Institut National Polytechnique de Grenoble, ENSPG, BP 46, 38402 St Martin d'H&es (France) ..... Hoop stress is always very small, whereas the absolute.
Materials Science and Engineering, A 167 (1993) 97-105
97
Thermal residual stresses in particle-reinforced viscoplastic metal matrix composites M. Surry, C. Teodosiu* and L. F. Menezes** Institut National Polytechnique de Grenoble, ENSPG, BP 46, 38402 St Martin d'H&es (France) (Received February 12, 1992; in revised form February 9, 1993)
Abstract A spherical symmetric thermoviscoplastic model is presented to calculate the thermal residual stresses in particlereinforced metal matrix composites. It is based on the assumption that the particles behave elastically whereas the matrix exhibits an elastoviscoplastic behaviour. An internal-variable-type constitutive equation is used to predict the behaviour of the matrix, which is taken as AI-1100. The model can predict the change of the stress and displacement fields within the matrix and the particles during cooling of the material as well as the thermal residual stresses induced at room temperature. The effects of cooling rate and volume fraction of particles are emphasized. The stresses on reheating from room temperature and during thermal treatment are calculated. It is shown in particular that holding the composite at moderate temperature ( ~ 200 °C) does not lead to substantial relaxation of the stresses. Aging treatment is thus usually carried out under a residual stress field which might influence the precipitation sequence and kinetics.
1. Introduction
Particle-reinforced aluminium matrix composites are now being extensively studied because they realize a good compromise between cost, manufacturing convenience and mechanical properties (stiffness, tensile strength and creep resistance). Despite the large number of studies devoted to these materials, their mechanical behaviour is not yet fully understood, and one of the problems still under investigation is the role played by residual stresses. These stresses are inherent in metal matrix composites because of the mismatch in thermal expansion of matrix and reinforcements. They develop on cooling after fabrication or thermomechanical treatment. Several analyses of residual stresses have been carried out, both experimentally, using neutron diffraction [1, 2] and X-ray diffraction (see review by Eigenmann [3]) and theoretically, by computer simulation (see for example refs. 4-7). Most of this work, however, has been concerned with continuous fibre systems or whiskers. Lee et al. [8] have proposed a model for evaluation of residual stresses around a *Present address: Laboratoire des Propri&rs Mrcaniques et Thermodynamiques des Matrriaux, UPR CNRS, Universit6 Paris-Nord, 93430 Villetaneuse, France. **Present address: Departamento de Engenharia Mecfinica, Faculdade de Cirncias e Tecnologia, Universidade de Coimbra, 3000 Coimbra, Portugal. 0921-5093/93/$6.00
spherical particle, but they have assumed no ratedependency for the plastic behaviour of the matrix. Our purpose here is to take account of this factor and to examine the effects of various material and thermal treatment parameters on the residual stresses in AI-SiC particle composites. A one-dimensional model is used, and the viscoplastic behaviour of the matrix is assumed to be described by an internal-variable-type constitutive equation, valid for the whole range of temperatures studied. Thermal treatment includes cooling from fabrication or solutionizing temperature together with reheating in order to simulate aging. The study of this treatment is of interest since it was found that. the presence of residual stresses and local plastic strain around the particles might influence precipitation kinetics and thus aging characteristics of the composites.
2. Thermomechanical model
2.1. Geometry and boundary conditions The material is modelled by considering a representative spherical matrix cell of radius R, which contains a single concentric spherical particle of radius r0, as shown in Fig. 1. It is assumed that the absolute temperature field is homogeneous and that its evolution is given by T(t) = T0 + Tt © 1993 - Elsevier Sequoia. All rights reserved
M. Su~ry et al.
98
ParSiC~~j~ Licle ~/MaEr~x
/
Thermal residual stresses in viscoplastic MMCs
AI 1100-0
Fig. 1. Schematic representation of the spherical model. R is the outer radius of the matrix and h~ is the radius of the particle.
where TO is the initial temperature, t is time and T is the temperature rate, which is considered constant during cooling and, eventually, on reheating from room temperature. The material is taken as stress-free at the temperature To, and t h e heating produced by plastic deformation is neglected. Spherical co-ordinates (r, 4, 0) are used throughout. Taking into account the spherical symmetry, all fields involved are independent of # and 0, and the only nonzero components of the displacement vector u and of the Cauchy stress tensor e are Ur=U(r,t),
Orr=Orr(r,t),
Ooo=a##=o##(r,t)
(1)
t>0
(3)
where ac is the thermal expansion coefficient of the composite, and can be either measured or estimated by the mixture rule, i.e.
a c = Vpap + ( 1 - Vp)a M
(5)
where D e, D th= aM/'l and DP denote the elastic, thermal and viscoplastic strain rate tensors respectively, and 1 is the second-order unit tensor. The viscoplastic deformation is assumed to be isochoric (tr D p = 0). Because there is no rotation and the strains are small with respect to unity, we can use the formalism of small deformations. In particular, the evolution equation for the Cauchy stress is given by the thermoelastic law
(6)
(2)
or subjected to a prescribed displacement
ur(R, t)= acR[T(t)- To]
D = D e + D th + D p
O = 2 ~ M ( D -- DP) + ( K M - 2 / t M / 3 ) ( t r D ) I - 3 K M a M T1
The outer boundary is either taken as stress-free, i.e.
Orr(R, t)= 0
thermoelastically. In contrast, the homologous temperature of the matrix varies from unity at t = 0 to -0.3 on cooling to room temperature, and hence the inelastic properties of the matrix must be taken into account over the whole temperature range. More precisely, we assume that the matrix has an isotropic thermoelastoviscoplastic behaviour, whose state is described by, in addition to a and T, a structural parameter s, to account for the isotropic work-hardening and recovery (see for example ref. 9). To specify the constitutive equations of the matrix, we first assume that the total strain rate tensor D can be decomposed as
where/u Mand K Mare the elastic shear and bulk moduli of the matrix. The viscoplastic strain rate is related to the stress deviator a' = o - ( 1/3)(tr a)l by the isotropic law n p=
3~P
26
or'
(7)
(4)
where vp = (r0/R)3 is the volume fraction of particles and ap, a M denote the coefficients of thermal expansion of the panicle and the matrix respectively. Equation (2) corresponds, in the axisymmetric case, to the vanishing of the mean value of the stress vector on the outer boundary, and hence to neglecting the interparticle interaction. In contrast, the boundary condition given by eqn. (3) approximates the influence of the interparticle interaction on the overall thermal expansion of the composite. The thermal expansion coefficients of the matrix and the particle are assumed to be constant during cooling or reheating, and equal to their average experimental values over the whole temperature range. 2.2. Constitutive behaviour Since the maximum temperature experienced by the particle is a significant distance from its decomposition temperature, we assume that the particle behaves
where 2
is the equivalent tensile stress and ~P = (2DP:Dp)I/2
(8)
is the equivalent tensile plastic strain, which is prescribed by a constitutive law of the form
~P=f(6, s, T)
(9)
The evolution of the structural parameter s is assumed to be governed by
~=g(6, s, T)=~Ph(6, s, T)
(10)
Following ref. 9, we use in our calculations the following specific forms for the functions f and h:
RgTIL
,11,
M. Su&y et al.
h(d, s, T) = holl - s/s*l ~ sign(1 - s/s*)
/
Thermal residual stresses in viscoplastic MMCs
(12)
where ,,
The finite element discretization of eqn. (14) (for details see ref. 10) yields the system of linear algebraic equations in the standard form
[k"]{Au}={Af "} ~R~T]J
(13)
is a "saturation value" of s, Rg is the gas constant and A, Q, ~, m, h 0, a, g and n are material parameters. The initial value so of s is assumed to be prescribed.
2.3. Time-integration procedure Assume that the state variables are known at time t,,. In order to determine their values at time t,,+~ = t,, +At, the evolution equations must be integrated over the time increment At. As shown by Teodosiu and Menezes [10], for small plastic strain increments (less than 1%), the evaluation of the integral of D p over the time step by a forward gradient procedure provides a time-marching scheme that meets the stability and efficiency requirements. For details of the forward gradient procedure we refer to Anand et al. [ 11 ], where this method has first been applied in conjunction with our constitutive framework (for a thermoviscoplastic approach without a structural parameter, see also Povirk et al. [7]).
(17)
where {Au} is the displacement increment, [k ~] is the (secant) stiffness matrix and {Af"} is the incremental nodal forces, both evaluated at time t,, and corresponding to the time increment At. Solution of the system for {Au} allows the updating of the configuration and of the state variables. The above algorithm has been implemented in the code T H E R E S [10] and is currently being extended to three-dimensional problems.
3. Results
The material properties used in the analysis are chosen to be representative of AI-1100-O matrix [9] and SiC particles [13]. The properties of the aluminium are
E M(in MPa)= 73 474 - 43.48 x (T-273) /~M (in MPa)= 27 041 - 17.057 x (T-273) aM = 0.287 X 10-4 K - 1 [14]
2. 4. Finite element implementation
and those of the particles are
For the model with spherical symmetry shown in Fig. 1, the virtual work principle reduces to
Ep = 41 x 104 MPa
r 0
/~p= 16.532
R
f o : d e r 2 d r + f o:der2dr=O 0
99
(14)
r 0
where de is the virtual strain field associated with a virtual displacement field, say du. For the particle we use the thermoelastic solution (see for example ref. 12), in which for 0 ~
b
o
Vp=0.15 '
-40 -50
Ld
o
2600C
c -30
I=- 100OC/s 0
5.00
420°C k-
vp=0.15
s"
PARTICLE
lO o
> '2 (3-
(l~m)
Fig. 2. Radial stress distribution in the particle and in the matrix as a function of temperature: the volume fraction of particles is 0.15 and the cooling rate is 100 K s- x.
Au
20
20K::
o
r = - 1oo ocl/s I
420°C
1 ,oo Vp=0.15
t =-lO0°C/s
D Otad
PARTICLE
0.00
-60
0 10 Radial
20
distance
30 (pm)
Fig. 3. Mean stress distribution in the particle and in the matrix as a function of temperature: the volume fraction of particles is 0.15 and the cooling rate is 100 K s- 1.
T h e radial stress is also compressive in the matrix, its absolute value increasing with decreasing temperature and radial distance. T h e stress components o # and a00 are positive, which leads to a rather complex variation of the mean stress (Fig. 3). This is negative at the interface and gradually increases to b e c o m e positive at the outer boundary. T h e equivalent stress in the matrix also increases with decreasing temperature and radial distance, the maximum value being close to 60 MPa at r o o m temperature. This stress is higher than the yield stress of the material (35 MPa) in the O condition, which can be explained by the fact that strain hardening occurs in the matrix during cooling, especially at
MATRIX
10 2O Radial distance (ffm)
i
30
Fig. 5. Equivalent plastic strain distribution in the matrix as a function of temperature: the volume fraction of particles is 0.15 and the cooling rate is 100 K s- 1.
low temperature, as a consequence of the plastic strain generated. Figure 5 shows the variation of the equivalent plastic strain generated in the matrix on cooling. Like the equivalent stress, it increases normally with decreasing temperature and decreasing radial distance, amounting to about 3% at r o o m temperature at the interface. This rather large value, as well as the steep variation close to the interface, explains the in situ high-voltage electron microscopy observations by Vogelsang et al. [15], who showed that dislocations emanated from the fibre-matrix interface on cooling and found a particularly high dislocation density near the interface. Figure 6 displays the variation with temperature of the radial,
M. SuOry et al. 60
Vp=0.1 5 T
/
Thermal residual stresses in viscoplastic MMCs
/
PARTICLE
101
MAT
100°C/ no
O 40 [2_
-10
/ / Vo=O. 15
~:
20 ¢= loooc/s
O 20 O
©
(b
k_
0
-30
-
5 -4o ©
4~
o -20
Od° -50 - a,,(R)=0
(h U)
,
,
-60~ c - ~
© L -40
0 5 - Rodial stress
-60 660
//I
520 Ternperoture
10
2O
Radial d i s t a n c e 20 C°C)
Fig. 7. Radial stress distribution in the particle and in the matrix for the boundary conditions used in the calculation.
Fig. 6. Variation of the equivalent tensile stress, mean stress and radial stress at the interface with temperature.
4O O
mean, and equivalent tensile stresses at the interface. It is clear that their absolute value increases rapidly when approaching room temperature. This variation is much slower at high temperature because of the rapid relaxation and the low value of the yield stress in this temperature range.
U3 U3 ®
3.3. Influence of particle volume fraction The particle volume fraction Vp is defined in the model by (ro/R) 3. ro is assumed to be constant, so that increasing Vp results in decreasing R. Figures 8 and 9 show the distributions of the mean stress and equivalent tensile stress in the matrix at room temperature for various particle volume fractions. For low vp, the mean stress is compressive close to, and tensile far away
PARTICLE
MATRIX
30 20
L
10
c © (])
0
4~ O9
-10
3.2. Influence of boundary conditions The above calculations were carried out assuming zero radial stress at r = R as a boundary condition. The displacement at this point results from the contraction of the composite on cooling. Another possible boundary condition is to assume that the coefficient of thermal expansion of the composite is simply given by the mixture rule, as mentioned in section 2.1. Figure 7 compares radial stress distribution at room temperature in the matrix and the particle for the two boundary conditions. The condition Orr(R, t)= 0 leads to slightly smaller absolute values of the radial stress. However, the overall agreement of the results given by the two conditions shows that the assumption of a stress-free outer boundary can be regarded as realistic insofar as the mixture rule can give a good estimate of the coefficient of thermal expansion of the composite.
30
(~m)
',,p=o.,5r /
~D -2o K?
~o) -so rY
I
.
vp=o.05 /
•
t = - lO0*C/s
-40 10 Radial
20 distance
30
40
(l.Jm)
Fig. 8. Residual mean stress distribution in the matrix as a function of the particle volume fraction: the cooling rate is
100 K s-J.
from, the particle. With increasing/tip, the extent of the compressed zone decreases, and for Vp= 0.3 the mean stress is completely tensile. In the case of the equivalent tensile stress, the situation is more simple: its distribution is independent of Vp, being maximum at the interface and decreasing with increasing radial distance (Fig. 9). The above results were obtained by assuming zero radial stress at r =R. It is interesting to compare these results again with those obtained by use of the mixture rule for the thermal expansion coefficient. Figure 10 compares the variation with Vp of the thermal expansion coefficient of the composite ac given by the
102
M. Su#ry et al.
/
Thermal residual stresses in viscoplastic MMCs O &
O
n
(O 03 L.
60
PARTICLE
MATRIX 03 o3
50
60
60
Vp=0.1 5
5O
k_
40
'8
~)
c-
fl) *~
30
c ®
4~ ff
0>
4J
20
c @
(3~)
o> 2o
10
'~ 0 ~) 0
10
20 distance
30
(iJm)
1
5-5 4 -
50
5 -
100
-6 D
T=-
'~ 0 Q) 0 n"
PARTICLE
Radial
MATRIX
10 20 30 distance ( p m )
IO0*C//s
•••x
ture rule
~c=u(R)/(R ~ ) ~ ' ~
x
0.15
2-
Fig. 11. Residual equivalent tensile stress distribution in the matrix as a function of the cooling rate.
3.0
O 2.5
rote (*C/s):
1 -
Q) 10 40
Fig. 9. Residual equivalent tensile stress distribution in the matrix as a function of the particle volume fraction: the cooling rate is 100 K s- ~.
?
Cooling
cr
t=-lOO"C/s Radial
T O o
40
2.0
0.00 Volume
0.15 fraction
0.30
Fig. 10. Change of the thermal expansion coefficient of the composite a c with particle volume fraction, a c is calculated either from the mixture rule or from the displacement at the outer radius of the matrix. mixture rule with that obtained assuming O,r(R , t)= 0. In the latter case a c is given simply by a c = u ( R ) / R 6 T , where u ( R ) is the radial displacement at r = R and 6 T = 6 4 0 K is the difference between To and room temperature. The good agreement between the two curves shows again that the two boundary conditions are approximately equivalent. 3.4. Influence of cooling rate After fabrication or thermal treatment of a composite, cooling to room temperature must be accom-
plished. Sometimes, rapid quenching is absolutely necessary to avoid any microstructural change (as in the case of solution treatment). Frequently, however, it is not possible to quench the material for practical reasons. It is then interesting to know what would be the influence of the cooling rate on residual stress state in a composite material. Calculations of the stress distributions at room temperature were carried out for various cooling rates from 0.15 K s -1 (typical of furnace cooling) to 100 K s-1 (typical of water quenching) (Figs, 11 and 12). T h e equivalent tensile stress decreases with decreasing cooling rate (Fig. 11), but the effect is very limited. For example, its value at the interface decreases from about 57 MPa to about 42 MPa when the cooling rate changes from 100 K s -1 to 0.15 K s -x. This slight variation can be explained by the fact that stresses build up mainly in the low temperature range ( < 3 0 0 °C, Fig. 6) where plastic behaviour is almost rate-independent. Changing the cooling rate thus affects only the stresses in the high-temperature region, but this effect is limited owing to the very low value of the stresses generated in this range. The same slight influence of the cooling rate is noticed for the mean stress (Fig. 12). The variation with cooling rate of the residual radial and hoop stresses at the interface is shown in Fig. 13. H o o p stress is always very small, whereas the absolute value of the radial stress decreases significantly only for small cooling rates.
M. Su~ry et al.
/
Thermal residual stresses in viscoplastic MMCs
3O PARTICLE © CL ,~j
2O
Ul ® 4-,
10
C
o ©
0
B
1- T=-O.15°C/s
D
73
-10
t=-loo*c/s
2-
®
60
MATRqX
vp-o ~5
/
50
/
Ck
10
Radial
O
(1) o o
20
o
10 0
03 03
10 ~
.
o3
B ~5
30
(~m
-30
1-heating rate=.15'C/s
-40
2-heating rate=5 "C/s
-50
3-heating rote = 1O0 'C/!
-60 660
Fig. 12. Residual mean stress distribution in the matrix for two extreme cooling rates.
n
40
C
20
distance
cooling rote=100"C/s
30
I I
-20 0
103
10
320
20
320
660
Temperature (°C)
Fig. 14. Radial stress at the interface as a function of temperature during cooling to room temperature at 100 K s-L and subsequent heating at various rates.
Hoop stress f
"J
0
O O
o -10 g) 4~
.c
-20
4--' O
-30
vp=OI15
O k_
-40 O 73
~ s t r e s s
-50
O
c~ -60 0
25 Cooling
50
75
rote
(°C/s)
1oo
Fig. 13. Variation of the residual hoop and radial stresses with cooling rate for a particle volume fraction of 0.15. It can be concluded from these results that only very low cooling rates reduce the residual stresses, and even this reduction is moderate. Consequently, residual stresses will be present irrespective of the cooling rate employed. 3.5. Variation of residual stresses on reheating Thermal treatment at moderate temperature is sometimes carried out in A1 alloys to increase the mechanical properties by age hardening. This treatment involves nucleation and growth of precipitates which can be greatly influenced by the stress state of the material. Recent T E M investigations showed indeed that the stresses generated around SiC particles in AI-Cu/SiC composites lead to inhomogeneous
precipitation in the matrix with a larger density of precipitates close to the particles [16]. Since the aging treatment takes place after quenching from the solutionizing temperature, it is thus of interest for determining the change on reheating of the stresses generated at room temperature during quenching. Moreover, since the treatment usually lasts for several hours, these stresses will relax during that time. It is then interesting to determine their relaxation rate during holding to the aging temperature. The solutionizing temperature is obviously lower than the melting temperature: in order to simplify the calculation, it is assumed here that they are identical. This assumption is fully justified by the very large relaxation rate of the stresses close to the solidus temperature so that the build-up of the stresses is negligible during cooling of the alloy from the solidus to the solutionizing temperature (see Fig. 6). The calculation is carried out using the parameters for commercially pure aluminium. Obviously, this material is not age-hardenable. However, since material parameters are not available for AI-Mg-Si or AI-Cu alloys, the calculation will at least give an estimate of the stress relaxation that will take place during an aging treatment of such materials. Figure 14 shows the variation with temperature (from 660 °C down to 20 °C and up to 660 °C) of the radial stress at the particle-matrix interface. Three heating rates are considered, from 0.15 K s- ~ to 100 K s -l. The radial stress varies sharply on reheating, vanishing at - 4 5 °C and reaching a maximum at a temperature (close to 100 °C) that depends slightly on
104
M. Su~ry et aL
/
Thermal residual stresses in viscoplastic MMCs
the heating rate. It decreases thereafter to approximately 0 at 6 6 0 °C. The maximum also depends on the heating rate, but its effect is rather small, which is consistent with the limited influence of the relaxation phenomena in this range of temperature. The change of sign of the radial stress at the interface is important, because it might lead to decohesion when the strength of the interface is not sufficiently large, or brittle reaction products are formed during fabrication. Another important result obtained from the calculations is that heating the composite by merely 25 °C leads to zero residual stresses. This might be interesting for the study of the influence of these stresses on particular properties or phenomena, such as elastic properties or age hardening. Age hardening involves heating the composite to a moderate temperature and holding it for sufficient time to induce precipitation. As an example, Fig. 15 shows the change of the mean stress in the matrix on reheating to 200 °C from room temperature at 5 K s -1 and holding at this temperature for various times. As before, the mean stress changes its sign on reheating, and its absolute value decreases slightly with increasing holding time. This small change is again due to the limited effect of the relaxation phenomena at 200 °C. Consequently, during an aging treatment, the matrix remains under stresses. These stresses are quite small, but they might be sufficient to influence the precipitate morphology. It must be remembered that the calculated stresses are valid for a commercially pure aluminium for which the yield stress in the annealed
/j
25
4. Discussion and conclusions
Residual stress and deformation fields in aluminium reinforced with SiC particles were calculated for various material and processing parameters by use of a finite element formulation. The matrix is considered to be thermoelastoviscoplastic, with a constitutive equation valid over the entire temperature range. In order to obtain quantitative predictions, several assumptions have been made, whose validity requires discussion. The particle has been assumed to be spherical and placed in the centre of a representative unit of matrix material. Real particles are usually angular, but the angular shape can significantly affect only the distribution of the stresses at the very close interface. Larger local plastic strains than for the ideal case can thus be
I 1 - a t 20"C
20
2 - a t 200°C, tm=O s 3 - a t 200"C, tin=300
"O- ' 1 5 0_
4 - a t 200°C,tm= 1200 5 - a t 200"C,tm=3600
10 (13 O3
condition is smaller than that for an age-hardenable alloy such as A1-Mg-Si or A1-Cu. The influence of the internal stresses on precipitation has been studied by Prangnell and Stobbs [16]: they observed predominant orientations of the 0' plates indicative of the existence of internal stresses close to the particles. They also noticed that the precipitation can be extremely inhomogeneous, reflecting the extent of the plastic zone around the particles. During heating to 200 °C and holding at this temperature, the plastic strain indeed increases, but only very slightly (Fig. 16); this is again indicative of the very limited relaxation that takes place at this temperature.
5
4.0
C 0
2 - a t 200°C,t~=0 s
X C
3-or
.t.0
5 - o [ 200*C,t,.- 3600
u) L)
2.0
5_ Cooling rote= 100*C/s
--5
4J C q)
Heating rote=5°C/s
/ PARTICLE
-15 0
10
t
2,3,4,5
Cooling rate = I O0°C/ Heating rote=5OC/s
1.0
up to 2OO°C
O >
up to 2QOeC
-10
200~C,t~=300 s
4 - a t 200oC,t~ - 1200
5
03 (D
0
1-at 20°C
O O
D ET Ld
I MATRIX
20
Radial d i s t a n c e
[
PARTICLE
0.0
30 (iJm)
Fig. 15. Mean stress distribution in the matrix at room temperature (1) and after reheating at 200 °C and holding at this temperature for various times (2-5): the cooling rate down to room temperature is 100 K s-l; the heating rate up to 200 °C is 5 K s -1.
0
10
MATRIX
2O
Radial distance
[
30
(iJm)
Fig. 16. Equivalent plastic strain in the matrix at room temperature (1) and after reheating at 200 °C and holding at this temperature for various times (2-5): the cooling rate down to room temperature is 100 K s-1; the heating rate up to 200 °C is 5Ks-L
M. Sukry et al.
/
Thermal residual stresses in viscoplastic MMCs
expected, but the model can give a good estimate of the real situation. The influence of the boundary conditions has been studied by taking either a stress-free outer surface or a prescribed displacement at the outer surface as imposed by the composite coefficient of thermal expansion ac given by the mixture rule. These conditions gave almost the same results. Other boundary conditions can be considered, particularly by taking for a c an experimentally measured value. Such a value would probably take account of the interaction between the particles, which is completely neglected in the present calculations. This interaction takes place even for relatively small volume fractions of particles, since for Vp= 0.05 the spacing between the particles is less than three times their diameter. A limitation of the analysis is to consider the matrix as a homogeneous isotropic material. In fact, the grain size of the matrix is usually comparable to the particle size, so the matrix would ideally be modelled as a multicrystal, taking account of the particular orientation of the grains close to the particles. Another simplifying assumption was made, i.e. that the temperature is uniform in the material during cooling. This condition is probably well satisfied under most circumstances, but high cooling rates should lead to temperature gradients in the material, thus changing the distribution of the residual stresses. Such an effect was taken into account by Povirk et al. [7] for the case of SiCw-6061 A1 composites; it was shown that the temperature becomes non-uniform within the unit cell considered in their calculation only for very high cooling rates of the order of 1000 K s-1. This phenomenon is thus negligible under most practical conditions. The model leads to some conclusions as to the influence of the material and processing parameters on stress and strain distributions generated upon cooling. (1) The interface plays an important role, since most of the stress components are greater at the interface than far away in the matrix. (2) The equivalent stress at the interface is greater than the initial yield stress of the matrix. This is due to strain-hardening generated in the material during cooling. (3) The plastic strain can be as high as 3% at the interface. Such a high value explains the greater dislocation density observed close to the interfaces.
105
(4) Cooling rates slightly affect the level of the residual stresses. A n y composite material would then contain residual stresses whatever the cooling conditions from the fabrication temperature. (5) Reheating of the composite after quenching to room temperature changes the sign of the radial stress, which becomes positive, and this can eventually induce some decohesion at the particle-matrix interface. (6) Holding time of the composite at moderate temperature (= 200 °C) does not lead to substantial relaxation of the stresses. Consequently, during an aging treatment, the material is still constrained and this might influence the precipitation sequence and kinetics. References 1 H. Lilholt and D. Juul Jensen, in J. Herriot (ed.), Composites Evaluation, Butterworth, Sevenoaks, 1987, p. 156. 2 E J. Whiters, D. Juul Jensen, H. Lilholt and W. M. Stobbs, in E L. Matthews et al. (eds.), Proc. ICCM6/ECCM2, Elsevier, London, 1987, p. 2.255. 3 B. Eigenmann, B. Scholtes and E. Macherauch, Mater. Sci. Eng. A, 118 (1989) 1. 4 R.J. Arsenault and M. Taya, Acta Metall., 35 (1987) 651. 5 E. Zywicz and D. M. Parks, Comp. Sci. Tech., 33 (1988) 295. 6 M. Vedula, R. N. Pangborn and R. A. Queeney, Composites, 19(1988) 133. 7 G. L. Povirk, A. Needleman and S. R. Nutt, Mater. Sci. Eng. A, 125(1990) 129. 8 J. K. Lee, Y. Y. Earmme, H. I. Aaronson and K. C. Russel, Metall. Trans. A, 11 (1980) 1837. 9 S. B. Brown, K. H. Kim and L. Anand, Int. J. Plast., 5 (1989) 95. 10 C. Teodosiu and L. F. Menezes, Rev. Roum. Sci. Techn. M&. Appl., 36 (3, 4) ( 1991 ) 243. 11 L. Anand, A. Lush, M. E Briceno and D. M. Parks, Report of Research in Mechanics of Materials, Dept. Mech. Eng., MIT, Boston, MA, 1985. 12 C. Teodosiu, H. Ribes and M. Su&y, in S. I. Andersen et al. (eds.), Proc. 9th Riso Symp. on Metallurgy and Materials, Roskilde, Denmark, 1988, p. 485. 13 R. W. Davidge, Mechanical Behaviour of Ceramics, Cambridge University Press, 1979. 14 Landolt-Bornstein, Zahlenwerte und Funktionen, Bd. H." Eigenschaftender Materie in ihren Aggregatzustiinden, Teil 1: Mechanisch-thermische Zustandsgr6flen, K. Sch~ifer and G.
Beggerow (eds.), Springer, Berlin, 1971, p. 239. 15 M. Vogelsang, R. J. Arsenault and R. M. Fisher, Metall. Trans. A, 17(1986) 379. 16 E B. Prangnell and W. M. Stobbs, in N. Hansen et al. (eds.), Proc. 12th Riso Symp. on Materials Science, Roskilde, Denmark, 1991, p. 603.
