FREE-EDGE EFFECTS A R O U N D H O L E S IN COMPOSITE LAMINATES by HEMAGUPTHA DHARMARAJ GOONETILLEKE B.Sc. (Eng) (Hons), The University of Sri Lanka, 1973 M.A.Sc, The University of British Columbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT T H E REQUIREMENTS DOCTOR FOR THE DEGREE OF OF PHILOSOPHY in T H E F A C U L T Y O F G R A D U A T E STUDIES Department of Metallurgical Engineering We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITSH NOVEMBER COLUMBIA 1986 ® Hemaguptha Dharmaraj Goonetilleke, 1986 OF In presenting degree at this the thesis in University of partial fulfilment of of department this or publication of thesis for by his or her representatives. DE-6(3/81) agree It this thesis for financial gain shall not Metallurgical _ Engineering _ The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date December 10, 1986. that the scholarly purposes may be permission. _ , Department of requirements British Columbia, I agree freely available for reference and study. I further copying the is for an advanced Library shall make it that permission for extensive granted by the understood that head of copying my or be allowed without my written ii ABSTRACT The recieved free-edge less attention free-edge than the effect straight around free-edge holes in problem. composite laminates has Mathematical analysis of 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 the literature interlaminar stress' distributions reasonably well. Experimental also agree with the present results. around holes under quasi-static combination A simple function problems associated are also discussed. the delamination found A n experimental study of the in damage development hole boundaries are found to be in good qualitative semi-quantitative of of loading for a number of different laminates is reported. The delamination observed at the agreement observations three correlation interlaminar with the development between these stress components of reliable methods results is also and a stress derived. The of delamination prediction iii TABLE OF CONTENTS : 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 3.4 interlaminar stresses 35 Interlaminar Stress Distribution 39 iv 3.4.1 Interlaminar normal stress a 3.5 3.6 4. 3.4.2 Interlaminar shear stress r 40 3.4.3 Interlaminar shear stress T 43 Solution of In-Plane Stresses 45 3.5.1 Isotropic Solution 45 3.5.1 Orthotropic Solution 47 Residual Thermal Stresses 49 Comparisons With Literature 60 4.1 60 4.2 5. 39 z Theoretical Comparisons 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 Experimental Comparisons 75 4.2.1 Whitcomb (1981) 77 4.2.2 Kress and Stinchcomb (1985) 80 Experimental Observations 126 5.1 Introduction 126 5.2 Experimental Procedure 127 5.3 5.2.1 Specimen Preparation 128 5.2.2 Testing and Observations 129 Results 131 5.3.1 [ 0 / 9 0 2 / ± 3 0 ] 2 laminates s and [±30/90 /02] 2 s 131 5.3.2 [ 0 2 / ± 4 5 ] s and [ ± 4 5 / 0 2 ] 5.3.3 [ 0 2 / ± 3 0 ] s laminate 5.3.4 [0/90] s and [90/0] 5.3.5 [45/0/-45/90] 5.4 6. Discussion s s s laminates 138 142 laminates laminate 144 146 149 Summary and Conclusions 219 References 223 vi LIST OF TABLES Page I. Analytical studies of free-edge interlaminar stresses around holes by different authors II. 24 In-plane ply elastic properties used by different authors in stress calculations III. In-plane 125 ply elastic properties used in the present stress calculations IV. Comparison of delamination with interfaces in a [02/902/ ± 3 0 ] V. 214 s interlaminar stresses at different graphite/epoxy specimen Comparison of delamination with interlaminar stresses at interfaces in a [ ± 3 0 / 9 0 2 ^ 2 ^ s graphite/epoxy specimen 215 different 217 vii LIST O F FIGURES Page 1.1. Delarnination at a hole boundary in a composite laminate. 2.1. Distribution [0/90] 2.2. s of T Q s obtained by several authors for a boron/epoxy laminate. Distribution of [0/90] Z 6 a 22 obtained z by different authors for a 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 57 3.8. Approximate distribution of r ,„ compatible with the assumed z near the free-edge. distribution of a . 58 z 3.9. Approximate distribution of 4.1. Effect of IIa on a z at z = near the free-edge. h in a [90/0] s laminate using Point Stress method. ( d = t ) 4.2. Effect of IIa on o z at z = h in a [90/0] Average Stress method. ( d = t ) 59 85 s laminate using 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] laminate, using s Modified Average stress method. ( d — t ) Effect of IIa on a z at z = h in a [90/0] laminate, using s Point Stress method. ( d = I ) Effect of IIa on a at z = Average Stress method. Effect of //a on a z h in a [90/0] s laminate using ( < / = / ) at z = h in a [90/0] s laminate using s laminate, using Modified Point Stress method. ( d = I ) Effect of IIa on a at z = h in a [90/0] Modified Average stress method. ( d = / ) Present results compared with numerical solution for [90/0] s laminate at z = h. Results compared = of normalized Modified Average with numerical solution for [90/0] Stress s method laminate at z h. Present results of T Q distribution at z = Z h in a [0/90] s laminate compared with the solution of Raju and Crews Present results of T Q distribution at z = Z h in a [90/0] s laminate compared with the solution of Raju and Crews Present results [0 /±30/T30] 2 Results compared of compared s laminate at z = normalized with with numerical laminate at z = 0. numerical solution for a 0. Modified solution Average for a Stress method [ 0 2 / i " 3 0 / T 30] s 4.15. Present results midpane of interlaminar in a [ 0 2 / ^ 4 5 / ^ 4 5 ] normal s stress distribution at laminate compared with the solution of Rybicki and Schmueser (1977). 4.16. Present results midpane of interlaminar in a [ 0 2 / ± 6 0 / T 6 0 ] s normal stress distribution at 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 midpane in a [ ± 4 5 / £ 4 5 / 9 0 2 ] s : normal stress distribution at laminate compared with the solution of Rybicki and Schmueser (1977). 4.19. Present results of interlaminar normal stress distribution at midpane in a [ ± 6 Q / T 6 0 / 9 0 2 ] s laminate compared with the solution of Rybicki and Schmueser (1977). 4.20. Present results compared with numerical solutions (Whitcomb, 1981) of interlaminar stress thickness in [45/90/-45/0] 4.21. s distributions laminate specimen, (in tension) Present results compared with numerical solutions (Whitcomb, 1981) of interlaminar thickness in [ 9 0 / ± 45/0] 4.22. (a)-(c). Present results stress s compared distributions 4.23. (a)-(c). Present results compared across laminate specimen, (in compression) with Tang's (1977) interlaminar stresses at z = 0 in a [0/90] with Tang's solution of laminate. s (1977) interlaminar stresses at z = 0 in a [ ± 4 5 ] 4.24. across s solution of laminate. C-scan records of various notched laminates after or compressive fatigue cycles. (Whitcomb, 1981). 10 7 tensile 4.25. Present solution of a distribution in a [ 0 / ± 4 5 / 0 ] z laminate. s (Whitcomb, 1981). 4.26. Present solution of a distribution z 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 [ 9 0 / ± 45/0] s specimen subjected to compression fatigue. 4.29. Radiographs of damage in a [0/90/± 45] sequential loading to (a). 0.9 a ^ 4.30. (a)-(c). Interlaminar stress distribution laminate s and (b). 1.05 in a after o^. [0/90/±45] laminate. s (Kress and Stinchcomb, 1985). 4.31. Radiographs of damage in a [45/90/-45/0] sequential loading to (a). 0.9 a^ t 4.32. (a)-(c). Interlaminar laminate s and (b). 1.3 after o^. stress distribution in a [45/90/-45/0] laminate. s (Kress and Stinchcomb, 1985). 4.33. Damage on [0/90/±45] 4.34. Damage 0/90, and 45/-45 interfaces of a -45/0 interfaces of a laminate. s on [45/90/-45/0] 90/45 45/90, s 90/-45 and laminate. 5.1. Specimen sections at the hole edge. 5.2. (a)-(e). Radiographs of damage in a [ O 2 / 9 O 2 / ± 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 ] 5.4. (a)-(f). Micrographs and replicas of sections showing delamination in a [02/902/±30] s laminate. 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 [ ± 3 0 / 9 0 2 / 0 2 ] 5.7. (a)-(g). Micrographs and replicas of sections showing delamination in a [ ± 30/902 /02] laminate. s laminate. s 167 170 5.8. Radiographs taken before loading a [ ±45/02]$ laminate. 177 5.9. Radiographs of damage in a [ 0 2 / ± 4 5 ] laminate. 178 5.10. Radiographs of damage in a [ ± 4 5 / 0 2 ] laminate. 179 s s 5.11. (a)-(c). Interlaminar stress distributions in a [ 0 2 / ± 4 5 ] laminate. 180 5.12. (a)-(c). Interlaminar stress distributions in a [ ± 4 5 / 0 2 ] s laminate. 183 5.13. 186 Fracture surfaces of [ 0 2 / ± 4 5 ] s and [ ± 4 5 / 0 2 ] s s laminates. 5.14. (a)-(c). Replicas of sections showing delamination at different angular locations in a [ ± 4 5 / 0 2 l s laminate. 5.15. (a)-(c). Micrographs of sections 187 showing angular locations in a [ 0 2 / ± 3 0 ] s delamination at different 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] laminate after sequential s 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 direction in [0/90] s taken and [90/0] at s 65° from the loading laminates. 5.22. (a)-(e). Radiographs of damage in a [45/0/-45/90] 203 s laminate after sequential loading to different stress levels. 5.23. (a)-(c). Interlaminar stress distributions in a [45/0/-45/90] 204 s laminate. 206 5.24. (a)-(c). Micrographs of sections showing delamination angular locations in a [45/0/-45/90] 5.25. s laminate with function. Comparison of delamination in [ ± 30/902/02]s laminate with stress combination 5.27. different laminate. Comparison of delamination in [ O 2 / 9 O 2 / i 3 0 ] stress combination 5.26. s at Plot of maximum function. delamination at each combination function for [ O 2 / 9 O 2 / ± 3 0 ] laminates. s angle vs. and [ ± 30/902 stress / 0 2ls 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 faculty members are also extended to my fellow graduate students in the Department of Metallurgical Engineering. The assistance and of the technical staff of this department is greatly appreciated. I Research Council, assistantship. am Ottawa for also grateful providing to the financial National support in Science the form and Engineering of a research 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 aircraft, automobiles, buildings and machinery are but much stronger components widely produced nowadays with for 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 provided the necessary high strength technological attainable background in for fine early fibers of some materials of composite development 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 initiating at a free-edge. mode of failure in such multilayered composites is delamination Delamination is the separation of the layers at the interface. Introduction The onset laminate of delamination behaviour, at including laminate failure. free-edges Around can holes have and significant other 2 influence internal on 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 of these three-dimensional stresses -commonly known as "Interlaminar components Stresses" - are found to be mainly responsible for delamination induced failure. Therefore the study o f interlaminar stresses in composite laminates is very important in understanding laminate behaviour. The orthotropic, and individual layers on a macroscopic level, (or may plies) be of a considered composite laminate homogeneous. A are 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 stress in that direction). This can be contrasted gross laminate 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 C L T . Though the gross laminate stresses satisfy the boundary conditions, the individual ply stresses do not stresses are generated phenomenon solids. unique themselves do so. In order to maintain equilibrium, interlaminar along ply interfaces within a very local region near the edges- a to composite laminates and not generally observed in homogeneous Introduction 3 The interlaminar stresses that develop in composite laminates exhibit rapid changes of gradient near the free-edges. interlaminar stresses extremely difficult are commonly used free-edges is one for the This has made analytical treatment of Numerical methods, such as finite elements, which purpose of evaluating interlaminar stresses along can be quite costly when used with curved free-edges. such free-edge commonly encountered. The straight The circular boundary complexity associated with the three-dimensional nature of the stress distribution coupled with the wide range of possible laminate and formulations loading configurations for each case. It require is here individual programming that simple approximate or methods mathematical 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 problem have been unsuccesful, as they have considered the effect for the hole 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 out-of-plane however, the plate stresses which decay to zero do not generate boundary layer stresses, of the type observed in laminate edges. Thus, it can be argued, that even in the non-homogeneous the exact solution must decay laminated plate the laminate stresses corresponding to without giving rise to any interlaminar interlaminar stresses observed at the laminate hole must therefore effects. The 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 of approach to determine the out-of-plane stresses. This method 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 cracking, general with or the stresses calculated. Delamination is rarely splitting, which lack of understanding must influence of the effect the laminate of the observed behaviour without matrix significantly. 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 using the present approach in chapter V. Good agreement is observed in terms of the predictions of delamination locations around the attempt to combine the three interlaminar with the hole and through stress components to the stresses calculated thickness. A first 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 failure reinforced in fiber laminated plates is of composite materials. One mode of 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 instrumental in precipitating delamination and subsequent strength The complex three-dimentional interlaminar states of stresses stress are that the exist interlaminar stresses are degrdation. out-of-plane along the edges components of 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. 8 Literature Survey 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 are and curved free-edges. usually the traction-free-edges of finite While the straight free-edges width laminates, curved free-edges treated are the boundaries of internal discontinuities. The circular hole provides the simplest form of a curved free-edge two geometries geometry. From an analytical point of view, one difference between the is that the straight free-edge effect can sometimes two-dimensional problem in laminate elasticity. The curved free-edge be treated as a effect on the other hand, is essentially a three-dimensional problem. 2.1 STRAIGHT FREE-EDGE EFFECTS:- Most of the studies on free-edge with straight free-edges. Mathematical analysis effects of free-edge have been on laminates 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 study the distribution of interlaminar shear stresses in the laminate as a set of anisotropic layers separated layers acting as adhesives between developed by neglecting the the laminate they modeled by isotropic layers, with anisotropic layers. The interlaminar normal stress in the 9 the the isotropic equilibrium equations 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. argued that the force and moment resultants which are statically equivalent They 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 confirmed by Rybicki (1971) using finite elements. An approach to design susceptible to delamination failure by interlaminar normal stress was later been laminates presented by Pagano and Pipes (1973) using the same concepts. Other Herakovich (1978) (1977) based upon based an approximate upon theories a perturbation extension of include the analysis, and Reissner's (1950) that solution by developed Hsu by and Pagano variational principle. While the solution by Hsu and Herakovich provided mathematical evidence for singular interlaminar shear stresses, Pagano's theory problem solving view point contained Among the no edge singularities, an many analytical approaches, advantage from a 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 treated as a homogeneous anisotropic body in equilibrium independent of the 10 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. ( K i m and Soni, 1984; Soni and Kim, 1986) However, the application of this method requires extensive computational time even for simple laminate configurations. More developed by Johnson recent and approximate Kemp (1985) and theories include that Conti by the and structural De model 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 extension of and formulation. the work by Pagano The model by Conti Pipes It provides a simple approximate stresses at a straight free-edge. (1973) method based to and De Paulis is an a simple polynomial on evaluate all the interlaminar 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 finite-width the distribution laminate, under of stresses uniform axial and displacements throughout a between the layer, extension. The results of their solution have shown that significant interlaminar shear stresses at laminate free-edges are allow shear transfer four layers of the laminate. They observed required to 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 allow Pipes and Pagano to prove the existence, nor predict the magnitude of not 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, very fine finite-element grids and improved programming 1985). With the use of 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, inconsistencies finite-element are found in as the pointed out published by Salamon numerical (1980), results. a Using number a of triangular grid to model the 0/90 interface region of a cross-ply laminate, Wang and Crossman (1977) revealed an apparently bounded and a singular normal stress in the [0/90] s normal stress in the [90/0] s laminate 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 elements in which traction-free-edge application of hybrid-stress finite conditions are satisfied exactly. Similar inconsistencies are found in the literature for other laminate configurations and interlaminar shear stress distributions. Literature Survey From presence the evidence found of elastic stress singularities can free-edge in the severely literature influence it is clear 12 that the numerical solutions of the 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 Choi the precise nature of the free-edge stress fields Wang and (1982) presented a solution procedure to evaluate the exact order of the stress singularity. Based on Lekhnitskii's (1963) complex-variable stress anisotropic elasticity, a rigorous mathematical free-edge functions solution was and basic developed relationships in determine the to 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 geometry as compared to a straight free-edge free-edges. complexity of the curved free edge , fewer studies have been done on curved 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 almost attention. Except all studies for the have been closed-form analytical solution given by numerical, mostly employing finite elements. Tang (1977), In an early study by Levy, Armen and Whiteside (1972) a composite finite element constructed of orthotropic membranes separated interlaminar shear stresses, t [±30] s t of their solution This model free- edges. was shear-resisting Results are edge around holes in [ ± 4 5 ] , nature by results first are s was used to calculate the at the free boron/epoxy laminates. Due to the given for the first ply interface and [0/90] not media s obtained developed by Puppo for and the interlaminar Evensen normal (1970) for stress straight Literature Survey 13 distribution. Extending the work by Rybicki (1971) Rybicki and Hopper (1973) developed a three-dimensional on straight finite-element free-edges, 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] reversing laminate. s the interlaminar plies stress According to does alter not distributions. their Of analysis, the a magnitudes, particular change but interest in in only their stacking sequence by changes the of work distribution of interlaminar normal stress at the mid-plane of a [90/0] the resulting magnitude at the free-edge s is sign the radial laminate. Though 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 stresses. Both determining and on an isoparametric displacement thick and thin laminates formulation to calculate the interlaminar of boron/epoxy have been considered in the stresses through the thickness and around the hole. Interlaminar normal tangential configurations. interlaminar Unfortunately shear their stresses results are can not calculated be for compared distributions obtained by them are not for ply interfaces cross-ply with and others, [±45] since s the but for locations one-sixth the ply thickness away from these interfaces. The influence of stacking sequence, laminate-thickness-to-hole lay-up angle and the ratio of diameter on laminate behaviour has been examined by Rybicki and Schmueser (1976, 1978) using a three-dimensional finite element analysis. Results are 14 Literature Survey obtained for number of different graphite/epoxy laminates under uniform stress loading. Attention is focused only on the distribution of the interlaminar normal stress, a , at the z 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 [0 /±6/Td] , 2 is 3 0 ° , 45° reduced [±d/^e/0 ] > s and 2 6 0 ° . In order and [90 /±6/l-e] [±8/ rd/% ] : s 2 to simplify the number of finite elements 2 s the computations where 6 s Rybicki and Schmueser 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 changed. More importantly, they observed that unlike the sign of a cannot always be predicted from the as the straight sign of the stacking sequence free-edge is case , 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 distributions were obtained for [0/90] thickness at the interface between normal and shear interface from hole the boundary crossed s and and [90/0] along stresses strongly suggest was observed at a straight s s radial plies. Through from the outside in both [0/90] what of elements near the hole boundary. Stress the the graphite/epoxy laminates, through the lines and thickness s the and [90/0] s free-edge. hole the the first ply laminates. This is quite As at distributions of interlaminar existence of singularities at Crossman (1977) found a non-singular behaviour in a in the [90/0] around mentioned earlier, different Wang and distribution through the thickness 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 the [0/90] laminate. The s [90/0] laminate s does not however show reversal. As shown later, this allows the use of equilibrium arguments in the [90/0] laminate, but not in the [0/90] s s this 15 very abrupt in calculating a z laminate. Raju and Crews obtained circumferential distributions of interlaminar normal stress which are compressive for most of the [0/90] s and [90/0] s region around the hole in both 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. economical According to the method for authors, estimating this approximate interlaminar stresses, procedure and a appears to be suitable alternative an to complex three-dimensional analyses of the hole problem. In investigated the a influence study of of fatigue interlaminar damage stresses around on holes, delamination Whitcomb in (1981) 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 boundary (1984) have recently region in a [0/90] s carried out interlaminar stress analysis of the hole 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 normal Raju from outside. According to them, the general shape of the stress stress distribution of their solution agrees roughly with and Crews (see interlaminar that obtained by 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 among different methods of the agreements observed of interlaminar stress calculations. Ericson et in the literature 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 ratio on interlaminar stress distributions in the [0/90] three-dimensional finite element analysis with a substructuring s laminate. technique thickness Using they have a 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 similar tangential interlaminar shear stress distribution. The shape of the to that obtained by Raju and Crews, with distribution is 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 boron/epoxy and [ ± 4 5 ] s tangential interlaminar shear stresses were found in [0/90] 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] shear s laminate, the general trends are similar. Tang's solution of the radial interlaminar 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 free-edge significant stresses along straight effort and has thus been curved boundaries. made The towards understanding 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 delamination and strength applied free-edge loading, behaviour possible only through experimental observations of in composite delamination is laminates. considered stresses. Many experimental investigations on free-edge to the prediction of the to Regardless of the be initiated by effects have therefore type of interlaminar been related 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 sequence has also been the subject of many of laminate strength on stacking 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 Browning, 1972) attribute the phenomenon of free-edge and Pipes, 1973; Whitney delamination to the presence of tensile interlaminar normal stresses, others (e.g., Whitcomb, 1981; Soni and K i m , 1986) consider it to Whitney and be the Browning product used of some, lamination or theory all, components to normal stress and predicted delamination in number determine of the of different interlaminar sign stresses. of interlaminar 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 Whitcomb conditions in areas where (1981) both the high. However in some specimens observed delamination under fatigue 19 loading interlaminar shear and tensile normal stresses were delamination occured due to high interlaminar shear when the normal stresses were found to be compressive. Soni and K i m (1986) tested a large number of laminates exhibiting a dominant interlaminar shear stress to characterize the onset of understanding delamination of the due to interaction static tensile between loading. different They interlaminar concluded stress that an 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 Rowlands stress presence and of in-plane stress concentrations Whiteside (1974) observed concentration near the hole a found reduction boundaries for near discontinuities. in strength a range of associated material Daniel, with high and stacking sequence variations. Alterations in the mode of failure of these laminates from catastrophic to noncatastrophic have been explained on stresses calculated near a straight free-edge. the basis of the changes in interlaminar Whitney and K i m (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 delamination locations and the stresses calculated using finite elements. delamination sectioning through the the specimen thickness at at each different angle. The angles around use of the ultrasonic fatigue induced The locations of hole were found by 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 response X-ray of laminates radiography after containing and specific numbers around the hole. It other Raju (1984). mechanics the damage non-destructive development techniques. Some of cycles to determine the is observations reported delamination agree with the and holes, This results approach that their specimens were using deplied on the location of initial release rate analysis by O'Brien free-edge point of view, which is fundamentally monitered distribution of damage in each ply of a strain energy considers was fatigue delamination different from the from a fracture 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 characterization of laminate static and fatigue need mode to be resolved failure before making progress observations of delamination onset and strengths. However, a number in this criterion which takes into account the the of problems direction. First, a reliable mixed influence of different interlaminar stress components is required. Along with such criteria, reliable experimental techniques are also needed secondary understood. to determine the damage mechanisms associated interlaminar strengths. Second, the influence of such as splitting and matrix cracking has not been Without such an understanding it would be difficult to analyse the well various Literature Survey interacting Finally, relationships improved between experimental composite laminates is essential. delamination techniques of and laminate assessing behaviour different modes including of 21 failure. damage in Tang 1977 Rybicki and Hopper 1973 Levy et al 1972 Fig. 2.1. Distribution of TQ Z obtained by several authors for a [0/90] . boron/epoxy laminate. s / ^ % N Ericson et al. 1972 v /o z Q x 10 N b CO V) £ o 00 o "o E o z o c E 5- Raju and Crews 1982 CM 6" Fig. 2.2. Distribution of a z obtained by different authors for a [0/90] graphite/epoxy laminate. o g = applied stress s 24 TABLE I . Analytical different studies of free-edge interlaminar stresses around holes by authors:- METHOD USED AUTHOR LAMINATES STUDIED Levey, Composite Finite [±45] , Elements [0/90] s boron/epoxy Hopper 3-D Finite Element [0/90] s boron/epoxy Barker 3-D Finite Elements [0/90] Armen and Whiteside (1972) Rybicki and [±30] s and s (1973) Dana and [±45] s boron/epoxy (1974) Tang (1977) and s Boundary Layer [0/90] Theory and boron/epoxy s [±45] s graphite/epoxy Rybicki and Schmueser 3-D Finite Elements [0/90] , [90/0] s s (1976, 1978) [±9/j-e/o ] , 2 s [±d/*d/90 ] and 2 s [90 /±6/3-9] 2 6 = s where 3 0 ° , 45° 60°. graphite/epoxy and 25 TABLLE I Contd., AUTHOR METHOD USED LAMINATES STUDIED Raju and Crews 3 - D Finite Elements (1982) [0/90] s and [90/0] graphite/epoxy Ericson, Persson, Singular Finite Carlsson and Elements [0/90] s graphite/epoxy [0/90] s graphite/epoxy Gustavsson (1984) Lucking, Hoa Sankar (1984) Sankar (1984) and 3-D Finite Elements s 26 CHAPTER m MATHEMATICAL ANALYSIS This chapter presents the mathematical analysis used to calculate the interlaminar stresses near the angle around the hole boundary of a composite laminate as a function of hole. The first section gives a general analysis. Subsequent description of the method of 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 approximated by a 2 - D problem. The laminate at the circular edge of the hole is 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 laminate thickness. The hypothesis ignores shear deformations entire of layers with respect to each other. The assumption is made that any line perpendicular to the mid-plane before deformation neither remains extension perpendicular nor contraction. to the As a mid-plane result, the after deformation, shear strains and associated it suffers 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 approximate straight edge laminates, the effects of interlaminar stresses near the the hole boundary are first neglected. These stresses are however determined later from equilibrium considerations. Mathematical Analysis 3.1 28 METHOD OF ANALYSIS:- Consider the X-Y-Z planes laminate shown in Fig. 3.1 coordinate system located at its center. parallel to XY and the fiber the origin of the The fibers in various layers all lie in 0 ^ of each orientation angle positive in the anticlockwise direction from with the layer is measured 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 above laminate, we compare its stress behaviour with that of a homogeneous similar laminate, geometry' under inplane traction. Unlike the the the plate of 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 giving rise to different layer stresses that can be found by the orientations application of classical lamination theory. The distribution of the thickness average in-plane stresses throughout the homogeneous determined from plate, the as well theories of as the plane non-homogeneous elasticity. The exact laminated plane plate, stress can be 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 non-homogeneous giving boundary is approached. This occurs in both the homogeneous and plates. In the homogeneous plate these in-plane stresses decay without 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 laminate stresses satisfy the not themselves do so. free-edge In order Analysis 29 boundary conditions the individual ply stresses do 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 hole is based on the above observations of the fundamental difference in the behaviour between the two plates. The plate stresses which decay to zero in the homogeneous plate do not generate non-homogeneous non-homogeneous the type laminated of out-of-plane plate. Since boundary the layer corresponding stresses observed in the laminate stresses in the 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 laminate above 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 analysis of the free-edge these excessive (or deviatoric) ply stresses in a free-body 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. employing the Contrary to his deviatoric ply stress components objection, in a free it can be shown that by 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 and Pipes (1973). In the generated straight free-edge problem, it is the Analysis transverse 30 ply stresses 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 modified stress additional ply stress concept, method. The following analysis and hence describes is referred the formulations to as of the 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 laminate approximating the circular edge at an angle 8 Fig. 3.2. As will from the Analysis 31 X axis is shown in 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 radial directions of the straight approximating laminate hole, a cylindrical edge laminates. The system coordinate coincide with system of coordinates chosen hole center. Thus the direction of r being away from the the is used is 8,r,z tangential and describe the to with the positive 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 laminate. One method of estimating a set of uniform laminate stresses for the edge laminate is to take it as that predicted by the edge straight 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 assumed r = to deviate from this solution within the region defined Analysis by the 32 boundary 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, the form Pagano and Pipes (1973) are considered appropriate. For TQ Z and T , RZ used. For a , z simple models of their approximate solution was 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 3.2 2-D APPROXIMATION be analysed stresses, which for are through-thickness-averaged on the laminate. laminate stresses are individual assumed The 33 OF T H E H O L E P R O B L E M : - Once the can Analysis layer to be stresses. It defined, the should uniform throughout be the straight edge laminate noted that laminate, are the laminate in fact, the in-plane stresses, statically equivalent to the actual stress system 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 degrees with respect longitudinal axis bending-membrane symmetric with to of the the tangential the fibers in individual direction, which laminate. This layers is considered symmetry property oriented (P ^-6) as the principal eliminates any 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°. where A = tA'^o . (iJ 1 = 1,2,6), (3.1) * 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; JLf Ll °1 = °6 : Ll °2 LI L = °r ; and summation is implied over the range of repeated inverse extensional stiffness matrix A^ are given by °6 LI = °rd f*s -"\\ ( 1 2 ) subscripts. The coefficients of the Mathematical Analysis A = u 1 34 (3.3) hi (f k = l 'J wheie (J . are the plane stress reduced stiffness coefficients of the k th layer, h is the 1 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 o. = k with a (/' = tf.e . 0 (IJ = 1,2,6), (3.4) 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 a z and the shear stress T essential concepts of this mechanism, as first suggested are summarized in Fig. 3.6. Interlaminar along the interfaces different layers free-edges. interface in order experiencing by the Pagano and Pipes, normal and shear stresses are generated to maintain the force transverse . The stresses The interlaminar normal stress changes and moment that are not equilibrium of balanced at the sign at some point along the 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 °2 = = r>k o 2i i Q e tQ'i.A'Jo 2i iJ J (3.5) 1 k the additional ply radial stress a 2 would be found component of this stress due to laminate radial stress, i.e; by substracting out the Mathematical Analysis &, 2 = o\2 36 tQ^-.A'Jo^i 2i i2 2 = ' ^ f A ; 'u + %> A (36) Which in the usual, expanded notation, can be written as, M2 l'6 22 '6 26 66^ A +Q A 1+Qk ^6 A 2 ^ Just as the transverse ply stress, o , in an uniaxially loaded y straight edge laminate generates the interlaminar stresses o z shear stress T and T , YZ the in-plane 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^ tangential stresses that gives rise to J Q induced by the laminate radial and . Since the k th ply shear stress T ^ Z is derived from (3.1) and (3.4) as; k Jk o 6 6i i = tQ^Ajh (ij = 1 k where in standard contracted notation a, o 1,2,6), k refers to r . ra (3.8) the additional ply shear k stress is obtained by substracting out the component due to laminate shear stress o ^ . Thus, o 6 6 = 6i 16 6 tQ t.(A~ a -A~ o ,) ij j 16 6 l ^6r 1 L ] L Mathematical Analysis = tQ^Ar/a ^ 37 A7 o ) 1 } (3.9) L 2 2 which is written in the expanded form as; M6 12 26 2'2 Q 66 2'6 A +Qk A + k The modified approach A ^ 1) (3-10) 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 6 2 k L °2-°2 = = tQ .A" o 2i ij j k 1 o„ z L (3.11) ' L 0 v k k where d ^ is now the ply radial stress deviation o in the k th ply, which in the r expanded form appear as; M2 n 22 2~2+Q 26 2~6^ A +Qk HQ Ajk ]2 Similarly J 6 + the A k Q A' k 22 J 2 6 + ply shear ^ A Q^A;/) stress tr^ deviations - o L r are (3.12) 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 applied decide on to the some representative approximate straight values edge described earlier, the laminate stresses ah O for the laminate. In , and laminate the a r stresses to point be stress method are derived from the 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 laminate stresses are and shear calculated stresses. by In averaging the average , stress method and over the a characteristic length / in the radial direction. These averages of the exact in-plane laminate straight stresses edge form the system laminate is subjected. of bi-axial stresses to Thus, in evaluating the which the equivalent expressions given in equations (3.7), (3.10), (3.12) and (3.14) these average values are used. Mathematical 3.4 3.4.1 INTERLAMINAR Analysis 39 STRESS DISTRIBUTION: Interlaminar Normal Stress a :The force and moment equivalent to the interlaminar stresses o resultants and r ^ z which are statically can now be determined through k simple equilibrium arguments using d . Let us first consider the interlaminar normal stress o . z 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 o The approximate appropriate solution suggested by Pagano and Pipes is z distribution. considered to be 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 of a z the moment equilibrium of the layers. The assumed uniform distribution over two thirds of the boundary layer width d, and its linear variation over the remaining distance requires the magnitudes of a while a m a' = o/5 o = o (z) m and o' m to be related as, (3.15) is given by, m m = 90 M(z) / 7d 2 (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) = Y <V*>*^ (- ) 2 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(zfwhere interface M(zj/ is the moment 2 o (2k- 21+1) (3.18) l 1=1 length r of the couple produced o = k m Interlaminar Shear Stress r k m at this interface is found as; 2k L d (2k-21+1) 7d i=i (3.19) l J r •- An approximate solution for the distribution of r in at the the k th and (k+1) th layers. Substituting this in (3.16), the maximum interlaminar normal stress a 3.4.2 h /2Z per unit C between = the case of the hole problem) along the ply interface (or, T RZ is also required. A solution, which is compatible with Pagano and Pipes approximate solution for a, z 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 oy oz = 0 | £ j = o + | L B (3.20) oz The second of the above equations can be rearranged in the following form r = y*W -[ l I?** 2 where < is the thickness of the laminate. Let us consider a plane z=const the top surface ply. If the basic form of the a y (- ) 3 21 within 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, o (z) = -Y Tf*dl 2 z Substituting the expression given in (3.22) for t o (z) z 1 = in the above equation, we obtain 1/2 z2 S ^J{t/2-%)di z <>y n 2 (3.23) Mathematical Analysis 42 which can be reduced to; o (z) z = l^(t/2-z) (3.24) 2 Equations (3.22) and (3.24) relate the distribution of interlaminar stresses r „_ and yz a. within the surface layer to that of a „. A n 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), yz T = (z) -777&TS°zV y (t/2-z) o <- > d 3 The integral on the right hand side of this equation under the o curve. The characteristic form of the T z is equivalent to the area distribution resulting from this analysis is shown in Fig. 3.8. The stresses are distributed layer width free-edge. d, with its maximum occurring at a distance The original a z 26 z over a boundary of 5^/18 from the 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 through force equilibrium analysis of suitable free-body diagrams. Thus, z=const Mathematical m = T where F(z) is the total shear mV = T force per 2 F unit < )' z Analysis 43 (- > d 3 length generated by the 27 r yz distribution on this plane. In satisfying the conditions of equilibrium it is found to be given by t/2 F(z) where oy(Z) is the = transverse ply problem, this becomes 5^, the S oft)di stress at (3.28) z= £. In the case of the additional ply radial stress or the hole 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 »' 1=1 (3.29) 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 3.4.3 Interlaminar Shear Stress k m = 2hz b d i=\ r l (3.30) T '~ XZ 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, TQ The distribution given by Whitney, Daniel Z and variation of the in-plane shear stress, such that, in the case of the hole problem). Pipes (1982) assumes a parabolic Mathematical xy = T T xy Analysis U -( y / d 44 (3.31) Equilibrium equations (3.20) of the theory of elasticity for a laminate.of thickness t require 1 1 / 2 = 5 = 2j / 2 Zn- ?„r* l ^ d Z t/2 , / r\ di (3.32) y 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 The integration in (3.33) can be replaced N is given by, by an algebraic summation, i f the stresses are to be determined at the ply interfaces. in-plane shear thickness h, stress k T xy of the k th < where r n ply remains constant through = f f * r Since the the ply (334 ml > is the maximum T _ shear stress generated at the A: th interface, between *z v 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 ^ employing the expression (3.34) to yield: from (3.10) or (3.14) and by Mathematical Analysis 45 (3.35) 3.5 S O L U T I O N OF I N - P L A N E STRESSES The approximate formulation of the hole problem presented last section was based on the evaluation of in-plane laminate in the 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 stresses OQ, o r laminate and r ^ , stresses thus evaluated are the thickness average in-plane 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 infinitely wide isotropic plate is given by Timoshenko and Goodier (1934) as, in an Mathematical Analysis °r = 2 T = - r0 where OQ, o r + \ <'+^> £ j(J-i^+^) «* 46 » sin 28 (3.36) 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 °e 1 7 0+ 1 = °e 5 1 d r a = i^7TTT21 (a+l) ' + { Similarly, the laminate stresses and + ( a+l -T; ^ c o s 2 6 * (3-37) used in the average stress methods are found to be given by 2 o L r = -^—^ {i 2(a+l) 2 - [ Lz* j c o s 2 6 } ( 3 3 8 ) (a+l) 2 and, L rd = _p 21 [(a+it-tf (a+1)3 V ' Mathematical 3.5.2 Analysis 47 Orthotropic Solution:- Let us consider an anisotropic plate of infinite extent with a circular hole of radius a whose contour is free subjected of external forces. The plate is 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 , a , •* y r v r at any point (x,y) with respect to these coordinate *y r v axes are given by Savin (1968) as, a = x 2 +2 Re [s^ p y = 2 Re [4>'(zj)+ V' (zj\ r x y = -2 Re [sj<p' ( ) r (zj] 2 (3.40) in the above expressions are given by, Z +x/\Z- -a (l + sy} 2 2 J ] = JB °0r*f 2(s s2) r where Sj and s (z^) *W 2(s s ) H Z 2 ) + sV Z] = -ma (Zj) 2 (zj) + s ^ ' o The complex functions 4> (zj) and \p (z^ t , Z +v/{Zj-a (l 2 2 + sy} (3.41) are the complex roots of the equation 2 l/~ c whose coefficients respect to the 16^ 2c + C^j s are the coordinate (3.41) are given by, ( 12 66> 2c +c s2 coefficients 26+ 2c of the 22 c plate = < ) 0 142 compliance matrix with axes. The complex variables Zj and Z 2 in (3.40) and Mathematical Zj = x + sjy Z = x + sy 2 The solutions Analysis 48 (3.43) 2 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 L P T as; U The solution are stresses now transformed to J = Cjj a , lA.. a Y (3.44) ij v polar and T y v coordinates resulting using the from the exact following stress transformation; a, r °e cos e 2sin 8 cos 8 - 2sin 8 cos 8 sin 6 cos 6 • sin 8 cos 6 sin 8 cos 8 cos 8-sin 8 2 — 2 sin 6 2 (3.45) 2 2 2 xy For orthotopic laminates employing the point stress method, the stresses o , OQ r 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 3.6 49 R E S I D U A L T H E R M A L STRESSES:- If from Analysis its cure a laminate is subjected temperature, thermal stresses are to a constant induced which temperature change alter the AT stress state everywhere, including near. the hole boundary. The changes that occur in the stress state are uniform throughout stresses are the laminate, except near the hole boundary, where interlaminar 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. If the net mid-plane strains in the straight edge laminate resulting from both mechanical and thermal loads are given by e , the ply stresses in each layer are found by LPT as; (3.46) 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, n (3.47) Mathematical Analysis SO where iV - are the force resultants defined as the force per unit length. Equation (3.47) is z usually written as N + Nj = i where N. T A e°. (3.48) i} are the thermal forces given by llj = hI k=l CTna^T l J Solving for the (3.49) 1 mid-plane strains and substituting in (3.46), the ply stresses in the k th ply are found to be given by, o k / = (£••[*' \N + N )ij j m m m The additional interlaminar normal stress o and z generated a.AT] J T m ply the radial (3.50) v k & r stress which interlaminar shear stress r by the tangential and shear laminate stresses ah. and r z to the is that which is „. This is found by u setting the laminate radial stress contributes / ru ( 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 on the other hand results from the laminate tangential and Z radial stresses OQ and and is thus found by setting N$ in the above expression to zero. Again, A ^ contributes to the additional ply shear stress. Thus in both cases, given 7 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 is substracted out from the ply radial stress a . The deviatoric r r laminate radial stress a ply shear stress f ^ L rd r on the otherhand is calculated by substracting the laminate shear k rd stress T „ from the ply shear stress T . Thus, in both cases, all of the terms in N or N n T contribute to the interlaminar stresses. m m m Mathematical Fig. 3.1. 3.2. laminate configuration. Straight edge approximation of a laminate hole. Analysis Mathematical Analysis 54 Mathematical Analysis 2-D Approximation of the hole problem How additional ply stresses due to lamination generate interlaminar stresses Calculation of larninate stresses for the straight edg e laminate using Point and Avera,ge stress methods Models of approxi.mate interlaminar Stress die>tributions in the trans\ erse direction r Exact plane stress solution of in-plane laminate stresses around the hole Fig. 3:5. Flow diagram of the method of analysis. 55 Mathematical Analysis 56 ^1 Mathematical Analysis 58 Mathematical Analysis Fig. 3.9. Approximate distribution of r near the free-edge. 59 60 C H A P T E R TV C O M P A R I S O N S WITH L I T E R A T U R E In this chapter, the results of the analytical technique presented in the previous Chapter literature. are Comparisons compared are made observations of delamination damage with theoretical with the obtained and calculations experimental of results interlaminar by several authors. from stresses the and A fair agreement was observed for a wide range of laminates, as discussed in the following sections. 4.1 THEORETICAL C O M P A R I S O N S : - 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 dimensional finite element associated programs are with the formulation and the high, such calculations offer the use of three most popular means of laminate stress analysis The Schmueser solutions given (1978), and Whitcomb (1981) by Raju and Crews (1982), Rybicki employ three dimensional finite element and 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 (1961) for isotropic elastic materials. The results stresses calculated using the methods Raju and Crews are layer theory developed by Reiss of these solutions are compared with described in the previous Chapter. The results of 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 This is considered to be an added refinement for which the stresses are calculated. 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 composites have rigorously been proven of interlaminar stress singularities in multilayered by Wang and Choi (1982). Such singularities make stress calculations only tend toward accuracy, without convergence. From a practical point of view, this makes somewhat difficult, realistic estimates of interlaminar stresses at and any attempts to improve the the free solution accuracy through edge 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 4.1.1 62 Raju and Crews (1982):- Interlaminar circular hole in [90/0] using a and [0/90] s three-dimensional formulation. stress distributions have We consider finite first the been calculated near a graphite/epoxy laminates by Raju and Crews, s element [90/0] s analysis, based on laminate subjected a displacement to a gross applied stress of o„ with the elastic properties as used by Raju and Crews (see Table II - page in Figures 4.1, 4.2, 4.3 and 125). Plotted interlaminar normal stress o Jo _ for the z 4.4 are z = h plane from the distributions of the point, average, 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 laminate characteristic shear stress distribution of o four methods z give length and to changing to shift to the essentially increase (from laminate l/a = 0 to circumferential smaller angles and increase similar values except that the 1), the stress increasing cause the in magnitude. A l l modified point and modified average approximations predict a sign change for 6 > 7 5 ° . Plotted in Figures 4.5, 4.6, 4.7 and 4.8 are the distributions of interlaminar normal stress ( o / o ) for the same laminate using a boundary layer z 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 characteristic length is increased (from lla = 0.1 to 0.5) o layer width. As the is seen to decrease in z magnitude. The contribution of the increasing laminate shear stress and the changing laminate circumferential boundary layer width; the results are similar in stress all is net four apparently result offset by being lower o methods except the effects along the for the sign of increased free-edge. change The behaviour predicted by the modified point and modified average stress methods. Comparison of the results of o discussed so far with z the finite element solution by Raju and Crews shows that the general characteristics of the stress distribution is best predicted laminate thickness. I = d = t where Thus, t is the in when calculating both / and interlaminar thickness of the laminate d are stresses, equal the to one condition 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 The modified methods of approximation on the change predict compressive stresses throughout that takes place at around other hand the region. do predict the sign 8 0 ° . 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 Raju [0/90] and Crews also present the laminate. These results are s magnitude distribution of o very similar to their and sign. The solution of the [90/0] present work for a 64 for a z values, both in s [0/90] would s be o equal in magnitude to the [90/0] solution, but opposite in sign. However close s inspection of the results in Raju and Crews shows that in the case of the [0/90] laminate, not only does the interlaminar normal stress distribution through s 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] laminate does not show this very abrupt reversal, and therefore The [0/90] s s allows comparison. 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 interface, but not at material properties. interface [90/0] the s the z = 0 Similarly, the [0/90] s interface, Raju where there and Crews data is no discontinuity in show that at the z = 0 laminate has an interlaminar stress of opposite sign to the laminate over most of the boundary. However, the mesh is very coarse at z = 0 interface, appears the h that the and problem no lies detailed not in results are reducing presented the approximation, but in the 2 - D equilibrium argument experimental evidence to back the 3-D by them. problem to Thus a it 2-D However, there is a host of equilibrium argument in the 2 - D case, and a great deal of use is made of it Returning to the solution of o z 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 one laminate thickness.t for occur is in the order of Henceforth we will use a value of one laminate thickness / and d in all comparisons. In order to better compare the distributions, the stresses are sometimes normalized with respect values, as shown in Fig. 4.10 for the [90/0] shapes of the to their maximum laminate. This shows clearly that the s essential characteristics of the distribution (eg., the location of the maximum stress and the range over which they are tensile) are predicted reasonably modified average absolute value stress aproximation. It of maximum o z for is this however lay-up important given by to Raju well by the note that the and Crews is approximately 3 times that obtained by the present approach. Raju distributions around the and hole Crews also present interlaminar shear for both [0/90] and s compared with the results of the modified average and 4.12. Shear stresses normalized with respect these Figures. Except for different signs, the T Q solution are identical for the two laminates, [90/0] s laminates. These Z are stress calculation in Figs. 4.11 to their maxima are Z stress T Q plotted in distributions of the finite element 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 6 = 6 7 ° . The angle at which the sign of the shear stress changes within a degree or two from that predicted by the finite element present solution, using the modified average it near is predicted solution. The 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 4.1.2 r^ r Rybicki and Schmueser (1978):- Using distribution of a interlaminar three-dimensional normal midplane was studied by Rybicki laminates of [90 /±8/+'6] 2 the type stress, o, finite around z a hole [0 /±8/^8] , 2 [±8/+8/0 ] , s 2 there singularity is expected. Since discontinuity a the laminate fairly in material coarse and s The material properties used in midplane no the 2 given in Table II (page 125). A l l results is at [±8/3-8/% ] s the analysis are where program, and Schmueser for a series of Graphite /epoxy where 8 is 3 0 ° , 45° and 6 0 ° . t s element mesh are for the properties, was used laminate and by thus Rybicki no and Schmueser, there is no indication of any such effects. Figures [02/±30/T30] approximate S 4.13 and 4.14 show the results for the laminate. Interlaminar normal stress is calculated using the present 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 o at z 8 > 60°. The modified average approximation shows the best agreement It is also clear that we consistently predict larger average t magnitudes than stress results the numerical results. normalized by the Figure 4.14 compares maximum stress with the modified numerical results, Rybicki and Schmueser modeled the (0$ and (90 ) plies as one material, and the (±8/^-8) plies as a single material witn effective modulus properties. 2 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 [ ± 3 0 / f 30/02] : not an exact s and Schmueser image as predicted magnitudes for the [ ± 3 0 / ? 3 0 / 0 2 ] s present the distribution for a lay-up, and though the sign of the distribution is reversed, it is mirror [02/±30/T30] also s by the present solution. They predict laminate that are roughly double those of the 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 methods at approximations. The modified stress first were difficult to explain in terms of physical reasoning. However, it was found later these methods that a sound physical argument also. It is based on the fact can be that the presented phenomenon to explain 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 stresses o k and T ^ r exact plane material stress solution at discontinuity homogeneous stresses predicted by the combined laminated plate theory (LPT) and in a given point the thickness within direction. a If laminate the material from were the truly through the thickness, as assumed by the plane stress solution, these would simply be the laminate stresses and r difference result between these values interlaminar stresses observed can therefore be respectively. The rv considered as the source in composite laminates. The modified stress of 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 agreement with numerical results, the other three methods 68 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]$ 45° and 6 0 ° . The results of the present calculation for the laminates of the type [±6/3-6/02]$ ^ [902/56/+'6] type of laminates where 6 is 3 0 ° , are not presented s here, since they exact mirror images of the results shown in these Figures. Although sign, the results of the finite element solution such mirror magnitudes. images, Specific but rather reference the same will are the of opposite for these lay-ups do not exhibit general be made shapes to these with results different in the stress 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 Stresses with the general character are tensile (or compressive) of the finite within the same element stress distribution. 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 = 3 5 ° , where the deviation is found to be largest For the [ ± 4 5 / + 4 5 / 0 2 ] s laminate, this deviation is much element stresses are approximately twice those for the [0 /±45/32 less since the finite 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 paragraph. As before, magnitudes, for the the comparison is [±60/T60/02] better, laminate s especially in 69 in terms of absolute 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 0°. remains tensile all around the hole and is minimum at z Although the magnitudes of the present solution are consistently that of the Finite element, [902/±30/T30] laminate, s it is not more than twice the image of the [ ± 3 0 / T 3 0 / 9 0 2 ] present solution would higher than at any angle. For the predict the exact minor laminate stresses, whereas the solution by Rybicki s and Schmueser, though of opposite sign, shows much less variation in magnitude as function of angle. Figures [ + 45/^45/902] s 4.18 and and [ ± 6 0 / + 60/ 9023 the variation of o s 4.19 show the results for the laminates. Although of difTerent magnitudes, around the hole predicted by the Finite element essentially similar to that for the [ ± 3 0 / T 3 0 / 9 0 2 ] s solution is 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 9 0 ° , 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] : reverse stacking order, the difference s laminate with the between the results becomes larger as the Comparisons with Literature angle increases. Although the same general shape is observed, the result exhibits a much less variation in magnitude [±60/T60/902] laminate S predict compressive o indicates predicted for tensile all [902/±60/^60] element difference between the results Although the finite element result would at laminate midplane for angles less than 2 8 ° , the present z result finite as function of angle. For the shown in Fig. 4.19 the remains approximately constant throughout 70 o for z angles, by this both region. solutions, However, for the compressive reverse stresses stacking order are in laminate. While the present solution yields an exact mirror image s 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 interlaminar was observed, stresses around holes demonstrating through the possibility of equilibrium considerations. evaluating The method seems to have some success in predicting the approximate shape and sign of the stress distribution. The results comparison could therefore reported by be made with o z them are only midplane o . z No 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 around in holes two Whitcomb different stacking [ 9 0 / ± 45/0] . The elastic properties s has analysed sequences; interlaminar namely stress distribution [45/90/-45/0] s of the zero deg plies of these laminates and 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 not include a complete show the distribution locations. These description of interlaminar of results o are and z TQ compared Z through with stresses around the thickness stresses calculated the at 71 hole, three using the but angular modified average stress method for the two stacking sequences in Figs. 4.20 and 4.21. The calculated values of a interfaces through the thickness at different z are shown in Fig. 4.20 for the [45/90/-45/0] laminate. The results of s 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 is predicted reasonably well at each interface, locating z correctly the interfaces with maximum stress. At 120° however, the poor especially with regard to absolute magnitudes. The sign of o at the second z interface from the element analysis, outer agreement is layer does not agree with that resulting from the finite although the basic shapes similar result was and 175° of the distributions (through the distribution. The thickness) roughly agree. A agreement interlaminar was better shear at stresses 90° obtained by observed for than 120°. at Whitcomb along T@ the Z At 90°, First the and high 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 the for these two interfaces third interface given by are close to zero. The maximum shear stress at 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 For the [90/ ± 45/0] laminate s shown in 72 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 near the z 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 element solution is relatively good for the interlaminar shear stress TQ Z through the laminate thickness. Except for the two outermost the finite distribution 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 boron/epoxy and for a [ ± 4 5 ] s are given for a [0/90] s laminate made of 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 TQ and T Z actual R Z are derived at 0/90 and 45/-45 interfaces. It is not clear what the dimensions parallel study 73 of the on the plate effect geometry of t/a used in the computations ratio on boundary-layer assumes a range of values from 0.01 to 0.03 for t/a, effects the plate are, but reported a here thickness-to-hole radius ratio. For interlaminar stresses compared with present the calculated those reported calculation seem to [0/90] by laminate s the modified average a circular stress by Tang in Figs. 4.22 (a)-(c). predict the same obtained by Tang for all stress components. signs are predicted containing general For o z hole, the approximation are The results of the shape of the distribution shown in Fig. 4.22 (a) the correctly. The maximum interlaminar normal stress is obtained almost at the same angular location as that given by Tang. A secondary peak in o , analogous to that observed by Tang approximately 5° off the loading direction, z is also obtained by the present calculation, but, at least 12° loading direction. At 0° and 90° the absolute values of o 15° away from the are z 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 as shown in Fig. 4.22 (b). The sign and relative magnitude of T Q Z distribution 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 and 90° from solutions. In the fact, loading direction T Q this is expected Z becomes along the 6=0° zero, as and 1:3. At 0 ° , 24° predicted 90° for by a both [0/90] s k lay-up, since the inplane. ply shear stress r ^ TQ Z is zero at these two locations and is a direct product due to the matching of the inplane ply shear stresses at Comparisons with Literature 74 the free edge. The distributions of the and magnitude. comparison [0/90] Within s the the interlaminar shear stress T N laminate in Fig. 4.22 (c) shows a similarity in shape one-quater of the exhibits two maxima, one near 0° from of present approximate hole boundary the distribution of T RZ and the other one close to 9 0 ° . This results 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 around most of the boundary, although the solution by RZ are roughly equal Tang predicts negative stresses in the range 20° to 4 5 ° . 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 maximum T R Z is also to note that the magnitude of the is approximately an order of magnitude less than the maximum T § shown in Fig. 4.22 (b). Thus, T component interesting for the [0/90] s RZ Z is found to be an insignificant interlaminar stress laminate. As will be seen later, this is true for many other laminates of practical interest The comparison of interlaminar normal stress results for [ + 45] s graphite/epoxy the 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 as seen in Z 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 in the [0/90] Z laminate, (see Fig. 4.22 (b).) The stress becomes slightly negative in s the Comparisons with Literature neighbourhood of 6 = 4 0 ° . TQ Z Unlike the result for [0/90] is observed at 8 = 0° and 90° for the [ ± 4 5 ] k high inplane ply shear stresses r ^ s s 75 laminate, high values of configuration, since there are 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 certain ambiguity exists in these results, t the results of the here. A and therefore, a direct comparison with 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 stresses at laminate free straight free-edge is generally accepted that the presence of high interlaminar edges cause delamination along ply interfaces. The literature on 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 t An attempt to contact the author and clarify the results was unsuccessfid. use 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 boundary may change the original stress field initiated at the hole 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 with circular holes and attributed the differences in strength of laminated to the differences plates 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 free-edge resulted in laminates alternate stacking sequence. weaker by 10 to 20 percent than Whitcomb (1981) studied fatigue damage the straight corresponding 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 the interlaminar stress analysis by O'Brien and Raju (1984). with Comparisons with Literature In the are compared with the following sections the stresses calculated 77 experimental results mentioned above 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 only qualitative. The altered extent which makes such comparisons 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 been made that took place. C-scan records of typical delamination locations have for the Comparisons of the and [45/0/-45/0] specimens after laminates affected tension or compression the difference preferentially at 0° in stacking sequence and 180 ° loading the [0/±45/0] s [0/±45/0] s of these two specimen delaminated from the loading direction, but the [45/0/-45/0] specimen delaminated uniformly around the hole. For the [ 0 / ± 4 5 / 0 ] normal cycles. delamination growth ( Fig. 4.24 ). It has been noted by Whitcomb that under tensile fatigue interlaminar fatigue C-scan records for the two orthotropic laminates reveals that the s 10 stress calculated using the present s approximate s specimen, the 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) compressive tensile o r stresses, In while contrast the those which results of fall inside the present the circle calculation represent for the Comparisons with Literature [45/0/-45/0] laminate (Fig. 4.26) predict tensile o s on either side of the hole and compressive o z within a small angular region z in regions above and below the hole. Since the interlaminar shear stress T Q and T R similar interface, Z for the two stacking sequences at 78 each distributions are essentially Z the delamination observation can be attributed to the difference in o difference in distribution. Thus, more delamination can be expected to occur along the loading direction, above and below the hole, for the [ 0 / ± 4 5 / 0 ] laminate than for the [45/0/-45/0] s s laminate. It has also been observed by Whitcomb that the sign of the loading also affected delaminated expected much more since the 4.26) would become stresses delamination growth. In particular, the [45/0/-45/0] would be extensively under compressive a at tension. stresses (shown inside the tensile under obtained compression than compressive fatigue an angle s specimen This can base circle in Fig. loading. The highest equivalent to be that at which tensile 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 [45/90/-45/0] given s by Whitcomb laminate subjected at to tension 9 0 ° , 120° and 160° for the [ 9 0 / ± 45/0] (Fig 4.28). For the [45/90/-45/0] angles s s 90°, fatigue 120° and 175° for ( Fig 4.27 ), and at the angles laminate subjected to compression fatigue laminate at 9 0 ° , 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 has the Z 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 where T Q Z is found to be maximum and delamination was observed by Whitcomb at the o z relatively high. 79 However, no edge of the hole at about 175° from the loading direction. The results of the present calculation for this angular location entire predict laminate low interlaminar thickness. It shear stresses and is however interesting compressive to note the o through z 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 normal stresses calculated at this angle from the free edge. Also, the to change through thickness compressive interlaminar sign and become interlaminar tensile away 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 o z calculated by the present approach and shown in Fig. 4.21. At 120°, the location of the shear ( T Q ) stress Z 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 have been stress distributions calculated by the made at three angular locations for present approach. Comparisons each quasi-isotropic laminate Comparisons with Literature 80 presented. Results of microscopic section studies are not presented by Whitcomb for the orthotropic laminates laminates. show that the However, the calculated examination interlaminar of C-scan records stress can still be of these related to the observed delamination. 4.2.2 Kress and Stinchcomb (1985):- In graphite/epoxy around damage laminates circular holes using a zinc study Kress during iodide on and the fatigue Stinchcomb cyclic tensile enhanced response of investigated two quasi-isotropic damage loading. Non-destructive X-radiography provided development inspection information continuous damage process during the fatigue life. The study is of special on of the 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 process specimens information made during each unloading and reloading produced 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 laminate delaminations 80% of its mean and were tensile Stinchcomb first detected strength. report in the Radiographs that in the 90/45 interface of the [0/90/±45] after loading to zinc-iodide infiltrated region for this laminate made after 90% and 105% stress loadings t s hole 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 laminate which is determined independently using several test specimens. the Comparisons with Literature direction on four locations around the hole, symmetrical with horizontal and vertical axes. No delamination appears to have 81 respect to initiated on the 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 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 at 0/90 and 90/45 interfaces the Z stresses are obtained 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 component r z of interlaminar shear is almost uniformly distributed around the hole thus having little influence on the location of delamination initiation. Delamination been observed specimens by Kress and to 60% of their initiation in the [45/90/-45/0] Stinchcomb on radiographs mean tensile strength. made They report laminate s has after loading the 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 ) delaminations clearly perpendicular indicates to the severe loading direction, with on no opposite visible sides of delamination the on hole, 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 largest at 45/90 and 90/-45 interfaces. The interlaminar shear T Q is also high for these interfaces, loacation. The same is true of T correlation between theoretical but nearly zero for in Fig. 4.32 (b) Z other 82 interfaces at this distribution shown in Fig. 4.32 (c). Thus good RZ 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 present stress agreement calculations. Successive delamination, predictions based on the though appears to be in 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 laminate ( Fig. 4.33 ) damage appears on 0/90, 90/45 to have initiated interfaces at an of the angle to [ 0 / 9 0 / ± 45] s 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 which continue across the interface upper and delaminations surround the compressive o near 90° lower edges of the shear stresses RZ hole. On the 45/-45 hole completely perhaps due to the fact that from the loading direction is offset by high interlaminar z shear stresses in the region. Over the remaining upper and lower parts of the hole boundary o is found to be highly tensile. z The delaminated regions in the generally smaller than in the above [ 0 / 9 0 / ± 45] to largely compressive interlaminar shear interlaminar stress normal distributions can [45/90/-45/0] considered the as hole. The overall equal for stacking sequences. Nevertheless, on 45/90, 90/-45 interfaces of the laminate delaminations appear slightly inclined to the normal and T Q maximum Z at diametrically opposite horizontal axis. A careful stresses occur at and T Q Z two ( Fig. 4.34 examination of the s ) interlaminar shear stress distributions of Figs. 4.32 (a) and (b) reveals that the positions diametrically opposite is high on either side at 90° z the [45/90/-45/0] positions to closely with the horizontal axis as the observed delaminations. T of o are laminate which can be attributed s stresses around be laminate s each RZ other, aligned shear stress too from the loading direction, intensifying the influence 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 sequential life also appear very much similar to that observed in 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 1 20.0 30.0 I,, 40.0 50.0 60.0 I ,.< " X"^^ 70.0 I 80.0 I l__ l/a = 0.00 /i/'*' l/a = 0 * 0 5 ^ . / \ l/a = 0 . 2 0 \ l/a = 1.00'N^ Fig. 4.1. Effect of l/a on o z METHOD / N / / ax i = h in a [90/0] s laminate. = Point Stress. B O U N D A R Y L A Y E R WIDTH = laminate thickness. Comparisons with Literature AVERAGE STRESS METHOD Angle, degrees 20.0 30.0 40.0 — — J 1— 50.0 — L _ 60.0 1 70.0 L 80.0 I/a = 0.00 l/a = 0.05^" \ \ l/a = 0.20 / \ l/o = 100 Fig. 4.2. Effect of l/a on o z METHOD at z = h in a [90/0] s laminate. = Average Stress. B O U N D A R Y L A Y E R WIDTH = laminate thickness. 86 Comparisons with Literature 87 MODIFIED POINT STRESS METHOD Angle, degrees 10.0 20.0 —I t... 30.0 40.0 50.0 •••J ....I 60.0 70.0 1 1 y^-i ----l/a = 0.05 ^ \ \ l/a = 1.00 Fig. 4.3. l/a = 0.20 v Effect of l/a on o z l/a = 0.00 7 ^ . / j _ 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 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. 88 Comparisons with Literature POINT STRESS METHOD Angle, degrees 10.0 • JL 20.0 ^ I 30.0 40.0 I I 50.0 I 60.0 70.0 I I l/a = 0.50 l/a = 0.20 l/a = 0.10 \ Fig. 4.5. Effect of l/a on o z 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. 80.0 _J ^ 89 Comparisons with Literature AVERAGE STRESS METHOD Angle, degrees 10.0 20.0 30.0 I 40.0 50.0 1 1 60.0 70.0 1 1 80.0 l/a = 0.50 *-\ ^- l/a = 0.20 l/a = 0.10 \ Fig. 4.6. Effect of l/a on o z 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. 90 Comparisons with Literature 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. 91 Comparisons with Literature 92 MODIFIED AVERAGE STRESS METHOD Angle, degrees 10.0 _J 20.0 I 30.0 I 40.0 L_ 50.0 60.0 I I 70.0 I , 80.0 J , r l/a = 0.50 ——— ^ l/a = 0.20 l/a = 0.10 Fig. 4.8. Effect of IIa 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 = 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 laminate at i = h. with numerical solution for [90/0] s Comparisons with Literature Fig. 4.11. Present results of TQ Z laminate (1982). compared with distribution at z = the solution of h in a Raju and [0/90] 95 s Crews Comparisons with Literature Fig. 4.12. Present results of TQ laminate (1982). compared Z with distribution at z = the solution of h in a [90/0] Raju and s Crews % Comparisons with Literature 97 Point Stress Average Stress Modified Point Stress Modified Average Stress Rybicki and Schmueser 1978 Fig. 4.13. Present results [02/±10/130] s compared with laminate at i = numerical 0. solution for a **** Fig. 4.14. Results of Modified Average Stress method compared with U 3 numerical solution for a [ 0 2 / ± 3 0 / T 3 0 ] a z s is normalized with respect to a (maxy z laminate at z = 0. ? vii* • i • fj'Ml||»|l III l r i | fO.f C.v,v~ X,- • fj j Lile-mure 99 Comparisons with Literature Fig. 4.16. Present midpane results in a of interlaminar [0 fi60/3-60] 2 s normal laminate solution of Rybicki and Schmueser (1977). stress distribution compared with 100 at the Comparisons with Literature Fig. 4.17. Present midpane results of interlaminar in a [ ± 3 0 / f 3 0 / 9 0 2 ] s : normal laminate solution of Rybicki and Schmueser (1977). stress distribution compared with . 101 at the Comparisons with Literature Fig. 4.18. Present midpane results • of interlarninar • normal in a [ ± 4 5 / ^ 4 5 / 9 0 2 l s laminate solution of Rybicki and Schmueser (1977). stress distribution compared with 102 at the Comparisons with Literature Fig. 4.19. Present midpane results of interlaminar in a [ ± 6 0 / ^ 60/902] normal s stress distribution laminate compared solution of Rybicki and Schmueser (1977). with - 103 at the Comparisons with Literature T 104 8 /°g 2 -0.6 -0.3 1 0.0 V 0.3 1 > 0.6-06 -0.3 00 0.3 0.6 1 1 » t 90 1 1 9 i -Sr- J- 120 y ii f— 4 1 1 175* J 1 -0.6 -0.3 0.0 0.3 0.6-0.6 -0.3 00 0.3 Whitcomb 1981 Present Solution! Fig. 4.20. Present 1981) results of compared interlaminar with numerical solutions (Whitcomb, stress thickness in [45/90/-45/0] s distributions across specimen, (in tension) laminate 06 Comparisons with Literature °z/°9 -0.6 -0.6 -0.3 0.0 -0.3 0.0 ' 0.3 0.3 r 6z/°g 0.6-0.6 -0.3 0.6-0.6 -0-3 \ 0.0 0-0 0.3 0.3 Whitcomb 1981 Present Solution Fig. 4.21. Present 1981) results of compared with numerical solutions (Whitcomb, interlaminar stress . distributions thickness in [ 9 0 / ± 45/0] s across specimen, (in compression) laminate 0.6 06 105 Comparisons with Literature Fig. 4.22 (a). Present results interlaminar laminate. compared normal stress with o z Tang's at z (1977) = solution of 0 in a [0/90] s Comparisons with Literature CO 6H Fig. 4.22 (b). Present results interlaminar laminate. compared shear stress with T g Tang's z a t z (1977) = solution of h i n a [0/90] s 107 Comparisons with Literature Fig. 4.22 (c). Present results compared with Tang's (1977) solution of interlaminar shear stress T laminate. RZ at z = h in a [0/90] s 108 Comparisons with Literature Fig. 4.23 (a). Present results interlaminar laminate. compared normal stress with o z Tang's at z (1977) = solution of 0 in a [±45] s 109 Comparisons with Literature Fig. 4.23 (b). Present results interlaminar laminate. compared shear stress with Tang's T Q at Z z (1977) = h 110 solution of in a [±45] s Comparisons with Literature Fig. 4.23 (c). Present results interlaminar laminate. compared shear stress with r rz Tang's at z = (1977) h in solution of a [±45] s Comparisons with Literature TENSION C O M P R l I ON ORTHOTROPIC ( 0/«4>'0>, la) Fig. 4.24. I0/s45/0l (45/0/-45/OI OUASI-ISOTROPIC C-scan records of various notched (45/0/ (c) laminates (0 after or compressive fatigue cycles. (Whitcomb, 1981). 4VGi I 10 7 tensile 112 Fig. 4.25. Present solution of o (Whitcomb, 1981). z distribution in a [ 0 / ± 4 5 / 0 ] s laminate. Comparisons with Literature Fig. 4.26. Present solution of o z laminate. (Whitcomb, 1981). distribution in a 114 [45/0/-45/0] s Comparisons with Literature Fig. 4.27. Delamination location for [45/90/-45/0] s specimen subjected to specimen to tension fatigue. (Whitcomb, 1981). Fig. 4.28. Delamination location for compression fatigue. [90/± 45/0] s (Whitcomb, 1981). subjected 115 4.29. Radiographs of damage in a [ 0 / 9 0 / ± 45] sequential loading to (a). 0.9 ^ and (b). 1.05 Kress and Stinchcomb (1985) s laminate o^. after Comparisons with Literature Fig. 4.30 (a). Interlaminar normal stress o z distribution in a [ 0 / 9 0 / ± 4 5 ] laminate. (Kress and Stinchcomb, 1985). 117 Comparisons with Literature Fig. 4.30 (b). Interlaminar shear stress T Q distribution in a [ 0 / 9 0 / ± 4 5 ] Z laminate. (Kress and Stinchcomb, 1985). 118 s Comparisons with Literature Fig. 4.30 (c). Interlaminar shear stress r r z distribution, in a [ 0 / 9 0 / ± 45] laminate. (Kress and Stinchcomb, 1985). s 119 Fig. 4.31. Radiographs of damage in a [45/90/-45/0] laminate after sequential loading to (a). 0.9 a ^ o^. s Kress and Stinchcomb (1985) and (b). 1.3 Comparisons with Literature Fig. 4.32 (a). Interlaminar [45/90/-45/0] normal s stress o z distribution in laminate. (Kress and Stinchcomb, 1985). a 121 Fig. 4.32 (b). Interlaminar [45/90/-45/0] shear s stress T Q Z distribution in laminate. (Kress and Stinchcomb, 1985). a Comparisons with Literature Fig. 4.32 (c). Interlaminar shear stress r r z distribution in [45/90/-45/0] laminate. (Kress and Stinchcomb, 1985). s 123 a Comparisons with Literature + Fig. 4.33. Damage on [0/90/±45] s 0/90, 90/45 and 45/-45 interfaces of a laminate. Kress and Stinchcomb (1985) X ** Fig. 4.34. Damage on 45/90, 90/-45 and -45/0 interfaces [45/90/-45/0] laminate. Kress and Stinchcomb (1985) s of a 124 125 T A B L E II. In-plane ply elastic properties used by different authors in stress calculations:- AUTHOR MATERIAL ELASTIC PROPERTIES Raju and Crews Graphite/epoxy (1982) E ll = 14.5 GPa 12 = 5.86 GPa 12 = 0.21 ll = 151.60 GPa v Graphite/epoxy (1977) E E 12 = 6.89 GPa 12 = 0.25 E ll = 140.0 GPa - E 22 = 14.0 GPa 12 5.9 GPa v Graphite/epoxy G V Tang (1977) Boron/epoxy G = 19.93 GPa - 12 - 12 V 12 4.99 GPa 0.21 137.90 GPa ll 22 G 211.68 GPa 22 12 E 0.21 - v E - 12 ll E E Graphite/epoxy 11.00 GPa 22 G Whitcomb (1981) 138.0 GPa 22 E G Rybicki and Schmueser — 14.48 GPa : - 5.86 GPa = 0.21 126 CHAPTER 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; A l 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 the prepreg tapes and the manufacturer's 127 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 ± 3 0 ] and [±30/902/02] AS/3501-6 laminates s graphite/epoxy and 8-ply material. In [02/±45] addition, a s and glass/epoxy [±45/02] laminates s laminate of s of [02/±30] s construction was also included in the investigation. The crossply and quasi-isotropic layups investigated are respectively the 4-ply laminates and 8-ply [45/0/-45/90] s [0/90] and s [90/0] s AS/3501-6 graphite/epoxy 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 PROCEDURE:- Uni-directional prepare the laminates described prepreg above. A tapes available in roll standard autoclave was form were used to used cure to 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 diamond cutting wheeel. Except were cut for the from the XAS/914 composite graphite/epoxy others were cut to a nominal width of 50 mm and central circular hole of 12.7 mm (0.5 in) diameter using a high speed diamond drill. When drilling plates a length using specimens, a all of 280 mm. A was drilled in each specimen 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 gritSpecimens 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 loaded up to failure had damage confined mostly to the central hole region. were Experimental Observations 5.2.2 129 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 were loaded in an lnstron testing machine. The specimens mounted using medium grade sand paper held between between the grips was approximately specimens. A cross head speed 170 mm grips the (6.75 the in) without specimens tabs were free span grips. The for the 50 mm wide 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 45 seconds at 30 positive-negative film beam. Once the kV and 10 mA X-ray laminate this was about current A Polaroid was placed behind the specimen in the path radiography was completed, the specimen was type of the reloaded 55 X-ray 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 delaminations at the hole radial planes around direction. A typical set to a micron finish found to finish to of and the determine of 6 microns great ply hole, of cuts is shown for potographic be delamination order on which interfaces the and matrix cracking appeared. A slow diamond cutter was used to cut sections along polished in for at different angles to the loading in Fig. 5.1. These sections were then microscopic examination, and later to 1 and replication work. Replication of the sections was use in cracking. further The establishing replicas were microscopic prepared by evidence making of surface impressions of the polished sections on .125 mm (.005 i n ) thick pieces of cellulose acetate sheets, sample and peeled off displayed moistened with allowing sufficient from the acetone. time replicating for After hardening, surface. a negative form of the applying When actual a the viewed mild pressure acetate sheet under surface, manifesting the on was gently microscope delaminations the in it 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 replicating the developed due acceptable fine acetate quickly geometry was to have a hole circular edge. Yet, a successful to difficulties involved finish, wetting and and evenly. Also, in enough replication polishing making an the big the for acetate access in technique could not curved imprint of the hardened easy surface to be an surface on cellulose sheet could not be straightened out for subsequent microscopic work, without inducing much damage to the imprint Experimental Observations 5.3 131 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 angle but different ply laminates are stacking sequences are presented first, discussed together. The results following which those of the of crossply the 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] and s [±30/90 /02 2 The [O2/9O2/ ± 3 0 ] s JS radiographs laminates:of damage around the hole in a 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 defective preparation. increased. from a possible weakness prepreg These material cracks or from grew slowly in the poor but zero-deg laminate ply, inherited construction continuously as the or from a specimen stress level was 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 [O2/9O2/±30] s laminate distribution of interlaminar stresses around a hole for the 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 9 0 ° , on both sides of the hole. For many interfaces the interlaminar shear TQ Z 70° and 110°, while r r z is found to be high around is found to be high around 5 5 ° , 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 T RZ and TQ Z found between the 30-deg and -30-deg plies. Experimental Observations 133 The specimen was sectioned at angles of 0 ° , 4 5 ° , 7 0 ° . 9 0 ° , 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 interface between the zero-deg delamination was seen at the second 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 § T RZ at and are obtained at fourth and fifth interfaces. Thus delamination can be expected 90/30 and present stress interface are 30/-30 interfaces calculations. The prior to any traces also explained by the of other At interface and between 30-deg place delamination on the observed existence of maximum TQ basis of at the and r Z interface despite the presence of relatively low tensile 90- Z r z the second at this o . z 7 0 ° , significant delamination was observed at the second zero-deg and 90-deg plies. The zero-deg plies, and a little at the ply cracking appeared fourth between 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 equal for all interfaces. At the fourth interface the 134 shear stress components are both high, making it the next favourable location for delamination. Delaminations observed at fourth and fifth at 90° from the loading direction interfaces from outside, and found to be were connected through 30-deg ply cracking. The interlaminar normal stress distribution in Fig. 5.3 predicts compressive o at z 90° for the entire laminate, increasing in magnitude towards the laminate midplane. However, at this angle a high, non-zero TQ obtained at the fifth interface between the 30-deg and -30-deg plies. TQ all other interfaces. maximum at the The interlaminar shear r , r z fourth on the other interface, and shows high values at the is is zero Z for Z hand, has its 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 , and that at the second 0/90 z to the peaks in TQ fourth Z and T RZ stresses. Slight delamination was also observed at the 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 inherent radiographs material defects of early stages of loading in Fig. 5.5 is a result of or poor specimen cracking on either side of the hole about 50° construction. from These include matrix the loading direction and a zero-deg ply crack at 170° on the right side. Damage due to loading was first Experimental Observations seen in the radiograph taken after 135 185 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 laminate due hole in [±30/902^2^% 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 around 90° from the loading direction. As in the case of [O2/9O2/±30] the interlaminar shear TQ and 110°. Similarly, r rz Z s centred laminate is found to be high for many interfaces around 70° 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 around 90° from [±30/902^2^% the loading lay-up generates tensile direction, whereas o z on each [O2/9O2/±30] s side generates compressive o in this region. This may have caused the delamination to initiate at z 90° from the loading direction in the [ ± 30/902 ^2^s shear stresses, though appears to be similar in laminate. The distribution of general for both laminates, is different for any given interface. The specimen was sectioned at angles of 0 ° , 4 5 ° , 7 0 ° , 90°, 110°, replicas 135° and at these 180° angles as before, are and the shown micrographs in Fig. 5.7. of polished sections and 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 TQ at the first 30/-30 interface and T Z interface RZ can thus be considered as the other hand, both TQ and T Z r z cause of observed become maximum at the at the second -30/90 delamination. On the fourth interface 90 and zero-deg plies causing it to delaminate, despite the presence between of relatively high compressive o at this location. z At 70° from the observed at the fourth interface from loading direction, severe delamination was 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. analysis Comparing these ( F i g . 5.6), experimental laminate TQ results. Z and r observations good correlation can While the r z are with the be found distribution of o both z results of the between the present stress analytical and is nearly zero throughout found to be very high and maximum at the 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 At 90° 137 from the loading direction, significant delamination at the first interface and a little at the second were observed. Except for the first interface, TQ Z highest the for interfaces, is found to be zero for all others, with thickness, o second interface neglegibly and small nearly values r half that everywhere on the other hand was for first and r z else the through the third laminate was tensile all throughout, increasing progressively in magnitude from z outer surfaces to midplane. Delamination at the first and second interfaces can thus be related to the presence of high interlaminar normal stresses at these locations. The high o z shear and tensile interlaminar 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 7 0 ° , severe delamination was observed at the interface, 110° from the loading direction. The interlaminar shear stresses TQ T r z fourth Z and 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 magnitude towards hole boundary the laminate delamination at the free-edge, been assisted by o z is compressive for all interfaces midplane. This may have prevented increasing in initiation of though away from the edge, delamination may have 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 The comparison between the experimental 138 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 [O2/9O2/±30] s and 215) and [±30/9fJ2/02] s V (page 217) respectively. These for tables the show two laminates 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 for TQ and T while for o gross applied stress. magnitudes of shear The stresses, values given ignoring the sign, Z are z both the absolute 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 may only be used to predict delamination initiation, since the change with initiation and propagation of damage comparisons have been made after at original stress state the hole. Nevertheless, the initiation and propagation of finite amounts of delamination achieveing good agreement between the results. 5.3.2 [02/±45] s and [±45/02] The s laminates:- 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. defects is shown in Fig. 5.8 for a specimen of [ ± 45 / 0 2 ] 139 configuration. The voids s 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 [ 0 2 / i " 4 5 ] specimen, s made after repeated loading to increasingly higher in-plane stresses. A large number of cracks in the 45-deg plies appeared zero-deg ply cracks were seen throughout on each side the specimen, while only two 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 increasing load. In contrast, the radiographs of [ ± 4 5 / 0 2 ] s to grow rapidly with 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 ] fact that a given stress level stress foT [±45/02]s (far- field) failure represented laminate stress for specimen. This was so, even in spite of the s than each a higher for laminate the fraction of the [02/±45] type was s mean laminate. determined by The failure mean monotonic tension tests of three specimens of each type. The maximum stress applied before making each radiograph mean failure stress. is given in Figs. 5.9 and 5.10 as a percentage of the 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 the found to be tensile for the [±45/02J around the S laminate. The hole is seen [02/±45] magnitude to increase of with this the laminate and compressive for s normal distance stress from at the any interface surface to that interface. On the other hand, the shear stress distributions in the two laminates are found the to be similar for any interface between similar plies. A close inspection of shear stress distributions distributions of both TQ Z shown and r rz in Figs. 5.11 and 5.12 show that are essentially similar in view of the two-fold symmetry associated with specimen and loading geometry. Thus, the only between interlaminar stress distributions of the that of a , z s and compressive for [ ± 4 5 / 0 2 ] s configurations. The excessive delamination observed in [ 0 2 / ± 4 5 ] Though delamination be attributed due to to this difference combined effects difference two laminate types is found to be which is mostly tensile for [ 0 2 / i 4 5 ] other, can therefore the in o z laminate over the s at the of interlaminar hole boundary. normal and stresses occurs in both laminates, the tensile normal stresses in [ 0 2 / ± 4 5 ] s shear 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 Furthermore, interface in the from the [02/±45] s outside between specimen, narrow zero-deg lips and extending vertical direction from the hole boundary, consisting of surface bounded by the zero-deg ply cracks tangent to the formed above and below the failure stress. At failure, complete separation +45-deg outward plies. in the zero-deg plies and hole, were seen. These lips hole when loaded to more than of these lips from 80% of its mean 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 TQ T ± 45-deg RZ are both maximum at the second interface between zero-deg plies, over most of the boundary in both stacking sequences. fracture surfaces of the two laminate narrow lips of a fractured while failure [±45/02] s across the [02/±45] hole s through types are Z and The appearence shown in Fig. 5.13. The and of long specimen are clearly visible in this figure, entire laminate thickness is seen for the revealed the specimen. Microscopic examination of polished sections existence of voids in the laminates. The presence of voids along ply interfaces made any comparison between through and stress distributions difficult. the thickness locations of delamination initiation This is evident from Fig. 5.14, which series of replicas of polished sections prepared from a [ ± 4 5 / 0 2 ] s shows a specimen, loaded to 265 MPa gross applied stress (42% of the mean failure stress). The sections are made at angles of 0 ° , 6 0 ° , 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 outside between delamination. Delamination of the first interface + 45-deg plies is clearly seen near the free-edge probably due to high TQ Z shear stress and non-compressive o z from at 60°. This is obtained at this angle as shown in Fig. 5.12. At 90° from the loading direction, damage is seen at almost every differentiate interface across the laminate thickness. It is clearly difficult to between any separation due to voids formed along ply interfaces and delamination due to high interlaminar stresses. Experimental Observations 5.3.3 [02/±30] s 142 laminate:- Laminates of [02/±30] s configuration were prepared using Scotchply-type 1003 glass/epoxy material system. Non-destructive detection of damage development using zinc-iodide enhanced this laminate, due transparency to high absorbancy X-ray radiography of X-rays was not by glass fibres. feasible with However, the 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 the specimen leaving damaged circular edge could therefore be seen regions less illuminated. Delamination at 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 within one of the 60° - 80° delamination. Delamination was first detected 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 delaminations could be seen between with these 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 found to be tensile at every interface around the 143 hole, except within a narrow angular region on each side of the hole. The interlaminar shear T Q on the other Z hand is high at second and third interfaces in the neighbourhood of 70° and 90° respectively. However, the high other component of interlaminar shear, T but largest at the second interface between most of the boundary. The importance of T zero-deg is not so and 30-deg plies for in predicting delamination at any RZ given interface is very little, since the distribution of r is nearly uniform around rz the hole. At observed. Although o 0° from the loading direction no delamination was is high at this location, T Q is zero throughout, except at z Z 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 7 0 ° , 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 segment near the free-edge. While o parts of zero-deg plies from the specimen is only slightiy tensile for every interface at z this angle, T Q is very high at the second interface between zero-deg and 30-deg Z plies. At the first interface T Q is nearly half that at the second and is- zero at Z 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 at the boundary. Z A similar result was obtained at 90° from the loading direction. Delamination of the third interface .between ± 3 0 - d e g plies was observed at is this angle, where magnitude of T Q Z T Q Z of is significantly the third interface high at this interface near its maximum. The but zero at every other interface. The observed delamination of Fig. 5.15 can thus be related to high T Q which may also have been supplemented by T Z 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 o . It is also interesting to note the high density of ply cracks z near the free-edge in this micrograph. In predicting delamination at the hole boundary in [ 0 2 / ± 3 0 ] glass/epoxy laminates the distribution of TQ those of o and T . z z [0790] s and [90/0] was found to be more significant than This may partly be due to the fact that the peaks in TQ RZ are higher than those of o 5.3.4 Z and s Z T . RZ laminates:- s The radiographs of damage around the hole due to sequential loading in 4-ply [0/90] s They show damage after and [90/0] s laminates are shown in Figs. 5.17 and 5.18. 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] specimen zero-deg ply cracks are visible in the radiograph taken after s 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 seen first observed in the [0/90] s specimen. Delaminations were 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 angle of approximately 60° - 6 5 ° boundary makes an with the loading direction. Subsequent delamination growth is found to occur along the zero-deg ply cracks. Experimental Observations The observed delamination in the [90/0] Fig. 5.18, is also similar to that just described. The 145 specimen, shown in s locations of delamination initiation at the hole boundary and its growth appear very much similar to that in the [0/90] specimen. s However, delamination seen in the [90/0] at any given stress level, the amount of specimen is less than that observed in the [0/90] s 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 , is found to be tensile for z the [0/90] laminate, s distributions of o 70° from the both around are the the first ply interface loading direction. They are found hole. and. the mid-plane. The at these interfaces remain tensile with their maxima occuring at z longitudinal and transverse laminate at to also symmetrical with axes. In contrast, the distributions of o z be compressive However, the basic through the shapes laminate of the respect to the in the [90/0] thickness, distributions and and s also 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 TQ T R Z Z and are finite at the first interface between the zero-deg and 90-deg plies. Their distributions laminate exhibit maxima at types. longitudinal and Since the transverse locations around the hole. around 70° distributions are axes, the from also above the loading direction, in both symmetrical with mentioned maxima respect occur to at the four Experimental Observations The delaminations observed in the two laminate 146 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] two laminates are s type. The interlaminar stress distributions obtained for the also similar except for the compressive o obtained z for the [90/0] laminate. s In the [0/90] laminate s all three stress components maxima at around 70° from the loading direction. In the [90/0] s exhibit 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 6 0 ° - 6 5 ° 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 direction. of a polished section The observed taken delamination of at the an angle first laminate mid-plane, must be due to the presence of 65° from ply interface, the rather loading than the 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 were centre of each loaded friction grips. in an specimen was 6.5 mm (0.25 in.) in diameter. Instron testing machine using a pair of The specimens 1-in. wide tensile 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 ± 4 5 - d e g plies are visible after stress. These cracks increase in length and number with 140 MPa increasing applied stress. Matrix cracking of the zero-deg ply begins to appear in the radiographs taken after loading to 200 MPa stress. Delamination radiograph taken after loading at 185 the hole MPa gross boundary first applied stress. appears Dark in the 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, a, z 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 o z at different interfaces increases with the distance to that interface from outside. The interlaminar shear TQ Z 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 TQ Z 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 distribution on the other hand predicts high stresses for regions near 45° N and 90° from the loading direction, T TZ distribution at the second interface reaches its maximum at 4 5 ° , while that at the first and third interfaces reach their maxima at 9 0 ° . The interlaminar stress distribution discussed above is sufficient to explain the experimental observations of delamination. The locations of observed delamination coincide with distributions of TQ Z the locations of stress peaks in TQ Z distribution. The 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 o location of delamination observed, o this angular location, while T and z z T seem RZ favourable for the particular is tensile throughout the laminate thickness at is significantly high for the first two interfaces RZ at this location. For the third interface between the 45-deg and 90-deg plies however, o z and T RZ are both found to be even higher though TQ Z 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 MPa sectioning of a specimen loaded to nearly 235 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 o z and relatively small T obtained RZ through the laminate thickness. Relatively high TQ Z for different interfaces shear stresses are obtained only for the first two interfaces at this location. At between zero-deg and 4 5 ° , delamination was -45-deg plies. This observed is in at the agreement second with interface the stress Experimental Observations distributions shown in Fig. 5.23. o calculated for this interface z a relatively high magnitude. is tensile and has T Q too is high for this interface at 45° Z loading direction. In addition, the highest value of r 149 from the at this angular location is r z also obtained for the second interface where delamination is observed. At 9 0 ° , delamination was observed between -45-deg and 90-deg plies. Here o mid-plane. The interlaminar shear r interface. However, the interface is very high and smaller to only that z at the along the third too has r z the highest value at this T Q component of interlaminar shear is found to be zero Z for the third interface at this angle. 5.4 DISCUSSION:- The comparisons interlaminar stresses calculated of experimental using the present observations approximate of delamination with 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 results appear to justify agreement the use between of theoretical approximate calculations methods of and stress experimental 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 undamaged at the preceded free-edge of an hole. Strictly, these stresses can only be used to predict delamination initiation undamaged by matrix hole. In cracking practice, or delamination splitting, which strictly, the stresses calculated for the undamaged at the alters hole the boundary original is frequently stress distribution, 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 / ^o -o ) 2 e r 2 + (o -o ) r + (o-o ) 2 z At + H T ^ + T ^ + T ^ 2 e a given interface and shear stress' components (viz; o near the and T ^ ) r 2 ) ] " 2 = k (5.1) hole boundary, the in-plane radial 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; [ o 2 z + 1( r 2 rz + r 2 6 z ) ] 1 / 2 = k (5.2) Experimental Observations Delamination is expected to occur when the 151 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 include the detrimental effect of tensile no influence interlaminar on delamination. In order normal stresses on to delamination equation (5.2) can be modified as; [ a\ | + 3( r o z z This towards delamination delamination due to investigators, (e.g., + 2 rz predicts initiation tensile Whitney that while 1 / = 2 interlaminar compressive normal Browning, ) ] 2 6z tensile interlaminar and r 1972; stresses stresses normal oppose has Pagano k been and (5.3) stresses it The observed Pipes, 1973) contribute onset by of many 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 different interfaces semi-quantitative and stresss combination function suggested here can be evaluated at angles observations around of the the extent hole. It is also possible of delamination with comparison is shown in Figs. 5.25 and 5.26 for the [ O 2 / 9 O 2 / ± 3 0 ] to compare the this criterion. Such a 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 The points on Figs. 5.25 and 5.26 have been obtained 152 by comparing the delamination observed at every interface through the laminate thickness with the corresponding value of the stress stresses calculated for the undamaged first delamination. Delamination at combination function. Strictly, the interlaminar hole can only be used to predict the onset of the 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 experimental involved in formulating the suggested stress function its agreement with results is very pleasing. Though one would not necessarily predict a linear relation in Fig. 5.27, the function (which is the observed driving force trend of an increasing calculated for initiation of delamination) stress combination leading to a larger measured delamination (which is truly a propagation effect) is to be expected. Experimental Observations Polished Surface Fig. 5.1. Sectioning of a specimen at the hole. 153 Experimental Observations 154 (c) 280 MPa Fig. 5.2. Radiographs of damage in a [O2/9O2/±30] laminate s 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 [02/90 /±30] s 2 stress laminate. a z distribution in a Fig. 5.3 (b). interlaminar [0 /902/±30] 2 shear s stress laminate. T BZ distribution in a Experimental Observations Fig. 5.3 (c). interlaminar shear stress r r [0 /90 /±30] 2 2 s laminate. distribution in 158 a Fig. 5.4 (a). Micrograph and [0 /90 /±30] 2 2 s replica showing laminate at 0 ° . delamination in a Experimental Observations 160 Fig. 5.4 (c). Micrograph and [0 /90 /±30] 2 2 s replica showing laminate at 70°. delamination in a Fig. 5.4 (d). Micrograph and [02/90 /±30] 2 s replica showing laminate at 9 0 ° . delamination in a Experimental Observations 163 Experimental Observations 164 Experimental Observations (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 165 Experimental (e) 550 MPa Observations 166 Experimental Observations Fig. 5.6 (a). Interlaminar normal [ ± 30/902 2 J S laminate. / 0 stress o z distribution in 167 a Experimental Observations Fig. 5.6 (b). Interlaminar [±30/90 /0 ] 2 2 s shear stress laminate. TQ Z distribution in 168 a Experimental Observations Fig. 5.6 (c). Interlaminar shear stress [± 30/902 23s laminate. /0 r r z distribution in 169 a Experimental Observations Fig. 5.7 (a). Micrograph and [±30/90 /0 ] 2 2 s replica showing laminate at 0 ° . delamination in Fig. 5.7 (b). Micrograph and [ ± 30/902 / 0 ] 2 s replica showing laminate at 4 5 ° . delamination in a Fig. 5.7 (c). Micrograph and [±30/902/0 ] 2 s replica showing laminate at 7 0 ° . delamination in a Fig. 5.7 (d). Micrograph and [ ± 30/90 /0 ] 2 2 s replica showing laminate at 9 0 ° . delamination in a Experimental Observations Fig. 5.7 (e). Micrograph and [±30/902/0 ] 2 s replica showing laminate at 110°. delamination in a 174 Experimental Fig. 5.7 (f). Micrograph and [±30/90 /0 ] 2 2 s replica showing laminate at 135°. delamination Observations in a 175 Fig. 5.7 (g). Micrograph and [ + 30/902/0 ]s 2 replica showing laminate at 180°. delamination in a Experimental Observations Fig. 5.8. Radiographs taken before loading a [ ± 4 5 / 0 2 ] s laminate. 177 (0 /±45) 2 Experimented Observations s IX 45° p lamjnatj nitiation 178 MPa 311 MPa 44% of uts 2 5 % of uts x splitti 0°|Ny Fig. 5.9. Radiographs of damage in a [ 0 2 / ± 4 5 ] s laminate. 178 (±45/02), Experimental Observations 178 MPa 27.8% of uts Fig. 5.10. Radiographs of damage in a [ ± 4 5 / 0 2 ] s laminate. 179 Fig. 5.11 (a). Interlaminar laminate. normal stress o z distribution in a [02/i~45] s Experimental Observations Fig. 5.11 (b). Interlaminar shear stress r Qz laminate. distribution in a [02/±45] 181 s Fig. 5.11 (c). Interlaminar laminate. normal stress r rz distribution in a [02/±45] s Experimental Observations Fig. 5.12 (a). Interlaminar normal stress o z laminate. distribution in a [±45/02J"s 183 Experimental Observations Fig. 5.12 (b). Interlaminar shear stress TQ Z laminate. distribution in a [±45/02] 184 s Experimental Observations Fig. 5.12 (c). Interlaminar laminate. shear stress T RZ distribution in a [±45/02] 185 s Experimental Observations Fig. 5.13. Fracture surfaces of [ 0 2 / ± 4 5 ] s and [ ± 45/C>2] laminates. s J86 Replicas of sections showing delamination at different locations in a [ ± 4 5 / 0 ] 2 (c). at 9 0 ° . s laminate (a), at 0° angular (b). at 60° Experimental Fig. 5.15. Micrographs of sections showing angular locations in a [ 0 2 / ± 3 0 ] 70° (c). at 90° s delamination Observations at laminate (a), at 0° different (b). at (c). at 90° Fig. 5.16 (a). Interlaminar laminate. normal stress a z distribution in a [02/±30j s Experimental Observations Fig. 5.16 (b). Interlaminar shear stress T QZ laminate. distribution in a [02/±30] 191 s Fig. 5.16 (c). Interlaminar laminate. shear stress T RZ distribution in a [fJ2/±30] s Experimental Observations (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. 193 Experimental Observations (d). 375 MPa. (e). 420 MPa. 194 Experimental Observations (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. 195 Experimental Observations (e). 420 MPa. 196 Experimental Observations Fig. 5.19 (a). Interlaminar laminate. normal stress a z distribution in a [0/90] 197 s Experimental Observations Fig. 5.19 (b). Interlaminar laminate. shear stress TQ Z distribution in a [0/90] 198 s Experimental Observations Fig. 5.19 (c). Interlaminar laminate. shear stress r r z distribution in a [0/90] 199 s Experimental Observations Fig. 5.20 (a). Interlaminar laminate. normal stress o z distribution in a [90/0] 200 s Experimental Observations 201 to Fig. 5.20 (b). Interlaminar laminate. shear stress TQ Z distribution in a [90/0] s Experimental Observations Fig. 5.20 (c). Interlaminar laminate. shear stress T RZ distribution in a [90/0] 202 s Fig. 5.21. Micrographs of sections direction in [0/90] s taken laminate. at 65° from the loading 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 (d) 210 MPa (e) 235 MPa. 205 Experimental Observations Fig. 5.23 (a). Interlaminar [45/0/-45/90] normal s laminate. stress a z distribution in 206 a Experimental Observations Fig. 5.23 (b). Interlaminar [45/0/-45/90] shear s stress laminate. TQ Z distribution in 207 a Experimental Observations Fig. 5.23 (c). Interlaminar [45/0/- 45/90] shear s laminate. stress r distribution in a 208 Fig. 5.24. Micrographs angular of sections locations in a (b). at 4 5 ° . (c). at 9 0 ° . showing delamination [45/0/-45/90] s laminate, at different (a), at 0°. Experimental Observations (c). at 90° Experimental Observations 211 p o 10 • • • • p fO • • • o or • • • p d tpODTJ • -B—^—B—B- • 0.0 50.0 100.0 T 150.0 200.0 —i Stress Combination Function MPa Fig. 5.25. Comparison of delamination stress combination function. in [ 0 2 / 9 0 ^ / ± 3 0 ] s laminate with 250.0 Experimental 212 Observations o O-i • p CO c o •— D c E • — q U3 a © c O E < q • • q c>. 0.0 -S—B- 50.0 T • • •o • CD 100.0 150.0 T 1 200.0 250.0 Stress Combination Function MPa Fig. 5.26. Comparison of delamination stress combination function. in [ ± 307902/O^Js laminate with 213 Experimental Observations p d-i p CO c o 1 E o 9 " <D o c o E < • p • • • • • o CN • • O d.0.0 T 50.0 100.0 1 T 150.0 200.0 250.0 Stress Combination Function MPa Fig. 5.27. Plot of maximum delamination at each combination function for [ O 2 / 9 O 2 / ± 3 0 ] laminates. s and angle vs. stress [± 30/902/0 ] 2 s TABLE HI. In-plane ply elastic properties used in the present stress calculations:- MATERIAL ELASTIC PROPERTIES Hercules AS1/3501-6 Graphite/epoxy E E 12 = 7.10 GPa 12 = 0.30 G Scotch-ply E Type-1003 E Graphite/epoxy 77 = 38.6 GPa 22 = 8.27 GPa I2 = 4.14 GPa 12 = 0.26 G v XAS/914 = 138.0 GPa 22 = 8.96 GPa v Glass/epoxy }1 E E 77 = 145.0 GPa 22 = 9.5 GPa 12 = 5.6 GPa 12 = 0.31 G v 215 TABLE IV. Comparison of delamination interfaces in a [0 /90 /±30] 2 Angle 0° 45° 70° Interface 2 az T 6z with s o interlaminar stresses at different graphite/epoxy specimen: rz T Function Delamination 1 1.4 -0.0 5.1 8.9 0.0 2 5.4 -0.0 10.1 18.3 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 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 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 90° 110° 135° Interface °z T 6z T rz Function Delamination 1 -0.9 0.0 -3.3 5.7 0.0 2 -3.6 0.1 -6.7 11.0 0.0 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 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 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] graphite/epoxy specimen: s Angle 0° Interface °z 6z T rz T Function 1 -3.5 -14.4 -13.1 33.5 0.0 2 -14.0 0.0 -26.1 43.0 0.3 3 -26.2 0.0 -19.6 21.6 0.7 4 -35.0 0.0 -13.1 0.0 0.3 5 -40.3 0.0 -6.6 0.0 0.0 1 2.2 24.1 8.2 44.1 5.0 2 -7.8 -4.9 -45.6 79.0 5.0 3 -35.4 -27.2 -57.6 104.5 0.7 4 - 69.5 - 49.6 -69.7 130.7 8.8 5 -97.5 -24.8 -34.8 0.0 0.0 6 -106.8 0.0 0.0 0.0 0.0 1 0.2 -86.7 0.9 150.2 0.0 2 8.2 -70.4 28.7 131.9 3.7 3 14.1 -98.5 -6.5 171.6 0.0 4 1.2 -126.7 -41.8 231.0 8.0 5 -15.6 -63.3 -20.9 114.5 0.0 6 -21.2 0.0 0.0 0.0 0.0 218 T A B L E V. (Contd.,) Angle 90° 110° 135° I: •face °z T 6z T rz 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 failure in fiber reinforced initiating composite at free-edges is an laminates. The important mode of 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 The results of the numerical results from the literature. approximate Comparisons method have have been 220 been compared made on the with 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 damage found that the presence modes may alter the interlaminar stress distributions to the of other interactive exent which makes these comparisons only qualitative. An experimental program was also undertaken to examine the nature of damage observed induced at the in a number hole boundaries of different by static loading. The delamination laminates was compared with damage 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 qualitative present agreement was stress delamination observed calculations. It was preceeded was by found by sectioning the laminate at the between these observed that matrix cracking. observations in most This of can hole. Good and the the laminates alter the results of the examined 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 relative amounts direction of delamination propagation. In spite of delamination observed at different interfaces of these problems, and at different the angles Summary and Conclusions 221 were in agreement with the stresses calculated. A mixed-mode failure criterion was also suggested. estimates of the relative amounts of delamination observed at different different laminates were compared with Semi-quantitative interfaces in two 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 generation stresses as calculated by LPT contribute to the of interlaminar stresses. O f the two possible methods available for calculating these stresses, the "modified stress method") more physically meaningful method, (called the leads to better agreement with numerical results in the literature. 2. Simple equilibrium arguments can be used successfully to relate the stresses to magnitudes interlaminar of interlaminar stresses. It stresses can around predict holes the when signs the and in-plane relative 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 mixed-mode difficult observations failure criterion of delamination. makes accurate The lack predictions of of a reliable delamination Summary and Conclusions 4. With the delamination present and approximate interlaminar technique stresses can simple be correlations attempted. 222 between 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. Conti P. and De Paulis A., (1985), " A Simple Model to Simulate the Interlaminar Stresses Generated near the free-Edge of a Composite Laminate", Delamination and Debonding of Materials, ASTM STP 876, Johnson W.S., Ed., pp. 35-51. Dana J.R. and Barker R . M . , (1974), "3-Dimensional analysis for the Stress-Distribution near Circular Holes in Laminated Composites", Report VPI-E-74-18, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, (NTIS A D 783 504/4GI) Daniel I.M., Rowlands R.E. A N D Whiteside J.B., (1974), "Effects of Material and Stacking Sequence on Behavior of Composite Plates with Holes", Experimental Mechanics, J. of the Society for Experimental Stress Analysis, Vol 14, pp. 1-9. Ericson K.., Persson M . , Carlsson L. and Gustavsson A., (1984) "On the Prediction of the initiation of Delamination in a [0/90] Laminate with a Circular Hole", J.Comp.Mat, Vol 18, pp. 495-506. s Foye R.L. and Baker D.J., (1970), "Design of Orthotropic Laminate", Presented at the 11 th Annual AIAA Structures, Structural Dynamics and Materials Gonf., Denver, Colorado, Goonetilleke H.D., Poursartip A . and Teghtsoonian E., (1985), "Estimating Interlaminar Stress Distributions Around Holes in Composite Laminates", Proceed, of Fifth Int. Confernce on Composite Materials, Harrigan Jr. W . C , Strife J. and Dhingra A . K . , Eds., I C C M V, San Diego, California, pp. 1233-1245. Herakovich C.T., Nagarkar A. and O'Brien D.A., (1979), "Failure Analysis of Composite Laminates with Free-Edges", Moderen Developments in Composite Materials and Structures, Vinson J.R., Ed., The American Society of Mechanical Engineers (ASME), N . Y . , pp. 53-66. Hsu P.W. and Herakovich C.T., (1977), "Edge Laminates", J.Comp.Mat, Vol 11, pp. 422-428. Effects in Angle-Ply Composite Hsu P.W. and Herakovich C.T., (1977), " A Perturbation Solution for Interlaminar Stresses in Composite Laminates", Composite Materials Testing and Design (Fourth Conf.), A S T M STP 617, American Society for Testing and Materials, pp. 296-316. 224 Isakson G . and Levy. A., (1971), "Finite-Element Analysis of Interlaminar Fibrous Composites", J. Comp. Mat, Vol 5, pp.273-275. Shear in Johnson E.R. and Kemp B . L , (1985), "Modeling the Stress Field in Laminated Composite Plates near Discontinuities", Composite Structures, Vol 3, pp. 145-166. Kim R.Y. and Soni S.R., (1984), "Experimental and Analytical Studies on the of Delamination in Laminated Composites", J. Comp. Mat, Vol 18, pp. 70-80. Onset Kress G.R. and Stinchcomb W.W., (1985), " Fatigue Response of Notched Graphite/Epoxy Laminates", Recent Advances in Composites in the United States and Japan, A S T M STP 864, Vinson J.R. and Taya M . , Eds., American Society for Testing and Materials, Philadelphia, pp. 173-196. Lekhnitskii S.G., Holden-Day, Inc., (1963), Theory of Elasticity of Anisotropic Elastic Body, Levy A., Armen H.Jr., and Whiteside J., (1971), "Elastic and Plastic Interlaminar Shear Deformation in Laminated Composites under Generalized Plane Stress", Proceed, of the Third Conf. on Matrix Methods of Structural Analysis, Wright Patterson A F B , (NTIS A D 737 508) Lucking W . M . , Hoa S.V. and Sankar T.S., (1984), "The Interlaminar Stresses of [0/90] Composite Laminates J. Comp. Mat, Vol 17, pp. 188-198. s Mau S.T., Tong P. and Pian T.H.H., (1972), "Finite Laminated Thick Plates", J. Comp. Mat, Vol 6, pp. 304-311. O'Brien T.K. and Delamination Around Structures, Structural Conf., Palm Springs, Effect of Geometry on with Circular Holes", Element Solutions for Raju I.S., (1984), "Strain-Energy-Release Rate Analysis of an Open Hole in Composite Laminates", A I A A / A S M E 25 th Dynamics and Materials Conf. and A I A A Dynamics Specialists California, AIAA-84-0961, pp. 526-536. Pagano N.J., (1974), "On the Calculation of Interlaminar Normal Stress in Composite Laminate", J. Comp. Mat., Vol 8, PP. 65-82. Pagano N.J., (1978), "Stress Fields in Composite Structures, Vol 14, pp. 385-400. Pagano N.J. and Pipes R.B., (1971), "The Laminate Strength", J.Comp.Mat, pp. 50-57. Laminates", Influence of Int Stacking J. Solids and Sequence on Pagano N.J. and Pipes R.B., (1973), "Some Observations on the Interlaminar Strength of Composite Laminates", Int. J. Mechanical Sciences, Vol 15, pp. 679-688. 225 Pagano N.J. and Soni S.R., (1983), "Global-Local Laminate Variational Model", Int. J. Solids and Structures, Vol 19, No. 3, pp. 207-228. Pipes R.B., Kaminski B.E. and Pagano N.J., (1973), "Influence of the Free-Edge upon the Strength of Angle-Ply Laminates", Analysis of the Test Methods for High Modulus Fibers and composites, A S T M STP 521, American Society for Testing and Materials, pp. 218-230 Pipes R.B. and Pagano N.J., (1970), "Interlaminar Stresses in Composite under Uniform Axial Extension", J. Comp. Mat, Vol 4, pp. 538-548. Laminates Poursartip. A., (1984), private communication. Puppo A . H . and Evensen H.A., (1970), "Interlaminar Shear in Laminated Composites Under Generalized Plane stress", J. Comp. Mat., Vol 4, pp. 204-220. Radford D.W., (1982), "Fracture Toughness of a Carbon Fibre-Epoxy Composite Material", M.A.Sc. Thesis, Dept. of Metallurgical Eng., University of British Columbia. Raju I.S. and Crews J.H.Jr., (1981), "Interlaminar Stress Singularities at a Straight Free Edge in Composite Laminates", J. Computers and Structures, Vol 14, No. 1/2, pp. 21-28. Raju I.S. and Crews Jr. J.H., (1982), "Three-Dimensional Analysis of [0/90] and [90/0] laminates With a Central Circular Hole", Composites Technology Review, V 4, No. 4, pp. 116-124. s s Reiss E.L., (1961), "Extension of a Thick Infinite Report of New York University, I M M - N Y U - 2 8 1 . Plate With a Circular Hole", Reissner E., (1950), "On a Variational Theorem in Elasticity", J. Mathematics Physics, Vol 29, pp. 90-99. Rybicki E.F., "Approximate Three-Dimensional Solutions for Symmetric under In-Plane Loading", J.Comp.MaL, Vol 5, July 1971, pp. 354-360. and Laminates Rybicki E.F. and Hopper A.T., (1973), "Analytical Investigation of Stress Concentrations Due to Holes in Fiber Reinforced Plastic Laminated Plates- Three Dimensional Models", Technical Report, A F M L - T R - 7 3 - 1 0 0 , Battelle, Columbus Laboratories, Columbus, Ohio, (NTTS A D 768 453) Rybicki E.F. and Schmueser D.W., (1976), "Three-Dimensional Finite Element Stress Analysis of Laminated Plates Containing a Circular Hole", Technical Report, A F M L - T R - 7 6 - 9 2 , Battelle, Columbus Laboratories, Columbus, Ohio, (NTIS A D A032 428) 226 Rybicki E.F. and Schmueser D.W., (1978), "Effect of Stacking Sequence and L a y - U p Angle on Free Edge Stresses around a Hole in a Laminated Plate under Tension", J. Comp. Mat, Vol 12, pp. 300-313. Salamon N.J., (1980), "An Assessment of the Interlaminar Stress Laminated Composites", J.Comp.Mat Supplement, Vol 14, pp. 177-194. Problem in Savin G.N., (1968), Stress Distribution around Holes, Translation of "Raspredeleniye Napryazheniy Okolo Otversity", "Navkova Dunka" Press, Kiev, National aeronautics and Space Administration Technical Translation N A S A TT F-607. Soni S.R. and K i m R.Y., (1986), "Delamination of Composite Laminates Stimulated by Interlaminar Shear", Composite Materials: Testing and Design (Seventh Conf.), A S T M STP 893, Whitney J.M., Ed., American Society for Testing and Materials, Philadelphia, pp. 286-307. Spilker R.L. and Chou S.C., (1980), "Edge Effects in Symmetric Composite Laminates: Importance of Satisfying the Traction-Free-Edge Condition", J. Comp. M a t , Vol 14, pp. 2-20. Tang S., (1975), " A Boundary Layer Theory-Part I: Laminated Composites in Plane stress", J.Comp. Mat, Vol 9, pp. 33-41. Tang S., (1977), "Interlaminar Stresses around Circular Cutouts in Composite under Tension", AIAA Journal, Vol 15, No. 11, pp. 1631-1637. Tang S. and Levy A., (1975), "A Boundary Layer Theory-Part Laminated Finite Strip", J.Comp. Mat, Vol 9, pp. 42-52. II: Plates Extension of Timoshenko S.P. and Goodier J.N., (1934), Theory of Elasticity., Engineering Societies Monographs, McGraw-Hill Book Company. Wang A.S.D. and Crossman F.W., (1977), "Some New Results on Edge Effects in Symmetric Composite Laminates", J. Comp. Mat, Vol 11, pp. 92-106. Wang A.S.D. and Crossman F.W., (1978), "Calculation of Edge Stresses Multi-Layer Laminates by Sub-Structuring", J.Comp.MaL, Vol 12, pp. 76-83. in Wang J.T.S. and Dickson J.N., (1978), "Interlaminar Stresses in Symmetric Composite Laminates", J.Comp.MaL, Vol 12, pp. 390-402. Wang S.S. and Choi I., (1982), "Boundary-Layer Effects in Composite Laminates: Part 1- Free-Edge Stress Singularities", A S M E J. of Applied Mechanics, vol 49, pp. 541-548. Wang S.S. and Choi I., (1982), "Boundary-Layer Effects in Composite Laminates: 227 Part 2- Free-Edge Stress Solutions and Basic Characteristics", ASME J. of Applied Mechanics, vol 49, pp. 549-560. Whitcomb J.D., (1981), "Experimental and Analytical Study of Fatigue Damage in Notched Graphite/Epoxy Laminates", Fatigue of Fibrous Composite Materials, A S T M STP 723, American Society of Testing and Materials, pp. 48-63. Whitcomb J.D. and Raju I.S., (1985), "Analysis of Interlaminar Stresses in Thick Composite Laminates with and without Edge Delamination", Delamination and Debonding of Materials, A S T M STP 876, Johnson W.S., Ed., American Society for Testing and Materials, pp. 69-94. Whitney J.M. and Browning C.E., (1972), Coupons", J.Comp.Mat, Vol 6, pp. 300-303. "Free-Edge Delamination of Tensile Whitney J.M. and K i m R.Y., (1977), "EfTect of Stacking Sequence on the Notched Strength of Laminated Composites", Composite Materials: Testing and Design (Fourth Conf.), A S T M STP 617, American Society for Testing and Materials, pp. 229-242. Whitney J.M., Daniel I.M. and Pipes R.B., (1982), Experimental Mechanics of Fiber Reinforced Composite Materials, Society for Experimental Stress Analysis, Monograph No. 4.
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Free-edge effects around holes in composite laminates
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Free-edge effects around holes in composite laminates Goonetilleke, Hemaguptha Dharmaraj 1986
pdf
Page Metadata
Item Metadata
Title | Free-edge effects around holes in composite laminates |
Creator |
Goonetilleke, Hemaguptha Dharmaraj |
Publisher | University of British Columbia |
Date Issued | 1986 |
Description | 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. |
Subject |
Laminated materials |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0103883 |
URI | http://hdl.handle.net/2429/27305 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mining Engineering |
Affiliation |
Applied Science, Faculty of Mining Engineering, Keevil Institute of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1987_A1 G66.pdf [ 27.17MB ]
- Metadata
- JSON: 831-1.0103883.json
- JSON-LD: 831-1.0103883-ld.json
- RDF/XML (Pretty): 831-1.0103883-rdf.xml
- RDF/JSON: 831-1.0103883-rdf.json
- Turtle: 831-1.0103883-turtle.txt
- N-Triples: 831-1.0103883-rdf-ntriples.txt
- Original Record: 831-1.0103883-source.json
- Full Text
- 831-1.0103883-fulltext.txt
- Citation
- 831-1.0103883.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0103883/manifest