In this paper, an extension of the Timoshenko model for plane beams is outlined, with the aim of describing, under the assumption of small displacements and strains, a class of dissipative mechanisms observed in cementitious materials. In the spirit of micromorphic continua, the modified beam model includes a novel kinematic descriptor, conceived as an average sliding relevant to a density of micro-cracks not varying along time. For the pairs of rough surfaces, in which such a distribution of micro-cracks is articulated, both an elastic deformation and a frictional dissipation are considered, similarly to what occurs for the fingers of the joints having a tooth saw profile. The system of governing differential equations, of the second order, is provided by a variational approach, endowed by standard boundary conditions. To this purpose, a generalized version of the principle of virtual work is used, in the spirit of Hamilton–Rayleigh approach, including as contributions: (i) the variation of the inner elastic energy, generated by the linear elasticity of the sound material and, in a nonlinear way, by the mutual, reversible deformation of the asperities inside the micro-cracks; (ii) the virtual work of the external actions consistent with the beam model, i.e., the distributed transversal forces and the moments per unit lengths; besides these two contributions, constituting the conservative part of the system, (iii) the dissipation due to friction specified through a smooth Rayleigh potential, entering a nonlinear dependence of viscous and Coulomb type on the sliding rate. Through a COMSOL Multiphysics"Equation missing" implementation, 1D finite element analyses are carried out to simulate structural elements subjected to three- and four-point bending tests with alternating loading cycles. The dissipation of energy is investigated at varying the model parameters, and the predictions turn out to be in agreement with preliminary data from an experimental campaign. The present approach is expected to provide a valuable tool for the quantitative and comparative assessment of the hysteresis cycles, favoring the robust design of cementitious materials.
