Numerical multiscale modelling of nonlinear elastowetting

We investigate here the static finite deformation of two (or more) three-dimensional nonlinear elastic solids which merge in a third medium, defining a triple line (TL). The total energy, accounting for elastic, surface and possible gravity potentials is then minimized numerically in order to solve two physical problems of interest: (i) a soft incompressible axisymmetric drop at rest on a stiffer substrate and (ii) a stiff drop at rest upon a softer substrate, both in situations of strong substrate surface tension effects. Because of the interplay between surface and bulk properties, the behavior of the TL is related to another smaller length-scale (typically micrometre scale) than the elastic solids length-scale (typically millimetre scale). A unified deformability parameter relating all scales is introduced and an adaptive finite element discretization of the domain is used. A first example approached by our model is the elasto-wetting problem in which a very soft drop sits on a stiffer elastic solid. The numerical predictions of our model are compared with the analytical results of the linear elastic theory. In particular, our computations show that Laplace pressure, together with the asymmetry of the solid surface tensions (between the wet and dry part of the interface), well account for the rotation of the cusp of the ridge. In the second example, motivated by indentation problems of a nonlinear elastic substrate by a stiff sphere, we analyse the interplay of the solid surface tensions and elasticity at small-scale. The numerical results also show that gravity might have an important effect and must be included in any micro-scale modelling.