In compression, buckling will occur preferentially if (P_cr / A) , the Buckling Stress, is less than the Yield Stress of the material. If the Buckling Stress is greater than the Yield stress the material will plastically deform without buckling. For a simple rod of material with a circular cross-section the buckling stress depends upon (L / r )² and the Yield Stress, s_y , has a constant value.

The diagram opposite shows the effect of
changing the ratio of sample length to sample
radius for two structural materials. The blue
curves represent the buckling failure, the
horizontal black lines the value of the Yield
Stress. The vertical black lines represent the
values of (L / r) at which the samples pass from
the Elastic/Plastic deformation mode to the
Buckling mode.

The stress at which buckling occurs is also dependent upon the boundary conditions of the sample

From: Hibbeler, Macmillan, (1993)