We measure the spatial distribution of the mechanical stress induced inside translucent glass/epoxy composites by means of polarization-sensitive optical coherence tomography. The Stokes parameters determined from two orthogonal polarization components of the backscattered light allow the internal stress to be identified in terms of its magnitude and principal direction based on a birefringence light scattering model of glass/epoxy composites. Measurement examples show the particular case of stress concentration near a through hole and the internal structural damages caused by excessive tensile loading.
©2003 Optical Society of America
Optical coherence tomography (OCT) has a relatively not long history of development, but it is emerging as a practical nondestructive means of acquiring highly depth-resolved images of translucent light-scattering objects by localizing the backscattered light by means of low coherence interferometry . The new tomographic technique finds many useful applications in various fields of science and engineering, especially in the biomedical area where high depth-resolution imaging is needed in diverse diagnostic testing . Principles of OCT can also be applied to the nondestructive testing of engineering materials such as polymer composites to examine their internal stresses and structures . In this investigation, we made a new attempt to measure the spatial distribution of mechanical stresses inside glass/epoxy composites, which aims to locate any structural defects and delaminated layers caused by fabrication imperfection or overloading.
Various types of composite materials are used in industry because of their advantageous mechanical properties such as high ratio of strength to weight. The spatial distribution of mechanical stresses inside composite materials may be visualized and quantified by means of conventional transmissible photoelasticity, but its use is limited to transparent and tenuous fiber composites . Glass/epoxy composites considered in this investigation are densely packed with thin glass fibers, and they yield so low transparency that the conventional method of photoelasticity testing fails to provide good quality birefringence fringes. An alternative way is to apply the principles of polarization-sensitive optical coherent tomography (PS-OCT), which is an evolved form of the original OCT. The PS-OCT technique enables to monitor not only the amplitude of the backscattered light but also the change of polarization caused by the internal medium of the specimen [5–7]. Therefore the birefringence field inside glass/epoxy composites may be mapped by analysis of the phase information obtained from the PS-OCT technique, which leads to determination of the spatial distribution of the mechanical stresses induced within the heterogeneous composites.
2. Polarization-sensitive optical coherence tomography
Figure 1 shows the schematic of the PS-OCT system configured in this investigation. It is basically a Michelson interferometer designed to detect the change of polarization of the backscattered light from the specimen. Short coherence interference provides depth-sectioning capability, which takes place by use of a beam of wide band spectrum. The reference beam focused on to a flat mirror is linearly polarized, and the test beam incident on the specimen is circularly polarized to avoid any particular polarization-dependency of backscattering from the composite specimen under examination. Interference between the backscattered test and reference waves is detected after separation into two orthogonal polarization components. The reference flat mirror moves backwards with a constant speed while interference is localized along the depth of the specimen within the short temporal coherence length given by the wide-band source light. A mechanical stage provides lateral scanning, which allows the complete two-dimensional map of the specimen to be obtained.
As for the hardware devices constituting the PS-OCT system, the source light is a super-luminescent diode of 900 µW output power, whose center wavelength is at λ0=684.5 nm and spectral bandwidth is 9.1 nm. In the reference arm, a quarter wave Fresnel-Rhomb prism is set to rotate the linear polarization of the incident light beam by 22.5° to the horizontal direction, so a total rotation of 45° is obtained after reflection for the flat mirror. In the test arm, the beam is made circularly polarized by use of another Fresnel-Rhomb prism set to rotate polarization by 45°. The movement of the reference flat mirror induces a 7 kHz Doppler frequency in both the detected interference signals of two orthogonal polarization components, so unwanted electric noises encountered in data sampling are removed using band-pass filters with a bandwidth of 200Hz centered at the Doppler frequency.
As both glass fibers and epoxy resin are of birefringence nature, the glass/epoxy composite yields photoelastic responses to incident light, and behaves as a birefringence phase retarder. Glass fibers within the composite are usually aligned to take uniaxial tensile loading along a single predetermined direction. Then the composite is subject to the state of plane stress, the resulting phase retardation δ between two orthogonal polarization light components is derived in the form of 
where z denotes the depth of the composite, σ is the magnitude difference between the two principal stresses induced in the xy-plane, and f L is the equivalent longitudinal material-fringe value for the heterogeneous composite. If the principal direction of the induced stress field has an angle φ to the horizontal polarization axis, the interference intensity signals of the horizontal and vertical polarization components of the PS-OCT system, denoted AH and AV, respectively, are modulated such as [5,6]
where R(z) is the internal reflectance of the composite specimen which usually attenuates drastically as the depth z increases. Δz is the optical path difference between the test and reference waves, k0 is the mean wave number given by the mean wavelength λ0 of the incident source light, i.e., k 0=2π/λ 0, and lc is the temporal coherence length of the source light. The result of Eq. (2) assumes that no depolarization takes place in the backscattered light and also the internal reflectance R(z) is not affected by polarization of light.
The Hilbert transformation of the measured signals enables to obtain the envelope profile and also the phase distribution of both AH and Av, which then lead to determination of the Stokes parameters that are given by 
where |AH |and |AV | are the amplitudes and ∠AH and ∠AV are the phases of AH and Av, respectively. Substituting Eq. (2) into Eq. (3) allows the Stokes parameters to be expressed in a more convenient form for analysis of photoelasticity such as
Note that among the Stokes parameters, S 0 is simply proportional to the internal reflectance R(z), and S 3 experiences a sinusoidal variation with a period in proportion to the magnitude of the internal mechanical stress induced. In addition, S 1 and S 2 provide information on the principal direction of the induced stress. The above photoelasticity relationship between the Stokes parameters and the stress is not limited to the particular case of uniaxial loading, and generally valid if the phase retardation δ and the principal direction of birefringence φ are formulated as the two-dimensional case of plane stress.
3. Photoelasticitytesting of glass/epoxy composites
The specimens tested in this investigation are glass/epoxy composites composed of multiple layers of prepregs made from glass fibers of 9 µm in diameter mixed with thermally curable epoxy resin with a volume fraction of 60 percent. The specimens underwent a two-step curing process in a vacuum oven: initially under a platen pressure of 0.56MPa during 1 hour at 80°C, and subsequently at high temperature elevated to 130 °C for 2.5 hours with a platen pressure of 0.88 MPa. The specimens were cut to a nominal size of 125 mm in length and 18 mm in width with a thickness of 0.76 mm. Table 1 lists mechanical and optical properties of the glass fibers and also the epoxy resin of the cured specimens named collectively UGN150. All the numerical data are those provided by the manufacturer, except that the material-fringe value of the glass fibers was adopted from Ref. . Besides, we directly measured the material-fringe values of the epoxy resin by applying standard tensile load test procedures.
Figure 2 shows two representative sets of 2-D images of all four depth-resolved Stokes parameters, which are obtained when the composite specimens are free from external loading (set ‘A’) and subject to uniaxial uniform tensile loading (set ‘B’). Each image covers a physical size of 1.0 mm in depth and 0.3 mm in width with a single pixel resolution of 8 µm×10 µm. The fifth image on the right for each set displays the mean values of the Stokes parameters averaged over the whole width of images. Even in the absence of external loading in set ‘A’, a slow oscillation is observed especially in S3 image, which indicates the presence of birefringence due to residual stress. Figure 3 shows an experimental relation between the magnitude of induced stress and the fringe period of S3 image, whose slope corresponds to the material-fringe value f L as defined in Eq. (4). The experimental value of f L was measured 116 kN/m fringe, which is in good agreement with the theoretical value of 128 kN/m fringe computed from the data in Table 1.
As a special case of birefringence stress measurement, we measured the stress concentration near a circular hole made on a specimen. Figure 4 shows 2-D images of the Stokes parameters measured along the line A-B on the specimen depicted in the same figure. The specimen was subject to uniaxial tension loading aligned to the fiber direction, whose nominal far stress reaches around 118 MPa.
Figure 5 presents the tensile stress derived from the acquired Stokes parameters of Fig. 4. As predicted in the theoretical stress distribution, the stress reaches maximum at the point where the line A-B intersects the hole boundary . The stress concentration factor, defined as K=σmax/σnorminal, was measured 1.9, which is a bit larger in comparison with a theoretical value of 4.3. We reason that the difference is due to the fact that the actual shape of the hole is not a perfectly circular due to damages caused in the drilling process of fabricating the hole.
The first Stokes parameter S0 of the PS-OCT system represents the magnitude of the intensity of the backscattered light. Thus, any abrupt change in the detected S0 indicates strong discontinuity in the refractive index caused by cracks or damages of the prepregs of the specimen. Along with S0, other Stokes parameters, S1, S2, and S3 can be used to detect structural changes that are not simply detected from S0. Figure 6 shows the PS-OCT images near a hole edge, which was damages due to heavy duty drilling. The acquired images reveal serious matrix cracks accompanying abrupt discontinuity in S0 and also the non-uniform residual stress distribution caused by high temperature induced during drilling process. Figure 7 compares two sets of images, which were taken before and after the structural failure of a composite sample. S0 image clearly reveals two severe delaminations of the top and the bottom surface after failure. Other Stokes parameter images of S2 and S3 show supplement information that there is no apparent level of stress inside two delaminated layers.
In addition to the one-dimensional tensile stress distribution near a circular holes presented in Fig. 5, our PS-OCT method allows obtaining the two-dimensional stress distribution in terms of the stress magnitude and the principle direction by analyzing the Stokes vector images. Since the stress magnitude is proportional to the slop of the phase term of S3 Stokes parameter as defined in Eq. (4), it is calculated from the instantaneous phase determined by the Hilbert transform of S3 parameter with subsequent numerical differentiation of the measured phase. Besides, S1 and S2 Stokes parameters provide the angle of the principle direction by taking the arc-tangent operation of S1/S2 following Eq. (4). Figure 8 shows a two-dimensional image of the stress distribution estimated from the Stokes parameters measured in Fig. 4. Before calculating the stress field, random noises in the Stokes parameters images were removed using median filtering. The spatial image of Fig. 8 reveals non-uniform distribution of stress through the depth direction, which is reasoned to arise due to micro-scale cracks generated within the specimen during fabrication the hole with drilling. In the far field from the hole, the stress distribution becomes uniform as expected. The principle direction appears to be close to 0 deg, which is in fact the direction of fibers parallel to the external loading. Actual calculation of the principle direction involves the arc tangent operation, which becomes unstable being affected by random phase noises when S2 Stokes parameter is small close to 0. To cope with the numerical instability, we neglected the principle direction angle value if the magnitude of S2 Stokes parameter is smaller than 0.1, and instead interpolated the angle using neighboring valid data, which causes the strip patterns observed in the principle direction image.
We performed a nondestructive testing of translucent glass-epoxy composites to map the inside stress distribution by means of the polarization-sensitive optical coherent tomography. The Stokes parameters determined from two orthogonal polarization components of the backscattered light allow the internal stress to be identified in terms of its magnitude and principal direction based on a birefringence light scattering model of glass/epoxy composites. Experimental results with densely packed glass/epoxy composites demonstrated that the polarization state changes are mostly dependent on the internal stress in spite of heavy scattering of light. Analysis of the cross-sectional images of the Stokes parameters obtained from the developed measurement system allows the internal stress distribution and also damages inside composites to be visualized in microscopic level.
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