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同角三角函数基本关系化简求证：1+2sinxcosx/cos²x-sin²x=1+tanx/1-tanx这么证行不行左边=(sinx+cosx)²/cos²x-sin²x=sin²x+2sinxcosx+2cos²x/cos²x-sin²x上下除cos²x =tan²x+2tanx+1/1-tanx=(tanx+1)²/1-tanx可是不等于右边...我这么化简哪里错啦?
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tanx存在,则 x≠±π/2+2kπ,k∈N,cos²x≠0左边=(sinx+cosx)²/（cos²x-sin²x）=（sin²x+2sinxcosx+cos²x）/（cos²x-sin²x）上下除cos²x =（tan²x+2tanx+1）/（1-tan²x）=(1+tanx)/（1-tanx）=右边做题过程中你把“=（tan²x+2tanx+1）/（1-tan²x）”的（1-tan²x）写成了1-tanx
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左边=(sinx+cosx)²/cos²x-sin²x=（sinx+cosx）/(cosx-sinx)=(1+tanx)/(1-tanx)=右边
你这个我会，可是我那个做法，哪个步骤错了？
不能上下除cos²x，因为不能确定是否为零
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3.已知函数f（x）=2cos2x+sin²x（1）求f（3分之派）的值（2）求f（x）的最大值和最小值2.已知α,β∈（0,派）,且tanα,tanβ是方程x²-5x+6=0的的两根（1）求α+β的值（2）求cos（α-β）的值
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1.f(x)=2cos2x+(1-cos2x)/2=3cos2x/2+1/2f(π/3)=-3/4+1/2=-1/4当x=kπ（k∈z）时f(x)最大值为2当x=kπ+π/2(k∈z)时f(x)最小值为-12.由tanα,tanβ是方程x²-5x+6=0的的两根得tanα+tanβ=5 tanβtanα=6tan(α+β)=(tanα+tanβ)/(1-tanβtanα)=-1tanα,tanβ同号,且大于零则α+β=3π/4cos(α-β)=cosαcosβ+sinαsinβ由tanα+tanβ=5得sinαcosβ+cosαsinβ=5cosαcosβ=sin(α+β)=√2/2由tanβtanα=6得sinαsinβ=6cosαcosβ而cos(α-β)=cosαcosβ+sinαsinβ=7cosαcosβ=7√2/10
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扫描下载二维码From Wikipedia, the free encyclopedia
This article does not
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by . Unsourced material may be challenged and . (June 2015) ()
Illustration of a unit circle. The variable t is an
Animation of the act of unrolling the circumference of a unit circle, a circle with radius of 1. Since C = 2πr, the circumference of a unit circle is 2π.
In , a unit circle is a
of . Frequently, especially in , the unit circle is the circle of radius one centered at the origin (0, 0) in the
in the . The unit circle is often denoted S1; the generalization to higher dimensions is the .
If (x, y) is a point on the unit circle's circumference, then |x| and |y| are the lengths of the legs of a
whose hypotenuse has length 1. Thus, by the , x and y satisfy the equation
{\displaystyle x^{2}+y^{2}=1.}
Since x2 = (-x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant.
The interior of the unit circle is called the open , while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.
One may also use other notions of "distance" to define other "unit circles", see the article on
for additional examples.
The unit circle can be considered as the , i.e., the set of
z of the form
{\displaystyle z=\,\mathrm {e} ^{it}\,=\cos(t)+i\sin(t)=\operatorname {cis} (t)}
for all t (see also: ). This relation represents . In , this is referred to as .
All of the trigonometric functions of the angle θ (theta) can be constructed geometrically in terms of a unit circle centered at O.
Sine function on unit circle (top) and its graph (bottom)
cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an
θ from the positive x-axis, (where counterclockwise turning is positive), then
{\displaystyle \cos(\theta )=x}
{\displaystyle \sin(\theta )=y}
The equation x2 + y2 = 1 gives the relation
{\displaystyle \cos ^{2}(\theta )+\sin ^{2}(\theta )=1.}
The unit circle also demonstrates that
are , with the identities
{\displaystyle \cos \theta =\cos(2\pi k+\theta )}
{\displaystyle \sin \theta =\sin(2\pi k+\theta )}
Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P(x1,y1) on the unit circle such that an angle t with 0 & t & π/2 is formed with the positive arm of the x-axis. Now consider a point Q(x1,0) and line segments PQ ⊥ OQ. The result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin(t) = y1 and cos(t) = x1. Having established these equivalences, take another radius OR from the origin to a point R(-x1,y1) on the circle such that the same angle t is formed with the negative arm of the x-axis. Now consider a point S(-x1,0) and line segments RS ⊥ OS. The result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π - t, R is at (cos(π - t),sin(π - t)) in the same way that P is at (cos(t),sin(t)). The conclusion is that, since (-x1,y1) is the same as (cos(π - t),sin(π - t)) and (x1,y1) is the same as (cos(t),sin(t)), it is true that sin(t) = sin(π - t) and -cos(t) = cos(π - t). It may be inferred in a similar manner that tan(π - t) = -tan(t), since tan(t) = y1/x1 and tan(π - t) = y1/-x1. A simple demonstration of the above can be seen in the equality sin(π/4) = sin(3π/4) = 1/√2.
When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the unit circle, these functions produce meaningful values for any -valued angle measure – even those greater than 2π. In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like
and  – can be defined geometrically in terms of a unit circle, as shown at right.
Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be calculated without the use of a calculator by using the .
The unit circle, showing
can be identified with points in the , namely the number a + bi is identified with the point (a, b). Under this identification, the unit circle is a
under multiplication, called the circle group; it is usually denoted T. On the plane, multiplication by cos θ + i sin θ gives a counterclockwise rotation by θ. This group has important applications in mathematics and science.[]
Unit circle in complex dynamics
{\displaystyle f_{0}(x)=x^{2}}
is a unit circle. It is a simplest case so it is widely used in study of dynamical systems.
Wikibooks has a book on the topic of:
in Wiktionary, the free dictionary.
: Hidden categories:Prof. LIU, C.T. (CSE), City University of Hong Kong
Center for Advanced Structural Materials, MBE Department, CSE
University Distinguished Professor (CSE)
Contact Information
Research Interests
Physical metallurgy and mechanical behaviour of metals, alloys, nanostructure materials, intermetallic compounds and bulk amorphous alloys
Microstructure and phase transformation
Alloy design of high-temperature structural materials, precious metal alloys, Ti-base alloys, metal-matrix composites
Innovative material processing
Academic Background
BS Degree in Mechanical Engineering
National Taiwan University, Taipei, Taiwan
MS Degree in Engineering and Materials Science
Brown University, Providence, Rhode Island
PhD Degree in Materials Science and Engineering
Brown University, Providence, Rhode Island
Professional Experience
2009 - 2011
Chair Professor
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong
2009 - 2011
Distinguished Professor
Materials Engineering, Auburn University, USA
2008 - present
Adjunct Professor
Institute for Materials Research, Tohoku University, Japan
2006 - 2008
Professor and Distinguished Research Professor
Department of Materials Science and Engineering, University of Tennessee, USA
2005 - 2006
Distinguished Research Professor
Department of Materials Science and Engineering, University of Tennessee, USA
1997 - 2005
Senior Corporate Fellow
Oak Ridge National Laboratory, Lockheed Martin Energy Research Corporation, USA
1985 - 1997
Corporate Fellow
Oak Ridge National Laboratory, Martin Marietta Energy Systems, Inc., USA
1983 - 2005
Group Leader
Alloying Behavior and Design Group, Metals and Ceramics Division, Oak Ridge National Laboratory, USA
1967 - 1983
Research Staff Member / Senior Research Staff Member
Metals and Ceramics Division, Oak Ridge National Laboratory, USA
Honours, Awards and Recognitions
First International Science and Technology Cooperation Award, Jiangsu Province Government, Feburary 23, 2013
Vice Editor of Science China: Technological Science sponsored by CAS and CNSF, October 26, 2012 (starting 2013)
Chairman of the Academic Committee, the State Key Laboratory for Advanced Metal and Materials, University of Science and Engineering, Beijing, June, 2012
Appointed as a member of the Editorial Board of Nano Energy
Editorial Board Member of Progress in Nature Science: Material International
Chairman of the Academic Committee, Jiangsu Advanced Material Research Institute, Danyang, Jiangsu
R&D 100 Award for Development of Alumina-Forming Austenitic Stainless Steel
2007 National Award for International Scientific and Technological Cooperation sponsored by National Office for Science and Technology Awards, P. R. China
Appointed as Adjunct Professor, World Premier International Research Center, Tohoku University
Awarded with Honorary Member of Japan Institute for Metals
Appointed as the Chairman of the Editors Board, Journal of Intermetallics
Appointed Honorary Distinguished Professor of Sichuan University, China
Received a National Friendship Award from the Office of Foreign Affairs, Chinese State Council
Elected as a Foreign Member of Chinese Academy of Engineering (CAE) (similar to Foreign Member of NAE in USA)
The Albert Sauveur Achievement Award Selection Committee, Committee Member, ASM
Appointed as a Governor of the Acta Materialia Board to represent ASM
Research on structural amorphous steels (published in PRL 92,
to -4, 2004) by ZP Lu, CT Liu, JR Thompson and WD Porter was selected as one of the Top Physics Stories in 2004 by APL
2004 Director's Award for Outstanding Individual Accomplishment in Science and Technology, Oak Ridge National Laboratory
Elected as a member of National Academy of Engineering (NAE) USA
Appointed as Honorary Professor at Central South University, Changsha, China
Appointed as Honorary Professor at Nanjing University of Science and Technology, Nanjing China
Elected as a Fellow of the World Technology Network
Identified as a World Highly Cited Researcher in Materials Science by ISI in the materials fields during the past 20 years
Selected as a Distinguished Inventor of Battelle Memorial Institute
Served on the Materials Technology Advisory Committee, Industrial Technology Research Institute, Taiwan
Served on the International Advisory Committee of Shanyang National Laboratory for Materials Science, China
The 2001 Acta Metallurgica Gold Medal, February, 2001 "for his leadership and outstanding achievements in science and technology of ordered intermetallic alloys"
Identified as an author in the list of 1000 Most Cited Physicists during 1981-June, 1997 by ISI
Appointed as a member of the Editorial Board for Journal of University of Science and Technology Beijing, Beijing, China
Committee member for the evaluation of research papers for the new Energy and Industrial Technology Development Organization Grant, Japan
Member of the Materials Technology Advisory Committee (MTAC) at Materials Research Laboratory, Industrial Technology Research Institute, Taiwan ()
Winner of the 1998 Sustained Outstanding Research Award in the Metals and Ceramics Sciences in the DOE 1998 Materials Science Research Competition
Received a Brown Engineering Alumni Medal Award, Brown University, May 1998
Appointed Chairman, ASM/TMS Distinguished Lecture in Materials & Society Selection Committee, February 1998
Appointed Chairman, TMS Fellow Award Committee, February, 1998
Senior Corporate Fellow, Oak Ridge National Laboratory, May 1997
An editor of J. the Chinese Institute of Engineers
Appointed as a Honorable Professor, Beijing University of Science and Technology, Beijing, China, April 1997
Received the Institute 1996 Best Paper Award in the category of Mechanical Behavior and Mechanical Characterization of Materials (with Y. Mishima and E. H. Lee) from Japan Institute of Metals (JIM), September, 1996
Appointed to NRC Panel on AFOSR Materials Science Proposal Reviews (metals expertise), National Research Council (NRC)
ORNL R&D Accomplishment Award for a breakthrough in alloy design of TiAl alloys
Fifth in worldwide ranking of highly cited authors publishing high-impact papers in all journals of Materials Science and Metallurgy during
[Science Watch, Vol. 6, No. 9, 1995, Institute for Scientific Information (ISI)]
Received the 1995 Buehler Technical Paper Merit Award for the best paper published in the Journal of Materials Characterization (with E. P. George and J. A. Horton from ORNL)
Fellow, TMS
NRC member of the Air Force Office of Scientific Research (AFOSR) Materials Science Review Panel, National Research Council (NRC)
Associate Editor, Materials Letters
Editor, Journal of Intermetallics
Received Outstanding Achievement Award from the Office of Special Applications, DOE
Distinguished Lecturer, invited by the National Science Council, Taiwan, R.O.C.
NSF committee member of site visit to Materials Research Lab (MRL) at MIT
DOE Outstanding Achievement Award to the Galileo RTG Team
I.R 100 Competition Award for "Ductile Iron Aluminides"
Appointed Principal Editor, Journal of Materials Research
E. O. Lawrence Award (established by the U.S. President), U.S. Department of Energy
Golden Acorn Award, the Inventor's Forum, Martin Marietta Energy Systems, Inc.
Corporate Fellow, Oak Ridge National Laboratory, Martin Marietta Energy Systems, Inc.
Recipient of the silver Jefferson Cup, a recognition given each year by Martin Marietta Corporation to a select few employees who have demonstrated excellence and have attained preeminence by virtue of superior performance
Scientist of the Year, Martin Marietta Energy Systems, Inc.
Distinguished Alumni Award, Brown University
Fellow, American Society for Metals
Winner of the Department of Energy's 1984 Materials Sciences Research Competition Award in the Significant Implication for Energy Technology: Metallurgy and Ceramics Category
Spacecrafts Voyage I and II Team Award by JPL/NASA
IoR 100 Award for Development of NiFe Aluminides
The International Precious Metals Institute's Highest Personal Accomplishment Award - The Honorary Platinum Membership
The first recipient of the Henry J. Albert Award for outstanding theoretical or experimental contributions to the metallurgy of precious metals, by the International Precious Metals Institute
IoR 100 Award for Development of New LRO Alloys
Pioneer II Saturn Mission Team Award by NASA
Pioneer/Jupiter Team Award by NASA
Professional Society Activities
Governor of Acta Materialia appointed by TMS, 2005
Chairman and member, The Fellow Committee, TMS,
Fellow, 1994
Chairman of the Acta Met/Hume Rothery Award Committee, TMS,
Chairman of the Alloy Phases Committee, TMS,
Secretary of the Alloy Phase Committee, TMS,
Member of the Acta Metallurgica/Hume Rothery Award Committee, TMS, 1990-
Chairman of the ASM Engineering Material Achievement Award (EMAA) Selection Committee, 2003
Fellow, 1984
Chairman and Member of the ASM/TMS Distinguished Lecture in Materials and Society Selection Committee 1997-98
Member of the ASM Historical landmarks Awards Selection Committee
Appointed as Vice Chair for the ASM 2002 Engineering Material Achievement Award (EMAA) Selection Committee, 2002
Appointed as Chair for the ASM 2003 Engineering Material Achievement Award (EMAA) Selection Committee, 2003
The Honorary Platinum Member (Fellow), International Precious Metals Institute, 1982
Member of the Award Committee
Principal Editor, Journal of Materials Research,
World Technology Network
Fellow, June 2003
JIM, Japan
Honorary Member, 2007
Patents: A total of 29 US patents
Recent Publications
Edited books: 26 books
Open literature publications
Sheng Guo, Chun Ng, C. T. Liu, “Anomalous solidification microstructures in Co-freeAlxCrCuFeNi2 High-Entropy Alloys”, J Alloy and Compounds 557, 77-81, 2013 (Jan.)
YR Wen, A. Hirata, ZW Zhang, T. Fujita, C. T. Liu, JH Jiang, MW Chen, “Microstructure characterization of Cu-rich nanoprecipitates in a Fe–2.5 Cu–1.5 Mn–4.0 Ni–1.0 Al multicomponent ferritic alloy”, Acta Mater. 61, , 2013 (April)
ZW Zhang, CT Liu, XL Wang, MK Miller, D Ma, G Chen, JR Williams, BA Chin, “Effect of proton irradiation on nanocluster precipitation in ferritic steel containing fcc alloying additions”, Acta Mater. 60, 12
G. Chen, X. L. Zhang, and C. T. Liu, “High strength and plastic strain of Mg-based bulk metallic glass composite containing in situ formed intermetallic phases”, Scripta Mater. 68, 150-153, Oct. 2012
G Chen, JL Cheng, and CT Liu, “Large-sized Zr-based bulk metallic glass composite with enhanced tensile properties” Intermetallics 28, 25-33, 2012
Yang, Y. and CT Liu, “Size effect on stability of shear
band propagation in bulk metallic glasses: An overview”, J. Mater. Sci. 47, 55-67, 2012
TG Nieh, Y. Yang, J. Lu, C. T. Liu, “Effect of surface modifications on shear banding and plasticity in metallic glasses: An overview”, Prog. Nature Science: Mater. Inter. 22, 355-363, 2012 (Oct.)
Hirata A, T. Fujita, C. T. Liu, M. W. Chen, “Characterization of oxide nanoprecipitates in an oxide dispersion strengthened 14YWT steel using aberration-corrected STEM”, Acta Mater. 60, , 2012 (Sept.)
Z. K. Teng, C. T. Liu, M. K. Miller, G. Ghosh, E. A. Kenik, S. Huang, P. K. Liaw., “Effects of precipitate microstructure on the room temperature ductility of NiAl-strengthened ferritic steels,”Materials Science and Engineering A., 2012, doi: 10.1016/j.msea.
Zhang Z. W., Liu CT, Wen Y, Hirata A, Guo S, Chen G, Chen M, Chin B., “Influence of aging and thermomechanical treatments on the mechanical properties of a nanocluster-strengthened ferritic steel”,
Mater. Trans. A,
43, p. 351-59, 2012
XL Wang, CT Liu, U. Keiderling, AS Stoica, L. Yang, MK Miller, CL Fu, D. Ma, and K. An, “Unual thermal stability of nano-structured ferritic alloys” J. Alloys and Compounds 529, 96-101, 2012
ZK Teng, CT Liu, MK Miller, G. Ghosh, EA Kenik, S. Huang, PK Liaw, “Room temperature ductility of NiAl-strengthened ferritic steels: Effect of precipitate microstructure”, Mater Sci. Engr. A 541, 22-27, 2012
Y Yang, JF Zeng, A Volland, JJ blabdin, S Gravier and CT Liu, “Fractal Growth of dense packing phase in annealed metallic glass imaged by hign resolution atomic force microscopy, Acta Mater. 60 (13-14), , 2012
Fan Cang, HG Yan, C. T. Liu, HQ Li, PK Liaw, Y Ren, T. Egami, “Changes in the atomic structure through glass transition observed by X-ray scattering”, Intermetallics 23, 111-115, 2012
Y Wen, Y. Liu, A. Hirata, F. Liu, T. Fujita, Y Dou, D. Liu, B. Liu, ,Z. Liu, C. T. Liu, “Innovative processing of high-strength and low cost ferritic steels strengthened by Y-Ti-O nanoclusters”, Mater. Sc. Engr. A 544, 59-69, May 2012
Z. K. Teng, F. Zhang, M. K. Miller, C. T. Liu, S. Huang, R. H. Tien, Y. T. Chou, Y. A. Chang, P. K. Liaw. “Thermodynamic modeling and experimental validation of the Fe-Al-Ni-Cr-Mo alloy system, ’’Materials Letters, -40
MR Chen, Y. Jiang, YH He, LW Lin, BY Huang, C. T. Liu, “Pore evolution regulation in synthesis of open pore structured Ti-Al intermetallic compoinds by solid diffusion”, J. alloy and Compounds 521, 12-15, April 2012
ZW Zhang, CT Liu, XL Wang, KC Littrell, MK Miller, K An, and BA Chin, “From embryos to precipitates: A study of nucleation and growth in a multicomponent ferritic steel”, Phys. Rev. B 84, 11)
Liu XJ, Xu Y., Lu ZP, Hui X., Chen GL, Zheng GP, Liu, C.T., “Atomic packing symmetry in the metallic liquid and glass states”, Acta Mater. 59, , 2011
Q, Liu CT, Yang. Y, Dong YD, and
Lu J, Atomic-scale structural evolution and stability of supercooled liquid of a Zr-based bulk metallic glass,
Phy. Rev. Lett. 106, 11
Liu ZY, Yang Y., Liu CT, “Size-affected shear band speed in bulk metallic glasses”, App.Phys. Lett .99, 11
Guo, S.; Ng, C.; Lu, J.; Liu, C. T., Effect of valence electron concentration on stability of fcc or bcc phase in high entropy alloys, J. Applied Physics 109(10), 11
Sheng Guo and C. T. Liu, ”Phase stability in high entropy alloys: formation of solid-solution phase or amorphous phase”, Prog. Nature Scie.: Mater. International, 21, 433-446, 2011
Wen YR, Liu Y, Liu DH, Tang B, Liu CT,
Microstructural evolution of ferritic steel powder during mechanical alloying with iron oxide, International Journal of Materials Research 102 (2), 160-167, 2011
Zhang ZW, Liu CT, Guo S, Cheng JL, Chen G, Fujita T, Chen MW, Chung YW, Vaynman S, Fine ME, Chin BA,
“Boron effects on the ductility of a nano-cluster-strengthened ferritic steel”,
Sci. Eng. A-Structural Materials Properties Microstructure and Processing 528, 855-859, 2011
Fan, C.; Ren, Y.; Liu, C. T., Liaw, P. K.; Yan, H. G.; Egami, T., Atomic migration and bonding characteristics during a glass transition investigated using as-cast Zr-Cu-Al. Physical Review B 83(19), 11
Liu ZY, Guo S, Liu XJ, Ye JC, Yang Y, Wang XL, Yang L, An K, Liu CT, Micromechanical characterization of casting-induced inhomogeneity in an Al0.8CoCrCuFeNi high-entropy alloy, Scripta Materialia 64, 868-871, 2011
Yang Y, JC Ye, J Lu, and CT Liu, “Dual character of stable shear bands in bulk metallic glasses”, Intermetallics 19, , 2011
Jang JSC, Wu KC, Jian SR, Hsieh PJ, Huang JC, and Liu CT, “A Ni-free Zr-based bulk metallic glass with remarkable plasticity”, J Alloy compounds 509, S109-114, 2011
Liu ZY, Yang Y, Guo S, Liu XJ, Lu J, Liu YH, Liu CT, Cooling rate effect on Young's modulus and hardness of a Zr-based metallic glass. Journal of Alloys and Compounds 509, , 2011
SL, He YH, Liu XL, Zhang QA, Zou J, Huang H, Song M, Huang BY, Liu CT, Du Y, Large-scale synthesis of tungsten single-crystal microtubes via vapor-deposition process. Journal of Crystal Growth 316 (1) 137-144, 2011
Dong, H. X.; He, Y. H.; Jiang, Y.; Wu, L.; Zou, J.; Xu, N. P.; Huang, B. Y.; Liu, C. T., Effect of Al content on porous Ni-Al alloys,
Mater. Sci. Engr. A -Structural Materials Properties Microstructure and Processing 528(13-14), , 2011
SL Wang, Y He, X Liu, H Huang, J Zou, M Song, BY Huang, and C. T. Liu, “Noval C/Cu Sheath/core nanostructures synthesized via low-temperature MOCVD”, Nanotechnology 22, 11
Gao, H. Y.; He, Y. H.; Shen, P. Z.; Zou, J.; Xu, N. P.; Jiang, Y.; Huang, B. Y.; Liu, C. T., Congenerous and heterogeneous brazing of porous FeAl intermetallics. Powder Metallurgy 54 (2), 142-147, 2011
L Wu, J Yao, H Dong, Y He, N Xu, J Zou, BY Huang, CT Liu, “The corrosion behavior of porous Ni3Al intermetallic materials in trong alkali solution”, Intermetallics 19, , 2011
H. X. Dong, T Lei, YH He, N. Xu, BY Huang, C. T. Liu, “Electrochemical performance of porous Ni3Al electrodes for hydrogen evolution reaction”, International J Hydrogen Energy 36, , 2011
Y. Jiang, YH He, BY Huang, J. Zou, H. Huang, NP Xu, C. T. Liu, “Criterion to control self-propagation high-temperature synthesis for porous Ti-Al intermetallics”, Powder Metall. 54, 404-407, 2011
L Huang, PK Liaw, C. T. Liu, Y Liu, JS Huang, “Microstructure evolution of TiAl+ W+B alloy”, Trans Nonferrous Met. Soc. China 21, 11
F. Liu, Yong Liu, Y. Wen, Y. Dou, D. Zhao, C. T. Liu, “Microstructures and mechanical properties of Fe-14Cr-3W-Ti-Y2O3 steel with 1 wt.% of Cu addition fabricated by a new method”, J. Nuclear materials 414, 422-425, 2011
JC Ye, J Lu, CT Liu, Q Wang and Y Yang, "Atomistic free-volume zones and inelastic deformation of metallic glasses", Nature Mater. 9, 619-623, 2010
Liu, XJ, Xu, Y, Hui, X, Lu, ZP, Li, F., Chen, GL, Lu, J, and Liu, CT, "Metallic liquids and glasses: Atomic order and global packing", Phy. Rev. Lett. 105, 10
YA Wu, YH Xiao, GL Chen, CT Liu and ZP Lu, "Bulk metallic glass composites with transformation-mediated work-hardening and ductility", Adv. Mater. 22, 10
Sheng Guo, ZP Lu, and CT Liu, "Identify the best glass forming ability criterion", Intermetallics 18, 883-888, 2010
Teng ZK, Liu CT, Ghosh G, Liaw PK, Fine ME, "Effects of Al on the microstructure and ductility of NiAl- strengthened ferritic steels at room temperature", Intermetallics 7
Liu YH, Liu CT, Gali A, Inoue A, Chen MW, "Evolution of shear bands and its correlation with mechanical response of a ductile Zr55Pd10Cu20Ni5Al10 bulk metallic glass", Intermetallics 5
ZW Zhang, CT Liu, S. Guo, JL Cheng, G Chen, Takeshi Fujita, MW Chen, YW Chung, S Vaynman, ME Fine, BA Chin, "Boron effects on the ductility of a nano-cluster-strengthened ferritic steel", Mater. Sci. Engr. A 528, 855-859, 2010
ZK Teng, MK Miller, G. Ghosh, CT Liu, S. Huang, KF Russsell, ME Fine, and PY Liaw, "Characterization of nanoscale NiAl-type precipitates in a ferritic steel by electron microscopy and atom probe tomography", Scripta Mater. 63, 61-64, 2010
Yang Y, Ye JC, Lu J, Liaw PK, Liu CT, "Characteristic length scales governing plasticity/brittleness of bulk metallic glasses at ambient temperature", Applied Physics Letters 1905
JL Cheng, G Chen, P Gao, CT Liu, and Y Li, "The critical cooling rate and microstructure evolution of Zr41.2Ti13.8Cu12.4Ni10Be22.5 composites by Bridgman solidification", Intermetallics 18,115-118, 2010
JL Cheng, G Chen, F Xu, YL Du, YS Li and CT Liu, "Correlation of the microstructure and mechanical properties of Zr-based in-situ bulk metallic glass", Intermetallics 18, , 2010
Dong HX, Jiang Y, He YH, Zou J, Xu NP, Huang BY, Liu CT, Liaw PK, "Oxidation behavior of porous NiAl prepared through reactive synthesis", Materials Chemistry and Physics
Dong HX, He YH, Zou J, Xu NP, Huang BY, Liu CT, "Effect of preheating treatment at 575 degrees C of green compacts on porous NiAl", Journal of Alloys and Compounds
PZ Shen, M Song, YH He, HY Gao, J Zou, HP Xu, BY Huang, CT Liu, "Synthesis and characterization of porous Fe-25wt.% Al alloys with controllable pore structure", Power Metallurgy and Metal Ceramics 49, 183-92, 2010
YH Liu, CT Liu, WH Wang, A. Inoue, T. Sakurai, and MW Chen, "Thermodynamic Origins of Shear Band Formation and the universal Scaling Law of Metallic Glass Strength", Phy. Rev, Lett. 103, 09
Cheng S, Stoica AD, Wang XL, Ren Y, Almer J, Horton JA, Liu CT, Clausen B, Brown DW, Liaw PK, Zuo L., "Deformation Crossover: From Nano- to Mesoscale",
Physical Review Letters
Ling Yang, Michael K Miller, Xun-Li Wang, CT Liu, AD Stoica, Dong Ma, J Almer, and D Shi, "Nanoscale solute partitioning in bulk metallic glasses", Adv. Mater. 21, 305-308, 2009
Jun Xu, CT Liu, MK Miller, and Hongmin Chen, "Nanocluster-associated vacancies in nanocluster-strengthened ferritic steel as seen via positron-lifetime spectroscopy", Phy. Rev. B 79, 020204 (R), 2009
CH Hsueh, H Bei, CT Liu, EP George, PF Becher, "Controlled normal/shear loading and shear fracture in bulk metallic glasses", Intermetallics 17, 802-810, 2009
JH Schneibel, CT Liu, MK Miller, MJ Mills, P. Sarosi, and M. Heilmaier, "Untrafine-grained nanocluster-strengthened alloys with unusually high creep strength", Scrip. Mater. 61, 793-96, 2009
G Chen, H.BEi, Y Cao, A Gali, CT Liu, and EP George, "Enhanced plasticity in a Zr-based bulk metallic glass composite with in situ formed intermetallic phases", App. Phy. Lett. 95, 09
Cang Fan, C. T. Liu, and H. G. Yan, "Mechanical properties of bulk metallic glasses at cryogenic temperatures", Modern Phy. Lett. 23, , 2009
Fan C, Liaw PK, Liu CT, "Atomistic model of amorphous materials", Intermetallics
Y Yamamoto, ML Santella, CT Liu, ND Evans, PJ Maziasz, MP Brady, "Evaluation of Mn substitution for in alumina-forming austenitic stainless steels", Mater. Sci. Eng. A 524, 176-185, 2009
XY Cheng, Yip-Wah Chung, Bo Zhao, B Hong, and CT Liu, "Effect of Atomic ordering on hydrogen dissociation on Ni3Fe surfaces", App. Phy. Lett. 94, 09
Dong HX, Jiang Y, He YH, Song M, Zou J, Xu NP, Huang BY, Liu CT, Liaw PK, "Formation of porous Ni-Al intermetallics through pressureless reaction synthesis", Journal of Alloys and Compounds
W Kai, IF Ren, RF Wang, PC Kao, and CT Liu, "The oxidation behavior of an Fe61B15Zr8Mo7Co5Y2Cr2 bulk metallic glass at 650C in various oxygen-containing environments", Intermetallics 17, 165-68, 2009
W Kai, IF Ren, RF Wang, PC Kao, and CT Liu, "The oxidation behavior of an Fe61B15Zr8Mo7Co5Y2Cr2 bulk metallic glass at 550-700C", Intermetallics 17, 205-210, 2009
S Wang, Y He, X Fang, J Zou, Y Wang, H Huang, PMFJ Costa, M Song, B Huang, Chain T Liu, PK Liaw, Y Bando, and D Golberg, "Structure and Field-Emission Properties of Sub-Micrometer-sized Tungsten-Whisker arrays fabricated by vapor deposition", Advanced Mater. 21, , 2009
H Huang, YQ Wu, SL Wang, YH He, J. Zou, BY Huang, CT LIU, "Mechanical properties of single crystal tungsten microwhiskers characterized by nanoindentation", Mater. Sci. Engr. A 523, 193-198, 2009
SCJ Jason, SR Jian, TH Li, JC Huang, CYA Tsao, and CT Liu, "Structural and mechanical characterizations of ductile Fe particles-reinforced MG-based bulk metallic glass composites", J. Alloys and Compounds 485, 290-294, 2009
Gao HY, He YH, Shen PZ, Zou J, Xu NP, Jiang Y, Huang BY, Liu CT, "Porous FeAl intermetallics fabricated by elemental powder reactive synthesis", Intermetallics 1
Gao HY, He YH, Shen PZ, Zou J, Xu NP, Jiang Y, Huang BY, Liu CT, "Effects of Al content on porous Fe-Al alloys", Powder Metallurgy
Wall JJ, Almer JD, Vogel SC, Liaw PK, Choo H, Liu CT, "Synchrotron X-ray scattering investigations of oxygen-induced nucleation in a Zr-based glass-forming alloy", Scripta Materialia
Wu Y, Chen GL, Hui XD, Liu CT, Lin Y, Shang XC, Lu ZP, "A quantitative link between microplastic instability and macroscopic deformation behaviors in metallic glasses", Journal of Applied Physics
Zhongwu Zhang, Guang Chen, Hongbin Bei, Feng Li, Feng Ye, Gualiang Chen, Chain-Tsuan Liu, "Directional recrystallization and microstructures of an Fe-6.5Wt%Si alloy", J. Mater. Res. 24 (8), 09
Jang JSC, Jian SR, Chang CF, Chang LJ, Huang YC, Li TH, Huang JC, Liu CT, "Thermal and mechanical properties of the Zr53Cu30Ni9Al8 based bulk metallic glass microalloyed with silicon", Journal of Alloys and Compounds
Liu Y, Wu H, He SW, Liu CT, Huang BY, "Casting effect on softening of metallic glasses", Journal of Alloys and Compounds
Shen PZ, He YH, Gao HY, Zou J, Xu NP, Jiang Y, Huang BY, Liu CT, "Development of a new graded-porosity FeAl alloy by elemental reactive synthesis", Desalination
Shen PZ, Song M, Gao HY, He YH, Zou J, Xu NP, Huang BY, Liu CT, "Structural characteristics and high-temperature oxidation behavior of porous Fe-40 at.%Al alloy",
Journal of Materials Science 3
Yao Jiang, Chuping Deng, Yuehui He, Yap Zhao, Nanping Xu, Jin Zou, Baiyun Huang, C. T. Liu, "Reactive synthesis of microporous titanium-aluminide membranes", Mater. Lett. 63, 22-24, 2009
Jang JSC, Tsao SF, Chang L, Huang JC, Liu CT, "Nanocrystallization of Zr61Al7.5Cu17.5Ni10Si4 metallic glass", Intermetallics
Jiang Y, Deng CP, He YH, Zhao Y, Xu NP, Zou J, Huang BY, Liu CT, "Reactive synthesis of microporous titanium-aluminide membranes", Materials Letters
CH Hsueh, H Bei, CT Liu, PF BEcher and EP George, "Shear fracture of bulk metallic glasses with controlled applied normal stresses", Scripa Mater. 59, 111-114, 2008
M. P. Brady, Y. Yamamoto, M. L. Santella, P. J. Maziasz, B. A. Pint, C. T. Liu, Z. P. Lu, H. Bei, "Development of Alumina-Forming Austenitic Stainless Steels for High-Temperature Structural Use", Journal of Metals (JOM), pp 12-18, July issue, 2008
ZP Lu, H Bei, Y Wu, G. L. Chen, EP George and CT Liu, "Oxygen effects on plastic deformation of a Zr-based bulk metallic glass", Appl. Phys. Lett. 92, 08
JJ Wall, CT Liu, WK Rhim, JJZ Li, PK Liaw, H Choo, and WL John, "Heterogeneous nucleation in a glass-forming alloy", Appl. Phys. Lett. 92, 08
Cang Fan, D Chen, PK Liaw, H Chou, C Benmore, J Siewenie, GL Chen, JX Xie, and CT Liu, "Influence of molten status on nanoquasicrystalline-forming Zr-based metallic Glasses", Appl. Phys. Lett. 93, 08
M.P. Brady, Y. Yamamoto, Z.P. Lu, P.J. Maziasz, C.T. Liu, B.A. Pint, and M.L. Santella, "Alumina-Forming Austenitics: A New Class of Heat-Resistant Stainless Steels", Stainless Steel World Magazine, pp 23-29, March 2008
ZP Lu, CT Liu, Y WU, H Tan, Y Li, GL Chen, "Composition effects on glass forming ability and its indicator γ", Intermetallics 16, 410-417, 2008
Y. Yamamoto, M. Takeyama, Z.P. Lu, C.T. Liu, N.D. Evans, P.J. Maziasz, and M.P. Brady, "Alloying effects on creep and oxidation resistance of austenitic stainless steel alloys employing intermetallic precipitates", Intermetallics 16, 453-462, 2008
XJ Liu, Chen GL, Hou HY, Hui XD, Lu ZP, Yao KF, CT Liu, "Atomistic mechanism for nanocrystallization in metallic glasses", Acta Mater 56(12), , 2008
XJ Wan, YX Chen, DD Shi, and CT Liu, "Effect of alloy stochiometry and boron doping on the H2 induced encironmental embrittlement of Ni3Fe intermetallics", Interemetallics 16, 550-553, 2008
SL Wang, Y. H. He, J. Zou, Y. Wang, H. Huang, BY Huang, CT Liu, and PK Liaw, "Catalytic Growth of Metallic tungsten whiskers based on the vapor-solid-solid mechanism", Nanotech. 19, 08
M.P. Brady, Y. Yamamoto, Z.P. Lu, P.J. Maziasz, C.T. Liu, B.A. Pint, and M.L. Santella, "Alumina-Forming Austenitics: A New Class of Heat-Resistant Stainless Steels", Chinese Journal of Nature (in Chinese) 30(3), pp 1-5, 2008
M. K. Miller, C. L. Fu, M. Krcmar, D. T. Hoelzer, and C. T. Liu, "Vacancies as a constitutive element for the design of nanocluster-strengthened ferritic steels (abstract in English and text in Chinese)", Shanghai Metals 30, No. 4, 1-6, 2008
ZW Zhang, G Chen, H Bei, F Ye, GL Chen, and CT Liu, "Improvement of magnetic properties of an Fe-6.5 wt. % Si alloy by directional recrystallization", App. Phy. Lett. 93, 08
MK Miller, CL Fu, M Krcmar, DT Hoelzer, and CT Liu, "Vacancies as a constitutive element for the design of nanocluster-strengthened ferritic steels", Front. Mater. Sci. China, DOI 10.-009-08
C. T. Liu, C. L. Fu, M. F. Chisholm, J. R. Thompson, Maja Kremar, and X.-L. Wang, "Magnetism and solid solution effects in NiAl (40% Al) alloys", Prog. Mater. Sci. 52, 352-370, 2007
Y. Yamamoto, M. P. Brady, Z. P. Lu, P. J. Maziasz, C. T. Liu, B. A. Pin, K. L. More, H. M. Meyer, and E. A. Payzant, "Creep-resistant, Al2O3-forming Austanitic Stainless Steels", SCIENCE 316, p. 433-436, 2007
L. Yang, X.-L. Wang, C. T. Liu, J. a. Fernandez-Beca, C. L. Fu, J. W. Richardson and D. Shi, "Neutron diffraction study of the structure and low-temperature phase transformation in ternary NiAl + M (M= Ni, Fe, Co) alloy", Scripta Mater. 56, 911-914, 2007
JH Schneibel, CT Liu, DT Hoelzer, MJ Mills, P Sarosi, T Hayashi, U Wendt and H. Heyse, "Development of porosity in an oxide dispersion-strengthened ferritic alloy containing nanoscale oxide particles", Scripta Materi. 57, , 2007
ZP Lu, Y Liu, and CT Liu, "Evaluation of Glass-Forming Ability", pp 87-115 in the book "Bulk Metallic Glass", Springer, 2007
Yuehui He, Yao Jiang, Nanping Xu, Jin Zou, Baiyun Huang, CT Liu and PK Liaw, "Fabrication of Ti-Al micro/nanometer-sized Porous alloys through the Kirkendall effect", Adv. Mater. 19, , 2007
Y. Liu, C. T. Liu, E. P. George, and X. Z. Wang, "The Soret effect in bulk metallic glasses", Intermetallics 15, 557-563 (2007)
ZP Lu, H. Bei, and CT Liu, "Recent progress in quantifying glass-forming ability of bulk metallic glasses", Intermetallics 15, 618-624, 2007
ZP Lu and CT Liu, "A scheme to design multicomponent bulk metallic glasses", Mater. Trans., JIM 48, , 2007
Shiliang Wang, Yuehui He, Jin Zou, Peng Cao, Yao Jiang, Baiyun Huang, CT. Liu, and PK Liaw, "Synthesis of tungsten oxide tapered needles with nanotips", J. Crystal Growth 303, 574-59, 2007
XH Du, JC Huang, CT Liu, and ZP Lu, "New criterion of glass forming ability for bulk metallic glasses", J. Appl. Phys. 101, 07
Y Yamamoto, MP Brady, ZP Lu, CT Liu, M Takeyama, PJ Maziasz, and BA Pint, "Alumina-forming austenitic stainless steels strengthened by Laves phase and MC carbide predipitates", Metall. Materi Trans. 38A, , 2007
XJ Liu, Chen G L, Hui X D, Hou H Y, Yao K F, CT Liu, "Growth mechanism from nano-ordered clusters to nanocrystals in a deeply undercooled melt of Zr-Ni-Ti metallic glass", J Appl Phys 102(6):
to -5, 2007
Cang Fan, P. K. Liaw, T. W. Wilson, H. Choo and C. T. Liu, "Structure and mechanical behavior of amorphous alloys and their in-situ composites", pp. 93-112 in the book "The world of bulk metallic glasses and their composites", Research Signpost, Kerala, India, 2007
G. Y. Wang, P. K. Liaw, Y. Yokoyama, W. H. Peter, B. Yang, M. Freels, R. A. Bucanan, C. T. Liu, C. R. Brooks, "Influence of air and vacuum environment on fatigue behavior of Zr-based bulk metallic glasses", J. Alloy and Compounds 434-435, 68-70, 2007
GY Wang, PK Liaw, Y. Yokoyama, A. Peker, WH Peter, B. Yang, M. Freels, ZY Zhang, V. Keppens, R. Hermann, RA Bucanan, CT Liu, and CR Brook, "Studying fatigue behavior and Poisson's ratio of bulk-metallic glasses", Intermetallics 15, 663-667, 2007
TH Hung, YC Chang, YN Wang, CW Tang, JN Kuo, HM Chen, YL Tsai, JC Huang, JSC Jang, and CT Liu, "Development of Mg based amorphous alloys with higher amounts of rare earth elements", Mater. Trans, JIM 48, 07
JSC Jang, CC Tseng, LJ Chang, CF Chang, WJ Lee, JC Huang, and CT Liu, "Glass forming ability and thermal properties of the Mg-based amorphous alloys with dual rare earth elements addition", Mater. Trans, JIM 48, 07
YP Deng, YF Guan, JD Fowlkes, SQ Wen, FX Liu, GM Pharr, PK Liaw, CT Liu, and PD Rock, "A combinatorial thin film sputtering approach for synthesizing and characterizing ternary ZrCuAl metallic glasses", Intermetallics 15, , 2007
Shiliang Wang, Yuehui He, Jin Zou, Yao Jiang, Jian Xu, Baiyun Huang, CT. Liu, and PK Liaw, "Synthesis of single crystal tungsten nanowires by nickel-catalyzed vapor-phase method at 850C", J. Crystal Growth 306, 433-436, 2007
M. L. Morrison, R. A. Buchanan, P. K. Liaw, B. A. Green, G. Y. Wang, C. T. Liu and J. A. Horton, "Four-Point-Bending-Fatigue Behavior of the Zr-based Vitreloy 105 Bulk Metallic Glass", Mater. Sci. Eng. A 467, 190-197, 2007
M. L. Morrison, R. A. Buchanan, P. K. Liaw, B. A. Green, G. Y. Wang, C. T. Liu and J. A. Horton, "Corrosion-Fatigue Studies of the Zr-based Vitreloy 105 Bulk Metallic Glass", Mater. Sci. Eng. A 467, 198-206, 2007
M. K. Miller, C. T. Liu, J. A. Wright, W. Tang and K. Hildal, "APT characterization of some iron-based bulk metallic gasses", J. Intermetallics 14, , 2006
J. S. C. Jang, L. J. Chang, T. H. Hung, J. C. Huang, and C. T. Liu, "Thermal stability and crystallization of Zr-Al-Cu-Ni based amorphous alloy added with born and silicon", J. Intermetallics 14, 951-957, 2006
Y. Liu, C. T. Liu, E. G. George, and X. Z. Wang, "Thermal diffusion and compositional inhomgeneity in cast Zr50Cu50 bulk metallic glass", Appl. Phys. Lett. 89, 06
B. Yang, C. T. Liu, and T. G. Nieh, "Unified equation for the strength of bulk metallic glasses", Appl. Phys. Lett. 88, 06
G. Y. Sun, G. Chen, C. T. Liu, and G. L. Chen, "Innovative processing and property improvement of metallic glass based composites", Script Mater. 55, 375-378, 2006
J.J. Wall JJ, C. Fan, P.K. Liaw, C.T. Liu and H. Choo, "A Combined Drop / Suction Casting Machine for the Manufacture of Bulk-Metallic Glass Materials", Review of Scientific Instruments 77, Art. No. 6)
C. Fan, H. Li, L. J. Kecskes, K. Tao, H. Choo, P. K. Liaw, and C. T. Liu, "Mechanical Behavior of Bulk Amorphous Alloys Reinforced by Ductile Particles in Cryogenic Temperature", Phys. Rev. Lett. 96, 06)
Z. P. Lu, J. Shen, D. W. Xing, J. F. Sun, and C. T. Liu, "Binary eutectic clusters and glass formation in ideal glass-formation liquids", Appl. Phys. Lett. 89, 06
Cang Fan, P. K. Liaw, T.W. Wilson, H. Choo, Y. F. Gao, and C. T. Liu, "Pair distribution function study and mechanical behavior of as-cast and structurally relaxed Zr-based bulk metallic glasses", Appl. Phys. Lett. 89, 06
Cang Fan, P. K. Liaw, T. W. Wilson, W. Dmowski, Th. Proffen, H. Choo, and C.T. Liu, "Structural Model for Bulk Amorphous Alloys", Appl. Phys. Lett. 89, 06
Cang Fan, P. K. Liaw, V. Haas, J.J. Wall, H. Choo, A. Inoue, and C. T. Liu, "Structures and Mechanical Behaviors of Zr55Cu35Al10 Bulk Amorphous Alloys at Ambient and Cryogenic Temperatures", Phys. Rev. B 74, 06
L. Xia, S. S. Fang, Q. Wang, Y. D. Dong, and C. T. Liu, "Thermodynamic modeling of glass formation in metallic glasses", App. Phys. Lett. 88, 06
B. Yang, C. T. Liu, T. G. Nieh, M. L. Morrison, P. K. Liaw, R. A. Buchanan, "Localized heating and fracture criterion for bulk metallic glasses", J. Mater. Res. 21(4), 915-922, 2006
G. L. Chen, X. L. Liu, D. Hui, H. Y. Hou, C. T. Liu, and J. Wadsworth, "Molecular dynamic simulations and atomic structures of amorphous materials", Appl. Phys. Lett. 88, 06
Y. Liu, L. F. Chen, H. P. Tang, C. T. Liu, and B. Y. Huang, "Design of powder metallurgy titanium alloys and composites", Mater. Sci. Engr. A 418, 25-35, 2006
Yuehui He, Shiliang Wang, Baiyub Huang, C. T. Liu, and P. K. Liaw, "Novel tungsten oxide microneedles with nanosized tips", Appl. Phys. Lett. 88, 06
Last modified on 20 March, 2013