Study of Cracking of Thin Glass Interposers Intended for Microelectronic Packaging Substrates

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1 Study of Cracking of Thin Glass Interposers Intended for Microelectronic Packaging Substrates Scott R. McCann 1,2, Yoichiro Sato 3, Venkatesh Sundaram 1,4, Rao R. Tummala 1,4,5, and Suresh K. Sitaraman 1,2,* 1 3D Systems Packaging Research Center Georgia Institute of Technology Atlanta, GA The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology 3 Asahi Glass Co., Ltd., Tokyo, Japan 4 School of Electrical and Computer Engineering, Georgia Institute of Technology 5 School of Materials Science and Engineering, Georgia Institute of Technology *contact suresh.sitaraman@me.gatech.edu Abstract Glass interposers have gained increased attention and interest in microelectronics industry since 20. This is because glass has a tailorable coefficient of thermal expansion (CTE), high mechanical rigidity, availability in large and thin panel form, low processing cost, smooth surface for fine line and space fabrication, and superior electrical properties. While thin glass panels offer such a plethora of benefits, there are several processing and reliability challenges that glass imposes. As a brittle material, glass has low fracture toughness and is prone to cracking. In a typical large-area glass panel processing, layers of dielectric polymers and conducting copper are sequentially deposited and patterned. Due to these temperature histories and difference in the CTE among different materials, the panel is subjected to process-induced residual stresses. When the panel is subsequently diced into smaller substrates, the glass could crack. This cracking is due to high residual stresses as well as dicing defects and possible delamination at the polymer-glass interface. This experimental and theoretical work aims to investigate thin glass cracking and understand the mechanics of such cracks, focusing on the stresses induced by build-up layers. As part of this work, glass panels of 150 x 150 mm size and 0 μm thickness were laminated on both sides with ZS-0 polymer of 22.5 μm thickness and cured. After the lamination process, 5 μm thickness of copper is then deposited through a semi-additive electroless and electrolytic plating processes. This process of polymer and copper are repeated to create a total of two metal layers on each side of the panel. The panel is then diced into 18.4 x 18.4 mm substrate coupons. Dicing defects are characterized using optical inspection. Cracking failures are documented. The unbroken substrates are thermal cycled between -40 and 125 C. In parallel to the experimental investigation, numerical models are created based on the sequential fabrication process. Copper material properties are obtained from literature as well as from in-house nanoindentation tests. Polymer properties are obtained from vendor data, and the stress-free temperature is obtained through experiments. Dicing process is simulated by inserting a vertical crack through the panel, and various dicing defects are introduced in the singulated substrate. Energy available for crack propagation of such defects is determined through fracture mechanics approach, and design guidelines to mitigate glass fracture during dicing and reliability testing are explored. Introduction Glass has gained traction recently as a possible candidate for microelectronic packaging substrate [1, 2]. This is because of some of its electrical and mechanical properties as well as its low cost potential [3]. In comparison to organics such as FR-4, glass is rigid, has a high glass transition temperature and tailorable CTE, provides a smooth surface for fine line lithography [4], and also provides good structural support. All of these properties lead to lower warpage [5] and fine pitch I/Os [6] while keeping the package thin. Glass is an electrical insulator with very low electrical loss and high resistivity. However, glass has poor thermal conductivity, and thus, thermal management should be addressed through other means such as through glass vias [7]. Also, glass is a brittle material that is prone to cracking. Sequential deposition and thermal processing of polymer and copper is carried out on glass panels to create glass interposers for microeletronic packages. These build up materials have different CTEs, creating thermo-mechanical stresses. Once fabricated, glass panels must be separated into individual interposers through dicing. Such singulation or dicing could create large enough defects, that when combined with stresses from redistribution layers (RDL), could lead to crack propagation ultimately resulting in glass interposer cracking. The goal of this work is to investigate why singulated glass interposers crack during dicing or subsequent thermal exposure. This paper also discusses options for reducing such cracks, and explores one such option in detail. Fabrication, Dicing, and Reliability Testing Glass interposers are fabricated using clean-room processes applicable to substrate fabrication. Fabrication begins with a 150 x 150 mm bare glass panel, which is cleaned and laminated with a layer of 22.5 μm polymer, ZEONIF TM ZS-0. The polymer is cured in a thermal oven at 180 ºC. Copper traces are deposited through semi-additive processes, which starts with an electroless copper seed layer, on top of which dry film photoresist is laminated. The photoresist is exposed and developed, and copper is then electroplated. The photoresist is then stripped and the seed layer is removed through etching, leaving 5 μm of copper, which is then annealed. This process is two sided, creating layers of polymer and copper traces on both sides /15/$ IEEE Electronic Components & Technology Conference

2 simultaneously. The processes are repeated to add a second layer of ZEONIF ZS-0 and copper traces on each side, for a total of four metal layers, which constitute the RDL. Finally, a dry film passivation is laminated, exposed, developed, and cured. Each panel has a six by six array of 18.4 x 18.4 mm interposers. Dicing is then used to singulate the glass panel into individual interposers, producing the structure seen in Figure 1, which shows an expanded schematic of a four metal layer glass interposer. For this work, dicing is done using blade dicing, though other options such as score and break, laser dicing, and laser ablation are available. For more information on the dicing process and optimization, see Wei s work [8]. In early test cases, cracking from the diced edge, through the glass core, occurred immediately after dicing. This is while samples are still in the dicing tool, attached to the tape. However, not all fabricated panels exhibit this failure, and the uncracked panels proceed to reliability testing. Such glass cracking upon dicing has been reported by Koizumi, who terms the cracking failure from the free edge SeWaRe [9]. Figure 1: Schematic of four metal layer glass interposer. For samples that did not crack immediately upon dicing, the surface roughness along the diced edge is measured using confocal microscopy. Data measured is shown in Figure 2. The original or unoptimized blade shows the maximum surface roughness. With subsequent improved or optimized dicing conditions, the surface roughness became smaller. The maximum roughness value measured is treated as the largest defect or flaw size present in the material. Such locations of maximum surface roughness are the prime locations for crack propagation. Surface Roughness [um] Unoptimized Blade (Bare Glass) Initial Blade Optimization (Disco) Pulsed Laser Dicing (Disco) BKM Blade (Disco) Figure 2: Surface roughness of different dicing methods as measured using confocal microscopy. Reliability testing includes preconditioning and temperature cycling testing (TCT). For preconditioning, there are three steps: bake, moisture soak, and reflow. Bake is done at 120 C for 24 hours. For the moisture soak, moisture sensitivity level 3 (MSL-3) from JEDEC Standard 020D.1 (60% RH at 30 C for 168 hours) is used []. However, the available humidity chamber was not capable of achieving the temperature target while maintaining the required humidity level, so the temperature was kept at about 43 C, making the test conditions somewhat more rigorous than normal MSL-3. Reflow is done three times with a peak temperature of 260 C. The time limits given in JESD22-A113F, the JEDEC Standard on Preconditioning of Non-hermetic Surface Mount Devices Prior to Reliability Testing [11], are followed. After preconditioning, temperature cycling of -40 to 125 C in onehour cycles with fifteen minute ramp times and fifteen minute dwells was performed, following test condition G in JESD22- A4D, the JEDEC Standard on Temperature Cycling [12]. Samples were inspected at the end of processing, after preconditioning, after 50, 250, 500, and 00 thermal cycles. Inspection was done using an optical microscope to inspect the edges and a C-mode scanning acoustic microscope (CSAM). From this inspection, crack progress, if any, was documented and analyzed. Initial Experimental Data The first set of samples was fabricated, diced, and reliability tested, as described in the Fabrication, Dicing, and Reliability Testing section. Figure 1 shows a schematic of the glass interposers fabricated and the details are presented in Table I. These samples had two polymer and two copper layers on each side of the substrate. Each polymer layer was 22.5 μm thick, and each copper layer was μm thick. The experimental results after dicing, preconditioning, and 50 temperature cycles are shown in Table II where this first set of samples is designated with a prefix 1. The samples are classified by using stop light colors as status indicators. Green indicates the expected signature after dicing, usually a pock mark pattern for blade dicing of glass, seen in Figure 3a. Yellow is worse than green and indicates interface failure; Figure 3b shows an example of interface failure. In some cases, interface delamination occurs, as in the figure, while in other cases, it does not, making it difficult to systematically measure the amount of interfacial delamination. While C- SAM is used, the amount of interfacial delamination is usually below the resolution of the available tool, making it difficult to collect meaningful data. Red means cracking of glass and an example is shown in Figure 3c. As described in Table II, the samples, 1-A4 and 1-A5, did not fail after dicing, showed some interfacial delamination after preconditioning, and the glass substrate cracked after 50 temperature cycles, the first inspection point during TCT. For the samples to pass the reliability criteria, they need to survive 00 thermal cycles. To achieve this, numerical models were developed to help understand the problem and design a solution, and based on the findings from the numerical simulations, new samples were built. 1939

3 Table I: Details for initial samples fabricated for model validation. Metal Layers Glass Thickness [μm] Polymer Thickness [μm] Copper thickness [μm] Passivation thickness [μm] Some researchers often use the glass transition temperature as the reference temperature in their simulations [13]. It should be mentioned that this type of matching the curvature (or warpage) to determine the reference temperature should be treated with caution, as the modulus, Poisson s ratio, and CTE of the polymer and the glass were not directly measured in this work; rather they were obtained from vendor datasheets. Similar to the polymer reference temperature, the reference temperature for annealed copper was obtained using curvature measurements of a glass substrate with two sides of polymer and one side of electroless and electroplated copper followed by annealing at 180 C. When warpage (or curvature) measurements were done at room temperature and compared against simulations, it was seen that a reference temperature of 8 C was applicable to copper. This reference temperature was validated using two different configurations where one configuration had a 0% copper plating on top of the polymer layer, while the second configuration had an 80% patterned copper on top of the polymer layer. The details of copper layout and the measured warpage contours are presented in Fig. 4. The reference temperature details are shown in Table III. It is seen that copper s reference temperature of 8 C is approximately the average of the electrolytic plating temperature, 40 C, and the annealing temperature, 180 C. Sample Table II: Experimental results from fabrication, dicing, and temperature cycling to validate modeling predictions. Sample Type Number 1 A4 1 A5 After dicing After Preconditioning After 50 cycles Green Green Yellow Yellow Red Red Figure 4: (a) schematic of one sided sample for curvature measurement with copper and (b) shadow moiré profile of surface. Table III: Average Curvature and Reference Temperatures for Build Up Materials Material Figure 3: Optical inspection of glass interposer edge to show failure classification: (a) expected dicing signature (green), (b) interface failure and/or minor cracking (orange), and (c) cracking failure (red). Stress Measurement through Warpage As a first step in the simulations, it is necessary to identify the reference temperature at which the substrate is flat without any warpage. In this work, such a reference temperature was obtained by measuring the curvature at room temperature using a shadow moiré tool and matching the curvature warpage against simulated value. Starting with a glass substrate laminated with a laminated polymer on one side, the average curvature along its two diagonals was measured to be 2.79 (1/m) using a shadow moiré tool. Similarly, when simulations were carried out starting with a glass substrate and a polymer, it was seen that a reference temperature of 162 C will produce a curvature of as 2.79 (1/m) at a room temperature of 27 C. It should be pointed out that the glass transition temperature of the polymer was 162 C, the same as the reference temperature. 0% 5um Copper (Annealed) 80% 8um Copper (Annealed) ZEONIF ZS-0 Average Curvature [1/m] Reference Temperature [ C] Numerical Model Introduction Using the reference temperatures, in combination with other measured material properties and material manufacturer data, finite-element models are created using ANSYSTM to predict the occurrence of glass cracking failure. Figure 5 shows an example of the 2D plane-strain finite element model. As seen, glass is the interposer core material, with copper and polymer as build up materials, and solder resist passivation on the top and bottom. Table IV shows the material properties used in the model. The model employs symmetry at the left hand side. A dicing-induced flaw was introduced on the free edge on the right, which can be placed 1940

4 , and are the stress and strain tensors, where respectively [14]. For a linear, brittle, isotropic material, such as glass, J is equal to the strain energy release rate, G [15]. The strain energy release rates obtained through the contour integral approach were cross checked using the Virtual Crack Closure Technique (VCCT) [16]. For the VCCT, the predetermined crack path was assumed to be straight forward into the glass for cracks in the glass. If the dicing induced defect happens to be at the interface between polymer and glass, then the propagation of the flaw or defect should be studied through interfacial fracture mechanics models. Also, interfacial cracks may continue to propagate as interfacial cracks or may kink into the glass. Interface cracks have been shown to kink into brittle materials [17], the criteria for doing so depends on the ratio of the critical energy release rate of the interface to the critical energy release rate of the material [18] and mode mix [19, 20]. Details of interfacial delamination and crack kink models are discussed in the Cracking Process and Failure Prediction section. Figure 5: Finite element model geometry schematic. Figure 6: (a) Finite element model mesh example and (b) zoomed in near defect with contour integral. Table IV: Material properties used in modeling. Material Glass Copper Polymer Solder Resist Modulus [GPa] CTE [ppm] / 95 anywhere along the thickness of the diced edge. This flaw or defect was assumed to be a perfectly sharp crack and oriented horizontally. A typical mesh is shown in Figure 6. The mesh was refined near the free edge with constant element size near the crack tip. A contour integral approach was used within the finite element software to calculate the energy release rate using J-integral, where J is defined as,, (2) Model to Simulate Dicing Dicing is usually done in the presence of water. When a glass interposer is exposed to water or moisture, it could cause interfacial delamination or glass cracking. During dicing, the crack propagation is likely due to a lower fracture toughness of glass in the presence of water. For example, the critical stress intensity factor of borosilicate glass drops from 0.8 MPa m in air to about 0.4 MPa m in the presence of water [21]. To investigate whether a crack will propagate through the glass horizontally during dicing operation in water, a second model was constructed to be compared to the model shown in Figure 5. The second model had two interposers, with a partial dicing cut inserted between them and a layer of contact elements with fixed out of plane displacement below the model to represent the dicing table, as shown in Figure 7. This second model is during dicing while the first model, described in the previous section, is after dicing. The model is otherwise identical to the first, with a horizontal defect in the glass, normal to the free edge created by dicing. Both models were run without solder resist passivation. Figure 7: Finite element model geometry for model simulating dicing. (1) where Γ is the counterclockwise curve surrounding the crack tip, x and y are in in-plane directions (as shown in Figure 6), u is the displacement vector, ds is the infinitesimal distance along the path, T are the tractions, and W is the strain-energy density defined as, The during dicing model had two crack tips. The larger crack, that is the cut made by the dicing saw, was vertical. If the vertical crack propagates, the panel is considered diced. This is the desired outcome of the dicing operation. Creating a vertical crack in the glass instead of abrading it with a blade is known to produce a much cleaner surface. The score and 1941

5 break method, which is commonly used to cut bare glass, does exactly that, by scoring the surface and then bending or locally heating the glass until it breaks. The second crack tip was a defect at the free edge of the glass due to dicing abrasion and is oriented horizontally into the glass. If this initial horizontal defect propagates into the glass, the sample will fail, and this will be an undesired outcome. During dicing model consisted of a 60 μm vertical dicing cut into the 0 μm thick glass with a 5 μm horizontal defect from the cut edge of the glass. It was seen that the energy release rate for crack propagation was J/m 2. This number is less than 2% different from the energy release rate for a horizontal crack in a fully diced glass substrate, as expected and as shown in Table V. Since the results from the two models varied by less than two percent, the during dicing model, which included a larger area, two crack tips, and contact elements for tape support, was computationally expensive, and thus, all further analyses were done using the after diced model. Table V: Comparison of energy release rates from models during dicing and after dicing. G total [J/m 2 ] During dicing After diced Next, to investigate glass cracking failure during dicing, the simplified models of fully-diced glass substrates with edge flaws were run to simulate the dicing conditions. The room temperature applied as a boundary condition because of the cooling water. The obtained energy release rates were compared to the critical energy release rate of borosilicate glass in water, 1.98 J/m 2. The energy release rate as a function of initial defect size is shown in Figure 8. Multiple models were constructed with a range of horizontal crack sizes and the energy release rate available upon polymer processing, copper annealing, and dicing was examined. An initial defect size as small as μm may be close enough to result in glass cracking failure for the 90 μm ZS-0, 40 μm copper sample. This explains the on-tape failure observed Figure 8: Energy release rate as a function of initial defect size during dicing. prior to this work, which was cut with the original or unoptimized blade. Also, as seen in Figure 8, if one were to use 40 um ZS-0 and 20 um copper, it is unlikely that the substrate will crack during dicing. To address this type of failure, blade dicing optimization work was done by Frank Wei and Disco [8]. Cracking Process and Failure Prediction Now that glass cracking during dicing has been addressed, glass cracking failure during reliability testing, such as the failures seen in samples 1-A4 and 1-A5, may be analyzed. To better understand such glass cracking under thermal cycling, scanning electron microscopy (SEM) was used on a sample after failure, shown in Figure 9. Figure 9a shows a schematic of a glass interposer corner with an interfacial delamination that kinks into the glass. Figure 9b shows the glass interposer, zoomed-out, under the SEM, with green and red boxes indicating regions of interest. Figure 9c shows a close up on the edge region (green box); Figure 9d shows a close up on corner region (red box), and Figure 9e shows a higher zoom on the region from Figure 9d. The polymer is labeled ZIF for ZS-0 in the pictures. In Figure 9c, d, and e, the bulk of the glass remained on the bottom after separation, leaving no glass or small amounts of glass on the top. The crack is seen to propagate at the glass-polymer interface and kink into the glass, but not into the polymer. From this observation, there are three different types of failure which occur subsequently: first, there is glass chipping from the free edge, created by dicing; second, bare regions of ZS- 0 are seen, indicating a failure at the glass-polymer interface; and third, cohesive cracking of the glass. To investigate the failure during reliability testing, simulations were run to see if interfacial delamination between glass and ZS-0 would kink into the glass to result in glass fracture. Thus, treating the adhesion failure as an initial crack, the numerical models shown in Figure 5 were modified to include a kink to the glass, using the notation shown in Figure. Following previous convention [17, 18], a is the crack length after kinking, ω is the kink angle, and the interfacial delamination length is much greater than the kinked crack length. A range of kink angles were run, which showed less than % difference between and 45 degree crack kinks. A representative kink is chosen to be μm long and at a 15 degree angle from the glass-polymer. The interfacial delamination size is varied and the results from such simulations are shown in Figure 11, in which the total energy release rate at -40 C is a function of various interfacial delamination sizes. The total energy release rate increases with larger defect sizes, as expected, and the energy release rate of the 90 μm ZS-0, 40 μm copper structure reaches the critical energy release rate at about 0 μm interfacial delamination. The energy release rate of the 40 μm ZS-0, 20 μm copper structure does not reach the critical energy release rate. Thus, a thicker build up leads to higher total energy release rate for any single initial defect size. The 90 μm polymer and 40 μm copper interposer is predicted to experience cracking failure, while the available energy of the thinner build up does not reach the critical energy release rate of glass. As long as the available energy does not reach the critical energy release rate, glass cracking failure should not 1942

6 occur. Thus the 40 μm polymer and 20 μm copper sample should not have cohesive glass cracking. Based on these results, 40 μm polymer and 20 μm copper was chosen as a potential solution. The data in Figure 8 and Figure 11 do not appear entirely smooth because the model includes variations such as copper pads and solder resist passivation openings, which lead to different localized stress patterns for each crack tip location. Figure : Schematic of crack kinking from glass-polymer interface into glass. Figure 11: Energy release rate at -40 C as a function of initial defect size for glass interposers with different build ups with interface failure. Experimental Demonstration of Solution The models suggest that with thinner polymer and copper layers, there will be less chances for cracking, and thus, new samples with polymer layers at μm thick for a total thickness of 40 μm and copper layers at 5 μm thick for a total thickness of 20 μm were fabricated, diced, and reliability tested, as described in the Fabrication, Dicing, and Reliability Testing section. This thinner structure, called Sample 2, still follows the schematic in Figure 1 and the details are presented in Table VI. The experimental results after dicing, preconditioning, 50 temperature cycles, and 00 temperature cycles for Sample 2 are shown in Table VII. Table VII uses the same status indicator color scheme as in Table II. Table VI: Details for samples fabricated for model validation. Figure 9: SEM of glass interposer after cracking failure: (a) schematic of interposer surface near edge showing SEM regions, (b) zoomed-out, (c) edge, (d) corner, and (e) corner zoomed-in Metal Layers Glass Thickness [μm] Polymer Thickness [μm] Copper thickness [μm] Passivation thickness [μm] Sample

7 Table VII: Experimental results from fabrication, dicing, and temperature cycling to validate modeling predictions. Sample Type - Number After dicing After Preconditioning After 50 cycles After 00 cycles 2 D4 Green Yellow Yellow Yellow 2 E5 Green Yellow Yellow Yellow Similar to the first round of samples, the new samples showed expected dicing signature after dicing and minor interface failure after preconditioning. However, at 50 temperature cycles, the new samples, 2-D4 and 2-E5, did not fail. These samples continued thermal cycling and passed 00 cycles without major failure. This is the result predicted by modeling, that the new samples would not fail despite the interfacial adhesion failure. Additional samples and other solution methods are being explored and tested, and the results from those studies will be reported in a future publication. Conclusions In this work, glass cracking failure was studied through finite-element modeling and experimental work including reliability testing. Glass cracking failure starts with dicing induced defects, which may grow due to interface adhesion failure. Glass cracks cohesively due to the existing defect or crack size and thermo-mechanical stresses induced by CTE mismatch. Energy release rate was used to predict the occurrence of crack propagation, and it was seen that thicker build up layers would increase the total energy release rate. Thinner build ups are seen to reduce the energy available for crack propagation such that glass cracking failure would not occur, even in the presence of a large initial defect. Based on this work, glass panels with four polymer layers of 40 μm total thickness and four copper layers of 20 μm total thickness have been successfully fabricated, diced, pre-conditioned, and thermal cycled over 00 cycles without any major failure or glass cracking. Additional experiments and simulations are currently underway to further enhance the robustness of glass substrates. Acknowledgments This work was supported by funding from the Low Cost Glass Interposers and Packages global industry consortium at Georgia Tech 3D Systems Packaging Research Center. The authors would like to thank Mikael Broas, Makoto Kobayashi, Xian Qin, Dr. Sathyanarayanan Raghavan, Dr. Vanessa Smet, Yutaka Takagi, Christine Taylor, Jialing Tong, and Laura Wambera for their valuable help and support. References 1. Y. Sato, S. Sitaraman, V. Sukumaran, B. Chou, J. Min, M. Ono, et al., "Ultra-Miniaturized and Surface-mountable Glass-based 3D IPAC Packages for RF Modules," presented at the ECTC, X. Qin, N. Kumbhat, V. Sundaram, and R. Tummala, "Highly-Reliable Silicon an Glass Interposers-to-Printed Wiring Board SMT Interconnections: Modeling, Design, Fabrication, and Reliability," presented at the ECTC, V. Sundaram, Y. Sato, T. Seki, Y. Takagi, V. Smet, M. Kobayashi, et al., "First Demonstration of a Surface Mountable, Ultra-Thin Glass BGA Package for Smart Mobile Logic Devices," presented at the ECTC, R. Furuya, F. Liu, H. Lu, H. Deng, T. Ando, V. Sundaram, et al., "2um RDL Wiring Using Dry Film Photoresists and 5um RDL via by Projection Lithography for Demonstration of Low Cost 2.5D Panel-based Glasss and Organic Interposers," presented at the ECTC, S. R. McCann, V. Sundaram, R. Tummala, and S. K. Sitaraman, "Flip-Chip on Glass for Low Warpag," in ECTC, V. Smet, M. Kobayashi, T. Wang, P. M. Raj, and R. Tummala, "A New Era in Manufacturable, Low- Temperature and Ultra-Fine Pitch Cu Interconnections and Assembly Without Solders," presented at the ECTC, S. Cho, Y. Joshi, V. Sundaram, Y. Sato, and R. Tummala, "Comparison of Thermal Performance between Glass and Silicon Interposers," presented at the ECTC, F. Wei, "Empirical Investigations on Die Edge Defects Reductions in Die Singulation Processes for Glass-Panel Based Interposers for Advanced Packaging," presented at the ECTC, N. Koizumi, "Basic Study of Packaging Structure Using Glass Material," in GIT, J. Standard, "J-STD-020D: Moisture/Reflow Sensitivity Classification for Nonhermetic Solid State Surface Mount Devices," J. Standard, "JESD22-A113: Preconditioning of Nonhermetic Surface Mount Devices Prior to Reliability Testing," J. Standard, "JESD22-A4-C temperature cycling," J. T. Tzeng, "Predictions and Experimental Verification of Residual Stresses in Thermoplastic Composite Cylinders," presented at the American Society for Composites 8th Technical Conference, J. R. Rice, "A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks," J. Applied Mechanics, vol. June, pp , J. R. Rice, P. C. Paris, and J. G. Merkle, "Some Further Results of J-Integral Analysis and Estimates," in ASTM STP 536, 1973, pp E. F. Rybickiand and M. F. Kanninen, "Finite element calculation of stress intensity factors by modified crack closure integral," Eng. Frac. Mech., vol. 9, pp , M.-Y. He and J. W. Hutchinson, "Kinking of a Crack Out of an Interface," J. Applied Mechanics, vol. 56, pp , M.-Y. He, A. Bartlett, A. G. Evans, and J. W. Hutchinson, "Kinking of a Crack Out of an Interface: Role of In-Plane Stress," J. American Ceramic Society, vol. 74, pp , A. Agrawal and A. M. Karlsson, "Obtaining Mode Mixity for a Bimaterial Interface Cracking Using the Virtual Crack Closure Technique," Int. J. Fracture, vol. 141, pp , J. Wang and C. Zhang, "Energy Release Rate and Phase Angle of Delaminatino in Sandwich Beams and Symmetric Adhesively Bonded Joints," I. J. Solids Structures, vol. 46, pp , S. M. Wiederhorn and L. H. Bolz, "Stress Corrosion and Static Fatigue of Glass," J. American Ceramic Society, vol. 53, pp ,

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