Measurement of Residual Stresses in Fiber Reinforced Composites Based on X-Ray Diffraction

B. Benedikt, P. K. Predecki, L. Kumosa, P. Rupnowski, M. Kumosa

Center for Advanced Materials and Structures Department of Engineering

University of Denver 2390 South York St., Denver, Colorado 80208

1. ABSTRACTX-ray diffraction (XRD) measurements have been made to determine residual stresses in embedded Al particles placed between the first and second ply of both unidirectional and fabric graphite/PMR-15 composite specimens. In order to further verify the results obtained, the effect of externally applied four-point bending on the measured strains and stresses in the Al inclusions was investigated. In the modeling part of this research, the state of stress in these inclusions was calculated using visco-elastic laminated plate theory and the Eshelby method. The numerical solutions for different concentrations of inclusions were subsequently compared with the experimental results. It has been shown that the thermal residual stresses in graphite/polyimide composites can be determined with reasonable accuracy by using X-ray diffraction from embedded inclusions in conjunction with the application of the Eshelby method.

2. INTRODUCTIONThe thermal residual stresses in composite materials are caused by the mismatch between coefficients of thermal expansion (CTE) of the matrix and the reinforcements. The magnitude of these residual stresses depends on the difference of CTE between the matrix and the reinforcements multiplied by the temperature change ∆T. Because of the fact that the CTE for the polymer is higher than the CTE for the graphite fibers, the generated thermal residual stresses are tensile in the matrix and compressive in the fibers. The proper evaluation of these tensile stresses in the matrix becomes particularly important when high temperature composites are considered since ∆T for these types of composites is high enough to generate residual stresses on the order of the tensile strength of the polyimide resin. Residual stresses inside embedded crystalline inclusions in polymer matrix composites were first measured by Predecki and Barrett [1] by performing XRD measurements. It was found that residual and applied strains transferred to various crystalline inclusions were high enough to be detected by X-ray diffraction. A new methodology for the evaluation of residual stresses in the polymer matrix of unidirectional polymer matrix composites based on XRD measurements of residual stresses in embedded crystalline inclusions has been recently developed by Benedikt et al. [2-3]. First, the residual thermal stresses in a unidirectional graphite/PMR-15 polyimide composite were determined by performing visco-elastic computations assuming interlaminar and intralaminar composite architectures. Second, the residual stresses in the Al inclusions were obtained through XRD measurements and from the application of the visco-elastic Eshelby method for multiple ellipsoidal inclusions. The residual stresses in the polymer matrix were extracted from the X-ray strains in the inclusions and subsequently compared with the strains obtained from the interlaminar and intralaminar analyses. The newly developed methodology has also been applied for the determination of the residual thermal stresses in 8 HS fabric graphite/PMR-15 composites, however, in the case of fabric composites, only interlaminar

Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45. 218 ISSN 1097-0002

This document was presented at the Denver X-ray Conference (DXC) on Applications of X-ray Analysis. Sponsored by the International Centre for Diffraction Data (ICDD). This document is provided by ICDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific community. All copyrights for the document are retained by ICDD. Usage is restricted for the purposes of education and scientific research. DXC Website – www.dxcicdd.com

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thermal residual stresses were determined since the actual stress field in the fabric plies is highly non-uniform and the authors are not aware of any model that could be used to accurately determine these stresses.

3. MATERIALS TESTED Both the unidirectional and fabric laminates were manufactured at the NASA Glenn Research Center in the form of 152 × 152 mm plates. Before curing, the Al inclusions were suspended in acetone and painted on top of the first unidirectional or fabric prepreg ply surface. After drying they were covered by the subsequent plies to form a laminate. In order to calculate the residual stresses numerically, the properties of all three phases (PMR-15, graphite fibers and Al inclusions) making up the composite system were needed. It was assumed that thermal and elastic properties of both graphite fibers and Al inclusions are time and temperature independent, while the PMR-15 matrix was modeled as a visco-elastic material with CTE depending upon temperature. Furthermore, Hashin’s equations [4] and the so-called bridging model [5-6] were used to determine the thermal and visco-elastic behavior of both unidirectional and fabric plies. The distributions of Al inclusions were determined using scanning electron microscopy (SEM). The details of SEM analysis are given in reference [2]. After the SEM analysis was performed it turned out that the average area fraction of Al inclusions in a cluster was 40% ± 7% and 46% ± 7.4% for unidirectional and fabric specimens, respectively. The examples of embedded Al inclusions are presented in Figure 1.

a

b

Figure 1. Examples of embedded Al inclusions, (a) cluster of interlaminar inclusions between unidirectional plies, (b) intralaminar inclusions in unidirectional composite

4. ANALYTICAL MODELS The goal of the present analysis is twofold. First: to find the thermal residual stresses in the embedded Al inclusions knowing the residual stress in the polymer matrix. Second: one can reverse this question and ask how to calculate the residual stresses in the matrix when the state of stress in the Al inclusions is known. Both problems can be solved using the Eshelby model [7] modified by Mori and Tanaka [8] to account for many inclusions. Since the polymer resin was assumed to be visco-elastic, only an approximate solution was obtained. It is important to note that the Eshelby equations can be used not only to determine the stress inside the Al inclusions due to change in temperature from the assumed stress free temperature (3150C) down to room temperature (RT) (300C), but also due to applied external forces. For the sake of brevity, the final Eshelby equations (1a-c) will be presented without derivation:

Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45. 219 ISSN 1097-0002

( )

( )∑

∑−

=

−

=

+−−+++−

=+−−+++

1

0

ReRe

1

0

*ReRe

)()()()1()()())()((

)()()()1()()(

f

i

BendTssRTM

f

i

BendTssRTI

iiiiiiSifC

iiiiiiSC

εεεεεεξξ

εεεεεε (1a)

( ) 0)()()( =−+ jjSVj TTf

R εεε , for j=0, 1, 2, … , f (1b)

( )∑−

=

+−−+++=1

0

*ReRe )()()()1()()()(f

i

BendTssRTIinclusion iiiiiiSCf εεεεεεσ (1c)

where S is the Eshelby tensor for spherical inclusions, CI is the stiffness tensor for the Al inclusions, CM is the time and temperature dependent stiffness tensor for the matrix, εT is the stress free transformation strain, εT* is the stress free transformation strain of the inhomogeneous inclusion, εR is the average strain caused by the presence of many inclusions, f is the number of steps used in numerical visco-elastic analysis, Vf is the average volume fraction of the inclusions, ξ is a so-called reduced time [2], and εRes is the average residual thermal strain in the matrix, which can be obtained using either laminated plate theory (interlaminar model) or the Eshelby equations for cylindrical inclusions (intralaminar model) together with the Hashin equations (unidirectional specimens) or the bridging model (fabric specimens). It was assumed and later verified that the number of embedded inclusions was small enough to neglect their influence on stress and strain distributions inside the plate far away from the embedded inclusions. The effect of externally applied bending was accounted for by the term εBend(i), which was equal to zero for all values of i except for i=f-1, since the bending was applied at RT. The stress inside the aluminum inclusions σinclusion was determined from equation 1c, provided all εRes(i), i = 1,…,f were calculated from equations 1a, b. On the other hand, knowing from the XRD the state of stress in the embedded Al inclusions is enough to calculate εRes without using laminated plate theory or the Eshelby method for cylindrical inclusions. In order to do so, the special case of the equations (1 a-c) was used with εRes(f) considered as an unknown. Since we are interested only in thermal residual stresses caused by the graphite fibers, the contribution to the measured stresses in the embedded inclusions due to the mismatch between the CTE of the aluminum and the polymer matrix should also be considered. The procedure of determination of the residual stresses from XRD data can be summarized as follows: 1.Use equations (1 a-c) to model the post curing cycle, assuming that both εRes and εBend are zero

for each i. The result is σInclusion, which corresponds to the thermal residual stress in the Al inclusions due to the mismatch of CTE between the Al inclusions and the PMR-15 matrix only

2.For a fixed value of the index i corresponding to T=30oC and σInclusion equal to the stress obtained from XRD minus the stress calculated in 1, use equations (1 a-c) to get εRes at T=30oC

3.Compare εRes calculated in 2 with εRes calculated from laminated plate theory

5. X-RAY EXPERIMENT As mentioned in the previous paragraph, the XRD measurements were carried out in order to determine the state of stress and strain in the embedded Al inclusions. It should also be emphasized that the embedded crystalline inclusions should serve as strain sensors, which are sufficiently sensitive to XRD but should not in principle substantially change the stress and strain distributions inside a composite plate. Thus a tradeoff had to be achieved between having sufficient diffracted intensity, and not influencing the stress and strain distributions. After

Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45. 220 ISSN 1097-0002

preliminary experiments it was determined that 4.6 mg/cm2 nominal surface loading of Al inclusions during lay-up was high enough to yield adequate diffraction intensity. It was also determined that this concentration of inclusions only slightly changed the A and D matrices used in the laminated plate theory. The method of determination of residual stresses by XRD, which was followed in the presented analysis, is thoroughly described in [9]. First, the lattice spacing d0 of the (422) planes for unstressed Al inclusions at RT was measured by dispersing the free inclusions on a quartz plate substrate and obtaining a diffraction pattern in a Siemens D500 diffractometer fitted with pseudo parallel-beam optics and a solid-state detector. Diffraction conditions are shown in Table 1. Also, the temperature inside the diffractometer chamber was monitored during each run. Knowing the position of the peak, d0 was obtained from Bragg’s law, where a pseudo-Voigt function was fitted to the experimental peak profile to get the Kα1 peak position. The direction of the diffraction vector was specified by two angles, φ and ψ in the usual way [9]. The d0 measurements were performed at a fixed value of φ and six different values of ψ from 0o to 45o. The lattice spacing was slightly different in each case, but this difference was not large enough to cause any substantial error. For each peak measurement d0 was corrected to a single temperature using the CTE for aluminum and then the average from all six measurements was calculated. This average was assumed to be an estimator of the true value of d0 at a given temperature.

Radiation Cu Kα1, λ=1.5406 A 2 θ Range (deg) 136-139.5 Step Size (deg) .02 Step Time (sec) 10 (for ψ ≤ 26.57), 5 (for ψ>26.57)

Divergence Slit (deg) 0.1 Soller Slit (deg) 0.15

Detector Energy, Window 8.04 keV, 340 eV Table 1. Diffraction conditions

50 mm by 16 mm specimens were cut from as the supplied composite plates with embedded Al inclusions using a slow cutting Buehler diamond saw. In order to reduce surface roughness, the specimens were lightly polished with 400-grit paper. Subsequently they were mounted either on a flat glass plate or in a four-point bend fixture. The specimen location was checked with a dial gage micro flat accurate to 0.025 mm. In order to monitor the strain caused by the four-point bending fixture, strain gages were mounted on the irradiated surface of the specimens between the inner two pins of the four-point bending fixture. XRD measurements were performed on these specimens at φ = 00 and φ = 900 and at six ψ angles from 00 to 450 for each φ angle. Taking all the non-diagonal components of εij equal to zero, which is consistent with the assumption that the coordinate system defined by the direction of the fibers is simultaneously the principal coordinate system for the embedded inclusions the fundamental equation of X-ray diffraction [9] could be put into a form:

332

33110

0,0,0 sin)( εψεεε ψφψφ +−=

−= =

= ddd

(2a)

Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45. 221 ISSN 1097-0002

332

33220

0,90,90 sin)( εψεεε ψφψφ +−=

−= =

= ddd

(2b)

Equations (2a-b) were used to determine the principal strains in the embedded Al inclusions. If both εφ=0,ψ and εφ=90,ψ vs. sin2(ψ) form straight lines, then the intercepts and slopes of these lines determine the principal strains. After the experiment had been performed, it turned out that all sin2(ψ) plots were linear within experimental error (see Figure 2). The normal strain ε33 was taken to be an average between εφ=0,ψ=0 and εφ=90,ψ=0. The slopes were determined using least squares linear fits. In order to determine the corresponding stresses, X-ray elastic constants for the aluminum 422 reflections were calculated by taking the mean of the Reuss and Voigt models [9], using literature values of the single crystal compliances. This procedure gave the Young’s modulus E and Poisson ratio ν values of 71 MPa and 0.35, respectively. The stresses were subsequently calculated from strains using isotropic Hooke’s law.

6. RESULTS AND DISCUSSION Examples of sin2(ψ) plots for three different values of axial bending strain are shown in Figures 2a-b. The instant conclusion one can draw from these figures is that the degree of scatter about the linear fits is quite variable. The amount of scatter was not related to the magnitude of the applied strain but was probably the result of a non-uniform distribution of particles. There was no persistent evidence of curvature in the sin2(ψ) plots, but if the curvature is present, it may be caused by the existence of non-diagonal components of the strain and stress tensors. It was assumed that the specimen's coordinate system was the same as the principal coordinate system for all inclusions, however the distribution of the inclusions is not uniform and not all inclusions are exactly spherical which might generate the non-diagonal components of both the strain and stress tensors. In other words, the principal coordinate system might change from one inclusion to another due to the changing shape and distribution of the inclusions.

y = 0.000898x - 0.000245R2 = 0.987728

-0.0003

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0 0.1 0.2 0.3 0.4 0.5sin2(ψ)

ε(ψ,ϕ)

a

y = 0.000352x - 0.000200R2 = 0.935485

-0.00025

-0.0002

-0.00015

-0.0001

-0.00005

0

0 0.1 0.2 0.3 0.4 0.5sin2(ψ)

ε(ψ,ϕ)

b

Figure 2. Example of plot of εϕψ as a function of sin2ψ for (a) ε = 900 µε, ϕ = 0o and (b) ε = 900 µε, ϕ = 90o

The final results of the XRD measurements are shown in Figure 3, where the strains in the aluminum inclusions embedded in both the unidirectional and fabric specimens are presented as a function of applied axial bending strain. It should be noted that for fabric specimens the strains along the direction of the fibers (ε11 and ε22) are almost the same as expected and the applied bending moment breaks down this symmetry (Figure 3b). For unidirectional specimens, ε11 and ε22 are markedly different regardless of the magnitude of the applied bending. Furthermore, it was observed that the strains in the embedded inclusions were changing linearly with the applied

Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45. 222 ISSN 1097-0002

bending strain for both unidirectional and fabric specimens. The stress field in the surrounding polymer matrix was subsequently calculated using the Eshelby model along with the Tanaka-Mori theorem [8]. It should be understood that this approach assumes that: all embedded inclusions are identical; there is perfect bonding between the matrix and the Al inclusions, and that they are randomly distributed. Once the average stress in the polymer matrix was calculated, it was compared with existing solutions, namely the laminated plate theory (interlaminar model) and the Eshelby model for cylindrical inclusions (intralaminar model). Because of the fact that the Eshelby approach cannot be applied in the case of fabric laminates, only the interlaminar model was considered. Furthermore, the SEM analysis revealed that a very few inclusions were located inside the fabric plies, therefore the measured XRD peak positions were due to reflections mainly from interlaminar inclusions. The residual stresses in the matrix obtained by using the methods mentioned are compared in Table 2. The errors were estimated from the uncertainty resulting from XRD measurements and the deviation of inclusion distributions.

y = 0.083x + 565.13

y = -0.0004x + 121.3

y = -0.0343x - 225.03

-400

-200

0

200

400

600

800

1000

0 500 1000 1500 2000 2500 3000Applied strain [10-6]

Stra

ins

insi

de A

l par

ticle

s [1

0-6]

ε11

ε22

ε33

a

y = -0.0954x - 415.61

y = 0.0355x + 485.73

y = 0.0672x + 504.72

-800-600-400-200

0200400600800

1000

0 1000 2000 3000Applied Strain [10-6]

Stra

ins

insi

de A

l par

ticle

s [1

0-6] ε11

ε22

ε33

b

Figure 3. Measured XRD strains in the embedded aluminum inclusions as a function of applied axial bending strain; (a) unidirectional specimens, (b) fabric specimens

σ11 [MPa]

σ22 [MPa]

σ33 [MPa]

Unidirectional composite Interlaminar model 48.3 31.9 0 Intralaminar model 48.1 14.6 14.6 X-ray and Eshelby 56 ± 8 40 ± 9 28 ± 8

Fabric Composite Interlaminar model 63 63 0 X-ray and Eshelby 64±9 65±11 30±9

Table 2. Average residual stresses in the polymer matrix

After the data presented in Table 2 had been analyzed it became clear that the majority of the embedded inclusions must be considered as interlaminar inclusions since the σ11 and σ22 stress components calculated from the interlaminar model agreed quite well with the corresponding stress components obtained from XRD measurements and the Eshelby model. However, the presence of non-zero tensile stresses along the x3 direction proves that the simple laminated plate theory that predicts σ33 to be zero everywhere inside the plate is also not adequate to precisely describe the thermal residual stress. The existence of a non-zero tensile stress σ33 in the polymer

Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45. 223 ISSN 1097-0002

resin can be explained by the SEM analysis, which shows that the boundaries of the plies are not flat and parallel to the surface of the plate. It can be shown using FEM that this generates tensile stresses through the thickness, and the remaining components of the stress tensor, namely σ11 and σ22, are changed only slightly [2].

7. CONCLUSIONS It was shown that the thermal residual stresses in the amorphous polymer resin in graphite/PMR-15 unidirectional and fabric composites could be determined based on XRD from embedded crystalline inclusions. However, in order to determine them accurately the visco-elastic properties of the matrix, the cool down cycle, and the shape and distribution of the inclusions must be known. Despite the fact that the distribution of the inclusions was not strictly controlled, the residual stresses obtained in the composite agreed quite well with the interlaminar assumptions. The X-ray strains in the inclusions have been found to be linearly dependent upon the applied strain caused by the four-point bending for both unidirectional and fabric specimens. The change in the stress components σ11 and σ22 inside the polymer matrix as a function of bending can be quite accurately predicted using laminate theory, however the σ33 strain component differs significantly from the numerical predictions.

8. ACKNOWLEDGEMENTS This research has been supported by the Air Force Office of Scientific Research and the NASA Glenn Research Center under grants F49620-97-1-0426 (AFOSR) and F49620-00-1-0159 (AFOSR and NASA).

9. REFERENCES 1. P. Predecki and C. S. Barrett, Stress Measurement in Graphite/Epoxy Composites by X-ray

Diffraction from Fillers, J. Comp. Mat., vol. 13, p. 61-71, 1979 2. B. Benedikt, M. Kumosa, P.K. Predecki, L. Kumosa, M. G. Castelli and J. K. Sutter, An

Analysis of Residual Thermal Stresses in a Unidirectional Graphite/PMR-15 Composite Based on the X-ray Diffraction Measurements, Composites Science and Technology (2001) in press.

3. B. Benedikt, P. K. Predecki, L. Kumosa, D. Armentrout, J. K. Sutter, M. Kumosa, The Effect of Bending Loads on Internal Stresses in Aluminum Inclusions Embedded in a Unidirectional Graphite/PMR-15 Composite Determined by X-ray Diffraction Measurements, Composites Science and Technology, in press

4. Z. Hashin, Analysis of Properties of Fiber Composites with Anisotropic Constituents, Journal of Applied Mechanics, vol. 46, p. 543, 1979

5. T. Ishikawa, T. W. Chou, Nonlinear Behavior of Woven Fabric Composites, Journal of Composite Materials, vol. 17, p. 399-413, 1983

6. T. Ishikawa, T. W. Chou, In-Plane Thermal Expansion and Thermal Bending Coefficients of Fabric Composites, Journal of Composite Materials, vol. 17, p. 92-104, 1983

7. T. Mura, Micromechanics of Defects in Solids, 2nd edition, Martinus Nijhoff Publishers, Dordrecht, 1987

8. T. Mori and K. Tanaka, Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions, Acta Metall., vol. 21, p. 571-574, 1973

9. I. C. Noyan and J. B. Cohen, Residual Stress. Measurement by Diffraction and Interpretation. Spriger-Verlag, New York, 1987

Copyright (c)JCPDS-International Centre for Diffraction Data 2002, Advances in X-ray Analysis, Volume 45. 224 ISSN 1097-0002