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eqnimpl

# eqnimpl - EN224 Linear Elasticity Division of Engineering...

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EN224: Linear Elasticity Division of Engineering 1.5 Field Equations Implied by the Fundamental System We now derive several auxiliary field equations which follow as a consequence of the field equations listed in the preceding section. These field equations will be useful when we begin to develop techniques for solving boundary value problems. Strain Equations of Compatibility Motivation : When we solve boundary value problems in linear elasticity, we sometimes solve for the stress field (for example, we may use an Airy stress function to generate a solution), and then need to deduce the displacement field. How do we do this? Alternatively, we might ask the question: are all equilibrium stress fields admissible solutions to linear elastic boundary value problems? The answer is, not necessarily. Given the stress, one could compute a distribution of strain in the solid. However, not all strain distributions can be derived from a displacement field. Below, we list the conditions necessary and sufficient to guarantee that a strain distribution can be derived from a displacement field. Let u be and on Then (1) Or Or Proof: Substitute into any of the above, and use smoothness of u, together with a liberal dose of algebra. Conversely, if is simply connected, and is a symmetric tensor field satisfying (1) on Then there exists a displacement field u satisfying Furthermore, given the strain field, we can compute the displacement field as follows: Generated by www.PDFonFly.com at 1/10/2010 9:36:07 PM URL: http://www.engin.brown.edu/courses/en224/eqnimpl/eqnimpl.html

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