# ICME Overview for Steel Reinforced Concrete

## Contents |

### Multiscale Overview of Transport and Crack Propagation Phenomena in Steel Reinforced Concrete

#### Summary

#### History Effects

In seeking to accurately simulate the coupled transport and crack development at work in reinforced concrete, a number of challenges arise from both the conceptual bridging of length scales as well as those issues peculiar to the complex nature of concrete itself. While history effects are certainly important to consider when modeling almost any material, the “living” nature of the cement binder in reinforced concrete makes accurate, multi-scale analysis impossible without such considerations. In treating the “life” of cement binder as beginning with the hydration process, the necessity of certain a priori knowledge becomes apparent. More specifically, variables like the Ca/Si and water/cement ratios of the binder, the presence of additives (like silica fume, fly ash, and blast furnace slag) in the mixture, the fineness and compaction of the mixture, the age of the structure, and the level of exposure to aggressive chemical species will have an effect on the characteristics of the binder at almost every length scale.#### Continuum Scale

In considering the behavior of reinforced concrete at the continuum level, we may treat it as being of at least four solid phases, those being the cement binder (CSH), carbonated binder, the aggregate, the steel reinforcement. In order to accurately simulate continuum behavior, we will require the mechanical properties (Young’s and bulk moduli, yield stresses, and thermal diffusion coefficients) for each of these phases and the bonding strength between these phases. For this, we will look to simulations in atomistics, DFT and Molecular Dynamics at micro and nano scales. As detailed in the work of Cerny and Ravnanikova ^{[1]} The volume fraction of each phase and the surface area of the aggregate and reinforcement within the binder must also be taken into account, and we may do so by looking to the meso scale.

Additionally, many aggressive chemical species in solution with moisture make up liquid and gaseous phases within the pore system and must be taken into account. Maekawa, Ishida, and Kishi^{[2]} did this by considering the sorptivity and moisture content of the concrete in the form of coefficients of the diffusive transport and convective transport terms at the surface of each or our solid phases in the general transport PDE. (Maekawa, et. al) Again, meso scale analysis can yield information regarding these variables.

#### Meso Scale

In investigating both crack growth and initiation and transport, we look to the meso-scale to “pass up” information regarding the pore structure of the cement binder. Variables like porosity and tortuosity of the binder as well as the pore surface area of the binder and void volume fraction of the solid phases may be ascertained for the cement RVE. In the case of the porosity, tortuosity, and pore surface area, these variables will be fed into the calculation of the sorptivity at the continuum level.^{[3]} Similarly, the void volume fraction of the solid phases observed at this level will affect the mechanical behavior with respect to crack growth and propagation.

The variables related to the pore structure of the cement binder are largely considered to be functions of the degree of hydration and inclusion of cement mixture additives. While a priori knowledge of the w/c and Ca/Si ratios are passed from the history along with details about the inclusion of additives in the binder and the fineness of the unhydrated grains, calculation of the degree of hydration will require information regarding the thermal and ionic diffusivity coefficients of the binder, aggregates, and reinforcement. We must look to the lower length scales to provide these. Furthermore, the mechanical variables like the moduli, and yield stresses similarly require delving into the lower length scales for their calculation.

#### Micro Scale

While porosity and tortuosity gleaned from meso-scale analysis are necessary for the calculation of convective terms in the transport equation, these variables are a consequence of hydration phenomena observed on scales of the order of 10-6m-10-10m. We are fed information regarding the grain size of unhydrated silicates, w/c and Ca/Si ratios, age, and use of additives from the history, which we may use in any number of available hydration models in order to simulate the hydration process of the cement binder. In doing so, we not only are able to calculate porosity, tortuosity, void volume fraction, and pore surface area to pass on to the meso-scale, but we are also able to generate information on the proposed crystal structure of the CSH hydration products.

Modeling the hydration process (and by extension to higher length scales, the pore structure) of cement is incomplete with just the above mentioned variables, however, and requires knowledge about the heat conductivity of the concrete. Being an exothermic reaction, hydration can lead to thermal expansion of the cement binder and the heat released during said reaction will evaporate moisture in the mixture. These can have adverse effects on the strength of the pore structure as they will contribute to the nucleation of cracks and the restriction of the hydration process in partially hydrated cement. And so, we must look to the nano-scale to provide us with thermal diffusion coefficients.

#### Nano Scale

Finally arriving at the bottom of the length scales, as shown in the work of Pellene, Lequeux, and van Damme, it is possible to calculate the moduli and thermal and moisture diffusion coefficients through the use of *ab initio* calculations, principles of energy minimization, and molecular dynamics simulations, all of which can be passed up to the continuum scale.^{[4]} DFT may be used to calculate the bond strength between the various solid phases and passed up as well. Also Faraday’s law may be used to analyze the corrosion in steel reinforcement, which can be fed up to the void volume fraction of that phase at the meso-scale.

### References

- ↑ Cerny, R. R., P. (2002). Transport Processes in Concrete. New York, NY, Spon Press.
- ↑ Maekawa, K., Ishida, T., and Kishi, T., “Multi-scale Modeling of Concrete Performance -Integrated Material and Structural Mechanics,” Journal of Advanced Concrete Technology, Vol. 1, No. 2, pp. 91–126, 2003.
- ↑ Maekawa, K., Ishida, T., and Kishi, T., “Multi-scale Modeling of Concrete Performance -Integrated Material and Structural Mechanics,” Journal of Advanced Concrete Technology, Vol. 1, No. 2, pp. 91–126, 2003.
- ↑ Pellene, R., Lequeux, N., and van Damme, H., “Engineering the bonding scheme in C-S-H: The Iono-Covalent framework,” Cement and Concrete Research, No. 38, pp. 159–174, 2008.