Biomedical Engineering Reference
In-Depth Information
a homogenization procedure. These methods utilize an explicit microstructure as a
means for both obtaining the macroscopic stress-strain behavior and for reconstruct-
ing the microscale stress-strain problem. A number of homogenization schemes have
been proposed, each with a specific use. In this section, a brief summary and discus-
sion of the preceding topics will be provided.
8.3.1 Continuum Mechanics
The presence of a continuum assumes that the length scale of microstructures is infin-
itesimally small in comparison to the macroscale, and that the deformation gradient,
and thus strain and stress, can be defined uniquely at every point within the domain.
This latter point implies an affine deformation map in which an infinitesimal line
element
dX
in the reference configuration is mapped to the current configuration
dx
:
dx
=
F
·
dX
+
X
0
,
(8.1)
in which
dX
is an infinitesimal material line element in the reference configuration,
dx
is the deformed version of the same infinitesimal material line element, and
X
0
represents a rigid body translation vector.
F
is the (nonsymmetric) deformation
gradient:
∂
x
F
=
X
.
(8.2)
∂
A number of second-order strain measurement tensors are computed from the defor-
mation gradient, including the right Cauchy deformation gradient (
C
), the Green-
Lagrange strain (
E
) and the engineering or infinitesimal strain (
e
):
F
T
F
C
=
,
(8.3)
1
2
(
E
=
C
−
1
) ,
(8.4)
(
T
1
2
e
=
F
−
1
)
+
(
F
−
1
)
.
(8.5)
The engineering strain is used extensively for linear elasticity, but is generally of
limited use for the finite deformations seen in biological tissues. A useful concept
in the study of aligned collagenous tissue is the notion of a unit vector to describe
the fiber direction, which is denoted
a
0
in the reference configuration. This fiber
vector is rotated and stretched by the deformation gradient,
ʻ
a
=
F
·
a
0
, where
ʻ
is the fiber stretch and
a
is a new unit vector describing the rotated orientation
of the fiber direction. The concept of strain invariants is of particular importance in
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