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Spherical indentation analysis of stress relaxation for thin film viscoelastic materials

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Abstract

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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References

  • Bufler H (1971) Theory of elasticity of a multilayered medium. J Elast 1:125–143

    Article  Google Scholar 

  • Cao Y, Ma D, Raabe D (2009) The use of flat punch indentation to determine the viscoelastic properties in the time and frequency domains of a soft layer bonded to a rigid substrate. Acta Biomater 5(1):240–248

    Article  Google Scholar 

  • Chadwick RS (2002) Axisymmetric indentation of a thin incompressible layer. SIAM J App Math 62(5):1520–1530

    Article  Google Scholar 

  • Chen WT (1971) Computation of stresses and displacements in a layered elastic medium. Int J Eng Sci 9:775–800

    Article  Google Scholar 

  • Chen WT, Engel PA (1972) Impact and contact stress analysis in multilayer media. Int J Solids Struct 8:1257–1281

    Article  Google Scholar 

  • Cox MAJ, Driessen NJB, Boerboom RA, Bouten CVC, Baaijens FPT (2008) Mechanical characterization of anisotropic planar biological soft tissues using finite indentation: experimental feasibility. J Biomat 41:422–429

    Google Scholar 

  • Darling EM, Zaucher S, Guilak F (2006) Viscoelastic properties of zonal articular chondrocytes measured by atomic force microscopy. Osteo Arth Cart 14:571–579

    Article  CAS  Google Scholar 

  • Darling EM, et al. (2007) Thin-layer model for viscoelastic, stress-relaxation testing of cells using atomic force microscopy: do cell properties reflect metastatic potential. Biophys J 92:1784–1791

    Article  CAS  Google Scholar 

  • Dhaliwal RS, Rau IS (1970) The axisymmetric Boussinesq problem for a thick elastic layer under a punch of arbitrary profile. Int J Engr Sci 8:843–856

    Article  Google Scholar 

  • Dimitriadis EK, et al. (2002) Determination of elastic moduli of thin layers of soft material using the atomic force microscope. Biophys J 82:2798–2810

    Article  CAS  Google Scholar 

  • Dintwa E, Tijskens E, Ramon H (2008) On the accuracy of the Hertz model to describe the normal contact of soft elastic spheres. Granul Matter 10:209–221

    Article  Google Scholar 

  • Domke J, Radmacher M (1998) Measuring the elastic properties of thin polymer films with the atomic force microscope. Langmuir 14:3320–3325

    Article  CAS  Google Scholar 

  • Ferry JD (1980) Viscoelastic properties of polymers, 3rd edn. Wiley, New York

    Google Scholar 

  • Findley WN, Onaran K (1976) Creep and relaxation of nonlinear viscoelastic materials. Dover Publications, New York

    Google Scholar 

  • Fretigny C, Chateauminois A (2000) Solution for the elastic field in a layered medium under axisymmetric contact loading. J Phys D: Appl Phys 40:5418–5428

    Article  Google Scholar 

  • Graham GAC, Sabin GCW (1973) The correspondence principle of linear viscoelasticity for problems that involve time-dependent regions. Int J Eng Sci 11(1):123–140

    Article  Google Scholar 

  • Greenwood JA (2010) Contact between an axisymmetric indenter and a viscoelastic half-space. Int J Mech Sci 52(6):829–835

    Article  Google Scholar 

  • Hertz H (1882) Uber die, Beruhrung fester, elastischer Korper. J Reine Angew Math 92:156–171

    Google Scholar 

  • Jardet V, Morel P (2003) Viscoelastic effects on the scratch resistance of polymers: relationship between mechanical properties and scratch properties at various temperatures. Prog Org Coat 48(2–4):322–331

    Article  Google Scholar 

  • Johnson KL (1985) Contact mechanics. CUP, Cambridge

    Book  Google Scholar 

  • Landau LD, Liftshitz EM (1986) Course of theoretical physics: theory of elasticity. Elsevier, Oxford

    Google Scholar 

  • Lee EH, Radok JRM (1960) The contact problem for viscoelastic bodies. J App Mech Trans ASME 27:438–444

    Article  Google Scholar 

  • Levental I, et al. (2010) A simple indentation device for measuring micrometer-scale tissue stiffness. J Phys Condens Matter 22:194120

    Article  CAS  Google Scholar 

  • Moreno-Flores S, et al. (2010) Stress relaxation and creep on living cells with the atomic force microscope: a means to calculate elastic moduli and viscosities of cell components. Nanotech 21:445101

    Article  Google Scholar 

  • Oyen ML (2006) Analytical techniques for indentation of viscoelastic materials. Philos Mag 86(33):5625–5641

    Article  CAS  Google Scholar 

  • Pailler-Mattei C, Bec S, Zahouani H (2008) In vivo measurements of the elastic mechanical properties of human skin by indentation tests. Med Eng Phys 30:599–606

    Article  CAS  Google Scholar 

  • Rizzo FJ, Shippy DJ (1971) An application of the correspondence principle of linear viscoelasticity theory. SIAM J Appl Math 21(2):321–336

    Article  Google Scholar 

  • Ting TCT (1966) The contact stresses between a rigid indenter and a viscoelastic half-space. J App Mech Trans ASME 27:845–854

    Article  Google Scholar 

  • Tu Y-O, Gazis DC (1964) The contact problem of a plate pressed between two spheres. Trans ASME J Appl Mech 31:659–666

    Article  Google Scholar 

  • Wakatsuki T, et al. (2001) Effects of cytochalasin D and latrunculin B on mechanical properties of cells. J Cell Sci 114(5):1025–1036

    CAS  Google Scholar 

  • Zhang CY, Zhang YW, Zeng KY (2004) Extracting the mechanical properties of a viscoelastic polymeric film on a hard elastic substrate. J Mater Res 19(10):3053–3061

    Article  CAS  Google Scholar 

Download references

Acknowledgments

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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Appendix

Appendix

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

$$ \eta_{n} k_{1n} \gamma+k_{1n} k_{2n} \gamma =\left( {k_{1n} +k_{2n}} \right)\sigma +\eta_{n} \sigma\quad\quad n=1\ldots N. $$
(25)

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

$$ \sigma =\sum\limits_{n=1}^{N} \sigma_{n} $$
(26)

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

$$\begin{array}{rll} \sigma (t)=\mathrm{\Psi} (t)\gamma_{0} &=&\sum\limits_{n=1}^{N} \left[\left( {\frac{k_{1n} k_{2n}} {k_{1n} +k_{2n}} } \right)\right.\\ && \qquad\left.+\left( {\frac{k_{1n}^{2}}{k_{1n} +k_{2n}} } \right)e^{-t/T_{n}} \right]\gamma_{0} \end{array} $$
(27)

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:

$$ \mathrm{\Psi} ( t )=\sum\limits_{n=1}^{N} \left[ {\left({\frac{k_{1n} k_{2n}} {k_{1n} +k_{2n}} } \right)+\left( {\frac{k_{1n}^{2}}{k_{1n} +k_{2n}} } \right)e^{-t/T_{n}} } \right] $$
(28)

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

$$ G( t )=\frac{1}{2}\sum\limits_{n=1}^{N} \left[ {\left( {\frac{k_{1n} k_{2n}} {k_{1n} +k_{2n}} } \right)+\left( {\frac{k_{1n}^{2}}{k_{1n} +k_{2n}} } \right)e^{-t/T_{n}} } \right]. $$
(29)

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

$$ G_{0} =G( 0 )=\sum\limits_{n=1}^{N} \frac{k_{1n}} {2}. $$
(30)

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

$$ G_{\infty} =G(\infty )=\frac{1}{2}\sum\limits_{n=1}^{N} \left( {\frac{k_{1n} k_{2n}} {k_{1n} +k_{2n}} } \right). $$
(31)

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

$$ \mathrm{\Psi} (t)=2G_{\infty} +2\sum\limits_{n=1}^{N} G_{n} e^{-t/T_{n}} . $$
(32)

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.

$$\begin{array}{rll} G(t)&=&\frac{1}{2}\left[ {\left({\frac{k_{11} k_{21}} {k_{11}+k_{21}} } \right)+\left( {\frac{k_{12} k_{22}} {k_{12} +k_{22}} } \right)+\left( {\frac{k_{13} k_{23}} {k_{13} +k_{23}} } \right)} \right] \\ &&+\frac{1}{2}\left( {\frac{k_{11}^{2}}{k_{11} +k_{21}}} \right)e^{-t/T_{1}} +\frac{1}{2}\left( {\frac{k_{12}^{2}}{k_{12} +k_{22}} } \right)e^{-t/T_{2}} \\ &&+\frac{1}{2}\left( {\frac{k_{13}^{2}}{k_{13} +k_{23}} } \right)e^{-t/T_{3}} . \end{array} $$
(33)

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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). https://doi.org/10.1007/s00397-013-0707-5

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