# Using indentation to characterize the poroelasticity of gels

Entry by Jianyu Li, AP225, Fall, 2010

## Contents

## Abstract

The paper reports using indentation to determine the mechanical properties of the gel such as elastic constants and permeability. The indenter is firstly pressed into a gel to a fixed depth. While the solvent in the gel migrates, the force on the indenter relaxes.

## Theory

The theory of poroelasticity is used in this paper:

- The displacement field <math>u_i(x_1,x_2,x_3,t)</math>.
- The strain <math>\epsilon_ij=(\partial u_i/\partial x_j+ \partial u_j/\partial x_i)/2</math>.
- The conservation of solvent molecules <math>\partial C/\partial t=-\partial J_k/\partial x_k</math>, C is the concentration of the solvent in the gel, <math>J_k</math> is the flux.
- The stress is so small that the increase of the volume of the gel equals the volume of the solvent absorbed,<math>\epsilon_kk=\omega(C-C_0)</math>,<math>\omega</math> is the volume per solvent molecule.
- The stress <math>\sigma_ij=2G[\epsilon_ij+\epsilon_kk\delta_ijv/(1-2v)]-\delta_ij(\mu-\mu_0)/\omega</math>,<math>G</math> is the shear modulus, <math>v</math> is the Poisson's ratio, <math>\mu</math> is the chemical potential of the solvent in the gel.
- The mechanical equilibrium <math>\partial \sigma_ij/\partial x_j=0</math>.
- The flux driven by the gradient of the chemical potential <math>J_i=-(k/\eta^2)\partial \mu/\partial x_i</math>, <math>k</math> is the permeability, <math>\eta</math> is the viscosity of the solvent.
- The diffusion equation <math>\partial C/\partial t=D\nabla^2C</math>, <math>D=[2(1-v)/(1-2v)]Gk/\eta</math>.

An indenter is pressed into the gel at a fixed depth <math>h</math>, the force starts to relax as a function of time, <math>F(t)</math>, as showed in Fig. 1.

- <math>t=0</math>,<math>C=C_0, \epsilon_kk=0</math>, the gel is like an incompressible elastic solid, <math>F(0)\propto G</math>.
- <math>t>0</math>, <math>\frac{F(t)-F(\infty)}{F(0)-F(\infty)}=g(\frac{Dt}{a^2})</math>, define the normalized time <math>\tau=Dt/a^2</math>,<math>g(\tau)</math> is dimensionless function specific to the shape of the indenter.
- <math>t>>0</math>, <math>/mu=/mu_0</math>, the gel is like a compressible elastic solid, <math>F(\infty)\propto G/[2(1-v)]</math>.

The poroelastic contact problems are solved by the finite element software ABAQUS.The calculation results are showed in Fig.2 and Fig. 3. Fig.2 lists the formulas of <math>F(0)</math> for different indenters. Fig.3 is the plot of <math>g(\tau)</math> for different indenters.

Based on the expressions for <math>F(0)</math> and <math>g(\tau)</math> in Fig.2, <math>G</math> is determined by <math>F(0)</math>, <math>v</math> is calculated from <math>\frac{F(0)}{F(\infty)}</math>. The measured curve <math>F(t)</math> is matched to Eq.(1), determines <math>D</math>.

## Experiment

- Materials: alginate hydrogel with diameter <math>30mm</math> and thickness <math>20mm</math>, water.
- Indenter: a conical aluminum indenter with half angle <math>\theta=70^o</math>.
- Load schedule: the indenter is pressed into the gel at a depth of <math>h=1,2,3mm</math> within <math>10s</math>, then held for hours.

## Result

- The relaxation curve depends on the depth of indentation, as illustrated in Fig. 4(a).
- After the force is normalized by <math>h^2</math>, the curves do not overlap, as Fig. 4(b). It indicates the relaxation mechanism is not viscoelasticity.
- The force and time are both normalized by <math>h^2</math>, so the curves collapse into one single one which is consistent with poroelasticity, showed in Fig. 4(c).

- In calculation, <math>G=27.9kPa, v=0.28, D=3.24*10^-8 m^2/s</math>.

## Summary

Based on the theory of poroelasticity, force relaxation curves are obtained in a simple form. Indentation is used to determine the elastic constants and permeability. Application of this method lies in the quantitative characterization of gels and soft tissues.