Biomedical Engineering Reference
In-Depth Information
A Gradient-Enhanced Continuum Damage
Model for Residually Stressed Fibre-Reinforced
Materials at Finite Strains
Tobias Waffenschmidt, César Polindara and Andreas Menzel
Abstract
Themodelling of damage effects inmaterials constitutes amajor 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 addi-
tively 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 vari-
able. 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 varia-
tional 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
resid-
ual 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.
Keywords
Gradient-enhanced damage
·
·
Large deformations
Finite element
·
·
method
Residual stresses
Anisotropic biological tissues
