The buckling of a soft elastic sample under growth or swelling has
highlighted a new interest in materials science, morphogenesis, and biology or
physiology. Indeed, the change of mass or volume is a common fact of any living
species, and on a scale larger than the cell size, a macroscopic view can help
to explain many features of common observation. Many morphologies of soft
materials result from the accumulation of elastic compressive stress due to
growth, and thus from the minimization of a nonlinear elastic energy. The
similarity between growth and compression of a piece of rubber has revived the
instability formalism of nonlinear elastic samples under compression, and in
particular Biot's instability. Here we present a modern treatment of this
instability in the light of complex analysis and demonstrate the richness of
possible profiles that an interface can present under buckling, even if one
restricts oneself to the two spatial dimensions. Special attention is given to
wrinkles, folds and cusps, a surprising observation in swelling gels or clays.
The standard techniques of complex analysis, nonlinear bifurcation theory and
path-independent integrals are revisited to highlight the role of physical
parameters at the origin of the observed patterns below and above the Biot
threshold.Comment: 40 pages. 8 figures 179 reference