Computational Materials Science 13 (1998) 56±60
Micromechanical simulation of crack growth in WC/Co using
embedded unit cells
S. H
onle
a
a,b,*
, S. Schmauder
a
Staatliche Materialpr
ufungsanstalt (MPA), Universit
at Stuttgart, Pfa€enwaldring 32, D-70569 Stuttgart, Germany
b
Max Planck Institut f
ur Metallforschung, Seestrasse 92, D-70174 Stuttgart, Germany
Abstract
Nowadays, hard metals are omnipresent as tool materials in industry. The material behavior is from decisive importance, from the safety point of view as well as from economical aspects. As a result of the manufacturing and coating
process cracks are introduced into the surface of hard metal tools. A micromechanical model has been developed in
order to simulate crack growth in a WC/Co hard metal. The elastic model consists of a unit cell with a cobalt island in a
carbide environment, which is embedded in a composite surrounding. The energy release rate is calculated for a crack
propagating along the symmetry plane of the model on a microscopic scale. The cobalt phase in¯uences the crack
driving force in an important way. The energy release rate of a crack approaching the cobalt phase increases, while it
decreases rapidly for the crack propagating towards the center of the cobalt island. Parametric studies were carried out
to determine the in¯uence of di€erent cobalt inclusion shapes and cobalt volume fractions on the energy release rate.
Moreover, the energy release rate is calculated for a unit cell with two square cobalt inclusions and compared to crack
propagation in a computational cell with a single inclusion. Ó 1998 Published by Elsevier Science B.V. All rights
reserved.
Keywords: Finite element method; Unit cell; Energy release rate; WC/Co hard metal
1. Introduction
Since the late 20s hard metals played an important role as tool materials for cutting processes
[1,2]. Hard metals are manufactured by sintering
of multiphase powders, conventionally a very hard
brittle ceramic phase and a ductile binder phase.
The material under observation is a WC/Co hard
metal, which is used for milling applications. For
better tool performance the WC/Co tool is coated
with a thin TiC layer [3,4].
*
Corresponding author. Tel.: +49 711 685 2701; fax: +49 711
685 2635; e-mail: [email protected]
Coated hard metal inserts contain cracks as a
result of the coating process [5,6]. Cobalt enriched
gradient zones (Fig. 1) [7] underneath the coating
prevent crack growth into the tool to a certain
extent. Subject of the present paper is the numerical simulation of crack advance and the local
material response of the WC/Co hard metal in a
fracture process.
2. Model descriptions
A micromechanical elastic two-dimensional FEmodel with elastic material properties has been
developed in order to simulate crack growth in
0927-0256/98/$ ± see front matter Ó 1998 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 7 - 0 2 5 6 ( 9 8 ) 0 0 0 4 5 - 7
S. H
onle, S. Schmauder / Computational Materials Science 13 (1998) 56±60
57
Table 1
Material properties for WC/Co composite and constituents
Ecomp ˆ 595 GPa
EWC ˆ 714 GPa
ECo ˆ 211 GPa
Vcomp ˆ 0.22
VWC ˆ 0.19
VCo ˆ 0.31
propagating along the symmetry plane of the cell is
calculated according to the procedure given in [8]
for di€erent crack lengths on a microscopic scale.
The dimensionless normalized energy release rate
G is calculated as follows [9]
GE
…1†
r2 h
with E, Young's modulus of the composite, r,
average stress in the model, h, height of the model.
Moreover, the geometry of the cobalt inclusion is
varied as well as the cobalt volume fraction and
the arrangement of the inclusion.
G ˆ
Fig. 1. Cobalt enriched zone (example marked) in a graded
WC/Co hard metal.
3. Results and discussion
WC/Co (Fig. 2). The model includes a non-selfconsistent unit cell with a cobalt island in a carbide
environment. The unit cell is embedded in an
elastic body with the average material properties
of the WC/Co composite, depending on the volume fraction of the material under consideration,
according to Fig. 2.
The elastic material properties of the composite
material and the constituents are listed in Table 1.
The energy release rate (ERR) G of a crack
In order to simulate the failure behavior of a
WC/Co hard metal, a micromechanical computational cell model was set up to calculate the ERR
for a crack propagating through a carbide-cobalt
cell (Fig. 2). Therefore, the model containing an
initial crack was loaded uniaxially with a strain
value related to the global fracture toughness KIc
ˆ 16.7 MPamÿ1=2 for WC/Co with a cobalt volume fraction of 16% [10]. The energy release rate
was calculated for di€erent crack lengths a in the
WC/Co-cell, varying from 0 to l, where l represents
the width of the cell.
3.1. Inclusion shape
Fig. 2. Scheme of an embedded two-dimensional unit cell model
(with prescribed displacement ~
u).
At ®rst, the ERR for a propagating crack was
calculated for di€erent cobalt inclusion shapes in a
volume element of hard metal with a given cobalt
volume fraction of 16% (Fig. 3). The cobalt phase
in¯uences the crack driving force in an important
way. The energy release rate of a crack approaching the cobalt phase increases strongly,
while it decreases rapidly for the crack propagating towards the center of the cobalt island (Fig. 4).
The cobalt phase absorbs a large amount of
58
S. H
onle, S. Schmauder / Computational Materials Science 13 (1998) 56±60
Fig. 3. Variation of cobalt inclusion shapes.
Fig. 5. Total energy consumption (TEC) for varying cobalt
inclusion shapes.
Fig. 4. Normalized elastic energy release rate (ERR) for varying
cobalt inclusion shapes.
cracking energy. Thus, such a crack may probably
be arrested in the cobalt phase.
All the curves exhibit a similar shape and the
maximum values of the elastic energy release rates
vary in a range of about 20% (Fig. 4). A crack is
found to be more attracted by a sharp-edged cobalt inclusion in front of the crack tip. This cobalt
inclusion arrangement can be related to cobalt
inclusions between carbide grains in realistic
structures with small ``opening angles''.
The total energy consumptions (TEC) of the
cracks, which are determined by integrating the
ERR curves, vary in a range of only a few percent
for the di€erent inclusion shapes (Fig. 5). Thus,
the cobalt inclusion shape in¯uences the attraction
of the crack tip in the elastic regime, but has a
negligible e€ect on the total elastic energy consumption of the crack propagating through the
computational cell.
Fig. 6. Normalized elastic energy release rate (ERR) for varying
volume fractions of a square cobalt inclusion.
3.2. Inclusion volume fraction
To show the in¯uence of the cobalt volume
fraction, a parametric study was carried out using
a computational cell with a centered square cobalt
inclusion. The volume fraction of the cobalt inclusion was varied from 8% to 32% which is a
typical regime for WC/Co hard metals [6].
The calculated energy release rates for varying
cobalt volume fractions are illustrated in Fig. 6.
The maximum value of the normalized energy release rate (Gmax ), which gives rise to the attraction
of the crack by the cobalt inclusion, is found to be
an increasing function with increasing cobalt volume fraction (Fig. 7). The total energy consumption (TEC) decreases linearly with increasing
cobalt content (Fig. 8). Thus, when propagating
S. H
onle, S. Schmauder / Computational Materials Science 13 (1998) 56±60
59
Fig. 7. Maximum ERR for varying volume fractions of a
square cobalt inclusion.
Fig. 9. Maximum normalized ERR for varying cobalt inclusion
tip angles.
Fig. 8. TEC for varying volume fractions of a square cobalt
inclusion.
Fig. 10. Normalized ERR for two square cobalt inclusions.
through a linear-elastic hard metal volume element, the crack is more attracted by a higher cobalt content while consuming less energy.
inclusions investigated, the crack is most attracted
by cobalt inclusions with an inclusion tip angle of
120°.
3.3. Inclusion tip angle
3.4. Multiple inclusions
Due to the fact of high maximum values of the
ERR related to sharp-edged cobalt inclusions
(Fig. 4) a parametric study was carried out, using
a hexagonal cobalt inclusion with a ®xed cobalt
volume fraction of 16%. The entrance angle a/2 of
the cobalt inclusion was varied from 30° to 80°. As
a consequence of the stress±strain distribution at
the crack tip the maximum value of the ERR is
found at an angle of about 60° which represents a
hexagon angle of 120° (Fig. 9). Thus, among the
The energy release rate was calculated for a unit
cell with two square cobalt inclusions in the crack
plane, where the overall cobalt volume fraction
was again set to 16%. According to the results
presented in Fig. 10 the energy release rate increases while the crack is approaching the ®rst
cobalt inclusion and decreases rapidly when
propagating through the ®rst inclusion in the same
way as in the calculation for a single inclusion. The
energy released by the crack in the ®rst inclusion is
60
S. H
onle, S. Schmauder / Computational Materials Science 13 (1998) 56±60
not in¯uenced by the second cobalt inclusion. The
total energy consumption for this crack has the
same value as in the case of a single cobalt inclusion with the same cobalt volume fraction.
4. Conclusions
The following conclusions can be drawn according to the presented results:
· A crack is most attracted by sharp-edged cobalt
inclusions in front of the crack tip.
· An inclusion tip angle of the cobalt island of
120° was found to be most bene®cial for attracting a crack.
· Unit cells with a higher content of cobalt are
more attractive to cracks, although less energy
is consumed when the crack propagates through
the cell.
· The arrangement of the cobalt inclusions in
crack direction has no important in¯uence on
the energy consumption of a propagating crack.
Acknowledgements
The authors gratefully acknowledge the ®nancial support by the European Commission for ®nancial support via the Brite-Euram Project
BRE2-8109 ``Design of new tool materials with a
structural gradient for milling applications''.
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