..V . :I‘...‘. .1 ¢,.. 4 . . . mes “58:9 w New 9:1ng unifiifiifi\tir\§i\\ \ 5” Wigglmllalll This is to certify that tile dissertation entitled Thermal Quench of Brittle Materials presented by Chin-Chen Chiu has been accepted towards fulfillment of the requirements for Ph.D. degree in Materials Science {Joana/u: Major professor Date CMEA'J— 371 /(7 9/ MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE L THERMAL QUENCH of BRITTLE MATERIALS by Chin-Chen Chiu A DISSERTATION Submitted to Michigan State University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in Materials Science Department of Metallurgy, Mechanics, and Materials Science 1991 654—‘6353A ABSTRACT Thermal Quench of Brittle Materials by Chin-Chen Chiu The objective of this dissertation was to study quench-induced mechanical property changes in brittle materials. In this study, commercial microscope glass slides (soda-lime glass) were quenched from above and below the glass's annealing temperature. The quenching media were water, motor oil, and air. The experimental work includes single-quench thermal shock testing, cyclic thermal shock testing (fatigue), retained strength measurement, residual stress measurement, elastic modulus and internal friction measurement, heat flow analysis using Differential Scanning Calorimeter (DSC), and thermal expansion measurement by ThermoMechanical Analysis (TMA). In conjunction with the experimentation, the theoretical analyses include: (1) thermoelastic and thermoviscoelastic stress calculations infinite glass plates, (2) residual stress analysis, including its influence on dynamic elastic modulus, (3) statistical study of subcritical crack growth effect on thermal shock resistance, and (4) quench- induced strength degradation in glass plates. Experimentally, there are significant differences in the effects of a thermal quench from above the annealing temperature when compared with thermal quenching from below the annealing temperature. For example, the viscous flow occurring above annealing temperature can relieve transient thermal stresses and decrease the probability of thermal shock damage. In addition to thermal quench of glass slides, a paper titled "Elastic modulus determination of coating layers as applied to layered composites" is included in the appendix of this dissertation. The appendix proposes two techniques, one based on the dynamic resonance method and the other based on the static loading method (four point bend), to measure the in-plane elastic modulus of SiC coating/graphite substrate composites. ACKNOWLEDGEMENTS I wish to express sincere appreciation to Dr. Eldon D. Case for his support, guidance, and willingness to share his time and knowledge throughout this research. Special thanks to my colleagues, C. Y. Lee, W. J. Lee, and Y. M. Kim for their extra time and efforts during this research. To my wife, shiow-ying, I thank for her support. iv TABLE OF CONTENTS ILISIP(IF IHUIEES LIST OF FIGURES 1. 2. INTRODUCTION LITERATURE REVIEW EXPERIMENTAL PROCEDURES EXPERIMENTAL RESULTS, THEORETICAL ANALYSIS, AND DISCUSSION (4.1) Thermoelastic and Thermoviscoelastic Stress Calculation and Residual Stress Analysis (4.1.1) Quench from below the annealing temperature (4.1.2) Quench from above the annealing temperature (4.1.3) Numerical calculation (4.2) Influence of Thermal Shock on Material Properties (4.2.1) Annealing temperature determination (4.2.2) Fracture strength (4.2.3) Thermal quench induced residual stresses (4.2.4) Effect of thermal quenching on elastic modulus and internal friction (4.2.5) Relaxation of the quench-induced residual stresses Page viii ix 10 14 14 17 21 38 4O 42 44 53 (4.3) Statistical Study of Subcritical Crack Growth Effect on Thermal Shock Resistance (4.3.1) Candidate fracture strength distribution (4.3.2) Subcritical crack growth (4.3.2.1) Evaluation of fracture strength degradation 58 67 69 (4.3.2.2) Experimental results of fracture strength degradation for single-quench thermal shock (4.3.2.3) Fracture strength degradation for cyclic thermal shock 72 82 (4. 4) Computer Simulation of Quench-Induced Strength Degradation in Glass Plates (4.4.1) Theoretical consideration (4.4.1.1) Initial strength distribution (4.4.1.2) No subcritical crack growth effect (4.4.1.3) Subcritical crack growth effect included (4.4.2) Computer simulation 5. SUMMARY AND CONCLUSIONS 6. LIST or REFERENCES 7. APPENDIX Appendix A: INFLUENCE OF RESIDUAL STRESSES ON THE MEASUREMENT OF THE DYNAMIC ELASTIC MODULUS References of Appendix A Appendix B: PHOTOELASTIC DETECTION OF CRACKS AND THE SUBSEQUENT RESTRICTIONS ON QUENCH CONDITIONS FOR CRACK-FREE SPECIMENS References of Appendix B Appendix. C: DETERMINATION OF DISTRIBUTION A IN THE STATISTICAL ANALYSIS OF RETAINED STRENGTH DISTRIBUTION Appendix D: INFLUENCE OF THE CRACK DENSITY NUMBER ON DEVELOPMENT OF RETAINED STRENGTH vi 87 87 88 91 94 108 111 119 129 130 131 132 134 Appendix E: ELASTIC MODULUS DETERMINATION OF COATING LAYERS AS APPLIED TO LAYERED CERAMIC COMPOSITES (1). Introduction (2). Theoretical Background (2.1) Static bend test (2.2) Dynamic resonance ( 3) . Experimental Procedure (3.1) Model composite beam preparation (3.2) SiC coating/graphite substrate composite specimens, monolithic graphite specimens, and free-standing SiC coatings (3.3) Elasticity measurement (4). Results and Discussion (4.1) Model composite beam elasticity results (4.2) SiC/Graphite composites and free-standing SiC layers elasticity (4.3) Comparisons between the model composite beam results and the 81C coating results (5). Conclusion References of Appendix E (XHTPUTTHIIRROEHUU! A. Thermoelastic and thermoviscoelastic stress calculation No. 1 No. 2 No. 3 B. Simulation of quench-induced strength degradation in glass plates No. 1 No. 2 No. 3 vii 136 136 137 137 141 149 149 150 151 153 153 158 160 162 164 166 168 170 173 174 178 LIST OF TABLES Table Number El. E2. E3. E4. Numerical values used in calculating the transient thermal stresses in microscope glass slides. Elastic modulus as a function of the quench medium temperature of quenched polymer glass reported by Vega et a1. [80]. Fracture strength data of annealed glass slide specimens fractured in three-point bend. Parameters for the normal, lognormal, and Weibull distribution functions, as calculated from maximum likelihood estimators. Statistics of the empirical fracture strength distribution function obtain from goodness-of-fit test (kolmogorov-smirnov test). Statistical evaluation of subcritical crack growth effect in quenched glass slides. Parameters required for computer simulation of retained strength degradation. Dynamic resonance measurements of the in-plane elastic moduli of two-layer glass/glass composites. Dynamic resonance measurements of the in-plane elastic moduli of the alumina in A1203/glass composites. Dynamic resonance measurements of the in-plane elastic moduli of the alumina in A1203/g1ass/A1203 composites. A comparison of in-plane elastic modulus data of SiC coatings. viii Page 20 50 65 66 66 86 90 156 159 159 161 LIST OF FIGURES Figure Number Page '1. A plot of retained strength versus of quench temperature difference according to a strength degradation model proposed by Hasselman [7]. 2 2. A plot of retained strength versus of quench temperature difference, a strength degradation model proposed by Lewis [38]. 8 3. Schematic of the furnace and apparatus for thermal shock fatigue testing. 13 4. Transient thermal stresses in a quenched glass plate. (A) thermal qugnching from below the annealing temperature, AT - 300 C, (B) thermal quenching from above the annealing temperature, AT - 620 C. The annealing temperature of the glass plate is taken as 580 0C and quench medium temperature of the glass plate is assumed to be 25 C. 22 5. Residual stress profile in a glass plate quenched from above the annealing temperature. The annealing temperature of the glass plate is taken as 580 oC and quench mediumotemperature of the glass plate is assumed to be 25 C. 24 6. Influence of AT on surface stresses. (A) thermoelastic stresses; (B) thermoviscoelastic stresses. The annealigg temperature of the glass plate is taken as 580 C and quench medium temperature of the glass plate is assumed to be 25 0C. 25 7. The maximum surface stresses versus AT in a quenched plate, as a function of the quench temperature difference AT. Note the divergence between the thermoelastic stress curve and the thermoviscoelastic stress curve that becomes evident for temperatures somewhat above the annealing temperature. 27 8. Influence of Biot' s modulus on surface stresses. (A) thermoelastic stresses, with AT - 300 0C. (B) thermoviscoelastic stresses, with AT - 6200 C. 29 9. Thermal shock damage map for a glass plate. The annealing temperature is assumed to be 580 oC and the temperature of the quenching medium is 25 C. 32 Figure Number 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. Schematic of a thermal shock induced strength degradation curve as inferred form the thermal shock damage map for a quenched glass plate. The T represents the annealing temperature of glassaplate, at which the minimum retained strength occurs. Relationship between residual surface stresses, initial glass temperature, and Biot's modulus for a quenched glass plate. Shifts in the thermal shock damage map that result from assuming different flexural fracture strength values for the glass plate. The curves show that the numerical technique presented in this study is stable with respect to changes in the assumed fracture strength. The dimensional change versus temperature for an annealed glass slide, as measured dilatometrically at a heating rate of 5 oC/minute. Aso indicated on the figure, from about 100 to 500 C, a nearly constant coefgicient of linear thermal expansion of 6. 48 X 10 /o C was measured. (A) Retained fracture strength of glass slides thermally shocked by water quench, oil quench, and air quench as measured in three point bend. (B) Residual stresses in quenched glass slides determined by Vicker's indentation. Relative decrease in elastic modulus as a function of the thermal quench of glass slides. Relative increase in internal friction Qo 1a a function of the thermal quench of glass slides. Surface cracks of glass slides quenched into a water bath. Figures a, b, and c correspond to AT-180, AT-230, and AT-400 C, respectively. Elastic modulus (A) and internal friction (B) versus thermal quench conditions. The ratio of C/C is used as an index of the severity of thermal quench conditions and residual stresses. The Young's modulus versus temperature for the annealed state (solid line) and the residually stressed state of a glass slide. Thermal flow measurement of glass slides using a Differential Scanning Calorimeter. X Page 33 35 36 39 41 45 46 48 51 54 56 Figure Number 21. 22. 23. 24. 25. 26. 27. 28. Fracture strength histogram for 239 annealed glass slide specimens fractured in three-point bend. Illustration of cumulative distribution functions and the empirical distribution function for the fracture strength data of annealed glass slides. Schematic of the evaluation of fracture strength degradation. Retained fracture strength of glass slides following a single quench into a room temperature water bath at: (a) AT - 150: C, (b) AT - 1600 C, (c) AT - 170: C, (d) AT - 180°C, (e) AT - 190° c, (f) AT - 200° c. The dashed curves in figures (a) through (e) represent the fracture strength distribution of annealed glass slides. A plot of retained fracture strength versus AT. The solid line represents the original strength data curve (single thermal shock), which becomes the dashed line after compensating for subcritical crack growth effects (see Table 6). (a) Influence of a cumulative number of thermal shock cycles on the retained fracture strength of the glass slides repeatedly shocked below ATc , where ATc is the critical quench temperaturec difference determined from single- quench testing (Figure 25). (b) Variation of retained fracture strength with respect to AT and the cumulative number of thermal shock cycles, N. (a) Computer simulation of crack growth length as a function of time for cracks extending subcritically. (b) Crack growth velocity as a function of time. Note that small differences in quench temperature difference can induce dramatic changes in the crack growth behavior. Computer simulation of the strength degradation of specimens shocked at: (a) AT - 140 C; (b) AT - 240 be (c) AT - 160 °c; (d) AT - 180 °c; (e) AT - 200 oC. The solid line represents the retained strength without subcritical crack growth effect. The dashed curve represents the retained strength with subcritical crack growth effect. xi Page 59 63 7O 73 81 84 95 97 Figure Number 29. 30. 31. A1. A2. A3. D1. El. E2. E3. A ATc distribution of one thousand pieces of annealed glass slide specimens, which was calculated from the initial flaw size using equation (29). The initial flaw size was randomly generated through computer program No. 4. A schematic of illustrating the strength degradation in a group of shocked specimens. A plot of retained strength versus AT. Solid line represents the retained strength data without subcritical crack growth effect. Dashed line is retained strength data with the subcritical crack growth effect. Residual stress profile in a glass plate, as evaluated numerically from thermoviscoelastic theory. Schematic of specimen suspension for the standing wave resonance technique of elastic modulus measurement. Influence of the thermal residual stresses on the stress-strain range of a standing wave. (a) Nonlinear stress-strain behavior of material. (b) Longitudinal stress due to vibration, where a represents the maximum stress-strain range. (c) Superposition of thermal residual stresses and longitudinal stress, where b is the maximum stress-strain range. Influence of crack density number, N, on mean retained strength. Stress-strain relationship in a composite beam subjected to a pure bending moment. (a) Cross section of composite beam; (b) strain development in axial direction; (c) stress in axial direction. Relationship between relative strain, K, relative thickness, R, and relative elastic modulus, Ec/E , for the in-plane modulus measurement of coatings. Illustration of the composite beam whose elastic modulus is determined by the dynamic resonance method. Page 104 105 107 120 122 126 135 138 142 144 Figure Number E4. E5. E6. E7. Cross section and dimensions of a composite beam with two coating layers. Schematic of the static four point bend apparatus used for the in-plane modulus measurement of SiC coatings. Illustration of two-layer and three layer-model composites. Influence of glue adhesion area fraction on the measured elastic modulus of two-layer glass/glass composite beams. xiii Page 147 153 154 157 1. INTRODUCTION Compared with metals, ceramics have good high-temperature properties, for example, chemical corrosion resistance, oxidation resistance, creep resistance, and mechanical strength. Thus, ceramic materials are promising candidates for use in severe thermal environments. However, ceramics are brittle and can be fractured by a thermomechanical loading. Consequently, thermal shock studies are basic to the characterization the reliability and performance of high-temperature structural ceramics. Typical thermal shock of ceramics involves quenching of hot specimens into a water bath below the material's annealing temperature [1-3], by which tremendous heat flow occurs and severe thermal stresses develop in the quenched component. The thermal stresses may cause the pre-existing cracks in ceramic components to propagate, which degrades the fracture strength of the shocked components relative to that of the non-shocked components [4-6]. As a result, the thermal shock resistance can be specified by the critical quench temperature difference, ATC, required for the fracture strength degradation. In 1969, Hasselman proposed a famous model for the calculation of ATc [7]. Hasselman also proposed that: (l) pre-existing cracks propagate when quenched above ATC, (2) cracks are stable (do not grow) below ATc, and (3) dynamic crack growth causes a discontinuous drop in the retained fracture strength at ATc (Figure 1). From 1969 on, Hasselman's model became a basis for further study of the strength degradation of quenched brittle materials. no shock damage Retained Strength T\ 0 thermal shock damage Quenching Temperature Difference (00) Figure l. A plot of retained strength versus of quench temperature difference according to a strength degradation model proposed by Hasselman [7]. [A ’ This is to certify that the dissertation entitled Thermal Quench of Brittle Materials presented by Chin-Chen Chiu has been accepted towards fulfillment of the requirements for Ph.D. degree in Materials Science Wood/UL Major professor Datea’bt‘élvfij: 3]. qu/ MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY 1 Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE l - ll |%=—= flit—"— MSU Is An Affirmative ActiorVEquaI Opportunity Inuitution cWMG-Df THERMAL QUENCH of BRITTLE MATERIALS by Chin-Chen Chiu A DISSERTATION Submitted to Michigan State University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in Materials Science Department of Metallurgy, Mechanics, and Materials Science 1991 '5’.) .2 JV 6 54—." 4 ABSTRACT Thermal Quench of Brittle Materials by Chin-Chen Chiu The objective of this dissertation was to study quench-induced mechanical property changes in brittle materials. In this study, commercial microscope glass slides (soda-lime glass) were quenched from above and below the glass's annealing temperature. The quenching media were water, motor oil, and air. The experimental work includes single-quench thermal shock testing, cyclic thermal shock testing (fatigue), retained strength measurement, residual stress measurement, elastic modulus and internal friction measurement, heat flow analysis using Differential Scanning Calorimeter (DSC), and thermal expansion measurement by ThermoMechanical Analysis (TMA). In conjunction with the experimentation, the theoretical analyses include: (1) thermoelastic and thermoviscoelastic stress calculations infinite glass plates, (2) residual stress analysis, including its influence on dynamic elastic modulus, (3) statistical study of subcritical crack growth effect on thermal shock resistance, and (4) quench- induced strength degradation in glass plates. Experimentally, there are significant differences in the effects of a thermal quench from above the annealing temperature when compared with thermal quenching from below the annealing temperature. For example, the viscous flow occurring above annealing temperature can relieve transient thermal stresses and decrease the probability of thermal shock damage. In addition to thermal quench of glass slides, a paper titled "Elastic modulus determination of coating layers as applied to layered composites" is included in the appendix of this dissertation. The appendix proposes two techniques, one based on the dynamic resonance method and the other based on the static loading method (four point bend), to measure the in—plane elastic modulus of SiC coating/graphite substrate composites. ACKNOWLEDGEMENTS I wish to express sincere appreciation to Dr. Eldon D. Case for his support, guidance, and willingness to share his time and knowledge throughout this research. Special thanks to my colleagues, C. Y. Lee, W. J. Lee, and Y. M. Kim for their extra time and efforts during this research. To my wife, shiow-ying, I thank for her support. iv TABLE OF CONTENTS ILISTPIIF TTUEEES LIST OF FIGURES 1. INTTKHNRJPIONI ILITTSUVDURTIIUEVIEWI EXPERIMENTAL PROCEDURES EXPERIMENTAL RESULTS, THEORETICAL ANALYSIS, AND DISCUSSION (4.1) Thermoelastic and Thermoviscoelastic Stress Calculation and Residual Stress Analysis (4.1.1) Quench from below the annealing temperature (4.1.2) Quench from above the annealing temperature (4.1.3) Numerical calculation (4.2) Influence of Thermal Shock on Material Properties (4.2.1) Annealing temperature determination (4.2.2) Fracture strength (4.2.3) Thermal quench induced residual stresses (4.2.4) Effect of thermal quenching on elastic modulus and internal friction (4.2.5) Relaxation of the quench-induced residual stresses Page viii ix 10 14 l4 17 21 38 40 42 44 53 (4.3) Statistical Study of Subcritical Crack Growth Effect on Thermal Shock Resistance (4.3.1) Candidate fracture strength distribution (4.3.2) Subcritical crack growth (4.3.2.1) Evaluation of fracture strength degradation 58 67 69 (4.3.2.2) Experimental results of fracture strength degradation for single-quench thermal shock (4. 3 .2 . 3) Fracture strength degradation for cyclic thermal shock 72 82 (4.4) Computer Simulation of Quench-Induced Strength Degradation in Glass Plates (4.4.1) Theoretical consideration (4.4.1.1) Initial strength distribution (4.4.1.2) No subcritical crack growth effect (4.4.1.3) Subcritical crack growth effect included (4.4.2) Computer simulation 5. SUMMARY AND CONCLUSIONS 6. LIST or REFERENCES 7. APPENDIX Appendix A: INFLUENCE OF RESIDUAL STRESSES ON THE MEASUREMENT OF THE DYNAMIC ELASTIC MODULUS References of Appendix A Appendix B: PHOTOELASTIC DETECTION OF CRACKS AND THE SUBSEQUENT RESTRICTIONS ON QUENCH CONDITIONS FOR CRACK-FREE SPECIMENS References of Appendix B Appendix. C: DETERMINATION OF DISTRIBUTION A IN THE STATISTICAL ANALYSIS OF RETAINED STRENGTH DISTRIBUTION Appendix D: INFLUENCE OF THE CRACK DENSITY NUMBER ON DEVELOPMENT OF RETAINED STRENGTH vi 87 87 88 91 94 108 111 119 129 130 131 132 134 Appendix E: ELASTIC MODULUS DETERMINATION OF COATING LAYERS AS APPLIED TO LAYERED CERAMIC COMPOSITES (1). Introduction (2). Theoretical Background (2.1) Static bend test (2.2) Dynamic resonance (3). Experimental Procedure (3.1) Model composite beam preparation (3.2) 810 coating/graphite substrate composite specimens, monolithic graphite specimens, and free-standing SiC coatings (3.3) Elasticity measurement (4). Results and Discussion (4.1) Model composite beam elasticity results (4.2) SiC/Graphite composites and free-standing SiC layers elasticity (4.3) Comparisons between the model composite beam results and the $10 coating results (5). Conclusion References of Appendix E > <8) __6__ ,, 1'1 Equation 11 thus describes the thermoelastic stresses in an infinite glass plate of thickness 22 quenched from below the annealing temperature. (4.1.2). Quench from above the annealing temperature When a glass plate is heated to near its annealing temperature, the viscosity of the glass decreases. At the annealing temperature, viscous flow is significant. For example, internal stress in glass is substantially relieved by viscous flow during a one hour heat treatment at the annealing temperature [63]. If a glass plate is quenched from above its annealing temperature, the transient thermal stresses are also influenced by viscous flow [64]. The mathematics of the thermal stresses in linear viscoelastic materials have been treated by several authors who account for the effect of viscous flow and stress relaxation [64-70]. Lee et a1. unified the mathematical framework and presented a thermoviscoelastic theory for the thermal stresses and residual stresses that arise when a glass plate is quenched symmetrically from both surfaces [64]. Upon comparison with experimental results, Narayanaswamy et a1. modified Lee's numerical calculation technique to bring the theoretical results into closer agreement with 18 experimental data [71]. Since Lee's theory can conveniently model the transient thermal stresses in a rapidly quenched glass plate, we use the formulations presented by Lee [64] and Narayanaswamy [71] in the present study. Linear elastic materials have a constant elastic modulus so that materials subjected to a given strain do not experience stress relaxation. However, linear viscoelastic materials exhibit stress relaxation for an isothermal deformation, which is described by [65,72] t a €k1(t') 0.. t = G.. t - t' dt' 12 U( > O 1Jkl< > a t! < > where aij’ ekl’ Gijkl(t)’ and t are the stress tensor, strain tensor, relaxation modulus tensor, and time, respectively. For viscoelastic plates subjected to a thermal quench, Lee proposed that the transient thermal stresses may be calculated from [64] 2 axx(Z,t) dz - 0 (13) O t a a (Z,t) a 3 R(£(Z,t) - €(Z,t')) e(t') - aT(Z,t') dt' (14) xx 0 a t’ where the reduced time, 5, expresses the temperature and time dependence of the viscoelastic material properties. The parameter, 6, is defined by [64,66,69] t €(Z.t) - <1>[T(Z.t')]dt' (15) 0 T(Z,t) is the temperature distribution in the quenched glass plate 19 (equation (8)). Lee indicated that the time shift factor T(T) for soda lime silica glass measured at a base temperature of 538 0C is [64] log10[(T)] - 0.03861(T - 538) (16) R(£) is an auxiliary modulus function associated with the relaxation modulus tensor, G, such that for a Maxwell solid R(§) is given by [66,67] H<£> E0 R<€> - exp(- fl s/ro) <17a> 3(l-v) Normalizing R(€) by the factor (Eo/ 3(1 - u) gives the unitless function R(E), where F(e) - H<€> exp<- B 6/10) (17b) where l + u fl - (17C) 3 (l - u) and H(€) is the Heaviside unit step-function. Eo and To are the instantaneous elastic modulus at t.- 0 and the relaxation time, respectively. Batteh [73] approximated R directly from Narayanaswamy's data for soda lime silica glass [71] as §(£) - exp(-s/700> (18) According to Narayanaswamy's modification [71], equation (14) is rewritten as 20 axx(Z,t) = ayy(Z,t) n E0 Z {{ 6(ti) - 6(ti_1) - a[T(Z,ti) - T(Z,ti_l)] i=1 €(z,ti) - € {- _ J 1 R<£ - 5') ds' } (19> £._1 1 Equations (13) and (19) constitute a set of nonlinear integral equations. To calculate the transient thermal stresses, we first numerically solve for £(Z,t) by combining equations (8), (15), and (16). The integral of f R d£' in equation (19) is then calculated using equation (18). Finally, the transient stress a(Z,t) and strain £(t) are calculated iteratively as a function of time t, using equations (13) and (19). The residual stresses are the steady-state values to which a(Z,t) converges if time becomes great enough, for example t - 50 seconds. The residual surface stresses correspond to the convergent values, a(2,t). Table 1 lists the required input values for the numerical analysis. Table 1. Numerical values used in calculating the transient thermal stresses in microscope slide glass specimens. All values were measured in this research, unless otherwise indicated. -6 o -1 thermal expansion (a) : 8 * 10 C [74,75] elastic modulus (E) : 70 GPa Poisson's ratio (u) : 0.25 [74,75] half thickness of specimen (2) : 0.001 _7 m2 _1 thermal diffusivity (a) : 4.8 * 10 m 5 [74,75] Biot's modulus (B) : 10 quenching medium temperature (T ) : 20 oC flexural fracture strength 0 : 101.4 MPa (4.1.3). Numerical calculation For a glass plate quenched from below its annealing temperature, numerical modeling of the thermoelastic stresses shows that compressive stress arises internally in the plate and that tensile stresses develop on the plate's surface (Figure 4(a)). The maximum tensile stress always appears on the surface of the glass plate. If a glass plate is quenched from above the annealing temperature, thermoviscoelastic stresses can induce viscous flow in the glass plate which in turn leads to complex stress fields and induces residual stresses in the plate (Figure 4(b) and Figure 5). The thermoelastic surface stresses versus time increase as AT increases (Figure 6(a)). The time evolution of the surface stresses that develop during quenching are keenly dependent on whether the plate is quenched from below or from above the annealing temperature. For quenching from below the annealing temperature, the thermoelastic stresses versus time increase monotonically as AT increases (Figure 6(a)). In contrast, for quenching form above the annealing temperature, the thermoviscoelastic stresses generally decrease as the temperature increases (Figure 6(b)). Figure 7 illustrates the relationship between the AT and the maximum surface tensile stresses. For AT < 500 0C, the glass plate acts as an elastic material and the stresses follow the solid line. According to equation (11), the maximum surface tensile stresses increase linearly with increasing AT. For AT > 580 0C, viscous flow of the glass becomes significant, resulting in a decrease of the maximum surface tensile stresses as a function of increasing AT (the dashed line in Figure 7). Between AT=500 oC and AT=580 0C, the two 21 22 150 Time (second) A: 0.005 B: 0.38 i C: 1.3 D: 2.92 Biot’s modulus = 10 Nl tn L \ 1 o- ‘c _D I Thermoelastic Stresses (MPa) Position on 2 Axis (mm) Figure 4(a). Transient thermal stresses in a quenched glass plate. Thermal quenching from below the annealing temperature, AT - 300 oC. The annealing temperature of the glass plate is taken as 580 oC and the quench medium temperature is assumed to be 25 C. 23 Thermovlscoelastic Stresses (MPa) .1 \ ..~ E O \\ F ’\ 225 A A: 0.08 B: 0.27 ‘B C: 0.64 150- D: 1.25 ,\ E: 2.16 C F: 3.43 75- Time (second) fi/ Biot’s modulus = 10 I I 0 Position on 2 Axis (mm) Figure 4(b). Transient thermal stresses in a quenched glass plate. Thermal quenching from above the annealing temperature, AT - 620 0C. plate is taken as 580 °C and the quench medium temperature is assumed to be 25 The annealing temperature of the glass Residual Stresses (MPa) 24 200 ”:3 . /A\ -e -2004 c A:AT = 620 00 Biot’s = 8 B:AT = 660 Biot’s = 8 -300 D can = 700 Biot’s = 5 DzAT = 700 Biot’s = 15 -400 r r . . . T . , . , . . -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Poistion on 2 Axis (mm) Figure 5. Residual stress profile in a glass plate quenched from above the annealing temperature. The annealing temperature of the glass plate is taken as 580 oC and the quench medium temperature is assumed to be 25 C. 25 250 4 AT=500 °c 7" ' lus= 10 n. 200. Biotsmodu E 400 m 1 8 § 150‘ 300 a; . .2 ‘65 100~ 200 2 a: . E '55 50_ 100 IE / 0 ...... .. ...... . n. . 0.01 0.1 1.0 10.0 Figure 6(a). Time (seconds) Influence of AT on surface thermoelastic stresses. The annealing temperature of the glass plate is taken as 580 C and the quench medium temperature is assumed to be 25 oC. Therefore, each curve corresponds to a quench from below the annealing temperature. 26 300 it? . a A g, 200- » U) o . 0! , 3 100-_______./ 1'7) . o 560 “Fr; 0* g ‘ 600 100- .9 ' 640 3 i g _200_ Biot’s modulus = 10 .2 AT = 680 0c I— ‘ ¥ '300 1 l l i l l 0.00001 0.0001 0.001 0.01 0.1 1.0 10.0 Time (seconds) Figure 6(b). Influence of AT on surface thermoviscoelastic stresses. The annealing temperature of the glass plate is taken as 580 oC and theoquench medium temperature is assumed to be 25 C. Each curve corresponds to a quench from above the annealing temperature, thus thermoviscoelastic theory is required to calculate the transient stresses. 27 E 400 . E - ——Thermoelast1c stresses 3; L --Thermoviscoelastic stresses 0 _ . g; 3001- 2 ' \ *u - \ (D _ \ \ 8 200- \ ‘5 - \ ‘l: . \ 3 \ m - \ E 100 - \ = I .§ - a O L 1 L l l t 4 l 1 1 1 L t 1 l l l E 0 200 400 600 800 Quenching Temperature Difference (°C) Figure 7. The maximum surface stresses versus AT in a quenched plate, as a function of the quench temperature difference AT. Note the divergence between the thermoelastic stress curve and the thermoviscoelastic stress curve that becomes evident for temperatures somewhat above the annealing temperature. Again, the temperature of the quenching medium is assumed to be 580 C. 28 curves superimpose. The thermal stresses illustrated in Figures 4, 6 and 7 are computed for a Biot's modulus equal to ten. However, Biot’s modulus influences the magnitude of thermal stresses such that as Biot's modulus increases, the maximum value of the residual tensile stress increases in the plate's interior and the transient tensile stress on the plate's surface also increases (Figure 5, 8(a), and 8(b)). Thermal quench conditions for a given material system can be described in terms of the surface heat transfer coefficient, quench temperature difference, and the specimen dimension. However, Biot's modulus can replace both the surface heat transfer coefficient and specimen size variables such that thermal quench conditions can be simply expressed by two variables, Biot's modulus and quench temperature difference. Thermal shock damage usually initiates at the regions of maximum transient tensile stress which occurs on the surface of quenched brittle components. Crack growth also may occur in the interior of tempered glass which develops sufficiently high residual stresses. To predict the thermal shock damage of glass plates quenched from above their annealing temperature, we assume that (1) during thermal quench, crack growth is dominated by the propagation of pre-existing surface cracks; (2) the surface layers in the initial cooling step approach elastic behavior; (3) the fracture strength is independent of temperature; and (4) after the thermal quench occurs, the residual tensile stress in the interior of the quenched glass plate may be large enough to propagate pre-existing flaws. Therefore, an appropriate shock damage criterion is that the 29 AT: 300°C Biot’s = 40 20 10 250 A 1 3 200- E a, . 3 m 150- 2 a . .2 '55 100- 2 a) q 3 3 50- J: I" . 0 0.01 Figure 8(a). V "VU'Ur V U V'IU'U' U T 0.1 1.0 10.0 Time (seconds) Influence of Biot's modulus on surface thermoelastic stresses, with AT - 300 oC. The annealing 0 temperature of the glass plate is taken as 580 C and the quench medium temperature is assumed to be 25 C. 30 400 350 - 300 - AT = 620 0c . Biot’s = 40 250 200 - d 150- 1 100‘ Thermovlscoelastic Stresses (M Pa) o ‘3 l . l \ \— \_ -50 0.00001 Figure 8(b). r r r F F l 0.0001 0.001 0.01 0.1 1.0 10.0 Time (seconds) Influence of Biot's modulus on surface thermoviscoelastic stresses, with AT - 620°C. The annealing temperature of the glass plate is taken as 580 0C and the quench medium temperature is assumed to be 25 C. 31 crack growth occurs if the magnitude of the transient surface stress or the residual interior tensile stress exceeds the fracture strength. From the thermal stress calculations#, we can construct a thermal shock damage map consisting of three curves, curve A for the thermoelastic stresses and curves B and C for the thermoviscoelastic stresses (Figure 9). The thermal quench conditions in the map are characterized by Biot's modulus and the quench temperature difference. When the thermal quench conditions correspond to a point above curves A and B, the maximum surface tensile stresses exceed the fracture strength and cracks propagate. If the quench conditions correspond to a point above the curve C, crack growth is driven by the residual interior tensile stress. When the quench conditions correspond to a point below the curves, there is no shock damage. The effect of viscous flow upon the stresses becomes obvious near the annealing temperature, which is about 580 0C in this case (see Figure 7 and 9). Thus, residual thermal stresses develop in glass plates quenched from above about 600 0C. For a thermal quench of a glass plate, three values of the critical quench temperature difference, ATc, may appear on the thermal shock damage map. The critical values of ATc may be calculated from the intersections of the three curves in the thermal shock damage map (Figure 9). For example, if the glass plate is quenched at B - 5, the critical quench temperature differences are ATc - 300 oC, ATc - 670 0C, and ATc - 780 0C. From the three The curves in Figure 9 were evaluated using Computer programs No. l and No. 3. The curves correspond to the thermal stress conditions in which the maximum transient stress is 101.2 MPa. Biot’s Modulus 32 30 ' No shock damage 20 - Thermal shock damage Residual 1 0 _ stresses I \\ I \I A fh‘c'~m- \ 3' No shock / damage 0 1 ' I . . 1 0 200 400 600 800 Quenching Temperature Difference (°C) Figure 9. Thermal shock damage map for a glass plate. The annealing temperature of the glass plate is taken as 580 o C and the quench medium temperature is assumed to be 25 °c. 33 ATc \ shock damage ATc \ No shock damage No shock idamage/A 1'c Thermal shock damage Retained Fracture Strength IRE§HCNJalSflfeEfiNBS Quenching Temperature Difference (°C) Figure 10. Schematic of a thermal shock induced strength degradation curve as inferred form the thermal shock damage map for a quenched glass plate. The T represents the ammealing temperature of glass plate, at shich the minimum retained strength occurs. 34 critical values, we can qualitatively infer the shape of the thermal quench strength degradation curve (see Figure 10). Glass plates quenched from below AT - 300 0C have no thermal shock damage and their fracture strength does not change. Above AT = 780 0C as well as between AT - 300 oC and AT - 670 oC, thermal shock damage reduces the retained fracture strength of the glass plate. When the initial specimen's temperature is near to the annealing temperature, Ta’ the retained strength is minimized. Shock damage disappears for 670 0C < AT < 780 oC. The gradual increase and eventual saturation in retained fracture strength for 580 0C < AT < 780 oC results from viscous flow of the glass. The viscous flow decreases the magnitude of thermal stresses, which in turn decreases the probability and the severity of the thermal shock damage. Thus, between 580 oC and 780 0C the glass plate can be tempered (strengthened) without thermal shock damage. Figure ll illustrates the relation among residual surface stresses, initial quenching temperature, and Biot's modulus. For thermal quench conditions corresponding to a given Biot's modulus, the residual surface stresses reach a saturation value with further increase of the initial glass temperature. Therefore, the residual- stress-induced increase in the retained fracture strength also saturates (Figure 11). In this study, thermal shock damage was analyzed for glass plates. Physically meaningful changes in the input parameters (Table l) for the numerical analysis only shifts the curves, without changing the essential characteristics of the thermal shock damage map. For example, in a plot of Biot's modulus versus the quenching Residual Stresses (MPa) 35 i 600 +- B: Biot’s modulus _ 8:20 400 — B=10 ’ 8:5 200 - . ;1 o I. L l L l l 1 1 500 600 700 800 Initial Temperature (°C) Figure ll. Relationship between residual surface stresses, initial glass temperature, and Biot's modulus for a quenched glass plate. 36 30 fracture = 160 Mpa . strength - 101.4 in 20 - 2 60 :3 13 (D . E .S” ‘6 a 10- 0 . 1 a 1 . 1 . 1 0 200 400 600 800 Quenching Temperature Difference (°C) Figure 12. Shifts in the thermal shock damage map that result from assuming different flexural fracture strength values for the glass plate. The curves show that the numerical technique presented in this study is stable with respect to changes in the assumed fracture strength. 37 temperature difference, changing the fracture stress from the experimentally determined value of 101.4 MPa downward to 60 MPa or upward to 160 MPa systematically shifts the numerical results with respect to the quenching temperature difference, but the general shape of the curve is largely maintained (Figure 12). Variations in specimen thickness, surface heat transfer coefficient, and thermal conductivity are encompassed by B, the Biot modulus (equation (9)). Figure 12 is quite important with respect to the numerical modeling presented in this paper and the possibility of extending our analysis to other viscoelastic materials. Numerical techniques can be unstable, in that small changes in the input data may cause large swings in the results [76], but (at least with regard to changes in the fracture stress) the technique presented here seems to be stable. Therefore, the map may apply also to those ceramic materials which are viscoelastic at high temperature. Gebauer et a1. [9,10] and Ohira et a1. [49] quenched aluminosilicate rods into an silicone oil bath over a range of different quench temperature differences and obtained a strength degradation curve similar to Figure 10. Mai [48] quenched TiC and WC in water bath at z 20 oC and also yielded a curve similar to Figure 10. Gebauer proposed that viscous flow in the glassy grain boundary phases might account for the observed strength changes. The analysis in the present study shows that the viscous flow of glass yields the same "shape" for the strength distribution curve as observed experimentally by Gebauer [9.10]. Although there are similarities between the behavior predicted in this study and the experimental results in the literature for polycrystalline ceramics quenched from high temperatures, additional 38 study is needed on the quench behavior of ceramics containing a significant glassy grain boundary phase. (4.2) The Influence of Thermal Quench on Mechanical Properties The experimental results on quenched microscope glass slides presented in this section will be compared with the theoretical analysis given in the previous section. Thermal shock damage and residual stresses are typically evaluated via fracture strength. However, elastic modulus and internal friction also can be useful indicators of microcracks and microstructural change [77-83]. Thus, in this study, fracture strength, elastic modulus, and internal friction were used to monitor the effect of thermal quenching on glass slides. In addition, the annealing temperature of the glass was determined from linear thermal expansion measurements. Quench-induced residual stresses were evaluated using the Vickers indentation technique. The annealing-induced residual stress relaxation was detected from a heat flow analysis using Differential Scanning Calorimeter (DSC). (4.2.1). Annealing temperature determination The linear thermal expansion rate of the glass slides was nearly constant up to 510 0C, but increased sharply for the interval between about 550 0C and 600 0C (Figure 13). With further heating the expansion rate decreased as the viscous flow of the glass slides became significant. From Figure 13, the annealing temperature of the glass was around 580 0C [63]. Dimension Change (11m) 39 60 Annealing _ Temperature 40— 6.48 um/m°C 2O - o I 1 l 1 l 1 l 1 l 1 l 1 O 1 00 200 300 400 500 600 700 Tern peratu re (° C) Figure 13. The dimensional change versus temperature for an annealed glass slide, as measured dilatometrically at a heating rate of 5 OC/minute. As indicated on the figure, from about 100 to 500 0C, a nearly constagtocoefficient of linear thermal expansion of 6.48 X 10 / C was measured. (4.2.2). Fracture strength In this study glass plates were quenched into three different media: water, oil, and air. The magnitude of the surface heat transfer coefficient, h, is such that h (water quench) > h (oil quench) > h (air quench) [61,84,85]. Thus, the Biot's modulus, B, is ranked as follows: B (water quench) > B (oil quench) > B (air quench), where B is defined as B - . (9) Typical thermal shock experiments involve quenching only from below the annealing temperature. In this study, we included thermal quenching from both above and below the annealing temperature. This wider range of temperatures is important since the observed retained fracture strength for specimens quenched from below the annealing temperature of 550 0C (for our specimens) can show very different trends from those specimens quenched from temperatures above the annealing temperature. In addition, the retained fracture strength behavior depends strongly on the quenching medium. For thermal quenching from below the glass annealing temperature, the retained fracture strength data indicate critical quench temperature differences of ATc z 165 oC and ATc z 400 0C for the water quench and the oil quench, respectively (Figure l4(a)). For air quenching from below the annealing temperature, the retained fracture strength does not change significantly, implying that thermal shock damage does not occur. These experimental results indicate that for thermal quenching from below the glass annealing temperature, a 40 41 400 l 200 - Retained Fracture Strength (MPa) 0 Water quench D Oil quench AAir quench Strength of annealed slides -_ _-. ‘- : ' 1' 1 1 1 ' l r r l ' 1 00 200 300 400 500 600 700 800 Quenching Temperature Difference (°C) Figure 14(a). Retained fracture strength of glass slides thermally shocked by water quench, oil quench, and air quench as measured in three point bend. 42 greater Biot's modulus leads to a smaller ATc, which agrees with the authorsf previous theoretical analysis and with the thermal shock theories of other researchers, such as Kingery [61]. The retained fracture strength for quenching from above the annealing temperature is also a function of the quenching medium. Thermal quenching from above the annealing temperature in air results in a retained fracture strength which gradually increases with increasing AT. For the oil quench, the retained fracture strength decreases at AT 2 400 0C but increases for AT > 630 0C, where the increases in the retained fracture strength result from quench-induced residual stresses. Water quenching always resulted in a drop in residual strength between AT 2 165 oC and AT z 680 0C. In this study, a maximum AT of 680 0C was not exceeded, since for AT > 680 0C, the specimens deformed macroscopically by viscous flow. (4.2.3). Thermal quench induced residual stresses Indentation measurements [56-58] on quenched glass slide specimens indicate that residual stresses arise in both the oil quench and the air quench for quenching from above the annealing temperature (Figure 14(b)). (Residual stress measurements were not attempted on specimens quenched into water from above the annealing temperature, since these specimens cracked extensively.) The oil quench corresponds to a greater Biot's modulus than the air quench, thus the oil quench induces higher residual stresses than the air quench. Although the experimentally determined residual stresses rise very rapidly with increasing AT (Figure l4(b)), a numerical 43 CiOil quench CB 140 - AAir quench a a I 2 » f V 100 " I (D . m I g : El 10 C) _ 60 — w - E :i . 2 . ’ A . m 20 — g] 113 . fifi' _. ___D.<. . _ 1 l 1 l 1 l 1 L L ‘1 1 L J _l 1— 200 100 200 300 400 500 600 700 e 0 Quenching Temperature Difference (°C) Figure 14(b). Residual stresses in quenched glass slides determined by Vicker's indentation. 44 analysis by the authors predicts that as AT continues to rise (for quenching from above the annealing temperature), the residual stress values will level off and saturate with further increases in AT. Viscous deformation of the specimens prevented us from determining whether or not a saturation in residual stress does occur for AT > 680 °c. (4.2.4). Effect of thermal quenching on elastic modulus and internal friction Measurements of elastic modulus and internal friction are nondestructive and reflect the total accumulation of microstructural damage. Figures 15 and 16 illustrate the relationship between elastic modulus, internal friction, and AT. As was the case for retained fracture strength, the water quench and oil quench curves change at AT z 165 oC and AT z 400 0C, respectively, indicating that thermal shock damage decreases the elastic modulus and increases the internal friction. For water quench, surface cracks became visible at AT > 165 0C, with more and more cracks accumulating as AT increased (Figure 16). Elastic modulus and internal friction continuously vary with increasing AT, reflecting an accumulation of cracks. However, the retained strength curves in Figure 14(a) are nearly constant despite the accumulation of surface cracks. Analysis of the retained strength behavior in this case (water quench, AT > 165 0C) is very complicated, since the accumulating cracks eventually form a network of cracks. One simple interpretation of the near constancy in the retained fracture strength is that the critical flaw size does not increase. However, classical fracture mechanics models of isolated, 45 10 So: Unquenched elastic modulus o. a? o o v- 18 2 p E11 0 -20 Ill \‘ ,' i 1 m 0 d: l ~— -30 — 00 B AAir quench ' DOiI quench -4o Orwatenqusnem L 1 1 1 . 1 1 1 O 100 200 300 400 500 600 700 800 Quenching Temperature Difference (C) Figure 15. Relative decrease in elastic modulus as a function of the thermal quench of glass slides. 46 1 000 OWater quench ' DOil quengh - O _ AAir quench °8° (0‘1—0'511/03‘ * 100% “ 0'51: Unquenched internal friction L Li I 1 l L, l 1 l L l Figure 16. Relative increase in internal friction Q.1 of the thermal quench of glass slides. 1 O 100 200 300 400 500 600 700 800 Quenching Temperature Difference (°C) as a function 47 non-interacting cracks in a brittle specimen are likely not adequate for the web-like crack field that develops at high AT in the water quench (Figure 17). The elastic modulus and internal friction are essentially constant for air-quenched specimens (Figures 15 and 16) quenched from below the annealing temperature. Above the annealing temperature, the elastic modulus and internal friction gradually change, and this change may be associated with quench-induced residual stresses (see Appendix A). The elastic modulus (internal friction) for oil- quenched specimen first decreases (increases) as AT increases from AT - 400 0C to AT - 500 0C, which is due to the increasing degree of thermal shock damage. [When 500 0c < At < 630 0c, the shock damage becomes great enough. Thus, the elastic modulus (internal friction) of shocked specimen is difficult to measure and the experimental data are absent between 500 0C < AT < 630 0C. When AT > 630 oC, viscous flow results in the decrease in the transient thermal stress. The degree of the shock damage decreases and the the experimental data appear in Figures 15 and 16 again. Changes in elastic modulus and internal friction as a function of static strain or as a function of thermal quenching have been reported for inorganic and polymer glasses. Mallinder et al. indicated that the elastic modulus of soda glass fiber was a function of static strain, E - Eo(l-S.lle), where Eo - 72.5 GPa [79]. The strain, 6, was induced via an external tensile load acting on both ends of a fiber about 0.02 mm in diameter. Vega et a1. [80] pointed out that thermal quenching into several media caused residual stresses and 48 Figure 17. Surface cracks of glass slides quenched into a water bath. gigures a, b, agd c correspond to AT - 180 C, AT-230 C, and AT-400 C, respectively. 49 decreased the elastic modulus of polymer glass (see Table 2). Pavlushkin et a1. [81] observed that the elastic modulus of an industrially tempered glass was lower by about 3 percent than that of annealed glass*. Room temperature vibrational pendulum measurements in chilled glasses gave an internal friction that was about 40 percent greater than the internal friction of annealed glass [82,83]. However, thermal history can affect the specific volume (the reciprocal density), viscosity, and other physical properties of a glass [70,86]. While thermal quenching does induce residual stress in the glass slides, it is unclear what role (if any) changes in physical properties such as specific volume (that also accompany the thermal quench) play in the observed simultaneous decrease in elastic modulus and increase in internal friction.** .As indicated by equation (4), the quenched-in residual surface stresses in the glass slides were determined as a function of C/Cr, using the Vickers indentation technique [56-58], where 2C is the indentation crack length for an annealed specimen and 2Cr is the indentation crack length for a residually stressed specimen (indented at a load identical to that used for the annealed slide). Thus, in this study the ratio of C/Cr is used as an index of the severity of * Quench temperature and quench media were not specified by Pavlushkin et al. [81], Fitzgerald [82], and Day et al. [83]. ** Thermal quenching results in compressive residual stresses in the specimen’s surface. However, the residual stresses do not affect the observed hardness. 50 Table 2. Elastic Modulus as a Function of the Quench Medium Temperature of Quenched Polymer Glass Reported by Vega et al. [80]. quenching rate Media Temp. (0C) Modulus (GPa) slow cool+ 30 1.7 water quench 30 1.5 slow cool 70 1.5 water quench 70 0.95 nitrogen quench 70 1.0 + slow cool - 0.01 oCés. ++ water quench - 160 C/s. thermal quench conditions and residual stresses. The experimentally determined values of residual stress, Sr’ ranged from Sr - O to Sr = 160 MPa (Figure 18). In this study, it was found that the elastic modulus and internal friction changes as a function of quench-induced residual stresses can be expressed as an empirical function of C/Cr (Figure 18) such that E-E _9 ° * 100% = - 4.35 [1 - (C/Cr) 1 (20a) Q - Q0 _3 * 100% - 100 [l - (C/Cr) ] (20b) where E and Q"1 represent the elastic modulus and the internal friction of the quenched glass slides. Eo and Q0.1 are elastic (E - EO)/Eo * 100 % 51 Residual Stresses (M Pa) 0 24 55 91 134 183 0.0\ 1 1 . . -2.0-4 ‘ A 73.: A \ A 4 0 \\ A . \.__ ‘ J A‘ A ‘ t -6.0 4 1 f r l ' l ' 1.0 1.2 1.4 1.6 1:8 2.0 Dimensionless Indentation Crack Length (C/Cr) Figure 18(a). Elastic modulus versus thermal quench conditions. The ratio of C/C is used as an index of the severity of thermal quench conditions and residual stresses. 52 Residual Stresses (MPa) 0 24 55 91 134 183 120 l 1 l - 1 1 A ‘ A \° 2, 90- ‘O' 1"“ s ‘,a”"”"”"“ 7° // A A C .. M > 60 // ‘ F ‘ / 'o° A‘ ’ 1 g/ A b 30 - A z A o e , . r . , . r e 1.0 1.2 1.4 1.6 1.8 2.0 Dimensionless Indentation Crack Length (C/Cr) Figure 18(b). Internal friction versus thermal quench conditions. The ratio of C/C is used as an index of the severity of thermal quench conditions and residual stresses. 53 modulus and internal friction of the annealed slides. Since the shock damage-free specimens with the residual stresses of from 100 MPa to 40 MPa could not be produced by either motor oil quench (SAE 20W 50) or by air quench (cooling in air), the data in Figure 18 forms two clusters (Appendix B). (4.2.5). Relaxation of the quench-induced residual stresses Evidence of the relaxation of quench-induced residual stresses can be obtained via both elasticity and calorimetric measurements. While the elastic modulus of an annealed glass slide decreases with temperature, the modulus versus temperature curve for an annealed glass slide is identical upon heating and cooling. The solid line in Figure 19 shows the reversibility of the elastic modulus for an annealed glass slide heated and cooled at a rate of one degree Celsius per minute. In contrast, a slide having a quenched-in residual stress does not show reversibility in the elastic modulus for the first heating and cooling cycle. For example, when the annealed glass slide in figure 8 was quenched into an oil medium at a quench temperature difference of AT - 680 0C, a residual surface stress of 140 MPa was induced in the slide, as measured by the Vickers indentation technique [56-58]. After quenching, room temperature elasticity measurements on the residually stressed slide showed a lower elastic modulus than the same slide had in its annealed state (Figure 19). When the residually stressed slide was heated from room temperature to approximately 300 oC (again at a rate of approximately 1 oC/minute), the modulus curve of the quenched slide remained below and approximately parallel to the 54 110 ’6 . D. g 1. m 100 2 :1 D O E m "or 90 1: :1 3 ‘ oTempered glass ' AAnpealed glass 1 1 1 8"o 100 200 500 400 500 600 Temperature (°C) Figure 19, The Young's modulus versus temperature for the annealed state (solid line) and the residually stressed state of a glass slide. 55 curve for the annealed state of the slide. Upon heating from 300 0C to a maximum test temperature of 600 0C, the modulus for the residually stressed state of the slide approached the modulus for the annealed state of the slide (Figure 19). Upon cooling, the modulus of the residually stressed state was slightly below and parallel to the modulus for the initial annealed state. Thus the heating and cooling curves for the residually stressed state form an open loop (the dashed curves in Figure 19). It should be noted that when the heating and cooling cycle was repeated for a second temperature cycle from room temperature to 600 oC and back to room temperature (not shown in Figure 19), the modulus versus temperature curve was reversible. Thus, the relatively high state of residual stress (140 MPa) apparently did affect the elastic modulus, but (as would be expected) the thermal cycling of the specimen from room temperature to 600 oC and back to room temperature at a rate of 1 oC/minute released the residual stresses to the extent that the modulus was reversible upon subsequent thermal cycling at the same heating and cooling rate. Appendix A discusses a model for the experimentally observed change in elastic modulus due to the residual stress state. Thermodynamically, residual stresses can be regarded as an unstable state, such that when a residually stressed glass plate is heated, the residual stresses are released. The release of the stored elastic strain energy associated with the residual stresses can be examined calorimetrically. Figure 20 illustrates a Differential Scanning Calorimetric (DSC) analysis of the heat flow as a function of temperature for the annealed and the residually Heat Flow (W/g) 56 0.00 —0.05 _ . tempered glass -o.1o~ —o.15- —0.20 - . l 1 l 1 l 1 l 1 L 1 l 1 ‘0'250 1 00 200 300 400 500 600 700 Temperature (°C) Figure 20. Thermal flow measurement of glass slides using a Differential Scanning Calorimeter. 57 stressed state of a single glass slide. The annealed and residually stressed states of the glass slide are represented by the solid and dashed lines, respectively. As was the case in the elasticity measurements discussed above, residual stresses in the DSC specimen were induced by a thermal quench of AT = 680 OC into a room temperature oil bath. The cross-hatched region between the curves of the annealed and the residually stressed states is a measure of the stored elastic strain energy released from the residual stresses in the glass slide. As indicated in Figure 20, the residual stress relaxation begins at about 300 0C, which roughly agrees with the onset of relaxation indicated by the elastic modulus measurements. However, the DSC measurements were made at a heating rate of 10 oC/minute, while the elasticity measurements were made at a heating and cooling rate of 1 oC/minute. Future studies should repeat the DSC and elasticity measurements on identically treated slides with identical heating and cooling rates during measurement. (4.3). Statistical study of the Effect of Subcritical Crack.Growth on Thermal Shock Resistance In this section subcritical crack growth is investigated by comparing the fracture strength distribution for annealed glass plates to that of quenched (thermally shocked) glass. In addition to single-quench thermal shock testing, subcritical crack growth was evaluated in terms of thermal shock fatigue damage (cyclic thermal shock). Results indicate that subcritical crack growth effects are observable in the shock testing of glass slides via systematic shifts in the retained strength distribution. (4.3.1) Candidate fracture strength distributions The strength distribution for a group of monolithic ceramic specimens is related to the distribution of pre-existing critical flaws in the specimens [4,5]. While the Weibull distribution function is typically used to describe fracture strength distributions [18,87], distributions other than the Weibull function may provide a better fit to the strength data in some cases [4,5,88,99]. For example, Doremus found that the normal distribution function fit the static strength data for Pyrex glass better than the Weibull distribution [4]. Shimokawa et al. reported that the lognormal function fit fatigue data for carbon fiber/epoxy matrix composite specimens better than the Weibull function did [88]. In this study, the fracture strength histogram for 239 annealed glass slide specimens is approximately symmetrical so that the normal distribution function is one of the natural candidates to 58 Observed Frequency 00 CD B) CD 59 40 10 50 70 90 1 10 130 1 50 Fracture Strength (M Pa) Figure 21. Fracture strength histogram for 239 annealed glass slide specimens fractured in three-point bend. 60 describe the data (Figure 21). Consequently, in this study the three candidate distribution functions used to analyze the distributions of the fracture strength for the annealed glass slides were the two-parameter Weibull, the normal, and the lognormal functions. The corresponding probability functions are characterized as follows: The Normal distribution is described by [90]: f(y) - 1 exp[- _1_ ( y - p )2 ] (21a) 0 (2 “)1/2 2 0 Y F(y) = f(t) dt (21b) A 1 n 11- _ 2 Y1 (21C) n i-l A n A 02 - .1. 2 < y, - #)2 <21d> n i-l where f(y), F(y), p, and 02 represent the probability density function (PDF), cumulative distribution function (CDF), mean, and variance, respectively. The L and 02 are the maximum likelihood estimators for p and 02. The Lognormal distribution with two parameters is characterized by [91]: f(y) - 1 expi - _i_ < l°g(Y) ' “ >2 ] <22a> y a (2 «)1/2 2 a 61 y F(y) - f(t) dt (22b) 0 A 1 n u - ___ X 1°g<_Z_)m'1 exp<-m> <23a> b b F(y) - 1 - exp(- (y/b>m> (23b) E(y) - b r(1 + l/m) (23c) Var(y) = b2[ F(l + 2/m) - (F(l + l/m))2 ] (23d) where E(y), Var(y), and F are the expected value, variance, and gamma function, respectively. Weibull parameters m and b can be calculated according to the maximum likelihood estimators [93]. First, m is calculated iteratively from equation (24a). The b is then computed using equation (24b). 62 n X y? ln(y.) . l 1 i=1 1 l n - ___ = ___ Z ln i=1 n Applying the maximum likelihood method [90-92] to the annealed glass slide fracture strength data (Table 3) yielded the statistical parameters for the normal, lognormal, and Weibull distribution functions (Table 4). To measure the discrepancy between the fracture strength data and the three candidate distribution functions, a goodness-of-fit test is required. The goodness of fit test employed in this study is based on the empirical distribution function (EDF) and the Kolmogorov-Smirnov test [93,94]. The EDF, Fn(y), is defined as [94] Fn(y) number of observation 5 y -w < y < m (25a) n Fn(y) = _i_ yi s y < y1+1 (25b) n Fn(y) - 0 y < Y1 (25c) Fn(y) - 1 yn s y <25d> y1 < y2 < y3,... < yn are the order statistics for the fracture strength data y1 for a random sample of size n. Fn(y) is the step function illustrated in Figure 22. The three continuous curves in Cumulative Distribution Function 63 1.0 ,1, --* 0.8 - lognormal -1 0.6 - \ .1 normal ' 0.4~ . / 0.2-1 ‘ Weibull - Ar" 7 0.0 W 70 90 110 130 Fracture Strength (MPa) Figure 22. Illustration of cumulative distribution functions and the empirical distribution function for the fracture strength data of annealed glass slides. 64 Figure 22 represent the cumulative distribution functions (GDP) of the normal, lognormal, and Weibull distributions with the parameters listed in Table 4. In the Kolmogorov-Smirnov test, a good fit requires that Dn be small, where Dn is defined in terms of the greatest vertical difference between the GDP and the EDF [93], such that D = m x(D+ D”) (26 ) n a n ’ n a + . Dn - max (Fn(yi) - F(yi)) - max ( _:_ - F(yi)) (26b) lSiSn lsiSn n - 1-1 Dn - max (F(yi) - ) (26c) lsiSn n The Kolmogorov-Smirnov statistic (for which the statistical parameters are estimated by maximum likelihood estimators) shows that the normal distribution corresponds to the smallest Dn' However, since the critical value in the Kolmogorov-Smirnov test (two-sided test at significance level a - 0.05) is 0.0878, the normal, lognormal, and Weibull distributions all fit this study's strength data for annealed glass slides about equally well (Table 5). For convenience, the normal distribution is used for the thermal shock resistance analysis in the next section. 65 Table 3. Fracture strength data of annealed glass slides. 8* f* S f S f 74.2 - 2 77.8 - 3 81.4 - 4 83.2 - 3 85.0 — 8 86.8 - 7 88.6 - 6 90.4 - 6 92.2 - 11 94.0 - 10 95.8 - 12 97.7 - 15 99.5 - 18 101.3 - 14 103.1 - 23 104.9 - 18 106.7 - 16 108.5 - 11 110.3 - 10 112.1 - 18 113.9 - 6 115.8 - 2 117.6 - 4 119.4 - 4 121.2 - 3 123.0 - 1 124.8 - 2 126.6 - 2 * S - fracture strength (MPa). * f - total number of the specimens which corresponds to the strength, S. 66 Table 4. Parameters for the normal, lognormal, and Weibull distribution functions, as calculated from maximum likelihood estimators [17-19]. normal : mean p2 - 101.38 MPa variance 0 - 104.90 lognormal : mean p2 - 4.6137 -2 variance 0 = 1.0619 X 10 Weibu11* : expected value E(y) = 101.17 MPa variance Var(y) = 127.33 Weibull b = 106.00 MPa parameters m = 10.834 * The cumulative distribution function of Weibull distribution is: F(y) - 1 - eXP(- (y/b>m) Table 5. Statisitics of the empirical fracture strength distribution function obtained from goodness+of fit test (Kolmogorov-Smirnov test). The statistics D , D , and D are defined by equations 26 (a) - 26 (c). n n 0+ 0' D n n n normal 0.0478 0.0685 0.0685 lognormal 0.0521 0.0867 0.0867 Weibull 0.0787 0.0374 0.0787 (4.3.2) Subcritical crack growth The literature does not agree on the significance of subcritical crack growth on thermal shock resistance in ceramics [7,51,52]. Therefore, it is appropriate here to review briefly the thermal shock literature as it applies to subcritical crack growth. In his 1969 model of thermal shock damage in ceramics [7], Hasselman proposed that: (1) pre-existing cracks propagate when quenched above a critical quench temperature difference, ATc, (2) cracks are stable (do not grow) below ATc, and (3) dynamic crack growth causes a discontinuous drop in the retained fracture strength at ATC. Subcritical crack growth's effects on thermal shock, which were not included in Hasselman's 1969 theory [7], are treated for single- quench thermal shock in a 1974 paper by Badaliance, Krohn, and Hasselman [51]. Subcritical crack growth was modeled by taking into account the propagation of pre-existing cracks for quench temperature differences below ATC. The numerical calculation yielded the critical quench temperature difference ATC - 147 0C. If subcritical crack growth was ignored, a ATC of 238 0C was obtained. Badaliance inferred that the large discrepancy (91 0C) was due to thermal shock induced subcritical crack growth. In modeling the ATC change due to subcritical crack growth, Badaliance used K0 - 0.248 1/2 for the purposes of numerical MPa 1111/2 and Kc - 0.749 MPa m calculations, where Kc is the critical stress intensity and K0 is the threshold for subcritical crack growth. The values of K0 and KC adopted by Badaliance result from static fatigue testing under a mechanical loading [95]. 67 68 In another key study of the effect of subcritical crack growth on thermal shock behavior, Ashizuka, Easler, and Bradt quenched heated borosilicate glass rods into a room temperature water bath [52]. The retained strength of the shocked borosilicate glass rods was measured in a liquid nitrogen bath and in a room temperature water bath. It was assumed that the moisture-free environment of the liquid nitrogen bath would provide "baseline" values of retained fracture strength, free from subcritical crack growth effects. Ashizuka calculated K associated with the appropriate AT by [52] 1 AT a E F(B) (27a) (1-V) KC = a Y c“2 (27b) where K1, E, a, v, C, and Y are the stress intensity factor, elastic modulus, thermal expansion, Poisson's ratio, flaw size, and geometric constant, respectively. The F(B) is a function of Biot's modulus, B [61]. To determine the inert strength, a, both the bend fixture and the specimens were immersed in a liquid nitrogen bath. The specimens were subsequently fractured in four-point bend at a crosshead speed of 0.5 mm/min and an approximate stressing rate of 93.3 MPa/min [52]. Using equations (5a) and (5b), Ashizuka inferred that KO 2 0.9 KC and the subcritical crack growth should be minor [52]. The technique differences in the Badaliance's and Ashizuka’s studies were that (1) Badaliance utilized the static fatigue data 69 to theoretically evaluate the quench-induced subcritical crack growth effect, and (2) Ashizuka experimentally measured fracture strength and then statistically infer the subcritical crack growth effect. Badaliance was engaged in theoretical evaluation and Ashizuka did experimentation. (4.3.2.1). Evolution of fracture strength degradation In our study, the fracture strength of annealed glass slides were characterized by a normal distribution. The evolution of the strength distribution for a group of annealed glass slides depended, for example, on whether or not subcritical crack growth was included in the thermal shock damage process. We discuss the evolution of the crack damage (A) neglecting subcritical crack growth, and (B) including subcritical crack growth. A. No subcritical crack growth effect: In 1983, Lewis proposed that since the critical flaw sizes of a group of brittle components were characterized by a distribution, the mean retained strength of quenched components should gradually decrease as AT increased [38]. Lewis's concept* [38] of the evolution of the fracture strength distribution is experimentally tested in this paper using a large number of glass microscope slides (239 slides were fractured to determine the as-annealed strength distribution and a total of 180 slides were fractured in the single quench tests). * The concept of the evolution of the fracture strength distribution for thermal shock problems was suggested in 1955 by Manson [18]. 70 unshocked specimens number of specimen thermal J Lshock fracture strength P “/l A Ta (3) distribution B A (W (b) , distribution-(lure (CO ///:\\\ ~//K\\\3 distribution B AsT distribution- A I d (d) A /~\ \ A‘A temperature: > ATb ATd >ATC >AT.a Figure 23. Schematic of the evaluation of fracture strength degradation. 71 If transient thermal stresses are mild enough (K < K thermal c for each specimen), then no strength degradation occurs and the initial fracture strength distribution is not altered (Figure 23(a). When thermal stresses are extremely severe (K > KC for every thermal slide in the total population of glass slides), the strength of each specimen drops as the critical flaws in the specimens extend (Figure 23(b)). When the shock severity is intermediate (K > KC for thermal some fraction of the slide population), the retained strength distribution breaks into two clusters and becomes bimodal (Figures 23(c) and 23(d)) [38]. In this study, the cluster with lower fracture strength is defined as distribution B (Figures 23(c) and 23(d)). The cluster with higher fracture strength is defined as distribution A. The strength will drop for that fraction of specimens for which Kthermal > KC since the critical flaws extend during thermal shock, while (neglecting subcritical crack growth) the strength remains unchanged for those specimens where Kthermal < KC. B. Subcritical crack growth effect included: If subcritical crack growth occurs during thermal shock, the evolution of retained fracture strength will differ from that suggested above, in that there will be additional crack growth regimes and an additional crack growth criterion. If K the critical flaws are subjected to "pop-in" Kthermal > c’ crack growth at the initial stage of the thermal shock process. Thus, no subcritical crack growth effect on strength degradation is expected (Figure 23(b)). If Kt < Ko for all slides, then no hermal 72 crack extension will occur by either subcritical or pop-in growth (Figure 23(a)). If Kc > K KO, then critical flaws will not experience thermal > pop-in growth, but they can extended subcritically. Since subcritical crack growth typically occurs at a relatively low velocity as compared to a pop-in type crack growth, we would expect subcritical crack growth to produce shifts in the mean strength of distribution A (dashed line in Figures 23(c) and 23(d)), as opposed to the drastic transformations in strength possible in pop-in crack growth. In order to test for systematic shifts in the retained strength distribution as a function of AT, we must first approximate the form of the initial strength distributions. Histograms of the retained fracture strength data (distribution A) for thermally shocked specimens are then compared to the normal distribution determined in section 3.1 (Figure 21) for the annealed specimens. (4.3.2.2). Experimental results of fracture strength degradation for single-quench thermal shock The glass microscope slides in a portion of this study were subjected to a single quench into a room temperature water bath. For specimens shocked at a quench temperature difference of 1500 C, the retained fracture strength distribution (represented by the solid curve in Figure 24a) shifts slightly to the left when compared to the strength distribution of the annealed glass slides (the dashed line in Figure 24a) (see Appendix C). This small shift to the left (toward lower strengths) in the retained strength 73 1O Observed Frequency (a) strength distribution of annealed glass slides distribution A :0Ltiiiéi‘iiiitiiiiiCHI AAAAAAAA“ Figure 24(a). 30 do 90 120 150 Fracture Strength (MPa) Retained fracture strength of glass slides following a single qgench into a room temperature water bath at AT - 150 C. 74 10 . (b) > 1 ‘3 C! d) a ‘ 8' .r. 5 - B « \ distributionA 5 distribution 8 in ‘ :o: a : C) ‘ 3:: :9 :— q 353 £53 I \ 1‘1 E E \ o_ ‘:.: 2.: ’A' AAAAAA I o 30 so 90 120 150 Fracture Strength (MPa) Figure 24(b). Retained fracture strength of glass slides following a single quench into a room temperature water bath at AT - 160° c. 75 1O . (c) > I o c g I a- distribution 8 d) «5 ‘ a! h- 9: i; :z; "' .0. 0 :1: a: s: u - o ;.; ¢n :9 a . :3; o - -- :: ... O O .........Q‘ § u§§§§§§§§§ \ O- A “““““““ F r 0 30 60 90 120 Fracture Strength (M Pa) Figure 24(c). Retained fracture strength of glass slides following a single quench into a room temperature water bath at AT - 170° c. 76 10 (d) >0 0 q C: 2 U' ‘ strength distribution of E 5 annealed glass slides '8 distribution B distribution A z :2: :2: 4’ a; :5 0' as .. a» n :- ~: :- ° :3 :1 :: Fracture Strength (M Pa) Figure 24(d). Retained fracture strength of glass slides following a single quench into a room temperature water bath at AT - 180° c. 77 10 _ . (e) > . ¢J I: w I a distribution B 0 .0. it: u d) E: d) to n (D , . 0 30 , 60 90 120 150 Fracture Strength (M Pa) Figure 24(e). Retained fracture strength of glass slides following a single quench into a room temperature water bath at AT - 190° c. 78 10 distribution B g“1620:9101(01031610161019!31033303301910): Observed Frequency 0| 1 Eééiéiéééiééééai :-n-noueseseszsa seseszszsmsn-me AAA‘A‘-A (f) Fracture Strength (MPa) Figure 24(f). Retained fracture strength of glass slides following a single quench into a room temperature water bath at AT - 200° c. 79 distribution is interpreted by the authors as indicative of the onset of strength degradation. The histograms of the retained fracture strength for glass slides shocked at AT's of 1600 C, 1700 C, 1800 C, and 1900 C show the development of a bimodal distribution (Figures 24(b) - 24(e)). Distribution B (which is similar to the distribution shown schematically in Figures 23(b) and 23(c)), represents the retained fracture strength of slides quenched severely enough to cause pop-in crack growth. In addition, distribution A shifts progressively to the left toward lower strength values, as compared with the strength distribution of annealed glass slides (dashed line). The experimental data in Figures 24(b) - 24(e) breaks into two clusters. The cluster with higher fracture strength is distribution A. The details for determining such distributions are given in Appendix C. Distribution A disappears in the retained strength data for a AT of 2000 C (Figure 24(f)). The absence of a distribution of type A suggests that each specimen experienced pop-in crack growth. Thus the entire strength distribution was converted into a distribution of type B, which shows a single mode located at relatively low strength values. The strength shift for distribution A was attributed to subcritical crack growth. Comparison of the mean strength of distribution A (Figures 24(a) - 24(e)) and the mean strength of annealed glass slides indicates that the strength shift Au increases monotonically from 4.28 MPa for a AT of 1500 C to 26.56 MPa for a AT of 190° 0 (Table 6). 80 The apparent temperature effect on subcritical crack growth agrees qualitatively with static crack propagation results for silica reported by Sakaguchi et a1. [96] and dynamic fatigue results by Ritter et a1. [97]. Sakaguchi et a1. tested compact tension specimens of fused quartz under static tensile stress in distilled water [96]. Ritter et al. measured the dynamic fatigue of indented soda-lime glass in distilled water using a ring-on-ring test fixture [97]. In both studies the subcritical crack-growth rate increased with increasing water temperature [96,97]. To compensate for the effects of subcritical crack growth, the retained strength data at each AT between 1500 C and 1900 C (Figures 24(a) - 24(e)) were shifted by An (see Table 6 and Figure 5). This shift in strength corresponds to a shift in critical quench temperature difference from the actual quench data ATc z 1750 C to a ”shifted" value ATc z 1900 C. In this paper, the AT corresponding to the 50 percent probability level of failure (see Figures 24(c) and 24(d)) was considered to be the critical quench temperature difference, ATC. The shift in ATc attributable to subcritical crack growth was much less than the subcritical crack growth-induced shift (about 91° C) that Badaliance et al. [52] inferred from their data and computations. The retained strength evolution for the glass slides shocked in this study indicates that subcritical crack growth does play a role in the thermal shock damage process. Fracture Strength (MPa) 81 150 o . . . . , . . , . . , . 50 100 150 200 250 Quenching Temperature Difference (°C) Figure 25. A plot of retained fracture strength versus AT. The solid line represents the original strength data curve (single thermal shock), which becomes the dashed line after compensating for subcritical crack growth effects (see Table 6). (4.3.2.3). Fracture strength degradation during cyclic thermal shock In addition to the single-quench testing, cyclic thermal shock of the glass slides was analyzed in terms of subcritical crack growth. For single quench testing, the retained fracture strength began to decrease at a AT of about 1500 C, with an increase in the magnitude of the error bars* for the strength degradation curve at about 1600 C (Figure 25). Under cyclic thermal shock conditions, thermal shock damage appeared at temperatures below 1500 C. The magnitude of the thermal-shock induced strength drop also increased as the number of thermal shock cycles increased (Figure 26(a)). * The fracture strength distribution of annealed glass slides in this study had a standard deviation 0 - 10.2 MPa (Figure 21). Thermal shock damage caused the strength distribution of shocked slides to form two clusters, with one cluster corresponding to slides that underwent pop-in growth and the other cluster corresponding to slides that underwent slow crack growth only (Figures 24(b) - 24(e)). Thus, when AT is large enough that the strength distribution becomes bimodal then the standard deviation of shocked slides becomes large in comparison with that of annealed slides. As an example, consider that only five specimens had been thermally shocked at a given AT and that three specimens underwent pop-in type crack growth and that the other two specimens experienced subcritical crack growth only. The error bar, which represents two standard deviation in the strength values, would be considerably larger in this case than the corresponding error bars for the as- annealed strength distribution or for the case where all specimens undergo only pop-in growth or only subcritical crack growth. 82 83 To make Figure 26(b) more readable, the error bars have been omitted. However, the magnitude of the error bars for Figure 26(b) are shown in the corresponding data points in Figures 25 and 26(a). The presence of thermal fatigue effects implies that pre-existing cracks can extend during each quench cycle, although the growth tends to saturate for the lower AT values. For example, the strength of the slides quenched at AT's of 1300 C and 1400 C tend toward a saturated damage level for the number of thermal cycles performed in this study, while the strength drops off precipitously for specimens shocked repeatedly at a AT of 150° 0. Therefore, Figures 26(a) and 26(b) demonstrate that subcritical crack growth can occur below the critical quench temperature difference (which corresponds to a stress intensity factor below Kc). Subcritical crack growth is a complex function of temperature and chemical environment. In addition, subcritical crack growth depends on K0 and Kc. For example, as Kc increases, pop-in crack growth decreases. Also, thermal stresses are very strong functions of time and position, thus these stresses are even more difficult to characterize than stresses in quasi-static loading experiments dealing with subcritical crack growth [35-37,9S-97]. The relatively small shift in ATc attributable to subcritical crack growth agrees qualitatively with Ashizuka's inference that subcritical crack growth was insignificant to the thermal shock resistance of borosilicate glass rods [52]. 150 84 .a CD (3 Fracture Strength (MPa) ‘r‘A‘AT = 130 OC 50 - 777‘" ' ‘ + 140 0c () r ' l '. T7 ' l ' 0 10 20 30 40 50 Thermal Shock cycle Figure 26(a). Influence of a cumulative number of thermal shock cycles on the retained fracture strength of the glass slides repeatedly shocked below AT , where AT is the critical quench temperature differgnce determined from single-quench testing. Fracture Strength (MPa) 85 150 _a CD CD 0 ........,... 50 100 150 200 250 Quenching Temperature Difference (°C) Figure 26(b). Variation of retained fracture strength with respect to AT and the cumulative number of thermal shock cycles, N. 86 Table 6. Thermal shock induced changes in the mean strength p and the standard deviation 0 for the retained strength distribution as a function of the quench temperature difference AT. The difference in mean strengths, Ap, measures the extent of slow (subcritical) crack growth. (Units of strength: MPa) total shocked part A of bimodal slow crack new distribution AT specimens distribution growth effect without slow 0 crack growth (C) A A A A A A A A a 0a An =101.38— “a = p + An AT- 0 101.38 10.2 AT-150 97.41 15.3 97.41 11.5 4.28 101.38 AT-160 73.3 34.1 94.25 14.7 7.13 80.43 AT-170 60.75 32.1 83.59 12.1 17.79 78.54 AT-180 58.1 28.8 80.13 13.7 21.25 79.35 AT-190 35 26 74.82 13.7 26.56 61.56 AT-200 22.5 10.7 0 * * Since all specimens for AT - 200 0C were subject to "pop-in" crack growth, the subcritical crack growth effect could not be evaluated. (4.4). Computer Simulation of Quench-Induced Strength Degradation in Glass Plates Hasselman in 1969 proposed a famous model for the calculation of ATc, indicating that crack growth at AT 2 ATC caused a discontinuous drop in the retained strength [7]. However, Lewis (1983) proposed that since the critical flaw size of a group of brittle components was characterized by a distribution, the mean retained strength of quenched components should exhibit a gradual decrease as AT increased [38]. In this section, we extend Lewis' qualitative idea by combining the initial strength distributions, Hasselman's theory (1969), and subcritical crack growth [95,98], to qualitatively model the strength degradation in a group of quenched soda-lime glass specimens. The consequent computer simulation was divided into two parts. Part I of the simulation neglected subcritical crack growth and part II of the simulation included subcritical crack growth. (4.4.1). Theoretical Consideration (4.4.1.1). Initial strength distributions Quench-induced fracture is a stochastic process that depends on the size and spatial distribution of pre-existing flaws. In this study, the initial fracture strength, S, of our group of glass specimens was characterized by a normal distribution with a mean of 101.38 MPa and a standard derivation of 10.2 MPa. (The experimental results were presented in a previous paper section). In the 87 88 computer model, the strength distribution used in the computer model was generated using an IMSL (International Mathematics and Statistics Library) random number generator. (The names of subroutines are RNNOR, SSCAL, and SADD, which are shown in Computer program No. 4.) The sample population contained one thousand data points. The corresponding critical crack size, a = a0, was then evaluated by K 2 a0 — 1r ° (23) where Kc and Y are the critical stress intensity factor and a geometrical parameter, respectively. Y was chosen as 1.1215, which is appropriate for a penny-shaped surface flaw where the specimen is loaded quasistatically in uniaxial tension. (4.4.1.2). No subcritical crack growth effect For a brittle component that experiences a single thermal shock, Hasselman proposed that pre-existing cracks could propagate if the quench temperature difference, AT - ATO, exceeded the critical value, ATc [7]. In Hasselman's model [7], only "one" specimen was considered and the pre-existing cracks in the one specimen were assumed to have uniform size. In the present study, the 1000 glass specimens included in computer model were assumed to have different critical flaw sizes, a - so, as determined from equation (23). However, the pre-existing cracks in an individual specimen were also assumed to have the uniform size a - a0, as Hasselman did. 89 Thus, the critical quench temperture difference, AT = ATc, for the individual specimens was calculated as [7] « 0 (1.21;)2 1/2 16 (l-u2)Na3 _1 2 AT - 1 + a / (29) 2 02E (1_V2) 9 (1-2u) where N is the crack number density in a component with crack length, a. G is the surface fracture energy required to form a unit area of new crack surface. E, u, and a are the elastic modulus, Poisson's ratio, and the thermal expansion coefficient, respectively. The required input parameters are listed in Table 7. Therefore, if ATo > ATc, the resulting "pop-in" crack length, a1, can be calculated using equation (29). Substituting AT - ATc into equation (29) gives two positive real solutions: the original crack length, a0, and a larger crack length, a Crack growth from ao to ql' a is quasi-static, based on Griffith's fracture criterion. ql Hasselman proposed that the pop-in cracks extend kinetically to a final length af, such that [7] 3(aATc)2E 16(1-u2)Na3 '1 16(l-v2)Na: ’1 l + o - 1 + 2 (1-2u) 9 (l-2u) 9 (l-2u) - 2 n N c (a: - a2 (30) f) Equation (29) is a crack instability criterion. Equation (30) determines the kinetic-propagation induced crack length. Although the pre-existing cracks kinetically grow from ao to a the quench f, 90 Table 7. Parameters required for computer simulation property magnitude units references elastic modulus (E) 70 GPa * Poisson's ratio (v) 0.18 * thermal expansion coefficient (a) 6.5 X 10.6 * fracture surface energy (G) 4.0 Pa in.2 [98] crack density (N) 2 X 1011 m-3 # specimen's thickness (22) 0.0012 m * surface heat transfer coefficient (h) 0.4 W/cm2 0C [61] thermal conductivity(C) 0.03 W/cm °c [95] activation energy (2*) 1.0 x 10‘6 J/mole [95] pre-exponential factor (V0) 2.9 X 104 m/s [95] crack growth constant (b) 0.11 [95] fatigue limit (KO) 0.248 MPa ml/2 [95] * The properties were measured in this study. # see Appendix D. 91 temperature difference, AT - ATO, may be great enough to continuously cause the cracks to quasi-statically propagate [7]. When AT - ATo is substituted in equation (29), a solution of crack length a is obtained. If the a is smaller than a , the 42 q2 f corresponding quasi-static crack propagation does not occur. As a result, the final crack length is af rather than aq2. If ad2 > af, the final crack length is a indicating that quasi-static crack q2’ growth occurs after the kinetical crack growth. The retained strength, S, is then transformed from the aq (or af) using equation (28). (4.4.1.3). Subcritical crack growth effect included For glass slides quenched into a water bath, we approximate the thermoelastic stresses in the slides by using the expression for the thermoelastic stresses in an infinite glass plate of thickness 22 (see page 16) a(Z,t) - a E AT(Z,t) Z 2 sin(6n) 2 exp(-a (6 /2) t) 1 - u n-l “ 5 + cos(6 ) sin(6 ) n n n sin(6 ) ( n - cos(6 2/£)) (10a) “___—’6 n n an tan(5n) - B n - 1, 2, 3, .... (10b) G) AT(Z,t) - ATo E: 2 sin(6n) cos(6n Z/l) n-l ] exp(-d(6n/£)2 t) 6 + sin(6 ) cos(6 ) n n n (3b) 92 AT(Z,t) - temperature gradient in a glass slide t - time d - thermal diffusivity 6 - the root of Eq. (10b) 2 - coordinate in the direction normal to glass plate surfaces 2 - half thickness of glass plate. 2 - 2 and Z - -2 on both plate surfaces B - Biot's modulus Badaliance, Krohn and Hasselman investigated the interrelation among the stress intensity factor, subcritical crack growth, and quasistatic crack propagation during thermal quench [51]. Pre- existing cracks are suspectable to subcritical crack growth during a thermal quench if the magnitude of the stress intensity factor, is KO < K where K0 is the threshold for Kthermal’ thermal’ subcritical crack growth. The crack growth may be described by [51,95,98] da/dt - vo exp[(-E* + b K 1)/RT] (31) therma , 1/2 Kthermal - 1.1215 a(£,t) (« a0) (32) where V0 and b are constants. E R, and T represent the activation *. energy, the universal gas constant, and the absolute temperature, respectively. (Table 7 lists the subcritical crack growth parameters.) Badaliance proposed that the pre-existing surface cracks were subjected to transient thermal stresses equivalent to the surface transient stresses. In the present study, we used Badalinace's approximation to model subcritical crack growth during thermal quench. 93 In this paper, we used equation (29) to determine whether or not pop-in crack growth occurs. If the initial quench temperature difference ATo exceeded ATc, pop-in crack growth was assumed to occur. (Equation (24) only predicts whether or not the crack growth occurs, without explaining when the crack growth occurs.) In the computer simulation of thermal quench damage in this paper, if pop-in crack growth did not occur, subcritical crack growth mau nor may not occur, subject to the following conditions: (a) (b) , (C) When K was smaller than K , no subcritical crack growth thermal 0 occurred. When Kthermal exceeded K0, the pre-ex1st1ng cracks grew according to equation (31). Crack propagation was simulated iteratively via equations (10a), (31), and (32). Subcritical crack growth in pre-existing cracks caused ATc to change with elapsed time. The transient average temperature, Ta’ of specimen also decreased with elapsed time. When the critically growing crack satisfied equation (29) (ATa > ATC), the cracks were allowed to grow according to equation (30) (substituting AT - ATa). Cracks that grow subcritically up to the critical flaw size (for the "current" thermoelastic stress state) can grow by pop-in rather than continuing to grow subcritically. In the computer simulation, when a crack grows subcritically to the critical flaw length, the computation of crack length shifts from the subcritical crack growth subroutine (represented by equation 31) to the pop-in growth subroutine. (d) The initial time and the final time of the thermal shock simulation were t - 0 second and t - 2.5 seconds, respectively. 94 At the end of the thermal shock duration, the retained fracture strength, 8, was transformed from the final crack length using equation (23). (4.4.2). Numerical Results and Discussion The computer simulation indicates that the evolution of quench- induced crack growth depends on whether or not subcritical crack growth effect is included in the thermal shock damage process. According to Hasselman's model (Equations (29) and (30)) and under the conditions in Table 7, a pre-existing crack with crack length a - 1.2X10-5m never grows at AT < 218 oC. However, the crack can grow subcritically at 106 0C < AT < 218 oC (because of K > KO). Figure 27(a) illustrates the variation of the crack length with time at different AT. Figure 27(b) shows the velocity of crack growth as a function of time. When AT - 185 0C and AT - 195 0C, the crack length increased 0.9 pm and 0.1 pm at the end of the quenching process, respectively (Figure 27). When AT - 205 oC and AT - 215 0C, the subcritically growing cracks become great enough that Griffith's failure criterion is finally satisfied and a pop-in crack growth occurs. According to the computer simulation, subcritical growth effect enhances thermal shock damage. A thermal shock may cuase the pre-existing crack to propagate, which degrades the fracture strength of the shocked specimens relative to that of the non-shocked specimens. Since the critical flaw size (initial fracture strength) of a group of brittle components is characterized by a distribution, thermal shock testing accordingly requires a large number of specimens to achieve a reliabe 95 19 ~ 215,“: 205°C g 17- 5' J '5 C” 5 _I 15- A! C3 2 . o . 13_ 195 c . 185°C 11 Arm...“ . ..fi..., . .. 0.001 0.01 0.1 Figure 27(a). Time (second) Computer simulation of crack length as a function of time for cracks extending subcritically. Note that small differences in quench temperature can induce dramatic changes in the crack growth behavior. Velocity (m/s) 96 1 E-2 15-3 '- 215 205 154 '- 1E-5 - 1E-6 - 185 1 E-7 1 EB (13) AT = 195 0c I 0.0001 0.001 0.01 0.1 Time (second) Figure 27(b). function of time for cracks extending subcritically. Computer simulation of crack growth velocity as a Number of Specimen 97 h) C: CD I Figure 28(a). (a) ‘ I 1 . 50 100 150 Retained Strength (M Pa) Computer simulation of the strgngth degradation of specimens shocked at AT - 140 C. The solid line represents the retained strength without subcritical crack growth effect. The dashed curve represents the retained strength with subcritical crack growth effect. Number of Specimen 98 600 (b) J \ 400 '- 200 J 0 ‘L I ' i 0 50 100 1 50 Retained Strength (MPa) Figure 28(b). Computer simulation of the strength degradation of specimens shocked at AT - 240 oC. 99 g 200- d) E (D d) CL 0) '5 , G, s: 100- E 3 a Z A ’ o - 7*rrér r o 50 100 Retained Strength (M Pa) (C) 150 Figure 28(c). Computer simulation of the strength degradation of specimens shocked at AT - 160 C. 100 (d) no subcritical 5 20° ‘ crack growth effect E C) 9 i Q -l (D l '6 i h l 8 100 ~ i E i I: z i l l l i o 1 ' k — r-‘M I 0 50 100 Retained Strength (MPa) 150 Figure 28(d). Computer simulation of the strength degradation of specimens shocked at AT - 180 C. Number of Specimen 101 (e) 400- f i i i i i i 200- i i i i. i i ’~\ 0 f 1" r —-"_" ‘I J 0 50 100 150 Retained Strength (MPa) Figure 28(c). Computer simulation of the strgngth degradation of specimens shocked at AT - 200 C. 102 statistical estimate. The computer simulation results illustrate the strength degradation development of 1000 quenched glass specimens (Figure 28). The solid curves represent no subcritical crack growth and dashed lines represent subcritical crack growth. For simplicity, we shall discuss the evolution of the fracture distribution for two cases: (A) neglecting subcritical crack growth, and (B) including subcritical crack growth. A. No Subcritical Crack Growth Effect: Equation (29) yields the critical quench temperature difference for a specimen with a given flaw size a - a0. For the thermal stress conditions listed in Table l, ATc will vary from specimen to specimen according to the critical flaw size. Figure 29 shows the distribution of ATc corresponding to 1000 glass specimens, which are transformed from the critical flaw size using equations 23 and 29. When 1000 glass specimens are shocked at AT - 140 0C, the quench temperature difference is smaller than the ATc for each specimen (see Figure 29). Thus, no strength degradation occurs and the initial fracture strength distribution is not altered (Figure 28(a)). When the glass specimens are shocked at AT - 240 0C, the quench temperature difference is greater than the-ATc for each specimen. Consequently, the strength of each specimen drops as the critical flaws in the specimens extend (Figure 28(b)). When AT - 160 oC, (AT - 180 0C, or AT - 200 0C), some fraction of specimens are subjected to pop-in crack growth. Thus, the retained strength distribution becomes bimodal (Figures 28(c), 28(d), and 28(e)). B. Subcritical Crack Growth Effect Included: If subcritical crack growth is present during thermal shock, the evolution of retained fracture strength will differ from that suggested above, in that there will be additional crack growth regimes and an additional crack growth criterion. When 1000 glass specimens are shocked at AT - 240 OC, all critical flaws are subjected to pop-in crack growth. Thus, subcritical crack growth will have no effect on strength degradation (Figure 28(b)). If the glass specimens are shocked at AT - 140 0C, no pop-in crack growth occurs. Furthermore, the transient thermal stress corresponding to AT - 140 0C is not great enough to significantly activate subcritical crack growth. At AT - 160 oC, AT - 180 0C, or AT - 200 OC, subcritical crack growth result in observable shifts in the retained fracture strength (dashed curves in Figures 28(c), 28(d), and 28(e)) in comparison with pop-in crack growth (solid curves). The computer simulation results qualitatively imply that the initial fracture distribution of a group of the specimens shocked at T - To can be divided into four domains (Figure 30). Since the ATG of each specimen in domain I is smaller than ATO, all specimens are subjected to pop-in crack growth. Specimens in domain J do not undergo pop-in crack growth initially. Later on, the specimens in domain J are activated from subcritical crack growth into pop-in crack growth. Specimens in domain P only experience subcritical crack growth. Specimens in domain Q do not accumulate thermal shock damage because of K < KO. (If subcritical crack growth were not considered, domains J and P would not exist. As a result, the arrows point out the development of retained strength 103 Number of Specimen 104 120 100- 80- 60- 40.. 20- i ' ~ 100 150 200 250 _ 300 Critical Quench Temperature Difference ( ATc) Figure 29. A ATc distribution of 1000 pieces of annealed glass slides, which was calculated from the initial flaw size using equation (29). The initial flaw size was randomly generated through computer program. 105 A J before quench i : Ii 5: G Li 1 , C o i §_ after quench tn ‘5 3 Q’ E p' , .3: i r Fracture Strength Figure 30. A schematic of illustrating the retained strength degradation in a group of shocked specimens. 106 distribution, turning into domains L', M’, P', and Q'. The populations of domains L, M, P, and Q are determined by the To, K0, and thermal stress conditions. Figure 31 is a plot of mean retained strength versus AT, which corresponds to the retained strength data represented for individual AT's given in the series of Figure 28a - 28e. The solid curve does not include subcritical crack growth while the dotted curve does include subcritical crack growth. The strength degradation curves exhibit a smooth strength decrease, rather than a discontinuous drop, in agreement with Lewis' concept. In the present study, the critical quench temperature difference, ATc, was considered to be the AT corresponding to the 50 percent probability level of failure. Thus, the solid line yields ATc z 200 oC and the dotted line gives ATc z 190 o C. Subcritical crack growth apparently augments the thermal shock damage, resulting in a decrease in ATC by about 10 oC. 107 150 i no subcritical crack growth effect \ with subcritical / \ crack growth effect ‘ «i 505 l. \ a shift ofATc \ Retained Strength (MPa) ' l 80 160 2110 Quenching Temperature Difference (°C) Figure 31. A plot of retained strength versus AT. Solid line ‘ represents the mean retained strength without subcritical crack growth effect. Dashed line is mean retained strength with the subcritical crack growth effect. The computer simulation data was presented in Figure 28. 5. SUMMARY and CONCLUSIONS Thermal quenching into a water bath from below the material's annealing temperature is the standard technique for thermal shock testing of brittle ceramics that behave viscoelastically above some elevated temperature [64-73]. The typical thermal stress calculations for such experiments are thermoelastic in nature [21- 37]. However, in practice thermal stress conditions for a ceramic component may include a broad quenching temperature ranges, such as thermal quench from above the annealing temperature. Thus, thermal stress calculations should include both thermoelastic theory and viscous flow effects. In this paper, a thermal shock damage map (Figure 9) was proposed to predict the effect of thermal quench on a glass plate, based on thermoelastic and thermoviscoelastic stress theory. The map indicates that quenching a glass plate may induce either thermal shock damage and/or residual stresses depending on the heat transfer characteristics of the quench bath, the initial material temperature, and material properties. The analysis also indicates that thermal quench from below the glass's annealing temperature increases the probability of thermal shock damage as the quench temperature difference AT increases. Viscous flow occurs when a glass is quenched from above the annealing temperature. Viscous flow reduces the transient surface tensile stress, although viscous flow leads to the eventual development of residual tensile stresses in the plate's interior. Thus, a damage-free tempered (quenched) glass plate will be restricted to a certain range of AT. 108 109 Commercial glass microscope slides were quenched from below and above the annealing temperature. The quenching media include air, oil, and water. The Biot's modulus, B, is ranked as follows: B (water quench) > B (oil quench) > B (air quench). Quenching from below the annealing temperature may lead to thermal shock damage. As the Biot's modulus decreased, the critical quench temperature difference increased. Shock damage decreased the retained fracture strength and elastic modulus, while it increased internal friction. Quenching from above the annealing temperature caused residual stresses and thermal shock damage. For a given quench temperature difference, a decrease in Biot's modulus reduced the magnitude of the quench-induced residual stress and prevented thermal shock damage. The trend in retained fracture strength data agrees with the theoretical analysis (see Figure 9 and reference 61). Thermal quench from above the annealing temperature, without accompanying thermal shock damage, decreases elastic modulus and increases internal friction (Appendix A). A statistical study of subcritical crack growth during thermal shock damage (for a temperature range of AT < ATC) was also performed in this study. As AT was varied in thermal shock experiments on the glass microscope slides, shifts in the retained fracture strength distributions indicate the relative contributions of subcritical (slow) crack growth (Figures 23 and 24). The initial step in the experimental assessment of subcritical crack growth in thermal shock was to determine the strength distribution of a population of unshocked specimens. In this study, a Kolmogorov- Smirnov goodness-of-fit statistic indicated that the normal, 110 lognormal, and Weibull distributions fit the fracture strength for unshocked, annealed slides about equally well. For convenience, a normal distribution was used for the thermal shock resistance analysis. The strength degradation observed during thermal shock fatigue tests qualitatively indicated that slow crack growth (subcritical crack growth) occurs below ATC (below KC). Non-negligible subcritical crack growth during thermal shock was also demonstrated experimentally via a statistical analysis of the retained fracture strength data for single-quenched glass slides. However, for the single-quenched glass slides, the AT0 for subcritical crack growth decreased by about 150 C, which is much less than the 910 C shift in ATc reported by Badaliance [51]. A computer model of thermal shock damage was presented, in which a modified form of Hasselman's theory [7] was used to map the initial flaw size (initial strength) into final crack lengths (retained strength). Subcritical crack growth was approximated by superimposing subcritical growth equations on Hasselman's theory. The input parameters used in the computer simulation are experimental data reported on soda-lime glass. The computer model qualitatively showed that because the initial strength of a group of glass specimens was characterized by a distribution, the strength degradation curve exhibited a gradual decrease with AT. Subcritical crack growth effect enhances thermal shock damage and decrease the critical quench temperature difference. Ji"U'H‘i‘lil'i'iiii[iiiiiigfliriiii|'-‘ 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 6. REFERENCES T. K. Gupta, "Strength Degradation and Crack Propagation in Thermally Shocked A1203", J. Amer. Ceram. Soc , 55 [5] 249-253 (1972). D. P. H. 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APPENDICES (APTTHHDIXIIK INFLUENCE OF RESIDUAL STRESSES ON THE MEASUREMENT OF THE DYNAMIC ELASTIC MODULUS Experimental results indicate that thermal quenching of a glass plate from above the annealing temperature can induce residual stresses in the plate and decrease the plate's dynamic resonant frequency (observed as the dynamic elastic modulus, Figure 18). A qualitative analysis, based on dynamic vibration theory and quench- induced nonlinear elasticity can explain the direct relationship between residual stress development and the resonant frequency decrease. First, the residual stress profile in tempered glass is determined numerically. Figure A-l illustrates the stress distribution calculated from thermoviscoelastic stress theory. The residual stress distribution is a complex function of initial glass temperature, thermal quench conditions, etc. However, the spatial distribution is symmetric and nearly parabolic. Therefore, we approximate the stress profile in terms of a polynomial 2n 2n-2 2n-4 0*(2) - A2n z + A2n-2 z + A2n-4 z + ... +:.Ao (A-la) The residual stress 0* is subject to the boundary conditions 0*(h/2) - 0*(- h/2) < 0. (A-lb) 119 120 1004 -1004 -300 D AT = 740 06 Residual stresses (M Pa) _ A:mms=1 50° B: Biot’s = 4 szms=8 D: Biot’s = 35 '700 l ' i ** r r’ ' r ' r— ‘ l -0.6 -0.4 -O.2 0.0 0.2 0.4 0.6 Position in 2 Axis (mm) V Figure Al. Residual stress profile in a glass plate, as evaluated numerically from thermoviscoelastic theory. 121 The equilibrium of forces and moments requires that /2 [h 0*(2) dz - 0 (A-lc) -h/2 /2 0*(2) 2 dz - 0 (A-ld) -h/2 where 0* and h represent the quench-induced residual stresses and the thickness of the glass specimen, respectively. In general, a fourth-order polynomial is sufficient to accurately model the residual stress distribution in the glass plates. For example, the calculated residual stress associated with a quench of AT - 740 oC and Biot's modulus - 15 can be fitted well by 0*(2) - -1864Z4 - 70722 + 134 (correlation coefficient - 0.997). Similarly, if we assume a nonlinear elasticity similar to that observed by Mallinder et al. the thermal strain distribution, 6*(2), can be approximated by the general form 9 2 4» 6*(2) - B4 2 + 32 z + Bo (A-2) The dynamic elastic modulus of materials may be measured by the standing wave resonance technique [14-16]. Therefore, we shall model the effect of the residual stress on the dynamic elastic modulus in terms of a free-free suspended vibrating bar. A free- free suspended vibrating bar (Figure A-2) is the test specimen used in the standing resonance technique [14-16]. According to the dynamics of beams, the governing equation (the Bernoulli-Euler beam equation) is [Al] 122 driver - pickup cofion / string t// /% Figure A2. Schematic of specimen suspension for the standing wave resonance technique of elastic modulus measurement. 4 2 E1 0 W(x,t) + a:_ a W(x,t) 0 (A-3a) 8x4 3 at2 The boundary conditions are 62W(0,t) 62W(L,t) - o - 0 (A-3b) 3x2 8x2 33W(0,t) 63W(L.t) ' - 0 - 0 (A-3c) 8x 3 8x 3 where E, p, a, L, and I represent the elastic modulus, density, cross sectional area, specimen length, and the second moment of inertia of the cross section of the beam with respect to the neutral axis. (In this derivation, we neglect the effects of shear deformation and rotary inertia, since their effects are not significant for the bar-shaped specimens considered here [A1]). The parameter g is the acceleration due to gravity. W is the transverse deflection of the beam, which is a function of time, t, and position along the longitudinal axis, X. Solving equation A-3 by separation of variables and applying the boundary conditions gives the characteristic equation [A1] Wh(x,t) - wh(x) (Clcos(flnt) + Czsin(fint)) (A-4a) - wn(x) (C3 cos(flnt + 0)) where C , C , w and p are defined as 1 2 n n 124 tan(o) - 02/01 (A-4b} wh(x) - cos(knx) + cosh(khx) cos(k L) - cosh(k L) - n “ (sin(k x) + sinh(k x)) (A-4c) sin(knL) - sinh(knL) n n p - k [El 3 1 1/2 (A-4d) and, in addition cos(knL) cosh(knL) - 1 (A-4e) C1” C2, and C3 are integer constants. Each Wn corresponds to the nth mode of the harmonic transverse vibration with the frequency fn - pn/2s. For n - 1, the fundamental vibrational mode, the first root of equation A-4e is le - 4.73004 [A1]. Consequently, the fundamental natural frequency of the glass specimen is given by g 1/2 f - 1 E I (A-Sa) 2 s a p Furthermore, 2 4 E 0.09652 f L p (A-Sb) h2 Equation (A-Sb) is useful for elastic modulus calculations of bar- shaped specimens, where E is expressed in terms of kgf/m2 (l kgf/m2 - 9.806 Pa). Experimentally, quenched glass specimens had lower resonance frequencies than the annealed glass specimens. Apparently, residual stresses in the tempered glass lead to the resonant frequency 125 decrease, and from equation (A-Sb), a decrease in resonant frequency implies a decrease in elastic modulus, E. Equation (A-S) (based on linear elasticity) can not explain the experimentally observed decrease in Young's modulus. On the other hand, the energy method applied to the dynamics of beams (Rayleigh's approximation method [A2]) can qualitatively account for the material nonlinearity. Mallinder et al. [8] indicated that soda glass exhibits I“ nonlinear elasticity under large deformations, such that E - E0 (1 - i 5.11 6), where e is the strain and E0 - 72.5 GPa. Meanwhile, Gogotsi ‘ et a1. [A3] and Swain [A4] also reported nonlinear stress-strain behavior in ceramic materials. Thus, if we assume that the residual i F stresses in tempered glass can also induce nonlinear effects, namely, 2* - so (1 - c e) - so ((1 - c (5* + 7)) (A46) where 6* is the thermal residual strain, 6 is the longitudinal strain due to beam vibration and C is a proportionality constant. Eo corresponds to the slope of the stress-strain curve near the origin (Figure A-3). E* is the elastic modulus as a function of the residual stresses. In this study, since the flexural deformation of a free-free suspended vibrating bar, 6, is small, the geometrical nonlinearity is neglected. The vibration induced deformations of the annealed glass and the tempered glass specimens are assumed to satisfy beam dynamics and classic beam theory. As a result, the observed elastic modulus of an annealed glass bar is E z E0. However, residual stresses may lead to nonlinear stress-strain behavior (Figure A-3) 126 (a) stress slope (b) a strain 1 1. 1 i stress in annealed glass tensile compressive l7" b ‘Ii 1 ——g—..—.—— , i ,__ I, ' ’l’ i -—' I \ + l — \ \s I \\ i 7‘ l ”T5.— thermal residual longitudinal stress stress in stress prof1le due to vibration tempered glass Figure A3. Influence of the thermal residual stresses on the stress- strain range of a standing wave. (a) Nonlinear stress- strain behavior of material. (b) Longitudinal stress due to vibration, where a represents the maximum stress- strain range. (c) Superposition of thermal residual stresses and longitudinal stress, where b is the maximum stress-strain range. 127 so that the natural frequency calculation should include nonlinear elasticity effect. Mechanical energy conservation requires the maximum kinetic energy to equal the maximum potential energy [A2]. Thus, T - U (A-7a) max max here, '1' — P a (9 )2 dx (A-7b) 0 2 g n E 1 2 u - (w..) dx. (A-7c) 0 2 n T and U represent the kinetic energy and the potential energy of the annealed glass specimen. W and W" are derivatives with respect to time and x, respectively. From equation (A-7), we obtain L r - p a 0; 52 sin2(fl t +0) w2 dx (A-8a) n n n 0 2 g E I 2 2 2 U - C3 cos (6 t +0) (w ") dx (A-8b) 0 2 “ Equating the maximum values of T and U yields an estimate of f2, the square of the fundamental natural frequency (n - 1) of annealed glass, such that 2 g [L a 1 (w")2 dx 2 5 0 f - - (A-9) 2 2 4 u 4 x IL p a V2 dx 0 128 Residual stresses change the effective vibrational potential energy in a quenched (tempered) glass bar. However, the functional form of the equation for kinetic energy does not change, and thus equation (A-7a) is rewritten as Tmax - (U*max - U*resid)' (A-lO) Ul is the maximum potential energy in a tempered glass bar undergoing vibration. U is the total potential energy arising *max from the internal thermal residual stresses and the longitudinal stress induced by beam vibration (Figure A-3). U*resid is the thermal residual strain energy of a tempered glass bar in the static or quiescent state. That is, U is the potential energy of the *resid bar when there are no vibrations due to the standing wave resonance measurement technique. Thus, we have U*max ' U*resid - /2 _ - b [(a + 6 E ) - a ] d6 dz dx i:i?h/2 i * * * /2 _ _ - b [11h J E (l - C(£* + 6)) e de dz dx (A-ll) 0 -h/2 ° where b is the width of the bar. Combining equations (A-2) and (A- 11) yields U*max - U*resid E I o ,, 2 3 4 3 2 - (Wu ) [I - C (1 34h 4» 32h +80) dx (A-12) 0 2 12 '26 129 Note that if 1 is the curvature of the beam with respect to the neutral axis, then 6 - 2/1, 1/1 - W", and fo*z dz - 0. Substituting equation (A-l2) with equation (A-10) and then comparing with equation (A-9) and (A-Sb), we get ‘19 - ___ - 1 - c (_3_ 34h4 + _3_ B2h2 +30) (A-13) E m "E‘ 112 20 N where EIn is the observed elastic modulus. f* and f are the fundamental natural frequencies of the tempered glass and of the annealed glass, respectively. Equation (A-13) indicates that, if a tempered glass specimen exhibits nonlinear elastic behavior, the observed elastic modulus may be a function of the residual stress state . References for Appendix A A1) E. Volterra and E. C. Zachmanoglou, p. 321, yihzgpipng, Charles E. Merrill Books Inc., Columbus, Ohio, (1965). A2) I. H. Shemes and C. L. Dyn. p. 334. Ensrsx_sns_£inite_nlsment E2th2sa_in_§£rsstsrsl_nsshaniss. Hemisphere Publishing 00.. New York. (1985). A3) C. A. Gogotsi, Y. L. Groushevsky, and K. K. Strelov, "The Significance of Non-elastic Deformation in the Fracture of Heterogeneous Ceramic Materials“, Ceramurgia International, 4 [2] 113-118, (1978). A4) M. V. Swain, 'R-Curve Behavior and Thermal Shock Resistance of Ceramics”, J. Am. Ceram. Soc., 73 [3] 621-28, (1990). .APTTEUIEX B PHOTOELASTIC DETECTION OF CRACKS AND THE SUBSEQUENT RESTRICTIONS ON QUENCH CONDITIONS FOR CRACK-FREE SPECIMENS Photoelasticity is an useful method for detecting the quench- induced cracks in a tempered glass plate. The cracks, which may be, say, 2 mm in length, are sometimes invisible to the unaided eye for specimens quenched from above the annealing temperature, although cracks of a similar length are typically detected much more easily in specimens quenched from below the annealing temperature. The difficulty in detecting cracks in glass slides quenched from above the annealing temperature likely stems from a decrease in crack opening displacement as a function of the surface compressive stresses that arise when glass plates are quenched from above their annealing temperature (Figures 18(a) and 18(b)). However, cracks locally disturb the residual stress field of the tempered glass specimens so that photoelastic stress ‘measurements yield fringe patterns that are characteristic of the stress release and redistribution [B1,BZ]. In the study of the effect of residual stresses on the observed elastic modulus changes, tempered glass specimens were examined photoelastically in order to eliminate the specimens with cracks. By not measuring the elastic modulus for specimens having detectable quench-induced cracks, we attempted to eliminate the crack-modulus effects and focus on the effect of residual stress upon the elastic modulus (see Appendix A also). 130 131 For quenching into room temperature oil (SAE 20W 50), about 1/5 of the specimens quenched at AT - 680 0C were eliminated from the elasticity measurement due to cracking, and about 4/5 of the specimens quenched at AT - 660 0C were eliminated because of cracking. Oil quenching for the AT interval 550 0C < AT < 640 oC did not provide a non-cracked specimen. For air quench (cooling freely in air), all quenched specimens successfully developed residual stresses without cracking, although due to the lower surface heat transfer coefficient, h, the air quench produced a much lower residual stress than did the oil quench. However, neither quenching route provided specimens with residual stresses, Sr, in the range of 40 MPa < Sr < 100 MPa. Thus, the data in Figure 18 form two clusters, the cluster of Sr < 40 MPa corresponds to the air Quenched specimens and the cluster where Sr > 100 MPa corresponds to the oil quenched specimens. References for Appendix B B1) C. W. Smith, 'Photoelasticity in Fracture Mechanics", Experimental Mechanics, 20 [11] 390-396, (1980). 82) A. Shukla and J. W. Dally, "A Photoelastic Study of Energy Loss During a Fracture Event”, Experimental Mechanics, 21 [4] 163-168, (1981). APPENDIX C DETERMINATION OF DISTRIBUTION A IN A STATISTICAL ANALYSIS OF RETAINED FRACTURE STRENGTH For certain ranges of quench temperature difference AT, the schematics in Figures 23(c) and 23(d) depict the retained strength distribution as separating into two clusters. In Figures 23(c) and 23(d), the two clusters (also labeled as distribution A and distribution B) are clearly distinguisable. However, in practice the distributions may not exhibit such a clear separation (Figure 24(b)). This appendix proposes a systematic way of approaching this problem. In order to proceed with the analysis, We assumed that: (1) The shocked specimens without pop-in crack growth (distribution A) exhibits a normal strength distribution similar to that of the annealed specimens, and (2) Subcritical crack growth only shifts distribution A, without changing its shape. Consequently, we describe distribution A by a normal distribution with the same standard deviation as the annealed specimens. In this paper, we determined distribution A as follows: (1) The retained strength data for n specimens were ranked in ascending order such that y1 < yz < y3 ..... < yr1 < ya. (2) Distribution A is a subset of the retained strength data. The subset consists of the ordered strength data for (n - j + 1) specimens, which is yd, yJfl, y‘m, ...., yrl, yn. The jth strength values is selected such that the standard deviation of the subset, ; , is approximately equal to, a, the standard deviation of the strength distribution for the annealed glass slides. In this study, a for the annealed slide glass 132 133 population was 10.2 MPa. (3) After determining the strength values to include in distribution A, the mean strength, ;A’ is then calculated. In Figure 5, the solid curves represent the normal distributions with mean ”A and standard deviation 0A for the (n - j + l) specimens that underwent subcritical crack growth. The dashed curves in Figure 5 illustrates the normal distribution for the (n - j + 1) specimens, but with the mean and standard deviation of the unshocked annealed glass slides (p - 101.38 MPa and a - 10.2 MPa). The dashed curve thus presents the distribution “as it would have been” without the shift in the strength distribution produced by subcritical crack growth in the glass slides. .APTTHUIEK D THE INFLUENCE OF CRACK NUMBER DENSITY ON STRENGTH DEGRADATION OF SHOCKED COMPONENTS The knowledge of the crack number density, N, in shocked components is required for the determination of ATc (see equation (29)). Since N is difficult to evaluate experimentally, N becomes an arbitrary variable in the computer simulation. Figure. D1 illustrates the effect of N on the strength degradation curve. The essential characteristics of strength degradation curve do not vary as N changes. In this study, N - 2 * 1011 1/m3 was adopted. 134 135 150 ’6‘ % v100- :5 CD C: 2 (75 13 _ 12 ,“5’ 50- N—2X10 ..‘3 G) at , 2x1011 7 2x109 0 2x10 . . . , . . , 80 160 240 Quenching Temperature Difference (°C) Figure D1. The effect of crack number density, N, on subcritical crack growth. APPENDIX E ELASTIC MODULUS DETERMINATION OF COATING LAYERS AS APPLIED TO LAYERED CERAMIC COMPOSITES This Appendix develops relationships for determining the in- plane elastic modulus of a coating by two experimental techniques: (1) dynamic resonance and (2) static bend. Dynamic resonance measurements on model two-layer and three-layer composite beams (consisting of bonded strips of alumina and glass) agree well with the relationships developed. In addition, the dynamic resonance and static bend techniques are applied to a SiC coating/graphite substrate composite, where the two methods give statistically similar results for the elastic modulus of the SiC coatings. 1. INTRODUCTION: Coatings as a protective layer may improve the surface properties of the substrate, such as the wear resistance and the high temperature corrosion resistance [E1,E2]. The stress-strain behavior [E2,E3], contact stress field [D4], integrated surface hardness [E5], coating delamination [E6,E7], cracking [E8], spalling, bending, and residual stress state of coated systems [E9- ElZ] are all functions of the elastic modulus of the coating. As a result, the in-plane elastic modulus of coatings is a basic parameter for characterizing coating performance. Watkins et a1. measured the elastic modulus of coatings using the four point bend test [E13]. King indicated that the modulus 136 137 could be estimated from the indentation test [E14], while other researchers determined the modulus by measuring the coating-induced curvature of the substrate [E15-El7]. Dynamic resonance is a simple technique to measure the elastic modulus of homogenous materials [E18,El9]. Dynamic resonance can also be useful in the measurement of the elastic moduli of coating/substrate composites if the formulae associated with the elastic modulus calculation are modified. This paper extends the resonance method to evaluate the in-plane elastic modulus of coatings, using the Bernoulli-Euler beam equation [E20,E21]. In addition, a static surface strain-bending moment method is also presented for determining the elastic moduli of coatings. 2.. THIEOEUTPICJUL BAKHQSRIMHND 2.1 Static bend test The neutral axis of a homogenous rectangular beam coincides with the centroid of cross section when the beam is subjected to symmetrical bending [E22]. If one of the beam's surfaces is coated with a material of different elastic modulus, the beam is no longer homogenous and the neutral axis shifts from the centroid of cross section of the composite beam [822]. (If the coating had the same modulus as the substrate, the neutral axis would still coincide with the centroid of the composite beam.) Thus, the in-plane elastic modulus of the coating can be evaluated from the shift of the neutral axis. Figure El(a) illustrates the cross section of the composite beam. The thickness of the coating and the substrate are 2c and is 138 Y i [1.1 i \ substrate tut—___]. (n geometric center- neutral axis“ z X [I I interface .1 .. “ coating \\\\\\ 40:31; F'5c"i a - E 6 (a) (b) (C) Figure E1. Stress-strain relationship in a composite beam subjected to a pure bending moment. (a) Cross section of composite beam; (b) strain development in axial direction; (c) stress in axial direction. 139 respectively. The distance from the neutral axis to the coating- substrate interface is 2. For convenience, assume that the coating has a larger modulus than the substrate. The neutral axis then deviates from the centroid and approaches the interface. In this study, perfect interfacial bonding is assumed for stress translation between coatings and substrate. When the beam is subjected to a pure bending moment, normal strain in the longitudinal direction develops (Figure El(b)) [E22], where the strain curve passes through the neutral axis. Based on the trigonemetric relations (Figure El(b)), surface strains cc and es in the coating and substrate satisfy the following equation e 28 - 2 - s - (E-la) e 2 + 2 c c or K 2 - 2 - ( ) (2 + 2 ) (E-lb) s ifjjir- s c where 5s K - - ___ 6 c K is the relative strain, which is always positive. Figure El(c) shows the normal stress in the longitudinal direction (the X axis in Figure El(c)). In the bent composite beam, the dependence of the normal stresses ac and as (where ”c" and "s" denote coating and substrate, respectively) upon the transverse coordinate Y is described by [E22] 140 (E-2) where r is the radius of curvature of the neutral axis, and Ec and Es are the elastic moduli of coatings and substrate. From equilibrium of the,axial forces [E22], the following equation has to be satisfied 28-2 -2 as dA + ac dA - 0 (E-3) -£ -1-2c where A is the cross sectional area. Substituting equation (E-2) into (E-3) yields 28:: - Eel: 2 E'l + 2 E l s s c c Combining equations (E-lb) and (E-4), we have KR + 2K - R Ec - Es R (E-S) 2R-K+1 here Is R- 2 c R is the relative thickness. The in-plane elastic modulus of the coating, Ec, can be calculated from equation (E-S). In practice, Es 141 can be determined in a separate experiment on the substrate material itself, before the coating is applied. Since equation (E-S) was derived assuming a pure bending moment, a test method involving pure bend is needed. A pure bending moment can arise from a four point bending fixture. Consequently, Ec can be evaluated from the surface strains, cc and 68, which can be detected using a strain gage attached to a specimen loaded in four-point bend. Figure E2 shows the relationship between relative strain, relative modulus and elastic modulus. For a given coating/substrate material system (EC/Es - constant), the relative strain approaches one with the increase of the relative thickness. If the experimental error due to the strain measurement is constant, the uncertainty of the measured modulus data increases with the increasing R. Consequently, a small R is recommended in measuring the elastic modulus of coatings. 2.2 Dynamic resonance In dynamic beam vibration theory, the Bernoulli-Euler beam equation can approximately describe beam vibrations [E20,E21,E23]. In the present study, the coatings' elastic modulus is calculated using the Bernoulli-Euler beam equation [E20,E21]. 84W(x,t) + ap 62W(x,t) _ E1 0 - (E-6) 6x4 3 at2 Equation (E-6) corresponds to the free, undamped vibration of a monolithic beam with constant cross sectional area, a. E, p, and I represent the elastic modulus, density, and the second moment 142 350-04 / .i E3 . Fi=200 3:100/ V = / g 10-01 R 5° 0 . _ 2 R—10 .2 1;; 1.0-5 2 : In 0.51 0 . .2 .5 . a 0.1 ' l r f ' F ' l fl 1 ' I ' l m 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Relative Strain (K) Figure E2. Relationship between relative strain, K, relative thickness, R, and relative elastic modulus, E /E , for the in-plane modulus measurement of coatings. 143 inertial of the cross section of the beam with respect to the neutral axis. The parameter g is the acceleration due to gravity. W is the transverse deflection of the beam, which is a function of time, t, and position along longitudinal axis, X. For a comoposite . beam, if the interface bonding of coating-substrate is perfect, equation (E-6) becomes 4 a p + 8 P (EcIc + E313) a W(x,t) + ( c c s s ) 82W(x,t) 0 (E-7) 4 3 2 6x at where the subscripts c and 8 refer to the material parameters associated with the coatings and substrate respectively. In practice, the (acpc+asps)/g term can be expressed as AD/g, according to a p + a p a p + a p c c s s _ (a + a ) c c s s c s (a + a ) g g c s - A .3. (122-8) 8 where A and D are the cross sectional area and average bulk density of the composite beam. Dynamic resonance modulus measurements were performed using rectangular composite beams with free ends (Figure E3). Since the bending moment and the shear force are zero at both free ends, the boundary conditions of the transverse vibration are given by 82W(0,t) 82W(L.t) - o - 0 (E-9a) 8x2 6x2 144 pickup driver cotton support‘—‘\\ thread ’ coating \ 9\\—-substrate Figure E3. Illustration of the composite beam whose elastic modulus is determined by the dynamic resonance method. 145 a3w(o,t) 63W(L,t) _ 0 - 0 (E-9b) 6x3 6x3 where L is the length of the composite beam. Solving equation (E-7) by the method of separation of variables and applying the boundary conditions gives the characteristic equation [E20,E21] Wh(x,t) - (Clcos(flnt) + Czsin(flnt)) cos(knx) + cosh(knx) cos(k L) - cosh(k L) - n “ (sin(k x) + sinh(khx)) sin(knL) - sinh(knL) “ (E-lO) where 2 g 1/2 fin - kn [(EcIc + EsIs) ] (E-ll) A D cos(knL) cosh(knL) - 1 (E-12) C1 and 02 are integer constants. Each Wn corresponds to a harmonic transverse vibration with the frequency, f -fin/2s. The n represents the nth mode of vibration. For n-l, the fundamental vibrational mode, the first root of equation (E-12) is le -4.73004 [E21,E22]. Consequently, the fundamental vibration frequency of the composite beam is given by k2 g f - 1 (EcIc + E818) 1/2 2s A D 11.15 EcIc + EsIs 1/2 (E-l3) 2 A D L 146 For a composite beam with the cross section shown in Figure El(a), the second moment of inertia with respect to the neutral axis is 13-1 2 Is - Y dA -2 (E-14) -2 .c.[ .2... -2-2 c Combining equations (E-4), (E-l3), and (E-14) gives 2 2 2 l 3(E 2 - E 2 ) EC - ___ 0.02413 £2140 (2c+2s) - E52: + s 8 ° ° 13 4 (E 2 + E 2 ) c s s c c (E-lS) Thus, for the elastic modulus of a "single-sided" coating (Figure El(a)), the in-plane elastic modulus of coatings can be evaluated using equation (E-15) in which the adopted units are Hertz, meter, and kilogram. The elastic moduli, Ec and E8, are expressed in terms of kgf/m2 (1 kgf/m2 - 9.806 Pa). Although Equation (E-15) is in implicit form, it can Be solved using a numerical iteration method. If a composite beam has coating layers on both the upper and lower substrate surfaces (Figure E4), the neutral axis still coincides with the centroid of cross section. As a result, the second moment of inertia is z /2 I - J‘ 12 dA (E-16a) 3 -2‘/2 147 coating _[ //////i//// 3/2 --ii-— 0 £5 substrate Fv Tf/////7//// £/2‘ / X .31 0 Figure E4. Cross section and dimensions of a composite beam with two coating layers. 148 (2 +2 )/2 2 /2 1c - I c 3 Y2 dA - I s Y2 dA (E-16b) -(2c+2S)/2 -2s/2 Combining Equation (E-13) and (E-l6), we get an expression for the coatings' in-plane modulus for a substrate with a coating on both the upper and lower faces (Figure E4). 1 E - [ 0.09652 f c 3 3 (2c+28) - 2s 2 4 3 L D (2c+2s) - E828 J (E-l7) (It is reemphasized that the unit of elastic modulus is kgf/mz.) Equation (E-17), in explicit form, may be evaluated directly, in constrast to the implicit form (equation (E-15)). When 28 becomes zero, equations (E-lS) and (E-17) become equation (E-18), which corresponds to the modulus Ec of a homogenous beam composed of the coating material 2 4 Ec _ 0.09652 f L 0 (E_18) 12 C Equation (E-18) is similar in form to the ASTM standard test method for the elastic modulus of a monolithic beam [E-l9], with the exception of the correction factor for effects of shear deformation and rotatory inertia [E19-E21,E23]. However, the importance of the correction factor is minor when the ratio of beam length to thickness exceeds thirty. For instance, the error is only 0.8% if the correction factor is neglected for a beam whose length to thickness ratio is thirty [E19]. 3. EXPERIMENTAL PROCEDURE The static and dynamic modulus measurements included in this study were performed on SiC coating/graphite substrate materials in addition to a number of glass/glass and alumina/glass model composite beam specimens that were prepared in order to more rigorously test the theory. 3.1 Model composite beam preparation The following three types of model composite beam specimens were prepared for this study: a two-layer glass/glass composite, a two-layer alumina/glass composite, and a three-layer alumina/glass/alumina composite. The glass layers in the composite beams were 7.6 cm X 0.8 cm X 0.127 cm strips cut by a low speed diamond saw from glass microscope slides. After annealing at 600 0C for 30 minutes in air in an electric furnace, the furnace power was shut off and the cut glass strips were allowed to free-cool to room temperature. The alumina layers in the model composite beams were 7.6 cm cm X 0.8 cm X 0.064 cm strips cut from an as-received alumina substrate (Saxonburg Ceramics Inc., Monroe, NC) with a mass.density of 3.707 gm/cm3 and an average grain size of about 18 pm. After sectioning, the alumina specimens were annealed at 1100 0C for 10 hours in hour in an electric furnace. Two layer glass/glass composite beam specimens were prepared by two adhesion methods: (1) by sintering and (2) by adhesion via gluing. Two annealed glass strips, placed so that their 7.6 cm X 0.8 cm faces were superimposed, were sintered at 740 0C for 30 minutes in air. During heating, the glass strips were set on a flat 149 150 aluminosilicate refractory board to minimize possible deformation of the glass strips. Super glue (Ross Adhesives, Conros Corporation, Detroit, MI) was also used to bond additional glass/glass two layer composites. The thermal expansion mismatch between glass slides and the alumina substrates made sintering difficult so that only glue bonding was feasible for the two and three layer alumina/glass and alumina/glass/alumina composite beams. Glued composite alumina/glass and alumina/glass/alumina composite beams were fabricated from the annealed alumina and glass strips described above. 3.2 Sic coating/graphite substrate composite specimens, monolithic graphite specimen, and free-standing Sic 2Coatings Using a low speed diamond saw, billets of SiC coating/graphite composite material were cut into composite beams 8 cm X 0.8 cm X 0.3 cm with coating on a single 8 cm X 0.8 surface. (The SiC coated graphite billets were prepared by a chemical vapor deposition technique by M. B. Miller, Material Technology Cororation, Dallas, Texas.) The uncoated surfaces of the graphite substrate were polished using 600 grit SiC polishing paper. One 8.57 cm X 0.805 cm X 0.285 cm graphite monolithic substrate specimen (without the 81C coating) was also cut from the as-received billets. Ten of the 816 coating/graphite composite beams were heated in air in an electric furnace at 550 0C until the graphite substrates were totally removed by oxidation in order to produce free-standing SiC layers from the coatings. The free-standing Sic coatings were 151 approximately 5 cm X 0.8 cm X 0.011 cm. X ray diffraction measurements between 10 degrees 20 and 75 degrees 20 indicated that the SiC layers was not modified significantly by the oxidation anneal. 3.3 Elasticity measurements The fundamental frequency of the transverse vibrational mode of the specimens was measured using the dynamic resonance method, which is described elsewhere [E18,El9]. The elastic moduli of the annealed glass, alumina strips (prior to bonding the strips together), and the monolithic graphite specimen were calculated according to equation (E-18). The moduli of the glass/glass and multilayer alumina/glass composite beams were determined according to equations (E-lS) and (E-l7). The in-plane elastic moduli of the 810 coatings were determined from the measured fundamental flexural frequency using equation (E-lS). The elastic moduli of the free- standing SiC coatings were measured by dynamic resonance from equation (E-l8). The in-plane elastic modulus of the SiC coating/graphite beam composites and the modulus of monolithic graphite specimen were also measured using a static bend test in a four point bend fixture with a 2.4 on inner span and a 7.2 cm outer span. Strain gauges (EA-06- 12582-350, Measurements Group Inc., Raleigh, NC) were attached parallel to the beam axes on the upper and lower faces on the composite beam (Figure 5). The strains were recorded and the elastic modulus for the model composite beams and for the 81C coatings were determined from equation (E-S). 152 Both the static (equation (E-5)) and the dynamic (equations (E- 15) and (E-l7)) techniques require accurate knowledge of the specimen dimensions and (when applicable) the coating thickness. Dimensions of the glass/glass and multilayer alumina/glass composites were determined to within i 0.004 cm using a micrometer. The average thickness of the SiC coatings on the sectioned SiC/graphite composites was measured with an optical microscope to an accuracy of i 5 pm. 4. RESULTS AND DISCUSSION Measurements of the in-plane elastic modulus of coatings in this study agree well with the relationships developed in section 2 of this paper for modulus determinations by dynamic resonance (equations (E-lS) and (E-l7)) and by static bend (equation (E-5)). Section 4.1 presents results of the elasticity measurements on the glass/glass, alumina/glass and alumina/glass/alumina model composite beams. Section 4.2 discusses the results of the measurements on the $10 coating/graphite composite beams and subsequent measurements on the free-standing 816 films produced from the composite beams via oxidation of the graphite substrate. 4.1 Model composite beam elasticity results For the two layer glass/glass model composite beams, the elastic modulus of each of the original glass strips (labeled as A and G in Figure E6) was measured prior to bonding. After bonding the composite beam via either gluing or sintering, Ec' the in-plane modulus of the composite beam was measured. Strip G was considered 153 strain gage coating substrate strain gage Figure E5. Schematic of the static four point bend apparatus used for the in-plane modulus measurement of SiC coatings. 154 a) Two-layer model composites A (G) G Alumina/glass and glass/glass composites b) Three-layer model composites >63> Alumina/glass/alumina composites A- alumina substrate. G- glass strip. Figure E6. Illustration of two-layer and three layer-model composites. 155 to be the "substrate" and strip A was treated as the "coating" for both the glued and sintered composite beams. Ec (the elastic modulus of the coating strip “A" in Figure E6) was computed from equation (E-lS) for each of the five glued glass/glass composite beams. The values of Ec obtained from equation (E-lS) agreed to within 1 1.2 percent with the modulus of strip "A" measured prior to bonding (Table El). However, Ec for five sintered glass/glass two- layer composites differed by up to 7.2 percent from the moduli of the corresponding "A" strips prior to bonding (Table E1). The larger deviation for the sintered glass composites may stem from slight deformations or stresses that can occur during sintering or during the cooling subsequent to sintering. (The two layer glass/glass specimens were sintered at 740 0C, which is considerably higher than the 600 degree anneal temperature of given the glass slides used in the other composite beam specimens). Since equations (E-lS) and (E-l7) assume perfect bonding, the effects of the imperfect adhesion also were explored. As expected, the measured in-plane modulus of the two layer glass/glass composite beams decreased monotonically (Figure E7) as the relative fraction of the glue-bonded interface decreased. (Since the glass specimens are transparent, the glue-bonded interface can be observed directly). Poor adhesion impedes the stress transfer between the coating and the substrate, decreasing the integrated elastic stiffness [E22]. As was the case for the glass/glass composites, the elastic modulus of the individual alumina and glass strips was measured prior to glue-bonding the alumina/glass and the alumina/glass/alumina composite beams. For the alumina/glass and 156 Table El. Dynamic resonance measurements of the in-plane elastic moduli of two-layer glass/glass composites (units: GPa) Substrate “Coating” In-plane modulus Deviation* glass G glass A of glass A glass/glass prepared by sintering 67.9 68 72.9 7.2% 66.7 69.4 73.7 6.2% 68.9 67.6 70.8 4.7% 67.8 67.1 67.7 0.9% 66.3 66.5 71.2 7.1% mean: 71.3i2.l prepared by glue adhesion 68.6 67.6 67.3 -0.4% 66.4 67.8 67.3 -0.7% 66.3 68.6 67.8 -l.2% 67.9 66.3 66.5 0.3% 67.8 67.9 68.6 1.0% mean: 67.6iO.9 mean: 67.5i0.7 * Deviation - (original modulus of glass A - in-plane modulus of glass A)/original modulus of glass A * 100%. 157 h C i N O l in-Plane Elastic Modulus (GPa) I v I I l V I V I 0 . 100 80 6o 40 20 0 Glue Adhesion Area Fraction (%) Figure E7. Influence of glue adhesion area fraction on the measured elastic modulus of two-layer glass/glass composite beams. 157 {a 80 * n. l 9, :. 3 60- 3 'U 9 2 40" 73 2 In in 20- C s . ‘l a. 5:. ' ‘ l V I ' I V l 100 ' 80 60 40 20 0 Glue Adhesion Area Fraction (%) Figure E7. Influence of glue adhesion area fraction on the measured elastic modulus of two-layer glass/glass composite beams. 158 alumina/glass/alumina beams, the alumina strips were considered to be the “coatings' while the intermediate glass layer (labeled as G in Figure E6) was considered be to the ”substrate” for purposes of the modulus calculations. For the four alumina/glass composite beams, the in-plane elastic moduli Ec of the alumina ”coatings“ computed from equation 15 were within 3 percent of the pre-bond modulus of the same alumina strips (Table E2). For the each of the three alumina/glass/alumina composite beams, the average modulus of the two alumina coating layers calculated from equation (E-17) agrees to within 1 percent with the average modulus for the corresponding pairs of alumina strips measured prior to bonding them to the glass substrate (Table E3). 4.2 SiC/graphite composites and free-standing sic layers elasticity The dynamic resonance and the static bend test gave 10.36 and 9.97 Gpa, respectively, for the elastic modulus of the monolithic (uncoated) graphite substrate. For the purpose of the subsequdnt calculations of the in-plane elastic moduli of the $10 coatings, we selected the statically-determined value of 9.97 GPa as 23' the elastic modulus of the graphite substrate. (Recall that, as discussed in Section 3.2, the graphite substrate specimen was cut from the same as-received billet of CVD SiC coated graphite as were the SiC/graphite composite specimens.) The in-plane elastic modulus of the 81C coatings on the commercially coated SiC/graphite composite beams was determined via both the static bend test and the dynamic resonance technique (Table C4). For ten SiC/graphite composite beams, dynamic resonance gave 159 Table E2. Dynamic resonance measurements of the in-plane elastic moduli of the alumina in Al 0 /glass composites 2 3 (units: GPa) Unbonded modulus of In-plane modulus Deviation* alumina strip of alumina 307.5 297.9 -3.1% 307.2 306.8 -0.1% 304.7 295.0 -3.1% 306.3 300.3 -1.9% mean: 306.4il.1 mean: 300.0:4.3 * Deviation - (unbounded modulus - in-plane modulus)/ unbounded modulus * 100% Table E3. Dynamic resonance measurements of the in-plane elastic moduli of the alumina in Al O /glass/Al 0 composites 2 3 2 3 (units: GPa) Unbonded modulus of In-plane modulus Deviation alumina strip (two strips of alumina per composite specimen) 303.6 308.9 308.9 1% 303.1 305.7 306.7 1% 309.3 306.3 305.2 -1% mean: 306.2:2.4 mean: 306.9il.5 .160 an average in-plane elastic modulus of 379.5 GPa, while the static bend test on the same ten SiC/graphite specimens gave an average modulus of 359.6 GPa. The mean elastic modulus of the ten monolithic (free-standing) SiC coatings was 367.0 GPa, as measured by the dynamic resonance method. The SiC coatings measured under three different test conditions (dynamic measurements on composite beams, static measurement on composite beams, and dynamic measurements on the free standing coatings) present somewhat different mean values for the elastic modulus of the SiC coating (Table E4). Therefore, further analysis is required to determine whether or not the differences in the means are statistically significant. Using the Krushal-Wallis test [24], the null hypothesis was that the three test conditions gave the same value for the elastic modulus of the $10 coating. At a significance level of a-0.05, there is no difference in the three data populations from which the samples are taken. Thus, there are no statistically significant differences among the 810 coatings' in-plane elastic modulus values, as determined in the three test conditions listed in Table E3. 4.3 Comparisons between the model composite beam results and the Sic coating results The elastic modulus data for the glass/glass composite beams shows a coefficient of variation CV (where CV is the standard deviation/mean) of only 0.01 (Table El). Considering the alumina/glass and the alumina/glass/alumina composite beams as a single group, the CV of the data for the in-plane modulus of the 161 Table E4. A comparison of in-plane elastic modulus data of SiC coatings (units: GPa) static bend test dynamic resonance method composite beam composite beam monolithic SiC coating layer 337.7 407.5 406.3 412.5 378.3 332.7 411.9 294.8 317.2 371.9 387.2 404.1 340.8 377.4 345.2 365.4 433.6 355.2 332.9 411.0 389.0 321.9 369.0 362.0 342.0 340.4 415.0 358.8 395.7 342.9 mean: 359.6i3l.6 mean: 379.5i39.2 mean: 367i34.3 162 alumina coatings was 0.016 (Tables E2 and E3). However, CV of the 81C coatings is about 0.098 (Table E4). The relatively large CV for the 810 coatings may be due to the uncertainty in the data for the $10 coating thickness. For the model composite beams, the thickness of the glass slides and alumina substrates is uniform to within about 0.127i0.0005 cm and 0.064i0.0005 cm, respectively. This thickness can be measured easily prior to bonding together the individual layers of the model composites. In contrast to the model composites, the billets of SiC graphite/composite materials, prepared by chemical vapor deposition, show a slight coating thickness gradient between the corner and the central surface. The SiC coating thickness ranges from 89 to 103 pm (extreme values), which is observed from all of the composite beams cut from the billets. 5. CONCLUSIONS Relationships have been developed for the determination of the in-plane elastic modulus of coatings by a dynamic resonance technique (equations (E-lS) and (E-17)) and by a static bend method (equation (E-5)). Dynamic resonance modulus measurements of model composite systems, including two-layer glass/glass composite beams (bonded by either gluing or by sintering) and of two-layer alumina/glass beams (bonded by glue) agree well with predictions of the coating modulus based on equation 15. The coating's elastic modulus for the three-layer alumina/glass/alumina composite beams are well described by equation (E-l7). The in-plane elastic modulus 163 of the SiC coatings on SiC coating/graphite composite beams were determined by both the dynamic resonance and the static bend techniques. The graphite substrate was then removed from the SiC/graphite beams, leaving a free-standing SiC layer. A Krushal- Wallis test showed no statistically significant differences among the three sets for the elastic modulus calculated for of the SiC coating (that is, for measurements done statically and dynamically on intact composite beams, as well as dynamic resonance on free- standing SiC layers). In this study, the ratio R of substrate thickness to coating thickness (28/2c) varied from 1.0 for the two layer glass/glass model composites to 2.0 for the alumina/glass two-layer composites, to about 27 in the case of the two-layer SiC/graphite composites. The ratio Es/Ec of the elastic moduli of the substrate and coating, respectively, varied from 1.0 for the glass/glass model composites, to about 0.2 for the alumina/glass model composites, to about 0.027 for the CVD SiC/graphite composites. Thus, for a considerable range of relative coating thickness and relative elastic moduli for layered ceramic composites, the equations and techniques presented in this paper provide a reasonably accurate means of determining the elastic modulus of the coating layer. For completeness, future studies should include measurements on which the coating modulus is less than the substrate modulus, although in many applications the materials of interest are as those treated here (that is, where the coating modulus is equal to or greater than the substrsate modulus). References for Appendix E E1) R. F. Bunshah, pp. 158, gggpingg, Noyes Publication, Park Ridge, New Jersey, (1982). E2) H. W. Grunling, K. Schneider, and L. Singheiser, ”Mechanical Properties of Coated Systems", Mater. Sci. and Eng., 88: 177- 189 (1987). E3) T. Shikama, H. Shinmo, M. Fukutomi, M. Fujitsuka, and M. Okada, “Mechanical Properties of Molybdenum Coated with Titanium Carbide Film“, J. mater. Sci., 18[10]: 3092-3098 (1983). E4) R. B. King and T. C. O'Sullivan, "Sliding Contact Stresses in a ' Two-Dimensional Layered Elastic Half-Space", Int. J. Solids Structures, 23[5]: 581-597 (1987). E5) A. K. Bhattacharya and W. D. Nix, ”Analysis of Elastic and Plastic Deformation Associated with Indentation Tastings of Thin Films on Substrates“, Int. J. Solids Structures, 24[12]: 1287-1298 (1988). E6) D. B. Marshall and A. G. Evans, ”Measurement of Adherence of Residually Stressed Thin Films by Indentation. I: Mechanics of Interface Delamination", J. Appl. Phys., 56[10]: 2632-2638 (1984). E7) A. G. Evans and J. W. Hutchinson, "On the Mechanics of Delamination and Spelling in Compressed Films", Int. J. Solids Structures, 20(5]: 455-466 (1984). ' E8) A. K. Sinha, H. J. Levinstein, and T. E. Smith, “Thermal Stresses and Cracking Resistance of Dielectric Films (SiN, Si N4 , and SiO ) on Si substrates", J. Appl. Phys., 49[4]: 24 3- 2426 (1978). E9) A. Brenner and S. Senderoff, "Calculation of Stress in Electrodeposits from the Curvature of a Plated Strip", Journal of Research of National Bureau of Standards, 42: 105-123 (1949). E10) P. H. Townsend and D. M. Barnett, “Elastic Relationships in Layered Composite Media with Approximation for the Case of Thin Films on a Thick Substrate", J. Appl. Phys. 62[ll]: 4438-4444 (1987). E11) A. V. Virkar, J. L. Huang, and R. A. Cutler, "Strengthening of Oxide Ceramics by Transformation-Induced Stresses", J. Am. Ceram. Soc., 70[3]: 164-70 (1987). E12) R. L. Mullen, R. C. Hendricks, and G. McDonald, "Interface Roughness effect on Stresses in Ceramic Coatings" , Ceram. Eng. Sci. Proc. ,8[7-8]: 559-571 (1987). 164 E13) E14) E15) E16) E17) E18) E19) E20) E21) E22) E23) E24) 165 T. R. Watkins, D. J. Green, and E. Rybe, ”Measurement of the In-Plane Young's Modulus and Residual Stresses in CVD SiC Coatings", presented at 9lst Annual Meeting of the American Ceramic Society, (1989). R. B. King, "Elastic Analysis of Some Punch Problems for a Layered Medium", Int. J. Solids Structures, 23[12]: 1657-1664 (1987). A. G. Vannie, “A Method for the Determination of the Stress in, and Young's Modulus of Silicon Nitride Passivation Layers”, Solid State Technology, 23[1]: 81-84 (1980). D. S. Williams, ”Elastic Stiffness and Thermal Expansion Coefficient of Boron Nitride Films”, J. Appl. Phys., 57[6]: 2340-2342 (1985). E. M. Corcoran, ”Determining Stresses in Organic Coatings Using Plate Beam Deflection", Journal of Paint Technology, 4l[538]: 635-640 (1969). E. Schreiber, O. L. Anderson, and N. Soga, pp. 82, filpgpig WNW. McGraw-Hill. New York. (1974). “Standard Testing Method for Young's Modulus, Shear Modulus, and Poission's Ratio for Glass and Glass-Ceramics by Resonance", ASTM, Designation: C623-71 (Reapproved 1981). E. Volterra and E. C. Zachmanoglou, pp. 321, Dynamip§_pf Vipxgpipng, Charles E. Merrill Books, Inc., Columbus, Ohio, (1965). S. K. Clark, pp.75-87, , Prentice-Hall, Inc., Englewood Cliffs, New Jersey, (1972). S. P. Timoshenko and D. H. Young, pp. 113, uppgpigls, Fourth Edition, Van Nostrand Reinhold 60., Princeton New York, (1962). C. W. D. Silva, “Dynamic Beam Model with Internal Damping Rotatory Inertia and Shear Deformation", AIAA Journal, l4[5]: 676-680 (1976). ' H. R. Neave and P. L. Worthington, pp. 243, Diggpihppipn;£;gg Iggpg, Unwin Hyman Ltd., London, (1988). COMPUTER PROGRAM 00000 00000 120 10 30 78 Computer program No. 1 This is a part of pgograms to calculate the "thermoelastic stress in glass plates". This program can evaluate the relationship between Biot's modulus and critical quench temperature difference if the fracture strength of specimen is pre—determined. B - Biot's modulus ; 28 - thickness of plate Ex - thermal expansion ; EL - elastic modulus CFS - critical fracture strength : DI - diffusivity TEMP - quench temperature difference ; TEMP - Tl - T0 T1 - initial temperature ; T0 - quenching medium temperature DO 78 L - l, 2 STREN - 60 + 80 *(L-l) DO 30 M = l , 20 TEMP - 80 + 20 * M Tl - 100 + 20 * M TO - 20 DO 10, N - l , 10 B - 0.1 * N**3 Bl - B KK - 1 IF (N .80. 1) THEN 83 - 0.05 ELSE B3 - 0.1 * (N-l)**3 END IF CALL CHIUCH(B,XG, TEMP, T1, T0) IF (XG .LT. STREN) THEN GO TO 10 ELSE BZ - (Bl + 83)/2 B - BZ END IF CALL CHIUCH(B,XG, TEMP, T1, T0) KK - KK + l IF((ABS(XG - STREN)/STREN .LT. 0.001) .OR. (KK .EQ. 20)) THEN WRITE (2,*) TEMP, B , KK, XG PRINT *, TEMP, B , KK , XG GO TO 30 ELSE IF(XG . GT. SIREN) THEN Bl - B2 B2 - (Bl + 83)/2 B - BZ ELSE B3 - 82 82 -(Bl + 83)/2 B - 82 END IF END IF GO TO 120 CONTINUE PRINT *, TEMP CONTINUE WRITE(2 ,*) STREN CONTINUE STOP END 166 Susi _a.- - ....— 1 I miss-11'5- u'.'\.r . - .' 167 subroutine chiuch(B,XG, TEMP, T1 , TO) REAL X(0:12) , T(205) INPUT OF SOME PARAMETERS H - 1.0E-3 DI - 4.8E-7 EX - 8.0E—6 EL -7.0E+10 eignvalue calculation using half—interval tech. ' X(M) * TAN(X(M)) = B DO 10 , E - O , 10 XMl - 0 + 3.14159 * E XM2 - 1.570795 + E * 3.14159 DO 20 , W - 1 , 50 m3 - (1041 + m2)/2 IF (ABS(XM1 - XM2) .LT. 0.0000003) THEN GO TO 15 . ELSE XMMl - XMl * TAN(XM1) - B XMM2 - XM2 * TAN(XM2) - B XMM3 - XM3 * TAN(XM3) - B END IF IF ((XMMl * XMM3) .GT. 0.0) THEN XMl - XM3 XM2 - XM2 ELSE XMl - XMl XM2 - XM3 END IF 20 CONTINUE lS X(E) - XM3 10 CONTINUE TO CALCULATE THERMOELASTIC STRESSES XG ' 0.0 DO 22 M - 1 , 200 XRR - 0.0 T(M) ' 0.000001 * l.l**M DO 60 , E - 0 , 10 XPP - EXP(-DI * (X(E)/H)**2 * T(M)) XA - SIN(X(E))/(X(E) + COS(X(E))* SIN(X(E))) XB - SIN(X(E))/X(E) - COS(X(E)*H/H) XRR - XPP * XA * XB + XRR 60 CONTINUE STRESS - 2 * EX*EL* TEMP/(l-0.25) * XRR/1000000 IF (STRESS .GT. XG) THEN XG - STRESS ELSE GO TO 80 END IF 22 CONTINUE 80 RETURN END 00 00000(‘)0 00 00000 000 Computer program No. 2 This is a part of pgograms to calculate the "transient thermoviscoelastic stress" B - Biot's modulus ; 2H - thickness of plate ; DI - diffusivity EL - elastic modulus CPS - critical fracture strength EX - thermal expansion TEMP - quench temperature difference Tl - initial temperature T(M) is real time I I o I CCT(L,M) is reduced time : L - REAL A(0:22,0:1302), STRESS(0:22,0:1302), STRAIN(0:1302) REAL T(Ozl302) , CCT(0:22,0:1302) , REAL X(0:12) ; TEMP - T1 - T0 T0 - quenching medium temperature position variable , TTT(0:22,0:1302) M = time variable A IS A SPECIAL FUNCTION WHICH IS USED IN THERMOELASTIC STRESS CAL. INPUT OF SOME PARAMETERS B - 10 H - 1.0E-3 DI - 4.8E-7 EX - 8.0E-6 EL =7.0E+10 TEMP - 670 T1 - 690 T0 - 20 To calculate the temperature distribution eignvalue calculation using half-interval tech. : X(M) DO 10 , E = 0 , 10 XMl - O + 3.14159 * E XM2 e 1.570795 + E * 3.14159 DO 20 , w - 1 , 50 XM3 — (XMl + XM2)/2 IF (ABS(XM1 - XM2) .LT. GO TO 15 ELSE XMMl - XMl * TAN(XM1) ‘ XMMZ a XM2 * TAN(XM2) XMM3 - XM3 * TAN(XM3) - END IF IF ((XMMl * XMM3) .GT. 0.0) XMl - XM3 XM2 - XM2 ELSE XMl - XMl XM2 - XM3 END IF 20 CONTINUE 15 X(E) ' XM3 10 CONTINUE B B 8 THEN 0.0000003) THEN To calculate the reduced time and temperature gradient Loop F is related to the position along plate thickness. The half thickness of glass plate includes 21 points. Real time is divided into 1300 points T(M) loop is real time. in 17 seconds. DO 40 , F - 0 , 20 CCT(F,0) - 0.0 TTT(F,0) - T1 STRESS(F,0) - 0.0 40 CONTINUE T(O) - 0.0 DO 30 , F - 0 , 20 z - H * F /20 Since we divide the half thickness to be 20 division, above has " / 20". * TAN(X(M)) the equation 2 is the corrodinate in the direction of plate thickness XSS - 0.0 168 0()0 (3 60 50 30 80 70 84 86 82 90 169 D0 50 , M - l , 1300 T(M) - 0.00001 * M**2 XRR - 0 DO 60 , E - 0 , 10 XPP - EXP(-DI ' (X(E)/H)**2 * T(M)) XA - SIN(X(E))* COS(X(E)* Z/ H) X8 = X(E)+ SIN(X(E))* COS(X(E)) XQQ - XA/XB XRR - XPP * XQQ + XRR CONTINUE TTT(F,M) - TO + TEMP * 2 * XRR XT - EXP(0.0889028* TTT(F,M-l))+ EXP(0.0889028* TTT(F,M)) CCT(F,M) - 1.68974E-21 * XT * (T(M) - T(M-l))/2 + CCT(F,M-1) CONTINUE CONTINUE To calculate the thermoviscoelastic stresses (1) To Calculate the A(J,L) ; a simplied function in calculating precedure. DO 70 , I - 1 , 1300 DO 80 , E - O , 20 A(F,I) - EXP((—CCT(F,I) + CCT(F, I-l))/l400) CONTINUE CONTINUE I ‘ GOO 000 J = l , 20 X0 - A(J,I) + A(J-l,I) + XQ - (TTT(J,I) - TTT(J,I-l)) * A(J,I) XB - (TTT(J-l,I) - TTT(J—l,I-1)) * A(J-1,I) - XA + XB + XR - (A(J,I))**2 * STRESS(J,I-l) XS - XC + (A(J-1,I))**2 * STRESS(J—l,I-l) + XS CONTINUE XA - STRAIN(I-l)* EL/(1—0.25)* XQ+ EX* EL/(l-0.25)* XR- XS XB - EL/(1-0.25)*XQ STRAIN(I) - XA/XB DO 86 , F - 0 , 20 XA - STRAIN(I)- STRAIN(I-l)- EX *(TTT(F,I)-TTT(F,I-l)) HH - EL/(1-0.25)* XA * A(F,I) STRESS(F,I) a HH + (A(F,I))**2 * STRESS(F,I-l) CONTINUE CONTINUE PRINT * , ' POSITION MPa ’ DO 90 , WK - 0 , 40 W - WK - 20 XL - w/20 * H * 1000 THE UNIT IS mm and MPa XG - STRESS(ABS(W),1300)/1000000 WRITE(1, *) XL, XG PRINT * , XL, XG CONTINUE STOP END 00000 0000000 Computer program No. 3 This is a part of programs to calculate the "thermoviscoelastic stress in glass plates". This objective of this program is to evaluate the relationshiop between Biot's modulus and critical quench temperature difference when the fracture strength of slides are pre-determined. ----thermoviscoelastic stress ------ B - Biot’s modulus : 2H = thickness of plate EX - thermal expansion ; EL - elastic modulus CFS - critical fracture strength : DI - diffusivity TEMP - quench temperature difference ; TEMP - Tl - T0 T1 - initial temperature : TO - quenching medium temperature T(M) is real time , M - time variable CCT(L,M) is reduced time ; L - position variable DO 78 L - 1, 2 STREN = 60 + 80 *(L—l) DO 30 M - 1 , 7 TEMP - 580 + 20 * M T1 - 600 + 20 * M T0 - 20 DO 10, N - 1 , 10 B = 0.2 * N**3 Bl - B KK - 1 IF (N .EQ. 1) THEN B3 - 0.05 ELSE B3 - 0.2 * (N—l)**3 END IF CALL CHIUCH(B,XG, TEMP, T1, T0) IF (XG .LT. STREN) THEN GO TO 10 ELSE B2 - (81 + 83)/2 B - 82 END IF 120 CALL CHIUCH(B,XG, TEMP, T1, T0) KK - KK + l IF((ABS(XG - STREN)/STREN .LT. 0.001) .OR. (KK .EQ. 14)) THEN WRITE (3,*) TEMP, B , KK, XG PRINT *, TEMP, B , KK , XG GO TO 30 ELSE IF(XG . GT. STREN) THEN Bl - 82 82 - (Bl + B3)/2 B - 82 ELSE BB - 82 82 -(Bl + 83)/2 B - BZ END IF END IF GO TO 120 10 CONTINUE PRINT *, TEMP 30 CONTINUE WRITE(3 ,*) STREN 78 CONTINUE STOP END subroutine chiuch(B,XG, TEMP, Tl , T0) REAL A(0:22,0:l302), STRESS(0:22,0:1302), STRAIN(0:1302) 170 171 REAL T(0:1302) , CCT(0:22,0:1302) , TTT(0:22,0:1302) REAL X(0:12) C A IS A SPECIAL FUNCTION WHICH IS USED IN THERMOELASTIC STRESS CAL. C INPUT OF SOME PARAMETERS H - l.0E-3 DI - 4.8E-7 EX - 8.0E-6 EL -7.0E+10 C To calculate the temperature distribution C eignvalue calculation using half-interval tech. C ' X(M) * TAN(X(M)) - B DO 10 , E - 0 , 10 XMl - 0 + 3.14159 * E XM2 - 1.570795 + E * 3.14159 DO 20 , W - l , 50 XM3 - (XMl + XM2)/2 IF (ABS(XM1 - XM2) .LT. 0.0000003) THEN GO TO 15 ELSE XMMl - XMl * TAN(XMl) - B XMMZ - XM2 * TAN(XMZ) - B XMM3 - XM3 * TAN(XM3) - B END IF IF ((XMMl * XMM3) .GT. 0.0) THEN XMl - XM3 XM2 - XM2 ELSE XMl - XMl XM2 - XM3 END IF 20 CONTINUE 15 X(E) - XM3 10 CONTINUE To calculate the reduced time and temperature gradient Loop F is related to the position along plate thickness. The half thickness of glass plate includes 21 points. T(M) loop is real time. Real time is divided into 1300 points in 17 seconds. DO 40 , F - 0 , 20 CCT(F,0) - 0.0 TTT(F,0) - Tl STRESS(F,O) - 0.0 40 CONTINUE T(O) - 0.0 DO 30 , F - 0 , 20 z - H * F /20 Since we divide the half thickness to be 20 division, the equation above has " / 20". z is the corrodinate in the direction of plate thickness XSS - 0.0 DO 50 , M - 1 , 1300A T(M) - 0.00001 * M**2 XRR - 0 DO 60 , E - 0 , 10 XPP - EXP(-DI * (X(E)/H)**2 * T(M)) XA SIN(X(E))* COS(X(E)* Z/ H) XB - X(E)+ SIN(X(E))* COS(X(E)) XQQ - XA/XB ' XRR - XPP * XQQ + XRR 60 CONTINUE TTT(F,M) - TO + TEMP * 2 * XRR XT - EXP(0.0889028* TTT(F,M-l))+ EXP(0.0889028* TTT(F,M)) CCT(F,M) - 1.68974E—21 * XT * (T(M) - T(M-l))/2 + CCT(F,M-l) 50 CONTINUE 30 CONTINUE C To calculate the thermoviscoelastic stresses 00000 000 000000000 00000 80 70 84 86 82 172 (1) To Calculate the A(J,L); a simplied function in calculating precedure. XG - 0.0 DO 70 , I - 1 , 1300 DO 80 , F - 0 , 20 A(F,I) a EXP((-CCT(F,I) + CCT(F, I-l))/1400) CONTINUE CONTINUE (2) STRAIN(O) DO 82 , I XQ - 0.0 XR - 0.0 XS - 0.0 DO 84 , J - l , 20 X0 - A(J.I) + A(J-l,I) + XQ XA - (TTT(J,I) - TTT(J,I-l)) * A(J,I) XB (TTT(J-l,I) - TTT(J—l,I-1)) * A(J-l,I) XR XA + XB + XR XC (A(J,I))**2 * STRESS(J,I-1) XS XC + (A(J-1,I))**2 *‘STRESS(J-1,I—l) + XS CONTINUE XA.- STRAIN(I-l)* EL/(1-0.25)* XQ+ EX* EL/(l-0.25)* XR— XS XB - EL/(l-0.25)*XQ STRAIN(I) - XA/XB DO 86 , F - O , 20 XA - STRAIN(I)- STRAIN(I-l)- EX *(TTT(F,I)-TTT(F,I-1)) HH - EL/(1-0.25)* XA * A(F,I) STRESS(F,I) - HH + (A(F,I))**2 * STRESS(F,I—l) CONTINUE XGGG - STRESS(20,I)/1000000 IF (XGGG .GT. XG) THEN XG - XGGG GO TO 82 ELSE IF(((XG-XGGG)/XG) .LT. 0.01) GO TO 82 GO TO 156 END IF CONTINUE . PRINT * , ' POSITION MPa ' DO 90 , WK - 0 , 40 W - WK - 20 XL - W/20 * H * 1000 THE UNIT IS mm and MPa XG - STRESS(ABS(W),l300)/1000000 WRITE(1, *) XL, XG PRINT * , XL, XG CONTINUE - 0.0 - l , 1300 DO 231 I - l , 1300 XGGG - STRESS(ZO, I)/1000000 IF (XGGG .GT. XG) THEN XG - XGGG END IF 231 CONTINUE 156 RETURN END 00000 00 10 Computer program No. 4 This is a part of programs is perform the "computer simulation of strength degradation of shocked glass plates" Purpose : to create a distribution of initial fracture strength (a distribution of initial crack flaw size) using the random number generater of IMSL program. (normal distribution) Statistic parameter: mean = 101.4. standard deviation - 10.24 Kc : critical intensity factor. Y : geometric parameter CHARACTER*15 CRACKLEN REAL X(1000) , 2(1000) , KC, XX Y - 1.1215 KC - 0.78E+6 random generation of fracture strength, X( ), using IMSL program DO 5 N - 1 , 1000 CALL RNNOR(1, XX) CALL SSCAL(l, 10.24, XX, 1) CALL SADD(1, 101.4, XX, 1) X(N) - XX CONTINUE OPEN (UNIT - 1, FILE = ’CRACKLEN’ , STATUS a 'NEW' ) transformation from X( ) to initial crack length, 2( ) DO 10 N - l , 1000 Z(N) - (Kc/Y/(X(N)* 1E+6))**2 / 3.14159 WRITE (1,*) Z(N) CONTINUE CLOSE (UNIT - 1) STOP END 173 0000 0000000 Computer program No. 5 This is a part of programs to perform the 'computer simulation of strength degradation of shocked glass plates’ Purpose : to simulate the retained strength degradation ----- (1) without subcritical crack growth Stress intensity factor Kc - 0.78E+6. geometric parameter Y - 1.1214 thermal expansion - EX. elastic modulus = E. Poission ratio - V. Fracture surface energy - G. crack density - XN. fatigue limit - KO quench temperature difference - DELTAT (arbitrary) ----- DATA FILE OF INITIAL CRACK LENGTH: CRACKLEN CHARACTER*15 CRACKLEN DOUBLE PRECISION X(lOOO), XF1(1000), XF2(1000) DOUBLE PRECISION 21(1000), 22(1000), 23(1000) DOUBLE PRECISION A, B, C, D, ZTl ,ZTZ ,ZT3 , W1, W2, W3 ,TC DOUBLE PRECISION TT, LENGTH INTEGER J, II(0:53) , I1(0:53) , I2(O:53) KC - 0.75E+6 Y - 1.1214 EX - 6.5E—6 E - 70E+9 V - 0.25 G - 4 XN - 2E +11 OPEN (UNIT - 1, FILE - 'CRACKLEN' , STATUS - ’OLD') DO 1 N - l , 1000 READ(1, *) X(N) 1 CONTINUE CLOSE (UNIT - l) The purpose of loop 888 is to repeatedly calculate the strength degradation related to different deltat. DO 777 , III - 1, 4 XN - 2E+5 * 100**III IF (III .EQ. 4) THEN XN - ZE+12 END IF DO 888 IJK - 1 , 10 DELTAT - 80 + (IJK - l) *20 PRINT * , ' DELTAT - ’, DELTAT The purpose of loop 3 and 4 is to clearn all variables in order for the repeated calculation of loop 888. DO 3 , N - 1 , 1000 Zl(N) - X(N) XF1(N) - 0.0 XF2(N) '0.0 z_2(m -o.o Z3(N) '0.0 3 CONTINUE DO 4 I W - 0 y 53 II(WN) ‘ 0.0 11(WN) ‘ 0.0 12(WN) - 0.0 4 CONTINUE 174 1375 C 1.1 : transformation from Zl( ) to final crack length, 22( ) C Zl( ) is the initical crack length B = 2* 3.14159* XN* G C - 164*(1-V**2) * XN /9/(1-2*V) D - 3.14159*G*(l-2*V)**2/2/E/EX**2/(l-v**2) definition of A, B, C and D, are given in Hasselman’s paper(l969) (3 C To find the final crack length due to kinetic behavior using C Hasselman’s Eq. (4) and (7). C Numercial technique: interval-halving mothod C Tc is the critical temperature for kinetic crack growth. K - 0 KK - 0 C K represents the total number of specimens without crack growth C KK is the total number with crack growth DO 30 M a 1 , 1000 To - D**0.5 *(1+ C*Zl(M)**3) / Zl(M)**O.5 A - 3*(EX *. Tc)**2 * E/2/(1- 2*V) IF (Tc .GT. DELTAT) THEN K - K + 1 501 Z2(K) - Zl(M) GO TO 30 END IF C The physical meaning of statement 501 is ’no crack growth’. C It means that ZZ( ) represents the part of intial flaws C without crack growth. CB After statment 502, the other part of initial crack 'grows'. 502 ZT1 - Zl(M) * 1.001 DO 20 N - 1 , 35 . CALL CHIUCH(A, B, C, Zl(M), ZT1, W1) IF (N .EQ. 1) THEN W3 - W1 ZT3 - ZT1 GOTO 20 ELSE IF (W1*W3 .GT. 0.0) THEN W3 - W1 ZT3 - ZT1 ZT1 - Zl(M) * 3**N GO TO 20 ELSE GO TO 25 END IF END IF 20 CONTINUE C The purpose of loop 20 is to find an interval in which C the solution exists. The solution will be calcualted in C loop 28 using interval-halving method. 25 DO 28 NM - 1 , 30 ZT2 - (ZT1 +ZT3)/2 CALL’CHIUCH(A, B, c, Zl(M), 2T2, w2) IF (w3*w2 .GT. 0.0) THEN w3 - W2 ZT3 - 2T2 ZT2 - (ZT1+ZT3)/2 ELSE w1 - W2 211 - ZT2 ZT2 - (ZT1 + zr3)/2 176 END IF IF (ABS(ZT1 - ZT3) .LT. 0.0000000001) GO TO 19 28 CONTINUE 19 XX - KK + l Z3(KK) - ZT2 30 CONTINUE C Z3( ) represents the final crack length due to kinetic C crack growth. C 1.2 : To find the final quasi—static crack length according to C to Eq. (4) DO 200 N - 1, KK TT - D**0.5*(1+C*Z3(KK)**3)/23(KK)**0.5 IF (TT .LT. DELTAT) THEN GO TO 477 ELSE GO TO 200 END IF C To find final crack length using interval-halving method 477 ZT1 - Z3(KK) * 1.001 DO 40 NN - l , 35 CALL CHIUCH(A, B, C, Z3(KK), ZT1, W1) IF (N .EQ. 1) THEN W3 - W1 ZT3 - ZT1 GO TO 40 ELSE . IF (W1*W3 .GT. 0.0) THEN W3 - W1 ZT3 - ZT1 ZT1 - Z3(KK) * 3**NN GO TO 40 ELSE GO TO 45 END IF END IF 40 CONTINUE 45 DO 48 M - 1, 30 ZT2 - (ZT1 + ZT3)/2 CALL CHIUCH(A, B, C, Z3(KK), ZT2, W2) IF (W3*W2 .GT. 0.0) THEN W3 ' W2 ZT3 - ZT2 ZT2 - (ZT1 + ZT3)/2 ELSE W1 - W2 ZT1 - ZT2 ZT2 - (ZT1 + ZT3)/2 END IF IF (ABS(ZT1 - ZT3) .LT. 0.0000000001) GO TO 44 48 CONTINUE 44 Z3(KK) - ZT2 200 CONTINUE 4 : transformation from 22( ) and Z3( ) to final strength, Xfl and XF2( ) , respectively. Sorting of retained fracture strength DO 46 N - 1 , 1000 DD - Kc/Y/(3.14159*X(N))**0.5 / 1E+6 000 46 50 155 60 157 49 71 72 73 888 777 177 J - DD/4 II(J) - II(J) + 1 CONTINUE XMEAN - 0.0 IF (K .EQ. 0) GO TO 155 DC 50 N - 1 , K XF1(N) - KC/Y/(3.14159*22(N))**0.5/1E+6 XMEAN - XMEAN + XF1(N) J - XFl(N)/4 Il(J) - II(J) +1 CONTINUE IF (KK .EQ. 0) GO TO 157 DO 60 N - l , KK XF2(N) - KC/Y/(3.14159*Z3(N))**0.5/1E+6 XMEAN - XMEAN + XF2(N) J - XF2(N)/4 12(J) - I2(J) +1 CONTINUE DO 49 WN - 0, 35 WRITE (3,*) WN*4+2 , II(WN) , II(WN) , I2(WN) CONTINUE XMEAN - XMEAN/(K+KK) VAR - 0.0 IF (K .EQ. 0) GOTO 72 DO 71 N - 1, K VAR - (XMEAN - XF1(N))**2 + VAR CONTINUE DO 73 N - 1, KK VAR - (XMEAN -XF2(N))**2 + VAR CONTINUE STAND - (VAR/(K+KK))**0.5 WRITE (3, *) WRITE (3, *) DELTAT, XN, XMEAN, STAND WRITE (3 , *) WRITE (3, *) CONTINUE CONTINUE STOP END SUBROUTINE CHIUCH(A, B, C, Zl , ZT , ZTRIAL) DOUBLE PRECISION A , B, C, 21, ZT, ZTRIAL ZTRIAL - A*(l/(1+C*Zl**3)-l/(1+C*ZT**3))-B*(ZT**2-Zl**2) RETURN END 0000000 1.1 Computer program No. 6 This is a part of programs to perform "computer simulation of strength degradation of shocked glass plates" Purpose : to simulate the retained strength degradation -- (2) with subcritical crack growth Stress intensity factor Kc - 0.75E+6. geometric parameter Y - 1.1215 thermal expansion - EX. 'elastic modulus - E. Poission ratio - V. Fracture surface energy - G. crack density - XN. fatigue limit - KO quench temperature difference - DELTAT (arbitrary) DATA FILE OF INITIAL CRACK LENGTH: CRACKLEN CHARACTER*15 CRACKLEN DOUBLE PRECISION X(1000), XF1(1000), XF2(1000) DOUBLE PRECISION Zl(1000), 22(1000), Z3(1000) DOUBLE PRECISION A, B, C, D, ZT1 ,ZT2 ,ZT3 , W1, W2, W3 ,TC DOUBLE PRECISION TT, LENGTH, FINAL, DELTAT INTEGER J, II(0:53) , Il(0:53) , I2(0z53) Kc = 0.75E+6 Y - 1.1215 EX - 6.5E-6 E - 70E+9 V - 0.25 G - 4. XN - 2E +11 OPEN (UNIT - 1, FILE - 'CRACKLEN’ , STATUS - ’OLD’) DO 1 N - 1 , 1000 READ(1, *) X(N) CONTINUE CLOSE (UNIT - l) The purpose of loop 888 is to repeatedly calculate the strength degradation related to different deltat. DO 888 IJK - 1 , 8 DELTAT = 100 + (IJK - l) *20 PRINT * , ’ DELTAT ' ’, DELTAT The purpose of loop 3 and 4 is to clearn all variables in order for the repeated calculation of loop 888. D0 3 , N - l , 1000 Zl(N) - X(N) XF1(N) ‘ 0.0 XF2(N) '0.0 22(N) -0.0 23(N) -0.0 CONTINUE DO 4 , WN - 0 , 53 II(WN) ' 0.0 II(WN) - 0.0 12(WN) - 0.0 CONTINUE transformation from Zl( ) to final crack length, 22( ) Zl( ) is the initical crack length B - 2* 3.14159* XN* G C - 16 *(1-v**2) * XN /9/(l-2*V) 178 0 0000 00 501 0000 502 20 25 28 179 D - 3.14159*G*(l—2*V)**2/2/E/EX**2/(1-V**2) definition of A, B, C and D, are given in Hasselman’s paper (1969) To find the final crack length due to kinetic behavior by means of Hasselman's Eq. (4) and (7). Numercial technique: interval-halving mothod Tc is the critical temperature for kinetic crack growth. K - 0 KK - O K represents the total number of specimens without crack growth KK is the total number with crack growth DO 30 M - 1 , 1000 Tc - D**0.5 *(1+ C*Zl(M)**3) / Zl(M)**0-5 A - 3*(EX * Tc)**2 * E/2/(l- 2*V) IF (Tc .GT. DELTAT) THEN K - K + 1 22(K) - Zl(M) GO TO 30 END IF The physical meaning of statement 501 is ’no crack growth’. It means that ZZ( ) represents the part of intial flaws without crack growth. , After statment 502, the other part of initial crack ’grows’. ZT1 - Zl(M) * 1.001 CALL CHIUCH(A, B, C, Zl(M), ZT1, W1) IF (N .EQ. 1) THEN W3 - W1 ZT3 - ZT1 GOTO 20 ELSE IF (W1*W3 .GT. 0.0) THEN W3 - W1 ZT3 - ZT1 ZT1 - Zl(M) * 3**N GO TO 20 ELSE GO TO 25 END IF END IF CONTINUE The purpose of loop 20 is to find an interval in which the solution exists. The solution will be calcualted in loop 28 using interval-halving method. DO 28 NM - 1 , 3o ZT2 - (ZT1 +ZT3)/2 CALL CHIUCH(A, B, c, Zl(M), ZT2, w2) IF (W3*W2 .GT. 0.0) THEN w3 - wz ‘ZT3 - ZT2 'ZT2 - (ZT1+ZT3)/2 ELSE W1 - W2 ZT1 - ZT2 ZT2 - (ZT1 + zr31/2 END IF IF (ABS(ZT1 - ZT3) .LT. 0.0000000001) GO TO 19 CONTINUE KK - xx + 1 180 Z3(KK) - ZT2 30 CONTINUE C 23( ) represents the final crack length due to kinetic crack C growth. C 1.2 : To find the final quasi-static crack length according to Eq. (4) DO 200 N - 1, KK TT - D**0.5*(1+C*Z3(KK)**3)/Z3(KK)**0.5 IF (TT .LT. DELTAT) THEN GO TO 477 ELSE GO TO 200 END IF C To find final crack length using interval-halving method 477 ZT1 - Z3(KK) * 1.001 DO 40 NN = 1 , 35 CALL CHIUCH(A, B, C, 23(KK), ZT1, W1) IF (N .EQ. 1) THEN ' W3 - W1 ZT3 - ZT1 GO TO 40 ELSE IF (W1*W3 .GT. 0.0) THEN W3 - W1 ZT3 - ZT1 ZT1 - 23(KK) * 3**NN GO TO 40 ELSE GO TO 45 , END IF END IF 40 CONTINUE 45 DO 48 M - l, 30 ZT2 - (ZT1 + ZT3)/2 CALL CHIUCH(A, B, C, 23(KK), ZT2, W2) IF (W3*W2 .GT. 0.0) THEN W3 - W2 ZT3 - ZT2 ZT2 - (ZT1 + ZT3)/2 ELSE W1 - W2 ZT1 - ZT2 ZT2 - (ZT1 + ZT3)/2 END IF IF (ABS(ZT1 - ZT3) .LT. 0.0000000001) GO TO 44 48 CONTINUE 44 Z3(KK) ‘ ZT2 200 CONTINUE THE SUBROUTINE HRONG is to study the effect of subcritical crack growth on the retained strength degradation. It means that 22( ) will enter the subsroutine. The chage in ZZ( ), resulting from the calculation of subroutine, is due to subcritical crack growth. 0000 NUMB - 0 DO 65 N - 1, K CALL HRONG(ZZ(N),A,B,C,D, DELTAT, FINAL, NUMB) 22(N) - FINAL 65 CONTINUE 000 0000000 (30 46 50 155 60 157 49 888 181 PRINT *, 'STUPID --- STUPID ----- STUPID ----- STUPID' PRINT *, NUMB transformation from ZZ( ) and Z3( ) to final strength, XF1( ) XF2( ) , respectively. Sorting of retained fracture strength DO 46 N - l , 1000 DD - KC/Y/(3.14159*X(N))**0.5/1E+6 J - DD/4 II(J) - II(J) + 1 CONTINUE IF (K .EQ. 0) GO TO 155 DO 50 N - 1 , K XF1(N) - Kc/Y/(3.14159*22(N))**0.5/1E+6 J - XFl(N)/4 I1(J) - I1(J) +1 CONTINUE IF (KK .EQ. 0) GO TO 157 D0 60 N - 1 , KK XF2(N) - Kc/Y/(3.14159*Z3(N))**0.5/1E+6 J - XF2(N)/4 - 12(J) - 12(J) +1 CONTINUE DO 49 WN - 0, 50 . WRITE (2,*) WN*4+2 , II(WN) , Il(WN) , 12(WN) CONTINUE WRITE (2 , *) WRITE (2, *) CONTINUE STOP END SUBROUTINE CHIUCH(A, B, C, 21 , ZT , ZTRIAL) DOUBLE PRECISION A , B, C, Z1, ZT, ZTRIAL ZTRIAL - A*(1/(1+C*21**3)-1/(1+C*ZT**3))-B*(ZT**2-Zl**2) RETURN END SUBROUTINE HRONG(CRACK, A,B,C,D, DELTAT, FINAL, NUMB) To consider the subcritical crack growth, we first calculate transient thermal stresses using thermoelastic theory. If K exceeds K0, the pre-existing cracks Z( ) has subcritical crack growth. Once the crack length satisfies the Griffith’s criterion, the crack grows. The temperature considered is the instaneneous surface temperature. If the crack length does not, the crack still grows according to subcritical crack growth. DOUBLE PRECISION X(0:7),CRACK, A, B, C, D,DELTAT, FINAL DOUBLE PRECISION XP,XQ,PP,QQ,RR,XP7,RR7,TEMP7, XM1,XM2,XM3 DOUBLE PRECISION ZT1, ZT2, ZT3, W1, W2, W3, TC REAL TEMP, STRESSl, STRE582 INTEGER M INPUT some parameters BIOT - 150 H - 0.6E-3 z - 0.6E=3 DI = 4.8E-7 EX - 6.5E-6 E - 70.0E+9 V - 0.25 TO calculate the eignvalue X( ), for X( )*TAN(X()) - Biot by neans of interval-half method 182 D0 310 , M - 0 , 5 XMl - 0 + 3.14159 * M XM2 - 1.570795 + M * 3.14159 DO 320 , N - 1 , 60 XM3 - (XMl + XM2)/2 XXMl - XMl * TAN(XMl) - BIOT XXMZ - XM2 * TAN(XMZ) - BIOT XXM3 - XM3 * TAN(XM3) - BIOT IF (ABS(XM1 - XM2) .LT. 0.00000001) THEN GO TO 315 ELSE IF (XXM1*XXM3 .GT. 0.0) THEN XMl - XM3 XM2 - XM2 ELSE XMl - XMl XM2 - XM3 END IF END IF 320 CONTINUE 315 X(M) - XM3 310 CONTINUE C TO calculate the thermal stress and Ko TIMEl - 0.0 STRESSZ - 0.0 4 DO 10 , AN - 1 , 500 TIME - 0.00001 * AN**2.4 STRESSI - 0.0 TEMP - 0.0 TEMP9 - 0.0 DO 20, M - 0 ,5 XP - SIN(X(M)) * COS(X(M)*Z/H) XQ - X(M) + SIN(X(M)) *COS(X(M)) PP - EXP(-DI*(X(M)/H)**2 * TIME) QQ ' SIN(X(M))/(X(M) + SIN(X(M))*COS(X(M))) RR - SIN(X(M))/X(M) - COS(X(M) * Z /H) STRESSl - STRESSl+ PP*QQ*RR*(2*EX*E*DELTAT/(l- V)) TEMP a TEMP + 2* DELTAT * XP /XQ * PP TEMP7 - 0.0 DO 12, L - l , ll 22 - H/lO * (L-l) XP7 - SIN(X(M)) * COS(X(M) * zz/H) RR7 - SIN(X(M))/X(M) - COS(X(M) * ZZ/H) TEMP7 - TEMP7 + 2*DELTAT *XP7 /XQ *PP The purpose of LOOP 12 is to find the average temperature, temp9, at instaneneous time of thermal shock 12 CONTINUE TEMP9 - TEMP9 + TEMP7 / 11 () (3 20 CONTINUE KI - 1.1214 * STRESSl * (3.14159 * CRACK)**0.5 C PRINT *, STRESSl , TEMP, TEMP9, KI IF((KI .LT. 0.248E+6).AND. (STRESSl .GT. STRESSZ)) THEN ‘STRESSZ - STRESSl TIMEl - TIME GO TO 10 END IF C It means that no subcritical crack growth occurs, but the crack C still has potential to grow in the near further. IF((KI .LT. 0.248E+6).AND.(STRESSI .LT. STRESSZ)) GO To 70 C It means that subcritical crack growth CAN NOT occur any more. 183 IF (KI .GT. 0.248E+5) THEN CRACKl - CRACK VELOCITY - EXP(10.3) * EXP((-1.088E+5 +0.11* KI)/ C (TEMP + 273)/8.3) C PRINT *, VELOCITY CRACK - VELOCITY * (TIME - TIMEl) + CRACK TIMEl - TIME STRESSZ - STRESSl C PRINT *, CRACK C It means that subcritical crack growth occurs. END IF IF(CRACK .GT. 0.1E-3 ) THEN CRACK - CRACKl NUMB - NUMB + 1 GO TO 70 END IF TC = D**O.5 * (1+C*CRACK**3)/CRACK**0.5 IF (TC .LT. TEMP9) GO TO 80 C It means that the crack length satisfies the Griffith failure C criterion. The program will go into the "pop—in" crack growth C loop again. 10 CONTINUE C The purpose of DO LOOP 80 is to find the final crack length due to C the ’pop-in crack growth length’ after the activation of the C subcritical crack growth. GO TO 70 80 PRINT *, VELOCITY , CRACK () () PRINT *, TC , TEMP9 80 ZT1 - CRACK * 1.0000000001 DO 30 N - 1 ,20 W1 - A*(l/(1+C*CRACK**3) - l/(1+C*ZT1**3)) -B*(ZT1**2- C CRACK**2) IF (N .EQ. 1) THEN W3 - W1 ZT3 - ZT1 GO TO 30 ELSE IF (W1*W3 .GT. 0.0) THEN W3 - W1 ZT3 - ZT1 ZT1 - CRACK * 1.5**N GO TO 30 ELSE GO TO 35 END IF END IF 30 CONTINUE GO TO 70 C IT means that the crack is long enough that no ’pop-in’ C crack growth can occur in such a situation. 35 DO 38 NM - 1, 20 ZT2 - (ZT1 + ZT3) /2 W2 - A*(1/(1+ C* CRACK**3) - 1/(1+ C*ZT2**3)) - C B*(ZT2**2 - CRACK**2) IF (W3*W2 .GT. 0.0) THEN W3 - W2 ZT3 - ZT2 38 29 70 100 184 ZT2 - (ZT1 + ZT3)/2 ELSE W1 - W2 ZT1 - ZT2 ZT2 - (ZT1 + ZT3)/2 END IF IF (ABS(ZT1 - ZT3) .LT. 0.0000001) GO TO 29 CONTINUE ' FINAL - ZT2 GO TO 100 FINAL - CRACK RETURN END