Victor B. Yakovlev National Research University MIET, Moscow , Russia yakovlev@miee.ru

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

yakovlev@miee.ru

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!