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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.

We study the state of stress and strain in a square plate containing an elliptic hole in the case of a porous elastic solid undergoing small strain, using a constitutive relation that has been put into place recently to describe the response of such solids undergoing small strains. We carry out the study by solving the problem numerically. We verify that our numerical solutions agree with those for the classical linearized elastic solid when certain appropriate material parameters are set to zero. We show that the stress concentration factor in the case of the porous elastic solid can be much higher, as much as 300% of the stress concentration in the case of the classical linearized elastic solid, when the aspect ratio is sufficiently small, depending on the values of certain material parameters. The difference between the stress concentration for the porous solid increases as the aspect ratio (the ratio of the major axis to the minor axis) decreases. By allowing the aspect ratio of the ellipse to go to zero, we can obtain the state of stress and strain adjacent to a crack in the square plate; however, in this limit, the strains would greatly exceed the assumption under which the constitutive theory is derived.

Most solid bodies are porous and in such bodies we expect the material properties to vary with porosity and hence with the density. Such bodies whose properties depend on the density cannot be described by the classical linearized elastic constitutive relation when they are undergoing small deformations, as the material moduli cannot depend on density. A constitutive relation which is valid in the small displacement gradient range wherein the material moduli are dependent on the density can be developed by linearizing implicit constitutive relations for describing elastic bodies introduced recently by Rajagopal [1]. In this paper, we determine the stress concentration due to a hole in a slab that is subject to biaxial loading in a body described by such a constitutive relation. We find that the stress concentration can be much as 147% higher than that for classical linearized elasticity for the range of loadings considered in the study.