# Shear Modulus of Compliant Biomaterials

It is important to understand the mechanical properties of biomaterials to be able to know the reciprocal relationship between function and properties. The mechanical properties of biomaterial will usually affect the functioning of the tissue and vice versa. Nanoindentation seems like a good choice to measure small scale mechanical properties such as the hierarchical structure of tissue over all relevant length scales.

However nanoindentation techniques developed for engineering materials do not work with biological materials because of two main reasons. First, there is a difficulty knowing the area of mutual contact between the punch and the tissue. Second, it is difficult to make relevant and reliable measurements of contact stiffness and damping given the compliant and viscoelastic nature of the test material. Here is a demonstration of how the iNano can be used to test biologic material using edible gelatin. The hardware, procedure and analyzes required for the testing are explained. Edible gelatin was used as it is quite similar to tissue, and is easily available to be prepared in a repeatable and controlled manner.

## General Theory of Complex Shear Modulus, G*

The constitutive relation that governs the viscoelasticity of biomaterials is

 (1) τ* = G*γ*,

wherein the complex shear modulus. G*, is the material property which relates complex shear stress , τ*, and complex shear strain, γ*. The complex shear stress may be expressed as an oscillation in time, t, having an amplitude, τo, and angular frequency, ω, as

 (2) τ* = τ0eiωt

In response to such a shear stress, we expect a complex shear strain, γ*, having amplitude γo , which also oscillates at an angular frequency of ω,but lags by a phase angle δ

(3) γ* = γo ei( ωt- δ) = γo ei ωt /ei δ

Rearranging Equation 1 to solve for G* and substituting the expressions for complex stress and strain from Equations 2 and 3 gives this expression for complex modulus

(4) G* = τ*/γ* = (τ0 / γo) ei δ = (τ0 / γo) [cosδ + isinδ]

Equation 4 clarifies the components of the complex modulus as:

(5a) G* = G’ + iG’’ , where
(5b) G’ = (τ0 / γo)cosδ , and
(5c) G’’ = (τ0 / γo)sinδ

Where G’ is known as the “shear storage modulus” and G’’ is known as the “shear loss modulus”. Finally, the ration G’’/G’ is called the “loss factor”, because it quantifies the ability of the material to damp out energy relative to the ability to store energy:

 (5d) Loss factor = G’’/G’ = tanδ

Macroscopically, G’, G’’ and tanδ are all measured using rheometry. Here we use instrumented indentation testing to measure comparable values of these very same properties on a much smaller scale.

## Measuring Complex Modulus by Instrumented Indentation

The experiment employs Sneddon’s contact solution to relate the shear storage modulus, G’, to contact stiffness, S, Poisson’s ratio, ν, and contact diameter, D.

 (6a) G’ = S(1- ν)/(2D)

Loubet et al. Say that in an indentation test, the shear loss modulus must be related to contact damping, Dsω, in an analogous way.

 (6b) G’’ = Dsω(1- ν)/(2D)

And the validity of this analogy is borne out by much experience. To measure the components of the complex shear modulus, one must know the contact diameter, the Poisson’s ratio, and the stiffness and damping of the contact.

The contact diameter is found by using a flat-ended cylinder or cone, as the indenter simply uses the punch face’s contact diameter, this doesn’t change during the test.

The Poisson’s ratio for gels and biomaterials is assumed at the value of incompressibility ν = 0.5, as water is the main component of such materials.

This leaves the problem of measuring contact stiffness and damping in measuring complex modulus. The changes are very small, leading to experimental challenges. The iNano produced by Nanochemicals Inc. reveals that the contact stiffness and damping are determined as the directly measured values of stiffness and damping, K and , less the contribution of the instrument, Ki and Diω

 (7a) S = K-KI (7b) Dsω = Dω - Diω

The iNano is designed to measure the stiffness and damping in a minimized and measurable manner. It includes an extra step where Ki and Diω are carefully measured. Once S and are determined as per equation 7, the storage and loss modulus can be determined by equation 6 and the loss factor is

 (8) tanδ = G’’/G’ = Dsω/S

## Experimental Method

### Sample Preparation

Food grade gelatin was made in double concentration using gelatin power from Knox Unflavoured Gelatine, Kraft Foods Group, Inc., USA. A pouch was mixed with 4 oz (118 ml) of boiling water to dissolve completely and samples were placed in holders filled to the brim with the gel solution. The samples were placed in air-tight containers to prevent dehydration as they set overnight. Before testing the sample was removed from the air right container and a thin piece of glass obtained from a section of a microscope slide coverslip, was floated on top of the gel to provide a stiff surface on which the instrument could engage the sample. Scotch tape, sticky side up, was adhered with epoxy to the edge of the sample holder to provide a surface for cleaning the indenter between testes.

### Equipment and Procedure

Table 1. Summary of Method Inputs.

Input Value Units
Punch Diameter 107.7 μm
Poisson’s Ratio 0.5 none
Pre-test Compression 10 μm
Target Frequency 145 Hz
Phase Change for Contact 0.5 degrees

Figure 1.

An iNano nanoindenter with a flat-ended cylindrical punch tip of diameter 107.7 nm, was used to perform ten indentations on the gel surface. Test sites were spaced by 400 μm to avoid mutual interaction. The test method “Complex Shear Modulus of Biomaterials” comprised of indentation tests with the following steps.

1. Self-calibration- the stiffness and damping of the instrument were measured under test conditions without being in contact with the sample.

2. Engagement- the whole actuator was moved down until the indenter touched the glass cover slip of the sample.

3. Approach – With the indenter over the gel, the actuator was moved down till contact with the gel was sensed by a shift in phase angle.

4. Pre-test compression – the flat face of the indenter was pressed into full contact with the gel.

5. Test – the indenter was vibrated in contact with the gel to measure the composite stiffness and damping (K and )

After each test the “Quick Touch” method was used to clean the indenter by pressing it into contact with the scotch tape mounted on the edge of the sample holder.

Figure 2. Still shots from real-time video of indentation process: (a) Flat-punch indenter approaching the sample surface; (b) Measuring the complex shear modulus of the gel.

## Results and Discussion

The smooth gel surface showed a circular mark, left by the flat face of the punch. This shows that the instrument sensed contact and applied the pre-test compression so that the full face of the indenter was in contact with the gel.

The ten tests showed values for storage modulus, G’, and loss modulus, G’’, was reasonable for the material tested. A higher than expected variation was seen for nanoindentation measurements of a smooth, uniform material. The extremely small values measured make the integrated self-calibration process is essential for accurate measurement.

Figure 3. Indentation residual mark after test.

The test to test variation may be attributed to measurement uncertainty and not true point to point variation in material properties. Repeated measurements of instrument stiffness gave a standard deviation of 0.25N/m. This is 20% of the mean contact stiffness. This degree of uncertainty fully accounts for the observed relative variation in G’ of 18%. The relative variation in loss modulus, G”, has a similar explanation: the standard deviation in instrument damping is 33% of the mean contact damping.

The influence of measurement uncertainty may be mitigated by using a larger punch to increase the size of the contact. From Equation 6, we see that the contact stiffness, S, depends proportionally on the storage modulus, G’, and the contact diameter, D. Thus, if we want to increase the value of the contact stiffness, relative to our uncertainty of 0.25 N/m, then we must increase the diameter of the punch. Doubling the size of the punch should cut in half the relative variation in G’. The same thing can be said of damping: doubling the diameter of the punch should cut in half the relative variation in G”.

Test G' G" Tan δ Contact Stiffness, S Contact Damping, Dsω
Pa Pa None N/m N/m
1 3163 802 0.25 1.36 0.35
2 2996 587 0.2 1.29 0.25
3 3458 820 0.24 1.49 0.35
4 3862 1611 0.42 1.66 0.69
5 2968 993 0.34 1.28 0.43
6 3081 496 0.16 1.33 0.21
7 2394 455 0.19 1.03 0.2
8 2485 545 0.22 1.07 0.24
9 2352 482 0.21 1.01 0.21
10 2103 324 0.15 0.91 0.14
Mean, X 2886.3 711.5 0.24 1.24 0.31
Std. Dev., σ 521 355.8 0.08 0.22 0.15
σ/X (%) 18.10% 50.00% 32.70% 18.10% 50.00%

## Conclusion

Food grade gelatine was used as a biomaterial substitute in the new test method, “Complex Shear Modulus of Biomaterials ” using the iNano system manufactured by Nanomechanics. Inc. The flat punch with diameter of only 100 μm, the shear storage modulus was G’ = 2.89 ± 0.52 kPa and the shear loss modulus was G” = 0.71 ± 0.36 kPa (N = 10). Using a larger-diameter punch would improve the relative uncertainty in measured properties, but at the sacrifice of spatial resolution in the measurement.

## References

1. Sneddon, I.N. (1965) The relation between and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3(1):47-57.
2. Oliver, W.C., Pharr, G.M. (1992) An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7(6):1564–1583.
3. Pharr, G.M., Oliver, W.C., Brotzen, F.R. (1992) On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation. J. Mater. Res. 7(3):613–617.
4. Loubet, J.L., Oliver, W.C., Lucas, B.N. (2000) Measurement of the loss tangent of low-density polyethylene with a nanoindentation technique. J. Mater. Res. 15(05):1195–1198.
5. Herbert, E.G., Oliver, W.C., Lumsdaine, A. (2009). Measuring the constitutive behavior of viscoelastic solids in the time and frequency domain using flat punch nanoindentation. J. Mater. Res. 24(3):626–637.

This information has been sourced, reviewed and adapted from materials provided by Nanomechanics, Inc, a KLA Corporation company.