Effect of Mass Distribution on Deformation of Three
Dimensional Open-Celled Porous Materials Under
Uniaxial Compression
Bridget Reardon, Mechanical Engineering, Clarkson University
Kathleen A. Issen, Professor, Mechanical Engineering, Clarkson University
Abstract
To further understand the microstructure of human trabecular bone, and its response
under uniaxial compression, the effect of mass distribution on deformation of three dimensional
porous materials was investigated. Researchers have used simple structures consisting of
cylindrical struts in a cubic arrangement to model trabecular bone. However, the struts connect
to form sharp corners, unlike actual bone, which contains fillet material at the strut junctions to
provide a smooth transition between the trabeculae. This study investigates how the fillet
geometry affects the specimen mechanical response by designing specimens with varying
amounts of fillet material and subjecting them to uniaxial compression testing. Images of the
testing are recorded and used to perform digital image correlation to examine areas of localized
strain within the specimen. Two of the specimens exhibited similar failure mechanisms:
buckling that induced shear banding. The specimen designed to mimic actual bone, with a large
fillet, exhibited a uniform microscopic buckling deformation, demonstrating that the fillet
material does change the mechanical deformation mode.
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Introduction
Human vertebra consists of an outer shell of cortical bone, with a porous internal
structure of cancellous bone, a structure made of interconnected struts called trabeculae. In order
to create bone implant materials that sufficiently reflect the mechanics of human bone, internal
cancellous bone must first be quantified and its material properties analyzed so all necessary
functions are performed by the implant material. Because of the variation in size and orientation
of the individual struts, bone is typically quantified using average parameters. Human specimens
have been tested, but the varying parameters make it difficult to compare the results between
specimens. Therefore, investigators have turned to modeling the cancellous bone, using
simplified structures, to either perform a Finite Element Analysis (FEA) or build a prototype to
test.
To model the cancellous bone, Jensen (1990) used cylindrical struts in a cubic
arrangement, seen in Figure 1. Similar models have been used by other researchers (Silva and
Gibson, 1997; Yeh and Keaveny, 1999; Yeh and Keaveny, 2001; Kim and Al-Hassani, 2002).
Figure 1: From Jensen (1990), cylindrical strut model of cancellous bone (left), and human bone image (right) with
fillet material in the joints (circled).
Images of actual bone seen in Figure 1 show a smooth transition from one strut to another as
opposed to the sharp corner in Jensen’s cylindrical strut model. This extra fillet material in the
nodes is only included in one of the studies (Kim et al., 2002). Kim et al. used a doubly tapered
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strut (Figure 2) to reduce the stress concentration caused by the sharp corner of the original
struts.
Figure 2: Doubly tapered strut designed by Kim et al. (2002) to remedy the sharp corner created by the joining of
struts.
This was created with a mathematical equation, but they do not discuss the reasoning behind this
specific design. Despite being included in this study, the fillet geometry has not been analyzed
or quantified in the human bone.
Smith (2006) observed a change in failure mode when material is redistributed in a two
dimensional hexagonal honeycomb under compression. Specimens with more material in the
junction failed perpendicular to the compression direction, while those with less failed along a
diagonal. Since material distribution has an effect in two dimensions, it could also affect the
response of three dimensional materials, such as the cylindrical strut models used to represent
cancellous bone. This research investigates the relationship between mass distribution and
deformation modes in a three dimensional open celled porous material that mimics models of
human cancellous bone. The fillet material in the junction of the cells is expected to change both
the method of deformation and the maximum load capabilities of specimens having the same
volume ratio. This will demonstrate the importance of quantifying the mass distribution in
cancellous bone samples investigated by researchers.
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Methods
Statistical parameters used to quantify bone were taken from a micro-CT scan of a core
sample from a human vertebra, to determine characteristics of the microstructure to be designed.
The unit cells were designed in CAD, based on models of 12 cylindrical struts connected in a
cube. The diameters of these cylinders were similar to the trabecular thickness given in the data,
and the length of the strut was similar to the trabecular spacing. The rest of the bone volume was
included by creating a fillet in each corner of the unit cell. This design is called the Basic cell,
shown below in Figure 2 on the left. Since there has been no documentation of quantifying the
fillet in actual bone, a circular fillet was created using trial and error to match the bone volume to
total volume ratio. That same volume ratio was matched by creating a unit cell without the fillet
material, but larger strut thicknesses (No Fillet cell, center of Figure 2). An intermediate cell,
with the same volume ratio was designed as well, using a larger strut and smaller fillet, called the
Small Fillet cell (Figure 2, right).
Figure 3: Unit cells. Basic, No Fillet, Small Fillet (left to right).
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Number
Volume
TbTh
Fillet
BV/TV
BS/BV
TbSp
Bone Specimen
0.1839
9.123
10.88
1.8313
BA1_1
84720.2
0.49
2.04 9.399144 3.36305
4.88
NF1_1
75686.97
0.994
0 9.156285 4.333099
4.7
TC1_1
75686.97
0.81
1.165 9.52113 3.76107
4.7
Table 1: Design characteristics of Basic, No Fillet, and Small Fillet specimens compared to bone characteristics.
Volume is the total specimen volume, TbTh is the trabecular thickness or strut diameter, Fillet is the fillet radius,
BV/TV is the ratio of bone volume to total volume, BS/BV is the surface area to bone volume ratio, TbSp is the
trabecular spacing, or strut length.
The unit cell designs were stacked to create a 9x9x9 array of unit cells to complete the
full specimen. A stereolithography rapid prototyping machine was used to manufacture the
specimens using the CAD designs to trace the solid part, curing liquid polymer with a laser, layer
by layer. Latex was stretched and attached to one side of each specimen, and a speckle painted
pattern was applied to the latex so images could be correlated after testing.
Uniaxial compression tests were performed on an MTS 809 machine under displacement
control. The crosshead was lowered at a rate of 0.002 mm per second, while the load and
displacement were recorded by a LabView VI. Images were recorded by Point Grey Research
Scorpion cameras every 2 seconds on the latex face, and an adjacent face of the specimen.
Following the test, force versus image and force versus displacement curves were created for
each specimen. The images were used to obtain strain maps of the specimen face through Vic2D
digital image correlation software. The software identifies groupings of speckles and tracks their
displacement throughout the test. These strain maps are used to identify areas of strain
localization in the main body of the specimen that may be invisible to the naked eye. Images of
the non-latex face are used to identify the microstructural deformation process that relates to the
surface strain maps.
Results and Discussion
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The peak load was higher in the specimens with similar deformation modes. The Small
Fillet and No Fillet specimens sustained loads up to 1900 to 2600 Newtons, whereas the Basic
cell only held 500 Newtons, as seen in Figure 4. The fillet material in the junction lead to a
lower peak load, thus the strut thickness may have more of a positive effect on load capabilities
than reinforcing at the junctions, which the fillet material does.
Load v Displacement
3000
2500
Load (N)
2000
BA1_2
1500
NF1_1
TC1_1
1000
500
0
0
0.5
1
1.5
2
2.5
3
Displacement (mm)
Figure 4: Load Displacement curve illustrates the varying peak loads from the lowest strut thickness cell (Basic,
blue) to the largest strut thickness (No Fillet, red). The Small Fillet strut thickness was slightly less than the No
Fillet, and the peak stress is also a bit less.
The Basic specimen also had a lower Young’s Modulus than the No Fillet specimen. The
No Fillet Young’s Modulus was approximately 70 MPa, whereas the Basic was approximately
35 MPa. The main output of FEA is maximum stress and Young’s Modulus. Thus the fact that
both of these were measured to be lower in the Basic specimen, which comes from the bone
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parameters, than in the No Fillet, which is similar to the previous models of trabecular bone,
illustrates a need to find appropriate models for bone, accounting for the mass distribution.
Despite the appearance of three different failure modes, the initiation of the failure in two
of the specimens was quite similar, beginning with buckling, which induced a shear band. The
struts at the failure locations were found to bend at the junctions, resulting in fracture in some
cases right at the junction. The Basic cell exhibited node rotation, which resulted in bending of
the struts. As seen in Figure 5, the specimen formed a uniform compaction pattern. This leads
to a curiosity about the transition from the uniform deformation to the compacting shear band.
Figure 5: Deformation modes in the Basic, No Fillet, and Small Fillet specimens. The No Fillet and Small Fillet
developed shear bands in the second and first rows respectively, whereas the Basic had uniform microscopic
bifurcation.
The mass distribution affected the deformation mechanism between the Basic cell and the
Small Fillet cell, but had little impact between the Small Fillet and No Fillet. How does the
failure mode transition between the Basic cell and Small fillet cell; is it gradual or a snap
transition? The intermediate fillet sizes will be created and tested to look for a transition point
between the shear band failure mode and the uniform failure mode.
Conclusions
The differing failure modes and varying strengths of the samples show that the mass
distribution within the structure affects its method of failure. This will be useful in modeling
human cancellous bone, which is commonly modeled by cylindrical struts in a cubic pattern,
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with no fillet material in the corners of the cells. The specimens without fillet material showed a
higher tendency toward buckling than did the ones with fillets.
By moving the material from struts into the junction, the maximum load capability and
the deformation mode of the structure change. The maximum load is lowered, but the failure
mode is a more stable deformation. This does however show that the FEA models performed by
other researchers may not be accurate because of the way that the trabecular junction is modeled.
A quantification of this junction may lead to more accurate ways of modeling trabecular bone,
and create better bone implant designs.
References
Jensen, K.S., L. Mosekilde, L. Mosekilde. “A Model of Vertebral Trabecular Bone Architecture
and its Mechanical Properties”. Bone 11. 1990. 417-423.
Kim, H.S., S.T.S. Al-Hassani. “A morphological model of vertebral trabecular bone”. Journal
of Biomechanics 35. 2002. 1101-1114.
Smith, Jason A. MS Thesis. Clarkson University, 2006.
Silva, M.J., L.J. Gibson. “Modeling the Mechanical Behavior of Vertebral Trabecular Bone:
Effects of Age-Related Changes in Microstructure”. Bone 21. 1997. 191-199.
Yeh, O.C., T.M. Keaveny. “Biomechanical Effects of Intraspecimen Variations in Trabecular
Architecture: A Three-Dimensional Finite Element Study”. Bone 25. 1999. 223-228.
Yeh, O.C., T.M. Keaveny. “Relative Roles of Microdamage and Microfracture in the
Mechanical Behavior of Trabecular Bone”. Journal of Orthopaedic Research 19. 2001.
1001-1007.
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