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%\lfoot{PR-14-03 - GEMS \#~101505966 - Schlumberger Confidential}

\begin{document}
\title{Initial Results for Nano-Indentation Simulations in Uintah-MPM}

\author{Jim Guilkey}
\affil{University of Utah, Department of Mechanical Engineering}

\date{}

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\maketitle

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\vspace{-.2 in}
\begin{abstract}

In support of experimental nano-indentation measurements being made at ARL-Adelphi, analogous
numerical simulations have been carried out in an attempt to give insight to the experimental
results.  The very first simulations done attempted to match the experimental conditions,
including a ``cube-corner" indentor, and an elastic-plastic with linear hardening specimen.
The force required to push the diamond indentor into the specimen was recorded and plotted.
While the simulations ran successfully, interpretation of the data is challenging.
In an attempt to better understand the results, a series of simpler simulations were run,
including different indentor shapes and test specimens that were elastic and some that were
elastic-perfectly plastic.
This report briefly describes those tests and the results, with the goal of better understanding
the results for these simpler situations, thereby preparing all involved to interpret the
more complex results in the future.  Functional agreement with the expected reaction force is
demonstrated as well.

\end{abstract}

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%\thanks{Perforating Research Report Number PR-14-03~\footnotemark - Schlumberger Confidential}
%footnotetext{Also GEMS Analysis Report \#~101505966}

\newpage

\section{Simulation Geometries and Boundary Conditions} \label{Geometry}

\subsection{Simulation Geometry}

In this report, only the simplified cases will be described.  For each, a small elastic specimen has a free top surface and is bounded by planes of symmetry on all other surfaces (bottom and sides).
The specimen has elastic properties approximating silicon, while the indentor has elastic properties 
approximating diamond (much stiffer).  The diamond indentor is pushed by a rigid piston for which
the displacement is controlled.  Separate indentor and piston materials were used so that the
friction contact algorithm in Uintah could be used to goven the interaction between the indentor and
specimen.  Specifically, this contact algorithm turns off the contact constraint once there is no
longer a normal traction representing compression between two materials, and their relative 
velocities are no longer toward each other.

Initial configurations for the three cases under consideration are shown in
Figures~\ref{ConstAreaInit}-\ref{LinAreaInit}.  These graphics show the initial particle
representation.  Particles are underscaled, slightly, to illustrate the resolution with which
the objects are represented.  Dark blue particles represent the rigid piston, teal (or light green)
particles are the diamond indentor, while the lime green and red particles are the specimen.
Red particles are the same material, just represented at a lower resolution to save computational
time away from the interaction zone.  Figure\ref{ConstAreaInit} shows a constant area indentor, the
cube-corner indenter in Figure~\ref{QuadAreaInit} has a cross-sectional area that varies as the
square of the depth of penetration, while Figure~\ref{LinAreaInit} depicts a cube-edge indentor, the
cross-sectional area of which increases linearly with depth of indentation.

\begin{figure}
  \centering
    \includegraphics[scale=.25]{Figures/ConstAreaInit.png}
  \caption{Initial configuration for the constant area indentor case.  In this case, a quarter
symmetry model was appropriate.  Particles are scaled to show the resolution with which the objects
are represented.  Particle color indicates as follows:  Blue is the rigid piston, teal is the diamond
indentor, green and red are the silicon sample material, where the particles shown in red
are simply larger (lower particle density) for computational efficiency.}
  \label{ConstAreaInit}
\end{figure}

\begin{figure}
  \centering
    \includegraphics[scale=.25]{Figures/QuadAreaInit.png}
  \caption{Initial configuration for the cube-corner indentor case.  Here, the cross sectional
area of the indentor increases quadratically with depth of penetration.
Particles are scaled to show the resolution with which the objects
are represented.  Particle color indicates as follows:  Blue is the rigid piston, teal is the diamond
indentor, green and red are the silicon sample material, where the particles shown in red
are simply larger (lower particle density) for computational efficiency.}
  \label{QuadAreaInit}
\end{figure}

\begin{figure}
  \centering
    \includegraphics[scale=.25]{Figures/LinAreaInit.png}
  \caption{Initial configuration for the constant area indentor case.  In this case, a quarter
symmetry model was appropriate.  Particles are scaled to show the resolution with which the objects
are represented.  Particle color indicates as follows:  Blue is the rigid piston, teal is the diamond
indentor, green and red are the silicon sample material, where the particles shown in red
are simply larger (lower particle density) for computational efficiency.}
  \label{LinAreaInit}
\end{figure}

\subsection{Boundary Conditions}

The boundary conditions on all but the top surface of the specimen are symmetry planes.  The piston
is given a constant velocity downward (-10 nm/nanosecond) for 2 nanoseconds at which time the
piston is briefly stopped and reversed.  This results in a 20\% indentation into the 100 nm thick
sample.

\section{Elastic Results} \label{Results}

Here we describe the results for elastic samples, first with a depiction of the geometry, followed
by graphs of the force on the indentor.

\subsection{Indented Geometry}

Figures~\ref{ConstAreaFull}-\ref{LinAreaFull} depict the particle configuration at maximum
penetration, which is 20\% of the thickness of the sample (maximum depth of the penetrator).

\begin{figure}
  \centering
    \includegraphics[scale=.25]{Figures/ConstAreaFull.png}
  \caption{Constant area configuration at full indentation.}
  \label{ConstAreaFull}
\end{figure}

\begin{figure}
  \centering
    \includegraphics[scale=.25]{Figures/QuadAreaFull.png}
  \caption{Quadratic area configuration at full indentation.}
  \label{QuadAreaFull}
\end{figure}

\begin{figure}
  \centering
    \includegraphics[scale=.25]{Figures/LinAreaFull.png}
  \caption{Linear area configuration at full indentation.}
  \label{LinAreaFull}
\end{figure}

\subsection{Reaction Force Plots}

A feature of the rigid contact algorithm is the output of the resultant force that the rigid 
object experiences.  This can be thought of as the force required to maintain the velocity
assigned to that object.  Figures~\ref{ConstAreaForce}-\ref{LinAreaForce} contain plots of the
force on the y-axis and time on the x-axis.  Time could be converted to indentor displacement 
very simply, and eventually will be, but for the purpose of this report, the independent variable
will remain time.

\begin{figure}
  \centering
    \includegraphics[scale=.4]{Figures/ConstAreaForce.png}
  \caption{Constant area configuration at full indentation.  Analytical expressions
are approximate fits only.}
  \label{ConstAreaForce}
\end{figure}

\begin{figure}
  \centering
    \includegraphics[scale=.4]{Figures/QuadAreaForce.png}
  \caption{Quadratic area configuration at full indentation.  Analytical expression is
an approximate fit to the loading portion of the data.}
  \label{QuadAreaForce}
\end{figure}

\begin{figure}
  \centering
    \includegraphics[scale=.4]{Figures/LinAreaForce.png}
  \caption{Linear area configuration at full indentation.  Analytical expression is
an approximate fit to the loading portion of the data.}
  \label{LinAreaForce}
\end{figure}

For the constant area indentor, the force vs. time plot looks very much as one would expect.  The
force magnitude increases linearly until the piston stops, and then during unloading, the force
magnitude decreases with the same slope, again, as expected.

(For the next two plots, there is significant noise in the unloading.  I have a couple of 
suspicions as to what may be causing this, and this will eventually be resolved.  For the
sake of the discussion here, the unloading portion of the curve can be ignored.)
For the cube-corner indentor, the cross-sectional area of indentor that is in contact with the sample goes up as the square of the depth of indentation.  Figure~\ref{QuadAreaForce} includes an approximate fit to the data (green curve) shows that 
the force magnitude is quadratically dependent on penetration depth during loading.

This result motivated the final configuration, in which the cross sectional area of the indentor that
is in contact with the sample increases linearly with depth of penetration.  There, in
Figure~\ref{LinAreaForce}, an approximate fit to the data is again drawn in green.  This time, the
force magnitude increases as time (depth) to the 3/2 power.

\subsection{Expected Functional Form}

Based on a lack of experience, the author had no solid expectations of the functional form of the
force experienced by the indentor.  However, the nano-indentation Wikipedia page (https://en.wikipedia.org/wiki/Nanoindentation)
revealed:

\begin{equation}
E_r=\frac{1}{\beta}\frac{\sqrt{\pi}}{2}\frac{S}{\sqrt{A_p(h_c)}}
\end{equation}

where $E_r$ is the ``reduced Young's Modulus" of the test specimen, $\beta$ is an order $1$ geometric
constant, $A_p(h_c)$ is the projected area of the indentor at depth $h_c$.  $S$ is the stiffness
and is proportional to the slope of the unloading curve, $\frac{dP}{dh}$.  Rearranging this, we have:

\begin{equation}
S=\frac{2 E_r \beta}{\sqrt{\pi}} {\sqrt{A_p(h_c)}}
\label{eqn2}
\end{equation}

Consider each of the three cases in light of Eq.~\ref{eqn2}.  For the constant area case,
$A_p(h_c)$ is a constant, and the slope of the loading (and unloading) curve is a constant.
For the cube-corner indentor, $A_p(h_c) = 2.598 h^2$, so we expect the slope of the unloading
curve to vary linearly with penetration, which it does (Figure~\ref{QuadAreaForce}).  Finally, for
cube-edge indentor, $A_p(h_c) = k_1 h$, and $P=k_2 h^{3/2}$ (Figure~\ref{LinAreaForce}, so $\frac{dP}{dh} = 3/2 k_2 h^{1/2}$.  Thus, for each of the cases considered, simulations are returning the
expected functional form of the force vs. indentation curve.


\section{Discussion}\label{Discussion}

Now that we have demonstrated that Uintah simulations generate results of the expected functional
form for loading, we are nearly ready to move to more complex material response.  Because moving to
plasticity will require using the unloading portion of the curve, resolving the noise during
unloading will be necessary.  We also need to translate the time domain currently used on the 
horizontal axis to an indentation, and ensure that we can extract a reliable value of Young's modulus
for the specimen.  At that point, we will feel confident in our methods and will be able to move
to more complicated material response.


\end{document}
