Theories of Growth

  • Marcelo EpsteinEmail author
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 262)


The modelling of processes of growth and the associated phenomena of remodelling, ageing and morphogenesis requires a rethinking and reformulation of the fundamental notions of material body, balance equations and constitutive theory. Some of these ideas are presented and treated with various degrees of detail.


  1. 1.
    Coleman BD, Gurtin ME (1967) Thermodynamics with internal state variables. J Chem Phys 47:597–613CrossRefGoogle Scholar
  2. 2.
    Cowin SC, Hegedus DH (1976) Bone remodelling I: theory of adaptive elasticity. J Elast 6:313–326CrossRefGoogle Scholar
  3. 3.
    DiCarlo A (2005) Surface and bulk growth unified. In: Steinmann P, Maugin GA (eds) Mechanics of material forces, advances in mechanics and mathematics, vol 11. Springer, US, pp 53–64CrossRefGoogle Scholar
  4. 4.
    Epstein M (2010) Kinetics of boundary growth. Mech Res Commun 37:453–457CrossRefGoogle Scholar
  5. 5.
    Epstein M (2015) Mathematical characterization and identification of remodeling, growth aging and morphogenesis. J Mech Phys Solids 84:72–84MathSciNetCrossRefGoogle Scholar
  6. 6.
    Epstein M, Elżanowski M (2007) Material inhomogeneities and their evolution. Springer, Berlin, HeidelbergzbMATHGoogle Scholar
  7. 7.
    Epstein M, Goriely A (2012) Self-diffusion in remodeling and growth. Z Angew Math Phys 63:339–355MathSciNetCrossRefGoogle Scholar
  8. 8.
    Epstein M, Maugin GA (2000) Thermomechanics of volumetric growth in uniform bodies. Int J Plast 6:951–978CrossRefGoogle Scholar
  9. 9.
    Eshelby JD (1951) The force on an elastic singularity. Phil Trans R Soc A 244:87–112MathSciNetCrossRefGoogle Scholar
  10. 10.
    Garikipati K, Arruda EM, Grosh K, Narayanan H, Calve S (2004) A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics. J Mech Phys Solids 52:1595–1625MathSciNetCrossRefGoogle Scholar
  11. 11.
    Goriely A (2017) The mathematics and mechanics of biological growth. Springer, New YorkCrossRefGoogle Scholar
  12. 12.
    Green AE, Rivlin RS (1964) On Cauchy’s equations of motion. Z Angew Math Phys 15:291–294MathSciNetCrossRefGoogle Scholar
  13. 13.
    Javadi M, Epstei M (2019) Invariance in growth and mass transport. Math Mech Solids, in press. Scholar
  14. 14.
    Kuhl E, Steinmann P (2003) Mass- and volume-specific views on thermodynamics for open systems. Proc R Soc A 459:2547–2568MathSciNetCrossRefGoogle Scholar
  15. 15.
    Marsden JE, Hughes TJR (1983) Mathematical foundations of elasticity. Prentice-Hall, Englewood Cliffs, NJ; Dover Publications, New York (1994)Google Scholar
  16. 16.
    MATLAB version (R2017b). The MathWorks Inc., Natick, Massachusetts (2017)Google Scholar
  17. 17.
    Menzel A, Kuhl E (2012) Frontiers in growth and remodeling. Mech Res Commun 42:1–14CrossRefGoogle Scholar
  18. 18.
    Misner CW, Thorne KS, Wheeler JA (1973) Gravitation. W.H. Freeman and Company, San FranciscoGoogle Scholar
  19. 19.
    Noll W (1963) La mécanique classique basée sur un axiome d’objectivité. In: Châtelet A, Destouches J-L (eds) La méthode axiomatique dans les mécaniques classiques et nouvelles. Gauthier-Villars, Paris, pp. 47–56; Reproduced in the foundations of mechanics and thermodynamics, selected papers by Walter Noll, pp 135–144. Springer, Berlin, Heidelberg (1974)Google Scholar
  20. 20.
    Noll W (1967) Materially uniform bodies with inhomogeneities. Arch Ration Mech Anal 27:1–32MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27:455–467CrossRefGoogle Scholar
  22. 22.
    Segev R (1996) Growing bodies and the Eshelby tensor. Meccanica 31:507–518CrossRefGoogle Scholar
  23. 23.
    Skalak R, Dasgupta G, Moss M, Otten E, Dullemeuer P, Vilmann H (1982) Analytical description of growth. J Theor Biol 94:555–577MathSciNetCrossRefGoogle Scholar
  24. 24.
    Taber LA (1995) Biomechanics of growth, remodeling, and morphogenesis. Appl Mech Rev 48:487–545CrossRefGoogle Scholar
  25. 25.
    Thompson DW (1917) On growth and form. Cambridge University Press, CambridgeGoogle Scholar
  26. 26.
    Truesdell C (1984) Rational thermodynamics, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  27. 27.
    Truesdell CA, Noll W (1965) The non-linear field theories of mechanics. In: Flügge S (ed) Handbuch der physik, Vol III/3. Springer, BerlinGoogle Scholar
  28. 28.
    Turing AM (1952) The chemical basis of morphogenesis. Phil Trans R Soc B 237:37–72MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wang C-C, Truesdell C (1973) Introduction to rational elasticity. Noordhoff International Publishing, LeydenzbMATHGoogle Scholar
  30. 30.
    Wolfram Research (2017) Inc., Mathematica, Version 11, ChampaignGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mechanical and Manufacturing EngineeringUniversity of CalgaryCalgaryCanada

Personalised recommendations