Projective Elasticity

Abstract

We present the foundations of a projective geometric theory of elasticity, as well as outline a few possible application possibilities. We give the description of the Cauchy stress and infinitesimal strain tensors compatible with coordinate description of projective geometry and derive their transformation rules under projective transformations. We identify these tensors with generalized conics and quadrics; which shows that the material law cannot be directly given as a correlation, and that the absence of normal stresses and normal strains are projective invariants. The transformation of the graph of the Airy stress function is given in a point-wise projective three dimensional way, which preserves Pucher's equation corresponding to a unidirectionally loaded membrane shell. The paper closes with an example considering tapered beams, showing how to find a statically possible stress distribution in trapezioidal plates relying on the Euler-Bernoulli beam theory for rectangular beams.