We consider the question of how bad is the geometry of a body in terms of the magnitude of the stresses induced in it.
A traditional stress concentration factor indicates, for a given loading condition, the ratio between the maximum of a component of the stress and the value of that component for an idealized geometry. We regard those nominal stresses as traction boundary conditions and generalize the notion of stress concentration factor in a number of ways. Firstly, we leave the material properties of the body open and consider the following question. Assume that one is performing a process of structural optimization by varying the distribution of material properties. What would be the smallest value of the stress concentration for the given load?
Next, we arrive at a purely geometric property of the body. Noting that one usually does not know the exact nature of the loading, we allow the force distribution to vary and define the generalized stress concentration factor as the supremum of optimal stress concentration factors over all applied forces. The stress object considered here contains not only the traditional stress tensor but also a self force that may be thought of as the reaction of a 3-dimensional foundation.
We prove that the generalized stress concentration factor is equal to the norm of the mapping that takes a Sobolev function defined on the interior of the body and extends it to the boundary. Finally, we present some properties of the generalized stress concentration factor.
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