\chapter{Investigating the dependence of the Anisotropy of Thermal Conductivity on Geometric properties}
\paragraph
In this chapter, series of targeted experiments are done to investigate the dependence of the anisotropy of thermal conductivity of porous ceramic materials on geometric properties of their microstructure. There are some important geometric parameters of microstructure that can affect the homogenized thermal conductivity. Porosity coefficient, pores orientation and also the density of pores are some of these important factors. Therefore, we are going to set up an experiment to study the rate of changes of thermal conductivity regarding these parameters.
\paragraph
To do so, a square will be used as an RVE with a rectangle in it. This rectangle which later will be called "cloud of porosity" has three specifications. First, it includes selected number of circles or ellipses which play the role of porosity in the experiments. Second, it will rotate 180 degree through each experiment. This rotation will help to investigate role of orientation of the porosity structure. Third the pores will be distributed randomly within the rectangle.
\paragraph
In order to investigate the effect of porosity coefficient, we will change the parameters such as geometry of pores which could be circle or ellipse, size and number of them and also dimensions of cloud of porosity . We can run several experiments with different porosity coefficients and show the results on the graphs. Also, for checking the role of orientation, we are going to rotate the cloud of porosity and find the related thermal conductivity in every selected angle. In this way we could depict the variations of the principle geometric axes with respect to the orthotropic axes and see the correlation between them. %It should be mentioned that all of these investigation could be done in one experiment. In other words after selecting the parameters which will affect the porosity coefficient, rotation of cloud of porosity will be done automatically.
\paragraph
Before starting to explain the experiments, it will be helpful to define what the principal geometric and anisotropy axes are.
%\subsubsection {Moment of Inertia for a rigid solid body}
%\begin{figure}[htb]
%\begin{center}
%\begin{tabular}[1]{c}
%\includegraphics[width=80mm]{moment.eps} \\
%\end{tabular}
%\caption{Rigid body in space with mass $m$ and differential volume $dv$}
%\label{moment}
%\end{center}
%\end{figure}
%\paragraph
%For a rigid body with mass $m$, density $\rho$, and volume $v$, as shown in Fig \ref{moment} the moments of inertia are defined as follow:
%\begin{equation}
%$\label{principal1}
%$I_{xx}= \int_v \rho (y^2 + z^2)dv,
%\end{equation}
%\begin{equation}
%\label{principal2}
%I_{yy}= \int_v \rho (z^2 + x^2)dv,
%\end{equation}
%\begin{equation}
%\label{principal3}
%I_{zz}= \int_v \rho (x^2 + y^2)dv,
%\end{equation}
%and the product of inertia:
%\begin{equation}
%\label{prinsi}
%I_{xy} = I_{yx} = \int_v (\rho xy)dv,
%\end{equation}
%\begin{equation}
%\label{prinsi1}
%I_{yz} = I_{zy} = \int_v (\rho zy)dv,
%\end{equation}
%\begin{equation}
%\label{prinsi2}
%I_{xz} = I_{xz} = \int_v (\rho xz)dv,
%\end{equation}
%\paragraph
%Now we consider the changes of moments and product of inertia due to the rotation of the coordinate axes. As shown in Fig (\ref{principle}), the origin of the coordinate axes is located at the fixed point O. In general the origin O is not the mass center C of the rigid body. From the definitions of the moments of inertia given in Equation \ref{principal1} and \ref{prinsi1}, it results that the moments of inertia cannot be negative. Furthermore,
%\begin{equation}
%\label{sum}
%I_{xx} + I_{yy} + I_{zz} = 2 \int_v \rho^2 dv
%\end{equation}
%\begin{equation}
%\label{sum2}
%r^2 = x^2 + y^2 + z^2
%\end{equation}
%\paragraph
%\begin{figure}[htb]
%\begin{center}
%\begin{tabular}[1]{c}
%\includegraphics[width=80mm]{inertia.eps} \\
%\end{tabular}
%\caption{Rotation of coordinate axes}
%\label{principle}
%\end{center}
%\end{figure}
%\paragraph
%The distance $r$ corresponding to any mass element $rdv$ of the rigid body does not change with a rotation of axes from $xyz$ to $x'y'z'$ (Fig \ref{principle}). Therefore the sum of the moments of inertia is invariant with respect to a coordinate system rotation. In terms of matrix notation, the sum of the moments of inertia is just the sum of the elements on the principal diagonal of the inertia matrix and is known as the trace of that matrix. So the trace of the inertia matrix is unchanged by a coordinate rotation, because the trace of any square matrix is invariant under an orthogonal transformation.
%\paragraph
%Next the products of inertia are considered. A coordinate rotation of axes can result in a change in the signs of the products of inertia. A 180 degree rotation about the x axis, for example, reverses the signs of $Ixy$ and $Ixz$, while the sign of $Iyz$ is unchanged. This occurs because the directions of the positive y and z axes are reversed. On the other hand, a 90 degree rotation about the x axis reverses the sign of $Iyz$. It can be seen that the moments and products of inertia vary smoothly with changes in the orientation of the coordinate system because the direction cosines vary smoothly. Therefore an orientation can always be found for which a given product of inertia is zero.
%\paragraph
%It is always possible to find an orientation of the coordinate system relative to a given rigid body such that all products of inertia are zero simultaneously, that is, the inertia matrix is diagonal. The three mutually orthogonal coordinate axes are known as principal axes in this case, and the corresponding moments of inertia are the principal moments of inertia.The three planes formed by the principal axes are called principal planes.
\paragraph
In this research we are considering two dimensional problems, so matrix notation for moment of inertia will be:
$\left[
\begin{array}{cc}
Ixx& Ixy \\
Ixy & Iyy \\
\end{array}
\right]$
and we could find the position of principal axes of geometry, using this equation:
\begin{equation}
\label{principal axes}
\alpha_ P = \frac{\arctan\frac {2\times I_{xy}}{I_{yy}-I_{xx}}}{2}
\end{equation}
Axes of anisotropy for the thermal conductivity will be defined in the same way, using thermal conductivity tensor components:
\begin{equation}
\label{orthotropy axes}
\alpha_ O = \frac {\arctan \frac {2\times k_{12h}}{k_{11h}-k_{22h}}}{2}
\end{equation}
As it was said before, we could investigate the affect of orientation of pores to the anisotropy of homogenized thermal conductivity, by comparing these two angles.
\paragraph
\section{Experiment A:}
\paragraph
In this experiment the geometry of the pores is selected to be ellipses with these dimensions: a = 0.04, b = 0.02. Also the dimension of the cloud of porosity is fixed to $(0.8\times0.15)$. Keeping these parameters untouched, we run the code for three different number of ellipses, 10, 30, 50 separately and plot the results as following. The porosity coefficient for these three case is: 0.006, 0.018, 0.028.
\begin{figure}[h!]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{cloud1-10.eps} &
\includegraphics[width=30mm]{cloud1-30.eps} &
\includegraphics[width=30mm]{cloud1-50.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Cloud of porosity for experiment A, geometry of pores are ellipses (a = 0.04, b = 0.02) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{cloud}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=100mm]{fig1-1.eps} \\
\end{tabular}
\caption{$k_{11}$ for Experiment A, geometry of pores are ellipses (a = 0.04, b = 0.02) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{result1.1.a}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=100mm]{fig1-4.eps} \\
\end{tabular}
\caption{$k_{22}$ for Experiment A, geometry of pores are ellipses (a = 0.04, b = 0.02) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{result1.1.b}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=100mm]{fig1-2.eps} \\
\end{tabular}
\caption{$k_{12}$ for Experiment A, geometry of pores are ellipses (a = 0.04, b = 0.02) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{result1.1.c}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=100mm]{fig1-3.eps} \\
\end{tabular}
\caption{correlation between geometric and orthotropic axes for Experiment A, geometry of pores are ellipses (a = 0.04, b = 0.02) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{result1.1.d}
\end{center}
\end{figure}
\subsection{Discussion}
\paragraph
Figures \ref{result1.1.a} and \ref{result1.1.b} show that thermal conductivity will decrease when porosity coefficient increases. Higher porosity coefficient means smaller amount of material which will result in smaller amount of thermal conductivity.
Also, as it can be seen, the changes for $k_{11}$ and $k_{22}$ are inverse which show the fact that by changing the porosity orientation, when the thermal conductivity increases in x direction, it will decrease in y direction and vice versa.
\paragraph
In addition, these figures show higher ranges of thermal conductivity for higher porosities. For example in Fig \ref{result1.1.c}, ranges of $k_{12}$ is between 0.0006 and 0.0032 when number of pores are 50 (porosity coefficient = 0.028), but it is between 0.00048 and 0.0013 when number of pores are 30 (porosity coefficient = 0.018). Same thing can be seen in Fig \ref{result1.1.b}, ranges of $k_{22}$ is between 0.96 and 0.97 for 50 number of pores while it is between 0.9755 and 0.98 for 30 number of pores. It can be interpreted that when we have larger porosity coefficients, we should expect larger changes in amount of thermal conductivity regarding to the changes of the orientation of the pores.
\paragraph
Fig \ref{result1.1.c} shows that regardless of changes of orientation, the off diagonal elements of thermal conductivity tensor ($k_{12}$), is very close to zero for materials having smaller amount of porosity or those which can be considered as homogeneous materials. With increasing the porosity coefficient, $k_{12}$ will increase and show more variation regarding to orientation changes as well. With a closer look to this graph, we can find that, the intersections of graphs are not exactly coinciding. It can be related to the random distribution of the pores which results in tiny differences in orientation of the porosity. [Please see Fig. \ref{cloud}].
\paragraph
In Fig \ref{result1.1.d}, we could track changing of the anisotropy of thermal conductivity with respect to orientation of the pores (principal geometric axes). It can be seen that, there is a strong correlation between them and only small deviation can be found when number of ellipses are equal to 10.
\paragraph
{\bf In the following, rest of the experiments are presented. In general, the trends in all the graphs are almost the same, but in case of observing any special case, it will be remarked.}
\paragraph
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experiment B}
\paragraph
In the next two experiments, we are going to investigate role of porosity coefficient. So we run the same test as experiment A, but only change dimensions of ellipses: a = 0.1, b = 0.05. Keeping these parameters untouched, we run the code for three different number of circles, 10, 30, 50 separately. The porosity coefficient in these three case is: 0.037, 0.082, 0.106. Results are plotted in Fig \ref{result2.1}.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{e2-10.eps} &
\includegraphics[width=30mm]{e2-30.eps} &
\includegraphics[width=30mm]{e2-50.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Cloud of porosity for experiment B, geometry of pores are ellipses (a = 0.1, b = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{cloudb}
\end{center}
\end{figure}
\paragraph
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{e2-1.eps} &
\includegraphics[width=80mm]{e2-2.eps} \\
(a) & (b)\\
\includegraphics[width=80mm]{e2-3.eps} &
\includegraphics[width=80mm]{e2-4.eps} \\
(c)& (d)\\
\end{tabular}
\caption{Results for Experiment B, geometry of pores are ellipses (a = 0.1, b = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{result2.1}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experiment C}
\paragraph
This experiment is the same as previous one, only we select bigger ellipses. Dimension of ellipses in this test is selected to be: a = 0.2, b = 0.1. Also the dimension of the cloud of porosity is fixed to $(0.8\times0.15)$. Keeping these parameters untouched, we run the code for three different number of circles, 10, 30, 50 separately. The porosity coefficient for these cases are: 0.11, 0.168, 0.185. Results are plotted in Fig \ref{result3.1}.\\
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{e3-10.eps} &
\includegraphics[width=30mm]{e3-30.eps} &
\includegraphics[width=30mm]{e3-50.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Cloud of porosity for experiment C, geometry of pores are ellipses (a = 0.2, b = 0.1) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{cloudc}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{e3-1.eps} &
\includegraphics[width=80mm]{e3-2.eps} \\
(a) & (b)\\
\includegraphics[width=80mm]{e3-3.eps}&
\includegraphics[width=80mm]{e3-4.eps} \\
(c) & (d)\\
\end{tabular}
\caption{Results for Experiment C, geometry of pores are ellipses (a = 0.2, b = 0.1) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{result3.1}
\end{center}
\end{figure}
{\bf Remark: The same trend as was explained in the observation part is seen in graph of these experiments.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experiment D}
\paragraph
The goal of next two experiment is to investigate the effect of density of the distribution of the pores. We select the same parameters as experiment B, but using bigger cloud of porosity in order to decrease density of pores distribution. So, the geometry of the pores is selected to be ellipse with the these dimensions: a = 0.1, b = 0.05. But dimension of cloud of porosity is changed to $(0.8\times0.4)$. Keeping these parameters untouched, we run the code for three different number of circles, 10, 30, 50 separately. In this case the porosity coefficients are: 0.038, 0.101, 0.147. Result of this experiment can be seen in Fig \ref{result4.1}.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{c4-10.eps} &
\includegraphics[width=30mm]{c4-30.eps} &
\includegraphics[width=30mm]{c4-50.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Cloud of porosity for experiment D, geometry of pores are ellipses (a = 0.1, b = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.4)$}
\label{cloudd}
\end{center}
\end{figure}
\paragraph
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{e4-1.eps} &
\includegraphics[width=80mm]{e4-2.eps} \\
(a) & (b)\\
\includegraphics[width=80mm]{e4-3.eps} &
\includegraphics[width=80mm]{e4-4.eps} \\
(c) & (d)\\
\end{tabular}
\caption{Results for Experiment D, geometry of pores are ellipses (a = 0.1, b = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.4)$}
\label{result4.1}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experiment E}
\paragraph
In this experiment, we continue the same trend as previous one and only change the dimension of the cloud of porosity to $(0.8\times0.6)$. Keeping other parameters untouched, we run the code for three different number of circles, 10, 30, 50 separately. The porosity coefficient in these cases are: 0.039, 0.107, 0.161 and results are plotted in Fig \ref{result5.1}.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{e5-10.eps} &
\includegraphics[width=30mm]{e5-30.eps} &
\includegraphics[width=30mm]{e5-50.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Cloud of porosity for experiment E, geometry of pores are ellipses (a = 0.1, b = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.6)$}
\label{cloude}
\end{center}
\end{figure}
\paragraph
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{e5-1.eps} &
\includegraphics[width=80mm]{e5-2.eps} \\
(a) & (b)\\
\includegraphics[width=80mm]{e5-3.eps} &
\includegraphics[width=80mm]{e5-4.eps} \\
(a) & (b)\\
\end{tabular}
\caption{Results for Experiment E, geometry of pores are ellipses (a = 0.1, b = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.6)$}
\label{result5.1}
\end{center}
\end{figure}
\section{Experiment F}
\paragraph
In the next five experiments, all the previous experiments will be tested on the circles as the geometry of the pores. This is done in order to investigate the effect of direction of the pores on the material behaviour in terms of thermal conductivity. (Ellipse is a kind of directional geometry while circles do not have direction).
\paragraph
Like Experiment A, B and C, cloud of porosity dimension is $(0.8\times0.15)$. Pores are circles with the radius of: R = 0.02. Keeping these parameters untouched, we run the code for three different number of circles, 10, 30, 50 separately. The porosity coefficient for these cases are: 0.004, 0.01, 0.015. The results can be found in Fig\ref{circle1.2}\\
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{c1-10.eps} &
\includegraphics[width=30mm]{c1-30.eps} &
\includegraphics[width=30mm]{c1-50.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Cloud of porosity for experiment F, geometry of pores are circles (R = 0.02) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{cloudf}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{cx2-1.eps} &
\includegraphics[width=80mm]{cx2-2.eps} \\
(a) & (b)\\
\includegraphics[width=80mm]{cx2-3.eps} &
\includegraphics[width=80mm]{cx2-4.eps} \\
(c) & (d)\\
\end{tabular}
\caption{Results for Experiment F, geometry of pores are circles (R = 0.02) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{circle1.2}
\end{center}
\end{figure}
According to Fig \ref{circle1.2}, almost the same trend as previous ones is seen. Range of $k_{11}$ and $k_{22}$ for circular pores are less than ellipsoidal ones. Also there are more perturbation in last graph for the smallest porosity coefficient.
\section{Experiment G}
\paragraph
In this experiment, the radius of the pores is selected to be R = 0.05 in order to increase the porosity coefficient. The dimension of the cloud of porosity is unchanged: $(0.8\times0.15)$, similar to condition which was defined in experiment F. We run the code for three different number of circles, 10, 30, 50 separately. The porosity coefficients for these cases are: 0.019, 0.05, 0.071. Results can be seen in Fig \ref{circle2.2}.\\
\begin{figure}[h!]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{c2_10.eps} &
\includegraphics[width=30mm]{c2_30.eps} &
\includegraphics[width=30mm]{c2_50.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Cloud of porosity for experiment G, geometry of pores are circles (R = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{cloudg}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{c2-1.eps} &
\includegraphics[width=80mm]{c2-2.eps} \\
(a) & (b)\\
\includegraphics[width=80mm]{c2-3.eps} &
\includegraphics[width=80mm]{c2-4.eps} \\
(c) & (d)\\
\end{tabular}
\caption{Results for Experiment G, geometry of pores are circles (R = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{circle2.2}
\end{center}
\end{figure}
{\bf Remark: There are very strong correlation between the assigned parameters and the graphs are smooth for both cases.}
\section{Experiment H}
\paragraph
In this experiment, larger circles are selected: R = 0.1. The dimension of the cloud of porosity is unchanged: $(0.8\times0.15)$. The porosity coefficient in these cases increases to: 0.068, 0.129, 0.154. The results of this part can be found in Fig \ref{circle3.2}.
\begin{figure}[h!]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{c3-10.eps} &
\includegraphics[width=30mm]{c3-30.eps} &
\includegraphics[width=30mm]{c3-50.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Cloud of porosity for experiment H, geometry of pores are circles (R = 0.1) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{cloudh}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{c3-1.eps} &
\includegraphics[width=80mm]{c3-2.eps} \\
(a) & (b)\\
\includegraphics[width=80mm]{c3-3.eps} &
\includegraphics[width=80mm]{c3-4.eps} \\
(c) & (d)\\
\end{tabular}
\caption{Results for experiment H, geometry of pores are circles (R = 0.1) and cloud of porosity dimension is: $(0.8 \times 0.15)$}
\label{circle3.2}
\end{center}
\end{figure}
\section{Experiment I}
\paragraph
In this experiment and the next one, the radius of pores is kept untouched, but we changed the dimension of cloud of porosity in order to check the effect of changing the density of distribution of the pores. Geometry of the pores is selected to be circle with radius of R = 0.05. Also the dimension of the cloud of porosity is selected to be: $(0.8\times0.4)$. We run the code for three different number of circles, 10, 30, 50 separately. The porosity coefficients for these cases are: 0.019, 0.055, 0.085. The results are plotted in Fig \ref{circle4.2}.\\
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{ci4-10.eps} &
\includegraphics[width=30mm]{ci4-30.eps} &
\includegraphics[width=30mm]{ci4-50.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Cloud of porosity for experiment I, geometry of pores are circles (R = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.4)$}
\label{cloudi}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{c4-1.eps} &
\includegraphics[width=80mm]{c4-2.eps} \\
(a) & (b)\\
\includegraphics[width=80mm]{c4-3.eps} &
\includegraphics[width=80mm]{c4-4.eps} \\
(c) & (d)\\
\end{tabular}
\caption{Results for Experiment I,geometry of pores are circles (R = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.4)$}
\label{circle4.2}
\end{center}
\end{figure}
{\bf Remark: As it is seen, lines which represent changes of the smallest porosity coefficient have more erratic changes.}
\section{Experiment J}
\paragraph
As the last experiment, the dimension of the cloud of porosity is changed to $(0.8\times0.6 )$. The geometry of the pores are circles with radius of R = 0.05. Keeping these parameters untouched, we run the code for three different number of circles, 10, 30, 50 separately. The porosity coefficient for these cases are: 0.02, 0.056, 0.089. Results can be found in Fig \ref{circle5.2}.\\
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{ci5-10.eps} &
\includegraphics[width=30mm]{ci5-30.eps} &
\includegraphics[width=30mm]{ci5-50.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Cloud of porosity for experiment J, geometry of pores are circles (R = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.6)$}
\label{cloudj}
\end{center}
\end{figure}
\paragraph
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{c5-1.eps} &
\includegraphics[width=80mm]{c5-2.eps} \\
(a) & (b)\\
\includegraphics[width=80mm]{c5-3.eps} &
\includegraphics[width=80mm]{c5-4.eps} \\
(c) & (d)\\
\end{tabular}
\caption{Results for Experiment J, geometry of pores are circles (R = 0.05) and cloud of porosity dimension is: $(0.8 \times 0.6)$}
\label{circle5.2}
\end{center}
\end{figure}
{\bf Remark: There are more erratic changes in these figures. It can be due to the large cloud of porosity which is defined for this experiment. Changing the orientation of this cloud of porosity can produce this chaos, it means that, when pores are more disperse, rotating of the RVE will result in larger differences in thermal conductivity in each direction. Therefore it can be said that not only the porosity coefficient can affect the material behavior, but also the pores dispersion can be effective.}
\section{Conclusion of the Experiments}
\paragraph
In experiments A, B and C, we gradually increase the porosity coefficient by increasing size of the pores and fixing the distribution density by keeping untouched the cloud of porosity dimension. We can see in Fig \ref{com1} that by increasing the porosity coefficient we will have smaller amount of thermal conductivity in every single angle of principal geometric axes. Meanwhile the off diagonal elements of thermal conductivity $k_{12}$, has both increasing and decreasing trends, depend on the angle of principal geometric axes. It means that the trends depends on the orientation of the porosity. The same trend can be seen for experiment G, H and I, when the geometry of pores is circle.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{fig1-1.eps} &
\includegraphics[width=80mm]{fig1-2.eps} \\
(a) & (b) \\
\includegraphics[width=80mm]{e2-1.eps} &
\includegraphics[width=80mm]{e2-3.eps} \\
(c) & (d) \\
\includegraphics[width=80mm]{e3-1.eps} &
\includegraphics[width=80mm]{e3-3.eps} \\
(e) & (f)\\
\end{tabular}
\caption{Comparison of the results by increasing the porosity coefficient, (a)and (b): $k_{11}$ and $k_{12}$ for porosity coefficients: 0.006, 0.018, 0.028. (c) and (d): $k_{11}$ and $k_{12}$ for porosity coefficients: 0.037,0.082,0.106. (e) and (f): $k_{11}$ and $k_{12}$ for porosity coefficients: 0.11, 0.168, 0.185.}
\label{com1}
\end{center}
\end{figure}
\paragraph
Also, for geometry of pores, we selected ellipses and circles in order to consider effect of pores shape on the material behavior. For clear investigation of this issue, we select results for the same porosity coefficient and also the same cloud of porosity dimension and plotted the results for both geometries in one graph. Fig \ref{com2} shows differences very well. We can see that with the same porosity coefficient (0.018) elements of thermal conductivity tensor for both shape are almost the same. It shows that shape of porosity can affect the homogenized thermal conductivity but the effect is not significant.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{com1_1.eps} &
\includegraphics[width=80mm]{com1_2.eps} \\
(a) & (b) \\
\includegraphics[width=80mm]{com1_3.eps} &
\includegraphics[width=80mm]{com1_4.eps} \\
(c) & (d) \\
\end{tabular}
\caption{Comparison of the results of two different pore geometry}
\label{com2}
\end{center}
\end{figure}
\paragraph
Dispersion of the pores is another parameter which was investigated in this chapter. In order to have clear conclusion about this parameter, we choose the results of experiment B, D and F for 30 numbers of ellipses and plotted them in Fig \ref{com3}. In these experiments geometry and dimension of the pores are selected to be the same, ellipses (a = 0.1 and b = 0.05), but we gradually decrease pores distribution density by increasing the dimension of cloud of porosity in this way: ($0.8\times0.15$), ($0.8\times0.4$), ($0.8\times0.6$).
\\ According to Fig. \ref{com3}, when pores are more disperse, range of changes in element of thermal conductivity tensor regard to the angle of anisotropy are smaller. In this case, material microstructure behaviour goes toward more homogeneity, therefore, it can be less affected by changes of anisotropy. Also according to Fig \ref{com3}, by decreasing the pores distribution density, no constant increasing or decreasing trend can be seen in the amount of thermal conductivity tensor elements and it depends on the orientation of the pores.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}[2]{cc}
\includegraphics[width=80mm]{com3-1.eps} &
\includegraphics[width=80mm]{com3-2.eps} \\
(a) & (b)\\
\includegraphics[width=80mm]{com3-3.eps} &
\includegraphics[width=80mm]{com3-4.eps} \\
(c) & (d)
\end{tabular}
\caption{Comparison of the results for different pores distribution }
\label{com3}
\end{center}
\end{figure}