A first order homogenization approach for structural elements: Beam kinematics

This contribution deals with a first order homogenization approach for structural elements with the focus on shear deformable beam elements. The homogenization theory itself is based on the well‐known Hill‐Mandel condition. Therefore, the numerical treatment leads to a coupled multiscale approach whose calculation can be performed with the so‐called FE2 scheme. As a result, not only cross‐sectional properties in the linear elastic case can be evaluated but also geometrical and physical nonlinearities can be considered. Since the mesoscale, which will be homogenized, is modelled by 3D brick elements, the considered global structure only needs to be periodic to allow a definition of a representative volume element (RVE). Thus, the procedure is not limited to pure cross‐sectional modelling, but is also able to consider inhomogeneities in the longitudinal direction of the system. As the beam kinematics do not describe the complete deformation gradient or strain tensor, the equilibrium condition introduces a length dependency of the homogenized shear stiffnesses and the stiffnesses which depend on them due to eccentricities. This is due to the fact that the averaged shear deformation of the RVE does not linearly depend on its length multiplied by the beam shear deformation. To overcome this problem, additional constraints are introduced which ensure this linear connection. They are chosen in a way that they do not contribute to the virtual internal work. Therefore, the Hill‐Mandel condition is not affected by them. With these assumptions, the length dependency is eliminated and the averaged shear deformation of the RVE equals the shear deformation of the beam kinematics. As a result, the method is able to reproduce well‐known cross‐sectional values known from the literature and is also capable to evaluate complex nonlinear systems. For the latter, the agreement of the results with those of continuum models is very satisfying.