On the deformation of elastic rods in a symmetric micromorphic theory

We investigate the deformation of an elastic isotropic rod in the framework of a simplified micromorphic theory introduced by Forest and Sievert. In contrast with the classical micromorphic model, which includes 18 elastic constants, this theory is characterized by constitutive equations which involve 6 constants and a material length scale parameter to describe microstructure-dependent size effects. First, we formulate the equilibrium problem of a rod subjected to a resultant force and resultant moment acting on its plane ends. Then, we generalize the method of construction of the solution avoiding a priori assumptions proposed by Iesan in classical elasticity. The method leads to the decomposition of the general problem into the basic problems of extension, bending, torsion and flexure. The analytical solutions are obtained in a closed form and reduced to their classical elasticity counterparts when the microstructure effects are suppressed. The results are useful to obtain explicit solutions when the shape of the cross section is assigned and are preliminary to the solution of the problem of cylinders loaded on a lateral surface such as the Almansi–Michel problem.