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Effects of shape and volume fraction of second phase on stress states in two-phase materials
Materials Science and Engineering A285 (2000) 144 – 150 www.elsevier.com/locate/msea
Effects of shape and volume fraction of second phase on stress states in two-phase materials Masaharu Kato, Toshiyuki Fujii, Susumu Onaka * Department of Inno6ati6e and Engineered Materials, Tokyo Institute of Technology, 4259 Nagatsuta, Yokohama 226 -8502, Japan
Abstract The effects of shape and volume fraction of a second phase on stress states and deformation behavior of two-phase materials are discussed. The second phase is treated as inhomogeneous spheroidal inclusions embedded in a matrix. Analytical expressions to describe the stress states in elastically and plastically deformed two-phase materials are obtained with the Eshelby method and the Mori–Tanaka concept of the ‘average stress’. The variation of the stress states with the change in the aspect ratio of the spheroidal second-phase is shown for various volume fractions. Considering that the second phase is also plastically deformable, the overall deformation behavior of the two-phase materials is discussed with the results obtained by the evaluation of the stress and strain distributions in the materials. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Two-phase material; Deformation behavior; Eshelby’s method; Mori and Tanaka’s theorem; Average stress; Inclusion problem
1. Introduction A plastically strong-second phase in a matrix often causes strengthening of materials. For designing twophase materials with desirable mechanical properties, theoretical considerations on stress states in the materials are needed to predict their deformation behavior. Within the framework of continuum mechanics, Tanaka and Mori [1] and Mori and Tanaka [2] have considered the stress states and presented theories, which satisfactorily explain plastic deformation behavior of the two-phase material such as dispersion-hardened alloys. Although they have considered various variables to characterize the two-phase materials, some simplifications included in previous calculations are better to be excluded for the prediction of deformation behavior of actual two-phase material [1,2]. In the present study, we deal with two subjects that have been simplified in the previous calculations [1,2]. One is the shape of the second phase. Only three special shapes, i.e. sphere, disk and needle, have been previously treated as possible shapes of the second phase. Instead of this simplification, we approximate the second-phase shape to general spheroids with various aspect ratios. * Corresponding author. Fax: +81-45-9245173. E-mail address: [email protected] (S. Onaka)
This is because consideration on intermediate shapes between the special shapes reveals important roles of the shape effects which can not be predicted from discussion on the special shape [3–5]. The other is the volume fraction of the second phase. Even if the second phase and matrix are elastically isotropic, the effects of volume fraction on the stress distribution in the twophase materials have not been shown by analytical expressions when the second phase has different elastic moduli from those of the matrix. This may be due to that complicated algebraic calculations are needed when we include interactions between elastically inhomogeneous inclusions. Relationships to describe the effects of the volume fraction derived in the present study are suitable for various analytical discussions on the stress states in two-phase materials. In the present study, we first summarize results given by the Eshelby method [6,7] and the Mori–Tanaka concept of the ‘average stress’ [2]. Then we derive relationships convenient for the evaluation of stress states in the two-phase materials when we include the above extensions. Although the second phase has often been assumed to be plastically non-deformable in theoretical considerations, metal–metal composites sometimes accompany plastic deformation of the second phase itself. In the present study, we finally discuss the overall deformation behavior of the two-phase materi-
0921-5093/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 0 6 3 9 - 0
M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
als including the above extensions and the effect of possible plastic deformation of the second phase.
2. The Eshelby method and the Mori – Tanaka concept of the ‘average stress’
145
solved for some special cases such as uniaxial applied stress and a purely dilatational eigenstrain [9,10]. Conducting rather tedious calculations, we can analytically obtain Gijkl and Jijkl. Two assumptions involved in these calculations are (i) isotropic C *ijkl and Cijkl ; and (ii) a spheroidal shape of V. When V is described on the x1 –x2 –x3 orthogonal coordinate system as
2.1. Uniform stress in an isolated inhomogeneous inclusion
x 21 + x 22 x 23 + 2 5 1, a2 c
Let us consider an isolated ellipsoidal inhomogeneity V with elastic moduli C *ijkl embedded in an infinitely extended matrix M with elastic moduli Cijkl. When C *ijkl is different from Cijkl the remote external stress s O ij applied to the two-phase material M+ V is disturbed. Eshelby has shown that the stress s A ij in V is obtained by solving the following equations [6,7]:
the calculated explicit expressions for Gijkl and Jijkl are shown in Appendix A as a function of the aspect ratio a= c/a of V and isotropic C *ijkl and Cijkl.
O sA ij =Cijkl (e kl + Skimno* mn* − o* kl*)
=C*ijkl (e O kl + Sklmno* mn*),
(1)
and O sO ij =Cijkle kl,
(2)
(7)
2.2. A6erage stresses in inhomogeneous inclusions When a material under s O ij contains many inhomogeneous inclusions V, the average stress in the matrix sij M and that in the inclusions sij V are evaluated using Mori and Tanaka’s concept of the ‘average stress’ [2] together with the equivalent inclusion idea of Eshelby [6,7]. When the volume fraction of V is denoted by Vf, sij M and sij V should satisfy Vfsij V + (1− Vf)sij = s O ij .
(9)
the equivalent Here, Sijkl is the Eshelby tensor and o ** ij eigenstrain of the fictitious homogeneous inclusion which reproduces the uniform stress s A kl in V. For simplicity, we introduce a forth-order tensor Gijkl which A relates s O kl to s ij as
When the uniform eigenstrain o is generated in the inclusions with an identical shape and orientation, sij V is given by solving the following simultaneous equations [11]:
O sA ij =Gijkls kl.
sij V = Cijkl (e D kl + Sklmno* mn** − o* kl**)
(3)
P ij
Gijkl depends on the shape of V, Cijkl and C *ijkl and satisfies Gijkl =Gjikl =Gijlk. Instead of Gijkl in Eq. (3), Wu has considered a tensor, which relates e O ij to the uniform elastic strain in V for the evaluation of elastic moduli of composite materials [8]. When V has a uniform eigenstrain o Pij and no external stress is applied to M+V, the uniform stress s Bij in V is also given by Eshelby [6,7]:
Using Gijkl and Jijkl in Section 2.1, Eqs. (9)–(11) are rewritten as
s Bij =Cijkl (Sklmno*mn −o*kl ) = C*ijkl (Sklmno*mn − o Pkl),
sij V = Gijkl skl M + Jijklo Pkl,
(4)
where o *ij is the equivalent eigenstrain of the fictitious homogeneous inclusion which reproduces s Bij in V. Introducing another forth-order tensor Jijkl, the relationship between s Bij and o Pij shown by Eq. (4) is rewritten as s =J o . B ij
P ijkl kl
(5)
Here, Jijkl depends on the shape of V, Cijkl and C *ijkl and satisfies Jijkl =Jklij =Jjikl =Jijlk. When s O ij is applied to M containing V with o Pij , the total stress sij in V B becomes the sum of s A ij and s ij . Hence, from Eqs. (3) and (5), sij is written as P sij = Gijkls O kl + Jijklo kl.
(6)
When C *ijkl and Cijkl are isotropic and the shape of V B is spheroidal, s A ij and s ij in Eqs. (1) and (4) have been
P = C*ijkl (e D kl + Sklmno* mn** − o kl),
(10)
where o *** is the eigenstrain in the equivalent inclusion ij and e D is the average elastic strain in the matrix which ij is related to sij M by sij M = Cijkle D kl.
(11)
(12)
and Vf(Gijkl skl M + Jijklo Pkl)+ (1− Vf)sij M = s O ij .
(13)
These equations can be solved analytically to obtain sij V and sij M using Gijkl and Jijkl in Appendix A. The resultant sij V and sij M are expressed as a P function of s O ij , o ij ,Vf, Cijkl, C * ijkl and the aspect ratio of a of V. 3. Stress states in plastically deformed two-phase material We consider plastic deformation in a two-phase material containing many spheroidal inhomogeneities V
M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
146
with an identical shape and orientation. The uniaxial O stress s O is considered as an external stress ap33 = s plied along the axis of rotation symmetry of the spheroidal V. We evaluate the stress distribution in the two-phase material when plastic deformation occurs both in the matrix M and in the inhomogeneities V. When uniform plastic strains in M and V are expressed by o P(M) and o P(V) , from symmetry and volume conij ij and o P(V) can be stancy, nonzero components of o P(M) ij ij rewritten as 1 o P(M) =o P(M) = − o P(M), 11 22 2
o P(M) =o P(M), 33 o
P(V) 33
=o
P(V)
,
o
P(V) 11
=o
P(V) 22
1 = − o P(V). 2
(14a)
1 o P11 = o P22 = Do and other o Pij =0. 2 (16)
The stresses in M and V which govern their plastic deformation are s33M − s11M and s33V −s11V, respectively, since sij M and sij V becomes axisymmetrical around the x3 axis. The use of the average stresses as the yield criteria has been confirmed from energy consideration [12]. The average stresses s33M − s11M and s33V − s11V are written as a function of Do as M s33M − s11M = s M 0 + g Do,
(17)
s33V − s11V = s + g Do,
(18)
V 0
(14b)
When the difference of the plastic strains between M and V is written as Do =o P(M) −o P(V),
o P33 = − Do,
(15)
the average stresses sij M in M and sij V in V are evaluated with Do. That is, sij M and sij V after the occurrence of the plastic strains o P(M) and o P(V) are ij ij reproduced by considering the generation of the uniform eigenstrain o Pij in V as much as
O Fig. 1. The variation of the normalized average stress s V with the 0 /s change in the aspect ratio a for (a) the elasticity harder V in M with m*= 3m; and (b) the softer V in M with m*= 0.5m. The Poisson ratios n* for V and n for M are assumed to be n*= n =1/3.
V
where O V (1−Vf)s M 0 + Vfs 0 = s ,
(19)
and (1− Vf)g M + Vfg V = 0
(20)
V are satisfied. The stresses s M 0 and s 0 in Eqs. (17) and (18) physically mean the average stresses in the elastically deformed two-phase material. We know from Eq. M (19) that an increase in s V 0 causes a decrease in s 0 . On the other hand, g M and g V in Eqs. (17) and (18) show the effects of o P(M) and o P(V) on the average stresses. ij ij From Eq. (20) note that g V is positive and g M is negative. Since g M is negative, s33M − s11M decreases with increasing Do under constant s O. This explains work hardening in composites containing plastically non-deformable phases [1,2]. Using Eqs. (12), (13) and (16) and Gijkl and Jijkl in M V V Appendix A, explicit expression for s V 0 , g , s 0 and g V in Eqs. (17) and (18) are obtained. Those for s 0 and g V are derived in Appendix B. From Eqs. (19) and (20), M V sM are obtained from s V 0 and g 0 and g , respectively. In Fig. 1a and b, the variation of the normalized O average stress s V with the change in the aspect ratio 0 /s a are indicated for various values of the volume fraction Vf. To construct Fig. 1a and b, the Poisson ratios n* for V and n for M are assumed to be n*=n= 1/3. Fig. 1a and b, respectively, show the results for the elastically harder V in M with m*= 3m and softer V in M with m*= 0.5m, where m* is the shear modulus of V O V M and m that of M. We have s M 0 B s B s 0 and s 0 \ O V s \ s 0 for the elastically harder and softer V, respecO tively. As shown in Fig. 1a and b, s V depends on 0 /s the aspect ratio a more strongly for smaller Vf. When O Vf is fixed, s V monotonically increases or decrease 0 /s with increasing a from unity to . The normalized O average stress s V becomes the largest for elastically 0 /s harder V and the smallest for elastically softer V when their shape is needle-like (a ). However, in the O range of aB1, s V does not change monotonically 0 /s but has an absolute maximum or minimum at around a=0.2.
M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
147
havior of the two-phase material with the variations of s33V − s11V and s33M − s11M. When V is elastically harder than M, the variations of s33M − s11M and s33V − s11V with Do can be represented V as shown in Fig. 4. s M Y and s Y are the yield stresses of M and V. When we assume hypothetically that the material is elastically deformed (o P(M) = o P(V) =0 and Do = 0) under s O, Fig. 4 shows s33M − s11M \s M Y and s33V − s11V B s V Y. These mean that initial plastic deformation occurs in M and Do increases while o P(V) = 0. When Do reaches Do1 in Fig. 4, V starts to deform plastically since s33V − s11V = s V Y is sa-
Fig. 2. The variation of g V/m with the change in the aspect ratio a for (a) the elastically harder V in M with m*= 3m; and (b) the softer V in M with m*= 0.5m. The Poisson ratios n* for V and n for M are assumed to be n* = n =1/3.
On the assumption of n* =n =1/3, the variation of g V/m for m*=3m and m* =0.5m are, respectively, shown in Fig. 2a and b. Comparing Fig. 2a with Fig. 1a, we find that the variation of g V/m are similar to O those of s V when V is elastically harder than M. In 0 /s O Fig. 3 a and b, the contour lines of s V and g V/m for 0 /s Vf = 0.25 are shown as a function of the ratio of elastic constants f= m*/m and a. As shown in Fig. 3a and b, O and g V/m becomes stronger the a dependence of s V 0 /s when V becomes elastically harder than M.
4. Deformation behavior of a two-phase material Let us suppose that elastically harder V with a certain value of Vf is embedded in M. For this case, we find from Figs. 1 – 3 that s33V −s11V becomes the smallest and hence s33M −s11M becomes the largest when the shape of V is an oblate spheroid with a : 0.2. If V is plastically non-deformable, this means that the back stress due to this V is the smallest and the effect of V to suppress plastic deformation of M becomes a minimum. On the other hand, when the elastically harder V is needle-like (a ), plastic deformation of M is most effectively suppressed. In this section, considering that V is also plastically deformable, we generally discuss the deformation be-
O V Fig. 3. The contour lines of (a) s V 0 /s ; and (b) g /m for Vf = 0.25 as a function of the ratio of elastic constants f =m*/m and the aspect ratio a. The Poisson ratios n* for V and n for M are assumed to be n* = n =1/3.
Fig. 4. The average stresses in M and V as a function of Do when the uniaxial external stress s O is applied to the two-phase material M +V.
148
M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
tisfied. However, this plastic deformation in V increases o P(V) and causes the decrease in Do. Hence, M and V thereafter continue to deform keeping the difference of the plastic strain as much as Do1. The continuous plastic deformation of the two-phase material occurs when M and V are perfectly plastic solids. If we include effects of inherent work hardening of M and V, the plastic deformation stops after the occurrence of a certain amount of plastic deformation. The above is a conclusion of the example shown in Fig. 4 where s33M − s11M \s M is satisfied for Do =Do1. If we have Y V s33M −s11M Bs M Y when s33V −s11V =s Y instead of the example shown in Fig. 4, the plastic deformation of M does not cause that in V and the deformation of the two-phase material stops under s O. V The yield stress s M Y and s Y and the variations of s33M −s11M and s33V −s11V with the change in Do determine the deformation behavior of the two phase material. Even if the combination of M and V is fixed, the deformation behavior of the two-phase material is affected by the shape and volume fraction of the second phase through the a and Vf dependencies of s M 0 , M V sV , g and g . In the present study, we did not 0 consider thermal residual stress of relaxation due to local plastic deformation in the two-phase materials. These effects on the deformation behavior have been treated with the Eshelby method in previous studies [13–15]. Mori and Wakashima have treated a case where only creep deformation of M occurs and discussed an equilibrium state of the two-phase material by energy consideration [16]. Their assumptions are that o P(V) = 0 and o P(M) changes so as to minimize the mechanical free energy of the two-phase material. The equilibrium state discussed by them is shown by the point A in Fig. 4. The equilibrium plastic strain of M as much as Do = − M sM and the resultant hydrostatic stress state in M 0 /g are the same as those obtained by the energy consideration [2,16].
5. Conclusions The effects of shape and volume fraction of a second phase on the stress states in the elastically and plastically deformed two-phase materials have been discussed. The second phase has been treated as inhomogeneous spheroidal inclusions embedded in a matrix. Using the Eshelby method and the Mori– Tanaka concept of the ‘average stress’, the expressions to describe the stress states in the deformed two-phase materials have derived in analytical form. The secondphase shape and volume-fraction dependencies of the stress and strain distributions in the deformed twophase materials have been emphasized. Considering that the second phase is also plastically deformable, the
overall deformation behavior of the two-phase materials has been discussed.
Acknowledgements This work has been partially supported by a Grantin-Aid for Scientific research (C) from the Ministry of Education, Science and Culture under Grant No. 1065687 and the Proposal-Based New Industry Creative Type Technology R&D Promotion Program from the New Energy and Industrial Technology Development Organization (NEDO) of Japan.
Appendix A. Expressions for the components of Gijkl and Jijkl For the spheroidal inhomogeneity V: (x 21 +x 22)/a 2 + x /c 2 5 1, non-zero components of Gijkl which relates s with s O ij in Eq. (3) and satisfies Gijkl = Gjikl =Gijlk are given as follows: 2 3 A ij
G1111 = G2222 =
1 m* − (2K*(Q3Dm −m) 2(m − 2DmS1212) 2R + m*(Q2DK −K)),
(A1a)
1 G3333 = − (K*(Q1Dm −m)+ 2m*(Q4DK −K)), R (A1b) G1122 = G2211 = − −
m* 2(m − 2DmS1212)
1 (2K*(Q3Dm −m)+ m*(Q2DK −K)) 2R
,
(A1c)
1 G1133 = G2233 = − (K*(Q1Dm −m)−m*(Q4DK −K)), R (A1d) 1 G3311 = G3322 = − (K*(Q3Dm −m)−m*(Q2DK −K)), R (A1e) G1212 =
m* , 2(m − 2DmS1212)
G2323 = G3131 =
m* , 2(m − 2DmS2323)
(A1f ) (A1g)
where DK = K−K*,
(A2a)
Dm =m− m*,
(A2b)
and
M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
R =(Q1Dm −m)(Q2DK − K)
J1111 = J2222
+2(Q3Dm −m)(Q4DK − K).
(A3)
Here K and m are the bulk and shear moduli of the matrix M, respectively, and K* and m* those for the inhomogeneity V. S1212, S2323 and Q1 to Q4 are components and their combinations of the Eshelby tensor. These are the function of the aspect ratio a and the Poisson ratio n of the matrix M and written as S1212 =
!
"
7−8n S(a) (7 − 8n) − −4(1 − 2n) , 24(1− n) 32p(1 −n) a2 (A4a)
!
=
!
1 (3K*KW + 4m*mU + 12K*Km*(Q4 −Q2)), 3R (A7b)
J1122 = −
mm*(2S1212 − 1) (m− 2DmS1212)
−
5−4n S(a) (5 + 8n) − +4(1 − 2n) , 6(1− n) 8p(1− n) a2 (A4c) 1+n 1 −2n a −1 − S(a) , 3(1 − n) 2p(1 − n) a2 (A4d)
Q3 = − S1133 +S3333
!
"
(2n −1) 5−4n S(a) = − +2(5 − n) , 6(1− n) 8p(1− n) a2
1 (3K*KW − 2m*mU 3R
+3K*Km*(Q4 − Q2)), 2
Q2 = 2S1133 + S3333 =
1 (3K*KW + m*mU − 6K*Km*(Q4 −Q2)) 3R (A7c)
J1133 = J2233 = −
"
(A7a)
J3333 = −
"
Q1 = S1111 +S1122 −2S3311
mm*(2S1212 − 1) 1 − (3K*KW + m*mU (m − 2DmS1212) 3R −6K*Km*(Q4 − Q2)),
1− 2n S(a) (2 − n) + +(1 + n) , S2323 = S3131 = 6(1 −n) 8p(1− n) a2 (A4b)
=
149
J1212 =
mm*(2S1212 − 1) , (m − 2DmS1212)
J2323 = J3131 =
mm*(2S2323 − 1) , (m− 2DmS2323)
(A7d) (A7e) (A7f )
where (A4e)
W=(Q1Dm −m)(Q2 − 1)+ 2(Q3Dm −m)(Q4 −1)
(A8)
U= (Q1 − 1)(Q2DK −K)+ 2(Q3 − 1)(Q4DK −K).
Q4 = S1111 +S1122 +S3311
(A9)
1+n 1 −2n a −1 + S(a) , 3(1−n) 4p(1− n) a2 2
=
(A4f )
where S(a) is a function of a and n given by
n
2 a2 3a S(a)= p (a 2 +2) − arccos(a) , 3 (1− a 2)2 (1 −a 2)1/2 (a B 1) S(a)=
(A5a)
4 p, 15
S(a)
(a= 1)
(A5b)
n
(B1) (a \ 1)
(A5c)
Among Q1 to Q4 we have the relationship written as (1 −2n)(Q1 − Q3) =(1 + n)(Q4 −Q2), (1+n) , (1−n)
V sV in Eq. (18) are written as 0 and g
{(A1A4 −A2A3)Vf +(2A3m+ 3A4K)(1 −Vf)}s O, {6Km(1− Vf)2 +(2A1m+ 3A4K)Vf(1 −Vf) +(A1A4 −A2A3)V 2f }
3a − 2 ln{a +(a 2 −1)1/2} , (a − 1)1/2
Q2 + 2Q4 =
V in Eq. Appendix B. Explicit expressions for s V 0 and g (18)
sV 0 =
2 a2 (a 2 +2) = p 2 3 (a −1)2
S1212, S2323, Q1 to Q4 and R in the above equations are the same with those in the equations for Gijkl.
gV= (1 − Vf){(A1A6 −A3A5)Vf +3A6K(1 −Vf)}m, {6Km(1 − Vf)2 +(2A1m+ 3A4K)Vf(1 −Vf) +(A1A4 −A2A3)V 2f }
(B2)
(A6a) (A6b)
For the spheroidal inhomogeneity V with uniform o Pij , non zero components of Jijkl which relates s Bij with o Pij in Eq. (5) and satisfies Jijkl =Jklij =Jjikl =Jijlk are written as follows:
where A1 =
3KK* {3m −Dm(Q1 + 2Q3)}, R
(B3a)
A2 =
4mK* Dm(Q3 − Q1), R
(B3b)
M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
150
A3 =
3Km* DK(Q2 −Q4), R
(B3c)
A4 =
2mm* {3K − DK(Q2 +2Q4)}, R
(B3d)
A5 =
12mm*K* (Q3 −Q1), R
(B3e)
A6 =
− 6mm* {(Q1 −1)(Q2DK − K) R +2(Q3 −1)(Q4DK − K)}.
(B3f )
Here, Q1 to Q4 and R are already given in Appendix A.
References [1] K. Tanaka, T. Mori, Acta Met. 18 (1970) 931. [2] T. Mori, K. Tanaka, Acta Met. 21 (1973) 571.
.
[3] S. Onaka, T. Fujii, M. Kato, Mech. Mater. 20 (1995) 329. [4] T. Fujii, M. Kato, S. Onaka, Scr. Mater. 34 (1996) 1529. [5] S. Onaka, Y. Suzuki, T. Fujii, M. Kato, Scr. Mater. 38 (1998) 783. [6] J.D. Eshelby, Proc. R. Soc. A241 (1957) 376. [7] J.D. Eshelby, Prog. Solid Mech. 2 (1961) 89. [8] T.T. Wu, Int. J. Solids Struct. 2 (1966) 1. [9] R.H. Edwards, J. Appl. Mech. Trans. ASME 18 (1951) 19. [10] M. Shibata, K. Ono, Acta Met. 26 (1978) 921. [11] T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, The Hague, 1987, p. 394. [12] T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, The Hague, 1987, p. 215. [13] P.J. Withers, W.M. Stobbs, O.B. Perdersen, Acta Met. 37 (1989) 3061. [14] A. Rotta, R.E. Bolmaro, Mater. Sci. Eng. A299 (1997) 182. [15] A. Rotta, R.E. Bolmaro, Mater. Sci. Eng. A299 (1997) 192. [16] T. Mori, K. Wakashima, in: R.J. Arsenault, D. Cole, T. Gross, G. Kostorz, P.K. Liaw, S. Parameswaran, H. Sizek (Eds.), Proceedings of the J. Weertman Symposium, The Minerals, Metals and Materials Society, 1996, pp. 401 – 407.
Effects of shape and volume fraction of second phase on stress states in two-phase materials Masaharu Kato, Toshiyuki Fujii, Susumu Onaka * Department of Inno6ati6e and Engineered Materials, Tokyo Institute of Technology, 4259 Nagatsuta, Yokohama 226 -8502, Japan
Abstract The effects of shape and volume fraction of a second phase on stress states and deformation behavior of two-phase materials are discussed. The second phase is treated as inhomogeneous spheroidal inclusions embedded in a matrix. Analytical expressions to describe the stress states in elastically and plastically deformed two-phase materials are obtained with the Eshelby method and the Mori–Tanaka concept of the ‘average stress’. The variation of the stress states with the change in the aspect ratio of the spheroidal second-phase is shown for various volume fractions. Considering that the second phase is also plastically deformable, the overall deformation behavior of the two-phase materials is discussed with the results obtained by the evaluation of the stress and strain distributions in the materials. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Two-phase material; Deformation behavior; Eshelby’s method; Mori and Tanaka’s theorem; Average stress; Inclusion problem
1. Introduction A plastically strong-second phase in a matrix often causes strengthening of materials. For designing twophase materials with desirable mechanical properties, theoretical considerations on stress states in the materials are needed to predict their deformation behavior. Within the framework of continuum mechanics, Tanaka and Mori [1] and Mori and Tanaka [2] have considered the stress states and presented theories, which satisfactorily explain plastic deformation behavior of the two-phase material such as dispersion-hardened alloys. Although they have considered various variables to characterize the two-phase materials, some simplifications included in previous calculations are better to be excluded for the prediction of deformation behavior of actual two-phase material [1,2]. In the present study, we deal with two subjects that have been simplified in the previous calculations [1,2]. One is the shape of the second phase. Only three special shapes, i.e. sphere, disk and needle, have been previously treated as possible shapes of the second phase. Instead of this simplification, we approximate the second-phase shape to general spheroids with various aspect ratios. * Corresponding author. Fax: +81-45-9245173. E-mail address: [email protected] (S. Onaka)
This is because consideration on intermediate shapes between the special shapes reveals important roles of the shape effects which can not be predicted from discussion on the special shape [3–5]. The other is the volume fraction of the second phase. Even if the second phase and matrix are elastically isotropic, the effects of volume fraction on the stress distribution in the twophase materials have not been shown by analytical expressions when the second phase has different elastic moduli from those of the matrix. This may be due to that complicated algebraic calculations are needed when we include interactions between elastically inhomogeneous inclusions. Relationships to describe the effects of the volume fraction derived in the present study are suitable for various analytical discussions on the stress states in two-phase materials. In the present study, we first summarize results given by the Eshelby method [6,7] and the Mori–Tanaka concept of the ‘average stress’ [2]. Then we derive relationships convenient for the evaluation of stress states in the two-phase materials when we include the above extensions. Although the second phase has often been assumed to be plastically non-deformable in theoretical considerations, metal–metal composites sometimes accompany plastic deformation of the second phase itself. In the present study, we finally discuss the overall deformation behavior of the two-phase materi-
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M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
als including the above extensions and the effect of possible plastic deformation of the second phase.
2. The Eshelby method and the Mori – Tanaka concept of the ‘average stress’
145
solved for some special cases such as uniaxial applied stress and a purely dilatational eigenstrain [9,10]. Conducting rather tedious calculations, we can analytically obtain Gijkl and Jijkl. Two assumptions involved in these calculations are (i) isotropic C *ijkl and Cijkl ; and (ii) a spheroidal shape of V. When V is described on the x1 –x2 –x3 orthogonal coordinate system as
2.1. Uniform stress in an isolated inhomogeneous inclusion
x 21 + x 22 x 23 + 2 5 1, a2 c
Let us consider an isolated ellipsoidal inhomogeneity V with elastic moduli C *ijkl embedded in an infinitely extended matrix M with elastic moduli Cijkl. When C *ijkl is different from Cijkl the remote external stress s O ij applied to the two-phase material M+ V is disturbed. Eshelby has shown that the stress s A ij in V is obtained by solving the following equations [6,7]:
the calculated explicit expressions for Gijkl and Jijkl are shown in Appendix A as a function of the aspect ratio a= c/a of V and isotropic C *ijkl and Cijkl.
O sA ij =Cijkl (e kl + Skimno* mn* − o* kl*)
=C*ijkl (e O kl + Sklmno* mn*),
(1)
and O sO ij =Cijkle kl,
(2)
(7)
2.2. A6erage stresses in inhomogeneous inclusions When a material under s O ij contains many inhomogeneous inclusions V, the average stress in the matrix sij M and that in the inclusions sij V are evaluated using Mori and Tanaka’s concept of the ‘average stress’ [2] together with the equivalent inclusion idea of Eshelby [6,7]. When the volume fraction of V is denoted by Vf, sij M and sij V should satisfy Vfsij V + (1− Vf)sij = s O ij .
(9)
the equivalent Here, Sijkl is the Eshelby tensor and o ** ij eigenstrain of the fictitious homogeneous inclusion which reproduces the uniform stress s A kl in V. For simplicity, we introduce a forth-order tensor Gijkl which A relates s O kl to s ij as
When the uniform eigenstrain o is generated in the inclusions with an identical shape and orientation, sij V is given by solving the following simultaneous equations [11]:
O sA ij =Gijkls kl.
sij V = Cijkl (e D kl + Sklmno* mn** − o* kl**)
(3)
P ij
Gijkl depends on the shape of V, Cijkl and C *ijkl and satisfies Gijkl =Gjikl =Gijlk. Instead of Gijkl in Eq. (3), Wu has considered a tensor, which relates e O ij to the uniform elastic strain in V for the evaluation of elastic moduli of composite materials [8]. When V has a uniform eigenstrain o Pij and no external stress is applied to M+V, the uniform stress s Bij in V is also given by Eshelby [6,7]:
Using Gijkl and Jijkl in Section 2.1, Eqs. (9)–(11) are rewritten as
s Bij =Cijkl (Sklmno*mn −o*kl ) = C*ijkl (Sklmno*mn − o Pkl),
sij V = Gijkl skl M + Jijklo Pkl,
(4)
where o *ij is the equivalent eigenstrain of the fictitious homogeneous inclusion which reproduces s Bij in V. Introducing another forth-order tensor Jijkl, the relationship between s Bij and o Pij shown by Eq. (4) is rewritten as s =J o . B ij
P ijkl kl
(5)
Here, Jijkl depends on the shape of V, Cijkl and C *ijkl and satisfies Jijkl =Jklij =Jjikl =Jijlk. When s O ij is applied to M containing V with o Pij , the total stress sij in V B becomes the sum of s A ij and s ij . Hence, from Eqs. (3) and (5), sij is written as P sij = Gijkls O kl + Jijklo kl.
(6)
When C *ijkl and Cijkl are isotropic and the shape of V B is spheroidal, s A ij and s ij in Eqs. (1) and (4) have been
P = C*ijkl (e D kl + Sklmno* mn** − o kl),
(10)
where o *** is the eigenstrain in the equivalent inclusion ij and e D is the average elastic strain in the matrix which ij is related to sij M by sij M = Cijkle D kl.
(11)
(12)
and Vf(Gijkl skl M + Jijklo Pkl)+ (1− Vf)sij M = s O ij .
(13)
These equations can be solved analytically to obtain sij V and sij M using Gijkl and Jijkl in Appendix A. The resultant sij V and sij M are expressed as a P function of s O ij , o ij ,Vf, Cijkl, C * ijkl and the aspect ratio of a of V. 3. Stress states in plastically deformed two-phase material We consider plastic deformation in a two-phase material containing many spheroidal inhomogeneities V
M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
146
with an identical shape and orientation. The uniaxial O stress s O is considered as an external stress ap33 = s plied along the axis of rotation symmetry of the spheroidal V. We evaluate the stress distribution in the two-phase material when plastic deformation occurs both in the matrix M and in the inhomogeneities V. When uniform plastic strains in M and V are expressed by o P(M) and o P(V) , from symmetry and volume conij ij and o P(V) can be stancy, nonzero components of o P(M) ij ij rewritten as 1 o P(M) =o P(M) = − o P(M), 11 22 2
o P(M) =o P(M), 33 o
P(V) 33
=o
P(V)
,
o
P(V) 11
=o
P(V) 22
1 = − o P(V). 2
(14a)
1 o P11 = o P22 = Do and other o Pij =0. 2 (16)
The stresses in M and V which govern their plastic deformation are s33M − s11M and s33V −s11V, respectively, since sij M and sij V becomes axisymmetrical around the x3 axis. The use of the average stresses as the yield criteria has been confirmed from energy consideration [12]. The average stresses s33M − s11M and s33V − s11V are written as a function of Do as M s33M − s11M = s M 0 + g Do,
(17)
s33V − s11V = s + g Do,
(18)
V 0
(14b)
When the difference of the plastic strains between M and V is written as Do =o P(M) −o P(V),
o P33 = − Do,
(15)
the average stresses sij M in M and sij V in V are evaluated with Do. That is, sij M and sij V after the occurrence of the plastic strains o P(M) and o P(V) are ij ij reproduced by considering the generation of the uniform eigenstrain o Pij in V as much as
O Fig. 1. The variation of the normalized average stress s V with the 0 /s change in the aspect ratio a for (a) the elasticity harder V in M with m*= 3m; and (b) the softer V in M with m*= 0.5m. The Poisson ratios n* for V and n for M are assumed to be n*= n =1/3.
V
where O V (1−Vf)s M 0 + Vfs 0 = s ,
(19)
and (1− Vf)g M + Vfg V = 0
(20)
V are satisfied. The stresses s M 0 and s 0 in Eqs. (17) and (18) physically mean the average stresses in the elastically deformed two-phase material. We know from Eq. M (19) that an increase in s V 0 causes a decrease in s 0 . On the other hand, g M and g V in Eqs. (17) and (18) show the effects of o P(M) and o P(V) on the average stresses. ij ij From Eq. (20) note that g V is positive and g M is negative. Since g M is negative, s33M − s11M decreases with increasing Do under constant s O. This explains work hardening in composites containing plastically non-deformable phases [1,2]. Using Eqs. (12), (13) and (16) and Gijkl and Jijkl in M V V Appendix A, explicit expression for s V 0 , g , s 0 and g V in Eqs. (17) and (18) are obtained. Those for s 0 and g V are derived in Appendix B. From Eqs. (19) and (20), M V sM are obtained from s V 0 and g 0 and g , respectively. In Fig. 1a and b, the variation of the normalized O average stress s V with the change in the aspect ratio 0 /s a are indicated for various values of the volume fraction Vf. To construct Fig. 1a and b, the Poisson ratios n* for V and n for M are assumed to be n*=n= 1/3. Fig. 1a and b, respectively, show the results for the elastically harder V in M with m*= 3m and softer V in M with m*= 0.5m, where m* is the shear modulus of V O V M and m that of M. We have s M 0 B s B s 0 and s 0 \ O V s \ s 0 for the elastically harder and softer V, respecO tively. As shown in Fig. 1a and b, s V depends on 0 /s the aspect ratio a more strongly for smaller Vf. When O Vf is fixed, s V monotonically increases or decrease 0 /s with increasing a from unity to . The normalized O average stress s V becomes the largest for elastically 0 /s harder V and the smallest for elastically softer V when their shape is needle-like (a ). However, in the O range of aB1, s V does not change monotonically 0 /s but has an absolute maximum or minimum at around a=0.2.
M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
147
havior of the two-phase material with the variations of s33V − s11V and s33M − s11M. When V is elastically harder than M, the variations of s33M − s11M and s33V − s11V with Do can be represented V as shown in Fig. 4. s M Y and s Y are the yield stresses of M and V. When we assume hypothetically that the material is elastically deformed (o P(M) = o P(V) =0 and Do = 0) under s O, Fig. 4 shows s33M − s11M \s M Y and s33V − s11V B s V Y. These mean that initial plastic deformation occurs in M and Do increases while o P(V) = 0. When Do reaches Do1 in Fig. 4, V starts to deform plastically since s33V − s11V = s V Y is sa-
Fig. 2. The variation of g V/m with the change in the aspect ratio a for (a) the elastically harder V in M with m*= 3m; and (b) the softer V in M with m*= 0.5m. The Poisson ratios n* for V and n for M are assumed to be n* = n =1/3.
On the assumption of n* =n =1/3, the variation of g V/m for m*=3m and m* =0.5m are, respectively, shown in Fig. 2a and b. Comparing Fig. 2a with Fig. 1a, we find that the variation of g V/m are similar to O those of s V when V is elastically harder than M. In 0 /s O Fig. 3 a and b, the contour lines of s V and g V/m for 0 /s Vf = 0.25 are shown as a function of the ratio of elastic constants f= m*/m and a. As shown in Fig. 3a and b, O and g V/m becomes stronger the a dependence of s V 0 /s when V becomes elastically harder than M.
4. Deformation behavior of a two-phase material Let us suppose that elastically harder V with a certain value of Vf is embedded in M. For this case, we find from Figs. 1 – 3 that s33V −s11V becomes the smallest and hence s33M −s11M becomes the largest when the shape of V is an oblate spheroid with a : 0.2. If V is plastically non-deformable, this means that the back stress due to this V is the smallest and the effect of V to suppress plastic deformation of M becomes a minimum. On the other hand, when the elastically harder V is needle-like (a ), plastic deformation of M is most effectively suppressed. In this section, considering that V is also plastically deformable, we generally discuss the deformation be-
O V Fig. 3. The contour lines of (a) s V 0 /s ; and (b) g /m for Vf = 0.25 as a function of the ratio of elastic constants f =m*/m and the aspect ratio a. The Poisson ratios n* for V and n for M are assumed to be n* = n =1/3.
Fig. 4. The average stresses in M and V as a function of Do when the uniaxial external stress s O is applied to the two-phase material M +V.
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M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
tisfied. However, this plastic deformation in V increases o P(V) and causes the decrease in Do. Hence, M and V thereafter continue to deform keeping the difference of the plastic strain as much as Do1. The continuous plastic deformation of the two-phase material occurs when M and V are perfectly plastic solids. If we include effects of inherent work hardening of M and V, the plastic deformation stops after the occurrence of a certain amount of plastic deformation. The above is a conclusion of the example shown in Fig. 4 where s33M − s11M \s M is satisfied for Do =Do1. If we have Y V s33M −s11M Bs M Y when s33V −s11V =s Y instead of the example shown in Fig. 4, the plastic deformation of M does not cause that in V and the deformation of the two-phase material stops under s O. V The yield stress s M Y and s Y and the variations of s33M −s11M and s33V −s11V with the change in Do determine the deformation behavior of the two phase material. Even if the combination of M and V is fixed, the deformation behavior of the two-phase material is affected by the shape and volume fraction of the second phase through the a and Vf dependencies of s M 0 , M V sV , g and g . In the present study, we did not 0 consider thermal residual stress of relaxation due to local plastic deformation in the two-phase materials. These effects on the deformation behavior have been treated with the Eshelby method in previous studies [13–15]. Mori and Wakashima have treated a case where only creep deformation of M occurs and discussed an equilibrium state of the two-phase material by energy consideration [16]. Their assumptions are that o P(V) = 0 and o P(M) changes so as to minimize the mechanical free energy of the two-phase material. The equilibrium state discussed by them is shown by the point A in Fig. 4. The equilibrium plastic strain of M as much as Do = − M sM and the resultant hydrostatic stress state in M 0 /g are the same as those obtained by the energy consideration [2,16].
5. Conclusions The effects of shape and volume fraction of a second phase on the stress states in the elastically and plastically deformed two-phase materials have been discussed. The second phase has been treated as inhomogeneous spheroidal inclusions embedded in a matrix. Using the Eshelby method and the Mori– Tanaka concept of the ‘average stress’, the expressions to describe the stress states in the deformed two-phase materials have derived in analytical form. The secondphase shape and volume-fraction dependencies of the stress and strain distributions in the deformed twophase materials have been emphasized. Considering that the second phase is also plastically deformable, the
overall deformation behavior of the two-phase materials has been discussed.
Acknowledgements This work has been partially supported by a Grantin-Aid for Scientific research (C) from the Ministry of Education, Science and Culture under Grant No. 1065687 and the Proposal-Based New Industry Creative Type Technology R&D Promotion Program from the New Energy and Industrial Technology Development Organization (NEDO) of Japan.
Appendix A. Expressions for the components of Gijkl and Jijkl For the spheroidal inhomogeneity V: (x 21 +x 22)/a 2 + x /c 2 5 1, non-zero components of Gijkl which relates s with s O ij in Eq. (3) and satisfies Gijkl = Gjikl =Gijlk are given as follows: 2 3 A ij
G1111 = G2222 =
1 m* − (2K*(Q3Dm −m) 2(m − 2DmS1212) 2R + m*(Q2DK −K)),
(A1a)
1 G3333 = − (K*(Q1Dm −m)+ 2m*(Q4DK −K)), R (A1b) G1122 = G2211 = − −
m* 2(m − 2DmS1212)
1 (2K*(Q3Dm −m)+ m*(Q2DK −K)) 2R
,
(A1c)
1 G1133 = G2233 = − (K*(Q1Dm −m)−m*(Q4DK −K)), R (A1d) 1 G3311 = G3322 = − (K*(Q3Dm −m)−m*(Q2DK −K)), R (A1e) G1212 =
m* , 2(m − 2DmS1212)
G2323 = G3131 =
m* , 2(m − 2DmS2323)
(A1f ) (A1g)
where DK = K−K*,
(A2a)
Dm =m− m*,
(A2b)
and
M. Kato et al. / Materials Science and Engineering A285 (2000) 144–150
R =(Q1Dm −m)(Q2DK − K)
J1111 = J2222
+2(Q3Dm −m)(Q4DK − K).
(A3)
Here K and m are the bulk and shear moduli of the matrix M, respectively, and K* and m* those for the inhomogeneity V. S1212, S2323 and Q1 to Q4 are components and their combinations of the Eshelby tensor. These are the function of the aspect ratio a and the Poisson ratio n of the matrix M and written as S1212 =
!
"
7−8n S(a) (7 − 8n) − −4(1 − 2n) , 24(1− n) 32p(1 −n) a2 (A4a)
!
=
!
1 (3K*KW + 4m*mU + 12K*Km*(Q4 −Q2)), 3R (A7b)
J1122 = −
mm*(2S1212 − 1) (m− 2DmS1212)
−
5−4n S(a) (5 + 8n) − +4(1 − 2n) , 6(1− n) 8p(1− n) a2 (A4c) 1+n 1 −2n a −1 − S(a) , 3(1 − n) 2p(1 − n) a2 (A4d)
Q3 = − S1133 +S3333
!
"
(2n −1) 5−4n S(a) = − +2(5 − n) , 6(1− n) 8p(1− n) a2
1 (3K*KW − 2m*mU 3R
+3K*Km*(Q4 − Q2)), 2
Q2 = 2S1133 + S3333 =
1 (3K*KW + m*mU − 6K*Km*(Q4 −Q2)) 3R (A7c)
J1133 = J2233 = −
"
(A7a)
J3333 = −
"
Q1 = S1111 +S1122 −2S3311
mm*(2S1212 − 1) 1 − (3K*KW + m*mU (m − 2DmS1212) 3R −6K*Km*(Q4 − Q2)),
1− 2n S(a) (2 − n) + +(1 + n) , S2323 = S3131 = 6(1 −n) 8p(1− n) a2 (A4b)
=
149
J1212 =
mm*(2S1212 − 1) , (m − 2DmS1212)
J2323 = J3131 =
mm*(2S2323 − 1) , (m− 2DmS2323)
(A7d) (A7e) (A7f )
where (A4e)
W=(Q1Dm −m)(Q2 − 1)+ 2(Q3Dm −m)(Q4 −1)
(A8)
U= (Q1 − 1)(Q2DK −K)+ 2(Q3 − 1)(Q4DK −K).
Q4 = S1111 +S1122 +S3311
(A9)
1+n 1 −2n a −1 + S(a) , 3(1−n) 4p(1− n) a2 2
=
(A4f )
where S(a) is a function of a and n given by
n
2 a2 3a S(a)= p (a 2 +2) − arccos(a) , 3 (1− a 2)2 (1 −a 2)1/2 (a B 1) S(a)=
(A5a)
4 p, 15
S(a)
(a= 1)
(A5b)
n
(B1) (a \ 1)
(A5c)
Among Q1 to Q4 we have the relationship written as (1 −2n)(Q1 − Q3) =(1 + n)(Q4 −Q2), (1+n) , (1−n)
V sV in Eq. (18) are written as 0 and g
{(A1A4 −A2A3)Vf +(2A3m+ 3A4K)(1 −Vf)}s O, {6Km(1− Vf)2 +(2A1m+ 3A4K)Vf(1 −Vf) +(A1A4 −A2A3)V 2f }
3a − 2 ln{a +(a 2 −1)1/2} , (a − 1)1/2
Q2 + 2Q4 =
V in Eq. Appendix B. Explicit expressions for s V 0 and g (18)
sV 0 =
2 a2 (a 2 +2) = p 2 3 (a −1)2
S1212, S2323, Q1 to Q4 and R in the above equations are the same with those in the equations for Gijkl.
gV= (1 − Vf){(A1A6 −A3A5)Vf +3A6K(1 −Vf)}m, {6Km(1 − Vf)2 +(2A1m+ 3A4K)Vf(1 −Vf) +(A1A4 −A2A3)V 2f }
(B2)
(A6a) (A6b)
For the spheroidal inhomogeneity V with uniform o Pij , non zero components of Jijkl which relates s Bij with o Pij in Eq. (5) and satisfies Jijkl =Jklij =Jjikl =Jijlk are written as follows:
where A1 =
3KK* {3m −Dm(Q1 + 2Q3)}, R
(B3a)
A2 =
4mK* Dm(Q3 − Q1), R
(B3b)
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150
A3 =
3Km* DK(Q2 −Q4), R
(B3c)
A4 =
2mm* {3K − DK(Q2 +2Q4)}, R
(B3d)
A5 =
12mm*K* (Q3 −Q1), R
(B3e)
A6 =
− 6mm* {(Q1 −1)(Q2DK − K) R +2(Q3 −1)(Q4DK − K)}.
(B3f )
Here, Q1 to Q4 and R are already given in Appendix A.
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.
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