Multi-Scale Mechanism Based Life Prediction of Polymer Matrix

MULTI-SCALE MECHANISM BASED LIFE PREDICTION OF POLYMER MATRIX

COMPOSITES FOR HIGH TEMPERATURE AIRFRAME APPLICATIONS

PRIYANK UPADHYAYA

SAMIT ROY, COMMITTEE CHAIR ANWARUL HAQUE JAMES P. HUBNER MARK E. BARKEY NITIN CHOPRA

A DISSERTATION

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Department of Aerospace Engineering and Mechanics in the Graduate School of The University of Alabama

TUSCALOOSA, ALABAMA

A multi-scale mechanism-based life prediction model is developed for high-temperature polymer matrix composites (HTPMC) for high temperature airframe applications. In the

first part of this dissertation the effect of Cloisite 20A (C20A) nano-clay compounding on the thermo-oxidative weight loss and the residual stresses due to thermal oxidation for a thermoset polymer bismaleimide (BMI) are investigated. A three-dimensional (3-D) micromechanics based finite element analysis (FEA) was conducted to investigate the residual stresses due to thermal oxidation using an in-house FEA code (NOVA-3D).

In the second part of this dissertation, a novel numerical-experimental methodology is outlined to determine cohesive stress and damage evolution parameters for pristine as well as isothermally aged (in air) polymer matrix composites. A rate-dependent viscoelastic cohesive layer model was implemented in an in-house FEA code to simulate the delamination initiation and propagation in unidirectional polymer composites before and after aging.

Double cantilever beam (DCB) experiments were conducted (at UT-Dallas) on both pristine and isothermally aged IM-7/BMI composite specimens to determine the model parameters.

The J-Integral based approach was adapted to extract cohesive stresses near the crack tip.

Once the damage parameters had been characterized, the test-bed FEA code employed a micromechanics based viscoelastic cohesive layer model to numerically simulate the DCB experiment. FEA simulation accurately captures the macro-scale behavior (load-displacement history) simultaneously with the micro-scale behavior (crack-growth history).

ii DEDICATION

This dissertation is dedicated to my sister Mrs. Ruchi Upadhyay.

iii LIST OF ABBREVIATIONS AND SYMBOLS

HTPMC High Temperature Polymer Matrix Composite

BMI Bismaleimide

C20A Cloisite 20A

DCB Double Cantilever Beam

FEA Finite Element Analysis

RVE Representative Volume Element

2-D Two Dimensional

3-D Three Dimensional

C Oxygen Concentration

Di j Orthotropic Diﬀusivity

R(C) Reaction Rate

W Weight of Specimen

DNC Diﬀusivity of C20A modiﬁed BMI

DBR Diﬀusivity of BMI

α Scalar Damage Parameter

δ Crack Opening Displacement

θ Rotation at the Load Pin

P Reaction Force at the Load Pin

Tg Glass Transition Temperature

iv Ti Traction

{ε} Strain Vector

{H} Hereditary Strain Vector

{σ} Stress Vector

[M(t)] Viscoelastic Stiﬀness Matrix

λcr Critical Principal Stretch

λ¯ Principal Stretch Measure

σr Radial Stress

σθ Hoop Stress

σcohesive Cohesive Stress

σ¯ ud Undamaged Stress

φ Oxidation State Variable

φox Threshold Value of Oxidation State Variable

ε Weight Fraction due to Oxidation in Air

γ Weight Fraction due to Oxidation in Inert Conditions

ζ Nano-clay Loading

ψ Aspect ratio of Nano-clay Platelets

v ACKNOWLEDGEMENTS

Foremost, my heartfelt gratitude goes to my advisor Dr. Samit Roy for his valuable guidance and continuous support at every step of this work, right from conception of the idea to its conclusion. I am grateful to Dr. Roy for providing me with this opportunity to study at The University of Alabama and for his constant support that was invaluable in the completion of this work. His sense of conﬁdence and level of trust in me has led me to the successful completion of my dissertation work.

Besides my advisor, I would like to thank Dr. Mark E. Barkey, Dr. Anwarul Haque,

Dr. James P. Hubner, and Dr. Nitin Chopra, for their encouragement, guidance and patience while serving as members of my doctoral committee.

I greatly appreciate the contribution by Dr. Hongbing Lu and Dr. Mohammad H.

Haque at UT-Dallas towards this study. I am especially thankful to Dr. Mohammad H.

Haque for conducting experiments, needed to validate my dissertation work.

I am also grateful to all the faculty members of the Department of Aerospace Engi- neering and Mechanics at The University of Alabama, who helped me a lot during my course work and imbibed a learning spirit in me. The blessings of my parents and the love and aﬀection of my sister (Ruchi) and brother (Vijay) has given me the necessary driving force to complete this work. In the end, a special thanks to all who could not be mentioned here, but have always had a positive inﬂuence in my life. The research project was funded by the

“Low Density Materials Program” of the AFOSR.

vi CONTENTS

ABSTRACT ...... ii

DEDICATION ...... iii

LIST OF ABBREVIATIONS AND SYMBOLS ...... iv

ACKNOWLEDGEMENTS ...... vi

LIST OF TABLES ...... x

LIST OF FIGURES ...... xi

1 INTRODUCTION ...... 1

1.1 Objective ...... 4

2 LITERATURE SURVEY ...... 6

2.1 Thermo-oxidative Behavior of Polymer Resin Systems ...... 6

2.2 Barrier Properties of Nanoparticles ...... 10

2.3 Damage Modeling for Composites ...... 15

2.4 Cohesive Layer Models ...... 22

3 MATHEMATICAL MODELING OF THERMAL OXIDATION ...... 25

3.1 Three Zone Oxidation Model ...... 26

3.2 Determination of Thermo-Oxidative Parameters ...... 27

3.2.1 Calculation of β ...... 28

3.2.2 Calculation of αR0 ...... 28

3.2.3 Calculation of φox ...... 29

vii 4 INFLUENCE OF NANO-CLAY ON THERMO-OXIDATIVE STABILITY AND MECHANICAL PROPERTIES OF BMI ...... 31

4.1 Materials and Manufacturing Process ...... 31

4.2 Thermo-Oxidative Aging Experiments ...... 32

4.2.1 Isothermal aging of BMI and C20A/BMI in air ...... 32

4.2.2 Isothermal aging of BMI and C20A/BMI in Argon ...... 35

4.2.3 Isothermal aging of BMI and C20A/BMI in 60% O2 ...... 36

4.3 Calculation of Thermo-oxidative Parameters from Experimental Data . . . . 41

4.4 Nano-indentation Test for Mechanical Properties ...... 42

4.5 Thermal Shrinkage Test ...... 44

5 FINITE ELEMENT SIMULATION OF RESIDUAL STRESSES DUE TO SHRINK- AGE IN IM-7/BMI COMPOSITE ...... 45

5.1 Finite Element Model ...... 45

5.2 Simulation Results ...... 49

6 MULTI-SCALE VISCOELASTIC COHESIVE LAYER MODEL FOR PREDICT- ING DELAMINATION IN HTPMC ...... 55

6.1 Viscoelastic Cohesive Layer Model ...... 55

6.2 Damage Evolution Law ...... 58

6.3 Model Calibration ...... 60

6.3.1 Flexure Experiment ...... 60

6.3.2 FEA simulation of ﬂexure experiment ...... 60

6.4 Double Cantilever Beam Experiment ...... 67

6.4.1 Specimen preparation and DCB specimen geometry ...... 67

6.4.2 Experimental method ...... 68

viii 6.5 Determination of Cohesive Stress using J-integral ...... 69

6.6 Estimation of Damage Evolution Law ...... 74

6.6.1 Determination of critical principal stretch λcr ...... 75

6.6.2 Determination of damage parameters α0 and m ...... 77

6.7 Numerical Simulation of DCB Experiments ...... 81

6.8 Sensitivity study on model parameters ...... 87

7 CONCLUSIONS ...... 90

7.1 Conclusions ...... 90

7.2 Future Work ...... 92

REFERENCES ...... 95

ix LIST OF TABLES

4.1 Oxidation parameters for neat BMI and C20A modiﬁed BMI ...... 42

4.2 Comparison of mechanical properties for neat and C20A modiﬁed BMI(3 wt%) 43

5.1 Material properties used in FEA simulation ...... 49

6.1 Elastic properties used in FEA modeling for un-aged and aged (for 3000 hours at 250 °C) IM-7/BMI unidirectional laminate ...... 62

6.2 Creep Compliance of BMI resin (Reference Temperature = 204 °C) . . . . . 64

6.3 Micromechanical damage parameters used in FEA modeling for un-aged and aged (for 3000 hours at 250 °C) IM-7/BMI unidirectional laminate ...... 65

6.4 Damage evolution law parameters used in FEA model for IM-7/BMI . . . . 80

6.5 Elastic properties of transversely isotropic IM-7/BMI lamina[Andrews and Garnich, 2008]...... 81

x LIST OF FIGURES

2.1 (a) Cross section of a G30-500/PMR-15 unidirectional composite after 2092 hours of isothermal aging in air at 288 °C (b) Close-up view of oxidized composite showing ﬁber matrix debond [Ripberger, Tandon, and Schoeppner, 2005]9

3.1 Schematic diagram showing three zones during thermal oxidation ...... 25

3.2 Oxidized and un-oxidized regions depicted in section view ...... 29

4.1 Neat and C20A modiﬁed BMI specimens ...... 32

4.2 Specimens inside the oven during isothermal aging experiment ...... 33

4.3 Weight loss data for BMI and 3% C20A compounded BMI at 250 °C in air . 33

4.4 Experimental set up for weight loss experiment in Argon and 60% O2 environment ...... 36

4.5 Weight loss data for BMI and 3% C20A compounded BMI at 250 °C in Argon 37

4.6 Weight loss data for BMI and 3% C20A compounded BMI at 250 °C in 60% O2 37

4.7 Comparison of weight loss data for BMI and 3% C20A compounded BMI at 250 °C in (a) air (b) argon (c) 60% O2 environment ...... 40

4.8 Measured shrinkage strains for neat BMI and C20A modiﬁed BMI ...... 43

5.1 (a) Hexagonal arrangement of fibers and (b) In-plane displacement boundary conditions on the RVE with fiber-fiber interaction ...... 46

5.2 3-D ﬁnite element mesh of RVE showing boundary conditions ...... 47

5.3 3-D contour showing oxidation state variable φ (neat BMI case) ...... 50

5.4 3-D contour showing radial strain εr (neat BMI case) ...... 51

5.5 3-D contour showing radial stress σr (neat BMI case) ...... 51

5.6 3-D contour showing hoop stress σθ (neat BMI case) ...... 52

xi 5.7 FEA prediction of oxidation state variable φ at the interface along z direction along ﬁber-matrix interface at diﬀerent aging times ...... 53

5.8 FEA prediction of σr at the interface along z direction along ﬁber-matrix interface at diﬀerent aging times ...... 53

5.9 FEA prediction of σθ at the interface along z direction along ﬁber-matrix interface at diﬀerent aging times ...... 54

6.1 Opening crack containing cohesive ligament (b) Reduction of RVE to cohesive zone by area averaging ﬁbril tractions [Allen and Searcy, 2001]...... 56

6.2 Three-point bending ﬁxture ...... 61

6.3 FEA mesh for ﬂexure test specimen of IM-7/BMI unidirectional laminate . . 62

6.4 FEA simulation showing inter-laminar delamination for ﬂexure test of un-aged IM-7/BMI unidirectional laminate ...... 65

6.5 Normalized load displacement curve for un-aged IM-7/BMI unidirectional laminate ...... 66

6.6 Normalized load displacement curve for aged IM-7/BMI unidirectional laminate 66

6.7 An image of the test conﬁguration for delamination type DCB specimen having pre-crack under initial loading ...... 67

6.8 Schematic diagram of DCB specimen ...... 69

6.9 J-Integral versus COD (δ) for pristine IM-7/BMI ...... 72

6.10 Cohesive stress versus COD (δ) for pristine IM-7/BMI ...... 73

6.11 J-Integral versus COD (δ) for aged (1000 hours at 260 °C in air) IM-7/BMI 74

6.12 Cohesive stress versus COD (δ) for aged (1000 hours at 260 °C in air) IM-7/BMI 75

6.13 Scalar damage parameter for pristine IM-7/BMI ...... 79

6.14 logα˙ versus logλ¯ for pristine IM-7/BMI ...... 80

6.15 Scalar damage parameter for isothermally aged (1000 hours at 260 °C in air) IM-7/BMI ...... 81

xii 6.16 logα˙ versus logλ¯ for isothermally aged (1000 hours at 260 °C in air) IM-7/BMI 82

6.17 Meshed DCB specimen of IM-7/BMI with boundary conditions ...... 83

6.18 Load versus displacement for pristine IM-7/BMI DCB specimen ...... 84

6.19 Crack-length versus displacement for pristine IM-7/BMI DCB specimen . . 85

6.20 Load versus displacement for isothermally aged (1000 hours at 260 °C) IM- 7/BMI DCB specimen ...... 85

6.21 Deformed and un-deformed contour plots showing εy for isothermally aged specimen at 260 °C clearly showing crack propagation from 70 mm to 74 mm at 450 sec ...... 86

6.22 Crack-length versus displacement for isothermally aged (1000 hours at 260 °C) IM-7/BMI DCB specimen ...... 86

6.23 Load versus displacement for pristine IM-7/BMI DCB specimen for diﬀerent λcr values ...... 88

6.24 Load versus displacement for pristine IM-7/BMI DCB specimen for diﬀerent α0 values ...... 89

6.25 Load versus displacement for pristine IM-7/BMI DCB specimen for diﬀerent m values ...... 89

xiii CHAPTER 1

INTRODUCTION

HTPMC used in high temperature applications such as supersonic inlet ducts, advanced fan casings and engine exhaust washed panels are known to have a limited life due to high thermo-mechanical loads and environmental degradation. Although there has been a considerable amount of work investigating the response of metals under such loading conditions, such studies are still not mature for HTPMC. The kind of coupled physical, chemical and mechanical response these HTPMC exhibit under extreme hygro-thermal loads makes it a very challenging problem. Recently, considerable insight has been gained for polymers

(particularly amorphous polymers) undergoing physical aging, chemical aging and strain dependent aging [Regnier and Guibe, 1997], [Bowles, Jayne, and Leonhardt, 1994], [Colin,

Marais, and Verdu, 2002], [Colin and Verdu, 2005], [Pochiraju and Tandon, 2006], [Pochi- raju, Tandon, and Schoeppner, 2008]. Although there are models to predict the physical aging behavior of polymers, the problem of thermo-oxidative aging of HTPMC under cyclic mechanical/ hygro-thermal loading has not been addressed adequately.

Many new resin systems for HTPMC capable of sustaining temperature in the range

200°C to 350°C are providing new opportunities for using HTPMC in propulsion, airframe and space structure components. Some of these HTPMC resin systems are PMR-15, AFR

700B, DMBZ-15, Avimid N, bismalimide (BMI) and HTM 512. PMR-15 is one of the most widely used resins for bypass ducts, nozzle ﬂaps, bushings, and bearings. However, PMR-15

1 is made from methylene dianiline (MDA), a known carcinogen and a liver toxin, and the

Occupational Safety and Health Administration (OSHA) imposes strict regulations on the handling of MDA during the fabrication of PMR-15 composites. Recent concerns about the safety of workers working to manufacture and repair PMR-15 components have led to the implementation of costly protective measures. These costs involved to avoid safety issues with PMR-15 have led to the development of new material systems. New resin systems are being developed to address both safety and other limitations in thermo-oxidative stability, hydrolytic stability and processing constraints of these resin systems. One of these new resins is BMI which is a polyimide with a 232°C maximum service temperature. BMIs have served on military aircraft for decades and into 5th generation ﬁghter planes. BMI is also used for exhaust wash areas on Boeing aircraft.

The material qualiﬁcation methodology currently followed by the aerospace industry is primarily empirical because design engineers are more comfortable with the enormous amount of experimental data available. For example, 16,000 coupon and element tests were required to specify the design allowables with the required ﬁdelity and reliability for the

F-22 Raptor high temperature program. Moreover, a very small insigniﬁcant change in the material property, manufacturing process or service load requires additional costly and exhaustive set of experiments to ensure the safety. In the current methodology, there is no space for advanced models for material assessment before insertion. Structural analysis done after insertion of the component using FEA does not incorporate the underlying material system behavior. Consequently, additional costly experiments must be conducted to augment the structural analysis to accurately capture the response of environmental factors such as temperature and moisture content especially when coupled with high strain-rate eﬀects.

2 Therefore, the primary hurdle to the introduction of a new improved HTPMC in design allowable database is the overwhelming amount of experimental testing that is required.

The laboratory testing based design methodology is a direct consequence of the necessity to address the uncertainty involved in manufacturing and to accurately estimate the performance and failure of a component under service conditions. Because of limited large-scale tests in the emerging design environment of HTPMC, it is critical that a robust mechanism based multi-scale model evolves to capture the response and the failure mechanisms involved in HTPMC.

Therefore, the motivation for the current research is our limited ability to use advanced composites in high temperature applications due to design requirements and knock- down factors. To fulfill these requirements we need to be able to predict end-of-life properties and durability of composites at elevated temperatures under mechanical and environmental loading conditions. Durability and degradation mechanisms in composite materials are fundamentally influenced by the fiber, matrix, and interphase regions that constitute the composite domain. The failure modes in composite laminates mainly involve fiber-matrix debond, delamination, fiber failure and matrix cracking. Three of these failure mechanisms, namely fiber-matrix debond, matrix cracking and delamination are studied in the current research. It is envisaged that the high residual stresses due to thermo-oxidative shrinkage lead to micromechanical damage. From the collective review of previous research work it was found that high radial stresses at the fiber-matrix interface are potentially responsible for fiber-matrix debond. Similarly in the matrix region, high hoop stresses are present which can lead to matrix cracking. To investigate the possibility of micromechanical damage due to fiber-matrix debond and matrix cracking, a coupled 3-D diffusion-reaction and stress sim-

3 ulation is conducted for a repetitive unit cell. Material system that is being investigated during this research work constitutes of BMI polymer matrix and IM-7 fiber. These materials are currently gaining a wide usage in aircraft structures, especially in airframe and engine inlet casings [Luo, Lu, Roy, and Lu, 2012]. Neat BMI exhibits linear viscoelastic or time- dependent behavior and IM-7 fiber is transversely isotropic in nature. But, the response of a composite differs significantly from that of its constituents, especially after thermo-oxidative degradation. After thermal oxidation, composite laminate starts to exhibit anisotropy in mechanical and oxygen diffusion properties. Therefore, it is important to have a design based on micromechanical analysis rather than on experiments to enhance the affordability of insertion of new material systems. In this context, the current research is an attempt to present a micromechanics based model that is capable of predicting performance and failure of structural components under mechanical loads in extreme environmental conditions. The objective of this research is presented in next section.

1.1 Objective

The goal of this research is to present a multi-scale mechanism based model that is capable of handling the anisotropy, thermo-oxidative degradation and rate eﬀects (due to viscoelasticity of the polymer matrix) to predict the response and degradation of HTPMC. It is envisioned that the proposed mechanism based multi-scale model will result in a HTPMC life prediction tool which will lead to the development of a reliable analysis-based design guideline for aerospace structures undergoing high strain rates and aggressive environmental loadings. Examples of the predictive capabilities of the model will be shown for both aged and un-aged test specimens made of IM-7/BMI. The model developed in this process could subsequently be incorporated in commercial FEA software, such as ABAQUS or ANSYS,

4 through user deﬁned subroutines to enable user-friendly large-scale structural life-predictions by the Air Force Research Laboratory (AFRL) and the aerospace industry.

Although, the current research was conducted on IM-7/BMI, our methodology is quite general and expected to be valid for similar high temperature thermoset polymer matrix composites. However, it is feasible that the degradation mechanism-based models developed in this work might not apply to a novel high-temperature polymer system due to diﬀerences in polymer morphology (for example, cross-linked thermoset versus linear thermoplastic polymers). In that case, new mechanism-based modeling tool-set will need to be developed for that speciﬁc family of polymers. The next chapter will cover the recent advances in

HTPMC durability research.

5 CHAPTER 2

LITERATURE SURVEY

2.1 Thermo-oxidative Behavior of Polymer Resin Systems

For eﬃcient introduction of HTPMC in the design allowable database, it is important to accurately predict the response of HTPMC under cyclic mechanical and environmental loading at elevated temperatures. The primary obstacle to the use of HTPMC is thermo- oxidative degradation and resulting damage evolution at elevated temperatures ranging between 200 °C to 300 °C, as experienced, for example, by engine exhaust-washed structures.

Aromatic polyimides have proven especially useful for these high temperature applications because of their thermal and oxidative stability.

In view of this fact, Regnier and Guibe [Regnier and Guibe, 1997] performed dynamic thermogravimetric (TGA) experiments on K736 BMI resin at several heating rates in different aging environments. These TGA experiments were performed in order to elucidate the thermal behavior and to obtain the data to study the degradation process. They concluded that the polymer degradation in air is the result of multiple mechanisms operating simultaneously.

In recent times, many researchers have tried to develop a model to capture the thermo- oxidative aging process. In most cases, it has been clearly established that matrix oxidation is the main aging process [Bowles and Nowak, 1988][Meador, Lowell, Cavano, and Herrera-

Fierro, 1996][Colin, Marais, and Favre, 1999][Colin et al., 2002]. However, there is not a

6 consensus on the possible contribution of fiber-matrix interface to the aging process. Usually, oxidation is diffusion limited process which leads to creation of an outer oxidized layer having different material properties than the inner pristine material. The oxidized region is typically characterized by increased density and embrittlement [Schoeppner, Tandon, and Pochiraju,

2008]. This can potentially lead to the development of cracks which provide additional pathways for oxygen diﬀusion and thereby assisting further oxidation into deeper regions

[Pochiraju and Tandon, 2006].

In general, the degradation of a polymer matrix composite depends on the rate of oxygen diﬀusion as well as the oxidation reaction rate itself. Bowles et al. [Bowles et al.,

1994] reported that weight loss rate for PMR-15/C6000 composites were less than that of both carbon ﬁber and the polymer individually. On the other hand, Alston [Alston, 1980] and

Wang et al. [Skontorp, Wong, and Wang, 1995] observed a higher weight loss for composite than for the neat polymer. Similar synergistic weight-loss data have been reported for

G30/PMR-15 composite by Tandon et al. [Pochiraju and Tandon, 2006],[Tandon, Pochiraju, and Schoeppner, 2006].

From a modeling perspective, Colin et al. [Colin and Verdu, 2005] proposed a kinetic model to predict the oxidized layer thickness for a BMI resin F655-2. Their strategy is based on a model which couples the O2 diﬀusion and reaction-consumption kinetics and provides access to the thickness distribution of all the chemical modiﬁcations involved in the aging process. They derived the mathematical representation of O2 reaction rate from the mechanistic scheme of branched radical chain reaction. Their model predictions of changes in weight density and oxidation zone thickness were in excellent agreement with the test results.

To extend their work for unidirectional composites they improved this model to predict

7 relative weight loss in T800H/BMI composites by assuming that the volatile formation results essentially from hydro-peroxide decomposition.

Pochiraju et al.[Pochiraju et al., 2008] studied the thermo-oxidative behavior of unidirectional of G30-500/PMR-15 composite. They found that the elastic modulus of PMR-15 is sensitive to temperature and oxidation state and it degrades by approximately 28% due to a temperature change from 25°C to 288°C. More recently [Tandon, Pochiraju, and Schoepp- ner, 2008], they concluded that the ﬁber axis is the preferred damage evolution direction, and the strong anisotropy in observed oxidation growth can be attributed to the ﬁber-matrix interface debond and/or matrix cracking.

Since thermoset polymers go through a high temperature curing cycle, cure shrinkage and mismatches in the coeﬃcient of thermal expansion of the ﬁbers and matrix during the composite cure process give rise to localized micromechanical residual stresses and damage.

Therefore, the highly stressed ﬁber-matrix interface regions and the interstitial (inter-ﬁber) matrix regions tend to oxidize and develop micro-mechanical damage at an accelerated rate.

Micro-mechanical damage can cause additional permeation paths for oxygen and moisture deep into the composite. Experiment conducted by Ripberger et al. conﬁrmed the initiation

ﬁber matrix debond due to thermal oxidation in unidirectional G30-500/PMR-15 composite at 288 °C in air as illustrated in the micrograph presented in Fig. 2.1.

Recently, Andrews and Garnich [Andrews and Garnich, 2008] concluded from FEA models that the fiber ends at laminate free-edges experience high radial and shear residual stresses at the interface that could cause fiber-matrix debond or matrix crack initiation. Con- sequently, the highly stressed fiber-matrix interface regions and the interstitial (inter-fiber) matrix regions tend to oxidize and develop micro-mechanical damage at an accelerated rate.

8 (a) (b)

Figure 2.1: (a) Cross section of a G30-500/PMR-15 unidirectional composite after 2092 hours of isothermal aging in air at 288 °C (b) Close-up view of oxidized composite showing ﬁber matrix debond [Ripberger et al., 2005]

Micro-mechanical damage can cause additional permeation paths for oxygen and moisture deep into the composite, thereby providing a synergistic degradation mechanism. Roy et al.[Roy, Wang, Park, and Liechti, 2006],[Roy and Singh, 2010] experimentally observed and modeled the synergistic ﬁber-matrix debond due to thermal oxidation of unidirectional IM-

7/PETI-5 composite at 288 °C in air. Upadhyaya et al.[Upadhyaya, Singh, and Roy, 2011] presented a multi-scale mechanism based model to predict the thermo-oxidative degradation in IM-7/PETI5. The multi-scale model incorporated micro-scale level damage such as inter-crosslink chain scission in a polymer due to isothermal aging and formulated an internal state variable which is a function of remaining crosslink density of oxidized polymer in unidirectional composite laminate to predict the degradation in inter-laminar shear strength.

Therefore, to improve the performance of HTPMC, the barrier to permeation of gas should be improved. In this context, nano-clay reinforced polymers show considerable promise. As an added bonus, reinforced polymers also exhibit improved mechanical properties. Next section presents a summary of related research done for enhance of barrier and mechanical properties of polymer matrix composites.

9 2.2 Barrier Properties of Nanoparticles

Inorganic particles, including carbon black, talc, and mica, have been compounded

with polymers as a reinforcing agent for many decades. Recently, dispersion of nano-clay

platelets has gained much interest for its ability to produce signiﬁcant improvements in prop-

erties by inclusion of only a small weight percent of nano-clay platelets. A vast majority

of the work done on nano-clay reinforced composites is focused on thermoplastic polymer

systems. For thermoplastic polymers, signiﬁcant increases in thermo-oxidative stability, spe-

ciﬁc strength, stiﬀness, and permeability, have been reported. For example, Hussain et al.

observed that a small addition ( 5 wt%) of dispersed nano-clay can result in an improve-

ment (by ∼100 %) in mechanical properties of thermoplastic [Hussain, Roy, Narasimhan,

Vengadassalam, and Lu, 2007]. Similarly, Yano et al. [Yano, Usuki, Okada, Kurauchi, and

Kamigaito, 1993], [Yano, Usuki, and Okada, 1997] studied thermoplastic polyimide systems

and found that well dispersed single silicate layers at 2-3 wt% in polyimide composites

improved stiﬀness (by ∼100%) and strength (by ∼50%). Similar improvements in tensile and barrier properties for polyimide/organoclay nanocomposites were found by Chang et al.

[Chang and An, 2002].

Tyan et al. [Tyan, Wei, and Hsieh, 2000] treated clay (0-7 wt%) with a reactive diamine, namely 4, 4’-oxydianilinediamine (ODA), which is used in synthesis of polyimide nanocomposites. Signiﬁcant increases in modulus (210% for 7 wt% nano-clay loading) and maximum stress (52% for 7 wt% nano-clay loading) were observed in ODA treated nanocomposites. Additionally, for nanocomposites, higher limits of elongation at break were observed as compared with that of the neat resin. Agag et al. [Agag, Koga, and Takeichi, 2001] con-

10 ducted a study focusing on both mechanical and thermal properties using similar materials.

They observed enhancement in tensile modulus by increasing the adhesion between polymer

matrix and nano-clay. An increment in Tg value was also reported for these polyimide ﬁlms.

Wang et al. [Wang, Wang, Wu, Chen, and He, 2005] studied the eﬀect of microstructure of nanocomposites on corresponding thermal and mechanical properties. The value of storage modulus (G’) was found to be 21% higher than that of neat polyimide in the glassy state.

In addition, an increase of 69% in the value of G’ was seen above Tg. However, no change in

Tg was reported.

In an eﬀort to develop materials that can withstand high level of stresses coupled

with extreme hygrothermal conditions, researchers at NASA initiated work on nano-clay

reinforced thermoset polymers [Abdalla, Dean, and Campbell, 2002]. The desired material

characteristics for such structural components are high Tg value, better thermal stability at

high service temperature, and better mechanical properties over a broad range of temper-

ature. The following paragraphs presents a summary of recent works on development of

thermoset polymer based (PMR, epoxy) nanocomposites.

Abdalla et al. [Abdalla et al., 2002] investigated the eﬀect of nano-clay platelets on

viscoelastic and mechanical properties of thermoset PMR-15. Dynamic mechanical anal-

ysis (DMA) testing showed that 2.5% (by weight) clay loading resulted in a signiﬁcant

improvement in ﬂexural modulus and strength with no reduction in the elongation at fail-

ure. However, doubling the clay loading (5%) resulted in degradation of ﬂexural properties.

Higher Tg were measured for all the nanocomposites compared to the neat PMR-15, with

the highest values obtained for the 5% clay loaded samples. Thermo-mechanical measure-

ments suggested that coeﬃcient of thermal expansion (CTE) improved for composites of

11 unmodiﬁed clay, but decreased for surfactant modiﬁed clays. Interestingly, no improvement in thermo-oxidative stability was observed for PMR-15, as measured by weight loss during isothermal aging at 288°C for 1000 hours.

Islam, et al. [Dean, Islam, Small, and Aldridge, Dean et al.] examined intercalation of both organically modified and unmodified montmorillonite clays with PMR-15 and observed an increase in Tg value of nanocomposites by approximately 28°C. Additionally, significant enhancement in thermo-oxidative stability substantiated by the reduction in weight loss (25%) recorded during isothermal aging experiments was reported in this research work.

Micrographs of these specimens suggested that an optimum clay loading exists for perfect exfoliation state in PMR-15. Nano-clay existed in a primarily exfoliated state for 1 wt% clay loading whereas 5 wt% clay loading showed nano-clay platelets in an intercalated state.

Similarly, Gintert et al. [Gintert, Jana, and Miller, 2007] have reported an optimum organic nano-clay exfoliation method to achieve high thermal stability, increased crosslink density, maximum clay exfoliation, and improved thermal properties in PMR-15 composites. Sim- ilar observations regarding exfoliation state in PMR-15 nanocomposites were reported by

Abdalla, et al [Abdalla et al., 2002].

Campbell and Scheiman [Campbell and Scheiman, 2002] investigated the thermo- oxidative stability of PMR-15 nanocomposites, specifically the effect of packing density and orientation of diamine on oligomer melt viscosity and oligomer crosslink enthalpy. Nanocom- posites of PMR-15 polyimide and a diamine modified silicate were prepared by adding the silicate to PMR- 15 resin. Their results showed that better dispersion of clay leads to enhanced thermo-oxidative stability. As discussed in section 2, PMR-15 composites generally degrade through oxidation of the surface layer followed by micro-cracking in the polymer

12 matrix [Meador et al., 1996]. The micro-cracks in turn allow permeation of oxygen into the bulk of the sample, thereby promoting further oxidative degradation. But, in nano-clay reinforced composites with exfoliated nano-clay platelets, better thermo-oxidative stability is observed because nano-clay platelets act as barriers and reduces the permeability and thus decreases bulk thermal oxidation.

Campbell et al. [Campbell, Johnston, Inghram, McCorkle, and Silverman, 2003] investigated the eﬀect of nano-clay on barrier properties of thermoplastic (BPADA-BAPP) and thermosetting (PMR-15) polyimide resin. Reductions in gas permeability and water absorption were observed in thermoplastic polyimide nanocomposites [Hussain et al., 2007].

The thermosetting polyimide showed a reduction in weight loss during isothermal aging in air at 288 °C. The addition of 5 wt% nano-clay to neat PMR-15 resin results in a 15% reduction in weight loss after aging for 1000 hours at 288 °C in air. While the weight loss of all samples aged in nitrogen was lower than those aged in air, addition of nano-clay reduced weight loss by 15% in each case. A greater decrease in weight loss was observed for carbon fabric reinforced composites. Exfoliation of only 1 wt% or 2 wt% nano-clay decreases the weight loss on oxidative aging by 25% and 20% for carbon-reinforced composites.

With a wide range of applications, including metal coatings, use in electronics/electrical components, high tension electrical insulators, ﬁber-reinforced plastic materials and structural adhesives epoxy is another conventional thermoset polymer. The wide applicabil- ity of epoxy has made it a very popular test platform for researchers conducting advanced studies such as nano-clay intercalation and exfoliation. In-situ polymerization of epoxy monomers, blending with monomer, and solution blending of monomer and clay using polar organic solvents has already been investigated. Brown et al. [Brown, Curliss, and Vaia,

13 2000] outlined the potential improvements in properties that can be obtained from epoxy resins with layered nano-clays. They concluded that layered nano-clays with proper organic surface modification compatible with the polymer matrix may result in enhanced ability to exfoliate clay layers and offer a significant increase in properties. Another study by Jiankun et al. [Jiankun, Yucai, Zongneng, and Xiao-Su, 2001] studied the intercalation and exfoliation behavior of organoclays in epoxy resin. Their work showed that the organically modified nano-clays can be easily intercalated by epoxy through a mild mixing at 70-80 °C to form a homogenous and stable organoclay/epoxy intercalated hybrid.

Messersmith and Giannelis [Messersmith and Giannelis, 1994] prepared an epoxy- silicate nanocomposite by dispersing an organically modiﬁed mica-type silicate (MTS) in an epoxy resin. The nanocomposite exhibited a broadened Tg at slightly higher temperature compared to the neat epoxy resin. Furthermore, the dynamic storage modulus of the nanocomposite containing 4% (by volume) silicate was approximately 58% higher in the glassy region and 450% higher in the rubbery plateau region as compared with the neat epoxy resin.

Pinnavaia and Lan [Lan and Pinnavaia, 1994] studied epoxy/clay nanocomposites in a rubbery state. They developed a method of clay treatment which is now commonly used.

Tensile tests conducted in this work showed that inclusion of the intercalated clay increased tensile strength and modulus signiﬁcantly over that of the pristine epoxy. In another study by Pinnavaia et al. [Lan, Kaviratna, and Pinnavaia, 1995] it was found that the extent of exfoliation of the clay in polymer depends on the accessibility of the epoxy and diamine to the clay galleries as well as the relative rates of intra and extra gallery network formation.

Preliminary mechanical measurements showed that exfoliated epoxy/clay nanocomposites

14 have higher moduli than intercalated clay composites. Similarly, there is evidence form research done by Choi et al. [Choi and Tamma, 2001] that exfoliated nano-clay platelets provide better thermo-oxidative stability by creating a more tortuous path for diﬀusion of oxygen into the polymer resin.

In summary, there is enough evidence supporting the idea of compounding nano- clay with thermoplastic as well as thermoset polymers to improve mechanical, thermal and, oxidative properties. In conclusion, it is now well established that properly exfoliated and dispersed nano-clay platelets can provide enhancement in both stiﬀness and strength of polymers with improved barrier properties. Recently, the costs involved to avoid safety issues with PMR-15 have led to the development of new material systems. One of these new resins is BMI resin which is a polyimide with a 232°C maximum service temperature. In the research presented in this dissertation, C20A nano-clay is compounded with BMI to study its eﬀect on mitigating thermo-oxidative degradation and residual stresses due to thermal oxidation in a thermoset polymer.

2.3 Damage Modeling for Composites

Continuous fiber-reinforced composites possess high strength and stiffness in the fiber direction. The overall mechanical behavior depends on the constituent properties as well as the microstructure of the composite. The deformation and damage mechanisms of composites are very different from the matrix material alone. Several possible failure mechanisms exist for fiber reinforced composites, such as interfacial fiber-matrix debonding, interlaminar delamination, matrix cracking, fiber breakage, fiber pull-out, and shear sliding of fibers.

Several techniques are available in the open literature to model the initiation and the propagation of damage in composite laminates. One way of modeling damage in com-

15 posites is based on continuum mechanics where reduced stiffness of damaged material is calculated with the help of effective damage parameters. In conventional damage models, reduction in load carrying capacity of the material is achieved through zeroing a few per- tinent stiffness terms or removing the failed elements from the FEM mesh. For example,

Chaboche [Chaboche, 1981] presented a stiﬀness reduction model in 1981. In this model,

3-D damage evolution was governed by an isotropic scalar damage parameter that represented a macroscopic measure of actual crack density. Later, in order to incorporate the material characteristics of constituents and the composite parameters such as volume fractions, ﬁber shape and arrangements, Lemaitre and Chaboche [Lemaitre and Chaboche, 1994] proposed a combined approach which used micromechanics based analysis for the thermo- elastoviscoplasticity of the composite with damage evolution. Likewise, Tay et al. [Tay, Tan,

Tan, and Gosse, 2005] proposed a method where property reduction is eﬀected by applying a set of external nodal forces such that the net internal nodal forces of elements adjacent to the damaged element are reduced or zeroed (for complete failure case).

Nemat-Nasser and Hori [Nemat-Nasser and Hori, 1999] presented a summary of various analytical models to predict eﬀective elastic properties of elastic solids with micro- cavities, micro-cracks and traction free surfaces. Similarly, Voyiadjis and Kattan [Voyiadjis and Kattan, 2006] applied continuum damage mechanics to composite materials within the framework of the theory of elasticity, and presented a directional data model of damage mechanics for composite materials using fabric tensors. In this micromechanics based model, the behavior of composite materials is related to the damage eﬀects through fabric tensors. To obtain the damaged properties of each constituent, damage mechanics was employed to each constituent separately. Then, proper homogenization approach was adopted to calculate the

16 damaged properties of composite. Also, a generalized thermodynamics based formulation of damage evolution was presented, which was used in solving diﬀerent numerical problems successfully.

Damage and failure of composite materials is inherently a multi-scale phenomenon, coupling diﬀerent scales of damage initiation and progression. The changes in the material structure, in general, are irreversible during the process of damage. Damage accumulation can take place under elastic deformation (high cycle fatigue), under elastic-plastic deformation (ductile plastic damage and low cycle fatigue) or under creep conditions (creep damage)

[Kachanov, 1986]. The choice of damage parameter, as a law, is not simple. It can be done either by a physical microstructural study, or by a direct generalization of experimental data. From application point of view, it is very important that the evolution of damage parameter should be simple enough and must have an evident mechanical sense [Kachanov,

1986]. In this regard, two types of damage evolution models, namely phenomenological and micromechanical, have been proposed in the literature for modeling failure of composite materials. One of the phenomenological damage evolution model was given by Chaboche

[Chaboche, 1981] in 1981. In 1987, by examining experimental data, Simo and Ju [Simo and Ju, 1987] concluded that the amount of micro-cracking at a particular strain level is sensitive to the applied loading rate (high strain rate) in dynamic environments. There- fore, they proposed a phenomenological rate sensitive damage model which required only one additional parameter. Matzenmiller et al. [Matzenmiller, Lubliner, and Taylor, 1995] presented a strain controlled continuum model wherein damage variables are introduced for the phenomenological treatment of the state of defects, and its implications on the degradation of the stiﬀness properties. To denote the loss area due to cracking within the material

17 ligament, they introduced two non-negative damage parameters quantifying the relative size of disk-like cracks. Similarly, Chan et al. [Chan, Cheng, Jie, and Chow, 2005] presented an anisotropic phenomenological damage model and damage criterion for localized necking.

To determine the damage parameter and hardening rule, experimentally measured material properties of damaged and pristine material were used in this model.

In all these models mentioned above, nature of phenomenological damage law is based on the macro level response of material and therefore, it did not explicitly account for the micro level damage. To study the eﬀect of micro level damage, Lene and Leguillon [Lene and

Leguillon, 1982] investigated the eﬀect of a tangential slip at the interface of the components

(fiber displacements in a matrix for instance). A linear law in terms of a scalar coefficient was used to describe this slip phenomenon. It was found that the equivalent medium is homogeneous, anisotropic material, and its elastic properties depend on the scalar parameter involved in the slip law. Similarly, in the work done by Wriggers et al. [Wriggers, Zavarise, and Zohdi, 1998], the damage variable is defined as the amount of open debonded interface area. It was concluded that the amount of debonded surface area should serve as a primary internal variable in a homogenized macroscopic constitutive model for damage in composites.

Fish et al. [Fish, Yu, and Shek, 1999] developed a non-local damage theory for brittle composite materials based on double-scale asymptotic expansion of damage. In this work, a closed-form expression relating local fields to the overall strains and damage has been derived. They introduced the concept of non-local phase fields (stress, strain, free-energy density, damage release rate, etc.) via weighting functions defined over the micro-phase. The capabilities of this model were confirmed by excellent numerical results.

Choi and Tamma [Choi and Tamma, 2001] conducted a ﬁnite element study and

18 modeling for the micromechanical predictions of the homogenized elastic properties of woven composites. Then, they used a ﬁnite element model in a micromechanical damage analysis to predict the damage initiation and propagation in the unit cell, and the resulting stress-strain curves. The completely damaged properties were taken to be 1% of the original properties in each constituent of woven fabric. Ju et al. [Ju, Ko, and Ruan, 2006] developed a micromechanical elastoplastic-damage formulation to predict the overall elastoplastic behavior and interfacial damage evolution in ﬁber-reinforced ductile matrix composites. In this work, concept of eigenstrains due to cylindrical inclusion in conjunction with ensemble area averaging was adopted to micromechanically estimate the overall damage accumulation.

Raghvan and Ghosh [Raghavan and Ghosh, 2005], have shown that the damage evolution in the microstructure of the composites signiﬁcantly aﬀects the material symmetry.

Also, different load paths would create a different damage evolution profile in the microstructure which will result in a different change in material symmetry. The anisotropic stiffness tensor will couple normal and shear strain components in the elastic energy expression in the

fixed coordinate system. Coupling terms reduces to vanishingly small numbers if principal damage coordinate system (PDCS) is used to express the stiffness and the initial material symmetry is retained and the damage effect tensor has a diagonal representation. Determi- nation of continuously evolving PDCS requires determination of second order damage tensor and its Eigen vectors for each load step. Nonlinear least square minimization method was used to obtain the six independent components of symmetric damage tensor. Subsequently,

Eigen vectors are obtained to get the rotation matrix which transforms the global coordinate system (GCS) to the PDCS.

To examine the evolution of PDCS with diﬀerent load histories, a micromechanical

19 analysis problem of a simple unit cell representative volume element (RVE) was used. For proportional loading (shear loading is proportional to the axial load and transverse axial load is zero) orientation of the PDCS with respect to the GCS is at some fixed angle and remains unchanged during loading and unloading. For non-proportional loading (axial and shear load is applied independently), in the first half of the loading when only axial load is applied keeping transverse load and shear to be zero, PDCS coincides with the GCS. For the second half of the loading (similar to the proportional loading) PDCS continuously rotates and gets a final orientation angle which is less than the angle obtained in the proportional loading case.

Most of the damage models discussed above do not account for the evolution of damage or the loading history. Only a few namely Voyiadjis and Kattan [Voyiadjis and

Kattan, 1992], Fish et al. [Fish et al., 1999], Kouznetsova et al. [Kouznetsova, Brekelmans, and Baaijens, 2001] and Raghvan and Ghosh [Raghavan and Ghosh, 2005] include damage evolution as well as loading history. Ignoring these factors can result in a signiﬁcant error for the problems with independent (non-proportional) axial and shear loading. The solution for this shortcoming is possible through simultaneous RVE-based microscopic-macroscopic analysis in each load step [Fish et al., 1999],[Feyel and Chaboche, 2000],[Massart, Peerlings, and Geers, 2007]. However, this approach is computationally very expensive since detailed micromechanical analysis is needed for each load step at every gauss point in all the elements of macrostructure.

To overcome the shortcoming of the parallel micro-macro analysis Raghvan et al.

[Raghavan and Ghosh, 2005],[Ghosh, Bai, and Raghavan, 2007] developed a computationally eﬃcient anisotropic homogenization based continuum damage mechanics (HCDM) model for

20 composites going through micro-structural damage. In this model ﬁber-matrix debonding is taken as the microstructural damage mechanism for a 2-D (two-dimensional) analysis using

Vornoi cell finite element model. In multi-scale modeling, the use of continuum damage mechanics in the non-critical regions can make the whole approach extremely efficient. Such models can avoid the extensive micromechanical analysis at each load step but the micromechanical analysis is needed in the close vicinity of dominant crack or localized instability for better prediction of catastrophic failure. In an effort to incorporate loading effects, Ghosh et al. modified their 2-D model [Ghosh, Ling, Majumdar, and Kim, 2000] to a 3-D model

[Jain and Ghosh, 2008] by introducing a PDCS which evolves with loading path.

The general form of CDM models introduce ﬁctitious stress acting on an eﬀective resisting area. The reduction in the original resisting area is caused due to the presence of micro cracks in the degraded material. Work done by Simo and Ju, 1987 [Simo and Ju,

1987] relates the effective fictitious stress to the actual Cauchy stress through a fourth order damage effect tensor. The hypothesis of equivalent elastic energy was used to evaluate the damage tensor and establish a relation in damaged and undamaged stiffness. This hypothesis specifically assumes that the elastic complimentary energy in a damaged material with the actual stress is equal to that in a hypothetical undamaged material with a fictitious effective stress.

Murakami [Murakami, 1988] worked to find a relation between second order damage tensor and the damage effect tensor. Since any arbitrary damage tensor results in an asymmetric effective stress tensor, Voyiadjis and Kattan [Voyiadjis and Kattan, 1996] suggested a representation to maintain the symmetric nature of effective stress tensor. Using this representation, the damaged stiffness is updated in terms of undamaged stiffness.

21 The anisotropic CDM model proposed by Raghavan and Ghosh [Raghavan and Ghosh,

2005] involving fourth order damage tensor introduces a damage evolution surface to delin- eate the interface between damaged and undamaged domains in the strain space. The surface equation involves a fourth order symmetric negative deﬁnite tensor which corresponds to the direction of the rate of stiﬀness degradation tensor and a damage state variable.

2.4 Cohesive Layer Models

It is now well-established that in the presence of a large fracture process zone near the crack tip, the basic assumptions of linear elastic fracture mechanics (LEFM) are no longer valid [Kanninen and Popelar, 1985]. Specifically, in some polymers, the occurrence of void nucleation and growth ahead of the crack-tip results in a damage (process) zone that is not traction free. Further, for a crack in a polymer matrix composite, fiber-bridging may also be present within the damage zone. Therefore, in such cases, a cohesive layer modeling approach would be more accurate in accounting for the nonlinear processes that occur within the “damage zone”. Cohesive zone model was first introduced by Barenblatt [Barenblatt,

1962] and Dugdale [Dugdale, 1960] in the 1960s. In the 1980s, application of cohesive zone models to determine strength of composites and adhesive joints were introduced by Bäcklund

[Backlund, 1981] and Stigh [Stigh, 1987]. To model the Interfacial debonding in composite laminates, Swaminathan et al. [Swaminathan, Pagano, and Ghosh, 2006] developed a 3-D micromechanical model in which the interface behavior is described by 3-D cohesive zone with bilinear traction boundary conditions. In this model, the interface is represented by cohesive springs of inﬁnitesimal length. Needleman [Needleman, 1987] and Stigh [Stigh,

1988] demonstrated how the cohesive zone model ﬁts within the scope of conventional stress analysis using FEA. Cohesive zone models have seen an almost explosive increase in use

22 and applications during recent years. With cohesive modeling, no additional properties are necessary to simulate crack growth. Only the cohesive law is needed to analyze both initiation and growth of a crack. Typically, cohesive elements in FEA codes follow a pre- defined traction separation law that simulates the crack initiation and propagation. Initially the traction across the interface increases with the displacement then drops after reaching a maximum value and eventually vanishes indicating the failure of cohesive elements. Three different zones are identified to represent these tractions (normal and tangential) in terms of effective opening displacement. In hardening region traction increases with the opening displacement and decreases in the softening region and eventually drops down to zero when cohesive springs fail which is when complete debonding occurs. The unloading pattern is same as the loading path in the hardening region but it changes for the softening region demonstrating the irreversible nature of damage process. Another advantage of cohesive zone models is that these models can simulate different types of failure mechanisms, such as fiber matrix debond and interlaminar delamination. This is also a drawback in modeling flexibility; namely if the fracture toughness changes with crack growth, a conventional cohesive law cannot capture this phenomenon by itself. Another major issue involved in this kind of modeling work is the scale of crack opening length which needs to be chosen very carefully.

In order to develop a cohesive zone model that does not require a prescribed traction- separation law, Allen and Searcy [Allen and Searcy, 2001] proposed a viscoelastic cohesive zone model and demonstrated the use of this model by numerically solving example problems with diﬀerent displacement boundary conditions and strain-rates. They also proposed a damage evolution law that was phenomenologically derived due to the absence of near-tip experimental data. In this context, the process zone ahead of the crack tip is usually very

23 small compared with the specimen size in most materials. Therefore, experimentally it is quite challenging to precisely determine the traction separation law in the cohesive zone. The work presented in this dissertation employs a modiﬁed version of the viscoelastic cohesive layer model prosed by Allen and Searcy [Allen and Searcy, 2001], but the damage evolution law is fully characterized based on actual data from experiments. Sorensen and Jacobsen

[Sorensen and Jacobsen, 2003] presented a review of existing experimental procedures to estimate the cohesive law and delineated two major approaches. The ﬁrst approach is to use a direct tension test, with the assumption that a uniform damage state evolves across the ligament. In reality, it is very diﬃcult to achieve uniform damage state in a ligament during direct tension test, and therefore this approach is impractical. The second approach is the J-integral approach where macro level J-integral data is used to extract the micro level constitutive (traction-separation) behavior in the process zone. Sorensen and Jacobsen

[Sorensen and Jacobsen, 2003] adapted the same approach to conduct their ongoing research.

Recently, Fuchs and Major [Fuchs and Major, 2011] used the J-integral approach to determine the cohesive zone models for glass-ﬁber reinforced composites and studied the eﬀect of loading direction on the constitutive cohesive law. In the work presented in this dissertation, the

J-integral approach is employed to determine the cohesive law for delamination behavior of

IM-7/BMI unidirectional laminates, before and after isothermal aging at 260 °C for 1000 hours. For this purpose, double cantilever beam (DCB) experiments were conducted to acquire the macro level J-integral data, and the displacement and strain ﬁelds in the process zone were obtained using digital image correlation (DIC). From experimental data, cohesive law and damage evolution parameters were determined and used in the viscoelastic cohesive layer model to simulate delamination growth.

24 CHAPTER 3

MATHEMATICAL MODELING OF THERMAL OXIDATION

In this chapter, the use of a diﬀusion-reaction mechanism to represent the time dependent thermo-oxidation behavior of a polymer material is presented. Based on the work done by Tandon et al. [Pochiraju and Tandon, 2006], a three-zone model representing three dis- tinct oxidation zones was used. A schematic diagram showing three oxidation zones, namely, un-oxidized, active, and fully oxidized zone is presented in Fig. 3.1. A polymer state variable

φ is used to represent the oxidation state in a particular oxidation zone quantitatively, indicating the availability of active polymer site for oxidation reaction. Polymer state variable varies from its maximum value of 1 (in un-oxidized zone) to its minimum threshold value of

φ = φox (in the fully oxidized zone). A thin active oxidation layer is present in between the oxidized and un-oxidized zone where φox < φ < 1, as shown in Fig. 3.1

Oxidized Active Un-oxidized

ϕ=ϕox 1>ϕ>ϕox ϕ=1 Three zone model with oxidation state parameter ϕ

Figure 3.1: Schematic diagram showing three zones during thermal oxidation

25 3.1 Three Zone Oxidation Model

The diﬀusion of oxygen in a polymer composite incorporating consumptive reaction

rate is governed by Fick’s law as

∂C ∂ 2C ∂ 2C ∂ 2C = D11 + D22 + D33 − R(C) (3.1) ∂t ∂x2 ∂y2 ∂y2

where, C(x,y,z,t) is the oxygen concentration, Di j are the orthotropic diﬀusivities, and R(C) is the polymer consumption rate. The rate of oxidation reaction is modeled with respect to the saturation reaction rate R0 when the reaction is not oxygen deprived. For BMI type polymer resins, the reaction rate reduces when the availability of oxygen reduces according to the equation,

R(C) = R0 f (c) (3.2)

here, R0 is the saturation reaction rate and f (C) is a function which simulates the situation when the amount of available oxygen is less than the saturation concentration, and is given by following expression [Pochiraju and Tandon, 2006]

2βC βC f (C) = 1 − (3.3) 1 + βC 2(1 + βC)

where βC is the normalized concentration and β is an oxidation reaction coeﬃcient. If it is

assumed that the polymer weight loss is directly proportional to the rate of reaction R(C),

then, dW ∝ −R(C) (3.4) dt

26 The polymer state variable φ is deﬁned as the ratio of the current weight of the polymer specimen to its original (un-oxidized) weight. Therefore time derivative of polymer state variable is proportional to the weight loss rate,

dφ dW ∝ (3.5) dt dt

Combining Eqns. 3.2, 3.4 and 3.5 gives,

dφ = −αR0 f (C) (3.6) dt where α is a proportionality constant. To model the thermo-oxidative behavior of a BMI type polymer, three oxidation reaction parameters (αR0, β and φox) need to be experimentally characterized. To determine these parameters, experimental data from isothermal aging of polymer specimen in conjunction with oxidation layer growth data is needed. The procedure followed to calculate each one of these parameters is outlined separately in the next section.

The aging parameters for nano-clay modiﬁed polymer can be determined in the same manner as described in next section.

3.2 Determination of Thermo-Oxidative Parameters

This section summarizes the methodology [Pochiraju and Tandon, 2006] adopted to determine the parameters needed to model thermo-oxidative behavior of thermoset polymer.

Deﬁnition of these parameters and the procedure adopted to extract each parameter from thermal aging experiments is outlined in following sections.

27 3.2.1 Calculation of β

To calculate β, weight loss data in two diﬀerent aging environments is required.

Combining Eqn. 3.2 and 3.3 for two diﬀerent oxygen concentrations C1 and C2, and equating

the reaction rates gives

2βC1 βC1 R(C1 2βC2 βC2 1 − = 1 − (3.7) 1 + βC1 2(1 + βC1) R(C2) 1 + βC2 2(1 + βC2)

Weight loss rate in a particular aging environments is proportional to the reaction

rate R(C). From the slopes of linear ﬁts to the weight loss versus time data in two diﬀerent

R(C1 aging environments, the ratio of reaction rates can be obtained. Here R(C1) and R(C2) R(C2) are reaction rates in diﬀerent aging environment having oxygen concentrations C1 and C2

R(C1 respectively. Substituting the values of ratio , and C1, C2 in Eqn. 3.7, a cubic equation R(C2)

in β is obtained to be solved for only one physically permissible (positive) value of β.

3.2.2 Calculation of αR0

Once the value of β is determined, to calculate αR0 Eqn. 3.5 and Eqn. 3.6 can be

rewritten for a particular aging environment with constant oxygen concentration C as

dW dt C (αR0)C = (3.8) f (C)

dW Now, to determine αR0, function f (C) and dt C is required. The slope of the linear

dW ﬁt to the weight loss versus time data provide dt C for a particular aging environment

and function f (C) can be determined by substituting the oxygen concentration C of that

dW particular aging environment and β in Eqn. 3.3. Substituting dt C and f (C) in Eqn.

28 Figure 3.2: Oxidized and un-oxidized regions depicted in section view

3.8, the value of αR0 is determined. It is imperative to note here that thermo-oxidative

parameter αR0 is independent of the aging environment chosen for this calculation. Also, it

should be noted that α and R0 cannot be obtained independently unless R0 is known.

3.2.3 Calculation of φox

The threshold value of polymer state variable φox is given by

V(ε − γ) φox = 1 − 1 (3.9) V0 + 2Va

In Eqn. 3.9, V is the original specimen volume, Vo is the oxidized volume, and Va is the active

volume for oxidation (refer to Fig. 3.2). The variables ε and γ are weight loss fractions due

to oxidation in air and inert atmosphere respectively. It is diﬃcult from the tested samples

to diﬀerentiate between the active zone and the oxidized zone; therefore, at this point active

volume Va is assumed to be eﬀectively zero.

For a specimen with initial length L, width W, and thickness T (see Fig. 3.2), if to is the thickness of the oxidized layer in each direction, then the volume of oxidized material Vo is given by

Vo = V − (L −to)(W −to)(T −to) (3.10)

29 where, V = LWT is the volume of the test specimen prior to thermal oxidation. From the test

data, ε, γ and Vo are functions of time for a particular oxygen concentration. Substituting speciﬁc values from aging experiments in Eqn. 3.9 we obtain φox as a function of time. To compensate for the eﬀect of material erosion and volumetric shrinkage, the φox value has to be obtained by averaging the φox predictions over a initial time period of ∼100 hours.

Results from isothermal aging experiments, and characterization of αR0, β and φox from

experimental data is presented in the next chapter.

30 CHAPTER 4

INFLUENCE OF NANO-CLAY ON THERMO-OXIDATIVE STABILITY AND

MECHANICAL PROPERTIES OF BMI

This chapter presents a summary of experiments conducted on neat as well as nano- clay (C20A) modiﬁed BMI. First, the materials and manufacturing process is presented. This is followed by isothermal aging experiment results, conducted in inert (argon) and oxidative environments (air and 60% oxygen) at 250 °C, close to the glass transition temperature of

BMI (Tg = 280 °C). As described in Chapter3, weight loss data from these experiments is used to extract the modeling parameters for neat and C20A modiﬁed BMI. The modeling parameters determined from these experiments will be used as input to the FEA model to predict residual stresses (refer to Chapter5) due to thermal oxidation in IM-7/BMI unidirectional composite.

4.1 Materials and Manufacturing Process

BMI resin system provided by Advanced Composites Group (ACG) was melted by keeping it inside a preheated oven at 125 °C. Melted resin was poured in beakers coated with mold release agent to fabricate 0.125 inch thick, 2 inch diameter, round “poker chip” specimens. Degassing of these specimens was carried out for 20 minutes in a preheated oven at 110 °C and at a pressure of 29 in. Hg. To avoid the boiling of resin, temperature had to be maintained precisely at 110 °C. Degassed specimens were kept for 6 hours inside the oven at 190 °C for curing. Cured specimens were taken out of the oven to cool down to room

31 Neat BMI 3 wt% C20A modified BMI

Figure 4.1: Neat and C20A modiﬁed BMI specimens

temperature. After polishing, these baseline BMI specimens were used for isothermal aging

experiment.

To study the eﬀect of nano-clay on barrier properties as well as mechanical properties,

BMI resin was compounded with 3% (by weight) of C20A nano-clay using a high-shear mixer.

High shear mixer helps in breakup of clay tactoids and thus aids in exfoliation. A similar

process as described above was followed to make these nano-modiﬁed BMI “poker chip”

specimens. One each of baseline resin and nano-compounded BMI specimen is shown in Fig.

4.1. Three replicate specimens each of baseline BMI resin and 3wt% C20A compounded

BMI were isothermally aged in three diﬀerent aging environments for more than 1000 hours

at 250 °C.

4.2 Thermo-Oxidative Aging Experiments

4.2.1 Isothermal aging of BMI and C20A/BMI in air

Three replicate specimens of each neat and C20A modiﬁed BMI were desiccated before starting the aging experiment. To conduct isothermal aging in air environment, these desiccated specimens were kept in a preheated oven at 250 °C as shown in Fig. 4.2. Weight

of each specimen was recorded carefully at regular intervals. Experimental data for average

32 Figure 4.2: Specimens inside the oven during isothermal aging experiment

% weight loss in air 12 Neat BMI 3%C20A/BMI 10

% Weight Loss 4

0 0 200 400 600 800 1000 1200 1400 1600 Time (hrs)

Figure 4.3: Weight loss data for BMI and 3% C20A compounded BMI at 250 °C in air

33 percentage weight loss due to thermal oxidation of BMI specimens in air at 250 °C is plotted

in Fig. 4.3. As can be observed from Fig. 4.3, the presence of nano-clay reduces oxidative

weight loss in air by approximately 40% as compared with the baseline. The weight loss

data for C20A modiﬁed resin was corrected for the presence of nano-clay platelets using a

rule of mixtures approach, assuming a nano-clay volume fraction of 2.0%, as shown below.

The total mass of the nano-clay modiﬁed specimen Mt, in terms of mass of individual

constituents can be written as

Mt = Mnr + Mnc (4.1)

where Mnr is the mass of neat resin and Mnc is the masses of nano-clay present in the

compounded specimen. Therefore, if ψ% nano-clay (by wt.) is compounded with neat

resin, the total mass is given by Mt = (1 + ψ)Mnr. Using the mass-density relationship and

rearranging terms in Eqn. 4.1, we can rewrite Eqn. 4.1 as

ρncVnc ψ = (4.2) ρnrVnr

where, ρnr and ρnc are the densities of neat resin and nano-clay respectively, and Vnr and

Vnc are absolute volumes of neat resin and nano-clay. From Eqn. 4.2, volume fraction of nano-clay χ in compounded specimens can be solved to obtain,

Vnc ψρnr χ = = (4.3) Vt ψρnr + ρnc

Here, Vt = Vnr +Vnc is the total volume of the specimen. The density of BMI and C20A nano-clay is 1.21 gm/m3 and 1.77 gm/m3, respectively. Substituting known values in Eqn.

34 4.3, the volume fraction of nano-clay χ is found to be 2.0%, for 3wt% nano-clay loading.

Assuming that nano-clay platelets does not go through thermal oxidation, weight loss data

for compounded specimens was corrected by dividing it by (1 − χ).

4.2.2 Isothermal aging of BMI and C20A/BMI in Argon

Typically, in an inert environment, post-polymerization and thermolysis are the two active mechanisms responsible for weight loss, whereas in an oxidizing environment thermo- oxidation is also present. Therefore, similar isothermal aging experiments were conducted in an inert atmosphere (e.g., Argon) in addition to the experiments conducted in oxidative atmosphere (air and 60% O2). The diﬀerence in the weight loss for oxygen rich environment and inert environment gives the weight loss due to oxidation alone. To avoid specimen interaction with the oxygen present in ambient air, specimens were kept inside a steel bag which was made air tight using clamps, as shown in Fig. 4.4. A narrow tube was inserted through a small opening into the steel bag and a positive ﬂow of argon gas was maintained throughout the aging experiment.

Averaged percentage weight loss data for baseline as well as C20A modified BMI in argon environment at 250 °C is presented in Fig. 4.5. As mentioned before, the average weight loss data for C20A modified resin was corrected for the presence of nano-clay platelets using a rule of mixtures approach. From Fig. 4.5, it is evident that the average weight loss in inert conditions for C20A modified BMI is significantly lower (∼27%) compared with that of neat BMI. Also, comparing the averaged percentage weight loss in inert conditions (Fig.

4.5) and air environment (Fig. 4.3), it is clear that polymer consumption due to thermal oxidation is notably less in inert conditions.

35 Tube

Clamps Steel bag

Figure 4.4: Experimental set up for weight loss experiment in Argon and 60% O2 environment

4.2.3 Isothermal aging of BMI and C20A/BMI in 60% O2

To calculate the thermal oxidation modeling parameters described by Tandon et al.

[Tandon et al., 2006] (refer to section3), weight loss data in two diﬀerent concentrations of oxygen is needed. Therefore, in addition to weight loss experiment in air (∼ 20% O2 concentration), customized gas cylinder containing 60% oxygen and 40% argon was ordered from

Airgas Inc. to conduct the thermo-oxidative aging experiment in oxygen rich environment at approximately 60% O2 concentration. Procedure outlined in Chapter 4.2.2 was followed to conduct these experiments. Fig. 4.6 presents average weight loss data for thermo-oxidative aging experiment performed at 250 °C in oxygen rich environment (60 % O2). It is clearly seen in Fig. 4.6 that C20A modiﬁed BMI shows signiﬁcant (∼33%) reduction in weight loss compared to baseline BMI, after 1000 hours.

Comparing the reduction in weight loss due to nano-clay in 60% O2 environment with

36 % weight loss in argon 3 Neat BMI 3% C20A/BMI 2.5

1.5 % Weight Loss 1

0.5

0 0 200 400 600 800 1000 1200 Time (hrs)

Figure 4.5: Weight loss data for BMI and 3% C20A compounded BMI at 250 °C in Argon

% weight loss in 60% O 2 10 Neat BMI 3% C20A/BMI 8

4 % Weight Loss

0 0 200 400 600 800 1000 1200 Time (hrs)

Figure 4.6: Weight loss data for BMI and 3% C20A compounded BMI at 250 °C in 60% O2

37 the reduction in weight loss in air environment, after similar aging time, it is observed that

the reduction is slightly lower for 60% case. Since the availability of oxygen for permeation

is higher in 60% O2 environment, the eﬀectiveness of nano-clay as a barrier is relatively low.

For comparison, averaged weight loss data in diﬀerent aging environments is plotted in Fig. 4.7. From Fig. 4.7(a), Fig. 4.7(b) and Fig. 4.7(c) it is evident that the weight loss for

C20A compounded BMI is signiﬁcantly less compared with the baseline BMI. For example, after 1200 hours of aging in air at 250 °C, C20A modiﬁed BMI shows 40% reduction in weight

loss. Similarly, C20A modiﬁed resin exhibits 27% lower weight loss than baseline resin after

1000 hours of aging in argon environment, and approximately 33% lower weight loss in 60%

oxygen environment after a similar duration of aging time. Comparing the weight loss data

in diﬀerent aging environments, it is evident that resin weight loss increases with the amount

of oxygen available in the aging environment, and that C20A nano-clay works eﬀectively as

a barrier to reduce oxygen permeation through the bulk resin and thereby reducing thermo-

oxidative weight loss. This can be explained through the kinetic model presented by Colin

et al. [Colin et al., 2002]. Six elementary steps involved in thermal aging process are

(I) Initiation λROOH + γRH ⇒ µR° + ηRO2° + H2O + νV

(II) Propagation R° + O2 ⇒ RO2°

(III) Propagation RO2° + RH ⇒ ROOH + R°

(IV) Termination R° + R° ⇒ inactive products

(V) Termination R° + RO2° ⇒ inactive products

(VI) Termination RO2° + RO2° ⇒ inactive products + O2

In this work, it is assumed that the weight loss comes mainly from the initiation step

(I). As described in this work, the peroxyl radicals (RO2°), formed from alkyl (R°) radical

38 oxidation, are a reactant in the propagation step (III) that creates hydro-peroxides (ROOH).

With higher availability of oxygen in environment, the alkyl radical reaction produces more peroxyl (RO2°) and hydro-peroxides (ROOH). As the hydro-peroxide concentration increases, the rate of initiation step (I) goes up and weight loss begins to increase via water and volatile emission products in this reaction step. The next section presents the calculation of thermo- oxidative parameters from experimental data for neat and C20A modiﬁed BMI.

39 (c) % weight loss in 60% O (a) % weight loss in air (b) % weight loss in argon 2 10 10 10 Neat BMI Neat BMI Neat BMI 3%C20A/BMI 3% C20A/BMI 3% C20A/BMI 8 8 8

6 6 6

40 4 4 4 % Weight Loss % Weight Loss % Weight Loss

2 2 2

0 0 0 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Time (hrs) Time (hrs) Time (hrs)

Figure 4.7: Comparison of weight loss data for BMI and 3% C20A compounded BMI at 250 °C in (a) air (b) argon (c) 60% O2 environment 4.3 Calculation of Thermo-oxidative Parameters from Experimental Data

Following the methodology described in Chapter3, thermo-oxidative parameters were

determined for baseline and C20A modiﬁed BMI from the isothermal aging data presented

in sections 4.2.1, 4.2.2 and 4.2.3. In order to determine β, weight loss data obtained at two

diﬀerent aging environments (air and 60% O2) are needed. The concentration of oxygen in air

3 3 and 60% O2 is 4.81 mol/m and 13.72 mol/m respectively. Since weight loss is proportional

R(C ) to the reaction rate (Eqn. 3.4), ratio of reaction rates 1 can be easily determined by R(C2)

comparing the weight loss data for these aging environments. From the weight loss data, the

R(C ) ratio 1 for baseline BMI was 0.70 and it was 0.47 for nano-clay modiﬁed BMI at 250 °C. R(C2)

Combining Eqn. 3.2 and Eqn. 3.3 for two diﬀerent oxygen concentrations C1 and C2 , and equating the reaction rates, results in a cubic equation in β with three roots of which only one has physically admissible (positive) value. The calculated values of β for baseline resin

(BR) and nano-clay modiﬁed resin (NC) are β BR = 0.122 m3/mol and β NC = 0.030 m3/mol,

respectively.

Once the value of β is known to us, αR0 can be determined by substituting f (C) and

dW the weight loss rate dt C in Eqn. 3.8 for any one of aging environments with known O2 concentration C. The values of αR0 for baseline resin and for nano-clay modiﬁed BMI are

BR −4 NC −3 αR0 = 9.34 × 10 gm/hr and αR0 = 1.22 × 10 gm/hr, respectively.

Eqn. 3.9 is invoked to calculate the thresh-hold value of the polymer state variable

(φox). From the test data, ε, γ and Vo are functions of time for a particular oxygen concentration. Substituting speciﬁc values in Eqn. 3.9, we obtain φox as a function of time. To compensate for the eﬀect of material erosion and volumetric shrinkage, the φox value was

41 Table 4.1: Oxidation parameters for neat BMI and C20A modiﬁed BMI

Parameter Neat BMI C20A modiﬁed BMI % Change

−4 −3 Reaction rate (αR0) 9.34×10 1.22×10 +30.7

Normalization parameter (β) 0.122 0.030 -75.3

Threshold oxidation state variable (φox) 0.639 0.712 +11.4

obtained by averaging the φox predictions of initial time period of ∼100 hours. φox value

obtained for baseline BMI is 0.639, and for C20A modiﬁed BMI it is 0.712. The oxidation

parameters for baseline BMI and C20A modiﬁed BMI are summarized in Table 4.1.

Oxidation parameter αR0 is the maximal rate of oxidation when oxygen is in excess.

The limit of oxygen concentration C, beyond which oxygen is said to be in excess is given by the reciprocal of oxidation parameter β. And, oxidation parameter φox deﬁnes the avail-

ability of polymer for consumption through oxidation reaction.Therefore, from the oxidation

parameters presented in Table 4.1, it is concluded that the oxidation reaction rate for C20A

modiﬁed BMI is higher compared with that of neat BMI. However, signiﬁcantly lower value

of β in conjunction with slightly higher value of φox for C20A modiﬁed BMI makes it very dif-

ﬁcult for oxidation reaction to achieve the saturation state. Therefore, overall lesser polymer

is consumed by the oxidation reaction which is corroborated by the weight loss experiment

results presented in section 4.2.

4.4 Nano-indentation Test for Mechanical Properties

Nano-indentation tests were performed on both neat as well as C20A modiﬁed BMI

(3wt%) to estimate the modulus and the hardness at UT-Dallas. A spherical tip (∼400 µm)

42 Table 4.2: Comparison of mechanical properties for neat and C20A modiﬁed BMI(3 wt%)

Property Neat BMI C20A modiﬁed BMI % Change

Elastic Modulus (GPa) 2.78 4.19 +50.7

Hardness (MPa) 26.0 42.0 +61.5

Comparison of shrinkage strain 1

0.8

0.6

0.4 Shrinkage Strain (%)

Neat BMI 0.2 3%C20A/BMI

0 0 100 200 300 400 500 600 Time (hrs)

Figure 4.8: Measured shrinkage strains for neat BMI and C20A modified BMI is used for these tests. For reliability, 20-25 tests at different locations in a specimen were conducted. A comparison of compressive elastic modulus and hardness for neat and C20A modified BMI are presented in Table 4.2. As can be observed from the data in Table 4.2, there is a 51% improvement in the compressive modulus, and 62% improvement in hardness due to C20A compounding. All indentation tests were performed at room temperature.

43 4.5 Thermal Shrinkage Test

Since thermoset polymers go through a high temperature curing cycle, cure shrinkage

and thermal oxidation give rise to localized micromechanical residual stresses and dam-

age. Therefore, thermal shrinkage tests were conducted on baseline and C20A modiﬁed

BMI(3wt%) specimens at UT-Dallas. These specimens were kept in a preheated oven at 250

°C and monitored by a digital camera through a glass window. Photographs taken during

the shrinkage experiment in conjunction with digital image correlation (DIC) technique were

employed to record the local strain ﬁelds. Averaged shrinkage strains over the test domains

are presented in Fig. 4.8 for baseline and C20A modiﬁed BMI.

It is evident from Fig. 4.8 that the long-term shrinkage strain for C20A modiﬁed

BMI (-0.7%) is signiﬁcantly (30%) lower as compared with the baseline resin (-1.0%). In

the current work, oxidative shrinkage in carbon ﬁber is assumed to be negligible at 250 °C and therefore the major contribution to oxidation induced residual stresses comes from the oxidation of polymer phase alone. These experimentally measured shrinkage strains due to thermal oxidation will be used as input to the micromechanics based numerical model presented in Chapter5.

44 CHAPTER 5

FINITE ELEMENT SIMULATION OF RESIDUAL STRESSES DUE TO SHRINKAGE

IN IM-7/BMI COMPOSITE

The experimental data presented in Chapter4 were used as input data to numerically investigate residual stresses due to thermo-oxidative shrinkage in a unidirectional lamina consisting of baseline resin and C20A modified resin as matrix material, respectively. A coupled finite element (FE) simulation of diffusion-reaction and stress analysis was carried out for this purpose. Thermal oxidation was modeled using a mechanism based multi-scale

ﬁnite element approach. The mechanism-based FEA model (NOVA-3D) uses a modiﬁed

Fick’s law of diﬀusion that includes a reaction term related to the rate of oxygen consumption due to the chemical reaction between the polymer and oxygen, as described in Chapter3.

The model incorporates the coupling between oxygen diﬀusion and polymer consumption process.

5.1 Finite Element Model

The FE model considers a RVE in which ﬁber and matrix are arranged in a hexagonal cross sectional array to maintain the periodicity of the RVE. From the symmetry of this ﬁber arrangement, the isolated two-dimensional (2-D) unit cell is depicted in Fig. 5.1(a) and the applied symmetry displacement boundary conditions for this analysis are shown in Fig.

5.1(b). This choice of RVE includes fiber-fiber stress interactions which would be ignored if a standard hexagonal RVE having a single fiber centered at the centroid of the hexagon was

45 r

Figure 5.1: (a) Hexagonal arrangement of fibers and (b) In-plane displacement boundary conditions on the RVE with fiber-fiber interaction used [Andrews and Garnich, 2008]. The carbon fiber radius is assumed to be 5 µm (micron) and the fiber volume fraction is 0.6. For this shrinkage analysis, the length of the model along the fiber axis is assumed to be equal to 10 times the fiber radius, i.e. 50 µm. In order to examine the stress variations at the fiber-matrix interface close to the free edge, a fully

3-D ﬁnite element model with 20-noded brick elements is constructed as shown in Fig. 5.2.

However, convergence of stress solution requires a ﬁne mesh at the free edge with small time increments (<10 sec). For this purpose a linear biased mesh along z direction (see Fig. 5.2) is used in this simulation with a total of 200,000 degrees of freedom.

For the stress analysis, one end of the RVE (at z = 50µm in Fig. 5.2, away from the free edge) is on rollers (zero displacement in the z-direction) to reﬂect symmetry, and the other end is free to deform (at the free edge, z = 0µm). In-plane boundary conditions for this model are depicted in Fig. 5.1(b). For the oxygen diﬀusion boundary conditions,

3 the RVE free edge is exposed to a speciﬁed oxygen concentration of C0 = 4.8mol/m i.e.

46 u2= 0

u1= 0

Face on rollers u3 = 0 z = 50 μm

u1= 0

Y O2 u = 0 2 X Z

Free end u3 = 0 z = 0 μm 3 C0 = 4.8 mol/m

Figure 5.2: 3-D ﬁnite element mesh of RVE showing boundary conditions the concentration of oxygen in air. Zero ﬂux (symmetry) boundary condition is applied at the opposite end of the RVE (at z = 50µm). The cure temperature for BMI is 190 °C, and isothermal oxidation experiments were conducted at 250 °C, therefore a thermal load of

∆T = 60°C is applied to this unit cell to simulate thermal expansion of the resin and fiber due to isothermal aging. While thermal expansion is assumed to occur in both fiber and matrix, oxygen diffusion (and oxidative shrinkage) is assumed to occur only in the resin matrix. In

47 the absence of experimental data, the ratio of oxygen diﬀusivity for oxidized to un-oxidized

material is assumed as 10, for both baseline and C20A modiﬁed BMI.

The ﬁber used in these simulations is IM-7 carbon ﬁber. All material properties used

in these simulations are tabulated in Table 5.1. As oxygen diﬀuses into the RVE and reacts

with the resin, the resin begins to shrink due to oxidation. The isotropic shrinkage strain

value is assumed to be proportional to the current oxidation state variable φ for the polymer.

The linear relation between shrinkage strain and oxidation state variable φ is,

1 − φ εs = εs0 (5.1) 1 − φox

Therefore, shrinkage strain is 0 when resin is un-oxidized (φ = 1) and attains a max-

imum value εs0 when the resin is completely oxidized (φ = φox). To calculate the diﬀusivity of the resin after nano-clay compounding, a three dimensional diﬀusion model proposed by

Liu et al. [Liu, Hoa, and Pugh, 2007] capturing the effect of tortuosity and aspect ratio of randomly oriented nano-clay platelets was used. Using this model, diffusivity for nano-clay compounded resin can be expressed in terms of diffusivity of neat resin by the following expression

DBR DNC = (5.2) 2ζψ 1 + 3π

where DNC and DBR are diﬀusivities for nano-clay compounded resin and baseline resin

respectively, the volume fraction of nano-clay is ψ, and ζ is the average aspect ratio of nano-

clay platelets. The average aspect ratio for C20A nano-clay is approximately 100. Eqn. 5.2

48 predicts a 40.4% reduction in the oxygen diﬀusivity of the BMI resin due to the presence of

C20A nano-clay, as listed in Table 5.1.

Table 5.1: Material properties used in FEA simulation

Property Fiber (IM-7) Neat BMI C20A modiﬁed BMI % Change

E1 = E2(GPa) 19.5 2.78 4.19 +50.7

E3(GPa) 276.0 2.78 4.19 -

G12(GPa) 5.74 1.03 1.55 -

G23(GPa) 70 1.03 1.55 -

ν12 0.7 0.35 0.35 -

ν23 0.28 0.35 0.35 -

−6 α1 = α2(10 /°C) 5.6 44.0 44.0 -

−6 α3(10 /°C) -0.4 44.0 44.0 -

D(µm2/s) - 6.0 3.578 -40.4

−4 −3 αR0 - 9.34×10 1.22×10 +30.7

β - 0.122 0.030 -75.3

φox - 0.639 0.712 +11.4

εs - -1.0% -0.7% -30.0

5.2 Simulation Results

Three dimensional contour plots for oxidation state variable φ, radial residual strain

εr, radial residual stress σr and residual hoop stress σθ for the baseline resin case are shown in Fig. 5.3, Fig. 5.4, Fig. 5.5 and Fig. 5.6 respectively. It is clear from the radial stress plot that the highest radial stress σr occurs near the free-edge at the ﬁber-matrix interface and is close to the ultimate tensile strength of BMI, which is around 70 MPa; indicating

49 Y

Z X

Figure 5.3: 3-D contour showing oxidation state variable φ (neat BMI case) that fiber-matrix debonding is imminent. Fig. 5.7 presents the oxidation state variable at the fiber-matrix interface plotted as a function of the normalized z-axis at the fiber-matrix interface for both baseline resin and nano-clay compounded case. It is evident from this

figure that over the same time period, the oxidative degradation for C20A modified BMI is significantly lower, compared with baseline BMI. Oxidative degradation is lower for the

C20A modiﬁed resin because oxygen molecules must travel through a more tortuous path due to the presence of nano-clay platelets and therefore less oxygen diﬀuses into the nano-clay

1−φ = 1 modiﬁed resin.It should be noted that 1−φox indicates a fully oxidized zone.

Fig. 5.8 and Fig. 5.9 present the normalized radial stress (σr) and hoop stress (σθ ) at the ﬁber-matrix interface (refer to Fig. 5.1 (b) for description of polar coordinates) plotted as a function of the normalized z-axis at the ﬁber-matrix interface, respectively. As can be

50 Y

Z X

Figure 5.4: 3-D contour showing radial strain εr (neat BMI case)

Z X

Figure 5.5: 3-D contour showing radial stress σr (neat BMI case)

51 Y

Z X

σθ

Figure 5.6: 3-D contour showing hoop stress σθ (neat BMI case) observed from Fig. 5.8 and Fig. 5.9 that these stresses reach their respective peak values close to the exposed fiber ends at the free edge, thereby making the free edge a more likely site for debond initiation. The FE simulations indicate that high radial stress (σr) that is close the strength of BMI is likely to cause fiber-matrix debonding, and this is corroborated by experimental data [Tandon et al., 2006]. Similarly, in the matrix part of unit cell really high hoop stresses are observed and such high hoop stresses can result in matrix cracking, again as experimentally observed by [Tandon et al., 2006]. As can be observed from Fig. 5.8 and Fig. 5.9 significant reductions in interfacial radial and hoop stresses (25%) is achieved due to nano-clay compounding of BMI. Therefore, it is evident that even a small amount (3 wt%) of nano-clay compounded with BMI significantly enhances the load carrying capacity and service life of a carbon/BMI composite.

52 Oxidation state (0: Un−oxidized 1: Fully Oxidized) 1 BR: 1200s NC: 1200s 0.8 BR: 600s NC: 600s

0.6 ox φ φ − 1 − 1 0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 z/z 0

Figure 5.7: FEA prediction of oxidation state variable φ at the interface along z direction along ﬁber-matrix interface at diﬀerent aging times

Radial stress σ r 80 BR: 1200s NC: 1200s 60 BR: 600s NC: 600s 40

20 (MPa) r σ 0

−20

−40 0 0.2 0.4 0.6 0.8 1 z/z 0

Figure 5.8: FEA prediction of σr at the interface along z direction along ﬁber-matrix interface at diﬀerent aging times

53 σ Hoop stress θ 60 BR: 1200s NC: 1200s 40 BR: 600s NC: 600s

20 (MPa) θ

σ 0

−20

−40 0 0.2 0.4 0.6 0.8 1 z/z 0

Figure 5.9: FEA prediction of σθ at the interface along z direction along ﬁber-matrix interface at diﬀerent aging times

54 CHAPTER 6

MULTI-SCALE VISCOELASTIC COHESIVE LAYER MODEL FOR PREDICTING

DELAMINATION IN HTPMC

This chapter focuses on the multi-scale viscoelastic cohesive layer damage model to predict the life of HTPMC under environmental as well as mechanical loads. First the details of the cohesive layer model including the damage evolution law proposed by Allen [Allen and

Searcy, 2001] are given. Then, a calibration of this model is performed by comparing the

FEA simulation results with the experimental data obtained from three point bending test and also from DCB experiments. A good agreement between experimental and FEA results is observed for both pristine and aged case. The cohesive law parameters used in this simulation are presented for aged as well as unaged IM-7/BMI.

6.1 Viscoelastic Cohesive Layer Model

For the present micromechanics model, a RVE has been idealized as shown in Fig.

6.1(a). The ﬁbrils (seen in Fig. 6.1(b)) are represented by right circular cylinders which exhibit the same linear viscoelastic properties as that of the surrounding bulk material.

Furthermore, the ﬁbrils are assumed only to carry spatially homogeneous uniaxial loads.

These ﬁbrils in the material ligament should exhibit some material orthotropy since the drawing of polymeric material from the bulk media often results in polymer chain alignment.

Unfortunately, the length scale of the ﬁbril makes it extremely diﬃcult to determine these constitutive properties experimentally. Thus, it is necessary to make isotropic assumption

55 (a) (b) Crack Tip Fibrils Material Ligament

Cohesive Zone Tip

Y Area where fibrils are no longer intact X Z

Figure 6.1: Opening crack containing cohesive ligament (b) Reduction of RVE to cohesive zone by area averaging ﬁbril tractions [Allen and Searcy, 2001] about the constitutive behavior of these ﬁbrils. Based on the work done by Allen et al.

[Allen and Searcy, 2001], the area averaged ﬁbril tractions Ti across the cross-sectional area of the cohesive layer RVE, shown in Fig. 6.1(a), may be expressed by,

1 Z fibril Ti = ti dA i = 1,2,3 (6.1) A A where, A is the cross-sectional area of the RVE.

Assuming uniform traction within each ﬁbril within the RVE, Eqn. 6.1 can be dis- cretized as, N Ak(t) fibril Ti = ∑ ti i = 1,2,3 (6.2) k=1 A

th where, Ak represents the cross-sectional area of the k fibril within the RVE, and N is the number of fibrils in the RVE. From continuum mechanics, components of surface traction vector at the mid-plane of each fibril can be related to the components of the stress tensor

56 by, fibril fibril σi jn j = ti (6.3) and σ¯i jn j = Ti

where, the over bar represents area averaged quantity (average stress in RVE) and n j are the rectangular Cartesian components of the direction cosines at the ﬁbril mid-plane.

Substituting Eqn. 6.3 in Eqn. 6.2, we obtain homogenized form of RVE stresses,

N Ak(t) fibril σ¯i jn j = ∑ σi j n j (6.4) k=1 A

or, ( N ) Ak(t) fibril σ¯i j − ∑ σi j n j = 0 (6.5) k=1 A

Since Eqn. 6.5 must hold for any arbitrary orientation of the ﬁbril mid-plane, there-

fore, N Ak(t) fibril σ¯i j = ∑ σi j (6.6) k=1 A

Deﬁning a continuum internal damage state variable α, representing the time varying

area fraction of the growing voids with respect to the cross-sectional area of the RVE,

N A − ∑ Ak(t) k=1 α(t) = (6.7) A

Eqn. 6.6 reduces to,

fibril σ¯i j = (1 − α(t))σi j (6.8)

As described in Roy and Reddy [Roy and Reddy, 1988], the multi-axial viscoelastic

57 stress-strain law for a polymer ﬁbril may be expressed in matrix notation as,

fibril {σ(t)} = [M(t)]({ε(t)} − {H(t)}) (6.9)

where, [M(t)] is a 6 × 6 matrix of time-dependent viscoelastic stiﬀness coeﬃcients, {ε(t)} is the vector containing the components of mechanical strains at time t, and {H(t)} contains

the hereditary (load history-dependent) strain components. Details regarding the derivation

of the [M] matrix and {H} vector from the viscoelastic convolution integral can be found

in [Roy and Reddy, 1988]. Combining Eqs. 6.8 and 6.9 gives the constitutive relationship

between rate-dependent area averaged viscoelastic stresses and strains within a cohesive RVE

ligament at an interlaminar interface, including evolving damage, strain rate, and moisture

and temperature eﬀects through time-temperature-moisture superposition principle,

{σ¯ (t)} = (1 − α(t))[M(t)]({ε(t)} − {H(t)}) (6.10)

6.2 Damage Evolution Law

In order to complete the model description it is necessary to construct an evolution law

for the continuum damage state variable. Consider the above case where we can approximate

the polymer ﬁbrils to be right cylinders with circular cross section. Due to Poisson’s eﬀect,

as these ﬁbrils elongate along the length direction they contract in the radial direction,

if the radius of a particular ﬁbril falls below a critical value that indicates the onset of

instability leading to failure. This critical value of radius should be considered as a material

property to deﬁne the cohesive damage law. Since experiments needed to quantify the critical

58 value of damage initiation radius of a ﬁbril are diﬃcult to perform, a phenomenological

damage evolution law is chosen. Since, the change of ﬁbril diameter as a function of time

is proportional to the applied principal-stretch along the polymer ﬁbril, a phenomenological

power-law based damage evolution law is adapted [Upadhyaya et al., 2011], given by,

  ¯ m ˙ dα α0λ λ ≥ 0 and α < 1 = (6.11) dt  0 λ˙ < 0 or α = 1

¯ where λ, is a principal-stretch measure within the RVE [Upadhyaya et al., 2011], and α0 and m are material constants that are assumed to be dependent on environmental conditions but independent of the applied strain rate. The value of the principal stretch measure at time t is given by Eqn. 6.12,

¯ λ(t) = λI(t) − λcr, λI(t) ≥ λcr (6.12)

where λI is the current value of the principal stretch and λcr is the critical value of principal stretch at damage initiation.

From numerical computation standpoint, within the cohesive layer RVE, damage initiation (i.e., initiation of voids/polymer ﬁbrils) is assumed to occur if the applied principal- stretch along the ﬁbril exceeds the critical value of the principal-stretch, that is λI(t) ≥ λcr.

The micromechanics based viscoelastic traction-separation behavior helps in overcoming the numerical instabilities encountered during simulations modeling delamination growth. An incremental form of a traction-displacement law was integrated numerically and implemented

59 in the ﬁnite element program to predict delamination growth in unidirectional composites.

Such simulations are prone to convergence issues due to the occurrence of elastic snap-

back instability. The sudden release of strain energy at the onset of failure, which causes

instability, is mitigated due to natural viscoelastic damping. Consequently, the instability

encountered at failure initiation with elastic cohesive elements is circumvented, and as an

added bonus, it enables tracking of the non-linear load versus displacement curve well beyond

peak load as illustrated in the next section.

6.3 Model Calibration

6.3.1 Flexure Experiment

Three-point bending tests were conducted on pristine as well as thermally aged IM-

7/BMI ([08]s) specimens at UT-Dallas. The test set up is shown in Fig. 6.2. Four replicate

specimens of each category (aged and un-aged) were tested using a 5567A series Instron

machine with a 30 kN load cell. The dimensions of these specimens were 50mm × 5.1mm ×

2.2mm.The beam span i.e. the distance between two supporting pins as shown in Fig. 6.2 is

44 mm. Macro level load displacement curves obtained from these ﬂexure experiments were used to illustrate the capabilities of multi axial viscoelastic cohesive layer model in modeling delamination behavior for unidirectional composites.

6.3.2 FEA simulation of ﬂexure experiment

A 2-D plane strain FEA model was used to simulate the unidirectional ﬂexure experiments. A 2-D FEA mesh was generated as shown in Fig. 6.3. The mesh consists of a total of

2325 eight-noded quadratic elements out of which 525 elements are viscoelastic cohesive layer elements. Seven layers of viscoelastic cohesive elements, along the length of the specimen, were used in the mesh to simulate interlaminar regions in both un-aged and aged IM-7/BMI

60 Figure 6.2: Three-point bending ﬁxture

laminate. Elastic properties and damage evolution parameters used in FEA modeling for

unoxidized and oxidized unidirectional IM-7/BMI are given in Table 6.1. Degradation in

elastic properties of IM-7/BMI for aged material is derived by comparing the initial slope

of the load displacement curve for aged and un-aged ﬂexure test results. After 3000 hours

of isothermal aging at 250 °C, a signiﬁcant (∼80 %) degradation in elastic properties is observed.

The viscoelastic properties for pristine and aged BMI, presented in Table 6.2, were acquired from work done by Luo et al. [Luo et al., 2012]. Using symmetry, only half of the

FEA mesh is shown in Fig. 6.3. The mesh reﬁnement close to the load and support pin is clearly seen in Fig. 6.3. Within the cohesive element, damage initiation is assumed to occur when the critical value of the von-Mises equivalent stress σvm,cr is exceeded (at each

element Gauss point). Following the prior work done by [Upadhyaya et al., 2011], the critical

61 Cohesive Layers Symmetry Line shown in red

Mesh refinement Mesh refinement at the load pin at the support pin

Figure 6.3: FEA mesh for ﬂexure test specimen of IM-7/BMI unidirectional laminate

Table 6.1: Elastic properties used in FEA modeling for un-aged and aged (for 3000 hours at 250 °C) IM-7/BMI unidirectional laminate

Un-aged IM-7/BMI Aged IM-7/BMI

E1(GPa) 162.0 45.0

E2(GPa) 9.7 1.1

G12(GPa) 5.9 0.22

G23(GPa) 3.5 1.45

ν12 0.32 0.32

ν23 0.40 0.4

62 von-Mises stress in the resin dominated interlaminar region is reduced with thermo-oxidative

degradation of the matrix. The critical von-Mises stress values for unaged and aged BMI

were found to be 40.2 MPa and 4.3 MPa respectively. After thermal oxidation the critical

von-Mises stress for BMI degrades by (∼90%). The micromechanical damage parameters

α0 and m were estimated by comparing the experimental and numerical load displacement

data. The micromechanical damage parameters used in this simulation are summarized in

Table 6.3 for pristine and aged IM-7/BMI.

The FEA model was able to accurately simulate the actual ﬂexure test results for

the un-aged and isothermally aged laminates. Both in the unaged as well as aged laminate,

upon application of bending load the matrix in the interlaminar region reaches the critical

von-Mises stress before ﬁber failure occurs, thereby causing the laminate to fail due to

interlaminar delamination. A contour plot for damage parameter α(t) for un-aged laminate is presented in Fig. 6.4 clearly showing the failed elements in the FEA mesh (shown deleted).

Fig. 6.5 and Fig. 6.6 show a comparison of the experimental load versus displacement curve versus FEA simulation for the three-point bend test for un-aged and aged specimen, respectively. Good agreement is observed for both aged and un-aged category, even beyond peak load. It should be noted that the viscoelastic cohesive layer used in the model assisted in attaining numerical stability during delamination failure, and the analysis could continue beyond the point of failure due to the mitigating eﬀect of viscous regularization. Note that failure load for aged case is only 40% of un-aged case.

It is very clear from these simulation results that the phenomenological damage evolution law given by Allen and Searcy [Allen and Searcy, 2001] enables us to model the experimental behavior of these test specimens computationally. However, the estimation

63 Table 6.2: Creep Compliance of BMI resin (Reference Temperature = 204 °C) Pristine BMI (σ = 24 MPa) Aged BMI at 195°C for 1500 hours (σ = 2.5 MPa) −5 −5 i τi(s) Ji(10 /MPa) τi(s) Ji(10 /MPa) 1 1×10−9 1.34 1×10−17 1.07 2 1×10−8 1.01 1×10−16 1.35 3 1×10−7 1.22 1×10−15 1.87 4 1×10−6 1.14 1×10−14 1.42 5 1×10−5 2.97 1×10−13 1.42 6 1×10−4 3.19 1×10−12 2.47 7 1×10−3 2.69 1×10−11 6.18 8 1×10−2 1.46 1×10−10 1.64 9 1×10−1 5.80 1×10−9 4.03 10 1×100 2.01 1×10−8 2.48 11 1×101 6.18 1×10−7 1.87 12 1×102 3.48 1×10−6 2.87 13 1×103 5.40 1×10−5 1.14 14 1×104 3.61 1×10−4 2.59 15 1×105 11.8 1×10−3 1.09 16 1×106 12.5 1×10−2 9.87 17 1×107 39.2 1×10−1 3.63 18 1×108 62.9 1×100 9.59 19 1×109 1.28 1×101 1.01 20 1×1010 227 1×102 1.76 21 1×1011 269 1×103 1.74 22 1×1012 273 1×104 1.06 23 1×1013 273 1×105 1.45 −5 −5 J0(10 /MPa) =24.2 J0(10 /MPa) =20.4

64 Table 6.3: Micromechanical damage parameters used in FEA modeling for un-aged and aged (for 3000 hours at 250 °C) IM-7/BMI unidirectional laminate

Un-aged IM-7/BMI Aged IM-7/BMI

α0 0.0033 0.09

m 0.8 0.8

Interlaminar delamination see deleted elements

Figure 6.4: FEA simulation showing inter-laminar delamination for ﬂexure test of un-aged IM-7/BMI unidirectional laminate

65 Flexure Test: Unaged IM−7/BMI 1.2 FEA Analysis 1 Experimental

0.8

0.6 Load (N)

0.4

0.2

0 0 0.5 1 1.5 2 2.5 3 Displacement (mm)

Figure 6.5: Normalized load displacement curve for un-aged IM-7/BMI unidirectional laminate

Flexure Test: Aged (3000 hrs at 250°C) IM−7/BMI 0.05

0.045

0.04

0.035

0.03

0.025

Load (N) 0.02

0.015 FEA Analysis 0.01 Experimental 0.005

0 0 0.2 0.4 0.6 0.8 1 1.2 Displacement (mm)

Figure 6.6: Normalized load displacement curve for aged IM-7/BMI unidirectional laminate

66 Figure 6.7: An image of the test conﬁguration for delamination type DCB specimen having pre-crack under initial loading of damage parameters α0 and m was done through curve ﬁtting of load deformation data.

This can be time consuming and the uniqueness of these parameters cannot be guaranteed.

Therefore, a more elegant methodology is presented here to calculate these parameters for

Mode-I type failure, entirely from the DCB test data. The next section presents the details of methodology adopted to extract cohesive law parameters from experimental data. First, the material and specimen geometry with DCB experiment details are presented. This is followed by the mathematical formulation to extract cohesive stresses and damage evolution parameters. Finally, numerical simulation results demonstrating the accuracy of micromechanics based cohesive layer model are presented in detail.

6.4 Double Cantilever Beam Experiment

6.4.1 Specimen preparation and DCB specimen geometry

A total of 16 plies of IM-7/BMI ([08]s) prepreg sheets were stacked to form a unidirectional composite laminate. The ﬁber volume fraction was 0.6. The specimens were cut from the composite panel using a diamond saw blade. The test conﬁguration is shown in

67 Fig. 6.7. The specimens were 140mm×14mm×2.34mm and have a 70 mm long pre-crack. A pre-crack in the ﬁber direction was prepared in the specimen for interlaminar delamination growth. To verify that no intralaminar damage was induced, scanning electron microscopy

(SEM) image was acquired on the open crack surface

A pair of piano hinges was glued to the specimen for gripping the DCB specimen during testing. The side edge of the DCB specimen was polished and sprayed with black paint as dots on white background that produced a speckle of good contrast for determining the crack position. To study the eﬀect of thermal oxidation on delamination toughness, isothermal aging of IM-7/ BMI composite panels was conducted inside a convection oven for approximately 1000 hours at 260 °C in air environment. After 1000 hours of aging at 260

°C, the coupons were removed from the oven and cooled down slowly to room temperature to conduct DCB experiments under ambient condition at a temperature of 23 ± 1 °C and a relative humidity of 35 ± 3%.

6.4.2 Experimental method

Mode-I fracture toughness experiments (ASTM D5528-01) were conducted on pristine (un-aged) unidirectional IM-7/BMI specimens at UT-Dallas, and also on IM-7/BMI specimens after thermo oxidative aging for 1000 hours at 260 °C. For each case at least four specimens were tested to verify the repeatability of test data. These tests were performed using an Instron 5969 dual column tabletop universal testing machine with a 50 kN load cell.

These tests were conducted in displacement control and the upper cross-head movement rate was 1 mm/min. A crack was allowed to grow until 25 mm of cross-head displacement was reached. The load-displacement data was recorded and digital images were taken using a

Nikon D7000 DSLR camera with 200 mm macro lens. The DIC code developed by Lu et

68 Figure 6.8: Schematic diagram of DCB specimen al. [Lu and Cary, 2000] was used to analyze the images taken during the experiment. To accurately determine the crack length, the digital image was converted to a grayscale and a negative was created to highlight the contrast due to the crack within the accuracy of the single pixel (representing about 25 µm) using an image processing software. The next section discusses the mathematical formulation to derive near-tip cohesive stresses from J-integral data.

6.5 Determination of Cohesive Stress using J-integral

The J-integral represents the strain energy release rate per unit fracture surface area in a material. The standard integral form deﬁning J is

Z dui J = Wdy − Ti dS (6.13) Γ dx

where Wis the strain energy density, Ti is the surface traction vector along the contour, u is the displacement ﬁeld and dS is an element along the integration path Γ .

To extract cohesive law through J-Integral, we need to choose two diﬀerent paths for J-integral calculation. First closed integration path should be an exterior one passing

69 through the points where external loads are applied during the experiment. The second

closed interior path should be chosen very close to the crack tip where cohesive stresses are

present. Later, independence of J-integral can be employed to equate the two J-integrals

and obtain the cohesive stress near the crack tip.

For a DCB specimen, employing Eqn. 6.13 to evaluate the J-integral along the exterior

closed path shown in Fig. 6.8 (shown red), we get [Anthony and Paris, 1988]

2Pθ J = (6.14) b

where, P is the reaction force at the loading pin location measured during the DCB experi-

ment, θ is rotation at the loading pin acquired through DIC calculation, and b is the width

of the specimen. Reaction force P and y displacement (v) at the load application point are

recorded during the DCB experiment and DIC technique is used to extract the rotation θ for the full time history of DCB test. Knowing P and θ we can evaluate the time history of

J-integral using Eqn. 6.14 and this experimental J-integral is subsequently used in extraction of cohesive law. To derive the cohesive stresses from J-integral, we choose the second closed interior path Γint very close to the debond tip as marked in green color in Fig. 6.8. The calculation of J-integral along the closed path Γint gives

Z Z Z δ ∂v ˙ cohesive dδ ˙ cohesive Jint = − Ty dS = − σ(δ,δ) dx = σ(δ,δ) dδ (6.15) Γint ∂x Γint dx 0

where Ty is surface traction in y direction, v is y direction displacement, δ is crack opening displacement (COD) and σ(δ,δ˙)cohesive is cohesive stress normal to the crack plane. Standard

70 relation between traction and stress ﬁeld is used in simplifying the above equation. It should be noted that cohesive stress not only depends on the separation but may also depends on its rate for viscoelastic materials. Further, small displacements and rotations are assumed.

As described by Fuchs and Major [Fuchs and Major, 2011], from Eqn. 6.15, for Mode-

I type failure, cohesive stresses can be evaluated by taking the ﬁrst derivative of J-integral with respect to COD (δ)

∂Jint σ(δ,δ˙)cohesive = (6.16) ∂δ

Under the assumption that the process zone is small and that the dissipation is small compared with total stored energy, by invoking path independence of J-integral, we can rewrite Eqn. 6.16 as

∂Jext σ(δ,δ˙)cohesive = (6.17) ∂δ

Therefore, if the variation of macro level J-integral Jext with micro level COD is known, cohesive law can easily be determined through Eqn. 6.17. From the experimental data, J-integral (J) for a DCB specimen was calculated by using Eqn. 6.14. The COD can be determined precisely from DIC calculations by measuring the diﬀerence of y displacements

(v) between two points (one above and one below) at the precise location of the crack tip in the unloaded state. To be consistent with the mathematical framework of the J-integral derivation, it is imperative to select these two points to be exactly at the initial crack tip, even though it may be diﬃcult to determine the exact location of the pre-crack. After calculating the J-integral from the DCB experimental data as a function of COD, a “smoothing spline

ﬁt” is obtained for J-integral versus COD data. The spline is then used to take the ﬁrst

71 derivative of J-integral with respect to COD (δ) at each data point i.e.

∂J σ cohesive = (6.18) ∂δ

Pristine: J−Integral vs. COD (δ) 600

500

400

300

J−Integral (N/m) 200 Experimental Data 100 Smoothing Spline Fit

0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 δ (mm)

Figure 6.9: J-Integral versus COD (δ) for pristine IM-7/BMI

In Fig. 6.9, J-integral versus COD (δ) experimental data as well as the smoothing spline ﬁt is plotted for pristine specimen. Similar plot for aged IM-7/BMI specimen is given in Fig. 6.11. J-integral curves typically have a sigmoidal shape as a function of COD, and reach a steady-state plateau that indicates that the cohesive zone is fully developed.

Data smoothing was performed to avoid the inﬂuence of measurement noise on cohesive stress estimation. For this purpose, “smoothing spline ﬁts” were applied on the J-Integral-

72 Pristine: Cohesive stress vs COD (δ) 30

(MPa) 15 cohesive σ 10

0 0 0.01 0.02 0.03 0.04 δ (mm)

Figure 6.10: Cohesive stress versus COD (δ) for pristine IM-7/BMI

COD data, using the spap2 function available in MATLAB (Matlab R2009b, TheMathWorks

Inc., Natik, US). It is important to note here that the crack starts to propagate when the cohesive stress reaches its peak value (see Fig. 6.10 and Fig. 6.12). From the DIC analysis of recorded images, the time when the crack starts to propagate for each specimen was determined carefully. The cohesive law for pristine specimen determined using Eqn. 6.18 is shown in Fig. 6.10. Similar cohesive law was determined for the aged specimen as shown in

Fig. 6.12.

The inter-ply region in a unidirectional laminate is polymer matrix dominated and therefore the strength of polymer corresponds to the maximum value of the cohesive stress, i.e. 29.2 MPa for pristine and 5.6 MPa for specimens aged for 1000 hours at 260 °C. The cohesive stress attains its peak value at 303 sec for the pristine case, and at 215 sec for specimens

73 Isothermally aged: J−Integral vs. COD(δ) 250

200

150

100 J−Integral (N/m)

Experimental Data 50 Smoothing Spline Fit

0 0 0.01 0.02 0.03 0.04 0.05 δ(mm)

Figure 6.11: J-Integral versus COD (δ) for aged (1000 hours at 260 °C in air) IM-7/BMI

aged for 1000 hours, respectively. Therefore, a signiﬁcant degradation of 80% in the peak

cohesive stress was observed after 1000 hours of thermo oxidative aging, and is corroborated

by the earlier crack initiation in the aged specimen. Next, the scalar damage parameter

α and damage evolution parameters were estimated by comparing the damaged (cohesive)

stresses to the undamaged (pristine) stresses. The methodology adapted to estimate the

viscoelastic damage evolution parameters is summarized in next section.

6.6 Estimation of Damage Evolution Law

As described in section 6.2, a principal-stretch based failure criterion is used for dam-

age initiation in the cohesive layer. The damage initiates as the local principal-stretch value

reaches the critical value of principal stretch λcr, which corresponds to the peak stress in the cohesive traction-separation law. The complete deﬁnition of this model requires three scalar-

74 Isothermally aged:Cohesive Stress vs. COD(δ) 6

(MPa) 3 cohesive σ 2

0 0 0.01 0.02 0.03 0.04 0.05 δ(mm)

Figure 6.12: Cohesive stress versus COD (δ) for aged (1000 hours at 260 °C in air) IM-7/BMI valued material parameters; critical principal-stretch value λcr, α0 and m. The principal-

stretch values were calculated from the displacement gradients recorded in the damage zone

from DIC image analysis. The right Cauchy-Green strain tensor (C) was calculated from

the strain gradients and the square root of eigenvalues for right Cauchy- Green strain tensor

gives the principal-stretches, as shown below.

6.6.1 Determination of critical principal stretch λcr

At any point within the RVE with displacement u, the displacement gradient u ⊗ ∇

∂ui = +u⊗∇ is a tensor with components ∂x j . The deformation gradient F is deﬁned as, F I , or

in Cartesian components

∂ui Fi j(t) = δi j + (6.19) ∂x j

75 where I is identity tensor and δi j is the Kronecker delta function. the right Cauchy-Green tensor C is given by [Malvern, 1969],

> C = F · F (6.20)

In Cartesian components, Eqn. 6.20 can be rewritten as

Ci j = FkiFk j (6.21)

Assuming a multiplicative decomposition of a constant deformation gradient tensor F in a homogeneous stretch U and a rotation R, F can be decomposed as,

F = R · U (6.22)

Substituting F in Eqn. 6.20 we get,

> > C = U · R · R · U (6.23)

> > Rotation R is proper orthogonal and it satisﬁes R · R = I, and U is symmetric (U = U), therefore Eqn. 6.23 simpliﬁes to

2 C = U (6.24)

The characteristic equation for the Eigenvalue problem in this case is,

c Ci j − λ δi j = 0, i, j = 1,2,3 (6.25)

76 c c c Solving Eqn. 6.25 yields eigenvalues λI , λII and λIII. Therefore, using Eqn. 6.24, the principal Eigenvalue of the stretch tensor U (principal stretch λI) can be obtained by taking positive square-root of the principal Eigenvalue of C,

q c λI = λI (6.26)

The principal-stretch value that corresponded to the crack initiation time and maxi-

mum cohesive stress for each case (pristine and isothermally aged) was taken as the critical

principal-stretch value (λcr) for each case.

6.6.2 Determination of damage parameters α0 and m

By deﬁnition, the cohesive stress history (see Fig. 6.12) obtained through J-integral

includes damage state after peak stress has been reached. To calculate the damage parameter

α(t), a comparison between the cohesive stresses for undamaged and damaged (cohesive)

material was performed. Strain data acquired from DIC calculations near the crack tip was

used to obtain the undamaged material stresses. For undamaged material, volume averaged

cohesive stresses are obtain through the following viscoelastic stress-strain relationship,

ud {σ¯ (t)} = [M(t)]({ε(t)} − {H(t)}) (6.27)

where {σ¯ (t)}ud, is the volume averaged stress in a viscoelastic cohesive layer without any

damage. Additional details of this viscoelasticity model can be found in Roy and Reddy

[Roy and Reddy, 1988]. The delamination zone is assumed to be resin rich and therefore

material properties of BMI are used for the calculation of undamaged stresses within the

77 cohesive layer. To calculate the viscoelastic stiﬀness coeﬃcients, viscoelastic properties of aged as well as pristine BMI were used from the creep experiments performed by Luo et al.

[Luo et al., 2012] (see Table 6.2), albeit aged at slightly diﬀerent conditions. Although the material under consideration is viscoelastic, stress strain behavior is fairly linear because the time duration for the DCB experiment is very small and material did not exhibit signiﬁcant stress relaxation within the time scale of the experiment.

Incorporating the induced damage through the internal damage parameter α(t), the governing equation for viscoelastic cohesive layer is given as [Upadhyaya et al., 2011],

cohesive {σ(t)} = (1 − α(t))[M(t)]({ε(t)} − {H(t)}) (6.28)

Therefore, combining Eqs. (6.27) and (6.28) in the cohesive layer, we can rewrite the governing viscoelastic damage law equation in terms of undamaged and damaged material stresses and solve for α(t) to obtain,

σ cohesive α(t) = 1 − (6.29) σ¯ ud

It is important to note that the damage parameter is meaningful only after damage initiation has taken place. In the current DCB experiments, maximum cohesive stress (29.2

MPa) is reached at 303 sec from the start of loading for the pristine specimen. Similarly, for aged case, maximum cohesive stress (5.6 MPa) is reached at 215 sec. Therefore, damage parameter calculation is performed only after the damage initiation time for each case.

In Figs. 6.13 and 6.15, complete data set for scalar damage parameter is plotted

78 Pristine: Damage Parameter α(t) 1.05

(t) 0.95 α

0.9

0.85

Damage Parameter 0.8 Experimental Data 0.75 Smoothing Spline Fit

0.7 300 325 350 375 400 425 450 Time (sec)

Figure 6.13: Scalar damage parameter for pristine IM-7/BMI

for pristine and aged cases, respectively. As can be seen from these ﬁgures, the damage

parameter increases with time and then plateaus at a value close to 1(complete separation),

which follows the basic deﬁnition of damage parameter α.

Since α follows a power-law behavior, damage law constants α0 and m can be deter-

mined by taking logarithm on both sides of the Eq. (6.11). Taking logarithm on both sides

of Eq. (6.11) we get

¯ logα˙ = logα0 + mlogλ (6.30)

After determining principal-stretch measure λ¯ , a smoothing spline ﬁt was used to

diﬀerentiate α with respect to time to obtain α˙ . A line was ﬁtted to logα˙ versus logλ¯ data

(after critical stress has been reached) and α0 and m were determined from the intercept

79 Pristine: log(α ˙ ) vs. log(λ¯) −2.1

−2.15 Experimental Data −2.2 Linear Fit −2.25

−2.3 ) α −2.35 log( ˙ −2.4

−2.45

−2.5

−2.55

−2.6 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2 log(λ¯)

Figure 6.14: logα˙ versus logλ¯ for pristine IM-7/BMI

Table 6.4: Damage evolution law parameters used in FEA model for IM-7/BMI

λcr α0 m

Pristine 1.0264 0.0507 0.45

Aged 1.0148 0.0415 0.83 and the slope of this linear fit. It is important to underline here that the intercept could not be found without extrapolating the linear fit beyond the limits of experimentally acquired data points. In Fig. 6.14 and Fig. 6.16, experimental value as well as the linear fit is plotted for pristine and isothermally aged specimens, respectively. The calculated damage evolution parameters are tabulated in Table 6.4. These material parameters were subsequently used in our FEA model to simulate the delamination growth for pristine as well as aged specimens.

Details of these numerical simulations are presented in next section.

80 Isothermally aged: Damage Patameter α (t) 1.02

(t) 0.98 α

0.96

0.94

Damage Parameter 0.92 Experimental Data 0.9 Smoothing Spline Fit

0.88 200 225 250 275 300 325 350 Time (sec)

Figure 6.15: Scalar damage parameter for isothermally aged (1000 hours at 260 °C in air) IM-7/BMI

6.7 Numerical Simulation of DCB Experiments

Based on specimen geometry shown in Fig. 6.7, a two dimensional (2-D) plane strain

FEA model using an in-house test-bed code (NOVA-3D) was used to simulate the DCB experiment, for both pristine and isothermally aged specimens. A 2-D FEA mesh was generated as shown in Fig. 6.17. The mesh consists of a total of 2694 eight-noded quadratic

Table 6.5: Elastic properties of transversely isotropic IM-7/BMI lamina[Andrews and Gar- nich, 2008]

E1(GPa) E2 = E3(GPa) G12(GPa) ν12 ν23

Pristine 174.0 12.1 9.1 0.36 0.45

Aged 104.0 7.3 5.5 0.36 0.45

81 Isothermally aged: log(α ˙ ) vs. log(λ¯) −2.8

−2.85

−2.9

−2.95 ) α −3 log( ˙ −3.05 Experimental Data −3.1 Linear Fit −3.15

−3.2 −2.1 −2 −1.9 −1.8 −1.7 log(λ¯)

Figure 6.16: logα˙ versus logλ¯ for isothermally aged (1000 hours at 260 °C in air) IM-7/BMI elements out of which 300 elements are viscoelastic cohesive layer elements. A layer of viscoelastic cohesive elements ahead of crack tip, along the length of the specimen were used in the mesh to simulate delamination in IM-7/BMI laminate. The thickness of cohesive layer elements was carefully chosen to match the thickness of the actual strain localization zone in the DCB specimens by observing at the high resolution images taken during the experiments. Cohesive layer thickness used in these simulations was 0.06 mm, which is very small compared to the overall specimen thickness (2.34 mm).

Micromechanical damage evolution parameters used in FEA modeling for IM-7/BMI are given in Table 6.4. Elastic properties of IM-7/BMI unidirectional lamina (ﬁber volume fraction Vf = 0.6) were taken from work published by Andrews et al. [Andrews and Garnich,

2008]. Signiﬁcant (∼40%) degradation was observed in the elastic material properties of IM-

82 Aluminum piano hinge

Figure 6.17: Meshed DCB specimen of IM-7/BMI with boundary conditions

7/BMI after 1000 hours of thermo-oxidative aging at 260 °C in air. This degradation was estimated by comparing the initial slope of load displacement curves for pristine and aged case. Elastic properties for pristine and aged unidirectional IM-7/BMI composite are given in Table 6.5.

As shown in Fig. 6.17, the piano hinge attached to the specimen is made of aluminum, and was modeled as such in the FEA simulation. The elastic modulus for aluminum is 70

GPa and Poisson’s ratio is 0.33. The cohesive layer is modeled as a viscoelastic matrix material with evolving damage. The viscoelastic properties for pristine and aged BMI were taken from work done by Luo et al.[Luo et al., 2012] and tabulated in Table 6.2. Fig. 6.17 shows the FEA mesh and boundary conditions used in numerical simulation of the DCB specimen, with a zoomed-in view of the mesh near the crack tip with viscoelastic cohesive

83 Pristine: Load vs. Displacement 50

Load (N) 20

Experimental Data 10 FEA Analysis

0 0 5 10 15 20 25 Displacement (mm)

Figure 6.18: Load versus displacement for pristine IM-7/BMI DCB specimen

elements. From the damage propagation standpoint, the scalar damage parameter α at each

gaussian integration point in each cohesive element was checked at every time step. An

element was deleted if the scalar damage parameter reached 1 at any gauss point in that

element. Ideally, such deleted element should not have any stiﬀness at all, but instead of

assigning these elements zero stiﬀness a very small stiﬀness was prescribed to avoid numerical

instabilities. The FEA model was able to accurately simulate the actual DCB experiment

as discussed in the next paragraph.

To compare the macro level load-displacement results for the DCB, Figs. 6.18 and

6.20 show a comparison of the experimental and simulated load versus displacement curves

for pristine specimens and specimen aged for 1000 hours at 260 °C, respectively. The FEA simulation results show very good agreement even beyond peak load. The saw-tooth pattern

84 Pristine: Crack Length vs. Time 45

15 Crack Length (mm)

10 Experimental Data 5 FEA Analysis 0 0 400 800 1200 1600 Time (sec)

Figure 6.19: Crack-length versus displacement for pristine IM-7/BMI DCB specimen

Isothermally aged: Load vs. Displacement 25

Load (N) 10

Experimental Data 5 FEA Analysis

0 0 5 10 15 20 25 Displacement (mm)

Figure 6.20: Load versus displacement for isothermally aged (1000 hours at 260 °C) IM- 7/BMI DCB specimen

85 Figure 6.21: Deformed and un-deformed contour plots showing εy for isothermally aged specimen at 260 °C clearly showing crack propagation from 70 mm to 74 mm at 450 sec

Isothermally aged: Crack Length vs. Time 50

Crack Length (mm) 15

10 Experimental Data FEA Analysis 5

0 0 400 800 1200 1600 Time (sec)

Figure 6.22: Crack-length versus displacement for isothermally aged (1000 hours at 260 °C) IM-7/BMI DCB specimen

86 seen in the simulated load displacement curve is due to successive failure of cohesive elements.

The applied load drops after a cohesive element fails, and then again increases until the next

element failure occurs.

Fig. 6.21 depicts the εy contour plot for aged case showing strain concentration near the crack tip. A deformed mesh plot depicting crack propagation due to interlaminar delamination at 450 sec is also shown in this ﬁgure (see deleted elements), and is compared with the undeformed plot (below) showing the original crack location. Comparison of experimentally measured crack propagation length versus time and from FEA analysis for the pristine and the aged specimen is plotted in Figs. 6.19 and 6.22, respectively. The viscoelastic cohesive layer based FEA analysis accurately captures the crack propagation history. Therefore, unlike other existing models, the present FEA model is capable of accurately modeling the macro level load-displacement behavior in conjunction with the micro level crack growth history with time, using only three material damage constants, λcr, α0, and m.

6.8 Sensitivity study on model parameters

The objective of this parametric study is to examine the eﬀect of variation in model

parameters λcr, α0 and m on the simulation results. For this purpose, diﬀerent FEA sim-

ulations were conducted, where one model parameter (for example λcr) was varied while

other two parameters (α0 and m) were held constant. These simulations were conducted for

pristine IM-7/BMI DCB specimen. Load displacement history for λcr sensitivity study is

presented in Figure 6.23. It is clear from Figure 6.23 that the peak of the load displacement

curve is very sensitive to the principal critical stretch value λcr. As evident from Figure

6.23, a lower value of λcr allows early damage initiation thereby reducing the load carrying

capacity of DCB specimen.

87 Sensitivity study (λ ): α =0.057, m=0.45 (Pristine) cr 0 60

30 Load (N) Experimental Data 20 λ = 1.0164 (FEA) cr λ = 1.0264 (FEA) 10 cr λ = 1.0364 (FEA) cr 0 0 5 10 15 20 25 Displacement (mm)

Figure 6.23: Load versus displacement for pristine IM-7/BMI DCB specimen for diﬀerent λcr values

Similar FEA simulations were repeated to investigate the α0 and m sensitivity. Sen- sitivity study for parameter α0 is depicted in Figure 6.24. Since rate of scalar damage parameter α(t) is directly proportional to α0 (refer to Eqn. 6.11), the shape of load displacement curve beyond peak load is governed by the value of α0 which is clearly seen in

Figure 6.24. As shown in Figure 6.25, variation in model parameter m does not have any noticeable eﬀect on load displacement curve. In the next chapter, key results of this doctoral work are outlined with a broader future scope.

88 Sensitivity study (α ): λ =1.0264, m=0.45 (Pristine) 0 cr 90 Experimental Data 80 α = 0.01 (FEA) 0 α = 0.03 (FEA) 70 0 α = 0.05 (FEA) 60 0

40 Load (N)

0 0 5 10 15 20 25 Displacement (mm)

Figure 6.24: Load versus displacement for pristine IM-7/BMI DCB specimen for diﬀerent α0 values

Sensitivity study (m): α =0.057, λ =1.0264 (Pristine) 0 cr 50

Load (N) 20

Experimental Data 10 m = 0.15 (FEA) m = 0.45 (FEA) m = 0.75 (FEA) 0 0 5 10 15 20 25 Displacement (mm)

Figure 6.25: Load versus displacement for pristine IM-7/BMI DCB specimen for diﬀerent m values

89 CHAPTER 7

CONCLUSIONS

7.1 Conclusions

Thermo oxidative aging experiments were conducted for BMI in three different oxygen aging environments. To enhance the thermal stability of BMI resin it was compounded with a small (3wt %) amount of surfactant modified C20A nano-clay. Similar thermo- oxidative aging experiments were conducted for nano-clay modified BMI. Weight loss in air environment for C20A modified BMI was significantly less (40%) as compared with the baseline resin. Parameters needed for the oxidation modeling were obtained from the test data for both baseline and C20A modified BMI. These parameters suggest that the C20A modified BMI oxidizes at a significantly slower rate compared with the baseline resin due to the barrier properties of nano-clay resulting in slower synergistic diffusion of oxygen.

Nano-indentation tests were conducted at UT-Dallas to estimate the elastic modulus and hardness of baseline as well as nano-modiﬁed BMI. Signiﬁcant improvements in elastic modulus (51%) and hardness (62%) were achieved by nano-compounding.

Micro-mechanics based coupled diffusion-reaction stress analysis FEA simulations were performed to estimate the residual stresses caused by the thermo-oxidative aging process. These models are fully three-dimensional and include fiber-fiber interaction. Presence of high radial stresses at the interface at the free-edge underscores the potential for fiber- matrix debonding. Elevated hoop stresses within the matrix near the free-edge can similarly

90 lead to matrix cracking. Due to a lack of data regarding interfacial strength of IM-7/BMI unidirectional composite, it was assumed that the interface is matrix dominated and we prescribed the matrix strength properties to the interface region. From our FEA prediction, a significant reduction in residual radial stress (25%) and hoop stress (25%) results from 3wt% nano-clay compounding of BMI. Even though the increase in modulus due to the presence of nano-clay (see Section 4.4) should elevate the residual stress level in the matrix, however the increase in barrier properties of the matrix in conjunction with the synergistic reduction in its shrinkage strain due to the presence of nano-clay platelets results in a significant overall reduction in residual stresses. Therefore, it is concluded that even a small amount (3wt%) of nano-clay compounding could significantly enhance the thermo-oxidative stability and load carrying capacity of a carbon/BMI composite by reducing oxygen diffusion and mitigating residual stresses due to thermo-oxidation.

In the present work, DCB testing conducted at UT-Dallas in conjunction with DIC technique was used to experimentally determine the coupled viscoelasticity-damage cohesive law for IM-7/BMI unidirectional composite. A novel numerical-experimental approach is proposed to determine the micro-scale cohesive layer properties for unidirectional composite from macro scale observations (J-Integral). Assuming the process zone to be small compared with the specimen size, the J-integral is diﬀerentiated with respect to COD to estimate the cohesive stresses in the damage zone ahead of crack tip. A digital camera in combination with DIC algorithm [Lu and Cary, 2000] is used to precisely measure the COD. Damage model used in this work is a principal-stretch based failure model and the critical principal- stretch for pristine BMI is found to be 1.0264 from DIC experiment. The critical stretch value for isothermally aged BMI is 1.0148.

91 The scalar damage parameters involved in the damage evolution law was estimated

by comparing damaged material stresses to the undamaged material stresses. Viscoelastic

cohesive zone model was employed in an in-house test-bed FEA code (NOVA-3D) to numer-

ically simulate the DCB experiment. Cohesive layer thickness was carefully chosen to match

the thickness of the actual strain localization zone in the DCB specimens by observing the

high resolution images taken during the experiments. A sensitivity analysis on mesh sizing

and step size (∆t) was conducted. A suitable combination of step size (∆t) and cohesive

element size in x direction was chosen to simulate the experiments successfully. It should

be noted that the viscoelastic cohesive layer used in the model assisted in attaining numeri-

cal stability during mode-I type (interlaminar delamination) failure, and the analysis could

continue beyond the point of failure due to the mitigating eﬀect of viscous regularization.

At the macro scale, the load displacement curve obtained numerically matches very well

with the experimental data. At the micro-scale, crack propagation length also shows good

agreement with experimentally measured values. Therefore, unlike other existing numer-

ical models, the present numerical-experimental model is capable of accurately capturing

the macro-scale behavior (load displacement history) simultaneously with the micro-scale

behavior (crack growth history), using only three material damage constants λcr, α0 and m.

7.2 Future Work

The promising results found in this work brought to light possibilities for further exploration of multi-scale mechanism based life prediction methods for polymer matrix composites for high temperature applications. However, the results presented in the current work combined with the collective reviews of previous work reinforce that the behavior of

92 polymer matrix nano-composites is not fully understood. In order to further the level of understanding in this area, some recommendations are listed below.

• As a next step, an investigation of thermo-oxidative behavior of ﬁber reinforced nano-

composites with nano-clay modiﬁed BMI as matrix material should be conducted to

fully understand the end-of-life properties and durability of composites at elevated

temperatures.

• In the 3D FEA model that simulated the residual stresses due to thermal shrinkage

strains, due to computer hardware and time limitations, zero shear at the free surface

(ﬁber end) could not be achieved. Similar issue was observed by Andrews et. al

[Andrews and Garnich, 2008], and they used the sub-modeling technique in ABAQUS

to achieve the zero shear at free surface. Although we used an in-house FORTRAN code

NOVA-3D for these numerical studies, the sub-modeling approach can be implemented

in this code to get better accuracy in the future.

• The present model is excellent in capturing mode-I (interlaminar delamination) type

failure but for completeness, it needs modiﬁcations to be able to simulate mixed mode

failure. Goutianos and Sorensen [Goutianos and Sorensen, 2012] have already presented

cohesive laws for modeling mixed mode fracture. Based on these cohesive laws, diﬀerent

scalar damage parameters can be derived and used to simulate mixed mode fracture

problems numerically. However, the mesh penetration due to high shear deformations

involved in mixed mode fracture cases need to be addressed adequately through the

use of interfacial contact elements beforehand.

• In the current model, we assumed that the viscoelastic constitutive relationship in-

93 corporates the strain rate effects and the damage parameters α0 and m, account for environmental effects. However, due to time constraints, we could not conduct any experiments to verify this partitioning. DCB tests should be conducted on aged specimens in different aging environments to verify the validity of this assumption.

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