  Nonlinear mechanics of solids is an exciting field that encompasses both
beautiful mathematics, such as the emergence of instabilities and the formation
of complex patterns, as well as multiple applications. Two-dimensional crystals
and van der Waals (vdW) heterostructures allow revisiting this field on the
atomic level, allowing much finer control over the parameters and offering
atomistic interpretation of experimental observations. In this work, we
consider the formation of instabilities consisting of radially-oriented
wrinkles around mono- and few-layer "bubbles" in two-dimensional vdW
heterostructures. Interestingly, the shape and wavelength of the wrinkles
depend not only on the thickness of the two-dimensional crystal forming the
bubble, but also on the atomistic structure of the interface between the bubble
and the substrate, which can be controlled by their relative orientation. We
argue that the periodic nature of these patterns emanates from an energetic
balance between the resistance of the top membrane to bending, which favors
large wavelength of wrinkles, and the membrane-substrate vdW attraction, which
favors small wrinkle amplitude. Employing the classical "Winkler foundation"
model of elasticity theory, we show that the number of radial wrinkles conveys
a valuable relationship between the bending rigidity of the top membrane and
the strength of the vdW interaction. Armed with this relationship, we use our
data to demonstrate a nontrivial dependence of the bending rigidity on the
number of layers in the top membrane, which shows two different regimes driven
by slippage between the layers, and a high sensitivity of the vdW force to the
alignment between the substrate and the membrane.
