- Email: [email protected]
Physical Properties of Polycrystalline Materials
13.
Physical Properties of Polycrystalline Materials
One of the important objectives of many texture investigations is the calculation of the anisotropy of physical properties of polycrystalline materials from the corresponding properties of the individual crystallites and parameters characteriz ing the polycrystalline material. Questions of this kind possess not only theoret ical, but also great practical interest. The directional dependence of magnetiz ation energy of transformer sheet or the earing behaviour of deep-drawing sheet are suggested. In some cases, as with the magnetization energy, the property of the polycrystalline material is given by the average value of the properties of the individual crystallites. It therefore depends not only on the single crystal properties, but also on the orientation distribution function f(g). In other cases (e.g. in the case of elastic and plastic properties) an additional interaction effect between crystallites, and thus a typical 'polycrystal effect', occurs. The properties of the polycrystalline materials can then also depend on the distribution of grain bound aries or in complete generality on the correlation of the orientations at different points of the material.
13.1.
Physical Properties of Single Crystals
By a physical property of a material one understands with complete generality the relation between two (or more) measurable quantities of the material which depends on the material24»220'291. Thus the relation between the volume and the mass of a material is characterized by its density, which is a property of the material. However, we are frequently concerned with essentially more complicated quantities, and therefore also essentially complicated properties of the material. Thus the diffracting power of a crystal depends in a complicated way on the direc tions of the incident and reflected beams. In this case we have thus to deal with a strongly directionally dependent property. We can thus with complete generality define a 'property' of a material by Y = E(X)
(13.1)
where Y and X are some quantities, usually orientation dependent, which are specifically connected to each other by the property E of the material.
Physical Properties of Polycrystalline Materials 295 13.1.1.
Representation by Tensors
A large class of directionally dependent physical quantities can be represented by tensors. By a tensor of nth. order E
= {%,..<„}
(13.2)
one understands a quantity with the components E^^ , which transforms to another coordinate system as the η-ίο\ά product of the coordinates. If we transform from one rectangular Cartesian coordinate system K, in which a point possesses the coordinates xi9 to another, K\ in which the point possesses the coordinates x'iy the following relation holds for the coordinates of the transformation: x\ = atjXj
(13.3)
The result is to be summed over the recurring indices according to the EINSTEIN convention. The components of the tensor in the coordinate system Kf are then given in terms of those in the coordinate system K by the relation E
U..-in
= aiihaHH ' ' ' aininEhh-in
(13*4)
(The summation signs of the n summations over the indices jx to jn are omitted.) If, now, the quantity Y in equation (13.1) is a tensor of mth order and the quantity X one of nth order and the function E describes a continuous relation between two quantities, one can thus expand it in a series: Y = E0 + Ex · X + E2 · X2 + ... + Er · X*
(13.5)
The powers Xr are tensors of (r · n)th order and the coefficients Er tensors of (r · n + m)th order. The tensors Er in this case describe the property concerned. Each term of this series thus has the general form y(m) _ £(^+η)χ(η)
(13.6)
or, in component notation *iii2...im
— •^iii a ..i m ?i72-..?n
?ι?2··.7η
(I**· · )
which is to be summed over all repeated indices. Equation (13.7) is a system of linear relations between the tensor components. Each component of the tensor Y is linearly dependent on each component of the tensor X. The scalar quantities, as, for example, the density, are naturally also included in this representation with m = n = 0. 13.1.2.
Representation by Surfaces
In addition to the representation of a physical property by tensors, another re presentation plays an important role — namely the representation by surfaces or, which is the same thing, by scalar functions of a direction in space302. For example, in a ferromagnetic crystal without the influence of an external magnetic field the
296
Physical Properties of Polycrystalline Materials
spontaneous magnetization lies in an easily magnetizable direction h0. If one forces this by an external magnetic field into another direction h, one must thus employ an energy (per unit volume) which depends on the direction h: E = E(h)=E(0,ß)
(13.8)
This representation of a property by one (or more) functions of a direction in space has the advantage of being particularly intuitive. The function E(h) must have the crystallographic symmetry — i.e. it must assume the same values for crystallographically equivalent directions ha. It can therefore be expanded in a series of spherical surface harmonics of the crystal symmetry (including the centre of inversion): oo M(l)
®{h) = Σ Σ eW(h) 1=0
(13.9)
μ=1
The coefficients ef here thus characterize the property concerned. Since the func tions kjf(h) will possess the complete crystal symmetry, symmetry conditions no longer exist between the coefficients ef in contrast to the components of a tensor (see, e.g., reference 24), which, in general, are not linearly independent. Both representations of a property, the surface and the tensor representation, can be transformed into each other. If in equation (13.8) we express the direction hy not by spherical angular coordinates Φ and /?, but by the components hv h2, hs, we can thus write E = E(h) = E{hv h2, \)
(13.10)
We can now expand E in a series of increasing powers of the h^: E(h) =E° + E\h{ + Elffch, + EfMh
+ ...
(13.11)
The quantities E^ and the products of \ therein are components of tensors. We have thus a special case of the general tensor representation (13.5). Conversely, every tensor can be represented by a certain number of scalar functions of a direction — i.e. by surfaces. The component ^83.. .3 =
%iPZ)% · · · aSjnEhH...jn
a
(13.12)
of the tensor E in the coordinate system K' depends only on the orientation of the direction X'3 of this coordinate system with respect to the crystal fixed coordinate system K, not on the orientation of the two other directions X[ and X'2. If Φ and β are the spherical angular coordinates of the direction X% in the crystal fixed coordinate system K, the function thus represents such a surface. Because of the commutability of the factors in the product a^jflzu ··· a3j m e ( l u a ^ i o n (13.12), the function Ε'^^Φ,β) depends only on the sum of all those components of the tensor E which are formed from Ej^ j by permutation of the indices. If the tensor E is symmetric with respect
Physical Properties of Polycrystalline Materials 297 to all these permutations of indices, the function Ε^^(Φ, β) thus already com pletely describes the tensor E. If the tensor E is not symmetric with respect to all permutations of the indices, one must thus include further linear combinations of tensor components, which also depend only on the orientation of X'% in the coor dinate system K. In this way, as was shown by WONDRATSCHEK 302 , every tensor can be expressed by a certain number of functions Ε{(Φ, β) of the direction X3. These functions must naturally satisfy the crystal symmetry. They can therefore be expressed in a series of symmetric spherical surface harmonics: n
M(l)
Ε&Φ,β) = Σ Σ ΦΡ{Φ, β)
(13.14)
1=0 μ=1
The coefficients efc then form a representation of the physical property, which is equivalent to the representation by the tensor components ^?1?ί...? · Thus, e.g., the elastic properties of a crystal can be represented by the directional dependence of the two quantities *is88(
and
[8,11η{Φ,β)+έ^ι(Φ,β)]
(13.15)
302
of YOUNG'S modulus and the cross-modulus , instead of the components «i?-w of the tensor of the elastic modulus, with respect to a crystal fixed coordinate system. As is known, a symmetric tensor of second order with components E^ can be described by a surface of second order: Ε(Φ,β)=Εφ& 13.1.3.
(13.16)
Representation by Functions of the Orientation g
If we expose a ferromagnetic crystal to the influence of an inhomogeneous magne tic field, the magnetization in different portions of the crystal thus lies in different crystallographic directions. The magnetization energy in this case depends on the complete orientation of a coordinate system associated with the magnetic field relative to the crystal axes and not solely on the orientation of an individual direction, as in the case of a homogeneous magnetic field. If this orientation is given by the rotation g with the EULER angles
(13.17)
Since the crystal symmetry (:) must be realized in the crystal fixed coordinate system and also the distribution of magnetization directions of the inhomogene ous magnetic field can possess a certain symmetry (\) (instrumental or external symmetry), we can expand the function E(g) in a series of symmetric generalized spherical harmonics according to these two symmetries: 00 M(l)
E(g) = Σ Σ
Nil)
Σ etvTr(g)
|=0 μ=1 ν=1
(13.18)
298
Physical Properties of Polycrystalline Materials
If t h e magnetic field is homogeneous, it t h u s possesses rotational symmetry with t h e field direction as symmetry axis. The generalized spherical harmonics of this symmetry are, however, identical with t h e spherical surface harmonics except for a factor, a n d equation (13.18) transforms, as it m u s t , into equation (13.9). Each component Ε[^Λ of t h e tensor E in t h e coordinate system Kf is a func tion of t h e orientation g of this coordinate system relative t o t h e crystal fixed coordinate system K. Accordingly, t h e tensor E is naturally represented trivially by a certain number of functions of t h e rotation g — namely t h e functions 2 ^ { (g). These functions depend on t h e components -Ε,-^...j of t h e tensor E in t h e crystal fixed coordinate system. If among t h e indices iv ...,in some are equal t o one a n other (and for n > 3 this must necessarily be t h e case), t h e 2£7l7>...7 appear in equation (13.4) only as sums over all permutations of t h e indices / of t h e correspond ing positions. If, now, t h e orientation dependence of a tensor component E'iiia^ {g) is known, t h e corresponding sum of components i£7l72...7 can t h u s be calculated therefrom. If t h e tensor E is symmetric with respect t o permutation of t h e indices concerned, it is thus completely described by t h e orientation dependence Ε[^^ {g) of one of its components. The tensor of t h e elastic compliance, for example, h a s t h e symmetry s
ijkl = sjikl — sklij
(13.19)
The component s
'i2w(9) = aiha2i^u^ushhUU
(13.20)
because of t h e commutability of t h e factors αί?· depends only on t h e sum of t h e tensor components 8
hUUU + *?i7i?47»
(13.21)
However, since because of t h e symmetry condition (13.19) these are equal, t h e function s'12SS(g) completely describes this tensor, which is t h u s represented by a single function of g. If t h e tensor is not symmetric with respect to those indices which have t h e same values in t h e component Ε[^Λ , one must use more of its components E'i'sj for its characterization. The transformation coefficients αί?· are t h e direction cosines of t h e coordinates X\ in t h e system of axes Xp They are t h u s functions of t h e rotation g which trans forms t h e coordinate system K into t h e system K'. They form a representation of t h e group of rotations g, which is equivalent t o t h e representation b y T™n(g) (see, e.g., reference 199). The a^ are therefore linearly expressible by generalized spherical harmonics of first degree (see equation 2.50 a n d Table 14.1): «*(?)=
'Σ
m——l
Σ
n=—l
μ™ν1ηΤΓ(9)
(13.22)
Physical Properties of Polycrystalline Materials
299
where t h e μίγη a n d vjn are given by Table 13.1. Now it is t r u e for t h e product of two generalized spherical harmonics (see Section 14.4) t h a t Tm^Trnini
=
g (l1l2m1mSi \l9m1 i=\h-h\
Table 13.1
+ m 2 ) (ΙΜι^
Tf>+™^+n*
| Ϊ, * i + n2)
(13.23)
THE COEFFICIENTS μίΐη AND v?n, BY WHICH THE TRANSFORMATION COEFFICIENTS a^ CAN BE REPRESENTED BY THE GENERALIZED SPHERICAL HARMONICS OF FD3ST DEGREE
ftim i
*j*
\m
\
-1 1
1
i
2 3
0
+1
0
+ 0
1
0
w i
0
0
-1 1
1 2 3
0
+1
+
i
0 0
1
w
0
The symbols in parentheses are t h e CLEBSCH—GORDAN coefficients. B y repeated application of formula (13.23) one can then express t h e product of t h e direction cosines as a series of generalized spherical harmonics:
aiiU ■ «;„·,... %ir = Σ
Σ
Σ «HMa · · · v ; k k ■ ■ ■ ?,) ΤΓ(9) (13.24)
1=0 m~—I n=—I The function $1283(0)> which characterizes t h e tensor of t h e elastic compliances, for example, thereby assumes t h e form
»'να»(9) = Σ
Σ
Σ αΓ(1233;^3η)8ΜιΜίΤΓη(9)
1=0 m= —I n= —I
13.2.
(13.25)
The Problem of Averaging
W e consider a n arbitrary tensor property E. Two quantities Y and X m a y be linked to each other b y i t :
Y = E·X
(13.26)
If, now, t h e material considered, which possesses t h e property E, is polycrystalline, the property tensor E is dependent on t h e orientation of t h e crystallites and, hence, on t h e position r within t h e material. This is also true for t h e two quantities X and Y. W e t h u s have t o write Y(r) = E(r) · X(r)
(13.27)
300 Physical Properties of Polycrystalline Materials The material may now be macroscopically homogeneous. We consider a volume V, which is large compared with the grain size. The average value of E(r) over this volume is then equal to the average value over the whole sample. We further assume that the quantities X(r) and Y(r) are macroscopically homogeneous. The average values over V are then likewise independent of the location of the region V within the whole sample. We denote this average value by a bar, so that, for example, for the average value of E it is true that E=-i
v
fE(r)dV v
(13.28)
The average values of Y and X are similarly defined. We now set Ε(Γ) = Ε + ΔΕ(Γ)
(13.29)
Y(r) = Ϋ + ΔΥ(Γ) X(r) = X + &X(r)
(13.30) (13.31)
With equation (13.28) it is then true that
t
f &E(r)dV = 0
(13.32)
Corresponding relations are also valid for ΔΥ and ΔΧ. If one substitutes equations (13.29) —(13.31) in equation (13.27), one thus obtains Y + ΔΥ(Γ) - E · X + E · ΔΧ(Γ) + ΔΕ(Γ) · X + ΔΕ(Γ) · ΔΧ(Γ)
(13.33)
If one forms the average value over V, one thus obtains with equation (13.32) Ϋ = E · X + - i [ ΔΕ(Γ) · ΔΧ(Γ) dV (13.34) V ff If we consider the averaged tensor X as the independent variable, the components of the quantity ΔΧ(Γ) must depend on those of X. For X = 0 it is also true that ΔΧ(Γ) = 0. Because of the linear relation (13.27) a linear relation between X and ΔΧ(Γ) is generally valid: ΔΧ(Γ) = U(r) · X
(13.35)
If X, and therefore also AX(r), are rath-order tensors, U(f) is thus a tensor of order 2n, which can depend on the values of E(r) at all positions r' of the whole sam ple 196 . If one substitutes equation (13.35) into equation (13.34), one thus obtains = [E + - i / A E ( r ) U ( r ) d F
(13.36)
The macroscopic quantities Y and X are related to each other in equation (13.36) in a similar way to the microscopic ones Y and X in equation (13.27). The quantity
Physical Properties of Polycrystalline Materials
301
in brackets, E = E + - i / ΔΕ(Γ) U(r) dV
(13.37)
is thus the corresponding material property of the polycrystalline (macroscopically homogeneous) material. It is generally different from the simple arithmetic average value E. The simple arithmetic average value E is, in general, relatively easy to calcu late, while the calculation of the additional quantity frequently leads to very great difficulties. For its calculation not only the orientation distribution function f(g) is necessary, but also an orientation correlation function which describes simul taneously the orientation g at the point r and the orientation g' at the point r \ Moreover, the additional quantity in equation (13.37) is frequently relatively small in comparison with E. The approximation E^E
'
(13.38)
is therefore frequently employed. One obtains a better approximation for the polycrystal property E than the average value E in the following way. If one solves the system of linear equations (13.26) for the components of the tensor X, one thus obtains X = E- 1 -Y
(13.39)
1
The tensor Er is another representation of the same property as that described in equation (13.26) by the tensor E. Examples of such tensor pairs E and E" 1 are the tensors of the electrical conductivity and the electrical resistivity or the elastic stiffness and the elastic compliance. If, now, by analogy with equation (13.28), one forms the average value of the tensor E""1:
IF 1 = 4" (E-\r)aV in general, Εφ
v4
(F1)"1
(13.40) (13.41)
The two averages (13.28) and (13.40) yield limiting values for the actual property of the polycrystalline material. Average (13.28) assumes X = const.
'
(13.42)
Thus, in general, the conditions for X are invalid at the grain boundaries, and the average (13.40) requires Y = const.
(13.43)
This, in general, violates the boundary condition for the quantity Y. The actual behaviour of the polycrystalline material will therefore lie between these two 21 Bunge, Texture analysis
302 Physical Properties of Polycrystalline Materials extremes, so that one can frequently set
E^i-tE + iE^)-1]
(13-44)
In elasticity theory equations (13.28), (13.40) and (13.44) describe the well-known approximations of HILL 1 5 2 , RETTSS244 and VOIGT 2 9 1 . An essentially more general form of the average construction was given by ALEXANDROW and AISENBEBG 13 , KRÖNER187'188, K N E E R 1 8 0 ' 1 8 1 and
13.3.
MORRIS213.
The Calculation of the Simple Mean Values E
We shall now discuss the calculation of the simple average value E for the three forms noted for the representation of a physical property. This means that we must average either the tensor components Eixiz%Ä ?ι?·2 >?· (equation 13.7) or the coef ficients ef (equation 13.9) or the coefficients efv (equation 13.18) over the volume V corresponding to equation (13.28). Since the property E(r) will depend only on the orientation g of the crystallite present at the point r, the integration over V can be carried out in two steps. We first integrate over all those volume elements dV which possess the orientation g, and then over all orientations g. We thereby obtain Ε=±φΕ(9) v
9
fdV
(13.45)
V(g)
If here we substitute for dV from equation (3.3), we obtain E = j>E{g)f{g)ag
(13.46)
The orientation distribution function f(g) thus appears as a weight function for the calculation of the average values of orientation-dependent physical properties. Its great practical importance does not least depend thereon. 13.3.1.
Tensor Representation
A property is described in a crystal fixed coordinate system KB by the rth-order tensor E with components 2£?1?·2...7·. It has the components E'iii2^ in the sample fixed coordinate system KA. The crystal coordinate system KB is produced from the sample coordinate system KA by the rotation g. KB will then naturally be transformed into KA by g~x, and, according to equations (13.4) and (13.24), it is true for the components Ε[χ{^Λ in the sample fixed system KA that
^Α...ν=έ
1=0
Σ
Σ
m=—ln=—l
Trig-1)EhU...ir
(13·4?)
Setting
Tf^g-1) = T?nm(g)
(13.48)
Physical Properties of Polycrystalline Materials 303 equation (13.47) transforms into
Ki,..ir=I
Σ 1=0
Σ ^m(hh-'-ir\hk''-h)Trn(g)^l1t...Jr
(13.49)
m——ln=~l
If we expand the weight function of the average value (equation 13.46), and thus the orientation distribution function/(gr), in a series of symmetric generalized spherical harmonics and express the symmetry of these functions by the symmetry coefficients Αψμ and A™v, we obtain oo M(l) N(l)
+1
-f I
Kg) = Σ Σ Σ
Σ
Σ ^Αψ^Αγτψ^)
1=0 μ=1 v=l
'
.
(i3.50)
m=—ln=—l
If, now, we average the components E'^^ of the tensor E over all orientations g in the sample fixed coordinate system according to equation (13.46), we obtain for the property tensor of the textured material with
EiXH...ir = «(h*2· · · V; hk· · ·ir) Eujt...jr *
(13.51)
M{1) J\T(Z)_
a{ixi2... ir; w2... jr) = Σ Σ Σ <(hh · · · \; hfe · · ·?V) Cf 1=0 μ=1
and
<(hH--.ir;hh.-4r)= Σ
(13.52)
ν=1
Σ
m=-ln=-l
ά1
i^f^^m(hh-'ir;hk-4r)^rMv " Γ -L
(13.53) Equation (13.51) expresses the components of the averaged property tensor E of a textured material by those of the single crystal with respect to the crystal fixed coordinate system. The averaging coefficients ά^ ... jr) according to equa tion (13.52) depend on the coefficients Of of the orientation distribution function. The quantities äf(i± ... jr) appearing in equation (13.52) are purely mathematical quantities. According to equation (13.53), they depend only on the crystal and sample symmetries, and according to equation (13.24), on the order of the ten sor E. They can thus generally be calculated. Further, an example of this is given below for the elastic properties (fourth-rank tensor properties). 13.3.2.
Surface Representation
The calculation of the average value E proceeds in a particularly simple way if one employs the surface representation Ε(Φ, β) of the property 2£35. The angles Φ and β are the spherical angular coordinates of the crystal direction h. In a crys tallite of orientation g the crystal direction h may point in the sample direction y. If we want to calculate the average value of the property E in the direction y over all orientations g that occur, E{y)=j>E{h)f{g)ag 21*
(13.54)
304 Physical Properties of Polycrystalline Materials we can carry out the average over g in two steps. First we average over all those orientations g for which the crystal direction h falls in the sample direction y. E(h) is thus naturally constant, and the average of the function f(g) over these orientations, according to equation (4.33), yields the general axis distribution function A(h, y) and, since the direction y is constant during the integration, the inverse pole figure of the direction y. By consideration of the normalization of this function, there results (13.55) E(y) =^-Φ E(h) A(h, y) ah = - L E{h) RJh) ah 4π 4τζ " For calculation of the average value E(y) in a sample direction y one thus needs only the inverse pole figure of this direction and not the complete orientation distribution function f(g). Herein lies the particular advantage of the surface representation of physical properties. If the average value E(y) for all sample directions y is required, the general axis distribution function A(h, y) is needed as the weight function. This function will therefore incidentally be denoted as the generalized inverse pole figure. If one substitutes for E(h) and A(h, y) the cor responding series expansions, * r
M(l)
(13·56)
m) = Σ Σ
and oo M(l) N(l)
ημν
(13 57)
A{h,y) = ίπ Σ Σ Σ « Τ Γ Τ *?"(*) ** ι=ο μ=ι ν=ι ^ -h ι one obtains r
_
·
N(l)
E{y) = Σ Σ Wiy)
(13.58)
1=0 v=l
with the averaged property coefficients
n
1 — 21 "Γ
M(l) Σ Of*
(13.59)
J- μ=1
If the orientation distribution of the crystallites is rotationally symmetric about the Z-axis of the sample coordinate system 34 , equation (13.57) thus trans forms into (see equation 5.24) oo M(l) 1 /
A(h9y) = A(h, Φ) = Σ
Σ
0
yw+Ίkr{h)
:
_
°ΐΡι[Φ)
(13,60)
We obtain in this case for the property of the polycrystalline material, which is likewise rotationally symmetric about the Z-axis, Ε(Φ)=
Σ%?ι{Φ) ι=ο
(13.61)
Physical Properties of Polycrystalline Materials 305
with
(l3 62)
^i^i-Wj^
-
13.3.2.1. Rotationally Symmetric Properties In many cases the property function E(h) is rotationally symmetric with respect to a certain crystal direction h0 (though the crystal naturally does not possess such symmetry). Thus, e.g., in the case of hexagonal crystals the magnetization energy will be assumed as rotationally symmetric about the hexagonal axis. If, therefore, we establish for the crystal direction h a new spherical angular coor dinate system©, ψ', such that its pole coincides with the direction h0, the property E(h) depends in this coordinate system only on the angle Θ. The integration in equation (13.55) can therefore be first carried out over the angle \p\ where Ε(Θ) is constant: S
E e iy)=Jzf ( )\f 4π Q
LÖ
*,(©,?')<¥
sin<9d<9
(13.63)
The integral in the brackets can, according to equation (4.137), be replaced by an integral over the pole figure Ρ^(Θ, ψ) belonging to the direction hQ:
E{y)
=έ ϊ
Ε{β)
■ 2π
[f ΡΛ · (Θ ' Ψ) Η s i n ( 9 d 0
(13.64)
The direction y is the pole of the spherical angular coordinate system Θ, %p. One sees that in the case of a rotationally symmetric property Ε(Θ) the average value E(y) depends only on the values of the h0 pole figure, where hQ is the axis of ro tational symmetry of the property concerned. It is also obviously clear that, because of the rotational symmetry of the property of the single crystal, the poly crystal property will not be altered if one rotates each crystal arbitrarily about its symmetry axis. The polycrystal property, therefore, can depend only on the spatial distribution of the axes of rotational symmetry — i.e. the hQ pole figure. We represent the pole figure by its series expansion: oo
PHJP,
ψ)=Σ J=0
JV(i)
Σ mho) k(0, ψ)
(13.65)
v=l
and expand the property function E(ß) in LEGENDRE polynomials: Ε(Θ) = Σ ήΡ,(β) (13.66) ι=ο Taking equation (14.200) into account, there then results from equation (13.64) _
°°
E(y) = Σ
N
e, l7
*
1V
®
Σ mho) M(y)
1=0 y2(2l + 1) v=i
(ΐ3-β7>
306 Physical Properties of Polycrystalline Materials This can be written __
oo Nil)
Ε{ν)=Σ Zeikiy)
(13.68)
1=0 v=l
with the averaged coefficients 1\ =
6l
.
Fj{ho)
(13.69)
which, as they must, depend only on the coefficients F\{hQ) of the h0 pole figure. If, also, the orientation distribution is rotationally symmetric (i.e. we are con cerned with a fibre texture), we express the pole figure according to equation (5.26) in a series of normalized LEGENDRE polynomials. The coefficients Ft thereby appearing differ from the coefficients F} by the normalization factor 1/|/2π of the rotationally symmetric spherical surface harmonics. However, since we also expanded the rotationally symmetric property function of the polycrystalline material in a series of LEGENDRE polynomials with the coefficients \ (equation 13.61), it is also true for these coefficients that 1/2(21 + 1) 13.3.3.
■*l(*o)
(13-70)
Representation by Orientation Functions
Finally, we have represented the property E by one or more functions of the ro tation #', which transforms the crystal fixed coordinate system KB into the external coordinate system K'. In the previously mentioned example of the magnetiz ation energy in an inhomogeneous magnetic field the coordinate system K' was considered to be fixed with respect to the magnetic field. The rotation g thus describes the orientation of the magnetic field with respect to the crystal axes. If g is the rotation which transforms the sample fixed system KA into KB, the rotation 9" = 9''9
(13.71)
thus transforms the sample fixed coordinate system KA into the external system K' {Figure 13.1). The property E (equation 13.18) is then represented as a func tion of this rotation as follows: r
Mil) Nil) +1
E{g') = E(g" ■ g^) = Σ Σ Σ
Σ *Τπ*(9") Tf^g)
(13.72)
1=0 μ=1 v = l s = — I
In order to obtain from this the average value over all orientations g, we must, according to equation (13.46), multiply by oo M(l) Nil)
+1
f(9) = Σ Σ Σ Σ crÄr'Trü) 1=0 μ=1 v=i n=—l
(13.73)
Physical Properties of Polycrystalline Materials
307
Figure 13.1 Specimen coordinate system KA, crystal coordinate system KB and external coordinate system K'
integrate over g, and obtain r
E{9") ==
M(l)
N(l)
Σ Σ Σ
1=0 μ=\
WltJl
1 ~2μν
_
(13.74)
v=l
Mil)
(13.75)
7-T-T Σ <ΗμλΟ}ν
Also, for this representation of the property E, the coefficients i f of the average value have thus been calculated from those of the individual crystallites and the texture coefficients C}v.. If the orientation distribution is rotationally symmetric about the Z-axis of the sample coordinate system, equation (13.73) thus transforms into oo
Mil)
f(g) = B(h) = Σ Σ CW(h)
(13.76)
1=0 μ=1
h is that crystal direction which coincides with the axis of rotational symmetry. The average value E(g") must in this case also be the same for all rotations g", which differ from each other only by a rotation about the axis of rotational sym metry. It can thus only depend on the orientation of the rotation axis — i.e. the Z-axis of the sample fixed coordinate system in the variable coordinate system K'. If we denote this by the vector z, we thus obtain for the average value _
_
r
Nil)
E(g") = E(z) = Σ Σ
:
eimz)
(13.77)
1=0 v=l
with
Mil) *l
Σ W
(13.78)
We have thus derived completely general relations between the simple average value of a physical property of polycrystalline material and the corresponding
308 Physical Properties of Polycrystalline Materials property of the single crystal. The relations are independent of which particular property is treated. They depend only on the form of the representation of the pro perty. In many cases the simple average value E is identical with the actual polycrystal property E. In many other cases equation (13.44) represents a sufficient, though only empirically based, approximation. Since, now, the calculation of the average value (2M-) from E*1 is formally identical with the calculation of E from E, the calculation of the polycrystal quantity E essentially reduces to the simple average value in these cases. 13.4.
Average Values of Special Properties
In the following the general formulae for the simple average values of physical properties will be applied to some special cases. 13.4.1.
Magnetization Energy in a Homogeneous Magnetic Field
The magnetization energy of ferromagnetic crystals in a homogeneous magnetic field sufficient for saturation will, in general, be given as a function of the direction of magnetization — i.e. in the surface representation of a physical property. Since, further, the magnetization direction in each individual crystallite in this case agrees with the exterior magnetic field, the magnetization energy of a polycrystalline material is equal to the simple average value of the energies of the individual crystallites. In this case it is thus true that equation (13.28) is exact. 13.4.1.1. Magnetization Energy of Cubic Crystals The magnetization energy of ferromagnetic cubic crystals can be described with sufficient accuracy by the expression E(h) = K^hlhl + h\h\ -H λ§Α§) + K6h\h\h\
(13.79)
The functions
(13.80)
(13.81)
and appearing here possess cubic symmetry. They are related to the cubic spherical harmonics in the manner described in Section 14.7.5. (See reference 193.) It follows therefrom that ...
1 h\(h)
1
(13.82)
and m ,fcv
1 k(h)
1 kl(h)
1
(13.83)
Physical Properties of Polycrystalline Materials 309 Substituting in equation (13.79), by omission of the constant term, there results E(h) =
k\(h)
KA ΚΛ 5 ^ 55
+
H(h) K6 n6
(13.84)
231
The series expansion of the property function (13.9) thus has only two terms with the coefficients 1
2. (13.85) n% 231 We now assume orthorhombic sample symmetry (sheet symmetry). The spherical surface harmonics of this symmetry are given by
el
=■
e
5 ~*~ 5 5
l
6
^ ) = 77= ^ ( φ ) ' *?(y) ==-±=Ρ?ν-1\Φ)οο*2(ν--1)γ y Δτι yn
(13.86)
By use of equations (13.85) and (13.86), we obtain from equation (13.58) with equation (13.59) Ε{Φ, γ) =
KA 9η4)/π
with 0\τΡ^Φ)
Ρ 4 (φ, γ)=η=
]/2
and
5L *ΆΦ,γ) + 13w
+ 55
6
| / π 231
F6(0,y)
+ <7412Ρ£(Φ) cos 2γ + Ο^Ρ\{Φ) cos 4y
(13.87)
(13.88)
Ρ 6 (φ, y) = - L 0 6 η Ρ 6 (Φ) + 0 6 12 Ρ|(Φ) cos 2 7 r
2
+ (7613Ρ|(Φ) cos 4y + ^614^β(Φ) cos 6y
(13.89)
Φ and y are the spherical angular coordinates of the direction of the magnetic field with respect to the coordinate system fixed in the sample. Since in the case of cubic symmetry only one linearly independent spherical surface harmonic of degree I fg 10 exists, the texture coefficients C}v in equations (13.88) and (13.89) can be determined by measurement of a single pole figure (see Section 4.2.2). From the magnetization energy (equation 13.87) one obtains the torque exer ted in the plane of the sheet Φ = 90° on a sample with circular cross-section in a homogeneous magnetic field: M = with and
ΰΕ(Φγ) 8y
K*
9ηΑ\/π
5 ^ 55
K(Y)' lSn j / π 231 Κ(γ) 6
F±(y) = 2C412Pf (90°) sin 2γ + 4<74l3P4t(90o) sin 4y
(13.90) (13.91)
Ffo) = 2C612Pf (90°) sin 2γ + 4Ο613Ρ|(90°) sin 4y + 6C614P§(90°) sin 6γ (13.92)
310
Physical Properties of Polycrystalline Materials
The torque curves for different Fe—Si samples were calculated by this method by SZPUNAB and OJANEN 2 7 4 , from textures determined by neutron diffraction and the coefficients Cf calculated therefrom. As Figure 13.2 shows, the agree ment between measured values and those calculated from the texture was rather good. The calculation of the average values of a physical property can also be carried out by direct integration of equation (13.46), if the orientation distribution f(g) is calculated — for example, not by methods based on series expansion, but, e.g., by the method of WILLIAMS. In this way the magnetic torque was calculated by HTJTCHINSON and SWIFT 158 » 159 from the texture (biaxial pole figures after WILLIAMS) for a rimmed steel cold-rolled 50% and annealed at 700 °C (Figure 13.3) and com pared with the experimental values. The agreement is similar to the calculation with the help of the coefficients from equations (13.90) —(13.92). This is also to be expected, since both methods are, of course, mathematically equivalent. The integration in equation (13.46) was carried out over 1296 orientation points.
measured curve
30
x
calculated points
20
i
10 0 -10 -.20
i
I
\
-30 -40 -50
Δ
1
I
I
I
I
L
10°. 20° 30° 40° 50° 60° 70° 80° 90° Angle to rolling direction Figure 13.2 Torque curve of an Fe—Si sheet calculated according to equation (13.90) from neutron diffraction texture measurements. After SZPUNAB and OJASTEN274
For rotationally symmetric textures (fibre, wire, rod) equation (13.87), in cor respondence with equations (13.61) and (13.62), simplifies to
**>~sVTi
Κχ
, Κ{
+ 55
^ φ )+έΐ/ίΙ^Ιϊ^ φ ) (13.93)
where C\ and C$ are the fibre texture coefficients defined according to equation (5.4).
Physical
Properties
of Poly cry stalline Materials
311
40l·
-40
—
Calculated
rom
texture
data
• Measured values 1 I I I 1 I 1 1 1 I 1 I 1 1 1 1 1 0° 10 20 30 40 50 60 70 8090°lQ0 110 120 130140 150 160 17θ180° Angle t o rolling d i r e c t i o n
Figure 13,3 Torque curve of a rimmed steel, cold-rolled 50% and annealed at 700°C, calculated from biaxial pole figures and measured according to H T J T C H I N S O N and SWIFT 1 5 8 ' 1 5 9
The magnetic hysteresis (power loss) of a magnetic material can likewise b e expressed b y a n expression of t h e form, of equation (13.79), though with other coefficients. HUTCKENSON a n d S W I F T 1 5 8 ' 1 5 9 used as approximation for t h e power
loss of a cubic crystal P(h) = A0 + A^hlhl
+ λξλξ + h\h\)
(13.94)
This expression was averaged over all orientations with t h e help of t h e calculated biaxial pole figure according t o equation (13.46). T h e result is represented in Figure 13.4. One sees t h a t t h e direction dependence is well reproduced. The coef ficients A0 a n d A1 were experimentally fitted. 13.4.1.2. Magnetization E n e r g y of Hexagonal Crystals F o r most cases t h e magnetization energy of hexagonal crystals can b e described with sufficient accuracy b y Ε(Φ) = K% sin 2 Φ + K± sin 4 Φ
(13.95)
Φ is t h e angle between t h e magnetization direction a n d t h e hexagonal axis. T h e energy is t h u s independent of t h e angle γ and can b e expressed b y two rotationally
312
Physical Properties of Polycrystalline Materials '
!
1
1
!
!
1
I
!
1
Ao = 1-02
A, =12-95 J
4 h
•
-/ΊίθΗζ :£
Αο = 0·91 · Αι = 9 97
3l·
%^^^ _^^50Hz
•
2h
•
o
•
"^
Ao=0-41 · A, = 6 -10
30H
~~ι
Γ · Me asure d v a I ues — Calculated from text ure data 0[
i
go
l
l
o1 Q
2o0
g3 o
1
1
1_
g4 o
g5 o
βgo
1 y go
1 ggo
g go
Angle to rolling direction Figure 13.4 Power loss in Fe—Si according to HUTCHINSON and SWIFT 158 ' 159 . The curve was calculated from biaxial pole figures; the points were measured. The coefficients A0 and Ax were determined from the experiment symmetric spherical surface harmonics of second and fourth order:
^=^μ\ι^φ-± 1 j/2^r 1/ 2 L 2
^
(13.96)
2
and in**
1(Φ) =
*
1
ι/9
ϊ7^τ[τ
Γ35
d
^
0Ο84φ
-Τ
15
, _
0Ο82φ
3
+Τ
(13.97)
By a transformation one obtains therefrom Sin2 0=jL-l.
3
"3
1/JL γ2π *£(Φ)
(13.98)
and sin4 φ =
Ji _ 1J yl_ γr 2π kl(0) + ± I/A γ2π ί\{Φ) zv 15
21 ]/ 5
' ' ' 35 1/ 9
(13.99)
By neglecting the constant terms there results Ε{Φ) = - y
T
pn
[TZ2
+ -#4j
ίο\{Φ) + ^Ύ\/2π
^Κ^Φ) (13.100)
Physical Properties of Polycrystalline Materials
313
The coefficients ef are thus given by e\ =
]^^[i*.+ir4 t-ir^M**
(13 101)
·
We now further assume orthorhombic sample symmetry and obtain for the aver aged energy, according to equations (13.58) and (13.59),
««» — Tyt
3
K
+UK
^(Φ,γ)+^·^Κ^(Φ,γ)
with
and
F2(0, γ) = - L <72ηΡ2(Φ) + 0212Ρ|(Φ) cos 2γ J/2
(13.102) (13.103)
Ρ 4 (Φ, y) = --L Ο^ΡάΦ) + £412Ρ£(Φ) cos 2y + <7413Ρ|(Φ) cos 4y J/2
(13.104) Since only one linearly independent hexagonal spherical surface harmonic of second and fourth orders exists, the coefficients in equations (13.103) and (13.104) can also be determined from a single pole figure, as in the case of cubic symmetry. 13.4.2.
The Remanence in Ferromagnetic Materials
In the previous section we have considered the energy which is necessary to bring about the spontaneous magnetization of a crystal in a specific crystallographic direction h. We now consider those directions, h^ for which the energy assumes a minimum value. Several cases are to be differentiated. In the first case exactly one such direction h0 (as well as its opposite, — h0) results: in the second case there result several different directions h^ for which the magnetization will be minimal. To the first group belong, e.g., many hexagonal materials; to the second, cubic materials. Finally, it can also occur that for a whole band of directions — e.g. all those perpendicular to a direction — a minimum energy will be assumed. If one thinks of a crystal magnetized to saturation in the direction h, the magnet ization thus lies in the direction ft. If, now, one removes the magnetic field, the magnetization will rotate into the nearest direction hi of minimum energy. The component of magnetization in the direction of the original field then amounts to 7 R = I8 cos
(13.105)
The component / R will be called the remanence; Is is the saturation magnetiz ation. We now assume that only a single direction h0 of minimum energy exists, and choose a spherical angular coordinate system Θ, \p, such that its pole Θ = 0 coincides with h 0 . We then obtain for the relative remanence in the case of original
314 Physical Properties of Polycrystalline Materials magnetization in the direction Θ, ψ 4(6>3 ip)/Is = | cos Θ |
(13.106)
In this case the remanence is thus a property of the crystal independent of the angle ψ. With respect to this property the crystal thus has rotational symmetry. If we set |cos6>| = i > ^ ( < 9 )
(13.107)
1=0
we thus obtain a series expansion corresponding to equation (13.66) for 7 R (0)// S . The coefficients ^ are given in Table 13.2. Table 13.2
COEFFICIENTS ez OF THE SERIES EXPANSION OF THE REMANENCE IN HEXAGONAL FERROMAGNETIC CRYSTALS
0 2 4 6
0.707107 0.395 285 -0.088 388 0.039 836
16 18 20 22
-0.005 909 0.004 692 -0.003 817 0.003 166
8 10 12 14
-0.022 777 0.014 767 -0.010 358 0.007 670
24 26 28 30
-0.002 668 0.002 279 -0.001970 0.001 719
We now consider a polycrystalline material and assume that the magnetization is not influenced by interactions with neighbouring crystallites. The remanence of the polycrystal is then equal to the sum of the remanences of the individual crystallites. If Φ and γ are the spherical angular coordinates of an arbitrary sample direction y, we thus obtain for the relative remanence in the case of magnetization in the direction (9, y according to equations (13.68) and (13.69) oo
N(l)
Ivlß> 7)lh = Σ Σ ^ϊ(Φ} Z=0r=l
γ)
(13.108)
with
?
=Fi^nr w,o)
(13 109)
'
The h\{0, γ) are the normalized spherical surface harmonics of the sample sym metry (in the case of orthorhombic sheet symmetry see, e.g., equation 13.86), and the Fl(h0) are the coefficients of the series expansion of the h0 pole figure, and thus the pole figure corresponding to the preferred direction of magnetization.
Physical Properties of Polycrystalline Materials
315
They can be directly determined from this pole figure according to equation (4.70) or, if this pole figure is not measurable, by means of the Cf from other measurable pole figures. In the case of rotationally symmetric textures equation (13.108) transforms into (see references 131, 264, 265) Ι*{Φ)
(13.110)
■ Σ *ιΡι(Φ) 1=0
with the coefficients ^ =
*i
(13.111)
■fi(fco)
|/2(2Z + 1)
The Fi(h0) are the coefficients of pole figures for fibre textures defined by equation (5.26). As one sees from Figure 13.5, the coefficients el decrease very rapidly with I. If, for example, one assumes a limit of 5%, one thus only needs to consider series terms up to sixth order69 . If, further, one also omits the sixth-order term and uses only the terms of second and fourth order, then, according to Figure 4.4, it is
J
I
I
1
L
1
1
1
M/Hcos θ|
1 . 5 % ^ 0
2 4
6
„ 8 10 12 14 16 18 20 22 24 26 28 Degree
I
Figure 13.5 The relative remanence |cos Θ\ of an hexagonal single crystal and the coefficients et of the series expansion in LEGENDRE polynomials69
316 Physical Properties of Polycrystalline Materials M(l = 4) = 1. The coefficients F\(hi)9 which are needed for calculation of the average values, can then be calculated from a single arbitrary pole figure. We finally further consider the case of several different preferred directions, as occurs, e.g., in the case of cubic crystals. If we further assume that, on removing the magnetic field, the magnetization changes into the nearest preferred direction h^ it is thus true for the relative remanence of the single crystal in different angu lar regions that cos (hh-^>
Ίπ(Φ,β)/Ι*
cos (hh2y
in region J5X in region B2
cos <Λ/ι7>
in region Bj
13
n
The region Bj is so defined that within it the angle is the smallest among the angles (Jthi). If we expand Ι^(Φ, β)/Ι& in a series of spherical surface harmonics of the crystal symmetry oo
ΙΛ(Φ,β)/ΙΒ
=Σ
M(l)
Σ *%(Φ,β)
(13.113)
it is thus true for the coefficients that tf = Σ *
f cos (hh{> Α*"(Φ, β) sin Φ άΦ άβ
(13.114)
Bi
If the directions hi are symmetrically equivalent, the integrals over the individual regions are equal. One thus obtains ef = n J cos (hh{> ί?μ(Φ, β) sin Φ άΦ άβ
(13.115)
n is the number of symmetrically equivalent regions, and Bx is an arbitrary one among them. The remanence of the polycrystalline material results therefrom according to equations (13.58) and (13.59). For cubic crystal symmetry and the preferred directions h{ = <100>, h{ = <110> and hi = <111> the regions Bt are shown in Figure 13.6. 13.4.3.
Tensor Properties of Second Order
A large number of physical properties can be described by tensors of second order — e.g. thermal expansion, optical refractive index and electrical conductivity. We will therefore calculate the orientation average value of such properties in general form (see references 34, 35, 271, 272). A symmetric tensor of second order can always be described by an ellipsoid, which we can relate to its principal axes: E(h) = Enh\ + E22hl + E^hi
(13.116)
/ \ -\
\ A B,
\V^L\
r—^
■ \
Γ
Λ
\ \
/
/Λ'
\ C^/\
\
///
w^ii^s! Siv V /V: ^\j \ \
~^"^^ \
Λ
]/
,Λ/
/-- A - -Be"!
7
~—
Figure 13.6 The regions Bi for cubic crystal symmetry, into which the magnetization transforms after removal of the field in the pertinent preferred direction h^ and indeed for the three cases h^ = {100}, h^ = {110} and h^ = {111}
318
Physical Properties of Polycrystalline Materials
With respect to such a property all crystals have at least orthorhombic symmetry. In the case of cubic symmetry all three coefficients are equal, and E is independent of the direction and thus the average value E is independent of the texture. If one sets hx = sm0eosß
(13.117)
Α2 = θΐηΦβΐη/ϊ
(13.118)
h3 = cos Φ
(13.119)
h\ = - ί (1 - cos2 Φ) (1 + cos 2β)
(13.120)
h\ = 4" C1 -
(13.121)
then
cos2 φ
a
) i1 -
cos 2
ß)
Al = c o s 2 0
(13.122)
These can be expressed by means of the orthorhombic spherical surface harmonics of second order:
Η(Φ) = Γ Τ ^ | / 4 T (3 cos2 Φ - !) 1
/ r
( 13 · 123 )
Q
Äf (Φ) = -=r ] / — — (1 - cos2 Φ) cos 2/9 J/π V ^ 4 One thereby obtains
(13.124)
Ä? = γ - γ j / 2 ^ |/-|- *|(Φ) + A |/~ |/JL ΐ|(φ, 0)
(13.125)
Ä|
(13 126)
=T
_
T^j^Tkl{0) ~\^Τ^φ'β)
*t = γ + γγ2π ]/ΎΗ(Φ)
·
(13.127)
This yields Ε(Φ,β)=Ύ(Ε11
+ ΕΆ + Εκ)
+ - 1 | / t o y - | - Λ1(Φ) (2£ 33 - -»11 - ^22)
+ "f- l/π 1/4 *Ι(Φ'β) {Εη ~ Ε™]
(13 128)
·
Physical Properties of Polycrystalline Materials
319
One thus obtains for the coefficients of the series expansion of the property func tion
-Tf^T'■ (8»„ -
JPU - Ä»)
(13.129)
el = -f-^Ί/τ<*" - E^
<13·130)
We further assume orthorhombic sample symmetry — i.e. sheet or foil. Then, according to equation (13.59), there result for the averaged coefficients
~4 = Ts | / T f(2^3S ~ En ~jE?22) °"+Ϋ* {En~ E^ °ft
(13,131)
~ 4 = i ]/Ύ t(2i?33 ~ Ei1 ~ Εί&) ^12 + ^{En ~ ^ C|21
(13 132)
·
One thereby finally obtains for the averaged quantity Ε(φ, γ)
+ ΤΓϋΤΤΓ Κ2^33 - *11 - ^22)
,
Ρ ΐ /
7 [( 2j ^33 — ^11 — ^22) ^ 2 2
15J/5 + b3{En - Ε22) Of2] ΡΙ{Φ) cos 2γ For fibre textures, according to equation (13.62), there results
* 2 = ά i/ir ^33 ~ ^11 ~ ^22) σ · + ^
(2?n
~ ^22) ^
(13.133)
(13,134)
It follows therefrom that Έ(Φ)=γ{Ε11
+ Ε* + Ew)
+ ^--jL· [(2^33 - Eu - E22) C\ + 53(2?n - E22) Of] Ρ2(Φ) ±0 }/2π (13.135) 13.4.3.1. Thermal Expansion of Uranium We will consider thermal expansion as an example of a tensor property of second order. In addition, because of its strong thermal anisotropy and because of its great technical importance, uranium has been the object of many experimental 22*
320
Physical Properties of Polycrystalline Materials
investigations. I t has been shown t h a t t h e thermal expansion of a polycrystalline uranium sample under certain conditions is given by the simple average value of t h e expansions of the individual crystallites 1 1 5 . I n t h e case of t h e strongly anisotropic expansion of uranium this is by no means self-evident. Thus, e.g., the ther mal expansion of a polycrystalline graphite sample differs noticeably from the simple average value 2 4 2 . If we set for the thermal expansion coefficients of uranium at 75 °C in the axis directions 2 8 , En = + 2 0 . 3 · 10- 6 ;
E22 = - 1 . 4 · 10~ 6 ;
Ezs = + 2 2 . 2 · 10~6 (13.136)
we thus obtain for the expansion of a polycrystalline sample with orthorhombic sample symmetry (sheet symmetry) in t h e direction Φ, γ according to equation (13.133) Έ(Φ, γ) · 10 s = 13.7 + [1.11C 2 U + 1.63C! 1 ] Ρ2(Φ) + [1.52C
(13.137)
Thus results according to equation (13.135) for t h e expansion of a sample with rotationally symmetric texture Ε(Φ) · 106 = 13.7 + [0.68C21 + l.OOCf ] Ρ2(Φ)
(13.138)
13.4.3.2. Growth of Uranium During Neutron Irradiation As a second example of a tensor property of second order we consider the dimen sional change of uranium during neutron irradiation. This property can be de scribed with sufficient accuracy by a tensor of second order 271 ' 272 , and t h e dimen sional change of a polycrystalline sample is given within t h e accuracy of measure m e n t by t h e simple average value, in contrast to graphite. The experimentally determined dimensional changes in t h e axis directions are En = -A
;
E22 = +A;
E^ = 0
(13.139)
According to equation (13.133) one thereby obtains for t h e length change in t h e direction Φ, γ of a polycrystalline sample with sheet symmetry Ε(Φ, γ) = -A
[0.löOCJPPa(Φ) + 0.206(7| 2 Ρ|(Φ) cos 2γ]
(13.140)
I t follows directly therefrom t h a t a sample with random orientation distribution shows no length change and t h a t t h e length change of a textured sample is in dependent of t h e coefficients C\λ and οψ. For a sample with rotationally symmetric texture the expansion in a direction which makes the angle Φ with the axis is given according to equation (13.135) by Ε(Φ) = -A
· 0.092(7|Ρ 2 (Φ)
(13.141)
Physical Properties of Polycrystalline Materials 3 2 1
13.4.4.
Elastic Properties
13.4.4.1. Cubic Crystal Symmetry We denote the stress tensor by a and the strain tensor by ε and limit ourselves to small deformations, so that a linear relation exists between the two tensors: ß<7 = smau
(13.142)
C
0ij = m^ki
(13.143)
In the notation used in equation (13.39)
(13.144)
Correspondingly, for the polycrystalline material «<ί = *ΜΛΙ
(13.145)
5ij = W i i
(13.146)
Also in this case it is naturally true that ~cmi = (km)-1
(13.147)
The poly crystal quantities s^u and c^u in equations (13.145) and (13.146) differ, in general, from the simple average values c ^ and si?·^. We set approximately kjki & cm = cjm
(13.148)
this thus corresponds to the approximation used by orientation. Likewise b
ijkl
«W = SW
corresponds to the approximation of RETJSS
VOIGT 2 9 1
for random crystal (13.149)
244
.
If the stress tensor is a simple tensile stress and if the grain boundaries are perpendicular to. the direction of a {Figure 13.7'a), then
4u = (cJwr1
< 13 · 150 )
whence, according to equation (13.44), *iW ™ T isW + sJßi] = SW
(13.151)
322
Physical Properties of Poly crystalline
Materials
▲
ε
CT= c o n s t "
Reuss
c:o n >t
Voigt
(a)
(b)
Figure 13.7 Validity of VOIGT'S and REUSS' approximation for two extremes of grain shape I t is customary to call this approximation for t h e elastic properties of poly crystal line materials t h e V O I G T — R E U S S — H I L L approximation. I t is well fulfilled for weakly anisotropic crystals, and t h e deviations from it still are small for stronger anisotropy 8 8 , 8 9 . The quantities s^ki and c^kl obey t h e symmetry conditions s
ijkl
=
s
jikl — sijlk
=
s
klij
(13.152)
One can therefore write t h e m in m a t r i x representation (Table 13,3). Because of t h e crystal symmetry a series of further relations exist between these quantities, which are illustrated for cubic s y m m e t r y b y t h e scheme in Figure 13.8 (e.g. reference 220). The cubic axes were t h u s used as t h e coordinate system. Tor isotropic materials t h e scheme in Figure 13.9 is valid. One sees t h a t in t h e case of cubic symmetry only one more linearly independent constant results t h a n in t h e isotropic case. W e denote t h e components of t h e tensor s, with respect t o t h e cubic axes, by s ^ ; we can t h u s decompose t h e tensor s^ki for cubic s y m m e t r y into an isotropic and an anisotropic p a r t in t h e following m a n n e r : s
m = SW + sJiW
(13.153)
s}jki denotes t h e isotropic p a r t according to Figure 13.9 and 5
im
ö
1122
- & ? 1212 .
(13.154)
Physical Properties ofPoly'crystalline Materials Table 13,3
MATRIX REPRESENTATION OF THE ELASTICITY TENSOR
ij
1 2 3 4 5 6
11 22 33 23 31 12
2 22
1 11
n H
m
323
4 23
3 33
δ 31
6 12
^1111
' 51H2
5
^1212
1211
a:: \ \ Fig. 13.8
Fig. 13.9
Figure 13.8 Elasticity tensor for cubic symmetry. The heavy points represent com ponents which can be different from zero; the components joined by a line are equal Figure 13.9 Elasticity tensor for isotropic materials. The crosses denote (1/2) ( s l l n — 5
1122;
is t h e amount of t h e anisotropy, and t h e tensor t^kl has t h e components hill
— ^2222 — ^3333 =
*
(13.155)
All other components are zero. Since t h e isotropic p a r t s?-w is orientation indepen dent, one obtains for t h e average value s
ijkl — sijkl — sijkl +
s
Jijkl
~~ sijkl +
s
&(hjkl ~
(13.156)
hjkl)
According to equation (13.51), t h e quantity t^n is given by hjkl — ä(ijM; nnnn) = a{ijkl)
(13.157)
(which is t o be summed over n from n = 1 t o n = 3). The a(ijkl, nnnn) are given b y equation (13.52). Because r = 4, there appear therein in addition t o CQ1 = 1 only t h e coefficients C}1, C}2, C4 3 . One t h u s obtains liki = äl^ijkl) where t h e af(ijkl) mation : af(ijkl)
+ a^{ijkl)
Cl1 + af{ijkl)
C412 + af{ijkl)
C\*
(13.158)
result from t h e quantities defined in equation (13.53) by sum = a%v(ijkl;
nnnn)
(13.159)
324 Physical Properties of Polycrystalline Materials For t h e components of the averaged tensor in R E U S S ' approximation, one thereby obtains *W = *Μ = sw + Sa&oHtfM) — t m + äl^{ijkl) + af{ijkl)
C412 + äf{ijkl)
0\λ
Of]
(13.160)
I n particular, for t h e case of random orientation distribution one obtains 4ki = *w = 4m + Sa&Piifil)
- kjku
(13.161)
«fin = 0·6*ιιιι + 0.45?122 + 0.8s? 212
(13.162)
sf122 = 0.2s°ull
+ 0.8s°1122 - 0As°1212
(13.163)
sf2l2 = 0.2s°ull
- 0.2·5?122 + 0.6s? 212
(13.164)
The three last quantities fulfil, as they must, t h e condition given in Figure 13.9. The ä%v(ijJcl) are purely mathematical quantities; they depend neither on t h e elastic constants of t h e single crystals nor on t h e orientation distribution. They can therefore be calculated with complete generality in t h e manner described b y equations (13.22) —(13.24). One t h u s obtains Table 13.4 (see reference 50). Table 13.4
THE AVERAGE COEFFICIENTS ä^v(ijkl) FOR CUBIC CRYSTAL SYMMETRY A N D ORTHORHOMBIC SAMPLE S Y M M E T R Y
ijkl
alHijM) - tijkl
a^iifil)
α\\ι^Η)
af{ijkl)
1111 2222 3333 1122 1133 2233 1212 1313 2323
-0.4 -0.4 -0.4 +0.2 +0.2 +0.2 +0.2 +0.2 +0.2
+0.021 818 +0.021 818 +0.058 182 +0.007 273 -0.029 091 -0.029 091 +0.007 273 -0.029 091 -0.029 091
-0.032 530 +0.032 530
+0.043 032 +0.043 032
0 0
-0.043 032
0
0
-0.043 032
+0.032 530 -0.032 530 +0.032 530 -0.032 530
0 0
0 0
I n exactly t h e same manner there result for t h e constants c^u. in t h e case of cubic s y m m e t r y with
tfjki = cm + cffm C
a
=
(13.165)
C
llll ~~ C1122 ~~ ^C1212
(13.166)
I t follows therefrom t h a t c
ijU
=
^ijkl — Cijkl + cdijJcl = CyM + Ca(tijH — tim)
<$W = cm = cw + ta&VWM) + af{ijkl)
- hm + ä^iijkl)
Cf + af{ijkl)
Cf]
(13.167)
C\l (13.168)
Physical Properties of Polycrystalline Materials
325
For random orientation distribution one further obtains cJw = 4a + cJafrW) - tijkl] (13.169) as well as the equations for cjju analogous to equations (13.162) —(13.164). One then also obtains the constants sjjki according to equation (13.150). The elastic compliances s^i of the polycrystal are calculated according to the approximation given by equation (13.151). One obtains YOUNG'S modulus E(y) in an arbitrary sample direction y from the elastic compliances s^i (see, e.g., reference 220): -Ξ— = 2/ismi + 2/P2222 + 2/P3333 + fyf 0|(S112a + 2s1212) E{y) + 2yf»f («use + 251818) + tyhKhwz + 2S2323)
(13.170)
modulus E(y) was calculated in this way for a Cu—Si sample deformed by shear spinning by DUBAND 1 2 3 . It is represented in Figure 13.10. If one sets
YOUNG'S
2/3 = °; 2/i = cosy; i/2 = siny the YOUNG'S modulus in the plane of the sheet is
E(y)
= smi
cos4 γ + 52222 sin4 γ +
»1212
(13.171)
+ -^i U M )ein»2y
(13.172)
Figure 13.10 Elasticity modulus E(y) as a function of the sample direction y, after DURAND 123 . The material studied is Cu—Si deformed by shear spinning
326 Physical Properties of Polycrystalline Materials The coefficients 0 | 1 = — 1 . 0 2 ± 0.15;
Cl2=
- 0 . 4 8 ± 0.15;
Cf =
-1.60 ± 0.08 (13.173)
were determined for a 9 0 % cold-rolled copper sheet from t h e first four pole figures (111), (200), (220), (311) 82 . One obtains therefrom t h e values given in Table 13.5 for t h e quantities t^i as well as t h e s^i in t h e three approximations V — R — H . Table 13.5
THE COMPONENTS OF THE TEXTURE TENSOR ί ^ AND THE ELASTIC COMPLIANCES OF A 9 0 % COLD-ROLLED COPPER SHEET IN THREE APPROXIMATIONS (IN 10~12 cm2 dyn -1 )
v
ijkl
s
ijkl
hjki ~~ km
Hjkl
s
1111 2222 3333 1122 1133 2233 1212 1313 2323
+ 1.49 +1.49 +1.49 -0.63 -0.63 -0.63 +0.33 +0.33 +0.33
-0.4753 -0.5067 -0.4596 +0.2612 +0.2141 +0.2455 +0.2612 +0.2141 +0.2455
+0.7960 +0.7503 +0.8190 -0.2486 -0.3174 -0.2716 +0.7139 +0.6451 +0.6909
+0.6293 +0.6074 +0.6403 -0.1832 -0.2161 -0.1942 +0.5182 +0.4708 +0.5013
ijkl
s
ij!d
+0.7126 +0.6788 +0.7297 -0.2159 -0.2667 -0.2329 +0.6160 +0.5580 +0.5961
If one inserts t h e last three columns into equation (13.172), one thus obtains t h e behaviour of t h e ^ - m o d u l u s in t h e plane of t h e sheet represented in Figure 13.11. One sees t h a t t h e curve Επ agrees well with t h e measurements of W E E R T S . The case of rotational symmetry is naturally obtained as a special case of orthorhombic symmetry. I t results from t h e general case when one sets
ci1
V
21+1 4π
20
CJ;
Cl2 = Cl* = 0
JO 40 50 60 Angle towards rolling direction
(13.174)
90°
Figure 13.11 Elasticity modulus of a cold-rolled copper sheet as a function of angle from the rolling direction .
Physical Properties of Polycrystalline Materials 327 One then obtains in place of equation (13.158) km = «S(*?«) + »iW») C\
(13.175)
The coefficients a\(ijkl) are given in Table 13.6 (see reference 243). Table 13.6
THE AVERAGE COEFFICIENTS
a\(ijkl)
FOR FIBRE TEXTURES
IN THE CASE OF CUBIC CRYSTAL SYMMETRY
ijkl
al(ijkl) -
1111 3333 1122 1133 1212
-0.4 -0.4 +0.2 +0.2 +0.2
tm
al(ijkl) +0.018 +0.049 +0.006 -0.024 +0.006
465 240 155 620 155
The VOIGHT^-REITSS—HILL approximation is the mean value of two unrealistic assumptions (see, e.g., Figure 13.7) without good theoretical basis. Practically, however, it agrees with the measured poly crystal values within a few per cent 88 ' 89 . Better-based approximations must, however, make explicit or implicit assump tions about the shape and arrangement (orientation correlation) of the crystal lites (see, e.g., references 29, 65, 66, 188). KRONER'S theory 187 assumes statist ically random orientation correlation. It was originally carried out for random orientation distribution. It was extended by KNEER 1 8 0 » 1 8 1 to rotationally sym metric textures and by MORRIS 213 » 215 to textures of cubic crystal and orthorhombic sample symmetry. To be sure, the calculation according to this theory requires a considerable calculation expenditure. Since the results also agree within about 2% with those of the V—R—H approximation, which is essentially simpler to calcu late, this approximation is used for nearly all practical calculations. 13.4.4.2. Orthorhombic Crystal Symmetry The averaging of elastic properties for orthorhombic (or higher) crystal symmetry was carried out by MORRIS 212 . Since the elastic anisotropy in this case can not be described by a single quantity sa or ca, one must begin with the general averaging formula (13.51). If we use the two-index notation for the components of the elastic tensor according to Table 13.3, equation (13.51) can thus be written with
smn = a{mnpq) sm
(13.176)
4
I
I
ä(mnpq) = Σ
Σ
Σ
afv(mnpq) Of
(13.177)
Z=0 A*=0(2) v=0(2)
(we have here used a different enumeration for μ and v). The coefficients afv were given by MORRIS 212 , for a different normalization of the coefficients (7f*\ They are converted to the normalization used here in Table 13.7.
328
I o
P
o Ü o
«
«
H P.
W
O H O
o CO E-i
fc H fc
O
O O
«
O
CO
525
H
S CO EH
Sz=5
H O O
«
s H l>
•8
iV
TH
CM
o
i*«a
T*
ri<
(M
^
O
rh
■*
T*
CM
O
CM
CM
O
CM CM
O
o o
O
CM CM
CM
CM
«Γ •«s»
X X X OH
X
X X
O Q
1
CM
CM
CD
CM
1
00
H
Q
TA
N
CM*
CM C* CO CD t O C i CM CD r H
CD
^ CM
00
00
CO
rH
I>
TH Ö
CD
TH
rH
CO
τ-i
rH
1
CO
CO
r-i
CO
CO
CM
1 1
CO
CM
CO
1
CO
CO
1 GO
CO
CO
CO
1
CD
1
CD
τμ CM
1 1
CM
C5
τ*4 CM
CM
1 CD CD CM rH
1 CM
I
CM
1
CM
CM
1
1
CM CM
oo
1 1
CM CO
1
CD
I
CM CM
co
1 CM
CM
CM
CM
CM
1 1
CM
1
CM
1 1
CM
CO CD
1 1
CM CO
CD
^
1
CM
^
CO
CD
CD
CM
CM
CM CM
TH
CO CM CM CM
"*
1 1
CD
CD
CM CM
CM CD
CD
CO CO CM CM
T*
CD
CM CM CM CM
1
rH rH CM CM
1
CM
1 CM
CM
CM
τμ
^
1 1 .
1
rtf
CM
CM CO TH vO
Ö
0 0 CO 0 5 vO TH CM
< CD OS 0 0 TH TJ 0 5 OS CO TH CM
rj< H
TH
CM* Ö
vO TH O O D - CM ι θ M M OS CM O
O
TH
Ö Ö Ö
M "tf O
TH
rH
TH
TH
1 TH
CM
1 1
CO CO CO
CM
1
CD
rH
00
CO CO
rH
I CO
CO
Ot>
rt
CM
"rft CM
1 11 I
CO
CO
X
X
CJ5
CO
1 CD
CD
CO CO CD
1
CD CD CM rH
I
CD
1 1
CD
CO
1
X ÖH
X OH
X ÖH
X
X
O
o o
X
"Si
1
o o
5S. »
1111 1122 1133 1112 1113 1123
CO
O0
CO
1 ^f4 CM
00
*^4 CM
CM CM vi rH
1
CD
CD rH
CO
τΗ/t CD
CD
1 ^Η^
1 1
CO
"tf
' "<*
"^H
^H CO
TH 1
T H
CO CM CO CO
^
"Ψ
CO rH CO CO
1
CD
CM rH CO CO
' CD
CO CO CO CO
1 1 CO
CM CM CO CO
CM CM Til 0 0 | r H r H CM
rH rH CO CO
rH r-i
1
r-i
1 1
r-i
1
1
oo
CM
1
1
CM
O
1
00 QO
1
CM CM
CM CM
1
CM CM
1 1 1
*rH
CXI CO CO rH CM ^ι T*H r-i
T* ^ ^
CXI CM CM
1 I
Ttf CM CM
1
0 0 CM CM CO CO
1
CXI 00 00
1
CO
1
00
CM 00 00
1 1
1
00 00
1 co
CM
CM 00 00
1
CM
1
CM CM
1 1
CM 00 00
r-i
r-i
r-<
1 1 r-i
1
1
1
CM
00
1
CM CM
1
1 1
1
00 00
CO CO 00 CM 00 00
Ti*
1 1 1
1
1 ιθ xO
ιθ
iO iO
IO
1
rH
rH r-i
1
r-i
rH
1 1
r-i
1 1 1
rH rH
CM
00
329
1
CM CM
00
CO CO 00 CM 00 00
1 I 1 1 00 "# I i
(M
1
o
O
1
r-i
iO
1
vO iO
1 lO iO
iO
1 rH 1
CM
1 1 1
r-i
r-i
1
CM CM CM 00 CM CM CO CO r-i r-i CO
1
rH
r-i
1
rH r-i
rH r-i
1 1
CM CM CM 00 00 00 00
CM CM CM 00
r-i
oo
r-l CM CO CM CO CO rH CM CO r-i r-i CM CO CO CO CO CO CO r-i r-i r-i r-i rH r-i
1 1
r-i
1 1
1
rH
CO (M CO CO
1
r-i
r^
1
CM CM
^μ
^
CO rH CO CO
1 1 1
CM CM CM
CM rH CO CO
CM O
r-i
00
1 "«# CM (M 1 1
CM CM
τμ CM CM
1 1
1 1 1
CM CM CM
CM CO CO rH r-i CM O iO iO tO iQ XO
1
rH CM © r-i
00 CM CM CO
1
CM CM
CM
CM CM
CM CM
CXI
1
CXI CM
1
1
1
1 1
o
rH rH
©
1 1
rH rH
o
1
©
1
CM 03
CO CO 00 CM 00 00
1
r-i
rH TH
1
rH
CO CO
1
CM CM
1 CM
CO CO (M rH
CM CM CM 00 00 00
r-\ CM r-i (M CM CM r-i rH
330 Physical Properties of Polycrystalline Materials 13.4.5. s
Plastic Anisotropy
We consider a volume element of a plastically deformed material with coordinates χχχ^χζ. During the plastic deformation the material undergoes a displacement with components %w2%. ^ t n e deformation is homogeneous {Figure 13.12), ui = etjXj
(13.178)
The deformation tensor e^ can be decomposed into a symmetric and an antisym metric portion, the latter simply describing a rigid rotation. The symmetric por tion of the deformation tensor is the strain tensor: (13.179)
£
ij — ~o" \eij "Γ eji)
Figure 13.12 Definition of a homogeneous displacement By suitable choice of the coordinate system it can be written in principal axis form. If one further assumes that the volume remains constant during the deformation, the sum of the diagonal terms must be equal to zero. One can thus bring the strain tensor into the form 1 0 0 -q 0 0
0 * 0 -(1-i).
(13.180)
η can thus be considered as the absolute amount of the deformation, while q characterizes the axial ratio of the deformation. This deformation is represented in Figure 13.13. The material offers a resistance to deformation, which may be described by the stress tensor σ^. The required deformation work dA is then given by dA = σϋ deij (13.181)
Physical Properties of Polycrystalline Materials 331 ▲
/
,
MV
(i-q)-n
/
►\
\
/
>
►
Figure 13.13 General deformation in the principal axis representation corresponding to the deformation tensor (13.180) In the principal axis representation one thus obtains άΑ = άη [σ η - er33 - q(a22 - σ^)] = M\q). άη
(13.182)
The factor M' is a measure of the deformation resistance of the material. If the material is crystalline, M' will thus also depend on the orientation of the principal axes with respect to the crystal axes. It may be described by the rotation g' which transforms the principal axis system of the strain tensor into the crystal coordinate system. The deformation resistance is thus generally given by άΑ = M\q, g') άη = M(q, g') r 0 άη
(13.183)
The function M'(q, g') can be experimentally determined. It can, with certain assumptions (e.g. in the framework of the TAYLOR theory 278 ), also be calculated53'60. Then τ 0 is the critical resolved shear stress in the crystallographically equivalent glide systems and M(q, g) is the Taylor factor, i f (0, g') is, for example, given in Figure 13.14 for cubic crystal symmetry and {111} slip in the framework of the TAYLOR theory. The case q = 0 represents the so-called plane-strain defor mation, which, in general, can also be assumed as an idealized deformation for rolling. Since the strain tensor has orthorhombic symmetry, the deformation resistance can be developed in a series of cubic orthorhombic generalized spherical harmonics: L
M{q, g') = Σ
M(l) N(l)
Σ
Z=0 μ=1
Σ »if (i) Tfig')-
(13.184)
ν=1
If M(q,gf) is numerically known, the coefficients mf(q) can be calculated in the manner described in Section 4.1.2. They are given in Table 13.8 for I ^ 10 for different values of q in the framework of the TAYLOR theory. Figure 13.15 shows the rapid decrease of these coefficients with the degree I for q = 0.
332
Physical Properties of Polycrystalline Materials
Figure 13.14 The TAYLOR factor i f (0, g) for 'plane strain' deformation q = 0 as a function of the crystal orientation relative to the principal axes of deformation^*0
We shall now calculate the average value M(q, g") of the deformation resistance M(q, g') for a polycrystalline material. We thereby permit the principal axes of the strain tensor not to coincide with the sample coordinate system. If the prin cipal axes of the strain tensor form the coordinate system K', the relations be-
4
©
Ö
Ö
CO
Ö
CM
o
i!
Öi
r-4
Ö
©
I s ii
as 00 rH CM CM CO 00
O
00 as CM 00 CO ιθ OS i—1 OS CO © o r-4 rH CM O
GO CO OS CM CO ιθ CO rH rH CM
CO CO t~ O CO
Ö
©
CO O D-
© ©
O© OS 00 *H CM rH CM
© >0 "■# OL CM CM CO CO CM
©
1
Ö
o
o
1
i-i *"# CO CO CO CO CD CM CM OS o r-i CM
ö ö ö ö I I
°. ^ ^ ^1.
i O -* CM O T * 00 O rH
r-4 ** rH C^ rH © "H CM OS "
Ö Ö
1
CO CO as
Ö
Ö Ö Ö 1 1
© o
1
CO OS o [>> rH 00 o
o as
Ö O
CO
1
o
O t> CO
1
00 CO CM r-\ rH
rA O
o o co o o o
I
CM "* i-i ^ xO O L © ©> ©' ©
1
1
©* ©
1
©' ©
as rH as © CM CO rH 00 rH rH rH CO ©'
1
© ©
co o©'
©
333
OS i— I © r-4 ©'
r-4 O 00 ©# ©"
rH rH C* t> r-\ rH © ©
O M ( rH xO O H © ©'
©'
O Ο0 vO
I
© © 1-4 Tj< CO 00 © © ©* ©
DOS -Ψ © ©'
I
00 ^ CO © ©'
©'
1
CO ΓOS © © Ή^ ©# © ©' ©*
CM CM CO' GO CO t>- CO rH rH CM O ©
©
rH O rH O
t> H C O 1-4 C O CM © © ©* ©*
(M (M O © ©'
|> CD H © ©'
©'
Ö O 1 C (M 'H © ©'
C5 ιθ O © ©'
00 O r-4 © ©'
©
©
©
71 00 CO © ©
l·· © ^ © ©
CO O I >© ©
©
CO CO CM CO
©
C* ^* rH
CO 00 r-4 GO OS CO r-4 O CM c3 GO rH rH r-4 rH CM © Ö © Ö Ö
H H CM (M i* r if © © © ©
D i—l *# © ©"
CO C O -rH © ©*
C O ^ T* ©# ©'
rH rH CO © ©'
CO ^ C M © ©'
00 r-4 © CO ©"
rH N ■~H © ©'
CM © ^ O CO rH © © ©' ©'
CO GO CO r-4
as CO CM ^44 CM rH CO © ©
xC 00 US N CM CO © © ©" ©'
r-\ © CO r-4
© ©
D- CM oo oo lO rH r-4 CO ©' ©'
CO r-i as τ4 © τ4 ©* ©
r H CO tas oo oo i—I τ4 ·*Φ © © © © ©' ©"
xO © rH © ©*
OS CM CO CM CO GO © rH © © © Ή ^ ©* ©" ©'
OO 00 CO © C M "^ rH lO i— i © rH CM ©' ©' ©'
CO r-4 C O © ©'
CM GS " * "Ψ © © © © ©' ©*
rH OS O © ©'
C^ " t1 H © ©
^ rH © 00 CO H © © © ©'
© CO H ©
CO
©
xC
©
^
LO xO t >- CO CT 5 I >- CD CO rH CD "H ^ — t "* CM © © # © r-4 © © ' ©" © ' © ' ©" W rH rH rH ©
i£5 CD CM © ©
CO
£> CD Γ© O
t>lH CO r-4 03 O r-\ CO CO i—1 o r-4 © © ©
©
r H lC lH CO CO I >00 ^ L> CD ^44 r-4
CM CO τΗ
rH "^
©
O
CM . CO
OO 00
"·#
rH
00
©
00
©
co co co
00
o
M i l
Tf< t^00 © Ö
CO CO (M 00 OS CO O
1
as 00 00 ©
CO rH "Φ iC D- CS
CO r-4 CO CO 00 CO 00
Ö Ö 1
CO co OS r-\ CM OS r>
o
CO
©
o CO
CO rH O
rH rH CO
[>O CM
CO CO
23 Bunge, Texture analysis
334
Physical Properties of Poly crystalline
Materials
1,6
4 6 8 10 Figure 13.15 The coefficients mf of the TAYLOR factors for I ^ 10 5 3
tween g, g' and g" are thus demonstrated in Figure 13.16. In contrast to the general case treated in Section 13.3.3, we have here used g' as rotation, which transforms the principal axis system into the crystal system, while there the opposite rotation was assumed. In place of equation (13.184) we must thus average the function L
1
Mil)
Nil)
M^g'- ) = Σ Σ Σ <{q) mg'-1) 1=0 μ=1
ν=1
(13.185)
which corresponds to equation (13.18) and thus to the assumptions in Section 13.3.3. The usual assumption will be made that the functions Tf are real in the cubicorthorhombic case. According to equation (13.74), for the average deformation resistance M(q, g") one then obtains L
Nil)
Nil)
M{q, g") = Σ Σ Σ mr(q) W ' ) 1=0 μ=1
f=l
(i3.i86)
Physical Properties of Polycrystalline Materials 335
Deformation Tensor
Figure 13,16 Relations between the crystal coordinate system ϋΓΒ, sample coordinate system K^ and principal axes system K' of the deformation tensor with M(l)
^\q)=w^Zi^{q)civ
(13.187)
The symmetry of the deformation tensor is the same as that of the sample — namely the orthorhombic symmetry. The functions T]fv(g") thus are orthorhombic in both coordinate systems. We consider, in particular, the case in which the JC3-direction of the strain tensor coincides with the Z-direction of the sample coordinate system. The rotation g" then has the form
(13.188)
The functions Tfv(g") can then be written Tf =
-l)oc
(13.189)
For the average deformation resistance one thereby obtains N{L)
__
M{q, oc) = Σ «*(£) °os 2(v - 1) oc
(13.190)
with the coefficients L
1
Σ a (q) = j=o Σ 2Z + 1 A=I v
KW
(13.191)
M(q, oc) is plotted as a function of q for oc = 0 and 45° in Figure 13,17 for a steel sheet79 whose texture coefficients Cf are contained in Table 13,9. Adefonnation tensor with the principal axes described by equation (13.188) corresponds to a 23*
336 Physical Properties of Polycrystalline Materials sample which is rotated through t h e angle oc from t h e rolling direction and is extended in t h a t direction, where t h e relative width and thickness decreases are given b y q and (1 — q), respectively [Figure 13.18). As one sees from Figure 13.17, t h e deformation resistance has a minimum for a certain width decrease q. If t h e sample [Figure 13.18) is elongated in its long direction by άη, without any restric tion t o t h e shape change in t h e two directions perpendicular thereto (free tension), t h e width contraction will be so selected t h a t M[q) is a minimum. The values of q for t h e minima read from Figure 13.17 are plotted in Figure 13.19 as a function of t h e angle oc a n d compared with experimental values, while Figure 13.20 shows Strain
3.5
ratio
R
0.111 0.250 0.429 0.667 1.0
1.5
2.33
4
9
3.4
3.3
-
et 9Xp(?r.
3.2
Min.
a = 0°/
3.1
3.0
o -J-—
=45
b^xp er.
0.1
0.2
MiΛ.>
0.3 0.4 0.5 0.6 0.7 0.8 Contraction ratio q
0.9
Figure 13.17 The average TAYLOR factor M{q, oc) as a function of the axial ratio q of the deformation53 Table 13.9
1 2
3 4 5 6
COEFFICIENTS C\V OF THE STEEL TEXTURE
1 = 4:
1= 6
1= 8
1 = 10
-1.4689 -0.4590 0.4973
2.6861 -1.2023 0.4641 -0.1406
-0.0707 0.2859 -0.6541 -0.4697 -0.2219
-1.0248 0.1910 0.0920 -0.2714 -0.0488 0.0202
Physical Properties of Polycrystalline Materials
337
t h e corresponding If-values. The q u a n t i t y
R =
(13.192)
1-0
which is frequently used as a measure of plastic anisotropy, is also given in Figures 13.19 and 13.20.
Figure 13.18 Orientation of a tensile specimen relative to the sample coordinate system KA
0°
10°
20° 30° 40° 50° 60° 70°, Angle to rolling direction
ft0°
90°
Figure 13.19 The transverse contraction ratio q determined from Figure 13.17 and measured, as well as the associated R-value of the plastic anisotropy for a 95% coldrolled steel sheet as a function of the sample angle79
338 Physical Properties of Polycrystalline Materials 9.2
!
1
[tons/sq. in
1 MY
9-0
L^C^™\^
" ^
5
8 8.6 I—
/ ^ e x Derime ital
(/>
I 8.4
wN ^
9.8 calcul atecl·^
3.2
^4
1 T
H 3.1 °
""^
H3.0
L—*^r> i
8.2
2.9
8.0 0°
10°
20°
30° 40° 50° Angle to rolling
60° 7Q° 80° direction
Figure 13.20 The measured and calculated average 95% cold-rolled steel sheet79
13.4.6.
The Reflectivity
TAYLOR
90°
factor M(qmin%9 oc) for a
of Crystallites for X-rays
It is also of interest to specify the diffracting power of crystals for X-ray beams, which, in general, is described with the help of the reciprocal lattice, in the repre sentation by functions of a direction. If an X-ray beam of wavelength λ falls on a crystal, it can thus be reflected by a lattice plane (hkl). The BRAGG relation X = 2dhklsm&hkl
(13.193)
is true for the reflection angle 2#MZ. The reflection can thus only occur for certain angles 2#ÄW. The usual reflection condition, which says that the normal to the reflecting lattice plane (hkl) must fall in the direction of the bisector of the angle between the incident beam and the reflected beam, is added thereto. If we consider the reflection for a specific angle 2#ÄK in the case of fixed incident and reflecting directions, and bring the crystal into all possible orientations, we thus see that reflections only occur when the normal to the (hkl) plane, or a symmetrically equivalent plane, falls in the direction of the angle bisector. We can thus specify the diffracting power for the reflection angle 2#Mj by a function of those crystal directions h which fall in the direction of the angle bisector. It is clear that this function has the character of a ό-function — i.e. is different from zero only in the direction of the normals of the symmetrically equivalent (hkl) planes. We denote it by Em(h) and set 36 ' 38 Em(h)
= S(hkl)E'hkl(h)
(13.194)
Physical Properties of Polycrystalline Materials 339
S(hkl) is a factor depending on hkl, and the function E'm(h) describes the orient ation dependence. We so normalize it that #^H(fc)dfc = l
(13.195)
We expand it in a series of spherical surface harmonics of the crystal symmetry: oo
Kki(h) = Σ
Μ(λ)
Σ
<#(*«) k$(h)
(13.196)
We obtain the coefficients e%(hkl) in the following manner. We multiply by k*ß(h) and integrate over all h. This yields j>E'm(h) k"(h) dh = eftftiU)
(13.197)
Since the function E' is different from zero only in the immediate vicinity of the points h = [hkl], the functions 1$(h) can, however, be regarded as constant within these small regions, so that we obtain with equation (13.195) e%(hkl) = k^(hkl) φ E'm(h) ah = if"(AM)
(13.198)
k%(hkl) denotes the value of the spherical surface harmonics k%(h) for the directions of the normals of the lattice plane (hkl) and its symmetric equivalents. We thus ob tain the representation of the diffracting power for the reflecting angle 2#MZ: oo Μ(λ) .
®hu{h) = S(hkl) Σ Σ *Γ(*«) *?(*)
(13.199)
Λ=0μ=0
The complete reflection behaviour of the crystal can thus be described by as many functions Eh1d(h) as different reflecting planes (hkl) occur (which are, strictly speaking, naturally infinitely many). If one thinks of the functions Em(h) as distributions on spheres of radii lldhkl, one thus obtains a point distribution in space, the so-called reciprocal lattice. For the representation used here, however, the association with spheres of radius l/dh]a is not necessary. According to equation (13.58), one obtains for the diffracting power of a polycrystalline material with the sample direction y as the angle bisector _
oo N(X)
Emiy)
= Σ Σ el(hkl) *Hy)
with the coefficients S(hl·])
~
This yields
(13.200)
λ=0ν=1
el{hkl) =
M
M
(13 201
2ΓΓΤ -5 www® oo Μ(λ) Ν(λ)
Em{y) = em
Σ Σ Σ
Λ=0 μ=1
ν=1
· >
Qßv
-ΪΓΊΓΛ 4Λ - r 1
^*") W
<13·202)
The averaged function Eh1d(y) is naturally essentially the (hkl) pole figure (see equation 4.35).
340
Physical Properties of Polycrystalline
Materials
Because of the ό-character of the function Ehkl(h), its series expansion (equa tion 13.199) includes infinitely many terms, in contrast to the series expansions of most other physical properties, which frequently contain only terms of very small order. This is the reason that texture determination from X-ray diffraction measurements considerably exceeds that from other physical properties as far as accuracy is concerned. 13.5.
Determination of the Texture Coefficients from Anisotropie Polycrystal Properties
We have derived relations between physical properties of polycrystals and those of the individual crystals together with the orientation distribution function. The relation is particularly simple when the property of the polycrystal is, given exactly or to sufficient approximation by the simple average value. If we use the representation of a property by functions of the orientation g (or, as a special case thereof, the representation by surfaces), for the relation between the coeffi cients efA of the single crystal and i f of the polycrystal (see equation 13.75) it is thus true that -μν _ 6l
=
1 ^ μλρλν 21 + 1 A~i 6* l '
1 ^ λ fg M(I) 1 ^ μ ^ N'(l) 1^ν<^Ν{1) 0 ^ I fg r
(crystal symmetry) (instrumental symmetry) (sample symmetry) (order of property) (13.203)
We have thereby first regarded the single crystal coefficients efλ and the texture coefficients C\v as given and calculated the polycrystal coefficients efv therefrom. We shall now regard the single-crystal and polycrystal coefficients as given and calculate the texture coefficients C\v from equation (13.203). (If, for example, we assume the diffracting power for X-ray beams as the crystal property, we thus obtain the method of texture determination from pole figures measurable by Xray diffraction described in previous sections.) A very important theorem first results. Many of the property functions lead to a very small value of r. Thus for tensor properties of second order r — 2, in the case of elastic properties r = 4, and for the magnetization energy of cubic crystals we have r = 6. For I > r all coefficients e^and consequently also all efv thus vanish. Equation (13.203) is therefore identically satisfied for arbitrary values of the texture coefficients Of1 with I > r. From the measurement of the anisotropy of a physical property one can thus in principle determine the texture coefficients at most up to a degree I = r, which is equal to the order of the property function considered. One can therefore regard a property which is represented by a function of lower order as 'long wavelength'. If one uses this to 'look into' the texture, there thus results a correspondingly poor 'resolving power'. One can no longer detect details of the texture which are finer than the resolving power. This is completely analogous to the dependence of the resolving power of optical instruments on the wavelength
Physical Properties of Potycrystalline
Materials
341
of t h e radiation used. I n particular, magnetic measurements t h u s allow determin ation of t h e texture to at most I = 6. For each pair of values I and v relation (13.203) represents a linear system of equations with M(l) unknowns C\v. F o r each value of μ one equation results, so t h a t the number of these is N'(l). I n order for t h e system of equations to have a unique solution, t h e number of equations m u s t b e a t least equal to t h e number of unknowns: N'{1) ^ M(l)
(13.204)
M(l) denotes t h e number of linearly independent spherical surface harmonics of t h e crystal symmetry and N'(l) t h a t of t h e symmetry in t h e coordinate system K'. In the previously considered example this was t h e symmetry of t h e distribution of magnetization directions in an inhomogeneous magnetic field. Since t h e property was investigated with t h e help of t h e variable coordinate system K\ we shall refer to this symmetry as 'instrumental' symmetry. Now, t h e higher t h e symmetry is, t h e smaller t h e number of linearly independent spherical surface harmonics is. Equation (13.204) means therefore in somewhat simplified form instrumental symmetry ^ crystal s y m m e t r y
(13.205) v
The solubility of t h e system of equations (13.203) for t h e C\ can t h u s be limited in two ways: firstly, t h a t beyond a certain order I = r all coefficients are zero; secondly, t h a t too few equations are available. I n m a n y cases t h e 'instrumental' s y m m e t r y is t h e symmetry of rotation — i.e. t h e property depends not on all three orientation parameters of t h e crystal, b u t only on t h e orientation with respect t o a direction (e.g. t h e direction of a homo geneous magnetic field). I t is independent with respect to a rotation about this direction. I n this important case t h e number of spherical surface harmonics of t h e instrumental symmetry is N'(l)=l
(13.206)
For each pair of values I a n d v we then have only one equation. Thus we must have M(l)^l
(13.207)
The allowable values of i max> according t o this condition are given for different crystal symmetries in Table 13.10. Only in t h e case of rotationally symmetric crystal s y m m e t r y is t h e resolving power not limited by equation (13.207) (orient ations of fibres or ellipsoids of rotation). For cubic and hexagonal symmetries Table 13.10
HIGHEST DEGBEE I = 1^^ is FULFILLED
FOR WHICH THE RELATION M(I) <; 1
Crystal trisymmetry: clinic
monoclinic
orthorhombic
tetragonal
hexagcnal
cubic
«max.
0
0
2
4
10
0
rotational
342
Physical Properties of Polycrystalline Materials
the resolving power is not more severely limited by equation (13.207) than it is with magnetic measurements anyway because of I fg 6. For crystals of lower symmetry equation (13.207) indicates the impossibility of texture determination from a single physical property. If one is to avoid the solution limit established by equation (13.207), one must thus measure more property functions. One then obtains one equation from each property for each pair of values Z, u. If 7 E is the number of different measured properties, in place of equation (13.207) there re sults the condition M(l)^IB
(13.208)
By measurement of several properties one can thus obtain as good a resolving power as by measurement of several pole figures, if one has suitably 'short wave length' properties — i.e. those whose series expansions contain terms with suffi ciently high values of L Now nearly all property functions are in this sense very 'long wavelength' with the exception of the diffraction effects of the crystal lat tice, which are very strongly angularly dependent and therefore contain terms of higher order in their series expansions. A quantitative texture determination with suitably good resolving power is, in general, therefore not possible from the aver age values of several physical properties. The reflection of light by etch pits is one of the few properties of medium 'wavelength' which can be used for texture determination. If one can so etch the crystallites that completely plane crystal surfaces result which are suitably large in comparison with the wavelength of light, the reflected light beam will not be broadened compared with the incident beam. The reflection curve will be very sharp and its series expansion will contain terms of higher order. In actuality, however, the etched crystal surfaces are not completely plane and also not always large in comparison with the wavelength of light. The reflection curve will be broadened, and its series expansion will no longer contain as many terms as in the ideal case, but still many more than, for example, the magnetic properties. If one therefore successively etches different crystal surfaces and measures the angular dependence of the reflection curve of the polycrystal, one can calculate the texture coefficients therefrom, as from X-ray diffraction measurements. Unfortunately, the exact form of the reflection curve depends on many factors which are very difficult to control, so that the 'property function' of the single crystal can only be inexactly determined. The coefficients efA are thus inexactly known and also the C\v can not be exactly calculated. Therefore to the best of our knowledge no quantitative texture determination has been made by this method. 13.6.
Determination of Single Crystal Properties from Polycrystal Measure ments If one knows the texture coefficients Cf and one has measured the orientation dependence of the polycrystal properties, one can thus use equation (13.203) to calculate the single crystal coefficients ef\ For each pair of values l9 μ one obtains
Physical Properties of Polycrystalline Materials 343 N(l) equations with M(l) unknowns. For a unique solution we must thus have N(l)^M(l)
(13.209)
This means, in general sample symmetry fg crystal symmetry
(13.210)
The sample symmetry must be lower than the crystal symmetry, since otherwise an additional symmetrization of the property will be produced by the orientation distribution. In the extreme case of spherical symmetry (thus random orientation distribution) one can not make any assertions about the orientation-dependent terms of the single crystal properties, since the poly crystal properties are quasiisotropic. In the case of rotationally symmetric textures (fibre textures) N(l) = 1. In order for a solution to be possible it is thus necessary that M{1) ^ 1
(13.211)
From Figure 4.4, as well as Table 13.10 in the preceding section, one thus recog nizes that in principle most properties of cubic and hexagonal crystals can already be calculated from measurements on fibre-textured samples. 13.7.
Textures with Equal Physical Properties
From equation (13.203), which gives the relation between single crystal and polycrystal coefficients, one recognizes that texture coefficients C}p with I > r have no influence on the poly crystal properties, r is the highest order occurring in the series expansion. We have further seen by some examples that in most cases r is very small. For tensor properties of second order r = 2; in the case of elastic properties r = 4; and in the case of magnetic properties of ferromagnetic cubic crystals r = 6. All textures which differ in their series expansions only in the coefficients with 7 > r therefore yield the same average value of the physical property considered. They are — for the properties estimated — equivalent. We therefore ask of the totality of all different textures, which are equivalent with respect to a specific physical property. We obtain them if we allow the coefficients Cf with I > r to traverse all possible value combinations, under the condition that for no choice of coefficients is the orientation distribution function f(g) allowed to assume nega tive values, since orientation distributions with negative frequencies are naturally physically meaningless. We must thus have oo M(l) N(l)
Μ = Σ Σ ΣΟΓΤΓ(9)^0 1=0 μ=1
(13.212)
v=l
Thus, if the orientation distribution consists of only a single orientation g, accord ing to equation (4.19), one obtains for the coefficients Cfv = (21 + 1) T*»v(g)
(13.213)
344 Physical Properties of Polycrystalline Materials If one allows g to traverse all possible orientations, one t h u s obtains value combin ations of t h e coefficients (7fv, which are all 'allowable' — i.e. t h e texture function f(g) is nowhere allowed to be negative. If one regards the Ofv as coordinates of a multidimensional space, t h e n t h e point corresponding to g traverses a certain region of this space. Each point of this region corresponds to an allowable value combination of t h e coefficients and t h u s to an allowable ' t e x t u r e ' ; however, all these 'textures' correspond exactly to single orientations. W e shall call t h e region in the coefficient space so obtained the 'single crystal region'. If the texture consists of two orientations g1 and g2 which are present with relative frequencies Yx and F 2 , one t h u s obtains for t h e coefficients Of = (21 + 1) [Vfiffa)
+ VtTffo)]
(13.214)
The point represented in the coefficient space thus lies on the straight line through the points 1 and 2. Since Vx and V2 can not be negative, it moreover lies between these two points. Every point so obtained also again corresponds to an allowable texture. W e can also form many-fold combinations with positive weights. I n this way, however, we obtain also all points of t h e coefficient space, because each texture is defined by its component orientations and the associated weights. We thus ob t a i n an essentially greater 'texture region' which includes the 'single crystal region'. I n addition to t h e points of t h e 'single crystal region' themselves, it also contains all straight lines between each two points of this region. Each point of this 'tex ture region' corresponds to an allowable value combination of the indices and thus to an allowable, non-negative orientation distribution function. If now we require t h a t the material possess certain values of a physical pro perty — for example, certain magnetic properties — a certain number of the coef ficients Cfv with l ig r are thereby fixed. W i t h the allowable 'texture region' in the coefficient space at hand, one can then directly recognize in which region the re maining parameters may still vary without the representing point leaving the 'texture region'. I n this way one obtains the totality of all textures which are equivalent with respect to a specific physical property.
13.7.1.
Fibre Textures of Ferromagnetic
Cubic
Materials
We shall illustrate the above with a simple example. We consider fibre textures of ferromagnetic cubic materials. If the texture consists only of a single component with the crystal direction h parallel to t h e fibre axis, t h e coefficients Of are thus given by Cf = 4nk
(13.215)
If we take the values of the first two cubic spherical harmonics k\(h) and k\(h) for different values of h from a table and introduce C\ and C\ as coordinates of a point in a rectangular coordinate system, the point thus sweeps out ( the 'single crystal region' {Figure 13.21).
Physical Properties of Poly crystalline Materials i
10
1 \\ n53
Ax
\22i\j57 I '
\
\
■
6
9 6
ftwt^^ |\\
I
L
\ ■
345
= 4jrkJ Te//t/re region
-7 6
5 4 ν7ί2 J
JJ7T
10-9-δ -7-6-5 -4\J
irty \
Ϊ
r r y ^ . . Kr4***»
4 i / ί 5 6 7 0 i 70 *
iff"}
55^i 203/
J/ng/e crystal region
moj „5
-9 -10 Figure 13.21 fibre texture
Coefficient diagram for the magnetization energy of cubic materials with
The convex envelope of this region is the total 'texture region' (or its projection in two dimensions). Value combinations of the coefficients C\ andCß which cor respond to a point in the 'single crystal region' can be realized by a single orient ation (which in the case of fibre textures is naturally not a single crystal, but con tains all orientations which transform into one another by rotations about the fibre axis). It can, however, also be realized by real textures. Points which lie in the 'texture region' but outside the 'single crystal region' can only be realized by two (or more) orientations. Since each point of the diagram corresponds to a value combination of the coefficients C\ and CQ and since the magnetization energy depends only on these two values, a completely determined magnetic anisotropy is associated with each point. Different textures which correspond to the same point have the same anisotropy. The zero point C\ — 0, CQ = 0 naturally corres ponds to random orientation distribution. Since the zero point lies in the 'single crystal region', quasi-isotropic behaviour can, however, also be generated by a single orientation, which, as one sees, lies in the neighbourhood of, the orient ation [124]. One can infer its exact orientation from the graphic representation of the two functions h\(0, ß) and h\{0, ß). One obtains the values φ0= One thus behaviour nizes that behaviour
74.5°; ßo = 26°
(13.216)
sees that a sharply developed texture can also yield quasi-isotropic with respect to a specific property. From Figure 13.21 one also recog in the general case (ϋΓ4 and K6 Φ 0) it is not possible to realize isotropic by a double fibre texture [411] -f- [100] in certain relative proportions.
346
Physical Properties of Polycrystalline Materials
However, this is possible by a threefold fibre texture [111] + [100] + [110]. One can then even generate any arbitrarily obtainable anisotropy behaviour by such a threefold fibre texture. If the second ferromagnetic anisotropy constant is zero, Ks = 0
(13.217)
the anisotropy of the poly crystal will thus be independent of C\. All points on a straight line parallel to the C\ axis of the diagram then yield the same anisotropy. One can therefore directly read off in which interval C\ may move, in order to get the texture of the anisotropy behaviour concerned. If KQ Φ 0 and one seeks from all textures those which contain no component of sixth order in their an isotropy, the representing point must thus lie on the C\ axis. The coefficient C\ can then vary on the interval - 3 . 5 8 ^ C\ ^ +4.40
(13.218)
All these textures yield an anisotropy without sixth-order components. 13.7.2.
Magnetic Anisotropy of an Fe—Si Sheet
An example of a general (not rotationally symmetric) texture was (of course without use of the coefficient diagram) considered by DUNN and WALTER 1 2 0 . The texture of an Fe—Si alloy was determined by X-ray diffraction measurements. Among others it possessed the components (305) [5, 21, 3], (017) [271], (110) [115]. The orientation (110) [001] only amounted to 0.4%. The magnetic torque curve cal culated from these orientations was in good general agreement with the measured curve. The torque curve could, however, be equally well explained by a texture with the components 28%(110) [001] + 72% random
(13.219)
By application of the coefficient diagram, however, one can construct arbitrarily many additional textures which will also all yield the same torque curve. 13.7.3.
Tensor Properties of Second Rank for Fibre Textures
We further consider tensor properties of second rank in the case of fibre textures of orthorhombic crystals. The anisotropy of the polycrystal quantities then de pends on the two texture coefficients C\ and Cf. For single orientations it is now true that G\ = ink\(0) == 2 γί^Ρ2(Φ)
(13.220)
Cf = 4π&|(Φ, β) = 4 γ^ Pf (Φ) cos 2β
(13.221)
If one now allows Φ and β to vary, the factor cos 2β thus assumes all values be tween — 1 and + 1 for each value of Φ. If one therefore plots ± 4 γπΡ%{Φ) versus 2 γ2π -Ρ2(Φ)> o n e ^ n u s obtains the limits of the single crystal region (Figure 13.22).
Physical Properties of Polycrystalline Materials
347
Since it is linearly confined, in these cases it coincides with the total texture region. Every anisotropy behaviour obtainable by a texture can in these cases thus be produced by a single ideal orientation — in particular, naturally also the isotropic behaviour. This results for an ideal orientation with the spherical angular coordinates Φ 0 = 55°
ßo = 45°
(13.222)
Which {hxh2h2} indices these coordinates correspond to naturally depends on the axis ratio of the orthorhombic unit cell. One recognizes from equation (13.135) that for given values Elv E22, E^ the anisotropy depends only on a linear combin ation of the coefficients C\ and C\. There are therefore a multiplicity of coefficients C\ and Of which yield the same anisotropy behaviour. The corresponding points lie on a straight line whose slope is determined by the constants E1V E22, E^. Thus follows, e.g., according to equation (13.138), a slope of —1.47 for the thermal ex pansion of uranium. The straight line through the zero point for this case is shown in Figure 13.22. All textures whose points are contained on this straight line thus yield isotropic behaviour. The anisotropy of a material with arbitrary texture is determined by the distance of the points from this straight line.
00
now
Figure 13.22 Coefficient diagram for tensor properties of second rank of orthorhombic materials with fibre texture
According to equation (13.141), the irradiation growth of uranium depends only on the texture coefficient C\. The lines of equal anisotropy are here thus parallel to the G\ axis. The coefficient diagram thus permits a survey of the anisotropic behaviour of polycrystals in a very simple way. Of course, we have only considered two very
348
Physical
Properties of PolycrystaUine
Materials
simple cases which allow two-dimensional representation. If more texture coef ficients enter into the anisotropy expression for the polycrystal, one is confronted with correspondingly multidimensional representations. One can, however, event ually use several two-dimensional projections which are also simply representable.
13.8.
Physical Meaning of the Coefficients Cf
The series expansion of the orientation distribution function f(g) (equation 4.7) can be understood as the function f(g) being composed of 'elementary portions' of the form Tiv(g). This decomposition of the actual distribution function into elementary terms is purely mathematical. The elementary terms Tiv(g), in general, have no physical meaning, and the coefficients (7fv which characterize the individual amounts of such elementary distribution functions thereby also have no physical meaning. However, one can associate a certain physical sense with the coefficients Of with small i-values. We consider the elastic anisotropy of a sheet. The Emodulus in the direction γ is then given by expression (13.172), which we can write as a FOUBIEB series:
1 Ε(γ)
E1 + E2 cos 2γ + Ez cos 4y
(13.223)
The coefficients Ei are then given by E
*= T
+ E{0°) + ^(45°) 1 ^7(0°)
1 [E{90°
E0 ^ 3 =
E(90°)
T
1 1 #(90°) ' E(0°)
+
(13.224) (13.225)
J0(45°)
(13.226)
The coefficients Ei thus characterize the average value, the 'second-order' term and the 'fourth-order' term, respectively, of the curve of the elastic modulus {Figure 13.11). If for sijkl in equation (13.172) we substitute the expression "8{.Μ = s^Mf the RETJSS approximation according to equation (13.160) with the values of Table 13.4, we thus obtain
r
* i =
(13.227)
®2 = -*αά12(1111) Cl2
(13.228)
E, = v * f ( l l l l ) 0413
(13.229)
s ull is thereby the expression (13.161) for a material with random orientation distri bution. If, as in Figure 13.11, the anisotropy is hot too great, a linear relation also
Physical Properties of Polycrystalliw Materials
349
exists between t h e coefficients C\v and t h e quantities E1 = - ί [£(90°) + ί!(0°) + 2Ε(4δ°)] <» W +.a1Cl1 #2
=
J _ [£(90°) -
Ä(0°)]
Ä 8 = - i - [2£(90°) + E(0°) -
** 2E(4b°)] &
(13.230)
« 2 C?f
(13.231)
a 3 C 4 13
(13.232)
E are t h e poly crystal values calculated according t o H I L L ' S approximation. The coefficients a1 are, however, no longer as simple expressions as in equations (13.227) —(13.229). They can, however, be easily calculated numerically according t o t h e algorithm of t h e H I L L ' S approximation. The dependence of t h e q u a n t i t y E1 on C}1 and Es on C\6 is represented in Figure 13.23 for differently textured copper sheets 7 2 , 7 8 . The coefficients C\v t h u s describe t h e texture dependence of £ zero-order', c second-order' and 'fourth-order' terms of the anisotropy of t h e elastic modulus. They have a physical meaning of their own in these cases. One obtains a similar linear relation for all properties whose directional dependence in a single crystal can be described in satisfactory approximation by a function of fourth order: for example, t h e magnetic properties, if one restricts t h e m t o t h e term of fourth order. Also, one can for m a n y purposes describe t h e plastic anisotropy by a fourth-order function as a satisfactory approximation. The zero-, seconda n d fourth-order terms of t h e anisotropy of these properties can then likewise be described by expressions of t h e t y p e of equations (13.230) —(13.232). Thereby
-16 -14 -12 -10 -08 - 0 6 - 0 . 4 -02
0
02
04
06 08 10
Texture coefficients c[v Figure 13.23 The isotropic part E1 and the constituents of fourfold symmetry E3 of elastic anisotropy of copper sheets with different textures as a function of texture coeffi cients Clv. The crosses correspond to measured values; the solid lines were calculated according to HILL'S approximation according to references 72 and 78 24
Bunge, Texture analysis
350
Physical Properties of Polycrystalline Materials 33.0
-l0- §psi
^6
32.5
Ei fl
32.0
31.5
-°y'
• -
.
·
--0.66
I -0 4
.
I
.
-0.2
^ /
-
/
•
1 0.2 o
--05
•
•
31.0
isotropic
5^ · /^ ·
1
1
1 1.5
»
1
2.0
Z.5
*
\ °1
—1.0 --1.5
30.5 30.0
pi 0—
4
2- fold symm.
--2.0-
Ei
_
β
^^
1.0
*i: 4- fold symm.
Λ 0 5
• 1 .5
I -0.3
>^
1
1
s
-0.1
1
1 0.3
1
1 0.5
•
0.5
1.0
I \0.1
-
*
«V
1.5
•Ν^ Figure 13.24 Relations between the isotropic parts and the constituents of twofold and fourfold symmetry of the anisotropy of the elastic modulus and the r-values of the plastic anisotropy for 35 samples of rolled and annealed carbon steels according to STICKELS and MOULD269
one immediately obtains a linear relation between the corresponding anisotropy quantities of different properties — e.g. the elastic, magnetic and plastic: %. - &*.) ~ Ä * . " JEW) ~ (^last. " ^last.) ~
*L
~-#iLg.
El.
~JCg.
' ^plast. J ~^piast. ^plast.
<ψ - Cf
~Cf
(13.233) (13.234) (13.235) 269
on a These relations were, for example, confirmed by STICKELS and MOULD large number of low-carbon steels. They compared the quantities EleL and -ß^ last ., where the r-value was used as a measure of plastic anisotropy. The relation given by equations (13.233)—(13.235) for the three terms of the elastic and plastic anisotropy is reproduced in Figure 13.24.
Physical Properties of Polycrystalline Materials
One of the important objectives of many texture investigations is the calculation of the anisotropy of physical properties of polycrystalline materials from the corresponding properties of the individual crystallites and parameters characteriz ing the polycrystalline material. Questions of this kind possess not only theoret ical, but also great practical interest. The directional dependence of magnetiz ation energy of transformer sheet or the earing behaviour of deep-drawing sheet are suggested. In some cases, as with the magnetization energy, the property of the polycrystalline material is given by the average value of the properties of the individual crystallites. It therefore depends not only on the single crystal properties, but also on the orientation distribution function f(g). In other cases (e.g. in the case of elastic and plastic properties) an additional interaction effect between crystallites, and thus a typical 'polycrystal effect', occurs. The properties of the polycrystalline materials can then also depend on the distribution of grain bound aries or in complete generality on the correlation of the orientations at different points of the material.
13.1.
Physical Properties of Single Crystals
By a physical property of a material one understands with complete generality the relation between two (or more) measurable quantities of the material which depends on the material24»220'291. Thus the relation between the volume and the mass of a material is characterized by its density, which is a property of the material. However, we are frequently concerned with essentially more complicated quantities, and therefore also essentially complicated properties of the material. Thus the diffracting power of a crystal depends in a complicated way on the direc tions of the incident and reflected beams. In this case we have thus to deal with a strongly directionally dependent property. We can thus with complete generality define a 'property' of a material by Y = E(X)
(13.1)
where Y and X are some quantities, usually orientation dependent, which are specifically connected to each other by the property E of the material.
Physical Properties of Polycrystalline Materials 295 13.1.1.
Representation by Tensors
A large class of directionally dependent physical quantities can be represented by tensors. By a tensor of nth. order E
= {%,..<„}
(13.2)
one understands a quantity with the components E^^ , which transforms to another coordinate system as the η-ίο\ά product of the coordinates. If we transform from one rectangular Cartesian coordinate system K, in which a point possesses the coordinates xi9 to another, K\ in which the point possesses the coordinates x'iy the following relation holds for the coordinates of the transformation: x\ = atjXj
(13.3)
The result is to be summed over the recurring indices according to the EINSTEIN convention. The components of the tensor in the coordinate system Kf are then given in terms of those in the coordinate system K by the relation E
U..-in
= aiihaHH ' ' ' aininEhh-in
(13*4)
(The summation signs of the n summations over the indices jx to jn are omitted.) If, now, the quantity Y in equation (13.1) is a tensor of mth order and the quantity X one of nth order and the function E describes a continuous relation between two quantities, one can thus expand it in a series: Y = E0 + Ex · X + E2 · X2 + ... + Er · X*
(13.5)
The powers Xr are tensors of (r · n)th order and the coefficients Er tensors of (r · n + m)th order. The tensors Er in this case describe the property concerned. Each term of this series thus has the general form y(m) _ £(^+η)χ(η)
(13.6)
or, in component notation *iii2...im
— •^iii a ..i m ?i72-..?n
?ι?2··.7η
(I**· · )
which is to be summed over all repeated indices. Equation (13.7) is a system of linear relations between the tensor components. Each component of the tensor Y is linearly dependent on each component of the tensor X. The scalar quantities, as, for example, the density, are naturally also included in this representation with m = n = 0. 13.1.2.
Representation by Surfaces
In addition to the representation of a physical property by tensors, another re presentation plays an important role — namely the representation by surfaces or, which is the same thing, by scalar functions of a direction in space302. For example, in a ferromagnetic crystal without the influence of an external magnetic field the
296
Physical Properties of Polycrystalline Materials
spontaneous magnetization lies in an easily magnetizable direction h0. If one forces this by an external magnetic field into another direction h, one must thus employ an energy (per unit volume) which depends on the direction h: E = E(h)=E(0,ß)
(13.8)
This representation of a property by one (or more) functions of a direction in space has the advantage of being particularly intuitive. The function E(h) must have the crystallographic symmetry — i.e. it must assume the same values for crystallographically equivalent directions ha. It can therefore be expanded in a series of spherical surface harmonics of the crystal symmetry (including the centre of inversion): oo M(l)
®{h) = Σ Σ eW(h) 1=0
(13.9)
μ=1
The coefficients ef here thus characterize the property concerned. Since the func tions kjf(h) will possess the complete crystal symmetry, symmetry conditions no longer exist between the coefficients ef in contrast to the components of a tensor (see, e.g., reference 24), which, in general, are not linearly independent. Both representations of a property, the surface and the tensor representation, can be transformed into each other. If in equation (13.8) we express the direction hy not by spherical angular coordinates Φ and /?, but by the components hv h2, hs, we can thus write E = E(h) = E{hv h2, \)
(13.10)
We can now expand E in a series of increasing powers of the h^: E(h) =E° + E\h{ + Elffch, + EfMh
+ ...
(13.11)
The quantities E^ and the products of \ therein are components of tensors. We have thus a special case of the general tensor representation (13.5). Conversely, every tensor can be represented by a certain number of scalar functions of a direction — i.e. by surfaces. The component ^83.. .3 =
%iPZ)% · · · aSjnEhH...jn
a
(13.12)
of the tensor E in the coordinate system K' depends only on the orientation of the direction X'3 of this coordinate system with respect to the crystal fixed coordinate system K, not on the orientation of the two other directions X[ and X'2. If Φ and β are the spherical angular coordinates of the direction X% in the crystal fixed coordinate system K, the function thus represents such a surface. Because of the commutability of the factors in the product a^jflzu ··· a3j m e ( l u a ^ i o n (13.12), the function Ε'^^Φ,β) depends only on the sum of all those components of the tensor E which are formed from Ej^ j by permutation of the indices. If the tensor E is symmetric with respect
Physical Properties of Polycrystalline Materials 297 to all these permutations of indices, the function Ε^^(Φ, β) thus already com pletely describes the tensor E. If the tensor E is not symmetric with respect to all permutations of the indices, one must thus include further linear combinations of tensor components, which also depend only on the orientation of X'% in the coor dinate system K. In this way, as was shown by WONDRATSCHEK 302 , every tensor can be expressed by a certain number of functions Ε{(Φ, β) of the direction X3. These functions must naturally satisfy the crystal symmetry. They can therefore be expressed in a series of symmetric spherical surface harmonics: n
M(l)
Ε&Φ,β) = Σ Σ ΦΡ{Φ, β)
(13.14)
1=0 μ=1
The coefficients efc then form a representation of the physical property, which is equivalent to the representation by the tensor components ^?1?ί...? · Thus, e.g., the elastic properties of a crystal can be represented by the directional dependence of the two quantities *is88(
and
[8,11η{Φ,β)+έ^ι(Φ,β)]
(13.15)
302
of YOUNG'S modulus and the cross-modulus , instead of the components «i?-w of the tensor of the elastic modulus, with respect to a crystal fixed coordinate system. As is known, a symmetric tensor of second order with components E^ can be described by a surface of second order: Ε(Φ,β)=Εφ& 13.1.3.
(13.16)
Representation by Functions of the Orientation g
If we expose a ferromagnetic crystal to the influence of an inhomogeneous magne tic field, the magnetization in different portions of the crystal thus lies in different crystallographic directions. The magnetization energy in this case depends on the complete orientation of a coordinate system associated with the magnetic field relative to the crystal axes and not solely on the orientation of an individual direction, as in the case of a homogeneous magnetic field. If this orientation is given by the rotation g with the EULER angles
(13.17)
Since the crystal symmetry (:) must be realized in the crystal fixed coordinate system and also the distribution of magnetization directions of the inhomogene ous magnetic field can possess a certain symmetry (\) (instrumental or external symmetry), we can expand the function E(g) in a series of symmetric generalized spherical harmonics according to these two symmetries: 00 M(l)
E(g) = Σ Σ
Nil)
Σ etvTr(g)
|=0 μ=1 ν=1
(13.18)
298
Physical Properties of Polycrystalline Materials
If t h e magnetic field is homogeneous, it t h u s possesses rotational symmetry with t h e field direction as symmetry axis. The generalized spherical harmonics of this symmetry are, however, identical with t h e spherical surface harmonics except for a factor, a n d equation (13.18) transforms, as it m u s t , into equation (13.9). Each component Ε[^Λ of t h e tensor E in t h e coordinate system Kf is a func tion of t h e orientation g of this coordinate system relative t o t h e crystal fixed coordinate system K. Accordingly, t h e tensor E is naturally represented trivially by a certain number of functions of t h e rotation g — namely t h e functions 2 ^ { (g). These functions depend on t h e components -Ε,-^...j of t h e tensor E in t h e crystal fixed coordinate system. If among t h e indices iv ...,in some are equal t o one a n other (and for n > 3 this must necessarily be t h e case), t h e 2£7l7>...7 appear in equation (13.4) only as sums over all permutations of t h e indices / of t h e correspond ing positions. If, now, t h e orientation dependence of a tensor component E'iiia^ {g) is known, t h e corresponding sum of components i£7l72...7 can t h u s be calculated therefrom. If t h e tensor E is symmetric with respect t o permutation of t h e indices concerned, it is thus completely described by t h e orientation dependence Ε[^^ {g) of one of its components. The tensor of t h e elastic compliance, for example, h a s t h e symmetry s
ijkl = sjikl — sklij
(13.19)
The component s
'i2w(9) = aiha2i^u^ushhUU
(13.20)
because of t h e commutability of t h e factors αί?· depends only on t h e sum of t h e tensor components 8
hUUU + *?i7i?47»
(13.21)
However, since because of t h e symmetry condition (13.19) these are equal, t h e function s'12SS(g) completely describes this tensor, which is t h u s represented by a single function of g. If t h e tensor is not symmetric with respect to those indices which have t h e same values in t h e component Ε[^Λ , one must use more of its components E'i'sj for its characterization. The transformation coefficients αί?· are t h e direction cosines of t h e coordinates X\ in t h e system of axes Xp They are t h u s functions of t h e rotation g which trans forms t h e coordinate system K into t h e system K'. They form a representation of t h e group of rotations g, which is equivalent t o t h e representation b y T™n(g) (see, e.g., reference 199). The a^ are therefore linearly expressible by generalized spherical harmonics of first degree (see equation 2.50 a n d Table 14.1): «*(?)=
'Σ
m——l
Σ
n=—l
μ™ν1ηΤΓ(9)
(13.22)
Physical Properties of Polycrystalline Materials
299
where t h e μίγη a n d vjn are given by Table 13.1. Now it is t r u e for t h e product of two generalized spherical harmonics (see Section 14.4) t h a t Tm^Trnini
=
g (l1l2m1mSi \l9m1 i=\h-h\
Table 13.1
+ m 2 ) (ΙΜι^
Tf>+™^+n*
| Ϊ, * i + n2)
(13.23)
THE COEFFICIENTS μίΐη AND v?n, BY WHICH THE TRANSFORMATION COEFFICIENTS a^ CAN BE REPRESENTED BY THE GENERALIZED SPHERICAL HARMONICS OF FD3ST DEGREE
ftim i
*j*
\m
\
-1 1
1
i
2 3
0
+1
0
+ 0
1
0
w i
0
0
-1 1
1 2 3
0
+1
+
i
0 0
1
w
0
The symbols in parentheses are t h e CLEBSCH—GORDAN coefficients. B y repeated application of formula (13.23) one can then express t h e product of t h e direction cosines as a series of generalized spherical harmonics:
aiiU ■ «;„·,... %ir = Σ
Σ
Σ «HMa · · · v ; k k ■ ■ ■ ?,) ΤΓ(9) (13.24)
1=0 m~—I n=—I The function $1283(0)> which characterizes t h e tensor of t h e elastic compliances, for example, thereby assumes t h e form
»'να»(9) = Σ
Σ
Σ αΓ(1233;^3η)8ΜιΜίΤΓη(9)
1=0 m= —I n= —I
13.2.
(13.25)
The Problem of Averaging
W e consider a n arbitrary tensor property E. Two quantities Y and X m a y be linked to each other b y i t :
Y = E·X
(13.26)
If, now, t h e material considered, which possesses t h e property E, is polycrystalline, the property tensor E is dependent on t h e orientation of t h e crystallites and, hence, on t h e position r within t h e material. This is also true for t h e two quantities X and Y. W e t h u s have t o write Y(r) = E(r) · X(r)
(13.27)
300 Physical Properties of Polycrystalline Materials The material may now be macroscopically homogeneous. We consider a volume V, which is large compared with the grain size. The average value of E(r) over this volume is then equal to the average value over the whole sample. We further assume that the quantities X(r) and Y(r) are macroscopically homogeneous. The average values over V are then likewise independent of the location of the region V within the whole sample. We denote this average value by a bar, so that, for example, for the average value of E it is true that E=-i
v
fE(r)dV v
(13.28)
The average values of Y and X are similarly defined. We now set Ε(Γ) = Ε + ΔΕ(Γ)
(13.29)
Y(r) = Ϋ + ΔΥ(Γ) X(r) = X + &X(r)
(13.30) (13.31)
With equation (13.28) it is then true that
t
f &E(r)dV = 0
(13.32)
Corresponding relations are also valid for ΔΥ and ΔΧ. If one substitutes equations (13.29) —(13.31) in equation (13.27), one thus obtains Y + ΔΥ(Γ) - E · X + E · ΔΧ(Γ) + ΔΕ(Γ) · X + ΔΕ(Γ) · ΔΧ(Γ)
(13.33)
If one forms the average value over V, one thus obtains with equation (13.32) Ϋ = E · X + - i [ ΔΕ(Γ) · ΔΧ(Γ) dV (13.34) V ff If we consider the averaged tensor X as the independent variable, the components of the quantity ΔΧ(Γ) must depend on those of X. For X = 0 it is also true that ΔΧ(Γ) = 0. Because of the linear relation (13.27) a linear relation between X and ΔΧ(Γ) is generally valid: ΔΧ(Γ) = U(r) · X
(13.35)
If X, and therefore also AX(r), are rath-order tensors, U(f) is thus a tensor of order 2n, which can depend on the values of E(r) at all positions r' of the whole sam ple 196 . If one substitutes equation (13.35) into equation (13.34), one thus obtains = [E + - i / A E ( r ) U ( r ) d F
(13.36)
The macroscopic quantities Y and X are related to each other in equation (13.36) in a similar way to the microscopic ones Y and X in equation (13.27). The quantity
Physical Properties of Polycrystalline Materials
301
in brackets, E = E + - i / ΔΕ(Γ) U(r) dV
(13.37)
is thus the corresponding material property of the polycrystalline (macroscopically homogeneous) material. It is generally different from the simple arithmetic average value E. The simple arithmetic average value E is, in general, relatively easy to calcu late, while the calculation of the additional quantity frequently leads to very great difficulties. For its calculation not only the orientation distribution function f(g) is necessary, but also an orientation correlation function which describes simul taneously the orientation g at the point r and the orientation g' at the point r \ Moreover, the additional quantity in equation (13.37) is frequently relatively small in comparison with E. The approximation E^E
'
(13.38)
is therefore frequently employed. One obtains a better approximation for the polycrystal property E than the average value E in the following way. If one solves the system of linear equations (13.26) for the components of the tensor X, one thus obtains X = E- 1 -Y
(13.39)
1
The tensor Er is another representation of the same property as that described in equation (13.26) by the tensor E. Examples of such tensor pairs E and E" 1 are the tensors of the electrical conductivity and the electrical resistivity or the elastic stiffness and the elastic compliance. If, now, by analogy with equation (13.28), one forms the average value of the tensor E""1:
IF 1 = 4" (E-\r)aV in general, Εφ
v4
(F1)"1
(13.40) (13.41)
The two averages (13.28) and (13.40) yield limiting values for the actual property of the polycrystalline material. Average (13.28) assumes X = const.
'
(13.42)
Thus, in general, the conditions for X are invalid at the grain boundaries, and the average (13.40) requires Y = const.
(13.43)
This, in general, violates the boundary condition for the quantity Y. The actual behaviour of the polycrystalline material will therefore lie between these two 21 Bunge, Texture analysis
302 Physical Properties of Polycrystalline Materials extremes, so that one can frequently set
E^i-tE + iE^)-1]
(13-44)
In elasticity theory equations (13.28), (13.40) and (13.44) describe the well-known approximations of HILL 1 5 2 , RETTSS244 and VOIGT 2 9 1 . An essentially more general form of the average construction was given by ALEXANDROW and AISENBEBG 13 , KRÖNER187'188, K N E E R 1 8 0 ' 1 8 1 and
13.3.
MORRIS213.
The Calculation of the Simple Mean Values E
We shall now discuss the calculation of the simple average value E for the three forms noted for the representation of a physical property. This means that we must average either the tensor components Eixiz%Ä ?ι?·2 >?· (equation 13.7) or the coef ficients ef (equation 13.9) or the coefficients efv (equation 13.18) over the volume V corresponding to equation (13.28). Since the property E(r) will depend only on the orientation g of the crystallite present at the point r, the integration over V can be carried out in two steps. We first integrate over all those volume elements dV which possess the orientation g, and then over all orientations g. We thereby obtain Ε=±φΕ(9) v
9
fdV
(13.45)
V(g)
If here we substitute for dV from equation (3.3), we obtain E = j>E{g)f{g)ag
(13.46)
The orientation distribution function f(g) thus appears as a weight function for the calculation of the average values of orientation-dependent physical properties. Its great practical importance does not least depend thereon. 13.3.1.
Tensor Representation
A property is described in a crystal fixed coordinate system KB by the rth-order tensor E with components 2£?1?·2...7·. It has the components E'iii2^ in the sample fixed coordinate system KA. The crystal coordinate system KB is produced from the sample coordinate system KA by the rotation g. KB will then naturally be transformed into KA by g~x, and, according to equations (13.4) and (13.24), it is true for the components Ε[χ{^Λ in the sample fixed system KA that
^Α...ν=έ
1=0
Σ
Σ
m=—ln=—l
Trig-1)EhU...ir
(13·4?)
Setting
Tf^g-1) = T?nm(g)
(13.48)
Physical Properties of Polycrystalline Materials 303 equation (13.47) transforms into
Ki,..ir=I
Σ 1=0
Σ ^m(hh-'-ir\hk''-h)Trn(g)^l1t...Jr
(13.49)
m——ln=~l
If we expand the weight function of the average value (equation 13.46), and thus the orientation distribution function/(gr), in a series of symmetric generalized spherical harmonics and express the symmetry of these functions by the symmetry coefficients Αψμ and A™v, we obtain oo M(l) N(l)
+1
-f I
Kg) = Σ Σ Σ
Σ
Σ ^Αψ^Αγτψ^)
1=0 μ=1 v=l
'
.
(i3.50)
m=—ln=—l
If, now, we average the components E'^^ of the tensor E over all orientations g in the sample fixed coordinate system according to equation (13.46), we obtain for the property tensor of the textured material with
EiXH...ir = «(h*2· · · V; hk· · ·ir) Eujt...jr *
(13.51)
M{1) J\T(Z)_
a{ixi2... ir; w2... jr) = Σ Σ Σ <(hh · · · \; hfe · · ·?V) Cf 1=0 μ=1
and
<(hH--.ir;hh.-4r)= Σ
(13.52)
ν=1
Σ
m=-ln=-l
ά1
i^f^^m(hh-'ir;hk-4r)^rMv " Γ -L
(13.53) Equation (13.51) expresses the components of the averaged property tensor E of a textured material by those of the single crystal with respect to the crystal fixed coordinate system. The averaging coefficients ά^ ... jr) according to equa tion (13.52) depend on the coefficients Of of the orientation distribution function. The quantities äf(i± ... jr) appearing in equation (13.52) are purely mathematical quantities. According to equation (13.53), they depend only on the crystal and sample symmetries, and according to equation (13.24), on the order of the ten sor E. They can thus generally be calculated. Further, an example of this is given below for the elastic properties (fourth-rank tensor properties). 13.3.2.
Surface Representation
The calculation of the average value E proceeds in a particularly simple way if one employs the surface representation Ε(Φ, β) of the property 2£35. The angles Φ and β are the spherical angular coordinates of the crystal direction h. In a crys tallite of orientation g the crystal direction h may point in the sample direction y. If we want to calculate the average value of the property E in the direction y over all orientations g that occur, E{y)=j>E{h)f{g)ag 21*
(13.54)
304 Physical Properties of Polycrystalline Materials we can carry out the average over g in two steps. First we average over all those orientations g for which the crystal direction h falls in the sample direction y. E(h) is thus naturally constant, and the average of the function f(g) over these orientations, according to equation (4.33), yields the general axis distribution function A(h, y) and, since the direction y is constant during the integration, the inverse pole figure of the direction y. By consideration of the normalization of this function, there results (13.55) E(y) =^-Φ E(h) A(h, y) ah = - L
M(l)
(13·56)
m) = Σ Σ
and oo M(l) N(l)
ημν
(13 57)
A{h,y) = ίπ Σ Σ Σ « Τ Γ Τ *?"(*) ** ι=ο μ=ι ν=ι ^ -h ι one obtains r
_
·
N(l)
E{y) = Σ Σ Wiy)
(13.58)
1=0 v=l
with the averaged property coefficients
n
1 — 21 "Γ
M(l) Σ Of*
(13.59)
J- μ=1
If the orientation distribution of the crystallites is rotationally symmetric about the Z-axis of the sample coordinate system 34 , equation (13.57) thus trans forms into (see equation 5.24) oo M(l) 1 /
A(h9y) = A(h, Φ) = Σ
Σ
0
yw+Ίkr{h)
:
_
°ΐΡι[Φ)
(13,60)
We obtain in this case for the property of the polycrystalline material, which is likewise rotationally symmetric about the Z-axis, Ε(Φ)=
Σ%?ι{Φ) ι=ο
(13.61)
Physical Properties of Polycrystalline Materials 305
with
(l3 62)
^i^i-Wj^
-
13.3.2.1. Rotationally Symmetric Properties In many cases the property function E(h) is rotationally symmetric with respect to a certain crystal direction h0 (though the crystal naturally does not possess such symmetry). Thus, e.g., in the case of hexagonal crystals the magnetization energy will be assumed as rotationally symmetric about the hexagonal axis. If, therefore, we establish for the crystal direction h a new spherical angular coor dinate system©, ψ', such that its pole coincides with the direction h0, the property E(h) depends in this coordinate system only on the angle Θ. The integration in equation (13.55) can therefore be first carried out over the angle \p\ where Ε(Θ) is constant: S
E e iy)=Jzf ( )\f 4π Q
LÖ
*,(©,?')<¥
sin<9d<9
(13.63)
The integral in the brackets can, according to equation (4.137), be replaced by an integral over the pole figure Ρ^(Θ, ψ) belonging to the direction hQ:
E{y)
=έ ϊ
Ε{β)
■ 2π
[f ΡΛ · (Θ ' Ψ) Η s i n ( 9 d 0
(13.64)
The direction y is the pole of the spherical angular coordinate system Θ, %p. One sees that in the case of a rotationally symmetric property Ε(Θ) the average value E(y) depends only on the values of the h0 pole figure, where hQ is the axis of ro tational symmetry of the property concerned. It is also obviously clear that, because of the rotational symmetry of the property of the single crystal, the poly crystal property will not be altered if one rotates each crystal arbitrarily about its symmetry axis. The polycrystal property, therefore, can depend only on the spatial distribution of the axes of rotational symmetry — i.e. the hQ pole figure. We represent the pole figure by its series expansion: oo
PHJP,
ψ)=Σ J=0
JV(i)
Σ mho) k(0, ψ)
(13.65)
v=l
and expand the property function E(ß) in LEGENDRE polynomials: Ε(Θ) = Σ ήΡ,(β) (13.66) ι=ο Taking equation (14.200) into account, there then results from equation (13.64) _
°°
E(y) = Σ
N
e, l7
*
1V
®
Σ mho) M(y)
1=0 y2(2l + 1) v=i
(ΐ3-β7>
306 Physical Properties of Polycrystalline Materials This can be written __
oo Nil)
Ε{ν)=Σ Zeikiy)
(13.68)
1=0 v=l
with the averaged coefficients 1\ =
6l
.
Fj{ho)
(13.69)
which, as they must, depend only on the coefficients F\{hQ) of the h0 pole figure. If, also, the orientation distribution is rotationally symmetric (i.e. we are con cerned with a fibre texture), we express the pole figure according to equation (5.26) in a series of normalized LEGENDRE polynomials. The coefficients Ft thereby appearing differ from the coefficients F} by the normalization factor 1/|/2π of the rotationally symmetric spherical surface harmonics. However, since we also expanded the rotationally symmetric property function of the polycrystalline material in a series of LEGENDRE polynomials with the coefficients \ (equation 13.61), it is also true for these coefficients that 1/2(21 + 1) 13.3.3.
■*l(*o)
(13-70)
Representation by Orientation Functions
Finally, we have represented the property E by one or more functions of the ro tation #', which transforms the crystal fixed coordinate system KB into the external coordinate system K'. In the previously mentioned example of the magnetiz ation energy in an inhomogeneous magnetic field the coordinate system K' was considered to be fixed with respect to the magnetic field. The rotation g thus describes the orientation of the magnetic field with respect to the crystal axes. If g is the rotation which transforms the sample fixed system KA into KB, the rotation 9" = 9''9
(13.71)
thus transforms the sample fixed coordinate system KA into the external system K' {Figure 13.1). The property E (equation 13.18) is then represented as a func tion of this rotation as follows: r
Mil) Nil) +1
E{g') = E(g" ■ g^) = Σ Σ Σ
Σ *Τπ*(9") Tf^g)
(13.72)
1=0 μ=1 v = l s = — I
In order to obtain from this the average value over all orientations g, we must, according to equation (13.46), multiply by oo M(l) Nil)
+1
f(9) = Σ Σ Σ Σ crÄr'Trü) 1=0 μ=1 v=i n=—l
(13.73)
Physical Properties of Polycrystalline Materials
307
Figure 13.1 Specimen coordinate system KA, crystal coordinate system KB and external coordinate system K'
integrate over g, and obtain r
E{9") ==
M(l)
N(l)
Σ Σ Σ
1=0 μ=\
WltJl
1 ~2μν
_
(13.74)
v=l
Mil)
(13.75)
7-T-T Σ <ΗμλΟ}ν
Also, for this representation of the property E, the coefficients i f of the average value have thus been calculated from those of the individual crystallites and the texture coefficients C}v.. If the orientation distribution is rotationally symmetric about the Z-axis of the sample coordinate system, equation (13.73) thus transforms into oo
Mil)
f(g) = B(h) = Σ Σ CW(h)
(13.76)
1=0 μ=1
h is that crystal direction which coincides with the axis of rotational symmetry. The average value E(g") must in this case also be the same for all rotations g", which differ from each other only by a rotation about the axis of rotational sym metry. It can thus only depend on the orientation of the rotation axis — i.e. the Z-axis of the sample fixed coordinate system in the variable coordinate system K'. If we denote this by the vector z, we thus obtain for the average value _
_
r
Nil)
E(g") = E(z) = Σ Σ
:
eimz)
(13.77)
1=0 v=l
with
Mil) *l
Σ W
(13.78)
We have thus derived completely general relations between the simple average value of a physical property of polycrystalline material and the corresponding
308 Physical Properties of Polycrystalline Materials property of the single crystal. The relations are independent of which particular property is treated. They depend only on the form of the representation of the pro perty. In many cases the simple average value E is identical with the actual polycrystal property E. In many other cases equation (13.44) represents a sufficient, though only empirically based, approximation. Since, now, the calculation of the average value (2M-) from E*1 is formally identical with the calculation of E from E, the calculation of the polycrystal quantity E essentially reduces to the simple average value in these cases. 13.4.
Average Values of Special Properties
In the following the general formulae for the simple average values of physical properties will be applied to some special cases. 13.4.1.
Magnetization Energy in a Homogeneous Magnetic Field
The magnetization energy of ferromagnetic crystals in a homogeneous magnetic field sufficient for saturation will, in general, be given as a function of the direction of magnetization — i.e. in the surface representation of a physical property. Since, further, the magnetization direction in each individual crystallite in this case agrees with the exterior magnetic field, the magnetization energy of a polycrystalline material is equal to the simple average value of the energies of the individual crystallites. In this case it is thus true that equation (13.28) is exact. 13.4.1.1. Magnetization Energy of Cubic Crystals The magnetization energy of ferromagnetic cubic crystals can be described with sufficient accuracy by the expression E(h) = K^hlhl + h\h\ -H λ§Α§) + K6h\h\h\
(13.79)
The functions
(13.80)
(13.81)
and appearing here possess cubic symmetry. They are related to the cubic spherical harmonics in the manner described in Section 14.7.5. (See reference 193.) It follows therefrom that ...
1 h\(h)
1
(13.82)
and m ,fcv
1 k(h)
1 kl(h)
1
(13.83)
Physical Properties of Polycrystalline Materials 309 Substituting in equation (13.79), by omission of the constant term, there results E(h) =
k\(h)
KA ΚΛ 5 ^ 55
+
H(h) K6 n6
(13.84)
231
The series expansion of the property function (13.9) thus has only two terms with the coefficients 1
2. (13.85) n% 231 We now assume orthorhombic sample symmetry (sheet symmetry). The spherical surface harmonics of this symmetry are given by
el
=■
e
5 ~*~ 5 5
l
6
^ ) = 77= ^ ( φ ) ' *?(y) ==-±=Ρ?ν-1\Φ)οο*2(ν--1)γ y Δτι yn
(13.86)
By use of equations (13.85) and (13.86), we obtain from equation (13.58) with equation (13.59) Ε{Φ, γ) =
KA 9η4)/π
with 0\τΡ^Φ)
Ρ 4 (φ, γ)=η=
]/2
and
5L *ΆΦ,γ) + 13w
+ 55
6
| / π 231
F6(0,y)
+ <7412Ρ£(Φ) cos 2γ + Ο^Ρ\{Φ) cos 4y
(13.87)
(13.88)
Ρ 6 (φ, y) = - L 0 6 η Ρ 6 (Φ) + 0 6 12 Ρ|(Φ) cos 2 7 r
2
+ (7613Ρ|(Φ) cos 4y + ^614^β(Φ) cos 6y
(13.89)
Φ and y are the spherical angular coordinates of the direction of the magnetic field with respect to the coordinate system fixed in the sample. Since in the case of cubic symmetry only one linearly independent spherical surface harmonic of degree I fg 10 exists, the texture coefficients C}v in equations (13.88) and (13.89) can be determined by measurement of a single pole figure (see Section 4.2.2). From the magnetization energy (equation 13.87) one obtains the torque exer ted in the plane of the sheet Φ = 90° on a sample with circular cross-section in a homogeneous magnetic field: M = with and
ΰΕ(Φγ) 8y
K*
9ηΑ\/π
5 ^ 55
K(Y)' lSn j / π 231 Κ(γ) 6
F±(y) = 2C412Pf (90°) sin 2γ + 4<74l3P4t(90o) sin 4y
(13.90) (13.91)
Ffo) = 2C612Pf (90°) sin 2γ + 4Ο613Ρ|(90°) sin 4y + 6C614P§(90°) sin 6γ (13.92)
310
Physical Properties of Polycrystalline Materials
The torque curves for different Fe—Si samples were calculated by this method by SZPUNAB and OJANEN 2 7 4 , from textures determined by neutron diffraction and the coefficients Cf calculated therefrom. As Figure 13.2 shows, the agree ment between measured values and those calculated from the texture was rather good. The calculation of the average values of a physical property can also be carried out by direct integration of equation (13.46), if the orientation distribution f(g) is calculated — for example, not by methods based on series expansion, but, e.g., by the method of WILLIAMS. In this way the magnetic torque was calculated by HTJTCHINSON and SWIFT 158 » 159 from the texture (biaxial pole figures after WILLIAMS) for a rimmed steel cold-rolled 50% and annealed at 700 °C (Figure 13.3) and com pared with the experimental values. The agreement is similar to the calculation with the help of the coefficients from equations (13.90) —(13.92). This is also to be expected, since both methods are, of course, mathematically equivalent. The integration in equation (13.46) was carried out over 1296 orientation points.
measured curve
30
x
calculated points
20
i
10 0 -10 -.20
i
I
\
-30 -40 -50
Δ
1
I
I
I
I
L
10°. 20° 30° 40° 50° 60° 70° 80° 90° Angle to rolling direction Figure 13.2 Torque curve of an Fe—Si sheet calculated according to equation (13.90) from neutron diffraction texture measurements. After SZPUNAB and OJASTEN274
For rotationally symmetric textures (fibre, wire, rod) equation (13.87), in cor respondence with equations (13.61) and (13.62), simplifies to
**>~sVTi
Κχ
, Κ{
+ 55
^ φ )+έΐ/ίΙ^Ιϊ^ φ ) (13.93)
where C\ and C$ are the fibre texture coefficients defined according to equation (5.4).
Physical
Properties
of Poly cry stalline Materials
311
40l·
-40
—
Calculated
rom
texture
data
• Measured values 1 I I I 1 I 1 1 1 I 1 I 1 1 1 1 1 0° 10 20 30 40 50 60 70 8090°lQ0 110 120 130140 150 160 17θ180° Angle t o rolling d i r e c t i o n
Figure 13,3 Torque curve of a rimmed steel, cold-rolled 50% and annealed at 700°C, calculated from biaxial pole figures and measured according to H T J T C H I N S O N and SWIFT 1 5 8 ' 1 5 9
The magnetic hysteresis (power loss) of a magnetic material can likewise b e expressed b y a n expression of t h e form, of equation (13.79), though with other coefficients. HUTCKENSON a n d S W I F T 1 5 8 ' 1 5 9 used as approximation for t h e power
loss of a cubic crystal P(h) = A0 + A^hlhl
+ λξλξ + h\h\)
(13.94)
This expression was averaged over all orientations with t h e help of t h e calculated biaxial pole figure according t o equation (13.46). T h e result is represented in Figure 13.4. One sees t h a t t h e direction dependence is well reproduced. The coef ficients A0 a n d A1 were experimentally fitted. 13.4.1.2. Magnetization E n e r g y of Hexagonal Crystals F o r most cases t h e magnetization energy of hexagonal crystals can b e described with sufficient accuracy b y Ε(Φ) = K% sin 2 Φ + K± sin 4 Φ
(13.95)
Φ is t h e angle between t h e magnetization direction a n d t h e hexagonal axis. T h e energy is t h u s independent of t h e angle γ and can b e expressed b y two rotationally
312
Physical Properties of Polycrystalline Materials '
!
1
1
!
!
1
I
!
1
Ao = 1-02
A, =12-95 J
4 h
•
-/ΊίθΗζ :£
Αο = 0·91 · Αι = 9 97
3l·
%^^^ _^^50Hz
•
2h
•
o
•
"^
Ao=0-41 · A, = 6 -10
30H
~~ι
Γ · Me asure d v a I ues — Calculated from text ure data 0[
i
go
l
l
o1 Q
2o0
g3 o
1
1
1_
g4 o
g5 o
βgo
1 y go
1 ggo
g go
Angle to rolling direction Figure 13.4 Power loss in Fe—Si according to HUTCHINSON and SWIFT 158 ' 159 . The curve was calculated from biaxial pole figures; the points were measured. The coefficients A0 and Ax were determined from the experiment symmetric spherical surface harmonics of second and fourth order:
^=^μ\ι^φ-± 1 j/2^r 1/ 2 L 2
^
(13.96)
2
and in**
1(Φ) =
*
1
ι/9
ϊ7^τ[τ
Γ35
d
^
0Ο84φ
-Τ
15
, _
0Ο82φ
3
+Τ
(13.97)
By a transformation one obtains therefrom Sin2 0=jL-l.
3
"3
1/JL γ2π *£(Φ)
(13.98)
and sin4 φ =
Ji _ 1J yl_ γr 2π kl(0) + ± I/A γ2π ί\{Φ) zv 15
21 ]/ 5
' ' ' 35 1/ 9
(13.99)
By neglecting the constant terms there results Ε{Φ) = - y
T
pn
[TZ2
+ -#4j
ίο\{Φ) + ^Ύ\/2π
^Κ^Φ) (13.100)
Physical Properties of Polycrystalline Materials
313
The coefficients ef are thus given by e\ =
]^^[i*.+ir4 t-ir^M**
(13 101)
·
We now further assume orthorhombic sample symmetry and obtain for the aver aged energy, according to equations (13.58) and (13.59),
««» — Tyt
3
K
+UK
^(Φ,γ)+^·^Κ^(Φ,γ)
with
and
F2(0, γ) = - L <72ηΡ2(Φ) + 0212Ρ|(Φ) cos 2γ J/2
(13.102) (13.103)
Ρ 4 (Φ, y) = --L Ο^ΡάΦ) + £412Ρ£(Φ) cos 2y + <7413Ρ|(Φ) cos 4y J/2
(13.104) Since only one linearly independent hexagonal spherical surface harmonic of second and fourth orders exists, the coefficients in equations (13.103) and (13.104) can also be determined from a single pole figure, as in the case of cubic symmetry. 13.4.2.
The Remanence in Ferromagnetic Materials
In the previous section we have considered the energy which is necessary to bring about the spontaneous magnetization of a crystal in a specific crystallographic direction h. We now consider those directions, h^ for which the energy assumes a minimum value. Several cases are to be differentiated. In the first case exactly one such direction h0 (as well as its opposite, — h0) results: in the second case there result several different directions h^ for which the magnetization will be minimal. To the first group belong, e.g., many hexagonal materials; to the second, cubic materials. Finally, it can also occur that for a whole band of directions — e.g. all those perpendicular to a direction — a minimum energy will be assumed. If one thinks of a crystal magnetized to saturation in the direction h, the magnet ization thus lies in the direction ft. If, now, one removes the magnetic field, the magnetization will rotate into the nearest direction hi of minimum energy. The component of magnetization in the direction of the original field then amounts to 7 R = I8 cos
(13.105)
The component / R will be called the remanence; Is is the saturation magnetiz ation. We now assume that only a single direction h0 of minimum energy exists, and choose a spherical angular coordinate system Θ, \p, such that its pole Θ = 0 coincides with h 0 . We then obtain for the relative remanence in the case of original
314 Physical Properties of Polycrystalline Materials magnetization in the direction Θ, ψ 4(6>3 ip)/Is = | cos Θ |
(13.106)
In this case the remanence is thus a property of the crystal independent of the angle ψ. With respect to this property the crystal thus has rotational symmetry. If we set |cos6>| = i > ^ ( < 9 )
(13.107)
1=0
we thus obtain a series expansion corresponding to equation (13.66) for 7 R (0)// S . The coefficients ^ are given in Table 13.2. Table 13.2
COEFFICIENTS ez OF THE SERIES EXPANSION OF THE REMANENCE IN HEXAGONAL FERROMAGNETIC CRYSTALS
0 2 4 6
0.707107 0.395 285 -0.088 388 0.039 836
16 18 20 22
-0.005 909 0.004 692 -0.003 817 0.003 166
8 10 12 14
-0.022 777 0.014 767 -0.010 358 0.007 670
24 26 28 30
-0.002 668 0.002 279 -0.001970 0.001 719
We now consider a polycrystalline material and assume that the magnetization is not influenced by interactions with neighbouring crystallites. The remanence of the polycrystal is then equal to the sum of the remanences of the individual crystallites. If Φ and γ are the spherical angular coordinates of an arbitrary sample direction y, we thus obtain for the relative remanence in the case of magnetization in the direction (9, y according to equations (13.68) and (13.69) oo
N(l)
Ivlß> 7)lh = Σ Σ ^ϊ(Φ} Z=0r=l
γ)
(13.108)
with
?
=Fi^nr w,o)
(13 109)
'
The h\{0, γ) are the normalized spherical surface harmonics of the sample sym metry (in the case of orthorhombic sheet symmetry see, e.g., equation 13.86), and the Fl(h0) are the coefficients of the series expansion of the h0 pole figure, and thus the pole figure corresponding to the preferred direction of magnetization.
Physical Properties of Polycrystalline Materials
315
They can be directly determined from this pole figure according to equation (4.70) or, if this pole figure is not measurable, by means of the Cf from other measurable pole figures. In the case of rotationally symmetric textures equation (13.108) transforms into (see references 131, 264, 265) Ι*{Φ)
(13.110)
■ Σ *ιΡι(Φ) 1=0
with the coefficients ^ =
*i
(13.111)
■fi(fco)
|/2(2Z + 1)
The Fi(h0) are the coefficients of pole figures for fibre textures defined by equation (5.26). As one sees from Figure 13.5, the coefficients el decrease very rapidly with I. If, for example, one assumes a limit of 5%, one thus only needs to consider series terms up to sixth order69 . If, further, one also omits the sixth-order term and uses only the terms of second and fourth order, then, according to Figure 4.4, it is
J
I
I
1
L
1
1
1
M/Hcos θ|
1 . 5 % ^ 0
2 4
6
„ 8 10 12 14 16 18 20 22 24 26 28 Degree
I
Figure 13.5 The relative remanence |cos Θ\ of an hexagonal single crystal and the coefficients et of the series expansion in LEGENDRE polynomials69
316 Physical Properties of Polycrystalline Materials M(l = 4) = 1. The coefficients F\(hi)9 which are needed for calculation of the average values, can then be calculated from a single arbitrary pole figure. We finally further consider the case of several different preferred directions, as occurs, e.g., in the case of cubic crystals. If we further assume that, on removing the magnetic field, the magnetization changes into the nearest preferred direction h^ it is thus true for the relative remanence of the single crystal in different angu lar regions that cos (hh-^>
Ίπ(Φ,β)/Ι*
cos (hh2y
in region J5X in region B2
cos <Λ/ι7>
in region Bj
13
n
The region Bj is so defined that within it the angle
ΙΛ(Φ,β)/ΙΒ
=Σ
M(l)
Σ *%(Φ,β)
(13.113)
it is thus true for the coefficients that tf = Σ *
f cos (hh{> Α*"(Φ, β) sin Φ άΦ άβ
(13.114)
Bi
If the directions hi are symmetrically equivalent, the integrals over the individual regions are equal. One thus obtains ef = n J cos (hh{> ί?μ(Φ, β) sin Φ άΦ άβ
(13.115)
n is the number of symmetrically equivalent regions, and Bx is an arbitrary one among them. The remanence of the polycrystalline material results therefrom according to equations (13.58) and (13.59). For cubic crystal symmetry and the preferred directions h{ = <100>, h{ = <110> and hi = <111> the regions Bt are shown in Figure 13.6. 13.4.3.
Tensor Properties of Second Order
A large number of physical properties can be described by tensors of second order — e.g. thermal expansion, optical refractive index and electrical conductivity. We will therefore calculate the orientation average value of such properties in general form (see references 34, 35, 271, 272). A symmetric tensor of second order can always be described by an ellipsoid, which we can relate to its principal axes: E(h) = Enh\ + E22hl + E^hi
(13.116)
/ \ -\
\ A B,
\V^L\
r—^
■ \
Γ
Λ
\ \
/
/Λ'
\ C^/\
\
///
w^ii^s! Siv V /V: ^\j \ \
~^"^^ \
Λ
]/
,Λ/
/-- A - -Be"!
7
~—
Figure 13.6 The regions Bi for cubic crystal symmetry, into which the magnetization transforms after removal of the field in the pertinent preferred direction h^ and indeed for the three cases h^ = {100}, h^ = {110} and h^ = {111}
318
Physical Properties of Polycrystalline Materials
With respect to such a property all crystals have at least orthorhombic symmetry. In the case of cubic symmetry all three coefficients are equal, and E is independent of the direction and thus the average value E is independent of the texture. If one sets hx = sm0eosß
(13.117)
Α2 = θΐηΦβΐη/ϊ
(13.118)
h3 = cos Φ
(13.119)
h\ = - ί (1 - cos2 Φ) (1 + cos 2β)
(13.120)
h\ = 4" C1 -
(13.121)
then
cos2 φ
a
) i1 -
cos 2
ß)
Al = c o s 2 0
(13.122)
These can be expressed by means of the orthorhombic spherical surface harmonics of second order:
Η(Φ) = Γ Τ ^ | / 4 T (3 cos2 Φ - !) 1
/ r
( 13 · 123 )
Q
Äf (Φ) = -=r ] / — — (1 - cos2 Φ) cos 2/9 J/π V ^ 4 One thereby obtains
(13.124)
Ä? = γ - γ j / 2 ^ |/-|- *|(Φ) + A |/~ |/JL ΐ|(φ, 0)
(13.125)
Ä|
(13 126)
=T
_
T^j^Tkl{0) ~\^Τ^φ'β)
*t = γ + γγ2π ]/ΎΗ(Φ)
·
(13.127)
This yields Ε(Φ,β)=Ύ(Ε11
+ ΕΆ + Εκ)
+ - 1 | / t o y - | - Λ1(Φ) (2£ 33 - -»11 - ^22)
+ "f- l/π 1/4 *Ι(Φ'β) {Εη ~ Ε™]
(13 128)
·
Physical Properties of Polycrystalline Materials
319
One thus obtains for the coefficients of the series expansion of the property func tion
-Tf^T'■ (8»„ -
JPU - Ä»)
(13.129)
el = -f-^Ί/τ<*" - E^
<13·130)
We further assume orthorhombic sample symmetry — i.e. sheet or foil. Then, according to equation (13.59), there result for the averaged coefficients
~4 = Ts | / T f(2^3S ~ En ~jE?22) °"+Ϋ* {En~ E^ °ft
(13,131)
~ 4 = i ]/Ύ t(2i?33 ~ Ei1 ~ Εί&) ^12 + ^{En ~ ^ C|21
(13 132)
·
One thereby finally obtains for the averaged quantity Ε(φ, γ)
+ ΤΓϋΤΤΓ Κ2^33 - *11 - ^22)
,
Ρ ΐ /
7 [( 2j ^33 — ^11 — ^22) ^ 2 2
15J/5 + b3{En - Ε22) Of2] ΡΙ{Φ) cos 2γ For fibre textures, according to equation (13.62), there results
* 2 = ά i/ir ^33 ~ ^11 ~ ^22) σ · + ^
(2?n
~ ^22) ^
(13.133)
(13,134)
It follows therefrom that Έ(Φ)=γ{Ε11
+ Ε* + Ew)
+ ^--jL· [(2^33 - Eu - E22) C\ + 53(2?n - E22) Of] Ρ2(Φ) ±0 }/2π (13.135) 13.4.3.1. Thermal Expansion of Uranium We will consider thermal expansion as an example of a tensor property of second order. In addition, because of its strong thermal anisotropy and because of its great technical importance, uranium has been the object of many experimental 22*
320
Physical Properties of Polycrystalline Materials
investigations. I t has been shown t h a t t h e thermal expansion of a polycrystalline uranium sample under certain conditions is given by the simple average value of t h e expansions of the individual crystallites 1 1 5 . I n t h e case of t h e strongly anisotropic expansion of uranium this is by no means self-evident. Thus, e.g., the ther mal expansion of a polycrystalline graphite sample differs noticeably from the simple average value 2 4 2 . If we set for the thermal expansion coefficients of uranium at 75 °C in the axis directions 2 8 , En = + 2 0 . 3 · 10- 6 ;
E22 = - 1 . 4 · 10~ 6 ;
Ezs = + 2 2 . 2 · 10~6 (13.136)
we thus obtain for the expansion of a polycrystalline sample with orthorhombic sample symmetry (sheet symmetry) in t h e direction Φ, γ according to equation (13.133) Έ(Φ, γ) · 10 s = 13.7 + [1.11C 2 U + 1.63C! 1 ] Ρ2(Φ) + [1.52C
(13.137)
Thus results according to equation (13.135) for t h e expansion of a sample with rotationally symmetric texture Ε(Φ) · 106 = 13.7 + [0.68C21 + l.OOCf ] Ρ2(Φ)
(13.138)
13.4.3.2. Growth of Uranium During Neutron Irradiation As a second example of a tensor property of second order we consider the dimen sional change of uranium during neutron irradiation. This property can be de scribed with sufficient accuracy by a tensor of second order 271 ' 272 , and t h e dimen sional change of a polycrystalline sample is given within t h e accuracy of measure m e n t by t h e simple average value, in contrast to graphite. The experimentally determined dimensional changes in t h e axis directions are En = -A
;
E22 = +A;
E^ = 0
(13.139)
According to equation (13.133) one thereby obtains for t h e length change in t h e direction Φ, γ of a polycrystalline sample with sheet symmetry Ε(Φ, γ) = -A
[0.löOCJPPa(Φ) + 0.206(7| 2 Ρ|(Φ) cos 2γ]
(13.140)
I t follows directly therefrom t h a t a sample with random orientation distribution shows no length change and t h a t t h e length change of a textured sample is in dependent of t h e coefficients C\λ and οψ. For a sample with rotationally symmetric texture the expansion in a direction which makes the angle Φ with the axis is given according to equation (13.135) by Ε(Φ) = -A
· 0.092(7|Ρ 2 (Φ)
(13.141)
Physical Properties of Polycrystalline Materials 3 2 1
13.4.4.
Elastic Properties
13.4.4.1. Cubic Crystal Symmetry We denote the stress tensor by a and the strain tensor by ε and limit ourselves to small deformations, so that a linear relation exists between the two tensors: ß<7 = smau
(13.142)
C
0ij = m^ki
(13.143)
In the notation used in equation (13.39)
(13.144)
Correspondingly, for the polycrystalline material «<ί = *ΜΛΙ
(13.145)
5ij = W i i
(13.146)
Also in this case it is naturally true that ~cmi = (km)-1
(13.147)
The poly crystal quantities s^u and c^u in equations (13.145) and (13.146) differ, in general, from the simple average values c ^ and si?·^. We set approximately kjki & cm = cjm
(13.148)
this thus corresponds to the approximation used by orientation. Likewise b
ijkl
«W = SW
corresponds to the approximation of RETJSS
VOIGT 2 9 1
for random crystal (13.149)
244
.
If the stress tensor is a simple tensile stress and if the grain boundaries are perpendicular to. the direction of a {Figure 13.7'a), then
4u = (cJwr1
< 13 · 150 )
whence, according to equation (13.44), *iW ™ T isW + sJßi] = SW
(13.151)
322
Physical Properties of Poly crystalline
Materials
▲
ε
CT= c o n s t "
Reuss
c:o n >t
Voigt
(a)
(b)
Figure 13.7 Validity of VOIGT'S and REUSS' approximation for two extremes of grain shape I t is customary to call this approximation for t h e elastic properties of poly crystal line materials t h e V O I G T — R E U S S — H I L L approximation. I t is well fulfilled for weakly anisotropic crystals, and t h e deviations from it still are small for stronger anisotropy 8 8 , 8 9 . The quantities s^ki and c^kl obey t h e symmetry conditions s
ijkl
=
s
jikl — sijlk
=
s
klij
(13.152)
One can therefore write t h e m in m a t r i x representation (Table 13,3). Because of t h e crystal symmetry a series of further relations exist between these quantities, which are illustrated for cubic s y m m e t r y b y t h e scheme in Figure 13.8 (e.g. reference 220). The cubic axes were t h u s used as t h e coordinate system. Tor isotropic materials t h e scheme in Figure 13.9 is valid. One sees t h a t in t h e case of cubic symmetry only one more linearly independent constant results t h a n in t h e isotropic case. W e denote t h e components of t h e tensor s, with respect t o t h e cubic axes, by s ^ ; we can t h u s decompose t h e tensor s^ki for cubic s y m m e t r y into an isotropic and an anisotropic p a r t in t h e following m a n n e r : s
m = SW + sJiW
(13.153)
s}jki denotes t h e isotropic p a r t according to Figure 13.9 and 5
im
ö
1122
- & ? 1212 .
(13.154)
Physical Properties ofPoly'crystalline Materials Table 13,3
MATRIX REPRESENTATION OF THE ELASTICITY TENSOR
ij
1 2 3 4 5 6
11 22 33 23 31 12
2 22
1 11
n H
m
323
4 23
3 33
δ 31
6 12
^1111
' 51H2
5
^1212
1211
a:: \ \ Fig. 13.8
Fig. 13.9
Figure 13.8 Elasticity tensor for cubic symmetry. The heavy points represent com ponents which can be different from zero; the components joined by a line are equal Figure 13.9 Elasticity tensor for isotropic materials. The crosses denote (1/2) ( s l l n — 5
1122;
is t h e amount of t h e anisotropy, and t h e tensor t^kl has t h e components hill
— ^2222 — ^3333 =
*
(13.155)
All other components are zero. Since t h e isotropic p a r t s?-w is orientation indepen dent, one obtains for t h e average value s
ijkl — sijkl — sijkl +
s
Jijkl
~~ sijkl +
s
&(hjkl ~
(13.156)
hjkl)
According to equation (13.51), t h e quantity t^n is given by hjkl — ä(ijM; nnnn) = a{ijkl)
(13.157)
(which is t o be summed over n from n = 1 t o n = 3). The a(ijkl, nnnn) are given b y equation (13.52). Because r = 4, there appear therein in addition t o CQ1 = 1 only t h e coefficients C}1, C}2, C4 3 . One t h u s obtains liki = äl^ijkl) where t h e af(ijkl) mation : af(ijkl)
+ a^{ijkl)
Cl1 + af{ijkl)
C412 + af{ijkl)
C\*
(13.158)
result from t h e quantities defined in equation (13.53) by sum = a%v(ijkl;
nnnn)
(13.159)
324 Physical Properties of Polycrystalline Materials For t h e components of the averaged tensor in R E U S S ' approximation, one thereby obtains *W = *Μ = sw + Sa&oHtfM) — t m + äl^{ijkl) + af{ijkl)
C412 + äf{ijkl)
0\λ
Of]
(13.160)
I n particular, for t h e case of random orientation distribution one obtains 4ki = *w = 4m + Sa&Piifil)
- kjku
(13.161)
«fin = 0·6*ιιιι + 0.45?122 + 0.8s? 212
(13.162)
sf122 = 0.2s°ull
+ 0.8s°1122 - 0As°1212
(13.163)
sf2l2 = 0.2s°ull
- 0.2·5?122 + 0.6s? 212
(13.164)
The three last quantities fulfil, as they must, t h e condition given in Figure 13.9. The ä%v(ijJcl) are purely mathematical quantities; they depend neither on t h e elastic constants of t h e single crystals nor on t h e orientation distribution. They can therefore be calculated with complete generality in t h e manner described b y equations (13.22) —(13.24). One t h u s obtains Table 13.4 (see reference 50). Table 13.4
THE AVERAGE COEFFICIENTS ä^v(ijkl) FOR CUBIC CRYSTAL SYMMETRY A N D ORTHORHOMBIC SAMPLE S Y M M E T R Y
ijkl
alHijM) - tijkl
a^iifil)
α\\ι^Η)
af{ijkl)
1111 2222 3333 1122 1133 2233 1212 1313 2323
-0.4 -0.4 -0.4 +0.2 +0.2 +0.2 +0.2 +0.2 +0.2
+0.021 818 +0.021 818 +0.058 182 +0.007 273 -0.029 091 -0.029 091 +0.007 273 -0.029 091 -0.029 091
-0.032 530 +0.032 530
+0.043 032 +0.043 032
0 0
-0.043 032
0
0
-0.043 032
+0.032 530 -0.032 530 +0.032 530 -0.032 530
0 0
0 0
I n exactly t h e same manner there result for t h e constants c^u. in t h e case of cubic s y m m e t r y with
tfjki = cm + cffm C
a
=
(13.165)
C
llll ~~ C1122 ~~ ^C1212
(13.166)
I t follows therefrom t h a t c
ijU
=
^ijkl — Cijkl + cdijJcl = CyM + Ca(tijH — tim)
<$W = cm = cw + ta&VWM) + af{ijkl)
- hm + ä^iijkl)
Cf + af{ijkl)
Cf]
(13.167)
C\l (13.168)
Physical Properties of Polycrystalline Materials
325
For random orientation distribution one further obtains cJw = 4a + cJafrW) - tijkl] (13.169) as well as the equations for cjju analogous to equations (13.162) —(13.164). One then also obtains the constants sjjki according to equation (13.150). The elastic compliances s^i of the polycrystal are calculated according to the approximation given by equation (13.151). One obtains YOUNG'S modulus E(y) in an arbitrary sample direction y from the elastic compliances s^i (see, e.g., reference 220): -Ξ— = 2/ismi + 2/P2222 + 2/P3333 + fyf 0|(S112a + 2s1212) E{y) + 2yf»f («use + 251818) + tyhKhwz + 2S2323)
(13.170)
modulus E(y) was calculated in this way for a Cu—Si sample deformed by shear spinning by DUBAND 1 2 3 . It is represented in Figure 13.10. If one sets
YOUNG'S
2/3 = °; 2/i = cosy; i/2 = siny the YOUNG'S modulus in the plane of the sheet is
E(y)
= smi
cos4 γ + 52222 sin4 γ +
»1212
(13.171)
+ -^i U M )ein»2y
(13.172)
Figure 13.10 Elasticity modulus E(y) as a function of the sample direction y, after DURAND 123 . The material studied is Cu—Si deformed by shear spinning
326 Physical Properties of Polycrystalline Materials The coefficients 0 | 1 = — 1 . 0 2 ± 0.15;
Cl2=
- 0 . 4 8 ± 0.15;
Cf =
-1.60 ± 0.08 (13.173)
were determined for a 9 0 % cold-rolled copper sheet from t h e first four pole figures (111), (200), (220), (311) 82 . One obtains therefrom t h e values given in Table 13.5 for t h e quantities t^i as well as t h e s^i in t h e three approximations V — R — H . Table 13.5
THE COMPONENTS OF THE TEXTURE TENSOR ί ^ AND THE ELASTIC COMPLIANCES OF A 9 0 % COLD-ROLLED COPPER SHEET IN THREE APPROXIMATIONS (IN 10~12 cm2 dyn -1 )
v
ijkl
s
ijkl
hjki ~~ km
Hjkl
s
1111 2222 3333 1122 1133 2233 1212 1313 2323
+ 1.49 +1.49 +1.49 -0.63 -0.63 -0.63 +0.33 +0.33 +0.33
-0.4753 -0.5067 -0.4596 +0.2612 +0.2141 +0.2455 +0.2612 +0.2141 +0.2455
+0.7960 +0.7503 +0.8190 -0.2486 -0.3174 -0.2716 +0.7139 +0.6451 +0.6909
+0.6293 +0.6074 +0.6403 -0.1832 -0.2161 -0.1942 +0.5182 +0.4708 +0.5013
ijkl
s
ij!d
+0.7126 +0.6788 +0.7297 -0.2159 -0.2667 -0.2329 +0.6160 +0.5580 +0.5961
If one inserts t h e last three columns into equation (13.172), one thus obtains t h e behaviour of t h e ^ - m o d u l u s in t h e plane of t h e sheet represented in Figure 13.11. One sees t h a t t h e curve Επ agrees well with t h e measurements of W E E R T S . The case of rotational symmetry is naturally obtained as a special case of orthorhombic symmetry. I t results from t h e general case when one sets
ci1
V
21+1 4π
20
CJ;
Cl2 = Cl* = 0
JO 40 50 60 Angle towards rolling direction
(13.174)
90°
Figure 13.11 Elasticity modulus of a cold-rolled copper sheet as a function of angle from the rolling direction .
Physical Properties of Polycrystalline Materials 327 One then obtains in place of equation (13.158) km = «S(*?«) + »iW») C\
(13.175)
The coefficients a\(ijkl) are given in Table 13.6 (see reference 243). Table 13.6
THE AVERAGE COEFFICIENTS
a\(ijkl)
FOR FIBRE TEXTURES
IN THE CASE OF CUBIC CRYSTAL SYMMETRY
ijkl
al(ijkl) -
1111 3333 1122 1133 1212
-0.4 -0.4 +0.2 +0.2 +0.2
tm
al(ijkl) +0.018 +0.049 +0.006 -0.024 +0.006
465 240 155 620 155
The VOIGHT^-REITSS—HILL approximation is the mean value of two unrealistic assumptions (see, e.g., Figure 13.7) without good theoretical basis. Practically, however, it agrees with the measured poly crystal values within a few per cent 88 ' 89 . Better-based approximations must, however, make explicit or implicit assump tions about the shape and arrangement (orientation correlation) of the crystal lites (see, e.g., references 29, 65, 66, 188). KRONER'S theory 187 assumes statist ically random orientation correlation. It was originally carried out for random orientation distribution. It was extended by KNEER 1 8 0 » 1 8 1 to rotationally sym metric textures and by MORRIS 213 » 215 to textures of cubic crystal and orthorhombic sample symmetry. To be sure, the calculation according to this theory requires a considerable calculation expenditure. Since the results also agree within about 2% with those of the V—R—H approximation, which is essentially simpler to calcu late, this approximation is used for nearly all practical calculations. 13.4.4.2. Orthorhombic Crystal Symmetry The averaging of elastic properties for orthorhombic (or higher) crystal symmetry was carried out by MORRIS 212 . Since the elastic anisotropy in this case can not be described by a single quantity sa or ca, one must begin with the general averaging formula (13.51). If we use the two-index notation for the components of the elastic tensor according to Table 13.3, equation (13.51) can thus be written with
smn = a{mnpq) sm
(13.176)
4
I
I
ä(mnpq) = Σ
Σ
Σ
afv(mnpq) Of
(13.177)
Z=0 A*=0(2) v=0(2)
(we have here used a different enumeration for μ and v). The coefficients afv were given by MORRIS 212 , for a different normalization of the coefficients (7f*\ They are converted to the normalization used here in Table 13.7.
328
I o
P
o Ü o
«
«
H P.
W
O H O
o CO E-i
fc H fc
O
O O
«
O
CO
525
H
S CO EH
Sz=5
H O O
«
s H l>
•8
iV
TH
CM
o
i*«a
T*
ri<
(M
^
O
rh
■*
T*
CM
O
CM
CM
O
CM CM
O
o o
O
CM CM
CM
CM
«Γ •«s»
X X X OH
X
X X
O Q
1
CM
CM
CD
CM
1
00
H
Q
TA
N
CM*
CM C* CO CD t O C i CM CD r H
CD
^ CM
00
00
CO
rH
I>
TH Ö
CD
TH
rH
CO
τ-i
rH
1
CO
CO
r-i
CO
CO
CM
1 1
CO
CM
CO
1
CO
CO
1 GO
CO
CO
CO
1
CD
1
CD
τμ CM
1 1
CM
C5
τ*4 CM
CM
1 CD CD CM rH
1 CM
I
CM
1
CM
CM
1
1
CM CM
oo
1 1
CM CO
1
CD
I
CM CM
co
1 CM
CM
CM
CM
CM
1 1
CM
1
CM
1 1
CM
CO CD
1 1
CM CO
CD
^
1
CM
^
CO
CD
CD
CM
CM
CM CM
TH
CO CM CM CM
"*
1 1
CD
CD
CM CM
CM CD
CD
CO CO CM CM
T*
CD
CM CM CM CM
1
rH rH CM CM
1
CM
1 CM
CM
CM
τμ
^
1 1 .
1
rtf
CM
CM CO TH vO
Ö
0 0 CO 0 5 vO TH CM
< CD OS 0 0 TH TJ 0 5 OS CO TH CM
rj< H
TH
CM* Ö
vO TH O O D - CM ι θ M M OS CM O
O
TH
Ö Ö Ö
M "tf O
TH
rH
TH
TH
1 TH
CM
1 1
CO CO CO
CM
1
CD
rH
00
CO CO
rH
I CO
CO
Ot>
rt
CM
"rft CM
1 11 I
CO
CO
X
X
CJ5
CO
1 CD
CD
CO CO CD
1
CD CD CM rH
I
CD
1 1
CD
CO
1
X ÖH
X OH
X ÖH
X
X
O
o o
X
"Si
1
o o
5S. »
1111 1122 1133 1112 1113 1123
CO
O0
CO
1 ^f4 CM
00
*^4 CM
CM CM vi rH
1
CD
CD rH
CO
τΗ/t CD
CD
1 ^Η^
1 1
CO
"tf
' "<*
"^H
^H CO
TH 1
T H
CO CM CO CO
^
"Ψ
CO rH CO CO
1
CD
CM rH CO CO
' CD
CO CO CO CO
1 1 CO
CM CM CO CO
CM CM Til 0 0 | r H r H CM
rH rH CO CO
rH r-i
1
r-i
1 1
r-i
1
1
oo
CM
1
1
CM
O
1
00 QO
1
CM CM
CM CM
1
CM CM
1 1 1
*rH
CXI CO CO rH CM ^ι T*H r-i
T* ^ ^
CXI CM CM
1 I
Ttf CM CM
1
0 0 CM CM CO CO
1
CXI 00 00
1
CO
1
00
CM 00 00
1 1
1
00 00
1 co
CM
CM 00 00
1
CM
1
CM CM
1 1
CM 00 00
r-i
r-i
r-<
1 1 r-i
1
1
1
CM
00
1
CM CM
1
1 1
1
00 00
CO CO 00 CM 00 00
Ti*
1 1 1
1
1 ιθ xO
ιθ
iO iO
IO
1
rH
rH r-i
1
r-i
rH
1 1
r-i
1 1 1
rH rH
CM
00
329
1
CM CM
00
CO CO 00 CM 00 00
1 I 1 1 00 "# I i
(M
1
o
O
1
r-i
iO
1
vO iO
1 lO iO
iO
1 rH 1
CM
1 1 1
r-i
r-i
1
CM CM CM 00 CM CM CO CO r-i r-i CO
1
rH
r-i
1
rH r-i
rH r-i
1 1
CM CM CM 00 00 00 00
CM CM CM 00
r-i
oo
r-l CM CO CM CO CO rH CM CO r-i r-i CM CO CO CO CO CO CO r-i r-i r-i r-i rH r-i
1 1
r-i
1 1
1
rH
CO (M CO CO
1
r-i
r^
1
CM CM
^μ
^
CO rH CO CO
1 1 1
CM CM CM
CM rH CO CO
CM O
r-i
00
1 "«# CM (M 1 1
CM CM
τμ CM CM
1 1
1 1 1
CM CM CM
CM CO CO rH r-i CM O iO iO tO iQ XO
1
rH CM © r-i
00 CM CM CO
1
CM CM
CM
CM CM
CM CM
CXI
1
CXI CM
1
1
1
1 1
o
rH rH
©
1 1
rH rH
o
1
©
1
CM 03
CO CO 00 CM 00 00
1
r-i
rH TH
1
rH
CO CO
1
CM CM
1 CM
CO CO (M rH
CM CM CM 00 00 00
r-\ CM r-i (M CM CM r-i rH
330 Physical Properties of Polycrystalline Materials 13.4.5. s
Plastic Anisotropy
We consider a volume element of a plastically deformed material with coordinates χχχ^χζ. During the plastic deformation the material undergoes a displacement with components %w2%. ^ t n e deformation is homogeneous {Figure 13.12), ui = etjXj
(13.178)
The deformation tensor e^ can be decomposed into a symmetric and an antisym metric portion, the latter simply describing a rigid rotation. The symmetric por tion of the deformation tensor is the strain tensor: (13.179)
£
ij — ~o" \eij "Γ eji)
Figure 13.12 Definition of a homogeneous displacement By suitable choice of the coordinate system it can be written in principal axis form. If one further assumes that the volume remains constant during the deformation, the sum of the diagonal terms must be equal to zero. One can thus bring the strain tensor into the form 1 0 0 -q 0 0
0 * 0 -(1-i).
(13.180)
η can thus be considered as the absolute amount of the deformation, while q characterizes the axial ratio of the deformation. This deformation is represented in Figure 13.13. The material offers a resistance to deformation, which may be described by the stress tensor σ^. The required deformation work dA is then given by dA = σϋ deij (13.181)
Physical Properties of Polycrystalline Materials 331 ▲
/
,
MV
(i-q)-n
/
►\
\
/
>
►
Figure 13.13 General deformation in the principal axis representation corresponding to the deformation tensor (13.180) In the principal axis representation one thus obtains άΑ = άη [σ η - er33 - q(a22 - σ^)] = M\q). άη
(13.182)
The factor M' is a measure of the deformation resistance of the material. If the material is crystalline, M' will thus also depend on the orientation of the principal axes with respect to the crystal axes. It may be described by the rotation g' which transforms the principal axis system of the strain tensor into the crystal coordinate system. The deformation resistance is thus generally given by άΑ = M\q, g') άη = M(q, g') r 0 άη
(13.183)
The function M'(q, g') can be experimentally determined. It can, with certain assumptions (e.g. in the framework of the TAYLOR theory 278 ), also be calculated53'60. Then τ 0 is the critical resolved shear stress in the crystallographically equivalent glide systems and M(q, g) is the Taylor factor, i f (0, g') is, for example, given in Figure 13.14 for cubic crystal symmetry and {111}
M{q, g') = Σ
M(l) N(l)
Σ
Z=0 μ=1
Σ »if (i) Tfig')-
(13.184)
ν=1
If M(q,gf) is numerically known, the coefficients mf(q) can be calculated in the manner described in Section 4.1.2. They are given in Table 13.8 for I ^ 10 for different values of q in the framework of the TAYLOR theory. Figure 13.15 shows the rapid decrease of these coefficients with the degree I for q = 0.
332
Physical Properties of Polycrystalline Materials
Figure 13.14 The TAYLOR factor i f (0, g) for 'plane strain' deformation q = 0 as a function of the crystal orientation relative to the principal axes of deformation^*0
We shall now calculate the average value M(q, g") of the deformation resistance M(q, g') for a polycrystalline material. We thereby permit the principal axes of the strain tensor not to coincide with the sample coordinate system. If the prin cipal axes of the strain tensor form the coordinate system K', the relations be-
4
©
Ö
Ö
CO
Ö
CM
o
i!
Öi
r-4
Ö
©
I s ii
as 00 rH CM CM CO 00
O
00 as CM 00 CO ιθ OS i—1 OS CO © o r-4 rH CM O
GO CO OS CM CO ιθ CO rH rH CM
CO CO t~ O CO
Ö
©
CO O D-
© ©
O© OS 00 *H CM rH CM
© >0 "■# OL CM CM CO CO CM
©
1
Ö
o
o
1
i-i *"# CO CO CO CO CD CM CM OS o r-i CM
ö ö ö ö I I
°. ^ ^ ^1.
i O -* CM O T * 00 O rH
r-4 ** rH C^ rH © "H CM OS "
Ö Ö
1
CO CO as
Ö
Ö Ö Ö 1 1
© o
1
CO OS o [>> rH 00 o
o as
Ö O
CO
1
o
O t> CO
1
00 CO CM r-\ rH
rA O
o o co o o o
I
CM "* i-i ^ xO O L © ©> ©' ©
1
1
©* ©
1
©' ©
as rH as © CM CO rH 00 rH rH rH CO ©'
1
© ©
co o©'
©
333
OS i— I © r-4 ©'
r-4 O 00 ©# ©"
rH rH C* t> r-\ rH © ©
O M ( rH xO O H © ©'
©'
O Ο0 vO
I
© © 1-4 Tj< CO 00 © © ©* ©
DOS -Ψ © ©'
I
00 ^ CO © ©'
©'
1
CO ΓOS © © Ή^ ©# © ©' ©*
CM CM CO' GO CO t>- CO rH rH CM O ©
©
rH O rH O
t> H C O 1-4 C O CM © © ©* ©*
(M (M O © ©'
|> CD H © ©'
©'
Ö O 1 C (M 'H © ©'
C5 ιθ O © ©'
00 O r-4 © ©'
©
©
©
71 00 CO © ©
l·· © ^ © ©
CO O I >© ©
©
CO CO CM CO
©
C* ^* rH
CO 00 r-4 GO OS CO r-4 O CM c3 GO rH rH r-4 rH CM © Ö © Ö Ö
H H CM (M i* r if © © © ©
D i—l *# © ©"
CO C O -rH © ©*
C O ^ T* ©# ©'
rH rH CO © ©'
CO ^ C M © ©'
00 r-4 © CO ©"
rH N ■~H © ©'
CM © ^ O CO rH © © ©' ©'
CO GO CO r-4
as CO CM ^44 CM rH CO © ©
xC 00 US N CM CO © © ©" ©'
r-\ © CO r-4
© ©
D- CM oo oo lO rH r-4 CO ©' ©'
CO r-i as τ4 © τ4 ©* ©
r H CO tas oo oo i—I τ4 ·*Φ © © © © ©' ©"
xO © rH © ©*
OS CM CO CM CO GO © rH © © © Ή ^ ©* ©" ©'
OO 00 CO © C M "^ rH lO i— i © rH CM ©' ©' ©'
CO r-4 C O © ©'
CM GS " * "Ψ © © © © ©' ©*
rH OS O © ©'
C^ " t1 H © ©
^ rH © 00 CO H © © © ©'
© CO H ©
CO
©
xC
©
^
LO xO t >- CO CT 5 I >- CD CO rH CD "H ^ — t "* CM © © # © r-4 © © ' ©" © ' © ' ©" W rH rH rH ©
i£5 CD CM © ©
CO
£> CD Γ© O
t>lH CO r-4 03 O r-\ CO CO i—1 o r-4 © © ©
©
r H lC lH CO CO I >00 ^ L> CD ^44 r-4
CM CO τΗ
rH "^
©
O
CM . CO
OO 00
"·#
rH
00
©
00
©
co co co
00
o
M i l
Tf< t^00 © Ö
CO CO (M 00 OS CO O
1
as 00 00 ©
CO rH "Φ iC D- CS
CO r-4 CO CO 00 CO 00
Ö Ö 1
CO co OS r-\ CM OS r>
o
CO
©
o CO
CO rH O
rH rH CO
[>O CM
CO CO
23 Bunge, Texture analysis
334
Physical Properties of Poly crystalline
Materials
1,6
4 6 8 10 Figure 13.15 The coefficients mf of the TAYLOR factors for I ^ 10 5 3
tween g, g' and g" are thus demonstrated in Figure 13.16. In contrast to the general case treated in Section 13.3.3, we have here used g' as rotation, which transforms the principal axis system into the crystal system, while there the opposite rotation was assumed. In place of equation (13.184) we must thus average the function L
1
Mil)
Nil)
M^g'- ) = Σ Σ Σ <{q) mg'-1) 1=0 μ=1
ν=1
(13.185)
which corresponds to equation (13.18) and thus to the assumptions in Section 13.3.3. The usual assumption will be made that the functions Tf are real in the cubicorthorhombic case. According to equation (13.74), for the average deformation resistance M(q, g") one then obtains L
Nil)
Nil)
M{q, g") = Σ Σ Σ mr(q) W ' ) 1=0 μ=1
f=l
(i3.i86)
Physical Properties of Polycrystalline Materials 335
Deformation Tensor
Figure 13,16 Relations between the crystal coordinate system ϋΓΒ, sample coordinate system K^ and principal axes system K' of the deformation tensor with M(l)
^\q)=w^Zi^{q)civ
(13.187)
The symmetry of the deformation tensor is the same as that of the sample — namely the orthorhombic symmetry. The functions T]fv(g") thus are orthorhombic in both coordinate systems. We consider, in particular, the case in which the JC3-direction of the strain tensor coincides with the Z-direction of the sample coordinate system. The rotation g" then has the form
(13.188)
The functions Tfv(g") can then be written Tf =
-l)oc
(13.189)
For the average deformation resistance one thereby obtains N{L)
__
M{q, oc) = Σ «*(£) °os 2(v - 1) oc
(13.190)
with the coefficients L
1
Σ a (q) = j=o Σ 2Z + 1 A=I v
KW
(13.191)
M(q, oc) is plotted as a function of q for oc = 0 and 45° in Figure 13,17 for a steel sheet79 whose texture coefficients Cf are contained in Table 13,9. Adefonnation tensor with the principal axes described by equation (13.188) corresponds to a 23*
336 Physical Properties of Polycrystalline Materials sample which is rotated through t h e angle oc from t h e rolling direction and is extended in t h a t direction, where t h e relative width and thickness decreases are given b y q and (1 — q), respectively [Figure 13.18). As one sees from Figure 13.17, t h e deformation resistance has a minimum for a certain width decrease q. If t h e sample [Figure 13.18) is elongated in its long direction by άη, without any restric tion t o t h e shape change in t h e two directions perpendicular thereto (free tension), t h e width contraction will be so selected t h a t M[q) is a minimum. The values of q for t h e minima read from Figure 13.17 are plotted in Figure 13.19 as a function of t h e angle oc a n d compared with experimental values, while Figure 13.20 shows Strain
3.5
ratio
R
0.111 0.250 0.429 0.667 1.0
1.5
2.33
4
9
3.4
3.3
-
et 9Xp(?r.
3.2
Min.
a = 0°/
3.1
3.0
o -J-—
=45
b^xp er.
0.1
0.2
MiΛ.>
0.3 0.4 0.5 0.6 0.7 0.8 Contraction ratio q
0.9
Figure 13.17 The average TAYLOR factor M{q, oc) as a function of the axial ratio q of the deformation53 Table 13.9
1 2
3 4 5 6
COEFFICIENTS C\V OF THE STEEL TEXTURE
1 = 4:
1= 6
1= 8
1 = 10
-1.4689 -0.4590 0.4973
2.6861 -1.2023 0.4641 -0.1406
-0.0707 0.2859 -0.6541 -0.4697 -0.2219
-1.0248 0.1910 0.0920 -0.2714 -0.0488 0.0202
Physical Properties of Polycrystalline Materials
337
t h e corresponding If-values. The q u a n t i t y
R =
(13.192)
1-0
which is frequently used as a measure of plastic anisotropy, is also given in Figures 13.19 and 13.20.
Figure 13.18 Orientation of a tensile specimen relative to the sample coordinate system KA
0°
10°
20° 30° 40° 50° 60° 70°, Angle to rolling direction
ft0°
90°
Figure 13.19 The transverse contraction ratio q determined from Figure 13.17 and measured, as well as the associated R-value of the plastic anisotropy for a 95% coldrolled steel sheet as a function of the sample angle79
338 Physical Properties of Polycrystalline Materials 9.2
!
1
[tons/sq. in
1 MY
9-0
L^C^™\^
" ^
5
8 8.6 I—
/ ^ e x Derime ital
(/>
I 8.4
wN ^
9.8 calcul atecl·^
3.2
^4
1 T
H 3.1 °
""^
H3.0
L—*^r> i
8.2
2.9
8.0 0°
10°
20°
30° 40° 50° Angle to rolling
60° 7Q° 80° direction
Figure 13.20 The measured and calculated average 95% cold-rolled steel sheet79
13.4.6.
The Reflectivity
TAYLOR
90°
factor M(qmin%9 oc) for a
of Crystallites for X-rays
It is also of interest to specify the diffracting power of crystals for X-ray beams, which, in general, is described with the help of the reciprocal lattice, in the repre sentation by functions of a direction. If an X-ray beam of wavelength λ falls on a crystal, it can thus be reflected by a lattice plane (hkl). The BRAGG relation X = 2dhklsm&hkl
(13.193)
is true for the reflection angle 2#MZ. The reflection can thus only occur for certain angles 2#ÄW. The usual reflection condition, which says that the normal to the reflecting lattice plane (hkl) must fall in the direction of the bisector of the angle between the incident beam and the reflected beam, is added thereto. If we consider the reflection for a specific angle 2#ÄK in the case of fixed incident and reflecting directions, and bring the crystal into all possible orientations, we thus see that reflections only occur when the normal to the (hkl) plane, or a symmetrically equivalent plane, falls in the direction of the angle bisector. We can thus specify the diffracting power for the reflection angle 2#Mj by a function of those crystal directions h which fall in the direction of the angle bisector. It is clear that this function has the character of a ό-function — i.e. is different from zero only in the direction of the normals of the symmetrically equivalent (hkl) planes. We denote it by Em(h) and set 36 ' 38 Em(h)
= S(hkl)E'hkl(h)
(13.194)
Physical Properties of Polycrystalline Materials 339
S(hkl) is a factor depending on hkl, and the function E'm(h) describes the orient ation dependence. We so normalize it that #^H(fc)dfc = l
(13.195)
We expand it in a series of spherical surface harmonics of the crystal symmetry: oo
Kki(h) = Σ
Μ(λ)
Σ
<#(*«) k$(h)
(13.196)
We obtain the coefficients e%(hkl) in the following manner. We multiply by k*ß(h) and integrate over all h. This yields j>E'm(h) k"(h) dh = eftftiU)
(13.197)
Since the function E' is different from zero only in the immediate vicinity of the points h = [hkl], the functions 1$(h) can, however, be regarded as constant within these small regions, so that we obtain with equation (13.195) e%(hkl) = k^(hkl) φ E'm(h) ah = if"(AM)
(13.198)
k%(hkl) denotes the value of the spherical surface harmonics k%(h) for the directions of the normals of the lattice plane (hkl) and its symmetric equivalents. We thus ob tain the representation of the diffracting power for the reflecting angle 2#MZ: oo Μ(λ) .
®hu{h) = S(hkl) Σ Σ *Γ(*«) *?(*)
(13.199)
Λ=0μ=0
The complete reflection behaviour of the crystal can thus be described by as many functions Eh1d(h) as different reflecting planes (hkl) occur (which are, strictly speaking, naturally infinitely many). If one thinks of the functions Em(h) as distributions on spheres of radii lldhkl, one thus obtains a point distribution in space, the so-called reciprocal lattice. For the representation used here, however, the association with spheres of radius l/dh]a is not necessary. According to equation (13.58), one obtains for the diffracting power of a polycrystalline material with the sample direction y as the angle bisector _
oo N(X)
Emiy)
= Σ Σ el(hkl) *Hy)
with the coefficients S(hl·])
~
This yields
(13.200)
λ=0ν=1
el{hkl) =
M
M
(13 201
2ΓΓΤ -5 www® oo Μ(λ) Ν(λ)
Em{y) = em
Σ Σ Σ
Λ=0 μ=1
ν=1
· >
Qßv
-ΪΓΊΓΛ 4Λ - r 1
^*") W
<13·202)
The averaged function Eh1d(y) is naturally essentially the (hkl) pole figure (see equation 4.35).
340
Physical Properties of Polycrystalline
Materials
Because of the ό-character of the function Ehkl(h), its series expansion (equa tion 13.199) includes infinitely many terms, in contrast to the series expansions of most other physical properties, which frequently contain only terms of very small order. This is the reason that texture determination from X-ray diffraction measurements considerably exceeds that from other physical properties as far as accuracy is concerned. 13.5.
Determination of the Texture Coefficients from Anisotropie Polycrystal Properties
We have derived relations between physical properties of polycrystals and those of the individual crystals together with the orientation distribution function. The relation is particularly simple when the property of the polycrystal is, given exactly or to sufficient approximation by the simple average value. If we use the representation of a property by functions of the orientation g (or, as a special case thereof, the representation by surfaces), for the relation between the coeffi cients efA of the single crystal and i f of the polycrystal (see equation 13.75) it is thus true that -μν _ 6l
=
1 ^ μλρλν 21 + 1 A~i 6* l '
1 ^ λ fg M(I) 1 ^ μ ^ N'(l) 1^ν<^Ν{1) 0 ^ I fg r
(crystal symmetry) (instrumental symmetry) (sample symmetry) (order of property) (13.203)
We have thereby first regarded the single crystal coefficients efλ and the texture coefficients C\v as given and calculated the polycrystal coefficients efv therefrom. We shall now regard the single-crystal and polycrystal coefficients as given and calculate the texture coefficients C\v from equation (13.203). (If, for example, we assume the diffracting power for X-ray beams as the crystal property, we thus obtain the method of texture determination from pole figures measurable by Xray diffraction described in previous sections.) A very important theorem first results. Many of the property functions lead to a very small value of r. Thus for tensor properties of second order r — 2, in the case of elastic properties r = 4, and for the magnetization energy of cubic crystals we have r = 6. For I > r all coefficients e^and consequently also all efv thus vanish. Equation (13.203) is therefore identically satisfied for arbitrary values of the texture coefficients Of1 with I > r. From the measurement of the anisotropy of a physical property one can thus in principle determine the texture coefficients at most up to a degree I = r, which is equal to the order of the property function considered. One can therefore regard a property which is represented by a function of lower order as 'long wavelength'. If one uses this to 'look into' the texture, there thus results a correspondingly poor 'resolving power'. One can no longer detect details of the texture which are finer than the resolving power. This is completely analogous to the dependence of the resolving power of optical instruments on the wavelength
Physical Properties of Potycrystalline
Materials
341
of t h e radiation used. I n particular, magnetic measurements t h u s allow determin ation of t h e texture to at most I = 6. For each pair of values I and v relation (13.203) represents a linear system of equations with M(l) unknowns C\v. F o r each value of μ one equation results, so t h a t the number of these is N'(l). I n order for t h e system of equations to have a unique solution, t h e number of equations m u s t b e a t least equal to t h e number of unknowns: N'{1) ^ M(l)
(13.204)
M(l) denotes t h e number of linearly independent spherical surface harmonics of t h e crystal symmetry and N'(l) t h a t of t h e symmetry in t h e coordinate system K'. In the previously considered example this was t h e symmetry of t h e distribution of magnetization directions in an inhomogeneous magnetic field. Since t h e property was investigated with t h e help of t h e variable coordinate system K\ we shall refer to this symmetry as 'instrumental' symmetry. Now, t h e higher t h e symmetry is, t h e smaller t h e number of linearly independent spherical surface harmonics is. Equation (13.204) means therefore in somewhat simplified form instrumental symmetry ^ crystal s y m m e t r y
(13.205) v
The solubility of t h e system of equations (13.203) for t h e C\ can t h u s be limited in two ways: firstly, t h a t beyond a certain order I = r all coefficients are zero; secondly, t h a t too few equations are available. I n m a n y cases t h e 'instrumental' s y m m e t r y is t h e symmetry of rotation — i.e. t h e property depends not on all three orientation parameters of t h e crystal, b u t only on t h e orientation with respect t o a direction (e.g. t h e direction of a homo geneous magnetic field). I t is independent with respect to a rotation about this direction. I n this important case t h e number of spherical surface harmonics of t h e instrumental symmetry is N'(l)=l
(13.206)
For each pair of values I a n d v we then have only one equation. Thus we must have M(l)^l
(13.207)
The allowable values of i max> according t o this condition are given for different crystal symmetries in Table 13.10. Only in t h e case of rotationally symmetric crystal s y m m e t r y is t h e resolving power not limited by equation (13.207) (orient ations of fibres or ellipsoids of rotation). For cubic and hexagonal symmetries Table 13.10
HIGHEST DEGBEE I = 1^^ is FULFILLED
FOR WHICH THE RELATION M(I) <; 1
Crystal trisymmetry: clinic
monoclinic
orthorhombic
tetragonal
hexagcnal
cubic
«max.
0
0
2
4
10
0
rotational
342
Physical Properties of Polycrystalline Materials
the resolving power is not more severely limited by equation (13.207) than it is with magnetic measurements anyway because of I fg 6. For crystals of lower symmetry equation (13.207) indicates the impossibility of texture determination from a single physical property. If one is to avoid the solution limit established by equation (13.207), one must thus measure more property functions. One then obtains one equation from each property for each pair of values Z, u. If 7 E is the number of different measured properties, in place of equation (13.207) there re sults the condition M(l)^IB
(13.208)
By measurement of several properties one can thus obtain as good a resolving power as by measurement of several pole figures, if one has suitably 'short wave length' properties — i.e. those whose series expansions contain terms with suffi ciently high values of L Now nearly all property functions are in this sense very 'long wavelength' with the exception of the diffraction effects of the crystal lat tice, which are very strongly angularly dependent and therefore contain terms of higher order in their series expansions. A quantitative texture determination with suitably good resolving power is, in general, therefore not possible from the aver age values of several physical properties. The reflection of light by etch pits is one of the few properties of medium 'wavelength' which can be used for texture determination. If one can so etch the crystallites that completely plane crystal surfaces result which are suitably large in comparison with the wavelength of light, the reflected light beam will not be broadened compared with the incident beam. The reflection curve will be very sharp and its series expansion will contain terms of higher order. In actuality, however, the etched crystal surfaces are not completely plane and also not always large in comparison with the wavelength of light. The reflection curve will be broadened, and its series expansion will no longer contain as many terms as in the ideal case, but still many more than, for example, the magnetic properties. If one therefore successively etches different crystal surfaces and measures the angular dependence of the reflection curve of the polycrystal, one can calculate the texture coefficients therefrom, as from X-ray diffraction measurements. Unfortunately, the exact form of the reflection curve depends on many factors which are very difficult to control, so that the 'property function' of the single crystal can only be inexactly determined. The coefficients efA are thus inexactly known and also the C\v can not be exactly calculated. Therefore to the best of our knowledge no quantitative texture determination has been made by this method. 13.6.
Determination of Single Crystal Properties from Polycrystal Measure ments If one knows the texture coefficients Cf and one has measured the orientation dependence of the polycrystal properties, one can thus use equation (13.203) to calculate the single crystal coefficients ef\ For each pair of values l9 μ one obtains
Physical Properties of Polycrystalline Materials 343 N(l) equations with M(l) unknowns. For a unique solution we must thus have N(l)^M(l)
(13.209)
This means, in general sample symmetry fg crystal symmetry
(13.210)
The sample symmetry must be lower than the crystal symmetry, since otherwise an additional symmetrization of the property will be produced by the orientation distribution. In the extreme case of spherical symmetry (thus random orientation distribution) one can not make any assertions about the orientation-dependent terms of the single crystal properties, since the poly crystal properties are quasiisotropic. In the case of rotationally symmetric textures (fibre textures) N(l) = 1. In order for a solution to be possible it is thus necessary that M{1) ^ 1
(13.211)
From Figure 4.4, as well as Table 13.10 in the preceding section, one thus recog nizes that in principle most properties of cubic and hexagonal crystals can already be calculated from measurements on fibre-textured samples. 13.7.
Textures with Equal Physical Properties
From equation (13.203), which gives the relation between single crystal and polycrystal coefficients, one recognizes that texture coefficients C}p with I > r have no influence on the poly crystal properties, r is the highest order occurring in the series expansion. We have further seen by some examples that in most cases r is very small. For tensor properties of second order r = 2; in the case of elastic properties r = 4; and in the case of magnetic properties of ferromagnetic cubic crystals r = 6. All textures which differ in their series expansions only in the coefficients with 7 > r therefore yield the same average value of the physical property considered. They are — for the properties estimated — equivalent. We therefore ask of the totality of all different textures, which are equivalent with respect to a specific physical property. We obtain them if we allow the coefficients Cf with I > r to traverse all possible value combinations, under the condition that for no choice of coefficients is the orientation distribution function f(g) allowed to assume nega tive values, since orientation distributions with negative frequencies are naturally physically meaningless. We must thus have oo M(l) N(l)
Μ = Σ Σ ΣΟΓΤΓ(9)^0 1=0 μ=1
(13.212)
v=l
Thus, if the orientation distribution consists of only a single orientation g, accord ing to equation (4.19), one obtains for the coefficients Cfv = (21 + 1) T*»v(g)
(13.213)
344 Physical Properties of Polycrystalline Materials If one allows g to traverse all possible orientations, one t h u s obtains value combin ations of t h e coefficients (7fv, which are all 'allowable' — i.e. t h e texture function f(g) is nowhere allowed to be negative. If one regards the Ofv as coordinates of a multidimensional space, t h e n t h e point corresponding to g traverses a certain region of this space. Each point of this region corresponds to an allowable value combination of t h e coefficients and t h u s to an allowable ' t e x t u r e ' ; however, all these 'textures' correspond exactly to single orientations. W e shall call t h e region in the coefficient space so obtained the 'single crystal region'. If the texture consists of two orientations g1 and g2 which are present with relative frequencies Yx and F 2 , one t h u s obtains for t h e coefficients Of = (21 + 1) [Vfiffa)
+ VtTffo)]
(13.214)
The point represented in the coefficient space thus lies on the straight line through the points 1 and 2. Since Vx and V2 can not be negative, it moreover lies between these two points. Every point so obtained also again corresponds to an allowable texture. W e can also form many-fold combinations with positive weights. I n this way, however, we obtain also all points of t h e coefficient space, because each texture is defined by its component orientations and the associated weights. We thus ob t a i n an essentially greater 'texture region' which includes the 'single crystal region'. I n addition to t h e points of t h e 'single crystal region' themselves, it also contains all straight lines between each two points of this region. Each point of this 'tex ture region' corresponds to an allowable value combination of the indices and thus to an allowable, non-negative orientation distribution function. If now we require t h a t the material possess certain values of a physical pro perty — for example, certain magnetic properties — a certain number of the coef ficients Cfv with l ig r are thereby fixed. W i t h the allowable 'texture region' in the coefficient space at hand, one can then directly recognize in which region the re maining parameters may still vary without the representing point leaving the 'texture region'. I n this way one obtains the totality of all textures which are equivalent with respect to a specific physical property.
13.7.1.
Fibre Textures of Ferromagnetic
Cubic
Materials
We shall illustrate the above with a simple example. We consider fibre textures of ferromagnetic cubic materials. If the texture consists only of a single component with the crystal direction h parallel to t h e fibre axis, t h e coefficients Of are thus given by Cf = 4nk
(13.215)
If we take the values of the first two cubic spherical harmonics k\(h) and k\(h) for different values of h from a table and introduce C\ and C\ as coordinates of a point in a rectangular coordinate system, the point thus sweeps out ( the 'single crystal region' {Figure 13.21).
Physical Properties of Poly crystalline Materials i
10
1 \\ n53
Ax
\22i\j57 I '
\
\
■
6
9 6
ftwt^^ |\\
I
L
\ ■
345
= 4jrkJ Te//t/re region
-7 6
5 4 ν7ί2 J
JJ7T
10-9-δ -7-6-5 -4\J
irty \
Ϊ
r r y ^ . . Kr4***»
4 i / ί 5 6 7 0 i 70 *
iff"}
55^i 203/
J/ng/e crystal region
moj „5
-9 -10 Figure 13.21 fibre texture
Coefficient diagram for the magnetization energy of cubic materials with
The convex envelope of this region is the total 'texture region' (or its projection in two dimensions). Value combinations of the coefficients C\ andCß which cor respond to a point in the 'single crystal region' can be realized by a single orient ation (which in the case of fibre textures is naturally not a single crystal, but con tains all orientations which transform into one another by rotations about the fibre axis). It can, however, also be realized by real textures. Points which lie in the 'texture region' but outside the 'single crystal region' can only be realized by two (or more) orientations. Since each point of the diagram corresponds to a value combination of the coefficients C\ and CQ and since the magnetization energy depends only on these two values, a completely determined magnetic anisotropy is associated with each point. Different textures which correspond to the same point have the same anisotropy. The zero point C\ — 0, CQ = 0 naturally corres ponds to random orientation distribution. Since the zero point lies in the 'single crystal region', quasi-isotropic behaviour can, however, also be generated by a single orientation, which, as one sees, lies in the neighbourhood of, the orient ation [124]. One can infer its exact orientation from the graphic representation of the two functions h\(0, ß) and h\{0, ß). One obtains the values φ0= One thus behaviour nizes that behaviour
74.5°; ßo = 26°
(13.216)
sees that a sharply developed texture can also yield quasi-isotropic with respect to a specific property. From Figure 13.21 one also recog in the general case (ϋΓ4 and K6 Φ 0) it is not possible to realize isotropic by a double fibre texture [411] -f- [100] in certain relative proportions.
346
Physical Properties of Polycrystalline Materials
However, this is possible by a threefold fibre texture [111] + [100] + [110]. One can then even generate any arbitrarily obtainable anisotropy behaviour by such a threefold fibre texture. If the second ferromagnetic anisotropy constant is zero, Ks = 0
(13.217)
the anisotropy of the poly crystal will thus be independent of C\. All points on a straight line parallel to the C\ axis of the diagram then yield the same anisotropy. One can therefore directly read off in which interval C\ may move, in order to get the texture of the anisotropy behaviour concerned. If KQ Φ 0 and one seeks from all textures those which contain no component of sixth order in their an isotropy, the representing point must thus lie on the C\ axis. The coefficient C\ can then vary on the interval - 3 . 5 8 ^ C\ ^ +4.40
(13.218)
All these textures yield an anisotropy without sixth-order components. 13.7.2.
Magnetic Anisotropy of an Fe—Si Sheet
An example of a general (not rotationally symmetric) texture was (of course without use of the coefficient diagram) considered by DUNN and WALTER 1 2 0 . The texture of an Fe—Si alloy was determined by X-ray diffraction measurements. Among others it possessed the components (305) [5, 21, 3], (017) [271], (110) [115]. The orientation (110) [001] only amounted to 0.4%. The magnetic torque curve cal culated from these orientations was in good general agreement with the measured curve. The torque curve could, however, be equally well explained by a texture with the components 28%(110) [001] + 72% random
(13.219)
By application of the coefficient diagram, however, one can construct arbitrarily many additional textures which will also all yield the same torque curve. 13.7.3.
Tensor Properties of Second Rank for Fibre Textures
We further consider tensor properties of second rank in the case of fibre textures of orthorhombic crystals. The anisotropy of the polycrystal quantities then de pends on the two texture coefficients C\ and Cf. For single orientations it is now true that G\ = ink\(0) == 2 γί^Ρ2(Φ)
(13.220)
Cf = 4π&|(Φ, β) = 4 γ^ Pf (Φ) cos 2β
(13.221)
If one now allows Φ and β to vary, the factor cos 2β thus assumes all values be tween — 1 and + 1 for each value of Φ. If one therefore plots ± 4 γπΡ%{Φ) versus 2 γ2π -Ρ2(Φ)> o n e ^ n u s obtains the limits of the single crystal region (Figure 13.22).
Physical Properties of Polycrystalline Materials
347
Since it is linearly confined, in these cases it coincides with the total texture region. Every anisotropy behaviour obtainable by a texture can in these cases thus be produced by a single ideal orientation — in particular, naturally also the isotropic behaviour. This results for an ideal orientation with the spherical angular coordinates Φ 0 = 55°
ßo = 45°
(13.222)
Which {hxh2h2} indices these coordinates correspond to naturally depends on the axis ratio of the orthorhombic unit cell. One recognizes from equation (13.135) that for given values Elv E22, E^ the anisotropy depends only on a linear combin ation of the coefficients C\ and C\. There are therefore a multiplicity of coefficients C\ and Of which yield the same anisotropy behaviour. The corresponding points lie on a straight line whose slope is determined by the constants E1V E22, E^. Thus follows, e.g., according to equation (13.138), a slope of —1.47 for the thermal ex pansion of uranium. The straight line through the zero point for this case is shown in Figure 13.22. All textures whose points are contained on this straight line thus yield isotropic behaviour. The anisotropy of a material with arbitrary texture is determined by the distance of the points from this straight line.
00
now
Figure 13.22 Coefficient diagram for tensor properties of second rank of orthorhombic materials with fibre texture
According to equation (13.141), the irradiation growth of uranium depends only on the texture coefficient C\. The lines of equal anisotropy are here thus parallel to the G\ axis. The coefficient diagram thus permits a survey of the anisotropic behaviour of polycrystals in a very simple way. Of course, we have only considered two very
348
Physical
Properties of PolycrystaUine
Materials
simple cases which allow two-dimensional representation. If more texture coef ficients enter into the anisotropy expression for the polycrystal, one is confronted with correspondingly multidimensional representations. One can, however, event ually use several two-dimensional projections which are also simply representable.
13.8.
Physical Meaning of the Coefficients Cf
The series expansion of the orientation distribution function f(g) (equation 4.7) can be understood as the function f(g) being composed of 'elementary portions' of the form Tiv(g). This decomposition of the actual distribution function into elementary terms is purely mathematical. The elementary terms Tiv(g), in general, have no physical meaning, and the coefficients (7fv which characterize the individual amounts of such elementary distribution functions thereby also have no physical meaning. However, one can associate a certain physical sense with the coefficients Of with small i-values. We consider the elastic anisotropy of a sheet. The Emodulus in the direction γ is then given by expression (13.172), which we can write as a FOUBIEB series:
1 Ε(γ)
E1 + E2 cos 2γ + Ez cos 4y
(13.223)
The coefficients Ei are then given by E
*= T
+ E{0°) + ^(45°) 1 ^7(0°)
1 [E{90°
E0 ^ 3 =
E(90°)
T
1 1 #(90°) ' E(0°)
+
(13.224) (13.225)
J0(45°)
(13.226)
The coefficients Ei thus characterize the average value, the 'second-order' term and the 'fourth-order' term, respectively, of the curve of the elastic modulus {Figure 13.11). If for sijkl in equation (13.172) we substitute the expression "8{.Μ = s^Mf the RETJSS approximation according to equation (13.160) with the values of Table 13.4, we thus obtain
r
* i =
(13.227)
®2 = -*αά12(1111) Cl2
(13.228)
E, = v * f ( l l l l ) 0413
(13.229)
s ull is thereby the expression (13.161) for a material with random orientation distri bution. If, as in Figure 13.11, the anisotropy is hot too great, a linear relation also
Physical Properties of Polycrystalliw Materials
349
exists between t h e coefficients C\v and t h e quantities E1 = - ί [£(90°) + ί!(0°) + 2Ε(4δ°)] <» W +.a1Cl1 #2
=
J _ [£(90°) -
Ä(0°)]
Ä 8 = - i - [2£(90°) + E(0°) -
** 2E(4b°)] &
(13.230)
« 2 C?f
(13.231)
a 3 C 4 13
(13.232)
E are t h e poly crystal values calculated according t o H I L L ' S approximation. The coefficients a1 are, however, no longer as simple expressions as in equations (13.227) —(13.229). They can, however, be easily calculated numerically according t o t h e algorithm of t h e H I L L ' S approximation. The dependence of t h e q u a n t i t y E1 on C}1 and Es on C\6 is represented in Figure 13.23 for differently textured copper sheets 7 2 , 7 8 . The coefficients C\v t h u s describe t h e texture dependence of £ zero-order', c second-order' and 'fourth-order' terms of the anisotropy of t h e elastic modulus. They have a physical meaning of their own in these cases. One obtains a similar linear relation for all properties whose directional dependence in a single crystal can be described in satisfactory approximation by a function of fourth order: for example, t h e magnetic properties, if one restricts t h e m t o t h e term of fourth order. Also, one can for m a n y purposes describe t h e plastic anisotropy by a fourth-order function as a satisfactory approximation. The zero-, seconda n d fourth-order terms of t h e anisotropy of these properties can then likewise be described by expressions of t h e t y p e of equations (13.230) —(13.232). Thereby
-16 -14 -12 -10 -08 - 0 6 - 0 . 4 -02
0
02
04
06 08 10
Texture coefficients c[v Figure 13.23 The isotropic part E1 and the constituents of fourfold symmetry E3 of elastic anisotropy of copper sheets with different textures as a function of texture coeffi cients Clv. The crosses correspond to measured values; the solid lines were calculated according to HILL'S approximation according to references 72 and 78 24
Bunge, Texture analysis
350
Physical Properties of Polycrystalline Materials 33.0
-l0- §psi
^6
32.5
Ei fl
32.0
31.5
-°y'
• -
.
·
--0.66
I -0 4
.
I
.
-0.2
^ /
-
/
•
1 0.2 o
--05
•
•
31.0
isotropic
5^ · /^ ·
1
1
1 1.5
»
1
2.0
Z.5
*
\ °1
—1.0 --1.5
30.5 30.0
pi 0—
4
2- fold symm.
--2.0-
Ei
_
β
^^
1.0
*i: 4- fold symm.
Λ 0 5
• 1 .5
I -0.3
>^
1
1
s
-0.1
1
1 0.3
1
1 0.5
•
0.5
1.0
I \0.1
-
*
«V
1.5
•Ν^ Figure 13.24 Relations between the isotropic parts and the constituents of twofold and fourfold symmetry of the anisotropy of the elastic modulus and the r-values of the plastic anisotropy for 35 samples of rolled and annealed carbon steels according to STICKELS and MOULD269
one immediately obtains a linear relation between the corresponding anisotropy quantities of different properties — e.g. the elastic, magnetic and plastic: %. - &*.) ~ Ä * . " JEW) ~ (^last. " ^last.) ~
*L
~-#iLg.
El.
~JCg.
' ^plast. J ~^piast. ^plast.
<ψ - Cf
~Cf
(13.233) (13.234) (13.235) 269
on a These relations were, for example, confirmed by STICKELS and MOULD large number of low-carbon steels. They compared the quantities EleL and -ß^ last ., where the r-value was used as a measure of plastic anisotropy. The relation given by equations (13.233)—(13.235) for the three terms of the elastic and plastic anisotropy is reproduced in Figure 13.24.