References
- I. Sevostianov and A. Giraud, Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape, Int. J. Eng. Sci., vol. 64, pp. 23–36, 2013. DOI: https://doi.org/10.1016/j.ijengsci.2012.12.004.
- Y. Davit, et al., Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare? Adv. Water Resource., vol. 62, pp. 178–206, 2013. DOI: https://doi.org/10.1016/j.advwatres.2013.09.006.
- D. Lukkassen and G. W. Milton, On hierarchical structures and reiterated homogenization. Proceedings of the International Conference in Honour of Jaak Peetre on his 65th birthday: Function Spaces, Interpolation Theory and Related Topics [Series: De Gruyter Proceedings in Mathematics], pp. 355–368, August 17–22, Lund, Sweden, 2000.
- R. Penta and A. Gerisch, The asymptotic homogenization elasticity tensor properties for composites with material discontinuities, Continuum Mech. Thermodyn., vol. 29, no. 1, pp. 187–206, 2017. DOI: https://doi.org/10.1007/s00161-016-x.
- A. Ramírez-Torres, S. D. Stefano, A. Grillo, R. Rodríguez-Ramos, J. Merodio, and J. Penta, Three scales asymptotic homogenization and its application to layered hierarchical hard tissues, Int. J. Solids Struct., vol. 130–131, pp. 190–198, 2018a. DOI: https://doi.org/10.1016/j.ijsolstr.2017.09.035.
- A. Ramírez-Torres, P. Penta, R. Rodríguez-Ramos, and A. Grillo, Homogenized out-of-plane shear response of three-scale fiberreinforced composites, Comput. Visual. Sci., vol. 20, no. 3–6, pp. 85–93, 2019a. DOI: https://doi.org/10.1007/s00791-018-0301-6.
- Z. Yang, Y. Sun, Y. Liu, T. Guan, and H. Dong, A three-scale asymptotic expansion for predicting viscoelastic properties of composites with multiple configuration, Eur. J. Mech. A. Solids., vol. 76, pp. 235–246, 2019b. DOI: https://doi.org/10.1016/j.euromechsol.2019.04.016.
- H. Dong, X. Zheng, J. Cui, Y. Nie, Z. Yang, and Z. Yang, High-order three-scale computational method for dynamic thermo-mechanical problems of composite structures with multiple spatial scales, Int. J. Solids Struct., vol. 169, pp. 95–121, 2019. DOI: https://doi.org/10.1016/j.ijsolstr.2019.04.017.
- D. Tsalis, G. Chatzigeorgiou, and N. Charalambakis, Homogenization of structures with generalized periodicity, Composites Part B. Eng., vol. 43, no. 6, pp. 2495–2512, 2012. DOI: https://doi.org/10.1016/j.compositesb.2012.01.054.
- D. Tsalis, T. Baxevanis, G. Chatzigeorgiou, and N. Charalambakis, Homogenization of elastoplastic composites with generalized periodicity in the microstructure, Int. J. Plasti., vol. 51, pp. 161–187, 2013. DOI: https://doi.org/10.1016/j.ijplas.2013.05.006.
- R. Penta, K. Raum, Q. Grimal, S. Schrof, and A. Gerisch, Can a continuous mineral foam explain the stiffening of aged bone tissue? A micromechanical approach to mineral fusion in musculoskeletal tissues, Bioinspir. Biomim., vol. 11, no. 3, pp. 035004, 2016. DOI: https://doi.org/10.1088/1748-3190/11/3/035004.
- A. Ramírez-Torres, et al., Effective properties of hierarchical fiber-reinforced composites via a three-scale asymptotic homogenization approach, Math. Mech. Solid., vol. 24, no. 11, pp. 3554–3574, 2019b. DOI: https://doi.org/10.1177/1081286519847687.
- G. Allaire and M. Briane, Multiscale convergence and reiterated homogenisation, Proc. R. Soc. Edin. Sec. A. Math., vol. 126, no. 2, pp. 297–342, 1996. DOI: https://doi.org/10.1017/S0308210500022757.
- A. Bensoussan, G. Papanicolau, and J.-L. Lions, Asymptotic Analysis for Periodic Structures, 1st ed., Vol. 5. Elsevier, North-Holland, 1978.
- D. Trucu, M. Chaplain, and A. Marciniak-Czochra, Three-scale convergence for processes in heterogeneous media, Applicable Anal., vol. 91, no. 7, pp. 1351–1373, 2012. DOI: https://doi.org/10.1080/00036811.2011.569498.
- Z. Yang, Y. Sun, J. Cui, and J. Ge, A three-scale asymptotic analysis for ageing linear viscoelastic problems of composites with multiple configurations, Appl. Mathe. Model., vol. 71, pp. 223–242, 2019a. DOI: https://doi.org/10.1016/j.apm.2019.02.021.
- O. L. Cruz-González, et al., An approach for modeling non-ageing linear viscoelastic composites with general periodicity, Composite Structures., vol. 223, pp. 110927, 2019. DOI: https://doi.org/10.1016/j.compstruct.2019.110927.
- Y.-M. Yi, S.-H. Park, and S.-K. Youn, Asymptotic homogenization of viscoelastic composites with periodic microstructures, Int. J. Solids Struct., vol. 35, no. 17, pp. 2039–2055, 1998. DOI: https://doi.org/10.1016/S0020-7683(97)00166-2.
- S. Liu, K.-Z. Chen, and X.-A. Feng, Prediction of viscoelastic property of layered materials, Int. J. Solids Struct., vol. 41, no. 13, pp. 3675–3688, 2004. DOI: https://doi.org/10.1016/j.ijsolstr.2004.01.015.
- A. B. Tran, J. Yvonnet, Q.-C. He, C. Toulemonde, and J. Sanahuja, A simple computational homogenization method for structures made of linear heterogeneous viscoelastic materials, Comput. Method. Appl. Mech. Eng., vol. 200, no. 45–46, pp. 2956–2970, 2011. DOI: https://doi.org/10.1016/j.cma.2011.06.012.
- V. R. Sherman, Y. Tang, S. Zhao, W. Yang, and M. A. Meyers, Structural characterization and viscoelastic constitutive modeling of skin, Acta Biomater., vol. 53, pp. 460–469, 2017. DOI: https://doi.org/10.1016/j.actbio.2017.02.011.
- A. N. Annaidh, K. Bruyère, M. Destrade, M. D. Gilchrist, and M. Otténio, Characterization of the anisotropic mechanical properties of excised human skin, J. Mech. Beh. Biomed. Materi., vol. 5, no. 1, pp. 139–148, 2012. DOI: https://doi.org/10.1016/j.jmbbm.2011.08.016.
- D. Malhotra, et al., Linear viscoelastic and microstructural properties of native male human skin and in vitro 3D reconstructed skin models, J. Mech. Behav. Biomed. Mater., vol. 90, pp. 644–654, 2019. DOI: https://doi.org/10.1016/j.jmbbm.2018.11.013.
- W. Yang, et al., On the tear resistance of skin, Nat. Commun., vol. 6, no. 6649, pp. 6, 2015. DOI: https://doi.org/10.1038/ncomms7649.
- R. Panchal, L. Horton, P. Poozesh, J. Baqersad, and M. Nasiriavanaki, Vibration analysis of healthy skin: toward a noninvasive skin diagnosis methodology, J. Biomed. Opt., vol. 24, no. 1, 2019. DOI: https://doi.org/10.1117/1.JBO.24.1.015001.
- I. Muha, A. Naegel, S. Sabine, A. Grillo, M. Heisig, and G. Wittum, Effective diffusivity in membranes with tetrakaidekahedral cells and implications for the permeability of human stratum corneum, J. Membr. Sci., vol. 368, no. 1-2, pp. 18–25, 2011. DOI: https://doi.org/10.1016/j.memsci.2010.10.020.
- J. M. Benítez and F. J. Montáns, The mechanical behavior of skin: Structures and models for the finite element analysis, Comput. Struct., vol. 190, pp. 75–107, 2017. DOI: https://doi.org/10.1016/j.compstruc.2017.05.003.
- Z. Hashin, Viscoelastic Behavior of Heterogeneous Media, J. Appl. Mech., vol. 32, no. 3, pp. 630–636, 1965. DOI: https://doi.org/10.1115/1.3627270.
- Q.-D. To, S.-T. Nguyen, G. Bonnet and M.-N. Vu, Overall viscoelastic properties of 2D and two-phase periodic composites constituted of elliptical and rectangular heterogeneities, Eur. J. Mech. A. Solids., vol. 64, pp. 186–201, 2017. DOI: https://doi.org/10.1016/j.euromechsol.2017.03.004.
- J. L. Auriault, C. Boutin, and C. Geindreau, Homogenization of Coupled Phenomena in Heterogenous Media, Wiley, New York, 2009.
- A. Ramírez-Torres, et al., An asymptotic homogenization approach to the microstructural evolution of heterogeneous media, Int. J. Non- Linear Mech., vol. 106, pp. 245–257, 2018b. DOI: https://doi.org/10.1016/j.ijnonlinmec.2018.06.012.
- R. M. Christensen, Theory of Viscoelasticity - 2nd Edition an Introduction, Academic Press, Cambridge, MA, 1982.
- S. Maghous and G. J. Creus, Periodic homogenization in thermoviscoelasticity: case of multilayered media with ageing, Int. J. Solids Struct., vol. 40, no. 4, pp. 851–870, 2003. DOI: https://doi.org/10.1016/S0020-7683(02)00549-8.
- E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics. Berlin: Springer-Verlag, 1980.
- D. Guinovart-Sanjuán, et al., Analysis of effective elastic properties for shell with complex geometrical shapes, Composite Struct., vol. 203, pp. 278–285, 2018. DOI: https://doi.org/10.1016/j.compstruct.2018.07.036.
- H. Joodaki and M. B. Panzer, Skin mechanical properties and modeling: A review, Proc. Inst. Mech. Eng. H., vol. 232, no. 4, pp. 323–343, 2018. DOI: https://doi.org/10.1177/0954411918759801.
- F. Xu and T. Lu, Introduction to Skin Bio-Thermo Mechanics and Thermal Pain, Vol. 7. New York: Springer, 2011.
- A. McBride, S. Bargmann, D. Pond, and G. Limbert, Thermoelastic modelling of the skin at finite deformations, J. Therm. Biol., vol. 62, pp. 201–209, 2016. DOI: https://doi.org/10.1016/j.jtherbio.2016.06.017.
- Z. Hashin, Theory of fiber reinforced materials. NASA Contractor Report, Washington DC, USA, Rep. NASA CR-1974, 1972.
- K. J. Hollenbeck, Invlap. M: A Matlab function for numerical inversion of Laplace transforms by the Hoog algorithm. 1998. http://www.isva.dtu.dk/staff/karl/invlap.html.
- W. Srigutomo, Gaver-Stehfest algorithm for inverse Laplace transform, 2006. www.mathworks.com/matlabcentral/fileexchange/9987.
- M. A. Araújo-Cavalcante and S. P. Cavalcanti-Marques, Homogenization of periodic materials with viscoelastic phases using the generalized FVDAM theory, Comput. Mater. Sci., vol. 87, pp. 43–53, 2014. DOI: https://doi.org/10.1016/j.commatsci.2014.01.053.
- M. A. Araújo-Cavalcante and S. P. Cavalcanti-Marques, Microstructure effects in wavy-multilayers with viscoelastic phases, Eur. J. Mech. A. Solids., vol. 64, pp. 178–185, 2017. DOI: https://doi.org/10.1016/j.euromechsol.2017.03.003.
- D. Gao, Y. Lei, and B. Yao, Analysis of dynamic tissue deformation during needle insertion into soft tissue, IFAC Proc., vol. 46, no. 5, pp. 684–691, 2013.
- N. Abolhassani, R. Patel, and M. Moallem, Experimental study of robotic needle insertion in soft tissue, Int. Congr. Ser., vol. 1268, pp. 797–802, 2004. DOI: https://doi.org/10.1016/j.ics.2004.03.110.
- P. A. Netti, D. A. Berk, M. A. Swartz, A. J. Grodzinsky, and R. K. Jain, Role of extracellular matrix assembly in interstitial transport in solid tumors, Cancer Res., vol. 60, no. 9, pp. 2497–2503, 2000.
- R. Penta and D. Ambrosi, The role of the microvascular tortuosity in tumor transport phenomena, J. Theoret. Biol., vol. 364, pp. 80–97, 2015. DOI: https://doi.org/10.1016/j.jtbi.2014.08.007.
- A. Ramírez-Torres, et al., The role of malignant tissue on the thermal distribution of cancerous breast, J. Theoret. Biol., vol. 426, pp. 152–161, 2017. DOI: https://doi.org/10.1016/j.jtbi.2017.05.031.
- L. Daridon, C. Licht, S. Orankitjaroen, and S. Pagano, Periodic homogenization for Kelvin-Voigt viscoelastic media with a Kelvin-Voigt viscoelastic interphase, Eur. J. Mech. A. Solids., vol. 58, pp. 163–171, 2016. DOI: https://doi.org/10.1016/j.euromechsol.2015.12.007.
- J. Sanahuja, Effective behaviour of ageing linear viscoelastic composites: Homogenization approach, Int. J. Solids Struct., vol. 50, no. 19, pp. 2846–2856, 2013. DOI: https://doi.org/10.1016/j.ijsolstr.2013.04.023.