Refined equations and buckling modes under four-point bending loading of the sandwich test specimen with composite facing layers and a transversely flexible core

A refined geometrically nonlinear theory of sandwich plates and shells with a flexible soft core and composite facing layers having low shear and compressive stiffness was introduced. It is based on the refinement of the shear model of S.P. Timoshenko taking into account transverse compression, as well as on the use of simplified three-dimensional equations of the theory of elasticity for a transversely flexible core. By integrating the equations over the transverse coordinate to describe the stress-strain state of the core, two two-dimensional unknown functions were introduced, which are transverse shear stresses, constants by thickness. For describing the static deformation process with high variability of the parameters of the stress-strain state of the core, two variants of two-dimensional geometrically nonlinear equations were derived. In the first one, the geometric nonlinearity was considered in the standard approximation by retaining the terms containing only the membrane forces in the facing layers. In the second variant additional geometrically nonlinear terms of a higher order of smallness were kept. Using the compiled equations, a geometrically and physically nonlinear problem of four-point bending of the sandwich specimen was formulated with regard to the physically nonlinear relationship between the transverse shear stresses and the corresponding shear strains in the facing layers. A numerical method stemming from the finite sum method (integrating matrix method) was developed for its solution and post-buckling behavior of specimen was investigated. It was shown that when the specimens are tested, their failure can be caused by the transverse shear buckling mode of the facing layer near the loading roller.