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@@ -1,358 +1,376 @@
\documentclass[a4paper, 12pt,oneside]{article}
%On peut changer "oneside" en "twoside" si on sait que le résultat sera recto-verso.
%Cela influence les marges (pas ici car elles sont identiques à droite et à gauche)
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% quelques symboles mathematiques en plus
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% a utiliser sur Linux/Windows
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% a utiliser avec UTF8
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%Très utiles pour les groupes mixtes mac/PC. Un fichier texte enregistré sous codage UTF-8 est lisible dans les deux environnement.
%Plus de problème de caractères accentués et spéciaux qui ne s'affichent pas
% a utiliser sur le Mac
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%Pour l'utilisation plus simple des unités et fractions
\usepackage{siunitx}
%Pour utiliser du time new roman... Comenter pour utiliser du ComputerModern
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%graphs
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%Début du document
\begin{document}
\title{\normalsize{Lab Work Report - Group N$^\circ$\\ XX - Experiment}}
\date{\normalsize{\today}}
\author{\normalsize{Name} 1\and \normalsize{Name 2}}
%Crée la page de titre
%\maketitle
%Ajoute la table des matières
%\tableofcontents
%Début du rapport à la page suivante
%\newpage
%De manière à ce que template latex ressemble au mieux au template word, on empêche latex de créer la page de titre et la créons à la main
%En taille de police 12, la commande \large donne une taille de police 14
%On utilise la commande \sffamily pour créer des caractères sans-serif
\begin{center}
\large\textbf{\sffamily Experiment N$^\circ$ D3: Plasticity of solids, tensile curve.}\\%
\large\sffamily Group N$^\circ$16: Ancarola Raffaele, Cincotti Armando\\%
\large\sffamily \today\qquad Sesé Sansa Elena\\%TODO cambiare nome assistente
\end{center}
% Introduction
\section{Introduction}
\section{About elasticity of metal alloys}\label{dislocations}
Appling a stress to a solid metal results in two different behaviours depending on the applied stress.
As the stress stays under a certain limit, called elastic limit, the metal change its shape in a reversible way.
As the stress passes this limit, the deformation doesn't stay reversible, as in the crystalline structure of the metal
appers different imperfections in the form of dislocations that modifies the shape of the crystalline structure (see figure \ref{}).
In order extend the elastic lymit of a metal, one can use an alloy of that metal with another
one, in order to create precipitates (groups of foreign molecules) in the alloy. In fact, these precipitates helps
increasing the elastic limit as they block dislocations in the metal, diminishing the number of new dislocation
formed by the stress. As in fluid solutions, precipitates appears as a saturation of the solute, depending on the temperature
of the solvent. The graph in Figure \ref{} shows the phase diagram of a metal alloy Al-Si0.7$\%$ showing that the solution is
solid between $620^\circ$C and $490^\circ$, causing the molecules of Silicium to randomly substitute Aluminium molecules.
Precipitates begin to appear on temperatures under $490^\circ$C. \cite{ref1}
%TODO inserire grafico fase.png
\section{Descritption of the experimental procedure}
The point of the experiment is to measure interesting elastic and plastic
caracteristics of metallic samples of the alloy Al-Si0.7$\%$
that passed trough different hardening and softening procedures.
\paragraph{Measurements.}
%TODO, inserire per questo paragrafo sensor.png
In order to take measurments while a sample get stressed at a costant speed, the given
experimental set up includes two sensors that measures the strenght $F$ [\si{\newton}]
applied to the sample and the stretch $\Delta L$ [\si{\metre}] (see Figure \ref{}). The measurements are taken as
electrical tensions $U_F$ [\si{\volt}] and $U_L$ [\si{\volt}], that can be converted, respectively, in strenght
knowing the coefficient of proportionnality $c_F = 0.002$ \si{\volt\per\newton}, and in lenght as
$c_L = 1$ \si{\volt\per\milli\metre} \cite{ref1}.
Finally, knowing the surface of the sample $S$ supposed constant through the manipulation, and its initial lenght $L_0$,
one can determine the applied stress $\sigma$ and the strain of the sample $\epsilon$ as given by the following relations:
\begin{equation}\label{sig}
\sigma = \frac{F}{S} = \frac{U_F}{c_FS}
\end{equation}
\begin{equation}\label{eps}
\epsilon = \frac{\Delta L}{L_0} = \frac{U_L}{c_LL_0}
\end{equation}
\paragraph{Tensile curve and interesting values for a metal alloy.}
%TODO inserire curve.png
The graph in Figure \ref{} shows the typical Tensile curve for a metal alloy
when a stress $\sigma$ is apllied to it. It can be observed that in a certain domain of stress, also called
elastic domain, the relation between $\sigma$ and $\epsilon$ stays linear. This relation is
called Hooke's law and described by equation (\ref{hook}) :
\begin{equation}\label{hook}
\sigma= E\epsilon
\end{equation}
where E is the Young's modulus of the material. In the elastic domain every deformation remains reversible.
As the stress passes a certain value it enters the plastic domain, where the deformation is no more reversible
and in the material appears as a reamaining strain $\epsilon_r \neq 0$ when $\sigma$ disappears.
The remanent strain $\epsilon_r$ for a given $\sigma_r$ in the plastic domain, fall on the intersection
between the $\epsilon$ axis and a line of slope $E$ passing through the point where the tensile curve
``touches'' $\sigma = \sigma_r$ line (see point II in Figure \ref{}).
The plastic domain ends at $\sigma = \sigma_{rup}$ as the maximal stress is reached.
After that it is possible that the metal continue his elongation before breaking,
and so the breaking strain $\epsilon_{rup}$ is re-defined as the remanent strain just before the metal break
(see Figure \ref{}).
As explained in the previous paragraph, tension and strain for a given sample can be collected as data
and displayed on a graph with a PC, in order to extract interesting information from it.
By a linear regression on the values in the elastic domain, it is possible to determine $E$ for the given sample
(see point I in Figure \ref{}).
Then, as it is really difficult to determine the limit for the elastic domain, using the measured $E$, $\sigma_{0.02}$
can be measured as the stress related to the strain $\epsilon_{0.02} = 0.02$. This is done in order to take $\sigma_{0.02}$
as limit for the elastic domain. To finish with, $\sigma_{rup}$ corresponds to the {\it maximum} of the curve
and the actual $\epsilon_{rup}$ can be extracted from data startig from the breaking point in the curve (last point of continuity)
with a line of slope E as done for $\sigma_{0.02}$.
{\bf Nota bene} The formulas used for the extraction of the interesting values are illustrated in the Annexes in section \ref{formula}.
\paragraph{Treating samples.}
In order to study how procedures of softening and hardening of a metal alloy modify its elastic and plastic
behaviours, three different kind of samples are studied:
\begin{itemize}
\item A non treated sample : stressed and studied as received.
\item A softened sample : baked in oven at $(550 \pm 5)^\circ$ C during one hour then thoughen up in water
\item A softened sample then hardened : once softened as for the previous sample, it is then annealed for a
given time $T$ at $(270 \pm 5)^\circ$ C and then thoughen up again.
\end{itemize}
\section{Results of the experiment and Discussions}
+\subsubsection{Analysing Tensile curves.}
+
+\begin{figure}[h]
+\hspace{-0.05\textwidth}
+\begin{subfigure}{0.49\textwidth}
+ \resizebox{\textwidth}{!}{
+ \input{graphs/cold.tex}
+ }
+ \caption{Elastic limit $\sigma_{0.02}$}
+ \label{elastic_lim}
+\end{subfigure}
+\hspace{0.02\textwidth}
+\begin{subfigure}{0.49\textwidth}
+ \resizebox{\textwidth}{!}{
+ \input{graphs/rup.tex}
+ }
+ \caption{Breaking stress $\sigma_{rup}$ and strain $\epsilon_{rup}$}
+ \label{break_lim}
+\end{subfigure}
+\caption{Determination of the limit points, taken for the non-treated sample}
+\label{limit_graph}
+\end{figure}
+
+In Figure \ref{elastic_lim} and \ref{break_lim} can be observed an exemple on how to determine the intersting values by analising the tensile
+curve of a studied sample. In Figure \ref{elastic_lim} can be observed
+two different slopes, the linear regression is done over the second slope, which is the most reasonable and also leads to the plastic domain.
+Once obtained $E$ by a linear fit on that slope, using that value one can determine $\sigma_{0.02}$ as illustrated in Figure \ref{elastic_lim}
+and $\epsilon_{rup}$ as illustrated in Figure \ref{break_lim}.
+
\begin{minipage}{\textwidth}
\hspace{-0.1\textwidth}
\begin{minipage}{0.5\textwidth}
As it can be observed in Figure \ref{strain_graph}, for each curve
it can be distinguished the {\it elastic} domain and the {\it plastic}
one.
Tempering the samples affects on both the elastic and plastic properties of the metal.
On Table \ref{restab} it can be observed that the {\it Young's modoulous}, which define
how a material resist to its own deformation, lower when the sample is softend during 1 hour in a
oven at $550^\circ$ C. Except for the sample hardenend during 15 minutes, the softening treatement caused all
the treated samples to have a lower their $E$, meaning they resist the less to their deformation. In fact the
softened sample has the lowest Young's modulous.
\end{minipage}
\hspace{0.02\textwidth}
\begin{minipage}{0.55\textwidth}
\resizebox{\textwidth}{!}{
\input{graphs/all.tex} % HERE
}
\captionof{figure}{Strain curve for every involved sample} % TODO
\label{strain_graph} % TODO
\end{minipage}
\end{minipage}
\begin{table}[H]
\centering
\begin{tabular}{c|c|c|c|c|}
\cline{2-5}
& $E$ [\si{\newton/\metre^2}] & $\sigma_{0.02}$ [\si{\newton/\metre^2}] & $\sigma_{rup}$ [\si{\newton/\metre^2}] & $\epsilon_{rup}$\\
\hline
\multicolumn{1}{|c|}{Non treated} & $(9.7 \pm 0.1) \cdot 10^9$ & $(2.6 \pm 0.1) \cdot 10^8$ & $(3.2 \pm 1.2) \cdot 10^8$ & $0.49 \pm 0.05$\\
\hline
\multicolumn{1}{|c|}{Softened} & $(2.7 \pm 0.1) \cdot 10^9$ & $(6.4 \pm 0.1) \cdot 10^7$ & $(1.7 \pm 0.6) \cdot 10^8$ & $0.40 \pm 0.06$\\
\hline
\multicolumn{1}{|c|}{Hardened 15 min} & $(1.3 \pm 0.1) \cdot 10^{10}$ & $(1.5 \pm 0.1) \cdot 10^8$ & $(2.0 \pm 0.7) \cdot 10^8$ & $0.36 \pm 0.03$\\
\hline
\multicolumn{1}{|c|}{Hardened 20 min} & $(5.7 \pm 0.1) \cdot 10^{9}$ & $(1.6 \pm 0.1) \cdot 10^8$ & $(2.1 \pm 0.8) \cdot 10^8$ & $0.41 \pm 0.04$ \\
\hline
\multicolumn{1}{|c|}{Hardened 25 min} & $(4.0 \pm 0.1) \cdot 10^{9}$ & $(1.8 \pm 0.1) \cdot 10^8$ & $(1.9 \pm 0.7) \cdot 10^8$ & $0.09 \pm 0.02$\\
\hline
\multicolumn{1}{|c|}{Hardened 30 min} & $(4.5 \pm 0.1) \cdot 10^{9}$ & $(1.5 \pm 0.1) \cdot 10^8$ & $(1.9 \pm 0.7) \cdot 10^8$ & $0.18 \pm 0.03$ \\
\hline
\end{tabular}
\caption{Interesting values measured for every sample}
\label{restab}
\end{table}
In the case of the the sample that was hardened during 15 minutes by looking at Figure \ref{}
it is difficult to say why its Young's modulous $E = (1.3 \pm 0.1) \cdot 10^{10}$
is that high. Another measurement for a sample treated the same way could have told us
if the hardening process really helped augmeting that much the Young's modulous of the sample.
\paragraph{About $\sigma_{0.02}$.}
As it appears on Table \ref{restab}, the elastic limit $\sigma_{0.02}$ decreases for the samples that were
treated for high temperature. In fact, the lower limit is observed in the softened sample (see also Figure \ref{strain_graph})
as it was heated at $(550 \pm 5)^\circ$ C and then thoughen up. This process of fast cooling blocks the state of the alloy
in the one he was at the previous temperature, as the metal don't have time to slowly cool by itself.
As explained in section \ref{dislocations}
-and shown on Figure \ref{}, at $(550 \pm 5)^\circ$ C the alloy doesn't present precipitates, resulting in a lower elastic limit.
+and shown on Figure \ref{}, at $(550 \pm 5)^\circ$ C the alloy doesn't have precipitates in it, resulting in a lower elastic limit.
Re-heating the sample also result in the apparition of new precipitates, in fact the hardened samples have a bigger limit then the softend one,
but lower then the non-treated one. This suggest that the sample we were given already passed through a hardening procedure, in order to get the
best elastic properties from it.
\paragraph{About $\sigma_{rup}$.}
As for the elastic limit, the hardening process affects also $\sigma_{rup}$ value. In fact an harder sample restits greater stresses, and the values
in Table \ref{restab} are quite consistent with this observation. This shows that more elastic metals also resists to greater stress, which is
trivial as $\sigma_{0.02} < \sigma_{rup}$. Although, it can be observed that $\sigma_{rup}$ for the softened sample approaches
the breaking limit of the other treated samples, and this can be explained by the fact that this sample presents a bigger plastic domain
as it is in fact softer than others samples.
\paragraph{About $\epsilon_{rup}$.}
-To finish with,
-
-\begin{figure}[h]
-\hspace{-0.05\textwidth}
-\begin{subfigure}{0.49\textwidth}
- \resizebox{\textwidth}{!}{
- \input{graphs/cold.tex}
- }
- \caption{Elastic limit $\sigma_{0.02}$}
- \label{elastic_lim}
-\end{subfigure}
-\hspace{0.02\textwidth}
-\begin{subfigure}{0.49\textwidth}
- \resizebox{\textwidth}{!}{
- \input{graphs/rup.tex} % HERE
- }
- \caption{Breaking stress $\sigma_{rup}$ and strain $\epsilon_{rup}$}
- \label{break_lim}
-\end{subfigure}
-\caption{Determination of the limit points, taken for the non-treated sample}
-\label{limit_graph}
-\end{figure}
-
-
-
+To finish with, $\epsilon_{rup}$ represents how much the sample have been elongated before breaking.
+By looking at these values on Table \ref{restab}, it is difficult to observe any tendency.
+Nonetheless, oddly enough it can be observed that the maximum strain is obtained
+for the non-treated sample which is also the hardest one. The softened sample should
+intuitively have the biggest $\epsilon_{rup}$, but in the studied case it broke just as it reached the maximum stress
+$\sigma_{rup}$ (see Figure \ref{}), in another measurement it could have continued its elogation before breaking. A strain so low is also due to the
+fact that its Young's modulous is the lowest.
+Here we conclude by saying that a statistic study over
+a lot more measurements could help observing a tendency in the results,
+and could help with more convincing observations.
+
+\subsection{Portevin-Le Chatelier effect}
\section{Conclusion}
\section{Annexes}
\subsection{Extraction of interesting values}\label{formula}
+%TODO mettici e commenta le formule e/o metodo per determinare i valori interessanti
+
\subsection{About the error}
\paragraph{Incertitude on distances}
All measured distances taken with the ruler have the incertitude
$\Delta x = 0.1$ \si{\centi\metre}.
\bigskip
\begin{itemize}
\item \textbf{Nota bene}: this incertitude is never shown on values written on tables because it's the same for all of them.
\end{itemize}
\section{Literature References}
% Bibliographie
\begin{thebibliography}{99}
\bibitem{ref1}
Notice des TP de physique D3 : Essais Traction.
\end{thebibliography}
\end{document}
diff --git a/labo/D3/tikzext/TemplateTPII-figure0.md5 b/labo/D3/tikzext/TemplateTPII-figure0.md5
index d927e5d..4150411 100644
--- a/labo/D3/tikzext/TemplateTPII-figure0.md5
+++ b/labo/D3/tikzext/TemplateTPII-figure0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {048EE0D8D16FA3467EFB5CA16F343EE2}%
+\def \tikzexternallastkey {F86773A3AD65328680B4CE686A8437C5}%
diff --git a/labo/D3/tikzext/TemplateTPII-figure1.md5 b/labo/D3/tikzext/TemplateTPII-figure1.md5
index 4150411..be8de0c 100644
--- a/labo/D3/tikzext/TemplateTPII-figure1.md5
+++ b/labo/D3/tikzext/TemplateTPII-figure1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {F86773A3AD65328680B4CE686A8437C5}%
+\def \tikzexternallastkey {B452EBE3383F9899FD957EE112A7AACF}%
diff --git a/labo/D3/tikzext/TemplateTPII-figure2.md5 b/labo/D3/tikzext/TemplateTPII-figure2.md5
index be8de0c..d927e5d 100644
--- a/labo/D3/tikzext/TemplateTPII-figure2.md5
+++ b/labo/D3/tikzext/TemplateTPII-figure2.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B452EBE3383F9899FD957EE112A7AACF}%
+\def \tikzexternallastkey {048EE0D8D16FA3467EFB5CA16F343EE2}%