Membranes

A membrane is assumed to be a perfectly flexible, thin elastic fabric, which is uniformly stretched in all directions by a tension which has a constant value N per unit length along any section or boundary. The deflection u (= u₃) satisfies the differential equation $$-N\left( u,11+u,22 \right)=-N\Delta u=p$$ where p is the lateral pressure. The traction across a cut is the product of the tension N and the derivative in the direction of the normal vector n = {n₁, n₂}T of the cut, $$t=N\frac{\partial u}{\partial n}=N\left( u,1{{n}_{1}}+u{{,}_{2}}{{n}_{2}} \right),$$ that is the N-fold normal derivative or N-fold slope. The close connection between the slope and the traction expresses Fig. 3.1. The greater the pressure the more the membrane will deflect and the greater the slope on the boundary and, therefore, also the traction t on the boundary.