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

3.2 Decoupled invariant formulation for orthotropic symmetry
To generalise the above formulation to orthotropic symmetry, a second local unit vector
m 0,
orthogonal to n 0,
is introduced. Its associated structural tensors in the material and
spatial configurations are respectively L m =m
0 0 Ä m0
and L m = m Äm.
Henceforth,
the convention that the Greek letter b stands for either n 0 or m 0 is adopted. The subscript
and superscript b are therefore used to distinguish each family of fibres and their
associated quantities: invariants, energy functions and constitutive parameters. One can
define the anisotropic invariants { I4 } b
b= and { I5 } b
b= :
1,2
1,2
4b = C : b
2 = lb =
2
3 2 lb,
5b
2 = C : b
I L J I L (7)
Following equations (5) and (6) four shear invariants are defined:
II - I 1
I
a a C L (8)
1 4b 5b 2
5b
1b = , 2b = :
4 b =
2
II 3 4b
lb
( I4b)
3.3 A strain energy function to represent the orthotropic behavior of skin
The motivation behind this formulation is to capture key mechanical characteristics of
skin which, across mammalian species, is known to behave like an orthotropic
hyperelastic material [12]. Skin is assumed to be made of two families of oriented
collagen fibres embedded in a soft isotropic solid matrix. Naturally, the constitutive
equations developed are also applicable to a wide range of biological and engineering
materials.
Collagen is the main load bearing constituent of the extra-cellular matrix. The
tropocollagen molecule represents a basic level of the structural hierarchy of
collagenous tissues. This long molecule is characterised by a triple helix made of three
collagen polypeptide chains wound around one another and strengthened by hydrogen
bonds [13]. Collagen molecules are further arranged into bundles to form microfibrils
which are the building blocks of the next level of the tissue hierarchy: the collagen
fibrils. Here, it is proposed to capture the non-Gaussian mechanics of collagen
microfibrils by using a worm-like chain model [14]. The basic idea behind the use of
entropic elasticity of macromolecules [15] to represent the mechanical behaviour of
collagenous tissues is that collagen molecules aggregated into microfibrils can be
viewed as flexible rods that bend smoothly under the influence of thermal fluctuations.
Collagen fibres are assumed to deform incompressibly and, when this assumption is
combined with a principle of affinity for the chains, one can relate the (macroscopic)
deviatoric stretch l b to the (nanoscopic) measure of the end-to-end chain length in the
reference and current configurations, respectively r0b and rb :
rb
l b = (9)
r0b
Departing from an 8-chain model representation [16], molecular chains are assumed to
be independent and only mechanically coupled to each other by their shear interactions
so no use of a network conformation is made [1].
v The existence of a strain energy y as the sum of a volumetric [ y ( J ) ], cross-fibre shear
[ 1 yˆ b( a 1b)
], along-the-fibre shear [ 2 ˆ l
yb( a 2b)
] and deviatoric fibre energies [ yb( l b)
] is
postulated: