The effect of surface bending and surface stress on the transmission of a vertical line force in soft materials

When the surface of a soft substrate that carries a constant in-plane residual stress is indented by a concentrated line force, its profile near the applied load is found to have a kink, which results from a local balance of the surface stresses and the imposed force. Although the local bulk stresses in the substrate no longer have a net contribution to this force balance, they nevertheless grow according to a weak logarithmic singularity with respect to distance from the line load. Here we study how a normal line load is transmitted across a solid surface that can provide additional resistance due to bending deformation; we present an exact closed-form solution. Our analysis shows that the ability of the surface to resist bending completely regularizes the stress field - it is continuously differentiable everywhere. In particular, the stress state in the elastic substrate is hydrostatic right underneath the line load if the material is incompressible. Its maximum value is directly proportional to the applied normal load and inversely proportional to the elasto-bending length. It also depends on a dimensionless parameter which is the ratio of the elasto-capillary length to the elasto-bending length.