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\begin{document}
\title{Modulus of Rupture of Ceramics and Bending of Sandwich Structures}
\author{Johan Rebby John}
\maketitle
\begin{abstract}
In this experiment we conducted bending tests on several different specimens of Aluminum as well as Ceramics. Using the data gathered from these tests as well as measurements we took of their primary dimensions, we calculated (for each specimen) modulus of rupture, flexure strain, Young's modulus, as well as specific strength and stiffness. These tests gave us insight into new characteristics of aluminum and ceramics that allowed us to better understand their applications in industry.
\end{abstract}
\pagebreak
\section{Introduction}
Our objective in this lab was to determine the Poisson’s ratio and Young’s Modulus for different specimens of Aluminum. These included high purity aluminum, aluminum honeycombed, and solid aluminum. To do this, we used raw data gathered from bending tests we conducted of these specimens with two different tensile testing machines. Using a micrometer, we also measured the dimensions of each of the specimens. The equations we used to calculate Modulus of Rupture, flexure strain, and the young’s modulus are all shown below. The MOR (modulus of rupture) is calculated using:
\begin{equation}
\sigma_(fb) = \frac{3LP_f}{2tw.^2}
\end{equation}
For flexure stain:
\begin{equation}
\epsilon_f = \frac{6wv}{L.^2}
\end{equation}
And for calculating the young’s modulus:
\begin{equation}
E_B = \frac{L.^3 m}{4tw.^3}
\end{equation}
\begin{equation}
Stiffness(Specific) = \frac{E_B}{Density}
\end{equation}
\begin{equation}
Strength(Specific) = \frac{P_f}{Density}
\end{equation}
In these equations, $P_f$ is the load at the fracture while $L$, $t$ and $w$ are the length, thickness, and width of each specimen respectively.
\section{Materials and Procedure}
We first measured the primary dimensions of each of the specimens using a micrometer A picture of the specimens are shown below.
\begin{figure}[H]
\centering
\title{Figure 1: Small Specimens}
\includegraphics[width=0.9\textwidth]{SmallSpec}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 2: Big Specimens}
\includegraphics[width=0.9\textwidth]{BigSpec}
\end{figure}
We then conducted bending tests using the INSTRON 5500R and 4500R. As more and more force was applied, we could visibly see the samples form a V-shaped bend in the middle.
\begin{figure}[H]
\centering
\title{Figure 3: Bending Test (Before)}
\includegraphics[width=0.9\textwidth, angle = 270]{Unbroken}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 4: Bending Test (After)}
\includegraphics[width=0.9\textwidth]{broken}
\end{figure}
\section{Results}
Our measurements of the primary dimensions using a micrometer is shown in the table below.
\begin{figure}[H]
\centering
\title{Figure 5: Measurements of Primary Dimensions}
\includegraphics[width=0.9\textwidth]{Table}
\end{figure}
\begin{figure}
\centering
\title{Figure 6: Plot of Aluminum (Big L): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{AlBigL}
\end{figure}
We plotted Flexure load vs Flexure displacement (extension) for each of the specimens using Microsoft Excel as shown below.
\begin{figure}[H]
\centering
\title{Figure 7: Plot of Aluminum (Big L): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{AlBigL}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 8: Plot of Aluminum (Big T): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{AlBigT}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 9: Plot of Aluminum (Small L): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{AlSmallL}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 10: Plot of Aluminum (Small T): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{AlSmallT}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 11: Plot of Ceramic (1-201151): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{Ceramic1}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 12: Plot of Ceramic (2-201151): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{Ceramic2}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 13: Plot of Ceramic (3-201151): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{Ceramic3}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 14: Plot of Ceramic (4-201151): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{Ceramic4}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 15: Plot of Ceramic (5-201151): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{Ceramic5}
\end{figure}
Our calculated values for MOR, Flexure Strain, and Young's Modulus is summarized in the table below. In addition to these results, we also have calculated specific strength and stiffness data for aluminum.
\begin{figure}[H]
\centering
\title{Figure 16: MOR, Flexure Strain, and Young's Modulus Values}
\includegraphics[width=0.9\textwidth]{Table2}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 17: Specific Strength and Specific and Stiffness Values}
\includegraphics[width=0.9\textwidth]{Table3}
\end{figure}
The M-values are the slopes of the elastic regions of the flexure load versus flexure strain plots. These were determined by finding trendlines of elastic regions of these plots as shown below.
\begin{figure}[H]
\centering
\title{Figure 18: Linear Fit of Aluminum (Big L): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{ALBigLfit}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 19: Linear Fit of Aluminum (Big T): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{AlBigTFit}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 20: Linear Fit of Aluminum (Small L): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{AlSmallLfit}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 21: Linear Fit of Aluminum (Small T): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{AlSmallTfit}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 22: Linear Fit of Ceramic (1-201151): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{Ceramic1fit}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 23: Linear Fit of Ceramic (2-201151): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{Ceramic2fit}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 24: Linear Fit of Ceramic (3-201151): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{Ceramic3fit}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 25: Linear Fit of Ceramic (4-201151): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{Ceramic4fit}
\end{figure}
\begin{figure}[H]
\centering
\title{Figure 26: Linear Fit of Ceramic (5-201151): Change in Flexure Load Over Flexure Extension}
\includegraphics[width=0.9\textwidth]{Ceramic5fit}
\end{figure}
\section{Discussion}
Our data overall looked very solid.Though not very obvious, there is most probably a bit of error in our results. There are many reasons why our data contains the error it does. Human error can always be a culprit either in the form of random or systematic error. In addition to this, our bending tests never reached visible fracture for any of our specimens despite they all had a V-shape. However, this is not as issue since, based on the shapes of the plots of load vs displacement, we can clearly say that we had practically reached the fracture point. The materials we tested had distinct similarities and differences. The aluminum specimens all had similar graph curves as did all the ceramic ones. In addition to this, the aluminum specimens had a defined yield strength and thus a clear elastic and plastic region. Whereas, the ceramic specimens only had elastic deformation as they clearly only had linear slopes for the flexure load versus flexure strain plots. Another interesting thing to note is that there was a huge variety in the values calculated for Aluminum (MOR, Modulus, Specific strength and stiffnes, etc...) while those for the Ceramic specimens were all reasonably within the same range of values. This may or may not be due to the inherent properties of aluminum versus the ceramics specimens.
\section{Conclusion}
This lab was very useful as it revealed that Ceramics materials seemed to share more similar characteristics (in terms of the closeness of the values we calculated for the different Ceramic specimens) as opposed to Aluminum specimens. Such information could be very useful for industry as it could point to how different materials can be used practically in conditions to maximize benefits they have. For example, from this lab, we could potentially conclude that more various ceramic specimens have consistent characteristics between them as opposed to Aluminum. Thus they could be used in situations that require stable materials. Aluminum, on the other hands, seems to have a plethora of characteristics for various conditions. Thus,. they could be used for all sorts of structures and applications that require flexibility. This is only one example and thus bending test could be very useful in determining other materials' ideal functions in industry.
\end{document}
