Skip to main content
SpringerLink
Log in

Anticavitation and Differential Growth in Elastic Shells

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

Elastic anticavitation is the phenomenon of a void in an elastic solid collapsing on itself. Under the action of mechanical loading alone typical materials do not admit anticavitation. We study the possibility of anticavitation as a consequence of an imposed differential growth. Working in the geometry of a spherical shell, we seek radial growth functions which cause the shell to deform to a solid sphere. It is shown, surprisingly, that most material models do not admit full anticavitation, even when infinite growth or resorption is imposed at the inner surface of the shell. However, void collapse can occur in a limiting sense when radial and circumferential growth are properly balanced. Growth functions which diverge or vanish at a point arise naturally in a cumulative growth process.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abeyaratne, R., Hou, H.-S.: Void collapse in an elastic solid. J. Elast. 26, 23–42 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alford, P.W., Humphrey, J.D., Taber, L.A.: Growth and remodeling in a thick-walled artery model: effects of spatial variations in wall constituents. Biomech. Model. Mechanobiol. 7, 245–262 (2007)

    Article  Google Scholar 

  3. Ben Amar, M., Goriely, A.: Growth and instability in elastic tissues. J. Mech. Phys. Solids 53, 2284–2319 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Ambrosi, D., Preziosi, L.: Cell adhesion mechanisms and stress relaxation in the mechanics of tumours. Biomech. Model. Mechanobiol. 8(5), 397–413 (2009)

    Article  Google Scholar 

  5. Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. A 306, 557–610 (1982)

    Article  MATH  ADS  Google Scholar 

  6. Batchelor, G.K.: An Introduction to Fluid Mechanics. Cambridge University Press, Cambridge (1967)

    Google Scholar 

  7. Burstrom, H.G.: Tissue tensions during cell elongation in wheat roots and a comparison with contractile roots. Physiol. Plant. 25(3), 509–513 (1971)

    Article  Google Scholar 

  8. Cardamone, L., Valentin, A., Eberth, J.F., Humphrey, J.D.: Origin of axial prestretch and residual stress in arteries. Biomech. Model. Mechanobiol. 8(6), 431–446 (2009)

    Article  Google Scholar 

  9. Casey, J., Naghdi, P.M.: A remark on the use of the decomposition F=F e F p in plasticity. Trans. ASME 47, 672–675 (1980)

    Article  MATH  Google Scholar 

  10. Chaudhry, H.R., Bukiet, B., Davis, A., Ritter, A.B., Findley, T.: Residual stresses in oscillating thoracic arteries reduce circumferential stresses and stress gradients. J. Biomech. 30(1), 57–62 (1997)

    Article  Google Scholar 

  11. Chen, Y., Hoger, A.: Constitutive functions of elastic materials in finite growth and deformation. J. Elast. 59, 175–193 (2000)

    Article  MATH  Google Scholar 

  12. Delfino, A., Stergiopulos, N., Moore, J.E., Meister, J.J.: Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation. J. Biomech. 30(8), 777–786 (1997)

    Article  Google Scholar 

  13. Dinwoodie, J.M.: Growth stresses in timber—a review of literature. Forestry 39(2), 162 (1966)

    Article  Google Scholar 

  14. Dollhofer, J., Chiche, A., Muralidharan, V., Creton, C., Hui, C.Y.: Surface energy effects for cavity growth and nucleation in an incompressible neo-hookean material—modeling and experiment. Int. J. Solids Struct. 41, 6111–6127 (2004)

    Article  MATH  Google Scholar 

  15. Fond, C.: Cavitation criterion for rubber materials: a review of void-growth models. J. Polym. Sci. B: Polym. Phys. 39, 2081–2096 (2001)

    Article  ADS  Google Scholar 

  16. Fung, Y.C.: What are the residual stresses doing in our blood vessels? Ann. Biomed. Eng. 19(3), 237–249 (1991)

    Article  MathSciNet  Google Scholar 

  17. Garikipati, K.: The kinematics of biological growth. Appl. Mech. Rev. 62, 030801 (2009)

    Article  ADS  Google Scholar 

  18. Gent, A.N., Lindley, P.B.: Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A 249, 195–205 (1958)

    ADS  Google Scholar 

  19. Goriely, A., Ben Amar, M.: On the definition and modeling of incremental, cumulative, and continuous growth laws in morphoelasticity. Biomech. Model. Mechanobiol. 6(5), 289–296 (2007)

    Article  Google Scholar 

  20. Goriely, A., Ben Amar, M.: Differential growth and instability in elastic shells. Phys. Rev. Lett. 94(19), 198103 (2005)

    Article  ADS  Google Scholar 

  21. Goriely, A., Moulton, D.E., Vandiver, R.: Elastic cavitation and differential growth in biological tissues. Preprint (2010)

  22. Goriely, A., Vandiver, R.: On the mechanical stability of growing arteries. IMA J. Appl. Math. (2010). doi:10.1093/imamat/hxq021

    MathSciNet  Google Scholar 

  23. Guillou, A., Ogden, R.W.: Growth in soft biological tissue and residual stress development. In: Mechanics of Biological Tissue, pp. 47–62. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  24. Gupta, A., Steigmann, D.J., Stölken, J.S.: On the evolution of plasticity and incompatibility. Math. Mech. Solids 12(6), 583–610 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Haughton, D.M.: On non-existence of cavitation in incompressible elastic membranes. Q. J. Mech. Appl. Math. 39(2), 289–296 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  26. Himpel, G., Kuhl, E., Menzel, A., Steinmann, P.: Computational modelling of isotropic multiplicative growth. Comput. Model. Eng. Sci. 8(2), 119–134 (2005)

    MATH  Google Scholar 

  27. Hoger, A.: On the determination of residual stress in an elastic body. J. Elast. 16(3), 303–324 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  28. Holzapfel, G.A., Ogden, R.W.: Modelling the layer-specific three-dimensional residual stresses in arteries, with an application to the human aorta. J. R. Soc. Interface 7, 787–799 (2010)

    Article  Google Scholar 

  29. Horgan, C.O., Polignone, D.A.: Cavitation in nonlinearly elastic solids: a review. Appl. Mech. Rev. 48(8), 471–485 (1995)

    Article  ADS  Google Scholar 

  30. Johnson, B.E., Hoger, A.: The dependence of the elasticity tensor on residual stress. J. Elast. 33(2), 145–165 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. Klisch, S.M., Van Dyke, T.J., Hoger, A.: A theory of volumetric growth for compressible elastic biological materials. Math. Mech. Solids 6, 551–575 (2001)

    Article  MATH  Google Scholar 

  32. Lopez-Pamies, O.: Onset of cavitation in compressible, isotropic, hyperelastic solids. J. Elast. 94, 115–145 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  33. Lubarda, V.A., Hoger, A.: On the mechanics of solids with a growing mass. Int. J. Solids Struct. 39, 4627–4664 (2002)

    Article  MATH  Google Scholar 

  34. McMahon, J., Goriely, A., Tabor, M.: Spontaneous cavitation in growing elastic membranes. Math. Mech. Solids 8, 471–485 (2008)

    Google Scholar 

  35. Naghdi, P.M.: A critical review of the state of finite plasticity. Z. Angew. Math. Phys. (ZAMP) 41(3), 315–394 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ogden, R.W.: Non-linear Elastic Deformation. Dover, New York (1984)

    Google Scholar 

  37. Pence, T.J., Tsai, H.: On the cavitation of a swollen compressible sphere in finite elasticity. Int. J. Non-Linear Mech. 40(2–3), 307–321 (2005)

    Article  MATH  ADS  Google Scholar 

  38. Pence, T.J., Tsai, H.: Swelling induced cavitation of elastic spheres. Math. Mech. Solids 11(5), 527 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  39. Pericak-Spector, K.A., Sivaloganathan, J., Spector, S.J.: An explicit radial cavitation solution in nonlinear elasticity. Math. Mech. Solids 7(1), 87–93 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  40. Peters, W.S., Tomos, A.D.: The history of tissue tension. Ann. Bot. (Lond.) 77(6), 657–665 (1996)

    Article  Google Scholar 

  41. Polignone, D.A., Horgan, C.O.: Cavitation for incompressible anisotropic nonlinearly elastic spheres. J. Elast. 33, 27–65 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  42. Rodriguez, E.K., Hoger, A., McCulloch, A.: Stress-dependent finite growth in soft elastic tissue. J. Biomech. 27, 455–467 (1994)

    Article  Google Scholar 

  43. Rodriguez, E.K., Hoger, A., McCulloch, A.: Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27, 455–467 (1994)

    Article  Google Scholar 

  44. Sivaloganathan, J.: On cavitation and degenerate cavitation under internal hydrostatic pressure. Proc. R. Soc. Lond. A 455, 3645–3664 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. Sivaloganathan, J., Spector, S.J.: On cavitation, configurational forces and implications for fracture in a nonlinearly elastic material. J. Elast. 67, 25–49 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  46. 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)

    MATH  Google Scholar 

  47. Taber, L.A.: Biomechanics of growth, remodeling and morphogenesis. Appl. Mech. Rev. 48, 487–545 (1995)

    Article  ADS  Google Scholar 

  48. Taber, L.A.: Biomechanical growth laws for muscle tissues. J. Theor. Biol. 193, 201–213 (1998)

    Article  Google Scholar 

  49. Taber, L.A., Eggers, D.W.: Theoretical study of stress-modulated growth in the aorta. J. Theor. Biol. 180, 343–357 (1996)

    Article  Google Scholar 

  50. Taber, L.A., Humphrey, J.D.: Stress-modulated growth, residual stress, and vascular heterogeneity. J. Biomech. Eng. 123, 528–535 (2001)

    Article  Google Scholar 

  51. Vaishnav, R.N., Vossoughi, J.: Residual stress and strain in aortic segments. J. Biomech. 20(3), 235–239 (1987)

    Article  Google Scholar 

  52. Vandiver, R., Goriely, A.: Tissue tension and axial growth of cylindrical structures in plants and elastic tissues. Europhys. Lett. (EPL) 84, 58004 (2008)

    Article  ADS  Google Scholar 

  53. Xiao, H., Bruhns, T.O., Meyers, A.: Elastoplasticity beyond small deformations. Acta Mech. 182, 31–111 (2006)

    Article  MATH  Google Scholar 

  54. Yuan, F.: Stress is good and bad for tumors. Nat. Biotechnol. 15(8), 722–723 (1997)

    Article  Google Scholar 

  55. Moulton, D.E., Goriely, A.: Circumferential buckling instability of a growing cylindrical tube. Preprint (2010)

Download references

Author information

Authors and Affiliations

  1. OCCAM, Mathematical Institute, University of Oxford, Oxford, UK

    Derek E. Moulton & Alain Goriely

Authors
  1. Derek E. Moulton
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alain Goriely
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alain Goriely.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moulton, D.E., Goriely, A. Anticavitation and Differential Growth in Elastic Shells. J Elast 102, 117–132 (2011). https://doi.org/10.1007/s10659-010-9266-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-010-9266-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation