Biomedical Engineering Reference
In-Depth Information
For a properly discretized RVE, the FE method can be used to perform the
homogenization. In the linear case, FE simulations are used to obtain the unknown
coefficients of the elasticity tensor. To obtain a full set of homogenized material
coefficients, a sufficient number of loading conditions must be applied. Depending
on the type of homogenization and the underlying material symmetry, this may
include simulated tensile testing in orthogonal directions and shear testing in
orthogonal shearing directions. For a unit cell with an orthotropic symmetry, a
total of 6 unique loading simulations must be performed to obtain the 9 inde-
pendent coefficients in the elasticity tensor [
77
]. For a FE simulation, the periodic
displacement boundary conditions (Eq.
11
) must be enforced explicitly [
247
]. This
can be achieved by converting the periodic boundary equations into a set of linear
constraint equations (e.g., via a master node approach) within the FE solver. The
application of periodic boundary conditions typically requires that the FE mesh has
identical nodal distributions on opposing faces and edges (i.e., the faces and edges
are conformal). For homogenizations that utilize a RVE that does not have con-
formal faces, other permissible boundary conditions must be used. These include
kinematic boundary conditions, traction boundary conditions and mixed boundary
conditions [
172
]. For these cases, the resulting homogenization is not exact.
Although homogenization methods have historically been applied to linear
material behavior and kinematics, they can also be applied to nonlinear materials
and nonlinear kinematics [
98
]. In the linear case, a finite number of loading
scenarios can be used to solve for the unknown coefficients. In the nonlinear case,
this methodology cannot be used because the functional form of the stress-strain
response is unknown. For example, there is no combination of loading scenarios
that can directly resolve whether a stress-strain response is quadratic, exponential
or some other function. Strain energy based approaches have been suggested that
curve fit an assumed functional response or populate a lookup table for interpo-
lation [
257
]. However, they have yet to find widespread use.
An attractive alternative for nonlinear homogenizations is the use of a micro-
mechanical model in combination with the appropriate boundary conditions. A
micromechanical model can be subjected to loading scenarios that are of interest
(e.g. uniaxial tensile loading of a tendon) in combination with periodic boundary
conditions, and the homogenized response can be examined. This has proven
useful in several studies that have sought to examine microscale forces and
structure-function relationships in ligaments and tendons [
186
,
205
]. See
Sect. 5.2
for a discussion of these models.
However, micromechanical models are limited to very specific loading sce-
narios and do not provide a general homogenized response. In order to provide a
more general homogenization, the FE
2
method has been proposed. FE
2
based
homogenization utilizes a nested FE problem that consists of a macroscale
boundary value problem and a microscale boundary value problem [
66
,
78
,
122
,
127
,
128
,
213
,
256
]. A macroscale mesh is defined in the normal fashion. When
the stress or elasticity tensor for the macroscale model are needed during the
nonlinear FE solution procedure, the macroscale deformation gradient is passed to
the microscale problem and a homogenization is performed on a discretized RVE
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