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Original Articles

Rethinking and researching the physical meaning of the standard linear solid model in viscoelasticity

Pages 2370-2385 | Received 27 Sep 2022, Accepted 05 Dec 2022, Published online: 04 Jan 2023

References

  • E. W. Billington and A. Tate, The Physics of Deformation and Flow, McGraw-Hill, New York, USA, 1981.
  • C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, 3rd ed., Springer-Verlag, Berlin, Germany, 2004.
  • Y. Feng, J. Goree, and B. Liu, Viscoelasticity of 2D liquids quantified in a dusty plasma experiment, Phys. Rev. Lett., vol. 105, no. 2, p. 025002, 2010. DOI: 10.1103/PhysRevLett.105.025002.
  • P.-H. Wu, D. R.-B. Aroush, A. Asnacios, W.-C. Chen, M. E. Dokukin, B. L. Doss, P. Durand-Smet, A. Ekpenyong, J. Guck, N. V. Guz, P. A. Janmey, J. S. H. Lee, N. M. Moore, A. Ott, Y.-C. Poh, R. Ros, M. Sander, I. Sokolov, J. R. Staunton, N. Wang, G. Whyte and D. Wirtz, Comparative study of cell mechanics methods, Nat Methods, vol. 15, no. 7, pp. 491–498, 2018. DOI: 10.1038/s41592-018-0015-1.
  • T. Hashidume and Q. Q. Ni, High-frequency viscoelasticities of CFRP structures by ultrasonic dynamic mechanical analysis, Compos. Struct., vol. 225, p. 111144, 2019. DOI: 10.1016/j.compstruct.2019.111144.
  • C. Y. Lin, Ramp-creep ultrasound viscoelastography for measuring viscoelastic parameters of materials, Materials, vol. 13, no. 16, p. 3593, 2020. DOI: 10.3390/ma13163593.
  • C. Y. Lin and K. V. Chang, Effects of loading and boundary conditions on the performance of ultrasound compressional viscoelastography: A computational simulation study to guide experimental design, Materials, vol. 14, no. 10, p. 2590, 2021. DOI: 10.3390/ma14102590.
  • N. Makris, The fractional derivative of the dirac delta function and additional results on the inverse laplace transform of irrational functions, Fractal Fract., vol. 5, no. 1, p. 18, 2021. DOI: 10.3390/fractalfract5010018.
  • R. Lakes and R. S. Lakes, Viscoelastic Materials, Cambridge University Press, New York, USA, 2009.
  • C. X. Deng, X. Hong, and J. P. Stegemann, Ultrasound imaging techniques for spatiotemporal characterization of composition, microstructure, and mechanical properties in tissue engineering, Tissue Eng. B Rev., vol. 22, no. 4, pp. 311–321, 2016. DOI: 10.1089/ten.TEB.2015.0453.
  • R. Xiao, H. Sun, and W. Chen, An equivalence between generalized Maxwell model and fractional Zener model, Mech. Mater., vol. 100, pp. 148–153, 2016. DOI: 10.1016/j.mechmat.2016.06.016.
  • J. Chang and O. Chaudhuri, Beyond proteases: Basement membrane mechanics and cancer invasion, J. Cell Biol., vol. 218, no. 8, pp. 2456–2469, 2019. DOI: 10.1083/jcb.201903066.
  • M. Puljiz and A. M. Menzel, Memory-based mediated interactions between rigid particulate inclusions in viscoelastic environments, Phys. Rev. E, vol. 99, no. 1-1, p. 012601, 2019.
  • A. Bonfanti, J. L. Kaplan, G. Charras, and A. Kabla, Fractional viscoelastic models for power-law materials, Soft Matter, vol. 16, no. 26, pp. 6002–6020, 2020. DOI: 10.1039/d0sm00354a.
  • Z. Yang, P. Wu, and W. Liu, Time-dependent behavior of laminated functionally graded beams bonded by viscoelastic interlayer based on the elasticity theory, Arch. Appl. Mech., vol. 90, no. 7, pp. 1457–1473, 2020. DOI: 10.1007/s00419-020-01677-4.
  • W. N. Findley, J. S. Lai, and K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials: With an Introduction to Linear Viscoelasticity, Dover Publications, New York, NY, 1976.
  • H. F. Brinson and L. C. Brinson, Polymer Engineering Science and Viscoelasticity: An Introduction, Springer, New York, NY, 2008.
  • A. Wineman, Nonlinear viscoelastic solids—a review, Math. Mech. Solids, vol. 14, no. 3, pp. 300–366, 2009. DOI: 10.1177/1081286509103660.
  • H. T. Banks, S. Hu, and Z. R. Kenz, A brief review of elasticity and viscoelasticity for solids, Adv. Appl. Math. Mech., vol. 3, no. 1, pp. 1–51, 2011. DOI: 10.4208/aamm.10-m1030.
  • Y. C. Fung, Biomechanics: Mechanical Properties of Living Tissues, Springer Science & Business Media, New York, NY, 2013.
  • X. Cui, X. Wu, M. Wan, B. Ma, and Y. Zhang, A novel constitutive model for stress relaxation of Ti-6Al-4V alloy sheet, Int. J. Mech. Sci., vol. 161, p. 105034, 2019.
  • C. Y. Lin, Alternative form of standard linear solid model for characterizing stress relaxation and creep: Including a novel parameter for quantifying the ratio of fluids to solids of a viscoelastic solid, Front. Mater., vol. 7, p. 11, 2020. DOI: 10.3389/fmats.2020.00011.
  • L. Meng and W. Chen, A new thermodynamically based model for creep and cyclic plasticity, Int. J. Mech. Sci., vol. 214, p. 106923, 2022. DOI: 10.1016/j.ijmecsci.2021.106923.
  • C. Y. Lin, Y. W. Shau, C. L. Wang, H. M. Chai, and J. H. Kang, Quantitative evaluation of the viscoelastic properties of the ankle joint complex in patients suffering from ankle sprain by the anterior drawer test, Knee Surg. Sports Traumatol. Arthrosc., vol. 21, no. 6, pp. 1396–1403, 2013. DOI: 10.1007/s00167-013-2459-2.
  • R. Ansari and M. K. Hassanzadeh-Aghdam, Micromechanical investigation of creep-recovery behavior of carbon nanotube-reinforced polymer nanocomposites, Int. J. Mech. Sci., vol. 115, pp. 45–55, 2016.
  • A. Yango, J. Schäpe, C. Rianna, H. Doschke, and M. Radmacher, Measuring the viscoelastic creep of soft samples by step response AFM, Soft Matter, vol. 12, no. 40, pp. 8297–8306, 2016. DOI: 10.1039/c6sm00801a.
  • K. A. Khan and R. K. A. Al-Rub, Time dependent response of architectured Neovius foams, Int. J. Mech. Sci., vol. 126, pp. 106–119, 2017. DOI: 10.1016/j.ijmecsci.2017.03.017.
  • P. Xu, Z. Zhou, T. Liu, S. Pan, X. Tan, and Z. Chen, The investigation of viscoelastic mechanical behaviors of bolted GLARE joints: Modeling and experiments, Int. J. Mech. Sci., vol. 175, p. 105538, 2020. DOI: 10.1016/j.ijmecsci.2020.105538.
  • C. Y. Lin and S. R. Lin, Investigating the accuracy of ultrasound viscoelastic creep imaging for measuring the viscoelastic properties of a single-inclusion phantom, Int. J. Mech. Sci., vol. 199, p. 106409, 2021. DOI: 10.1016/j.ijmecsci.2021.106409.
  • C. Y. Lin and W. C. Chen, How complex viscoelastic behaviors within a viscoelastic three-layer structure affect the measurement accuracy of ultrasound viscoelastic creep imaging, Mech. Adv. Mater. Struct., 2022. DOI: 10.1080/15376494.2022.2049933.
  • M. H. Hajmohammad, M. B. Azizkhani, and R. Kolahchi, Multiphase nanocomposite viscoelastic laminated conical shells subjected to magneto-hygrothermal loads: Dynamic buckling analysis, Int. J. Mech. Sci., vol. 137, pp. 205–213, 2018. DOI: 10.1016/j.ijmecsci.2018.01.026.
  • M. H. Hajmohammad, R. Kolahchi, M. S. Zarei, and A. H. Nouri, Dynamic response of auxetic honeycomb plates integrated with agglomerated CNT-reinforced face sheets subjected to blast load based on visco-sinusoidal theory, Int. J. Mech. Sci., vol. 153, pp. 391–401, 2019.
  • T. Mukhopadhyay, S. Adhikari, and A. Batou, Frequency domain homogenization for the viscoelastic properties of spatially correlated quasi-periodic lattices, Int. J. Mech. Sci., vol. 150, pp. 784–806, 2019. DOI: 10.1016/j.ijmecsci.2017.09.004.
  • H. Guo, F. Shang, X. Tian, and T. He, An analytical study of transient thermo-viscoelastic responses of viscoelastic laminated sandwich composite structure for vibration control, Mech. Adv. Mater. Struct., vol. 29, no. 2, pp. 171–181, 2020.
  • M. Di Paola, L. Galuppi, and G. R. Carfagni, Fractional viscoelastic characterization of laminated glass beams under time-varying loading, Int. J. Mech. Sci., vol. 196, p. 106274, 2021. DOI: 10.1016/j.ijmecsci.2021.106274.
  • A. Mohamed, S. Yousef, T. Hashem, and M. A. Abdelnaby, Microstructure and modeling of uniaxial mechanical properties of polyethersulfone nanocomposite ultrafiltration membranes, Int. J. Mech. Sci., vol. 204, p. 106568, 2021. DOI: 10.1016/j.ijmecsci.2021.106568.
  • R. Selvaraj, E. S. P, S. Kannan, G. Manickam, and A. Ankush, Mechanical characterization and dynamic behavior of 3D printed composite sandwich panel featuring hybrid semi-active core through experimental and numerical approach, Mech. Adv. Mater. Struct., 2021. DOI: 10.1080/15376494.2021.1962456.
  • L. Viviani, M. Di Paola, and G. Royer-Carfagni, A fractional viscoelastic model for laminated glass sandwich plates under blast actions, Int. J. Mech. Sci., vol. 222, p. 107204, 2022. DOI: 10.1016/j.ijmecsci.2022.107204.
  • L. Garcia Cucalon, E. Rahmani, D. N. Little, and D. H. Allen, A multiscale model for predicting the viscoelastic properties of asphalt concrete, Mech. Time-Depend Mater., vol. 20, no. 3, pp. 325–342, 2016. DOI: 10.1007/s11043-016-9303-2.
  • P. Liu, Q. Xing, D. Wang, and M. Oeser, Application of linear viscoelastic properties in semianalytical finite element method with recursive time integration to analyze asphalt pavement structure, Adv. Civ. Eng., vol. 2018, pp. 1–15, 2018. DOI: 10.1155/2018/9045820.
  • O. Sirin, D. K. Paul, M. S. Khan, E. Kassem, and M. K. Darabi, Effect of aging on viscoelastic properties of asphalt mixtures, J. Transp. Eng., B: Pavements, vol. 145, no. 4, p. 04019034, 2019. DOI: 10.1061/JPEODX.0000137.
  • J. Huang and Y. Sun, Viscoelastic analysis of the damping asphalt mixtures (DAMs) made with a high content of asphalt rubber (AR), Adv. Civ. Eng., vol. 2020, pp. 1–12, 2020. DOI: 10.1155/2020/8826926.
  • Z. Wang, N. Guo, X. Yang, and S. Wang, Micromechanical prediction model of viscoelastic properties for asphalt mastic based on morphologically representative pattern approach, Adv. Mater. Sci. Eng., vol. 2020, pp. 1–12, 2020. DOI: 10.1155/2020/7915140.
  • M. Jafari, A. Babazadeh, and M. Shahri, The role of stress sensitivity of modified binders with the same linear viscoelastic properties in evaluating rutting resistance of asphalt mixtures, Int. J. Pavement Eng., 2022. DOI: 10.1080/10298436.2022.2027417.
  • C. Boutin, K. Viverge, and S. Hans, Dynamics of contrasted stratified elastic and viscoelastic plates-application to laminated glass, Compos. B: Eng., vol. 212, p. 108551, 2021. DOI: 10.1016/j.compositesb.2020.108551.
  • Z. M. Pawlak and A. Denisiewicz, Identification of the fractional Zener model parameters for a viscoelastic material over a wide range of frequencies and temperatures, Materials, vol. 14, no. 22, p. 7024, 2021. DOI: 10.3390/ma14227024.
  • T. Tao, G. Zhao, J. Zhai, and S. Ren, A strong adaptive piecewise model order reduction method for large-scale dynamical systems with viscoelastic damping, Mech. Syst. Sig. Process, vol. 164, p. 108203, 2022. DOI: 10.1016/j.ymssp.2021.108203.
  • E. van Ruymbeke, S. Coppola, L. Balacca, S. Righi, and D. Vlassopoulos, Decoding the viscoelastic response of polydisperse star/linear polymer blends, J. Rheol., vol. 54, no. 3, pp. 507–538, 2010. DOI: 10.1122/1.3368729.
  • R. Hsissou, A. Bekhta, O. Dagdag, A. El Bachiri, M. Rafik, and A. Elharfi, Rheological properties of composite polymers and hybrid nanocomposites, Heliyon, vol. 6, no. 6, p. e04187, 2020. DOI: 10.1016/j.heliyon.2020.e04187.
  • Y. Wang, L. Shang, P. Zhang, X. Yan, K. Zhang, S. Dou, J. Zhao and Y. Li, Measurement of viscoelastic properties for polymers by nanoindentation, Polym. Test., vol. 83, p. 106353, 2020. DOI: 10.1016/j.polymertesting.2020.106353.
  • P. Christöfl, C. Czibula, M. Berer, G. Oreski, C. Teichert, and G. Pinter, Comprehensive investigation of the viscoelastic properties of PMMA by nanoindentation, Polym. Test., vol. 93, p. 106978, 2021. DOI: 10.1016/j.polymertesting.2020.106978.
  • M. Mahiuddin, D. Godhani, L. Feng, F. Liu, T. Langrish, and M. A. Karim, Application of Caputo fractional rheological model to determine the viscoelastic and mechanical properties of fruit and vegetables, Postharvest Biol. Technol., vol. 163, p. 111147, 2020. DOI: 10.1016/j.postharvbio.2020.111147.
  • O. K. Ozturk and P. S. Takhar, Physical and viscoelastic properties of carrots during drying, J. Texture Stud., vol. 51, no. 3, pp. 532–541, 2020. DOI: 10.1111/jtxs.12496.
  • M. Qiao, G. Xia, T. Cui, Y. Xu, X. Gao, Y. Su, Y. Li, H. Fan, Effect of moisture, protein, starch, soluble sugar contents and microstructure on mechanical properties of maize kernels, Food Chem., vol. 379, p. 132147, 2022. DOI: 10.1016/j.foodchem.2022.132147.
  • D. Dalecki, K. P. Mercado, and D. C. Hocking, Quantitative ultrasound for nondestructive characterization of engineered tissues and biomaterials, Ann. Biomed. Eng., vol. 44, no. 3, pp. 636–648, 2016. DOI: 10.1007/s10439-015-1515-0.
  • W. Kim, V. L. Ferguson, M. Borden, and C. P. Neu, Application of elastography for the noninvasive assessment of biomechanics in engineered biomaterials and tissues, Ann. Biomed. Eng., vol. 44, no. 3, pp. 705–724, 2016. DOI: 10.1007/s10439-015-1542-x.
  • X. Hong, J. P. Stegemann, and C. X. Deng, Microscale characterization of the iscoelastic properties of hydrogel biomaterials using dual-mode ultrasound elastography, Biomaterials, vol. 88, pp. 12–24, 2016. DOI: 10.1016/j.biomaterials.2016.02.019.
  • T. E. Brown, B.J. Carberry B.T. Worrell, O.Y. Dudaryeva, M.K. McBride, C.N. Bowman, and K.S. Anseth, Photopolymerized dynamic hydrogels with tunable viscoelastic properties through thioester exchange, Biomaterials, vol. 178, pp. 496–503, 2018. DOI: 10.1016/j.biomaterials.2018.03.060.
  • X. Hong, R. T. Annamalai, T. S. Kemerer, C. X. Deng, and J. P. Stegemann, Multimode ultrasound viscoelastography for three-dimensional interrogation of microscale mechanical properties in heterogeneous biomaterials, Biomaterials, vol. 178, pp. 11–22, 2018. DOI: 10.1016/j.biomaterials.2018.05.057.
  • A. Ghorbanoghli and K. Narooei, A new hyper-viscoelastic model for investigating rate dependent mechanical behavior of dual cross link self-healing hydrogel, Int. J. Mech. Sci., vol. 159, pp. 278–286, 2019. DOI: 10.1016/j.ijmecsci.2019.06.019.
  • T. H. Jovic, G. Kungwengwe, A. C. Mills, and I. S. Whitaker, Plant-derived biomaterials: A review of 3D bioprinting and biomedical applications, Front. Mech. Eng., vol. 5, p. 19, 2019. DOI: 10.3389/fmech.2019.00019.
  • Y. M. Efremov, T. Okajima, and A. Raman, Measuring viscoelasticity of soft biological samples using atomic force microscopy, Soft Matter, vol. 16, no. 1, pp. 64–81, 2020. DOI: 10.1039/c9sm01020c.
  • S. Tang, B. M. Richardson, and K. S. Anseth, Dynamic covalent hydrogels as biomaterials to mimic the viscoelasticity of soft tissues, Prog. Mater. Sci., vol. 120, p. 100738, 2021. DOI: 10.1016/j.pmatsci.2020.100738.
  • C. Y. Lin, J. H. Kang, C. L. Wang, and Y. W. Shau, Relationship between viscosity of the ankle joint complex and functional ankle instability for inversion ankle sprain patients, J. Sci. Med. Sport, vol. 18, no. 2, pp. 128–132, 2015. DOI: 10.1016/j.jsams.2014.02.009.
  • C. Y. Lin, Y. W. Shau, C. L. Wang, and J. H. Kang, Modeling and analysis of the viscoelastic response of the ankle ligament complex in inversion ankle sprain, Ann. Biomed. Eng., vol. 43, no. 9, pp. 2047–2055, 2015. DOI: 10.1007/s10439-014-1240-0.
  • J. Palacio-Torralba, S. Hammer, D.W. Good, S. A. McNeill, G.D. Stewart, R.L. Reuben and Y. Chen, Quantitative diagnostics of soft tissue through viscoelastic characterization using time-based instrumented palpation, J. Mech. Behav. Biomed. Mater., vol. 41, pp. 149–160, 2015.
  • M. Baghban and A. Mojra, Early relaxation time assessment for characterization of breast tissue and diagnosis of breast tumors, J. Mech. Behav. Biomed. Mater., vol. 87, pp. 325–335, 2018. DOI: 10.1016/j.jmbbm.2018.07.037.
  • K. Park, G. E. Lonsberry, M. Gearing, A. I. Levey, and J. P. Desai, Viscoelastic properties of human autopsy brain tissues as biomarkers for Alzheimer’s diseases, IEEE Trans. Biomed. Eng., vol. 66, no. 6, pp. 1705–1713, 2018.
  • B. Zhou and X. Zhang, Comparison of five viscoelastic models for estimating viscoelastic parameters using ultrasound shear wave elastography, J. Mech. Behav. Biomed. Mater., vol. 85, pp. 109–116, 2018. DOI: 10.1016/j.jmbbm.2018.05.041.
  • A. Nabavizadeh, M. Bayat, V. Kumar, A. Gregory, J. Webb, A. Alizad and Mostafa Fatemi, Viscoelastic biomarker for differentiation of benign and malignant breast lesion in ultra-low frequency range, Sci. Rep., vol. 9, no. 1, pp. 1–12, 2019. DOI: 10.1038/s41598-019-41885-9.
  • H. Zhang, Y. Guo, Y. Zhou, H. Zhu, P. Wu, K. Wang, L. Ruan, M. Wan and M. F. Insana, Fluidity and elasticity form a concise set of viscoelastic biomarkers for breast cancer diagnosis based on Kelvin–Voigt fractional derivative modeling, Biomech. Model Mechanobiol., vol. 19, no. 6, pp. 2163–2177, 2020. DOI: 10.1007/s10237-020-01330-7.
  • Y. Park, C.A. Best, K. Badizadegan, R.R. Dasari, M.S. Feld, T. Kuriabova, M.L. Henle, A.J. Levine and G. Popescu, Measurement of red blood cell mechanics during morphological changes, Proc. Natl. Acad. Sci. USA, vol. 107, no. 15, pp. 6731–6736, 2010. DOI: 10.1073/pnas.0909533107.
  • C. Rianna and M. Radmacher, Comparison of viscoelastic properties of cancer and normal thyroid cells on different stiffness substrates, Eur. Biophys. J., vol. 46, no. 4, pp. 309–324, 2017. DOI: 10.1007/s00249-016-1168-4.
  • Y. H. Chim, L. M. Mason, N. Rath, M. F. Olson, M. Tassieri, and H. Yin, A one-step procedure to probe the viscoelastic properties of cells by Atomic Force Microscopy, Sci. Rep., vol. 8, no. 1, pp. 1–12, 2018. DOI: 10.1038/s41598-018-32704-8.
  • D. Huang, Y. Huang, Y. Xiao, X. Yang, H. Lin, G. Feng, X. Zhu, X. Zhang, Viscoelasticity in natural tissues and engineered scaffolds for tissue reconstruction, Acta Biomater., vol. 97, pp. 74–92, 2019.
  • Y. Abidine, A. Giannetti, J. Revilloud, V. M. Laurent, and C. Verdier, Viscoelastic properties in cancer: From cells to spheroids, Cells, vol. 10, no. 7, p. 1704, 2021. DOI: 10.3390/cells10071704.
  • S. Abuhattum, D. Mokbel, P. Müller, D. Soteriou, J. Guck, and S. Aland, An explicit model to extract viscoelastic properties of cells from AFM force-indentation curves, Iscience, vol. 25, no. 4, p. 104016, 2022. DOI: 10.1016/j.isci.2022.104016.
  • D. L. Chen, P. F. Yang, and Y. S. Lai, A review of three-dimensional viscoelastic models with an application to viscoelasticity characterization using nanoindentation, Microelectron. Reliab., vol. 52, no. 3, pp. 541–558, 2012. DOI: 10.1016/j.microrel.2011.10.001.
  • X. Q. Zhou, D. Y. Yu, X. Y. Shao, S. Q. Zhang, and S. Wang, Research and applications of viscoelastic vibration damping materials: A review, Compos. Struct., vol. 136, pp. 460–480, 2016. DOI: 10.1016/j.compstruct.2015.10.014.
  • A. Serra-Aguila, J. M. Puigoriol-Forcada, G. Reyes, and J. Menacho, Viscoelastic models revisited: Characteristics and interconversion formulas for generalized Kelvin–Voigt and Maxwell models, Acta Mech. Sin., vol. 35, no. 6, pp. 1191–1209, 2019. DOI: 10.1007/s10409-019-00895-6.
  • P. E. Rouse, Jr., A theory of the linear viscoelastic properties of dilute solutions of coiling polymers, J. Chem. Phys., vol. 21, no. 7, pp. 1272–1280, 1953. DOI: 10.1063/1.1699180.
  • Y. Li, S. Tang, M. Kröger, and W. K. Liu, Molecular simulation guided constitutive modeling on finite strain viscoelasticity of elastomers, J. Mech. Phys. Solids, vol. 88, pp. 204–226, 2016.
  • M. Shahidi, B. Pichler, and C. Hellmich, Viscous interfaces as source for material creep: A continuum micromechanics approach, Eur. J. Mech. - A/Solids, vol. 45, pp. 41–58, 2014. DOI: 10.1016/j.euromechsol.2013.11.001.
  • M. Shahidi, B. Pichler, and C. Hellmich, Interfacial micromechanics assessment of classical rheological models. I: Single interface size and viscosity, J. Eng. Mech., vol. 142, no. 3, p. 04015092, 2016. DOI: 10.1061/(ASCE)EM.1943-7889.0001012.
  • M. Shahidi, B. Pichler, and C. Hellmich, Interfacial micromechanics assessment of classical rheological models. II: Multiple interface sizes and viscosities, J. Eng. Mech., vol. 142, no. 3, p. 04015093, 2016. DOI: 10.1061/(ASCE)EM.1943-7889.0001013.
  • M. Shahidi, B. Pichler, and C. Hellmich, How interface size, density, and viscosity affect creep and relaxation functions of matrix-interface composites: A micromechanical study, Acta Mech., vol. 227, no. 1, pp. 229–252, 2016. DOI: 10.1007/s00707-015-1429-9.
  • A. Ahmadian, F. Ismail, S. Salahshour, D. Baleanu, and F. Ghaemi, Uncertain viscoelastic models with fractional order: A new spectral tau method to study the numerical simulations of the solution, Commun. Nonlinear Sci. Numer. Simul., vol. 53, pp. 44–64, 2017. DOI: 10.1016/j.cnsns.2017.03.012.
  • M. Niedziela and J. Wlazło, Notes on computational aspects of the fractional-order viscoelastic model, J. Eng. Math., vol. 108, no. 1, pp. 91–105, 2018. DOI: 10.1007/s10665-017-9911-0.
  • M. A. Matlob and Y. Jamali, The concepts and applications of fractional order differential calculus in modeling of viscoelastic systems: A primer, Crit. Rev. Biomed. Eng., vol. 47, no. 4, pp. 249–276, 2019
  • H. H. Sherief and M. A. El-Hagary, Fractional order theory of thermo-viscoelasticity and application, Mech. Time-Depend Mater., vol. 24, no. 2, pp. 179–195, 2020. DOI: 10.1007/s11043-019-09415-2.
  • J. Guo, Y. Yin, and G. Peng, Fractional-order viscoelastic model of musculoskeletal tissues: Correlation with fractals, Proc. R. Soc. A, vol. 477, no. 2249, p. 20200990, 2021. DOI: 10.1098/rspa.2020.0990.
  • R. Christensen, Theory of Viscoelasticity: An Introduction, Elsevier, Amsterdam, The Netherlands, 2012.
  • G. Y. Zhou, S. T. Tu, F. Z. Xuan, and Z. Wang, Viscoelastic model to describe mechanical response of compact heat exchangers with plate-foam structure, Int. J. Mech. Sci., vol. 53, no. 12, pp. 1069–1076, 2011. DOI: 10.1016/j.ijmecsci.2011.08.012.
  • Y. H. Chen, Y. Jia, F. Yang, C. C. Huang, and S. Lee, Boussinesq type solution for a viscoelastic thin film on an elastic substrate, Int. J. Mech. Sci., vol. 117, pp. 79–92, 2016. DOI: 10.1016/j.ijmecsci.2016.08.009.
  • K. Rajabi and S. Hosseini-Hashemi, Application of the generalized Hooke’s law for viscoelastic materials (GHVMs) in nonlocal free damped vibration analysis of viscoelastic orthotropic nanoplates, Int. J. Mech. Sci., vol. 124, pp. 158–165, 2017.
  • F. L. Li, C. Zhang, and Y. S. Wang, Analysis of the effects of viscosity on the SH-wave band-gaps of 2D viscoelastic phononic crystals by Dirichlet-to-Neumann map method, Int. J. Mech. Sci., vol. 195, p. 106225, 2021. DOI: 10.1016/j.ijmecsci.2020.106225.
  • Z. Wang, J. Zhang, Z. Jiang, W. Xiong, and Z. Mao, A transient and time lag deformation alternating-coupling micro elastohydrodynamic lubrication model, Int. J. Mech. Sci., vol. 210, p. 106744, 2021. DOI: 10.1016/j.ijmecsci.2021.106744.
  • A. Plaseied and A. Fatemi, Deformation response and constitutive modeling of vinyl ester polymer including strain rate and temperature effects, J. Mater. Sci., vol. 43, no. 4, pp. 1191–1199, 2008. DOI: 10.1007/s10853-007-2297-z.
  • H. Takagi, M. Takahashi, R. Maeda, Y. Onishi, Y. Iriye, T. Iwasaki and Y. Hirai, Analysis of time dependent polymer deformation based on a viscoelastic model in thermal imprint process, Microelectron. Eng., vol. 85, no. 5–6, pp. 902–906, 2008. DOI: 10.1016/j.mee.2008.01.018.
  • Q. Ge, K. Yu, Y. Ding, and H. J. Qi, Prediction of temperature-dependent free recovery behaviors of amorphous shape memory polymers, Soft Matter, vol. 8, no. 43, pp. 11098–11105, 2012. DOI: 10.1039/c2sm26249e.
  • K. Yu, T. Xie, J. Leng, Y. Ding, and H. J. Qi, Mechanisms of multi-shape memory effects and associated energy release in shape memory polymers, Soft Matter, vol. 8, no. 20, pp. 5687–5695, 2012. DOI: 10.1039/c2sm25292a.
  • S. D. Solares, Nanoscale effects in the characterization of viscoelastic materials with atomic force microscopy: Coupling of a quasi-three-dimensional standard linear solid model with in-plane surface interactions, Beilstein J. Nanotechnol., vol. 7, no. 1, pp. 554–571, 2016. DOI: 10.3762/bjnano.7.49.
  • H. Yu, R. Kongsmo, N. Patil, J. He, D. W. Breiby, and Z. Zhang, On determining the Poisson’s ratio of viscoelastic polymer microparticles using a flat punch test, Int. J. Mech. Sci., vol. 128, pp. 150–158, 2017.
  • C. Ganser, C. Czibula, D. Tscharnuter, T. Schöberl, C. Teichert, and U. Hirn, Combining adhesive contact mechanics with a viscoelastic material model to probe local material properties by AFM, Soft Matter, vol. 14, no. 1, pp. 140–150, 2018. DOI: 10.1039/C7SM02057K.
  • Y. W. Huang, W. S. Lee, Y. F. Chuang, W. Cao, F. Yang, and S. Lee, Time-dependent deformation of artificial muscles based on Nylon 6, Mater. Sci. Eng. C Mater. Biol. Appl., vol. 98, pp. 445–451, 2019. DOI: 10.1016/j.msec.2018.12.118.
  • Z. Pei, L. Wang, P. Wu, J. Zhang, and D. Zhou, Analytical solution of deformations for two-layer Timoshenko beams glued by a viscoelastic interlayer, Math. Prob. Eng., vol. 2019, pp. 1–15, 2019. DOI: 10.1155/2019/7620816.
  • L. Bazli, M. H. Bagherian, M. Karrabi, F. Abbassi‐Sourki, and H. Azizi, Effect of starch ratio and compatibilization on the viscoelastic behavior of POE/starch blends, J. Appl. Polym. Sci., vol. 137, no. 29, p. 48877, 2020. DOI: 10.1002/app.48877.
  • G. Zhao, J. Xu, Y.Feng, J. Tang, Y. Chen, S. Xin, X. Jian, S. Li, S. Zhang and J. Xu, A rate-dependent cohesive zone model with the effects of interfacial viscoelasticity and progressive damage, Eng. Fract. Mech., vol. 248, p. 107695, 2021. DOI: 10.1016/j.engfracmech.2021.107695.
  • U. Güven, Effects of lateral inertia on the group velocities and specific damping capacities of viscoelastic polymeric rods, Int. J. Impact Eng., vol. 162, p. 104156, 2022. DOI: 10.1016/j.ijimpeng.2022.104156.
  • T. M. Shazly, N. Artzi, F. Boehning, and E. R. Edelman, Viscoelastic adhesive mechanics of aldehyde-mediated soft tissue sealants, Biomaterials, vol. 29, no. 35, pp. 4584–4591, 2008. DOI: 10.1016/j.biomaterials.2008.08.032.
  • Z. Feng, D. Seya, T. Kitajima, T. Kosawada, T. Nakamura, and M. Umezu, Viscoelastic characteristics of contracted collagen gels populated with rat fibroblasts or cardiomyocytes, J. Artif. Organs, vol. 13, no. 3, pp. 139–144, 2010. DOI: 10.1007/s10047-010-0508-x.
  • A. Tirella, G. Mattei, and A. Ahluwalia, Strain rate viscoelastic analysis of soft and highly hydrated biomaterials, J. Biomed. Mater. Res. A., vol. 102, no. 10, pp. 3352–3360, 2014. DOI: 10.1002/jbm.a.34914.
  • L. Cacopardo, N. Guazzelli, R. Nossa, G. Mattei, and A. Ahluwalia, Engineering hydrogel viscoelasticity, J. Mech. Behav. Biomed. Mater., vol. 89, pp. 162–167, 2019. DOI: 10.1016/j.jmbbm.2018.09.031.
  • H. W. Wu, T. Kuhn, and V. T. Moy, Mechanical properties of L929 cells measured by atomic force microscopy: Effects of anticytoskeletal drugs and membrane crosslinking, Scanning, vol. 20, no. 5, pp. 389–397, 2006. DOI: 10.1002/sca.1998.4950200504.
  • W. R. Trickey, G. M. Lee, and F. Guilak, Viscoelastic properties of chondrocytes from normal and osteoarthritic human cartilage, J. Orthop. Res., vol. 18, no. 6, pp. 891–898, 2000. DOI: 10.1002/jor.1100180607.
  • E. J. Koay, A. C. Shieh, and K. A. Athanasiou, Creep indentation of single cells, J. Biomech. Eng., vol. 125, no. 3, pp. 334–341, 2003. DOI: 10.1115/1.1572517.
  • E. A. Peeters, C. W. Oomens, C. V. Bouten, D. L. Bader, and F. P. Baaijens, 2005. Viscoelastic properties of single attached cells under compression.
  • E. H. Zhou, C. T. Lim, and S. T. Quek, Finite element simulation of the micropipette aspiration of a living cell under-going large viscoelastic deformation, Mech. Adv. Mater. Struct., vol. 12, no. 6, pp. 501–512, 2005. DOI: 10.1080/15376490500259335.
  • C. T. Lim, E. H. Zhou, and S. T. Quek, Mechanical models for living cells—a review, J. Biomech., vol. 39, no. 2, pp. 195–216, 2006. DOI: 10.1016/j.jbiomech.2004.12.008.
  • T. C. Skalak and G. W. Schmid-Schönbein, Viscoelastic properties of microvessels in rat spinotrapezius muscle, J. Biomech. Eng., vol. 108, no. 3, pp. 193–200, 1986. DOI: 10.1115/1.3138602.
  • D. Bessems, C. G. Giannopapa, M. C. Rutten, and F. N. van de Vosse, Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels, J. Biomech., vol. 41, no. 2, pp. 284–291, 2008. DOI: 10.1016/j.jbiomech.2007.09.014.
  • M. Siami, K. Jahani, and M. Rezaee, Identifying the parameters of viscoelastic model for a gel-type material as repre-sentative of cardiac muscle in dynamic tests, Proc. Inst. Mech. Eng. H, vol. 235, no. 10, pp. 1205–1216, 2021. DOI: 10.1177/09544119211025868.
  • I. I. Argatov, Mathematical modeling of linear viscoelastic impact: Application to drop impact testing of articular cartilage, Tribol. Int., vol. 63, pp. 213–225, 2013. DOI: 10.1016/j.triboint.2012.09.015.
  • J. P. Quiroga, W. Wilson, K. Ito, and C. C. van Donkelaar, The effect of loading rate on the development of early damage in articular cartilage, Biomech. Model. Mechanobiol., vol. 16, no. 1, pp. 263–273, 2017. DOI: 10.1007/s10237-016-0815-0.
  • S. Li, A. G. Patwardhan, F. M. Amirouche, R. Havey, and K. P. Meade, Limitations of the standard linear solid model of intervertebral discs subject to prolonged loading and low-frequency vibration in axial compression, J. Biomech., vol. 28, no. 7, pp. 779–790, 1995. DOI: 10.1016/0021-9290(94)00140-y.
  • K. M. Groth and K. P. Granata, The viscoelastic standard nonlinear solid model: Predicting the response of the lumbar intervertebral disk to low-frequency vibrations, Trans. ASME, J. Biomech. Eng., vol. 130, no. 3, p. 031005, 2008. DOI: 10.1115/1.2904464.
  • Manlio Tassieri, M. Laurati, D.J. Curtis, D.W. Auhl, S. Coppola, A. Scalfati, K. Hawkins, P. R. Williams and J.M. Cooper, i-Rheo: Measuring the materials’ linear viscoelastic properties “in a step”!, J. Rheol., vol. 60, no. 4, pp. 649–660, 2016. DOI: 10.1122/1.4953443.
  • J. A. Moreno‐Guerra, I. C. Romero‐Sánchez, A. Martinez‐Borquez, M. Tassieri, E. Stiakakis, and M. Laurati, Model‐free rheo‐AFM probes the viscoelasticity of tunable DNA soft colloids, Small, vol. 15, no. 42, p. 1904136, 2019. DOI: 10.1002/smll.201904136.
  • W. M. Lai, D. H. Rubin, D. Rubin, and E. Krempl, Introduction to Continuum Mechanics, Butterworth-Heinemann, Oxford, UK, 2009.
  • W. F. Walker, F. J. Fernandez, and L. A. Negron, A method of imaging viscoelastic parameters with acoustic radiation force, Phys. Med. Biol., vol. 45, no. 6, pp. 1437–1447, 2000. DOI: 10.1088/0031-9155/45/6/303.
  • P. N. Wells and H. D. Liang, Medical ultrasound: Imaging of soft tissue strain and elasticity, J. R. Soc. Interface, vol. 8, no. 64, pp. 1521–1549, 2011. DOI: 10.1098/rsif.2011.0054.
  • R. S. Lakes and R. Vanderby, 1999. Interrelation of creep and relaxation: A modeling approach for ligaments.
  • C. Y. Lin, Y. C. Chen, C. H. Lin, and K. V. Chang, Constitutive equations for analyzing stress relaxation and creep of viscoelastic materials based on standard linear solid model derived with finite loading rate, Polymers, vol. 14, no. 10, p. 2124, 2022. DOI: 10.3390/polym14102124.
  • K. C. Lee, J. Siegel, S.E.D. Webb, S. Lévêque-Fort, M.J. Cole, R. Jones, K. Dowling, M.J. Lever and P.M.W. French, Application of the stretched exponential function to fluorescence lifetime imaging, Biophys. J., vol. 81, no. 3, pp. 1265–1274, 2001.
  • D. C. Johnston, Stretched exponential relaxation arising from a continuous sum of exponential decays, Phys. Rev. B, vol. 74, no. 18, p. 184430, 2006. DOI: 10.1103/PhysRevB.74.184430.
  • T. J. Miao and J. Tang, Characterization of charge carrier behavior in photocatalysis using transient absorption spectroscopy, J. Chem. Phys., vol. 152, no. 19, p. 194201, 2020. DOI: 10.1063/5.0008537.