Constitutively optimal governing equations for higher-grade elastic beams

International audienceA method is proposed herein to build beam equations for materials featuring higher-grade elasticity. As it is based on the minimization of the constitutive equation gap, static admissibility conditions are taken into account so that it naturally converges to the usual beam equations resulting from Cauchy elasticity when the beam dimensions are large enough. The method is exemplified, for Euler-Bernoulli beams, on first-strain gradient and second strain gradient elasticity, which yield local and non-local beam equations, respectively. The solutions of these equations are computed for tensile and bending loads over a wide range of beam dimensions in order to assess the role of the different grades on the global behavior of simple mechanical structures. Under first-strain gradient elasticity, the proposed approach extends