Biomedical Engineering Reference
In-Depth Information
6.3 Structurally motivated constitutive models - elastic regime
In this section, we summarize recent structural constitutive models for the elastic be-
haviour of arteries. A number of references are available on this subject (e.g. [54]).
Here, we focus on models which explicitly include fibre orientation that will be used
in the discussion of damage in the next section. For simplicity of discussion, we will
not include the residual stress or the viscoelastic effects. Further, we concentrate on
the passive response of the arterial wall and in particular will not discuss the active
response due to smooth muscle. A review of models for active contributions from
smooth muscle cells can be found in [54] including early phenomenological mod-
elling by Rachev and Hayashi [89] and by Zulliger et al. [125]. Humphrey and col-
laborators include an anisotropic contribution from passive smooth muscle in their
constitutive models of the arterial wall (e.g. [110]) in addition to an active response.
In this work, the arterial wall is treated as an incompressible, hyperelastic multi-
mechanism material with independent passive contributions from collagen and elas-
tin and smooth muscle. The stress tensor will be decomposed into a constraint re-
sponse that does no work as well as an extra stress contribution
σ
E
that can do work,
σ
=
−
pI
+
σ
E
(6.1)
where
p
is the Lagrange multiplier arising from the constraint of incompressibility.
Assuming the material response to be hyperelastic it then follows that,
W
σ
:
D
=
σ
E
:
D
=
(6.2)
where
W
is the strain energy per unit volume in the unloaded configuration
κ
0
,
D
is the symmetric part of the velocity gradient, and the overdot is used to denote the
material derivative. We consider the strain energy to consist of the sum of strain
energies from an isotropic mechanism
W
iso
and an anisotropic mechanism
W
aniso
=
+
W
W
iso
W
aniso
(6.3)
and then use (6.2) and (6.3) to define corresponding contributions to the Cauchy
stress tensor
σ
,
σ
E
=
σ
iso
+
σ
aniso
(6.4)
with
W
iso
,
W
aniso
.
σ
iso
:
D
=
σ
aniso
:
D
=
(6.5)
Both collagen fibres and passive smooth muscle will contribute to the anisotropic
mechanism. These mechanisms will in general have different constitutive responses
and different unloaded configurations. The arterial collagen is in a wavy or crimped
state in the unloaded artery and is gradually recruited to load bearing as the vessel
is strained. Hence, multiple reference configurations will be needed to identify the
kinematic state for recruitment. Further, damage to the elastin and collagen fibres
may occur independently and prior to failure of the arterial wall. The explicit treat-
ment of these multiple mechanisms makes it possible to model damage and failure
