# Effect of Different Gradings of Lightweight Aggregates on the Properties of Concrete
## 3 Numerical Modeling and Characterization
### Numerical Modeling with a Dense Packing Algorithm
#### 3.1.2 Numerical Simulations  
The material properties of the virtual specimens, the elastic modulus, compressive strength, and thermal conductivity are numerically computed using the FE simulation. For the calculation, ABAQUS commercial software [37] (v2016, Dassault Systemes, Velizy-Villacoublay Cedex, France) is used. In particular, a concrete damaged plasticity (CDP) model in the ABAQUS package is used here to examine the mechanical responses. Detailed formulations for mechanical and thermal analysis can be found in [38; 39] (mechanical) and [40; 41] (thermal), and only general formulations are given in this paper.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Types** & **Matrix (Mortar)** & **Liaver® 4 mm** & **Liaver® 2 mm** & **Liaver® 1 mm** & **Liaver® 0.5 mm** \\ \hline
**Thermal Conductivity (W/m/K)** & 0.31 & 0.070 & 0.073 & 0.075 & 0.080 \\
**Density (kg/m\({}^{3}\))** & 1047 & 310 & 340 & 350 & 450 \\
**Specific Heat (I/kg/K)** & 960 & 1200 & 1150 & 1140 & 1100 \\ \hline
**Elastic Modulus (GPa)** & 8.15 & 0.76 & 0.79 & 0.80 & 0.84 \\
**Poisson’s Ratio** & 0.3 & 0.37 & 0.37 & 0.36 & 0.36 \\
**Yield Strength (MPa)** & 40.78 & 1.2 & 2.2 & 2.4 & 2.9 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Material properties of lightweight aggregates (Liaver®) and matrix [19].  
Figure 3: Sample of virtual specimens with different gradings (note: each color denotes spherical Liaver® with different sizes. The total volume of Liaver® particles in each specimen is about 73%.).  
For the mechanical analysis, the governing equation for the stress-strain relationship controlled by a scalar damaged elasticity is described as:  
\[\mathbf{\sigma}=(1-d)\mathbf{D}_{0}^{el}\colon(\mathbf{\epsilon}-\mathbf{e}^{pl})=\mathbf{D}_{0} \colon(\mathbf{\epsilon}-\mathbf{e}^{pl}), \tag{2}\]  
where \(\mathbf{\sigma}\) is the stress tensor, and \(d\) is the scalar stiffness degradation variable that can take values from 0 (undamaged state) to 1 (fully damaged state). \(\mathbf{D}_{0}^{el}\), \(\mathbf{D}_{0}\), \(\mathbf{\epsilon}\), and \(\mathbf{e}^{pl}\) are the initial elastic modulus, the degraded elastic stiffness, the total strain tensor, and the plastic strain tensor, respectively.  
For the boundary conditions, a displacement boundary condition is imposed on the top surface of the samples, and the opposite surface is set to be a fixed boundary condition. The required input parameters for the simulation, such as elastic modulus, yield compressive strength, and dilation angle, are selected from the analytical formulations in [38; 39; 42].  
To investigate heat transfer characteristics of the specimens, the required governing equation with considering the heat loss parameter can be modified and described as:  
\[\frac{\partial T}{\partial t}=\frac{k}{\rho C}\left(\frac{\partial^{2}T}{ \partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{ \partial z^{2}}\right)-\lambda\cdot T^{*}, \tag{3}\]  
where \(T\) [K], \(T^{*}\) [K], and \(t\) [s] are the temperature, the surrounding temperature, and the time, respectively. \(k\) [W/m/K] is the thermal conductivity, \(\rho\) [kg/m\({}^{3}\)] is the mass density, \(C\) [J/g/K] is the specific heat, and \(\lambda\) [1/s] is the heat loss coefficient. Equation (3) is integrated for the weak form of the governing equation. Then, the effective heat flux and the effective thermal conductivity can be calculated using Fourier law [40; 41].  
The input parameters, such as material thermal conductivity, density, and specific heat, are measured using the Hot Disk device [33]. Boundary conditions for the simulation are imposed on the top and bottom surfaces of the specimens as: a constant temperature (60 \({}^{\circ}\)C) on the top surface, heat loss on the opposite surface with the heat loss coefficient of 1.3 [1/s], and the surrounding temperature of 23 \({}^{\circ}\)C. Other remaining surfaces are assumed to have no heat loss.