4D Numerical Analysis of Scaffolds: A New Approach - Tissue Engineering: Computer Modeling, Biofabrication and Cell Behavior - page 85

density for an isotropic, incompressible, hyperelastic material is determined by two
invariants. The first and second invariants in uniaxial tension are given by:

2
λ

λ 2 I C = + (19)

1
λ 2 +

II C = 2 λ (20)

where λ is the axial stretch ( λ = 1. For the Neo-Hookean

incompressible hyperelastic solid, the stored energy function is given by:

1 + ε ), that satisfies λ ≥

μ 1
2

W = (I C − 3 ) (21)

where μ 1 > 0 is the material property, usually called the shear modulus. An exten-
sion of this model is the Mooney-Rivlin incompressible hyperelastic solid, which
stored energy function has the form:

μ 1
2

μ 2
2

W = (I C − 3 ) + (II C − 3 ) (22)

with two material properties μ 1 and μ 2 > 0. Higher order stored energy functions
may be considered to describe the experimental data, such as a reduced 2nd order
stored energy function, that includes a mixed term with both invariants of the right
Cauchy-Green stretch tensor and an extra material constant μ 3 , which the stored
energy function has the form:

μ 1
2

μ 2
2

μ 3
6

=

(I C −

+

(II C −

+

(I C − 3 )(II C −

W 3 ) 3 ) 3 ) (23)

Considering the equations above, the axial nominal stress for the three models,
Neo-Hookean ( σ NH ), Mooney-Rivlin ( σ MR ) and reduced second order ( σ 2 nd red ),
will be given by:

μ 1 λ

1
λ 2

σ NH = − (24)

μ 1 λ μ 2 1

1
λ 2

1
λ 3

σ MR = − + − (25)

μ 3 ) λ μ 3 ) 1 μ 3 λ 2

1
λ 2

1
λ 3

1
λ 4

σ 2 nd red

= (μ 1 − − + (μ 2 − − + − (26)

According to Soares et al. [ 39 ], the model constitutive material parameters de-
pend on degradation time. The material parameters are considered to be material
functions of degradation damage instead of material constants. For fibers of a blend
of PLA-PCL (90:10), it was determined that only the first material parameter μ 1
varies linearly with hydrolytic damage (as defined in Eq. ( 9 )) [ 53 ].
From Fig. 10 , one can see that the hyperelastic material models fit well the mea-
sured storage energy, for all the degradation steps up to 8 weeks (about 50 % of
damage). The experimental data of storage energy was calculated by measuring