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\begin{document}

\Large
\noindent \textbf{Problem Set 2} \\
\normalsize
ME 444: Continuum Mechanics \\
D.~H.\ Kelley \\
\numpoints~points \\

\begin{questions}

    \question[2] Show that if $\bm{F}$ is an arbitrary rank-2 tensor, then $\bm{F} \bm{F}^T$ and $\bm{F}^T \bm{F}$ are symmetric tensors. 

    \question[2] Show that the only possible real eigenvalues of an orthogonal tensor $\bm{Q}$ are $\lambda = \pm 1$. 

    \question Consider an arbitrary vector $\bm{a}$ and an arbitrary rank-2 tensor $\bm{F}$, and let $\bm{F}^S$ and $\bm{F}^A$ denote the symmetric and antisymmetric parts of $\bm{F}$, respectively. 
        \begin{parts}
            \part[2] Show that $ \bm{a} \cdot \left( \bm{F}^A \bm{a} \right) = 0 $. 
            \part[2] Show that $ \bm{a} \cdot \left( \bm{F}^S \bm{a} \right) = \bm{a} \cdot \left( \bm{F} \bm{a} \right) $. 
        \end{parts}

    \question Suppose $c_i = \epsilon_{ijk} a_j b_k$, where 
                \[ [a_j] = \left[ \begin{array}{c}
                    1 \\
                    2 \\
                    3
                \end{array} \right] \]
        and 
                \[ [b_k] = - \left[ \begin{array}{c}
                    3 \\
                    2 \\
                    1
                \end{array} \right] \]
        are order-1 tensors. The form of $c_i$ implies it to be a pseudotensor. 
        \begin{parts}
            \part[1] Calculate the components $c_i$. 
        \part[3] Consider a coordinate transformation comprised of a rotation and no parity inversion, according to the transformation matrix 
            \[ \left[ Q_{ij} \right] = \left[
        \begin{array}{ccc}
            1 & 0 & 0 \\
            0 & \sqrt{3}/2 & -1/2 \\
            0 & 1/2 & \sqrt{3}/2
        \end{array} \right]. \]
        Calculate the components of all three quantities in the transformed coordinates: $c_i^\prime$, $a_j^\prime$, and $b_k^\prime$. Then, calculate $\epsilon_{ijk} a_j^\prime b_k^\prime$ and compare it to $c_i^\prime$, commenting briefly. 
    \part[3] Now consider another transformation from the original (unprimed) coordinate system, comprised instead of a parity inversion and no rotation, according to the transformation matrix $-\delta_{ij}$. Calculate the components of all three quantities in the transformed coordinates: $c_i^{\prime\prime}$, $a_j^{\prime\prime}$, and $b_k^{\prime\prime}$. Then, calculate $\epsilon_{ijk} a_j^{\prime\prime} b_k^{\prime\prime}$ and compare it to $c_i^{\prime\prime}$, commenting briefly. 
        \end{parts}

\question Let $\bm{A}$ be an arbitrary rank-2 tensor. 
    \begin{parts}
        \part[2] \label{IIA} Given $II_A = \frac{1}{2} \left( A_{ii} A_{jj} - A_{ij} A_{ji} \right)$, show that $II_A = \frac{1}{2} \left( \left( \tr \bm{A} \right)^2 - \tr \bm{A}^2 \right)$. 
        \part[2] Using the result of~\ref{IIA} and the Cayley-Hamilton theorem, deduce that 
            \[ III_A = \frac{1}{6} \left( \left( \tr \bm{A} \right)^3 - 3 \tr \bm{A} \tr \bm{A}^2 + 2 \tr \bm{A}^3 \right). \]
    \end{parts}
    
    \question[3] The components of $\bm{A}$ are
        \[ \left[ A_{ij} \right] = \left[
        \begin{array}{ccc}
            2 & 2 & 0 \\
            -1 & 1 & 0 \\
            0 & 0 & -1
        \end{array} \right]. \]
        Show that $\bm{A}$ is invertible and find its inverse. 
%         Show that $\bm{A}$ is invertible and find its polar decompositions.

    \question Let $\phi$, $\bm{u}$, and $\bm{T}$ be arbitrary differentiable tensor fields of rank 0, 1, and 2, respectively. Show that
        \begin{parts}
            \part[1] $ \grad \left(\phi \bm{u} \right) = \bm{u} \otimes \grad \phi + \phi \grad \bm{u} $ 
            \part[1] $ \ddiv \left( \phi \bm{T} \right) = \bm{T} \grad \phi + \phi \ddiv \bm{T} $ 
        \end{parts}

    \question[3] Let $\bm{u}$ be a continuously differentiable vector field. Determine the axial vector of the antisymmetric tensor field $\bm{A} = \grad \bm{u} - \left( \grad \bm{u} \right)^T $. 

    \question The positions $\bm{x(t;\bm{X})}$ of the particles in a body are given by 
        \begin{eqnarray*}
            x_1 & = & X_1 + kt X_3, \\
            x_2 & = & X_2 + kt X_3, \\
            x_3 & = & X_3 - kt \left( X_1 + X_2 \right),
        \end{eqnarray*}
        where $k$ is a positive constant, $t$ is time, and $\bm{X}$ are the initial positions. Let $\bm{dX} = \alpha \hat{\bm{e}}_1 $, $\bm{dY} = \beta \left( \hat{\bm{e}}_1 + \hat{\bm{e}}_2 \right) /2$, and $\bm{dZ} = \gamma \hat{\bm{e}}_3$ be three material lines in the reference configuration (\emph{not} basis vectors).
        \begin{parts}
            \part[2] Find the deformation gradient $\bm{F}$. 
            \part[2] Find the area $dA$ and unit normal $\hat{\bm{N}}$ to the area which is spanned by $\bm{dX}$ and $\bm{dY}$. 
            \part[2] Find the area $da$ and unit normal $\hat{\bm{n}}$ to the area which is spanned by $\bm{dx}$ and $\bm{dy}$ (where $\bm{dx}$ and $\bm{dy}$ are the first two material lines, in the current configuration). 
            \part[2] Find the volume $dV$ spanned by $\bm{dX}$, $\bm{dY}$, and $\bm{dZ}$. 
            \part[2] Find the volume $dv$ spanned by $\bm{dx}$, $\bm{dy}$, and $\bm{dz}$ (where $\bm{dz}$ is the third material line, in the current configuration). 
        \end{parts}

    \question Consider the motion
        \begin{eqnarray*}
            x_1 & = & kt + X_1 \\
            x_2 & = & k X_1^2 t^2 + X_2 \\
            x_3 & = & X_3,
        \end{eqnarray*}
        where the Lagrangian coordinates $X_i$ designate the position of a particle at time $t=0$ and the Eulerian coordinates $x_i$ designate the position of a particle at $t>0$.
        \begin{parts}
            \part[5] Find the velocity $\bm{v}$ and acceleration $\bm{a}$ in both a Lagrangian and Eulerian description.
            \part[2] If in an Eulerian description there is a temperature field $\theta = Ax_1+Bx_2$ (where $A$ and $B$ are constants), find the material derivative $\frac{D\theta}{Dt}$. 
        \end{parts}

    \question[4] Consider the motion which has the velocity field 
        \begin{eqnarray*}
            v_1 & = & kx_1 \\
            v_2 & = & -kx_2 \\
            v_3 & = & 0 \, ,
    \end{eqnarray*}
    where $k$ is a scalar constant and $\bm{x}$ are the current particle positions. Determine the positions $x_i$ and velocity components $v_i$ in the Lagrangian description. 

    \question[2] What topic on this problem set was easy? What topic was hard? What topic was impossible? 

\end{questions}

\vspace{2em}
\textbf{Some hints:}

\begin{itemize}

    \item When calculating the Eulerian coordinates $x_i$ from a velocity field $v_i$, it's often easiest to do the integral by moving the differentials to opposite sides, then gathering like terms. For example, consider
\[
    v_1 = \frac{dx_1}{dt} = 3 t^2 x_1.
\]
Rearranging,
\[ 
\frac{dx_1}{x_1} = 3t^2 dt.
\]
Integrating,
\[
\ln x_1 = t^3 + C,
\]
where $C$ is a constant. Exponentiating both sides gives
\[
x_1 = \exp(t^3) \exp(C),
\]
and since we require $x_1(t=0) = X_1$, 
\[
x_1 = X_1 \exp(t^3).
\]

\item The volume ratio is given by
\[
\frac{v}{V} = \det \bm{F},
\]
where $\bm{F}$ is the deformation gradient tensor. 

\end{itemize}

\end{document}