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Stresses around fiber ends at free and embedded ply edges
Composites Science and Technology 68 (2008) 3352–3357
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Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Stresses around fiber ends at free and embedded ply edges E.W. Andrews a,*, M.R. Garnich b a b
Firehole Technologies, Wyoming Technology Business Center, 1000 E. University Avenue, Laramie, WY 82071, USA Department of Mechanical Engineering, 1000 E. University Avenue, University of Wyoming, Laramie, WY 82071, USA
a r t i c l e
i n f o
Article history: Received 3 April 2008 Received in revised form 15 August 2008 Accepted 4 September 2008 Available online 12 September 2008 Keywords: A. Polymer–matrix composites B. Thermomechanical properties C. Finite element analysis C. Stress concentrations
a b s t r a c t Using the finite element method, the stress state around fiber ends at a free surface in a unidirectional fiber composite was studied. The material is subjected to a thermal load consistent with a cryogenic application as well as ply level transverse tensile loading associated with thermal stresses in a laminate. At the free end the fiber–matrix interface sees tensile radial stresses as well as shear stresses. This makes the fiber end a likely site for interface debonding (crack initiation). These initial cracks can then link up and grow to become full through-thickness matrix cracks. The effect that an epoxy coating over the fiber ends has on the local stress distribution was also considered. These capped models simulate fiber ends embedded within a laminate. The epoxy cap has the effect of suppressing the interface stresses at the fiber end. The significance of these results in relation to cryogenic cycling tests using coupon specimens with free and embedded edges is discussed. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Carbon fiber reinforced plastic (CFRP) materials are candidate materials for cryogenic pressure vessel applications. When such materials are subjected to temperature variations from a given stress-free temperature, substantial thermal stresses can result. These can arise at the ply level due to different ply orientations in a laminate, as well as at the fiber/matrix level due to different thermal expansion coefficients for the constituents. The stresses can cause matrix cracking, which is significant for pressure vessel applications because of the leakage that may result. Subjecting flat coupons to cyclic cryogenic immersion is one approach for assessing the resistance of various CFRP material systems to this transverse microcracking, e.g. [1]. However, the presence of free edges in such samples complicates the interpretation of the results for application to cryogenic fluid storage tanks. In fact, mechanical fatigue and cryogenic cycling tests have indicated that transverse cracks initiate at the free edges and propagate inward, e.g. [2,3]. In cryogenic fluid storage tanks truly free edges (where cut fiber ends are exposed) will generally not be present. Cut fiber ends can be present in tanks made using filament winding or fiber placement techniques but these ends are generally embedded or potted in the matrix. Therefore the applicability of results and conclusions from the flat laminate coupon tests to actual cryogenic storage tanks can only be qualitative. To study this issue, finite element simulations were performed to study the stress distribution around the fiber end at the free * Corresponding author. Present address: 800 S 9th Street, Laramie, WY 82070, USA. E-mail address: [email protected] (E.W. Andrews). 0266-3538/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2008.09.001
edge. This work was done in conjunction with an experimental study described separately [4]. The stress state in the vicinity of the fiber end is compared to the stress state away from the fiber ends (in the ‘‘bulk”). Two types of models were generated: (1) models with free edges and (2) models in which the edge is capped with a variable thickness layer of matrix material. The case of a capped edge simulates the situation where fiber ends are embedded in a laminate. The influence of this cap on the stresses was assessed. The models are idealized, assuming perfect bonding of the fibers, perfectly flat free surfaces and periodic distributions of fibers. In actual coupon specimens, the surfaces are actually quite flat as they are polished to facilitate microscopic inspection for cracks. A specific CFRP material system was considered, namely IM7/5250-4, and the fiber volume fraction was varied to assess the influence on the stresses. For the case of bare fiber ends subjected to cooling, it is understood that singular tensile stresses exist at the fiber–matrix interface as the fiber end is approached, e.g. [5]. Therefore this location is a likely area for crack initiation. It has been demonstrated numerically using an axisymmetric model that if the fiber end is covered with an adequately thick cap of matrix material, the tensile singularity disappears [6]. These results considered a single fiber volume fraction (Vf = 0.35), which is lower than typical CRFP materials. It has also been shown that for the case of a model composite, consisting of stainless steel rods in a polyester matrix, thermal cycling results in substantial interface crack growth and a cap of matrix material eliminates this cracking [7]. In the tests on these stainless steel/polyester composites, no transverse load was applied. In the case of a composite laminate, transverse ply stresses will also be present due to the coefficient of thermal expansion
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E.W. Andrews, M.R. Garnich / Composites Science and Technology 68 (2008) 3352–3357
(CTE) mismatch between different plies. For the problem of a fiber composite subjected to a transverse load, it is known that the interfacial stresses are singular at the free end, e.g. [8]. The stresses are singular even with a cap of matrix material (e.g. [9]). This singularity complicates the experimental measurement of the fiber– matrix interface strength and innovative specimen designs have been used to extract the interface strength, e.g. [9]. For the case of a bare fiber end subjected to a transverse load and thermal loading, an elasticity solution was derived by Folias et al. [10]. In the present study, the combination of these two problems (thermal loading and transverse tension) was considered. This is important for CFRP laminates since transverse tensile stresses generally occur even in the absence of mechanical loads. The available results from the literature show that for the case of thermal loading a tensile singularity exists at the fiber end, and a coating of matrix material has the effect of suppressing the singularity. These previous studies did not include transverse tension and were axisymmetric, thus neglecting the influence of adjacent fibers. 2. Finite element simulations: methods Consider a unidirectional lamina and assume that the fibers are arranged in a repeating hexagonal pattern, as shown in Fig. 1a. Because of the CTE difference between the fibers and the matrix, a temperature change will cause thermal stresses at the microstructural level. In addition, a transverse tensile load is applied to simulate ply level stresses generated by the mismatch in CTE for the differently oriented plies. In the present study a [90/0]s laminate was assumed. Given the lamina properties for this simple laminate the lamina transverse tensile stress rt generated due to a temperature change DT can be evaluated as
rt ¼
E1 E2 ða2 a1 ÞDT E1 þ E2 ð1 þ 2m12 Þ
ð1Þ
where E, a and m represent the elastic modulus, CTE and Poisson’s ratio, respectively. The subscripts indicate direction with the fibers assumed to be oriented along the 1-direction, while the 2-direction is transverse to the fibers in the plane of the laminate. Given the transverse stress, the transverse strain for this laminate is evaluated as
a
et ¼
rt E2
þ m12
rt
ð2Þ
E1
From the symmetries of this fiber arrangement and the nature of the applied loading, the unit cell indicated in Fig. 1a and shown with boundary conditions in Fig. 1b was isolated for analysis. The model is necessarily three dimensional and extends some distance parallel to the fiber axes as shown in Fig. 2. One end (representing the bulk material, away from the free edge) is on rollers (no deformation in the z-direction) and one end is free to deform, representing the bare fiber ends. The model was made adequately long to correctly recover bulk results away from the end. To analyze this problem the submodeling capability in ABAQUS was used. A typical global model and the two submodels are shown in Fig. 2. Note that the global model is cropped so that the full length of the model in the z-direction, denoted as L, is not shown. In both the global models and the submodels, linear, reduced integration brick elements (C3D8R) were used. The meshes are biased along the z-direction to improve the resolution of the stresses near the fiber end. Fiber volume fractions Vf equal to 0.45, 0.6 and 0.75 were simulated. Although a fiber volume fraction of 0.75 is high for a fiber composite, the intent was to explore the influence of Vf on this problem. Therefore values 0.15 above and below a typical value of 0.60 were considered. In these simulations the length of the first submodel in the z-direction was set to be one tenth of the global model length L. The submodel was defined to extend a radial distance 0.75d from the fiber–matrix interface both into the matrix and the fiber where the distance d indicated in Fig. 2. The second submodel had all these dimensions set to be exactly one half of those in the first submodel. The dashed lines on the models are intended to approximately represent the region occupied by the subsequent submodel. The same mesh density was used for all three fiber volume fractions. The mesh density along the fiber direction (z-direction) is most important for capturing the rapidly varying stresses as the fiber end is approached. To give the reader a sense of the element size, and how it compares to the fiber diameter, for the case Vf = 0.6 in the global model the element at the fiber end has a z length of 0.017D where D is the fiber diameter. Again considering the case Vf = 0.6, in the first submodel this length is 0.0038D while in the second submodel it is reduced by a factor of 2 to 0.0019D.
b
Remains horizontal
y
Remains vertical
x
θ
Fig. 1. (a) Schematic depicting assumed hexagonal fiber packing with unit cell shown by the dashed box. (b) Boundary conditions on unit cell and definition of angle h.
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E.W. Andrews, M.R. Garnich / Composites Science and Technology 68 (2008) 3352–3357 Table 2 Matrix mechanical properties Matrix type
5250-4
Modulus, E (GPa) Poisson’s ratio, m Expansion coefficient, a (106/°C)
6.1a 0.35 44.0
a
Fig. 2. Finite element meshes, fiber volume fraction = 60%. Arrows indicate progression from global model, first submodel and second submodel.
The composite material system IM7/5250-4 was considered. This material has been the subject of recent cryogenic cracking studies, e.g. [11,12]. The matrix is assumed to be isotropic in mechanical properties and thermal expansion, while the fibers are assumed to be transversely isotropic in mechanical properties and thermal expansion. The matrix Young’s modulus and thermal expansion coefficient were obtained from manufacturer data sheets [13]. The Poisson’s ratio was assumed to be 0.35, a typical value for epoxy matrix materials [14]. The thermal expansion and mechanical properties of the IM7 fiber were obtained from Kulkarni and Ochoa [15]. The fiber properties are assumed to be temperature independent over the range of interest. Ply data for transverse stiffness [16] at a temperature of 196 °C along with a finite element micromechanics model [17] was used to estimate a value for the matrix Young’s modulus at a temperature of 196 °C. The value determined was 6.1 GPa, an increase of about 35% over the room temperature value of 4.6 GPa. For the thermal expansion, lacking data for the variation in thermal expansion with temperature the room temperature value was used. This approximation worked well for another matrix material, 977-3 epoxy [18]. The fiber and matrix material properties used in the finite element simulations are summarized in Tables 1 and 2. The required ply properties were calculated for each volume fraction using a finite element micromechanics model [17]. For the case Vf = 0.6 the micromechanics predictions are compared with the experimental data [16] in Table 3. The modulus E2 is the same for both cases since the value for the cryogenic Young’s modulus of the matrix was estimated by increasing the Young’s modulus in the micromechanics analysis until it matched the experimental data. However, the good overall agreement suggests Table 1 Fiber mechanical properties Fiber type
IM7
Longitudinal modulus, E1 (GPa) Transverse modulus, E2 (GPa) Shear modulus G12 (GPa) Poisson’s ratio, m12 Transverse shear modulus G23 (GPa) Poisson’s ratio, m23 Longitudinal CTE, a1 (106/°C) Transverse CTE, a2 (106/°C)
276 19.5 70.0 0.28 5.735 0.7 0.4 5.6
Estimated from ply data.
that the assumed mechanical and thermal properties for the fiber and matrix are reasonable. The same micromechanics analysis was used to determine the required ply properties for the cases Vf = 0.45 and 0.75. These results are shown in Table 4. To simulate embedded fiber ends, a layer of matrix material capping the fiber ends was modeled as shown in Fig. 3. The cap thickness shown is 0.5D. It was found that thicker caps had little effect on the results. In these simulations the surface of the cap is traction free. The temperature change was set at 383 °C. This corresponds to a test temperature of 196 °C (liquid nitrogen) and a stress-free temperature of 187 °C [12]. From Eqs. (1) and (2) the calculated transverse tensile strains are 1.089%, 0.849% and 0.601%, respectively, for Vf = 0.45, 0.60 and 0.75. A displacement boundary condition in the x-direction is imposed to simulate these strain levels. 3. Finite element simulations: results 3.1. Bulk stresses Radial normal stresses at the fiber–matrix interface far from the fiber ends (in the bulk) are shown in Fig. 4. These graphs are generated by defining a node path circumferentially along the fiber– matrix interface at the end of the model opposite the free surface (i.e., a distance L from the free surface) and extracting the predicted radial stress along this path. The angle h in this figure refers to that defined in Fig. 1. Two groups of three curves are presented. One group is for the transverse tensile loading and one for the thermal loading, and each group shows the results for the three fiber vol-
Table 3 Comparison of measured [16] and predicted lamina properties, Vf = 0.60 Lamina
IM7/5250-4, experimental
IM7/5250-4, micromechanics
Longitudinal modulus, E1 (GPa) Transverse modulus, E2 (GPa) Shear modulus G12 (GPa) Poisson’s ratio, m12 Longitudinal expansion coefficient, a1 (106/°C) Transverse expansion coefficient, a2 (106/°C)
174 12.1 9.0 0.36 0.25
168 12.1 8.11 0.305 0.29
21.1
24.5
Table 4 Predicted lamina properties for Vf = 0.45 and 0.75 Lamina
IM7/5250-4, micromechanics, Vf = 0.45
IM7/5250-4, micromechanics, Vf = 0.75
Longitudinal modulus, E1 (GPa) Transverse modulus, E2 (GPa) Shear modulus G12 (GPa) Poisson’s ratio, m12 Longitudinal expansion coefficient, a1 (106/°C) Transverse expansion coefficient, a2 (106/°C)
128
209
10.3 5.6 0.315 0.825
14.3 13.4 0.295 0.0493
32.2
17.0
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E.W. Andrews, M.R. Garnich / Composites Science and Technology 68 (2008) 3352–3357
Fig. 3. Finite element meshes for cap models, fiber volume fraction = 60%. Arrows indicate progression from global model, first submodel and second submodel.
3.2. Fiber end stresses: bare end To expose the effect of the fiber end (free surface), attention is focused on the interfacial radial and r–z shear stress components. In order to compare the different volume fractions and study the influence of the cap, average radial and shear stresses were calculated. In this calculation the angular variation (at a given z position along the length of the fiber) is eliminated and the average stress rave is defined by the following equation:
rave ¼
a
2
p
Z p=2
rðhÞdh
ð3Þ
0
100 75
Radial Stress (MPa)
ume fractions. The radial thermal stress is generally compressive, as the matrix contracts and compresses the fiber. The average compressive stress decreases as the volume fraction is increased and actually becomes tensile over some regions of the interface for the case Vf = 0.75. This observation is significant as even lower volume fraction materials will contain regions of high fiber volume fraction as a result of randomness of the fiber volume fraction. The angular variation in stress is due to effects of interactions with adjacent fibers. The stress due to the transverse tension peaks at h = 0° and is tensile everywhere. The amplitude of the peak stress decreases with increasing fiber volume fraction. In the combined problem, the thermal radial stress reduces the peak stress and generates regions of compression at the fiber–matrix interface.
Tension
50 25 Thermal
0 -25 -50
Vf = 0.45 Vf = 0.60
-75
Vf = 0.75
-100 0
0.2
0.4
0.6
0.8
1
Distance from Fiber End (Normalized)
b
400
200
Radial Stress (MPa)
Radial Stress (MPa)
Vf = 0.45
Tension
150
Vf = 0.60 Vf = 0.75
100 50 0
Vf = 0.45 Vf = 0.60
300
Vf = 0.75
200
Tension
100
0
Thermal
-50
-100
Thermal
0
-100 0
30
Angle θ
60
Fig. 4. Radial stresses in the bulk.
90
0.02
0.04
0.06
0.08
0.1
Distance from Fiber End (Normalized) Fig. 5. Radial Stresses at fiber–matrix interface as fiber end is approached for the case of a bare fiber end. (a) Global model results and (b) submodel results.
3356
a
120 Vf = 0.45 Vf = 0.60
Shear Stress (MPa)
100
Radial Stress (MPa)
a
E.W. Andrews, M.R. Garnich / Composites Science and Technology 68 (2008) 3352–3357
Vf = 0.75
Thermal
80 60 40
Tension
20
100 50 0 -50 -100
Combined Tension Thermal
-150 -200 -0.2
-0.1
0 0
0.1
0.2
0.3
0.4
0
0.1
0.2
Distance (Normalized)
0.5
Distance from Fiber End (Normalized)
b 120
Shear Stress (MPa)
b
Vf = 0.45
Shear Stress (MPa)
100
Vf = 0.60 Vf = 0.75
Thermal
80 60 40
Tension
200
100
50
0 -0.2
20
Combined Tension Thermal
150
-0.1
0
0.1
0.2
Distance (Normalized) 0 0
0.1
0.2
0.3
0.4
0.5
Distance from Fiber End (Normalized) Fig. 6. Shear stresses at fiber–matrix interface as fiber end is approached for the case of a bare fiber end. (a) Global model results and (b) submodel results.
Fig. 7. (a) Radial stresses and (b) Shear stresses at fiber–matrix interface in the vicinity of the cap, Vf = 0.45.
play a role in local fiber/matrix debonding as has been observed experimentally [2]. 3.3. Fiber end stresses: capped end
The above integral is approximated using the finite element results. The averaged radial stress and shear stress at the interface as a function of distance from the free surface are plotted in Figs. 5 and 6, respectively. In these plots the origin (x = 0) corresponds to the free surface and distance along the fiber is normalized by the fiber diameter. First consider the radial stresses (global model) shown in Fig. 5a. Away from the fiber end the radial stresses approach a constant value. As the end is approached the radial stresses first decrease and then rapidly increase. The submodel results shown in Fig. 5b indicate the substantial increase in stress that occurs as the mesh is refined as is typical for a stress singularity. The thermal load resulted in much larger stress variations for the same finite element mesh. The peak value and spatial extent of the singularity were found to increase with decreasing volume fraction. In Fig. 6 the shear stress is zero away from the fiber end but increases and then decreases as the fiber end is approached. The results indicate that decreasing the element size increases the peak stress and moves the peak closer to the fiber end. The shear stresses are singular, in this idealized model, in that the peak stress increases without bound as the element size decreases. However, due to the stress-free boundary condition the stress must approach zero at the fiber end. Higher stresses are induced by the thermal loading, and the amplitude and spatial extent of the stresses increase with decreasing volume fraction. These stresses (radial and shear) local to the fiber ends can reasonably be expected to
Models as shown in Fig. 3 were used to simulate the effect of a layer of matrix material (cap) that embeds the fiber ends below the traction free surface. Only results for a cap of thickness 0.5D are presented as they were insensitive to increased thickness. The averaged radial and shear stresses were calculated as before. The results for the volume fractions Vf = 0.45 and 0.75 are plotted in Figs. 7 and 8, respectively. These results are from the second submodel. In each figure, part (a) and (b) show averaged radial and shear stresses, respectively. Negative values of the x-axis represent positions inside the cap of matrix material. First consider the radial stresses for the case Vf = 0.45 plotted in Fig. 7a. For the case of pure tensile loading the stress is nearly constant except for the step change and highly localized tensile singularity at the fiber end. It is known that for the case of a capped or embedded fiber end subjected to transverse tension there is a singularity but it is reduced compared to the case of a bare fiber end, e.g. [9]. The combined stresses show that the compressive singularity due to thermal loading overwhelms the tensile singularity due to global tension. The shear stresses shown in Fig. 7b indicate that the shear stresses with a cap are similar to those for the bare fiber end (see Fig. 6). The results for Vf = 0.60 are simply intermediate to those for Vf = 0.45 (Fig. 7) and Vf = 0.75 (Fig. 8) and are not shown. In all cases the cap is effective in completely suppressing the tensile singularity, while the singularity in shear stress persists. This suggests that for
E.W. Andrews, M.R. Garnich / Composites Science and Technology 68 (2008) 3352–3357
Radial Stress (MPa)
a
100 50 0 -50 Combined Tension Thermal
-100
-150 -200 -0.2
-0.1
0
0.1
0.2
Distance (Normalized)
Shear Stress (MPa)
b
200 Combined Tension Thermal
150
the associated transverse tension in a balanced cross-ply laminate. The models that included a cap showed a suppression of the tensile singularities while the shear stress singularities persist. Since free edges appear to play a strong role in the initiation of matrix ply cracks it is relevant to better understand the stress state associated with polished edges in the coupon level testing. The stress analysis results presented here suggest that for a bare fiber end formation of interface cracks at the fiber ends is very likely as both shear and tensile (peel) stress singularities are present. Such interface damage combined with cyclic loading could then lead to formation of a complete matrix ply crack in the presence of ply level transverse tensile stresses. The analysis showed that the tensile (peel) stress at the fiber–matrix interface is effectively suppressed by a coating of matrix material over the fiber ends. The experimental results of Garnich and Dalgarno [4] indicate that this is sufficient to substantially mitigate the formation of matrix ply cracking during thousands of thermal cycles even though the interface shear stress singularity is still present. Acknowledgements This work was supported by the United States Department of Defense through the Missile Defense Agency under contract number FA9453-04-C-0286. Jeffry Welsh of the Air Force Research Laboratory Space Vehicles Directorate (AFRL/VS) was the POC for the effort. Mr. Dan Milligan assisted with some of the data analysis.
100
50
0 -0.2
3357
-0.1
0
0.1
0.2
Distance (Normalized) Fig. 8. (a) Radial stresses and (b) shear stresses at fiber–matrix interface in the vicinity of the cap, Vf = 0.75.
the case of a capped fiber end, debonding would have to be driven by the shear stress alone. In a companion effort [4] a problem similar to that considered here was studied experimentally. In that work cryogenic cycling tests were performed on coupon specimens with a layer of matrix material coating the laminate edges. It was observed that the coating has a very significant effect on reducing matrix ply cracking in a cyclic thermal environment, which suggests that the cracking was initiated primarily by the tensile stress. The situation of a ply end embedded within a laminate, as in ply drop offs, is more complicated than the problem considered here. However, it is expected that embedded fiber ends will show a suppression of tensile stress similar to that seen in this problem. This suggests that the crack density in an actual tank can be substantially less than that observed in coupon specimens subjected to the same thermal cycling history. 4. Conclusions Finite element simulations of the stress field around fiber ends have been presented for several fiber volume fractions. The models are three dimensional and include adjacent fiber interactions. The models were subjected to a thermal load as well as a transverse tensile load consistent with the thermal loading of a cross-ply laminate. The effect of a cap or layer of matrix material embedding the fiber ends was also considered. Singular shear stresses and singular tensile stresses are present at the fiber–matrix interface near the free surface, making debonding likely for severe thermal loads. The stress singularities due to the thermal loading of a ply were more severe than those due to
References [1] Bechel VT, Camping JD, Kim RY. Cryogenic/elevated temperature cycling induced leakage paths in PMCs. Compos B Eng 2005;36:171–82. [2] Lafarie-Frenot MC, Henaff-Gardin C, Gamby D. Matrix cracking induced by cyclic ply stresses in composite laminates. Compos Sci Technol 2001;61:2327–36. [3] Yokozeki T, Aoki T, Ishikawa T. Fatigue growth of matrix cracks in the transverse direction of CFRP laminates. Compos Sci Technol 2002;62:1223–9. [4] Garnich MR, Dalgarno RW. Cryogenic microcracking in IM7/5250-4 laminates: edge effects and crack density saturation. 48th AIAA/ASCE/AHS/ASC structures, structural dynamics, and materials conference proceedings. Honolulu, Hawaii: AIAA; April 2007. [5] Abedian A, Szyszkowski W. Influence of the free surface on the thermal stresses in unidirectional composites. Compos A 1997;28:573–9. [6] Abedian A, Szyszkowski W, Yannacopoulos S. Effects of surface geometry of composites on thermal stress distribution: a numerical study. Compos Sci Technol 1999;59:41–54. [7] Abedian A, Yannacopoulos S, Szyszkowski W. Effects of free surface geometry on the damage initiation in unidirectional composites due to temperature cycling; an experimental investigation. Can Metall Q 2000;39:99–112. [8] Li CC, Folias ES. Edge effect of a carbon fiber meeting a surface. Mech Mater 1991;12:267–78. [9] Tandon GP, Kim RY, Warrier SG, Majumdar BS. Influence of free edge and corner singularities on interfacial normal strength: application in model unidirectional composites. Compos B 1999;30:115–34. [10] Folias ES, Hohn M, Nicholas T. Predicting crack initiation in composite material systems due to a thermal expansion mismatch. Int J Fract 1998;93:335–49. [11] Bechel VT, Kim RY. Damage trends in cryogenically cycled carbon/polymer composites. Compos Sci Technol 2004;64:1773–84. [12] Bechel VT, Fredin MB, Donaldson SL, Kim RY, Camping JD. Effect of stacking sequence on micro-cracking in a cryogenically cycled carbon/bismaleimide composite. Compos A 2003;34:663–72. [13] http://www.cytec.com/engineered-materials/products/Datasheets/ CYCOM%20RTM%205250-4.pdf; accessed Nov 2007. [14] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates. Compos Sci Technol 1998;58:1011–22. [15] Kulkarni R, Ochoa O. Transverse and longitudinal CTE measurements of carbon fibers and their impact on interfacial residual stresses in composites. J Compos Mater 2006;40:733–54. [16] Kim RY, Rice BP, Donaldson SL. Microcracking in composite laminates under thermal environments: 150 to 196 °C. Proceedings of the 47th SAMPE international symposium. Long Beach, CA: SAMPE; 2002. p. 833–42. [17] Garnich MR, Karami G. Finite element micromechanics for stiffness and strength of wavy fiber composites. J Compos Mater 2004;38:273–92. [18] Choi S, Sankar B. Micromechanics analysis of composite laminates at cryogenic temperatures. J Compos Mater 2006;40:1077–91.
Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Stresses around fiber ends at free and embedded ply edges E.W. Andrews a,*, M.R. Garnich b a b
Firehole Technologies, Wyoming Technology Business Center, 1000 E. University Avenue, Laramie, WY 82071, USA Department of Mechanical Engineering, 1000 E. University Avenue, University of Wyoming, Laramie, WY 82071, USA
a r t i c l e
i n f o
Article history: Received 3 April 2008 Received in revised form 15 August 2008 Accepted 4 September 2008 Available online 12 September 2008 Keywords: A. Polymer–matrix composites B. Thermomechanical properties C. Finite element analysis C. Stress concentrations
a b s t r a c t Using the finite element method, the stress state around fiber ends at a free surface in a unidirectional fiber composite was studied. The material is subjected to a thermal load consistent with a cryogenic application as well as ply level transverse tensile loading associated with thermal stresses in a laminate. At the free end the fiber–matrix interface sees tensile radial stresses as well as shear stresses. This makes the fiber end a likely site for interface debonding (crack initiation). These initial cracks can then link up and grow to become full through-thickness matrix cracks. The effect that an epoxy coating over the fiber ends has on the local stress distribution was also considered. These capped models simulate fiber ends embedded within a laminate. The epoxy cap has the effect of suppressing the interface stresses at the fiber end. The significance of these results in relation to cryogenic cycling tests using coupon specimens with free and embedded edges is discussed. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Carbon fiber reinforced plastic (CFRP) materials are candidate materials for cryogenic pressure vessel applications. When such materials are subjected to temperature variations from a given stress-free temperature, substantial thermal stresses can result. These can arise at the ply level due to different ply orientations in a laminate, as well as at the fiber/matrix level due to different thermal expansion coefficients for the constituents. The stresses can cause matrix cracking, which is significant for pressure vessel applications because of the leakage that may result. Subjecting flat coupons to cyclic cryogenic immersion is one approach for assessing the resistance of various CFRP material systems to this transverse microcracking, e.g. [1]. However, the presence of free edges in such samples complicates the interpretation of the results for application to cryogenic fluid storage tanks. In fact, mechanical fatigue and cryogenic cycling tests have indicated that transverse cracks initiate at the free edges and propagate inward, e.g. [2,3]. In cryogenic fluid storage tanks truly free edges (where cut fiber ends are exposed) will generally not be present. Cut fiber ends can be present in tanks made using filament winding or fiber placement techniques but these ends are generally embedded or potted in the matrix. Therefore the applicability of results and conclusions from the flat laminate coupon tests to actual cryogenic storage tanks can only be qualitative. To study this issue, finite element simulations were performed to study the stress distribution around the fiber end at the free * Corresponding author. Present address: 800 S 9th Street, Laramie, WY 82070, USA. E-mail address: [email protected] (E.W. Andrews). 0266-3538/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2008.09.001
edge. This work was done in conjunction with an experimental study described separately [4]. The stress state in the vicinity of the fiber end is compared to the stress state away from the fiber ends (in the ‘‘bulk”). Two types of models were generated: (1) models with free edges and (2) models in which the edge is capped with a variable thickness layer of matrix material. The case of a capped edge simulates the situation where fiber ends are embedded in a laminate. The influence of this cap on the stresses was assessed. The models are idealized, assuming perfect bonding of the fibers, perfectly flat free surfaces and periodic distributions of fibers. In actual coupon specimens, the surfaces are actually quite flat as they are polished to facilitate microscopic inspection for cracks. A specific CFRP material system was considered, namely IM7/5250-4, and the fiber volume fraction was varied to assess the influence on the stresses. For the case of bare fiber ends subjected to cooling, it is understood that singular tensile stresses exist at the fiber–matrix interface as the fiber end is approached, e.g. [5]. Therefore this location is a likely area for crack initiation. It has been demonstrated numerically using an axisymmetric model that if the fiber end is covered with an adequately thick cap of matrix material, the tensile singularity disappears [6]. These results considered a single fiber volume fraction (Vf = 0.35), which is lower than typical CRFP materials. It has also been shown that for the case of a model composite, consisting of stainless steel rods in a polyester matrix, thermal cycling results in substantial interface crack growth and a cap of matrix material eliminates this cracking [7]. In the tests on these stainless steel/polyester composites, no transverse load was applied. In the case of a composite laminate, transverse ply stresses will also be present due to the coefficient of thermal expansion
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(CTE) mismatch between different plies. For the problem of a fiber composite subjected to a transverse load, it is known that the interfacial stresses are singular at the free end, e.g. [8]. The stresses are singular even with a cap of matrix material (e.g. [9]). This singularity complicates the experimental measurement of the fiber– matrix interface strength and innovative specimen designs have been used to extract the interface strength, e.g. [9]. For the case of a bare fiber end subjected to a transverse load and thermal loading, an elasticity solution was derived by Folias et al. [10]. In the present study, the combination of these two problems (thermal loading and transverse tension) was considered. This is important for CFRP laminates since transverse tensile stresses generally occur even in the absence of mechanical loads. The available results from the literature show that for the case of thermal loading a tensile singularity exists at the fiber end, and a coating of matrix material has the effect of suppressing the singularity. These previous studies did not include transverse tension and were axisymmetric, thus neglecting the influence of adjacent fibers. 2. Finite element simulations: methods Consider a unidirectional lamina and assume that the fibers are arranged in a repeating hexagonal pattern, as shown in Fig. 1a. Because of the CTE difference between the fibers and the matrix, a temperature change will cause thermal stresses at the microstructural level. In addition, a transverse tensile load is applied to simulate ply level stresses generated by the mismatch in CTE for the differently oriented plies. In the present study a [90/0]s laminate was assumed. Given the lamina properties for this simple laminate the lamina transverse tensile stress rt generated due to a temperature change DT can be evaluated as
rt ¼
E1 E2 ða2 a1 ÞDT E1 þ E2 ð1 þ 2m12 Þ
ð1Þ
where E, a and m represent the elastic modulus, CTE and Poisson’s ratio, respectively. The subscripts indicate direction with the fibers assumed to be oriented along the 1-direction, while the 2-direction is transverse to the fibers in the plane of the laminate. Given the transverse stress, the transverse strain for this laminate is evaluated as
a
et ¼
rt E2
þ m12
rt
ð2Þ
E1
From the symmetries of this fiber arrangement and the nature of the applied loading, the unit cell indicated in Fig. 1a and shown with boundary conditions in Fig. 1b was isolated for analysis. The model is necessarily three dimensional and extends some distance parallel to the fiber axes as shown in Fig. 2. One end (representing the bulk material, away from the free edge) is on rollers (no deformation in the z-direction) and one end is free to deform, representing the bare fiber ends. The model was made adequately long to correctly recover bulk results away from the end. To analyze this problem the submodeling capability in ABAQUS was used. A typical global model and the two submodels are shown in Fig. 2. Note that the global model is cropped so that the full length of the model in the z-direction, denoted as L, is not shown. In both the global models and the submodels, linear, reduced integration brick elements (C3D8R) were used. The meshes are biased along the z-direction to improve the resolution of the stresses near the fiber end. Fiber volume fractions Vf equal to 0.45, 0.6 and 0.75 were simulated. Although a fiber volume fraction of 0.75 is high for a fiber composite, the intent was to explore the influence of Vf on this problem. Therefore values 0.15 above and below a typical value of 0.60 were considered. In these simulations the length of the first submodel in the z-direction was set to be one tenth of the global model length L. The submodel was defined to extend a radial distance 0.75d from the fiber–matrix interface both into the matrix and the fiber where the distance d indicated in Fig. 2. The second submodel had all these dimensions set to be exactly one half of those in the first submodel. The dashed lines on the models are intended to approximately represent the region occupied by the subsequent submodel. The same mesh density was used for all three fiber volume fractions. The mesh density along the fiber direction (z-direction) is most important for capturing the rapidly varying stresses as the fiber end is approached. To give the reader a sense of the element size, and how it compares to the fiber diameter, for the case Vf = 0.6 in the global model the element at the fiber end has a z length of 0.017D where D is the fiber diameter. Again considering the case Vf = 0.6, in the first submodel this length is 0.0038D while in the second submodel it is reduced by a factor of 2 to 0.0019D.
b
Remains horizontal
y
Remains vertical
x
θ
Fig. 1. (a) Schematic depicting assumed hexagonal fiber packing with unit cell shown by the dashed box. (b) Boundary conditions on unit cell and definition of angle h.
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E.W. Andrews, M.R. Garnich / Composites Science and Technology 68 (2008) 3352–3357 Table 2 Matrix mechanical properties Matrix type
5250-4
Modulus, E (GPa) Poisson’s ratio, m Expansion coefficient, a (106/°C)
6.1a 0.35 44.0
a
Fig. 2. Finite element meshes, fiber volume fraction = 60%. Arrows indicate progression from global model, first submodel and second submodel.
The composite material system IM7/5250-4 was considered. This material has been the subject of recent cryogenic cracking studies, e.g. [11,12]. The matrix is assumed to be isotropic in mechanical properties and thermal expansion, while the fibers are assumed to be transversely isotropic in mechanical properties and thermal expansion. The matrix Young’s modulus and thermal expansion coefficient were obtained from manufacturer data sheets [13]. The Poisson’s ratio was assumed to be 0.35, a typical value for epoxy matrix materials [14]. The thermal expansion and mechanical properties of the IM7 fiber were obtained from Kulkarni and Ochoa [15]. The fiber properties are assumed to be temperature independent over the range of interest. Ply data for transverse stiffness [16] at a temperature of 196 °C along with a finite element micromechanics model [17] was used to estimate a value for the matrix Young’s modulus at a temperature of 196 °C. The value determined was 6.1 GPa, an increase of about 35% over the room temperature value of 4.6 GPa. For the thermal expansion, lacking data for the variation in thermal expansion with temperature the room temperature value was used. This approximation worked well for another matrix material, 977-3 epoxy [18]. The fiber and matrix material properties used in the finite element simulations are summarized in Tables 1 and 2. The required ply properties were calculated for each volume fraction using a finite element micromechanics model [17]. For the case Vf = 0.6 the micromechanics predictions are compared with the experimental data [16] in Table 3. The modulus E2 is the same for both cases since the value for the cryogenic Young’s modulus of the matrix was estimated by increasing the Young’s modulus in the micromechanics analysis until it matched the experimental data. However, the good overall agreement suggests Table 1 Fiber mechanical properties Fiber type
IM7
Longitudinal modulus, E1 (GPa) Transverse modulus, E2 (GPa) Shear modulus G12 (GPa) Poisson’s ratio, m12 Transverse shear modulus G23 (GPa) Poisson’s ratio, m23 Longitudinal CTE, a1 (106/°C) Transverse CTE, a2 (106/°C)
276 19.5 70.0 0.28 5.735 0.7 0.4 5.6
Estimated from ply data.
that the assumed mechanical and thermal properties for the fiber and matrix are reasonable. The same micromechanics analysis was used to determine the required ply properties for the cases Vf = 0.45 and 0.75. These results are shown in Table 4. To simulate embedded fiber ends, a layer of matrix material capping the fiber ends was modeled as shown in Fig. 3. The cap thickness shown is 0.5D. It was found that thicker caps had little effect on the results. In these simulations the surface of the cap is traction free. The temperature change was set at 383 °C. This corresponds to a test temperature of 196 °C (liquid nitrogen) and a stress-free temperature of 187 °C [12]. From Eqs. (1) and (2) the calculated transverse tensile strains are 1.089%, 0.849% and 0.601%, respectively, for Vf = 0.45, 0.60 and 0.75. A displacement boundary condition in the x-direction is imposed to simulate these strain levels. 3. Finite element simulations: results 3.1. Bulk stresses Radial normal stresses at the fiber–matrix interface far from the fiber ends (in the bulk) are shown in Fig. 4. These graphs are generated by defining a node path circumferentially along the fiber– matrix interface at the end of the model opposite the free surface (i.e., a distance L from the free surface) and extracting the predicted radial stress along this path. The angle h in this figure refers to that defined in Fig. 1. Two groups of three curves are presented. One group is for the transverse tensile loading and one for the thermal loading, and each group shows the results for the three fiber vol-
Table 3 Comparison of measured [16] and predicted lamina properties, Vf = 0.60 Lamina
IM7/5250-4, experimental
IM7/5250-4, micromechanics
Longitudinal modulus, E1 (GPa) Transverse modulus, E2 (GPa) Shear modulus G12 (GPa) Poisson’s ratio, m12 Longitudinal expansion coefficient, a1 (106/°C) Transverse expansion coefficient, a2 (106/°C)
174 12.1 9.0 0.36 0.25
168 12.1 8.11 0.305 0.29
21.1
24.5
Table 4 Predicted lamina properties for Vf = 0.45 and 0.75 Lamina
IM7/5250-4, micromechanics, Vf = 0.45
IM7/5250-4, micromechanics, Vf = 0.75
Longitudinal modulus, E1 (GPa) Transverse modulus, E2 (GPa) Shear modulus G12 (GPa) Poisson’s ratio, m12 Longitudinal expansion coefficient, a1 (106/°C) Transverse expansion coefficient, a2 (106/°C)
128
209
10.3 5.6 0.315 0.825
14.3 13.4 0.295 0.0493
32.2
17.0
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E.W. Andrews, M.R. Garnich / Composites Science and Technology 68 (2008) 3352–3357
Fig. 3. Finite element meshes for cap models, fiber volume fraction = 60%. Arrows indicate progression from global model, first submodel and second submodel.
3.2. Fiber end stresses: bare end To expose the effect of the fiber end (free surface), attention is focused on the interfacial radial and r–z shear stress components. In order to compare the different volume fractions and study the influence of the cap, average radial and shear stresses were calculated. In this calculation the angular variation (at a given z position along the length of the fiber) is eliminated and the average stress rave is defined by the following equation:
rave ¼
a
2
p
Z p=2
rðhÞdh
ð3Þ
0
100 75
Radial Stress (MPa)
ume fractions. The radial thermal stress is generally compressive, as the matrix contracts and compresses the fiber. The average compressive stress decreases as the volume fraction is increased and actually becomes tensile over some regions of the interface for the case Vf = 0.75. This observation is significant as even lower volume fraction materials will contain regions of high fiber volume fraction as a result of randomness of the fiber volume fraction. The angular variation in stress is due to effects of interactions with adjacent fibers. The stress due to the transverse tension peaks at h = 0° and is tensile everywhere. The amplitude of the peak stress decreases with increasing fiber volume fraction. In the combined problem, the thermal radial stress reduces the peak stress and generates regions of compression at the fiber–matrix interface.
Tension
50 25 Thermal
0 -25 -50
Vf = 0.45 Vf = 0.60
-75
Vf = 0.75
-100 0
0.2
0.4
0.6
0.8
1
Distance from Fiber End (Normalized)
b
400
200
Radial Stress (MPa)
Radial Stress (MPa)
Vf = 0.45
Tension
150
Vf = 0.60 Vf = 0.75
100 50 0
Vf = 0.45 Vf = 0.60
300
Vf = 0.75
200
Tension
100
0
Thermal
-50
-100
Thermal
0
-100 0
30
Angle θ
60
Fig. 4. Radial stresses in the bulk.
90
0.02
0.04
0.06
0.08
0.1
Distance from Fiber End (Normalized) Fig. 5. Radial Stresses at fiber–matrix interface as fiber end is approached for the case of a bare fiber end. (a) Global model results and (b) submodel results.
3356
a
120 Vf = 0.45 Vf = 0.60
Shear Stress (MPa)
100
Radial Stress (MPa)
a
E.W. Andrews, M.R. Garnich / Composites Science and Technology 68 (2008) 3352–3357
Vf = 0.75
Thermal
80 60 40
Tension
20
100 50 0 -50 -100
Combined Tension Thermal
-150 -200 -0.2
-0.1
0 0
0.1
0.2
0.3
0.4
0
0.1
0.2
Distance (Normalized)
0.5
Distance from Fiber End (Normalized)
b 120
Shear Stress (MPa)
b
Vf = 0.45
Shear Stress (MPa)
100
Vf = 0.60 Vf = 0.75
Thermal
80 60 40
Tension
200
100
50
0 -0.2
20
Combined Tension Thermal
150
-0.1
0
0.1
0.2
Distance (Normalized) 0 0
0.1
0.2
0.3
0.4
0.5
Distance from Fiber End (Normalized) Fig. 6. Shear stresses at fiber–matrix interface as fiber end is approached for the case of a bare fiber end. (a) Global model results and (b) submodel results.
Fig. 7. (a) Radial stresses and (b) Shear stresses at fiber–matrix interface in the vicinity of the cap, Vf = 0.45.
play a role in local fiber/matrix debonding as has been observed experimentally [2]. 3.3. Fiber end stresses: capped end
The above integral is approximated using the finite element results. The averaged radial stress and shear stress at the interface as a function of distance from the free surface are plotted in Figs. 5 and 6, respectively. In these plots the origin (x = 0) corresponds to the free surface and distance along the fiber is normalized by the fiber diameter. First consider the radial stresses (global model) shown in Fig. 5a. Away from the fiber end the radial stresses approach a constant value. As the end is approached the radial stresses first decrease and then rapidly increase. The submodel results shown in Fig. 5b indicate the substantial increase in stress that occurs as the mesh is refined as is typical for a stress singularity. The thermal load resulted in much larger stress variations for the same finite element mesh. The peak value and spatial extent of the singularity were found to increase with decreasing volume fraction. In Fig. 6 the shear stress is zero away from the fiber end but increases and then decreases as the fiber end is approached. The results indicate that decreasing the element size increases the peak stress and moves the peak closer to the fiber end. The shear stresses are singular, in this idealized model, in that the peak stress increases without bound as the element size decreases. However, due to the stress-free boundary condition the stress must approach zero at the fiber end. Higher stresses are induced by the thermal loading, and the amplitude and spatial extent of the stresses increase with decreasing volume fraction. These stresses (radial and shear) local to the fiber ends can reasonably be expected to
Models as shown in Fig. 3 were used to simulate the effect of a layer of matrix material (cap) that embeds the fiber ends below the traction free surface. Only results for a cap of thickness 0.5D are presented as they were insensitive to increased thickness. The averaged radial and shear stresses were calculated as before. The results for the volume fractions Vf = 0.45 and 0.75 are plotted in Figs. 7 and 8, respectively. These results are from the second submodel. In each figure, part (a) and (b) show averaged radial and shear stresses, respectively. Negative values of the x-axis represent positions inside the cap of matrix material. First consider the radial stresses for the case Vf = 0.45 plotted in Fig. 7a. For the case of pure tensile loading the stress is nearly constant except for the step change and highly localized tensile singularity at the fiber end. It is known that for the case of a capped or embedded fiber end subjected to transverse tension there is a singularity but it is reduced compared to the case of a bare fiber end, e.g. [9]. The combined stresses show that the compressive singularity due to thermal loading overwhelms the tensile singularity due to global tension. The shear stresses shown in Fig. 7b indicate that the shear stresses with a cap are similar to those for the bare fiber end (see Fig. 6). The results for Vf = 0.60 are simply intermediate to those for Vf = 0.45 (Fig. 7) and Vf = 0.75 (Fig. 8) and are not shown. In all cases the cap is effective in completely suppressing the tensile singularity, while the singularity in shear stress persists. This suggests that for
E.W. Andrews, M.R. Garnich / Composites Science and Technology 68 (2008) 3352–3357
Radial Stress (MPa)
a
100 50 0 -50 Combined Tension Thermal
-100
-150 -200 -0.2
-0.1
0
0.1
0.2
Distance (Normalized)
Shear Stress (MPa)
b
200 Combined Tension Thermal
150
the associated transverse tension in a balanced cross-ply laminate. The models that included a cap showed a suppression of the tensile singularities while the shear stress singularities persist. Since free edges appear to play a strong role in the initiation of matrix ply cracks it is relevant to better understand the stress state associated with polished edges in the coupon level testing. The stress analysis results presented here suggest that for a bare fiber end formation of interface cracks at the fiber ends is very likely as both shear and tensile (peel) stress singularities are present. Such interface damage combined with cyclic loading could then lead to formation of a complete matrix ply crack in the presence of ply level transverse tensile stresses. The analysis showed that the tensile (peel) stress at the fiber–matrix interface is effectively suppressed by a coating of matrix material over the fiber ends. The experimental results of Garnich and Dalgarno [4] indicate that this is sufficient to substantially mitigate the formation of matrix ply cracking during thousands of thermal cycles even though the interface shear stress singularity is still present. Acknowledgements This work was supported by the United States Department of Defense through the Missile Defense Agency under contract number FA9453-04-C-0286. Jeffry Welsh of the Air Force Research Laboratory Space Vehicles Directorate (AFRL/VS) was the POC for the effort. Mr. Dan Milligan assisted with some of the data analysis.
100
50
0 -0.2
3357
-0.1
0
0.1
0.2
Distance (Normalized) Fig. 8. (a) Radial stresses and (b) shear stresses at fiber–matrix interface in the vicinity of the cap, Vf = 0.75.
the case of a capped fiber end, debonding would have to be driven by the shear stress alone. In a companion effort [4] a problem similar to that considered here was studied experimentally. In that work cryogenic cycling tests were performed on coupon specimens with a layer of matrix material coating the laminate edges. It was observed that the coating has a very significant effect on reducing matrix ply cracking in a cyclic thermal environment, which suggests that the cracking was initiated primarily by the tensile stress. The situation of a ply end embedded within a laminate, as in ply drop offs, is more complicated than the problem considered here. However, it is expected that embedded fiber ends will show a suppression of tensile stress similar to that seen in this problem. This suggests that the crack density in an actual tank can be substantially less than that observed in coupon specimens subjected to the same thermal cycling history. 4. Conclusions Finite element simulations of the stress field around fiber ends have been presented for several fiber volume fractions. The models are three dimensional and include adjacent fiber interactions. The models were subjected to a thermal load as well as a transverse tensile load consistent with the thermal loading of a cross-ply laminate. The effect of a cap or layer of matrix material embedding the fiber ends was also considered. Singular shear stresses and singular tensile stresses are present at the fiber–matrix interface near the free surface, making debonding likely for severe thermal loads. The stress singularities due to the thermal loading of a ply were more severe than those due to
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