Supercritical and subcritical buckling bifurcations for a compressible hyperelastic slab subjected to compression

• A non-linear correction for the Euler buckling formula is obtained for a compressible slab.
• A criterion is provided to determine the bifurcation type: supercritical or subcritical.
• Approximate analytical solutions for post-buckling bifurcations are obtained.
• An analytical formula for the post-buckling amplitude is obtained.

We study the buckling bifurcation of a compressible hyperelastic slab under compression with sliding–sliding end conditions. The combined series-asymptotic expansions method is used to derive the simplified model equations. Linear bifurcation analysis yields the critical stress value of buckling, which gives a non-linear correction to the classical Euler buckling formula. The correction is due to the geometrical non-linearities coupled with the material non-linearities. Then through non-linear bifurcation analysis, the approximate analytical solutions for the post-buckling deformations are obtained. The amplitude of buckling is expressed explicitly in terms of the aspect ratio, the incremental dimensionless engineering stress, the mode of buckling and the material constants. Most importantly, we find that both supercritical and subcritical buckling could occur for compressible materials. The bifurcation type depends on the material constants, the geometry of the slab and the mode numbers.