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Unformatted text preview: Chapter 2 Deformation and Strain In this chapter we explain how deformation is described precisely . This gives us the framework to model the deformation of a metal beam for example, which is holding up a building (although we won’t do this problem here). 2.1 Two Coordinate Systems Consider the following two problems involving the measurement of velocity. The first is a pipe with fluid moving through it. The pipe may bend, and we’d like to model the velocity of the fluid as it moves through the pipe. An experimentalist may measure the rate at which the total mass passes through a particular cross-section. Given the density, the mean velocity can then be calculated. Note that there is no interest in measuring the velocity of a particular particle, nor is it necessary to know which particle (molecule) is going where. In fact, the experimentalist is measuring the velocity of different particles at a given location as a function of time. The second scenario is a basketball base board attached to a pole. As a basketball is bounced off of it, it vibrates and we’d like to know the velocity as a function of time. In this case, we want to know the velocity of each location of the baseboard, i.e. we fix a point, then measure the velocity of that point . These two scenarios indicate the necessity of two different coordinate systems. We refer to them as the Lagrangian or material coordinate system (used for the basketball base board) named because the coordinate system is associated with the material (base board) and the Eulerian or spatial coordinate system, named because the coordinate system is fixed in space (where the velocity is measured in the pipe). Consider Figure 2.1, which depicts an object which has been deformed over time. At time t = 0 we have a “particle”, X , at spatial location x ( X , 0), so that the position is a function of which “particle” we are looking at. At time t = t > 0 the particle X has moved to a new location x ( X , t ), so that the position is a function of the particular particle and the time. The two coordinate systems may have the same base vectors ( E i = e i ), or they may be different. The two coordinate systems are as follows: 31 32 CHAPTER 2. DEFORMATION AND STRAIN E e E e 2 1 2 1 t=0 t=t >0 X X x x ( X ,t) ( X ,t=0) Figure 2.1: The Lagrangian and Eulerian coordinate systems Lagrangian or Material coordinates: This is the coordinate system typically used in modeling solids. The independent variables are time, t , and the particle, or position of the particle at some reference time (usually t = 0), X . Thus if T is temperature, T ( X , t ) is the temperature of the particle at position X at the reference time, for all time. This formulation would be used for example, if the thermal degradation of a material is a function of what temperature it has been in its past. In this coordinate system we think of ourselves as fixed to a particle, and the indices are typically denoted by upper case letters, v = v I E I...
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- Fall '06