residual stress characterization for laminated...

RESIDUAL STRESS CHARACTERIZATION FOR LAMINATED COMPOSITES

By

SHAO-CHUN LIU

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1999

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ACKNOWLEDGMENTS

The author would like to thank Dr. Peter Ifju for his continuous support and

advice, and the other members of his graduate committee, Dr. C.-T. Sun, Dr. Cristescu,

Dr. Shankar, Dr. Vu-Quoc, Dr. Jenkins, Dr. Beatty and Dr. Brennan, for their guidance.

Thanks also go to Mr. Ron Brown for his helpful instruction in the machine shop, and his

colleagues Xiaokai Niu, Brian Kilady, Ali Abdel-Hadi, Leishan Chen, Jongyoon Ok,

Brian Wallace, Oung Park, and Scott Ettinger for their friendship and help.

The author would like to acknowledge the generous sponsorship of the National

Science Foundation (NSF) under the grant number CMS 9502678, and the National

Aeronautics and Space Administration (NASA) Space Grant Consortium of Florida.

Finally, the author would like to express the most sincere gratitude to his parents,

wife, and children for their patience and understanding in the occasions without the

presence of their son, husband and father.

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TABLE OF CONTENTSpage

ACKNOWLEDGMENTS................................................................................................... ii

LIST OF TABLES............................................................................................................... v

LIST OF FIGURES............................................................................................................ vi

ABSTRACT ........................................................................................................................ x

CHAPTERS

1 INTRODUCTION......................................................................................................... 1Background ................................................................................................................... 1Literature Review .......................................................................................................... 2

Experimental Techniques ........................................................................................ 3Destructive methods .......................................................................................... 3Nondestructive methods .................................................................................... 5Shrinkage measurement of polymers ................................................................ 7

Analytical and Computational Modeling ................................................................ 9Elastic modeling................................................................................................ 9Viscoelastic modeling ..................................................................................... 10

Research Objectives .................................................................................................... 12

2 HIGH TEMPERATURE MOIRÉ INTERFEROMETRY AND THE CUREREFERENCING METHOD .......................................................................................15High Sensitivity Moiré Interferometry ........................................................................15Non-Room Temperature Moiré Interferometry with the Optical Thermal

Chamber ................................................................................................................19Optical Thermal Chamber .....................................................................................20

General description..........................................................................................20Chamber calibration........................................................................................22

Cure Referencing Method ...........................................................................................27The Principle.........................................................................................................28Grating Replication ...............................................................................................28Composite Specimen Fabrication..........................................................................32Experimental Setup and Procedure.......................................................................33

Tuning of optical setup....................................................................................33Thermal loading for composite laminate specimens.......................................35

Data Analysis and Results.....................................................................................37Significance of the Tests.............................................................................................49

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3 POST-GEL CHEMICAL SHRINKAGE OF THERMOSET RESINS....................... 51Introduction ................................................................................................................. 51Experimental Methodology......................................................................................... 52Specimen Preparation and Experimental Procedure ................................................... 52

Master Grating Selection....................................................................................... 54Refining of Replication Procedure ........................................................................ 58

In-plane and Out-of-plane Deformation Measurement ............................................... 58Results and Discussion................................................................................................ 63Conclusion................................................................................................................... 71

4 VISCOELASTIC CONSTITUTIVE MODELING FOR POLYMER MATRIXLAMINATED COMPOSITES.................................................................................... 73Introduction ................................................................................................................. 73Standard Linear Solid Model and Correspondence Principle for Linear

Viscoelastic Materials ........................................................................................... 74Time-Temperature Superposition Principle ................................................................ 75Thermal Chamber Manufacture and Test Fixture Design........................................... 76Specimen Preparation and Testing Plan...................................................................... 78

Gage and Accessory Selection .............................................................................. 79Specimen Preparation............................................................................................ 80Experimental Procedure and Testing Plan ............................................................ 81

Results and Data Analysis ........................................................................................... 84General Behaviors of Materials............................................................................. 84

Loading phase.................................................................................................. 85Holding (creep) phase...................................................................................... 86Unloading and recovery phases....................................................................... 92

Curves Fitting for the Creep Tests ........................................................................ 92Model Formation....................................................................................................... 108

5 DISCUSSION AND CONCLUSION ....................................................................... 111Application of Linear Viscoelastic Model for Residual Stress Relaxation............... 111

Overview ............................................................................................................. 111Experiment Procedure and Results...................................................................... 112

Concluding Remarks ................................................................................................. 122

APPENDICES

A THE MATHCAD FILE FOR THE DATA ANALYSIS OF 90-DEGREECOMPOSITE TENSILE SPECIMENS WITH THREE-ELEMENTSTANDARD LINEAR SOLID MODEL.................................................................. 124

B THE MATHCAD FILE FOR THE DATA ANALYSIS OF 90-DEGREECOMPOSITE TENSILE SPECIMENS WITH FOUR-ELEMENTSTANDARD LINEAR SOLID MODEL.................................................................. 131

REFERENCES................................................................................................................ 152

BIOGRAPHICAL SKETCH........................................................................................... 158

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LIST OF TABLES

Table page

1 Components of residual strain in different lay-ups of laminate specimens............... 502 Names and vendors of the resins used....................................................................... 533 Chemical curing shrinkage for five different resins. ................................................. 644 Time-Temperature Superposition shift factors (in log units), and reference

temperature (Tr) is 100 °C.......................................................................................1075 Comparison of apparent residual strains between newly made composite specimens

and two-year old specimens....................................................................................1166 Residual stresses calculated by linear viscoelastic model and laminate theory for the

same batch of composite specimens........................................................................121

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LIST OF FIGURES

Table page

1 NASA’s X-34 rocket plan...........................................................................................12 Schematic diagram of moiré interferometry..............................................................163 Photograph of a four-beam interferometer ................................................................184 Schematic diagram of the environmental chamber system setup..............................215 Photograph of the environmental chamber system setup..........................................226 Null field fringe patterns of distortion calibration. (a) U field without the chamber,

(b) V field without the chamber, (c) U field with the chamber, (d) V field with thechamber .....................................................................................................................24

7 Calibration tests in five different temperature settings. The curves are obtained fromthermal couple readings inside the remote chamber, and the markers are obtainedfrom analyzing moiré fringe patterns of aluminum control specimens which are themore precise representation of the specimen surface temperature............................26

8 The coefficient of thermal expansion for aluminum alloy 6061. The twodiscontinuities at -60 °C and 100 °C are due to different fitting functions fordifferent temperature ranges......................................................................................27

9 General grating replication procedure.......................................................................2910 Grating replication procedure for CRM ....................................................................3111 Vacuum bag lay-up for CRM ....................................................................................3312 AS4/3501-6 composite curing profile.......................................................................3413 Schematic diagram of a four-beam moiré interferometer .........................................3514 Collimation of incident beams. When the incident beams are perfectly collimated,

the frequency of the virtual grating will be space-wise constant (fA= fB = fC). Themoiré pattern due to the interference between virtual grating and the referencegrating will be insensitive to the position of the reference grating ...........................36

15 Fringe patterns for [0]16 unidirectional laminate at different temperatures...............3816 Fringe patterns for [02/902]2s symmetric cross-ply laminate at different temperatures

...................................................................................................................................4017 Fringe patterns for [03/90]2s symmetric unbalanced cross-ply laminate at different

temperatures..............................................................................................................4218 Fringe patterns for [0 2/452]2s symmetric angle-ply laminate at different temperatures

...................................................................................................................................4419 The strain-temperature curves of [1016] unidirectional laminated composite...........4720 The strain-temperature curves of [02/902]2s cross-ply laminated composite.............4821 The strain-temperature curves of [0/903]2s unbalanced cross-ply laminated composite

...................................................................................................................................4822 The strain-temperature curves of [02/452]2s angle-ply laminated composite.............49

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23 Apparatus to produce silicone rubber specimen molds: an aluminum pipe, analuminum rod, a piece of release-film-covered glass, and a specimen mold ........... 54

24 The original procedure for neat resin specimen replication ...................................... 5525 Distorted fringe patterns of a PC10-C specimen on the silicone rubber master grating

................................................................................................................................... 5626 Fringe patterns of a PC10-C specimen on the PC10-C master grating before

separation................................................................................................................... 5727 Illustrations for making (a) room temperature cure, (b) UV cure, and (c) high

temperature cure specimen gratings .......................................................................... 5928 Schematic diagram of a Fizeau interferometer.......................................................... 6129 Experimental setup for out-of-plane deformation measurement (Fizeau

interferometer). (a) Overview; (b) Close-up look of Fizeau interferometer and a neatresin specimen ........................................................................................................... 62

30 A high temperature specimen grating after being separated from its master grating ....................................................................................................................................... 63

31 Typical shrinkage vs. time curves for each kind of specimens ................................. 6632 In-plane chemical shrinkage and out-of-plane deformation fringe patterns for a

PC10-C epoxy specimen ........................................................................................... 6733 In-plane chemical shrinkage and out-of-plane deformation fringe patterns for a UV

cure resins (SR 448:SR 205=60:40).......................................................................... 6834 In-plane chemical shrinkage and out-of-plane deformation fringe patterns of 3501-6

epoxy at room temperature and 140 °C.....................................................................6935 Residual strain vs. temperature of AS4/3501-6 [0]16 laminate in transverse direction

...................................................................................................................................7236 Apparent strain of 3501-6 neat resin specimens vs. temperature..............................7237 Spring-dashpot arrangement for standard linear solid model....................................7438 Construction of the long-term “master curve” at reference temperature from short-

term testing data at different temperature..................................................................7639 Thermal chamber and test fixture for uniaxial tensile tests.......................................7740 Composite prepreg panel with Teflon release film template for applying strain gages.

The template for making 10-degree specimens is shown in this picture...................7841 Experiment setup from several viewpoints...............................................................8242 Testing profiles for (a) 0-degree specimens, (b) 10-degree and 90-degree specimens

...................................................................................................................................8443 Stress-strain curves for 0-degree specimens at different temperature settings during

loading phase. From Fig. 43 through 45, all strain values are the average of front andback strain gage readings...........................................................................................87

44 Stress-strain curves for 90-degree specimens at different temperature settings duringloading phase.............................................................................................................88

45 Stress-strain curves for 10-degree specimens at different temperature settings duringloading phase.............................................................................................................89

46 Strain-time curves for 0-degree specimens at different temperature settings duringholding phase.............................................................................................................90

47 Strain-time curves for 90-degree specimens at different temperature settings duringholding phase.............................................................................................................91

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48 Shear strain-time curves for 10-degree specimens at different temperature settingsduring holding phase ................................................................................................. 91

49 Estimating the values of initial guess for curve fitting.............................................. 9450 Curve fitting and experimental data for the longitudinal strain of 90-degree

specimens during the holding phase at different temperatures. (a) 22 to 80 °C; (b)100 to 177 °C; (c) 195 °C..........................................................................................94

51 Curve fitting and experimental data for the shear strain of 10-degree specimensduring the holding phase at different temperatures. (a) 22 to 100 °C; (b) 120 to 195°C...............................................................................................................................96

52 Estimating the values of initial guess for curve fitting with the fourth element .......9753 Curve fitting with four element model and experimental data for the longitudinal

strains of 90-degree specimens during the holding phase at different temperatures.(a) 22 to 80 °C; (b) 100 to 177 °C; (c) 195 °C..........................................................98

54 Curve fitting with four element model and experimental data for the shear strains of10-degree specimens during the holding phase at different temperatures. (a) 22 to100 °C; (b) 120 to 195 °C .......................................................................................100

55 log (S22(t)) versus log (t) for AS4/3501-6 unidirectional composite laminate atdifferent temperatures..............................................................................................102



58 The tensile creep compliance S22(t) master curve for 3501-6/AS4 unidirectionallaminate specimens in transverse fiber direction. Reference temperature is at 22 °C..................................................................................................................................104

59 The shear creep compliance S66(t) master curve for 3501-6/AS4 unidirectionallaminate specimens. Reference temperature is also at 22 °C. ................................105

60 The tensile creep compliance S22(t) master curve for 3501-6/AS4 unidirectionallaminate specimens in transverse fiber direction with consideration of physicalaging. Reference temperature is now at 100 °C. ....................................................106

61 The shear creep compliance S66(t) master curve for 3501-6/AS4 unidirectionallaminate specimens with consideration of physical aging. Reference temperature isalso at 100 °C. .........................................................................................................107

62 Comparison between directly calculated creep compliance S22(t) and curve-fittingS22(t). The curve represents long-term creep test is also shown to compare with themaster curve obtained from momentary creep tests (short-term creep). .................109

63 Fringe patterns for [016]T unidirectional composite laminate panel. Time lag frommanufacture date tlag= 702 days...............................................................................113

64 Fringe patterns for [02/902]2s balanced cross-ply composite laminate panel. tlag= 702days..........................................................................................................................114

65 Fringe patterns for [03/90]2s unbalanced cross-ply composite laminate panel. tlag=702 days...................................................................................................................114

66 Fringe patterns for [02/452]2s angle-ply composite laminate panel..........................115

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67 Continuous and piecewise S22 master curve in logarithm scale. In all the followingdiagrams, piecewise master curves are completely covered by continuous mastercurves and reference temperature is 22 °C..............................................................117

68 Continuous and piecewise S66 master curve in logarithm scale..............................11869 Continuous and piecewise S12 master curve in logarithm scale..............................11870 S22(t) for time from 0 to 107.78 seconds (702 days)..................................................11971 S66(t) for time from 0 to 107.78 seconds (702 days)..................................................12072 S12(t) for time from 0 to 107.78 seconds (702 days)..................................................120

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

RESIDUAL STRESS CHARACTERIZATION FOR LAMINATED COMPOSITES

By

Shao-Chun Liu

December 1999

Chairman: Dr. Peter G. IfjuMajor Department: Aerospace Engineering, Mechanics and Engineering Science

With increasing applications of advanced laminated composites, process-induced

residual stress has drawn more and more attention in recent years. Efforts have been

devoted to understanding residual stress both quantitatively and qualitatively.

In the current study, a novel technique called the Cure Referencing Method was

developed which has the capability for measuring the residual stress on the symmetric

laminated composite plates. It can also differentiate residual stress into two components:

one is due to the mismatch of the coefficient of thermal expansion, the other is caused by

the matrix chemical curing shrinkage.

The chemical curing shrinkage of the polymer matrix was investigated in further

detail. A technique was developed to measure the post-gel chemical curing shrinkage

which is the portion of curing shrinkage that really induces the residual stress in the

polymer matrix composites.

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Time-dependent material property is another issue associated with polymer matrix

composite materials. The data of several short-term tensile creep tests run at different

temperature were used to construct a linear viscoelastic model for describing the behavior

of the composites over a long period of time. It was found that physical aging of the

polymer matrix needs to be taken into account in order to have a more accurate

representation of the long-term behavior. A fair agreement was obtained between the

result of the long-term creep test and the master curve constructed from several

momentary creep tests.

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CHAPTER 1INTRODUCTION

Background

Significant engineering advances in structural design have historically depended

on materials. Today there is a need for high strength, lightweight materials for a wide

range of applications such as aerospace, civil infrastructure, automotive, sporting goods,

etc. As a result, fiber reinforced composites such as graphite/epoxy are currently

receiving a great deal of attention due to their favorable mechanical properties. For

example, the X-34 rocket plane (Fig. 1), a reuseable orbital plane for one of NASA’s

research projects, uses graphite/epoxy composites for both its primary and secondary

structures [1]. With the innovative use of these

materials, researchers will be able to reduce

the operating cost per unit payload by an order

of magnitude. In order to thoroughly utilize

these materials to their highest potential,

every aspect of their behavior must be

understood. One aspect that requires

characterization is the influence of the Fig. 1 NASA’s X-34 rocket plan

manufacturing process on the mechanical behavior of the material itself. Since many of

these materials require a high temperature cure, residual stresses often occur in the final

2

structure. These stresses are caused by thermal expansion mismatch of the constituents

and from chemical shrinkage of the polymers, and are inherently difficult to measure and

characterize. This work documents efforts to measure and characterize residual stress in

a common graphite/epoxy system (AS4 graphite fiber /3501-6 epoxy).

Literature Review

Residual stresses, sometimes called “process-induced stresses,” in composite

materials are always a serious issue. We can consider the residual stresses as a pre-load

to the manufactured structure. Depending on how we utilize the composite materials, this

pre-load can be useful. However, it usually significantly degrades the material strength

[2]. A great deal of work has been done to understand, measure, and try to reduce

residual stresses in composites. Among the different kinds of composites, such as

particulate composite, laminated composite, and woven composite, and among the

different kinds of material compositions, such as glass beads, carbon fibers, metal matrix,

and polymer matrix, it is the fiber reinforced polymer matrix laminated composite that we

are particularly interesting in. This kind of composite, sometimes called fiber reinforced

plastics (FRP), is the most used in advance applications such as in the aerospace industry.

For this type of composite, residual stresses exist on two different scales [3]: the

fiber scale and the ply scale. Several researchers have established analytical or

computational micromechanics models to obtain the residual stresses on the fiber scale.

However, so far no attempt has been made to experimentally measure the residual

stresses on the fiber scale for real composite structural elements. In this work, we focus

on the ply-scale residual stress measurement and modeling. Based on that, the available

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literature will be divided into three different classifications: experimental approach,

analytical approach, and computational approach.

Experimental Techniques

For measuring residual stresses, there are two main categories of experimental

techniques, destructive methods and nondestructive methods.

Destructive methods

As it states in the name, the destructive methods render the composites unusable

after testing. After measurements are taken by using destructive methods, the specimens

are usually no longer suitable to be structural elements. The hole-drilling method,

cutting/sectioning method, and first ply failure method are several common methods in

this category. They all involve taking a portion out of the specimen to create a free

surface and release the stresses on the surface.

The hole-drilling method is a very popular method to measure residual stresses. It

was originally developed for isotropic materials [4,5] and later for orthogonal materials

[6]. Traditionally, a strain gage rosette is used with the hole-drilling method to record the

strain relief around the hole. However, in the case of composite laminates, we have

verified that the attenuation of the stress relief around the hole dissipates in a few ply

thicknesses. Hence, strain gages cannot be used on composite residual stress

measurement because of spatial resolution limitations. Even the whole-field strain

measurement method, such as high sensitivity moiré interferometry, may not have

adequate spatial resolution to measure the deformation due to the drilling. Furthermore,

Schajer and Yang [7] showed that the generalization of elastic relations from isotropic

materials to orthotropic materials, conventionally used for hole-drilling method, is not

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valid. For different degree of orthotropy, it is necessary to make some correction in the

compliance matrix for gage calibration.

Similar to hole-drilling methods, cutting methods utilized the idea that producing

a free surface or separating plies releases residual stresses [8-10]. Lee et al. and

Gascoigne used moiré interferometry to record the change of the displacement field

before and after cutting. They first replicated a diffraction grating on the lateral surface of

the composite specimen, and then used a diamond saw to cut either parallel or

perpendicular to the ply interface. The surface residual stresses and the intralaminar

residual stresses can be obtained respectively. Lee and Czarnek [11] went one step

further: With the data obtained from moiré interferometry and a finite element method

(FEM) post-processor, they claimed that the distribution of the residual strain could be

accessed within a single ply. Sunderland et al. introduced another variant of cutting

methods called “successive grooving.” The authors applied strain gages on one side of a

thin composite laminate specimen. On the opposite side, they used a diamond saw to cut

a groove and let the saw incrementally advance. By monitoring the change of strain and

equilibrium conditions, using classical laminate theory, the internal stresses (residual

stresses) in each layer can be calculated. The cutting method, like the hole-drilling

method, also encounters the problem that the stress field around the cut diminishes

rapidly for the composite laminates. The strain reading may not be accurate enough to

represent the deformation due to the stress relief.

Ply sectioning is another destructive method reported in the literature [12-14].

Sectioning could be achieved by precise machining or embedding a separating film inside

the laminate (that Måson and Seferis called “process simulated laminate (PSL)”). After

5

the ply removal and stress relief, the unbalanced stresses create out-of-plane deformation

on the laminates. Residual stresses were then calculated from laminate theory. The

drawback of this method is the difficulty of ply-separation. Among these works, Lee et

al. [8], Gascoigne [9], and Joh et al. [14] chose moiré interferometry as the tool for strain

measurement and obtained the displacement contour map around the cuts or sections.

Those results become valuable information for verification and comparison to the results

of finite element method (FEM).

Hahn [15] and Kim and Hahn [16] introduced the first ply-failure method. They

claimed that the swelling due to moisture absorption could be used to compensate for the

effect of curing stress due to CTE mismatch between the matrix and the fibers. When the

specimens absorbed the moisture with different degrees of saturation, the tensile strength

of the specimen also varied. The relationship between the strength and the swelling could

be correlated to obtain the residual stresses of composites. However, waiting for the

specimens to become saturated takes a long period of time, and the authors’ goals were to

understand fracture initiation of composites rather than measuring residual stresses.

Nondestructive methods

Although the nondestructive methods are more favorable for mechanical testing,

the existing nondestructive methods for residual stress measurement are somewhat

limited as explained in the following section.

Measuring the warpage (i.e. the curvature) of an unsymmetrical composite

laminate is a widely used method [17-20]. There are two methods to produce warped

laminates. The most used method involves producing an unsymmetrical laminate. Upon

cooling the laminate warps as stresses increase. The second method is to produce a

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symmetric laminate and machine off layers from one side to relieve the residual stresses

thus producing a warped laminate. Of course, this will be classified as a destructive

method. With classical laminate theory, the curvature can be related to the moment

resultant on the laminates; thus, from the stress-strain relationship, the residual stresses

can be calculated. The disadvantage of this method is that it is very unusual to make a

structural component out of an unsymmetrical laminated composite; therefore, the usage

of warpage measurement is greatly limited when dealing with practical situations.

Imbedding strain gages into laminates and allowing them to go through the

manufacture procedure while monitoring the strain change is another nondestructive

method. Two kinds of strain gages have been used for this application: traditional strain

gages [21] and optical fiber strain gages [22]. In order to maintain the reference for strain

measurement, the strain gages and strain indication instruments need to be connected to

each other throughout the whole composite curing procedure. Although the strain gages

are small (thin), their presence acts to redistribute the residual stress in the composite.

Even the optical fibers, which are even smaller in dimension than traditional strain gages,

have a diameter 10 to 100 times larger than the diameter of a carbon fiber, thus disturbing

the natural order of the composite.

X-ray diffraction was utilized to measure the residual stress of composites by

including metal filler particles into the matrix material [23-24]. Although the diameter of

the filler particles has been reported to be on the same order as the diameter of fibers, the

issue is that fillers are distributed throughout the composite and affect the mechanical

properties of the composite globally rather than locally as in imbedded strain sensor

methods. Additionally, the relationship between residual stress and the results from X-

7

ray diffraction need to be carefully correlated in order to obtain meaningful information.

Madhukar, Kosuri, and Bowles [25] measured the residual stress by monitoring

the tension development in a single graphite fiber imbedded in epoxy during the curing

process. In this paper, the authors clearly demonstrate the interaction between chemical

shrinkage and thermal expansion, and proposed the way to optimize the curing cycle to

reduce the residual stresses. Nevertheless, those measurements were not taken on a real

part or structural element; consequently, the usefulness of the technique is greatly reduced

but can still yield valuable information in an academic sense.

In order to overcome the drawbacks in the methods mentioned above, Ifju et al.

[26] proposed a novel technique to measure the residual stress for symmetric fiber-

reinforced composite laminates using high sensitivity moiré interferometry. A moiré

diffraction grating is attached to the composite panel during the manufacture process.

After cure and after separation from the tool the diffraction grating will deform with the

laminate and thus record the dimensional change. By comparing the specimen grating

with the reference grating on the tool, the strain information of the composite can be

retrieved. However, to obtain the residual stresses from residual strains, the authors used

linear elastic laminate theory, which may be an over-simplified theory for most of the

polymer matrix composites (PMCs). In their conclusion, they claimed possible errors due

to neglecting viscoelasticity in PMCs (the analysis will be discussed in more details in

Chapter 2).

Shrinkage measurement of polymers

Once a method has been established to measure the total residual stress, in order

to reduce or even eliminate them, it is necessary to identify and characterize the sources

8

of these residual stresses. Most of the literature assumes that the chemical shrinkage of

polymer matrix and the differences in the thermal expansion/contraction of each

composition (the coefficient of thermal expansion (CTE) mismatch) during the

manufacture procedure are the two primary mechanisms inducing residual stress.

However, there is limited literature that addresses these two issues individually.

There are two types of polymers: thermoplastic and thermoset. In this study, the

discussion will be limited to thermoset polymer resin, which is the matrix material of our

composite system. For thermoset resin, polymer chemical shrinkage occurs when the

polymer chains start to crosslink. As crosslinking of the polymer advances, the polymer

chains bond together with strong covalent bonds. The polymer becomes consolidated and

stiffer, and the specific volume of the polymer shrinks. Several papers have been

published on the measurement of polymer chemical shrinkage. Hodges et al. [27], and

Snow and Armistead [28] used glass-bulb dilatometers to measure the volumetric change

of thermoplastic and thermoset resins during the curing process. Essentially, they used a

glass bulb connected to a capillary tube. The apparatus was partially filled with the resin

to be tested and the confining fluid (usually mercury or silicone oil). Then the bulb was

put into a temperature-controlled bath. The level of the resin was monitored while it was

curing. The specific volume versus time profile was obtained.

Russell [29] illustrated a different dilatometer. Instead of a glass bulb, the author

used a piezometer cell connected to a metal bellow. When the resin in the piezometer

cell underwent a change in volume (the author also measured the volumetric change for

laminate prepregs), the deflection of the bellow was calibrated to record the difference in

specific volume. Both Hodges et al. and Russell utilized the volumetric measurement as

9

a tool to monitor the progress of the chemical reaction and compared the result with

different methods such as viscosity measurement, differential scanning calorimetry

(DSC), and dynamic mechanical analysis (DMA). Their purpose was different from

strain measurement and did not attempt to relate their results to residual stress

information.

In 1995, Wang et al. [30] presented a unique method to measure the thermal

induced strains during cure and cooling for epoxy resins. By using an epoxy-coated thin

aluminum film and a quartz displacement probe, the chemical shrinkage and thermally

expansion of the neat resin were derived from the deflection of the aluminum film and

Timoshenko beam theory, and monitored in-situ. The authors did not attempt to correlate

the results to the residual stresses in composite materials.

Analytical and Computational Modeling

In addition to the above-mentioned experimental efforts, there have been

numerous mathematical models developed to determine residual stresses in composite

materials.

Elastic modeling

Hahn and Pagano [31] proposed a stress-strain-temperature relationship to model

the process-induced residual stress based on classical laminated theory. This is the first

attempt in the literature to assess the residual stresses of composites with a mathematical

model. However, since this was the first effort on this topic, the authors made several

assumptions, which have proved to be invalid. They assumed the stress-free state was

located at the end of highest temperature (cure temperature) stage. Now we know the

stress-free state (the gel point) is usually at the beginning or prior to the final hold stage.

10

Their thermal-elastic model was correct for the cooling stage, but apparently could not

describe the viscoelastic behavior of the polymer at the cure temperature holding stage.

Also, the authors only considered the mechanical and thermal contributions and

completely neglected the irreversible chemical shrinkage of the polymer matrix materials.

Bogetti and Gillespie [32] used a one dimensional heat conduction equation and

finite difference analysis to stimulate the curing process of a thick thermoset composite

laminate. According to their parametric study, input values for volumetric shrinkage

significantly influenced the prediction of the residual stress distribution. This study gives

insight into how important the experimental shrinkage measurement can be in the residual

stresses. Additionally, their work did not take viscoelasticity into account.

Viscoelastic modeling

Strictly speaking, almost all commonly used PMCs more or less exhibit some

viscoelastic properties. When the service temperature of the composite is high, it is

usually necessary to cure the composite at an even higher temperature to achieve the

desired thermal stability at the service temperature. In this case, the viscoelastic

behaviors, such as stress relaxation and creep, will be highly accelerated due to the time-

temperature dependence of polymer matrix [33]. Since the mid-60s, the viscoelastic

properties of composites started to gain attention. Around the mid-60s, Schapery [34-37]

published several papers on the constitutive modeling of viscoelastic media under the

influence of temperature. In Schapery’s 1967 paper, by utilizing the corresponding

principle, the author clearly defined the general form of the stress-strain relationship for

anisotropic composite materials with a linear viscoelastic and thermorheologically simple

assumption. However, that work did not address the issue of residual stress in

11

composites, neither did it consider the effect on the composition of anisotropic composite

materials.

Weitsman [38] investigated the effect of the temperature profile on the curing

cycle both analytically and experimentally and attempt to obtain an optimal cooling path

to minimize the residual stresses. Harper and Weitsman [39,40] also employed the

corresponding principle by implementing the viscoelastic properties of the resin.

However, their work was only limited to the cooling stage of the curing cycle.

White and Hahn [41,42] made an in-depth discussion of the curing process model.

In the article, a model called LamCure was presented which combined the cure kinetics

with the residual stress model. Differential Scanning Calorimetry (DSC) was used to

obtain the parameters needed for the cure kinetics model, and several tensile and creep

tests were performed on partially cured and fully cured composite specimens to obtain

viscoelastic mechanical properties. Nevertheless, the authors neglected the contribution

of chemical shrinkage to the final residual stress. They claimed that at high temperature,

like the cure temperature of the composite, the viscoelastic properties of the material

dominates, and that stress relaxation was evident. The stress due to chemical shrinkage

would relax rapidly and cannot be developed. The present author believes that this is not

true. According to Liu et al. [43], the experiments show that, depending on the lay-up of

the composite laminate, there are from 2 to 22 percent of final residual stress induced by

the chemical shrinkage.

Wang et al. [44] used linear thermoviscoelastic laminate theory to calculate the

residual stresses and warpage of unsymmetrical woven-glass/epoxy laminates. The

formulation of stress-strain relationship was presented. Unlike White and Hahn’s

12

approach, the authors did not consider the matrix and fiber separately and did not

examine the cure kinetics of the system. Consequently, the model was not able to depict

the evolution of curing in terms of degree of cure. Instead, the time coordinate along a

certain temperature profile was used to describe the evolution of the chemical reaction.

Once the temperature profile was determined, the degree of cure should be a unique

function of time.

Kim and White [45] modeled relaxation modulus of 3501-6 epoxy as a

thermorheologically complex material. Shift functions were also obtained. The

experiments were done by using a dynamic mechanical analyzer (DMA). With the

information of the shift functions the creep master curves were obtained and the time-

dependent viscoelastic moduli could be formulated accordingly. At about the same time,

Adolf and Martin [46] presented a comprehensive constitutive model for a cross-linked

polymer to calculate the process-induced stresses. Experimental data for the viscoelastic

material properties and the cure kinetic model parameters were required for the

calculation. However, these papers used the data obtained from matrix resin specimens

to model the whole composite material system (fibers and matrix). It is doubtful that the

models could represent the composite systems well.

Research Objectives

In this work, there are two main objectives proposed.

First, to develop a series of methods that can measure the residual strains and

quantify the components of residual strains resulting from different mechanisms. The

mismatch of CTE between the fibers and matrix, and chemical curing shrinkage of

13

thermoset resins are two mechanisms we consider in this study. In order to achieve these

two purposes, this technique must allow us to perform the measurement in the elevated

temperature environments and be capable for “recording” the strain development during

the composites (and/or their matrix resins’) curing process. With the literature review

performed, modifying and improving the current methodology of moiré interferometry

are the most suitable for characterizing process induced residual stresses of polymer

matrix composites. There are many existing techniques that can be used to measure

residual strain of composites or the chemical curing shrinkage of polymers, but none of

them can accomplish both tasks at the same time. Moreover, not all of the chemical

shrinkage contributes to the residual stresses. The new technique is able to measure only

the portion of the shrinkage, which occurs after the gel point and is accountable for

residual stresses. In this work, the technique called “cure referencing method” (CRM),

based on high sensitivity moiré interferometry and an optical thermal chamber, is

implemented to achieve our first goal and is addressed in the following chapters.

Second, using a viscoelastic constitutive equation to describe the stress-strain

behavior of a polymer matrix composite laminate. Classical laminated theory and time-

temperature superposition (TTS) principle [47] were employed to construct the equation.

The long-term material properties, the elements in the creep compliance matrix, were

measured through a series of tensile tests on 0-degree longitudinal, 90-degree transverse,

and 10-degree off-axial unidirectional composite laminate specimens. To obtain the

long-term properties in a short-term test, another thermal chamber was used to elevate the

temperature of the testing environment to accelerate the time effect. With this

14

viscoelastic model, the relaxation of the residual stress can be predicted and compared

with the experimental measurement by the cure referencing method.

15

CHAPTER 2HIGH TEMPERATURE MOIRÉ INTERFEROMETRY AND THE CURE

REFERENCING METHOD

High Sensitivity Moiré Interferometry

Moiré interferometry is an optical method that utilizes moiré phenomenon of

optical interference to measure the deformation of objects. Geometric moiré, shadow

moiré, high sensitivity moiré interferometry, or even micro moiré interferometry are all

based on the same principle—the interference between two gratings of similar frequency.

Although the methodology of moiré interferometry is not new, the book published by

Post et al. in 1994 [48] opened a new era for the high sensitivity moiré interferometry.

Figure 2 illustrates the principle of high sensitivity moiré interferometry. The

angle of incidence α of the light beams into specimens can be determined by the

wavelength of coherent light source (λ), the frequency of virtual grating (f) and the

following equation

αλ

sin2 ⋅=f (2.1)

The frequency of specimen grating (fs) is required to be equal to f / 2 in order to

produce fringe patterns in the direction normal to the specimen surface. When the

specimen grating is undeformed, the reference virtual grating can be tuned by adjusting

the angle of the two incident light beams, so that it perfectly overlaps with the original

16

master gratings or undeformed specimen grating. In this case, if we view from the

camera, there will be no interference fringe. This is called null-field. However, when the

specimen is deformed in the x-y plane, the two first-order diffracted light beams carry the

distorted wave fronts and the interference between them will produce a fringe pattern.

α

1st order diffraction from incident beam 1 interference patterns

captured by camera

Incident beam 1

Incid

ent b

eam

2

0th order diffraction (reflection)from incident beam 2

0th order diffraction (reflection)from incident beam 1

1st order diffraction from incident beam 2

Specimen withdiffraction grating,grating frequency fs

Virtual reference grating,grating frequency f

Fig. 2 Schematic diagram of moiré interferometry

Once the fringe patterns are photographed, the in-plane displacement field can be

extracted according to the equations

),(1

),(

),(1

),(

yxNf

yxV

yxNf

yxU

y

x

⋅=

⋅=(2.2)

17

or with selected gage length, two in-plane normal strains and the in-plane shear strain can

be also calculated by the equations:

)(1

1

1

x

N

y

N

f

y

N

f

x

N

f

yxxy

yy

xx

∆∆

+∆

∆⋅=

∆∆

⋅=

∆∆⋅=

γ

ε

ε

(2.3)

where U (x, y) and V (x, y) are the displacements in x and y direction at the point of

interest (x, y) with respect to the chosen origin. Nx and Ny are the fringe numbers in x and

y direction from the origin to the point of interest. εx and εy are the normal strains in x

and y direction, and γxy is the shear strain. ∆x and ∆y are the selected gage length, ∆Nx and

∆Ny are the fringe order difference over the selected gage length respectively. The actual

sensitivity of the setup depends on the wavelength of our light source and the frequency

of reference grating. The optical setup in this study employs a 20 mWatt, class IIIb,

Helium-Neon laser (wavelength λ = 632.8 nm) from UNIPHASE, and the frequency of

the reference grating is 2,400 lines per mm (60,960 lines per inch). From equations (2.1)

and (2.2), we obtain the incident angle of 49.4 degrees, and the displacement sensitivity

of 0.417 µm per fringe order. Figure 3 is the photography of the moiré interferometer

setup used for residual stress characterization. A four beam system is employed to obtain

both U-field and V- field images. The laser beam is squeezed into a single mode optical

fiber using a laser coupler to produce a point light source at the fiber tip of the other end.

The divergenet light from the point light source is caught and collimated by a parabolic

mirror, and split into four beams by four 45-degree flat mirrors.

18

Fig. 3 Photograph of a four-beam interferometer

There are several advantages of high sensitivity moiré interferometry over

traditional geometric moiré or the other displacement/strain measuring methods. With the

much higher grating frequency than geometric moiré, it can achieve very high sensitivity.

Also the spatial resolution and signal-to-noise ratio are superior. From the pictures in the

next few chapters, even with very high density of fringes, the fringes are still well defined

and in excellent contrast. Whole field measurement is another advantage of moiré

interferometry. As in Fig. 2, the addition of two incident light beams in the y direction

will enable us to measure the horizontal displacement and shear strain components. This

capability makes moiré interferometry a very favorable experimental tool for finite

element analysis (FEA) validation. Material properties or boundary conditions required

for FEA input can be obtained or corrected. The displacement contours or strain contours

19

from moiré fringe patterns can also be used to compare with the computational results.

The most important advantage of moiré interferometry in this study is the long-

term capability of monitoring the strain field. Establishing a reference condition at some

point in time is the key for long-term strain measurements. The reference condition for

moiré interferometry can be obtained by tuning the master grating (parent grating) to a

specific specimen grating. The procedure of specimen grating replication will be

elaborated on a later section. The master grating is usually made on the substrate material

that can be treated as totally rigid within the time frame we are considering. Therefore, as

long as the master grating is not damaged or lost, the subsequent measurement can always

be performed. Unlike most of other methods we mentioned in the literature review, the

specimen will loose its reference once removed or disconnected from the experiment

apparatuses. One more benefit from its robust reference condition, moire interferometry is

also an outstanding method for investigating the hygrothermal effects on polymer matrix

composites. In the current study, thermal load was applied on composite laminate

specimens, but the moisture effect was not considered. The humidity effect is considered

negligible since we retain all specimens in a desiccant box.

Non-Room Temperature Moiré Interferometry with the Optical Thermal Chamber

Moiré interferometry can be much more versatile if the usable temperature range

can be extended well below and above room temperature. There are two issues

associated with non-room temperature moiré interferometry: First, one must create an

environment that can perform the moiré tests. For simplicity, a thermal chamber only for

retaining specimens (not entire interferometer optical setup) was designed and

20

manufactured. This issue is discussed in the next section. Second, the specimen gratings

must survive the specific environment without degradation or distortion. The second

issue can be resolved by carefully choosing the procedure and/or the materials used for

grating replication. This issue will be discussed later in this chapter.

Optical Thermal Chamber

In order to perform the moiré tests in a specific thermal environment, an optical

thermal chamber needed to be constructed to fulfill several requirements. First, after

mounted in the chamber, the specimen position must be adjustable. Second, distortion

caused by the window glass must be minimized in order to prevent errors. With these

conditions in mind, our first thermal chamber was constructed.

General description

To extend the thermal application of moiré interferometry, an environmental

chamber with a wide temperature range capability was necessary. To meet the

requirements of moiré testing, a remote thermal chamber was built and connected to a

thermal chamber (model number EC12 from Sun Electronic Systems, Inc.). The EC12

thermal chamber (will be referred as “the oven” hereafter) has a maximum allowable

temperature range of -173 °C to +315 °C when it is connected to a low pressure liquid

nitrogen tank, and serves as a heating or cooling source for the remote chamber. The

hot/cool air is drawn to the remote chamber by a 4-inch diameter centrifugal tangential

duct fan and flexible thermal tubing (McMaster-Carr Supply Company, Sure-Flow

silicone-coated fiberglass hoses, temperature range -60 °C to 260 °C) insulated with glass

fiber. The outer walls of the chamber were constructed with 1/4 inch thick aluminum

plates, and had dimensions of 165mm x 165mm x 165mm (6 ½” x 6 ½ ” x 6 ½ ”). For

21

chamber insulation, the interior of the remote chamber is lined with one-inch-thick high

temperature calcium silicate board (McMaster-Carr Supply Company, max. temp. 927

°C, thermal conductivity 0.8 Btu @ 427 °C), thus making the inner dimensions 102 mm x

102mm x 102mm (4” x 4” x 4”). The chamber window consisted of three pieces of 1/4”

thick glass with wedged aluminum spacers in between to reduce optical noise due to

multiple reflections. The inner most layer of window was made of low expansion

BorofloatTM glass to avoid cracking due to the high temperature gradient through the

thickness during heating or cooling. The system setup is shown in Fig. 4 schematically

and is photographed in Fig. 5.

Fig. 4 Schematic diagram of the environmental chamber system setup

22

Fig. 5 Photograph of the environmental chamber system setup

A thermal couple probe is extended from the oven through the air duct into the

remote chamber. By attaching the thermal couple onto the specimen surface, we are able

to set the temperature profile (thermal history) of specimens in the remote chamber.

Chamber calibration

There are two main concerns for the calibration of the environmental chamber.

First of all, since moiré interferometry is an optical method, any optical element we put

into the optical path will alter the results. The window glass of the chamber can be a

source of problems. To document the effect of the window glass, an aluminum coated

silicone rubber grating on 6061-T3 aluminum alloy substrate was made as a control

specimen for calibration. The aluminum control specimen was first tuned to null field

without the chamber (without glass), and a picture of the fringe pattern was taken. Then,

23

we replaced the regular specimen holder by the chamber. By only adjusting the position

of the specimen and without changing the interferometer, the new fringe patterns show

the distortion caused by the window glass. Figure 6 (a), (b), (c), and (d) show the pictures

of null field patterns of a control specimen without and with window glass. The size of

the square gauge mark at the center of the specimen is 9 mm by 9 mm, and there are

three-fringe difference in U field and one fringe difference in V field across the whole

area. According to this experiment, the distortion is small but not negligible. We can

minimize this problem by positioning the master grating inside the chamber when we

tune the moiré interferometer for the reference position. Also when replacing master

grating with the specimen, the tuning procedure needs to be done inside the chamber as

well.

The second concern of using the chamber is the temperature stability. To perform

the calibration, the aluminum control specimen was again put into the remote chamber.

Figure 7 shows the temperature change inside the remote chamber in the first 100 seconds

after we turned off the circulation fan to obtain a vibration-free condition, which is

essential for photographing moiré fringe patterns. From high temperature to low

temperature several settings were tested. The curves represent the thermal couple reading

from the oven with the probe almost touching the specimen, and the data points are the

results from analyzing moiré patterns. As we expected, the larger the temperature

difference (from room temperature), the faster the temperature drops (for high

temperature) or increases (for low temperature). Moreover, the actual temperature we

obtained from the fringe patterns was more stable than the temperature reading from the

24

(b)

(d)

(a)

(c)

Fig. 6 Null field fringe patterns of distortion calibration. (a) U field without the chamber, (b) V fieldwithout the chamber, (c) U field with the chamber, (d) V field with the chamber

25

thermal couple, since the air inside the chamber dissipates/absorbs heat much faster than

the specimen does. Also the thermal mass of the thermocouple was much smaller than

that of the specimen.

It is important that we make a correction for the coefficient of thermal expansion

(CTE) as a function of temperature for our aluminum alloy control specimen during

analysis. The CTE of the aluminum alloy is a function strongly depending on the

temperature [49] within our testing range (from -130 °C to +175 °C). From Fig. 8, we can

see the CTE at -130 °C is only about 62.5% of that at +175 °C. The actual temperature (

Ti ) on the control specimen is obtained from solving the following integration equation

dl l T dTl

l

T

Ti i

0 00∫ ∫= ⋅α ( ) (2.4)

here T0 is the room temperature, l0 is the original gauge length at room temperature, α(T)

is the function of the CTE in terms of temperature T, and

dll

li

0∫ = ∆l = fringe number N × 0.417 µm (2.5)

is obtained from fringe patterns. If we look at 175 °C data points in Fig. 7 where the

temperature drops fastest, the average temperature change in 10 seconds is 0.4 °C, which

corresponds to 1.24 µε on an AS graphite/epoxy laminate [50] in transverse fiber

direction, and 48 µε on the neat epoxy resin. These strain values are in about the same

order of magnitude as the sensitivity limit of our moiré interferometer, and the average

temperature change we used was obtained from our aluminum control specimen, which

has relatively high thermal conductivity. Therefore, the actual temperature change on

26

laminate specimens or neat resin specimens is less, and the strain error would be even

smaller.

Fig. 7 Calibration tests in five different temperature settings. The curves areobtained from thermal couple readings inside the remote chamber, and themarkers are obtained from analyzing moiré fringe patterns of aluminumcontrol specimens which are the more precise representation of thespecimen surface temperature.

27

Fig. 8 The coefficient of thermal expansion for aluminum alloy 6061. The twodiscontinuities at -60 °C and 100 °C are due to different fitting functions fordifferent temperature ranges.

Cure Referencing Method

With all the features stated before, moiré interferometry, combined with a novel

technique of grating replication during the composite specimen fabricating process,

constitutes a suitable methodology for the measurement of the process induced residual

strain. This methodology is called Cure Referencing Method (CRM). In order to explain

the concept of CRM better, first the principle of CRM will be given, and the following

sections will give a brief step-by-step description of how to perform residual stress

measurement by CRM.

28

The Principle

In moiré interferometry, displacement of the specimen is measured by the

deformation of the diffraction grating on its surface. For CRM, the grating is attached to

the composite specimen during the curing process, at the stress-free state. The stress-free

state exists before the matrix resin starts to gel. Because the ultra-low expansion substrate

material of the diffraction grating also serves as a tool for the composite curing

procedure, the diffraction grating remains undeformed until the curing procedure is

completed, and the laminate specimens are separated from the ultra-low expansion tool.

Before resin gelation, the epoxy resin is still free to flow around fibers and unable to carry

stress. At that point, the matrix is solidified and the grating is rigidly attached to the

laminate surface. Residual stresses arise from this point, but the in-plane dimensions of

specimen and the frequency of specimen grating remain the same due to the constraint

from the rigid tool.

Once the specimen and grating are separated from the master grating tool, the

accumulated residual stresses will deform the specimen as well as the diffraction grating

attached to it. If the interferometer is first tuned to the null field using the master grating

tool, and by replacing the master grating with specimen grating, the relative displacement

between the master grating and specimen grating can be measured accordingly.

Grating Replication

In order to perform a moiré test, a diffraction grating need to be replicated onto

the specimen surface. This is usually done by transferring the grating from a master

grating onto the specimen surface with an adhesive layer. Figure 9 shows the general

procedure for grating replication. Step 1: pour a pool of adhesive onto the master grating,

29

and slowly lower the specimen into the pool avoiding air bubbles. Step 2: apply uniform

pressure on the top of the specimen and clean up excessive adhesive. Step 3: separate the

specimen from the master grating after adhesive is cured.

VSHFLPHQ

XQFXUHGDGKHVLYH

PDVWHU�JUDWLQJ

PHWDOOLFILOP

(a) Step 1

ZHLJKWFRWWRQ�VZDE

XQFXUHGDGKHVLYH

(b) Step 2

FXUHGDGKHVLYH

PDVWHU�JUDWLQJ

PHWDOOLFILOP

VSHFLPHQDQG�JUDWLQJ

(c) Step 3

Fig. 9 General grating replication procedure

Like a regular moiré interferometry test, CRM starts from the replication of

diffraction grating. The difference is that the replication procedure takes place during the

30

composite manufacturing process. To survive the harsh environment of high temperature

and high pressure during the curing process, an alternate approach for grating replication

was developed and adopted [51]. In this procedure, all gratings were replicated on a kind

of ultra low expansion glass called Astrosital to maintain the dimensional stability of

gratings. First, we start with a silicone rubber (GE RTV 615) master grating. A 1,200

lines/mm, phase type, crossed line photo-resist master grating (A) was used for

replicating the silicone rubber master grating (B) in this study (refer to Fig. 10).

Second, an intermediate grating (C) made of Shell Epon 862 and curing agent W

was replicated from the silicone rubber master grating (B). For the best result the master

grating, substrate material, the epoxy compounds were heated to 130 °C, and the epoxy

was degassed in a vacuum oven. The intermediate grating was cured at 130 °C for ten

hours. After separation from the master grating, two layers of aluminum films with

diluted Kodak Photo-flo in between [48] were vacuum deposited on the intermediate

grating (C’).

This intermediate grating (C’) became the master grating of the grating tool for

autoclaving. The epoxy for replication of the grating tool is the same epoxy as the matrix

resin in our composite prepreg, 3501-6 epoxy from Hexcel. After it was cured at 130 °C

for ten hours and post-cured at 177 °C for two hours, the grating tool (D) was separated

from the intermediate grating, and two layers of aluminum were vacuum deposited on the

tool (D’) like the intermediate grating.

Finally, a thin layer of 3501-6 epoxy was cast on the top of the two layers of

aluminum on the tool grating with a special tool wrapped by Teflon film [51]. The tool

grating (D”) now is ready for the autoclave procedure.

31

Fig. 10 Grating replication procedure for CRM

32

Composite Specimen Fabrication

Four different prepreg lay-up sequences, [0]16 unidirectional, [02/902]2s cross-ply,

[0/903]2s unbalanced cross-ply, and [02/452]2s angle-ply were prepared for each autoclave

specimen fabrication. The original size of the panels were 6 inches (152.4 mm) square.

The prepreg panels were then put into a vacuum bag (Airtech International Inc.,

WN1500) together with release films (Airtech International Inc., Release Ease 234TFP-1

and 234TFNP), bleeder and breather fabrics (Airtech International Inc., Airweave Super

10), and the master grating tool. The vacuum bag lay-up for the laminate specimens is

illustrated in Fig. 11. The whole vacuum bag assembly was then put into an

electronically heated autoclave oven (Baron Blackesleee Inc., Model BAC-24, max.

pressure 110 psi, max. temperature 650 °F) for curing.

For AS4/3501-6 graphite/epoxy pregreg, the manufacturer's recommended curing

profile shown in Fig. 12 was used [52]. First, the temperature was raised at 5 °F per

minute to the first dwell stage of 225 °F for one hour accompanied with 15 psi pressure

outside the vacuum bag, and 30 inches Hg of vacuum inside the vacuum bag. The main

purpose of the first stage is to keep the matrix resin in a low-viscosity condition and allow

vacuum to eliminate voids inside the pregreg. After one hour in the first dwell stage, the

temperature is raised again at the same rate, 5 °F per minute, to the final curing

temperature of 350 °F. The outer pressure is then increased to 100 psi, and the vacuum

inside the vacuum bag is released to atmosphere when the outer pressure reaches 30 psi.

This final curing stage, with the high temperature and the high pressure conditions, allows

the crosslinking of matrix resin to develop and the consolidation of the whole composite

system to complete. After six hours in the final curing stage, we start

33

Fig. 11 Vacuum bag lay-up for CRM

decreasing the temperature also at 5 °F per minute until the autoclave temperature drops

to 100 °F. Pressure is released when the temperature reaches 175 °F. Finally, the

vacuum bag is removed from the autoclave, and the composite specimens are removed

from the vacuum bag and separated from the grating tool.

Experimental Setup and Procedure

Three batches and four different laminate lay-up specimens as mentioned before

for each batch were manufactured.

Tuning of optical setup

The total process induced residual strains were first measured at room

temperature. A four-beam moiré interferometer was used for this measurement.

A schematic diagram of the optical setup is shown in Fig. 13. The interferometer

is first tuned with respect to the intermediate grating, which has exactly the same pattern

as theundeformed specimen grating (unlike the grating tool, which is the mirror image of

the specimen grating due to the in-perfection of perpendicularity of the crossed-line

34

0

(°F)

50

100

150

200

250

300

350

400

1hr.

6 hr.s

10

20

30

40

50

60

7080

90

100

(psi)

Time

Temperature

Presure

vacuum (30 in. Hg)

vacuum released

Fig. 12 AS4/3501-6 composite curing profile

grating). The interferometer was tuned so that two incident beams were perfectly

collimated [51] (Fig. 14), the virtual grating was perfectly registered with the intermediate

grating, and a null field pattern appeared on the camera screen.

Once the null field image was obtained, the reference grating was removed, and

replaced by the specimen grating. The optical setting of the interferometer is required to

remain unchanged since the virtual grating is now our reference grating. The specimen

grating was tuned until the 0th order diffraction (reflection) light beams converged back

to the point light source on the tip of the optical fiber.

For the moiré tests at elevated temperature, the laminate specimens were cut down

to 3 inches (76.2 mm) square by a diamond impregnated saw to fit into the chamber. The

interferometer was first tuned to the null field with respect to the specimen grating at the

room temperature inside the thermal chamber. When the temperature inside the chamber

35

Fig. 13 Schematic diagram of a four-beam moiré interferometer

increased (closer to stress-free temperature), the fringe density also increased. By doing

so, if the specimen has 1% strain, the error can be easily calculated as 0.99%.

Thermal loading for composite laminate specimens

Starting from the room temperature, the specimen gratings were heated at a rate of

3 °C per minute to 15 °C increments. After the set temperature was reached, the

temperature was held for 15 minutes to allow the chamber and specimen grating to reach

thermal equilibrium. Then the oven and the circulation fan for the thermal chamber were

both turned off to obtain a vibration-free condition for photographing the moiré fringe

patterns. Both the U-field (horizontal) and V-field (vertical) images were taken in a

timely fashion to avoid a serious drop of temperature inside the chamber and specimen

36

perfect collimation

convergence

divergence

A B C

A B C

f f fA = = B C

f f fA > > B C

A B C

f f fA < < B C

Fig. 14 Collimation of incident beams. When the incident beams are perfectlycollimated, the frequency of the virtual grating will be space-wise constant(fA= fB = fC). The moiré pattern due to the interference between virtualgrating and the reference grating will be insensitive to the position of thereference grating.

37

grating. KODAK black and white 35mm TMAX 400 roll film and a single-lens reflex

(SLR) camera without a lens were used for photography. After photography, the

temperature was raised again to the next 15 °C increment until the curing temperature,

177 °C, was reached. The same procedure was repeated for cooling from 177 °C to room

temperature. Rotation carrier fringes were applied to several fringe patterns to make the

images more readable. Totally, twelve specimens, three autoclave runs, four different

lay-ups of laminates per run were tested. Figures 15 through 18 are some typical fringe

patterns taken with the thermal chamber during the tests at various temperature settings.

The circular marks shown on those pictures were drawn using a compass with 12.7mm

(1/2 inch) radius. The directions of the crossed marks were aligned with the fiber

direction and transverse fiber direction.

Data Analysis and Results

The process induced residual strain of [0]16 unidirectional specimen, {εuni}, was

directly obtained from moiré fringe images, and was treated as the result of free thermal

contraction and matrix resin chemical shrinkage, since all its fibers orient in the same

direction and there is no mutual constraint between plies. Equation (2.6) describes this

relationship.

shrinkage

yuni

xuni

thermal

yuni

xuni

yuni

xuni

xyuni

yuni

xuni

uni

+

=

=

=

000

}{ _

_

_

_

_

_

_

_

_

εε

εε

εε

γεε

ε (2.6)

The apparent strains {εlam}, which were obtained from the moiré fringe patterns of

the other three lay-ups of laminate specimens, were actually the result of free thermal

38

(a) -field, 23.2 °CU (b) -field, 23.2 °CV

(c) -field+ rotation, 23.2 °CU (d) -field+rotation, 23.2 °CV

(e) -field, 70 °CU (f) -field, 70 °CV

[0 ]16 T

0° fiber direction

Fig. 15 Fringe patterns for [0]16 unidirectional laminate at different temperatures.

39

[0 ]16 T

0° fiber direction

Fig. 15--continued

40

(a) -field, 23.2 °CU (b) -field, 23.2 °CV

(c) -field+rotation, 85 °CU (d) -field+rotation, 85 °CV

(e) -field, 177 °CU (f) -field, 177 °CV

0° fiber direction

[0 /90 ]2 2 2s

Fig. 16 Fringe patterns for [02/902]2s symmetric cross-ply laminate at differenttemperatures.

41

(g) -field+rotation, 100 °C(cooling)

U

(i) -field+rotation, 23.5 °C(cooling)

U

(h) -field+rotation, 100 °C(cooling)

V

(j) -field+rotation, 23.5 °C(cooling)

V

0° fiber direction

[0 /90 ]2 2 2s

Fig. 16--continued

42

(a) -field+rotation, 23.8 °CU

(c) -field+rotation, 100 °CU

(e) -field, 177 °CU

(b) -field+rotation, 23.8 °CV

(d) -field+rotation, 100 °CV

(f) -field, 177 °CV

0° fiber direction

[0 /90]3 2s

Fig. 17 Fringe patterns for [03/90]2s symmetric unbalanced cross-ply laminate atdifferent temperatures.

43

(g) -field, 115 °C(cooling)

U

(i) -field+rotation, 40 °C(cooling)

U

(h) -field, 115 °C(cooling)

V

(j) -field+rotation, 40 °C(cooling)

V

0° fiber direction

[0 /90]3 2s

Fig. 17--continued

44

(a) -field, 22.5 °CU

(c) -field, 100 °CU

(e) -field+rotation, 177 °CU

(b) -field, 22.5 °CV

(d) -field, 100 °CV

(f) -field+rotation, 177 °CV

[0 ]2 2s/452

0° fiber direction

Fig. 18 Fringe patterns for [02/452]2s symmetric angle-ply laminate at differenttemperatures.

45

(i) -field+rotation, 23.2 °C(cooling)

U (j) -field+rotation, 23.2 °C(cooling)

V

(g) -field+rotation, 115 °C(cooling)

U (h) -field+rotation, 115 °C(cooling)

V

[0 ]2 2s/452

0° fiber direction

Fig. 18—continued

46

contraction, matrix chemical shrinkage after transformation to the specific orientation

superimposed with the deformation due to residual stresses (constraint in between

adjacent plies). This relationship can be written as the following:

}{}{][}{ i.e.

}{}{][}{

1

1

lamunikkresidual

kresidualuniklam

T

T

εεε

εεε

−⋅=

−⋅=

−

−

(2.7)

[T]k-1 is the transformation matrix for the specific orientation of the kth ply.

Therefore, the residual stress in the kth ply {σresidual}k can be calculated from the residual

strain matrix {εresidual}k, the transformation matrix [T]k-1, and the laminate stiffness matrix

[Q] by the following expression:

}){}{]([][][][}{ 11lamunikkkkresidual TTQT εεσ −⋅⋅⋅⋅= −− (2.8)

More detailed explanation and analysis can be found in the previous work [53, 54].

To achieve another objective, separating the two components of residual strain

due to thermal contraction and matrix chemical shrinkage, the fringe patterns at every

temperature setting were analyzed. By gradually bringing the specimens back to the

stress-free temperature, the residual strain due to the thermal contraction was

compensated by thermal expansion. The amount of strain remaining on the specimens

would be solely due to the chemical shrinkage of matrix resin. Since the null fields were

tuned with respect to specimen gratings at room temperature, the strain values obtained

from the elevated temperature fringe patterns need to be subtracted from the total strains.

With the data points at different temperatures, and the assumption of constant coefficient

of thermal expansion (CTE), the CTE’s of composite laminate are the slope of strain

47

versus temperature curves. Figures 19 through 22 are the strain-temperature curves for all

four kinds of laminate specimens. The longitudinal fiber directions (0-degree direction)

are aligned with x- direction.

Fig. 19 The strain-temperature curves of [1016] unidirectional laminated composite

48

Fig. 20 The strain-temperature curves of [02/902]2s cross-ply laminated composite

Fig. 21 The strain-temperature curves of [0/903]2s unbalanced cross-ply laminatedcomposite

49

Fig. 22 The strain-temperature curves of [02/452]2s angle-ply laminated composite

Significance of the Tests

As seen in Fig. 19 through 22, at the cure temperature, those residual strain

components in the transverse direction (fiber dominated direction) did not come back to

zero. The difference is about 22.3% of the total residual strain for unidirectional

specimens, 3.2% for balanced cross ply specimens, 12.1% for the unbalanced cross ply

specimens, and 21.4% for the angle ply specimens. Table 1 summarizes the CTE’s and

the portion of residual strain due to matrix chemical shrinkage of the four different

composite specimens in both longitudinal and transverse fiber directions.

50

% of residual strain dueto chemical shrinkage

(%)

---

22.3

2.0

5.3

4.5

12.1

3.2

21.4

Residual strain duematrix chemicalshrinkage (µε)

---

1083.1

8.4

20.5

11.1

118.2

6.7

646.8

Residual strain due tothermal contraction (µε)

47.7

3768.4

406.6

365.0

232.5

857.8

203.3

2371.6

CTE(µε/°C)

0.31

24.47

2.64

2.37

1.51

5.57

1.32

15.40

Average totalresidual strain @

RT (µε)

36.1

4851.5

415.0

385.5

243.6

976.0

210.0

3018.4

Directions

longitudinal

transverse

longitudinal

transverse

longitudinal

transverse

longitudinal

transverse

Table 1 Components of residual strain in different lay-ups of laminate specimens

[0]16

[02/902]2s

[0/903]2s

[02/452]2s

51

CHAPTER 3POST-GEL CHEMICAL SHRINKAGE OF THERMOSET RESINS

Introduction

In Chapter Two, the results from the high temperature moiré tests indicate the

importance of characterizing the matrix chemical shrinkage of the polymer composite. It

is necessary to develop a more comprehensive and in-depth investigation of the curing

shrinkage. To measure the curing shrinkage of the matrix resin is the goal in this chapter.

For thermoset polymers, the crosslinking of the polymer chains result in chemical

shrinkage during the curing process. As the curing process progresses, the polymer

chains continue crosslinking until the asymptotic value of crosslinking density is reached

at the particular ambient temperature and pressure condition [55]. However, only the

shrinkage occurring after the solidification will contribute to the residual stresses in

composite materials. Although the post-gel shrinkage is only a small portion of total

curing shrinkage, we have proven the influence of post-gel shrinkage cannot be ignored.

Before gelation, the polymer resin is in its fluid state and is not capable of carrying

mechanical stress. After the gel point, the resin begins to solidify and stresses build up

between the fibers and matrix. Therefore, the shrinkage that occurs after the gel point is

more meaningful to the residual stress characterization. This chapter describes a new

technique, which focuses on measuring only the post-gel chemical shrinkage of thermoset

resins, and was developed particularly for the purpose of residual stress characterization.

52

Experimental Methodology

In these experiments, shrinkage measurement was taken using moiré

interferometry [48]. As mentioned in Chapter 2, one big advantage of moiré

interferometry is its long-term capability of strain measurement due to the ease of

preserving its reference condition. In this chapter, the advantage is exploited again to

record the deformation caused by resin chemical shrinkage during the curing process. An

investigation on the distortion of master grating was also documented to further support

the cure referencing method (CRM) developed previously.

The interferometer and optical setup used in the Chapter 2 was used again for

these shrinkage measurements, but this time, neat resin specimens were produced instead

of the composite laminate specimens. To demonstrate the usage of this newly developed

technique, five different resins (one room temperature cure epoxy, three ultra-violet light

cure arcrylates, and one high temperature cure epoxy) were used to produce neat resin

specimens. The names of the resins and their vendors are given in Table 2 below. The

high temperature cure epoxy was chosen to be the same epoxy of the composite system

used for the CRM.

Specimen Preparation and Experimental Procedure

The neat resin specimen gratings were directly replicated from master gratings

during the manufacture process of the neat resin specimens. In the process, the specimen

gratings solidify at the gel point of the resins. As the curing process continues, chemical

shrinkage and thermal effects cause stresses within the specimens. When the specimen is

53

Table 2 Names and vendors of the resins used

Roomtemperature cure UV cure1 High temperature

cure

Resins PC 10-CSR 3482, SR 205

2, Nadic methyl

anhydride3, camphorquinone4, and N,N-Dimethyl-p-toluidine4

3501-6

VendorMeasurementsGroup, Inc.

Sartormer Company, Inc. and Sigma-Aldrich Corporation

HexcelCorporation

1 Three different compositions were used for mixing the UV cure resins. The weight ratios are: (1)SR348:SR205 = 60:40, (2) SR348:SR205:Nadic methyl anhydride = 54:36:10, and (3)SR348:SR205:Nadic methyl anhydride = 48:32:20. Camphorquinone or N,N-Dimethyl-p-toluidine waslight sensitizing chemicals.2 SR348, ethoxylated bis-phenol A dimetharcrylate esters (Bis-MEPP), and SR205, triethylene glycoldimethacrylate from Sartormer Company, Inc.3 Nadic methyl anhydride from Sigma-Aldrich Corporation.4 Camphorquinone and N,N-Dimethyl-p-toluidine from Sigma-Aldrich Corporation.

separated from the master grating, the specimen undergoes deformation by the internal

stresses. From the specimen grating deformation we can measure the specimen

deformation from the gel point of the specimen to any time after grating separation

regardless the loading history of the specimens. This is the same principle as explored by

the CRM.

A section of an aluminum rod (60mm H x 25.4mm diameter) were put into a

section of an aluminum pipe (60mm H x 42.1mm OD, 37.5mm ID) and placed on a piece

of glass covered with release film to create a hollow cylindrical cavity for making neat

resin specimen molds. Two-part GE RTV627 silicone rubber compounds were mixed

and poured into the cavity to make hollow cylindrical specimen molds (Fig. 23).

The specimen grating replication takes place during the specimen curing process.

The specimen molds, with 25.4 mm inner diameter, and 6.4 mm (for room temperature

and high temperature epoxy) or 3.2 mm (for UV resins) height, were placed on top of the

54

Fig. 23 Apparatus to produce silicone rubber specimen molds: an aluminum pipe,an aluminum rod, a piece of release-film-covered glass, and a specimenmold

master gratings (as shown in Fig. 24). After degassing in a vacuum jar or a vacuum oven

(for high temperature cure epoxy), the resins were poured into the reservoir formed by the

master grating and silicone rubber molds (also preheated for high temperature epoxy).

The smooth and seamless contact between the silicone rubber molds and master gratings

prevented the resin from leaking. After curing, the neat resin specimens with diffraction

grating directly engraved on the surface can be separated easily from specimen molds

because of the weak adhesion of silicone rubber.

Master Grating Selection

For the first trial, GE RTV615 silicone rubber was used to make master gratings

because of its easy handling, transparency, and large working temperature range.

However, the experimental results showed that silicone rubber gratings are too compliant;

consequently, the master gratings were distorted by the shrinkage of the specimens during

the curing process. Figure 25 shows the distorted fringe patterns of a PC10-C specimen

55

Fig. 24 The original procedure for neat resin specimen replication

56

before it was separated from the master silicone rubber grating. The images were taken

from the back of the master grating. We can clearly see that the specimen area (inside the

circle) produced a lot of fringes due to distortion compared to the master grating area

(outside the circle) which is essentially a null field. Similar distortion occurred even

when the master gratings were made of a commonly used Envirotex Lite epoxy [48]

(from ETI Environmental Technology Inc.).

U field V field

Fig. 25 Distorted fringe patterns of a PC10-C specimen on the silicone rubbermaster grating

Much stiffer PC10-C epoxy was then chosen to make master gratings for room

temperature curing and UV curing resins, and 3501-6 epoxy from Hexcel Corporation

was chosen to make the high-temperature master gratings. Figure 26 shows the fringe

patterns before a PC10-C specimen was separated from the PC10-C master grating. The

pictures were also taken from the back of PC10-C master grating. The optical noise is

due to the multiple reflection through the glass substrate. In-plane rigid body rotation

was applied to the specimen in order to see the null field better. From the pictures with

rotation carrier fringes, the fringe density is very uniform both inside and outside the

57

circle. This indicates no distortion of the master gratings. The high-temperature master

gratings were specially made on ultra low expansion (ULE, Corning code 7971, tooling

grade) glass substrate to avoid error caused by thermal deformation of regular glass.

Corning ULE is more expensive than Astrosital used in the CRM, but has much better

optical properties allowing us to tune the interferometer through the substrate.

(a) U field (b) V field

(c) U field + rotation (d) V field + rotation

Fig. 26 Fringe patterns of a PC10-C specimen on the PC10-C master grating beforeseparation

58

Refining of Replication Procedure

Great caution was taken to avoid out-of-plane deformation of specimen gratings,

which can introduce considerable errors on the in-plane measurements. The keys to

reducing the out-of-plane deformation are having similar boundary conditions for both

the tops and bottoms of specimens and separating the specimens from the top and bottom

gratings/substrates at the same time.

After considerable trial and error, a standard procedure for replicating neat resin

specimen grating was developed to achieve the conditions stated above. The PC-10C

specimen gratings were made with two aluminized PC-10C master gratings on both the

tops and bottoms. For UV cure resins, the tops of the specimens were covered by a thin

piece of glass with a thin layer of GE RTV615 uncured silicone rubber compound in

between instead of an aluminized grating. This enables the UV light source to reach the

resin while allowing easy separation. For the high temperature cure specimen gratings

ULE glass substrates with two layers of aluminum were used on the top of specimens.

This also guaranteed the separation of the top covers from the epoxy specimens. Figure

27 illustrates the finalized procedure for specimen grating replication.

In-plane and Out-of-plane Deformation Measurement

Once the specimens were separated from their master gratings, they were

positioned on a grating alignment tool as described in the reference [48]. Two crossed-

line grating directions were determined and marked by razor blades. Then, the specimen

gratings were positioned in front of a moiré interferometer, which was tuned with the

master grating for the individual specimen. Both the U-field and V-field moiré fringe

59

patterns were captured using Polaroid 9cm × 12cm instant films Polapan 52 or 57. The

average of in-plane linear shrinkage can be obtained by analyzing the fringe patterns with

the equations (2.2) and (2.3).

(a) room temperature cure specimen

(b) UV cure specimen

(c) high temperature cure specimen

Fig. 27 Illustrations for making (a) room temperature cure, (b) UV cure, and (c)high temperature cure specimen gratings

60

For 3501-6 epoxy specimens, the strain values obtained at room temperature are in

fact the combination of their chemical shrinkage and thermal contraction. Therefore, in

order to extract the pure chemical shrinkage from the total strain values, the specimens

were placed into a pre-heated optical thermal chamber, which allowed moiré tests to be

performed at high/low temperature. The optical thermal chamber is the same chamber we

used for the CRM in Chapter 2. The temperature in the chamber was first set ten degrees

higher (T’) than the temperature that the fringe patterns were going to be taken (T) so that

after the specimens were put into the chamber, the designated temperature (T) can be

quickly reach. Fringe patterns were taken five minutes after the temperature reached the

setting (T) to avoid further corsslinking of the neat resin specimens. The specimens were

removed from the chamber as soon as the pictures were taken for the same reason. Two

elevated temperatures not near the glass transition temperature (Tg ≈ 195 °C in this case)

[56] of 3501-6 epoxy were chosen for photography. From these tests, the coefficient of

thermal expansion (CTE) of pure 3501-6 epoxy can be obtained. The curing shrinkage of

the epoxy was calculated by extrapolating the strain-temperature curves to the specimen

curing temperature.

For the verification of out-of-plane deformation, Fizeau interferometry was used.

Fizeau interferometry is a two-beam interferometry system. The biggest advantage of

Fizeau interferometer is the simplicity of the experimental setup and usage. An identical

He-Ne laser, optical fiber, laser coupler, and parabolic mirror as specified in Chapter 2

was used for the collimated light source. Figure 28 is the schematic diagram of a Fizeau

interferometer and Fig. 29 (a) and (b) are photographs of the experimental setup. An

optical-flat 10-degree wedge was used in the optical system to produce a reference

61

optically flat wave front. If the specimen has out-of-plane deformation, the wave front of

the reflected light from the specimen will become warped. When this warped wave front

interferes with the flat wave front reflected from one surface of the wedge, constructive

and destructive interference will occur. This forms a fringe pattern that represents the

topography of the specimen surface. The relative out-of-plane displacement can be

calculated by the equation (3.1) [48]

θλ

cos2

),(),(

⋅⋅= yxN

yxw (3.1)

where w(x,y) is the out-of-plane displacement, λ is the wavelength of the light source,

N(x,y) is the fringe order at a particular point (x,y), and θ is half of the angle between

incident light beam and reflected light beam.

collimated light source

2θ

specimen

optical-flatwedge

two waves interfere toproduce fringe patterns

reflection from wedge’sfront surface (will notgo into camera)

to lens andcamera

Fig. 28 Schematic diagram of a Fizeau interferometer

62

camera

laser

parabolic mirror

shutter controller

optical-flat wedge ( behind)specimen in

(a)

(b)

Fig. 29 Experimental setup for out-of-plane deformation measurement (Fizeauinterferometer). (a) Overview; (b) Close-up look of Fizeau interferometerand a neat resin specimen

63

Results and Discussion

At least four specimen gratings were produced for each kind of resin. Figure 30 is

a picture of a 3501-6 epoxy specimen grating and its ULE master grating after separation.

As seen in the picture, the vacuum-deposited aluminum films on the master grating and

the cover glass are now transferred to the specimen. This makes separation easier and

also improves the diffraction efficiency of the specimens. Among three different

thermoset resins, the high temperature cure, 3501-6 epoxy had the largest chemical curing

shrinkage upon separation (Table 3). One of the reasons is due to the high cross-link

density between polymer chains of high temperature cure epoxy.

Fig. 30 A high temperature specimen grating after being separated from its mastergrating

64

COV (%)

9.26

10.60

Average

-341

-644

Side B

-306

-645

Specimen #4

Side A

-295

-568

Side B

-339

-653

Specimen #3

Side A

-333

-680

Side B

-355

-750

Specimen #2

Side A

-381

-711

Side B

-366

-559

Specimen #1

Side A

-379

-593

PC10-C epoxy

upon separation

fully relaxed

COV (%)

4.27

12.87

5.70

7.03

0.98

3.95

Average

-1178

-2670

-1313

-3169

-1324

-3071

Specimen #4

-1244

-3031

-1222

-3057

-1319

-3200

Specimen #3

-1161

-2203

-1353

-3003

-1312

-2999

Specimen #2

-1124

-2730

-1391

-3121

-1324

-3144

Specimen #1

-1183

-2719

-1284

-3495

-1340

-2940

upon separation

fully relaxed

upon separation

fully relaxed

upon separation

fully relaxed

UV resins

48:32:20

54:36:10

60:40

COV (%)

11.06

13.21

Average

-1433

-1053

Specimen #4

-1239

-856

Specimen #3

-1536

-1114

Specimen #2

-1372

-1177

Specimen #1

-1586

-1065

3501-6 epoxy

upon separation

fully relaxed

Table 3 Chemical curing shrinkage for five different resins

(Unit is in µε)

65

When we separated the specimens from the grating molds, there was an

immediate contraction. As time past (2~3 months), the rate of contraction reduced to

near zero. Then we repeated the measurement again, but the results somehow surprised

us. Room temperature and UV cured specimens exhibited significant creeping as

expected. On the other hand, for 3501-6 epoxy specimens, the strain values remain about

the same or even decrease (specimens expanded) (also see Fig. 31). The reason why the

high temperature cure epoxy acts differently from the other kinds of resin is not fully

understood at this point. One possible reason is the hygro effect due to humidity, which

has the opposite effect of chemical shrinkage on specimens. Although we store all the

specimens inside a dry box, the relative humidity is at least 22% at all times. When we

open the box to access the specimens or when the specimens were outside the dry box for

measurement, the humidity is even higher. Room temperature cure and UV cure resins

should exhibit a similar effect, but were probably compensated by the large amount of

creep deformation. After the in-plane strains were measured, a Fizeau interferometer

was used to check the out-of-plane deformation. The fringe patterns showed that the out-

of-plane deformation was on the order of 20 wavelengths over the entire specimen

surface (25.4mm diameter), but only 1 to 3 wavelengths over the gage length (9.2mm

square). Therefore, the effect of the out-of-plane deformation on the in-plane curing

shrinkage is considered negligible. Figure 32 through 34 shows some typical fringe

patterns for each different kind of resin.

66

Fig. 31 Typical shrinkage vs. time curves for each kind of specimens

67

front, U field back, U field

front, V field back, V field

front, W field back, W field

Fig. 32 In-plane chemical shrinkage and out-of-plane deformation fringe patternsfor a PC10-C epoxy specimen.

68

U field V field

W field

Fig. 33 In-plane chemical shrinkage and out-of-plane deformation fringe patternsfor a UV cure resins (SR 448:SR 205=60:40).

69

U field V field

W field

(a) at room temperature

Fig. 34 In-plane chemical shrinkage and out-of-plane deformation fringe patterns of3501-6 epoxy at room temperature and 140 °C.

70

U field V field

(b) at 140 °C

Fig. 34-continued

In order to validate the methodology for the high temperature cure epoxy, the

following analysis was performed. From previous work [51], the strain-temperature

curves (in the transverse fiber direction) of 16-ply AS4/3501-6 unidirectional laminates

were measured as shown in Fig. 35. The strain-temperature curves of our current 3501-6

epoxy specimens are shown in Fig. 36. According to Agarwal and Broutman [3], using

the rule of mixture, the transverse CTE of a unidirectional laminate (αT), with fiber

volume fractions greater than 25%, can be represented by

mmmffT VV ⋅⋅++⋅= αναα )1( (3.2)

where αf and αm are the CTE of fiber and matrix material, Vf, and Vm are the fiber and

matrix volume fractions, and νm is the Poisson ratio of the matrix material. Using the

71

values αf ≈ 0, Vf = 0.65, and the νm = 0.36 from the manufacturer, and substituting average

αm = 52.4×10-6 m/m/°C from Fig. 36, we can obtain

6109.24 −×=Tα m/m/°C.

This value has excellent agreement with the average αT value, 25.0 × 10-6 m/m/°C, which

is calculated from Fig. 35.

Conclusion

A simple and effective technique was developed to measure the post-gel chemical

curing shrinkage of polymers, which plays an important role in characterizing the residual

stresses of polymer matrix composite materials. The curing shrinkage can be accurately

measured without knowing the exact gel point during the curing process. However, care

must be taken to avoid out-of-plane deformation of the specimen which will cause errors

in the in-plane shrinkage measurement. It has also been determined that depending on the

resin system, stress relaxation would dramatically influence the results of measurement

both at the room temperature and elevated temperature.

72

Fig. 35 Residual strain vs. temperature of AS4/3501-6 [0]16 laminate in transversedirection.

Fig. 36 Apparent strain of 3501-6 neat resin specimens vs. temperature

73

CHAPTER 4VISCOELASTIC CONSTITUTIVE MODELING FOR POLYMER MATRIX

LAMINATED COMPOSITES

Introduction

In the previous work [26, 51], we have been able to demonstrate the CRM to be

an effective tool for measuring residual strain caused by the manufacturing process of

polymer matrix composite laminates. In order to calculate the residual stresses from the

residual strains, a constitutive model of the laminate specimens is required. The authors

have used a simple model—linear classical lamination theory, to deduce the residual

stresses. For an orthotropic composite laminate plate, the stress-strain relation can be

represented and further simplified by Hooke’s law and classical lamination theory.

Equation 4.1or 4.2 describes this relationship for a special orthotropic and transverse

isotropic plate subjected to plane stress condition.

00

0

0

12

2

1

66

2212

1211

12

2

1

⋅

=

εεε

σσσ

C

CC

CC

(4.1)

⋅

=

12

2

1

66

2212

1211

12

2

1

00

0

0

σσσ

εεε

S

SS

SS

(4.2)

here Cij are the elements of stiffness matrix, Sij are the elements of compliance matrix.

74

Also from the study conducted in Chapter 3, it is desirable to employ a more

sophisticated model such as a linear viscoelastic constitutive model to describe the stress

relaxation behavior of polymer matrix composites (PMC) when time dependency is

involved. It is our goal to obtain the constitutive equation that can take into the account

of viscoelastic effects and replace the material constants in [C] matrix or [S] matrix of

equations (4.1) and (4.2) with time dependent functions.

Standard Linear Solid Model and Correspondence Principle for Linear ViscoelasticMaterials

To describe the behavior of a viscoelastic body, linear spring and linear dashpot

elements are usually used to construct the constitutive models [33, 57]. The Kelvin-Voigt

model (single spring-dashpot parallel arrangement) or Maxwell model (single spring-

dashpot series arrangement) alone is easy to formulate but is usually insufficient to

represent a material system. Standard linear solid model with the arrangement shown

below (Fig. 37) was the model we first chose for the AS4/3501-6 composite system.

k0

k1

Fig. 37 Spring-dashpot arrangement for standard linear solid model

The governing equations can be obtained from solving the following ordinary

differential equation:

01111

100 and , εεεεεµε −=⋅+⋅=⋅ kdt

dk (4.3)

75

where ε is the total strain of the assembly, k0, k1 are the spring constants of spring

elements, and µ1 is the viscosity of the dashpot element. Hence, when the whole

assembly is subjected to an instant constant load σ0 at isothermal condition, the solution

for the above equation becomes

)1()(1

0

0

0 τσσεt

ekk

t−−⋅+= (4.4)

here, τ = 1 / µ1 is the relaxation time. By finding the constants k0, k1 and µ1 in the

equation (4.4), the elements in the time-dependent creep compliance matrix can be

defined. In the current study, only orthotropic, transverse isotropic composite laminates

were considered. For this kind of material, four independent material constant are

required for constructing the creep compliance matrix, which are S11(t), S12(t), S22(t), and

S66(t). Uni-axial tensile tests were used for generating necessary data for analytical

modeling. Three different unidirectional tensile specimen configurations were chosen for

fabrication—[0]16, [90]16, and [10]16.

Time-Temperature Superposition Principle

When linear viscoelastic materials are subjected to a creep test or stress relaxation

test, a relationship between increasing ambient temperature and acceleration in time scale

exists. For thermorheologically simple viscoelastic materials, this relationship is even

more simplified. The long-term relaxation (or creep) behavior at a reference temperatures

can be constructed by shifting the short-term relaxation/creep curves at different

temperature parallel to the time axis (Fig. 38). This relationship to describe the

“speeding-up” behavior of the particular materials is called time-temperature

76

superposition (TTS) principle [50]. Equation (4.5a) or (4.5b) formulates this relationship

with a temperature-dependent shift factor aT.

)()( riTi TaT λλ ⋅= (4.5a)

)()( riTi TaT ρρ ⋅= (4.5b)

where λi = µi / ki is the relaxation time for ith Maxwell element, and ρi = µi / ki is the

retardation time for ith Kelvin-Voigt element in a multi-element model. In our standard

solid linear model, the subscript i vanishes, and only one λ and one ρ remain.

log(time) log(time)

log(

cree

p co

mpl

ianc

e)

log(

cree

p co

mpl

ianc

e)

T1

T2

T3

T4

aT2

aT3

aT4

Fig. 38 Construction of the long-term “master curve” at reference temperature fromshort-term testing data at different temperature.

Thermal Chamber Manufacture and Test Fixture Design

In order to utilize the time-temperature superposition principle, high temperature

tensile tests were designed to obtain the necessary data. The high temperature

environment was achieved by a newly designed and built thermal chamber. The regular

grips for tensile tests have large thermal mass; therefore, the thermal chamber was

designed to have just enough space to accommodate the active and dummy specimens,

77

necessary strain gage wiring, thermal couples, extension bars from grips, and two pairs of

small clamps. Two pinholes were drilled on each piece of the clamps. One is for the pin

connecting to the extension bar, and one is for the screw that goes through the pinholes

on the specimens and to fasten the clamps (Fig. 39).

Fig. 39 Thermal chamber and test fixture for uniaxial tensile tests

The chamber has folded aluminum sheet interior, 3/4” (19.1 mm) thick inner wall

made of high temperature calcium silicate insulation board (McMaster-Carr Supply

Company, max. temp. 927 °C, thermal conductivity 0.8 Btu @ 427 °C), and half inch

(12.7 mm) thick outer wall made of high strength calcium silicate board (McMaster-Carr

Supply Company, max. temp. 760 °C, thermal conductivity 1.16 Btu @ 427 °C). The

final dimensions of this chamber are 5.25” x 5.25” x 16” (133.4 mm x 133.4 mm x 406.4

78

mm) outside, and 2.5” x 2.5” x 13.5” (63.5 mm x 63.5 mm x 342.9 mm) inside. All the

pieces were bonded together with silicone sealant and connected to the EC12 oven used

in Chapter 2 by similar air ducts and the same circulation fan. This chamber was tested to

a have sustained working temperature of up to 210 °C.

Specimen Preparation and Testing Plan

Specimens were again prepared using the AS4/3501-6 unidirectional prepreg from

Hexcel Corporation. The aerial weight of the prepreg was 145, and the fiber volume

fraction was 65%. The prepreg was cut and stacked to sixteen plies. High temperature

strain gages (SK-06-250BA-500, Measurements Group, Inc.) were positioned at the

desirable location with Teflon sheet templates onto both sides of the prepreg panels (Fig.

40).

Fig. 40 Composite prepreg panel with Teflon release film template for applyingstrain gages. The template for making 10-degree specimens is shown in thispicture.

79

Gage and Accessory Selection

High temperature, high electrical resistance gages were chosen based on several

reasons. First, for the long-term experiments run at the elevated temperature, high

temperature resistance is essential. The SK series strain gages have a working

temperature range from 230 °C to -269 °C (450 °F to -452 °F). Matching solder (450-

20S-25) and wires (330-FTE) both from Measurement Group, Inc. were chosen to

accommodate the high temperature environment for our tests. Second, 500 ohm

resistance gages were selected to reduce the heat accumulation, since graphite/epoxy

composite does not have good thermal conductivity. Finally, for the high temperature

testing environment, high temperature adhesive was necessary to ensure proper load

transfer from the test coupons to the strain gages. High temperature adhesive usually

requires high temperature cure. However, in order to reduce the alternation of specimen

properties caused from the thermal history of strain gage adhesive curing, the best way to

attach the strain gages was to apply strain gages during the composite specimen

manufacturing process. Because the gages were attached to the composite laminates

during the curing process, there was no need to go through another thermal cycle.

For 0-degree and 90-degree specimens, a total of four gages were applied on the

surfaces (0° and 90° on both front and back surfaces) to obtain the normal strain and the

Poisson ratio. For 10-degree specimens, a total of six gages were applied on the surfaces

(0°, 90°, and 45° on both front and back surfaces) in order to obtain the shear strain and

shear modulus.

80

Specimen Preparation

The composite laminate lay-up was the same for the three different specimen

configurations (16 plies, unidirectional), but the specimens were cut along three different

angles to produce the three specimen types. After the prepreg lay-up and strain gage

application was completed, the same vacuum bag lay-up and the same autoclave curing

profile as stated in Chapter 2 for the CRM was used to cure the composite laminates.

After the autoclave curing process, the individual strain gages were inspected and tested

to ensure proper working condition. The composite panels were then cut using a low

speed diamond saw (Isomet, Bulter, Inc.) to prevent significant edge effect and

subsequent errors due to machining according to ASTM standard 3039/3039M [58].

Water was added as the coolant for cutting. Composite panels were mounted on a

platform which could slide down along a track parallel to the cutting blade. The

specimens were trimmed into the final dimensions of 1” x 10” (25.4 mm x 254 mm) for

tensile testing.

To prevent early failure of the unidirectional tensile specimens, end tabs were

bonded to the ends of the specimens. The end tabs were made of same composite

material system (AS4/3501-6) with a [02/902]2s cross-ply configuration with a dimension

of 1” x 1.5” (25.4 mm x 38.1 mm) without beveling. Shell Epon 828 resin and curing

agent 9552 were used as the bonding adhesive. In the trial tests the epoxy was able to

maintain adequate strength at the highest temperature setting, 195 °C, without end tab

failure.

After the epoxy for the end tabs was fully cured, holes for pins were drilled at the

center of end tabs through the specimens. A 3/8” (9.5 mm) outer diameter diamond core

81

drill bit (part no. 2868A25, McMaster-Carr Supply Company), and the matching adapter

were used to drill the pinholes. A slow drilling rate was used to prevent over heating of

specimens and tools. This also reduced the chance of delamination between end tabs and

specimens during drilling.

After drilling the holes, specimens were ready for strain gage wiring. Another

identical specimen was used as a dummy specimen in a half Wheatstone bridge

configuration since there is no self-temperature-compensation gage available for the

graphite/epoxy composite material system. Although it was more difficult than room

temperature application, high temperature solder and wire was required for our tests.

Strain gages were connected to a strain gage conditioner (2100 System, Measurements

Group, Inc.), where the Wheatstone bridge circuit was completed, excitation voltage was

supplied, and the output signals were amplified. The excitation voltage was 3.5 volts to

prevent local overheating. The gain setting varied depending on the gage orientation with

respect to the specimen fiber direction and the maximum strain values. After being

amplified by the strain gage conditioner, the output signals were input into the controller

of the testing machine for data acquisition.

Experimental Procedure and Testing Plan

The specimens and testing fixtures were assembled, and the thermal chamber and

EC12 oven were installed on a MTS servo hydraulic testing machine. The test machine

has both axial and torsional loading capability, and was controlled by a digital TestStar II

controller. Testing profiles and data acquisition were programmed, executed, and

recorded on a Compaq Deskpro 6000 computer. Figure 41(a) through (c) show the

experiment setup from several different viewpoints.

82

(a) MTS testing machine and EC12 oven

(b) TestStar II digital controller, strain gage conditioner, and the work station

Fig. 41 Experiment setup from several viewpoints

83

(c) thermal chamber, test fixture assembly, and strain gage conditioner

Fig. 41--continued

After several trial tests, load control creep tests were chosen to obtain time

dependent material properties. The thermal chamber, active and dummy specimens were

heated from room temperature to the desired temperature setting at the rate of 2.8 °C per

minute (5 °F /min.). The temperature inside the chamber was controlled by the controller

of EC12 oven with an extended thermal couple probe. Another thermal couple was

extended from the chamber to a thermocouple-to-analog converter (TAC80B-K, Omega

Engineering, Inc.), and connected to TestStar II testing machine controller as an input

signal for monitoring and recording the temperature during tests. These two

thermocouples (both CO1-K foil-terminal, Omega Engineering, Inc.) were stacked

together and directly attached to the surface of specimens with high temperature tape.

Test loading profiles are shown in Fig. 42. The loading rate was 2 lbf per second for all

three kinds of specimens. Once the load reached the final load level, the specimens were

held at that load value for 60 minutes, and then unloaded at the same loading rate. For 0-

84

degree specimens, final load was 700 lbf. For 90-degree specimens, 400 lbf. (about 80%

of the strength) was used for room temperature, 40 °C, 60 °C, and 80 °C settings, 250 lbf.

(about 50% of the strength) for 100 °C, 120 °C, 140 °C, 160 °C, and 177 °C settings, 50

lbf. (about 10% of the strength) for 195 °C setting in order to obtain enough strain

response. For 10-degree specimens, 150 lbf. final load was used. After unloading, the

specimens were held at 0 lbf. load for 30 minutes (0-degree) or 60 minutes (10- and 90-

degree) for recovery. During loading and unloading, the data sampling rate was 1 Hz,

and was 0.2 Hz for holding and recovery. At least four tests were continuously performed

in sequence for each temperature settings, and one specimen was repeatedly used for the

same temperature setting.

load

load

l oad

ing

l oad

ing

un loa di ng

u nlo ad in g

holding

holding

recovering recovering

time

(a) (b)time

60 min.

60 min. 60 min.30 min.

Fig. 42 Testing profiles for (a) 0-degree specimens, (b) 10-degree and 90-degreespecimens

Results and Data Analysis

General Behaviors of Materials

This section will describe the general behavior of our specimens during the creep

tests. For convenience, we will describe the testing procedure as loading, holding (creep),

unloading, and recovery phases.

85

Loading phase

For 0-degree specimens, the loading stress-strain curves look perfectly linear

(elastic), since this is the fiber-dominated direction. Also the curves did not show

noticeable temperature dependency. At the different temperatures, when the specimens

were subject to the same amount of load, they all had about the same value of strain (see

Fig. 43).

From Fig. 44 we can see that the stress-strain curves for the 90-degree specimens

deviated from the room temperature straight line as the temperature increased. There is

an obvious change in the elastic moduli in the transverse fiber direction (axial loading

direction) (from 1.54 Msi at room temperature to 0.93 Msi at 195 °C), since this is matrix

dominated direction.

For the 10-degree specimens, the axial and transverse strains were obtained

directly from readings of the strain gages. The shear strain values were calculated real-

time by the TestStar II controller during the tests with the following equation derived

from the strain gage rosette relationship [59].

°°° ⋅−⋅+⋅−= 9045012 598.0879.1282.1 εεεγ (4.6)

here γ12 is the shear strain in the 10-degree plane (fiber direction), ε0° is the read-out of

the strain gage along the loading direction, ε90°, ε45° are the read-outs of the strain gages in

the 90° and 45° loading direction respectively. Considering the 10-degree specimens

(Fig. 45), the specimens were still within the elastic range, but no matter axial or

transverse normal strains, or shear strain in 10-degree direction, they all exhibited

temperature dependence.

86

Holding (creep) phase

After the loading stages, the specimens were held at constant force for 60 minutes.

The longitudinal strain values for the 0-degree specimens remained essentially the same,

since these are fiber-dominated and we can assume that graphite fibers are elastic within

the testing temperature range (see Fig. 46).

For the 90-degree specimens (Fig. 47), even at room temperature, the longitudinal

strain had a 1% of increase (from 3370 µε to 3404 µε), and at 177 °C the increase was

24.5% (from 2823 µε to 3516 µε). At 100 °C, one specimen failed during the hold at the

final load of 400 lbf. Also another failed during loading at 195 °C when the load

exceeded 230 lbf.

Similar behaviors were observed for 10-degree specimens. There is a 2.6%

increase in shear strain value in 10-degree specimen at room temperature over the holding

period, and 7.0% at 177 °C (Fig. 48).

87

Fig. 43 Stress-strain curves for 0-degree specimens at different temperature settingsduring loading phase. From Fig. 43 through 45, all strain values are theaverage of front and back strain gage readings.

88

Fig. 44 Stress-strain curves for 90-degree specimens at different temperaturesettings during loading phase.

89

Fig. 45 Stress-strain curves for 10-degree specimens at different temperaturesettings during loading phase.

90

Fig. 45-continued

Fig. 46 Strain-time curves for 0-degree specimens at different temperature settingsduring holding phase.

91

Fig. 47 Strain-time curves for 90-degree specimens at different temperature settingsduring holding phase.

Fig. 48 Shear strain-time curves for 10-degree specimens at different temperaturesettings during holding phase.

92

Unloading and recovery phases

The unloading stress-strain curves remain linear and return to zero (the origin) for

the 0-degree specimens. For the 90-degree specimens, there was a considerable amount

of strain that remained when the stress returned to zero at every temperature setting.

Some of the strain gradually returned to zero during the one-hour recovery phase,

depending on the load level and temperature. The 10-degree specimens showed a similar

behavior. At the higher temperature, there was a large amount of shear strain remained

when the specimens were unloaded to zero. During the recovery phase, the shear strain

values increased (negative during the hold sequence) and even passed zero to positive.

Since shear strains were calculated from longitudinal and transverse strains, the drift in

the strain gage readings were enlarged due to the formula which we used to calculated

shear strains.

Curves Fitting for the Creep Tests

After tests, the cross-sectional area of each specimen was measured again to

ensure the change in cross-sectional area would not produce a noticeable increasing of

stress. The raw data for the holding phase of each test was extracted and imported into

MathSoft MATHCAD 8 program using its “READPRN” function, where all the curve

fitting and analysis were performed. It was necessary to make correction for the strain

values of 90-degree specimens due to the transverse sensitivity of strain gages and highly

biased strain field. The following equations were employed for making this correction

[60].

93

)K(K1

K1

)K(K1

K1

xxtyy2t

t0yy

yytxx2t

t0xx

εενε

εενε

′⋅−′⋅−

⋅−=

′⋅−′⋅−

⋅−=(4.7)

Here, ε’xx and ε’yy are the apparent strains from strain gage reading in the longitudinal and

transverse directions respectively, Kt is the transverse-sensitive factor of gages, ν0 =

0.285 is the Poisson’s ratio of the material used for gage calibration, and εxx and εyy are

the true strains in two directions after correction. It was found that the errors associated

with the transverse sensitivity of the strain gages could be as large as 35% of the final

values of creep compliance S12(t).

As stated in the previous section, a three-element standard solid linear model was

selected initially to describe the material behavior in the matrix dominated orientations.

In the MATHCAD program, the function “genfit” allowed us to fit an arbitrary function

to the data. By solving the governing equation of the standard solid linear model, the

function for fitting the data was

[ ])tvexp(1v)t( 010 ⋅−−⋅+= εε (4.8)

here ε0 = ε(0) is the strain value when the holding stage started, v0 and v1 are the

constants which curve fitting was used to solve for. With the non-linear fitting function,

v0 and v1 are very sensitive to the initial guess. The values of initial guess were obtained

from the experimental curves as illustrated in Fig. 49. After several iterations, when the

difference between the final values and the initial values was less than 1%, the iteration

process was terminated. The MATHCAD file for the curve fitting is attached in

94

Appendix A. The curve fitting results and experimental data are shown in Fig. 50 for 90-

degree specimens and in Fig. 51 for 10-degree specimens.

time

stra

in

v1

v0

1

Fig. 49 Estimating the values of initial guess for curve fitting.

(a)

Fig. 50 Curve fitting and experimental data for the longitudinal strain of 90-degreespecimens during the holding phase at different temperatures. (a) 22 to 80°C; (b) 100 to 177 °C; (c) 195 °C.

95

(b)

(c)Fig. 50--continued

96

(a)

(b)

Fig. 51 Curve fitting and experimental data for the shear strain of 10-degreespecimens during the holding phase at different temperatures. (a) 22 to 100°C; (b) 120 to 195 °C.

As seen in Fig. 50 and Fig. 51, the fitting curves do not match the experimental

data very well especially when the testing temperature was high. In order to improve the

97

fitting, a dashpot element was added to the model as the fourth element which is in series

with the other three elements. The MATHCAD file for curve fitting with the fourth

element added is attached in Appendix B. By solving the governing equation, the fitting

function used for MATHCAD “genfit” function was

[ ] tv)tvexp(1v)t( 2010 ⋅+⋅−−⋅+= εε (4.9)

where ε0, v0 and v1 are defined the same as in equation (4.8), and v2 is proportional to the

reciprocal of the viscosity, µ2, of Newtonian fluid in the additional dashpot. The initial

guess for v2 values is also obtained from the plots of experimental data by estimating the

slope of the final portion of the strain versus time curves during the hold sequence (see

Fig. 52). A similar iteration process was also applied to obtain the final values of v0, v1,

and v2.

time

stra

in

v1

∆ε∆t

v0

1

v =2 ∆t∆ε

k0

k1

Fig. 52 Estimating the values of initial guess for curve fitting with the fourthelement.

98

From Fig. 53 and Fig. 54, we can clearly see that the curve fit with the four-

element model fits the experimental data better compared to the fit with the three-element

model. Additionally, the fit is good, even when the test temperatures were high.

For the 0-degree specimens, the strain response in longitudinal direction (fiber

direction) was considered as a linear function (first order polynomial) of time. In

MATHCAD, functions “regress” and “interp” are used to find this polynomial.

(a)

Fig. 53 Curve fitting with four element model and experimental data for thelongitudinal strains of 90-degree specimens during the holding phase atdifferent temperatures. (a) 22 to 80 °C; (b) 100 to 177 °C; (c) 195 °C.

99

(b)

(c)Fig. 53--continued

100

(a)

(b)

Fig. 54 Curve fitting with four element model and experimental data for the shearstrains of 10-degree specimens during the holding phase at differenttemperatures. (a) 22 to 100 °C; (b) 120 to 195 °C.

101

A similar curve fitting procedure was applied for determining the Poisson’s ratio

versus time functions, ν21(t) (from 90-degree specimens) and ν12(t) (from 0-degree

specimens) at different temperatures. After all of the strain-versus-time functions (ε (t))

were obtained, the creep compliance matrix elements S22(t), S66(t), and S11(t) were

calculated by dividing ε (t) by the applied stress, and S12(t), and S21(t) were calculated

from the equations

)()()()()()( 211112222112 tStSttSttS =⋅−=⋅−= νν (4.9)

The statement, S12(t) = S21(t), comes from our orthotropic and transverse isotropic

material property assumptions. Figure 55 trough 57 show creep compliance S22(t), S12(t),

and S66(t) on a logarithmic scale. The calculation for the creep compliance matrix

element S22(t), and later the construction of the master curves for these elements are also

included in the files in Appendix B (90-degree specimens). For 10-degree specimens and

0-degree specimens, the data analysis were performed in the similar manner.

Using a thermorheologically simply assumption, we now shift the creep

compliance curves, S22(t) and S66(t), at the different temperatures along the horizontal

axis (time axis) until the initial values of the higher temperature curves meet the values of

the previous temperature setting curves. By connecting all curves at the different

temperatures into a single curve, we construct the master curve of the creep compliance

elements over a long period of time at a specific reference temperature (room temperature

(22 °C) in our experiments and calculation). Figures 58 and 59 show the master curves

for creep compliance matrix elements, S22(t) and S66(t). In both cases, noticeable kinks

occur when the curves at two different temperatures intersect.

102

log (psi-1) Creep Compliance S12(t) vs. Time (curve fit)

log(Time) (log(sec))

Fig. 55 log (S22(t)) versus log (t) for AS4/3501-6 unidirectional composite laminateat different temperatures.




103




This indicates an inadequate assumption of the thermorheologically simple material

properties. Additionally, when time value becomes very large, the v2 term in the fitting

function also becomes very large, which makes the fitting curves unreasonable.

To solve this problem, physical aging of 3501-6 epoxy is taken into account, since

most of our tests were done at the temperature much below the Tg of 3501-6 epoxy

(except at 195 °C). Physical aging is the process of reduction in free volume of

amorphous polymers after they are quenched to below their Tg and slowly return to the

thermal equilibrium state [50, 57]. Some work has been performed [61-65] to investigate

the influence of this phenomenon on the viscoelastic behavior of polymers. Since the

mobility of polymer chains decrease due to less and less free space available, the elastic

moduli of polymers tend to increase in general. As a result, the creep compliance versus

104

log (psi-1) Creep Compliance S22(t) Master Curve vs. Time


Fig. 58 The tensile creep compliance S22(t) master curve for 3501-6/AS4unidirectional laminate specimens in transverse fiber direction. Referencetemperature is at 22 °C.

105



Fig. 59 The shear creep compliance S66(t) master curve for 3501-6/AS4unidirectional laminate specimens. Reference temperature is also at 22 °C.

time curves in the matrix-dominated direction need to be shifted along the vertical axis.

By doing so, the new master curves are shown in Fig. 60 and Fig. 61. They look much

smoother and more reasonable with the combination of horizontal and vertical shift. The

shift amount for each temperature setting is listed in Table 4.

To compare with the curve fitting results, the creep compliance matrix elements

were also calculated point by point directly from the experimental data. From Fig. 62, the

curve fitting functions of the creep compliance matrix elements agreed with the values of

creep compliance directly calculated from experimental data fairly well.

Finally, to validate the creep compliance master curves which were constructed

from momentary creep tests at different temperatures, another creep test was performed

106



Fig. 60 The tensile creep compliance S22(t) master curve for 3501-6/AS4unidirectional laminate specimens in transverse fiber direction withconsideration of physical aging. Reference temperature is now at 100 °C.

107



Fig. 61 The shear creep compliance S66(t) master curve for 3501-6/AS4unidirectional laminate specimens with consideration of physical aging.Reference temperature is also at 100 °C.

Table 4 Time-Temperature Superposition shift factors (in log units), and referencetemperature (Tr) is 100 °C.

Creep compliance T-Tr (°C) Horizontal shift Vertical shift

S22 0 0 0

20 1.22 -0.0074

40 2.53 -0.0185

60 3.55 -0.0417

77 4.23 -0.0836

S66 0 0 0

20 0.86 0.0042

40 1.57 0.0079

60 2.25 0.0250

77 3.12 0.0623

108

specimen. The testing temperature was chosen to be a common application temperature,

100 °C, for this composite system. The load holding duration was increased from one

hour to seven days and was held at 250 lbf. In the first 20 minutes, data points were taken

every 5 seconds. From the 21st minute to the first five hours, sampling time was changed

to once every 60 seconds. Finally, from the sixth hour to the end of test, data points were

taken every 20 minutes.

As shown in Fig. 62, in the first three days of the long-term test, the result has

moderate agreement with the master curve constructed from the short-term creep tests. A

vertical shift in the log(creep compliance) axis was necessary to compensate the different

age of specimens, which were cut from the same piece of composite plate but tested at

different times. From around the fourth day of the test, the strain values separated.

Therefore, the data after 259,200 seconds (three days) were truncated.

Model Formation

Based on the creep compliance master curves constructed for unidirectional

laminated composites, and corresponding principle of linear viscoelastic materials, we

now will formulate the stress-strain relationship to describe the viscoelastic behaviors of

the composite laminate. First the equation (4.2) becomes

⋅

=

)t(

)t(

)t(

)t(00

0)t()t(

0)t()t(

)t(

)t(

)t(

12

2

1

66

2212

1211

12

2

1

σσσ

εεε

S

SS

SS

(4.10)

109

Fig. 62 Comparison between directly calculated creep compliance S22(t) and curve-fitting S22(t). The curve represents long-termcreep test is also shown to compare with the master curve obtained from momentary creep tests (short-term creep).

110

All terms in the equation are replaced by time-dependent functions except for S11.

From the experimental data, we can assume that S11(t) becomes a time-independent

constant. Second, these functions, in the material 1-2 coordinate system, can be

transformed into an arbitrary x-y coordinate system (usually the loading directions) by the

transformation matrix [T] according to the following equations

[ ] [ ]

⋅⋅

⋅=

−

)t(

)t(

)t(

)t(00

0)t()t(

0)t()t(

)t(

)t(

)t(

xy

y

x

66

2212

12111

xy

y

x

σσσ

εεε

T

S

SS

SS

T (4.11a)

or

⋅

=

)t(

)t(

)t(

)t()t()t(

)t()t()t(

)t()t()t(

)t(

)t(

)t(

xy

y

x

662616

262212

161211

xy

y

x

σσσ

εεε

SSS

SSS

SSS

(4.11b)

where

[ ]

−−−=

θθθθθθθθθθ

θθθθ

22

22

22

sincoscossincossin

cossin2cossin

cossin2sincos

T

By substituting the residual stress values we calculated from the Cure Referencing

Method, the strain functions of time can be calculated. In the next chapter, the relaxation

of residual stresses will be examined, and Cure Referencing Method was used again to

validate the time-dependent viscoelastic model for AS4/3501-6 composite laminates.

111

CHAPTER 5DISCUSSION AND CONCLUSION

Application of Linear Viscoelastic Model for Residual Stress Relaxation

Overview

In the last few chapters, research has been conducted to measure and characterize

the process-induced residual stresses and strains of the laminated polymer matrix

composite system—AS4/3501-6. A method called the “Cure Referencing Method”

(CRM) was developed, tested, and validated [66] to measure the residual strain for the

composites. During the analysis for calculating residual stresses from residual strains,

linear classical laminate theory was used.

Another method was developed to characterize the chemical curing shrinkage of

the polymer matrix. The residual strain due to the manufacture process can then be

separated into two parts: residual strain due to the CTE mismatch and residual strain due

to matrix chemical curing shrinkage.

A linear viscoelastic model was also constructed to describe the long-term

behaviors of the specific composite system based on the short-term (momentary) creep

tests at several different temperatures and time-temperature superposition principle.

Physical aging of polymer matrix was also taken into account to obtain smoother master

curves for the elements of the creep compliance matrix.

112

In this section, additional experiments were performed to verify our empirical

creep compliance master curves and the viscoelastic stress-strain relation. These tests

were done by using moiré interferometry. The composite specimens were from the same

batch of specimens which were been used in Chapter Two. After the manufacturing

process, these specimens were used for the CRM to measure the residual stresses. Since

moiré interferometry has the capability for long-term evaluation for in-plane deformation

of an object, we have stored the specimen gratings and their master gratings for about two

years (702 days = 107.78 seconds) in order to perform the necessary tests.

Specimens were kept in a desiccant box and room temperature ambient

environment. The fluctuation of room temperature is neglected in this study. The effect of

humidity is not believed to be negligible, but in the current study we have tried to

eliminate the influence as much as possible with laboratory humidity control and

desiccant substance. Further investigation can be found elsewhere [67]. The Astrocital

master gratings were kept in a cabinet to prevent damage from dust. The substrate of the

master gratings were considered dimensionally stable (no deformation) during this time

interval.

Experiment Procedure and Results

The optical arrangement was the same as describe in Chapter Two (please refer to

Fig. 13 in Chapter Two). At room temperature, the moiré interferometer was first tuned

with the intermediate epoxy master gratings for each individual specimen. After the null

fields were attained, master gratings were replaced by corresponding composite

specimens. After adjusting the position of the composite specimens, fringe patterns were

photographed using Polaroid instant sheet film. Figure 63 through 66 are the photographs

113

of fringe patterns for four composite specimens with different lay-up configurations. All

pictures were taken at room temperature.

Fig. 63 Fringe patterns for [016]T unidirectional composite laminate panel. Time lagfrom manufacture date tlag= 702 days.

114

0° fiber direction

(a) -field, 22 °CU (b) -field, 22 °CV

[0 /90 ]2 2 2s

Fig. 64 Fringe patterns for [02/902]2s balanced cross-ply composite laminate panel.tlag= 702 days.

(a) -field, 22 °CU (b) -field, 22 °CV

0° fiber direction

[0 /90]3 2s

Fig. 65 Fringe patterns for [03/90]2s unbalanced cross-ply composite laminate panel.tlag= 702 days.

115

Fig. 66 Fringe patterns for [02/452]2s angle-ply composite laminate panel

Since the diffraction gratings were attached to composite specimens via epoxy

grating films during manufacturing process, there was a concern about the integrity of the

gratings and specimens. From those photographs we can see that after almost two years,

the diffraction gratings on composite specimens are still very high quality. Most of the

116

grating area shows a very uniform fringe pattern without abnormal distortion. Grating

delaminated was observed in very small area only and primarily around the edge.

With s similar analysis was perfromed before, the residual strain information

could be extracted from the fringe patterns. Table 5 summarizes and compares the

apparent strains of four different composite specimens when they were just made (initial

time, ti = October 8, 1997) and about two years later (final time, tf = September 10, 1999).

Table 5 Comparison of apparent residual strains between newly made compositespecimens and two-year old specimens.

[016]T [02/902]2s [03/90]2s [02/452]2s

Date 0° 90° 0° 90° 0° 90° 0° 90° shear

10/08/97 -31.2 -4650.6 -428.1 -393.7 -241.1 -992.5 -200.1 -3034.8 2854.3

09/10/99 0 -4662.5 -369.4 -369.4 -192.1 -804.4 -147.8 -3037.2 1653.2

(unit in µε)

With the creep compliance master curves and strain values at the time tf, we can

calculate residual stresses of the laminated composite for a particular lay-up configuration

at the time tf by the following equation:

difff12

f2

f1

f1

residualf12

f2

f1

)t(

)t(

)t(

]T[)]t(Q[]T[

)t(

)t(

)t(

⋅⋅⋅=

−

γεε

τσσ

(5.1)

where

117

−⋅−⋅−

−⋅−

−⋅

=

)t(

100

0)t()t()t(

)t(

)t()t()t(

)t(

0)t()t()t(

)t(

)t()t()t(

)t(

)]t(Q[

f66

f2

12f22f11

f11

f2

12f22f11

f12

f2

12f22f11

f12

f2

12f22f11

f22

f

S

SSS

S

SSS

SSSS

S

SSS

S

(5.2)

and

⋅−

=

−

uni

f2

f11

lam

fxy

fy

fx

difff12

f2

f1

0

)t(

)t(

]T[

2

)t()t(

)t(

)t(

)t(

)t(

εε

γεε

γεε

(5.3)

From the piecewise master curves in Chapter Four, we can connect them into a

continuous master curve in logarithmic scale. Figure 67 through 69 show the continuous

master curves together with the piecewise master curves.



Fig. 67 Continuous and piecewise S22 master curve in logarithm scale. In all thefollowing diagrams, piecewise master curves are completely covered bycontinuous master curves and reference temperature is 22 °C.

118



Fig. 68 Continuous and piecewise S66 master curve in logarithm scale.



Fig. 69 Continuous and piecewise S12 master curve in logarithm scale.

119

With the continuous master curves on logarithmic scale, we can obtain a single

curve for each creep compliance matrix element as a function of time (S22(t), S66(t), and

S12(t)) over a long period of time (see Fig. 70, Fig. 71, and Fig. 72).

psi-1 Creep Compliance S22(t) Master Curve vs. Time

Time (sec)

Fig. 70 S22(t) for time from 0 to 107.78 seconds (702 days).

120


Time (sec)



Time (sec)


121

Now we can substitute the values of S22(t), S66(t), S12(t) and strain values from our

laminate specimens at t = 107.78 seconds into equations (5.1), (5.2), and (5.3) to obtain the

residual stresses at the final time. Again, using MATHCAD 8, the results are shown

below in Table 6.

Table 6 Residual stresses calculated by linear viscoelastic model and laminatetheory for the same batch of composite specimens. Unit is in psi.

Balanced Cross-ply, [02/902]2s

σx σy γxy

t = 0 t = 107.78 t = 0 t = 107.78 t = 0 t = 107.78

0° plies -6.582⋅103 -5.981⋅103 6.446⋅103 6.554⋅103 0 0

90° plies 6.485⋅103 6.554⋅103 -5.866⋅103 -5.981⋅103 0 0

0° plies -6.582⋅103 -5.981⋅103 6.446⋅103 6.554⋅103 0 0

90° plies 6.485⋅103 6.554⋅103 -5.866⋅103 -5.981⋅103 0 0

Unbalanced Cross-ply, [03/90]2s

σx σy γxy

t = 0 t = 107.78 t = 0 t = 107.78 t = 0 t = 107.78

0° plies -2.868⋅103 -2.403⋅103 5.63⋅103 5.95⋅103 0 0

90° plies 6.641⋅103 5.95⋅103 -1.851⋅104 -1.515⋅104 0 0

0° plies -2.868⋅103 -2.403⋅103 5.63⋅103 5.95⋅103 0 0

90° plies 6.641⋅103 5.95⋅103 -1.851⋅104 -1.515⋅104 0 0

Angle-ply, [02/452]2s

σx σy γxy

t = 0 t = 107.78 t = 0 t = 107.78 t = 0 t = 107.78

0° plies -2.886⋅103 -2.433⋅103 2.454⋅103 2.478⋅103 9.138⋅103 5.293⋅103

45° plies 8.956⋅103 3.188⋅103 -9.195⋅103 -1.531⋅104 -2.562⋅103 -9.237⋅103

0° plies -2.886⋅103 -2.433⋅103 2.454⋅103 2.478⋅103 9.138⋅103 5.293⋅103

45° plies 8.956⋅103 3.188⋅103 -9.195⋅103 -1.531⋅104 -2.562⋅103 -9.237⋅103

122

The residual stress calculation show considerable difference between initial time

and final time. In matrix dominated directions (90° plies and shear direction for angle-ply

specimens) the difference is even larger. This result is somehow surprising. Although

3501-6 epoxy is brittle and considered exhibiting more elastic behavior rather than

viscoelastic behavior, we can still clearly see that its viscoelastic properties have a large

influence on the process induced residual stresses.

Concluding Remarks

In conclusion, throughout this study, we have been trying to measure and

understand the process induced residual stresses for advanced laminated composites.

Numerous researchers have indicated that the residual stresses are innately difficult to

measure and characterize. With the newly developed methodology, Cure Referencing

Method (CRM), we were able to demonstrate the full advantages of high sensitive moiré

interferometry.

The most important feature of moiré interferometry for our application is its long-

term capability. In this study, we showed that the CRM can not only effectively measure

the residual stresses at the time when the composite laminates were manufactured, but

also well after.

Our study also proved that the chemical curing shrinkage of polymer matrix

contributes to a large portion of residual strains. With the aid of an optical thermal

chamber, the CRM enabled us to separate the residual stresses into two components:

stresses resulting from the CTE mismatch and stresses resulting from matrix chemical

shrinkage. Another method allowed us to accurately measure the post-cure

123

polymerization shrinkage of neat resins. The way by which the diffraction gratings were

transferred from master gratings to neat resin specimens eliminates the need of knowing

the gel point in polymerization process. Although the chemical shrinkage does not

completely relax during manufacturing procedure, the viscoelastic effects still cannot be

ignored, even for a brittle thermoset epoxy, such as Hexcel 3501-6.

The AS4/3501-6 composite system does not have a notable viscoelastic behavior

at room temperature in a short period of time. Therefore, the concern is how to accelerate

the time scale and still accurately represent the real material properties. With several

momentary tensile creep tests performed and considering physical aging of the polymer,

the time-temperature superposition principle allowed us to construct a function of time

for each material property.

In the future, this research can be further investigated. The moisture effect is

always a serious issue especially for polymer matrix composite systems. Our research

indicated that there is different level of influence of chemical shrinkage in different fiber

orientations or different ply sequence. Physical aging of polymer matrix alternates the

material properties considerably, and is a potential factor of our experimental error which

cannot be ignored.

124

APPENDIX ATHE MATHCAD FILE FOR THE DATA ANALYSIS OF 90-DEGREE COMPOSITE

TENSILE SPECIMENS WITH THREE-ELEMENT STANDARD LINEAR SOLIDMODEL

import data files:i 0 1, 680..cr90_22_1 READPRN "90_22c_2.txt"( ) cr90_40_1 READPRN "90_40c_1.txt"( )

cr90_22_2 READPRN "90_22c_3.txt"( ) cr90_40_2 READPRN "90_40c_4.txt"( )














average three test results:

cr220< > cr90_22_1

0< >cr90_22_2

0< >cr90_22_3

0< >

3

cr221< > cr90_22_1

1< >cr90_22_2

1< >cr90_22_3

1< >

3

cr400< > cr90_40_1

0< >cr90_40_2

0< >cr90_40_3

0< >

3

cr401< > cr90_40_1

1< >cr90_40_2

1< >cr90_40_3

1< >

3

cr600< > cr90_60_1

0< >cr90_60_2

0< >cr90_60_3

0< >

3

125

cr601< > cr90_60_1

1< >cr90_60_2

1< >cr90_60_3

1< >

3

cr801< > cr90_80_1

1< >cr90_80_2

1< >cr90_80_3

1< >

3

cr800< > cr90_80_1

0< >cr90_80_2

0< >cr90_80_3

0< >

3

cr1000< > cr90_100_1

0< >cr90_100_2

0< >cr90_100_3

0< >

3

cr1001< > cr90_100_1

1< >cr90_100_2

1< >cr90_100_3

1< >

3

cr1200< > cr90_120_1

0< >cr90_120_2

0< >cr90_120_3

0< >

3

cr1201< > cr90_120_1

1< >cr90_120_2

1< >cr90_120_3

1< >

3

cr1400< > cr90_140_1

0< >cr90_140_2

0< >cr90_140_3

0< >

3

cr1401< > cr90_140_1

1< >cr90_140_2

1< >cr90_140_3

1< >

3

cr1600< > cr90_160_1

0< >cr90_160_2

0< >cr90_160_3

0< >

3

cr1601< > cr90_160_1

1< >cr90_160_2

1< >cr90_160_3

1< >

3

cr1770< > cr90_177_1

0< >cr90_177_2

0< >cr90_177_3

0< >

3

cr1771< > cr90_177_1

1< >cr90_177_2

1< >cr90_177_3

1< >

3

cr1951< > cr90_195_1

1< >cr90_195_2

1< >cr90_195_3

1< >

3

cr1950< > cr90_195_1

0< >cr90_195_2

0< >cr90_195_3

0< >

3

curve fitting for each temperature:• 22 °C

A 0.0774 T0400

A

no_eta22 t v,( )

cr221< >

0 v1 1 ev0 t.

.

v1 t. ev0 t.

.

1 ev0 t.

vg_cr2.053 10

3.

3.307 105.

p22 genfit cr220< >

cr221< >, vg_cr, no_eta22,

126

k0_22T0

cr221< >

0

k1_22T0

p221

ρ221

p220

• 40 °C

A 0.0768 T0400

A

no_eta40 t v,( )

cr401< >

0 v1 1 ev0 t.

.

v1 t. ev0 t.

.

1 ev0 t.

vg_cr1.466 10

3.

4.331 105.

p40 genfit cr400< >


k0_40T0

cr401< >

0

k1_40T0

p401

ρ401

p400

• 60 °C

A 0.0772 T0400

A

no_eta60 t v,( )

cr601< >

0 v1 1 ev0 t.

.

v1 t. ev0 t.

.

1 ev0 t.

vg_cr1.305 10

3.

4.932 105.

p60 genfit cr600< >


k0_60T0

cr601< >

0

k1_60T0

p601

ρ601

p600

• 80 °C

A 0.0779 T0400

A

no_eta80 t v,( )

cr801< >

0 v1 1 ev0 t.

.

v1 t. ev0 t.

.

1 ev0 t.

vg_cr1.327 10

3.

8.344 105.

p80 genfit cr800< >


k0_80T0

cr801< >

0

k1_80T0

p801

ρ801

p800

• 100 °C

A 0.0775 T0250

A

127

no_eta100 t v,( )

cr1001< >

0 v1 1 ev0 t.

.

v1 t. ev0 t.

.

1 ev0 t.

vg_cr1.656 10

3.

6.235 105.

p100 genfit cr1000< >

cr1001< >, vg_cr, no_eta100,

k0_100T0

cr1001< >

0

k1_100T0

p1001

ρ1001

p1000

• 120 °C

A 0.0774 T0250

A

no_eta120 t v,( )

cr1201< >

0 v1 1 ev0 t.

.

v1 t. ev0 t.

.

1 ev0 t.

vg_cr1.379 10

3.

1.003 104.


cr1201< >, vg_cr, no_eta120,

k0_120T0

cr1201< >

0

k1_120T0

p1201

ρ1201

p1200

• 140 °C

A 0.0756 T0250

A

no_eta140 t v,( )

cr1401< >

0 v1 1 ev0 t.

.

v1 t. ev0 t.

.

1 ev0 t.

vg_cr1.335 10

3.

1.384 104.


cr1401< >, vg_cr, no_eta140,

k0_140T0

cr1401< >

0

k1_140T0

p1401

ρ1401

p1400

• 160 °C

A 0.0771 T0250

A

no_eta160 t v,( )

cr1601< >

0 v1 1 ev0 t.

.

v1 t. ev0 t.

.

1 ev0 t.

vg_cr1.082 10

3.

2.78 104.


cr1601< >, vg_cr, no_eta160,

128

k0_160T0

cr1601< >

0

k1_160T0

p1601

ρ1601

p1600

• 177 °C

A 0.0764 T0250

A

no_eta177 t v,( )

cr1771< >

0 v1 1 ev0 t.

.

v1 t. ev0 t.

.

1 ev0 t.

vg_cr1.283 10

3.

6.448 104.


cr1771< >, vg_cr, no_eta177,

k0_177T0

cr1601< >

0

k1_177T0

p1771

ρ1771

p1770

• 195 °C

A 0.0777 T050

A

no_eta195 t v,( )

cr1951< >

0 v1 1 ev0 t.

.

v1 t. ev0 t.

.

1 ev0 t.

vg_cr1.542 10

3.

4.325 104.


cr1951< >, vg_cr, no_eta195,

k0_195T0

cr1951< >

0

k1_195T0

p1951

ρ1951

p1950

generate curves:r 0 10, 3450..g22 r( ) no_eta22 r p22,( )0 g40 r( ) no_eta40 r p40,( )0

g60 r( ) no_eta60 r p60,( )0 g80 r( ) no_eta80 r p80,( )0




129

Experimental vs. Curve Fitting (Three Element Model)Strain (in/in)

Time (sec)


Time (sec)

130


Time (sec)

131

APPENDIX BTHE MATHCAD FILE FOR THE DATA ANALYSIS OF 90-DEGREE COMPOSITE

TENSILE SPECIMENS WITH FOUR-ELEMENT STANDARD LINEAR SOLIDMODEL

import data filesi 0 1, 680..cr90_22_1 READPRN "90_22c_2.txt"( ) cr90_40_1 READPRN "90_40c_1.txt"( )















week100 READPRN "90_100c_1w.txt"( )

average three test results:• time axis

cr220< > cr90_22_1

0< >cr90_22_2

0< >cr90_22_3

0< >

3

cr400< > cr90_40_1

0< >cr90_40_2

0< >cr90_40_3

0< >

3

cr600< > cr90_60_1

0< >cr90_60_2

0< >cr90_60_3

0< >

3

cr800< > cr90_80_1

0< >cr90_80_2

0< >cr90_80_3

0< >

3

132

cr1000< > cr90_100_1

0< >cr90_100_2

0< >cr90_100_3

0< >

3

cr1200< > cr90_120_1

0< >cr90_120_2

0< >cr90_120_3

0< >

3

cr1400< > cr90_140_1

0< >cr90_140_2

0< >cr90_140_3

0< >

3

cr1600< > cr90_160_1

0< >cr90_160_2

0< >cr90_160_3

0< >

3

cr1770< > cr90_177_1

0< >cr90_177_2

0< >cr90_177_3

0< >

3

cr1950< > cr90_195_1

0< >cr90_195_2

0< >cr90_195_3

0< >

3

• corrected strains

cr221< > cr90_22_1

1< >cr90_22_2

1< >cr90_22_3

1< >

3

cr401< > cr90_40_1

1< >cr90_40_2

1< >cr90_40_3

1< >

3

cr601< > cr90_60_1

1< >cr90_60_2

1< >cr90_60_3

1< >

3

cr801< > cr90_80_1

1< >cr90_80_2

1< >cr90_80_3

1< >

3

cr1001< > cr90_100_1

1< >cr90_100_2

1< >cr90_100_3

1< >

3

cr1201< > cr90_120_1

1< >cr90_120_2

1< >cr90_120_3

1< >

3

cr1401< > cr90_140_1

1< >cr90_140_2

1< >cr90_140_3

1< >

3

cr1601< > cr90_160_1

1< >cr90_160_2

1< >cr90_160_3

1< >

3

cr1771< > cr90_177_1

1< >cr90_177_2

1< >cr90_177_3

1< >

3

cr1951< > cr90_195_1

1< >cr90_195_2

1< >cr90_195_3

1< >

3

• Poisson ratio

cr222< > cr90_22_1

2< >cr90_22_2

2< >cr90_22_3

2< >

3

cr402< > cr90_40_1

2< >cr90_40_2

2< >cr90_40_3

2< >

3

cr602< > cr90_60_1

2< >cr90_60_2

2< >cr90_60_3

2< >

3

133

cr802< > cr90_80_1

2< >cr90_80_2

2< >cr90_80_3

2< >

3

cr1202< > cr90_120_1

2< >cr90_120_2

2< >cr90_120_3

2< >

3

cr1402< > cr90_140_1

2< >cr90_140_2

2< >cr90_140_3

2< >

3

cr1602< > cr90_160_1

2< >cr90_160_2

2< >cr90_160_3

2< >

3

cr1772< > cr90_177_1

2< >cr90_177_2

2< >cr90_177_3

2< >

3

cr1002< > cr90_100_1

2< >cr90_100_2

2< >cr90_100_3

2< >

3

cr1952< > cr90_195_1

2< >cr90_195_2

2< >cr90_195_3

2< >

3

• data smoothing for the Poisson ratioM22 medsmooth cr22

2< >11, M40 medsmooth cr40

2< >11,

M60 medsmooth cr602< >

11, M80 medsmooth cr802< >

11,



11,



11,



11,

r 0 10, 3450..

curve fitting for each temperature:• 22 °C

A 0.0774 T0400

A

w_eta22 t v,( )

cr221< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

vg_cr

2.421 103.

2.984 105.

1.334 109.

ν_22 t v,( )

cr222< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

ν_vg_cr

4.2 103.

1.3 103.

4.2 107.

p22 genfit cr220< >

cr221< >, vg_cr, w_eta22, ν_p22 genfit cr22

0< >M22, ν_vg_cr, ν_22,

134

k0_22T0

cr221< >

0

k1_22T0

p221

ρ221

p220

η0_22T0

p222

s22_22i

cr221< >

i

T0

• 40 °C

A 0.0768 T0400

A

w_eta40 t v,( )

cr401< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

vg_cr

2.568 103.

2.978 105.

4.68 109.

ν_40 t v,( )

cr402< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

ν_vg_cr

2.8 103.

1.9 103.

2.6 107.

p40 genfit cr400< >


0< >M40, ν_vg_cr, ν_40,

k0_40T0

cr401< >

0

k1_40T0

p401

ρ401

p400

η0_40T0

p402

s22_40i

cr401< >

i

T0

• 60 °C

A 0.0772 T0400

A

w_eta60 t v,( )

cr601< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

vg_cr

3.021 103.

3.154 105.

6.588 109.

ν_60 t v,( )

cr602< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

ν_vg_cr

3.5 103.

1.6 103

1.3 107.

p60 genfit cr600< >


0< >M60, ν_vg_cr, ν_60,

135

k0_60T0

cr601< >

0

k1_60T0

p601

ρ601

p600

η0_60T0

p602

s22_60i

cr601< >

i

T0

• 80 °C

A 0.0779 T0400

A

w_eta80 t v,( )

cr801< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

vg_cr

3.911 103.

5.136 105.

1.277 108.

ν_80 t v,( )

cr802< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

ν_vg_cr

1.7 103.

2.5 103.

1.7 107.

p80 genfit cr800< >


0< >M80, ν_vg_cr, ν_80,

k0_80T0

cr801< >

0

k1_80T0

p801

ρ801

p800

η0_80T0

p802

s22_80i

cr801< >

i

T0

• 100 °C

A 0.0775 T0250

A

w_eta100 t v,( )

cr1001< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

vg_cr

4.289 103.

4.282 105.

8.123 109.

ν_100 t v,( )

cr1002< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

ν_vg_cr

5.8 103.

2.4 103.

3.2 107.

136


cr1001< >, vg_cr, w_eta100,

ν_p100 genfit cr1000< >

M100, ν_vg_cr, ν_100,

k0_100T0

cr1001< >

0

k1_100T0

p1001

ρ1001

p1000

η0_100T0

p1002

s22_100i

cr1001< >

i

T0

• 120 °C

A 0.0774 T0250

A

w_eta120 t v,( )

cr1201< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

vg_cr

2.298 103.

7.342 105.

9.892 109.

ν_120 t v,( )

cr1202< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

ν_vg_cr

7.2 103.

1.6 103.

4.8 107.


cr1201< >, vg_cr, w_eta120,


M120, ν_vg_cr, ν_120,

k0_120T0

cr1201< >

0

k1_120T0

p1201

ρ1201

p1200

η0_120T0

p1202

s22_120i

cr1201< >

i

T0

• 140 °C

A 0.0756 T0250

A

w_eta140 t v,( )

cr1401< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

vg_cr

3.168 103.

8.812 105.

1.898 108.

137

ν_140 t v,( )

cr1402< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

ν_vg_cr

4.1 103.

1.75 103.

6.2 107.


cr1401< >, vg_cr, w_eta140,


M140, ν_vg_cr, ν_140,

k0_140T0

cr1401< >

0

k1_140T0

p1401

ρ1401

p1400

η0_140T0

p1402

s22_140i

cr1401< >

i

T0

• 160 °C

A 0.0771 T0250

A

w_eta160 t v,( )

cr1601< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

vg_cr

3.148 103.

1.522 104.

4.327 108.

ν_160 t v,( )

cr1602< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

ν_vg_cr

3.9 103.

3.4 103.

1.0 107.


cr1601< >, vg_cr, w_eta160,


M160, ν_vg_cr, ν_160,

k0_160T0

cr1601< >

0

k1_160T0

p1601

ρ1601

p1600

η0_160T0

p1602

s22_160i

cr1601< >

i

T0

• 177 °C

A 0.0764 T0250

A

138

w_eta177 t v,( )

cr1771< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

vg_cr

2.813 103.

4.143 104.

8.712 108.

ν_177 t v,( )

cr1772< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

ν_vg_cr

3.1 103.

5.4 103.

1.4 107.


cr1771< >, vg_cr, w_eta177,


M177, ν_vg_cr, ν_177,

k0_177T0

cr1771< >

0

k1_177T0

p1771

ρ1771

p1770

η0_177T0

p1772

s22_177i

cr1771< >

i

T0

• 195 °C

A 0.0777 T050

A

w_eta195 t v,( )

cr1951< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

vg_cr

2.922 103.

3.194 104.

4.569 108.

ν_195 t v,( )

cr1952< >

0 v1 1 ev0 t.

. v2 t.

v1 t. ev0 t.

.

1 ev0 t.

t

ν_vg_cr

1.1 102.

7.6 103.

1.2 107.


cr1951< >, vg_cr, w_eta195,


M195, ν_vg_cr, ν_195,

k0_195T0

cr1951< >

0

k1_195T0

p1951

ρ1951

p1950

η0_195T0

p1952

139

s22_195i

cr1951< >

i

T0

generate curves:r 0 10, 3450.. j 0 1, 590..g22 r( ) w_eta22 r p22,( )0 g40 r( ) w_eta40 r p40,( )0

g60 r( ) w_eta60 r p60,( )0 g80 r( ) w_eta80 r p80,( )0




Experimental vs. Curve Fitting (Four Element Model)Strain (in/in)

Time (sec)

140


Time (sec)


Time (sec)

141

calculate creep compliance matrix elements:

S22_22 t( )1

k0_22

1

k1_221 e

t

ρ22. 1

η0_22t.

ν21_22 t( ) ν_22 t ν_p22,( )0 S12_22 t( ) ν21_22 t( ) S22_22 t( ).

S22_40 t( )1

k0_40

1

k1_401 e

t

ρ40. 1

η0_40t.

ν21_40 t( ) ν_40 t ν_p40,( )0 S12_40 t( ) ν21_40 t( ) S22_40 t( ).

S22_60 t( )1

k0_60

1

k1_601 e

t

ρ60. 1

η0_60t.

ν21_60 t( ) ν_60 t ν_p60,( )0 S12_60 t( ) ν21_60 t( ) S22_60 t( ).

S22_80 t( )1

k0_80

1

k1_801 e

t

ρ80. 1

η0_80t.

ν21_80 t( ) ν_80 t ν_p80,( )0 S12_80 t( ) ν21_80 t( ) S22_80 t( ).

S22_100 t( )1

k0_100

1

k1_1001 e

t

ρ100. 1

η0_100t.

ν21_100 t( ) ν_100 t ν_p100,( )0 S12_100 t( ) ν21_100 t( ) S22_100 t( ).

S22_120 t( )1

k0_120

1

k1_1201 e

t

ρ120. 1

η0_120t.

ν21_120 t( ) ν_120 t ν_p120,( )0 S12_120 t( ) ν21_120 t( ) S22_120 t( ).

S22_140 t( )1

k0_140

1

k1_1401 e

t

ρ140. 1

η0_120t.

ν21_140 t( ) ν_140 t ν_p140,( )0 S12_140 t( ) ν21_140 t( ) S22_140 t( ).

S22_160 t( )1

k0_160

1

k1_1601 e

t

ρ160. 1

η0_140t.

ν21_160 t( ) ν_160 t ν_p160,( )0 S12_160 t( ) ν21_160 t( ) S22_160 t( ).

S22_177 t( )1

k0_177

1

k1_1771 e

t

ρ177. 1

η0_177t.

ν21_177 t( ) ν_177 t ν_p177,( )0 S12_177 t( ) ν21_177 t( ) S22_177 t( ).

S22_195 t( )1

k0_195

1

k1_1951 e

t

ρ195. 1

η0_195t.

ν21_195 t( ) ν_195 t ν_p195,( )0 S12_195 t( ) ν21_195 t( ) S22_195 t( ).

142

the long-term test (one week):

A_1w 1.012 0.074. P_1w 250 T0_1wP_1w

A_1w

j 0 1, 1005..

S22_100wj

week1001< >

j

T0_1w

(psi-1) Creep Compliance S22(t) vs. Time (curve fit)

Time (sec)

143

log (psi-1) Log Creep Compliance S22(t) vs. Log Time (curve fit)


log (psi-1) Log Creep Compliance S22(t) vs. Log Time (experimental)


144

(psi-1) Creep Compliance S12(t) vs. Time (curve fit)

Time (sec)

log (psi-1) Log Creep Compliance S12(t) vs. Log Time (curve fit)


145

log (psi-1) Creep Compliance S22(t) Master Curve (no vertical shift)


log (psi-1) Creep Compliance S22(t) Master Curve (with vertical shift)


146

log (psi-1) Creep Compliance S12(t) Master Curve (with vertical shift)


connect piecewise master curves S22(t) into a single curvei 0 690..

t 1 1.1, 101.41.. t1 1 1.1, 10

1.17.. t2 1 1.1, 101.19.. t3 1 1.1, 10

1.12..

t4 1 1.1, 100.79.. t5 1 1.1, 10

0.91.. t6 1 1.1, 100.89.. t7 1 1.1, 10

3.35..m 0 1, 1150..

101.41 0( )

1

0.11 248.04= m1 0 1, 247.. 0 247 247=

logs22m1 log S22_22 m1 0.1. 1( )( )

logtm1 log m1 0.1. 1( )

102.58 1.41( )

1

0.11 138.911= m2 248 249, 248 137.. 248 137 385=

logs22m2 log S22_40 m2 248( ) 0.1. 1( )( ) 0.0054

logtm2 log m2 248( ) 0.1. 1( ) 1.41

103.77 2.58( )

1

0.11 145.882= m3 386 387, 386 144.. 386 144 530=

logs22m3 log S22_60 m3 386( ) 0.1. 1( )( ) 0.0101

logtm3 log m3 386( ) 0.1. 1( ) 2.58

104.89 3.77( )

1

0.11 122.826= m4 531 532, 531 121.. 531 121 652=

147

logs22m4 log S22_80 m4 531( ) 0.1. 1( )( ) 0.0205

logtm4 log m4 531( ) 0.1. 1( ) 3.77

105.68 4.89( )

1

0.11 52.66= m5 653 654, 653 51.. 653 51 704=

logs22m5 log S22_100 m5 653( ) 0.1. 1( )( ) 0.0397

logtm5 log m5 653( ) 0.1. 1( ) 4.89

106.59 5.68( )

1

0.11 72.283= m6 705 706, 705 71.. 705 71 776=

logs22m6 log S22_120 m6 705( ) 0.1. 1( )( ) 0.0475

logtm6 log m6 705( ) 0.1. 1( ) 5.68

107.48 6.59( )

1

0.11 68.625= m7 777 778, 777 69.. 777 69 846=

logs22m7 log S22_140 m7 777( ) 0.1. 1( )( ) 0.0589

logtm7 log m7 777( ) 0.1. 1( ) 6.59

101.5

1

0.11 307.228= m8 847 848, 847 306.. 847 306 1.153 10

3.=

logs22m8 log S22_160 m8 847( ) 0.1. 1( )( ) 0.0823

logtm8 log m8 847( ) 0.1. 1( ) 7.48

log (psi-1) Single Function for Creep Compliance log(S22(t))


148

(psi-1) Single Function for Creep Compliance S22(t)

Time(sec)

connect piecewise master curves S12(t) into a single curvei 0 690..

t 1 1.1, 101.05.. t1 1 1.1, 10

1.19.. t2 1 1.1, 100.98..

t3 1 1.01, 100.95.. t4 1 1.1, 10

0.72.. t5 1 1.1, 100.87..

t6 1 1.1, 101.22.. t7 1 1.1, 10

1.5..m 0 1, 984..

101.05 0( )

1

0.11 103.202= m1 0 1, 102.. 0 102 102=

logs12m1 log S12_22 m1 0.1. 1( )( )

logtm1 log m1 0.1. 1( )

101.19

1

0.11 145.882= m2 103 104, 103 144.. 103 144 247=

logs12m2 log S12_40 m2 103( ) 0.1. 1( )( ) 0.0397

logtm2 log m2 103( ) 0.1. 1( ) 1.05

100.98

1

0.11 86.499= m3 248 249, 248 85.. 248 85 333=

149

logs12m3 log S12_60 m3 248( ) 0.1. 1( )( ) 0.0459

logtm3 log m3 248( ) 0.1. 1( ) 2.24

100.95

1

0.11 80.125= m4 334 335, 334 79.. 334 79 413=

logs12m4 log S12_80 m4 334( ) 0.1. 1( )( ) 0.0774

logtm4 log m4 334( ) 0.1. 1( ) 3.22

100.72

1

0.11 43.481= m5 414 415, 414 42.. 414 42 456=

logs12m5 log S12_100 m5 414( ) 0.1. 1( )( ) 0.0196

logtm5 log m5 414( ) 0.1. 1( ) 4.17

100.87

1

0.11 65.131= m6 457 458, 457 64.. 457 64 521=

logs12m6 log S12_120 m6 457( ) 0.1. 1( )( ) 0.0002

logtm6 log m6 457( ) 0.1. 1( ) 4.89

101.22

1

0.11 156.959= m7 522 523, 522 155.. 522 155 677=

logs12m7 log S12_140 m7 522( ) 0.1. 1( )( ) 0.0149

logtm7 log m7 522( ) 0.1. 1( ) 5.76

101.5

1

0.11 307.228= m8 678 679, 678 306.. 678 306 984=

logs12m8 log S12_160 m8 678( ) 0.1. 1( )( ) 0.0401

logtm8 log m8 678( ) 0.1. 1( ) 6.98

150

log (psi-1) Single Function for Creep Compliance log(S12(t))


(psi-1) Single Function for Creep Compliance S12(t)

Time(sec)

compare with long term test (reference temperature at 100 °C):t 1 2, 10

1.22.. t1 1 10, 101.31.. t2 1 10, 10

1.02..

t3 1 2, 100.75.. t4 1 2, 10

2.5..

151

i 1 2, 10.. j 0 1, 590.. k 1 2, 579..

log (psi-1) Creep Compliance S22(t) Master Curve (curve fit, experimental, and long term)


152

REFERENCES

1. “X-34,” High-Performance Composites. May/June, p. 9 (1999).

2. Pagano, N. J. and Hahn, H. T., “Evaluation of Composite Curing Stresses,”Composite Materials: Testing and Design (Fourth Conference), ASTM STP 617, pp.317-329 (1977).

3. Agarwal, B. D. and Broutman, L. J., Analysis and Performance of Fiber Composites,Second Edition, John Wiley & Sons, Inc., New York (1990).

4. Mathar, J., “Determination of Initial Stress by Measuring the Deformations AroundDrilled Holes,” Transactions of ASME, Vol. 56, p.249, (1934).

5. Rendler, N. J. and Vigness, I., “Hole-drilling Strain-gage Method of MeasuringResidual Stresses,” Experimental Mechanics, Vol. 6, pp. 577-586, December (1966).

6. Bert, C. W. and Thompson, G. L., “A Method for Measuring Planar Residual Stressesin Rectangularly Orthotropic Materials,” Journal of Composite Materials, Vol. 2, No.2, April, pp. 244-253 (1968).

7. Schajer, G. S. and Yang, L., “Residual-stress Measurement in Orthotropic MaterialsUsing the Hole-drilling Method,” Experimental Mechanics, Vol. 34, No. 4, pp. 324-333 (1994).

8. Lee, J., Czarnek, R., and Guo, Y., “Interferometric Study of Residual Strains in ThickComposites,” Proceedings of the 1989 SEM Conference on Experimental Mechanics,Cambridge, MA, pp. 356-364 (1989).

9. Gascoigne, H. E., “Residual Surface Stresses in Laminated Cross-ply Fiber-epoxyComposite Materials,” Experimental Mechanics, Vol. 34, No. 1, pp. 27-36 (1994).

10. Sunderland, P., Yu, W. J., and Månson, J. E., "A Technique for the Measurement ofProcess-Induced Internal Stresses in Polymers and Polymer Composites,”Proceedings of ICCM-10, Vol. 3, pp. 125-132 (1995).

11. Lee, J. and Czarnek, R., “Measuring Residual Strains in Composites Laminates UsingMoiré Interferometry,” Proceedings of the 1991 SEM Conference on ExperimentalMechanics, Milwaukee, WI, pp. 405-415 (1991).

153

12. Chapman, T. J., Gillespie, J. W., and Pipes, R. B., "Prediction of Process-InducedResidual Stresses in Thermoplastic Composites,” Journal of Composite Materials,Vol. 24, pp. 616-643 (1990).

13. Månson, J. E. and Seferis, J. C., “Process Simulated Laminate (PSL)” A Methodologyto Internal Stress Characterization in Advanced Composite Materials,” Journal ofComposite Materials, Vol. 26, No. 3, pp. 405-431 (1992).

14. Joh, D., Byun, K. Y., and Ha, J., “Thermal Residual Stresses in Thick Graphite/EpoxyComposite Laminates-Uniaxial Approach,” Experimental Mechanics, Vol. 33, No. 1,pp. 70-76 (1993).

15. Hahn, H. T., “Residual Stresses in Polymer Matrix Composite Laminates,” CompositeMaterials, Vol. 10, pp. 265-277 (1976).

16. Kim, R. Y. and Hahn, H. T., “Effect of Curing Stresses on the First Ply-failure inComposite Laminates,” Journal of Composite Materials, Vol. 13, January, pp. 2-16(1979).

17. Jeronimidis, G. and Parkyn, A. T., "Residual Stresses in Carbon Fiber-ThermoplasticMatrix Laminates,” Journal of Composite Materials, Vol. 22, May, pp. 401-415(1988).

18. Hyer, M. W., “Calculation of Room-Temperature Shapes of UnsymmetricLaminates,” Journal of Composite Materials, Vol. 15, July, pp. 296-310 (1981).

19. Kim, K. S., Hahn, H. T., and Croman, R. B., "The Effect of Cooling Rate on ResidualStresses in a Thermoplastic Composite,” Journal of Composites Technology &Research, Vol. 11, No. 2, pp. 47-52 (1989).

20. Fakuda, H., Takahashi, K., and Toda, S., “Thermal Deformation of Anti-SymmetricLaminates at Cure,” Proceedings of ICCM-10, Vol. 3, pp. 141-148 (1995).

21. Unger, W. J. and Hansen, J. S., “The Effect of Thermal Processing on Residual StrainDevelopment in Unidirectional Graphite Fiber Reinforced PEEK,” Journal ofComposite Materials, Vol. 27, No. 1, pp. 59-81 (1993).

22. Carman, G. P. and Sendechyj, G. P., “Review of the Mechanics of Embedded OpticalSensors,” Journal of Composites Technology & Research, JCTRER, Vol. 17, No. 3,pp. 183-193 (1995).

23. Predecki, P. and Barrett, C. S., “Stress Measurement in Graphite/Epoxy Compositesby X-Ray Diffraction from Fillers,” Journal of Composite Materials, Vol. 13, pp. 61-71 (1979).

154

24. Fenn, R. H., Jones, A. M., and Wells, G. M., “X-Ray Diffraction Investigation ofTriaxial Residual Stresses in Composite Materials,” Journal of Composite Materials,Vol. 27, No. 14, pp. 1338-1351 (1993).

25. Madhukar, M. S., Kosuri, R. P., and Bowles, K. J, "Reduction of Curing InducedFiber Stresses by Cure Cycle Optimization in Polymer Matrix Composites,”Proceedings of ICCM-10, Vol. 3, pp. 157-164 (1995).

26. Ifju, G. P., Kilday, B. C., Liu, S.-C., and Niu, X., “Measurement of Residual Stress inLaminated Composites,” Proceedings of VIII International Congress on ExperimentalMechanics and Experimental/Numerical Mechanics in Electronic Packaging,Nashville, Tennessee, pp.198-199 (1996).

27. Hodges, J., Yates, B., Darby, M. I., Wostenholm, G. H., Clemmet, J. F., and Keates,T. F., “Residual Stresses and the Optimum Cure Cycle for an Epoxy Resin,” Journalof Materials Science, Vol. 24, pp. 1984-1990 (1989).

28. Snow, A. W. and Armistead, J. P., “A Simple Dilatometer for Thermoset CureShrinkage and Thermal Expansion Measurements,” Journal of Applied PolymerScience, Vol. 52, pp. 401-411 (1994).

29. Russell, J. D., “Cure Shrinkage of Thermoset Composites,” SAMPE Quarterly, Vol.24, No. 2, pp.28-33 (1993).

30. Wang, H. B., Yang, Y. G., Yu, H. H., and Sun, W. M., “Assessment of ResidualStresses During Cure and Cooling of Epoxy Resin,” Polymer Engineering andScience, Vol. 35, No. 23, pp. 1895-1898 (1995).

31. Hahn, H. T. and Pagano, N. J., “Curing Stresses in Composite Laminates,” Journal ofComposite Materials, Vol. 9, pp. 91-106 (1975).

32. Bogetti, T. A. and Gillespie, J. W. Jr., “Process-Induced Stress and Deformation inThick-Section Thermoset Composite Laminates,” Journal of Composite Materials,Vol. 26, No. 5, pp. 626-660 (1992).

33. Schapery, R. A., “Viscoelastic Behavior and Analysis of Composite Materials,” inMechanics of Composite Materials, G. P. Sendeckyi, ed., Academic Press, New York,pp. 85-167 (1974).

34. Schapery, R. A., “Application of Thermodynamics to Thermomechanical, Fracture,and Birefringent Phenomena in Viscoelastic Media,” Journal of Applied Physics, Vol.35, pp. 1451-1465 (1964).

35. Schapery, R. A., “A Method of Viscoelastic Stress Analysis Using Elastic Solutions,”Journal of the Franklin Institute, Vol. 279, pp.268-289 (1965).

155

36. Schapery, R. A., “Stress Analysis of viscoelastic Composite Materials,” Journal ofComposite Materials, Vol. 1, pp.228-267 (1967).

37. Schapery, R. A., “Thermal Expansion Coefficient of Composite Materials Based onEnergy Principles,” Journal of Composite Materials, Vol. 2, No. 3, July, pp. 380-404(1968).

38. Weitsman, Y., “Optimal Cool-Down in Linear Viscoelasticity,” Journal of AppliedMechanics, Vol. 47, March, pp. 35-39 (1980).

39. Harper, B. D. and Weitsman, Y., “Residual Stresses in an Unsymmetrical Cross-PlyGraphite/Epoxy Laminate,” AIAA Journal, Vol.19, pp. 325-332 (1981).

40. Harper, B. D. and Weitsman, Y., “On the Effects of Environmental Conditioning onResidual Stresses in Composite Laminates,” International Journal of Solids andStructures, Vol. 21, No. 8, pp. 907-926 (1985).

41. White, S. R. and Hahn, H. T., “Process Modeling of Composite Materials: ResidualStress Development during Cure. Part I. Model Formulation,” Journal of CompositeMaterials, Vol. 26, No. 16, pp. 2402-2422 (1992).

42. White, S. R. and Hahn, H. T., “Process Modeling of Composite Materials: ResidualStress Development during Cure. Part II. Experimental Validation,” Journal ofComposite Materials, Vol. 26, No. 16, pp. 2423-2453 (1992).

43. Liu, S.-C., Niu X., and Ifju, P., “Residual Stress Characterization by MoiréInterferometry.” Proceedings of the SEM Spring Conference of Experimental andApplied Mechanics and Experimental/Numerical Mechanics in Electronic PackagingIII, Houston, Texas, pp.176-178 (1998).

44. Wang, T.-M., Daniel, I. M., and Gotro, J. T., “Thermoviscoelastic Analysis ofResidual Stresses and Warpage in Composite Laminates,” Journal of CompositeMaterials, Vol. 26, No. 6, pp. 883-899 (1992).

45. Kim, Y. K. and White, S. R., “Process- Induced Stress Relaxation Analysis ofAS4/3501-6 Laminate,” Proceedings of the American Society for Composites, TenthInternational Conference on Composite Materials(ICCM10), Whistler, Canada, pp.281-292 (1995).

46. Adolf, D. and Martin, J. E., “Calculation of Stresses in Crosslinking Polymers,”Journal of Composite Materials, Vol. 30, No. 1, pp. 13-34 (1996).

47. Ferry, J. D., Viscoelasic Elastic Properties of Polymers, 2nd ed., John Wiley & Sons,New York (1970).

48. Post, D., Han, B., and Ifju, P. G., High Sensitivity Moiré: Experimental Analysis forMechanics and Materials, Springer Verlag, New York (1994).

156

49. ALCOA Aluminum Handbook, Aluminum Company of American, Pittsburgh (1967).

50. Gibson, R. F., Principles of Composite Material Mechanics, McGraw-Hill, Inc., NewYork (1994).

51. Ifju, P. G., Kilday, B. C., Niu, X., Liu, S.-C., and Ettinger, S., “Residual StrainMeasurement in Composites using the Curing Referencing Method,” to be publishedby Experimental Mechanics.

52. Boll, D. J., “Curing of 3501-6 Composite Systems,” Hercules Advanced Materials &Systems Company, Magna, February (1992).

53. Kilday, B. C., “Residual Stress Measurement of AS4/3501-6 Laminates Using MoiréInterferometry,” Master’s Thesis, University of Florida, Gainesville, May (1997).

54. Ifju, P., Kilday, B. C., Niu, X., and Liu, S.-C., “A Novel Method to Measure ResidualStresses in Laminated Composites,” Journal of Composites, Vol. 33, No. 16, pp.1511-1524 (1999).

55. Mallick, P. K., Fiber-Reinforced Composites—Materials, Manufacturing, andDesign, 2nd ed., Marcel Dekker, Inc., New York (1988).

56. Kim, Y. K., and White, S. R., “Stress Relaxation Behavior of 3501-6 Epoxy ResinDuring Cure,” Polymer Engineering and Science, Vol. 36, No. 23, pp. 2851-2862(1996).

57. Ferry, J. D., Viscoelastic Properties of Polymers, 3rd ed., John Wiley & Sons, NewYork (1980).

58. ASTM, 1997 Annual Book of ASTM Standards, Section 15, Vol. 15.03, pp. 98-108.

59. Chamis, C. C. and Sinclair, J. H., “Ten-deg Off-axis Test for Shear Properties in FiberComposites,” Experimental Mechanics, Vol. 17, No. 9, pp. 339-346 (1977).

60. Dally, J. W. and Riley, W. F., Experimental Stress Analysis, 3rd ed., McGraw-Hill,Inc., New York (1991).

61. Struik, L. C. E., "Physical Aging in Plastics and other Glassy Materials," PolymerEngineering and Science, Vol. 17, pp. 165-173 (1977).

62. Struik, L. C. E., Physical Aging in Amorphous Polymers and Other Materials,Elsevier, Amsterdam (1978).

63. Janas, V. F. and McCullough, R. L., "The Effect of Physical Aging on theViscoelastic Behavior of a Thermoset Polyester," Composites Science andTechnology, Vol. 30, pp. 99-118 (1987).

157

64. Ogale, A. A. and McCullough, R. L., "Physical Aging Characteristics of PolyesterEther Ketone," Composites Science and Technology, Vol. 30, pp. 137-148 (1987).

65. Sullivan, J. L., "Creep and Physical Aging of Composite," Composites Science andTechnology, Vol. 39, pp. 207-232 (1990).

66. Niu, X., Ifju, P. G., and Liu, S.-C., “Prediction of Process Induced Residual StressesUsing the Cure Referencing Method (CRM),” Proceedings of the SEM AnnualConference on Theoretical, Experimental and Computational Mechanics,Cincinnati, pp.739-742 (1999).

67. Niu, X. “Process Induced Residual Stresses and Dimensional Distortion in AdvancedLaminated Composites,” Ph. D. dissertation, University of Florida, Gainesville,August (1999).

158

BIOGRAPHICAL SKETCH

Shao-Chun Liu was born in 1967, Tainan, Taiwan, the Republic of China. He

obtained his Bachelor of Science degree from the Department of Physics at the National

Central University in 1989, where he had the most enjoyable memory of student life. He

served in Marine Corps of the R.O.C. for two years and came to U.S. in 1991.

He spent one year in the Intensive English Institute at the University of Illinois at

Urban-Champaign to improve his language, and arrived Gainesville, Florida in 1992 and

started his graduate study at the University of Florida.

Under the guidance of Dr. Chang-Tsan Sun, he finished the master’s program in

1994 and continued his doctoral study under the supervision of Dr. Peter Ifju. His field of

interest is residual stress of advanced laminated composites and experimental stress

analysis. In 1999, he received his Doctor of Philosophy degree.

residual stress characterization for laminated...

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