Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams

This paper covers a large variety of theoretical generic beam models including some small length scale terms. Strain gradient elasticity and Eringen’s nonlocal elasticity models are applied to beam mechanics including Euler–Bernoulli, Timoshenko and higher-order shear beam models. The buckling and vibration behaviour of these generalized shear beam models is investigated for pinned–pinned boundary conditions. The variational formulation of these enriched beam models is given leading to consistent variationally-based boundary conditions. The paper first starts with the axial behaviour of gradient or nonlocal elasticity bars. The beam behaviour is then analyzed using a unified framework, where the kinematics classification is presented from a generalized gradient constitutive law. It is shown that higher-order shear beam models can be classified in a common gradient elasticity Timoshenko theory, whatever the shear strain distribution assumptions over the cross section. We show the kinematics equivalence between Bickford–Reddy higher-order shear beam model and Shi–Voyiadjis higher-order shear beam model, even if both models are statically not equivalent (from the stress calculation). This equivalence is highlighted on buckling and vibrations results. The model valid for macrostructures is generalized for micro or nanostructures using some nonlocal and gradient theories to account for small scale effects, in the axial and in the bending directions. We both use the Eringen’s based integral theory and the gradient theory to derive the buckling and vibration differential equations. These two theories can be connected using a generalized hybrid nonlocal law. Eringen’s model is compared to a stress gradient model, whereas the gradient elasticity theory is typically a strain gradient theory. The nonlocal framework is also developed in a variational consistent framework, for bending, vibrations and buckling configurations. The nonlocality is shown to be equivalent to higher-order inertia modelling for the dynamics analysis. Buckling and vibrations solutions are presented for the nonlocal higher-order beam/column models with pinned–pinned boundary conditions. We finally analyse the main characteristics of both nonlocal and gradient theories to capture the small scale effects for micro and nanostructures. Stiffening or softening effect of gradient or nonlocal elasticity models are discussed for the buckling and the vibrations analyses.