### Added a place for the Documentation

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 \documentclass{article} \usepackage{amsmath} \usepackage{graphicx} \usepackage{subcaption} \usepackage[a4paper,left=1.1in,right=1.1in,top=1.1in,bottom=1.1in]{geometry} \renewcommand{\familydefault}{\sfdefault} \begin{document} \section*{Correction of Normal Load by Lid Displacement}\label{correction-of-normal-load-by-lid-displacement} Because the tie rods which measure the shear force produce an angle that is dependent on the current lid position, the normal force $F_N$ measured by the machine is not equal to the effective normal stress $F_{Neff}$. It has to be corrected with respect to the currently measured shear force $F_{S1}$ and the current angle $\beta$, that is dependent on the current lid position $d_{lid}$. \begin{figure}[h!] \centering \includegraphics[width = .7\columnwidth]{RST_Angle} \caption{Formation of an angle between the lid and the sensor.} \end{figure} For this we use a simple decomposition of the resulting force $F_R=F_{s1}$ into the effective shear force $F_S$ and the normal force $F_N$ excerted by pulling on the lid at an oblique angle $\beta$: \begin{equation} F_R = \sqrt{F_S^2 + F_N^2} \end{equation} \begin{equation} F_S = F_R \, cos \, \beta \end{equation} \begin{equation} F_N = F_R \, sin\, \beta \end{equation} The angle is defined by the length of the tie rods $l_{rod} = 412 \,mm$ and the lid displacement $d_{lid}$ with respect to the reference point at zero lid displacement $d_0 = +\,9\,mm$. The usual range of lid displacement is a few millimeters upwards (negative) and downwards (positive). \begin{equation} \beta = sin^{-1} \, \frac{(d_0-d_{lid})}{l_{rod}} \end{equation} \begin{figure}[h!] \centering \begin{subfigure}[t]{0.45\textwidth} \includegraphics[width = \textwidth]{angle_vs_lid} \caption{Angle as a function of the lid position.} \end{subfigure} ~ \begin{subfigure}[t]{0.45\textwidth} \includegraphics[width = \textwidth]{correction_factors} \caption{Factors needed to compensate the difference between measured and effective forces.} \end{subfigure} \end{figure} The angle only shows a very small variation between 2$^\circ$ and -1.5$^\circ$, where negative shows a downward slope and positive an upward slope, in contrast to the definition of $d_{lid}$. As a result, the normal force component of the resulting force is changing from a very small positive value into a small negative value. This shows in correspondence to the position of the lid, that if the lid is higher than the sensor ($d_{lid} > d_0$) then it pushes down onto the sample resulting in a higher normal stress and vice versa. The shear stress component has a maximum when $d_{lid}=d_0$ because here the tie rods are parallel to the direction of shear. The above graph shows the factors by which the measured forces (or stresses) have to be corrected. The machine does this for the internal determination of the normal load that is applied onto the lid, but the measurement is done using an external controller which only measures the pure signal. This leads to the following corrections with $F_R$ being the measured shear force and $F_N$ the measured normal force: \begin{itemize} \item Effective normal force: \begin{itemize} \item $F_{Neff} = F_N - F_R \, c_{FN}$ \end{itemize} \item Effective shear force: \begin{itemize} \item $F_{Seff} = F_R \, c_{FS}$ \end{itemize} \end{itemize} \end{document}

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