A size-dependent model of strain gradient elastic laminated micro-beams with a weak adhesive layer

A size-dependent model for a laminated micro-beam with a soft adhesive is developed in the framework of strain gradient elasticity theory. The layered beam is constituted by two Euler–Bernoulli strain gradient elastic isotropic beams, joined through a strain gradient spring-type contact law at the adhesive level. The governing bending and extensional equations and boundary conditions are obtained by using the variational principle. The differential system shows a coupling between the flexural and axial behaviors of the upper and lower beams, due to the presence of interface terms related to shear and peeling stresses. Two benchmark problems have been presented through their closed-form solutions, namely a simply-supported laminated beam subjected to a uniform distributed load and a mode 2-type loading configuration of a layered axially deformable beam. Size effects and non-local phenomena, due to high strain concentrations, are highlighted. Though simple in their features, the examples prove the efficiency of the proposed approach in designing micro-scale layered beams.