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Dislocations punched-out around a short fiber in a short fiber metal matrix composite subjected to uniform temperature change
Acra meraIl. Vol. 35, No. 1, pp. 155462, 1987 Printed in Great Britain. All rights reserved
Copyright
DISLOCATIONS PUNCHED-OUT AROUND FIBER IN A SHORT FIBER METAL MATRIX SUBJECTED TO UNIFORM TEMPERATURE
0
OOOI-6160/87 $3.00 + 0.00 1987 Pergamon Journals Ltd
A SHORT COMPOSITE CHANGE
M. TAYA Department of Mechanical Engineering, FU-10, University of Washington, Seattle, WA 98195, U.S.A. and T. MORI Department of Materials Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuda. Midori-ku, Yokohama 227, Japan (Received 6 January 1986; in revised form 27 March 1986)
Abstract-A thermal expansion mismatch between short fiber and matrix in a short fiber metal matrix composite subjected to a uniform temperature change is simulated by arrays of prismatic dislocation loops. The prismatic dislocation loops so generated will be punched-out around a short fiber with the punching distance c’. Then an analytical model is constructed to compute c’ and the stress field in and around a short fiber. Finally, the effect of several parameters on c’ is studied.
R&mm&A l’aide d’arrangements de boucles de dislocation prismatiques, nous simulons une difference d’expansion thermique entre une fibre courte et la matrice, dans un composite a matrice metalhque et a fibres courtes soumis a une variation uniforme de temperature. Les boucles de dislocation prismatiques ainsi produites autour d’une fibre sont dloignees dune distance c’. Nous contruisons done un modele analytique pour calculer c’ et le champ de contrainte a l’interieur et autour dune fibre courte. Nous Btudions enfin l’effet de plusieurs parametres sur c’.
Zusammeufassung-Mit Anordnungen prismatischer Versetzungsschleifen wird die durch thermische Ausdehnung erzeugte Fehlpassung zwischen einer kurzen Faser und der Matrix in einer mit kurzen Fasern verstiirkten Metallmatrix, welche einer einheitlichen Temperaturanderung unterworfen ist, simuliert. Die erzeugten prismatischen Versetzungsschleifen werden von der kurzen Faser aus bis zu einem Abstand c’ getrieben. Ein analytisches Model1 wird aufgestellt, mit dem c’ und das Spannungsfeld innerhalb und aul3erhalb der kurzen Faser berechnet werden kiinnen. Schlieglich wird der Einflug einiger Parameter auf c’ untersucht.
1. INTRODUCTION When a short fiber metal matrix composite (MMC) is subjected to a temperature change AT, dislocations are generated around short fibers due to the mismatch of the coefficients of thermal expansion (CTE) between the short fiber and matrix. Vogelsang et al. [1] have observed such dislocations around a short fiber in a Sic Whisker/Al composite in an in situ TEM study. We have also observed such dislocations
around the end of a Sic Whisker in a Sic Whisker/ 2124 Al composite (Fig. 1). Mura and Taya [2] have recently studied analytically the stress field in and around a short fiber embedded in the metal matrix by replacing the above mismatch strain by the Somigliana surface dislocations [3]. These surface dislocations, however, are more likely to be relaxed by punching. Punching of arrays of prismatic dislocation loops adhered to the surface of a spherical particle was studied by Tanaka et al. [4] with the aim of
understanding the breakdown of the linear workhardening of a dispersion hardening alloy [5]. The model used by Tanaka et al. is based on Eshelby’s
equivalent inclusion method [6]. In this paper we will focus on the punching distance of the prismatic dislocation loops that simulate the thermal expansion mismatch strain between a short fiber and the metal matrix, and also on the stress field in and around a short fiber. To this end, a Sic Whisker/Al composite will be used as a target short fiber MMC. For the difference of CTEs between a Sic whisker and an aluminum matrix is quite large. We will state our analytical model and the formulation to compute the punching distance and the resulting stress field in Section 2. Then the results of the punching distance and the stress field are given as well as the discussion in Section 3. Finally the conclusion is stated in Section 4. 155
156
TAYA and MORI:
PUNCHED
DISLOCATIONS
IN METAL MATRIX COMPOSITES
of the short fiber, respectively. The domain of punched-out dislocations is also simulated by another prolate spheroid (a,) with the major axis being c’ while the minor axis is set equal to the diameter of the short fiber a. In the punched-out (or relaxed) state, two different types of dislocation loops exist, one in the boundary of domain zf, and the other in that of a*. It is assumed in our model that the above two types of dislocation loops are smeared out in each domain, thus they are replaced by the following eigenstrains [I (1)
Fig. 1. TEM photo showing dislocations around a fiber-end in Sic/AI composite.
(2) 2. ANALYTICAL MODEL AND FORMULATION
The surface dislocations (or mismatch strain) due to the difference in CTE between a short fiber and the matrix are simulated by arrays of prismatic dislocation loops [Fig. 2(a)] which will be punched-out [Fig. 2(b)]. Figure 2 represents the case of a Sic whisker embedded in the matrix metal (aluminum) which is cooled down from some reference temperature by bT (to). The CTEs of the SiC whisker and aluminum are denoted by (orand s, respectively, and of course ur < c(,. A short fiber is modelled in Fig. 2 by a prolate spheroid (0,) with the major and minor axes being c and a which correspond to the length and diameter
where a*=(a,-h)bZ’.
(3)
Let the stiffness tensors of the matrix and fiber be Cij, and CbEI, respectively. For simplicity we assume that both the matrix and fiber are isotropic in stiffness and thermal expansion. It is noted that the anisotropy of the fiber can also be solved by the present model. With equations (1) and (21, the problem of Fig. 2 is changed to that of an inhomogeneous inclusion Q, with Cikl and e? + ep surrounded by an inclusion with C,, and ep which in turn is embedded in an infinite matrix.
L x3
(0)
XI
(bf
Fig. 2. Arrays of prismatic dislocation loops that were caused by the mismatch of CTE between fiber and
matrix (a) wiil be punched-out (bl.
TAYA and MORI:
2.1. Punching of prismatic dislocation loops The criterion to determine the punching distance c’ is given by [4] --=-au aw (4)
acf
ad
where U is the total potential energy of Fig. 2(b) and W is the total work done by the motion of the dislocation loops over the punching distance c’. Hence, the remaining task is to compute U and W. It is assumed in the following formulation that the volume fraction of fiber f, is reasonably small to facilitate the computation, hence the interaction between fibers is neglected. We note in passing that the case of a finite volume fraction of fiber can be formulated by the present model though the actual computation will become more vigorous. 2.1.1. Total potential energy U. The total potential energy U of a short fiber MMC will be obtained here by use of Eshelby’s equivalent inclusion method. Since the composite is not subjected to any external force, the total potential energy U is equal to the total elastic strain energy of Fig. 2. Therefore a,,(eij -e;)dV (5) s Ll where oij, eij and et are components of the internal stress, total strain and any non-elastic strain (eigenstrain), respectively, D denotes the volume of the composite, and dV is the volume element. Before simplifying equation (5) further, we will obtain first the formula to compute the internal stress field. Let us consider an inclusion Q, with C,,, and e? which is embedded in the matrix without fiber a,. Then the resulting internal stress G’, is given by
panying the change in the elastic constant of a, from C,,, to CLk, and the introduction of eit in R,, and is a fictitious eigenstrain and related to e:, as eklI** e:, = S’klrnnemnI** . of, can be easily obtained (9) as uh =
- e::)
(6)
where e:, = $,,ez
(7)
where ei, and Si,,,,, are the internal strain caused by this inclusion a2 and Eshelby’s tensor related to @, and the superscript 2 is not “square”, but denotes the quantity related to Q,. The repeated indices are to be summed over 1, 2 and 3. This rule will be used throughout this paper unless otherwise noted. For e$ given by equation (2) we can compute the stress field 0,: within R, easily. Now we introduce an inhomogeneous inclusion Iz, with CL,, and ef: inside a,. Let the stress field caused by this introduction of R, be denoted by e,:. Then the total stress 0,: + 0,: in a, is given by g,f + 0: = C,:,, (Szmez - e$ + ei, - ek:) = C,,,, (Gm,e$
-e$+e:,-e::*)
(8)
where equations (6) and (7) as well as Eshelby’s equivalent inclusion method and Mura’s reformulation for a combined equivalent condition [7] were used, e:, is the additional total strain in fi, accom-
(9)
from equations
C~lkdSh&!$*
-
ei?*)
(8) and
(10)
after solving for eL:* in equation (8). Now we turn to the total potential energy U and will show below that U defined by equation (5) can be reduced to the formula expressed in terms of given eigenstrains ef;r and ey. Since j,a,,e,jdV = 0 [7], and e$+ and e2,*are defined only in domains Q, - R, and Q, respectively, U is reduced to U = -i
sD
cr,,ezdV
1 =--s 2 02- n,
-k
U =i
c’, = G,&:,
157
PUNCHED DISLOCATIONS IN METAL MATRIX COMPOSITES
= --
I
1 2
(or, + oy)e’,* dV
9,‘“:’ +oi)(ef:+e2,*)dV o,!, ey dV - i(Q, - R,)aZ, e2,*
where a, - !,2, and Q, denote the volume of the domains Q, - R, and Sz,, respectively. By the Tanaka-Mori theory [8], the volume integral [the first term in equation (1 l)] can be expressed by
I o2-a1
of, ey dV = R,Cijk,(S:,m, - SL,,,,,)e!,t*.e2,*. (12)
After substituting equation (12) into (11) and using equation (9) we obtain U = - if ,Ci,k,(S:k,e::* -tf201:
e5,
-fflC,,,(S:,,.e~* - if,a;
- e$*)ef:
ey
-eLT*)ef:
(13)
where U is expressed per unit volume, f, and I; are the volume fractions of 0, and R,, respectively and they are related by fz = (c’/c) f, In equation (13) only unknown is ef:* which will be solved from equation (8) 2.1.2. Work done by the motion of prismatic dislocation loops W. The evaluation of W is basically the same as that used by Tanaka et al. [4] and it is described briefly below. Denoting the coordinate along the x,-axis [Fig. 2(b)] by Z, the average work w by the motion of a dislocation loop at x1 = Z during the punching can be expressed as w=2na(l-grZt--1)bk
TAYA and MORI:
158
PUNCHED
DISLOCATIONS
where k and b are the friction stress and Burgers’ vector of a dislocation. Then the total work done by the motion of 2N dislocation loops at Z, W, becomes t 2xa ( 1-L $)‘“Znk n-1
W=2
- ,)bk
IN METAL MATRIX COMPOSITES
where Izand p are Lame’s constants of the matrix and S, the kronecker delta. Hence as/2~ is given by
(14)
a:r -=-
42
2P
2P
where Z,, is the location of the nth dislocation loop. By taking the limiting sequence, N + co and keeping Nb = CD! *, we arrive at
I;a*
,=2/;2lra(I-$rZ(;-+*dZ
=-
4xkac2 3
c’ ---I ( c
+s:,,,- 1
and the other components
a*.
of stress vanish. In the
(15) above derivation, equation (2) was used. v in equa-
)
W can be expressed per unit volume as W=p,f,k
(19)
“‘-1 ( c
ci* )
(16)
where & = c/a which can be called as fiber aspect ratio. 2.1.3. Calculation of c’. After having solved for ejj** from equation (8) and using equations (6) and (7), U [equation (t3)] can be expressed in terms of known quantities, Cijke, CLk,, e,$@,et?, f,, 8, and c’. Then we substitute U so obtained and W given by equation (16) into equation (4) to arrive at (17) where ~1is the shear modulus of the matrix and H(r) is a function of t( = c//c) and other parameters related to the elastic constants of the matrix and fiber and the fiber aspect ratio. The detailed expression of H(5) is given in Appendix A. The punching distance c’ (or c) is solved numerically from equation (17) for given k/p and AT, the results of which will be discussed in Section 3.
tion (19) is Poisson’s ratio of the matrix. The explicit expression of the Eshelby’s tensor S,,, is given in Appendix B where aspect ratio fl should be replaced by b2 = c’/a for the above equations. Similarly at is obtained from equation (10) as
4 42 -=-=
I
$g
2P 3.4
ai3 -=
2P i
&(SII”
(Sf*,,
+
+
G22
s1,2*
+
+
54311
s:,,,
-
-
1)
1)
1_V2”)w,33+S:333+-
+
1)
(1
S&
1 e$* I
(20)
2.2. Stress field in and around a short fiber
The formula to compute the stress field in and around a short fiber has been given in Section 2.1 for the punched-out state (relaxed state). Here we will obtain the formula to compute the stress field for the non-relaxed state, i.e. c = c’ for comparison of the stress field between the relaxed and non-relaxed states. The locations of the state of stress that will be computed are shown by A, B and C in Fig. 3(a) for the relaxed state and in Fig. 3(b) for the non-relaxed state, since the computation of the stress at these points are straight forward. For completeness, we will first summarize the formula for the relaxed state and then derive the formula for the non-relaxed state below. 2.2.1. Stem field for relaxed state. The stress field within R, [shaded by vertical lines in Fig. 3(a)] r_r before a, is introduced, can be obtained from equation (6) and (7) and C,,, given by Cijk/=
AOljskl
+
P(6ik6j,+
dj16kj)
(18)
(al (b) Fig. 3. Stresses are calculated at points A, B and C for the relaxed state (a) and the non-relaxed state (b).
TAYA and MORI:
and other components of stress vanish, where e::* and e::* are solved from equation (8), and S’,,, can be computed from Appendix B with @= & = c/a. The stress field within the fiber (i.e. at point A) o$ is uniform and given by a$=a$+oZ,.
(21)
Next we will obtain the stress just outside &, i.e. at points B and C [Fig. 3(a)]. The stress just outside S&can be obtained by use of jump condition [7,9, lo] and the explicit expressions for the stresses at B (0:) and C (0:) are given by
1 + (1 - v)
(22) where equation (2) was used. 22.2. Stress field for the non-relaxed state. For the non-relaxed state, R2 becomes identical to Q, [Fig. 3(b)], leading to the simplification of the formula to compute the stress field, (6,). Eshelby’s equivalent inclusion method in this case yields for domain G, Gij = C$,(Si,meg
Several values of fiber aspect ratios and AT are used to examine their effects on the punching distance c’, the results of which will be discussed in the following subsection (3.1). 3.1. Punching distance C’ First the effect of AT on c’ was examined and the results are shown in Fig. 4. It is obvious from Fig. 4 that the punching distance c’ increases with (AT1 and punching becomes impossible for the temperature change less than about -55°C. It should be noted, however, that the present model is independent of actual temperature, thus for example it cannot account for creep behavior of the matrix and fiber. Next we have computed c’ for several values of fiber aspect ratios, I/d = 1.5-50, and plotted the results of c’ in Fig. 5. It follows from Fig. 5 that the larger the value of l/d is, the smaller the value of c’/c becomes and it converges to 1 at about l/d = 27. Namely for given parameters, punching cannot take place for the fiber with l/d > 27.
1 2.0
t
SiClAP
Composite
I/d =5 12. = 0.001 I*
C’
c 1.5 -
- e$ - ei:)
= Cij,,(S:Jmeg - ek:*)
(23)
where eg* is a fictitious eigenstrain which is constant in a, and vanishes outside R,. The non-vanishing stress components of Cs, within the fiber [at point A in Fig. 3(b)] are r$‘, = 6ig, and (?$. It should be noted here that equation (8) for c = c’ is reduced to equation (23) since S&, = S&,. This also confirms the validity of the model for the relaxed state. The stresses just outside the fiber at points B and C [Fig. 3(b)] are computed by the formula similar to equation (22). 3.
159
PUNCHED DISLOCATIONS IN METAL MATRIX COMPOSITES
1.0
1 - 200
-100
-300
4T (“Cl
Fig. 4. c/c’ vs AT for Sic/Al composite with l/d = 5 and K/P = 0.001.
SIC/AL
Composite
AT=-200’C,
;=O.OOl
NUMERICAL RESULTS AND DISCUSSION
On the basis of the formulation developed in the previous section, we have computed the punching distance c’ and the stress field. The thermomechanical and geometrical parameters used in this computation are given in Table 1.
Table 1. The thermo-mechanical
and geometrical parameters
Parameters
Unit
Al matrix
SIC whisker
Young’s modulus Poisson’s ratio CTE Fiber aspect ratio
GPa
47.5 0.3 23.6
427 0.17 4.3 5
x 10-6/Y
C’ c
IO
20 Fiber Aspect
Fig. S. c/c’ vs I/d for Sic/Al composite and ~/,u = 0.001.
30 Roth,
$
with AT = -200°C
160
TAYA and MORI:
PUNCHED DISLOCATIONS IN METAL MATRIX COMPOSITES
The above results can be easily understood by considering a simplified situation, Cfikl = C,, and e$ = (txr- a,)AT with other components of e$ being zero. Then, after punching, the total potential energy U is given by
10.0 Si C/AI
Composite
I l/d=5
U = &a2c2(e$)2/ct.
AT= -200°C C’
The generalized punching
c
force fp is defined as
fp=-$. fr takes the largest value of 5.0
at the beginning of punching (c’ = c) and decreases as punching proceeds. The generalized retarding force, fR,against punching is defined as
which is given by equation (15) as 4nkac
1.0
fa=T4.
I
(27)
I
10‘3
I
I
10-4
10-5
I
10-6
K
From equation
cc
(26) and (27), and for c $ a
& < cfPLX 4 =(1fa ‘-x-
a
00 P
;
-=I.
(28)
This is the physical reason for the decrease in the punching distance for a long fiber. Namely, the potential energy decrease by punching is less than the energy dissipation required for punching when a fiber is sufficiently large. From the above result, it appears that one must consider a different mechanism of stress relaxation for a long fiber composite. One obvious mechanism that one can consider is the uniform plastic deformation throughout the matrix. Then, the average
l/d=5,
f=
0.001,
Fig. 6. c/c’ vs K/F for SiC/AI composite with I/d = 5 and AT = -200°C.
stress in the matrix is given by - f,aij,where crij is the stress inside a fiber when only a single fiber is present in an infinitely extended medium. Even when eigenstrains are purely dilatational as in the case of thermal strains, (ujj 1> 1u1I I = 1q2 I for a fiber. Thus, the stress state in the matrix favors the occurrence of plastic deformation. It is apparent that this plastic deformation relaxes the internal stress induced by a thermal expansion mismatch. This type of stress
AT=-200DC
C’
c 1.5 -
I.01 0
/-r 5
15
IO
20
Ef i,
Fig. 7. c/c’ vs Ef/E,,, for Sic/Al composite with I/d = 5, K/P = 0.001 and AT = -200°C.
TAYA and MORI:
PUNCHED DISLOCATIONS IN METAL MATRIX COMPOSITES Table 2. Stresses at points
5 2P
x
lo-*
A
For 11 -0.225 -0.225 0.0035
B C
relaxed state 22 -0.225 0.198 osm35
3.2.Stress field The stress field is computed for the relaxed (punched-out) and non-relaxed states by use of the parameters given in Table 1. For these parameters, the value of c’/c was computed from equation (17), i.e. c’jc = 1.573. The state of stress is focused on several points, A, B, and C for the relaxed state [Fig. 3(a)] and for the non-relaxed state [Fig. 3(b)]. The results of the state of stress at these points are given in Table 2 where all the stresses are normalized
A, B and C (see Fig. 3) 11
33 -0.84 0.069 -0.84
relaxation is particularly favored for a long fiber composite. This is because 1c33- Q,,) increases as the aspect ratio of fibers (aligned) increases, and it is proportional to the Bow stress of the matrix yielded. The other parameters that affect the punching distance are the ratio of friction stress to the shear modulus of the matrix, k/p and the fiber to matrix stiffness ratio, E,/E,,,. The effect of parameter k/p on c’ is shown in Fig. 6. Figure 6 indicates that the larger the value of k/p, the smaller that of C’/C becomes. For a practical metal matrix composite system, 1 < E,/E,,, < 20, we have studied the effect of EJE,,, with the above range of c’/c and the results are shown in Fig. 7. It follows from Fig. 7 that c’/c is sensitive to EJE,,, for the smaller values of E,IE,,,, but it becomes much less sensitive to EJE,,, for the large values of E,/E,.
For non-relaxed 22
-0.245 -0.245 -0.149
state 33 -1.184 0.119 -0.184
-0.245 0.213 -0.149
Acknowledgements-This work was supported partially by a Grant from Honda R&D Company to the University of Delaware. We are thankful to Mr W. G. Patterson of University of Delaware for his technical assistance in taking TEM pictures and also to Professor R. J. Arsenault for his comments on TEM pictures.
REFERENCES I. M. Vogelsang, R. J. Arsenault and R. M. Fisher. Metall. Trans. In press. 2. T. Mura and M. Taya, Recent Advances in Composites in the United States and Japan (edited by J. R. Vinson and M. Taya), ASTM, STP864, p. 209 (1985). 3. R. J. Asaro, Znt. J. Eng. Sci. 13, 271 (1975). 4. K. Tanaka, K. Narita and T. Mori, Acta metall. 20,297 (1972). 5. M. F. Ashby, Phil. Mag. 14, 1157 (1966). 6. J. D. Eshelby, Proc. R. Sot. A241, 376 (1957). 7. T. Mura, Micromechanics ofDefects in Soli&. Martinus Nijhoff, The Hague (1982). 8. K. Tanaka and T. Mori, J. Elasticity 2, 199 (1972). 9. L. J. Walpole, Proc. R. Sot. A252, 561 (1959). 10. Y. Takao and M. Taya, J. appf. Mech. 52, 806 (1985). 11. J. D. Eshelby, Proc. R. Sot. A252, 561 (1959).
12. Y. Mikata, Ph.D dissertation, Univ. of Delaware (1984).
APPENDIX H(5) = - $
by 2~ The values of the stresses at point C for the relaxed state are approximate. For the stresses at C were computed from ofi and the jump condition and the effect of of, was neglected due to the fact that the punching distance c’ is considered to be reasonably large and the effect of crf, on the stress field outside R, decreases as O(l/]c’-cl*) [ll, 121. It is obvious from Table 2 that the magnitudes of the stresses for the relaxed state are always smaller than those for the non-relaxed state. 4.
+
0
8‘42 ag
A: H(c)
{A,(D,& - D&z) + A,@‘#,, -D,&,)]
CD241
-
W2,)
CONCLUSION
The punching of prismatic dislocation loops caused by the thermal expansion mismatch between the matrix and short fiber in a short fiber MMC was studied by use of Eshelby’s model. It was found in this study that the larger the values of 1AhT] and k/p, the larger the punching distance c’ becomes, but it decreases with the increase in the fiber aspect ratio. The stress fields for the relaxed (punched-out) and non-relaxed (non punched-out) states were also computed, indicating that the relaxed state gives rise to the stress field of smaller magnitude compared with the non-relaxed state.
161
where
A,=--
1 1 - 2v
s:,23 + &
(G,,
- 1)
162
TAYA and MORI:
PUNCHED
DISLOCATIONS
_(X+F)
E
II -
E
I2
-(Sl,ll+sfn2)+~S:311+~
by 3 B2 S 11”= S,rr -_ - q1 _ v) (82 _ 1)
P
pa
x
P
Y
%a+-S:333+- I-2v 2P
1 821
=
i
cc,,
B22
=
2P
+
~I,**
+
%,I,)
x
+2
i Wll33
IN METAL MATRIX COMPOSITES
+
G,,,)
+
is:,,,
E S2 I 2v 3311 1-2v 0 cc l-v +
-
I-2v
Q,= ~(St,,,+S1,22+S:l,,-')+SI,,,+St,*2-l Q2 = &,r,+++,-1)
2233=
--
2(1 -v)(/32-
S 3311 -S -
3322 =
-
-
1)
W) 1
and where x =A’--1. ji =/l’-/1.
1 b2
S 1133- -S
1
(A3)
In the above equations, L and p are Lame’s constants of the matrix, If and pf are Lame’s constants of the fiber, v is Poisson’s ratio of the matrix, S& and St,, are Eshelby’s tensors related to a, and f&, respectively and they are given explicitly in Appendix B, and r = c’jc. APPENDIX B: ESHELBY’S TENSOR S,j,, The Eshelby’s tensors that are used in this study are given
+2(1-v)
2(1 -v)
1 ‘-2v+m}
3 * - 2v + 2(/32- 1) g
where
g
=&MB’- I)‘“-~s’+bj
and where /I is the ratio of the major axis to the minor axis of a prolate spheroid with the major axis taken along the x3-axis. Hence for fZ,, p, = c/u and for &., /I2= c’/a should be used.
Copyright
DISLOCATIONS PUNCHED-OUT AROUND FIBER IN A SHORT FIBER METAL MATRIX SUBJECTED TO UNIFORM TEMPERATURE
0
OOOI-6160/87 $3.00 + 0.00 1987 Pergamon Journals Ltd
A SHORT COMPOSITE CHANGE
M. TAYA Department of Mechanical Engineering, FU-10, University of Washington, Seattle, WA 98195, U.S.A. and T. MORI Department of Materials Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuda. Midori-ku, Yokohama 227, Japan (Received 6 January 1986; in revised form 27 March 1986)
Abstract-A thermal expansion mismatch between short fiber and matrix in a short fiber metal matrix composite subjected to a uniform temperature change is simulated by arrays of prismatic dislocation loops. The prismatic dislocation loops so generated will be punched-out around a short fiber with the punching distance c’. Then an analytical model is constructed to compute c’ and the stress field in and around a short fiber. Finally, the effect of several parameters on c’ is studied.
R&mm&A l’aide d’arrangements de boucles de dislocation prismatiques, nous simulons une difference d’expansion thermique entre une fibre courte et la matrice, dans un composite a matrice metalhque et a fibres courtes soumis a une variation uniforme de temperature. Les boucles de dislocation prismatiques ainsi produites autour d’une fibre sont dloignees dune distance c’. Nous contruisons done un modele analytique pour calculer c’ et le champ de contrainte a l’interieur et autour dune fibre courte. Nous Btudions enfin l’effet de plusieurs parametres sur c’.
Zusammeufassung-Mit Anordnungen prismatischer Versetzungsschleifen wird die durch thermische Ausdehnung erzeugte Fehlpassung zwischen einer kurzen Faser und der Matrix in einer mit kurzen Fasern verstiirkten Metallmatrix, welche einer einheitlichen Temperaturanderung unterworfen ist, simuliert. Die erzeugten prismatischen Versetzungsschleifen werden von der kurzen Faser aus bis zu einem Abstand c’ getrieben. Ein analytisches Model1 wird aufgestellt, mit dem c’ und das Spannungsfeld innerhalb und aul3erhalb der kurzen Faser berechnet werden kiinnen. Schlieglich wird der Einflug einiger Parameter auf c’ untersucht.
1. INTRODUCTION When a short fiber metal matrix composite (MMC) is subjected to a temperature change AT, dislocations are generated around short fibers due to the mismatch of the coefficients of thermal expansion (CTE) between the short fiber and matrix. Vogelsang et al. [1] have observed such dislocations around a short fiber in a Sic Whisker/Al composite in an in situ TEM study. We have also observed such dislocations
around the end of a Sic Whisker in a Sic Whisker/ 2124 Al composite (Fig. 1). Mura and Taya [2] have recently studied analytically the stress field in and around a short fiber embedded in the metal matrix by replacing the above mismatch strain by the Somigliana surface dislocations [3]. These surface dislocations, however, are more likely to be relaxed by punching. Punching of arrays of prismatic dislocation loops adhered to the surface of a spherical particle was studied by Tanaka et al. [4] with the aim of
understanding the breakdown of the linear workhardening of a dispersion hardening alloy [5]. The model used by Tanaka et al. is based on Eshelby’s
equivalent inclusion method [6]. In this paper we will focus on the punching distance of the prismatic dislocation loops that simulate the thermal expansion mismatch strain between a short fiber and the metal matrix, and also on the stress field in and around a short fiber. To this end, a Sic Whisker/Al composite will be used as a target short fiber MMC. For the difference of CTEs between a Sic whisker and an aluminum matrix is quite large. We will state our analytical model and the formulation to compute the punching distance and the resulting stress field in Section 2. Then the results of the punching distance and the stress field are given as well as the discussion in Section 3. Finally the conclusion is stated in Section 4. 155
156
TAYA and MORI:
PUNCHED
DISLOCATIONS
IN METAL MATRIX COMPOSITES
of the short fiber, respectively. The domain of punched-out dislocations is also simulated by another prolate spheroid (a,) with the major axis being c’ while the minor axis is set equal to the diameter of the short fiber a. In the punched-out (or relaxed) state, two different types of dislocation loops exist, one in the boundary of domain zf, and the other in that of a*. It is assumed in our model that the above two types of dislocation loops are smeared out in each domain, thus they are replaced by the following eigenstrains [I (1)
Fig. 1. TEM photo showing dislocations around a fiber-end in Sic/AI composite.
(2) 2. ANALYTICAL MODEL AND FORMULATION
The surface dislocations (or mismatch strain) due to the difference in CTE between a short fiber and the matrix are simulated by arrays of prismatic dislocation loops [Fig. 2(a)] which will be punched-out [Fig. 2(b)]. Figure 2 represents the case of a Sic whisker embedded in the matrix metal (aluminum) which is cooled down from some reference temperature by bT (to). The CTEs of the SiC whisker and aluminum are denoted by (orand s, respectively, and of course ur < c(,. A short fiber is modelled in Fig. 2 by a prolate spheroid (0,) with the major and minor axes being c and a which correspond to the length and diameter
where a*=(a,-h)bZ’.
(3)
Let the stiffness tensors of the matrix and fiber be Cij, and CbEI, respectively. For simplicity we assume that both the matrix and fiber are isotropic in stiffness and thermal expansion. It is noted that the anisotropy of the fiber can also be solved by the present model. With equations (1) and (21, the problem of Fig. 2 is changed to that of an inhomogeneous inclusion Q, with Cikl and e? + ep surrounded by an inclusion with C,, and ep which in turn is embedded in an infinite matrix.
L x3
(0)
XI
(bf
Fig. 2. Arrays of prismatic dislocation loops that were caused by the mismatch of CTE between fiber and
matrix (a) wiil be punched-out (bl.
TAYA and MORI:
2.1. Punching of prismatic dislocation loops The criterion to determine the punching distance c’ is given by [4] --=-au aw (4)
acf
ad
where U is the total potential energy of Fig. 2(b) and W is the total work done by the motion of the dislocation loops over the punching distance c’. Hence, the remaining task is to compute U and W. It is assumed in the following formulation that the volume fraction of fiber f, is reasonably small to facilitate the computation, hence the interaction between fibers is neglected. We note in passing that the case of a finite volume fraction of fiber can be formulated by the present model though the actual computation will become more vigorous. 2.1.1. Total potential energy U. The total potential energy U of a short fiber MMC will be obtained here by use of Eshelby’s equivalent inclusion method. Since the composite is not subjected to any external force, the total potential energy U is equal to the total elastic strain energy of Fig. 2. Therefore a,,(eij -e;)dV (5) s Ll where oij, eij and et are components of the internal stress, total strain and any non-elastic strain (eigenstrain), respectively, D denotes the volume of the composite, and dV is the volume element. Before simplifying equation (5) further, we will obtain first the formula to compute the internal stress field. Let us consider an inclusion Q, with C,,, and e? which is embedded in the matrix without fiber a,. Then the resulting internal stress G’, is given by
panying the change in the elastic constant of a, from C,,, to CLk, and the introduction of eit in R,, and is a fictitious eigenstrain and related to e:, as eklI** e:, = S’klrnnemnI** . of, can be easily obtained (9) as uh =
- e::)
(6)
where e:, = $,,ez
(7)
where ei, and Si,,,,, are the internal strain caused by this inclusion a2 and Eshelby’s tensor related to @, and the superscript 2 is not “square”, but denotes the quantity related to Q,. The repeated indices are to be summed over 1, 2 and 3. This rule will be used throughout this paper unless otherwise noted. For e$ given by equation (2) we can compute the stress field 0,: within R, easily. Now we introduce an inhomogeneous inclusion Iz, with CL,, and ef: inside a,. Let the stress field caused by this introduction of R, be denoted by e,:. Then the total stress 0,: + 0,: in a, is given by g,f + 0: = C,:,, (Szmez - e$ + ei, - ek:) = C,,,, (Gm,e$
-e$+e:,-e::*)
(8)
where equations (6) and (7) as well as Eshelby’s equivalent inclusion method and Mura’s reformulation for a combined equivalent condition [7] were used, e:, is the additional total strain in fi, accom-
(9)
from equations
C~lkdSh&!$*
-
ei?*)
(8) and
(10)
after solving for eL:* in equation (8). Now we turn to the total potential energy U and will show below that U defined by equation (5) can be reduced to the formula expressed in terms of given eigenstrains ef;r and ey. Since j,a,,e,jdV = 0 [7], and e$+ and e2,*are defined only in domains Q, - R, and Q, respectively, U is reduced to U = -i
sD
cr,,ezdV
1 =--s 2 02- n,
-k
U =i
c’, = G,&:,
157
PUNCHED DISLOCATIONS IN METAL MATRIX COMPOSITES
= --
I
1 2
(or, + oy)e’,* dV
9,‘“:’ +oi)(ef:+e2,*)dV o,!, ey dV - i(Q, - R,)aZ, e2,*
where a, - !,2, and Q, denote the volume of the domains Q, - R, and Sz,, respectively. By the Tanaka-Mori theory [8], the volume integral [the first term in equation (1 l)] can be expressed by
I o2-a1
of, ey dV = R,Cijk,(S:,m, - SL,,,,,)e!,t*.e2,*. (12)
After substituting equation (12) into (11) and using equation (9) we obtain U = - if ,Ci,k,(S:k,e::* -tf201:
e5,
-fflC,,,(S:,,.e~* - if,a;
- e$*)ef:
ey
-eLT*)ef:
(13)
where U is expressed per unit volume, f, and I; are the volume fractions of 0, and R,, respectively and they are related by fz = (c’/c) f, In equation (13) only unknown is ef:* which will be solved from equation (8) 2.1.2. Work done by the motion of prismatic dislocation loops W. The evaluation of W is basically the same as that used by Tanaka et al. [4] and it is described briefly below. Denoting the coordinate along the x,-axis [Fig. 2(b)] by Z, the average work w by the motion of a dislocation loop at x1 = Z during the punching can be expressed as w=2na(l-grZt--1)bk
TAYA and MORI:
158
PUNCHED
DISLOCATIONS
where k and b are the friction stress and Burgers’ vector of a dislocation. Then the total work done by the motion of 2N dislocation loops at Z, W, becomes t 2xa ( 1-L $)‘“Znk n-1
W=2
- ,)bk
IN METAL MATRIX COMPOSITES
where Izand p are Lame’s constants of the matrix and S, the kronecker delta. Hence as/2~ is given by
(14)
a:r -=-
42
2P
2P
where Z,, is the location of the nth dislocation loop. By taking the limiting sequence, N + co and keeping Nb = CD! *, we arrive at
I;a*
,=2/;2lra(I-$rZ(;-+*dZ
=-
4xkac2 3
c’ ---I ( c
+s:,,,- 1
and the other components
a*.
of stress vanish. In the
(15) above derivation, equation (2) was used. v in equa-
)
W can be expressed per unit volume as W=p,f,k
(19)
“‘-1 ( c
ci* )
(16)
where & = c/a which can be called as fiber aspect ratio. 2.1.3. Calculation of c’. After having solved for ejj** from equation (8) and using equations (6) and (7), U [equation (t3)] can be expressed in terms of known quantities, Cijke, CLk,, e,$@,et?, f,, 8, and c’. Then we substitute U so obtained and W given by equation (16) into equation (4) to arrive at (17) where ~1is the shear modulus of the matrix and H(r) is a function of t( = c//c) and other parameters related to the elastic constants of the matrix and fiber and the fiber aspect ratio. The detailed expression of H(5) is given in Appendix A. The punching distance c’ (or c) is solved numerically from equation (17) for given k/p and AT, the results of which will be discussed in Section 3.
tion (19) is Poisson’s ratio of the matrix. The explicit expression of the Eshelby’s tensor S,,, is given in Appendix B where aspect ratio fl should be replaced by b2 = c’/a for the above equations. Similarly at is obtained from equation (10) as
4 42 -=-=
I
$g
2P 3.4
ai3 -=
2P i
&(SII”
(Sf*,,
+
+
G22
s1,2*
+
+
54311
s:,,,
-
-
1)
1)
1_V2”)w,33+S:333+-
+
1)
(1
S&
1 e$* I
(20)
2.2. Stress field in and around a short fiber
The formula to compute the stress field in and around a short fiber has been given in Section 2.1 for the punched-out state (relaxed state). Here we will obtain the formula to compute the stress field for the non-relaxed state, i.e. c = c’ for comparison of the stress field between the relaxed and non-relaxed states. The locations of the state of stress that will be computed are shown by A, B and C in Fig. 3(a) for the relaxed state and in Fig. 3(b) for the non-relaxed state, since the computation of the stress at these points are straight forward. For completeness, we will first summarize the formula for the relaxed state and then derive the formula for the non-relaxed state below. 2.2.1. Stem field for relaxed state. The stress field within R, [shaded by vertical lines in Fig. 3(a)] r_r before a, is introduced, can be obtained from equation (6) and (7) and C,,, given by Cijk/=
AOljskl
+
P(6ik6j,+
dj16kj)
(18)
(al (b) Fig. 3. Stresses are calculated at points A, B and C for the relaxed state (a) and the non-relaxed state (b).
TAYA and MORI:
and other components of stress vanish, where e::* and e::* are solved from equation (8), and S’,,, can be computed from Appendix B with @= & = c/a. The stress field within the fiber (i.e. at point A) o$ is uniform and given by a$=a$+oZ,.
(21)
Next we will obtain the stress just outside &, i.e. at points B and C [Fig. 3(a)]. The stress just outside S&can be obtained by use of jump condition [7,9, lo] and the explicit expressions for the stresses at B (0:) and C (0:) are given by
1 + (1 - v)
(22) where equation (2) was used. 22.2. Stress field for the non-relaxed state. For the non-relaxed state, R2 becomes identical to Q, [Fig. 3(b)], leading to the simplification of the formula to compute the stress field, (6,). Eshelby’s equivalent inclusion method in this case yields for domain G, Gij = C$,(Si,meg
Several values of fiber aspect ratios and AT are used to examine their effects on the punching distance c’, the results of which will be discussed in the following subsection (3.1). 3.1. Punching distance C’ First the effect of AT on c’ was examined and the results are shown in Fig. 4. It is obvious from Fig. 4 that the punching distance c’ increases with (AT1 and punching becomes impossible for the temperature change less than about -55°C. It should be noted, however, that the present model is independent of actual temperature, thus for example it cannot account for creep behavior of the matrix and fiber. Next we have computed c’ for several values of fiber aspect ratios, I/d = 1.5-50, and plotted the results of c’ in Fig. 5. It follows from Fig. 5 that the larger the value of l/d is, the smaller the value of c’/c becomes and it converges to 1 at about l/d = 27. Namely for given parameters, punching cannot take place for the fiber with l/d > 27.
1 2.0
t
SiClAP
Composite
I/d =5 12. = 0.001 I*
C’
c 1.5 -
- e$ - ei:)
= Cij,,(S:Jmeg - ek:*)
(23)
where eg* is a fictitious eigenstrain which is constant in a, and vanishes outside R,. The non-vanishing stress components of Cs, within the fiber [at point A in Fig. 3(b)] are r$‘, = 6ig, and (?$. It should be noted here that equation (8) for c = c’ is reduced to equation (23) since S&, = S&,. This also confirms the validity of the model for the relaxed state. The stresses just outside the fiber at points B and C [Fig. 3(b)] are computed by the formula similar to equation (22). 3.
159
PUNCHED DISLOCATIONS IN METAL MATRIX COMPOSITES
1.0
1 - 200
-100
-300
4T (“Cl
Fig. 4. c/c’ vs AT for Sic/Al composite with l/d = 5 and K/P = 0.001.
SIC/AL
Composite
AT=-200’C,
;=O.OOl
NUMERICAL RESULTS AND DISCUSSION
On the basis of the formulation developed in the previous section, we have computed the punching distance c’ and the stress field. The thermomechanical and geometrical parameters used in this computation are given in Table 1.
Table 1. The thermo-mechanical
and geometrical parameters
Parameters
Unit
Al matrix
SIC whisker
Young’s modulus Poisson’s ratio CTE Fiber aspect ratio
GPa
47.5 0.3 23.6
427 0.17 4.3 5
x 10-6/Y
C’ c
IO
20 Fiber Aspect
Fig. S. c/c’ vs I/d for Sic/Al composite and ~/,u = 0.001.
30 Roth,
$
with AT = -200°C
160
TAYA and MORI:
PUNCHED DISLOCATIONS IN METAL MATRIX COMPOSITES
The above results can be easily understood by considering a simplified situation, Cfikl = C,, and e$ = (txr- a,)AT with other components of e$ being zero. Then, after punching, the total potential energy U is given by
10.0 Si C/AI
Composite
I l/d=5
U = &a2c2(e$)2/ct.
AT= -200°C C’
The generalized punching
c
force fp is defined as
fp=-$. fr takes the largest value of 5.0
at the beginning of punching (c’ = c) and decreases as punching proceeds. The generalized retarding force, fR,against punching is defined as
which is given by equation (15) as 4nkac
1.0
fa=T4.
I
(27)
I
10‘3
I
I
10-4
10-5
I
10-6
K
From equation
cc
(26) and (27), and for c $ a
& < cfPLX 4 =(1fa ‘-x-
a
00 P
;
-=I.
(28)
This is the physical reason for the decrease in the punching distance for a long fiber. Namely, the potential energy decrease by punching is less than the energy dissipation required for punching when a fiber is sufficiently large. From the above result, it appears that one must consider a different mechanism of stress relaxation for a long fiber composite. One obvious mechanism that one can consider is the uniform plastic deformation throughout the matrix. Then, the average
l/d=5,
f=
0.001,
Fig. 6. c/c’ vs K/F for SiC/AI composite with I/d = 5 and AT = -200°C.
stress in the matrix is given by - f,aij,where crij is the stress inside a fiber when only a single fiber is present in an infinitely extended medium. Even when eigenstrains are purely dilatational as in the case of thermal strains, (ujj 1> 1u1I I = 1q2 I for a fiber. Thus, the stress state in the matrix favors the occurrence of plastic deformation. It is apparent that this plastic deformation relaxes the internal stress induced by a thermal expansion mismatch. This type of stress
AT=-200DC
C’
c 1.5 -
I.01 0
/-r 5
15
IO
20
Ef i,
Fig. 7. c/c’ vs Ef/E,,, for Sic/Al composite with I/d = 5, K/P = 0.001 and AT = -200°C.
TAYA and MORI:
PUNCHED DISLOCATIONS IN METAL MATRIX COMPOSITES Table 2. Stresses at points
5 2P
x
lo-*
A
For 11 -0.225 -0.225 0.0035
B C
relaxed state 22 -0.225 0.198 osm35
3.2.Stress field The stress field is computed for the relaxed (punched-out) and non-relaxed states by use of the parameters given in Table 1. For these parameters, the value of c’/c was computed from equation (17), i.e. c’jc = 1.573. The state of stress is focused on several points, A, B, and C for the relaxed state [Fig. 3(a)] and for the non-relaxed state [Fig. 3(b)]. The results of the state of stress at these points are given in Table 2 where all the stresses are normalized
A, B and C (see Fig. 3) 11
33 -0.84 0.069 -0.84
relaxation is particularly favored for a long fiber composite. This is because 1c33- Q,,) increases as the aspect ratio of fibers (aligned) increases, and it is proportional to the Bow stress of the matrix yielded. The other parameters that affect the punching distance are the ratio of friction stress to the shear modulus of the matrix, k/p and the fiber to matrix stiffness ratio, E,/E,,,. The effect of parameter k/p on c’ is shown in Fig. 6. Figure 6 indicates that the larger the value of k/p, the smaller that of C’/C becomes. For a practical metal matrix composite system, 1 < E,/E,,, < 20, we have studied the effect of EJE,,, with the above range of c’/c and the results are shown in Fig. 7. It follows from Fig. 7 that c’/c is sensitive to EJE,,, for the smaller values of E,IE,,,, but it becomes much less sensitive to EJE,,, for the large values of E,/E,.
For non-relaxed 22
-0.245 -0.245 -0.149
state 33 -1.184 0.119 -0.184
-0.245 0.213 -0.149
Acknowledgements-This work was supported partially by a Grant from Honda R&D Company to the University of Delaware. We are thankful to Mr W. G. Patterson of University of Delaware for his technical assistance in taking TEM pictures and also to Professor R. J. Arsenault for his comments on TEM pictures.
REFERENCES I. M. Vogelsang, R. J. Arsenault and R. M. Fisher. Metall. Trans. In press. 2. T. Mura and M. Taya, Recent Advances in Composites in the United States and Japan (edited by J. R. Vinson and M. Taya), ASTM, STP864, p. 209 (1985). 3. R. J. Asaro, Znt. J. Eng. Sci. 13, 271 (1975). 4. K. Tanaka, K. Narita and T. Mori, Acta metall. 20,297 (1972). 5. M. F. Ashby, Phil. Mag. 14, 1157 (1966). 6. J. D. Eshelby, Proc. R. Sot. A241, 376 (1957). 7. T. Mura, Micromechanics ofDefects in Soli&. Martinus Nijhoff, The Hague (1982). 8. K. Tanaka and T. Mori, J. Elasticity 2, 199 (1972). 9. L. J. Walpole, Proc. R. Sot. A252, 561 (1959). 10. Y. Takao and M. Taya, J. appf. Mech. 52, 806 (1985). 11. J. D. Eshelby, Proc. R. Sot. A252, 561 (1959).
12. Y. Mikata, Ph.D dissertation, Univ. of Delaware (1984).
APPENDIX H(5) = - $
by 2~ The values of the stresses at point C for the relaxed state are approximate. For the stresses at C were computed from ofi and the jump condition and the effect of of, was neglected due to the fact that the punching distance c’ is considered to be reasonably large and the effect of crf, on the stress field outside R, decreases as O(l/]c’-cl*) [ll, 121. It is obvious from Table 2 that the magnitudes of the stresses for the relaxed state are always smaller than those for the non-relaxed state. 4.
+
0
8‘42 ag
A: H(c)
{A,(D,& - D&z) + A,@‘#,, -D,&,)]
CD241
-
W2,)
CONCLUSION
The punching of prismatic dislocation loops caused by the thermal expansion mismatch between the matrix and short fiber in a short fiber MMC was studied by use of Eshelby’s model. It was found in this study that the larger the values of 1AhT] and k/p, the larger the punching distance c’ becomes, but it decreases with the increase in the fiber aspect ratio. The stress fields for the relaxed (punched-out) and non-relaxed (non punched-out) states were also computed, indicating that the relaxed state gives rise to the stress field of smaller magnitude compared with the non-relaxed state.
161
where
A,=--
1 1 - 2v
s:,23 + &
(G,,
- 1)
162
TAYA and MORI:
PUNCHED
DISLOCATIONS
_(X+F)
E
II -
E
I2
-(Sl,ll+sfn2)+~S:311+~
by 3 B2 S 11”= S,rr -_ - q1 _ v) (82 _ 1)
P
pa
x
P
Y
%a+-S:333+- I-2v 2P
1 821
=
i
cc,,
B22
=
2P
+
~I,**
+
%,I,)
x
+2
i Wll33
IN METAL MATRIX COMPOSITES
+
G,,,)
+
is:,,,
E S2 I 2v 3311 1-2v 0 cc l-v +
-
I-2v
Q,= ~(St,,,+S1,22+S:l,,-')+SI,,,+St,*2-l Q2 = &,r,+++,-1)
2233=
--
2(1 -v)(/32-
S 3311 -S -
3322 =
-
-
1)
W) 1
and where x =A’--1. ji =/l’-/1.
1 b2
S 1133- -S
1
(A3)
In the above equations, L and p are Lame’s constants of the matrix, If and pf are Lame’s constants of the fiber, v is Poisson’s ratio of the matrix, S& and St,, are Eshelby’s tensors related to a, and f&, respectively and they are given explicitly in Appendix B, and r = c’jc. APPENDIX B: ESHELBY’S TENSOR S,j,, The Eshelby’s tensors that are used in this study are given
+2(1-v)
2(1 -v)
1 ‘-2v+m}
3 * - 2v + 2(/32- 1) g
where
g
=&MB’- I)‘“-~s’+bj
and where /I is the ratio of the major axis to the minor axis of a prolate spheroid with the major axis taken along the x3-axis. Hence for fZ,, p, = c/u and for &., /I2= c’/a should be used.