Buckling analysis of a reinforced sandwich column using the Bloch wave theory

International audienceThis paper deals with sandwich structures whose core layer is made of a homogeneous foam periodically strengthened by orthogonal reinforcements. Beside traditional sandwiches which generally display satisfactory specific flexural properties but fatally insubstantial stiffnesses in the through-thickness direction, 3D reinforced sandwich materials provide optimal out-of-plane mechanical properties. Despite this, buckling remains one of the major failure modes of such structures and, compared to the case of traditional sandwiches, both global and local buckling phenomena are more complicated in presence of transverse reinforcements. Indeed, in most cases, the modal deformed shapes involve simultaneously the skins and the reinforcements in an intricate way. The main feature of these buckling modes is periodicity, but the typical wave length appears to be generally different from the characteristic length between reinforcements. However, it is possible to investigate such periodic modes on a simple unit cell by using the so-called Bloch wave theory. In this work, an efficient procedure is defined so as to deal with the buckling behavior of a sandwich column with periodic orthogonal reinforcements. First, a numerical method is implemented in the framework of the commercial software Abaqus. The evaluation of the critical strains is performed on a unit cell: an initial average compressive strain is enforced, then natural frequencies are computed and the critical strains are deduced by extrapolation of the previous eigenvalues. A Python program is developed so as to automate these successive calculation steps and a Fortran program is also needed (within Abaqus) in order to cope with the two real and imaginary problems to be solved due to the Bloch periodic conditions. Furthermore, an exact analytical solution of this problem is obtained in the particular case of a reinforced sandwich with no foam core (for simplicity purposes). The analytical and numerical solutions obtained with a unit cell model are finally compared to the results of numerical computations performed on a complete beam with an arbitrary number of cells, for validation purposes. The critical strains/displacements are found to be in very good agreement and the buckling modes rebuilt from the real and imaginary components of the unit cell modal solutions perfectly coincide with the buckling modes of the complete beam obtained through a linearized buckling analysis