Biomedical Engineering Reference
In-Depth Information
Unit Cell
RVE
Fig. 6 Comparison of a unit cell and an RVE. For materials with a periodic microstructure, such
as a lattice of spheres embedded in a matrix material (left), a unit cell (middle left) can be defined
that describes the microscale geometry. For the case of media with random microstructures
(middle right), a volume element representative of the microstructure, called an RVE (right), can
be defined
appropriate boundary conditions and then simulated macroscopic loading is used
to compute the effective material response. For a periodic unit cell, the exact
homogenized effective material properties are obtained. If the RVE is statistically
representative of the material microstructure, the ''apparent material properties''
are obtained [
99
,
172
].
The concept of homogenization is based upon the Hill principle [
98
], which
states that the volume averaged strain energy at the macroscale is equal to the
volume averaged strain energy at the microscale (i.e., energy is conserved):
h
r : e
i ¼ hi
:
hi
:
ð
10
Þ
Special boundary conditions must be applied to satisfy the Hill condition. For a
periodic unit cell, they are periodic boundary conditions [
172
,
247
]. The periodic
boundary conditions enforce the constraints that opposing faces of the unit cell
must deform identically, and that the traction forces on opposing faces must be
antiperiodic [
172
,
211
,
247
]:
u
k
þ
x
ðÞ
u
k
x
ðÞ¼
e
0
x
þ
x
ð Þ
t
k
þ
x
ðÞ¼
t
k
x
ðÞ
on C
;
ð
11
Þ
where u
k+
and u
k-
are the displacements on opposing faces and t
k+
and t
k-
are
traction forces on opposing faces (both on the boundary C), e
0
is the applied strain
and x
+
and x
2
are the position vectors on opposing faces.
Historically, homogenizations have been primarily used to analyze linear
material behavior, with the effective coefficients of the linear elasticity tensor
being computed. Analytical methods, which obtain homogenized coefficients via
closed form solutions, have been applied to problems that feature simple RVE
geometries (e.g. a homogenization of annulus fibrosis [
254
], also refer to [
122
] for
a summary of such methods in engineering applications). However, homogeni-
zation techniques based on analytical methods lack the ability to address the
complex 3D microstructural features in ligament and tendon. Thus, methods based
on FE discretization are particularly appealing.
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