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A Gradient-Enhanced Continuum Damage Model for Residually Stressed Fibre-Reinforced Materials at Finite Strains

Waffenschmidt, Tobias; Polindara, César and Menzel, Andreas LU (2015) In Biomedical Technology 74. p.19-40
Abstract
The modelling of damage effects in materials constitutes a major challenge in various engineering-related disciplines. However, the assumption of purely local continuum damage formulations may lead to ill-posed boundary value problems and—with regard to numerical methods such as the finite element method—to mesh-dependent solutions, a vanishing localised damage zone upon mesh refinement, and hence physically questionable results. In order to circumvent these deficiencies, we present a non-local gradient-enhanced damage model at finite strains. We additively compose the hyperelastic constitutive response at local material point level of an isotropic matrix and of an anisotropic fibre-reinforced material. The inelastic constitutive response... (More)
The modelling of damage effects in materials constitutes a major challenge in various engineering-related disciplines. However, the assumption of purely local continuum damage formulations may lead to ill-posed boundary value problems and—with regard to numerical methods such as the finite element method—to mesh-dependent solutions, a vanishing localised damage zone upon mesh refinement, and hence physically questionable results. In order to circumvent these deficiencies, we present a non-local gradient-enhanced damage model at finite strains. We additively compose the hyperelastic constitutive response at local material point level of an isotropic matrix and of an anisotropic fibre-reinforced material. The inelastic constitutive response is characterised by a scalar [1– d]-damage model, where we assume only the anisotropic elastic part to damage. Furthermore, we enhance the local free energy by a gradient-term. This term essentially contains the gradient of the non-local damage variable which we introduce as an additional global field variable. In order to guarantee the equivalence between the local and non-local damage variable, we incorporate a penalisation term within the free energy. Based on the principle of minimum total potential energy, we obtain a coupled system of variational equations. The associated non-linear system of equations is symmetric and can conveniently be solved by standard incremental-iterative Newton-Raphson schemes or arc-length-based solution methods. As a further key aspect, we incorporate residual stresses by means of a multiplicative composition of the deformation gradient. As a three-dimensional finite element example, we study the material degradation of a fibre-reinforced tube subjected to internal pressure. This highlights the mesh-objective and constitutive properties of the model and illustratively underlines the capabilities of the formulation with regard to biomechanical application such as the simulation of arteries. (Less)
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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
Gradient-enhanced damage, Large deformations, Finite element method, Residual stresses, Anisotropic biological tissues
in
Biomedical Technology
editor
Lenarz, Thomas; Wriggers, Peter; and
volume
74
pages
19 - 40
publisher
Springer
external identifiers
  • scopus:84921739502
ISSN
1613-7736
ISBN
978-3-319-10980-0
DOI
10.1007/978-3-319-10981-7_2
language
English
LU publication?
yes
id
7a70988d-e1a6-4a3b-9012-5b8f9a7842f7 (old id 8521490)
date added to LUP
2016-02-08 15:53:08
date last changed
2017-07-02 04:10:31
@inbook{7a70988d-e1a6-4a3b-9012-5b8f9a7842f7,
  abstract     = {The modelling of damage effects in materials constitutes a major challenge in various engineering-related disciplines. However, the assumption of purely local continuum damage formulations may lead to ill-posed boundary value problems and—with regard to numerical methods such as the finite element method—to mesh-dependent solutions, a vanishing localised damage zone upon mesh refinement, and hence physically questionable results. In order to circumvent these deficiencies, we present a non-local gradient-enhanced damage model at finite strains. We additively compose the hyperelastic constitutive response at local material point level of an isotropic matrix and of an anisotropic fibre-reinforced material. The inelastic constitutive response is characterised by a scalar [1– d]-damage model, where we assume only the anisotropic elastic part to damage. Furthermore, we enhance the local free energy by a gradient-term. This term essentially contains the gradient of the non-local damage variable which we introduce as an additional global field variable. In order to guarantee the equivalence between the local and non-local damage variable, we incorporate a penalisation term within the free energy. Based on the principle of minimum total potential energy, we obtain a coupled system of variational equations. The associated non-linear system of equations is symmetric and can conveniently be solved by standard incremental-iterative Newton-Raphson schemes or arc-length-based solution methods. As a further key aspect, we incorporate residual stresses by means of a multiplicative composition of the deformation gradient. As a three-dimensional finite element example, we study the material degradation of a fibre-reinforced tube subjected to internal pressure. This highlights the mesh-objective and constitutive properties of the model and illustratively underlines the capabilities of the formulation with regard to biomechanical application such as the simulation of arteries.},
  author       = {Waffenschmidt, Tobias and Polindara, César and Menzel, Andreas},
  editor       = {Lenarz, Thomas and Wriggers, Peter},
  isbn         = {978-3-319-10980-0},
  issn         = {1613-7736},
  keyword      = {Gradient-enhanced damage,Large deformations,Finite element method,Residual stresses,Anisotropic biological tissues},
  language     = {eng},
  pages        = {19--40},
  publisher    = {Springer},
  series       = {Biomedical Technology},
  title        = {A Gradient-Enhanced Continuum Damage Model for Residually Stressed Fibre-Reinforced Materials at Finite Strains},
  url          = {http://dx.doi.org/10.1007/978-3-319-10981-7_2},
  volume       = {74},
  year         = {2015},
}