L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
Liang Donga , Arthur Heuerb , Harold Kahnb , Zhen Lib , Vikram Deshpandec , Haydn Wadleya
a Department
of Materials Science and Engineering, University of Virginia, Charlottesville, Virginia, USA
of Materials Science and Engineering, Case Western Reserve University, Cleveland Ohio, USA
c Engineering Department, Trumpington Street, University of Cambridge, UK
b Department
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Strong cellular lattices with nitro-carburized
stainless steel hollow trusses
Collinear lattice structures have been fabricated from
304 stainless steel hollow tubes via an alternating collinear
lay-up process followed by bonding using a vacuum brazing method. By varying the tube wall thickness from 50 to
203 lm, square and diamond lattice topologies with relative
densities between 0.03 and 0.11 were manufactured in this
way and their compressive mechanical responses were
characterized. A low temperature nitro-carburization treatment was then performed on a second set of the collinear
lattice cores at a temperature of 440 8C for 20 h. The treatment created a thin (10 to 20 lm thick) extremely hard
(*12 GPa) surface layer on the interior and external surfaces of the hollow trusses. This significantly increased the
compressive buckling resistance of individual trusses.
Compressive strength enhancements (compared with untreated counterparts in the annealed (as brazed) condition)
varied from 1.2 for thick walled tubes to 3.8 when the wall
thickness was decreased to about twice the hardened layer
depth. The moduli and strengths of all the lattices were
found to increase with lattice relative density, and were
well predicted by micromechanical models. The lowest relative density (thinnest wall) nitro-carburized hollow truss
collinear lattice structures exhibited a specific compressive
strength significantly higher than that of any other cellular
metal reported to date. Nitro-carburized stainless steel collinear lattices therefore appear to be promising candidates for
the cores of lightweight sandwich panels intended for elevated temperature and/or multifunctional applications.
Keywords: Cellular lattice; Nitro-carburization; 304 stainless steel; Compressive strength
1. Introduction
Low density cellular metals have been widely used for the
cores of sandwich panel structures intended for load supporting applications [1 – 6]. They have also been investigated for numerous multifunctional applications such as
heat exchange [7, 8], dynamic load protection [9 – 11], and
for acoustic damping [12]. Many types of cellular materials
have been developed to meet the needs of these various applications. They include open and closed cell foams made
from various polymers, metals and ceramics [1, 13], stronger honeycomb topology materials [14, 15], as well as
lattice structures with tetrahedral [16 – 18], pyramidal
[18 – 23], 3D-Kagome [18, 24, 25], octet truss [26], woven
metal textile [6, 27, 28] or collinear [29] topologies, and
simpler to fabricate, prismatic corrugations [30, 31]. The
design of an optimal sandwich structure for one of these applications begins with selection of a material system. It then
distributes the allowable panel mass per unit area between
its core and faces, and then chooses a core cell topology
that, in combination with material properties, avoids panel
failure under the intended loading environment. For example, if an edge clamped sandwich panel must support large
centrally applied bending loads, it is essential that the core
has sufficient through thickness compressive and transverse
shear elastic moduli and strengths to maintain face sheet separation throughout loading process.
The mechanical properties of a cellular material are determined by its relative density, by its parent material mechanical properties and by the cell topology. For example,
a metallic lattice made from elastic–perfectly plastic trusses
that fail by plastic yielding, has a compressive collapse
strength, rc , given by [30];
rc ¼ Rrys q
ð1Þ
where rys is the yield strength of the solid material from
which the lattice is made, q is the relative density of the lattice and R is a lattice topology dependent \strength coefficient".
The compressive strength coefficient describes the efficiency with which an applied compressive force is resolved
into forces within the lattice structure. For a square honeycomb core with low aspect ratio webs all aligned in the
loading direction, R = 1. However, for tetrahedral and pyramidal lattices, the trusses are inclined to the loading direction and R = sin2x, where x as the inclination angle between the trusses and the loading direction [30]. Figure 1
shows a collinear lattice in either a square, Fig. 1a, or 458
rotated diamond orientation, Fig. 1b. The lattice topology
dependent strength coefficients, R, of such collinear lattices
can be found by analyzing the unit cell, Fig. 1c. For the unit
cell of a square orientation collinear lattice, Fig. 1a, half of
the trusses have x = 908 with the remainder having x =
08. Therefore, the effective value of sin2x = 0.5. For a unit
cell of a diamond orientation lattice, Fig. 1b, all the trusses
have x = 458, and the effective value of sin2x = 0.5 again.
It is important to note that for a finite width diamond orientation lattice, an edge effect should be taken into consideration as some trusses are not connected to both loading faces,
leading to a reduction in the load supporting area. The compressive strength of finite width samples with the diamond
1
L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
is elastic modulus of the parent material and k is a factor accounting for the rotational stiffness of the truss ends (k = 2
for built-in end conditions and 1 for pin-jointed ends). For
a circular cross-section hollow tube truss, the area moment
of inertia, I is a function of the tubes cross sectional geometry:
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I¼
p 4
a a4i Þ
4 o
ð4Þ
where 2 ao and 2 ai are the outer and inner diameters of the
tube. For thin walled tubes (where ai approaches ao ), the
buckling strength can be substantially higher than that of
the equivalent area solid strut since the area moment of inertia ratio of a tube to a solid counterpart of equal area is
a2o þ a2i
.
a2o a2i
If the lattice is made from a material with a non-zero
strain-hardening rate, the trusses continue to deform after
plastic yielding until the onset of inelastic buckling. In this
case, the compressive strength of the lattice is determined
by the inelastic global buckling strength of the truss, rIE ,
which was first predicted by Engesser tangent modulus theory [33, 34]:
rIE ¼
p2 k2 IEt
Al2
ð5Þ
where Et is the tangent modulus of the parent material at the
inelastic bifurcation stress [33]. It is determined as the modulus at the intersection point of the true stress–true strain
curve and that defined by Eq. (5). The lattice compressive
strength in this case is given by:
2 2 p k IEt
q
rc ¼ RrIE q ¼ R
Al2
Fig. 1. Sketches of (a) the square and (b) diamond orientated collinear
lattice structures. (c) shows a unit cell of a square collinear lattice with
hollow trusses.
orientation therefore depends upon their length to height
(aspect) ratio since struts at the sides do not connect the
top and bottom of the sample. The strength of such finite
samples can be predicted by multiplying the strength given
by Eq. (1) by the factor [1 ð1=S tan xÞ] where S is the
length to height (aspect) ratio of the sample [32].
If the trusses are sufficiently slender, and especially if
made of a low elastic modulus (high yield strain) material,
they elastically buckle prior to plastic yielding, and the
compressive strength of the lattice is obtained by replacing
rys in Eq. (1) with the Euler (global) buckling stress:
rE ¼
p2 k2 IEs
Al2
The lattice strength is then given by:
2 2 p k IEs
q¼R
q
rc ¼ RrE Al2
ð2Þ
ð3Þ
where I, A, and l are the area moment of inertia, the crosssectional area, and the length of the truss respectively; Es
2
ð6Þ
Engesser tangent modulus theory predicts the critical stress
of an axially loaded column when lateral deflection is initiated. It defines the lower bound inelastic global buckling
stress of such a compressed column since the load bearing
capacity of an imperfection free column could increase with
increasing lateral deflection. The upper bound to the column buckling stress is defined by a reduced modulus theory
[33], in which Et in Eq. (5) is replaced by the reduced modEt
ulus Er ¼ pffiffiffi4E
ffi sp
ffiffiffi 2 .
ð Es þ Et Þ
Shanley [34] showed the critical inelastic global buckling stress to be a function of a buckling columns transverse
displacement and located between the critical load predicted by the tangent modulus theory (the lower bound)
and the reduced modulus (upper bound) theory. Shanley’s
equation for the inelastic global buckling critical stress
(rSIE ) is given by:
2
3
6
rSIE ¼ rIE 6
41 þ
7
1
7
ao 1 þ Et =Es 5
þ
v0 1 Et =Es
ð7Þ
where rIE is the critical inelastic global buckling stress defined by Engesser tangent modulus model (Eq. (5)), 2 ao is
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L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
the tube outer diameter, Et is the tangent modulus at an
axial strain e, and v0 is the lateral deflection. Geometry dictates that the lateral deflection and axial strain of the tube
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
are related. To a first order apprixmation, v0 = 2l 2e e2
pffiffiffiffiffiffiffiffiffiffiffiffi
for k = 1 (pin-joint) end conditions and 2l e e2 for k = 2
ends.
Equations (1), (6) and (7) provide a rationale for the design of high specific strength cellular structures from hollow tubes. If the strut slenderness is sufficiently low that
yielding controls the strength, increasing the specific
strength of the material used to fabricate the lattice increases the strut/lattice compressive strength. However,
there will eventually come a point where the yield strength
is so high that strut buckling intervenes, and the elastic
modulus and second moment of the struts then govern the
strength.
For a short and very thin-walled tube loaded in compression, failure will not be caused by global buckling as described by elastic (Euler) or inelastic buckling, but rather
involves local bending of the tube walls; a failure mode referred to as local buckling [35, 36]. The most important
shape-parameter governing the local buckling stress of cylindrical surfaces is the tube diameter-to-wall thickness ratio Dm /t, where t is the wall thickness and Dm refers to the
tube’s mean diameter:
Dm ¼
a2o a2i
t
ð8Þ
The tube length can also affect the critical stress to some extent, but in most structural applications the members are
long enough that length effects can be neglected. The critical local buckling stress in the elastic range is given by:
rlE ¼
Kc E s
Dm =t
ð9Þ
2
where Kc ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, and m is Poisson’s ratio; for a ma3ð1 m2 Þ
terial with m = 0.3, Kc therefore equals 1.2. However, numerous studies have shown that this theoretical value
cannot be achieved experimentally because the buckling
behavior of thin walled tubes is highly sensitive to initial
imperfections. In practice Kc is determined by fitting to experimental data, and typically varies between 0.24 and 0.8
[34, 37]. In the inelastic buckling regime, Es in Eq. (9) is
usually replaced by an effective modulus equal to the geometric mean of the elastic and tangent moduli to give a conservative estimate for the critical inelastic local buckling
stress:
pffiffiffiffiffiffiffiffiffi
Kc E s E t
rlIE ¼
ð10Þ
Dm =t
Hollow tube lattice design concepts has been previously investigated by fabricating collinear lattice structures from
304 stainless steel hollow tubes with both the square and
the (458 rotated) diamond orientation topologies, Fig. 1a
and b [29]. The relative density of such hollow truss lattices
depends upon the outer (2ao ) an inner (2ai ) tube diameters,
and the unit cell length, l, Fig. 1c. The relative density, q,
of the lattice is the ratio of the volumes of solid material
(Vs ) within a unit cell to that of the unit cell (Vc ):
q ¼
Vs pða2o a2i Þ
¼
2ao l
Vc
ð11Þ
The hollow tube lattice therefore enables independent control of the cell size (length) and the lattice relative density
by varying the dimensions of the truss (the diameter and
wall thickness of the hollow truss) and/or the spacing between trusses. The use of hollow trusses substantially increases the resistance to buckling [29], thereby offsetting
the usually rapid drop in lattice strength as the relative density decreased and truss buckling became the dominant
mode of compressive failure.
The highest specific compressive strength structures
were achieved when the failure mechanism was inelastic
buckling of the trusses, and the strength was then well predicted by Eq. (5) or (7), with the scaling factor R = sin2x,
and a built-in node end condition (k = 2) [29]. However,
while such lattices had a higher strength coefficient than
other cellular metal topologies (in part a consequence of
the high tangent modulus of annealed 304 stainless steel),
and exhibited excellent mechanical energy absorption, the
full benefit of the high second moment area of hollow struts
was truncated by the low yield strength of the annealed
struts. For this alloy, the solid material’s strength cannot
be increased by subsequent heat treatment.
Metallic foams and periodic micro-truss cellular lattice
structures with hybrid struts have been recently fabricated
by electroplating Ni based alloys on the metal foams [38]
and electroless Ni deposition and/or Ni electroplating (depending on the relative density) on microlattices produced
by the self-propagating polymer waveguide process [39] or
on rapidly prototyped polymer lattice preforms [40 – 42].
This approach takes advantage of the large strength increase that can be obtained by grain size reduction into the
nm-scale, and by the fact that the electrodeposited material
is positioned far away from the neutral (bending) axis of
the coated struts or ligaments of the cellular material. The
inner polymer preform can also be subsequently removed
[43, 44], leaving just the deposited coating in the form of a
hollow truss lattice, which further improves the specific
strength and stiffness of the lattice. However, while the
electroplating approach has excellent throwing power
(non-line of sight coverage), a coating thickness of only
120 nm to 3 lm has so far been applied [43]. The strength
of such thin walled metallic lattices with struts of mmlength scale was then governed by elastic modes of collapse, and could not fully exploit the coating metal’s high
yield strength.
Recent studies have demonstrated the possibility of
greatly increasing the strength of some austenitic stainless
steel by developing very high (colossal) super saturations
of carbon and/or nitrogen in a thin (25 – 70 lm thick) layer
on its surface [45, 46]. This is achieved using low temperature \paraequilibrium" conditions to inhibit carbide formation, and can result in increase in local strength by factors of
5 or more in some alloy systems. The process has also been
shown to greatly improve pitting corrosion and fatigue
loading resistance [45 – 48]; both of which can be problematic for high specific surface area cellular materials in
some loading environments. Since the processing tempera3
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L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
tures for this hardening process are low (around 400 8C) and
the wall thicknesses of tubes used can be made comparable
to the interstitial element diffusion distances [49], the possibility exists of making collinear lattices of exceptional
compressive strength. This approach compliments recent
studies of gas-phase aluminized 3D woven and braided
Ni – Cr wires that have also demonstrated a substantial increase in the strength due to on the formation of c’ precipitates within the wires [50].
A recent numerical study [51] has investigated the effect
of such a surface carburization effect upon the compressive
response of a pyramidal lattice made from stainless steel
tubes. The numerical results confirm that carburization
treatments could significantly increase the peak strength of
the tubes, but would have a relatively small effect upon the
deformation mode as long as the slenderness ratio is not
too high. At very high slenderness ratios, the tubes collapse
by an elastic buckling, and the peak stress is then insensitive to the degree of carburization. Here, we have fabricated
collinear lattice structures in an identical manner to those
reported in the previous study [29], and have investigated
their mechanical properties before and after nitro-carburization. We show that the lattice compressive strength is increased by almost a factor of four as the tube wall thickness
decreases, and investigate the mechanisms of tube collapse
that are responsible for the improved mechanical response.
2. Lattice fabrication and nitro-carburization
Type 304 stainless steel hollow tubes (Vita Needle Corp.,
Needham, MA, USA) were used to fabricate the collinear
lattices. The tubes were identical to those used in a previous
study [29], and had the same outer diameter of 1.473 mm
and wall thicknesses of 50, 101, 127 and 203 lm. The metal
tubes were collinearly aligned in layers with the aid of a
BN-coated stainless steel assembly tool, Fig. 2a. The orientation of the tube layers was alternated to create both the
square and diamond lattice architectures. The layers were
bonded by vacuum brazing using NICROBRAZ alloy 51
(Wall Colmonoy Corp., Madison Heights, MI, USA) of
nominal composition Ni-25Cr-10P-0.03C (wt.%), Fig. 2b.
Vacuum brazing was performed at 1 020 8C for 1 hr at a
base pressure of *10–2 Pa. Additional details of the vacuum brazing process can be found in Ref. [29].
The bonded lattices were removed from the assembly
tool and cut to size using electro-discharge machining,
Fig. 2c and d. For the square lattice, their length L was
*51.5 mm, their width W was *29.5 mm, and their height
H was *16.5 mm. For the diamond lattice, L was
*64 mm, W was *29.5 mm, and H was *22 mm. The
center-to-center cylinder spacing of the tubes, l was kept
constant at 5 mm. For the range of inner and outer tube
diameters used here, this choice of strut length resulted in
lattices with relative densities of 0.03, 0.06, 0.073, and
Fig. 2. Collinear lattice fabrication procedure showing (a) the stacking arrangement for the hollow truss lattice structures, (b) brazing of the square
and diamond orientation lattices, (c) photograph of a square lattice sample, and (d) a photograph of a diamond orientation hollow truss lattice after
cutting to shape.
4
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L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
0.11 as the wall thickness increased from 50 to 203 lm. Table 1 summarizes the geometric parameters of the hollow
tubes and the corresponding relative density of collinear lattice structures made from them.
The lattices were then subjected to a low temperature nitro-carburization treatment performed at 440 8C for 20 h.
During nitro-carburization a 0.20 slpm (standard liters per
minute) NH3 + 0.20 slpm CO + 0.90 H2 + 0.90 slpm N2
gas flow was used. A surface activation process was performed before the nitro-carburization treatment to break
down surface oxide layers that otherwise imped interstitial
atom entry into the stainless steel. This was conducted at
440 8C under gas flows of 0.20 slpm HCl + 1.8 slpm N2 for
3 hr. All of the processes were performed at the atmospheric pressure. After nitro-carburization, 2.5 mm thick
304 stainless steel face sheets were adhesively attached
to the lattices with HYSOLE-120HPTM (LoctiteBrand,
Westlakes, OH, USA) high strength epoxy. Curing of the
epoxy adhesive was performed at room temperature for
12 hr. Individual tubes of each wall thickness were also nitro-carburized in the same manner.
A cross-sectional scanning electron microscope image of
a nitro-carburized tube with a wall thickness of 101 lm is
shown in Fig. 3. A 20 lm thick nitro-carburized layer can
be clearly seen on the external surface of the tube. A thin region about 8 lm thick at the surface of the layer had a different microstructure to the rest of the carburized layer. A
thinner, *10 lm thick, nitro-carburized layer can also be
seen on the inside of the tube. This difference is attributed
to the reduced reactant gas flow rates inside the tubes. A
50 lm thick 304 stainless steel tube was also subjected to
the same nitro-carburization treatment and used to characterize the depth and degree of interstitial hardening in this
alloy. Nano-hardness tests again indicated the case hardening depth of the treated alloy was *20 lm on the outer side
and 10 lm on the inner side where gas flow was more restricted, Fig. 4a. The volume fraction of the case hardened
Fig. 3. Cross-sectional SEM image of a nitro-carburized 304 stainless
steel tube with a 101 lm wall thickness. The case hardened layers can
be seen on both the inner and outer surfaces of the tube.
Fig. 4. (a) Hardness and Young’s modulus and (b) C, N atomic concentration data for a nitro-carburized 304 stainless steel tube with a
50 lm wall thickness.
Table 1. 304 stainless steel tube parameters and corresponding relative density of lattice structures.
Tube diameter
Relative density of lattice,
(q)
Outer diameter, 2ao
(mm)
Inner diameter, 2ai
(mm)
Wall thickness, t
(mm)
1.473
1.473
1.473
1.473
1.067
1.219
1.270
1.373
0.203
0.127
0.102
0.050
0.110
0.073
0.060
0.030
5
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L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
layers for each of the nitro-carburized tubes was calculated
and is summarized in Table 2.
The nitrogen and carbon composition–depth profiles
were determined on cross-sectioned specimens by Auger
electron spectroscopy (AES) using a PHI-680 scanning Auger microprobe (Physical Electronics, Chanhassen, MN,
USA). Before the acquisition of each AES line scan, about
50 nm of material was sputtered off the cross-sectional
sample by an Ar+ plasma to remove hydrocarbon contaminants and native oxides. During acquisition of the AES line
scans, dynamic Ar+ sputtering was carried out to avoid carbon accumulation within the chamber. This was needed because the overlap of oxygen and Cr peaks could induce uncertainty in the absolute Cr content. The corresponding
carbon and nitrogen concentration profiles are shown in
Fig. 4b.
During the nitro-carburizing process, the nitrogen- and
carbon-containing gas species were introduced into the furnace simultaneously, and the nitrogen and carbon interstitials entered the steel simultaneously. However, nitrogen
and carbon mutually increase the chemical potential of the
other interstitial, and therefore the total interstitial concentration at any depth is limited. Since carbon exhibits a greater diffusivity in austenitic stainless steel than nitrogen, the
result is an accumulation of carbon interstitials deeper into
the steel, and an accumulation of nitrogen interstitials closer to the surface [49]. This higher nitrogen content appears
to be linked to the change in appearance of the layer near
the surface evident in Fig. 3.
ening resulting in the Cauchy flow stress eventually reaching almost 1 GPa just before failure. Both the elastic modulus and 0.2 % offset yield strength are summarized in
Table 2 for all the tube samples. The nitro-carburized tubes
exhibited no measurable increase in elastic modulus, but
had a substantially higher yield strength compared to the
annealed tubes, this difference increased as the tube wall
thickness was reduced to 50 lm. While the nitro-carburized
tubes had higher yield strengths than the annealed material,
their strain hardening rates and especially fracture strains
were lower, and the tubes failed to attain the tensile strength
of the annealed alloy; a consequence of the brittle nature of
the case hardened layer.
The uniaxial compressive stress versus strain responses
of the nitro-carburized stainless steel hollow tubes of different wall thicknesses were also determined and are compared to those of their annealed counterparts in Fig. 6. The
tubes had an aspect ratio l=2ao &3.4 (where l is the tube
length = 5 mm, and 2ao the tube outer diameter =
1.473 mm); similar to that of the lattice unit cell. The collapse strengths of all the single tubes are summarized in Table 2. The nitro-carburization treatment increased the compressive strength of all the tubes, Fig. 6. However, the
increase was greatest for the thinnest walled tubes; the
3. Single tube mechanical properties
3.1. Experimental characterization
In order to later analyze the lattice response, the uniaxial
tensile responses of both annealed and nitro-carburized
304 stainless steel tubes were first determined according to
the ASTM A370-14 standard. Metal plugs were snug-fitted
in both ends of the tubular specimens to permit the testing
machine jaws to grip the specimens properly without crushing or creating a stress concentration; all the specimens then
failed outside the grip areas. The tensile responses measured at a strain rate of 10–4 s–1 are shown in Fig. 5. The
stress–strain responses of the annealed tubes with different
wall thicknesses were the same up to failure at a plastic
strain of *38 %. In the annealed state, the 304 stainless
steel alloy had a Young’s modulus of 182 GPa, a low yield
strength of 240 MPa, but exhibited substantial strain hard-
Fig. 5. True tensile stress–strain responses of nitro-carburized 304
stainless steel tubes with different wall thicknesses. The response of
the annealed 304 stainless steel is included for comparison.
Table 2. Properties (volumetric fraction of the treated alloy, 0.2 % offset yield strength, Young’s modulus, and ultimate strength) of the
nitro-carburized 304 stainless steel tubes with different wall thicknesses. Properties of the annealed tubes have been included for comparison (data in brackets; stiffness and yield strength were determined from tensile testing).
Nitro-carburized/Annealed
t (lm)
Vtreated (%)
rys (MPa) (Tension/Compression)
ES (GPa)
Tensile strength rUTS (MPa)
Compressive strength rUTS (MPa)
6
50
61
761/812 (240)
181 (178)
830 (1 003)
1 786 (380)
102
30
523/496 (240)
185 (184)
721 (1 003)
1 119 (517)
127
24
475/428 (240)
172 (182)
728 (1 003)
1 006 (607)
203
15
375/341 (240)
176 (181)
751 (1 003)
855 (836)
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L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
Fig. 6. Compressive stress–strain responses of (a) annealed and (b) nitro-carburized 304 stainless steel tubes of different wall thicknesses with a
length of 5 mm. Photographs showing the buckling modes of each wall thickness tube are also shown.
7
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L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
50 lm thick tube strength was increased by about a factor of
5 over that of the annealed counterpart. The compressive
strength of the nitro-carburized tubes decreased with wall
thickness because of the reduced volume fraction of the
case hardened layers. While the 203 lm thick treated tubes
had a similar collapse strength to their annealed counterparts, their yield strength was still about a factor of 2 higher.
Photographs of the annealed and nitro-carburized tubes
after their peak strength was achieved are shown to the right
of Fig. 6a and b. The buckling mode of the annealed tubes
evolved from a local inelastic buckling mode towards a global inelastic buckling mechanism as the annealed tube wall
thickness increased. Figure 6a, shows the collapse mechanisms for annealed tubes (l = 5 mm) of different wall thicknesses. For the thinnest wall tubes with t = 50 lm buckling
occurred by multi-lobe diamond local inelastic buckling.
As the thickness was increased to t = 101 lm, collapse
occurred by two-lobe diamond local inelastic buckling.
Further increases in tube wall thickness (t = 127 and
203 lm) resulted in a mixed mode of two-lobe diamond local inelastic buckling and global inelastic buckling. These
observations agree well with the numerically predicted collapse mechanism map of 304 stainless steel vertical tubes
[36]. Examination of Fig. 6b indicates that the collapse
mechanisms of the nitro-carburized tubes were multi-lobe
diamond local inelastic buckling for t = 50 lm, axisymmetric local inelastic buckling for t = 101 lm, two-lobe
diamond local inelastic buckling for t = 127 lm, and a
mixed mode of two-lobe diamond local inelastic buckling
and global inelastic buckling for t = 203 lm.
tained can be greater than for typical micro-indentation
(by a factor of up to two for metals) [53]. The strength of
the annealed alloy in the tube center was therefore reduced
by the factor 1/1.23 for the ultimate strength calculation.
The same study has also shown that the hardness of an intrinsically hard material suffers a minimal indenter size effect [53], and therefore no reduction factor was used for
the case hardened layer’s ultimate strength. A Ramberg–
Osgood model [54] (see Appendix) was fitted to the elasto–plastic model stress–strain curve for the case hardened
layer, Fig. 7a, and we simplified the analysis by assuming
an ultimate strain equal to that of the annealed alloy
(*0.38); even though the case hardened layer is expected
to have a much smaller strain to failure.
The predicted compressive stress–strain responses of the
nitro-carburized vertical tubes obtained using the effective
properties given in Fig. 7a, are shown in Fig. 7b. Good
agreement was seen between the predicted compressive
3.2. Vertical tube buckling analysis
A nitro-carburized tube can be regarded as a composite
consisting of an annular region of annealed condition material sandwiched between two rings of a much harder, but
identical modulus material. The inner tube (nitro-carburized) layer is taken to be 10 lm thick while that at the outer
surface was 20 lm in thickness. The uniaxial compressive
response of such a composite can be analyzed with a Voigt
composite model which assumes equal strain in each layer,
and sufficient shear strength at the interface so that no slip
will occur during deformation. Such a composite model
for the case hardened tube element was also successfully
used in a recent numerical study of an analogous problem
[51]. This simple rule of mixture approach predicts well
the tensile stress–strain curve of a unidirectional fiber composite in the fiber direction, even beyond the elastic deformation region [52].
Since the case hardened layer had an average hardness of
about 9 GPa, Fig. 4a, (by using the usual practice for converting hardness to strength by dividing by 3) we take its
peak strength to be 3 GPa, and use an elasto–plastic stress–
strain response shown in Fig. 7a to represent it. The
Young’s modulus was taken to be 220 GPa; consistent with
nano-indentation measurements. It is noted that the ultimate
strength of the annealed alloy is about 1 000 MPa, Fig. 4a.
However, the measured hardness of the annealed condition
alloy at the center of the tube wall was about 3.7 GPa, corresponding to an ultimate strength about 1 233 MPa. We
attribute the higher strength to the indenter size effect; for
the low loads used in nano-indentation, the hardness ob8
Fig. 7. (a) Material models for the true stress–strain responses of the
case hardened layers. (b) Predicted (solid curve) stress vs strain responses of the nitro-carburized tubes with different wall thicknesses.
Experimental (dashed) compressive stress–strain responses were included for comparison.
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L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
strength (solid lines) and the measurements, up to the critical buckling stress of the samples. The analysis indicates
that in all cases, the buckling strength during tube compression was attained after the initiation of plastic yielding of
the case hardened metal. Careful examination of the simulated stress versus strain responses, Fig. 7b, reveals the presence of two slope changes (yield points), which were also
less clearly observed in the tube compression measurements. The first slope change corresponds to the inner annealed alloy annular ring yielding while the second resulted
from yield of the case hardened layers. While the lower
yield points on the compressive stress–strain curves are
consistent with the yield points measured during tensile
tests, Fig. 5, the subsequent tensile responses were different
from those in compression due to the influence of the brittle
case hardened layers. We note that since the post-buckling
behavior cannot be addressed by this simple model, the
compressive stress–strain curves for the post-buckling regime are incorrectly predicted.
Using the constitutive responses for the annealed and nitro-carburized tubes (in the composite cylinders approximation), Fig. 7a, with either the Engesser (Eq. (5)) or Shanley (Eq. (7)) predicted global inelastic buckling stress
enables the critical buckling stress of all the tubes to be predicted as a function of the tube wall thickness. These predictions for both end clamping conditions (k = 1 and 2) are
plotted against tube wall thickness and compared to the experimental data in Fig. 8. It is noted that as the single tube
ends were fixed during the experiments, the k = 2 prediction should be closest to the experimental end conditions.
Figure 8 also includes the local inelastic buckling predictions based upon Eq. (10) which converge with the local
elastic buckling stresses for the thinnest walled nitro-carburized tube.
In the annealed condition, Fig. 8a the measured buckling
strengths and modes shown in the photographs accompanying Fig. 6 are consistent with the local inelastic buckling
model predictions (using Kc = 0.35 which best fitted the
data) for the three tubes whose wall thicknesses were less
than about 140 lm. For the thickest wall tube, the buckling
stress predicted by the Shanley (k = 2) global inelastic buckling model was very slightly lower than that of a local inelastic buckle. Though this predicted stress better matched the
experimental data point, an annealed tube of this wall thickness is in the vicinity of the transition point between the two
failure mechanisms, and the buckling failure mode is therefore defined as a mixed mode, consistent with the buckling
mode evident in the photograph of Fig. 6a.
A similar analysis for the nitro-carburized tubes is shown
in Fig. 8b. The composite cylinder approximated stress versus strain response of the nitro-carburized tubes was first
obtained, and the local or global buckling strengths were
subsequently determined using corresponding buckling criteria (Eqs. (2), (5), (7), (9) and (10)). The Kc values used for
local buckling stress calculation of the nitro-carburized
tubes (those with walls thicker than 30 lm) were obtained
as the volumetric average of the three layers by assuming
Kc = 0.8 for the case hardened layer and 0.35 for the center
(annealed condition) alloy. Since the two case hardened
layers had a combined thickness of 30 lm, predictions for
tubes with a wall thickness of 30 lm or less, were assumed
to consist of only case hardened material and shading is
used to denote this region on Fig. 8b.
The hardness and ultimate strength of a case hardened
tube with wall thickness less than 30 lm were determined
from the hardness vs depth profiles from both the outer
and inner sides of the wall, Fig. 4. To simplify the analysis,
it was assumed that the case hardened material exhibited
the elasto–plastic response shown in Fig. 7a and the inelastic buckling stress of the tube then equaled its plastic yield
stress. The elastic buckling stresses were determined by
Eq. (9) assuming Kc = 0.8. The buckling model predictions
for the nitro-carburized tubes have been separated into two
regimes located above and below a wall thickness of
30 lm. Below a wall thickness of 30 lm, the fully nitro-carburized tubes are predicted to fail by local elastic buckling
at a stress that rises linearly with wall thickness to 3 GPa
at a thickness of 30 lm.
As the wall thickness was increased beyond 30 lm, the
nitro-carburized tubes were at first equally well predicted
to collapse by either inelastic local or global inelastic buckling. However, for the thickest tube wall sample, the lowest
strength buckling mode was inelastic global buckling con-
Fig. 8. Predicted (curves) and measured (open and black diamonds)
peak compressive strength of (a) annealed and (b) nitro-carburized
304 stainless steel tubes vs. wall thickness. In (b) the Shanley predictions are denoted by a blue dashed (k = 1) or solid (k = 2) line while
those of the Engesser models are shown in black.
9
L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
sistent with the photographic observations, Fig. 6b and the
measured critical stresses. The compressive strength of the
thickest wall tube was better predicted by the Engesser global inelastic buckling stress (k = 2) than by Shanley global
inelastic buckling stress (k = 2). The Engesser model predicts the critical stress when lateral deflection begins [33];
if imperfection or defects exist, the compressive stress
would begin to decrease; otherwise, the compressive stress
could continue to increase till the Shanley stress is approached [34].
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4. Lattice compression results
4.1. Mechanical properties
The lattice structures were tested in unconfined compression in the thickness (H) direction at ambient temperature
and a strain rate of 5 · 10–4 s–1. The measured load cell
force was used to calculate the nominal stress applied to
the face sheet area of the structures. While this provided a
good estimate for the stress supported by the square oriented lattice, it underestimates the real stress in the diamond-oriented samples, Fig. 1b, since only a part of the
sample top surface is connected by a strut to its bottom.
The effective area of the cellular structure in the diamond
oriented case was only 0.67 that of the face sheets. In the
experiments to follow, we report the apparent strength
using the full cross-sectional area (L W) of the sample to
compute the stress. The nominal through thickness strain
was measured using a laser extensometer to monitor the
displacement between the face sheets; the initial thickness
of the cellular structure within the face sheets was used as
the gage length. Photographs of the samples were taken periodically during testing to characterize the buckling modes
of both the annealed and nitro-carburized lattices.
Typical compressive nominal stress versus strain responses for the nitro-carburized lattice structures with
square and diamond orientations are shown in Figs. 9 and
Fig. 9. Comparison of the compressive responses of the square topology annealed and nitro-carburized lattice structures with relative densities of
(a) 3.0 %, (b) 6.0 %, (c) 7.3 %, and (d) 11.0 %.
10
L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
additional flow stress peaks before final rapid hardening
due to core densification (truss impingement) after a plastic
strain of 30 to 50 %. The initial compressive stiffness and
the peak strength of the annealed and nitro-carburized collinear lattices are summarized in Tables 3 and 4 respectively.
No apparent change in lattice stiffness was observed after
nitro-carburization treatment. However in all cases, the
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10. The compressive responses of annealed lattice counterparts are included for comparison. The nitro-carburized lattices exhibited characteristics typical of many cellular
metal structures [17], including a region of elastic response,
plastic yield followed by hardening, achievement of a peak
strength (here controlled by inelastic buckling) terminated
by a large drop in flow stress and followed by a series of
Fig. 10. Comparison of the compressive responses of the annealed and nitro-carburized diamond orientation lattice structures with relative densities of (a) 3.0 %, (b) 6.0 %, (c) 7.3 % and (d) 11.0 %.
Table 3. Stiffness and strength of annealed 304 stainless steel collinear lattice structures.
Relative
density
0.030
0.060
0.073
0.110
Square topology
Diamond topology
Stiffness
(GPa)
Peak strength
(MPa)
Stiffness
(GPa)
Peak strength
(MPa)
3.15
6.06
7.35
11.70
6.42
14.67
22.48
37.30
1.57
3.14
3.86
5.45
4.28
10.38
14.81
22.96
11
L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
Table 4. Stiffness and strength of nitro-carburized 304 stainless steel collinear lattice structures.
Relative
density
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0.030
0.060
0.073
0.110
Square topology
Diamond topology
Stiffness
(GPa)
Peak strength
(MPa)
Stiffness
(GPa)
Peak strength
(MPa)
3.28
6.80
7.83
12.05
23.65
28.52
35.70
42.75
1.59
4.08
4.51
5.88
16.20
22.39
25.17
28.00
compressive strength of the nitro-carburized lattices was
significantly higher than annealed lattice counterparts, and
the square lattice had a higher peak strength than its diamond counterpart of identical relative density.
4.2. Buckling mechanisms
A sequence of photographs showing the progressive collapse of square collinear lattices with a relative density of
3 % in both the annealed and nitro-carburized conditions
are shown in Fig. 11. In both cases the vertical tubes had a
free length of 15 mm and were bonded to the face sheets
through which the compressive load was applied with a
thick layer of adhesive. Four rows of horizontal tubes
crossed the vertical trusses, but two were embedded in the
adhesive bond with the face sheets. The unit cell length
(and width) was 5 mm.
Examination of Fig. 11a shows that the q ¼ 0:03 annealed square lattice (whose tube walls were 50 lm in
thickness) failed by cooperative local inelastic buckling of
the trusses parallel to the loading direction. This was initiated mid-way along the vertical tube trusses at a strain
of slightly less than 1.4 %, and had propagated across the
entire row of vertical struts when the strain had reached
5 %. The tubes collapsed by a local inelastic buckling mode
at a strain corresponding to the first peak in strength seen in
the stress versus strain response of this lattice, Fig. 9a. As
deformation progressed beyond the peak load to a strain of
10 %, the sample compression was accommodated by continued buckling in the center row of cells. Local inelastic
buckling then initiated near the center of the vertical trusses
in the bottom row of cells, and had again spread across the
entire sample once the strain reached 20 %. The onset of
this second cooperative buckling process coincided with
the second peak in the stress versus strain response, Fig. 9a.
A similar peak in strength, but at a strain of 30 % was observed when the third row of trusses buckled.
The deformation sequence of a square nitro-carburized
collinear lattice of the same relative density is shown in
Fig. 11b. Initiation of local inelastic buckling again coincided with the first peak in strength at a strain of 0.6 %,
Fig. 9a and usually midway along the vertical tubes (within
the mid-plane cell row). However, in this case, the location
of the buckling sites was near the nodal region where a horizontal tube crossed the sample. Like the annealed sample,
further compression was accommodated by compression
of the center row of cells until a strain of 15 % was reached,
whereupon the vertical trusses of either the top or bottom
row of cells began to buckle by a similar mode. As the
12
strain continued to increase towards 30 %, the deformation
of the newly buckled trusses caused the height of the originally horizontal tubes to be reduced causing an inclination of
the two center horizontal tubes. Some of the buckled regions also underwent fracture. Figure 11 shows that neither
the annealed or carburized square lattices exhibited a lateral
Poisson strain during compression.
Annealed square lattices with relative densities of 0.06
and 0.073 exhibited similar deformation characteristics to
the q ¼ 0:03 sample during compressive loading, Fig. 12a.
However, as the relative density was increased to 0.11, the
failure mode changed from local inelastic buckling to global buckling of the vertical tube struts, Fig. 13a. In Fig. 13a,
views of the side (angle I) and the front (angle II) of the
sample are shown at a strain of 11 % where the first peak
in strength was observed, Fig. 9d. Further straining to a
plastic strain of 25 % was accommodated by cooperative
rotation of the top and bottom vertical trusses, with an associated lateral displacement. No fracture of individual
trusses was observed in any of the annealed lattice structures.
As the relative density of the square nitro-carburized lattice increased to q ¼ 0:06 and then 0.073, a mixed mode
of buckling developed with both local and global inelastic
buckling components, Fig. 12b. The peak in strength,
Fig. 9b and c, coincided with global inelastic buckling of
the vertical tubes and was followed by local inelastic buckling of the vertical trusses as deformation progressed. The
mechanism of collapse in the q ¼ 0:11 nitro-carburized
square lattice was global plastic buckling, Fig. 13b of the
vertical tube struts, and was very similar to that seen in the
annealed sample. However, the peak in strength of the nitro-carburized lattices occurred at a lower strain than annealed counterparts, Fig. 9d. No fracture of individual
trusses was observed in the square nitro-carburized lattices
with relative densities of 0.06, 0.073 and 0.11.
Photographs of diamond lattice structures with a relative
density of 0.03 taken during compressive loading are shown
in Fig. 14a at several strains. Analogous results for a density of 0.06 are shown in Fig. 12c. The annealed diamond
lattices with q ¼ 0.03, 0.06 and 0.073 all failed by inelastic
local buckling of individual trusses between nodes at the
first peak in strength, Fig. 10a, b and c. As deformation progressed, locally buckled trusses continued to buckle until a
zigzag layer of diamond unit cells collapsed and trusses of
adjacent layers made contact. This was accompanied by a
rapid drop in flow stress. An adjacent layer of diamond unit
cells then began to buckle and was accompanied by a second peak in strength and rapid drop in flow stress after-
L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
The peak compressive strengths of nitro-carburized and
annealed lattice structures are shown in Fig. 16a for the
square lattices and for the diamond orientations in Fig. 16b.
It is clear that the nitro-carburzied process has significantly
increased the lattice peak strength. The ratio of the nitrocarburized to annealed peak strengths (strength improvement factor), is also plotted on Fig. 16a and b. For thinnest
wall thickness lattices, almost four-fold increases in
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wards. Such a process was repeated until the lattice had
completely collapsed. The annealed lattice core with
q ¼ 0:11 deformed in a similar way, but the individual
trusses failed by mixed mode of local and global buckling,
Fig. 15a. The nitro-carburized diamond lattice cores exhibited similar deformation characteristics as their annealed
lattice counterparts, but peak in strength occurred at a smaller strain.
Fig. 11. Photographs showing the deformation characteristics of q = 3.0 % annealed (a) and nitro-carburized (b) square orientation lattice structures at different plastic strains.
13
L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
strength were observed. These figures also show that the
strength ratio decreased with increasing relative density; a
consequence of the decreasing cross sectional area fraction
of the case hardened layer.
5. Lattice buckling analysis
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5.1. Lattice strength
The lattice strengths can be predicted by analyzing the unit
cell shown in Fig. 1c. For the unit cell of a square lattice,
Eq. (1) was used to predict the strengths, where rys was
taken as either the yield strength of the solid material for
the plastic yielding failure mechanism, or replaced by
Eqs. (2), (5), (7), (9) and (10), for the corresponding local
and global buckling mechanisms. Since the single vertical
tube properties have already been analyzed in Section 3.2,
the square lattice responses can be simply obtained by multiplying the single vertical tube properties, the lattice relative density and the topology dependent strength coefficient, R = 0.5 for the square lattice.
For a diamond lattice unit cell, the tube elements have an
inclined angle, x = 458. When an end clamped inclined
tube is placed under compression loading, the bending moments at the ends introduce shear forces, and the total force,
F applied along the vertical direction is given by [55]:
1
F ¼ FA sin x þ FS cos x ¼ pffiffiffi ðFA þ FS Þ
2
ð12Þ
12I
where FS is shear force at the ends given by FS ¼ FA 2 .
Al
The diamond lattice unit cell compressive strength is
therefore;
FA
FS
FA
3p 4
4
rc ¼
1þ
1þ
¼
D
d
4ao l
FA
4ao l
16l2
1
3p 4
4
¼ qrA 1 þ
ð13Þ
D
d
2
16l2
where rA is the axial collapse strength of a single tube defined by Eqs. (2), (5), (7), (9) and (10), for the corresponding local or global buckling mechanisms. Equation 13 can
be used to predict the compressive strength of a diamond
lattice without an edge effect. However, it is important to
recall that the diamond lattice test samples had a low L/H
(aspect) ratio which reduced their apparent strength by a
factor of 0.67. The stress predicted by Eq. (13) is therefore
multiplied by 0.67 to make comparisons with the experimental measurements. The compressive strength of the diamond lattice test samples is therefore given by:
3p 4
4
qrA
rc ¼ 0:33 1 þ
ð14aÞ
D
d
16l2
It is noted here that Eq. (14a) was obtained by assuming
clamped (k = 2) ends; for pin-jointed inclined tubes (k =
1), bending moment at the ends is negligible, and the predicted compressive strength of the diamond lattice test samples would then be given by:
rc ¼ 0:33 qrA
ð14bÞ
Fig. 12. Photographs showing the deformation characteristics of q = 6.0 % annealed ((a) and (c)) and nitro-carburzied ((b) and (d)) collinear lattice
structures at a plastic strain of 15 %.
14
L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
5.2. Comparison of predictions with measurements
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The predicted strength (normalized by the product of the
annealed alloy yield strength and colinear lattice relative
density) is plotted in Fig. 17a and c for the normalized lat-
tice. The predicted strengths have been reduced by the aspect ratio dependent factor (0.67) for the diamond oriented
test samples used here. The plastic yielding mechanism predicts a structural efficiency parameter that is independent of
relative density, and is shown by a broken line in both fig-
Fig. 13. Photographs showing the deformation characteristics of q = 11.0 % annealed and nitro-carburized square orientation lattice structures at
different plastic strains.
15
L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
elastic buckling. As the relative density increases local inelastic buckling can occur (when q > 0:02). As the relative
density increased, the predicted local inelastic buckling
strength eventually exceeds the Engesser and Shanley
(k = 2) predicted inelastic buckling strengths.
The measured compressive strength coefficients obtained by normalizing the (finite aspect ratio) sample compressive strength by the product of the annealed stainless
steels yield strength and colinear lattice relative density
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ures. The critical stress to activate the local buckling mechanisms is denoted by red lines in the two figures. It can
be seen that the strength predicted by the Engesser and by
the Shanley (global) inelastic buckling mechanisms significantly exceeded the plastic yielding predicted strength. The
predicted global inelastic buckling strength of cellular
structures with built-in (k = 2) nodes also significantly exceeded that of the pin-jointed case. At low relative densities, q < 0:02, core collapse is predicted to occur by local
Fig. 14. Photographs showing the deformation characteristics of q = 3.0 % annealed and nitro-carburized diamond orientation lattice structures at
different plastic strains.
16
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L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
are summarized in Table 5, and are overlaid on the predictions shown in Fig. 17a and c. The normalized strength of
the annealed lattices increased from 0.6 to 1.4 as their relative density was increased from 0.03 to 0.11. In all cases
these measured strengths exceeded the predicted strengths
for plastic yielding consistent with the observation of a
yield point in the stress–strain responses before attainment
of the peak strength, Figs. 9 and 10. At low relative densities (
q < 0:08) the peak strengths of both lattice oreintations are consistent with model predictions for failure by local inelastic buckling of the trusses, as observed during the
q > 0:08; they are close
tests. For relative densities above to the Engesser model predictions for global inelastic buckling with a built-in end node condition. These predicted
mechanisms correlate well with the optical observations of
the buckling failure mechanisms, Figs. 11 and 12. Their
normalized strengths increased with relative density for
the samples studied here (with fixed strut lengths) because
the increase in density was achieved by an increase in tube
wall thickness which increased the tubes resistance to buckling more rapidly than its increase of lattice density.
Note that examination of Fig. 17a and c appears to indicate that the diamond lattice structures have lower strength
coefficients than square lattice counterparts. However, this
is a consequence of the sample (aspect ratio dependent)
knock-down factor of 0.67 that was applied to the predictions of the diamond lattices. It also evident that the measured diamond lattice strengths are slightly lower than the
model predictions for q > 0:03. However, the global inelastic buckling (k = 2 case) and the local buckling model predictions presented in Fig. 17c were obtained by assuming
the inclined tube struts have a built-in end condition, and
transverse displacement was constrained at both ends.
However, the actual compression tests were performed
without lateral confinement, and small transverse displacements of the nodes occurred during deformation. In addition, the adhesive nodes have a rotational resistance that
lay between k = 1 and 2. The bending moments at the strut
ends are therefore lower than the model predictions when
k = 2 was assumed.
The predicted strength coefficients of the nitro-carburized collinear lattices (again normalized by the product of
the annealed alloy yield strength and colinear lattice relative density) and reduced by a factor of 0.67 for the diamond oriented samples, are shown in Fig. 17b and d. The
predictions of the models have been seperated into two
parts with q ¼ 0:018 as the boundary (which corresponds
to a truss tube wall thickness of 30 lm). When q < 0:018,
Fig. 15. Photographs showing the deformation characteristics of q = 11.0 % annealed and nitro-carburized diamond orientation lattice structures at
different plastic strains.
17
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L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
the individual trusses are fully nitro-carburized, and the
stress–strain relationship was modeled as elasto–plastic,
Fig. 7a, with a yield strength of 3 GPa. The inelastic global
buckling stress then coincided with the plastic yielding
stress. However, in this regime, the local elastic buckling
stress is predicted to dominate the response at low density.
The stress to activate this increased with relative density
until it reached the global inelastic buckling stress at
q 0:016.
q > 0:018, the compressive stress–strain curves of
For individual tubes were modeled using the Voight approximation and the mechanical properties are summarized in
Fig. 7. The local inelastic buckling model, as well as the
Engesser and Shanley inelastic global buckling models predicted nonlinear decreases of the critical strength with increasing relative density. The local inelastic buckling model predictions are lower than the Engesser and Shanley
inelastic buckling model predictions (k = 2), until q approached 0.07; thereafter the local buckling model predictions began to increase in a nonlinear manner with increasing relative density, and exceeded the Shanley inelastic
buckling (k = 2) predictions at q 0:09.
The sample test data have been overlaid on the predictions. After the nitro-carburization treatement, the measured compressive strength coefficients increased by a factor of between 2 and 4; increasing from 1.1 to 3.2 as their
relative density was decreased, (i. e. with increasing tube
wall thickness), Fig. 17b and c. At low relative density, they
were substantially stronger than their annealed counterparts. However, this difference decreased as the tube wall
thickness increased. The measured compressive strength
coefficients agree well with the model predictions, and the
optically observed failure mechanisms. Lattices failed by
local inelastic buckling of trusses at low relative density
(q ¼ 0:03), and a mixed mode of local and global inelastic
buckling of truss members at intermediate relative densities
(q = 0.06 and 0.073). At a relative density q ¼ 0:11 collapse was by global inelastic buckling of trusses. Again,
the nodal rotational resistance of the tested samples lay
somewhere between k = 1 and 2, consistent with the observed constrained node rotation. The rapid rise in lattice
strength as the tube wall thickness was decreased is a direct
result of the increase in effective strength of the nitro-carburized tube walls which delayed the onset of the inelstic
buckling operative in the samples tested here.
6. Concluding remarks
Fig. 16. Peak compressive strength of the nitro-carburized and annealed lattice structures as a function of relative density. Model predictions of peak compressive strength are provided for comparison. Peak
strength improvement factors of nitro-carburized lattices over the annealed lattice counterparts are also summarized.
18
Collinear lattice structures with 304 stainless steel hollow
trusses were fabricated with an alternating collinear lay-up
process and metallurgically bonded by a vacuum brazing
method. The dimensions of the cylinders, the wall thickness
of hollow trusses and spacing between the trusses enabled
independent control of the cell size and the relative density
of the structure. Square and diamond topologies with relative densities from 0.03 to 0.11 were manufactured. Low
temperature nitro-carburization treatment was then performed on the collinear lattice cores at a temperature of
440 8C for 20 hrs. The nitro-carburization treatment created
a thin but extremely hard surface layer on the truss members, which significantly improved the collapse strength of
the individual trusses, and enabled a compressive strength
enhancement by a factor ranging from 1.2 to 3.8 compared
with the annealed lattice counterparts. Inelastic buckling
of the trusses was the dominant failure mode; however,
such buckling might be delayed by filling tubes with lightweight polymer or metal foams: though it is unclear if this
additional mass might be better deployed by increasing the
tube wall thickness.
The compressive strength of the annealed and nitro-carburized lattices are compared with other types of stainless
steel cellular structures such as woven textiles [28], pyramidal lattices [19, 20] and square honeycomb cores [14], in
Fig. 18. The predicted diamond lattice strengths in this case
have not been multiplied by the sample aspect ratio dependent factor of 0.67 so they represent the response of large
aspect ratio (wide) samples in which almost all the struts
are in contact with both the upper and lower face sheets.
The experimental data were therefore scaled by 1/0.67 to
represent the wide sample strength limit. Examination of
L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
The nitro-carburized square and diamond lattices exibited significantly improved strength coefficients over the
annealed lattice counterparts. At low relative density where
the wall thickness becomes comparable to the depth of the
case hardened layers, they are superior to all other topolo-
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the data shows that the annealed lattices are substantially
stronger than woven textiles and their strength factors (R),
and are very competitive with the pyramidal and square
honeycomb structures made from annealed 304 stainless
steel.
Fig. 17. Predicted and measured peak compressive strength coefficients of collinear lattice structures vs. relative density, (a) square annealed lattice, (b) square nitro-carburized lattice, (c) diamond annealed lattice and (d) diamond nitro-carburized lattice. Note the difference in scales between
the separate figures.
Table 5. Comparison of the compressive strength coefficients of nitro-carburized collinear lattices made from 304 stainless steel hollow
tubes with the annealed lattice counterparts.
Compressive strength coefficient (R)
Relative
density
Nitro-carburized
square
Annealed
square
Nitro-carburized
diamond
Annealed
diamond
0.030
0.060
0.073
0.110
3.28
1.98
2.04
1.62
0.89
1.02
1.28
1.41
2.25
1.55
1.44
1.06
0.59
0.72
0.85
0.87
19
L. Dong et al.: Strong cellular lattices with nitro-carburized stainless steel hollow trusses
International Journal of Materials Research downloaded from www.hanser-elibrary.com by Carl Hanser Verlag on October 28, 2015
For personal use only.
gies. For example, at a relative density of 0.03, the nitrocarburized hollow truss structures are 6 times stronger than
a square honeycomb and 7 time stronger than a pyramidal
truss structure constructed from annealed 304 stainless
steel. Low relative density nitro-carburized hollow truss
lattice structures made by the alternating collinear lay up
process appear to exhibit a higher specific compressive
strength than any other cellular metal topologies reported
to date. It is noted that other topologies reported in Fig. 18
could, in principle also be nitro-carburized and their
strength improved. It is also noted that the superior corrosion and fatigue resistance of the nitro-carburized stainless
steel may have beneficial consequences for some applica-
Fig. 18. (a) The compressive strength vs. density of various 304 stainless steel cellular structures. (b) Comparison of the peak compressive
strength coefficients of nitro-carburized hollow truss lattices with annealed lattice counterparts and other types of 304 stainless steel cellular structure topologies as a function of the relative density.
20
tions. The nitro-carburized 304 stainless steel collinear lattices therefore appear to be promising candidates for lightweight sandwich cores intended for elevated temperature
and multifunctional applications.
We are grateful to the DARPA MCMA program (Grant Number
W91CRB-10-1-005) for the financial support of this work. The program was overseen by Dr Judah Goldwasser of DARPA.
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(Received June 17, 2015; accepted September 3, 2015)
Correspondence address
Liang Dong
Department of Materials Science and Engineering
University of Virginia
Charlottesville, VA, 22903
USA
Tel.: +1-434-982-5678
E-mail: [email protected]
Bibliography
DOI 10.3139/146.111312
Int. J. Mater. Res. (formerly Z. Metallkd.)
106 (2015) E; page 1 – 21
# Carl Hanser Verlag GmbH & Co. KG
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Appendix
Modified Ramberg–Osgood fitting
The true tensile stress–strain curve has been separated in
two parts and fitted separately.
Annealed condition
(1) r < r0:2 , n = 28, E0 = 182 GPa, r0:2 = 240 MPa, Ramberg–Osgood fitting
n
r
r
e¼
þ 0:002
E0
r0:2
(2) r > r0:2 , a modified Ramberg–Osgood fitting [53] is
applied by moving both vertical and horizontal axes from
the origin to a selected point which enables a smooth transition between the two fitting curves.
e1 ¼ 0:0025, r1 ¼ 235:6 MPa, E1 = 6.745 GPa, n = 2.05,
eu ¼ 0:365, ru = 1 000 MPa, a modified Ramberg–Osgood
fitting
r r1
ru r1
r r1 n
þ eu e1 e e1 ¼
E1
E1
ru r1
Case hardened layer stress–strain
(1) r < r0:2 , n = 3.5, E0 = 220 GPa, r0:2 = 1 500 MPa,
Ramberg–Osgood fitting
n
r
r
þ 0:002
e¼
E0
r0:2
(2) r > r0:2 , e1 ¼ 0:01, r1 ¼ 1851 MPa, E1 = 1.41 GPa,
n ¼ 15, eu ¼ 0:38, ru ¼ 3000 MPa, a modified Ramberg–
Osgood fitting
r r1
ru r1
r r1 n
þ eu e1 e e1 ¼
E1
E1
ru r1
21