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\begin{document}

\Large
\noindent \textbf{Problem Set 4} \\
\normalsize
ME 444: Continuum Mechanics \\
D.~H.\ Kelley \\
\numpoints~points \\

\begin{questions}

    \question \begin{parts} % Mase:1970 5.14 and 5.12
            \part[2] Determine the form of the energy equation if the stress tensor at a point is $T_{ij} = \left( -p+\lambda D_{kk} \right) \delta_{ij} + 2\mu D_{ij}$ and the heat conduction obeys the Fourier law $q_i = -k \frac{\partial}{\partial x_i} \theta $. Here $p$, $\lambda$, and $\mu$, and $k$ are constants, $D_{ij}$ is the strain rate tensor, and $\theta$ is the temperature. 
            \part[2] Determine the stress power $T_{ij} D_{ij}$ if 
                \[ \bm{D} = 
                \left[
                \begin{array}{ccc}
                    1 & 6 & 4 \\
                    6 & 3 & 2 \\
                    4 & 2 & 5
                \end{array}
                \right]~\mathrm{s}^{-1}.
                \]
        \end{parts}

    \question[2] Given Young's modulus $E_Y = 103$~GPa and Poisson's ratio $\nu = 0.34$, find the Lam\'e constants $\lambda$ and $\mu$. % Lai:2009 5.9 

    \question[2] If the components of strain on a material element of structural steel are % Lai:2009 5.11
        \begin{eqnarray*}
            E_{11} &=& 36 \times 10^{-6}, \\
            E_{12} &=& 12 \times 10^{-6}, \\
            E_{13} &=& 30 \times 10^{-6}, \\
            E_{22} &=& 40 \times 10^{-6}, \\
            E_{23} &=& 0, \, \mathrm{and}\\
            E_{33} &=& 25 \times 10^{-6}, \\
        \end{eqnarray*}
        find the stress components, assuming the steel behaves as an elastic material. For structural steel, $\lambda = 119.2$~GPa and $\mu = 79.2$~GPa. 

    \question An isotropic elastic body with $E_Y = 207$~GPa and $\mu = 79.2$~ GPa has a uniform state of stress given by % Lai:2009 5.13
        \[ \bm{T} = 
        \left[
        \begin{array}{ccc}
            100 & 40 & 60 \\
            40 & -200 & 0 \\
            60 & 0 & 200
        \end{array}
        \right] \, \mathrm{MPa}.
        \]
        \begin{parts}
            \part[3] What are the strain components? 
            \part[2] What is the total change of volume for a five-centimeter cube of the material? 
        \end{parts}

    \question Consider a linearly elastic medium. Assume the following form for the displacement field: % Lai:2009 5.23
        \begin{eqnarray*}
            u_1 &=& \varepsilon \left( \sin \beta \left( x_3-ct \right)+\alpha \sin \beta \left( x_3+ct \right) \right), \\
            u_2 = u_3 &=& 0
        \end{eqnarray*}
        \begin{parts}
            \part[2] Is this elastic wave longitudinal or transverse? 
            \part[2] In what direction does this wave propagate? 
            \part[4] Find the strains and stresses, and determine under what conditions the equations of motion are satisfied in the absence of body forces. 
            \part[2] Now suppose that there is a boundary at $x_3 =0$ that is traction-free. Under what conditions will the above motion satisfy this boundary condition for all time? 
            \part[2] Now suppose that there is also a traction-free boundary at $x_3 = l$. What further conditions will be imposed on the above motion to satisfy this boundary condition for all time? 
        \end{parts}

    \question % Lai:2009 5.30
        \begin{parts}
            \part[2] Write a displacement field for an infinite train of longitudinal waves propagating in the direction of $3 \hat{\bm{e}}_1 + 4 \hat{\bm{e}}_2$, where $\hat{\bm{e}}_1$ and $\hat{\bm{e}}_2$ are Cartesian unit vectors. 
            \part[2] Write a displacement field for an infinite train of transverse waves propagating in the direction of $3 \hat{\bm{e}}_1 + 4 \hat{\bm{e}}_2$ and polarized in the $x_1 x_2$ plane, where $\hat{\bm{e}}_1$ and $\hat{\bm{e}}_2$ are Cartesian unit vectors. 
        \end{parts}

    \question A transverse elastic wave of amplitude $\varepsilon_1$ comes to a traction-free plane boundary. If Poisson's ratio is $\nu = 1/3$, determine the amplitudes and angles of reflection of the reflected waves if the angle of incidence is % Lai 5.32
        \begin{parts}
            \part[3] $\alpha_1 = 0$ 
            \part[3] $\alpha_1 = 15^\circ$ 
        \end{parts}

    \question A longitudinal elastic wave is incident on a boundary $x_2=0$ with incident angle $\alpha_1$ with the $x_2$ axis. 
        \begin{parts}
            \part[5] Show that if the boundary is fixed, in general there are two reflected waves, one longitudinal and the other transverse (also polarized with the incident plane $x_1 x_2$). Also find the amplitude ratio of reflected and incident elastic waves. % Lai 5.35
            \part[5] If instead the boundary is traction-free, find the amplitude ratio of reflected and incident elastic waves. % Lai 5.36
        \end{parts}

    \question[2] Show that $R^{-1}$ (where $R$ is the radial distance from the origin) is a harmonic function~--- i.e., it satisfies the Laplace equation $\nabla^2 R^{-1} = 0$. 

    \question[5] The stresses in the Boussinesq problem in cylindrical coordinates $(r,\varphi,z)$ are given by
        \begin{eqnarray*}
            T_{rr} &=& -\frac{F_z}{2\pi}\left( \frac{3r^2z}{R^5} - \frac{1-2\nu}{R(R+z)} \right) \\
            T_{\theta \theta} &=& -\frac{F_z (1-2\nu)}{2\pi} \left( -\frac{z}{R^3} + \frac{1}{R(R+z)} \right) \\
            T_{zz} &=& -\frac{F_z}{2\pi} \frac{3z^3}{R^5} \\
            T_{rz} &=& -\frac{F_z}{2\pi} \frac{3rz^2}{R^5} \\
            T_{r \theta} &=& 0 \\
            T_{\theta z} &=& 0.
        \end{eqnarray*}
        Express the stresses in Cartesian coordinates. % Lai 5.87

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

\end{questions}
\end{document}