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static value problems

static value problems - EN224 Linear Elasticity Division of...

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EN224: Linear Elasticity Division of Engineering 3. 3D Static Boundary Value Problems Objective: Find elastostatic states in 3D solids with prescribed boundary conditions This is very difficult to do in general! A few useful techniques: (1) Represent the solution in terms of harmonic potentials . We can then generate elastostatic states from any harmonic function, and if we re very lucky we find the solution to the problem we are interested in. Many useful solutions have been found this way, but the chances of finding a new one are small! (2) Try superposition and the method of images. This will only solve rather simple problems, but it s worth a shot. (3) Use the reciprocal theorem. This is a good way of solving dislocation problems. If you re very lucky it will give you a general solution for your region, but you need to solve a tricky singular problem first. (4) Transforms . This is the most powerful approach, as it gives you a formal procedure to follow. Examples include Fourier transforms (good for half - space problems and for the infinite solid), Hankel transforms (good for axisymmetric problems), Mellin transforms (good for quarter - space problems), among others. (5) Approximate the solution . If the 3D boundary value problem can t be solved, one can sometimes make progress by reducing the problem to 2D. Examples include plane problems in elasticity, Saint Venant torsion, plates and shell theory.

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