LIBRARY Michigan Stair: i University PLACE lN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE :c—i _J fiF—‘I MSU Is An Affirmative Action/Equal Opportunity Institution cztcimmunflt THE DEPENDENCE OF THE EFFECTIVE YOUNG’S MODULUS OF GLASS/EPOXY AND GLASS/GLUE LAMINATES ON ADHESION AREA AND GLUE BOND THICKNESS By KIYONG LEE A THESIS Submitted to MICHIGAN STATE UNIVERSITY in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Materials Science and Mechanics 1993 TEE DEPENDENCE OF THE EFFECTIVE YOUNO'E IODULUE OE GLASS/EPOXY AND GLASS/GLUE LINIEIIES OI ADEESIOE AREA AND GLUE BOND THICKNESS BY KIYONG LEE The effects of adhesives on the effective elastic modulus of adhered soda-lime-silica microscope glass slides were investigated using three different kinds of adhesives: super glue, epoxy cement and epoxy resin. The specimens were thus three-layer composites, with a bond layer sandwiched between two glass slides. A sonic resonance technique was used to determine the elastic moduli of single slides, glass slide/glue composite specimens and epoxy resin specimens. The prismatic bar-shaped specimens were suspended horizontally from a driver and a pick-up transducers. The fundamental flexural frequencies of specimens were used to calculate the elastic modulus. The change of Young's modulus of adhered soda-lime-silica microscope slides was observed as a function of adhesion area percent. Young's moduli decreased continuously with decreasing the adhesion area. For an area percent adhesion less than 30 to 35 percent the Young’s moduli decreased relatively rapidly with a decrease in the adhesion area. The effective Young's modulus of the laminates changed from about 70 GPa for 100 percent area coverage of adhesive down to about 15 GPa for an area coverage of about 1 percent for a single glue spot. For specimens having a 100 percent area coverage of adhesive, the effective Young's modulus of the laminates changed by about 4 GPa as the glue bond thickness ranged from 0.025 mm to 0.275 mm. TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES 1. INTRODUCTION AND REVIEW OF LITERATURE 1.1. 1.2. 1.3. 1.4. 'External Crack' model 'Rule of Mixtures (ROM)' Model 'Dynamic Beam Vibration' Model Description of Current study and Outline of Remaining Text 2. EXPEEINENTAL PROCEDURE 2.1. 2.1.1. 2.1.2. 2.2. 2.3. 2.4. 2.4.1. 2.4.2. 2.4.3. 2.5. 2.5.1. 2.5.2. 2.6. 2.7. Materials Microscope Slides Adhesives The Measurement Of Mass The Measurement Of Thickness Applying Adhesives And Preparing Glass Slide/Glue Composite Specimens Applying 'Super Glue' Applying 'Epoxy Cement' Applying 'Epoxy Resin' The Measurement Of Area Fraction Of Glue Template Method For Small Circle-Shaped Glue Spots Grid Counting Method For Irregularly-Shaped Glue Spots The Measurement Of Elastic Modulus Making Epoxy Resin Specimens 3. RESULTS AND DISCUSSION 3.1. 3.1.1. 3.1.2. 3.2. 3.2.1. 3.2.2. 3.2.2.1. 3.2.2.2. 3.2.3. 3.2.4. 3.2.5. 3.2.6. Effects of Adhesion Area, Number of Glue Spots, and Glue Bond Thickness The Effects of Super Glue Adhesion on Elastic Modulus The Effects of Epoxy Cement Adhesion on Elastic Modulus The Effects of Epoxy Resin Adhesion on Elastic Modulus The Effect of Adhesion Area on Elastic Modulus The Effect of Glue Bond Thickness on Elastic Modulus Glass Slide/Epoxy Resin Composite Specimens having a Fixed Glue Composition but Different Adhesion Areas Glass Slide/Epoxy Resin Composite Specimens of Differing Glue Composition and Fixed Adhesion Areas The Effect of Epoxy Resin Composition on Elastic Modulus Experimentally obtained Elastic Moduli, Densities and Poisson's Ratios of Epoxy Resin Specimens Comparison of Rule of Mixtures and Dynamic Beam Vibration models Change of Effective Young's Modulus of Glass Slide/ Glue Specimen with respect to Glue-Bond Thickness Ranges at Fixed Adhesion Areas iii Page 77 77 86 3.2.7. 3.2.9. 3.2.10. Possible Physical Mechanisms for the Difference between the Measured Modulus of the Glass Slide/Glue Composite Specimens and the Predictions of the ROM and Dynamic Modulus Models Possible Changes in Effective Young's Modulus of a Bond Layer for the Difference between the Experimentally Determined Modulus and the Moduli Predicted from the ROM Model Dependence of the ROM and the Dynamic Modulus Models on the Relative Glue Bond Thickness and Comparison with Experimental Results Possible Changes in Effective Young's Modulus of a Glass Slide/Glue Specimen for the Difference between the Measured Moduli and the Moduli Predicted from the Dynamic Modulus Model Consideration of Insufficient Bonding as a Possible Factor for Deviation of Measured Elastic Moduli from the Moduli predicted by ROM Model General Trends in Effective Young's Modulus on Adhesion 4. SUNNAR! AND CONCLUSIONS The Effect of Adhesion Area on Elastic Modulus The Effect of Number of Glue Spots on Elastic Modulus The Effect of Glue Bond Thickness on Elastic Modulus The Effect of Epoxy Resin Composition on Elastic Modulus The Comparison of the ROM model and the Dynamic Beam Vibration model Future Work and Practical Applications of This Study 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. Appendix A. Appendix B. B-l. 8-2. B-3. 8-4. 8-5. 3-6. B-7e 8—9 e 8-10 e Calculation of Adhesion Area by a Template Method Experimentally Determined Fundamental Frequency and the corresponding Young's Modulus of each glass slide/glue composite specimen The experimental data for the glass slide/super glue composite specimens having one glue spot The experimental data for the glass slide/super glue composite specimens having two glue spot The experimental data for the glass slide/super glue composite specimens having three glue spot The experimental data for the glass slide/super glue composite specimens having five glue spots The experimental data for the glass slide/epoxy cement composite specimens having one glue spot The experimental data for the glass slide/epoxy cement composite specimens having three glue spots The experimental data for the glass slide/epoxy resin composite specimen having three glue spots of 50% resin and 50 i hardener The experimental data for the glass slide/epoxy resin composite specimens adhered by the epoxy resin of 35% resin and 65‘ hardener The experimental data for the glass slide/epoxy resin composite specimens adhered by the epoxy resin of 65% resin and 35$ hardener The experimental data for the glass slide/epoxy resin composite specimens adhered by the epoxy resin of 80% resin and 20t hardener LIST OF NEFENENCES iv 91 93 95 98 106 109 120 120 121 122 123 124 124 126 126 128 130 131 133 134 135 136 139 140 141 142 LIST OF TABLES Table Number Table Table Table Table Table Table Table Table Table Table Table 1. 9. 10. 11. Classification of the glass slide/super glue composite specimens according to the number of glue spots and the range of glue thickness. Classification of the glass slide/epoxy cement composite specimens according to the number of glue spots. Classification of the glass slide/epoxy resin composite specimens according to the composition of the glue and the range of glue thickness. Classification of the glass slide/epoxy resin composite specimens with 100 t adhesion area. These specimens were included in the experimental comparison of the ROM and the Dynamic Beam Vibration models (Section 3.3.5). For the sonic resonance method, the relative position of nodes according to the vibrational mode. Comparison of experimentally obtained elastic moduli, densities and Poisson's ratios with reference values. Young's moduli (GPa) of glass slide/epoxy resin composite specimens obtained from the regression curves in Figure 33. The table shows representative Young's moduli of glass slide/epoxy resin composite specimens having a given adhesion areas and a glue bond thickness within a given range of glue bond thickness. Elastic modulus of quenched polymer glass as a function of the quench medium temperature [19]. Pressure derivatives of elastic constants of MgO at 23°C, NaCl, KCl at 22°C, and Quartz at 25°C [20-22]. Young's modulus of epoxy resin layer in a glass slide/epoxy resin composite specimen to compensate for the ROM model. Comparison of measured Young's moduli of a glass slide layer with Young's moduli of the glass slide layer calculated from the Dynamic modulus model required to obtain the measured effective Young's moduli. Page 28 30 33 33 41 74 87 92 94 96 104 Table Table Table Table Table Table Number Page 12. Comparison of measured Young's moduli of a glue bond 105 layer with Young's moduli of the glue bond layer calculated from the Dynamic modulus model required to obtain the measured effective Young's moduli. 13. Measured Young's moduli and moduli predicted from 107 the ROM and the Dynamic Modulus models as a function of relative thickness obtained from the corresponding regression curves in Figure 38. 14. Young's moduli obtained from a regression curve in 108 Figure 28 and the difference between the Young's moduli, Em" for adhesion area of 100% and the Young's moduli, E, for adhesion area ranging from 0% to 100%. 15. Effective Young's modulus calculated from the ROM 114 model and comparison with the experimental modulus values based on Figure 49. 16. Differences between the data of the super glue, the 115 epoxy cement and the epoxy resin shown in Table 15. vi LIST OF FIGURES Figure Number Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Figure 8. Figure 9. Figure 10. Figure 11. Elliptical Internal Cracks (a), External Cracks (b) under uniaxial loading [1]. Composite specimen composed of layers 1, 2 and a bond layer for Rule of Mixtures model [5], where uniaxial tension is assumed for the elastic modulus determination. Composite specimen composed of layer 1, 2 and a bond layer for Dynamic Beam Vibration model [5], where the specimen is driven at flexural resonance to determine the elastic modulus. Photograph of glass slide/epoxy resin composite specimen which shows three glue spots and points marked for the measurement of glue bond thickness. Change of normalized mass of various glues (o, o Elmer's School Glue; D, s Cement For Plastic Models; A, A Epoxy Cement; 0, 0 Super Glue) as a function of time in hours after gluing. Relative position(s) of glue spot(s): (a) one glue spot, (b) two glue spots, (c) three glue spots, (d) five glue spots, where leength of specimen and W=width of specimen. Photograph of epoxy resin (i.e. "Quick Setting Epoxy Adhesive”), a 7.65 cm x 7.65 cm piece of standard notepad paper and a stick for mixing and applying. Template used for measurement of the area fraction of the smaller circular glue spots on the composite specimens. Grid paper used for measurement of the area fraction of irregularly shaped larger glue spots on the composite specimens. Photograph of the sonic resonance apparatus. (A glass slide/glue specimen is suspended in the air) A block diagram of the sonic resonance apparatus [15]. vii Page 13 20 22 25 29 35 36 38 39 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Number 12. Photograph of epoxy resin specimens. The ratios of resin and hardener used for each specimen are 50%:508, 65%:35% and 80%:205, respectively. 13. For a super glue bond layer, the effect of the number of super glue spots on Young's modulus as a function of adhesion area(\). The curves represent a least-squares best fit to equation 31. 14. For a super glue bond layer, the effect of the number of super glue spots on Young’s modulus as a function of adhesion area(%). The curves represent a least-squares best fit to equation 32. 15. For a super glue bond layer, the effect of the number of super glue spots on Young's modulus as a function of adhesion area(%). The curves represent a least-squares best fit to equation 33. 16. For a super glue bond layer, the effect of the super glue on Young's modulus as a function of adhesion area(%) with the data for specimens having 2, 3 and 5 glue spots lumped together. The curve represents a least-squares best fit to equation 31. 17. For a super glue bond layer, the effect of the super glue on Young's modulus as a function of adhesion area(%) with the data for specimens having 2, 3 and 5 glue spots lumped together. The curve represents a least-squares best fit to equation 32. 18. For a super glue bond layer, the effect of the super glue on Young's modulus as a function of adhesion area(t) with the data for specimens having 2, 3, 5 glue spots lumped together. The curve represents a least-squares best fit to equation 33. 19. For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(%) for specimens having one super glue spot. The curves represent a least -squares best fit to equation 31. 20. For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(%) for specimens having one super glue spot. The curves represent a least -squares best fit to equation 32. 21. For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(t) for specimens having one super glue spot. The curves represent a least -squares best fit to equation 33. 22. For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(§) for specimens having two and three super glue spots. The curves represent a least-squares best fit to equation 31. viii Page 46 49 50 51 53 54 55 56 57 58 59 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Number 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(%) for specimens having two and three super glue spots. The curves represent a least-squares best fit to equation 32. For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(%) for specimens having two and three super glue spots. The curves represent a least-squares best fit to equation 33. For an epoxy cement bond layer, the effect of epoxy cement on Young's modulus as a function of adhesion area(%) for specimens having one and three glue spots. The curves represent a least-squares best fit to equation 31. For an epoxy cement bond layer, the effect of epoxy cement on Young's modulus as a function of adhesion area(%) for specimens having one and three glue spots. The curves represent a least-squares best fit to equation 32. For an epoxy cement bond layer, the effect of epoxy cement on Young's modulus as a function of adhesion area(%) for specimens having one and three glue spots. The curves represent a least-squares best fit to equation 33. For an epoxy resin bond layer, the effect of epoxy resin on Young's modulus as a function of adhesion area(%). The curve represents a least-squares best fit to equation 31. For an epoxy resin bond layer, the effect of epoxy resin on Young’s modulus as a function of adhesion area(%). The curve represents a least-squares best fit to equation 32. For an epoxy resin bond layer, the effect of epoxy resin on Young’s modulus as a function of adhesion area(%). The curve represents a least-squares best fit to equation 33. For an epoxy resin bond layer, the effect of glue bond thickness of epoxy resin on Young's modulus as a function of adhesion area(t). The curves represent a least-squares best fit to equation 31. For an epoxy resin bond layer, the effect of glue bond thickness of epoxy resin on Young's modulus as a function of adhesion area(t). The curves represent a least-squares best fit to equation 32. For an epoxy resin bond layer, the effect of glue bond thickness of epoxy resin on Young's modulus as a function of adhesion area(t). The curves represent a least-squares best fit to equation 33. ix Page 60 61 63 64 65 67 68 69 71 72 73 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Number 34. 36. 37. 41. 42. For an epoxy resin bond layer, the effect of composition of epoxy resin on Young's modulus as a function of relative glue bond thickness. The curves represent a least-squares best fit to equation 34. Comparison of ROM and Dynamic Beam Vibration models for epoxy bonds made from an initial composition of 50% resin and 50% hardener. The curves represent a least-squares best fit to equation 36. Comparison of ROM and Dynamic Beam Vibration models for epoxy bonds made from an initial composition of 65% resin and 35% hardener. The curves represent a least-squares best fit to equation 36. Comparison of ROM and Dynamic Beam Vibration models for epoxy bonds made from an initial composition of 80% resin and 20% hardener. The curves represent a least—squares best fit to equation 36. Comparison of experimentally determined moduli with the moduli predicted from the Rule of Mixtures and Dynamic Beam Vibration models as a function of relative glue bond thickness for 50% resin and 50% hardener. The curves represent a least-squares best fit to equation 37. Comparison of experimentally determined moduli with the moduli predicted from the Rule of Mixtures and Dynamic Beam Vibration models as a function of relative glue bond thickness for 65% resin and 35% hardener. The curves represent a least-squares best fit to equation 37. Comparison of experimentally determined moduli with the moduli predicted from the Rule of Mixtures and Dynamic Beam Vibration models as a function of relative glue bond thickness for 80% resin and 20% hardener. The curves represent a least-squares best fit to equation 37. Change of Young's modulus with respect to glue-bond thickness ranges (R,:0.025-0.075 m, R¢:0.125-0.175 m, R,:0.225-0.275 m) at adhesion area percent, A, ranging from 0% to 40%. The data were obtained from the three regression curves using equation 33 in Figure 33. Change of Young's modulus with respect to glue-bond thickness ranges (R,:0.025-0.075 m, R2:0.125-0.175 m, R,:0.225-0.275 m) at adhesion area percent, A, ranging from 50% to 100%. The data were obtained from the three regression curves using equation 33 in Figure 33. Page 75 79 80 81 82 83 84 88 89 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Number 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. Change of the measured and the calculated Young's modulus with respect to glue-bond thickness ranges (R,:0.025-0.075 m, R230.125-0.l75 m, R,:0.225- 0.275 mm) at 100% adhesion area. The calculated data were obtained from the three regression curves using equation 33 in Figure 33. Calculated Young's moduli of epoxy resin bond layer to compensate for the Rule of Mixtures model. Comparison of experimentally determined Young's modulus (.) and calculated Young’s modulus from the Dynamic Modulus model (———) and the ROM model ( ----- ) for 50% resin and 50% hardener. Comparison of experimentally determined Young's modulus (.) and calculated Young's modulus from the Dynamic Modulus model (——-) and the ROM model ( ----- ) for 65% resin and 35% hardener. Comparison of experimentally determined Young's modulus (.) and calculated Young's modulus from the Dynamic Modulus model (—-—) and the ROM model ( ----- ) for 80% resin and 20% hardener. The general trend in Young's modulus between super glue adhered specimens and epoxy cement adhered specimens for one glue spot as a function of adhesion area(%). The general trend in Young's modulus of glass slide /super glue, epoxy cement, and epoxy resin composite specimens having two or more glue spots as a function of adhesion area(%). The effect of the glue bond thickness on Young's modulus between the super glue and the epoxy resin adhered composite specimens as a function of adhesion area(%). Change in elastic modulus of a glass slide for the super glue and the epoxy cement adhered composite specimens. Change in elastic modulus of a glass slide for the super glue and the epoxy resin adhered composite specimens. xi Page 90 97 99 100 101 110 111 112 118 119 1. INTRODUCTION AND REVIEW OF LITERATURE The primary motivation for this research was to explore the feasibility of using sonic resonance elasticity measurements to non— destructively assess the integrity of adhesive bonds in laminate composites. For example, one could envision using sonic resonance measurements to monitor (in-situ) the time-evolution of the degradation of a bond layer in a laminate composite heated to temperatures at which the adhesive degrades. Alternatively, one could use elasticity measurements as a quality-control tool to evaluate bond-layer defects (such as incomplete bonding) that might occur during processing of laminate composites. However, before elasticity measurements can be employed to assess bond integrity, one must first understand how the experimentally-determined elasticity values change as a function of variables such as the relative bond adhesion area and the bond thickness. This study seeks to help establish the basic understanding needed to realize the potential of elastic modulus measurements of laminate composites (especially in the arena of the analysis of bond- phase defects or bond degradation). As a model laminate composite specimen, we chose to use two glass microscope slides bonded by a variety of adhesives. Glass microscope slides were appropriate for this study because glass microscope slides are: (l) readily available, (2) relatively inexpensive, in part due to the fact that microscope slides may be employed in the as-received state, without dimensioning, grinding, or polishing, (3) comparatively uniform (in the as-received state) in terms of external slide dimensions and elastic moduli, (3) brittle (which is important since the primary area of interest for the author is ceramics and ceramic composites). In addition to the relation of this research to damage assessment in brittle laminate composites, this study has application to the subject of external cracks in materials (Figure 1). An external crack may be defined loosely as a crack which penetrates a solid, leaving only a single, unbroken internal ligament surrounded by a continuous, surface-breaking cracks (often considered in fracture mechanics) in which the cracked area is (typically) small compared to the specimen cross-section. Composite specimens in this study which were adhered only by a single glue spot centered on a long-transverse specimen face might model such an external crack (Figure 1). Specimens adhered by two or more glue spots might model delamination cracks. However, it must be emphasized that the specimens used in this study would only be appropriate to model planar cracks located at the midplane of a prismatic bar. The specimen would not be appropriate to model other crack geometries, crack face orientations, or spatial distributions of cracks. In this paper, the effects of adhesives on the effective elastic modulus of adhered soda-lime-silica microscope glass slides were investigated using three different kinds of adhesives. The area fraction, thickness, and composition of adhesives affected the measured elastic modulus for glass slide/glue composite specimens. Also a 'Rule of Mixtures' model and a 'Dynamic Beam Vibration' model were compared with experimental results of the glass slide/glue composite specimens containing a layer of 100 percent adhesion area sandwiched between two glass slide layers. Before discussing the experimental procedure and results of this study, we shall briefly review the 'Rule of Mixtures' model, the 'Dynamic Beam Vibration' model and an 'External Crack' model in order to compare differences and similarities between our study and such models. The following reviews shall show the assumptions and results of each model. The glass slide/glue composite specimens used in this study can be considered as a three-layer composite composed of two slide glasses and adhesive. Therefore the models considered shall be confined to those models which deal with three-layer composites. 1.1. 'External Crack' model Remeny and Cook introduced an 'external crack' model to estimate the effect of strongly interacting cracks which is very common in rocks with a high crack density [1]. In this section, Kemeny and Cook's assumptions for the external crack model and the stress intensity factor calculation shall be reviewed for two dimensional internal cracks and external cracks (Figure 1). Kemeny and Cook [1) considered a random distribution of flat, internal cracks or external cracks in a linear elastic, isotropic and homogeneous medium. Narrow elliptical internal cracks and external cracks were considered [1], the surfaces of which were assumed to be friction-free. Assuming plane strain, the intrinsic Young's modulus, E, can be related to the effective Young’s modulus, E, for a solid containing cracks under a uniaxial stress a (l, 2]. _¢9V - -£f! + A?’ 1;; 23 (1) where V - volume of the body containing the cracks A! - increase in strain energy due to the presence of the cracks. For a two dimensional elastic body, 0" the additional strain energy due to the existence of a single internal crack of length 2c is given by [1, 3] ————-D’Q -————”Q 2c ' C 2a 2W [4 (a) (b) Figure l. Elliptical Internal Cracks (a), External Cracks (b) under uniaxial loading [1]. 2(1'V2) c[ 2 2 Km2 . 3 Jo. r u (l-V) c ( ) where E = intrinsic Young's modulus v - Poisson's ratio K,-=crack tip stress intensity factor for opening mode F5 - crack tip stress intensity factor for shearing mode Km 8 crack tip stress intensity factor for tearing mode of deformation. For a single external crack, the lower and upper limits of integration in equation 2 are replaced by a and a+c, where a is the contact length for the external crack, and c is the length of the external crack surrounding the contact area [1]. A plate of width 2W, height h, and unit thickness containing either an internal crack of length 2c or an external crack of contact length 2a gives the following crack tip stress intensity factors under the uniaxial tension 0 at an angle 0 to the sides of height h [1, 4]. For an internal crack [1, 4] K, = o no sin’O (38) Kn = o no sine cosO c << w (3b) x”, . 0. (3C) For an external crack [1, 4] K, - 02" sinzfi (4a) na Ru 2 fl sine cosO a << W (4b) «a Km = o. (40) Substituting equations 3a-3c and 4a-4c into equation 2 and integrating gives the additional strain energies due to a single internal crack or single external crack under uniform stress, respectively, as [1] 046) - ”€202 sin’fl L22). (5) U.(9) = BW’O’ sin’O ln [ (fl (1-v’)] . (5) a En Kemeny and Cook [I] assumed that N internal cracks or external cracks are randomly distributed in the body with a mean internal crack length squared or a mean external crack contact length squared . Taking a cylindrical average from zero to 2n, the average of sin’fl, denoted as =l/2. Also making the approximation that wac for a</N) - effective external crack contact length (E2 =- /N) . Substituting equations 7 and 8 into equation 1 gives the effective Young's modulus for a body containing a random distribution of either internal MIMI MIMI where m The cracks or external cracks, respectively, as [1] 1 a (9) 1 + «mu-v2) 1 (1°) 1 + %m 1n[(1+%)(1-v=)] Néi/v crack density parameter - S/E external crack shape parameter which characterizes the relative amount of contact per unit area. effects of varying the two parameters m and n can be predicted from equations 9 and 10. For both internal cracks and external cracks, as the crack density approaches infinity, the effective Young's modulus approaches zero. For the external cracks, while the crack density m is held constant, the effective Young's modulus decreases with the decrease of the relative crack contact area n [1]. A key assumption for the Kemeny and Cook model [1] is that material undergoing a plane strain under a uniaxial stress is linear elastic, isotropic and homogeneous. Also, Kemeny and Cook considered the flat, friction free, elliptical internal cracks or external cracks to be randomly distributed within the body. In our study, Young's modulus of glass slide/glue composite specimens was calculated from free vibration. However, the area surrounding the glue spots can be considered as external cracks, so has relevance to the current study for the determination of the effective modulus of the glue layer. 1.2. 'Rule of Mixtures (ROM)' Model In this study the glass slide/glue composite specimens contained a glue layer between two glass slides. Thus, the glass slide/glue composite specimen configuration can be considered as a three layered composite. Therefore, this review shall concentrate on the ROM model for estimation of effective elastic modulus of a three layer composite [5). The main assumptions for the ROM model shall be noted and compared with the physical reality of the glass slide/glue composite specimens included in our study. When a load is unidirectionally applied to a three layer composite (Figure 2), the strain of each layer is equal to the effective strain, a“, of the composite [5. 6]: Figure 2. Load i [it i .————-Layer 1 \\\\\\\\\\\\\\\\\\ Bond.Layer Layer 2 \ \\\\\\\\ \\\\\\\\\\\\\\\\\\ Load Composite specimen composed of layers 1, 2 and a bond layer for Rule of Mixtures model [1], where uniaxial tension is assumed for the elastic modulus determination. For this study, layers 1 and 2 were glass microscope slides. cu = 2,, = a, = ‘47 (11) where subscripts '11', '12' and 'b’ refer to layers 1, 2 and bond layer sandwiched between the layers 1 and 2, respectively. Equation 11 is based on the assumption of perfect interfacial bonding between layer 1, layer 2 and bond layer, such that there is no sliding of a layer on an adjacent layer [6]. Assuming elastic behavior of each layer, the stresses are given by [5. 5] on 3 311 8:1 (12a) 0,2 = E,2 3:2 (12b) o, = E, s, (12c) Accordingly, the load on each layer can be given by [5] Pu ’ 0,, Au 3 511311311 (13‘) Pt: ' an A:2 ' Bazaar: (135) Pb = as Ab " 3535A» (13°) Where A", A,2 and A, are the cross-sectional areas of the layers 1, 2 and bond layer, respectively. 10 The effective load applied to the composite, Pd - P" + Pa + P, and effective cross sectional area of the composite, Ad I A" + A” + A” such that [5, 6] Pa ' 0.934 a 011311 I 012312 I 0&5» ° (14) Therefore, [5, 6] A A odzo,,_£+on__‘3+o,._’. A A A d d d (15) 3‘71: Va +0” V12+Ob Vb where V", V,2 and V), are the volume fractions of the layers 1, 2 and the bond layer. Differentiating equation 15 with respect to strain yields do do do do A . " V + _2 V + _.‘ v . ds do u do u do b (16) With the assumption of elastic behavior in each layer, don/dc, don/dc and dob/d: can be represented by the corresponding elastic moduli. As a result, E30“, the effective elastic modulus of the three layer composite [5. 6] Em, = suvu + Suva + my, . (17) 11 The principal assumptions of the ROM model are perfect interfacial bonding between layers and the linear elastic behavior of each layer. Linear elastic behavior means that the slope of stress-strain curve of each layer, do/ds, is linear. This linearity is typically applicable for glass or ceramic composites [6]. 1.3. 'Dynamic Beam Vibration' model In the dynamic beam vibration model, beam vibrations can be described approximately by the Bernoulli-Euller beam equation [7, 8]. For the free, undamped vibration of a monolithic bar, the Bernoulli- Euler beam equation is given by [5, 7-9] srflifl+££flilflso 18 ax‘ G at2 ( ) where E Young's modulus I a the second moment of inertia of the cross section of the bar with respect to the neutral axis W - transverse deflection of the bar, which is a function of position x along longitudinal axis and time t A - cross sectional area of the bar p - density of the bar G s acceleration due to gravity. For a three-layer composite in which a bond layer is sandwiched between layers 1 and 2 (Figure 3) and assuming perfect interfacial bonding 12 Layer 1 Neutral Plane Bond Layer Layer 2 Figure 3. Composite specimen composed of layer 1, 2 and a bond layer for Dynamic Beam Vibration model [5], where the specimen is driven at flexural resonance to determine the elastic modulus. As in figure 1, layers 1 and 2 were glass microscope slides for this study. 13 between the layer 1 or 2 and the bond layer, equation 18 becomes [5, 9] a‘W(x,t) + (AuPuIAuPerAst) 52V(x,t) I=0 (Eula +EDIIZ+EDID) ax‘ G at: (19) where subscripts '11', '12' and 'b' refer to layers 1, 2 and bond layer, respectively. In equation 19, the (A,,p,,+A,,pu+A,p,) term can be expressed as Acpc, where Ac is cross sectional area of the composite (A‘sAu+Aa+A,) and pc is the average density of the composite p,=(A,,pu+A,,p,2+Abp,)/(A,,+A,,+A,,) [5, 9). For free-free suspension of a bar, the bending moments and the shearing forces must be zero at both ends of the bar, such that the boundary conditions for the transverse vibration can be given by [5, 7- 9]: bending moments alarm,” _ 32W(L,t) _ (EI),_:2_ 0 , (BIL—.55.. 0 for t z 0 (20‘) shearing forces 8’W(0,t) g B’W(L,t) a (EILT 0 , (FELT 0 for t 2 0 (20b) where L = length of the bar. 14 Applying the above boundary conditions to equation 19 gives the fundamental flexural resonance frequency of the three-layer composite [5, 7’9] F 8 11.1528 [(511 In I 312 It: I Eb 1'06]; (21) 1'.2 Aci’c 4.4.-4 d, I" - I y2 till 3 [(d, - d)2d,, + (d, - d)d,2, + ._". w (22a) 4,-4 3 -d (,3 1,,- f y’dA=[..'.2+d,2,d+d,,d2 w (225) 4-4. 3 4" d3 I,- f yids-[J—d3d+d,d2]w (22c) 4 3 where 6", dc - thickness of layers 1 and 2, respectively dk‘- thickness of bond layer y’ - position along transverse axis W :- width of the bar. From the equilibrium of the axial forces, ‘edu-J .4 4-4 I oudA+ f andA‘lrf o,dA=0 (23) 4-4 4-5 4 where on - normal stress of layer 1 (0H - Euy/r) 15 on - normal stress of layer 2 (on - Fay/r) o. a normal stress of bond layer (a. - Eg/r) .4 ll position along transverse axis '1 ll radius of curvature of the neutral axis. From the equilibrium condition given in equation 23, the distance, d, from the neutral axis to the interface between the layer 1 or 2 and the bond layer is calculated in terms of known values [5, 10], d g (Eu an2 I 25:1 dz: db "’ Eb do: " 5:: duz) (24) (23,, d" + 23, d, + 23,, an) Substituting equation 24 into 22, in turn, 22 into 21, the fundamental flexural resonance frequency can be calculated. The effective elastic modulus, Euwfl, of a three-layer composite is given by [5] - (s I +5 1’ +3 I) 530m” 11 u u 12 b s . (25) (I,,+I,,+Ib) In the dynamic modulus model, perfect interfacial bonding between two layers and bond layer is a key assumption. 16 1.4. Description of Current Study and Outline of Remaining Text The current study investigates the effects of adhesion on the effective Young's modulus, glass slide/glue composite specimens. The effective Young's modulus of each specimen was determined by the experimental procedure which shall be discussed in section 2 'Experimental Procedure'. In section 2, the materials and the procedures used to fabricate the specimens and the technique to determine the effective Young's moduli shall be discussed in detail. Section 3 'Results and Discussion' includes an analysis of the experimental results in terms of several empirical equations. The effects of adhesion area, the number of glue spots, and the glue bond thickness on the effective Young's modulus of the glass slide/glue specimens shall be discussed in detail. Also, the experimental modulus values shall be compared with the values predicted from the ROM and the Dynamic modulus models discussed in sections 1.2 and 1.3. 17 2. EXPERINENTAL PROCEDURE In this study microscope glass slides were glued to make glass slide/glue composite specimens using three types of adhesives. The adhesives were 'Sure Shot Super Glue (Devcon Corp., Wood Dale, IL, made in Japan)’, 'Elmer's Epoxy Cement (Borden Inc., HPPG, Columbus, Ohio, 43215)’ and ’Quick Setting Epoxy Adhesive (Super Glue Corporation, Hollis, N.Y., 11423)'. As will be discussed in the following sections each type of adhesive was applied on glass slides and the thickness and mass of glass slides, adhesives and final glass slide/glue composite specimens were measured. The area fraction of adhesive applied on each glass slide/glue composite specimen was determined using two different methods, template method and grid counting method. For epoxy resin (i.e. 'Quick Setting Epoxy Adhesive'), glass slide/epoxy resin composite specimens were prepared with 100 percent adhesion area but differing compositions of resin and hardener in order to investigate the effect of the composition on the effective Young's modulus of the three layer composite. Also, five epoxy resin specimens were made of epoxy resin itself. Different compositions of resin and hardener were incorporated in the epoxy resin specimens in order to measure the epoxies' intrinsic Young's modulus and investigate the effect of composition on the epoxy resin specimens. For all the epoxy resin specimens included in this study, the sonic resonance technique was used to determine Young's modulus through free-free suspension of specimen. 18 2.1. Materials 2.1.1. Microscope Slides Soda-lime-silica glass microscope slides made by 'Erie Scientific Company' (Model No. 2954-F, Division of Sybron Corp., Portsmouth Industrial Park, Portsmouth, N.H. 03801) were used for this study (Figure 4). The approximate mass of individual slides was 5.7 grams and the dimension was 7.62 cm x 2.54 cm (3 inches by 1 inch) with an approximate thickness of 1.2 mm. Individual glass slides had square edges (Beveled edge glass slides are available but were not included in this study). 2.1.2. Adhesives Before performing the experimental portion of this study, four different kinds of adhesives were considered. The four adhesives were (1) 'Sure Shot Super Glue', (2) 'Elmer's Epoxy Cement', (3) 'Cement For Plastic Models (No. 3501, The Testor Corporation, Rockford, IL 61108)’, and (4) 'Elmer's School Glue (Borden Inc., Dept CP, Columbus, Ohio 43215)’. The large change of mass of adhesives with the lapse of time for experimental period make the exact mass measurements of specimens difficult and may result in some errors in measuring the elastic modulus. To select the most appropriate adhesives, the mass change of each adhesive was measured as a function of time. A total of eight glass slide/glue composite specimens was made by adhering two glass slides 19 Figure 4. Photograph of glass slide/epoxy resin composite specimen which shows three glue spots and points marked for the measurement of glue bond thickness. 20 using each adhesive. One hour after gluing the mass of each glass slide/glue composite specimen was measured. During a ninety two hour period the mass of each glass slide/glue composite specimen was remeasured a total of fifteen times. The mass measurements were then normalized with respect to the initial mass. Figure 5 shows the mass as a function of time. The data was fit by a least-squares procedure to an equation of the form an, - 1 - xllnuzzr) (26) where am a normalized mass of glue - mass of glue at one hour after gluing/mass of glue at time T (hours) after gluing K, and K, - constants The mass of glass slide/glue composite specimens adhered by super glue (i.e. 'Sure Shot Super Glue') and epoxy cement (i.e. 'Elmer's Epoxy Cement') changed by 0.0003 g in four days (Figure 5). The mass of 'Elmer's School Glue' and 'Cement For Plastic Models' adhesives changed by up to 0.0125 g in four days because of volatile components in the adhesive (Figure 5). As a result of their mass stability with respect to time, super glue and epoxy cement were selected as adhesives to be included in our study. After selecting the adhesives, epoxy resin (i.e. 'Quick Setting Epoxy Adhesive') was added as a desired adhesive for further experiment. 21 LI 1 l 1 l 1 l 1 I 1 l 1 l 4 l 1 J 1 l 1 4 +- ‘ A A A A 1.0 ”a"? - .. fl 5-1:“.-. 33‘5 ”-5.5, 5 {figffiffm'ju- U) 0.9% ‘ - 2 . .1 _ O 2 O.8_l “m\\\ . _ .- ‘~~ ; . [I] O 7- O ‘*§T=:~ . t :3 ' “‘~ ‘Eb-\‘ r-J . ““533:: 39-- e P U 0.6“ i“:g:::: ------ ’_ - s ““““ 91-02:: ----- E»... Q 'l s s s ---- m ..... " [3;] 0.5- o - u. . - ._‘i 0.44 o _ <3 .4 O o .— 52; 0.3‘ o o o o o _ O ‘ . Z 02- _ 0.1- - 0.0 I I I I I l I I I I I l I I I I I l I O 10 20 30 4O 50 60 7O 80 90 100 TIME AFTER GLUING (hours) Figure 5. Change of normalized mass of various glues (o, o Elmer's School Glue; D, I Cement For Plastic Models; a, 1 Epoxy Cement; 0, 4 Super Glue) as a function of time in hours after gluing. In this figure, normalized glue mass refers to the ratio of the initial mass of the glue to the glue mass at a later time. The curves represent a least-squares best fit to equation 26. (— x,-0.1696, Kg'0.8292, r-O.9720 and K.-0.1014, R1-0.5818, r-0.9720; --- K.=0.1017, 1930.7908, r-0.9896 and x,-o.1120, K2-0.6509, r-0.9565; K,=0.0027, lip-0.6144, r-0.6281 and lip-0.0014, K,-14.8621, r-0.0834; -' -° It'll-0.0012, K,-4.0128, r-0.2358 and XII-0.0012, lip-4.0128, r-0.2358) 22 2.2. The Measurement Of Mass Before gluing, the mass of individual glass slides was measured using an electronic balance (Model No. A 210 P, Sartorius Corp., 140 Wilbur Pl., Bohemia, N.Y.). This balance is accurate to within t0.0001 gram. Super glue, epoxy cement and epoxy resin adhesives had different setting times. Two criteria were used to determine the setting time for the adhesives. First, a shear force was applied to a glass slide/glue composite specimen. If the slides did not move with respect to one another due to the shear couple (which was applied by hand), then the lack of shear was taken as one indication that the adhesive had set. Secondly, when the glue extruded from the edge of a glass slide/glue composite specimen was no longer sticky, then this was taken as another indication that the glue had set. If both criteria (lack of shearing and lack of stickiness) were satisfied, then the adhesive was considered to be set. For adhesion area less than about sixty five percent (in glass slide/glue composite specimens having three glue spots), there was typically no extruded glue from the glass slide/glue composite specimen edges. In these cases, then the shearing criterion was the single criterion used to determine setting time. The setting time of each adhesive determined in this manner was about one hour for the super glue and twenty four hours for the epoxy cement. Thus, the mass of the glass slide/super glue and the glass slide/epoxy cement composite specimens was measured in at least one hour and twenty four hours after gluing, respectively. On the other hand, the setting time of the epoxy resin was variable depending on the ratio of resin and hardener in the epoxy. For an epoxy composition of 50 percent resin and 50 percent hardener the 23 setting time was four hours and for other epoxy compositions of resin and hardener (20%:80%, 35%:65%, 65%:35%, and 80%:20%) the setting time was one to two days. Therefore, the mass was measured in at least four hours for the glass slide/glue composite specimens made by epoxy resin of 50 percent resin and 50 percent hardener and in at least two days for the glass slide/glue composite specimens made by epoxy resin of the other compositions. 2.3. The Measurement Of Thickness A micrometer (Model No. M115-25, MITUTOYO, made in Japan) was used to measure the thickness of individual glass slides, adhesives and glass slide/glue composite specimens. The micrometer can measure lengths to within $0.001 mm. First, the position(s) of a point(s) where a glue spot(s) was to be made was determined by eye and marked on a pair of glass slides using a dark-colored permanent marker having a fine point (Figure 4). Schema of the glue spot (marked points) are shown in Figure 6. After marking the point(s), the thickness of individual slides was measured at the marked point(s). For glass slide/glue composite specimens measured at two or more points, the thickness was averaged. Within one hour after measuring the mass of glass slide/glue composite specimens (Section 2.2.) the thickness of glass slide/glue composite specimens was measured at the marked points and averaged. As a result, the precise thickness of glue could be determined from the difference between the thickness of glass slide/glue composite specimen and the summed thickness of two glass slides. 24 F +1 :1/2 w Fl/ZL 1/4LI 1/4L (a) (b) 1/2 w; 0 O I<—>l la—H H—vl 1/6 L 1/6 L 1/6 L Figure 6. Relative position(s) of glue spot(s): (a) one glue spot, (b) two glue spots, (c) three glue spots, (d) five glue spots, where L-length of specimen and W-width of specimen. 2S 2.4. Applying Adhesives and Preparing Glass Slide/Glue Composite Specimens 2.4.1. Applying 'Super Glue' Effects of the glue area fraction, the number of glue spots and the thickness of glue on the Young's modulus were investigated using super glue. For the investigation, glass slide/super glue composite specimens with different glue area fraction, number of glue spots and thickness of glue were made by applying one, two, three or five glue spots on one slide (Figure 6). Within a few seconds after glue spots were made, the second slide was placed upon the first slide where the super glue had been applied, resulting in a glass slide/super glue composite specimen (Figure 4). Within a few seconds after gluing, some pressure (about 20 Newtons to 118 Newtons) was applied on the glass slide/super glue composite specimen using the investigator's two thumbs. As the mass of glue and the pressure applied to the slide increased, the larger the area fraction of glue was and the thinner the glue was after bonding. Therefore, with each glue application, different mass of glue and pressure were used to obtain glass slide/super glue composite specimens with various area fraction of glue between 0 percent and 100 percent and with glue bond thicknesses within one of the three following thickness ranges, 0.010 mm - 0.012 mm, 0.019 mm - 0.021 mm and 0.028 mm - 0.030 mm. One hundred and forty six glass slide/super glue composite specimens were made by this procedure but only sixty seven glass slide/super glue composite specimens had a glue thickness that was within one of the above thickness ranges. An additional twenty one 26 glass slide/super glue composite specimens with the desired area fraction of glue were selected to investigate the effect of adhesion area even though the thicknesses of the glass slide/super glue composite specimens were not in the above three thickness ranges. The total of eighty eight glass slide/super glue composite specimens used in this study are listed in Table 1. 2.4.2. Applying 'Epoxy Cement’ The epoxy cement was composed of two components, resin and hardener, which were contained in two separate tubes. To mix the epoxy, ribbons of the resin and the hardener of approximately the same length were squeezed onto a 7.65 cm x 7.65 cm piece of standard notepad paper (Figure 7). The resin and hardener were mixed using a small rod. Epoxy glue with the composition of approximately 50 percent resin and 50 percent hardener was made. In order to investigate only the effects of area fraction and number of glue spots, one or three glue spots were applied on one slide. Within a few seconds the second slide was put on the slide where the glue was applied. Pressure was applied to the glass slide/epoxy cement composite specimen in the same manner as for the super glue. The twenty glass slide/epoxy cement composite specimens made by this technique are listed in Table 2. 27 Table 1. Classification of the glass slide/super glue composite specimens according to the number of glue spots and the range of glue thickness. NUMBER OF GLUE SPOTS RANGE OF GLUE NUMBER OF SPECIMENS THICKNESS (mm) (TOTAL 88) One glue spot 0.010 - 0.012 7 0.019 - 0.021 11 0.028 - 0.031 6 others 4 Two glue spots 0.010 - 0.012 3 0.019 - 0.021 8 0.028 - 0.030 0 others 2 Three glue spots 0.010 - 0.012 12 0.019 - 0.021 6 0.028 - 0.030 others 6 Five glue spots 0.010 - 0.012 1 0.019 - 0.021 0 0.028 - 0.030 0 others 4 * 0.010 - 0.012 1 0.019 - 0.021 2 0.028 - 0.030 0 others 5 * Specimens having adhesion area greater than 90% which could not be classified according to the number of glue spots. 28 Figure 7. Photograph of epoxy resin (i.e. "Quick Setting Epoxy Adhesive"), a 7.65 cm x 7.65 cm piece of standard notepad paper and a stick for mixing and applying. 29 Table 2. Classification of the glass slide/epoxy cement composite specimens according to the number of glue spots. NUMBER OF GLUE SPOTS NUMBER OF SPECIMENS (TOTAL 20) One glue spot 10 Three glue spots 10 30 2.4.3. Applying 'Epoxy Resin' In order to investigate the effect of adhesive composition as well as the effects of the area fraction and the thickness of the adhesive, adhesives of various compositions of resin and hardener were made. First, using a balance (Serial No. 2155, E.H.Sargent & Co., Mettler instrument corp., Hightstown N.J.) with the accuracy of 0.0001 gram, the mass was measured for the small piece of paper onto which the resin and the hardener was to be squeezed for mixing. Then, the resin was squeezed onto the paper and the mass was remeasured. In order to obtain five different compositions of resin and hardener (20%:80%, 35%:65%, 50%:50%, 65%:35% and 80%:20%), the mass of the hardener was calculated and the required mass was squeezed on the same piece of paper using the balance. The resin and hardener was mixed on the paper using a small stick so that the desired compositions of adhesive was obtained. The resin, hardener, the notepad paper and stick used for mixing in this study are shown in Figure 7. Within ten minutes after mixing the resin and the hardener, three glue spots were made on one slide using a stick. The positions of the glue spots were selected in the same manner as in super glue. Then, the second slide was put on the slide where the glue was applied, resulting in a glass slide/epoxy resin composite specimen (Figure 4). The selected thickness ranges of the epoxy adhesive were 0.025 - 0.075 mm, 0.125 - 0.175 mm and 0.225 - 0.275 mm. Pressure applied by the investigator's thumbs was used to obtain desired thickness and the various area fractions of adhesive. One hundred and ten glass slide/epoxy resin composite specimens were made with three glue spots or 100 percent coverage of glue. However, four glass slide/epoxy resin composite specimens resulted in misalignments between 31 the two slides after gluing (That is, the "top" and "bottom" slides were not coincident). The misaligned glass slide/epoxy resin composite specimens were excluded from the investigation. In addition, ten glass slide/epoxy resin composite specimens glued with the adhesive composition of 20 percent resin and 80 percent hardener were excluded because the fundamental flexural frequencies had amplitudes that were too low and peaks that were so broad that accurate modulus measurements could not be made. Data for a total of ninety six glass slide/epoxy resin composite specimens was included in this investigation (Tables 3 and 4). 2.5. The Measurement Of Area Fraction Of Glue Two techniques were used for the measurement of the glass slide/glue composite specimen's glue area fraction. The selection of the measurement technique depended on the shape of glued area in each glass slide/glue composite specimen. 2.5.1. Template Method For Small Circle-Shaped Glue Spots When small amounts of glue were applied and two glass slides were glued together for smaller than about sixty five percent area of glue (in glass slide/glue composite specimens having three glue spots), the glue spots tended to spread in circle, regardless of what type of glue was used. For the circular glue spots a template (Rum Sung, No 001, pencil allowance 0.5 mm, made in Korea) having forty one circles of 32 Table 3. Classification of the glass slide/epoxy resin composite specimens according to the composition of the glue and the range of glue thickness. COMPOSITION OF GLUE RANGE OF GLUE NUMBER OF SPECIMENS (RESIN : HARDENER) THICKNESS (mm) (Total 70) 50% : 50% 0.025 - 0.075 16 50% : 50% 0.125 - 0.175 13 50% : 50% 0.225 - 0.275 14 50% : 50% others 27 Table 4. Classification of the glass slide/epoxy resin composite specimens with 100 % adhesion area. These specimens were included in the experimental comparison of the ROM and the Dynamic Beam Vibration models (Section 3.3.5). COMPOSITION OF GLUE AREA FRACTION OF NUMBER OF SPECIMENS (RESIN : HARDENER) GLUE (8) (TOTAL 39) 20% : 80% 100 0 35% : 65% 100 8 50% : 50% 100 13 65% : 35% 100 10 80% : 20% 100 8 33 various sizes from 1.5 mm to 35 mm was used to measure the glue area fraction. The size increment between adjacent circles on the template was either 0.5 mm or 1 mm (Figure 8). The template was placed over each of the glass slide/glue composite specimen's glue spots. The size of each glue spot was determined from the template circle that most closely matched the glue spot diameter. As a result, a somewhat accurate glue area fraction could be calculated over the area of glass slide/glue composite specimen (See Appendix A). 2.5.2. Grid Counting Method For Irregularly-Shaped Glue Spots For heavier glue masses and larger glue area fractions, the glued area tended to be irregular which meant that the template method mentioned in section 2.5.1. could no longer be applied. Therefore, a grid counting method which employed translucent grid paper ruled into 1/8” squares was used to measure glue area fractions (Figure 9). First, a line was traced along the edge of glued area on a glass slide/glue composite specimen itself with dark-colored pen. Then, the glass slide/glue composite specimen was put under the paper mentioned above. The traced line was re-drawn on the translucent grid paper in order to count the number of squares within the glued area. In counting the number of squares, the squares that the trace line intersected were counted as half-squares. As a result, the glue area fraction could be calculated from the ratio of the number of grid squares within glued area to total number of grid squares covered by the glass slide/glue composite specimen surface. The surface area of a glass slide/glue composite specimen 7.62 cm x 2.54 cm (3 inches by 1 inch) is equivalent 34 PENCIL ALLOWANCE 0. 5” mm Figure 8. Template used for measurement of the area fraction of the smaller circular glue spots on the composite specimens. The diameter in millimeters is marked below each circle. (The measuring unit of each circle is millimeter.) 35 Figure 9. Grid paper used for measurement of the area fraction of irregularly shaped larger glue spots on the composite specimens. 36 to 192 grid squares. 2.6. The Measurement Of Elastic Modulus In this study, the sonic resonance technique was used to determine the elastic moduli of single slides, glass slide/glue composite specimens and epoxy resin specimens [11]. The apparatus included a rectangular specimen suspended horizontally from a driver and a pick-up transducers by cotton threads (shown in Fig 10 and schematically in Figure 11). An electrical signal was generated by the frequency synthesizer (3325A Synthesizer/Function Generator, Hewlett-Packard). The signal was transmitted to a piezoelectric driver transducer (Model No. 62-1, Astatic Corp., Conneaut, Ohio) and converted to mechanical movement. The specimen then vibrated the suspending cotton thread. The resulting mechanical movement generated in the specimen was transmitted to the pick-up transducer through the cotton threads. The pick-up transducer converted the mechanical movement to an electrical signal. The electrical signal was then filtered and amplified (4302 Dual 24DB/Octave Filter-Amplifier made by Ithaco, Ithaca, N.Y.). The filtered and amplified signal was then fed into an oscilloscope (V-llOOA, 100MHz Oscilloscope made by Hitachi, Japan), a voltmeter (8050A Digital Multimeter made by Fluke, Everett, WA.) and a counter (5314A, Universal Counter made by Hewlett-Packard). Mechanical resonant frequencies can be found by monitoring the digital voltmeter and oscilloscope while changing the frequency of an electrical signal. In order to determine the elastic modulus of 37 Figure 10. Photograph of the sonic resonance apparatus. (A glass slide/glue specimen is suspended in the air) 38 Frequency Universal Synthesizer Oscilloscope Voltmeter Counter Trigger Filter fs‘ Amplifier Driver ick—up Transducer Transducer Cotton Cotton Thread Specimen Thread / / L Figure 11. A block diagram of the sonic resonance apparatus [15]. 39 specimen, one must identify the vibrational modes of the specimen (12, 13]. The resonant mode identification was performed using a steel wire to probe for the locations of nodes and antinodes. Nodes have no vertical displacement and antinodes have a maximum displacement. The location of nodes and antinodes are unique to a specific vibrational mode (Table 5 [12]), thus once the position of the nodes and antinodes are known, then the vibrational mode has been uniquely identified. When a steel wire is placed upon a nodal position, the amplitude of the resonant frequency changes little since there is no vertical displacement at the node. When the wire is placed away from the nodal position, the amplitude decreases because the wire suppresses the mechanical vibration. In this study, the amplitude of a vibrational frequency was measured while the damping wire was moved in 2 mm increments from the left edge of specimen to the right edge. The locations of nodes and antinodes at selected resonant frequencies were determined by plotting amplitude versus relative position of wire. Comparing the node and antinode location information (found via the steel probe wire) with Table 5, the vibrational mode of each resonant frequency was identified. The fundamental flexural frequencies and torsional frequencies of specimens thus were determined at room temperature in air. The elastic moduli and shear moduli could be calculated from these resonance frequencies by using the following equations [14-16]. Elastic modulus, E, of a rectangular specimen is given by [14-16] 40 Table 5. For the sonic resonance method, the relative position of nodes according to the vibrational mode. L is the length of the rectangular specimen. MODES OF VIBRATION FLEXURAL MODES TORSIONAL MODES Fundamental 0.224 L 0.500 L 0.776 L First Overtone or 0.132 L 0.250 L Harmonic* 0.500 L 0.750 L 0.868 L * First overtone refers to the flexural vibration, while the first harmonic refers to the torsional vibration. 41 2 E = 0.94642 L‘ p SM r,“ DZ (27) _ 0.94642 L‘ m 5,,“ pi, 03w where L = length of the specimen p a mass density of the material S":x = the shape factor for flexural vibration of prismatic bars Fflex = fundamental flexural resonant frequency D = cross sectional dimension in the direction of vibration m = mass of the specimen W = width of the specimen. The shape factor, Sm“, for the rectangular specimen is the function of the specimen dimensions and Poisson's ratio given by [14-16] s = 1 + 6.585(1 + 0.07521: + 0.8109v2)(2)2 - 0.868(2)‘ "“ L L 8.34(1 + 0.202311 + 2.173v2)(.‘£)‘ (23) 1 + 6.338(1 + 0.14081v + 1.53112)(2)2 L where v = Poisson's ratio. The shear modulus, G, of a rectangular specimen is given by [14-16] 42 F G = 411295.473"): (29) where L = length of the specimen p = mass density of the material SW, = the shape factor for torsional vibration of prismatic bars FW, = torsional resonant frequency N’ 2 an integer (which is unity for the fundamental mode). The shape factor, Sm”, for the rectangular specimen is given by [14-16] [1+ (%)2) [1+0.0085N2(%’)2] s = -0.06(1L..W)‘-5(‘T‘Z-l)2 tars 3o 4-2.521(£)1- 1'991 ( ) exp(lt’—’)+1 where W = width of the specimen t = thickness of the specimen. 2.7. Making Epoxy Resin Specimens To investigate the effect of glue composition on the measured elastic moduli and to compare the modulus results to the Rule of Mixtures and to the Dynamic Beam Vibration models, epoxy resin specimens were fabricated using five different epoxy glue compositions including the ratios of resin and hardener of 20%:80%, 35%:65%, 50%:50%, 65%:35% and 80%:20%. 43 A rectangular plastic box of dimension 94.5 mm x 27.4 mm, height 13.5 mm, and wall thickness 1.8 mm was used as a mold for the epoxy resin specimens. The mass of the plastic-box mold was measured, then the desired quantity of resin was squeezed into the mold and the total mass was measured. From the net mass of resin, the required mass of hardener for the desired composition was calculated and squeezed into the same mold. Then, the resin and hardener were mixed to a homogeneous state in the box, using a small stick. Within 3 minutes after mixing, the mold was placed in a vacuum chamber pumped by a roughing pump (Welch Duo-Seal Vacuum Pump, Model No. 1402, Serial No. 121054, Sargent-Welch Scientific Co., 7300 North Linder Avenue, Skokie, IL, 60076). The vacuum helped to remove pores which appeared during the mixing of the resin and the hardener. After pumping for about twenty seconds, the mold was removed of the vacuum chamber to check the existence of large pores. In case of there were still many large pores in the epoxy, the mold was placed in the chamber again. The vacuum chamber was then pumped down one or two more times until the large pores were removed. The setting time was different from specimen to specimen. The molds were removed from the five epoxy resin specimens when the glue was no longer sticky. After about three weeks the molds of three out of the five epoxy resin specimens having epoxy resin composition of the ratios of resin and hardener of 50%:50%, 65%:35%, and 80%:20% were removed by grinding. Two epoxy resin specimens with the epoxy resin compositions of 20%:80% and 35%:65% still were not set even after two months. Thus only three epoxy resin specimens having the ratios of resin and hardener of 50%:50%, 65%:35%, 80%:20% were used as a part of the comparison of ROM and Dynamic Beam Vibration models. 44 The three final epoxy resin specimens are shown in Figure 12. The epoxy resin resulted from mixing of white resin and yellow hardener, so that we can see from Figure 12 that the more content of the resin was included in a epoxy resin specimen, the whiter the color of the epoxy resin specimen was. 45 Figure 12. Photograph of epoxy resin specimens. The ratios of resin and hardener used for each specimen are 50%:50%, 65%:35% and 80%:20%, respectively. 46 3. RESULTS AND DISCUSSION For the effect of glue area fraction or glue bond thickness on elastic modulus, three candidate equations were used, 5' = Em [1 — c,xC‘] (31) E = 5100 [1 ' C3 Exp(C‘X)] (32) E = c5 [1 - Céxi] (33) where E = Elastic modulus of glass slide/glue composite specimen Ema= average elastic modulus of glass slide/glue composite specimen of 100 percent adhesion area (Appendix B) 0,, C2, C3, C,, C, and C6 = constants X = l - Area fraction of glue. For the super glue and the epoxy cement, the glass slide/glue composite specimens having 100 percent adhesion area could not be fabricated because three to eight irregularly-shaped pores with sizes of about 0.5 mm up to 8 mm were always included within the glue bond layer. Thus, the elastic moduli of the glass slide/glue composite specimens having greater adhesion area than 90% were averaged to obtain Em, (Appendix B). The three equations 31-33 were used to analyze our experimental results. As will be discussed in the following sections, equations 31 47 and 32 gave the best descriptions of the effect of adhesion area on the observed Young's modulus. Equation 33 gave the best description of the effect of glue bond thickness on the effective Young's modulus. The success of equation 33 in describing the glue bond thickness effect may be related to the fact that the modulus at x=0 (where x is l-A and A=glue bond area) is given by C5, where C, is a fitted parameter. In contrast, the value of modulus for x=0 in equation 31 and 32 was Em" which was average elastic modulus of glass slide/glue composite specimen of 100 percent adhesion area based on the experimentally obtained elastic modulus at 100% adhesion area (Appendix B). 3.1. Effects of Adhesion Area, Number of Glue Spots, and Glue Bond Thickness 3.1.1. The Effects of Super Glue Adhesion on Elastic Modulus The effects of adhesion area, number of glue spots and glue bond thickness on the Young’s modulus of glass slide/super glue composites were studied using super glue. Figures 13-15 illustrate the Young's modulus as a function of adhesion area according to the number of glue spots. Without regard to the number of glue spots, Young's modulus of adhered glass slides decreased continuously with decreasing the adhesion area. The Young's modulus for glass slide/super glue composite specimens having one glue spot changed as a function of total adhered area in a different manner from that for glass slide/super glue composite specimens having two, three and five glue spots. For two or more glue 48 .-_. a-.. .-. m ..... .e b. ‘- ‘1 ‘1 Q. .~- Q.‘. ”a? 93 60- ..... g 50- :: i Q 40.- 0 ONE GLUE SPOT _ g . 0 TWO GLUE SPOTS o h- m 30- 1 THREE GLUE SPOTS O S .. c: - 11 FIVE GLUE SPOTS O ‘9‘“. O o - :3 20- - --------- ‘cl=o.7a7a c2=2.0959 r=0.9718 o - E- . -. -------- ‘ C1=0.3585 c2=4.9727 r=0.9632 10'- '''''' c1=0.4099 C2=5.0131 r=0.9693 _ " / c1=0.3323 c2=5.0955 r=0.9884 ' O T l I l fi I I I I I f I I l I I I I I 100 90 80 70 60 50 40 30 20 10 O ADHESION AREA (7.) Figure 13. For a super glue bond layer, the effect of the number of super glue spots on Young's modulus as a function of adhesion area(%). The curves represent a least-squares best fit to equation 31. 49 80 I l l l l l l l 1 l l I l 1 L4 1 l l l | Super Glue- A 70 _.-.-.-IT'““"'9"‘“ 01162» ~ - - _. c0 - .............................. p. ‘ " . $ 60- """"""""""" V q S 50- S . Q 40- 0 ONE GLUE SPOT g . a Two GLUE SPOTS o 1. m 30- 1 THREE GLUE SPOTS O — - 80 6)‘. 0 L23 - x FIVE GLUE SPOTS 8211 o . :3 201 - ---------- C3=0.0394 c4=3.0599 r=0.9418 ‘93 - E- - .—-—----~° C3=0.0009 C4=6.0983 r=0.9698 10- ...... C3=0.0012 C4=5.8489 r=0.9’737 f ‘ / 03:0.0010 C4=5.8050 r=0.98'79 O I I fi I T l I I I I I l fir I I r I r I 100 90 80 70 60 50 40 30 20 10 O ADHESION AREA (70) Figure 14. For a super glue bond layer, the effect of the number of super glue spots on Young's modulus as a function of adhesion area(%). The curves represent a least-squares best fit to equation 32. 50 3C¥1 4 20- 4 10-q q YOUNG'S MODULUS (GPa) A C) 1 O a Two GLUE SPOTS o n- . THREE GLUE SPOTS 0 e P x FIVE GLUE SPOTS O ‘99.; o - ......... (35:68.3463 C6=O.8497 r=0.9494 ‘93 - - C5=7O.5256 C6=O.267O r=0.9122 ’ '—-‘ C5=7O.6358 C6=O.3697 r=0.9510 "- -' / C5=7O.0561 C6=O.2612 r=0.9792 1 J l L, Super Glue . ONE GLUE SPOT _ O 100 Figure 15. I I'I'I'I'I'T'l'l'jj 90 80 7O 60 50 4O 30 20 10 O ADHESION AREA (‘75) For a super glue bond layer, the effect of the number of super glue spots on Young's modulus as a function of adhesion area(t). The curves represent a least-squares best fit to equation 33. - 51 spots, the three curves nearly superimpose (Figures 13-15). For the glass slide/super glue composite specimens having two, three, and five glue spots, the Young's modulus versus area fraction of three curves decreased relatively rapidly for area fractions less than about 30 to 35 percent (Figures 13-15). Due to similar trends for the Young's modulus data for the glass slide/super glue composite specimens having two, three and five glue spots were plotted using the same symbol and analyzed using a least squares best fit program (Figures 16-18). To investigate the effect of glue bond thickness, the data for glass slide/super glue composite specimens having one glue spot were plotted separately according to a pre-selected range of glue bond thickness (Figures 19-21). The analysis and plotting of the data for the glass slide/super glue composite specimens having two and three glue spots were performed in the same manner as for the data of the glass slide/super glue composite specimens having one glue spot (Figures 22-24). However, the effect of the super glue bond thickness on the measured elastic modulus of the glass slide/super glue composite specimens is not pronounced. The weak dependence of Young's modulus upon the thickness of the super glue bond may result from the relative thinness of the super glue bonds compared to the glass slide thickness (See Table l for the thickness of the super glue bonds and Table 3 for the thickness of the epoxy bonds). 52 6C¥e 30-! 20- YOUNG'S MODULUS (GPa) .p C) l 10-4 O l 1 l . 8C)! 1 1 Super Glu r 70 4* in #- r' 5* TWO, THREE AND FIVE GLUE SPOTS _' /’ C1=O.4102 C2=5.7098 r=0.9575 _ 100 Figure 16. I I I l I I I I I l I I I I I l I l I 90 80 7O 60 5O 4O 30 20 10 O ADHESION AREA (‘76) For a super glue bond layer, the effect of the super glue on Young's modulus as a function of adhesion area(t) with the data for specimens having 2, 3 and 5 glue spots lumped together. The curve represents a least-squares best fit to equation 31. 53 80 1 l 1 l 1 l 1 l 1 L 1 l 1 L 1 l 1 l 1 . Super Glu . 504 YOUNG'S MODULUS (GPa) .p O l .4 31!- 30- _ 20— - 10" x TWO. THREE AND FIVE GLUE SPOTS *- ‘ / C3=O.OOOS C4=6.5221 r=O.9649 ' O I I I l I l I I I I r l I j I f f l I 100 90 80 70 60 50 4O 30 20 10 O ADHESION AREA (70) Figure 17. For a super glue bond layer, the effect of the super glue on Young's modulus as a function of adhesion area(t) with the data for specimens having 2, 3 and 5 glue spots lumped together. The curve represents a least-squares best fit to equation 32. S4 l 1 I 1 YOUNG'S MODULUS (GPa) Super Glu _. *- L 9* TWO, THREE AND FIVE GLUE SPOTS - / C5=71.2766 C6=O.3402 r=0.9017 " 100 Figure 18. I I'I'l‘l'lfil'lfif'l' 90 80 7O 60 50 4O 30 20 10 O ADHESION AREA (70) For a super glue bond layer, the effect of the super glue on Young’s modulus as a function of adhesion area(%) with the data for specimens having 2, 3, S glue spots lumped together. The curve represents a least-squares best fit to equation 33. SS l 1 l 1 l 1 l 1 l 1 l 1 1 SJ 1 1 l 1 YOUNG'S MODULUS (GPa) #- C) l 104 / E100=7O.05 C1=O.7656 C2=1.6592 r=0.9‘908\\ ' -’ -------- (3100:6993 C1=1.2843 C2=3.8884 r=O9939 ’ Super Glu . ‘~ “ ‘ ...... \ \ “ 3* 0.010 —— 0.012 mm A 0.019 — 0.021 mm 0 0.028 - 0.030 mm -——— [9100:7020 (31:0.8'773 C2=2.2453 r=O.9’263 ‘"' 100 Figure 19. I I'I'IFI‘F'I'I‘ITI' 9080706050403020100 ADHESION AREA (70) For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(%) for specimens having one super glue spot. The curves represent a least-squares best fit to equation 31. 56 8C) 1 l 1 1 1, l 1 l 1 l 1 l 1 l 1 l 1 l 1 . Super Glu . I— 13‘ . 0- .. 8 00 D _ —J . :3 Q — o E . (I) 30— *< 0.010 —- 0.012 mm p - - 1 ‘\\A g - A 0.019 - 0.021 mm ' :2 20‘ o 0.028 - 0.030 mm - o . _ _ _ K, \ . > / Elmo—70.05 C3—O.0553 C4—2.6836 r:0.9610\\ 10'- ————— E100=7O.20 C3=0.0365 C4=3.2970 r='-O.9511\ " ‘ - ---------- E100=69.93 C3=O.0024 c4=7.0025 r=q.9810 O I I I I I T I l I I I l I l I l I I I 100 90 so 70 60 50 4O 30 20 10 O ADHESION AREA (70) Figure 20. For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(t) for specimens having one super glue spot. The curves represent a least-squares best fit to equation 32. 57 1 l 1 l 1 l 1 J 1 SJ 1 l 1 l 1 I 1EEJ 1 —1----——--- ‘~‘ ~~ ‘ ‘ Super Glue . "‘. -- ‘ ‘s YOUNG'S MODULUS (GPa) A C) l ‘1 ‘5‘ \\ ‘ 30- * 0.010 —- 0.012 mm r- d 1 0.019 — 0.021 mm L 207 O 0.028 — 0.030 mm — ‘ / C5=67.0114 C6=O.8186 r=0.9548 P 10" """" C5=68.3317 C6=O.9764 r=0.9592 - ‘ - --------- C5=71.1264 C6=0.9381 r=0.9920 O I I I T I I I I I I I I I I I I I I I 100 90 80 '70 60 50 4O 30 20 10 O ADHESION AREA (70) Figure 21. For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(%) for specimens having one super glue spot. The curves represent a least-squares best fit to equation 33. 58 l l l i l l l l l l l l l l 1 l l J l SCFH YOUNG'S MODULUS (GPa) 18> C) 1 Super Glu . A 0.010 — 0.012 mm - 0.019 - 0.021 mm ' """" E100=7O.05 C1=O.3945 C2=4.’7711 r=0.9833 "' / E100=7O.20 C1=O.3343 C2=4.2232 r=0.9776 Figure 22. I r I I I I I I ‘I I fl I I I I I I I I 90 80 7O 60 50 4O 30 20 10 O ADHESION AREA (7.) For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(%) for specimens having two and three super glue spots. The curves represent a least-squares best fit to equation 31. 59 L 1 l I l J L l l l l l l l l l I 1 YOUNG'S MODULUS (GPa) .p O 1 10d Super Glu . A 0.010 — 0.012 mm - 3" 0.019 — 0.021 mm ' """" E100=7O.05 C3=0.0016 C4=5.5348 r=0.9857 '— / EIOO=7O.20 C3=O.OOZO €425.1706 r=0.9816 O 100 Figure 23. I I I I I I I j I I 17 I I I I I I I f 90 80 7O 60 50 4O 30 20 10 O ADHESION AREA (70) For a super glue bond layer, the effect of glue bond thickness of super glue on Young's modulus as a function of adhesion area(%) for specimens having two and three super glue spots. The curves represent a least-squares best fit to equation 32. 60 80 1 l 1 l 1 l 1 l 1 l 1 l 1 L411 1 1 l 1 . Super Glue . 7:? Cl. 8 U) :3 ._‘1 1:) C3 0 E ‘ . 00 130-‘ F {3 .J _. 2: :3 20% A 0.010 — 0.012 mm - S I we 0.019 - 0.021 mm - 101 ————— C5=71.4643 C6=O.3613 r=0.9554 - ‘ / C5=7O.9941 C6=O.2996 r=0.9482 b O I I I I I I I I I I I I I I I I I I I 100 90 80 7O 6O 50 4O 30 20 10 O ADHESION AREA (70) Figure 24. For a super glue bond layer, the effect of glue bond thickness of super glue on Young’s modulus as a function of adhesion area(%) for specimens having two and three super glue spots. The curves represent a least-squares best fit to equation 33. 61 3.1.2. The Effects of Epoxy Cement Adhesion on Elastic Modulus The effect of area fraction of the glue on elastic modulus was investigated using epoxy cement. Data from 10 glass slide/epoxy cement composite specimens having one glue spot and 10 glass slide/epoxy cement composite specimens having three glue spots show that as the adhesion area decreases, the elastic modulus decreases (Figures 25-27). A similar experimental result was obtained for the glass slide/super glue composite specimens (Section 3.1.1.). The elastic moduli of the glass slide/epoxy cement composite specimens having one glue spot or three glue spots changed as a function of adhesion area in a manner very similar to that of the super glue. As was the case with the super-glue bonded specimens, the data for the epoxy-bonded specimens was fit to equations 31-33. 3.2. The Effects of Epoxy Resin Adhesion on Elastic “odulus The effects of adhesion area, glue bond thickness and composition of epoxy resin were investigated. In addition, the ROM and the Dynamic Beam Vibration models were compared using the glass slide/epoxy resin composite specimens having a 100 percent adhesion area. 3.2.1. The Effect of Adhesion Area on Elastic lbdulus Without considering the effect of glue bond thickness, the Young's moduli of seventy glass slide/epoxy resin composite specimens with the 62 l I I 1 l L L l J 1 80 Epoxy Cement . 7:? 0.. 8 U) :3 .4 :3 Q Q 2 ([3 30-1 O\\\\\ _ Z O ‘9‘ :3 20- 0 ONE GLUE SPOT onc" E3 4 x THREE GLUE SPOTS 104 ————— C1=O.8000 c2=1.4544 r=0.9516 - ' / C1=O.4205 C2=4.5380 r=O.9661 ' O I I I T I I I I I I I I I I fi I I I I 100 90 80 7O 60 50 4O 30 20 10 O ADHESION AREA (70) Figure 25. For an epoxy cement bond layer, the effect of epoxy cement on Young's modulus as a function of adhesion area(t) for specimens having one and three glue spots. The curves represent a least-squares best fit to equation 31. 63 l 1 L47] YOUNG'S MODULUS (GPa) 0 ONE GLUE SPOT ‘1 oo- 1. THREE GLUE SPOTS '''''' C3=O.0854 C4=2.284O r=0.8895 - / C3=O.0022 c4=5.2595 r=0.9773 ' Epoxy Cement, 100 Figure 26. I l'l'l'l'f‘l'l'lTl' 90 80 7O 60 50 4O 30 20 10 O ADHESION AREA (70) For an epoxy cement bond layer, the effect of epoxy cement on Young's modulus as a function of adhesion area(t) for specimens having one and three glue spots. The curves represent a least-squares best fit to equation 32. 64 / Ia-r\\ vJ—h —_ .F—wa). 7.....1~Z\~AV\W Fig1 L 1 l 1 I 1 l 1 80 4 . Epoxy Cement. 70- }? q E; (N)- ? 50- 4 d :3 C3 140-4 C) E .1 a) 130-4 :2 . :3 20" 0 ONE GLUE SPOT ‘9 oc— 8 - 1. THREE GLUE SPOTS - 10‘ ————— C5=62.3218 C6=O.8554 r=0.8537 ‘~- q / C5=71.1766 C6=O.3830 r=O.9335 ' O I I I I I I I I I I I I I I I I I I I 100 90 80 7O 60 50 4O 30 20 10 O ADHESION AREA (7.) Figure 27. For an epoxy cement bond layer, the effect of epoxy cement on Young's modulus as a function of adhesion area(%) for specimens having one and three glue spots. The curves represent a least-squares best fit to equation 33. 65 composition of 50 percent resin and 50 percent hardener were plotted with respect to adhesion area as shown in Figures 28-30. Using a least squares best-fit procedure, the data was fit to equations 31-33 (Figure 28-30). As in the results for super glue and epoxy cement, the Young's modulus of glass slide/epoxy resin composite specimens decreased with the decrease of adhesion area. 3.2.2. The Effect of Glue Bond Thickness on Elastic Modulus The effect of glue bond thickness on Young's modulus are apparent through two different types of data classification: (1) fixed glue composition but varying adhered areas and (2) fixed adhesion area but varying glue composition. 3.2.2.1. Glass Slide/Epoxy Resin Composite Specimens having a Fixed Glue Composition but Different Adhesion Areas Seventy glass slide/epoxy resin composite specimens out of a total of ninety six were made with a fixed epoxy composition of 50 percent resin and 50 percent hardener, but having differing adhesion area fractions. Using a least-squares best-fit program the Young's modulus versus area fraction of glue data was fit to equations 31-33. When the data is sorted into three glue-bond thickness ranges (0.225 - 0.275 mm, 0.125 - 0.175 mm, and 0.025 - 0.075 mm), the Young's modulus for each thickness range decreases as a function of area 66 80' l l l l l l l l 1 l l l L l l l l J I Epoxy Res1n_ 7o ’8 at % 60-1 8 50- g 4 Q _ o 40 2 q 03 130-1 '- L29 - . :3 120- - o . >_, F' IOJ x THREE GLUE SPOTS r ‘ / C1=O.4610 C2=3.8187 r=0.9342 ' O I T I l I I I r I I I I I r I I I I I 100 90 so 70 60 50 4o 30 20 1o 0 ADHESION AREA (‘75) Figure 28. For an epoxy resin bond layer, the effect of epoxy resin on Young's modulus as a function of adhesion area(%). The curve represents a least-squares best fit to equation 31. 67 l l 80 1 L l l l I l l 1 I l l l l l L E q Epoxy ReSIn_ 7:? o. 8 U) :3 _J :3 c: o 2 S0 o . . g goJ .. o . . >1 10'- x THREE GLUE SPOTS - ‘ / C3=O.OO46 C4=4.6462 r=0.9405 ' O I I I I I I I I j I I I I I I l j I I 100 90 80 7o 60 50 4o 30 20 1o 0 ADHESION AREA (70) Figure 29. For an epoxy resin bond layer, the effect of epoxy resin on Young's modulus as a function of adhesion area(t). The curve represents a least-squares best fit to equation 32. 68 l 1 l 1 l 1 l 1 l 1 l 1 l 1 l l 1 Epoxy ‘Resm _ YOUNG'S MODULUS (GPa) .n C) l / C5=68.7016 C6=O.4260 r=0.9230 9“ THREE GLUE SPOTS — 100 Figure 30. I I I I I I I I I I I I I I I 1 I I I 90 80 '70 60 50 4O 3O 20 10 O ADHESION AREA (‘75) For an epoxy resin bond layer, the effect of epoxy resin on Young's modulus as a function of adhesion area(%). The curve represents a least-squares best fit to equation 33. 69 fraction of glue (Figures 31-33). In addition, for each thickness range, the elastic modulus decreases as the glue bond thickness increases. Qualitatively, the observed decrease in Young's modulus with an increase in glue bond thickness can be understood in terms of a sandwich "layer" model. The outer two layers of each glass slide/epoxy resin composite specimen are glass slides having a Young's modulus of approximately 70.53 i 0.32 GPa (average value calculated from 140 individual glass slides). In contrast, the elastic modulus of the epoxy layer is only 2.977 to 3.428 GPa (Table 6). Thus the considerably lower modulus of the epoxy indicates that as the relative thickness of the epoxy bond increases, the overall modulus of the glass slide/epoxy resin composite specimen should decrease. However, for the glass slide/epoxy resin composite specimens having a glue bond that cover less than 100 percent of bonded surface, the quantitative analysis of the effect of the epoxy bond thickness is not straightforward. 3.2.2.2. Glass Slide/Epoxy Resin Composite Specimens of Differing Glue Composition and Fixed Adhesion Areas Thirty nine specimens out of the total of ninety six epoxy bonded specimens were made with five different epoxy compositions but with 100 percent adhesion area (See Section 2.4.3.). The elastic modulus decreased with increasing epoxy bond thickness for each of the four different epoxy compositions (Figure 34). 70 4 l 1 l 1 l 1 1 L4,] 1 l 1 L 1 l 1 l Epoxy Resin, YOUNG'S MODULUS (GPa) 1.. 0.025 - 0.075 mm "— 0 0.125 - 0.175 mm - A 0.225 — 0.275 mm L- / E100=69.54 €120.4206 C2=4.2106 r=0.9836 - ————— E100=68.42 C1=O.4905 c2=4.2013 r=0.9821 " --------- ' E100=67.29 C1=O.5196 C2=3.6292 r=0.9839 100 Figure 31. I I I I I I T I I I I I I I I I I I I 90 80 7O 60 50 4O 30 20 10 O ADHESION AREA (70) For an epoxy resin bond layer, the effect of glue bond thickness of epoxy resin on Young's modulus as a function of adhesion area(%). The curves represent a least-squares best fit to equation 31. 71 1 l 1 l 1 1 1 l 1 l 1 l 1 l 1 L l 1 Epoxy Resin _ YOUNG'S MODULUS (GPa) 11 0.025 - 0.075 mm "“- 0 0.125 - 0.175 mm - A 0.225 — 0.275 mm - / E100=69.54 c3=o.0029 c4=5.0222 r=0.9895 - —————— E100=68.42 c3=o.0035 C4=4.9661 r=0.98'70 " -------- E100=6729 C3=O.0067 C4=4.3852 r=0.9851 * Figure 32. I I j I r I I I I I I I I I I I I I I 90 80 '70 60 5O 4O 30 20 10 O ADHESION AREA (70) For an epoxy resin bond layer, the effect of glue bond thickness of epoxy resin on Young's modulus as a function of adhesion area(%). The curves represent a least-squares best fit to equation 32. 72 l 1 l 1 l 1 l 1 l 1 l L l 1 l 1 l 1 l 80* Epoxy Resin, ”a? O. 8 00 :3 A :3 a o 2 U) 30—4 9* 0.025 - 0.075 mm P CZ? 1 0 0.125 - 0.175 mm P :3 20- A 0.225 — 0.275 mm P S. ‘ / C5=7O.7877 C6=O.3964 r=0.9656 " 10‘ ————— C5=68.7752 C6=O.4486 r=0.9687 ‘ - _-..-.—---«, (35:67‘2661 (36:0.4838 r=0.9746 " O I I I I I I I I I I I j I I Ifi I I I 100 90 80 70 60 50 4o 30 20 1o 0 ADHESION AREA (70) Figure 33. For an epoxy resin bond layer, the effect of glue bond thickness of epoxy resin on Young's modulus as a function of adhesion area(s). The curves represent a least-squares best fit to equation 33. 73 Table 6. Comparison of experimentally obtained elastic moduli, densities and Poisson's ratios with reference values. COMPOSITION ELASTIC DENS ITY POISSON ' S Reference (Resin : MODULUS (GPa) (gm/cm3) RATIO Hardener) 50% : 50% 2.977 1.133 0.27 This study 65% : 35% 3.174 1.136 0.29 This study 80% : 20% 3.428 1.164 0.31 This study * 2.7 - 4.1 ** 0.34 [17] *** 3.0 - 6.0 1.1 - 1.4 0.38 - 0.4 [18] * Specific composition not specified, material listed as "cured epoxy resins" ** Mass density not specified *** Specific composition not specified, material listed as "epoxy resins". 74 1 l l l l l l l l l l l l b “ YOUNG'S MODULUS (GPa) ~ Epox Resin H P u. ‘5. “~.. ‘~ _. ‘ s \ s \ § “ “ . § Q.. s 5‘ "~.. ResinzHardener=80120 ResinzHardener=65135 3* ResinzHardener=50250 *- 0 ResinzHardener=35265 - ------ A1=71.3894 A2=—27.6232 r=0.6428 _ - - A1=70.6693 A2=—26.6098 r=0.5820 ------ A1=69.8205 A2=‘25.1504 r=0.3258 — A1=71.0777 A2=-47.9523 1:0.7947 0.00 Figure 34. I I I I I I I I I I I I I 0.02 0.04 0.06 0.08 0.10 0.12 0.14 RELATIVE GLUE BOND THICKNESS For an epoxy resin bond layer, the effect of composition of epoxy resin on Young's modulus as a function of relative glue bond thickness. The curves represent a least-squares best fit to equation 34. 75 3.2.3. The Effect of Epoxy Resin Composition on Elastic Modulus For the investigation of the effect of epoxy resin composition on Young's modulus, thirty nine glass slide/epoxy resin composite specimens with varying compositions of resin and hardener were fabricated (Section 2.4.3.). The effect of epoxy resin composition on the elastic modulus was analyzed in terms of the linear relation ta II A, + Aztk (34) Elastic Modulus where E A, and A2 = constants tn relative thickness of the glue bond thickness of the glue bond/total thickness of the glass slide/glue composite specimen. The data for the various glue composition specimens were fit to equation 34 using a least-squares best-fit program. As the content of resin in the glue decreased from 80 percent to 35 percent, the elastic modulus decreased over the entire range of glue bond thickness except for 35 percent resin and 65 percent hardener composition (Figure 34). The effect of glue composition on Young's modulus can be reconfirmed directly from the data about the epoxy resin specimens shown in Table 6. 76 3.2.4. Experimentally obtained Elastic Moduli, Densities and Poisson's Ratios of Epoxy Resin Specimens The Young's moduli of epoxy resin specimens were determined using the sonic resonance technique. The densities and Poisson's ratios also were calculated. Table 6 compares the experimental mass density and elastic modulus data with reference values [17, 18]. As the content of resin in the glue increases, the elastic modulus increases (Table 6). Also, the experimental value of Young's moduli and densities obtained in this study are reasonable when compared to the values from two different references. 3.2.5. Comparison of Rule of Mixtures and Dynamic Beam Vibration models The Rule of Mixtures and the Dynamic Beam Vibration models were compared to data for glass slide/epoxy resin composite specimens having an 100 percent adhesion area. The glass slide/epoxy resin composite specimens with 100 percent adhesion area can be considered as a continuous glue bond layer between two glass slide layers. The elastic moduli of such glass slide/epoxy resin composite specimens were calculated using equation 17 for the ROM model and equation 25 for the Dynamic Beam Vibration model. The relative differences, 5“, between the experimentally determined Young’s modulus, ng, and the predicted Young's modulus, E, from the ROM and Dynamic beam vibration models were calculated as 77 51 = ._____ . (35) Plotting 6R as a function of tn, the relative thickness of the glue bond results in an approximately linear relationship for both the ROM and Dynamic beam vibration models (Figures 35-37). The 6R versus ta relations (which show opposite slopes for the ROM and Dynamic beam vibration models) was fit to the relationship E -l? - (___...) = Bl + 321,. (36) where Bl and B: are constants. The moduli calculated from equations 17 and 25 were compared with the experimentally obtained Young’s moduli using equation 35 and the moduli obtained from a least-squares fit to equation 36 (see Figures 35- 37 for 50%, 65%, 80% resin, respectively). Also, the calculated moduli were directly compared with the measured Young's moduli using the regression curves fit to the linear relationship given by (Figures 38-40 for 50%, 65%, and 80% resin, respectively) E = B3 + B,tR (37) Where E =- elastic modulus (GPa) B3 and B4 are constants tn: relative thickness. 78 1 111 1 l 4 l 1 0.08 0.06J 0.04- l 1 l 1 q 507. Resin:50% Hardener Epoxy Resin _ _____ ~. I I I r I I I I I I I 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Figure 35. RELATIVE GLUE BOND THICKNESS Comparison of ROM and Dynamic Beam Vibration models for epoxy bonds made from an initial composition of 50% resin and 50% hardener. The curves represent a least-squares best fit to equation 36. b—- Dynamic Beam Vibration model: B.--0.002S, 8280.5106, r=0.7446 and """ Rule of Mixtures model: B(=-0.0012, 323-004959, r-0.7603) 79 1 L 1 1 1 1 . 65% Resinz357o Hardener L 1 l 1 Epoxy Resin fi — O . O6 -‘ ‘ ”1‘46”- .— d a \"\ 1“ A. —0.08 I l I I I I 1 I I ' 1 ‘ 0.00 0.02 0.04 0.06 0.08 0.10 0.12 RELATIVE GLUE BOND THICKNESS Figure 36. Comparison of ROM and Dynamic Beam Vibration models for epoxy bonds made from an initial composition of 65% resin and 35% hardener. fit to equation 36. 8.80.0094, 8280.2590, r-0.3763 and """ Rule of Mixtures model: 80 8.80 . 0189 , 82I-O . 8209 , r80 . 9108) The curves represent a least-squares best b—— Dynamic Beam Vibration model: l 008 1 I n l 1 l 1 I 80% Resinz207o Hardener Epoxy Resin _ ’0.08 I I I I I I I I I I IA 0.00 0.02 0.04 0.06 0.08 0.10 0.12 RELATIVE GLUE BOND THICKNESS Figure 37. Comparison of ROM and Dynamic Beam Vibration models for epoxy bonds made from an initial composition of 80% resin and 20% hardener. The curves represent a least-squares best fit to equation 36. b—— Dynamic Beam Vibration model: B,--0.0122, B;-0.4210, r-0.6951 and """" Rule of Mixtures model: B.=-0.0112, B;--0.SB92, r80.8361) 81 YOUNG'S MODULUS (GPa) I | l l l l J l 1 I l '72 5070 Resin : 50% Hardener - e kfi L n O O ’70—.“'-. \\\\\\\\\ . . . . . —' ~~“‘}4i“‘ - ‘‘‘‘‘ l \‘A “““ ‘ 68- * ix~ * “““ 1~\\ A a "'X ‘ “\““~\“‘~‘ x \w 66-1 9*" ‘ .. ai‘n P 9" ROM model xxx} 64— A Experimental data ,. 3K ' Dynamic Modulus model 62J ---------- ROM model : 33:70.4715 B4=—67.0177 r=O.9773 F . ----- Experimental: B3=70.6193 B4=—34.2256 r=0.6851 _ / Dynamic : 83:70.5045 B4=—0.4327 r=0.0016 60 r l r l j l I l I l I 0.00 0.02 0.04 0.06 0.08 0.10 0.12 RELATIVE GLUE BOND THICKNESS Figure 38. Comparison of experimentally determined moduli with the moduli predicted from the Rule of Mixtures and Dynamic Beam Vibration models as a function of relative glue bond thickness for 50% resin and 50% hardener. The curves represent a least-squares best fit to equation 37. (— Dynamic modulus model: 33:70.5045, B‘s-0.4327, r80.0016, ----- Experimental data: 8,870.6193, B4--34.2256, r-O.6851, ROM model: 33:70.4715, sen-67.0177. 130.9773) 82 72 1 I 1 I I l l l 1 l n 65% Resin : 35% Hardener I“e "' e e L e __e_1r_______$ l ...... ‘ ' 0‘3 70. ““““ fiftx‘ “““““““ . O J “““: .......... ‘ ‘ ‘ ’- v ‘ K ~~~~~~~~~~~~ U) 68- ‘ ““4;— D J ‘ r-J “3g b :3 *~ 9 66- '" CD ,\ 2 4 9H . (I) 9“ ROM model is 64~ ‘ E . ’V - Z xperlmental data a: a“ :3 ‘ 0 Dynamic Modulus model a; S 62.: """"""" ROM model : 83:71.5284 B4=—75.8832 r=O.9835 .— ‘ """" Experimental: 83:70.2689 B4=—20.0842 r=O.4245 l. / Dynamic : 83:70.9455 B4=—3.2559 r=0.0849 60 l I j l I I I l I I r 0.00 0.02 0.04 0.06 0.08 0.10 0.12 l”ii-stirs 39. RELATIVE GLUE BOND THICKNESS Comparison of experimentally determined moduli with the moduli predicted from the Rule of Mixtures and Dynamic Beam Vibration models as a function of relative glue bond thickness for 65% resin and 35% hardener. The curves represent a least-squares best fit to equation 37. (-—— Dynamic modulus model: B,-70.9455, 83-34559, r-0.0849, ----- Experimental data: 3,370.2689, B.=-20.0842, r80.4245, ' 'ROM model: 83:71.5284, B‘--7S.8832, r=O.9835) 83 72 l l l l J l 1 l l l l l~~ 80% Resin : 20% Hardener ‘ ““““““““ at 4 _a a fif—L—’ ~- . ““~~4- ‘ E ’70- -- ~~~~~~ 4 ‘‘‘‘‘ - . ~~~~~~~~~ A .. 8 .. ‘ ~~~~. ~~~~~~~ ~~ ‘~~4 m 68“ BK ‘ ‘~~3( ‘ l- :3 "“ L —J ‘ . D 3" g 66- .. - 2 . . CD ROM model ~ _ fi... _ g 64 Experimental data 39* 3 * Dynamic Modulus model "we: 8 62- - ---------- ROM model : 83:70.5169 B4=-65.0101 r=O.9930 _ ---- Experimental: 83:71.3897 B4=-27.6106 r=0.6410 / Dynamic. : 83:70.5744 B4=1.O515 r=0.0353 60 I l I I r I I I I I I 0.00 0.02 0.04 0.06 0.08 0.10 0.12 RELATIVE GLUE BOND THICKNESS Fwiszure 40. Comparison of experimentally determined moduli with the moduli predicted from the Rule of Mixtures and Dynamic Beam Vibration models as a function of relative glue bond thickness for 80% resin and 20% hardener. The curves represent a least-squares best fit to equation 37. (— Dynamic modulus model: 8,870.5744, 891.0515, r-0.0353, ----- Experimental data: 83-71.3897, B.--27.6106, r-O.6410, non model; 83:70.5169, B‘s-65.0101, r=0.9930) 84 As the relative glue bond thickness increases, the deviation of the experimentally obtained Young's modulus increased with respect to the predictions of both the Dynamic Beam Vibration model and the ROM modelu'igures 35-40). The curves for the deviation, 6,, for the two models have similar slopes but the opposite algebraic signs. In addition, the deviation, 6., goes to zero as the relative glue bond thickness approaches zero. The deviation of the measured elastic moduli from the moduli predicted by the ROM and Dynamic modulus models may involve imperfect interfacial bonding between two glass layers and inelastic behavior of epoxy glue layer. The glass slides, however, certainly do behave elastically. Also, unlike the Ron model, the loading in this study was not a uniaxial loading but a free-free suspension vibration by sonic resonance technique. However, the sonic resonance modulus measurement technique employed in this study is appropriate to the assumptions made for the Dynamic modulus model. Another reason for the deviation between the experimental values and the values predicted from the Dynamic modulus model could stem from the large difference in stiffness of the glass slide layers and the glue bond layer. For our laminated glass slide/glue composite specimens, the difference in stiffness can give rise to a piecewise linear (as opposed to a linear) variation of inplane displacement through the thickness [19] . Thus, the Euler-Bernoulli assumptions are less valid as the glue bond thickness increases, such that the effect is increased with increasing the glue bond thickness. As the relative glue bond thickness approaches zero, both models “9m to successfully describe the Young's modulus of three-layer cmposite . 85 3.2.6. Change of lffective Young's lodulus of Glass Slide/Glue Specimens with respect to Glue-Bond Thickness Ranges at Fixed Adhesion Areas The data included in the three regression curves in Figure 33 were analyzed with respect to three glue-bond thickness ranges and adhesion area percent ranging from 0‘ to 100‘ (Table 7, Figures 41 and 42). Also, for 100‘ adhesion area the Young's moduli calculated from equation 33 were compared to the measured data (Figure 43). For the entire range of adhesion area fraction, the Young's modulus (decreased with the increase of the glue bond thickness (Figures 41 and ‘42). Also, each of the curves showed approximately the same slope. In addition, as the adhesion area increased from 0‘ to 100% by increments <3! 10‘, the Young's modulus difference between two adjacent curves decreased from 10 GPa (going from as adhesion area to 10‘ adhesion area) t1: 0.03 GPa (going from 90‘ adhesion area to 100% adhesion area) (Table ‘7, Figure 41 and Figure 42). The slopes of the curves for the calculated Young's modulus and the measured Young's modulus are very similar (Figure 43). As a result, for adhesion areas less than 100%, the predicted 'chang's modulus from the ROM and the dynamic modulus models will show the similar deviation from the measured Young's modulus as discussed in Section 3.2.5. 86 Table 7. Young's moduli (GPa) of glass slide/epoxy resin composite specimens obtained from the regression curves in Figure 33. The table shows representative Young's moduli of glass slide/ epoxy resin composite specimens having a given adhesion areas and a glue bond thickness within a given range of glue bond thickness. ADHESION I assess or GLUE sown THICKNESS AREA (1!) R.(0.025-0.075m) m(0.125-0.175m) R,(O.225-O.27Sm) 100 70.79 68.78 67.27 90 70.76 68.74 67.23 80 70.56 68.53 67.01 70 70.03 67.94 66.39 60 68.99 66.80 65.18 50 67.28 64.92 63.20 40 64.73 62.11 60.24 30 61.16 58.19 56.10 20 56.42 52.97 50.60 10 I 50.33 46.28 43.54 0 I 42.73 37.92 34.72 87 l J, 11 70 Epoxy Resm ‘EE *‘N§§‘§~‘~§"‘E“~*~___“_____‘--~* % 60- \\ A=407o _ m A=30z D d F —J :3 g 50... A=20% .— 2 m 4 r a; A=10% g 40- - o >.. - A=O% ' 30 1 F l R1 R2 R3 Piglare 41. RANGE OF GLUE BOND THICKNESS Change of Young's modulus with respect to glue-bond thickness ranges (R.:0.025-0.075 m, R,:O.125-O.175 m, i§:0.225-O.275 mm) at adhesion area percent, A, ranging from 0% to 40‘. The data were obtained from the three regression curves using equation 33 in Figure 33. 88 r Epoxy Resin 70- - 7:? m D 8 . (3 68- -— *4 A=IOO% g A=90% <3 - = (NZ _ 2 A—707 {/3 661 - o g . A=607. l o >— 64- — A=50% l l l R1 R2 R3 Figure 42. RANGE OF GLUE BOND THICKNESS Change of Young's modulus with respect to glue-bond thickness ranges (R.:0.025-0.075 mm, R,:O.125-O.1‘75 m, I§:O.225-O.275 mm) at adhesion area percent, A, ranging from 50% to 100$. The data were obtained from the three regression curves using equation 33 in Figure 33. 89 I I J 71 Epoxy Resm ’5 ‘35 70- _. V m D D .. 3 Calculated Data For A=100% Q — x" _ O 69 2 {/1 ~ . 0 CR :3 68' Measured Data For A=lOO% — >.. 67 F l l Figur'e 43. RANGE OF GLUE BOND THICKNESS Change of the measured and the calculated Young's modulus with respect to glue-bond thickness ranges (R.:0.025-0.075 mm, R2:0.125-O.17S mm, R,:O.225-O.275 m) at 100% adhesion area. The calculated data were obtained from the three regression curves using equation 33 in Figure 33. 90 3.2.7. Possible Physical Mechanisms for the Difference between the Weasured Iodulus of the Class Slide/Glue Composite Specimens and the Predictions of the ROI and Dynamic Iodulus models We shall consider two types of possible mechanisms that could potentially explain the differences (Figures 35-40) between the experimental data and the theoretical predictions of the ROM and dynamic modulus models. The two mechanisms are: (l) A change in the elastic modulus of the glue bond layer itself, perhaps due to residual stresses induced by shrinkage of the bond layer, or (2) an adhesion area less than the 100 percent adhesion area assumed in the experiment. (Note that the ROM and dynamic modulus, models were only applied to the case where the experimentally determined adhesion area was nominally 100 percent, or nearly 100 percent (Figures 35-40)). If the actual adhesion area is less than 100 percent, the effective modulus of the specimens would be lower than predicted by the ROM and dynamic modulus models. However, the feasibility of a residual—stress induced change in the modulus of the bond layer must be considered in terms of the relative modulus changes observed in a literature. We shall briefly review stress (or pressure) induced changes in elastic modulus, using examples from the thermal quenching of a polymer glass [20] and the pressure induced changes in stiffness for crystalline ceramics [21-23]. Vega and Bogus [20] measured residual stresses in terms of residual Optical birefringence induced into polymer glasses by thermal quenching into several media. Vega and Bogue reported a decrease of elastic mC> (cm) 99.5 11.8340 2.550 0.018 2426.4 69.83 99.0 11.8530 2.552 0.018 2438.0 70.44 98.7 11.9316 2.568 0.019 2447.9 70.16 98.7 12.0782 2.664 0.017 2547.8 68.92 98.4 11.8806 2.556 0.017 2437.3 70.24 97.1 11.8478 2.555 0.011 2435.9 70.05 96.6 11.8429 2.545 0.007 2421.4 70.00 96.1 11.7362 2.533 0.020 2419.4 70.25 59.4 11.4795 2.478 0.020 2332.5 68.21 38.3 11.4576 2.486 0.019 2298.0 65.45 34.7 11.8428 2.565 0.019 2387.8 66.49 29.0 12.0438 2.599 0.011 2395.7 65.43 28.0 11.8429 2.567 0.019 2380.6 65.94 27.3 11.4510 2.478 0.020 2247.2 63.15 27.1 11.8697 2.571 0.020 2359.5 64.62 23.0 11.8910 2.575 0.017 2349.4 63.88 21.2 11.8746 2.575 0.020 2336.7 63.11 16.0 12.0086 2.591 0.012 2300.7 60.73 11.5 11.4885 2.477 0.011 2091.9 54.97 10.0 11.8774 2.562 0.017 2194.7 56.53 7.2 11.8447 2.568 0.019 2105.0 51.50 Average elastic modulus at 1004 adhesion area, Em - 69.99 GPa. 130 8-3. The experimental data for the glass slide/super glue composite specimens having three glue spot. ADHESION MASS THICKNESS GLUE BOND FUNDAMENTAL YOUNG'S AREA THICKNESS FREQUENCY MODULUS _£L_ Jr”) (m) 4m) -(Hz) (GPaIL-_ 99.5 11.8340 2.550 0.018 2426.4 69.83 99.0 11.8530 2.552 0.018 2438.0 70.44 98.7 12.0782 2.664 0.017 2547.8 68.92 98.7 11.9316 2.568 0.019 2447.9 70.16 98.4 11.8806 2.556 0.017 2437.3 70.24 97.1 11.8478 2.555 0.011 2435.9 70.05 96.6 11.8429 2.545 0.007 2421.4 70.00 96.1 11.7362 2.533 0.020 2419.4 70.25 89.0 11.8931 2.562 0.011 2446.4 70.34 87.0 11.9224 2.571 0.018 2455.3 70.28 86.7 11.9412 2.581 0.021 2467.8 70.29 76.0 12.0052 2.581 0.011 2470.5 70.82 74.5 11.6531 2.533 0.020 2419.9 69.78 72.0 11.9372 2.573 0.012 2434.0 68.99 64.0 11.9553 2.583 0.018 2441.4 68.72 60.0 12.0637 2.600 0.019 2461.9 69.13 44.0 12.0853 2.603 0.011 2430.2 67.25 42.3 11.8533 2.567 0.019 2411.2 67.70 31.0 12.0660 2.602 0.010 2350.5 62.89 30.7 11.8929 2.568 0.018 2325.7 63.13 14.3 11.8593 2.562 0.012 2216.1 57.55 13.8 11.2377 2.434 0.012 2061.6 55.04 10.6 11.2443 2.442 0.020 2087.6 55.92 8.9 11.2318 2.420 0.009 2001.0 52.73 6.5 11.2378 2.421 0.010 1982.2 51.71 6.1 11.2155 2.431 0.014 1936.3 48.64 131 ADHESION MASS THICKNESS GLUE BOND FUNDAMENTAL YOUNG'S AREA THICKNESS FREQUENCY MODULUS 4‘) (91‘1“) (ml (Hz) (39‘) 6.0 12.0213 2.585 0.010 2043.0 48.27 3.6 11.1885 2.410 0.010 1881.4 47.02 3.0 11.1662 2.415 0.019 1949.8 50.09 2.9 11.1566 2.412 0.011 1871.0 46.31 2.4 11.1908 2.420 0.012 1840.6 44.16 0.35 11.1512 2.420 0.016 1635.0 34.95 Average elastic modulus at 1005 adhesion area, E“m - 69.99 GPa. 132 8-4. The experimental data for the glass slide/super glue composite specimens having five glue spots. ADHESION AREA ALL... 99.5 11.8340 2.550 0.018 2426.4 69.83 99.0 11.8530 2.552 0.018 2438.0 70.44 98.7 12.0782 2.664 0.017 2547.8 68.92 98.7 11.9316 2.568 0.019 2447.9 70.16 98.4 11.8806 2.556 0.017 2437.3 70.24 97.1 11.8478 2.555 0.011 2435.9 70.05 96.6 11.8429 2.545 0.007 2421.4 70.00 96.1 11.7362 2.533 0.020 2419.4 70.25 16.9 11.2400 2.425 0.013 2175.0 61.96 13.7 11.2526 2.430 0.016 2100.8 57.52 11.4 11.2239 2.423 0.012 2084.8 56.99 10.2 11.2488 2.431 0.013 2102.1 57.50 7.0 11.2802 2.433 0.014 2035.2 53.91 Average elastic modulus at 100% adhesion area, Em,==69.99 GPa. 133 8-5. The experimental data for the glass slide/epoxy cement composite specimens having one glue spot. ADHESION MASS YOUNG'S AREA MODULUS 93.8 12.0099 2.629 0.111 2525.7 70.07 77.9 11.3689 2.568 0.179 2471.3 68.13 75.0 11.3028 2.518 0.130 2414.9 68.61 E 52.1 11.2273 2.533 0.142 2158.9 53.51 38.0 11.9162 2.699 0.167 2014.4 40.87 p 23.7 11.8574 2.631 0.092 1678.2 30.47 L 14.9 11.6075 2.587 0.106 1449.8 23.42 9.9 11.1338 2.456 0.056 1323.4 21.87 6.5 11.4778 2.511 0.051 1300.5 20.38 3.0 11.0725 2.404 0.010 1206.8 19.29 0.84 11.0951 2.432 0.031 1223.5 19.19 Average elastic modulus at 100‘ adhesion area, Em - 70.07 GPa. 134 B-6. The experimental data for the glass slide/epoxy cement composite specimens having three glue spots. ADHESION MASS FUNDAMENTAL YOUNG'S AREA FREQUENCY MODULUS 0) (gram) _ __L’2:.L («3111.)= 93.8 12.0099 2.629 0.111 2525.7 70.07 78.1 11.3265 2.494 0.112 2392.6 69.46 48.9 11.2234 2.470 0.079 2308.9 65.98 33.2 11.2076 2.458 0.069 2249.9 63.86 22.0 11.1322 2.427 0.052 2173.9 61.15 19.5 11.1725 2.437 0.048 2123.4 57.84 12.9 11.1012 2.404 0.031 2060.5 56.38 8.4 11.1524 2.428 0.046 1975.5 50.53 4.9 11.1350 2.401 0.019 1917.5 49.15 1.0 11.8871 2.580 0.033 1843.9 39.11 Average elastic modulus at 100‘ adhesion area, Em :- 70.07 GPa. 135 8-7. The experimental data for the glass slide/epoxy resin composite specimen having three glue spots of 50! resin and 50 t hardener. ADHESION MASS THICKNESS GLUE BOND FUNDAMENTAL YOUNG' S AREA THICKNESS FREQUENCY MODULUS (3) (gram) (mm) (mm) (32) (9931, 100.0 12.2718 2.745 0.163 2613.8 67.36 100.0 11.8679 2.657 0.205 2554.7 68.62 100.0 12.4155 2.827 0.247 2693.3 66.24 100.0 11.4081 2.500 0.090 2394.0 69.54 100.0 11.8415 2.609 0.088 2502.6 69.40 100.0 11.7618 2.601 0.145 2492.7 69.02 100.0 12.1338 2.780 0.305 2673.7 67.09 100.0 12.0773 2.748 0.279 2647.6 67.80 100.0 11.7657 2.621 0.153 2518.4 68.87 100.0 12.4723 2.810 0.227 2694.4 67,82 100.0 11.4094 2.498 0.090 2391.1 69.55 100.0 12.1581 2.678 0.116 2570.6 69.52 100.0 11.7183 2.555 0.075 2440.5 69.54 97.9 12.2605 2.754 0.196 2613.9 66.65 91.9 12.4416 2.834 0.256 2679.1 65.20 88.3 11.6821 2.574 0.115 2463.8 69.10 87.8 12.1726 2.718 0.169 2607.9 68.52 86.7 12.2673 2.787 0.234 2660.9 66.68 84.4 11.6875 2.541 0.064 2430.0 69.91 80.2 12.3047 2.823 0.260 2666.6 64.63 75.8 12.1792 2.702 0.129 2574.8 68.02 73.3 12.5443 2.942 0.370 2837.9 65.93 69.5 11.6766 2.573 0.112 2454.8 68.65 64.7 11.5846 2.519 0.052 2386.9 68.62 63.3 12.1068 2.719 0.162 2554.1 65.29 57.3 12.1575 2.759 0.189 2563.6 63.22 136 YOUNG’S MODULUS 54.7 11.6453 2.745 0.330 2629.3 64.68 52.1 11.6333 2.537 0.059 2375.5 66.81 49.7 11.7023 2.618 0.147 2442.2 64.64 48.9 11.8819 2.800 0.328 2636.5 62.52 48.9 11.6016 2.530 0.058 2358.6 66.23 48.8 12.0290 2.677 0.119 2490.6 64.64 46.7 11.7326 2.716 0.263 2523.8 61.99 43.4 11.7529 2.664 0.197 2565.1 67.97 38.8 11.6982 2.710 0.249 2532.9 62.66 38.7 12.0026 2.827 0.295 2652.5 62.11 31.1 11.5324 2.511 0.044 2274.5 62.62 30.5 11.6820 2.677 0.204 2347.3 55.76 30.1 12.1782 2.860 0.281 2600.4 58.50 29.4 12.0169 2.682 0.116 2437.3 61.49 28.0 11.6341 2.722 0.256 2459.0 57.96 27.3 12.0528 2.703 0.124 2391.0 57.98 26.7 11.6491 2.615 0.138 2351.8 59.88 26.1 12.6239 2.673 0.206 2341.7 60.23 25.6 11.5588 2.533 0.063 2277.3 61.30 22.7 11.9785 2.643 0.074 2330.7 58.57 21.9 11.9528 2.814 0.278 2425.6 52.44 21.8 11.6575 2.635 0.157 2324.3 57.20 21.5 11.2844 2.493 0.077 2208.7 59.04 21.4 11.5432 2.651 0.197 2346.5 56.69 18.1 11.9958 2.646 0.079 2293.8 56.62 15.6 11.2752 2.615 0.206 2213.2 51.32 14.3 11.2412 2.461 0.053 2099.1 55.22 14.2 11.5655 2.576 0.107 2182.7 53.57 13.3 11.9852 2.722 0.154 2237.0 48.88 11.6 11.2720 2.654 0.249 2079.8 43.34 137 ' Ant-I‘lm' .‘-. ~. I. a .a“ ADHESION sass mamas 01.0: sons manna YOUNG ' s AREA FREQUENCY MODULUS (4) (gram) (8:) (GPa) 9.7 11.9506 2.608 0.036 2156.4 52.06 8.7 11.3072 2.738 0.326 2143.1 42.04 7.3 11.4689 2.499 0.044 3036.0 50.78 5.8 11.9349 2.630 0.064 2055.9 46.08 5.6 11.9635 2.627 0.054 2074.6 47.20 5.4 11.5123 2.641 0.176 2036.7 43.08 5.1 11.5704 2.612 0.139 2015.9 43.85 5.0 11.2447 2.666 0.261 1950.0 37.50 4.5 11.5350 2.657 0.187 2010.9 41.32 3.8 11.5757 2.698 0.221 1918.2 36.04 2.9 11.5329 2.691 0.220 1946.9 37.28 2.5 11.5901 2.610 0.131 1808.5 35.43 2.2 11.8963 2.601 0.045 2350.8 40.33 2.2 11.9417 2.606 0.040 1949.2 42.60 Average elastic modulus at 1004 adhesion area, Em - 68.49 GPa. 138 3-8. The experimental data for the glass slide/epoxy resin composite specimens adhered by the epoxy resin of 354 resin and 658 hardener. ADHES ION MASS THICKNESS GLUE BOND FUNDAMENTAL YOUNG ' S AREA MODULUS m (722L— (cps) 100.0 11.7782 2.605 0.134 2478.5 68.02 100.0 11.8840 2.645 0.175 2516.0 67.56 100.0 12.0646 2.740 0.272 2612.0 66.79 100.0 11.8365 2.625 0.159 2504.8 68.39 100.0 12.1636 2.770 0.301 2652.4 67.20 100.0 12.1220 2.781 0.312 2644.3 65.56 100.0 12.4970 2.884 0.347 2737.2 64.66 100.0 12.1103 2.756 0.285 2617.3 66.15 100.0 11.8631 2.731 0.323 2579.0 64.38 100.0 11.8093 2.691 0.276 2547.0 65.33 139 B-9. The experimental data for the glass slide/epoxy resin composite specimens adhered by the epoxy resin of 654 resin and 35% hardener. YOUNG'S MODULUS GPA ADHESION AREA 8 100.0 11.3725 2.478 0.066 2380.4 70.38 100.0 11.5281 2.570 0.166 2465.7 68.62 100.0 11.7434 2.670 0.261 2572.3 67.84 100.0 11.8515 2.730 0.327 2644.7 67.71 100.0 11.8343 2.713 0.309 2647.6 69.04 100.0 11.6810 2.639 0.234 2555.8 68.99 100.0 11.8027 2.690 0.278 2611.7 68.73 100.0 11.7506 2.683 0.273 2579.0 67.25 100.0 11.7111 2.6380 0.221 2541.2 68.46 100.0 11.5302 2.571 0.160 2461.5 68.32 140 8-10. The experimental data for the glass slide/epoxy resin composite specimens adhered by the epoxy resin of 808 resin and 204 hardener. ADHES ION MASS GLUE BOND FUNDAMENTAL YOUNG ’ S AREA THICKNESS FREQUENCY MODULUS (5) (Hz) (GPa) 100.0 11.4276 2.505 0.091 2411.4 70.25 100.0 11.5407 2.561 0.156 2480.1 70.31 100.0 11.4156 2.520 0.118 2430.7 70.21 100.0 11.7169 2.670 0.262 2569.8 67.71 100.0 11.7646 2.676 0.271 2591.8 68.61 100.0 11.8715 2.714 0.306 2641.6 69.09 100.0 11.5624 2.564 0.158 2473.0 69.80 100.0 11.6677 2.617 0.203 2522.0 68.81 141 10. 11. 12. 13. LIST OF REFERENCES J. Remeny and N. G. W. Cook, "Effective Moduli, Non-linear Deformation and Strength of a Cracked Elastic Solid”, J. Rock Mech. Min. Sci. & Geomech. Abstr., 23[2]: 107-118, 1986. Walsh J. B. "The effect of cracks on the uniaxial elastic compression of rock", J. Geophys. Res., 70: 399-411, 1965. G. R. Irwin, ”Analysis of stresses and strains near the ends of a crack traversing a plate“, J. Appl. Mech., 24: 361-364, 1957. G. C- 81h. Eandb22k_2f_Strees_12§en§itx_zasters. Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, 1973. Eldon D. Case and Youngman Kim, "The Effect of Surface Limited Microcracks on The Effective Young's Modulus of Ceramics, I. B. D. Agarwal and L. J. Eroutman, pp. 20-26 in Agglygig_ggg 2erf2rmanse_ef_£iber_§9mngsites. Wiley. New York. 1980. E. Volterra and E. C. Zachmanoglou, pp. 321-322 in Qyngmigg_gfi gibggtiggg, Charles E. Merrill Books, Inc., Columbus, OH, 1965. s. K. Clark. pp. 75-87 in nxnsmi2a_ef_ggntinngns_nlements. Prentice Hall, Inc., Englewood Cliffs, NJ, 1972. C. C. Chiu and E. D. Case, "Elastic Modulus Determination of Coating Layers as Applied to Layered Ceramic Composite", Mat. Sci. and Eng., A132: 39-47, 1991. s. P. Timoshenko and D. H. Young, pp. 113-115 in fit;§gg§h_gfi nagggiglg, Fourth ed., Van Nostrand Reinhold Co., Princeton, NY, 1962. F. Forster, "Ein neues Messverfahren zur Bestimmug des Elastiziatmoduls und der Dampfung' (A new Method for Determination of Modulus of Elasticity and Damping): Zeitschrift fur Metallkunde, 29[4]: 109-115, 1937. E. Schreiber, 0. L. Anderson, and N. Soga, Chapter 4 in 313331; Qenatants_and_2hsir_ueasursments. MCGrIw-Hill. New York. 1974. S. Spinner and W. E. Tefft, "A Method for Determining Mechanical Resonance Frequencies and for Calculating Elastic Moduli from These Frequencies", ASTM Proc., 61: 1221-1238, 1961. 142 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. Won Jae Lee, "Thermal Fatigue in Ceramics and Ceramic Matrix Composites”, A Dissertation Submitted to Michigan State University for the Degree of Doctor of Philosophy, Department of Metallurgy, Mechanics and Materials Science, 1991. G. Pickett, "Equations for Computing Elastic Constants from Flexural and Torsional Resonant Frequencies of Vibration of Prism and Cylinder”, ASTM Proc., 45: 846-865, 1945. D. P. H. Rasselman, 'B d la u sona t Fr one o ngtgggglgg_£;i§mg, Carborundum Co., Niagara Falls, NY, 1961. Henry Lee. pp- 6-24 in Handheek_ef_£nezx_seeins. Macraw-Hill. NY 1967. Derek Hull. pp- 29 in 88_Intr9dn2ti9n_tg_99m22§ite_nateziala. Cambridge Solid State Science Series, NY, 1981. Prof. Averill, Assist. Professor, MSU, personal communication, March, 1993. Javier De La Vega and Donald C. Bogus, "Mechanical Properties and Residual Stresses in Non-Equilibrium Glasses", Chem. Eng. Comm., 53: 23-31, 1987. Orson L. Anderson and P. Andreatch, Jr., "Pressure Derivatives of Elastic Constants of Single-Crystal Mgo at 23° and -195.8°C", J. Amer. Cer. Soc., 49[8]: 404-409, R. A. Bartels and D. E. Elastic Constants of NaCl and RC1 at 295°K and 195°R", J. Phys. Chem. Solids, M. J. McSkimin, 26: 537-549, Schuele, 1965. 1966. "Pressure Derivatives of the P. Andreatch, Jr., and R. N. Thurston, "Elastic Moduli of Quartz versus Hydrostatic Pressure at 25° and -195.8°C", J. Appl. Phys., 36(5): 1624-1632, 1965. F. P. Mallinder and 8. Classes, C. C. Chiu, 5(4): 91-103, A. Proctor, ”Elastic Constants of Fused Silica as a Function of Large Tensile Strain”, Phys. and Chem. 1964. "Thermal Quench of Brittle Materials”, A Dissertation nd Submitted to Michigan State University for the Degree of Doctor of Philosophy, Department of Metallurgy, Mechanics and Materials Science, 1991. 143