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Computational homogenization of diffusion in three-phase mesoscale concrete

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Abstract

A three dimensional (3D) mesoscale model of concrete is presented and employed for computational homogenization in the context of mass diffusion. The mesoscale constituents of cement paste, aggregates and interfacial transition zone (ITZ) are contained within a statistical volume element (SVE) on which homogenization is carried out. The model implementation accounts for ITZ anisotropy thereby the diffusivity tensor depends on the normal of the aggregate surface. The homogenized response is compared between 3D and 2D SVEs to study the influence of the third spatial dimension, and for varying mesoscale compositions to study the influence of aggregate content on concrete diffusivity. The computational results show that the effective diffusivity of 3D SVEs is about 40 % greater than 2D SVEs when ITZ is excluded for the SVE, and 17 % when ITZ is included. The results are in agreement with the upper Hashin–Shtrikman bound when ITZ is excluded, and close to the Taylor assumption when ITZ is included.

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Notes

  1. i.p.= intersection point.

  2. Superscripts \(M\) denotes macroscale, whereas \(s\) denotes subscale.

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Acknowledgments

This research was financially supported by The Swedish Research Council, which is gratefully acknowledged. The computations were performed on resources at Chalmers Centre for Computational Science and Engineering (C\(^3\)SE) provided by the Swedish National Infrastructure for Computing (SNIC).

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Chalmers University of Technology, Göteborg, 412 96, Sweden

    Filip Nilenius & Karin Lundgren

  2. Department of Applied Mechanics, Chalmers University of Technology, Göteborg, 412 96, Sweden

    Fredrik Larsson & Kenneth Runesson

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  1. Filip Nilenius
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Correspondence to Filip Nilenius.

Voigt bound

Voigt bound

\(V_\mathrm{ITZ}\) is given by

$$\begin{aligned} V_\mathrm{ITZ}&= \frac{4}{3}\pi (r+t)^3 - \frac{4}{3}\pi r^3 \nonumber \\&=\{t\ll r\} \approx \frac{16\pi }{3}r^2t, \end{aligned}$$
(46)

where \(t\) and \(r\) are the thickness of the ITZ and the aggregate radius, respectively, which yields

$$\begin{aligned} \frac{V_\mathrm{ITZ}}{V_\mathrm{a}} = \frac{4t}{r} \quad \Rightarrow \quad n_\mathrm{ITZ} = \frac{4t}{r}n_\mathrm{a}= n_\mathrm{a}\sum _{i=1}^N \frac{4t}{r_i}f_i. \end{aligned}$$
(47)

The Voigt bound then takes the form

$$\begin{aligned} D_\mathrm{eff}^\text {Voigt}(n_\mathrm{a})&= n_\mathrm{a}D_\mathrm{a}+ n_\mathrm{cp}D_\mathrm{cp}+ n_\mathrm{ITZ} D_\mathrm{ITZ} \nonumber \\&= n_\mathrm{a}D_\mathrm{a}+n_\mathrm{cp}D_\mathrm{cp}+ n_\mathrm{a}\sum _{i=1}^N \frac{4f_i}{r_i}\hat{D}_\mathrm{ITZ} \nonumber \\&= \{n_\mathrm{a}+ n_\mathrm{cp}\approx 1 \} \nonumber \\&= n_\mathrm{a}D_\mathrm{a}+ (1-n_\mathrm{a}) D_\mathrm{cp}+ n_\mathrm{a}\sum _{i=1}^N \frac{4f_i}{r_i}\hat{D}_\mathrm{ITZ} \nonumber \\&= n_\mathrm{a}D_\mathrm{a}+ D_\mathrm{cp}+ n_\mathrm{a}\sum _{i=1}^N\left( \frac{4f_i}{r_i}\hat{D}_\mathrm{ITZ} - f_i D_\mathrm{cp}\right) \nonumber \\&= \{D_\mathrm{a}=0~{\mathrm{cm}^{2} \mathrm{s}^{-1}}\} \nonumber \\&= D_\mathrm{cp}+n_\mathrm{a}\sum _{i=1}^N\left( \frac{4f_i}{r_i}\hat{D}_\mathrm{ITZ} - f_i D_\mathrm{cp}\right) . \end{aligned}$$
(48)

This bound is valid for \(\hat{D}_\mathrm{ITZ} = 0\) but not for high values of \(n_\mathrm{a}\) while \(\hat{D}_\mathrm{ITZ} > 0\) since \(D_\mathrm{eff}^\text {Voigt}(n_\mathrm{a}=1,\hat{D}_\mathrm{ITZ} > 0)\ne D_\mathrm{a}\).

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Nilenius, F., Larsson, F., Lundgren, K. et al. Computational homogenization of diffusion in three-phase mesoscale concrete. Comput Mech 54, 461–472 (2014). https://doi.org/10.1007/s00466-014-0998-0

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