An implicit constitutive relation for describing the small strain response of porous elastic solids whose material moduli are dependent on the density

In this short note, we develop a constitutive relation that is linear in both the Cauchy stress and the linearized strain, by linearizing implicit constitutive relations between the stress and the deformation gradient that have been put into place to describe the response of elastic bodies (Rajagopal, KR. On implicit constitutive theories. Applications of Mathematics 2003; 28: 279-319), by assuming that the displacement gradient is small. These implicit equations include the classical linearized elastic constitutive approximation as well as some classes of constitutive relations that imply limiting strain in tension, as special subclasses. Moreover, the constitutive relations that are developed allow the material moduli to depend on the density; thus, they can be used to describe the response of porous materials, such as porous metals, bone, rocks, and concrete undergoing small deformations.

Automated summary

In this note, a constitutive relation linear in both the Cauchy stress and linearized strain is developed by linearizing implicit constitutive relations between stress and deformation gradient. The developed constitutive relations include classic linearized elastic constitutive approximation and some classes implying limiting strain in tension, as special subclasses, with material moduli dependent on density. This allows for the description of responses in porous materials undergoing small deformations, such as porous metals, bone, rocks, and concrete.