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Elastic strain energies of sphere, plate and needle inclusions
MATERIALS SCIENCE & ENGINEERING ELSEVIER
Materials Science and Engineering A211 (1996) 95-103
A
Elastic strain energies of sphere, plate and needle inclusions Masaharu Kato a, Toshiyuki Fujii a, Susumu Onaka b aDepartment of Materials" Science and Engineering, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 226, Japan bDepartment of Engineering Physics and Mechanics, Faculty of Engineering, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-01, Japan Received 18 September 1995; in revised form 30 October 1995
Abstract The inclusion problem of linear isotropic elasticity is applied to discuss the orientation and shape dependencies of the elastic strain energies of sphere, plate and needle inclusions with general misfit strains. The strain-energy minimization criterion is adopted and difference in elastic constants between the inclusion and the surrounding medium is taken into account. Regardless of the types of the misfit strains, it is generally found that the plate shape is most favorable for soft inclusions. For hard inclusions, however, the minimum-energy shape depends on the signs of the three principal misfit strains, a sphere when the three misfit strains have the same sign and a needle when they have different signs. Some other new and general characteristics on the strain energies of the coherent inclusions are newly found. Stresses and strain energies of incoherent inclusions after the occurrence of plastic accommodation are also evaluated. Keywords: Elasticity; Micromechanics; Elastic strain energy; Inclusion; Misfit strains; Coherency
I. Introduction Elastic strain energy plays an important role in determining the morphology and crystallography of second phases, e.g. precipitates, martensites, etc. Using Eshelby's inclusion problem of isotropic elasticity [1,2], we have studied the elastic strain energy of a general inhomogeneous spheroidal inclusion with principal misfit strains of a tetragonal type [3,4]. The cases for both purely dilatational (isotropic) and purely deviatoric misfit strains have been included in the analysis and favorable shape and orientation have been predicted for an inclusion with given elastic constants and misfit strains using the strain-energy minimization criterion. Inclusions with more special shapes, such as a plate and a needle, have also been discussed separately [5,6]. In this study, by considering only three representative inclusion shapes, i.e., sphere, plate and needle, elastic strain energies of coherent and incoherent inclusions will be evaluated. Different from the previous studies, however, the case for more general misfit strains (other than the tetragonal type) will be treated. Efforts will be placed on the derivation of general solutions as analytical as possible. 0921-5093/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved SSDI 0921-5093(95)10091 - 1
In reality, not only the elastic strain energy but also other factors, such as interracial energy or nucleation and growth kinetics, affect the morphology of second phases. Since this study deals only with the elastic strain energy, it does not aim at the prediction of actual second-phase morphology. Rather it extracts and evaluates the genuine contribution of the elastic strain energy as one of the important factors in determining the shape and orientation of various second phases embedded in infinitely extended matrices.
2. Calculation procedure To solve the general inclusion problem within the framework of isotropic elasticity, we will treat all the following parameters (a) to (c) as variables.
2.1. Inclusion shape (sphere, plate and needle) On the (~], ~2, ~'3) coordinates (will be abbreviated as the d coordinates) fixed to the inclusion, the inclusion shape will be described by the following spheroid:
96
M. Kato et al. / Materials Science and Engineering A211 (1996) 95 103
~2 + ~2
~32< 1
(1)
+ a-7 -
where the positive a~ and a3 are the half-lengths of the principal axes of the spheroid. Three specific shapes corresponds to (i) al = a3, for a sphere (ii) al >>a3 and a3---~0, for a plate (iii) al << a 3 and a 3 ~ oo, for a needle. 2.2. Misfit strains
Principal misfit strains of the inclusion are expressed on the (Xl, x2, x3) coordinates (the x coordinates) fixed to the surrounding medium as e~t, e~ and e~. In our previous studies [3,4,6], only the tetragonal type of misfit strains, i.e. e~ = e~ ~ e~, was considered for simplicity. In this study, we will treat more general cases. Without the loss of generality, we can assume that
(2)
_
holds. In general, the elastic strain energies of plate and needle inclusions with given misfit strains depend on their orientation, i.e. on the relative orientation relationship between the ~ and x coordinates. The components of the ~ (i = 1, 2, 3) directions (with the ~3-axis being perpendicular to the plate normal or parallel to the needle direction) are expressed on the x coordinates as (see Fig. 1) 41//[ cos 0 cos¢, cos 0 sin ¢, - sin 0]~ ~2//[ - sin ¢, cos ¢, O]x ~3//[ sin 0 cos
¢, sin 0 sin ¢, cos
(3)
O]x
X3
t
""-'--...
" ' / ~ 3
¢ (a)
The subscripts x and ~ denote that the components are expresssed on the x and ~ coordinates, respectively. When the principal misfit strains e y (i = 1, 2, 3) are described on the ~ coordinates, their components e~ can be obtained by a simple tensor conversion as
/ =[
~M cos2O cos2q~ + e~t cos20 sin2¢ + e~t sin20, (~y - e~~t) cos0 sin~ cos4~,
(t2M ~:~t)cos0 sinq~ cos¢, ~:i~4sin2q~ + e~t cos2qS, \egsinOcosOcose¢+e.~tsinOcosOsinZd)-e~sinOcosO, (E~ ~:i'a) sin0 sin0 cos~b. e~ sin0 cos0 cos2¢ + e~~ sin0 cos0 sin2q~ e~~t sin0 cos0~ (e~ - ~:~) sin0 sinq~ cos¢ ] r,iw sin20 cos2~b + e~ sin20 sin24 + ~:~ cos20 /,~
(4)
2.3. Difference in elastic constants between the inclusion and the surrounding medium
As in the previous studies [3-6], we introduce the ratio f of elastic constants as a positive dimensionless quantity: C**t = fCukl
(5)
where C*~t and Cokt are the isotropic elastic constants of the inclusion and the surrounding medium, respectively. Eq. (5) means that the shear modulus/~* of the inclusion is f times larger than the shear modulus/t of the surrounding (/1" = f / t ) while the Poisson ratio v for both is assumed to be the same. If f > 1, the inclusion is elastically harder than the surrounding and the converse is the case for 0 < f < 1. Using Eshelby's solutions of the inclusion problem [1,2], the stresses a t inside the inclusion and the elastic strain energy W assigned per unit volume of the inclusion can be calculated as [7]. o-~.= Cijkl(Sklmnemn * * -- ~ l ) : Co'kl(Sklmne~mn - e~'l)
(6)
W = - (1/2)a~e~
(7)
Here, Sklmn are the so-called Eshelby tensors and e,j are the fictitious equivalent eigenstrains [1,2,7]. The explicit forms of e* for the three representative inclusion shapes were calculated analytically by solving the simultaneous Eq. (6) and are listed in Appendix.
0 .....
(b)
Fig. 1. Definition of the x coordinates of the surrounding medium and the ~ coordinates of the plate (a) and needle (b) inclusions. ~3-axis is perpendicular to the plate in (a) and parallel to the needle in (b).
3. G e n e r a l c h a r a c t e r i s t i c s for the strain e n e r g y o f coherent inclusions
In the following, by choosing various combinations of the principal misfit strains elM, e~t and e~, the strain energies W s, W p and W u for sphere, plate and needle inclusions, respectively, will be calculated as a function of orientation (0 and ¢), the elastic constant r a t i o f a n d the Poisson ratio v. Different from the case of the tetragonal misfit strains [3-6], both angles 0 and ¢ are
M. Kato et al. / Materials Science and EngineeringA211 (1996) 95-103 needed to express the orientation of an inclusion with more general misfit strains. Before conducting individual calculations, it is beneficial to recall or derive some general conclusions of the inclusion problem. (a) If the elastic constants are homogeneous ( f = 1) and misfit strains are purely dilatational (isotropic, i.e. 6~ = 8~ = 6~), the elastic strain energy is independent of the inclusion shape [1-4,7]. (b) The elastic strain energy of a plate inclusion depends only on its elastic constants and is independent of the elastic constants of the surrounding medium [5,7]. The general form of the elastic strain energy of a plate inclusion is known to be independent of 8~3 (i = 1, 2, 3) and is written as
W P = f t l { (8~')2 A-(6422)2A- 2v6~'6422 •
17 v
}
4- 2(6f2) 2
ically known as scalar-invariant, that is, their values are independent of the choice of coordinates on which the tensor components e~ are described. This explains the orientation-independent strain energy of the needle inclusions at f ~ 0 . Furthermore, since f/l =/~*, Eq. (9) indicates that the strain energy at f--, 0 depends only on the elastic constants of the inclusions. As f approaches infinity, however, the equivalent eigenstrains of a spherical inclusion approaches a finite value. This means that; (d) As f approaches infinity, the elastic strain energy of a spherical inclusion approaches a constant value independent of f. The analytical form of the f-independent strain energy W S ( f ~ ~) for a spherical inclusion at f ~ c~ can be obtained using the equivalent eigenstrains and Eqs. (5)-(7) as
(8) wS(U---~ (30) - - 45 -~-
where e t has been defined in Eq. (4). Since f # is equal to the shear modulus #* of the inclusion, the conclusion (b) is verified. Because of this, the elastic strain energy of a plate inclusion with a given orientation is always a linear function of f. We find from Eq. (8) that the strain energy becomes zero if 6~ = 6~22= ~('12= 0 is satisfied. In other words, when the 4~-42 plane parallel to the plate is unstrained, the strain energy becomes zero regardless of the values of other misfit strain components. Since this case has already been discussed [5], we will not treat it in this study. In addition to the above well-known conclusions, some new ones for the sphere and needle can be derived. They are related to the two extreme cases of f approaching either zero (voids) or infinity (perfectly rigid inclusions). For f - ~ 0 , we find from the expressions of the equivalent eigenstrains in the Appendix and Eqs. (5) (7) that the following general conclusion holds: (c) As f approaches zero, the elastic strain energy of a needle inclusion with any misfit strains becomes orientation independent and identical to that of a spherical inclusion with the same misfit strains. The explicit form of the elastic strain energies W S ( f ~ 0) and wN(f--~ 0) of both spherical and needle inclusions at f ~ 0 becomes
¢ + 822 ~ -~- 633) ~ 2
k --
~ 2{611622
- (64,)2}]
--
[5(1
- - I/)(gl~l -~- 8~2 ~ - 8 4 3 ) 2
(7 - 5v){8~ie~2 + 8 2~2 8 3~3
-~- 8"~ 3 3 6 1~ 1
(e~:) 2 - (8~3): - (64,)2}]
(10)
Furthermore, the equivalent eigenstrains for a needle inclusion in the Appendix further indicates that a finite value at f - ~ oo is assured only when the 643 component is zero, i.e., (e) As f approaches infinity, the elastic strain energy of a needle inclusion approaches a constant value independent o f f only when the normal misfit strain (643) parallel to the needle direction (the ¢3 direction) is zero. The fact that 6~33is zero physically means that the ¢3 direction remains undistorted by the occurrence of the misfit strains. For this to be the case, we immediately notice that one of the three principal misfit strains must either be zero or have a different sign from the other two. Otherwise, no direction in the three-dimensional space can be undistorted. The explicit form of the elastic strain energy w N ( f --~ 0 0 , 843 = 0) for a needle inclusion at f--* ~ and 8343= 0 is obtained as follows. wN(U---+ 00, 6~33= O) - - _ 2/t _
3 --4v
[(1
- - V)(81~I -4- 8 2~2 ) 2 - - 6 1~1 6 2~2 -~- ( 8 1x2 ) 2
+ (6~3)2 + (8~,) 2 + 2(1 -- 2v){(62~3)2 + (61,)2}] (11)
m S ( f -'-)"O) = w N ( f --'~O)
= f p [ [~ (1e -l, v
97
~ ~
~ ~-(8,~:):
--~ 6 2 2 6 3 3 -[- 8 3 3 6 1 1
(6~):
(9)
Here, the terms e~, + 8~2 + 63~3 and 6~, 6~22+ 82¢2843 + 83436~, - (e~2)2 - (62~3)z - (84,) 2 are mathemat-
In the following analysis, we will refer to the above conclusions (a) to (e) whenever appropriate. As in the previous studies [ 3 6 ] , the inclusions with only the above misfit strains in them will be referred to as coherent inclusions. F o r easier demonstration, the misfitted coherent inclusions will be classified into two groups, one with principal misfit strains of the same sign (unmixed) and the other with those of different signs (mixed).
M. Kate et al. / Materials Science and Engineering A211 (1996) 95-103
98 103
!"1
. . . . . . . .
I
. . . . . . . .
I
'
' ''''"1
'
¢
jJ
CM
Sphere
CO
~t
,,
Plate
=t 102
Needle
"" •s •• /
~ 101 m
does not hold, the strain energy of a needle becomes infinitely large as f approaches infinity. When an inclusion is softer than the surrounding ( f < l), the plate shape has the smallest strain energy and when it is harder, the spherical shape has the smallest strain energy. These characteristics are of course in agreement with those in the previous studies [7 9]. It is useful to have analytical solutions for some special cases. Table 1 lists these solutions for stresses in the inclusions and strain energies for f--* 0, 1 and oe.
' ' ..... I
•J
/
/
/
/ /
/
tt-
~
10 °
o) o
."
t~ 10 -1 III
M--
M :
[;M=
4.2. General unm&ed misfit strains
8
•s
10-20.01 . . . . . . .0.1 . ' . . . . . . . . 1I . . . . . . . .10 , . . . . . . .100 ., Ratio
of elastic
constants,
f
Fig. 2. f dependence of the elastic strain energy of sphere, plate and needle inclusions with purely dilatational (isotropic) misfit strains of e lM = g 2M __ M --C 3 =~.
4. Coherent inclusions with unmixed misfit strains
Let us first consider the coherent inclusions with unmixed misfit strains. As mentioned above, conclusion (e) cannot be satisfied for unmixed misfit strainsl Therefore, the elastic strain energy of a needle inclusion becomes infinitely large as f approaches infinity. Among the unmixed misfit strains, purely dilatational (isotropic) ones are the simplest and have been discussed most frequently. This case will be examined briefly and then more general cases will be considered.
If one of the three unmixed misfit strains has a different value, specific orientations of plate and needle inclusions to give the minimum values W,, of the elastic strain energy can be found, although they are not necessarily determined uniquely [5,6]. For example, if O
f
Stress (0"¢j)
Energy (W)
Sphere
0
all
3f(1 + v)
:
0"22 = 0"33
2f(1 + v) (1 --2v)
4.1. Purely dilatational misfit strains
By setting e M = e M = e M = e (e, a small quantity), we have analytically calculated the elastic strain energies for the three different shapes using Eqs. (5)-(7) and those in the Appendix. Since the misfit strains are isotropic, the strain energies are independent of the plate and needle orientations and are expressed as WS =
6f(1 + v) {f(1 + v) + 2(1 - 2v)}/re2
1
-
Plate
+
V)/,/e2
All
0"11 :
0
f ( f + 3)(1 + v) - { - ~ (i ---2v)} #e2
0-33 =
--
4/re
2f(1 + v)
l_-G57ff)
2ill + v)
0
2f(1 + v) = _ (l_2v),Ue (l+v)
0"~1= 0"22
(14)
6,tt,~ 2
O"11 = 0"22 ~ 0"33
(13)
1--v
WN
2(1+v) 2 ~ ~-~-/te
3(1 _---~v)/Ze 0"22 :
0",, = 0"2 = 0"33 :
(12) Needle
Wp _ 2f(1
0-1 I = 0"22 = 0-33
4(1 + v) O0
(1 - v)~e
(1-2v) 2(1 ) 2 ~ -+ vv-f/~
2(1 + v)
Fig. 2 shows these elastic strain-energy curves (in units of/~g2) as a function o f f The Poisson ratio has been chosen as v = 1/3. We immediately notice that conclusions (a) to (d) are satisfied. Since conclusion (e)
0"33 - OO
- - / / C
(1 - v)
0-1J = 0-22 = - 2 ( 1
0"33
=
2
~ --2v~/ze
+v),ue
- 2 f ( 1 + v)/ze
f ( 1 + v)/xe 2
2
99
M. Kato et al. / Materials Science and Engineering A211 (1996) 95-103 .'"1
.-I
........
I
........
]
,
, f,l,ll I
,
, ,,,,rf I
Sphere
10 3 .......
Plate Needle •
10 2 •
.
/
."
/
Fujii [5] applied the strain-energy criteria to analyse the orientation of plate inclusions with general mixed misfit strains of e ~ < e ~ <0
(15)
J
/
is satisfied, the minimum-energy plate normal ~3e on the x coordinates becomes irrational and is described as
J e-
101 _-c"
,,/ O
///
10 °
/'"
~P
=
M
C,,I
,.-~-- 2 ~ V(~3
)
__
el
(16) )
J
with the minimum energy of
~ =2
,,,/"
(E3 __ M ' O,
=
-
Wem=f/~(1 + v)(e~) 2 =/~*(1 + v)(e~4)2 = (E*/2)(e~4) 2 (17)
uJ 10-1 -
,,
,d
,
0.01
, ,,,,,,I
0.1
,
, ,,,,,,[
,
1
, ,,,,,,I
,
, ,,,,,d
10
100
Ratio of elastic constants, f Fig. 3. f dependence on the elastic strain energy of sphere, plate and needle inclusions with unmixed principal misfit strains of c~ = ~, ~ = 2e, and eM 3 = 3e. It is the case of the three unmixed principal misfit strains being different (e.g. 0 < elm < e ~ < e~t) that a unique plate and needle orientations can be found for Wm, i.e. a plate perpendicular to the x3-axis and a needle parallel to the xl-axis. Fig. 3 shows the result of such a case for eft = e, e~t = 2e, g 3M = 3e and v = 1/3. It can be seen that the conclusions (b), (c) and (d) prevail. Moreover, although conclusion (a) is no more satisfied, the general appearance of the curves in Fig. 3 remains similar to that of Fig. 2, that is, the plate and the sphere have the smallest strain energy for very soft and very hard inclusions, respectively, and the strain energy of the needle approaches infinity as f - - . oc.
where /~* and E* are respectively the shear modulus and Young's modulus of the plate inclusion. Because of the conclusion (b), the ~P3 direction in Eq. (16) is independent o f f and is a function of only v for given misfit strains. It is found that the minimum-energy orientation (0 = 28.2 ° and ~b = 0 ° for -g3 in Eq. (3)) and the energy of the plate inclusions used to obtain Fig. 4 indeed satisfy Eqs. (16) and (17), respectively. As shown by Dahmen and Westmacott [11] and by Kato and Fujii [5], for the tetragonal misfit strains of e~4 < 0 < e ~ t = e ~ and e l M = e ~ t < 0 < e M 3 , the minimum-energy plate normal can be any direction that lies on the surface of a cone making the following angle c~ from the x3- and yl-axes, respectively: tan c~=
~/ (ey + ve~') (e~ + ve~)
2"1'
'
.......
I
Fig. 4 shows the Wm VS. f curves for inclusions with mixed misfit strains: e ~ = - e , e ~ t = e , e ~ = 2 e and v = 1/3. In addition to the conclusions (b), (c) and (d), conclusion (e) is also satisfied in this case. Because of this, some interesting characteristics can be found. Although the plate shape remains most preferable for soft inclusions, as in the case of the unmixed misfit strains, not the sphere but the needle becomes most preferable for hard inclusions.
........
i
........
f
........
I
Sphere
~a 10 2
5. Coherent inclusions with mixed misfit strains
(18)
g
.......
Plate
....
Needle
," , ,"
,,'"
101
//~,7 ¢/~" fJ"
100
-
7 st
t-
O .m
/
10 -1
,'" s"
e~ = e
,"
sM=2s
s"
ILl 10-2
5.1. Plate inclusions with m i x e d misfit strains
The o p t i m u m orientations of plate inclusions have been discussed in detail using both strain-energy and geometry approaches [5,10-14]. As a generalization of the work by D a h m e n and Westmacott [11], K a t o and
d
,
0.01
.......
I
0.1
........
I
1
........
I
,
, ,,,,,,I
10
R a t i o of e l a s t i c c o n s t a n t s ,
100 f
Fig. 4. f dependence of the elastic strain energy of sphere, plate and needle inclusions with mixed principal misfit strains of e~t = e,e~t =e, and e~ =2e
M. Kato et al. / Materials Science and Eng&eering A211 (1996) 95-103
100
v '"1
. . . . . . . .
I
I
. . . . . . . .
'
'
'''"'1
. . . . . . . .
I
90
70
=
_ 0.:,'
o
=
"\
eM =
~
•
50
e~
40
eft
(19)
A similar situation arises when a needle has mixed and different principal misfit strains. We have found that the minimum-energy needle direction ~u for eft < 0 < e~t < e ~ is o f the type 0 = 90 ° in Eq. (3), i.e.
80 v
tan f l ( f ~ ~ ) = / xl
3N - [ cos q~,v, sin ~bu, 0]
where with f (e~ = figure found
30 20 10
(20)
q)N is a function o f f and v. The variation o f q~N and v for the case o f the misfit strains o f Fig. 4 - e, e ~ = e and e~4 = 2e is shown in Fig. 6. This is similar to Fig. 5 and (fiN for f ~ ~ is again to be independent o f v:
tan (fiN(f---+ (30) = ~ -
e~ eftt
(21)
0 Hd
. . . . . . . .
I
. . . . . . . .
I
. . . . . . . .
I
,
,
,,,r,,I
0.01 0.1 1 10 100 Ratio of elastic constants, f Fig. 5, fand v dependencies of the half-apex angle fl from the xl-axis to give the minimum strain energy of a needle inclusion with tetragonal and mixed misfit strains of eft = - e, e~ = e, and e~ = e. On the contrary, if the misfit strains are such that the inequality Eq. (15) c a n n o t be satisfied, the m i n i m u m energy orientation o f the plate inclusion becomes rational [5].
5.2. Needle inclusions with m i x e d misfits strains
In c o m p a r i s o n with the plate, the minimum-energy orientation o f the needle behaves in a m o r e complex manner. This is obviously due to the fact that the elastic strain energy o f the needle depends on the elastic constants o f both the inclusion and the surrounding medium. K a t o et al. [6] have f o u n d that for the case o f tetragonal misfit strains, say eft < e ~ = e ~ , the minim u m - e n e r g y needle directions (the ~3 direction) lie on the surface o f a cone m a k i n g an angle fl from the xj-axis. Different f r o m c~ (for a plate) in Eq. (18), fl (for a needle) is a function o f b o t h f and v for given misfit strains [6] I. The variation o f the half-apex angle fl with f f o r eft = - e , e2M = e ~ = e a n d 0 < v < 0 . 5 is graphically shown in Fig. 5. It is interesting to find that as f increases, fl either decreases or increases depending on whether v is smaller or larger than 0.25. At the limit o f f - - , ~ , fl becomes 45 ° irrespective o f the values o f v. The value fl(f--+oo) o f fl at f o o v is f o u n d to be independent o f v and is written as [6] For the case of purely deviatoric misfit strains, e.g., - eft/2 = e~ = e.~, Kato et al. [6] have shown that regardless of the values of v, the angle fl always decreases as f increases and approaches a constant value of 54.7° given by Eq. (19) at f--. ~. In this case, fl is independent o f f only when v = 0.5.
As shown previously [5,6,15,16], the surface o f the cone with its half-apex angle f l ( f ~ ~ ) o f Eq. (19) is made o f a set o f invariant-line (IL) directions and the u direction defined in Eq. (20) describes the IL direction on the plane perpendicular to the x3-axis. In other words, the minimum-energy criterion for the needle inclusions predicts the same orientation as that predicted by the purely geometrical I L criterion only for the special case o f f ~ ~ . F o r other cases, the minim u m - e n e r g y direction deviates from the I L direction because o f the Poisson effect that is not taken into account in the purely geometrical theories. It should be recalled here that for plate inclusions, the plate-normal direction o f Eq. (18) and the I L direction coincide each other only when the Poisson ratio v is set equal to zero [5,11]. Therefore, the role o f NI l
i
~ iI:lll
I
i
,
: ILlll
'
' ''''"1
'
' ''''"1
90 80
v=0
8M= E;
70
C~=2C
60 6~ o~ 50 z
0.25
40
-e. 30 20
0.5
10 0
,
, ,,,,,,I
,
, ,,i,,,]
,
, ,,,,,,I
,
,
,,,,,,I
100 0.1 1 10 0.01 Ratio of elastic constants, f Fig. 6. f and v dependencies of the angle ~]~N t O give the minimum strain energy of a needle inclusion with general mixed misfit strains of e f t = - e , e ~ =e, and eM3 =2e"
101
M. Kato et al./ Materials Science and Engineering A211 (1996) 95-103
the Poisson ratio in determining the minimum-energy orientation is different between needle and plate inclusions.
'"1
. . . . . .
(23)
A
~
where c~ have been defined in Eq. (4). The occurrence of the purely hydrostatic stress state necessarily requires that all the off-diagonal components of cijA must satisfy e A = -- c~. (for i # j )
(24)
By replacing the diagonal components of c~ in the right-hand side of the equations in the Appendix with e A and using Eq. (24), new equivalent eigenstrains c A* can be calculated as a function of unknown cS . Then, in addition to a~ caused by c~, o-~ caused by e A can be formally obtained from Eq. (6) by replacing c~ and 0. , respectively. The stresses a0r after c*. with e A 0. and e A* '- and a a0. and can the accommodation are the sum of o-~. be expressed as a function of known c~. and still unknown e A . aT
=
~
A = o-r6~j
O" ij -I- o" ij
(25)
The elastic strain energy WA of the incoherent inclusion now becomes WA = -- (1/2)~ic T,T__ A ¢ eijA ) O. __ ( 1 / 2 ) ( a ~ - + aO.)(e~/+ -
-
........
I
........
I
0.4
~
0.4
/
f
/
~
v'''v=01/3
~/
~ C
//////1, / , /
"0.4
/4/ //
o
0
,~ll
........
).01 Ratio
I
........
0.1 of
I
eedle
........
1 elastic
I
I
........
10
100
constants,
f
Fig. 7. f and v dependenciesof the strain energies of fully incoherent sphere and needle inclusions.
(22)
Therefore, the total eigenstrains (misfit strains plus accommodated strains) eT is written as - -
I
05
Observing the micromechanics approach throughout in this paper, we define the incoherent inclusions as those after the full occurrence of local plastic accommodation that reduces the strain energy of the original coherent inclusions [5,6,17]. It has been proved [17] that the incoherent inclusions have hydrostatic stresses of the type a~ = O-T60. (O'r: a constant, b0.: Kronecker's delta) in them. As an extension of the previous study [5,6], the hydrostatic stress aT and the elastic strain energy WA of the incoherent inclusions will be evaluated. Since the volume is conserved by plastic deformation, the additional eigenstrains e A in an inclusion caused by the plastic accommodation must satisfy [17]
,T
........
0.6
6. Incoherent inclusions
cA - el', + cA2 + C~3 = 0
"1
0.5
(26)
The plastic accommodation is considered to occur until the strain energy WA becomes minimum. This together with the condition of Eq. (22) leads to [17] 0 WA/Oe A, - 8 WA/0cA3 = 0
(27)
8 WA/C~CA2-- 8 WA/8cA3 = 0
(28)
From Eqs. (27) and (28), the unknown diagonal components of e A (and, thus, aijA and WA) can be obtained. It was found that both the calculated hydrostatic stress ~rr and the elastic strain energy WA are the functions of c[~ + c~2 + c r • Since this trace, say A, is a scalar-invariant quantity and is equal to elm + C2 M + C~ = A, the calculated aT and WA for three kinds of inclusions are listed below using A (i) sphere (al = a3) as =
4f(1 + v) 3{f(1 + v) + 2(1 - 2v)} ffA
2f(l + v) WAS-- 3{f(1 + V) + 2(1 -- 2V)} /~A2
(ii)
plate (al >> a3,
a3 -~
(29) (30)
0)
a~-=0
(31)
W~ = 0
(32)
(iii) needle (al << a3, a3 --* oo) 2f(1 + v) aN = -- 2f(1 + V) + 3(1 -- 2V)/zA
(33)
f(1 + v) wN = 2f(1 + V) + 3(1 -- 2V)/~A2
(34)
The accommodated strain energies WA of Eqs. (30) and (34) are graphically shown as a function o f f and v in Fig. 7. It can be seen that regardless of the values of v, the strain energies of not only sphere but also needle inclusions saturate as f approaches infinity. (This energy saturation of the incoherent needle inclusion ira-
102
M. Kato et al. / Materials Science and Engineering A211 (1996) 95 103
plies that the plastic accommodation occurs so as to satisfy conclusion (e). That is, no misfit exists along the needle direction.) The asymptotic values of WA are (2/3)/tA 2 for the sphere and (1/2)/zA 2 for the needle. For v = 0.5, the energies are independent of the ratio of elastic constants, as can be found from Eqs. (30) and Eq. (34). Furthermore, for any value of v, the strain energy of an incoherent needle is always smaller than that of an incoherent sphere. Since the strain energy of an incoherent plate is zero (Eq. (32)), this result means that the inequality of W~ < W N < WA s always holds for A va 0. This corresponds to the generalization of the results in earlier studies [18,19]. It should be noted that all the above energies WA are independent of the orientation of the inclusion. This comes from the assumption that e,jA can be chosen appropriately so that the resultant strain energy takes its possible minimum. In reality, however, some restrictions may exist for the choice of e A . For example, if only a single slip system is allowed as the A the fully incoherent state plastic accommodation e~/, cannot in general be attained. (As an exception, one can point out the so-called crystallographic theories of martensitic transformations [10,12,13,20 22]. In these theories, Eq. (31) and (32) are realized by the occurence of a single accommodation slip system only when a plate takes a special orientation.) Such incomplete accommodation results in partial reduction of the strain energy and the inclusion in this state may be called semicoherent. Naturally, many possible energy states, with values between the coherent and fully incoherent states, can be conceived for semicoherent inclusions. It is interesting to find from Eqs. (30), (32) and (34) that although the strain energy of any plate inclusion can be reduced all the way to zero, the strain energies of incoherent sphere and needle inclusions cannot become zero unless the original misfit strains are of a purely deviatoric type, i.e. e~t + e~ + e ~ = A = 0. In other words, it is only for the case of the purely deviatoric misfit strains that the plastic accommodation of Eq. (22) can cancel out the original misfit strains completely. For such a case, the incoherent inclusion behaves just like a void [17].
7. Summary and conclusions Using linear isotropic elasticity, the elastic states of sphere, plate and needle inclusions with various types of misfit strains were examined. The following general conclusions and results were obtained. (1) When a coherent inclusion is much softer than the surrounding medium, a plate shape has the smallest elastic strain energy regardless of the types of the misfit strains. However, when it is much harder, either a
sphere or a needle has the smallest strain energy depending on whether the signs of the three principal misfit strains are unmixed or mixed. (2) As the elastic constants of the inclusion approach zero, the elastic strain energy of a coherent needle becomes less sensitive to its orientation and asymptotically approaches that of a coherent sphere. (3) At the limit of perfectly rigid inclusions, the elastic strain energy of a coherent plate becomes infinitely large but that of a coherent sphere takes a finite value. The elastic strain energy of a coherent needle becomes either infinite of finite depending on whether the signs of the three principal misfit strains are unmixed or mixed. (4) The optimum orientations of plate and needle inclusions predicted by the present strain-energy approach are in general different from those predicted by purely geometrical approaches. However, they become identical under certain hypothetical cases, that is, when the Poisson ratio is set equal to zero for a plate and when the inclusion is perfectly rigid for a needle. (5) By the occurrence of perfect plastic accommodation, a coherent inclusion becomes fully incoherent. The stresses in the incoherent inclusion become hydrostatic and the deviatoric components of the misfit strains completely vanish. The elastic strain energy takes the smallest value of zero for a plate followed in order by a needle and a sphere.
Acknowledgements This research was supported by a Grant-in-Aid from the Ministry of Education, Science and Culture under Grant No. 07455278.
Appendix A: Equivalent eigenstrains Since the Eshelby tensors Sokt are known for a given shape of an ellipsoidal inclusion, by solving the simultaneous equations of Eq. (6) under the assumption of Eq. (5), the equivalent eigenstrains e* can be obtained as a function of actual misfit strains e~. The results for the three different shapes of inclusions are summarized below. I. Sphere e*1 = f ( 1 -- v) 4
[2f(4 - 5v) + (7 - 5v) t-f(1 + v) + 2(1 - 2v)J L'l~2 - -
15f(1 - v)e~2 2J(4 - 5v) + (7 - 5v)
Other e* are obtained by the cyclic permutation of indices.
M. Kato et al./ Materials Science and Engineering A211 (1996) 95-103
II. Plate
ST! = f S ~ "l ,
(1
82*:2-J~ -- g422' 'g~3 --
References - f )Vv ( g ~ l -'k ~2) + 8~3, 1 - --
[1] [2] [3] [4] [5] [6]
81.2 --_f812 , 4 823. = 8}3, ~= 831. = 831¢
Ill. Needle eI~l = f ( 1 -- V) { f ( 5 - - 4 V ) + (3 -- 4V)}e~ + ( f - - 1)(1 -- 4V)e~:2l
I
j
-
-
..~_/( 1 -- f) P~ 3
f + (1 - 2v) ~ 2 = / ( 1 -- V)
[{f(5-4v)+(3-4v)Ie~:2+Of-1)(1-4v)e~l + A 1 - f)ve~33 f + (1 - 2v)
e*3
i3,
e*,- 1 +f
e*:2 -
103
4f(1 - v)e~2 f(3 - 4v) + 1'
e*3- 1 +f'
J.D. Eshelby, Proc. R. Soc. London, Ser. A, 241 (1957) 376. J.D. Eshelby, Prog. Solid Mech., 2 (1961) 89. S. Onaka, T. Fujii and M. Kato, Mech. Mater., 20 (1995) 329. T. Fujii, S. Onaka and M. Kato, Scr. Metall. Mater., in press. M. Kato and T. Fujii, Acta Metall. Mater., 42 (1994) 2929. M. Kato, T. Fujii and S. Onaka, Acta Metall. Mater., 44 (1996) 1263. [7] T. Mura, Micromechanies of Defects in Solids, 2nd edn., Martinus Nijhoff, The Hague, 1987. [8] F. Laszlo, J. Iron Steel Inst., 164 (1950) 5. [9] D.M. Barnett, J.K. Lee, H.I. Aaronson and K.C. Russell, Ser. Metall., 8 (1974) 1447. [10] T. Mura, T. Mori and M. Kato, J. Mech. Phys, Solids, 24(1976) 305. [11] U. Dahmen and K.H. Westmacott, Acta Metall., 34 (1986) 475. [12] M. Kato and M. Shibata-Yanagisawa, J. Mater. Sei., 25 (1990) 194. [13] M. Shibata-Yanagisawa and M. Kato, Mater. Trans., JIM, 31 (1990) 18. [14] D. Duly, Acta Metall., 41 (1993) 1559. [15] U. Dahmen, Acta Metall., 30 (1982) 63. [16] U. Dahmen, P. Ferguson and K.H. Westmacott, Acta Metall., 32 (1984) 803. [17] T. Mori, M. Okabe and T. Mura, Acta Metall., 28 (1980) 319. [18] F.R.N. Nabarro, Proc. R. Soc. London, Ser. A, 175(1940) 519. [19] J.K. Lee, D.M. Barnett and H.L. Aaronson, Metall. Trans,, 8A (1977) 963. [20] M.S. Wechsler, D.S. Lieberman and T.A. Read, Trans. Am. Inst. Min. Engr., 197 (1953) 1503. [21] J.S. Bowles and J.K. Mackenzie, Acta Metall., 2 (1954) 129. [22] J.K. Mackenzie and J.S. Bowles, Aeta Metall., 2 (1954) 138.
Materials Science and Engineering A211 (1996) 95-103
A
Elastic strain energies of sphere, plate and needle inclusions Masaharu Kato a, Toshiyuki Fujii a, Susumu Onaka b aDepartment of Materials" Science and Engineering, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 226, Japan bDepartment of Engineering Physics and Mechanics, Faculty of Engineering, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-01, Japan Received 18 September 1995; in revised form 30 October 1995
Abstract The inclusion problem of linear isotropic elasticity is applied to discuss the orientation and shape dependencies of the elastic strain energies of sphere, plate and needle inclusions with general misfit strains. The strain-energy minimization criterion is adopted and difference in elastic constants between the inclusion and the surrounding medium is taken into account. Regardless of the types of the misfit strains, it is generally found that the plate shape is most favorable for soft inclusions. For hard inclusions, however, the minimum-energy shape depends on the signs of the three principal misfit strains, a sphere when the three misfit strains have the same sign and a needle when they have different signs. Some other new and general characteristics on the strain energies of the coherent inclusions are newly found. Stresses and strain energies of incoherent inclusions after the occurrence of plastic accommodation are also evaluated. Keywords: Elasticity; Micromechanics; Elastic strain energy; Inclusion; Misfit strains; Coherency
I. Introduction Elastic strain energy plays an important role in determining the morphology and crystallography of second phases, e.g. precipitates, martensites, etc. Using Eshelby's inclusion problem of isotropic elasticity [1,2], we have studied the elastic strain energy of a general inhomogeneous spheroidal inclusion with principal misfit strains of a tetragonal type [3,4]. The cases for both purely dilatational (isotropic) and purely deviatoric misfit strains have been included in the analysis and favorable shape and orientation have been predicted for an inclusion with given elastic constants and misfit strains using the strain-energy minimization criterion. Inclusions with more special shapes, such as a plate and a needle, have also been discussed separately [5,6]. In this study, by considering only three representative inclusion shapes, i.e., sphere, plate and needle, elastic strain energies of coherent and incoherent inclusions will be evaluated. Different from the previous studies, however, the case for more general misfit strains (other than the tetragonal type) will be treated. Efforts will be placed on the derivation of general solutions as analytical as possible. 0921-5093/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved SSDI 0921-5093(95)10091 - 1
In reality, not only the elastic strain energy but also other factors, such as interracial energy or nucleation and growth kinetics, affect the morphology of second phases. Since this study deals only with the elastic strain energy, it does not aim at the prediction of actual second-phase morphology. Rather it extracts and evaluates the genuine contribution of the elastic strain energy as one of the important factors in determining the shape and orientation of various second phases embedded in infinitely extended matrices.
2. Calculation procedure To solve the general inclusion problem within the framework of isotropic elasticity, we will treat all the following parameters (a) to (c) as variables.
2.1. Inclusion shape (sphere, plate and needle) On the (~], ~2, ~'3) coordinates (will be abbreviated as the d coordinates) fixed to the inclusion, the inclusion shape will be described by the following spheroid:
96
M. Kato et al. / Materials Science and Engineering A211 (1996) 95 103
~2 + ~2
~32< 1
(1)
+ a-7 -
where the positive a~ and a3 are the half-lengths of the principal axes of the spheroid. Three specific shapes corresponds to (i) al = a3, for a sphere (ii) al >>a3 and a3---~0, for a plate (iii) al << a 3 and a 3 ~ oo, for a needle. 2.2. Misfit strains
Principal misfit strains of the inclusion are expressed on the (Xl, x2, x3) coordinates (the x coordinates) fixed to the surrounding medium as e~t, e~ and e~. In our previous studies [3,4,6], only the tetragonal type of misfit strains, i.e. e~ = e~ ~ e~, was considered for simplicity. In this study, we will treat more general cases. Without the loss of generality, we can assume that
(2)
_
holds. In general, the elastic strain energies of plate and needle inclusions with given misfit strains depend on their orientation, i.e. on the relative orientation relationship between the ~ and x coordinates. The components of the ~ (i = 1, 2, 3) directions (with the ~3-axis being perpendicular to the plate normal or parallel to the needle direction) are expressed on the x coordinates as (see Fig. 1) 41//[ cos 0 cos¢, cos 0 sin ¢, - sin 0]~ ~2//[ - sin ¢, cos ¢, O]x ~3//[ sin 0 cos
¢, sin 0 sin ¢, cos
(3)
O]x
X3
t
""-'--...
" ' / ~ 3
¢ (a)
The subscripts x and ~ denote that the components are expresssed on the x and ~ coordinates, respectively. When the principal misfit strains e y (i = 1, 2, 3) are described on the ~ coordinates, their components e~ can be obtained by a simple tensor conversion as
/ =[
~M cos2O cos2q~ + e~t cos20 sin2¢ + e~t sin20, (~y - e~~t) cos0 sin~ cos4~,
(t2M ~:~t)cos0 sinq~ cos¢, ~:i~4sin2q~ + e~t cos2qS, \egsinOcosOcose¢+e.~tsinOcosOsinZd)-e~sinOcosO, (E~ ~:i'a) sin0 sin0 cos~b. e~ sin0 cos0 cos2¢ + e~~ sin0 cos0 sin2q~ e~~t sin0 cos0~ (e~ - ~:~) sin0 sinq~ cos¢ ] r,iw sin20 cos2~b + e~ sin20 sin24 + ~:~ cos20 /,~
(4)
2.3. Difference in elastic constants between the inclusion and the surrounding medium
As in the previous studies [3-6], we introduce the ratio f of elastic constants as a positive dimensionless quantity: C**t = fCukl
(5)
where C*~t and Cokt are the isotropic elastic constants of the inclusion and the surrounding medium, respectively. Eq. (5) means that the shear modulus/~* of the inclusion is f times larger than the shear modulus/t of the surrounding (/1" = f / t ) while the Poisson ratio v for both is assumed to be the same. If f > 1, the inclusion is elastically harder than the surrounding and the converse is the case for 0 < f < 1. Using Eshelby's solutions of the inclusion problem [1,2], the stresses a t inside the inclusion and the elastic strain energy W assigned per unit volume of the inclusion can be calculated as [7]. o-~.= Cijkl(Sklmnemn * * -- ~ l ) : Co'kl(Sklmne~mn - e~'l)
(6)
W = - (1/2)a~e~
(7)
Here, Sklmn are the so-called Eshelby tensors and e,j are the fictitious equivalent eigenstrains [1,2,7]. The explicit forms of e* for the three representative inclusion shapes were calculated analytically by solving the simultaneous Eq. (6) and are listed in Appendix.
0 .....
(b)
Fig. 1. Definition of the x coordinates of the surrounding medium and the ~ coordinates of the plate (a) and needle (b) inclusions. ~3-axis is perpendicular to the plate in (a) and parallel to the needle in (b).
3. G e n e r a l c h a r a c t e r i s t i c s for the strain e n e r g y o f coherent inclusions
In the following, by choosing various combinations of the principal misfit strains elM, e~t and e~, the strain energies W s, W p and W u for sphere, plate and needle inclusions, respectively, will be calculated as a function of orientation (0 and ¢), the elastic constant r a t i o f a n d the Poisson ratio v. Different from the case of the tetragonal misfit strains [3-6], both angles 0 and ¢ are
M. Kato et al. / Materials Science and EngineeringA211 (1996) 95-103 needed to express the orientation of an inclusion with more general misfit strains. Before conducting individual calculations, it is beneficial to recall or derive some general conclusions of the inclusion problem. (a) If the elastic constants are homogeneous ( f = 1) and misfit strains are purely dilatational (isotropic, i.e. 6~ = 8~ = 6~), the elastic strain energy is independent of the inclusion shape [1-4,7]. (b) The elastic strain energy of a plate inclusion depends only on its elastic constants and is independent of the elastic constants of the surrounding medium [5,7]. The general form of the elastic strain energy of a plate inclusion is known to be independent of 8~3 (i = 1, 2, 3) and is written as
W P = f t l { (8~')2 A-(6422)2A- 2v6~'6422 •
17 v
}
4- 2(6f2) 2
ically known as scalar-invariant, that is, their values are independent of the choice of coordinates on which the tensor components e~ are described. This explains the orientation-independent strain energy of the needle inclusions at f ~ 0 . Furthermore, since f/l =/~*, Eq. (9) indicates that the strain energy at f--, 0 depends only on the elastic constants of the inclusions. As f approaches infinity, however, the equivalent eigenstrains of a spherical inclusion approaches a finite value. This means that; (d) As f approaches infinity, the elastic strain energy of a spherical inclusion approaches a constant value independent of f. The analytical form of the f-independent strain energy W S ( f ~ ~) for a spherical inclusion at f ~ c~ can be obtained using the equivalent eigenstrains and Eqs. (5)-(7) as
(8) wS(U---~ (30) - - 45 -~-
where e t has been defined in Eq. (4). Since f # is equal to the shear modulus #* of the inclusion, the conclusion (b) is verified. Because of this, the elastic strain energy of a plate inclusion with a given orientation is always a linear function of f. We find from Eq. (8) that the strain energy becomes zero if 6~ = 6~22= ~('12= 0 is satisfied. In other words, when the 4~-42 plane parallel to the plate is unstrained, the strain energy becomes zero regardless of the values of other misfit strain components. Since this case has already been discussed [5], we will not treat it in this study. In addition to the above well-known conclusions, some new ones for the sphere and needle can be derived. They are related to the two extreme cases of f approaching either zero (voids) or infinity (perfectly rigid inclusions). For f - ~ 0 , we find from the expressions of the equivalent eigenstrains in the Appendix and Eqs. (5) (7) that the following general conclusion holds: (c) As f approaches zero, the elastic strain energy of a needle inclusion with any misfit strains becomes orientation independent and identical to that of a spherical inclusion with the same misfit strains. The explicit form of the elastic strain energies W S ( f ~ 0) and wN(f--~ 0) of both spherical and needle inclusions at f ~ 0 becomes
¢ + 822 ~ -~- 633) ~ 2
k --
~ 2{611622
- (64,)2}]
--
[5(1
- - I/)(gl~l -~- 8~2 ~ - 8 4 3 ) 2
(7 - 5v){8~ie~2 + 8 2~2 8 3~3
-~- 8"~ 3 3 6 1~ 1
(e~:) 2 - (8~3): - (64,)2}]
(10)
Furthermore, the equivalent eigenstrains for a needle inclusion in the Appendix further indicates that a finite value at f - ~ oo is assured only when the 643 component is zero, i.e., (e) As f approaches infinity, the elastic strain energy of a needle inclusion approaches a constant value independent o f f only when the normal misfit strain (643) parallel to the needle direction (the ¢3 direction) is zero. The fact that 6~33is zero physically means that the ¢3 direction remains undistorted by the occurrence of the misfit strains. For this to be the case, we immediately notice that one of the three principal misfit strains must either be zero or have a different sign from the other two. Otherwise, no direction in the three-dimensional space can be undistorted. The explicit form of the elastic strain energy w N ( f --~ 0 0 , 843 = 0) for a needle inclusion at f--* ~ and 8343= 0 is obtained as follows. wN(U---+ 00, 6~33= O) - - _ 2/t _
3 --4v
[(1
- - V)(81~I -4- 8 2~2 ) 2 - - 6 1~1 6 2~2 -~- ( 8 1x2 ) 2
+ (6~3)2 + (8~,) 2 + 2(1 -- 2v){(62~3)2 + (61,)2}] (11)
m S ( f -'-)"O) = w N ( f --'~O)
= f p [ [~ (1e -l, v
97
~ ~
~ ~-(8,~:):
--~ 6 2 2 6 3 3 -[- 8 3 3 6 1 1
(6~):
(9)
Here, the terms e~, + 8~2 + 63~3 and 6~, 6~22+ 82¢2843 + 83436~, - (e~2)2 - (62~3)z - (84,) 2 are mathemat-
In the following analysis, we will refer to the above conclusions (a) to (e) whenever appropriate. As in the previous studies [ 3 6 ] , the inclusions with only the above misfit strains in them will be referred to as coherent inclusions. F o r easier demonstration, the misfitted coherent inclusions will be classified into two groups, one with principal misfit strains of the same sign (unmixed) and the other with those of different signs (mixed).
M. Kate et al. / Materials Science and Engineering A211 (1996) 95-103
98 103
!"1
. . . . . . . .
I
. . . . . . . .
I
'
' ''''"1
'
¢
jJ
CM
Sphere
CO
~t
,,
Plate
=t 102
Needle
"" •s •• /
~ 101 m
does not hold, the strain energy of a needle becomes infinitely large as f approaches infinity. When an inclusion is softer than the surrounding ( f < l), the plate shape has the smallest strain energy and when it is harder, the spherical shape has the smallest strain energy. These characteristics are of course in agreement with those in the previous studies [7 9]. It is useful to have analytical solutions for some special cases. Table 1 lists these solutions for stresses in the inclusions and strain energies for f--* 0, 1 and oe.
' ' ..... I
•J
/
/
/
/ /
/
tt-
~
10 °
o) o
."
t~ 10 -1 III
M--
M :
[;M=
4.2. General unm&ed misfit strains
8
•s
10-20.01 . . . . . . .0.1 . ' . . . . . . . . 1I . . . . . . . .10 , . . . . . . .100 ., Ratio
of elastic
constants,
f
Fig. 2. f dependence of the elastic strain energy of sphere, plate and needle inclusions with purely dilatational (isotropic) misfit strains of e lM = g 2M __ M --C 3 =~.
4. Coherent inclusions with unmixed misfit strains
Let us first consider the coherent inclusions with unmixed misfit strains. As mentioned above, conclusion (e) cannot be satisfied for unmixed misfit strainsl Therefore, the elastic strain energy of a needle inclusion becomes infinitely large as f approaches infinity. Among the unmixed misfit strains, purely dilatational (isotropic) ones are the simplest and have been discussed most frequently. This case will be examined briefly and then more general cases will be considered.
If one of the three unmixed misfit strains has a different value, specific orientations of plate and needle inclusions to give the minimum values W,, of the elastic strain energy can be found, although they are not necessarily determined uniquely [5,6]. For example, if O
f
Stress (0"¢j)
Energy (W)
Sphere
0
all
3f(1 + v)
:
0"22 = 0"33
2f(1 + v) (1 --2v)
4.1. Purely dilatational misfit strains
By setting e M = e M = e M = e (e, a small quantity), we have analytically calculated the elastic strain energies for the three different shapes using Eqs. (5)-(7) and those in the Appendix. Since the misfit strains are isotropic, the strain energies are independent of the plate and needle orientations and are expressed as WS =
6f(1 + v) {f(1 + v) + 2(1 - 2v)}/re2
1
-
Plate
+
V)/,/e2
All
0"11 :
0
f ( f + 3)(1 + v) - { - ~ (i ---2v)} #e2
0-33 =
--
4/re
2f(1 + v)
l_-G57ff)
2ill + v)
0
2f(1 + v) = _ (l_2v),Ue (l+v)
0"~1= 0"22
(14)
6,tt,~ 2
O"11 = 0"22 ~ 0"33
(13)
1--v
WN
2(1+v) 2 ~ ~-~-/te
3(1 _---~v)/Ze 0"22 :
0",, = 0"2 = 0"33 :
(12) Needle
Wp _ 2f(1
0-1 I = 0"22 = 0-33
4(1 + v) O0
(1 - v)~e
(1-2v) 2(1 ) 2 ~ -+ vv-f/~
2(1 + v)
Fig. 2 shows these elastic strain-energy curves (in units of/~g2) as a function o f f The Poisson ratio has been chosen as v = 1/3. We immediately notice that conclusions (a) to (d) are satisfied. Since conclusion (e)
0"33 - OO
- - / / C
(1 - v)
0-1J = 0-22 = - 2 ( 1
0"33
=
2
~ --2v~/ze
+v),ue
- 2 f ( 1 + v)/ze
f ( 1 + v)/xe 2
2
99
M. Kato et al. / Materials Science and Engineering A211 (1996) 95-103 .'"1
.-I
........
I
........
]
,
, f,l,ll I
,
, ,,,,rf I
Sphere
10 3 .......
Plate Needle •
10 2 •
.
/
."
/
Fujii [5] applied the strain-energy criteria to analyse the orientation of plate inclusions with general mixed misfit strains of e ~ < e ~ <0
(15)
J
/
is satisfied, the minimum-energy plate normal ~3e on the x coordinates becomes irrational and is described as
J e-
101 _-c"
,,/ O
///
10 °
/'"
~P
=
M
C,,I
,.-~-- 2 ~ V(~3
)
__
el
(16) )
J
with the minimum energy of
~ =2
,,,/"
(E3 __ M ' O,
=
-
Wem=f/~(1 + v)(e~) 2 =/~*(1 + v)(e~4)2 = (E*/2)(e~4) 2 (17)
uJ 10-1 -
,,
,d
,
0.01
, ,,,,,,I
0.1
,
, ,,,,,,[
,
1
, ,,,,,,I
,
, ,,,,,d
10
100
Ratio of elastic constants, f Fig. 3. f dependence on the elastic strain energy of sphere, plate and needle inclusions with unmixed principal misfit strains of c~ = ~, ~ = 2e, and eM 3 = 3e. It is the case of the three unmixed principal misfit strains being different (e.g. 0 < elm < e ~ < e~t) that a unique plate and needle orientations can be found for Wm, i.e. a plate perpendicular to the x3-axis and a needle parallel to the xl-axis. Fig. 3 shows the result of such a case for eft = e, e~t = 2e, g 3M = 3e and v = 1/3. It can be seen that the conclusions (b), (c) and (d) prevail. Moreover, although conclusion (a) is no more satisfied, the general appearance of the curves in Fig. 3 remains similar to that of Fig. 2, that is, the plate and the sphere have the smallest strain energy for very soft and very hard inclusions, respectively, and the strain energy of the needle approaches infinity as f - - . oc.
where /~* and E* are respectively the shear modulus and Young's modulus of the plate inclusion. Because of the conclusion (b), the ~P3 direction in Eq. (16) is independent o f f and is a function of only v for given misfit strains. It is found that the minimum-energy orientation (0 = 28.2 ° and ~b = 0 ° for -g3 in Eq. (3)) and the energy of the plate inclusions used to obtain Fig. 4 indeed satisfy Eqs. (16) and (17), respectively. As shown by Dahmen and Westmacott [11] and by Kato and Fujii [5], for the tetragonal misfit strains of e~4 < 0 < e ~ t = e ~ and e l M = e ~ t < 0 < e M 3 , the minimum-energy plate normal can be any direction that lies on the surface of a cone making the following angle c~ from the x3- and yl-axes, respectively: tan c~=
~/ (ey + ve~') (e~ + ve~)
2"1'
'
.......
I
Fig. 4 shows the Wm VS. f curves for inclusions with mixed misfit strains: e ~ = - e , e ~ t = e , e ~ = 2 e and v = 1/3. In addition to the conclusions (b), (c) and (d), conclusion (e) is also satisfied in this case. Because of this, some interesting characteristics can be found. Although the plate shape remains most preferable for soft inclusions, as in the case of the unmixed misfit strains, not the sphere but the needle becomes most preferable for hard inclusions.
........
i
........
f
........
I
Sphere
~a 10 2
5. Coherent inclusions with mixed misfit strains
(18)
g
.......
Plate
....
Needle
," , ,"
,,'"
101
//~,7 ¢/~" fJ"
100
-
7 st
t-
O .m
/
10 -1
,'" s"
e~ = e
,"
sM=2s
s"
ILl 10-2
5.1. Plate inclusions with m i x e d misfit strains
The o p t i m u m orientations of plate inclusions have been discussed in detail using both strain-energy and geometry approaches [5,10-14]. As a generalization of the work by D a h m e n and Westmacott [11], K a t o and
d
,
0.01
.......
I
0.1
........
I
1
........
I
,
, ,,,,,,I
10
R a t i o of e l a s t i c c o n s t a n t s ,
100 f
Fig. 4. f dependence of the elastic strain energy of sphere, plate and needle inclusions with mixed principal misfit strains of e~t = e,e~t =e, and e~ =2e
M. Kato et al. / Materials Science and Eng&eering A211 (1996) 95-103
100
v '"1
. . . . . . . .
I
I
. . . . . . . .
'
'
'''"'1
. . . . . . . .
I
90
70
=
_ 0.:,'
o
=
"\
eM =
~
•
50
e~
40
eft
(19)
A similar situation arises when a needle has mixed and different principal misfit strains. We have found that the minimum-energy needle direction ~u for eft < 0 < e~t < e ~ is o f the type 0 = 90 ° in Eq. (3), i.e.
80 v
tan f l ( f ~ ~ ) = / xl
3N - [ cos q~,v, sin ~bu, 0]
where with f (e~ = figure found
30 20 10
(20)
q)N is a function o f f and v. The variation o f q~N and v for the case o f the misfit strains o f Fig. 4 - e, e ~ = e and e~4 = 2e is shown in Fig. 6. This is similar to Fig. 5 and (fiN for f ~ ~ is again to be independent o f v:
tan (fiN(f---+ (30) = ~ -
e~ eftt
(21)
0 Hd
. . . . . . . .
I
. . . . . . . .
I
. . . . . . . .
I
,
,
,,,r,,I
0.01 0.1 1 10 100 Ratio of elastic constants, f Fig. 5, fand v dependencies of the half-apex angle fl from the xl-axis to give the minimum strain energy of a needle inclusion with tetragonal and mixed misfit strains of eft = - e, e~ = e, and e~ = e. On the contrary, if the misfit strains are such that the inequality Eq. (15) c a n n o t be satisfied, the m i n i m u m energy orientation o f the plate inclusion becomes rational [5].
5.2. Needle inclusions with m i x e d misfits strains
In c o m p a r i s o n with the plate, the minimum-energy orientation o f the needle behaves in a m o r e complex manner. This is obviously due to the fact that the elastic strain energy o f the needle depends on the elastic constants o f both the inclusion and the surrounding medium. K a t o et al. [6] have f o u n d that for the case o f tetragonal misfit strains, say eft < e ~ = e ~ , the minim u m - e n e r g y needle directions (the ~3 direction) lie on the surface o f a cone m a k i n g an angle fl from the xj-axis. Different f r o m c~ (for a plate) in Eq. (18), fl (for a needle) is a function o f b o t h f and v for given misfit strains [6] I. The variation o f the half-apex angle fl with f f o r eft = - e , e2M = e ~ = e a n d 0 < v < 0 . 5 is graphically shown in Fig. 5. It is interesting to find that as f increases, fl either decreases or increases depending on whether v is smaller or larger than 0.25. At the limit o f f - - , ~ , fl becomes 45 ° irrespective o f the values o f v. The value fl(f--+oo) o f fl at f o o v is f o u n d to be independent o f v and is written as [6] For the case of purely deviatoric misfit strains, e.g., - eft/2 = e~ = e.~, Kato et al. [6] have shown that regardless of the values of v, the angle fl always decreases as f increases and approaches a constant value of 54.7° given by Eq. (19) at f--. ~. In this case, fl is independent o f f only when v = 0.5.
As shown previously [5,6,15,16], the surface o f the cone with its half-apex angle f l ( f ~ ~ ) o f Eq. (19) is made o f a set o f invariant-line (IL) directions and the u direction defined in Eq. (20) describes the IL direction on the plane perpendicular to the x3-axis. In other words, the minimum-energy criterion for the needle inclusions predicts the same orientation as that predicted by the purely geometrical I L criterion only for the special case o f f ~ ~ . F o r other cases, the minim u m - e n e r g y direction deviates from the I L direction because o f the Poisson effect that is not taken into account in the purely geometrical theories. It should be recalled here that for plate inclusions, the plate-normal direction o f Eq. (18) and the I L direction coincide each other only when the Poisson ratio v is set equal to zero [5,11]. Therefore, the role o f NI l
i
~ iI:lll
I
i
,
: ILlll
'
' ''''"1
'
' ''''"1
90 80
v=0
8M= E;
70
C~=2C
60 6~ o~ 50 z
0.25
40
-e. 30 20
0.5
10 0
,
, ,,,,,,I
,
, ,,i,,,]
,
, ,,,,,,I
,
,
,,,,,,I
100 0.1 1 10 0.01 Ratio of elastic constants, f Fig. 6. f and v dependencies of the angle ~]~N t O give the minimum strain energy of a needle inclusion with general mixed misfit strains of e f t = - e , e ~ =e, and eM3 =2e"
101
M. Kato et al./ Materials Science and Engineering A211 (1996) 95-103
the Poisson ratio in determining the minimum-energy orientation is different between needle and plate inclusions.
'"1
. . . . . .
(23)
A
~
where c~ have been defined in Eq. (4). The occurrence of the purely hydrostatic stress state necessarily requires that all the off-diagonal components of cijA must satisfy e A = -- c~. (for i # j )
(24)
By replacing the diagonal components of c~ in the right-hand side of the equations in the Appendix with e A and using Eq. (24), new equivalent eigenstrains c A* can be calculated as a function of unknown cS . Then, in addition to a~ caused by c~, o-~ caused by e A can be formally obtained from Eq. (6) by replacing c~ and 0. , respectively. The stresses a0r after c*. with e A 0. and e A* '- and a a0. and can the accommodation are the sum of o-~. be expressed as a function of known c~. and still unknown e A . aT
=
~
A = o-r6~j
O" ij -I- o" ij
(25)
The elastic strain energy WA of the incoherent inclusion now becomes WA = -- (1/2)~ic T,T__ A ¢ eijA ) O. __ ( 1 / 2 ) ( a ~ - + aO.)(e~/+ -
-
........
I
........
I
0.4
~
0.4
/
f
/
~
v'''v=01/3
~/
~ C
//////1, / , /
"0.4
/4/ //
o
0
,~ll
........
).01 Ratio
I
........
0.1 of
I
eedle
........
1 elastic
I
I
........
10
100
constants,
f
Fig. 7. f and v dependenciesof the strain energies of fully incoherent sphere and needle inclusions.
(22)
Therefore, the total eigenstrains (misfit strains plus accommodated strains) eT is written as - -
I
05
Observing the micromechanics approach throughout in this paper, we define the incoherent inclusions as those after the full occurrence of local plastic accommodation that reduces the strain energy of the original coherent inclusions [5,6,17]. It has been proved [17] that the incoherent inclusions have hydrostatic stresses of the type a~ = O-T60. (O'r: a constant, b0.: Kronecker's delta) in them. As an extension of the previous study [5,6], the hydrostatic stress aT and the elastic strain energy WA of the incoherent inclusions will be evaluated. Since the volume is conserved by plastic deformation, the additional eigenstrains e A in an inclusion caused by the plastic accommodation must satisfy [17]
,T
........
0.6
6. Incoherent inclusions
cA - el', + cA2 + C~3 = 0
"1
0.5
(26)
The plastic accommodation is considered to occur until the strain energy WA becomes minimum. This together with the condition of Eq. (22) leads to [17] 0 WA/Oe A, - 8 WA/0cA3 = 0
(27)
8 WA/C~CA2-- 8 WA/8cA3 = 0
(28)
From Eqs. (27) and (28), the unknown diagonal components of e A (and, thus, aijA and WA) can be obtained. It was found that both the calculated hydrostatic stress ~rr and the elastic strain energy WA are the functions of c[~ + c~2 + c r • Since this trace, say A, is a scalar-invariant quantity and is equal to elm + C2 M + C~ = A, the calculated aT and WA for three kinds of inclusions are listed below using A (i) sphere (al = a3) as =
4f(1 + v) 3{f(1 + v) + 2(1 - 2v)} ffA
2f(l + v) WAS-- 3{f(1 + V) + 2(1 -- 2V)} /~A2
(ii)
plate (al >> a3,
a3 -~
(29) (30)
0)
a~-=0
(31)
W~ = 0
(32)
(iii) needle (al << a3, a3 --* oo) 2f(1 + v) aN = -- 2f(1 + V) + 3(1 -- 2V)/zA
(33)
f(1 + v) wN = 2f(1 + V) + 3(1 -- 2V)/~A2
(34)
The accommodated strain energies WA of Eqs. (30) and (34) are graphically shown as a function o f f and v in Fig. 7. It can be seen that regardless of the values of v, the strain energies of not only sphere but also needle inclusions saturate as f approaches infinity. (This energy saturation of the incoherent needle inclusion ira-
102
M. Kato et al. / Materials Science and Engineering A211 (1996) 95 103
plies that the plastic accommodation occurs so as to satisfy conclusion (e). That is, no misfit exists along the needle direction.) The asymptotic values of WA are (2/3)/tA 2 for the sphere and (1/2)/zA 2 for the needle. For v = 0.5, the energies are independent of the ratio of elastic constants, as can be found from Eqs. (30) and Eq. (34). Furthermore, for any value of v, the strain energy of an incoherent needle is always smaller than that of an incoherent sphere. Since the strain energy of an incoherent plate is zero (Eq. (32)), this result means that the inequality of W~ < W N < WA s always holds for A va 0. This corresponds to the generalization of the results in earlier studies [18,19]. It should be noted that all the above energies WA are independent of the orientation of the inclusion. This comes from the assumption that e,jA can be chosen appropriately so that the resultant strain energy takes its possible minimum. In reality, however, some restrictions may exist for the choice of e A . For example, if only a single slip system is allowed as the A the fully incoherent state plastic accommodation e~/, cannot in general be attained. (As an exception, one can point out the so-called crystallographic theories of martensitic transformations [10,12,13,20 22]. In these theories, Eq. (31) and (32) are realized by the occurence of a single accommodation slip system only when a plate takes a special orientation.) Such incomplete accommodation results in partial reduction of the strain energy and the inclusion in this state may be called semicoherent. Naturally, many possible energy states, with values between the coherent and fully incoherent states, can be conceived for semicoherent inclusions. It is interesting to find from Eqs. (30), (32) and (34) that although the strain energy of any plate inclusion can be reduced all the way to zero, the strain energies of incoherent sphere and needle inclusions cannot become zero unless the original misfit strains are of a purely deviatoric type, i.e. e~t + e~ + e ~ = A = 0. In other words, it is only for the case of the purely deviatoric misfit strains that the plastic accommodation of Eq. (22) can cancel out the original misfit strains completely. For such a case, the incoherent inclusion behaves just like a void [17].
7. Summary and conclusions Using linear isotropic elasticity, the elastic states of sphere, plate and needle inclusions with various types of misfit strains were examined. The following general conclusions and results were obtained. (1) When a coherent inclusion is much softer than the surrounding medium, a plate shape has the smallest elastic strain energy regardless of the types of the misfit strains. However, when it is much harder, either a
sphere or a needle has the smallest strain energy depending on whether the signs of the three principal misfit strains are unmixed or mixed. (2) As the elastic constants of the inclusion approach zero, the elastic strain energy of a coherent needle becomes less sensitive to its orientation and asymptotically approaches that of a coherent sphere. (3) At the limit of perfectly rigid inclusions, the elastic strain energy of a coherent plate becomes infinitely large but that of a coherent sphere takes a finite value. The elastic strain energy of a coherent needle becomes either infinite of finite depending on whether the signs of the three principal misfit strains are unmixed or mixed. (4) The optimum orientations of plate and needle inclusions predicted by the present strain-energy approach are in general different from those predicted by purely geometrical approaches. However, they become identical under certain hypothetical cases, that is, when the Poisson ratio is set equal to zero for a plate and when the inclusion is perfectly rigid for a needle. (5) By the occurrence of perfect plastic accommodation, a coherent inclusion becomes fully incoherent. The stresses in the incoherent inclusion become hydrostatic and the deviatoric components of the misfit strains completely vanish. The elastic strain energy takes the smallest value of zero for a plate followed in order by a needle and a sphere.
Acknowledgements This research was supported by a Grant-in-Aid from the Ministry of Education, Science and Culture under Grant No. 07455278.
Appendix A: Equivalent eigenstrains Since the Eshelby tensors Sokt are known for a given shape of an ellipsoidal inclusion, by solving the simultaneous equations of Eq. (6) under the assumption of Eq. (5), the equivalent eigenstrains e* can be obtained as a function of actual misfit strains e~. The results for the three different shapes of inclusions are summarized below. I. Sphere e*1 = f ( 1 -- v) 4
[2f(4 - 5v) + (7 - 5v) t-f(1 + v) + 2(1 - 2v)J L'l~2 - -
15f(1 - v)e~2 2J(4 - 5v) + (7 - 5v)
Other e* are obtained by the cyclic permutation of indices.
M. Kato et al./ Materials Science and Engineering A211 (1996) 95-103
II. Plate
ST! = f S ~ "l ,
(1
82*:2-J~ -- g422' 'g~3 --
References - f )Vv ( g ~ l -'k ~2) + 8~3, 1 - --
[1] [2] [3] [4] [5] [6]
81.2 --_f812 , 4 823. = 8}3, ~= 831. = 831¢
Ill. Needle eI~l = f ( 1 -- V) { f ( 5 - - 4 V ) + (3 -- 4V)}e~ + ( f - - 1)(1 -- 4V)e~:2l
I
j
-
-
..~_/( 1 -- f) P~ 3
f + (1 - 2v) ~ 2 = / ( 1 -- V)
[{f(5-4v)+(3-4v)Ie~:2+Of-1)(1-4v)e~l + A 1 - f)ve~33 f + (1 - 2v)
e*3
i3,
e*,- 1 +f
e*:2 -
103
4f(1 - v)e~2 f(3 - 4v) + 1'
e*3- 1 +f'
J.D. Eshelby, Proc. R. Soc. London, Ser. A, 241 (1957) 376. J.D. Eshelby, Prog. Solid Mech., 2 (1961) 89. S. Onaka, T. Fujii and M. Kato, Mech. Mater., 20 (1995) 329. T. Fujii, S. Onaka and M. Kato, Scr. Metall. Mater., in press. M. Kato and T. Fujii, Acta Metall. Mater., 42 (1994) 2929. M. Kato, T. Fujii and S. Onaka, Acta Metall. Mater., 44 (1996) 1263. [7] T. Mura, Micromechanies of Defects in Solids, 2nd edn., Martinus Nijhoff, The Hague, 1987. [8] F. Laszlo, J. Iron Steel Inst., 164 (1950) 5. [9] D.M. Barnett, J.K. Lee, H.I. Aaronson and K.C. Russell, Ser. Metall., 8 (1974) 1447. [10] T. Mura, T. Mori and M. Kato, J. Mech. Phys, Solids, 24(1976) 305. [11] U. Dahmen and K.H. Westmacott, Acta Metall., 34 (1986) 475. [12] M. Kato and M. Shibata-Yanagisawa, J. Mater. Sei., 25 (1990) 194. [13] M. Shibata-Yanagisawa and M. Kato, Mater. Trans., JIM, 31 (1990) 18. [14] D. Duly, Acta Metall., 41 (1993) 1559. [15] U. Dahmen, Acta Metall., 30 (1982) 63. [16] U. Dahmen, P. Ferguson and K.H. Westmacott, Acta Metall., 32 (1984) 803. [17] T. Mori, M. Okabe and T. Mura, Acta Metall., 28 (1980) 319. [18] F.R.N. Nabarro, Proc. R. Soc. London, Ser. A, 175(1940) 519. [19] J.K. Lee, D.M. Barnett and H.L. Aaronson, Metall. Trans,, 8A (1977) 963. [20] M.S. Wechsler, D.S. Lieberman and T.A. Read, Trans. Am. Inst. Min. Engr., 197 (1953) 1503. [21] J.S. Bowles and J.K. Mackenzie, Acta Metall., 2 (1954) 129. [22] J.K. Mackenzie and J.S. Bowles, Aeta Metall., 2 (1954) 138.