# The Influence of Interlock Loss between Rebar and Concrete on Bond Performance of RC Member
## 4 FE Analysis for Reproduction on Beam Loading Experiment
### Segregated Modeling of Mortal and Course Aggregate Due to Fine Mesh Size  
In order to clearly reproduce mechanical interlock between rebar lug and concrete, the authors reproduced the shape of threaded bars by hexahedral elements. To achieve that, a small mesh size of 2 mm minimum was set. This mesh size deviated from the applicability of the original concrete model that described the averaged behavior of concrete as the composite material of aggregate and cement paste. Thus, the aggregate and mortar parts of the concrete elements were segregated into concrete elements in order to reproduce the actual condition as closely as possible while there was room for consideration of equivalent softening behavior in homogenized concrete model. Comparison between the homogenizedconcrete modeling and segregated modeling in the structural behavior of beam is shown in Figure A1 in Appendix A.  
In the experiment, the volume ratio of coarse aggregate in the concrete was 36%. Based on that, 36% of concrete elements were randomly set to the aggregate element, and the other concrete elements were set to the mortar elements as shown in Figure 8. Table 4 lists the material property values applied in the aggregate and mortar elements. Because limestone was used as the coarse aggregate in the experiment, the general values of compressive strength (140 MPa) and tensile strength (10 MPa) of limestone aggregate were used for the aggregate elements [34]. Since the fracture of aggregate was brittle, the parameters were set so that tensile bearing stress after occurrence of crack drops sharply with large stiffening parameter while concrete constitutive law was applied for aggregate elements. For the elastic modulus of the aggregate element, the estimated value obtained from the density of aggregate with Equation (9) was used [39].  
\[E_{ag}=\left(2.35\gamma_{ag}-5.78\right)\times 10^{5} \tag{9}\]  
where \(E_{ag}\) is the elastic modulus (MPa) and \(\gamma_{ag}\) is the density of aggregate (g/cm\({}^{3}\)). The calculated elastic modulus of course aggregate had almost same value as that of general limestone aggregate [40].  
The compressive strength and tensile strength taken from the concrete material test results shown in Table 3 were applied for the element properties of the mortar element because the strength of mortal was dominant for concrete failure in normal strength concrete. Elastic modulus of mortal was lower than that of concrete in general and [PERSON]'s equation for two-phase composite materials, shown in Equation (10), was used for determining the elastic modulus of mortal [41,42].  
\[E_{c}=E_{m}\frac{\left(1-V_{g}\right)E_{m}+\left(1+V_{g}\right)E_{g}}{\left(1 +V_{g}\right)E_{m}+\left(1-V_{g}\right)E_{g}} \tag{10}\]  
where \(E_{c},\ E_{m},\ E_{g}\) is the elastic modulus of concrete, mortar, and coarse aggregate, respectively, and \(V_{g}\) is the volume ratio of coarse aggregate in concrete. The elastic modulus of mortal element \(E_{m}\) could be obtained by Equation (10) from the measured elastic modulus of the concrete \(E_{c}\) and the elastic modulus of the aggregate \(E_{g}\) estimated by Equation (9) in this study. Strictly speaking, it was necessary to consider the influences of the interfacial transition zone between the aggregate and mortar elements, and the contact conditions between aggregate elements according to the aggregate diameter and the aggregate shape. However, in this analytical model, these influences were considered to be small because the damage was concentrated on the mortar elements, which had a lower elastic modulus and strength than aggregates.