## 4 Properties Evaluation Using Experiments and Numerical Methods
### Numerical Simulation for Thermal and Mechanical Analysis  
The material properties of the foamed samples are evaluated using the 3D micro-CT images and FE simulations. The ABAQUS package [39] is used for the simulations, and detailed descriptions of the FE formulations for both thermal and mechanical analysis are in [40; 41]; only a brief description is presented in this paper. The modified governing equation for 3D heat flow analysis by considering heat loss is described as follows:  
\[\frac{\partial T}{\partial t}=\frac{k}{\rho C}\left(\frac{\partial^{2}T}{ \partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{ \partial z^{2}}\right)-\lambda\cdot T^{*} \tag{4}\]  
where \(T\) (K) is the temperature, \(T^{*}\) (K) is the surrounding temperature and \(t\) (s) is the time. \(k\) (W/m/K) is the thermal conductivity; \(\rho\) (kg/m\({}^{3}\)) is the mass density; \(C\) (J/g/K) is the specific heat; \(\lambda\) (1/s) is the heat loss coefficient. A weak form of the FE simulation can be obtained by integrating the governing differential equation in Equation (4). The effective heat flux can be calculated by averaging the elemental heat flux, and the effective thermal conductivity is computed from the Fourier law as:  
\[k_{(n)}=q_{(n)}L/\bigtriangleup T_{(n)} \tag{5}\]  
where \(\bigtriangleup T_{(n)}\), \(q_{(n)}\) and \(L\) denote the temperature difference along the heat flow direction, the effective heat flux of the sample and the characteristic length, respectively. The directional component is denoted as \(n\). The mandatory input parameters for the heat transfer analysis, such as the bulk (solid) thermal conductivity and specific heat of the foamed materials, are obtained from the experiments using the Hot Disk machine [27] and the analytical formulation in [2]; the required parameters are given in Table 2. For the boundary conditions, a constant temperature (60 \({}^{\circ}\)C) is applied to the top surface in each vertical direction (\(x,y\) and \(z\)). Heat loss is allowed only in the bottom surface, while other remaining surfaces are assumed to have no heat loss. Here, the heat loss coefficient (\(\lambda\)) and the surrounding temperature of the sample are set as 1.6 (1/s) and 22 \({}^{\circ}\)C, respectively. The thermal conductivities for all samples are computed in the \(x,y\) and \(z\) directions. The compressive strengths of the foamed samples are also numerically computed. The ABAQUS package including the concrete damage plasticity (CDP) model is utilized to evaluate the behavior of the foamed samples with different densities and pore distributions. Details of the FE formulations are \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Specimens** & **LithoPore®100** & **LithoPore®200** & **LithoPore®450** & **LithoPore®600** \\ \hline
**Thermal conductivity (W/m/K)** & 0.05 & 0.08 & 0.14 & 0.24 \\
**Density (kg/m\({}^{3}\))** & 100 & 200 & 450 & 600 \\
**Specific heat (J/g/K)** & 435 & 380 & 250 & 178 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The input parameters for heat analysis. Figure 8: Hot Disk sensor and its operation process for evaluating thermal properties. presented in [42], and only a brief description is shown in this paper. The governing equation for the stress-strain relation of the CDP model, controlled by a scalar damaged elasticity, is given as follows:  
\[\sigma=(1-d)\mathbf{D}_{0}^{el}\colon(\mathbf{\epsilon}-\mathbf{e}^{pl})=\mathbf{D}_{0}\colon( \mathbf{\epsilon}-\mathbf{e}^{pl}) \tag{6}\]  
where \(\sigma\) is the stress tensor and \(d\) is the scalar stiffness degradation variable, which can take values from zero (undamaged material) to one (fully damaged material). \(\mathbf{D}_{0}^{el}\), \(\mathbf{D}_{0}\), \(\mathbf{\epsilon}\) and \(\mathbf{e}^{pl}\) denote the initial elastic modulus, the degraded elastic stiffness, the total strain tensor and the plastic strain tensor, respectively.