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A planar model study of creep in metal matrix composites with misaligned short fibres
Acta metall, mater. Vol. 41, No. 10, pp. 2973 2983, 1993
0956-7151/93 $6.00 + 0.00 Copyright ~ 1993 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
A P L A N A R M O D E L S T U D Y OF CREEP IN M E T A L M A T R I X COMPOSITES WITH M I S A L I G N E D SHORT FIBRES N. J. S O R E N S E N Materials Department, Riso National Laboratory, DK 4000 Roskilde, Denmark (.Received 7 October 1992; in revised form 20 January I993)
Abstract--The effect of fibre misalignment on the creep behaviour of metal matrix composites is modelled, including hardening behaviour (stage 1), dynamic recovery and steady state creep (stage 2) of the matrix material, using an internal variable constitutive model for the creep behaviour of the metal matrix. Numerical plane strain results in terms of average properties and detailed local deformation behaviour up to large strains are needed to show effects of fibre misalignment on the development of inelastic strains and the resulting over-all creep resistance of the material. The creep resistance for the composite is markedly reduced by the fibre misalignment and the time needed to reach an approximate steady state is elongated due to the strain induced rotation of the short fibres in the matrix.
1. INTRODUCTION A basic understanding of the creep behaviour of metals reinforced by short high-stiffness ceramic fibres is important for a range of applications of these new materials in automotive industry, aircraft and aerospace construction where high temperature strength and thrust to weight ratio are key design parameters. Theoretical models based on finite element solutions of the field equations of continuum plasticity have been used by different authors to predict the inelastic properties of idealized metal matrix composites. Room temperature properties of short fibrereinforced metal matrix composites have been studied in this context by Christman et al. [1]. Their study included plane strain models for simple clusters of short fibres and axisymmetric models of an idealized periodic array of hexagonal cylinders, each consisting of the metal (aluminium) with a short fibre embedded in the centre (a configuration with aligned fibre ends). Tvergaard [2] used a different axisymmetric finite element model to predict the properties of short fibre reinforced metal matrix composites with different amounts of shifting ("staggering") of the neighbouring short fibres. A full 3D finite element model was used by Levi and Papazian [3] to analyze the two basic configurations of the short fibres termed "aligned" and "staggered", respectively. Recently the axisymmetric models [1] and [2] were compared with full 3D finite element models by Horn [4], who also studied the response under transverse loading. Finite element models of the elastic plastic response of continuous fibre-reinforced metal matrix composites have been presented by Brockenbrough et al. [5] and Sorenson [6]. Creep behaviour of short fibre metal matrix composites has been studied with numerically analysis by Dragone and Nix [7] in an analysis of the aligned and
staggered short-fibre configurations. Effects of the reinforcement shape and morphology have been modelled by Bao et al. [8] and Povirk et al. [9] used the axisymmetric model of the periodic array of aligned hexagonal cylinders to analyze creep in thermally cycled AI/SiC composites. In all of the models [1-9] it is assumed that the short fibres are perfectly parallel. In real materials, however, the short fibres are not perfectly parallel and the material behaviour of the composite can be expected to vary with the degree of misalignment. In the present paper a plane strain model of misaligned short 2-dimensional fibres is analyzed as a model problem for the real 3D geometry found in whisker-reinforced metal matrix composites. The plane strain assumption, which is made to reduce the computational requirement, means that the fibres are really continuous in the direction perpendicular to the plane. The results can be used for qualitative predictions of the effects of fibre misalignment on the behaviour of short fibre reinforced metal matrix composites, which is the topic here. The behaviour of metal matrix composites under creep conditions is complicated by the fact that the matrix is highly constrained by the reinforcement and thus large variations in the stress and strain fields in the metal matrix are found. This highly nonuniform stress distribution leads to strain rates ranging from traditional creep, dominated by dislocation glide through climb, to "hot-working" where strain hardening, dynamic recovery and in some cases dynamic recrystallisation are important. Furthermore, special scale dependent hardening phenomena may play a role Pedersen [10]. The present paper uses an internal variable constitutive model proposed by Brown et al. [11] to describe the behaviour of the metal matrix, whereas the reinforcement is modelled as an elastic material. The internal variable model does not account for scale
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SORENSEN: PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
effects. However, it has the advantage over standard power law creep that it accounts for primary creep and dynamic recovery in addition to the steady state creep. Furthermore, the assumption of a perfect interfacial bond between the fibre and the matrix is made and thus no interracial decohesion is allowed in the model. 2. P R O B L E M
FORMULATION
A periodic 2-dimensional array of tilted short fibres (see Fig. 1) is subjected to a plane strain creep analysis to be carried out here. The characteristic unit cell chosen to be analyzed has the initial dimensions A0 and B0, while d is the width of the inclusion and the tilt-angle is denoted by ~p. The unit cell is characterized by the aspect ratio ctc = Bo/Ao and the fibre aspect ratio is ~f = l / d . The volume fractions of short fibres (10 is M f = AoB--~o. (1) The material of Fig. 1 is loaded to the average stress Y'z in the x2-direction at the start. From this time on the creep deformations are simulated by a numerical analysis of the unit cell in Fig. 1. In this analysis a Cartesian coordinate system, with x ~and x 2 denoting coordinate directions, is used as reference in a convected coordinate Lagrangian formulation of the field equations, where gij and G o are the components of the metric tensors of the initial and current configurations, respectively, with determinants g and G, and qij = (Gij - go)~2 are the covariant components of the Lagrangian strain tensor. The contravariant displacement components on the reference base vectors are denoted by u i, and the contravariant components of the Cauchy stress tensor on the current coordinates are tr U while the corresponding components of the Kirchhoff stress tensor are given by r ~ = x / ~ a 0. With the current dimensions of the unit cell of Fig. 1 denoted by A and B in the x ~ and x2-directions, respectively, the boundary conditions are u~=0,
~=0,
xl=0
~=A,
72=0,
x l=Ao
fi2=0,
71=0,
x 2=0
~2 = B,
~ 1 = O,
X 2 =B o
U3 = ~3 = 0,
0 ~ xl~
do,
0 ~ x 2 ~Bo
conditions, only a small reduction in the computational effort is obtained by choosing the smallest unit cell (i.e. half of the cell defined above). 3. C O N S T I T U T I V E
MODEL FOR THE MATRIX MATERIAL
In order to model the material response of the composite under creep conditions an internal variable constitutive model is used to describe the behaviour of the matrix material (see Brown et aL [ll]). The constitutive model is basically the steady state hyperbolic sine creep equation of Garofalo [13], generalized to account for strain hardening and dynamic recovery by the introduction of a single scalar variable, the so-called deformation resistance, here denoted by s. The internal variable repressnts an average isotropic resistance to macroscopic plastic flow offered by the underlying isotropic strengthening mechanisms, such as dislocation density, solid solution strengthening or subgrain and grain size effects. The constitutive model has the following form
(o.)1 , m
•C = A l e x p - ~ Ee
, =(h01-
slnh ~ s
~ . ~s i g n ( 1 - ~ . ) ) ~ c
(3)
(4)
with s*=s
~
e
~exp
O
~
.
(5)
•c is the effective creep In the flow equation (3), ee strain-rate, 0 is the absolute temperature, and the yon Mises stress is tre= ~ s ° / 2 , where s o = z ° - z ~ G ° / 3 ,_ Ao_, I-
-I
I
i/
X2
(2)
where T ~are the components of the nominal surface tractions, and (') denotes the time derivative. The increments in the cell dimensions are prescribed using a Rayleigh-Ritz procedure (see Tvergaard [12]), so that the average stress, Z2, has a constant value while the average stress in the x i-direction is zero. The unit cell chosen in this problem formulation is not the smallest possible, since a rotational symmetry exists around the centre of each of the cells shown in Fig. 1. However, due to the more complicated boundary
x,
Fig. 1. Periodic array of tilted short fibres in a two-dimensional material.
SORENSEN: PLANARMODELOF CREEP IN METALMATRIX COMPOSITES is the Kirchhoff stress deviator. In the evolution equations (4-5) for the internal variable, the quantity s* represents a saturation value of s associated with a given temperature/strain rate pair. The material parameters in these constitutive equations are: A j, Q, m, 4, h0, a, g and n. In this model study the values of these parameters are chosen to be At = 1.91 - 107 s ~, Q = 175 kJ/mol, ~ = 7.0, m = 0.233, g = 18.9 MPa, n = 0.0705, h0 = 1.1 G P a and a = 1.3. The basic behaviour of the constitutive model is shown on Fig. 2(a,b), where the unreinforced metal is loaded in plane strain tension at the starting time by a stress 5:2 = a0, where a0 = 20 MPa. After 13,500 s the load is reduced to 75% of the initial stress level. The upper curve in Fig. 2(a), shows the development of the deformation resistance from the initial value s o = 20 M P a to the saturation level at the given stress. Following the load drop the deformation resistance slowly decreases towards a new saturation level. After the increase in the load at 38,000 s the deformation resistance grows until it reaches the saturation level seen prior to the load drop. The corresponding development in the strain for this loading is seen in Fig. 2(b), where the first part of the curve corresponds to normal creep behaviour with the material entering
2.00
(a)
1.75
1.50 1.25 1.00
}-2/(I0
0.75 0.50
i
10000
0
,
l
i
20000
30000
40000
t i m e (s)
a steady state (stage 2) after a short period of primary creep (stage 1). After the load drop the recovery effects dominate and the initial low strain rate slowly increases, indicating a strain softening. After reloading part of a typical creep response is seen; first a short hardening period until a steady creep state is reached with the same characteristic creep rate as found prior to the load drop. Using the constitutive model described above the creep strain rate 0 c is taken to be the effective creep strain rate (3) generalized to multiaxial strain via the stress deviator •c 2
1.50
#
(b)
1.00
0.75 ~
(b)
Fig. 3. Meshes used for the analysis of composites with tilted short fibres for ~f = 5 and./'= 0.10. (a) ~, = 4, ~0 - 15% (b)%=2,~=25.
s,j
0 c = ~e - - - . 3 rr~
true strain 1,25
(a)
0.25
The total strain rate is expressed as the sum of the elastic part and the creep part
o,,: O~ + 0 ~
f'; = R'Jkt(Okt- Oct). 0
(6)
(7)
The elastic stress-strain relationship is taken to be of the form f~ = R~ik~O~t so that the constitutive relation in the presence of creep and elastic deformation can be written as
-
0.50
0.00
2975
'
'
'
'
10000
20000
30000
40000
(8)
Here, the elastic instantaneous moduli are taken to be
time (s) Fig. 2. Basicbehaviourofthe constitutivemodelappliedfor the matrix material. The development of the deformation resistance corresponding to the load drop and reloading (a) and the strain in the tensile direction (b). The deformation resistance and stressed in (a) are divided by the starting values so and a0, respectively. (From Sorensen [14],)
with E and v denoting Young's modulus and Poisson's ratio respectively, and the Jaumann rate of the Kirchhoff stress tensor is
~(i =ii/ + (Gi~zJt + Gilrlk)#kt"
(10)
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SORENSEN:
PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES 4. METHOD OF ANALYSIS
true stroin 0.1 O0
rP=7'5° =
0.080
Approximate solutions for the deformations in the metal matrix composite are here obtained by a linear incremental finite element method based on the Lagrangian formulation of the field equations. The measure of deformation is the Lagrangian strain tensor which can be expressed in terms of the covariant displacement components u~ by
Oo
0.060 0.040 0.020
r/q = ½(u¢i + uj,, "4- u~ukj ).
0.000 0
2000
4000
6000
8000
t i m e (s)
Fig. 4. Overall creep strain [(E2)= ln(B/Bo) ] vs time for various angles of tilting of the short fibres. (:£ = 4, ~r = 5 and f = 0.10).
(a)
(11)
10000
(b)
Here (),i denotes the covariant derivative in the reference coordinate system. The equilibrium equation used in the incremental formulation of the boundary value problem is the incremental principle
(c)
Fig. 5. Contours of effective plastic strains ceP (a), yon Mises stresses (b) and deformation resistance (c). ~o = 0 °, c~¢= 4, (E2) =0.103. (The numbers in b, c are in units of MPa.)
SORENSEN:
PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
2977
1/.
(a)
(b)
(c)
Fig. 6. Contours of effective plastic strains e~ (a), von Mises stresses and deformation resistance (c). q) = 15, c~ = 4, (E2) =0.109. (The numbers in b, c are in units of MPa.) of virtual work, which is an expansion of the principle of virtual work in the formulation of Budiansky [15], solely involving integrals over the original known configuration of the (undeformed) material. To lowest order the incremental equation is
fv{A~qij+z'JAu~.~ukj}dV=fATi6u~dS
quadrilaterals, each built up of four triangular, plane strain, linear displacement elements. Examples of meshes are shown in Fig. 3(a,b) for the materials with cell aspect ratios c~c = 4 and ~c = 2, respectively. The rate tangent modulus method of Peirce et al. [16] has been used to increase the stable step size of the incremental procedure (see Appendix A). 5. NUMERICAL RESULTS
where V and S are the reference volume and surface of the region analyzed, and the stress and strain increments during the time increment At are Ar ii = f°At and Ar/i] : O(]At, respectively. In the incremental finite element solution the mesh used over the region shown in Fig. 1 consists of
Figure 4 shows plane strain creep curves for the different materials with different amount of tilting of the short fibres (Fig. 1). The true average strain in the loading direction [(e2) = ln(B/Bo)] is here plotted as a function of the time. The load is set equal to the stress Y~2= 20 MPa, at the time t = 0 s and the initial value of the deformation resistance is chosen to be
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S I ~ R E N S E N :PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
true stroin 0.10
¢p=25o
¢p=12.5 ° ~o= oo
0.08 0.06 0,04 0.02 0.00
1000
2000 time
3000
4000
5000
(s)
Fig. 7. Overall creep strain [(£2)= ln(B/Bo)] vs time for various angles of tilting of the short fibres. ~c = 2, ctr -- 5 and f = 0.10. So = 20 MPa (to allow for a stage 1 description in the present strain rate range). The values E = 59 GPa and v =0.35 are used for Young's modulus and Poisson's ratio for the matrix, respectively, and the corresponding quantities for the elastic inclusions are Ef = 470 GPa and Vr= 0.21. The short fibres are taken to be purely elastic throughout the deformation history and the volume fraction of short fibres is f = 0.1 in all the cases analyzed. The creep curves shown in Fig. 4 correspond to materials with ~c = 4 and ~f = 5. The results in Fig. 4 shown that a reduction of 44% in the time needed to reach 10% overall straining
(a)
(b)
results if the short fibres are tilted 15° towards one another. The material with ~p = 7.5 ° also has a lower creep resistance than the material with parallel fibres (tp = 0°) corresponding to a reduction of 27% in the time needed to reach 10% overall strain. The material with parallel short fibres almost reaches a steady state creep situation at the end of the loading period when (E2) is around 0.1. By contrast, the material with q~ = 15° is rather far from entering a steady state within in the strain interval analyzed, as seen in Fig. 4. It is noted that the hydrostatic stress level in the matrix decreases for increasing fibre tilting, and the higher creep rate for tilted fibres is a result of this plastic constraint. The contours of constant effective plastic strain (E~) in Fig. 5(a) illustrate the highly symmetric deformation pattern found in materials with aligned parallel short fibres. This configuration is stable in the sense that there is no change in the fibre orientation as the creep deformation proceeds. The largest deformation shown in Fig. 5(a) is localized around the fibres, where the strain reaches values higher than E~ = 1.1 close to the corners of the fibres. Along the sides of the fibre the plastic strains decrease from the high values found close to the corners to quite low values along the longitudinal and transverse planes of symmetry. Low plastic strains are also seen along the sides of the unit cell, as a result of the aligned distribution, and particular low strain regions (E~p = 0.05) are found along the cell side just above the
(c)
Fig. 8. Contours of effective plastic strains E~ (a), von Mises stresses (b) and deformation resistance (c). ~o =0 °, ~c=2, (~2)=0.106. (The numbers in b, c are in units of MPa.)
SORENSEN: PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
2979
(°'° 0,0]
(a)
(b)
(c)
Fig. 9. Countours of effective plastic strains ~P (a), von Mises stresses (b) and deformation resistance (c). ~P= 25°, ~c = 2, (E2) = 0.107. (The numbers in b, c are units of MPa.) short fibre, reflecting the choice of cell, the fibre aspect ratio, and the plane strain assumption. The largest von Mises stresses concentrate around the corners of the short fibre, as seen in Fig. 5(b), leading to the large plastic deformations seen in these regions in Fig. 5(a). The variation in the von Mises stress indicates that the build-up of quite high stresses is possible in some areas of the matrix partly due to local hardening. The local hardening in the matrix can be described by the picture of the local resistance towards plastic deformation, shown in Fig. 5(c) in terms of contours of constant deformation resistance (s). The deformation resistance grows during the creep of the matrix from the initial level (so = 20 MPa) to the levels illustrated in Fig. 5(c). The effect of tilting the fibres 15° out of the stable parallel position changes the deformation pattern as seen in Fig. 6(a), showing contours of constant effective plastic strains at the overall strain (E2)= 0.109. During the deformation of the material to this overall strain the angle of tilting changes from the initial value q~ = 15° to the current angle 12.4c. Large deformations are still seen in the regions close to the corners of the short fibres. In the regions of the matrix where the corners of the neighbouring fibres are at the largest distance (compare with Fig. 1) the largest deformations are found, whereas corresponding smaller deformations are found in the regions where the corners are closest to one another. AM 4 1 1 0
M
In the materials with misaligned fibres the gradual change in the constraint on the matrix, resulting from the rotation of the inclusions towards the stable parallel configuration gives rise to less concentrated stresses around the corners of the short fibres. This is evident from the contours of constant von Mises stresses shown in Fig. 6(b), compared with the corresponding contours for the material with parallel fibres [Fig. 5(b)]. The local hardening of the matrix illustrated in terms of contours of constant deformation resistance in Fig. 6(c) also shows a different picture from the parallel fibre material. In the matrix the resistance against plastic deformation has here grown to the levels of 34-36 MPa in large regions resulting from the higher local creep strain rates. In order to study composites with a higher degree of fibre misalignment in terms of the simple periodic configuration in Fig. l, a representative cell with a smaller aspect ratio is chosen. In Fig. 7 plane strain creep curves are shown for composites with the cell aspect ratio ~c = 2 and fibre aspect ratio ~f = 5 and different angles of tilting (the same loading conditions as above was used, i.e. Y'~2 = 20 MPa at the time t = 0 s). For composites with this rather low cell aspect ratio misalignment also has a marked effect on reducing the creep strength. For the material where the angle of tilting ~0 = 25 '~ the time needed to reach 10% overall strain is 53% smaller than the time needed to reach 10% overall strain for the material with parallel fibres (and same fibre aspect ratios). For
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SORENSEN: PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
the material with ¢p =12.5 ° the corresponding reduction in the time needed to reach 10% overall strain is 44%. All of the composites with the cell aspect ratios, ~c = 4 have a larger creep resistance than the materials with the lower cell aspect ratio ~¢ = 2, but the materials with the higher cell aspect ratio are more sensitive to fibre misorientation. The lower creep resistance found for the materials with the smaller aspect ratio (see Fig. 7) is explained by the fact that the interaction between the fibres in the transverse direction is reduced by the relatively larger (transverse) fibre distance. The contours of constant plastic strains seen in Fig. 8(a) show a large region between the fibres in the transverse direction where the effective plastic strain has reached values in the interval between 0.1 and 0.3. Similar regions were seen in Fig. 5(a) but the region is larger for the material with smaller aspect ratio shown in Fig. 8(a). The von Mises stresses seen in Fig. 8(b) are in the interval from 14-16MPa in the area between the fibres and peak values around 18 MPa are seen near the corners of the short fibres. Compared with the unreinforced metal, where the von Mises stress would be 17.3 MPa under these loading conditions, it is seen that all of the matrix, except the small regions very close to the ends of the short fibres, is carrying a smaller load. The contours of constant deformation resistance seen in Fig. 8(c) show a somewhat larger resistance against inelastic deformation in the regions
1.0
of the matrix close to the cell sides than seen in the material with larger cell aspect ratio at the same overall strain level. The larger deformation resistance in the composite with ctc = 2 shows that in this fibre configuration the constraint against inelastic deformation is larger in the initial creep stage than was the case for the material with ~c = 4. Figure 9(a) shows contours of constant effective plastic strain in the composite with the initial angle of tilting of the short fibres is ~0 = 25 °, at the overall strain level (e2)=0.103. Again it is seen that the largest deformations appear at the corners facing the direction of the rotation of the short fibres (counter clockwise). At this point of overall strain the short fibres have rotated to the current angle of 20.9 ° from the parallel position. The fibre tilting has reduced the constraint imposed on the matrix markedly, giving rise to peak plastic strains of less than half the peak strains seen in the composite with parallel fibres. The distribution of the von Mises stresses shown on the contour plot in Fig. 9(b) for the composite with ~a = 25 ° differs from that seen on Fig. 8(b) for the material with parallel fibres. Especially, the stresses along the cell sides have become quite large in a small region just below the line between the closest corners of neighbouring short fibres• Above the line between the closest fibre corners quite low stresses are seen, as in the previous stress pictures, but due to the rather high angle of tilting a large stress gradient exists. The
1.0
/0,9.,/
l°,q
fo~~
~N.~i:':':':':i:i:i:':': :i:i::;i:i:i:i:!:!:i10 5 f lO
/1,0
(a)
(b)
Fig. 10. Contours of constant values of the ratio s/s* in a material with parallel fibres (a) and tilted fibres (b). ctc = 2, (~2)=0.106.
SORENSEN: PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
1/.,0-180-
120~ ~k. A 108~ '40/ 6O
2981
~
40 -180 20 O0 8O
o
I0~ 60 -80 ,100 120 ~" . . . . 120
140-180" ~'~.~ -11.0 180 -
(a)
(b)
Fig. 11. Contours of maximal principal stress in the short fibres of a material with parallel fibres (a) and tilted fibres (b). ~o = 2Y, ~c = 2, (E2)= 0.106. (The numbers in Fig. a, b are in units of MPa.)
hardening behaviour is also changed by the large amount of tilting of the short fibres as seen in Fig. 9(c). The deformation resistance, s, is quite high just below the line between the closest fibre corners and very low just above. Thus, a barrier between a soft, easily deformable zone and a harder one carring more load is formed by the approaching fibre ends. The development of inelastic deformation in the composites depends on several different parameters, e.g. the angle of fibre tilting, the choice of aspect ratios, and the loading conditions. The material with parallel fibres is therefore in a different creep state than one with tilted fibres. This situation is illustrated in Fig. 10(a,b) by the plots of constant values of the ratio s/s*, which is smaller than unity when hardening is dominating (s is increasing towards s*), equals unity when the material is in steady state creep, and is greater than unity when recovery is taking place. In the regions of the matrix at the longest sides of the inclusion [Fig. 10(a)] the ratio s/s* equals 0.8, indicating that this region is still in a state dominated by hardening (stage 1). In large regions the material is in a steady state and close to the corners the ratio is slightly gerater than unity, indicating that here recovery effects (strain softening) dominate. In the material where the fibres are tilted to the angle ~o = 25 °, the ratio s/s* is now smaller just above the line between the closest corners of neighbouring fibres, than that at the cell side, reflecting the hardening dominated
inelastic deformation. Below this line a balance between hardening and recovery results in a steady state material behaviour, indicated by the fact that the value of the ratio between the current value of the deformation resistance and the saturation value is close to unity. Also in the material with tilted fibres small regions close to the corners are slightly dominated by recovery effects (i.e. s/s* is greater than unity). Figure ll(a) shows the distribution of the load carried by the short fibres in terms of the maximal principal stress for the material with parallel fibres (~c = 2 ) after inelastic deformation to the overall strain (E2) = 0.106. Values of the maximal principal stresses in the interval 60-80 MPa are seen in the largest part of the short fibre, with the highest stresses in the corners, due to the perfect bond between the fibres and the matrix. For the material with tilted fibres (¢p = 2 5 °, c~c=2) in Fig. l l(b) the central regions of the short fibre are subjected to smaller stresses, in the interval 40-60 MPa. The largest principal stresses are now found in the corners facing the direction of rotation (counter clockwise) of the short fibre. 6. DISCUSSION Based on a planar model the present analysis shows some effects of short fibre tilting away from the
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SORENSEN: PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
parallel position in an idealised regular array of aligned short fibres. The behaviour of matrix composites with short fibres is in reality determined by the 3 dimensional deformation state in these materials. However, analyses based on a 2 dimensional deformation state provide a simplified view, which is useful in a qualitative understanding of the material behaviour, and plane strain models have been used in several studies of 3D phenomena (analysis of clustering of short fibres [1] or studies based on crystal plasticity [17] are often based on the plane strain assumption). In a resent study of creep in a metal matrix composite Sorensen et al. [18] found good qualitative agreement between 2D and 3D F E M predictions of clustering of the short fibres. The present analysis gives results for various angles of misalignment of the short fibres in two types of composites differing only by the cell aspect ratio. Materials processed by extrusion are often modelled by cell aspect ratios approximately equal to the fibre aspect ratio, whereas materials processed by methods which lead to less parallel short fibres, such as squeeze casting, are better modelled by a smaller cell aspect ratio. The two choices of aspect ratios also reflect the variations in a single material, where a small aspect ratio could be most realistic in one region, whereas a larger aspect ratio could be the best fit in another. The results given by the cell model analysis thus reflect local behaviour in the regions where the boundary conditions resemble those specified for the cell. In all the composites it is seen that during the creep deformation of the composite the fibres rotate towards the more stable parallel position. The fibre tilting gives a reduction in the creep strength, due to less constraining effect imposed by the tilted fibres on the matrix. The current angle of rotation, during the creep to a given value of overall strain, increases with the initial angle of misalignment. Following this fibre rotation is a considerable straining of the matrix in the regions close to the corners of the short fibres in the direction of rotation. Based on the planar analyses given here there does not seem to be a larger possibility for the tilted fibres to break during the initial stage of the creep of the composite, than for the materials with the parallel fibre configuration. Decohesion is, however, not allowed for in the model presented here, and an analysis involving failure mechanisms would have to be carried out to determine the possibility of debonding or fibre fracture. The internal variable model used in this analysis provides a possibility of a combined description of the phenomena of strain hardening, steady state behaviour and strain softening (recovery) in the metal matrix. In fact, the state of the metal varies throughout the matrix and some regions reach a steady state while other areas are still dominated by the hardening (stage 1 creep). The use of phenomenological constitutive equations to represent the metal matrix is of course
an approximation to the real behaviour, since the size of the unit cell analyzed corresponds to dimensions on the order of a few microns, and thus single crystal behaviour may be more dominant. Predictions for metal matrix composites based on models using isotropic hardening have been compared with models using crystal plasticity by Needleman and Tvergaard [17], who found that difference between the two model approaches was associated with the strongly localized flow induced by the high strain concentrations at the sharp whisker edges. Experimental studies by Barlow and Hansen [19] showed increased dislocation activity at the fibre ends, indicating a qualitative accordance with the predictions of large strains at the corners of the inclusions. The present analysis represents a qualitative analysis, where the effects of misalignment on the overall behaviour and local variations in matrix behaviour are considered, without accounting for issues such as slip plane orientation vs whisker orientation or dislocation effects such as bowing between the short fibres, etc. Acknowledgement--The helpful discussions with Professor
Viggo Tvergaard of the Technical University of Denmark are very much appreciated. REFERENCES
1. T. Christman, S. Suresh and A. Needleman, Acta metall. 37, 3029 (1988). 2. V. Tvergaard, Acta metall, mater. 39, 184 (1990). 3. A. Levy and J. M. Papazian, Metall. Trans. 21A, 411 (1990i. 4. C. L. Horn, J. Mech. Phys. Solids 40, 991 (1992). 5. J. R. Brockenbrough, S. Suresh and H. A. Wienecke, Acta metall, mater. 39, 735 (1990). 6. N. Sorensen, Int. J. Solids Struct. 29, 867 (199l). 7. T. L. Dragone and W. D. Nix, in Metal & Ceramic Matrix Composites: Processing, Modelling and Mechanical Behaviour (edited by R. B. Bhagat, A. H. Clauer,
8. 9. 10. 11. 12. 13. 14.
P. Kumar and A. M. Ritter), pp. 367-380. The Minerals, Metals & Materials Soc. (1990). G. Bao, J. W. Hutchinson and R. M. McMeeking, Acta metall, mater. 39, 1871 (1990). G. L. Povirk, S. R. Nutt and A. Needleman, Scripta metall, mater. 26, 461 (1992). O. B. Pedersen, in Proc. 7th Int. Syrup. on continuum Models of Discrete Systmes (CMDS 7), Paderborn, Germany (1992). To be published. S. B. Brown, K. H. Kim and L. Anand Int. J. Plasticity 2, 95 (1989). V. J. Tvergaard, Mech. Phys. Solids 24, 291 (1976). F. Garofalo, Trans. metall. Soc. A.1.M.E. 227,351 (1963). N. Sorensen, in Proc. 13th Ris~ Int. Syrup. on Materials Science: Modelling of Plastic Deformation and its Engineering Applications (edited by S. I. Andersen et al.), Riso
15. 16. 17. 18. 19.
National Laboratory, Roskilde, Denmark, pp. 457-465 (1992). See also Szrensen, N. Ph.D thesis (1993). To be published. B. Budiansky, Problems of Hydrodynamics and Continuum Mechanics, p. 77. SIAM, Philadelphia (1969). D. Peirce, C. F. Shih and A. Needleman, Comput. Struct. 18, 875 (1984). A. Needleman and V. Tvergaard, J. appl. Mech. To be published. N. Sorensen, A. Needleman and V. Tvergaard, Mater. Sci. Engng A158, 129 (1992). C. Barlow and N. Hansen Acta metall, mater. 39, 1971 (1991).
SORENSEN: APPENDIX
P L A N A R M O D E L OF CREEP IN M E T A L M A T R I X COMPOSITES with
A
A brief description of the rate tangent modulus method for isotropic elastic~iscoplastic solids with the internal variable constitutive equations of Brown et al. [11], following the basic formulation of Peirce et al. [16], is given here. By the introduction of an interpolation parameter ,9 the current increment of the effective plastic strain can be written as ~ = (1 - `9)~e "p(,>+`gEe "0(,+ a,>
(A1)
M,;/ =
3
i~ =
i°~(') + O A t - - / ~ Os
' - _ _ go(,)]
\
oe~ )
1 - ~AtQ
6~At +
0 s~
~,At.
+Si~(E~--E~ )
and the effective stress increment is expressed as 3s de = MVt~ij- M , s _ -,s 7p 2 ae
(A4)
oat~ (A2)
In the linear incremental procedure a first order approximation of ~¢ in terms of g~ is used on the form
,~=s
+ ,uMOflo
where
1
c~ae
E so ___
2 l + v a e"
Following this the effective plastic strain rate is expressed as
with i~(t) expressed by the Taylor expansion
i~(,+a,) = g~
2983
(A3)
-
-
(AS)
~TAtQ
and Q=-~M
s 8i~
8i~ Oi
ao Oao
8s Oive
v-'j-+--.
(A6)
Using the expressions for the effective plastic strain rate given above the constitutive equations take the following form i~ = (R ~*',_ i.tM'J Mkl)Okl
-
/t o 1 --
8AtQ
(A7)
0956-7151/93 $6.00 + 0.00 Copyright ~ 1993 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
A P L A N A R M O D E L S T U D Y OF CREEP IN M E T A L M A T R I X COMPOSITES WITH M I S A L I G N E D SHORT FIBRES N. J. S O R E N S E N Materials Department, Riso National Laboratory, DK 4000 Roskilde, Denmark (.Received 7 October 1992; in revised form 20 January I993)
Abstract--The effect of fibre misalignment on the creep behaviour of metal matrix composites is modelled, including hardening behaviour (stage 1), dynamic recovery and steady state creep (stage 2) of the matrix material, using an internal variable constitutive model for the creep behaviour of the metal matrix. Numerical plane strain results in terms of average properties and detailed local deformation behaviour up to large strains are needed to show effects of fibre misalignment on the development of inelastic strains and the resulting over-all creep resistance of the material. The creep resistance for the composite is markedly reduced by the fibre misalignment and the time needed to reach an approximate steady state is elongated due to the strain induced rotation of the short fibres in the matrix.
1. INTRODUCTION A basic understanding of the creep behaviour of metals reinforced by short high-stiffness ceramic fibres is important for a range of applications of these new materials in automotive industry, aircraft and aerospace construction where high temperature strength and thrust to weight ratio are key design parameters. Theoretical models based on finite element solutions of the field equations of continuum plasticity have been used by different authors to predict the inelastic properties of idealized metal matrix composites. Room temperature properties of short fibrereinforced metal matrix composites have been studied in this context by Christman et al. [1]. Their study included plane strain models for simple clusters of short fibres and axisymmetric models of an idealized periodic array of hexagonal cylinders, each consisting of the metal (aluminium) with a short fibre embedded in the centre (a configuration with aligned fibre ends). Tvergaard [2] used a different axisymmetric finite element model to predict the properties of short fibre reinforced metal matrix composites with different amounts of shifting ("staggering") of the neighbouring short fibres. A full 3D finite element model was used by Levi and Papazian [3] to analyze the two basic configurations of the short fibres termed "aligned" and "staggered", respectively. Recently the axisymmetric models [1] and [2] were compared with full 3D finite element models by Horn [4], who also studied the response under transverse loading. Finite element models of the elastic plastic response of continuous fibre-reinforced metal matrix composites have been presented by Brockenbrough et al. [5] and Sorenson [6]. Creep behaviour of short fibre metal matrix composites has been studied with numerically analysis by Dragone and Nix [7] in an analysis of the aligned and
staggered short-fibre configurations. Effects of the reinforcement shape and morphology have been modelled by Bao et al. [8] and Povirk et al. [9] used the axisymmetric model of the periodic array of aligned hexagonal cylinders to analyze creep in thermally cycled AI/SiC composites. In all of the models [1-9] it is assumed that the short fibres are perfectly parallel. In real materials, however, the short fibres are not perfectly parallel and the material behaviour of the composite can be expected to vary with the degree of misalignment. In the present paper a plane strain model of misaligned short 2-dimensional fibres is analyzed as a model problem for the real 3D geometry found in whisker-reinforced metal matrix composites. The plane strain assumption, which is made to reduce the computational requirement, means that the fibres are really continuous in the direction perpendicular to the plane. The results can be used for qualitative predictions of the effects of fibre misalignment on the behaviour of short fibre reinforced metal matrix composites, which is the topic here. The behaviour of metal matrix composites under creep conditions is complicated by the fact that the matrix is highly constrained by the reinforcement and thus large variations in the stress and strain fields in the metal matrix are found. This highly nonuniform stress distribution leads to strain rates ranging from traditional creep, dominated by dislocation glide through climb, to "hot-working" where strain hardening, dynamic recovery and in some cases dynamic recrystallisation are important. Furthermore, special scale dependent hardening phenomena may play a role Pedersen [10]. The present paper uses an internal variable constitutive model proposed by Brown et al. [11] to describe the behaviour of the metal matrix, whereas the reinforcement is modelled as an elastic material. The internal variable model does not account for scale
2973
2974
SORENSEN: PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
effects. However, it has the advantage over standard power law creep that it accounts for primary creep and dynamic recovery in addition to the steady state creep. Furthermore, the assumption of a perfect interfacial bond between the fibre and the matrix is made and thus no interracial decohesion is allowed in the model. 2. P R O B L E M
FORMULATION
A periodic 2-dimensional array of tilted short fibres (see Fig. 1) is subjected to a plane strain creep analysis to be carried out here. The characteristic unit cell chosen to be analyzed has the initial dimensions A0 and B0, while d is the width of the inclusion and the tilt-angle is denoted by ~p. The unit cell is characterized by the aspect ratio ctc = Bo/Ao and the fibre aspect ratio is ~f = l / d . The volume fractions of short fibres (10 is M f = AoB--~o. (1) The material of Fig. 1 is loaded to the average stress Y'z in the x2-direction at the start. From this time on the creep deformations are simulated by a numerical analysis of the unit cell in Fig. 1. In this analysis a Cartesian coordinate system, with x ~and x 2 denoting coordinate directions, is used as reference in a convected coordinate Lagrangian formulation of the field equations, where gij and G o are the components of the metric tensors of the initial and current configurations, respectively, with determinants g and G, and qij = (Gij - go)~2 are the covariant components of the Lagrangian strain tensor. The contravariant displacement components on the reference base vectors are denoted by u i, and the contravariant components of the Cauchy stress tensor on the current coordinates are tr U while the corresponding components of the Kirchhoff stress tensor are given by r ~ = x / ~ a 0. With the current dimensions of the unit cell of Fig. 1 denoted by A and B in the x ~ and x2-directions, respectively, the boundary conditions are u~=0,
~=0,
xl=0
~=A,
72=0,
x l=Ao
fi2=0,
71=0,
x 2=0
~2 = B,
~ 1 = O,
X 2 =B o
U3 = ~3 = 0,
0 ~ xl~
do,
0 ~ x 2 ~Bo
conditions, only a small reduction in the computational effort is obtained by choosing the smallest unit cell (i.e. half of the cell defined above). 3. C O N S T I T U T I V E
MODEL FOR THE MATRIX MATERIAL
In order to model the material response of the composite under creep conditions an internal variable constitutive model is used to describe the behaviour of the matrix material (see Brown et aL [ll]). The constitutive model is basically the steady state hyperbolic sine creep equation of Garofalo [13], generalized to account for strain hardening and dynamic recovery by the introduction of a single scalar variable, the so-called deformation resistance, here denoted by s. The internal variable repressnts an average isotropic resistance to macroscopic plastic flow offered by the underlying isotropic strengthening mechanisms, such as dislocation density, solid solution strengthening or subgrain and grain size effects. The constitutive model has the following form
(o.)1 , m
•C = A l e x p - ~ Ee
, =(h01-
slnh ~ s
~ . ~s i g n ( 1 - ~ . ) ) ~ c
(3)
(4)
with s*=s
~
e
~exp
O
~
.
(5)
•c is the effective creep In the flow equation (3), ee strain-rate, 0 is the absolute temperature, and the yon Mises stress is tre= ~ s ° / 2 , where s o = z ° - z ~ G ° / 3 ,_ Ao_, I-
-I
I
i/
X2
(2)
where T ~are the components of the nominal surface tractions, and (') denotes the time derivative. The increments in the cell dimensions are prescribed using a Rayleigh-Ritz procedure (see Tvergaard [12]), so that the average stress, Z2, has a constant value while the average stress in the x i-direction is zero. The unit cell chosen in this problem formulation is not the smallest possible, since a rotational symmetry exists around the centre of each of the cells shown in Fig. 1. However, due to the more complicated boundary
x,
Fig. 1. Periodic array of tilted short fibres in a two-dimensional material.
SORENSEN: PLANARMODELOF CREEP IN METALMATRIX COMPOSITES is the Kirchhoff stress deviator. In the evolution equations (4-5) for the internal variable, the quantity s* represents a saturation value of s associated with a given temperature/strain rate pair. The material parameters in these constitutive equations are: A j, Q, m, 4, h0, a, g and n. In this model study the values of these parameters are chosen to be At = 1.91 - 107 s ~, Q = 175 kJ/mol, ~ = 7.0, m = 0.233, g = 18.9 MPa, n = 0.0705, h0 = 1.1 G P a and a = 1.3. The basic behaviour of the constitutive model is shown on Fig. 2(a,b), where the unreinforced metal is loaded in plane strain tension at the starting time by a stress 5:2 = a0, where a0 = 20 MPa. After 13,500 s the load is reduced to 75% of the initial stress level. The upper curve in Fig. 2(a), shows the development of the deformation resistance from the initial value s o = 20 M P a to the saturation level at the given stress. Following the load drop the deformation resistance slowly decreases towards a new saturation level. After the increase in the load at 38,000 s the deformation resistance grows until it reaches the saturation level seen prior to the load drop. The corresponding development in the strain for this loading is seen in Fig. 2(b), where the first part of the curve corresponds to normal creep behaviour with the material entering
2.00
(a)
1.75
1.50 1.25 1.00
}-2/(I0
0.75 0.50
i
10000
0
,
l
i
20000
30000
40000
t i m e (s)
a steady state (stage 2) after a short period of primary creep (stage 1). After the load drop the recovery effects dominate and the initial low strain rate slowly increases, indicating a strain softening. After reloading part of a typical creep response is seen; first a short hardening period until a steady creep state is reached with the same characteristic creep rate as found prior to the load drop. Using the constitutive model described above the creep strain rate 0 c is taken to be the effective creep strain rate (3) generalized to multiaxial strain via the stress deviator •c 2
1.50
#
(b)
1.00
0.75 ~
(b)
Fig. 3. Meshes used for the analysis of composites with tilted short fibres for ~f = 5 and./'= 0.10. (a) ~, = 4, ~0 - 15% (b)%=2,~=25.
s,j
0 c = ~e - - - . 3 rr~
true strain 1,25
(a)
0.25
The total strain rate is expressed as the sum of the elastic part and the creep part
o,,: O~ + 0 ~
f'; = R'Jkt(Okt- Oct). 0
(6)
(7)
The elastic stress-strain relationship is taken to be of the form f~ = R~ik~O~t so that the constitutive relation in the presence of creep and elastic deformation can be written as
-
0.50
0.00
2975
'
'
'
'
10000
20000
30000
40000
(8)
Here, the elastic instantaneous moduli are taken to be
time (s) Fig. 2. Basicbehaviourofthe constitutivemodelappliedfor the matrix material. The development of the deformation resistance corresponding to the load drop and reloading (a) and the strain in the tensile direction (b). The deformation resistance and stressed in (a) are divided by the starting values so and a0, respectively. (From Sorensen [14],)
with E and v denoting Young's modulus and Poisson's ratio respectively, and the Jaumann rate of the Kirchhoff stress tensor is
~(i =ii/ + (Gi~zJt + Gilrlk)#kt"
(10)
2976
SORENSEN:
PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES 4. METHOD OF ANALYSIS
true stroin 0.1 O0
rP=7'5° =
0.080
Approximate solutions for the deformations in the metal matrix composite are here obtained by a linear incremental finite element method based on the Lagrangian formulation of the field equations. The measure of deformation is the Lagrangian strain tensor which can be expressed in terms of the covariant displacement components u~ by
Oo
0.060 0.040 0.020
r/q = ½(u¢i + uj,, "4- u~ukj ).
0.000 0
2000
4000
6000
8000
t i m e (s)
Fig. 4. Overall creep strain [(E2)= ln(B/Bo) ] vs time for various angles of tilting of the short fibres. (:£ = 4, ~r = 5 and f = 0.10).
(a)
(11)
10000
(b)
Here (),i denotes the covariant derivative in the reference coordinate system. The equilibrium equation used in the incremental formulation of the boundary value problem is the incremental principle
(c)
Fig. 5. Contours of effective plastic strains ceP (a), yon Mises stresses (b) and deformation resistance (c). ~o = 0 °, c~¢= 4, (E2) =0.103. (The numbers in b, c are in units of MPa.)
SORENSEN:
PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
2977
1/.
(a)
(b)
(c)
Fig. 6. Contours of effective plastic strains e~ (a), von Mises stresses and deformation resistance (c). q) = 15, c~ = 4, (E2) =0.109. (The numbers in b, c are in units of MPa.) of virtual work, which is an expansion of the principle of virtual work in the formulation of Budiansky [15], solely involving integrals over the original known configuration of the (undeformed) material. To lowest order the incremental equation is
fv{A~qij+z'JAu~.~ukj}dV=fATi6u~dS
quadrilaterals, each built up of four triangular, plane strain, linear displacement elements. Examples of meshes are shown in Fig. 3(a,b) for the materials with cell aspect ratios c~c = 4 and ~c = 2, respectively. The rate tangent modulus method of Peirce et al. [16] has been used to increase the stable step size of the incremental procedure (see Appendix A). 5. NUMERICAL RESULTS
where V and S are the reference volume and surface of the region analyzed, and the stress and strain increments during the time increment At are Ar ii = f°At and Ar/i] : O(]At, respectively. In the incremental finite element solution the mesh used over the region shown in Fig. 1 consists of
Figure 4 shows plane strain creep curves for the different materials with different amount of tilting of the short fibres (Fig. 1). The true average strain in the loading direction [(e2) = ln(B/Bo)] is here plotted as a function of the time. The load is set equal to the stress Y~2= 20 MPa, at the time t = 0 s and the initial value of the deformation resistance is chosen to be
2978
S I ~ R E N S E N :PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
true stroin 0.10
¢p=25o
¢p=12.5 ° ~o= oo
0.08 0.06 0,04 0.02 0.00
1000
2000 time
3000
4000
5000
(s)
Fig. 7. Overall creep strain [(£2)= ln(B/Bo)] vs time for various angles of tilting of the short fibres. ~c = 2, ctr -- 5 and f = 0.10. So = 20 MPa (to allow for a stage 1 description in the present strain rate range). The values E = 59 GPa and v =0.35 are used for Young's modulus and Poisson's ratio for the matrix, respectively, and the corresponding quantities for the elastic inclusions are Ef = 470 GPa and Vr= 0.21. The short fibres are taken to be purely elastic throughout the deformation history and the volume fraction of short fibres is f = 0.1 in all the cases analyzed. The creep curves shown in Fig. 4 correspond to materials with ~c = 4 and ~f = 5. The results in Fig. 4 shown that a reduction of 44% in the time needed to reach 10% overall straining
(a)
(b)
results if the short fibres are tilted 15° towards one another. The material with ~p = 7.5 ° also has a lower creep resistance than the material with parallel fibres (tp = 0°) corresponding to a reduction of 27% in the time needed to reach 10% overall strain. The material with parallel short fibres almost reaches a steady state creep situation at the end of the loading period when (E2) is around 0.1. By contrast, the material with q~ = 15° is rather far from entering a steady state within in the strain interval analyzed, as seen in Fig. 4. It is noted that the hydrostatic stress level in the matrix decreases for increasing fibre tilting, and the higher creep rate for tilted fibres is a result of this plastic constraint. The contours of constant effective plastic strain (E~) in Fig. 5(a) illustrate the highly symmetric deformation pattern found in materials with aligned parallel short fibres. This configuration is stable in the sense that there is no change in the fibre orientation as the creep deformation proceeds. The largest deformation shown in Fig. 5(a) is localized around the fibres, where the strain reaches values higher than E~ = 1.1 close to the corners of the fibres. Along the sides of the fibre the plastic strains decrease from the high values found close to the corners to quite low values along the longitudinal and transverse planes of symmetry. Low plastic strains are also seen along the sides of the unit cell, as a result of the aligned distribution, and particular low strain regions (E~p = 0.05) are found along the cell side just above the
(c)
Fig. 8. Contours of effective plastic strains E~ (a), von Mises stresses (b) and deformation resistance (c). ~o =0 °, ~c=2, (~2)=0.106. (The numbers in b, c are in units of MPa.)
SORENSEN: PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
2979
(°'° 0,0]
(a)
(b)
(c)
Fig. 9. Countours of effective plastic strains ~P (a), von Mises stresses (b) and deformation resistance (c). ~P= 25°, ~c = 2, (E2) = 0.107. (The numbers in b, c are units of MPa.) short fibre, reflecting the choice of cell, the fibre aspect ratio, and the plane strain assumption. The largest von Mises stresses concentrate around the corners of the short fibre, as seen in Fig. 5(b), leading to the large plastic deformations seen in these regions in Fig. 5(a). The variation in the von Mises stress indicates that the build-up of quite high stresses is possible in some areas of the matrix partly due to local hardening. The local hardening in the matrix can be described by the picture of the local resistance towards plastic deformation, shown in Fig. 5(c) in terms of contours of constant deformation resistance (s). The deformation resistance grows during the creep of the matrix from the initial level (so = 20 MPa) to the levels illustrated in Fig. 5(c). The effect of tilting the fibres 15° out of the stable parallel position changes the deformation pattern as seen in Fig. 6(a), showing contours of constant effective plastic strains at the overall strain (E2)= 0.109. During the deformation of the material to this overall strain the angle of tilting changes from the initial value q~ = 15° to the current angle 12.4c. Large deformations are still seen in the regions close to the corners of the short fibres. In the regions of the matrix where the corners of the neighbouring fibres are at the largest distance (compare with Fig. 1) the largest deformations are found, whereas corresponding smaller deformations are found in the regions where the corners are closest to one another. AM 4 1 1 0
M
In the materials with misaligned fibres the gradual change in the constraint on the matrix, resulting from the rotation of the inclusions towards the stable parallel configuration gives rise to less concentrated stresses around the corners of the short fibres. This is evident from the contours of constant von Mises stresses shown in Fig. 6(b), compared with the corresponding contours for the material with parallel fibres [Fig. 5(b)]. The local hardening of the matrix illustrated in terms of contours of constant deformation resistance in Fig. 6(c) also shows a different picture from the parallel fibre material. In the matrix the resistance against plastic deformation has here grown to the levels of 34-36 MPa in large regions resulting from the higher local creep strain rates. In order to study composites with a higher degree of fibre misalignment in terms of the simple periodic configuration in Fig. l, a representative cell with a smaller aspect ratio is chosen. In Fig. 7 plane strain creep curves are shown for composites with the cell aspect ratio ~c = 2 and fibre aspect ratio ~f = 5 and different angles of tilting (the same loading conditions as above was used, i.e. Y'~2 = 20 MPa at the time t = 0 s). For composites with this rather low cell aspect ratio misalignment also has a marked effect on reducing the creep strength. For the material where the angle of tilting ~0 = 25 '~ the time needed to reach 10% overall strain is 53% smaller than the time needed to reach 10% overall strain for the material with parallel fibres (and same fibre aspect ratios). For
2980
SORENSEN: PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
the material with ¢p =12.5 ° the corresponding reduction in the time needed to reach 10% overall strain is 44%. All of the composites with the cell aspect ratios, ~c = 4 have a larger creep resistance than the materials with the lower cell aspect ratio ~¢ = 2, but the materials with the higher cell aspect ratio are more sensitive to fibre misorientation. The lower creep resistance found for the materials with the smaller aspect ratio (see Fig. 7) is explained by the fact that the interaction between the fibres in the transverse direction is reduced by the relatively larger (transverse) fibre distance. The contours of constant plastic strains seen in Fig. 8(a) show a large region between the fibres in the transverse direction where the effective plastic strain has reached values in the interval between 0.1 and 0.3. Similar regions were seen in Fig. 5(a) but the region is larger for the material with smaller aspect ratio shown in Fig. 8(a). The von Mises stresses seen in Fig. 8(b) are in the interval from 14-16MPa in the area between the fibres and peak values around 18 MPa are seen near the corners of the short fibres. Compared with the unreinforced metal, where the von Mises stress would be 17.3 MPa under these loading conditions, it is seen that all of the matrix, except the small regions very close to the ends of the short fibres, is carrying a smaller load. The contours of constant deformation resistance seen in Fig. 8(c) show a somewhat larger resistance against inelastic deformation in the regions
1.0
of the matrix close to the cell sides than seen in the material with larger cell aspect ratio at the same overall strain level. The larger deformation resistance in the composite with ctc = 2 shows that in this fibre configuration the constraint against inelastic deformation is larger in the initial creep stage than was the case for the material with ~c = 4. Figure 9(a) shows contours of constant effective plastic strain in the composite with the initial angle of tilting of the short fibres is ~0 = 25 °, at the overall strain level (e2)=0.103. Again it is seen that the largest deformations appear at the corners facing the direction of the rotation of the short fibres (counter clockwise). At this point of overall strain the short fibres have rotated to the current angle of 20.9 ° from the parallel position. The fibre tilting has reduced the constraint imposed on the matrix markedly, giving rise to peak plastic strains of less than half the peak strains seen in the composite with parallel fibres. The distribution of the von Mises stresses shown on the contour plot in Fig. 9(b) for the composite with ~a = 25 ° differs from that seen on Fig. 8(b) for the material with parallel fibres. Especially, the stresses along the cell sides have become quite large in a small region just below the line between the closest corners of neighbouring short fibres• Above the line between the closest fibre corners quite low stresses are seen, as in the previous stress pictures, but due to the rather high angle of tilting a large stress gradient exists. The
1.0
/0,9.,/
l°,q
fo~~
~N.~i:':':':':i:i:i:':': :i:i::;i:i:i:i:!:!:i10 5 f lO
/1,0
(a)
(b)
Fig. 10. Contours of constant values of the ratio s/s* in a material with parallel fibres (a) and tilted fibres (b). ctc = 2, (~2)=0.106.
SORENSEN: PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
1/.,0-180-
120~ ~k. A 108~ '40/ 6O
2981
~
40 -180 20 O0 8O
o
I0~ 60 -80 ,100 120 ~" . . . . 120
140-180" ~'~.~ -11.0 180 -
(a)
(b)
Fig. 11. Contours of maximal principal stress in the short fibres of a material with parallel fibres (a) and tilted fibres (b). ~o = 2Y, ~c = 2, (E2)= 0.106. (The numbers in Fig. a, b are in units of MPa.)
hardening behaviour is also changed by the large amount of tilting of the short fibres as seen in Fig. 9(c). The deformation resistance, s, is quite high just below the line between the closest fibre corners and very low just above. Thus, a barrier between a soft, easily deformable zone and a harder one carring more load is formed by the approaching fibre ends. The development of inelastic deformation in the composites depends on several different parameters, e.g. the angle of fibre tilting, the choice of aspect ratios, and the loading conditions. The material with parallel fibres is therefore in a different creep state than one with tilted fibres. This situation is illustrated in Fig. 10(a,b) by the plots of constant values of the ratio s/s*, which is smaller than unity when hardening is dominating (s is increasing towards s*), equals unity when the material is in steady state creep, and is greater than unity when recovery is taking place. In the regions of the matrix at the longest sides of the inclusion [Fig. 10(a)] the ratio s/s* equals 0.8, indicating that this region is still in a state dominated by hardening (stage 1). In large regions the material is in a steady state and close to the corners the ratio is slightly gerater than unity, indicating that here recovery effects (strain softening) dominate. In the material where the fibres are tilted to the angle ~o = 25 °, the ratio s/s* is now smaller just above the line between the closest corners of neighbouring fibres, than that at the cell side, reflecting the hardening dominated
inelastic deformation. Below this line a balance between hardening and recovery results in a steady state material behaviour, indicated by the fact that the value of the ratio between the current value of the deformation resistance and the saturation value is close to unity. Also in the material with tilted fibres small regions close to the corners are slightly dominated by recovery effects (i.e. s/s* is greater than unity). Figure ll(a) shows the distribution of the load carried by the short fibres in terms of the maximal principal stress for the material with parallel fibres (~c = 2 ) after inelastic deformation to the overall strain (E2) = 0.106. Values of the maximal principal stresses in the interval 60-80 MPa are seen in the largest part of the short fibre, with the highest stresses in the corners, due to the perfect bond between the fibres and the matrix. For the material with tilted fibres (¢p = 2 5 °, c~c=2) in Fig. l l(b) the central regions of the short fibre are subjected to smaller stresses, in the interval 40-60 MPa. The largest principal stresses are now found in the corners facing the direction of rotation (counter clockwise) of the short fibre. 6. DISCUSSION Based on a planar model the present analysis shows some effects of short fibre tilting away from the
2982
SORENSEN: PLANAR MODEL OF CREEP IN METAL MATRIX COMPOSITES
parallel position in an idealised regular array of aligned short fibres. The behaviour of matrix composites with short fibres is in reality determined by the 3 dimensional deformation state in these materials. However, analyses based on a 2 dimensional deformation state provide a simplified view, which is useful in a qualitative understanding of the material behaviour, and plane strain models have been used in several studies of 3D phenomena (analysis of clustering of short fibres [1] or studies based on crystal plasticity [17] are often based on the plane strain assumption). In a resent study of creep in a metal matrix composite Sorensen et al. [18] found good qualitative agreement between 2D and 3D F E M predictions of clustering of the short fibres. The present analysis gives results for various angles of misalignment of the short fibres in two types of composites differing only by the cell aspect ratio. Materials processed by extrusion are often modelled by cell aspect ratios approximately equal to the fibre aspect ratio, whereas materials processed by methods which lead to less parallel short fibres, such as squeeze casting, are better modelled by a smaller cell aspect ratio. The two choices of aspect ratios also reflect the variations in a single material, where a small aspect ratio could be most realistic in one region, whereas a larger aspect ratio could be the best fit in another. The results given by the cell model analysis thus reflect local behaviour in the regions where the boundary conditions resemble those specified for the cell. In all the composites it is seen that during the creep deformation of the composite the fibres rotate towards the more stable parallel position. The fibre tilting gives a reduction in the creep strength, due to less constraining effect imposed by the tilted fibres on the matrix. The current angle of rotation, during the creep to a given value of overall strain, increases with the initial angle of misalignment. Following this fibre rotation is a considerable straining of the matrix in the regions close to the corners of the short fibres in the direction of rotation. Based on the planar analyses given here there does not seem to be a larger possibility for the tilted fibres to break during the initial stage of the creep of the composite, than for the materials with the parallel fibre configuration. Decohesion is, however, not allowed for in the model presented here, and an analysis involving failure mechanisms would have to be carried out to determine the possibility of debonding or fibre fracture. The internal variable model used in this analysis provides a possibility of a combined description of the phenomena of strain hardening, steady state behaviour and strain softening (recovery) in the metal matrix. In fact, the state of the metal varies throughout the matrix and some regions reach a steady state while other areas are still dominated by the hardening (stage 1 creep). The use of phenomenological constitutive equations to represent the metal matrix is of course
an approximation to the real behaviour, since the size of the unit cell analyzed corresponds to dimensions on the order of a few microns, and thus single crystal behaviour may be more dominant. Predictions for metal matrix composites based on models using isotropic hardening have been compared with models using crystal plasticity by Needleman and Tvergaard [17], who found that difference between the two model approaches was associated with the strongly localized flow induced by the high strain concentrations at the sharp whisker edges. Experimental studies by Barlow and Hansen [19] showed increased dislocation activity at the fibre ends, indicating a qualitative accordance with the predictions of large strains at the corners of the inclusions. The present analysis represents a qualitative analysis, where the effects of misalignment on the overall behaviour and local variations in matrix behaviour are considered, without accounting for issues such as slip plane orientation vs whisker orientation or dislocation effects such as bowing between the short fibres, etc. Acknowledgement--The helpful discussions with Professor
Viggo Tvergaard of the Technical University of Denmark are very much appreciated. REFERENCES
1. T. Christman, S. Suresh and A. Needleman, Acta metall. 37, 3029 (1988). 2. V. Tvergaard, Acta metall, mater. 39, 184 (1990). 3. A. Levy and J. M. Papazian, Metall. Trans. 21A, 411 (1990i. 4. C. L. Horn, J. Mech. Phys. Solids 40, 991 (1992). 5. J. R. Brockenbrough, S. Suresh and H. A. Wienecke, Acta metall, mater. 39, 735 (1990). 6. N. Sorensen, Int. J. Solids Struct. 29, 867 (199l). 7. T. L. Dragone and W. D. Nix, in Metal & Ceramic Matrix Composites: Processing, Modelling and Mechanical Behaviour (edited by R. B. Bhagat, A. H. Clauer,
8. 9. 10. 11. 12. 13. 14.
P. Kumar and A. M. Ritter), pp. 367-380. The Minerals, Metals & Materials Soc. (1990). G. Bao, J. W. Hutchinson and R. M. McMeeking, Acta metall, mater. 39, 1871 (1990). G. L. Povirk, S. R. Nutt and A. Needleman, Scripta metall, mater. 26, 461 (1992). O. B. Pedersen, in Proc. 7th Int. Syrup. on continuum Models of Discrete Systmes (CMDS 7), Paderborn, Germany (1992). To be published. S. B. Brown, K. H. Kim and L. Anand Int. J. Plasticity 2, 95 (1989). V. J. Tvergaard, Mech. Phys. Solids 24, 291 (1976). F. Garofalo, Trans. metall. Soc. A.1.M.E. 227,351 (1963). N. Sorensen, in Proc. 13th Ris~ Int. Syrup. on Materials Science: Modelling of Plastic Deformation and its Engineering Applications (edited by S. I. Andersen et al.), Riso
15. 16. 17. 18. 19.
National Laboratory, Roskilde, Denmark, pp. 457-465 (1992). See also Szrensen, N. Ph.D thesis (1993). To be published. B. Budiansky, Problems of Hydrodynamics and Continuum Mechanics, p. 77. SIAM, Philadelphia (1969). D. Peirce, C. F. Shih and A. Needleman, Comput. Struct. 18, 875 (1984). A. Needleman and V. Tvergaard, J. appl. Mech. To be published. N. Sorensen, A. Needleman and V. Tvergaard, Mater. Sci. Engng A158, 129 (1992). C. Barlow and N. Hansen Acta metall, mater. 39, 1971 (1991).
SORENSEN: APPENDIX
P L A N A R M O D E L OF CREEP IN M E T A L M A T R I X COMPOSITES with
A
A brief description of the rate tangent modulus method for isotropic elastic~iscoplastic solids with the internal variable constitutive equations of Brown et al. [11], following the basic formulation of Peirce et al. [16], is given here. By the introduction of an interpolation parameter ,9 the current increment of the effective plastic strain can be written as ~ = (1 - `9)~e "p(,>+`gEe "0(,+ a,>
(A1)
M,;/ =
3
i~ =
i°~(') + O A t - - / ~ Os
' - _ _ go(,)]
\
oe~ )
1 - ~AtQ
6~At +
0 s~
~,At.
+Si~(E~--E~ )
and the effective stress increment is expressed as 3s de = MVt~ij- M , s _ -,s 7p 2 ae
(A4)
oat~ (A2)
In the linear incremental procedure a first order approximation of ~¢ in terms of g~ is used on the form
,~=s
+ ,uMOflo
where
1
c~ae
E so ___
2 l + v a e"
Following this the effective plastic strain rate is expressed as
with i~(t) expressed by the Taylor expansion
i~(,+a,) = g~
2983
(A3)
-
-
(AS)
~TAtQ
and Q=-~M
s 8i~
8i~ Oi
ao Oao
8s Oive
v-'j-+--.
(A6)
Using the expressions for the effective plastic strain rate given above the constitutive equations take the following form i~ = (R ~*',_ i.tM'J Mkl)Okl
-
/t o 1 --
8AtQ
(A7)