  We consider mechanically-induced pattern formation within the framework of a
growing, planar, elastic rod attached to an elastic foundation. Through a
combination of weakly nonlinear analysis and numerical methods, we identify how
the shape and type of buckling (super- or subcritical) depend on material
parameters, and a complex phase-space of transition from super-to subcritical
is uncovered. We then examine the effect of heterogeneity on buckling and
post-buckling behaviour, in the context of a heterogeneous substrate adhesion,
elastic stiffness, or growth. We show how the same functional form of
heterogeneity in different properties is manifest in a vastly differing
post-buckled shape. A form of inverse problem is then considered: whether
different functional forms could produce the same shape, and if this could be
detected. Finally, a fourth form of heterogeneity, an imperfect foundation, is
incorporated and shown to have a more dramatic impact on the buckling
instability, a difference that can be qualitatively understood via the weakly
nonlinear analysis.
