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Penetration of CFRP laminates by cylindrical indenters Ursenbach, Daniel O. 1995

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P E N E T R A T I O N OF C F R P L A M I N A T E S B Y C Y L I N D R I C A L I N D E N T E R S by D A N I E L O. U R S E N B A C H B.Sc. Mechanical Engineering, University of Calgary, 1993 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Metals and Materials Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1995 © Daniel O. Ursenbach, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of fAc^h Cu^J j/lahr/etls "bhcji/ie&r-,\nt^ The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract The main focus of this thesis has been towards understanding the local penetration behaviour of a C F R P laminate undergoing quasi-static out-of-plane loading. This information is necessary for predicting dynamic impact behaviour. A n experimental program was performed that characterized the damage mechanisms occurring during penetration and confirmed the validity of model assumptions. The principal test consisted of a 7.62 mm diameter blunt-nosed cylindrical indenter penetrating various thicknesses of C F R P coupons supported over a rectangular opening. Even though some additional experimental data is yet required for a complete study, the necessary characterization tests have been identified, and, in some cases, developed. A simple analytical model to describe the quasi-static penetration process was developed giving an approximate prediction of the force-displacement behaviour. This model was based upon actual physical phenomena and used an approximate energy method to predict the displacement profile of the delaminated plate. Good correlation was found between the predicted delamination sizes and those measured using C-scans. ii Table o f Contents Abstract ii Table of Contents iii List of Tables v List of Figures vi Acknowledgments xi Chapter One - Introduction 1 Background 1 Impact Velocity Regimes 3 Projectile Geometry 3 Material Properties 5 Present Work 6 Chapter Two - Experimental Results 13 Background 13 Static Penetration Test 17 Elastic Bending and Indentation 20 Matrix Cracking 20 Delamination Onset 22 Post Delamination Loading 26 Total Delamination Area Calculations 27 Delaminated Plate Stiffness 29 Delaminated Plate Load-Strain Behaviour 30 Fracture Mechanics Considerations 31 Delamination in Tough Systems 34 Rear Surface Matrix Splitting 34 Fibre Breakage 35 Plugging 36 Friction 37 Large Circular Opening Static Penetration Tests 39 Influence of Geometric Constants 39 Impact Test 41 iii Impact Test Results 42 Ballistic Tests 45 Summary 45 Chapter Three - Analytical Modelling and Model Verification 78 Background 78 Present Model 84 Model Description 85 Model Formulation 86 Stage I - Elastic Bending and Indentation 87 Stage II - Delaminated Plate 96 Stage III - Friction 108 Summary 109 Chapter Four - Conclusions and Future Work 125 Summary 125 Future Work 128 References 131 Appendix A 135 Appendix B 137 Exact Solutions 137 Approximate Solution 139 Results " 140 iv L is t of Tables Table 2.1 Summary of static and impact tests performed on the T300H/F593. 47 Table 2.2 Summary of static and impact tests performed on the IM7/8551-7 (taken from Delfosse [1994]). 47 Table 2.3 Summary of strain gauge tests performed on the T300H/F593 system. 48 Table 2.4 Summary of measured delamination diameters 49 Table 2.5 Mode I and II Critical Strain Energy Release Rates for Various Brittle Carbon Fibre Composite Materials (taken from Daniel and Ishai [1994]) 49 Table 2.6 Measured Load-Displacement and Load-Strain Parameters 50 Table 2.7 Energy and Velocity Required for Delamination and Perforation of 3 x 5" Plates, mprojectile = 0.311 kg 50 Table 3.1 Material Parameters Used as Input for Indentation Model 92 Table 3.2 Loading Conditions for Kirchoff and Whitney-Pagano Solutions 92 v L is t o f Figures Figure 1.1 Standard velocity classifications for foreign object impacts. 9 Figure 1.2 Failure modes in laminated composites resulting from impact at various velocities (based on Hoskin and Baker [1986]). 9 Figure 1.3 Global / local modelling concept. 10 Figure 1.4 Static and impact load-displacement curves for 6.15 mm IM7/8551 -7. 10 Figure 1.5 Static (top) and impact (bottom) penetration (velocity = 29.6 m/s) of 5.03 mm thick T300H/F593 C F R P plates. 11 Figure 1.6 Ballistic/high velocity penetration of IM7/8551-7, 4.2 g blunt nosed projectile at 261 m/s. 12 Figure 2.1 Comparison of energies required for static or dynamic perforation of a C F R P laminate using different indenter nose shapes (from Delfosse and Poursartip [1995]). 51 Figure 2.2 Hardened steel punch dimensions (all dimensions in mm). 51 Figure 2.3 Experimental setup for 3 x 5" static penetration test. 52 Figure 2.4 Typical load-displacement curves for brittle and tough C F R P systems in the static penetration test. 53 Figure 2.5 Idealized load-displacement curve for brittle matrix C F R P static penetration test. 53 Figure 2.6 Idealized load-displacement curve for tough matrix C F R P static penetration test. 54 Figure 2.7 Load-displacement curves for T300H/F593, 3 x 5" opening static penetration tests for 4 thicknesses. 54 Figure 2.8 Detailed idealized load-displacement curve for brittle matrix C F R P static penetration test. 55 Figure 2.9 Effect of removing local indentation from static penetration test. 55 Figure 2.10 Cross-sectional micrograph of a delaminated 5.03 mm thick T300H/F593. This shows the measured edge of the projected delamination radius which coincided with the crack tip viewed in the cross-section (not seen). 56 Figure 2.11 A schematic description of two basic impact damage growth mechanisms of laminated composites (from Choi , W u and Chang [1991]). 56 Figure 2.12 Frames from video recording of static penetration test - T300H/F593 system. The first frame shows the unloaded specimen. The second and vi third frames, 33 ms apart, show the abrupt change in curvature at the onset of delamination. 57 Figure 2.13 Pulse echo ultrasound (PEUS) image of a statically loaded specimen just after delamination load drop. T300H/F593 system, 5.03 mm thick plate. The image represents aplate area of 101.8 x 101.8 mm. The darkness of the image represents the intensity of the signal. 58 Figure 2.14 Strain gauge placement for the static penetration test. 58 Figure 2.15 Load/unload cycle for a 5.03 mm thick T300H/F593 with strain gauge readings for top and bottom surfaces. 59 Figure 2.16 Load/unload cycle for a 5.03 mm thick T300H/F593 with strain gauge readings for top surface. 59 Figure 2.17 Displacement profile measured across the short length of a 3.35 mm statically loaded T300H/F593 plate. 60 Figure 2.18 Peak forces encountered before the onset of delamination for 3 different opening sizes. 60 Figure 2.19 Assumed delamination pattern used to calculate total area of delamination. 61 Figure 2.20 A n example of delaminations and fibre breakage in a C F R P laminate with a stacking sequence of [45/0/-45/90] n s (different from the present system) caused by static or dynamic out-of-plane loading (from Delfosse [1994]). 61 Figure 2.21 Total calculated area of delamination for T300H/F593 system using two different assumed delamination patterns. 62 Figure 2.22 Unload/reload paths for 5.03 mm thick T300H/F593 system. 62 Figure 2.23 Post delamination load-strain curves for determining slope value. 63 Figure 2.24 Measured slopes of the load-strain curve as a function of plate thickness, linearly extrapolated to other thicknesses. 63 Figure 2.25 Strain gauge readings on the bottom surface in various orientations during static penetration test. 64 Figure 2.26 Idealized load-displacement curve showing energy lost during delamination. 64 Figure 2.27 Energy lost at delamination onset as a function of thickness during static penetration test. 65 Figure 2.28 Calculated energy lost per unit area of delamination for statically loaded specimens. 65 vii Figure 2.29 P E U S Images of delamination growth in the tough IM7/8551-7 system corresponding to different unload/load cycles seen in the graph (taken from Delfosse [1994]). 66 Figure 2.30 Recorded failure strains during static loading tests. 67 Figure 2.31 Strain to failure and energy lost due to fibre breakage for 6.66 mm thick specimen. s 67 Figure 2.32 Cross sectional micrograph of a 5.33 mm T300H/F593 specimen after final failure by plugging. 68 Figure 2.33 Final plugging failure load as a function of thickness. 68 Figure 2.34 Idealized post failure/friction model showing various stages of plug pushout. 69 Figure 2.35 Peak frictional forces of the plug and the penetrator against the inside of the cut-out hole. 69 Figure 2.36 Peak frictional shear stesses. 70 Figure 2.37 Experimental undamaged and damaged plate stiffnesses for two back plate opening sizes. 70 Figure 2.38 Experimental load-strain slopes for two back plate opening sizes and linear extrapolation. 71 Figure 2.39 Projected delamination diameter as a function of thickness for two back plate opening sizes. 71 Figure 2.40 Load-displacement curves for static penetration tests over a 25.4 mm (1") opening for different thicknesses (from Delfosse and Poursartip [1995]). 72 Figure 2.41 Load-displacement curves for static penetration tests over a 25.4 mm (1") opening for different thicknesses. Forces and displacements are normalized with respect to a plate thickness of 6.15 mm (from Delfosse and Poursartip [1995]). 72 Figure 2.42 Effect of back plate opening size on the load-displacement behaviour of 5.03 mm thick T300H/F593 specimen. 73 Figure 2.43 Gas-gun schematic and cross-section of projectile/load cell. 74 Figure 2.44 Static and impact load-displacement curves for a 6.15 mm thick IM7/8551-7 plate. 75 Figure 2.45 Static and impact load-displacement curves for a 5.03 mm thick T300H/F593 plate. 75 Figure 2.46 P E U S images of a static (left) and an impact (right) penetrated specimen. The velocity of the impacted specimen was 29.6 m/s, well viii above perforation velocity. Each image represents a 101.6 x 101.6 mm section of plate. 76 Figure 2.47 Comparison of projected delamination diameters for static and impact loaded T300H/F593 specimens. 76 Figure 2.48 Cross sectional micrograph of an impact loaded and delaminated 5.03 mm thick T300H/F593 specimen. 77 Figure 2.49 Cross sectional micrograph of an impact penetrated 5.03 mm thick T300H/F593 specimen. 77 Figure 3.1 Coordinate system for beam deflection calculation (from Howard et al. [1993]). I l l Figure 3.2 Schematic load/displacement plot (from Howard et al. [1993]). 111 Figure 3.3 Delamination propagation model for Davies and Robinson model (from Davies and Robinson [1992]). 111 Figure 3.4 Circular plate model for delamination analysis (from Ho-Cheng and Dharan [1990]). 112 Figure 3.5 Three stages describing the compression, delamination and buckling of a thin orthotropic layer attached to a thick isotropic plate and subjected to axial compression in the y direction (from Chai and Babcock [1985]). 112 Figure 3.6 Rule of mixtures analysis of stiffness loss (from O'Brien [1982]). 113 Figure 3.7 Delamination size and stiffness data accumulation during quasi-static loading (taken from O'Br ien [1982]). 113 Figure 3.8 Detailed idealized load-displacement curve for a brittle C F R P laminate. 114 Figure 3.9 Model schematic for a damaged circular isotropic plate. 114 Figure 3.10 Indentation by a rigid flat-ended cylindrical punch (taken from Hi l l s et al. [1993]). 115 Figure 3.11 Static indentation curve. 115 Figure 3.12 Comparison of static indentation curve with an empirical power law and the analytical expression. 116 Figure 3.13 Pressure distribution profile predicted by indentation model for a rigid flat-ended cylindrical punch. 116 Figure 3.14 Calculated stiffness using elastic plate bending equations compared with experimental results. 117 Figure 3.15 Calculated load-strain ratios using elastic plate bending equations compared with experimental results. 117 Figure 3.16 Theoretical shear stress equilibrium at the edge of the contact area (from Sjoblom [1987]). 118 IX Figure 3.17 Peak force at delamination onset for statically loaded T300H/F593 plates as a function of plate thickness. 118 Figure 3.18 Critical shear force (as defined by Jackson and Poe [1993]) as a function of plate thickness for T300H/F593 system. 119 Figure 3.19 Predicted critical loads at the onset of delamination compared with experimental results. 119 Figure 3.20 Possible displacement profile functions for undamaged plate deflection model. 120 Figure 3.21 Possible displacement profile functions for hybrid plate deflection model. 120 Figure 3.22 Delamination size prediction compared with experimental values from C-scans. 121 Figure 3.23 Calculated undamaged and fully delaminated bending rigidities. 121 Figure 3.24 Damaged and undamaged bending rigidities. 122 Figure 3.25 Delamination size prediction for two back plate opening sizes compared with experimental values from C-scans. 122 Figure 3.26 Predicted and experimental load-strain slope values. 123 Figure 3.27 Predicted plate displacement profile compared with witness plate profile. 123 Figure 3.28 Predicted and experimental peak plugging force. 124 Figure 3.29 Idealized friction model during stage III loading. 124 Figure 4.1 Kevlar and S-2 glass force-displacement curves for quasi-static punch test over 25.4 mm diameter opening. 130 Figure B . l Nomenclature for exact and approximate solutions 140 Figure B.2 Nomenclature for the integrated ring-loaded exact solution. 141 Acknowledgments I would first like to acknowledge my supervisor, Dr. Reza Vaziri, for his wise and patient guidance. His example of analytical thinking and awareness of detail have been inwrought on my mind. His constant encouragement and determined hard work have been cast into my character. I would like to thank Drs. Daniel Delfosse and Anoush Poursartip who have brought engineering principles and experimental results to life. The able hands and practical eye of Mr. Roger Bennett and Mr. Serge Milaire bridged the gap between vague ideas and significant results. Many thanks to all the members of the Composites Group who have made the time and effort to become my friends. Your moral support and practical advice are greatly appreciated. The research funds provided by the Canadian Natural Sciences and Engineering Research Council (NSERC), the materials provided by Dr. H. Razi of the Boeing Company, Seattle, and the test facilities at Integrated Technologies, Inc. in Bothel, Washington, USA, are gratefully acknowledged. The support of the Canadian Department of National Defence is also acknowledged. And to my family and a patient God who have moulded me from the potter's clay. xi Chapter One Introduction 1.1 Background In the rapidly expanding field of composite materials, a greater knowledge of impact damage behaviour is required. While a satisfactory understanding of the behaviour of metals under impact conditions has been developed, little of this knowledge is applicable to fibre reinforced plastics. Anisotropic and heterogeneous conditions, as well as complex local damage states, prohibit the use of established metal based models. The requirement for a precise understanding of this behaviour is increased by the crucial areas of application in which fibre reinforced composites (FRC's ) are found in today's industry. The replacement of traditional metals in aerospace and military applications by F R C ' s has been pervasive. The high stiffness-to-weight or strength-to-weight ratio of carbon and glass fibre composite structures has made them attractive as next generation production materials. Understanding the influence of impact damage mechanisms on the response of these composite structures is important for their effective design against foreign object impact. This is further accentuated when considering the weak through-thickness shear strength of the composite material in relation to its in-plane properties. 1 The experimental observations and analysis of quasi-static loading behaviour presented in this thesis provide a simple means to predict impact damage on F R C structures. This ability comes from the identification of the similar damage mechanisms and plate behaviour occurring under both quasi-static and impact conditions. The approach of using quasi-static tests to predict impact behviour is the basis for a forthcoming A S T M standard on impact damage resistance. The complex nature of F R C s results in a number of possible damage modes during impact. These damage mechanisms include indentation, matrix cracking, delamination, fibre shearing (cutting) and fibre tensile failure (see Abrate [1991] and [1994]). The degree to which each of these modes is observed during impact is a function of projectile velocity and the geometries of the target and projectile. The first step in model development, therefore, is to identify general categories of impact behaviour according to the velocity, geometry and material properties of the target and projectile. It should be noted that many studies simply turn to the energy of the projectile to categorize impact behaviour. However, this approach does not consider such factors as the rate effects of the material constituents or the mass or shape of the projectile. The absolute projectile velocity wi l l determine the behaviour of the local structural constituents in the target under impact. The global plate deflection behaviour, on the other hand, is determined by the projectile velocity relative to the reflecting impact pressure waves in the target. Therefore the dimensions of the target therefore also become important. The recognition of these factors result in a wide range of load-deflection behaviours. 2 These target behaviours have been categorized below in terms of general velocity regimes. This is followed by a brief discussion of the influence which the material and geometric conditions have over target response. 1.1.1 Impact Velocity Regimes The dynamic characteristics of the impact event can determine a variety of target responses. Referring to the diagram in Figure 1.1, low and intermediate velocity impacts, are usually considered to be less than 100 m/s. They involve global deflection of the structure as flexural waves travel to the boundaries and back many times during the impact event. These situations may occur in maintenance operations when a tool is dropped on a structural member, or during operating conditions, such as an airplane being struck by runway debris during takeoff. The damage in the case of a non-penetrating or partially penetrating event wi l l be made up of delaminations, matrix cracks and some tensile fibre breakage. Examples of this are shown in the bottom two diagrams of Figure 1.2 for projectiles of a constant mass. Damage from a high velocity impact event w i l l be much more localized and involve a smaller delamination area as the projectile w i l l shear through the structure. This can be seen in the top diagram of Figure 1.2. This type of impact would be a concern for in-flight conditions for aircraft, engine casings and ballistic protection of armoured vehicles. Hypervelocity impacts (greater than 1 km/s) occur regularly in space applications. 1.1.2 Projectile Geometry Delfosse [1994] showed that the geometry of the projectile tip has considerable influence over the impact event. It was found that a conical tipped indenter encountered least 3 resistance in pliable fibre composites such as Kevlar™ and Spectra™. Delfosse claimed that the conical tip tended to slice through the material in this case. In the stiffer carbon and glass fibre composites, however, the cone shape met with much more resistance since the conical shape needed to push the stiff fibres laterally in order to proceed. The flat nosed cylindrical indenter was found to be the greatest threat to these structural carbon and glass fibre composites. The sharp corners would shear through the material, cutting out a plug in front of the flat surface. In order to provide a conservative damage tolerance analysis for a structural composite, it is therefore necessary to look at the worse-case scenario of a flat nosed cylindrical projectile. Pierson [1994] established an effective model to predict the penetration behaviour of a conical tipped steel indenter through a laminated C F R P plate. Because the impact of a conical indenter tends to stay very localized due to the cutting and ploughing action of the cone, the model successfully incorporated an analysis which separates local and global effects. The separation of local and global effects is possible since the damage caused by the impact of foreign objects wi l l , in many cases, be fairly localized in relation to the overall structure. B y separating the local penetration event from the overall dynamic plate bending, the entire penetration process can be modeled as a superposition. Figure 1.3 shows the concept used by Pierson whereby a local damage model would be used to calculate the resisting force experienced by a projectile of a given geometry and velocity. This force, in turn, can be used 4 to calculate the displacement of the plate in a given time step and the progression of damage within the local zone. This method was also suggested by Sjoblom et al. [1988]. To model the penetration of a blunt nosed cylindrical indenter, Pierson used a quasi-static punch test to record the load-displacement behaviour of the penetrator as it passes through the laminate. He again used the local-global model, using this load-displacement curve of the quasi-static punch test for the local damage model portion of the plate. This approach is cumbersome, as it requires a quasi-static punch experiment for every combination of plate dimension, lay-up, thickness, impactor size, nose shape and related material properties for which the prediction is desired. It is further complicated by the fact that the delaminations occurring during penetration are of the order of the plate planar dimensions. This creates a significant reduction in overall plate stiffness not accounted for in the global plate model. 1.1.3 Material Properties Other factors affecting target behaviour involve the material properties. One of the most consequential is the rate sensitive nature of both glass fibres (see Harding and Welsh [1983]) and polymeric fibres, such as Spectra™ and aramid. The strain-rate sensitivity makes it more difficult to generalize target behaviour under different impact velocities. Carbon/graphite fibres, on the other hand, are rate insensitive over a wide range of strain rates (also Harding and Welsh [1983]) and are therefore much more attractive in terms of modelling. 5 1.2 Present Work Taking the above considerations into account, a general approach to model development becomes apparent. For relatively strain-rate insensitive materials, such as carbon fibre-reinforced plastics (CFRP's), it is possible to carry out quasi-static punch tests to establish the local load-displacement relationship. In turn, this can be used to describe the dynamic penetration response. This is possible because of the many similarities which exist between the out-of-plane static and dynamic behaviour of CFRP's . To illustrate this, Figure 1.4, taken from a previous study, shows the load-displacement curves for quasi-static and dynamic penetration of IM7/8551-7 C F R P by blunt-nosed cylindrical indenters. The major difference which can be noted between these two curves is the initial bending stiffness, represented by the initial slope of the curve. However, more important are the key similarities in the penetration events. Figure 1.4 shows that the peak forces attained during penetration are virtually equivalent. Figure 1.5 compares the cross-sectional views of these penetrated specimens and reveals that the local damage mechanisms of the two targets are also very similar. Both reveal delamination away from the impact site, transverse fibre shearing due to plug formation, and tensile fibre failure on the rear surface. Figure 1.6 shows that a ballistic impact event also exhibits similar damage behaviour. It should be noted that the focus of this study is the resulting impact damage rather than residual properties of the structure. Traditionally the use of the actual impact damage as a design criteria is known as damage resistance. The damage resistance criteria considers the peak forces occuring during the penetration event as a measure of target performance and is the current basis in forthcoming A S T M standards. On the other hand, the study of the 6 residual properties of the impacted structure is known as damage tolerance. Damage tolerance uses the impact energy of the event when comparing the ability of a structure to perform after impact. This is seen in the compression after impact (CAI) standards currently available (BSS 7260 or N A S A Reference Publication 1142). The degree of detail and the end-use application environment of any given model w i l l dictate the type of model infrastructure required. Previous studies often focused on either an analytical/numerical or an empirical approach to modelling the impact response of composite materials. Empirical models require a large number of tests to cover the various combinations of laminate thickness, punch diameter and plate boundary conditions. Without an understanding of how these variables affect the response, this approach becomes little more than a curve fitting exercise. A t the other end of the spectrum, analytical/numerical approaches can often be too cost or labour intensive i f only a cursory analysis is required. Numerical models, in particular, require an accurate description of the constitutive behaviour. This entails formulation and extensive material characterization tests. The present study attempts to f i l l the requirement for an effective local penetration model of a blunt-nosed cylindrical indenter. Using experimental methods to obtain detailed information on the penetration behaviour, the foundation for a rigorous analytical model is developed. This model would establish a working balance between the two spectrums of engineering modelling. The experimental program is explained in detail in Chapter 2. The background to the quasi-static penetration tests is discussed. Experimental results are given which describe the 7 various damage mechanisms occurring during penetration. These results are presented as a function of plate thickness and compared to low and intermediate velocity tests. The load-displacement curve is divided into three main stages of penetration characterized by their particular damage state. Chapter 3 presents the development of the analytical model and compares the model predictions with the experimental results. The three stages of the load-displacement relation identified in Chapter 2 are used as a basis for model formulation. Each stage is represented by either a simple closed-form solution or a semi-empirical formula. Model accuracy is deduced through parametric evaluation and comparison to experimental results. Chapter 4 assesses the present state of the model development and identifies future areas of concentration in both analysis and experimentation. The limits of the model developed herein are discussed, and possible applications are suggested. 8 1000 m/s 4-100 m/s 10 m/s 1 m/s 1x10-3 m/s 4-Hypervelocity High Velocity / Ballistic Intermediate Velocity Low Velocity Quasi-static Figure 1.1 Standard velocity classifications for foreign object impacts. 4 Through penetration — small damage zone Some delamination —' Some loose fibre ends (a) High velocity impact damage Local fibre/resin crushing Some delamination —' x —Fibre fracture on back face (b) Intermediate velocity impact damage j , — Impact zone - little if any damage Extensive delamination in conical region (c) Low velocity impact damage Figure 1.2 Failure modes in laminated composites resulting from impact at various velocities (based on Hoskin and Baker [1986]). 9 II t F(t) Figure 1.3 Global / local modelling concept. TJ ro O 25000 20000 15000 10000 5000 IM7/8551-7 6.15 mm thick plate Static Test • Impact Test - 26.9 m/s -+-6 10 Displacement [mm] Figure 1.4 Static and impact load-displacement curves for 6.15 mm IM7/8551-7. 10 Figure 1.5 Static (top) and impact (bottom) penetration (velocity = 29.6 m/s) of 5.03 mm thick T300H/F593 CFRP plates. 11 Figure 1.6 Ballistic/high velocity penetration of IM7/8551-7, 4.2 g blunt nosed projectile at 261 m/s. 12 Chapter Two Experimental Results 2.1 Background The use of quasi-static punch tests to determine both material characteristics and damage mechanisms related to impact behaviour may seem questionable at first. However, studies have shown that the information garnered from these tests can be quite useful both qualitatively and quantitatively. Not only is it easier to observe the material behaviour and damage mechanisms during loading and interrupted evaluation, but values such as peak load and energy can often be used to predict impact behaviour. Lee and Sun [1993] carried out a series of quasi-static punch tests on a Hercules AS4/3501-6 graphite/epoxy quasi-isotropic laminate using a blunt-nosed cylindrical indenter. They found that the initial progression of damage during penetration consisted of matrix cracks caused by shear stress and tensile bending stress. Following this was delamination and eventually the formation of a plug as the indenter sheared through the material. Friction forces then had to be overcome as the penetrator pushed out the remaining plug. Using di-iodo-butane (DIB) enhanced X-radiography, they were able to determine the location and magnitude of interior damage. 13 Upon comparing this damage with that sustained in subsequent dynamic tests of the same geometry, Sun and Potti [1993] found that the dynamic failure modes were very similar to those of static cases. The projectile velocities ranged from 20 to 150 m/s. They noted that the dynamic case showed the same formation and pushout of a plug along with some fibre breakage on the back side of the specimen. While not having the advantage of an instrumented impact projectile to obtain the dynamic force-displacement behaviour, they were able to compare perforation energies. These showed a very close correlation for the thinner laminates (2 mm and 4.1 mm) and diverged only slightly for the thicker plates (6.1 mm and 8.1 mm). Delfosse et al. [1995] compared impacts on C F R P plates by 25.4 mm diameter hemispherical tipped impactors that had the same incident energy, but with widely varying velocities, i.e. different masses. He also compared the impact behaviour with quasi-static punch tests. He found that while the damage mechanisms were similar in nature from one velocity range to the next, the magnitude of each mechanism differed. For example, he found that at higher velocity there tends to be less global plate deformation, and thus less delamination, but increasing fibre breakage. Jenq et al. [1994] performed a series of quasi-static punch tests on plain woven glass/epoxy specimens using a hemispherical tipped indenter. These tests were done in order to investigate the progressive damage modes of the targets and to obtain the punch load-displacement relation. They observed the same progression of damage as Lee and Sun [1993], i.e. elastic plate deflection, matrix cracking followed by an abrupt delamination resulting in a significant loss of load carrying capacity. N o plugging behaviour was seen at 14 this point. This was most likely due to the round shape of the indenter. They did, however, find pronounced fibre breakage and pushing out of the fibres. Six high speed impact tests were then conducted with a pneumatic gun at incident velocities ranging from 140 to 200 m/s. These tests were done in order to compare the damage mechanisms and energy loss with the static tests. Upon inspection it was noted that the major damage pattern in the impacted specimens was similar to that in the quasi-statically penetrated specimens. However, the area of the rhombus-shaped delamination in the high speed impacted specimens was determined to be about nine times greater than that of the quasi-statically penetrated specimens. Using the target's static material properties it was further found that the predicted ballistic limit was about 24% in error. This was due to the rate sensitive nature of the glass/epoxy system. Once the target's elastic moduli were increased by an arbitrary value of two times the static value in the simulation, a good agreement was found. Zhu et al. [1992-2] also found discrepancies in predicted dynamic results due to rate sensitivity when they compared quasi-static and dynamic penetration behaviour of woven Kevlar/polyester laminates. They used plates of varying thickness and cylindro-conical impactors for their study. Although the observed damage mechanisms progressed in the same order, they found that the material parameters were of little use unless some correction could be made for the rate sensitive nature of the material. For example, instead of using the static compressive yield stress in their analysis, they performed dynamic compression tests using Hopkinson-bar experiments. They further noted that as the speed of the impact event increased, the amount of global plate deflection became less important. 15 In order to carry out a rigorous experimental study which gives the most conservative results, it is necessary to examine the various test variables with respect to their damage threat. Blunt nosed projectiles represent the most dangerous threat to C F R P laminates and are therefore the appropriate choice for a worst-case analysis. Delfosse and Poursartip [1995] compared the energy required for perforation over a range of velocities for conical, hemispherical and blunt nosed projectiles, and found that the blunt nosed, projectile required the lowest energy for perforation. Figure 2.1 shows that this discrepancy actually diminishes as the impact velocity increases. Lee and Sun [1993] and Sun and Potti [1993] carried out static punch tests using a blunt nosed, 14.5 mm diameter punch on 10 cm by 10 cm quasi-isotropic AS4/3501-6 graphite/epoxy laminates rigidly fixed around the edges. The load-displacement curve of the penetration process revealed two large drops in load. The load drops were identified as delamination failure due to the shear force exerted by the punch and final failure by a plugging/cutting mechanism. These damage mechanisms were also characteristic of the present study. Cristescu et al. [1975] studied delamination growth on 0-90 crossply fibreglass roving epoxy plates impacted by a blunt nosed cylindrical impactor at intermediate velocity. Their examination revealed an entirely different source of delamination, namely the cylindrical plug delaminating the layer below it as it cut through the previous layer. It could be hypothesized that this mechanism occurred differently in the glass-fibre composites because of the lower overall plate stiffness and the ductile nature of the fibres in comparison to the graphite/epoxy composites. Delamination continued on that level until the plug was able to 16 completely cut through the deflecting lamina. Each successive layer would have more time to delaminate to a larger degree as the velocity of the projectile gradually reduced. They noted that the bottom laminae would fail in tension because the cutting action of the projectile had been severely reduced by the plug formed in front of it. 2.2 Static Penetration Test To form an experimental database for this investigation, a large number of composite plates were tested under out-of-plane static loading conditions to simulate foreign object impact events. In particular, a quasi-static penetration or punch test was used to gain an overall understanding of the penetration behaviour. The quasi-static punch tests were displacement controlled tests, using a hardened steel cylindrical punch of 7.6 mm diameter (Figure 2.2). These tests were performed on an Instron universal testing machine. The load acting on the specimen was recorded with a 20 000 lbs (89 000 N) load cell and the displacement of the crosshead was measured with a linear voltage displacement transducer ( L V D T ) resting on the pressure plate (see Figure 2.3). A C F R P material, T300H/F593, with a relatively brittle matrix was the focus for the test program. The tensile failure strain of the T300H fibres, as measured in a unidirectional lamina tensile test, was 1.6% (±0.2%). The laminates used had a lay-up of [-45/90/45/0]n s with n equal to 2, 3, 4 and 5 corresponding to plate thicknesses of 3.35 mm, 5.03 mm, 6.66 mm and 8.30 mm, respectively. A tough matrix system material, IM7/8551-7, with a lay-up of [-45/90/45/0/0]4 s and a nominal thickness of 6.15 mm was tested in a previous study under the same conditions. 17 The tests were initially performed on plates that were simply supported over a 25.4 mm diameter circular opening. This configuration allowed for the observation of the local load-displacement behaviour with minimum contribution from bending effects. The main drawback of this test was that the delaminations formed during the test extended to the boundaries. This no longer simulated a "real life" scenario in which imbedded delaminations are constrained by the surrounding undamaged material. Therefore further tests were carried out on rectangular plates with outer dimensions of 101.6 mm by 152.4 mm (4" x 6") and placed over a rectangular opening of 76.2 mm by 127.0 mm (3" x 5"), as shown in Figure 2.3. This configuration was chosen to conform with Boeing's fixture conditions for standard impact, and compression after impact tests (Boeing Standard Specification B S S 7260). This test became the "test standard" upon which both modelling and further tests would focus. Summaries of all the tests performed on these materials can be seen in Table 2.1 and Table 2.2. Further static penetration tests using 139.7 mm (5.5") diameter circular plates over a 114.3 mm (4.5") circular opening were deemed necessary in order to validate assumptions made in the development of the analytical model. It also assured that the relatively short width of the rectangular specimen was not causing any boundary effects on the delamination growth. The T300H/F593 system was chosen for the "simplicity" of its response to impact. Carbon fibre composites have a relatively low strain-rate sensitivity, making correlation with the dynamic impact event straightforward. The choice of a brittle system over a tough system allowed for the decoupling of the damage mechanisms. Figure 2.4 shows the difference between the load-displacement behaviour of a relatively brittle matrix system (T300H/F593) 18 and a tough matrix system (IM7/8551-7). Figure 2.5 and Figure 2.6 give an idealized schematic representation of the same systems, indicating the major damage mechanisms. A s can be seen, the major difference between the systems occurs at the onset of delamination, indicated by a large drop in load in the brittle case and only a change of slope for the tough system. This actually represents the different delamination growth mechanisms for the two systems. In the brittle case, delamination occurs instantaneously and grows very little during subsequent loading, whereas the tough system experiences delamination growth. This decoupling of damage mechanisms in the brittle material and the resulting analytical simplification that becomes apparant later led to a decision to carry out first round testing and analysis on the brittle matrix system. The simply supported boundary condition was also chosen for simplicity, as it is easily repeatable for testing. A built-in edge condition is difficult to achieve experimentally, and much variation can exist from test to test when this is attempted. A summary of the measured load-displacement curves for the T300H/F593 laminated C F R P composite plates for the rectangular testing conditions is shown in Figure 2.7. Obvious trends and consistencies can be related to the idealized schematic in Figure 2.5. Stage I is characterized by the elastic loading stage followed by a sharp drop in load. A lower stiffness and a second sharp drop, indicating complete failure, characterize Stage II. In Stage III the penetrator pushes out the remaining material in front of the penetrator. These stages are described in detail below using information gathered from interrupted static tests followed by post-mortem characterization tests. The post-mortem tests included deplies, C-scans, pulse-echo ultrasonics and metallographic investigations. 19 2.2.1 Elastic Bending and Indentation In the first part of the detailed idealized curve of Figure 2.8, segment O to A represents a combination of the elastic bending of the plate and the elastic/plastic indentation at the surface under the indenter. In order to separate the bending response of the plate from the local indentation, a separate "static indentation" test was performed. This test was done by placing a small (25 mm x 25 mm) coupon of the 6.66 mm thick C F R P on a flat steel plate in the Instron. The specimen was then loaded using the same method as the static penetration tests. It was assumed that the indentation behaviour of the 6.66 mm coupon would be representative of the range of thicknesses. Load-displacement data from the static indentation test was then subtracted from the static penetration tests resulting in a pure plate bending curve (see Figure 2.9). This curve was compared with current plate bending theories. 2.2.2 Matrix Cracking The first type of damage encountered during the static penetration event is matrix cracking. Just prior to the first drop in load at point A (Figure 2.8), small "ticks" were heard from the specimen with no change in the load-displacement behaviour. These matrix cracks pass transversely through the various laminae and are the starting points for the delaminations occurring at point A . Specimens loaded to delamination point A and then unloaded, were cross-sectioned, polished, and viewed under a microscope. A s can be seen in Figure 2.10, a shear crack emanates from the contact periphery of the flat indenter. This is obviously the point where 20 stresses are highest due to the singularity of stresses at the contact periphery. This crack passes from the point of contact at a slight angle through the entire first sublaminate (i.e. one grouping of a [-45/90/45/0] lay-up). Subsequent matrix cracks can be seen passing through single laminae at sharper angles than the initial shear crack. This inclined angle indicates that these matrix cracks are induced by shear stresses rather than bending stresses which induce a vertical matrix crack. The majority of researchers identify two different types of matrix cracks occurring during static or dynamic impact. Virtually all of them conclude that these are the initiation mechanisms of delamination, i.e. it is not possible to have delaminations in the composite material without first having the presence of matrix cracks. Jih and Sun [1993] and Choi , Downs and Chang [1991], among others, define the two types of matrix cracks as transverse shear and bending cracks. Transverse shear cracks develop slightly away from the impact site at an angle of approximately 45 degrees. This inclination is a result of a combination of interlaminar shear stress and transverse normal stress (see Figure 2.11) imparted by the impacting projectile. Appearing on the bottom layers of the laminate are vertical matrix cracks resulting from the high in-plane tensile stresses due to the bending of the plate. The matrix cracks themselves start somewhere inside the lamina and extend to the laminae interfaces as the stresses increase. Not until they reach the laminae interfaces do the delaminations begin, at which point Choi , Downs and Chang refer to them as "critical matrix cracks". The load (or energy) at which the matrix cracks reach their critical size, thus determining the onset of delamination and stiffness reduction, is of utmost interest. Therefore it is necessary 21 to understand how the material and geometric parameters affect this value. The importance of this area of research is just now being realized and studied. It has been identified as a very complex problem due to the many variations possible in a composite material. Obviously the lay-up sequence and orientation of the fibres affect the interlaminar shear strength as well as the stress distribution. Both Jih and Sun and Choi , Downs and Chang found that residual stresses from the curing of the composite during manufacture can greatly affect the critical load or energy of matrix crack formation.. But this in turn is also a function of the lay-up sequence and the thermal coefficient mismatch. Choi , Downs and Chang found that even small changes in the curing cycle can greatly affect the residual thermal stresses and thus the onset of matrix cracking. A s can be seen in Figure 2.11, a further condition which may appear are the presence of "micro-cracks". These are an offshoot of the delaminations and extend vertically from the delamination surfaces. They were only discovered by Choi , Downs and Chang after microscopic investigation with X-ray cameras. Since they are much smaller in size relative to other damage and do not affect the stiffness or strength of the overall laminate, they are usually disregarded in any analysis. 2.2.3 Delamination Onset The drop in load which occurs at point A (Figure 2.8) is a result of a sudden delamination. During the loading of the specimen, this appears as the sudden formation of a dimple, or change in curvature, of the loaded plate accompanied by a loud, audible "crack". Two frames from a video camera recording of the static penetration test, taken 33 ms apart, show the 22 sudden nature of this event in Figure 2.12. The change in curvature was measured using both strain gauges and copper witness plates. The sudden formation and presence of a delamination was proven from C-scans and pulse-echo ultrasonic scans (PEUS - performed by Integrated Technologies, Inc. of Bothel, Washington) performed on specimens loaded and unloaded just before and just after the load drop. N o damage was observed just prior to the load drop while sizable damage was viewed after, as is represented by the dark circular region of Figure 2.13. 2.2.3.1 Strain gauges To get a more complete understanding of the time dependent plate displacement behaviour during the loading process, strain gauge measurements were recorded at different locations on the plate surface. Experimental techniques for observing plate behaviour during impact of composites have often included the use of strain gauges. This technique has been used in the past to either gather data about the plate deformation profile or measure the magnitude and location of stress waves. Prasad et al. [1994] placed gauges on the distal side of their AS4/3502 graphite/epoxy and IM7/5260 graphite/bismaleimide material systems. The gauges were placed both directly under the impact site and at arbitrary locations in order to test their analytical prediction of plate transient response. They found that for this non-penetrating event, the measured strain magnitudes were very sensitive to the precise location of impact in relation to the gauges. Davies and Zhang [1995] used gauges to record the onset of matrix splitting on the backface. Postulating that matrix splitting on the distal side was responsible for an initial drop in 23 stiffness, gauges were placed at right angles to the fibres. They then recorded the time and strain level at which splitting took place. K i m et al. [1995] used arrays of strain gauges to determine delamination size in Fiberite T300/976 plates of various lay-ups during in-plane compression loading tests. The resulting magnitudes and directions of strain measurements indicated the size of the delaminations and their location, both in-plane and out-of-plane. W u et al. [1994] proposed an inverse analytical method for determining the force-deflection behaviour of an impacted T300/976 graphite/epoxy using strain gauges distributed over the surface of the laminate to measure the resulting stress waves. Calder and Goldsmith [1971] used a combination of strain gauges and displacement gauges to measure the displacement profile of an impacted 2024-0 aluminium plate. In the present study, four different strain gauge configurations with various dimensions were used for the measurements. A summary of the strain gauge tests is provided in Table 2.3. A lower than normal input voltage of approximately 1.5 V was necessary due to the small planar dimensions of the gauge and the poor heat dissipating characteristics of the C F R P . Each of the gauges completed a 1/4 wheatstone bridge circuit, and data was collected and reduced on a P C . The strain gauges were placed so that the maximum strain directly under the impactor and the growth nature of the delamination could be determined. Three different strain gauges placed as shown in Figure 2.14 resulted in the strain-time curves of Figure 2.15 and Figure 2.16 (super-imposed on a load-time curve). The test consisted of three load-unload cycles on a 5.03 mm thick plate. Figure 2.15 shows the strain 24 data for the back side gauge directly under the impactor as well as the top gauge closest to the impactor (both in the fibre direction). A s can be seen at the onset of delamination, the strain gauge on the back surface of the plate shows a jump in tensile strain, indicating an increase in curvature. The top mounted gauge (top gauge #1) makes a complete reversal of strain from compression to tension, indicating a change in plate curvature from concave to convex (see inset, Figure 2.8). Unfortunately the back side gauge failed on the third cycle and no further data was available. The top gauge #2 (Figure 2.16) also indicates the reversal in strain in the second cycle, although the maximum strain values are lower. During the third cycle top gauge #1 sees a drop in tensile strain as the soft zone apparently grows slightly and the inflection point moves further outward. This is also witnessed by top gauge #2 as the tensile strain increases at this point. A summary of all tests performed with strain gauges and their locations relative to the centre point is recorded in Table 2.3. 2.2.3.2 Witness Plates To obtain a more complete picture of the plate displacement profile after the onset of delamination, it was decided to use a witness plate on the back face of the C F R P plate during loading. A 0.15 mm thick Teflon sheet was placed between the specimen plate and the 0.40 mm thick copper witness plate. Initial trials were conducted to ensure that the addition of the witness plate added no measurable stiffness. The specimen plate was loaded to a point just after delamination and then unloaded. The witness plate was then placed on the resting plate of a drill press with an x-y coordinate table and measured at increments using a dial gauge mounted in the chuck. This provided coordinates in three dimensions. Maximum 25 displacements of the witness plate at the centre point and L V D T were compared and it was found that a small degree of elastic springback had occurred in the copper witness plate. A typical displacement profile is shown in Figure 2.17 for the 3.35 mm thick C F R P . The maximum displacement shown here is 2.75 mm whereas the L V D T measured a maximum displacement of 3.44 mm. The effect which this elastic springback has on the overall representation of the plate displacement profile is unknown. The experimental plate profile measured from the witness plate is compared with the predicted model in Chapter 3. 2.2.3.3 Peak Loads Figure 2.18 shows the peak load at delamination for three opening sizes as a function of thickness. It is important to note that the size of the opening or areal plate dimension has an insignificant effect on the peak load at delamination. This was also found by Davies and Robinson [1992]. It can be postulated then, that the stress field created by the contact force of the impactor has a significantly greater influence on delamination onset (crack propagation) than the plate bending stresses. 2.2.4 Post Delamination Loading To determine the delamination behaviour after its initial formation, cyclic load/unload tests were carried out on the four different specimen thicknesses. A t each unload cycle the specimen was removed and C-scanned carefully, with projected delamination borders etched directly onto the specimen. The specimen was then reloaded in the Instron, returning to the previous loading path. Each specimen was C-scanned 2 or 3 times in this manner after 26 delamination (from point B to D , Figure 2.8) and it was found that there was never more than 5 - 10% increase in projected delamination diameter. To validate the accuracy of the C-scan measurements, two C-scanned specimens were sectioned and viewed under a microscope to see how the actual delamination crack compared to the scanned measurement. Figure 2.10 shows a schematic of the cross-section. The measured delamination radius from the C-scan coincided with the largest delamination near the distal side of the plate. W u and Springer [1988-1] also found that the C-scan data and cross-sectional views were fairly consistent, although they found a tendency for the cross-sectionally measured delamination to be slightly larger. They stated that this might have been a result of the sectioning/cutting process. Further validation was obtained when comparing the C-scanned delamination size measured from the P E U S scan of a 5.03 mm thick specimen (Figure 2.13). The P E U S scan gave delamination diameters in the x and y directions of 47.6 and 51.8 mm, respectively, whereas the corresponding C-scans gave dimensions of 44.7 and 53.3 mm. This represents a difference of 3 - 6%. 2.2.5 Total Delamination Area Calculations A n important result found in this investigation was the actual delamination pattern in the cross-section. It showed that the delaminations get smaller in diameter as they approach the top surface, forming a truncated cone envelope (see Figure 2.19). This information is important for two reasons. First, the geometry of these delaminations w i l l define the approximate boundary between the damaged and undamaged portion of the plate. This is 27 significant for modelling considerations. Secondly, the information is necessary in order to calculate the total delamination area. This would allow the prediction of the energy released per unit area of delamination of the damaged specimen once the energy dissipated through damage is known. B y further practise with the C-scan technique, it was possible to locate the delaminations with respect to the top and bottom surface. It should be remembered that the C-scan wi l l indicate only the first free surface reached by the ultrasonic waves. A more exact determination of the geometric variables for calculating the size of the truncated cone was thus possible. It was found that in most cases that the angle of inclination of the cone was approximately 45 degrees. In order to calculate the total area of delamination, it was also necessary to understand the in-plane pattern of the delamination. In previous studies (Delfosse [1994], for example) it has been shown that the delaminations are linked from ply to ply by matrix cracks running along the fibres. The damage thus forms a double helix of delaminations (see Figure 2.20), switching direction at the (non-delaminated) middle layer. For a quasi-isotropic laminate, there is an equivalent of one complete (360 degree) delamination for every sublaminate (i.e. one grouping of a [-45/90/45/0] lay-up). This information, combined with the dimensions of the truncated cone provide all the necessary information to calculate the total area of delamination. The derived expression for delamination area is (2.1) 28 where N is the number of sublaminates and b is the radius of the bottom (largest) delamination measured with the C-scan. Table 2.4 gives a summary of the measured delamination diameters of the top and bottom delaminations where possible. Figure 2.21 shows the total calculated delamination areas based on a truncated cone assumption, as described above, as well as a simple cylindrical shape. The cylindrical shape area calculation assumes that the delaminations are transversely aligned with each delamination in each sublaminate being the same size as the bottom delamination, forming a cylinder in three dimensions. A s can be seen, this assumption of "projected area" is not unreasonable. A calculation was also made of the total area of matrix cracks between laminae i.e. the vertical cracks extending from one layer to the next in the staircase/double helix pattern. It was found that the total area of the matrix cracks amounted to only 3 - 7 % of the total area of delamination and matrix cracks. 2.2.6 Delaminated Plate Stiffness Figure 2.7 shows that there exists a "secondary stiffness" for the entire plate after the occurrence of delamination (curve B D in Figure 2.8). This overall reduced plate stiffness can be regarded as the result of superposing a restrained circular section of lower modulus into a large undamaged plate. Since the delamination grows very little here, this stiffness is also linear. It should be noted that this approximation is appropriate for the thinnest three of the four plate thicknesses, but the loading path of the 8.30 mm plate contains a number of 29 large load drops in the post-delamination region, making a linear approximation difficult in this case. It is also interesting to note the load/unload cycle behaviour. Figure 2.22 shows the apparent stiffening effect as the unloaded, delaminated structure (5.03 mm thickness) is re-loaded to its previous load path. Since this departure from the load path occurs only after delamination, it is most likely a reflection of layer interaction. A s the specimen is unloaded, the delaminated layers are allowed to relax and move apart. During the re-load, the delaminated layers are pressed back together to the point where the free surfaces come into contact with each other and become effectively bonded. The structure then continues on the previous linear load path behaving as a "solid" laminate. 2.2.7 Delaminated Plate Load-Strain Behaviour A further observation which was made during the testing procedure was the constant slope of strain (measured on the rear surface - see Figure 2.14) after delamination had occurred. Figure 2.23 shows the strain-load curve for the 5.03 mm thick plate. Here we see that, like the displacement stiffness, the slope of the load-strain curve reduces after delamination. Note that for the test on the specimen indicated as SVD1 in the figure, the load-strain measurement was taken from the first part of an unload cycle. Average measurements of the delaminated load-strain slope values for the 5.03 and 6.66 mm thick plates are plotted in Figure 2.24 as a function of plate thickness. Successful measurements from the 3.35 and 8.30 mm were not obtained as the strain gauges on these specimens broke before the delamination onset. The 30 slope of the load-strain curve represents a characteristic property of the test specimen which is used in the analysis in the next chapter. It should be noted that even though the support conditions were not symmetric about the point of impact, the value of strain directly underneath the impactor on the back surface was equivalent in all directions. Figure 2.25 shows the load-displacement and strain-displacement curves for the 5.03 mm thick plate where two strain gauges were placed perpendicular to each other on the back surface under the impactor. Their values are very close, with the difference being less than the experimental error present between two other tests with the same configuration. 2.2.8 Fracture Mechanics Considerations 2.2.8.1 Determining an Energy Balance of Delamination Insight into the delamination behaviour and the overall plate response can be obtained by carrying out an energy balance on the delamination event. This may in turn identify useful parameters for modelling purposes. Through previously discussed experimental observation, it is apparent that the energy lost during the sudden drop in load on the load-deflection curve (see Figure 2.26) is related to the formation of new damage. The damage mechanisms into which this energy could be dissipated are matrix cracking and delamination, both of which are simply the formation of new surfaces. Typical energy losses such as vibration, heat, inelastic behaviour of the projectile or the support, are also possible. No fibre breakage is seen at this early stage and is 31 therefore not considered. The energy lost during delamination, AW, was measured from the load-displacement curves for all four thicknesses. Figure 2.27 shows the results of these measurements as a function of plate thickness. To carry out a proper energy balance of the delamination event, it was necessary to account for all the energy transferred during the event. The sudden drop in load and the accompanying loud crack which occurred at delamination onset reveal that the crack growth mechanism for the brittle system is unstable. This instability made it difficult to account for the energy dissipation during crack formation. However, it was recognized that the delamination growth became more and more stable as the thickness of the plate increased. Figure 2.7 shows that the drops in load from the point of delamination onset to delamination arrest became less severe and more frequent in the loading path of the 8.30 and 6.66 mm plates. In a stable crack growth situation virtually all of the energy is used in the creation of new surfaces. A critical energy release rate, Gc, can be calculated by dividing the elastic energy lost in the delamination process by the total surface area of delamination. This value is constant and unique to the material for a stable crack growth situation. Figure 2.28 shows the result of this type of calculation for the present system where the energy lost at delamination (AW) is divided by the total area of delamination as a function of plate thickness. The calculation is performed using both area calculations defined previously. If there was a stable crack growth situation, the values in Figure 2.28 would be constant for varying thicknesses. A s can be seen the data set is not constant but decreases asymptotically. 32 It seems likely that the values for the thinner plates are much higher because the elastic energy stored in the plates is not all available for crack growth and is dissipated elsewhere. The asymptotic nature of the curve supports our first assumption that the crack growth is becoming more stable as plate thickness increases, and that the extra energy released during the unstable event is diminishing. One would expect that the value to which the energy per unit of damage area is converging should be the critical energy release rate for the material. These energy values seem to converge at a value of approximately 200 - 300 J/m 2 , which places it roughly between the values for mode I and II energy release rates of brittle matrix systems as seen in Table 2.5. This may indicate a mixed mode crack propagation. 2.2.8.2 Mode of Delamination Fracture Current research generally shows that delaminations due to out-of-plane loading form from a combination of mode I and II type fracture (due to normal and shear stresses, respectively), but seem to be dominated by mode II fracture. Razi and Kobayashi [1993] argued from their experimental and finite element analysis that mode II is clearly predominant and that mode I may be neglected altogether. They also noted, however, that the critical energy release rate for crack arrest was found to be about 75% of the critical crack initiation energy. Choi , W u and Chang [1991] suggested from their finite element analysis that it is a combination of both mode I and II. In the model proposed by W u and Springer [1988-2], however, it was postulated that fracture was due solely to mode I opening. 33 2.2.9 Delamination in Tough Systems A s reported by Delfosse [1994], the delamination in the tough material system grows slowly over the loading path B C (Figure 2.6). This would be considered a stable crack growth as opposed to the unstable crack growth mechanism exhibited by the brittle system. Figure 2.29 shows the P E U S scans of the IM7/8551-7 which was tested in a similar manner to the brittle system above, i.e. damage measurements at various stages of loading. Even though in this instance the material was supported over a 25.4 mm diameter hole and the delamination size was of the order of the plate size, the growth pattern is still evident. Another interesting difference found in the loading of the tough system, was the presence of top surface cracks propagating away from the impact site in the long side direction on either side of the indenter. It was estimated that these cracks passed through two or three layers at the top surface and appeared to grow with a stable propagation pattern during the loading cycle from B to C. The influence of these cracks and their effect on the overall energy absorption during impact should be the source of further study. 2.2.10 Rear Surface Matrix Splitting It was noted that during the static loading of some of the plates the strain gauge on the rear surface would fail far in advance of the fibre failure strain. This occurred even though the gauges in most cases were aligned with the fibres. Since this happened at different points along the load-displacement curve for different thicknesses, it was not related to either a critical load or a displacement, but most likely to the strain on the rear surface. It was realized after reviewing the test data, that the point at which this occurred was consistently at 34 or above the same level of strain of approximately 0.7 - 0.9 % strain. This strain level corresponds to the failure strain of the epoxy. Figure 2.30 shows the strain at failure for each of the tests. It was thus postulated that the matrix was splitting between the fibre tows due to plate bending. In the second set of tests over the 4.5" diameter circular opening, a new set of gauges were acquired that were very narrow. These were placed in the middle of the fibre tow, so as to avoid premature splitting. While the problem of gauge splitting still occurred, using the narrow gauges placed as accurately as possible within the tow reduced the number of premature gauge failures. 2.2.11 Fibre Breakage The next damage mechanism which occurs in the static penetration loading cycle is that of small amounts of fibre breakage on the rear surface, denoted as the drop in load at point C in the schematic of Figure 2.8. Figure 2.31 shows the load-displacement curve for the 6.66 mm thick plate, with the drop in load indicated on the diagram being caused by the rear 4 plies of the laminate splitting in tension. This was confirmed by visual inspection and the strain readings. To further understand this fibre breakage mechanism, a similar specimen was unloaded right after this drop in load and examined using a deply inspection technique. This is done by cutting a strip 50 cm wide (to include the damage zone) out of the specimen and heating it at 400° C for 2 hours. This effectively burns off the resin and allows the individual laminae to be separated. The laminae were then inspected to determine the damage mechanisms present. 35 In the test in question, only the bottom 5 plies showed damage in the form of tensile fibre breakage. The previous specimen which was loaded right through to final failure showed that the last 4 layers failed by tensile fibre breakage and the rest by a shear plugging mechanism. Delfosse et al. [1995] uses a technique of calculating the energy lost to fibre breakage by measuring the cross-sectional area of the split fibres. This area is multiplied by a fibre fracture energy. Using this technique, and a fibre fracture energy value of 155 kJ /m 2 (also from Delfosse et al.), it was found that the energy lost in this load drop should be 1.9 J. This is compared to a loss of 3.0 J, which is the energy calculated from the loss in area under the load-displacement curve at this point (Figure 2.31). The extra energy released in the load drop may have been due to small delamination increments as a result of a load redistribution occurring with fibre failure. The energy dissipated in this instance of fibre breakage (3.0 J) was roughly 6 - 10% of the total static perforation energy (30.6 J for 6.66 mm thick plate). 2.2.12 Plugging Point D (Figure 2.8) corresponds to the maximum deflection of the penetrator just before complete failure. This failure is a result of a damage mechanism known as plugging. The plug is the circular section of material cut out in front of a blunt nosed projectile as it perforates a stiff plate. Figure 2.32 shows a cross-section of the T300H/F593 just after the drop in load from D to E . A s can be seen, a clear crack surface passes through the thickness of the plate, with some interaction of fibres and layers, creating a high friction interaction. The tough IM7/8551-7 system shows the same plugging behaviour. Cristescu et al. 36 discovered this plugging mechanism in woven 0-90 degree ply fibreglass roving-epoxy plates. Lee and Sun [1993] and Sun and Potti [1993] reported this behaviour for AS4/3501-6 graphite/epoxy. They also report that on the last few layers, part of the plugged surface remained attached to the rear surface creating a " l i d " . It was seen that plugging commences at the top surface of the laminate with the shear cracks seen in Figure 2.10. These cracks first appeared before or at delamination formation. Since no. growth of this shear crack was seen during the loading of the plate after delamination and prior to the plugging load, it can be assumed that the plug formation is an unstable crack growth event. This is also seen by the large amount of energy released through noise and load loss during the load drop from D to E (Figure 2.8). Lee and Sun [1993] witnessed the presence of this sudden failure, and even used an unstable crack growth mechanism in their finite element analysis to represent the behaviour. Figure 2.33 shows the plugging failure load as a function of plate thickness for the three backplate opening sizes. A s with the delamination load, the plate areal dimension seems to have little effect on this final load. In the case of the T300H/F593 system, it was found that the peak plugging load and the delamination load were virtually the same. This is merely a coincidence as other systems do not display this behaviour. 2.2.13 Friction The distance which the plug is pushed through at the initial failure point is dependent on the fixed displacement of the impactor. The drop in load from D to E in Figure 2.8 does not go right to zero, but is stopped because of the friction that remains between the plug and the 37 sides of the cut out hole. A more detailed schematic of the friction forces is shown in Figure 2.34. It was observed that the plug was progressively ejected while the load drops from E to F and the surface area of contact between the two decreases. The plug was completely ejected at point F and the remaining load represents the friction existing between the impactor and the laminate. The remaining load value remains constant for the present test conditions, but would decrease to zero in the case of a free moving projectile. This would be represented by the drop from point G to H . Frictional forces at points E and F are seen in Figure 2.35 for the T300H/F593 system over the 25.4 mm opening. The shear stresses between the plug and the coupon at point E were calculated from these forces. It was assumed that the plate sprang back completely and that the length of the plug still in contact with the plate was equal to the plate thickness minus the penetrator displacement at that point. The results of these calculations, as well as the shear stresses of the penetrator in contact with the plate after point F, can be seen in Figure 2.36. This figure confirms that this shear stress is more or less constant for all thicknesses, and that the shear stress of the plug in contact with the plate is much higher than that of the steel penetrator. It was observed for the tests on the larger backplate openings that the peak force (point E) was much less than that for the 25.4 mm (1") opening, but the final penetrator friction force (line FG) remained the same. 38 2.3 Large Circular Opening Static Penetration Tests Table 2.6 gives a summary of the results for the 114.3 mm (4.5") diameter circular opening tests including the load-displacement stiffnesses and the load-strain ratios. These tests were performed in order to better validate the model which assumes a circular boundary condition. N o test was possible for the 6.66 mm laminate as there was not enough material. The experimental values are compared with the data from the rectangular support tests in Figure 2.37 and Figure 2.38. The measured delamination sizes are compared in Figure 2.39. A s can be seen, the delaminations on the panels tested over the larger circular opening were only slightly larger in all cases. 2.4 Influence of Geometric Constants A s part of a related series of tests, the influence of geometric conditions on experimental results was explored. The influence of plate thickness, indenter diameter and back plate opening size were studied by Delfosse and Poursartip [1995] using the T300H/F593 and IM7/8551-7 systems. Figure 2.40 shows the four thicknesses of the T300H/F593 statically penetrated over a 25.4 mm diameter opening. To determine the overall effect of thickness on the load-displacement behaviour, the load-displacement data were normalized with respect to an arbitrary thickness of 6.15 mm through the following relationships, with hN set equal to 6.15 mm PN=P hN \hplate J (2.2) 39 ( u V (2-3) wN =w \hplale J A best fit was found when setting n = 1.2 for the force normalization and n = 0.3 for the displacement. The graph of the normalized data can be seen in Figure 2.41. The best fit for the tough system (IM7/8551-7) was found with n = 1.1 for the force and 0.5 for the displacement. Combining the force and displacement ratios, the variation of the energy absorbed during penetration can be determined. The energy was found to not vary linearly with plate thickness, but rather to the power of approximately 1.5 (i.e. n = 1.2 + 0.3). This would mean that it would require 50% more energy to perforate a plate of thickness 2h than to perforate two plates of thickness h. The influence of indenter nose diameter was then explored by Delfosse and Poursartip on the IM7/8551-7 system using five indenter diameters, ranging from 0.22" to 0.7" (5.59 to 17.78 mm). A similar normalization procedure was followed, yielding results of n = 1.0 for force and n = -0.3 for displacement. The normalizations were based on the 0.3" (7.62 mm) diameter indenter. In this case, the energy absorbed would vary with penetrator diameter to the power of 0.7. It was seen that penetrators smaller than 0.22" caused failure different than that of plugging. Figure 2.42 shows the effect of back plate opening size in the static penetration test. The increase in bending stiffness is quite evident as the opening size decreases. A s stated previously, the peak forces in each case are very similar. In order to develop a characterization test to obtain force parameters without any bending effects, a punch test 40 using a small hole just slightly larger than the diameter of the indenter was attempted. Unfortunately this was unsuccessful and resulted in much higher forces, possibly because of the material wedging between the penetrator and back plate. 2.5 Impact Test In order to understand the limits of applying quasi-static penetration information to an impact event, it was necessary to perform a series of intermediate velocity impact experiments. This phase of the testing program involved dynamic impact tests of the C F R P plates over the same rectangular opening as the static tests. A n instrumented low mass impact testing system (LMI) (Figure 2.43) was used which allowed comparison of damage mechanisms, peak loads and other penetration behaviours with static tests. The system, as previously described by Delfosse et al. [1993], consisted of a horizontal gas gun with an instrumented projectile weighing approximately 0.3 kg. The projectile, shown in Figure 2.43, incorporated a 22 k N (5kip) piezoelectric load cell ( P C B Piezotronics, model 208M88) connected to a Tektronix T D S 420, 150 M H z , 8 bit digital storage oscilloscope. The connection was made with a long wire that uncoiled from a rod as the projectile traveled down the launcher tube. In order to avoid a re-hit due to the build up of pressure in the tube, two relief valves were opened shortly (50 msec) after the shot to relieve the back pressure. Also described by Delfosse et al. is the data reduction procedure for the raw data taken from the oscilloscope. The raw data of the low mass impacts contain oscillations from the resonance of the impactor at its natural frequency of 6.8 kHz . These oscillations were selectively eliminated by performing a forward fast fourier transform (FFT) of the recorded 41 signal into the frequency domain. The spectral coefficients were then multiplied by a correction factor which was obtained by optimizing the signals obtained from impacts on a large steel cylinder. The steel cylinder was chosen in order to guarantee that the recorded signal was not contaminated by structural oscillations of the target. This corrected signal was then converted into a filtered force-displacement curve by an inverse F F T . This filtering treatment is equivalent to an attenuation of the projectile response in the range of its natural frequency. Only corrected force-displacement curves were used in the data evaluations discussed here. 2.5.1 Impact Test Results In order to verify the applicability of using quasi-static results to predict the behaviour of an impact event, it was necessary to determine i f the damage induced in the C F R P panel was of the same nature and quantity for an event of equal energy. The peak forces and the delamination size and behaviour should be equivalent in order for our model to be valid. 2.5.1.1 Perforation Impact Tests A tough 6.15 thick mm C F R P panel was impacted at 26.9 m/s, far exceeding the velocity required for perforation. A s can be seen in Figure 2.44, the loading path was very different. However, the peak force encountered in the test, 20 000 N , was very similar to the peak forces found for this thickness in the static test (22 000 N) . The brittle system in Figure 2.45, however, shows the impact on a 5.03 mm thick brittle T300H/F593 panel at 29.6 m/s. A s can be seen, the peak load was almost twice as high in the impact case. 42 It is interesting to note in Figure 2.44 that although the initial dynamic stiffness does not follow the static penetration curve, it does overlap the curve for the static indentation test. This indicates that the penetration event is occurring so quickly that the impacting flexural stress wave does not have time to go to the plate boundary and reflect back, and the penetration is governed only by the local indentation behaviour. This correlation could be a useful empirical tool for predicting impact stiffness behaviour. A pulse echo ultrasound (PEUS) image of the perforated T300H/F593 specimen is seen in Figure 2.46 along with that of a statically penetrated specimen of the same thickness. The darkness of the image is related to the intensity of the signal reflected off a given delamination surface. A s can be seen, the resulting delamination behaviour was dissimilar to that of the static penetration test. It is postulated that the large area of delamination in the impact test occurred after the plate had already failed. The penetrator pushed out the last few layers with the friction force of the exiting penetrator. This hypothesis was proven by performing non-penetrating impact tests. 2.5.1.2 Delamination Impact Tests In order to compare the delamination areas of both static and dynamic events during Stage II loading, the coupons were impacted with just enough incident energy to cause delamination, but not enough energy to cause plugging. Projectile velocities were calculated from the static load-deflection information assuming that the energies required to reach delamination and plugging were approximately the same for both static and impact events. The static load-deflection curve was integrated first to obtain 43 the energy required to create delamination. The curve was again integrated to determine the energy required to perforate the specimen. These values, along with the actual velocities used to reach delamination, are shown in Table 2.7. It was first necessary to calibrate the instrumented projectile with the launch pressure for the gas gun. This was done by firing a number of test rounds at various pressures at a dummy target to obtain a gas gun pressure/velocity curve. Predictions of delamination energy were fairly accurate. In most cases delamination was observed after one shot of the gas gun, except for the case of the 5.03 mm plate, which was not delaminated until the fourth attempt. Figure 2.47 compares the delamination size of the impact and the static tests performed on the rectangular frame. It can be seen that the delaminations'produced by impact are slightly larger than those produced by static loading, but are definitely within the same magnitude range. The delamination size for the 5.03 mm thick plate is an anomaly and is due to just one arm of the delamination near the bottom surface extending to the boundary. The nature of the delaminations are also very similar. The cross-sectional view in Figure 2.48 should be compared with that of the statically induced delamination cross-section in Figure 2.10. A s can be seen, both the nature and extent of the delaminations are comparable. Figure 2.49 shows the plugging failure of the same material for an impact at 29.6 m/s. This failure is seen to be similar in nature to the static plugging failure seen in Figure 2.32. 44 2.6 Ballistic Tests A series of ballistic tests were performed at the TNO/Prins Maurits Laboratory ( P M L ) in the Netherlands on the 8.30 mm thick panels. These tests included high speed x-ray photographs of the 4.1 g blunt-nosed projectile as it passed through the plate. Initial reports included both projectile velocity- and displacement-time curves of the event divided into 8 time steps. A final report is yet to be published. Results from the test gave a ballistic limit velocity for this projectile of between 183 and 196 m/s, resulting in a perforation energy of 69 - 79 J. To compare this with the static perforation energy, the static load-displacement curve was integrated, giving an energy value of 48 J (Table 2.7) to reach the plugging failure load. Conservative friction calculations (as wi l l be described in Chapter 3) show that another 17 J would be used up in ejecting the remainder of the plug as well as the projectile. This gives a total static penetration energy of 65 J, very close to the experimental ballistic perforation energy. 2.7 Summary The static penetration behaviour of a C F R P laminate by a blunt-nosed projectile was explored in terms of three separate loading stages separated by delamination and plugging mechanisms. The damage and peak loads encountered in impacted specimens were found to be similar in intermediate velocity tests. The effects of material and structural properties on the load-displacement behaviour were also evaluated. 45 It was shown that information from a static penetration test could be used to predict the behaviour in a low to intermediate velocity impact test. The peak forces encountered in both cases were similar as were the delamination areas. The experimental results provide the necessary information for the model development discussed in the next chapter. The three separate loading stages which were identified in the study provide a framework on which the model is built. 46 Table 2.1 Summary of static and impact tests performed on the T300H/F593. Material Date Thickness Run* Support Coupon Impactor Impactor Impactor Test Velocity Impact Perforation Energy [J] [mm] Dim. [in] Dim. [in] Dia. [in] Shape Mass [kg] [m/s] Energy [J] Damage Friction Total T300H/F593 21/7/94 3.30 T32-sd1 1 " - circ 3x3 0.3" flat static defl. [-45/90/45/0]ns 19/8/94 3.30 T32-sd2 1 " - circ 3x3 0.3" flat static defl. •T3x" 2/2/95 3.30 T32-sd55 3x5 4x6 0.3" flat static defl. 10/2/95 3.30 T32-sd60 3x5 4x6 0.3" flat static defl. 30/5/95 3.30 T32-sd62 3x5 4x6 0.3" flat static defl. 20/7/95 3.30 T32-sr1 4.5" - circ 5.5" - circ 0.3" flat static defl. 18/7/95 3.30 T32-11 3x5 4x6 0.3" flat 0.311 gas gun 8.350 10.842 21/7/94 5.00 T33-sd1 1 " - circ 3x3 0.3" flat static defl. 26/8/94 5.00 T33-sd51 3x5 4x6 0.3" flat static defl. 31/8/94 5.00 T33-sd52 3x5 4x6 0.3" flat static defl. 1/11/94 5.00 T33-sdv1 3x5 4x6 0.3" flat static defl. 31/1/95 5.00 T33-sd53 3x5 4x6 0.3 " flat static defl. 7/2/95 5.00 T33-sd57 3x5 4x6 0.3 " flat static defl. 7/2/95 5.00 T33-sd58 3x5 4x6 0.3 " flat static defl. 30/5/95 5.00 T33-sd63 3x5 4x6 0.3" flat static defl. 10/7/95 5.00 T33-sdr1 4.5" - circ 6x6 0.3" flat static defl. 14/7/95 5.00 T33-sdr2 4.5" - circ 5.5" - circ 0,3" flat static defl. 4/11/94 5.00 T33-i-F1 3x5 4x6 0.3 " flat 0.305 gas gun 29.600 133.658 18/7/95 5.00 T33-i2 3x5 4x6 0.3" flat 0.311 gas gun 11.200 19.506 21/7/94 6.70 T34-sd1 1 " - circ 3x3 0.3" flat static defl. 31/1/95 6.70 T34-sd54 3x5 4x6 0.3" flat static defl. 2/2/95 6.70 T34-sd56 3x5 4x6 0.3" flat ' static defl. 4/2/95 6.70 T34-si51 rigid 2x2 0.3 " flat static ind. 10/2/95 6.70 T34-sd59 3x5 4x6 0.3" flat static defl. 18/7/95 6.70 T34-H 3x5 4x6 0.3" flat 0.311 gas gun 10.130 15.957 21/7/94 8.30 T35-sd1 1 " - circ 3x3 0.3" flat static defl. 13/2/95 8.30 T35-sd61 3x5 4x6 0.3 " flat static defl. 20/7/95 8.30 T35-sdr1 4.5" - circ 5.5" - circ 0.3" flat static defl. 18/7/95 8.30 T35-H 3x5 4x6 0.3" flat 0.311 gas gun 11.41 20.244 Table 2.2 Summary of static and impact tests performed on the IM7/8551-7 (taken from Delfosse [1994]). Material Date Thickness Run* Support Coupon Impactor Impactor Impactor Test Velocity Impact Perforation Energy [J] [mm] Dim. [in] Dim. [in] Dia. [in] Shape Mass[kg] [m/sl Energy [J] Damage Friction Total IM7/8551-7 10/2/94 6.15 qs-5i02f rigid 2x2 0.2" flat static ind. [-45/90/45/0/0]4s 8/3/93 6.15 qs-si-f4 rigid 2x2 0.3" flat static ind. "quasi" 10/2/94 6.15 qs-si05f rigid 2x2 0.5" flat static ind. 10/2/94 6.15 qs-sd02f 1" - circ 3x3 0.2" flat static defl. 61 16 77 19/4/93 6.15 qs-sf5-f 1" - circ 2x2 0.3" flat static defl. 60 16 76 11/8/93 6.15 qs-sf-f7 1" - circ 3x3 0.3" flat static defl. 16/1/94 6.15 qs-sd-f6 1"- circ 3x3 0.3" flat static defl. 16/1/94 6.15 qs-sd-f7 1" - circ 3x3 0.3" flat static defl. 16/1/94 6.15 qs-sd-f8 V'-circ 3x3 0.3" flat static defl. 31/5/93 6.15 qs-sf-f 3x5 4x6 0.3" flat static defl. 68 16 84 16/1/94 6.15 qs-sd-f1 3x5 4x6 0.3" flat static defl. 22/8/94 6.15 qs-sd015f 1" - circ 3x3 0.15" flat static defl. 71 17 88 22/8/94 6.15 qs-sd022f 1" - circ 3x3 0.22" flat static defl. 71 17 88 10/2/94 6.15 qs-sd05f 1" - circ 3x3 0.5" flat static defl. 71 17 88 22/8/94 6.15 qs-sd07f 1" - circ 3x3 0.7" flat static defl. 71 17 88 22/7/94 3.20 qs-sd-ht 1" - circ 3x3 0.3" flat static defl. 71 17 88 20/7/94 12.30 qs-sd-2t 1" - circ 3x3 0.3" flat static defl. 71 17 88 20/5/93 6.15 qs-i-f5 1 " - circ 2x2 0.3" flat 0.308 gas gun 31.95 157.20 65 17 82 14/6/93 6.15 qs-i-f6 1 "- circ 3x3 0.3" flat 0.308 gas gun 28.08 121.43 65 17 82 7/12/92 6.15 qs-i-f1 3x5 4x6 0.3" flat 0.320 gas gun 20.50 67.24 7/12/92 6.15 qs-i-f2 3x5 4x6 0.3" flat 0.320 gas gun 26.90 115.78 78 17 95 1/7/93 6.15 qs-b-f5 1 " - circ 3x3 0.3 " flat 0.0042 ballistic 293.00 180.28 99 15/7/93 6.15 qs-b-(6 1 " - circ 3x3 0.3 " flat 0.0042 ballistic 261.00 143.05 93 5/8/93 6.15 qs-b-f7 1 " - circ 3x3 0.3 " flat 0.0042 ballistic 167.00 58.57 9/8/93 6.15 qs-b-f8 1 " - circ 3x3 0.3 " flat 0.0042 ballistic 189.00 75.01 4 7 Table 2.3 Summary of strain gauge tests performed on the T300H/F593 system. Gauaes Gauae 1 Gauqe 2 Gauae 3 Gauae 4 Test Location Direction Tvpe Dist fmml Location Direction Tvpe Dist fmml Location Direction Tvpe Dist fmml Location Direction Tvpe Dist fmml T33-SD53 B F B 0/4.0 B P B 0/10.4 T F B 10.5/.5 T33-SDV1 T F B 9.2/1.7 T F B 21.3/3.9 T F B 31.7/1. B F B .8/2.1 T34-SD56 B F L 1.6/0 B F L 5.3/3.6 T34-SD59 B F B 4.0/1.8 T34-SD54 B F B 4.0/0 B P B 0/12.2 T F B 9.8/1.4 T32-SD55 B F D 0/0 B P D 1.9/0 T33-SD57 B X1 D .7/0 B Y1 D .7/0 B X2 L 7.7/0 B Y2 L 0/6.5 T32-SD60 B F L .4/1.5 T35-SD61 B F D .7/0 B P D .7/0 T33-SDR1 B F T 0/0 B F T 0/4.9 T33-SDR2 B F T 0/0 T32-SR1 B F T 0/0 Leaend Location B = Bottom T = Top Direction F = Fibre Direction • y P = P Tvoe erpendicular to Fibre g (HBM tie /MM FP-f)fl-nfi?AK-1?nt B = B I = I D = Double (MM EA-06-062TT-120) T - T h i n / M M P A . f T A . ' S ' V i n C L I O m Distance from Centre Point fibre dirWperp. fibre dirn or x/y 48 Table 2.4 Summary of measured delamination diameters Bottom Delamintation I Top Delamination I Date Thicknes R u n * Support Coupon Initial Delam Dia[mm] Final Delam Dia[mm] Initial Delam Dia[mm] Final Delam DiafmrnJ [mm] Dim. [in] Dim. [in] Minimum Maximum Avg. [mm] Minimum Maximum Avg. [mm] Minimum Maximum Minimum Maximum 21/7/94 3.30 T32-sd1 1 " - circ 3x3 19/8/94 3.30 T32-sd2 1 " - circ 3x3 2/2/95 3.30 T32-sd55 3x5 4x6 44.0 57.0 50.5 10/2/95 3.30 T32-sd60 3x5 4x6 30.0 34.0 32.0 34.0 36.0 35.0 30/5/95 3.30 T32-sd62 3x5 4x6 28.0 33.0 30.5 20/7/95 3.30 T32-sr1 4.5" - circ 5.5" - circ 33.0 43.0 38.0 30.0 30.0 18/7/95 3.30 T32-i1 3x5 4x6 33.0 46.0 39.5 18.0 25.0 21/7/94 5.00 T33-sd1 1 " - circ 3x3 26/8/94 5.00 T33-sd51 3x5 4x6 31/8/94 5.00 T33-sd52 3x5 4x6 1/11/94 5.00 T33-sdv1 3x5 4x6 54.0 55.0 54.5 31/1/95 5.00 T33-sd53 3x5 4x6 42.0 55.0 48.5 7/2/95 5.00 T33-sd57 3x5 4x6 52.0 65.0 58.5 7/2/95 5.00 T33-sd58 3x5 4x6 34.0 42.0 38.0 30/5/95 5.00 T33-sd63 3x5 4x6 40.0 43.0 41.5 23.0 24.0 10/7/95 5.00 T33-sdr1 4.5" - circ 6x6 42.0 55.0 48.5 44.0 55.0 49.5 14/7/95 5.00 T33-sdr2 4.5" - circ 5.5" - circ 48.0 50.0 49.0 50.0 62.0 56.0 4/11/94 5.00 T33-i-F1 3x5 4x6 18/7/95 5.00 T33-i2 3x5 4x6 60.0 70.0 65.0 30.0 40.0 I 21/7/94 6.70 T34-sd1 1 " - circ 3x3 31/1/95 6.70 T34-sd54 3x5 4x6 2/2/95 6.70 T34-sd56 3x5 4x6 74,0 84.0 79.0 55.0 57.0 4/2/95 6.70 T34-si51 rigid 2x2 10/2/95 6.70 T34-sd59 3x5 4x6 59.0 60.0 59.5 65.0 70.0 67.5 35.0 40.0 42.0 50.0 18/7/95 6.70 T34-i1 3x5 4x6 57.0 66.0 61.5 21/7/94 8.30 T35-sd1 1 " - circ 3x3 13/2/95 8.30 T35-sd61 3x5 4x6 52.0 70.0 61.0 87.0 93.0 90.0 20/7/95 8.30 T35-sdr1 4.5" - circ 5.5" - circ 62.0 80.0 71.0 90.0 96.0 93.0 23.0 25.0 30.0 40.0 18/7/95 8.30 T35-M 3x5 4x6 72.0 77.0 74.5 34.0 46.0 Table 2.5 Mode I and II Critical Strain Energy Release Rates for Various Brittle Carbon Fibre Composite Materials (taken from Daniel and Ishai [1994]) Material Type of Test Strain Energy Release Rate - Gie [J/m2] Mode I T300/5208 D C B 103 D C B 88 AS4/3501-6 D C B 198 D C B 190 H T D C B 189 AS4/3502 D C B 160 Mode II T300/5208 C L S 433 C L S 154 T300/914 E N F 518 C B E N 496 AS1/3501-6 E N F 458 AS4/3502 E L S 543 E N F 587 49 Table 2.6 Measured Load-Displacement and Load-Strain Parameters 3 x 5" Plate Plate Thickness [mm] K0 - Undamaged Stiffness [kN/m] K - Damaged Stiffness [kN/m] Ke - Damaged Load-Strain Slope [kN/strain] 3.35 1671 1770 359* 5.03 4440 3200 625 6.66 8460 3670 883 8.30 14700 4200 1143* 4.5" Round Plate 3.35 2060 1800 675 5.03 3290 2780 832 6.66 6000* 4000* 980* 8.30 11600 4700 1150* * Indicates interpolated/extrapolated value used for model input Table 2.7 Energy and Velocity Required for Delamination and Perforation of 3 x 5" Plates, m p r o j e c t i l c = 0.311 kg Plate Thickness [mm] Determined from static load-deflection curve EdelamU] EperfU] ^delcm [m/s] Vperf[m/s] Experimental V [m/s] E[J] 3.35 8.4 13.5 7.37 9.32 8.35 10.84 5.03 8.8 20.2 7.52 11.40 11.20 19.51 6.66 10.4 30.6 8.16 14.03 10.13 15.96 8.30 13.3 48.0 9.25 17.57 11.41 20.24 50 Figure 2.1 Comparison of energies required for static or dynamic perforation of a CFRP laminate using different indenter nose shapes (from Delfosse and Poursartip [1995]). 0 . 3 " Diameter 16.50 7.62 9.40 30 .00 m = 24.9 g Figure 2.2 Hardened steel punch dimensions (all dimensions in mm). 51 12.7 mm Diameter Flat Indentor Brittle Matrix Tough Matrix 0 1 2 Punch Displacement [mm] 3 4 5 6 7 Figure 2.4 Typical load-displacement curves for brittle and tough CFRP systems in the static penetration test. Brittle Matrix Elastic T J co o Delamination Damaged Friction Punch Displacement Figure 2.5 Idealized load-displacement curve for brittle matrix CFRP static penetration test. 5 3 Tough Matrix T3 CO O Elastic Damaged Friction • Delamination Plugging Punch Displacement Figure 2.6 Idealized load-displacement curve for tough matrix CFRP static penetration test. 18000 16000 14000 12000 10000 8000 6000 4000 2000 T300H/F593 3 x 5" opening h = 3.35 mm h = 5.03 mm h = 6.66 mm - - - h = 8.30 mm 3 4 5 Punch Displacement [mm] Figure 2.7 Load-displacement curves for T300H/F593, 3 x 5" opening static penetration tests for 4 thicknesses. 54 o Point A Point B III Damage Mechanism A - Delamination C - Fibre Breakage D - Plugging Punch Displacement Figure 2.8 Detailed idealized load-displacement curve for brittle matrix CFRP static penetration test. 14000 Deflection [mm] Figure 2.9 Effect of removing local indentation from static penetration test. 55 Figure 2.10 Cross-sectional micrograph of a delaminated 5.03 mm thick T300H/F593. This shows the measured edge of the projected delamination radius which coincided with the crack tip viewed in the cross-section (not seen). Impact Damage Growth Mechanism Figure 2.11 A schematic description of two basic impact damage growth mechanisms of laminated composites (from Choi, Wu and Chang [1991]). 56 Figure 2.12 Frames from video recording of static penetration test - T300H/F593 system. The first frame shows the unloaded specimen. The second and third frames, 33 ms apart, show the abrupt change in curvature at the onset of delamination. 57 Figure 2.13 Pulse echo ultrasound (PEUS) image of a statically loaded specimen just after delamination load drop. T300H/F593 system, 5.03 mm thick plate. The image represents a plate area of 101.8 x 101.8 mm. The darkness of the image represents the intensity of the signal. Top View Side View (section AA) A ' • D e l a m i n a t i o n Fibre Direct ion Figure 2.14 Strain gauge placement for the static penetration test. 58 •o O 9000 8000 7000 6000 5000 4000 -f 3000 2000 -t 1000 0 -•- Load Bottom Strain Gauge - a - Top Strain Gauge #1 T300H/F593 5.03 mm thick plate Final Plugging Failure 0.02 0.015 0.01 + 0.005 c 2 co o -0.005 Q . E o O CD CO O CM O CN ^J" CD CO CN CN CN CN CN Time [s] Figure 2.15 Load/unload cycle for a 5.03 mm thick T300H/F593 with strain gauge readings for top and bottom surfaces. 9000 Delamination Final Plugging Failure Time [s] Figure 2.16 Load/unload cycle for a 5.03 mm thick T300H/F593 with strain gauge readings for top surface. 59 Figure 2.17 Displacement profile measured across the short length of a 3.35 mm statically loaded T300H/F593 plate. 18000 16000 + 14000 12000 2 10000 ro _3 8000 + 6000 4000 + 2000 0 0 * Peak Delam Force - 3" x 5" o Peak Delam Force -1" circular • Peak Delam Force - 4.5" circular T300H/F593 4 5 Plate Thickness [mm] Figure 2.18 Peak forces encountered before the onset of delamination for 3 different opening sizes. 60 Section A - A Figure 2.19 Assumed delamination pattern used to calculate total area of delamination. Ply #5 Ply #6 Delamination 45° Fibre breakage Figure 2.20 An example of delaminations and fibre breakage in a CFRP laminate with a stacking sequence of [45/0/-45/90]ns (different from the present system) caused by static or dynamic out-of-plane loading (from Delfosse [1994]). 61 Figure 2.21 Total calculated area of delamination for T300H/F593 system using two different assumed delamination patterns. 1 2 0 0 0 T 3 0 0 H / F 5 9 3 Punch Displacement [mm] Figure 2.22 Unload/reload paths for 5.03 mm thick T300H/F593 system. 62 Figure 2.23 Post delamination load-strain curves for determining slope value. 1200 1000 E E E 800 •2 600 CO or c 'ro W 400 TS ro o 200 T300H/F593 3" x 5" Opening 3 4 5 6 Plate Thickness [mm] Figure 2.24 Measured slopes of the load-strain curve as a function of plate thickness, linearly extrapolated to other thicknesses. 63 Figure 2.25 Strain gauge readings on the bottom surface in various orientations during static penetration test. -o CD O Elastic Damaged Friction Indenter D i s p l a c e m e n t Figure 2.26 Idealized load-displacement curve showing energy lost during delamination. 64 c o E ™ o Q 5 + 1 + T300H/F593 3 x 5 " Opening -+-3 4 5 6 Plate Thickness [mm] Figure 2.27 Energy lost at delamination onset as a function of thickness during static penetration test. 900 E 800 c o ro c ra a> O 700 + 600 * 500 3 a> a. 400 4-300 g 200 + E? 100 + a c UJ T300H/F593 3 x 5 " opening o delta W/(Area Cone) • delta W/(Area Cylinder) 3 4 5 6 Plate Thickness [mm] Figure 2.28 Calculated energy lost per unit area of delamination for statically loaded specimens. 6 5 25000 20000 £ 15000 o £ 10000 5000 IM7/8551-7 1" circular opening Static deflection test State 1 State 2 • State 3 State 4 2 4 6 D i s p l a c e m e n t ( m m ) 8 State. 2 ;State:-S State4 Will iS4 I M F * • • f Figure 2.29 PEUS Images of delamination growth in the tough IM7/8551-7 system corresponding to different unload/load cycles seen in the graph (taken from Delfosse [1994]). 66 0.018 0.016 0.014 .= 0.012 2 4-1 V) £ 0.01 ^ 0.008 ro O 0.006 + 0.004 0.002 0 T32-SR1 Fibre Failure Strain Matrix Failure Strain T300H/F593 T32-SD60 T32-SD55 T33-SD53 T33-SDV1 T33-SD57 T33-SDR1 T33-SDR2 T34-SD56 T34-SD59 T34-SD54 T35-SD61 T e s t Figure 2.30 Recorded failure strains during static loading tests. Figure 2.31 Strain to failure and energy lost due to fibre breakage for 6.66 mm thick specimen. 67 Figure 2.32 Cross sectional micrograph of a 5.33 mm T300H/F593 specimen after final failure by plugging. 20000 18000 16000 r 14000 + o £ 12000 | •§, 10000 O) a. 8000 -= 6000 --m LL 4000 2000 0 * 1" opening A 3 x 5" opening « 4.5" opening T300H/F593 - f-3 4 5 6 Plate Thickness [mm] Figure 2.33 Final plugging failure load as a function of thickness. 68 Figure 2.34 Idealized post failure/friction model showing various stages of plug pushout. Figure 2.35 Peak frictional forces of the plug and the penetrator against the inside of the cut-out hole. 6 9 Plate Thickness [mm] Figure 2.36 Peak frictional shear stesses. Figure 2.37 Experimental undamaged and damaged plate stiffnesses for two back plate opening sizes. 70 1.20E+06 1.00E+06 | 8.00E+05 •5 6.00E+05 W 4.00E+05 2.00E+05 O.OOE+00 • Ke Experimental - 1 " circular opening • Ke Experimental - 3 x 5 " opening T300H/F593 -+- -t-3 4 5 6 Plate Thickness [mm] Figure 2.38 Experimental load-strain slopes for two back plate opening sizes and linear extrapolation. 80.0 70.0 60.0 50.0 40.0 4-T300H/F593 a> 30.0 20.0 o Rectangular Opening - 3 x 5 " a Circular Opening - 4.5" 10.0 0.0 -f-3 4 5 6 Plate Thickness [mm] Figure 2.39 Projected delamination diameter as a function of thickness for two back plate opening sizes. 71 T300H/F593 1" circular opening h = 3.35 mm h = 5.03 mm h = 6.66 mm h = 8.30 mm Displacement [mm] Figure 2.40 Load-displacement curves for static penetration tests over a 25.4 mm (1") opening for different thicknesses (from Delfosse and Poursartip [1995]). Figure 2.41 Load-displacement curves for static penetration tests over a 25.4 mm (1") opening for different thicknesses. Forces and displacements are normalized with respect to a plate thickness of 6.15 mm (from Delfosse and Poursartip [1995]), 72 12000 10000 8 0 0 0 •o 6 0 0 0 CO o 4 0 0 0 2 0 0 0 -1 " round open ing - 3 x 5 " o p e n i n g 4 . 5 " round open ing 2 3 4 Punch Displacement [mm] Figure 2.42 Effect of back plate opening size on the load-displacement behaviour of 5.03 mm thick T300H/F593 specimen. 73 Figure 2.43 Gas-gun schematic and cross-section of projectile/load cell. 74 25000 0 1 2 3 4 5 6 7 8 9 10 Displacement [mm] Figure 2.44 Static and impact load-displacement curves for a 6.15 mm thick IM7/8551-7 plate. 16000 T300H/F593 5.03 mm thick plate - - Impact Test - 29.6 m/s Static Penetration Test Static Indentation 2 3 Displacement [mm] Figure 2.45 Static and impact load-displacement curves for a 5.03 mm thick T300H/F593 plate. 75 Figure 2.46 PEUS images of a static (left) and an impact (right) penetrated specimen. The velocity of the impacted specimen was 29.6 m/s, well above perforation velocity. Each image represents a 101.6 x 101.6 mm section of plate. 80.0 70.0 4 60.0 E, $> 50.0 v E A Q 40.0 c o '•4-» c 30.0 £ ro o Q 20.0 10.0 0.0 T300H/F593 3 x 5" opening -+- -+- -+-• Static Load o Impact Load -t-0.00 1.00 2.00 3.00 4.00 5.00 6.00 Plate Thickness [mm] 7.00 8.00 9.00 Figure 2.47 Comparison of projected delamination diameters for static and impact loaded T300H/F593 specimens. 76 Figure 2.48 Cross sectional micrograph of an impact loaded and delaminated 5.03 mm thick T300H/F593 specimen. Figure 2.49 Cross sectional micrograph of an impact penetrated 5.03 mm thick T300H/F593 specimen. 77 Chapter Three Analytical Modelling and Model Verification 3.1 Background The analytical modelling of damaged laminated composites has been treated in various sources. In general, we are interested in those models in which composite materials encounter stiffness reductions due to damage. We are also interested in inverse analytical methods to determine the loading path. Studies dealing with modelling of delaminated composite plates often turn to a fracture mechanics approach to determine stiffness reduction and plate response. Delamination itself can be defined as a crack between two separated laminates, or layers of fibre. Empirical studies have been made to investigate the relationship between the loss in stiffness and the growth of delamination (work of fracture). Howard et al. [1993] showed this by means of a laminated four-point bend test which delaminates between the rear surface plies as shown in Figure 3.1. They use the Griffith formulation of strain energy release rate G,-c (where i denotes either mode I, II, or III) based on the load-displacement graph (Figure 3.2) for a two-crack front beam specimen. A s the limit of Pj tends to P2, P2 dC Gic= — •— (3.1) ,c Ab da 7 8 where b denotes the crack width, a, the crack length, and C, the laminate compliance. Davies and Robinson [1992] derived an expression for G for a centrally notched isotropic beam loaded transversely at the centre in a simply supported condition (Figure 3.3). They used exact beam bending equations and the energy balance equation to calculate a strain energy relation in terms of geometric variables, modulus and load. Applying this to plates, they found that for a notched plate with a single central delamination where E is the isotropic modulus, h, the plate thickness and v, the Poisson's ratio of the material. Note that this formula is independent of the delamination or plate size. Ho-Cheng and Dharan [1990] derived an expression of this same isotropic form for the case of a dril l bit pushing out the last layer of a composite material. They combined a classical plate bending formula for a circular plate with a strain energy calculation. From this they were able to calculate the critical load at the onset of crack propagation as \_dU_ b da (3.2) 9P2(l-v2) (3.3) 64n2Eh3 (3.4) with the nomenclature as shown in Figure 3.4. 79 To assess the critical in-plane compressive strength of a near-surface interlaminar defect (Figure 3.5), Chai and Babcock [1985] combine an elastic stability and a fracture energy formulation. In their case, the areal delamination shape was elliptical making it necessary to define energy release rates in both major, and minor axes, a and b, respectively. These are given as G-;=-±.SJL Ct=-±M (3.5) nb da na db Using the elastic stability of the problem to obtain a further relationship, an admissible displacement function was assumed in conjunction with a Rayleigh-Ritz approach. Unknown coefficients were determined from the requirement of a stationary potential energy and an experimentally known buckling strain, s b, was required as input. Knowing this and the critical energy release rate, Gc, they were able to inversely deduce the loading behaviour of a given problem. Zhu et al. [1992-1] built on the quasi-static transverse loading problem of Davies and Robinson [1992] and Ho-Cheng and Dharan [1990] but used the more complex solution technique of Chai and Babcock [1985]. Considering a conical projectile pushing through a laminated Kevlar composite, they found experimentally that the back face layers were pushed out in a mode I delamination. In their opinion, this failure progression was initiated by a reflected pressure wave during impact. The energy release rate formulations of Chai and Babcock [1985] were simplified for the case of a circular delamination. Taking the relation for elastic plate bending from Timoshenko and Woinowsky-Krieger [1959] and the V o n Karman strain-displacement relations for strain, 8 , and curvature, K , they determined an 80 equation for the strain energy of the plate in terms of the laminate stiffness coefficients, central deflection w0, and delamination dimensions, a and b. This was then used to establish the strain energy release rates in the two directions. Another relationship for the geometry and loading conditions of Chai and Babcock was established using the Castigliano theorem. Knowing that the force to resist penetration by the sublaminate could be expressed as ^0 = dU (3.6) 8WQ they were able to employ the strain energy of the plate to reduce the number of unknown variables in the problem. Both the maximum delamination size and maximum plate deflection could then be calculated once the values for the critical energy release rate and maximum resistance of the sublaminate, Gc and Pc, respectively, were known. O'Brien [1982] also derived a closed-form equation for the strain energy release rate associated with delamination using the energy lost in tensile stiffness reduction. This formulation was for edge delamination growth in in-plane tension loaded laminated composites (Figure 3.6). Using a rule of mixtures approach for the stiffness reduction of an edge delaminated specimen, he showed that 7 7 I 7 7 * 7 7 \ A ^ „ (3.7) h=\E -^iAMJ-T + hLAM ' A where E is the resulting reduced modulus, E* is the stiffness of a laminate completely delaminated along one or more interfaces, ELAM is the stiffness of undamaged laminate from 81 laminate plate theory and A and A* are the delaminated and total interfacial areas respectively. This formula was confirmed experimentally by comparing the measured delamination size during loading with the moduli determined from the stress-strain curve (Figure 3.7). B y incorporating the energy release rate, G, with the nominal strain s, through the strain energy density relation where Vis the volume, O 'Br ien was able to calculate a critical energy release rate. This was found to be This value of critical energy release rate was an independent material parameter and could therefore be used for any given lay-up or geometry to calculate the critical strain, s c, once the value of E* was found from a separate stress analysis. The finite element method has been an important tool to model impact behaviour. While many advances have been made in this area, it does not yet provide a rigorous representation of the complicated damage mechanisms which occur in composite materials. Lee and Sun [1993] used the finite element method to model the quasi-static punch problem. This allowed a detailed analysis of the various stress and strain states during the process. Material constants which are strain rate sensitive could then be modified independently in order to customize the model for dynamic conditions. G = -(3.8) G - ^ ( E L 4 M ~ E ) (3.9) 82 Lee and Sun performed an axisymmetric analysis on a quasi-isotropic AS4/3501-6 using the commercial finite element code, M A R C . The onset of stiffness reduction due to damage was taken to be matrix cracking, dominated by the maximum stress perpendicular to the fiber direction. The strength of the transverse lamina, in this case, was deduced using a Weibull strength theory, asserting that the probability of failure is related to the volume of stressed material. It was further assumed that the onset of matrix cracking coincided with the onset of delamination. Since experiments by Lee and Sun were performed with a clamped boundary and a relatively small plate to impactor diameter ratio, the delaminations always extended to the boundaries. This simplified the model, making it unnecessary to include any crack propagation criteria. The overall reduced stiffness of the plate was then represented by a reduced shear modulus value based on a simple analysis for a split plate. The formation of the plug, however, was treated from a fracture mechanics point of view. The strain energy release rate was calculated by a crack closure integral scheme at the crack tip. This analysis could not predict the load value at plug initiation, making it necessary to obtain this value from a punch test. In most cases, the delamination does not extend to the boundary and it is necessary to implement a fracture mechanics approach in the analysis to accurately represent the loss of stiffness. For example, in modeling 2-D delaminations in impacted laminated plates, G i m [1994] represented the undelaminated region by a single layer of plate elements while the delaminated region was modeled by two layers of plate elements containing a delamination at the interface. In addition, G i m modified the virtual crack closure technique and implemented 83 it into the code so that the strain energy release rate could be computed using the plate elements. Davies and Zhang [1995] extended the work of Davies and Robinson [1992] by implementing their energy release rate formula (Equation 3.3) into a version of the finite element code, FE77. Using 8-noded Mind l in curvilinear quadrilateral elements, they included a non-linear Hertzian spring between the plate and the indenter and a stiffness degradation model found through a 3-D finite element brick analysis. They performed tests with pre-embedded delamination sites to test their fracture mechanics formulation. In summary, composite penetration models range from analytical/numerical to empirically based approaches. A s stated in the Introduction, empirical models suffer from the requirement of a large number of tests to cover the various combinations of laminate thickness, punch diameters and plate boundary conditions. Without an understanding of how these variables affect the response, this approach is little more than a curve fitting exercise. If only a cursory analysis is desired analytical/numerical approaches can often be too cost or labour intensive. Numerical models, in particular, require an accurate description of the constitutive behaviour which entails formulation and extensive material characterization tests. The present study seeks to find a simple and balanced analytical solution to the penetration process. 3.2 Present Model For the current investigation it was desired to generate a phenomenological model representing local quasi-static penetration behaviour. The first requirement of the quasi-static 84 penetration model was the use of simple, closed form solutions. The second requirement was that it be adaptable to any new experimental data, other test geometries and material properties. It was hoped at the outset that the local static penetration model could be implemented into a dynamic global plate deflection model to represent impact behaviour. The decoupled nature of the brittle load-displacement behaviour lends itself well to a building block approach to model development. The first step was to define the model in terms of the three main regions representing the different stages of target failure during projectile penetration as discussed in the previous chapter. Referring to the schematic in Figure 3.8, this allowed each region to be modelled by separate closed form expressions. The models representing these stages of penetration are described briefly here and followed by a detailed derivation of the developed model. 3.3 Model Description Stage I : Since the C F R P plate is undamaged and elastic, its bending response can be described by the Whitney-Pagano equations (Whitney and Pagano [1970]) for a simply supported laminated plate subjected to a central point load. These equations account for the combined influence of bending and shear deformation of the plate and lead to accurate predictions of the plate bending stiffness, K0. Local elastic indentation is modeled using a method described by Hi l l s et al. [1993] which integrates a ring load over a surface for a given projectile profile on a half space. Stage II: Here we account for the fact that the central delaminated (damaged) zone is softer than the rest of the plate, and thus exhibits a local dishing behaviour as shown in the inset of 85 Figure 3.8. We assume that the overall response of the delaminated plate is also elastic, but with a reduced bending stiffness. Because the experiments indicated that the delaminated zone did not grow noticeably once initiated at point A , we treat the problem as a fixed size damaged zone within an otherwise undamaged plate. For simplicity, and based on the observation that the delaminated zone is circular, we model the delamination as an isotropic circular plate of radius b and bending rigidity D embedded in a concentric simply-supported, isotropic circular plate of radius a and bending rigidity D0, as shown in Figure 3.9. This equivalent circular plate is chosen such that its effective central bending stiffness is equal to that of the original undamaged rectangular laminate, K0. Having defined the problem, we use the average measured values of the reduced stiffness, K, of the damaged plate, and the strain on the rear surface to obtain more information about the deformation profile of the soft zone. These experimental measurements are then combined with an approximate Rayleigh-Ritz type analysis to estimate the size of the damage zone, 2b, and its effective bending rigidity, D. The success of our modelling approach is ultimately judged by comparing the predicted damage size with that measured using C-scan. Stage III: The behaviour in this stage can be modelled by assuming a constant friction stress to act on the surface of the plug, thus resulting in a linear reduction in load as the plug is ejected. 3.4 Model Formulation The following section deals with the detailed formulation of the engineering model. Each stage of the loading curve is represented by a closed form solution and results are compared 86 with experimental data. In cases where more than one applicable model exist, results of each are discussed. Final conclusions are discussed in Chapter 4. 3.4.1 Stage I - Elastic Bending and Indentation In the undamaged elastic region of Stage I, the impact problem can simply be broken down into two distinct responses, global deformation of the target mid-plane and local indentation of the target surface. The response is given by A,,=a + w (3.10) where AP is the projectile displacement, a is the indentation and w is the target deflection. 3.4.1.1 Indentation The static indentation curve can be represented using established indentation laws. A Hertzian type relationship is given by P = ka" (3.11) where the exponent n is equal to 1.5 (after Hertz's law) and the indentation stiffness, k, would need to be fitted to the experimental curve. It should be noted that Hertz's law was developed for a spherical indenter shape, and its application is therefore limited in this case. A purely analytical expression is offered by Hil ls et al. [1993], who derive a relationship for a flat penetrator indenting an elastic surface (Figure 3.10) as 4 c a P = -r (3.12) 87 where 1-v A = + (3.13) and the moduli and \x2 are the respective shear moduli of the penetrator and target and v, and v 2 the respective Poisson's ratios. This model assumes a rigid penetrator since the alternative solution for a deforming flat indenter violates the assumption of a half-space and is not closed-form. Using the corresponding values in Table 3.1, this model is implemented into the overall representation of quasi-static penetration through Equation 3.10. The weakness of this model lies in the fact that it seems to be sensitive to the transverse shear modulus, fj,2, of the target material, which is only known approximately for the present system. Figure 3.11 shows the comparison of the indentation curve with a Hertzian type relationship where the exponent n is equal to 1.5 (Equation 3.11) and the indentation stiffness k is fitted to the experimental curve. The relationship appears to follow quite closely the actual indentation behaviour. Even though Hertz's law was developed for a spherical indenter shape, there is clearly a strong correlation here. Figure 3.12 shows the comparison between the experimental static indentation curve and the curve predicted by Hil ls et al. [1993] in Equation 3.12 using input parameters from Table 3.1. A s stated, the model is sensitive to the out-of-plane shear modulus parameter of the target material, which in our case is unknown. The in-plane shear modulus, however, is 4.6 GPa, Results 88 and provides a rough approximation for this value. Model lines for a shear modulus of 2, 4 and 6 GPa are shown in Figure 3.12. A s can be seen, the value of 2 GPa gives the most accurate fit. Due to what may be an initial stiffening effect of the experimental curve, the model line needs to be offset by approximately 0.1 mm in order to match the data. This stiffening effect could be accounted for by the compaction of the matrix or a roughness on the surface not allowing complete contact. The solution given by Hi l ls et al. also provides the relationship for the pressure seen under the penetrator. This is given by and can be seen graphically in Figure 3.13 for a load of 8000 N , which is the approximate critical load for delamination of the 5.03 mm thick plate. Note the infinite stress result at the edges which corresponds to the shear crack failures seen in the cross-sectional views of Figures 2.10 and 2.48. 3.4.1.2 Elastic Bending The next step in this stage is to predict the undamaged plate stiffness, i.e. the load-displacement relationship, K0=P/w. Using accepted plate theories, we have been able to determine this elastic response quite accurately. P (3.14) 89 Kirchoff Theory The Kirchoff plate analysis for isotropic plates is directly applicable to laminates. The analysis of these plates requires a few basic assumptions. It is required, first of all , that the thickness of the laminate be small compared to the in-plane dimensions and that the displacements, in turn, are small compared to the thickness. The laminae are assumed to be perfectly bonded so that the strain field is continuous across layer interfaces. The laminae are considered to be in a state of plane stress, and the transverse strain e z z is negligible. The analysis presented here is taken from Pierson [1994] and is exact for specially orthotropic conditions. The static solution for a simply supported rectangular laminate turns out to be a double Fourier sinusoidal series where a and b are the plate dimensions in the x and y directions, DH are the components of the bending stiffness matrix from classical laminate plate theory, and pmn is defined by the loading condition, three possibilities of which are defined in Table 3.2. Surface strains in this case are defined as oo oo Pmn sin(wTDc/a)sin(mry/6) w D | 1 ( f ) 4 + 2 ( D 1 2 + 2 D 6 6 ) © ¥ + D 2 2 K ) 4 (3.15) 71 hd2w Y and zy = hd2w 2 by2 (3.16) 2 dx and can be compared with strain gauge readings. 90 Whitney-Pagano Theory Laminated composite materials tend to be much weaker in the transverse direction than traditional homogeneous materials used in engineering applications. In order to compensate for this, Whitney and Pagano [1970] improved on the Kirchoff theory by taking into account transverse shear effects. For a simply supported edge condition the two shear rotations, around the x and y axes, respectively, and the displacement function, are: co oo xVx = 2Z2ZUmn cos(mn x/a)sm(nn y/b) (3.17) m n CO CO XVy=2Z2ZVmn sin(/727l x/a) C0s(«7T y/b) (3.18) m n GO 00 W = Z Z ^ i n « S m ( / " 7 I : , [ ; / a ) s i n ( W 7 t y/b) (3.19) m n where Umn, Vmn, and Wmn and related constants are defined in Appendix A . The plate central stiffness is then easily defined as in the Kirchoff theory by K0=P/w. Using the shear rotations of Equations 3.17 and 3.18, we may also determine the strains on the surface of the plate using the relationships « . = - ^ (3-20) 2 ox V = - - ^ ' (3-21) y 2 by • ' which can be compared to experimental strain gauge readings. 91 Results Plate stiffnesses for the T300H/F593 were calculated using both Kirchoff and Whitney-Pagano theory. Resulting stiffnesses for point loads are shown in Figure 3.14 along with the experimental stiffnesses (with indentation removed). A s can be seen, the Whitney-Pagano stiffness gives a far more accurate representation of the plate stiffness. Convergence studies of the Kirchoff and Whitney-Pagano models were necessary to determine the number of terms required for proper accuracy. The Kirchoff model required approximately 16 terms to converge and the Whitney-Pagano model required approximately 45 terms for reliable accuracy. Another validation for this model was to compare the measured strain on the backface during this stage with that calculated using Equations 3.16, 3.20 and 3.21. Figure 3.15 shows a summary of the load-strain slopes calculated by both the Kirchoff and Whitney-Pagano equations for the four thicknesses. Again, the Whitney-Pagano equations give a very accurate representation. 3.4.1.3 Peak Force at Delamination The point along the load-displacement curve at which the onset of delamination occurs has been a topic of research for many investigations. Very often this point is predicted by the impact energy as a function of plate thickness, since the measure of energy lends itself easily to experimental impact studies. Recalling, however, that the shape of the initial matrix cracks leading to delamination indicate a shear stress induced failure, it may be more appropriate to describe damage initiation in terms of force rather than energy. 92 Using a model described by Sjoblom [1987] for a low velocity impact event, the shear stress due to contact force through the thickness of the laminate is approximately distributed as shown in Figure 3.16. A balance of forces results in the relationship P = 2nrhTare (3-22) The Hertzian contact rule (for the hemispherically-tipped indenter) gives the relation P = Kca15 (3-23) where Kc is the contact stiffness and a is the depth of indentation. Assuming that the delamination onset is the result of a critical shear stress, and that the shear strength remains constant through the thickness, these relationships are then combined with a geometric relationship as a third equation. Sjoblom [1987] solved these for the case of a spherical indenter, and derived an equation of the form Pmii=C.h» (3-24) which indicates the point of delamination initiation. For his analytical solution of the spherical indenter, Sjoblom found that n= 1.5. The initiation force, Pinit, was also given as a function of diameter, indentation stiffness, and shear strength. The parameter n in these cases is a reflection of the degree to which the material and structural properties influence the target response. He determined that the critical load is less strongly affected by punch diameter (n = .75) and indentation stiffness (n = -0.5), but equally affected by shear strength (i.e. n = 1.5). 93 This formula is independent of the support size because of the assumption that the shear forces generated due to local contact are responsible for the damage initiation. In the more general, and perhaps more applicable, derivation of Davies and Robinson [1992], the critical delamination force is again found to be a function of the plate thickness with the same exponent, n= 1.5, in Equation (3.10). Their derivation is based on a crack of arbitrary size imbedded in a point loaded circular isotropic plate and uses an energy release rate value, G , to determine the onset of crack growth. The resulting analytical expression is where E, an isotropic modulus, w i l l represent both an extensional and a bending modulus i f the laminate in question is stacked quasi-isotropically and there is a sufficiently large number of layers. It should be noted that this equation is independent of both boundary size and delamination radius, making it a purely analytical expression and not requiring an empirical calibration. Jackson and Poe [1993] established a relationship between a "critical shear force" and laminate thickness. This shear force is a function of the delamination size, and is defined as where b is the delamination radius. This value varied as a function of plate thickness as (3.25) q:= 2nb (3.26) n (3.27) 94 with n varying between 1 and 1.5 associated with a constant value of the average shear stress and a constant energy release rate of mode II fracture, respectively. Rearranging this to be in the same form as Equation 3.11 then gives P,nil=C2bh" (3.28) It should be noted that finite element analyses have also been used to determine maximum force at delamination. For example, Lee and Sun [1993] used a maximum shear strain theory in the impacted region to determine the point where delamination occurs. Results Figure 3.17 shows the peak loads at delamination for the three sizes of boundaries as a function of plate thickness. Two possible equations of the same form as Equation (3.10) are shown superimposed on the data for n = 1 and n = 1.5. It seems that a closer correlation is present for n = 1.5. Figure 3.18 uses the alternative relationships offered by Jackson and Poe [1993] in Equation 3.27. The delamination radius, b, is measured as the projected radius of the delamination site using a C-scan apparatus. A s can be seen, the data seem to follow a very linear trend (n = 1 for Equation 3.27) but do not intersect the origin. Values determined here for Qr seem to match the same range of values determined by Jackson and Poe for various graphite/epoxy laminates. The purely analytical expression to predict the onset of delamination from Davies and Robinson [1992], as given by Equation 3.25, was derived in such a way that it is independent of delamination radius and is thus free from any material characterization test requirements. The equivalent isotropic E is obtained by using an average of the flexural moduli in the x-95 and y- directions. Solutions for this formula are shown in Figure 3.19 for different values of G, namely 200, 500 and 800 J/m . A s can be seen, this expression tends to underestimate the peak force by a factor of 2 - 5. 3.4.2 Stage II - Delaminated Plate A s is shown in the schematic of Figure 3.8, Stage II is characterized by an approximately linear behaviour after delamination. According to experimental observation, the plate at this point contains a circular delaminated zone of fixed size. Truly, then, the lowered stiffness in this stage is a reflection of a combination of an undamaged outer plate and an embedded circular soft zone. If we assume that the entire plate was circular and that both undamaged and damaged regions continued to behave elastically, the following would represent the compatibility condition at the plate centre: _L = _!_ + ! (3.29) Kr K0 K where KT is the stiffness of the entire plate in stage II, K0 is the stiffness of the undamaged portion of the plate, and K, the lowered stiffness of the circular "soft" zone. The stiffness relationship could also be derived in terms of the respective displacements of both parts of the deflecting plate due to a given load, P: K = (3.30) wQ(D0) + w(D) 96 where D0 and D are the isotropic bending rigidities of the undamaged and damaged plate, respectively. The key issue in modelling this stage is to successfully represent the behaviour of the damaged zone and its effect on the behaviour of the entire plate. 3.4.2.1 Initial Attempts The first attempts at modelling the damaged zone involved a representation of only the delaminated area as a circular isotropic plate with a clamped boundary condition. The strain and displacement output were then summed with that found for the elastic, undamaged plate determined from Equation (3.10). Both exact and approximate closed form solutions of this problem were taken from Timoshenko and Woinowsky-Krieger [1959] and yielded varying results, none of which were adequate. These attempts are briefly summarized in Appendix B . 3.4.2.2 Present Model The initial attempts were not successful because they either did not allow for strain calculations on the plate surface (because of the presence of logarithmic terms) and/or did not represent accurately the true conditions of a "hybrid" plate, i.e. the displacement and strain fields of the damaged and undamaged zones were not coupled. Furthermore, the nature of the aforementioned formulations made it difficult to adjust them readily to different loading ' and boundary conditions. It was therefore necessary to completely re-define the plate model from the beginning, developing an approximate energy method solution unique to the present scenario. 97 The first requirement of this model was to represent the entire hybrid plate by one closed form solution, defining a continuous displacement field. For simplicity it was decided to represent the entire plate, both damaged and undamaged parts, in Stage II as isotropic. A s a result of these assumptions, it was also necessary to approximate the plate as a circular disk. This last assumption is adequate according to the experimental results of Chapter 2 where it was found that the strain on the bottom surface directly beneath the indenter was equal in all directions. The ability to completely define the quasi-static response in the elastic stage using the Whitney-Pagano solution enabled an easy derivation of the isotropic parameters for the undamaged portion of the hybrid plate. The damaged zone would also be modelled elastically, but with a stiffness lower than that of the undamaged region as shown in Figure 3 .8 . Undamaged Plate Stiffness A n undamaged plate bending stiffness K0 is determined from the Whitney-Pagano solution for a rectangular laminated plate and applied to an undamaged isotropic circular plate with simply supported edges. The value of K0 ( = Plw) is then used to determine a bending rigidity, D0, for the equivalent circular plate. The Rayleigh-Ritz approach, requires an approximate displacement profile to represent the deformation behaviour. The first requirement of the displacement function is that it must satisfy the simply supported edge condition. The function also needs to be twice differentiable so that both slopes and strains (from the first and second derivatives, 98 respectively) could be obtained. Needless to say, it is also desired that the chosen function be as simple as possible. Two possible displacement functions fitting these requirements are w = w, f 2\ r V A J (3.31 (a)) w = WQ s i n \2a) (3.31(b)) These are shown in Figure 3.20 and are compared to the exact plate solution for a point load as given by Timoshenko and Woinowsky-Krieger [1959]. Equation 3.31 (b) proved to be inadequate because of the difficulties associated with carrying out the integration and differentiation in the subsequent analysis. Equation 3.31 (a) was found to satisfy all the requirements. The total strain energy stored in bending of the undamaged plate is given by Vv as rdr (3.32) where W(r) is defined as W(r) = d w 2 1 + — (dw\ dr2 r2 K.dr J + -2v dw d w r dr dr2 (3.33) The potential energy of a patch load representing the indenter of radius c is given by J'2TC re ( ? \ V A J rdr dQ (3.34) 99 where q = P/nc2v*> the load intensity corresponding to the punch force, P. The equilibrium condition requires that the total energy of the system be stationary with respect to the first variation in the displacement field, i.e. 8 (Vu+Q) = 0 (3.35) dw0 Knowing K0=P/w0, the undamaged bending rigidity is found to be K0(2a2-c2) D0=^^- (3.36) 0 16TC(1 + V ) where a is the outer radius of the undamaged plate. Hybrid Plate Model This same approach is then extended for the hybrid (combined damaged and undamaged) plate. Using the same displacement function for the undamaged plate, it is necessary to choose a function to represent the embedded "soft" zone of unknown stiffness and diameter. Not knowing the exact nature of the edge condition at the interface, two different possibilities are examined. A general polynomial series of the following form is used to approximate the deformation profile w = wm+c]r + c2r2+c3r3+--- ;0<r<b (3.37) where wm is the total central deflection. The coefficients, ch o f Equation 3.37 are solved in terms of w0 for both quadratic and cubic approximations. The cubic shape necessitates continuity, from the damaged to undamaged region, of the displacement, w, and the slope, 100 dw/dr. The quadratic shape requires only continuity of w. Figure 3.21 shows the profile of the possible mathematical formulas. The total strain energy of the system is obtained by the summation of the damaged and undamaged regions; namely K = V0 + V = | o 2 f ^  [ W(r) rdr + ^ jV(r) rdr dO (3.38) where W(r) is the same as in Equation 3.33. The potential energy due to the external load, Q, is again given by Equation 3.34. Since there are now two displacement fields, represented by w0 and wm, the minimum total energy of the system with respect to the first variation in both fields is (VT + a) = 0 and (VT + Q.) = 0 (3.39) 3v0 dwm This operation results in two algebraic equations in terms of a, b, c, D0, D, w0, wm and P. Now, P, a and c define the loading condition and are known quantities, and D0 is known from Equation 3.36. Since we now have 2 equations and 4 unknowns (b, D, w0, and wm), we resort to the experimental measurements for additional information. In particular, we make use of the measured central plate deflection, wm, and the measured rear surface strain, eb. B y virtue of the assumed linear elastic behaviour in Stage II, this strain varies linearly with the load, i.e. P = Ke zb where Ke is the slope of the load-strain curve in the damaged state. A s shown in Chapter 2, the strain gauge data were found to be in keeping with this linear variation. 101 Finally, using the above information eliminates the two variables w0 and wm leaving only 2 unknowns, namely, b and D that can be solved for explicitly using Equation 3.39. Determination of Damaged Bending Stiffness. D At this point, no rigorous solution has been developed to represent the damaged bending stiffness, D. Development of this parameter would be key in making the model less empirical. A s a lower bound to this stiffness, however, the solution for a completely delaminated circular plate with no restraints across the layer interfaces is found. It is known from the experimental results that a delamination area totaling the circular projected area of the damage zone occurs for each sublaminate. This can be represented by an isotropic model of n stacked plates is used. This model is very conservative, as it does not take into account the constraints at the delamination boundary, the staircase helical pattern of linked delaminations, or the friction present between plates. The friction action in this case would also include a great deal of "fibre bridging" between layers, i.e. the extension of the high modulus fibres from one layer to the next. The bending moduli (in x and y directions) for one sublaminate are first determined using classical laminate plate theory. From each of these an isotropic bending stiffness is calculated by F h3 A = h , n 2 (3.40) ' 1 2 ( l - v 2 ) 102 The bending stiffness for the entire plate, then, is simply n-Dt. The average of these bending stiffnesses as a function of plate thickness is shown in Figure 3.23 and is compared to the undamaged isotropic bending stiffness. Results Since the working model for stage II loading was developed simultaneously with the experimental program, it was feasible to carry out a rigorous parametric study. Using the measured slopes of K and Ke as input parameters to the model, the predicted delamination sizes were found for both cubic and quadratic approximations of the damage zone. Figure 3.22 shows these predictions along with the experimentally measured delamination sizes. The experimental data here were taken from the tests on the 76.2 mm x 127 mm (3" x 5") rectangular specimens. The radius, a, in the analysis was therefore approximated to be half the length of the long side, 63.5 mm. The measured delamination diameter of the 8.30 mm thick plate can be disregarded as the diameter has approached and interfered with the boundary of the plate. A s can be seen, the relationship using the quadratic approximation is fairly accurate, with the cubic approximation overestimating the delamination size by a factor of approximately 1.5-2. This follows from the fact that the condition of slope continuity in the cubic model is too restrictive. The cubic shape requires a larger delamination size in order to reach the appropriate measured strain at the centre. The slope discontinuity at the interface, produced by a quadratic approximation for the damaged profile, reflects more accurately the physical behaviour at the delamination boundary. 103 These predicted and experimental trends of the delamination diameter as a function of thickness were also observed by Cantwell and Morton [1991] for low velocity impact loading of C F R P plates. A by-product of the analysis was the prediction of the damaged bending stiffness of the soft zone, shown in Figure 3.24 for different plate thicknesses. Also shown, is the corresponding undamaged bending rigidity, D0. Note that the values of D0 for the circular plate follow closely the principal orthotropic values Dn and D22 for the original laminate. A s stated previously, we have no accurate experimental measure for the damaged bending rigidity, D. However, the predicted value is bounded at the upper limit by D0 and at the lower limit by the bending rigidity of the completely delaminated plate model. While the focus of the modelling was the Boeing standard 4 x 6" specimen, it was deemed necessary to confirm that our assumption of a circular plate for the model was not creating a large discrepancy in the results. Figure 3.25 includes the experimental and model predictions of delamination size, b, for the 114.3 mm (4.5") diameter test frame which are quite reasonable. These test results are a better reflection of the model accuracy, as the value chosen for plate size, a, is no longer arbitrary as it was in the previous test case. The appropriateness of the model may also be appraised by reversing the solution process. Using the experimental delamination size as an input variable, the slope of the load-strain curve, Ke, can be determined as seen in Figure 3.26. Since a minimum and maximum delamination diameter is recorded during the C-scan process, an upper and lower limit is calculated shown by the shaded bands in the figure. While the values are within the same 104 magnitude, it is difficult to completely assess the performance of the model without a more complete set of strain data. Post-Delamination Displacement Profile Figure 3.27 compares the predicted plate displacement profile with the experimental profile from the copper witness plate. The predicted delamination size, b, and damaged plate stiffness, K, are used as input parameters in the model equations. To eliminate the effect of elastic springback in the comparison, the load input parameter is taken as the force on the load-displacement curve corresponding to the maximum witness plate deflection (2.75 mm). It should be noted that while the plate displacement along the short side of the rectangular specimen is being measured here, the model is predicting a displacement curve for a round plate. Because of this discrepancy it was necessary to adjust the damaged rigidity so that the maximum displacements matched. In this case it can be seen that the model predicts the essential form of the profile quite accurately. Load-Strain Curve as a Function of Plate Thickness Unfortunately the lack of data points for the slopes of the load-strain curves resulted in a linear approximation through the available points to determine the remaining input constants (see Figures 2.24 and 2.38). A cursory examination of the analytical equations was performed to perhaps determine how this strain value should vary as a function of thickness. The slope of the load-strain curve, Ke, turns out to be a function of three constants, b, D, and hA. In turn, both b and D are also functions of h. B y manipulating the solution constants, it is found that the slope of the load-strain curve, Ke, varies with b only slightly and b can be 105 2 disregarded. If D is taken to vary with h , as would be the case for an isotropic plate, we find that Ke varies with h . Unfortunately, this assumes a constant isotropic modulus, E, for all thicknesses, which is not the case for the calculated values of D. A n y further attempt to establish this relationship via the model equations without any further detailed experimentation is premature. However, parametric studies using various combinations of experimental input and output values revealed that the unknown function was very close to linear. Fracture Mechanics Approach For an incremental growth in delamination, there must be a balance of the change in elastic energy stored and the external work added to the system with the work of creating new crack surfaces. B y carrying out such an energy balance of the event, a critical energy release rate, Gc, that is characteristic to the material can be calculated. Standardized tests have been developed to determine this value, as reviewed in Daniel and Ishai [1994]. Table 2.5 shows some of these values for brittle systems during mode I and II fracture. The application of a critical energy release rate in the analysis w i l l be discussed in the following formulation. The model equations can be simply solved in terms of energy release rate instead of the slope of the load-strain curve of the delaminated plate, Ke. Even without knowing the mode of fracture, a general energy release rate may be determined using the basic definition of energy release rate AWF yAx (3.41) 106 where AWp is the work of fracture, y is the crack front width and Ax denotes the increase in crack length. For our system we take the work of fracture, AWF, as the energy lost to delamination, or the area under the curve between the damaged and undamaged stiffnesses as seen in Figure 2.26. The limitations of using this value for the present material system is discussed in Chapter 2. In terms of our analytical expressions, G would be given as 1 dVT (3-42) 2nnb db where n represents the number of stacked delaminated layers, assuming a transversely aligned delamination configuration, i.e. a cylinder of damage as discussed in Chapter 2. The model is then re-derived in terms of G rather than Ke, and solved again for D and delamination size, b. It should be noted that the critical force, Pinit, at which delamination occurs, is now required as an input variable and that there are 4 possible roots for the delamination size, b. Two of these can be rejected immediately, as they are negative. Results Solving the equations for an energy release rate value, G, gives two possible positive roots for delamination size prediction as a function of thickness. It was found that for reasonable values of G, the smaller root predicts a delamination size which is an order of magnitude too small. The larger root, on the other hand, is unstable. For example, a change from 400 to 470 J/m for the energy release rate of the 5.03 mm plate produces a variation in delamination radius from 8.8 to 21.6 mm. Ab- lim ->0 AW; F 2nnbAb 107 Plugging Lee and Sun [1993] model the plug formation in a finite element analysis using fracture mechanics, but they do not have an analytical solution to predict the onset. With the present set of data, it seems that an equation of the form of Equation (3.10) would provide a semi-empirical prediction capability for the time being. Results The peak failure load at the onset of plugging can be predicted only by means of a calibrated empirical curve. The power law relationship, as with the delamination onset, fits the experiments quite well as seen in Figure 3.28. Note that the exponent, 1.5, is the same as for the delamination power law. In this case, the peak loads are essentially the same as the critical delamination loads but this is only a coincidence and should not be used as an empirical relationship. 3.4.3 Stage III - Friction After the tip of the penetrator has fully perforated the laminate and has reached the back surface, the remaining forces (points E, F in Figure 2.8) are due to friction between the penetrator and the side of the laminate. A s the static model is to be the basis for an impact or ballistic penetration model for a projectile of finite length, the present model should reflect the same condition, i.e. a penetrator of fixed length. The proposed model for stage III is the same as outlined in the previous chapter and shown again in Figure 3.29. Empirical curves, such as those shown in Figures 2.35 and 2.36 for the T300H/F593 would supply the frictional 108 forces encountered at points E and F. The thickness of the plate and the length of the projectile, supplied as a user input, would determine the displacements of points F, G and H. However, once the laminate has failed at point D, the forces encountered by the penetrator are of little interest. The energy absorbed by these friction forces is more important as this can amount to a substantial part of the perforation energy. The energy absorbed is simply calculated as the area under the curve from point E to H by 2 F (5E , h~\ + /--(3.43) where P and 8 are the loads and displacements, respectively, / is the length of the projectile and h is the thickness of the laminate. 3.5 Summary Closed form analytical solutions were identified to represent each of the loading stages of the static penetration event. The solutions were compared to the experimental results and good correlation was found in most cases. A n approximate energy solution successfully represented the hybrid nature of the delaminated plate. More experimental data would be required to further expand and calibrate the models. 109 Table 3.1 Material Parameters Used as Input for Indentation Model c - Penetrator Radius [mm] Projectile Properties - Steel v, p., - [GPa] Target Properties - T300H/F593 v 2 u 2 - [GPa] 3.81 0.3 79.0 0.3 2.0 Table 3.2 Loading Conditions for Kirchoff and Whitney-Pagano Solutions Loading Condition Pmn uniform ] 62P o ; m,n = 1,3,5,... TC mn patch \6pn . mnt . nnr\ . mnc . nnd 2 sin sin sin sin TC mn a b a b point 4P . mnt . nnr\ — sin sin ab a b 110 P / 2 , P12 Figure 3.1 Coordinate system for beam deflection calculation (from Howard et al. [1993]). Displacement of rollers Figure 3.2 Schematic load/displacement plot (from Howard et al. [1993]). Figure 3.3 Delamination propagation model for Davies and Robinson model (from Davies and Robinson [1992]). I l l Figure 3.4 Circular plate model for delamination analysis (from Ho-Cheng and Dharan [1990]). Figure 3.5 Three stages describing the compression, delamination and buckling of a thin orthotropic layer attached to a thick isotropic plate and subjected to axial compression in the.y direction (from Chai and Babcock [1985]). 112 LAMINATED la) TOTALLY DELAMINATED . (b) PARTIALLY DELAMINATED (C) Figure 3.6 Rule of mixtures analysis of stiffness loss (from O'Brien [1982]). [+30/±30/90/90] GRAPHITE/EPOXY LAMINATE Figure 3.7 Delamination size and stiffness data accumulation during quasi-static loading (taken from O'Brien [1982]). 113 T J CD O 0 1 -*\ Point A Point B Damage Mechanisms A - Delamination C - Fibre Breakage D - Plugging P u n c h D i s p l a c e m e n t Figure 3.8 Detailed idealized load-displacement curve for a brittle CFRP laminate. q = IX C t t t t t D Do Figure 3.9 Model schematic for a damaged circular isotropic plate. 114 Figure 3.10 Indentation by a rigid flat-ended cylindrical punch (taken from Hills et al. [1993]). Displacement [mm] Figure 3.11 Static indentation curve. 115 Displacement [mm] Figure 3.12 Comparison of static indentation curve with an empirical power law and the analytical expression. Edge of Indenter Punch 1400 1200 Load = 8000 N 1000 800 600 400 4-200 -6- h--1 0 1 Radial Distance [mm] -4 -2 Figure 3.13 Pressure distribution profile predicted by indentation model for a rigid flat-ended cylindrical punch. 116 30000 E E o 25000 20000 15000 10000 5000 T300H/F593 3" x 5" Opening • * •• Experimental - » Kirchoff Theory -e—Whitney-Pagano Theory 3 4 5 6 Plate Thickness [mm] Figure 3.14 Calculated stiffness using elastic plate bending equations compared with experimental results. 2.50E+06 „ 2.00E+06 E E 1 E £ • 1.50E+06 ra or — 1.00E+06 w •a o - 1 5.00E+05 4-0.00E+00 T300H/F593 3" x 5" opening n .•' / - / / / .- / • / / / - - o - - Experimental —a— Kirchoff Theory — B — Whitney Pagano Theory A •/ / / X i i i t 0 1 3 4 5 6 Plate Thickness [mm] Figure 3.15 Calculated load-strain ratios using elastic plate bending equations compared with experimental results. 117 1 1 1 Figure 3.16 Theoretical shear stress equilibrium at the edge of the contact area (from Sjoblom [1987]). •a is o 25000 20000 15000 10000 5000 * Peak Delam Force - 3" x 5" c Peak Delam Force - 1" circular • Peak Delam Force - 4.5" circular P= 1794 *h P= 750 * hA(3/2) T300H/F593 3 4 5 6 Plate Thickness [mm] Figure 3.17 Peak force at delamination onset for statically loaded T300H/F593 plates as a function of plate thickness. 118 90000 3 4 5 6 Plate Thickness [mm] Figure 3.18 Critical shear force (as defined by Jackson and Poe [1993]) as a function of plate thickness for T300H/F593 system. c 25000 20000 15000 10000 5000 * Peak Delam Force - 3x5 o Peak Delam Force -1" • Peak Delam Force - 4.5" P= 1794*h • - - P= 750*h A(3/2) -o--G=200 J/m A 2 - * - - G = 500 J/m A 2 - A - - G = 800 J/m A 2 T300H/F593 Davies and Robinson Model Plate Thickness [mm] Figure 3.19 Predicted critical loads at the onset of delamination compared with experimental results. 119 Figure 3.20 Possible displacement profile functions for undamaged plate deflection model. Figure 3.21 Possible displacement profile functions for hybrid plate deflection model. 120 70 60 T300H/F593 3 x 5" opening s= 50 40 -•— Experimental •« - Quadratic Model Cubic Model 30 --| 20 Q . 10 Not Valid 3 4 5 6 Plate Thickness [mm] Figure 3.22 Delamination size prediction compared with experimental values from C-scans. Figure 3.23 Calculated undamaged and fully delaminated bending rigidities. 121 Figure 3.24 Damaged and undamaged bending rigidities. ;ure 3.25 Delamination size prediction for two back plate opening sizes compared with experimental values from C-scans. 122 1.40E+06 1.20E+06 | 1.00E+06 E E. 2 8.00E+05 o n 6.00E+05 •a 4.00E+05 2.00E+05 4-O.OOE+00 • <r • • Ke Experimental - 4.5" Round - * - - Ke Experimental - 3 x 5 " 4.5" Round Model Prediction T300H/F593 3 x 5" Model Prediction 3 4 5 6 7 Plate Thickness [mm] Figure 3.26 Predicted and experimental load-strain slope values. 0.003 0.0025 \ 0.002 c •B 0.0015 Q 0.001 0.0005 A T300H/F593 'e. 3 x 5" opening Load = 3836 N @ w = 2.75 mm Delamination Radius = 14 mm \' \ Experimental / Profile Delamination Model \ Undamaged Plate Model X. \ \ \ 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Radius [m] 0.045 Figure 3.27 Predicted plate displacement profile compared with witness plate profile. 123 25000 20000 15000 10000 4-5000 T 3 0 0 H / F 5 9 3 * Peak Plug Force - 1" round a Peak Plug Force - 3 x 5 " o Peak Plug Force - 4.5" round P= 750 * hA(3/2) - ^ P = 1926 * h 4 5 6 Plate Thickness [mm] Figure 3.28 Predicted and experimental peak plugging force. H Indenter Displacement H Figure 3.29 Idealized friction model during stage III loading. 124 Chapter Four Conclusions and Future Work 4.1 Summary The main focus of this thesis has been to gain a general understanding of the quasi-static penetration behaviour in C F R P laminates by flat nosed cylindrical indenters. The framework for the development of a simple analytical model to describe the static penetration of C F R P plates was built with this understanding. This can be used to provide the local structural parameters required for projectile impact modelling. The penetration process was first classified according to velocity and damage mechanisms. The methodology by which the quasi-static punch test could be used for dynamic impact prediction was identified. The experimental program discussed in Chapter 2 revealed that the penetration event can be divided into three separate stages distinguished by their unique damage mechanisms. The first stage was characterized by a local indentation of the penetrator into the surface of the plate as well as the overall elastic bending of the plate. The second stage began with the sudden growth of delamination on several planes through the thickness of the plate. The presence of these delaminations caused an overall reduction in the stiffness of the entire plate 125 and an abrupt change in the displacement profile of the plate. The third stage was initiated by the final failure of the target through a plugging mechanism in which the penetrator cut through the thickness of the plate. A s the penetrator pushed out the remaining plug and passed through the plate, significant frictional forces were overcome. These observations were corroborated, in part, by the reviewed literature. Chapter 3 discussed a simple, analytical approach using closed-form solutions to model the behaviour of the brittle C F R P system. Stage I plate deflection was predicted accurately with the Whitney-Pagano equations. The equations given by Hi l ls et al. [1993] for modelling local indentation behaviour gave a good approximation but were found to be sensitive to the target out-of-plane shear modulus. The application of this model should be explored further, including a determination of the target out-of-plane shear modulus. A reasonable empirical formula of the same form as the Hertz law for spherical indenters is also a useable representation, necessitating only one calibration test to determine the indentation stiffness of Equation 3.11. The end of Stage I, signaled by the sudden onset of delamination, is best represented by an empirical formula. This formula takes on the same form as the indentation law with the same exponential power of 1.5. It also requires a calibrating static penetration test to determine the stiffness constant. Both the literature and the present experimental data set confirmed that this approach gave the best fit. The use of a critical shear force, as determined by Jackson and Poe [1993], did not fit the present data set but should be explored for a tough matrix system composite. The purely analytical solution of Davies and Robinson [1992] gave peak 126 forces 2 - 5 times below the experimental data. This approach is attractive because of its simplicity and should also be looked at further. The approximate energy method used for representing the post-delamination behaviour of the plate during Stage II of the loading process has been the most substantial advance of the present model. Requiring parameter inputs of both the plate stiffness and the load-strain ratio, the model predicts accurately the value of projected delamination radius. Experimental techniques involving the strain gauges on the rear surface have gradually been refined to the point where repeatable testing is possible. A combination of gauge dimensions, placement and bonding techniques are important factors. Other methods of retrieving this data, such as Moire interferometry and off-surface bonded gauges, should certainly be looked at. While witness plates gave a good overall displacement profile, their reliability for determining accurate curvatures is suspect because of elastic spring back after unloading. The model equations were resolved in order to replace the parameter of the slope of the load-strain curve with the more traditional material property of energy release rate. The problem with this approach is the unstable crack growth behaviour present during delamination onset. Far more energy is released than goes into crack surface formation, especially for the thinner plates. The solution was also found to be highly sensitive to energy release rates above 350 J/m . This approach merits further study, particularly for tougher materials where stable crack growth is present. A n y work in this area should be preceded by an accurate determination of both the mode I and II values of energy release rate for the material. 127 In order to make the Stage II modelling rigorous, it w i l l be necessary to develop an analytical expression for the bending rigidity, D (Equation 2.38). This should be possible using plate theory and the delamination pattern formation presently available from experimental observation. The conditions leading to the onset of plugging are not well understood. Only an empirical curve of the same form as Equation 3.24 is available to predict this event. This is clearly an area which merits further research. Stage III is characterized by the drop in load to a non-zero value after final failure of the laminated plate. The level of the frictional force resulting from the plug/plate interface is not predictable by an analytical approach. It was found, however, that an approximately constant shear stress is present for any plate thickness. This is the case for both the plug/plate and penetrator/plate interface. Once these stresses have been established, the energy absorbed by friction for any plate thickness can be determined. 4.2 Future Work The application of the current local constitutive model to a dynamic global analysis is not direct. It has been shown that the delamination growth during impact is not a localized effect. There is significant degradation of the overall stiffness of the laminate, preventing the coupling of a global dynamic model. This may not be the case for ballistic/high velocity impacts and tests would be required to see i f the damage is localized in this case. The static penetration model w i l l , however, be a useful basis for a finite element analysis where the calculated size and rigidity of the damaged zone may be incorporated directly into the mesh. 128 Future work should also include extending the present model to handle tough C F R P systems. This would involve a time-dependent change in the overall delaminated stiffness of the plate as the delamination grows during Stage II loading (see Figure 2.6). A complete set of related experiments covering a range of plate thicknesses would be required in parallel with this analysis. The tough system would also be a useful testbed to further develop the fracture mechanics approach attempted in Chapter 3. With a reliable model to represent C F R P materials, work should commence on other structural and protective F R C ' s . Difficulties arise immediately when confronted with the dissimilar nature of the force-displacement curves compared to C F R P ' s . Figure 4.1 shows the static penetration curves for Kevlar and S-2 glass over a 25.4 mm diameter opening. While this complex behaviour seems to present a daunting task, the methods developed during the current study can be applied directly to determine the necessary empirical data and model formulation. Additional tests to characterize the strain rate effects of these materials would also be required. Fortunately the methods and technologies for this are available (e.g. the split Hopkinson pressure bar). In summary, a promising foundation for a rigorous engineering model has been developed. A greater understanding of projectile impact has been made available by determining the behaviour of a laminated composite plate during static penetration. The predicted load-displacement behaviour of the static penetration test can provide the necessary parameters for modelling the local behaviour due to low velocity and ballistic impact. 129 Figure 4.1 Kevlar and S-2 glass force-displacement curves for quasi-static punch test over 25.4 mm diameter opening. 1 3 0 References Abrate, S., "Impact on Laminated Composite Materials", Appl Mech Rev, vol . 44, no. 4, 1991, pp. 155-190. Abrate, S., "Impact on Laminated Composites: Recent Advances", Appl Mech Rev, vol . 47, no. 11, 1994, pp. 517-544. 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Y . , Wu , H.T. , and Chang, F. , " A New Approach Toward Understanding Damage Mechanisms and Mechanics of Laminated Composites Due to Low-Veloci ty Impact: Part II -Analysis", Journal of Composite Materials, vol . 25, 1991, pp. 1012-1038. Cristescu, N . , Malvern, L . E . , and Sierakowski, R . L . , "Failure Mechanisms in Composite Plates Impacted by Blunt-Ended Penetrators", Foreign Object Impact Damage to Composites, 1975, pp. 159-172. Daniel, I . M . and Ishai, O., "Engineering Mechanics of Composite Materials". 1994, Oxford University Press, Davies, G . A . O . and Robinson, P., "Prediciting Failure by Debonding/Delamination", 1 s t edition, 1992, pp. 5-1-5-28, A G A R D Conference Proceedings 530, 74th meeting of the A G A R D Structures and Materials Panel, North Atlantic Treaty Organization, France. Davies, G . A . O . and Zhang, X . , "Impact Damage Prediction in Carbon Composite Structures", Int. J. 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Delfosse, D . , Poursartip, A . , Coxon, B.R.,, and Dost, E.F. , "Non-Penetrating Impact Behaviour of C F R P at L o w and Intermdediate Velocities", In: Composite Materials: Fatigue and Fracture (Fifth V o l u m e ! ed. R. H . Martin, 1995, A S T M STP 1230, American Society for Testing and Materials, Philidelphia. Garg, A . C . , "Delamination-A Damage Mode In Composite Structures", Engineering Fracture Mechanics, vol . 29, 1988, pp. 557-584. G im, C . K . , "Plate Finite Element Modeling O f Laminated Plates", Computers & Structures, vol. 52, no. 1, 1994, pp. 157-168. Harding, J. and Welsh, L . M . , " A Tensile Testing Technique for Fiber-reinforced Composites at Impact Rates of Strain", J. Mat. Sci., vol . 18, 1983, pp. 1810-1826. Hi l l s , D . A . , Nowel l , D. , and Sackfield, A . , "Mechanics of Elastic Contacts". 1 s t edition, 1993, pp. 182-193, Butterworth-Heinemann Ltd. Oxford. Ho-Cheng, H . and Dharan, C . K . H . , "Delamination During Dri l l ing in Composite Laminates", Journal of Engineering for Industry, vol . 112, 1990, pp. 236-239. Hoskin, B . C . and Baker, A . A . , "Composite Materials for Aircraft Structures". A I A A Education Series, 1986, American Institute of Aeronautics and Astronautics, Inc., New York. Howard, S.J., Phillipps, A . J . , and Clyne, T .W. , "The Interpretation of Data from the Four-Point Bend Delamination Test to Measure Interfacial Fracture Toughness", Composites, no. 2, 1993, pp. 103-112. Jackson, W . C . and Poe, C C , "The Use of Impact Force as a Scale Parameter for the Impact Response of Composite Laminates", Journal of Composites Technology & Research, vol . 15, no. 4, 1993, pp. 282-289. 132 Jenq, S.T., Jing, H.S. , and Chung, C , "Predicting The Ballistic L imi t For Plain Woven Glass/Epoxy Composite Laminate", Int. J. Impact Engng, vol . 15, no. 4, 1994, pp. 451-464. Jih, C.J. and Sun, C.T., "Prediction of Delamination in Composite Laminates Subjected to L o w Velocity Impact", Journal of Composite Materials, vol . 27, no. 7, 1993, pp. 684-700. K i m , K . , Segall, A . , and Springer, G . , S., "The Use O f Strain Measurements For Detecting Delaminations In Composite Laminates", Composite Structures, vol . 23, 1995, pp. 75-84. Lee, S.R. and Sun, C.T., " A Quasi-Static Penetration Model for Composite Laminates", Journal of Composite Materials, vol . 27, no. 3, 1993, pp. 251-271. O'Brien, T .K . , "Characterization of Delamination Onset and Growth in a Composite Laminate", Journal in Composite Materials, 1982, pp. 140-167. Pierson, M . O . , "Modelling the Impact Behaviour of Fiber Reinforced Composite Materials", Master's Thesis, 1994, The University of British Columbia. Prasad, C . B . , Ambur, D.R. , and Starnes, J.H.J., "Response O f Laminated Composite Plates To Low-Speed Impact by Different Impactors", AIAA Journal, vol . 32, no. 6, 1994, pp. 1270-1277. Razi , H . and Kobayashi, A . S . , "Delamination in Cross-Ply Laminated Composite Subjected to Low-Velocity Impact", AIAA Journal, vol . 31, no. 8, 1993, pp. 1498-1502. Sjoblom, P., "Simple Design Approach Against Low-Veloci ty Impact Damage", vol . 32, 1987, pp. 529-539, 32nd International S A M P E Symposium, S A M P E , Covina, California. Sjoblom, P., Hartness, J.T. and Cordell, T . M . , "On Low-Veloci ty Impact Testing of Composite Materials", Journal of Composite Materials, vol . 22, no. 1, 1988, pp. 30-52. Sun, C.T., "High Velocity Impact and Penetration of Thick Composite Laminates", A final report submitted to the U.S. Army Research Office, 1994, Purdue University, School of Aeronautics and Astronautics, West Lafayette, Indiana, Grant No . DAAL03-91-G-0013 . Sun, C T . and Potti, S.V., "High Velocity Impact and Penetration of Composite Laminates", vol . 4, 1 s t edition, 1993, pp. 157-165, Proceedings of I C C M - 9 , Ninth International Conference on Composite Materials, Woodhead Publishing, University of Zaragoza. Timoshenko, S.P. and Woinowsky-Krieger, S., "Theory of Plates and Shells". 1959, McGraw-Hi l l . Whitney, J . M . and Pagano, N . J . , "Shear Deformation in Heterogeneous Anisotropic Plates", J. of Applied Mechanics, 1970, pp. 1031-1036. Wu, E . , Yeh, J., and Yen, C , "Impact on Composite Laminated Plates: A n Inverse Method", Int. J. Impact Engng, vol . 15, no. 4, 1994, pp. 417-433. 133 Wu, H.T. and Springer, G.S., "Measurements of Matrix Cracking and Delamination Caused by Impact on Composite Plates", Journal of Composite Materials, vol . 22, 1988, pp. 518-532. Wu , H.T. and Springer, G.S., "Impact Induced Stresses, Strains, and Delaminations in Composite Plates", Journal of Composite Materials, vol . 22, 1988, pp. 533-559. Zhu, G . , Goldsmith, W., and Dharan, C . K . H . , "Penetration of Laminated Kevlar B y Projectiles-II. Analytical Model" , Int. J. Solids Structures, vol . 29, no. 4, 1992, pp. 421-436. Zhu, G . , Goldsmith, W., and Dharan, C . K . H . , "Penetration O f Laminated Kevlar B y Projectiles -1 . Experimental Investigation", Int. J. Solids Structures, vol . 29, no. 4, 1992, pp. 399-420. 134 Appendix A Whitney-Pagano Solution The shear rotations and displacement functions for the Whitney-Pagano solution are given by: V x = X £ u„m C O S ( O T T I x/a) sin(«7i y/b) (A. 1) m n OO 00 W y = X X sin(/M7t x/a) cos(/77t y/b) (A.2) 00 CO w = XS^ ffw si n(W 7 T x/a)sin(rat (A.3) where a and 6 are plate dimensions in the x and y directions, respectively, and JJ^jL[XL2,-L22Lu)Pmn { A A ) y J ^ 3 h^)Pmn ( A 5 ) K„,=- ^ — (A.6) 135 The constant,pm n, is determined by the loading condition in Table 3.2. Other variables are defined as: I n = Du(mn/a)2 +D66(nn/bf + kA55 Ln=(Dn+D66)(mn/a)(nn/b) L]3=kA55(mn/a) rAj^ =D 6 6 (wTc/a ) 2 +D 2 2 («7 t / 6 ) 2 +AA 4 4 L23 = kAi4(nn/b) Z33 = kA55(mn/a)2 + AA 4 4(«7t/Z>) 2 A - ^11(^-22^33 _ ^23) ~ ^12(^2^33 --^23-^13) + A 3 (-^ 12-^ 23 ~ ^22^13) (A.8) where the [D] and [A] are the bending and extensional matrices as derived from classical laminate plate theory. The variable k is a parameter introduced following the Mind l in isotropic theory and is equal to 0.833 for any material. 136 Appendix B Initial Attempts, Post Delamination Modelling The initial approach to model the delaminated area was to assume that the delamination zone could be represented by a circular isotropic plate with a clamped edge condition i.e. a zone of lower stiffness embedded in an undamaged outer plate. To test the validity of this approach, the strain at the back surface of the plate would be calculated and added to that calculated by an undamaged plate deflection. This value would then be compared to the experimentally measured value of strain. Exact Solutions Point Loaded Clamped Circular Plate Timoshenko and Woinowsky-Krieger [1959] give the exact solution to a clamped plate loaded at the centre as where the variables are defined as in Figure B .1 . Recalling that strain at any point through the thickness of the plate is given as Pr (B.1) w = 8 T I D (B.2) 137 where z is the distance from the centreline, it can be seen immediately that at r = 0, Equation B.2 contains infinite terms. This type of model, therefore, is inadequate for our purposes, as there is no way to find the strain at the back face for r = 0 to compare with experimental results. Patch Loaded Clamped Circular Plate A n exact solution which contains no logarithmic terms (thus allowing differentiation to determine strain values) is given by Timoshenko for a patch loaded plate. The derivation begins with two displacement functions of a line (ring) loaded circular isotropic plate with a built-in edge condition. With the variables defined in Figure B.2 , the first function gives the displacement value for c < r < b, P 8 T I D ,,2 2x b2 +C2 , 2 2 M R 2b b (B.3) while the second displacement function is valid for 0 < r < c 87t£> (c2+r2)log^ + 2bz (B.4) B y setting P = 2ncqdc and integrating the line load over the inner portion of the circle from 0 to c, one obtains a relationship for a patch loaded clamped circular isotropic plate. In this case the strain can be found at the bottom surface as _ _ / z d2w Jr(hending) ~ ^ ^ ^^.2 ' (B.5) where h is equal to the plate thickness. This gives 138 h P ( e 2 4nD (B.6) V Note that this is no longer a function of r, i.e. the strain is constant everywhere under the patch load. We can therefore compare the predicted strain with that found in experiment for a given displacement or load. Strain results from this model are discussed below. Approximate Solution Timoshenko and Woinowsky-Krieger [1959] also present an approximate solution for a uniformly loaded clamped plate undergoing large displacements. It is modified in our case to a patch load of radius c. Using a virtual displacement method and the approximate displacement profile a general formula is derived which also takes into account the effect of in-plane (radial) displacements. This allows the calculation of-both bending and membrane strains, where the membrane strain due to bending is given by This feature is not necessary, however, since the point of interest is always the strain at the centre point for which the membrane strain vanishes due to the assumed displacement field. Results from this model are discussed below. (B.7) (B.8) 139 Results The values for surface strain at the centre point were calculated from the exact solution (Equation B.6) and calculated strain of the approximate solution (Equation B.7). For a load of 8000 N on the 5.03 mm plate, the experimental strain was found to be 1.1%, while the exact solution of Equation B.6 yielded 3.2% strain and the strain from approximate solution of Equation B.7 gave 5.8%. The results would have been higher yet i f the strain due to the deflection of the undamaged portion were included. Obviously these models were showing too high of a curvature. Since the size of the delamination was fixed, and the bending stiffness was adjusted to give the experimental stiffness of the damaged zone, this therefore indicated that the assumed boundary conditions were incorrect. b a w • r Figure B.1 Nomenclature for exact and approximate solutions 140 Figure B.2 Nomenclature for the integrated ring-loaded exact solution. 141 

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