ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

3. METHODS
When modelling the mechanics of collagen fibre-rich tissues, a classic assumption is to
consider the tissue as a composite material made of one or several families of (oriented)
collagen fibres embedded in a highly compliant isotropic solid matrix composed mainly
of proteoglycans. The preferred fibre alignment is defined by the introduction of a socalled
structural tensor which appears as an argument of the strain energy function [2].
Generally, a principle of linear superposition of the fibre and matrix strain energies is
also assumed [3-6] but, for most studies, and by construction, fails to account for the
fibre-fibre and fibre-matrix interactions which can be highly relevant in certain
conditions (e.g. warping and sliding of the knee cruciate ligaments). To date, only few
studies have integrated these shear interactions into constitutive formulations; for
selected examples see [7-10]. In this section finite strain constitutive equations for
continuum transversely isotropic and orthotropic biological materials are introduced.
3.1 Decoupled invariant formulation for transverse isotropy symmetry
Following standard usage in continuum mechanics, one denotes by I, the second-order
T
identity tensor, F, the gradient of the deformation, C = F F, the right Cauchy-Green
deformation tensor and its principal invariants:
1 2 2
I1 = C : I, I2 = ( I1 - C : I) , I3 = det(
C ) = J
(1)
2
A first direction of anisotropy is introduced by considering a local unit vector n 0
characterising the local fibre direction in the material (or reference) configuration). One
can define a structural tensor L n = n
0 0 Än0and
the two following associated invariants:
2 2
I 4( n0) = C : Ln = l = Fn 0 0 / n,
I5(
n
0
0) = C : L
n
n
(2)
0
where n is the local fibre direction in the spatial configuration and L n = n Än,
its
associated structural tensor.
The starting point of the decoupled transversely isotropic constitutive formulation is to
decompose the deformation gradient according to Lu et al. [11]:
T
which leads to Cˆ = Fˆ F: ˆ
é
ˆ ê
F = ê
êë 1
l n0
ù
1 ú
( I - Ln) + L n ú
l ú n0
úû
(3)
2
3
ˆ J
C =
l n0 æ ö
ç
-1
ç 1 ÷
C + ç 1 - ÷
ç 3÷
L
ç l ÷ n
çè n ÷ ø
(4)
This kinematic split permits the definition of two tensorial invariants [11]:
II 1 4( 0) - I
1
5( 0)
a1(
( ˆ) ˆ
0) trace é - det ( ) ù n n
n = ê C C I - L =
ë n0
úû
II
(5)
I
a = ˆ : =
2( n ) n
5( n )
2 I
4( n )
0
3 4( n )
2 C L 0
(6)
0 0
Finally one obtains a set of four decoupled invariants { J, l , a , a }
I = n0 1( n0) 2( n0)
which
leads to four mutually orthogonal stress tensors. This effectively decouples the four
deformation modes associated with these invariants.
0
0