Mathematical details of the flow-grain predictor of grain orientation

In the late 1970s, a finite element model was being developed at Colorado State University to rationally predict the stress distribution in a piece of wood, containing a knot, loaded in uniaxial tension. The mesh generation associated with this model used the theory of laminar flow (flow-grain analogy) about an elliptical obstruction as its theoretical basis. The knot, located in the longitudinal-tangential plane of the wood, was analogized as an elliptical obstruction in a fluid stream. Though the basic mathematical formulation for the theory has been available since at least the early 1950s, a detailed derivation of how to use the flow-grain model to determine the localized grain deviation has not appeared in the literature. In this technical note, the process for determining the grain deviation at any point in the plane of the wood containing the knot is made explicit.

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