Fatigue Loading
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 22 october 2013 om 10:00 uur
door
Rafiullah KHAN
Master of Science in Mechanical Engineering, North West Frontier Province University of Engineering & Technology Peshawar, Pakistan
Dr.ir. R.C. Alderliesten, copromotor
Samenstelling promotiecommissie:
Rector Magnificus, Voorzitter
Prof. dr. ir. R. Benedictus, Technische Universiteit Delft, promotor Dr. ir. R.C. Alderliesten, Technische Universiteit Delft, copromotor Prof. dr. R. Curran, Technische Universiteit Delft
Prof.dr.ir. R. Marissen, Technische Universiteit Delft Prof.dr.ir. K. van Breugel, Technische Universiteit Delft Prof.dr. W. van Paepegem, Universiteit Gent
Dr. C. Rans, Carleton University
ISBN: 978-90-88917-15-8
Copyright © 2013 by Rafiullah khan
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the author.
Dedicated to
my Teachers, Family and Friends
Delamination Growth in Composites under Fatigue Loading
By
Rafiullah Khan
Fiber reinforced composites are attractive for aerospace applications due to high specific strength and stiffness. Their use has been gradually increased to 50% by weight of the aircraft over past decades. As a consequence, modern aircraft utilize composites in the primary structures like wing skin and fuselage. The use of composites in primary structures has increased the need for reliable strength assessment methodologies.
Composites are inherent to various damage types of which delamination is the most severe type of damage. Delaminations may grow due to fatigue resulting in the stress redistribution and potentially leading to structural failure, thus making fatigue an important design concern. Damage tolerance of aircraft structures is a key aspect in maintenance and safety of aircraft. For damage tolerant design of structures, the development of accurate delamination growth assessment tools is necessary.
Delamination growth is affected by both cyclic and monotonic part of the fatigue load cycle. The effect of monotonic part is known as stress ratio (ratio of minimum to maximum cyclic stress) effect on delamination growth, and it has been extensively studied in the literature. Chapter 2 provides a detailed review of the literature concerning the stress ratio effect on delamination growth.
The literature review shows that previous studies empirically relate delamination growth to a driving force parameter that seems not based on physical mechanisms. Studies are present where mechanisms of delamination growth have been investigated; however there is a lack of efforts to link these quantitatively to delamination growth models.
The objective of this thesis is the development of a mechanistic model for delamination growth that is based on the observed delamination mechanisms and the effects of monotonic and cyclic loadings in fatigue. The thesis is based on the hypothesis that both monotonic and cyclic loading affect fracture surface formation, which can be used for delamination growth characterization. The secondary objective of the thesis is the characterization of fracture surfaces for the effect of monotonic and cyclic loading. To limit the scope, delamination growth under mode I fatigue has been investigated in the thesis.
The approach of the thesis is experimental. Delamination growth is characterized experimentally both on macroscopic and microscopic levels, as described in chapter 3. Fatigue tests were performed on double cantilever beam (DCB) specimens to investigate delamination growth behavior under different stress ratios. Specimens were made from cured laminates of M30SC/DT120 carbon/epoxy prepregs. Crack closure during delamination
of the fracture surfaces was performed using scanning electron microscopy. Width tapered DCB (WTDCB) specimens were used for the delamination growth tests under fatigue with constant monotonic and cyclic load during delamination extension.
Results of the fatigue tests and microscopy are presented in chapter 4. The delamination growth rate has been related to the strain energy release rate (SERR). The SERR range has been defined such that it resembles the correct analogous to the stress intensity factor (SIF) range. For constant SERR range, the delamination growth rate is higher for higher stress ratios. Crack closure was observed to occur for the lowest stress ratio applied in the tests. Fractographic analysis of the fracture surfaces revealed broken fibers, loose fibers, hackles and striations. The striations and hackles on the fracture surfaces of WTDCB specimens were quantitatively analyzed for different combinations of monotonic load and cyclic load amplitudes. It was observed that striation spacing increased with monotonic and cyclic load. The hackle length increased with monotonic load, but decreased with the cyclic load amplitude.
Crack closure and fiber bridging marginally explain the stress ratio effect on delamination growth, as discussed in chapter 5. Crack closure increases the effective minimum load at crack tip at the lower stress ratio only. This results in higher effective stress ratio at the crack tip. In this case, the SERR range was corrected for crack closure. By plotting delamination growth rate against corrected SERR range, the data shifted to the region with higher stress ratios. To illustrate the effect of crack closure in 3D representation, delamination growth rate was plotted against SERR range and maximum SERR. It was observed that the data corrected for crack closure shifted to the higher stress ratio region, while remaining on the same crack resistance surface.
It was further observed that fiber bridging decreases the delamination growth rate. The stress ratio remains the same. It was observed that fiber bridging affects both minimum and maximum loads during fatigue resulting in same stress ratio as without fiber bridging. In a 3D representation of delamination growth rate versus SERR range and maximum SERR, the data was observed to shift to the lower delamination growth rate region due to fiber bridging. The experimental results showed that delamination growth is not a unique function of SERR range, but also depend on the stress ratio. This implies that delamination growth depends on both cyclic and monotonic loads. A two parameter model for delamination growth was developed based on the observation of the effect of cyclic and monotonic load on the fracture surfaces. Chapter 6 describes the mechanism of delamination growth and the development of the mechanistic two parameter model for delamination growth prediction. The two parameter components in the model are superimposed rather than multiplied in agreement with the superposition of the effects of cyclic and monotonic loads observed with microscopic features on the fracture surfaces. The two parameter model for delamination growth represents a crack resistance surface for the material in the 3D coordinates of delamination growth rate versus SERR range and maximum SERR.
vii
The model has been implemented using data from the delamination growth experiments. The surface fitting tool of the commercial software MATLAB was used to obtain the equation. To validate the model, experimental data was taken from the literature. The predictions with the model and the reported experimental observations were observed to be in good agreement. The current model is different from previous models in that the relation between delamination growth and correlating parameters is no longer a simple fit of the experimental data by regression. The fit is rather an educated fit based on the observed contribution of monotonic and cyclic load components on fracture mechanisms. The two parameters in the model are superimposed to describe contribution of the load components. In previous two parameter models the terms were multiplied without justification using the physics of delamination growth.
The conclusions of the thesis are summarized in chapter 8. It can be concluded that the effect of monotonic load on delamination growth is not fully explained by crack closure and fiber bridging. The delamination growth should be characterized using both monotonic and cyclic load components. These load components affects delamination growth at microscopic level independent of one another. The two parameter terms in the model are added in conjunction to the superposition of the effects of these parameters on microscopic features. It is concluded that the model can be extended to the delamination growth in different modes of fracture.
Nomenclature ... xiii
Abbreviations ... xvii
Chapter 1 Introduction ... 1
Chapter 2 Literature review ... 7
2.1 Introduction ... 7
2.2 Classification of the investigations ... 8
2.2.1 Effect of stress ratio on mode I fatigue delamination growth... 8
2.2.2 Effect of stress ratio on mode II fatigue delamination growth ... 15
2.2.3 Effect of stress ratio on mode III fatigue delamination growth ... 20
2.2.4 Effect of stress ratio on mixed fatigue delamination growth ... 21
2.3 Discussion ... 23
2.4 Conclusions ... 25
Chapter 3 Experiments and data evaluation ... 31
3.1 Introduction ... 31
3.2 Effect of stress ratio on mode I delamination growth ... 31
3.3 Crack closure effect on mode I delamination growth ... 36
3.4 Effect of fiber bridging on mode I delamination growth ... 37
3.5 Effect of width tapered DCB configuration ... 42
3.6 Microscopy of mode I fatigue delamination growth ... 45
Chapter 4 Experimental results ... 51
4.1 Introduction ... 51
4.2 Mode I fatigue delamination growth ... 51
x
4.2.4 Mode I fatigue delamination growth in WTDCB specimens ... 61
4.3 Microscopy results ... 64
4.3.1 SEM Examination of fracture surfaces for the effect of stress ratio ... 64
4.3.2 Roughness measurements of fracture surfaces using LCSM... 66
4.3.3 In-situ SEM examination of the DCB specimen edges during monotonic loading ... 67
4.3.4 SEM examination of the edges of the DCB specimens after fatigue tests ... 69
4.3.5 Discussion of the SEM examination for the stress ratio effect ... 71
4.3.6 SEM examination of the fracture surfaces of WTDCB specimens ... 72
4.4 Discussion ... 80
Chapter 5 Effect of crack closure and fiber bridging on mode I fatigue
delamination growth ... 83
5.1 Introduction ... 83
5.2 Effectof crack closure on mode I fatigue delamination growth ... 83
5.3 Effect of fiber bridging on mode I fatigue delamination growth ... 87
5.4 Conclusions ... 92
Chapter 6 Two parameter model for delamination growth ... 95
6.1 Introduction ... 95
6.2 Mechanism of delamination growth ... 96
6.2.1 Hackle formation during matrix decohesion ... 97
6.2.2 Striation formation during fiber matrix decohesion ... 99
6.3 Monotonic and cyclic load contribution related to fractographic features ... 100
6.4 Development of two-parameter model principle ... 101
xi
6.4.2 Relation of striation spacing with cyclic and monotonic loading ... 106
6.4.3 Macroscopic delamination growth ... 107
6.5 Implementation of the model ... 114
6.6 Verification of the model with data sets from the literature ... 122
6.6.1 Verification with data from Hojo‟s work ... 122
6.6.2 Verification with data from Bathias work ... 127
6.6.2 Verification with data from Mall‟s work ... 131
6.7 Discussion of the model ... 135
Chapter 7 Discussion ... 139
7.1 Aspects related to general delamination characterization ... 139
7.1.1 Opening modes ... 140
7.1.2 Monotonic and cyclic load contributions... 144
7.1.3 Ply orientations ... 145
7.2 Delamination growth experiments ... 147
7.2.1 Selection of experiment and specimen ... 147
7.2.2 Considering fiber bridging ... 148
7.3 Fractographic evaluation... 149
7.4 Characterization of delamination ... 150
7.4.1 Similitude principles ... 150
7.4.2 Plotting delamination resistance data ... 151
7.4.3 Boundaries of crack resistance surface in mode I delamination growth ... 151
7.4.4 Effects of crack closure and fiber bridging on delamination growth ... 152
7.5 Mechanistic model for delamination ... 153
7.6 Contribution of the present work to the stress ratio effect evaluation ... 154
xii
8.1 Characterization of delamination growth ... 161
8.2 Effect of crack closure and fiber bridging on delamination growth ... 161
8.3 Fractographic observations ... 161
8.4 Mechanistic model for delamination growth ... 162
Appendix A Strain energy release rate. ... 163
A.1 Strain energy release rate ... 163
A.2 Derivation of strain energy release rate range ΔGs ... 164
A.3 Strain energy release rate range in case of crack closure ... 165
Appendix B Fatigue delamination growth tests results. ... 167
B.1 Effect of stress ratio on delamination growth rate ... 167
B.2 Crack closure tests... 169
B.3 Fatigue delamination growth tests for fiber bridging effect investigation ... 171
B.4 Delamination growth tests under constant Gmax and constant ΔGs using width tapered DCB specimens... 176
Appendix C Microscopy results ... 187
C.1 Hackles ... 187 C.2 Striations ... 200
Samenvatting ... 209
Acknowledgements ... 213
Curriculum vitae ... 215
List of Publications ... 217
xiii
N
OMENCLATURE
Symbol
Description
Unit
A a ac Δa B b C Co da/dN E1 E2 Gcloase Gmax Gmin ΔG ΔGs ΔGeff ΔGI ΔGII Gmax GImax GIImax GIIImax
Power law constant Delamination length Cutting thread position Delamination extension Plate width
Width of double cantilever beam specimen Compliance
Compliance offset
Delamination growth rate Longitudinal Young‟s modulus Transverse Young‟s modulus SERR at crack closure Maximum SERR Minimum SERR
Arithmetic SERR range SERR range
SERR range corrected for crack closure SERR range under mode I
SERR range under mode II Maximum SERR
Maximum SERR under Mode I Maximum SERR under Mode II Maximum SERR under Mode III
[-] [mm] [mm] [mm] [m] [mm] [m/N] [-] [m/cycle] [GPa] [GPa] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2]
xiv Imin GIImin GIIImin Gth GIth G IIth Gc GIc GIIc Kmax KImax KIImin Kth KIth KIIth Kc KIIc ΔK KIc ΔKI ΔKII ΔKeq k L N P
Minimum SERR under Mode II Minimum SERR under Mode III Threshold SERR
Threshold SERR under Mode I Threshold SERR under Mode II Critical SERR
Critical SERR under Mode I Critical SERR under Mode II Maximum SIF
Maximum SIF under Mode I Minimum SIF under Mode II Threshold SIF
Threshold SIF under Mode I Threshold SIF under Mode II Critical SIF
Critical SIF under Mode II SIF range
Critical SIF under Mode I SIF range under Mode I SIF range under Mode II
Equivalent stress intensity factor range Taper of WTDCB specimen Hackle length Number of cycles Load [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [J/m2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [MPa.m1/2] [-] [mm] [-] [N]
xv R s Sl Sm σ δ γ υ12 ∏ Stress ratio Striation space
Compliance of fully open crack Compliance of segment
Stress
Displacement
Stress ratio parameter Poison‟s ratio for 12 plane Potential energy [-] [mm] [m/N] [m/N] [Pa] [mm] [-] [-] [J]
COD DCB FBG LCSM LEFM SEM SERR SIF UD WTDCB
Crack opening displacement Double cantilever Beam Fiber bragg grating
Laser confocal scanning microscope Linear elastic fracture mechanics Scanning electron microscope Strain energy release rate Stress intensity factor Unidirectional
C
HAPTER
1
I
NTRODUCTION
Fiber reinforced polymer composites are attractive for aerospace applications because of their exceptional strength and stiffness-to-density ratios. Aerospace industries use composites to lower weight of aircraft structures to increase their fuel efficiency. The composite components used in military and commercial aircrafts are horizontal and vertical stabilizers, wing skins, fin boxes, flaps, spoilers, doors, elevator elements, rudders and other parts [1].
Initially composites were used only in secondary aircraft structures and its use was limited to about 2% by weight of the aircraft [2]. However with improved material and knowledge, composites are now being adopted in primary aircraft structures. Modern aircrafts like the Boeing 787 and the Airbus A 350 have wing skins and fuselage made of composites. The weight percentage of composites in these aircrafts is 50-53 respectively with an increased fuel efficiency of 20-23% compared to similar sized aircraft utilizing aluminium [3-4].
2
Figure 1.1: Illustration of use of composites in Boeing 787 [5] (a) and A 350 [6] (b)
The use of composites in primary aerospace structures has increased the need for a higher reliable design. Composites are inherent to various damage types including fiber breakage, delamination and micro cracking in the matrix. The occurrence of damage in composite structures can never be entirely avoided. The structures should be deigned to function safely despite the presence of damage, a concept known as damage tolerance [7]. The damage tolerance analysis of a composite structure is based on the initial damage size, damage growth and residual strength of the structure after damage growth [8].
Delamination is the most severe type of all types of damages. The strength and stiffness of composite structures reduce due to delamination, potentially leading to structural failure [9]. The causes of delamination are bad layups of plies during manufacturing, low velocity impact of tools during assembling and service, overstressing or fatigue. Fatigue is a major cause of delamination growth in composite structures, making it a primary design concern. For the adoption of damage tolerance design approaches in primary composite structures in aerospace applications, the development of accurate fatigue delamination growth assessment tools is necessary.
The delamination growth under fatigue loading should be described using known load cases. Adopting similitude principles this implies that delamination data is needed for isolated cases (mode I and mode II) in relation to the driving forces that come from the applied load. In fatigue this is known to be the cyclic part and the monotonic part of the load cycle. This has been extensively investigated and reported in the literature [10-14], but what seems remarkable that most of these studies empirically relate the delamination growth to a driving force parameter that seems not based on physical mechanisms. Some only use the monotonic load as reference (using maximum strain energy release rate (SERR), Gmax) [11-12], others
use the cyclic portion (SERR range, ∆G) [15-16] and all relate the different curves to the effect of the stress ratio. At some point people realise that both monotonic and cyclic load
3
contribute and they propose two-parameter models [10, 17], but the formulation seems to lack the substantiation from physical mechanisms.
The motivation to this research is that damage tolerance can only be addressed well if the physics in damage characterization and prediction are well acknowledged.
The hypothesis for the research presented in this thesis is that the damage growth is described by both the cyclic and the monotonic part of the load cycles, but in a coherent way and substantiated with the microscopic delamination mechanisms. This hypothesis implies that any combination of monotonic and cyclic load yields a growth rate characteristic for the interface, where depending on the representation (3D graph, 2D graph) the stress ratio effect should be attributed to that characteristics resistance.
The primary objectives of the thesis are
Analysis of the mechanism behind the effects of monotonic and cyclic loading in delamination growth.
Development of a mechanistic model for assessing delamination growth in composites under different stress ratios.
According to the above hypothesis, it means that the microscopic mechanisms should be characterized in order to attribute the delamination formation to either the monotonic or the cyclic load, or a combination of the two load components.
As a consequence, the secondary objective of the thesis is the characterization of fracture surfaces for the effect of monotonic and cyclic loading.
The role of crack closure and fiber bridging have been reported in the literature for the stress ratio effect [18] or the monotonic load effect on delamination growth. These mechanisms are also addressed in the thesis in context of the above hypothesis.
The approach of the thesis is experimental. The delamination growth is characterized both at macroscopic and microscopic levels. On macroscopic level, fatigue experiments are performed to characterize delamination growth rates under varying monotonic and cyclic loadings to generate data sets for model implementation. The mechanism of crack closure and fiber bridging are investigated for the explanation of the monotonic loading effect on delamination growth. On microscopic level, the fracture surfaces under different monotonic and cyclic loadings have been analyzed for the mechanism of delamination growth. The delamination growth rate under different monotonic and cyclic loadings is linked to the microscopic feature formation for the development of a mechanistic model.
To keep the focus, the thesis is limited to mode I fatigue delamination growth. Single composite systems have been used in the experiments. The fatigue experiments are performed under pure tension at room temperature and atmospheric pressure conditions.
Next chapter is review of the literature studies covering investigations of the stress ratio effect on delamination growth. The review covers delamination growth under all fracture modes. Prominent literature studies are discussed in detail. Chapter 3 describes the experiments performed for generating data sets for modeling delamination growth. Two main experimental programs were performed. First is the characterization of delamination growth and second is
4
the microscopy of fracture surfaces. Chapter 4 presents experimental results. In this chapter, observations of the fatigue experiments, delamination growth rates versus SERR data sets and microscopy results are presented. Chapter 5 presents analysis of the crack closure and fiber bridging and their role in the effect of stress ratio on delamination growth. In chapter 6, a two- parameter mechanistic model is developed for the delamination growth under fatigue. The model has been implemented using experimental results and validated using case studies from the literature. Chapter 7 is the discussion about the contribution of the developed model and its significance over previous models. Chapter 8 presents the conclusion of this thesis.
References
1. Muzamdar, S.K., Composites manufacturing : materials, product, and process engineering,
CRC Press LLC. 2001.
2. Quiter, A., Composites in Aerospace Applications, IHS ESDU, USA. www.ihsesdu.com. 3. Airbus information, A350 XWB: SHAPING EFFICIENCY. www.airbus.com, 2012. 4. Hale J., Boeing 787, from the Ground Up. QTR_4,06, AERO, www.Boeing.com. 5. FREISSINET S., www.1001crash.com, 2011.
6. McConnell V. P., Past is prologue for composite repair. www.reinforcedplastics.com, 2011. 7. Sierakowski, R.L. and G.M. Newaz, Damage Tolerance in Advanced Composites. 1995:
Technomic Publishing Company.
8. Rodi R., The residual strength failure sequence in fiber metal laminates, PhD Thesis. 2012, Aerospace Faculty Tu Delft Nederland.
9. Harris, B., Fatigue in composites. CRC Press, Washington DC.
10. Hojo M., Tanaka K., Gustafson C-G., Hayashi R., Effect of stress ratio on near-threshold
propagation of delamination fatigue cracks in unidirectional CFRP. Compos Sci Technol,
1987. 29: p. 19.
11. Mall S., Ramamurthy G., Rezaizdeh M. A. , Stress Ratio Effect on Cyclic Debonding in
Adhesively Bonded Composite Joints. Composite Structures, 1987. 8: p. 15.
12. Martin R. H., Murri B., Characterization of Mode I and Mode II Delamiantion Growth and
Thresholds in AS4/PEEK Composites. Composite Materials: Testing and Design (Ninth
Volume), ASTM STP 1059, S. P. Garbo, Ed., ASTM Philadelphia, 1990, pp. 251-270, 1990. 13. Russel A.J., Street K. N,, The Effect of Matrix Toughness on Delamination: Static and fatigue
Fracture under Mode II Shear Loading of Graphite Fiber Composites. Toughened
Composites, ASTM STP 937, N. J. Johnson, Ed., American Society for Testing and Materials, Philadelphia,1987, pp-275-294, 1987.
14. Sutton A.S., Fatigue Crack Propagation in an Epoxy Polymer. Engineering Fracture Mechanics, 1974. 6: p. 8.
5 15. Gustafson C.G, Hojo M., Delamination fatigue crack-growth in unidirectional graphite epoxy
laminates. journal of Reinforced Plastics and Composite, 1987. 6(1): p. 16.
16. M. Beghini L..B., P. Forte., Experimental investigation on the influence of crack front to fiber
orientation on fatigue delamination growth rate under mode II. Composite Scince and
Technology, 2006. 66: p. 7.
17. Jia J, D.J., Study of load ratio for mode-I fatigue fracture of wood–FRP–bonded interfaces. Journal of Composite Materials, 2004. 38(14): p. 30.
18. Ritchie R.O., Mechanisms of Fatigue Crack Propagation in Metals, Ceramics and
Composites: Role of Crack Tip Shielding. Material Sience and Engineering, 1988. A103: p.
15-28.
C
HAPTER
2
L
ITERATURE
R
EVIEW
Abstract
In this chapter review of the previous studies on fatigue stress ratio effect on delamination growth in composites is presented. This chapter provides a basis for the motivation of the research presented in the thesis by pointing gap in the investigations. The previous studies have been classified according to fracture modes. Prominent studies are discussed in detail.
2.1 Introduction
Delamination and delamination growth under fatigue loading are of concern in composite structures. The strength and the stiffness of composite structures reduce due to delamination, which can lead to loss of structural integrity and potentially failure of the structure. In order to increase the fatigue life and reliability of composite structures, researchers have extensively investigated the delamination growth in composites under fatigue in past few decades [1-19]. The delamination grows under different modes of fatigue loadings namely, mode I (tensile), mode II (shear), mode III (transverse shear) or combination of these, as shown in figure 2.1.
Figure 2.1: Modes of opening (a) mode I /tensile mode (b) mode II /shear mode (c) mode III/
transverse shear mode
Under all modes of fatigue, the monotonic and cyclic loads, both are known to affect the delamination growth behavior. If the applied stress range is used for the delamination growth rate characterization, Smax (maximum stress), Smin (minimum stress), Smean (mean stress) or the
stress ratio (ratio of minimum to maximum stress, figure 2.2), is left to describe the effect of monotonic loading. In the presentations of damage growth against stress range ∆S, strain energy release rate (SERR) range ∆G or stress intensity factor (SIF) range ∆K, the stress ratio is known to affect the curves with higher stress ratios at the left side and lower stress ratios at the right side of the plot. The stress ratio effect is characterized by an increase in delamination growth rate with increasing stress ratio for the same cyclic load range.
In order to implement the effect of stress ratio in delamination growth prediction models, the understanding of the effect is required. Many researchers dedicated their research in this regard [1-2, 4-5, 11, 20-38]. The previous studies cover the investigation of the effect for
8
different modes of fracture, different materials and environments. Various delamination growth models have been proposed by researchers for the delamination growth prediction. Some studies also investigate the mechanism for the effect of stress ratio.
This chapter presents a comprehensive review of the leading investigations on stress ratio effect in delamination growth. Section 2.2 classifies the investigations. Section 2.3 is a general discussion of the investigations. The conclusions of the review are presented in section 2.4.
Figure 2.2: Definition of the fatigue stress cycle
2.2 Classification of the investigations
The previous investigations have been classified according to fracture modes i.e. mode I, mode II, mode III and combination of these. The investigations for each mode have been listed in tables accordingly in the following sections. These tables list the author, material system and the proposed model for delamination growth prediction.
2.2.1 Effect of stress ratio on mode I fatigue delamination growth
The delamination grows under mode I fatigue under tension loading. The crack is closed under mode I compression, resulting in no delamination growth. The range of stress ratio for delamination growth under mode I is illustrated in figure 2.3 by plotting stress amplitude Sa
against Sm. The stress ratio range for delamination growth is -∞<R<1. In the
tension-compression region i.e. -∞<R<0, the delamination grows only in the tension part of the fatigue cycle with effective stress ratio R=0.
9
Figure 2.3: Illustration of the relation between stress ratio R, the stress amplitude Sa and the
mean stress Sm for mode I
The previous studies of the stress ratio effect on mode I fatigue delamination have generally focused on the characterization of material behavior. Table 2.1 lists the previous studies. The prominent investigations are discussed in depth in the following paragraphs.
Bathias et.al [2] investigated the effect of stress ratio on mode I delamination growth in Brochier/1452 laminates. The material was tested under three different stress ratios equal to 0.01, 0.4 and 0.5 respectively. The delamination growth rate was related to the range of SERR
ΔG defined as Gmax - Gmin. For same ΔG, the delamination growth was higher for higher stress
ratio. A model was proposed for delamination growth prediction in terms of ΔG as given in table 2.1.
The model proposed by Bathias is stress ratio specific. For each stress ratio, the parameters of the model should be determined through experiments. The use of ΔG can mislead the investigation of the stress ratio effect. Rans et al. [39] demonstrated that ΔG is a similitude of the combination of cyclic load amplitude and mean cyclic load. The stress ratio effect is not correctly represented due to specific combination of the two loadings in ΔG formulation .The use of ΔG results in an inconsistent stress ratio effect presentation for different materials as given in table 2.2. The fracture surfaces were analyzed for stress ratio effect in the study. Striation marks have been reported at the lowest stress ratio.
10
Table 2.1: Overview of literature on the effect of stress ratio effect in mode I fatigue
delamination growth
Author Material system Delamination growth model Validation data
stress ratio range Reference
Bathias C. et al. Brochier fabric/1452 𝑑𝑎 𝑑𝑁= 𝐴𝛥𝐺 𝑛 0.01, 0.4 and 0.5 [2] Hojo M. et. al T300/914, T300/#2500 𝑑𝑎 𝑑𝑁= 𝐴 ∆𝐾𝑒𝑞 𝑛 0.1, 0.2, 0.3, 0.5 and 0.7 [5] Hojo M. et. al T300/914, T300/#2500 𝑑𝑎 𝑑𝑁= 𝐴 ∆𝐾𝑒𝑞 𝑛 0.1, 0.2, 0.3 and 0.5 [24] Hojo M. et. al FiberT300,T800/ Matrix 3601, #3631 𝑑𝑎 𝑑𝑁= 𝐴 ∆𝐾𝑒𝑞 𝑛 0.2 and 0.5 [27] Hojo M. et. al alumina fiber/ bisphenol 𝑑𝑎 𝑑𝑁= 𝐴 ∆𝐾𝑒𝑞 𝑛 0.1 and 0.5 [32] Mall S. et. al T300/5208 adhesive EC 3445 𝑑𝑎 𝑑𝑁= 𝐴𝛥𝐺 𝑛 0.1, 0.5 and 0.75 [20] Gustafson C.G. et. al T300/914C 𝑑𝑎 𝑑𝑁= 𝐴𝛥𝐺 𝑛 0.1, 0.3 and 0.5 [4] Martin R. H. and Murri B. AS4/PEEK da dN=A 𝐺max 𝐵 1 − 𝐺𝑡 𝐺𝑚𝑎𝑥 𝐷1 1 − 𝐺𝑚𝑎𝑥𝐺 𝑐 𝐷2 0.1 and 0.5 [25] Atodaria D. R. et. al E-glass fabric/fiberite 977-3 da dN=B ∆ G 1-γ . G average γ p 0.1, 0.3 and 0.5 [30] Dalmas and Laksimi
E-glass/ES 70 No model proposed 0.01 and 0.3 [40]
Jia J. et al. wood-FRP 𝑑𝑎
𝑑𝑁= 𝐵∆𝐺𝑒𝑞
𝑚 0.1, 0.3 and 0.5 [34]
Khan R.
11
Table 2.2: Effect of increasing stress ratio on delamination growth with using ΔG as
correlating parameter
Author Material system Effect on da/dN Reference
Hojo M. T300/914, No effect [5]
Hojo M. T300/#2500 Decrease [5]
Bathias C. Brochier fabric/1452 Increase [2]
Mall S. T300/5208, adhesive EC 3445 No effect [20]
Khan R. M30SC/DT120 Decrease [19]
The approach of ΔG = Gmax - Gmin was followed by other researchers for delamination growth
characterization. Mall [20] investigated the effect of stress ratio on debonding of T300/5208-EC3445, adherend-adhesive system under mode I fatigue. After comparing the results of the debond growth rates versus Gmax and ΔG, Mall observed that the data collapses on a single
curve in case of ΔG. The ΔG was thus proposed as the controlling parameter for the debond growth rate. An equation similar to Bathias was proposed for the prediction of fatigue debonding.
Gustafson [4] compared stress ratio effect on mode I delamination growth in T300/914C laminates using Gmax and ΔG as correlating parameters. The results were similar to Mall, i.e.
the delamination growth rates under different stress ratios collapsed on a single curve in case of ΔG. A delamination growth prediction model was proposed in terms of ΔG. In this study crack closure was investigated. The crack closure was observed for the tests under stress ratio equal to 0.1. For this test, the SERR range was corrected for the crack closure using ΔGeff =
Gmax - Gop, where Gop was determined from load at crack opening in the fatigue cycle. The
source of crack closure was not investigated in the study.
Hojo [5] investigated the stress ratio effect on mode I delamination growth in T300/914 and T300/#2500 laminates. Test results were compared using Gmax, ΔG and ΔK as correlating
parameters. In case of ΔG, the stress ratio effect was negligible in the power law region for T300/914 laminates, while a small dependency was observed in the threshold region. For T300/#2500 laminates, the stress ratio dependency was obvious for all three correlating parameters i.e. Gmax, ΔG and ΔK. In order to merge different stress ratio curves into a single
curve, Hojo proposed a correlating parameter ΔKeq thatwas defined as:
∆𝐾𝑒𝑞 = ∆𝐾 1 − 𝑅 −𝛾 = ∆𝐾1−𝛾𝐾 𝑚𝑎𝑥
𝛾
(2.1)
Where γ (0 ≤ γ ≤ 1) is an empirical parameter used to account for the relative contribution of
cyclic stress and maximum stress in the delamination growth. The γ was determined by plotting ΔK against the stress ratio parameter (1-R) at given delamination growth rates as shown in figure 2.4 and fitting a straight line to a selected curve using equation
12
∆𝐾 = ∆𝐾0 1 − 𝑅 𝛾 (2.2)
Where ∆K0 is the SIF extrapolated to R=0 at a given delamination growth rate.
Figure 2.4: Plotting ΔK versus (1-R) for various delamination growth rates [5]
By examining test data sets from literature, it was observed that the value of γ is not unique for different delamination growth rates. In different literature studies, different γ values has been reported which implies that γ is not a material constant [41]. The consequence of a variable γ is that the delamination growth rates under different stress ratios only merge at the specific delamination growth rate that was used for γ evaluation. In another study undertaken by Hojo et al. [27], this discrepancy became obvious that is illustrated by figure 2.5. The figure shows the delamination growth versus ΔKeq for T300/#3601 and T800/#3631. For
T300/#3601, different stress ratio curves come closer only at the lowest delamination growth rates because it was used for γ calculation. At higher delamination growth rates, the effect is present and distinct curves exist for different stress ratios.
Hojo proposed the following empirical model for delamination growth prediction in terms of
ΔKeq
𝑑𝑎
𝑑𝑁 = 𝐴 𝛥𝐾𝑒𝑞 𝑛
(2.3)
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Figure 2.5: Crack growth rate versus ΔKeq for T300/#3601 and T800/#3631 at various stress
ratios [27]
The equation 2.3 is two-parameter empirical model in terms of Kmax and ΔK. These two
parameters are multiplied in the study, however no physical mechanism has been devised for this multiplication.
In his study, Hojo observed crack closure in fatigue test under lowest stress ratio i.e. 0.1. The source of crack closure was not investigated. The value of ΔK was corrected for crack closure, which shifted test data to higher stress ratio region in da/dN versus ΔK plot.
Hojo adopted approach of ΔKeq in other studies [24, 27, 32]. In [24], Hojo investigated the
effect of air and water environment on mode I delamination growth under different stress ratios. In [27], effect of the matrix resin growth was investigated. Test specimens were made from T300/3601, T300/3631, T800/3601 and T800/3601 prepregs. In [32], the effect of temperature was investigated. The tests were performed at room temperature and at a temperature equal to 77 K. In all of the above studies, Hojo proposed equation 2.3 for the prediction of mode I delamination growth under different stress ratios.
Jia [34] investigated the stress ratio effect in wood/fiber reinforce polymer laminates under mode I fatigue. Jia extended Hojo‟s approach of ΔKeq, to SERR. An equivalent SERR range
ΔGeq was proposed as correlating parameter. Following model was proposed for the
14
𝑑𝑎
𝑑𝑁 = 𝐵 𝛥𝐺𝑒𝑞 𝑚
(2.4)
where B and m are the power law parameters that depend on the material.
Three formulation of ΔGeq were compared by fitting the experimental data under different
stress ratios. These formulation were
∆𝐺𝑒𝑞 = ∆𝐺 1−𝛾 𝐺𝑚𝑒𝑎𝑛𝛾 (2.5) ∆𝐺𝑒𝑞 = ∆𝐺 1−𝛾 𝐺𝑚𝑖𝑛
𝛾
(2.6) ∆𝐺𝑒𝑞 = 𝛥𝐺/ (1 − 𝑅) 1−𝛾 (2.7)
In above equations Gmean and Gmin are the mean and minimum SERR. The ΔG =Gmax - Gmin
and γ is the material parameter, representing contribution of the loading parameters in delamination growth. The value of γ was different in each of the above equation. Based on the maximum value of the coefficient of determination for equation 2.5, Jia defined ΔGeq
according to equation 2.5. Jia model for delamination growth rate is empirical. The study lacks the link between model and physical mechanism of delamination growth.
Atodaria et al. [30] proposed a two parameter model for the prediction of mode I delamination growth at various stress ratios using the SERR approach. The proposed model is given by the following equation
𝑑𝑎
𝑑𝑁 = 𝐵 ∆ 𝐺 1−𝛾
. 𝐺 𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝛾 𝑝 (2.8)
Where B, γ and p, are the equation parameters determined by iterations to best fit the experimental data. The γ indicates the relative influence of √Gaverage and Δ√G on
delamination growth rate.
The Δ√G=√Gmax-√Gmin and √Gaverage was defined as
𝐺 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = 1/𝑛 𝐺𝑚𝑎𝑥 𝐺 𝑤
𝐺𝑡
1/𝑤
(2.9)
In the above equation n is the number of divisions of the fatigue cycle between Kmax and Kth
(Figure 2.6 [30]), w is the weighting factor that was based on the hypothesis that in each fatigue cycle the crack grows in a slow but progressive manner as the stress increase from minimum to maximum. The growth is lower at lower stresses and higher at higher stresses in a given cycle due to which the SERR at each division should be weighted differently. The w was calculated by iterations to best fit the experimental data.
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The model proposed by Atodaria requires five equation parameters to be determined by iteration to best fit the experimental data. In his study, Atodaria didn‟t provided physical evidence for the progressive crack growth in a fatigue cycle.
Figure 2.6: Division of a fatigue cycle in n equal divisions for calculating the weight average
SIF [30]
2.2.2 Effect of stress ratio on mode II fatigue delamination growth
The mechanism of delamination growth under mode II fatigue is different from mode I. In mode II, there is no crack closure effect, the delamination grows under both positive and negative values of the stress ratio and the growth is affected by the shear friction of the fracture surfaces that result in cusp and roller formation on the fracture surfaces [42]. The effect of stress ratio in mode II is also different from mode I [43]. Using several case studies of mode II delamination growth from the literature, Rans et al. [39] demonstrated that the effect of the stress ratio become negligibly small when the delamination growth rate is correlated SERR range defined as ΔGs= (√Gmax-√Gmin)2. In mode I, a significant stress ratio
effect was observed for ΔGs.
The relation of stress ratio and cyclic load amplitude is illustrated in figure 2.7 for the mode II fatigue delamination growth. The delamination growth behaviour is same on the two sides of the symmetry line (R=-1) as shown in the figure.
The previous studies of the stress ratio effect on mode II delamination growth are listed in table 2.3. The prominent investigations are discussed in the following paragraphs.
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Figure 2.7: Illustration of the relation between stress ratio R, the stress amplitude Sa and the
mean stress Sm for mode II
Russel and Street [21] investigated the stress ratio effect on mode II fatigue delamination growth in different material systems as mentioned in table 2.3. Two stress ratios, 0 and -1, were used in the fatigue tests. The delamination growth was characterized using ΔGII as the
correlating parameter that was defined as ΔGII = Gmax for R=0 and ΔGII =Gmax + Gmin for R=
-1. For the same ΔGII, the delamination growth was higher for R= -1 in all materials tested.
Russel and Street have explicitly taken ΔGII as the similitude for the cyclic load as shown by
figure 2.8. The figure shows the load cycles and the corresponding SERR cycles under the stress ratios 0 and -1.
Figure 2.8: Relation between cyclic load and cyclic SERR for R=0 and R= -1, the + and –
signs refer to the shear direction [21]
The approach of ΔG was followed by other researchers in a similar way to Russel and Street. Mall [22] investigated mode II stress ratio effect in debonding of T300/5208/EC3445. The
17
specimens were tested under two stress ratios equal to -1 and 0.1. The delamination growth rate was higher for R= -1 test for the same ΔG.
Matsubara [35] investigated mode II delamination growth in unidirectional tape and satin-woven fabric laminates under different stress ratios. The SERR range was used as correlating parameter. The SERR range was defined by following equation:
𝛥𝐺𝑠 = 𝐺𝐼𝐼𝑚𝑎𝑥(1 − 𝑅)2 (2.10)
The above definition is analogous to the range of SIF ΔK=Kmax - Kmin. Equation 2.10 can be
written in terms of Gmax and Gmin as
𝛥𝐺𝑠 = ( 𝐺𝑚𝑎𝑥 − 𝐺𝑚𝑖𝑛)2 (2.11)
For some case studies, Rans [39] reported that using ΔGs, the stress ratio effect for mode II
becomes negligibly small. However in his study, Matsubara observed significant stress ratio effect for T300/5208/EC3445 .The delamination growth curves were distinct for each stress ratio.
Tanaka and Tanaka [41] investigated mode II delamination growth behaviour under different stress ratios in T300H/3631 laminates. They proposed an empirical model for the prediction of delamination growth that characterizes the effect of stress ratio by developing empirical relations for the Paris equation exponent. The model is given by the following equation
𝑑𝑎
𝑑𝑁 = 𝑉𝐿 𝐾𝐼𝐼𝑚𝑎𝑥/𝐾𝐼𝐼𝑐
𝑛 (2.12)
Where VL is the delamination growth rate at threshold, n is the Paris equation parameter and
KIIC is the critical SIF. The n was defined by the following equation
𝑛 = 𝑙𝑜𝑔 𝑉𝐻/𝑉𝐿
𝑙𝑜𝑔 1−𝑅 𝐾𝐼𝐼𝑐/∆𝐾𝐼𝐼𝑡 (2.13)
Where VH is the critical delamination growth rate and ∆KIIth is the range of SIF at threshold.
The main assumption in the derivation of equation 2.12 was that the ΔKIIth is constant
regardless of the stress ratio. This assumption was based on the experimental observation in the study where the delamination growth curves converged at a delamination growth equal to 1e-9 m per cycles as shown in figure 2.9a. Figure 2.9b is a schematic representation of delamination growth rate under different stress ratios showing the convergence of the different curves at ΔKth.
In Tanaka‟s model, the assumption of unique ΔKth for different stress ratios cannot be
generalized for all materials. There are cases in the literature where ΔKth has different values
at the threshold delamination growth rate. This imply that the curves for different stress ratios either do not converge to a single ΔKth or converges at multiple values of ΔKth. Matsuda [29]
conducted mode II delamination growth tests on T800H/3900-2 and observed that the value of
ΔKth was not unique for different stress ratios as shown in figure 2.10. The dotted and the
solid lines in the figure represent the tests under room temperature, while the markers represent the tests performed at a temperature equal to 77K. From the figure, it is evident that
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the value of ΔKth is different for both types of tests under different temperatures. The above
examination implies that Tanaka‟s model can‟t be generalized for all material types.
Table 2.3: Overview of the literature on the stress ratio effect in mode II fatigue
delamination growth
Author Material system Delamination growth model Validation data
stress ratio range Reference
Russel and Street AS1/3501-6,AS4/2220-3, C6000/F155, AS4/Apc2 𝑑𝑎 𝑑𝑁= 𝐴 𝛥𝐺 𝑛 -1 and 0 [21] Mall S. et al. T300/5208,adhesive EC 3445 𝑑𝑎 𝑑𝑁= 𝐴 𝛥𝐺 𝑛 -1 and 0.1 [22] Martin
and Murri AS4/PEEK
da dN=A 𝐺max 𝐵 1 − 𝐺𝑡 𝐺𝑚𝑎𝑥 𝐷1 1 − 𝐺𝑚𝑎𝑥 𝐺𝑐 𝐷2 0.1 and 0.5 [25] Gambone
L. AS4/3501-6 No model proposed
-1,-0.5, 0, 0.14, 0.25, 0.33, 0.4 and 0.5 [26] Lin C. T. and Kao P. W carbon fiber reinforced aluminium laminate
No model proposed 0.1 and 0.5 [11]
Tanaka and Tanaka T300H/3631 𝑑𝑎 𝑑𝑁= 𝑉𝐿 𝐾𝐼𝐼𝑚𝑎𝑥/𝐾𝐼𝐼𝑐 𝑛 -1,-0.5, 0.2, 0.5 and 0.5 [41] Matsuda S. et al. T800H/3900-2 𝑑𝑎 𝑑𝑁= 𝐴 𝛥𝐾 𝑛 0.1 and 0.5 [29] Hojo M. et al. alumina fiber/ bisphenol 𝑑𝑎 𝑑𝑁= 𝐴 𝛥𝐾 𝑛 0.1 and 0.5 [32] Matsubara G. et al. Unidirectional tape and satin-woven fabric,T-Glass, E-glass 𝑑𝑎 𝑑𝑁= 𝐴 𝛥𝐺 𝑛 -1,-0.5, 0.1, and 0.5 [35] Kawashita
L. F. et al. IM7/8552 No model proposed 0.1, 0.3 and 0.5 [36] Allegri G. et al. IM7/ 8552 𝑑𝑎 𝑑𝑁= 𝐶 𝐺𝐼𝐼𝑚𝑎𝑥/𝐺𝐼𝐼𝑐 2𝑏/(1−𝑅2) 0.1, 0.3 and 0.5 [37] Anderson J. * 𝑑𝑎 𝑑𝑁 = 𝐶 ∆𝐾−∆𝐾𝑡 𝐾𝑐−𝐾𝑚 𝑏 * [33] *Theoretical approach
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Figure 2.9: Crack propagation rate versus ΔKII under different stress ratios [41],(a) Experimental data (b), Schematic representation [41]
Figure 2.10: Relation between crack propagation rate and stress intensity range ΔKII for
T800H/3900-2 at room temperature and 77 K [29]
Anderson [33] developed an empirical model for the prediction of delamination growth. The model implicitly includes the stress ratio effect by modifying the correlating parameter of the Paris equation for the effect. The model was represented by the following equation
𝑑𝑎 𝑑𝑁 = 𝐶 ∆𝐾−∆𝐾𝑡 𝐾𝑐−𝐾𝑚 𝑏 (2.14)
Where C and b are power law parameters, Kc is the fracture toughness for a given loading
mode, Km is the mean applied SIF, and ΔKth is the threshold SIF range.
Anderson implicitly linked the empirical model to the damage formation ahead of delamination crack tip. It was assumed that the damage formation is a function of cyclic and monotonic load. Literature studies [5, 29, 41] were used for the validation of the developed
20
model. He reported a reasonable agreement between the model prediction and the experimental data of the listed references however, Allegri [37] recently examined the model and pointed its discrepancy in delamination prediction for some material systems.
2.2.3. Effect of stress ratio on mode III fatigue delamination growth
The fatigue delamination growth under mode III (transverse shearing) has been investigated less compared to mode I and II. The relation of stress ratio, Sa and Sm in mode III is same to
the mode II.
The effect of the stress ratio in mode III delamination growth was investigated by Donaldson et al. [23] in AS4/3502 laminates. Donaldson also used Gmax and ΔG = Gmax - Gmin for
characterizing mode III fatigue delamination growth.
The stress ratio effect in mode III rate resembles to mode II when Gmax is used as the
correlating parameter. Figure 2.11 shows comparison of the stress ratio effect on mode III and mode II delamination growths using Gmax as correlating parameter. The curves for different
stress ratios are converged at higher delamination rates. Donaldson proposed the Paris equation in terms of Gmax for the prediction of delamination growth rate under mode III.
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Figure 2.11: (a) Delamination growth rate versus GIIImax under mode III [23] (b)
Delamination growth rate versus GIImax/Gc under mode II [37] 2.2.4 Effect of stress ratio on mixed mode (I+II) fatigue delamination growth
In mixed mode (I + II) delamination, the individual modes can contribute from 0-100%. The stress ratio affects the delamination growth in mode I and mode II differently as mentioned in the previous sections. The mode ratio (GI/GII) is thus an important parameter to analyze the
stress ratio effect in case of mixed mode delamination growth. The literature studies [1, 4, 20] have generally used crack lap shear (CLS) specimens for characterization of the mixed mode delamination growth under different stress ratios. The mode ratio changes with delamination extension in CLS specimens. The analysis of stress ratio effect is complicated due to varying mode ratio in CLS specimens. Table 2.4 lists the previous studies reporting the stress ratio effect in mixed mode delamination growth.
Mall et al. [20] investigated mixed mode delamination growth under different stress ratios in T300/5208, adhesive EC 3445. The mode ratio was 0.03-0.38 in the fatigue tests. The results were correlated to GTmax= GImax+GIImax and ΔGT = (ΔGImax- ΔGImin) + (ΔGIImax- ΔGIImin). The
difference between delamination growth rates under different stress ratios was reduced in case of ΔGT as compared to GTmax, due to which Mall proposed that ΔGT is the controlling
parameter for the delamination growth. The stress ratio effect analyses become more complex using ΔGT due to mixing the effects of monotonic and cyclic loadings and individual
mode contributions in the mixed mode.
Gustafson et al. [4] investigated the stress ratio effect on mixed mode delamination growth in T300/914C material system. The mode ratio for the tests was 0.366-0.446. Similar to Mall, Gustafson also proposed ΔGT as the correlating parameter for delamination growth
characterization.
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Table 2.4: overview of literature on the stress ratio effect in mixed mode (I+II) fatigue
delamination growth
Author Material system Delamination growth model Validation data stress ratio range Mode ratio range Reference Mall S. et al. T300/5208,adhesive EC 3445 𝑑𝑎 𝑑𝑁= 𝐴 ∆𝐺𝐼+ ∆𝐺𝐼𝐼 𝑛 0.01, 0.1, 0.5 and 0.75 0.03-0.38 [20] Gustafson C.G. et. al T300/914C 𝑑𝑎 𝑑𝑁= 𝐴 ∆𝐺𝐼+ ∆𝐺𝐼𝐼 𝑛 0.1, 0.3 and 0.5 0.366-0.446 [4] Schön * 𝑑𝑎 𝑑𝑁= 𝐷∆𝐺𝑟 𝑛 * 0-1 [31]
* Theoretical approach was used
Schön [31] developed an empirical model for prediction of the mixed mode fatigue delamination growth under different stress ratios. The model was based on the assumption that the threshold SERR ΔGr,th is independent of the stress ratio. In the model, the stress ratio
is represented by a parameter Q that is equal to R when -1 ≤ R ≤1 and 1/R when |R|>1. Schön proposed the following equation for delamination growth prediction
𝑑𝑎
𝑑𝑁= 𝐷∆𝐺𝑟
𝑛 (2.15)
Where D and n are equation parameters. The effect of stress ratio was included in the model by modifying the exponent n of equation 2.15 as
𝑛 = 𝑙𝑜𝑔 𝑑𝑎 𝑑𝑁 𝑡−𝑙𝑜𝑔 𝑑𝑎 𝑑𝑁 𝑐 𝑙𝑜𝑔 ∆𝐺𝑟 ,𝑡 −𝑙𝑜𝑔 𝐺𝑚𝑎𝑥 ,𝑐 1−𝑄2 (2.16)
Where Gmax,c is the critical energy release rate, (da/dN)c is the critical delamination growth
rate and (da/dN)th is the threshold delamination growth rate. In equation 2.15, the parameter
D was defined as:
𝐷 =
𝑑𝑎 𝑑𝑁 𝑡 ∆𝐺𝑟 ,𝑡
𝑛 (2.17)
Schön validated the model using literature studies [21, 25, 27, 41].
In Schön‟s model, the assumption of taking ΔGr,th independent of the stress ratio cannot be
23
in AS1/3501-6, AS4/2220-3, C6000/F155 and AS4/Apc2 composites in mode II delamination growth. Similarly, Mall [22] observed the effect of stress ratio on ΔGth in T300/5208,
adhesive EC 3445 under mode II. Hojo [5] and Bathias [2] observed a stress ratio effect on
ΔGth in T300/#2500 and Brochier-fabric/1452 respectively in mode I delamination growth.
The literature survey shows that the previous studies for the mixed mode stress ratio effect lack the approach to separate the effects coming from individual modes contributions. The individual modes have different mechanism of delamination growth and the stress ratio affect the delamination growth differently in each mode. The analysis of the stress ratio effect is thus more complex in mixed mode than the single mode delamination growth.
2.3 Discussion
The stress ratio effect on fatigue delamination growth has been extensively studied in the literature. Due to complex nature of the delamination phenomenon, there is a lack of understanding about characterizing the effect. The common thing in the previous studies is the use of such correlating parameter in delamination growth models that eliminate stress ratio effect. It can be stated from the literature survey, that the researchers generally believed that stress ratio has no physical effect on delamination growth mechanism and the apparent effect in delamination growth characterization can be eliminated by using a proper correlating parameter. For this purpose, the delamination growth data was correlated to ΔK, ΔG, ΔGs,
Kmax and Gmax. Paris equations were proposed as delamination growth models in terms of one
of the above parameter.
The above parameters affect the delamination growth analysis in different ways. The use of SIF for delamination growth characterization is not appropriate. In composites, the complexity of the stress field at delamination tip at an interface of two potentially dissimilar materials/layers makes the evaluation of SIF difficult. Alternatively, researchers adopted the approach of SERR which is easily determined for composites. In general the SERR, as a correlating parameter, is either used as Gmax or the SERR range ∆G=Gmax-Gmin. For
delamination growth under fatigue, the Gmax fails to include the effect of the ∆G, related to the
minimum load in the applied load cycle. The ∆G has been adopted as analogous to the ∆K, which is incorrect as mentioned in section 2.2.2. The correct analogous of the ∆K in terms of SERR is ΔGs (equation 2.11), that gives same results to ∆K. The behavior of delamination
growth with increasing stress ratio was analyzed using ΔGs for the literature studies. These
results were compared with ΔK. The observations are presented in table 2.5. The table shows
that delamination growth behavior is similar with increasing stress ratio for both parameters. The effective stress ratio at the crack tip in mode I is influenced by crack shielding mechanisms. Crack closure due to plasticity or surface roughness, fiber bridging or ligament bridging shields crack tip in some materials [44]. The cyclic loading at the crack tip is affected due to crack shielding, that also change the effective stress ratio. Literature survey reveals that few studies investigated crack closure. Hojo et al. [5] and Gustafson et al. [4] conducted tests to determine the crack closure effect in mode I fatigue delamination growth. Crack closure was observed at lower stress ratios in both studies. In Hojo‟s study, ∆K was corrected for crack closure that increased the effective stress ratio. Gustafson used ∆G in his investigation and the results were corrected for crack closure effect.
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Table 2.5: Effect of increasing stress ratio on delamination growth using ΔGs and ΔK as
correlating parameters
Material
Effect on da/dN at constant
ΔGs=(√Gmax-√Gmin)2
Effect on da/dN at constant
ΔK Reference
T300/914, Increase Increase [5]
T300/#2500 Increase Increase [5]
Brochier fabric/1452 Increase Increase [2]
T300/5208, adhesive EC 3445
Increase Increase [20]
M30SC/DT120 Increase Increase [37]
The observed crack closure in the above studies was small to fully explain the stress ratio effect in a same way as it explains stress ratio effect in metal fatigue growth [45-46]. The difference in the behavior of metal crack and delamination with respect to crack closure may be attributed to the source of crack closure in these two types of material systems. The previous studies lack the investigation of the source of crack closure in delamination growth. In general, the previous delamination growth models are either empirically developed or lacks the link between fracture mechanisms and delamination growth. In some mode I delamination growth studies, the mechanism for the effect of stress ratio was investigated using SEM (scanning electron microscopy) as discussed in section 2.2.2. Bathias and Laksimi [2] observed striations on the fracture surfaces in lower stress ratio tests. Hojo et al. [5] observed that the fatigue fracture surface features at higher stress ratios resembles to the mode I static fracture features.
The fatigue delamination growth depends on both monotonic and cyclic part of the fatigue cycle [47]. This implies that at micro level, the fracture surface is accordingly affected by these two components of the fatigue load cycle. The previous studies generally investigated stress ratio effect on fracture mechanisms using double cantilever beam (DCB) specimens. As stress ratio is the ratio of minimum to maximum cyclic load, the delamination growth and fracture surface does not change due to stress ratio, rather it changes due to change in monotonic and cyclic loads. In a fatigue tests using DCB specimen (no matter displacement control or load control), stress ratio remains constant, however monotonic and cyclic load changes with delamination extension. This implies that resulting fracture surface topology does not remain constant at a specific stress ratio throughout delamination length in DCB
25
specimens. The study of the fracture surface for the effect of monotonic and cyclic load is thus improper using DCB specimens.
The mechanism of mode II delamination growth has been investigated in some previous studies. The delamination growth models developed in the previous studies were empirically developed without linking to the micromechanism of the mode II delamination growth. Tanaka and Tanaka [41] investigated the delamination tip and the fracture surfaces under different mode II stress ratios. The delamination tip was different in positive stress ratio test than the negative stress ratio tests as shown in figure 2.12. For the positive stress ratios, the main crack was accompanied by small microcracks that were caused by principle normal stresses at an angle of 450 to the fiber direction in the plies. In case of negative stress ratios the main crack propagates along a zigzagging path due to principle stress reversal causing X-shaped microcracks. Tanaka also reported a remarkable difference between the fracture surfaces under positive and negative stress ratios. The pitch of river markings was much smaller for negative stress ratios than positive stress ratios.
The above discussion shows that the stress ratio has an influence on the fracture mechanisms for the mode I and mode II delamination growths. Practically this influence is the result of the effects of monotonic and cyclic loads. Efforts should be thus focused to study effect of these loadings on the micromechanisms.
In the previous studies, empirical two-parameter models have been proposed for delamination growth that either implicitly or explicitly characterize the stress ratio effect on delamination growth. The use of two-parameter models for fatigue delamination growth characterization is more appropriate than using single parameter model, because the fatigue spectrum is fully described by any two parameters of Gmax, Gmin, Gmean ∆Gs and R. For empirical two-parameter
models in literature, the number of equation parameters is more than single parameter models. Atodaria‟s [30] model need five constants that should be determined from experimental data fitting. Hojo‟s [5] model need four constants. The number of equation constants can be reduced by developing mechanistic two-parameter models.
Figure 2.12: SEM micrograph of mode II fatigue crack under (a) R=0.2 (b) R=-1 [41]
2.4 Conclusions
Significant number of studies has been dedicated to the investigation of the stress ratio effect on delamination growth. From the literature survey following conclusions are drawn:
