Deformation of an elastic substrate due to a sessile drop

Abstract

The small deformation of a horizontal elastic substrate due to an axisymmetric sessile drop resting on the substrate is calculated for exact hydrostatic interfacial shapes computed by solving the Young–Laplace equation using an iterative method. The drop height, base radius, and top interfacial curvature are documented in terms of the ratio of the capillary length to the equivalent drop radius defined with respect to the drop volume. The substrate is subjected to a uniform discoidal load over the drop base due to the elevated interior pressure, and a singular ring load with a normal and possibly a tangential component around the contact line. The relative contributions of the Laplace pressure and drop weight to the normal load over the drop base are delineated in terms of the drop size. Once the interfacial shapes are available, the substrate deformation field is reconstructed from three modular kernels originating from the axisymmetric versions of the Boussinesq and Cerruti solutions for a semi-infinite elastic medium. The effect of gravity causing the drop to deviate from the spherical shape has an important influence on the magnitude and profile of the substrate deformation. Numerical results illustrate the effect of spreading of the vertical component of the capillary force and the significance of the tangential component of the capillary force around the contact line.