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Journal of Nonlinear Science

, Volume 26, Issue 4, pp 929–978 | Cite as

A Geometric Theory of Nonlinear Morphoelastic Shells

  • Souhayl Sadik
  • Arzhang Angoshtari
  • Alain Goriely
  • Arash YavariEmail author
Article

Abstract

Many thin three-dimensional elastic bodies can be reduced to elastic shells: two-dimensional elastic bodies whose reference shape is not necessarily flat. More generally, morphoelastic shells are elastic shells that can remodel and grow in time. These idealized objects are suitable models for many physical, engineering, and biological systems. Here, we formulate a general geometric theory of nonlinear morphoelastic shells that describes both the evolution of the body shape, viewed as an orientable surface, as well as its intrinsic material properties such as its reference curvatures. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell, the so-called material manifold. Geometric quantities attached to the surface, such as the first and second fundamental forms, are obtained from the metric of the three-dimensional body and its evolution. The governing dynamical equations for the body are obtained from variational consideration by assuming that both fundamental forms on the material manifold are dynamical variables in a Lagrangian field theory. In the case where growth can be modeled by a Rayleigh potential, we also obtain the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. We apply these ideas to obtain stress-free growth fields of a planar sheet, the time evolution of a morphoelastic circular cylindrical shell subject to time-dependent internal pressure, and the residual stress of a morphoelastic planar circular shell.

Keywords

Bulk growth Morphoelasticity Shell Nonlinear elasticity Geometric mechanics Residual stress 

Mathematics Subject Classification

74Axx 74Fxx 74Lxx 

Notes

Acknowledgments

SS was supported by a Fulbright Grant. AG is a Wolfson/Royal Society Merit Award Holder and acknowledges support from a Reintegration Grant under EC Framework VII. We thank M.F. Shojaei for his help with some of the numerical examples. This research was partially supported by AFOSR – Grant No. FA9550-12-1-0290 and NSF—Grant No. CMMI 1042559 and CMMI 1130856.

References

  1. Ambrosi, D., Guana, F.: Stress-modulated growth. Math. Mech. Solids 12(3), 319–342 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Ambrosi, D., Ateshian, G., Arruda, E., Cowin, S., Dumais, J., Goriely, A., Holzapfel, G., Humphrey, J., Kemkemer, R., Kuhl, E., Olberding, J., Taber, L., Garikipati, K.: Perspectives on biological growth and remodeling. J. Mech. Phys. Solids 59(4), 863–883 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Angoshtari, A., Yavari, A.: Differential complexes in continuum mechanics. Arch. Ration. Mech. Anal. 216(1), 193–220 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Aron, H.: Das gleichgewicht und die bewegung einer unendlich dünnen, beliebig gekrümmten elastischen schale. J. Reine Angew. Math. 78, 136–174 (1874)MathSciNetGoogle Scholar
  5. Amar, MBen, Goriely, A.: Growth and instability in elastic tissues. J. Mech. Phys. Solids 53(10), 2284–2319 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bonnet, O.: Mémoire sur la théorie des surfaces applicables sur une surface donnée. J. l’École Polytech. 24, 209–230 (1865)Google Scholar
  7. Chien, W.-Z.: The intrinsic theory of thin shells and plates i. Q. Appl. Math. 1, 297–327 (1943)MathSciNetGoogle Scholar
  8. Chladni, E.F.F.: Die Akustik. Breitkopf & Härtel, Wiesbaden (1830)Google Scholar
  9. Chuong, C., Fung, Y.: Residual stress in arteries. In Frontiers in Biomechanics, pp. 117–129. Springer, (1986)Google Scholar
  10. Coddington, A., Levinson, N.: Theory of Ordinary Differential Equations. International series in pure and applied mathematics. Tata McGraw-Hill, New York (1955)zbMATHGoogle Scholar
  11. Cosserat, E., Cosserat, F.: Théories des Corps Déformables. Hermann, Paris (1909)zbMATHGoogle Scholar
  12. Cox, D.: Galois Theory. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts.Wiley, (2012). ISBN 9781118218426Google Scholar
  13. Delsanto, P.P., Guiot, C., Degiorgis, P.G., Condat, C.A., Mansury, Y., Deisboeck, T.S.: Growth model for multicellular tumor spheroids. Appl. Phys. Lett. 85(18), 4225–4227 (2004)CrossRefGoogle Scholar
  14. Dervaux, J., Ciarletta, P., Amar, MBen: Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the föppl-von kármán limit. J. Mech. Phys. Solids 57(3), 458–471 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New Jersey (1976)zbMATHGoogle Scholar
  16. do Carmo, M.: Riemannian Geometry [translated by F. Flahetry from the 1988 Portuguese edition]. Mathematics: Theory & Applications. Birkhäuser Boston, (1992). ISBN 1584883553Google Scholar
  17. Eckart, C.: The thermodynamics of irreversible processes. iv. the theory of elasticity and anelasticity. Phys. Rev. 73(4), 373 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Efrati, E., Sharon, E., Kupferman, R.: Elastic theory of unconstrained non-euclidean plates. J. Mech. Phys. Solids 57(4), 762–775 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ericksen, J.L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1(1), 295–323 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Fox, D., Raoult, A., Simo, J.: A justification of nonlinear properly invariant plate theories. Arch. Ration. Mech. Anal. 124(2), 157–199 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55(11), 1461–1506 (2002a)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Friesecke, G., Müller, S., James, R.D.: Rigorous derivation of nonlinear plate theory and geometric rigidity. Comptes Rendus Math. 334(2), 173–178 (2002b)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Friesecke, G., James, R.D., Mora, M.G., Müller, S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by gamma-convergence. Comptes Rendus Math. 336(8), 697–702 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Fung, Y.: On the foundations of biomechanics. J. Appl. Mech. 50(4b), 1003–1009 (1983)CrossRefGoogle Scholar
  25. Fung, Y.: What are the residual stresses doing in our blood vessels? Ann. Biomed. Eng. 19(3), 237–249 (1991)MathSciNetCrossRefGoogle Scholar
  26. Fung, Y.-C.: Stress, strain, growth, and remodeling of living organisms. In Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids, pp. 469–482. Springer, (1995)Google Scholar
  27. Fusi, L., Farina, A., Ambrosi, D.: Mathematical modeling of a solid-liquid mixture with mass exchange between constituents. Mathe. Mech. Solids 11(6), 575–595 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Geitmann, A., Ortega, J.K.: Mechanics and modeling of plant cell growth. Trends Plant Sci. 14(9), 467–478 (2009)CrossRefGoogle Scholar
  29. Germain, S.: Recherches sur la théorie des surfaces élastiques. Mme. Ve. Courcier, Paris (1821)Google Scholar
  30. Goriely, A., Amar, MBen: Differential growth and instability in elastic shells. Phys. Rev. Lett. 94(19), 198103 (2005)CrossRefGoogle Scholar
  31. Green, A., Zerna, W.: The equilibrium of thin elastic shells. Q. J. Mech. Appl. Math. 3(1), 9–22 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Han, H., Fung, Y.: Residual strains in porcine and canine trachea. J. Biomech. 24(5), 307–315 (1991)CrossRefGoogle Scholar
  33. Helmlinger, G., Netti, P.A., Lichtenbeld, H.C., Melder, R.J., Jain, R.K.: Solid stress inhibits the growth of multicellular tumor spheroids. Nat. Biotechnol. 15(8), 778–783 (1997)CrossRefGoogle Scholar
  34. Hicks, N.J.: Notes on differential geometry. Van Nostrand mathematical studies, no.3. Van Nostrand Reinhold Co., (1965). ISBN 9780442034108Google Scholar
  35. Hori, K., Suzuki, M., Abe, I., Saito, S.: Increased tumor tissue pressure in association with the growth of rat tumors. Jpn. J. Cancer Res.: Gann 77(1), 65–73 (1986)Google Scholar
  36. Hsu, F.-H.: The influences of mechanical loads on the form of a growing elastic body. J. Biomech. 1(4), 303–311 (1968)CrossRefGoogle Scholar
  37. Humphrey, J.: Cardiovascular Solid Mechanics: Cells, Tissues, and Organs. Springer, (2002). ISBN 9780387951683Google Scholar
  38. Ivey, T.A., Landsberg, J.M.: Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems. American Mathematical Society, Providence (2003)CrossRefzbMATHGoogle Scholar
  39. Kadianakis, N., Travlopanos, F.: Kinematics of hypersurfaces in riemannian manifolds. J. Elast. 111(2), 223–243 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. Kirchhoff, G.: Über das gleichgewicht und die bewegung einer elastischen scheibe. J. Reine Angew. Math. 40, 51–88 (1850)MathSciNetCrossRefGoogle Scholar
  41. Koiter, W.T.: On the nonlinear theory of thin elastic shells. I- Introductory sections. II—Basic shell equations. III—Simplified shell equations. K. Ned. Akad. van Wet., Proc., Ser. B 69(1), 1–54 (1966)MathSciNetGoogle Scholar
  42. Kondaurov, V., Nikitin, L.: Finite strains of viscoelastic muscle tissue. J. Appl. Math. Mech. 51(3), 346–353 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  43. Kröner, E.: Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Arch. Ration. Mech. Anal. 4(1), 273–334 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  44. Le Dret, H., Raoult, A.: Le modèle de membrane non linéaire comme limite variationnelle de l’élasticité non linéaire tridimensionnelle. Comptes Rendus l’Acad. Scie.Sér. 1, Math. 317(2), 221–226 (1993)Google Scholar
  45. Lods, V., Miara, B.: Analyse asymptotique des coques en flexion non linéairement élastiques. Comptes Rendus l’Acad. Scie.Sér. 1, Math. 321(8), 1097–1102 (1995)MathSciNetGoogle Scholar
  46. Lods, V., Miara, B.: Nonlinearly elastic shell models: a formal asymptotic approach ii. the flexural model. Arch. Ration. Mech. Anal. 142(4), 355–374 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  47. Love, A.E.H.: The small free vibrations and deformation of a thin elastic shell. Philos. Trans. R. Soc. Lond. A 179, 491–546 (1888)CrossRefzbMATHGoogle Scholar
  48. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1892)zbMATHGoogle Scholar
  49. Lu, J., Papadopoulos, P.: A covariant constitutive description of anisotropic non-linear elasticity. 51(2):204–217, (2000). ISSN 0044-2275Google Scholar
  50. Lubarda, V., Hoger, A.: On the mechanics of solids with a growing mass. Int. J. Solids Struct. 39(18), 4627–4664 (2002)CrossRefzbMATHGoogle Scholar
  51. Lubarda, V.A.: Constitutive theories based on the multiplicative decomposition of deformation gradient: thermoelasticity, elastoplasticity, and biomechanics. Appl. Mech. Rev. 57(2), 95–108 (2004)CrossRefGoogle Scholar
  52. Marsden, J., Hughes, T.: Mathematical Foundations of Elasticity, Dover Civil and Mechanical Engineering Series. Dover, London (1983)Google Scholar
  53. Marsden, J.E., Ratiu, T.: Introd. Mech. Symmetry. Springer, New York (1994)CrossRefGoogle Scholar
  54. Mathieu, E.: Mémoire sur le mouvement vibratoire des cloches. Gauthier-Villars (1882)Google Scholar
  55. McMahon, J., Goriely, A., Tabor, M.: Nonlinear morphoelastic plates I: genesis of residual stress. Math. Mech. Solids 16(8), 812–832 (2011a)MathSciNetCrossRefzbMATHGoogle Scholar
  56. McMahon, J., Goriely, A., Tabor, M.: Nonlinear morphoelastic plates II: exodus to buckled states. Math. Mech. Solids 16(8), 833–871 (2011b)MathSciNetCrossRefzbMATHGoogle Scholar
  57. Miara, B.: Nonlinearly elastic shell models: a formal asymptotic approach i. the membrane model. Arch. Ration. Mech. Anal. 142(4), 331–353 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  58. Naghdi, P.: Foundations of elastic shell theory. In: Sneddon, I.N., Hill, K. (eds.) Progress in Solid Mechanics, vol. 4, pp. 1–90. North Hollande Publishing Cy, Amsterdam (1963)Google Scholar
  59. Nishikawa, S.: Variational Problems in Geometry, volume 205 of Iwanami series in modern mathematics. American Mathematical Society, (2002). ISBN 9780821813560Google Scholar
  60. Novozhilov, V.: The theory of thin shells [translated by P. G. Lowe from the 1951 Russian edition]. P. Noordhoff, (1964)Google Scholar
  61. Olsson, T., Klarbring, A.: Residual stresses in soft tissue as a consequence of growth and remodeling: application to an arterial geometry. Eur. J. Mech.A/Solids 27(6), 959–974 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  62. Omens, J.H., Fung, Y.-C.: Residual strain in rat left ventricle. Circ. Res. 66(1), 37–45 (1990)CrossRefGoogle Scholar
  63. Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys. 51, 032902 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  64. Pezzulla, M., Shillig, S.A., Nardinocchi, P., Holmes, D.P.: Morphing of geometric composites via residual swelling. Soft. Matter. 11, 5812–5820 (2015a)CrossRefGoogle Scholar
  65. Pezzulla, M., Smith, G.P., Nardinocchi, P., Holmes, D.P.: Geometry and mechanics of thin growing bilayers. pp. 1–5, (2015b). arXiv:1509.05259v2
  66. Pollack, J.B., Hubickyj, O., Bodenheimer, P., Lissauer, J.J., Podolak, M., Greenzweig, Y.: Formation of the giant planets by concurrent accretion of solids and gas. Icarus 124(1), 62–85 (1996)CrossRefGoogle Scholar
  67. Polyanin, A., Zaitsev, V.: Handbook of Nonlinear Partial Differential Equations. CRC Press, (2004). ISBN 9780203489659Google Scholar
  68. Rodriguez, E.K., Hoger, A., McCulloch, A.D.: Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27(4), 455–467 (1994)CrossRefGoogle Scholar
  69. Sadik, S., Yavari, A.: Geometric nonlinear thermoelasticity and the time evolution of thermal stresses. Math. Mech. Solids (2015). doi: 10.1177/1081286515599458
  70. Sadik, S., Yavari, A.: On the origins of the idea of the multiplicative decomposition of the deformation gradient. Math. Mech. Solids. (2016). doi: 10.1177/1081286515612280,
  71. Sanders Jr., J.L.: Nonlinear theories for thin shells. Technical report technical report no. 10, DTIC Document, (1961)Google Scholar
  72. Silberberg, J.S., Barre, P.E., Prichard, S.S., Sniderman, A.D.: Impact of left ventricular hypertrophy on survival in end-stage renal disease. Kidney Int. 36(2), 286–290 (1989)CrossRefGoogle Scholar
  73. Skalak, R.: Growth as a finite displacement field. In: Proceedings of the IUTAM Symposium on Finite Elasticity, pp. 347–355. Springer, (1982)Google Scholar
  74. Skalak, R., Dasgupta, G., Moss, M., Otten, E., Dullemeijer, P., Vilmann, H.: Analytical description of growth. J. Theor. Biol. 94(3), 555–577 (1982)MathSciNetCrossRefGoogle Scholar
  75. Skalak, R., Zargaryan, S., Jain, R.K., Netti, P.A., Hoger, A.: Compatibility and the genesis of residual stress by volumetric growth. J. Math. Biol. 34(8), 889–914 (1996)CrossRefzbMATHGoogle Scholar
  76. Stojanović, R., Djurić, S., Vujošević, L.: On finite thermal deformations. Arch. Mech. Stosow. 1(16), 103–108 (1964)MathSciNetGoogle Scholar
  77. Synge, J.L., Chien, W.Z.: The intrinsic theory of elastic shells and plates. In: von Kármán anniv. vol., pp. 103–120. Cal. Inst. Tech., Pasadena, (1941)Google Scholar
  78. Taber, L.A.: Biomechanics of growth, remodeling, and morphogenesis. Appl. Mech. Rev. 48(8), 487–545 (1995)CrossRefGoogle Scholar
  79. Takamizawa, K., Matsuda, T.: Kinematics for bodies undergoing residual stress and its applications to the left ventricle. J. Appl. Mech. 57(2), 321–329 (1990)CrossRefGoogle Scholar
  80. Verpoort, S.: The geometry of the second fundamental form: curvature properties and variational aspects. PhD thesis, Katholieke Universiteit Leuven, (2008)Google Scholar
  81. Yavari, A.: A geometric theory of growth mechanics. J. Nonlinear Sci. 20(6), 781–830 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  82. Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal. 205(1), 59–118 (2012a)MathSciNetCrossRefzbMATHGoogle Scholar
  83. Yavari, A., Goriely, A.: Weyl geometry and the nonlinear mechanics of distributed point defects. Proc. R. Soc. A 468, 3902–3922 (2012b)MathSciNetCrossRefGoogle Scholar
  84. Yavari, A., Goriely, A.: Nonlinear elastic inclusions in isotropic solids. Proc. R. Soc. A: Math., Phys. Eng. Sci. 469(2160), 20130415 (2013a)MathSciNetCrossRefGoogle Scholar
  85. Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear disclination mechanics. Math. Mech. Solids 18(1), 91–102 (2013b)MathSciNetCrossRefGoogle Scholar
  86. Yavari, A., Goriely, A.: The geometry of discombinations and its applications to semi-inverse problems in anelasticity. Proc. R. Soc. A 470, 20140403 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  87. Yavari, A., Goriely, A.: The twist-fit problem: finite torsional and shear eigenstrains in nonlinear elastic solids. Proc. R. Soc. A 471, 20150596 (2015)CrossRefGoogle Scholar
  88. Yavari, A., Marsden, J.E., Ortiz, M.: On spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 042903 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Souhayl Sadik
    • 1
  • Arzhang Angoshtari
    • 2
  • Alain Goriely
    • 3
  • Arash Yavari
    • 1
    • 4
    Email author
  1. 1.School of Civil and Environmental EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of Civil and Environmental EngineeringThe George Washington UniversityWashingtonUSA
  3. 3.Mathematical InstituteUniversity of OxfordOxfordUK
  4. 4.The George W. Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA

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