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Evaluation of the room-temperature strength of oxide fibres produced by the internal-crystallization method
Composites Science and Technology 59 (1999) 1977±1981
Evaluation of the room-temperature strength of oxide ®bres produced by the internal-crystallization method V.M. Kiiko, S.T. Mileiko* Solid State Physics Institute of the Russian Academy of Sciences, Chernogolovka Moscow District, 142432 Russia Received 27 August 1998; accepted 9 April 1999
Abstract Oxide ®bres produced by the internal crystallization route have been used in a variety of matrices to yield heat-resistant composites. These ®bres are crystallized in channels previously made in the molybdenum matrix by diusion bonding of an assembly of molybdenum foils and wires, so that their cross-section has an unusual shape. Hence, a special procedure is necessary for evaluating the ®bre strength. The procedure developed includes looping a ®bre around a series of the rigid cylinders of successively smaller diameters and calculating the tensile strength by using a procedure based on Weibull statistics. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Fibres; A. Oxides; B. Strength; B. Defects; C. Statistics
1. Introduction Oxide ®bres produced by the internal crystallization method (ICM) [1,2] (see also Chapter 14 in Ref. 3) can now be extracted from a molybdenum matrix and subsequently used in a variety of matrices to yield heatresistant composites [4,5]. These ®bres are crystallized in channels previously-made in the molybdenum matrix by the diusion bonding of an assembly of molybdenum foils and wires, so that the ®bre cross-section has the shape shown in Fig. 1. Sizes of the cross-section d and t vary within any batch of ®bres as a consequence of inevitable variations in wire diameter and winding pitch. Measurment of the ®bre strength by testing separate ®bres in tension therefore presents a rather dicult problem because of diculties of alignment of the ®bre specimen in a testing machine. Certainly measuring the ®bre strength in a corresponding matrix is the best way characterizing the ®bre if the characteristics are to be used in evaluation of the composite strength (see Refs. 3 and 6 for the reasons). However to make a quick estimate of the ®bre tensile strength for the purpose of comparison of various ®bres or ®bres from dierent batches, etc., simple testing methods are required. One such method was suggested by Siemers et al. [7] who * Corresponding author. E-mail address: [email protected] (S.T. Mileiko)
tested silicon carbide ®bres in bending by looping them around a series of rigid cylinders of successively smaller diameters and then calculated the tensile strength by a procedure based on Weibull statistics. In the present paper, a modi®ed scheme of the `loop test' is used and an interpretation of the experimental results to account for special features of the ®bres under consideration is suggested. 2. Experimental procedure and interpretaion of the experimental data In Ref. 7 a set of ®bres of a constant diameter and a constant length laid parallel were looped around a series of rigid cylinders of successively smaller diameters to ®nd the ultimate strain of the ®bre and calculate the Weibull parameters of the ®bre strength by the use of wellknown procedures. When dealing with ICM ®bres which are characterized by cross-sections of sizes d and t which are not constant, even within one batch of the ®bres, both the experimental technique and interpretation of the data need to be modi®ed. This has been now done. 2.1. Experimental technique An arrangement of the rigid cylinder and ®bre enclosed between two molybdenum foil covers is shown
0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(99)00054-8
1978
V.M. Kiiko, S.T. Mileiko / Composites Science and Technology 59 (1999) 1977±1981
2.2. Interpretation of the data The maximum ®bre stress corresponding to the rigid cylinder radius is calculated according to E
Fig. 1. Cross-section of a typical ICM ®bre.
in Fig. 2. Since the ®bre height d varies, it is impossible to test a number of ®bres in parallel. Hence, just one ®bre is tested. First, a ®bre is bent over a rigid steel cylinder of a suciently large radius, R. The number of ®bre breaks is counted. If the number of breaks is small enough, that means that the ratio of the average distance between neighboring breaks to some characteristic dimension of the ®bre cross-section is larger than 10 which makes it possible to neglect end eects. The ®bre is then bent over a cylinder of smaller radius and the new total number of breaks is counted. The process is repeated until the average distance between the breaks becomes less than about 10. The ®bre cross-section parameters, d and t, are measured by observing a ®bre under an optical microscope at low magni®cation.
d 2R
where E is the Young's modulus of the ®bre material. To a ®rst approximation, we assume to be the ®bre bending strength at a length equal to the average distance between the ®bre breaks. Such a procedure obviously makes the ®bre strength in a subsequent analysis higher than the actual strength. However, using a series of cylinders with a small increment of radius makes the error suciently small. Variation of the cross-section parameters of the ®bres calls for the use of either volume or surface as a geometrical parameter in the Weibull distribution to describe the statistical characteristics of the ®bre strength. We start with the volume hypothesis. Taking the Weibull distribution in the form ! 1 r dV 1 P ; V 1 ÿ exp ÿ Vo V o we obtain the scale dependence for the ®bre strength as ÿ1= V 2 o Vo where and o are the mean strength values for volumes V and Vo , respectively. It is convenient to introduce a constant characteristic ®bre length, lo , and choose Vo as Vo lo Aav
Fig. 2. A scheme for testing the ®bre by looping it around a rigid cylinder.
3
where Aav is the average cross-sectional area in a ®bre batch under test. Our goal is to determine the value of from the strength scatter of ®bres of constant volume, Vo , and then, on the basis of the known mean value of the strength for that volume to obtain the scale dependence of the mean ®bre strength. To determine , we use well-known approximation 1:2=; being the variation coecient of the ®bre strength. Since we have experimental points as a set of (V; ) values, we need to use a step-by-step approximation method. We choose, as a ®rst approximation, a typical value of , say 3, and then use Eq. (2) to calculate the strength values at Vo . The set of strength values obtained yields a value of the standard deviation and, consequently, a value of . The value of yields a new value of , and the procedure is then repeated. Normally, three or ®ve iterations are sucient, provided the ®rst
V.M. Kiiko, S.T. Mileiko / Composites Science and Technology 59 (1999) 1977±1981
1979
p d2 b d ÿ 1 ÿ 2 ; Fo ÿ ; and y=h 2 4 where t=d and d is the wire diameter. Hence, following the procedure outlined in Ref. 8 we obtain M MVICM p 1= ÿ1= 1 ÿ ÿ 1 ÿ 2 d : 4 0
Fig. 3. Values of M, which connect Weibull characteristic stresses 0b and ot for ICM ®bres versus Weibull parameter .
approximation for the value of has been properly chosen. A set of Weibull parameters, that is ; o ; Vo is obtained as a result of the calculations. It is convenient to replace the notation o with ob to stress the relationship of o to bending. Hence, we actually obtain then parameter set ; ob ; Vo and now wish to calculate ; ot ; Vo where ot relates the statistical characteristics to the tensile strength of the ®bres. A model that links the bending, ob , and tensile, ot , characteristic stresses for a brittle material is presented elsewhere [8]. A general relationship is ot MV ob
4
where MV is determined by comparison of the probabilities for tension and pure bending, namely ! l 5 P ; l 1 ÿ exp ÿ lo o and ! l 1 h dx b d P ; V 1 ÿ exp ÿ lo Fo o 0 0
6
respectively. For the ®bre cross-section shown in Fig. 1 we have
7
Strictly speaking, the relationship we shall use is aected by the shape of the ®bre in terms of the ratio t=d. So the procedure can be considered as approximate since the values of t and d vary even within one ®bre batch. If we assume that characteristic defects are located on the ®bre surface or in its vicinity, then the iteration procedure the set of strength characteristics, to obtain that is ; ob ; So , remains as described above for the volume hypothesis, with Eq. (3) being replaced with So lo Pav where Pav is an average value of the cross-section perimeter. To obtain ot we use the corresponding relationship found in Ref. 8 that is
1 P ÿ1= 4 h 1 2 ÿ1= 2
M MSICM
8
The dependencies given by Eqs. (7) and (8) are shown in Fig. 3. It is clear that the larger the value of the Weibull parameter , the closer to each other are the values of ob and ot . The eect of is not so pronounced. 3. Experimental We present here experimental data to illustrate the application of the method. Results of its systematic application to be published elsewhere [9].
Fig. 4. A photograph of a ®bre of ICM type prepared for counting the number of breaks.
1980
V.M. Kiiko, S.T. Mileiko / Composites Science and Technology 59 (1999) 1977±1981
A batch of the Al2O3±Al5Y3O12 eutectic ®bre produced by the internal-crystallization route were tested according to the scheme described above. A view of a ®bre looped around a rigid cylinder and then un-looped to count the number of breaks is shown in Fig. 4. The results of calculations of the set of Weibull parameters
are given in Table 1. The experimental data are given in Figs. 5 and 6. Scale dependencies of the ®bre strength are also plotted. In order to make their appearance more conventional, the equation
Table 1 The Weibull parameters for a batch of Al2O3±Al5Y3O12 eutectic ®bres. The Young's modulus is assumed to be 400 Gpa
is used in Fig. 5 instead of Eq. (2). Similarly, the following equation is plotted in Fig. 6:
Volume defect hypothesis
Surface defect hypothesis
0b (MPa)
0t (MPa)
Vo (mm3)
0b (MPa)
0t (MPa)
So (mm2)
3.67
1249
857.5
0.0320
3.64
844.8
379.1
10.054
lAAv ÿ1= l o Vo
lPAv ÿ1= : l o So
9
10
Note that both values of o , namely ob and ot , are used in the ®gures. 4. Conclusions A technique of ®bre testing by looping them around a series of rigid cylinders of successively smaller diameters appears to be suitable, after minor modi®cation, for the strength evaluation of oxide ®bres produced by the internal-crystallization method. The ®bre tensile strength is obtained by using a procedure based on Weibull statistics. Acknowledgements
Fig. 5. The experimental data (¯exural strength) and calculated scale dependencies of the ¯exural and tensile strengths of the Al2O3± Al5Y3O12 eutectic ®bres (Block K195). The volume hypothesis is assumed. The Weibull parameters are given in Table 1.
The work has been performed under ®nancial support of the International Science and Technology Center (Project no. 507-97) and INTAS-RFBR (Project no. 950599). A contribution to the experimental work done by Mr. A.A. Kolchin, Dr. G.K. Strukova, and Mrs. T.I. Boromychenko is gratefully acknowledged. References
Fig. 6. The experimental data (¯exural strength) and calculated scale dependencies of ¯exural and tensile strengths of the Al2O3±Al5Y3O12 eutectic ®bres (Block K195). The surface hypothesis is assumed. The Weibull parameters are given in Table 1.
[1] Mileiko ST, Kazmin VI. Crystallization of ®bres inside a matrix: a new way of fabrication of composites. J Mater Sci 1992; 27:2165±72. [2] Mileiko ST, Kazinin VI. Structure and mechanical properties of oxide ®bre reinforced metal matrix composites produced by the internal crystallization method. Comp Sci Technol 1992;45:209±20. [3] Mileiko ST. Metal and ceramic based composites. Amsterdam: Elsevier, 1997. [4] Mileiko ST. Novel oxide ®bres to reinforce metal, intermetallic and ceramic matrices. In: Advanced Multilayered and FibreReinforced Composites, Proc. of NATO Advance Res. Workshop, Kiev, Ukraine, June 1997. Dordrecht: Kluwer Academic, 1998. p. 333±52. [5] Mileiko ST. Heat-resistant oxide-®bre composites. Presented at ECCM-8, Naples, 1998. [6] Mileiko ST. Oxide ®bres. In: Watt WW, Perov BV, editors. Strong Fibres Handbook of Composites, Vol. 1. Amsterdam: North-Holland, 1985. p. 87±114.
V.M. Kiiko, S.T. Mileiko / Composites Science and Technology 59 (1999) 1977±1981 [7] Siemers PA, Mehati RL, Moran H. A comparison of the uniaxial tensile and pure bending strength of SiC ®laments. J Mater Sci 1988;23:1329±33. [8] Mileiko ST. On relationships between strength values obtained in tension and bending. In preparation.
1981
[9] Mileiko ST, Kiiko VM, Sarkissyan NS, Starostin MYu, Gvozdeva SI, Kolchin AA, Strukova GK. Microstructure and properties of A12O3±Al5Y3O12 ®bres produced by an internal crystallization route. Comp Sci Technol 1999;59(11):1763±72.
Evaluation of the room-temperature strength of oxide ®bres produced by the internal-crystallization method V.M. Kiiko, S.T. Mileiko* Solid State Physics Institute of the Russian Academy of Sciences, Chernogolovka Moscow District, 142432 Russia Received 27 August 1998; accepted 9 April 1999
Abstract Oxide ®bres produced by the internal crystallization route have been used in a variety of matrices to yield heat-resistant composites. These ®bres are crystallized in channels previously made in the molybdenum matrix by diusion bonding of an assembly of molybdenum foils and wires, so that their cross-section has an unusual shape. Hence, a special procedure is necessary for evaluating the ®bre strength. The procedure developed includes looping a ®bre around a series of the rigid cylinders of successively smaller diameters and calculating the tensile strength by using a procedure based on Weibull statistics. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Fibres; A. Oxides; B. Strength; B. Defects; C. Statistics
1. Introduction Oxide ®bres produced by the internal crystallization method (ICM) [1,2] (see also Chapter 14 in Ref. 3) can now be extracted from a molybdenum matrix and subsequently used in a variety of matrices to yield heatresistant composites [4,5]. These ®bres are crystallized in channels previously-made in the molybdenum matrix by the diusion bonding of an assembly of molybdenum foils and wires, so that the ®bre cross-section has the shape shown in Fig. 1. Sizes of the cross-section d and t vary within any batch of ®bres as a consequence of inevitable variations in wire diameter and winding pitch. Measurment of the ®bre strength by testing separate ®bres in tension therefore presents a rather dicult problem because of diculties of alignment of the ®bre specimen in a testing machine. Certainly measuring the ®bre strength in a corresponding matrix is the best way characterizing the ®bre if the characteristics are to be used in evaluation of the composite strength (see Refs. 3 and 6 for the reasons). However to make a quick estimate of the ®bre tensile strength for the purpose of comparison of various ®bres or ®bres from dierent batches, etc., simple testing methods are required. One such method was suggested by Siemers et al. [7] who * Corresponding author. E-mail address: [email protected] (S.T. Mileiko)
tested silicon carbide ®bres in bending by looping them around a series of rigid cylinders of successively smaller diameters and then calculated the tensile strength by a procedure based on Weibull statistics. In the present paper, a modi®ed scheme of the `loop test' is used and an interpretation of the experimental results to account for special features of the ®bres under consideration is suggested. 2. Experimental procedure and interpretaion of the experimental data In Ref. 7 a set of ®bres of a constant diameter and a constant length laid parallel were looped around a series of rigid cylinders of successively smaller diameters to ®nd the ultimate strain of the ®bre and calculate the Weibull parameters of the ®bre strength by the use of wellknown procedures. When dealing with ICM ®bres which are characterized by cross-sections of sizes d and t which are not constant, even within one batch of the ®bres, both the experimental technique and interpretation of the data need to be modi®ed. This has been now done. 2.1. Experimental technique An arrangement of the rigid cylinder and ®bre enclosed between two molybdenum foil covers is shown
0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(99)00054-8
1978
V.M. Kiiko, S.T. Mileiko / Composites Science and Technology 59 (1999) 1977±1981
2.2. Interpretation of the data The maximum ®bre stress corresponding to the rigid cylinder radius is calculated according to E
Fig. 1. Cross-section of a typical ICM ®bre.
in Fig. 2. Since the ®bre height d varies, it is impossible to test a number of ®bres in parallel. Hence, just one ®bre is tested. First, a ®bre is bent over a rigid steel cylinder of a suciently large radius, R. The number of ®bre breaks is counted. If the number of breaks is small enough, that means that the ratio of the average distance between neighboring breaks to some characteristic dimension of the ®bre cross-section is larger than 10 which makes it possible to neglect end eects. The ®bre is then bent over a cylinder of smaller radius and the new total number of breaks is counted. The process is repeated until the average distance between the breaks becomes less than about 10. The ®bre cross-section parameters, d and t, are measured by observing a ®bre under an optical microscope at low magni®cation.
d 2R
where E is the Young's modulus of the ®bre material. To a ®rst approximation, we assume to be the ®bre bending strength at a length equal to the average distance between the ®bre breaks. Such a procedure obviously makes the ®bre strength in a subsequent analysis higher than the actual strength. However, using a series of cylinders with a small increment of radius makes the error suciently small. Variation of the cross-section parameters of the ®bres calls for the use of either volume or surface as a geometrical parameter in the Weibull distribution to describe the statistical characteristics of the ®bre strength. We start with the volume hypothesis. Taking the Weibull distribution in the form ! 1 r dV 1 P ; V 1 ÿ exp ÿ Vo V o we obtain the scale dependence for the ®bre strength as ÿ1= V 2 o Vo where and o are the mean strength values for volumes V and Vo , respectively. It is convenient to introduce a constant characteristic ®bre length, lo , and choose Vo as Vo lo Aav
Fig. 2. A scheme for testing the ®bre by looping it around a rigid cylinder.
3
where Aav is the average cross-sectional area in a ®bre batch under test. Our goal is to determine the value of from the strength scatter of ®bres of constant volume, Vo , and then, on the basis of the known mean value of the strength for that volume to obtain the scale dependence of the mean ®bre strength. To determine , we use well-known approximation 1:2=; being the variation coecient of the ®bre strength. Since we have experimental points as a set of (V; ) values, we need to use a step-by-step approximation method. We choose, as a ®rst approximation, a typical value of , say 3, and then use Eq. (2) to calculate the strength values at Vo . The set of strength values obtained yields a value of the standard deviation and, consequently, a value of . The value of yields a new value of , and the procedure is then repeated. Normally, three or ®ve iterations are sucient, provided the ®rst
V.M. Kiiko, S.T. Mileiko / Composites Science and Technology 59 (1999) 1977±1981
1979
p d2 b d ÿ 1 ÿ 2 ; Fo ÿ ; and y=h 2 4 where t=d and d is the wire diameter. Hence, following the procedure outlined in Ref. 8 we obtain M MVICM p 1= ÿ1= 1 ÿ ÿ 1 ÿ 2 d : 4 0
Fig. 3. Values of M, which connect Weibull characteristic stresses 0b and ot for ICM ®bres versus Weibull parameter .
approximation for the value of has been properly chosen. A set of Weibull parameters, that is ; o ; Vo is obtained as a result of the calculations. It is convenient to replace the notation o with ob to stress the relationship of o to bending. Hence, we actually obtain then parameter set ; ob ; Vo and now wish to calculate ; ot ; Vo where ot relates the statistical characteristics to the tensile strength of the ®bres. A model that links the bending, ob , and tensile, ot , characteristic stresses for a brittle material is presented elsewhere [8]. A general relationship is ot MV ob
4
where MV is determined by comparison of the probabilities for tension and pure bending, namely ! l 5 P ; l 1 ÿ exp ÿ lo o and ! l 1 h dx b d P ; V 1 ÿ exp ÿ lo Fo o 0 0
6
respectively. For the ®bre cross-section shown in Fig. 1 we have
7
Strictly speaking, the relationship we shall use is aected by the shape of the ®bre in terms of the ratio t=d. So the procedure can be considered as approximate since the values of t and d vary even within one ®bre batch. If we assume that characteristic defects are located on the ®bre surface or in its vicinity, then the iteration procedure the set of strength characteristics, to obtain that is ; ob ; So , remains as described above for the volume hypothesis, with Eq. (3) being replaced with So lo Pav where Pav is an average value of the cross-section perimeter. To obtain ot we use the corresponding relationship found in Ref. 8 that is
1 P ÿ1= 4 h 1 2 ÿ1= 2
M MSICM
8
The dependencies given by Eqs. (7) and (8) are shown in Fig. 3. It is clear that the larger the value of the Weibull parameter , the closer to each other are the values of ob and ot . The eect of is not so pronounced. 3. Experimental We present here experimental data to illustrate the application of the method. Results of its systematic application to be published elsewhere [9].
Fig. 4. A photograph of a ®bre of ICM type prepared for counting the number of breaks.
1980
V.M. Kiiko, S.T. Mileiko / Composites Science and Technology 59 (1999) 1977±1981
A batch of the Al2O3±Al5Y3O12 eutectic ®bre produced by the internal-crystallization route were tested according to the scheme described above. A view of a ®bre looped around a rigid cylinder and then un-looped to count the number of breaks is shown in Fig. 4. The results of calculations of the set of Weibull parameters
are given in Table 1. The experimental data are given in Figs. 5 and 6. Scale dependencies of the ®bre strength are also plotted. In order to make their appearance more conventional, the equation
Table 1 The Weibull parameters for a batch of Al2O3±Al5Y3O12 eutectic ®bres. The Young's modulus is assumed to be 400 Gpa
is used in Fig. 5 instead of Eq. (2). Similarly, the following equation is plotted in Fig. 6:
Volume defect hypothesis
Surface defect hypothesis
0b (MPa)
0t (MPa)
Vo (mm3)
0b (MPa)
0t (MPa)
So (mm2)
3.67
1249
857.5
0.0320
3.64
844.8
379.1
10.054
lAAv ÿ1= l o Vo
lPAv ÿ1= : l o So
9
10
Note that both values of o , namely ob and ot , are used in the ®gures. 4. Conclusions A technique of ®bre testing by looping them around a series of rigid cylinders of successively smaller diameters appears to be suitable, after minor modi®cation, for the strength evaluation of oxide ®bres produced by the internal-crystallization method. The ®bre tensile strength is obtained by using a procedure based on Weibull statistics. Acknowledgements
Fig. 5. The experimental data (¯exural strength) and calculated scale dependencies of the ¯exural and tensile strengths of the Al2O3± Al5Y3O12 eutectic ®bres (Block K195). The volume hypothesis is assumed. The Weibull parameters are given in Table 1.
The work has been performed under ®nancial support of the International Science and Technology Center (Project no. 507-97) and INTAS-RFBR (Project no. 950599). A contribution to the experimental work done by Mr. A.A. Kolchin, Dr. G.K. Strukova, and Mrs. T.I. Boromychenko is gratefully acknowledged. References
Fig. 6. The experimental data (¯exural strength) and calculated scale dependencies of ¯exural and tensile strengths of the Al2O3±Al5Y3O12 eutectic ®bres (Block K195). The surface hypothesis is assumed. The Weibull parameters are given in Table 1.
[1] Mileiko ST, Kazmin VI. Crystallization of ®bres inside a matrix: a new way of fabrication of composites. J Mater Sci 1992; 27:2165±72. [2] Mileiko ST, Kazinin VI. Structure and mechanical properties of oxide ®bre reinforced metal matrix composites produced by the internal crystallization method. Comp Sci Technol 1992;45:209±20. [3] Mileiko ST. Metal and ceramic based composites. Amsterdam: Elsevier, 1997. [4] Mileiko ST. Novel oxide ®bres to reinforce metal, intermetallic and ceramic matrices. In: Advanced Multilayered and FibreReinforced Composites, Proc. of NATO Advance Res. Workshop, Kiev, Ukraine, June 1997. Dordrecht: Kluwer Academic, 1998. p. 333±52. [5] Mileiko ST. Heat-resistant oxide-®bre composites. Presented at ECCM-8, Naples, 1998. [6] Mileiko ST. Oxide ®bres. In: Watt WW, Perov BV, editors. Strong Fibres Handbook of Composites, Vol. 1. Amsterdam: North-Holland, 1985. p. 87±114.
V.M. Kiiko, S.T. Mileiko / Composites Science and Technology 59 (1999) 1977±1981 [7] Siemers PA, Mehati RL, Moran H. A comparison of the uniaxial tensile and pure bending strength of SiC ®laments. J Mater Sci 1988;23:1329±33. [8] Mileiko ST. On relationships between strength values obtained in tension and bending. In preparation.
1981
[9] Mileiko ST, Kiiko VM, Sarkissyan NS, Starostin MYu, Gvozdeva SI, Kolchin AA, Strukova GK. Microstructure and properties of A12O3±Al5Y3O12 ®bres produced by an internal crystallization route. Comp Sci Technol 1999;59(11):1763±72.