Constitutively optimal governing equations for higher-grade elastic beams

A method is proposed herein to build beam equations for materials featuring higher-grade elasticity. As it based on the minimization of constitutive equation gap, static admissibility conditions are taken into account so that naturally converges usual resulting from Cauchy elasticity when dimensions large enough. The exemplified, Euler–Bernoulli beams, first-strain gradient and second-strain elasticity, which yield local non-local equations, respectively. solutions these computed tensile bending loads over a wide range in order assess role different grades global behavior simple mechanical structures. Under approach extends validity obtained previously or loadings. Considering this additionally allows distinguish two regimes, depending whether driven by surface (it denote regime as ecto-elastic regime) bulk. It may also model chemo-mechanical couplings. finally suggests affect stiffness same dimensional range, both should always be considered simultaneously analyzing experimental results.