1. Introduction
Carbon fiber reinforced polymer matrices have been employed in an ever-increasing way in aircraft structures due to their excellent properties, such as specified strength and stiffness, as well as their abilities that allow designers to tailor the layups to obtain an optimized structure for particular applications, etc. However, polymer matrix composites materials have high impact damage susceptibilities which are inevitable throughout the manufacturing process and service life of the airplane structures [
1]. Low velocity impacts (LVI) on composites materials could cause such damage that it is nearly invisible from the outer surface but cause extensive inner damage. This results in damage that is difficult to be detect by eyes but with a significant reduction in strength, as much as 60% [
2]. Therefore, the mechanical properties of the composite materials with barely visible impact damage (BVID) or impact damage of cut-off energy often are used as the material allowable values of the aircraft structures in accordance with the federal aviation administration regulation 25.571 [
3] and the advisory circular 20-107B [
4].
Extensive research has been conducted to understand on the effects of low-velocity impact damage on the residual strength of composite laminates, both from an experimental and a numerical focus. For example, Cantwell and Morton [
5] provided a comprehensive review of the impact response of continuous fiber-reinforced composites a long time ago, and the effects of varying the properties of the fiber, matrix and interphase are examined, as well as the role of target geometry and loading rate on the dynamic response. Ali et al. [
6] also presented a review of different strategies dealing with development of new composite materials investigated by several research groups that can be used to mitigate the low velocity impact damage in laminated composites. However, because of the importance and complexity associated with impact and following mechanical loading process, the compression-after-impact is still one of the hottest topics for composite materials. Lin and Wass [
7] carried drop-tower tests and experimentally investigated the low velocity impact on a shear dominated composite laminate with stacking sequence [45/0/
45/90]
3s. He et al. [
8] investigated the effects of material parameters, and constitutive relationships on the dynamic mechanical response of composite laminates subjected to high- and low-velocity impacts. Additionally, they also studied the role of impactor shape on the damage area of the adhesive layer and displacement of the center of the laminated plates. Umair et al. [
9] conducted low velocity drop-weight impact tests on polymeric composites and investigated the maximum force, displacement and impact energy absorption during the impact events.
In the cases of numerical simulations, two independent numerical damage models were often used to simulate the failure of the composite materials [
10]. The first simulation constitutes the progressive failure analysis (PFA) considering the damage initiation criteria using Hashin’s theory and damage evolution based on the energy criterion. The second simulation constitutes the cohesive zone model (CZM), which was used to simulate failure of composite material caused by the delamination phenomenon. The compressive failure process of the composite materials subjected to impact damage were simulated by scholars following the two above-mentioned methods. For example, Tan et al. [
11] developed a high-fidelity three-dimensional composite damage model to predict both low-velocity impact damage and CAI strength of composite laminates. The intralaminar damage model accounts for physically based tensile and compressive failure mechanisms of the fibers and matrix, when subjected to a three-dimensional stress state. Cohesive behavior was employed to model the interlaminar failure between plies with a bi-linear traction–separation law. Abdulhamid et al. [
12] investigated the impact and CAI of a composite laminate using a discrete ply model. Their study highlights the importance of modeling intra-ply matrix cracking for impact simulation. Sun and Hallett [
13] tested two similar quasi-isotropic laminates and investigated the key driving mechanisms and damage evolution of the compressive failure of laminated composites containing BVID using compression after low-velocity impact and indentation tests. The effect of ply-blocking and influence of factors, such as impact energy, delamination area and surface indentation, on compressive failure was studied. Rozylo et al. [
14] presented a simplified model of damage of composite plates (SDM) subjected to low-velocity impact testing, and the damage model assumes reduced thickness of laminate plies versus impact energy. Ouyang et al. [
15] proposed an equivalent damage model to quickly predict the CAI strength of composite laminates. Progressive failure analysis and the virtual crack closure technique (VCCT) are used to simulate the intra-laminar and inter-laminar failure of impacted laminates under axial compression.
These studies provide valuable evidence for the composite structure design of the upper wing panel and the fuselage belly, where compression is the primary loading. However, these achievements are not appropriate to the webs of the composite spars, ribs and the sides of the fuselage, where in-plane shear is the primary loading. In plane shear-after-impact (ShAI) properties of the composite laminates are essential to the design of these structures.
There are many distinct standardized test methods to obtain the in-plane shear properties of the composite materials. Some of the most well-known test methods are Iosipescu shear (ASTM D 5379), ±45° tension shear (ASTM D 3518), two-rail and three-rail shear (ASTM D 4255), double-notched shear (ASTM D 3846), V-Notched Beam (ASTM D5379), torsion of a thin tube (ASTM D 5448), torsion of a solid rod (ASTM E 143), V-notched rail shear (ASTM 7078), picture frame shear (ASTM D 8067), and so on. These in-plane shear test methods are applied and discussed in many studies [
16,
17,
18,
19,
20,
21], in addition to these, the Arcan and modified Arcan test methods were also applied by some scholars to obtain the in-plane pure shear properties of the composite materials [
22,
23,
24,
25]. However, most of these above-mentioned methods are limited to the specimen geometry, layups and shear stress area. The pure shear stress region lies solely in a quite narrow area between the notches according to the Iosipescu shear and all the other notched shear methods, and this makes them inappropriate to test the shear-after-impact responses. The ±45° Tensile Shear test is characterized by off-axis tensioning of a rectangular cross ply lamina, typically at a ±45° orientation and thus, cannot be applied to the composite laminates with complex layup in the actual application. In addition, due to the fixed boundary conditions of the end clamps, a very complex stress state is imparted into the specimen once tensioned. The torsional shear of a thin-walled circular tube is considered the ideal shear test from the applied mechanics viewpoint, but it also presents many challenges such as high specimen manufacturing cost and difficulties of conducting the test. Furthermore, it cannot account for the property variations induced by the different manufacturing process between the flat specimens and tubes.
The above-mentioned methods have been applied to test the in-plane shear properties of the composite materials, but only a few of them can provide a large area in shear stress, which is necessary to investigate the material’s ShAI properties. Among these, Souza et al. [
26] executed three-rail shear tests on the fiber-reinforced laminated composite laminates with barely visible impact damage and studied the shear-after-impact performance. However, the two-rail or three-rail shear methods produced simple shear stress but not pure shear stress in the specimens and thus, significant normal stresses are present in the test section. Whereas, in the picture frame shear test, a square composite sheet is clamped with four rigid arms on all four sides and these arms are pivoted together at each corner, which can represent the real geometric features and loading conditions of the webs that are surrounded by chords and stiffeners on the four sides. The two diagonally opposite corners of the jig are then displaced using a mechanical testing machine and thus, a quite uniform shear stress state within most of the area of the specimen can be produced. The picture frame test method has been applied in some research to study the ShAI behaviors of the stiffened composite panels at a subcomponent level [
27,
28,
29,
30,
31]. However, to the best of the authors’ knowledge, very few studies have been published on getting the ShAI properties of the composite materials at a coupon level using the picture frame test method. According to the building block approach [
4], the material data at the coupon level are necessary to obtain the material allowable values and they are the basis of the airplane structural design.
In this study, a set of methods combined with theoretical, experimental and simulation work is developed and proposed to obtain the ShAI properties of the composite laminates in pure shear stress at a couple level. The buckling load and material failure load of the balanced and symmetric laminate are compared, and on this basis, the layup and sizes of the tested specimen are defined. Subsequently, impact tests of the composite laminate specimens with different impact energies and picture frame tests of the impacted specimens are conducted to investigate the ShAI properties. Furthermore, a finite element simulation method has been developed to analyze the failure process.
2. Specimen Sizing
The buckling load of the in-plane shear specimens should be higher than the material failure load in order to avoid global buckling during the loading process. Additionally, the specimens should be large enough to eliminate the interactive effects between the impact damage and the specimens’ edges. The shear buckling load of an orthotropic laminate with symmetric and balanced layups can be expressed as Equation (1).
where the
D11 and
D22 are the relative values in the ABD matrix of the composite laminate,
b is the length of the shorter side of the rectangular specimen,
Ks is the shear buckling factor and its value can be obtained in accordance with the reference [
32].
Through calculation, the buckling and material failure load values of the orthotropic laminates with different number of plies (layups are [45/0/
45/90]
ns, where
n = [1, 2, 3]) and different specimen sizes can be obtained as shown in
Figure 1. The mechanical parameters of the unidirectional lamina are listed in
Table 1, and three different failure criteria, Tsai-Hill, Tsai-Wu [
33], and maximum shear strain (MSE) are used to predict the failure load of the composite laminates.
From
Figure 1, it can be seen that the laminates with ply numbers of less than 16 are not appropriate to test the in-plane shear-after impact properties of the composite materials. As for the laminates with 24 plies, when the length of the specimen’s side is shorter than 171 mm, the buckling load will be higher than the material failure loads obtained from all the three criteria. With the ply number of the laminate continuing to increase, for example 32 plies, the requirement on the testing machine will be greatly increased due to the higher failure load and further, the manufacturing costs of the specimens and jigs will also rise greatly. Thus, with the combined consideration of the material and manufacturing costs, the stacking layups of the specimen was decided to be [45/0/
45/90]
3s and the area of testing section was decided to be 150 mm × 150 mm. Through calculation, the failure load based on maximum shear strain criterion and the buckling load of this specimen are 2250 N/mm and 2955 N/mm, respectively.
5. Finite Element Modelling and Simulations
Two finite element models of the composite laminate were established using the commercial finite element code, Abaqus [
37]. The first model is the buckling analysis model of the composite laminate and the other model is used to analyze the composite laminate subjected to impact loading and shear-after-impact loading. A linear perturbation-Buckling algorithm was used to analyze the buckling load and buckling mode of the composite specimen under shear loading. A two-step explicit algorithm was used to simulate both the impact and the shear-after-impact loading processes. The time of the impact step and shear-after-step are 0.005 s and 0.2 s, respectively. The loading rate of 10 mm per second can eliminate the dynamic loading effect since the simulated responses of the undamaged laminate obtained by the explicit algorithm and static algorithm are similar.
5.1. Meshes and Boundary Conditions
The meshes of finite element model of the composite laminate plate under impact and shear-after-impact are shown in
Figure 8. The model is composed of one laminate plate, one tup, one supporting plate at the bottom of the laminate plate and eight plates at the four sides of the laminate plate, and all their dimensions are consistent with the test specimens and jigs. Eight-node in-plane reduced integral continuum shell elements with enhanced hourglass control and reduced integration, SC8R, were used to model the plies of the composite laminate plate to avoid the shear locking problem and to increase the computing efficiency. The modelling method of one element per ply lamina of the composite laminate provides a reasonable approximation of the through-thickness stresses of the composite laminates. The 0° axis of the composite laminate is along the horizontal direction and the ply angles are in a counterclockwise arrangement, which means the 45° direction is the direction from the bottom left to the upper right. There is a layer of zero mm thick eight-node three-dimensional cohesive elements, COH3D8, between two adjacent layers of SC8R elements, and these COH3D8 elements share nodes with the SC8R elements on both sides to simulate the delamination failure. The mesh density in the center of the composite laminate is finer than other areas in order to simulate the impact damage and the shear failure correctly, and in the meantime, save the simulation cost.
As shown in
Figure 9, the composite laminate plate was partitioned by five planes in horizontal and vertical directions, respectively, and the center of the composite laminate was partitioned by two inclined planes that are ±45˚ with the horizontal plane and by four circles with diameters of 4 mm, 8 mm, 16 mm, and 24 mm, respectively. The numbers inside the symbols “()” are the serial numbers of the edges which have special seeding method as listed in
Table 2, and the global mesh density of the laminate plate is 5.3 mm.
There are 100,928 SC8R and 55,915 COH3D8 in the meshes of the composite plate in total, and the mesh size in the center of the specimen is about 0.3 mm. The supporting jig was simulated as a discrete rigid plate with the four side length values being 150 mm with a 125 × 75 mm notch in the center, which is the same as the jig’s dimension defined in the test standards, ASTM D 7136. The tup was simulated as an analytical rigid surface with a diameter of 16 mm and a mass of 6.277 kg. Two hundred and ten linear quadrilateral elements of type S4R were used to simulate each steel plates in the picture frame test jig and the thickness of the S4R element were set to be 10 mm. The pivot relationships between the steel plates were defined using the MPC Pin function in the Abaqus.
In the impact loading step of the impact and ShAI loading model, the composite laminate was fixed all around and a predefined velocity perpendicular to the impacting surface of the composite laminate plate was set to the tup. A surface-to-surface contact relationship was used to define the contact relationships between the rigid tup and the composite laminate. “Hard” contact was used to define the normal interaction properties of the contact to avoid penetration during the analysis process. Penalty friction formulation were used to define the tangential interaction properties of the contact. The friction coefficient between the composite skin and the steel impactor was set to be 0.1. In the following shear loading step, the tup was fixed, and the bottom left corner of the test jig was fixed, whereas the upper right corner was applied with a displacement of 3 mm along the diagonal direction.
In the buckling analysis model, the tup was suppressed, and the bottom left corner of the test jig was fixed whereas the upper right corner was applied with a unit load along the diagonal direction.
5.2. Constitutive Models and Material Parameters
(1) Intralaminar failure Criterion
Progressive damage modeling was used to model the damage propagation process of the composite laminate and degradation of the strength under impact loading and shear loading. The Hashin failure criteria [
38] were used to evaluate the damage initiation of the composite material, as shown in
Table 3. The stress of the lamina model was calculated by classical composite laminate plate theory. In this model, the parameter α in
Table 3 is set to be the default value of 0, and the point at which damage initiates is determined by measuring when each of several damage criteria is met. Each damage criteria compares the different measured stresses against the strength properties of the respective material. Losses in strength and stiffness of composite laminates were considered as failures. The failure modes of fiber damage fracture, matrix damage fracture and delamination failure could be drawn from the simulation.
An energy-based method was employed to simulate the damage evolution after damage was initiated [
37]. Strain softening, which means the load carrying ability decreases as the strain increases, was used to model how the material will respond after damage has already occurred. The response is computed using
, where ε is the strain and
is the damaged elasticity matrix, as shown in Equation (10).
In this equation,
D is a combination of variables as shown in Equation (11), while
df,
dm, and
ds are damage state variables for the fiber, matrix, and shear, respectively.
When damage has been initiated in a particular mode, the equivalent stress-equivalent displacement relationship controls how the material properties will degrade. This damage variable varies from 0, as damage initiates, to 1, as the deformation energy equals to the fracture energies of the composite material in fiber direction and transverse direction.
(2) Interlaminar damage
The cohesive zone model approach was used to predict onset and propagation of interlaminar cracks, such as delamination between the adjacent plies of composite and debonding between the stiffener and skin. The cohesive zone model approach combines both, a strength-based and a fracture mechanics-based criteria for the prediction of interlaminar failure. The interface started to degrade once the interface strength reached the mechanical properties of the interface. The degradation of the interface properties involves energy dissipation. Once the energy dissipated by surface area is equal to the fracture toughness of the interface, a new free area has been created.
The cohesive zone model approach assumes that all the dissipation mechanisms occurring at the fracture process zone can be lumped to an interface or surface coincident with the crack plane. The mechanical behavior of this interface is representative of all the nonlinear mechanisms occurring within the fracture process zone and it is usually formulated in terms of a traction–separation law, also called cohesive law.
Ye’s criterion [
39] was used as strength degradation initiation criterion to predict the initiation of the degradation process, which is written as Equation (12).
where
,
,
are the normal traction and out of plane shear traction, and
,
,
are the normal and shear interface strengths. The 〈⬚〉 is the Macaulay symbol, it is written as Equation (13). It represents that the compressive through-thickness stress has no influence on the failure of the interface, and the interface is considered to be in a pure mode II crack state.
The interface of adjacent plies would be fully degraded when the energy dissipated equals the fracture toughness. The B-K criterion [
40] was implemented to predict the damage propagation under mixed-mode loading conditions, which is shown as Equation (14). The value of η, which is the power exponent, is set to be 1.45.
where G
IC, G
IIC and G
IIIC refer to the critical fracture energies required to cause failure in the normal, the first and the second shear direction, respectively.
In ABAQUS, the damage factor
D (SDEG: Scalar stiffness degradation, 0 ≤
D ≤ 1) is introduced to characterize the degree of damage for the cohesive element [
39]. The stiffness coefficient (
K) in damage evolution is expressed by Equation (15).
where
K0 is the stiffness of complete material. When
D equals to 0, it means that the material does not yield or has just begun to yield; when
D equals to 1, it means the material has been damaged and the load carrying capacity is lost. The material parameters of the cohesive elements were listed in
Table 4.