# Numerical Assessment and Repair Method of Runway
## 3 Numerical Simulation (LS-DYNA M&S)
### RHT Concrete Model and Material  
Concrete is a brittle material made up of cement paste, small aggregates, and large aggregates. The RHT (Riedel-Hiermaier-Thoma) model is a material model that was introduced in 2004 using ANSYS AUTODYN by [PERSON], [PERSON] as a hydrocode that can well represent such brittle materials. After its development, this material model was added as one of the LS-DYNA material models. The RHT Concrete (RHTC) model is specific toward concrete among many brittle materials. It is a macro model that possesses the ability to well describe the dynamic strength of concrete in response to an impact or outside pressure [23]. Additionally, since it considers plastic behavior instead of elastic behavior, it can be used to model high speed impact distribution more accurately than other models. Even though you must input more than 30 parameters to use the RHTC model, [PERSON]'s CONC-35 parameter set can be used for concrete with 35 MPa compressive strength. The RHTC model is a model that combines shear and pressure. It uses Herrmann applied equation of state (EOS) to describe the hydrostatic behavior of concrete. Figure 3 represents the p-\(\alpha\) equation of state graph for a porous material. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Symbol** & **Units** & **Value** \\ \hline Young’s modulus & E & GPa & 27 \\ Compressive strength & f\({}_{\text{c}}\) & MPa & 3.5 \\ Shear modulus & & GPa & 16.7 \\ Density & \(\rho\) & kg/m\({}^{3}\) & 2314 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material properties of concrete slab for the model. Figure 2: Field test M&S model. In the RHTC model, the material that fills up the void space of the matrix is represented using the polynomial equation of state shown in Equation (4) below. \[p=\left\{ \begin{array}{c}A_{1}\mu+A_{2}\mu^{2}+A_{2}\mu^{3}+(B_{0}+B_{1}\mu) \rho_{0}e\ if\ \mu\geq 0\\ T_{1}\mu+T_{2}\mu^{2}+B_{0}\rho_{0}e\ if\ \mu<0\end{array} \right. \tag{4}\]  
where  
\(p\) = Pressure;  
\(A_{1}\), \(A_{2}\), \(A_{3}\), \(B_{0}\), \(B_{1}\), \(T_{1}\), \(T_{2}\) = Constant;  
\(e\) = Internal energy. For the porous material calculation, there needs to be additional formula regarding the porosity of the material which follows the Herrmann formula shown in Equation (5). This formula is a pressure-dependent equation that was used in AUTODYN's RHTC model. \[\alpha(p)=1+\big{(}\alpha_{p}-1\big{)}\bigg{(}\frac{p_{comp}-p}{p_{comp}-p_{ crush}}\bigg{)}^{n} \tag{5}\]  
where  
\(\alpha\) = Porosity ratio;  
\(p\) = compaction pressure;  
\(p_{crush}\) = Pore crush pressure (Initial compaction pressure);  
\(p_{comp}\) = Solid compaction pressure. In LS-DYNA, porosity is represented by time depended Equation (6) below. \[\alpha(t)=\max\bigg{(}1,\min\bigg{(}\alpha_{0},\ min_{s\leq t}\bigg{(}1+( \alpha_{0}-1)\bigg{(}\frac{p_{comp}-p(s)}{p_{comp}-p_{crush}}\bigg{)}^{n}\bigg{)} \bigg{)}\bigg{)} \tag{6}\]  
The RHTC model represents fracture surface using yield strength, breaking strength, and residual strength. As the strain material increases past the breaking point of the material, the hardness, tensile strength, and resistance increase greatly, resulting in strain hardening. This strain hardening parameter was obtained using the researched concrete strain hardening information and applying it to a mathematical model. Yield strength is defined as a part of breaking strength with a cap, and this breaking strength is described by Equation (7). Figure 3: Schematic description of the p-equation of state ([PERSON] et al., 1969). \[Y_{c}(p^{*})=f_{c}\times\Big{[}A\times\Big{(}p^{*}-p_{spall}{}^{*}\times F_{rate}( \dot{\epsilon})\Big{)}^{n}\Big{]},\ p_{spall}{}^{*}=f_{t}/f_{c} \tag{7}\]  
where  
\(f_{c}\) = the material's uniaxial compressive strength;  
\(f_{t}\) = the material's tensile strength;  
\(p^{*}\) and \(p_{spall}{}^{*}\) = pressure and spall strength, respectively;  
\(\dot{\epsilon}\) = strain rate;  
\(n\) = porosity exponent;  
\(F_{rate}(\dot{\epsilon})\) = function of the DIF (Dynamic Increase Factor).