A gradient-enhanced continuum damage model with application to fibre-reinforced tissues at finite strains

A non-local gradient-based damage formulation within a geometrically non-linear set- ting is presented. The hyperelastic constitutive response at local material point level is governed by a strain energy function which is additively composed by an isotropic neo-Hookean matrix and by an anisotropic fibre-reinforced material based on the model proposed by T. Gasser, R. Ogden, and G. Holzapfel

A gradient-enhanced continuum damage model for fibre-reinforced materials

Modelo de daño no local definido a materiales fibrados. Este modelo se aplica al estudio de problemas típicos en la biomecánica de los tejidos blandos, como las paredes arteriales

A microsphere-based remodelling formulation for anisotropic biological tissues

Biological tissues possess the ability to adapt according to the respective local loading conditions, which results in growth and remodelling phenomena. The main goal of this work is the development of a new remodelling approach that, on the one hand, reflects the alignment of fibrous soft biological tissue with respect to representative loading directions. On the other hand, the continuum approach proposed is based on a sound micro-mechanically motivated formulation. To be specific, use of a worm-like chain model is made to describe the behaviour of long-chain molecules as present in, for instance, collageneous tissues. The extension of such a one-dimensional constitutive equation to the three-dimensional macroscopic level is performed by means of a microsphere formulation. Inherent with the algorithmic treatment of this type of modelling approach, a finite number of unit vectors is considered for the numerical integration over the domain of the unit sphere. As a key aspect of this contribution, remodelling is incorporated by setting up evolution equations for the referential orientations of these integration directions. Accordingly, the unit vectors considered now allow interpretation as internal variables, which characterize the material's anisotropic properties. Several numerical studies underline the applicability of the model that, moreover, nicely fits into iterative finite element formulations so that general boundary value problems can be solved

Extremal states of energy of a double-layered thick-walled tube – application to residually stressed arteries

Various biological tissues are designed to optimally support external loads for complex geometries and mechanobiological structures. This results in complex microstructures of such materials. The design of, for instance, (healthy) arteries, which are in the focus of this work, is characterised by a residually stressed fibre-reinforced multi-layered composite with highly non-linear elastic response. The complex interaction of material properties with the geometry and residual stress effects enables the optimal support under different blood pressures, respectively blood flow, within the vessel. The fibres reinforcing the arterial wall, as well as residual stresses present in the vessel, strongly influence its overall behaviour and performance. Turn-over and remodelling processes of the collagenous fibres occurring in the respective layers – either resulting from natural growth phenomena or from artificially induced changes in loading condition such as stent deployment – support the optimisation of the multi-layered composite structure of arteries for the particular loading conditions present in the artery. Within this contribution, the overall energetic properties of an artery are discussed by means of the inflation, bending and extension of a double-layered cylindrical tube. Different states of residual stresses and different fibre orientations are considered so that, for instance, representative fibre angles that result in extremal states of the total potential energy can be identified. In view of turn-over and remodelling processes, these orientations are considered to constitute preferred directions of fibre alignment. In summary, the main goal of this work is to calculate optimal material, structural and loading parameters by concepts of energy-minimisation. Several numerical studies show that the obtained values – such as the fibre orientations, the residual axial stretch and the opening angle – are in good agreement with respective physiological parameters reported in the literature

Application of an anisotropic growth and remodelling formulation to computational structural design

A classical structural optimisation problem consists of a problem-specific objective function which has to be minimised in consideration of particular constraints with respect to design and state variables. In this contribution we adopt a conceptually different approach for the design of a structure which is not based on a classical optimisation technique. Instead, we establish a constitutive micro-sphere-framework in combination with an energy-driven anisotropic microstructural growth formulation, which was originally proposed for the simulation of adaptation and remodelling phenomena in hard biological tissues such as bones. The goal of this contribution is to investigate this anisotropic growth formulation with a special emphasis on its application to structural design problems. To this end, four illustrative three-dimensional benchmark-type boundary value problems are discussed and compared qualitatively with the results obtained by classical structural optimisation strategies. The simulation results capture the densification effects and clearly identify the main load bearing regions. It turns out, that even though making use of this conceptually different growth formulation as compared to the procedures used in a classical structural optimisation context, we identify qualitatively very similar structures or rather regions of densification. Moreover, in contrast to common structural optimisation strategies, which mostly aim to optimise merely the size, shape or topology, our formulation also contains the improvement of the material itself, which apart from the structural improvement results in the generation of problem-specific local material anisotropy and textured evolution. (C) 2012 Elsevier Ltd. All rights reserved

A computational framework for modelling damage-induced softening in fibre-reinforced materials - Application to balloon angioplasty

A computational framework for modelling damage-induced softening in fibre-reinforced materials is presented. The main aspect of this framework is the proposed non-local gradient-enhanced continuum damage formulation. At the material level, the elastic constitutive behavior is defined by a hyperelastic functional including a volumetric and an isochoric contribution. The isochoric contribution is subdivided into three contributions associated to three different phases i=0,1,2. Phase 0 is represented by an incompressible neo-Hookean material, whereas phases 1 and 2 are represented by an exponential format that accounts for the stretching along two preferred anisotropy directions, i.e. two fibre families. Furthermore, a 1-di-type damage function, is introduced to reproduce the loss of stiffness in each phase i. Following the ideas discussed in Dimitrijeciv and Hackl (2008) Waffenschmidt et al. (2014), and references cited therein, the model is built around the enhancement of the local free energy function by means of terms that contain the referential gradients of the non-local damage variables ϕ(symbol)i The inclusion of these terms ensures an implicit regularisation of the finite element implementation. A finite element implementation of the non-local gradient-enhanced continuum damage model is presented. To this end we develop an 8-noded Q1Q1P0 hexahedral element following a variational approach, in order to efficiently model the quasi-incompressible behaviour of the hyperelastic material. This element is implemented in Abaqus by means of a user subroutine UEL. Three boundary value problems are studied: an anisotropic plate with a hole, a balloon angioplasty and a full-3D artery-like tube. These computational experiments serve to illustrate the main capabilities of the proposed model