singular soluions

singular soluions - EN224 Linear Elasticity Division of...

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EN224: Linear Elasticity Division of Engineering 3.2 Singular solutions for the infinite solid. Our first 3D boundary value problems will be the simplest: we will derive certain important solutions for an infinite solid. Although one rarely encounters infinite solids in practice, these solutions are useful because one can play clever games with them. For example, we could find a distribution of singular states within an infinite solid that will generate a solution for a bounded region: this is the basis of the boundary element method. We will also find that many important solutions in micromechanics can be derived from the basic singular states, including fields for inclusions, cracks, and dislocations. We will also use this opportunity to illustrate a few of the techniques that are available for solving boundary value problems. Concentrated Force in an Infinite Solid (Kelvin State) This solution is so important that we will derive it using two different methods, just to be sure! We begin by stating the boundary value problem carefully. We will see that the strain energy associated with the solution is unbounded, so we have to take special care to ensure that the solution is well posed. Find with Generated by www.PDFonFly.com at 1/10/2010 9:42:17 PM URL: http://www.engin.brown.edu/courses/en224/singular/singular.html
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where denotes any region enclosing the origin. It may be shown that these conditions are necessary and sufficient to determine our solution uniquely; for now, we will assume that this is the case. Solution obtained by guesswork We can generate 3D elastostatic states by substituting any harmonic function into the potential representations outlined in the preceding section. If we are lucky, we will find a potential that generates the solution we are interested in. Examine the list of Boussinesq potentials listed in the preceding section. Solution A generates solutions with , while solutions E, F, G generate solutions with We would not expect the point force solution to have either of these characteristics, so it makes sense to try to generate our solution from representations B, C, or D. We are looking for a potential function that generates displacements that decay as 1/R. We are looking for a potential that is harmonic everywhere except the origin. We could try as our first guess Amazingly, this turns out to work: it generates the solution for a point force acting in the direction at the origin. We need to check that all the conditions listed above are satisfied, and compute the value of C. Evidently, (c) and (d) are both satisfied. To compute the value of C and to check condition (a), let us evaluate the integral over a spherical surface centered at the origin. First, compute the stress: Note , whence Introduce a spherical polar coordinate system: to see that Generated by www.PDFonFly.com at 1/10/2010 9:42:17 PM URL: http://www.engin.brown.edu/courses/en224/singular/singular.html
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To satisfy take So that the appropriate potential is
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singular soluions - EN224 Linear Elasticity Division of...

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