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Void growth in a viscous metal
~Z~7225/~l/~l~~l2soz.M)IO Copyright@ 1981Pergamon Press Ltd.
Inf. 1. EngngSci. Vol. 19, pp. 1083494, 1981 Printed in Great Britain. All rights reserved
VOID GROWTH IN A VISCOUS METALt MINORU TAYA Department of Mechanical and Aerospace Engineering, University of Delaware, Newark, DE 19711,U.S.A. and ETHAN D. SEIDEL Department of Statistics, Princeton University, Princeton, NJ 08540,U.S.A. Abstract-An investigation is made of void growth in a viscous metal. First, the effective viscosity k and expansion viscosity K of a viscous body containing spherical voids is estimated. Then the growth of a typical void embedded in a compressible viscous body with p and K is studied. The analytical tool used in this study is a combination of Eshelby’s equivalent inclusion method and Mori-Tanaka’s back stress analysis. By Mori-Tanaka’s back stress analysis the interaction among voids which is enhanced at larger volumn fractions of void can be taken into account in our analysis.
I. INTRODUCTION ANY DUCTILE metal containing secondary particles gives rise to the initiation and growth of micro voids upon straining. Under the assumption of quasi-static deformation the mechanism of growth in a plastically deforming matrix has been investigated theoretically by a number of researchers, for example [l-5]. McClintock[l] and Rice and Tracey[21 showed that a triaxial stress plays an important role in void growth. However, their models[l, 23 did not include the interaction among voids and tend to estimate the larger failure strain er than that observed experimentally. On the other hand, Tracey[3] and Needleman[4] included the interaction to conclude that it accelerates void growth. Nemat-Nasser and Taya[S] studied numerically the effect of the lateral constraint on void growth and the mode of plastic deformation between voids and arrived at the same conclusion as Tracey and Needleman when the lateral constraint (triaxial stress) is large. However, the models used in [3-51 are 2-dimensional and render a limited applicability. Apart from the quasi-static deformation discussed above, materials of our interest are ductile and deform at high strain-rate (i - lo4 set-’ or more). The flow stress of such materials is given by u = (Yt pi, where a and p are constants for a certain range of E[6]. For typical ductile metals such as copper and aluminum [6,7], the order of (Yis much smaller than 2~; and a can be neglected. Thus the constitutive equation of a viscous metal is further simplified to a viscous fluid type: g = 2yi. Eshelby[8] showed that the inclusion problem in a viscous fluid can be solved in the same manner as that in an elastic body assuming that the inertia effect is neglected. Based on Eshelby’s equivalent inclusion method, Bilby et al. [9], Howard and Brierley [lo] and Budiansky et al. [l I] solved the growth of an ellipsoidal inhomogeneity embedded in a viscous fluid, and the void growth problem is merely a special case in their analysis. In their models the matrix surrounding an ellipsoidal inhomogeneity (or void) was assumed to be incompressible. This assumption can be considered valid only for a small volume fraction of void. For a large volume fraction of void, 5% or more, the surrounding matrix becomes compressible. In other words, the interaction among voids and that between a void and the outer boundary is enhanced as a volume fraction of void increases. The above interaction can be taken into account by an average back stress which can be again evaluated in terms of the eigenstrains as Mori and Tanaka didll21. Mori-Tanaka’s back stress analysis has been successfully applied to several inhomogeneity problems by Taya et al. [13-151. The present paper deals with the void growth problem in a compressible viscous material under the assumption that the inertia effect is neglected. The analytical tool used in the present tThis research was supported by the University of Delaware Research Foundation. 1083
1084
M. TAYA and E. D. SEIDEL
study is a combination of Eshelby’s equivalent inclusion method and Mori-Tanaka’s back stress analysis. We describe first the method of estimating the effective viscosity p and expansion viscosity K of a viscous body containing numerous voids in Section 2. Then the problem of void growth in a compressible viscous body with p and K will be formulated in Section 3, and the results will be discussed in the last section. 2. COMPUTATION
OF THE EFFECTIVE VISCOSITY, AND POISSON’S RATIO
EXPANSION
VISCOSITY
Our analytical method here and in the following section is based on Eshelby’s equivalent inclusion method. The justification of applying Eshelby’s method to the inhomogeneity problems in a viscous fluid has been made in [&lo]. Hence Eshelby’s equivalent inclusion method will be used without referred to fundamental equations inherent in a low Reynold number fluid dynamics. In order to compute the effective viscosity p and expansion viscosity K, we assume that all voids are spherical of the same size so that the infinite body containing voids become isotropic if it were treated as a homogeneous material. 2.1 The effective viscosity p Consider an infinite viscous body D which contains spherical voids and is subjected to applied shear strain-rate e& = e& = y” along x2- and x3 surfaces as shown in Fig. 1. Let domain of voids be denoted by R and that of a matrix becomes D-C The viscosity of matrix is denoted by po. We assume that the average of the stress disturbance due to all fl is given by in matrix [12-161
the the the the
where c+ijand gij are the average disturbances of deviatoric stress and strain-rate in the matrix, respectively. Then we add a single void to the composite system D (the matrix and voids). Eshelby’s equivalent inclusion method yields in D 2po(ei + ~ij + e, - eQ) = 0
(2)
2/J&eij + 4 - e fi)
(3)
CTfj
=
where gij and eij are the disturbances of deviatoric stress and strain-rate due to this single void, respectively, and eQ is called “eigenstrain” and defined as having a non-zero value in n and
Fig. 1. A theoretical model for computing the effective viscosity p.
Void growth in a viscous metal
1085
being zero in D - R. Since /Dar, d V = 0, we obtain (1
-f)C?ij
+f(Uij)* = 0
(4)
where f is a volume fraction of void and ( )o denotes the volume average in C!. From eqns (3) and (4) we have Zijt f( eij - e Q)= 0.
(5)
According to ~shelby~S], e;j is related to ef as
where Sirk,is called the Eshelby’s tensor and is a function of the geometry of fi and the matrix material properties. For a given applied strain-rate y. (Fig. l), eigenstrain e& is obtained from eqns (Z), (5) and (6) *-
e32 -
YO
*-
(7)
e23 (1 -
f)(l
-
=3232)’
In order to obtain the effective viscosity p, we use the equivaIence of the energy dissipation 4~: = 4fioyz0+ f4poroeT2.
(8)
The detailed derivation of eqn (8) is given in Appendix A. From eqns (7) and (8), the effective viscosity of a viscous body containing voids, p is obtained as f-p--
5 f 3(l -f)
(9)
where S3232 = l/S for a sphericaf void embedded in an incompressible matrix. It should be noted here that by the original Eshelby’s method without MartTanaka’s back stress analysis the formula corresponding to eqn (9) becomes
Thus the denominator in the second term of eqn (9), (1 - f) takes care of the interaction between voids which is enhanced at a larger volume fraction of void. 2.2 The efective expansion viscosity K The calculation model for obtaining K is different from that in Fig. 1 such that each void is given a vofumetric strain-rate ~7~~. Then the increase in the macroscopic vofumetric strain-rate, h is given by A = f&k. (11) Eshelby’s equivalent inclusion method yields the stress in R Uij
=
-
SijPo= A&j(Z& - e&) t Z~O(~j- et)
(12)
where PO is the pressure in Q and becomes zero in the case of void. Since ho in eqn (12) is Lame’s constant of the incompressible matrix, we must have (13) Thus the imposed volumetric strain-rate in R can be replaced by the equivaient inclusion with
1086
M. TAYA
andE.
D. SEIDEL
the eigenstrain e$ = Zkk.For a given eigenstrain e$. the total energy dissipation W per unit volume is given by [12, 161 w = 401 - f)CLO lS(l_Vo)[4(e~12+e~~~+rZ1~)+(5~o+I)
In the previous equation the interaction among inclusions comes in through the factor (1 -f)[12], and v. is the matrix Poisson’s ratio. From eqns (11) and (13) the non-vanishing eigenstrain in our problem is e:, = e$ = e& = 3f’
(15)
A substitution of eqn (15) and v. = l/2 into eqn (14) yields the energy dissipation per unit volume as w = 4cLo(1- f
3f
w
(16)
If a viscous body containing numerous voids is treated as a homogeneous material with volumetric strain-rate A, then the corresponding energy dissipation per unit volume W can be expressed as W=KA’
where
K
(17)
is the effective expansion viscosity. By equating eqn (16) to eqn (17) we obtain 118)
K=4po:f- f, When a volume fraction of void
f is small, eqn (18) can be approximated as
K=4cLo.
(19)
3f
Thus we have recovered the formula obtained by Batchelor[l7]. 2.3 The effective Poisson’s ratio u For the computation of void growth in the next section, it is convenient to define the effective Poisson’s ratio (Y) of a porus viscous body which is assumed to be isotropic. As in the case of isotropic linearly elastic body, ZJis related to k and K as (3K-2/.l)
(20)
' = 2(3Ktp)'
A substitution of eqns (9) and (18) into eqn (20) yields 2p-I v=4p
P=
(1-f)
Tf.
f]l5(1-f,
(21) 1
It should be noted in eqn (21) that v becomes l/2 when f = 0.
Void growth in a 3. COMPUTATION
viscousmetal
1087
OF VOID GROWTH
A theoretical model for void growth is basically the same as that used in the previous section (Fig. 1) except for the far field boundary condition and the shape of void as shown in Fig. 2, which shows the current configuration under the applied strain-rate. Focusing on a typical ellipsoidal void (for example, located in the center in Fig. 2), we consider a mixture of pure matrix and the other voids (indicated by dashed curves) as a homogeneous, isotropic and compressible viscous material. Denote the domain of this void and the infinite body by R and 0, respectively. Hence the domain of the compressible matrix becomes II-Q. We denote the major (along x3-axis) and minor (along x1-, x2 - axes) of the ellipsoid by c and a, respectively. A factor t in Fig. 2 takes value on 1, 0, - Y which corresponds to the cases of triaxial strain, uniaxial strain and uniaxial tension, respectively. eA
t
-
X3
J-
X2
XI
-ten
J feA Fig. 2. A theoretical model for the void growth.
Following Eshelby[8,9], the equivalent inclusion method yields in fz A&j(e&
+
ekk -
efk)+ 2&e: +
eij
-
(22)
ef) = 0
where A = K-213~~ p and K are already defined in the previous section by eqns (9) and (18), respectively, ef is the applied strain-rate, eij is the strain-rate disturbed by Q and et is the eigenstrain defined in Q and becomes zero in D-fl. As before, the following relation holds
where Sijk/is a function of c, a and v and its explicit expression is given in Appendix B. We formulate two cases separately: (i) 5 = - V,0 and (ii) 5 = 1. 3.1 When(=-vand 0 In this case the shape of a deforming void is proiate spheroid and the state of stress and strain-rate in Q can be considered to be transversely isotropic; et1 = ezz and ef, = ef2. Setting i = j in (22) and noting that the value of 3A + 2~ remains finite for f # 0, we obtain eD,+e, -et=&
(24)
With eqn (23) and 6?pp ’ = (1-f 2[)e,+ eqn (24) becomes 20
-&II
-
&22-
S33defrr
+ (1 - %%333
-
S3333).
4
= (I+
WeA-
(25)
Settling ii = 33 in eqn (22) and using eqns (23) and (24), we obtain 2S3311e:1+(S3333_l)e:3=-eA*
WJ)
1088
M. TAYA and E. D. SEIDEL
We solve for el;and e& in eqns (25) and (26) to obtain * -__!B eA A
(27)
ell-
* -2 B
e33-
eA
A
(28)
where A = (1 - S,,,, - S,,zz)(l B,
= IS,,,,
t
60
-
s3333)
-
2&,33&3,,
S3333N
(29)
B2=1-S,,,,-S,22+%?33,,.
The total strain-rate in 0 is given by
= 5eA + b!GIII+ SIi22)eTI f S1133eT3
c - =
43
f
(30)
e33
c
=
eA f
2&3,,et,
+
(31)
s3333e&
where a super dot stands for a time derivative. From eqns (27)-(31), the rates of void growth in x, or x2-direction, d/a and in x3-direction C!/Ccan be obtained as (32) _i
=
1 + @,S33,,
+ B2S3333)
A
c
eA
(33)
1 .
When f becomes zero; the surrounding matrix being strictly incompressible, the cases of 5 = 0 and - v are reduced to the cases of triaxial tension and uniaxial tension (v = l/2), respectively, and are worked out by Budiansky et al.[ll]. In the case of 5 = 0 the void growth rate becomes infinite upon applied strain and corresponds to the ordinate in Fig. 7. Whereas the case of [ = - l/2 (uniaxial tension) can be formulated explicitely to yield Lila and (:/c as
ri -=
--f
a
1 2
($,I,
+
S,,22)&
+
S,133B2
I
eA
; = {1 + 2&3,,B, + Sm&kA
(35)
where B
=
’
B
=
2
c,,
(;G*+c,2)
(C,,G-
y
c,,c*,,
&2,+c,,)
(G2C2,
-
CllC22)
h-2+
A3 i 2 4(pz_l)lo I
c,2=-~+{_I+qc (P - 1)
(34)
1
2 4(p - 1)I O
1089
Voidgrowth in a viscousmetal
(34) and where 0, lo and S‘,, are defined in Appendix B. The values of Silk,must be obtained from
Appendix B by substituting l/2 for Y. 3.2 When 5 = I In this case the mode of deformation is spherical symmetric
eli = e22= e33
e7, = ef2 = ef3 = e*.
(37)
From eqns (22), (23) and (37), we have 1 eh = (1 - s*,,i -2s,1**)ek
(38)
Hence the rates of void growth, ci/a( = i/c) are obtained as 1
d E -=-= 0
c
(I-
sm,-%*2)
(39)
h.
For a given volume fraction of void and a constant &?A we can integrate directly eqn (39) to obtain a
-=exp a0
i
(l-S,,fiAt2S,,**)
I
Gw
where eAt is the total strain applied at time t, and a0 is the initial radius of a spherical void. 4.NUMERICALRESULTS
ANDDISCUSSIONS
In order to simulate the void growth in a ductile metal deforming at strain-rate, we use the following data eA= lo4 see-’ eAtf =
0.4
tr = 4 x to-’ set where fr is the duration of the constant strain-rate applied eA.Three different values off are also used as a parameter: f = 0.05,O.l and 0.3. We also consider three kinds of loading conditions, (i) triaxial strain (5 = l), (ii) uniaxial strain (5 = 0) and (iii) uniaxial tension (5 = - v), As a measure of void growth, natural tensile strains, ln(c/co) and ln(u/~~) are used with the initial shape of a void being spherical. In order to compute ln(c/co) and Malao), we use incremental method with 50 steps except for ,_$ = 1, for which analytical results are given by eqn (37). The results for 5 = 1, 0 and - v are plotted as solid curves for In(c/co) and dashed curves for ln(u/uo) in Figs. 3-5, respectively. It should be noted in Fig. 3 that the results for In(c/co) and ln(a/ao) coincide due to spherical symmetry. It follows from Figs. 3-5 that the rate of increase in ln(a/uo) and MC/co) is largest for LJ= 1 (triaxial strain) and smallest for 5 = - v (uniaxial tension). The effect of f on ln(a/ao) and ln(c/co) becomes significant for .$= 1 and 0, while this effect has less influence on In(uluo) and ln(cfco) as seen from Fig. 5 where the case of incompressible surrounding matrix (‘j = 0) is also shown.
M. TAYA and E. D. SEIDEL
1091
Void growth in a viscous metal
0.1
0.2
I
I
0.3
0.4
Fig. 5. In(n/aO)(dashed curves) and In(c/c,) (solid curves) vs e,.,t for 6 = -
Fig. 6. In(V/V,J vs e,& for .$= 1 (triaxial strain). UES Vol. 19, No. S-D
eAt
o
(uniaxial tension).
1092
M. TAYA and E. D. SEIDEL
It has been well observed experimentally[l8,19] that the void growth in the necked region of a tensile specimen is enhanced more significantly than that in the parallel region. The state of deformation in the necked region can be simulated by the case of uniaxial straining (5 = 0) (Fig. 4). Thus the void in the necked region can grow transversely as well as along the straining direction. This has been also observed by Nemat-Nasser and Taya in their model study[5]. The transverse void growth leads to the coalescense of two or more neighboring voids. In order to investigate the void growth as a volume increase, we have plotted ln( Vl V,) vs t~,~t for cases of ,$= 1,O and - v in Figs. 6-8, respectively, where V and V, are the void volume at the current and the initial stages, respectively. In these figures the case of f = 0 which has been recently studied by Budiansky et al.[ll] is also shown. It is seen from Figs. 6-8 that the increase in the void volume In( V/ V,,) becomes larger as f decreases for the cases of 5 = I and 0, but for the case of 5 = - u (uniaxial tension) the above trend becomes reverse as seen from Fig. 8.
Fig. 7. In( Vi V,) vs e,4t for .$= 0 (uniaxial strain).
Fig. 8. In( V/V,) vs e,t for 5 = - v (uniaxial tension)
Void growth in a viscous metal
1093
The proceeding discussions are led to the following conclusions: (i) The present model based on a surrounding matrix predicts smaller rate of void growth for the cases of 5 = 1 and 0 compared with that by the incompressible model[ll], but for the case of 6 = - v (uniaxial tension) the void growth rate by the present model becomes larger than that by the incompressible model. (ii) The effect of volume fraction of void f on the void growth is more significant for the cases of t = 1 and 0 and less for the case of 5 = - v. (iii) The present model can be applicable to the dynamic case if the applied strain-rate remains constant. However, the state of constant strain-rate of order 10“set-’ or higher is difficult to achieve experimentally. Hence the justification of the present model still remains to be made in the future. REFERENCES [1] F. A. McCLINTOCK, 1. Appl. Mech. 35,363 (1%8). [2] J. R. RICE and D. M. TRACEY, J. Mech. Phys. Solids 17,201 (1969). [3] D. M. TRACEY, Engag Fract. Mech. 3,301 (1971). [4] A. NEEDLEMAN, L Appl. Mech. 39,964 (1972). [S] S. NEMMAT-NASSER and M. TAYA, Jntl. J. Solids Structures 16,483 (1980). [6] A. KUMAR and R. G. KUMBLE, J. Appl. Phys. 40(9), 3475(1969). [7] A. KUMAR, F. E. HAUSER and J. E. DORN, Acta Metall. 16, 1189(1968). [S] J. D. ESHELBY, Proc. R. Sot. Land. A241,376 (1975). [9] B. A. BILBY, J. D. ESHELBY and A. K. KUNDU, Techtonophysics 28,265 (1975). [lo] I. C. HOWARD and P. BRIERLY, ht. J. Engng Sci. 14, 1151(1976). [ll] B. BUDIANSKY, J. W. HUTCHINSON and S. SLUTSKY, In Mechanics of Solids, The Rodney Hill 60th AnniuersnryVolume (Edited by H. Cl.Hopkins and M. J. Sewell). Pergamon Press, Oxford (1981). [12] T. MORI and K. TANAKA, Acta Metall. 21,571 (1973). [13] M. TAYA and T. MURA, 1. Appl. Mech. To be Published. [14] M. TAYA and T. W. CHOU, Intl. J. So/ids Structures.To be published. [l5] M. TAYA and 1‘. W. CHOU, Proc. Japan-U.S. Conf. on Composite Materials (Edited by K. Kawata), Tokyo, Japan (1981). [I61 T. MURA, Micromechanics of Defect in Solids, Noordhoff, Holland. To be published. [17] G. K. BATCHELOR, An Introduction to Fluid Dynamics, p. 253. Cambridge University Press, New York (1970). [ill] D. P. BAUER and S. J. BLESS, Report AFML-TR-79-4021(1979). [19] T. B. COX and J. R. LOW, Jr., Metall. Trans. 5, 1157(1974). (Received 2 October 1980)
APPENDIX A The total energy dissipation in an infinite body containing voids, E is given by E=
(ci t uii)(e$t tij t eij)du. ID
The integrand in eqn (Al) can be expanded as
(Uft uij)(i)(e$ t 2ijt eij)=
ufef t
Uf(&j+
eij) t uij(e; t
Zij t eij).
642)
We note that the third term in eqn C.42) upon integration over D vanishes by use of Gauss’ divergence theorem and the following relations ef = I/Z(U~~ t I&)
t?ij= l/2( iii t qi, Q
uijnj
=
I/2(Vu
t Uj.i)
=O on jD(
oij.j=0
643)
in D
(A4) 645)
where (DIis the outer boundary of D. The second term in eqn (A2) can be rewritten as flf(Zijt eij)= uK2ij t eij- e$) t cTfe$ = 2p0e&(t?ij t eij- e$) t u$ez = p.eO I, Df &* u 1,
(‘46)
Where eqn (3) is used to derive eqn (h6). The first term on the right hand side of eqn (A6) vanishes upon integration over D since JDuijd V = 0.
1094
M. TAYA
and E. D. SEIDEL
Thus eqn (Al) is reduced to E = u$e:
t f&:
(A?
where the above E denotes the energy dissipation per unit volume. If we consider the composite body D (matrix and voids) as a homogeneous viscous body subjected to the applied strain-rate -m then the corresponding energy dissipation per unit volume is given by E =4&
(A81
By equating eqn (A7) to eqn (AS), we can obtain the effective viscosity p. APPENDIX
B
The Eshelby’s tensors Sii,., are given in [LX].We present them in a more convenient form below. The suffices ijkl in the Eshelby’s tensors are consistent with the coordinate systems (Figs. I and ?). I. When Cl is sphere 7 J,,,,=
S 2z*?=S3333=-
S,,2? = &11=
(7 - 5U) l5(l -v)
(I -Sv) SN, = - l5(l _ r, (Bl)
2. When fI is prolate spheriod along x,-axis
sIlll=S**~~=~[3jlt~)t(l-2’-~)‘“] S 33”=‘+(~_“)(pz_~) S ,1:2=S221i=&
1
( l-2v+(B2_1)
[It&-
1
3PZ
I r.
11
&jtl-2s
IO
____-+‘(“-(I
I St,,3 = S?233= *(I _ V) S 33,
___ I 4(lev)
p’
P2
I
___’
I = s3322 = 2(1
/32 _ I
4
- ,I-,,+_!L I
_ v) r
-2V))41
82 _ 1
+I4
B2_l !
t
3
p,_1+N-2~0)
-1
where
p=L
2P
IO= n(/3(p2 CP - 1)
a
- I)“’ - cash-’ fl}
43 11
iB2)
Inf. 1. EngngSci. Vol. 19, pp. 1083494, 1981 Printed in Great Britain. All rights reserved
VOID GROWTH IN A VISCOUS METALt MINORU TAYA Department of Mechanical and Aerospace Engineering, University of Delaware, Newark, DE 19711,U.S.A. and ETHAN D. SEIDEL Department of Statistics, Princeton University, Princeton, NJ 08540,U.S.A. Abstract-An investigation is made of void growth in a viscous metal. First, the effective viscosity k and expansion viscosity K of a viscous body containing spherical voids is estimated. Then the growth of a typical void embedded in a compressible viscous body with p and K is studied. The analytical tool used in this study is a combination of Eshelby’s equivalent inclusion method and Mori-Tanaka’s back stress analysis. By Mori-Tanaka’s back stress analysis the interaction among voids which is enhanced at larger volumn fractions of void can be taken into account in our analysis.
I. INTRODUCTION ANY DUCTILE metal containing secondary particles gives rise to the initiation and growth of micro voids upon straining. Under the assumption of quasi-static deformation the mechanism of growth in a plastically deforming matrix has been investigated theoretically by a number of researchers, for example [l-5]. McClintock[l] and Rice and Tracey[21 showed that a triaxial stress plays an important role in void growth. However, their models[l, 23 did not include the interaction among voids and tend to estimate the larger failure strain er than that observed experimentally. On the other hand, Tracey[3] and Needleman[4] included the interaction to conclude that it accelerates void growth. Nemat-Nasser and Taya[S] studied numerically the effect of the lateral constraint on void growth and the mode of plastic deformation between voids and arrived at the same conclusion as Tracey and Needleman when the lateral constraint (triaxial stress) is large. However, the models used in [3-51 are 2-dimensional and render a limited applicability. Apart from the quasi-static deformation discussed above, materials of our interest are ductile and deform at high strain-rate (i - lo4 set-’ or more). The flow stress of such materials is given by u = (Yt pi, where a and p are constants for a certain range of E[6]. For typical ductile metals such as copper and aluminum [6,7], the order of (Yis much smaller than 2~; and a can be neglected. Thus the constitutive equation of a viscous metal is further simplified to a viscous fluid type: g = 2yi. Eshelby[8] showed that the inclusion problem in a viscous fluid can be solved in the same manner as that in an elastic body assuming that the inertia effect is neglected. Based on Eshelby’s equivalent inclusion method, Bilby et al. [9], Howard and Brierley [lo] and Budiansky et al. [l I] solved the growth of an ellipsoidal inhomogeneity embedded in a viscous fluid, and the void growth problem is merely a special case in their analysis. In their models the matrix surrounding an ellipsoidal inhomogeneity (or void) was assumed to be incompressible. This assumption can be considered valid only for a small volume fraction of void. For a large volume fraction of void, 5% or more, the surrounding matrix becomes compressible. In other words, the interaction among voids and that between a void and the outer boundary is enhanced as a volume fraction of void increases. The above interaction can be taken into account by an average back stress which can be again evaluated in terms of the eigenstrains as Mori and Tanaka didll21. Mori-Tanaka’s back stress analysis has been successfully applied to several inhomogeneity problems by Taya et al. [13-151. The present paper deals with the void growth problem in a compressible viscous material under the assumption that the inertia effect is neglected. The analytical tool used in the present tThis research was supported by the University of Delaware Research Foundation. 1083
1084
M. TAYA and E. D. SEIDEL
study is a combination of Eshelby’s equivalent inclusion method and Mori-Tanaka’s back stress analysis. We describe first the method of estimating the effective viscosity p and expansion viscosity K of a viscous body containing numerous voids in Section 2. Then the problem of void growth in a compressible viscous body with p and K will be formulated in Section 3, and the results will be discussed in the last section. 2. COMPUTATION
OF THE EFFECTIVE VISCOSITY, AND POISSON’S RATIO
EXPANSION
VISCOSITY
Our analytical method here and in the following section is based on Eshelby’s equivalent inclusion method. The justification of applying Eshelby’s method to the inhomogeneity problems in a viscous fluid has been made in [&lo]. Hence Eshelby’s equivalent inclusion method will be used without referred to fundamental equations inherent in a low Reynold number fluid dynamics. In order to compute the effective viscosity p and expansion viscosity K, we assume that all voids are spherical of the same size so that the infinite body containing voids become isotropic if it were treated as a homogeneous material. 2.1 The effective viscosity p Consider an infinite viscous body D which contains spherical voids and is subjected to applied shear strain-rate e& = e& = y” along x2- and x3 surfaces as shown in Fig. 1. Let domain of voids be denoted by R and that of a matrix becomes D-C The viscosity of matrix is denoted by po. We assume that the average of the stress disturbance due to all fl is given by in matrix [12-161
the the the the
where c+ijand gij are the average disturbances of deviatoric stress and strain-rate in the matrix, respectively. Then we add a single void to the composite system D (the matrix and voids). Eshelby’s equivalent inclusion method yields in D 2po(ei + ~ij + e, - eQ) = 0
(2)
2/J&eij + 4 - e fi)
(3)
CTfj
=
where gij and eij are the disturbances of deviatoric stress and strain-rate due to this single void, respectively, and eQ is called “eigenstrain” and defined as having a non-zero value in n and
Fig. 1. A theoretical model for computing the effective viscosity p.
Void growth in a viscous metal
1085
being zero in D - R. Since /Dar, d V = 0, we obtain (1
-f)C?ij
+f(Uij)* = 0
(4)
where f is a volume fraction of void and ( )o denotes the volume average in C!. From eqns (3) and (4) we have Zijt f( eij - e Q)= 0.
(5)
According to ~shelby~S], e;j is related to ef as
where Sirk,is called the Eshelby’s tensor and is a function of the geometry of fi and the matrix material properties. For a given applied strain-rate y. (Fig. l), eigenstrain e& is obtained from eqns (Z), (5) and (6) *-
e32 -
YO
*-
(7)
e23 (1 -
f)(l
-
=3232)’
In order to obtain the effective viscosity p, we use the equivaIence of the energy dissipation 4~: = 4fioyz0+ f4poroeT2.
(8)
The detailed derivation of eqn (8) is given in Appendix A. From eqns (7) and (8), the effective viscosity of a viscous body containing voids, p is obtained as f-p--
5 f 3(l -f)
(9)
where S3232 = l/S for a sphericaf void embedded in an incompressible matrix. It should be noted here that by the original Eshelby’s method without MartTanaka’s back stress analysis the formula corresponding to eqn (9) becomes
Thus the denominator in the second term of eqn (9), (1 - f) takes care of the interaction between voids which is enhanced at a larger volume fraction of void. 2.2 The efective expansion viscosity K The calculation model for obtaining K is different from that in Fig. 1 such that each void is given a vofumetric strain-rate ~7~~. Then the increase in the macroscopic vofumetric strain-rate, h is given by A = f&k. (11) Eshelby’s equivalent inclusion method yields the stress in R Uij
=
-
SijPo= A&j(Z& - e&) t Z~O(~j- et)
(12)
where PO is the pressure in Q and becomes zero in the case of void. Since ho in eqn (12) is Lame’s constant of the incompressible matrix, we must have (13) Thus the imposed volumetric strain-rate in R can be replaced by the equivaient inclusion with
1086
M. TAYA
andE.
D. SEIDEL
the eigenstrain e$ = Zkk.For a given eigenstrain e$. the total energy dissipation W per unit volume is given by [12, 161 w = 401 - f)CLO lS(l_Vo)[4(e~12+e~~~+rZ1~)+(5~o+I)
In the previous equation the interaction among inclusions comes in through the factor (1 -f)[12], and v. is the matrix Poisson’s ratio. From eqns (11) and (13) the non-vanishing eigenstrain in our problem is e:, = e$ = e& = 3f’
(15)
A substitution of eqn (15) and v. = l/2 into eqn (14) yields the energy dissipation per unit volume as w = 4cLo(1- f
3f
w
(16)
If a viscous body containing numerous voids is treated as a homogeneous material with volumetric strain-rate A, then the corresponding energy dissipation per unit volume W can be expressed as W=KA’
where
K
(17)
is the effective expansion viscosity. By equating eqn (16) to eqn (17) we obtain 118)
K=4po:f- f, When a volume fraction of void
f is small, eqn (18) can be approximated as
K=4cLo.
(19)
3f
Thus we have recovered the formula obtained by Batchelor[l7]. 2.3 The effective Poisson’s ratio u For the computation of void growth in the next section, it is convenient to define the effective Poisson’s ratio (Y) of a porus viscous body which is assumed to be isotropic. As in the case of isotropic linearly elastic body, ZJis related to k and K as (3K-2/.l)
(20)
' = 2(3Ktp)'
A substitution of eqns (9) and (18) into eqn (20) yields 2p-I v=4p
P=
(1-f)
Tf.
f]l5(1-f,
(21) 1
It should be noted in eqn (21) that v becomes l/2 when f = 0.
Void growth in a 3. COMPUTATION
viscousmetal
1087
OF VOID GROWTH
A theoretical model for void growth is basically the same as that used in the previous section (Fig. 1) except for the far field boundary condition and the shape of void as shown in Fig. 2, which shows the current configuration under the applied strain-rate. Focusing on a typical ellipsoidal void (for example, located in the center in Fig. 2), we consider a mixture of pure matrix and the other voids (indicated by dashed curves) as a homogeneous, isotropic and compressible viscous material. Denote the domain of this void and the infinite body by R and 0, respectively. Hence the domain of the compressible matrix becomes II-Q. We denote the major (along x3-axis) and minor (along x1-, x2 - axes) of the ellipsoid by c and a, respectively. A factor t in Fig. 2 takes value on 1, 0, - Y which corresponds to the cases of triaxial strain, uniaxial strain and uniaxial tension, respectively. eA
t
-
X3
J-
X2
XI
-ten
J feA Fig. 2. A theoretical model for the void growth.
Following Eshelby[8,9], the equivalent inclusion method yields in fz A&j(e&
+
ekk -
efk)+ 2&e: +
eij
-
(22)
ef) = 0
where A = K-213~~ p and K are already defined in the previous section by eqns (9) and (18), respectively, ef is the applied strain-rate, eij is the strain-rate disturbed by Q and et is the eigenstrain defined in Q and becomes zero in D-fl. As before, the following relation holds
where Sijk/is a function of c, a and v and its explicit expression is given in Appendix B. We formulate two cases separately: (i) 5 = - V,0 and (ii) 5 = 1. 3.1 When(=-vand 0 In this case the shape of a deforming void is proiate spheroid and the state of stress and strain-rate in Q can be considered to be transversely isotropic; et1 = ezz and ef, = ef2. Setting i = j in (22) and noting that the value of 3A + 2~ remains finite for f # 0, we obtain eD,+e, -et=&
(24)
With eqn (23) and 6?pp ’ = (1-f 2[)e,+ eqn (24) becomes 20
-&II
-
&22-
S33defrr
+ (1 - %%333
-
S3333).
4
= (I+
WeA-
(25)
Settling ii = 33 in eqn (22) and using eqns (23) and (24), we obtain 2S3311e:1+(S3333_l)e:3=-eA*
WJ)
1088
M. TAYA and E. D. SEIDEL
We solve for el;and e& in eqns (25) and (26) to obtain * -__!B eA A
(27)
ell-
* -2 B
e33-
eA
A
(28)
where A = (1 - S,,,, - S,,zz)(l B,
= IS,,,,
t
60
-
s3333)
-
2&,33&3,,
S3333N
(29)
B2=1-S,,,,-S,22+%?33,,.
The total strain-rate in 0 is given by
= 5eA + b!GIII+ SIi22)eTI f S1133eT3
c - =
43
f
(30)
e33
c
=
eA f
2&3,,et,
+
(31)
s3333e&
where a super dot stands for a time derivative. From eqns (27)-(31), the rates of void growth in x, or x2-direction, d/a and in x3-direction C!/Ccan be obtained as (32) _i
=
1 + @,S33,,
+ B2S3333)
A
c
eA
(33)
1 .
When f becomes zero; the surrounding matrix being strictly incompressible, the cases of 5 = 0 and - v are reduced to the cases of triaxial tension and uniaxial tension (v = l/2), respectively, and are worked out by Budiansky et al.[ll]. In the case of 5 = 0 the void growth rate becomes infinite upon applied strain and corresponds to the ordinate in Fig. 7. Whereas the case of [ = - l/2 (uniaxial tension) can be formulated explicitely to yield Lila and (:/c as
ri -=
--f
a
1 2
($,I,
+
S,,22)&
+
S,133B2
I
eA
; = {1 + 2&3,,B, + Sm&kA
(35)
where B
=
’
B
=
2
c,,
(;G*+c,2)
(C,,G-
y
c,,c*,,
&2,+c,,)
(G2C2,
-
CllC22)
h-2+
A3 i 2 4(pz_l)lo I
c,2=-~+{_I+qc (P - 1)
(34)
1
2 4(p - 1)I O
1089
Voidgrowth in a viscousmetal
(34) and where 0, lo and S‘,, are defined in Appendix B. The values of Silk,must be obtained from
Appendix B by substituting l/2 for Y. 3.2 When 5 = I In this case the mode of deformation is spherical symmetric
eli = e22= e33
e7, = ef2 = ef3 = e*.
(37)
From eqns (22), (23) and (37), we have 1 eh = (1 - s*,,i -2s,1**)ek
(38)
Hence the rates of void growth, ci/a( = i/c) are obtained as 1
d E -=-= 0
c
(I-
sm,-%*2)
(39)
h.
For a given volume fraction of void and a constant &?A we can integrate directly eqn (39) to obtain a
-=exp a0
i
(l-S,,fiAt2S,,**)
I
Gw
where eAt is the total strain applied at time t, and a0 is the initial radius of a spherical void. 4.NUMERICALRESULTS
ANDDISCUSSIONS
In order to simulate the void growth in a ductile metal deforming at strain-rate, we use the following data eA= lo4 see-’ eAtf =
0.4
tr = 4 x to-’ set where fr is the duration of the constant strain-rate applied eA.Three different values off are also used as a parameter: f = 0.05,O.l and 0.3. We also consider three kinds of loading conditions, (i) triaxial strain (5 = l), (ii) uniaxial strain (5 = 0) and (iii) uniaxial tension (5 = - v), As a measure of void growth, natural tensile strains, ln(c/co) and ln(u/~~) are used with the initial shape of a void being spherical. In order to compute ln(c/co) and Malao), we use incremental method with 50 steps except for ,_$ = 1, for which analytical results are given by eqn (37). The results for 5 = 1, 0 and - v are plotted as solid curves for In(c/co) and dashed curves for ln(u/uo) in Figs. 3-5, respectively. It should be noted in Fig. 3 that the results for In(c/co) and ln(a/ao) coincide due to spherical symmetry. It follows from Figs. 3-5 that the rate of increase in ln(a/uo) and MC/co) is largest for LJ= 1 (triaxial strain) and smallest for 5 = - v (uniaxial tension). The effect of f on ln(a/ao) and ln(c/co) becomes significant for .$= 1 and 0, while this effect has less influence on In(uluo) and ln(cfco) as seen from Fig. 5 where the case of incompressible surrounding matrix (‘j = 0) is also shown.
M. TAYA and E. D. SEIDEL
1091
Void growth in a viscous metal
0.1
0.2
I
I
0.3
0.4
Fig. 5. In(n/aO)(dashed curves) and In(c/c,) (solid curves) vs e,.,t for 6 = -
Fig. 6. In(V/V,J vs e,& for .$= 1 (triaxial strain). UES Vol. 19, No. S-D
eAt
o
(uniaxial tension).
1092
M. TAYA and E. D. SEIDEL
It has been well observed experimentally[l8,19] that the void growth in the necked region of a tensile specimen is enhanced more significantly than that in the parallel region. The state of deformation in the necked region can be simulated by the case of uniaxial straining (5 = 0) (Fig. 4). Thus the void in the necked region can grow transversely as well as along the straining direction. This has been also observed by Nemat-Nasser and Taya in their model study[5]. The transverse void growth leads to the coalescense of two or more neighboring voids. In order to investigate the void growth as a volume increase, we have plotted ln( Vl V,) vs t~,~t for cases of ,$= 1,O and - v in Figs. 6-8, respectively, where V and V, are the void volume at the current and the initial stages, respectively. In these figures the case of f = 0 which has been recently studied by Budiansky et al.[ll] is also shown. It is seen from Figs. 6-8 that the increase in the void volume In( V/ V,,) becomes larger as f decreases for the cases of 5 = I and 0, but for the case of 5 = - u (uniaxial tension) the above trend becomes reverse as seen from Fig. 8.
Fig. 7. In( Vi V,) vs e,4t for .$= 0 (uniaxial strain).
Fig. 8. In( V/V,) vs e,t for 5 = - v (uniaxial tension)
Void growth in a viscous metal
1093
The proceeding discussions are led to the following conclusions: (i) The present model based on a surrounding matrix predicts smaller rate of void growth for the cases of 5 = 1 and 0 compared with that by the incompressible model[ll], but for the case of 6 = - v (uniaxial tension) the void growth rate by the present model becomes larger than that by the incompressible model. (ii) The effect of volume fraction of void f on the void growth is more significant for the cases of t = 1 and 0 and less for the case of 5 = - v. (iii) The present model can be applicable to the dynamic case if the applied strain-rate remains constant. However, the state of constant strain-rate of order 10“set-’ or higher is difficult to achieve experimentally. Hence the justification of the present model still remains to be made in the future. REFERENCES [1] F. A. McCLINTOCK, 1. Appl. Mech. 35,363 (1%8). [2] J. R. RICE and D. M. TRACEY, J. Mech. Phys. Solids 17,201 (1969). [3] D. M. TRACEY, Engag Fract. Mech. 3,301 (1971). [4] A. NEEDLEMAN, L Appl. Mech. 39,964 (1972). [S] S. NEMMAT-NASSER and M. TAYA, Jntl. J. Solids Structures 16,483 (1980). [6] A. KUMAR and R. G. KUMBLE, J. Appl. Phys. 40(9), 3475(1969). [7] A. KUMAR, F. E. HAUSER and J. E. DORN, Acta Metall. 16, 1189(1968). [S] J. D. ESHELBY, Proc. R. Sot. Land. A241,376 (1975). [9] B. A. BILBY, J. D. ESHELBY and A. K. KUNDU, Techtonophysics 28,265 (1975). [lo] I. C. HOWARD and P. BRIERLY, ht. J. Engng Sci. 14, 1151(1976). [ll] B. BUDIANSKY, J. W. HUTCHINSON and S. SLUTSKY, In Mechanics of Solids, The Rodney Hill 60th AnniuersnryVolume (Edited by H. Cl.Hopkins and M. J. Sewell). Pergamon Press, Oxford (1981). [12] T. MORI and K. TANAKA, Acta Metall. 21,571 (1973). [13] M. TAYA and T. MURA, 1. Appl. Mech. To be Published. [14] M. TAYA and T. W. CHOU, Intl. J. So/ids Structures.To be published. [l5] M. TAYA and 1‘. W. CHOU, Proc. Japan-U.S. Conf. on Composite Materials (Edited by K. Kawata), Tokyo, Japan (1981). [I61 T. MURA, Micromechanics of Defect in Solids, Noordhoff, Holland. To be published. [17] G. K. BATCHELOR, An Introduction to Fluid Dynamics, p. 253. Cambridge University Press, New York (1970). [ill] D. P. BAUER and S. J. BLESS, Report AFML-TR-79-4021(1979). [19] T. B. COX and J. R. LOW, Jr., Metall. Trans. 5, 1157(1974). (Received 2 October 1980)
APPENDIX A The total energy dissipation in an infinite body containing voids, E is given by E=
(ci t uii)(e$t tij t eij)du. ID
The integrand in eqn (Al) can be expanded as
(Uft uij)(i)(e$ t 2ijt eij)=
ufef t
Uf(&j+
eij) t uij(e; t
Zij t eij).
642)
We note that the third term in eqn C.42) upon integration over D vanishes by use of Gauss’ divergence theorem and the following relations ef = I/Z(U~~ t I&)
t?ij= l/2( iii t qi, Q
uijnj
=
I/2(Vu
t Uj.i)
=O on jD(
oij.j=0
643)
in D
(A4) 645)
where (DIis the outer boundary of D. The second term in eqn (A2) can be rewritten as flf(Zijt eij)= uK2ij t eij- e$) t cTfe$ = 2p0e&(t?ij t eij- e$) t u$ez = p.eO I, Df &* u 1,
(‘46)
Where eqn (3) is used to derive eqn (h6). The first term on the right hand side of eqn (A6) vanishes upon integration over D since JDuijd V = 0.
1094
M. TAYA
and E. D. SEIDEL
Thus eqn (Al) is reduced to E = u$e:
t f&:
(A?
where the above E denotes the energy dissipation per unit volume. If we consider the composite body D (matrix and voids) as a homogeneous viscous body subjected to the applied strain-rate -m then the corresponding energy dissipation per unit volume is given by E =4&
(A81
By equating eqn (A7) to eqn (AS), we can obtain the effective viscosity p. APPENDIX
B
The Eshelby’s tensors Sii,., are given in [LX].We present them in a more convenient form below. The suffices ijkl in the Eshelby’s tensors are consistent with the coordinate systems (Figs. I and ?). I. When Cl is sphere 7 J,,,,=
S 2z*?=S3333=-
S,,2? = &11=
(7 - 5U) l5(l -v)
(I -Sv) SN, = - l5(l _ r, (Bl)
2. When fI is prolate spheriod along x,-axis
sIlll=S**~~=~[3jlt~)t(l-2’-~)‘“] S 33”=‘+(~_“)(pz_~) S ,1:2=S221i=&
1
( l-2v+(B2_1)
[It&-
1
3PZ
I r.
11
&jtl-2s
IO
____-+‘(“-(I
I St,,3 = S?233= *(I _ V) S 33,
___ I 4(lev)
p’
P2
I
___’
I = s3322 = 2(1
/32 _ I
4
- ,I-,,+_!L I
_ v) r
-2V))41
82 _ 1
+I4
B2_l !
t
3
p,_1+N-2~0)
-1
where
p=L
2P
IO= n(/3(p2 CP - 1)
a
- I)“’ - cash-’ fl}
43 11
iB2)