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Microstructural model of intergranular fracture during tensile tests
Journal of
Journal of Materials ProcessingTechnology60 (1996) 305-310
ELSEVIER
Materials Processing Technology
Microstructural model of intergranular fracture during tensile tests J. Lu, J.A. Szpunar
Department of Mining and Metallurgy, McGill University 3450 University Street, Montreal, Quebec, Canada, H3A 2.4 7
Abstract
A microstructural model of intergranular fracture in textured materials is presented. In this model, the material is represented by a twodimensional microstructure with non-regular polygonal grains which represents material's texture and grain shape measured in experiments or calculated from Monte Carlo simulations. The grain boundary character, grain boundary energy, and fracture stress are assigned to each grain boundary according the grain boundary character distribution, lntergranular fracture susceptibility is analyzed by defining the probability of fmding a continuous path along the grain boundaries which are intrinsically susceptible to fracture. In this analysis the orientations of the grain boundary with respect to the applied or residual tensile stress axis is considered. The probability of intergranular fracture for each grain boundary depends on the intergranular fracture resistance, the interface orientation relative to the stress axis, and a value of the tensile stress acting on the grain boundary. The crack arrest distance and the fracture toughness are calculated in terms of the frequency of low-energy grain boundaries, fracture stress of low-energy grain boundary, angle distribution of grain bound ary interfaces, and anisotropy of grain shape. The results indicate that the fracture toughness increases and the crack arrest distance decreases dramatically with increasing the frequency of the low-energy grain boundaries. Lowering the grain boundary energy can improve the fracture toughness and decrease the crack arrest distance. The angle distribution of grain boundary interfaces and the grain shape factor are also very effective in controlling the fracture toughness. High fracture toughness of polycrystalline materials is related to the presence of a high frequency of low-energy boundaries which are resistant to fracture. The best fracture toughness for brittle materials can be achieved by controlling the frequencies of the low-energy grain boundaries, the grain boundary character, and the boundary inclination.
Keywords: Low-energy grain boundary, Intergranular fracture, Grain boundary character distribution, Crack arrest distance, Fracture stress, Fracture toughness.
1. Introduction It is well known that grain boundaries strongly affect the physical, chemical and mechanical properties of polycrystalline materials. Accordingly, a great deal of efforts has been devoted to understanding the structure, energetic, and properties of grain boundaries [1-5]. Grain boundary is also an important source of fracture in polycrystal and intrinsic or extrinsic brittleness of polycrystalline materials is mostly due to intergranular fracture. Watanabe [6] first introduced the concept of "grain boundary design and control," with the purpose of improving various properties of polycrystalline materials through enhancing the frequency of low-energy boundaries in the grain boundary character distribution (GBCD). An enhancement of the fraction of low energy, fracture resistant boundaries is essential to the toughening by grain boundary design a control [7]. In general, low angle (~1) boundaries and symmetrical Y~3 boundaries (twins) are particularly strong, low P~, high angle boundaries, as a group, are less strong, and high ~, high angle boundaries for brittle materials Ni3A1 are weak [8]. The premise of "grain
0924-0136/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved PI10924-0136 (96) 02346-1
boundary design and control" [9] is that the toughness of a polycrystalline material that fails intergranularly will increase with to the increase in the fraction of strong boundaries. Therefore it is essential that the types of strong boundaries be identified in a statistically significant manner in the first place [8]. In this report, we present the modeling results of the influence of grain boundary on intergranular fracture arrest in brittle materials. In the model, the material is represented by a twodimensional microstructure with non-regular polygonal grains which can be obtain from experiments or Monte Carlo simulation. Microstructural effects are included by varying the average grain sizes, grain boundary energies, grain boundary distributions, and grain boundary geometry distributions.
2. Microstructural model
The idea of Markov chain fracture model is based on a simple geometric model which was recently proposed to evaluate the potential effects of grain boundary design and control on
306
,1. Lu, J.A. Szpunar /Journal of Materials Processmg Technology 60 ('1996) 305-310
intergranular fracture and intergranular stress corrosion [3]. In this simple geometric model, the susceptibility of a bulk material to failure by stress corrosion cracking is defined in terms of the probability of propagating a crack, by the combined action of stress and corrosion, through a l-mite distance. This length defines a limit, beyond which, crack propagation would continue to component failure even in the absence of either active grain boundary paths or a corrosive environment; its magnitude is primarily dependent on the specific stress state and wall thickness of the component. Intergranular stress corrosion cracking susceptibility can thus be considered in terms of the probability of finding a continuous path, existing the distance of the critical crack length, consisting entirely of intrinsically susceptible grain boundary segments, and each of which is favorably oriented to the direction of the applied or residual tensile stress. A susceptible grain boundary segment can be defined as a crystal interface (planar), which as a result of its structural and/or chemical characteristics, is prone to enhanced corrosion (relative to the lattice) in the specific environment, and/or preferential sliding or fracture under the local operative stress condition. In the Markov chain fracture model, however, a realistic geometric microstructure and a realistic grain boundary character distribution were considered. To represent the microstructure, we have adopted the Monte-Carlo simulation method developed by Anderson and Srolovitz et al. [10-11], and accordingly the microstructure is described by a matrix. In results obtained from the Monte-Carlo simulation we defined the grain boundary as the direct link between the nearest triple points within a grain and then the grain boundary character and energy are assigned to each grain boundary according the grain boundary distribution. The grain boundary character distribution could also be obtained from the experiment or the one which could be used to obtain the desire properties of polycrystalline materials. In addition, the probability of intergranular fracture at a given triple junction is calculated according to the fracture resistances of grain boundaries at this triple junction. On the basis of geometric considerations, intergranular crack arrest (i.e., crack blunting or transition to transgranular crack propagation) can be considered to occur at a triple junction when the probability of crack continuation along either of the two available intergranular paths becomes negligible. In the Markov chain fracture model, by considering both orientations of the grain boundary plane relative to the stress axis and intrinsic character of grain boundary, the probability of crack front extension for a grain boundary at a given triple junction can be shown to be given by,
1, P
~,(0)-> ~aB
exp(~(0) _ 1), ~(0)<~oB
(1)
O'GB
applied tensile stress on this grain boundary (i.e., this grain boundary path will allow forward advance of the crack propagation). When the fracture resistance of the grain boundary is larger than the applied tensile stress on the grain boundary, there is still a chance of forward advance of the crack propagation according to the calculation from Eq. (1). For a given grain boundary character distribution, the propagation of a crack through a polycrystalline microstructure started at the triple junction in the surface of the sample. This was necessary since, prior to the development of the crack, the sample was elastically homogeneous. The crack begins by propagating along the low cohesion grain boundary network until it finds itself in a situation in which both grain boundary paths at the triple points do not allow forward advance of the crack propagation, i.e., crack arrested. The vertical distance from this triple point to the start point in the surface is called crack arrested distance. If the crack path propagates through the whole sample, the crack arrested distance is equal to the sample's width and this sample is definite failure. We have to calculate all crack paths with possible initial crack at triple points in the surface of the sample. In addition, for a given grain boundary character distribution, there exist many possible configurations for each grain boundary in the microstructural sample. Therefore in order to get very good statistic simulation results we have to calculate as many configurations as possible. In this investigation, we have calculated 80 configurations by using 80 different initial random seeds. In our experience, every calculated result converges to a value when 60 configurations are used. In the calculations we assume (i) no transgranular fracture and (ii) the maximum value of the applied tensile stress on the grain boundary is half of the fracture resistance of single crystal. The final normalized crack arrest distance required to bhmt 99% of intergranular fracture is calculated by
z.
do
:
do J ~ . L,y
where do is the average grain size. The first sum is taken over all possible initial cracks at triple points in the surface area and the second sum is taken over 80 different configurations of grain boundary in the microstructural sample. To investigate the fracture process in polycrystalline materials we also have to deal with the energetics of intergranular fracture and focus on energy change between the grain boundary and the surface of fractured bicrystal specimen. According to the energetics, when the intergranular fracture occurs and the plastic deformation is involved, the driving the crack propagation energy y is given by[8]
y = 2 y ~ - YGB + Yp where o(O) is the applied tensile stress on the grain boundary, 0 is the angle between the grain boundary plane and the stress axis, oas is the fracture resistance of the grain boundary. The probability of crack extension along a given grain boundary is one when the fracture resistance of this grain bolmdary is less than the
(2)
(3)
where ¥~ is the surface energy of the exposed grain boundary, and ¥GB the energy of the preexisting grain boundary, and Yv the plastic energy associated with the propagation of the microcrack.
J. Lu, J.A. Szpunar/Journal of Materials Processing Technology 60 (1996) 305-310
It is clear that the energy 7 is smaller in the case of absence of plastic deformation, than in the case where plastic deformation is present. Moreover, it should be noted that as the contribution of yp tO y decreases, in other words, as condition for the intergranular fracture is more ideal, a stronger dependence of fracture energy "/ on the grain boundary energy YcB is expected. This is an indication that the intergranular fracture becomes more important [9, 12]. Therefore, in brittle materials the fracture mode is predominantly intergranular. From Eq.(3) when intergranular fracture occurs in the absence of plastic deformation the fracture toughness of brittle material can be defined by the following expression:
(4)
and with a reference fracture toughness ~r~f = 1~1 for ¥~B=2y=. Here N is the total number of fractured grain boundaries in the crack paths.
3. Results and Discussion
We have carried out Monte Carlo simulations in 2-d with different time steps. Figure 1 shows grain size distribution as determined from this cross-section from a 2-d microstructure with 1141 grains and 3265 grain boundaries obtained from Monte Carlo simulation after 10,000 MC steps (see Figure 2). The shape of the grain size distribution function for this simulation agrees remarkably well with the experiment.
15.0
i0.0
~ 5.0
0.0 --' .0
-0.5
t.0
0.5
Loglo ( d / d 0 ) Figure 1 Grain size distribution as determined from crosssections of the microstmcture with 1141 grains and 3265 grain boundaries. Here do is the average grain size. Examples of the propagation of a crack through a polycrystalline microstructure with different fractions of lowenergy grain boundaries in a random orientation distribution of grain boundary planes are shown in Figure 2. In this case a small
307
pre-crack was nucleated at the triple point in the bottom of the sample microstructure prior to straining the sample. This was necessary since, prior to the development of the crack, the sample was elastically homogeneous. The crack begins by propagating along the low cohesion grain boundary network until it finds itself in a situation in which the grain boundary turns out too strong to be broken by the load. At this point, the driving force for crack propagation along the grain boundary network is greatly reduced, and the crack is arrested. Since the grain boundaries are weak compared with the interior of the grain, the stress field of the crack causes a new crack to nucleate at the site along grain boundary. This figure only demonstrates one crack propagation until it is arrested. The result also shows the crack arrested distance decreases dramatically with increasing the low-energy grain boundaries. Since the values of the crack arrested distance and other fracture parameters depend sensitively on the details of the microstructure in the vicinity of the pre-crack, they are statistically calculated by using 80 different initial random seeds for the random orientation distribution of grain boundary planes. Figure 3 shows the normalized crack arrest distance required to blent 99% of all crack, as a function of the low-energy grain boundary fraction. As shown in this figure, with increasing the fraction of low-energy grain boundaries by only 10% the crack length can be reduced considerably. The effect of the fracture resistance of low-energy grain boundary can also be seen in Figure 3. Increasing the fracture resistance of low-energy grain boundary can also reduce the crack arrest distance. The higher the fracture resistance of low-energy grain boundary the less low-energy grain boundaries are required to arrest the crack propagation within a certain distance, for example, a critical length of component failure. For a low fracture resistance of low-energy grain boundary the crack propagation cannot be arrested within the critical distance even there have high percentage of low-energy of grain boundaries. This critical length defines a limit, beyond which crack propagation would continue to component failure even in the absence of either active grain boundary paths or a corrosive environment [3, 13-14]. In our case if we define five grain diameters as the critical length and more than 25% of grain boundaries are of low-energy with the highest fracture resistance, the cracks formed initially cannot propagate further because of the presence of fewer random boundaries and that the polycrystalline material will show a higher ductility. This result agrees with experimental and theoretical results [15]. The grain boundary energy ¥6B is known to depend on the type and the structure of grain boundary, i.e., the boundary misorientation and inclination described by the crystallographic and geometrical parameters. We can expect that there exists a dependence of the fracture energy on the type and structure of the grain boundary. The "t'6~ is a measure of this dependence and the surface energy Ys does not change with crystallographic orientation as much as YcB. This leads to the possibility of theoretical analysis of structure-dependent intergranular fracture. It is evident that the low-energy grain boundaries are expected to need more energy to break than the high-energy grain boundaries. In other words, the low-energy grain boundaries are more resistant to fracture [9]. The influence of the low-energy grain boundary fraction and the fracture resistance of low-energy grain boundary
308
3". Lu, J.A. Szpunar / Journal of Materials Processing Technology 60 (1996) 305-310
on fracture toughness of intergranular cracks in the random orientation distribution of grain boundary plane is shown in Figure 4. Increasing the low-energy grain boundaries can increase the fracture toughness of brittle materials during the intergranular fracture. The brittle materials can also be toughened by enhancing the fracture resistance of low-energy grain boundary.
fL=O.1
(c)
fL=O.O
(a)
fL:O.3
(d)
fL=o.05
(b)
Figure 2 Examples of crack paths of intergranular fracture with various fractions of low-energy grain boundaries fL in a random orientation distribution of the grain boundary planes. Assume o~JoB=0.40 and OR=O.O10m
fL = 0.4
(e)
309
.Z Lu, flA. Szpunar /Journal of Materials Processing Technology 60 (1996) 305-310
30[
0.00
Palumbo et al's Calculation
I
25'
© --
20
G -A -* -L
° --
= 0.80 aGB/0"B = 0 . 6 0 ~B/O-B = 0.40 qB/a~ = 0.30 %B/~rB = 0 . 1 5
• --
~/~rB = 0.05
~/aB
I z,r
10
5
0 0.0
I 0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
0.6
0.8
.0
fL
Figure 3 Normalized crack arrest distance (L/do) required to blunt 99% of intergranular cracks as a function of fraction of low-energy grain boundaries fL with different grain boundary fracture resistances o6B/o B in a random orientation distribution of grain boundary planes. Assume the random grain boundary fracture resistance OR=0.010B. Here o B is the fracture resistance of a perfect bulk crystal.
Figure 4 Fracture toughness ~/,~f of intergranular cracks as a function of fraction of low-energy grain boundaries fL with different grain boundary energies rLs (=0.5~'GJYs) in a random orientation distribution of grain boundary planes. Assume the random grain boundary energy YR=0.99*2Ys. Here ¥s is the surface energy of an exposed grain boundary.
These results suggest the importance of grain boundary design and control through material processing, whereby considerable decreasing intergranular crack propagation distance may be achieved through only moderate increases in the fraction of lowenergy grain boundary with high fracture resistance in the grain boundary character distribution of conventional brittle polycrystalline materials.
is related to the presence of a high frequency of low-energy boundaries which are resistance to fracture. The grain boundary design and control by manipulating the grain boundary character distribution (GBCD) are important to fracture toughness improvement of the brittle polycrystal materials.
4. Conclusion
This work was supported by the Natural Science and Engineering Research Council of Canada. The Hydro-Quebec Fellowship to support one of the authors (J. Lu) is acknowledged.
The intergranular fracture in brittle materials has been modeled as function of the grain boundary character distribution. The calculations have predicted that with increasing the fraction of low energy grain boundaries, the fracture toughness increases while the crack arrest distance decreases. The intergranular crack formed initially cannot propagate further because of the presence of fewer random boundaries and that the polycrystalline materials will show high ductility if we define five grain diameters as the critical length and more than 25% of grain boundaries are of lowenergy with the highest fracture resistance, such as E3 twin grain boundaries, etc. High fracture toughness of fme-grained materials
Acknowledgements
References
[1] [2] [3] [4] [5]
L.C. Lim and R. Raj, Acta Metall., 32, 1177 (1984). J. Don and S. Majundar, Acta Metall., 34, 961 (1986). G. Palumbo, P.J. King, K.T. Aust, U. Erb, and P.C. Lichtenberger, Script Metall. et Mater., 25, 1775 (1991). L.C. Lira and T. Watanabe, Acta Metall., 38, 2507 (1990). J.F. Lustko, D. Wolf, S. Yip, S.R. Phillpot, and T. Nguyen, Phys. Rev. B38, 11572 (1988).
310
[6] [7] [8] [9] [10]
J. Lu, J.A. Szpunar / Journal of Materials Processing Technology 60 (1996) 305-310
T. Watanabe, Res. Mechanica, 11, 47 (1984). T. Watanabe, Mater. Forum, 11, 284 (1988). H. Lin and D.P. Pope, Acta Metall., 41, 553 (1993). T. Watanabe, Mater. Forum, 46, 25 (1989). M.P. Anderson, D.J. Srolovitz, G.S. Grest, and P.S. Sahni, Acta Metall., 32, 783 (1984). [11] D.J. Srolovitz, M.P. Anderson, P.S. Sahni, and G.S. Grest, Acta Metall., 32, 793 (1984).
[12] T. Watanabe, Trans. Japan Inst. Met., 27, 73 (1986). [13] K.T. Aust, in Karl T. Aust International Symposium on Grain Boundary Engineering, U. Erb and G. Palumbo, eds., Kingston, Ontario, Canada, 197 (1993). [14] K.T. Aust and G. Palumbo, in Structure and Property Relationship for Interfaces, J.L. Waiter, A.H. King and K. Tangr, eds., ASM, 3 (1991). [15] T. Watanabe, Mater. Forum, 126-128, 295 (1993).
Journal of Materials ProcessingTechnology60 (1996) 305-310
ELSEVIER
Materials Processing Technology
Microstructural model of intergranular fracture during tensile tests J. Lu, J.A. Szpunar
Department of Mining and Metallurgy, McGill University 3450 University Street, Montreal, Quebec, Canada, H3A 2.4 7
Abstract
A microstructural model of intergranular fracture in textured materials is presented. In this model, the material is represented by a twodimensional microstructure with non-regular polygonal grains which represents material's texture and grain shape measured in experiments or calculated from Monte Carlo simulations. The grain boundary character, grain boundary energy, and fracture stress are assigned to each grain boundary according the grain boundary character distribution, lntergranular fracture susceptibility is analyzed by defining the probability of fmding a continuous path along the grain boundaries which are intrinsically susceptible to fracture. In this analysis the orientations of the grain boundary with respect to the applied or residual tensile stress axis is considered. The probability of intergranular fracture for each grain boundary depends on the intergranular fracture resistance, the interface orientation relative to the stress axis, and a value of the tensile stress acting on the grain boundary. The crack arrest distance and the fracture toughness are calculated in terms of the frequency of low-energy grain boundaries, fracture stress of low-energy grain boundary, angle distribution of grain bound ary interfaces, and anisotropy of grain shape. The results indicate that the fracture toughness increases and the crack arrest distance decreases dramatically with increasing the frequency of the low-energy grain boundaries. Lowering the grain boundary energy can improve the fracture toughness and decrease the crack arrest distance. The angle distribution of grain boundary interfaces and the grain shape factor are also very effective in controlling the fracture toughness. High fracture toughness of polycrystalline materials is related to the presence of a high frequency of low-energy boundaries which are resistant to fracture. The best fracture toughness for brittle materials can be achieved by controlling the frequencies of the low-energy grain boundaries, the grain boundary character, and the boundary inclination.
Keywords: Low-energy grain boundary, Intergranular fracture, Grain boundary character distribution, Crack arrest distance, Fracture stress, Fracture toughness.
1. Introduction It is well known that grain boundaries strongly affect the physical, chemical and mechanical properties of polycrystalline materials. Accordingly, a great deal of efforts has been devoted to understanding the structure, energetic, and properties of grain boundaries [1-5]. Grain boundary is also an important source of fracture in polycrystal and intrinsic or extrinsic brittleness of polycrystalline materials is mostly due to intergranular fracture. Watanabe [6] first introduced the concept of "grain boundary design and control," with the purpose of improving various properties of polycrystalline materials through enhancing the frequency of low-energy boundaries in the grain boundary character distribution (GBCD). An enhancement of the fraction of low energy, fracture resistant boundaries is essential to the toughening by grain boundary design a control [7]. In general, low angle (~1) boundaries and symmetrical Y~3 boundaries (twins) are particularly strong, low P~, high angle boundaries, as a group, are less strong, and high ~, high angle boundaries for brittle materials Ni3A1 are weak [8]. The premise of "grain
0924-0136/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved PI10924-0136 (96) 02346-1
boundary design and control" [9] is that the toughness of a polycrystalline material that fails intergranularly will increase with to the increase in the fraction of strong boundaries. Therefore it is essential that the types of strong boundaries be identified in a statistically significant manner in the first place [8]. In this report, we present the modeling results of the influence of grain boundary on intergranular fracture arrest in brittle materials. In the model, the material is represented by a twodimensional microstructure with non-regular polygonal grains which can be obtain from experiments or Monte Carlo simulation. Microstructural effects are included by varying the average grain sizes, grain boundary energies, grain boundary distributions, and grain boundary geometry distributions.
2. Microstructural model
The idea of Markov chain fracture model is based on a simple geometric model which was recently proposed to evaluate the potential effects of grain boundary design and control on
306
,1. Lu, J.A. Szpunar /Journal of Materials Processmg Technology 60 ('1996) 305-310
intergranular fracture and intergranular stress corrosion [3]. In this simple geometric model, the susceptibility of a bulk material to failure by stress corrosion cracking is defined in terms of the probability of propagating a crack, by the combined action of stress and corrosion, through a l-mite distance. This length defines a limit, beyond which, crack propagation would continue to component failure even in the absence of either active grain boundary paths or a corrosive environment; its magnitude is primarily dependent on the specific stress state and wall thickness of the component. Intergranular stress corrosion cracking susceptibility can thus be considered in terms of the probability of finding a continuous path, existing the distance of the critical crack length, consisting entirely of intrinsically susceptible grain boundary segments, and each of which is favorably oriented to the direction of the applied or residual tensile stress. A susceptible grain boundary segment can be defined as a crystal interface (planar), which as a result of its structural and/or chemical characteristics, is prone to enhanced corrosion (relative to the lattice) in the specific environment, and/or preferential sliding or fracture under the local operative stress condition. In the Markov chain fracture model, however, a realistic geometric microstructure and a realistic grain boundary character distribution were considered. To represent the microstructure, we have adopted the Monte-Carlo simulation method developed by Anderson and Srolovitz et al. [10-11], and accordingly the microstructure is described by a matrix. In results obtained from the Monte-Carlo simulation we defined the grain boundary as the direct link between the nearest triple points within a grain and then the grain boundary character and energy are assigned to each grain boundary according the grain boundary distribution. The grain boundary character distribution could also be obtained from the experiment or the one which could be used to obtain the desire properties of polycrystalline materials. In addition, the probability of intergranular fracture at a given triple junction is calculated according to the fracture resistances of grain boundaries at this triple junction. On the basis of geometric considerations, intergranular crack arrest (i.e., crack blunting or transition to transgranular crack propagation) can be considered to occur at a triple junction when the probability of crack continuation along either of the two available intergranular paths becomes negligible. In the Markov chain fracture model, by considering both orientations of the grain boundary plane relative to the stress axis and intrinsic character of grain boundary, the probability of crack front extension for a grain boundary at a given triple junction can be shown to be given by,
1, P
~,(0)-> ~aB
exp(~(0) _ 1), ~(0)<~oB
(1)
O'GB
applied tensile stress on this grain boundary (i.e., this grain boundary path will allow forward advance of the crack propagation). When the fracture resistance of the grain boundary is larger than the applied tensile stress on the grain boundary, there is still a chance of forward advance of the crack propagation according to the calculation from Eq. (1). For a given grain boundary character distribution, the propagation of a crack through a polycrystalline microstructure started at the triple junction in the surface of the sample. This was necessary since, prior to the development of the crack, the sample was elastically homogeneous. The crack begins by propagating along the low cohesion grain boundary network until it finds itself in a situation in which both grain boundary paths at the triple points do not allow forward advance of the crack propagation, i.e., crack arrested. The vertical distance from this triple point to the start point in the surface is called crack arrested distance. If the crack path propagates through the whole sample, the crack arrested distance is equal to the sample's width and this sample is definite failure. We have to calculate all crack paths with possible initial crack at triple points in the surface of the sample. In addition, for a given grain boundary character distribution, there exist many possible configurations for each grain boundary in the microstructural sample. Therefore in order to get very good statistic simulation results we have to calculate as many configurations as possible. In this investigation, we have calculated 80 configurations by using 80 different initial random seeds. In our experience, every calculated result converges to a value when 60 configurations are used. In the calculations we assume (i) no transgranular fracture and (ii) the maximum value of the applied tensile stress on the grain boundary is half of the fracture resistance of single crystal. The final normalized crack arrest distance required to bhmt 99% of intergranular fracture is calculated by
z.
do
:
do J ~ . L,y
where do is the average grain size. The first sum is taken over all possible initial cracks at triple points in the surface area and the second sum is taken over 80 different configurations of grain boundary in the microstructural sample. To investigate the fracture process in polycrystalline materials we also have to deal with the energetics of intergranular fracture and focus on energy change between the grain boundary and the surface of fractured bicrystal specimen. According to the energetics, when the intergranular fracture occurs and the plastic deformation is involved, the driving the crack propagation energy y is given by[8]
y = 2 y ~ - YGB + Yp where o(O) is the applied tensile stress on the grain boundary, 0 is the angle between the grain boundary plane and the stress axis, oas is the fracture resistance of the grain boundary. The probability of crack extension along a given grain boundary is one when the fracture resistance of this grain bolmdary is less than the
(2)
(3)
where ¥~ is the surface energy of the exposed grain boundary, and ¥GB the energy of the preexisting grain boundary, and Yv the plastic energy associated with the propagation of the microcrack.
J. Lu, J.A. Szpunar/Journal of Materials Processing Technology 60 (1996) 305-310
It is clear that the energy 7 is smaller in the case of absence of plastic deformation, than in the case where plastic deformation is present. Moreover, it should be noted that as the contribution of yp tO y decreases, in other words, as condition for the intergranular fracture is more ideal, a stronger dependence of fracture energy "/ on the grain boundary energy YcB is expected. This is an indication that the intergranular fracture becomes more important [9, 12]. Therefore, in brittle materials the fracture mode is predominantly intergranular. From Eq.(3) when intergranular fracture occurs in the absence of plastic deformation the fracture toughness of brittle material can be defined by the following expression:
(4)
and with a reference fracture toughness ~r~f = 1~1 for ¥~B=2y=. Here N is the total number of fractured grain boundaries in the crack paths.
3. Results and Discussion
We have carried out Monte Carlo simulations in 2-d with different time steps. Figure 1 shows grain size distribution as determined from this cross-section from a 2-d microstructure with 1141 grains and 3265 grain boundaries obtained from Monte Carlo simulation after 10,000 MC steps (see Figure 2). The shape of the grain size distribution function for this simulation agrees remarkably well with the experiment.
15.0
i0.0
~ 5.0
0.0 --' .0
-0.5
t.0
0.5
Loglo ( d / d 0 ) Figure 1 Grain size distribution as determined from crosssections of the microstmcture with 1141 grains and 3265 grain boundaries. Here do is the average grain size. Examples of the propagation of a crack through a polycrystalline microstructure with different fractions of lowenergy grain boundaries in a random orientation distribution of grain boundary planes are shown in Figure 2. In this case a small
307
pre-crack was nucleated at the triple point in the bottom of the sample microstructure prior to straining the sample. This was necessary since, prior to the development of the crack, the sample was elastically homogeneous. The crack begins by propagating along the low cohesion grain boundary network until it finds itself in a situation in which the grain boundary turns out too strong to be broken by the load. At this point, the driving force for crack propagation along the grain boundary network is greatly reduced, and the crack is arrested. Since the grain boundaries are weak compared with the interior of the grain, the stress field of the crack causes a new crack to nucleate at the site along grain boundary. This figure only demonstrates one crack propagation until it is arrested. The result also shows the crack arrested distance decreases dramatically with increasing the low-energy grain boundaries. Since the values of the crack arrested distance and other fracture parameters depend sensitively on the details of the microstructure in the vicinity of the pre-crack, they are statistically calculated by using 80 different initial random seeds for the random orientation distribution of grain boundary planes. Figure 3 shows the normalized crack arrest distance required to blent 99% of all crack, as a function of the low-energy grain boundary fraction. As shown in this figure, with increasing the fraction of low-energy grain boundaries by only 10% the crack length can be reduced considerably. The effect of the fracture resistance of low-energy grain boundary can also be seen in Figure 3. Increasing the fracture resistance of low-energy grain boundary can also reduce the crack arrest distance. The higher the fracture resistance of low-energy grain boundary the less low-energy grain boundaries are required to arrest the crack propagation within a certain distance, for example, a critical length of component failure. For a low fracture resistance of low-energy grain boundary the crack propagation cannot be arrested within the critical distance even there have high percentage of low-energy of grain boundaries. This critical length defines a limit, beyond which crack propagation would continue to component failure even in the absence of either active grain boundary paths or a corrosive environment [3, 13-14]. In our case if we define five grain diameters as the critical length and more than 25% of grain boundaries are of low-energy with the highest fracture resistance, the cracks formed initially cannot propagate further because of the presence of fewer random boundaries and that the polycrystalline material will show a higher ductility. This result agrees with experimental and theoretical results [15]. The grain boundary energy ¥6B is known to depend on the type and the structure of grain boundary, i.e., the boundary misorientation and inclination described by the crystallographic and geometrical parameters. We can expect that there exists a dependence of the fracture energy on the type and structure of the grain boundary. The "t'6~ is a measure of this dependence and the surface energy Ys does not change with crystallographic orientation as much as YcB. This leads to the possibility of theoretical analysis of structure-dependent intergranular fracture. It is evident that the low-energy grain boundaries are expected to need more energy to break than the high-energy grain boundaries. In other words, the low-energy grain boundaries are more resistant to fracture [9]. The influence of the low-energy grain boundary fraction and the fracture resistance of low-energy grain boundary
308
3". Lu, J.A. Szpunar / Journal of Materials Processing Technology 60 (1996) 305-310
on fracture toughness of intergranular cracks in the random orientation distribution of grain boundary plane is shown in Figure 4. Increasing the low-energy grain boundaries can increase the fracture toughness of brittle materials during the intergranular fracture. The brittle materials can also be toughened by enhancing the fracture resistance of low-energy grain boundary.
fL=O.1
(c)
fL=O.O
(a)
fL:O.3
(d)
fL=o.05
(b)
Figure 2 Examples of crack paths of intergranular fracture with various fractions of low-energy grain boundaries fL in a random orientation distribution of the grain boundary planes. Assume o~JoB=0.40 and OR=O.O10m
fL = 0.4
(e)
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.Z Lu, flA. Szpunar /Journal of Materials Processing Technology 60 (1996) 305-310
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25'
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= 0.80 aGB/0"B = 0 . 6 0 ~B/O-B = 0.40 qB/a~ = 0.30 %B/~rB = 0 . 1 5
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~/~rB = 0.05
~/aB
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I 0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
0.6
0.8
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fL
Figure 3 Normalized crack arrest distance (L/do) required to blunt 99% of intergranular cracks as a function of fraction of low-energy grain boundaries fL with different grain boundary fracture resistances o6B/o B in a random orientation distribution of grain boundary planes. Assume the random grain boundary fracture resistance OR=0.010B. Here o B is the fracture resistance of a perfect bulk crystal.
Figure 4 Fracture toughness ~/,~f of intergranular cracks as a function of fraction of low-energy grain boundaries fL with different grain boundary energies rLs (=0.5~'GJYs) in a random orientation distribution of grain boundary planes. Assume the random grain boundary energy YR=0.99*2Ys. Here ¥s is the surface energy of an exposed grain boundary.
These results suggest the importance of grain boundary design and control through material processing, whereby considerable decreasing intergranular crack propagation distance may be achieved through only moderate increases in the fraction of lowenergy grain boundary with high fracture resistance in the grain boundary character distribution of conventional brittle polycrystalline materials.
is related to the presence of a high frequency of low-energy boundaries which are resistance to fracture. The grain boundary design and control by manipulating the grain boundary character distribution (GBCD) are important to fracture toughness improvement of the brittle polycrystal materials.
4. Conclusion
This work was supported by the Natural Science and Engineering Research Council of Canada. The Hydro-Quebec Fellowship to support one of the authors (J. Lu) is acknowledged.
The intergranular fracture in brittle materials has been modeled as function of the grain boundary character distribution. The calculations have predicted that with increasing the fraction of low energy grain boundaries, the fracture toughness increases while the crack arrest distance decreases. The intergranular crack formed initially cannot propagate further because of the presence of fewer random boundaries and that the polycrystalline materials will show high ductility if we define five grain diameters as the critical length and more than 25% of grain boundaries are of lowenergy with the highest fracture resistance, such as E3 twin grain boundaries, etc. High fracture toughness of fme-grained materials
Acknowledgements
References
[1] [2] [3] [4] [5]
L.C. Lim and R. Raj, Acta Metall., 32, 1177 (1984). J. Don and S. Majundar, Acta Metall., 34, 961 (1986). G. Palumbo, P.J. King, K.T. Aust, U. Erb, and P.C. Lichtenberger, Script Metall. et Mater., 25, 1775 (1991). L.C. Lira and T. Watanabe, Acta Metall., 38, 2507 (1990). J.F. Lustko, D. Wolf, S. Yip, S.R. Phillpot, and T. Nguyen, Phys. Rev. B38, 11572 (1988).
310
[6] [7] [8] [9] [10]
J. Lu, J.A. Szpunar / Journal of Materials Processing Technology 60 (1996) 305-310
T. Watanabe, Res. Mechanica, 11, 47 (1984). T. Watanabe, Mater. Forum, 11, 284 (1988). H. Lin and D.P. Pope, Acta Metall., 41, 553 (1993). T. Watanabe, Mater. Forum, 46, 25 (1989). M.P. Anderson, D.J. Srolovitz, G.S. Grest, and P.S. Sahni, Acta Metall., 32, 783 (1984). [11] D.J. Srolovitz, M.P. Anderson, P.S. Sahni, and G.S. Grest, Acta Metall., 32, 793 (1984).
[12] T. Watanabe, Trans. Japan Inst. Met., 27, 73 (1986). [13] K.T. Aust, in Karl T. Aust International Symposium on Grain Boundary Engineering, U. Erb and G. Palumbo, eds., Kingston, Ontario, Canada, 197 (1993). [14] K.T. Aust and G. Palumbo, in Structure and Property Relationship for Interfaces, J.L. Waiter, A.H. King and K. Tangr, eds., ASM, 3 (1991). [15] T. Watanabe, Mater. Forum, 126-128, 295 (1993).