Mechanics of Saturated Colloidal Packings: A Comparison of Two Models

Abstract

A successful prediction of the response of poroelastic material to external forces depends critically on the use of appropriate constitutive relation for the material. The most commonly used stress–strain relation that captures the behavior of poroelastic materials saturated with a liquid is that proposed by Biot (J Appl Phys 12:155–164, 1941). It is a linear theory akin to the generalized Hooke’s law containing material constants such as the effective bulk and shear modulus of the porous material. However, the effective elastic coefficients are not known a priori and need to be determined either via separate experiments or by fitting predictions with measurements. The main cause for this drawback is that Biot’s theory does not account for the microstructural details of the system. This limitation in Biot’s model can be overcome by utilizing the constitutive relation proposed by Russel et al. (Langmuir 5(24):1721–1730, 2008) for the case of colloidal packings. We show that in the linear limit, the constitutive relation proposed by Russel and coworkers is equivalent to that of Biot. The elastic coefficients obtained from such a linearization are related to the micro-structural details of the packing such as the particle modulus, the packing concentration and the nature of packing, thereby enabling a more effective utilization of Biot’s model for problems in the linear limit. The derivation ignores surface forces between the particles, which makes the results also applicable to particles whose sizes are beyond the colloidal range. We compare the predictions of Biot’s model to those of the linearized model of Russel and coworker’s for two different one-dimensional model problems: fluid outflow driven by an applied mechanical load, also termed as the consolidation problem, and wave propagation in a saturated colloidal packing.