Acta Mechanica Solida Sinica

, Volume 25, Issue 5, pp 483–492 | Cite as

Biomechanical Modeling of Surface Wrinkling of Soft Tissues with Growth-Dependent Mechanical Properties

  • Yanping Cao
  • Yi Jiang
  • Bo Li
  • Xiqiao Feng


This paper explores growth induced morphological instabilities in biological soft materials. In view of that the growth of a living tissue not only changes its geometry but also can alter its mechanical properties, we suggest a refined volumetric growth model incorporating the effects of growth on the mechanical properties of materials. Analogy between this volumetric growth model and the conventional thermal stress model is addressed for both small and finite deformation problems, which brings great ease for the finite element analysis based on the suggested model. Examples of growth induced surface wrinkling behavior in soft composites, including core-shell soft cylinders and three-layered soft tissues, are explored. The results and discussions foresee possible applications of the model in understanding the correlation between the morphogenesis and growth of soft biological tissues (e.g. skins and tumors), as well as in evaluating the deformation and surface instability behavior of soft artificial materials induced by swelling/shrinkage.

Key words

volumetric growth soft tissues surface instability morphogenesis 


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  1. [1]
    Humphrey, J.D., Continuum biomechanics of soft biological tissues. Proceedings of the Royal Society, 2003, A459: 3–46.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Li, B., Cao, Y.P., Feng, X.Q. and Gao, H., Mechanics of morphological instabilities and surface wrinkling in soft materials: A review. Soft Matter, 2012, 8: 5728–5745.CrossRefGoogle Scholar
  3. [3]
    Ben Amar, M. and Goriely, A., Growth and instability in elastic tissues. Journal of the Mechanics and Physics of Solids, 2005, 53: 2284–2319.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Taber, L.A., Biomechanics of growth, remodeling, and morphogenesis. Applied Mechanics Reviews, 1995, 48: 487–545.CrossRefGoogle Scholar
  5. [5]
    Dervaux, J. and Ben Amar, M., Buckling condensation in constrained growth. Journal of the Mechanics and Physics of Solids, 2011, 59: 538–560.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Anderson, A.R.A., Weaver, A.M., Cummings, P.T. and Quaranta, V., Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell, 2006, 127: 905–915.CrossRefGoogle Scholar
  7. [7]
    Rodriguez, E.K., Hoger, A. and McCulloch, A.D., Stress-dependent finite growth in soft elastic tissues. Journal of Biomechanics, 1994, 27: 455–467.CrossRefGoogle Scholar
  8. [8]
    Humphrey, J.D. and Rajagopal, K.R., A constrained mixture model for growth and remodeling of soft tissues. Mathematical Models and Methods in Applied Sciences, 2002, 12: 407–430.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Ambrosi, D., Preziosi, L. and Vitale, G., The insight of mixtures theory for growth and remodeling. Zeitschrift fur Angewandte Mathematik und Physik, 2010, 61: 177–191.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Goriely, A. and Ben Amar, M., Differential growth and instability in elastic shells. Physical Review Letters, 2005, 94: 198103.CrossRefGoogle Scholar
  11. [11]
    Li, B., Jia, F., Cao, Y.P., Feng, X.Q. and Gao, H., Surface wrinkling patterns on a core-shell soft sphere. Physical Review Letters, 2011, 106: 234301.CrossRefGoogle Scholar
  12. [12]
    Li, B., Cao, Y.P. and Feng, X.Q., Growth and surface folding of esophageal mucosa: a biomechanical model. Journal of Biomechanics, 2011, 44: 182–188.CrossRefGoogle Scholar
  13. [13]
    Li, B., Cao, Y.P., Feng, X.Q. and Gao, H., Surface wrinkling of mucosa induced by volumetric growth: theory, simulation and experiment. Journal of the Mechanics and Physics of Solids, 2011, 59: 758–774.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Jin, L.H., Cai, S.Q. and Suo, Z.G, Creases in soft tissues generated by growth. Europhysics Letters, 2011, 95: 64002.CrossRefGoogle Scholar
  15. [15]
    Savin, T., N.Kurpios, N.A., Shyer, A.E., Florescu, P., Liang, H.Y., Mahadevan, L. and Tabin, C.J., On the growth and form of the gut. Nature, 2011, 476: 57–63.CrossRefGoogle Scholar
  16. [16]
    Kuwazuru, O., Saothong, J. and Yoshikawa, N., Mechanical approach to aging and wrinkling of human facial skin based on the multistage buckling theory. Medical Engineering and Physics, 2008, 30: 516–522.CrossRefGoogle Scholar
  17. [17]
    Zhou, J., Kim, H.Y. and Davidson, L.A., Actomyosin stiffens the vertebrate embryo during crucial stages of elongation and neural tube closure. Development, 2009, 136(4): 677–688.CrossRefGoogle Scholar
  18. [18]
    Biot, M.A., Internal instability of anisotropic viscous and viscoelastic media under initial stress. Journal of the Franklin Institute, 1965, 279(2): 65–82.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Ogden, R.W., Non-linear Elastic Deformations. Dover, New York, 1984.zbMATHGoogle Scholar
  20. [20]
    Abaqus. Abaqus analysis user’s manual, version 6.8, 2008.Google Scholar
  21. [21]
    Ansys. Ansys analysis user’s manual, 2008.Google Scholar
  22. [22]
    Stojanovic, R., Djuric, S., Vujosevic, L., On finite thermal deformations, Archiwum Mechaniki Stosowanej, 1964, 16: 103–108.MathSciNetGoogle Scholar
  23. [23]
    Meissonnier, F.T., Busso, E.P. and O’Dowd, N.P., Finite element implementation of a generalized non-local rate-dependent crystallographic formulation for finite strains. International Journal of Plasticity, 2001, 17: 601–640.CrossRefGoogle Scholar
  24. [24]
    Stoop, N., Wittel, F.K., Ben Amar, M., Muller, M.M. and Herrmann, H.J., Self-contact and instabilities in the anisotropic growth of elastic membranes. Physical Review Letters, 2010, 105: 068101.CrossRefGoogle Scholar
  25. [25]
    Cao, Y.P. and Hutchinson, J.W., From wrinkles to creases in elastomers: the instability and imperfection-sensitivity of wrinkling. Proceedings of the Royal Society, 2012, A468: 94–115.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Cao, Y.P. and Hutchinson, J.W., Wrinkling phenomena in neo-Hookean film/substrate bilayers. Journal of Applied Mechanics, 2012, 79, 031019.CrossRefGoogle Scholar
  27. [27]
    Jia, F., Cao, Y.P., Liu, T.S., Jiang, Y., Feng, X.Q. and Yu, S.W., Wrinkling of a bilayer resting on a compliant substrate. Philosophical Magazine, 2012, 92: 1554–1568.CrossRefGoogle Scholar
  28. [28]
    Cao, Y.P., Li, B. and Feng, X.Q., Surface wrinkling and folding of core-shell soft cylinders. Soft Matter, 2012, 8: 556–562.CrossRefGoogle Scholar
  29. [29]
    Magnenat-Thalmann, N., Kalra, P., Leveque, J.L., Bazin, R., Batisse, D. and Querleux, B., A computational skin model: fold and wrinkle formation. IEEE Transactions on Information Technology in Biomedicine, 2002, 6: 317–322.CrossRefGoogle Scholar
  30. [30]
    Flynn, C. and McCormack, B.A.O., Simulating the wrinkling and aging of skin with a multi-layer finite element model. Journal of Biomechanics, 2010, 43: 442–448.CrossRefGoogle Scholar
  31. [31]
    Tracqui, P., Biophysical models of tumour growth. Reports on Progress in Physics, 2009, 72: 056701.CrossRefGoogle Scholar
  32. [32]
    Cristini, V., Frieboes, H.B., Gatenby, R., Caserta, S., Ferrari, M. and Sinek, J., Morphologic instability and cancer invasion. Clinical Cancer Research, 2005, 11: 6772–6779.CrossRefGoogle Scholar
  33. [33]
    Wirtz, D., The physics of cancer: the role of physical interactions and mechanical forces in metastasis. Nature Reviews Cancer, 2011, 11: 512–522.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2012

Authors and Affiliations

  1. 1.Institute of Biomechanics and Medical Engineering, Department of Engineering MechanicsTsinghua UniversityBeijingChina

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