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HIPing conditions for processing of metal matrix composites using the continuum theory for sintering—I. Theoretical analysis
Acfa mater.Vol. 44, No. 2,pp.707-713,1996
Pergamon 0956-7151(95)00179-4
Elsevier Science Ltd Copyright 0 1996 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00
HIPING CONDITIONS FOR PROCESSING OF METAL MATRIX COMPOSITES USING THE CONTINUUM THEORY FOR SINTERING-I. THEORETICAL ANALYSIS E. OLEVSKY’, H. J. DUDEK*t and W. A. KAYSSER’ ‘Max-Planck-Institute 70569 Stuttgart and
for Metal Research, ‘Institute for Materials Linder Hiiher,
Powder Metallurgy Laboratory, HeisenbergstraBe 5, Research, German Aerospace Research Establishment, D-51 140 KSln, Germany
(Received 2 December 1994; in revised form 24 March 1995) Abstract-The model of hot isostatic pressing fibre-reinforced metal matrix composites is elaborated. On the basis of the rheological relationships obtained, taking into consideration the structural anisotropy and large void imperfections, the kinetics of porosity and imperfection parameters’ evolution are analysed. The algorithm of the calculation for the distribution of micro-stresses and deformations around fibres, is elaborated for the cases of one- and two-layers fibre coatings.
of “spontaneous” densification (sintering), caused by the surface tension in the main, were considered separately in many cases. The continuum theory of sintering [16], permitting the description of the nonlinear-viscous flow of porous bodies, represents the basis for the unification of the above-mentioned approaches into the general model of consolidation in porous media. From the modelling point of view, the two problems are essential for the consideration of the deformation behaviour of the afore-mentioned fibrereinforced MMC. The first one, takes into consideration the anisotropic mechanical (in particular viscous) properties. In the present work, the example of the application of the continuum theory of sintering, developed for the case of modelling the deformation of a transversal-isotropic body, is proposed. The second one, takes into account the large imperfections (large-scale pores) which influence the shrinkage kinetics during HIPing [18]. From the topological point of view, these imperfections cause “the heterogeneity of the phase of void”, because the porous structure becomes bimodal in their presence. The modelling of the densification of the bimodal materials is a separate problem. The pores of different scales have different shrinkage intensities and, in this connection, can hinder the densification of each other. A number of attempts of solving the given problem are known to have been undertaken for the cases of hot isostatic pressing [19,20] and sintering [21,22]. In the present paper the approach proposed in Ref. [22] is developed taking into consideration the solid phase anisotropic viscous properties and external forces influence.
1. INTRODUCTION
The ability to make parts and components by hot isostatic pressing (HIP), of fibre reinforced metal matrix composites (MMC), is seen as a necessary additional process step to enable technology to further exploit the advantageous property characteristics of these materials. The industrial objective is the development of a cost-effective route for making parts and components, out of MMCs, by HIP. This can be achieved by developing an understanding of the hot deformation characteristics of MMC materials, so leading to the formulation of a predictive mathematical model, capable of selecting the optimum hot isostatic pressing conditions to achieve the best engineering properties possible. Mathematical modelling of HIPing processes is part of a rather wide field of deformation process modelling in porous media. In the last 15-20 yr the methods of continuum mechanics have been intensively used in this field. The theory of plasticity of porous media and ideologically close approaches [l-4], were successfully used for the solution of the problem of cold working by compression of powders and porous materials. Recently, the models came to be widely used in practice for describing the densification in porous media, taking into consideration a temperature treatment. The analysis of hot isostatic pressing [5S121, as well as sintering [13-171, should be related to this field of work. At the same time, the consolidation processes influenced by external forces and processes
tTo whom
all correspondence
should
be addressed. 707
708
OLEVSKY et al.: HOT ISOSTATIC PRESSING CONDITIONS-I 2. CONTINUUM THEORY OF SINTERING AS THE THEORETICAL BASIS FOR THE MODEL CONSIDERATION
In the framework of the theory investigated, a porous medium is considered as the continuum whose point is a representative volume, large enough in comparison with a pore (particle) and small enough in comparison with the size of a porous object as a whole. At the same time, a porous medium is considered as a two-phase material, including the phase of substance (“solid phase”) and the phase of void (pores). The solid phase, in turn, can be a multiphase material. The continuum theory of sintering ascertains the following relationship (rheological equation), between stresses bij and strain rates t,, in a porous body (see Appendix) o(@)
a.-=~((P4j+($ ‘J
-~cp)ts,)+P,s,
(1)
ment, when the surface tension is negligibly small, compared to the stresses corresponding to externally applied forces. And the case when all the items are present can correspond to sintering under pressure. The effective strain rate is connected with the parameters of a deformation state as in the following
~ Jq(fI)j2 + W=J&j
W)e2.
(2)
Here C and d are the first invariant of the strain rate tensor (the rate of volume change) and the second invariant of the strain rate deviator (the rate of shape change), respectively. P and y can be expressed through the tensor components of the strain rates P = t,s,, 1’= ((6, - es,j/3)(P, - P&,/3))“2.
(3)
The equivalent effective stress 6, as determined from equation (l), is a function of I& The relationship for o( I@ is identical with the relationship for the dependence “stress intensity-strain rate intensity” in the matrix phase. For the linear-viscous properties of a solid phase
where ~(0) and $(e) are functions of porosity 0, being analogous to the bulk and shear viscosity moduli; P, is an effective Laplace stress (“effective” means that the parameter pointed out is also a function of porosity and corresponds to the result 0 = 2% I@ (4) of the surface tension’s influence upon the porous object as a whole); G,Kronecker’s symbol; 0 and @’ where I],, is the shear viscosity of a solid phase. are the equivalent effective stress and strain rates, Herewith equation (1) can be transformed to respectively; e-the rate of volume change [see o,=2~o(rpP,j+(~ -$p)es,j) + PLSij. (5) equation (3)]. Parameter m was suggested as a generalization of For ideal-plastic properties of a solid phase the Odquist parameter for a porous material. We can obtain I$’as a spatial average intensity of the strain bij = z(J (6) rate of the substance over the representative element where z0 is a solid phase yield limit. [12]. If we determine the correlation between the Then equation (1) has the form average intensity of stress rcM)and strain rate y(M)in solid phase by the formula rcM)=f(vtM)), then ti = F-‘((F(j(M))))
J1_ez, o”=&qq2
(Cp’, + ($ - fq)asij) + Pr8,j. (7)
where aF(vc”))/& =f(jcM)), ( ) = designation of In the general case of the nonlinear-viscous properties spatial averaging, t = time. of a solid phase, the relationship Q = e(w) can be The relationship between I&’and t generalizes the rather complicated. One of the possible examples is rheological properties of a solid phase (substance) to the following degree relationship the properties of the porous object as a whole. It can be shown that expression (1) is relevant for the description of a wide circle of possible technological where A, 0 < n < 1 are rheological constants. processes of porous medial treatment. The item in the It is evident that the limiting cases of n = 0 and left-hand part (aij) corresponds to the influence of the n = 1, for such a material, correspond to its idealexternal forces. The first item in the right-hand part plastic [equation (6)] and linear-viscous [equation (4)] corresponds to the stresses caused by the internal properties. resistance of a solid phase (e.g. to the stresses initiated in a substance of a porous body by the influence of externally applied forces and surface tension); 3. THE MODEL REPRESENTATION OF THE the second item in the right-hand part (PLSij) OBJECT UNDER INVESTIGATION corresponds to the influence of the surface tension. The case of gi, = 0 corresponds to the condition of The volume considered (Fig. 1) contains the free sintering-only surface tension is the moving oriented fibres, with a radius of R,,,coated with a factor for deformation; the case of P, x 0 can correlayer of metal matrix of the radius R,,.There are spond to cold treatment by pressure or to hot treatsmall pores, whose radius is rp, between the coated
OLEVSKY et al.:
HOT ISOSTATIC PRESSING CONDITIONS-I
Fig. 1. Schematic representation of the coated fibres arrangement before HIPing.
fibres. Apart from the small pores, the material also contains large-scale imperfections, with a radius of which are the consequences of “packing R, > VP, mistakes”-an absence of some fibres. The average distance between the imperfection centres is taken as 2R,. The proposed representation permits the reduction of the current HIPing problem to the problem of external hot isostatic compression of a hollow porous
cylinder (Fig. 2) whose internal radius is R,, the external radius is R, and the porosity 0 is caused by the presence of pores, of the radius rp, in the cylindrical layer. The properties of the substance of this porous cylindrical layer are transversal-isotropic, corresponding to its composite content, including the linear-viscous metal matrix which contains the cylindrical fibres. These properties are influenced by the fibre radius and concentration values, coating thickness, etc.
4. ANISOTROPIC-VISCOUS
MODEL
Equation (1) represents a rheological relationship for sintering a material with an isotropic solid phase. For a general case of anisotropic properties of a solid phase, the constitutive relationships of the continuum theory of sintering have the form
where C,,, are components of viscosity tensor which should depend on porosity. In the case of transversal-isotropic properties, the stress components for each point of the investigation object are frz = C& Fig.
2.
Model
representation of investigation.
the
object
under
+ c*,t,
+ C,&
+ PL
urp= czp e, + c,, P, + c,, d, + PL or = c,, P, + C,,L, + c,, c, + P,
(8)
OLEVSKY et al.:
710
HOT ISOSTATIC PRESSING CONDITIONS-I
bi = stress tensor component; di = strain rates tensor component; and C, = viscosity coefficient [23]. For an axisymmetric flow [24] V, = d z,
V, = ar + b r
The imperfection kinetics is determined on the basis of equation (9) b vrlr=R,=uR,+R=R,.
(15)
I
(9)
Using the known values of a and b in equation (14), we have
where Vi = flow velocities; a, b, d = constants.
(16)
In view of equation (9), the strain rate components become
if R,$ R,, and lPLl < IPI, equation simplified to
,+av, aZ
A
‘=pk
=d
w
&-=u
4 __
ar
1
G - Gql
b
k = (C,, + c~(p)c2z - 2C& - c,, - c,,’
r2
Integration
where ei are strain rates. It follows from equations (8) and (10) that
~c=Czqd+u(C,,+C,,)+&,,-C,,)+P,
grn= G,d + 4C,, + Cq,)
Pkdt
Rlin < r,,,i”exp( -1 (11)
have the form
conditions
-J
cr, = 0
(12)
Therefore, have two
(19)
(and 6, = 0).
instead of equation
(20)
(14), we now only
(PLi - P)R;R;
where P = isostatic pressure; PLi = Laplace stress on the surface of the imperfection. Substituting equation (11) into equation (12), we obtain
b = (C,, - C,,)(R:
- R:)
(PL - Pri)R: + (P - PL)R: (C,, + C,,)(R: - R:) ’
a= C,,d + 2C,,u = P - PL
Pk dt).
By considering the fibres as rigid, it is possible to assume the first of the boundary conditions of equation (12) in the simplified form
Gr = 0
4r=R2 = p
(18)
where: At = time of HIP, R,, = initial imperfection radius, R,(t) = current imperfection radius. Let R, < rmin. This is a criterion for the elimination of imperfection, where r,, is some small constant. Then equation (18) can be written as
o, = C,,d + 2C,,a + P,
-;w~r-cq,)fPL.
(17)
of equation (17) gives ln[&]=
The boundary
(16) may be
(21)
Since lPLl < IPI, IPLiI < IPI
C~,d+u(C,,+C,,)+~(C,,-C,,)=P,i-P,
- PR2R2 b=(C,,-C,,);R;-R;)
C,,d+u(C,+Cqq)--$C,.-C,,)=P-P,.(13)
PR; a = (C,, + C,,)(R:
The solution of equation (13) gives
u=
b=
- R:)’
(22)
((P - PM: - (PLI- PdR:)G - (f’ - PdR: - R:)‘G, (R:--R:)[(C,,+C,,)C,,--Cf,l PLi-
P
c,, - c,,
R:R; R:
d=((P-P,)R:-(P,i-P,)R:)C=,-(P-P~)(R:-RR:)(C,,+C,,) (R: - R:)KC,,
+ C&L
-
2'3
(14)
OLEVSKY et al.: In this case equation
(16) can be rewritten -2PR,
Analogous
to equation
5. POROSITY FACTOR
as
R;C,,
RI = (C$ - CZ,,)(R;
(23)
- Rf)’
(15)
F’,,=,,=aR,+;=d,.
(24) 2
From equations (22) and (24) the expression for the evolution of the distance between the imperfection centres follows
d = -PRK,,(R:
+ R:) - C,,(R: - R:)l
2
CC&
-
CZ,,HR:
]16,251
9m= he (1 - Q2
-R:)
C,, = psr + k,, (26)
C,, = k,, - ~“sr are
The previous reasoning did not cover a character of the dependence between the properties of the “regular” part of the material and porosity. As has been mentioned earlier, the coefficients C, must depend upon porosity which changes during HIPing. It was noted that the matrix is itself the heterophase material. It includes the pores-packing cavities between the coated fibres. Therefore, q,,, and v, must depend upon porosity 0
(25)
Consider the layer R, - R, as a composite with rigid inclusions in the form of cylinders. Following [23], we obtain
where p and k respectively. In view of [23]
shear
and
711
HOT ISOSTATIC PRESSING CONDITIONS-I
bulk
moduli,
2-30 v, = 4
(30)
qMe = shear viscosity of a corresponding metal matrix solid phase. Here, the taking into consideration of porosity is carried out with the procedure of changing the matrix properties at each separately considered time step. In this sense the linearity of the medium properties is not violated. At the same time, the correspondence between the effective scalar invariants of stresses (0) and strain rates (m), as has already been noted in C
k,,=k,+T+ l/h
-km
+
l/3(/+ -
PL,)I+ (1 - cMkn + 4/3pL,) (27)
where the indices F and m correspond to the properties of inclusions (fibres) and matrix, respectively; c = concentration of inclusions (c = (Rfi,,/RMJ2). Taking into consideration that pF+ M and k,+ 00, we have from k,, = &, + (l/3 + c)/J,,, p.r = k +q,31r
m
From
m
]k,(l + 2~) + 1/3pL,(7 + 8~11.(28)
the hydrodynamic km =
analogy
%I 3(1 - 2v,)
r/Ill pm = 2(1 + v,) where q, = viscosity of Me-matrix, v, = Poisson’s ratio. Taking into consideration that the used relationships [23] are relevant for the case of small c, apply the self-consistent method [25] (l-iterative), in accordance with which the obtained values of the heterophase material characteristics in equation (28) are ascribed to its matrix, and repeat the calculations in accordance with equation (28). This means the following scheme of the calculations: p,, k,*p&, kO,,=v!n> k,hvc,,k,r.
the item 2, is also linear for a porous material in the case of linear-viscous properties of a solid phase. It should be noted that, for the derivation of the expression for Poisson’s ratio in equation (30), the self-consistent method was used. For high-porous bodies this is not rigorous, if the “phase of void” is a predominant one. Therefore, for high porosity values, the last expression in equation (30) looses its meaning, and in particular, it cannot be used when 0 > 213, e.g. when Poisson’s ratio becomes negative. For this case, equation (30) should be modified [25]. However, the investigation object-MMC compositecannot be considered as a high porous material for the case studied, therefore the form of the last expression in equation (30) does not limit the application of the model. From the mass-conservation law it follows [16]
(31) P = C, + i, + eq (P, = 0) = volume change where rate [equation (3)]. Substitute equation (22) into equation (31)
&2PR:
R:-R:C,,+C,,
1-e (32)
712
OLEVSKY
et al.:
HOT ISOSTATIC
The differential equations (23) (25) and (32) form the close set which can be solved taking into consideration equations (26)-(30). The following expression for the calculation of the radius of a regular pore, obtained on the basis of the continuum theory of sintering, can be added to this set [22]
PRESSING
CONDITIONS-I
used as for the case considered above. The values obtained are substituted into equation (27) for the calculation of the effective viscosity coefficients. For the calculation of the deformation (radial, tangential) distributions, equation (10) is integrated with respect to time cr=b
lMe(1
-
0)’
Taking into consideration that qMe is a function of temperature [26], the HIP temperature cycle can be involved in the model, thus influencing the kinetics of the porosity in equation (32), the pore radius in equation (33), the imperfection radius in equation (23) and the distance between imperfections in equation (25). 6. ESTIMATION
OF STRESS DISTRIBUTION AROUND FIBRES
The formulae of equation (11) represent the function of the stress distribution around imperfections. For fibres we should use the following expression [24] (for radial stress) bf
err =
(1 + .);‘1 - 2vJ
(1 + v&z’+ (3v, - 1) -
r2
1
. (34)
Here Q and v, are viscosity and Poissons ratio of the coating, respectively; r = radial coordinate, uf and b’= constants which can be found from the boundary conditions uf-bf/r21,=,,,=0 orIr=nuc = oFrn
Or2 =[ x
(1 + .,,“(‘;
x
- 2v,,)
(1 fv,,)af+(3v,,-
2v,,)
(l+r&;+(3v,,-1)s
1):
1, 1,
R,, G r G R,
Rci < r.
(36)
Here R, = a radial coordinate of the intercoating boundary. Additional boundary conditions af - bf/Rzi = ai - bi/RH oqIr=R,
eV=I
P,dt
(38)
= %*lr=iQ.
where L, and t, are the radial and tangential deformations, respectively [the expressions for P, and P, are analogous to equation (lo)]. 7. CONCLUSIONS
On the basis of the continuum theory of sintering, the mathematical model for HIPing of fibre reinforced metal matrix composites has been built. The model takes into consideration the solid phase anisotropic properties, as well as an influence of porosity. The model includes the consideration of the porous structure, of the matrix, as bimodal when taking into account the influence of the large imperfections upon the densification kinetics. The relationships obtained permit the calculation of the evolution of porosity, the imperfection radius and the distance between the neighbouring imperfections during HIPing. The algorithm of the calculation, of stress, strain and strain rate distributions around the fibres for the cases of one- and two-layer coatings, is elaborated. The calculation scheme elaborated is used for analysing the HIPing conditions of the Sic-fibre reinforced titanum alloys [26].
(35)
where oy” = average radial stress in accordance with an average value of equation (11). For a two-layer coating [26], we should use two functions for 0, (lower index corresponds to the number of coating) Or1= (1 + .,:t;
&dt;
‘P
i, = P
(37)
In the case of a two-layer coating the solid phase is already a three-phase material. Therefore, for the internal coating and fibres, equation (28) should be
REFERENCES 1. H. A. Kuhn and C. L. Downey, Znt. J. Powder Metall. 7, 15 (1971). 2. R. J. Green, Znt. J. Mech. Sci. 14, 215 (1972). 3. S. Shima and M. Oyane, In?. J. Mech. Sci. 18, 285 (1976). 4. A. L. Gurson, J. Engng Mater. Technol. 99, 2 (1977). Li, M. F. Ashby and K. E. Easterling, 5. W.-B. Acta metall. 35, 2831 (1987). 6. T. Shinke, T. Soh, T. Nakagawa and A. Nohara, Steel Engng Rep. 37, 2932 (1987). 7. M. Abouaf and J. L. Chenot et al., Znt. J. Num. Meth. Engng 25, 191 (1988). 8. J. M. Duva and P. D. Crow, Acta metall. 40, 3 1 (1992). 9. J. Besson and M. Abouaf, J. Am. Ceram. Sot. 75,2165 (1992). 10. D. M. Elzey and H. N. G. Wadley, Acta metall. 41,2297 (1993). Il. E. Olevsky, M. Shtem and V. Skorohod, Hot Zsostatic Pressing 93 (edited by L. Delaey and H. Tas), p. 45. Elsevier, Amsterdam (1994). 12. A. Maximenko, E. Olevsky, M. Panfilov and M. Shtern, Hot Zsostatic Pressing 93 (edited by L. Delaey and H. Task D. 61. Elsevier, Amsterdam (1994). 13. E. Old&y and V. Skorohod, Technological and Construction Plasticity of Porous Materials. IPMS NAS Ukraine, 97 (in Russian) (1988). 14. H. Riedel, _ Ceramic ._. Powder ~~~Science ZZZ, p. 619. American Ceramic Society, Westerville, Ohio (1990).
OLEVSKY
et al.: HOT ISOSTATIC
Mech. Gran. Mater. Powder Syst. 15. R. M. McMeeking, ASME 37, 51 (1992). E. A. Olevsky and M. B. Shtern, 16. V. V. Skorohod, Sci. Sinter. 23, 79 (1991). 17. A. C. F. Cocks, Acta metall. 42, 2191 (1994). 18. A. Frisch, W. A. Kaysser and G. Petzow, Proc. of World Conj. on Powder Metallurgy, p. 237. London (1990). 19. S. Nair and J. K. Tien, Metall. Trans. A18A, 97 (1987). Acta metall. 37, 1645 20. K. H. Li and P. D. Funkenbusch, (1989). 21. G. W. Scherer, J. Am. Cerum. Sot. 67, 709 (1984). 22. E. Olevsky and R. Rein, Proc. of 13th Int. Plansee-Sem (edited by H. Bildstein and R. Ecks), Vol. 1, p. 972. Metallwerk Plansee, Reutte (1993). Mechanics of Composite Materials. 23. R. M. Christensen, Wiley, New York (1979). 24. L. D. Landau and E. M. Lifshitz, Theoretical Physics, Vol. VII-Theory of Elasticity, pp. 33-34. Nauka, Moscow (1987). Rheological Basis of Theory of Sintering. 25. V. Skorohod, Naukova Dumka, Kiev (in Russian) (1972). 26. E. Olevsky, H. J. Dudek and W. A. Kaysser, Acta metall. 44, 715 (1996).
NOMENCLATURE A = C,,, = c= P= P,, = k, = k,, = k, = n= P =
power-law creep constant viscosity tensor component fibre concentration rate of volume change strain rate tensor component bulk viscosity module of a fibre bulk viscosity module bulk viscosity module of a composite matrix power-law creep exponent hydrostatic pressure applied (P < 0 for isostatic compression) P, = effective Laplace stress P, = Laplace stress on the imperfection surface R, = current imperfection radius R,,, = initial imperfection radius R, = distance between imperfections R,, = fibre radius R,, = coating radius R,, = radial coordinate of the intercoating boundary for the criterion of the elimination of rmln = constant imperfection rr, = current pore radius I’.c= radial velocity W = equivalent effective strain rate of a porous material y = rate of shape change jc”) = average intensity of strain rates in a solid phase 6,j = Kronecker’s symbol t,, to = radial and tangential deformation of a material around fibre r),, = shear viscosity of a solid phase qc, (qc2) = shear viscosity of the first (second) coating rlMe= shear viscosity of a metal matrix solid phase qrn = shear viscosity of a composite matrix 0 = porosity
PRESSING nr = p,, = p, = v,, (v,r) = v, = blj = oy” = o(m) r0 rc”) cp, $
= = = =
CONDITIONS-I
713
shear viscosity module of a fibre shear viscosity module shear viscosity module of a composite matrix Poisson’s ratio of the first (second) coating Poisson’s ratio of a composite solid phase stress tensor component average radial stress in the porous matrix around a fibre equivalent effective stress of a porous material yield limit of a solid phase average intensity of stresses in a solid phase functions of porosity.
APPENDIX
Derivation of the Constitutive Relationships Consolidation of a Porous Body [I61 In accordance with the principles dynamics, the equation cr,e, - $
of irreversible
fir thermo-
B@,, 0) = 0
-
(Al)
where F = free energy of the system, fi = rate of energy dissipation, t = time, serves as the basis for the model. Herewith aF -=P,i
(A-3
at
where P, is an effective Laplace stress, t! = volume change rate. The dissipation of energy is conditioned by the processes, taking place in a substance. However, if the description corresponds to the macro-level, then fi has to depend on the averaged parameter of the solid phase flow. The effective strain rate I$’[equation (2)] is taken as such a parameter. Taking into consideration that I? is a quasihomogeneous function, it is possible to represent (Al) in the form
&a' c7,, - PL 6, - *z
“,, = 0. (-43) “3 1, ( From this equality, by virtue of the principles of irreversible thermodynamics, equation (1) follows. Equation (1) represents the relation between a stress tensor component oil and strain rate tensor component P, in a generalized frame of reference. For example, for axisymmetrical coordinate system (with coordinates r, cp, z) the radial stress can be written:
=F(($
+j)P,+(+
P,, P,, L?~= the corresponding tensor. For cp and + the following q = (1 - 0)2, (0 = porosity).
-fcp)(P,+&,))+P, components expressions
of
(A4) the
strain
can be used [25]
* = 2ff?(l - e)/3e
(A5)
Pergamon 0956-7151(95)00179-4
Elsevier Science Ltd Copyright 0 1996 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00
HIPING CONDITIONS FOR PROCESSING OF METAL MATRIX COMPOSITES USING THE CONTINUUM THEORY FOR SINTERING-I. THEORETICAL ANALYSIS E. OLEVSKY’, H. J. DUDEK*t and W. A. KAYSSER’ ‘Max-Planck-Institute 70569 Stuttgart and
for Metal Research, ‘Institute for Materials Linder Hiiher,
Powder Metallurgy Laboratory, HeisenbergstraBe 5, Research, German Aerospace Research Establishment, D-51 140 KSln, Germany
(Received 2 December 1994; in revised form 24 March 1995) Abstract-The model of hot isostatic pressing fibre-reinforced metal matrix composites is elaborated. On the basis of the rheological relationships obtained, taking into consideration the structural anisotropy and large void imperfections, the kinetics of porosity and imperfection parameters’ evolution are analysed. The algorithm of the calculation for the distribution of micro-stresses and deformations around fibres, is elaborated for the cases of one- and two-layers fibre coatings.
of “spontaneous” densification (sintering), caused by the surface tension in the main, were considered separately in many cases. The continuum theory of sintering [16], permitting the description of the nonlinear-viscous flow of porous bodies, represents the basis for the unification of the above-mentioned approaches into the general model of consolidation in porous media. From the modelling point of view, the two problems are essential for the consideration of the deformation behaviour of the afore-mentioned fibrereinforced MMC. The first one, takes into consideration the anisotropic mechanical (in particular viscous) properties. In the present work, the example of the application of the continuum theory of sintering, developed for the case of modelling the deformation of a transversal-isotropic body, is proposed. The second one, takes into account the large imperfections (large-scale pores) which influence the shrinkage kinetics during HIPing [18]. From the topological point of view, these imperfections cause “the heterogeneity of the phase of void”, because the porous structure becomes bimodal in their presence. The modelling of the densification of the bimodal materials is a separate problem. The pores of different scales have different shrinkage intensities and, in this connection, can hinder the densification of each other. A number of attempts of solving the given problem are known to have been undertaken for the cases of hot isostatic pressing [19,20] and sintering [21,22]. In the present paper the approach proposed in Ref. [22] is developed taking into consideration the solid phase anisotropic viscous properties and external forces influence.
1. INTRODUCTION
The ability to make parts and components by hot isostatic pressing (HIP), of fibre reinforced metal matrix composites (MMC), is seen as a necessary additional process step to enable technology to further exploit the advantageous property characteristics of these materials. The industrial objective is the development of a cost-effective route for making parts and components, out of MMCs, by HIP. This can be achieved by developing an understanding of the hot deformation characteristics of MMC materials, so leading to the formulation of a predictive mathematical model, capable of selecting the optimum hot isostatic pressing conditions to achieve the best engineering properties possible. Mathematical modelling of HIPing processes is part of a rather wide field of deformation process modelling in porous media. In the last 15-20 yr the methods of continuum mechanics have been intensively used in this field. The theory of plasticity of porous media and ideologically close approaches [l-4], were successfully used for the solution of the problem of cold working by compression of powders and porous materials. Recently, the models came to be widely used in practice for describing the densification in porous media, taking into consideration a temperature treatment. The analysis of hot isostatic pressing [5S121, as well as sintering [13-171, should be related to this field of work. At the same time, the consolidation processes influenced by external forces and processes
tTo whom
all correspondence
should
be addressed. 707
708
OLEVSKY et al.: HOT ISOSTATIC PRESSING CONDITIONS-I 2. CONTINUUM THEORY OF SINTERING AS THE THEORETICAL BASIS FOR THE MODEL CONSIDERATION
In the framework of the theory investigated, a porous medium is considered as the continuum whose point is a representative volume, large enough in comparison with a pore (particle) and small enough in comparison with the size of a porous object as a whole. At the same time, a porous medium is considered as a two-phase material, including the phase of substance (“solid phase”) and the phase of void (pores). The solid phase, in turn, can be a multiphase material. The continuum theory of sintering ascertains the following relationship (rheological equation), between stresses bij and strain rates t,, in a porous body (see Appendix) o(@)
a.-=~((P4j+($ ‘J
-~cp)ts,)+P,s,
(1)
ment, when the surface tension is negligibly small, compared to the stresses corresponding to externally applied forces. And the case when all the items are present can correspond to sintering under pressure. The effective strain rate is connected with the parameters of a deformation state as in the following
~ Jq(fI)j2 + W=J&j
W)e2.
(2)
Here C and d are the first invariant of the strain rate tensor (the rate of volume change) and the second invariant of the strain rate deviator (the rate of shape change), respectively. P and y can be expressed through the tensor components of the strain rates P = t,s,, 1’= ((6, - es,j/3)(P, - P&,/3))“2.
(3)
The equivalent effective stress 6, as determined from equation (l), is a function of I& The relationship for o( I@ is identical with the relationship for the dependence “stress intensity-strain rate intensity” in the matrix phase. For the linear-viscous properties of a solid phase
where ~(0) and $(e) are functions of porosity 0, being analogous to the bulk and shear viscosity moduli; P, is an effective Laplace stress (“effective” means that the parameter pointed out is also a function of porosity and corresponds to the result 0 = 2% I@ (4) of the surface tension’s influence upon the porous object as a whole); G,Kronecker’s symbol; 0 and @’ where I],, is the shear viscosity of a solid phase. are the equivalent effective stress and strain rates, Herewith equation (1) can be transformed to respectively; e-the rate of volume change [see o,=2~o(rpP,j+(~ -$p)es,j) + PLSij. (5) equation (3)]. Parameter m was suggested as a generalization of For ideal-plastic properties of a solid phase the Odquist parameter for a porous material. We can obtain I$’as a spatial average intensity of the strain bij = z(J (6) rate of the substance over the representative element where z0 is a solid phase yield limit. [12]. If we determine the correlation between the Then equation (1) has the form average intensity of stress rcM)and strain rate y(M)in solid phase by the formula rcM)=f(vtM)), then ti = F-‘((F(j(M))))
J1_ez, o”=&qq2
(Cp’, + ($ - fq)asij) + Pr8,j. (7)
where aF(vc”))/& =f(jcM)), ( ) = designation of In the general case of the nonlinear-viscous properties spatial averaging, t = time. of a solid phase, the relationship Q = e(w) can be The relationship between I&’and t generalizes the rather complicated. One of the possible examples is rheological properties of a solid phase (substance) to the following degree relationship the properties of the porous object as a whole. It can be shown that expression (1) is relevant for the description of a wide circle of possible technological where A, 0 < n < 1 are rheological constants. processes of porous medial treatment. The item in the It is evident that the limiting cases of n = 0 and left-hand part (aij) corresponds to the influence of the n = 1, for such a material, correspond to its idealexternal forces. The first item in the right-hand part plastic [equation (6)] and linear-viscous [equation (4)] corresponds to the stresses caused by the internal properties. resistance of a solid phase (e.g. to the stresses initiated in a substance of a porous body by the influence of externally applied forces and surface tension); 3. THE MODEL REPRESENTATION OF THE the second item in the right-hand part (PLSij) OBJECT UNDER INVESTIGATION corresponds to the influence of the surface tension. The case of gi, = 0 corresponds to the condition of The volume considered (Fig. 1) contains the free sintering-only surface tension is the moving oriented fibres, with a radius of R,,,coated with a factor for deformation; the case of P, x 0 can correlayer of metal matrix of the radius R,,.There are spond to cold treatment by pressure or to hot treatsmall pores, whose radius is rp, between the coated
OLEVSKY et al.:
HOT ISOSTATIC PRESSING CONDITIONS-I
Fig. 1. Schematic representation of the coated fibres arrangement before HIPing.
fibres. Apart from the small pores, the material also contains large-scale imperfections, with a radius of which are the consequences of “packing R, > VP, mistakes”-an absence of some fibres. The average distance between the imperfection centres is taken as 2R,. The proposed representation permits the reduction of the current HIPing problem to the problem of external hot isostatic compression of a hollow porous
cylinder (Fig. 2) whose internal radius is R,, the external radius is R, and the porosity 0 is caused by the presence of pores, of the radius rp, in the cylindrical layer. The properties of the substance of this porous cylindrical layer are transversal-isotropic, corresponding to its composite content, including the linear-viscous metal matrix which contains the cylindrical fibres. These properties are influenced by the fibre radius and concentration values, coating thickness, etc.
4. ANISOTROPIC-VISCOUS
MODEL
Equation (1) represents a rheological relationship for sintering a material with an isotropic solid phase. For a general case of anisotropic properties of a solid phase, the constitutive relationships of the continuum theory of sintering have the form
where C,,, are components of viscosity tensor which should depend on porosity. In the case of transversal-isotropic properties, the stress components for each point of the investigation object are frz = C& Fig.
2.
Model
representation of investigation.
the
object
under
+ c*,t,
+ C,&
+ PL
urp= czp e, + c,, P, + c,, d, + PL or = c,, P, + C,,L, + c,, c, + P,
(8)
OLEVSKY et al.:
710
HOT ISOSTATIC PRESSING CONDITIONS-I
bi = stress tensor component; di = strain rates tensor component; and C, = viscosity coefficient [23]. For an axisymmetric flow [24] V, = d z,
V, = ar + b r
The imperfection kinetics is determined on the basis of equation (9) b vrlr=R,=uR,+R=R,.
(15)
I
(9)
Using the known values of a and b in equation (14), we have
where Vi = flow velocities; a, b, d = constants.
(16)
In view of equation (9), the strain rate components become
if R,$ R,, and lPLl < IPI, equation simplified to
,+av, aZ
A
‘=pk
=d
w
&-=u
4 __
ar
1
G - Gql
b
k = (C,, + c~(p)c2z - 2C& - c,, - c,,’
r2
Integration
where ei are strain rates. It follows from equations (8) and (10) that
~c=Czqd+u(C,,+C,,)+&,,-C,,)+P,
grn= G,d + 4C,, + Cq,)
Pkdt
Rlin < r,,,i”exp( -1 (11)
have the form
conditions
-J
cr, = 0
(12)
Therefore, have two
(19)
(and 6, = 0).
instead of equation
(20)
(14), we now only
(PLi - P)R;R;
where P = isostatic pressure; PLi = Laplace stress on the surface of the imperfection. Substituting equation (11) into equation (12), we obtain
b = (C,, - C,,)(R:
- R:)
(PL - Pri)R: + (P - PL)R: (C,, + C,,)(R: - R:) ’
a= C,,d + 2C,,u = P - PL
Pk dt).
By considering the fibres as rigid, it is possible to assume the first of the boundary conditions of equation (12) in the simplified form
Gr = 0
4r=R2 = p
(18)
where: At = time of HIP, R,, = initial imperfection radius, R,(t) = current imperfection radius. Let R, < rmin. This is a criterion for the elimination of imperfection, where r,, is some small constant. Then equation (18) can be written as
o, = C,,d + 2C,,a + P,
-;w~r-cq,)fPL.
(17)
of equation (17) gives ln[&]=
The boundary
(16) may be
(21)
Since lPLl < IPI, IPLiI < IPI
C~,d+u(C,,+C,,)+~(C,,-C,,)=P,i-P,
- PR2R2 b=(C,,-C,,);R;-R;)
C,,d+u(C,+Cqq)--$C,.-C,,)=P-P,.(13)
PR; a = (C,, + C,,)(R:
The solution of equation (13) gives
u=
b=
- R:)’
(22)
((P - PM: - (PLI- PdR:)G - (f’ - PdR: - R:)‘G, (R:--R:)[(C,,+C,,)C,,--Cf,l PLi-
P
c,, - c,,
R:R; R:
d=((P-P,)R:-(P,i-P,)R:)C=,-(P-P~)(R:-RR:)(C,,+C,,) (R: - R:)KC,,
+ C&L
-
2'3
(14)
OLEVSKY et al.: In this case equation
(16) can be rewritten -2PR,
Analogous
to equation
5. POROSITY FACTOR
as
R;C,,
RI = (C$ - CZ,,)(R;
(23)
- Rf)’
(15)
F’,,=,,=aR,+;=d,.
(24) 2
From equations (22) and (24) the expression for the evolution of the distance between the imperfection centres follows
d = -PRK,,(R:
+ R:) - C,,(R: - R:)l
2
CC&
-
CZ,,HR:
]16,251
9m= he (1 - Q2
-R:)
C,, = psr + k,, (26)
C,, = k,, - ~“sr are
The previous reasoning did not cover a character of the dependence between the properties of the “regular” part of the material and porosity. As has been mentioned earlier, the coefficients C, must depend upon porosity which changes during HIPing. It was noted that the matrix is itself the heterophase material. It includes the pores-packing cavities between the coated fibres. Therefore, q,,, and v, must depend upon porosity 0
(25)
Consider the layer R, - R, as a composite with rigid inclusions in the form of cylinders. Following [23], we obtain
where p and k respectively. In view of [23]
shear
and
711
HOT ISOSTATIC PRESSING CONDITIONS-I
bulk
moduli,
2-30 v, = 4
(30)
qMe = shear viscosity of a corresponding metal matrix solid phase. Here, the taking into consideration of porosity is carried out with the procedure of changing the matrix properties at each separately considered time step. In this sense the linearity of the medium properties is not violated. At the same time, the correspondence between the effective scalar invariants of stresses (0) and strain rates (m), as has already been noted in C
k,,=k,+T+ l/h
-km
+
l/3(/+ -
PL,)I+ (1 - cMkn + 4/3pL,) (27)
where the indices F and m correspond to the properties of inclusions (fibres) and matrix, respectively; c = concentration of inclusions (c = (Rfi,,/RMJ2). Taking into consideration that pF+ M and k,+ 00, we have from k,, = &, + (l/3 + c)/J,,, p.r = k +q,31r
m
From
m
]k,(l + 2~) + 1/3pL,(7 + 8~11.(28)
the hydrodynamic km =
analogy
%I 3(1 - 2v,)
r/Ill pm = 2(1 + v,) where q, = viscosity of Me-matrix, v, = Poisson’s ratio. Taking into consideration that the used relationships [23] are relevant for the case of small c, apply the self-consistent method [25] (l-iterative), in accordance with which the obtained values of the heterophase material characteristics in equation (28) are ascribed to its matrix, and repeat the calculations in accordance with equation (28). This means the following scheme of the calculations: p,, k,*p&, kO,,=v!n> k,hvc,,k,r.
the item 2, is also linear for a porous material in the case of linear-viscous properties of a solid phase. It should be noted that, for the derivation of the expression for Poisson’s ratio in equation (30), the self-consistent method was used. For high-porous bodies this is not rigorous, if the “phase of void” is a predominant one. Therefore, for high porosity values, the last expression in equation (30) looses its meaning, and in particular, it cannot be used when 0 > 213, e.g. when Poisson’s ratio becomes negative. For this case, equation (30) should be modified [25]. However, the investigation object-MMC compositecannot be considered as a high porous material for the case studied, therefore the form of the last expression in equation (30) does not limit the application of the model. From the mass-conservation law it follows [16]
(31) P = C, + i, + eq (P, = 0) = volume change where rate [equation (3)]. Substitute equation (22) into equation (31)
&2PR:
R:-R:C,,+C,,
1-e (32)
712
OLEVSKY
et al.:
HOT ISOSTATIC
The differential equations (23) (25) and (32) form the close set which can be solved taking into consideration equations (26)-(30). The following expression for the calculation of the radius of a regular pore, obtained on the basis of the continuum theory of sintering, can be added to this set [22]
PRESSING
CONDITIONS-I
used as for the case considered above. The values obtained are substituted into equation (27) for the calculation of the effective viscosity coefficients. For the calculation of the deformation (radial, tangential) distributions, equation (10) is integrated with respect to time cr=b
lMe(1
-
0)’
Taking into consideration that qMe is a function of temperature [26], the HIP temperature cycle can be involved in the model, thus influencing the kinetics of the porosity in equation (32), the pore radius in equation (33), the imperfection radius in equation (23) and the distance between imperfections in equation (25). 6. ESTIMATION
OF STRESS DISTRIBUTION AROUND FIBRES
The formulae of equation (11) represent the function of the stress distribution around imperfections. For fibres we should use the following expression [24] (for radial stress) bf
err =
(1 + .);‘1 - 2vJ
(1 + v&z’+ (3v, - 1) -
r2
1
. (34)
Here Q and v, are viscosity and Poissons ratio of the coating, respectively; r = radial coordinate, uf and b’= constants which can be found from the boundary conditions uf-bf/r21,=,,,=0 orIr=nuc = oFrn
Or2 =[ x
(1 + .,,“(‘;
x
- 2v,,)
(1 fv,,)af+(3v,,-
2v,,)
(l+r&;+(3v,,-1)s
1):
1, 1,
R,, G r G R,
Rci < r.
(36)
Here R, = a radial coordinate of the intercoating boundary. Additional boundary conditions af - bf/Rzi = ai - bi/RH oqIr=R,
eV=I
P,dt
(38)
= %*lr=iQ.
where L, and t, are the radial and tangential deformations, respectively [the expressions for P, and P, are analogous to equation (lo)]. 7. CONCLUSIONS
On the basis of the continuum theory of sintering, the mathematical model for HIPing of fibre reinforced metal matrix composites has been built. The model takes into consideration the solid phase anisotropic properties, as well as an influence of porosity. The model includes the consideration of the porous structure, of the matrix, as bimodal when taking into account the influence of the large imperfections upon the densification kinetics. The relationships obtained permit the calculation of the evolution of porosity, the imperfection radius and the distance between the neighbouring imperfections during HIPing. The algorithm of the calculation, of stress, strain and strain rate distributions around the fibres for the cases of one- and two-layer coatings, is elaborated. The calculation scheme elaborated is used for analysing the HIPing conditions of the Sic-fibre reinforced titanum alloys [26].
(35)
where oy” = average radial stress in accordance with an average value of equation (11). For a two-layer coating [26], we should use two functions for 0, (lower index corresponds to the number of coating) Or1= (1 + .,:t;
&dt;
‘P
i, = P
(37)
In the case of a two-layer coating the solid phase is already a three-phase material. Therefore, for the internal coating and fibres, equation (28) should be
REFERENCES 1. H. A. Kuhn and C. L. Downey, Znt. J. Powder Metall. 7, 15 (1971). 2. R. J. Green, Znt. J. Mech. Sci. 14, 215 (1972). 3. S. Shima and M. Oyane, In?. J. Mech. Sci. 18, 285 (1976). 4. A. L. Gurson, J. Engng Mater. Technol. 99, 2 (1977). Li, M. F. Ashby and K. E. Easterling, 5. W.-B. Acta metall. 35, 2831 (1987). 6. T. Shinke, T. Soh, T. Nakagawa and A. Nohara, Steel Engng Rep. 37, 2932 (1987). 7. M. Abouaf and J. L. Chenot et al., Znt. J. Num. Meth. Engng 25, 191 (1988). 8. J. M. Duva and P. D. Crow, Acta metall. 40, 3 1 (1992). 9. J. Besson and M. Abouaf, J. Am. Ceram. Sot. 75,2165 (1992). 10. D. M. Elzey and H. N. G. Wadley, Acta metall. 41,2297 (1993). Il. E. Olevsky, M. Shtem and V. Skorohod, Hot Zsostatic Pressing 93 (edited by L. Delaey and H. Tas), p. 45. Elsevier, Amsterdam (1994). 12. A. Maximenko, E. Olevsky, M. Panfilov and M. Shtern, Hot Zsostatic Pressing 93 (edited by L. Delaey and H. Task D. 61. Elsevier, Amsterdam (1994). 13. E. Old&y and V. Skorohod, Technological and Construction Plasticity of Porous Materials. IPMS NAS Ukraine, 97 (in Russian) (1988). 14. H. Riedel, _ Ceramic ._. Powder ~~~Science ZZZ, p. 619. American Ceramic Society, Westerville, Ohio (1990).
OLEVSKY
et al.: HOT ISOSTATIC
Mech. Gran. Mater. Powder Syst. 15. R. M. McMeeking, ASME 37, 51 (1992). E. A. Olevsky and M. B. Shtern, 16. V. V. Skorohod, Sci. Sinter. 23, 79 (1991). 17. A. C. F. Cocks, Acta metall. 42, 2191 (1994). 18. A. Frisch, W. A. Kaysser and G. Petzow, Proc. of World Conj. on Powder Metallurgy, p. 237. London (1990). 19. S. Nair and J. K. Tien, Metall. Trans. A18A, 97 (1987). Acta metall. 37, 1645 20. K. H. Li and P. D. Funkenbusch, (1989). 21. G. W. Scherer, J. Am. Cerum. Sot. 67, 709 (1984). 22. E. Olevsky and R. Rein, Proc. of 13th Int. Plansee-Sem (edited by H. Bildstein and R. Ecks), Vol. 1, p. 972. Metallwerk Plansee, Reutte (1993). Mechanics of Composite Materials. 23. R. M. Christensen, Wiley, New York (1979). 24. L. D. Landau and E. M. Lifshitz, Theoretical Physics, Vol. VII-Theory of Elasticity, pp. 33-34. Nauka, Moscow (1987). Rheological Basis of Theory of Sintering. 25. V. Skorohod, Naukova Dumka, Kiev (in Russian) (1972). 26. E. Olevsky, H. J. Dudek and W. A. Kaysser, Acta metall. 44, 715 (1996).
NOMENCLATURE A = C,,, = c= P= P,, = k, = k,, = k, = n= P =
power-law creep constant viscosity tensor component fibre concentration rate of volume change strain rate tensor component bulk viscosity module of a fibre bulk viscosity module bulk viscosity module of a composite matrix power-law creep exponent hydrostatic pressure applied (P < 0 for isostatic compression) P, = effective Laplace stress P, = Laplace stress on the imperfection surface R, = current imperfection radius R,,, = initial imperfection radius R, = distance between imperfections R,, = fibre radius R,, = coating radius R,, = radial coordinate of the intercoating boundary for the criterion of the elimination of rmln = constant imperfection rr, = current pore radius I’.c= radial velocity W = equivalent effective strain rate of a porous material y = rate of shape change jc”) = average intensity of strain rates in a solid phase 6,j = Kronecker’s symbol t,, to = radial and tangential deformation of a material around fibre r),, = shear viscosity of a solid phase qc, (qc2) = shear viscosity of the first (second) coating rlMe= shear viscosity of a metal matrix solid phase qrn = shear viscosity of a composite matrix 0 = porosity
PRESSING nr = p,, = p, = v,, (v,r) = v, = blj = oy” = o(m) r0 rc”) cp, $
= = = =
CONDITIONS-I
713
shear viscosity module of a fibre shear viscosity module shear viscosity module of a composite matrix Poisson’s ratio of the first (second) coating Poisson’s ratio of a composite solid phase stress tensor component average radial stress in the porous matrix around a fibre equivalent effective stress of a porous material yield limit of a solid phase average intensity of stresses in a solid phase functions of porosity.
APPENDIX
Derivation of the Constitutive Relationships Consolidation of a Porous Body [I61 In accordance with the principles dynamics, the equation cr,e, - $
of irreversible
fir thermo-
B@,, 0) = 0
-
(Al)
where F = free energy of the system, fi = rate of energy dissipation, t = time, serves as the basis for the model. Herewith aF -=P,i
(A-3
at
where P, is an effective Laplace stress, t! = volume change rate. The dissipation of energy is conditioned by the processes, taking place in a substance. However, if the description corresponds to the macro-level, then fi has to depend on the averaged parameter of the solid phase flow. The effective strain rate I$’[equation (2)] is taken as such a parameter. Taking into consideration that I? is a quasihomogeneous function, it is possible to represent (Al) in the form
&a' c7,, - PL 6, - *z
“,, = 0. (-43) “3 1, ( From this equality, by virtue of the principles of irreversible thermodynamics, equation (1) follows. Equation (1) represents the relation between a stress tensor component oil and strain rate tensor component P, in a generalized frame of reference. For example, for axisymmetrical coordinate system (with coordinates r, cp, z) the radial stress can be written:
=F(($
+j)P,+(+
P,, P,, L?~= the corresponding tensor. For cp and + the following q = (1 - 0)2, (0 = porosity).
-fcp)(P,+&,))+P, components expressions
of
(A4) the
strain
can be used [25]
* = 2ff?(l - e)/3e
(A5)