An Euler–Bernoulli second strain gradient beam theory for cantilever sensors

Abstract This paper derives an Euler–Bernoulli beam theory for isotropic elastic materials based on a second strain gradient description. As such a description has been proved to allow for the definition of surface tension for solids, the equations satisfied by a beam featuring a through-thickness cohesion modulus gradient are established in order to describe the behaviour of micro cantilever sensors. Closed-form solutions are given for mechanical and chemical loadings. It is then shown that the involved material parameters seem virtually identifiable from full-field measurements and that the shape of the displacement field resulting from a chemical loading depends on the cantilever’s thickness as well as on the material parameters. This makes such a theory potentially able to explain some of the experimental results found in the literature.