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A simple mechanical model to predict fracture and yield strengths of particulate two-phase materials
Computational Materials Science 15 (1999) 11±21
A simple mechanical model to predict fracture and yield strengths of particulate two-phase materials Zhonghua Li a, S. Schmauder b, M. Dong a b
b
Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200030, People's Republic of China Staatliche Materialpr ufugsanstalt (MPA), Universit at of Stuttgart, Pfaenwaldring 32, D-70569 Stuttgart, Germany Received 28 October 1998; received in revised form 28 January 1999; accepted 11 February 1999
Abstract A simple mechanical model is developed and the following basic characteristics of fracture and yield strengths of particulate two-phase materials are predicted based on this model: (1) The rule of mixture can be approximately used in the particulate two-phase materials with ®ne particles; (2) For a material of a soft (hard) matrix with hard (soft) particles, the mixture law is upper (lower) bound, the lower (upper) bound is dependent on the ratio of strength and volume fraction of the constituent phases; (3) Fine (coarse) size of particle phase increases the strength of the material of a soft (hard) matrix with hard (soft) particles; (4) For mixed microstructure, the real transition of strength during change between soft and hard matrix should vary in a range of the upper bound for the material of a hard matrix with soft particles and the lower bound for that of a soft matrix with hard particles. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Two-phase material; Fracture and yield strengths; Mixture law; Lower and upper bounds
1. Introduction A number of technically important alloys consist of soft and hard phases. Examples of this type of alloy are two-phase steels [1±4,7,8] and aÿb titanium alloys [5,6]. As a group, these alloys oer useful combinations of high strength and good ductility, and have been extensively studied in the past, both experimentally [1±8] and numerically [5,6,10,22,28±32]. One of the unswerving eorts in these studies is to develop a theoretical model to predict the composite ¯ow behavior from those of the constituent phases [5,9±23]. In principle, these theoretical models fall into three categories, i.e., (1) micro mechanical model based on the dislocation
theory [9], (2) continuum mechanical models [10±13] and (3) empirical mixture laws [6,15±22]. A notable feature of the ¯ow behavior of twophase materials is that when the volume fraction of the second particle phase reduces and approaches zero, the ¯ow behavior of the two-phase material approximates that of the single matrix-phase material. Due to this simple and trivial rule, experimental data of two-phase materials always scatter along the line which links the cases of two single component phases, i.e., the line de®ned by the rule of mixture raÿb fa ra fb rb ; where raÿb is the overall average stress of the twophase material, ra ; rb and fa ; fb are the stresses
0927-0256/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 9 9 ) 0 0 0 1 4 - 2
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Z. Li et al. / Computational Materials Science 15 (1999) 11±21
measured in bulk and volume fractions of the indicated phases, respectively. The rule of mixture was derived based on the assumption of interphase isostrain condition. It is, therefore, an exact analytical solution only for longitudinal loaded long-®bre composites with elastic constituents. In general, the rule of mixture is not expected to be applicable for the particulate reinforced composites for the simple reason that the isostrain condition rarely exists. However, several investigations [1±4] showed that for the particulate two-phase materials, the mixture law can be approximately applied, while others [5±9] showed that the mixture law is not applicable. For these dierent arguments based on the magnitude of the scattering bound, there is no strict demarcation, because nobody has developed a thorough demonstration for the necessary condition of the mixture law and clari®ed the rule that the scattering bound should be followed for the particulate two-phase materials. Because all the tested data scatter along the line de®ned by the mixture law, much eort has been paid to make a rationalization of the mixture law in order to ®t the tested data. Ankem and Margolin [5] introduced an additional interaction term between soft and hard phases. Fischmeister and Karlsson [17] proposed an intermediate mixture law, in which the relevant stresses (or strains) are the average values of in situ phases, not the bulk ones. Li and Gu [15] introduced plastic constraint factors to modify the bulk ¯ow stresses used in the mixture law. Siegmund et al. [22] introduced a stereological parameter to incorporate the eect of geometrical continuity of the constituent phases. Rizk and Bourell [16] introduced an additional dislocation contribution to the ferrite yield strength due to martensite transformation in the mixture law of yield stress for dual phase steels. Although these eorts provide, from dierent respects, some insight into understanding the ¯ow behavior of two-phase materials, these modi®ed mixture laws are dicult to use in practice, because all of these modi®cations are based on an understanding of the involute interactions between soft and hard phases, which depends on the details of the microstructure (volume fraction, size, shape, continuity of the hard and soft phases) and the properties of the component phases.
Due to the complex microstructure of twophase materials, some idealized and restricted assumptions must be made to develop a theoretical model. However, if the developed model based on these assumptions does not contain the major characterizing parameters which dominate the macro ¯ow behavior of two-phase materials, then the model cannot outline the fundamental characters of two-phase materials. From numerous experimental investigations [1±8] and FE-simulations [28±32], it has been known that in two-phase materials, the ratio of strengths of the soft and hard phases, volume fraction, size, shape and continuity of the component phases are the dominant parameters. But up to now, none of the theoretical models contains all these parameters. The present study focuses our attention on developing a simple mechanical model to predict the most important strength parameters, namely, yield and fracture strengths of two-phase materials, and provides simple upper and lower bounds and their dependence on particle size, continuity and strength ratio of the constituent phases. 2. Local virtual fracture model Fig. 1 shows a particulate two-phase material subjected to uniaxial stress rc . Because of the inhomogeneous stress distributions in the two phases, the stress at a local place in the two-phase material, such as at section AB, may take the lead in reaching their own fracture strengths rfm and rfp , where the subscripts m and p indicate the matrix and the second-phase particles, respectively. If the applied stress rc increases, the fracture of section AB may not take place due to a constraint caused by the surrounding material. The stresses acting on this section will not be remarkably changed during subsequent loading, because rfm and rfp are the ultimate strengths before fracture. Therefore, section AB can always be replaced by a ®ctitious crack with closure stresses rfm and rfp as shown in Fig. 1(b). For a homogeneous material, it is obvious that the closure stresses on the virtual crack surfaces must be equal to the applied stress rc , which is independent of the deformation behavior of the
Z. Li et al. / Computational Materials Science 15 (1999) 11±21
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Fig. 1. Schematic representation of the mechanical model: (a) the stresses in matrix and particle at section AB reach their respective fracture strengths; (b) mechanical model equivalent to (a).
material. This condition is equivalent to closing a Grith crack. It is therefore reasonable to assume that the condition of closing the virtual crack with inhomogeneous closure stresses could be concluded by means of closing the Grith crack. Thus, the stress intensity factor KIA (or KIB ) at the virtual crack tip A (or B) produced by rc , rfm and rfm should be zero. Otherwise, it will result in the stresses ahead of the virtual crack tips being larger than the closure stresses. This is in contradiction with our assumption that the stresses ahead of the crack tips are always less than the closure stresses. This condition should be necessary even when the virtual crack may be embedded in plastic stressstrain ®eld. Therefore, the model should be applicable both in brittle and ductile two-phase materials. During further loading, the stresses in matrix and particles ahead of the virtual crack tips will increase and reach rfm and rfp , which result in the growth of the virtual crack. Thus, there are two possibilities; one is that the stresses rfm and rfp acting on the surfaces of the advanced virtual crack can oset the stress intensity factor caused by applied stress, and the crack keeps closed. Further propagation of the virtual crack needs to increase the applied stress. The other possibility is that there must exist a critical virtual crack length
ac , at which the stresses rfm and rfp on the surfaces of the virtual crack cannot oset the stress intensity factor produced by the applied stress rc . In this case, the crack is unstable. Therefore, there exists a critical applied stress rfc corresponding to the critical virtual crack length ac . It is obvious that rfc is the fracture strength of the two-phase material and may be determined by using the equilibrium condition of the ®ctitious crack at its critical state. It should be noted that for predicting the yield and fracture strengths of the composites, the closure stress should be de®ned as eective stress in the component phase. It is lower than the stress component in loading direction in a soft matrix, and is higher than that in hard particles due to stress triaxiality [15,29]. Because the interphase stresses are self-equilibrated stress system, from the overall average eect on the equilibrium condition of the virtual crack, it is reasonable to assume that the eective stresses could approximately replace the stress components in the loading direction. The other assumption is the mean closure eective stresses used on the virtual crack surfaces, neglecting the local ¯uctuations in the stresses, for the reason that it is the mean stress which determines much of the mechanical behavior of the
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Z. Li et al. / Computational Materials Science 15 (1999) 11±21
composites. The local ¯uctuations in stress will not, since they average to zero, give rise to a remarkable macroscopic ¯ow. It will result in local ¯ow (microplasticity), and this is responsible for an early deviation from the initial linear elastic strain response to the applied loading [33,34]. Some micro failure mechanisms which will result in stress-free surfaces in the material, such as, microcracks, void nucleation and growth, interface debonding, are not regarded in this model. 3. Derivation of fracture strength As shown in Fig. 2, the particles divide the critical crack length ac into n equal segments, the lengths occupied by the particle and matrix in each segment are equal to ac Vp =n and ac Vm =n, respectively, where Vp and Vm are the volume fractions of the matrix and particle. Each segment is called a fracture element. In fracture element i, the coordinates c; e; d of the starting and end points of the particle and matrix are
iÿ1 ac n iÿ1 ac a ac V p d n n i d ac n Obviously, Eq. (a) is based on the particles being equiaxed and having the same size and distance from each other. The equilibrium condition [27] is given as p KIB rfc pac d r r n Z X rfp ac x ac ÿ x dx ÿ p ÿ x pa a ac x c c i1 c
ÿ
c e Z n X i1
0:
d
rfm p pac
r r ac x ac ÿ x dx ac ÿ x ac x b
The ®rst and second terms in the two integrals represent the stress intensity factors produced by rfp and rfm in the intervals 0 6 x 6 ac and ÿac 6 x 6 0, respectively. Integrating Eq. (b) and substituting the values of c; d and e from Eq. (a), we have n 2 fX i ÿ 1 Vp f arccos rc ÿ rp n p i1 n iÿ1 ÿ arccos n n X 2 i i ÿ 1 Vp : arccos ÿ arccos ÿ rfm n p n n i1 1
Fig. 2. Schematic representation of the equilibrium condition of the ®ctitious crack at critical state.
Eq. (1) is the general expression of the fracture strength for two-phase materials. When rfp > rfm , it indicates the case of a soft matrix with hard particles, and rfp < rfm indicates the case of a hard matrix with soft particles. For homogenous material (Vp 0; rfm rfp ; n i 1), we have rfc rfm . This is a simple stress equilibrium equation, which is always satis®ed during the whole elastic-plastic loading process before actual crack occurs. For two-phase materials with coarse microstructure, fracture may occur at a critical crack
Z. Li et al. / Computational Materials Science 15 (1999) 11±21
length ac of only one fracture element, i.e., n 1; then Eq. (1) can be simpli®ed as 2 f rm ÿ rfp arccos Vp : 2 rfc rfp p For two-phase materials with ®ne microstructure, the critical crack length of ac may include a large number of fracture elements. We take n ! 1 to represent the limiting case. If n ! 1, Eq. (b) may be written as p KIB rfc pac r r Zac p ac x ac ÿ x dx c ÿ p pac ac ÿ x ac x 0
in which dx ac =n; p is the average stress at dx. p can be exactly represented by Vp rfp Vm rfm due to dx being in®nitely small. Substituting p into Eq. (c) gives for n ! 1 rfc Vm rfm Vp rfp :
3
Eq. (3) indicates that if the size of the particles is very small, as could occur in the alloys strengthened by dispersion of hard particles, the strength of the two-phase material tends to follow the rule of mixture. If the particles are rigid, the rfp in Eqs. (1)±(3) should be replaced by the interface strength between matrix and particles. For the other limiting case of rfp 0, i.e., a matrix material with holes, it can be seen from Eqs. (1) and (2) that the material strength is greater than that predicted by use of the reduction of the net section (Eq. (3)). 4. Yielding strength of two-ductile phase materials Due to stress inhomogenous in two phases, local yielding may take place even though macro applied stress rc is less than the yield stress of the soft component phase as shown in FE-analyses [28,29]. The local yielding region is surrounded by elastic medium and the plastic deformation in this area is small before macro yielding occurs. Therefore, the stress in the local yielding region should approximately keep constant for the component phases with obvious yield point.
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Though elastic response of the material will slightly deviate from elastic limit due to local yielding, macro yield strength of the material in practice is always de®ned when extensive yielding occurs. Thus, we assume that the stresses in the particle and matrix at section AB (Fig. 1(a)) take the lead in reaching their own yield strengths, ryp and rym , respectively. They will approximately keep constant during loading in the range of rc < ryc , where ryc is the yield stress of the twoductile phase material. The action of section AB may be replaced by a crack with closure stresses ryp and rym , i.e., rfm and rfp in Fig. 1(b) can be replaced by rym and ryp , respectively. During further loading, the virtual crack grows and more materials at the surfaces of the virtual crack enter yielding and the crack keeps closure. Macro yielding of the two-phase material takes place at the moment, at which the virtual crack is unstable. Therefore, the yield stress of the two-ductile phase material, ryc , can be determined by using the equilibrium condition of the ®ctitious crack at its critical state. Because the stresses in matrix and particles ahead of the virtual crack tips are assumed to be less than rym and ryp , respectively, this model is within the framework of linear elastic fracture mechanics and should be reasonable for estimating yield stress of the material. Using the model of Fig. 2 and replacing rfm and f rp by rym and ryp , we can get the general expression of the yield stress of two-ductile phase materials: n 2 X i ÿ 1 Vp arccos ryc ÿ ryp n p i1 n iÿ1 ÿ arccos n n 2 yX i i ÿ 1 Vp : arccos ÿ arccos ÿ rm n p n n i1 4 When n 1 it gives 2 y ryc ryp rm ÿ ryp arccos Vp : p
5
When n ! 1, we have ryc Vm rym Vm rym :
6
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It should be noted that for most two-phase materials, the component phases do not show an obvious physical yielding stresses and exhibit a power hardening behavior. In this case, the yield strengths ryp and rym in Eqs. (4)±(6) should be replaced by the proof yield stress, e.g., the stresses at 0.2% (or 0.1%) plastic strain. In a strict sense, the proof yield strengths of the component phases on the surfaces of the virtual crack increase with increasing load in contrast to the earlier assumption. However, because the yield strength of the composite is also de®ned as a proof stress of 0.2% (or 0.1%) plastic strain, the stresses of the component phases on the section of the virtual crack are not remarkably enhanced during further loading until the overall plastic strain of the composite reaches 0.2% or 0.1%, i.e., Eqs. (4)±(6) can approximately be used for the two-phase materials with power hardening component phases as corroborated by experimental results shown in Section 6. 5. Basic behavior of yielding and fracture strength predicted by the model Figs. 3 and 4 show the variations of the normalized stress rfc =rfm (or ryc =rym with volume fraction of particles for the case of rfp =rfm (or ryp =rym 3.0, 5.0, 8.0 and rfp =rfm (or ryp =rym 0.8, 0.5, 0.0, respectively. In the case of rfp =rfm (or rrp =rym > 1, i.e. hard particles in a soft matrix, Eqs. (3) and (6) (the rule of mixture) give the upper bound and Eqs. (2) and (5) give the lower bound, Fig. 3. On the contrary (Fig. 4), Eqs. (2) and (5) give the upper bound and the mixture law gives the lower bound for the material of a hard matrix with soft particles. Figs. 3 and 4 show that for a given volume fraction of particles, the ®ne particles enhance the strength of the material of a soft matrix with hard particles and reduce the strength of the material of a hard matrix with soft particles. This in¯uence of the particle size predicted by the above equations is evidently reasonable, because both the reinforcement of a soft matrix by hard particles and the strength reduction of a hard matrix by soft particles are related to the interaction area between particle and matrix per
Fig. 3. Variation of the normalized fracture (or yielding) strength with volume fraction of particles for the case of rfp =rfm or ryp =rym 3:0, 5.0, 8.0 in Eqs. (1) and (4) for dierent numbers (or sizes) of particles (small n-values correspond to larger particles).
unit volume. For a given particle volume fraction, the interaction area per unit volume increases with the increasing number of particles, i.e., the reinforcement of a soft matrix by hard particles and the strength reduction of a hard matrix by soft particles increase with decreasing size of the particles. Fig. 5 describes the eect of the normalized size of particles on the normalized strength for a hard matrix with soft particles and a soft matrix with hard particles. In Fig. 5, d0 is de®ned as the size of the particle corresponding to n 1 in Eq. (1) or Eq. (4) and d corresponding to n 2; 3; 4 . . .. It can be seen from Fig. 5 that the reinforcement by hard particles in a soft matrix and strength reduction by soft particles in a hard matrix are approximately proportional to (d/d0 )1=2 . It can be concluded from Figs. 3±5 that the strength ratio of the hard and soft phases determine the possible range of the upper and lower strength bounds while the real range is dependent
Z. Li et al. / Computational Materials Science 15 (1999) 11±21
on the size of the particles. Fine particles are desired in the case of hard particles in a soft matrix while coarse particles are desired in the case of soft particles in a hard matrix.
Fig. 4. Variation of the normalized fracture (or yielding) strength with volume fraction of particles for the case of rfp =rfm or ryp =rym 0:8, 0.5, 0.0 in Eqs. (1) and (4).
Fig. 5. Eect of the normalized particle size on the strength normalized to the strength from the law of mixture (r f indicates fracture strength, r y indicates yielding strength).
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6. Applications and discussion Figs. 6 and 7 show the fracture and 0.2% yielding strengths for six kinds of equiaxed a + b Ti±Mn alloys with dierent volume fractions of bphase, which are obtained by annealing these alloys in the a + b ®eld at 973 K for 6, 24, 200 and 2130 h, followed by water quenching [5]. Since the ®nal heat treatment temperature is the same, chemical compositions of the component phases in these alloys are constant. Alloys 1 and 2 are amatrix with about 1.5% and 17±21% b-particles, respectively. Alloys 3 and 4 are a + b mixture microstructure, but in alloy 3 the a-phase (58±66%)
Fig. 6. Comparison of the experimental fracture strength of a ÿ b Ti±Mn alloys [5] with the theoretical model.
Fig. 7. Comparison of the experimental 0.2% yielding strength of a ÿ b Ti±Mn alloys [5] with the theoretical model.
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Z. Li et al. / Computational Materials Science 15 (1999) 11±21
tends to be more continuous, and in alloy 4 the bphase (about 60%) tends to be more continuous. Alloys 5 and 6 are b-matrix with 16±22% and 2± 3% a-particles, respectively. The size of the particle phase in these alloys increases with annealing time. As shown in Figs. 6 and 7 for alloys 1±3, most of the fracture and yielding strengths vary within the range of the upper and lower bounds de®ned by the mixture law and Eq. (2) or Eq. (5) for the case of a soft matrix with hard particles. The fracture and yielding strengths for the alloys with ®ne particles tend to approach the upper bound (the rule of matrix) and that with coarse particles tend to approach the lower bound. The fracture and yielding strengths of alloy 4 vary in the range of the upper bound and lower bounds de®ned by Eq. (2) or Eq. (5) for the case of a hard matrix with soft particles and for a soft matrix with hard particles, respectively, and it appears that the strength is independent of the particle size. The fracture and yielding strengths of alloys 5 and 6 (Figs. 6 and 7) vary essentially within the range of upper bound and lower bound for the case of a hard matrix with soft particles. Fracture strength and yield strength for the alloy with coarse particles tend to approach the upper bound and that with ®ne particles tend to approach the lower bound as predicted by the present model. Fig. 8 shows the 0.2% yielding strength for a series of dual phase steels, which are produced by quenching Fe±Mn±C steels with carbon ranging from 0.06 to 0.52 pct. from a variety of temperatures [1]. The alloys when quenched from one temperature contain dierent volume fractions of martensite but with constant ferrite and martensite compositions (i.e., strength). When the volume fraction of martensite is less than about 40%, most of the tested data are within the range predicted by the upper and lower bounds for the case of a soft matrix with hard particles (Fig. 8), while when the volume fraction of the martensite is larger than about 65%, the majority of the tested data can be described by the upper and lower bounds for the case of a hard matrix with soft particles, but quite a number of the tested data are not within this range due to large scattering of the experimental results for a given quenching temperature. For
Fig. 8. Comparison of the experimental 0.2% yielding strength of Fe±Mn±C alloys [1] with the theoretical model.
steels with about 40±65% martensite, all experimental results fall between the upper bound for the case of a hard matrix with soft particles and the lower bound for the case of a soft matrix with hard particles. Because the transition of matrix phase is very likely performed in this range or the material has a mixed microstructure with one phase being more continuous than the other one, the fracture and yield strengths exhibit great scattering bound and independence of particle size. The ultimate strength of these dual phase steels follow the same rule as the 0.2% yielding strength(not shown here). Davies [1] compared their results with other ®ne grained VAN80 and coarse grained Fe±Mn±C dual phase steels and found that up to 50 pct. martensite, the strength of coarse grained materials falls below the ®ne ones in agreement with our prediction and within 50±70 pct. martensite, the tensile strength is independent of the grain size. Fig. 9 shows the fracture strength of 12CrModual phase steels with dierent volume fractions of martensite tested at low temperatures [26]. As shown in this ®gure, the variations of the fracture strengths with martensite volume fraction follow the rule as predicted by the present model. It can be seen from Figs. 6±9 that the strength of two-phase material shows an S-shaped variation during changing in microstructure topology (soft matrix + particles, duplex, hard matrix + particles), which can be outlined by the present model. The S-shaped variation is also observed experi-
Z. Li et al. / Computational Materials Science 15 (1999) 11±21
Fig. 9. Comparison of the experimental fracture strength of 12CrMo dual phase steels [18] with the theoretical model.
mentally in ferrite-martensite two-phase steels [7] and predicted by the modi®ed mixture law in which the in¯uence of phase continuity is incorporated [7,22]. If the two-phase materials with dierent volume fractions of second phase, such as dual phase steels, were obtained by a material quenched at dierent temperatures, the martensite contents are varied. The higher the volume fraction of martensite, the lower the carbon contents in the martensite. Since the strength of the martensite is very carbon dependent [1,24,25], the corresponding terms in Eqs. (1)±(6)) should be a function of carbon content, i.e., martensite volume fraction. Fig. 10 shows a comparison between constant strength of hard phase and the strength of the hard phase decreasing linearly with volume fraction. As shown by the solid line in Fig. 10, the mixture law in this case is not a straight line. Therefore, even if the tested data lie in one side of the simple straight mixture law, it does not mean that they disobey the strength rule of two-phase materials predicted by the present model. It should be noted that if Eqs. (1) and (4) are used to predict the fracture and yield strengths of a two-phase material, it needs to determine the value of the fracture element n, which is dependent on the critical crack length ac and particle size. They are not de®ned explicitly in the present model. Though they could be determined by mechanical testing and microstructure observation (e.g. Kunio
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Fig. 10. Comparison between constant strength of hard phase and the strength of the hard phase decreasing with volume fraction in the theoretical model.
et al. [35] found that for the dual phase steels with dierent martensite volume fractions, the critical crack length (2ac ) contains 4±8 particles, i.e., n 2±4), it falls short in practical application because they are more complicated than doing simple tensile testing. No matter what kind of a model is used to predict the tensile strength based on some parameters to be determined by experiment, it will fall short in practical use, because the tensile experiment is the simplest one. Therefore, the most important thing for model development lies in successfully revealing the dependence of the tensile strength of the material on its microstructure parameters. That is important for optimization of strength and microstructure of the material. As compared with the experimental results, the present model serves the purpose well. It should be also noted that the present model is suitable for two-phase materials with approximately equiaxed second phase particles, i.e., the shape of the particle is not taken into account. The eect of the aspect of the particles on the strength of two-phase materials will be discussed elsewhere. 7. Summary The basic characters of the fracture and yielding strengths for two-phase materials predicted by the
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The matrix phase dominates the trend of the strength to deviate from the rule of mixture, and dominates the manner of the eect of the particle size on the strength. The condition that the rule of mixture can be applied in particulate two-phase materials is that the size of particles tends to be in®nitely small. Theoretically, the rule of mixture can therefore not be used in the real two-phase materials. However, the rule of mixture can provide a satisfying strength estimation when the volume fraction and the size of the particle phase are small in two-phase materials. References
Fig. 11. Schematic description of the strength character of twophase materials predicted by the present model.
[1] [2] [3] [4] [5] [6]
present model can be schematically summarized in Fig. 11: For a two-phase composite of a soft matrix with hard particles, the rule of mixture represents the upper bound, and Eq. (2) or Eq. (5) describe the lower bound; strength increases with decreasing particle size. In case of a hard matrix with soft particles, Eq. (2) or Eq. (5) describe the upper bound, and the rule of mixture represents the lower bound; strength increases with increasing particle size. For the mixed microstructure (volume fractions between about 40±60%), the real transition of strength during the change between soft and hard matrix should vary between the upper and lower bounds de®ned by Eqs. (2) and (5) for the cases of rfp < rfm (or ryp < rym ) and for the case of rfp > rfm (or ryp > rym ), respectively. A similar trend has been recently found by Sauter [36] for the strain partitioning onto the phases in the transition regime from one to the other matrix phases in AgNi-composites. The strength ratio of the hard and soft phases governs the possible range of the upper and lower bound, and the real range of the scattering band is dependent on the size of the particles in the actual material.
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Z. Li et al. / Computational Materials Science 15 (1999) 11±21 [24] L.F. Ramos, D.K. Matlock, G. Krauss, Metall. Trans. A 10A (1979) 259±261. [25] C.L. Magee, R.G. Davies, Acta Metall. 19 (1971) 345±354. [26] Y.L. Wang, T.H. Shao, D.Y. Liu, Mater. Mech. Engrg. (in Chinese) 9 (1985) 52±56. [27] D. Brock, Elementary Engineering Mechanics, Leyden, 1974. [28] Z. Li et al., Acta Metall. Sinica 23A (1987) 103±117. [29] Z. Li, H. Gu, Metall. Trans. 21A (1990) 717±732. [30] S. Ankem, H. Margolin, Metall. Trans. 13A (1982) 595± 601. [31] M. Dong, S. Schmauder, Acta Metall. Mater. 44 (1996) 2465±2478.
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[32] T. Christman, A. Needlemen, S. Suresh, Acta Metall. 37 (1989) 3029±3050. [33] P.J. Withers, W.M. Stobbs, O.B. Pedersen, Acta Metall. 37 (1989) 3061±3084. [34] L.M. Brown, W.M. Stobbs, Philos. Mag. 23 (1971) 1185± 1199. [35] T. Kunio, M. Shimizu, J.K. Yamada, J.K. Kim, H. Hirusawa, Fracture Mechanics of Ductile and Tough Materials and Its Applications to Energy Related Structures, 1979, p. 221. [36] M. Sautter, Ph.D Thesis, University of Stuttgart, Germany, 1995.
A simple mechanical model to predict fracture and yield strengths of particulate two-phase materials Zhonghua Li a, S. Schmauder b, M. Dong a b
b
Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200030, People's Republic of China Staatliche Materialpr ufugsanstalt (MPA), Universit at of Stuttgart, Pfaenwaldring 32, D-70569 Stuttgart, Germany Received 28 October 1998; received in revised form 28 January 1999; accepted 11 February 1999
Abstract A simple mechanical model is developed and the following basic characteristics of fracture and yield strengths of particulate two-phase materials are predicted based on this model: (1) The rule of mixture can be approximately used in the particulate two-phase materials with ®ne particles; (2) For a material of a soft (hard) matrix with hard (soft) particles, the mixture law is upper (lower) bound, the lower (upper) bound is dependent on the ratio of strength and volume fraction of the constituent phases; (3) Fine (coarse) size of particle phase increases the strength of the material of a soft (hard) matrix with hard (soft) particles; (4) For mixed microstructure, the real transition of strength during change between soft and hard matrix should vary in a range of the upper bound for the material of a hard matrix with soft particles and the lower bound for that of a soft matrix with hard particles. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Two-phase material; Fracture and yield strengths; Mixture law; Lower and upper bounds
1. Introduction A number of technically important alloys consist of soft and hard phases. Examples of this type of alloy are two-phase steels [1±4,7,8] and aÿb titanium alloys [5,6]. As a group, these alloys oer useful combinations of high strength and good ductility, and have been extensively studied in the past, both experimentally [1±8] and numerically [5,6,10,22,28±32]. One of the unswerving eorts in these studies is to develop a theoretical model to predict the composite ¯ow behavior from those of the constituent phases [5,9±23]. In principle, these theoretical models fall into three categories, i.e., (1) micro mechanical model based on the dislocation
theory [9], (2) continuum mechanical models [10±13] and (3) empirical mixture laws [6,15±22]. A notable feature of the ¯ow behavior of twophase materials is that when the volume fraction of the second particle phase reduces and approaches zero, the ¯ow behavior of the two-phase material approximates that of the single matrix-phase material. Due to this simple and trivial rule, experimental data of two-phase materials always scatter along the line which links the cases of two single component phases, i.e., the line de®ned by the rule of mixture raÿb fa ra fb rb ; where raÿb is the overall average stress of the twophase material, ra ; rb and fa ; fb are the stresses
0927-0256/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 9 9 ) 0 0 0 1 4 - 2
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Z. Li et al. / Computational Materials Science 15 (1999) 11±21
measured in bulk and volume fractions of the indicated phases, respectively. The rule of mixture was derived based on the assumption of interphase isostrain condition. It is, therefore, an exact analytical solution only for longitudinal loaded long-®bre composites with elastic constituents. In general, the rule of mixture is not expected to be applicable for the particulate reinforced composites for the simple reason that the isostrain condition rarely exists. However, several investigations [1±4] showed that for the particulate two-phase materials, the mixture law can be approximately applied, while others [5±9] showed that the mixture law is not applicable. For these dierent arguments based on the magnitude of the scattering bound, there is no strict demarcation, because nobody has developed a thorough demonstration for the necessary condition of the mixture law and clari®ed the rule that the scattering bound should be followed for the particulate two-phase materials. Because all the tested data scatter along the line de®ned by the mixture law, much eort has been paid to make a rationalization of the mixture law in order to ®t the tested data. Ankem and Margolin [5] introduced an additional interaction term between soft and hard phases. Fischmeister and Karlsson [17] proposed an intermediate mixture law, in which the relevant stresses (or strains) are the average values of in situ phases, not the bulk ones. Li and Gu [15] introduced plastic constraint factors to modify the bulk ¯ow stresses used in the mixture law. Siegmund et al. [22] introduced a stereological parameter to incorporate the eect of geometrical continuity of the constituent phases. Rizk and Bourell [16] introduced an additional dislocation contribution to the ferrite yield strength due to martensite transformation in the mixture law of yield stress for dual phase steels. Although these eorts provide, from dierent respects, some insight into understanding the ¯ow behavior of two-phase materials, these modi®ed mixture laws are dicult to use in practice, because all of these modi®cations are based on an understanding of the involute interactions between soft and hard phases, which depends on the details of the microstructure (volume fraction, size, shape, continuity of the hard and soft phases) and the properties of the component phases.
Due to the complex microstructure of twophase materials, some idealized and restricted assumptions must be made to develop a theoretical model. However, if the developed model based on these assumptions does not contain the major characterizing parameters which dominate the macro ¯ow behavior of two-phase materials, then the model cannot outline the fundamental characters of two-phase materials. From numerous experimental investigations [1±8] and FE-simulations [28±32], it has been known that in two-phase materials, the ratio of strengths of the soft and hard phases, volume fraction, size, shape and continuity of the component phases are the dominant parameters. But up to now, none of the theoretical models contains all these parameters. The present study focuses our attention on developing a simple mechanical model to predict the most important strength parameters, namely, yield and fracture strengths of two-phase materials, and provides simple upper and lower bounds and their dependence on particle size, continuity and strength ratio of the constituent phases. 2. Local virtual fracture model Fig. 1 shows a particulate two-phase material subjected to uniaxial stress rc . Because of the inhomogeneous stress distributions in the two phases, the stress at a local place in the two-phase material, such as at section AB, may take the lead in reaching their own fracture strengths rfm and rfp , where the subscripts m and p indicate the matrix and the second-phase particles, respectively. If the applied stress rc increases, the fracture of section AB may not take place due to a constraint caused by the surrounding material. The stresses acting on this section will not be remarkably changed during subsequent loading, because rfm and rfp are the ultimate strengths before fracture. Therefore, section AB can always be replaced by a ®ctitious crack with closure stresses rfm and rfp as shown in Fig. 1(b). For a homogeneous material, it is obvious that the closure stresses on the virtual crack surfaces must be equal to the applied stress rc , which is independent of the deformation behavior of the
Z. Li et al. / Computational Materials Science 15 (1999) 11±21
13
Fig. 1. Schematic representation of the mechanical model: (a) the stresses in matrix and particle at section AB reach their respective fracture strengths; (b) mechanical model equivalent to (a).
material. This condition is equivalent to closing a Grith crack. It is therefore reasonable to assume that the condition of closing the virtual crack with inhomogeneous closure stresses could be concluded by means of closing the Grith crack. Thus, the stress intensity factor KIA (or KIB ) at the virtual crack tip A (or B) produced by rc , rfm and rfm should be zero. Otherwise, it will result in the stresses ahead of the virtual crack tips being larger than the closure stresses. This is in contradiction with our assumption that the stresses ahead of the crack tips are always less than the closure stresses. This condition should be necessary even when the virtual crack may be embedded in plastic stressstrain ®eld. Therefore, the model should be applicable both in brittle and ductile two-phase materials. During further loading, the stresses in matrix and particles ahead of the virtual crack tips will increase and reach rfm and rfp , which result in the growth of the virtual crack. Thus, there are two possibilities; one is that the stresses rfm and rfp acting on the surfaces of the advanced virtual crack can oset the stress intensity factor caused by applied stress, and the crack keeps closed. Further propagation of the virtual crack needs to increase the applied stress. The other possibility is that there must exist a critical virtual crack length
ac , at which the stresses rfm and rfp on the surfaces of the virtual crack cannot oset the stress intensity factor produced by the applied stress rc . In this case, the crack is unstable. Therefore, there exists a critical applied stress rfc corresponding to the critical virtual crack length ac . It is obvious that rfc is the fracture strength of the two-phase material and may be determined by using the equilibrium condition of the ®ctitious crack at its critical state. It should be noted that for predicting the yield and fracture strengths of the composites, the closure stress should be de®ned as eective stress in the component phase. It is lower than the stress component in loading direction in a soft matrix, and is higher than that in hard particles due to stress triaxiality [15,29]. Because the interphase stresses are self-equilibrated stress system, from the overall average eect on the equilibrium condition of the virtual crack, it is reasonable to assume that the eective stresses could approximately replace the stress components in the loading direction. The other assumption is the mean closure eective stresses used on the virtual crack surfaces, neglecting the local ¯uctuations in the stresses, for the reason that it is the mean stress which determines much of the mechanical behavior of the
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Z. Li et al. / Computational Materials Science 15 (1999) 11±21
composites. The local ¯uctuations in stress will not, since they average to zero, give rise to a remarkable macroscopic ¯ow. It will result in local ¯ow (microplasticity), and this is responsible for an early deviation from the initial linear elastic strain response to the applied loading [33,34]. Some micro failure mechanisms which will result in stress-free surfaces in the material, such as, microcracks, void nucleation and growth, interface debonding, are not regarded in this model. 3. Derivation of fracture strength As shown in Fig. 2, the particles divide the critical crack length ac into n equal segments, the lengths occupied by the particle and matrix in each segment are equal to ac Vp =n and ac Vm =n, respectively, where Vp and Vm are the volume fractions of the matrix and particle. Each segment is called a fracture element. In fracture element i, the coordinates c; e; d of the starting and end points of the particle and matrix are
iÿ1 ac n iÿ1 ac a ac V p d n n i d ac n Obviously, Eq. (a) is based on the particles being equiaxed and having the same size and distance from each other. The equilibrium condition [27] is given as p KIB rfc pac d r r n Z X rfp ac x ac ÿ x dx ÿ p ÿ x pa a ac x c c i1 c
ÿ
c e Z n X i1
0:
d
rfm p pac
r r ac x ac ÿ x dx ac ÿ x ac x b
The ®rst and second terms in the two integrals represent the stress intensity factors produced by rfp and rfm in the intervals 0 6 x 6 ac and ÿac 6 x 6 0, respectively. Integrating Eq. (b) and substituting the values of c; d and e from Eq. (a), we have n 2 fX i ÿ 1 Vp f arccos rc ÿ rp n p i1 n iÿ1 ÿ arccos n n X 2 i i ÿ 1 Vp : arccos ÿ arccos ÿ rfm n p n n i1 1
Fig. 2. Schematic representation of the equilibrium condition of the ®ctitious crack at critical state.
Eq. (1) is the general expression of the fracture strength for two-phase materials. When rfp > rfm , it indicates the case of a soft matrix with hard particles, and rfp < rfm indicates the case of a hard matrix with soft particles. For homogenous material (Vp 0; rfm rfp ; n i 1), we have rfc rfm . This is a simple stress equilibrium equation, which is always satis®ed during the whole elastic-plastic loading process before actual crack occurs. For two-phase materials with coarse microstructure, fracture may occur at a critical crack
Z. Li et al. / Computational Materials Science 15 (1999) 11±21
length ac of only one fracture element, i.e., n 1; then Eq. (1) can be simpli®ed as 2 f rm ÿ rfp arccos Vp : 2 rfc rfp p For two-phase materials with ®ne microstructure, the critical crack length of ac may include a large number of fracture elements. We take n ! 1 to represent the limiting case. If n ! 1, Eq. (b) may be written as p KIB rfc pac r r Zac p ac x ac ÿ x dx c ÿ p pac ac ÿ x ac x 0
in which dx ac =n; p is the average stress at dx. p can be exactly represented by Vp rfp Vm rfm due to dx being in®nitely small. Substituting p into Eq. (c) gives for n ! 1 rfc Vm rfm Vp rfp :
3
Eq. (3) indicates that if the size of the particles is very small, as could occur in the alloys strengthened by dispersion of hard particles, the strength of the two-phase material tends to follow the rule of mixture. If the particles are rigid, the rfp in Eqs. (1)±(3) should be replaced by the interface strength between matrix and particles. For the other limiting case of rfp 0, i.e., a matrix material with holes, it can be seen from Eqs. (1) and (2) that the material strength is greater than that predicted by use of the reduction of the net section (Eq. (3)). 4. Yielding strength of two-ductile phase materials Due to stress inhomogenous in two phases, local yielding may take place even though macro applied stress rc is less than the yield stress of the soft component phase as shown in FE-analyses [28,29]. The local yielding region is surrounded by elastic medium and the plastic deformation in this area is small before macro yielding occurs. Therefore, the stress in the local yielding region should approximately keep constant for the component phases with obvious yield point.
15
Though elastic response of the material will slightly deviate from elastic limit due to local yielding, macro yield strength of the material in practice is always de®ned when extensive yielding occurs. Thus, we assume that the stresses in the particle and matrix at section AB (Fig. 1(a)) take the lead in reaching their own yield strengths, ryp and rym , respectively. They will approximately keep constant during loading in the range of rc < ryc , where ryc is the yield stress of the twoductile phase material. The action of section AB may be replaced by a crack with closure stresses ryp and rym , i.e., rfm and rfp in Fig. 1(b) can be replaced by rym and ryp , respectively. During further loading, the virtual crack grows and more materials at the surfaces of the virtual crack enter yielding and the crack keeps closure. Macro yielding of the two-phase material takes place at the moment, at which the virtual crack is unstable. Therefore, the yield stress of the two-ductile phase material, ryc , can be determined by using the equilibrium condition of the ®ctitious crack at its critical state. Because the stresses in matrix and particles ahead of the virtual crack tips are assumed to be less than rym and ryp , respectively, this model is within the framework of linear elastic fracture mechanics and should be reasonable for estimating yield stress of the material. Using the model of Fig. 2 and replacing rfm and f rp by rym and ryp , we can get the general expression of the yield stress of two-ductile phase materials: n 2 X i ÿ 1 Vp arccos ryc ÿ ryp n p i1 n iÿ1 ÿ arccos n n 2 yX i i ÿ 1 Vp : arccos ÿ arccos ÿ rm n p n n i1 4 When n 1 it gives 2 y ryc ryp rm ÿ ryp arccos Vp : p
5
When n ! 1, we have ryc Vm rym Vm rym :
6
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Z. Li et al. / Computational Materials Science 15 (1999) 11±21
It should be noted that for most two-phase materials, the component phases do not show an obvious physical yielding stresses and exhibit a power hardening behavior. In this case, the yield strengths ryp and rym in Eqs. (4)±(6) should be replaced by the proof yield stress, e.g., the stresses at 0.2% (or 0.1%) plastic strain. In a strict sense, the proof yield strengths of the component phases on the surfaces of the virtual crack increase with increasing load in contrast to the earlier assumption. However, because the yield strength of the composite is also de®ned as a proof stress of 0.2% (or 0.1%) plastic strain, the stresses of the component phases on the section of the virtual crack are not remarkably enhanced during further loading until the overall plastic strain of the composite reaches 0.2% or 0.1%, i.e., Eqs. (4)±(6) can approximately be used for the two-phase materials with power hardening component phases as corroborated by experimental results shown in Section 6. 5. Basic behavior of yielding and fracture strength predicted by the model Figs. 3 and 4 show the variations of the normalized stress rfc =rfm (or ryc =rym with volume fraction of particles for the case of rfp =rfm (or ryp =rym 3.0, 5.0, 8.0 and rfp =rfm (or ryp =rym 0.8, 0.5, 0.0, respectively. In the case of rfp =rfm (or rrp =rym > 1, i.e. hard particles in a soft matrix, Eqs. (3) and (6) (the rule of mixture) give the upper bound and Eqs. (2) and (5) give the lower bound, Fig. 3. On the contrary (Fig. 4), Eqs. (2) and (5) give the upper bound and the mixture law gives the lower bound for the material of a hard matrix with soft particles. Figs. 3 and 4 show that for a given volume fraction of particles, the ®ne particles enhance the strength of the material of a soft matrix with hard particles and reduce the strength of the material of a hard matrix with soft particles. This in¯uence of the particle size predicted by the above equations is evidently reasonable, because both the reinforcement of a soft matrix by hard particles and the strength reduction of a hard matrix by soft particles are related to the interaction area between particle and matrix per
Fig. 3. Variation of the normalized fracture (or yielding) strength with volume fraction of particles for the case of rfp =rfm or ryp =rym 3:0, 5.0, 8.0 in Eqs. (1) and (4) for dierent numbers (or sizes) of particles (small n-values correspond to larger particles).
unit volume. For a given particle volume fraction, the interaction area per unit volume increases with the increasing number of particles, i.e., the reinforcement of a soft matrix by hard particles and the strength reduction of a hard matrix by soft particles increase with decreasing size of the particles. Fig. 5 describes the eect of the normalized size of particles on the normalized strength for a hard matrix with soft particles and a soft matrix with hard particles. In Fig. 5, d0 is de®ned as the size of the particle corresponding to n 1 in Eq. (1) or Eq. (4) and d corresponding to n 2; 3; 4 . . .. It can be seen from Fig. 5 that the reinforcement by hard particles in a soft matrix and strength reduction by soft particles in a hard matrix are approximately proportional to (d/d0 )1=2 . It can be concluded from Figs. 3±5 that the strength ratio of the hard and soft phases determine the possible range of the upper and lower strength bounds while the real range is dependent
Z. Li et al. / Computational Materials Science 15 (1999) 11±21
on the size of the particles. Fine particles are desired in the case of hard particles in a soft matrix while coarse particles are desired in the case of soft particles in a hard matrix.
Fig. 4. Variation of the normalized fracture (or yielding) strength with volume fraction of particles for the case of rfp =rfm or ryp =rym 0:8, 0.5, 0.0 in Eqs. (1) and (4).
Fig. 5. Eect of the normalized particle size on the strength normalized to the strength from the law of mixture (r f indicates fracture strength, r y indicates yielding strength).
17
6. Applications and discussion Figs. 6 and 7 show the fracture and 0.2% yielding strengths for six kinds of equiaxed a + b Ti±Mn alloys with dierent volume fractions of bphase, which are obtained by annealing these alloys in the a + b ®eld at 973 K for 6, 24, 200 and 2130 h, followed by water quenching [5]. Since the ®nal heat treatment temperature is the same, chemical compositions of the component phases in these alloys are constant. Alloys 1 and 2 are amatrix with about 1.5% and 17±21% b-particles, respectively. Alloys 3 and 4 are a + b mixture microstructure, but in alloy 3 the a-phase (58±66%)
Fig. 6. Comparison of the experimental fracture strength of a ÿ b Ti±Mn alloys [5] with the theoretical model.
Fig. 7. Comparison of the experimental 0.2% yielding strength of a ÿ b Ti±Mn alloys [5] with the theoretical model.
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Z. Li et al. / Computational Materials Science 15 (1999) 11±21
tends to be more continuous, and in alloy 4 the bphase (about 60%) tends to be more continuous. Alloys 5 and 6 are b-matrix with 16±22% and 2± 3% a-particles, respectively. The size of the particle phase in these alloys increases with annealing time. As shown in Figs. 6 and 7 for alloys 1±3, most of the fracture and yielding strengths vary within the range of the upper and lower bounds de®ned by the mixture law and Eq. (2) or Eq. (5) for the case of a soft matrix with hard particles. The fracture and yielding strengths for the alloys with ®ne particles tend to approach the upper bound (the rule of matrix) and that with coarse particles tend to approach the lower bound. The fracture and yielding strengths of alloy 4 vary in the range of the upper bound and lower bounds de®ned by Eq. (2) or Eq. (5) for the case of a hard matrix with soft particles and for a soft matrix with hard particles, respectively, and it appears that the strength is independent of the particle size. The fracture and yielding strengths of alloys 5 and 6 (Figs. 6 and 7) vary essentially within the range of upper bound and lower bound for the case of a hard matrix with soft particles. Fracture strength and yield strength for the alloy with coarse particles tend to approach the upper bound and that with ®ne particles tend to approach the lower bound as predicted by the present model. Fig. 8 shows the 0.2% yielding strength for a series of dual phase steels, which are produced by quenching Fe±Mn±C steels with carbon ranging from 0.06 to 0.52 pct. from a variety of temperatures [1]. The alloys when quenched from one temperature contain dierent volume fractions of martensite but with constant ferrite and martensite compositions (i.e., strength). When the volume fraction of martensite is less than about 40%, most of the tested data are within the range predicted by the upper and lower bounds for the case of a soft matrix with hard particles (Fig. 8), while when the volume fraction of the martensite is larger than about 65%, the majority of the tested data can be described by the upper and lower bounds for the case of a hard matrix with soft particles, but quite a number of the tested data are not within this range due to large scattering of the experimental results for a given quenching temperature. For
Fig. 8. Comparison of the experimental 0.2% yielding strength of Fe±Mn±C alloys [1] with the theoretical model.
steels with about 40±65% martensite, all experimental results fall between the upper bound for the case of a hard matrix with soft particles and the lower bound for the case of a soft matrix with hard particles. Because the transition of matrix phase is very likely performed in this range or the material has a mixed microstructure with one phase being more continuous than the other one, the fracture and yield strengths exhibit great scattering bound and independence of particle size. The ultimate strength of these dual phase steels follow the same rule as the 0.2% yielding strength(not shown here). Davies [1] compared their results with other ®ne grained VAN80 and coarse grained Fe±Mn±C dual phase steels and found that up to 50 pct. martensite, the strength of coarse grained materials falls below the ®ne ones in agreement with our prediction and within 50±70 pct. martensite, the tensile strength is independent of the grain size. Fig. 9 shows the fracture strength of 12CrModual phase steels with dierent volume fractions of martensite tested at low temperatures [26]. As shown in this ®gure, the variations of the fracture strengths with martensite volume fraction follow the rule as predicted by the present model. It can be seen from Figs. 6±9 that the strength of two-phase material shows an S-shaped variation during changing in microstructure topology (soft matrix + particles, duplex, hard matrix + particles), which can be outlined by the present model. The S-shaped variation is also observed experi-
Z. Li et al. / Computational Materials Science 15 (1999) 11±21
Fig. 9. Comparison of the experimental fracture strength of 12CrMo dual phase steels [18] with the theoretical model.
mentally in ferrite-martensite two-phase steels [7] and predicted by the modi®ed mixture law in which the in¯uence of phase continuity is incorporated [7,22]. If the two-phase materials with dierent volume fractions of second phase, such as dual phase steels, were obtained by a material quenched at dierent temperatures, the martensite contents are varied. The higher the volume fraction of martensite, the lower the carbon contents in the martensite. Since the strength of the martensite is very carbon dependent [1,24,25], the corresponding terms in Eqs. (1)±(6)) should be a function of carbon content, i.e., martensite volume fraction. Fig. 10 shows a comparison between constant strength of hard phase and the strength of the hard phase decreasing linearly with volume fraction. As shown by the solid line in Fig. 10, the mixture law in this case is not a straight line. Therefore, even if the tested data lie in one side of the simple straight mixture law, it does not mean that they disobey the strength rule of two-phase materials predicted by the present model. It should be noted that if Eqs. (1) and (4) are used to predict the fracture and yield strengths of a two-phase material, it needs to determine the value of the fracture element n, which is dependent on the critical crack length ac and particle size. They are not de®ned explicitly in the present model. Though they could be determined by mechanical testing and microstructure observation (e.g. Kunio
19
Fig. 10. Comparison between constant strength of hard phase and the strength of the hard phase decreasing with volume fraction in the theoretical model.
et al. [35] found that for the dual phase steels with dierent martensite volume fractions, the critical crack length (2ac ) contains 4±8 particles, i.e., n 2±4), it falls short in practical application because they are more complicated than doing simple tensile testing. No matter what kind of a model is used to predict the tensile strength based on some parameters to be determined by experiment, it will fall short in practical use, because the tensile experiment is the simplest one. Therefore, the most important thing for model development lies in successfully revealing the dependence of the tensile strength of the material on its microstructure parameters. That is important for optimization of strength and microstructure of the material. As compared with the experimental results, the present model serves the purpose well. It should be also noted that the present model is suitable for two-phase materials with approximately equiaxed second phase particles, i.e., the shape of the particle is not taken into account. The eect of the aspect of the particles on the strength of two-phase materials will be discussed elsewhere. 7. Summary The basic characters of the fracture and yielding strengths for two-phase materials predicted by the
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Z. Li et al. / Computational Materials Science 15 (1999) 11±21
The matrix phase dominates the trend of the strength to deviate from the rule of mixture, and dominates the manner of the eect of the particle size on the strength. The condition that the rule of mixture can be applied in particulate two-phase materials is that the size of particles tends to be in®nitely small. Theoretically, the rule of mixture can therefore not be used in the real two-phase materials. However, the rule of mixture can provide a satisfying strength estimation when the volume fraction and the size of the particle phase are small in two-phase materials. References
Fig. 11. Schematic description of the strength character of twophase materials predicted by the present model.
[1] [2] [3] [4] [5] [6]
present model can be schematically summarized in Fig. 11: For a two-phase composite of a soft matrix with hard particles, the rule of mixture represents the upper bound, and Eq. (2) or Eq. (5) describe the lower bound; strength increases with decreasing particle size. In case of a hard matrix with soft particles, Eq. (2) or Eq. (5) describe the upper bound, and the rule of mixture represents the lower bound; strength increases with increasing particle size. For the mixed microstructure (volume fractions between about 40±60%), the real transition of strength during the change between soft and hard matrix should vary between the upper and lower bounds de®ned by Eqs. (2) and (5) for the cases of rfp < rfm (or ryp < rym ) and for the case of rfp > rfm (or ryp > rym ), respectively. A similar trend has been recently found by Sauter [36] for the strain partitioning onto the phases in the transition regime from one to the other matrix phases in AgNi-composites. The strength ratio of the hard and soft phases governs the possible range of the upper and lower bound, and the real range of the scattering band is dependent on the size of the particles in the actual material.
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