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Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension
Composites: Part A 35 (2004) 849–859 www.elsevier.com/locate/compositesa
Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension Jung-Ryul Leea,*, J. Molimarda, A. Vautrina, Y. Surrelb a
Department of Mechanical and Materials Engineering, Ecole Nationale Supe´rieure des Mines de Saint-Etienne, 158 Cours Fauriel, Saint-Etienne Cedex 2, 42023, France b BNM, INM/CNAM, 292 rue Saint Martin, 75141 Paris, France
Abstract Digital phase-shifting grating shearography has been applied for the investigation of the tensile behaviour of carbon/epoxy plain-weave fabric composite with a small waviness. Experimental analyses were performed for the two following configurations: a single lamina and an iso-phase double laminate. The yarn crimp effects such as the tension/bending and tension/in-plane shear couplings were concentrated on the resin rich regions for the single lamina. In the latter case, the yarn crimp effects were still significant because although the transverse shear strain due to the local bending effect of yarns is a little constrained by the other neighboring layer, the degree of the constraint was certainly insufficient to degenerate the local bending effect. q 2004 Elsevier Ltd. All rights reserved. Keywords: A. Fabrics/textiles; B. Anisotropy; C. Laminate mechanics; D. Non-destructive testing-optical full-field method
1. Introduction Since the weaving architectures provide some beneficial properties over the unidirectional tape laminates such as improved resistance to impact damage or delamination, woven composites have gained a considerable interest as construction and repairing materials in transport industries and civil structures. However, the in-plane properties are reduced because of the undulating yarns. One of the attempts to alleviate the loss of the in-plane properties is the use of the fabric with a small waviness, which is experimentally analysed in this paper. As for numerical approaches, the classical laminate plate theory is not in place for the fabric composite owing to the yarn crimp effects and hence the analysis about the stress distribution and deformation have been accomplished by using finite element methods [1 – 5] or various theoretical models [5,6]. In particular, Ito et al. [5] reported the mechanical moduli and the tensile behaviour according to the waviness of the single lamina and the iso-phase and out-of-phase laminates using two-dimensional models in the length and thickness directions and Woo et al. [6] investigated the moduli of the laminate using a threedimensional model according to the waviness and the phase * Corresponding author. Tel.: þ33-4-77-42-0048; fax: þ 33-4-77-420249. E-mail address: [email protected] (J.-R. Lee). 1359-835X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2004.01.022
shift between the layers. On the other hand, the complex yarn architecture does not permit with ease any quantitative experimental analysis even if the increasing use of woven composites still requires a comprehensive knowledge of their mechanical behaviours. This is because the classical pointwise sensors are not appropriate to analyse the complex architecture. Therefore, optical full-field methods such as photogrammetry analysis [4] and classical moire´ interferometry [7] have been utilized for this material. In the present paper, a plain-weave fabric lamina with a very small waviness was tested by digital phase-shifting grating shearography and the results are compared with an isophase double laminate. Grating shearography is a combination of three techniques, which are a phase-shifting technique, shearography and diffraction grating metrology. The introduction of a phase-shifting technique allows the quantitative and automated measurement. The shearography is mostly insensitive to vibration and thus there is not in need of the stringent vibration isolation. For the purpose of isolating strain, the shearography technique does not require numerical differentiation because the strain is directly the function of the displacement derivatives to be obtained by the optical differentiation. The use of the artificial grating provides an excellent signal-to-noise ratio (SNR) and a low laser power requirement. The former lead to excellent performances in the aspect of spatial resolution and the latter make it possible the easy realization of four
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illumination directions for six mechanical measurands, which are three in-plane strains, in-plane rotation and two directional slopes of a deformed object surface.
2. Description of digital phase-shifting grating shearography Fig. 1 shows grating shearography setup: a 150-mm diameter collimated beam obtained from 10-mW He –Ne laser (l ¼ 632:8 nm) illuminates the front surface of the specimen covered with a diffraction grating-1200 lines/mm (pitch ¼ 833.3 nm) and a classical three-mirror setup [8]. A screen located in front of the collimating lens1 allows only one part of the beam to pass (A, B, C or D). The diffracted beam is focused by the lens2, and then sheared in
Michelson interferometer. The Michelson interfermeter has a PZT-actuated mirror (3-PZT device PSH 1z NV, Piezosystem Jena), which is capable of tilting the mirror. Therefore, we can adjust the shear distance in the xðyÞ direction by tilting the mirror around the yðxÞ axis. The sheared and diffracted beams generate a fringe pattern in the image plane. The real image is focused by the lens3 on a rotating semi-transparent glass plate to remove speckle noise. The formed image on the glass plate is observed by a 752 £ 582 CCD camera equipped with a standard lens system (VCL-12 YM). A temporal phase-shifting technique is introduced for the quantitative phase determination of the acquired intensity field with a fringe pattern. The intensity field can be rearranged in the form of Iði; jÞ ¼ kIl½1 þ mði; jÞcos fði; jÞ
Fig. 1. Grating shearography system: (a) optical arrangement, (b) three-mirror setup, (c) diffraction grating [10].
ð1Þ
J.-R. Lee et al. / Composites: Part A 35 (2004) 849–859
851
Fig. 2. Image acquisition and processing procedure to obtain phase change maps.
where ði; jÞ is the pixel coordinates, kIl is the average intensity, m is the contrast, and f is the phase map in each load step. The phase-shifting algorithm used in this paper is the windowed discrete Fourier transform [9]. The phase shift is p=2 and seven phase-shifted intensity samples are employed for the evaluation of one phase map. The resulting form with a triangular window is represented by tan f ¼
ðI0 2 I6 Þ 2 3ðI2 2 I4 Þ 2ðI1 þ I5 Þ 2 4I3
ð2Þ
The whole image processing including this phase determination method is composed of several steps until raw measurands. Eight phase maps at each load step are determined by the combination between four directional illuminations and two directional shear distances as
presented in Fig. 2. Successively, we calculate eight phase change maps for each combination between the reference state ðrÞ and deformed state ðdÞ; i.e. Dfx;x ¼ fdx;x 2 frx;x ;Df2x;x ;Dfy;x ;Df2y;x ;Dfx;y ;Df2x;y ;Dfy;y ;Df2y;y ; where the first index represent each direction of the sensitivity vector ðg ¼ ko 2 ki Þ and the second one indicates the shearing direction. This phase change map is the raw measurand of the phase-shifting shearography. As shown in Fig. 3, the six mechanical measurands are next isolated at each measuring step by using the values of the incidence angle ðu ¼ 49:418Þ; the wavelength of the laser and the applied shear distances ðDx ¼ DyÞ: The shear distance is a function of the relative angle between the two mirrors inside the Michelson interferometer. The shear distances of this Michelson interferometer-based
Fig. 3. Mechanical setup: tensile test machine and specimens (in mm).
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Fig. 4. Post-image processing procedure to isolate mechanical measurands.
Fig. 5. Numerical filtering: (a) correspondence between the strain map before and after filtering and the fabric mesh, (b) effect of the Gaussian and sine/cosine separable median filtering.
J.-R. Lee et al. / Composites: Part A 35 (2004) 849–859
shearography are precisely evaluated by a grid method with a spatial phase-shifting technique. Practically, the shear distances of about 100 mm are used in this paper. More information about grating shearography can be found in Refs. [10,11].
3. Mechanical experiment The aim of this experimental study is to analyse heterogeneous strain fields in fabrics under uniaxial tension. A one-ply lamina as the fundamental construction
853
to stack a fabric laminate was first investigated. The object was a T700S/M10 12K plain-weave carbon fabric (48192, Hexcel Corporation), having the fiber and resin tensile moduli of 230 and 3.2 GPa and the waviness ðhy =2aÞ of 0.0078. In Fig. 3, one unit cell without its resin for the clarity of drawing presents the mesostructure of the fabric. A unit cell consists of two half-warp yarns and two half-fill yarns, and the warp yarns undulate crossing over and under the fill yarns. The size of one unit cell is about 8 £ 8 mm2 and the inspecting zone contains six unit cells. In the case of only one-ply fabric lamina a pure resin region caused by loose weaving is clearly identified
Fig. 6. Tensile strain maps and strain profiles along x1 - and x2 -lines parallel to the loading axis.
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Fig. 7. Local bending effect induced by the stretch of the undulating warp yarn.
in the middle of each unit cell. Fig. 3 also presents the tensile test machine and the specimen. For the second experiment, the first and the second plies in Fig. 3 were stacked so as to have iso-phase between the layers. The thickness of the single lamina and the iso-phase double laminate was 0.25 and 0.52 mm, respectively. In order to obtain the same configuration in the aspect of the yarn architecture both specimens were cut along the boundaries of the warp yarns. Even if the same product was used, the sizes of unit cell were little by little different and thus the width of specimen was a little different, 30.66 mm for the single lamina and 29.1 mm for the double laminate. Consequently, the applied loads were divided by the respective widths with the aim of comparing the two experiments. Each grating was glued in the middle of the front surface of each specimen. A displacement was imposed on the movable jaw in the tensile test machine. The load was controlled using
a classical load cell and the respective specimens were loaded in the four steps.
4. Post-image processing During the tensile test, the grating will act as a sensor because the surface deformation of the specimen induced by external load is digitized in the form of a phase change corresponding to the change of the attached grating. Due to the feature of the excellent SNR of grating shearography, the phase change maps can be directly converted into the six displacement derivative maps of Fig. 4 without filtering. This is possible due to the quasi-plane wavefront diffracted from the grating and the optical temporal filtering by the use of a semi-transparent rotating glass plate. The displacement derivative maps are next filtered and unwrapped. As shown in Fig. 4, the four-displacement
Fig. 8. Local x- and y-slope maps at 8 N/mm load step.
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derivative maps are directly converted into the four mechanical measurands, tensile strain, transverse strain, x- and y-slope maps. The other two measurands are isolated by the combination of the two remaining displacement derivative maps, ›u=›y and ›v=›x: As for filtering, the two-dimensional low-pass filter with a Gaussian kernel is applied to the six-displacement derivative maps because Gaussian kernel is more efficient
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than a box kernel in the aspects of the high frequency rejection, the preservation of spatial resolution and the small distortion of signal. The energy of the displacement derivative maps is primarily concentrated on its lowfrequency components because the mechanical behaviour of the specimen induces the high spatial correlation among neighboring pixels. On the other side, the energy of such degradation causes of phase map as wideband random
Fig. 9. Transverse strain maps and strain profiles along y0 -line perpendicular to the loading axis.
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optical and electronic noise or speckle noise is typically more spread out over the frequency domain. By cutting off the high-frequency components while preserving the lowfrequency signal, this Gaussian low-pass filtering suppresses a large amount of noise at the expense of reducing a small amount of signal. A separable median filter with a horizontal n £ 1 kernel and a vertical 1 £ n kernel is also applied if a filtered image still has some salt-and-pepper noise, because it can remove efficiently the impulsive noise without distortion for the local linear data in the onedimensional kernel. The salt-and-pepper noise should be removed before the step of the phase unwrapping, because it makes the unwanted phase jump during the line-by-line unwrapping process used in this study.
Although the line-by-line unwrapping process is the simplest algorithm, it is enough to unwrap the measurand map due to its excellent SNR. Another advantage of the linear line-by-line processing conserves the spatial resolution of the wrapped image. For example, the tensile strain map in the case of the single lamina obtained by these post-image processing procedures is shown in Fig. 5a, which also shows the correspondence between the strain map and the fabric mesh. Fig. 5b shows the filtering effect on a cross-section along x0 line in Fig. 5a. The dominant strain information was not changed. Finally, all mechanical measurands post-processed in both experiments for the single lamina and the iso-phase double laminate have the spatial resolution of
Fig. 10. Shear strain maps and strain profiles along x3 - and x4 -line parallel to the loading axis.
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Fig. 11. In-plane shear deformation of the warp and the fill yarns.
about 0.8 mm. This means the measuring area comprises about 600 pieces of three-elements stacked rosette with the gage area of 0.8 £ 0.8 mm2. 5. Results First, we here outline the results of the experimental analysis of the single lamina. Fig. 6 presents the postprocessed tensile strain map at each load step. The tensile
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strain of the fill yarn in the matrix dominant direction is higher than the value of the warp yarn in the fiber dominant direction. Two crescent-shape strain concentration regions occur between the vertical centerline of the fill yarn and its vertical borders. The two tensile strain concentration regions on the fill yarn are represented in the line profile graphs of Fig. 6 by two peaks. However, tensile strain decreases significantly from the beginning line of the warp yarn, which is the beginning of the fiber dominant region. The lowest tensile strain value occurs not in the middle of the warp yarn but in the resin rich region. Such a result on the warp yarn gives rise to two valleys in the line profile graphs. This is due to the local bending effect induced by the stretch of the warp yarn. The tension/bending coupling effect caused by the local bending effect of the undulating yarns is quite significant when dealing with one-ply woven composites because there is no adjacent ply, which can constrain the flexural deformation. Fig. 7 provides a rough schematic view of the local bending effect in the single lamina. In isolating the average tensile strain, the strain induced by local flexural deformation is compressive at the zone D –E and tensile at the zone E – F. Both the C – D and F – G zones existing visibly in the fabric with a large unit cell remain unchanged before and after deformation. The effect is more remarkable since the regions where the local bending effect occurs are resin rich regions. Therefore, two peaks and two valleys appear on the fill yarn and on the warp yarn, respectively, as already shown in Fig. 6. Actually, as presented in the local x- and y-slope maps in Fig. 8, warp yarns go down and fill yarns go up by reason of stretching the specimen. To characterize the local slope behaviours within the fabric cells, the global slopes
Fig. 12. Local x- and y-slope maps of the iso-phase double laminate at 8 N/mm load step.
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(averaged values) were subtracted from the original maps. The sudden changes of the behaviours occur at the horizontal and vertical crossings of the warp and the fill yarns. The fundamental interest of the slope mapping is clearly enlightened by this analysis. If the previous analyses about the tensile strain and local slope maps were reasonable, there should be considerable transverse tensile strain against Poisson’s effect on the ascended fill yarn. As shown in Fig. 9, the fill yarns are in tension. The line profiles on the warp yarn at each step are also shown in Fig. 9 and there are two distinct valleys on the warp yarn. The flexural deformation induced by the local bending effect and the resin rich region can also explain the heterogeneity of transverse strain maps even if they differ from the tensile strain maps in that the compression by Poisson’s effect governs the transverse strain maps. Fig. 10 presents the in-plane shear strain maps and their line profiles along the x3 - and x4 -line parallel to loading axis. It is shown that shear strains reach high magnitudes in the vicinity of the six-pure resin regions. This is the tension/ in-plane shear coupling effect induced by the local bending effect. In focusing now on the upper fill yarn, it is clear that the two diagonally opposite corners, which correspond to pure resin regions, undergo positive shear strains while the other two corners sustain negative shear strain values. It is easy to get a qualitative outline of the global shear behaviour of the yarns when the fabric is loaded in tension along the orthographic axes. Fig. 11 displays some basic results. In both the warp and fill yarn, the local areas with no shear are naturally located on the symmetry axes and are symbolized as squares surrounded by zeros while the nonzero areas are located in the corners. The diagonally opposite corners of each yarn are the same sign. Signs of the four corners in the warp yarn have the reverse signs as compared with the fill yarn.
6. Discussions 6.1. Comparison with the iso-phase double fabric laminate
Fig. 13. Comparison of the in-plane strain maps between the single lamina and the iso-phase double laminate at the same averaged surface tensile strain of 1290m1.
The x- and y-slope maps in Fig. 12 show that the local bending effect such as in the single lamina still exists in the iso-phase double laminate because its configuration allows the up and down movement of the yarns. Fig. 13 presents the comparison in the in-plane strain maps between the two cases at the same surface tensile strain of 1290m1; which was obtained by averaging over the six unit cells, respectively. In the tensile strain maps, the two peaks and the two valleys that have been in the fill and warp yarns of the single lamina smoothened in the iso-phase double laminate. Similarly, the strain concentrations in the transverse and shear strain maps were also dispersed. These alleviations of the tension/bending and the tension/inplane shear couplings are because two neighboring layers constrain on each other. It should be here noted that
J.-R. Lee et al. / Composites: Part A 35 (2004) 849–859
the respective measurands after post-image processing in both experiments were controlled so as to have the same spatial resolution.
7. Conclusion Digital phase-shifting grating shearography has been applied for the investigation of the elementary behaviour of carbon/epoxy plain-weave fabric composite with large unit cells under uniaxial tension. Experimental analyses have been performed on the surfaces of a single lamina and an iso-phase double laminate. We begun to perform the experimental analysis with the single lamina as a basic element and then the results were compared and discussed with the second results about the iso-phase double laminate. The final measurands in each specimen, i.e. the surface slopes and in-plane strain fields made it possible to catch an in-depth understanding of the mechanical behaviour of the fabrics under tension. In both cases the tension/bending and tension/in-plane shear coupling effects revealed and their cause was the local bending effects of the undulating yarns caused from their configurations allowing the transverse shear strain ð1zx Þ: As for the number of the layers, the tension/bending and tension/in-plane shear coupling effects smoothened in the iso-phase double laminate because the other layer works as the boundary condition of one side of the first layer. It should be noted that these measurements were possible because the 0.8-mm spatial resolution was small enough to follow these yarn crimp effects. Further experimental works
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are planned to understand the configurations of an out-ofphase double laminate and multi-layer laminates.
References [1] Whitcomb J. Three-dimensional stress analysis of plain weave composites. In: ASTM STP 1110, Philadelphia, PA; 1991. p. 417– 38. [2] Chapman C, Whitcomb J. Effect of assumed tow architecture on predicted moduli and stresses in plain weave composites. J Compos Mater 1995;29(16):2134– 59. [3] Naik NK, Ganesh VK. Failure behavior of plain weave fabric laminates under on-axis uniaxial tensile loading. II. Analytical predictions. J Compos Mater 1996;30(16):1779–822. [4] Nishiyabu K, Uetsuji Y, Masaru Z, Prodromou A, Huysmans G, Verpoest I. Strain fields on plain woven fabric composites under onaxis tensile load: comparison of experimental and numerical results. In: Proceeding of ECCM-9 Conference, Brighton, UK; 2000. [5] Ito M, Chou TW. Elastic moduli and stress field of plain-weave composites under tensile loading. Compos Sci Technol 1997;57: 787–800. [6] Woo KS, Whitcomb J. Effects of fiber tow misalignment on the engineering properties of plain weave textile composites. Compos Struct 1997;37(3/4):343–55. [7] Post D, Han B, Ifju P. High sensitivity Moire´: experimental analysis for mechanics and materials. Berlin: Springer; 1994. p. 313–29. [8] Post D, Han B, Ifju P. High sensitivity Moire´: experimental analysis for mechanics and materials. Berlin: Springer; 1994. p. 136–226. [9] Surrel Y. Fringe analysis. In: Rastogi PK, editor. Photomechanics. Berlin: Springer; 1999. p. 55 –102. [10] Bulhak J, Surrel Y. Grating shearography. In: SPIE3744, Pultusk, Poland; 1999. p. 506 –15. [11] Lee JR, Molimard J, Vautrin A, Surrel Y. Application of grating shearography and speckle shearography to mechanical analysis of composite material. Composites, Part A 2004; in press.
Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension Jung-Ryul Leea,*, J. Molimarda, A. Vautrina, Y. Surrelb a
Department of Mechanical and Materials Engineering, Ecole Nationale Supe´rieure des Mines de Saint-Etienne, 158 Cours Fauriel, Saint-Etienne Cedex 2, 42023, France b BNM, INM/CNAM, 292 rue Saint Martin, 75141 Paris, France
Abstract Digital phase-shifting grating shearography has been applied for the investigation of the tensile behaviour of carbon/epoxy plain-weave fabric composite with a small waviness. Experimental analyses were performed for the two following configurations: a single lamina and an iso-phase double laminate. The yarn crimp effects such as the tension/bending and tension/in-plane shear couplings were concentrated on the resin rich regions for the single lamina. In the latter case, the yarn crimp effects were still significant because although the transverse shear strain due to the local bending effect of yarns is a little constrained by the other neighboring layer, the degree of the constraint was certainly insufficient to degenerate the local bending effect. q 2004 Elsevier Ltd. All rights reserved. Keywords: A. Fabrics/textiles; B. Anisotropy; C. Laminate mechanics; D. Non-destructive testing-optical full-field method
1. Introduction Since the weaving architectures provide some beneficial properties over the unidirectional tape laminates such as improved resistance to impact damage or delamination, woven composites have gained a considerable interest as construction and repairing materials in transport industries and civil structures. However, the in-plane properties are reduced because of the undulating yarns. One of the attempts to alleviate the loss of the in-plane properties is the use of the fabric with a small waviness, which is experimentally analysed in this paper. As for numerical approaches, the classical laminate plate theory is not in place for the fabric composite owing to the yarn crimp effects and hence the analysis about the stress distribution and deformation have been accomplished by using finite element methods [1 – 5] or various theoretical models [5,6]. In particular, Ito et al. [5] reported the mechanical moduli and the tensile behaviour according to the waviness of the single lamina and the iso-phase and out-of-phase laminates using two-dimensional models in the length and thickness directions and Woo et al. [6] investigated the moduli of the laminate using a threedimensional model according to the waviness and the phase * Corresponding author. Tel.: þ33-4-77-42-0048; fax: þ 33-4-77-420249. E-mail address: [email protected] (J.-R. Lee). 1359-835X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2004.01.022
shift between the layers. On the other hand, the complex yarn architecture does not permit with ease any quantitative experimental analysis even if the increasing use of woven composites still requires a comprehensive knowledge of their mechanical behaviours. This is because the classical pointwise sensors are not appropriate to analyse the complex architecture. Therefore, optical full-field methods such as photogrammetry analysis [4] and classical moire´ interferometry [7] have been utilized for this material. In the present paper, a plain-weave fabric lamina with a very small waviness was tested by digital phase-shifting grating shearography and the results are compared with an isophase double laminate. Grating shearography is a combination of three techniques, which are a phase-shifting technique, shearography and diffraction grating metrology. The introduction of a phase-shifting technique allows the quantitative and automated measurement. The shearography is mostly insensitive to vibration and thus there is not in need of the stringent vibration isolation. For the purpose of isolating strain, the shearography technique does not require numerical differentiation because the strain is directly the function of the displacement derivatives to be obtained by the optical differentiation. The use of the artificial grating provides an excellent signal-to-noise ratio (SNR) and a low laser power requirement. The former lead to excellent performances in the aspect of spatial resolution and the latter make it possible the easy realization of four
850
J.-R. Lee et al. / Composites: Part A 35 (2004) 849–859
illumination directions for six mechanical measurands, which are three in-plane strains, in-plane rotation and two directional slopes of a deformed object surface.
2. Description of digital phase-shifting grating shearography Fig. 1 shows grating shearography setup: a 150-mm diameter collimated beam obtained from 10-mW He –Ne laser (l ¼ 632:8 nm) illuminates the front surface of the specimen covered with a diffraction grating-1200 lines/mm (pitch ¼ 833.3 nm) and a classical three-mirror setup [8]. A screen located in front of the collimating lens1 allows only one part of the beam to pass (A, B, C or D). The diffracted beam is focused by the lens2, and then sheared in
Michelson interferometer. The Michelson interfermeter has a PZT-actuated mirror (3-PZT device PSH 1z NV, Piezosystem Jena), which is capable of tilting the mirror. Therefore, we can adjust the shear distance in the xðyÞ direction by tilting the mirror around the yðxÞ axis. The sheared and diffracted beams generate a fringe pattern in the image plane. The real image is focused by the lens3 on a rotating semi-transparent glass plate to remove speckle noise. The formed image on the glass plate is observed by a 752 £ 582 CCD camera equipped with a standard lens system (VCL-12 YM). A temporal phase-shifting technique is introduced for the quantitative phase determination of the acquired intensity field with a fringe pattern. The intensity field can be rearranged in the form of Iði; jÞ ¼ kIl½1 þ mði; jÞcos fði; jÞ
Fig. 1. Grating shearography system: (a) optical arrangement, (b) three-mirror setup, (c) diffraction grating [10].
ð1Þ
J.-R. Lee et al. / Composites: Part A 35 (2004) 849–859
851
Fig. 2. Image acquisition and processing procedure to obtain phase change maps.
where ði; jÞ is the pixel coordinates, kIl is the average intensity, m is the contrast, and f is the phase map in each load step. The phase-shifting algorithm used in this paper is the windowed discrete Fourier transform [9]. The phase shift is p=2 and seven phase-shifted intensity samples are employed for the evaluation of one phase map. The resulting form with a triangular window is represented by tan f ¼
ðI0 2 I6 Þ 2 3ðI2 2 I4 Þ 2ðI1 þ I5 Þ 2 4I3
ð2Þ
The whole image processing including this phase determination method is composed of several steps until raw measurands. Eight phase maps at each load step are determined by the combination between four directional illuminations and two directional shear distances as
presented in Fig. 2. Successively, we calculate eight phase change maps for each combination between the reference state ðrÞ and deformed state ðdÞ; i.e. Dfx;x ¼ fdx;x 2 frx;x ;Df2x;x ;Dfy;x ;Df2y;x ;Dfx;y ;Df2x;y ;Dfy;y ;Df2y;y ; where the first index represent each direction of the sensitivity vector ðg ¼ ko 2 ki Þ and the second one indicates the shearing direction. This phase change map is the raw measurand of the phase-shifting shearography. As shown in Fig. 3, the six mechanical measurands are next isolated at each measuring step by using the values of the incidence angle ðu ¼ 49:418Þ; the wavelength of the laser and the applied shear distances ðDx ¼ DyÞ: The shear distance is a function of the relative angle between the two mirrors inside the Michelson interferometer. The shear distances of this Michelson interferometer-based
Fig. 3. Mechanical setup: tensile test machine and specimens (in mm).
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J.-R. Lee et al. / Composites: Part A 35 (2004) 849–859
Fig. 4. Post-image processing procedure to isolate mechanical measurands.
Fig. 5. Numerical filtering: (a) correspondence between the strain map before and after filtering and the fabric mesh, (b) effect of the Gaussian and sine/cosine separable median filtering.
J.-R. Lee et al. / Composites: Part A 35 (2004) 849–859
shearography are precisely evaluated by a grid method with a spatial phase-shifting technique. Practically, the shear distances of about 100 mm are used in this paper. More information about grating shearography can be found in Refs. [10,11].
3. Mechanical experiment The aim of this experimental study is to analyse heterogeneous strain fields in fabrics under uniaxial tension. A one-ply lamina as the fundamental construction
853
to stack a fabric laminate was first investigated. The object was a T700S/M10 12K plain-weave carbon fabric (48192, Hexcel Corporation), having the fiber and resin tensile moduli of 230 and 3.2 GPa and the waviness ðhy =2aÞ of 0.0078. In Fig. 3, one unit cell without its resin for the clarity of drawing presents the mesostructure of the fabric. A unit cell consists of two half-warp yarns and two half-fill yarns, and the warp yarns undulate crossing over and under the fill yarns. The size of one unit cell is about 8 £ 8 mm2 and the inspecting zone contains six unit cells. In the case of only one-ply fabric lamina a pure resin region caused by loose weaving is clearly identified
Fig. 6. Tensile strain maps and strain profiles along x1 - and x2 -lines parallel to the loading axis.
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Fig. 7. Local bending effect induced by the stretch of the undulating warp yarn.
in the middle of each unit cell. Fig. 3 also presents the tensile test machine and the specimen. For the second experiment, the first and the second plies in Fig. 3 were stacked so as to have iso-phase between the layers. The thickness of the single lamina and the iso-phase double laminate was 0.25 and 0.52 mm, respectively. In order to obtain the same configuration in the aspect of the yarn architecture both specimens were cut along the boundaries of the warp yarns. Even if the same product was used, the sizes of unit cell were little by little different and thus the width of specimen was a little different, 30.66 mm for the single lamina and 29.1 mm for the double laminate. Consequently, the applied loads were divided by the respective widths with the aim of comparing the two experiments. Each grating was glued in the middle of the front surface of each specimen. A displacement was imposed on the movable jaw in the tensile test machine. The load was controlled using
a classical load cell and the respective specimens were loaded in the four steps.
4. Post-image processing During the tensile test, the grating will act as a sensor because the surface deformation of the specimen induced by external load is digitized in the form of a phase change corresponding to the change of the attached grating. Due to the feature of the excellent SNR of grating shearography, the phase change maps can be directly converted into the six displacement derivative maps of Fig. 4 without filtering. This is possible due to the quasi-plane wavefront diffracted from the grating and the optical temporal filtering by the use of a semi-transparent rotating glass plate. The displacement derivative maps are next filtered and unwrapped. As shown in Fig. 4, the four-displacement
Fig. 8. Local x- and y-slope maps at 8 N/mm load step.
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derivative maps are directly converted into the four mechanical measurands, tensile strain, transverse strain, x- and y-slope maps. The other two measurands are isolated by the combination of the two remaining displacement derivative maps, ›u=›y and ›v=›x: As for filtering, the two-dimensional low-pass filter with a Gaussian kernel is applied to the six-displacement derivative maps because Gaussian kernel is more efficient
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than a box kernel in the aspects of the high frequency rejection, the preservation of spatial resolution and the small distortion of signal. The energy of the displacement derivative maps is primarily concentrated on its lowfrequency components because the mechanical behaviour of the specimen induces the high spatial correlation among neighboring pixels. On the other side, the energy of such degradation causes of phase map as wideband random
Fig. 9. Transverse strain maps and strain profiles along y0 -line perpendicular to the loading axis.
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optical and electronic noise or speckle noise is typically more spread out over the frequency domain. By cutting off the high-frequency components while preserving the lowfrequency signal, this Gaussian low-pass filtering suppresses a large amount of noise at the expense of reducing a small amount of signal. A separable median filter with a horizontal n £ 1 kernel and a vertical 1 £ n kernel is also applied if a filtered image still has some salt-and-pepper noise, because it can remove efficiently the impulsive noise without distortion for the local linear data in the onedimensional kernel. The salt-and-pepper noise should be removed before the step of the phase unwrapping, because it makes the unwanted phase jump during the line-by-line unwrapping process used in this study.
Although the line-by-line unwrapping process is the simplest algorithm, it is enough to unwrap the measurand map due to its excellent SNR. Another advantage of the linear line-by-line processing conserves the spatial resolution of the wrapped image. For example, the tensile strain map in the case of the single lamina obtained by these post-image processing procedures is shown in Fig. 5a, which also shows the correspondence between the strain map and the fabric mesh. Fig. 5b shows the filtering effect on a cross-section along x0 line in Fig. 5a. The dominant strain information was not changed. Finally, all mechanical measurands post-processed in both experiments for the single lamina and the iso-phase double laminate have the spatial resolution of
Fig. 10. Shear strain maps and strain profiles along x3 - and x4 -line parallel to the loading axis.
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Fig. 11. In-plane shear deformation of the warp and the fill yarns.
about 0.8 mm. This means the measuring area comprises about 600 pieces of three-elements stacked rosette with the gage area of 0.8 £ 0.8 mm2. 5. Results First, we here outline the results of the experimental analysis of the single lamina. Fig. 6 presents the postprocessed tensile strain map at each load step. The tensile
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strain of the fill yarn in the matrix dominant direction is higher than the value of the warp yarn in the fiber dominant direction. Two crescent-shape strain concentration regions occur between the vertical centerline of the fill yarn and its vertical borders. The two tensile strain concentration regions on the fill yarn are represented in the line profile graphs of Fig. 6 by two peaks. However, tensile strain decreases significantly from the beginning line of the warp yarn, which is the beginning of the fiber dominant region. The lowest tensile strain value occurs not in the middle of the warp yarn but in the resin rich region. Such a result on the warp yarn gives rise to two valleys in the line profile graphs. This is due to the local bending effect induced by the stretch of the warp yarn. The tension/bending coupling effect caused by the local bending effect of the undulating yarns is quite significant when dealing with one-ply woven composites because there is no adjacent ply, which can constrain the flexural deformation. Fig. 7 provides a rough schematic view of the local bending effect in the single lamina. In isolating the average tensile strain, the strain induced by local flexural deformation is compressive at the zone D –E and tensile at the zone E – F. Both the C – D and F – G zones existing visibly in the fabric with a large unit cell remain unchanged before and after deformation. The effect is more remarkable since the regions where the local bending effect occurs are resin rich regions. Therefore, two peaks and two valleys appear on the fill yarn and on the warp yarn, respectively, as already shown in Fig. 6. Actually, as presented in the local x- and y-slope maps in Fig. 8, warp yarns go down and fill yarns go up by reason of stretching the specimen. To characterize the local slope behaviours within the fabric cells, the global slopes
Fig. 12. Local x- and y-slope maps of the iso-phase double laminate at 8 N/mm load step.
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(averaged values) were subtracted from the original maps. The sudden changes of the behaviours occur at the horizontal and vertical crossings of the warp and the fill yarns. The fundamental interest of the slope mapping is clearly enlightened by this analysis. If the previous analyses about the tensile strain and local slope maps were reasonable, there should be considerable transverse tensile strain against Poisson’s effect on the ascended fill yarn. As shown in Fig. 9, the fill yarns are in tension. The line profiles on the warp yarn at each step are also shown in Fig. 9 and there are two distinct valleys on the warp yarn. The flexural deformation induced by the local bending effect and the resin rich region can also explain the heterogeneity of transverse strain maps even if they differ from the tensile strain maps in that the compression by Poisson’s effect governs the transverse strain maps. Fig. 10 presents the in-plane shear strain maps and their line profiles along the x3 - and x4 -line parallel to loading axis. It is shown that shear strains reach high magnitudes in the vicinity of the six-pure resin regions. This is the tension/ in-plane shear coupling effect induced by the local bending effect. In focusing now on the upper fill yarn, it is clear that the two diagonally opposite corners, which correspond to pure resin regions, undergo positive shear strains while the other two corners sustain negative shear strain values. It is easy to get a qualitative outline of the global shear behaviour of the yarns when the fabric is loaded in tension along the orthographic axes. Fig. 11 displays some basic results. In both the warp and fill yarn, the local areas with no shear are naturally located on the symmetry axes and are symbolized as squares surrounded by zeros while the nonzero areas are located in the corners. The diagonally opposite corners of each yarn are the same sign. Signs of the four corners in the warp yarn have the reverse signs as compared with the fill yarn.
6. Discussions 6.1. Comparison with the iso-phase double fabric laminate
Fig. 13. Comparison of the in-plane strain maps between the single lamina and the iso-phase double laminate at the same averaged surface tensile strain of 1290m1.
The x- and y-slope maps in Fig. 12 show that the local bending effect such as in the single lamina still exists in the iso-phase double laminate because its configuration allows the up and down movement of the yarns. Fig. 13 presents the comparison in the in-plane strain maps between the two cases at the same surface tensile strain of 1290m1; which was obtained by averaging over the six unit cells, respectively. In the tensile strain maps, the two peaks and the two valleys that have been in the fill and warp yarns of the single lamina smoothened in the iso-phase double laminate. Similarly, the strain concentrations in the transverse and shear strain maps were also dispersed. These alleviations of the tension/bending and the tension/inplane shear couplings are because two neighboring layers constrain on each other. It should be here noted that
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the respective measurands after post-image processing in both experiments were controlled so as to have the same spatial resolution.
7. Conclusion Digital phase-shifting grating shearography has been applied for the investigation of the elementary behaviour of carbon/epoxy plain-weave fabric composite with large unit cells under uniaxial tension. Experimental analyses have been performed on the surfaces of a single lamina and an iso-phase double laminate. We begun to perform the experimental analysis with the single lamina as a basic element and then the results were compared and discussed with the second results about the iso-phase double laminate. The final measurands in each specimen, i.e. the surface slopes and in-plane strain fields made it possible to catch an in-depth understanding of the mechanical behaviour of the fabrics under tension. In both cases the tension/bending and tension/in-plane shear coupling effects revealed and their cause was the local bending effects of the undulating yarns caused from their configurations allowing the transverse shear strain ð1zx Þ: As for the number of the layers, the tension/bending and tension/in-plane shear coupling effects smoothened in the iso-phase double laminate because the other layer works as the boundary condition of one side of the first layer. It should be noted that these measurements were possible because the 0.8-mm spatial resolution was small enough to follow these yarn crimp effects. Further experimental works
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are planned to understand the configurations of an out-ofphase double laminate and multi-layer laminates.
References [1] Whitcomb J. Three-dimensional stress analysis of plain weave composites. In: ASTM STP 1110, Philadelphia, PA; 1991. p. 417– 38. [2] Chapman C, Whitcomb J. Effect of assumed tow architecture on predicted moduli and stresses in plain weave composites. J Compos Mater 1995;29(16):2134– 59. [3] Naik NK, Ganesh VK. Failure behavior of plain weave fabric laminates under on-axis uniaxial tensile loading. II. Analytical predictions. J Compos Mater 1996;30(16):1779–822. [4] Nishiyabu K, Uetsuji Y, Masaru Z, Prodromou A, Huysmans G, Verpoest I. Strain fields on plain woven fabric composites under onaxis tensile load: comparison of experimental and numerical results. In: Proceeding of ECCM-9 Conference, Brighton, UK; 2000. [5] Ito M, Chou TW. Elastic moduli and stress field of plain-weave composites under tensile loading. Compos Sci Technol 1997;57: 787–800. [6] Woo KS, Whitcomb J. Effects of fiber tow misalignment on the engineering properties of plain weave textile composites. Compos Struct 1997;37(3/4):343–55. [7] Post D, Han B, Ifju P. High sensitivity Moire´: experimental analysis for mechanics and materials. Berlin: Springer; 1994. p. 313–29. [8] Post D, Han B, Ifju P. High sensitivity Moire´: experimental analysis for mechanics and materials. Berlin: Springer; 1994. p. 136–226. [9] Surrel Y. Fringe analysis. In: Rastogi PK, editor. Photomechanics. Berlin: Springer; 1999. p. 55 –102. [10] Bulhak J, Surrel Y. Grating shearography. In: SPIE3744, Pultusk, Poland; 1999. p. 506 –15. [11] Lee JR, Molimard J, Vautrin A, Surrel Y. Application of grating shearography and speckle shearography to mechanical analysis of composite material. Composites, Part A 2004; in press.