uniqueness

uniqueness - EN224: Linear Elasticity Division of...

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EN224: Linear Elasticity Division of Engineering 2. Theorems of Linear Elasticity We proceed to prove several useful theorems which follow as a consequence of the structure of the field equations of linear elasticity, and which are useful in interpreting or constructing solutions to boundary value problems. 2.1 Superposition Begin with the simplest case. Let be a region, and let be a time interval. Suppose that Then We have followed Sternberg’s notation here: denotes a set S of displacements, strains and stresses, while denotes the set of all displacement, strain and stress fields which satisfy the field equations of linear elasticity with body force b, density and compliance tensor field C on the region during the time interval . We will use this notation frequently. Thus, our theorem states that if and satisfy the field equations on the regionw ith body force and , respectively, then the linear combination satisfies the field equations with body force The proof is trivial: it follows from the linearity and homogeneity of the field equations. 2.2 Uniqueness It probably comes as no surprise that solutions to linear elastic boundary and initial value problems are unique. We proceed to show that this is indeed usually the case. Generated by www.PDFonFly.com at 1/10/2010 9:38:27 PM URL: http://www.engin.brown.edu/courses/en224/superpos/superpos.html#unique
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Elastodynamics (Neumann) Let be a region with boundary and let be a time interval. Suppose With And Then To prove this, consider . From the result stated in 2.1, it follows that with and Now, consider the rate of work done on the solid by the external loads associated with S. Evidently because body forces vanish within the solid, while either velocities or tractions vanish on the boundary of the solid. From the power identity Assume that C is such that a strain energy density may be defined (can you remember the conditions on C for this to be the case?) Then Generated by www.PDFonFly.com at 1/10/2010 9:38:27 PM URL: http://www.engin.brown.edu/courses/en224/superpos/superpos.html#unique
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Hence Integrating Observe that so the constant of integration must vanish. Furthermore, and are both positive definite,so that . Finally, we conclude that so that the strains, stresses and displacements associated with S must vanish. Thus , as required. Note that for elastodynamics, the displacement field is unique even for traction boundary value problems. This is not the case for elastostatic states. Observe also that our proof has one weakness: we assumed that the volume integrals in the power identity converge. Occasionally, we deal with linear elastic states which have an unbounded strain energy (examples include point forces, and dislocations). Additional constraints are required to ensure that these states are unique. We also assumed that the boundaries
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uniqueness - EN224: Linear Elasticity Division of...

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