Pure bending in non-linear micropolar elasticity

Large-deformation pure cylindrical bending of a homogeneous strip of uniform finite thickness with micropolar (Cosserats’) material constitution is analysed and related both to the large-deformation solution of classical (Cauchy’s) continuum theory and that of the small-deformation (linear) micropolar theory, known to exhibit a size effect. For a micropolar generalisation of the semi-linear material, the closed-form solution is obtained using a semi-inverse approach, which is equivalent to the solution in classical elasticity, possessing the same relationship between the curvature and the distribution of stress components and exhibiting the same softening effect relative to the linear solution. A couple stress of constant amount over the cross section develops in addition, in proportion to the curvature and the squared ratio between the characteristic bending length of the material and the thickness of the specimen and linear in Poisson’s ratio, in the same way as in the linear micropolar elasticity. The compound effect of non-linear softening and micropolar stiffening is analysed in detail with the view of providing a theoretical background for development of architected materials with internal structure for large-displacement applications. In particular, the size-effect estimate supplied by the linear analysis is shown to be in error, which builds up rapidly in highly deformed materials with relatively large characteristic bending length.