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Poroelastic relaxation indentation of thin layers of gels

Yuhang Hu's picture

We develop a method of poroelastic relaxation indentation (PRI) to characterize thin layers of gels.  The solution to the time-dependent boundary-value problem is obtained in a remarkably simple form, so that the force-relaxation curve obtained by indenting a gel readily determines all the poroelastic constants of the gel—the shear modulus, Poisson’s ratio, and the effective diffusivity.  The method is demonstrated with a layer of polydimethylsiloxane immersed in heptane.

The paper is accepted for publication by J. Appl. Phys, and can be downloaded from:


Rui He's picture

Hello, Yuhang, I have read your paper, it's a good method, and I wonder whether your method can be used to determine the poroelastic constants of the saturated soil ?

Yuhang Hu's picture

Hi Rui, theoretically it works for saturated soil material. Just several points need to be considered.

In our model, the individual solvent molecules and solid skeleton are assumed to be incompressible. The volume change of the material is due to the change of solvent concentration.

No inelastic deformation has been taken into consideration in our method. 

Rui He's picture

Hi, Yuhang, thanks very much for your reply. When comes to soil material, I still have some questions

1. There're only three constants in poroelastic gels, however, there're four poroelastic constants (Lame constant and two more Biot's constants) in soil material, and would you explain what "effective diffusivity" correspond to in soil?

2. In your experiment, the indentation can be treated as purely elastic, however, as you can imagine, lots of soil materials are very soft and inelastic (such as plastic) properties can easily comes out. Do you have any idea of how to treat this problem?

Yuhang Hu's picture

Hi Rui,

1. The poroelasticity theory which was orignially developed By Terzaghi and later genaralized by Biot has been studied for many years. There are many versions to define the poroelastic constants. But in general, there are five independent constants including the kinetic constant (diffusivity or permeability).  In gel theory it is usually assumed that the individual polymer chains and solvent molecules are incompressible, so we are left with three paramters. For soil material, Darcy's law is addapted as kinetic model.  The effective diffusivity is D=2(1-v)Gk/(1-2v)η, where k is the so-called permeability and η is the viscosity of the fluid. Details about how the expression is derived, you can refer to the following paper:

Jinhwan Yoon, Shengqiang Cai, Zhigang Suo, and Ryan C. Hayward. Poroelastic swelling kinetics of thin hydrogel layers: Comparison of theory and experiment. Soft Matter 6, 6004-6012 (2010).


2. The inelastic deformation so fas has not been considered in the indentation method. About theory, it has long been studied for soil material. You may refer to the following book:

O. Coussy, Poromechanics, (2004)

Rui He's picture

Hi, Yuhang, thanks very much for your reply, and I have read the paper and book you mentioned, they're very helpful, thanks!

I could not find the void ratio or porosity that was used in your study. Could you please what values you used for void ratio?

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