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Meso-scopic model for progressive failure of resin impregnated yarns in composite overwrapped pressure vessels (COPVs)
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Meso-scopic model for progressive failure of resin impregnated yarns in composite overwrapped pressure vessels (COPVs) Jiakun Liu∗, Stuart Leigh Phoenix Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca NY, 14850, United States
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Keywords: Composite Overwrap Pressure Vessels (COPVs) Finite element modeling Composite materials Progressive failure Stress concentration factor
a b s t r a c t Composite Overwrapped Pressure Vessels (COPVs) are widely used in space launch vehicles and the automotive industry. Due to the potential for catastrophic loss caused by failure of COPVs, improved understanding of its mechanical and damage growth behavior is necessary to ensure its safe operation. In this paper, we described a developed mesoscopic model with the capability of modeling tows/yarns with inserted, in-ply interface elements and interface layers as resin rich zones. Furthermore, the model realizes 1) arbitrary winding angle build-up of overwrap plies, 2) controllable detailed dimensions, and 3) user defined material behavior for all regions. These features are critical for potential studies of in-situ and parametric overwrap failure in COPVs. In the present work, stress profiles and progressive failure around a single tow break are shown by case studies involving various interface and damage evolution properties. The investigation indicates that the strength and damage evolution law of interface elements both have direct effect on the stress concentration factors and damage evolution around a tow break/fracture. Resulting physical phenomena and stress overload profile are discussed as well.
1. INTRODUCTION As part of an overall project on multiscale modeling of COPV failure, the current work focuses on mesoscale behavior, where in-situ, progressive damage modeling of the overwrap is carried out. The manufacturing process of prepreg tows often results in resin rich zones between tows and layers. This often forms mechanically weak interfaces when load is applied, leading to in-ply micro cracks and interply debonding. Separation of individual tows and tow breakage, as well as interlayer matrix cracking and debonding is observed in failed COPVs, such as shown in Figs. 1a, b as well as in [1] and [2]. Additionally, micrographs of sectioned overwraps and similar prepreg layered materials directly support such phenomena [3]. Such damage patterns are the fundamental motivation for the model described in the paper. Due to the nature of the overwrap at mesoscale as discussed above, damage evolution and failure of the overwrap are a combination and interaction of different types of failure modes: tow breakage, interface damage between tows, and interface damage between layers. Historically, COPV progressive failure has not been considered a credible failure mode in aerospace applications. The development of a damage control plan and adherence to the controls for handling mitigate mechanical damage which may occur. Historically, aerospace applications have been for single use operations. For launch vehicles, the cumulative time under pressure is usually very small such that progres-
∗
sive damage is minimal. The Space Shuttle program [4] was an exception for a reuse vehicle. The COPVs in the Orbiter vehicle are fabricated with Kevlar® as compared with carbon graphite fiber for most current applications. The COPVs on the International Space Station (ISS) have been used for an extended period of time. To mitigate the risk of composite stress rupture, the ISS program reduced the maximum pressure on ISS pressure vessels. As such, these tanks operate with a higher effective factor of safety. The existing industry standard for COPV, ANSI/AIAA S-081B-2018 [5] does not specifically address progressive failure. The standard establishes minimum factor of safety which envelopes such conditions for ultimate load. It also sets requirements on cycle life. However, the standard does not address the time dependent response of the COPV that would be experienced from such operational use. As the NewSpace economy develops, reusable spacecraft and launch vehicles are the norm. Additionally, NASA has announced for several new programs that are beyond Low Earth Orbit, including the Orion Multipurpose Crew Vehicle, Lunar Gateway and Artemis. For these programs, progressive damage may become an important failure mode which is not covered by existing design requirements. The primary driver of failure is high tensile stresses in the fiber direction of an overwrap layer that ultimately causes single tow breaks. These breaks result in overloading of adjacent tows in the same layer, while at the same time causing tow interface debonding. As the debonding length extends, overload ratios decrease, however, the effective over-
Corresponding author. E-mail addresses: [email protected] (J. Liu), [email protected] (S.L. Phoenix).
https://doi.org/10.1016/j.jsse.2019.10.007 Received 19 April 2019; Received in revised form 28 October 2019; Accepted 28 October 2019 Available online xxx 2468-8967/© 2019 International Association for the Advancement of Space Safety. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: J. Liu and S.L. Phoenix, Meso-scopic model for progressive failure of resin impregnated yarns in composite overwrapped pressure vessels (COPVs), Journal of Space Safety Engineering, https://doi.org/10.1016/j.jsse.2019.10.007
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while in contrast it will mostly cause interface cracks and even splitting if top and bottom layers are 90° Winding Angle: Filament winding leads to complex local mechanical and geometry features throughout the overwrap, where various combinations of winding build-up are formed in different locations. For example, fiber angles of all layers tend to 90° around polar bosses, whereas around cylinder regions, fiber angles are determined by layers’ winding angles. The complexity of winding build-up will directly affect local load transfer and interaction between different damage modes. Statistical strength distribution: Composite materials tend to follow Weibull strength distributions with shape and size effect [8], this brings in more complicated local load sharing around a tow break. In brief, a tow break may or may not result in nearby tow breaks, thus a tow break may not lead to a local cluster of several breaks that lead to failure of the overwrap. Stress State/Location: The stress states and location also significantly affect the failure evolution. For example, when a COPV is pressurized in service, the innermost surface of overwrap attaching to the metallic liner is in high pressure. A tow break occurring here will likely be clamped by the surface adhesion as well as the high compression stress. This may prevent the tow break from opening, thus high and concentrated overload around such a kind of break is expected. In contrast, the outermost surface has zero compressive stress. A tow break here will result in a tow/yarn pumped out [2], instead of generating high overload to the neighbors. The above factors are critical and could not be investigated without a model containing such features. However, no such model is available in the literature. Typically, the composite overwrap is treated as homogenized with effective properties computed based on classical laminate theories. Given these facts, a mesoscale model containing these features is needed to fill this gap.
Fig. 1a. Spherical COPV made of an aluminum liner and Kevlar®/epoxy overwrap. Left is an intact one, and on the right is the overwrap removed from a failed vessel subjected to a burst test.
2. Finite element overwrap model Fig. 1b. Close-up of tow breaks in crossing tow bands.
The finite element model is generated in commercial software ABAQUS® [9] environment with in-house developed pre-processing code written in Python. The in-house code performs all essential steps needed for a finite element simulation, namely: generating tow, tow interface and layer interface as assembled parts; defining and assigning materials to corresponding parts; creating meshes; defining load and boundary conditions; and assigning interactions such as tie constraints between tows and tow interfaces. User material properties and behavior used in the present paper are written in Fortran as a UMAT for static solver in Abaqus. The mesoscopic model is a square shaped stacking sequence of layers of arbitrary winding angles in which users define necessary geometric parameters and winding profiles. More specifically, the edge dimension of layers, the winding angle, interface width and tow/yarn width of each layer, and layer interface thickness are defined. Upon users’ definition, each layer with tows at specific winding angles containing inserted interface elements among them are generated. All layers are stacked together with small thickness interface layers inserted between them. As for tow breaks, user-controlled predefined ultra-thin lines tangential to the fiber direction are inserted in each layer across the width of tows, since tow fracture is tangential to transverse interface crack/debonding. In the present paper, stress profile and concentration factors around one single tow break in one layer will be investigated. Therefore, a mesoscopic model with only one predefined tow break located at the center is presented. Fig. 2a shows a single layer of 20 × 20 × 0.125 mm, with tow width of 2.5 mm, tow interface thickness of 0.01 mm, and winding angle of 30° These are typical dimensions of winding bands that build up overwraps, where the width of band is usually about 25 mm (1 inch), and there are about ten to twenty tow/yarns within a band. The predefined tow break line is located at the geometric center of the layer which will ensure
load length on neighboring tows expands, thus increasing the risk of neighboring tow failures at statistically occurring weak points. These different kinds of local stress distribution will play an important role in the progressive damage and failure of overwrap. In general, they are affected by the following factors: Tow Interface: As discussed previously, interfaces between tows are considered as resin rich zones, which are formed when individual tow units are drawn through an epoxy resin during their manufacture prior to being wound onto the COPV in bands. When a tow breaks and locally unloads, the interface surrounding it is immediately subjected to increased shear stress as it transfers this lost load to adjacent tow units, resulting in overloads on these tows. Motivated by the study of stress profiles around single fiber breaks at the microscale [6][7], one can expect that overload profiles around tow breaks are strongly influenced by the strength and shear load carrying capacity of tow interfaces. Namely, a stronger tow interface will result in higher stress concentration and a more localized overload region, while a weak interface will result in longer regions of tow debonding and possibly milder overloads. Layer Interface: Like the in-ply tow interface, layer interface properties are expected to affect the load transfer both in a layer and between different layers. The layer interface will transfer the contact and clamping stress from the top and bottom layers, thus when a layer has tow and interface damage, the overload will transfer to the other layers. For various winding angles, such overload will likely lead to different damage modes. For example, a tow/yarn break in a 0° layer will result in overload in fiber direction without causing in-ply matrix cracks if the top and bottom layers are 0°, 2
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Fig. 2a. Single layer of 30° with one potential tow break at the geometry center.
Fig. 2b. Three layers of [ − 30°, 0°, 30°], with one potential tow break at the center of 0° layer.
that the tow breaks immediately throughout the tow/yarn width at the specific predefined line. The thickness of such a line is controllable, and a relatively small value (1.0e−4 mm used in this paper) is applied to ensure material integrity until the tow reaches its tensile strength. For a multi-layer model, arbitrary number of layers with specific winding angles can be generated based on users’ input. Fig. 2b shows a model containing three layers with a winding build-up of [ − 30°, 0°, 30°]. The layer interface is created as a uniform layer of 0.01mm thickness and inserted between layers. The parts in Fig. 3 are set to partially transparent for better illustration. Once all parts are generated, the code also commands the software to assign either built-in or user-defined materials, constrain all contact pairs and interaction properties between parts, then define loads and boundary conditions, etc. This kind of process enables the potential comprehensive parametric study and design optimization in the future.
3.1. Tow/yarn section Like typical composites, mechanical behavior of tows is orthotropic elastic with brittle failure. In the model, only failures at predefined tow break lines are activated. The regular tow sections are defined as purely elastic without any failure evaluation. This will not affect the progressive damage and failure modes of overwrap as long as predefined breaks and interfaces are reasonably defined such that any kind of failure will be in the modes of tow breaks, interface debonding, or more commonly a mixture of them. Fig. 3a shows the stress-strain curve of tow in fiber and transverse directions, where 𝜎 u , 𝜀u ,𝜀f are tensile strength, ultimate strain, failure strain, respectively. Response in shear directions has brittle linear elastic feature as well. 3.2. Tow interface
3. Material properties Though the interface of tows is considered as a resin rich zone, it makes little sense to model it with the response and strength of pure resin. Both mechanical and failure properties of tow and layer interfaces
Mechanical properties and damage behavior will be briefly discussed for each material section. 3
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Fig. 3a. Stress-strain curve of tow in axial directions.
Fig. 3b. Stress-strain curve of tow interface in shear directions.
Table 1a Material Properties of tow.
are determined based on the available test datasheet. For example, in [10], strength of overwrap material in normal and in-plane shear directions are reported, from which the strengths of tow interface could be partially determined. In general, the interface is isotropic elastic with linear degradation in normal directions. The constitutive response in shear direction is more complicated and directly affects the overload profile around tow breaks. However, real testing of the complete interface shear behavior is difficult and thus need modifications. At present, the stress-strain curve in shear directions is shown in Fig. 3b, where yielding and sliding effects are implemented, and 𝜎 y , 𝜀y , 𝜀u , 𝜎 r , 𝜀f are ultimate stress, initial and final yielding strain, residual (frictional) stress, and failure strain, respectively. Degrading curves (𝜀u → 𝜀r , 𝜀b → 𝜀f ) are added to help improve numerical convergence behavior. It is worth mentioning that the values of yielding strain, residual stress and ultimate strain are subjective, since testing on these features are nearly impractical for a real overwrap. Likewise, strength in several directions (such as thru-thickness or out-of-plane shear) are not impractical to test as well, thus in order to fully define the damage profile in a continuum-based response, empirical values are implemented. Such treatment turns out to have little effect on the damage response. It is potentially more efficient to use cohesive elements with tractionseparation response for interfaces, since only the strength in the normal and shear directions are needed, this will be adopted in future studies for comparison.
E1 125 GPa ε𝑢11 0.0196 G12 4.8 GPa 𝑓 𝛾12 20
E2 3.6 GPa ε𝑢22 0.089 G13 4.8 GPa 𝑓 𝛾13 20
E3 3.6 GPa ε𝑢33 0.089 G23 2.0 GPa 𝑓 𝛾23 20
𝜈 12 0.33 ε𝑓11 500 𝑢 𝛾12 0.2
𝜈 13 0.33 ε𝑓22 80 𝑢 𝛾13 0.2
𝜈 23 0.4 ε𝑓33 80 𝑢 𝛾23 0.2
3.3. Layer interface Layer interfaces, like tow interfaces, are implemented as small thickness elements with user defined continuum-based mechanical and damage response. Details are neglected here considering the length of the paper.
4. Numerical simulation In the finite element model, eight-node brick element (C3D8) is used for all sections. The properties of tow and tow interface are shown in Table 1a-b, where 1, 2, and 3 represent fiber/tow, in-plane transverse and through-thickness direction, respectively. 4
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Fig. 4. Toughening coefficients to scale ultimate yielding and failure strain of tow interface in shear.
Notice that several values, as discussed above, require reasonable empirical input to complete the continuum-based material damage modeling. The values should yield response that matches real physics and do not affect the failure modes of overwrap. For example, the failure strain of tow in fiber direction is set to 500.0. This is the about the smallest value found to be large enough to get converged results in Static General step in ABAQUS/Standard. This value will result in a tow break opening of 1.0e−4 mm × 500.0 = 50 um, which is still relatively small and in agreement with the magnitude of final separation based on critical energy release rate. Besides, considering the bridging effect of fiber and matrix inside tow/yarns when breaking, a 50 um opening of the fracture surface when a tow unit completely fails is reasonable. Similarly, the defined ultimate and failure strain in other directions will not actually affect the failure modes and evolution in the model, since the interfaces around a potential predefined tow break will take the role and break in advance. Again, the interface can be modelled with cohesive elements with traction-separation response, which requires fewer properties compared to the continuum-based material response. In the present paper, the latter technique has been adapted to tow interface elements. The dimension of layer is 50 × 50 × 0.125 mm, with a tow width of 2.5mm and winding angle of 0° A displacement based uniaxial tensile load along the fiber/tow direction is applied at one side, while the other side is fixed. The final tensile load is 1 mm. This will give the entire layer a 2% strain in the fiber direction which is slightly above the failure strain of tows: 1.96%. For the following three case studies, we investigate different tow interface strengths. As shown in Fig. 4, two toughening coefficients 𝜑u , 𝜑b , are introduced to modify the ultimate yielding and sliding strain of interface elements. The properties used in first case are the same as in Table 1b. Namely, for Model 4.1 𝜑u = 𝜑b = 1.
4.1. Single tow break in one layer Firstly, one single tow break is located at the center of a layer is generated, Fig. 5a-d are the deformation and overload profiles around the tow break at different stages. Overload ratio is computed as stress concentration factor (SCF), defined as the ratio of current stress in tensile (fiber) direction to the tensile strength of tows. In the following figures, all deformation is the real scale (deform scale factor is set to 1.0), except the close-up screenshots of tow break line. Fig. 5a is the moment (increment) when tow breaks, interface elements just next to the tow break tip have failed, and the interface debonding has not occurred yet. At this stage, the stress concentration is highly localized near the tow break tip, with max SCF of around 1.167, and the recover length of stress is around two times the tow width. Fig. 5b shows the tow break line opens up immediately following tow breakage, the interface starts debonding, the maximum SCF slightly drops from 1.167 to 1.117, and the overload region expands. In Fig. 5c, as the load slightly increases, the debonding length of interface keeps growing due to high shear stress generated between tows, leading to sliding of broken tow accompanied by interface debonding. The overload region keeps expanding and SCF is more uniformly distributed, forming a narrow half-elliptical shaped area along the adjacent tows. Comparing to the numerical studies of stress concentration profile around a single fiber at microscale in [11], the effective overload region around a tow/yarn break is more localized and remains closer to the tow interface, thus only a narrow region along and between the damaged interface will be overloaded. This agrees with intuition, since the shearing capability is relatively much weaker than the shear carrying and transferring capability of matrix between fibers at microscale. Fig. 5d is the moment when interface debonding extend all the way to the edge of model, with the broken tow completely fails and becomes an individual unconnected unit. In this single layer model, there are no constraints from the top or bottom layers, which resemble the case of the outermost layer. In this case a broken tow will likely result in the failed tow/yarn pumping out, instead of continuously generating overload to the surroundings.
Table 1b Material properties of tow interface. E1 3.6 GPa ε𝑢11 0.044 G12 2.0 GPa 𝑦 𝛾12 0.04 𝑦 𝛾13 0.02 𝑦 𝛾23 0.02
E2 3.6 GPa ε𝑢22 0.044 G13 2.0 GPa 𝑢 𝛾12 2.0 𝑢 𝛾13 1.0 𝑢 𝛾23 1.0
E3 3.6 GPa ε𝑢33 0.044 G23 2.0 GPa 𝑟 𝛾12 2.2 𝑟 𝛾13 1.1 𝑟 𝛾23 1.1
𝜈 12 0.0 ε𝑓11 4.44
𝜈 13 0.0 ε𝑓22 4.44
𝜈 23 0.0 ε𝑓33 4.44
𝑏 𝛾12 80.0 𝑏 𝛾13 40.0 𝑏 𝛾23 40.0
𝑓 𝛾12 80.8 𝑓 𝛾13 40.4 𝑓 𝛾23 40.4
𝑦 𝜎12 80 MPa 𝑦 𝜎13 40 MPa 𝑦 𝜎23 40 MPa
4.2. Single break in one layer, with stronger interface 𝑟 𝜎12 40 MPa 𝑟 𝜎13 20 MPa 𝑟 𝜎23 20 MPa
To qualitatively study the effect of tow interface strength, the yielding strain and failure strain are increased to ten times the strain in case one, namely the toughening coefficients are 𝜑u = 𝜑b = 10. This will result in a much stronger interface. The deformation and overload profile are as shown in Fig. 6a-d. 5
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Fig. 5a. Deformation and SCF profile of model 4.1: At the moment tow breaks. Close-up look of tow break line is shown with a deformation scale factor of 5.
Fig. 5b. Deformation and SCF profile of model 4.1: Interface starts debonding.
Fig. 5c. Deformation and SCF profile of model 4.1: Interface debonding length grows, overload region expands.
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Fig. 5d. Deformation and SCF profile of model 4.1: Interface totally fails across the broken tow.
Fig. 6a. Deformation and SCF profile of model 4.2: At the moment tow breaks. Close-up look of tow break line is shown with a deformation scale factor of 5.
As shown in Fig. 6a, at the moment the tow breaks, the max SCF is just slightly higher than in Model 4.1. The stress recover length is also about two times the tow width. However, since more energy is required to cause interface failure, a delayed interface debonding and tow break opening is observed, as in Fig. 6b, compared to Model 4.1 in which the same amount of tensile load is applied. The localized maximum SCF is also noticeably higher (1.405 to 1.16) around break tips. Upon yielding, the interface starts to degrade and only the residual stress accounting for frictions between tows exists. In Fig. 6c, higher SCF around end of interface debonding is observed. The overload region at this stage is also wider than the previous case. Finally, unlike Model 4.1 with the weaker interface and the same amount of final load (2% uniaxial tensile), the interface does not debond all the way to the end (as shown in Fig. 6d). A higher load would be needed to cause the entire interface to debond.
4.3. Interface with lower and longer residual stress The comparison between Model 4.1 and Model 4.2 already indicates the significant effects of tow/yarn interface properties on the failure evolution. In Model 4.3, the residual stress of interface 𝜎 r is reduced from 40 MPa to 4 MPa, which is supposed to be closer to the real frictional stress between tows. The ultimate failure strain 𝜀f is set to 8000, which is ten times the value in Model 4.2 and gives: 𝜑u = 10, 𝜑b = 100, 𝜎 r = 4.0 MPa, 𝜎 y = 80.0MPa. This way, the total energies needed to break the interface are equivalent in Model 4.2 and Model 4.3. Results at the moment when the tow breaks and at the final load are shown in Fig. 7a-b, in which the max SCF drops to 1.075, while the effective region is not as locally concentrated as previous cases. This is due to the significantly reduced residual stress of interface before debonding. However, as shown in Fig. 7b, due to the large value of 𝜀f , the interface
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Fig. 6b. Deformation and SCF profile of model 4.2: Interface starts to debond, overload region expands.
Fig. 6c. Deformation and SCF profile of model 4.2: Interface keeps debonding.
Fig. 6d. Deformation and SCF profile of model 4.2: Interface keeps debonding and approaching the edge, but does not fail all the way to the edge.
along the broken tow does not debond (completely fail) even at the final load. Instead, the interface keeps its residual stress accounted for as friction. Thus, the overload region keeps its shape with a small max SCF of around 1.05. The comparison between Model 4.2 and 4.3 indicates that the tow/yarn interface behavior plays an important role and directly affects the stress redistribution and failure evolution of a broken tow. Though the energy required to cause interface debonding is equivalent between those two models, the detailed stress-strain curve (residual and failure
strain) lead to noticeably different overload profiles and stress concentration factors. 5. Summary and conclusions In this work, we briefly discussed several important features in the damage and failure evolution of overwrap in COPVs. Then a developed finite element model was presented. This model realizes the simulation and study of in-plane tow/yarn breakage, tow/yarn interfaces and inter8
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Fig. 7a. Deformation and SCF profile of model 4.3: At the moment tow breaks. Close-up look of tow break line is shown with a deformation scale factor of 5.
Fig. 7b. Deformation and SCF profile of model 4.3: At final load, interface does not debond, it remains active with residual stress to account for the friction, resulting in a wide and gentle overload region.
ply interfaces, as well as arbitrary winding build-up of multiple layers, which has not been presented in the literature yet. Then we showed the modeling of a single tow break in one layer. Results indicate that the model can properly simulate the essential damage phenomena and evolution of tow/yarn break and interface debonding. Numerical evaluation of overload (SCF) was discussed. A case study of three different interface strengths was presented. It was shown that the damage evolution is directly affected by detailed interface behavior, especially the residual stress, yielding and ultimate failure strain of interface. The progressive failure in composites is the result of interaction and accumulation of multi-mode failures, which will be further investigated in future studies. More specifically, behavior and evolution of multiple tow breakages and correlated interface damages will be introduced into the model. In this paper, the main focus is to study the stress concentration profile and propagation of a single tow break and how these phenomena are affected by the mechanical response of tow interface. Current and future work includes: 1) Investigation on the effect of a single tow break in various winding and stress states, with improved user material profiles. 2) Comprehensive study of progressive damage evolution of overwrap based on the model, where the interaction and
evolution between various failure modes will be investigated. 3) Generation of potential database to serve as constitutive laws for analysis and modeling of COPVs at component scale. References [1] P.B. McLaughlan, L.R. Grimes-Ledesma, Composite overwrapped pressure vessels, Nasa/Sp–2011–573 (2011) 1–20 http://www.sti.nasa.gov. [2] L. Grimes-Ledesma, S.L. Phoenix, H. Beeson, T. Yoder, N. Greene, Testing of carbon fiber composite overwrapped pressure vessel stress-rupture lifetime, in: Am. Soc. Compos. - 21st Tech. ConS Am. Soc. Compos., 2006, pp. 205–218. [3] L.K. Grunenfelder, T. Centea, P. Hubert, S.R. Nutt, Effect of room-temperature outtime on tow impregnation in an out-of-autoclave prepreg, Compos. Part A Appl. Sci. Manuf. 45 (2013) 119–126 https://doi.org/10.1016/j.compositesa.2012.10.001. [4] M. Kezirian, K. Johnson, S.L. Phoenix, Composite overwrapped pressure vessels (COPV): flight rationale for the space shuttle program, AIAA Sp. Conf. Expo (2011) https://doi.org/10.2514/6.2011-7363. [5] , American Institute of Aeronautics and Astronautics Standard: Space Systems—Composite Overwrapped Pressure Vessels (ANSI/AIAA S-081B-2018), March, 20, 2018 https://doi.org/10.2514/4.105425. [6] I.J. Beyerlein, S.L. Phoenix, A.M. Sastry, Comparison of shear-lag theory and continuum fracture mechanics for modeling fiber and matrix stresses in an elastic cracked composite lamina, Int. J. Solids Struct. 33 (1996) 2543–2574 https://doi.org/10.1016/0020-7683(95)00172-7. [7] I.J. Beyerlein, S.L. Phoenix, Stress concentrations around multiple fiber breaks in an elastic matrix with local yielding or debonding using quadratic 9
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influence superposition, J. Mech. Phys. Solids. 44 (1996) 1997–2039 https://doi.org/10.1016/S0022-5096(96)00068-3. [8] S.L. Phoenix, M. Ibnabdeljalil, C.Y. Hui, Size effects in the distribution for strength of brittle matrix fibrous composites, Int. J. Solids Struct. 34 (1997) 545–568 https://doi.org/10.1016/S0020-7683(96)00034-0. [9] , ABAQUS/ Standard User’s Manual, v.6.11, Dassault Systems®, Rhode Island, 2011.
[10] Toray Carbon Fibers America Inc. Torayca®t700s technical data sheet no.cfa-005. Technical report, 6 Hutton Centre Drive, Santa Ana, CA 92707. [11] R. Ganesh, S. Sockalingam, J.W. Gillespie, Dynamic effects of a single fiber break in unidirectional glass fiber-reinforced polymer composites: effects of matrix plasticity, Compos. Mater. 52 (2018) 1873–1886 https://doi.org/10.1177/0021998317737604.
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Journal of Space Safety Engineering journal homepage: www.elsevier.com/locate/jsse
Meso-scopic model for progressive failure of resin impregnated yarns in composite overwrapped pressure vessels (COPVs) Jiakun Liu∗, Stuart Leigh Phoenix Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca NY, 14850, United States
a r t i c l e
i n f o
Keywords: Composite Overwrap Pressure Vessels (COPVs) Finite element modeling Composite materials Progressive failure Stress concentration factor
a b s t r a c t Composite Overwrapped Pressure Vessels (COPVs) are widely used in space launch vehicles and the automotive industry. Due to the potential for catastrophic loss caused by failure of COPVs, improved understanding of its mechanical and damage growth behavior is necessary to ensure its safe operation. In this paper, we described a developed mesoscopic model with the capability of modeling tows/yarns with inserted, in-ply interface elements and interface layers as resin rich zones. Furthermore, the model realizes 1) arbitrary winding angle build-up of overwrap plies, 2) controllable detailed dimensions, and 3) user defined material behavior for all regions. These features are critical for potential studies of in-situ and parametric overwrap failure in COPVs. In the present work, stress profiles and progressive failure around a single tow break are shown by case studies involving various interface and damage evolution properties. The investigation indicates that the strength and damage evolution law of interface elements both have direct effect on the stress concentration factors and damage evolution around a tow break/fracture. Resulting physical phenomena and stress overload profile are discussed as well.
1. INTRODUCTION As part of an overall project on multiscale modeling of COPV failure, the current work focuses on mesoscale behavior, where in-situ, progressive damage modeling of the overwrap is carried out. The manufacturing process of prepreg tows often results in resin rich zones between tows and layers. This often forms mechanically weak interfaces when load is applied, leading to in-ply micro cracks and interply debonding. Separation of individual tows and tow breakage, as well as interlayer matrix cracking and debonding is observed in failed COPVs, such as shown in Figs. 1a, b as well as in [1] and [2]. Additionally, micrographs of sectioned overwraps and similar prepreg layered materials directly support such phenomena [3]. Such damage patterns are the fundamental motivation for the model described in the paper. Due to the nature of the overwrap at mesoscale as discussed above, damage evolution and failure of the overwrap are a combination and interaction of different types of failure modes: tow breakage, interface damage between tows, and interface damage between layers. Historically, COPV progressive failure has not been considered a credible failure mode in aerospace applications. The development of a damage control plan and adherence to the controls for handling mitigate mechanical damage which may occur. Historically, aerospace applications have been for single use operations. For launch vehicles, the cumulative time under pressure is usually very small such that progres-
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sive damage is minimal. The Space Shuttle program [4] was an exception for a reuse vehicle. The COPVs in the Orbiter vehicle are fabricated with Kevlar® as compared with carbon graphite fiber for most current applications. The COPVs on the International Space Station (ISS) have been used for an extended period of time. To mitigate the risk of composite stress rupture, the ISS program reduced the maximum pressure on ISS pressure vessels. As such, these tanks operate with a higher effective factor of safety. The existing industry standard for COPV, ANSI/AIAA S-081B-2018 [5] does not specifically address progressive failure. The standard establishes minimum factor of safety which envelopes such conditions for ultimate load. It also sets requirements on cycle life. However, the standard does not address the time dependent response of the COPV that would be experienced from such operational use. As the NewSpace economy develops, reusable spacecraft and launch vehicles are the norm. Additionally, NASA has announced for several new programs that are beyond Low Earth Orbit, including the Orion Multipurpose Crew Vehicle, Lunar Gateway and Artemis. For these programs, progressive damage may become an important failure mode which is not covered by existing design requirements. The primary driver of failure is high tensile stresses in the fiber direction of an overwrap layer that ultimately causes single tow breaks. These breaks result in overloading of adjacent tows in the same layer, while at the same time causing tow interface debonding. As the debonding length extends, overload ratios decrease, however, the effective over-
Corresponding author. E-mail addresses: [email protected] (J. Liu), [email protected] (S.L. Phoenix).
https://doi.org/10.1016/j.jsse.2019.10.007 Received 19 April 2019; Received in revised form 28 October 2019; Accepted 28 October 2019 Available online xxx 2468-8967/© 2019 International Association for the Advancement of Space Safety. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: J. Liu and S.L. Phoenix, Meso-scopic model for progressive failure of resin impregnated yarns in composite overwrapped pressure vessels (COPVs), Journal of Space Safety Engineering, https://doi.org/10.1016/j.jsse.2019.10.007
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while in contrast it will mostly cause interface cracks and even splitting if top and bottom layers are 90° Winding Angle: Filament winding leads to complex local mechanical and geometry features throughout the overwrap, where various combinations of winding build-up are formed in different locations. For example, fiber angles of all layers tend to 90° around polar bosses, whereas around cylinder regions, fiber angles are determined by layers’ winding angles. The complexity of winding build-up will directly affect local load transfer and interaction between different damage modes. Statistical strength distribution: Composite materials tend to follow Weibull strength distributions with shape and size effect [8], this brings in more complicated local load sharing around a tow break. In brief, a tow break may or may not result in nearby tow breaks, thus a tow break may not lead to a local cluster of several breaks that lead to failure of the overwrap. Stress State/Location: The stress states and location also significantly affect the failure evolution. For example, when a COPV is pressurized in service, the innermost surface of overwrap attaching to the metallic liner is in high pressure. A tow break occurring here will likely be clamped by the surface adhesion as well as the high compression stress. This may prevent the tow break from opening, thus high and concentrated overload around such a kind of break is expected. In contrast, the outermost surface has zero compressive stress. A tow break here will result in a tow/yarn pumped out [2], instead of generating high overload to the neighbors. The above factors are critical and could not be investigated without a model containing such features. However, no such model is available in the literature. Typically, the composite overwrap is treated as homogenized with effective properties computed based on classical laminate theories. Given these facts, a mesoscale model containing these features is needed to fill this gap.
Fig. 1a. Spherical COPV made of an aluminum liner and Kevlar®/epoxy overwrap. Left is an intact one, and on the right is the overwrap removed from a failed vessel subjected to a burst test.
2. Finite element overwrap model Fig. 1b. Close-up of tow breaks in crossing tow bands.
The finite element model is generated in commercial software ABAQUS® [9] environment with in-house developed pre-processing code written in Python. The in-house code performs all essential steps needed for a finite element simulation, namely: generating tow, tow interface and layer interface as assembled parts; defining and assigning materials to corresponding parts; creating meshes; defining load and boundary conditions; and assigning interactions such as tie constraints between tows and tow interfaces. User material properties and behavior used in the present paper are written in Fortran as a UMAT for static solver in Abaqus. The mesoscopic model is a square shaped stacking sequence of layers of arbitrary winding angles in which users define necessary geometric parameters and winding profiles. More specifically, the edge dimension of layers, the winding angle, interface width and tow/yarn width of each layer, and layer interface thickness are defined. Upon users’ definition, each layer with tows at specific winding angles containing inserted interface elements among them are generated. All layers are stacked together with small thickness interface layers inserted between them. As for tow breaks, user-controlled predefined ultra-thin lines tangential to the fiber direction are inserted in each layer across the width of tows, since tow fracture is tangential to transverse interface crack/debonding. In the present paper, stress profile and concentration factors around one single tow break in one layer will be investigated. Therefore, a mesoscopic model with only one predefined tow break located at the center is presented. Fig. 2a shows a single layer of 20 × 20 × 0.125 mm, with tow width of 2.5 mm, tow interface thickness of 0.01 mm, and winding angle of 30° These are typical dimensions of winding bands that build up overwraps, where the width of band is usually about 25 mm (1 inch), and there are about ten to twenty tow/yarns within a band. The predefined tow break line is located at the geometric center of the layer which will ensure
load length on neighboring tows expands, thus increasing the risk of neighboring tow failures at statistically occurring weak points. These different kinds of local stress distribution will play an important role in the progressive damage and failure of overwrap. In general, they are affected by the following factors: Tow Interface: As discussed previously, interfaces between tows are considered as resin rich zones, which are formed when individual tow units are drawn through an epoxy resin during their manufacture prior to being wound onto the COPV in bands. When a tow breaks and locally unloads, the interface surrounding it is immediately subjected to increased shear stress as it transfers this lost load to adjacent tow units, resulting in overloads on these tows. Motivated by the study of stress profiles around single fiber breaks at the microscale [6][7], one can expect that overload profiles around tow breaks are strongly influenced by the strength and shear load carrying capacity of tow interfaces. Namely, a stronger tow interface will result in higher stress concentration and a more localized overload region, while a weak interface will result in longer regions of tow debonding and possibly milder overloads. Layer Interface: Like the in-ply tow interface, layer interface properties are expected to affect the load transfer both in a layer and between different layers. The layer interface will transfer the contact and clamping stress from the top and bottom layers, thus when a layer has tow and interface damage, the overload will transfer to the other layers. For various winding angles, such overload will likely lead to different damage modes. For example, a tow/yarn break in a 0° layer will result in overload in fiber direction without causing in-ply matrix cracks if the top and bottom layers are 0°, 2
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Fig. 2a. Single layer of 30° with one potential tow break at the geometry center.
Fig. 2b. Three layers of [ − 30°, 0°, 30°], with one potential tow break at the center of 0° layer.
that the tow breaks immediately throughout the tow/yarn width at the specific predefined line. The thickness of such a line is controllable, and a relatively small value (1.0e−4 mm used in this paper) is applied to ensure material integrity until the tow reaches its tensile strength. For a multi-layer model, arbitrary number of layers with specific winding angles can be generated based on users’ input. Fig. 2b shows a model containing three layers with a winding build-up of [ − 30°, 0°, 30°]. The layer interface is created as a uniform layer of 0.01mm thickness and inserted between layers. The parts in Fig. 3 are set to partially transparent for better illustration. Once all parts are generated, the code also commands the software to assign either built-in or user-defined materials, constrain all contact pairs and interaction properties between parts, then define loads and boundary conditions, etc. This kind of process enables the potential comprehensive parametric study and design optimization in the future.
3.1. Tow/yarn section Like typical composites, mechanical behavior of tows is orthotropic elastic with brittle failure. In the model, only failures at predefined tow break lines are activated. The regular tow sections are defined as purely elastic without any failure evaluation. This will not affect the progressive damage and failure modes of overwrap as long as predefined breaks and interfaces are reasonably defined such that any kind of failure will be in the modes of tow breaks, interface debonding, or more commonly a mixture of them. Fig. 3a shows the stress-strain curve of tow in fiber and transverse directions, where 𝜎 u , 𝜀u ,𝜀f are tensile strength, ultimate strain, failure strain, respectively. Response in shear directions has brittle linear elastic feature as well. 3.2. Tow interface
3. Material properties Though the interface of tows is considered as a resin rich zone, it makes little sense to model it with the response and strength of pure resin. Both mechanical and failure properties of tow and layer interfaces
Mechanical properties and damage behavior will be briefly discussed for each material section. 3
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Fig. 3a. Stress-strain curve of tow in axial directions.
Fig. 3b. Stress-strain curve of tow interface in shear directions.
Table 1a Material Properties of tow.
are determined based on the available test datasheet. For example, in [10], strength of overwrap material in normal and in-plane shear directions are reported, from which the strengths of tow interface could be partially determined. In general, the interface is isotropic elastic with linear degradation in normal directions. The constitutive response in shear direction is more complicated and directly affects the overload profile around tow breaks. However, real testing of the complete interface shear behavior is difficult and thus need modifications. At present, the stress-strain curve in shear directions is shown in Fig. 3b, where yielding and sliding effects are implemented, and 𝜎 y , 𝜀y , 𝜀u , 𝜎 r , 𝜀f are ultimate stress, initial and final yielding strain, residual (frictional) stress, and failure strain, respectively. Degrading curves (𝜀u → 𝜀r , 𝜀b → 𝜀f ) are added to help improve numerical convergence behavior. It is worth mentioning that the values of yielding strain, residual stress and ultimate strain are subjective, since testing on these features are nearly impractical for a real overwrap. Likewise, strength in several directions (such as thru-thickness or out-of-plane shear) are not impractical to test as well, thus in order to fully define the damage profile in a continuum-based response, empirical values are implemented. Such treatment turns out to have little effect on the damage response. It is potentially more efficient to use cohesive elements with tractionseparation response for interfaces, since only the strength in the normal and shear directions are needed, this will be adopted in future studies for comparison.
E1 125 GPa ε𝑢11 0.0196 G12 4.8 GPa 𝑓 𝛾12 20
E2 3.6 GPa ε𝑢22 0.089 G13 4.8 GPa 𝑓 𝛾13 20
E3 3.6 GPa ε𝑢33 0.089 G23 2.0 GPa 𝑓 𝛾23 20
𝜈 12 0.33 ε𝑓11 500 𝑢 𝛾12 0.2
𝜈 13 0.33 ε𝑓22 80 𝑢 𝛾13 0.2
𝜈 23 0.4 ε𝑓33 80 𝑢 𝛾23 0.2
3.3. Layer interface Layer interfaces, like tow interfaces, are implemented as small thickness elements with user defined continuum-based mechanical and damage response. Details are neglected here considering the length of the paper.
4. Numerical simulation In the finite element model, eight-node brick element (C3D8) is used for all sections. The properties of tow and tow interface are shown in Table 1a-b, where 1, 2, and 3 represent fiber/tow, in-plane transverse and through-thickness direction, respectively. 4
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Fig. 4. Toughening coefficients to scale ultimate yielding and failure strain of tow interface in shear.
Notice that several values, as discussed above, require reasonable empirical input to complete the continuum-based material damage modeling. The values should yield response that matches real physics and do not affect the failure modes of overwrap. For example, the failure strain of tow in fiber direction is set to 500.0. This is the about the smallest value found to be large enough to get converged results in Static General step in ABAQUS/Standard. This value will result in a tow break opening of 1.0e−4 mm × 500.0 = 50 um, which is still relatively small and in agreement with the magnitude of final separation based on critical energy release rate. Besides, considering the bridging effect of fiber and matrix inside tow/yarns when breaking, a 50 um opening of the fracture surface when a tow unit completely fails is reasonable. Similarly, the defined ultimate and failure strain in other directions will not actually affect the failure modes and evolution in the model, since the interfaces around a potential predefined tow break will take the role and break in advance. Again, the interface can be modelled with cohesive elements with traction-separation response, which requires fewer properties compared to the continuum-based material response. In the present paper, the latter technique has been adapted to tow interface elements. The dimension of layer is 50 × 50 × 0.125 mm, with a tow width of 2.5mm and winding angle of 0° A displacement based uniaxial tensile load along the fiber/tow direction is applied at one side, while the other side is fixed. The final tensile load is 1 mm. This will give the entire layer a 2% strain in the fiber direction which is slightly above the failure strain of tows: 1.96%. For the following three case studies, we investigate different tow interface strengths. As shown in Fig. 4, two toughening coefficients 𝜑u , 𝜑b , are introduced to modify the ultimate yielding and sliding strain of interface elements. The properties used in first case are the same as in Table 1b. Namely, for Model 4.1 𝜑u = 𝜑b = 1.
4.1. Single tow break in one layer Firstly, one single tow break is located at the center of a layer is generated, Fig. 5a-d are the deformation and overload profiles around the tow break at different stages. Overload ratio is computed as stress concentration factor (SCF), defined as the ratio of current stress in tensile (fiber) direction to the tensile strength of tows. In the following figures, all deformation is the real scale (deform scale factor is set to 1.0), except the close-up screenshots of tow break line. Fig. 5a is the moment (increment) when tow breaks, interface elements just next to the tow break tip have failed, and the interface debonding has not occurred yet. At this stage, the stress concentration is highly localized near the tow break tip, with max SCF of around 1.167, and the recover length of stress is around two times the tow width. Fig. 5b shows the tow break line opens up immediately following tow breakage, the interface starts debonding, the maximum SCF slightly drops from 1.167 to 1.117, and the overload region expands. In Fig. 5c, as the load slightly increases, the debonding length of interface keeps growing due to high shear stress generated between tows, leading to sliding of broken tow accompanied by interface debonding. The overload region keeps expanding and SCF is more uniformly distributed, forming a narrow half-elliptical shaped area along the adjacent tows. Comparing to the numerical studies of stress concentration profile around a single fiber at microscale in [11], the effective overload region around a tow/yarn break is more localized and remains closer to the tow interface, thus only a narrow region along and between the damaged interface will be overloaded. This agrees with intuition, since the shearing capability is relatively much weaker than the shear carrying and transferring capability of matrix between fibers at microscale. Fig. 5d is the moment when interface debonding extend all the way to the edge of model, with the broken tow completely fails and becomes an individual unconnected unit. In this single layer model, there are no constraints from the top or bottom layers, which resemble the case of the outermost layer. In this case a broken tow will likely result in the failed tow/yarn pumping out, instead of continuously generating overload to the surroundings.
Table 1b Material properties of tow interface. E1 3.6 GPa ε𝑢11 0.044 G12 2.0 GPa 𝑦 𝛾12 0.04 𝑦 𝛾13 0.02 𝑦 𝛾23 0.02
E2 3.6 GPa ε𝑢22 0.044 G13 2.0 GPa 𝑢 𝛾12 2.0 𝑢 𝛾13 1.0 𝑢 𝛾23 1.0
E3 3.6 GPa ε𝑢33 0.044 G23 2.0 GPa 𝑟 𝛾12 2.2 𝑟 𝛾13 1.1 𝑟 𝛾23 1.1
𝜈 12 0.0 ε𝑓11 4.44
𝜈 13 0.0 ε𝑓22 4.44
𝜈 23 0.0 ε𝑓33 4.44
𝑏 𝛾12 80.0 𝑏 𝛾13 40.0 𝑏 𝛾23 40.0
𝑓 𝛾12 80.8 𝑓 𝛾13 40.4 𝑓 𝛾23 40.4
𝑦 𝜎12 80 MPa 𝑦 𝜎13 40 MPa 𝑦 𝜎23 40 MPa
4.2. Single break in one layer, with stronger interface 𝑟 𝜎12 40 MPa 𝑟 𝜎13 20 MPa 𝑟 𝜎23 20 MPa
To qualitatively study the effect of tow interface strength, the yielding strain and failure strain are increased to ten times the strain in case one, namely the toughening coefficients are 𝜑u = 𝜑b = 10. This will result in a much stronger interface. The deformation and overload profile are as shown in Fig. 6a-d. 5
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Fig. 5a. Deformation and SCF profile of model 4.1: At the moment tow breaks. Close-up look of tow break line is shown with a deformation scale factor of 5.
Fig. 5b. Deformation and SCF profile of model 4.1: Interface starts debonding.
Fig. 5c. Deformation and SCF profile of model 4.1: Interface debonding length grows, overload region expands.
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Fig. 5d. Deformation and SCF profile of model 4.1: Interface totally fails across the broken tow.
Fig. 6a. Deformation and SCF profile of model 4.2: At the moment tow breaks. Close-up look of tow break line is shown with a deformation scale factor of 5.
As shown in Fig. 6a, at the moment the tow breaks, the max SCF is just slightly higher than in Model 4.1. The stress recover length is also about two times the tow width. However, since more energy is required to cause interface failure, a delayed interface debonding and tow break opening is observed, as in Fig. 6b, compared to Model 4.1 in which the same amount of tensile load is applied. The localized maximum SCF is also noticeably higher (1.405 to 1.16) around break tips. Upon yielding, the interface starts to degrade and only the residual stress accounting for frictions between tows exists. In Fig. 6c, higher SCF around end of interface debonding is observed. The overload region at this stage is also wider than the previous case. Finally, unlike Model 4.1 with the weaker interface and the same amount of final load (2% uniaxial tensile), the interface does not debond all the way to the end (as shown in Fig. 6d). A higher load would be needed to cause the entire interface to debond.
4.3. Interface with lower and longer residual stress The comparison between Model 4.1 and Model 4.2 already indicates the significant effects of tow/yarn interface properties on the failure evolution. In Model 4.3, the residual stress of interface 𝜎 r is reduced from 40 MPa to 4 MPa, which is supposed to be closer to the real frictional stress between tows. The ultimate failure strain 𝜀f is set to 8000, which is ten times the value in Model 4.2 and gives: 𝜑u = 10, 𝜑b = 100, 𝜎 r = 4.0 MPa, 𝜎 y = 80.0MPa. This way, the total energies needed to break the interface are equivalent in Model 4.2 and Model 4.3. Results at the moment when the tow breaks and at the final load are shown in Fig. 7a-b, in which the max SCF drops to 1.075, while the effective region is not as locally concentrated as previous cases. This is due to the significantly reduced residual stress of interface before debonding. However, as shown in Fig. 7b, due to the large value of 𝜀f , the interface
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Fig. 6b. Deformation and SCF profile of model 4.2: Interface starts to debond, overload region expands.
Fig. 6c. Deformation and SCF profile of model 4.2: Interface keeps debonding.
Fig. 6d. Deformation and SCF profile of model 4.2: Interface keeps debonding and approaching the edge, but does not fail all the way to the edge.
along the broken tow does not debond (completely fail) even at the final load. Instead, the interface keeps its residual stress accounted for as friction. Thus, the overload region keeps its shape with a small max SCF of around 1.05. The comparison between Model 4.2 and 4.3 indicates that the tow/yarn interface behavior plays an important role and directly affects the stress redistribution and failure evolution of a broken tow. Though the energy required to cause interface debonding is equivalent between those two models, the detailed stress-strain curve (residual and failure
strain) lead to noticeably different overload profiles and stress concentration factors. 5. Summary and conclusions In this work, we briefly discussed several important features in the damage and failure evolution of overwrap in COPVs. Then a developed finite element model was presented. This model realizes the simulation and study of in-plane tow/yarn breakage, tow/yarn interfaces and inter8
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Fig. 7a. Deformation and SCF profile of model 4.3: At the moment tow breaks. Close-up look of tow break line is shown with a deformation scale factor of 5.
Fig. 7b. Deformation and SCF profile of model 4.3: At final load, interface does not debond, it remains active with residual stress to account for the friction, resulting in a wide and gentle overload region.
ply interfaces, as well as arbitrary winding build-up of multiple layers, which has not been presented in the literature yet. Then we showed the modeling of a single tow break in one layer. Results indicate that the model can properly simulate the essential damage phenomena and evolution of tow/yarn break and interface debonding. Numerical evaluation of overload (SCF) was discussed. A case study of three different interface strengths was presented. It was shown that the damage evolution is directly affected by detailed interface behavior, especially the residual stress, yielding and ultimate failure strain of interface. The progressive failure in composites is the result of interaction and accumulation of multi-mode failures, which will be further investigated in future studies. More specifically, behavior and evolution of multiple tow breakages and correlated interface damages will be introduced into the model. In this paper, the main focus is to study the stress concentration profile and propagation of a single tow break and how these phenomena are affected by the mechanical response of tow interface. Current and future work includes: 1) Investigation on the effect of a single tow break in various winding and stress states, with improved user material profiles. 2) Comprehensive study of progressive damage evolution of overwrap based on the model, where the interaction and
evolution between various failure modes will be investigated. 3) Generation of potential database to serve as constitutive laws for analysis and modeling of COPVs at component scale. References [1] P.B. McLaughlan, L.R. Grimes-Ledesma, Composite overwrapped pressure vessels, Nasa/Sp–2011–573 (2011) 1–20 http://www.sti.nasa.gov. [2] L. Grimes-Ledesma, S.L. Phoenix, H. Beeson, T. Yoder, N. Greene, Testing of carbon fiber composite overwrapped pressure vessel stress-rupture lifetime, in: Am. Soc. Compos. - 21st Tech. ConS Am. Soc. Compos., 2006, pp. 205–218. [3] L.K. Grunenfelder, T. Centea, P. Hubert, S.R. Nutt, Effect of room-temperature outtime on tow impregnation in an out-of-autoclave prepreg, Compos. Part A Appl. Sci. Manuf. 45 (2013) 119–126 https://doi.org/10.1016/j.compositesa.2012.10.001. [4] M. Kezirian, K. Johnson, S.L. Phoenix, Composite overwrapped pressure vessels (COPV): flight rationale for the space shuttle program, AIAA Sp. Conf. Expo (2011) https://doi.org/10.2514/6.2011-7363. [5] , American Institute of Aeronautics and Astronautics Standard: Space Systems—Composite Overwrapped Pressure Vessels (ANSI/AIAA S-081B-2018), March, 20, 2018 https://doi.org/10.2514/4.105425. [6] I.J. Beyerlein, S.L. Phoenix, A.M. Sastry, Comparison of shear-lag theory and continuum fracture mechanics for modeling fiber and matrix stresses in an elastic cracked composite lamina, Int. J. Solids Struct. 33 (1996) 2543–2574 https://doi.org/10.1016/0020-7683(95)00172-7. [7] I.J. Beyerlein, S.L. Phoenix, Stress concentrations around multiple fiber breaks in an elastic matrix with local yielding or debonding using quadratic 9
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influence superposition, J. Mech. Phys. Solids. 44 (1996) 1997–2039 https://doi.org/10.1016/S0022-5096(96)00068-3. [8] S.L. Phoenix, M. Ibnabdeljalil, C.Y. Hui, Size effects in the distribution for strength of brittle matrix fibrous composites, Int. J. Solids Struct. 34 (1997) 545–568 https://doi.org/10.1016/S0020-7683(96)00034-0. [9] , ABAQUS/ Standard User’s Manual, v.6.11, Dassault Systems®, Rhode Island, 2011.
[10] Toray Carbon Fibers America Inc. Torayca®t700s technical data sheet no.cfa-005. Technical report, 6 Hutton Centre Drive, Santa Ana, CA 92707. [11] R. Ganesh, S. Sockalingam, J.W. Gillespie, Dynamic effects of a single fiber break in unidirectional glass fiber-reinforced polymer composites: effects of matrix plasticity, Compos. Mater. 52 (2018) 1873–1886 https://doi.org/10.1177/0021998317737604.
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