# Experimental Study on a Prediction Model of the Shrinkage and Creep of Recycled Aggregate Concrete
## 4 Creep and Shrinkage Model of Recycled Aggregate Concrete
### Creep Increasing Coefficient of Attached Mortar  
[PERSON] et al. [18] considered that natural aggregate was replaced by mass or volume percentage when RAC was prepared by traditional methods. Compared with ordinary concrete, a certain amount of cement mortar is wrapped on the surface of the recycled aggregate. The content of natural aggregate in the RAC prepared by the conventional method is decreased and the content of the total mortar is increased. The elastic modulus of the component is decreased, and the shrinkage and the amount of creep are increased. It is assumed that the RCA is a two-phase material composed of residual cement mortar and natural aggregate, and the mortar in the RAC consists of residual cement mortar and new cement mortar on the aggregate surface. Based on the creep test results of RAC, the residual cement mortar content of RCA was taken as the research object to establish the creep prediction model of RAC.  
\[C_{\text{AM}}=\frac{C_{\text{RAC}}}{C_{\text{NAC}}}=(\frac{V_{\text{NM}}^{ \text{RAC}}+V_{\text{RM}}^{\text{RAC}}}{1-V_{\text{RCA}}^{\text{RAC}}})^{1.33}, \tag{1}\]  
\[V_{\text{NM}}^{\text{RAC}}=1-V_{\text{RCA}}^{\text{RAC}}. \tag{2}\]  
Simultaneous equations:  
\[C_{\text{AM}}=(\frac{V_{\text{NM}}^{\text{RAC}}+V_{\text{AM}}^{\text{RAC}}}{ 1-V_{\text{NCA}}^{\text{NAC}}-V_{\text{RCA}}^{\text{NAC}}})^{1.33}. \tag{3}\]  
According to the composition of RCA, the following equation holds:  
\[v_{\text{RCA}}=v_{\text{AM}}+v_{\text{OVA}},\ \ \ m_{\text{RCA}}=m_{\text{AM}}+m_{ \text{OVA}}, \tag{4}\]  
\[m_{\text{RCA}}=v_{\text{RCA}}\times\rho_{\text{RCA}},\ \ \ m_{\text{AM}}=v_{\text{AM}} \times\rho_{\text{AM}}, \tag{5}\]  
\[m_{\text{OVA}}=v_{\text{OVA}}\times\rho_{\text{OVA}}. \tag{6}\]  
In the equation, \(V\) is volume (m\({}^{3}\)); \(m\) is quality (kg); \(\rho\) is apparent density (kg/m\({}^{3}\)).  
The parameter \(M_{AM}\) is the mass ratio of the attached mortar and the RCA. \(M_{AM}=m_{AM}/m_{RCA}\), and the following equation is derived from the above equation:  
\[\rho_{\text{AM}}=\frac{m_{\text{AM}}}{v_{\text{AM}}}=\frac{v_{\text{RCA}} \times p_{\text{RCA}}\times M_{\text{AM}}}{v_{\text{RCA}}-\frac{(1-M_{\text{AM }})^{\text{pRCA}}\times p_{\text{RCA}}}{\rho_{\text{OVA}}}}=\frac{M_{\text{AM} }}{\frac{1}{\rho_{\text{OVA}}}-\frac{1-M_{\text{AM}}}{\rho_{\text{OVA}}}}, \tag{7}\]and \(v_{\rm AM}=v_{\rm RCA}\times M_{\rm AM}\times\frac{\rho_{\rm RCA}}{\rho_{\rm AM}}\).  
Simultaneous equations:  
\[v_{\rm AM}=v_{\rm RCA}\times[1-(1-M_{\rm AM})\frac{\rho_{\rm RCA}}{\rho_{\rm OVA }}], \tag{8}\]  
\[V_{\rm AM}=V_{\rm RCA}\times[1-(1-M_{\rm AM})\frac{\rho_{\rm RCA}}{\rho_{\rm OVA }}]. \tag{9}\]  
Generally, the replacement rate of RCA is defined as follows:  
\[r=\frac{m_{\rm RCA}^{\rm RAC}}{m_{\rm RCA}^{\rm RAC}+m_{\rm NCA}^{\rm RAC}}. \tag{10}\]  
The volume ratio of natural coarse aggregate and RCA in RAC is \(R\).  
\[R=\frac{v_{\rm NCA}^{\rm RAC}}{v_{\rm RCA}^{\rm RAC}}=\frac{V_{\rm NCA}^{\rm RAC }}{V_{\rm RCA}^{\rm RAC}}=\frac{(m_{\rm RCA}^{\rm RAC}+m_{\rm NCA}^{\rm RAC}) \times(1-r)/\rho_{\rm NCA}}{(m_{\rm RCA}^{\rm RAC}+m_{\rm NCA}^{\rm RAC}) \times r/\rho_{\rm RCA}}=\frac{1-r}{r}\times\frac{\rho_{\rm RCA}}{\rho_{\rm NCA}} \tag{11}\]  
Equation (3) can be rewritten as  
\[C_{\rm AM}=\left(\frac{1-V_{\rm RCA}\times[R+(1-M_{\rm AM})\frac{\rho_{\rm RCA }}{\rho_{\rm OVA}}]}{1-V_{\rm RCA}^{\rm RAC}\times(1+R)}\right)^{1.33}. \tag{12}\]  
In order to simplify the expression, the approximate \(\rho_{RCA}=\rho_{OVA}\) is considered, and the relationship between \(C_{AM}\) and the RCA replacement rate r is as follows:  
\[C_{\rm AM}=\left(\frac{1-V_{\rm RCA}^{\rm RAC}\times[\frac{1-r}{r}\times\frac {\rho_{\rm RCA}}{\rho_{\rm NCA}}+(1-M_{\rm AM})]}{1-V_{\rm RCA}^{\rm RAC} \times(1+\frac{1-r}{r}\times\frac{\rho_{\rm RCA}}{\rho_{\rm NCA}})}\right)^{1.33}. \tag{13}\]  
When \(r=100\%\), it is all RAC:  
\[C_{\rm AM}=\left(\frac{1-V_{\rm RCA}^{\rm RAC}(1-M_{\rm AM})}{1-V_{\rm RCA}^{ \rm RAC}}\right)^{1.33}. \tag{14}\]