% This is a sample LaTeX file and a template for Masters' Theses at WPI. % It complies (more or less) to the standards given by the Library % (as of February 1999) % % Feel free to use this file, but I give no guarantee for its compliance % to standards % % % The lengths (textheight, width etc.) are fine-tuned for ps1, ps2, and ps3, % but seem to be somewhat dependent on the machine you are using to compile, % the date, time, moon phase, the weather, and other quantum effects. % You may have to change \oddsidemargin a little, but it's about 98% correct. % % Also, the spacing is correct (doublespacing with footnotes correctly % singlespaced). Curiously, the font size is not specified in the % regulations. So feel free to change it, but the majority of theses % that I have seen is written in 12 point font. % % As for the inclusion of graphics, I recommend the methods specified % in ``latexguide.ps'' off the CS-GSO Website. You can use other % methods including copy and paste with a photocopier, but I think % using the graphicx package is the easiest. % % Have fun and good luck with the thesis. % % % % % % % The preamble % % % 12 point font, and your thesis is a ``report'' to LaTeX \documentclass[12pt]{report} % this enables correct linespacing and graphics inclusion via %``\includegraphics'' %\usepackage{setspace} \usepackage{graphicx} \usepackage{epsfig,amsmath,amsthm} % leave 1.5in margin to the left and 1in margin to the other % sides. Don't print page number in the margin (but rather above it) \setlength{\textheight}{8.63in} \setlength{\textwidth}{5.9in} \setlength{\topmargin}{-0.2in} \setlength{\oddsidemargin}{0.3in} \setlength{\evensidemargin}{0.3in} \setlength{\headsep}{0.0in} \newcommand{\np}{\nabla\hat{\varphi}} \newcommand{\p}{\hat{\varphi}} \newcommand{\e}{{\varepsilon}} \renewcommand{\c}{\chi^\nu} \newtheorem{defin}{Definition} \newtheorem{thm}{Theorem} \newtheorem{guess}{Remark} \newtheorem{guesses}{Remarks} % Start to write \begin{document} % First things first: The Titlepage % This is the recommended format by the library % % Define \brk as a command for leaving a little vertical space. Makes % the titlepage easier to read - normally, this is NOT GOOD LATEX % STYLE!!! % \newcommand{\brk}{\vspace*{0.18in}} % No page number on the title page \thispagestyle{empty} %*********************************************************************** \title {\Large {Optimal design of a two-layered elastic strip subjected to transient loading}} \author{Your full name} \date{May 2004} \maketitle %************************************************************************ \newpage % This is the command for doublespacing when you use the setspace % package % Please do NOT use \baselinestretch, this will mess up everything, % cause earthquakes, tornados and lots of questions for me... % If you need a singlespaced paragraph (BAD STYLE!!!), use % \singlespacing or \onehalfspacing and enclose it together with the % paragraph in braces {\singlespacing This is my text... blah blah blah} % %\doublespacing % Now you can start to be creative. % First, you need an abstract. % Fortunately, LaTeX has thought of that, so it's very easy: % \begin{abstract} In this thesis we study stress propagation in two-layered elastic medium and derive equations .... \end{abstract} % From here on, we need Roman page numbers according to the library % regulations. So let's assign those. \pagenumbering{roman} % or {Roman} if you like them capitalized \vspace*{\fill} %\doublespacing % The next thing is the Preface (``Acknowledgments''). % No standard environment for that, so we'll format it by hand. % \begin{center} \textbf{\Large Acknowledgments} \end{center} Optional \vspace{7in} \clearpage %\doublespacing % Now comes the Table of Contents, really easy in LaTeX. you never % have to worried about it. (Think of all the hours you would % have wasted in Word getting this thing updated without crashing % the system) :). \tableofcontents % THAT'S IT. REALLY. Everything else is automatic. No formatting, no headline. % All predefined. % Now - just as easy - the List of Figures. % This will catch all objects enclosed in \begin{figure}\end{figure} % statements. % \listoffigures % There is also a list of tables, if you have any. % This will catch all objects enclosed in \begin{table}\end{table} % statements. %\listoftables \bibliographystyle{plain} % And we need a clear separation between preface and text, otherwise % the numbering gets confused. \clearpage % And now - tataa - the text. % This is the place to become really creative. % From here on, we need arabic numbering again and we need to start % from 1. \pagenumbering{arabic} \setcounter{page}{1} % % Since this is a ``report'', the topmost level of hierarchy is % ``Chapter'', not section as you may be used to. Chapters are % enumerated starting from 1, so Sections are 1.1, Subsections are % 1.1.1, subsubsections don't get numbers. (You can change that, if % you want them to be called 1.1.1.1) % %CHAPTER 1 INTRODUCTION %------------ \chapter{Introduction} %BACKGROUND %------------------ \section{Definitions and Notation} A composite material can be defined simply as a material that consists of two (or more) identifiably distinct constituent materials. -istropic material: An isotropic material is ... -homogeneous material: An homogeneous material is ... -tensor: a tensor is ... -positive definite matrix: a real symmetric matrix is called positive definite if all of its eigenvalues are positive. -symmetric matrix: An N$\times$N matrix is symmetric if and only if $A = A'$. -eigenvalues and eigenvectors: -inverse of a square matrix: Let $A$ be an $n\times n$ matrix. If there exists an $n\times n$ matrix $B$ such that $AB = I$ and $BA =I$, then $A$ is said to be invertible and $B$ is called the inverse of $A$, denoted $A^{-1}$. \section{Problem Formulation and Prior Work} We consider one-dimensional wave propagation in an isotropic elastic strip of unit length, made of two layers. The layer interface is located at some fixed position $x_0$, where $0\leq x_0\leq 1$. The right face of the strip is subjected to a stress loading $p$, while the left face is fixed~(Figure~\ref{bvpgen}). .... %-------------------------- \section{Extended Abstract} In this thesis we study the mathematical theory developed to express the properties of composite materials, applied to heat conductors. We discover how the properties of a composite material change with its microgeometry and learn about special composites which insure optimal performance by minimizing or maximizing the heat dissipated in a structure. Our work is organized as follows:\\ In Chapter 2 we investigate and describe material designs which provide linear solutions to the steady state heat equation with linear boundary conditions. We are able to show that for the same linear boundary conditions, any isotropic, anisotropic or two phase material design (described in detail in Case 2 and Case 3 of Section 2.3), provide the same linear temperature distribution in the design domain $\Omega$. \\ We then proceed in Chapter 3, ... \vspace{2in}\\ The Matlab programs used to generate our graphical and numerical results, are included at the end of this thesis. \section{Analytical Solutions for Stress Wave Propagation and Optimal Design} We consider one-dimensional wave propagation in an isotropic elastic strip of unit length, made of two layers. The layer interface is located at some fixed position $x_0$, where $0\leq x_0\leq 1$. The right face of the strip is subjected to a stress loading $p$, while the left face is fixed~(Figure~\ref{bvpgen}). % Your picture file should be of the form: filename.eps % If your picture file is of another form, for example filename.bmp, then you should open it with Adobe Photoshop and save it as filename.eps % Please type your filename below and get rid off the comment sign (%) in front of centerline \begin{figure}[htbp] %Figure 1. %\centerline{\epsfig{file=filename.eps,width=14pc,angle=0}} \caption{Two-layered finite strip under transient load.} \label{bvpgen} \end{figure} The distribution of the material properties along the strip, insures the same travel time for the waves through each layer. This implies the following relation between the wave speeds $c_1$ and $c_2$ in each layer: \begin{eqnarray} t^{\ast}=\frac{1-x_{0}}{c_1}=\frac{x_0}{c_2} \label{tequal} \end{eqnarray} Our goal is to find a design that provides the smallest stress amplitude during the wave propagation along the strip. We choose our design parameter $\alpha$ to represent the impedance ratio between layer 1 and layer 2. The density and elastic modulus along the strip are denoted by the piece-wise constant functions $\rho_\alpha(x)$ and $E_\alpha(x)$. Let $\sigma_{\alpha}(x,t)$ represent the value of the stress at position $x$, time $t$, and design parameter $\alpha$. We formulate our optimal design problem as, \begin{eqnarray} {\bf P^{\ast}}=\inf_{\alpha>0}\ \ \sup_{0\leq x\leq 1,\ 0\leq t<+\infty}\sigma_\alpha (x,t), \label{opt1gen} \end{eqnarray} subject to the initial/boundary-value problem, \begin{eqnarray} \left\{\begin{array}{ll} \rho\ \frac{\partial^2 u}{\partial t^2} = \frac{\partial (E \frac{\partial u}{\partial x})}{\partial x} & \ \\ & \ \\ \sigma (1,t) = E \frac{\partial u}{\partial x} (1,t)=pH(t), \ \ u(0,t) = 0 & \\ & \ \\ u(x,0) = \frac{\partial u}{\partial t} (x,0) = 0 & \end{array}\right. \label{IBCgen} \end{eqnarray} Here, the material properties $\rho\equiv \rho_\alpha$ and $E\equiv E_\alpha$, are chosen to insure the same travel time through each layer, while $u(x,t)$ represents the displacement at $(x,t)$ and $H(t)$ represents the Heaviside function. \\ The problem (\ref{opt1gen})-(\ref{IBCgen}), can be easily converted to the case of two layers of equal thickness and wave speed, by replacing the spatial variable $x$ with the new variable $\xi=\int_{0}^{x}\frac{ds}{c(s)}$, and using condition (\ref{tequal}). Here, $c\equiv c(s)$ is the piece-constant wave speed function, taking values $c_1$ and $c_2$ in each layer respectively. As a result, the wave equation (\ref{IBCgen}.1) becomes: \begin{eqnarray} z\ \frac{\partial^2 u}{\partial t^2} = \frac{\partial (z \frac{\partial u}{\partial \xi})}{\partial \xi}, \end{eqnarray} where $z=z(\xi)$ represents the characteristic impedance given by the piece-wise constant function: \begin{eqnarray} z(\xi)=\left\{\begin{array}{ll} z_1, & \mbox{where\ } \frac{1}{2}< \xi \leq 1\\ & \\ z_2 & \mbox{where\ } 0\leq \xi \leq \frac{1}{2}. \end{array}\right. \label{distb1} \end{eqnarray} \begin{figure}[htbp] %Figure 1. %\centerline{\epsfig{file=2layer.eps,width=14pc,angle=0}} \caption{Two-layered finite strip under transient load.} \label{bvp} \end{figure} The wave speed now becomes constant in each layer ($c=c_1=c_2$), while $z_1$ and $z_2$ are positive, and $\alpha=\frac{z_1}{z_2}$. Putting everything together, we re-formulate our optimal design problem, as: \begin{eqnarray} {\bf P}=\inf_{\alpha>0}\ \ \sup_{0\leq \xi\leq 1,\ 0\leq t<+\infty}\sigma_\alpha (\xi,t), \label{opt1} \end{eqnarray} subject to the initial/boundary-value problem, \begin{eqnarray} \left\{\begin{array}{ll} z\ \frac{\partial^2 u}{\partial t^2} = \frac{\partial (z \frac{\partial u}{\partial \xi})}{\partial \xi} & \ \\ & \ \\ \sigma (1,t) = E \frac{\partial u}{\partial \xi} (1,t)=pH(t), \ \ u(0,t) = 0 & \\ & \ \\ u(\xi,0) = \frac{\partial u}{\partial t} (\xi,0) = 0. & \end{array}\right. \label{IBC} \end{eqnarray} As demonstrated later in the paper, solving problem $\bf P$, becomes equivalent to solving the physical problem $\bf P^{\ast}$, because the only essential condition that influences the stress wave propagation along the strip, is the equal travel time through each layer. Therefore, from now on, any conclusions made for the case of two layers of equal length (Figure~\ref{bvp}), will apply to the general physical case (Figure~\ref{bvpgen}) with layers of different lengths.\\ \\ For our two-layered strip (Figure~\ref{bvp}), the propagation of the stress wave from layer 1 to layer 2, can be expressed by the following relations, \begin{eqnarray} \begin{array}{l} %\sigma_{ |_{\begin{array}{l} % \mbox{\small back}\\ %\mbox{\small shock} %\end{array}}}=\sigma _{|_{\begin{array}{l} % \mbox{\small front}\\ %\mbox{\small shock} %\end{array}}}+[\sigma], \sigma^{-}=\sigma^{+}+[\sigma], \hspace{0.5in} [\sigma]_T = \frac{2}{1+{\alpha}}\ [\sigma]_I,\hspace{0.5in} [\sigma]_R = \frac{1-{\alpha}}{1+{\alpha}}\ [\sigma]_I. \end{array} \label{stress} \end{eqnarray} Here, $\sigma^{-}$ and $\sigma^{+}$ represent the stress values behind and ahead of the discontinuity, while $[\sigma]$ represents the stress jump. The subscripts refer to I (incident), T (transmitted), and R (reflected) wave. The above relations, see \cite{Mey}, express the continuity conditions at the layer interface, and will be our main reference in calculating the stress values throughout this paper.\\ In the special case of a homogeneous design, when $\alpha=1$, using (\ref{stress}) and the method of characteristics, one can easily derive the time history profile for the stress. In the region ahead and behind the stress discontinuity propagating along the strip, the stress takes the following values respectively, \begin{eqnarray}\left\{\begin{array}{ll} 0\mbox{ and }p, & \mbox{during the time interval } (0,\frac{1}{c})\\ & \\ p\mbox{ and }2p, & \mbox{during the time intervals } (\frac{2k-1}{c},\frac{2k}{c})\\ & \\ 2p\mbox{ and }p, & \mbox{during the time intervals } (\frac{2k}{c},\frac{2k+1}{c}) \end{array}\right.\ \ \mbox{for} \ k=1,2\hdots\infty. \end{eqnarray} This implies that, \begin{eqnarray} \max_{0<\xi<1,\ 00}\ \ \sup_{0\leq \xi\leq 1,\ 0\leq t<+\infty}\sigma_\alpha (\xi,t)\leq \max_{0\leq \xi\leq 1,\ 0 \leq t<+\infty}\sigma_{1} (\xi,t)=2p. \label{part1} \end{eqnarray} Based on the results given later in this section, we prove that the reverse inequality in (\ref{part1}) holds, and conclude that the homogeneous design is an optimal design for problem ${\bf P}$. We also notice that no design with design parameter $\alpha< 1$, can be optimal, since for such designs the stress amplitude always exceeds the value $2p$. Indeed, following the first transmitted wave, and using (\ref{stress}) and the method of characteristics, we derive that in the region behind the stress discontinuity, the stress will take values: \[\sigma(\xi,t) %{ |_{\begin{array}{l} %\mbox{\small behind}\\ %\mbox{\small shock} %\end{array}}} = \frac{2}{1+{\alpha}}\ p>p, \ \ \ \mbox{ where } \frac{1}{2c}1$. For a given $\alpha$, one can apply (\ref{stress}) and the method of characteristics for the case of two layers of equal or unequal length (Figure~\ref{xtdiagram}.a,b). \begin{figure}[ht] %Figure 1a. %\centerline{\epsfig{file=x_transform.eps,width=14pc,angle=0} %\hspace{0.7in}\epsfig{file=x_tnew.eps,width=14pc,angle=0}} %\centerline{\small a) Equal layer length\hspace{1.75in}b) Unequal layer length} \caption{Lagrangian diagram of stress waves.} \label{xtdiagram} \end{figure} In the region ahead and behind the stress discontinuity in layer 1, the stress takes the following values respectively, \begin{eqnarray} \left\{\begin{array}{ll} 0\mbox{ and }p & \mbox{during the time interval } (0,\frac{1}{2c})\\ & \\ p\mbox{ and }T_{\alpha,1} & \mbox{during the time interval } (\frac{1}{2c},\frac{1}{c})\\ & \\ T_{\alpha,k}\mbox{ and }p & \mbox{during the time intervals } (\frac{k}{c},\frac{k}{c}+\frac{1}{2c})\\ & \\ p\mbox{ and }T_{\alpha,k+1} & \mbox{during the time intervals } (\frac{k}{c}+\frac{1}{2c},\frac{k+1}{c}) \end{array}\right.\ \ \mbox{for } k=1,2,...\infty, \label{4} \end{eqnarray} where the following recurrence relation among the stress values $T_{\alpha,k}$ holds: \begin{eqnarray}\left\{\begin{array}{l} T_{\alpha,k}=\beta(\alpha)\cdot (T_{\alpha,k-1}-T_{\alpha,k-2})+T_{\alpha,k-3},\ \ k=4,5\hdots\infty, \\ \\ T_{\alpha,1}=\frac{2}{(\alpha+1)}p,\ \ T_{\alpha,2}=\frac{8\alpha p}{(\alpha+1)^2}, \ \ T_{\alpha,3}=\frac{2(3\alpha-1)^2 p}{(\alpha+1)^3}, \end{array}\right. \label{recurr} \end{eqnarray} where $\beta(\alpha)=\frac{3\alpha-1}{\alpha+1}$. The recurrence relation given in (\ref{recurr}) is linear, homogeneous and with constant coefficients. Its corresponding characteristic equation, \begin{eqnarray*} T^k-\beta(\alpha)\cdot T^{k-1}+\beta(\alpha)\cdot T^{k-2}-T^{k-3}=0, \end{eqnarray*} has three distinct roots. After further calculations and simplifications we find the solution to be $T_{\alpha,k}=p\ [1-\cos (k\varphi(\alpha))]$, where $\varphi(\alpha)= \arctan\frac{2\sqrt{\alpha}}{\alpha-1}$, and $\alpha>1$. From here, one can derive the following bounds for the stress amplitude in layer 1, \begin{eqnarray} 0\leq\sup_{\frac{1}{2}<\xi<1,\ 0 1$. Due to the expression for $S_{\alpha,k}$, the stress amplitude in layer 2, will either reach or exceed the value $2p$, as indicated below, \begin{eqnarray} \sup_{0<\xi<\frac{1}{2},\ 00}\sup_{0\leq \xi\leq 1,\ 0\leq t<+\infty}\sigma_{\alpha} (\xi,t)= \max_{0\leq \xi\leq 1,\ 0\leq t<+\infty}\sigma_{1} (\xi,t)=2p. \label{maxtot} \end{eqnarray} In conclusion, in this work, we solve the problem of minimizing the stress amplitude for a discrete, two-layered elastic strip of equal length, subjected to transient loading. Our results provide explicit formulas for the stress propagation and optimal designs in one dimension and apply for a two-layered elastic strip of unequal length. %CHAPTER 2 %--------- \chapter{Linear Solutions to the Steady State Heat Equation} In this chapter we investigate and describe material designs which provide linear solutions to the steady state heat equation with linear boundary conditions. \section{Numerical Results} \chapter{Appendix} \section{$G$-convergence} Here we include some background on $G$-convergence \section{$H$-convergence} Here we include some background on $H$-convergence, see \cite{mt1}. % Another version of Appendix \appendix \chapter{} \section{$G$-convergence} Here we include some background on $G$-convergence \section{$H$-convergence} Here we include some background on $H$-convergence, see \cite{mt1}. %bibliography %----------------- \begin{thebibliography}{99} \bibitem{blp} Copson E. 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S. {\em Applications of Composite Materials,}American Society For Testing And Materials, (1973). \bibitem{Pironneau} Hoskin B. C. {\em Composite Materials for Aircraft Structures,} AIAA Education Series,(1986). \bibitem{mt1} Author {\em $H$-convergence,}... \end{thebibliography} \end{document}