Hardness and fracture toughness are some of the most important mechanical properties. Here, we propose a simple model that uses only the elastic properties to calculate the hardness and fracture toughness. Its accuracy is checked by comparison with other available models and experimental data for metals, covalent and ionic crystals, and bulk metallic glasses. We found the model to perform well on all datasets for both hardness and fracture toughness, while for auxetic materials (i.e., those having a negative Poisson’s ratio), it turned out to be the only model that gives reasonable hardness. Predictions are made for several materials for which no experimental data exist.
REFERENCES
1.
I.
Maslenikov
, A.
Useinov
, A.
Birykov
, and V.
Reshetov
, “Reducing the influence of the surface roughness on the hardness measurement using instrumented indentation test
,” IOP Conf. Ser. Mater. Sci. Eng.
256
, 012003
(2017
). 2.
C. M.
Sung
and M.
Sung
, “Carbon nitride and other speculative super hard materials
,” Mater. Chem. Phys.
43
, 1
–18
(1996
). 3.
A. R.
Oganov
and A. O.
Lyakhov
, “Towards the theory of hardness of materials
,” J. Superhard Mater.
32
, 143
(2010
). 4.
D. M.
Teter
, “Computational alchemy: The search for new superhard materials
,” MRS Bull.
23
(1
), 22
–27
(1998
). 5.
X.
Chen
, H.
Niu
, D.
Li
, and Y.
Li
, “Modeling hardness of polycrystalline materials and bulk metallic glasses
,” Intermetallics
19
, 1275
–1281
(2011
). 6.
F.
Gao
, J.
He
, E.
Wu
, S.
Liu
, D.
Yu
, D.
Li
, S.
Zhang
, and Y.
Tian
, “Hardness of covalent crystals
,” Phys. Rev. Lett.
91
, 015502
(2003
). 7.
A.
Šimůnek
and J.
Vackář
, “Hardness of covalent and ionic crystals: First-principle calculations
,” Phys. Rev. Lett.
96
, 085501
(2006
). 8.
K.
Li
, X.
Wang
, F.
Zhang
, and D.
Xue
, “Electronegativity identification of novel superhard materials
,” Phys. Rev. Lett.
100
, 235504
(2008
). 9.
A. O.
Lyakhov
and A. R.
Oganov
, “Evolutionary search for superhard materials: Methodology and applications to forms of carbon and
,” Phys. Rev. B
84
, 092103
(2011
). 10.
H.
Niu
, S.
Niu
, and A. R.
Oganov
, “Simple and accurate model of fracture toughness of solids
,” J. Appl. Phys.
125
, 065105
(2019
). 11.
L. C.
Allen
, “Electronegativity is the average one-electron energy of the valence-shell electrons in ground-state free atoms
,” J. Am. Chem. Soc.
111
, 9003
(1989
). 12.
A. R.
Franco
, Jr., G.
Pintaude
, A.
Sinatora
, C. E.
Pinedo
, and A. P.
Tschiptschin
, “The use of a Vickers indenter in depth sensing indentation for measuring elastic modulus and Vickers hardness
,” Mater. Res.
7
(3
), 483
–491
(2004
). 13.
J. P.
Perdew
, K.
Burke
, and M.
Ernzerhof
, “Generalized gradient approximation made simple
,” Phys. Rev. Lett.
77
, 3865
(1996
). 14.
P. E.
Blochl
, “Projector augmented-wave method
,” Phys. Rev. B
50
, 17953
(1994
). 15.
G.
Kresse
and J.
Hafner
, “Ab initio molecular dynamics for liquid metals
,” Phys. Rev. B
47
, 558(R)
(1993
). 16.
G.
Kresse
and J.
Furthmüller
, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
,” Phys. Rev. B
54
, 11169
(1996
). 17.
R.
Hill
, “The elastic behaviour of a crystalline aggregate
,” Proc. Phys. Soc.
65
, 349
(1952
). 18.
R. A.
Andrievski
, “Superhard materials based on nanostructured high-melting point compounds: Achievements and perspectives
,” Int. J. Refract. Met. Hard Mater.
19
, 447
(2001
). 19.
V. L.
Solozhenko
, D.
Andrault
, Y. L.
Godec
, and M.
Mezouar
, “Ultimate metastable solubility of boron in diamond: Synthesis of superhard diamondlike
,” Phys. Rev. Lett.
102
, 015506
(2009
). 20.
The Science of Hardness Testing and its Research Applications, edited by J. H. Westbrook and H. Conrad (American Society for Metals, Materials Park, Ohio, 1973).
21.
V. L.
Solozhenko
, O. O.
Kurakevych
, and A. R.
Oganov
, “On the hardness of a new boron phase, orthorhombic -
,” J. Superhard Mater.
30
, 428
(2008
). 22.
V.
Domnich
, S.
Reynaud
, R. A.
Haber
, and M.
Chhowalla
, “Boron carbide: Structure, properties, and stability under stress
,” J. Am. Ceram. Soc.
94
(11
), 3605
–3628
(2011
). 23.
D. W.
He
, Y. S.
Zhao
, L.
Daemen
, J.
Qian
, D. T.
Shen
, and T. W.
Zerda
, “Boron suboxide: As hard as cubic boron nitride
,” Appl. Phys. Lett.
81
, 643
–645
(2002
). 24.
I.
Yonenaga
and T.
Suzuki
, “Indentation hardnesses of semiconductors and a scaling rule
,” Philos. Mag. Lett.
82
, 535
–542
(2002
). 25.
S. G.
Huang
, K.
Vanmeensel
, O.
Van der Biest
, and J.
Vleugels
, “Binderless WC and WC-VC materials obtained by pulsed electric current sintering
,” Int. J. Refract. Met. H
26
, 41
–47
(2008
). 26.
M.
Hebbache
, L.
Stuparević
, and D.
Živković
, “A new superhard material: Osmium diboride
,” Solid State Commun.
139
(5
), 227
–231
(2006
). 27.
F.
Fernandes
, T.
Christiansen
, K.
Dahl
, and M.
Somers
, “Growth of vanadium carbide by halide-activated pack diffusion
,” in Proceedings of Heat Treatment and Surface Engineering (2013), pp. 295
–301
.28.
Q.
Gu
, G.
Krauss
, and W.
Steurer
, “Transition metal borides: Superhard versus ultra-incompressible
,” Adv. Mater.
20
(19
), 3620
–3626
(2008
). 29.
D.
Ferro
, S. M.
Barinov
, J. V.
Rau
, A.
Latini
, R.
Scandurra
, and B.
Brunetti
, “Vickers and Knoop hardness of electron beam deposited ZrC and HfC thin films on titanium
,” Surf. Coat. Technol.
200
, 4701
–4707
(2006
). 30.
S.
Cardinal
, A.
Malchere
, V.
Garnier
, and G.
Fantozzi
, “Microstructure and mechanical properties of TiC-TiN based cermets for tools application
,” Int. J. Refract. Met. H
27
, 521
–527
(2009
). 31.
L.
Ćurković
, V.
Rede
, K.
Grilec
, and A.
Mulabdić
, “Hardness and fracture toughness of alumina ceramics,” in Proceedings of the Conference on Materials, Processes, Friction and Wear, Vela Luka, Croatia (Croatian Society for Materials and Tribology
, 2007
).32.
J.
Chen
, V. V.
Struzhkin
, Z.
Wu
, M.
Somayazulu
, J.
Qian
, S.
Kung
, A. N.
Christensen
, Y.
Zhao
, R. E.
Cohen
, H.
Mao
, and R. J.
Hemley
, “Hard superconducting nitrides
,” Proc. Natl. Acad. Sci. U.S.A.
102
(9
), 3198
–3201
(2005
). 33.
A.
Nino
, A.
Tanaka
, S.
Sugiyama
, and H.
Taimatsu
, “Indentation size effect for the hardness of refractory carbides
,” Mater. Trans.
51
, 1621
–1626
(2010
). 34.
M. D.
Drory
, J. W.
Ager
III, T.
Suski
, I.
Grzegory
, and S.
Porowski
, “Hardness and fracture toughness of bulk single crystal gallium nitride
,” Appl. Phys. Lett.
69
, 4044
–4046
(1996
). 35.
S. E.
Grillo
, M.
Ducarroir
, M.
Nadal
, E.
Tournie
, and J. P.
Fanrie
, “Nanoindentation of Si, GaP, GaAs and ZnSe single crystals
,” J. Phys. D Appl. Phys.
35
(1
), 3015
(2002
). 36.
V.
Domnich
, Y.
Aratyn
, W. M.
Kriven
, and Y.
Gogotsi
, “Temperature dependence of silicon hardness: Experimental evidence of phase transformation
,” Rev. Adv. Mater. Sci.
17
, 33
–41
(2008
).37.
I.
Yonenaga
, Y.
Ohkubo
, M.
Deura
, K.
Kutsukake
, Y.
Tokumoto
, Y.
Ohno
, A.
Yoshikawa
, and X. Q.
Wang
, “Elastic properties of indium nitrides grown on sapphire substrates determined by nano-indentation: In comparison with other nitrides
,” AIP Adv.
5
(7
), 077131
(2015
). 38.
R. K.
Banerjee
and P.
Feltham
, “Deformation and fracture of germanium single crystals
,” J. Mater. Sci.
9
, 1478
–1482
(1974
). 39.
T. H.
Fang
, W. J.
Chang
, and C. M.
Lin
, “Nanoindentation characterization of ZnO thin films
,” Mater. Sci. Eng. A
452–453
, 715
(2006
). 40.
J. W.
Cha
, S. C.
Hwang
, and E. S.
Lee
, “Evaluation of surface machinability using ultra-precision lapping process with IED
,” J. Mech. Sci. Technol.
23
, 1194
–1201
(2009
). 41.
A. K. M. A.
Shawon
and S.-C.
Ur
, “Mechanical and thermoelectric properties of bulk AlSb synthesized by controlled melting, pulverizing and subsequent vacuum hot pressing
,” Appl. Sci.
9
, 1609
(2019
). 42.
D.
Maske
, M.
Deshpande
, R.
Choudhary
, and D.
Gadkari
, “Lattice constant and hardness of InSb:Bi bulk crystals grown by vertical directional solidification
,” AIP Conf. Proc.
1728
, 020496
(2016
).43.
A.
Chmel
, A.
Dunaev
, I.
Shcherbakov
, and A.
Sinani
, “Luminescence from impact- and abrasive-damaged ZnS ceramics
,” Proc. Str. Int.
9
, 3
–8
(2018
). 44.
A. S.
Borshchevski
, N. A.
Goryunova
, and N. K.
Takhtare
, “Investigation of the microhardness of some semiconductors with a structure of sphalerite
,” J. Tech. Phys.
27
, 1408
–1413
(1957
).45.
Handbook of Chemistry and Physics, 85 ed., edited by D. R. Lide (CRC Press, 2004), Vol. 85.
46.
47.
C.
Ascheron
, H.
Neumann
, G.
Kühn
, and C.
Haase
, “Microhardness of ZnSe and its change by proton implantation
,” Cryst. Res. Technol.
24
(12
), 1275
–1279
(1989
). 48.
V.
Keryvin
, V. H.
Hoang
, and J.
Shen
, “Hardness, toughness, brittleness and cracking systems in an iron-based bulk metallic glass by indentation
,” Intermetallics
17
, 211
–217
(2009
). 49.
W. H.
Wang
, “Correlations between elastic moduli and properties in bulk metallic glasses
,” J. Appl. Phys.
99
, 093506
(2006
). 50.
D. H.
Xu
, G.
Duan
, W. L.
Johnson
, and C.
Garland
, “Formation and properties of new Ni-based amorphous alloys with critical casting thickness up to 5 mm
,” Acta Mater.
52
, 3493
–3497
(2004
). 51.
Y.
Zhang
, D. Q.
Zhao
, M. X.
Pan
, and W. H.
Wang
, “Glass forming properties of Zr-based bulk metallic alloys
,” J. Non-Cryst. Solids
315
, 206
–210
(2003
). 52.
S.
Li
, R. J.
Wang
, M. X.
Pan
, D. Q.
Zhao
, and W. H.
Wang
, “Formation and properties of RE (RE = Y, Ce, La, Pr, Nd, Gd, Tb, Dy, Ho and Er) bulk metallic glasses
,” J. Non-Cryst. Solids
354
, 1080
(2008
).53.
J.
Dagdelen
, J.
Montoya
, M.
De Jong
, and K.
Persson
, “Computational prediction of new auxetic materials
,” Nat. Commun.
8
, 323
(2017
). 54.
D. M.
Hatch
, S.
Ghose
, and J. L.
Bjorkstam
, “The - phase transition in AlPO cristobalite: Symmetry analysis, domain structure and transition dynamics
,” Phys. Chem. Miner.
21
, 67
–77
(1994
). 55.
R. G.
Munro
, S. W.
Freiman
, and T. L.
Baker
, Fracture Toughness Data for Brittle Materials
(NIST
, 1998
).56.
M.
Weber
, Handbook of Optical Materials, Laser and Optical Science and Technology
(Taylor and Francis
, 2002
).57.
G.
Anstis
, P.
Chantikul
, B. R.
Lawn
, and D.
Marshall
, “A critical evaluation of indentation techniques for measuring fracture toughness I, direct crack measurements
,” J. Am. Ceram. Soc.
64
, 539
(1981
). 58.
I.
Tanaka
, H.-J.
Kleebe
, M. K.
Cinibulk
, J.
Bruley
, D. R.
Clarke
, and M.
Rühle
, “Calcium concentration dependence of the intergranular film thickness in silicon nitride
,” J. Am. Ceram. Soc.
77
, 911
(1994
). 59.
P.
Lemaitre
, “Fracture toughness of germanium determined with the Vickers indentation technique
,” J. Mater. Sci. Lett.
7
, 895
(1998
). 60.
G.
Michot
, A.
George
, A.
Chabli-Brenac
, and E.
Molva
, “Fracture toughness of pure and in-doped GaAs
,” Scr. Metal.
22
, 1043
(1988
). 61.
F.
Ericson
, S.
Johansson
, and J. A.
Schweitz
, “Hardness and fracture toughness of semiconducting materials studied by indentation and erosion techniques
,” Mater. Sci. Eng. A
105
, 131
(1988
). 62.
M. J.
Tafreshi
, K.
Balakrishnan
, and R.
Dhanasekaran
, “Microhardness and optical studies on CdS single crystals grown by sublimation and hydrogen transport techniques
,” Mater. Res. Bull.
30
, 1387
(1995
). 63.
N.
Nrita
, K.
Higashida
, and S.
Kitano
, “Dislocation distribution around a crack tip and the fracture toughness in NaCl crystals
,” Scr. Metal.
21
, 1273
(1987
). 64.
J. D.
Rigney
and J. J.
Lewandowsk
, “Fracture toughness of monolithic nickel aluminide intermetallics
,” Mater. Sci. Eng. A
149
, 143
(1992
). 65.
P.
Specht
and P.
Neumann
, “Fracture planes and toughness of stoichiometric FeAl single crystals
,” Intermetallics
3
, 365
(1995
). 66.
J.
Ast
, B.
Merle
, K.
Durst
, and M.
Goken
, “Fracture toughness evaluation of NiAl single crystals by microcantilevers—A new continuous J-integral method
,” J. Mater. Res.
31
, 3786
(2016
). 67.
W.
Gerberich
, S.
Venkataraman
, J.
Hoehn
, and P.
Marsh
, Structural Intermetallics
(TMS
, Warrendale, PA
, 1993
).68.
X. Y.
Cheng
, X. Q.
Chen
, D. Z.
Li
, and Y. Y.
Li
, “Computational materials discovery: The case of the W-B system
,” Acta Cryst. C
70
, 85
(2014
). 69.
A. G.
Kvashnin
, H. A.
Zakaryan
, C.
Zhao
, Y.
Duan
, Y. A.
Kvashnina
, C.
Xie
, H.
Dong
, and A. R.
Oganov
, “New tungsten borides, their stability and outstanding mechanical properties
,” J. Phys. Chem. Lett.
9
, 3470
–3477
(2018
). 70.
G. N.
Greaves
, A. L.
Greer
, R. S.
Lakes
, and T.
Rouxel
, “Poisson’s ratio and modern materials
,” Nat. Mat.
10
, 823
(2011
). 71.
D. P.
Wang
, D. Q.
Zhao
, D. W.
Ding
, H. Y.
Bai
, and W. H.
Wang
, “Understanding the correlations between Poisson’s ratio and plasticity based on microscopic flow units in metallic glasses
,” J. Appl. Phys.
115
(12
), 123507
(2014
). 72.
S. F.
Pugh
, “Relations between the elastic moduli and the plastic properties of polycrystalline pure metals
,” Philos. Mag.
45
, 823
–843
(1954
).73.
A. G.
Kvashnin
, Z.
Allahyari
, and A. R.
Oganov
, “Computational discovery of hard and superhard materials
,” J. Appl. Phys.
126
(4
), 040901
(2019
). © 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.