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FE-modelling of the deformation behaviour of WC-Co alloys
COMPUTATIONAL MATERIALS SCIENCE
Computational Materials Science 1 (1993) 213-224 Elsevier
FE-modelling of the deformation behaviour of WC-Co alloys M.H. Poech, H.F. Fischmeister, D. Kaute 1 and R. Spiegler 2 Max-Planck-lnstitut fiir Metallforschung, Institut fiir Werkstoffwissenschaft, Seestrasse 92, D-7000 Stuttgart 1, Germany Received 6 January 1993 Accepted 22 February 1993
A systematic study has been made by Finite Element (FE) modelling on the initial elastic and plastic stress-strain behaviour of a WC-Co alloy in terms of microstructural parameters such as volume fraction, shape, and arrangement of Co areas. Factors which may affect the accuracy of the FE calculations, such as mesh geometry, mesh density and model size, are considered in some detail. The trends shown by the FE simulations are in agreement with published experimental data. Deviations of the FE-calculated flow curves from experiment at large strains are explained in terms of incipiant damage to the carbide skeleton.
I. Introduction
WC-Co hard metals have often been chosen as model materials for experimental and theoretical studies on the microstructure-property relations of composite materials [1-11]. The Finite Element Method (FEM) has been found useful for the type of micro-mechanical modelling required here, and it has been applied repeatedly to WC-Co hard metals [12-17]. This paper describes a systematic FEM study on the initial yield behaviour of a WC-Co alloy, with special attention to methodical aspects. The following points are critical in preparing an adequate FE model on the basis of a random cut-out from the real microstructure of a twophase material: (1) Correct representations are needed of the in situ properties of both phases; in the particular case of WC-Co alloys, the small size of the Co regions raises the strength of the Co phase above that of the bulk state. 1 Now at University of Cambridge, Department of Engineering, Trumpington Street, GB-Cambridge CB2 1PZ, UK. 2 Now at Th. Goldschmidt AG, Goldschmidtstrasse 100, D4300 Essen 1, Germany.
(2) The FE model must adequately represent the features of the microstructural geometry; in the particular case of the WC-Co alloys, this geometry is characterized by mutual penetration of the phases, a feature which is clearly apparent only in three dimensions. (3) The FE model must make optimum use of the limited computing capacity available. As a rule, the requirements for three-dimensional models of actual microstructures are prohibitive; even for two-dimensional models, the mesh density must be carefully adapted to the expected strains in the various phase regions. An approach to item (1) is given in ref. [18], where the in situ flow properties of the Co binder phase are derived from considerations of both the flow and the fracture behaviour of typical WC-Co alloys, and it is shown that the main contributions are solid solution strengthening of the Co by W and C and the limitation of dislocation slip length by the W C / C o boundaries. The latter can be correlated to the mean free path in the binder phase in terms of a Hall-Petch relation. The WC phase is assumed to remain in its elastic regime during deformation of the composite; consequently, it can be described with its elastic constants. Neglecting the anisotropy of WC, its be-
0927-0256/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
214
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
haviour is adequately described by its Young's modulus and its Poisson's ratio. It will be seen that the assumption of purely elastic behaviour of the WC phase becomes doubtful at increasing plastic strain. As long as the carbide phase remains in its elastic regime, the plastic strain partitions entirely to the Co regions. The distribution of local stress and strain in the microstructure is strongly inhomogeneous. The large gradients of strain require a high mesh density to avoid overestimating the stiffness of the material. This effect is practically negligible as long as both phases are in their elastic regime, but it becomes very important as soon as one phase enters the plastic regime. This has consequences both for the accuracy with which the geometry of the microstructure is modelled (item (2)), and for optimizing the use of the available computer capacity (item (3)). Both aspects will be discussed in the light of studies with systematically varied mesh densities, and practical rules will be given for optimizing the mesh density in relation to the contents of the microstructure cut-out being modelled. Since three-dimensional modelling of real microstructures seems to be out of reach of present computing capacities, the present study is limited to two-dimensional models representing a cut-out from a plane section of the hard metal microstructure.
\ / /
/
\
Fig. 1. Periodical type of mesh. For the cut-outs of fig. 3 a total number of 2048 elements has been used. consisted of up to 3000 elements, CPU times were as high as 15 h on a C O M P A R E X 8 / 8 9 and up to 6 h on an IBM 3090. Two different types of mesh were used: (i) one based on a strictly periodical lattice (fig. 1), as used by [20-23], and (ii) a more realistic representation of the microstructure of the type introduced by the authors' group [24] (fig. 2). For both types, comfortable procedures for mesh generation with a digitizing tablet were developed. In the periodical lattice the microstructure can easily be varied without changing the volume fractions, by simply reassigning the material properties (WC or Co) for equal numbers of suitably selected elements. This procedure was used to vary the roundness of Co areas as shown in fig. 3. Its main disadvantages, of course, are the rather
2. Finite element models
All calculations described below were performed with the FE code L A R S T R A N [19] which allows for nonlinearity both in geometrical and material response. Triangular plane strain elements with linear displacement field (TRIP3P) were selected in order to minimize CPU time for a given mesh density. A recent study comparing the performance of different element types has shown that linear elements require much finer meshes with accordingly longer CPU times to correctly reproduce the strain distribution than elements with, for instance, an incomplete biquadratic displacement field (QUAP8P). For the microstructure models considered here, which
Fig. 2. "Realistic" FE mesh, accurate representation of the microstructural geometry.
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
215
unrealistic appearance of the "microstructure" and the fact that mesh densities are equal in both phases. With the second type of mesh the Co areas are covered with smaller elements, allowing improved coverage of the regions undergoing plastic deformation, while the greater size of the elements in the WC areas helps to save CPU time. Plane strain boundary conditions were used throughout to account for the strong constraint arising between the rigid WC and the deforming Co phase. In plane strain loading the response of a structure is stiffer than in uniaxial loading because greater hydrostatic stresses develop; this has to be kept in mind when the numerical results are compared with experiments which are
A
3 Fig. 4. Variation in arrangement (1, 2, 3). The smaller cut-outs indicated in 1 were used for the calculations with increased mesh densities (cf. fig. 6).
C Fig. 3. Variation in shape (A, B, C).
usually done in uniaxial stress. Periodic boundary conditions with mirror symmetry at the edges are imposed to keep the edges straight and parallel during loading. The microstructure cut-outs which can be modelled with reasonable computer capacity today are uncomfortably small. Fig. 4 shows three cut-outs selected from micrographs of an actual alloy. By comparing the properties predicted for these three cut-outs, it will be possible to estimate the number of such cut-outs that must be modelled and averaged to obtain meaningful resuits. To obtain as much information as possible from the FE models, each of the cut-outs shown
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
216
Table 1 Mechanical properties used for F E simulation
WC Co
E [GPa]
v[ ]
% [MPa]
o"s [MPa]
Ec [ ]
714 211
0.194 0.310
540
1540
0.0948
in figs. 3 and 4 was loaded both in the vertical and in the horizontal direction, and mean values of Young's modulus E, Poisson's ratio v, and the yield stress at 0.02% and 0.1% plastic strain were evaluated from both the "vertical" and the "horizontal" stress-strain curves of each cut-out. Because of the small size of the cut-outs, loading in different directions produced slightly different stress-strain curves. This apparent anisotropy will be characterized by the ratio of the respective yield stresses. The stress-strain behaviour in compression of a comparable hard metal has been measured by [4]. We represent the mechanical properties of the Co phase by a Voce-type formula: o- = o-y + (Os- o y ) ( 1 - exp{-epl//ec}) with the parameters given in table 1, where the effects of solid solution and Hall-Petch strengthening on the in situ properties of the Co phase are already included, as suggested in [18]. Table 1 also states the elastic constants used for the WC phase.
Table 2 Microstructural parameters of the FE models and the W C - C o alloy FEModel A B C 1 2 3
V v c o [ ] Lco[p.m] 0.195
0.165
Alloy 10C
0.165
ffco[ ] /2co[ l M c o [ ]
0.704 0.802 1.23 0.665 0.819 0.914
0.444 0.496 0.821 0.408 0.358 0.495
1.01 1.12 1.14 0.93 0.89 1.29
0.325 0.313 0.119 0.331 0.392 0.346
0.84
0.49
1
0.31
The shape factor F is defined by 4xrA F = - p-2
'
where A designates the area and P the perimeter of each Co region. In determining F for the FE models, the "ghost particles" created by the mirror symmetry at the boundaries were taken into account. The anisotropy 12 is equal to the ratio of phase boundary intersections on horizontal and vertical test lines of equal length [26]. The matricity parameter M [27] characterizes the degree to which a given phase acts as the matrix in a microstructure. The geometry of the phase regions is explored by an image operation called skeletonization, which traces the midway paths
3. Quantitative characterization of microstructures The cut-outs to be modelled were taken randomly from scanning electron micrographs of an alloy with 10 wt.% of Co and coarse WC grains with well-developed angularity. The microstructure of this alloy had been carefully characterized by quantitative metaUography in a study of crack propagation in WC-Co alloys (alloy 10C in ref. [25]). The microstructural parameters determined were the Co volume fraction Vv, the mean free path L, the mean shape factor F, the anisotropy 12 and the matricity M of the Co phase as given in table 2.
Fig. 5. Microstructural cut-out (1) of the W C - C o alloy with skeleton lines of W C and Co.
217
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
between the boundaries of a phase region, as illustrated in fig. 5. For a given phase a, the matricity M is given by the total length of the skeleton lines of that phase (s~), relative to the length of the skeleton lines of both phases (s~ + St3):
M.
Sa Sa + S~
Thus M can also be considered as a measure of the total length of continuous paths through the phase: it has its maximum value ( M , = 1) when the phase forms a fully continuous matrix surrounding isolated spherical inclusions of the other phase (s~ = 0). It is small (ideally zero) when the phase is present in the form of compact inclusions; it increases when the inclusions become elongated, and it increases further when they form extended branches. This is an aspect of phase shape which may be expected to be highly relevant to the mechanical behaviour of the Co phase in the composite. We note that M is defined as a characteristic of the two-dimensional section through the microstructure seen in the microscope; in relation to the real microstructure, it lacks of information pertaining to the third dimension.
4. Results and discussion
The methodological object of the study was to assess in a quantitative manner the effects of mesh density and mesh type (figs. 1, 2), of model size, of local differences of the phase arrangement in different cut-outs randomly taken from a given microstructure, and of loading a given cutout in different directions. After studying these methodological points, attention was turned to the influence of the microstructural parameters, notably volume fraction, shape and arrangement of the phase regions. As continuum mechanics is not sensitive to absolute size, the effect of size can only be accounted for by way of the H a l l Petch strengthening of the Co phase, as discussed above; therefore, size-effects are not considered in this work.
4.1. M e s h density
The FE simulation of a given cut-out with different mesh densities revealed a strong effect of element density on yield stress. The elastic response, on the other hand, is hardly affected. A meaningful measure of mesh density D in models adapted to the microstructure of W C - C o alloys must obviously be related to the average size of the Co phase regions, and it should be normalized with respect to the Co volume fraction. In addition, the total number of elements must be increased when the number of Co regions in the cut-out increases. A suitable parameter for mesh density can be defined as follows:
N oE o O
~
-
fcoL2 " Here NCo, L,co and fco denote the number of elements in the plastically deforming Co areas, the mean free path and the volume fraction of Co, respectively. L 2 is the total area covered by the model. For an increase of mesh density D from 15 to 60, the calculated 0.1% yield strength of the same microstructural cut-out was found to decrease by 13.5%, while Young's modulus was changed only by 0.75%. Refinement of the mesh soon makes the number of elements in a given cut-out too large. Therefore, a second, smaller cut-out, and successively a third one, each with one-half the edge length of the preceding one were modelled (fig. 4, indicated in model 1), 4000,
o..
~p~ =0.1% 3000
~
2500
. . . . . .
ucr--a
oO
•u
2000
o
= 0.02 %
1500 ~ _ ~ 1000
0
0........ O. . . . . . . . . . . . . . . . . . .
100
.
.
.
200 300 400 500 FE mesh density D Fig. 6. Influence of FE mesh density on calculated yield strength values, closed symbols: FE results, open symbols: correction for mesh density included (see text).
M.H. Poech et aL / FE-modelling of the deformation behaviour of WC-Co alloys
218
Table 3 Coefficients to estimate the error of FE-calculated yield strength with element density (element type TRIP3P)
R p 0.02% R p o.1%
a
b
0.8222 2.0115
0.6073 0.7423
using three different mesh densities for each. The results are shown in fig. 6. The three cut-outs (model 1) have slightly different volume fractions and, of course, distinctly different phase arrangements, which result in different absolute magnitudes of the calculated yield strengths. Despite these differences, fig. 6 clearly shows that the calculated yield strengths converge to a constant value at very high mesh densities, while at low mesh densities the results are systematically high. This overestimation of stiffness is expected because the inhomogeneous distribution of plastic strains cannot be correctly reproduced by large elements. Fitting a simple formula to the curvature of the line segments in fig. 6, the relative error due to insufficient element density can be expressed as follows: °'calc
1 + aD -°.
O'sat
Values of the constants a and b for the type of mesh employed here are given in table 3. As a sensible compromise with respect to computation time, a mesh density corresponding to D = 100 was chosen for all further work, accepting an overestimation of yield strength by 5% at 0.02% plastic strain, and of 6.6% at 0.1% strain. Mesh densities above D = 500 would be required to bring the error below 2%. Since CPU time is approximately proportional to the square of the number of nodes, the computation time would increase by a factor of 25, which does not appear justified for the small gain in accuracy. 4.2. Loading in different directions In reality, there is no preferred direction in the microstructure of a sintered WC-Co alloy, and consequently, there should be no anisotropy in
the loading response of the models. However, the FE models are limited to small cut-outs (about 15 WC grains). A "vertical" and a "horizontal" load will encounter different phase arrangements. While these differences should average out in sufficiently large regions of an inherently isotropic structure, small cut-outs must be expected to present different stress-strain curves in different loading directions, and the magnitude of the difference will tell us something about how far we are from a sufficient model size. Figure 7 shows the ratio of calculated yield stresses for the cut-outs of figs. 3 and 4 in two perpendicular loading directions. The periodic lattice type of mesh seems to be less sensitive to the loading direction, which may be due to the simplification which it imposes on the microstructure. For the "realistic" mesh, the differences are uncomfortably large, indicating that calculated properties should be averaged over a large number of cut-outs if quantitative reliability is to be achieved. In fig. 7 the yield stress anisotropy is plotted against the geometrical anisotropy of the cut-outs, which is characterized by the ratio of phase boundaries intersected by horizontal and vertical test lines, 12. Although the two quantities correlate roughly, the geometrical anisotropy characterized by 12 is obviously not the proper key to the mechanical anisotropy. This is not really surprising because 12 in fact characterizes the anisotropy of phase boundary surface [26], or,
1.3 ~ 1.2
,
~._o 1.1
d
0.9 0.8 0.8
. 0.9
.
1
.
.
1.1
1.2
1.3
Anisotropy
Fig. 7. Influence of geometric anisotropy on yield strength
ratio.
219
M.H. Poech et al. / FE-modelling of the deformation behauiour of WC-Co alloys
equivalently, the ratio of the mean free Co paths in the two directions, neither of which can be expected to be the sole determinant of composite yield stress. More important is probably the phase arrangement. For instance, a soft response will be expected when a cut-out presents continuous bands of soft phase at 45 ° to the loading direction. In unfavourable cases, the mirror symmetry at the cell boundaries can create "ghost particles" of large size and exotic shape (e.g. dog bones) which block the deformation in the soft bands of the model. It has been demonstrated [28-30] that such a blockage makes the loading response stiffer. In addition to the apparent anisotropy of the small cut-outs, there is, of course, a real anisotropy in any two-dimensional F E model, since the model actually corresponds to a bundle of prismatic lengths of infinite length in the direction perpendicular to the plane of the microstructure. This is a serious draw-back of two-dimensional models, which presently cannot be overcome without prohibitive effort. Rather surprisingly, it has been found that two-dimensional models reproduce certain essential features of the mechanical behaviour of W C - C o alloys quite well, one example being mechanical hysteresis [13]. However, one would be well advised to exercise great caution when applying two-dimensional FE models to materials whose microstructures are modulated in three dimensions. Our present appreciation is that they serve well for exploring trends but should not be expected to reproduce mechanical properties in a quantitative manner. Fortunately, there are some problems where two-dimensional models are fairly appropriate to the physical situation, as in the case of crack propagation across Co regions between carbide grains discussed in ref. [17]. Exact comparisons between two- and three-dimensional calculations may be made for microstructural features of simple geometry, and it may be possible to develop a feeling for the usefulness of two-dimensional calculations from such comparisons. A valid comparison between F E calculations and experiments can be made in the case of transverse loading of a fiber composite, which is described in another paper from our group [29].
4.3. Volume fraction
The volume fractions of the soft and hard phases have a strong influence on the yield strength of composites. This influence can be described roughly by the well-known rules of mixture; for small variations of the volume fraction, it can be taken to be linear. The influence of volume fraction was studied by slightly shifting some nodes in the FE mesh, keeping the adjustments small enough to leave the microstructural geometry essentially unaffected. One percent reduction of the Co volume fraction was found to increase the yield strength of the composite (at 0.1% strain) by about 8%. This strong effect can be understood, recalling that the WC phase, which has the dominating volume fraction, undergoes no plastic deformation in the model. A semi-analytical calculation based on the "strain compatible series model" developed by Poech and Fischmeister (SCS) [18,31] makes predictions which are virtually identical with the FE results, as shown in fig. 8. 4. 4. Shape of phase regions
Using an FE mesh of the periodic lattice type (fig. 1), the shape of the phase regions in a cut-out is easily altered by switching elements
Calculated Yield Strength of WC-Co 5000
",
[
i
", a_
4000
- .
~_ 3 0 0 0
.
.
.
.
. * •
.
SCS SCS FEM FEM
0.02% 0.1% 0.02% 0.1%
"0
c
2000
n -o
1000
O 0
10
20
30
40
v o l u m e f r a c t i o n C o [%]
Fig. 8. Influence of Co volume fraction on yield strength, comparison of the FE results with a semi-analyticalapproach (scs).
220
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys 4000
4500 'E Q..
~" 3000
4000 JE:
O3 E
2000
~o 3500 1o
,= 1000
[ - ~ i
0
o~ 3000 c5 2500 0.3
i
0.05 0.1 Plastic Strain [%]
0.15
Fig. 9. Influence of Co shape on the plastic deformation behaviour.
from one phase to the other, as exemplified in fig. 4. Sharp corners or edges in the microstructure will cause stress and strain concentrations which in turn cause yielding at lower stresses than in a structure with more rounded phase regions. This was demonstrated already in one of the earliest papers on FE modelling of two-phase microstructures [32], and similar results are obtained for the progressive rounding of the microstructures considered in this study, as seen in fig. 9. As expected, the softer response of structures with more angular phase regions continues during further deformation. Defining parameters which characterize the mean shape of phase regions in microstructures is difficult. The shape factor F defined in section 3 is popular for characterizing the shape of isolated phase regions. Despite its sensitivity to the size distribution of the phase regions [33], it was used in this study as a first approach, for lack of a better parameter, to characterize the shape of the isolated Co regions which appear in plane sections of the W C - C o alloy, cf. table 2. Figure 10 shows the expected trend for the series of progressively rounded microstructures modelled with periodic lattice nets (A, B, C in fig. 3). The data points for the three cut-outs shown in fig. 4, which were modelled with meshes that allowed a realistic rendering of the phase geometry, conform to this trend, but only very roughly. We must conclude that the mean shape factor ff is not a good descriptor of phase shape in those
~de
I
I
I [] period~,almesh I
i
i
i
~
=
0.4
0.5
0.6
0.7
0.8
0.9
Mean Shape Factor
Fig. 10. Influence of the mean shape factor of Co regions on the yield strength. aspects which are relevant to mechanical behaviour. 4.5. Cut-outs with different phase arrangement
Different cut-outs randomly taken from a given microstructure, such as items 1, 2 and 3 in fig. 4, will present different phase arrangements, and they will in general have slightly different volume fractions. This is true also for the cut-outs considered here. To isolate the effect of phase arrangement from that of volume fraction, the FE models were brought to a uniform value of Co volume fraction (cf. table 2) by slightly shifting appropriate nodes of the net in a way which essentially preserved the original geometry of the microstructure. Figure 11 shows the effect of the variations of phase arrangement on the mechani4000 3000
E ¢o 2000 (,9
1000 L
0
0.05
.
i
0.1 0.15 Plastic Strain [o%]
0.2
Fig. 11. The influence of phase arrangement on the plastic deformation behaviour.
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
cal behaviour. It is small enough to allow representative results to be obtained by averaging over a limited number of such cut-outs.
4500
J ¢
CK
2Sv~K 2SvK ~ + Sv,~ '
where Sw~ is the surface per unit volume of boundaries between phases ~ and K. In Gurland's contiguity parameter, the boundaries between adjacent grains of the same phase play a large role, and for the WC phase in W C - C o alloys, this is in fact the dominant contribution. Since our models pertain to early stages of deformation, during which the carbide grain boundaries are assumed to remain intact, we prefer a parameter which characterizes the carbide skeleton without reference to individual carbide grains. This can be done with the recently developed "matricity parameter" [27] defined in section 3. In W C - C o microstructures, the case where the carbide forms a fully continuous matrix (Mwc = 1), while the cobalt is present in the form of isolated inclusions ( M c o = 0), will represent the upper bound for yield strength; the reverse case (Mwc = 0, M c o = 1) will represent the lower bound. A semi-analytical model calculation recently presented [35] predicts these bounds to be as far apart as 6069 and 1144 MPa (at 0.1% plastic strain, using the mechanical property data given in table 1 and the volume fraction of the microstructures considered here, 16.5 vol.%). This emphasizes the importance of a meaningful quantitative characterization of the phase arrangement in any attempt to predict mechanical properties of W C - C o alloys.
"realistic"model periodicalmesh
4000
4.6. Quantitative characterization of the phase arrangement: the matricity parameter In looking for a way to rationalize the effects illustrated by our sample calculations with different phase arrangements (fig. 11), we assume that they are caused by variations in the degree of which stiff phase regions are linked to form a stiff skeleton. Gurland [7] has introduced the concept of "contiguous volume" for this purpose, which is based on his definition [34] of the contiguity C of regions of one phase:
221
3500
3000
d 2500
. 0.1
0
.
. 0.2
.
0.3
0.4
0.5
Matricity Fig. 12. Influence of Co matricity on yield strength.
The matricity value of the carbide phase in each of the cut-outs of figs. 3 and 4 has been determined (cf. table 2). A plot of yield strength (at 0.1% strain) against matricity, cf. fig. 12, shows that this parameter seems to be a good descriptor for the influence of phase arrangement on yield strength at constant volume fraction. Further support for the relevance of the matricity parameter to mechanical behaviour is found when considering the distribution of local strains in the various cut-outs. Figure 13 shows the cumulative frequency distribution of strain levels in all elements belonging to the Co phase in the three cut-outs of fig. 3. These distributions reflect the range of strain hardening of the Co phase when the composite is loaded to a given level, equal for all three cut-outs. Characterizing the distributions by their standard deviations, a corre-
100 o>,
8O
o-
60
c
u_
4o E
20
0 0
0
0.5
1
1.5 ~p~[%]
2
2.5
3
Fig. 13. Cumulative frequency distribution of plastic strains in cut-outs A, B, and C.
M.H. Poech et aL / FE-modelling of the deformation behaviour of WC-Co alloys
222
0.8
j
8 •~, .£ "5 a~
0.6
~._~
o£3 t~'
0.4
n,0 0
I
i
i
0.1
0.2
0.3
0.4
Matricity
Fig. 14. Width of the plastic strain distribution (standard deviation of plastic strains normalized to the mean value) with Co matricity.
lation can be made with the microstructural geometry, which again is characterized by the matricity parameter. This is shown in fig. 14. As expected, the more branched-out the Co phase appears, the more inhomogeneous is its deformation. Extrapolating the correlation back to M = 0 predicts a state of fully homogeneous plastic strain, in agreement with Eshelby's solution for spherical inclusions [36].
4.7. Comparison with experimental stress-strain curves
The stress-strain behaviour in compression of an alloy corresponding to our models has been
4000
4,~/~,~4,~. ~
WC-16.5 vol% Co
~ ,,4"//~ P"
3000 to
I
studied by Doi [4]. Figure 15 summarizes the stress-strain curves of all the FE simulations discussed above and shows the comparison with Doi's experiment, whose data have been converted to the plane strain loading condition as described in the appendix. The comparison shows that the FE models predict a stress-strain behaviour for the 10C alloy which is not too far from reality, despite the fact that the models were two-dimensional. In the representation shown here (which is in terms of total rather than plastic strain), the spread resulting from the shape alterations shown in fig. 3, from the choice of different cut-outs (fig. 4) and from loading in two perpendicular directions is not too wide. Most of the models are initially somewhat too soft in comparison to the experiment, but the discrepancy is not drastic. The main shortcoming of the calculations is that they do not reproduce the decrease of the strain hardening rate after initial plastification. A possible interpretation of this discrepancy, which, in fact, we find quite likely, is based on the observations of [2,37,38] which indicate that the carbide skeleton looses its integrity by cracking (very likely at the grain boundaries) at relatively small composite strains. In addition, ample evidence has been brought forth for plastic processes in the carbide phase [38-40]. Both points would bring the stiffness of the carbide skeleton appreciably below the purely elastic behaviour which has been assumed in setting up the FE models. Finally, the residual thermal stresses which are present in WC-Co alloys [41] contribute to the early stages of plastic deformation [42]. Their effects, too, have been excluded from the present calculations. In this light, we feel that the agreement between the FE models and the experiments may be considered quite satisfactory.
20'00 FEMGo-shape [ FEMCo-arrangement Experiment(Doiet al.)
I
1000 0
0
I
i
i
i
0.2
0.4
0.6
0.8
Strain
[%]
1
Fig. 15. Summary of the FE simulations in comparison with experiment.
5. Concluding remarks
FE modelling has limitations which result from mesh density, boundary effects, and sampling errors due to the small size of the microstructure cut-outs. These limitations have been illustrated; some of them are tolerable, and the rest can be overcome by proper computing procedures and
M.H. Poechet al. / FE-modellingof the deformationbehaviourof WC-Co alloys suitable design of the models. In particular, as has been demonstrated in other work [17,43], the effects of the displacement conditions at the boundaries of the model can be reduced by surrounding the two-phase microstructure with an FE continuum which is given the overall properties of the composite. In the foreseeable future, FE models of real microstructures will have to remain two-dimensional, and this might be feared to be a real shortcoming. As pointed out in the paper, caution suggests that two-dimensional models should be used mainly to develop an understanding of trends, and to derive pointers to important features of mechanical behaviour, rather than for accurate predictions on an absolute scale. However, experience shows that in many cases, twodimensional FE models of real microstructures come surprisingly close to the behaviour of actual materials [6,12,13,24,32]. The results shown in fig. 15 of the present paper may be added to this list. Encouragement is also taken from their extremely good agreement with the results of semianalytical models recently presented [31,35] which are based on simplified, but three-dimensional geometries. Experience also indicates that two-dimensional models serve well whenever the properties of the phases making up the microstructure, and consequently the deformations undergone by the phases, are not too dissimilar [35]. The special promise of FE models is that they ought to be able to clarify the effects of phase arrangement or microstructural geometry, which so far have largely escaped modelling attempts. A large part of this problem lies in developing good quantitative descriptors for the microstructures. The present study shows that conventional descriptors, such as the area:perimeter ratio used in the traditional shape factor, may have little relevance to mechanical behaviour. An approach which may improve this situation has been demonstrated in the section on matricity. A second part of the problem lies in the loss of information incurred whenever a microstructure is reduced to a few characteristic mean numbers such as volume fraction, size, and a mean shape parameter. However, there is hope that nature works to our advantage in that it allows only limited
223
variability for microstructures generated for given classes of materials with given processing routes, as, for instance, for W C - C o alloys produced by liquid phase sintering of powders. The obvious "family likeness" of large groups of microstructures holds out the hope that microstructural modelling of the properties of two-phase materials will allow fairly accurate property predictions.
Appendix: unidirectional loading in plane strain The calculation procedure to convert unidirectional stress-strain data to plane strain given an isotropic material has already been presented by ref. [13]. A re-evaluation has revealed some small errors the correction of which will be given below. The state of unidirectional loading in plane strain can be characterized by e Z = 0 and ~ry = 0. With the assumption of proportional loading plasticity theory gives:
Eepl 1,'+-k
O"z
2O'erf
crx
l + - -EEpl o'e~f
With the unidirectional stress-plastic strain data (treff - %0 and the elastic constants (E, u) we get for the remaining strain components e x and %:
( 1-uk ex=O" x
, ~
epl
)
+ 2°-elf ( 2 - k )
( u ( l + k )E Ey=°'x
,
%~ ) 2~reff(l+k) -
Using the definition of effective stress (creff) by Von Mises the stress in loading direction o-~ evaluates to O'¢ff
~rx- v / l _ k + k 2 The global stress and strain tensors are defined by these equations so that the uniaxial stressstrain curves can simply be converted to the plane strain loading conditions.
224
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
References [1] J. Gurland and J.T. Norton, Trans. AIME (1952) 1051. [2] C. Nishimatsu and J. Gurland, Trans. ASM 52 (1960) 469. [3] H. Fischmeister and H.E. Exner, Arch. Eisenhiittenwes. 37 (1966) 499. [4] H. Doi, Y. Fujiwara, K. Miyake and Y. Oosawa, Met. Trans. 1 (1970) 1417. [5] H.E. Exner and J. Gurland, Powder Metallurgy 13 (1970) 13. [6] H.F. Fischmeister and B. Karlsson, Z. Metallkde. 68 (1977) 311. [7] H.C. Lee and J. Gurland, Mat. Sci. Eng. 33 (1978) 125. [8] J.R. Pickens, J. Gurland, Mat. Sci. Eng. 33 (1978) 135. [9] H.F. Fischmeister, in: Science of Hard Materials, eds. R.K. Viswanadham, D. Rowcliff and J. Gurland (Plenum Press, New York, 1983)) pp. 1-45. [10] L.S. Sigl and H.F. Fischmeister, Acta Metall. 36 (1988) 887. [11] D.B. Marshall, W.L. Morris, B.N. Cox and M.S. Dadkhah, J. Amer. Ceram. Soc. 73 (1990) 2938. [12] B.O. Jaensson and B.O. Sundstr6m, Mat. Sci. Eng. 9 (1972) 217. [13] B.O. Sundstr6m, Mat. Sci. Eng. 12 (1973) 265. [14] A.B. Ljungberg, C. Chatfield, M. Hehenberger, and B. Sundstr6m, 2. Int. Conf. on the Science of Hard Materials, eds. E.A. Almond, C.A. Brookes and R. Warren (Adam Hilger, Bristol, 1986) pp. 619-630. [15] L.S. Sigl and S. Schmauder, Int. J. Fract. 36 (1988) 305. [16] R. Spiegler, S. Schmauder and H.F. Fischmeister, in: Finite Elements in Engineering Applications (INTES, Stuttgart, 1990) pp. 21-42. [17] R.S. Spiegler and H.F. Fischmeister, Acta Metall. Mater. 40 (1992) 1653. [18] M.H. Poech, H. Fischmeister and R. Spiegler, J. Hard Materials 2 (1991) 197. [19] LARSTRAN, LASSO Engineering, Stuttgart, Germany (1992). [20] S. Ankem and H. Margolin, Met. Trans. 13A (1982) 595.
[21] [22] [23] [24]
S. Ankem and H. Margolin, Met. Trans. 13A (1982) 603. A. Saigal and J.T. Berry: AFS Trans. 101 (1985) 699. S. Ankem and H. Margolin, J. Metals (1986) 25. H.F. Fischmeister, J.-O. Hj~ilmered, B. Karlsson, G. Linden and B. Sundstr6m, Proc. 3rd Int. Conf. on the Strength of Metals and Alloys, Cambridge, vol. 1 (Institute of metals, London, 1973) pp. 621-625. [25] L.S. Sigl and H.E. Exner, Met. Trans. 18A (1987) 12991308. [26] E.E. Underwood, in: Quantitative Stereology (AddisonWesley, Reading, 1969). [27] M.H. Poech and D. Ruhr, Sonderb~inde der Prakt. Metallographie (1992), to be published. [28] J.R. Brockenbrough, S. Suresh and H.A. Wienecke, Acta Metall. Mater. 39 (1991) 735. [29] C. Dietrich, M.H. Poech, H.F. Fischmeister, and S. Schmauder, Comput. Mater. Sci. 1 (1993) 195. [30] M. Sautter, C. Dietrich, M.H. Poech, S. Schmauder and H.F. Fischmeister, Comput. Mater. Sci. 1 (1993) 225. [31] M.H. Poech and H.F. Fischmeister, Acta metall, mater. 40, No. 3 (1992) 487-494. [32] B. Karlsson and B.O. Sundstr6m, Mat. Sci. Eng. 16 (1974) 558. [33] H.F. Fischmeister, Z. Metallkde. 65 (1974) 558. [34] J. Gurland, Trans. Met. Soc. AIME 236 (1966) 642. [35] M.H. Poech, Scripta Metall. Mater. 27 (1992) 1027. [36] J.D. Eshelby, Proc. Roy. Soc. A241 (1957) 376. [37] B. Roebuck and E.A. Almond, Conf. on Recent Advances in Hard Metals Production, Loughborough Univ., Paper 28 (1979). [38] J. Kolaska, W. Schlump and H. Grewe, Z. Metallkde. 75 (1984) 535. [39] R. Arndt, Z. Metallkdel 63 (1972) 274. [40] B, Roebuck and A.E. Almond, Int. Met. Rev. 33 (1988) 90. [41] R. Spiegler, S. Schmauder and H.E. Exner, J. Hard Mater. 3 (1992) 143. [42] D.B. Zahl, R.M. McMeeking: Acta Metall. Mater. 39 (1991) 1117. [43] C. Dietrich, M.H. Poech and H.F. Fischmeister, to be published.
Computational Materials Science 1 (1993) 213-224 Elsevier
FE-modelling of the deformation behaviour of WC-Co alloys M.H. Poech, H.F. Fischmeister, D. Kaute 1 and R. Spiegler 2 Max-Planck-lnstitut fiir Metallforschung, Institut fiir Werkstoffwissenschaft, Seestrasse 92, D-7000 Stuttgart 1, Germany Received 6 January 1993 Accepted 22 February 1993
A systematic study has been made by Finite Element (FE) modelling on the initial elastic and plastic stress-strain behaviour of a WC-Co alloy in terms of microstructural parameters such as volume fraction, shape, and arrangement of Co areas. Factors which may affect the accuracy of the FE calculations, such as mesh geometry, mesh density and model size, are considered in some detail. The trends shown by the FE simulations are in agreement with published experimental data. Deviations of the FE-calculated flow curves from experiment at large strains are explained in terms of incipiant damage to the carbide skeleton.
I. Introduction
WC-Co hard metals have often been chosen as model materials for experimental and theoretical studies on the microstructure-property relations of composite materials [1-11]. The Finite Element Method (FEM) has been found useful for the type of micro-mechanical modelling required here, and it has been applied repeatedly to WC-Co hard metals [12-17]. This paper describes a systematic FEM study on the initial yield behaviour of a WC-Co alloy, with special attention to methodical aspects. The following points are critical in preparing an adequate FE model on the basis of a random cut-out from the real microstructure of a twophase material: (1) Correct representations are needed of the in situ properties of both phases; in the particular case of WC-Co alloys, the small size of the Co regions raises the strength of the Co phase above that of the bulk state. 1 Now at University of Cambridge, Department of Engineering, Trumpington Street, GB-Cambridge CB2 1PZ, UK. 2 Now at Th. Goldschmidt AG, Goldschmidtstrasse 100, D4300 Essen 1, Germany.
(2) The FE model must adequately represent the features of the microstructural geometry; in the particular case of the WC-Co alloys, this geometry is characterized by mutual penetration of the phases, a feature which is clearly apparent only in three dimensions. (3) The FE model must make optimum use of the limited computing capacity available. As a rule, the requirements for three-dimensional models of actual microstructures are prohibitive; even for two-dimensional models, the mesh density must be carefully adapted to the expected strains in the various phase regions. An approach to item (1) is given in ref. [18], where the in situ flow properties of the Co binder phase are derived from considerations of both the flow and the fracture behaviour of typical WC-Co alloys, and it is shown that the main contributions are solid solution strengthening of the Co by W and C and the limitation of dislocation slip length by the W C / C o boundaries. The latter can be correlated to the mean free path in the binder phase in terms of a Hall-Petch relation. The WC phase is assumed to remain in its elastic regime during deformation of the composite; consequently, it can be described with its elastic constants. Neglecting the anisotropy of WC, its be-
0927-0256/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
214
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
haviour is adequately described by its Young's modulus and its Poisson's ratio. It will be seen that the assumption of purely elastic behaviour of the WC phase becomes doubtful at increasing plastic strain. As long as the carbide phase remains in its elastic regime, the plastic strain partitions entirely to the Co regions. The distribution of local stress and strain in the microstructure is strongly inhomogeneous. The large gradients of strain require a high mesh density to avoid overestimating the stiffness of the material. This effect is practically negligible as long as both phases are in their elastic regime, but it becomes very important as soon as one phase enters the plastic regime. This has consequences both for the accuracy with which the geometry of the microstructure is modelled (item (2)), and for optimizing the use of the available computer capacity (item (3)). Both aspects will be discussed in the light of studies with systematically varied mesh densities, and practical rules will be given for optimizing the mesh density in relation to the contents of the microstructure cut-out being modelled. Since three-dimensional modelling of real microstructures seems to be out of reach of present computing capacities, the present study is limited to two-dimensional models representing a cut-out from a plane section of the hard metal microstructure.
\ / /
/
\
Fig. 1. Periodical type of mesh. For the cut-outs of fig. 3 a total number of 2048 elements has been used. consisted of up to 3000 elements, CPU times were as high as 15 h on a C O M P A R E X 8 / 8 9 and up to 6 h on an IBM 3090. Two different types of mesh were used: (i) one based on a strictly periodical lattice (fig. 1), as used by [20-23], and (ii) a more realistic representation of the microstructure of the type introduced by the authors' group [24] (fig. 2). For both types, comfortable procedures for mesh generation with a digitizing tablet were developed. In the periodical lattice the microstructure can easily be varied without changing the volume fractions, by simply reassigning the material properties (WC or Co) for equal numbers of suitably selected elements. This procedure was used to vary the roundness of Co areas as shown in fig. 3. Its main disadvantages, of course, are the rather
2. Finite element models
All calculations described below were performed with the FE code L A R S T R A N [19] which allows for nonlinearity both in geometrical and material response. Triangular plane strain elements with linear displacement field (TRIP3P) were selected in order to minimize CPU time for a given mesh density. A recent study comparing the performance of different element types has shown that linear elements require much finer meshes with accordingly longer CPU times to correctly reproduce the strain distribution than elements with, for instance, an incomplete biquadratic displacement field (QUAP8P). For the microstructure models considered here, which
Fig. 2. "Realistic" FE mesh, accurate representation of the microstructural geometry.
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
215
unrealistic appearance of the "microstructure" and the fact that mesh densities are equal in both phases. With the second type of mesh the Co areas are covered with smaller elements, allowing improved coverage of the regions undergoing plastic deformation, while the greater size of the elements in the WC areas helps to save CPU time. Plane strain boundary conditions were used throughout to account for the strong constraint arising between the rigid WC and the deforming Co phase. In plane strain loading the response of a structure is stiffer than in uniaxial loading because greater hydrostatic stresses develop; this has to be kept in mind when the numerical results are compared with experiments which are
A
3 Fig. 4. Variation in arrangement (1, 2, 3). The smaller cut-outs indicated in 1 were used for the calculations with increased mesh densities (cf. fig. 6).
C Fig. 3. Variation in shape (A, B, C).
usually done in uniaxial stress. Periodic boundary conditions with mirror symmetry at the edges are imposed to keep the edges straight and parallel during loading. The microstructure cut-outs which can be modelled with reasonable computer capacity today are uncomfortably small. Fig. 4 shows three cut-outs selected from micrographs of an actual alloy. By comparing the properties predicted for these three cut-outs, it will be possible to estimate the number of such cut-outs that must be modelled and averaged to obtain meaningful resuits. To obtain as much information as possible from the FE models, each of the cut-outs shown
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
216
Table 1 Mechanical properties used for F E simulation
WC Co
E [GPa]
v[ ]
% [MPa]
o"s [MPa]
Ec [ ]
714 211
0.194 0.310
540
1540
0.0948
in figs. 3 and 4 was loaded both in the vertical and in the horizontal direction, and mean values of Young's modulus E, Poisson's ratio v, and the yield stress at 0.02% and 0.1% plastic strain were evaluated from both the "vertical" and the "horizontal" stress-strain curves of each cut-out. Because of the small size of the cut-outs, loading in different directions produced slightly different stress-strain curves. This apparent anisotropy will be characterized by the ratio of the respective yield stresses. The stress-strain behaviour in compression of a comparable hard metal has been measured by [4]. We represent the mechanical properties of the Co phase by a Voce-type formula: o- = o-y + (Os- o y ) ( 1 - exp{-epl//ec}) with the parameters given in table 1, where the effects of solid solution and Hall-Petch strengthening on the in situ properties of the Co phase are already included, as suggested in [18]. Table 1 also states the elastic constants used for the WC phase.
Table 2 Microstructural parameters of the FE models and the W C - C o alloy FEModel A B C 1 2 3
V v c o [ ] Lco[p.m] 0.195
0.165
Alloy 10C
0.165
ffco[ ] /2co[ l M c o [ ]
0.704 0.802 1.23 0.665 0.819 0.914
0.444 0.496 0.821 0.408 0.358 0.495
1.01 1.12 1.14 0.93 0.89 1.29
0.325 0.313 0.119 0.331 0.392 0.346
0.84
0.49
1
0.31
The shape factor F is defined by 4xrA F = - p-2
'
where A designates the area and P the perimeter of each Co region. In determining F for the FE models, the "ghost particles" created by the mirror symmetry at the boundaries were taken into account. The anisotropy 12 is equal to the ratio of phase boundary intersections on horizontal and vertical test lines of equal length [26]. The matricity parameter M [27] characterizes the degree to which a given phase acts as the matrix in a microstructure. The geometry of the phase regions is explored by an image operation called skeletonization, which traces the midway paths
3. Quantitative characterization of microstructures The cut-outs to be modelled were taken randomly from scanning electron micrographs of an alloy with 10 wt.% of Co and coarse WC grains with well-developed angularity. The microstructure of this alloy had been carefully characterized by quantitative metaUography in a study of crack propagation in WC-Co alloys (alloy 10C in ref. [25]). The microstructural parameters determined were the Co volume fraction Vv, the mean free path L, the mean shape factor F, the anisotropy 12 and the matricity M of the Co phase as given in table 2.
Fig. 5. Microstructural cut-out (1) of the W C - C o alloy with skeleton lines of W C and Co.
217
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
between the boundaries of a phase region, as illustrated in fig. 5. For a given phase a, the matricity M is given by the total length of the skeleton lines of that phase (s~), relative to the length of the skeleton lines of both phases (s~ + St3):
M.
Sa Sa + S~
Thus M can also be considered as a measure of the total length of continuous paths through the phase: it has its maximum value ( M , = 1) when the phase forms a fully continuous matrix surrounding isolated spherical inclusions of the other phase (s~ = 0). It is small (ideally zero) when the phase is present in the form of compact inclusions; it increases when the inclusions become elongated, and it increases further when they form extended branches. This is an aspect of phase shape which may be expected to be highly relevant to the mechanical behaviour of the Co phase in the composite. We note that M is defined as a characteristic of the two-dimensional section through the microstructure seen in the microscope; in relation to the real microstructure, it lacks of information pertaining to the third dimension.
4. Results and discussion
The methodological object of the study was to assess in a quantitative manner the effects of mesh density and mesh type (figs. 1, 2), of model size, of local differences of the phase arrangement in different cut-outs randomly taken from a given microstructure, and of loading a given cutout in different directions. After studying these methodological points, attention was turned to the influence of the microstructural parameters, notably volume fraction, shape and arrangement of the phase regions. As continuum mechanics is not sensitive to absolute size, the effect of size can only be accounted for by way of the H a l l Petch strengthening of the Co phase, as discussed above; therefore, size-effects are not considered in this work.
4.1. M e s h density
The FE simulation of a given cut-out with different mesh densities revealed a strong effect of element density on yield stress. The elastic response, on the other hand, is hardly affected. A meaningful measure of mesh density D in models adapted to the microstructure of W C - C o alloys must obviously be related to the average size of the Co phase regions, and it should be normalized with respect to the Co volume fraction. In addition, the total number of elements must be increased when the number of Co regions in the cut-out increases. A suitable parameter for mesh density can be defined as follows:
N oE o O
~
-
fcoL2 " Here NCo, L,co and fco denote the number of elements in the plastically deforming Co areas, the mean free path and the volume fraction of Co, respectively. L 2 is the total area covered by the model. For an increase of mesh density D from 15 to 60, the calculated 0.1% yield strength of the same microstructural cut-out was found to decrease by 13.5%, while Young's modulus was changed only by 0.75%. Refinement of the mesh soon makes the number of elements in a given cut-out too large. Therefore, a second, smaller cut-out, and successively a third one, each with one-half the edge length of the preceding one were modelled (fig. 4, indicated in model 1), 4000,
o..
~p~ =0.1% 3000
~
2500
. . . . . .
ucr--a
oO
•u
2000
o
= 0.02 %
1500 ~ _ ~ 1000
0
0........ O. . . . . . . . . . . . . . . . . . .
100
.
.
.
200 300 400 500 FE mesh density D Fig. 6. Influence of FE mesh density on calculated yield strength values, closed symbols: FE results, open symbols: correction for mesh density included (see text).
M.H. Poech et aL / FE-modelling of the deformation behaviour of WC-Co alloys
218
Table 3 Coefficients to estimate the error of FE-calculated yield strength with element density (element type TRIP3P)
R p 0.02% R p o.1%
a
b
0.8222 2.0115
0.6073 0.7423
using three different mesh densities for each. The results are shown in fig. 6. The three cut-outs (model 1) have slightly different volume fractions and, of course, distinctly different phase arrangements, which result in different absolute magnitudes of the calculated yield strengths. Despite these differences, fig. 6 clearly shows that the calculated yield strengths converge to a constant value at very high mesh densities, while at low mesh densities the results are systematically high. This overestimation of stiffness is expected because the inhomogeneous distribution of plastic strains cannot be correctly reproduced by large elements. Fitting a simple formula to the curvature of the line segments in fig. 6, the relative error due to insufficient element density can be expressed as follows: °'calc
1 + aD -°.
O'sat
Values of the constants a and b for the type of mesh employed here are given in table 3. As a sensible compromise with respect to computation time, a mesh density corresponding to D = 100 was chosen for all further work, accepting an overestimation of yield strength by 5% at 0.02% plastic strain, and of 6.6% at 0.1% strain. Mesh densities above D = 500 would be required to bring the error below 2%. Since CPU time is approximately proportional to the square of the number of nodes, the computation time would increase by a factor of 25, which does not appear justified for the small gain in accuracy. 4.2. Loading in different directions In reality, there is no preferred direction in the microstructure of a sintered WC-Co alloy, and consequently, there should be no anisotropy in
the loading response of the models. However, the FE models are limited to small cut-outs (about 15 WC grains). A "vertical" and a "horizontal" load will encounter different phase arrangements. While these differences should average out in sufficiently large regions of an inherently isotropic structure, small cut-outs must be expected to present different stress-strain curves in different loading directions, and the magnitude of the difference will tell us something about how far we are from a sufficient model size. Figure 7 shows the ratio of calculated yield stresses for the cut-outs of figs. 3 and 4 in two perpendicular loading directions. The periodic lattice type of mesh seems to be less sensitive to the loading direction, which may be due to the simplification which it imposes on the microstructure. For the "realistic" mesh, the differences are uncomfortably large, indicating that calculated properties should be averaged over a large number of cut-outs if quantitative reliability is to be achieved. In fig. 7 the yield stress anisotropy is plotted against the geometrical anisotropy of the cut-outs, which is characterized by the ratio of phase boundaries intersected by horizontal and vertical test lines, 12. Although the two quantities correlate roughly, the geometrical anisotropy characterized by 12 is obviously not the proper key to the mechanical anisotropy. This is not really surprising because 12 in fact characterizes the anisotropy of phase boundary surface [26], or,
1.3 ~ 1.2
,
~._o 1.1
d
0.9 0.8 0.8
. 0.9
.
1
.
.
1.1
1.2
1.3
Anisotropy
Fig. 7. Influence of geometric anisotropy on yield strength
ratio.
219
M.H. Poech et al. / FE-modelling of the deformation behauiour of WC-Co alloys
equivalently, the ratio of the mean free Co paths in the two directions, neither of which can be expected to be the sole determinant of composite yield stress. More important is probably the phase arrangement. For instance, a soft response will be expected when a cut-out presents continuous bands of soft phase at 45 ° to the loading direction. In unfavourable cases, the mirror symmetry at the cell boundaries can create "ghost particles" of large size and exotic shape (e.g. dog bones) which block the deformation in the soft bands of the model. It has been demonstrated [28-30] that such a blockage makes the loading response stiffer. In addition to the apparent anisotropy of the small cut-outs, there is, of course, a real anisotropy in any two-dimensional F E model, since the model actually corresponds to a bundle of prismatic lengths of infinite length in the direction perpendicular to the plane of the microstructure. This is a serious draw-back of two-dimensional models, which presently cannot be overcome without prohibitive effort. Rather surprisingly, it has been found that two-dimensional models reproduce certain essential features of the mechanical behaviour of W C - C o alloys quite well, one example being mechanical hysteresis [13]. However, one would be well advised to exercise great caution when applying two-dimensional FE models to materials whose microstructures are modulated in three dimensions. Our present appreciation is that they serve well for exploring trends but should not be expected to reproduce mechanical properties in a quantitative manner. Fortunately, there are some problems where two-dimensional models are fairly appropriate to the physical situation, as in the case of crack propagation across Co regions between carbide grains discussed in ref. [17]. Exact comparisons between two- and three-dimensional calculations may be made for microstructural features of simple geometry, and it may be possible to develop a feeling for the usefulness of two-dimensional calculations from such comparisons. A valid comparison between F E calculations and experiments can be made in the case of transverse loading of a fiber composite, which is described in another paper from our group [29].
4.3. Volume fraction
The volume fractions of the soft and hard phases have a strong influence on the yield strength of composites. This influence can be described roughly by the well-known rules of mixture; for small variations of the volume fraction, it can be taken to be linear. The influence of volume fraction was studied by slightly shifting some nodes in the FE mesh, keeping the adjustments small enough to leave the microstructural geometry essentially unaffected. One percent reduction of the Co volume fraction was found to increase the yield strength of the composite (at 0.1% strain) by about 8%. This strong effect can be understood, recalling that the WC phase, which has the dominating volume fraction, undergoes no plastic deformation in the model. A semi-analytical calculation based on the "strain compatible series model" developed by Poech and Fischmeister (SCS) [18,31] makes predictions which are virtually identical with the FE results, as shown in fig. 8. 4. 4. Shape of phase regions
Using an FE mesh of the periodic lattice type (fig. 1), the shape of the phase regions in a cut-out is easily altered by switching elements
Calculated Yield Strength of WC-Co 5000
",
[
i
", a_
4000
- .
~_ 3 0 0 0
.
.
.
.
. * •
.
SCS SCS FEM FEM
0.02% 0.1% 0.02% 0.1%
"0
c
2000
n -o
1000
O 0
10
20
30
40
v o l u m e f r a c t i o n C o [%]
Fig. 8. Influence of Co volume fraction on yield strength, comparison of the FE results with a semi-analyticalapproach (scs).
220
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys 4000
4500 'E Q..
~" 3000
4000 JE:
O3 E
2000
~o 3500 1o
,= 1000
[ - ~ i
0
o~ 3000 c5 2500 0.3
i
0.05 0.1 Plastic Strain [%]
0.15
Fig. 9. Influence of Co shape on the plastic deformation behaviour.
from one phase to the other, as exemplified in fig. 4. Sharp corners or edges in the microstructure will cause stress and strain concentrations which in turn cause yielding at lower stresses than in a structure with more rounded phase regions. This was demonstrated already in one of the earliest papers on FE modelling of two-phase microstructures [32], and similar results are obtained for the progressive rounding of the microstructures considered in this study, as seen in fig. 9. As expected, the softer response of structures with more angular phase regions continues during further deformation. Defining parameters which characterize the mean shape of phase regions in microstructures is difficult. The shape factor F defined in section 3 is popular for characterizing the shape of isolated phase regions. Despite its sensitivity to the size distribution of the phase regions [33], it was used in this study as a first approach, for lack of a better parameter, to characterize the shape of the isolated Co regions which appear in plane sections of the W C - C o alloy, cf. table 2. Figure 10 shows the expected trend for the series of progressively rounded microstructures modelled with periodic lattice nets (A, B, C in fig. 3). The data points for the three cut-outs shown in fig. 4, which were modelled with meshes that allowed a realistic rendering of the phase geometry, conform to this trend, but only very roughly. We must conclude that the mean shape factor ff is not a good descriptor of phase shape in those
~de
I
I
I [] period~,almesh I
i
i
i
~
=
0.4
0.5
0.6
0.7
0.8
0.9
Mean Shape Factor
Fig. 10. Influence of the mean shape factor of Co regions on the yield strength. aspects which are relevant to mechanical behaviour. 4.5. Cut-outs with different phase arrangement
Different cut-outs randomly taken from a given microstructure, such as items 1, 2 and 3 in fig. 4, will present different phase arrangements, and they will in general have slightly different volume fractions. This is true also for the cut-outs considered here. To isolate the effect of phase arrangement from that of volume fraction, the FE models were brought to a uniform value of Co volume fraction (cf. table 2) by slightly shifting appropriate nodes of the net in a way which essentially preserved the original geometry of the microstructure. Figure 11 shows the effect of the variations of phase arrangement on the mechani4000 3000
E ¢o 2000 (,9
1000 L
0
0.05
.
i
0.1 0.15 Plastic Strain [o%]
0.2
Fig. 11. The influence of phase arrangement on the plastic deformation behaviour.
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
cal behaviour. It is small enough to allow representative results to be obtained by averaging over a limited number of such cut-outs.
4500
J ¢
CK
2Sv~K 2SvK ~ + Sv,~ '
where Sw~ is the surface per unit volume of boundaries between phases ~ and K. In Gurland's contiguity parameter, the boundaries between adjacent grains of the same phase play a large role, and for the WC phase in W C - C o alloys, this is in fact the dominant contribution. Since our models pertain to early stages of deformation, during which the carbide grain boundaries are assumed to remain intact, we prefer a parameter which characterizes the carbide skeleton without reference to individual carbide grains. This can be done with the recently developed "matricity parameter" [27] defined in section 3. In W C - C o microstructures, the case where the carbide forms a fully continuous matrix (Mwc = 1), while the cobalt is present in the form of isolated inclusions ( M c o = 0), will represent the upper bound for yield strength; the reverse case (Mwc = 0, M c o = 1) will represent the lower bound. A semi-analytical model calculation recently presented [35] predicts these bounds to be as far apart as 6069 and 1144 MPa (at 0.1% plastic strain, using the mechanical property data given in table 1 and the volume fraction of the microstructures considered here, 16.5 vol.%). This emphasizes the importance of a meaningful quantitative characterization of the phase arrangement in any attempt to predict mechanical properties of W C - C o alloys.
"realistic"model periodicalmesh
4000
4.6. Quantitative characterization of the phase arrangement: the matricity parameter In looking for a way to rationalize the effects illustrated by our sample calculations with different phase arrangements (fig. 11), we assume that they are caused by variations in the degree of which stiff phase regions are linked to form a stiff skeleton. Gurland [7] has introduced the concept of "contiguous volume" for this purpose, which is based on his definition [34] of the contiguity C of regions of one phase:
221
3500
3000
d 2500
. 0.1
0
.
. 0.2
.
0.3
0.4
0.5
Matricity Fig. 12. Influence of Co matricity on yield strength.
The matricity value of the carbide phase in each of the cut-outs of figs. 3 and 4 has been determined (cf. table 2). A plot of yield strength (at 0.1% strain) against matricity, cf. fig. 12, shows that this parameter seems to be a good descriptor for the influence of phase arrangement on yield strength at constant volume fraction. Further support for the relevance of the matricity parameter to mechanical behaviour is found when considering the distribution of local strains in the various cut-outs. Figure 13 shows the cumulative frequency distribution of strain levels in all elements belonging to the Co phase in the three cut-outs of fig. 3. These distributions reflect the range of strain hardening of the Co phase when the composite is loaded to a given level, equal for all three cut-outs. Characterizing the distributions by their standard deviations, a corre-
100 o>,
8O
o-
60
c
u_
4o E
20
0 0
0
0.5
1
1.5 ~p~[%]
2
2.5
3
Fig. 13. Cumulative frequency distribution of plastic strains in cut-outs A, B, and C.
M.H. Poech et aL / FE-modelling of the deformation behaviour of WC-Co alloys
222
0.8
j
8 •~, .£ "5 a~
0.6
~._~
o£3 t~'
0.4
n,0 0
I
i
i
0.1
0.2
0.3
0.4
Matricity
Fig. 14. Width of the plastic strain distribution (standard deviation of plastic strains normalized to the mean value) with Co matricity.
lation can be made with the microstructural geometry, which again is characterized by the matricity parameter. This is shown in fig. 14. As expected, the more branched-out the Co phase appears, the more inhomogeneous is its deformation. Extrapolating the correlation back to M = 0 predicts a state of fully homogeneous plastic strain, in agreement with Eshelby's solution for spherical inclusions [36].
4.7. Comparison with experimental stress-strain curves
The stress-strain behaviour in compression of an alloy corresponding to our models has been
4000
4,~/~,~4,~. ~
WC-16.5 vol% Co
~ ,,4"//~ P"
3000 to
I
studied by Doi [4]. Figure 15 summarizes the stress-strain curves of all the FE simulations discussed above and shows the comparison with Doi's experiment, whose data have been converted to the plane strain loading condition as described in the appendix. The comparison shows that the FE models predict a stress-strain behaviour for the 10C alloy which is not too far from reality, despite the fact that the models were two-dimensional. In the representation shown here (which is in terms of total rather than plastic strain), the spread resulting from the shape alterations shown in fig. 3, from the choice of different cut-outs (fig. 4) and from loading in two perpendicular directions is not too wide. Most of the models are initially somewhat too soft in comparison to the experiment, but the discrepancy is not drastic. The main shortcoming of the calculations is that they do not reproduce the decrease of the strain hardening rate after initial plastification. A possible interpretation of this discrepancy, which, in fact, we find quite likely, is based on the observations of [2,37,38] which indicate that the carbide skeleton looses its integrity by cracking (very likely at the grain boundaries) at relatively small composite strains. In addition, ample evidence has been brought forth for plastic processes in the carbide phase [38-40]. Both points would bring the stiffness of the carbide skeleton appreciably below the purely elastic behaviour which has been assumed in setting up the FE models. Finally, the residual thermal stresses which are present in WC-Co alloys [41] contribute to the early stages of plastic deformation [42]. Their effects, too, have been excluded from the present calculations. In this light, we feel that the agreement between the FE models and the experiments may be considered quite satisfactory.
20'00 FEMGo-shape [ FEMCo-arrangement Experiment(Doiet al.)
I
1000 0
0
I
i
i
i
0.2
0.4
0.6
0.8
Strain
[%]
1
Fig. 15. Summary of the FE simulations in comparison with experiment.
5. Concluding remarks
FE modelling has limitations which result from mesh density, boundary effects, and sampling errors due to the small size of the microstructure cut-outs. These limitations have been illustrated; some of them are tolerable, and the rest can be overcome by proper computing procedures and
M.H. Poechet al. / FE-modellingof the deformationbehaviourof WC-Co alloys suitable design of the models. In particular, as has been demonstrated in other work [17,43], the effects of the displacement conditions at the boundaries of the model can be reduced by surrounding the two-phase microstructure with an FE continuum which is given the overall properties of the composite. In the foreseeable future, FE models of real microstructures will have to remain two-dimensional, and this might be feared to be a real shortcoming. As pointed out in the paper, caution suggests that two-dimensional models should be used mainly to develop an understanding of trends, and to derive pointers to important features of mechanical behaviour, rather than for accurate predictions on an absolute scale. However, experience shows that in many cases, twodimensional FE models of real microstructures come surprisingly close to the behaviour of actual materials [6,12,13,24,32]. The results shown in fig. 15 of the present paper may be added to this list. Encouragement is also taken from their extremely good agreement with the results of semianalytical models recently presented [31,35] which are based on simplified, but three-dimensional geometries. Experience also indicates that two-dimensional models serve well whenever the properties of the phases making up the microstructure, and consequently the deformations undergone by the phases, are not too dissimilar [35]. The special promise of FE models is that they ought to be able to clarify the effects of phase arrangement or microstructural geometry, which so far have largely escaped modelling attempts. A large part of this problem lies in developing good quantitative descriptors for the microstructures. The present study shows that conventional descriptors, such as the area:perimeter ratio used in the traditional shape factor, may have little relevance to mechanical behaviour. An approach which may improve this situation has been demonstrated in the section on matricity. A second part of the problem lies in the loss of information incurred whenever a microstructure is reduced to a few characteristic mean numbers such as volume fraction, size, and a mean shape parameter. However, there is hope that nature works to our advantage in that it allows only limited
223
variability for microstructures generated for given classes of materials with given processing routes, as, for instance, for W C - C o alloys produced by liquid phase sintering of powders. The obvious "family likeness" of large groups of microstructures holds out the hope that microstructural modelling of the properties of two-phase materials will allow fairly accurate property predictions.
Appendix: unidirectional loading in plane strain The calculation procedure to convert unidirectional stress-strain data to plane strain given an isotropic material has already been presented by ref. [13]. A re-evaluation has revealed some small errors the correction of which will be given below. The state of unidirectional loading in plane strain can be characterized by e Z = 0 and ~ry = 0. With the assumption of proportional loading plasticity theory gives:
Eepl 1,'+-k
O"z
2O'erf
crx
l + - -EEpl o'e~f
With the unidirectional stress-plastic strain data (treff - %0 and the elastic constants (E, u) we get for the remaining strain components e x and %:
( 1-uk ex=O" x
, ~
epl
)
+ 2°-elf ( 2 - k )
( u ( l + k )E Ey=°'x
,
%~ ) 2~reff(l+k) -
Using the definition of effective stress (creff) by Von Mises the stress in loading direction o-~ evaluates to O'¢ff
~rx- v / l _ k + k 2 The global stress and strain tensors are defined by these equations so that the uniaxial stressstrain curves can simply be converted to the plane strain loading conditions.
224
M.H. Poech et al. / FE-modelling of the deformation behaviour of WC-Co alloys
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