On Saint-Venant's principle in a poroelastic arch-like region

In this paper we consider the state of plane strain in an elastic material with voids occupying a curvilinear strip as an arch-like region described by R:a<r<b,0<h<x, where r and θ are polar coordinates and a, b, and ɷ (<2π) are prescribed positive constants. Such a curvilinear strip is maintained in equilibrium under self-equilibrated traction and equilibrated force applied on one of the edges, whereas the other three edges are traction free and subjected to zero volumetric fraction or zero equilibrated force. In fact, we study the case when one right or curved edge is loaded. Our aim is to derive some explicit spatial estimates describing how some appropriate measures of a specific Airy stress function and volume fraction evolve with respect to the distance to the loaded edge. The results of the present paper prove how the spatial decay rate varies with the constitutive profile. For the problem corresponding to a loaded right edge, we are able to establish an exponential decay estimate with respect to the angle θ. Whereas for the problem corresponding to a loaded curved edge, we establish an algebraical spatial decay with respect to the polar distance r, provided the angle ɷ is lower than the critical value π√2. The intended applications of these results concern various branches of medicine as for example the bone implants.