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Free-edge effects around holes in composite laminates Goonetilleke, Hemaguptha Dharmaraj 1986

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F R E E - E D G E EFFECTS A R O U N D HOLES IN COMPOSITE LAMINATES by H E M A G U P T H A D H A R M A R A J G O O N E T I L L E K E B.Sc. (Eng) (Hons), The University of Sri Lanka, 1973 M.A.Sc, The University of British Columbia, 1982 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T O F THE REQUIREMENTS F O R THE D E G R E E OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES Department of Metallurgical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY O F BRITSH C O L U M B I A N O V E M B E R 1986 ® Hemaguptha Dharmaraj Goonetilleke, 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. _ , M e t a l l u r g i c a l Engineering Department of _ _ The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date December 10, 1986. DE-6(3/81) ii ABSTRACT The free-edge effect around holes in composite laminates has recieved less attention than the straight free-edge problem. Mathematical analysis of free-edge stresses around holes have mostly been numerical. The present work develops a simple approximate solution of the hole problem which allows for low cost computation. The method assumes that only the deviations of the ply stresses from the homogeneous plate solution of in-plane stresses around holes contribute to the interlaminar effects. It is then possible to use an equilibrium argument to calculate the interlaminar stresses at the hole boundary. The results obtained show good agreement with numerical results from the literature for a wide range of laminates, predicting the general shapes and signs of interlaminar stress' distributions reasonably well. Experimental observations of delamination found in the literature also agree with the present results. An experimental study of the damage development around holes under quasi-static loading for a number of different laminates is reported. The delamination observed at the hole boundaries are found to be in good qualitative agreement A simple semi-quantitative correlation between these results and a stress combination function of the three interlaminar stress components is also derived. The problems associated with the development of reliable methods of delamination prediction are also discussed. TABLE OF CONTENTS iii : Page Abstract ii Table of Contents iii List of Tables vi List of Figures vii Acknowledgement xiii Chapter 1. Introduction 1 2. Literature Survey 7 2.1 Straight Free-edge effects 8 2.2 Curved Free-Edge Effects 12 3. Mathematical Analysis 26 3.1 Method of Analysis 28 3.2 2 -D Approximation of the hole problem 33 3.3 Free-body Analysis of Interlaminar Stresses 34 3.3.1 Transverse ply Stresses generating interlaminar stresses 35 3.4 Interlaminar Stress Distribution 39 iv 3.4.1 Interlaminar normal stress a z 39 3.4.2 Interlaminar shear stress r 40 3.4.3 Interlaminar shear stress T 43 3.5 Solution of In-Plane Stresses 45 3.5.1 Isotropic Solution 45 3.5.1 Orthotropic Solution 47 3.6 Residual Thermal Stresses 49 4. Comparisons With Literature 60 4.1 Theoretical Comparisons 60 4.1.1 Raju and Crews (1982) 62 4.1.2 Rybicki and Schmueser (1978) 66 4.1.3 Whitcomb (1981) 70 4.1.4 Tang (1977) 72 4.2 Experimental Comparisons 75 4.2.1 Whitcomb (1981) 77 4.2.2 Kress and Stinchcomb (1985) 80 5. Experimental Observations 126 5.1 Introduction 126 5.2 Experimental Procedure 127 5.2.1 Specimen Preparation 128 5.2.2 Testing and Observations 129 5.3 Results 131 5.3.1 [ 0 2 / 9 0 2 / ± 3 0 ] s and [ ± 3 0 / 9 0 2 / 0 2 ] s laminates 131 5.3.2 [ 0 2 / ± 4 5 ] s and [ ± 4 5 / 0 2 ] s laminates 138 5.3.3 [ 0 2 / ± 3 0 ] s laminate 142 5.3.4 [0/90] s and [90/0] s laminates 144 5.3.5 [45/0/-45/90] s laminate 146 5.4 Discussion 149 6. Summary and Conclusions 219 References 223 vi LIST OF TABLES Page I. Analytical studies of free-edge interlaminar stresses around holes by different authors 24 II. In-plane ply elastic properties used by different authors in stress calculations 125 III. In-plane ply elastic properties used in the present stress calculations 214 IV. Comparison of delamination with interlaminar stresses at different interfaces in a [02/902/ ± 3 0 ] s graphite/epoxy specimen 215 V. Comparison of delamination with interlaminar stresses at different interfaces in a [±30 /902^2^s graphite/epoxy specimen 217 vii LIST OF FIGURES Page 1.1. Delarnination at a hole boundary in a composite laminate. 6 2.1. Distribution of T Q Z obtained by several authors for a [0/90] s boron/epoxy laminate. 22 2.2. Distribution of a z obtained by different authors for a [0/90] s graphite/epoxy laminate. 23 3.1. Laminate configuration. 52 3.2. Straight edge approximation of a laminate hole. 52 3.3. Stresses used in point stress approximation. 53 3.4. Stresses used in average stress approximation. 54 3.5. Flow diagram of the method of analysis. 55 3.6. Free-body equilibrium diagram. 56 3.7. Distribution of a z near the free-edge. 57 3.8. Approximate distribution of r ,„ compatible with the assumed distribution of az. 58 3.9. Approximate distribution of near the free-edge. 59 4.1. Effect of IIa on az at z = h in a [90/0] s laminate using Point Stress method. ( d = t ) 85 4.2. Effect of IIa on o z at z = h in a [90/0] s laminate using Average Stress method. ( d = t ) 86 Effect of II a on a at z = h in a [90/0] s laminate using Modified Point Stress method. ( d — t ) Effect of II a on at z = h in a [90/0] s laminate, using Modified Average stress method. ( d — t ) Effect of IIa on a z at z = h in a [90/0] s laminate, using Point Stress method. ( d = I ) Effect of IIa on a at z = h in a [90/0] s laminate using Average Stress method. ( < / = / ) Effect of //a on a z at z = h in a [90/0] s laminate using Modified Point Stress method. ( d = I ) Effect of IIa on a at z = h in a [90/0] s laminate, using Modified Average stress method. ( d = / ) Present results compared with numerical solution for [90/0] s laminate at z = h. Results of normalized Modified Average Stress method compared with numerical solution for [90/0] s laminate at z = h. Present results of T Q Z distribution at z = h in a [0/90] s laminate compared with the solution of Raju and Crews Present results of T Q Z distribution at z = h in a [90/0] s laminate compared with the solution of Raju and Crews Present results compared with numerical solution for a [ 0 2 / ± 3 0 / T 3 0 ] s laminate at z = 0. Results of normalized Modified Average Stress method compared with numerical solution for a [02/ i"30/T 30] s laminate at z = 0. 4.15. Present results of interlaminar normal stress distribution at midpane in a [ 0 2 / ^ 4 5 / ^ 4 5 ] s laminate compared with the solution of Rybicki and Schmueser (1977). 4.16. Present results of interlaminar normal stress distribution at midpane in a [ 0 2 / ± 6 0 / T 6 0 ] s laminate compared with the solution of Rybicki and Schmueser (1977). 4.17. Present results of interlaminar normal stress distribution at midpane in a [ ± 3 0 / : P 3 0 / 9 0 2 ] s laminate compared with the solution of Rybicki and Schmueser (1977). 4.18. Present results of interlaminar normal stress distribution at midpane in a [ ± 4 5 / : £ 4 5 / 9 0 2 ] s laminate compared with the solution of Rybicki and Schmueser (1977). 4.19. Present results of interlaminar normal stress distribution at midpane in a [±6Q/T60 /902] s laminate compared with the solution of Rybicki and Schmueser (1977). 4.20. Present results compared with numerical solutions (Whitcomb, 1981) of interlaminar stress distributions across laminate thickness in [45/90/-45/0] s specimen, (in tension) 4.21. Present results compared with numerical solutions (Whitcomb, 1981) of interlaminar stress distributions across laminate thickness in [90/± 45/0] s specimen, (in compression) 4.22. (a)-(c). Present results compared with Tang's (1977) solution of interlaminar stresses at z = 0 in a [0/90] s laminate. 4.23. (a)-(c). Present results compared with Tang's (1977) solution of interlaminar stresses at z = 0 in a [ ± 4 5 ] s laminate. 4.24. C-scan records of various notched laminates after 10 7 tensile or compressive fatigue cycles. (Whitcomb, 1981). 4.25. Present solution of a z distribution in a [ 0 / ± 4 5 / 0 ] s laminate. (Whitcomb, 1981). 4.26. Present solution of az distribution in a [45/0/-45/0] s laminate. (Whitcomb, 1981). 4.27. Delamination location for [45/90/-45/0] s specimen subjected to tension fatigue. 4.28. Delamination location for [90/± 45/0] s specimen subjected to compression fatigue. 4.29. Radiographs of damage in a [0 /90 /± 45] s laminate after sequential loading to (a). 0.9 a ^ and (b). 1.05 o ^ . 4.30. (a)-(c). Interlaminar stress distribution in a [ 0 / 9 0 / ± 4 5 ] s laminate. (Kress and Stinchcomb, 1985). 4.31. Radiographs of damage in a [45/90/-45/0] s laminate after sequential loading to (a). 0.9 a^t and (b). 1.3 o ^ . 4.32. (a)-(c). Interlaminar stress distribution in a [45/90/-45/0] s laminate. (Kress and Stinchcomb, 1985). 4.33. Damage on 0/90, 90/45 and 45/-45 interfaces of a [ 0 / 9 0 / ± 4 5 ] s laminate. 4.34. Damage on 45/90, 90/-45 and -45/0 interfaces of a [45/90/-45/0] s laminate. 5.1. Specimen sections at the hole edge. 5.2. (a)-(e). Radiographs of damage in a [O2/9O2/ ± 3 0 ] s laminate after sequential loading to different stress levels. 5.3. (a)-(c). interlaminar stress distribution in a [ O 2 / 9 O 2 / ± 3 0 ] s laminate. 5.4. (a)-(f). Micrographs and replicas of sections showing delamination in a [ 0 2 / 9 0 2 / ± 3 0 ] s laminate. xi 5.5.(a)-(e). Radiographs of damage in a [±30/902/02]$ laminate after sequential loading to different stress levels. 165 5.6. (a)-(c). Interlaminar stress distributions in a [±30/902/02] s laminate. 167 5.7. (a)-(g). Micrographs and replicas of sections showing delamination in a [± 30/902 /02] s laminate. 170 5.8. Radiographs taken before loading a [ ±45/02]$ laminate. 177 5.9. Radiographs of damage in a [ 0 2 / ± 4 5 ] s laminate. 178 5.10. Radiographs of damage in a [ ± 4 5 / 0 2 ] s laminate. 179 5.11. (a)-(c). Interlaminar stress distributions in a [ 0 2 / ± 4 5 ] s laminate. 180 5.12. (a)-(c). Interlaminar stress distributions in a [±45/02]s laminate. 183 5.13. Fracture surfaces of [ 0 2 / ± 4 5 ] s and [ ± 4 5 / 0 2 ] s laminates. 186 5.14. (a)-(c). Replicas of sections showing delamination at different angular locations in a [±45/02ls laminate. 187 5.15. (a)-(c). Micrographs of sections showing delamination at different angular locations in a [ 0 2 / ± 3 0 ] s laminate. 188 5.16. (a)-(c). Interlaminar stress distributions in a [ 0 2 / ± 3 0 ] s laminate. 190 5.17. (a)-(e). Radiographs of damage in a [0/90]s laminate after sequential loading to different stress levels. 193 5.18. (a)-(e). Radiographs of damage in a [90/0] s laminate after sequential loading to different stress levels. 195 5.19. (a)-(c). Interlaminar stress distributions in a [0/90] s laminate. 197 5.20. (a)-(c). Interlaminar stress distributions in a [90/0] s laminate. 200 5.21. Micrographs of sections taken at 65° from the loading direction in [0/90] s and [90/0] s laminates. 203 5.22. (a)-(e). Radiographs of damage in a [45/0/-45/90] s laminate after sequential loading to different stress levels. 204 5.23. (a)-(c). Interlaminar stress distributions in a [45/0/-45/90] s laminate. 206 5.24. (a)-(c). Micrographs of sections showing delamination at different angular locations in a [45/0/-45/90] s laminate. 5.25. Comparison of delamination in [ O 2 / 9 O 2 / i 30] s laminate with stress combination function. 5.26. Comparison of delamination in [ ± 30/902/02]s laminate with stress combination function. 5.27. Plot of maximum delamination at each angle vs. stress combination function for [ O 2 / 9 O 2 / ± 3 0 ] s and [ ± 30/902 / 0 2 l s laminates. xiii ACKNOWLEDGEMENT I wish to express my sincere gratitude to Professor E. Teghtsoonian for his useful advice and guidance throughout the course of this study. I am also greatly indebted to Professor A. Poursartip for his help and guidance throughout this work, and much more for his encouragement which I so often needed. Thanks are also extended to my fellow graduate students and faculty members in the Department of Metallurgical Engineering. The assistance of the technical staff of this department is greatly appreciated. I am also grateful to the National Science and Engineering Research Council, Ottawa for providing financial support in the form of a research assistantship. 1 CHAPTER I INTRODUCTION Composite materials possess many attractive properties which make them useful in a large number of present day applications. -There are many advantages in using composites over conventional materials. Lighter but much stronger components for aircraft, automobiles, buildings and machinery are widely produced nowadays with fiber reinforced composite materials. The variety of combinations and arrangements of fibers and matrices combined with' the concept of lamination have provided opportunities for tailoring materials and structures to meet a wide range of design requirements. The high strength attainable in fine fibers of some materials provided the necessary technological background for early development of composite materials. Large numbers of these fibers are bonded together in suitable matrix materials to form useful structural materials. Among these, the laminated composites form an important class of materials; each laminate consisting of several layers of fibers stacked together in a particular sequence. Within a given layer the fibers are parallel and uniformly distributed. The different layers, on the other hand, are oriented in a number of different directions to give the laminate the desired mechanical and thermal properties. It is these materials that the present work is primarily concerned with. One mode of failure in such multilayered composites is delamination initiating at a free-edge. Delamination is the separation of the layers at the interface. Introduction 2 The onset of delamination at laminate free-edges can have significant influence on laminate behaviour, including failure. Around holes and other internal discontinuities delamination can propagate in any direction as shown in Fig. 1.1, depending on the local stress distribution. Delamination is considered to result from a complex three-dimensional stress state found along the edges of composite laminates. The out-of-plane components of these three-dimensional stresses -commonly known as "Interlaminar Stresses" - are found to be mainly responsible for delamination induced failure. Therefore the study of interlaminar stresses in composite laminates is very important in understanding laminate behaviour. The individual layers (or plies) of a composite laminate are orthotropic, and on a macroscopic level, may be considered homogeneous. A laminate comprised of several such layers thus contain discontinuities in material properties through the thickness. This gives rise to different layer stresses, as found from the classical lamination theory (CLT), when the laminate is under external loads. ( A ply exhibiting a higher stiffness in a given direction carries a greater percentage of the gross laminate stress in that direction). This can be contrasted with the case of a homogeneous plate which exhibits a nearly uniform stress distribution as a function of thickness. The stresses that develop within a laminated composite, in regions sufficiently remote from boundaries, satisfy the requirements of lamination theory. Near boundaries, or other geometrical discontinuities, the behaviour of a laminate may deviate significantly from that predicted by CLT. Though the gross laminate stresses satisfy the boundary conditions, the individual ply stresses do not themselves do so. In order to maintain equilibrium, interlaminar stresses are generated along ply interfaces within a very local region near the edges- a phenomenon unique to composite laminates and not generally observed in homogeneous solids. Introduction 3 The interlaminar stresses that develop in composite laminates exhibit rapid changes of gradient near the free-edges. This has made analytical treatment of interlaminar stresses extremely difficult Numerical methods, such as finite elements, which are commonly used for the purpose of evaluating interlaminar stresses along straight free-edges can be quite costly when used with curved free-edges. The circular boundary is one such free-edge commonly encountered. The complexity associated with the three-dimensional nature of the stress distribution coupled with the wide range of possible laminate and loading configurations require individual programming or mathematical formulations for each case. It is here that simple approximate methods of estimating interlaminar stresses can be helpful. The present work is an attempt to develop such a technique that can predict the nature of the interlaminar stress distribution around holes in composite laminates. Previous attempts to develop approximate methods for the hole problem have been unsuccesful, as they have considered the effect of all the in-plane ply stresses. On the other hand, the method proposed in this work assumes that the only components of ply stresses that contribute to the interlaminar effect are the deviations of the lamination theory ply stresses from the gross laminate stresses, which are calculated using the exact plane stress solution. The physical basis for this argument is as follows; In order to satisfy the traction free boundary conditions the respective laminate stresses of the exact plane stress solution must decay to zero, as the hole boundary is approached. This is similar to a homogeneous plate containing a circular hole. In the homogeneous plate however, the plate stresses which decay to zero do not generate out-of-plane boundary layer stresses, of the type observed in laminate edges. Thus, it can be argued, that even in the non-homogeneous laminated plate the laminate stresses corresponding to the exact solution must decay without giving rise to any interlaminar effects. The interlaminar stresses observed at the laminate hole must therefore be related to the ply stresses (predicted by the combined laminated plate theory and exact plane stress solution) Introduction 4 which are in excess of the overall laminate stresses, (determined by the exact solution). It was thus conceived that these excessive (or deviatoric) ply stresses can be used in a free-body analysis of the free-edge to determine the out-of-plane stresses. This method of approach was partly inspired by observations in the literature, as noted by Salamon (1980) in his review, that the sign of interlaminar normal stress around a hole is not open to intuitive equilibrium arguments. Contrary to his objection, it can be shown that the hole problem can be approximated by a 2-dimensional formulation, and that simple equilibrium arguments may be used wherever their use would be allowed in an equivalent straight edge problem. The present work describes the formulation of this method of approach in detail, in chapter III. A number of simplifying assumptions are made which do not significantly affect the calculations. The results are compared with numerical results from the literature as described in chapter IV. The method shows fair agreement for a wide range of laminates. It is found to have reasonable success in predicting the general shapes and signs of interlaminar stress distributions. As the ultimate purpose of free-edge stress analysis is to predict delamination initiation, it is important that experimental observations of delamination be compared with the stresses calculated. Delamination is rarely observed without matrix cracking, or splitting, which must influence the laminate behaviour significantly. The general lack of understanding of the effect of the interaction of different interlaminar stress components on delamination onset complicates the picture even more. Thus the cost effectiveness of more complex formulations of the free-edge stress analysis become questionable, and an approximate method becomes more attractive and reasonable. The results of the present calculations are compared with experimental observations of delamination and damage growth found in the literature in chapter IV. As the amount of experimental data in the open literature is limited and Introduction 5 fragmented, an experimental program was undertaken. The delamination pattern observed at the hole boundaries under quasi-static loading is compared with the stresses calculated using the present approach in chapter V. Good agreement is observed in terms of the predictions of delamination locations around the hole and through the thickness. A first attempt to combine the three interlaminar stress components to predict delamination initiation is presented. The associated problems in correlating experimental observations with theoretical results, be they exact, numerical, or approximate are discussed. Introduction 6 7 CHAPTER D LITERATURE SURVEY The failure characteristics of composite laminates have been an area of major interest in the design and development of composite materials. One mode of failure in fiber reinforced laminated plates is delamination initiating at a free-edge. Delamination at free-edges is observed under many different loading conditions, especially in compression. Experimental and analytical investigations found in the literature (Foye and Baker, 1970; Pipes, Kaminski, and Pagano, 1973; Pagano and Pipes, 1973; Soni and Kim, 1986) indicate the importance of interlaminar stresses in understanding delamination initiated failure. The dependence of laminate strength on the detailed stacking sequence of specific layer orientations has been explained by considerations of interlaminar stresses. (Pagano and Pipes, 1971; Whitney and Browning, 1972; Bjeletich, Crossman, and Warren, 1977; Whitney and Kim, 1977) It has been argued that the interlaminar stresses are instrumental in precipitating delamination and subsequent strength degrdation. The interlaminar stresses are the out-of-plane components of complex three-dimentional states of stress that exist along the edges of composite laminates. Such stress states are considered to result from the presence and interaction' of geometric discontinuties of the laminate and material discontinuities through the thickness. A three-dimensional stress state is found only within narrow boundary layer regions near the free-edges, and is therefore, known as the "free-edge effect" in composite laminates. Literature Survey 8 The significance of the free-edge effect has long been recognized as one of the most important aspects in laminate behaviour. The free edge problem in composite laminates has been investigated for both straight free-edges and curved free-edges. While the straight free-edges treated are usually the traction-free-edges of finite width laminates, curved free-edges are the boundaries of internal discontinuities. The circular hole provides the simplest form of a curved free-edge geometry. From an analytical point of view, one difference between the two geometries is that the straight free-edge effect can sometimes be treated as a two-dimensional problem in laminate elasticity. The curved free-edge effect on the other hand, is essentially a three-dimensional problem. 2.1 STRAIGHT F R E E - E D G E EFFECTS: -Most of the studies on free-edge effects have been on laminates with straight free-edges. Mathematical analysis of free-edge stresses in finite width laminates under uniform axial stress or strain has been the subject of extensive research. (Pipes and Pagano, 1970; Puppo and Evensen, 1970; Pagano and Pipes, 1971; Pagano, 1974; Tang, 1975; Tang and Levy, 1975; Hsu and Herakovich, 1977; Wang and Crossman, 1977, 1978; Pagano, 1978; Wang and Dickson, 1978; Spilker and Chou, 1980; Wang and Choi, 1982; Pagano and Soni, 1983; Johnson and Kemp, 1985; Conti and De Paulis, 1985) t According to Pagano (1978), analytical studies of the free-edge problem may be classified into two general catagories: approximate theories and numerical solutions. The approximate theory proposed by Puppo and Evensen (1970) was one of the first analyses performed for finite width laminates under generalized plane stresses. In order to t Because of the large number of publications available only a few important ones are listed here. Further references are found in those cited here. Literature Survey 9 study the distribution of interlaminar shear stresses in the laminate they modeled the laminate as a set of anisotropic layers separated by isotropic layers, with the isotropic layers acting as adhesives between the anisotropic layers. The equilibrium equations developed by neglecting the interlaminar normal stress in the laminate' were solved to obtain the interlaminar shear stresses. In order to obtain approximate analytical solutions of the two interlaminar shear stresses as well as the normal stress Tang (1975) employed a boundary-layer theory, developed by Reiss and Locke (1961) for isotropic elastic plates, to laminated composites. Using this theory Tang and Levy (1975) obtained results for the same laminate configuration considered by Puppo and Evensen (1970) and by Pipes and Pagano (1970), and observed good agreement The nature of the interlaminar stress distribution under uniform axial strain was treated by Pagano and Pipes (1971) using equilibrium arguments. They argued that the force and moment resultants which are statically equivalent to the interlaminar stresses on planes perpendicular to the thickness direction can be determined through simple equilibrium statements. The concepts of this work have subsequently been confirmed by Rybicki (1971) using finite elements. An approach to design laminates susceptible to delamination failure by interlaminar normal stress was later presented by Pagano and Pipes (1973) using the same concepts. Other approximate theories include the solution by Hsu and Herakovich (1977) based upon a perturbation analysis, and that developed by Pagano (1978) based upon an extension of Reissner's (1950) variational principle. While the solution by Hsu and Herakovich provided mathematical evidence for singular interlaminar shear stresses, Pagano's theory contained no edge singularities, an advantage from a problem solving view point Among the many analytical approaches, this ply mechanics technique developed by Pagano and extended later by Pagano and Soni (1983) yields very good estimates of interlaminar stresses near the free-edge. On a local basis, each layer is Literature Survey 10 treated as a homogeneous anisotropic body in equilibrium independent of the laminate. The laminates are studied on a global level using an assumed displacement model. This "Global-Local Laminate Variational Model" has recently been applied successfully to predict the onset of delamination in a qualitative fashion. (Kim and Soni, 1984; Soni and Kim, 1986) However, the application of this method requires extensive computational time even for simple laminate configurations. More recent approximate theories include the structural model developed by Johnson and Kemp (1985) and that by Conti and De Paulis (1985). Although the model of Johnson and Kemp is similar to the one developed by Pagano (1978), it has four dependent variables less per layer than Pagano's model, leading to a significant reduction in computational cost The model by Conti and De Paulis is an extension of the work by Pagano and Pipes (1973) based on a simple polynomial formulation. It provides a simple approximate method to evaluate all the interlaminar stresses at a straight free-edge. The results of an experimental program reported by them have shown that their model can predict approximate values of interlaminar stresses at the onset of delamination which are in qualitative agreement The use of numerical methods in the study of free-edge stresses was first made by Pipes and Pagano (1970). They used the method of finite difference to examine the distribution of stresses and displacements throughout a four layer, finite-width laminate, under uniform axial extension. The results of their solution have shown that significant interlaminar shear stresses at laminate free-edges are required to allow shear transfer between the layers of the laminate. They observed that this edge effect is only restricted to a boundary layer region approximately equal to one laminate thickness. The approximate nature of the • finite difference solution did not allow Pipes and Pagano to prove the existence, nor predict the magnitude of free-edge Literature Survey II stress singularities. However, most of the work following that placed emphasis on assessing the singular behaviour of stresses in regions close to the free-edge. The use of Finite element technique has since been quite popular among many authors (e.g., Rybicki, 1971; Isakson and Levy, 1971; Mau, Tong and Pian, 1972; Wang and Grossman, 1977; Spilker and Chou, 1980; Raju and Crews, 1981; Whitcomb and Raju, 1985). With the use of very fine finite-element grids and improved programming techniques there has been increasing evidence to suggest the singular nature of the boundary-layer stress field. The resulting interlaminar stresses are found to rise continuously with decreasing element size. Hence, accurate calculations of interlaminar stresses at the very edge, where they appear to display a singular behaviour, have not been possible. Convergence of solution has not been obtained even with more complex element stiffness formulations. However, as pointed out by Salamon (1980), a number of inconsistencies are found in the published numerical results. Using a triangular finite-element grid to model the 0/90 interface region of a cross-ply laminate, Wang and Crossman (1977) revealed an apparently bounded normal stress in the [90/0] s laminate and a singular normal stress in the [0/90] s laminate. Other investigators (e.g., Wang and Dickson, 1978; Herakovich, Nagarkar and O'Brien, 1979) claimed singular normal stresses for the first ply interface in both laminates. Spilker and Chou (1980) on the other hand, were able to show that all interlaminar stress components converge to finite values near the free-edge. In direct conflict with the notion of stress singularity, they showed that interlaminar normal stresses for the above laminates converge to finite magnitudes at the first ply interface. The solution was obtained through application of hybrid-stress finite elements in which traction-free-edge conditions are satisfied exactly. Similar inconsistencies are found in the literature for other laminate configurations and interlaminar shear stress distributions. Literature Survey 12 From the evidence found in the literature it is clear that the presence of elastic stress singularities can severely influence numerical solutions of the free-edge problem. Highly localized singular stresses make interpretation and comparison of various numerical results dubious, especially in the vicinity of the free-edge. In order to understand the precise nature of the free-edge stress fields Wang and Choi (1982) presented a solution procedure to evaluate the exact order of the stress singularity. Based on Lekhnitskii's (1963) complex-variable stress functions and basic relationships in anisotropic elasticity, a rigorous mathematical solution was developed to determine the free-edge stress singularity for both cross-ply and angle-ply laminates. They found that the magnitude of the free-edge stress singularity is a function of only material elastic constants and fiber orientations of adjacent plies in the laminate. 2.2 CURVED FREE-EDGE EFFECTS:-Because of the inherent complexity of the curved free edge geometry as compared to a straight free-edge , fewer studies have been done on curved free-edges. The three-dimensional nature of this problem has made mathematical analysis more difficult. The circular hole, the simplest of this kind, is the one which has received most attention. Except for the closed-form analytical solution given by Tang (1977), almost all studies have been numerical, mostly employing finite elements. In an early study by Levy, Armen and Whiteside (1972) a composite finite element constructed of orthotropic membranes separated by shear-resisting media was used to calculate the interlaminar shear stresses, t Results are given for the first ply interface at the free edge around holes in [ ± 4 5 ] s , [ ± 3 0 ] s and [0/90] s boron/epoxy laminates. Due to the nature of their solution results are not obtained for the interlaminar normal stress t This model was first developed by Puppo and Evensen (1970) for straight free- edges. Literature Survey 13 distribution. Extending the work by Rybicki (1971) on straight free-edges, Rybicki and Hopper (1973) developed a three-dimensional finite-element analysis based on a complementary energy formulation. They analyzed the stress distributions around holes in a number of different lay-ups made of boron/epoxy. Fig. 2.1 compares the results of their calculations of tangential interlaminar shear stress with that given by Levy et al. for a [0/90] s laminate. According to their analysis, a change in stacking sequence by reversing the plies does not alter the magnitudes, but only changes the sign of interlaminar stress distributions. Of particular interest in their work is the radial distribution of interlaminar normal stress at the mid-plane of a [90/0] s laminate. Though the resulting magnitude at the free-edge is finite (which is expected, as there is no material discontinuity at the mid-plane), a steep gradient in the distribution is observed very near the boundary. The radial distributions calculated at angles of 0 ° , 30° and 90° from the loading direction also show a significant variation of this behaviour around the hole. Dana and Barker (1974) employed a three-dimensional finite element analysis based on an isoparametric displacement formulation to calculate the interlaminar stresses. Both thick and thin laminates of boron/epoxy have been considered in determining the stresses through the thickness and around the hole. Interlaminar normal and tangential interlaminar shear stresses are calculated for cross-ply and [ ± 4 5 ] s configurations. Unfortunately their results can not be compared with others, since the distributions obtained by them are not for ply interfaces but for locations one-sixth the ply thickness away from these interfaces. The influence of stacking sequence, lay-up angle and the ratio of laminate-thickness-to-hole diameter on laminate behaviour has been examined by Rybicki and Schmueser (1976, 1978) using a three-dimensional finite element analysis. Results are Literature Survey 14 obtained for number of different graphite/epoxy laminates under uniform stress loading. Attention is focused only on the distribution of the interlaminar normal stress, az, at the laminate mid-plane. Numerical results are based on seven laminates with different lay-up angles and seven additional laminates obtained by changing the stacking sequence. These included 4-layer cross-ply laminates and a series of laminates with stacking sequences of the type [02/±6/Td]s, [±d/^e/02]s> [±8/:rd/%2]s and [902/±6/l-e]s where 6 is 30°, 45° and 60° . In order to simplify the computations Rybicki and Schmueser reduced the number of finite elements required by "smearing" the ( ± 6 V T 0 ) group of plies as a single material with effective modulus properties. They observed that in general, the stress distributions change sign as well as magnitude as the stacking sequence is changed. More importantly, they observed that unlike the straight free-edge case , the sign of a cannot always be predicted from the sign of the moment away from the circular hole by using equilibrium considerations with remote applied stresses. Most of the analyses described so far deal with relatively coarse elements which may not have yielded accurate results near singularities. In order to obtain accurate interlaminar stresses, Raju and Crews (1982) employed a three-dimensional finite element anlysis with a very fine arrangement of elements near the hole boundary. Stress distributions were obtained for [0/90] s and [90/0] s graphite/epoxy laminates, through the thickness at the hole boundary and along radial lines and around the hole at the interface between the crossed plies. Through the thickness distributions of interlaminar normal and shear stresses strongly suggest the existence of singularities at the first ply interface from the outside in both [0/90] s and [90/0] s laminates. This is quite different from what was observed at a straight free-edge. As mentioned earlier, Wang and Crossman (1977) found a non-singular behaviour in a distribution through the thickness in the [90/0] s laminate. Raju and Crews also observed an abrupt reversal in sign of the distribution of a through the laminate thickness at 90° from the loading direction, in Literature Survey 15 the [0/90] s laminate. The [90/0] s laminate does not however show this very abrupt reversal. As shown later, this allows the use of equilibrium arguments in calculating az in the [90/0] s laminate, but not in the [0/90] s laminate. Raju and Crews obtained circumferential distributions of interlaminar normal stress which are compressive for most of the region around the hole in both [0/90] s and [90/0] s laminates. On the other hand, the tangential interlaminar shear stress distributions are found to be identical for both laminates, but opposite in sign. Numerical results which compare well with these distributions have also been obtained by the same authors using a different approach. They used tangential strains calculated from an exact two-dimensional solution for the laminate hole, as input to a series of straight edge laminates approximating the hole boundary. Each straight edge laminate was then analyzed by a simpler finite element model of the plane normal to the edge to obtain stresses at the free-edge. According to the authors, this approximate procedure appears to be an economical method for estimating interlaminar stresses, and a suitable alternative to complex three-dimensional analyses of the hole problem. In a study of fatigue damage around holes, Whitcomb (1981) investigated the influence of interlaminar stresses on delamination in graphite/epoxy laminates. Results of a conventional three-dimentional finite element analysis are given for quasi-isotropic laminates of [90/ + 45/0] s and [45/90/-45/0] s configurations. Interlaminar normal and shear (tangential) stress distributions through the thickness at the edge of the hole are given for each laminate at three different angles. They found that delamination due to fatigue is more likely in areas where both the interlaminar shear and tensile normal stresses are high. Ericson, Persson, Carlsson and Gustavsson (1984), and Lucking, Hoa and Sankar (1984) have recently carried out interlaminar stress analysis of the hole boundary region in a [0/90] s composite laminate. In order to overcome the difficulties Literature Survey 16 due to stress singularities, singular Finite elements have been used by Ericson et al. This requires the determination of the order of stress singularity around the hole. The technique developed by Wang and Choi (1982) for straight edge laminates was used by dividing the hole edge into straight edge segments. Using the same material properties as used by Raju and Crews (1982) they obtained interlaminar stress distributions at the first ply interface from outside. According to them, the general shape of the interlaminar normal stress stress distribution of their solution agrees roughly with that obtained by Raju and Crews (see Fig. 2.2). Despite their claim of rough agreement, the signs are predicted incorrectly in regions near 0 ° , 30° and 90° from the loading direction. Their values are also found to be two orders of magnitude smaller than that predicted by Raju and Crews. This is attributed by Ericson et al. to the coarseness of the mesh used by them and the weak singularity of their solution. Considering the accuracy claimed by these authors of the method employed, the agreement of their solution with that of Raju and Crews seems poor. This is typical of the agreements observed in the literature among different methods of interlaminar stress calculations. Ericson et al. also present results of interlaminar shear stress distributions, which are not however, detailed enough to be compared with others. Lucking et al. examined the effect of hole radius-to-plate thickness ratio on interlaminar stress distributions in the [0/90] s laminate. Using a three-dimensional finite element analysis with a substructuring technique they have been able to obtain stress results for the hole boundary region. The radial distributions of the tangential interlaminar shear and the normal stresses are found to decay rapidly within about one laminate thickness from the hole edge. The radial interlaminar shear stress on the other hand is found to have a less steep gradient The stress distributions at the first ply interface around the hole obtained by these authors can be compared with those given by Raju and Crews, in spite of the slight difference in material properties. The interlaminar normal stress distribution has the same general shape reaching a maximum Literature Survey 17 around 60° from the loading direction. However the magnitude of the maximum stress is found to be less than one-third the value given by Raju and Crews. The same is true of the tangential interlaminar shear stress distribution. The shape of the distribution is similar to that obtained by Raju and Crews, with its maximum occuring around 68° from the load axis. However the magnitude of the maximum shear stress is also about one-third of the value given by Raju and Crews. The above discussion shows clearly the dependence of numerical results on the element mesh refinement and the particular method used. At a singularity, the accurate value of the stress is always sensitive to the mesh refinement Except for the work by Tang (1977) closed form analytical solutions are not generally available for the hole problem. Extending the work on straight free-edges (Tang, 1975) to curvilinear boundaries, he obtained solutions for the hole problem in a limited number of infinitely wide composite laminates. In agreement with Levy et al. (1971) and Rybicki and Hopper (1973), comparatively large tangential interlaminar shear stresses were found in [0/90] s boron/epoxy and [ ± 4 5 ] s graphite/epoxy laminates. Although the peaks and valleys of the distributions are not quite lined up at the same locations as shown in Fig. 2.1 for the [0/90] s laminate, the general trends are similar. Tang's solution of the radial interlaminar shear stress also shows similar agreement with that of Levy et al. However, the distribution of the mid-plane interlaminar normal stress for the same laminate does not agree with that of Rybicki and Hopper. It is also found by Tang that the size of the boundary layer region within which interlaminar stresses are generated is directly proportional to the radius to thickness ratio. A significant effort has thus been made towards understanding free-edge stresses along straight and curved boundaries. The work described above is summerized in Table I (page 24) for the convenience of the reader. Efforts to determine realistic values of interlaminar stresses have frequently been made difficult by the presence Literature Survey 18 of stress singularities, the strength of which varies around hole boundaries. Verification of different analytical methods is therefore possible only through experimental observations of delamination and strength behaviour in composite laminates. Regardless of the type of applied loading, free-edge delamination is considered to be initiated by interlaminar stresses. Many experimental investigations on free-edge effects have therefore been related to the prediction of the onset of delamination. (Foye and Baker, 1970; Whitney and Browning, 1972; Pagano, 1973; Whitcomb, 1981; Kim and Soni, 1984) Once initiated, a delamination can propagate rapidly across the laminate under increasing load, reducing the overall laminate strength. Since the stacking sequence of a laminate can have an influence on the associated interlaminar stress field, the dependence of laminate strength on stacking sequence has also been the subject of many experimental investigations. (Pagano and Pipes, 1971; pipes, Kaminski and Pagano, 1973; Daniel, Rowlands and Whiteside, 1974; Whitney and Kim, 1977; Bjeletich, Crossman and Warren, 1977; Kress and Stinchcomb, 1985) It has been reported by many that the laminate stacking sequence can affect the static strength as well as the fatigue life. While many investigators (e.g., Pagano and Pipes, 1973; Whitney and Browning, 1972) attribute the phenomenon of free-edge delamination to the presence of tensile interlaminar normal stresses, others (e.g., Whitcomb, 1981; Soni and Kim, 1986) consider it to be the product of some, or all, components of interlaminar stresses. Whitney and Browning used lamination theory to determine the sign of interlaminar normal stress and predicted delamination in number of different laminates. Pagano and Pipes presented a failure hypothesis to characterize the delamination mode of failure, such that the design of a delamination specimen can be made on the basis of interlaminar normal stress distribution. They assumed an approximate form of the interlaminar normal stress distribution across the laminate width. Literature Survey 19 Whitcomb (1981) observed delamination under fatigue loading conditions in areas where both the interlaminar shear and tensile normal stresses were high. However in some specimens delamination occured due to high interlaminar shear when the normal stresses were found to be compressive. Soni and Kim (1986) tested a large number of laminates exhibiting a dominant interlaminar shear stress to characterize the onset of delamination due to static tensile loading. They concluded that an understanding of the interaction between different interlaminar stress components is necessary for accurate prediction of delamination. In the straight edge problem the free-edge stress analysis would help determine the interfacial locations of the onset of delamination. In the hole problem the situation is much more complicated, having to determine the locations of delaminations through the thickness, as well as the angle around the hole. This is further complicated by the presence of in-plane stress concentrations found near discontinuities. Daniel, Rowlands and Whiteside (1974) observed a reduction in strength associated with high stress concentration near the hole boundaries for a range of material and stacking sequence variations. Alterations in the mode of failure of these laminates from catastrophic to noncatastrophic have been explained on the basis of the changes in interlaminar stresses calculated near a straight free-edge. Whitney and Kim (1977) also attempted to characterize the tensile strengths of quasi-isotropic laminates containing holes, on the basis of interlaminar stresses calculated near a straight free-edge. They observed tensile failure without any delamination at the hole boundary. Whitcomb (1981) observed good agreement between fatigue induced delamination locations and the stresses calculated using finite elements. The locations of delamination through the thickness at different angles around the hole were found by sectioning the specimen at each angle. The use of ultrasonic C-scan and X-ray radiography helped them find the direction of subsequent delamination growth. This could Literature Survey 20 not be predicted using stress analysis based on the undamaged specimen, since the altered stress distribution can change the direction of delamination propagation. He thus pointed out the need for better stress analysis techniques capable of calculating stresses after damage develops at the boundary. In a recent study by Kress and Stinchcomb (1985) on the fatigue response of laminates containing holes, the damage development was monitered using X-ray radiography and other non-destructive techniques. Some specimens were deplied after specific numbers of cycles to determine the distribution of damage in each ply around the hole. It is reported that their observations on the location of initial delamination agree with the results of a strain energy release rate analysis by O'Brien and Raju (1984). This approach considers free-edge delamination from a fracture mechanics point of view, which is fundamentally different from the interlaminar stress criteria used by others. All the above experimental works have reported some amount of matrix cracking and splitting prior to or during delamination growth. This is found to be a major obstacle to the development of accurate methods of interlaminar stress calculations. It is clear that the verification of the validity of different analytical methods is possible through the experimental observations of delamination onset and the characterization of laminate static and fatigue strengths. However, a number of problems need to be resolved before making progress in this direction. First, a reliable mixed mode failure criterion which takes into account the influence of different interlaminar stress components is required. Along with such criteria, reliable experimental techniques are also needed to determine the associated interlaminar strengths. Second, the influence of secondary damage mechanisms such as splitting and matrix cracking has not been well understood. Without such an understanding it would be difficult to analyse the various Literature Survey 21 interacting relationships between delamination and laminate behaviour including failure. Finally, improved experimental techniques of assessing different modes of damage in composite laminates is essential. Tang 1977 Rybicki and Hopper 1973 Levy et al 1972 Fig. 2.1. Distribution of TQZ obtained by several authors for a [0/90] s . boron/epoxy laminate. / ^ % N Ericson et al. 1972 vz/oQ x 10 N b CO V) £ o 00 o "o E o z o c E 5-CM 6" Raju and Crews 1982 Fig. 2.2. Distribution of a z obtained by different authors for a [0/90] s graphite/epoxy laminate. og = applied stress 24 TABLE I . Analytical studies of free-edge interlaminar stresses around holes by different authors:-A U T H O R M E T H O D USED LAMINATES STUDIED Levey, Armen and Composite Finite Whiteside (1972) Elements [ ± 4 5 ] s , [ ± 3 0 ] s and [0/90] s boron/epoxy Rybicki and Hopper (1973) Dana and Barker (1974) 3 -D Finite Element 3 -D Finite Elements [0/90] s boron/epoxy [0/90] s and [ ± 4 5 ] s boron/epoxy Tang (1977) Boundary Layer Theory [0/90] s boron/epoxy and [ ± 4 5 ] s graphite/epoxy Rybicki and Schmueser (1976, 1978) 3-D Finite Elements [0/90] s , [90/0] s [±9/j-e/o2]s, [±d/*d/902]s and [902/±6/3-9]s where 6 = 30°, 45° and 60°. graphite/epoxy 25 T A B L L E I Contd., A U T H O R Raju and Crews (1982) Ericson, Persson, Carlsson and Gustavsson (1984) Lucking, Hoa and Sankar (1984) Sankar (1984) M E T H O D USED 3 -D Finite Elements Singular Finite Elements 3 -D Finite Elements LAMINATES STUDIED [0/90] s and [90/0] s graphite/epoxy [0/90] s graphite/epoxy [0/90] s graphite/epoxy 26 CHAPTER m MATHEMATICAL ANALYSIS This chapter presents the mathematical analysis used to calculate the interlaminar stresses near the hole boundary of a composite laminate as a function of angle around the hole. The first section gives a general description of the method of analysis. Subsequent sections are devoted to a detailed, step by step description of the formulations involved. The method involves the reduction of the 3 -D hole problem to an equivalent approximate 2 -D problem. The laminate at the circular edge of the hole is approximated by a series of straight edge laminates which are simple to analyse mathematically. The aim is to reduce the hole problem to the familiar Pagano and Pipes (1973) form, and to use simple approximate solutions to predict the interlaminar stresses at the boundary. In particular, the method assumes that the components of ply stresses that contribute to the interlaminar effects are the deviations of the ply radial and shear stresses -induced by the lamination of dissimilar plies- from the gross laminate stresses near the hole boundary. These stresses are used in a free-body analysis of the free-edge to determine the out-of-plane stresses. The laminates considered in the present analysis are assumed to be plates of infinite extent They contain circular holes, whose countours are free of external forces. If the laminates are of finite width, the adjacent straight free edges disturb the Mathematical Analysis 27 elastic fields around the holes, so that, exact solutions of in-plane laminate stresses within the theory of plane elasticity are not feasible. In the present analysis, however, it is assumed that the dimensions of the plate and hole are such that conditions of infinite plate width are met Such an assumption allows for exact solutions of inplane laminate stresses and subsequent evaluation of interlaminar effects near holes. For simplicity, only uniaxial loading of laminates is considered here. Load is applied along the laminate principal longitudinal axis at infinity. This results in a uniform distribution of laminate stresses, in regions sufficiently remote from the hole boundary. Closer to the hole, a non-uniform in-plane laminate stress field exists. In the present analysis, the assumptions of classical lamination theory are taken to be valid throughout the laminate. Classical lamination theory based on the Kirchhoff hypothesis assumes a displacement field which is uniform across the entire laminate thickness. The hypothesis ignores shear deformations of layers with respect to each other. The assumption is made that any line perpendicular to the mid-plane before deformation remains perpendicular to the mid-plane after deformation, and it suffers neither extension nor contraction. As a result, the shear strains associated with the thickness direction and the normal strain in the thickness direction become zero. For thin plates, such as the individual layers of a laminated composite these assumptions result in the existence of a plane stress state. Thus, in establishing constitutive relations for the approximate straight edge laminates, the effects of interlaminar stresses near the hole boundary are first neglected. These stresses are however determined later from equilibrium considerations. Mathematical Analysis 28 3.1 METHOD OF ANALYSIS:-Consider the laminate shown in Fig. 3.1 with the origin of the X-Y-Z coordinate system located at its center. The fibers in various layers all lie in planes parallel to XY and the fiber orientation angle 0 ^ of each layer is measured positive in the anticlockwise direction from the X axis. The laminate contains a central circular hole of radius a, and is subjected to a tensile stress a along the X axis. © In order to understand the origin of interlaminar stresses in the above laminate, we compare its stress behaviour with that of a homogeneous plate of similar geometry' under inplane traction. Unlike the laminate, the homogeneous plate contains no material discontinuities through the thickness. The stress distribution at a given point within the homogeneous plate is nearly uniform in the thickness direction. The laminate on the other hand, consists of several layers of fibers of different orientations giving rise to different layer stresses that can be found by the application of classical lamination theory. The distribution of the thickness average in-plane stresses throughout the homogeneous plate, as well as the non-homogeneous laminated plate, can be determined from the theories of plane elasticity. The exact plane stress solution of elasticity gives a complete description of the changing stress field in the vicinity of the hole boundary, which satisfies the traction free boundary conditions. The in-plane radial and shear stress components of the exact plane stress solution must accordingly, decay to zero as the hole boundary is approached. This occurs in both the homogeneous and non-homogeneous plates. In the homogeneous plate these in-plane stresses decay without giving rise to any out-of-plane stresses at the circular edge. In the non-homogeneous laminated plate however, a different behaviour is observed. Though the thickness average Mathematical Analysis 29 laminate stresses satisfy the free-edge boundary conditions the individual ply stresses do not themselves do so. In order to satisfy the boundary conditions and maintain equilibrium, interlaminar stresses are generated along ply interfaces around the hole. The present approach of estimating interlaminar stresses around the hole is based on the above observations of the fundamental difference in behaviour between the two plates. The plate stresses which decay to zero in the homogeneous plate do not generate the type of out-of-plane boundary layer stresses observed in the non-homogeneous laminated plate. Since the corresponding laminate stresses in the non-homogeneous laminated plate must also decay to zero according to the exact plane stress solution, it can be argued that the interlaminar stresses observed in the laminate must arise from the respective ply stresses which are in excess of the above laminate stresses. Components of these ply stresses equal to the corresponding laminate stresses are expected to behave according to the plane stress solution, without generating interlaminar stresses. Thus, the interlaminar stresses are generated only by the remaining ply stresses which are in excess of the gross laminate stresses predicted by the plane stress solution. It is then possible to use these excessive (or deviatoric) ply stresses in a free-body analysis of the free-edge to determine the interlaminar stresses. This method of approach was partly inspired by the observations in the literature, as noted by Salamon (1980) in his review, that the sign of interlaminar normal stress around a hole is not open to inuitive equilibrium arguments. Contrary to his objection, it can be shown that by employing the deviatoric ply stress components in a free body equilibrium analysis the hole problem can be approximated by a 2 - D formulation. In developing the above arguments it was first thought that the only ply stresses which contribute to the interlaminar effects are the ply radial and shear stresses induced by the lamination of dissimilar plies. This is analogous to the concept of using transverse ply stresses in the straight free-edge problem, as developed by Pagano Mathematical Analysis 30 and Pipes (1973). In the straight free-edge problem, it is the transverse ply stresses generated by the applied axial load on the laminate that give rise to the interlaminar effects. If additional transverse loads are superposed on the laminate, they will not affect the interlaminar stresses, as these loads are balanced by additional transverse ply stresses at the edge. In the same vein, it is not the laminate radial and shear stresses calculated by the exact plane stress solution that give rise to interlaminar stresses around a hole, but it is the additional ply radial and shear stresses due to lamination that do so. Away from the hole, the sum of these stresses through the thickness is zero, and there is no perturbation from the exact plane stress solution. At the hole boundary, however, these ply radial stresses must have decayed to zero from the values predicted by the combined laminated plate theory (LPT) and exact plane stress solution. The out-of-plane stresses will have increased to maintain force and moment equilibrium. The additional ply sresses due to lamination can thus be used in a free-body analysis to predict the interlaminar stresses around the hole. The application of the above concepts in a 2 - D equilibrium analysis results in values which are in poorer agreement with numerical results than those resulting from the previous approach. Furthermore, the use of deviatoric ply stresses rather than the additional ply stresses has a stronger justification on the basis of their physical reasoning. However, the method which employs deviatoric ply stresses was conceived after the evolution of the additional ply stress concept, and hence is referred to as the modified stress method. The following analysis describes the formulations of both approaches, though only the modified method is used in a greater part of the subsequent work. Thus it appears that when the ply stress components are correctly chosen the simple equilibrium argument holds even in the case of the hole problem. In the present analysis the laminate at the circular edge of the hole is approximated by a series of small straight edge laminates. A typical straight edge Mathematical Analysis 31 laminate approximating the circular edge at an angle 8 from the X axis is shown in Fig. 3.2. As will be shown later, the stress distribution along the radial plane in the laminate is related to the distribution of stresses within this straight edge laminate. Since the principal directions of the approximating laminate coincide with the tangential and radial directions of the hole, a cylindrical coordinate system is used to describe the straight edge laminates. The system of coordinates chosen is 8,r,z with the positive direction of r being away from the hole center. Thus the longitudinal and transverse directions of the straight edge laminate correspond to the tangential and radial directions respectively. The equivalent straight edge laminate just described is assumed to possess the stress distribution along the radial plane. The character of the stress field in the vicinity of the traction-free boundary can then be determined from an analysis of stresses in the straight edge laminate. We will begin by assuming that this straight edge laminate is subjected to biaxial stress loading, defined by a system of in-plane stresses which is uniform across the radial plane. The distribution of in-plane laminate stresses around a hole can be determined from the theories of plane elasticity. A detailed account of the exact solution of in-plane stresses for this problem is given in Sec. 3.5 using both isotropic and orthotropic theories. The stresses resulting from this solution are found to be non-uniform along the radial direction. We shall however require a set of uniform in-plane laminate stresses, derived from the non-uniform stresses, to be applied to the straight edge laminate. One method of estimating a set of uniform laminate stresses for the straight edge laminate is to take it as that predicted by the exact plane stress solution, at a point, distance / away from the boundary in the radial direction as shown in Fig. 3.3. Outside a circular boundary of radius a+l the in-plane ply stresses are predicted by the combined laminated plate theory and the exact plane stress solution. These stresses are Mathematical Analysis 32 assumed to deviate from this solution within the region defined by the boundary r = a+l. The in-plane stresses of the exact solution at this point will then become the uniform in-plane laminate stresses to which the equivalent straight edge laminate is subjected. For convenience, we will refer to this approach as the "Point Stress Method", and the distance / as the characteristic length. Alternatively, we may calculate the average values of in-plane stresses predicted by the exact plane stress solution over a certain radial distance /, and use them on the straight edge laminate ( see Fig. 3.4. ). The laminate is then subjected to a set of uniform in-plane laminate stresses given by the average values of stresses calculated over the characteristic length. This method of solution will be referred to as the "Average Stress Method". In order to evaluate the stresses at the free-edge, simple models of the form Pagano and Pipes (1973) are used. For a z , their approximate solution was considered appropriate. For TQZ and TRZ, approximate solutions were obtained through the use of equilibrium equations. The method of analysis described above is shown in the form of a flow diagram in Fig. 3.5. A detailed description of each step in the formulation is given in the following sections. The residual thermal stresses which affect the interlaminar stress distribution are also treated separately in the last section. Mathematical Analysis 33 3.2 2-D APPROXIMATION OF T H E H O L E P R O B L E M : -Once the laminate stresses are defined, the straight edge laminate can be analysed for individual layer stresses. It should be noted that the laminate stresses, which are assumed to be uniform throughout the laminate, are in fact, the through-thickness-averaged in-plane stresses, statically equivalent to the actual stress system on the laminate. The individual ply stresses, resulting from the application of such laminate stresses, can be computed using the classical laminated plate theory. The straight edge laminate is still symmetric with the fibers in individual layers oriented (P ^-6) degrees with respect to the tangential direction, which is considered as the principal longitudinal axis of the laminate. This symmetry property eliminates any bending-membrane coupling from the straight edge laminate and produces, as a result of the above mentioned bi-axial stresses, only stretching and shearing of the mid-plane. The mid-plane strains e°. of the straight edge laminate associated with these deformations are given by; e°. = tA'^o1. (iJ = 1,2,6), (3.1) where A * are the coefficients of the inverse extensional stiffness matrix of the straight edge laminate, a*, are the components of the uniform in-plane laminate stresses and t is the laminate thickness. Standard contracted notation has been used in (3.1) where, referring to Fig. 3.2; Ll JLf Ll L LI LI f*s -"\\ °1 = °6 : °2 = °r ; °6 = °rd ( 1 2 ) and summation is implied over the range of repeated subscripts. The coefficients of the inverse extensional stiffness matrix A^ are given by Mathematical Analysis 34 Au = hi (f (3.3) 1 k = l 'J wheie (J1. are the plane stress reduced stiffness coefficients of the k th layer, h is the ply thickness, and n is the total number of plies in the laminate. The individual ply stresses of the laminate can now be determined from lamination theory, as ok. = tf.e0. (IJ = 1,2,6), (3.4) with a (/' = 1,2,6), referring to the k th ply tangential, radial, and shear stresses respectively. It is assumed that plane stress lamination theory is recovered in the central region of the straight edge laminate, so that the ply stresses at a radial distance / are given by equation (3.4), / being the characteristic length mentioned earlier. We assume that the lamination ply stresses that exist at a radial distance / can be predicted by LPT, though, moving towards the hole boundary they may change in a manner not predicted by LPT. It is these stresses that are used in a free-body analysis of the free-edge to determine interlaminar stresses. 3.3 FREE-BODY ANALYSIS OF INTERLAMINAR STRESSES:-As shown in the previous sections the hole problem can be reduced to an equivalent 2-D problem through the use of approximating straight edge laminates. Within a given straight edge laminate the stress field is two dimensional and a function of only two space variables, that is r and z. While the stresses derived in equation (3.4) can vary through the laminate thickness, the radial and shear components of these ply stresses decay to zero as the hole boundary is approached, to satisfy the traction free boundary conditions. In the process interlaminar stresses are generated which reach a Mathematical Analysis 35 maximum at or near the hole boundary. The following is a description of the origin of interlaminar stresses and the free-body equilibrium analysis in estimating these stresses. 3.3.1 Transverse Ply Stresses Generating Interlaminar Stresses:-In the classical straight edge problem under uniaxial loading, it is the transverse ply stresses generated by the applied axial load on the laminate that give rise to interlaminar normal stress az and the shear stress T . The essential concepts of this mechanism, as first suggested by the Pagano and Pipes, are summarized in Fig. 3.6. Interlaminar normal and shear stresses are generated along the interfaces in order to maintain the force and moment equilibrium of different layers experiencing transverse stresses that are not balanced at the free-edges. The interlaminar normal stress changes sign at some point along the interface to produce a pure couple which balances the moment due to transverse ply and interlaminar shear stresses. In the biaxial straight edge laminate which approximate the circular edge, it is the additional ply radial stresses due to lamination that give rise to the above mentioned interlaminar stresses. The additional ply radial stresses which generate interlaminar stresses result from the laminate tangential and shear stress components. Since the k th ply radial stress a ^  is derived from equations (3.1) and (3.4) as; k r>k o °2 = Q2iei = tQ'i.A'Jo1 (3.5) 2i iJ J k the additional ply radial stress a 2 would be found by substracting out the component of this stress due to laminate radial stress, i.e; Mathematical Analysis 36 &, = o\- tQ^-.A'Jo^i 2 2 2i i2 2 = ' ^ f A ; + A'u %> (36) Which in the usual, expanded notation, can be written as, M2Al'6+Q22A2'61+Qk26A66^ ^6 ^ Just as the transverse ply stress, o y , in an uniaxially loaded straight edge laminate generates the interlaminar stresses oz and T YZ, the in-plane shear stress T gives rise to the interlaminar shear stress r ^ . The in-plane ply shear stresses generated by the applied axial load must decay to zero near the free edges to satisfy the traction free boundary conditions. In the hole problem, the approximating straight edge laminate is under bi-axial stress loading, and therefore, it is the additional ply shear stress r ^  induced by the laminate radial and tangential stresses that gives rise to J QZ . Since the k th ply shear stress T ^  is derived from (3.1) and (3.4) as; k Jk o 6 6i i = tQ^Ajh1 (ij = 1,2,6), (3.8) k k where in standard contracted notation a, refers to r . the additional ply shear o ra k stress is obtained by substracting out the component due to laminate shear stress o^. Thus, o 6 6 6i 16 6 = tQlt.(A~1aL-A~]oL,) ^6r ij j 16 6 Mathematical Analysis 37 = tQ^Ar/a1^ A72}oL2) (3.9) which is written in the expanded form as; M6A12+Qk26A2'2+Qk66A2'61) ^ (3-10) The modified approach of calculating interlaminar stresses on the basis of ply stress deviations is described in a similar manner. Here the ply stress deviations from the thickness average laminate stresses are assumed to be responsible for the out-of-plane stresses. In the case of the ply radial stresses the deviations are found by substracting out the laminate radial stress from the ply radial stresses. Thus, „k k L 62 = °2-°2 = tQk0.A"1oL- oL„ (3.11) 2i ij j z v ' k k where d ^  is now the ply radial stress deviation o r in the k th ply, which in the expanded form appear as; M2An+Qk22A2~2+Qk26A2~6^ ^ HQk]2Aj-6J+ Qk22A2'6J+ Q^A;/) tr^ - oLr (3.12) Similarly the ply shear stress deviations are found by substracting out the laminate shear stress from individual ply shear stresses. In the contracted notation this can be written as, Mathematical Analysis 38 (3.13) which in the expanded form appear as; (3.14) The distribution of in-plane laminate stresses resulting from the exact plane stress solution is non-uniform along the radial direction. We must therefore decide on some representative values for the laminate stresses to be applied to the approximate straight edge laminate. In the point stress method described earlier, the laminate stresses ah , and a are derived from the O r rd exact plane stress solution at a distance / from the hole boundary. These stresses are then used in the expression given in equation (3.7) and (3.10) to obtain the additional ply radial and shear stresses, or, in equation (3.12) and (3.14) to obtain the deviatoric ply radial and shear stresses. In the average stress method the laminate stresses are calculated by averaging , and over a characteristic length / in the radial direction. These averages of the exact in-plane laminate stresses form the system of bi-axial stresses to which the equivalent straight edge laminate is subjected. Thus, in evaluating the expressions given in equations (3.7), (3.10), (3.12) and (3.14) these average values are used. Mathematical Analysis 39 3.4 I N T E R L A M I N A R S T R E S S D I S T R I B U T I O N : 3.4.1 Interlaminar Normal Stress a :-The force and moment resultants which are statically equivalent to the interlaminar stresses oz and r ^ can now be determined through k simple equilibrium arguments using d . Let us first consider the interlaminar normal stress oz. The characteristic form of its distribution over a width d at z=const is shown for an uniaxially loaded laminate in Fig. 3.7.(a). (Pagano and Pipes, 1973) The distribution shown is confined to a narrow boundary layer region, of dimension comparable to the laminate thickness, and possesses a singularity at the intersection of the ply interface and the free-edge. The prediction of interlaminar normal stress at the free edge must be based upon an approximate solution of oz distribution. The approximate solution suggested by Pagano and Pipes is considered to be appropriate for the present analysis and is shown in Fig. 3.7.(b). As was pointed out earlier, the resultant of this stress distribution must be a pure couple in order to satisfy the moment equilibrium of the layers. The assumed uniform distribution of a z over two thirds of the boundary layer width d, and its linear variation over the remaining distance requires the magnitudes of a m and o'm to be related as, a' = o/5 (3.15) while a m is given by, o m = om(z) = 90 M(z) / 7d2 (3.16) where M(z) is the moment per unit length of the couple produced by the above distribution. The magnitude of this couple can be found in terms of the transverse Mathematical Analysis 40 ply stresses through free-body equilibrium analysis. Thus, M(z) = Y2 <V*>*^  (3-17) z where a i s the transverse ply stress at z = £ in an uniaxially loaded straight edge laminate. In the case of a bi-axially loaded straight edge laminate considered earlier, this becomes 5 ,^ the additional ply radial stress or the deviatoric ply radial stress. Since the value of this stress is constant through the thickness of any one layer, and we are at present interested only in the interfacial stress distributions, the integration in (3.17) can be replaced by an algebraic summation as; M(zf- = h2/2Z ol(2k- 21+1) (3.18) 1=1 r where M(zj/C is the moment per unit length of the couple produced at the interface between the k th and (k+1) th layers. Substituting this in (3.16), the k maximum interlaminar normal stress a at this interface is found as; m 2 k ok = L dl(2k-21+1) (3.19) m 7dJ i=i r 3.4.2 Interlaminar Shear Stress r •-An approximate solution for the distribution of r (or, TRZ in the case of the hole problem) along the ply interface is also required. A solution, which is compatible with Pagano and Pipes approximate solution for az, can be obtained through the use of equilibrium equations. The reduced form of the equilibrium equations for a laminate, in which, the stress components are Mathematical Analysis 41 independent of x gives, *LSX + | I 2 S = 0 oy oz | £ j + | L B = o (3.20) oz The second of the above equations can be rearranged in the following form ry*W = -l[2 I ? * * (3-21) where < is the thickness of the laminate. Let us consider a plane z=const within the top surface ply. If the basic form of the a y distribution in the y direction is assumed to remain same through the thickness of the ply, then Also, from the third of the equilibrium equations (3.20) we have, oz(z) = -Y2 Tf*dl (3.23) Substituting the expression given in (3.22) for t in the above equation, we obtain 1/2 z2n oz(z) = S ^J{t/2-%)di 1 z <>y2 Mathematical Analysis 42 which can be reduced to; oz(z) = l ^ ( t / 2 - z ) 2 (3.24) Equations (3.22) and (3.24) relate the distribution of interlaminar stresses r „_ and yz a. within the surface layer to that of a „. An expression for T in terms of a , * y yz z can now be obtained through the integration of (3.24), T? • Tph?i°*(z)dy ( 3 - 2 5 ) and by substitution of this result in (3.22), Tyz(z) = -777&TS°zVdy <3-26> (t/2-z) o z The integral on the right hand side of this equation is equivalent to the area under the oz curve. The characteristic form of the T distribution resulting from this analysis is shown in Fig. 3.8. The stresses are distributed over a boundary layer width d, with its maximum occurring at a distance of 5^/18 from the free-edge. The original az distribution and the characteristic form of the o y distribution resulting from the preceeding analysis are also included. It must be noted that the distribution of T shown here is y^ for a plane z=const located within the surface ply. Nevertheless, it is reasonable to assume that the character of this distribution remains unaltered through the entire laminate thickness. In order to simplify the calculation of the maximum stress, we will also approximate this to a triangular distribution as shown by the dashed line. The maximum value of T w can now be determined for any plane z=const through force equilibrium analysis of suitable free-body diagrams. Thus, Mathematical Analysis 43 Tm = TmV = 2 F<z)'d (3-27> where F(z) is the total shear force per unit length generated by the r yz distribution on this plane. In satisfying the conditions of equilibrium it is found to be given by t/2 F(z) = S oft)di (3.28) where oy(Z) is the transverse ply stress at z= £. In the case of the hole problem, this becomes 5^, the additional ply radial stress or the deviatoric ply radial stress of the approximating, bi-axially loaded straight edge laminate. Since the value of & is constant through the thickness of any one layer, the total shear force per unit length at an interface can be given as F(zf = hL » ' (3.29) 1=1 r Thus the maximum T RZ generated at the interface between the A: th and (k+1) th layers is obtained by substituting the above in (3.27), as; r k = 2hz bl (3.30) m d i=\ r 3.4.3 Interlaminar Shear Stress TXZ'~ If a simplified distribution of the ply shear stress, T , as shown in Fig. (3.9) is assumed across the laminate width a relation can be derived for the interlaminar shear stress T^ (or, TQZ in the case of the hole problem). The distribution given by Whitney, Daniel and Pipes (1982) assumes a parabolic variation of the in-plane shear stress, such that, Mathematical Analysis 44 Txy = Txy U -( y / d (3.31) Equilibrium equations (3.20) of the theory of elasticity for a laminate.of thickness t require 1 / 2 Zn-1 / 2 ? „ r * = 5 l ^ d Z t/2 , = 2j / r\ydi (3.32) This shows that the resulting distribution of r ^  is linear, with the maximum stress occuring at the free edge. The value of the maximum shear stress T N is given by, The integration in (3.33) can be replaced by an algebraic summation, if the stresses are to be determined at the ply interfaces. Since the k in-plane shear stress T of the k th ply remains constant through the ply xy thickness h, < = f fmlr* (334> where r is the maximum T v_ shear stress generated at the A: th interface, between n *z the it th and the (k+1) th layers. The Qz component of the interlaminar shear stress at the k hole boundary is thus computed by evaluating f ^ from (3.10) or (3.14) and by employing the expression (3.34) to yield: Mathematical Analysis 45 (3.35) 3.5 SOLUTION OF I N - P L A N E STRESSES The approximate formulation of the hole problem presented in the last section was based on the evaluation of in-plane laminate stresses near the hole boundary region. The assumption that the laminates are of infinite width makes the use of the exact plane stress solutions of the classical elasticity theory possible, in evaluating these stresses. The laminate stresses thus evaluated are the thickness average in-plane stresses OQ, or and r ^ , statically equivalent to the actual stress system on the laminate. The laminates may behave as single isotropic layers or anisotropic layers, depending on the effective elastic moduli. For instance, quasi-isotropic laminates exhibit nearly isotropic properties and therefore the in-plane stresses around holes in such laminates can be determined from the isotropic solution presented here. The other laminates considered here do behave as single orthotropic materials, requiring an orthotropic solution to determine the in-plane stresses. For those laminates, the anisotropic elastic solution put forward by Savin (1968), which is presented in this section was used. 3.5.1 Isotropic Solution The stress distribution around a hole of radius a in an infinitely wide isotropic plate is given by Timoshenko and Goodier (1934) as, Mathematical Analysis 46 °r = £ 2 + \ <'+^> «* » Tr0= - j(J-i^+^) sin 28 (3.36) where OQ, or and T^ are the plane stress components with respect to a system of polar coordinates defined by r and 8. Referring to Fig. 3.2, 8 is measured in the direction shown, while r is defined as the radial distance measured from the center of the hole. The stress p is applied at infinity along the X direction. In the point stress methods, the straight edge laminate stresses are taken to be that of (3.36) evaluated at r = a+l. In the average stress methods the expressions (3.36) are integrated over the transition length to obtain the average laminate stresses. The stress of the straight edge laminate in the tangential direction is thus found by, L 1 0 + 1 °e = 7 5 ° e d r 1 a = i^7TTT-{ ' + + ( - T ; ^ c o s 2 6 * (3-37) 21 (a+l) a+l Similarly, the laminate stresses and used in the average stress methods are found to be given by 2 2 oL = -^—^ {i - [ Lz* j c o s 2 6 } ( 3 3 8 ) r 2(a+l) (a+l)2 and, L = _p [(a+it-tf rd 21 (a+1)3 V ' Mathematical Analysis 47 3.5.2 Orthotropic Solution:-Let us consider an anisotropic plate of infinite extent with a circular hole of radius a whose contour is free of external forces. The plate is subjected to tension p along the X axis at infinity. The dimensions of the plate and hole are such that the conditions of infinite plate width are assumed. The plate geometry and the coordinate system are as shown in Fig. 3.1. The in-plane stress components a r , a v , r r v at any point (x,y) with respect to these coordinate •* y *y axes are given by Savin (1968) as, 2 , 2 ax = p +2 Re [s^ (zj) + s ^ ' (z^) oy = 2 Re [4>'(zj)+ V' (zj\ r x y = -2 Re [sj<p' (Z]) + s2V (zj] (3.40) The complex functions 4> (zj) and \p (z^ in the above expressions are given by, t(Zj) = -ma * W 2(srs2) ZJ+x/\Z-]-a2(l + sy} H Z 2 ) = JB °0r*f 2(srs2) Z2+v/{Zj-a2(l + sy} (3.41) where Sj and s2 are the complex roots of the equation cl/~ 2c16^ + (2c12+c66>s2- 2c26+ c22 = 0 <142) whose coefficients C^j s are the coefficients of the plate compliance matrix with respect to the coordinate axes. The complex variables Zj and Z2 in (3.40) and (3.41) are given by, Mathematical Analysis 48 Zj = x + sjy Z2 = x + s2y (3.43) The solutions of equations (3.40) through (3.43) yields the in-plane stresses around a hole in a composite laminate, when the coefficients of the laminate elastic compliance matrix are determined from LPT as; Cjj = lA.. U ij J (3.44) The stresses a Y , a v and T y v resulting from the exact solution are now transformed to polar coordinates using the following stress transformation; a, r °e — sin26 cos2e sin26 cos26 • sin 8 cos 6 sin 8 cos 8 2sin 8 cos 8 - 2sin 8 cos 8 cos28-sin28 xy (3.45) For orthotopic laminates employing the point stress method, the stresses o r, OQ and T ^  derived at a radial distance / from the hole boundary are taken as the uniform in-plane laminate stresses of the approximating straight edge laminate. For the average stress methods however, a closed form solution of the laminate stresses is not available. The stresses are averaged numerically over the radial distance /. In the present work Simpson's Rule is applied, which fits a second order polynomial into the radial distribution. The accuracy of the method improves greatly if a large number of intervals is used. Mathematical Analysis 49 3.6 RESIDUAL T H E R M A L STRESSES:-If a laminate is subjected to a constant temperature change AT from its cure temperature, thermal stresses are induced which alter the stress state everywhere, including near. the hole boundary. The changes that occur in the stress state are uniform throughout the laminate, except near the hole boundary, where interlaminar stresses are present These interlaminar stresses can still be derived using the methods suggested in Sec. 3.1., if the thermal effects are included in the approximate straight edge laminate stress analysis. Once the laminate stresses due to applied mechanical loads are defined for the approximating straight edge laminate, as outlined in Sec. 3.1 (using the exact plane stress solutions), it can be analysed for individual layer stresses incorporating both mechanical and thermal effects. The following treatment is for a typical straight edge laminate approximating the circular, edge at the hole boundary. from both mechanical and thermal loads are given by e , the ply stresses in each layer are found by LPT as; where a.j are the coefficients of the various layers with respect to the laminate principal axes, and A T is the temperature rise from the curing temperature. The other terms in (3.46) have the same meanings as described earlier in Sec. 3.2. Summing (3.46) through all the plies we obtain, If the net mid-plane strains in the straight edge laminate resulting (3.46) n (3.47) Mathematical Analysis SO where iVz- are the force resultants defined as the force per unit length. Equation (3.47) is usually written as Ni + Nj = Ai}e°. (3.48) T where N. are the thermal forces given by k=l l J 1 llj = h I CTna^T (3.49) Solving for the mid-plane strains and substituting in (3.46), the ply stresses in the k th ply are found to be given by, ok = (£••[*' \Nm + NT)- a.AT] (3.50) / ij j m m m J v / k The additional ply radial stress & which contributes to the r interlaminar normal stress oz and the interlaminar shear stress r r z is that which is generated by the tangential and shear laminate stresses ah. and „. This is found by u ru setting the laminate radial stress ( or, correspondingly the force resultant ) in T equation (3.50) to zero. But N ^, which is not allowed for by the plane stress solution k remains. Similarly, the additional ply shear stress f ^ which contributes to the interlaminar shear stress T Q Z on the other hand results from the laminate tangential and radial stresses OQ and and is thus found by setting N$ in the above expression to zero. Again, A 7 ^ contributes to the additional ply shear stress. Thus in both cases, given that the thermal stresses are always additional to the plane elasticity solution, all three thermal stress components contribute to each of the interlaminar stresses. In the modified stress methods the deviatoric ply stresses are found by first calculating the ply stresses due to all three components of laminate stresses from (3.50), and then substracting out the corresponding laminate stress. Thus, when calculating Mathematical Analysis 51 the deviatoric ply radial stress 5 , (allowing for both mechanical and thermal loads) the L k laminate radial stress a is substracted out from the ply radial stress a . The deviatoric r r r ply shear stress f ^  on the otherhand is calculated by substracting the laminate shear L k stress T „ from the ply shear stress T n . Thus, in both cases, all of the terms in Nm rd rd m T or N contribute to the interlaminar stresses. m Mathematical Analysis Fig. 3.1. laminate configuration. 3.2. Straight edge approximation of a laminate hole. Mathematical Analysis 54 Mathematical Analysis 55 2-D Approximation of the hole problem How additional ply stresses due to lamination generate interlaminar stresses Calculation of lar the straight edg Point and Avera, ninate stresses for e laminate using ge stress methods Models of approxi Stress die in the trans\ .mate interlaminar >tributions rerse direction Exact plane stress solution of in-plane laminate stresses around the hole Fig. 3:5. Flow diagram of the method of analysis. Mathematical Analysis 56 ^1 Mathematical Analysis 58 Mathematical Analysis 59 Fig. 3.9. Approximate distribution of r near the free-edge. 60 CHAPTER TV COMPARISONS WITH LITERATURE In this chapter, the results of the analytical technique presented in the previous Chapter are compared with theoretical and experimental results from the literature. Comparisons are made with the calculations of interlaminar stresses and observations of delamination damage obtained by several authors. A fair agreement was observed for a wide range of laminates, as discussed in the following sections. 4.1 THEORETICAL COMPARISONS:-There are very few numerical solutions for the hole problem in the literature, especially when compared to the large number of solutions for the straight edge problem. This is mainly due to the complexity of the hole problem. The stress fields around holes are fully three dimensional and functions of all three space variables. Solution of this problem, therefore, requires the use of numerical methods, such as finite element Although the costs associated with the formulation and the use of three dimensional finite element programs are high, such calculations offer the most popular means of laminate stress analysis The solutions given by Raju and Crews (1982), Rybicki and Schmueser (1978), and Whitcomb (1981) employ three dimensional finite element stress Comparisons with Literature 61 analysis to calculate the stresses around the hole. The closed-form analytical solution used by Tang (1977) employs an extension of a boundary layer theory developed by Reiss (1961) for isotropic elastic materials. The results of these solutions are compared with stresses calculated using the methods described in the previous Chapter. The results of Raju and Crews are compared first, since they present both interlaminar normal and shear stress distributions. The finite element mesh used in their formulation is increasingly finer closer to the free-edge at the ply interface for which the stresses are calculated. This is considered to be an added refinement that has greatly improved the accuracy of their calculations. In addition, the results of Raju and Crews are especially of interest, in that along with their 3-D solution they present a reduced 2 - D numerical solution which is the numerical method equivalent to the present approach. It must be pointed out that although the accuracy of finite element stress analysis can be effectively increased by element mesh refinement, it is not possible to achieve this at the very edge of a ply interface, where interlaminar stresses appear to display a singular nature. The presence of interlaminar stress singularities in multilayered composites have rigorously been proven by Wang and Choi (1982). Such singularities make stress calculations only tend toward accuracy, without convergence. From a practical point of view, this makes realistic estimates of interlaminar stresses at the free edge somewhat difficult, and any attempts to improve the solution accuracy through element mesh refinement (or other means) are superfluous. While the issue of convergence to the classical elasticity solution for a laminated structure remains unresolved, efforts to develop realistic descriptions of interlaminar stress fields are often being made. The present work is an attempt to estimate the relative magnitudes of interlaminar stresses and predict the general nature of the stress distribution around a circular hole. As such, the comparisons are made on the basis of relative changes that take place in the magnitudes and signs of interlaminar stresses as a function of angle around the hole. Comparisons with Literature 62 4.1.1 Raju and Crews (1982):-Interlaminar stress distributions have been calculated near a circular hole in [90/0] s and [0/90] s graphite/epoxy laminates by Raju and Crews, using a three-dimensional finite element analysis, based on a displacement formulation. We consider first the [90/0] s laminate subjected to a gross applied stress of o„ with the elastic properties as used by Raju and Crews (see Table II - page 125). Plotted in Figures 4.1, 4.2, 4.3 and 4.4 are the distributions of interlaminar normal stress o Jo _ for the z = h plane from the point, average, z 6 modified point and modified average stress methods. A fixed boundary layer width of one laminate thickness was used for the results shown in these Figures. For each method, the effect of assuming different characteristic lengths / is shown. The effect of varying the characteristic length, while keeping the boundary layer width d at one laminate thickness is similar in all four cases. Since the thickness of the laminate considered by Raju and Crews is 0.2 times the hole radius a, the ratio of d/a remains at 0.2. With l/a = 0, that is using the stresses on the hole boundary only, all four methods reduce to the same solution, and we are considering only the effect of the laminate circumferencial stress. As we allow the characteristic length to increase (from l/a = 0 to 1), the increasing laminate shear stress and changing laminate circumferential stress cause the distribution of o z to shift to the smaller angles and increase in magnitude. Al l four methods give essentially similar values except that the modified point and modified average approximations predict a sign change for 6 > 75° . Plotted in Figures 4.5, 4.6, 4.7 and 4.8 are the distributions of interlaminar normal stress ( o z / o ) for the same laminate using a boundary layer Comparisons with Literature 63 width d equal to the characteristic length /. The effects of varying the characteristic length together with the boundary layer width is shown in these Figures for each different method in contrast to the case for fixed boundary layer width. As the characteristic length is increased (from lla = 0.1 to 0.5) oz is seen to decrease in magnitude. The contribution of the increasing laminate shear stress and the changing laminate circumferential stress is apparently offset by the effects of increased boundary layer width; the net result being lower o along the free-edge. The results are similar in all four methods except for the sign change behaviour predicted by the modified point and modified average stress methods. Comparison of the results of oz discussed so far with the finite element solution by Raju and Crews shows that the general characteristics of the stress distribution is best predicted when both / and d are equal to one laminate thickness. Thus, in calculating interlaminar stresses, the condition I = d = t where t is the thickness of the laminate is always satisfied (unless otherwise noted) throughout the rest of the work. In Fig. 4.9, results from the four different approximations for / = d = 0.2a are compared with the finite element solution. The value of 0.2a corresponds to one laminate thickness for the Raju and Crews geometry. Agreement is fairly good over most of the quadrant, though both the point and average methods predict compressive stresses throughout the region. The modified methods of approximation on the other hand do predict the sign change that takes place at around 80°. It must be emphasized that the z = h plane is an interface between the 0° and 90° plies, and that the Raju and Crews solution indicates a singularity at this interface. Although the present solution cannot cope with a singularity, it appears to predict the same general shape, though with consistently lower magnitudes. Comparisons with Literature 64 Raju and Crews also present the distribution of oz for a [0/90] s laminate. These results are very similar to their [90/0] s values, both in magnitude and sign. The solution of the present work for a [0/90] s would be o equal in magnitude to the [90/0] s solution, but opposite in sign. However close inspection of the results in Raju and Crews shows that in the case of the [0/90] s laminate, not only does the interlaminar normal stress distribution through the thickness of the laminate become singular as it approaches the z = h interface but it also changes sign over a very short distance before the interface. The [90/0] s laminate does not show this very abrupt reversal, and therefore allows comparison. The [0/90] s behaviour appears not to be a 3 -D effect, but a result of having a very fine mesh in the region of interest. This is shown by the fact that their finite element 2-D results agree very well with their 3 -D results. It appears that it is possible, even in the 2 -D case, to have out-of-plane stresses at the free-edge different in sign to what an equlibrium argument would suggest Similar results exist in the straight free-edge literature. For example results from Wang and Crossman (1977) for cross-ply laminates show the same sign change behaviour at the z - h interface, but not at the z = 0 interface, where there is no discontinuity in material properties. Similarly, the Raju and Crews data show that at the z = 0 interface the [0/90] s laminate has an interlaminar stress of opposite sign to the [90/0] s laminate over most of the boundary. However, the mesh is very coarse at the z = 0 interface, and no detailed results are presented by them. Thus it appears that the problem lies not in reducing the 3-D problem to a 2 - D approximation, but in the 2 -D equilibrium argument However, there is a host of experimental evidence to back the equilibrium argument in the 2 - D case, and a great deal of use is made of i t Returning to the solution of oz for the [90/0] s laminate, we find that it is the modified average stress results which exhibit the best agreement Comparisons with Literature 65 when both / and d are equal to one laminate thickness. It has been reported by many, that the distance over which interlaminar effects occur is in the order of one laminate thickness.t Henceforth we will use a value of one laminate thickness for / and d in all comparisons. In order to better compare the shapes of the distributions, the stresses are sometimes normalized with respect to their maximum values, as shown in Fig. 4.10 for the [90/0] s laminate. This shows clearly that the essential characteristics of the distribution (eg., the location of the maximum stress and the range over which they are tensile) are predicted reasonably well by the modified average stress aproximation. It is however important to note that the absolute value of maximum oz for this lay-up given by Raju and Crews is approximately 3 times that obtained by the present approach. Raju and Crews also present interlaminar shear stress T Q Z distributions around the hole for both [0/90] s and [90/0] s laminates. These are compared with the results of the modified average stress calculation in Figs. 4.11 and 4.12. Shear stresses normalized with respect to their maxima are plotted in these Figures. Except for different signs, the T Q Z distributions of the finite element solution are identical for the two laminates, which is also true of the results obtained by the modified average stress method. The finite element solution predicts the maximum stress at about 75° while the present solution predicts it near 6 = 67°. The angle at which the sign of the shear stress changes is predicted within a degree or two from that predicted by the finite element solution. The present solution, using the modified average stress approximation, thus appears to predict the same general shape of the distribution, though the absolute value of the maximum stress given by finite element solution is approximately 5.3 times that of t As noted by Pagano and Pipes, this is also in agreement with a (loose) interpretation of Saint Venant's principle, since the L.T. stresses on any plane y = const, and extending throughout the entire thickness dimension are self-equilibrating. Comparisons with Literature 66 the present solution. The increase in this ratio of absolute magnitudes over that for the normal stress may partly be due to the stronger stress singularity observed by Raju and Crews for r ^ r 4.1.2 Rybicki and Schmueser (1978):-Using a three-dimensional finite element program, the distribution of interlaminar normal stress, o z , around a hole at the laminate midplane was studied by Rybicki and Schmueser for a series of Graphite /epoxy laminates of the type [02/±8/^8]s, [±8/+8/02]s, [±8/3-8/%2]s and [902/±8/+'6]s where 8 is 30°, 45° and 60° . t The material properties used in the analysis are given in Table II (page 125). All results are for the laminate midplane where there is no discontinuity in material properties, and thus no singularity is expected. Since a fairly coarse mesh was used by Rybicki and Schmueser, there is no indication of any such effects. Figures 4.13 and 4.14 show the results for the [ 0 2 / ± 3 0 / T 3 0 ] S laminate. Interlaminar normal stress is calculated using the present approximate methods with / = d = 1.2a which corresponds to one laminate thickness for all of the above lay-ups considered by Rybicki and Schmueser. Fig. 4.13 compares the computed results of o Jo _ of the four different methods with that of Rybicki and Schmueser. Agreement is reasonable with all four approximate methods predicting a sign change for oz at 8 > 60°. The modified average approximation shows the best agreement It is also clear that we consistently predict larger magnitudes than the numerical results. Figure 4.14 compares the modified average stress results normalized by the maximum stress with numerical results, t Rybicki and Schmueser modeled the (0$ and (902) plies as one material, and the (±8/^-8) plies as a single material witn effective modulus properties. Comparisons with Literature 67 which are also normalized. As evident from this Figure the general shapes of o z distribution around the hole are in rough agreement Rybicki and Schmueser also present the distribution for a [ ± 3 0 / : f 30/02] s lay-up, and though the sign of the distribution is reversed, it is not an exact mirror image as predicted by the present solution. They predict magnitudes for the [ ± 3 0 / ? 30/02] s laminate that are roughly double those of the [ 0 2 / ± 3 0 / T 3 0 ] s lay-up. Since the modified average stress method seems to predict the stress distribution better than the other three methods, this method will be used to calculate the interlaminar stresses in future, unless otherwise noted. It was thought initially (Goonetilleke, Poursatip and Teghtsoonian, 1985) that the point and average stress methods constituted the most satisfactory approximations. The modified stress methods at first were difficult to explain in terms of physical reasoning. However, it was found later that a sound physical argument can be presented to explain these methods also. It is based on the fact that the phenomenon of free-edge effect is found to occur only in multilayered laminates where there is discontinuity in material properties, and not in homogeneous solids in general. The different ply k k stresses o r and T ^  predicted by the combined laminated plate theory (LPT) and exact plane stress solution at a given point within a laminate result from the material discontinuity in the thickness direction. If the material were truly homogeneous through the thickness, as assumed by the plane stress solution, these stresses would simply be the laminate stresses and respectively. The r rv difference between these values can therefore be considered as the source of interlaminar stresses observed in composite laminates. The modified stress methods use these deviatoric ply stresses to calculate the interlaminar stresses. It is however important to note that though the modified average stress method shows the best Comparisons with Literature 68 agreement with numerical results, the other three methods are not always worse. With some laminates, the distribution is predicted nearly as well by one or more of the other three methods. Figures 4.15 through 4.19 show the results of the modified average stress calculations for the laminates of the type [02/5 6/^6]$ where 6 is 45° and 60°,and also for the [±6/36/902]$ type of laminates where 6 is 30°, 45° and 60°. The results of the present calculation for the laminates of the type [±6/3-6/02]$ ^ [902/56/+'6]s are not presented here, since they are the exact mirror images of the results shown in these Figures. Although of opposite sign, the results of the finite element solution for these lay-ups do not exhibit such mirror images, but rather the same general shapes with different stress magnitudes. Specific reference will be made to these results in the following paragraphs when comparing the curves in Figs. 4.15 to 4.19. Figure 4.15 compares the results for the [ f ^ / i : 4 5 / ^ 4 5 ^ laminate. Except for the hump observed at the centre, the present solution compares reasonbly well with the general character of the finite element stress distribution. Stresses are tensile (or compressive) within the same angular ranges as that predicted by the finite element solution. The absolute values are of the same order of magnitudes, although the present result is approximately 4 times that of the finite element solution at 6 = 35°, where the deviation is found to be largest For the [ ± 4 5 / + 4 5 / 0 2 ] s laminate, this deviation is much less since the finite element stresses are approximately twice those for the [02/±45/3- 45] s . Fig. 4.16 shows the results for the [ 0 2 / ± 6 0 / T 6 0 ] s laminate. As it is a quasi-isotropic laminate, inplane stresses are calculated using the isotropic solution. The comparison between the calculated results and the finite element result is very much similar to that of [ 0 2 / ± 4 5 / T 4 5 ] s which is described in the previous Comparisons with Literature 69 paragraph. As before, the comparison is better, especially in terms of absolute magnitudes, for the [ ± 6 0 / T 6 0 / 0 2 ] s laminate in which the previous stacking sequence is reversed. The results shown in Fig. 4.17 for the [±30/ : F-30/902]s laminate were also obtained using the isotropic solution. For this lay-up the shape of the stress distribution resulting from the Finite element analysis is predicted well by the present solution, o z remains tensile all around the hole and is minimum at 0 ° . Although the magnitudes of the present solution are consistently higher than that of the Finite element, it is not more than twice at any angle. For the [ 9 0 2 / ± 3 0 / T 3 0 ] s laminate, the present solution would predict the exact minor image of the [ ± 3 0 / T 3 0 / 9 0 2 ] s laminate stresses, whereas the solution by Rybicki and Schmueser, though of opposite sign, shows much less variation in magnitude as function of angle. Figures 4.18 and 4.19 show the results for the [ + 45/^45/902]s and [ ± 6 0 / + 60/ 9023 s laminates. Although of difTerent magnitudes, the variation of o around the hole predicted by the Finite element solution is essentially similar to that for the [ ± 3 0 / T 3 0 / 9 0 2 ] s laminate. This is also true of the present results, which compare reasonbly well with the finite element results. For the [ ± 4 5 / T 4 5 / 9 0 2 ] s laminate a good agreement was observed. At 0° , where the stresses are smallest, the value predicted by the present approach is only four times the numerical result and at 90°, where they are largest, it is less than 1.5 times the latter. The Finite element solution would predict the interlaminar normal stress at 90° to be approximately 65% of the applied stress, whereas the present solution estimates this at 92%. For the [902^ 45/:P45]s laminate with the reverse stacking order, the difference between the results becomes larger as the Comparisons with Literature 70 angle increases. Although the same general shape is observed, the finite element result exhibits a much less variation in magnitude as function of angle. For the [ ± 6 0 / T 6 0 / 9 0 2 ] S laminate shown in Fig. 4.19 the difference between the results remains approximately constant throughout Although the finite element result would predict compressive oz at laminate midplane for angles less than 28°, the present result indicates tensile oz for this region. However, compressive stresses are predicted for all angles, by both solutions, for the reverse stacking order in [ 9 0 2 / ± 6 0 / ^ 6 0 ] s laminate. While the present solution yields an exact mirror image of the result shown in Fig. 4.19, the finite element solution predicts a distribution that increases less rapidly than its counterpart In all twelve lay-ups given by Rybicki and Schmueser a good qualitative agreement was observed, demonstrating the possibility of evaluating interlaminar stresses around holes through equilibrium considerations. The method seems to have some success in predicting the approximate shape and sign of the stress distribution. The results reported by them are only midplane o z . No comparison could therefore be made with o z distribution at other interfaces or in the thickness direction. 4.1.3 Whitcomb (1981):-In an experimental and analytical study of fatigue damage in graphite/epoxy laminates Whitcomb has analysed interlaminar stress distribution around holes in two different stacking sequences; namely [45/90/-45/0] s and [90/± 45/0] s . The elastic properties of the zero deg plies of these laminates are given in Table II (page 125). Using a conventional three-dimensional finite element analysis he has analysed the region around the hole and compared the delamination locations with the stress distributions. The analytical results reported in his work do Comparisons with Literature 71 not include a complete description of interlaminar stresses around the hole, but show the distribution of oz and T Q Z through the thickness at three angular locations. These results are compared with stresses calculated using the modified average stress method for the two stacking sequences in Figs. 4.20 and 4.21. The calculated values of a z through the thickness at different interfaces are shown in Fig. 4.20 for the [45/90/-45/0] s laminate. The results of Whitcomb's finite element analysis are shown by the solid curves in these diagrams. Comparison is good at angles 90° and 175° from the loading direction. The sign and relative magnitude of o z is predicted reasonably well at each interface, locating correctly the interfaces with maximum stress. At 120° however, the agreement is poor especially with regard to absolute magnitudes. The sign of o z at the second interface from the outer layer does not agree with that resulting from the finite element analysis, although the basic shapes of the distributions (through the thickness) roughly agree. A similar result was observed for T @ Z distribution. The agreement was better at 90° and 175° than at 120°. At 90° , the high interlaminar shear stresses obtained by Whitcomb along the First and second interfaces are closely predicted by the present approach. At the third interface the stress changes sign and becomes negative, although the present result indicates only a slightly negative value, at 120° however, the signs are in poor agreement for the first two interfaces from outside. Nevertheless, the stresses calculated by the present method for these two interfaces are close to zero. The maximum shear stress at the third interface given by the finite element solution is predicted reasonably closely by the present calculations. Finally at 175°, a good agreement was observed in the sign and relative magnitude of interlaminar shear stress through the thickness. Comparisons with Literature 72 For the [90/ ± 45/0] s laminate shown in Fig.4.21 a qualitatively good agreement was observed. For the angles 90° and 120° from the loading direction the present calculation overestimates the magnitude of o z near the midplane, but estimates accurately the stresses at every interface for 160°. The signs and shapes of the distribution are predicted reasonably well for all three angles considered. The agreement between the computed results and the finite element solution is relatively good for the interlaminar shear stress TQZ distribution through the laminate thickness. Except for the two outermost interfaces at 120° from the loading direction, the signs are correctly predicted all throughout Even at these two interfaces the stresses calculated by the present method are nearly zero. The high interlaminar shear stresses predicted for the second interface at 90° and for the third interface at 120° are in good agreement, at 175° from the loading direction, the shear stresses become vanishingly small through the entire laminate thickness as predicted by both solutions. 4.1.4 Tang (1977):-In this work an extension of a boundary-layer theory, developed by Reiss (1961) for isotropic elastic plates, is used to obtain an analytical solution for the interlaminar stresses in laminated composites. The approach is based on a stress formulation. Results are given for a [0/90] s laminate made of boron/epoxy and for a [ ± 4 5 ] s laminate of graphite/epoxy. The elastic properties of the material used in the analysis are listed in Table II (page 125). The interlaminar normal stress o z is calculated at the midplane of the above laminate constructions, while the two shear stress components Comparisons with Literature 73 T Q Z and T R Z are derived at 0/90 and 45/-45 interfaces. It is not clear what the actual dimensions of the plate geometry used in the computations are, but a parallel study on the effect of t/a ratio on boundary-layer effects reported here assumes a range of values from 0.01 to 0.03 for t/a, the plate thickness-to-hole radius ratio. For the [0/90] s laminate containing a circular hole, the interlaminar stresses calculated by the modified average stress approximation are compared with those reported by Tang in Figs. 4.22 (a)-(c). The results of the present calculation seem to predict the same general shape of the distribution obtained by Tang for all stress components. For oz shown in Fig. 4.22 (a) the signs are predicted correctly. The maximum interlaminar normal stress is obtained almost at the same angular location as that given by Tang. A secondary peak in o z , analogous to that observed by Tang approximately 5° off the loading direction, is also obtained by the present calculation, but, at least 12° - 15° away from the loading direction. At 0° and 90° the absolute values of oz are almost equal, though at other angles lying in between they seem to differ by varying amounts. The results of the present calculation were particularly good with regard to the general shape of the interlaminar shear stress T Q Z distribution as shown in Fig. 4.22 (b). The sign and relative magnitude of T Q Z calculated by the present method is in good agreement with that given by Tang along the entire hole boundary. The absolute values of the present solution and the boundary-layer solution for the hole boundary are at a constant ratio of about 1:3. At 0 ° , 24° and 90° from the loading direction T Q Z becomes zero, as predicted by both solutions. In fact, this is expected along the 6=0° and 90° for a [0/90] s k lay-up, since the inplane. ply shear stress r ^ is zero at these two locations and T Q Z is a direct product due to the matching of the inplane ply shear stresses at Comparisons with Literature 74 the free edge. The comparison of the interlaminar shear stress T N distributions of the [0/90] s laminate in Fig. 4.22 (c) shows a similarity in shape and magnitude. Within one-quater of the hole boundary the distribution of T RZ exhibits two maxima, one near 0° and the other one close to 90° . This results from the present approximate solution as well as Tang's boundary-layer solution, though the exact angular location of each maximum predicted by the two solutions differ by about 5 ° . Further, the magnitudes of the two maxima resulting from these solutions are at the same ratio. The absolute values of T RZ are roughly equal around most of the boundary, although the solution by Tang predicts negative stresses in the range 20° to 45° . However, it is important to note that within this region T RZ is predicted to be less than 1% of the gross applied stress. It is also interesting to note that the magnitude of the maximum T R Z is approximately an order of magnitude less than the maximum T § Z shown in Fig. 4.22 (b). Thus, T RZ is found to be an insignificant interlaminar stress component for the [0/90] s laminate. As will be seen later, this is true for many other laminates of practical interest The comparison of interlaminar normal stress results for the [ + 45] s graphite/epoxy laminate shows poor agreement The results of the present calculation and the distribution of o z obtained by Tang are shown in Fig. 4.23 (a). Nevertheless, a good agreement was observed in the distribution of T Q Z as seen in Fig. 4.23 (b). Except for the difference in magnitude, the present solution agrees quite well with the boundary-layer solution given by Tang. The magnitudes are at a ratio of approximately 1:3 along the entire hole boundary. This is exactly the same ratio that was found to exist between the magnitudes of T Q Z in the [0/90] s laminate, (see Fig. 4.22 (b).) The stress becomes slightly negative in the Comparisons with Literature 75 neighbourhood of 6 = 4 0 ° . Unlike the result for [0/90] s laminate, high values of T Q Z is observed at 8 = 0° and 90° for the [ ± 4 5 ] s configuration, since there are k high inplane ply shear stresses r ^ at these two locations. Figure 4.23 (c) presents the results of the interlaminar shear stress component r r z . The solution given by Tang is also reproduced here. A certain ambiguity exists in these results, t and therefore, a direct comparison with the results of the present analysis is not attempted. Instead, the results of the present work are shown separately in this figure. Here the stresses are negative in the region marked by the minus (-) sign. The overall magnitudes of the stresses seem to compare quite well. The results would agree well if the stresses obtained by Tang in the region indicated by the plus( + ) sign are in fact negative, and the loading direction is parallel to the horizontal axis. The direction of loading in the present work is parallel to the vertical axes of the plots. 4.2 EXPERIMENTAL COMPARISONS :-It is generally accepted that the presence of high interlaminar stresses at laminate free edges cause delamination along ply interfaces. The literature on straight free-edge problem (Foye and Baker, 1970; Whitney and Browning, 1972; Soni and Kim, 1986) clearly indicates the importance of either or both interlaminar normal and shear stresses in predicting delamination. But, the exact form of the correlation that exists between interlaminar stress components and delamination is still not known precisely. The necessary conditions or failure criteria for delamination initiation and propagation are not, however, clearly established due to number of reasons. Among them are the difficulties associated with experimental detection and quantifiable assesment of delamination, the use t An attempt to contact the author and clarify the results was unsuccessfid. Comparisons with Literature 76 of a large number of specimen and loading geometries -making it a formidable task to conduct complete three-dimensional analysis of inplane and interlaminar behaviour-, and the presence of a number of different interactive failure modes which may be operative simultaneously. Related studies on the hole problem are rare and limited because of the greater complexity associated with theoretical analysis. Delamination prediction based on such analysis is even more limited since any type of damage initiated at the hole boundary may change the original stress field significantly. Nevertheless, delamination initiated failure modes under uniaxial tension of composite plates with circular holes are studied by a number of authors. Daniel, Rowlands, and Whiteside (1974), for example, investigated the influence of ply stacking sequence on the strength of laminated plates with circular holes and attributed the differences in strength to the differences in the state of interlaminar stresses near the boundary. Stacking sequences associated with tensile interlaminar normal stresses or high interlaminar shear stresses calculated at the straight free-edge resulted in laminates weaker by 10 to 20 percent than the corresponding alternate stacking sequence. Whitcomb (1981) studied fatigue damage development around holes in graphite/epoxy laminates by examining fatigue loaded specimens for damage type and location using light microscopy, ultrasonic C-scans and X-ray radiography. Delamination and ply cracking were found to be the dominant types of fatigue damage. Comparison of observed delamination with finite element stress analysis indicated that both interlaminar normal and shear stresses must be considered to explain the observed delamination. In a similar study by Kress and Stinchcomb (1985) X-ray radiography and a deply technique were used to determine the distribution of damage in each ply around the hole in two quasi-isotropic grphite/epoxy laminates subjected to tension fatigue. Their observations on the locations of initial delaminations in these two laminates agreed with the interlaminar stress analysis by O'Brien and Raju (1984). Comparisons with Literature 77 In the following sections the experimental results mentioned above are compared with the stresses calculated using the present approach. The observed delamination damage is compared with the results of the present approach which considers delamination as the only damage mode. The presence of other interactive damage modes may, however, alter the stress distribution to the extent which makes such comparisons only qualitative. The altered stress distribution, accompanied by delamination and other damage growth, can possibly initiate delamination at new locations or change the direction of delamination propagation. 4.2.1 Whitcomb (1981):-Damage development around holes in two orthotropic and two quasi-isotropic laminates under both tension and compression fatigue was investigated by Whitcomb. Delamination and ply cracking were found to be the primary modes of damage that took place. C-scan records of typical delamination locations have been made for the specimens after 10 tension or compression fatigue cycles. Comparisons of the C-scan records for the two orthotropic laminates [ 0 / ± 4 5 / 0 ] s and [45/0/-45/0] s reveals that the difference in stacking sequence of these two laminates affected the delamination growth ( Fig. 4.24 ). It has been noted by Whitcomb that under tensile fatigue loading the [ 0 / ± 4 5 / 0 ] s specimen delaminated preferentially at 0° and 180 ° from the loading direction, but the [45/0/-45/0] s specimen delaminated uniformly around the hole. For the [ 0 / ± 4 5 / 0 ] s specimen, the interlaminar normal stress calculated using the present approximate technique is found to be tensile for almost all interfaces above and below the hole, as shown in Fig.4.25. The curves which lie outside the base circle of this polar plot indicate (positive) tensile stresses, while those which fall inside the circle represent compressive o r In contrast the results of the present calculation for the Comparisons with Literature 78 [45/0/-45/0] s laminate (Fig. 4.26) predict tensile oz within a small angular region on either side of the hole and compressive oz in regions above and below the hole. Since the interlaminar shear stress T Q Z and T R Z distributions are essentially similar for the two stacking sequences at each interface, the difference in delamination observation can be attributed to the difference in o distribution. Thus, more delamination can be expected to occur along the loading direction, above and below the hole, for the [ 0 / ± 4 5 / 0 ] s laminate than for the [45/0/-45/0] s laminate. It has also been observed by Whitcomb that the sign of the loading also affected delamination growth. In particular, the [45/0/-45/0] s specimen delaminated much more extensively under compression than tension. This can be expected since the compressive a stresses (shown inside the base circle in Fig. 4.26) would become tensile under compressive fatigue loading. The highest tensile stresses would be obtained at an angle equivalent to that at which extensive delamination is observed in the C-scans. Specimens of quasi-isotropic laminates were sectioned and examined for delamination locations in the thickness direction. Micrographs of these sections are given by Whitcomb at angles 90° , 120° and 175° for the [45/90/-45/0] s laminate subjected to tension fatigue ( Fig 4.27 ), and at angles 90° , 120° and 160° for the [90 /± 45/0] s laminate subjected to compression fatigue (Fig 4.28). For the [45/90/-45/0] s laminate at 90°, delaminations were observed at 90/-45 interfaces where the interlaminar shear stress T Q Z calculated by the present approach is found to be maximum. The stresses shown are those calculated at the edge of the hole and normalized with respect to the absolute value of gross axial stress (see Fig. 4.20). T Q Z has the same maximum value at neighbouring 45/90 interface too, but the interlaminar normal stress calculated for this interface is found to be slightly lower. At 120°, delaminations were observed along -45/0 interfaces, Comparisons with Literature 79 where T Q Z is found to be maximum and oz relatively high. However, no delamination was observed by Whitcomb at the edge of the hole at about 175° from the loading direction. The results of the present calculation for this angular location predict low interlaminar shear stresses and compressive oz through the entire laminate thickness. It is however interesting to note the delamination of -45/0 interfaces away from the edge in the corresponding micrograph. The form of the o z distribution along an interface, infact, causes the compressive interlaminar normal stresses calculated at this angle to change sign and become tensile away from the free edge. Also, the through thickness interlaminar shear stress T RZ distribution resulting from the present analysis is found to have its maximum at this interface between -45-deg and zero-deg plies. In the [90/± 45/0] s laminate subjected to compression fatigue ( Fig 4.28 ) delamination was observed at 90° from the loading direction between the 45-deg plies. These delaminations can be associated with coincidental peaks in the shear stress T Q Z and high tensile normal stress oz calculated by the present approach and shown in Fig. 4.21. At 120°, the location of the shear ( T Q Z ) stress peaks shifted to the adjoining interface between -45-deg and zero-deg plies causing delamination at that interface. Here the magnitude of the tensile interlaminar normal stress too remained high and near its maximum which occured at the mid-plane. At 160° from the loading direction delamination was observed at the midplane between the zero-deg plies. At this angle, the tensile interlaminar normal stress was found to be maximum at the mid-plane while the shear stresses remained low in magnitude through the entire thickness. The delamination locations obtained by section studies compare well with the stress distributions calculated by the present approach. Comparisons have been made at three angular locations for each quasi-isotropic laminate Comparisons with Literature 80 presented. Results of microscopic section studies are not presented by Whitcomb for the orthotropic laminates. However, the examination of C-scan records of these laminates show that the calculated interlaminar stress can still be related to the observed delamination. 4.2.2 Kress and Stinchcomb (1985):-In a study on the fatigue response of two quasi-isotropic graphite/epoxy laminates Kress and Stinchcomb investigated damage development around circular holes during cyclic tensile loading. Non-destructive inspection of damage using zinc iodide enhanced X-radiography provided information on the continuous damage process during the fatigue life. The study is of special interest here because in addition to the information on damage growth during fatigue life it provides information on damage initiation. Data on early stages of damage development have been obtained by Kress and Stinchcomb by sequential loading of test specimens to progressively higher loads. Zinc-iodide enhanced X-radiographs of the specimens made during each unloading and reloading process produced information on damage initiation. Matrix cracks were the first to appear at lower stress levels, before any delaminations were detected at slightly higher stress levels, in both laminate types. Kress and Stinchcomb report that in the [ 0 / 9 0 / ± 4 5 ] s laminate delaminations were first detected in the 90/45 interface after loading to 80% of its mean tensile strength. Radiographs of the zinc-iodide infiltrated hole region for this laminate made after 90% and 105% stress loadings t are reproduced in Fig. 4.29. Delamination at the hole boundary appears at an angle to the loading t The stress is expressed as a percentage of the mean tensile strength of the laminate which is determined independently using several test specimens. Comparisons with Literature 81 direction on four locations around the hole, symmetrical with respect to the horizontal and vertical axes. No delamination appears to have initiated on either side of the hole perpendicular to the loading direction, or, above or below the hole parallel to the loading direction. The stress solution for this laminate using the present approximate technique yields compressive interlaminar normal stresses within a small angular region on either side of the hole and tensile stresses around the rest of the hole. Figures 4.30 (a)-(c) show the stress solutions for this laminate obtained by the present method. High interlaminar shear T Q Z stresses are obtained at 0/90 and 90/45 interfaces at roughly the same angle on four locations around the hole where delaminations are observed. It is nearly zero on either side across the horizontal diameter and above and below the hole. The high interlaminar shear stresses at 90/45 interfaces are now supplemented by relatively high interlaminar normal stresses in the vicinity of the delaminated regions. The apparent delamination of the interfaces between 90-deg and 45-deg plies thus appears to be governed by high interlaminar normal and (T Q Z) shear stresses. The r r z component of interlaminar shear is almost uniformly distributed around the hole thus having little influence on the location of delamination initiation. Delamination initiation in the [45/90/-45/0] s laminate has been observed by Kress and Stinchcomb on radiographs made after loading the specimens to 60% of their mean tensile strength. They report that delaminations were first detected in the 45/90 and 90/-45 interfaces. The radiographs provided in the published work ( made after 90% and 130% stress loadings as shown in Fig. 4.31 ) clearly indicates severe delaminations on opposite sides of the hole, perpendicular to the loading direction, with no visible delamination on top or bottom of the hole. These observations agree with the results of the present stress calculations, shown in Figs. 4.32 (a)-(c). The interlaminar normal stress is tensile only within a small angular region perpendicular to the loading direction, and is Comparisons with Literature 82 largest at 45/90 and 90/-45 interfaces. The interlaminar shear T Q Z in Fig. 4.32 (b) is also high for these interfaces, but nearly zero for other interfaces at this loacation. The same is true of T RZ distribution shown in Fig. 4.32 (c). Thus good correlation between theoretical and experimental results are observed for this laminate. The absence of any visible delamination above and below the hole parallel to the loading direction can be seen as a result of having low interlaminar shear stresses and compressive interlaminar normal stresses in this region. The progressive damage development in the two laminate types under constant amplitude tension-tension fatigue is shown by Kress and Stinchcomb through a series of radiographs taken at different times in the loading history. Matrix cracking and delamination seemed to be the dominant modes of damage that occured. Delamination at early stages of fatigue life is very much similar to that observed in sequential static loading, conforming to the predictions based on the present stress calculations. Successive delamination, though appears to be in agreement with analytical results, is much more widespread and influenced by the matrix cracks. In order to determine the shape and size of the delaminated region on a given interface the authors have used the deply technique. The essential features of the delamination zone for each deplied layer determined for the two laminate types are shown in Figs.4.33 and 4.34. In spite of the possible stress alteration associated with these delaminations ( and ply cracking ), the damage on each interface can be compared qualitatively with the stresses shown in Figs. 4.30 and 4.32. The damage on 0/90, 90/45 interfaces of the [0 /90/± 45] s laminate ( Fig. 4.33 ) appears to have initiated at an angle to the loading direction and propagated towards the upper and lower edges of the hole. This can Comparisons with Literature 83 be expected on the basis of high interlaminar shear and normal stresses found at an angle to the loading direction, and, relatively high normal and T RZ shear stresses which continue across the upper and lower edges of the hole. On the 45/-45 interface delaminations surround the hole completely perhaps due to the fact that compressive o z near 90° from the loading direction is offset by high interlaminar shear stresses in the region. Over the remaining upper and lower parts of the hole boundary o z is found to be highly tensile. The delaminated regions in the [45/90/-45/0] s laminate are generally smaller than in the above [0 /90 /± 45] s laminate which can be attributed to largely compressive interlaminar normal stresses around the hole. The overall interlaminar shear stress distributions can be considered as equal for the two stacking sequences. Nevertheless, on 45/90, 90/-45 interfaces of the [45/90/-45/0] s laminate delaminations appear at diametrically opposite positions ( Fig. 4.34 ) slightly inclined to the horizontal axis. A careful examination of the interlaminar normal and T Q Z shear stress distributions of Figs. 4.32 (a) and (b) reveals that the maximum stresses occur at positions diametrically opposite to each other, aligned closely with the horizontal axis as the observed delaminations. T RZ shear stress too is high on either side at 90° from the loading direction, intensifying the influence of oz and T Q Z in the near region. Damage on -45/0 interface on the other hand appears at four locations around the hole. Although the interlaminar normal stress is compressive, each of the two shear stress components exhibits peak values almost at same locations around the hole. The observations on the location of initial delaminations in the two laminate types agree with the present stress analysis. Delamination growth at early stages in fatigue life also appear very much similar to that observed in sequential static loading. Despite the changes in stress distribution due to existing Comparisons with Literature 84 damage, interlaminar stresses of the present analysis correlate well with the general form of delamination observed at different interfaces. Comparisons with Literature 85 POINT STRESS METHOD Angle, degrees 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 1 I,, ,.< I I I l _ _ " l/a = 0.00 X " ^ ^ l/a = 0 * 0 5 ^ . /i/'*' / \ l/a = 0 . 2 0 \ N / / l/a = 1.00'N^ / Fig. 4.1. Effect of l/a on oz ax i = h in a [90/0] s laminate. M E T H O D = Point Stress. B O U N D A R Y L A Y E R WIDTH = laminate thickness. Comparisons with Literature 86 AVERAGE STRESS METHOD Angle, degrees 20.0 30.0 40.0 — — J 1 — 50.0 60.0 70.0 80.0 — L _ 1 L I/a = 0.00 \ l/a = 0.05^" \ \ l/o = 100 l/a = 0.20 / Fig. 4.2. Effect of l/a on oz at z = h in a [90/0] s laminate. M E T H O D = Average Stress. B O U N D A R Y L A Y E R WIDTH = laminate thickness. Comparisons with Literature 87 MODIFIED POINT STRESS METHOD Angle, degrees 10.0 20.0 30.0 40.0 50.0 60.0 70.0 —I t... •••J ....I 1 1 y ^ - i - - - - -^ l/a = 0.05 j l /a = 0.00 \ l/a = 0.20 ^ _ \ 7 l /a = 1.00 v . / Fig. 4.3. Effect of l/a on oz at z = h in a [90/0] s laminate. M E T H O D = Modified Point Stress. B O U N D A R Y L A Y E R WIDTH = laminate thickness. Comparisons with Literature 88 MODIFIED AVERAGE STRESS METHOD Angle, degrees Fig. 4.4. Effect of l/a on o z at z = h in a [90/0] s laminate. M E T H O D = Modified Average stress. B O U N D A R Y L A Y E R WIDTH = laminate thickness. Comparisons with Literature 89 POINT STRESS METHOD Angle, degrees 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 • J L ^ I I I I I I _ J ^ l /a = 0.50 l /a = 0.20 l /a = 0.10 \ Fig. 4.5. Effect of l/a on oz at z = h in a [90/0] s laminate. M E T H O D = Point Stress. B O U N D A R Y L A Y E R WIDTH = Characteristic length. Comparisons with Literature 90 AVERAGE STRESS METHOD Angle, degrees 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 I 1 1 1 1 l/a = 0.50 * - \ ^ -l/a = 0.20 l/a = 0.10 \ Fig. 4.6. Effect of l/a on oz at i = h in a [90/0] s laminate. M E T H O D = Average Stress. B O U N D A R Y L A Y E R WIDTH - Characteristic length. Comparisons with Literature 91 MODIFIED POINT STRESS METHOD Angle, degrees l /a = 0.20 l/a = 0.10 \ Fig. 4.7. Effect of l/a on a z at z = h in a [90/0] s laminate. M E T H O D = Modified Point Stress. B O U N D A R Y L A Y E R WIDTH = Characteristic length. Comparisons with Literature 92 MODIFIED AVERAGE STRESS METHOD Angle, degrees 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 _ J I I L_ I I I , J , r l/a = 0.50 — — — ^ l/a = 0.20 l/a = 0.10 \ Fig. 4.8. Effect of IIa on oz at z = h in a [90/0] s laminate. M E T H O D = Modified Average stress. B O U N D A R Y L A Y E R WIDTH = Characteristic length. Comparisons with Literature 93 Point Stress Average Stress Modified Point Stress Modified Average Stress Raju and Crews 1982 Fig. 4.9. Present results compared with numerical solution for [90/0] s laminate at i = h. Comparisons with Literature 95 Fig. 4.11. Present results of TQZ distribution at z = h in a [0/90] s laminate compared with the solution of Raju and Crews (1982). Comparisons with Literature % Fig. 4.12. Present results of TQZ distribution at z = h in a [90/0] s laminate compared with the solution of Raju and Crews (1982). Comparisons with Literature 97 Point Stress Average Stress Modified Point Stress Modified Average Stress Rybicki and Schmueser 1978 Fig. 4.13. Present results compared with numerical solution for a [02/±10/130]s laminate at i = 0. **** Fig. 4.14. Results of Modified Average Stress method compared with U 3 numerical solution for a [ 0 2 / ± 3 0 / T 3 0 ] s laminate at z = 0. ? az is normalized with respect to az(maxy vii* Lile-mure 99 • i • lfj'ri|Ml||»|l III fO.f C.v,v~ X , -• fj j Comparisons with Literature 100 Fig. 4.16. Present results of interlaminar normal stress distribution at midpane in a [ 0 2 f i60 /3-60] s laminate compared with the solution of Rybicki and Schmueser (1977). Comparisons with Literature . 101 Fig. 4.17. Present results of interlaminar normal stress distribution at midpane in a [ ± 3 0 / : f 3 0 / 9 0 2 ] s laminate compared with the solution of Rybicki and Schmueser (1977). Comparisons with Literature 102 Fig. 4.18. Present results • of interlarninar • normal stress distribution at midpane in a [ ± 4 5 / ^ 4 5 / 9 0 2 l s laminate compared with the solution of Rybicki and Schmueser (1977). Comparisons with Literature - 103 Fig. 4.19. Present results of interlaminar normal stress distribution at midpane in a [ ± 6 0 / ^ 60/902] s laminate compared with the solution of Rybicki and Schmueser (1977). Comparisons with Literature 104 T82/°g -0.6 -0.3 0.0 0.3 0.6-06 -0.3 0 0 0.3 0.6 90 1 1 V > 1 1 » t 1 1 9 120 175* i J -- S r -y i 1 i f — J 1 4 1 -0.6 -0.3 0.0 0.3 0.6-0.6 -0.3 00 0.3 0 6 Whitcomb 1981 Present Solution! Fig. 4.20. Present results compared with numerical solutions (Whitcomb, 1981) of interlaminar stress distributions across laminate thickness in [45/90/-45/0] s specimen, (in tension) Comparisons with Literature 105 °z/°9 ' r6z/°g \ -0.6 -0.3 0.0 0.3 0.6-0.6 -0.3 0.0 0.3 0.6 -0.6 -0.3 0.0 0.3 0.6-0.6 -0-3 0-0 0.3 0 6 Whitcomb 1981 Present Solution Fig. 4.21. Present results compared with numerical solutions (Whitcomb, 1981) of interlaminar stress . distributions across laminate thickness in [90/± 45/0] s specimen, (in compression) Comparisons with Literature Fig. 4.22 (a). Present results compared with Tang's (1977) solution of interlaminar normal stress oz at z = 0 in a [0/90] s laminate. Comparisons with Literature 107 CO 6 H Fig. 4.22 (b). Present results compared with Tang's (1977) solution of interlaminar shear stress T g z a t z = h i n a [0/90] s laminate. Comparisons with Literature 108 Fig. 4.22 (c). Present results compared with Tang's (1977) solution of interlaminar shear stress T RZ at z = h in a [0/90] s laminate. Comparisons with Literature 109 Fig. 4.23 (a). Present results compared with Tang's (1977) solution of interlaminar normal stress oz at z = 0 in a [ ± 4 5 ] s laminate. Comparisons with Literature 110 Fig. 4.23 (b). Present results compared with Tang's (1977) solution of interlaminar shear stress T Q Z at z = h in a [ ± 4 5 ] s laminate. Comparisons with Literature Fig. 4.23 (c). Present results compared with Tang's (1977) solution of interlaminar shear stress r rz at z = h in a [ ± 4 5 ] s laminate. Comparisons with Literature 112 T E N S I O N C O M P R l I ON O R T H O T R O P I C ( 0/«4>'0>, (45/0/-45/OI la) O U A S I - I S O T R O P I C I0/s45/0l (c) (45/0/ 4VGi (0 I Fig. 4.24. C-scan records of various notched laminates after 10 7 tensile or compressive fatigue cycles. (Whitcomb, 1981). Fig. 4.25. Present solution of o z distribution in a [ 0 / ± 4 5 / 0 ] s laminate. (Whitcomb, 1981). Comparisons with Literature 114 Fig. 4.26. Present solution of o z distribution in a [45/0/-45/0]s laminate. (Whitcomb, 1981). Comparisons with Literature 115 Fig. 4.27. Delamination location for [45/90/-45/0] s specimen subjected to tension fatigue. (Whitcomb, 1981). Fig. 4.28. Delamination location for [90/± 45/0] s specimen subjected to compression fatigue. (Whitcomb, 1981). 4.29. Radiographs of damage in sequential loading to (a). 0.9 Kress and Stinchcomb (1985) a [0 /90 /± 45] s laminate after ^ and (b). 1.05 o ^ . Comparisons with Literature 117 Fig. 4.30 (a). Interlaminar normal stress oz distribution in a [0 /90 /±45] laminate. (Kress and Stinchcomb, 1985). Comparisons with Literature 118 Fig. 4.30 (b). Interlaminar shear stress T Q Z distribution in a [0/90/±45] s laminate. (Kress and Stinchcomb, 1985). Comparisons with Literature 119 Fig. 4.30 (c). Interlaminar shear stress r r z distribution, in a [0 /90 /± 45] s laminate. (Kress and Stinchcomb, 1985). Fig. 4.31. Radiographs of damage in a [45/90/-45/0] s laminate after sequential loading to (a). 0.9 a ^  and (b). 1.3 o^. Kress and Stinchcomb (1985) Comparisons with Literature 121 Fig. 4.32 (a). Interlaminar normal stress o z distribution in a [45/90/-45/0] s laminate. (Kress and Stinchcomb, 1985). Fig. 4.32 (b). Interlaminar shear stress T Q Z distribution in a [45/90/-45/0] s laminate. (Kress and Stinchcomb, 1985). Comparisons with Literature 123 Fig. 4.32 (c). Interlaminar shear stress r r z distribution in a [45/90/-45/0] s laminate. (Kress and Stinchcomb, 1985). Comparisons with Literature 124 + Fig. 4.33. Damage on 0/90, 90/45 and 45/-45 interfaces of a [0/90/±45] s laminate. Kress and Stinchcomb (1985) X ** Fig. 4.34. Damage on 45/90, 90/-45 and -45/0 interfaces of a [45/90/-45/0]s laminate. Kress and Stinchcomb (1985) 125 T A B L E II. In-plane ply calculations:-A U T H O R Raju and Crews (1982) Rybicki and Schmueser (1977) Whitcomb (1981) Tang (1977) elastic properties used by M A T E R I A L Graphite/epoxy Graphite/epoxy Graphite/epoxy Boron/epoxy Graphite/epoxy different authors in stress ELASTIC PROPERTIES Ell — 138.0 GPa E22 = 14.5 GPa G12 = 5.86 GPa v12 = 0.21 Ell = 151.60 GPa E22 11.00 GPa G12 = 6.89 GPa v12 = 0.25 Ell = 140.0 GPa -E22 = 14.0 GPa G12 5.9 GPa V12 - 0.21 Ell - 211.68 GPa E22 = 19.93 GPa G12 - 4.99 GPa v12 - 0.21 Ell 137.90 GPa E22 : 14.48 GPa G12 - 5.86 GPa V12 = 0.21 126 C H A P T E R V E X P E R I M E N T A L OBSERVATIONS 5.1 INTRODUCTION:-An experimental programme was undertaken to prepare a series of composite laminates and examine them for delamination damage around circular holes under quasi-static loading. The main objectives were to (i) ascertain the nature of damage induced at hole boundaries by static loading (ii) correlate delamination damage with interlaminar stresses calculated using the approximate methods described in Chapter III. Composite laminates of differing ply orientation and stacking sequence, to include angle ply, cross ply and quasi-isotropic laminates, were used in the investigation; Al l laminates were constructed at U.B.C. except for the quasi-isotropic XAS/914 graphite/epoxy laminates (Poursartip, 1984). The laminates were layed up using either the 305 mm (12-in) wide Magnamite AS/3501-6 graphite/epoxy prepreg tapes or the same width 3M Scotchply-type 1003 glass/epoxy prepreg tapes to form panels of dimension 280- by 190-mm (11 by 7.5 in). The subsequent tests were carried out almost entirely on laminates made out of graphite/epoxy material system. Due to experimental difficulties associated with non-destructive detection of delamination damage, the use of Scothply glass/epoxy material system was later abandoned. Typical material properties of Experimental Observations 127 the prepreg tapes and the manufacturer's recommended cure cycles are found in Radford (1982). The in-plane ply elastic properties, used to calculate the interlaminar stresses at hole boundaries, are listed in Table III (page 214) for each material system. The angle ply laminates investigated are the 12-ply [Ojl^Ojl ± 30] s and [ ± 3 0 / 9 0 2 / 0 2 ] s laminates and 8-ply [ 0 2 / ± 4 5 ] s and [ ± 4 5 / 0 2 ] s laminates of AS/3501-6 graphite/epoxy material. In addition, a glass/epoxy laminate of [ 0 2 / ± 3 0 ] s construction was also included in the investigation. The crossply and quasi-isotropic layups investigated are respectively the 4-ply [0/90] s and [90/0] s AS/3501-6 graphite/epoxy laminates and 8-ply [45/0/-45/90] s construction of XAS/914 graphite/epoxy system. The experimental procedure described in Sec. 5.2 briefly outlines the fabrication of laminates from prepreg material and the preparation of test specimens from these panels. It also provides a description of the test programme employed. The results are reported and analysed in Sec. 5.3 for each individual laminate separately. The experimental observations are compared with analytical results to determine the validity of the present approximate technique. 5.2 EXPERIMENTAL P R O C E D U R E : -Uni-directional prepreg tapes available in roll form were used to prepare the laminates described above. A standard autoclave was used to cure the prepregged material. After the completion of the cure cycle the completed laminates were removed from the autoclave. In order to avoid any edge effects due to higher resin content, at least half an inch wide strips were cut off from all four sides of the plate. Later analysis showed a high void content in most of the laminates. This is considered to be due to moisture absorption by the prepreg material during storage. However, since we are interested only in damage initiation and not in the failure, these laminates were Experimental Observations 128 considered to be satisfactory. 5.2.1 Specimen Preparation :-Specimens were cut from the composite plates using a diamond cutting wheeel. Except for the XAS/914 graphite/epoxy specimens, all others were cut to a nominal width of 50 mm and a length of 280 mm. A central circular hole of 12.7 mm (0.5 in) diameter was drilled in each specimen using a high speed diamond drill. When drilling the holes, the specimens were clamped tightly between two glass sheets to prevent any delamination at hole edges. The drill travel through the thickness was also set at a very low speed. Water coolant was used to prevent any overheating of the drill bit. The holes thus produced were of good quality, possessing damage free smooth edges. This was later confirmed by zinc iodide enhanced X-ray radiography prior to loading. The straight free edges on either side of the specimen width were smootheried by mechanical polishing on wheels upto 400 grit-Specimens which were examined during the early part of the experimental programme were tabbed at the ends to prevent any possible grip failure. This was later considered to be unnecessary, because most of the specimens were loaded only up to a fraction of the failure load. Even those which were loaded up to failure had damage confined mostly to the central hole region. Experimental Observations 129 5.2.2 Testing and Observations :-After checking for any initial delamination introduced by the machining process, the specimens were tested quasi-statically for damage development around the holes. Using a pair of 2-in wide tensile friction grips the specimens were loaded in an lnstron testing machine. The specimens without tabs were mounted using medium grade sand paper held between the grips. The free span between the grips was approximately 170 mm (6.75 in) for the 50 mm wide specimens. A cross head speed of 0.5 mm/min (0.02 in/min) was used in both loading and unloading. The specimens were loaded sequentially to higher loads until delamination was introduced at hole boundaries. After loading a specimen to a particular load value it was immediately unloaded to half that value. While keeping the load on the specimen constant at this value, the hole was covered with scotch tapes on both sides, and X-ray opaque zinc iodide solution was injected into the cavity formed by the tape and the circular free edge. The specimen was left under load for at least half an hour allowing enough time for zinc iodide to penetrate into cracks and delaminations. It was then removed from the machine and cleaned thoroughly of residual zinc iodide on the surface, before making the radiographs. The X-ray radiographs of the central hole region were made by exposing the specimens to an X-ray beam, for a length of time depending on the specimen thickness. For a typical 8-ply graphite/epoxy laminate this was about 45 seconds at 30 kV and 10 mA X-ray current A Polaroid type 55 positive-negative film was placed behind the specimen in the path of the X-ray beam. Once the radiography was completed, the specimen was reloaded in the Experimental Observations 130 Instron to a higher load, and the above process repeated. At selected times in the loading sequence, the specimens were also sectioned at the hole in order to determine on which interfaces the delaminations and matrix cracking appeared. A slow diamond cutter was used to cut sections along radial planes around the hole, at different angles to the loading direction. A typical set of cuts is shown in Fig. 5.1. These sections were then polished to a finish of 6 microns for microscopic examination, and later to 1 micron finish for potographic and replication work. Replication of the sections was found to be of great use in further establishing microscopic evidence of delamination and ply cracking. The replicas were prepared by making surface impressions of the polished sections on .125 mm (.005 in) thick pieces of cellulose acetate sheets, moistened with acetone. After applying a mild pressure on the sample and allowing sufficient time for hardening, the acetate sheet was gently peeled off from the replicating surface. When viewed under the microscope it displayed a negative form of the actual surface, manifesting delaminations in the form of thin walls of acetate. Attempts to replicate the circular free edge, while the specimen was still under load, met with little success. One reason for employing a wide specimen geometry was to have a hole big enough for easy access in replicating the circular edge. Yet, a successful replication technique could not be developed due to difficulties involved in polishing the curved surface to an acceptable fine finish, wetting and making an imprint of the surface on cellulose acetate quickly and evenly. Also, the hardened acetate sheet could not be straightened out for subsequent microscopic work, without inducing much damage to the imprint Experimental Observations 131 5.3 RESULTS:-The results of the experimental programme are presented separately for laminates with different ply orientations. The results for laminates with the same ply orientations but different stacking sequences are discussed together. The results of the angle ply laminates are presented first, following which those of the crossply and quasi-isotropic laminates are given. In each case, comparisons have been made with the corresponding stress calculations. Although the stresses calculated can only predict the onset of delamination, comparisons have been made between the extent of initial delamination and the stresses, even after significant damage growth. 5.3.1 [02 /902/±30] s and [±30 /90 2 / 02 J S laminates:-The radiographs of damage around the hole in a [O2/9O2/ ± 3 0 ] s laminate under sequential static loading are shown in Fig. 5.2. The first damage in the specimen appeared after loading to about 50 MPa. Two matrix cracks of the zero-deg plies, extending outward from the hole boundary, appeared about 25° from the loading direction above and below the hole. These cracks were on the same side of the hole with respect to the specimen vertical centre line. A third matrix crack of the zero-deg ply was also visible, on the other side of the hole about 55° from the loading direction. There were strong indications that these cracks resulted from a possible weakness in the zero-deg ply, inherited from a defective prepreg material or from poor laminate construction or specimen preparation. These cracks grew slowly but continuously as the stress level was increased. Experimental Observations 132 After repeated loading to higher stress levels, cracks emanating from the hole boundary appeared in the 90-deg plies. Several of these cracks were seen on each side of the hole in the radiograph made after loading to 280 MPa stress. At the same time, zero-deg ply cracks tangent to each side of the hole were also seen to emerge. They grew in length and became more visible as the load was increased, while 90-deg ply cracks increased in length and number spreading over a wider segment of the hole boundary on either side. Signs of delamination initiation were first observed in the radiograph made after loading to 370 MPa stress level. Dark regions representing delamination were visible in the radiograph at ah angle of at least 70° from the loading direction. With increasing load these regions extended in size, spreading over the hole boundary on either side, covering approximately the region between 50° and 130°. The specimen was sectioned at the hole after loading to 470 MPa, when the gross damage around the hole was as shown in the final radiograph of Fig. 5.2. The distribution of interlaminar stresses around a hole for the [ O 2 / 9 O 2 / ± 3 0 ] s laminate are shown in Fig. 5.3. The stresses shown are those calculated for the particular specimen geometry resulting from an applied stress of 470 MPa and the residual thermal stresses, normalized with respect to the applied stress. High tensile interlaminar normal stresses are obtained around 50° and 130° from the loading direction, which become compressive at 90°, on both sides of the hole. For many interfaces the interlaminar shear TQZ is found to be high around 70° and 110°, while r r z is found to be high around 55°, 90° and 125° from the loading direction. Thus high interlaminar stresses are obtained within the same angular region in which much of the damage and delamination took place. The compressive interlaminar normal stresses on opposite sides of the hole are perhaps counteracted by high TRZ and TQZ found between the 30-deg and -30-deg plies. Experimental Observations 133 The specimen was sectioned at angles of 0 ° , 45°, 70° . 90° , 110°, 135° and 180° from the loading direction. Fig. 5.4 shows the microscopic sections and replicas for each angle. For the angles 0° and 180° these micrographs appeared very much similar to each other, with little or no delamination. This can be expected on the basis of low interlaminar normal stresses and negligibly small shear stresses found at these angles. (see Fig. 5.3.) Only those micrographs of the 0° are included in Fig. 5.4. At 45° from the loading direction delamination was observed at fourth and fifth interfaces from the surface, corresponding to 90/30 and 30/-30 interfaces respectively. Comparatively little delamination was seen at the second interface between the zero-deg and 90-deg plies. The calculated stresses at 45° show higher interlaminar normal stresses at fourth and fifth interfaces than at other interfaces, with the exception of the midplane. Although a is maximum the shear stress components are both zero at the midplane, while finite values of T § Z and T RZ are obtained at fourth and fifth interfaces. Thus delamination can be expected at 90/30 and 30/-30 interfaces prior to any other place on the basis of the present stress calculations. The traces of delamination observed at the second interface are also explained by the existence of maximum TQZ and r r z at this interface despite the presence of relatively low tensile o z . At 70°, significant delamination was observed at the second interface between zero-deg and 90-deg plies, and a little at the fourth between 90- and 30-deg plies. The zero-deg ply cracking appeared previously in the radiographs can also be seen on these micrographs, close to the hole boundary on both sides of the laminate. The interlaminar shear stresses calculated at 70° are both maximum at the second interface from outside making it a favourable location for delamination. Note that the normal stresses calculated at this angle are nearly Experimental Observations 134 equal for all interfaces. At the fourth interface the shear stress components are both high, making it the next favourable location for delamination. Delaminations at 90° from the loading direction were observed at fourth and fifth interfaces from outside, and found to be connected through 30-deg ply cracking. The interlaminar normal stress distribution in Fig. 5.3 predicts compressive oz at 90° for the entire laminate, increasing in magnitude towards the laminate midplane. However, at this angle a high, non-zero TQZ is obtained at the fifth interface between the 30-deg and -30-deg plies. TQZ is zero for all other interfaces. The interlaminar shear r r z , on the other hand, has its maximum at the fourth interface, and shows high values at the fifth and third interfaces. The delamination at the second 0/90 interface at 110° from the loading direction can be related to interlaminar shear stresses, both of which show their maximum at this interface. The normal stress is again nearly equal for all interfaces. Slight delamination is observed at fourth and fifth interfaces corresponding to 90/30 and 30/-30 respectively. At 135°, delamination at the fifth 30/-30 interface can be related to the peak in o z , and that at the second 0/90 to the peaks in TQZ and TRZ stresses. Slight delamination was also observed at the fourth interface where the interlaminar normal stress is nearly as high as that at the fifth interface. We now consider the [ ± 30/902 ^ 2^s laminate and study the damage development under sequential static loading. The minute amount of damage visible in the radiographs of early stages of loading in Fig. 5.5 is a result of inherent material defects or poor specimen construction. These include matrix cracking on either side of the hole about 50° from the loading direction and a zero-deg ply crack at 170° on the right side. Damage due to loading was first Experimental Observations 135 seen in the radiograph taken after 185 MPa of gross applied stress. These were 90-deg ply cracking on the right side of the hole, which also started on the left side at a higher stress level. At 275 MPa, zero-deg ply cracks appeared on each side tangent to the hole boundary. With increasing load they extended in length in either direction, while the 90-deg ply cracks increased in length as well as in number. The first delamination appeared in the radiograph taken after 365 MPa of applied stress. Delamination was observed on each side of the hole at 90° from the loading direction. Subsequent loading to higher stress levels caused the delaminated regions to spread over a wide area. The specimen was sectioned after reaching 550 MPa applied stress and after making the final radiograph of Fig. 5.5. The interlaminar stresses around the hole in [±30/902^2^% laminate due to an applied stress of 550 MPa and residual thermal stresses are shown in Fig. 5.6. High tensile normal stresses are obtained in regions centred around 90° from the loading direction. As in the case of [O2/9O2/±30] s laminate the interlaminar shear TQZ is found to be high for many interfaces around 70° and 110°. Similarly, rrz is also found to be high around 55°, 90° and 125° from the loading direction. Thus one significant difference between the two stacking sequences is that the [±30/902^2^% lay-up generates tensile oz on each side around 90° from the loading direction, whereas [O2/9O2/±30] s generates compressive o z in this region. This may have caused the delamination to initiate at 90° from the loading direction in the [ ± 30/902 ^ 2^s laminate. The distribution of shear stresses, though appears to be similar in general for both laminates, is different for any given interface. The specimen was sectioned at angles of 0 ° , 45°, 70°, 90°, 110°, 135° and 180° as before, and the micrographs of polished sections and replicas at these angles are shown in Fig. 5.7. At 0 ° , no delamination was Experimental Observations 136 apparent The interlaminar stresses were either compressive or negligibly small. At 45°, delamination was observed at the first and fourth interfaces, and perhaps a little at the second. It is interesting to note that much of this delamination has taken place away from the free edge and that the interlaminar normal stresses at this angle were compressive for all interfaces, being nearly zero at the first and highest at the midplane. Delamination can therefore be expected to occur on interfaces closer to the outside surface if supplemented by shear stresses. The high interlaminar shear TQZ at the first 30/-30 interface and TRZ at the second -30/90 interface can thus be considered as the cause of observed delamination. On the other hand, both TQZ and T r z become maximum at the fourth interface between 90 and zero-deg plies causing it to delaminate, despite the presence of relatively high compressive o z at this location. At 70° from the loading direction, severe delamination was observed at the fourth interface from outside surface. Relatively little delamination was observed at the second between -30-deg and 90-deg plies. The zero-deg ply cracking observed in the radiographs can also be seen clearly on polished sections. The crack seen here runs through the four centre plies, close to the hole boundary, between the delaminated surfaces. Some 90-deg ply cracks are also visible in the photographs. Comparing these observations with the results of the present stress analysis (Fig. 5.6), good correlation can be found between the analytical and experimental results. While the distribution of o z is nearly zero throughout the laminate TQZ and r r z are both found to be very high and maximum at the fourth interface between 90 and zero-deg plies, causing it to delaminate more than any other interface. These two components of interlaminar shear were also high at the second interface making it the next possible location for delamination initiation. Experimental Observations 137 At 90° from the loading direction, significant delamination at the first interface and a little at the second were observed. Except for the first interface, TQZ is found to be zero for all others, r r z on the other hand was highest for the second interface and nearly half that for the first and third interfaces, with neglegibly small values everywhere else through the laminate thickness, o z was tensile all throughout, increasing progressively in magnitude from outer surfaces to midplane. Delamination at the first and second interfaces can thus be related to the presence of high interlaminar shear and tensile interlaminar normal stresses at these locations. The high o z derived for those interfaces near the midplane are not sufficiently aided by additional interlaminar shear stresses to cause any delamination near the midplane. As at 70° , severe delamination was observed at the fourth interface, 110° from the loading direction. The interlaminar shear stresses TQZ and T r z are both very high and maximum at this location. Although of relatively small magnitude a z is also slightly tensile at this interface. At 135°, delamination away from the free-edge can be seen on the first and fourth interfaces. Here again, the shear stress components are both maximum at the fourth, and high at the first interfaces. However, the interlaminar normal stress at the hole boundary is compressive for all interfaces increasing in magnitude towards the laminate midplane. This may have prevented initiation of delamination at the free-edge, though away from the edge, delamination may have been assisted by o z which changes sign and become tensile in this region. At 180° from the loading direction no delamination was observed. This is in agreement with neglegibly small shear stresses and compressive o, obtained for all interfaces across the entire laminate thickness. Experimental Observations 138 The comparison between the experimental observations of delamination and analytical solutions of interlaminar stresses made so far, have all been descriptive. In order to present them in a direct, concise manner the results are tabulated in tables IV (page 215) and V (page 217) for the two laminates [ O 2 / 9 O 2 / ± 3 0 ] s and [±30 /9 f J2 /02 ] s respectively. These tables show the relative extent of delamination observed at different interfaces, at each angle of sectioning. The amount of delamination is estimated for a given interface, on a scale of 0 to 10, zero being no delamination and 10 being the complete separation of the plies which form that interface. For convenience, the interfaces are numbered beginning from the outmost, such that the interface nearest the outside surface is designated one (1) and the midplane of a 12-ply laminate six (6). The interlaminar stresses shown in the tables are the stresses at the free-edge, normalized with respect to the gross applied stress. The values given for TQZ and T are the absolute magnitudes of shear stresses, ignoring the sign, while for o z both positive and negative values are given indicating tensile and compressive normal stresses. The stresses calculated by the present approximate technique are those for the undamaged hole that exist at the free-edge. Strictly, these stresses can only be used to predict delamination initiation, since the original stress state may change with initiation and propagation of damage at the hole. Nevertheless, comparisons have been made after the initiation and propagation of finite amounts of delamination achieveing good agreement between the results. 5.3.2 [ 0 2 / ± 4 5 ] s and [ ± 4 5 / 0 2 ] s laminates:-The radiographs taken of the above laminates, before loading, indicated the existence of fabrication defects in every specimen. These included voids and ply cracking that appeared around the hole. A typical radiograph with such Experimental Observations. 139 defects is shown in Fig. 5.8 for a specimen of [ ± 45 / 02 ] s configuration. The voids in the specimens were visible in polished sections prepared after sequential loading. In spite of the defects observed in those laminates, tests were carried out to investigate the influence of interlaminar stresses on delamination at hole boundaries. Fig. 5.9 shows a series of radiographs of a [02/ i"45] s specimen, made after repeated loading to increasingly higher in-plane stresses. A large number of cracks in the 45-deg plies appeared throughout the specimen, while only two zero-deg ply cracks were seen on each side tangent to the hole boundary. A significant amount of delamination was observed above and below the hole, in the region bounded by the zero-deg ply cracks. The extent of the delaminated region and the lengths of the zero-deg ply cracks were seen to grow rapidly with increasing load. In contrast, the radiographs of [ ± 4 5 / 0 2 ] s specimens exhibited much less damage around the hole, as shown in Fig. 5.10 for a specimen loaded to same stress levels. Though fewer in number, matrix cracks of the zero-deg and 45-deg plies still appeared in the radiographs. Delamination was also observed above and below the hole as before, within the same angular region bounded by two zero-deg ply cracks tangent to the hole. However, the gross amount of delamination (and ply cracking) observed in the [ ± 4 5 / 0 2 ] $ specimen at a given stress was much less than that observed in the [ 0 2 / ± 4 5 ] s specimen. This was so, even in spite of the fact that a given stress level represented a higher fraction of the mean failure stress foT [ ± 4 5 / 0 2 ] s laminate than for the [ 0 2 / ± 4 5 ] s laminate. The mean (far- field) failure stress for each laminate type was determined by monotonic tension tests of three specimens of each type. The maximum stress applied before making each radiograph is given in Figs. 5.9 and 5.10 as a percentage of the mean failure stress. Experimental Observations 140 In order to compare the above observations with interlaminar stresses, the stress distributions at the hole boundary obtained for each laminate type are shown in Figs. 5.11 and 5.12. Except within a narrow angular region on each side of the hole perpendicular to the loading direction, the interlaminar normal stresses are found to be tensile for the [ 0 2 / ± 4 5 ] s laminate and compressive for the [ ± 4 5 / 0 2 J S laminate. The magnitude of this normal stress at any interface around the hole is seen to increase with the distance from the surface to that interface. On the other hand, the shear stress distributions in the two laminates are found to be similar for any interface between similar plies. A close inspection of the shear stress distributions shown in Figs. 5.11 and 5.12 show that the distributions of both TQZ and rrz are essentially similar in view of the two-fold symmetry associated with specimen and loading geometry. Thus, the only difference between interlaminar stress distributions of the two laminate types is found to be that of a z , which is mostly tensile for [ 0 2 / i 4 5 ] s and compressive for [ ± 4 5 / 0 2 ] s configurations. The excessive delamination observed in [ 0 2 / ± 4 5 ] s laminate over the other, can therefore be attributed to this difference in o z at the hole boundary. Though delamination due to combined effects of interlaminar normal and shear stresses occurs in both laminates, the tensile normal stresses in [ 0 2 / ± 4 5 ] s laminate lead to extensive delamination at the hole boundary. Microscopic examination of the circular edge in delaminated specimens revealed that, in both laminates, much of the delamination took place at the second interface from the outside between zero-deg and +45-deg plies. Furthermore, in the [ 0 2 / ± 4 5 ] s specimen, narrow lips extending outward in the vertical direction from the hole boundary, consisting of surface zero-deg plies and bounded by the zero-deg ply cracks tangent to the hole, were seen. These lips formed above and below the hole when loaded to more than 80% of its mean failure stress. At failure, complete separation of these lips from the rest of the Experimental Observations 141 specimen occured at 0/45 interface near the hole region. These observations agree with the results of the present stress calculations. The interlaminar shear TQZ and TRZ are both maximum at the second interface between zero-deg and ± 45-deg plies, over most of the boundary in both stacking sequences. The appearence of fracture surfaces of the two laminate types are shown in Fig. 5.13. The long narrow lips of a fractured [ 0 2 / ± 4 5 ] s specimen are clearly visible in this figure, while failure across the hole through entire laminate thickness is seen for the [ ± 4 5 / 0 2 ] s specimen. Microscopic examination of polished sections revealed the existence of voids in the laminates. The presence of voids along ply interfaces made any comparison between through the thickness locations of delamination initiation and stress distributions difficult. This is evident from Fig. 5.14, which shows a series of replicas of polished sections prepared from a [ ± 4 5 / 0 2 ] s specimen, loaded to 265 MPa gross applied stress (42% of the mean failure stress). The sections are made at angles of 0 ° , 60°, and 90° from the loading direction. Although major delamination of the second interface from outside (between -45-deg and zero-deg plies) can be seen on 0° and 60° replicas, the presence of voids make it unclear which other interfaces suffer delamination. Delamination of the first interface from outside between + 45-deg plies is clearly seen near the free-edge at 60°. This is probably due to high TQZ shear stress and non-compressive oz obtained at this angle as shown in Fig. 5.12. At 90° from the loading direction, damage is seen at almost every interface across the laminate thickness. It is clearly difficult to differentiate between any separation due to voids formed along ply interfaces and delamination due to high interlaminar stresses. Experimental Observations 142 5.3.3 [ 0 2 / ± 3 0 ] s laminate:-Laminates of [ 0 2 / ± 3 0 ] s configuration were prepared using Scotchply-type 1003 glass/epoxy material system. Non-destructive detection of damage development using zinc-iodide enhanced X-ray radiography was not feasible with this laminate, due to high absorbancy of X-rays by glass fibres. However, the transparency properties of this material aided in visual detection of damage at the hole boundary. A source of light placed behind the specimen illuminated undamaged regions of the specimen leaving damaged regions less illuminated. Delamination at the circular edge could therefore be seen as dark areas when viewed from the front Specimens were loaded sequentially until damage was introduced at the hole boundary. Matrix cracking of the zero-deg ply, tangent to the hole, was observed prior to any delamination. Delamination was first detected within one of the 60° - 80° Regions measured from the loading direction. Dark regions of delaminations were then seen to extend gradually in size with increasing load. Careful examination of the circular edge revealed delaminations at the edge as very fine cracks running between the plies. Lines of cracks associated with these delaminations could be seen between zero-deg and 30-deg plies, within the same angular regions mentioned above. Micrographs of sections taken at 0 ° , 70° and 90° from the loading direction are shown in Fig. 5.15, for a specimen loaded to 233 MPa of gross applied stress. The interlaminar stresses at the hole boundary for this laminate, resulting from both an applied stress of 233 MPa and residual thermal stresses, are shown in Fig. 5.16. The values plotted in these figures are the stresses normalized with respect to the applied stress. The normal stresses shown in Fig. 5.16 a are Experimental Observations 143 found to be tensile at every interface around the hole, except within a narrow angular region on each side of the hole. The interlaminar shear T Q Z on the other hand is high at second and third interfaces in the neighbourhood of 70° and 90° respectively. However, the other component of interlaminar shear, T is not so high but largest at the second interface between zero-deg and 30-deg plies for most of the boundary. The importance of T RZ in predicting delamination at any given interface is very little, since the distribution of r rz is nearly uniform around the hole. At 0° from the loading direction no delamination was observed. Although oz is high at this location, T Q Z is zero throughout, except at the third interface from outside where it is still relatively small, r r z is maximum at the second and zero at the mid plane, with intermediate values at others. At 70° , delamination was observed at the second interface between zero-deg and 30-deg plies. This delamination and the zero-deg ply cracks tangent to the hole fully separated parts of zero-deg plies from the specimen segment near the free-edge. While oz is only slightiy tensile for every interface at this angle, T Q Z is very high at the second interface between zero-deg and 30-deg plies. At the first interface T Q Z is nearly half that at the second and is- zero at others. Since the interlaminar shear T RZ is also small at this angle, the observed delamination can be related to the presence of high T Q Z at the boundary. A similar result was obtained at 90° from the loading direction. Delamination of the third interface .between ±30-deg plies was observed at this angle, where T Q Z of the third interface is near its maximum. The magnitude of T Q Z is significantly high at this interface but zero at every other interface. The observed delamination of Fig. 5.15 can thus be related to high T Q Z which may also have been supplemented by T at the boundary. The net influence Experimental Observations 144 of these stresses in predicting delamination may also have offset any negative effects due to compressive oz. It is also interesting to note the high density of ply cracks near the free-edge in this micrograph. In predicting delamination at the hole boundary in [ 0 2 / ± 3 0 ] s glass/epoxy laminates the distribution of TQZ was found to be more significant than those of oz and TRZ. This may partly be due to the fact that the peaks in TQZ are higher than those of oz and TRZ. 5.3.4 [0790] s and [90/0] s laminates:-The radiographs of damage around the hole due to sequential loading in 4-ply [0/90] s and [90/0] s laminates are shown in Figs. 5.17 and 5.18. They show damage after the same stress loadings in the two laminate types. No damage is visible at the begining in either specimen. The first zero-deg ply crack appears in the [90/0] s specimen, after loading to 165 MPa stress. In the [0/90] s specimen zero-deg ply cracks are visible in the radiograph taken after 250 MPa stress loading. Here, the cracks are seen on each side of the hole tangent to the boundary, and found to be associated with many 90-deg ply cracks. However, these 90-deg ply cracks are not visible in the corresponding micrograph of the [90/0] s specimen. Subsequent loading to higher stress levels increased the lengths of the zero-deg ply cracks in both laminates. Delamination at the hole boundary was first observed in the [0/90] s specimen. Delaminations were seen at four locations around the hole boundary, symmetrical with respect to the longitudinal and transverse axes. As seen in Fig. 5.17, their location on the boundary makes an angle of approximately 60° -65° with the loading direction. Subsequent delamination growth is found to occur along the zero-deg ply cracks. Experimental Observations 145 The observed delamination in the [90/0] s specimen, shown in Fig. 5.18, is also similar to that just described. The locations of delamination initiation at the hole boundary and its growth appear very much similar to that in the [0/90] s specimen. However, at any given stress level, the amount of delamination seen in the [90/0] s specimen is less than that observed in the [0/90] s specimen. The interlaminar stresses at the hole boundary due to an applied stress of 420 MPa are shown in Figs. 5.19 and 5.20 for the two laminates. The stresses shown are those for the 0/90 interface and the mid-plane normalized with respect to the maximum applied stress. The contribution of laminate residual thermal stresses are also included. The interlaminar normal stress, o z , is found to be tensile for the [0/90] s laminate, both at the first ply interface and. the mid-plane. The distributions of o z at these interfaces remain tensile with their maxima occuring at 70° from the loading direction. They are also symmetrical with respect to the longitudinal and transverse axes. In contrast, the distributions of oz in the [90/0] s laminate are found to be compressive through the laminate thickness, and also around the hole. However, the basic shapes of the distributions and their magnitudes remain same for both laminate types. The interlaminar shear stress distributions, on the other hand, are same for both laminate types as shown in Figs. 5.19 and 5.20. Both TQZ and T R Z are finite at the first interface between the zero-deg and 90-deg plies. Their distributions exhibit maxima at around 70° from the loading direction, in both laminate types. Since the distributions are also symmetrical with respect to the longitudinal and transverse axes, the above mentioned maxima occur at four locations around the hole. Experimental Observations 146 The delaminations observed in the two laminate types show good qualitative agreement with the stress distributions described above. Although the general appearance of delamination is similar in both laminates, less delamination is observed in the [90/0] s type. The interlaminar stress distributions obtained for the two laminates are also similar except for the compressive o z obtained for the [90/0] s laminate. In the [0/90] s laminate all three stress components exhibit maxima at around 70° from the loading direction. In the [90/0] s laminate the two shear stress components exhibit their peaks at this location, ln agreement with these results the delaminations observed are also seen at an angle of 60°-65° from the loading direction in both laminates. As with the stress distribution, the delaminations on the hole boundary are also found to be symmetrically located at four positions around the hole. In both laminates delamination was observed at the first interface from outside between the zero-deg and and 90-deg plies. Fig. 5.21 shows a typical micrograph of a polished section taken at an angle of 65° from the loading direction. The observed delamination of the first ply interface, rather than the laminate mid-plane, must be due to the presence of interlaminar shear stresses at this interface. 5.3.5 [45/0/-45/90] s laminate:-Quasi-isotropic XAS/914 graphite/epoxy laminates were used to prepare tensile specimens of 18 mm width and 100 mm length. The hole drilled at the centre of each specimen was 6.5 mm (0.25 in.) in diameter. The specimens were loaded in an Instron testing machine using a pair of 1-in. wide tensile friction grips. Experimental Observations 147 The radiographs of damage introduced at the hole boundary of a specimen due to sequential static loading is shown in Fig. 5.22. No damage is visible in the radiograph taken after loading to about 115 MPa of gross stress. Matrix cracking of the 90-deg ply and ±45-deg plies are visible after 140 MPa stress. These cracks increase in length and number with increasing applied stress. Matrix cracking of the zero-deg ply begins to appear in the radiographs taken after loading to 200 MPa stress. Delamination at the hole boundary first appears in the radiograph taken after loading 185 MPa gross applied stress. Dark regions of delamination appears at four locations around the hole, each making an angle of nearly 75° with the loading direction. These regions become more prominent and clearer in the subsequent radiographs. The final radiograph of Fig. 5.22 was taken after loading the specimen to 235 MPa stress. the distribution of interlaminar stresses calculated for this specimen geometry is shown in Fig. 5.23. The distribution of interlaminar normal and shear stresses are calculated for a maximum applied stress of 235 MPa. The normalized stresses also include the effects due to residual thermal stresses. The interlaminar normal stress, az, distribution of Fig. 5.23 remains tensile over a significant part of the hole boundary. High o stresses are obtained in the region around 90° from the loading axis. Also the magnitude of oz at different interfaces increases with the distance to that interface from outside. The interlaminar shear TQZ distribution of Fig. 5.23 predicts high stresses only for the first two interfaces from outside, that between 45-deg and zero-deg plies and zero-deg and -45-deg plies. As seen in this figure, the peaks in the distribution of TQZ at these interfaces are obtained in regions near 75° from the loading direction. Four of these peaks are found around the hole, in different quadrants. Experimental Observations 148 each making approximately the same angle with the loading axis. The interlaminar shear T N distribution on the other hand predicts high stresses for regions near 45° and 90° from the loading direction, T TZ distribution at the second interface reaches its maximum at 45° , while that at the first and third interfaces reach their maxima at 90°. The interlaminar stress distribution discussed above is sufficient to explain the experimental observations of delamination. The locations of observed delamination coincide with the locations of stress peaks in TQZ distribution. The distributions of TQZ shown in Fig. 5.23 reach high magnitudes at either the first or the second interface near the regions where delamination is observed (Fig. 5.22). Also, the distribution of both oz and T RZ seem favourable for the particular location of delamination observed, o z is tensile throughout the laminate thickness at this angular location, while T RZ is significantly high for the first two interfaces at this location. For the third interface between the 45-deg and 90-deg plies however, oz and TRZ are both found to be even higher though TQZ is predicted to be relatively small. Thus, delamination is likely to occur even at the third interface in regions near 90° from the loading direction. Microscopic sectioning of a specimen loaded to nearly 235 MPa stress was done in order to compare the results in the thickness direction. Micrographs of sections taken at angles of 0 ° , 45° and 90° are shown in Fig. 5.24. At 0 ° , no delamination was observed. This can be expected on the basis of slightly compressive oz and relatively small T RZ obtained for different interfaces through the laminate thickness. Relatively high TQZ shear stresses are obtained only for the first two interfaces at this location. At 45°, delamination was observed at the second interface between zero-deg and -45-deg plies. This is in agreement with the stress Experimental Observations 149 distributions shown in Fig. 5.23. oz calculated for this interface is tensile and has a relatively high magnitude. T Q Z too is high for this interface at 45° from the loading direction. In addition, the highest value of r r z at this angular location is also obtained for the second interface where delamination is observed. At 90°, delamination was observed along the third interface between -45-deg and 90-deg plies. Here oz is very high and smaller to only that at the mid-plane. The interlaminar shear r r z too has the highest value at this interface. However, the T Q Z component of interlaminar shear is found to be zero for the third interface at this angle. 5.4 DISCUSSION:-The comparisons of experimental observations of delamination with interlaminar stresses calculated using the present approximate methods show good qualitative agreement. Tne locations of delamination around the hole and in the thickness direction compare well with the interlaminar stresses calculated for each laminate. The relative amounts of delamination observed at different interfaces and at different angles in a given laminate are also in agreement with the stresses calculated. The agreement between theoretical calculations and experimental results appear to justify the use of approximate methods of stress calculations in predicting delamination. Yet, a number of related problems remain unresolved. First, the stresses calculated by the present methods are those that exist at the free-edge of an undamaged hole. Strictly, these stresses can only be used to predict delamination initiation at the undamaged hole. In practice, delamination at the hole boundary is frequently preceded by matrix cracking or splitting, which alters the original stress distribution, strictly, the stresses calculated for the undamaged hole can not then be used to predict Experimental Observations 150 delamination initiation. At present, no solution is available which allows for changes in the stress distribution due to initial damage. Only those solutions which incorporate the effects due to damage development would provide the necessary mathematical background for predicting the propagation of delamination and other forms of damage. Even here, the approximate methods of stress calculations, such as that presented in this work may prove to be useful. Second, a reliable mixed-mode failure criterion is required in order to compare, quantitatively or semi-quantitatively the stress distributions with observed delamination. A method is required to combine all three interlaminar stress components into a single failure criterion. Since delamination occurs at the ply interface, where the strengths are mainly governed by the bonding matrix material, a criterion that predicts failure in an isotropic medium may be used. As a first approximation, the Von Mises yield criterion can be modified to generate a stress combination function. Using all six stress components, Von Mises criterion can be written as / 2 ^ o e - o r ) 2 + (or-oz)2 + (o-oe)2 + H T ^ + T ^ + T ^ 2 ) ] " 2 = k (5.1) At a given interface near the hole boundary, the in-plane radial and shear stress' components (viz; or and T ^ ) become zero. The in-plane stress O Q , which is finite at the ply interface, can also be neglected since the reinforcing plies on either side carry a greater percentage of the gross laminate tangential stress. The matrix material at the interface near the hole boundary is assumed to have tangential stresses that are small compared to the interlaminar stresses. Thus by equating all in-plane stress components to zero equation (5.1) can be reduced as; [ oz 2 + 1( r r z 2 + r 6 z 2 ) ] 1 / 2 = k (5.2) Experimental Observations 151 Delamination is expected to occur when the stress combination function on the left hand side of equation (5.2) reaches a critical value. It predicts that the sign of interlaminar normal stress has no influence on delamination. In order to include the detrimental effect of tensile interlaminar normal stresses on delamination equation (5.2) can be modified as; [ az\ oz | + 3( rrz 2 + r6z 2 ) ] 1 / 2 = k (5.3) This predicts that tensile interlaminar normal stresses contribute towards delamination initiation while compressive stresses oppose it The onset of delamination due to tensile interlaminar normal stresses has been observed by many investigators, (e.g., Whitney and Browning, 1972; Pagano and Pipes, 1973) However, according to equation (5.3), the net influence of interlaminar shear stress components on delamination seems to be higher than that of the normal stress component The stresss combination function suggested here can be evaluated at different interfaces and angles around the hole. It is also possible to compare the semi-quantitative observations of the extent of delamination with this criterion. Such a comparison is shown in Figs. 5.25 and 5.26 for the [ O 2 / 9 O 2 / ± 3 0 ] s and [ ± 30/902 ^2^s laminates described in section 5.3.1. The observations of the extent of delamination (Table IV and V, pages 215-218) are compared with the stress combination function of equation 5.3, evaluated for different interfaces and angles around the hole. For ease of comparison, the same scale is used in both plots. Although there is considerable scatter, the extent of delamination is seen to increase with the value of the above stress function. There are many overlapping points on the horizontal stress axis with zero delamination, but most of these correspond to low values of the stress function. The points on the two plots also overlap very closely. Experimental Observations 152 The points on Figs. 5.25 and 5.26 have been obtained by comparing the delamination observed at every interface through the laminate thickness with the corresponding value of the stress combination function. Strictly, the interlaminar stresses calculated for the undamaged hole can only be used to predict the onset of the first delamination. Delamination at any interface at a given angle will influence the subsequent interlaminar stress distribution at that angle significantly. A more discriminating comparison is to plot the length of the longest delamination at each angle as a function of the predicted stresses for that interface at that angle. Fig. 5.27 shows such a plot of the maximum delamination in both the [ 0 2 / 9 0 ^ / ± 3 0 ] s and [ ± 30/902/02] s laminates as a function of the stress combination function. A linear correlation between the extent of delamination and the magnitude of the stress function is clearly observed. Considering the approximations involved in formulating the suggested stress function its agreement with experimental results is very pleasing. Though one would not necessarily predict a linear relation in Fig. 5.27, the observed trend of an increasing calculated stress combination function (which is the driving force for initiation of delamination) leading to a larger measured delamination (which is truly a propagation effect) is to be expected. Experimental Observations 153 Polished Surface Fig. 5.1. Sectioning of a specimen at the hole. Experimental Observations 154 (c) 280 MPa Fig. 5.2. Radiographs of damage in a [O2/9O2/±30]s laminate after sequential loading to (a) 50 MPa (b) 185 MPa (c) 280 MPa (d) 370 MPa (e) 470 MPa Fig. 5.3 (a), interlaminar normal stress a z distribution in a [ 0 2 / 9 0 2 / ± 3 0 ] s laminate. Fig. 5.3 (b). interlaminar shear stress TBZ distribution in a [ 0 2 / 9 0 2 / ± 3 0 ] s laminate. Experimental Observations 158 Fig. 5.3 (c). interlaminar shear stress [ 0 2 / 9 0 2 / ± 3 0 ] s laminate. rr distribution in a Fig. 5.4 (a). Micrograph and replica showing delamination in a [ 0 2 / 9 0 2 / ± 3 0 ] s laminate at 0 ° . Experimental Observations 160 Fig. 5.4 (c). Micrograph and replica showing delamination in a [ 0 2 / 9 0 2 / ± 3 0 ] s laminate at 70°. Fig. 5.4 (d). Micrograph and replica showing delamination in a [ 0 2 / 9 0 2 / ± 3 0 ] s laminate at 90°. Experimental Observations 163 Experimental Observations 164 Experimental Observations 165 (c) 275 MPa Fig. 5.5. Radiographs of damage in a [ ± 30/902 / 0 2 ] s laminate after sequential loading to (a) 50 MPa (b) 185 MPa (c) 275 MPa (d) 365 MPa (e) 550 MPa Experimental Observations 166 (e) 550 MPa Experimental Observations 167 Fig. 5.6 (a). Interlaminar normal stress o z distribution in a [ ± 30/902 / 0 2 J S laminate. Experimental Observations 168 Fig. 5.6 (b). Interlaminar shear stress TQZ distribution in a [±30/902/02]s laminate. Experimental Observations 169 Fig. 5.6 (c). Interlaminar shear stress r r z distribution in a [± 30/902 /023s laminate. Experimental Observations Fig. 5.7 (a). Micrograph and replica showing delamination in [ ± 3 0 / 9 0 2 / 0 2 ] s laminate at 0 ° . Fig. 5.7 (b). Micrograph and replica showing delamination in a [ ± 30/902 / 0 2 ] s laminate at 45°. Fig. 5.7 (c). Micrograph and replica showing [ ± 3 0 / 9 0 2 / 0 2 ] s laminate at 70° . delamination in a Fig. 5.7 (d). Micrograph and replica showing delamination in a [ ± 30/90 2 /02]s laminate at 90° . Experimental Observations 174 Fig. 5.7 (e). Micrograph and replica showing delamination in a [ ± 3 0 / 9 0 2 / 0 2 ] s laminate at 110°. Experimental Observations 175 Fig. 5.7 (f). Micrograph and replica showing [ ± 3 0 / 9 0 2 / 0 2 ] s laminate at 135°. delamination in a Fig. 5.7 (g). Micrograph and replica showing delamination in a [ + 30/902/0 2]s laminate at 180°. Experimental Observations 177 Fig. 5.8. Radiographs taken before loading a [ ± 4 5 / 0 2 ] s laminate. Experimented Observations 178 (0 2/±45) s 178 MPa 25% of uts IX 45° p lamjnatj nitiation 311 MPa 44% of uts x splitti 0°|Ny Fig. 5.9. Radiographs of damage in a [ 0 2 / ± 4 5 ] s laminate. (±45/02), Experimental Observations 179 178 MPa 27.8% of uts Fig. 5.10. Radiographs of damage in a [ ± 4 5 / 0 2 ] s laminate. Fig. 5.11 (a). Interlaminar normal stress oz distribution in a [02/i~45] s laminate. Experimental Observations 181 Fig. 5.11 (b). Interlaminar shear stress rQz distribution in a [ 0 2 / ± 4 5 ] s laminate. Fig. 5.11 (c). Interlaminar normal stress r rz distribution in a [ 0 2 / ± 4 5 ] s laminate. Experimental Observations 183 Fig. 5.12 (a). Interlaminar normal stress oz distribution in a [±45/02J"s laminate. Experimental Observations 184 Fig. 5.12 (b). Interlaminar shear stress TQZ distribution in a [ ± 4 5 / 0 2 ] s laminate. Experimental Observations 185 Fig. 5.12 (c). Interlaminar shear stress T RZ distribution in a [ ± 4 5 / 0 2 ] s laminate. Experimental Observations J86 Fig. 5.13. Fracture surfaces of [ 0 2 / ± 4 5 ] s and [ ± 45/C>2]s laminates. Replicas of sections showing delamination at different angular locations in a [ ± 4 5 / 0 2 ] s laminate (a), at 0° (b). at 60° (c). at 9 0 ° . Experimental Observations Fig. 5.15. Micrographs of sections showing delamination at different angular locations in a [ 0 2 / ± 3 0 ] s laminate (a), at 0° (b). at 70° (c). at 90° (c). at 90° Fig. 5.16 (a). Interlaminar normal stress az distribution in a [ 0 2 / ± 3 0 j s laminate. Experimental Observations 191 Fig. 5.16 (b). Interlaminar shear stress TQZ distribution in a [ 0 2 / ± 3 0 ] s laminate. Fig. 5.16 (c). Interlaminar shear stress T RZ distribution in a [fJ2/±30] s laminate. Experimental Observations 193 (c). 335 MPa. Fig. 5.17. Radiographs of damage in a [0/90] s laminate after sequential loading to (a).165 MPa. (b). 250 MPa. (c). 335 MPa. (d). 375 MPa. (e). 420 MPa. Experimental Observations 194 (d). 375 MPa. (e). 420 MPa. Experimental Observations 195 (c). 335 MPa. Fig. 5.18. Radiographs of damage in a [90/0] s laminate after sequential loading to (a).165 MPa. (b). 250 MPa. (c). 335 MPa. (d). 375 MPa. (e). 420 MPa. Experimental Observations 196 (e). 420 MPa. Experimental Observations 197 Fig. 5.19 (a). Interlaminar normal stress a z distribution in a [0/90] s laminate. Experimental Observations 198 Fig. 5.19 (b). Interlaminar shear stress TQZ distribution in a [0/90] s laminate. Experimental Observations 199 Fig. 5.19 (c). Interlaminar shear stress r r z distribution in a [0/90] s laminate. Experimental Observations 200 Fig. 5.20 (a). Interlaminar normal stress o z distribution in a [90/0] s laminate. Experimental Observations 201 to Fig. 5.20 (b). Interlaminar shear stress TQZ distribution in a [90/0] s laminate. Experimental Observations 202 Fig. 5.20 (c). Interlaminar shear stress T RZ distribution in a [90/0] s laminate. Fig. 5.21. Micrographs of sections taken at 65° from the loading direction in [0/90] s laminate. Experimental Observations (c) 185 MPa Fig. 5.22. Radiographs of damage in a [45/0/-45/90] s laminate after sequential loading to (a) 115 MPa (b) 140 MPa (c) 185 MPa (d) 210 MPa (e) 235 MPa. Experimental Observations 205 (d) 210 MPa (e) 235 MPa. Experimental Observations 206 Fig. 5.23 (a). Interlaminar normal stress a z distribution in a [45/0/-45/90] s laminate. Experimental Observations 207 Fig. 5.23 (b). Interlaminar shear stress T Q Z distribution in a [45/0/-45/90] s laminate. Experimental Observations 208 Fig. 5.23 (c). Interlaminar shear stress [45/0/- 45/90] s laminate. r distribution in a Fig. 5.24. Micrographs of sections showing delamination at different angular locations in a [45/0/-45/90] s laminate, (a), at 0 ° . (b). at 4 5 ° . (c). at 9 0 ° . Experimental Observations (c). at 90° Experimental Observations 211 p o 10 • p fO • • • • o o r • • • • • • p d tpODTJ • 0.0 - B — ^ — B — B - T 50.0 100.0 150.0 200.0 Stress Combination Function MPa — i 250.0 Fig. 5.25. Comparison of delamination in [ 0 2 / 9 0 ^ / ± 3 0 ] s laminate with stress combination function. Experimental Observations 212 o O - i • p CO c o •— D c • — E a © q U3 c O E < q q c>. -S—B-• • • • •o • C D T T 1 250.0 0.0 50.0 100.0 150.0 200.0 Stress Combination Function MPa Fig. 5.26. Comparison of delamination in [ ± 307902/O^Js laminate with stress combination function. Experimental Observations 213 p d-i p CO c o 1 9 E " o <D o c o E < p • • • • • • o CN • • O d. T T 1 250.0 0.0 50.0 100.0 150.0 200.0 Stress Combination Function MPa Fig. 5.27. Plot of maximum delamination at each angle vs. stress combination function for [O2/9O2/ ± 3 0 ] s and [ ± 3 0 / 9 0 2 / 0 2 ] s laminates. TABLE HI. In-plane ply elastic properties used in the present stress calculations:-MATERIAL ELASTIC PROPERTIES Hercules E}1 = 138.0 GPa AS1/3501-6 E22 = 8.96 GPa Graphite/epoxy G12 = 7.10 GPa v12 = 0.30 Scotch-ply E77 = 38.6 GPa Type-1003 E22 = 8.27 GPa Glass/epoxy GI2 = 4.14 GPa v12 = 0.26 XAS/914 E77 = 145.0 GPa Graphite/epoxy E22 = 9.5 GPa G12 = 5.6 GPa v12 = 0.31 215 o TABLE IV. Comparison of delamination with interlaminar stresses at different interfaces in a [02/902/±30]s graphite/epoxy specimen: Angle Interface a z T6z Trz Function Delamination 1 1.4 -0.0 5.1 8.9 0.0 2 5.4 -0.0 10.1 18.3 0.0 0° 3 12.6 -0.0 16.6 31.3 0.0 4 23.1 -0.0 23.0 46.1 0.8 5 32.4 -13.1 11.5 44.3 1.0 6 35.5 -0.0 0.0 35.5 0.3 1 7.8 20.1 29.3 62.0 0.0 2 31.4 40.2 58.5 126.9 1.7 45° 3 60.1 22.4 48.7 110.6 0.0 4 83.6 4.5 38.9 107.6 2.5 5 106.5 24.8 46.8 140.6 1.8 6 119.1 0.0 -0.0 119.1 0.0 1 5.0 53.5 18.6 98.2 0.0 2 19.9 106.9 37.2 197.1 4.0 70° 3 31.6 83.9 6.3 149.2 0.0 4 26.7 61.0 • -24.6 116.9 2.5 5 14.0 -12.5 -22.6 46.9 1.7 6 8.0 0.0 0.0 8.0 0.0 216 T A B L E TV. (Contd.,) Angle Interface °z T6z Trz Function Delamination 1 -0.9 0.0 -3.3 5.7 0.0 2 -3.6 0.1 -6.7 11.0 0.0 90° 3 -19.1 0.1 -51.1 86.4 0.8 4 -58.3 0.1 -95.4 154.7 3.3 5 -96.6 -98.2 -47.7 162.6 3.0 6 -109.4 0.0 -0.0 0.0 0.0 1 5.0 -53.5 18.6 98.2 0.0 2 19.9 -106.9 37.2 197.1 4.7 110° 3 31.6 -83.9 6.3 149.2 0.0 4 26.7 -61.0 -24.6 116.9 0.5 5 19.6 -73.4 -1.9 128.7 1.0 6 19.1 -0.0 0.0 19.1 0.3 1 7.8 -20.1 29.3 62.0 0.0 2 31.4 -40.2 58.5 126.9 0.7 135° 3 60.1 -22.4 48.7 110.6 0.0 4 83.6 -4.5 38.9 107.6 0.3 5 91.9 20.2 -7.9 99.3 3.2 6 89.8 -0.0 -0.0 89.8 0.7 217 TABLE V. Comparison of delamination with interlaminar stresses at different interfaces in a [± 30/902/02]s graphite/epoxy specimen: Angle Interface °z T6z Trz Function 1 -3.5 -14.4 -13.1 33.5 2 -14.0 0.0 -26.1 43.0 0° 3 -26.2 0.0 -19.6 21.6 4 -35.0 0.0 -13.1 0.0 5 -40.3 0.0 -6.6 0.0 0.0 0.3 0.7 0.3 0.0 1 2.2 24.1 2 -7.8 -4.9 3 -35.4 -27.2 4 - 69.5 - 49.6 5 -97.5 -24.8 6 -106.8 0.0 8.2 44.1 5.0 -45.6 79.0 5.0 -57.6 104.5 0.7 -69.7 130.7 8.8 -34.8 0.0 0.0 0.0 0.0 0.0 1 0.2 -86.7 2 8.2 -70.4 3 14.1 -98.5 4 1.2 -126.7 5 -15.6 -63.3 6 -21.2 0.0 0.9 150.2 0.0 28.7 131.9 3.7 -6.5 171.6 0.0 -41.8 231.0 8.0 -20.9 114.5 0.0 0.0 0.0 0.0 218 T A B L E V. (Contd.,) Angle I: 90° 110° 135° •face °z T6z Trz Function Delamination 1 14.8 -116.2 55.2 223.3 7.7 2 59.2 -0.1 110.5 200.3 3.5 3 105.0 -0.1. 60.4 148.3 0.8 4 123.9 -0.1 10.4 125.2 0.3 5 128.1 -0.0 5.2 128.4 0.0 6 129.5 0.0 -0.0 129.5 0.0 1 -14.4 28.9 -53.8 104.8 0.5 2 -41.0 4.9 -45.6 67.9 2.7 3 -68.6 27.2 -57.6 86.4 0.3 4 -102.7 49.6 -69.7. 106.6 2.0 5 -130.7 24.8 -34.8 0.0 0.3 6 -140.1 -0.0 0.0 0.0 0.3 6 -42.0 -0.0 0.0 0.0 0.0 1 -3.5 -14.4 -13.1 33.5 0.0 2 -14.0 0.0 -26.1 43.0 0.8 3 -26.2 0.0 -19.6 21.6 0.0 4 -35.0 -0.0 -13.1 0.0 0.3 5 -40.3 -0.0 -6.6 0.0 0.0 6 -42.0 0.0 0.0 0.0 0.3 219 CHAPTER VI SUMMARY AND CONCLUSIONS Delamination initiating at free-edges is an important mode of failure in fiber reinforced composite laminates. The experimental and analytical investigations found in the literature indicate the importance of interlaminar stresses in understanding delamination initiated failure. The analytical treatment of interlaminar stresses is made difficult by the presence of singularities at laminate free-edges. The accurate, though still approximate, solution of interlaminar stresses thus require the use of numerical methods, such as finite elements, which can be quite costly when used for curved free- edges. In this work, a simple approximate technique to predict the sign and relative magnitude of interlaminar stresses around a hole in a laminate has been presented. The method assumes that the components of ply stresses that conribute to interlaminar effects are the deviations of the lamination theory ply stresses from the gross laminate stresses. The reasoning for the existence of interlaminar stresses around a hole in a laminated plate is explained on the basis of the difference in stress behaviour in a laminate, as opposed to an equivalent homogeneous plate. Furthermore, the deviations of the laminate ply sresses from the homogeneous solution can be used as input to simple equilibrium arguments, in a reduced 2-D formulation of the hole problem. Summary and Conclusions 220 The results of the approximate method have been compared with numerical results from the literature. Comparisons have been made on the basis of relative changes that take place in the magnitudes and signs of interlaminar stresses as a function of angle around the hole, and through the thickness. The method shows fair agreement for a wide range of laminates. • Comparisons have also been made with observations of delamination damage reported in the literature. Damage development around holes due to tension and compression fatigue, and quasi-static loading showed good agreement with the results of the present stress calculations. It has been found that the presence of other interactive damage modes may alter the interlaminar stress distributions to the exent which makes these comparisons only qualitative. An experimental program was also undertaken to examine the nature of damage induced at the hole boundaries by static loading. The delamination damage observed in a number of different laminates was compared with interlaminar stresses calculated using the present approximate methods. The non-destructive detection of damage at the hole boundary using zinc-iodide enhanced X-ray radiography helped reveal the locations of delamination around the hole. Through the thickness locations of delamination at specific angular positions were found by sectioning the laminate at the hole. Good qualitative agreement was observed between these observations and the results of the present stress calculations. It was observed that in most of the laminates examined delamination was preceeded by matrix cracking. This can alter the original stress ditribution around the hole making delamination prediction based on stress calculations of the undamaged hole somewhat unreliable. The altered stress distribution, accompanied by delamination and other damage, growth can possibly initiate delamination at new locations or change the direction of delamination propagation. In spite of these problems, the relative amounts of delamination observed at different interfaces and at different angles Summary and Conclusions 221 were in agreement with the stresses calculated. A mixed-mode failure criterion was also suggested. Semi-quantitative estimates of the relative amounts of delamination observed at different interfaces in two different laminates were compared with this criterion. A good correlation was observed between the stress combination function and the maximum delamination observed at each angle. These results clearly justify the use of approximate methods of stress analysis for predicting delamination in composite laminates. The main conclusions which can be drawn from the present study are; 1. Only some of the in-plane stresses as calculated by LPT contribute to the generation of interlaminar stresses. Of the two possible methods available for calculating these stresses, the more physically meaningful method, (called the "modified stress method") leads to better agreement with numerical results in the literature. 2. Simple equilibrium arguments can be used successfully to relate the in-plane stresses to interlaminar stresses. It can predict the signs and relative magnitudes of interlaminar stresses around holes when the responsible ply stress components are correctly chosen. 3. Delamination at hole boundaries is rarely observed without matrix cracking, which must influence the laminate behaviour considerably. Yet, the results of the present stress calculations appear to be in good qualitative agreement with the experimental observations of delamination. The lack of a reliable mixed-mode failure criterion makes accurate predictions of delamination difficult Summary and Conclusions 222 4. With the present approximate technique simple correlations between delamination and interlaminar stresses can be attempted. Experimental assessment of delamination and its treatment in the context of general damage development remain major obstacles to the development of such correlations. 223 References:-Bjeletich J.G., Crossman F.W. and Warren W.J., (1977), "The Influence of Stacking Sequence on Failure Modes in Quasi-Isotropic Graphite-Epoxy Laminates", Failure Modes in Composites IV, Proceed, of T M S - A I M E / A S M , Cornie J.A. and Crossman F.W., Eds., Chicago, pp. 118-137. 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