Biological materials possess the ability to adapt according to their particular loading conditions. The course will focus on the computational continuum modelling of such phenomena, which often are denoted as growth and remodelling. As soft biological tissues—such as skin, tendons, muscles, vessels—usually undergo large deformation and possess a highly fibrous microstructure, the fundamental continuum framework is based on finite anisotropic elasticity. Combination of the specific constitutive models with the finite element method then allows to simulate general boundary value problems. Apart from standard continuum theories, enhanced by the concept of a multiplicative decomposition, we will also discuss the application of so-called computational micro-sphere models, whereby we throughout restrict ourselves to purely mechanical formulations.

Basic concepts of continuum mechanics will be introduced and discussed by means of one-dimensional continua. The key idea consists in summarizing fundamentals present in the subsequent lectures. Apart from the underlying kinematics, fundamental balance equations will be elaborated with a particular focus on inelasticity. In addition to analytical derivations, the numerical treatment of the representative proto-type model problem will be discussed in detail so that all quantities—as needed for the implementation on the so-called local finite element level—are addressed. As an outlook towards the growth and remodelling formulations elaborated in the subsequent lectures, an isotropic density-based growth model is investigated as well.

A.J. Roberts. A one-dimensional introduction to continuum mechanics. World Scientific, 1994.

J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer, 1998.

C.R. Jacobs, M.E. Levenston, G.S. Beaupré, J.C. Simo, and D.R. Carter. Numerical instabilities in bone remodelling simultaneous: the advantages of a node-based finite element approach. J. Biomechanics, 28(4):449–459, 1995.

E. Kuhl, A. Menzel, and P. Steinmann. Computational modeling of growth—a critical review, a classification of concepts and two new consistent approaches. Comput. Mech., 32:71–88, 20

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22.02.2010 |
Basic concepts of one-dimensional |

Lectures 3 and 4 deal with the extension of the two previous lectures to applications in three-dimensional space. To set the stage, basic kinematics are reviewed. Particular emphasis will be placed on the modelling of anisotropy at finite deformations—for instance based on the introduction of structural tensors—whereby both the elastic part as well as the inelastic part possess anisotropic properties. These fundamental concepts nowadays became a standard for the modelling of hard and in particular soft biological tissues.

J.P. Boehler, editor. Applications of Tensor Functions in Solid Mechanics. Number 292 in CISM Courses and Lectures. Springer, 1987.

J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer, 1998.

A. Menzel and P. Steinmann. A view on anisotropic finite hyper-elasticity. Euro. J. Mech. A/Solids, 22:71–87, 2003.

A. Menzel and P. Steinmann. On the spatial formulation of anisotropic multiplicative elasto-plasticity. Comput. Methods Appl. Mech. Engrg., 192:3431–3470, 2003.

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23.02.2010 |
Basic concepts of three-dimensional |

Biological materials exhibit the ability to adapt according to their particular loading conditions. Lectures 5 and 6 will place emphasis on the phenomenological modelling of such phenomena, whereby similar concepts as previously discussed in the context of finite inelasticity are adopted. To be specific, we focus on formulations based on solely displacement degrees of freedom embedded into the theory of open systems. The evolution equations of the internal variables, however, are rather different than for example for elastoplastic continua and, in general, time dependent as establish for visco-elastic materials. Moreover, the mass of a biological tissue may not be conserved which should be accounted for in representative constitutive models. Apart from such growth effects, the lectures will additionally include a discussion on so-called remodelling or rather turnover as reflected by fibre reorientation.

E. Kuhl and P. Steinmann. Theory and numerics of geometrically nonlinear open system mechanics. Int. J. Numer. Methods Engng., 58:1593–1615, 2003.

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.

A. Menzel. Modelling of anisotropic growth in biological tissues—a new approach and computational aspects. Biomech. Model. Mechanobio., 3(3):147–171, 2005.

A. Menzel. A fibre reorientation model for orthotropic multiplicative growth— configurational driving stresses, kinematics-based reorientation, and algorithmic aspects. Biomech. Model. Mechanobio., 6(5):303–320, 2007.

S. Imatani and G.A. Maugin. A constitutive model for material growth and its application to three-dimensional finite element analysis. Mech. Res. Comm., 29:477–483, 2002.

L.A. Taber. Biomechanics of growth, remodelling, and morphogenesis. ASME Appl. Mech. Rev., 48(8):487–545, 1995.

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24.02.2010 |

Biological tissues possess—as many or almost all other materials—different mechanical properties at different scales of observation. Here, emphasis will be placed on the micro-mechanical properties of the fibres of the biological tissue considered. In particular, their distribution in space will be accounted for via so-called orientation distribution functions (odf). Conceptually speaking, an odf constitutes an extension of the previously discussed concept of structural tensors. Such an odf can either be prescribed as a constant distribution or may evolve in time, which reflects remodelling phenomena. Within these two lectures, we focus on the combination of basic odf-concepts with a so-called micro-sphere approach. The latter allows interpretation as a computational multi-scale approach as the constitutive function is evaluated by means of integration over the underlying micro-sphere. As an advantage, this micro-sphere approach conveniently allows to extend physically sound one-dimensional constitutive equations to the three-dimensional case.

E. Kuhl, A. Menzel, and K. Garikipati. On the convexity of transversely isotropic chain network models. Phil. Mag., 86(21–22):3241–3258, 2006.

C. Miehe, S. Göktepe, and F. Lulei. A micro-macro approach to rubber-like materials—part I: the non-affine micro-sphere model of rubber elasticity, J. Mech. Phys. Solids, 52:2617–2660, 2004.

V. Alastrué, M.A. Martínez, M. Doblaré, and A. Menzel. Anisotropic micro-sphere-based finite elasticity applied to blood vessel modelling. J. Mech. Phys. Solids, 57:178–203, 2009.

A. Menzel and T. Waffenschmidt. A micro-sphere-based remodelling formulation for Manisotropic biological tissues. Phil. Trans. R. Soc. A, 367:3499–3523, 2009.

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25.02.2010 |
Introduction to the micro-sphere model. |

The models previously discussed, in general, nicely fit into iterative finite element formulations. As a starting point of these two final lectures, related issues and aspects relevant for the implementation of the finite elasticity, growth and remodelling formulations are addressed. As an outlook we aim at presenting an introduction the concept of configurational mechanics. Conceptually speaking, this framework reflects the mechanics of material inhomogeneities and, accordingly, becomes useful to either detect these or to describe their evolution. While the presentation will at this stage be referred to a rather general continuum mechanical setting, it can be applied to all of the topics discussed above such as inelasticity, growth, and remodelling.

C. Miehe. Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity. Comput. Methods Appl. Mech. Engrg., 134:223–240, 1996.

M. Ekh and A. Menzel. Efficient iteration schemes for anisotropic hyperelasto-plasticity. Int. J. Numer. Methods Engng., 66:707–721, 2006.

G.A. Maugin. Material Inhomogeneities in Elasticity, volume 3 of Applied Mathematics and Mathematical Computation. Chapman & Hall, 1993.

M.E. Gurtin. Configurational Forces as Basic Concept in Continuum Physics, volume 137 of Applied Mathematical Sciences. Springer, 2000.

P. Steinmann. Application of material forces to hyperelastostatic fracture mechanics. I. Continuum mechanical setting. Int. J. Solids Struct., 37:7371–7391, 2000.

A. Menzel, R. Denzer, and P. Steinmann. On the comparison of two approaches to compute material forces for inelastic materials. Application to single-slip crystal-plasticity. Comput. Methods Appl. Mech. Engrg., 193(48–51):5411–5428, 2004.

A. Menzel and P. Steinmann. On configurational forces in multiplicative elastoplasticity. Int. J. Solids Struct., 44(13):4442–4471, 2007.

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26.02.2010 |
Computational aspects and an introduction |
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