Detailed Morphology Modeling and Residual Stress Evaluation in Tri-axial Braided Composites

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1 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<br>17th 4-7 May 2009, Palm Springs, California AIAA Detailed Morphology Modeling and Residual Stress Evaluation in Tri-axial Braided Composites Timothy Breitzman 1 and David H. Mollenhauer. 2 Air Force Research Laboratory, WPAFB, OH Endel V. Iarve 3 University of Dayton Research Institute, Dayton, OH and Eric G. Zhou 4 University of Dayton Research Institute, Dayton, OH Local strain fields in tri-axial braided composites arising in the vicinity of a saw cut due to the release of thermal processing stresses were experimentally measured by using moiré interferometry technique and compared to those obtained by 3D stress analysis. The independent mesh method (IMM) and digital chain methods were used to perform the 3D stress analysis and the composite tow morphology modeling respectively. A significant degree of morphological detail was required to achieve good comparison with experimental data. Three degrees of refinement were produced in direction of matching the actual morphology of the tri-axial braded composite which was tested experimentally. These levels of refinement included (i) correct tow path angle and curvature variation based on braid parameters (ii) addition of the effect of compaction process during the cure stage and (iii) addition of the surface sanding affects during moiré test preparation stage. The results obtained by using IMM were able to capture sharp variations of the strain components observed by using the moiré interferometric technique both in terms of spatial distribution and magnitude and provide accurate evaluation of the residual strain levels in the triaxial braided composites. C I. Introduction OMPOSITE materials with complex fiber architectures have broad applications in the various aerospace applications and have been studied significantly in the past decades 1-3. Recently, several new 3-D stress analysis techniques have being proposed more recently. Belytchko, et al. 4 extended the idea of X-FEM, which was originally proposed for mesh-independent crack modeling to analysis of heterogeneous media with phase boundaries not aligned with FE mesh lines. In this formulation, an arbitrary mesh, e.g. uniform, is created analogous to the voxel method. However, the elements overlapping the phase boundary are enriched by adding the degrees of freedom responsible for the strain field jump at the phase boundary instead of adjusting the element stiffness properties. Other methods such as the Independent Mesh Method (IMM) 5 and the domain superposition method 6,7,8 combine direct finite element meshing and voxel methodology features. These methods are based on direct and independent discretization of individual fiber tows as well as matrix (the latter can often be subdivided uniformly) and differ in the way these tows are connected to each other and the entire volume of fibers and matrix without distinction. In the IMM, the shape functions of the matrix displacement approximation are reduced by excluding all functions entirely covered by tows and modifying the integration domain of the shape functions partially covered by tows, as described below. All connections between contacting fiber tows and fiber and matrix are applied by using the penalty function method. In the domain superposition techniques, both the matrix and the fiber tow 1 Materials Research Engineer, AFRL/RXBC, 2941 Hobson Way, member. 2 Senior Materials Research Engineer, AFRL/RXBC, 2941 Hobson Way, non-member. 3 Distinguished Research Engineer, Multi-Scale Composites & Polymers, 300 College Park Drive, member. 4 Research Engineer, Multi-Scale Composites & Polymers, 300 College Park Drive, member. 1 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

2 displacement approximation functions are simultaneously used without modification. The method described in 7 is similar to the s-element method proposed by Fish 6, where one is required to calculate the cross integrals of both sets of shape functions in the fiber tow domains. The domain superposition method by Jiang and Hallett 8 proposes using a different approach. In this case, nodal constraints in the fiber volume are applied that require the tow displacement to be equal to that of the matrix. The fiber domain elastic properties are then modified by subtracting the matrix stiffness tensor. No connections between the fiber tows are required in methods 6-8 since all displacement constraints are imposed through the matrix. Methods 5-7 posses a significant degree of robustness and can tolerate small errors in fiber tow geometry with their common weakness being the approximate representation of the strain field jump at the phase interface. Only the X-FEM-based methodology allows for accurate representation of these jumps. However, besides requiring geometry definitions free of geometric errors, it also requires multiple and possibly very complex enrichment schemes to accommodate multiple fiber tow junctures and sharp corners, which might be difficult to parameterize upfront. This paper is devoted to describing an effort to develop a robust textile composite analysis method capable of capturing volume average fields as well as local stress-strain variations on the fiber tow level. In the present paper, we apply the IMM method 5,9 to predictions of residual stress release in a triaxially-braided composite with comparison to measurements obtained using moiré interferometry, which show good quantitative agreement. These predictions are based on tow morphology obtained by using the digital element chain technique of Zhou, et al. 10,11 and simulating the preform compaction at the processing stage. In addition effective stiffness computation for a plain weave RVE, for which the solution can also be readily obtained by using standard FE methodologies, is reported for validation purposes. I. Triaxially-Braided Unit Cell Consider an RVE of a triaxially-braided composite as shown in Figure 1. The RVE occupies a space L by W by H in the xyz-coordinates, and its entire volume, less the fiber tows, is filled with matrix. Here and below we consider the fiber tows as a unidirectional material with constant fiber volume fraction and the material stiffness symmetry axis continuously changing along its length. Figure 1: Coordinate axis, fiber tow geometry and matrix contour of a triaxially-braided RVE. In the case shown in Figure 1, load was applied in the y-direction as follows and is representative of a variety of boundary conditions that could be applied. Displacement boundary conditions are applied at the lateral edges y=0 and y=w along with symmetry conditions on the plane z=0 and rigid body constraint, so that u y (x,0,z)=0, u y (x,w,z)=u 0 (1) u z (x,y,0)=0, u x (0,y,z)=0, u x (L,y,z)=0 2

with traction-free boundary conditions applied to complement (1) on all surfaces. As seen in Figure 1, the edges of the tows were trimmed to conform to the edges of the RVE.

3 with traction-free boundary conditions applied to complement (1) on all surfaces. As seen in Figure 1, the edges of the tows were trimmed to conform to the edges of the RVE. In the following, we will consider the solution of the boundary value problem (1) for the RVE in Figure 1 by using IMM. We first perform 3-D spline approximation of displacements in the yarns, which are represented by volumes which can be parametrically mapped onto a unit cube. Then, we consider the matrix and its basis reduction as well as the integration of the shape functions excluding the volume of the fiber tows. At last, the displacement continuity is enforced by using the penalty function approach. Please refer to references 5,9 for procedure description. II. Plain Weave Quarter Unit Cell An idealized quarter model of the plain weave RVE, shown in Figure 2, was used to predict macroscopic stiffness and CTE values for a 2-D woven material. The constitutive relations in the global coordinates are written as =C(x)( -e(x)), (1) where and are the stress and strain tensor, and C(x) and e(x) are the stiffness matrix and nonmechanical strain tensor coefficients, so that, for curved fiber tows, C(x)=T 3 T 2 T 1 C uni T T 1T T 2T T 3, (2) e(x)= T T 1T T 2T T 3e uni, where T i are rotation matrices around x i axis and are defined in terms of f i vector components, and tensors C uni and e uni are constant stiffness and thermal-expansion coefficients of the unidirectional composite. The unidirectional properties used in the model were E 1 =135GPa, E 2 =E 3 =10.5GPa, 12 = 13 =0.30, 23 =0.45, G 12 =G 13 =6.3GPa, G 23 =3.6GPa, 1 = 1.1x10-6 / C, and 2 = 3 =18.3x10 6 / C. The matrix properties were E=3.45GPa, =0.35 and =4.68x10-5 / C. Because the model represented a quarter unit cell of the plain weave (not a full period), antisymmetric boundary conditions were applied in the plane for the shear loading cases according to Whitcomb 12 and are described as follows. For the σ xx, σ yy, σ zz, and thermal load cases, u x =0 on the x=0 and x=1 surfaces, and u y =0 on the y=0 and y=1 surfaces. In the σ xy load case, u x =0 on the surfaces y=0 and y=1, while u y =0 on the surfaces x=0 and x=1. For the σ xz load case, u z =0 on the x=0 and x=1 surfaces, while u y =0 on the y=0 and y=1 surfaces. Similarly, for the σ yz load case, u x =0 on the x=0 and x=1 surfaces, and u z =0 on the y=0 and y=1 surfaces. For all load cases, periodic displacement conditions were enforced on the z=0 and z=1 surfaces. In addition, rigid body motion was prevented by fixing all displacements at the point (x,y,z)=(0,0,0). The mesh density and subdivision in the fiber tows are identical in the case of the IMM solution and the standard FE analysis, as shown in Figure 2. In the case of IMM, two mesh densities in the matrix were examined: 8 intervals in all three directions and 16 intervals in all three directions. The results of effective properties prediction are shown in Figure 3. The effective properties predicted by using IMM and direct FE simulation are in good agreement. For an explanation of the k-integration cubes, see 5,9. (a) (b) 3

4 Figure 2: Quarter cell plain weave shown with minimum matrix mesh density of 8 by 8 by 8 for (a) IMM model and (b) FEA model. (a) (b) (c) (d) Figure 3: Average properties calculated by using the IMM for a quarter cell plain-weave model with matrix mesh densities of (8 by 8 by 8) dashed lines and (16 by 16 by 16) solid lines. Note: data points at k=18 are the converged FE solution results. III. Residual Stress Relief in a Triaxially-Braided Composite Understanding the residual stress distribution and magnitude is an important milestone on the road to strength and failure prediction in textile and nontextile composites. In previous efforts, the full-field deformation measurement method, moiré interferometry, was used to evaluate the redistribution of strain in laminated composites resulting from the release of residual stresses. 13 In that work, the authors cut a thin slot into the free edge of a composite, recorded the strain redistribution, and favorably compared the results with 3-D stress analysis. In the present work, similar ideas were applied to a triaxially-braided composite. The surface of the composite was lightly sanded with 600 grit sandpaper. A moiré interferometry diffraction grating was replicated onto the surface of composite. Using a diamond wafering saw, a 0.6-mm-wide slit was then saw cut perpendicular to the surface, as shown in Figure 4. The relief of residual stress as a result of such a cut will cause deformation of the grating. Alterations to the moiré fringe pattern were then recorded and analyzed to produce surface strain distributions. 4

5 Subsequent calculations were also performed by using the IMM and compared to experimental results along the two lines shown in Figure 4. Figure 4: Schematic and micrograph of the triaxially-braided composite examined with Moiré interferometry. Note: the red box indicates the region examined in the test, and the dotted lines show the location of a subsequent quantitative comparison. To aid in the understanding of the importance of morphology accuracy, two digitally simulated tow geometries were used for the analysis. The first geometry examined was a single layer of an uncompacted, triaxially-braided composite obtained via the methods described in 10. The second tow morphology examined, a fivelayer braid, was compacted as described by Zhou, et al 11 to simulate the preform compaction and layer nesting during the manufacturing process. From this simulated compacted morphology, only the top layer was used in the subsequent stress analyses. The cross-sectional views of these two morphologies, a third morphology discussed subsequently, and an x-ray computed tomography slice of the actual composite cross section, are displayed in Figure 5

5. Clearly, the morphology of the braided composite in any given cross section reflects a significant degree of randomness and can be compared to the respective digitally predicted geometries based

6 5. Clearly, the morphology of the braided composite in any given cross section reflects a significant degree of randomness and can be compared to the respective digitally predicted geometries based on key features rather than pointwise. Thus, the fiber tow cross sections of the uncompacted geometry, Figure 5a, have clearly much more rounded shapes than those of the compacted, Figure 5b, and those of the experimental in Figure 5d. The compacted geometry also produces much smaller gaps between the fiber tows in the in-plane view (not shown here), which is in good agreement with the experimental observations. (a) (b) (c) (d) Figure 5: (a) Uncompacted, (b) compacted, and (c) compacted-sanded geometry of the fiber tows compared to (d) an experimentally observed 5-layer cross section obtained with x-ray computed tomography (top layer digitally enhanced). An additional aspect taken into account in the IMM analysis was surface sanding and the uneven compaction of the actual specimen, resulting in a third tow morphology examined in the modeling effort. As mentioned above, the application of the moiré grating requires surface preparation, which resulted in the removal of excess resin from the surface of the composite and a slight reduction of the fiber tow thickness of the surface tows. Additionally, the compaction process during manufacturing more aggressively compacted the top layer tows than was represented in the predicted morphology. As a result, the tow geometries shown in Figure 5d are flatter on the surface than those shown in Figure 5b. As a further modification to the tow geometries of Figure 5b, a virtual sanding plane was created that effectively removed 0.09 mm from the top layer, producing the geometry shown in Figure 5c, which is clearly more representative of the actual morphology. 6

7 Thus, three numerical results are compared to experiment, namely 1) uncompacted tows, 2) compacted tows, and 3) compacted sanded tows. The details of the modeling process for these three numerical models are described generally as follows. Initially, results were obtained from a specimen of the size shown in Figure 4 that was subjected to free expansion from a T of 155 C without the saw cut present. The boundary conditions were u x =0 at x=0, u z =0 at x=z=0, and u y =0 at y=0. Then additional results were obtained from a specimen subjected to the same T with the slot as shown in Figure 4 with the same boundary conditions as before and the additional boundary conditions u x =0 at x=l and u z =0 at x=l & z=0. The final results presented below are the difference between these two numerical results. Physically, the saw cut was represented as a rectangular tow extruded through the region of interest. This rectangular tow was specified in the model input file to cut out any matrix and tow shape function crossing its boundary, while not being attached to the severed ends of the cut matrix or tow shape functions. In this manner, the saw cut was represented without modifying the original, uncut mesh lines and data structure, thus enabling easy subtraction of the initial and cut model results. All three tow morphologies were modeled in this manner. In addition, for the compacted sanded model, a surface rectangular tow was included that acted as a virtual sanding layer in the same manner as the virtual saw cutting tow. Material properties for the matrix were E=3.5 GPa, =0.33, and =30.5x10-6 / C, while for the unidirectional tows they were E 1 =135 GPa, E 2 =E 3 =10.5 GPa, 12 = 13 =0.30, 23 =0.45, G 12 =G 13 =6.3 GPa, G 23 =3.6 GPa, 1 = 1.1x10-6 / C, and 2 = 3 =18.3x10 6 / C. The modeled length in the x-direction (L) equaled mm, and the width in the z-direction (W) equaled 6.00 mm. The uncompacted morphology model was 1.01 mm thick, the compacted morphology model was 0.81 mm thick, and with the virtual sanding layer removed, the compacted sanded morphology model was 0.72 mm thick. Two in-plane strain components on each of the data lines in Figure 4 were examined. Figure 6 shows a comparison between the measured and predicted distribution of the zz and xx strain components along the saw cut. The experimentally measured strain distributions in all cases are characterized with well-localized high strain bands corresponding to the gaps between the fiber tows. This is especially pronounced along data line 1 ( above the cut, Figure 4). The IMM predictions obtained with the uncompacted geometry yield quite smeared values of the zz strain component, while the compacted and compacted sanded geometries yield, in most cases, remarkably good agreement with the experimental data. It is important to note that a significant degree of automation is also achieved between generating the input information for the stress analysis program BSAM and the digital chain element-based geometry generator, allowing examination of multiple cases with convenience. (a) 7

8 (b) Figure 6: In-plane strain variation along the saw cut edges: (a) corresponds to data line 1 and (b) to data line 2 shown in Figure 4. CONCLUSIONS Accurate prediction of average elastic properties by IMM was demonstrated and compared to results obtained by using traditional FE. The predicted effective properties are accurate in all cases for sufficiently high integration cube parameter k=4 and 8. Local strain fields in triaxially-braided composites arising in the vicinity of a saw cut due to the release of thermal residual stresses was performed. Three fiber tow morphologies corresponding to uncompacted, compacted, and compacted sanded specimen geometries were modeled by IMM. In both cases the geometric information was produced by the digital chain element modeling method, Zhou, et al. 10,11. The results obtained by using IMM were able to capture sharp variations of the strain components observed by using the Moiré interferometric technique and demonstrated robustness of the IMM to handle various complex fiber tow geometry configurations. Acknowledgments The work was funded by the Air Force Research Laboratory under University of Dayton Research Institute Contract FA D5052 References 1. Bogdanovich, A.E., Three-dimensional Continuum Micro-, Meso and Macro-mechanics of Textile Composites, Proc. 8 th Int. Conf on Textile Composites (TEXCOMP-8), Nottingham, Oct Bogdanovich, A.E. and Pastore, C.M., Mechanics of textile and laminated composites, London: Chapman & Hall; Lomov, S.V, Ivanov, D.S., Verpost, I., Zako, M., Kurashiki, T., Nakai, H., and Hirosawa, S., Meso-FE Modeling of Textile Composites: Road Map, Data Flow and Algorithms," 16 th International Conference on Composite Materials, Kyoto, Japan, July, 2007, Paper Pl07 4. Belytchko, T., Parimi, C., Moes, N., Sukumar, N., and Usui, S., Structured Extended Finite Element Methods for Solids Defined by Implicit Surfaces, Int. J Num Methods in Engineering, vol. 56, pp , Iarve, E.V., Mollenhauer, D.H., Zhou, E., and Whitney, T.J., Stress Analysis In Complex Fiber Architecture Composites: Independent Mesh Method, Finite Element Modeling of Textiles And Textile Composites, St- Petersburg, Russia, September, Fish, J., Yu, Q., and Shek, K., Computational Damage Mechanics for Composite Materials Based on Mathematical Homogenization, International Journal for Numerical Methods in Engineering, vol. 45, pp ,

9 7. Nakai, H., Kurashiki, T., and Zako, M., Individual Modeling of Composite Materials with Mesh Superposition Method Under Periodic Boundary Condition, 16 th International Conference on Composite Materials, Kyoto, Japan, July, 2007, Paper TukM1=03 8. Jiang, W-G., Hallett, S.R., and Wisnom, M.R., Development of Domain Superposition Technique for the Modeling of Woven Fabric Composites, book chapter in Mechanical Response of Composites, P.P. Camanho, et al. (eds.), Springer, Netherlands, Iarve, E.V. Zhou, G., Whitney, T. Mollenhauer, D. and Breitzman, T. (2008), Independent Mesh Method Based Prediction of Local and Volume Average Fields in Textile Composites, 49 th SDM AIAA, Schaumburg, Illinois, April, Zhou, G., Sun, X., and Wang, Y., Multi-Chain Digital Element Analysis in Textile Mechanics, Composites Science and Technology, vol. 64, pp , Zhou, G., Mollenhauer, D., and Iarve, E.V., Micro-Geometric Modeling of Textile Preforms with Vacuum Bag Compression: An Application of Multi-Chain Digital Element Technique, 49 th SDM AIAA, Schaumburg, Illinois, April, Tang, X., and Whitcomb, J.D., General Techniques for Exploiting Periodicity and Symmetries in Micromechanics Analysis of Textile Composites, J. Composite Materials, vol. 37, pp , Schoeppner, G.A., Mollenhauer, D.H., and Iarve, E.V., Prediction and Measurement of Residual Strains for a Composite Bonded Joint, Mechanics of Composite Materials, vol. 40, no. 2, pp ,

ANALYSIS AND MEASUREMENT OF SCARF-LAP AND STEP-LAP JOINT REPAIR IN COMPOSITE LAMINATES

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