SAINT-VENANTS PRINCIPLE IN LINEAR ISOTROPIC ELASTICITY FOR INCOMPRESSIBLE OR NEARLY INCOMPRESSIBLE MATERIALS

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 elastosta tics for a cylinder. For the semi-infinite cylinder with traction-free lateral surface and self-equilibrated loads at the near end, it has b een shown that the stresses decay exponentially from the end and resul ts are available for the estimated decay rate, which is a lower bound for the exact decay rate. These results are, however, generally conser vative in that they underestimate the exact decay rate. Another shortc oming, which motivated the present investigation, is that the estimate d decay rates tend to zero as the Poisson's ratio nu tends to the Valu e 1/2. Thus for the limiting case of an incompressible material, these methods fail to establish exponential decay. The purpose of the prese nt paper is to remedy this defect. In particular, an exponential decay estimate is established with estimated decay rate independent of Pois son's ratio. Thus, in particular, the results here hold in the incompr essible limit as nu --> 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 nu and that stress decay estimates for th e cylinder problem with estimated decay rates dependent on nu are, in fact, to be expected. However, in the absence of such results that do not deteriorate as nu --> 1/2, we obtain here an estimated decay rate that is independent of nu.