Availableonlineatwww.sciencedirectcom ScienceDirect E噩≈RS ELSEVIER Joumal of the European Ceramic Society 27(2007)2103-2110 www.elsevier.comlocate/jeurceramsoc Influence of individual thermal shock parameters on stress generated in silicon nitride and its prediction Ernest gondar , Miroslav rosko a jan Slovak University of Technology, Faculry of Mechanical Engineering, Pionierska 15, 83102 Bratislava, Slovak Republic Received 19 February 2006: received in revised form 12 July 2006; accepted 21 July 2006 Available online 2 October 200 Influence of repeated thermal shock on the stress generated in silicon nitride, was determined by a new testing method. This method allows verification of temperature and stress progress obtained from puter simulation. Input parameters were temperature, temperature difference, heating and cooling time Output parameters were the mean stress and stress peaks of specific cycles. Two methods were used to compare the influence-a newly defined parameter of infuence(PI)and a least square method. The results show a dominant influence of the temperature value which is higher than the influence of temperature difference. The least squares method was also used to predict the value of stress, with coefficient of determination higher than 0.95 o 2006 Elsevier Ltd. All rights reserved. Keywords: Thermal shock resistance; Testing: Si3N4 1. Introduction lute values of heating and cooling temperature and the heating and cooling time. The aim of our research is also the prediction Silicon nitride is known for its outstanding properties such of stress values, based on statistical analysis as hardness, abrasion resistance, chemical stability, creep resis- tance, high-temperature strength(up to 1400C)and its resis- 2. Experimental tance to thermal shocks There are several methods used to test the resistance of silicon nitride to thermal shocks, among Silicon nitride specimens were prepared by cold pressing and which the indentation-quench method is one of the most widely then hot pressing in nitrogen atmosphere. Activating densifica- used for technical ceramics with high resistance to thermal tion aids for silicon nitride were AlO3 and Y2O3 with a mass shocks--since 1996 to the present time. 2. A number of mod- ratio corresponding to 10% of YAG. The hot pressing of the ifications of this method have been introduced to lower the experimental material was performed on a laboratory hot press time consumption especially when dealing with repeated ther- with a special construction for the heating body.(Fig. 1).This mal shocks.In the Department of Materials and Technologies graphite body consists of two sections(1)and their electrical of the Faculty of Mechanical Engineering of the Slovak Univer- inputs(2)end with semicircular segments(3).These segments sity of Technology, a new testing method has been developed provide shielding of main electrical inputs(4) against heat radi and optimized' together with a computer simulation of the ated from the body temperature and stress conditions in the tested material.This The prepared specimens were of 2 mm thickness with 8mm simulation makes it possible to evaluate the influence of ther- diameter. In the middle of the specimens, cracks were initiated mal shock parameters on the stress, generated in the specimen. using a Vicker's indentor. These cracks were used to determine These parameters include the temperature difference, the abso- the fracture toughness and were also important for the testing initiated using a hemispherical indentor. o Sed also for indents, method itself. A similar method is being Corresponding author. Tel. +421 244455087: fax: +421 2 4445509 The depth profile of cracks was determined by horizont E-mail address: ernest gondar@stub.sk(E Gondar) erial sectioning of the material as well as from the observation 0955-2219/S-see front matter o 2006 Elsevier Ltd. All rights reserved. doi: 10. 1016/j-jeurceramsoc. 2006.07.024
Journal of the European Ceramic Society 27 (2007) 2103–2110 Influence of individual thermal shock parameters on stress generated in silicon nitride and its prediction Ernest Gondar a,∗, Miroslav Rosko a, Jan Somorcik b a Slovak University of Technology, Faculty of Mechanical Engineering, Pionierska 15, 831 02 Bratislava, Slovak Republic b Comenius University, Faculty of Mathematics, Physics and Informatics, Mlynska dolina, 842 48 Bratislava, Slovak Republic Received 19 February 2006; received in revised form 12 July 2006; accepted 21 July 2006 Available online 2 October 2006 Abstract Influence of repeated thermal shock on the stress generated in silicon nitride, was determined by a new testing method. This method allows verification of temperature and stress progress obtained from a computer simulation. Input parameters were temperature, temperature difference, heating and cooling time. Output parameters were the mean stress and stress peaks of specific cycles. Two methods were used to compare the influence—a newly defined parameter of influence (PI) and a least square method. The results show a dominant influence of the temperature values, which is higher than the influence of temperature difference. The least squares method was also used to predict the value of stress, with coefficient of determination higher than 0.95. © 2006 Elsevier Ltd. All rights reserved. Keywords: Thermal shock resistance; Testing; Si3N4 1. Introduction Silicon nitride is known for its outstanding properties such as hardness, abrasion resistance, chemical stability, creep resistance, high-temperature strength (up to 1400 ◦C) and its resistance to thermal shocks. There are several methods used to test the resistance of silicon nitride to thermal shocks, among which the indentation-quench method is one of the most widely used for technical ceramics with high resistance to thermal shocks—since 19961 to the present time.2,3 A number of modifications of this method have been introduced to lower the time consumption especially when dealing with repeated thermal shocks.4–6 In the Department of Materials and Technologies of the Faculty of Mechanical Engineering of the Slovak University of Technology, a new testing method has been developed and optimized7 together with a computer simulation of the temperature and stress conditions in the tested material.8 This simulation makes it possible to evaluate the influence of thermal shock parameters on the stress, generated in the specimen. These parameters include the temperature difference, the abso- ∗ Corresponding author. Tel.: +421 2 44455087; fax: +421 2 44455091. E-mail address: ernest.gondar@stuba.sk (E. Gondar). lute values of heating and cooling temperature and the heating and cooling time. The aim of our research is also the prediction of stress values, based on statistical analysis. 2. Experimental Silicon nitride specimens were prepared by cold pressing and then hot pressing in nitrogen atmosphere. Activating densification aids for silicon nitride were Al2O3 and Y2O3 with a mass ratio corresponding to 10% of YAG. The hot pressing of the experimental material was performed on a laboratory hot press with a special construction for the heating body9 (Fig. 1). This graphite body consists of two sections (1) and their electrical inputs (2) end with semicircular segments (3). These segments provide shielding of main electrical inputs (4) against heat radiated from the body. The prepared specimens were of 2 mm thickness with 8 mm diameter. In the middle of the specimens, cracks were initiated using a Vicker’s indentor. These cracks were used to determine the fracture toughness and were also important for the testing method itself. A similar method is being used also for indents, initiated using a hemispherical indentor.10 The depth profile of cracks was determined by horizontal serial sectioning of the material, as well as from the observation 0955-2219/$ – see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jeurceramsoc.2006.07.024
2104 E Gondar et al Joumal of the European Ceramic Society 27(2007)2103-2110 ew method of testin resistance of silicon nitride to repeated ther cks. (1)Specimen, (2 nt,(3)crack, (4)punch, (5) diameterof the punch(5 mm) is equal to the hole diameter of the sealing. This way, the only mechanical loading of the specimen is shear acting under the perimeter of the punch. Mechanical loading acting on the indent and the cracks is negligible. The Fig.1.Graphite heating body of the laboratory hot press. (1)Sections of the type of mechanical loading can be changed by modifying the body.(2)electrical input to the heating body. (3)semicircular segments and (4) punch diameter. The method is described more in detail in Refs main electrical input [14, 15]. Test chamber of the device is made of steel, the transpar ent front wall, which allows observation, is made of temperature of fracture areas. The sectioning took place in direction parallel resistant glass Rear wall, through which leads the induction coil with the surface of the specimen, contrary to a previously used is made of thermally insulating material based on aramide fibers method,2 where the direction of sectioning was perpendicular Because the punch is made of molybdenum, the chamber must be to the surface Low speed of grinding discs, as well as rich lubri- evacuated before the test. During testing, protective gas(argon) ation ensure that the crack size does not increase during the is passing through the test chamber. Simulation and verification ectioning. The depth profile is needed to specify the critical of temperature and stress progress has been introduced in Ref point for crack growth, which is located on the crack, under the [15] The principle of the new testing method(protected by a parameters of the test. The reference parameters are as or deformed zone(Fig. 2) The new testing method allows modification of various input patent)is shown in Fig 3. Specimens of circular cross-section lows: heating temperature Th=1100C, cooling temperature (1)are being used in this test, with cracks (3) formerly initiated Tc=500C, heating time th=16s and cooling time tc=6 by Vicker's indentor. The specimen is placed with its damaged These values provide the most intense thermal shock that can be side(2)facing downwards and this side is constantly(without achieved using our available equipment. For simplicity, a new interruptions) cooled by water. Its opposite side is cyclically method of description of input parameters has been introduced heated using a molybdenum punch(4). The punch is heated by Using this method, the reference parameters would be described an induction coil and is loaded by a mass of 6 kg. Its weight pre- as 1100/500-16/6 vents the cooling water from leaking from under the sealing The The stress progress for these parameters for a specimen with thickness of 20 mm(in the critical point--Fig. 2)is shown in Fig. 4. It can be seen, that the amplitude even in the first cycle is low, hence we can use the mean stress as representative for the entire stress progress. Thus, the first step was the evaluation of the influence of input parameters on the mean stress in a 20 mm thick specimen In Ref [15], specimens with thickness of 2 mm were subject of a more thorough research and verification. The stress progress for this specimen thickness is shown in Fig. 5. It can be seen that even after 200 cycles the stress is not stabilized. This fact deformed zone makes the evaluation of parameter influence more complicated. To facilitate the reproducibility of the results, it is necessary to study not only the mean stress, but also the value of stress in Fig. 2. Depth profile of the crack concrete cycles. This case is more common when dealing with
2104 E. Gondar et al. / Journal of the European Ceramic Society 27 (2007) 2103–2110 Fig. 1. Graphite heating body of the laboratory hot press. (1) Sections of the body, (2) electrical input to the heating body, (3) semicircular segments and (4) main electrical input. of fracture areas. The sectioning took place in direction parallel with the surface of the specimen,11 contrary to a previously used method,12 where the direction of sectioning was perpendicular to the surface. Low speed of grinding discs, as well as rich lubrication ensure that the crack size does not increase during the sectioning. The depth profile is needed to specify the critical point for crack growth, which is located on the crack, under the deformed zone (Fig. 2). The principle of the new testing method (protected by a patent13) is shown in Fig. 3. Specimens of circular cross-section (1) are being used in this test, with cracks (3) formerly initiated by Vicker’s indentor. The specimen is placed with its damaged side (2) facing downwards and this side is constantly (without interruptions) cooled by water. Its opposite side is cyclically heated using a molybdenum punch (4). The punch is heated by an induction coil and is loaded by a mass of 6 kg. Its weight prevents the cooling water from leaking from under the sealing. The Fig. 2. Depth profile of the crack. Fig. 3. Principle of the new method of testing the resistance of silicon nitride to repeated thermal shocks. (1) Specimen, (2) indent, (3) crack, (4) punch, (5) sealing, and (6) support. diameter of the punch (5 mm) is equal to the hole diameter of the sealing. This way, the only mechanical loading of the specimen is shear acting under the perimeter of the punch. Mechanical loading acting on the indent and the cracks is negligible. The type of mechanical loading can be changed by modifying the punch diameter. The method is described more in detail in Refs. [14,15]. Test chamber of the device is made of steel, the transparent front wall, which allows observation, is made of temperature resistant glass. Rear wall, through which leads the induction coil, is made of thermally insulating material based on aramide fibers. Because the punch is made of molybdenum, the chamber must be evacuated before the test. During testing, protective gas (argon) is passing through the test chamber. Simulation and verification of temperature and stress progress has been introduced in Ref. [15]. The new testing method allows modification of various input parameters of the test. The reference parameters are as follows: heating temperature Th = 1100 ◦C, cooling temperature Tc = 500 ◦C, heating time th = 16 s and cooling time tc = 6 s. These values provide the most intense thermal shock that can be achieved using our available equipment. For simplicity, a new method of description of input parameters has been introduced. Using this method, the reference parameters would be described as 1100/500-16/6. The stress progress for these parameters for a specimen with thickness of 20 mm (in the critical point15—Fig. 2) is shown in Fig. 4. It can be seen, that the amplitude even in the first cycle is low, hence we can use the mean stress as representative for the entire stress progress. Thus, the first step was the evaluation of the influence of input parameters on the mean stress in a 20 mm thick specimen. In Ref. [15], specimens with thickness of 2 mm were subject of a more thorough research and verification. The stress progress for this specimen thickness is shown in Fig. 5. It can be seen that even after 200 cycles the stress is not stabilized. This fact makes the evaluation of parameter influence more complicated. To facilitate the reproducibility of the results, it is necessary to study not only the mean stress, but also the value of stress in concrete cycles. This case is more common when dealing with
ondar et al. Journal of the European Ceramic Society 27(2007)2103-2110 Necessary condition when evaluating PI is that only one input parameter can be changed at a time. Other parameters must remain constant. The parameter of influence is only suitable for comparing of parameters'influence, it cannot be used to determine the absolute influence of a parameter. The evaluation using PI has been developed because we could not find a suit able widely used method. There was a possibility of using the ANOVA method 6(analysis of variance), but this method only evaluates, whether or not a parameter has infuence on the out- put quantity. It does not compare the infuence of the parameters Another disadvantage is, that it can only be used for cases with a finite number of values Our aim was not only to analyze the results, but also to predict the value of stress in the critical point. Based on the analysis of Fig. 4. Stress progress in critical point of a 20 mm thick specimen. our results, most suitable method was the least squares method Using this method, we were able to obtain the parameters'esti mates in a linear regression model in the form o=aTh+bTc+cth+ dte +e+E (2) where a, b, c, d and e are regression coefficients and e is the methods commonly used for statistical analysis of results. I7 e normally distributed random error. This method is one of 3. Results and discussion Results for 20 mm thick specimen are shown in Table 1. First 0020002500300035004000 line, containing reference parameters, is marked bold Changed 5001000 parameter is introduced in the third column, with its percentual change given in the fourth one. Most intense shocks, which can Fig. 5. Stress progress in critical point of a 2 mm thick specimen. be experimentally achieved, are caused by reference parame- ters. For this reason the changed input parameters caused lower practical experiments, because mean stress is not responsible intensity of thermal shock. The influence of heating temperature for unstable crack growth. We evaluated the influence of input was studied by decreasing Th by 100C, maintaining constant parameters on the stress peaks in the 4th and 20th cycle temperature difference, hence the cooling temperature had to be For both specimen thicknesses, the influence of input param- lowered of the same amount as well(simulations 2 and 3).The eters on the stress can be described by a so-called parameter influence of temperature difference was evaluated by increasing of influence(PD). The PI is a ratio of the change of stress to the cooling temperature while keeping the heating temperature the change of input parameter. This change(expressed in %)is constant(simulations 4 and 5). The influence of heating and understood with respect to the reference parameters. Thus, when cooling time was studied by increasing both quantities, lower- looking for example for the influence of heating temperature on ing thus again the intensity of thermal shock (simulations 6-9) the value of stress. the pi will be calculated as The next two columns of the table contain the values of the mean △σ[% stress in critical point and the percentual change with respect to (1) the value obtained at reference parameters. Parameter of influ △i[%] ence(Pl) is given in the last column. From Table 1 it can be Table Calculation of Pl from simulations output for a 20 mm thick specimen Simulation number Changed paramet Change of the parameter(%) Change of omean(%) 1100/500-16/6 28.97 1000/400-16/6 Th=1000°C 13.92 1530 900300-16/6 Th=900°C 32.46 1100/600-16/6 △T=500°C 0.370 1100700-166 △T=400°C 32.53 0.369 1100/500-20/6 h=20s 1100/500-25/6 0.004 t=10s 66.7 0.1 0.002
E. Gondar et al. / Journal of the European Ceramic Society 27 (2007) 2103–2110 2105 Fig. 4. Stress progress in critical point of a 20 mm thick specimen. Fig. 5. Stress progress in critical point of a 2 mm thick specimen. practical experiments, because mean stress is not responsible for unstable crack growth. We evaluated the influence of input parameters on the stress peaks in the 4th and 20th cycle. For both specimen thicknesses, the influence of input parameters on the stress can be described by a so-called parameter of influence (PI). The PI is a ratio of the change of stress to the change of input parameter. This change (expressed in %) is understood with respect to the reference parameters. Thus, when looking for example for the influence of heating temperature on the value of stress, the PI will be calculated as: PI = σ[%] Th[%] (1) Necessary condition when evaluating PI is that only one input parameter can be changed at a time. Other parameters must remain constant. The parameter of influence is only suitable for comparing of parameters’ influence, it cannot be used to determine the absolute influence of a parameter. The evaluation using PI has been developed because we could not find a suitable widely used method. There was a possibility of using the ANOVA method16 (analysis of variance), but this method only evaluates, whether or not a parameter has influence on the output quantity. It does not compare the influence of the parameters. Another disadvantage is, that it can only be used for cases with a finite number of values. Our aim was not only to analyze the results, but also to predict the value of stress in the critical point. Based on the analysis of our results, most suitable method was the least squares method. Using this method, we were able to obtain the parameters’ estimates in a linear regression model in the form: σ = aTh + bTc + cth + dtc + e + ε (2) where a, b, c, d and e are regression coefficients and ε is the normally distributed random error. This method is one of the methods commonly used for statistical analysis of results.17 3. Results and discussion Results for 20 mm thick specimen are shown in Table 1. First line, containing reference parameters, is marked bold. Changed parameter is introduced in the third column, with its percentual change given in the fourth one. Most intense shocks, which can be experimentally achieved, are caused by reference parameters. For this reason the changed input parameters caused lower intensity of thermal shock. The influence of heating temperature was studied by decreasing Th by 100 ◦C, maintaining constant temperature difference, hence the cooling temperature had to be lowered of the same amount as well (simulations 2 and 3). The influence of temperature difference was evaluated by increasing the cooling temperature while keeping the heating temperature constant (simulations 4 and 5). The influence of heating and cooling time was studied by increasing both quantities, lowering thus again the intensity of thermal shock (simulations 6–9). The next two columns of the table contain the values of the mean stress in critical point and the percentual change with respect to the value obtained at reference parameters. Parameter of influence (PI) is given in the last column. From Table 1 it can be Table 1 Calculation of PI from simulations output for a 20 mm thick specimen Simulation number Input parameters Changed parameter Change of the parameter (%) σmean Change of σmean (%) PI 1 1100/500-16/6 – – 28.97 – 2 1000/400-16/6 Th = 1000 ◦C 9.1 25.43 13.92 1.530 3 900/300-16/6 Th = 900 ◦C 18.2 21.87 32.46 1.784 4 1100/600-16/6 T = 500 ◦C 16.7 30.76 6.18 0.370 5 1100/700-16/6 T = 400 ◦C 33.3 32.53 12.29 0.369 6 1100/500-20/6 th = 20 s 25 29.00 0.1 0.004 7 1100/500-25/6 th = 25 s 56.3 28.98 0.03 0.001 8 1100/500-16/8 tc = 8 s 33.3 29.01 0.14 0.004 9 1100/500-16/10 tc = 10 s 66.7 29.00 0.1 0.002
210 E Gondar et al Joumal of the European Ceramic Society 27(2007)2103-2110 0,3690,37 20量 0001000400020004 ange of th change of te change ofaT change dTh change af AT 6. Graphical representation of PI for a 20 mm thick specimen, the two Fig. 7. Graphical representation of PI for a 20 mm thick specimen, the two most columns showing the infiuence of change of heating temperature Th rightmost columns showing the influence of change of average temperature Tavg seen that the reference parameters, which we stated as the most ntense ones. caused a lower value of mean stress in most simula tions(4-9). However, the reference parameters were determined as most intense not for mean stress, but for stress peak Graphical representation of the results is shown in Fig. 6. It can be seen that the heating and cooling time has generally the lowest influence on the mean stress pi is low because a significant change of these parameters caused only a negligible of Omean. Most important conclusion from this series of simulations is that the temperature difference does not have the most significant influence on the resulting stress. Pi reached its highest value with the change of heating temperature. The difference between the two values(1.53 and 1.784)represents 16.6%, which indicates only a relative suitability of Pl change of th change of aT change of Tavg change of Th The main reason for such a high PI in the case of change of heating temperature is probably that in fact two parameters Fig 8. Graphical representation of Pl on mean stress for a 2 mm thick specimen (heating and cooling temperature) were changed at a time in order to maintain a constant temperature difference(Table 1 lines 2 and 3). However, during the definition of PI we declared progress in this case, more simulations were performed for this eter can thickness. The results for mean stress can be seen in table 2 which describes the change of both temperatures is in this and Fig 8. The change of both heating and cooling temperature case simply their average value Tavg. Parameters Tmax, Tmin simulations 2-7)is again expressed also with the average tem- and Tavg are characteristic for thermal fatigue. The aim of our perature Tavg as an input parameter. The results for stress peaks work, however, is analysis of parameters within a small number in the 4th and 20th cycle are shown in Figs. 9 and 10 thermal fatigue, although our testing method is suitable also of parameters on the mean stress and on stress peaks in the 4th for this case. Let us replace lines 2 and 3 of the previous table. and 20th cycle. The only exceptionis the cooling time, especially Simulation number Input parameters Changed parameter Change of the parameter(o) Change of mean(%) 1100/500-16/6 Tavg=800°C 28.97 Tawg=700°C 900/300-166 New results are shown in Fig. 7. The influence of temper- atures, expressed by their average value, is lower. However, when comparing its influence on the mean stress and on the also the average temperature has more significant infuence on stress in fourth cycle( Figs. and 9). However, the influence is the mean stress than the temperature difference. The difference in both cases low, hence, this difference cannot be considered between the values of Pl(1.1l and 1.3)again casts doubt up decisive the Pl method as an absolute of parameters'influence The dominant influence of temperatures has been confirmed Next series of simulations was performed for a specimen also when taking into account the average temperature as an with thickness of 2 mm. Because of a more complicated stress input parameter. This can be considered a significant addition to
2106 E. Gondar et al. / Journal of the European Ceramic Society 27 (2007) 2103–2110 Fig. 6. Graphical representation of PI for a 20 mm thick specimen, the two rightmost columns showing the influence of change of heating temperature Th. seen that the reference parameters, which we stated as the most intense ones, caused a lower value of mean stress in most simulations (4–9). However, the reference parameters were determined as most intense not for mean stress, but for stress peaks. Graphical representation of the results is shown in Fig. 6. It can be seen that the heating and cooling time has generally the lowest influence on the mean stress, PI is low because a significant change of these parameters caused only a negligible change of σmean. Most important conclusion from this series of simulations is that the temperature difference does not have the most significant influence on the resulting stress. PI reached its highest value with the change of heating temperature. The difference between the two values (1.53 and 1.784) represents 16.6%, which indicates only a relative suitability of PI. The main reason for such a high PI in the case of change of heating temperature is probably that in fact two parameters (heating and cooling temperature) were changed at a time in order to maintain a constant temperature difference (Table 1, lines 2 and 3). However, during the definition of PI we declared that only one parameter can be changed at a time. A parameter which describes the change of both temperatures is in this case simply their average value Tavg. Parameters Tmax, Tmin and Tavg are characteristic for thermal fatigue. The aim of our work, however, is analysis of parameters within a small number of repeated thermal shocks, which cannot be considered as thermal fatigue, although our testing method is suitable also for this case. Let us replace lines 2 and 3 of the previous table. Simulation number Input parameters Changed parameter Change of the parameter (%) σmean Change of σmean (%) PI 1 1100/500-16/6 Tavg = 800 ◦C – 28.97 – 2 1000/400-16/6 Tavg = 700 ◦C 12.5 25.43 13.92 1.11 3 900/300-16/6 Tavg = 600 ◦C 25 21.87 32.46 1.30 New results are shown in Fig. 7. The influence of temperatures, expressed by their average value, is lower. However, also the average temperature has more significant influence on the mean stress than the temperature difference. The difference between the values of PI (1.11 and 1.3) again casts doubt upon the PI method as an absolute of parameters’ influence. Next series of simulations was performed for a specimen with thickness of 2 mm. Because of a more complicated stress Fig. 7. Graphical representation of PI for a 20 mm thick specimen, the two rightmost columns showing the influence of change of average temperature Tavg. Fig. 8. Graphical representation of PI on mean stress for a 2 mm thick specimen. progress in this case, more simulations were performed for this thickness. The results for mean stress can be seen in Table 2 and Fig. 8. The change of both heating and cooling temperature (simulations 2–7) is again expressed also with the average temperature Tavg as an input parameter. The results for stress peaks in the 4th and 20th cycle are shown in Figs. 9 and 10. The results show only small difference between the influence of parameters on the mean stress and on stress peaks in the 4th and 20th cycle. The only exception is the cooling time, especially when comparing its influence on the mean stress and on the stress in fourth cycle (Figs. 8 and 9). However, the influence is in both cases low, hence, this difference cannot be considered decisive. The dominant influence of temperatures has been confirmed also when taking into account the average temperature as an input parameter. This can be considered a significant addition to
ondar et al. Journal of the European Ceramic Society 27(2007)2103-2110 2107 Table 2 Calculation of Pl from simulations output for a 2 mm thick specimen Simulation number ThC°C Changed parameter Change of the parameter(%) Change of omean(%) 95.625 800200-16/6 27. 61319 1314 T=850°C 22.7 66.847 900300-16/6 Th=900° 18.2 72752 1314 950350-166 Tb=950°C 17.94 1319 1000/400-16/6 Th=1000°C 9.1 84.188 1314 Th=1050°C 4.5 89906 5.98 678901234561890223 1100550-166 0.346 1100600-16/6 101.126 0.344 1100650-166 103424 8.16 1100700-16/6 106.627 11.51 036 1100750-166 41.7 109.378 14.3 0.345 1100500-167 16 95.166 0.48 0.029 1100500-16/8 33.3 0. 14 0.004 1100500-169 50 95.03 61 0.012 1100500-16/10 95.378 0.26 1100500-16/l1 95.331 0.31 0o4 1100500-16/12 95.288 0.35 0.004 1100500-18/6 12.5 0.04 0.003 1100/500-20/6 95.7 0.08 0.003 1100/500-22/6 95.731 0 0.003 1100/500-24/6 h=24s 50 95.75 0.1 0.003 1100/500-25/6 h=25s 0.15 0.003 the""approach"used so far. Since The temperature difference has a point located on the axis of the specimen, 0.6 mm under the been considered not only as dominant, but as the only parameter heated side of the specimen. of thermal shock when evaluating the resistance of technical The number of simulations performed allows a statistical ceramics to thermal shocks.2.6, 18-4 nalysis of the results. Using least squares method, we obtained The possibility to determine the influence of parameters on the parameters' estimates in the linear regression model for the the stress becomes very complicated in the case when the mean mean stress in critical point and the resulting equations is stress in unstable(Fig. 11), because it is difficult to find a relevant value of the stress. The stress progress shown in Fig. 1l is from Omean=0.059Th+0.055Tc+0.031th-0054]e+2.866 (3) 1。8 4320 且 man change of at change of Tawgchange cf change af th change of te change of aT change of Tavgchange of Th Graphical representation of PI on stress peaks in the fourth cycle for a Fig. 10. Graphical representation of Pl on stress peaks in the 20th cycle for a thick specimen. 2 mm thick specimen
E. Gondar et al. / Journal of the European Ceramic Society 27 (2007) 2103–2110 2107 Table 2 Calculation of PI from simulations output for a 2 mm thick specimen Simulation number Th ( ◦C) Changed parameter Change of the parameter (%) σmean Change of σmean (%) PI 1 1100/500-16/6 – 95.625 2 800/200-16/6 Th = 800 ◦C 27.3 61.319 35.88 1.314 Tavg = 500 ◦C 37.5 0.957 3 850/250-16/6 Th = 850 ◦C 22.7 66.847 30.09 1.326 Tavg =550 ◦C 31.3 0.961 4 900/300-16/6 Th = 900 ◦C 18.2 72.752 23.92 1.314 Tavg = 600 ◦C 25 0.957 5 950/350-16/6 Th = 950 ◦C 13.6 78.470 17.94 1.319 Tavg = 650 ◦C 18.8 0.954 6 1000/400-16/6 Th = 1000 ◦C 9.1 84.188 11.96 1.314 Tavg = 700 ◦C 12.5 0.957 7 1050/450-16/6 Th = 1050 ◦C 4.5 89.906 5.98 1.329 Tavg = 750 ◦C 6.3 0.949 8 1100/550-16/6 T = 550 ◦C 8.3 98.375 2.88 0.346 9 1100/600-16/6 T = 500 ◦C 16.7 101.126 5.75 0.344 10 1100/650-16/6 T = 450 ◦C 25 103.424 8.16 0.326 11 1100/700-16/6 T = 400 ◦C 33.3 106.627 11.51 0.346 12 1100/750-16/6 T = 350 ◦C 41.7 109.378 14.38 0.345 13 1100/500-16/7 tc = 7 s 16.7 95.166 0.48 0.029 14 1100/500-16/8 tc = 8 s 33.3 95.487 0.14 0.004 15 1100/500-16/9 tc = 9 s 50 95.038 0.61 0.012 16 1100/500-16/10 tc = 10 s 66.7 95.378 0.26 0.004 17 1100/500-16/11 tc = 11 s 83.3 95.331 0.31 0.004 18 1100/500-16/12 tc = 12 s 100 95.288 0.35 0.004 19 1100/500-18/6 th = 18 s 12.5 95.666 0.04 0.003 20 1100/500-20/6 th = 20 s 25 95.701 0.08 0.003 21 1100/500-22/6 th = 22 s 37.5 95.731 0.11 0.003 22 1100/500-24/6 th = 24 s 50 95.756 0.14 0.003 23 1100/500-25/6 th = 25 s 56.3 95.768 0.15 0.003 the “approach” used so far. Since The temperature difference has been considered not only as dominant, but as the only parameter of thermal shock when evaluating the resistance of technical ceramics to thermal shocks.2,6,18–20 The possibility to determine the influence of parameters on the stress becomes very complicated in the case when the mean stress in unstable (Fig. 11), because it is difficult to find a relevant value of the stress. The stress progress shown in Fig. 11 is from Fig. 9. Graphical representation of PI on stress peaks in the fourth cycle for a 2 mm thick specimen. a point located on the axis of the specimen, 0.6 mm under the heated side of the specimen. The number of simulations performed allows a statistical analysis of the results. Using least squares method, we obtained the parameters’ estimates in the linear regression model for the mean stress in critical point and the resulting equations is: σmean = 0.059Th + 0.055Tc + 0.031th − 0.054tc + 2.866 (3) Fig. 10. Graphical representation of PI on stress peaks in the 20th cycle for a 2 mm thick specimen.
E Gondar et al Joumal of the European Ceramic Society 27(2007)2103-2110 y=00138x+9515 00263 R2=0.9945 04000 ● heating time cooling time Fig. 13. Linearity of time influence on mean stress. Fig. 11. Stress progress in a point located on the axis of 2 mm thick specimen. of these input parameters on the mean stress. Low value of the 0.6mm under its heated side R-squared value of cooling time is caused by dispersion of the results, which corresponds also to the dispersion of PI for this The results show a very good linearity--coefficient of deter- case(Fig. 8 and Table 2, rows 13-18). In absolute values, how- mination for this model was 0.999855721 hence the ever, this dispersion is negligil ble. as the maximum difference can be considered reliable without having to"filter out"any between the results is 0.449MPa errors using for example robust models. Graphical represen- High correlation is maintained also when dealing with stress tation of Eq. (3)would require a multidimensional graph. We in a specific cycle. The resulting equations for the 4th and 20th cycle are stated below along with their coefficients of determ parameters influence: a plot of stress versus one input parame- tion. Despite lower coefficient of determination in these cycles ter, while keeping the others constant. The linearity of influence its value is high enough to prove the equations reliable of average temperature(at AT=const)and cooling temperature (at Th=const)is shown in Fig. 12(for mean stress), along with 04thcycle =0.422Th+0.017Tc-0418th-2683te+25.686, the equations of their linear trend models and their coefficients coefficient of determination=0.997279754 of determination. The equations in this figure and the next one contain variables x and y instead of the real physical quantities This is a formal decision to prevent units mismatch. The slope 020th cycle =2.288Th+1.9Te-1 158-9798tc+104.59, of these lines also demonstrates the influence of the parameters. coefficient of determination =0.965192316 Again we must realize that the case of Th=const represents in fact the influence of AT. It proves true again that the influence The linearity of input parameters influence in the fourth cycle of AT is lower than the influence of the absolute values of heat- is shown if Figs. 14 and 15. In this case there is an evident ing and cooling temperature. There is an unexpected influence increase of the influence of cooling time, which corresponds to of increasing cooling temperature(at Th=const)on the mean the results of PI in this cycle(Fig 9). The slope of the cooling stress Increasing of cooling temperatures causes lower temper- time's trend line is negative, but this does not indicate a lower ature difference which should lead to lower values of mean influence-the influence is expressed by deviation of the trend stress line from its horizontal position. The influence of cooling tem- The linearity of influence of heating and cooling time is perature(influence of AD)on stress peaks in the fourth cycle is shown in Fig. 13 and again demonstrates a negligible influence lower than its influence on the mean stress. This corresponds to y=0.0548X+68,203 R2m0.9987 y=0,1166X+28,127 ◆ average temperature△T= const A cooling temperature Th=const A cooing temperature Th=con 00450500550600650700750800850 400450500550600650700750800850 Temperature [C] Temperature{°C Fig. 12. Linearity of temperature influence on mean stress. Fig. 14. Linearity of temperature influence on stress peaks in the fourth cycle
2108 E. Gondar et al. / Journal of the European Ceramic Society 27 (2007) 2103–2110 Fig. 11. Stress progress in a point located on the axis of 2 mm thick specimen, 0.6 mm under its heated side. The results show a very good linearity—coefficient of determination for this model was 0.999855721, hence the equation can be considered reliable without having to “filter out” any errors using for example robust models.21 Graphical representation of Eq. (3) would require a multidimensional graph. We chose a different approach to demonstrate the linearity of the parameters influence: a plot of stress versus one input parameter, while keeping the others constant. The linearity of influence of average temperature (at T = const) and cooling temperature (at Th = const) is shown in Fig. 12 (for mean stress), along with the equations of their linear trend models and their coefficients of determination. The equations in this figure and the next ones contain variables x and y instead of the real physical quantities. This is a formal decision to prevent units mismatch. The slope of these lines also demonstrates the influence of the parameters. Again we must realize that the case of Th = const represents in fact the influence of T. It proves true again that the influence of T is lower than the influence of the absolute values of heating and cooling temperature. There is an unexpected influence of increasing cooling temperature (at Th = const) on the mean stress. Increasing of cooling temperatures causes lower temperature difference, which should lead to lower values of mean stress. The linearity of influence of heating and cooling time is shown in Fig. 13 and again demonstrates a negligible influence Fig. 12. Linearity of temperature influence on mean stress. Fig. 13. Linearity of time influence on mean stress. of these input parameters on the mean stress. Low value of the R-squared value of cooling time is caused by dispersion of the results, which corresponds also to the dispersion of PI for this case (Fig. 8 and Table 2, rows 13–18). In absolute values, however, this dispersion is negligible, as the maximum difference between the results is 0.449 MPa. High correlation is maintained also when dealing with stress in a specific cycle. The resulting equations for the 4th and 20th cycle are stated below along with their coefficients of determination. Despite lower coefficient of determination in these cycles, its value is high enough to prove the equations reliable: σ4th cycle = 0.422Th + 0.017Tc − 0.418th − 2.683tc + 25.686, coefficient of determination = 0.997279754 (4) σ20th cycle = 2.288Th + 1.9Tc − 1.158th − 9.798tc + 104.59, coefficient of determination = 0.965192316 (5) The linearity of input parameters influence in the fourth cycle is shown if Figs. 14 and 15. In this case there is an evident increase of the influence of cooling time, which corresponds to the results of PI in this cycle (Fig. 9). The slope of the cooling time’s trend line is negative, but this does not indicate a lower influence—the influence is expressed by deviation of the trend line from its horizontal position. The influence of cooling temperature (influence of T) on stress peaks in the fourth cycle is lower than its influence on the mean stress. This corresponds to Fig. 14. Linearity of temperature influence on stress peaks in the fourth cycle.
E. Gondar et al. /Journal of the European Ceramic Society 27 (2007)2103-2110 estimate the value of stress in this particular cycle also for input 0080 y03302x+126.33 arameters that have not been simulated y=1,9017x+13089 The aim of this paper is an analysis of thermal shock param eters' influence in a point, which is critical for crack growth ◆ heating time oooling time under the conditions of above described loading Stress analysis was performed also for other 10 points located on vertical axis 25 30 of the specimen, from the heated side down to the cooled side Time [s Presentation of these results. however would increase the vol- ume of this paper too significantly and is beyond the scope of Fig. 15. Linearity of time influence on stress peaks in the fourth cycle the presented topic The results of our analysis are generally applicable for any umber of loading cycles In the case of a specimen with 2 mm width, the value of stress peaks in the critical point becomes stabilized after 16 cycles, 5 hence thermal loading with number y=0.1148X+51367 of cycles higher than this can be already considered as ther mal fatigue. Based on general characteristics and terminology of thermal fatigue it is possible to define maximum stress, min imum stress and the mean stress ● average temperature(△T= const) Cooing temperature(Th=const) 4. Conclusion The influence of repeated thermal shock parameters on stress, 400 450 500 550 600 650 700 750 800 850 generated in silicon nitride was evaluated using parameter of influence(PI)and statistical model. The results show clearly a dominant influence of temperature, followed by the influence of Fig. 16. Linearity of temperature influence on stress peaks in the 20th cycle. temperature difference, and heating and cooling time. This result the results of PI(Figs. 8 and 9). The coefficients of determination on the stress peaks in specific cycle uence on mean stress and again exceed 0.95 The statistical model, obtained using least squares method Figs. 16 and 17 demonstrate the linearity of input parame- howed very high coefficient of determination. The highest value ters influence in the 20th cycle. The influence of temperature is was reached by the equation for mean stress. Despite lower coef again dominant and the coefficients of determination are over ficients of equation for stress peaks in the 4th and 20th cycle, its 0.99. The main difference between the 4th and 20th cycle is value is high enough to prove the prediction reliable. The model the influence of the cooling temperature at Th=const (influence confirmed the surprisingly lower influence of temperature dif- of An). The influence of cooling temperature has an expected ference on the stress and a very low influence of heating and progress. Intersection of both lines in Fig. 16 is not of impor- cooling time tance, it simply indicates that the stress peak in 20th cycle is It is hence important, at least in the conditions of loading for the simulations 1100/750 and 1050/450 almost the same described in this article, to take into account not only the temper (137. 253 MPa and 137.464 MPa, respectively) The influence of heating and cooling time on the stress peaks cooling temperature when evaluating the resistance of technical is in both 4th and 20th cycle negative, which is logical, because ceramics to repeated thermal shocks. The heating and cooling the intensity of thermal shock decreases with increasing time. time should also not be neglected From the point of view of reproducibility of the results, it is important to note the high values of reliability, represented by fatigle nted results are applicable also for conditions of thermal the coefficient of determination. Using the least squares method, we can easily obtain a model equation for any cycle and then References a 1. Ande J. Indentation thermal shock test for ceram- 40958x+144.64 ics..Am. Ceran.Soc.,1996,79,1509-1514 R2=0.9958 2. Koh, Y H, Kim, H. w, Kim, H. E and Halloran, J, Thermal shock resis- ance of fibrous monolithic Si3 N4/BN ceramics. J. Eur. Ceram. Soc., 2004 24.2339-2347 the thermal shock resistance of alumina through the addition of submicron- sized aluminium nitride particles. J. Eur. Ceram Soc., 2004, 24, 2293-2301 T Vedula, V.R., Green, D J, Hellmann, J. R. and Segall, A. E, Test method- ology for thermal shock characterization of ceramics. J. Mater Sci., 1998, Fig. 17. Linearity of time influence on stress peaks in the 20th cycle. 33.5427-5432
E. Gondar et al. / Journal of the European Ceramic Society 27 (2007) 2103–2110 2109 Fig. 15. Linearity of time influence on stress peaks in the fourth cycle. Fig. 16. Linearity of temperature influence on stress peaks in the 20th cycle. the results of PI (Figs. 8 and 9). The coefficients of determination again exceed 0.95. Figs. 16 and 17 demonstrate the linearity of input parameters influence in the 20th cycle. The influence of temperature is again dominant and the coefficients of determination are over 0.99. The main difference between the 4th and 20th cycle is the influence of the cooling temperature at Th = const (influence of T). The influence of cooling temperature has an expected progress. Intersection of both lines in Fig. 16 is not of importance, it simply indicates that the stress peak in 20th cycle is for the simulations 1100/750 and 1050/450 almost the same (137.253 MPa and 137.464 MPa, respectively). The influence of heating and cooling time on the stress peaks is in both 4th and 20th cycle negative, which is logical, because the intensity of thermal shock decreases with increasing time. From the point of view of reproducibility of the results, it is important to note the high values of reliability, represented by the coefficient of determination. Using the least squares method, we can easily obtain a model equation for any cycle and then Fig. 17. Linearity of time influence on stress peaks in the 20th cycle. estimate the value of stress in this particular cycle also for input parameters that have not been simulated. The aim of this paper is an analysis of thermal shock parameters’ influence in a point, which is critical for crack growth under the conditions of above described loading. Stress analysis was performed also for other 10 points located on vertical axis of the specimen, from the heated side down to the cooled side. Presentation of these results, however, would increase the volume of this paper too significantly and is beyond the scope of the presented topic. The results of our analysis are generally applicable for any number of loading cycles. In the case of a specimen with 2 mm width, the value of stress peaks in the critical point becomes stabilized after 16 cycles,15 hence thermal loading with number of cycles higher than this can be already considered as thermal fatigue. Based on general characteristics and terminology of thermal fatigue it is possible to define maximum stress, minimum stress and the mean stress. 4. Conclusion The influence of repeated thermal shock parameters on stress, generated in silicon nitride was evaluated using parameter of influence (PI) and statistical model. The results show clearly a dominant influence of temperature, followed by the influence of temperature difference, and heating and cooling time. This result was the same for evaluation of the influence on mean stress and on the stress peaks in specific cycles. The statistical model, obtained using least squares method, showed very high coefficient of determination. The highest value was reached by the equation for mean stress. Despite lower coef- ficients of equation for stress peaks in the 4th and 20th cycle, its value is high enough to prove the prediction reliable. The model confirmed the surprisingly lower influence of temperature difference on the stress and a very low influence of heating and cooling time. It is hence important, at least in the conditions of loading described in this article, to take into account not only the temperature difference, but primarily the absolute values of heating and cooling temperature when evaluating the resistance of technical ceramics to repeated thermal shocks. The heating and cooling time should also not be neglected. Presented results are applicable also for conditions of thermal fatigue. References 1. Andersson, T. and Rowcliffe, D. J., Indentation thermal shock test for ceramics. J. Am. Ceram. Soc., 1996, 79, 1509–1514. 2. Koh, Y. H., Kim, H. W., Kim, H. E. and Halloran, J., Thermal shock resistance of fibrous monolithic Si3N4/BN ceramics. J. Eur. Ceram. Soc., 2004, 24, 2339–2347. 3. Nieto, M. I., Martinez, R., Mazerolles, L. and Baudin, C., Improvement in the thermal shock resistance of alumina through the addition of submicronsized aluminium nitride particles. J. Eur. Ceram. Soc., 2004, 24, 2293–2301. 4. Vedula, V. R., Green, D. J., Hellmann, J. R. and Segall, A. E., Test methodology for thermal shock characterization of ceramics. J. Mater. Sci., 1998, 33, 5427–5432.
211 E Gondar et al Joumal of the European Ceramic Society 27(2007)2103-2110 5. Schaus, M. and Pohl, M, Nd-YaG-Laser simulated thermal shock and ther- 14. Hlava, T, Thermal and stress analysis of silicon nitride subjected to repeated mal fatigue behaviour. Metall. 1998. 5. 464. thermal shocks. PhD Thesis, SiF STU Bratislava, 2005. 6. Absi, J. and Glandus, J. C, Improved method for severe thermal shock 15. Gondar, E, Hlava, T and Rosko, M, Verification of the stresses developed testing of ceramics by water quenching. J. Eur. Ceram. Soc., 2004, 24, in silicon nitride by repeated thermal shocks. J. Eur. Ceram. Soc., in press 2835-2838 corrected proof. 7. E, The testing method of resistance of silicon nitride based technical 16. Turner, J. R. and Thayer, J, Introduction to Analysis of variance: ceramics to thermal loading. Conferment project, SiFSTU Bratislava, 1998 Design, Analysis In ion. Sage Publications, Thousand Oaks, 8. Rosko, M, Stress analysis in silicon nitride subjected to a combination of 2001 thermal shocks and mechanical loading. In Proceedings of JUNIORMAT 17 Casella, G and Berger, R L. Statistical Inference. Duxbury Press, 2001 05,2005,pp.67-70 18. Hasselman, D. P. H, Unified theory of thermal shock fracture initiation 9. Pulc, V, Gondar, E and Misik, P. Graphitic oven. Patent AO229412, CSSR and crack propagation in brittle ceramics. J. A. Ceram. Soc., 1969, 52, 15 April 1986 600-604 10. Pulc, V. and Svec, P, Interaction of diamond indentor and silicon nitride- 19. Tancret, F, Comments on"thermal shock resistance of yttria-stabilized zir- based ceramics in the region of elastic-plastic deformation In Proceeding onia with palmqvist indentation cracks, J. Eur Ceram. Soc., in press, of Material in Engineering Practise 98, 1998, Pp. 196-202. corrected proo 11. Gondar, E, Rosko, M. and Zemankova, M, Study of depth profile of 20. Pettersson, P, Johnsson, M. and Shen, Z, Parameters for measuring the indented cracks in silicon nitride Metallic Mater. 2005. 43. 124-133 thermal shock of ceramic materials with an indentation-quench 12. Lube, T, Indentation crack profiles in silicon nitride. J. Eur: Cera. Soc., Eur Ceram Soc.,2002,22,1883-1889. 2001,21,211-218 21. Hampel, F R, Ronchetti, E. M, Rousseeuw, P J and Stahel, w.A., Robust 13. Pulc, V. Gondar, E and Svec, P, Testing method of the resistance of tech- Statistics: The Approach Based on Influence Functions. wiley, New York, nical ceramics to thermal fatigue. Patent Pending 518-2 1986
2110 E. Gondar et al. / Journal of the European Ceramic Society 27 (2007) 2103–2110 5. Schaus, M. and Pohl, M., Nd-YaG-Laser simulated thermal shock and thermal fatigue behaviour. Metall, 1998, 5, 464. 6. Absi, J. and Glandus, J. C., Improved method for severe thermal shock testing of ceramics by water quenching. J. Eur. Ceram. Soc., 2004, 24, 2835–2838. 7. Gondar, E., The testing method of resistance of silicon nitride based technical ceramics to thermal loading. Conferment project, SjF STU Bratislava, 1998. 8. Rosko, M., Stress analysis in silicon nitride subjected to a combination of thermal shocks and mechanical loading. In Proceedings of JUNIORMAT 05, 2005, pp. 67–70. 9. Pulc, V., Gondar, E. and Misik, P., Graphitic oven. Patent AO 229412, CSSR, 15 April 1986. 10. Pulc, V. and Svec, P., Interaction of diamond indentor and silicon nitridebased ceramics in the region of elastic–plastic deformation. In Proceedings of Material in Engineering Practise 98, 1998, pp. 196–202. 11. Gondar, E., Rosko, M. and Zemankova, M., Study of depth profile of indented cracks in silicon nitride. Metallic Mater., 2005, 43, 124–133. 12. Lube, T., Indentation crack profiles in silicon nitride. J. Eur. Ceram. Soc., 2001, 21, 211–218. 13. Pulc, V., Gondar, E. and Svec, P., Testing method of the resistance of technical ceramics to thermal fatigue. Patent Pending 518-2001. 14. Hlava, T., Thermal and stress analysis of silicon nitride subjected to repeated thermal shocks. PhD Thesis, SjF STU Bratislava, 2005. 15. Gondar, E., Hlava, T. and Rosko, M., Verification of the stresses developed in silicon nitride by repeated thermal shocks. J. Eur. Ceram. Soc., in press, corrected proof. 16. Turner, J. R. and Thayer, J., Introduction to Analysis of Variance: Design, Analysis & Interpretation. Sage Publications, Thousand Oaks, 2001. 17. Casella, G. and Berger, R. L., Statistical Inference. Duxbury Press, 2001. 18. Hasselman, D. P. H., Unified theory of thermal shock fracture initiation and crack propagation in brittle ceramics. J. Am. Ceram. Soc., 1969, 52, 600–604. 19. Tancret, F., Comments on “thermal shock resistance of yttria-stabilized zirconia with palmqvist indentation cracks”, J. Eur. Ceram. Soc., in press, corrected proof. 20. Pettersson, P., Johnsson, M. and Shen, Z., Parameters for measuring the thermal shock of ceramic materials with an indentation-quench method. J. Eur. Ceram. Soc., 2002, 22, 1883–1889. 21. Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A., Robust Statistics: The Approach Based on Influence Functions. Wiley, New York, 1986