COMMUNICATION
DOI: 10.1002/adem.201000145
Titanium Alloy Lattice Structures with Millimeter Scale Cell
Sizes**
By Pimsiree Moongkhamklang and Haydn N. G. Wadley*
Titanium sandwich panels with cellular cores of a uniform 1–5 mm diameter open cell size are well
suited for impact energy absorption and cross flow heat exchange applications. Periodic cellular
structures (lattices) made from high specific strength, high temperature alloys are preferred for these
multifunctional uses. A diffusion bonding method has been applied here to make cellular lattice
structures from a Ti–6A1–4V alloy. To illustrate the approach, lattice structures with both square and
diamond collinear topologies, a 2 mm open cell size, and a relative density of 15% were made from
254 mm diameter titanium alloy wires. These structures were found to have a compressive strength of
40 5 MPa that was controlled by plastic yield followed by buckling of the struts. The cellular
structures have been brazed to titanium alloy face sheets to create sandwich panel structures that
appear well suited for multifunctional applications up to 420 8C.
Cellular metals with stochastic cell structures (foams)[1] and
related materials with periodic (lattice) cell topologies[2] have
attracted significant interest because of their multifunctionality.[3] When configured as the core of a sandwich panel, they
provide efficient structural load support during bending[4–6]
while also enabling other functions such as cross flow heat
exchange,[7–9] acoustic damping,[10,11] and impact energy
absorption.[12–14]
The earliest foams were made from easily cast alloys based
upon aluminum and copper[15,16] but recent developments are
leading to the emergence of materials made from steels and
other higher strength alloys.[17–19] While foam structures with
either open or closed cell topologies (and occasionally with
both) are relatively easily made with a small (millimeter and
even submillimeter) cell size,[20] they deform by strut bending
and are therefore weak.[21] Periodic structures with lattice or
honeycomb cores have been shown to offer much higher
specific strengths because strut stretching dominates their
response to stress.[22,23]
Deshpande and Fleck[24] have shown that the specific
strength of a stretch dominated lattice structure usually
[*] Prof. H. N. G. Wadley, P. Moongkhamklang
Department of Materials Science and
Engineering, University of Virginia
Charlottesville, VA 22904-4745, USA
E-mail: [email protected]
[**] Acknowledgements, The study was supported by the Office of
Navel Research (ONR) and monitored by Dr. David Shifler
under grant number N00014-07-1-0114.
ADVANCED ENGINEERING MATERIALS 2010, 12, No. 11
depends upon the specific strength of the material used to
make it, the relative density and a geometric factor that
measures the efficiency with which the lattice supports
stress. Triangulated structures with the (tetrahedral) octet
truss structure[25] (the pyramidal) lattice block arrangement
of trusses,[26] and 3D kagome structures[27] are the most
efficient topologies for load support. They can be fabricated
by investment casting[28–30] and by sheet stamping/forming
combined with either spot welding,[31] brazing,[32] or
bonding.[33] However, it is difficult to fabricate low
density versions of these structures with cell diameters
below 10 mm.
Small cell sizes are of interest for compact heat exchange
and for impact mitigation where they then provide a more
uniform deformation response. Small cell size periodic
structures can be made from metal wires by laying up 2D
weaves or 0/90 collinear wire arrays and brazing.[34] The
mechanical response of structures made this way from
stainless steels have been analyzed in compression, shear,
and in bending[5] and shown to provide only slightly reduced
strengths compared to optimal topologies. The best of these
stainless steel structures give compressive strengths of
10–20 MPa at a density of 0.5 Mg m3.
Here, we describe a diffusion bonding process that has
been developed for making small cell size collinear lattices
from Ti–6Al–4V wires of much higher specific strength than
stainless steel. It is similar to a method recently proposed for
the making of cellular materials from titanium coated SiC
monofilaments.[35] We show that the compressive strength of
these titanium structures is double that of equivalent steel
structures making them well suited for situations where
ß 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
wileyonlinelibrary.com
1111
COMMUNICATION
P. Moongkhamklang and H. N. G. Wadley/Titanium Alloy Lattice Structures with Millimeter Scale . . .
efficient load support, impact mitigation, or heat exchange are
required at temperatures up to 420 8C.
Experimental
Collinear lattices with square and diamond orientations
were fabricated from 254 mm diameter Ti–6Al–4V wires (ATI
Wah Chang, Frackville, PA). The Ti–6Al–4V wires were
stacked in an orthogonal pattern within a wire alignment tool,
Figure 1, which had been spray coated with boron nitride (GE
Advanced Ceramics, Lakewood, OH) to prevent sticking.
Alignment was achieved by a set of 1.59 mm diameter
uniformly spaced stainless steel pins inserted into holes in
a stainless steel base plate. Since the ‘‘as received’’ wires had
significant curvature, five pairs of alignment pins were used to
align each wire. The spacing between each pair of the pins
defined the center-to-center truss (unit cell) spacing
l ¼ 2.03 mm resulting in an assembled (but not bonded) lattice
relative density, r0 ¼ 9.8%.
Once assembly of the Ti–6Al–4V wire lay-up was complete,
a dead weight was used to apply a force of 7.5 g force
(7.4 102 N) to each truss–truss node contact. The assembly
was then diffusion bonded by placing it in a vacuum furnace
at a base pressure of 107 Torr and heating to 900 8C for 6 h.
The lattice structures were removed from the tooling and cut
to shape using wire electro-discharge machining. Square
topology lattice samples were machined so that their struts
were aligned at 0 and 908 to the rectangular sample surfaces.
The samples were four cells high and six cells in length with a
width to height ratio, W/H ¼ 1. Diamond topology lattices
were rotated by 458 and cut to an aspect ratio L/H ¼ 5, with
three cells in the core height direction and a W/H ¼ 1.
Fig. 1. Schematic showing assembly sequence to make a Ti–6Al–4V lattice.
1112
http://www.aem-journal.com
The relative density of the diffusion bonded collinear
lattice structure, r, is the ratio of volumes of the solid material
within a unit cell of structure to that of the unit cell. It is given
by [35]:
a p
r¼
;
(1)
2b sinð2vÞ l
where a and l are the wire radius and the inter-wire spacing
(i.e., the unit cell length), b is the diffusion bonding
coefficient defined as W/W0, where W and W0 are the lattice
widths prior to and after the diffusion bonding process, and
v is the half angle of the truss orthogonal pattern (458 for both
the square and diamond lattices).[35] The measured diffusion
bonding coefficient, b, of the Ti–6Al–4V lattice was 0.64 and
so after diffusion bonding, the lattice had a relative density
r ¼ 15.3%.
Sandwich panel test structures were made by vacuum
brazing the Ti–6Al–4V lattices to 2 mm thick Ti–6Al–4V face
sheets using TiCuNi-601 paste braze alloy (Lucas-Milhaupt
Inc., Cudahy, WI). The assemblies were first heated at
20 8C min1 to 550 8C, and held for 5 min (to volatize and
remove the polymer binder), and then heated to 975 8C for
30 min at a base pressure of 107 Torr. Photographs of the
sandwich test structures with both the square and diamond
lattice cores are shown in Figure 2. The images reveal a small
misalignment of the trusses (and the truss–truss nodes) that
resulted from truss wire waviness.
Fig. 2. Examples of the as-manufactured Ti–6Al–4V specimens used for compression
testing (r ¼ 15.3%): front view of (a) square and (b) diamond, and side view of (c) square
and (d) diamond lattice core sandwich structures. The small misalignments of the lattice
resulted from curvature of the wire.
ß 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ADVANCED ENGINEERING MATERIALS 2010, 12, No. 11
P. Moongkhamklang and H. N. G. Wadley/Titanium Alloy Lattice Structures with Millimeter Scale . . .
Results and Discussion
The through thickness, compressive stress–strain responses
for square, and diamond lattice structures with r ¼ 15.3%
(ro ¼ 9.8%, b 0.64) are shown in Figure 3(a) and (b),
respectively. Both lattices exhibited characteristics typical of
many cellular metal structures, including a region of elastic
response, plastic yielding (over a small strain range), a peak in
the strength (at 15–20% strain range), a large post-peak stress
drop followed by a plateau region, and finally hardening
associated with densification. Unload/reload tests confirmed
that plastic yielding had occurred prior to attainment of the
peak in compressive stress.
Images of Ti–6Al–4V lattice truss structures at different
level of compression are shown in Figure 4. As the square
lattice was compressed, the trusses parallel to the loading
direction plastically yielded and then cooperatively buckled
near their midpoint. Truss buckling was only observed after
attainment of the peak stress. The buckling half-wavelength
corresponded with the total core height. In the diamond
lattice, the trusses also yielded and then buckled, but
not always in the same direction. Some trusses buckled in
the 1-3 plane while some buckled in the 1–2 plane. The
buckling half-wavelength varied from l to 6l throughout
the sample.
The Young’s modulus and compressive strength of square
and diamond lattices are plotted against r in Figure 5. Both the
stiffness and strength of the square lattices were higher than
those of the diamond lattices. Approximate analytical
expressions for the through thickness compressive stiffness
and strength of a collinear and a textile lattice made from an
elastic ideally plastic, monolithic solid material have been
previously developed.[34,36,37]
For a diamond collinear lattice made of straight struts, the
out-of-plane compressive stiffness is given by:[37]
1
H
1
Es r;
Ec ¼
(2)
4
L
where L and H are the length and height of the lattice core, Es
and r are the Young’s modulus of the Ti–6Al–4V wire and the
relative density of the diamond core, respectively. Queheillalt
et al.[34] have shown that the waviness in textile lattices results
in a knockdown in both stiffness and strength compared to the
collinear lattices. Since the trusses in the Ti–6Al–4V lattices
were also wavy, it is more appropriate to model their
compressive stiffness and strength with the models developed
for the textile lattices. The truss waviness reduces its axial
stiffness, and the effective modulus, Ew, of a wavy strut with a
wave amplitude equal to the wire diameter can be written:[34]
Ew ¼
1
1:09 þ 8ð1 þ nÞða=lÞ2
(3)
where n, a, and l are the Poisson’s ratio of the wire material, the
wire radius and the lattice unit cell length, respectively.
Substituting n ¼ 0.33, a ¼ 127 mm, and l ¼ 2.03 mm in
Equation 3 gives Ew/Es ¼ 0.88. The compressive stiffness of
a lattice made of wavy struts can then be
estimated by replacing Es by Ew in Equation 2.
Figure 5(b) shows that experimental data is
well approximately by the wavy textile
estimate. The relevant dimensional parameters and material properties used in the
models are summarized in Table 1.
The yield strength of a diamond collinear
lattice with straight struts is given by:[37]
1
H
sc ¼
(4)
1
syr
2
L
Fig. 3. Compressive stress–strain response for Ti–6Al–4V (a) square and (b) diamond lattices with r ¼ 15.3%.
The inserts show the stress–strain response at small values of strain. Plastic yielding was observed prior to the
peak stress in both cases.
ADVANCED ENGINEERING MATERIALS 2010, 12, No. 11
Es
where sy is the yield strength of the
constituent trusses (assumed straight). For
a wavy truss, the load carrying capacity is
again reduced and the collapse strength, sw,
of a wavy strut with wave amplitude equal to
ß 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
http://www.aem-journal.com
1113
COMMUNICATION
The lattice structures were tested in compression at
ambient temperature at a nominal strain rate of 2 104 s1.
The measured load cell force was used to calculate the
nominal stress applied to the sandwich, and the nominal
through thickness strain was obtained from a laser
extensometer positioned about the sample centerline. The
gage length for the strain measurement was the core
height, H.
To characterize the solid materials mechanical properties,
Ti–6Al–4V wires were subjected to the same thermal cycles
used to fabricate the Ti–6Al–4V sandwich structures and
then tested in uniaxial tension at room temperature at a
strain rate of 2 104 s1. These Ti–6Al–4V alloy wires
could be adequately approximated as an elastic–plastic solid
with Young’s modulus, Es ¼ 108 3.4 GPa and an average
0.2% offset yield strength sy ¼ 956 46.2 MPa. Two strain
regions of linear hardening modulus (the tangent modulus
Et ds/de) were observed after yield. In region I,
0.01 e 0.03 (where e ¼ 0.01 was the yield strain, ey),
Et,I ¼ 6.32 0.2 GPa. In region II, 0.03 e 0.1 (where
e ¼ 0.10 was the strain at tensile failure, " ),
Et,II ¼ 1.53 0.1 GPa. The ultimate tensile strength had an
average value sUTS ¼ 1179 48 MPa.
COMMUNICATION
P. Moongkhamklang and H. N. G. Wadley/Titanium Alloy Lattice Structures with Millimeter Scale . . .
where leb and k are the length of the column
and the column end constraint, respectively.
The lowest strength buckling mode of the
metallic lattice under uniaxial compression
corresponds to struts of length l buckling as a
pin joint strut. This gives leb ¼ l and k ¼ 1.
The compressive strength of the lattice is then
estimated by replacing sy by seb in
Equation 4. The strut waviness effect has
not been explored in detail. However, if the
wires are assumed to buckle as pinned-end
struts in the 1–3 plane, the waviness in the
1–2 plane is expected to have only a minor
effect upon the elastic buckling strength of
the textile lattice.[34]
For the square collinear lattice with
straight trusses, the out-of-plane compressive stiffness and yield strength are given
by:[34,39]
1
Ec ¼ Es r
2
(7)
and
1
s c ¼ s y r;
2
(8)
respectively. When the lattice collapses
by elastic cooperative buckling of the
constituent struts, Figure 4(a), the lattice
compression strength is given by:[40]
3
2
62
sc ¼ 4 2 þ
n
3
p2 1 4br
p
7 2
3
3 þ5b Es r
(9)
where n ¼ number of unit cells between the
faces. The compressive modulus and plastic
yield strength of the textile square lattice can
be estimated by replacing Es and sy by Ew and
sw, respectively, in Equations 7 and 8.
Figure 5(a) shows that the experimental data
Fig. 4. Sequence of photographs showing the progressive deformation of: (a) square lattice truss core;
for the compressive stiffness of square lattices
(b) diamond lattice truss core, made with Ti–6Al–4V wires during tests in out-of-plane compression. The
strains at which photographs were taken are labeled.
is in good agreement with predictions for the
textile lattices. The compressive strengths of
both the square and diamond lattices,
a fiber diameter (2a) is given by:[34]
Figure 5(c) and (d), are slightly overpredicted. The models
assume that the trusses buckle in the 1–3 plane so that the strut
s w ¼ 0:782s y :
(5)
waviness (in the 1–2 plane) effect is negligible. However, truss
buckling was observed in both the 1–3 and 1–2 planes.
The plastic yield strength of the wavy truss structure can
Therefore, the truss waviness may also result in a reduction of
then be estimated by replacing sy by sw in Equation 4.
the truss buckling strength, seb, and hence a reduction in the
The lattice collapses by elastic buckling of the constituent
lattice compressive strength.
struts if the Euler buckling strength, seb, of the constituent
The compressive strength of the Ti–6Al–4V lattices in both
struts is less than or equal to their plastic yield strength.
the square and diamond topologies is between 35 and 45 MPa
The Euler buckling strength of a cylindrical solid column is
and is predicted to be controlled by plastic yield consistent
given by:[38]
with observations of non-recoverable strains near the peak in
k2 p2 Es a2
strength. In this regime, the lattice strength is predicted to
s eb ¼
;
(6)
4ðleb Þ2
scale linearly with the yield strength of the wires and the
1114
http://www.aem-journal.com
ß 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ADVANCED ENGINEERING MATERIALS 2010, 12, No. 11
P. Moongkhamklang and H. N. G. Wadley/Titanium Alloy Lattice Structures with Millimeter Scale . . .
Conclusions
– A method for making titanium sandwich panels with millimeter-scale
titanium lattice cores has been developed. The method involves laying up
linear arrays of titanium alloy wires,
alternating the direction of consecutive
layers, and using diffusion bonding to
join the wires at the nodes where they
intersect. The relative density of the
lattice was controlled by the diameter
Fig. 5. Experimental data and analytical predictions for the Young’s modulus of (a) Ti–6Al–4V square and
of the wires, the spacing between the
(b) diamond lattices and the compressive strength of (c) square and (d) diamond lattices. The solid line marks the
wires in each collinear layer (cell
dominant failure mode as a function of the lattice relative density.
length), and the degree of diffusion
bonding.
– The out-of-plane modulus of square Ti–6Al–4V lattice
lattice relative density r. The models predict that elastic
structures scaled linearly with relative density r. The
buckling controls the strength when the relative densities
compressive peak strength was controlled by plastic
becomes less than 13%. When lattices fail by elastic truss
yielding and varied also with r.
buckling, the lattice compressive strength decreases rapidly
– The out-of-plane modulus of diamond TMC lattice strucwith r3 .
tures was about half that of the square lattice and also
The compressive strength of these Ti–6Al–4V textile lattices
scaled linearly with r. The compressive strength of the
is superior to that of the Ti–6Al–4V lattice block structure
diamond lattice was controlled by plastic yielding where
(31 MPa) of approximately the same relative density[30]
it is predicted to scale linearly with r.
– These titanium lattices may provide interesting high
Table 1. Lattice truss parameters used in the micromechanical models for predictions of
temperature multifunctional opportunities.
the lattice compressive stiffness and strength.
Geometrical parameters/mechanical properties
Symbol
Value
Radius of the Ti–6Al–4V wire truss [mm]
Lattice unit cell length [mm]
Aspect ratio of the diamond lattice
Angle of the trusses to the face sheets
in the diamond lattice [8]
Number of unit cells between the faces
of the square lattices
Diffusion bonding coefficient
Young’s modulus of the Ti–6Al–4V wire [GPa]
Poisson’s ratio of the Ti–6Al–4V alloy[a]
Yield strength of the Ti–6Al–4V wire [MPa]
a
l
L/H
v
127
2.03
5
45
n
4
b
Es
n
sy
0.64
108
0.33
956
[a] Typical value.[41]
ADVANCED ENGINEERING MATERIALS 2010, 12, No. 11
Received: April 19, 2010
Final Version: June 21, 2010
Published online: September 13, 2010
[1] H. N. G. Wadley, G. Gilmer, W. Barker, MRS Symp. Proc.
2000, 616.
[2] H. N. G. Wadley, Philos. Trans. R Soc. A 2006, 364, 31.
[3] A. G. Evans, J. W. Hutchinson, N. A. Fleck, M. F. Ashby,
H. N. G. Wadley, Prog. Mater. Sci. 2001, 46, 309.
[4] H. G. Allen, Analysis and Design of Structural Sandwich
Panels, Pergamon Press, Oxford 1969.
ß 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
http://www.aem-journal.com
1115
COMMUNICATION
and are comparable to Ti–6Al–2Sn–4Zr–2Mo
lattice block structures (40 MPa).[29] At a
given lattice topology and relative density,
the titanium lattice is approximately two
times weaker than the lattice made of
titanium coated SiC fibers[40] but 15–20 times
less costly.
Titanium alloy lattices with a relative
density that results in plastic yield controlled collapse are expected to retain good
dimensional stability during loading at
temperatures up to 420 8C.[41] They therefore provide interesting high temperature
multifunctional opportunities.
COMMUNICATION
P. Moongkhamklang and H. N. G. Wadley/Titanium Alloy Lattice Structures with Millimeter Scale . . .
[5] H. N. G. Wadley, N. A. Fleck, A. G. Evans, Compos. Sci.
Technol. 2003, 63, 2331.
[6] F. W. Zok, S. A. Waltner, Z. Wei, H. J. Rathbun, R. M.
McMeeking, A. G. Evans, Int. J. Solids Struct. 2004, 41,
6249.
[7] T. W. Clyne, I. O. Golosnoy, J. C. Tan, A. E. Markaki,
Philos. Trans. R Soc. A 2006, 364, 125.
[8] J. Tian, T. Kim, T. J. Lu, H. P. Hodson, D. T. Queheillalt,
H. N. G. Wadley, Int. J. Heat Mass Transfer 2004, 47, 3171.
[9] J. Tian, T. J. Lu, H. P. Hodson, D. T. Queheillalt, H. N. G.
Wadley, Int. J. Heat Mass Transfer 2007, 50, 2521.
[10] S. V. Belov, V. N. Antsiferov, A. S. Terekhin, S. K.
Balantsev, A. M. Beklemyshev, A. P. Kunevich, Powder
Metall. Met. Ceram. 1986, 25, 891.
[11] S. Kishimoto, Z. Song, N. Shinya, JSME Int. J. Series A
2003, 46, 447.
[12] D. T. Queheillalt, D. J. Sypeck, H. N. G. Wadley, Mater.
Sci. Eng. A, 2002, 323, 138.
[13] S. Yu, J. Liu, M. Wei, Y. Luo, X. Zhu, Y. Liu, Mater. Design
2009, 30, 87.
[14] H. Zhao, I. Elnasri, H. J. Li, Adv. Eng. Mater. 2006, 8, 877.
[15] J. Banhart, Prog. Mater. Sci. 2001, 46, 559.
[16] H. N. G. Wadley, Cellular Metals Manufacturing: An
Overview of Stochastic and Periodic Concepts, in: (Eds:
J. Barnhart, M. Ashby, N. Fleck), Cellular Metals and
Metal Foaming Technology, Ver MIT, Bremen (Germany)
2001, p. 137.
[17] C. Park, S. R. Nutt, Mater. Sci. Eng. A 2000, 288, 111.
[18] D. J. Sypeck, P. A. Parrish, H. N. G. Wadley, Novel
Hollow Powder Porous Structures, in (Eds: D. S.
Schwartz, D. S. Shih, H. N. G. Wadley, A. G. Evans),
Porous and Cellular Materials for Structural Applications,
Vol. 521, Mater. Res. Soc. Symp. Proc. Warrendale, PA
1998, p. 205.
[19] C. Park, S. R. Nutt, Mater. Sci. Eng. A 2001, 297, 62.
[20] H. N. G. Wadley, Int. J. Solids Struct. 2002, 4, 726.
[21] L. J. Gibson, M. F. Ashby, Cellular Solids: Structure and
Properties, Cambridge University Press, Cambridge
1997.
[22] F. Côté, V. S. Deshpande, N. A. Fleck, A. G. Evans, Mater.
Sci. Eng. A 2004, 380, 272.
[23] N. Wicks, J. W. Hutchinson, Mech. Mater. 2004, 36, 739.
1116
http://www.aem-journal.com
[24] V. S. Deshpande, N. A. Fleck, Int. J. Solids Struct. 2001, 38,
6275.
[25] W. Sanders, Metallic Materials With High Structural Efficiency, Lightweight Cellular Metals With High Structural
Efficiency, Vol. 146, Springer, Netherlands 2004.
[26] J. Zhou, P. Shrotriya, W. O. Soboyejo, Mech. Mater. 2004,
36, 723.
[27] Y. H. Lee, B. K. Lee, I. Jeon, K. J. Kang, Acta Mater. 2007,
55, 6084.
[28] S. Chiras, D. R. Mumm, A. G. Evans, N. Wicks, J. W.
Hutchinson, K. Dharmasena, H. N. G. Wadley, S.
Fichter, Int. J. Solids Struct. 2002, 39, 4093.
[29] Q. Li, E. Y. Chen, D. R. Bice, D. C. Dunand, Adv. Eng.
Mater. 2008, 10, 939.
[30] Q. Li, E. Y. Chen, D. R. Bice, D. C. Dunand, Metall. Mater.
Trans. A 2008, 39A, 441.
[31] C. Yungwirth, H. N. G. Wadley, J. O’Connor, A.
Zakraysek, V. S. Deshpande, Int. J. Impact Eng. 2008,
35, 920.
[32] H. N. G. Wadley, K. P. Dharmasena, Y. Chen, P. Dudt,
D. Knight, R. Charette, K. Kiddy, Int. J. Impact Eng. 2008,
35, 1102.
[33] D. T. Queheillalt, H. N. G. Wadley, Mater. Design 2009,
30, 1966.
[34] D. T. Queheillalt, V. S. Deshpande, H. N. G. Wadley,
J. Mech. Mater. Struct. 2007, 2, 1657.
[35] P. Moongkhamklang, D. M. Elzey, H. N. G. Wadley,
Compos. Part A: Appl. Sci. Manuf. 2008, 39, 176.
[36] D. J. Sypeck, H. N. G. Wadley, J. Mater. Res. 2001, 16,
890.
[37] M. Zupan, V. S. Deshpande, N. A. Fleck, Eur. J. Mech.
A-Solids 2004, 23, 411.
[38] S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability,
McGraw-Hill Book Company, Inc, New York
1961.
[39] D. T. Queheillalt, H. N. G. Wadley, Acta Mater. 2005, 53,
303.
[40] P. Moongkhamklang, V. S. Deshpande, H. N. G. Wadley,
Acta Mater. 2010, 58, 2822.
[41] (Eds: R. Boyer, G. Welsch, E. W. Collings), Material
Properties Handbook: Titanium Alloys, ASM International,
Materials Park, OH 1994.
ß 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ADVANCED ENGINEERING MATERIALS 2010, 12, No. 11