# Viscoelastic Behavior of Polymer-Modified Cement Pastes: Insight from Downscaling Short-Term Macroscopic Creep Tests by Means of Multiscale Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Modeling the Hydration-Induced Evolutions of the Non-Aging Creep Properties of Polymer-Modified Cement Pastes

#### 2.1. Micromechanical Representation of Polymer-Modified Cement Pastes

#### 2.2. Viscoelastic Phase Properties

#### 2.3. Homogenization of the Viscoelastic Properties of Polymer-Modified Cement Paste

## 3. Identification of Polymer Creep Properties Based on Macroscopic Creep Tests

#### 3.1. Hourly-Repeated Ultra-Short Creep Tests on Polymer-Modified Cement Pastes

#### 3.2. Universal Polymer Creep Properties

#### 3.3. Age-Dependent Polymer Creep Properties

## 4. Conclusions

- The pronounced creep activity of polymer-modified cement paste can be explained by an isochoric power-law-type creep behavior of the polymers, whereby the shear creep modulus of the polymers is two orders of magnitude smaller than that of the hydrates.
- The creep behavior of the polymer particles inside hydrating cement paste is not universal. The creep activity of the polymer particles decreases significantly as hydration proceeds. The underlying physical mechanism is very likely related (i) to self-desiccation resulting from the (water-consuming) hydration reaction and (ii) to the associated continuous decrease of the internal relative humidity in cement pastes [62].
- The experimentally-observed macroscopic creep behavior of hydrating polymer-modified cement pastes can be satisfactorily reproduced when considering that the power-law creep exponent of the polymer particles is age-independent and, thus, constant. In that case, the shear creep modulus of the polymer particles was found to follow a bilinear trend during the first week after production. Considering that both the shear creep modulus and the creep exponent of the polymers are age-dependent and, thus, evolving functions, the agreement between modeling results and experiments can be further improved, but at the cost of considerable additional computational efforts.
- As for future research regarding the viscoelastic behavior of polymer-modified cement pastes, it is desirable to monitor the evolution of the internal relative humidity. This will provide the necessary physical background to interpret the self-desiccation-induced changes of the creep behavior of the polymers.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

${\mathbb{C}}_{j}$ | elastic stiffness tensor of phase j |

clin | cement |

$cp$ | cement paste |

cyl | cylindrical |

${E}_{c,cp}^{\mathrm{exp}}$ | experimentally-determined creep modulus of cement paste |

exp | experimentally-determined |

${f}_{j}^{hf}$ | hydrate foam-related volume fraction of phase j |

${f}_{j}^{cp}$ | cement paste-related volume fraction of phase j |

hyd | hydrates |

$hf$ | hydrate foam |

$\mathbb{I}$ | fourth-order identity tensor |

${\mathbb{I}}^{\mathrm{vol}},{\mathbb{I}}^{\mathrm{dev}}$ | volumetric and deviatoric parts of the fourth-order identity tensor |

${J}_{cp}^{\mathrm{exp}}$, ${J}_{cp}^{\mathrm{exp}}$ | experimentally-determined and model-predicted uniaxial creep function of cement paste |

${J}_{v,cp}^{\mathrm{exp}}$, ${J}_{v,cp}^{\mathrm{exp}}$ | viscous parts of ${J}_{cp}^{\mathrm{exp}}$, ${J}_{cp}^{\mathrm{exp}}$ |

$\mathbb{J}$ | fourth-order creep tensor function |

${k}_{j}$ | bulk modulus of phase j |

mod | model-predicted |

P1, P2 | Polymer 1, Polymer 2 |

p | complex variable in the LC domain |

${\mathbb{P}}_{m}$ | Hill tensor with shape m, $m\in \left\{\mathrm{cyl},\phantom{\rule{4.pt}{0ex}}\mathrm{sph}\right\}$ |

pol | polymer |

$\mathbb{R}$ | fourth-order relaxation tensor function |

$\mathbb{S}$ | fourth-order Eshelby tensor |

sph | spherical |

t | chronological time |

${t}_{\mathrm{ref}}$ | reference time |

${\beta}_{j}$ | power-law creep exponent of phase j |

${\beta}_{cp}^{\mathrm{exp}}$ | experimentally-determined power-law exponent of cement paste |

$\Gamma $ | gamma function |

$\delta $ | Kronecker delta |

${\epsilon}_{j}$ | strain of phase j |

$\vartheta $ | zenith angle |

${\mu}_{j}$ | shear modulus of phase j |

${\mu}_{c,j}$ | shear creep modulus of phase j |

$\xi $ | degree of hydration |

${\sigma}_{j}$ | stress of phase j |

$\tau $ | time instant of loading |

$\phi $ | azimuth angle |

${(\u2022)}^{*}$ | LC transform of quantity (•) |

## Appendix A. Phase Volume Fractions

**Figure A1.**Evolution of “cement paste”-related phase volume fractions as a function of the hydration degree for (

**a**) plain cement paste with $w/c$ = 0.40, for (

**b**) polymer-modified cement paste with $w/c$ = 0.40, $p/c$ = 0.10 and for (

**c**) polymer-modified cement paste with $w/c$ = 0.40, $p/c$ = 0.10, and ${f}_{air}^{cp}$ = 0.048.

## Appendix B. Hill Tensor Expressions in the LC Space

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**Figure 1.**Micromechanical representation of polymer-modified cement pastes after [9]; the two-dimensional sketches refer to three-dimensional representative volume elements (RVEs).

**Figure 2.**Compressive loading prescribed during the ultra-short creep tests: (

**a**) prescribed maximum forces applied to the plain and polymer-modified cement pastes; and (

**b**) load history applied to the plain cement paste at an age of 35 h, including a holding plateau with a duration of 180 s.

**Figure 3.**Results from hourly-repeated three-minute-long creep testing for selected hydration degrees $\xi $: creep strain evolutions divided by the plateau stress, (

**a**) of the plain cement paste reference with $w/c$ = 0.40 and (

**b**,

**c**) of the two polymer-modified cement pastes with $w/c$ = 0.40 and $p/c$ = 0.10.

**Figure 4.**Comparison of experimentally-determined and model-predicted viscous strains of polymer-modified cement pastes P1 (

**a**) and P2 (

**b**) at material ages of 40 h, 60 h and 130 h for constant polymer creep properties.

**Figure 5.**Mean errors according to Equation (17) for age-independent power-law creep exponents within ${\beta}_{pol}\in \{0.1,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.1,1.3\}$ and for optimized age-dependent shear creep moduli for paste P1 (

**a**) and paste P2 (

**b**).

**Figure 6.**Comparison of experimental and computed creep strains of polymer-modified cement pastes P1 (

**a**) and P2 (

**b**) at ages of 40 h, 60 h and 130 h, considering age-dependent shear creep moduli and constant power-law creep exponents.

**Figure 7.**Evolutions of the shear creep modulus with respect to material age, while considering time-independent power-law exponents ${\beta}_{P1}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{\beta}_{P2}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.6$, for paste P1 (

**a**) and paste P2 (

**b**).

**Figure 8.**Evolutions of the shear creep modulus and the power-law creep exponent with respect to material age, for paste P1 (

**a**,

**c**) and paste P2 (

**b**,

**d**).

**Figure 9.**Comparison of experimental and computed creep strains of polymer-modified cement pastes P1 (

**a**) and P2 (

**b**) at ages of 40 h, 60 h and 130 h, considering both age-dependent shear creep moduli and age-dependent power-law creep exponents.

**Table 1.**Elastic properties and mass densities of the material phases in polymer-modified cement pastes.

Phase | Bulk Modulus | Shear Modulus | Mass Densities | Reference |
---|---|---|---|---|

(GPa) | (GPa) | (g cm${}^{-3}$) | ||

Air | ${k}_{\mathrm{air}}$ = 0 | ${\mu}_{\mathrm{air}}$ = 0 | ${\rho}_{\mathrm{air}}=0$ | |

Water | ${k}_{\mathrm{water}}$ = 0 | ${\mu}_{\mathrm{water}}$ = 0 | ${\rho}_{\mathrm{water}}=1.00$ | |

Hydrates | ${k}_{\mathrm{hyd}}$ = 18.69 | ${\mu}_{\mathrm{hyd}}$ = 11.76 | ${\rho}_{\mathrm{hyd}}=2.073$ | [23,29,43] |

Cement | ${k}_{\mathrm{cem}}$ = 116.70 | ${\mu}_{\mathrm{cem}}$ = 53.80 | ${\rho}_{\mathrm{cem}}=3.15$ | [33,44] |

Polymer P1 | ${k}_{\mathrm{P}1}$ = 3.97 | ${\mu}_{\mathrm{P}1}$ = 0.85 | ${\rho}_{\mathrm{P}1}=1.03$ | [8,9] |

Polymer P2 | ${k}_{\mathrm{P}2}$ = 4.02 | ${\mu}_{\mathrm{P}2}$ = 0.86 | ${\rho}_{\mathrm{P}2}=1.02$ | [8,9] |

**Table 2.**Mean errors $\u03f5$ according to Equation (17) for universal and age-dependent polymer creep properties.

Shear Creep Modulus | Power-Law Creep Exponent | Polymer | Mean Error $\mathit{\u03f5}$ |
---|---|---|---|

universal | universal | P1 | $1.709\times {10}^{-6}$ |

universal | universal | P2 | $1.927\times {10}^{-6}$ |

age-dependent | universal | P1 | $2.665\times {10}^{-7}$ |

age-dependent | universal | P2 | $2.857\times {10}^{-7}$ |

age-dependent | age-dependent | P1 | $1.975\times {10}^{-7}$ |

age-dependent | age-dependent | P2 | $2.227\times {10}^{-7}$ |

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**MDPI and ACS Style**

Göbel, L.; Königsberger, M.; Osburg, A.; Pichler, B.
Viscoelastic Behavior of Polymer-Modified Cement Pastes: Insight from Downscaling Short-Term Macroscopic Creep Tests by Means of Multiscale Modeling. *Appl. Sci.* **2018**, *8*, 487.
https://doi.org/10.3390/app8040487

**AMA Style**

Göbel L, Königsberger M, Osburg A, Pichler B.
Viscoelastic Behavior of Polymer-Modified Cement Pastes: Insight from Downscaling Short-Term Macroscopic Creep Tests by Means of Multiscale Modeling. *Applied Sciences*. 2018; 8(4):487.
https://doi.org/10.3390/app8040487

**Chicago/Turabian Style**

Göbel, Luise, Markus Königsberger, Andrea Osburg, and Bernhard Pichler.
2018. "Viscoelastic Behavior of Polymer-Modified Cement Pastes: Insight from Downscaling Short-Term Macroscopic Creep Tests by Means of Multiscale Modeling" *Applied Sciences* 8, no. 4: 487.
https://doi.org/10.3390/app8040487