We report measurements of the elastic properties of Zr-based bulk metallic glasses,\ud
Zr52.5Cu17.9Ni14.6Al10Ti5, Zr50Cu30Ni10Al10, and Zr50Cu40Al10 between 5 K and\ud
300 K. Both the shear and longitudinal modulus have been measured as a function of\ud
temperature, allowing accurate determination of the Poisson’s ratio and the related\ud
ratio of bulk modulus to shear modulus, K/G. These data make it possible to assess the\ud
influence of the alloy’s composition on the mechanical properties and enable an\ud
evaluation of the correlation between the elastic moduli and the ductility of the alloys

Inoue, Akihisa

Keppens, Veerle

Liaw, Peter K.

Yokoyama, Yoshihiko

Zhang, Zhiying

Name not available

Elastic properties of Zr-based bulk metallic glasses studied
by resonant ultrasound spectroscopy
Zhiying Zhang, Veerle Keppens,a) and Peter K. Liaw
Department of Materials Science and Engineering, The University of Tennessee,
Knoxville, Tennessee 37996
Yoshihiko Yokoyama and Akihisa Inoue
Institute for Material Research, Tohoku University, Sendai City 980-8577, Japan
(Received 15 June 2006; accepted 18 September 2006)
We report measurements of the elastic properties of Zr-based bulk metallic glasses,
Zr52.5Cu17.9Ni14.6Al10Ti5, Zr50Cu30Ni10Al10, and Zr50Cu40Al10 between 5 K and
300 K. Both the shear and longitudinal modulus have been measured as a function of
temperature, allowing accurate determination of the Poisson’s ratio and the related
ratio of bulk modulus to shear modulus, K/G. These data make it possible to assess the
influence of the alloy’s composition on the mechanical properties and enable an
evaluation of the correlation between the elastic moduli and the ductility of the alloys.
I. INTRODUCTION
Bulk metallic glasses (BMGs) with a unique combi-
nation of mechanical and physical properties are cur-
rently emerging as a new class of metallic materials for
structural and functional use. The structural use of
BMGs, however, encounters a technical difficulty in that
most BMGs exhibit low ductility during plastic deforma-
tion under tension. To solve this material problem,
considerable effort has been devoted to the alloy devel-
opment of “ductile” BMGs. Whereas the brittleness of
crystalline metals is known to be correlated to the ratio of
the elastic shear modulus to the bulk modulus, G/K,1 a
similar assessment for metallic glasses was not available
until recently, when Lewandowski and colleagues re-
ported a universal correlation between the energy of frac-
ture and the Poisson’s ratio for bulk metallic glasses,
with brittle glasses displaying a low Poisson’s ratio.2
This confirms earlier results from Schroers and cowork-
ers who concluded that a large Poisson’s ratio and a low
glass transition temperature might be regarded as indica-
tors of the ductile character of a bulk metallic glass and
could therefore be used as a means of identifying ductile
bulk metallic glasses.3 The importance of the Poisson’s
ratio in the study of glasses was also pointed out by
Novikov and Sokolov,4 who showed that the ratio of
instantaneous shear to bulk modulus G/K in glasses, or,
alternatively, the Poisson’s ratio , is linked to the fra-
gility of the glass forming liquid, an important parameter
used to evaluate the glass forming ability (GFA) of
glasses. Even though the latter might not hold for
BMGs,5 these references suggest that a systematic study
of the elastic properties and thus the Poisson’s ratio of
metallic glasses is expected to yield important informa-
tion about their mechanical properties.
Zr-based alloys have received wide interest because of
their superior GFA, high strength, and low Young’s
modulus (80–100 GPa), allowing these alloys to be used
in a variety of applications.6–9 Previously reported meas-
urements of the mechanical properties of Zr-based alloys
include tensile and compressive tests at room tempera-
ture,10 quasi-static uniaxial compression tests,11 tensile
tests,12 Charpy impact tests9,13 as well as fatigue behav-
ior measurements.12,14,15 The present work reports meas-
urements of the elastic moduli of Zr-based BMGs
Zr52.5Cu17.9Ni14.6Al10Ti5, Zr50Cu30Ni10Al10, and
Zr50Cu40Al10, using resonant ultrasound spectroscopy.
Both the shear and longitudinal modulus have been
measured as a function of temperature (5 K–300 K),
allowing accurate determination of the Poisson’s ratio
and the related ratio of shear modulus to bulk modulus,
G/K. These data make it possible to assess the influence
of the alloy composition on the mechanical properties
and enable an evaluation of the correlation between the
elastic moduli and the ductility of the alloys.
II. EXPERIMENTAL DETAILS
Zr-based BMGs, Zr5 2 . 5Cu1 7 . 9Ni1 4 . 6Al1 0Ti5 ,
Zr50Cu30Ni10Al10 and Zr50Cu40Al10 were prepared by
arc-melting pure Zr, Cu, Ni, and Al elements in argon
atmosphere.9,11,13 The glass transition temperature of
a)Address all correspondence to this author.
e-mail: vkeppens@utk.edu
DOI: 10.1557/JMR.2007.0040
J. Mater. Res., Vol. 22, No. 2, Feb 2007 © 2007 Materials Research Society364
these alloys is 686 K, 708 K, and 706 K, respectively.
Samples of approximately 4 × 4 × 4 mm3 were cut from
the ingots and then polished into rectangular parallelepi-
peds. Each sample was flat-mounted between the two
transducers of the resonant ultrasound spectroscopy
(RUS) probe. RUS is a novel technique developed by
Migliori and coworkers at Los Alamos National Labora-
tory for determining the complete elastic tensor of a solid
by measuring its free-body resonances.16 The mechanical
resonances can be calculated for a sample with known
dimensions, density, and elastic tensor. In a RUS experi-
ment, the mechanical resonances of a freely vibrating
solid of known shape are measured, and an iteration pro-
cedure is used to “match” the measured lines with the
calculated spectrum. This allows determination of all
elastic constants of the solid from a single frequency
scan, which clearly indicates a main advantage of RUS:
there is no need for separate measurements to probe dif-
ferent moduli, and multiple sample remounts and tem-
perature sweeps are avoided. Another advantage lies in
the ability of RUS to work with small samples: whereas
conventional techniques can demand a sample size up to
a centimeter, RUS measurements can be made on milli-
meter-sized samples.
Measurements as a function of temperature were per-
formed using a specially designed probe that fits in a
physical property measurements system from Quantum
Design (San Diego, CA). Because BMGs are isotropic,
only two independent elastic constants, typically the lon-
gitudinal modulus L and the shear modulus G, must be
determined, and the bulk modulus K, Young’s modulus
E, and Poisson’s ratio  can be calculated using the fol-
lowing equations
K =
3L − 4G
3 , (1)
E =
G3L − 4G
L − G , (2)
v =
L − 2G
2L − G =
3K − 2G
6K + 2G . (3)
III. RESULTS AND DISCUSSIONS
The upper panel of Fig. 1 shows the temperature
dependence of the shear modulus G for two different
samples of the Zr-based alloy Zr50Cu30Ni10Al10, and il-
lustrates how the modulus increases with decreasing tem-
perature. The small difference between both samples is
most likely due to a minor difference in density, the
sample with the higher density (6.88 g/cm3) having a
slightly higher modulus compared with the sample with
the lower density (6.86 g/cm3). The temperature depend-
ence of the shear modulus can be modeled quite well
using the so-called Varshni function, cij (T) c0ij − s/(et/T −
1), with T the temperature, c0ij the elastic constant at 0 K,
and s and t fitting parameters.17 This function was shown
by Varshni to describe the temperature dependence of
the elastic constants of many simple substances and
characterizes to some extent “normal” elastic behavior.
A similar temperature dependence is observed in the
other Zr-based glasses Zr52.5Cu17.9Ni14.6Al10Ti5 and
Zr50Cu40Al10, illustrated in the bottom panel of Fig. 1.
The figure clearly illustrates how all three alloys follow
the “normal” Varshni-behavior, and at the same time
shows how the elastic moduli of the alloys reflect
changes in composition: whereas the shear moduli of
Zr50Cu40Al10 and Zr50Cu30Ni10Al10 are very similar, the
shear modulus of Zr52.5Cu17.9Ni14.6Al10Ti5 is signifi-
cantly lower over the temperature range from 5 K to
300 K. Longitudinal moduli (L), bulk moduli (K), and
Young’s moduli (E), summarized in Table I, show the
same trend, which is in good agreement with the results
obtained by tensile and compression tests at room tem-
perature for alloys with comparable composition (i.e.,
93 GPa for Zr50Cu40Al10,12 89 GPa for Zr50Cu40Al10,12
FIG. 1. (a) Shear modulus G as a function of temperature for two
samples of Zr50Cu30Ni10Al10 and (b) shear modulus G as a function of
temperature for Zr50Cu30Ni10Al10 (□), Zr50Cu40Al10 (◊), and
Zr52.5Cu17.9Ni14.6Al10Ti5 (). The solid lines represent a fit using the
Varshni model, with parameters s  0.7 GPa and t  100 K. c0 
36.0 GPa and 35.9 GPa for Zr50Cu30Ni10Al10, c0  35.7 GPa for
Zr50Cu40Al10, and c0  34.0 GPa for Zr52.5Cu17.9Ni14.6Al10Ti5.
TABLE I. Shear modulus (G), longitudinal modulus (L), bulk modu-
lus (K), and Young’s modulus (E) at room temperature for Zr-based
alloys Zr52.5Cu17.9Ni14.6Al10Ti5, Zr50Cu30Ni10Al10, and Zr50Cu40Al10.
Zr52.5Cu17.9Ni14.6Al10Ti5 Zr50Cu30Ni10Al10 Zr50Cu40Al10
G (GPa) 32.4 34.4 34.0
L (GPa) 160 167 166
K (GPa) 117 121 121
E (GPa) 89 94 93
Z. Zhang et al.: Elastic properties of Zr-based bulk metallic glasses studied by resonant ultrasound spectroscopy
J. Mater. Res., Vol. 22, No. 2, Feb 2007 365
and 86 GPa for Zr52.5Cu17.9Ni14.6Al10Ti5).11 Recent at-
tempts to “predict” the elastic moduli of novel BMGs
have explored to what extent the elastic constants of the
glasses can be regarded as a weighted average of the
moduli of the constituent crystalline elements. These cal-
culations are based on the concept of the property of a
glass being an average, restricting consideration to sys-
tems in which all the constituent elements are metallic.
The averages of the moduli can be weighted by atomic
weight18–20 or volume fraction,21 with the latter being
more successful. It needs, however, to be pointed out that
both averaging techniques must be used with caution, as
it has been shown that they sometimes fail to provide an
accurate estimate for BMGs.22 Table II shows quite good
agreement between the experimental values of the elastic
moduli and the calculated moduli using Xc XiciVi/Vm
where Xi, ci and Vi are the modulus, atomic fraction, and
volume per atom for the i-th constituent element,23 Vm
and Xc are the measured average atomic volume and the
calculated modulus of the metallic glass, and the summa-
tion is over all n elements of which the glass is composed.
The elastic moduli for our Zr-based glasses follow the
trend of previously reported data for Zr-based alloys with
slightly different compositions,20,24 summarized in Table
III. It is worth pointing out that, among the Zr-based
glasses, the alloys used in the present study have the
largest reported K/G values. This is reflected in a rela-
tively high Poisson’s ratio for all three alloys, reaching
0.373 in Zr52.5Cu17.9Ni14.6Al10Ti5 at room temperature.
Figure 2 shows the temperature dependence of the Pois-
son’s ratio  as well as the ratio of bulk modulus to shear
modulus K/G. The high Poisson’s ratios (and K/G values)
correlate with the enhanced ductility observed in these
alloys,9,13 confirming the suggested link between the
elastic moduli and the ductility of the alloys.
IV. SUMMARY
The elastic moduli of three Zr-based bulk metallic
glasses, Zr52.5Cu17.9Ni14.6Al10Ti5, Zr50Cu30Ni10Al10,
and Zr50Cu40Al10 were measured between 5 K and 300 K
using resonant ultrasound spectroscopy. The temperature
dependence of the moduli follow normal “Varshni” be-
havior, and the values at room temperature follow the
trend of other Zr-based alloys with slightly different
compositions. The enhanced ductility of the present al-
loys is reflected by their larger Poisson’s ratio, confirm-
ing the suggested relationship between ductility and elas-
tic moduli of metallic glasses.
ACKNOWLEDGMENTS
Valuable discussions with R.P. Hermann and M.L.
Morrison are gratefully acknowledged. This work is
TABLE II. Comparison of experimental density and elastic constants at room temperature with the calculated values based on the volume fraction
of the constituent elements.
Zr52.5Cu17.9Ni14.6Al10Ti5 Zr50Cu30Ni10Al10 Zr50Cu40Al10
Experiment Calculated Experiment Calculated Experiment Calculated
 (g/cm3) 6.63 6.56 6.88 6.78 6.86 6.79
G (GPa) 32.4 38.3 34.4 37.9 34.0 36.3
K (GPa) 117 105 121 105 121 103
E (GPa) 89 89 94 88 93 84
 0.373 0.336 0.370 0.339 0.371 0.341
TABLE III. Room temperature elastic moduli and Poisson’s ratio for
various Zr-based metallic glasses.
G
(GPa)
K
(GPa)
E
(GPa) K/G 
Zr50Cu40Al10 23.0 121 93 3.56 0.372
Zr50Cu30Ni10Al10 34.4 121 94 3.52 0.370
Zr52.5Cu17.9Ni14.6Al10Ti5 32.4 117 89 3.61 0.373
aZr41Ti14Cu12.5Ni10Be22.5 37.4 114.8 101.3 3.06 0.353
aZr46.75Ti8.25Cu7.5Ni10Be27.5 35.2 113.4 95.7 3.22 0.359
aZr53Ti5Cu20Ni12Al10 32.1 106.8 87.6 3.32 0.363
aZr48Nb8Cu12Be24Fe8 35.2 113.4 95.7 3.22 0.359
aZr57Nb5Cu15.4Ni12.6Al10 32.0 107.7 87.3 3.37 0.365
aZr65Cu15Ni10Al10 31.0 106.7 83.0 3.52 0.367
aData from Ref. 20.
FIG. 2. (a) The ratio of bulk modulus to shear modulus K/G and (b)
the Poisson’s ratio  as a function of temperature for Zr50Cu30Ni10Al10
(□), Zr50Cu40Al10 (◊), and Zr52.5Cu17.9Ni14.6Al10Ti5 ().
Z. Zhang et al.: Elastic properties of Zr-based bulk metallic glasses studied by resonant ultrasound spectroscopy
J. Mater. Res., Vol. 22, No. 2, Feb 2007366
partially supported by the National Science Foundation,
Grant No. DMR-0506292.
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J. Mater. Res., Vol. 22, No. 2, Feb 2007 367


Elastic properties of Zr-based bulk metallic glasses studied\ud
by resonant ultrasound spectroscopy

We report measurements of the elastic properties of Zr-based bulk metallic glasses, Zr52.5Cu17.9Ni14.6Al10Ti5, Zr50Cu30Ni10Al10, and Zr50Cu40Al10 between 5 K and 300 K. Both the shear and longitudinal modulus have been measured as a function of temperature, allowing accurate determination of the Poisson’s ratio and the related ratio of bulk modulus to shear modulus, K/G. These data make it possible to assess the influence of the alloy’s composition on the mechanical properties and enable an evaluation of the correlation between the elastic moduli and the ductility of the alloys

University of Tennessee, Knoxville: Trace

University of Tennessee, KnoxvilleTrace: Tennessee Research and CreativeExchangeMaterials Science & Engineering Publications andOther Works Materials Science & EngineeringFebruary 2007Elastic properties of Zr-based bulk metallic glassesstudied by resonant ultrasound spectroscopyZhiying ZhangVeerle KeppensUniversity of Tennessee, Knoxville, vkeppens@utk.eduPeter K. LiawYoshihiko YokoyamaAkihisa InoueFollow this and additional works at: http://trace.tennessee.edu/utk_matepubsPart of the Materials Science and Engineering CommonsThis Article is brought to you for free and open access by the Materials Science & Engineering at Trace: Tennessee Research and Creative Exchange. Ithas been accepted for inclusion in Materials Science & Engineering Publications and Other Works by an authorized administrator of Trace: TennesseeResearch and Creative Exchange. For more information, please contact trace@utk.edu.Recommended CitationZhang, Zhiying; Keppens, Veerle; Liaw, Peter K.; Yokoyama, Yoshihiko; and Inoue, Akihisa, "Elastic properties of Zr-based bulkmetallic glasses studied by resonant ultrasound spectroscopy" (2007). Materials Science & Engineering Publications and Other Works.http://trace.tennessee.edu/utk_matepubs/8Elastic properties of Zr-based bulk metallic glasses studiedby resonant ultrasound spectroscopyZhiying Zhang, Veerle Keppens,a) and Peter K. LiawDepartment of Materials Science and Engineering, The University of Tennessee,Knoxville, Tennessee 37996Yoshihiko Yokoyama and Akihisa InoueInstitute for Material Research, Tohoku University, Sendai City 980-8577, Japan(Received 15 June 2006; accepted 18 September 2006)We report measurements of the elastic properties of Zr-based bulk metallic glasses,Zr52.5Cu17.9Ni14.6Al10Ti5, Zr50Cu30Ni10Al10, and Zr50Cu40Al10 between 5 K and300 K. Both the shear and longitudinal modulus have been measured as a function oftemperature, allowing accurate determination of the Poisson’s ratio and the relatedratio of bulk modulus to shear modulus, K/G. These data make it possible to assess theinfluence of the alloy’s composition on the mechanical properties and enable anevaluation of the correlation between the elastic moduli and the ductility of the alloys.I. INTRODUCTIONBulk metallic glasses (BMGs) with a unique combi-nation of mechanical and physical properties are cur-rently emerging as a new class of metallic materials forstructural and functional use. The structural use ofBMGs, however, encounters a technical difficulty in thatmost BMGs exhibit low ductility during plastic deforma-tion under tension. To solve this material problem,considerable effort has been devoted to the alloy devel-opment of “ductile” BMGs. Whereas the brittleness ofcrystalline metals is known to be correlated to the ratio ofthe elastic shear modulus to the bulk modulus, G/K,1 asimilar assessment for metallic glasses was not availableuntil recently, when Lewandowski and colleagues re-ported a universal correlation between the energy of frac-ture and the Poisson’s ratio for bulk metallic glasses,with brittle glasses displaying a low Poisson’s ratio.2This confirms earlier results from Schroers and cowork-ers who concluded that a large Poisson’s ratio and a lowglass transition temperature might be regarded as indica-tors of the ductile character of a bulk metallic glass andcould therefore be used as a means of identifying ductilebulk metallic glasses.3 The importance of the Poisson’sratio in the study of glasses was also pointed out byNovikov and Sokolov,4 who showed that the ratio ofinstantaneous shear to bulk modulus G/K in glasses, or,alternatively, the Poisson’s ratio , is linked to the fra-gility of the glass forming liquid, an important parameterused to evaluate the glass forming ability (GFA) ofglasses. Even though the latter might not hold forBMGs,5 these references suggest that a systematic studyof the elastic properties and thus the Poisson’s ratio ofmetallic glasses is expected to yield important informa-tion about their mechanical properties.Zr-based alloys have received wide interest because oftheir superior GFA, high strength, and low Young’smodulus (80–100 GPa), allowing these alloys to be usedin a variety of applications.6–9 Previously reported meas-urements of the mechanical properties of Zr-based alloysinclude tensile and compressive tests at room tempera-ture,10 quasi-static uniaxial compression tests,11 tensiletests,12 Charpy impact tests9,13 as well as fatigue behav-ior measurements.12,14,15 The present work reports meas-urements of the elastic moduli of Zr-based BMGsZr52.5Cu17.9Ni14.6Al10Ti5, Zr50Cu30Ni10Al10, andZr50Cu40Al10, using resonant ultrasound spectroscopy.Both the shear and longitudinal modulus have beenmeasured as a function of temperature (5 K–300 K),allowing accurate determination of the Poisson’s ratioand the related ratio of shear modulus to bulk modulus,G/K. These data make it possible to assess the influenceof the alloy composition on the mechanical propertiesand enable an evaluation of the correlation between theelastic moduli and the ductility of the alloys.II. EXPERIMENTAL DETAILSZr-based BMGs, Zr5 2 . 5Cu1 7 . 9Ni1 4 . 6Al1 0Ti5 ,Zr50Cu30Ni10Al10 and Zr50Cu40Al10 were prepared byarc-melting pure Zr, Cu, Ni, and Al elements in argonatmosphere.9,11,13 The glass transition temperature ofa)Address all correspondence to this author.e-mail: vkeppens@utk.eduDOI: 10.1557/JMR.2007.0040J. Mater. Res., Vol. 22, No. 2, Feb 2007 © 2007 Materials Research Society364these alloys is 686 K, 708 K, and 706 K, respectively.Samples of approximately 4 × 4 × 4 mm3 were cut fromthe ingots and then polished into rectangular parallelepi-peds. Each sample was flat-mounted between the twotransducers of the resonant ultrasound spectroscopy(RUS) probe. RUS is a novel technique developed byMigliori and coworkers at Los Alamos National Labora-tory for determining the complete elastic tensor of a solidby measuring its free-body resonances.16 The mechanicalresonances can be calculated for a sample with knowndimensions, density, and elastic tensor. In a RUS experi-ment, the mechanical resonances of a freely vibratingsolid of known shape are measured, and an iteration pro-cedure is used to “match” the measured lines with thecalculated spectrum. This allows determination of allelastic constants of the solid from a single frequencyscan, which clearly indicates a main advantage of RUS:there is no need for separate measurements to probe dif-ferent moduli, and multiple sample remounts and tem-perature sweeps are avoided. Another advantage lies inthe ability of RUS to work with small samples: whereasconventional techniques can demand a sample size up toa centimeter, RUS measurements can be made on milli-meter-sized samples.Measurements as a function of temperature were per-formed using a specially designed probe that fits in aphysical property measurements system from QuantumDesign (San Diego, CA). Because BMGs are isotropic,only two independent elastic constants, typically the lon-gitudinal modulus L and the shear modulus G, must bedetermined, and the bulk modulus K, Young’s modulusE, and Poisson’s ratio  can be calculated using the fol-lowing equationsK =3L − 4G3 , (1)E =G3L − 4GL − G , (2)v =L − 2G2L − G =3K − 2G6K + 2G . (3)III. RESULTS AND DISCUSSIONSThe upper panel of Fig. 1 shows the temperaturedependence of the shear modulus G for two differentsamples of the Zr-based alloy Zr50Cu30Ni10Al10, and il-lustrates how the modulus increases with decreasing tem-perature. The small difference between both samples ismost likely due to a minor difference in density, thesample with the higher density (6.88 g/cm3) having aslightly higher modulus compared with the sample withthe lower density (6.86 g/cm3). The temperature depend-ence of the shear modulus can be modeled quite wellusing the so-called Varshni function, cij (T) c0ij − s/(et/T −1), with T the temperature, c0ij the elastic constant at 0 K,and s and t fitting parameters.17 This function was shownby Varshni to describe the temperature dependence ofthe elastic constants of many simple substances andcharacterizes to some extent “normal” elastic behavior.A similar temperature dependence is observed in theother Zr-based glasses Zr52.5Cu17.9Ni14.6Al10Ti5 andZr50Cu40Al10, illustrated in the bottom panel of Fig. 1.The figure clearly illustrates how all three alloys followthe “normal” Varshni-behavior, and at the same timeshows how the elastic moduli of the alloys reflectchanges in composition: whereas the shear moduli ofZr50Cu40Al10 and Zr50Cu30Ni10Al10 are very similar, theshear modulus of Zr52.5Cu17.9Ni14.6Al10Ti5 is signifi-cantly lower over the temperature range from 5 K to300 K. Longitudinal moduli (L), bulk moduli (K), andYoung’s moduli (E), summarized in Table I, show thesame trend, which is in good agreement with the resultsobtained by tensile and compression tests at room tem-perature for alloys with comparable composition (i.e.,93 GPa for Zr50Cu40Al10,12 89 GPa for Zr50Cu40Al10,12FIG. 1. (a) Shear modulus G as a function of temperature for twosamples of Zr50Cu30Ni10Al10 and (b) shear modulus G as a function oftemperature for Zr50Cu30Ni10Al10 (□), Zr50Cu40Al10 (◊), andZr52.5Cu17.9Ni14.6Al10Ti5 (). The solid lines represent a fit using theVarshni model, with parameters s  0.7 GPa and t  100 K. c0 36.0 GPa and 35.9 GPa for Zr50Cu30Ni10Al10, c0  35.7 GPa forZr50Cu40Al10, and c0  34.0 GPa for Zr52.5Cu17.9Ni14.6Al10Ti5.TABLE I. Shear modulus (G), longitudinal modulus (L), bulk modu-lus (K), and Young’s modulus (E) at room temperature for Zr-basedalloys Zr52.5Cu17.9Ni14.6Al10Ti5, Zr50Cu30Ni10Al10, and Zr50Cu40Al10.Zr52.5Cu17.9Ni14.6Al10Ti5 Zr50Cu30Ni10Al10 Zr50Cu40Al10G (GPa) 32.4 34.4 34.0L (GPa) 160 167 166K (GPa) 117 121 121E (GPa) 89 94 93Z. Zhang et al.: Elastic properties of Zr-based bulk metallic glasses studied by resonant ultrasound spectroscopyJ. Mater. Res., Vol. 22, No. 2, Feb 2007 365and 86 GPa for Zr52.5Cu17.9Ni14.6Al10Ti5).11 Recent at-tempts to “predict” the elastic moduli of novel BMGshave explored to what extent the elastic constants of theglasses can be regarded as a weighted average of themoduli of the constituent crystalline elements. These cal-culations are based on the concept of the property of aglass being an average, restricting consideration to sys-tems in which all the constituent elements are metallic.The averages of the moduli can be weighted by atomicweight18–20 or volume fraction,21 with the latter beingmore successful. It needs, however, to be pointed out thatboth averaging techniques must be used with caution, asit has been shown that they sometimes fail to provide anaccurate estimate for BMGs.22 Table II shows quite goodagreement between the experimental values of the elasticmoduli and the calculated moduli using Xc XiciVi/Vmwhere Xi, ci and Vi are the modulus, atomic fraction, andvolume per atom for the i-th constituent element,23 Vmand Xc are the measured average atomic volume and thecalculated modulus of the metallic glass, and the summa-tion is over all n elements of which the glass is composed.The elastic moduli for our Zr-based glasses follow thetrend of previously reported data for Zr-based alloys withslightly different compositions,20,24 summarized in TableIII. It is worth pointing out that, among the Zr-basedglasses, the alloys used in the present study have thelargest reported K/G values. This is reflected in a rela-tively high Poisson’s ratio for all three alloys, reaching0.373 in Zr52.5Cu17.9Ni14.6Al10Ti5 at room temperature.Figure 2 shows the temperature dependence of the Pois-son’s ratio  as well as the ratio of bulk modulus to shearmodulus K/G. The high Poisson’s ratios (and K/G values)correlate with the enhanced ductility observed in thesealloys,9,13 confirming the suggested link between theelastic moduli and the ductility of the alloys.IV. SUMMARYThe elastic moduli of three Zr-based bulk metallicglasses, Zr52.5Cu17.9Ni14.6Al10Ti5, Zr50Cu30Ni10Al10,and Zr50Cu40Al10 were measured between 5 K and 300 Kusing resonant ultrasound spectroscopy. The temperaturedependence of the moduli follow normal “Varshni” be-havior, and the values at room temperature follow thetrend of other Zr-based alloys with slightly differentcompositions. The enhanced ductility of the present al-loys is reflected by their larger Poisson’s ratio, confirm-ing the suggested relationship between ductility and elas-tic moduli of metallic glasses.ACKNOWLEDGMENTSValuable discussions with R.P. Hermann and M.L.Morrison are gratefully acknowledged. This work isTABLE II. Comparison of experimental density and elastic constants at room temperature with the calculated values based on the volume fractionof the constituent elements.Zr52.5Cu17.9Ni14.6Al10Ti5 Zr50Cu30Ni10Al10 Zr50Cu40Al10Experiment Calculated Experiment Calculated Experiment Calculated (g/cm3) 6.63 6.56 6.88 6.78 6.86 6.79G (GPa) 32.4 38.3 34.4 37.9 34.0 36.3K (GPa) 117 105 121 105 121 103E (GPa) 89 89 94 88 93 84 0.373 0.336 0.370 0.339 0.371 0.341TABLE III. Room temperature elastic moduli and Poisson’s ratio forvarious Zr-based metallic glasses.G(GPa)K(GPa)E(GPa) K/G Zr50Cu40Al10 23.0 121 93 3.56 0.372Zr50Cu30Ni10Al10 34.4 121 94 3.52 0.370Zr52.5Cu17.9Ni14.6Al10Ti5 32.4 117 89 3.61 0.373aZr41Ti14Cu12.5Ni10Be22.5 37.4 114.8 101.3 3.06 0.353aZr46.75Ti8.25Cu7.5Ni10Be27.5 35.2 113.4 95.7 3.22 0.359aZr53Ti5Cu20Ni12Al10 32.1 106.8 87.6 3.32 0.363aZr48Nb8Cu12Be24Fe8 35.2 113.4 95.7 3.22 0.359aZr57Nb5Cu15.4Ni12.6Al10 32.0 107.7 87.3 3.37 0.365aZr65Cu15Ni10Al10 31.0 106.7 83.0 3.52 0.367aData from Ref. 20.FIG. 2. (a) The ratio of bulk modulus to shear modulus K/G and (b)the Poisson’s ratio  as a function of temperature for Zr50Cu30Ni10Al10(□), Zr50Cu40Al10 (◊), and Zr52.5Cu17.9Ni14.6Al10Ti5 ().Z. Zhang et al.: Elastic properties of Zr-based bulk metallic glasses studied by resonant ultrasound spectroscopyJ. Mater. Res., Vol. 22, No. 2, Feb 2007366partially supported by the National Science Foundation,Grant No. DMR-0506292.REFERENCES1. S.F. 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Elastic properties of Zr-based bulk metallic glasses studied by resonant ultrasound spectroscopy

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Elastic properties of Zr-based bulk metallic glasses studied\ud by resonant ultrasound spectroscopy

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