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Modeling of transformation toughening in brittle composites
Scripta METALLURGICA et MATERIALIA
Vol.
28, pp. 1393-1398, 1993 Printed in the U.S.A.
Pergamon Press Ltd
MODELING OF TRANSFORMATION TOUGHENING IN BRITTLE COMPOSITES S. P. Chen l, R. LeSar2 and A. D. Rollett 3 Theoretical Division 1, Center for Materials Science2, Materials Science and Technology Division3 Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
( R e c e i v e d F e b r u a r y 23, 1993) ( R e v i s e d March 22, 1993)
Transformation toughening by partially stabilized zirconia (PSZ) or tetragonal zirconia particles (TZP) has been used to increase the fracture toughness of brittle ceramics and intermemllics such as A1203, Y203, MoSi2 since the discovery of the transformation toughening of 7_.rO2by Garvie et al in 1975 [1-6]. Continuum and thermodynamic approaches to understanding the transformation toughening phenomena have been successful in predicting the qualitative trend of the volume fraction, f, and transformation strain dependence, e T, but quantitative agreement and the effects of the microstmcttme are not well described [7,8]. We present here the modeling results of wansformation toughening of brittle composites using a discrete micro-mechanical model (DMM) [9]. In the model, the material is represented by a two-dimensional triangular array of nodes connected by elastic springs. Microstrucmral effects arc included by varying the spring parameters for the bulk, gr#'n boundaries, and transforming particles. Using the decrease of the local stress around the crack to monttor the change of the fracture toughness, AK, we find that there is a linear dependence of toughness on the volume fraction of the transforming particles. We also find that composites with smaller transformation stresses and smaller particles are tougher than the composites with larger transformation stresses and larger particles. We have demonstrated that the discrete micromechanical model can serve as a quantitative basis for future rational design of brittle composite materials. Discrete micro-mechanical model We used a "ball-and-spring" discrete micromechanical model [9, 13, 14] to include the effects of microstructure and second-phase particle toughening phenomena. We treat the system as a two-dimensional triangular lattice of nodes, each of which is connected to its six nearest neighbors by elastic springs. The unit-less energy of this system is given by N
E = 1/2 5". i=l
6
Yj=l
kij (rij- r0ij) 2
(1)
where kij is the force constant and t0ij is the equilibrium length of the spring connecting node i to node j. Fracture occurs when the spring length, rij, exceeds the breaking length, rbij. The unstressed topology of the model is shown in Figures l(a) and l(b).
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The microsnucture is generated with a Ports model [ 10], which has been shown to yield realistic grain-size distributions [11]. A typical section is shown in Fig. l(b). The line marking the region between grains is a grain boundary. S represents a second phase particle. The key to the utility of the model is that springs can be assigned different parameters depending on whethe~ they connect nodes within the same grain, nodes in different grains, or nodes connected to second-phase particles. Thus, we can model competing fracture mechanisms. We assume that all bulk gra~s axe alike and that there is no orienmtional dependence in the grain-boundary proL~'des. Thus, all bulk springs (e. g. the 1-1 interaction in Fig. 1Co))ate defined by the three spring parameters kb, rob, and rbb. The properties of springs crossing grain boundaries (e.g., 1-2 in Fig. l(b)) axe given by kgb, r0gb, and rbgb . Similarly, springs connected to second-phase particles (e.g., S-1 in Fig l(b)) have properties k s, r0s, and rbs . Transformation of a second-phase particle is modeled by irreversibly increasing the equilibrium spring length tOs to r0,T s when the mean stress on the second-phase particle exceeds a prescribed critical stress, Oc. For all the calculations described here, kb=kgbfks=l =r0b=rOgb=r0 s. Based on a typical experimental value for ZrO2, we assume that there is a 1% increase in the length of the springs (eT=0.01) connected to second phase particles upon transformation, i.e., r0,Ts=l.01. The breaking strain for the bulk bonds is rbb=l.O05 and for the grain boundaries and second phase particles rbgb-~bs=l.003. These choices of the strength at interfaces can be related to the calculated cohesive strength at grain boundaries and general interfaces [15, 16]. We note that this choice of breaking strains ensures that there should be a mixture of trans- and intergranular fracuLre [12-14]. Finally, we choose a mean stress criterion for transformations with critical stresses Oci of 0.0005, 0.0010, and 0.0015 (for i equal to 1, 2 and 3, respectively) to test the effect of ~c on the enhancement of the fracture toughness. Also the second phase particles are randomly distributed from 0 vol. % up to 45 vol. % in the matrix with a particle size of 1 node (labelled as Si in the figures, where i repres~ting composites with a transformation stress of rico or cluster of 7 nodes (labelled as Li in the figures) to study the microslrnctural effects on the fracture toughness. The average grain size shown in Fig. 2 is 35 nodes. Therefore the relative linear size of the secondphase particle to the average grain size, Rp/g, is about 0.17 and 0.45, respectively.
grain bomd~ry (gb)
second-phaseparticle I
freesurface /~/ • 'C
boundary =
1
/"--'.4~ applied
crack /
freesurface
(x)
su~in (a)
1. I A A
1 ~
o
o\
,
1
1 ~2
l
1 - ~ 2 1
2
2
/
2
2 2
2 2
(b)
Fig. l(a). Topology of discrete micromechanical model. The top and bottom are free surfaces and the sides have periodic boundary conditions. An initial crack of varying length is created at the bottom edge. The system is su'ained by pulling in the x-direction. (b). Typical section of lattice of nodes. The lines drawn between areas with different numbers define grain boundaries. S represents a second-phase particle. The calculations proceed as follows. An initial dislribution of grain boundaries and second-phase particles is assigned to the system, which is then strained in the x-direction (Fig. l(a)) by a finite amount, 0.0001. The energy of the system is minimized with respect to the positions of the nodes. If the mean stress on any secondphase particles is greater than Oc, then we transform the particle with the largest stress and minimize the energy with respect to positions again. We iterate until no more particles transform. We then check to see if any springs have a length greater than the txeaking length. If so, we ineve~'sibly break the spring with the largest ratio of rij to ~)ij and minimize the energy again. We iterate until no more springs break. We then check to see if any particles
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transform, and so on. Once no more particles transform and no more springs break, the strain is increased and the procedure is repeated. Results and Discussion
The fractured polycrystals with no particles ~ d partially fl"actu~i polycr.y,stal s width small pa~i'cles .... t (f=0.28) and large particles (f=0.30) are shown in Fl.g. 2 (a), (b).~.a .~c),resp~.uve~y. [nc po~ycrys~s ~ .u~_u second-phase particles crack along the grain boundaries a~er an mmaJ propagauon o~ mmsgranmar crao: trsg. 2(a)). The composites with 28% ~ ~ volume fraction of small and large second-phase .pa~."cles do not . fracture along grain boundaries (Fig. (b), (c)). The change of the fracture path is aocompamed by an increase m fracture toughness, AK (Fig. 3 and 4), as the transforming second-phase particles shield the crack by exerting compressive stresses on the original crack.
.....
~
(a) f=0.0
+ ~g~'" :~',~
+ ~7'
(b) $2 f----0.28
" ~+ ~ + : ;
""
(c) L2 f=0.30
Fig. 2: (a) Polycrystals with 0% second-phase particles fracture along grain boundaries after a short propagation of a transgranular crack. (b) Composites with 28% small transformable second-phase particles ($2) and (c) with 30% large particles (1,2) are much tougher than the original polycrystals. All snapshots of the polycrystals under stresses were taken at the strain step of 0.0018. Light gray areas are untransformed particles, solid lines are grain boundaries, wide gray lines are cracks, and dark spots are transformed particles. Because of the small size of the simulation system (50x50 nodes), the stress measured over the whole composite represents the local stress around the crack (2x10 nodes) (Fig. 3). The local stress, OL, for composites with second phase particles is related to the local stress intensity, KL, as
oL(r) =
KL ............ (~ r) 1/2
(2)
F(0, ~)
where, KL is the local stress intensity, r is the distance from the crack tip, and F(O, ~) is the angular dependence [2]. The same formula can be written between the far-field stress intensity, K** and stress, o**. The stress-strain curve of the system with 0%, 28% (small) and 30% (large) particles is shown in Fig.
4
p.
,e showa
7
.ss
~hi~ldin~ occurs due to tile secona phase parllCles, ixte loom nu~n~ mt~,*n**3, ~ ............ , "• 7 ---:--° ~'-- -'r'k, ,-~,v/,,i ©v,tpmwn~ cracklxl comvletelv at a strain level of 0.0018 (Fsg. zta)), lne enncau lntcnslty, ~ Jr,,.. v,,~ . . . . ., . . . . . . . . . *. • " n~trix. stress at the fracture is 0.00157. Therefore, thts value ~s propomonal to the Ke of the untougbened brittle
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K**
Figure 3. A schematic plot of the stress field, a, as a fun~on of the distance, r, from the crack tip. The stress field can he described by the local and far field stress intensities,KL and K ~ Thick linerepresents actualsUess and thinlinesrepresentlocaland farfields. The distance rc rmrks the transition from local field to far field behavior.
L
rc
r
As seen from Fig. 4(a), we find that the system with 28% and 30% small/large second phase particles have lower local stress than the one for the matrix without second phase particles. The reduction of the stress over the block of nodes can he monitored as a function of the number of particles transformed as the strain (or stress) is applied (Fig. 4(b)). The increase of the fracture toughness is described by
t~K = K,,o- K L
= a (o~- O'L)= a (-AOxx)
(3)
where ct is a constant that depends on the simulation cell. This correlation of the stress change and the AK enable us to quantify the change of the fracture toughness, AK, unambiguously compared to the previous use of effective compliance or width of the transformation zone [9]. The dependence of AK on the volume fraction, f, the critical stresses, ~c, and particle sizes, Rp/g, are plotted in Fig. 4(c) and (d). The effects of the transformation stress, ac, on the change of fracture toughness were studied. We found thatcomposites with a smallertransformationstress¢~c(Sl, 2 and LI, 2 in Fig.4), increasethe toughness much fasterthan composites with largertransformationstress($3 and L3 in Fig.4(b),(c)).W e found thatthe increase of the fracturetoughness,AK, isproportionalto the volume fraction,f,of the transformingsecond phase particles and roughly proportionalto the inverseof the n'ansformafionstress,¢~c-I for SI and $2 (orLI and L2) composites as describedin the followingequation [2,7,8]:
AK=
cfac "I = c' f (h)I/2
(4)
where c and c' are parameters that are linear in the transformation strain [2], e T, but are independent of the transformation stress, Oc, and the volume fraction of the transforming phase, f. The transition from critical stress dependence, ac -1, to the process zone size dependence, h 1/2, of AK is straightforward from eq. (2). The AK value for composites with the largest ac ($3 and L3 in Fig 3(c) and (d)) are lower than the value derived from eq. (4).Therefore,eq. (4) overestimatesA K when Oc is largeand is not validfor the materialswith high ac. The largest increase in AK/Kc is about 2 to 3 in the wesent calculations, which indicates that the parameters are in the right range for most of the composites toughened by 7-,rO2in intetmetallics, oxides or silicides such as alumina or MoSi2 [4]. The microcracking which usually accompanies the transfona~ed particles, due to the local d ilnta6on (eT>0), is the sang as previous modeling [9] and is commonly seen in experiments [3]. For larger particle size composites, sometimes only partial transformation is achieved. This partial transformation
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phenomenon (hence less particles transformed as seen in Fig. 2(c) and 4(b)) can explain the lower fracture toughness increase for the composites with larger particles. This result also indicates that for the same amount of secoad phase pmigles the composites with mine randomly disaibuted particles (the ones with small particles) are tougher (by 10 to 2 0 ~ ) than the ones that are clustered in clumps [3].
o.o1~
0.0015 -a:fffiO.O
-b:S2 f,ffiO.28
/ / -
- b:S2 f = 0 . 2 8
a~'~
0.001
0.001
H
t~ .
0.0006
-C
-0.001 -0
0.0005 0.001 0.0015 E
~.oo2
-
50
100
150
200
Nt
(a)
(h)
3"
•
dK~ct~
0
dg/Kc~"
J Sl ~
J
2
2
1
1
"= ..y.z
.<
SS~L.. 0.0
0.1
0.2
0.3
0.4
odl 0.0
O.S
f
(c)
0.1
0.2
0.3
0.4
0.5
f*
(d)
Fig. 4: (a) The stress-strain curves of Fig. 2. (b) The dependence of the stress change, -AOxx (or equivalently, AK) on the number of particles that have wansformed. This Fig. indicates that we are in a subcrificai wansfo~tion toughening regime. (c): The composite with a smaller wansformatiou stress for the transformation to occur in the second phase materials (oc1=0.0005: solid circles S1) is tougher than the composite with a larger transformation stress (Oc2=0.0010: solid squares $2; oc3=0.015 : solid triangles $3). The composites with smaU particle sizes (S, Rp/g=0.17) are also tougher than the composites with larger particle sizes (L, Rp/g=0.45) (solid symbols). (d) The AK is plotted vs normalized volume fraction, f*--fi(ocl/oci), i=1, 2 or 3. Results for composites $1 and $2 agree with eq.(4), while results from composites $3 do not agree with ~1. (4).
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Using the discrete micromechanical model (DMM), we found three major points in the present studies of the subcriticai transformation toughening in brittle composites. (1): The incw.a~ in fracture toughness is proportional to the volume fraction, f, of the second phase transforming particles. (2): The increase in fi'actme toughness is inversely proportional to the critical transformation stress for composites with small transformation stresses. However, this relation is not valid for composites with large fie. Reducing the critical stress of the transformation is the most effective way to increase the fracture toughness. Optimization of the fracture toughness may be achieved by changing the alloy contents (e.g. Y203) of 7--'02 particles as seen for MoSi2 composites [4]. (3): The composites with smaller particles are tougher than the composites with larger particles. This effect is more pronounced for composites with a higher volume fraction of second-phase particles. Acknowledgement
We would like to acknowledge J. J. Petrovic, I. Reimanis, and G. A. Baker, Jr. for helpful discussions and the support of the Advanced Industrial Concepts Materials program of the U. S. Department of Energy. References
1. R. C. Garvie, R. H. J. Hannink and R. T. Pascoe, Nature (London), 258, 703 (1975). 2. A. G. Evans, J. Am. Ceram. Soc., 73 (2) 187-206 (1990), and references cited therein. 3. D. J. Careen, R. H. J. Hannink, and M. V. Swain, "Transformation Toughening of Ceramics", CRC Press, Boca Raton, Florida (1989). 4. J. J. Petrovic, A. K. Bhattacharya, R. E. Honnell, T. E. Mitchell, R. K. Wade, and K. J. McClellan, Mat. Sci. and Eng., A155, 259 (1992). 5. M. V. Swain and L. R. F. Rose, J. Am. Ceram. Soc., 69 (7) 511-518 (1986). 6. R. Dauskhardt, D. K. Veirs and R. O. Ritchie, J. Am. Ceram. Soc., 72 (7) 1124-1130 (1989). 7. R. M. McMeeking and A. G. Evans, J. Am. Ceram. Soc., 65 (5) 242-247 (1982). 8. B. Budiansky, J. W. Hutchinson, and J. C. Lambropolous, Int. J. Solids Struct., 19 (4) 337-355 (1983); D. M. Stump, Phil. Mag. A, 64, 879 (1991). 9. R. LeSar, A. D. Rollett, and D. J. Srolovitz, Mat. Sci. and Eng. A155, 267 (1992). 10. F. Y. Wu, Rev. Mod. Phys., 54, 235 (1982). 11. M. P. Anderson, D. J. Srolovitz, G. S. Grest, and P. S. Sahni, Acta MetaU., 32, 783 (1984). 12. W. H. Yang, D. J. Srolovitz, G. N. Hassold and M. P. Anderson, in "Simulation and Theory of Evolving Microstructures," Edited by M. P. Anderson and A. D. RoUett, (Minerals, Metals, and Materials Society, Warrcndale, PA, 1990), 277. 13. S. Ling and M. P. Anderson, MRS Proc. "Materials Theory and Modelling" (1993). 14. L. Monette and M. P. Anderson, Modeling in Mat. Sci. and Eng., (1993). 15. S. P. Chen, Phil. Mag. A, 66, I (1992); Mater. Sci. Engng., B, 6, 113 (1990). 16. S. P. Chen, A. F. Voter, A. M. Boring, R. C. Albers, and P. J. Hay, Scripta MetaU., 23, 217 (1989); J. Mater. Res., 5, 955 (1990).
Vol.
28, pp. 1393-1398, 1993 Printed in the U.S.A.
Pergamon Press Ltd
MODELING OF TRANSFORMATION TOUGHENING IN BRITTLE COMPOSITES S. P. Chen l, R. LeSar2 and A. D. Rollett 3 Theoretical Division 1, Center for Materials Science2, Materials Science and Technology Division3 Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
( R e c e i v e d F e b r u a r y 23, 1993) ( R e v i s e d March 22, 1993)
Transformation toughening by partially stabilized zirconia (PSZ) or tetragonal zirconia particles (TZP) has been used to increase the fracture toughness of brittle ceramics and intermemllics such as A1203, Y203, MoSi2 since the discovery of the transformation toughening of 7_.rO2by Garvie et al in 1975 [1-6]. Continuum and thermodynamic approaches to understanding the transformation toughening phenomena have been successful in predicting the qualitative trend of the volume fraction, f, and transformation strain dependence, e T, but quantitative agreement and the effects of the microstmcttme are not well described [7,8]. We present here the modeling results of wansformation toughening of brittle composites using a discrete micro-mechanical model (DMM) [9]. In the model, the material is represented by a two-dimensional triangular array of nodes connected by elastic springs. Microstrucmral effects arc included by varying the spring parameters for the bulk, gr#'n boundaries, and transforming particles. Using the decrease of the local stress around the crack to monttor the change of the fracture toughness, AK, we find that there is a linear dependence of toughness on the volume fraction of the transforming particles. We also find that composites with smaller transformation stresses and smaller particles are tougher than the composites with larger transformation stresses and larger particles. We have demonstrated that the discrete micromechanical model can serve as a quantitative basis for future rational design of brittle composite materials. Discrete micro-mechanical model We used a "ball-and-spring" discrete micromechanical model [9, 13, 14] to include the effects of microstructure and second-phase particle toughening phenomena. We treat the system as a two-dimensional triangular lattice of nodes, each of which is connected to its six nearest neighbors by elastic springs. The unit-less energy of this system is given by N
E = 1/2 5". i=l
6
Yj=l
kij (rij- r0ij) 2
(1)
where kij is the force constant and t0ij is the equilibrium length of the spring connecting node i to node j. Fracture occurs when the spring length, rij, exceeds the breaking length, rbij. The unstressed topology of the model is shown in Figures l(a) and l(b).
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The microsnucture is generated with a Ports model [ 10], which has been shown to yield realistic grain-size distributions [11]. A typical section is shown in Fig. l(b). The line marking the region between grains is a grain boundary. S represents a second phase particle. The key to the utility of the model is that springs can be assigned different parameters depending on whethe~ they connect nodes within the same grain, nodes in different grains, or nodes connected to second-phase particles. Thus, we can model competing fracture mechanisms. We assume that all bulk gra~s axe alike and that there is no orienmtional dependence in the grain-boundary proL~'des. Thus, all bulk springs (e. g. the 1-1 interaction in Fig. 1Co))ate defined by the three spring parameters kb, rob, and rbb. The properties of springs crossing grain boundaries (e.g., 1-2 in Fig. l(b)) axe given by kgb, r0gb, and rbgb . Similarly, springs connected to second-phase particles (e.g., S-1 in Fig l(b)) have properties k s, r0s, and rbs . Transformation of a second-phase particle is modeled by irreversibly increasing the equilibrium spring length tOs to r0,T s when the mean stress on the second-phase particle exceeds a prescribed critical stress, Oc. For all the calculations described here, kb=kgbfks=l =r0b=rOgb=r0 s. Based on a typical experimental value for ZrO2, we assume that there is a 1% increase in the length of the springs (eT=0.01) connected to second phase particles upon transformation, i.e., r0,Ts=l.01. The breaking strain for the bulk bonds is rbb=l.O05 and for the grain boundaries and second phase particles rbgb-~bs=l.003. These choices of the strength at interfaces can be related to the calculated cohesive strength at grain boundaries and general interfaces [15, 16]. We note that this choice of breaking strains ensures that there should be a mixture of trans- and intergranular fracuLre [12-14]. Finally, we choose a mean stress criterion for transformations with critical stresses Oci of 0.0005, 0.0010, and 0.0015 (for i equal to 1, 2 and 3, respectively) to test the effect of ~c on the enhancement of the fracture toughness. Also the second phase particles are randomly distributed from 0 vol. % up to 45 vol. % in the matrix with a particle size of 1 node (labelled as Si in the figures, where i repres~ting composites with a transformation stress of rico or cluster of 7 nodes (labelled as Li in the figures) to study the microslrnctural effects on the fracture toughness. The average grain size shown in Fig. 2 is 35 nodes. Therefore the relative linear size of the secondphase particle to the average grain size, Rp/g, is about 0.17 and 0.45, respectively.
grain bomd~ry (gb)
second-phaseparticle I
freesurface /~/ • 'C
boundary =
1
/"--'.4~ applied
crack /
freesurface
(x)
su~in (a)
1. I A A
1 ~
o
o\
,
1
1 ~2
l
1 - ~ 2 1
2
2
/
2
2 2
2 2
(b)
Fig. l(a). Topology of discrete micromechanical model. The top and bottom are free surfaces and the sides have periodic boundary conditions. An initial crack of varying length is created at the bottom edge. The system is su'ained by pulling in the x-direction. (b). Typical section of lattice of nodes. The lines drawn between areas with different numbers define grain boundaries. S represents a second-phase particle. The calculations proceed as follows. An initial dislribution of grain boundaries and second-phase particles is assigned to the system, which is then strained in the x-direction (Fig. l(a)) by a finite amount, 0.0001. The energy of the system is minimized with respect to the positions of the nodes. If the mean stress on any secondphase particles is greater than Oc, then we transform the particle with the largest stress and minimize the energy with respect to positions again. We iterate until no more particles transform. We then check to see if any springs have a length greater than the txeaking length. If so, we ineve~'sibly break the spring with the largest ratio of rij to ~)ij and minimize the energy again. We iterate until no more springs break. We then check to see if any particles
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transform, and so on. Once no more particles transform and no more springs break, the strain is increased and the procedure is repeated. Results and Discussion
The fractured polycrystals with no particles ~ d partially fl"actu~i polycr.y,stal s width small pa~i'cles .... t (f=0.28) and large particles (f=0.30) are shown in Fl.g. 2 (a), (b).~.a .~c),resp~.uve~y. [nc po~ycrys~s ~ .u~_u second-phase particles crack along the grain boundaries a~er an mmaJ propagauon o~ mmsgranmar crao: trsg. 2(a)). The composites with 28% ~ ~ volume fraction of small and large second-phase .pa~."cles do not . fracture along grain boundaries (Fig. (b), (c)). The change of the fracture path is aocompamed by an increase m fracture toughness, AK (Fig. 3 and 4), as the transforming second-phase particles shield the crack by exerting compressive stresses on the original crack.
.....
~
(a) f=0.0
+ ~g~'" :~',~
+ ~7'
(b) $2 f----0.28
" ~+ ~ + : ;
""
(c) L2 f=0.30
Fig. 2: (a) Polycrystals with 0% second-phase particles fracture along grain boundaries after a short propagation of a transgranular crack. (b) Composites with 28% small transformable second-phase particles ($2) and (c) with 30% large particles (1,2) are much tougher than the original polycrystals. All snapshots of the polycrystals under stresses were taken at the strain step of 0.0018. Light gray areas are untransformed particles, solid lines are grain boundaries, wide gray lines are cracks, and dark spots are transformed particles. Because of the small size of the simulation system (50x50 nodes), the stress measured over the whole composite represents the local stress around the crack (2x10 nodes) (Fig. 3). The local stress, OL, for composites with second phase particles is related to the local stress intensity, KL, as
oL(r) =
KL ............ (~ r) 1/2
(2)
F(0, ~)
where, KL is the local stress intensity, r is the distance from the crack tip, and F(O, ~) is the angular dependence [2]. The same formula can be written between the far-field stress intensity, K** and stress, o**. The stress-strain curve of the system with 0%, 28% (small) and 30% (large) particles is shown in Fig.
4
p.
,e showa
7
.ss
~hi~ldin~ occurs due to tile secona phase parllCles, ixte loom nu~n~ mt~,*n**3, ~ ............ , "• 7 ---:--° ~'-- -'r'k, ,-~,v/,,i ©v,tpmwn~ cracklxl comvletelv at a strain level of 0.0018 (Fsg. zta)), lne enncau lntcnslty, ~ Jr,,.. v,,~ . . . . ., . . . . . . . . . *. • " n~trix. stress at the fracture is 0.00157. Therefore, thts value ~s propomonal to the Ke of the untougbened brittle
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K**
Figure 3. A schematic plot of the stress field, a, as a fun~on of the distance, r, from the crack tip. The stress field can he described by the local and far field stress intensities,KL and K ~ Thick linerepresents actualsUess and thinlinesrepresentlocaland farfields. The distance rc rmrks the transition from local field to far field behavior.
L
rc
r
As seen from Fig. 4(a), we find that the system with 28% and 30% small/large second phase particles have lower local stress than the one for the matrix without second phase particles. The reduction of the stress over the block of nodes can he monitored as a function of the number of particles transformed as the strain (or stress) is applied (Fig. 4(b)). The increase of the fracture toughness is described by
t~K = K,,o- K L
= a (o~- O'L)= a (-AOxx)
(3)
where ct is a constant that depends on the simulation cell. This correlation of the stress change and the AK enable us to quantify the change of the fracture toughness, AK, unambiguously compared to the previous use of effective compliance or width of the transformation zone [9]. The dependence of AK on the volume fraction, f, the critical stresses, ~c, and particle sizes, Rp/g, are plotted in Fig. 4(c) and (d). The effects of the transformation stress, ac, on the change of fracture toughness were studied. We found thatcomposites with a smallertransformationstress¢~c(Sl, 2 and LI, 2 in Fig.4), increasethe toughness much fasterthan composites with largertransformationstress($3 and L3 in Fig.4(b),(c)).W e found thatthe increase of the fracturetoughness,AK, isproportionalto the volume fraction,f,of the transformingsecond phase particles and roughly proportionalto the inverseof the n'ansformafionstress,¢~c-I for SI and $2 (orLI and L2) composites as describedin the followingequation [2,7,8]:
AK=
cfac "I = c' f (h)I/2
(4)
where c and c' are parameters that are linear in the transformation strain [2], e T, but are independent of the transformation stress, Oc, and the volume fraction of the transforming phase, f. The transition from critical stress dependence, ac -1, to the process zone size dependence, h 1/2, of AK is straightforward from eq. (2). The AK value for composites with the largest ac ($3 and L3 in Fig 3(c) and (d)) are lower than the value derived from eq. (4).Therefore,eq. (4) overestimatesA K when Oc is largeand is not validfor the materialswith high ac. The largest increase in AK/Kc is about 2 to 3 in the wesent calculations, which indicates that the parameters are in the right range for most of the composites toughened by 7-,rO2in intetmetallics, oxides or silicides such as alumina or MoSi2 [4]. The microcracking which usually accompanies the transfona~ed particles, due to the local d ilnta6on (eT>0), is the sang as previous modeling [9] and is commonly seen in experiments [3]. For larger particle size composites, sometimes only partial transformation is achieved. This partial transformation
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TRANSFORMATION TOUGHENING
1397
phenomenon (hence less particles transformed as seen in Fig. 2(c) and 4(b)) can explain the lower fracture toughness increase for the composites with larger particles. This result also indicates that for the same amount of secoad phase pmigles the composites with mine randomly disaibuted particles (the ones with small particles) are tougher (by 10 to 2 0 ~ ) than the ones that are clustered in clumps [3].
o.o1~
0.0015 -a:fffiO.O
-b:S2 f,ffiO.28
/ / -
- b:S2 f = 0 . 2 8
a~'~
0.001
0.001
H
t~ .
0.0006
-C
-0.001 -0
0.0005 0.001 0.0015 E
~.oo2
-
50
100
150
200
Nt
(a)
(h)
3"
•
dK~ct~
0
dg/Kc~"
J Sl ~
J
2
2
1
1
"= ..y.z
.<
SS~L.. 0.0
0.1
0.2
0.3
0.4
odl 0.0
O.S
f
(c)
0.1
0.2
0.3
0.4
0.5
f*
(d)
Fig. 4: (a) The stress-strain curves of Fig. 2. (b) The dependence of the stress change, -AOxx (or equivalently, AK) on the number of particles that have wansformed. This Fig. indicates that we are in a subcrificai wansfo~tion toughening regime. (c): The composite with a smaller wansformatiou stress for the transformation to occur in the second phase materials (oc1=0.0005: solid circles S1) is tougher than the composite with a larger transformation stress (Oc2=0.0010: solid squares $2; oc3=0.015 : solid triangles $3). The composites with smaU particle sizes (S, Rp/g=0.17) are also tougher than the composites with larger particle sizes (L, Rp/g=0.45) (solid symbols). (d) The AK is plotted vs normalized volume fraction, f*--fi(ocl/oci), i=1, 2 or 3. Results for composites $1 and $2 agree with eq.(4), while results from composites $3 do not agree with ~1. (4).
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TRANSFORMATION TOUGHENING
Vol.
28, No. 11
Using the discrete micromechanical model (DMM), we found three major points in the present studies of the subcriticai transformation toughening in brittle composites. (1): The incw.a~ in fracture toughness is proportional to the volume fraction, f, of the second phase transforming particles. (2): The increase in fi'actme toughness is inversely proportional to the critical transformation stress for composites with small transformation stresses. However, this relation is not valid for composites with large fie. Reducing the critical stress of the transformation is the most effective way to increase the fracture toughness. Optimization of the fracture toughness may be achieved by changing the alloy contents (e.g. Y203) of 7--'02 particles as seen for MoSi2 composites [4]. (3): The composites with smaller particles are tougher than the composites with larger particles. This effect is more pronounced for composites with a higher volume fraction of second-phase particles. Acknowledgement
We would like to acknowledge J. J. Petrovic, I. Reimanis, and G. A. Baker, Jr. for helpful discussions and the support of the Advanced Industrial Concepts Materials program of the U. S. Department of Energy. References
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