# Space-time Gaussian models for evaluating corrosion-induced damages in reinforcing bars

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## Abstract

Chloride-induced steel reinforcing bars corrosion is a problem of worldwide importance because, in addition to massive industrialization, a majority of human population inhabit marine atmosphere zones where salinity is the main source of built infrastructure deterioration. Reinforced concrete structures vulnerable to chloride-induced corrosion potentially experience two stages of damage, namely non-structural and structural. Till date, with regard to non-uniform corrosion, there is very limited information regarding the significant role played by the non-structural stage of damage in influencing the progression of structural damage. The present work examines the process of non-uniform chloride ingress into concrete due to the existence of the steel bars and its influence on the spatial and temporal variations in the depth of corrosion penetration. Mathematical models in the form of Gaussian functions are proposed to represent space-time variation in corrosion initiation, spatial spread of corrosion penetration depth and corroded bar morphologies, and are compared with literature data. A numerical inverse analysis procedure is introduced to predict space-time evolution of non-structural damage from experimentally observed spatial spread of structural damage. Further, applicability of superposition approach for the evaluation of corrosion-induced stresses around the steel–concrete interface of a corner reinforcing bar is explored and shown to introduce spatial and temporal errors while evaluating structural damages.

## Keywords

Concrete structures chlorides corrosion damage modelling## Nomenclature

- \( C_{d} \)
cover depth (cm)

- \( d_{r} \)
diameter of the original rebar (mm)

- \( d_{cr} \)
corroding rebar diameter (mm)

- \( d_{cr}^{\hbox{min} } \)
minimum diameter of the corroding rebar (mm)

- \( D_{{c,{\text{Ref}}}} \)
reference chloride diffusion coefficient (m

^{2}/s)- \( H_{env} \)
relative humidity in the environment

- \( i_{\text{corr}} \)
corrosion rate (μA/cm

^{2})- \( p_{d} \)
corrosion penetration depth (mm)

- \( p_{d}^{a} \)
corrosion penetration depth at \( a \) (mm)

- \( p_{d}^{b} \)
corrosion penetration depth at \( b \) (mm)

- \( p_{d}^{\hbox{min} } \)
minimum corrosion penetration depth (mm)

- \( p_{d}^{\hbox{max} } \)
maximum corrosion penetration depth (mm)

- \( t \)
exposure period (years)

- \( T_{i} \)
time-to-corrosion initiation (years)

- \( T_{env} \)
temperature in the environment (K)

- \( T_{\,i}^{\,1} \)
time to first active corrosion (years)

- \( T_{\,i}^{\,\hbox{max} } \)
maximum value of time-to-corrosion initiation (years)

- \( {w \mathord{\left/ {\vphantom {w b}} \right. \kern-0pt} b} \)
water-to-binder ratio

- \( \theta_{a} \)
angle between rebar center and \( a \) (degree)

- \( \theta_{b} \)
angle between rebar center and \( b \) (degree)

- \( \theta_{\hbox{min} } \)
minimum angle with rebar center (degrees)

- \( \theta_{\hbox{max} } \)
maximum angle with rebar center (degrees)

- \( \kappa_{{\alpha_{L} }} \)
binding isotherm constant (mL pore solution/g sample)

- \( \kappa_{{\beta_{L} }} \)
binding isotherm constant (mL pore solution/mg Cl)

- \( \kappa_{{\alpha_{F} }} \)
binding isotherm constant (mL pore solution/g sample)

- \( \kappa_{{\beta_{F} }} \)
binding isotherm constant

- \( \eta_{1} \)
regression coefficient

- \( \eta_{2} \)
regression coefficient

- \( \lambda_{h} \)
moisture diffusion coefficient

- \( \mu_{h} \)
moisture diffusion coefficient

- \( \nu_{h} \)
moisture diffusion coefficient

- \( \upsilon_{1} \)
regression parameter

- \( \upsilon_{2} \)
regression parameter

- \( \chi_{1} \)
regression parameter

- \( \chi_{2} \)
regression parameter

- \( \omega \)
angular velocity (degree/year)

- \( \omega_{\hbox{max} } \)
maximum angular velocity (degree/year)

## Notes

### Acknowledgement

The support of Endeavour Research Fellowship from the Australian Government, Department of Education and Training is gratefully acknowledged.

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