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Elastic state and orientation of plate-shaped inclusions
Acta metall, mater. Vol. 42, No. 9, pp. 2929-2926, 1994 Copyright © 1994ElsevierScienceLtd 0956-7151(94)E0098-2 Printed in Great Britain. All rights reserved 0956-7151/94 $7.00+ 0.00
Pergamon
ELASTIC STATE A N D ORIENTATION OF PLATE-SHAPED INCLUSIONS M. KATO and T. FUJII Department of Materials Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 227, Japan (Received 12 July 1993; in revised form 21 February 1994) Abstract--The inclusion problem of linear elasticity is applied to predict the crystallography of misfitted plate-shaped inclusions with isotropic and anisotropic elastic constants. Habit-plane orientations are considered to be determined so as to minimize the elastic strain energy. When plastic accommodation occurs in the inclusion, the strain energy reduces and the habit-plane orientation changes. Similarities and differences between the present energy consideration and previous purely geometrical analyses are elucidated. The implication and significance of "incoherent" inclusions are discussed within the framework of micromechanics.
1. INTRODUCTION D a h m e n and Westmacott [1] proposed a geometrical criterion to predict the habit-plane orientations of plate-shaped precipitates with tetragonal misfit strains. The implication of this study was further discussed by Duly [2] in association with the strain energy minimization criterion. However, as they pointed out, there have been only several efforts to compare the basic ideas and assumptions between approaches based on the theory of elasticity and on purely geometrical theories such as the invariantline theory [3] and the theories of martensite cyrstallography [4-6]. The purpose of this study is to (i) fully discuss energetics and crystallography of plate-shaped precipitates using the inclusion problem of linear elasticity, (ii) compare and contrast the elasticity theory and the geometrical theory, and (iii) clearly define the term "incoherent" inclusion in conjunction with the elasticity theory of micromechanics. 2. BASIC PRINCIPLES FOR THE ANALYSIS In this study, the shape of an inclusion or a precipitate embedded in an infinitely large matrix is approximated as an ellipsoid. That is, on (¢1, ~2, ~3) cartesian coordinates fixed to the ellipsoid, the inclusion can be described as ~ / a 2 + ~ / a ~ + ~2/a2 ~< 1
(1)
where al, az and a 3 are half-lengths of the principal axes of the ellipsoid. For plate-shaped inclusions which are the main concern in this study, we assume that a~ = a 2 ~ a 3 and a3-+0. Principal misfit strains in the inclusion are expressed on the (x I II[100], x2 II[010], x3 ]1[001]) cartesian coordinates of the matrix crystal
as E~, E2 and E3. For the later analysis, we assume without the loss of generality that the inequality of EI~
2929
2930
KATO and FUJII: ELASTIC STATE OF PLATE-SHAPED INCLUSIONS
°
v
Fig. 1. Geometry of the (~l, ~2, ~3) coordinates of the flat ellipsoid, the (Xl, x2, x3) coordinates of the matrix on which the principal misfit strains El, E2 and E3are described and the orientation of the plate normal V.
as long as the plate-shaped inclusions [equation (1) with al = a2 >>a3 and a3 --*0] is considered, it has been proved [10] that the elastic strain energy is independent of the matrix elastic constants. Therefore, the following calculation using the inclusion problem is exactly accurate within the framework of linear elasticity. The usage of linear elasticity inevitably forces us to adopt the infinitesimal-deformation (ID) approach to discuss the current problem. This approach makes it possible for us to calculate the elastic state of the precipitate. However, the rigorous finite-deformation (FD) approach should be adopted if purely geometrical considerations, such as those based on the invariant-line theory [3] and the invariant-plane theory [4-6], of the morphology and crystallography of plate-shaped inclusions are to be conducted. In the F D analysis, not the strain components Ei (i = 1, 2 or 3) but the distortion components qi ( = 1 + e~) are used and the sequential occurrence of distortions and rotations is expressed by the multiplication of distortion and rotation matrices. On the other hand, the approximate ID analysis expresses the sequential occurrence as the summation of strain and rotation matrices. Since the summation of matrices is commutable, superposition of strains (and stresses) is possible and the analysis becomes much simpler. In the following, the comparison between the ID and F D approaches will be made whenever necessary. It may be argued that the application of the inclusion problem shown below is valid only for misfitted "coherent" precipitates. For example, Dahmen and Westmacott [1] have shown that a misfitted precipitate loses its coherency and becomes "semicoherent" when plastic deformation occurs in the precipitate to accommodate the misfit stresses. Therefore, if one wishes, the following elasticity analysis based on the ID approach in Section 3 may be regarded as that for "coherent" precipitates. However, in the framework of the inclusion problem,
both the misfit strains and the additional accommodation strains by plastic deformation belong to the so-called eigenstrains [11] or stress-free transformation strains [7]. Therefore, even after the occurrence of plastic accommodation, the inclusion problem can still be applied as it is. With this in mind, we will denote the inclusions either as "misfitted inclusions" (rather than coherent inclusions) or as "accommodated inclusions" (rather than semicoherent or incoherent inclusions). The implication of the "incoherent" inclusion will be discussed independently in Section 6.
3. MISFITTED PLATE INCLUSIONS
3.1. Isotropic elasticity When a plate-shaped inclusion of equation (1) with the plate normal of equation (2) has the principal misfit strains E~, E2 and E3 on the (Xl,X2,X3) coordinates, the associated elastic strain energy can be calculated using Eshelby's solutions. First, the adoption of equation (2) makes it possible to obtain all the components of directional cosines connecting the (XI,X2,X3) and (~1,~2'¢3) coordinates. Then, the misfit strain components e~ described on the (¢1, ¢2, ¢3) coordinates can be calculated using the well-known matrix algebra. Eshelby's solutions for a fiat inclusion of equation (1) with al = a2 ~ a3 and a3---~0 give the elastic strain energy W per unit volume as [7, 11] W= ~
{e21 q- e~2 + 2vej,e22 + 2(1 - v)e22}
(3)
where # is the shear modulus and v is Poisson's ratio of the inclusion and the summation convention over repeated indices is adopted. By varying tk and 2 in equation (2) from 0 ° to 90 °, the orientations of the habit plane normal V to give the minimum values of W were searched by setting v = 1/3. The calculated results for twelve examples of principal misfit strain combinations are summarized in Table 1. It can be found even intuitively that when all the principal misfit strains have the same (unmixed) sign but different values, the elastic strain energy becomes minimum when the plate habit plane contains the principal directions of two misfit strains whose absolute values are smaller than the remaining one. These are the cases for No. 1 and No. 2 in Table 1. For cases where two or three of unmixed misfit strains have the same value, the isotropic elasticity sometimes fails to predict a unique orientation of plate habit planes, as exemplified in No. 4 and No. 5. Even f o r mixed misfit strains, the isotropic elasticity cannot predict a unique orientation of plate habit planes when two principal misfit strains have the same value. Dahmen and Westmacott [1] have shown that for E, = E3 < 0 < ~2 (No. 6 in Table 1), the habit plane normal V to give the minimum strain
KATO and FUJII:
ELASTIC STATE OF PLATE-SHAPED INCLUSIONS
2931
energy can be any direction which lies on the surface of a cone making an angle 0 from the x2 direction. This 0 can be written as tan 0 = { -e~ (1 + v)t(e: + ve~ )}~/~.
(4)
As pointed out by D a h m e n and Westmacott [1], this 0 is slightly different from 0IL for the invariant line, tan 0~L = (--e~/t2) 1/~ [l]'L In order for equation (4) to be valid, the inequality - w I < e2 must be satisfied. As shown in the counter-example of No. 8 in Table 1, if this condition is not satisfied, the V direction can be any direction on the x3-xl[[[ (010)] plane (corresponding to 0 = 90°). We can easily extend equation (4) for more general cases of ~ < ~3 < 0 < ~ and ,t < 0 < ~3 < £2The habit plane normals V for these cases (No. 9 and No. 10) are irrational and they are perpendicular to the x3 II[001] direction. Therefore, as noted previously for tetragonal misfit strains [1, 2], it can be said that the predicted habit planes contain the x3 II[001] direction along which the misfit strain ~3 takes an intermediate value between ~ and ~2. Keeping this in mind, rotation of the ( X l , X 2 , X 3 ) coordinates by 0 about the x3 axis (see Fig. 3) results in the misfit strain components of
(cos0 sin0i)(! 0 0/cos0 sin0i) -sin0
cos0
0
0
e~ 0 ~ / s i n 0
0 e3/\
/~cos-O+e~sln 0 =|(e2--et)sin0 cos0
\
0
o
'r, 8 o .,-
@
]
cos0
0
(eZ-el)sinOcosO eisin20-+t2c0s20
o
N°NN
0
Oo)
oo
E
"
~3
(5) To minimize the elastic strain energy under the condition that the habit plane contains the x3 axis, it can be seen from equation (3) that the following quantity must be minimized {~1 COS2 0 "4- t 2
8
8
o
sin 2 0} 2 + ~
+ 2v(t I cos 2 0 + ~2 sin2 0)'3 = {(t~ cos 2 0 + '2 sin2 0) + w3} 2 + (1 -- v2)e~. (6) I
The minimum of equation (6) occurs when the inside of the brackets becomes zero, or tan 0 = { - (~1 -4-
V£3)/(£2 -[- ~ 3 ) }
1/2"
(7)
._~
I
c~~
~
o
:rE~ I-
I
r4 -- ~,~ , - ~ I
This is a generalization of equation (4) and it is obvious that equation (7) reduces to equation (4) tNote that this angle 0m was derived using the ID approach. The rigorous FD approach results in the corresponding expression of tan 0t~ = {(1 -r/~)/(r/22 - 1)}~/~ [3, 12, 13], where ~/~= 1 + ~ and q2 = I q--e2. When 1~11 and [e21 are much smaller than unity, the two OIL angles coincide.
I
I I I I
2932
KATO and FUJII:
ELASTIC STATE OF PLATE-SHAPED INCLUSIONS strains are such that the inequality (10) cannot be satisfied, habit plane normals become rational. In No. 11 and No. 12 in Table 1, examples of this case are also included.
3.2. Anisotropic elasticity
t;l= E3< 0
As an extension of Eshelby's problem, it has been shown [9] that when an anisotropic ellipsoidal inclusion has eigenstrains (misfit strains) E,.., total constrained strains 7ik inside the inclusion are uniform and are given by ~ik = SikmnEmn"
(11)
Fig. 2. When two of the three mixed misfit strains are the same, the predicted habit-plane n0rmals lie on a cone making the angle 0 from either the x~ (for E1< 0 < E3 = E:) or x 2 (for e I = E3 < 0 < e2) axis.
Siam, is written for the plate-shaped inclusion of equation (1) as
when E~ = E3. F r o m equation (7), the habit plane normal is calculated as
where Cjt.,. are the anisotropic elastic constants of the inclusion, the vector V and its components are defined in equation (2), and N,j(V) and D(V) are respectively the cofactor and the determinant of the following matrix K 0
V = [{ -(E, +
vE3)/(~2
-
El)} ~/2,
{(E2 +
vE3)/(E2
El)} ~/~, 0]. (8)
--
Equation (8) again gives an irrational orientation of the habit plane and the calculated habit plane normals for No. 9 and No. 10 in Table 1 indeed agree with those obtained from equation (8). F r o m equations (3), (6) and (7), the minimum strain energy Wmi. in this case is calculated as Wmin
_
/t
1 -- V
(1--v:)(~
2
E
2
= 2 ~3
Kij = Cipjq Vp Vq.
(12)
(13)
The stresses am. inside the inclusion and the elastic strain energy W per unit volume are then calculated as
am, = Cm,ik(Yik -- Etk)
(14)
W = -- 2~¢Tmn~'mn"
(15)
and (9)
where E is Young's modulus of the inclusion. Similar to the case of equation (4), it is found that in order for equations (7) and (8) to be valid, the following inequality must be satisfied el < --vE3 < E2.
Cj,m. v, Sikmn = 2D(V) {VkN'j(V) + V'Nkj(V)}
(10)
This imposes another condition for the occurrence of an irrational habit plane orientation. If the misfit Xo
v Fig. 3. Predicted orientation of the plate inclusion for E~< 0 < E3 < E2 and E~< E3 < 0 < E2. The habit plane normal V of equation (8) lies on the xj-x2 plane and the angle 0 satisfies equation (7).
Similar to the case of the isotropic elasticity, the orientation of the habit plane normal V to give the minimum strain energy was numerically calculated. F o r this purpose, cubic elastic constants of iron (Cll = 24.31, C l : = 13.81, C44 = 12.19 x 10J°N/m 2 using the Voigt notation) and molybdenum (C11 =46.0, C l : = 17.6, C,~= l l . 0 × 101°N/m 2) were chosen. Table 1 also lists the calculated results of the habit-plane normals for the anisotropic cases. The elastic anisotropy ratio A = 2 C ~ / ( C H - C ~ 2 ) is greater than unity (2.32) for iron and smaller than unity (0.77) for molybdenum. This means that the elastically soft directions are the (100)type in the former and the (111)-type in the latter [141. As can be seen in Table l, the adoption of the elastic anisotropy leads to the unique determination of the habit-plane orientation for any combination of misfit strains. F o r the tetragonal misfit strains o f e 1 = E3 < 0 < E2 (No. 6), Wen et al. [15] and D a h m e n and Westmacott [1] also considered the role of the cubic elastic anisotropy in determining the habit-plane orientation. They concluded that when A > 1, the habit-plane orientation should be of the type { h k 0} with h / k ~ (--El/E2) 1/2 and when
KATO and FUJII: ELASTIC STATE OF PLATE-SHAPED INCLUSIONS A < 1, it should be of the type {hkh} with h/k ~ 2( -- El /2e2 ) l/z. We find from No. 6 in Table 1 that the types of the habit-plane orientations are in agreement with their prediction. Although their analysis is sound in this aspect, their approximation is not necessarily accurate enough. F o r example, application of their calculation to our case of El=E3= --0.1 and E2=0.2 (No. 6) results in the predicted habit-plane normals [0.577,0.816,0] and [0, 0.816, 0.577] for iron and [0.408,0.816,0.408] for molybdenum. These are about 7 ° different from those calculated accurately and shown in Table 1. This difference may become important when comparison is attempted between the calculation and experimental results. The difference actually arises from the fact that such terms as (--El/E2) 1/2 and 2(--EI/2E2) 1/2 were derived only from geometrical considerations and no energetics nor elastic anisotropy was considered. This point will be discussed in detail later.
4. ACCOMMODATED PLATE INCLUSIONS As mentioned in Section 2, the above inclusion micromechanics can apply equally well after the occurrence of plastic accommodation provided that the accommodation occurs uniformly in the inclusion. It is found from equation (3) that elastic strain energy becomes minimum (zero) if ell = e22 = el2 = 0
(16)
is satisfied. This condition means that the ~-~2 plane is unstrained and constitutes the necessary condition of the invariant-plane deformation [13]. The most well-known theories based on the invariantplane deformation criterion are probably those of martensite crystallography [4-6] in which the habit plane of a plate martensite is assumed to be an invariant plane. In the FD-based theories, two types of eigendistortions in the plate are considered, the misfit distortion (the Bain deformation) and the lattice-invariant accommodation shear distortion either by slip or by twinning. It is known from the ID analysis [16, 17] that although the Bain deformation alone cannot attain the zero strain energy state, the combination of the Bain deformation and the additional accommodation shear can. Therefore, the minimization (zero) of the strain energy is tacitly assumed when using the invariant-plane deformation criterion, although the FD-based theories are purely geometrical. As an application of the studies on the inclusion problem of micromechanics, Kato and ShibataYanagisawa [18, 19] have completely reformulated the FD-based theories of martensite crystallography in analytical form using the ID approach. Their analysis for the general misfit strains together with the (101)[10T] lattice-invariant slip predicts the habit plane orientation of (derived from
2933
equation (20) in Ref. [19]) V = [{ - ( E , + E2)/(E3 -- E~ -- E~)} ~/~,
{(E3/(E3 -- E, -- E2)}'/2, 0].
(17)
For the tetragonal misfit strains (such as the Bain strains) of El < 0 < e3 = Ez this reduces to [18] V = [{(E 1 + E2)/E 1 }1/2, (_E2/EI)I/2, 0].
(18)
Note that the strain energy minimization criterion [equation (16)] has been used to derive equations (17) and (18). Equations (17) and (18) are the approximate solutions of those obtained by the FDbased theories of martensite crystallography. If the solutions become imaginary, the invariant-plane deformation condition cannot be satisfied. Dahmen and Westmacott [1] have considered an accommodated "semicoherent" plate precipitate with tetragonal misfit strains and slip. They have concluded that the habit plane contains (i) the invariant line at the intersection of the cone of unextended lines with the slip plane, and (ii) a tensile axis on the plane of the isotropic misfit with the highest Schmid factor for the slip system. A simple calculation with the same tegragonal misfit strains, slip on (101) and the [001] tensile axis shows that the above geometrical consideration predicts exactly the same habit plane orientation as that in equation (18), as shown in Fig. 4. It is therefore concluded that the criterion of strain energy minimization has been tacitly taken into account in the geometrical analysis of the semicoherent plate by Dahmen and Westmacott. Dahmen and Westmacott have further discussed that the predicted {hk 0}-type habit planes in x3//[ool]
x2 ii [o I o]
Fig. 4. [100] stereographic projection showing the habit plane (HP) and its normal V of an accommodated plate inclusion defined by the [001] tensile axis and the intersection P of the cone of invariant lines with the (101) slip plane for E~
2934
KATO and FUJII: ELASTIC STATE OF PLATE-SHAPED INCLUSIONS
equation (18) stem from the fact that only the slip plane but not the slip direction was specified in their analysis. However, this is not true. The analysis by Kato and Shibata-Yanagisawa [18] has revealed that the {h k 0}-type habit planes, shown in equations (17) and (18), are the genuine result of the ID approximation of the originally FD-based theories of martensite crystallography. If the rigorous FDbased analysis is conducted, the habit plane becomes a {h k/}-type, as in the original theories of martensite crystallography [4-6]. In the present study, elasticity has been used extensively to discuss habit-plane orientations of plate-shaped inclusions. Comparing the habit-plane orientation of equation (8) with that of equations (17) and (18), one may feel it rather strange that Poisson's ratio appears only in equation (8). In short, this comes from the fact that the minimum elastic strain energy Wm~n,such as that calculated in equation (9), has a definite positive value for the plate of equation (8) (the misfitted plate), whereas it completely vanishes to be zero for the plate of equations (17) and (18) (the accommodated plate). It is only for such a special case of zero strain energy that the predicted habit-plane orientation using the elasticity approach coincides with that using the purely geometrical approach provided that the ID-based approximate analysis is conducted in the latter approach. In general, misfit strains alone cannot realize the situation of zero strain energy. The occurrence of additional accommodation strains in the inclusion is necessary to reduce the strain energy, possibly all the way to zero. 5. SOME EXAMPLES OF T H E PRESENT
ANALYSES Referring to the experimental observations discussed by Dahmen and Westmacott [1], let us demonstrate how the present analyses can be applied to explain experimental observations. The b.c.t. ~" Fet6N2 precipitates in b.c.c, iron have the principal misfit strains of £~ = - 0 . 0 0 2 3 , £3 = - 0 . 0 0 2 3 and £2 = 0.0971 [1]. Since the strain components are relatively small, this corresponds to the case of the misfitted plate inclusions in Section 3. F o r simplicity, we will adopt the analysis based on the isotropic elasticity. In this case, since E1 < - w 3 < E2 [equation (10)] is satisfied, equation (8) can be used. Assuming that v = 1/3, we have V = [0.176, 0.984, 0]. This habit plane is about 10° away from (010) and is in excellent agreement with the experimental observation [20]. To treat the case of accommodated plane inclusions such as f.c.c. 3" Fe4N precipitates in b.c.c. Fe or h.c.p. Mo2C precipitates in b.c.c, molybdenum [1], the application of the rigorous FD-based theories of martensite cyrstallography is more preferable, as
discussed in detail in the previous section. However, just to demonstrate that ID-based solutions are not necessarily useless, we consider the case of the Fe4N precipitates with the principal misfit strains of El = --0.0643, £3 = --0.0643 and E2= 0.3233 [1]. Direct application of equation (17) leads to the habit-plane orientation of V = [0.895, 0.446, 0]. We find that this is in close agreement with the observed (940) habit plane [1].
6. INCOHERENT INCLUSIONS Since the words "semicoherent" and "incoherent" are often used for the accommodated inclusions, the meaning and implication of the "incoherent" inclusions are discussed below from the micromechanics point of view. in their study on micromechanics, Lee and Johnson [21] have just assumed that an incoherent inclusion has hydrostatic stress in it. Later, Mori et al. [22] discovered the physical meaning of the hydrostatic stress. We will use the results of the latter work for the following analysis. Let E~ be the general misfit strains in an inclusion. (in the present study, Ell = El, e22 = E2, £33 ~- £3 and other £~ are zero.) As the additional plastic accommodation which occurs in the inclusion, not only slip or twinning but also diffusion can be considered. If these accommodation processes occur uniformly in the inclusion, the volume of the inclusion will not change. Therefore, regardless of the actual processes, the additional plastic accommodation can be expressed as new eigenstrains A£~ which satisfies AEii = AEII + AE22 + AE33 = 0.
(19)
The occurrence of the two kinds of the eigenstrains creates stresses tro (by e,~) and Aao (by AEo) in the inclusion, where trii = Cijkt (Skt,,~ emn -- £kt)
(20)
Atr ij = Cijkl( Sklmn AEmn -- AEk/).
(21)
and
The elastic strain energy W per unit volume is W=
1 U+ Aau)(£ U+ A£U) - ~(a
= - ½(au£,J + Aa~*ij + tr0 Ae~ + Atr,yA£~). (22) We consider that the accommodation processes occur so as to decrease the strain energy of equation (22). As the accommodation process proceeds, the inclusion loses its coherency and it becomes "fully incoherent" when the situation of the minimum strain energy is reached [22]. From equations (20)-(22), the minimum of the strain energy W, Wmin, is found to occur when the following equation is satisfied 6 W = - (tr~ + Atrij)S A£U= 0
(23)
KATO and FUJII: ELASTIC STATE OF PLATE-SHAPED INCLUSIONS where 6 W is a change in W due to an arbitrary small change 6AE0. in AEU. Equation (23) under the condition of equation (19) is satisfied irrespective of the changes of 6A% when or0.+ Aa0. = ¢r60.
(24)
where ¢r is a constant and 60 is Kronecker's delta. It can be seen from equation (24) that the fully incoherent inclusions have hydrostatic stresses in them, as assumed by Lee and Johnson [21]. This is an important conclusion which can be derived rigorously as above. In other words, the fully incoherent inclusions behave just like gas-filled cavities. Another important conclusion derived from the above discussion is that the hydrostatic stress a and, thus, the m i n i m u m strain energy Wmin depend on the shape of the inclusion. For example, as derived in Appendix, Eshelby's inclusion problem of isotropic elasticity shows that cr and Wmin for the following three special shapes of the ellipsoidal inclusions become: (i) Sphere (al = a2 = a3) 4/z(l + v ) ,
Wmin__ 2/z(1 + v) (E, + E2 + E3) 2. 9(1 -- V)
(25)
2935
inclusions or precipitates are predicted by adopting the inclusion problem of linear elasticity. The strain energy minimization criterion has been used for this purpose. The main f n d i n g s and conclusions are summarized as follows. 1. The predicted habit-plane orientations are in general irrational and for the case of isotropic inclusions, they are dependent on Poisson's ratio. 2. Using the isotropic elasticity, a general condition for the realization of the irrational habit-plane orientation has been derived. 3. When anisotropic elasticity is adopted, the unique habit-plane orientations can be found even for the cases of tetragonal misfit strains. 4. When the plastic accommodation occurs in the misfitted plate inclusion, the elastic strain energy can be minimized all the way to zero. This case corresponds to the invariant-plane deformation treated by the theories of martensite crystallography and also to the "semicoherent" inclusion discussed by D a h m e n and Westmacott. 5. F r o m the viewpoint of micromechanics, fully incoherent inclusions are defined as those with hydrostatic stresses in them. For plate inclusions, the hydrostatic stresses and, thus, elastic strain energy become zero. Under this special case, the present energy approach and the previous geometrical approach predict identical habit-plane orientations.
(ii) Needle (al = a2 <~ a3, a3---~oo) 0"--
REFERENCES
2/~(1 + v) - (El -}- E2 -{- E3) , 5 -- 4v
Wmin -- # ( l Jr- ~) (El -~- E2 -~- E3)2"
5 - 4v
(26)
(iii) Plate (a~ = a2 >>a3, a3 -~0) a=0,
Wmi. = 0 .
(27)
Comparison between equations (9) and (27) reveals that the strain energy is indeed reduced (all the way to zero) by the plastic accommodation. Since the stresses and elastic strain energy of the incoherent plate inclusion are zero after the complete accommodation, the plate inclusion behaves as if it is a wedge crack. It is now clear that the semicoherent plate precipitates discussed by D a h m e n and Westmacott [1] and the martensite plates satisfying the invariant-plane condition [4-6] correspond exactly to this case. In other words, they are fully incoherent from the viewpoint of micromechanics. Furthermore, it is only for this case that the purely geometrical analysis and the analysis based on elasticity lead to the same prediction of habit-plane orientations. 7. SUMMARY AND CONCLUSIONS In this study, the habit-plane orientations for various misfitted and accommodated plate-shaped AMM 42/9--C
I. U. Dahmen and K. H. Westmacott, Acta metall. 34, 475 (1986). 2. D. Duly, Acta metall, mater. 41, 1559 (1993). 3. U. Dahmen, Acta metall. 30, 63 (1982). 4. M. S. Wechsler, D. S. Lieberman and T. A. Read, Trans. Am. Inst. Min. Engrs 197, 1503 (1953). 5. J. S. Bowles and J. K. Mackenzie, Acta metall. 2, 129 (1954). 6. J. K. Mackenzie and J. S. Bowles, Acta metall. 2, 138 (1954). 7. J. D. Eshelby, Proc. R. Soc. A 241, 376 (1957). 8. J. D. Eshelby, Prog. Solid Mech. 2, 89 (1961). 9. N. Kinoshita and T. Mura, Physica status solidi 25, 759 (1971). 10. J. K. Lee, D. M. Barnett and H. I. Aaronson, Metall. Trans. 8A, 963 (1977). 11. T. Mura, Micromechanics of Defects in Solids, 2nd Edn. Martinus Nijhoff, The Hague (1987). 12. M. Kato, Mater. Sci. Engng A 146, 205 (1991). 13. M. Kato, Mater. Trans. Japan. Inst. Metals 33, 89 (1992). 14. J. F. Nye, Physical Properties of Crystals, p. 145, Oxford Univ. Press, London (1957). 15. S. H. Wen, E. Kostlan, M. Hong, A. G. Khachaturyan and J. W. Morris Jr, Acta metall. 29, 1247 (1981). 16. T. Mura, T. Mori and M. Kato, J. Mech. Phys. Solids 24, 305 (1976). 17. A. G. Khachaturyan, Theory of Structural Transformations in Solids, Chap. I1. Wiley, New York (1983). 18. M. Kato and M. Shibata-Yanagisawa, J. Mater. Sci. 25, 194 (1990). 19. M. Shibata-Yanagisawa and M. Kato, Mater. Trans. Japan. Inst. Metals 31, 18 (1990). 20. P. Ferguson, U. Dahmen and K. H. Westmacott, Scripta metall. 18, 57 (1984).
2936
K A T O and FUJII:
ELASTIC STATE O F PLATE-SHAPED INCLUSIONS
21. J. K. Lee and W. C. Johnson, Acta metall. 26, 541 (1978). 22. T. Mori, M. Okabe and T. Mura, Acta metall. 28, 319 (1980).
indices. Therefore, the strain energy W in equation (22) become: 1
2#
W= -[aoE o=
15(1 - v )
x {4(AE~ + AE22+ AE~ + 2El AE, + 2E2 AE2 + 2E3 AE3) APPENDIX
+ (1 + 5v)(AE l AE2 + A~2 AE3 + AE3 AEI
Taking the case of the spherical inclusion, a and Wmin in equations (25) will be derived below. Inclusions with other shapes can be treated in a similar manner. The Eshelby tensors Sak~ for the spherical inclusion are written as:
SIIII
: $2222 = $3333 =
Using equation (19), equation (23) can be written as: ~W
a~-e3) ~AE2=0"
AEi = {(E2 + e3) -- 2El }/3, (A1)
Ae2 = {(E3 + El) -- 2E2}/3 ,
and other Sijkl are zero. From equations (20), (21) and (A1) together with ~II=EI, E22=E2, E33=E3, AEII=AEI, A£22 = AE2, A£33 = A£3, the stresses become: 2# o'li - {8El + (1 + 5V)(E2 + E3)}, (A2) 15(1 - v )
AE3 = {(El + E2) -- 2E3}/3.
SI2t2 = S2323 = $3131
Aall =
2,
- {8AEI + ( 1 + 5v)(AE2+AE3) }. 15(I - v )
(A5)
For equation (A5) to be satisfied regardless of 6AEI and t~AE2, the terms in the parentheses must be zero. This together with equations (A4) and (A5) leads to:
5v 15(I - v)' 4 - - 5v 15(1 -- v)'
(A4)
SW=( c~WBAE,0A~E3)6AEI+(t~-E2
7 -- 5v 15(1 -- v~' 1 -
S1122 = S2233 = S3311 ~'~
"[- E l AE 2 + E2 AE I + E2 AE 3 -~- E3 AE 2 -4- e 3 AE I -~- E I A~3)}.
(A3)
Other non-zero stress components, a22, t~33, A~722 and Aa33, can be obtained by the cyclic permutation of the
(A6)
From equations (A2), (A3) and (A6), we obtain: 4#(1 + v ) , , ~ ] _----~ tel + e2 + ~3)oU
a~+Aa~
(A7)
and W~n = --~(% + Aaa)(E a + AEa) _
2#(1 + v) (~1 + E2 + E3)2. 9(1 - v)
(A8)
Pergamon
ELASTIC STATE A N D ORIENTATION OF PLATE-SHAPED INCLUSIONS M. KATO and T. FUJII Department of Materials Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 227, Japan (Received 12 July 1993; in revised form 21 February 1994) Abstract--The inclusion problem of linear elasticity is applied to predict the crystallography of misfitted plate-shaped inclusions with isotropic and anisotropic elastic constants. Habit-plane orientations are considered to be determined so as to minimize the elastic strain energy. When plastic accommodation occurs in the inclusion, the strain energy reduces and the habit-plane orientation changes. Similarities and differences between the present energy consideration and previous purely geometrical analyses are elucidated. The implication and significance of "incoherent" inclusions are discussed within the framework of micromechanics.
1. INTRODUCTION D a h m e n and Westmacott [1] proposed a geometrical criterion to predict the habit-plane orientations of plate-shaped precipitates with tetragonal misfit strains. The implication of this study was further discussed by Duly [2] in association with the strain energy minimization criterion. However, as they pointed out, there have been only several efforts to compare the basic ideas and assumptions between approaches based on the theory of elasticity and on purely geometrical theories such as the invariantline theory [3] and the theories of martensite cyrstallography [4-6]. The purpose of this study is to (i) fully discuss energetics and crystallography of plate-shaped precipitates using the inclusion problem of linear elasticity, (ii) compare and contrast the elasticity theory and the geometrical theory, and (iii) clearly define the term "incoherent" inclusion in conjunction with the elasticity theory of micromechanics. 2. BASIC PRINCIPLES FOR THE ANALYSIS In this study, the shape of an inclusion or a precipitate embedded in an infinitely large matrix is approximated as an ellipsoid. That is, on (¢1, ~2, ~3) cartesian coordinates fixed to the ellipsoid, the inclusion can be described as ~ / a 2 + ~ / a ~ + ~2/a2 ~< 1
(1)
where al, az and a 3 are half-lengths of the principal axes of the ellipsoid. For plate-shaped inclusions which are the main concern in this study, we assume that a~ = a 2 ~ a 3 and a3-+0. Principal misfit strains in the inclusion are expressed on the (x I II[100], x2 II[010], x3 ]1[001]) cartesian coordinates of the matrix crystal
as E~, E2 and E3. For the later analysis, we assume without the loss of generality that the inequality of EI~
2929
2930
KATO and FUJII: ELASTIC STATE OF PLATE-SHAPED INCLUSIONS
°
v
Fig. 1. Geometry of the (~l, ~2, ~3) coordinates of the flat ellipsoid, the (Xl, x2, x3) coordinates of the matrix on which the principal misfit strains El, E2 and E3are described and the orientation of the plate normal V.
as long as the plate-shaped inclusions [equation (1) with al = a2 >>a3 and a3 --*0] is considered, it has been proved [10] that the elastic strain energy is independent of the matrix elastic constants. Therefore, the following calculation using the inclusion problem is exactly accurate within the framework of linear elasticity. The usage of linear elasticity inevitably forces us to adopt the infinitesimal-deformation (ID) approach to discuss the current problem. This approach makes it possible for us to calculate the elastic state of the precipitate. However, the rigorous finite-deformation (FD) approach should be adopted if purely geometrical considerations, such as those based on the invariant-line theory [3] and the invariant-plane theory [4-6], of the morphology and crystallography of plate-shaped inclusions are to be conducted. In the F D analysis, not the strain components Ei (i = 1, 2 or 3) but the distortion components qi ( = 1 + e~) are used and the sequential occurrence of distortions and rotations is expressed by the multiplication of distortion and rotation matrices. On the other hand, the approximate ID analysis expresses the sequential occurrence as the summation of strain and rotation matrices. Since the summation of matrices is commutable, superposition of strains (and stresses) is possible and the analysis becomes much simpler. In the following, the comparison between the ID and F D approaches will be made whenever necessary. It may be argued that the application of the inclusion problem shown below is valid only for misfitted "coherent" precipitates. For example, Dahmen and Westmacott [1] have shown that a misfitted precipitate loses its coherency and becomes "semicoherent" when plastic deformation occurs in the precipitate to accommodate the misfit stresses. Therefore, if one wishes, the following elasticity analysis based on the ID approach in Section 3 may be regarded as that for "coherent" precipitates. However, in the framework of the inclusion problem,
both the misfit strains and the additional accommodation strains by plastic deformation belong to the so-called eigenstrains [11] or stress-free transformation strains [7]. Therefore, even after the occurrence of plastic accommodation, the inclusion problem can still be applied as it is. With this in mind, we will denote the inclusions either as "misfitted inclusions" (rather than coherent inclusions) or as "accommodated inclusions" (rather than semicoherent or incoherent inclusions). The implication of the "incoherent" inclusion will be discussed independently in Section 6.
3. MISFITTED PLATE INCLUSIONS
3.1. Isotropic elasticity When a plate-shaped inclusion of equation (1) with the plate normal of equation (2) has the principal misfit strains E~, E2 and E3 on the (Xl,X2,X3) coordinates, the associated elastic strain energy can be calculated using Eshelby's solutions. First, the adoption of equation (2) makes it possible to obtain all the components of directional cosines connecting the (XI,X2,X3) and (~1,~2'¢3) coordinates. Then, the misfit strain components e~ described on the (¢1, ¢2, ¢3) coordinates can be calculated using the well-known matrix algebra. Eshelby's solutions for a fiat inclusion of equation (1) with al = a2 ~ a3 and a3---~0 give the elastic strain energy W per unit volume as [7, 11] W= ~
{e21 q- e~2 + 2vej,e22 + 2(1 - v)e22}
(3)
where # is the shear modulus and v is Poisson's ratio of the inclusion and the summation convention over repeated indices is adopted. By varying tk and 2 in equation (2) from 0 ° to 90 °, the orientations of the habit plane normal V to give the minimum values of W were searched by setting v = 1/3. The calculated results for twelve examples of principal misfit strain combinations are summarized in Table 1. It can be found even intuitively that when all the principal misfit strains have the same (unmixed) sign but different values, the elastic strain energy becomes minimum when the plate habit plane contains the principal directions of two misfit strains whose absolute values are smaller than the remaining one. These are the cases for No. 1 and No. 2 in Table 1. For cases where two or three of unmixed misfit strains have the same value, the isotropic elasticity sometimes fails to predict a unique orientation of plate habit planes, as exemplified in No. 4 and No. 5. Even f o r mixed misfit strains, the isotropic elasticity cannot predict a unique orientation of plate habit planes when two principal misfit strains have the same value. Dahmen and Westmacott [1] have shown that for E, = E3 < 0 < ~2 (No. 6 in Table 1), the habit plane normal V to give the minimum strain
KATO and FUJII:
ELASTIC STATE OF PLATE-SHAPED INCLUSIONS
2931
energy can be any direction which lies on the surface of a cone making an angle 0 from the x2 direction. This 0 can be written as tan 0 = { -e~ (1 + v)t(e: + ve~ )}~/~.
(4)
As pointed out by D a h m e n and Westmacott [1], this 0 is slightly different from 0IL for the invariant line, tan 0~L = (--e~/t2) 1/~ [l]'L In order for equation (4) to be valid, the inequality - w I < e2 must be satisfied. As shown in the counter-example of No. 8 in Table 1, if this condition is not satisfied, the V direction can be any direction on the x3-xl[[[ (010)] plane (corresponding to 0 = 90°). We can easily extend equation (4) for more general cases of ~ < ~3 < 0 < ~ and ,t < 0 < ~3 < £2The habit plane normals V for these cases (No. 9 and No. 10) are irrational and they are perpendicular to the x3 II[001] direction. Therefore, as noted previously for tetragonal misfit strains [1, 2], it can be said that the predicted habit planes contain the x3 II[001] direction along which the misfit strain ~3 takes an intermediate value between ~ and ~2. Keeping this in mind, rotation of the ( X l , X 2 , X 3 ) coordinates by 0 about the x3 axis (see Fig. 3) results in the misfit strain components of
(cos0 sin0i)(! 0 0/cos0 sin0i) -sin0
cos0
0
0
e~ 0 ~ / s i n 0
0 e3/\
/~cos-O+e~sln 0 =|(e2--et)sin0 cos0
\
0
o
'r, 8 o .,-
@
]
cos0
0
(eZ-el)sinOcosO eisin20-+t2c0s20
o
N°NN
0
Oo)
oo
E
"
~3
(5) To minimize the elastic strain energy under the condition that the habit plane contains the x3 axis, it can be seen from equation (3) that the following quantity must be minimized {~1 COS2 0 "4- t 2
8
8
o
sin 2 0} 2 + ~
+ 2v(t I cos 2 0 + ~2 sin2 0)'3 = {(t~ cos 2 0 + '2 sin2 0) + w3} 2 + (1 -- v2)e~. (6) I
The minimum of equation (6) occurs when the inside of the brackets becomes zero, or tan 0 = { - (~1 -4-
V£3)/(£2 -[- ~ 3 ) }
1/2"
(7)
._~
I
c~~
~
o
:rE~ I-
I
r4 -- ~,~ , - ~ I
This is a generalization of equation (4) and it is obvious that equation (7) reduces to equation (4) tNote that this angle 0m was derived using the ID approach. The rigorous FD approach results in the corresponding expression of tan 0t~ = {(1 -r/~)/(r/22 - 1)}~/~ [3, 12, 13], where ~/~= 1 + ~ and q2 = I q--e2. When 1~11 and [e21 are much smaller than unity, the two OIL angles coincide.
I
I I I I
2932
KATO and FUJII:
ELASTIC STATE OF PLATE-SHAPED INCLUSIONS strains are such that the inequality (10) cannot be satisfied, habit plane normals become rational. In No. 11 and No. 12 in Table 1, examples of this case are also included.
3.2. Anisotropic elasticity
t;l= E3< 0
As an extension of Eshelby's problem, it has been shown [9] that when an anisotropic ellipsoidal inclusion has eigenstrains (misfit strains) E,.., total constrained strains 7ik inside the inclusion are uniform and are given by ~ik = SikmnEmn"
(11)
Fig. 2. When two of the three mixed misfit strains are the same, the predicted habit-plane n0rmals lie on a cone making the angle 0 from either the x~ (for E1< 0 < E3 = E:) or x 2 (for e I = E3 < 0 < e2) axis.
Siam, is written for the plate-shaped inclusion of equation (1) as
when E~ = E3. F r o m equation (7), the habit plane normal is calculated as
where Cjt.,. are the anisotropic elastic constants of the inclusion, the vector V and its components are defined in equation (2), and N,j(V) and D(V) are respectively the cofactor and the determinant of the following matrix K 0
V = [{ -(E, +
vE3)/(~2
-
El)} ~/2,
{(E2 +
vE3)/(E2
El)} ~/~, 0]. (8)
--
Equation (8) again gives an irrational orientation of the habit plane and the calculated habit plane normals for No. 9 and No. 10 in Table 1 indeed agree with those obtained from equation (8). F r o m equations (3), (6) and (7), the minimum strain energy Wmi. in this case is calculated as Wmin
_
/t
1 -- V
(1--v:)(~
2
E
2
= 2 ~3
Kij = Cipjq Vp Vq.
(12)
(13)
The stresses am. inside the inclusion and the elastic strain energy W per unit volume are then calculated as
am, = Cm,ik(Yik -- Etk)
(14)
W = -- 2~¢Tmn~'mn"
(15)
and (9)
where E is Young's modulus of the inclusion. Similar to the case of equation (4), it is found that in order for equations (7) and (8) to be valid, the following inequality must be satisfied el < --vE3 < E2.
Cj,m. v, Sikmn = 2D(V) {VkN'j(V) + V'Nkj(V)}
(10)
This imposes another condition for the occurrence of an irrational habit plane orientation. If the misfit Xo
v Fig. 3. Predicted orientation of the plate inclusion for E~< 0 < E3 < E2 and E~< E3 < 0 < E2. The habit plane normal V of equation (8) lies on the xj-x2 plane and the angle 0 satisfies equation (7).
Similar to the case of the isotropic elasticity, the orientation of the habit plane normal V to give the minimum strain energy was numerically calculated. F o r this purpose, cubic elastic constants of iron (Cll = 24.31, C l : = 13.81, C44 = 12.19 x 10J°N/m 2 using the Voigt notation) and molybdenum (C11 =46.0, C l : = 17.6, C,~= l l . 0 × 101°N/m 2) were chosen. Table 1 also lists the calculated results of the habit-plane normals for the anisotropic cases. The elastic anisotropy ratio A = 2 C ~ / ( C H - C ~ 2 ) is greater than unity (2.32) for iron and smaller than unity (0.77) for molybdenum. This means that the elastically soft directions are the (100)type in the former and the (111)-type in the latter [141. As can be seen in Table l, the adoption of the elastic anisotropy leads to the unique determination of the habit-plane orientation for any combination of misfit strains. F o r the tetragonal misfit strains o f e 1 = E3 < 0 < E2 (No. 6), Wen et al. [15] and D a h m e n and Westmacott [1] also considered the role of the cubic elastic anisotropy in determining the habit-plane orientation. They concluded that when A > 1, the habit-plane orientation should be of the type { h k 0} with h / k ~ (--El/E2) 1/2 and when
KATO and FUJII: ELASTIC STATE OF PLATE-SHAPED INCLUSIONS A < 1, it should be of the type {hkh} with h/k ~ 2( -- El /2e2 ) l/z. We find from No. 6 in Table 1 that the types of the habit-plane orientations are in agreement with their prediction. Although their analysis is sound in this aspect, their approximation is not necessarily accurate enough. F o r example, application of their calculation to our case of El=E3= --0.1 and E2=0.2 (No. 6) results in the predicted habit-plane normals [0.577,0.816,0] and [0, 0.816, 0.577] for iron and [0.408,0.816,0.408] for molybdenum. These are about 7 ° different from those calculated accurately and shown in Table 1. This difference may become important when comparison is attempted between the calculation and experimental results. The difference actually arises from the fact that such terms as (--El/E2) 1/2 and 2(--EI/2E2) 1/2 were derived only from geometrical considerations and no energetics nor elastic anisotropy was considered. This point will be discussed in detail later.
4. ACCOMMODATED PLATE INCLUSIONS As mentioned in Section 2, the above inclusion micromechanics can apply equally well after the occurrence of plastic accommodation provided that the accommodation occurs uniformly in the inclusion. It is found from equation (3) that elastic strain energy becomes minimum (zero) if ell = e22 = el2 = 0
(16)
is satisfied. This condition means that the ~-~2 plane is unstrained and constitutes the necessary condition of the invariant-plane deformation [13]. The most well-known theories based on the invariantplane deformation criterion are probably those of martensite crystallography [4-6] in which the habit plane of a plate martensite is assumed to be an invariant plane. In the FD-based theories, two types of eigendistortions in the plate are considered, the misfit distortion (the Bain deformation) and the lattice-invariant accommodation shear distortion either by slip or by twinning. It is known from the ID analysis [16, 17] that although the Bain deformation alone cannot attain the zero strain energy state, the combination of the Bain deformation and the additional accommodation shear can. Therefore, the minimization (zero) of the strain energy is tacitly assumed when using the invariant-plane deformation criterion, although the FD-based theories are purely geometrical. As an application of the studies on the inclusion problem of micromechanics, Kato and ShibataYanagisawa [18, 19] have completely reformulated the FD-based theories of martensite crystallography in analytical form using the ID approach. Their analysis for the general misfit strains together with the (101)[10T] lattice-invariant slip predicts the habit plane orientation of (derived from
2933
equation (20) in Ref. [19]) V = [{ - ( E , + E2)/(E3 -- E~ -- E~)} ~/~,
{(E3/(E3 -- E, -- E2)}'/2, 0].
(17)
For the tetragonal misfit strains (such as the Bain strains) of El < 0 < e3 = Ez this reduces to [18] V = [{(E 1 + E2)/E 1 }1/2, (_E2/EI)I/2, 0].
(18)
Note that the strain energy minimization criterion [equation (16)] has been used to derive equations (17) and (18). Equations (17) and (18) are the approximate solutions of those obtained by the FDbased theories of martensite crystallography. If the solutions become imaginary, the invariant-plane deformation condition cannot be satisfied. Dahmen and Westmacott [1] have considered an accommodated "semicoherent" plate precipitate with tetragonal misfit strains and slip. They have concluded that the habit plane contains (i) the invariant line at the intersection of the cone of unextended lines with the slip plane, and (ii) a tensile axis on the plane of the isotropic misfit with the highest Schmid factor for the slip system. A simple calculation with the same tegragonal misfit strains, slip on (101) and the [001] tensile axis shows that the above geometrical consideration predicts exactly the same habit plane orientation as that in equation (18), as shown in Fig. 4. It is therefore concluded that the criterion of strain energy minimization has been tacitly taken into account in the geometrical analysis of the semicoherent plate by Dahmen and Westmacott. Dahmen and Westmacott have further discussed that the predicted {hk 0}-type habit planes in x3//[ool]
x2 ii [o I o]
Fig. 4. [100] stereographic projection showing the habit plane (HP) and its normal V of an accommodated plate inclusion defined by the [001] tensile axis and the intersection P of the cone of invariant lines with the (101) slip plane for E~
2934
KATO and FUJII: ELASTIC STATE OF PLATE-SHAPED INCLUSIONS
equation (18) stem from the fact that only the slip plane but not the slip direction was specified in their analysis. However, this is not true. The analysis by Kato and Shibata-Yanagisawa [18] has revealed that the {h k 0}-type habit planes, shown in equations (17) and (18), are the genuine result of the ID approximation of the originally FD-based theories of martensite crystallography. If the rigorous FDbased analysis is conducted, the habit plane becomes a {h k/}-type, as in the original theories of martensite crystallography [4-6]. In the present study, elasticity has been used extensively to discuss habit-plane orientations of plate-shaped inclusions. Comparing the habit-plane orientation of equation (8) with that of equations (17) and (18), one may feel it rather strange that Poisson's ratio appears only in equation (8). In short, this comes from the fact that the minimum elastic strain energy Wm~n,such as that calculated in equation (9), has a definite positive value for the plate of equation (8) (the misfitted plate), whereas it completely vanishes to be zero for the plate of equations (17) and (18) (the accommodated plate). It is only for such a special case of zero strain energy that the predicted habit-plane orientation using the elasticity approach coincides with that using the purely geometrical approach provided that the ID-based approximate analysis is conducted in the latter approach. In general, misfit strains alone cannot realize the situation of zero strain energy. The occurrence of additional accommodation strains in the inclusion is necessary to reduce the strain energy, possibly all the way to zero. 5. SOME EXAMPLES OF T H E PRESENT
ANALYSES Referring to the experimental observations discussed by Dahmen and Westmacott [1], let us demonstrate how the present analyses can be applied to explain experimental observations. The b.c.t. ~" Fet6N2 precipitates in b.c.c, iron have the principal misfit strains of £~ = - 0 . 0 0 2 3 , £3 = - 0 . 0 0 2 3 and £2 = 0.0971 [1]. Since the strain components are relatively small, this corresponds to the case of the misfitted plate inclusions in Section 3. F o r simplicity, we will adopt the analysis based on the isotropic elasticity. In this case, since E1 < - w 3 < E2 [equation (10)] is satisfied, equation (8) can be used. Assuming that v = 1/3, we have V = [0.176, 0.984, 0]. This habit plane is about 10° away from (010) and is in excellent agreement with the experimental observation [20]. To treat the case of accommodated plane inclusions such as f.c.c. 3" Fe4N precipitates in b.c.c. Fe or h.c.p. Mo2C precipitates in b.c.c, molybdenum [1], the application of the rigorous FD-based theories of martensite cyrstallography is more preferable, as
discussed in detail in the previous section. However, just to demonstrate that ID-based solutions are not necessarily useless, we consider the case of the Fe4N precipitates with the principal misfit strains of El = --0.0643, £3 = --0.0643 and E2= 0.3233 [1]. Direct application of equation (17) leads to the habit-plane orientation of V = [0.895, 0.446, 0]. We find that this is in close agreement with the observed (940) habit plane [1].
6. INCOHERENT INCLUSIONS Since the words "semicoherent" and "incoherent" are often used for the accommodated inclusions, the meaning and implication of the "incoherent" inclusions are discussed below from the micromechanics point of view. in their study on micromechanics, Lee and Johnson [21] have just assumed that an incoherent inclusion has hydrostatic stress in it. Later, Mori et al. [22] discovered the physical meaning of the hydrostatic stress. We will use the results of the latter work for the following analysis. Let E~ be the general misfit strains in an inclusion. (in the present study, Ell = El, e22 = E2, £33 ~- £3 and other £~ are zero.) As the additional plastic accommodation which occurs in the inclusion, not only slip or twinning but also diffusion can be considered. If these accommodation processes occur uniformly in the inclusion, the volume of the inclusion will not change. Therefore, regardless of the actual processes, the additional plastic accommodation can be expressed as new eigenstrains A£~ which satisfies AEii = AEII + AE22 + AE33 = 0.
(19)
The occurrence of the two kinds of the eigenstrains creates stresses tro (by e,~) and Aao (by AEo) in the inclusion, where trii = Cijkt (Skt,,~ emn -- £kt)
(20)
Atr ij = Cijkl( Sklmn AEmn -- AEk/).
(21)
and
The elastic strain energy W per unit volume is W=
1 U+ Aau)(£ U+ A£U) - ~(a
= - ½(au£,J + Aa~*ij + tr0 Ae~ + Atr,yA£~). (22) We consider that the accommodation processes occur so as to decrease the strain energy of equation (22). As the accommodation process proceeds, the inclusion loses its coherency and it becomes "fully incoherent" when the situation of the minimum strain energy is reached [22]. From equations (20)-(22), the minimum of the strain energy W, Wmin, is found to occur when the following equation is satisfied 6 W = - (tr~ + Atrij)S A£U= 0
(23)
KATO and FUJII: ELASTIC STATE OF PLATE-SHAPED INCLUSIONS where 6 W is a change in W due to an arbitrary small change 6AE0. in AEU. Equation (23) under the condition of equation (19) is satisfied irrespective of the changes of 6A% when or0.+ Aa0. = ¢r60.
(24)
where ¢r is a constant and 60 is Kronecker's delta. It can be seen from equation (24) that the fully incoherent inclusions have hydrostatic stresses in them, as assumed by Lee and Johnson [21]. This is an important conclusion which can be derived rigorously as above. In other words, the fully incoherent inclusions behave just like gas-filled cavities. Another important conclusion derived from the above discussion is that the hydrostatic stress a and, thus, the m i n i m u m strain energy Wmin depend on the shape of the inclusion. For example, as derived in Appendix, Eshelby's inclusion problem of isotropic elasticity shows that cr and Wmin for the following three special shapes of the ellipsoidal inclusions become: (i) Sphere (al = a2 = a3) 4/z(l + v ) ,
Wmin__ 2/z(1 + v) (E, + E2 + E3) 2. 9(1 -- V)
(25)
2935
inclusions or precipitates are predicted by adopting the inclusion problem of linear elasticity. The strain energy minimization criterion has been used for this purpose. The main f n d i n g s and conclusions are summarized as follows. 1. The predicted habit-plane orientations are in general irrational and for the case of isotropic inclusions, they are dependent on Poisson's ratio. 2. Using the isotropic elasticity, a general condition for the realization of the irrational habit-plane orientation has been derived. 3. When anisotropic elasticity is adopted, the unique habit-plane orientations can be found even for the cases of tetragonal misfit strains. 4. When the plastic accommodation occurs in the misfitted plate inclusion, the elastic strain energy can be minimized all the way to zero. This case corresponds to the invariant-plane deformation treated by the theories of martensite crystallography and also to the "semicoherent" inclusion discussed by D a h m e n and Westmacott. 5. F r o m the viewpoint of micromechanics, fully incoherent inclusions are defined as those with hydrostatic stresses in them. For plate inclusions, the hydrostatic stresses and, thus, elastic strain energy become zero. Under this special case, the present energy approach and the previous geometrical approach predict identical habit-plane orientations.
(ii) Needle (al = a2 <~ a3, a3---~oo) 0"--
REFERENCES
2/~(1 + v) - (El -}- E2 -{- E3) , 5 -- 4v
Wmin -- # ( l Jr- ~) (El -~- E2 -~- E3)2"
5 - 4v
(26)
(iii) Plate (a~ = a2 >>a3, a3 -~0) a=0,
Wmi. = 0 .
(27)
Comparison between equations (9) and (27) reveals that the strain energy is indeed reduced (all the way to zero) by the plastic accommodation. Since the stresses and elastic strain energy of the incoherent plate inclusion are zero after the complete accommodation, the plate inclusion behaves as if it is a wedge crack. It is now clear that the semicoherent plate precipitates discussed by D a h m e n and Westmacott [1] and the martensite plates satisfying the invariant-plane condition [4-6] correspond exactly to this case. In other words, they are fully incoherent from the viewpoint of micromechanics. Furthermore, it is only for this case that the purely geometrical analysis and the analysis based on elasticity lead to the same prediction of habit-plane orientations. 7. SUMMARY AND CONCLUSIONS In this study, the habit-plane orientations for various misfitted and accommodated plate-shaped AMM 42/9--C
I. U. Dahmen and K. H. Westmacott, Acta metall. 34, 475 (1986). 2. D. Duly, Acta metall, mater. 41, 1559 (1993). 3. U. Dahmen, Acta metall. 30, 63 (1982). 4. M. S. Wechsler, D. S. Lieberman and T. A. Read, Trans. Am. Inst. Min. Engrs 197, 1503 (1953). 5. J. S. Bowles and J. K. Mackenzie, Acta metall. 2, 129 (1954). 6. J. K. Mackenzie and J. S. Bowles, Acta metall. 2, 138 (1954). 7. J. D. Eshelby, Proc. R. Soc. A 241, 376 (1957). 8. J. D. Eshelby, Prog. Solid Mech. 2, 89 (1961). 9. N. Kinoshita and T. Mura, Physica status solidi 25, 759 (1971). 10. J. K. Lee, D. M. Barnett and H. I. Aaronson, Metall. Trans. 8A, 963 (1977). 11. T. Mura, Micromechanics of Defects in Solids, 2nd Edn. Martinus Nijhoff, The Hague (1987). 12. M. Kato, Mater. Sci. Engng A 146, 205 (1991). 13. M. Kato, Mater. Trans. Japan. Inst. Metals 33, 89 (1992). 14. J. F. Nye, Physical Properties of Crystals, p. 145, Oxford Univ. Press, London (1957). 15. S. H. Wen, E. Kostlan, M. Hong, A. G. Khachaturyan and J. W. Morris Jr, Acta metall. 29, 1247 (1981). 16. T. Mura, T. Mori and M. Kato, J. Mech. Phys. Solids 24, 305 (1976). 17. A. G. Khachaturyan, Theory of Structural Transformations in Solids, Chap. I1. Wiley, New York (1983). 18. M. Kato and M. Shibata-Yanagisawa, J. Mater. Sci. 25, 194 (1990). 19. M. Shibata-Yanagisawa and M. Kato, Mater. Trans. Japan. Inst. Metals 31, 18 (1990). 20. P. Ferguson, U. Dahmen and K. H. Westmacott, Scripta metall. 18, 57 (1984).
2936
K A T O and FUJII:
ELASTIC STATE O F PLATE-SHAPED INCLUSIONS
21. J. K. Lee and W. C. Johnson, Acta metall. 26, 541 (1978). 22. T. Mori, M. Okabe and T. Mura, Acta metall. 28, 319 (1980).
indices. Therefore, the strain energy W in equation (22) become: 1
2#
W= -[aoE o=
15(1 - v )
x {4(AE~ + AE22+ AE~ + 2El AE, + 2E2 AE2 + 2E3 AE3) APPENDIX
+ (1 + 5v)(AE l AE2 + A~2 AE3 + AE3 AEI
Taking the case of the spherical inclusion, a and Wmin in equations (25) will be derived below. Inclusions with other shapes can be treated in a similar manner. The Eshelby tensors Sak~ for the spherical inclusion are written as:
SIIII
: $2222 = $3333 =
Using equation (19), equation (23) can be written as: ~W
a~-e3) ~AE2=0"
AEi = {(E2 + e3) -- 2El }/3, (A1)
Ae2 = {(E3 + El) -- 2E2}/3 ,
and other Sijkl are zero. From equations (20), (21) and (A1) together with ~II=EI, E22=E2, E33=E3, AEII=AEI, A£22 = AE2, A£33 = A£3, the stresses become: 2# o'li - {8El + (1 + 5V)(E2 + E3)}, (A2) 15(1 - v )
AE3 = {(El + E2) -- 2E3}/3.
SI2t2 = S2323 = $3131
Aall =
2,
- {8AEI + ( 1 + 5v)(AE2+AE3) }. 15(I - v )
(A5)
For equation (A5) to be satisfied regardless of 6AEI and t~AE2, the terms in the parentheses must be zero. This together with equations (A4) and (A5) leads to:
5v 15(I - v)' 4 - - 5v 15(1 -- v)'
(A4)
SW=( c~WBAE,0A~E3)6AEI+(t~-E2
7 -- 5v 15(1 -- v~' 1 -
S1122 = S2233 = S3311 ~'~
"[- E l AE 2 + E2 AE I + E2 AE 3 -~- E3 AE 2 -4- e 3 AE I -~- E I A~3)}.
(A3)
Other non-zero stress components, a22, t~33, A~722 and Aa33, can be obtained by the cyclic permutation of the
(A6)
From equations (A2), (A3) and (A6), we obtain: 4#(1 + v ) , , ~ ] _----~ tel + e2 + ~3)oU
a~+Aa~
(A7)
and W~n = --~(% + Aaa)(E a + AEa) _
2#(1 + v) (~1 + E2 + E3)2. 9(1 - v)
(A8)