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Münster 09/2013

Evolutionary Solid Bodies - Growth and Remodelling of Biological Tissues


Andreas Menzel

French-German Summer School on Evolutionary Solid Bodies - Growth and Remodelling of Biological Tissues, Münster, 08.-14.09.2013



Biological materials possess the ability to adapt according to their particular loading conditions. The course will focus on the computational continuum modeling of such phenomena, which often are denoted as growth and remodeling. As soft biological tissues – such as skin, tendons, muscles, vessels – usually undergo large deformations and possess a highly fibrous microstructure, continuum-mechanics-based modeling frameworks are commonly based on finite anisotropic elasticity. Combination of the specific constitutive models with the finite element method then allows to simulate general boundary value problems. In addition to standard continuum theories, enhanced by the concept of a multiplicative decomposition, the application of so-called computational micro-sphere models will be discussed in the course with application to growth as well as remodeling with focus on hard and soft biological tissues.



  1. Contents
  2. Introduction
  3. Anisotropic finite elasticity
  4. Surface, mass and volume growth
  5. Remodelling
  6. Micro-sphere framework
  7. Appendix



  1. Implementation of a growth model based on mass density evolution
  2. Implementation of a growth model based on a multiplicative decomposition

Specific References

  1. A. Menzel, E. Kuhl, Frontiers in growth and remodeling, Mech. Res. Comm., 42:1-14, 2012
  2. A. Menzel, Modelling of anisotropic growth in biological tissues, Biomechan. Model. Mechanobiol., 3(3):147-171, 2005
  3. A. Menzel, A fibre reorientation model for orthotropic multiplicative growth, Biomechan. Model. Mechanobiol., 6(5):303-320, 2007
  4. A. Menzel, T. Waffenschmidt, A micro-sphere-based remodelling formulation for anisotropic biological tissues, Phil. Trans. R. Soc. A., 367(1902):3499-3523, 2009
  5. T. Waffenschmidt and A. Menzel, Application of an anisotropic growth and remodelling formulation to computational structural design, Mech. Res. Comm., 42:77-86, 2012.
  6. T. Waffenschmidt, A. Menzel and E. Kuhl, Anisotropic density growth of bone – a computational micro-sphere approach, Int. J. Solids Struc., 49(14): 1928-1946, 2012
  7. E. Kuhl, A. Menzel and P. Steinmann, Computational modeling of growth, Comput. Mech., 32(1-2):71-88, 2003
  8. E. Kuhl, R. Maas, G. Himpel, and A. Menzel, Computational modelling of arterial wall growth - attempts towards a patient specific simulation based on computer tomography. Biomechan. Model. Mechanobiol., 6(5):321–331, 2007

General References


  1. J.P. Boehler, editor. Applications of Tensor Functions in Solid Mechanics. Number 292 in CISM Courses and Lectures. Springer, 1987.
  2. J. Bonet and R.D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge, 2008.
  3. S.C. Cowin and S.B. Doty, Tissue Mechanics, Springer, 2007.
  4. J.D. Humphrey, Cardiovascular Solid Mechanics, Springer, 2002.
  5. A.J. Roberts. A one-dimensional introduction to continuum mechanics. World Scientific, 1994.
  6. J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer, 1998