A non-classical Bernoulli-Euler beam model based on a simplified micromorphic elasticity theory

Abstract A new non-classical Bernoulli-Euler (B-E) beam model is developed using a simplified micromorphic elasticity theory. This micromorphic theory, which contains 7 independent material constants, is first proposed by simplifying the classical Eringen-Mindlin micromorphic theory for isotropic linear elastic materials, which includes 18 elastic constants. The new B-L beam model is then formulated by applying the simplified micromorphic theory and employing a variational approach based on Hamilton's principle, which leads to the determination of the equations of motion and boundary conditions simultaneously. The newly developed beam model contains four elastic constants to account for micro-deformations of material particles and one material length scale parameter to describe microstructure-dependent size effects. The current micromorphic beam model reduces to its classical elasticity-based counterpart when the microstructure effects are not considered. To illustrate the new beam model, static bending and buckling of a cantilever beam and wave propagation in an infinitely long beam are analytically solved by directly applying the model. For the static bending problem, it is revealed that the beam deflection varies with the microstructure-dependent material constants, which is significant for very thin beams. For the buckling problem, the critical buckling load is found to change with the microstructural constants and beam thickness. For the wave propagation problem, a band gap is seen to exist for both the longitudinal and transverse waves.