Saint-Venant's principle in linear isotropic elasticity for incompressible or nearly incompressible materials

Abstract

One of the unresolved issues on Saint-Venant's principle concerns the energy decay estimates established in the literature for the traction boundary-value problem of three-dimensional linear isotropic elastostatics for a cylinder. For the semi-infinite cylinder with traction-free lateral surface and self-equilibrated loads at the near end, it has been shown that the stresses decay exponentially from the end and results were obtained for the estimated decay rate, which is a lower bound for the exact decay rate. These results are, however, generally conservative in that they underestimate the exact decay rate. Another shortcoming, which motivated the present investigation, is that the estimated decay rates tend to zero as the Poisson's ratio ν tends to the value 1/2. Thus for the limiting case of an incompressible material, these methods fail to establish exponential decay. The purpose of the present paper is to remedy this defect. In particular, an exponential decay estimate is established with estimated decay rate independent of Poisson's ratio. Thus, in particular, the results here hold in the incompressible limit as ν → 1/2. An alternative treatment directly for the incompressible case has been given recently. It should be noted that the stresses in the three-dimensional traction boundary-value problem do depend on Poisson's ratio ν and that stress decay estimates for the cylinder problem with estimated decay rates dependent on ν are, in fact, to be expected. However, in the absence of such results that do not deteriorate as ν → 1/2, we obtain here an estimated decay rate that is independent of ν.