Consistent Euler–Bernoulli beam theory in statics for gradient elasticity based on Laplacians of stress and strain

The Euler–Bernoulli beam bending theory in engineering mechanics assumes that the material behavior is isotropic elastic and that plane cross sections remain plane and rigid. It is well-known that this theory suffers from inconsistencies that, e.g., the shear strain is always vanishing, whereas the shear stress does not vanish. In recent work, consistent Euler–Bernoulli beam theories in classical and explicit gradient elasticities were accomplished by assuming the constitutive response to be anisotropic elastic, subject to internal constraints. This approach is extended in the present paper to get consistent Euler–Bernoulli beam theory for gradient elasticity based on Laplacians of stress and strain. The developed beam theory is employed to discuss bending of cantilever beams.