The article considers the basic mechanism of damage to sandwich panels, which is the local instability of the compressed panel facing. The fac-ing is connected to a shear-deformable core, which is treated as an infinite substrate in most instability problems. The paper addresses the prob-lem of determining the critical force in the case of a core whose properties (elastic moduli) differ in compression and tension. The distinction be-tween two zones (compression, tension) is substantively justified because the experimentally determined moduli of core elasticity in compres-sion and tension always differ from each other. This research provides equations for determining the critical force using the energy method. The examples illustrate the procedure for determining the critical load and the importance of the discussed issue. The obtained results indicate that the physical non-linearity of the core material influences the load value that causes local facing instability. The effect of this non-linearity has so far been overlooked in the literature in the context of sandwich panel mechanics. The presented solutions are of practical importance because they allow for the assessment of the sensitivity of the critical force and wrinkling stress to changes in core and facing physical properties.