Numerical solution of the beam pure bending problem in the framework of the theory of elastic materials with voids

The beam pure bending problem is solved numerically in the framework of the theory of elastic materials with voids, also called the elastic micro-dilatation theory. This theory describes a particular case of Mindlin’s type media with microstructure and contains only non-zero micro-dilatations. The physical meaning of the model involves a more comprehensive description of the stress-strain state of porous media, in which the volume pore fraction changes in response to applied external loads. In this paper we consider a generalized variant of the theory with surface effects. The finite element method is used to find a solution to the beam pure bending problem. The semi-inverse analytical micro-dilatation solution is compared with the FEM solution. It is shown that, unlike the analytical solution, in the FEM solution all the components of the stress-strain tensors are not equal to zero. There are also self-equilibrated normal stress components in the width direction and shear stress components in the cross section perpendicular to the neutral plane. In the numerical solution, all the stress boundary conditions are accurately satisfied on the free surfaces of the beam. Comparison of analytical and numerical simulation results shows that the analytical solution allows one to obtain sufficiently precise estimates to predict the influence of non-classical scale effects and surface effects on the effective stiffness and stress state of porous beams.