Constitutive Modelling of Skin Mechanics

  • Georges LimbertEmail author
Part of the Studies in Mechanobiology, Tissue Engineering and Biomaterials book series (SMTEB, volume 22)


The objective of this chapter is to provide a structured review of constitutive models of skin mechanics valid for finite deformations, with special emphasis on state-of-the-art anisotropic formulations which are essential in most advanced modelling applications. The fundamental structural and material characteristics of the skin, necessary for understanding its mechanics and for the formulation of constitutive equations, are briefly presented.


Acknowledgements/Funding Statement

The author would like to gratefully acknowledge the financial support he has received over the last few years to support research on skin biophysics and applications from the Royal Society, The Royal Academy of Engineering, The British High Commission in South Africa, EPSRC, Procter & Gamble, L’Oréal, Roche and the US Air Force. He would also like to thank Dr. Anton Page of the Biomedical Imaging Unit at the University of Southampton and Mr. Sandy Monteith of Gatan UK for respectively preparing the skin sample for serial block-face imaging and for organising the electron microscopy acquisition at Gatan USA in San Diego.


  1. 1.
    Burns T et al (2004) Rook’s textbook of dermatology, 7th edn. Blackwell Science, OxfordGoogle Scholar
  2. 2.
    Silver FH, Siperko LM, Seehra GP (2003) Mechanobiology of force transduction in dermal tissue. Skin Res Technol 9(1):3–23Google Scholar
  3. 3.
    Dandekar K, Raju BI, Srinivasan MA (2003) 3-D finite-element models of human and monkey fingertips to investigate the mechanics of tactile sense. J Biomech Eng-Trans ASME 125(5):682–691Google Scholar
  4. 4.
    Xu F, Lu T (2011) Introduction to skin biothermomechanics and thermal pain. Springer, Heidelberg, p 414Google Scholar
  5. 5.
    Limbert G (2017) Mathematical and computational modelling of skin biophysics – a review. Proc R Soc A Math Phys Eng Sci 473(2203):1–39MathSciNetzbMATHGoogle Scholar
  6. 6.
    Jor JWY et al (2013) Computational and experimental characterization of skin mechanics: identifying current challenges and future directions. Wiley Interdiscip Rev Syst Biol Med 5(5):539–556Google Scholar
  7. 7.
    Benítez JM, Montáns FJ (2017) The mechanical behavior of skin: structures and models for the finite element analysis. Comput Struct 190:75–107Google Scholar
  8. 8.
    Li W (2015) Modelling methods for in vitro biomechanical properties of the skin: a review. Biomed Eng Lett 5(4):241–250MathSciNetGoogle Scholar
  9. 9.
    Fung YC (1981) Biomechanics: mechanical properties of living tissues. Springer, New YorkGoogle Scholar
  10. 10.
    Humphrey JD (2003) Continuum biomechanics of soft biological tissues. Proc R Soci A Math Phys Eng Sci 459(2029):3–46MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lanir Y (2016) Multi-scale structural modeling of soft tissues mechanics and mechanobiology. J Elast 129(1–2):7–48MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hamed J, Matthew BP (2018) Skin mechanical properties and modeling: a review. Proc Inst Mech Eng Part H J Eng Med 232:323–343. CrossRefGoogle Scholar
  13. 13.
    Shimizu H (2007) Shimizu’s textbook of dermatology. Hokkaido University Press - Nakayama Shoten, Sapporo, p 564Google Scholar
  14. 14.
    Buganza Tepole A, Kuhl E (2014) Computational modeling of chemo-bio-mechanical coupling: a systems-biology approach toward wound healing. Comput Methods Biomech Biomed Eng 19:13–30Google Scholar
  15. 15.
    Kvistedal YA, Nielsen PMF (2009) Estimating material parameters of human skin in vivo. Biomech Model Mechanobiol 8(1):1–8Google Scholar
  16. 16.
    Lanir Y (1987) Skin mechanics. In: Skalak R, Chien S (eds) Handbook of bioengineering. McGraw-Hill, New YorkGoogle Scholar
  17. 17.
    Vierkötter A, Krutmann J (2012) Environmental influences on skin aging and ethnic-specific manifestations. Dermato-endocrinol 4(3):227–231Google Scholar
  18. 18.
    Silver FH, Freeman JW, DeVore D (2001) Viscoelastic properties of human skin and processed dermis. Skin Res Technol 7(1):18–23Google Scholar
  19. 19.
    Limbert G (2014) State-of-the-art constitutive models of skin biomechanics. In: Querleux B (ed) Computational biophysics of the skin. Pan Stanford, Singapore, pp 95–131Google Scholar
  20. 20.
    Marieb EN, Hoehn K (2010) Human anatomy & physiology, 8th edn. Pearson International Edition, San Francisco, p 1114Google Scholar
  21. 21.
    Chan LS (1997) Human skin basement membrane in health and autoimmune diseases. Front Biosci 2:343–352Google Scholar
  22. 22.
    Leyva-Mendivil MF et al (2015) A mechanistic insight into the mechanical role of the stratum corneum during stretching and compression of the skin. J Mech Behav Biomed Mater 49(0):197–219Google Scholar
  23. 23.
    Leyva-Mendivil MF et al (2017) Skin microstructure is a key contributor to its friction behaviour. Tribol Lett 65(1):12Google Scholar
  24. 24.
    Biniek K, Levi K, Dauskardt RH (2012) Solar UV radiation reduces the barrier function of human skin. Proc Natl Acad Sci USA 109(42):17111–17116Google Scholar
  25. 25.
    Wu KS, van Osdol WW, Dauskardt RH (2006) Mechanical properties of human stratum corneum: effects of temperature, hydration, and chemical treatment. Biomaterials 27(5):785–795Google Scholar
  26. 26.
    Ciarletta P, Ben Amar M (2012) Papillary networks in the dermal-epidermal junction of skin: a biomechanical model. Mech Res Commun 42:68–76Google Scholar
  27. 27.
    Burgeson RE, Christiano AM (1997) The dermal-epidermal junction. Curr Opin Cell Biol 9:651–658Google Scholar
  28. 28.
    Ribeiro JF et al (2013) Skin collagen fiber molecular order: a pattern of distributional fiber orientation as assessed by optical anisotropy and image analysis. PLoS One 8(1):e54724Google Scholar
  29. 29.
    Gosline J et al (2002) Elastic proteins: biological roles and mechanical properties. Philos Trans R Soc Lond B Biol Sci 357(1418):121–132Google Scholar
  30. 30.
    Sherratt MJ (2013) Age-related tissue stiffening: cause and effect. Adv Wound Care 2(1):11–17Google Scholar
  31. 31.
    Langer K (1861) Zur Anatomie und Physiologie der Haut. Über die Spaltbarkeit der Cutis. Sitzungsbericht der Mathematisch-naturwissenschaftlichen Classe der Wiener Kaiserlichen Academie der Wissenschaften Abt, p 44Google Scholar
  32. 32.
    Langer K (1978) On the anatomy and physiology of the skin: I. The cleavability of the cutis. Br J Plast Surg 31(1):3–8Google Scholar
  33. 33.
    Langer K (1978) On the anatomy and physiology of the skin: II. Skin tension (with 1 figure). Br J Plast Surg 31(2):93–106Google Scholar
  34. 34.
    Ní Annaidh A et al (2011) Characterization of the anisotropic mechanical properties of excised human skin. J Mech Behav Biomed Mater 5(1):139–148Google Scholar
  35. 35.
    Alexander H, Cook TH (1977) Accounting for natural tension in the mechanical testing of human skin. J Invest Dermatol 69:310–314Google Scholar
  36. 36.
    Flynn C, Stavness I, Lloyd J, Fels S (2015) A finite element model of the face including an orthotropic skin model under in vivo tension. Comput Methods Biomech Biomed Eng 18:571–582. CrossRefGoogle Scholar
  37. 37.
    Deroy C et al (2016) Non-invasive evaluation of skin tension lines with elastic waves. Skin Res Technol 23:326–335Google Scholar
  38. 38.
    Rosado C et al (2016) About the in vivo quantitation of skin anisotropy. Skin Res Technol 23:429–436. CrossRefGoogle Scholar
  39. 39.
    Wan Abas WAB (1994) Biaxial tension test of human skin in vivo. Biomed Mater Eng 4:473–486Google Scholar
  40. 40.
    Ní Annaidh A et al (2012) Automated estimation of collagen fibre dispersion in the dermis and its contribution to the anisotropic behaviour of skin. Ann Biomed Eng 40(8):1666–1678Google Scholar
  41. 41.
    Ottenio M et al (2015) Strain rate and anisotropy effects on the tensile failure characteristics of human skin. J Mech Behav Biomed Mater 41:241–250Google Scholar
  42. 42.
    Kvistedal YA, Nielsen PMF (2004) Investigating stress-strain properties of in-vivo human skin using multiaxial loading experiments and finite element modeling. In: Proceedings of the 26th annual international conference of the IEEE engineering in medicine and biology society, vols 1–7, 26, pp 5096–5099Google Scholar
  43. 43.
    Batisse D et al (2002) Influence of age on the wrinkling capacities of skin. Skin Res Technol 8(3):148–154Google Scholar
  44. 44.
    Delalleau A et al (2006) Characterization of the mechanical properties of skin by inverse analysis combined with the indentation test. J Biomech 39:1603–1610Google Scholar
  45. 45.
    Diridollou S et al (2000) In vivo model of the mechanical properties of the human skin under suction. Skin Res Technol 6(4):214–221Google Scholar
  46. 46.
    Dobrev Hq (2000) Use of Cutometer to assess epidermal hydration. Skin Res Technol 6(4):239–244Google Scholar
  47. 47.
    Hendriks FM et al (2003) A numerical-experimental method to characterize the non-linear mechanical behaviour of human skin. Skin Res Technol 9(3):274–283Google Scholar
  48. 48.
    Weickenmeier J, Jabareen M, Mazza E (2015) Suction based mechanical characterization of superficial facial soft tissues. J Biomech 48(16):4279–4286Google Scholar
  49. 49.
    Pensalfini M et al (2018) Location-specific mechanical response and morphology of facial soft tissues. J Mech Behav Biomed Mater 78(Suppl C):108–115Google Scholar
  50. 50.
    Müller B et al (2018) A novel ultra-light suction device for mechanical characterization of skin. PLoS One 13(8):e0201440Google Scholar
  51. 51.
    Tonge TK et al (2013) Full-field bulge test for planar anisotropic tissues: Part I – Experimental methods applied to human skin tissue. Acta Biomater 9(4):5913–5925Google Scholar
  52. 52.
    Geerligs M et al (2011) Linear shear response of the upper skin layers. Biorheology 48(3–4):229–245Google Scholar
  53. 53.
    Geerligs M et al (2011) In vitro indentation to determine the mechanical properties of epidermis. J Biomech 44:1176–1181Google Scholar
  54. 54.
    Lamers E et al (2013) Large amplitude oscillatory shear properties of human skin. J Mech Behav Biomed Mater 28:462–470Google Scholar
  55. 55.
    Lanir Y, Fung YC (1974) Two-dimensional mechanical properties of rabbit skin—II: Experimental results. J Biomech 7:171–182Google Scholar
  56. 56.
    Wong WLE, Joyce TJ, Goh KL (2016) Resolving the viscoelasticity and anisotropy dependence of the mechanical properties of skin from a porcine model. Biomech Model Mechanobiol 15(2):433–446Google Scholar
  57. 57.
    Veronda DR, Westmann R (1970) Mechanical characterization of skin – finite deformations. J Biomech 3:111–124Google Scholar
  58. 58.
    Marino M (2016) Molecular and intermolecular effects in collagen fibril mechanics: a multiscale analytical model compared with atomistic and experimental studies. Biomech Model Mechanobiol 15(1):133–154Google Scholar
  59. 59.
    Spencer AJM (1984) Constitutive theory for strongly anisotropic solids. In: Spencer AJM (ed) Continuum theory of the mechanics of fibre-reinforced composites. Springer, Vienna, pp 1–32Google Scholar
  60. 60.
    Šolinc U, Korelc J (2015) A simple way to improved formulation of FE2 analysis. Comput Mech 56(5):905–915MathSciNetzbMATHGoogle Scholar
  61. 61.
    Saeb S, Steinmann P, Javili A (2016) Aspects of computational homogenization at finite deformations: a unifying review from Reuss’ to Voigt’s bound. Appl Mech Rev 68(5):050801–050801-33Google Scholar
  62. 62.
    Leyva-Mendivil MF et al (2017) Implications of multi-asperity contact for shear stress distribution in the viable epidermis – an image-based finite element study. Biotribology 11:110–123Google Scholar
  63. 63.
    Young PG et al (2008) An efficient approach to converting three-dimensional image data into highly accurate computational models. Philos Trans R Soc A Math Phys Eng Sci 366(1878):3155–3173MathSciNetGoogle Scholar
  64. 64.
    Limbert G et al (2010) Trabecular bone strains around a dental implant and associated micromotions—a micro-CT-based three-dimensional finite element study. J Biomech 43(7):1251–1261Google Scholar
  65. 65.
    Linder-Ganz E et al (2007) Assessment of mechanical conditions in sub-dermal tissues during sitting: a combined experimental-MRI and finite element approach. J Biomech 40(7):1443–1454Google Scholar
  66. 66.
    Limbert G et al (2013) On the mechanics of bacterial biofilms on non-dissolvable surgical sutures: a laser scanning confocal microscopy-based finite element study. Acta Biomater 9(5):6641–6652Google Scholar
  67. 67.
    Leyva-Mendivil MF, Lengiewicz J, Limbert G (2017) Skin friction under pressure. The role of micromechanics. Surf Topogr: Metrol Prop 6:014001Google Scholar
  68. 68.
    Limbert G, Kuhl E (2018) On skin microrelief and the emergence of expression micro-wrinkles. Soft Matter 14(8):1292–1300Google Scholar
  69. 69.
    Limbert G (2018) Investigating the influence of relative humidity on expression microwrinkles. J Aesthet Nurs 7(4):204–207Google Scholar
  70. 70.
    Gerhardt LC et al (2008) Influence of epidermal hydration on the friction of human skin against textiles. J R Soc Interface 5(28):1317–1328Google Scholar
  71. 71.
    Adams MJ, Briscoe BJ, Johnson SA (2007) Friction and lubrication of human skin. Tribol Lett 26(3):239–253Google Scholar
  72. 72.
    Derler S et al (2009) Friction of human skin against smooth and rough glass as a function of the contact pressure. Tribol Int 42(11–12):1565–1574Google Scholar
  73. 73.
    Kwiatkowska M et al (2009) Friction and deformation behaviour of human skin. Wear 267(5–8):1264–1273Google Scholar
  74. 74.
    Wolfram LJ (1983) Friction of skin. J Soc Cosmet Chem 34:465–476Google Scholar
  75. 75.
    Stupkiewicz S, Lewandowski MJ, Lengiewicz J (2014) Micromechanical analysis of friction anisotropy in rough elastic contacts. Int J Solids Struct 51(23–24):3931–3943Google Scholar
  76. 76.
    Goldstein B, Sanders J (1998) Skin response to repetitive mechanical stress: a new experimental model in pig. Arch Phys Med Rehabil 79(3):265–272Google Scholar
  77. 77.
    Budday S, Kuhl E, Hutchinson JW (2015) Period-doubling and period-tripling in growing bilayered systems. Philos Mag(Abingdon) 95(28–30):3208–3224Google Scholar
  78. 78.
    Cao Y, Hutchinson JW (2012) From wrinkles to creases in elastomers: the instability and imperfection-sensitivity of wrinkling. Proc R Soc A Math Phys Eng Sci 468:94–115MathSciNetzbMATHGoogle Scholar
  79. 79.
    Weickenmeier J, Jabareen M (2014) Elastic–viscoplastic modeling of soft biological tissues using a mixed finite element formulation based on the relative deformation gradient. Int J Numer Methods Biomed Eng 30(11):1238–1262Google Scholar
  80. 80.
    Li W, Luo XY (2016) An invariant-based damage model for human and animal skins. Ann Biomed Eng 44(10):3109–3122Google Scholar
  81. 81.
    Buganza Tepole A et al (2011) Growing skin: a computational model for skin expansion in reconstructive surgery. J Mech Phys Solids 59(10):2177–2190MathSciNetzbMATHGoogle Scholar
  82. 82.
    Vermolen FJ, Gefen A, Dunlop JWC (2012) In vitro “wound” healing: experimentally based phenomenological modeling. Adv Eng Mater 14(3):B76–B88Google Scholar
  83. 83.
    Sherratt JA, Dallon JC (2002) Theoretical models of wound healing: past successes and future challenges. C R Biol 325(5):557–564Google Scholar
  84. 84.
    Buganza Tepole A (2017) Computational systems mechanobiology of wound healing. Comput Methods Appl Mech Eng 314:46–70MathSciNetzbMATHGoogle Scholar
  85. 85.
    Marsden JE, Hughes TJR (1994) Mathematical foundations of elasticity. Dover, New York, p 556Google Scholar
  86. 86.
    Holzapfel GA (2000) Nonlinear solid mechanics. A continuum approach for engineering. Wiley, Chichester, p 470zbMATHGoogle Scholar
  87. 87.
    Boehler L (1978) de comportement anisotrope des milieux continus. J Méc 17(2):153–190MathSciNetzbMATHGoogle Scholar
  88. 88.
    Limbert G, Taylor M (2002) On the constitutive modeling of biological soft connective tissues. A general theoretical framework and tensors of elasticity for strongly anisotropic fiber-reinforced composites at finite strain. Int J Solids Struct 39(8):2343–2358zbMATHGoogle Scholar
  89. 89.
    Spencer AJM (1992) Continuum theory of the mechanics of fibre-reinforced composites. Springer, New YorkGoogle Scholar
  90. 90.
    Criscione JC et al (2000) An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity. J Mech Phys Solids 48(12):2445–2465zbMATHGoogle Scholar
  91. 91.
    Criscione JC, Douglas AS, Hunter WC (2001) Physically based strain invariant set for materials exhibiting transversely isotropic behavior. J Mech Phys Solids 49(4):871–897zbMATHGoogle Scholar
  92. 92.
    Holzapfel GA, Ogden RW (2016) On fiber dispersion models: exclusion of compressed fibers and spurious model comparisons. J Elast 129(1–2):49–68MathSciNetzbMATHGoogle Scholar
  93. 93.
    Lanir Y (1983) Constitutive equations for fibrous connective tissues. J Biomech 16(1):1–22Google Scholar
  94. 94.
    Gasser TC, Ogden RW, Holzapfel GA (2006) Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface 3(6):15–35Google Scholar
  95. 95.
    Li K, Ogden RW, Holzapfel GA (2018) A discrete fibre dispersion method for excluding fibres under compression in the modelling of fibrous tissues. J R Soc Interface 15(138)Google Scholar
  96. 96.
    Li K, Ogden RW, Holzapfel GA (2018) Modeling fibrous biological tissues with a general invariant that excludes compressed fibers. J Mech Phys Solids 110:38–53MathSciNetGoogle Scholar
  97. 97.
    Alastrué V et al (2009) Anisotropic micro-sphere-based finite elasticity applied to blood vessel modelling. J Mech Phys Solids 57(1):178–203zbMATHGoogle Scholar
  98. 98.
    Holzapfel GA et al (2015) Modelling non-symmetric collagen fibre dispersion in arterial walls. J R Soc Interface 12(106)Google Scholar
  99. 99.
    Sáez P et al (2012) Anisotropic microsphere-based approach to damage in soft fibered tissue. Biomech Model Mechanobiol 11(5):595–608Google Scholar
  100. 100.
    Ogden RW (2016) Nonlinear continuum mechanics and modelling the elasticity of soft biological tissues with a focus on artery walls. In: Holzapfel GA, Ogden RW (eds) Lecture notes from the summer school “Biomechanics: trends in modeling and simulation, September, 2014, Graz. Springer, HeidelbergGoogle Scholar
  101. 101.
    Winitzki S (2003) Uniform approximations for transcendental functions. In: Kumar V et al (eds) Computational science and its applications—ICCSA 2003: Proceedings of international conference, Part I, Montreal, 18–21 May 2003. Springer, pp 780–789Google Scholar
  102. 102.
    Ogden RW (1984) Non-linear elastic deformations. Ellis Horwood, West SussexzbMATHGoogle Scholar
  103. 103.
    Jansen LH, Rottier PB (1958) Some mechanical properties of human abdominal skin measured on excised strips: a study of their dependence on age and how they are influenced by the presence of striae. Dermatologica 117:65–83Google Scholar
  104. 104.
    Shergold OA, Fleck NA, Radford D (2006) The uniaxial stress versus strain response of pig skin and silicone rubber at low and high strain rates. Int J Impact Eng 32(9):1384–1402Google Scholar
  105. 105.
    Delalleau A et al (2008) A nonlinear elastic behavior to identify the mechanical parameters of human skin in vivo. Skin Res Technol 14(2):152–164Google Scholar
  106. 106.
    Lapeer RJ, Gasson PD, Karri V (2010) Simulating plastic surgery: from human skin tensile tests, through hyperelastic finite element models to real-time haptics. Prog Biophys Mol Biol 103(2–3):208–216Google Scholar
  107. 107.
    Yeoh OH (1993) Some forms of the strain energy function for rubber. Rubber Chem Technol 66(5):754–771Google Scholar
  108. 108.
    Ogden RW (1972) Large deformation isotropic elasticity – correlation of theory and experiment for compressible rubberlike solids. Proc R Soc Lond A Math Phys Sci 328(1575):567zbMATHGoogle Scholar
  109. 109.
    Ogden RW (1972) Large deformation isotropic elasticity – correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc Lond A Math Phys Sci 326(1567):565zbMATHGoogle Scholar
  110. 110.
    Shergold OA, Fleck NA (2004) Mechanisms of deep penetration of soft solids, with application to the injection and wounding of skin. Proc R Soc A Math Phys Eng Sci 460(2050):3037–3058zbMATHGoogle Scholar
  111. 111.
    Lim J et al (2011) Mechanical response of pig skin under dynamic tensile loading. Int J Impact Eng 38(2):130–135Google Scholar
  112. 112.
    Evans SL, Holt CA (2009) Measuring the mechanical properties of human skin in vivo using digital image correlation and finite element modelling. J Strain Anal Eng Des 44(5):337–345Google Scholar
  113. 113.
    Flynn C, Taberner A, Nielsen P (2011) Modeling the mechanical response of in vivo human skin under a rich set of deformations. Ann Biomed Eng 39(7):1935–1946Google Scholar
  114. 114.
    Flynn C et al (2013) Simulating the three-dimensional deformation of in vivo facial skin. J Mech Behav Biomed Mater 28(0):484–494Google Scholar
  115. 115.
    Flory PJ (1969) Statistical mechanics of chain molecules. Wiley, ChichesterGoogle Scholar
  116. 116.
    Kuhl E et al (2005) Remodeling of biological tissue: mechanically induced reorientation of a transversely isotropic chain network. J Mech Phys Solids 53:1552–1573MathSciNetzbMATHGoogle Scholar
  117. 117.
    Kratky O, Porod G (1949) Röntgenuntersuchungen gelöster Fadenmoleküle. Recl Trav Chim Pays-Bas Belg 68:1106–1122Google Scholar
  118. 118.
    Bischoff JE, Arruda EA, Grosh K (2002) A microstructurally based orthotropic hyperelastic constitutive law. J Appl Mech Trans ASME 69(5):570–579zbMATHGoogle Scholar
  119. 119.
    Bischoff JE, Arruda EM, Grosh K (2004) A rheological network model for the continuum anisotropic and viscoelastic behavior of soft tissue. Biomech Model Mechanobiol 3(1):56–65Google Scholar
  120. 120.
    Garikipati K et al (2004) A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics. J Mech Phys Solids 52(7):1595–1625MathSciNetzbMATHGoogle Scholar
  121. 121.
    Flynn C, McCormack BAO (2008) A simplified model of scar contraction. J Biomech 41(7):1582–1589Google Scholar
  122. 122.
    Flynn CO, McCormack BAO (2009) A three-layer model of skin and its application in simulating wrinkling. Comput Methods Biomech Biomed Engin 12(2):125–134Google Scholar
  123. 123.
    Kuhl E, Holzapfel GA (2007) A continuum model for remodeling in living structures. J Mater Sci 42(21):8811–8823Google Scholar
  124. 124.
    Kuhn W (1936) Beziehungen zwischen Molekühlgrösse, statistischer Molekülgestalt und elastischen Eigenschaften hochpolymerer Stoffe. Kolloid Z 76:258–271Google Scholar
  125. 125.
    Arruda EM, Boyce MC (1993) A three-dimensional constitutive model for the large stretch behavior of rubber elastic-materials. J Mech Phys Solids 41(2):389–412zbMATHGoogle Scholar
  126. 126.
    Cohen A (1991) A Padé approximant to the inverse Langevin function. Rheol Acta 30(3):270–273Google Scholar
  127. 127.
    Nguessong AN, Beda T, Peyraut F (2014) A new based error approach to approximate the inverse langevin function. Rheol Acta 53(8):585–591Google Scholar
  128. 128.
    Jedynak R (2015) Approximation of the inverse Langevin function revisited. Rheol Acta 54(1):29–39Google Scholar
  129. 129.
    Marchi BC, Arruda EM (2015) An error-minimizing approach to inverse Langevin approximations. Rheol Acta 54(11):887–902Google Scholar
  130. 130.
    Darabi E, Itskov M (2015) A simple and accurate approximation of the inverse Langevin function. Rheol Acta 54(5):455–459Google Scholar
  131. 131.
    Bischoff JE, Arruda EM, Grosh K (2000) Finite element modeling of human skin using an isotropic, nonlinear elastic constitutive model. J Biomech 33(6):645–652Google Scholar
  132. 132.
    Dunn MG, Silver FH, Swann DA (1985) Mechanical analysis of hypertrophic scar tissue: structural basis for apparent increased rigidity. J Invest Dermatol 84(1):9–13Google Scholar
  133. 133.
    Belkoff SM, Haut RC (1991) A structural model used to evaluate the changing microstructure of maturing rat skin. J Biomech 24(8):711–720Google Scholar
  134. 134.
    Gunner CW, Hutton WC, Burlin TE (1979) The mechanical properties of skin in vivo—a portable hand-held extensometer. Br J Dermatol 100(2):161–163Google Scholar
  135. 135.
    Meijer R, Douven LFA, Oomens CWJ (1999) Characterisation of anisotropic and non-linear behaviour of human skin in vivo. Comput Methods Biomech Biomed Eng 2(1):13–27Google Scholar
  136. 136.
    Jor JWY et al (2011) Estimating material parameters of a structurally based constitutive relation for skin mechanics. Biomech Model Mechanobiol 10(5):767–778Google Scholar
  137. 137.
    Flynn C, McCormack BAO (2008) Finite element modelling of forearm skin wrinkling. Skin Res Technol 14(3):261–269Google Scholar
  138. 138.
    Flynn CO, McCormack BAO (2010) Simulating the wrinkling and aging of skin with a multi-layer finite element model. J Biomech 43(3):442–448Google Scholar
  139. 139.
    Limbert G, Middleton J (2005) A polyconvex anisotropic strain energy function. Application to soft tissue mechanics. In: ASME summer bioengineering conference, VailGoogle Scholar
  140. 140.
    Itskov M, Ehret AE, Mavrilas D (2006) A polyconvex anisotropic strain-energy function for soft collagenous tissues. Biomech Model Mechanobiol 5(1):17–26Google Scholar
  141. 141.
    Itskov M, Aksel N (2004) A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function. Int J Solids Struct 41(14):3833–3848MathSciNetzbMATHGoogle Scholar
  142. 142.
    Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48MathSciNetzbMATHGoogle Scholar
  143. 143.
    Tonge TK, Voo LM, Nguyen TD (2013) Full-field bulge test for planar anisotropic tissues: Part II – A thin shell method for determining material parameters and comparison of two distributed fiber modeling approaches. Acta Biomater 9(4):5926–5942Google Scholar
  144. 144.
    Buganza Tepole A, Gosain AK, Kuhl E (2014) Computational modeling of skin: using stress profiles as predictor for tissue necrosis in reconstructive surgery. Comput Struct 143:32–39Google Scholar
  145. 145.
    Flynn C, Rubin MB, Nielsen P (2011) A model for the anisotropic response of fibrous soft tissues using six discrete fibre bundles. Int J Numer Methods Biomed Eng 27(11):1793–1811zbMATHGoogle Scholar
  146. 146.
    Ankersen J et al (1999) Puncture resistance and tensile strength of skin simulants. Proc Inst Mech Eng Part H J Eng Med 213(H6):493–501Google Scholar
  147. 147.
    Flynn C, Rubin MB (2012) An anisotropic discrete fibre model based on a generalised strain invariant with application to soft biological tissues. Int J Eng Sci 60:66–76MathSciNetzbMATHGoogle Scholar
  148. 148.
    Limbert G (2011) A mesostructurally-based anisotropic continuum model for biological soft tissues—decoupled invariant formulation. J Mech Behav Biomed Mater 4(8):1637–1657Google Scholar
  149. 149.
    Bischoff JE, Arruda EM, Grosh K (2002) Finite element simulations of orthotropic hyperelasticity. Finite Elem Anal Des 38(10):983–998zbMATHGoogle Scholar
  150. 150.
    Lu J, Zhang L (2005) Physically motivated invariant formulation for transversely isotropic hyperelasticity. Int J Solids Struct 42(23):6015–6031zbMATHGoogle Scholar
  151. 151.
    Korelc J, Šolinc U, Wriggers P (2010) An improved EAS brick element for finite deformation. Comput Mech 46(4):641–659zbMATHGoogle Scholar
  152. 152.
    Gautieri A et al (2011) Hierarchical structure and nanomechanics of collagen microfibrils from the atomic scale up. Nano Lett 11:757–766Google Scholar
  153. 153.
    Sun YL et al (2002) Direct quantification of the flexibility of type I collagen monomer. Biochem Biophys Res Commun 295(2):382–386Google Scholar
  154. 154.
    Groves RB et al (2013) An anisotropic, hyperelastic model for skin: experimental measurements, finite element modelling and identification of parameters for human and murine skin. J Mech Behav Biomed Mater 18(0):167–180Google Scholar
  155. 155.
    Weiss JA, Maker BN, Govindjee S (1996) Finite element implementation of incompressible transversely isotropic hyperelasticity. Comput Methods Appl Mech Eng 135:107–128zbMATHGoogle Scholar
  156. 156.
    Yang W et al (2015) On the tear resistance of skin. Nat Commun 6:6649Google Scholar
  157. 157.
    Sherman VR, Yang W, Meyers MA (2015) The materials science of collagen. J Mech Behav Biomed Mater 52:22–50Google Scholar
  158. 158.
    Sherman VR et al (2017) Structural characterization and viscoelastic constitutive modeling of skin. Acta Biomater 53:460–469Google Scholar
  159. 159.
    Wang S et al (2012) Mechanics of epidermal electronics. J Appl Mech 79(3):031022–031022Google Scholar
  160. 160.
    Barbenel JC, Evans JH (1973) The time-dependent mechanical properties of skin. J Invest Dermatol 69(3):165–172Google Scholar
  161. 161.
    Pereira JM, Mansour JM, Davis BR (1990) Analysis of shear-wave propagation in skin – application to an experimental procedure. J Biomech 23(8):745–751Google Scholar
  162. 162.
    Pereira JM, Mansour JM, Davis BR (1991) Dynamic measurement of the viscoelastic properties of skin. J Biomech 24(2):157–162Google Scholar
  163. 163.
    Lanir Y (1979) The rheological behavior of the skin: experimental results and a structural model. Biorheology 16:191–202Google Scholar
  164. 164.
    Wu JZ et al (2006) Estimation of the viscous properties of skin and subcutaneous tissue in uniaxial stress relaxation tests. Biomed Mater Eng 16(1):53–66Google Scholar
  165. 165.
    Khatyr F et al (2004) Model of the viscoelastic behaviour of skin in vivo and study of anisotropy. Skin Res Technol 10(2):96–103Google Scholar
  166. 166.
    Boyer G et al (2009) Dynamic indentation on human skin in vivo: ageing effects. Skin Res Technol 15(1):55–67MathSciNetGoogle Scholar
  167. 167.
    Boyer G et al (2007) In vivo characterization of viscoelastic properties of human skin using dynamic micro-indentation. Annu Int Conf IEEE Eng Med Biol Soc 1–16:4584–4587Google Scholar
  168. 168.
    Goh KL, Listrat A, Béchet D (2014) Hierarchical mechanics of connective tissues: integrating insights from nano to macroscopic studies. J Biomed Nanotechnol 10(10):2464–2507Google Scholar
  169. 169.
    Redaelli A et al (2003) Possible role of decorin glycosaminoglycans in fibril to fibril force transfer in relative mature tendons—a computational study from molecular to microstructural level. J Biomech 36(10):1555–1569Google Scholar
  170. 170.
    Kearney SP et al (2015) Dynamic viscoelastic models of human skin using optical elastography. Phys Med Biol 60(17):6975–6990Google Scholar
  171. 171.
    Lokshin O, Lanir Y (2009) Viscoelasticity and preconditioning of rat skin under uniaxial stretch: microstructural constitutive characterization. J Biomech Eng 131(3):031009–031010Google Scholar
  172. 172.
    Lokshin O, Lanir Y (2009) Micro and macro rheology of planar tissues. Biomaterials 30(17):3118–3127Google Scholar
  173. 173.
    Fung YC (1973) Biorheology of soft tissues. Biorheology 10:139–155Google Scholar
  174. 174.
    Ehret A (2011) Generalised concepts for constitutive modelling of soft biological tissues. PhD Thesis RWTH Aachen University, pp 1–230Google Scholar
  175. 175.
    Balbi V, Shearer T, Parnell WJ (2018) A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation. Proc R Soc A Math Phys Eng Sci 474(2217):20180231MathSciNetzbMATHGoogle Scholar
  176. 176.
    Bischoff J (2006) Reduced parameter formulation for incorporating fiber level viscoelasticity into tissue level biomechanical models. Ann Biomed Eng 34(7):1164–1172Google Scholar
  177. 177.
    Pioletti DP et al (1998) Viscoelastic constitutive law in large deformations: application to human knee ligaments and tendons. J Biomech 31(8):753–757Google Scholar
  178. 178.
    Coleman BD, Noll W (1961) Foundations of linear viscoelasticity. Rev Mod Phys 3(2):239–249MathSciNetzbMATHGoogle Scholar
  179. 179.
    Limbert G (2004) Development of an advanced computational model for the simulation of damage to human skin. Welsh Development Agency (Technology and Innovation Division) – FIRST Numerics, Cardiff, pp 1–95Google Scholar
  180. 180.
    Limbert G, Middleton J (2004) A transversely isotropic viscohyperelastic material: application to the modelling of biological soft connective tissues. Int J Solids Struct 41(15):4237–4260zbMATHGoogle Scholar
  181. 181.
    Limbert G, Middleton J (2005) An anisotropic viscohyperelastic constitutive model of the posterior cruciate ligament suitable for high loading-rate situations. In: IUTAM symposium on impact biomechanics: from fundamental insights to applications. DublinGoogle Scholar
  182. 182.
    Limbert G, Middleton J (2006) A constitutive model of the posterior cruciate ligament. Med Eng Phys 28(2):99–113Google Scholar
  183. 183.
    Reese S, Govindjee S (1998) A theory of finite viscoelasticity and numerical aspects. Int J Solids Struct 35:3455–3482zbMATHGoogle Scholar
  184. 184.
    Lubarda VA (2004) Constitutive theories based on the multiplicative decomposition of deformation gradient: thermoelasticity, elastoplasticity and biomechanics. Appl Mech Rev 57:95–108Google Scholar
  185. 185.
    Vassoler JM, Reips L, Fancello EA (2012) A variational framework for fiber-reinforced viscoelastic soft tissues. Int J Numer Methods Eng 89(13):1691–1706MathSciNetzbMATHGoogle Scholar
  186. 186.
    Nguyen TD, Jones RE, Boyce BL (2007) Modeling the anisotropic finite-deformation viscoelastic behavior of soft fiber-reinforced composites. Int J Solids Struct 44(25–26):8366–8389zbMATHGoogle Scholar
  187. 187.
    Nedjar B (2007) An anisotropic viscoelastic fibre–matrix model at finite strains: continuum formulation and computational aspects. Comput Meth Appl Mech Eng 196(9–12):1745–1756zbMATHGoogle Scholar
  188. 188.
    Flynn C, Rubin MB (2014) An anisotropic discrete fiber model with dissipation for soft biological tissues. Mech Mater 68:217–227Google Scholar
  189. 189.
    Hollenstein M, Jabareen M, Rubin MB (2013) Modeling a smooth elastic–inelastic transition with a strongly objective numerical integrator needing no iteration. Comput Mech 52(3):649–667MathSciNetzbMATHGoogle Scholar
  190. 190.
    Holzapfel GA, Gasser TC (2001) A viscoelastic model for fiber-reinforced composites at finite strains: continuum basis, computational aspects and applications. Comput Methods Appl Mech Eng 190(34):4379–4403Google Scholar
  191. 191.
    Pena E et al (2007) An anisotropic visco-hyperelastic model for ligaments at finite strains. Formulation and computational aspects. Int J Solids Struct 44(3–4):760–778zbMATHGoogle Scholar
  192. 192.
    Pena E et al (2008) On finite-strain damage of viscoelastic-fibred materials. Application to soft biological tissues. Int J Numer Methods Eng 74(7):1198–1218zbMATHGoogle Scholar
  193. 193.
    Ehret AE, Itskov M, Weinhold GW (2009) A micromechanically motivated model for the viscoelastic behaviour of soft biological tissues at large strains. Nuovo Cimento Della Societa Italiana Di Fisica C-Geophysics and Space Physics 32(1):73–80Google Scholar
  194. 194.
    Gasser TC, Forsell C (2011) The numerical implementation of invariant-based viscoelastic formulations at finite strains. An anisotropic model for the passive myocardium. Comput Methods Appl Mech Eng 200(49-52):3637–3645MathSciNetzbMATHGoogle Scholar
  195. 195.
    Simo JC (1987) On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput Methods Appl Mech Eng 60:153–173zbMATHGoogle Scholar
  196. 196.
    Muñoz MJ et al (2008) An experimental study of the mouse skin behaviour: damage and inelastic aspects. J Biomech 41(1):93–99Google Scholar
  197. 197.
    Edsberg LE et al (1999) Mechanical characteristics of human skin subjected to static versus cyclic normal presures. J Rehabil Res Dev 36(2):133–141Google Scholar
  198. 198.
    Ehret AE, Itskov M (2009) Modeling of anisotropic softening phenomena: application to soft biological tissues. Int J Plast 25:901–919zbMATHGoogle Scholar
  199. 199.
    Ehret AE et al (2011) Porcine dermis in uniaxial cyclic loading: sample preparation, experimental results and modeling. J Mech Mater Struct 6(7–8):1125–1135Google Scholar
  200. 200.
    Volokh KY (2007) Prediciton of arterial failure based on a microstructural bi-layer fiber-matrix model with softening. In: Proceeding of the ASME summer bioengineering conference – 2007, pp 129–130Google Scholar
  201. 201.
    Volokh KY (2011) Modeling failure of soft anisotropic materials with application to arteries. J Mech Behav Biomed Mater 4(8):1582–1594Google Scholar
  202. 202.
    Volokh KY (2014) On irreversibility and dissipation in hyperelasticity with softening. J Appl Mech Trans ASME 81(7):074501Google Scholar
  203. 203.
    Mazza E et al (2005) Nonlinear elastic-viscoplastic constitutive equations for aging facial tissues. Biomech Model Mechanobiol 4(2–3):178–189Google Scholar
  204. 204.
    Mazza E et al (2007) Simulation of the aging face. J Biomech Eng Trans ASME 129(4):619–623Google Scholar
  205. 205.
    Rubin MB, Bodner SR (2002) A three-dimensional nonlinear model for dissipative response of soft tissue. Int J Solids Struct 39(19):5081–5099zbMATHGoogle Scholar
  206. 206.
    Mihai LA, Woolley TE, Goriely A (2018) Stochastic isotropic hyperelastic materials: constitutive calibration and model selection. Proc R Soc A Math Phys Eng Sci 474(2211):201708MathSciNetzbMATHGoogle Scholar
  207. 207.
    Lee T et al (2018) Propagation of material behavior uncertainty in a nonlinear finite element model of reconstructive surgery. Biomech Model Mechanobiol b(6):1857–1873Google Scholar
  208. 208.
    Azencott C-A et al (2017) The inconvenience of data of convenience: computational research beyond post-mortem analyses. Nat Methods 14:937Google Scholar
  209. 209.
    Buehler MJ (2006) Large-scale hierarchical molecular modeling of nanostructured biological materials. J Comput Theor Nanosci 3(5):603–623Google Scholar
  210. 210.
    Rim JE, Pinsky PM, van Osdol WW (2009) Multiscale modeling framework of transdermal drug delivery. Ann Biomed Eng 37(6):1217–1229Google Scholar
  211. 211.
    Bancelin S et al (2015) Ex vivo multiscale quantitation of skin biomechanics in wild-type and genetically-modified mice using multiphoton microscopy. Sci Rep 5:17635Google Scholar
  212. 212.
    Liu W, Röckner M (2015) Stochastic partial differential equations: an introduction, 1st edn. Springer, New York, p 272zbMATHGoogle Scholar
  213. 213.
    Kamiński M (2007) Generalized perturbation-based stochastic finite element method in elastostatics. Comput Struct 85(10):586–594Google Scholar
  214. 214.
    Kirchdoerfer T, Ortiz M (2016) Data-driven computational mechanics. Comput Methods Appl Mech Eng 304:81–101MathSciNetzbMATHGoogle Scholar
  215. 215.
    Oishi A, Yagawa G (2017) Computational mechanics enhanced by deep learning. Comput Methods Appl Mech Eng 327:327–351MathSciNetzbMATHGoogle Scholar
  216. 216.
    Barber D (2012) Bayesian reasoning and machine learning. Cambridge University Press, Cambridge, p 697zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.National Centre for Advanced Tribology at Southampton (nCATS) --- Bioengineering Science Research Group, Department of Mechanical Engineering, Faculty of Engineering and Physical SciencesUniversity of SouthamptonSouthamptonUK
  2. 2.Laboratory of Biomechanics and Mechanobiology, Division of Biomedical Engineering, Department of Human Biology, Faculty of Health SciencesUniversity of Cape TownObservatorySouth Africa

Personalised recommendations