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Surface Stresses/Couples in Solid and the Neumann Triangle Balance of Forces

Chung-Yuen Hui Field of Theoretical and Applied Mechanics, Dept. of Mechanical and Aerospace Engineering, Cornell University


Elasto-capillarity phenomena – solid deformation driven by liquid surface tension – have been extensively studied, but are distinct from phenomena driven by solid surface stress . A characteristic length scale that describes the deformation of a solid due to surface tension is given by the ratio between the surface stress and elastic modulus (E), /E. For stiff solids such as metals and ceramics, this length is smaller than inter-atomic distances. However, for soft materials such as elastomers and gels with E ranging from tens of Pa to several MPa, /E ranges from tens of nm to hundreds of µm. Effects of this large value of /E can be seen in different contexts. For example, surface stress constraints the faithful reproduction of soft surfaces in replica molding. The adhesion and contact mechanics of small hard particles on soft elastic substrates is known to be modulated by surface stress. Elastic deformation and surface stresses can alter the wetting mechanics of liquid drops on compliant substrates. The equilibrium configuration of a liquid drop on a deformable soft substrate is no longer a material property determined by the Young–Dupré equation ; but instead, by two separate conditions : a local balance of interface stresses (Neumann triangle of forces) and configuration energy balance. I will show an interesting example : when two identical liquid drops separated by a thin elastic film are place right on top of each other (one on either side of the film), they repel, eventually reaching an equilibrium configuration where there is a small overlap.
In this talk I focus on a classical problem closely related to the Neumann triangle of forces. A line force acting on a soft elastic solid, say due to the surface tension of a liquid drop, can cause significant deformation and the formation of a kink at the point of force application. The formation of kink regularizes the classical solution of linear elasticity which predicts that the displacement underneath the line load is unbounded. Analysis based on linearized elasticity theory shows that sufficiently close to its point of application, the applied force is borne entirely by the surface stress, not by the elasticity of the substrate ; this local balance of three forces is called Neumann’s Triangle. However, it is not difficult to imagine realistic properties for which this force balance cannot be satisfied. For example, if the line force corresponds to surface tension of water, the numerical values of solid-vapor and solid-liquid surface stresses can easily be such that their sum in insufficient to balance the applied force. In such cases conventional (or naïve) Neumann triangle of surface forces must break down. Here we study how force balance is rescued from the breakdown of naïve Neumann’s Triangle by a combination of (a) large hyper-elastic deformations of the underlying bulk solid, and (b) increase in surface stress due to surface elasticity (surface stiffening).

I will also raise the possibility that surfaces of solids can also store energy in bending. An important example is lipid bilayer. I will demonstrate that the ability for a surface to resist bending completely regularizes the stress and strain field underneath the line load – it is continuously differentiable everywhere.