Research Article
Open Access
Expected Ideal Solution and Verified
By Numerical Analysis for a Triangular
Unit Cell Model
Jeongho Choi*
School of Mechanical Engineering, Kyungnam University, Masanhappo-gu, Changwon-si,
Gyeongsangnam-do 51767, Republic of Korea
*Corresponding author: Jeongho Choi, School of Mechanical Engineering, Kyungnam University, 7 Kyungnamdaehak-ro, Masanhappo-gu,
Changwon-si, Gyeongsangnam-do 51767, Republic of Korea, Tel: +82-55-249-2210; E-mail:
@
Received: April 04, 2018; Accepted: April 18, 2018; Published: April 23, 2018
Citation: Choi J (2018) Expected Ideal Solution and Verified By Numerical Analysis for a Triangular Unit Cell Model. Int J Adv Robot Automn 3(2): 1-10. DOI:
10.15226/2473-3032/3/2/00133
Abstract
This study aims to provide critical information regarding the
development of ultralight materials with tailored properties using a
triangular unit cell model. Three different types of triangular unit cells
(or triangular prism unit cells, e.g. OST, OTT, and CTT) were studied
via finite element analysis to calculate and compare their relative
density, stiffness, and strength. The relative stiffness and strength
were extracted under a compression load. The ideal solutions were
calculated using commercial finite element analysis software and then
compared to the ideal Gibson–Ashby solution. The relative Young’s
modulus and the relative yield strength for the three different unit
cell models are shown to differ from the ideal solution of the Gibson–
Ashby theory. In conclusion, the effective stiffness of the OTT and
CTT is higher than that of the Gibson-Ashby model. When the relative
density is more than 0.1, the OTT and CTT have higher yield strength.
OTT and CTT have a higher plastic strength when the relative density
is more than 0.03–0.05. For aerospace or automobile use, further
study is needed to find the optimum truss and aperture sizes and
shape of the unit model.
Keywords: Cellular solids; Effective medium; Relative properties;
Triangular unit cell;
Introduction
Periodic Cellular Metals (PCMs), which are defined by the
Gibson–Ashby ideal solutions, are important research and
development topics [1,2]. PCMs are composed of open or closed
unit cells. The study of the closed or open unit cells is an important
step in the development and the prediction of the properties of
a new PCM structure. The study of the closed or open unit cells
is an important step in the development and the prediction of
the properties of a new PCM structure. Engineers and scientists
have developed several applications of PCMs. Choi, et al. [3,4,5,6]
created a unit cell model with a truss and developed a complex
structure and unit cell models such as corrugated wire mesh
laminate and pinwheel trusses. The development of a new unit
cell model is the most important aspect for building a sandwich
core structure. Gumruk, et al. [7] studied heat protection, thermal
insulation, and packaging for the automotive and shipping
industries. Rejab, et al. [8], Zhang, et al. [9], Jeong, et al. [10],
and Schaedler, et al. [11] showed other applications such as
sound insulation, battery electrodes, catalyst supports, acoustics,
vibration or shock energy damping, filters with broader contact
areas, such as those in chemical reactions, and packaging with
energy absorption. Sypeck, [12,13] and Wadley, [14] showed
that lattice truss topologies have the potential to be created as
triangular, diamond prismatic, Navtruss, a tetrahedral lattice, a
pyramidal lattice, three-dimensional Kagome, diamond textile,
diamond collinear, and square collinear. Hutchinson, et al. [15]
proposed an advanced structure referred to as a microlattice,
which was created by HRL Laboratories. This structure was
lighter than other structures because it used advanced technology.
In addition, Linul, et al. [16] studied open cell forms using
numerical analysis for different sizes of models. Serban, et al. [17]
investigated open-cell foams through numerical analysis. Thus,
several scientists and researchers have focused on developing
better structures by applying PCM structure unit cells in various
fields.
The advantages of trusses are applicability, low weight,
cost effectiveness, versatility, open-cell construction, and
environmental toughness. Additionally, they are potentials such
as structural strength, blast protection, ballistic protection,
and heat exchange. Low weight implies functionality, such as
carrying larger payloads, providing additional storage, reducing
fuel consumption, and allowing for higher speed. The low
weight and multifunctional attributes provide time and cost
effectiveness. Additionally, core topology, cell size, or relative
density is optimized depending on the application. Open-cell
topologies or periodic microtruss structures allow for the use of
additional materials for increasing ballistic resistance. Industrial
companies, such as Cellular Materials International Inc. [18] and
Boeing, [19], have shown that microtruss structures can provide
protection from environmental problems such as corrosion
in marine applications, the high-heat conditions of space, and
reentry vehicle applications.
The disadvantages of trusses are that truss models are
created using conventional materials, which are monolithic and
heavy with high densities. Truss models must consider bonding
techniques. That is, crossed points in truss models are not easily
bonded. The material properties of truss models may change
because of the heat treatment during bonding. In addition, the
bonding procedure is time consuming and expensive. To solve
these problems, the use of advanced techniques, such as threedimensional
printing, is required.
Liu, et al. [20] studied effective elastic moduli of triangular
lattice materials theoretically and then they checked the ideal
solution with finite element analysis. And Ptochos, et al. [21]
reported mechanical properties of micro-lattice structures by
calculated analytically.
This study investigates the ideal solutions and simulations
for truss unit cell models, which are defined as Opened Trusswall
Triangular (OTT), Opened Solid-wall Triangular (OST), and
Closed Truss-wall Triangular (CTT) models. Replacing solid walls
with truss structures can result in weight reduction. In other
words, the OTT model is a structure in which the solid wall of the
OST model is replaced by a truss. The CTT model is a structure
with an open face covered by a truss. The relative modulus
and relative compressive yield strength obtained for the OST,
OTT, and CTT models are compared to the Gibson–Ashby ideal
solutions which describe relative stiffness of a model correlated
with relative density of cell geometry. The goal of this study is
to determine the effective stiffness and effective strength for the
truss-wall triangular unit cell models as a function of the relative
density.
Ideal Solution
Unit Cell Model
The unit cell is a fundamental concept for creating a sandwich
structure. This implies that a unit cell model is a repeated
configuration. For a truss-wall unit cell model, relative stiffness is
proportional to the square, which infer cell geometry, of relative
density and relative strength is proportional to relative density
to the power of 3/2 on a log–log scale, as shown by Gibson, et al.
[2]. That is,
$$\begin{array}{l}\frac{{E}^{*}}{{E}_{s}}={C}_{1}{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{2}\\ and\text{(1)}\\ \frac{{\sigma}^{*}}{{\sigma}_{s}}={C}_{2}{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{\frac{3}{2}}\end{array}$$
Where C and C_{2} are constant. These infer the Eq.1 is moving
left or right and the power means slope.
Relative Density
Relative density (ρ*/ρs) is proportional to the inverse of
relative volume (Vs/V*), which is the ratio of the volume of a solid
wall to the volume of the cellular structure.
The unit cell model is based on a solid wall model. The model
is expected to have the same correlation between volume and
density as a power law, and it uses parameters such as diameter,
opening width, and the total number of trusses in the directions
of length, width, and height. This model has the same relationship
between relative density and relative volume as the OST or OTT
models.
$$\frac{{\rho}^{*}}{{\rho}_{s}}\propto \frac{{V}_{s}}{{V}^{*}}\text{(2)}$$
Opened Solid-Wall Triangular (OST) Model
An ideal solution was derived for the relative density of the
triangular prism, and this is shown in figure 1. The volume of the
applied material is
Figure 1: Schematic OST model (h = height; l = length, width; t = thickness)
$${V}_{s}=\frac{\sqrt{3}}{8}ht\left(l-\frac{t}{4}\right)\text{(3)}$$
The volume of the foam is
$${V}^{*}=\frac{\sqrt{3}}{8}{l}^{2}h\text{(4)}$$
The relative density for a triangular prism is shown in Eq.5
and Eq.6. It means l-t/4) is not equal to zero but positive value in
Eq.3 The ratio of thickness and length satisfies the condition that
thickness must be smaller than four times the length. To compare
this with the OST model, it is assumed that the thickness in the
solid-wall triangular model is equal to the diameter of the trusstriangular
wall. These are coming from Gibson, et al. [2].
$$\frac{{\rho}^{*}}{{\rho}_{s}}=\frac{t}{l},t<l\text{(5)}$$
$$\frac{{\rho}^{*}}{{\rho}_{s}}=\frac{d}{l},d<l\text{(6)}$$
Opened Truss-Wall Triangular (OTT) Model
An ideal solution was derived for the relative density for the
OTT model, which is referred to as a truss triangular honeycomb,
and this is shown in Figure 2. This model is based on the OST
Figure 2: Diagram of OTT model
model shown in figure 1. For the OTT model, the thickness (t) is
equal to the truss diameter (d). The height (h) is equal to m(d+w),
and l is equal to n(d+w), where n and m are the total number of
trusses in the directions of width and length, respectively, and w
is space.
Thus, the volume of the applied material for the OTT model is
$${V}_{s}=\frac{3}{4}\pi {d}^{2}\left(n+m\right)\left(d+w\right)=\frac{\sqrt{3}}{8}\left(nm{\left(d+w\right)}^{2}d-\frac{m}{4}\left(d+w\right){d}^{2}\right)\text{(7)}$$
The volume of the foam for the OTT model is
$${V}^{*}=\left(\frac{\sqrt{3}}{8}\right){n}^{2}m{\left(d+w\right)}^{3}\text{(8)}$$
If n = m = C = a constant, then
$${V}_{s}=\frac{\sqrt{3}}{8}\left({C}^{2}d{\left(d+w\right)}^{2}-\frac{C{d}^{2}}{4}\left(d+w\right)\right)\text{(9)}$$
$${V}^{*}=\left(\frac{\sqrt{3}}{8}\right){C}^{3}{\left(d+w\right)}^{3}\text{(10)}$$
Then, the relative density for the OTT model is
$$\frac{{\rho}^{*}}{{\rho}_{s}}=\frac{d}{4}\left(\frac{4C\left(d+w\right)-d}{\left(C{\left(d+w\right)}^{2}\right)}\right),d<4C\left(d+w\right)\text{(11)}$$
Closed Truss-Wall Triangular (CTT) Model
The CTT model was originally derived from the OST and OTT
models. The difference is that the truss model is added at the top
and bottom surface, as shown in figure 3
$${V}_{s}={V}_{S\_OTT}+{V}_{S\_TOP}+{V}_{S\_BOTTOM}\text{(12)}$$
Figure 3: Drawing CTT model, which is composed of two kinds of struts,
where l = m(d+w) and h = n(d+w)
Where,
$${V}_{S\_OTT}=\frac{3}{4}\pi {d}^{2}\left(n+m\right)\left(d+w\right)\text{(13)}$$
There are two cases of the triangular truss model, viz., the top
side model and the bottom side model, as shown in figure 4. It is
assumed that both cases are the same.
Figure 4: Illustration of CTT model, which is composed of two kinds of
struts (b = (d+w)/cosθ, where θ= 60 degrees)
When the inside angle, θ, is 60 degrees, the top and bottom
configuration is as shown in figure 4. Thus, the volume of the
applied material is
$${V}_{S\_TOP}={V}_{S\_BOTTOM}=\left(1-\frac{2}{\sqrt{3}}\right)\frac{\pi {d}^{2}}{2}{m}^{2}\left(d+w\right)\text{(14)}$$
Thus, the total volume of the applied material is
$${V}_{S}=\frac{1}{4}\pi {d}^{2}\left(d+w\right)\left(3\left(n+m\right)+{m}^{2}\left(4-\frac{8}{\sqrt{3}}\right)\right)\text{(15)}$$
In addition, the total volume of the foam is
$${V}^{*}=\left(\frac{\sqrt{3}}{4}\right){m}^{2}n{\left(d+w\right)}^{3}\text{(16)}$$
Then, the relative density for the CTT model is
$$\frac{{\rho}^{*}}{{\rho}_{s}}=\pi \left(\frac{3\left(n+m\right)+{m}^{2}\left(4-\frac{8}{\sqrt{3}}\right)-d}{\sqrt{3}{m}^{2}n}\right)\left(\frac{{d}^{2}\left(d+w\right)}{{\left(d+w\right)}^{3}}\right)\text{(17)}$$
If n = m = C = a constant, then
$$\frac{{\rho}^{*}}{{\rho}_{s}}=\pi \left(\frac{4\left(\sqrt{3}-1\right)C+6\sqrt{3}}{3{C}^{2}}\right)\left(\frac{{d}^{2}}{{\left(d+w\right)}^{2}}\right)\text{(18)}$$
For the numerical analysis of the model, each parameter is
defined using a value. The thickness, t, of the solid wall is equal
to the diameter of the truss, d, while the length of the solid wall,
l, is equal to the total number of trusses, n, multiplied by truss
diameter, d, and opening width, w. All parameters are fixed as
constants: d = 0.5 mm, w = 0.5 mm, n = 10, l = n(d+w) = 10 and
C = 1. Therefore, the relative density for the CTT model is higher
than that for the OTT model by 91.5%. Table 1 shows the relative
densities and percentage difference for the solid-wall, OST, and
truss-wall models, OTT and CTT. Because crossed trusses are
added in Top or bottom side at CTT model, the density of the foam
is higher than the density of the material, (ρ*= 4.44 ρs).
Table 1: Relative densities and percentage difference
Shape of unit cell model |
ρ* / ρ_{s} |
Percentage difference
between OTT and CTT (%) |
OST |
OTT |
CTT |
Triangular |
0.05 |
0.305 |
4.440 |
91.5 |
The primary objective of model analysis is to evaluate the
effective stiffness and strength for truss-wall unit cell models
in corrugations. Then, these are compared with the effective
stiffness and strength for solid-wall unit cell models. Thus, it is
found that solid-wall unit cell models can be stiffer and stronger
than truss-wall unit cell models when they use the same material.
Finite element analysis
ABAQUS software (v6.11) was used for the finite element
analysis of the OTT model, and it was tested for the OST and
CTT models. The applied model consists of the following four
diameters: d = 1 mm, 2 mm, 3 mm, and 4 mm. The width, length,
and height were fixed at 20 mm each for the fourfold truss model
shown in figure 5. Thus, for the truss model, the applied length
and width are equal to 20 mm. However, the applied height is
20 mm because it is half the height of the truss model shown in
figure 5. The boundary conditions for the model shown in figure
5 are Ux = Uy = Uz = 0, Rx = Ry = Rz = free for the bottom surface,
Ux = Uz = 0, Uy = 5 mm downward, and Rx = Ry = Rz = free for the
top surface. The applied boundary condition is simple. That is, the
bottom surface is fixed and the top surface moves downward by
5 mm in the vertical direction. This boundary condition is used
to measure the reaction force at the bottom. Figure 5 shows the
Figure 5: Dimensions of (a) regular opened solid-wall triangular (OST)
model, (b) equilateral opened truss-wall triangular (OTT) model, and
(c) equilateral closed truss-wall triangular (CTT) model. The arrow infers
an applied compression force
model and boundary condition. The rotation over the top and
bottom surfaces is free.
Meshing
It is a static analysis of the C3D10 mesh type which is a general
purpose 10-node tetrahedral element (4 integration points).
The material used is AISI304 stainless steel with an initial yield
strength of 215 MPa, an ultimate strength of 505 MPa, a density of
8.0 kg/cm^{3}, a Young’s modulus of 200 GPa, and a Poisson’s ratio
of 0.29 [22].
Each model has a different total number of nodes and
elements, which are listed in Appendix. A. The models have
different diameters: 1 mm, 2 mm, 3 mm, and 4 mm. In addition, to
analyze each model, the size of increments varies from 0.5 mm to
1.3 mm. Thus, the maximum number of incremental steps is 1000
when increment size is 0.1. The size of increments ranges from a
minimum of 1e-5 to a maximum of 0.1.
Simulation for Elastic and Yielding Stress
A simulation was used to analyze twelve different models.
The simulated models are for different truss diameters (1 mm,
2 mm, 3 mm, and 4 mm) within a space that is fixed at 4 mm. As
seen in each figure, the common result for the four models is that
the straight vertical truss exhibits the maximum stress while the
middle trusses exhibit different distributions of stress. In other
words, as the vertical truss deforms, the middle trusses support
it to resist deformation. The primary reason for this is the size of
the space. As the truss diameter increases, effective stress, elastic
modulus, and yield strength also increase.
According to the computational analysis of the OST model
with plate thickness 4mm shown in figure 6, the points with the
minimum stress are close to the top, inside, and middle edges.
This implies that as thickness increases, these edges have the
lowest strength and are the points likely to exhibit crushing. In
other words, stress is allocated from top to bottom. The maximum
stress is distributed from the top vertex to the bottom apex in a
diagonal direction. This is represented by a darker color on the
middle surface in the case of 4-mm thickness. The other cases
show the same maximum stress distribution. Therefore, the OTT
and CTT models use the OST model as the fundamental volume.
That is, the OTT and CTT models use the same truss diameter as
the thickness of a plate in the OST model.
The OTT model has an open area at the top and bottom, and
trusses are setup only on the wall side. This model converges
from the OST model with trusses. That is, the total volume for the
OTT model is the same as that for the OST model. The OTT model
is generated using crossed trusses with an opening width.
Figure 6: Stress distribution for OST model (Ux = Uy = Uz = 0, Rx = Ry =
Rz = free at bottom, Ux = Uz = 0, and Uy = 1 mm downward)
Figure 7 shows the simulated models with diameters 4 mm
within a fixed opening space between the trusses. As the truss
diameter increases, the wall side is filled with the trusses and
the open space between trusses disappears. Thus, applied load
affects vertical trusses and does not affect the horizontal truss.
The horizontal truss supports the vertical trusses.
Figure 7: Stress distribution for OST model (Ux = Uy = Uz = 0, Rx = Ry =
Rz = free at bottom, Ux = Uz = 0, and Uy = 1 mm downward)
The truss without a space is connected to all other trusses
directly and behaves like the truss in the OST model. Therefore,
the space in the OTT model is one of the significant factors for
reducing external loads.
For the CTT model with diameter 4mm shown in figure 8, the
mesh size is 1 and the approximate number of elements per circle
is 8.
Figure 8: Stress distribution for CTT model (Ux = Uy = Uz = 0, Rx = Ry
= Rz = free at bottom, Ux = Uz = 0, Uy = 1 mm downward, and Rx = Ry =
Rz = free at top)
In the CTT model, the top and bottom are covered with crossed
trusses with a fixed open space. The size of the crossed trusses is
the same as that of the truss applied at the wall. The CTT model
has the same stress distribution as that of the vertical trusses
because the horizontal truss only supports the vertical trusses.
In addition, the top and bottom walls are covered with crossed
trusses. However, this does not affect stress, owing to the opening
space between trusses. The same phenomenon as that observed
for the OTT model is confirmed for the CTT model. In other
words, the truss with the largest diameter and no truss space
in the CTT model behaves like a plate in the OST model shown
in figure 9. This phenomenon can be confirmed for the top- or
bottom-crossed truss with a diameter of 4 mm, as shown in figure
10. Therefore, the space in the CTT model is an important factor
for reducing external loads, as in the OTT model.
Figure 9: Compressive stress against strain for OST model
Figure 10: Data for compressive stress against strain for CTT model
Results
Each model provides the common result that stiffness
increases with truss diameter. The stiffness for the OST model is
the highest. The stiffness for the CTT model is higher than that for
the OTT model. From the results as effective stress as a function
of effective strain for the OST, OTT, and CTT models, it is observed
for all models that stress increases with truss diameter within a
fixed opening area.
Figure 9 shows effective stress as a function of effective strain
for the OST model. Figure10 and figure 11 show effective stress
as a function of effective strain for the CTT and OTT models,
respectively. All data are summarized in Table 3 and Table 4.
Figure 11: Compressive stress-strain for OTT model
Figure 8 shows the stress distribution for the CTT model, for
truss diameters 4 mm. Stress increases with truss diameter. It
shows that truss diameter is the primary factor for withstanding
a high reaction force. Thus, initial stress increases with truss
diameter. The initial yield loads for each model are shown in table
2.
Table 2: Initial yield loads
Type of
unit cell |
d
(mm) |
Initial yield loads
(N) |
OST |
1 |
14649 |
2 |
32400 |
3 |
31204 |
4 |
61284 |
CTT |
1 |
2326 |
2 |
7597 |
3 |
23521 |
4 |
38753 |
OTT |
1 |
1457 |
2 |
10160 |
3 |
23351 |
4 |
42159 |
The solid-wall unit cell models have relative densities ranging
from 0.049 to 0.198. The relative densities for the CTT unit cell
model range from 0.029 to 0.207. The relative densities for the
OTT unit cell model range from 1.016 to 10.835. The relative
densities are different because they are correlated with the
inverse relative volume. Thus, among the three models, the OST
model has the lowest relative density, and the OTT model has the
highest relative density. Table 3 lists the values of relative density
and relative Young’s modulus. Effective Young’s modulus, which
is denoted as E*, follows Hook’s law. It is equal to applied yield
strength divided by initial strain. Thus, the effective Young’s
modulus for OST range from 47 GPa to 559 GPa, while that for
CTT range from 14 GPa to 230 GPa and OTT range from 8 GPa
to 245 GPa. The reason that the calculated Young’s modulus of
the foam (E*=559 GPa) is higher than the Young’s modulus of the
base material (Es = 200 GPa) is a structural properties within a
limited volume. Base material infers applied material property
but the Young’s modulus of the foam means geometrical property.
According to Wang, et al. [23], there are two kinds of yield
strengths: initial yield strength, and yield strength at 25% strain.
The effective yield strength, σ*, when yielding begins is equal to
the load carrying capacity divided by the application surface area.
It ranges from 73.3 MPa to 306.4 MPa for OST, from 9.4 MPa to
193.8 MPa for CTT, and from 7.3 MPa to 210.8 MPa for OTT. When
yielding occurs at 25% strain, yield strength, which is denoted
by σ0.25, ranges from 49.7 MPa to 564.4 MPa for OST, from 2.7
MPa to 266.6 MPa for CTT, and from 8.37 MPa to 263.9 MPa for
OTT. The applied material is SST304 stainless steel, and its yield
strength is 215 MPa. Thus, the ratio of relative density to initial
Table 3: Relative elastic modulus
Type of
unit cell |
d (mm) |
E* (GPa) |
ρ* / ρ_{s} |
E*/E_{s} |
OST |
1 |
47 |
0.049 |
0.22 |
2 |
194 |
0.099 |
0.90 |
3 |
300 |
0.148 |
1.40 |
4 |
559 |
0.198 |
2.60 |
CTT |
1 |
14 |
0.029 |
0.07 |
2 |
59 |
0.086 |
0.27 |
3 |
126 |
0.147 |
0.58 |
4 |
230 |
0.207 |
1.07 |
OTT |
1 |
8 |
1.016 |
0.04 |
2 |
61 |
3.483 |
0.28 |
3 |
147 |
6.856 |
0.68 |
4 |
245 |
10.835 |
1.14 |
Table 4: Relative elastic modulus
Type of
unit cell |
d (mm) |
Initial
yield strength |
Yield strength
at 25% strain |
Relative yield strength |
σ * (MPa) |
σ _{0.25} (MPa) |
σ */ σ _{ys} |
σ _{0.25} / σ _{ys} |
OST |
1 |
73.3 |
49.7 |
0.341 |
0.231 |
2 |
162.0 |
171.0 |
0.753 |
0.788 |
3 |
156.0 |
393.7 |
0.726 |
1.831 |
4 |
306.4 |
564.4 |
1.425 |
2.625 |
CTT |
1 |
9.4 |
2.7 |
0.054 |
0.013 |
2 |
38.0 |
45.4 |
0.177 |
0.211 |
3 |
117.6 |
214.0 |
0.547 |
0.995 |
4 |
193.8 |
266.6 |
0.901 |
1.240 |
OTT |
1 |
7.3 |
8.37 |
0.034 |
0.011 |
2 |
50.8 |
49.3 |
0.236 |
0.229 |
3 |
116.8 |
207.7 |
0.543 |
0.966 |
4 |
210.8 |
263.9 |
0.980 |
1.227 |
yield strength ranges from 0.049 to 0.198 for OST, from 0.029
to 0.207 for CTT, and from 1.016 to 10.835 for OTT. The ratio of
relative density to yield strength at 25% strain ranges from 0.231
to 2.626 for OST, from 0.013 to 1.240 for CTT, and from 0.011 to
1.227 for OTT.
Based on relative yield strength, the percentage difference
between initial yield strength and the yield strength at 25%
shown in table 4 indicates that the yield strength at 25% strain is
higher than initial yield strength.
Discussion
For the analysis of the truss wall corrugated structure, the
OST, CTT, and OTT unit cell models were tested and validated
through computational analysis. The stiffness and strength for
the models were based on the theory of Gibson and Ashby, [2],
which states that stiffness is proportional to the square of density,
and strength is proportional to density raised to a power of 3/2.
The results of the finite element analysis of the OST, CTT, and OTT
models are in good agreement with those of the open-cell theory
of Gibson and Ashby. That is,
$$\begin{array}{l}\frac{{E}^{*}}{{E}_{s}}={C}_{1}{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{2}\\ and\text{(18)}\\ \frac{{\sigma}^{*}}{{\sigma}_{s}}={C}_{2}{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{\frac{3}{2}}\end{array}$$
Thus, the OST, CTT, and OTT models are considered as
periodic cellular structures.
Figure 12 shows relative elastic modulus as a function of
relative density. Relative stiffness is proportional to relative
density to a power of 1.73 for the OST model, 1.4 for the CTT
model, and 1.46 for the OTT model. That is,
Figure 12: Data for the relative elastic modulus against relative density
$$\begin{array}{l}{\left(\frac{{E}^{*}}{{E}_{s}}\right)}_{OST}\approx 42.6{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{1.7},\\ {\left(\frac{{E}^{*}}{{E}_{s}}\right)}_{OTT}\approx 93.2{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{2.6},\text{(20)}\\ {\left(\frac{{E}^{*}}{{E}_{s}}\right)}_{CTT}\approx 9.0{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{1.4}\end{array}$$
However, this power is 1.5 in Gibson and Ashby’s ideal
solution. Therefore, the CTT and OTT models are closer to Gibson
and Ashby’s ideal solution than the OST model.
Figure 13 shows the relative compressive yield strength
at initial yield strain as a function of relative density. Relative
compressive yield strength is proportional to relative density to a
power of 0.97 for the OST model, 0.68 for the CTT model, and 0.70
for the OTT model. This power is 2.0 in Gibson and Ashby’s ideal
solution. Therefore, the OTT model is close to Gibson and Ashby’s
ideal solution, and the CTT model is reasonably close to it. That is,
Figure 13: Data for the relative compressive yield strength against relative
density
$$\begin{array}{l}{\left(\frac{{\sigma}^{*}}{{\sigma}_{s}}\right)}_{OST}\approx 0.15{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{0.97},\\ {\left(\frac{{\sigma}^{*}}{{\sigma}_{s}}\right)}_{OTT}\approx 0.18{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{0.4},\text{(20)}\\ {\left(\frac{{\sigma}^{*}}{{\sigma}_{s}}\right)}_{CTT}\approx 0.23{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{0.7}\end{array}$$
Figure 14 shows the compressive yield strength at plastic
strain as a function of relative density. Relative compressive
yield strength is proportional to relative density to a power of
1.8 for the OST model: approximately 2.5 for the CTT model, and
approximately 3.7 for the OTT model. This power is 2.0 in Gibson
and Ashby’s ideal solution. That is,
Figure 14: Data for the relative compressive yield strength against relative
density
$$\begin{array}{l}{\left(\frac{{\sigma}^{*}}{{\sigma}_{s}}\right)}_{OST@plastic}\approx 51.2{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{1.8},\\ {\left(\frac{{\sigma}^{*}}{{\sigma}_{s}}\right)}_{OTT@plastic}\approx 831.9{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{3.7},\text{(21)}\\ {\left(\frac{{\sigma}^{*}}{{\sigma}_{s}}\right)}_{CTT@plastic}\approx 78.8{\left(\frac{{\rho}^{*}}{{\rho}_{s}}\right)}^{2.5}\end{array}$$
Therefore, the OST, OTT, and CTT models show significant
differences from Gibson and Ashby’s ideal solution under this
condition.
Conclusion
For the OST, OTT, and CTT, the effective stiffness, the
compressive yield strength, and compressive strength at the
plastic range were analyzed using a computational analysis. Each
model is based on a different shape of the triangular prism. OST
refers to plates applied into the wall, OTT to a truss applied into
the wall, and CTT to a truss applied into the wall and into the top
or bottom side.
The simulation predicts the ideal solution to ensure effective
stiffness and effective strength, as shown in Equation (19), (20),
and (21). Thus, OTT is shown to be stiffer than CTT, OTT has
greater yield strength than CTT, and OTT is stronger at plastic
than CTT. The ideal solutions for all models are plotted using the
Gibson–Ashby theory, and it can predict the effective stiffness
of OTT and CTT to be stiffer, and the yield strength is similar
within a density 0.2 approximately. If the density of OTT or CTT
is greater than 0.2, then the prediction for the yield strength via
Gibson-Ashby is higher. If the density of OTT or CTT is less than
0.2, then the predicted yield strength via Gibson-Ashby is lower.
If the density of OTT or CTT is more than 0.03–0.05 (approx.),
then the plastic strength predicted by Gibson-Ashby is lower than
OTT or CTT. If the density of OTT or CTT is less than 0.03–0.05
(approx.), then the plastic strength predicted by Gibson-Ashby
is higher than OTT or CTT. Thus, the stiffness of OTT and CTT is
higher than the ideal Gibson-Ashby solution. The yield strength of
OTT and CTT is higher when the relative density is greater than
0.1, and the plastic strength of OTT and CTT is higher when the
relative density is greater than 0.03–0.05.
In conclusion, the effective stiffness of the OTT and CTT is
greater than the ideal Gibson-Ashby solution. When the relative
density is greater than 0.2, the effective yield strength of the OTT
and CTT is higher than the ideal Gibson-Ashby solution. When
the relative density is greater than 0.03–0.05, the effective plastic
strength of the OTT and CTT is higher than an ideal Gibson-Ashby
solution.
For aerospace or automotive applications, optimizing various
parameters – such as width, length, height, diameter, aperture,
and corrugation angle – is an important research subject insofar
as the mechanical properties of truss-wall models depend on
these parameters. In the near future, a new method to create PCM
models without joining points may be devised using well-known
techniques, such as three-dimensional printing and casting.
Appendix A
For the solid-wall triangular (OST model), the mesh size is
1 and the approximate number of elements per circle is 8. The
maximum deviation factor for curvature control is 0.1. The total
number of nodes and elements are shown in table A, applying the
C3D10 tetrahedral mesh type.
For the OTT model, the mesh size is 1 and the approximate
number of elements per circle is 8. The maximum deviation factor
for curvature control is 0.1. The C3D10 tetrahedral mesh type is
applied. The maximum number of increments for the analysis is
1000, with an increment size of 0.1, a minimum increment size of
1e-5, and a maximum increment size of 0.1.
For the CTT model with diameter 4mm shown in figure 8,
the mesh size is 1 and the approximate number of elements per
circle is 8. The maximum deviation factor for curvature control
is 0.1. The total number of nodes and elements is shown in table
A. The C3D10 tetrahedral mesh type is applied. The maximum
number of increments for the analysis is 1000, with an increment
size of 0.1, a minimum increment size of 1e-5, and a maximum
increment size of 0.1.
Table A.1: Nodes and elements
Type of model |
d (mm) |
Nodes |
Elements |
OST |
1 |
61363 |
36041 |
2 |
88062 |
55562 |
3 |
119712 |
79067 |
4 |
160301 |
108923 |
OTT |
1 |
120934 |
70031 |
2 |
114860 |
70821 |
3 |
153399 |
99909 |
4 |
235538 |
159174 |
CTT |
1 |
149610 |
86512 |
2 |
140476 |
86936 |
3 |
180533 |
118296 |
4 |
115255 |
76445 |
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