Stress concentration due to the presence of a hole within the context of elastic bodies

Many rocks, metals, and concrete are porous, in fact most materials are porous. This would then imply that their properties depend on the density. In this report, we develop a constitutive relation to describe the response of elastic bodies that are linear in both the stress and the linearized strain with the material moduli depending on the density. Such a model is not possible within the context of the classical theory of linearized elasticity but is possible within the context of the implicit theory for elastic bodies that has been developed. The constitutive relations discussed in this paper can be useful to describe the response of porous elastic bodies in the small displacement gradient regime. Using these constitutive relations, we study the stress concentration due to the presence of a circular hole in a plate due to uniaxial extension. We find that the stress concentration factor can be significantly different from that in the case of the classical linearized elastic solid.