The present manuscript documents our first experiences with a computational model for stress-induced arterial wall growth and in-stent restenosis related to atherosclerosis. The underlying theoretical framework is provided by the kinematics of finite growth combined with open system thermodynamics. The computational simulation is embedded in a finite element approach in which growth is essentially captured by a single scalar-valued growth factor introduced as internal variable on the integration point level. The conceptual simplicity of the model enables its straightforward implementation into standard commercial finite element codes. Qualitative studies of stress-induced changes of the arterial wall thickness in response to balloon angioplasty or stenting are presented to illustrate the features of the suggested growth model. First attempts towards a patient-specific simulation based on realistic artery morphologies generated from computer tomography data are discussed.

SUMMARYTransverse isotropy is realized by one characteristic direction-for instance, the fibre direction in fibrereinforced materials. Commonly, the characteristic direction is assumed to be constant, but in some cases-for instance, in the constitutive description of biological tissues, liquid crystals, grain orientations within polycrystalline materials or piezoelectric materials, as well as in optimization processes-it proves reasonable to consider reorienting fibre directions. Various fields can be assumed to be the driving forces for the reorientation process, for instance, mechanical, electric or magnetic fields. In this work, we restrict ourselves to reorientation processes in hyper-elastic materials driven by principal stretches.The main contribution of this paper is the algorithmic implementation of the reorientation process into a finite element framework. Therefore, an implicit exponential update of the characteristic direction is applied by using the Rodriguez formula to express the exponential term. The non-linear equations on the local and on the global level are solved by means of the Newton-Raphson scheme. Accordingly, the local update of the characteristic direction and the global update of the deformation field are consistently linearized, yielding the corresponding tangent moduli. Through implementation into a finite element code and some representative numerical simulations, the fundamental characteristics of the model are illustrated.

For the modelling of biomaterials with changing mass one can distinguish between a coupling of mass changes and deformations at the constitutive level, see e.g. [3], and a coupling at the kinematic level, see e.g. [4]. The constitutive coupling is typically realised by weighting the free energy function with the relative density. Such an ansatz enables the simulation of density changes and is generally used to model hard tissues. The kinematic coupling is characterised by a multiplicative split of the deformation gradient into an elastic part and a growth part. Such an ansatz enables the simulation of mass changes as changes in density and volume and is appropriate to model soft tissues. In this contribution the agreements and disagreements of both approaches should be presented by means of a transversely isotropic material and studied by numerical examples. For a more detailed description of the discussed topic, the reader is also referred to [2]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Transversely isotropic materials are commonly described by one characteristic direction in the material configuration, which commonly is assumed to be constant. However, for some applications it makes sense to consider a reorientation of the characteristic direction, as for instance the orientation of biological materials adapts to the mechanical loading, see e.g. KUHL ET AL. [4]. Other examples are piezoelectric materials, which reorient due to electric or mechanical loading, as described in SMITH [7], and liquid crystals changing their orientation on account of electric or magnetic fields, see ERICKSEN [1]. Furthermore simulations with a reorientation of the characteristic direction can be used in the context of optimisation of composites. In this contribution we restrict ourselves to the modelling of hyper-elasticity and assume a time dependent reorientation of the characteristic direction with respect to the principal strain directions, see for instance MENZEL [5]. Thereby we concentrate on the numerical implementation into a finite element code. Constitutive EquationsTransverse isotropy can be described by the characteristic direction n A or respectively by the structural tensor A = n A ⊗ n A . In this contribution we assume a reorienting characteristic direction n A . Thus, insertion of the standard free energy density ψ 0 = ψ 0 (C, A) and the Piola-Kirchhoff stresses S = 2∂ C ψ 0 into the dissipation inequality yields the reduced dissipation inequalitywith the extra entropy source S 0 satisfying the second law of thermodynamics, see for instance HIMPEL ET AL. [2]. Analogous to the director in the theory of shells a rotation of the characteristic direction n A is assumed, so that its evolution becomeṡFor the definition of the angular velocity ω an alignment with the principal strain directions, see e.g. MENZEL [5], the principal stress directions, see e.g. HARITON ET AL.[3], the gradients of the electric or magnetic field or other characteristic directions is thinkable. In this contribution we concentrate on a reorientation in terms of the principal strain direction. It can be shown that the critical state of free energy can be reached for coaxial stress and strain tensors. The Piola-Kirchhoff stresses for transverse isotropy can be depicted asdepending on the scalars S i=1,...5 = ∂ Ii ψ. Thus it can easily be observed, that for an alignment of n A with one of the principal strain directions, the Piola-Kirchhoff stresses S and the right Cauchy-Green tensor C become coaxial. Hence, following to MENZEL [5] we assume an alignment of n A with the maximum principal strain directionTo avoid drilling rotation the angular velocity is defined orthogonal to n A and n Cwith the material parameter t acting as time relaxation parameter. Implementation into a Finite Element CodeFor the implementation into a finite element code the characteristic direction has been introduced as internal variable. Due to the EULER-theorem exp(−ε · ω ∆t) is a rotation about ω by the angle ω ∆t. Thus we apply an implicit exponential map fo...