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A fast and efficient numerical prediction of compression after impact (CAI) strength of composite laminates and structures
Thin–Walled Structures 148 (2020) 106588
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A fast and efficient numerical prediction of compression after impact (CAI) strength of composite laminates and structures Tian Ouyang a, Rui Bao a, *, Wei Sun b, a, Zhidong Guan a, **, Riming Tan a a b
School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China Unmanned Aerial Vehicle Technology Institute, The Third Academy of CASIC, Beijing, 100074, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Composite laminate Impact damage CAI strength Finite element analysis Stiffened panel
In this paper, an equivalent damage model is proposed to quickly predict the compression after impact (CAI) strength of composite laminates and stiffened panels. Low-velocity impact (LVI) and CAI tests at various energy levels are carried out on laminates. Based on the measured impact damage sizes, the model simplifies the damage area into concentric circles with different stiffness and strength properties by adopting the soft inclusion method. Progressive failure analysis and virtual crack closure technique (VCCT) are used to simulate the intra-laminar and inter-laminar failure of impacted laminates under axial compression in commercial finite element soft ware ABAQUS. The predicted failure modes and CAI strength of laminates are in good agreement with the experiment results from this paper and the literature at various impact energies, which proves the validity of the model. Whilst the numerical model achieves high computational efficiency, the effect of mass scaling on computational efficiency and accuracy is evaluated, which provides a reference for the parameter choice in engineering application. The model is further applied to predict the compressive failure load of the stiffened composite panel with skin bay impact damage as a practical application, showing this modelling approach is also suitable to composite structures.
1. Introduction Carbon fiber reinforced polymer (CFRP) is widely used in aircraft structures due to its excellent specific strength/stiffness, designability and fatigue resistance. However, the poor mechanical properties in the out-of-plane direction make CFRP quite sensitive to the inevitable lowvelocity impact (LVI) by foreign objects during manufacturing, service and maintenance. The impact load will induce multiple damages inside the composite materials while leaving a permanent dent on the surface. These damages, including matrix cracks, delamination and fiber breakage can result in a significant reduction in residual compression strength and stiffness of composites [1,2]. For composite structures in aircraft, it is must be ensured that the structure can sustain the loads after LVI to guarantee flight safety. Therefore, compression after impact (CAI) behaviors of composites attract the attention of scholars. Many experimental researches have been performed to understand the CAI failure mechanism. It is found that CAI failure can be influenced by factors such as impact energies [3], laminate geometric parameters [4], ply sequences [5], material systems
[6], and impact locations [7]. With the development and improvement of finite element (FE) method, CAI strength prediction through numerical simulation is more and more popular as a low cost analysis method. In order to predict the failure load of composite laminates after LVI, two successive analysis steps can be employed in a single FE model, that is, predicting the LVI damage in the first step, and then analyzing the compression process in the second step. This method has been adopted by many scholars. Three dimensional FE models were constructed by Caputo [8] and Liu [9] et al. to predict both LVI damage and CAI strength of composite laminates. Rivallant et al. [10] proposed a unique model based on the discrete ply model. The matrix cracks and delamination happening in LVI and CAI were simulated by inserting cohesive elements in and between plies. A continuum shell based modeling approach was introduced by Thorsson and Waas et al. [11–13]. The enhanced Schapery theory material model [14] and discrete cohesive zone model [15] were implemented in the FE model to capture the intra-lamina and inter-laminar failure modes both in LVI and CAI. The robustness of their model was proved by accurately predicting experiment results of laminates with different layups. Sun
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (R. Bao), [email protected] (Z. Guan). https://doi.org/10.1016/j.tws.2019.106588 Received 16 October 2019; Received in revised form 29 December 2019; Accepted 29 December 2019 Available online 6 January 2020 0263-8231/© 2020 Elsevier Ltd. All rights reserved.
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and Hallett [16] established a high-fidelity FE model to study the key driving mechanisms and damage evolution of the compressive failure of laminates with barely visible impact damage. Their numerical result shows that the initial delamination grows at the interfaces near the back face and then into the undamaged cone due to sublaminates buckling. After the entire damage region is delaminated and buckled, it propa gates rapidly and unstably in the lateral direction, leading to the final CAI failure. Thin ply laminates were studied through numerical simu lations by Soto et al. [17]. They found that the CAI process exhibited a brittle behavior with almost no damage growth before failure. Delami nation and especially the fiber constitutive law shape are important for accurate predictions while matrix cracks can be negligible. Abir et al. [18,19] built FE models to investigate the effects of stacking sequences, sublaminate scaling and ply clustering on CAI behaviors, and the rela tionship between different failure mechanisms and CAI strength was explored. However, although the above successive analysis steps method can make the simulation reveal the details of damage growth and final failure in CAI processes more factually, it requires detailed modelling and long CPU time. To avoid the high computational costs, the equiv alent damage model based on actual impact damage is a feasible approach for CAI strength prediction. In the analytical prediction method proposed by Esrail et al. [20], fiber damage and delamination regions introduced by LVI were simplified to concentric elliptical inclusions with different material pa rameters. Rayleigh-Ritz method was employed to calculate the stress to determine the local failure, and then a progressive failure analysis fol lowed until the entire laminate failed. Rozylo et al. [21] presented a simplified damage model (SDM) for quick CAI strength predictions. The model characterized the decrease in material strength after LVI by reducing the thickness of plies in the damage region according to the impact energy. Good agreement was achieved between the numerically predicted CAI strength and the experiment results. As for composite structures with larger size, the advantage of the equivalent damage model becomes more obvious. Based on the SDM, Debski et al. [22] analyzed the buckling, post-buckling and failure behaviors of impacted composite struts. Wang et al. [23] predicted the onset of buckling and collapse load of the pre-damaged stiffened panel by degrading the ma terial properties of the impact damage area. Moreover, both Li [24] and Sun [25] et al. have developed phenomenological mechanical FE models for composite stiffened panels with stiffener impact damage. It can be seen that the simplification of the impact damage makes it more prac tical and efficient to estimate the residual strength of the structure. Although many models have been proposed by scholars, there is still a lack of simplified models that take into account the effect of sub laminate buckling, which in some cases [16,26] may be a crucial factor, on CAI strength of composite laminates and structures. The purpose of this paper is to develop a phenomenological-based simplified damage model that can predict the CAI failure load of com posite structures quickly and effectively. LVI and CAI tests at different energies were performed on laminates. The impact-induced interlaminar and intra-laminar damage were both introduced into the FE model by means of the equivalent damage method. Standard CAI experiment results of two material systems from this paper and literature [10] respectively were used to validate the model, and the differences in failure modes between the two types of laminates were analyzed. Then in order to provide a reference for engineering application, the effect of mass scaling on computational efficiency and accuracy was discussed. Finally, this modelling technique was further applied to a stiffened panel with skin bay impact as a full structural application example.
parameters are shown in Table 1. The stacking sequence of the laminate was [45/-45/0/-45/0/45/0/45/0/45/90/45/-45/0/45/-45/90/0]S. Besides, in order to ensure that the proposed equivalent damage model is applicable to laminates with different material systems and ply sequences, standard CAI experiment results of T700/M21 carbon fiber/ epoxy resin laminates reported by Rivallant et al. [10] are also used for model validation, which will be discussed in section 5.2. The LVI tests were conducted according to ASTM D7136 standard [27]. The impactor was made of steel with a total mass of 5.477 kg and the head was a 12.7 mm diameter hemisphere. To study the variation trend of compression failure behaviors with the impact energy, LVI damage at different energy levels was introduced into the laminates. The applied impact energy as well as the number of specimens of each group are summarized in Table 2. Altogether 21 specimens were contained in the tests. After the impact, the dent depth of each specimen was measured immediately by a digital dial gauge, and then an ultrasonic C-scan equipment was employed to detect the internal damage. The CAI tests were carried out on an Instron 8802 electro-hydraulic servo test machine, of which the maximum load is �250 kN. According to ASTM D7137 standard [28], the specimen was placed in a special fixture for end compression loading until failure, and the loading rate was 1 mm/min. 3. Experiment results 3.1. LVI experiment results LVI experiment results can be divided into two cases according to the damage degree of the dent area. As typically shown in Fig. 1(a), when the impact energy is 22 J, the average depth of the dent is 0.44 mm, and no visible fiber or matrix damage is found around the dent. With the impact energy decreasing, the dent gradually becomes harder to observe. However, for specimens impacted by energy higher than 22 J, ma terial damage is obvious in the dent area. As shown in Fig. 1(b), when the impact energy is 26 J, the dent tends to be circular with an average depth of 0.94 mm. At this time, a small quantity of fiber breakage and matrix cracks begin to appear around the dent. With the rise of energy, a larger number of fiber fractures and matrix cracks are caused by impact, making the dent diameter and depth significantly increased. When the energy grows to 34 J, the average dent depth of the laminates is 2.67 mm. The trend of dent depth with impact energies is shown in Fig. 2. As can be seen from the figure, the slope of the dent depth-impact energy curve changes obviously between 22 J and 26 J due to the occurrence of fiber breakage, which is consistent with the “knee point” phenomenon mentioned in Refs. [5,29]. Generally, before the knee point, there is almost no fiber failure but matrix cracks and delamination. When the impact energy exceeds the knee point, large amounts of fiber breakage Table 1 Material parameters of T300/QY8911 for numerical simulation. Property E1 E2 G12
ν12
XT XC YT YC S Gft Gfc GIC GIIC, GIIC t
2. Specimens and experiments The specimen was a 150 mm � 100 mm rectangular laminate with the thickness of 4.2 mm. It was manufactured of T300/QY8911 carbon fiber/bismaleimide resin unidirectional prepreg, and the material 2
Values Longitudinal Young’s modulus Transverse Young’s modulus In-plane shear modulus Poisson’s ratio Longitudinal tensile strength Longitudinal compressive strength Transverse tensile strength Transverse compressive strength Shear strength Longitudinal tensile fracture toughness Longitudinal compressive fracture toughness Interfacial normal fracture toughness Interfacial shear fracture toughness Ply thickness
135 GPa 8.8 GPa 4.47 GPa 0.33 1548 MPa 1226 MPa 55.5 MPa 110.5 MPa 89.9 MPa 48 N/mm 24 N/mm 0.2 N/mm 0.5 N/mm 0.12 mm
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identical, both of which are large areas of longitudinal splits caused by stress concentration at the end of specimens. This result shows that the compressive bearing capacity of the laminate will not be affected when the impact energy is lower than the delamination threshold value. The trend of CAI strength with impact energies is shown in Fig. 4. It can be seen that the CAI strength decreases significantly by about 41.6% after delamination occurs in the specimen. However, as the impact en ergy continues to increase, the change of CAI strength gradually be comes smooth, making the entire curve present a “step” form. This phenomenon was also observed and reported in Refs. [5,6].
Table 2 Impact energy and the number of specimens of each group. Group
Pristine
L1
L2
L3
L4
L5
L6
L7
Impact energy/J Number of specimens
N/A 1
9 2
12 3
18 3
22 3
26 3
30 3
34 3
4. Numerical modelling technique It is always a complex problem to appropriately introduce the LVI damage into the model for accurate prediction of CAI strength. In this paper, a phenomenological-based method, considering both the intralaminar damage and the number and shape of delamination, was developed to simplify the impact-induced damage of the laminate. Different simplification of the intra-laminar damage was performed according to the damage degrees, while multiple delamination was replaced by some severe delamination. The intra- and inter-laminar failure were simulated by Hashin criterion and virtual crack closure technique, respectively.
Fig. 1. Typical dent appearance of the specimen after (a) 22 J and (b) 26 J impact.
4.1. Equivalent method for impact damage According to the post-impact morphology of all specimens, the damage area is simplified into two concentric circles, and the whole laminate is divided into three typical regions (see in Fig. 5). Region I is the dent area. As mentioned above, for specimens before the knee point, the main intra-laminar damage in the dent area is matrix damage; while for specimens after the knee point, severe fiber and matrix damage occur in the dent area. The region between the boundary of Region I and the outer fringe of the simplified circular delamination is Region II. Since the initiation of delamination is mostly caused by matrix cracking [31], the main intra-laminar damage in this region is assumed to be matrix damage. Region III is an undamaged area. Since the structure of an impact damage is quite complex, the effect on the material properties by every individual crack is actually not possible to measure. According to the soft inclusion model [1], a feasible approach to characterize the fiber and matrix damage in Region I and II is to reduce their macro material parameters. Therefore, for specimens after the knee point, based on the material degradation method pro posed in Refs. [24,32], the modulus and strength parameters in Region I are set to 5% of their pristine values. For specimens before the knee point, damage patterns in Region I and II are identical; thus the same degradation method was adopted for both of them. The modulus and strength parameters related to the matrix (i.e. parameters other than longitudinal ones) are set to 20% of their pristine values [24,32]. In addition to the degradation of material properties, sublaminates buckling caused by impact-induced delamination under axial compres sion should also be considered. According to the results of Ref. [33], delamination with a smaller size has less effect on the sublaminates buckling behavior of the laminate. On this basis, multiple delamination cracks can be approximated by some severe delamination to improve computational efficiency. Moreover, it should also be mentioned that LVI-induced delamination can be connected through the matrix crack between plies [31,34], and thus form a larger continuous crack. Ac cording to the previous experimental study [35], the present studied laminate specimen can form six groups of three-connected de laminations that are spaced by about 5 plies in turn along the thickness direction after LVI (see in Fig. 6). Therefore, the impact delamination inside the laminate can be preliminarily simplified to these six de laminations. However, delaminations in six interfaces still result in high computational costs. In order to further improve the computational ef ficiency, it is necessary to select the most dangerous one or two
Fig. 2. The trend of dent depth with various impact energies.
will occur in the laminate. Fig. 3 shows a comparison of projected delamination area measured by C-scan in selected specimens with different impact energies. It can be seen that there is a delamination threshold energy between 9 J and 12 J. Delamination can be detected in the laminate only when the impact energy is higher than the threshold value [30]. Regardless of the matrix split, the delamination shape is roughly circular. As the impact energy grows, the delamination area gradually increases (12 J–26 J) at first and then tends to level off (26 J–34 J). The reason for this phenomenon is that fiber failure begins to occur at 26 J impact, and further increased energy is largely dissipated by fiber breakage. The arithmetic average dent depth and projected delamination area of each group are summa rized in Table 3. It should be noted here that matrix splits are not taken into account in the delamination area due to the fact that they usually occur only in the last one or two plies at the rear side and have little effect on the global stiffness and strength of the laminate. 3.2. CAI experiment results Table 4 shows the average CAI strength of each group. Except that the coefficient of variation (CV) of L3 group is 10.6%, CV of the other groups are all within 5%, reflecting that the experiment results have good repeatability and reliability on the whole. In addition, the average CAI strength of L1 group is very close to the ultimate compression strength of the pristine specimen. The failure modes of them are 3
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Fig. 3. Collection of C-scan images of the specimens with various impact energies. (Different colors represent the thickness from which the scanning ultrasonic wave gets reflected). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) Table 3 Arithmetic average dent depth and delamination area of each group. Group
Average dent depth/mm (CV)
Average delamination area/mm2 (CV)
L1 L2 L3 L4 L5 L6 L7
0.02 0.13 (4.9%) 0.45 (16.0%) 0.44 (11.2%) 0.94 (15.2%) 1.75 (4.6%) 2.67 (16.8%)
Not detected 641 (1.6%) 778 (5.2%) 1268 (8.0%) 1544 (13.4%) 1552 (10.6%) 1689 (8.4%)
Table 4 Average CAI strength of each group. Group
Average CAI strength/MPa (CV)
Group
Average CAI strength/MPa (CV)
Pristine L1 L2 L3
476 474 277 (4.8%) 249 (10.6%)
L4 L5 L6 L7
238 209 205 195
(3.7%) (2.7%) (2.1%) (1.6%)
Fig. 4. The trend of CAI strength with various impact energies.
much higher load. Hence, delamination in the FE model is finally ar ranged at I2&5. Generally, after the low-velocity impact, permanent out-of-plane deformation will be formed in the impacted area. In the subsequent compression test, this local deformation will introduce additional bending moment, affecting the sublaminates buckling behavior of the laminate [36]. In the FE model, the out-of-plane deformation is char acterized by node offset in the dent area (see in Fig. 8). It should be noted here that in the actual situation, the impact-induced out-of-plane deformation exists both in the dent area and the delamination area (Region II). However, according to the numerical simulation, the pro posed FE model is insensitive to the out-of-plane deformation in Region II when there is already an out-of-plane deformation in the dent area. Thus in order to make the model better applied in engineering practice, the out-of-plane deformation modeling in Region II is ignored.
delaminations from Fig. 6 for introducing into the FE model. To this end, a pre-numerical calculation was performed. Six numerical models, of which the delamination is at I1, I2, I3, I1&6 (considering the symmetry of the stacking sequence), I2&5 and I3&4 respectively were built, and the numerical results are shown in Fig. 7. As can be seen from the figure, the numerical model with delamination at I2&5 shows the lowest residual strength, which means that this arrangement has the greatest influence on the CAI response of the laminate. This phenomenon is reasonable because, when the delami nation is closer to the surface, although the sublaminate is more prone to buckling, the delamination propagation is not that easy to occur due to the lower peeling stress induced by the buckling of the thinner sub laminate. This, as a result, leads to a higher failure load of the laminate. However, when the delamination is close to the mid-plane, the signifi cant increase of the thickness makes sublaminate buckling quite difficult to occur, and thus results in delamination propagation taking place at a 4
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Fig. 5. Schematic diagram of typical damage morphology of laminates after LVI (take the 22 J specimen as an example).
4.2. Damage model 4.2.1. Intra-laminar damage In two-dimensional cases, Hashin criterion [37] is often employed to determine the initiation of intra-laminar damage in composite laminates [38,39]. Four independent damage modes are considered in the crite rion, which are described as follows: � �2 � � σ 11 τ12 2 þ �1 (1) XT S
Fig. 6. Approximate distribution of the connected delaminations through the thickness.
�
σ 11
�2
�
σ 22
�2 þ
YT �
(2)
�1
XC
�τ �2 12
S
��� �2 YC YC 2S
σ 22
�1
(3)
� � � � � σ 22 2 τ12 2 1 þ þ �1 2S S
(4)
where σ 11 and σ 22 represent the stresses parallel and perpendicular to the fiber direction, respectively, while τ12 is the in-plane shear stress. Once the damage initiation criterion is satisfied, the composites undergo material property degradation. The constitutive relation of a damaged ply is defined as: (5)
σ ¼ Cd ε 2 Fig. 7. Residual strength comparison of laminates with various delamina tion locations.
Cd ¼
14 1 D
� 1� df E1 df ð1 dm Þν12 E2 0
1
3
�
df ð1 dm Þν21 E1 ð1 dm ÞE2 0
ð1
0 5 0 ds ÞG12 D (6)
where D ¼ 1 ð1 df Þð1 dm Þν12 ν21 ; df, dm and ds are damage vari ables associated with fiber damage, matrix damage and shear damage, respectively. Meanwhile, ds is not independent and can be expressed as: � ds ¼ 1 1 df ð1 dm Þ (7) A linear damage evolution law was adopted in the model. After the initiation of damage, the stiffness of the material progressively decreases with the increase of damage variables until enough energy is absorbed for complete failure. 4.2.2. Inter-laminar damage model Virtual crack closure technique (VCCT) [40], as an effective method to calculate the energy release rate (ERR), has been applied to predict inter-laminar failure in a lot of literature [41,42]. Different from the traction-separation law based cohesive zone model which requires some empirical parameters [43,44], the VCCT model does not introduce any special elements and uses only fracture toughness parameters; thus it was employed in this paper. In the VCCT model, the energy released by the propagation of a crack is assumed to be equal to the energy required to close the crack;
Fig. 8. FE models of specimens (a) before the knee point and (b) after the knee point.
5
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therefore, the ERR can be calculated in one step by the node force at the crack tip and the relative displacement between the nearest node pair to the crack tip. Initial bonded nodes are released when the following criterion is satisfied: Gequiv �1 GC
(8)
where Gequiv and GC are the equivalent and critical ERR, respectively. The critical ERR is calculated by B–K criterion [45]: �η � GII þ GIII GC ¼ GIC þ ðGIIC GIC Þ (9) GI þ GII þ GIII where GIC, GIIC and GIIIC are the critical ERR for mode I, mode II and mode III crack propagation, and their values are 0.2 N/mm, 0.5 N/mm and 0.5 N/mm, respectively; η is constant and is set as 1.45 according to Ref. [46].
Fig. 10. Initial bonded nodes of the FE model.
constraint”. In addition, according to the recommendation of Ref. [47], Rayleigh damping was introduced into the material so as to reduce the possible vibration in the simulation. The damping parameters were set as α ¼ 3000 and β ¼ 0 [29]. The boundary condition of the model was consistent with that of the experiment. The anti-buckling knife edges as well as end clamping blocks were simply simulated by constraining the corresponding degree of freedom of nodes. A compression displacement load was applied through a reference point by the “equation constraint” at the longitu dinal end of the laminate, and a “smooth step” was adopted to reduce dynamic effects. Besides, according to Ref. [48], mass scaling and higher loading rate were used to reduce the run time. The time increment was set to be no less than 10 6 s (errors < 1.5% compared with no mass scaling models) and the loading speed was fixed at 3.5 mm/s. Kinetic energy (ALLKE) and internal energy (ALLIE) of the model were recorded during the computation, and the proportion of ALLKE to ALLIE was less than 0.2%. The CPU time using 16 Xeon E5-2640 v2 cores (2 GHz) was about 1 h.
4.3. Modelling strategy Since no damage was observed in laminates impacted by 9 J, and their compression strength was consistent with that of the pristine specimen, only groups with the impact energy higher than 9 J (L2~L7 groups) were modeled in commercial FE software ABAQUS/Explicit to predict the CAI strength. In order to reduce the computational scale, a mesh convergence analysis was carried out beforehand. As shown in Fig. 9, loaddisplacement curves obtained by various mesh densities coincide with each other at the loading stage, and only the failure load decreases with the increase of the element size. Models with element sizes of 1 mm and 1.5 mm show 1.1% and 3.9% lower CAI strength respectively compared to the model with a 0.75 mm element size, which can be considered small. To achieve the balance between computational efficiency and accuracy, an element size of 1.5 mm was used in the model. Approximately 20000 continuum shell elements (SC8R) were created to discretize the laminate. Material properties used in the simulation is shown in Table 1. Based on the previous analysis, delam ination was arranged between the 10th and 11th plies and between the 25th and 26th plies with the same area as the projected delamination (see the average area in Table 3). The average dent diameters of L2~L7 groups were 4 mm, 5 mm, 6 mm, 8 mm, 10 mm and 13 mm, respectively while the average dent depth is shown in Table 3. Besides, considering that delamination mainly propagates along the transverse direction under compression, delamination growth was set to occur only within the 100 � 100 mm2 region (see in Fig. 10). Adjacent surfaces of the rest two 25 � 100 mm2 regions were joined together by using the “tie
5. Model validation and application 5.1. CAI strength prediction of T300/QY8911 laminates The comparison of the CAI strength between the experimental and numerical results are shown in Fig. 11. The predicted compression strength after different impact energies are 283 MPa, 270 MPa, 241 MPa, 210 MPa, 198 MP and 188 MPa, respectively, and the errors relative to the average experiment values are 2.2%, 8.4%, 1.3%, 0.5%, 3.4% and 3.6%, respectively. These results indicate that the
Fig. 9. Predicted load-displacement curves by 12 J model with various mesh densities.
Fig. 11. Comparison of CAI strength between experimental and numeri cal results. 6
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proposed FE models achieve accurate predictions of CAI strength of laminates in the range of 12 J–34 J. The typical CAI failure mode of the specimens is exhibited by the L2 group as an illustrative example in Fig. 12(a). As can be seen from the figure, significant sublaminates buckling appears on both the impacted and non-impacted sides of the specimen, but no fiber breakage is observed on the surfaces. In accordance with the experiment results, the FE model also shows obvious sublaminates buckling; meanwhile, no fiber failure takes place in the model except for a small amount of fiber breakage in the dent area. It can be seen from Fig. 12(b) that the compression load stops increasing after the delamination growth occurs, and then the delamination rapidly propagates to the edges of the lami nate. This phenomenon indicates that the failure of the model is due to delamination growth induced by the sublaminates buckling, and this growth is generally unstable. Based on the above results, it seems that for laminates with brittle resin (like QY8911), the low mode I and mode II matrix fracture toughness make the CAI failure inclined to be caused by propagation of delamination induced by sublaminates buckling. Thus, the effective characterization of the impact-induced delamination plays an important role in predicting CAI strength of such laminates.
Table 5 Material properties of T700/M21 for numerical simulation [10,49]. Property
Values
Elastic properties
ET1 ¼ 130 GPa; EC1 ¼ 100 GPa; E2 ¼ 7.7 GPa; G12 ¼ 4.8 GPa; ν12 ¼ 0.3
Strength Fracture toughness Interface properties
XT ¼ 2080 MPa; XC ¼ 1250 MPa; YT ¼ 60 MPa; YC ¼ 290 MPa; S ¼ 110 MPa Gft ¼ 133 N/mm; Gfc ¼ 10 N/mm; Gmt ¼ 0.5 N/mm; Gmc ¼ 1.6 N/mm GIC ¼ 0.5 N/mm; GIIC ¼ 1.6 N/mm; η ¼ 1.45
Table 6 Impact damage sizes of the T700/M21 laminates [10]. Impact energy/J
Dent depth/ mm
Dent diameter/ mm
Delamination diameter/ mm
6.5 17 26.5 29.5
0.06 0.23 0.52 0.70
3 5 7 10
19.4 34.0 40.3 41.0
equivalent damage model was also arranged at these two locations. Fig. 13 shows the comparison of CAI strength between the experi ment results of Ref. [10] and the predicted values of the FE model proposed in this paper. As can be seen from the figure, from 6.5 J to 29.5 J, the errors between experimental and numerical results are 4.3%, 12.3%, 6.5% and 5.3%, respectively. The results of equivalent damage models are in good agreement with the experiment results except that the 17 J model has an error of more than 10%. The reason for the un derestimate of compression strength after 17 J impact may be the dispersion of a single experiment result since the prediction value of the model in Ref. [10] was also lower than the experiment result. Taking 17 J and 29.5 J models as illustrative examples, the CAI failure processes simulated by numerical models are basically consistent with the experimental phenomena observed in Ref. [10]. As shown in Fig. 14(a), for the 17 J model, no fiber breakage is found on the surface with the increase of compression stress. However, when the stress in creases to 213 MPa, the model suddenly ruptures due to an unstable crack propagating from the impact area to the edges of the laminate. As for the 29.5 J model (see in Fig. 14(b)), the scenario is not the same. Progressive crack propagation is observed in the model. At 121 MPa, fiber breakage occurs due to the buckling of sublaminates around the impact area. Then, the breakage of the fiber also in turn increases the sublaminates buckling. When the crack propagates to the edges of the
5.2. CAI strength prediction of T700/M21 laminates In this section, based on the standard LVI experiment results reported by Rivallant et al. [10], the equivalent damage model was used to pre dict the CAI strength of laminates. The specimens were made of T700/M21 carbon fiber/epoxy resin unidirectional prepreg with a ply thickness of 0.26 mm and a stacking sequence of [02/452/902/-452]S. The material properties of T700/M21 are shown in Table 5. The damage sizes introduced by a 16 mm diameter impactor at different energies in Ref. [10] are extracted and summarized as shown in Table 6. It should be noted that dent diameters are not given in the literature and hence data in the table are estimated values. According to the experimental phenomenon in Ref. [50], delamination can be sup pressed at the local interface close to the impact side due to the direct compression of the impactor. Thus, the dent area can be estimated by taking the impact point as the center to make a circle tangent to the surrounding delamination. Besides, based on the experiment results described in the literature, laminates under 6.5 J and 17 J impact were before the knee point, while laminates under 26.5 J and 29.5 J impact were after the knee point. In addition, Abir et al. [19] have conducted numerical analysis using the same stacking sequence and material sys tem, and their results show that the largest impact delamination occurs at the interfaces of 90� /-45� and 45� /90� . Thus, delamination in the
Fig. 12. (a) Comparison of failure modes between the experiment result and FEM simulation; (b) Final failure dominated by transverse delamination propagation in large area. 7
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Fig. 13. CAI results comparison: (a) results from Ref. [10], (b) prediction by the proposed FE method. (Experiment values may have small errors since they are read from the figure by the author).
Fig. 14. Predicted CAI failure processes of (a) 17 J model and (b) 29.5 J model.
laminate, the final rupture of the model happens. Based on the above phenomena, it can be seen that there are two failure scenarios of the laminates. When the impact energy is low, the initial delamination is small and thus the sublaminates buckling is also slight. As a result, the final failure of the laminate is caused by the un stable crack propagation at high CAI stress. When the impact energy is high, the increase of the initial delamination area makes the sub laminates buckle easily. The sublaminates buckling and crack propa gation develop together and induce the final failure of the laminate. Thus the CAI strength decreases.
It is obvious that the above failure scenarios are different from those in section 5.1. This is mainly because laminates studied in this section is made of M21 epoxy resin, which has higher mode I and mode II matrix fracture toughness compared to QY8911 bismaleimide resin; thus delamination propagation in large area is relatively not easy to occur. Based on the numerical results, it can be concluded that when predicting CAI strength of laminates with high matrix fracture toughness, not only the impact-induced delamination, but also the fiber compression prop erty parameters have a quite important influence on the accuracy of the prediction. Besides, the above results also show that the presented 8
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equivalent damage model is effective. 5.3. Effect of mass scaling In engineering practice, designers usually hope to obtain the CAI strength of composites as soon as possible to evaluate the damage tolerance performance. Therefore, it is important to take account of both computational accuracy and efficiency. For explicit analysis, mass scaling and reduction of time period are unavoidable to achieve an economical solution. According to Ref. [47], for quasi-static simulations incorporating rate-independent materials, mass scaling and reduction of time period yield similar results. Thus, only the effect of mass scaling on numerical results is discussed below. One approach to perform mass scaling in ABAQUS/EXPLICIT is to define a target time increment. The L2~L7 models were simulated with different time increments, and the results are shown in Table 7. It can be seen that the predicted CAI strength increases with the increase of time increments. The results of no mass scaling are quite close to those of time increments with 1 � 10 6 s (errors < 1.5%), while the CPU time is reduced from 8–12 h to approximately 1 h, which significantly improves the computational efficiency. When the time increment increases to 5 � 10 6 s, the CPU time is further reduced to about 0.3 h. At this time, an obvious error appears in L3 model, but other models stay a good pre diction accuracy. With the time increment increasing to 1 � 10 5 s, the run time decreases to 0.1–0.2 h, and errors of all models except L3 are still within 10%. However, when the time increment is set as 5 � 10 5 s, a significant dynamic effect appears in the simulation. The predicted CAI strength of all models are considerably higher, with errors of more than 50%. Based on the above results, it can be seen that the time increment set as 1 � 10 6 s can greatly improve the computational efficiency while maintaining high accuracy. Setting the time increment as 1 � 10 5 s can reduce the CPU time to about 0.1–0.2 h, but there is an error of up to 16.1%, which needs to be weighed in practical application. However, further increasing the time increment is no longer applicable since the prediction values become significantly larger.
Fig. 15. The FE model of the L-stiffened composite panel with skin bay impact damage.
geometric dimensions and stacking sequences information can be found in Ref. [35]. After 27 J impact on the skin bay of the panel, the measured average dent depth, dent diameter and projected delamination diameter were 1.29 mm, 10 mm and 54 mm, respectively. An element size of 1.5 mm was used in the delamination growth area and there were about 52000 SC8R elements in the model. The boundary condition of the model (see in Fig. 15) was that all six degrees of freedom were constrained at the fixed end while all degrees of freedom except Uy were constrained at the loading end. The “equation constraint” was used to apply the displace ment load to the panel through a reference point, and the loading rate was 3.5 mm/s. Since no stiffener-skin debonding occurred in the test, this damage was not considered in the FE model. The CPU time of the simulation was approximately 2 h. Numerical results are compared with the experiment results from Ref. [35]. As shown in Fig. 16, the load-displacement curve obtained by FE simulation is in good agreement with the experiment results. The curves keep a linear growth until the ultimate failure, indicating that no global buckling of the panel occurs in the compression process. The predicted failure load is 210 kN, with the error of 7.9% compared to the average experiment value of 228 kN. Fig. 17 shows the comparison of failure modes between the FE simulation and the experiment. As can be seen from Fig. 17(a), the skin of the stiffened panel shows obvious sublaminates buckling near the impact point, and a large area of delamination also appears at the free edge of the skin. Moreover, fiber failure occurs only at the dent area. These phenomena are consistent with the experiment results shown in Fig. 17(b) and confirm that the failure of the stiffened panel with skin
5.4. Application of the model in composite structures In aircraft structures, stiffened composite panels are extensively applied in many components, such as the fuselage, wings and horizon tal/vertical tails. When a stiffened panel is subjected to LVI in the skin bay, the damage pattern is similar to that of the laminate. Therefore, the previously validated equivalent damage model was applied in this sec tion to predict the CAI failure load of stiffened composite panels. Fig. 15 shows the model of the L-stiffened composite panel with skin bay impact damage. The skin stacking sequence and material properties of the panel are consistent with the T300/QY8911 laminates. More detailed Table 7 Numerical results by various target time increments. Target time increment (s)
L2
No limit
282/ 1.8/ 12.3 283/ 2.2/1.3 289/ 4.3/0.3
L3
L4
L5
L6
L7
CAI strength (MPa)/error (%)/CPU time (h)
1 � 10
6
5 � 10
6
1 � 10
5
294/ 6.1/0.2
5 � 10
5
419/ 51.3/ 0.06
267/ 7.2/ 11.9 270/ 8.4/1.2 286/ 14.9/ 0.3 289/ 16.1/ 0.2 417/ 67.5/ 0.06
239/ 0.4/ 10.3 241/ 1.3/1.1 245/ 2.9/0.3
207/1.0/8.7
198/3.4/8.3
188/3.6/7.9
210/ 0.5/1.0 213/ 1.9/0.2
198/3.4/1.0 199/2.9/0.2
188/3.6/1.0 194/0.5/0.2
252/ 5.9/0.2
217/ 3.8/0.1
218/ 6.3/0.1
214/ 9.7/0.1
386/ 62.2/ 0.06
375/ 79.4/ 0.06
372/ 81.5/ 0.06
368/ 88.7/ 0.06
Fig. 16. Load-displacement curves of specimens and FEM simulation for the Lstiffened composite panel. 9
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(2) For the studied M21 epoxy resin matrix laminates, the mode I and mode II matrix fracture toughness are high, and delamination growth is relatively not easy to occur after the buckling of sub laminates. When the impact energy is low, the impact-induced delamination is small and thus the sublaminates buckling is also slight. As a result, the CAI failure of the laminate is due to the sudden breakage of fibers. When the impact energy is high, the increase of the impact-induced delamination area makes the sublaminates buckle easily. The CAI failure of the laminate is affected by the interaction of the sublaminates buckling and the fiber breakage. Thus, the characterization of impact-induced delamination and the fiber compression property parameters are both important factors affecting the accuracy of CAI strength prediction for such laminates with high matrix fracture toughness. (3) The proposed model shows that small mass scaling (1 � 10 6 s) can greatly improve the computational efficiency (1 h) while maintaining the accuracy. A further increase of the mass scaling can reduce the CPU time to 0.1–0.3 h, but increase the compu tational errors; thus it needs to be weighed in practical application. In addition, the equivalent damage model was further successfully applied to predict the compression failure load of stiffened composite panels with skin bay impact damage, showing that this modelling approach is also applicable to composite structures. To sum up, the equivalent damage model proposed in this paper takes approximately 1 h and 2 h respectively for CAI failure load prediction of the studied laminates and stiffened panels, which is very suitable for the need of rapid residual compression strength assessment of impacted structures in engineering practice.
Fig. 17. Failure modes of the L-stiffened composite panel: (a) FEM simulation; (b) experiment results [35].
bay impact damage is dominated by delamination propagation in large area transversely as a result of sublaminates buckling. In general, the FEM and experiment results are in good agreement on the failure load and dominant failure modes. These results show that the equivalent damage method proposed in this paper is applicable to fast prediction of the compression failure load of composite stiffened panels with skin bay impact damage. In addition, for composite stiffened panels with impact damage at other locations, the compression failure load can also be predicted by introducing the corresponding stiffener-skin debonding damage or stiffener damage into the model.
CRediT authorship contribution statement Tian Ouyang: Conceptualization, Methodology, Investigation, Visualization, Writing - original draft. Rui Bao: Supervision, Resources. Wei Sun: Investigation, Validation, Writing - review & editing. Zhidong Guan: Resources. Riming Tan: Investigation, Formal analysis, Data curation.
6. Conclusions In this paper, LVI and CAI tests were carried out on CFRP unidirec tional laminates made of bismaleimide resin under different energy levels. Delamination threshold energy and knee point energy were observed in LVI tests. In subsequent CAI tests, the results show that the impact before threshold energy does not affect the compression strength of laminates, while the residual strength will be seriously weakened once the delamination occurs. However, after the threshold energy, the change of CAI strength with the impact energy gradually levels off. On the basis of the experiment results, an equivalent damage FE model that can predict the CAI strength of laminates quickly and effi ciently was built. By using progressive failure analysis and VCCT, the CAI strength and failure modes of laminates of two material systems at different energy levels were predicted. The numerical results are consistent with the experiment results, which verifies the validity of the model. Conclusions obtained through the FE analysis are as follows:
Acknowledgements The first author would like to thank Dr. Zengshan Li for his kind help in the experiment. The project is partially supported by Beijing Munic ipal Science and Technology Commission via project No. Z191100004619006. References [1] A. Sj€ ogren, A. Krasnikovs, J. Varna, Experimental determination of elastic properties of impact damage in carbon fibre/epoxy laminates, Composer Part AAppl. Sci. Manuf. 32 (2001) 1237–1242. [2] R. Craven, S. Pindoria, R. Olsson, Finite element study of compressively loaded fibres fractured during impact, Compos. Sci. Technol. 69 (2009) 586–593. [3] S. Rivallant, C. Bouvet, E.A. Abdallah, B. Broll, J. Barrau, Experimental analysis of CFRP laminates subjected to compression after impact: the role of impact-induced cracks in failure, Compos. Struct. 111 (2014) 147–157. [4] M. Remacha, S. S� anchez-S� aez, B. L� opez-Romano, E. Barbero, A new device for determining the compression after impact strength in thin laminates, Compos. Struct. 127 (2015) 99–107. [5] T.A. Sebaey, E.V. Gonz� alez, C.S. Lopes, N. Blanco, P. Maimí, J. Costa, Damage resistance and damage tolerance of dispersed CFRP laminates: effect of the mismatch angle between plies, Compos. Struct. 101 (2013) 255–264. [6] D.D.R. Cartie, P.E. Irving, Effect of resin and fibre properties on impact and compression after impact performance of CFRP, Composer Part A-Appl. Sci. Manuf. 33 (2002) 483–493. [7] S.I. Thorsson, S.P. Sringeri, A.M. Waas, B.P. Justusson, M. Rassaian, Experimental investigation of composite laminates subject to low-velocity edge-on impact and compression after impact, Compos. Struct. 186 (2018) 335–346.
(1) For the studied QY8911 bismaleimide resin matrix laminates, the mode I and mode II matrix fracture toughness are low, and the impact-induced delamination is easy to propagate under the peeling stress caused by sublaminates buckling under compres sion. From 12 J to 34 J, the CAI failure of laminates is all due to the delamination propagation in large area. Therefore, when CAI prediction is carried out on such laminates with low matrix fracture toughness, the effective characterization of the impactinduced delamination has an important influence on the accu racy of the numerical results. 10
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Full length article
A fast and efficient numerical prediction of compression after impact (CAI) strength of composite laminates and structures Tian Ouyang a, Rui Bao a, *, Wei Sun b, a, Zhidong Guan a, **, Riming Tan a a b
School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China Unmanned Aerial Vehicle Technology Institute, The Third Academy of CASIC, Beijing, 100074, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Composite laminate Impact damage CAI strength Finite element analysis Stiffened panel
In this paper, an equivalent damage model is proposed to quickly predict the compression after impact (CAI) strength of composite laminates and stiffened panels. Low-velocity impact (LVI) and CAI tests at various energy levels are carried out on laminates. Based on the measured impact damage sizes, the model simplifies the damage area into concentric circles with different stiffness and strength properties by adopting the soft inclusion method. Progressive failure analysis and virtual crack closure technique (VCCT) are used to simulate the intra-laminar and inter-laminar failure of impacted laminates under axial compression in commercial finite element soft ware ABAQUS. The predicted failure modes and CAI strength of laminates are in good agreement with the experiment results from this paper and the literature at various impact energies, which proves the validity of the model. Whilst the numerical model achieves high computational efficiency, the effect of mass scaling on computational efficiency and accuracy is evaluated, which provides a reference for the parameter choice in engineering application. The model is further applied to predict the compressive failure load of the stiffened composite panel with skin bay impact damage as a practical application, showing this modelling approach is also suitable to composite structures.
1. Introduction Carbon fiber reinforced polymer (CFRP) is widely used in aircraft structures due to its excellent specific strength/stiffness, designability and fatigue resistance. However, the poor mechanical properties in the out-of-plane direction make CFRP quite sensitive to the inevitable lowvelocity impact (LVI) by foreign objects during manufacturing, service and maintenance. The impact load will induce multiple damages inside the composite materials while leaving a permanent dent on the surface. These damages, including matrix cracks, delamination and fiber breakage can result in a significant reduction in residual compression strength and stiffness of composites [1,2]. For composite structures in aircraft, it is must be ensured that the structure can sustain the loads after LVI to guarantee flight safety. Therefore, compression after impact (CAI) behaviors of composites attract the attention of scholars. Many experimental researches have been performed to understand the CAI failure mechanism. It is found that CAI failure can be influenced by factors such as impact energies [3], laminate geometric parameters [4], ply sequences [5], material systems
[6], and impact locations [7]. With the development and improvement of finite element (FE) method, CAI strength prediction through numerical simulation is more and more popular as a low cost analysis method. In order to predict the failure load of composite laminates after LVI, two successive analysis steps can be employed in a single FE model, that is, predicting the LVI damage in the first step, and then analyzing the compression process in the second step. This method has been adopted by many scholars. Three dimensional FE models were constructed by Caputo [8] and Liu [9] et al. to predict both LVI damage and CAI strength of composite laminates. Rivallant et al. [10] proposed a unique model based on the discrete ply model. The matrix cracks and delamination happening in LVI and CAI were simulated by inserting cohesive elements in and between plies. A continuum shell based modeling approach was introduced by Thorsson and Waas et al. [11–13]. The enhanced Schapery theory material model [14] and discrete cohesive zone model [15] were implemented in the FE model to capture the intra-lamina and inter-laminar failure modes both in LVI and CAI. The robustness of their model was proved by accurately predicting experiment results of laminates with different layups. Sun
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (R. Bao), [email protected] (Z. Guan). https://doi.org/10.1016/j.tws.2019.106588 Received 16 October 2019; Received in revised form 29 December 2019; Accepted 29 December 2019 Available online 6 January 2020 0263-8231/© 2020 Elsevier Ltd. All rights reserved.
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and Hallett [16] established a high-fidelity FE model to study the key driving mechanisms and damage evolution of the compressive failure of laminates with barely visible impact damage. Their numerical result shows that the initial delamination grows at the interfaces near the back face and then into the undamaged cone due to sublaminates buckling. After the entire damage region is delaminated and buckled, it propa gates rapidly and unstably in the lateral direction, leading to the final CAI failure. Thin ply laminates were studied through numerical simu lations by Soto et al. [17]. They found that the CAI process exhibited a brittle behavior with almost no damage growth before failure. Delami nation and especially the fiber constitutive law shape are important for accurate predictions while matrix cracks can be negligible. Abir et al. [18,19] built FE models to investigate the effects of stacking sequences, sublaminate scaling and ply clustering on CAI behaviors, and the rela tionship between different failure mechanisms and CAI strength was explored. However, although the above successive analysis steps method can make the simulation reveal the details of damage growth and final failure in CAI processes more factually, it requires detailed modelling and long CPU time. To avoid the high computational costs, the equiv alent damage model based on actual impact damage is a feasible approach for CAI strength prediction. In the analytical prediction method proposed by Esrail et al. [20], fiber damage and delamination regions introduced by LVI were simplified to concentric elliptical inclusions with different material pa rameters. Rayleigh-Ritz method was employed to calculate the stress to determine the local failure, and then a progressive failure analysis fol lowed until the entire laminate failed. Rozylo et al. [21] presented a simplified damage model (SDM) for quick CAI strength predictions. The model characterized the decrease in material strength after LVI by reducing the thickness of plies in the damage region according to the impact energy. Good agreement was achieved between the numerically predicted CAI strength and the experiment results. As for composite structures with larger size, the advantage of the equivalent damage model becomes more obvious. Based on the SDM, Debski et al. [22] analyzed the buckling, post-buckling and failure behaviors of impacted composite struts. Wang et al. [23] predicted the onset of buckling and collapse load of the pre-damaged stiffened panel by degrading the ma terial properties of the impact damage area. Moreover, both Li [24] and Sun [25] et al. have developed phenomenological mechanical FE models for composite stiffened panels with stiffener impact damage. It can be seen that the simplification of the impact damage makes it more prac tical and efficient to estimate the residual strength of the structure. Although many models have been proposed by scholars, there is still a lack of simplified models that take into account the effect of sub laminate buckling, which in some cases [16,26] may be a crucial factor, on CAI strength of composite laminates and structures. The purpose of this paper is to develop a phenomenological-based simplified damage model that can predict the CAI failure load of com posite structures quickly and effectively. LVI and CAI tests at different energies were performed on laminates. The impact-induced interlaminar and intra-laminar damage were both introduced into the FE model by means of the equivalent damage method. Standard CAI experiment results of two material systems from this paper and literature [10] respectively were used to validate the model, and the differences in failure modes between the two types of laminates were analyzed. Then in order to provide a reference for engineering application, the effect of mass scaling on computational efficiency and accuracy was discussed. Finally, this modelling technique was further applied to a stiffened panel with skin bay impact as a full structural application example.
parameters are shown in Table 1. The stacking sequence of the laminate was [45/-45/0/-45/0/45/0/45/0/45/90/45/-45/0/45/-45/90/0]S. Besides, in order to ensure that the proposed equivalent damage model is applicable to laminates with different material systems and ply sequences, standard CAI experiment results of T700/M21 carbon fiber/ epoxy resin laminates reported by Rivallant et al. [10] are also used for model validation, which will be discussed in section 5.2. The LVI tests were conducted according to ASTM D7136 standard [27]. The impactor was made of steel with a total mass of 5.477 kg and the head was a 12.7 mm diameter hemisphere. To study the variation trend of compression failure behaviors with the impact energy, LVI damage at different energy levels was introduced into the laminates. The applied impact energy as well as the number of specimens of each group are summarized in Table 2. Altogether 21 specimens were contained in the tests. After the impact, the dent depth of each specimen was measured immediately by a digital dial gauge, and then an ultrasonic C-scan equipment was employed to detect the internal damage. The CAI tests were carried out on an Instron 8802 electro-hydraulic servo test machine, of which the maximum load is �250 kN. According to ASTM D7137 standard [28], the specimen was placed in a special fixture for end compression loading until failure, and the loading rate was 1 mm/min. 3. Experiment results 3.1. LVI experiment results LVI experiment results can be divided into two cases according to the damage degree of the dent area. As typically shown in Fig. 1(a), when the impact energy is 22 J, the average depth of the dent is 0.44 mm, and no visible fiber or matrix damage is found around the dent. With the impact energy decreasing, the dent gradually becomes harder to observe. However, for specimens impacted by energy higher than 22 J, ma terial damage is obvious in the dent area. As shown in Fig. 1(b), when the impact energy is 26 J, the dent tends to be circular with an average depth of 0.94 mm. At this time, a small quantity of fiber breakage and matrix cracks begin to appear around the dent. With the rise of energy, a larger number of fiber fractures and matrix cracks are caused by impact, making the dent diameter and depth significantly increased. When the energy grows to 34 J, the average dent depth of the laminates is 2.67 mm. The trend of dent depth with impact energies is shown in Fig. 2. As can be seen from the figure, the slope of the dent depth-impact energy curve changes obviously between 22 J and 26 J due to the occurrence of fiber breakage, which is consistent with the “knee point” phenomenon mentioned in Refs. [5,29]. Generally, before the knee point, there is almost no fiber failure but matrix cracks and delamination. When the impact energy exceeds the knee point, large amounts of fiber breakage Table 1 Material parameters of T300/QY8911 for numerical simulation. Property E1 E2 G12
ν12
XT XC YT YC S Gft Gfc GIC GIIC, GIIC t
2. Specimens and experiments The specimen was a 150 mm � 100 mm rectangular laminate with the thickness of 4.2 mm. It was manufactured of T300/QY8911 carbon fiber/bismaleimide resin unidirectional prepreg, and the material 2
Values Longitudinal Young’s modulus Transverse Young’s modulus In-plane shear modulus Poisson’s ratio Longitudinal tensile strength Longitudinal compressive strength Transverse tensile strength Transverse compressive strength Shear strength Longitudinal tensile fracture toughness Longitudinal compressive fracture toughness Interfacial normal fracture toughness Interfacial shear fracture toughness Ply thickness
135 GPa 8.8 GPa 4.47 GPa 0.33 1548 MPa 1226 MPa 55.5 MPa 110.5 MPa 89.9 MPa 48 N/mm 24 N/mm 0.2 N/mm 0.5 N/mm 0.12 mm
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identical, both of which are large areas of longitudinal splits caused by stress concentration at the end of specimens. This result shows that the compressive bearing capacity of the laminate will not be affected when the impact energy is lower than the delamination threshold value. The trend of CAI strength with impact energies is shown in Fig. 4. It can be seen that the CAI strength decreases significantly by about 41.6% after delamination occurs in the specimen. However, as the impact en ergy continues to increase, the change of CAI strength gradually be comes smooth, making the entire curve present a “step” form. This phenomenon was also observed and reported in Refs. [5,6].
Table 2 Impact energy and the number of specimens of each group. Group
Pristine
L1
L2
L3
L4
L5
L6
L7
Impact energy/J Number of specimens
N/A 1
9 2
12 3
18 3
22 3
26 3
30 3
34 3
4. Numerical modelling technique It is always a complex problem to appropriately introduce the LVI damage into the model for accurate prediction of CAI strength. In this paper, a phenomenological-based method, considering both the intralaminar damage and the number and shape of delamination, was developed to simplify the impact-induced damage of the laminate. Different simplification of the intra-laminar damage was performed according to the damage degrees, while multiple delamination was replaced by some severe delamination. The intra- and inter-laminar failure were simulated by Hashin criterion and virtual crack closure technique, respectively.
Fig. 1. Typical dent appearance of the specimen after (a) 22 J and (b) 26 J impact.
4.1. Equivalent method for impact damage According to the post-impact morphology of all specimens, the damage area is simplified into two concentric circles, and the whole laminate is divided into three typical regions (see in Fig. 5). Region I is the dent area. As mentioned above, for specimens before the knee point, the main intra-laminar damage in the dent area is matrix damage; while for specimens after the knee point, severe fiber and matrix damage occur in the dent area. The region between the boundary of Region I and the outer fringe of the simplified circular delamination is Region II. Since the initiation of delamination is mostly caused by matrix cracking [31], the main intra-laminar damage in this region is assumed to be matrix damage. Region III is an undamaged area. Since the structure of an impact damage is quite complex, the effect on the material properties by every individual crack is actually not possible to measure. According to the soft inclusion model [1], a feasible approach to characterize the fiber and matrix damage in Region I and II is to reduce their macro material parameters. Therefore, for specimens after the knee point, based on the material degradation method pro posed in Refs. [24,32], the modulus and strength parameters in Region I are set to 5% of their pristine values. For specimens before the knee point, damage patterns in Region I and II are identical; thus the same degradation method was adopted for both of them. The modulus and strength parameters related to the matrix (i.e. parameters other than longitudinal ones) are set to 20% of their pristine values [24,32]. In addition to the degradation of material properties, sublaminates buckling caused by impact-induced delamination under axial compres sion should also be considered. According to the results of Ref. [33], delamination with a smaller size has less effect on the sublaminates buckling behavior of the laminate. On this basis, multiple delamination cracks can be approximated by some severe delamination to improve computational efficiency. Moreover, it should also be mentioned that LVI-induced delamination can be connected through the matrix crack between plies [31,34], and thus form a larger continuous crack. Ac cording to the previous experimental study [35], the present studied laminate specimen can form six groups of three-connected de laminations that are spaced by about 5 plies in turn along the thickness direction after LVI (see in Fig. 6). Therefore, the impact delamination inside the laminate can be preliminarily simplified to these six de laminations. However, delaminations in six interfaces still result in high computational costs. In order to further improve the computational ef ficiency, it is necessary to select the most dangerous one or two
Fig. 2. The trend of dent depth with various impact energies.
will occur in the laminate. Fig. 3 shows a comparison of projected delamination area measured by C-scan in selected specimens with different impact energies. It can be seen that there is a delamination threshold energy between 9 J and 12 J. Delamination can be detected in the laminate only when the impact energy is higher than the threshold value [30]. Regardless of the matrix split, the delamination shape is roughly circular. As the impact energy grows, the delamination area gradually increases (12 J–26 J) at first and then tends to level off (26 J–34 J). The reason for this phenomenon is that fiber failure begins to occur at 26 J impact, and further increased energy is largely dissipated by fiber breakage. The arithmetic average dent depth and projected delamination area of each group are summa rized in Table 3. It should be noted here that matrix splits are not taken into account in the delamination area due to the fact that they usually occur only in the last one or two plies at the rear side and have little effect on the global stiffness and strength of the laminate. 3.2. CAI experiment results Table 4 shows the average CAI strength of each group. Except that the coefficient of variation (CV) of L3 group is 10.6%, CV of the other groups are all within 5%, reflecting that the experiment results have good repeatability and reliability on the whole. In addition, the average CAI strength of L1 group is very close to the ultimate compression strength of the pristine specimen. The failure modes of them are 3
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Fig. 3. Collection of C-scan images of the specimens with various impact energies. (Different colors represent the thickness from which the scanning ultrasonic wave gets reflected). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) Table 3 Arithmetic average dent depth and delamination area of each group. Group
Average dent depth/mm (CV)
Average delamination area/mm2 (CV)
L1 L2 L3 L4 L5 L6 L7
0.02 0.13 (4.9%) 0.45 (16.0%) 0.44 (11.2%) 0.94 (15.2%) 1.75 (4.6%) 2.67 (16.8%)
Not detected 641 (1.6%) 778 (5.2%) 1268 (8.0%) 1544 (13.4%) 1552 (10.6%) 1689 (8.4%)
Table 4 Average CAI strength of each group. Group
Average CAI strength/MPa (CV)
Group
Average CAI strength/MPa (CV)
Pristine L1 L2 L3
476 474 277 (4.8%) 249 (10.6%)
L4 L5 L6 L7
238 209 205 195
(3.7%) (2.7%) (2.1%) (1.6%)
Fig. 4. The trend of CAI strength with various impact energies.
much higher load. Hence, delamination in the FE model is finally ar ranged at I2&5. Generally, after the low-velocity impact, permanent out-of-plane deformation will be formed in the impacted area. In the subsequent compression test, this local deformation will introduce additional bending moment, affecting the sublaminates buckling behavior of the laminate [36]. In the FE model, the out-of-plane deformation is char acterized by node offset in the dent area (see in Fig. 8). It should be noted here that in the actual situation, the impact-induced out-of-plane deformation exists both in the dent area and the delamination area (Region II). However, according to the numerical simulation, the pro posed FE model is insensitive to the out-of-plane deformation in Region II when there is already an out-of-plane deformation in the dent area. Thus in order to make the model better applied in engineering practice, the out-of-plane deformation modeling in Region II is ignored.
delaminations from Fig. 6 for introducing into the FE model. To this end, a pre-numerical calculation was performed. Six numerical models, of which the delamination is at I1, I2, I3, I1&6 (considering the symmetry of the stacking sequence), I2&5 and I3&4 respectively were built, and the numerical results are shown in Fig. 7. As can be seen from the figure, the numerical model with delamination at I2&5 shows the lowest residual strength, which means that this arrangement has the greatest influence on the CAI response of the laminate. This phenomenon is reasonable because, when the delami nation is closer to the surface, although the sublaminate is more prone to buckling, the delamination propagation is not that easy to occur due to the lower peeling stress induced by the buckling of the thinner sub laminate. This, as a result, leads to a higher failure load of the laminate. However, when the delamination is close to the mid-plane, the signifi cant increase of the thickness makes sublaminate buckling quite difficult to occur, and thus results in delamination propagation taking place at a 4
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Fig. 5. Schematic diagram of typical damage morphology of laminates after LVI (take the 22 J specimen as an example).
4.2. Damage model 4.2.1. Intra-laminar damage In two-dimensional cases, Hashin criterion [37] is often employed to determine the initiation of intra-laminar damage in composite laminates [38,39]. Four independent damage modes are considered in the crite rion, which are described as follows: � �2 � � σ 11 τ12 2 þ �1 (1) XT S
Fig. 6. Approximate distribution of the connected delaminations through the thickness.
�
σ 11
�2
�
σ 22
�2 þ
YT �
(2)
�1
XC
�τ �2 12
S
��� �2 YC YC 2S
σ 22
�1
(3)
� � � � � σ 22 2 τ12 2 1 þ þ �1 2S S
(4)
where σ 11 and σ 22 represent the stresses parallel and perpendicular to the fiber direction, respectively, while τ12 is the in-plane shear stress. Once the damage initiation criterion is satisfied, the composites undergo material property degradation. The constitutive relation of a damaged ply is defined as: (5)
σ ¼ Cd ε 2 Fig. 7. Residual strength comparison of laminates with various delamina tion locations.
Cd ¼
14 1 D
� 1� df E1 df ð1 dm Þν12 E2 0
1
3
�
df ð1 dm Þν21 E1 ð1 dm ÞE2 0
ð1
0 5 0 ds ÞG12 D (6)
where D ¼ 1 ð1 df Þð1 dm Þν12 ν21 ; df, dm and ds are damage vari ables associated with fiber damage, matrix damage and shear damage, respectively. Meanwhile, ds is not independent and can be expressed as: � ds ¼ 1 1 df ð1 dm Þ (7) A linear damage evolution law was adopted in the model. After the initiation of damage, the stiffness of the material progressively decreases with the increase of damage variables until enough energy is absorbed for complete failure. 4.2.2. Inter-laminar damage model Virtual crack closure technique (VCCT) [40], as an effective method to calculate the energy release rate (ERR), has been applied to predict inter-laminar failure in a lot of literature [41,42]. Different from the traction-separation law based cohesive zone model which requires some empirical parameters [43,44], the VCCT model does not introduce any special elements and uses only fracture toughness parameters; thus it was employed in this paper. In the VCCT model, the energy released by the propagation of a crack is assumed to be equal to the energy required to close the crack;
Fig. 8. FE models of specimens (a) before the knee point and (b) after the knee point.
5
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therefore, the ERR can be calculated in one step by the node force at the crack tip and the relative displacement between the nearest node pair to the crack tip. Initial bonded nodes are released when the following criterion is satisfied: Gequiv �1 GC
(8)
where Gequiv and GC are the equivalent and critical ERR, respectively. The critical ERR is calculated by B–K criterion [45]: �η � GII þ GIII GC ¼ GIC þ ðGIIC GIC Þ (9) GI þ GII þ GIII where GIC, GIIC and GIIIC are the critical ERR for mode I, mode II and mode III crack propagation, and their values are 0.2 N/mm, 0.5 N/mm and 0.5 N/mm, respectively; η is constant and is set as 1.45 according to Ref. [46].
Fig. 10. Initial bonded nodes of the FE model.
constraint”. In addition, according to the recommendation of Ref. [47], Rayleigh damping was introduced into the material so as to reduce the possible vibration in the simulation. The damping parameters were set as α ¼ 3000 and β ¼ 0 [29]. The boundary condition of the model was consistent with that of the experiment. The anti-buckling knife edges as well as end clamping blocks were simply simulated by constraining the corresponding degree of freedom of nodes. A compression displacement load was applied through a reference point by the “equation constraint” at the longitu dinal end of the laminate, and a “smooth step” was adopted to reduce dynamic effects. Besides, according to Ref. [48], mass scaling and higher loading rate were used to reduce the run time. The time increment was set to be no less than 10 6 s (errors < 1.5% compared with no mass scaling models) and the loading speed was fixed at 3.5 mm/s. Kinetic energy (ALLKE) and internal energy (ALLIE) of the model were recorded during the computation, and the proportion of ALLKE to ALLIE was less than 0.2%. The CPU time using 16 Xeon E5-2640 v2 cores (2 GHz) was about 1 h.
4.3. Modelling strategy Since no damage was observed in laminates impacted by 9 J, and their compression strength was consistent with that of the pristine specimen, only groups with the impact energy higher than 9 J (L2~L7 groups) were modeled in commercial FE software ABAQUS/Explicit to predict the CAI strength. In order to reduce the computational scale, a mesh convergence analysis was carried out beforehand. As shown in Fig. 9, loaddisplacement curves obtained by various mesh densities coincide with each other at the loading stage, and only the failure load decreases with the increase of the element size. Models with element sizes of 1 mm and 1.5 mm show 1.1% and 3.9% lower CAI strength respectively compared to the model with a 0.75 mm element size, which can be considered small. To achieve the balance between computational efficiency and accuracy, an element size of 1.5 mm was used in the model. Approximately 20000 continuum shell elements (SC8R) were created to discretize the laminate. Material properties used in the simulation is shown in Table 1. Based on the previous analysis, delam ination was arranged between the 10th and 11th plies and between the 25th and 26th plies with the same area as the projected delamination (see the average area in Table 3). The average dent diameters of L2~L7 groups were 4 mm, 5 mm, 6 mm, 8 mm, 10 mm and 13 mm, respectively while the average dent depth is shown in Table 3. Besides, considering that delamination mainly propagates along the transverse direction under compression, delamination growth was set to occur only within the 100 � 100 mm2 region (see in Fig. 10). Adjacent surfaces of the rest two 25 � 100 mm2 regions were joined together by using the “tie
5. Model validation and application 5.1. CAI strength prediction of T300/QY8911 laminates The comparison of the CAI strength between the experimental and numerical results are shown in Fig. 11. The predicted compression strength after different impact energies are 283 MPa, 270 MPa, 241 MPa, 210 MPa, 198 MP and 188 MPa, respectively, and the errors relative to the average experiment values are 2.2%, 8.4%, 1.3%, 0.5%, 3.4% and 3.6%, respectively. These results indicate that the
Fig. 9. Predicted load-displacement curves by 12 J model with various mesh densities.
Fig. 11. Comparison of CAI strength between experimental and numeri cal results. 6
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proposed FE models achieve accurate predictions of CAI strength of laminates in the range of 12 J–34 J. The typical CAI failure mode of the specimens is exhibited by the L2 group as an illustrative example in Fig. 12(a). As can be seen from the figure, significant sublaminates buckling appears on both the impacted and non-impacted sides of the specimen, but no fiber breakage is observed on the surfaces. In accordance with the experiment results, the FE model also shows obvious sublaminates buckling; meanwhile, no fiber failure takes place in the model except for a small amount of fiber breakage in the dent area. It can be seen from Fig. 12(b) that the compression load stops increasing after the delamination growth occurs, and then the delamination rapidly propagates to the edges of the lami nate. This phenomenon indicates that the failure of the model is due to delamination growth induced by the sublaminates buckling, and this growth is generally unstable. Based on the above results, it seems that for laminates with brittle resin (like QY8911), the low mode I and mode II matrix fracture toughness make the CAI failure inclined to be caused by propagation of delamination induced by sublaminates buckling. Thus, the effective characterization of the impact-induced delamination plays an important role in predicting CAI strength of such laminates.
Table 5 Material properties of T700/M21 for numerical simulation [10,49]. Property
Values
Elastic properties
ET1 ¼ 130 GPa; EC1 ¼ 100 GPa; E2 ¼ 7.7 GPa; G12 ¼ 4.8 GPa; ν12 ¼ 0.3
Strength Fracture toughness Interface properties
XT ¼ 2080 MPa; XC ¼ 1250 MPa; YT ¼ 60 MPa; YC ¼ 290 MPa; S ¼ 110 MPa Gft ¼ 133 N/mm; Gfc ¼ 10 N/mm; Gmt ¼ 0.5 N/mm; Gmc ¼ 1.6 N/mm GIC ¼ 0.5 N/mm; GIIC ¼ 1.6 N/mm; η ¼ 1.45
Table 6 Impact damage sizes of the T700/M21 laminates [10]. Impact energy/J
Dent depth/ mm
Dent diameter/ mm
Delamination diameter/ mm
6.5 17 26.5 29.5
0.06 0.23 0.52 0.70
3 5 7 10
19.4 34.0 40.3 41.0
equivalent damage model was also arranged at these two locations. Fig. 13 shows the comparison of CAI strength between the experi ment results of Ref. [10] and the predicted values of the FE model proposed in this paper. As can be seen from the figure, from 6.5 J to 29.5 J, the errors between experimental and numerical results are 4.3%, 12.3%, 6.5% and 5.3%, respectively. The results of equivalent damage models are in good agreement with the experiment results except that the 17 J model has an error of more than 10%. The reason for the un derestimate of compression strength after 17 J impact may be the dispersion of a single experiment result since the prediction value of the model in Ref. [10] was also lower than the experiment result. Taking 17 J and 29.5 J models as illustrative examples, the CAI failure processes simulated by numerical models are basically consistent with the experimental phenomena observed in Ref. [10]. As shown in Fig. 14(a), for the 17 J model, no fiber breakage is found on the surface with the increase of compression stress. However, when the stress in creases to 213 MPa, the model suddenly ruptures due to an unstable crack propagating from the impact area to the edges of the laminate. As for the 29.5 J model (see in Fig. 14(b)), the scenario is not the same. Progressive crack propagation is observed in the model. At 121 MPa, fiber breakage occurs due to the buckling of sublaminates around the impact area. Then, the breakage of the fiber also in turn increases the sublaminates buckling. When the crack propagates to the edges of the
5.2. CAI strength prediction of T700/M21 laminates In this section, based on the standard LVI experiment results reported by Rivallant et al. [10], the equivalent damage model was used to pre dict the CAI strength of laminates. The specimens were made of T700/M21 carbon fiber/epoxy resin unidirectional prepreg with a ply thickness of 0.26 mm and a stacking sequence of [02/452/902/-452]S. The material properties of T700/M21 are shown in Table 5. The damage sizes introduced by a 16 mm diameter impactor at different energies in Ref. [10] are extracted and summarized as shown in Table 6. It should be noted that dent diameters are not given in the literature and hence data in the table are estimated values. According to the experimental phenomenon in Ref. [50], delamination can be sup pressed at the local interface close to the impact side due to the direct compression of the impactor. Thus, the dent area can be estimated by taking the impact point as the center to make a circle tangent to the surrounding delamination. Besides, based on the experiment results described in the literature, laminates under 6.5 J and 17 J impact were before the knee point, while laminates under 26.5 J and 29.5 J impact were after the knee point. In addition, Abir et al. [19] have conducted numerical analysis using the same stacking sequence and material sys tem, and their results show that the largest impact delamination occurs at the interfaces of 90� /-45� and 45� /90� . Thus, delamination in the
Fig. 12. (a) Comparison of failure modes between the experiment result and FEM simulation; (b) Final failure dominated by transverse delamination propagation in large area. 7
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Fig. 13. CAI results comparison: (a) results from Ref. [10], (b) prediction by the proposed FE method. (Experiment values may have small errors since they are read from the figure by the author).
Fig. 14. Predicted CAI failure processes of (a) 17 J model and (b) 29.5 J model.
laminate, the final rupture of the model happens. Based on the above phenomena, it can be seen that there are two failure scenarios of the laminates. When the impact energy is low, the initial delamination is small and thus the sublaminates buckling is also slight. As a result, the final failure of the laminate is caused by the un stable crack propagation at high CAI stress. When the impact energy is high, the increase of the initial delamination area makes the sub laminates buckle easily. The sublaminates buckling and crack propa gation develop together and induce the final failure of the laminate. Thus the CAI strength decreases.
It is obvious that the above failure scenarios are different from those in section 5.1. This is mainly because laminates studied in this section is made of M21 epoxy resin, which has higher mode I and mode II matrix fracture toughness compared to QY8911 bismaleimide resin; thus delamination propagation in large area is relatively not easy to occur. Based on the numerical results, it can be concluded that when predicting CAI strength of laminates with high matrix fracture toughness, not only the impact-induced delamination, but also the fiber compression prop erty parameters have a quite important influence on the accuracy of the prediction. Besides, the above results also show that the presented 8
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equivalent damage model is effective. 5.3. Effect of mass scaling In engineering practice, designers usually hope to obtain the CAI strength of composites as soon as possible to evaluate the damage tolerance performance. Therefore, it is important to take account of both computational accuracy and efficiency. For explicit analysis, mass scaling and reduction of time period are unavoidable to achieve an economical solution. According to Ref. [47], for quasi-static simulations incorporating rate-independent materials, mass scaling and reduction of time period yield similar results. Thus, only the effect of mass scaling on numerical results is discussed below. One approach to perform mass scaling in ABAQUS/EXPLICIT is to define a target time increment. The L2~L7 models were simulated with different time increments, and the results are shown in Table 7. It can be seen that the predicted CAI strength increases with the increase of time increments. The results of no mass scaling are quite close to those of time increments with 1 � 10 6 s (errors < 1.5%), while the CPU time is reduced from 8–12 h to approximately 1 h, which significantly improves the computational efficiency. When the time increment increases to 5 � 10 6 s, the CPU time is further reduced to about 0.3 h. At this time, an obvious error appears in L3 model, but other models stay a good pre diction accuracy. With the time increment increasing to 1 � 10 5 s, the run time decreases to 0.1–0.2 h, and errors of all models except L3 are still within 10%. However, when the time increment is set as 5 � 10 5 s, a significant dynamic effect appears in the simulation. The predicted CAI strength of all models are considerably higher, with errors of more than 50%. Based on the above results, it can be seen that the time increment set as 1 � 10 6 s can greatly improve the computational efficiency while maintaining high accuracy. Setting the time increment as 1 � 10 5 s can reduce the CPU time to about 0.1–0.2 h, but there is an error of up to 16.1%, which needs to be weighed in practical application. However, further increasing the time increment is no longer applicable since the prediction values become significantly larger.
Fig. 15. The FE model of the L-stiffened composite panel with skin bay impact damage.
geometric dimensions and stacking sequences information can be found in Ref. [35]. After 27 J impact on the skin bay of the panel, the measured average dent depth, dent diameter and projected delamination diameter were 1.29 mm, 10 mm and 54 mm, respectively. An element size of 1.5 mm was used in the delamination growth area and there were about 52000 SC8R elements in the model. The boundary condition of the model (see in Fig. 15) was that all six degrees of freedom were constrained at the fixed end while all degrees of freedom except Uy were constrained at the loading end. The “equation constraint” was used to apply the displace ment load to the panel through a reference point, and the loading rate was 3.5 mm/s. Since no stiffener-skin debonding occurred in the test, this damage was not considered in the FE model. The CPU time of the simulation was approximately 2 h. Numerical results are compared with the experiment results from Ref. [35]. As shown in Fig. 16, the load-displacement curve obtained by FE simulation is in good agreement with the experiment results. The curves keep a linear growth until the ultimate failure, indicating that no global buckling of the panel occurs in the compression process. The predicted failure load is 210 kN, with the error of 7.9% compared to the average experiment value of 228 kN. Fig. 17 shows the comparison of failure modes between the FE simulation and the experiment. As can be seen from Fig. 17(a), the skin of the stiffened panel shows obvious sublaminates buckling near the impact point, and a large area of delamination also appears at the free edge of the skin. Moreover, fiber failure occurs only at the dent area. These phenomena are consistent with the experiment results shown in Fig. 17(b) and confirm that the failure of the stiffened panel with skin
5.4. Application of the model in composite structures In aircraft structures, stiffened composite panels are extensively applied in many components, such as the fuselage, wings and horizon tal/vertical tails. When a stiffened panel is subjected to LVI in the skin bay, the damage pattern is similar to that of the laminate. Therefore, the previously validated equivalent damage model was applied in this sec tion to predict the CAI failure load of stiffened composite panels. Fig. 15 shows the model of the L-stiffened composite panel with skin bay impact damage. The skin stacking sequence and material properties of the panel are consistent with the T300/QY8911 laminates. More detailed Table 7 Numerical results by various target time increments. Target time increment (s)
L2
No limit
282/ 1.8/ 12.3 283/ 2.2/1.3 289/ 4.3/0.3
L3
L4
L5
L6
L7
CAI strength (MPa)/error (%)/CPU time (h)
1 � 10
6
5 � 10
6
1 � 10
5
294/ 6.1/0.2
5 � 10
5
419/ 51.3/ 0.06
267/ 7.2/ 11.9 270/ 8.4/1.2 286/ 14.9/ 0.3 289/ 16.1/ 0.2 417/ 67.5/ 0.06
239/ 0.4/ 10.3 241/ 1.3/1.1 245/ 2.9/0.3
207/1.0/8.7
198/3.4/8.3
188/3.6/7.9
210/ 0.5/1.0 213/ 1.9/0.2
198/3.4/1.0 199/2.9/0.2
188/3.6/1.0 194/0.5/0.2
252/ 5.9/0.2
217/ 3.8/0.1
218/ 6.3/0.1
214/ 9.7/0.1
386/ 62.2/ 0.06
375/ 79.4/ 0.06
372/ 81.5/ 0.06
368/ 88.7/ 0.06
Fig. 16. Load-displacement curves of specimens and FEM simulation for the Lstiffened composite panel. 9
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Thin-Walled Structures 148 (2020) 106588
(2) For the studied M21 epoxy resin matrix laminates, the mode I and mode II matrix fracture toughness are high, and delamination growth is relatively not easy to occur after the buckling of sub laminates. When the impact energy is low, the impact-induced delamination is small and thus the sublaminates buckling is also slight. As a result, the CAI failure of the laminate is due to the sudden breakage of fibers. When the impact energy is high, the increase of the impact-induced delamination area makes the sublaminates buckle easily. The CAI failure of the laminate is affected by the interaction of the sublaminates buckling and the fiber breakage. Thus, the characterization of impact-induced delamination and the fiber compression property parameters are both important factors affecting the accuracy of CAI strength prediction for such laminates with high matrix fracture toughness. (3) The proposed model shows that small mass scaling (1 � 10 6 s) can greatly improve the computational efficiency (1 h) while maintaining the accuracy. A further increase of the mass scaling can reduce the CPU time to 0.1–0.3 h, but increase the compu tational errors; thus it needs to be weighed in practical application. In addition, the equivalent damage model was further successfully applied to predict the compression failure load of stiffened composite panels with skin bay impact damage, showing that this modelling approach is also applicable to composite structures. To sum up, the equivalent damage model proposed in this paper takes approximately 1 h and 2 h respectively for CAI failure load prediction of the studied laminates and stiffened panels, which is very suitable for the need of rapid residual compression strength assessment of impacted structures in engineering practice.
Fig. 17. Failure modes of the L-stiffened composite panel: (a) FEM simulation; (b) experiment results [35].
bay impact damage is dominated by delamination propagation in large area transversely as a result of sublaminates buckling. In general, the FEM and experiment results are in good agreement on the failure load and dominant failure modes. These results show that the equivalent damage method proposed in this paper is applicable to fast prediction of the compression failure load of composite stiffened panels with skin bay impact damage. In addition, for composite stiffened panels with impact damage at other locations, the compression failure load can also be predicted by introducing the corresponding stiffener-skin debonding damage or stiffener damage into the model.
CRediT authorship contribution statement Tian Ouyang: Conceptualization, Methodology, Investigation, Visualization, Writing - original draft. Rui Bao: Supervision, Resources. Wei Sun: Investigation, Validation, Writing - review & editing. Zhidong Guan: Resources. Riming Tan: Investigation, Formal analysis, Data curation.
6. Conclusions In this paper, LVI and CAI tests were carried out on CFRP unidirec tional laminates made of bismaleimide resin under different energy levels. Delamination threshold energy and knee point energy were observed in LVI tests. In subsequent CAI tests, the results show that the impact before threshold energy does not affect the compression strength of laminates, while the residual strength will be seriously weakened once the delamination occurs. However, after the threshold energy, the change of CAI strength with the impact energy gradually levels off. On the basis of the experiment results, an equivalent damage FE model that can predict the CAI strength of laminates quickly and effi ciently was built. By using progressive failure analysis and VCCT, the CAI strength and failure modes of laminates of two material systems at different energy levels were predicted. The numerical results are consistent with the experiment results, which verifies the validity of the model. Conclusions obtained through the FE analysis are as follows:
Acknowledgements The first author would like to thank Dr. Zengshan Li for his kind help in the experiment. The project is partially supported by Beijing Munic ipal Science and Technology Commission via project No. Z191100004619006. References [1] A. Sj€ ogren, A. Krasnikovs, J. Varna, Experimental determination of elastic properties of impact damage in carbon fibre/epoxy laminates, Composer Part AAppl. Sci. Manuf. 32 (2001) 1237–1242. [2] R. Craven, S. Pindoria, R. Olsson, Finite element study of compressively loaded fibres fractured during impact, Compos. Sci. Technol. 69 (2009) 586–593. [3] S. Rivallant, C. Bouvet, E.A. Abdallah, B. Broll, J. Barrau, Experimental analysis of CFRP laminates subjected to compression after impact: the role of impact-induced cracks in failure, Compos. Struct. 111 (2014) 147–157. [4] M. Remacha, S. S� anchez-S� aez, B. L� opez-Romano, E. Barbero, A new device for determining the compression after impact strength in thin laminates, Compos. Struct. 127 (2015) 99–107. [5] T.A. Sebaey, E.V. Gonz� alez, C.S. Lopes, N. Blanco, P. Maimí, J. Costa, Damage resistance and damage tolerance of dispersed CFRP laminates: effect of the mismatch angle between plies, Compos. Struct. 101 (2013) 255–264. [6] D.D.R. Cartie, P.E. Irving, Effect of resin and fibre properties on impact and compression after impact performance of CFRP, Composer Part A-Appl. Sci. Manuf. 33 (2002) 483–493. [7] S.I. Thorsson, S.P. Sringeri, A.M. Waas, B.P. Justusson, M. Rassaian, Experimental investigation of composite laminates subject to low-velocity edge-on impact and compression after impact, Compos. Struct. 186 (2018) 335–346.
(1) For the studied QY8911 bismaleimide resin matrix laminates, the mode I and mode II matrix fracture toughness are low, and the impact-induced delamination is easy to propagate under the peeling stress caused by sublaminates buckling under compres sion. From 12 J to 34 J, the CAI failure of laminates is all due to the delamination propagation in large area. Therefore, when CAI prediction is carried out on such laminates with low matrix fracture toughness, the effective characterization of the impactinduced delamination has an important influence on the accu racy of the numerical results. 10
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