30 Strain Stiffening Effects of Soft Viscoelastic Materials in Inertial Microcavitation 179 Table 30.1 Material properties used in the present study Symbol Value Symbol Value Symbol Value Symbol Value ρ 1060 kg/m3 γ 5.6 ×10−2 N/m D 24.2 ×10−6 m2/s A 5.3 ×10−5 W/mK2 c 1430m/s κ 1.4 Ks 0.55W/m B 1.17 ×10−2 W/mK p∞ 101.3 kPa Cp, v 1.62 kJ/kg · K pref 1.17 ×10 8 kPa T∞ 298.15K Cp, g 1.00 kJ/kg · K Tref 5200K Results and Discussıon Following Barajas et al [8], a fifth-order explicit Dormand-Prince Runge-Kutta method with adaptive step-size control is used to evolve the governing equations forward in time. Quantitative values of material parameters for the surrounding medium and the bubble contents used in this study are summarized in Table 30.1. Least squares error (LSE) is defined by minimizing the perpendicular offset between experimental data points and the numerical simulations. Minimizing the discrete LSE in the R-t curves gives the best estimate of the viscoelastic properties for each polyacrylamide gel. Figure 30.2b shows the characterized viscoelastic properties of the tested stiff & soft polyacrylamide hydrogels using neo-Hookean Kelvin-Voigt and Fung Kelvin-Voigt models. We find that for both stiff & soft polyacrylamide hydrogels, the nonlinear neo-Hookean Kelvin-Voigt model fitting results show that the shear moduli obtained during cavitation are stiffer than their quasi-static counterparts (see Fig. 30.2b). On the other hand, when we account for strain stiffening by replacing the traditional neo-Hookean spring in the nonlinear KelvinVoigt model with a constitutive relation inspired by the Fung model [9], we recover the quasi-static shear modulus at long time scale. We also find that by applying the Fung Kelvin-Voigt model, the accuracy of the measured dynamic material viscosity is also improved compared to the neo-Hookean Kelvin-Voigt formulation. Conclusions In this paper, we use inertial microcavitation rheometry (IMR) experiments to characterize viscoelastic properties of stiff & soft polyacrylamide hydrogels undergoing large, finite deformations (|Err| > 0.05) at strain-rates of up to 10 6 s−1. We explore the strain stiffening effects of these two types of hydrogels by implementing Fung Kelvin-Voigt model. Compared with our previous neo-Hookean Kelvin-Voigt model, we find the accuracy of the measured dynamic material viscosity (in the order of 10−1 Pa·s) is also improved. Acknowledgement We gratefully acknowledge the funding support from the Office of Naval Research (Dr. Timothy Bentley) under grants N000141612872 and N000141712058. JY acknowledges discussions and help from Prof. David L Henann and Dr. Jonathan B Estrada. References 1. Maxwell, A., et al.: Noninvasive thrombolysis using pulsed ultrasound cavitation therapy – histotripsy. Ultrasound Med. Biol. 35(12), 1982– 1994 (2009) 2. Venugopalan, V., et al.: Role of laser-induced plasma formation in pulsed cellular microsurgery and micromanipulation. Phys. Rev. Lett. 88(7), 078103 (2002) 3. Xu, Z., et al.: High speed imaging of bubble clouds generated in pulsed ultrasound cavitational therapy – histotripsy. IEEE Trans Ultrason. Ferroelect. Freq. Control. 54(10), 2091–2101 (2007) 4. Meaney, D., Smith, D.: Biomechanics of concussion. Clin. Sports Med. 30, 19–31 (2011) 5. Nyein, M., et al.: In silico investigation of intracranial blast mitigation with relevance to military traumatic brain injury. Proc. Natl. Acad. Sci. U. S. A. 107(48), 20703–20708 (2011) 6. Ramasamy, A., et al.: Blast-related fracture patterns: a forensic biomechanical approach. J. R. Soc. Interface. 8(58), 689–698 (2010) 7. Estrada, J.B., Barajas, C., Henann, D.L., Johnsen, E., Franck, C.: High strain-rate soft material characterization via inertial cavitation. J. Mech. Phys. Solids. 112, 291–317 (2018) 8. Barajas, C., Johnsen, E.: The effects of heat and mass diffusion on freely oscillating bubbles in a viscoelastic, tissue-like medium. J. Acoust. Soc. Am. 141(2), 908–918 (2017) 9. Fung, Y.C.: Biomechanics: mechanical properties of living tissues. Springer Science & Business Media, Berlin (2013) 10. Chen, D.T.N., Wen, Q., Janmey, P.A., et al.: Rheology of soft materials. Annu. Rev. Condens. Matter. Phys. 1(1), 301–322 (2010) 11. Keller, J.B., Miksis, M.: Bubble oscillations of large amplitude. J. Acoust. Soc. Am. 68, 628–633 (1980) 12. Prosperetti, A.: The thermal behavior of oscillating gas bubbles. J. Fluid Mech. 222, 587–616 (1991)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==