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Spherical indentation analysis of stress relaxation for thin film viscoelastic materials

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The mechanical testing of thin layers of soft materials is an important but difficult task. Spherical indentation provides a convenient method to ascertain material properties whilst minimising damage to the material by allowing testing to take place in situ. However, measurement of the viscoelastic properties of these soft materials is hindered by the absence of a convenient yet accurate model which takes into account the thickness of the material and the effects of the underlying substrate. To this end, the spherical indentation of a thin layer of viscoelastic solid material is analysed. It is assumed that the transient mechanical properties of the material can be described by the generalised standard linear solid model. This model is incorporated into the theory and then solved for the special case of a stress relaxation experiment taking into account the finite ramp time experienced in real experiments. An expression for the force as a function of the viscoelastic properties, layer thickness and indentation depth is given. The theory is then fitted to experimental data for the spherical indentation of poly(dimethyl)siloxane mixed with its curing agent to the ratios of 5:1, 10:1 and 20:1 in order to ascertain its transient shear moduli and relaxation time constants. It is shown that the theory correctly accounts for the effect of the underlying substrate and allows for the accurate measurement of the viscoelastic properties of thin layers of soft materials.

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The Z030 Mechanical Tester, MicroXAM2 Interferometer and NanoWizard II Atomic Force Microscope used in this research were obtained through the Birmingham Science City: Innovative Uses for Advanced Materials in the Modern World (West Midlands Centre for Advanced Materials Project 2), with support from Advantage West Midlands and partly funded by the European Regional Development Fund. This research was undertaken within the FP7-NMP NANOBIOTOUCH project (contract no. 228844). The authors are grateful for the financial support provided by the EC and the Science City Research Alliance.

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Correspondence to David Cheneler.



The relaxation operator has to encompass all the linear viscoelastic behaviour of the material. A classical approach to the modelling of the linear viscoelastic behaviour of real materials is based on the mechanical analogy with the response of combinations of springs and dashpots. These models are useful for representing materials whose relaxation does not occur at a single time but in a set of times, perhaps due to being made of molecular segments of different lengths or affected by different relaxation processes. A constitutive model that can predict a distribution of both creep and stress relaxation phenomena and so give a realistic representation of viscoelastic solid materials is the generalised standard linear solid model (Ferry 1980). This model consists of N single elements in parallel whereby each element comprised a spring in series with a Kelvin–Voigt element as shown in Fig. 13:

Fig. 13
figure 13

A schematic of the generalised standard linear solid model comprised of N single elements connected in parallel

The stress–strain relationship for each element is given by

$$ \eta_{n} k_{1n} \gamma+k_{1n} k_{2n} \gamma =\left( {k_{1n} +k_{2n}} \right)\sigma +\eta_{n} \sigma\quad\quad n=1\ldots N. $$

The strain is the same for each element; therefore, the total stress acting on the system is

$$ \sigma =\sum\limits_{n=1}^{N} \sigma_{n} $$

The stress can, therefore, be solved for the case of a step change in strain, γ 0, to give

$$\begin{array}{rll} \sigma (t)=\mathrm{\Psi} (t)\gamma_{0} &=&\sum\limits_{n=1}^{N} \left[\left( {\frac{k_{1n} k_{2n}} {k_{1n} +k_{2n}} } \right)\right.\\ && \qquad\left.+\left( {\frac{k_{1n}^{2}}{k_{1n} +k_{2n}} } \right)e^{-t/T_{n}} \right]\gamma_{0} \end{array} $$

where \(T_{n}={\eta }_{n} /(k_{1n} +k_{2n} )\) are the relaxation time constants. Therefore, the relaxation operator for the material can be written as a Prony series:

$$ \mathrm{\Psi} ( t )=\sum\limits_{n=1}^{N} \left[ {\left({\frac{k_{1n} k_{2n}} {k_{1n} +k_{2n}} } \right)+\left( {\frac{k_{1n}^{2}}{k_{1n} +k_{2n}} } \right)e^{-t/T_{n}} } \right] $$

The shear relaxation modulus defined as \(G(t)={\sigma (t)} \left /\right . {( {2\gamma _{0}} )}\) (Oyen 2006) for a material described by Eq. 28 is

$$ G( t )=\frac{1}{2}\sum\limits_{n=1}^{N} \left[ {\left( {\frac{k_{1n} k_{2n}} {k_{1n} +k_{2n}} } \right)+\left( {\frac{k_{1n}^{2}}{k_{1n} +k_{2n}} } \right)e^{-t/T_{n}} } \right]. $$

There are two important limits to Eq. 29, namely when t = 0, G(0) describes the instantaneous shear modulus of the material, i.e. the shear modulus experienced during an impulsively applied strain, and is given by

$$ G_{0} =G( 0 )=\sum\limits_{n=1}^{N} \frac{k_{1n}} {2}. $$

The other limit is when t = ∞. In this case, G (∞)gives the relaxed shear modulus, i.e. the steady state modulus experienced when a strain has been held constant for sufficient time that the stress is also constant. G (∞) is given by

$$ G_{\infty} =G(\infty )=\frac{1}{2}\sum\limits_{n=1}^{N} \left( {\frac{k_{1n} k_{2n}} {k_{1n} +k_{2n}} } \right). $$

Using this nomenclature and using G n to indicate the coefficients to the exponential functions in Eq. 29, Eq. 28 is simplified to

$$ \mathrm{\Psi} (t)=2G_{\infty} +2\sum\limits_{n=1}^{N} G_{n} e^{-t/T_{n}} . $$

In the text, when a three element model is being used to fit to the data, it is being assumed that a shear modulus given by Eq. 29 when N = 3, as given in full in Eq. 33, describes the mechanical behaviour of the material.

$$\begin{array}{rll} G(t)&=&\frac{1}{2}\left[ {\left({\frac{k_{11} k_{21}} {k_{11}+k_{21}} } \right)+\left( {\frac{k_{12} k_{22}} {k_{12} +k_{22}} } \right)+\left( {\frac{k_{13} k_{23}} {k_{13} +k_{23}} } \right)} \right] \\ &&+\frac{1}{2}\left( {\frac{k_{11}^{2}}{k_{11} +k_{21}}} \right)e^{-t/T_{1}} +\frac{1}{2}\left( {\frac{k_{12}^{2}}{k_{12} +k_{22}} } \right)e^{-t/T_{2}} \\ &&+\frac{1}{2}\left( {\frac{k_{13}^{2}}{k_{13} +k_{23}} } \right)e^{-t/T_{3}} . \end{array} $$

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Cheneler, D., Mehrban, N. & Bowen, J. Spherical indentation analysis of stress relaxation for thin film viscoelastic materials. Rheol Acta 52, 695–706 (2013).

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