Biomedical Engineering Reference
In-Depth Information
chapter we will be assuming that our thick elastic structure is
•
homogeneous, i.e., the material properties do not depend on x,and
•
isotropic, i.e., the response of the material deformation is the same in all
directions.
Additionally, we will be assuming that
•
the displacementgradient is small (i.e., rd 1).
Under these assumptions, one of the simplest constitutive models for the mechanical
behavior of linearly elastic structures, called the linearized Saint Venant-Kirchhoff
model, takes the following form:
S
D .r
d
C .r
d
/
T
/ C .r
d
/
I
;
(2.52)
Here, and are the Lamé constants, accounting the compression and distortion
of the structure, respectively.
Writing a constitutive model for the behavior or elastic structures in general is a
bit more involving. Arterial walls are, in fact, nonlinear. The linear approximation
written above is good as long as the displacement gradient and displacement are
not too large, which in the blood flow application means displacement not larger
than roughly 5 % of the reference radius of an artery. A typical displacement in a
healthy artery under normal physiological conditions is between 5 and 10 %. Thus,
many physiological and pathophysiological situations can exceed the linearly elastic
regime. Depending on what types of questions is one trying to answer, linear or
nonlinear models may be appropriate.
A typical assumption in biomedical literature on soft tissue mechanics is that
arterial walls behave as a hyperelastic material. This means that the relationship
between stress and strain in the structure can be written as the derivative of the
energy density function with respect to strain. More precisely, if we denote by
•
…—the second Piola-Kirchhoffstress tensor,
•
E—the Green-Lagrangestrain tensor,and
•
W —the energydensityfunction,
then, for a hyperelasticmaterial
@W
@E
.E/:
….E/ D
What is the relationship between the first and second Piola-Kirchhoff stress
tensors
S
and …, and between the Green-Lagrange strain tensor E and displacement
d? To explain these relationships we need to recall the notion of deformation.
For each point x 2
S
belonging to an undeformed, reference configuration
S
, deformation is a mapping ' which to each point x 2
S
associates a point
'.x/ D x C d.x/,whered denotes the displacement of x. Deformation gradient
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