Mechanics of Saturated Colloidal Packings: A Comparison of Two Models

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.