# Rebar Shape Time-Evolution During a Reinforced Concrete Corrosion Test: An Electrochemical Model
## 3 Rebar Shape Model
### Rebar Shape Time-Evolution  
The rebar corrosion causes metal loss and the reduction of the radius of the rebar. The radius loss is not uniform along the rebar perimeter because the current density is not uniform. On the other hand, corrosion products precipitate as oxides that are bulkier than the original metal [2]. In this paper, it is accepted that these corrosion products remain attached to the bar, giving rise to a metal rebar narrower than the original but covered with a layer of oxides that gives the whole a volume greater than the original. The remaining metal radius \(r_{m}\) and the radius of the rebar as a whole \(r_{ox}\) (including the oxide layer) are determined in this section. Both depend on the angle \(\theta\) since the current density \(j\) depends on it.  
Let \(d\theta\) be a differential rebar angle. The surface subtended by it is \(L_{z}r_{m}d\theta\), where \(L_{z}\) is the rebar length. The current traversing this surface is \(jL_{z}r_{m}d\theta\), and the circulated charge during an interval \(dt\) is \(jL_{z}r_{m}d\theta dt\). [PERSON]'s constant \(F\) gives the charge of a mole of electrons, then circulated moles of electrons are represented by \(\frac{jL_{z}r_{m}}{F}d\theta dt\). It is related with moles of dissolved metal by the oxidation reaction stoichiometry. Being \(n\), the moles of electrons needed to dissolve one mole of metal, the moles of dissolved metal during the \(dt\) interval are \(\frac{jL_{z}r_{m}}{nF}d\theta dt\). Using metal molar mass \(M\) and metal density \(\rho\), the volume of metal lost is \(-\frac{jML_{z}r_{m}}{nF\rho}d\theta dt\). Additionally, this volume can be expressed geometrically as \(L_{z}r_{m}d\theta dr_{m}\). Equating both expressions, the following is obtained for the metal radius variation.  
\[dr_{m}=-\frac{j(\theta,t)Mdt}{nF\rho} \tag{5}\]  
That can be integrated to obtain the metal radius.  
\[r_{m}(\theta,t)=\frac{D}{2}-\frac{M}{nF\rho}\int_{0}^{t}j(\theta,t)dt \tag{6}\]  
where \(\frac{D}{2}\) is the initial rebar radius. The oxidation reaction considered is shown below [2].  
\[\mathrm{Fe(s)}\rightarrow\mathrm{Fe^{2+}(ac)+2e^{-}} \tag{7}\]  
For which \(n=2\). The used values of the other constants are \(M=55.85\) g/mol, \(F=96,485\) C/mol, and \(\rho=7.85\) g/cm\({}^{3}\)[33].  
The length of the metal perimeter through which corrosion takes place does not remain constant and can be calculated as:  
\[s(\theta,t)=\int_{0}^{\theta}r_{m}(\theta,t)d\theta \tag{8}\]  
Radius of the rebar as a whole, including the oxide layer, \(r_{ox}\), is now obtained. Let \(d\theta\) be a differential rebar angle, where the variation of radii during an interval \(dt\) is \(dr_{m}\) for the metal and \(dr_{ox}\) for the whole rebar including the oxide layer. Then, the variation of metal volume \(dV_{m}\) and variation of the oxide layer volume \(dV_{ox}\) are geometrically given by:  
\[\begin{array}{c}dV_{m}=L_{z}r_{m}d\theta dr_{m}\\ dV_{ox}=L_{z}r_{ox}d\theta dr_{ox}-L_{z}r_{m}d\theta dr_{m}\end{array} \tag{9}\]  
The volume occupied by the oxides is considered to be \(\alpha\) times the volume occupied by the original metal. The value \(\alpha=2\)[28] has been considered. Then:  
\[dV_{ox}=-\alpha dV_{m} \tag{10}\]  
Using Equation (9) in this case and integrating, the radius of the rebar including the oxide layer \(r_{ox}\) is obtained using the formula below.  
\[r_{ox}(\theta,t)=\sqrt{\frac{D^{2}}{4}-(\alpha-1)\bigg{(}\frac{D^{2}}{4}-r_{m }^{2}(\theta,t)\bigg{)}} \tag{11}\]  
This radius gives the shape of the rebar that could be used in mortar cracking mechanical models.