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10.3 VALIDATION 207

(a) (b) (c)
Benchmark 1 RVE and respective repetitions r n×r n, being r n =1, 2, ..., 10

(d) (e) (f)

Benchmark 2 RVE and respective repetitions r n×r n, being r n =1, 2, ..., 10

(g) (h) (i)
Realistic RVE and respective repetitions r n r n,being r n =1, 2, ..., 10
FIG. 10.8 Model set used to validate the behavior of the methodology used to define the fabric tensor.

10.3.3 Structural Application

This homogenization methodology is intended to be used as an improvement of existing methodologies, usually
used in highly heterogeneous problem domains. Thus, it was necessary to compare the structural response of a het-
erogeneous material domain with the structural response of the corresponding homogenized material domain, whose
mechanical properties were obtained using the proposed methodology. That is, to concede the different problem
domains used in this study to be equivalent, the mechanical properties of the homogenous domain had to be defined
using the information of the heterogeneous models problem domain. This allows defining equivalent models despite
the different levels of heterogeneousness.
The heterogeneous RVE was defined using a heterogeneous domain (Fig. 10.1C). This problem domain (Fig. 10.12B)is
complex and is formed by two different materials, the trabecular bone and the void space. The homogeneous RVE
(Fig. 10.12A) was defined by a homogeneous domain, discretized by a set of uniformly distributed nodes and integration
points, with the same homogenized material properties acquired using the described methodology. The RVEs were con-
structed with an L L dimension. To define the problem, a displacement of 0.1 L at the nodes of the top layer, y ¼ L,
was imposed. The nodes at x ¼ 0and x ¼ L were constrained on the Ox direction, u ¼ 0, and the nodes at y ¼ 0and y ¼ L
were constrained on the Oy direction, v ¼ 0. Two different numerical approaches, the FEM and the NNRPIM, were used
the compare the mechanical behavior of these RVEs. The integration mesh constructed within the FEM is fundamentally
different from the integration mesh constructed with the NNRPIM formulation, resulting in very different positions of
the integration points. Thus, in order to compare the stress field obtained with the two different RVEs, the concept of von
h
h
Misses homogenized stress, σ eff ,is used.The σ eff , defined by Eq. (10.6), allows combining the stress field in one scalar
value, facilitating the comparison of different models. In this equation, n Q represents the number of integration points,
which discretize the problem domain, that do not belong to the vicinity of the domain boundary, typically 2%, avoiding
the inaccurate stress concentrations that appear near the domain boundary.
n Q
1 X
σ h eff ¼ σ x i eff (10.6)
ðÞ
n Q
i¼1

I. BIOMECHANICS