OR682: Individual Project 4




	If a metal rod is stretched then, at room temperature, it will at first behave elastically (like a rubber band) but eventually, at some position along the rod, a neck will form and then the rod will snap. Once a neck begins to form, the stress in the rod will drop dramatically, as illustrated in the left-hand figure below. (If you like, you could simulate this behavior with bubble gum.) At temperatures near absolute zero, however, the rod will behave differently, as in the right-hand figure. An oscillatory pattern of stresses is observed (due to the thermal properties of the metal) before necking occurs. A useful model for this oscillatory behavior has been proposed, in the form of a pair of ordinary differential equations: $\begin{eqnarray}\sigma' &=& \Sigma_0 \left\{ 1 - \exp\left[ { {\beta \theta + ... ... + \mu \sigma (\Sigma_0 - \sigma')] \equiv f_2(\theta,\sigma) \end{eqnarray}$ Here $\sigma(t)$ represents stress, $\theta(t)$ represents temperature, and t is a (scaled) time variable. The constants $\Sigma_0$ , $\alpha$ , $\beta$ , and $\mu$ are composite parameters that combine information about the temperature and the metal. These equations are only used to describe the behavior of stress corresponding to the thermal properties of the metal, and do not describe the (mechanical) necking that occurs at higher stress levels. Depending on the choices of the constants, the equations can generate four qualitatively different types of solutions. The goal of this project is to examine these solutions. The differential equations are stiff, and so must be solved using an appropriate solver. I want you to use ode15s from version 5 of Matlab. We will assume throughout that $\beta = 25$ . The remaining constants will be determined from the values of $\Sigma_0$ , $\alpha$ , and $\nu$ . Then $\mu$ is defined by $\mu = \nu / \beta$ . C Grade: Solve the differential equations with S₀ = 1.6 x 10^-2 , $\alpha = 210$ , and $\nu = 700$ . Initialize the system with $\sigma(0) = \theta(0) = 0$ . Integrate from t = 0 to t = 200. Plot $(t,\sigma(t))$ and $(t,\theta(t))$ (on the same graph); in addition, plot $(\theta(t),\sigma(t))$ . Determine the time values corresponding to the first local maximizers of $\theta(t)$ and $\sigma(t)$ . Discuss your results. How are stress and temperature related? B Grade: Derive the ``steady state'' solution to the system of equations by setting $\sigma' = \theta' = 0$ and solving for $\sigma_s$ and $\theta_s$ . Is your solution from the C-grade section converging to the steady-state solution? How sensitive is the steady-state solution to changes in the parameters? (Answer this question both numerically and analytically.) Derive the Jacobian of the system at steady state: $\begin{displaymath}A = \left. \pmatrix{ { {\partial f_1} \over {\partial \sigma... ...r} \right\vert _{ {\sigma=\sigma_s} \atop {\theta=\theta_s} } \end{displaymath}$ Prove or verify that the entries in the Jacobian matrix are $\begin{eqnarray}A_{1,1} &=& {\displaystyle - \, { {\alpha \Sigma_0} \over {1+... ... - \, { {1 + \theta_s(1 - \beta)} \over {(1+\theta_s)^4} } } \end{eqnarray}$ and that the formulas for $T = \hbox{Tr}(A)$ and $D = \det(A)$ are $\begin{eqnarray}T &= {\displaystyle - \, { {1 + \alpha \Sigma_0 (1+\theta_s)^... ...gma_0(\alpha + \beta \mu \Sigma_0)} \over {(1+\theta_s)^4} } } \end{eqnarray}$ A Grade: Modify your software so that it uses the formulas for the Jacobian. Is there any difference in the performance of the software? The qualitative behavior of the solutions to the differential equations depends on the solutions to $\begin{displaymath}\omega^2 - T \omega + D = 0. \end{displaymath}$ (That is, whether the roots are distinct or not, and real or not.) There are four possibilities, depending on the sign of T and the sign of T ² - 4D. These possibilities can be analyzed in terms of the values of $(\alpha \Sigma_0)$ and $(\nu/\alpha)$ . What four kinds of behavior are possible? How does the numerical solution behave in borderline cases? You can examine T and T ² - 4D graphically, by varying $\Sigma_0$ and $\nu$ . Deadlines: You should aim to have this project completed by the end of November.


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