A Gradient-Enhanced Continuum Damage Model for Residually Stressed Fibre-Reinforced Materials at Finite Strains - Biomedical Technology - page 23

˕

which requires the first variation of the total potential energy with respect to and

ˆ

to vanish, i.e.

a 0 i ,ʺ) = ʴ ˕ ʠ( ˕ , F ,ˆ, ∇ X ˆ ; 0 ∀ ʴ ˕ , (10)

a 0 i ,ʺ) = ʴ ˆ ʠ( ˕ , F ,ˆ, ∇ X ˆ ; 0 ∀ ʴˆ . (11)

Taking into account that ʴ ˕ W int = ʴ ˕ ʠ int and ʴ ˕ W ext =− ʴ ˕ ʠ ext , we obtain a

coupled system of variational equations, i.e.

ʴ ˕ W = ʴ ˕ W int − ʴ ˕ W ext = 0 ∀ ʴ ˕ , (12)

ʴ ˆ W = ʴ ˆ W int − ʴ ˆ W ext = 0 ∀ ʴˆ , (13)

where the internal and external contributions are given in spatial form as

b

t

ʴ ˕ W int = ˃ :∇ x ʴ ˕ d v, ʴ ˕ W ext = · ʴ ˕ d v + · ʴ ˕ d a , (14)

B t B t

∂ B t

ʴ ˆ W int = y ·∇ x ʴˆ d v, ʴ ˆ W ext = y ʴˆ d v. (15)

B t B t

Here the Cauchy stress and the vectorial damage quantity y are related to flux

terms, whereas the body force b and the scalar damage quantity y are associated to
source terms. They are defined as

˃

F − 1

b

J − 1 ˃ = ∂ F ʨ · cof ( ), =− ∂ ˕ ʨ vol , (16)

F − 1 J − 1 y = ∂ ∇ X ˆ ʨ · cof ( ), y =− ∂ ˆ ʨ. (17)

Relations ( 14 ) and ( 15 ) provide the basis for the finite element discretisation in
Sect. 4 .

3 Constitutive Relations

In this section, we first review the hyperelastic constitutive equations adopted on
the basis of an isotropic neo-Hookean relation and an anisotropic exponential part.
These relations characterise the elastic anisotropic response of the fibre-reinforced
material. Secondly, we specify the gradient-enhanced, non-local contribution to the
free energy, followed by the continuum damage formulation.

3.1 Hyperelastic Part of the Free Energy

From Sect. 2.2 , we recall the local free energy density ʨ loc ,Eq.( 1 ), to be additively

composed of an isotropic part ʨ iso , representing the isotropic matrix material, and