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MODERN APPROACHES IN PROPERTY ANALYSIS OF POLYCRYSTALLINE TEXTURED MEDIA IN GEO- AND MATERIAL SCIENCE BY NEUTRON DIFFRACTION DATA. Victor B. Yakovlev National Research University MIET, Moscow , Russia [email protected]. LEVELS OF DESCRIPTION IN GEOSCIENCE. Seismic Instrumentations. - PowerPoint PPT Presentation
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MODERN APPROACHES IN PROPERTY ANALYSIS OF POLYCRYSTALLINE TEXTURED MEDIA IN GEO- AND MATERIAL SCIENCE BY
NEUTRON DIFFRACTION DATA
Victor B. YakovlevNational Research University MIET, Moscow, Russia
Seismic Instrumentations
Acoustic Emission
Neutron Diffraction
LEVELS OF DESCRIPTION IN GEOSCIENCE
KEY PROBLEMS TO DESCRIPTION OF POLYCRYSTALLINE ROCKS
MACROLEVEL<ij(r)>=c*
ijkl(r)<kl(r)>
MICROLEVELij(r)=cijkl(r)kl(r)
ij(r)=Kijkl(r)<kl(r)>
ij(n)=Kijkl(n)<kl(r)>
TEXTURE
TEXTUREFORMATION
EFFECTIVE CHARACTERISTICS AND RELATED PROBLEMS
HISTORICAL BACKGROUND
GENERAL SINGULAR APPROXIMATION OF RANDOM FIELDS
)()( rr fLu ))(( kijkljil cL r
)()(cc rr fuL )( cckijkljil cL
(1)
(2)
Equilibrium equation of inhomogeneous and comparison media
)()(c rr uLuL
Solution of (1) in terms of deformations
1111)(,)( d)()()( rrrrr mnklmnljikij cG mnklmnijklij cQ (3)
)(c r iklkilGL Introduce Green tensor as
After transforms
)())()(())()(()( 111 rrrrrr cQIcQI (4)
111* ))()(())()()(( rrrrr cQIcQIcc
111 ))()(())()(()( rrrrr cQIcQIK
111 ))()(()())()(()()( rrrrrrr cQIccQIcK
)(rQ – integral tensor operator rrr d)()( )()(,)( sljikijkl GgQ
Direct evaluation leads to
jkliijkl ag ()() dAnna iljkiklj1
4
1
mncinlmil nncA
dddl
nl
nl
n sin,cos1
,sinsin1
,cossin1
33
22
11
PROBLEMS OF AVERAGING
212121221 dddsin),,(),,(8
1),,(
afa
1.
2.
3.
4.
),,( 21 f – Crystallographic ODF
ODF of polycrystalline Quartz
Plot 1 – Longitudinal wave in monocrystalline quartzPlot 2 – Voight approximationPlot 3 – averaged Hashin-Shtricman boundsPlot 4 – Reuss approximationPlot 5 – Transverse wave in monocrystalline quartz
VELOCITIES OF THE LONGITUDINAL WAVE IN TEXTURED POLYCRYSTALLINE QUARTZ
Matrix 10 5 2,5
Inclusion 100 50 25
11c 12c 66c
Symmetry
Disk, Cubic 29,57 29,57 11,45 11,45 6,61 6,61
Hexagonal 33,35 18,76 14,19 8,95 4,66 9,58
Tetragonal 27,10 33,77 9,52 12,15 7,46 4,66
Sphere, Isotropic 22,01 22,01 10,10 10,10 5,96 5,96
Fiber, Cubic 26,47 26,47 10,29 10,29 5,68 5,68
Hexagonal 20,60 34,99 9,80 10,14 5,82 5,40
Tetragonal 28,94 20,60 10,59 10,01 5,61 5,82
11c
33c
12c
23c
44c
66c
01,0,1 321 lll
1321 lll
100,1 321 lll
EFFECTIVE CHARACTERISTIC OF THE MATRIX REINFORCED COMPOSITE
44
1211
2c
ccAx
1 – Cubic, 2 – Tetragonal, 3 – Hexagonal symmetry of effective properties
Dependence of the anisotropy of the effective properties from 3l
DISTRIBUTION OF STRESS FIELDS ON THE SURFACE OF THE CRYSTALLITE IN POLYCRYSTALLINE TEXTURED QUARTZ
Dependence of the operators of concentration of stresses and strains from the rotation in the olivine polycrystalline sample with effective characteristics
DISTRIBUTION OF STRESS AND STRAIN FIELDS WITHIN THE CRYSTALLITE IN POLYCRYSTALLINE TEXTURED OLIVINE
PREFERED ORIENTATIONS OF CRYSTALLOGRAPHIC AXIS OF CRYSTALLITES IN OLIVINE ROCKS UNDER HYDROSTATIC PRESSURE
Blue color designates concentration of crystallites with preferred orientations
MODELING OF DEFORMATION TEXTURE
1. Crystallites in the polycrystal orientate under external stress-strain
condition
2. Local energy in preferred orientations of crystallites leads to minimumMathematical formulation
Algorithm of modeling
1. Split all Euler space on elementary volumes2. All knots are crystallites with Euler coordinates3. Evaluate local energy of crystallites4. Rotate every crystallites on one step in decreasing energy direction5. Repeat step 3 and 4
Relative local energy of quartz crystallites under external stress: axis, shift, hydrostatic pressure
E
EE
),,( 21
PREFERED ORIENTATIONS OF CRYSTALLOGRAPHIC AXIS OF CRYSTALLITES IN OLIVINE ROCKS
Experimental data (SKAT diffractometer) Model calculation after 8 iterations (external hydrostatic pressure)
Thank for your attention!