading and nonlinear creep. In case of a mature and dry material, the quasi-static test duration is significantly smaller than the characteristic time of creep. At early ages and in the naturally wet state, however, the characteristic times of creep is on the same order of magnitude or even smaller than the characteristic time of loading: ð53Þ

Since loading in a strength test is increased monotonously until the material fails, more than half of the test duration refers to load levels which are larger than 40% of the quasi-static strength, i.e. to the nonlinear creep regime.

2

0 −5

loading up to nucleation of the first macrocrack, and macrocrack propagation through the sample. In order to discuss these periods of time in the framework of the separation of time-scales principle, their orders of magnitude are estimated next:

mature−age and dry ::::::::::t prop ≪t nucl ≪t creep early−age and naturally wet…t prop ≪t creep b t ≈ nucl

test results sprop = h / 2 sprop = h

197

0

1

2

3

4

. logarithmic strain rate: log ( ε × 1s) [–] Fig. 13. Measured dynamic strength increase factors of mortar from Grote et al. [2] and corresponding model predictions obtained from evaluating Eq. (48) for specimen properties listed in Table 6 (sprop = crack propagation length; h = specimen height).

• In case of early-age testing of naturally wet samples, i.e. if the characteristic time of creep is smaller than the characteristic time of loading – see second line in Eq. (53) – significant nonlinear creep strains develop and this is an indicator for significant damage accumulation, reducing the initially available strength of the material. The smaller the loading rate, the more time is available for damage to evolve, and the smaller is the finally reached strength. Vice versa, the larger the loading rate, the less time is available for damage to evolve, and the larger is the finally reached strength.

I. Fischer et al. / Cement and Concrete Research 58 (2014) 186–200

• A mature and dry cement paste, in turn, is rather creep-inactive. Therefore, even quasi-static test durations do not result in accumulation of significant damage accompanying nonlinear creep. Therefore, the measured strength will be – in good approximation – independent of the loading rate and equal to the initial strength of the material. In high-dynamic testing, the duration of the test up to nucleation of the first macrocrack is (on the same order of magnitude as or even) smaller than the time required for a crack to propagate through the tested specimen, and the characteristic time of creep is even larger than the latter: t ≪t creep t nucl b ≈ prop

ð54Þ

Considering that the speed of crack propagation is finite, crack nucleation and failure of the material are not simultaneous processes. Based on our experimental observations, we envision the following processes. Even when the first crack (i) nucleates at a position of a local microstructural strength minimum and (ii) starts to propagate in the loading direction, the material beside the crack remains intact. Therefore the loading of the specimen can be further increased. Under the progressively increasing load level, in turn, additional cracks nucleate at other positions within the specimen. Also these cracks start to propagate, predominantly in the loading direction, such that “columns of material” form gradually between the cracks. This line of argumentation implies that a high-dynamically loaded specimen is, finally, destroyed into many small pieces; and this is consistent with experimental observations. The model (43) envisions (i) that the material between the cracks behaves linear elastic, and (ii) the load carrying capacity of the specimen is reached once the first crack has propagated though the entire specimen. In this context, it is interesting to discuss the buckling risk of the material columns between the cracks. With progress of time, the length of the columns increases (due to crack propagation), the characteristic width of the columns decreases (due to nucleation of further cracks), and the loading of the columns increases. All three processes increase the buckling risk of the columns, but as long as the columns remain stable, the loading of the specimen can be further increased. Notably, the high-dynamic buckling resistance of brittle rod-like structures significantly decreases with decreasing loading rate [31]. Therefore, buckling of material columns at small loading rates might take place before the first crack has propagated through the entire specimen, and this would explain why the model overestimates some dynamic strength increase factors at the beginning of the high-dynamic loading regime, see Figs. 12 and 13. Finally, we note that pore water is well-known to influence the performance of cementitious materials in compression tests [1,7,8,32–34]. There are two counteracting effects, discussed next. • On the one hand, water has the potential to increase the apparent strength of the material. If water cannot elude from being loaded during a compression test, a pore pressure is activated, i.e. water takes over a part of the macroscopically imposed loads, such that the solid skeleton is less loaded compared to the situation without pore pressure. This raises the expectation that a wet cement paste exhibits a larger strength than a comparable dry cement paste, in particular in the high-dynamic regime. • On the other hand, water has the potential to decrease the apparent strength of the material. With increasing water content, namely, also the creep activity of the material increases. This way, the hereestablished conceptual link between creep and damage implies that increasing water content reduces the measured strength of cement paste, at least in a quasi-static test.

5. Conclusions We carried out experiments to study the increase of compressive strength of 2-day-old cement paste with increasing loading rate in the quasi-static regime of stress rates ranging from 10− 2 MPa/s to 10+2 MPa/s, corresponding to an elastic strain rate regime ranging from 10−6s −1 to 10−2 s−1. From our tests and the related engineering mechanics analysis, we conclude the following. • In the tests on 2-day-old cement pastes, we observed mode I splitting failure, typically on one single plane. • The measured strength values can be satisfactorily fitted by combining a nonlinear creep model for monotonous load increase at constant stress rate with an ultimate strain criterion. • This provides the motivation to re-interpret the loading-rate sensitivity of the compressive strength of creep-active cement paste, in the regime of quasi-static loading rates: experimentally accessible strength values decrease with decreasing loading rate, because the test duration increases and this provides the possibility for nonlinear creep-related damage mechanisms to reduce the initial strength of the material. • When using the initial strength of the material as the reference for normalizing strength values determined with different loading rates, we arrive at a “strength influence factor” which is smaller than 1 in the quasi-static regime, see Fig. 14 for a schematic representation. In a second testing campaign, we studied the increase of compressive strength of 6-month-old and oven-dried cement paste with increasing loading rate from the quasi-static value 2.10−5 s−1 up to the high-dynamic value 2.9·10+3 s−1. From our tests and the related engineering mechanics analysis, we conclude the following. • Test results indicate no significant strengthening effect up to strain rates amounting to 2.10+2 s−1. • The high-dynamic tests destroyed the specimens into many small fragments. Inspection of their fracture planes showed that, also in the high-dynamic tests, cracks were propagating in the direction of uniaxial compressive loading. • An elasto-brittle model predicts measured dynamic strength increase factors qualitatively and quantitatively very satisfactorily, not only for the here-tested cement paste, but also for experimental data on mortar taken from the literature [2]. • The model suggests that high-dynamic strength increase is a structural effect, influenced by the size of the specimen and by the region at which the first cracks nucleates. If the first crack nucleates close to the middle of the specimen, the sample will fall in pieces at a load level which is by a factor of up to 2 smaller compared to the ultimate

strain rate: ε [1/s] 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103 104

strength influence factor: fc /fc,ini [–]

198

3

2

1

0 −6

−5

−4

−3

−2

−1

0

1

2

3

. logarithmic strain rate: log ( ε × 1s) [–]

4

Fig. 14. Schematic illustration of “strength influence factor” as a function of loading rate: the initial material strength (which is likely to be observable in the approximate regime of loading rates between ε˙ ¼ 10−1 to ε˙ ¼ 10þ1 ) is used for normalizing strength values.

I. Fischer et al. / Cement and Concrete Research 58 (2014) 186–200

load associated with the situation in which the first crack nucleates close to one of the loaded end surfaces. This explains at least part of the scatter which is typical for high-dynamic uniaxial compression testing of cementitious materials, see Table 4 and [2]. Comparing wet/creep-active with dry/creep-inactive cementitious materials, we draw the following conclusions. • In the regime of quasi-static loading rates which can be applied with classical laboratory equipment, the strength of dry/creep-inactive cementitious materials is independent of the loading rate, while the strength of wet/creep-active cementitious materials decreases with decreasing loading rate. • In the regime of high-dynamic loading rates which are typically applied with Split Hopkinson Pressure Bars, creep is unlikely to influence the performance of a specimen, because the characteristic duration of a high-dynamic test is significantly smaller than the characteristic time of creep. However, high dynamic loading is expected to activate significant pore water pressures such that wet cementitious materials are likely to sustain larger ultimate loads than dry materials. Acknowledgments The second author very cordially thanks (i) Hon.-Prof. Dr. Alfred Vogel for establishing the contact to the third author, which was the basis for the subsequent collaboration, (ii) Dr. Gilles Chanvillard for the interesting discussions regarding the experimental activities of Rossi et al. [15], and (iii) Prof. Christian Hellmich for the interesting discussions regarding compression tests. In addition, valuable help of the laboratory staff both at Vienna University of Technology and at the Institute Saint Louis is gratefully acknowledged. Appendix A. The effect of friction on strength and failure mode of cement paste and concrete In the sequel, we discuss two failure modes observed in compressive strength testing of cementitious materials: (i) the here-observed “axial splitting” and (ii) crack propagation along multiple axisymmetric planes which are inclined with respect to the axis of loading. Our aim is to relate these failure modes both to boundary conditions and to geometric properties of the specimen, allowing us to associate different failure modes with different types of strength values. A uniaxial compression test requires loading by means of uniform normal tractions [20], but inevitable friction in the interfaces between the load-applying system and the specimen results in undesired shear tractions. According to the principle of St. Venant [35], shear stresses decrease inside the specimen with increasing distance from the load plate. They are significant in the vicinity of the load plate, up to a distance from the latter, which is typically envisioned to be equal to the characteristic dimension of the contact area between load plate and specimen. This implies for a standard compression test on a cubical sample that significant shear stresses prevail in the entire cube volume. Their effect is somewhat similar to lateral confining pressure, and this explains why a cube compression test delivers a “shear-enhanced strength” which is by typically 20% larger than the genuine “uniaxial material strength”. In addition, the shear stresses induce cracking on axisymmetric planes which are inclined with respect to the axis of loading, for a more detailed discussion see, e.g., [20]. The genuine “uniaxial material strength” is accessible, e.g., with cylindrical specimens exhibiting a the length-to-diameter ratio amounting to 2. The central part of such a cylinder, namely, is practically free of undesired shear stresses such that the sought uniaxial stress state is achieved. First cracks typically appear close to the center of the specimen, and axial splitting is the related failure mode. As for cement paste, axial splitting involves a few or even just one splitting plane, see, e.g., the here-reported experiments, or the test campaign

199

performed to study the early-age strength evolution of young cement paste samples illustrated in Fig. 10 of [22]. As for concrete, axial splitting involves multiple splitting planes, see, e.g., Fig. 5.1 of [20], or the computer tomography images of [36]. The latter images provide evidence that, at the peak load, the first cracks propagate also in the loading direction. Later in the post-peak softening regime, cracks localize in a bandlike damage accumulation zone which transforms finally into a shear band [36]. Localization is very likely to be influenced by the concrete aggregates, and this explains why shear band formation is not observed when testing cement pastes. The different number of splitting planes observed in cement paste and concrete results from specific microstructural properties, described next. Concrete aggregates are well-known to be covered by interfacial transition zones (ITZs) exhibiting a thickness of typically 15 to 30 μm [37]. ITZs consist of cement paste, but they exhibit a larger porosity as well as smaller strength and stiffness than the bulk cement paste matrix in a greater distance from the aggregate surface. Crack initiation in concrete takes place in the regions of the ITZs, with crack planes aligned predominantly with the direction of axial loading. This is observed well before the ultimate load carrying capacity of concrete is reached, see, e.g., [38] and references therein. Due to the non-catastrophic nature of this process, ITZ-induced cracking is a distributed phenomenon. In other words, a sample of concrete exhibits many defects when approaching the peak load, and this explains axial splitting along multiple planes. For cement paste, in turn, interfacial transition zones are not reported, and lack of distributed ITZ cracks results in axial splitting involving a few or just one single crack plane. Summarizing, we note that axial splitting is the typical failure mode associated with the genuine uniaxial strength of cementitious materials. Failure modes comprising inclined cracks either stem from frictioninduced shear stresses (resulting in a “shear-enhanced strength” larger than the “uniaxial material strength”), or they refer to the post-peak failure mechanism [36]. In a high-dynamic uniaxial compression test, performed with a Split Hopkinson Pressure Bar, the specimen must be (i) short enough as to achieve a uniform stress field in the axial direction [39], and (ii) long enough as to avoid significant friction effects which would result – similar to the quasi-static situation described above – in an apparent increase of strength [40]. Both requirements were met in the here-reported tests. Since specimens are frequently “pulverized” in high-dynamic compression tests, a-posteriori determination of crack plane orientations is a challenging task. From the above explanations, one might expect that cement pastes will exhibit a smaller number of crack planes than concretes, and this might be the reason why inspection of material fragments allowed us to identify that cracks were predominantly propagating in load direction. Details of failure modes in other studies remain, unfortunately, rather unclear. Still, we note that the successful application of our high-dynamic strength model to mortar tests by Grote et al. [2] suggest, that the physical processes in cement paste and mortar appear to be similar. As for concrete, however, one might speculate that the ITZ-covered aggregates will influence not only the quasi-static behavior (see above), but also the high-dynamic behavior. In this context, it is noteworthy that cracking along an ITZ represents cracking within cement paste, such that the modeling ideas presented herein should be applicable. A more detailed study, however, is not the focus of the present paper, but provides motivation for future work. References [1] P.H. Bischoff, S.H. Perry, Compressive behaviour of concrete at high strain rates, Mater. Struct. 24 (6) (1991) 425–450. [2] D.L. Grote, S.W. Park, M. Zhou, Dynamic behavior of concrete at high strain rates and pressures: I. Experimental characterization, Int. J. Impact Eng. 25 (9) (2001) 869–886. [3] Z. Xu, H. Hao, H.N. Li, Experimental study of dynamic compressive properties of fibre reinforced concrete material with different fibres, Mater. Des. 33 (1) (2012) 42–55. [4] J.W. Tedesco, C.A. Ross, Strain-rate-dependent constitutive equations for concrete, J. Pressure Vessel Technol. Trans. ASME 120 (4) (1998) 398–405. [5] G. Gary, P. Bailly, Behaviour of quasi-brittle material at high strain rate. Experiment and modelling, Eur. J. Mech. A. Solids 17 (3) (1998) 403–420.

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