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.

在这个简短的说明中，我们通过线性化应力和变形梯度之间的隐式本构关系来开发一个在柯西应力和线性化应变中都是线性的本构关系，这些关系已经被用来描述弹性体的响应 (Rajagopal , KR. 论隐式本构理论.数学应用2003; 28: 279–319)，假设位移梯度很小。这些隐式方程包括经典的线性化弹性本构近似以及某些类型的本构关系，这些本构关系意味着张力中的限制应变，作为特殊的子类。此外，所建立的本构关系允许材料模量取决于密度；因此，它们可用于描述多孔材料（例如多孔金属、骨骼、岩石和混凝土）的小变形响应。