J.Am. Ceran.Soe.,88[]3104-310902005 Dol:lo.l.1551-2916.2005.00559x journal C) 2005 The American Ceramic Society Investigation of Anelastic Creep Recovery in SiC Whisker- Reinforced Alumina Composites G. C. Quan, K. T. Conlon, and D.S. Wilkinson Department of Materials Science and Engineering, McMaster University, Hamilton, oN, Canada LSS 4L7 Anelastic strain recovery has been studied in SiC whisker-rei Il. Materials and Experimental Procedures forced alumina composites following tensile creep The magni tude of the recovered strains were 10-3 with 20 vol% SiC Alumina matrix composites with 10, 20, and 30 vol% SiC whiskers upon unloading from 74 MPa at 1673 K, while no such hikers were prepared from premixed powders supplied by behavior was observed with 10 vol% SiC whiskers. A strain re- ACMC(see Acknowledg using ceramIc powder covery mechanism based on hertzian contact deformation with- rocessing followed by uniaxial hot The nixed (or particle)network is wders were first dispersed in distilled water at which appears to be consistent with the experimental observa- oH=2 and ball milled for 0.5 h. The heteroflocculated slurries tions. Two models (for whiskers and particles) have been devel- were then slip cast into rectangular-shaped billets for the sub- oped which seem to predict the magnitude of recoverable strain quent hot pressing. The details of the processing routes can be reasonably well found elsewhere. cReep experiments were conducted in tension Coading direction perpendicular to the hot-pressing axis)using a methodology similar to that followed by Carroll et al. L. Introduction S IGNIFICANT anelastic strain recovery following creep has long been observed in some ceramic materials, the presence of Ill. Estimation of Peak broadening because of whisker which can be used to better understand creep mechanisms. -9 Suppose that whisker bending is indeed the origin of anelastic vol%SiC whisker-reinforced alumina in four-point bending tests strain recovery and that the change in the (lll) Sic peak where the recovered strain was as large width is determined primarily by the extent of whisker bending loading from 50.5 MPa at 1773 K. However, no such behavior among all possible causes. It follows that during forward creep, vas observed in 5 vol% whisker-reinforced alumina. Anelasti- whiskers would undergo bending deformation, and thus, result city in whisker-reinforced ceramics is commonly attributed to the formation of a whisker network that exhibits a low effective ever, the reverse process takes place wherein the extent of whis elastic stiffness. In order to sustain the high degree of anelastic ker bending would decrease with a concomitant decrease in peal strain recovery upon load removal, strain energy has to be ac- width. This hypothesis is schematically illustrated in Fig. 1. It is cumulated in the constrained whisker network by the elastic portant to recognize the reversibility of this process, which bending of whiskers that are not free to rotate about their con- requires that the amount of whisker bending gained during for tact points. 6. 8.9 Gu et al. 10-13 reported that a composite with ward creep be the same as that reduced during recover 15 vol% whiskers showed significant anelastic recovery(10-3) The extent of peak broadening upon a certain degree of was essentially independent of the total forward creep stran lo o pon load removal at 1723 K, and that the recoverable strain whisker bending is estimated with the followi 1. There is only hydrostatic (or dilatational) distortion of the strain at which the load was removed ) They also found that unit cells after bending and Braggs law is still applicable for the a smaller whisker aspect ratio resulted in more strain recovery bent crystal (1.5x 10)and more extensive transient creep behavior. This 2. The amount of bending is small and the outer fiber mac. is a somewhat perplexing result as a high aspect ratio should be restrains are equal to the extreme lattice strains. associated with more whisker-bending strain according to por- Broadening of the line profile of diffraction peaks from po- ter's model, thus resulting in greater strain recovery. Furthe lycrystals can be caused by the small size of coherently diffract sic particulate-reinforced alumina but not with dilute (5 volo microstrains within the grains(strain effect). This is termed composites. This was ascribed to an inclusion network effect but physical broadening. In addition to this there is an inherent in- the underlying physical reason was not sought. As no bending strumental broadening. because of the effect of slit widths effect is associated with spherical particles, there must be a dif- sample size, penetration into the sample, and imperfect focusing, ferent mechanism responsible for the observed strain recovery covery in tensile mode, assessing the peak broadening effect as- tering angle, Wze for a pure diffraction peak (i.e. withow at- behavior therein. This study is aimed at studying the creep re- In the absence of crystallite size effect, the variance in scat sociated with whisker-bending deformation, and estimating the strumental broadening) is given as magnitude of the recoverable creep strain based on a newly pro- H3=(20-(0)24m() S M. Wiederhorncontributing editor This uscript No 20101. Received July 9, 2004: approved February 2, 2005. Council of Can ade rseg unding from the Natural Science and Engi- the [lll] direction. Because of the high density tion. the long-range periodicity along the other three off-axis <lll) directions has been Author to whom correspondence should be addressed. e-mail: wilkinson mcmaster.ca destroyed. 3104
Investigation of Anelastic Creep Recovery in SiC Whisker-Reinforced Alumina Composites G. C. Quan, K. T. Conlon, and D. S. Wilkinsonw Department of Materials Science and Engineering, McMaster University, Hamilton, ON, Canada L8S 4L7 Anelastic strain recovery has been studied in SiC whisker-reinforced alumina composites following tensile creep. The magnitude of the recovered strains were B103 with 20 vol% SiC whiskers upon unloading from 74 MPa at 1673 K, while no such behavior was observed with 10 vol% SiC whiskers. A strain recovery mechanism based on Hertzian contact deformation within a percolating whisker (or particle) network is suggested, which appears to be consistent with the experimental observations. Two models (for whiskers and particles) have been developed which seem to predict the magnitude of recoverable strain reasonably well. I. Introduction SIGNIFICANT anelastic strain recovery following creep has long been observed in some ceramic materials,1–7 the presence of which can be used to better understand creep mechanisms.7–9 Porter6 first reported significant anelastic strain recovery in 15 vol% SiC whisker-reinforced alumina in four-point bending tests where the recovered strain was as large as B2 103 upon unloading from 50.5 MPa at 1773 K. However, no such behavior was observed in 5 vol% whisker-reinforced alumina. Anelasticity in whisker-reinforced ceramics is commonly attributed to the formation of a whisker network that exhibits a low effective elastic stiffness. In order to sustain the high degree of anelastic strain recovery upon load removal, strain energy has to be accumulated in the constrained whisker network by the elastic bending of whiskers that are not free to rotate about their contact points.6,8,9 Gu et al. 10–13 reported that a composite with 15 vol% whiskers showed significant anelastic recovery (B103 ) upon load removal at 1723 K, and that the recoverable strain was essentially independent of the total forward creep strain (i.e. the strain at which the load was removed). They also found that a smaller whisker aspect ratio resulted in more strain recovery (B1.5 103 ) and more extensive transient creep behavior. This is a somewhat perplexing result as a high aspect ratio should be associated with more whisker-bending strain according to Porter’s model, thus resulting in greater strain recovery. Furthermore, they observed similar strain recovery behavior in 15 vol% SiC particulate-reinforced alumina but not with dilute (5 vol%) composites. This was ascribed to an inclusion network effect but the underlying physical reason was not sought. As no bending effect is associated with spherical particles, there must be a different mechanism responsible for the observed strain recovery behavior therein. This study is aimed at studying the creep recovery in tensile mode, assessing the peak broadening effect associated with whisker-bending deformation, and estimating the magnitude of the recoverable creep strain based on a newly proposed mechanism. II. Materials and Experimental Procedures Alumina matrix composites with 10, 20, and 30 vol% SiC whiskers were prepared from premixed powders supplied by ACMC (see Acknowledgments) using ceramic powder colloidal processing followed by uniaxial hot pressing. The pre-mixed composite powders were first dispersed in distilled water at pH 5 2 and ball milled for 0.5 h. The heteroflocculated slurries were then slip cast into rectangular-shaped billets for the subsequent hot pressing. The details of the processing routes can be found elsewhere.14 Creep experiments were conducted in tension (loading direction perpendicular to the hot-pressing axis) using a methodology similar to that followed by Carroll et al. 15 III. Estimation of Peak Broadening Because of Whisker Bending Suppose that whisker bending is indeed the origin of anelastic strain recovery and that the change in the (111) SiCz16 peak width is determined primarily by the extent of whisker bending among all possible causes. It follows that during forward creep, whiskers would undergo bending deformation, and thus, result in (111) SiC peak broadening. When the load is removed, however, the reverse process takes place wherein the extent of whisker bending would decrease with a concomitant decrease in peak width. This hypothesis is schematically illustrated in Fig. 1. It is important to recognize the reversibility of this process, which requires that the amount of whisker bending gained during forward creep be the same as that reduced during recovery. The extent of peak broadening upon a certain degree of whisker bending is estimated with the following assumptions: 1. There is only hydrostatic (or dilatational) distortion of unit cells after bending and Bragg’s law is still applicable for the bent crystal. 2. The amount of bending is small and the outer fiber macrostrains are equal to the extreme lattice strains. Broadening of the line profile of diffraction peaks from polycrystals can be caused by the small size of coherently diffracting domains (size effect) and by a non-uniform distribution of microstrains within the grains (strain effect). This is termed physical broadening. In addition to this there is an inherent instrumental broadening,17 because of the effect of slit widths, sample size, penetration into the sample, and imperfect focusing, etc. In the absence of crystallite size effect, the variance in scattering angle, W2y for a pure diffraction peak (i.e. without instrumental broadening) is given as18,19 W2y ¼ ð Þ 2y h i 2y 2 D E ¼ 4 tan2 y e2 hkl (1) Journal J. Am. Ceram. Soc., 88 [11] 3104–3109 (2005) DOI: 10.1111/j.1551-2916.2005.00559.x r 2005 The American Ceramic Society 3104 S. M. Wiederhorn—contributing editor This work was supported by the research funding from the Natural Science and Engineering Research Council of Canada (NSERC). w Author to whom correspondence should be addressed. e-mail: wilkinso@mcmaster.ca Manuscript No. 20101. Received July 9, 2004; approved February 2, 2005. z Each SiC whisker is a single crystal with cubic structure and the whisker axis is parallel to the [111] direction. Because of the high density of planar defects along this growth direction, the long-range periodicity along the other three off-axis /111S directions has been destroyed.
ovember 2005 Anelastic Creep Recovery Neutral axis (111)sic Strain (111)SiC一 Fig. 2. Simplified geometry of whisker bending diminishes; it is approximated that D sell where elu is the RMS strain, i. e. the standard deviation of the where 20 is the scattering angle and normal distribution. Now, the physical broadening because of whisker bending is expressed as FWHM=2√2m2tane (dnl) This model predicts that the magnitude of physical broadening fore, the Gaussian-type physical broadening in terms of the Is proportional to the fiber diameter and inversely related full-width at half-maximum (FWHM)can be ressed as the radius of bending curvature. Assuming a simple three- follows. point bending geometry as shown in Fig. 2(b), the radius of curvature can be estimated as followin wHM4=2V2h2√Wm=42h2n0√() The inhomogene mally assumed to Therefo on a Gaussian distribution, Pnkda) FWHMp=7V2In2 tan 0(n2 P()=√2e Figure 3 shows the calculated physical peak broadening function of whisker bending using Eqs.(9)and (11). Model I where enk is the root mean square(RMS)strain, i.e. assumes a Gaussian-type strain distribution and Model 2,a uniform strain distribution. As shown in the figure, the latter predicts greater change in peak width, for instance, predicting 0.015 change in FWHM compared with -0.009 by Model I In developing this figure we have assumed that the mid-point the distribution of the strain field around the deflection of the whisker is equal to its diameter. We believe tha strain,. Now, the fraction of crystallites this corresponds to a rather large creep strain for whiskers with aspect ratios typical of those in the composites we have studied. evaluated by the integral Such a small change in peak width was found difficult to detect using a conventional neutron diffractometer because of the lim- P(e1,e2)=/p(e)d ited instrumental resolution. In this case we would require the use of X-ray diffraction from a synchrotron or high-energy c(2)-r(m)0 pulsed neutron diffraction. However, it is likely that relaxation processes during creep will lead to a decrease in the peak width, which has been observed in Sic platelet-reinforced alumina composites. 2 This will counterbalance the increase in peak width because of whisker bending. Therefore, even with a high- In the case of whisker bending. it is convenient to follow the er instrumental resolution, separating the contributions because plane that is perpendicular to the whisker axis, i.e. (lll) plane as of whisker bending alone may be difficult shown in Fig. 2(a). The outer fiber strain calculated by contin uum mechanics is D/2p where D is the fiber diameter and p the radius of the curvature. For a small degree of bending, this IV. Creep Recovery Tests and Discussion strain is assumed to equal the maximum bending strain of the (ll) plane at the fiber surface in tension Figure 4 shows the tensile creep-stress relaxation curves for alu- mina reinforced with 10, 20, and 30 vol% SiC whiskers. Signi gmax=D-dhu-dn ficant anelastic strain recovery is seen with addition of 20 vol% SiC whiskers and a similar behavior was found with 30 vol% dir SiC whiskers while no such recovery is associated with 10 vol% reinforcement. This conforms to the suggestion that the onset where dil is the lattice spacing at the neutral plane and din the of significant anelastic strain recovery corresponds to a critical maximum(outer fiber) spacing. Note that the"volume fraction" whisker volume fraction required to form an interconnecting f crystallites whose lattice spacing is at maximum or minimum article network 6. 8, 9 Wilkinson estimated this threshold to be
where 2y is the scattering angle and ehkl ¼ dhkl h i dhkl h i dhkl (2) is the fractional change of spacing of d for {hkl} plane. Therefore, the Gaussian-type physical broadening in terms of the full-width at half-maximum (FWHM) can be expressed as follows: FWHMp ¼ 2 ffiffiffiffiffiffiffiffiffiffi 2 ln 2 p ffiffiffiffiffiffiffiffiffi W2y p ¼ 4 ffiffiffiffiffiffiffiffiffiffi 2 ln 2 p tan y ffiffiffiffiffiffiffiffiffiffiffi e2 hkl q (3) The inhomogeneous strain field is normally assumed to take on a Gaussian distribution, Phkl(e): PhklðeÞ ¼ 1 ffiffiffiffiffi 2p p ehkl exp ehkl h i ehkl ffiffiffi 2 p ehkl 2 " # (4) where ehkl is the root mean square (RMS) strain, i.e. ehkl ¼ ffiffiffiffiffiffiffiffiffiffiffi e2 hkl q (5) which describes the distribution of the strain field around the average lattice strain, /ehklS. Now, the fraction of crystallites whose lattice strain along (hkl) falls between e1 and e2 can be evaluated by the integral20 Phklðe1; e2Þ ¼ Z e2 e1 pðeÞ de ¼ 1 2 erf e2 h i ehkl ffiffiffi 2 p ehkl erf e1 h i ehkl ffiffiffi 2 p ehkl (6) In the case of whisker bending, it is convenient to follow the plane that is perpendicular to the whisker axis, i.e. (111) plane as shown in Fig. 2(a). The outer fiber strain calculated by continuum mechanics is D/2r where D is the fiber diameter and r the radius of the curvature. For a small degree of bending, this strain is assumed to equal the maximum bending strain of the (111) plane at the fiber surface in tension, emax ¼ D 2r ¼ dt 111 dn 111 dn 111 (7) where d111 n is the lattice spacing at the neutral plane and d111 t the maximum (outer fiber) spacing. Note that the ‘‘volume fraction’’ of crystallites whose lattice spacing is at maximum or minimum diminishes; it is approximated that D 2r ¼ 3e111 (8) where e111 is the RMS strain, i.e. the standard deviation of the normal distribution. Now, the physical broadening because of whisker bending is expressed as FWHMp ¼ 2 3 ffiffiffiffiffiffiffiffiffiffi 2 ln 2 p tan y D r (9) This model predicts that the magnitude of physical broadening is proportional to the fiber diameter and inversely related to the radius of bending curvature. Assuming a simple threepoint bending geometry as shown in Fig. 2(b), the radius of curvature can be estimated as following:21 r L2 8Dh (10) Therefore FWHMp ¼ 16 3 ffiffiffiffiffiffiffiffiffiffi 2 ln 2 p tan y DhD L2 (11) Figure 3 shows the calculated physical peak broadening as a function of whisker bending using Eqs. (9) and (11). Model 1 assumes a Gaussian-type strain distribution and Model 2, a uniform strain distribution. As shown in the figure, the latter predicts greater change in peak width, for instance, predicting B0.0151 change in FWHM compared with B0.0091 by Model 1. In developing this figure we have assumed that the mid-point deflection of the whisker is equal to its diameter. We believe that this corresponds to a rather large creep strain for whiskers with aspect ratios typical of those in the composites we have studied. Such a small change in peak width was found difficult to detect using a conventional neutron diffractometer because of the limited instrumental resolution.14 In this case we would require the use of X-ray diffraction from a synchrotron or high-energy pulsed neutron diffraction. However, it is likely that relaxation processes during creep will lead to a decrease in the peak width, which has been observed in SiC platelet-reinforced alumina composites.22 This will counterbalance the increase in peak width because of whisker bending. Therefore, even with a higher instrumental resolution, separating the contributions because of whisker bending alone may be difficult. IV. Creep Recovery Tests and Discussion Figure 4 shows the tensile creep-stress relaxation curves for alumina reinforced with 10, 20, and 30 vol% SiC whiskers. Significant anelastic strain recovery is seen with addition of 20 vol% SiC whiskers, and a similar behavior was found with 30 vol% SiC whiskers while no such recovery is associated with 10 vol% reinforcement. This conforms to the suggestion that the onset of significant anelastic strain recovery corresponds to a critical whisker volume fraction required to form an interconnecting particle network.6,8,9 Wilkinson8 estimated this threshold to be D ρ (111)SiC Neutral axis ρ ∆h O (a) (b) L Fig. 2. Simplified geometry of whisker bending. (111) SiC Time Strain Fig. 1. Schematic of peak width changes according to the whisker bending model. November 2005 Anelastic Creep Recovery 3105
3106 Journal of the American Ceramic Societ Quan et a Vol. 88. No. I Radius of bending curvature(um) Model 1 001200090000 40002000 0.018 model 1 0.016 model 2 80.014 D=0.5 um L=10 um 0.010 Model 2 s0008 P 0.000 4 Fig 3. The calculated full-width at half-maximum as a function of whisker bending. If the probability of finding lattice strain between a and E+de is P(e)dE, Model I assumes a Gaussian-type distribution of E and Model 2 a uniform around 12 vol% for composites in which whiskers are mode timated by considering the force equilibrium during external ately textured because of uniaxial hot pressi tensile loading because of rotation of the whiskers. wilkinson According to the whisker-bending model, upon loading, and Pompe showed that for this loading configuration the the whiskers within a percolative network will undergo bending stress concentration factor is independent of the whisker vol- deformation, storing elastic strain energy. This energy is to be ume fraction. Figure 6(a)shows a highly simplified unit cell for released when the load is removed, leading to a high degree of calculating the contact forces. If we assume the surrounding strain recovery . Therefore, high whisker aspect ratio should matrix induces rotation of the misaligned whisker, then at the be associated with more strain recovery. However, the study by contact points normal forces are generated. The magnitude of Gu et al. showed exactly the opposite, i. e the composite re- these contact forces can be estimated by comparing the moment inforced by shorter whiskers was shown to recover more strain. formed by the shear stress acting along the matrix whisker in Moreover, it was found in the same study that an alumina m terface(see Fig. 7) trix reinforced with 15 vol% spherical Sic particles also exhib- ited comparable strain recovery. This raises the question as to whether a fibrous inclusion geometry is key to having such an M,=4 LR cos odo= 4LRt elastic strain recovery An important clue can be inferred from the observation of 2LRESin 20 (13) elastic strain recovery in Sic particulate-reinforced alumina composites. As mentioned earlier, this system showed significant where R and L is radius and average segment length of whiskers strain recovery only when the inclusion content reached about o is the angle around the whisker axis. 0 is the misorientation 15 vol%, suggesting a network effect. Also the magnitude of the angle, and e is the applied tensile stress related to the local nor recovered strain in this system is comparable with that recorded in an alumina composite with the same content of Sic whiskers al stress o by a local stress tration factor k. such that RΣ, where k 26,6 he average degree of mis- For percolating elastic spheres, the most probable explanation alignment between the whisker the loading axis. This mo- for achieving a high degree of strain accumulation is localized ment is counter balanced by elastic deformation between particles because of Hertzian con- contact forces Fp. whereby moment because of the tact whereby a large strain can be achieved under low stress when spheres or cylinders are pressed into one another.25-2Ac- Mn= FpL 14 cording to Landau and Lishitz, for two spherical bodies contact, the center-to-center particle separation, h, scales with load f as from which the local contact force is determined to be h= constant x F2/3 (12 Fn= 2R2 sin 20 (15) Figure 5(a)illustrates this relation for spherical particles in The displacement because of contact deformation is given as24 contact. In fact. it has been shown that this relation remains true for any finite bodies in contact. Therefore, Hertzian contact deformation should be applicable for whiskers(or cylindrical rods) as well, as shown schematically in Fig. 5(b) h (16) elusive task, the main reason being the difficulty of estimating Then the strain in the loading direction is given by the local contact force. Here we present two simple models de- eloped for different reinforcement geometry, namely, sphere and whisker E= 0 For the whisker network loaded in-plane (i.e, perpendicular to the hot-pressing axis), local contact forces can be roughly es- 2R21-y)13 ' It is assumed that the surrounding matrix is easily deformed plastically at the testing (sin 20) 0 (17)
around 12 vol% for composites in which whiskers are moderately textured because of uniaxial hot pressing. According to the whisker-bending model,6,8 upon loading, the whiskers within a percolative network will undergo bending deformation, storing elastic strain energy. This energy is to be released when the load is removed, leading to a high degree of strain recovery.6,9,11 Therefore, high whisker aspect ratio should be associated with more strain recovery. However, the study by Gu et al. 13 showed exactly the opposite, i.e. the composite reinforced by shorter whiskers was shown to recover more strain. Moreover, it was found in the same study that an alumina matrix reinforced with 15 vol% spherical SiC particles also exhibited comparable strain recovery. This raises the question as to whether a fibrous inclusion geometry is key to having such anelastic strain recovery. An important clue can be inferred from the observation of anelastic strain recovery in SiC particulate-reinforced alumina composites. As mentioned earlier, this system showed significant strain recovery only when the inclusion content reached about 15 vol%, suggesting a network effect. Also the magnitude of the recovered strain in this system is comparable with that recorded in an alumina composite with the same content of SiC whiskers. For percolating elastic spheres, the most probable explanation for achieving a high degree of strain accumulation is localized elastic deformation between particles because of Hertzian contacty whereby a large strain can be achieved under low stress when spheres or cylinders are pressed into one another.23–25 According to Landau and Lishitz,24 for two spherical bodies in contact, the center-to-center particle separation, h, scales with load F as h ¼ constant F2=3 (12) Figure 5(a) illustrates this relation for spherical particles in contact. In fact, it has been shown that this relation remains true for any finite bodies in contact.24 Therefore, Hertzian contact deformation should be applicable for whiskers (or cylindrical rods) as well, as shown schematically in Fig. 5(b). Prediction of the magnitude of the recoverable strain is an elusive task, the main reason being the difficulty of estimating the local contact force. Here we present two simple models developed for different reinforcement geometry, namely, sphere and whisker. For the whisker network loaded in-plane (i.e., perpendicular to the hot-pressing axis), local contact forces can be roughly estimated by considering the force equilibrium during external tensile loading because of rotation of the whiskers. Wilkinson and Pompe9 showed that for this loading configuration the stress concentration factor is independent of the whisker volume fraction. Figure 6(a) shows a highly simplified unit cell for calculating the contact forces. If we assume the surrounding matrix induces rotation of the misaligned whisker, then at the contact points normal forces are generated. The magnitude of these contact forces can be estimated by comparing the moment formed by the shear stress acting along the matrix whisker interface (see Fig. 7): Mt ¼ 4 Z p=2 0 LR2 t coso do ¼ 4LR2 t ¼ 2LR2Ssin 2y (13) where R and L is radius and average segment length of whiskers, o is the angle around the whisker axis, y is the misorientation angle, and S is the applied tensile stress related to the local normal stress s by a local stress concentration factor k, such that s 5 kS, where k ¼ 1 2 sin 2y, y being the average degree of misalignment between the whiskers and the loading axis.9 This moment is counter balanced by another moment because of the contact forces Fp, whereby Mn ¼ FpL (14) from which the local contact force is determined to be Fp ¼ 2R2 S sin 2y (15) The displacement because of contact deformation is given as24 h ¼ F2=3 p 9 2 1 n2 E 2 1 R " #1=3 (16) Then the strain in the loading direction is given by e ¼ 2h L tan y ¼ 2R L 3 ffiffiffi 2 p Sð1 n2Þ E " #2=3 ðsin 2yÞ 2=3 tan y (17) P(ε) σ = ε*/3 −ε* ε* 0.0 0.1 0.2 0.3 0.4 0.5 −ε* ε* 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 Radius of bending curvature (µm) model 1 model 2 D=0.5 µm L=10 µm Change in FWHM (Degree) Mid-point Displacement (µm) 12000 10000 8000 6000 4000 2000 P(ε) Model 1 Model 2 Fig. 3. The calculated full-width at half-maximum as a function of whisker bending. If the probability of finding lattice strain between e and e1de is P(e) de, Model 1 assumes a Gaussian-type distribution of e and Model 2 a uniform distribution of e. y It is assumed that the surrounding matrix is easily deformed plastically at the testing temperature. 3106 Journal of the American Ceramic Society—Quan et al. Vol. 88, No. 11
ovember 2005 Anelastic Creep Recovery 3107 (a)0006 10 vol% SIC Pa 59 MPa 1400°C (b)0.006 20 yol% si ematic illustration of Hertzian contact deformation between and (b) rods in contact. 18 MPa approximated as follows: 0.003 Total 74 MPa Li cos 0i (18) 1400° where n is the Using Eqs (c)0.006 30 voL% SiC eTotal_2=/2(2R'Esin20 2/2(1-v2)/E)1/R"sine, ∑=1Lcos 0.005 59 MP 18 MPa here 0.000 150000200000 Time( inin20)3)=∑P0)snoi20)3△9(2l) Fig 4. Tensile creep stress relaxation curve for (a) 10 vol%,(b)20 vol%, and (c)30 vol% SiC whisker-reinforced alumina composites. os)=∑P(e)cos△ Assuming∑=75MPa,〈θ)=20°,RL=0.1,v=0.127 nd e=427 Gpa the unit-cell model predicts a recoverable strain of 4. x 10. It is noteworthy that this model predicts increasing strain with a decrease in whisker aspect ratio, which is n accord with the experimental results of Gu et al. It is also E teresting that there is no explicit dependence on volume frac- tion. this follows from the stress concentration factor k which depends only on the average whisker misalignment 0. As ex- plained by wilkinson and Pompe this result applies only when the loading axis lies in the plane for which the whiskers are ∠日 nearly aligned, as we have done in the experimental work dis cussed here. If one loads the sample normal to the plane of whisker alignment. then the stress concentration factor scales For a percolating path of whiskers along the loading dire tion. the total strain because of contact deformation can Fig.6.Ⅲ lustra plies only to isotropic materials, the transversely isotropic elastic con- on constrained
Assuming S 5 75 MPa, /yS 5 201, z R/L 5 0.1,J n 5 0.127,ww and E 5 427 Gpa,ww the unit-cell model predicts a recoverable strain of B4.4 104 . It is noteworthy that this model predicts increasing strain with a decrease in whisker aspect ratio, which is in accord with the experimental results of Gu et al. 13 It is also interesting that there is no explicit dependence on volume fraction. This follows from the stress concentration factor k which depends only on the average whisker misalignment y. As explained by Wilkinson and Pompe9 this result applies only when the loading axis lies in the plane for which the whiskers are nearly aligned, as we have done in the experimental work discussed here. If one loads the sample normal to the plane of whisker alignment, then the stress concentration factor scales as 1/f2/3, where f is the whisker volume fraction. For a percolating path of whiskers along the loading direction, the total strain because of contact deformation can be approximated as follows: eTotal ¼ Pn i¼1 P 2hi sin yi n i¼1 Li cos yi (18) where n is the number of contact points. Using Eqs. (15) and (16), it becomes eTotal ¼ Pn i¼1 2ð2R2Ssin2yiÞ 2=3 9 2 ð1n2Þ=E 2 1=R h i1=3 sinyi Pn i¼1Lcosyi (19) Therefore, eTotal ¼2R L 3 ffiffiffi 2 p Sð1n2Þ E " #2=3 ðsin2yiÞ 2=3 sinyi D E h i cosyi (20) where sinyiðsin2yiÞ 2=3 D E¼ Xm i¼1 PðyiÞsinyiðsin2yiÞ 2=3 Dyi (21) h i cosyi ¼ Xm i¼1 PðyiÞcosyiDyi (22) Σ Σ τ τ Σ Σ θi θi Fp Fp (a) (b) Fig. 6. Illustration of load transfer because of the far-field stress acting on constrained (a) fibers and (b) spherical particles through contact deformation. F F = h F ∝ h3/2 F F (a) (b) Fig. 5. Schematic illustration of Hertzian contact deformation between (a) spheres and (b) rods in contact. 0 50000 100000 150000 200000 0.000 0.001 0.002 0.003 0.004 0.005 0.006 1400°C 10 vol.% SiC 18 MPa 59 MPa Creep Strain 0.000 0.001 0.002 0.003 0.004 0.005 0.006 Creep Strain Time (s) 0 50000 100000 150000 200000 Time (s) 0 50000 100000 150000 200000 Time (s) 1400°C 1400°C 18 MPa 74 MPa 20 vol.% SiC 0.000 0.001 0.002 0.003 0.004 0.005 0.006 18 MPa 59 MPa 30 vol.% SiC Creep Strain (a) (b) (c) Fig. 4. Tensile creep stress relaxation curve for (a) 10 vol%, (b) 20 vol%, and (c) 30 vol% SiC whisker-reinforced alumina composites. z This is based on neutron diffraction measurement.14 J This corresponds to three contact points per whisker. wwAs Eq. (16) applies only to isotropic materials, the transversely isotropic elastic constants of the whisker were replaced by the average isotropic elastic constants.26 November 2005 Anelastic Creep Recovery 3107
Journal of the American Ceramic Societ Quan et a (a)00020 0.0018 P(e)=2/(Random) 00016 P(e)(Measured 0.0014 Network Model b0.0010 =20°- 0.0004 0.0002 Applied Stress, E(MPa) Fig. 7. Geometry for calculating the moment because of the shear (b) 0.0050 stress on th 0.0045 =- upper bound°) ∑P(0)△02=1 (23) P(0)is the whisker orientation distribution function learnt from the fiber texture analysis and m is the 0.0015 angles covered during whisker texture measurement. a random whisker distribution, i.e. P(0)=2/I, the tot Gu et 0.0005 =0.15 Totl__6(5/6)(2/3)R3v22(1-2 Applied Stress, E(MPa) Fig 8. Computed total strain (including both the rapidly recovered where r(X)=or-le-dr elastic portion upon load drop and the time-dependent portion) because Figure 8(a)shows the total strain as a function of the whisker network. The hatched area covers the recovered strains that stress assuming R/L=0. 1. It is seen that the results ar have been measured in this study (b) for a percolative spherical particle same order of magnitude with those observed durin network and in tension(current study ). It is noted the unit cell model with a uniform misalignment angle equal to the av V. Conclusions rage misorientation predicts lower strains as compared with the ach in which the distrib In this work we first estimated the peak broadening because of considered whisker bending, which appears to be too small to be detected A similar approach can be used for a spherical partic by conventional diffraction methods. Significant anelastic strain is under compression. Figure 5(b) illustrates the ge- recovery was observed following tensile creep of the composite ometry of such particles in contact. Under the applied stress containing 20 and 30 vol% SiC whiskers and the magnitude of the average contact force, Fp, will develop around the particles the recovered strains is consistent with those measured in pre- at the loading direction, which is expressed as vious work using four-point bending creep tests. This, combi with a lack of significant strain recovery with 10 vol% SiC Fp (25) whiskers, demonstrates that the whisker network is key to hav- particles to the far-field stress and R the average particle rad the ing a high degree of anelastic strain recovery. The effect of whisker aspect ratio on the amount of recoverable strain and the Now. the total strain can be written comparable creep strain recovery observed in Sic particulate reinforced alumina composite may suggest that the local Hertz- ian contact deformation between "hard" inclusions is more likely the underlying mechanism for anelastic creep recovery. The two simple models developed for whisker and particulate networks seem to predict the magnitude of recoverable strain For randomly distributed particles, it becomes Total 3∑r(1 We gratefully acknowledge the supply of composite powders from Dr J. F. Figure 8(b)shows the calculated contact strain versus applied Rhodes of the Advanced Composite Materials Corporation, Greer, SC, and the help stress and the factor, s. The calculation indicates that the re- from Mrs. Constance Barry with composite processing is very much appreciated. coverable creep strain in Sic particulate-reinforced alumina composite, i.e. <10-3, corresponds to 5 A.25. This may suggest that the average local contact force responsible for in- Reference clusion contact deformation is small, presumably because the rons and J. K. Tien."Crex shear portion of the local stress facilitates rearrangement of the particle uch a way that the local contact str ell and K H. G. Ashbee. ""High Temperature Creep of Lithium s-Ceramics: Part 2 Compression Creep and Recovery. "J Mater. 973)
and Xm i¼1 PðyiÞDyi ¼1 (23) P(yi) is the whisker orientation distribution function that can be learnt from the fiber texture analysis and m is the number of angles covered during whisker texture measurement. Assuming a random whisker distribution, i.e. P(yi) 5 2/p, the total strain is given as eTotal ¼6 7 Gð5=6ÞGð2=3Þ ffiffiffi p p R L 3 ffiffiffi 2 p Sð1n2Þ E " #2=3 (24) where GðXÞ¼R 1 0 t X1et dt Figure 8(a) shows the total strain as a function of the applied stress assuming R/L 5 0.1. It is seen that the results are in the same order of magnitude with those observed during creep in bending10–13 and in tension (current study). It is noted the unitcell model with a uniform misalignment angle equal to the average misorientation predicts lower strains as compared with the approach in which the distribution of whisker orientation is considered. A similar approach can be used for a spherical particle network that is under compression. Figure 5(b) illustrates the geometry of such particles in contact. Under the applied stress S, the average contact force, Fp, will develop around the particles at the loading direction, which is expressed as Fp ¼ SzpR2 (25) where z is a factor relating the average local contact force on the particles to the far-field stress and R the average particle radius. Now, the total strain can be written as eTotal ¼ 1 2 3Szpð1 n2Þ ffiffiffi 2 p E 2=3 ðcos yiÞ 5=3 D E h i cos yi (26) For randomly distributed particles, it becomes eTotal ¼ ffiffiffi 3 p p3=2 15Gð2=3ÞGð5=6Þ 3Szpð1 n2Þ ffiffiffi 2 p E 2=3 (27) Figure 8(b) shows the calculated contact strain versus applied stress and the factor, z. The calculation indicates that the recoverable creep strain in SiC particulate-reinforced alumina composite,13 i.e. B103 , corresponds to z 0.25. This may suggest that the average local contact force responsible for inclusion contact deformation is small, presumably because the shear portion of the local stress facilitates rearrangement of the particle network in such a way that the local contact strain is minimized. V. Conclusions In this work we first estimated the peak broadening because of whisker bending, which appears to be too small to be detected by conventional diffraction methods. Significant anelastic strain recovery was observed following tensile creep of the composite containing 20 and 30 vol% SiC whiskers, and the magnitude of the recovered strains is consistent with those measured in previous work using four-point bending creep tests. This, combined with a lack of significant strain recovery with 10 vol% SiC whiskers, demonstrates that the whisker network is key to having a high degree of anelastic strain recovery. The effect of whisker aspect ratio on the amount of recoverable strain and the comparable creep strain recovery observed in SiC particulate reinforced alumina composite may suggest that the local Hertzian contact deformation between ‘‘hard’’ inclusions is more likely the underlying mechanism for anelastic creep recovery. The two simple models developed for whisker and particulate networks seem to predict the magnitude of recoverable strain reasonably well. Acknowledgments We gratefully acknowledge the supply of composite powders from Dr. J. F. Rhodes of the Advanced Composite Materials Corporation, Greer, SC, and the help from Mrs. Constance Barry with composite processing is very much appreciated. References 1 R. M. Arons and J. K. Tien, ‘‘Creep and Strain Recovery in Hot-Pressed Silicon Nitride,’’ J. Mater. Sci., 15, 2046–58 (1980). 2 R. Morrell and K. H. G. Ashbee, ‘‘High Temperature Creep of Lithium Zinc Silicate Glass–Ceramics: Part 2 Compression Creep and Recovery,’’ J. Mater. Sci., 8, 1271–7 (1973). τ τ ω dω Fig. 7. Geometry for calculating the moment because of the shear stress on the whisker surface. 0 25 50 75 100 125 150 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018 0.0020 Network Model Unit Cell Model P(θ)=2/π (Random) P(θ) (Measured ) Contact Strain Applied Stress, Σ (MPa) 0 25 50 75 100 125 150 Applied Stress, Σ (MPa) 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 ζ = φ (upper bound ) φ = 0.15 φ = 0.3 Gu et al. φ = 0.15 Contact Strain ζ= 0.4 ζ= 0.3 ζ= 0.2 (a) (b) =25° =20° Fig. 8. Computed total strain (including both the rapidly recovered elastic portion upon load drop and the time-dependent portion) because of local contact deformation as a function of applied stress for (a) whisker network. The hatched area covers the recovered strains that have been measured in this study (b) for a percolative spherical particle network. 3108 Journal of the American Ceramic Society—Quan et al. Vol. 88, No. 11
ovember 2005 Anelastic Creep Recovery J.W. Holmes, "Tensile Creep Behavior of a Hot-Pressed SiC Fiber-Reinforced 1G.C.Quan,“ Creep of Sic w.Jones, Tensile Creep and ID. F. Carroll, S M. Wiederhorn, and D E. Roberts, "Technique for Tensile rower 3 egor of a stc-Fiber-Si3N4 Composite,J. Am. Ceran Creep &. Nutt. Defects in Silicon Carbide Whiskers. "J. Am. Ceram. Soc., 67[6] Testing of Ceramics, J. Am. Ceram. Soc., 72 91 1610-4(1989). F. F. Lange. D. R. Clark, and B. I. Davis, "Compressive Creep of Si3 N4/MgO 428-31(1984) A B.E. Warren, X-Ray Diffraction. Addison Wesley Pub. Co., Reading. MA. ker-Reinforced eW. R. Cannon and s. aig, " creep recovery Mechanisms" 2. 38i1-92 in Diffacto,metry l: istakes and sarnan "proc. Pens, soc., 81. 4-6 1965) Bradt, et al. Plenum Press, New P. Klug and L. E. Alexander, x-ray Diffraction Procedures, 2nd edition, pp.618-708.Wle D. S. Wilkinson, ""Creep Mechanisms in Multiphase Ceramic Materials. en mes rememtsof the re:idual simes D S wilkinson and w. Pompe. ""Creep and Anelastic Recovery of whisker es in AlyO,(CeO2) Ceramic Composites, "J. Am. Ceran. Soc., 77 [6] Gu. J.R. Porter and T.G. Langdon, "Anelastic Cree A. H. Cottrell, The Mechanical Properties of Matter. 126pp, John wiley Alumina Reinforced with Sic Whiskers or Particulates, "Ceran. Trans., 46, 307- R. Ham-Su "Creep of Silicon Carbide-Reinforced Alumina"; Ph.D. Disser- w.Z. Gu, J.R. Porter, and T. G. Langdon. ""Evidence for Anelastic Creep us Papers, J. Math.(Crelle's J. 92, 156-71(1881). 7167981419 2L. D Landau and E. M. Lishitz, Theory of Elasticity, 3rd editi Z Gu, J.R. Porter, and T.G. Langdon. ""Significance of Anelasticity During Creep of an Alumina Composite Reinforced with Silicon Carbide, "Ke Timoshenko and J N. Goodier, Theory of Elasticity, pp. 409-14. McGraw w. Gu, J. R. Porter, and T G. Langdon, "An Investigation of Anelastic Creep Majumdar, D. Kupperman, and J. Singh, ""Determination of Residual Recovery in SiC Whisker-and Particulate-Reinforced Alumina, "Ceram. Eng. Sci. Thermal Stresses in a SiC-Al-O3 Composite Using Neutron Diffraction, "J.Am. Proe,16[242-52(1995) Ceran.Soc,7l085863(1988)
3 J. W. Holmes, ‘‘Tensile Creep Behavior of a Hot-Pressed SiC Fiber-Reinforced Si3N4 Composite,’’ J. Mater. Sci., 26, 1808–14 (1991). 4 J. W. Holmes, Y. H. Park, and J. W. Jones, ‘‘Tensile Creep and Creep-Recovery Behavior of a SiC–Fiber–Si3N4 Composite,’’ J. Am. Ceram. Soc., 76 [5] 1281–93 (1993). 5 F. F. Lange, D. R. Clark, and B. I. Davis, ‘‘Compressive Creep of Si3N4/MgO Alloys: Part 2, Source of Viscoelastic Effect,’’ J. Mater. Sci., 15, 611–5 (1980). 6 J. R. Porter, ‘‘Dispersion Processing of Creep-Resistant Whisker-Reinforced Ceramic-Matrix Composites,’’ Mater. Sci. Eng., A107 [1–2] 127–32 (1989). 7 W. R. Cannon and S. Haig, ‘‘Creep Recovery Mechanisms’’; pp. 381–92 in Plastic Deformation of Ceramics, Edited by R. C. Bradt, et al. Plenum Press, New York, 1995. 8 D. S. Wilkinson, ‘‘Creep Mechanisms in Multiphase Ceramic Materials,’’ J. Am. Ceram. Soc., 81 [2] 275–99 (1998). 9 D. S. Wilkinson and W. Pompe, ‘‘Creep and Anelastic Recovery of Whiskerand Platelet-Reinforced Ceramics,’’ Acta Mater., 46 [4] 1357–69 (1998). 10W. Z. Gu, J. R. Porter, and T. G. Langdon, ‘‘Anelastic Creep Recovery of Alumina Reinforced with SiC Whiskers or Particulates,’’ Ceram. Trans., 46, 307– 18 (1994). 11W. Z. Gu, J. R. Porter, and T. G. Langdon, ‘‘Evidence for Anelastic Creep Recovery in Silicon Carbide Whisker-Reinforced Alumina,’’ J. Am. Ceram. Soc., 77, 1679–81 (1994). 12W. Z. Gu, J. R. Porter, and T. G. Langdon, ‘‘Significance of Anelasticity During Creep of an Alumina Composite Reinforced with Silicon Carbide,’’ Key. Eng. Mater., 104–107 [2] 873–80 (1995). 13W. Gu, J. R. Porter, and T. G. Langdon, ‘‘An Investigation of Anelastic Creep Recovery in SiC Whisker- and Particulate-Reinforced Alumina,’’ Ceram. Eng. Sci. Proc., 16 [1] 242–52 (1995). 14G. C. Quan, ‘‘Creep of SiC Whisker-Reinforced Alumina Composites’’; Ph.D. Dissertation, McMaster University, Hamilton, Canada, 2004. 15D. F. Carroll, S. M. Wiederhorn, and D. E. Roberts, ‘‘Technique for Tensile Creep Testing of Ceramics,’’ J. Am. Ceram. Soc., 72 [9] 1610–4 (1989). 16S. R. Nutt, ‘‘Defects in Silicon Carbide Whiskers,’’ J. Am. Ceram. Soc., 67 [6] 428–31 (1984). 17B. E. Warren, X-Ray Diffraction. Addison Wesley Pub. Co., Reading, MA, 1969. 18A. J. C. Wilson, ‘‘On Variance as a Measure of Line Broadening in Diffractometry II: Mistakes and Strain,’’ Proc. Phys. Soc., 81, 41–6 (1963). 19H. P. Klug and L. E. Alexander, X-ray Diffraction Procedures, 2nd edition, pp. 618–708. Wiley, New York, 1980. 20X. L. Wang, C. R. Hubbard, K. B. Alexander, P. F. Becher, J. A. FernandezBaca, and S. Spooner, ‘‘Neutron Diffraction Measurements of the Residual Stresses in Al2O3–ZrO2 (CeO2) Ceramic Composites,’’ J. Am. Ceram. Soc., 77 [6] 1569–75 (1994). 21A. H. Cottrell, The Mechanical Properties of Matter, 126pp., John Wiley & Sons Inc., New York, 1964. 22R. Ham-Su, ‘‘Creep of Silicon Carbide-Reinforced Alumina’’; Ph.D. Dissertation, McMaster University, Hamilton, Ontario, Canada, 1997. 23H. Herzt, ‘‘Miscellaneous Papers,’’ J. Math.(Crelle’s J.), 92, 156–71 (1881). 24L. D. Landau and E. M. Lishitz, Theory of Elasticity, 3rd edition, pp. 26–31. Pergamon Press, Oxford, 1986. 25S. Timoshenko and J. N. Goodier, Theory of Elasticity, pp. 409–14. McGrawHill, New York, 1951. 26S. Majumdar, D. Kupperman, and J. Singh, ‘‘Determination of Residual Thermal Stresses in a SiC-Al2O3 Composite Using Neutron Diffraction,’’ J. Am. Ceram. Soc., 71 [10] 858–63 (1988). & November 2005 Anelastic Creep Recovery 3109
Copyright of Journal of the American Ceramic Society is the property of Blackwell Publishing Limited. The copyright in an individual article may be maintained by the author in certain cases. Content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder,'s express written permission However, users may print, download, or email articles for individual use
