Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles
The uniaxial and hydrostatic deformations of martensitic structure B19’ of NiTi shape memory alloy are studied using first-principles calculations. The bulk and Young’s moduli and the theoretical strength under uniaxial tension and hydrostatic loading are computed from crystal response to applied de...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2008
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| Cite this: | Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles / P. Sestak, M. Cerny, J. Pokluda // Проблемы прочности. — 2008. — № 1. — С. 20-23. — Бібліогр.: 7 назв. — англ. |
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| author | Sestak, P. Cerny, M. Pokluda, J. |
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| citation_txt | Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles / P. Sestak, M. Cerny, J. Pokluda // Проблемы прочности. — 2008. — № 1. — С. 20-23. — Бібліогр.: 7 назв. — англ. |
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| description | The uniaxial and hydrostatic deformations of martensitic structure B19’ of NiTi shape memory alloy are studied using first-principles calculations. The bulk and Young’s moduli and the theoretical strength under uniaxial tension and hydrostatic loading are computed from crystal response to applied deformations. The behavior of angle β of the B 19’ structure was investigated along the whole deformation path. The computed values of Young’s moduli are compared with available experimental results. The results obtained complement and extend the already known characteristics of NiTi alloy.
Проведено исследование одноосной и гидростатической деформаций мартенситной структуры B19’ сплава с памятью формы NiTi по методу расчетов ab initio. Величины объемного модуля упругости, модуля Юнга и теоретической прочности при одноосном растяжении и гидростатическом нагружении рассчитаны по реакции кристалла на приложенные деформации. Изучен характер изменения угла β структуры B19’ по всей траектории деформации. Выполнено сопоставление расчетных значений модуля Юнга с имеющимися экспериментальными результатами. Полученная информация дополняет и расширяет уже известные данные о характеристиках сплава NiTi.
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| first_indexed | 2025-12-07T18:39:33Z |
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UDC 539. 4
E la s t ic P r o p e r t ie s o f B 1 9 ’ S tr u c tu r e o f N iT i A l lo y u n d e r U n ia x ia l a n d
H y d r o s ta t ic L o a d in g f r o m F ir s t P r in c ip le s
P . S estak ,1a M . C ern y ,1b and J . P o k lu d a 1c
1 Institute o f Physical Engineering, Faculty o f Mechanical Engineering, Brno University o f
Technology, Brno, Czech Republic
a sestak@kn.vutbr.cz, b cerny.m@fme.vutbr.cz, c pokluda@fme.vutbr.cz
The uniaxial and hydrostatic deformations o f martensitic structure B 19’ o f NiTi shape memory alloy
are studied using first-principles calculations. The bulk and Young’s moduli and the theoretical
strength under uniaxial tension and hydrostatic loading are computed from crystal response to
applied deformations. The behavior o f angle 5 o f the B 19’ structure was investigated along the
whole deformation path. The computed values o f Young’s moduli are compared with available
experim ental results. The results obtained com plem ent and extend the already known
characteristics o f NiTi alloy.
K eyw o rd s : N iT i, B 1 9 ’, shape m em ory alloy, first principles, ab-initio, elastic properties,
theoretical strength, uniaxial and hydrostatic deformation.
In trod uction . The shape m em ory alloys (SM A ) are important materials for many
industrial as w e ll as m edical applications ow ing to their shape m em ory effect. This effect
is connected w ith a transformation betw een martensitic and austenitic structure, w hich can
be started by an external pressure or temperature. There are several types o f the
transformations, depending on a particular alloy.
The nickel-titanium alloy is one o f the m ost important types o f SM A. It is w idely
used in m edicine (stents, bone im plants, etc.). The N iT i alloy can transform from the
m onoclin ic B 1 9 ’ (m artensitic) to cubic B 2 (austenitic) structure and v ice versa. A n
extensive overview o f a current state o f the art can be found in the paper b y Otsuka and
Ren [1].
The aim o f this work is to com pute the Young E and bulk B m oduli and the
theoretical strengths under uniaxial and hydrostatic (isotropic) loading. The Young
m odulus has been com puted for three crystallographic directions (E [100], E [010], and
E [001]) parallel to prim itive translation vectors (r1, r2 , and ^ ) show n in Fig. 1. The
dependence o f the 5 angle (betw een r1 and r 3 vectors) on the applied uniaxial
deform ation has been also investigated.
[Û10] N
r„
-- rx [1ÛQ]
Fig. 1. The martensitic structure (B19’) o f NiTi alloy with marked crystallographic directions.
© P. SESTÂK, M. CERNŸ, J. POKLUDA, 2008
20 ISSN 0556-171X. Проблемы прочности, 2008, № 1
mailto:sestak@kn.vutbr.cz
mailto:cerny.m@fme.vutbr.cz
mailto:pokluda@fme.vutbr.cz
Elastic Properties o f B 1 9 ’ Structure o f NiTi Alloy
E lectron ic S tru ctu re C om pu tations. A ll quantities o f interest are com puted from
the total energy E total o f the system at hand as a function o f an appropriate deform ation
and from the H ellm an-Feynm an stress tensor
The total energies and stresses w ere com puted b y the A binit program code. A binit is
a great tool for electronic structure calculations developed by the team o f Prof. X avier
G onze at the U niversity Catholique de Louvain [2, 3], w hich is distributed under the G N U
General Public L icense. Another additional package including pseudo-potentials [4]
together w ith its generators, manuals, tutorials, exam ples, etc. is available at [5].
The calculations w ere perform ed using G G A norm -conserving pseudo-potentials and
the cu to ff energy w as set to 1000 eV for com putations o f elastic m oduli and 800 eV for
theoretical strength evaluation. The solution w as considered to be self-consistent w hen the
energy difference o f three consequent iterations becam e sm aller than 0.1 f ie V for
com putation o f elastic properties and 1.0 i e V for the theoretical strength.
During the uniaxial deform ation the structure m ust be relaxed in order to a llow the
Poisson contraction. The relaxations w ere m ade by an external procedure utilizing the
Hellm an-Feynm an stress tensor [6 ] com puted by the A binit code.
C om p u tation o f E la stic M od u li and T h eoretica l S trength . The Y oung’s m odulus
can be com puted as
1 d E total
E = V 0 “ i 2- , (1)
where £ is the relative uniaxial deform ation, £ = a /a 0 — L Sim ilarly, the uniaxial stress
can be evaluated as
_ uni 1 d E total
V0 d£ (2 )
The bulk m odulus and the hydrostatic stress were calculated according to the
relations
1 d Etotal
B = (3)
and
a hyd = ± ^ to to L ,
V0 dv (4)
where v is the relative volum e v = V0 / V — I Both stresses approach their m axim a at the
points o f inflection o f E totai ( e ) or E totai ( v ) dependences. I f no other instability appears
before reaching the points, their m axim um values specify the corresponding theoretical
strength values o id .
R esu lts. The experim ental and ab-initio values o f prim itive translation vectors and
angle o f the B 1 9 ’ structure are displayed in Table 1. The ab-initio results predict slightly
larger values than those found in experim ent. This overestim ating o f translation vectors is
a typical effect o f G G A pseudopotential type but the related errors are sm aller than 5%.
Table 2 contains the com puted values o f the Y oung’s and bulk m oduli o f the
martensitic B 1 9 ’ and austenitic B2 structures. A s can be seen, the Y oung’s m oduli o f B 1 9 ’
are higher than those o f the austenitic B2 structure for all the directions studied. These
results are in contrast w ith experim ental and finite elem ent m ethod (FEM ) values, w hich
predict the Y oung’s m odulus o f martensite to about one-third to one-half o f that o f
austenite [7].
This can be explained by the fact that the experim ental data were m easured on
polycrystalline sam ples at finite temperatures, w hereas the atom istic m odel does not allow
ISSN 0556-171X. npoôëeubi npounocmu, 2008, № 1 21
P. Sestdk, M. Cerny, and J. Pokluda
for any shear deformation o f martensite variants, and describes a hom ogeneous deformation
o f a single crystal at the absolute zero temperature. On the other hand, the bulk m odulus
for the B 1 9 ’ structure is low er than that for the B 2 structure. Table 3 contains values o f the
theoretical strength for all applied deform ations.
T a b l e 1
The Experimental [1] and Ab-Initio Crystallographic Data for B19’ Structure
Parameter Experimental Ab initio
ao , A 2.889 3.007
^ , A 4.120 4.121
c^ A 4.622 4.813
5 , deg 96.8 100.6
V , A 3 54.63 58.61
T a b l e 2
The Elastic Moduli o f the B19’ and B2 Structures, as Obtained
from Present Ab-Initio Calculations along with Experimental [7] Results
Structure\moduli E [100], GPa E [010], GPa E [001], GPa B , GPa
B 19’ (ab-initio) 96 124 126 137
B2 (ab-initio) 72 72 72 155
B2 (FEM) 69 69 69 -
T a b l e 3
Computed Values o f the Theoretical Strength of the B19’ and B2 Structures
Structure\theoretical
strength
a d [100], GPa a uf [010], GPa a uf [001], GPa aP,Ga
B19’ (ab-initio) 19.0 27.5 20.7 22.1
B2 (ab-initio) - - - 24.0
- 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
Relative deform ation
Fig. 2. The angle 5 as a function o f applied deformation. The detail o f the range close to the
nondeformed state is depicted at the bottom right.
Under an applied deform ation, the angle 5 changes as a result o f the relaxation
procedure. The dependence o f the 5 angle on the applied deform ation is show n in Fig. 2.
22 ISSN 0556-171X. npo6neMbi npouHocmu, 2008, № 1
Elastic Properties o f B 1 9 ’ Structure o f NiTi Alloy
The points related to the theoretical strength are also marked (the inflection point in the
dependence E total ( e ) resp. E total ( v ) by dashed vertical lines. One can see that the angle
i grow s for all the deform ation paths except for the [010] direction. N ote that the [010]
direction is perpendicular to vectors ri and r 3 ; therefore, the decrease in the i angle
during the [01 0 ] deform ation is to be expected.
C onclusions. The uniaxial and hydrostatic deform ation o f martensitic structure B 1 9 ’
o f N iT i shape m em ory alloy w as studied using first-principles calculations. The results o f
com putations o f Y oung’s and bulk m oduli are as follow s: E [100] = 96 GPa, E [010] =
124 GPa, E [001] = 126 GPa, and B = 137 GPa. During the uniaxial and hydrostatic
deform ation the theoretical strengths w ere com puted for all cases o f applied deformations:
a u f [100] = 19.0 GPa, a u f [010] = 27.5 GPa, a u f [001] = 20 .7 GPa, and a h f =
22.1 GPa. The Young m oduli o f B 1 9 ’ are higher than those o f the austenitic B 2 structure
for all the directions studied and the bulk m odulus for the B 1 9 ’ structure is low er than that
for the B2 structure. The angle i grow s for all deform ation paths except for the [010]
direction.
Acknowledgments. This research was supported by research projects GA 106/05/H008 and
MSM 0021630518.
1. K. Otsuka and X. Ren, Prog. Mater. Sci., 50, 511 (2005).
2. X. Gonze, J.-M. Beuken, R. Caracas, et al., Comp. Mater. Sci., 25, 478 (2002).
3. X. Gonze, G.-M. Rignanese, M. Verstraete, et al., Zeit. Kristallogr., 220, 558 (2005).
4. M. Fuchs and M. Scheffler, Comp. Phys. Communicat., 119, 67 (1999).
5. http://www.abinit.org
6. D. R. Hamann, X. Wu, K. M. Rabe, and D. Vanderbilt, Phys. Rev. B, 71 (2005).
7. X. M. Wang and Z. F. Yue, Comp. Mater. S ci, 39, 697 (2007).
Received 28. 06. 2007
ISSN 0556-171X. n poôëeu u npouHocmu, 2008, № 1 23
http://www.abinit.org
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| id | nasplib_isofts_kiev_ua-123456789-48236 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T18:39:33Z |
| publishDate | 2008 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Sestak, P. Cerny, M. Pokluda, J. 2013-08-17T10:46:29Z 2013-08-17T10:46:29Z 2008 Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles / P. Sestak, M. Cerny, J. Pokluda // Проблемы прочности. — 2008. — № 1. — С. 20-23. — Бібліогр.: 7 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/48236 539. 4 The uniaxial and hydrostatic deformations of martensitic structure B19’ of NiTi shape memory alloy are studied using first-principles calculations. The bulk and Young’s moduli and the theoretical strength under uniaxial tension and hydrostatic loading are computed from crystal response to applied deformations. The behavior of angle β of the B 19’ structure was investigated along the whole deformation path. The computed values of Young’s moduli are compared with available experimental results. The results obtained complement and extend the already known characteristics of NiTi alloy. Проведено исследование одноосной и гидростатической деформаций мартенситной структуры B19’ сплава с памятью формы NiTi по методу расчетов ab initio. Величины объемного модуля упругости, модуля Юнга и теоретической прочности при одноосном растяжении и гидростатическом нагружении рассчитаны по реакции кристалла на приложенные деформации. Изучен характер изменения угла β структуры B19’ по всей траектории деформации. Выполнено сопоставление расчетных значений модуля Юнга с имеющимися экспериментальными результатами. Полученная информация дополняет и расширяет уже известные данные о характеристиках сплава NiTi. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles Исследование упругих свойств структуры B19’ сплава NiTi при одноосном и гидростатическом нагружении при использовании метода ab initio Article published earlier |
| spellingShingle | Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles Sestak, P. Cerny, M. Pokluda, J. Научно-технический раздел |
| title | Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles |
| title_alt | Исследование упругих свойств структуры B19’ сплава NiTi при одноосном и гидростатическом нагружении при использовании метода ab initio |
| title_full | Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles |
| title_fullStr | Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles |
| title_full_unstemmed | Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles |
| title_short | Elastic properties of B19’ structure of NiTi alloy under uniaxial and hydrostatic loading from first principles |
| title_sort | elastic properties of b19’ structure of niti alloy under uniaxial and hydrostatic loading from first principles |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/48236 |
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