Composite Structures 214 (2019) 73–82 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Finite element simulation of damage in ﬁber metal laminates under high velocity impact by projectiles with diﬀerent shapes T ⁎ Qian Zhua, Chao Zhanga, , Jose L. Curiel-Sosab, Tinh Quoc Buic, Xiaojing Xua a School of Mechanical Engineering, Jiangsu University, Zhenjiang, China Department of Mechanical Engineering, The University of Sheﬃeld, Sheﬃeld, UK c Department of Civil and Environmental Engineering, Tokyo Institute of Technology, Tokyo, Japan b A R T I C LE I N FO A B S T R A C T Keywords: Fiber metal laminates High velocity impact Damage prediction Delamination Finite element analysis Fiber metal laminates (FMLs) have been widely used in many high-tech industries as protective structures because of their excellent impact resistance. The damage constitutive model that accurately characterizes the complex damage modes and failure processes of FMLs is the base to study the ballistic impact problems by numerical simulation. In this paper, a nonlinear ﬁnite element model based on continuum damage mechanics is established to investigate the damage modes and failure mechanisms of carbon ﬁber reinforced aluminum laminates (CRALLs) under high velocity impact. Johnson-Cook material model and a 3D rate-dependent constitutive model are applied to identify the in-plane damage in aluminum and ﬁber composite layers respectively; cohesive elements governed by bilinear traction-separation constitutive model are implemented to simulate the inter-laminar delamination induced by impact. The ballistic performance and damage characteristics of CRALLs under high velocity impact by projectiles with diﬀerent shapes are studied in detail. The obtained numerical results correlate well with the available experimental data thus validates the proposed ﬁnite element model, which also provides an appropriate reference for numerical studies of high velocity impact issues in other FMLs. 1. Introduction Fiber Metal Laminates (FMLs) are a type of hybrid composites formed by bonding metal sheets and ﬁber composite layers together and curing at a certain temperature and pressure. FMLs combine the performance advantages of both metals and composite materials and possess excellent comprehensive mechanical properties, such as outstanding fatigue, fracture and impact resistance. Because of these merits, FMLs have been considered as a solution for damage tolerance and weight reduction of primary load-bearing and protective structures in the aerospace and other high-tech industries. For the favorable engineering applications, it is necessary to characterize and evaluate the mechanical response of FMLs under typical loading conditions. The quasi-static mechanical properties [1–6] and low velocity impact performance [7–14] have been studied extensively. However, studies on high velocity impact behaviors of FMLs are rather rare in the literature. Ballistic impact experiment is the most direct approach to analyze the high velocity impact performance and damage characteristics of FMLs. Santiagoa et al. [15] thoroughly explored the impact performance of a new group of FMLs, named as TFMLs, based on an aluminum alloy and a self-reinforced polypropylene (SRPP). Xu et al. [16] experimentally investigated the ballistic penetration resistance ability of the composite laminates and FMLs targets impacted by different projectiles. Abdullah and Cantwell [17] compared the perforation resistance of various FMLs by determining the speciﬁc perforation energy of each laminate system. It was observed that the multi-layer laminates oﬀer a superior perforation resistance to the sandwich laminates for a given target thickness. Moreover, the main energy absorption mechanisms of these laminates are membrane stretching, plastic deformation and shear fracture in the metal sheets and ﬁber failure, delamination and ductile tearing in the composite layers. Zarei et al. [18] conducted the ballistic tests of GLARE using one stage gas gun with the target plate protected in a square clamp. It was found that the ballistic limit of GLARE impacted by a conical projectile is lower than that of a ﬂat projectile and the speciﬁc perforation energy increases with the decreasing of thickness of aluminum layer. Ballistic impact experiments generally are expensive, time-consuming and severely restricted by the practical test conditions. In contrast, less limitation happens in the ballistic impact modeling using ﬁnite element method. Finite element modeling is not only capable for ⁎ Corresponding author at: Department of Mechanical Design, School of Mechanical Engineering, Jiangsu University, 301 Xuefu Road, Zhenjiang 212013, Jiangsu, China. E-mail address: [email protected] (C. Zhang). https://doi.org/10.1016/j.compstruct.2019.02.009 Received 25 November 2018; Received in revised form 13 January 2019; Accepted 4 February 2019 Available online 05 February 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved. Composite Structures 214 (2019) 73–82 Q. Zhu et al. T ∗ = (T − Tr )/(Tm − T ) the detailed analysis of the penetration process and damage mechanisms in the extremely short impact period, but also very convenient for parametrical discussion with diﬀerent structural parameters and boundary conditions, thus has been widely adopted in the study of impact matters of FMLs. Guan et al. [19] developed an explicit ﬁnite element model to analyze the impact resistance of FMLs under high velocity impact. Ahmadi et al. [20] and Yaghoubi et al. [21] numerically studied the inﬂuence of thickness on the high velocity impact behaviors of FMLs. Tang et al. [22] examined the eﬀect of thermal loading on the high velocity impact resistance of FMLs at diﬀerent ambient temperatures. Based on a new self-reinforced thermoplastic composite, Santiagoa et al. [23] studied the modeling of impact loading on FMLs. It was concluded that placing a thicker SRPP layer on the back of the laminates can improve the impact performance while it is more valuable to place thicker aluminum layers in the middle of the laminates. The aforementioned numerical simulation works mainly focus on the ballistic resistance and energy absorption of FMLs while the analysis of damage characteristics of FMLs under high velocity impact is very limited. Besides, even carbon ﬁber is not sensitive to strain rate, FMLs always show obvious rate-dependent characteristics [24,25] under high velocity impact, which have not been studied well until now. The main objective of this paper is to propose a nonlinear ﬁnite element model and apply it to explore the damage modes and failure mechanisms of carbon ﬁber reinforced aluminum laminates (CRALLs) under high velocity impact by projectiles with diﬀerent shapes. Johnson-Cook material model dependent on strain, strain rate and temperature, 3D rate-dependent constitutive model, and bilinear traction-separation constitutive model are respectively used to predict the in-plane damage and inter-laminar delamination occurred in the aluminum sheets, ﬁber composite layers and corresponding interfaces under impact loading. All of these constitutive models are coded by a user material subroutine VUMAT and implemented based on the ABAQUS/Explicit platform. The numerical prediction results, ballistic limit velocity, initial-residual velocity curves and failure modes of the CRALLs are examined and investigated in detail. The accuracy and effectiveness of the present ﬁnite element model is validated with respect to the available experimental data. This work can also be considered as an appropriate reference to numerically study the high velocity impact issues in other FMLs. where Tr is the room temperature and Tm is the melting point of the material. The linear accumulation is used in the Johnson-Cook model to consider the failure behavior in the material deformation history. It not only involves the change of failure strain with stress state, strain rate and temperature, but also reﬂects the accumulation of failure during the deformation process. The plastic failure strain εf is expressed by the stress state, strain rate and the temperature as: εf = [D1 + D2 exp(σ ∗)][1 + D4lnε ∗̇ ][1 + D5 T ∗] (3) whereσ ∗ is the mean stress normalized by the eﬀective stress; D1–D5 are failure parameters, which can be obtained from the experimental data. The damage parameter D is an accumulation value. When this damage parameter reaches the unity, the damage of metal component occurs. The parameter D is determined by D= ∑ Δεeq εf (4) where Δεeq is the equivalent plastic strain increment in an integration cycle. 2.2. Damage model of composite material The dynamic properties of ﬁber reinforced composites are a mixture related to the dynamic properties of the component materials. Generally, the ﬁber is assumed as transversely isotropic linear-elastic material and the matrix is regards as isotropic viscoelastic material. In this study, according to the microstructure of ﬁber reinforced composites and the existing research results [27,28], the constitutive relationship of ﬁber reinforced composites under high strain rate is derived. Then, combined with the dynamic failure criteria, the dynamic damage model of ﬁber reinforced composites for high velocity impact simulation is established. 2.2.1. Constitutive relationship of unidirectional composite Karim et al. [28] presented the expressions of relaxation modulus of unidirectional ﬁber reinforced composites at high strain rate as follows: ( t t ) ⎧ E11 (t ) = Ef 1 Vf + Vm Em + E1 e− θe1 + E2 e− θe2 ⎪ ⎪ Ef 2 Em −Mt + Re−Nt ⎪ E22 (t ) = E V ' + E V ' + Qe m f f2 m ⎪ Gf 12 Gm ⎨ G (t ) = + Q12 e−M12 t + R12 e−N12 t ' ⎪ 12 Gf 12 Vm + Gm V 'f ⎪ Gf 23 Gm ⎪ G23 (t ) = + Q23 e−M23 t + R23 e−N23 t ' Gf 23 Vm + Gm V 'f ⎪ ⎩ 2. Material damage constitutive models Under high velocity impact loading, FMLs have many kinds of damage modes, such as metal shear failure, ﬁber tensile failure, matrix cracking and interface delamination. The application of appropriate failure criteria and constitutive equations to simulate the mechanical behavior of metal, ﬁber composite, and interface is the key point for the high velocity impact simulation of FMLs. (5) In the above equations, the ﬁrst item on the right hand of the above formula describe the linear elastic behavior of the material, while the remaining two items reﬂect the strain rate eﬀect of the material. Ef 1 and Ef 2 are the ﬁber axial and transverse elastic modulus; Gf 12 and Gf 23 are the ﬁber axial and transverse shear modulus; Em and Gm are the matrix elastic modulus and shear modulus; Vf and Vm are the ﬁber and matrix volume fractions; V f' = Vf and Vm' = 1 − Vf are the modiﬁed ﬁber and matrix volume fractions. E1 and E2 are the elastic parameters in Maxwell element; θe1 and θe2 are the relaxation time of Maxwell element; M, N, Q, R, M12, N12, Q12, R12, M23, N23, Q23, R23 are the hysteresis parameters in Maxwell element, and their expressions are provided in detail in Ref. [28]. According to the generalized Hooke's law, the constitutive relationship of orthotropic material is expressed as 2.1. Damage model of metal Johnson-Cook model [26] is commonly used for metal structures in the impact simulation problems. It includes the eﬀects of strain, strain rate and temperature on stress, and has a wide range of application in engineering ﬁeld. Johnson-Cook model is a visco-plastic model consisting of two parts: the constitutive law and the failure criterion. The eﬀective stress is expressed as: σ = (A + Bε n )(1 + C lnε ∗̇ )(1 − T ∗m) (2) (1) where A is the yield stress, B and n are the strain hardening eﬀect parameters. These three parameters can be obtained by ﬁtting the plastic section of the static tensile curves. C is the strain rate eﬀect parameter; ε is the equivalent plastic strain; ε ∗̇ is the normalized effective plastic strain rate; m is the thermal softening exponent; T ∗ is a dimensionless temperature and can be determined by: ε = Sσ = E −1Uσ The inverse of the above equation is 74 (6) Composite Structures 214 (2019) 73–82 Q. Zhu et al. (7) σ = Cε = U−1Eε properties and expressed as: where S and C are the compliance matrix and stiﬀness matrix, respectively. And U and E −1 are deﬁned as: − μ 21 − μ31 ⎡ 1 ⎢− μ12 − μ32 1 ⎢ 1 U = ⎢− μ13 − μ 23 ⎢ 0 0 0 ⎢ 0 0 0 ⎢ 0 0 0 ⎣ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 ⎡1/ E11 ⎢ 0 1/ E22 0 0 ⎢ 0 0 1/ E33 0 ⎢ = 0 0 1/ G23 ⎢ 0 ⎢ 0 0 0 0 ⎢ 0 0 0 ⎣ 0 0⎤ 0⎥ 0⎥ ⎥ E −1 0⎥ 0⎥ 1⎥ ⎦ 0 0 ⎤ 0 0 ⎥ 0 0 ⎥ ⎥ 0 0 ⎥ 1/ G31 0 ⎥ ⎥ 0 1/ G12 ⎦ |ε ̇| S = S0 ⎛1 + c ln ⎞ ε0̇ ⎠ ⎝ ⎜ 2.3. Damage model of interface Delamination is the critical failure mode of FMLs under high velocity impact. In this work, cohesive elements are introduced to simulate the inter-laminar delamination at the interface of adjacent metal-composite and composite-composite layers. The traction stress and separation displacement of the nodes on the interface are governed by traction-separation description. Because the interface is very thin, only a normal traction and two shear tractions are assumed acting on it. The constitutive relationship of interface is given by (8) t k 0 0 ⎤ ⎧ δ1 ⎫ t ⎧ 1⎫ ⎡ 1 t2 = ⎢ 0 k2 0 ⎥ δ2 ⎨t ⎬ ⎢ ⎥⎨ ⎬ ⎩ 3 ⎭ ⎣ 0 0 k3 ⎦ ⎩ δ3 ⎭ ⎧ ∫0 E11 (t − τ ) ε ̇ (τ ) dτ ⎫ ⎪ ⎪ t ⎪ ∫0 E22 (t − τ ) ε ̇ (τ ) dτ ⎪ ⎪ ⎪ t ⎪ ∫0 E22 (t − τ ) ε ̇ (τ ) dτ ⎪ σ ⎧ 11 ⎫ ⎪ σ22 ⎪ ⎪ σ33 ⎪ = U−1 ⎨ σ23 ⎬ ⎨ ∫t G (t − τ ) ε ̇ (τ ) dτ ⎬ ⎪ ⎪ 0 23 ⎪ σ31 ⎪ ⎪ σ12 ⎪ ⎪ ∫t G (t − τ ) ε ̇ (τ ) dτ ⎪ ⎩ ⎭ ⎪ ⎪ 0 12 ⎪ ∫t G (t − τ ) ε ̇ (τ ) dτ ⎪ 12 ⎭ ⎩ 0 (9) ⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟ (10) 2 ⎟ (11) 2 2 2 σ + σ3 ⎞ 1 σ σ 2 Fmt = ⎛ 2 + ⎜⎛ 2 ⎟⎞ (σ23 − σ2 σ3) + ⎛ 12 ⎞ + ⎛ 13 ⎞ = 1 ⎝ YT ⎠ ⎝ S12 ⎠ ⎝ S13 ⎠ ⎝ S23 ⎠ ⎜ ⎟ ⎜ ⎟ (12) Matrix compressive failure (σ2 + σ3 < 0) 2 d= 2 σ + σ3 ⎞ ⎤ σ2 + σ3 + 1 (σ 2 − σ σ ) + ⎛ σ12 ⎞ ⎛ Yc ⎞ =⎛ 2 +⎡ 2 3 23 2 ⎢ 2S23 − 1⎥ Yc S 2 S23 23 ⎝ S12 ⎠ ⎠ ⎝ ⎠ ⎦ ⎣⎝ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ 2 ⎜ ⎟ 2 ⎜ ⎟ (18) δmf (δmmax − δm0 ) δmmax (δmf − δm0 ) (d ∈ [0, 1]) (19) ⎟ δmmax Here, the maximum relative displacement is deﬁned as δmmax = max{δmmax , δm} owing to the irreversibility of damage. The constitutive relationship of the interface considering the damage state is as follows 2 σ + ⎛ 13 ⎞ = 1 ⎝ S13 ⎠ ⎟ where G I , G II and G III are the current energy release rates while G IC , G IIC and G IIIC are the critical fracture energies of mode I, II and III, respectively. After damage initiation, a scalar damage variable d is used to represent the linear damage evolution of interface, namely Matrix tensile failure (σ2 + σ3 ≥ 0) Fmc (17) ⎛ G I ⎞ + ⎛ G II ⎞ + ⎛ G III ⎞ = 1 ⎝ G IC ⎠ ⎝ G IIC ⎠ ⎝ G IIIC ⎠ σ F fc = ⎛ 1 ⎞ = 1 X ⎝ c⎠ 2 2 where N, S and T are the tensile and shear strengths of interface, respectively. An interactive power law of energy [30] is used to determine the complete failure displacement δmf , namely ⎜ ⎟ 2 ⎛ 〈t1 〉 ⎞ + ⎛ t2 ⎞ + ⎛ t3 ⎞ = 1 ⎝T ⎠ ⎝S⎠ ⎝ N ⎠ 2 ⎜ (16) where the symbol < > represents the Macaulay operator. A quadratic nominal stress criterion [30] is employed to determine the damage initiation displacement δm0 , namely Fiber compressive failure (σ1 < 0) ⎜ 〈δ1 〉2 + δ22 + δ32 δm = 2 σ σ σ F ft = ⎛ 1 ⎞ + α ⎛ 12 ⎞ + α ⎛ 13 ⎞ = 1 ⎝ XT ⎠ ⎝ S12 ⎠ ⎝ S13 ⎠ (15) where t is the traction stress, δ is the separation displacement, and k represents the initial stiﬀness of the interface. It is known that the interface delamination is always happened under mixed-mode conditions. To involve the combination of normal and shear separations, the eﬀective relative displacement, δm, is introduced as 2.2.2. Failure criteria of composite material At high velocity impact, the failure criteria are critical for simulating damage in composite laminates. By investigating the failure behavior in specimens with diﬀerent ﬁber orientations, Hashin [29] concluded that there are only two main failure mechanisms: ﬁber dominated failure and matrix dominated failure. For unidirectional composite, there are four typical failure modes, which are governed by the following expressions: Fiber tensile failure (σ1≥0) 2 (14) where S is the strength at the current strain rate, ε ̇ is the corresponding strain rate, S0 is the reference strength corresponding to the reference strain rate ε0̇ and c is the modiﬁcation factor which can be obtained by ﬁtting the test data. Considering that the unidirectional ﬁber reinforced composites have the same mechanical properties in the 2 and 3 directions, the 3D rate dependent constitutive relationship can be expressed as: 2 ⎟ ⎟ (13) In the above equations, XT , Xc , YT , and Yc are axial tensile, axial compression, transverse tensile, and transverse compressive strengths respectively; S12, S13 and S23 are the shear strengths; α is the shear modiﬁcation factor. The composite laminates have obvious strain rate eﬀect during high velocity impact. Considering this eﬀect, the strength parameters of unidirectional composite are modiﬁed by the rate dependent strength t̂ ⎧ 0 0 ⎤ ⎡ k1 0 0 ⎤ ⎧ δ1 ⎫ ⎪ 1⎫ ⎪ ⎡1 − d 〈t1〉/ t1 t2 ̂ = ⎢ 0 1 − d 0 ⎥ ⎢ 0 k2 0 ⎥ δ2 ⎨ ⎬ ⎢ ⎢ ⎥⎨ ⎬ ⎪ t3 ⎪ 0 1−d ⎥ ̂ ⎣ ⎦ ⎣ 0 0 k3 ⎦ ⎩ δ3 ⎭ ⎩ ⎭ (20) where the symbol ^ represents actual stress in the damaged cohesive element. 75 Composite Structures 214 (2019) 73–82 Q. Zhu et al. 3.2. Contact deﬁnition and material properties The available technique of ‘Surface-to-Surface Contact’ is adopted to deﬁne the contact response between the projectile and the FMLs. The penalty contact method with a ﬁnite sliding formulation is selected to calculate the contact force during the ballistic penetration. The material properties of aluminum alloy 2024-T3, unidirectional composite and interface used in the ﬁnite element simulation are cited from Refs. [16,28,31–34] and summarized in Tables 1–3. Fig. 1. Geometric parameters of projectiles with diﬀerent shapes [16]. 3.3. Material degradation and element deletion 3. Nonlinear ﬁnite element model During ballistic penetration process, when any failure happens, the materials will lose their load-bearing capacity in certain mode. In view of this, the corresponding material properties need to be degraded. In this paper, an instantaneous elastic constants reduction rule, given in Table 4, is introduced to the degradation model of ﬁber composites. For the aluminum sheet, if the damage parameter reaches 1, the element will be removed from the simulation model; for the interface, if scalar damage variable is larger than 0.99, the damaged cohesive element will be deleted; and for ﬁber composites, if the two damage variables representing ﬁber and matrix failure modes in an element are both equal to 1, the element will be deleted. 3.1. Finite element modeling and boundary conditions In order to analyze the ballistic performance and damage characteristics of FMLs under high velocity impact, a nonlinear ﬁnite element model is established based on ABAQUS/Explicit. The available experimental data of carbon ﬁber reinforced aluminum laminates (CRALL2/1) under high velocity impact provided in Ref. [16] are selected for our comparison purpose. Three steel projectiles with ﬂat, hemisphere and sharp noses are used and the geometric parameters of these three projectiles are schematically shown in Fig. 1. Since the deformation of the projectile is small and negligible during the penetration process, the projectile is considered as a rigid body (Discrete rigid) and the four-node rigid shell elements (R3D4) are used for mesh discretization in the simulation. The concentrated mass (30 g) of the projectile and the initial impact velocity are imposed on the reference point of the projectile. The projectile’s moment of inertia is not necessary in this normal ballistic impact simulation because the friction between the projectile and the target plate is ignored here. The CRALL2/1 with a diameter of 100 mm and thickness of 2.48 mm are built from 2 layers of 0.8 mm aluminum 2024-T3 sheets, one layer of 0.8 mm T700S CFRPs (8 plies of 0.1 mm each with a stacking sequence [0/90]2s) and two 0.04 mm adhesive layers. In the CRALL2/1, since each layer has its special properties and the ﬁber orientations are diﬀerent in each ply of composite laminates, the failure response would occur distinctly. To explore the exact damage characteristics, separate plies are established to model the aluminum sheets and unidirectional composite layers in the laminates. Each layer of aluminum and composites is discretized by hourglass controlled eight node solid elements (C3D8R) with reduced integration. In order to effectively simulate the delamination of FMLs under high velocity impact, cohesive elements (COH3D8) are introduced at the interfaces between aluminum and composite and between adjacent plies of laminates. The thickness of metal-composite interface is 0.04 mm and zero-thickness cohesive elements are used for the composite-composite interface. During high velocity impact penetration, the deformation and damage of the target plate mainly take place in the contact area. Therefore, ﬁner mesh is employed at the impact zone and the mesh density decreases gradually from the impact zone to the boundary of the target plate. This mesh not only ensures the accuracy of numerical results, but also improves the computational eﬃciency of the model. Certainly, the ﬁnite element mesh here is also ﬁne enough to get the convergence computation results. A ﬁxed boundary condition is imposed along the periphery of the CRALL2/1, and all degrees of freedom are constrained to zero to replicate the experimental clamping conditions. Besides, the original distance between the outer surface of the projectile and the front face of the CRALL2/1 before contact is set as zero. The ﬁnite element model of CRALL2/1 under high velocity impact is displayed in Fig. 2. Here, the model consists of 149, 142 nodes, 69, 860 C3D8R elements and 62, 874 COH3D8 elements. 4. Numerical results and discussion 4.1. Veriﬁcation of ﬁnite element model To verify the accuracy and eﬀectiveness of the proposed ﬁnite element model, the predicted numerical results of FMLs under ballistic impact by projectiles with diﬀerent shapes (ﬂat, hemisphere and sharp nose) are hence compared with the available experimental data obtained from Ref. [16]. The ballistic limits Vb of the FMLs predicted by the numerical models with three diﬀerent projectiles, CRALL2/1-F, CRALL2/1-H and CRALL2/1-S, are listed in Table 5. Here, the ballistic limit is deﬁned as the maximum impact velocity that the projectile does not pass through the target. The errors of CRALL2/1-H and CRALL2/1-S models are very small while it is a little higher for CRALL2/1-F. The relatively larger error appeared in CRALL2/1-F could be attributed to the following factors. First, the material parameters adopted in the simulation come from diﬀerent reference source and some of them might not be accurate. Second, the instantaneous stiﬀness reduction is an engineering method without much speciﬁc physical basis, and then the stiﬀness reduction rule implemented here may not be appropriate. Besides, the small deviation of impact velocity in experiment, the thermal eﬀect in the aluminum sheet, the friction between the projectile and target plate, and the manufacturing defects of the specimen, which are not involved in the simulation model. Fig. 3 illustrates the comparison of initial-residual velocity curves between numerical results and experimental data for the three projectiles. Overall, the numerical prediction results are somewhat different from the actual experimental data, but acceptable consistency is recorded, which veriﬁes that the proposed model is reasonable. By comparing Fig. 3(a), (b) and (c), it can be found that CRALL2/1 has the strongest impact resistance to the ﬂat-nosed projectile, followed by the hemisphere-headed projectile, and the least is the sharp-nosed projectile with the same projectile mass. Besides, when the initial impact velocity increases gradually, the residual velocity of the projectile tends to be very similar. Therefore, it can be concluded that in the higher velocity range, the inﬂuence of projectile shape on the impact residual velocity becomes not signiﬁcant, which is also consistent with the available experimental investigation. The comparison of numerical and experimental failure modes [16] of CRALL2/1 under high velocity impact with the three projectiles are demonstrated in Fig. 4. The front, back and cross-sectional proﬁles of CRALL2/1 after penetration are provided and the initial impact 76 Composite Structures 214 (2019) 73–82 Q. Zhu et al. Fig. 2. Finite element model of CRALL2/1 under high velocity impact. Table 1 Material properties of aluminum 2024-T3 [16,31]. Table 4 Material stiﬀness reduction rule [33]. Material properties Value Material properties Value ρ (kg/m3) E (GPa) μ A (MPa) B (MPa) C n m 2750 72.2 0.35 369 684 0.0083 0.73 1.7 Tr (k) Tm (k) D1 D2 D3 D4 D5 293 775 0.112 0.123 1.5 0.007 0 Vf Failure mode Stiﬀness reduction Fiber tensile failure Fiber compressive failure Matrix tensile failure Matrix compressive failure Ef1 Ef2 Gf12 Gf23 Em Gm 0.01 0.01 1 1 0.2 0.2 0.2 0.2 0.01 0.01 0.2 0.2 1 1 0.2 0.2 1 1 0.01 0.01 1 1 0.01 0.01 Table 5 Experimental and numerical results for the ballistic limits of FMLs. Table 2 Material properties of unidirectional composite [16,28,32]. Material properties Value Material properties Value Ef1 (GPa) Ef2 = Ef3 (GPa) Gf12 = Gf13 (GPa) Gf23 (GPa) Em (GPa) E1 (GPa) θe1 (ms) E2 (GPa) θe2 (ms) Gm (GPa) G1 (GPa) θg1 (ms) 230 15 2.35 24 2.31 0.971 0.041 0.104 121,000 0.857 0.401 0.077 G2 (GPa) θg2 (ms) ρ (kg/m3) Vf μ12 = μ13 μ23 c XT (MPa) XC (MPa) YT (MPa) YC (MPa) S (MPa) 0.041 12,000 1570 0.6 0.25 0.38 0.1 2050 1050 71 132 75 Value ρ (kg/m3) k (GPa) N (Mpa) S = T (Mpa) GⅠC (J/m2) GⅡC = GIIIC (J/m2) 1440 1000 80 183 0.3 0.6 Experiment [16] (m/s) prediction (m/s) Error CRALL2/1-F CRALL2/1-H CRALL2/1-S 85.2 80.9 70.5 108.7 84.4 67.8 27.58% 4.33% 3.83% 4.2. Analysis of ballistic penetration process High velocity impact is considered as a transient dynamic process and the response of FMLs is governed by the local behavior of the material in the neighboring of the impact zone. FMLs have diﬀerent failure mechanisms at diﬀerent stages of impact. The main failure modes are metal shear failure, ﬁber tensile failure, delamination and matrix cracking. By using the ﬁnite element modeling, the deformation, damage propagation and delamination can be analyzed in detail throughout the ballistic penetration process. Figs. 5–7 illustrate the penetration process of CRALL2/1 impacted by three projectiles with diﬀerent shapes at the velocity of 150 m/s. The impact contact lasts < 300 μs, and diﬀerent failure modes occur, promote and couple with each other during the penetration process. The detailed impact processes are analyzed as below. Fig. 5 illustrates the ballistic penetration process of ﬂat-nosed projectile in the CRALL2/1 target. At t = 50 μs, because the contact area between the projectile and target is very large and the plasticity of aluminum sheet is higher than that of composites, the convex zone on the back of the target is relatively smooth and the tensile stress in the composite layers have not caused much ﬁber tensile fracture. However, on the front of the target plate, the ballistic impact loading has obvious shear eﬀect on the aluminum sheet. Therefore, as shown in Fig. 5(b), a plug column is formed in the front aluminum sheet due to much shear failure occurs in the contact zone along the edge of the projectile. Fiber shear and tensile failure happen in the middle of composites layers and the aluminum sheet on the back of the target deform apparently with much tensile tear failure and a small amount of shear failure. The projectile penetrates the target plate in a very short time. With the penetration of the projectile, many impact fragments are produced in Table 3 Material properties of interface cohesive elements [33,34]. Material properties Code velocities of ﬂat, hemisphere and sharp-nosed projectiles are 109.6 m/s, 106.9 m/s and 102.8 m/s respectively. The present numerical simulation model not only considers the inﬂuence of strain rate on the mechanical properties of aluminum sheets and carbon ﬁber reinforced composites, but also uses the cohesive elements to eﬀectively simulate the inter-laminar delamination. The numerical simulation results of the lateral deformation and failure state of the target plates are in good agreement with the experimental results, which again veriﬁes the efﬁciency of the proposed model. 77 Composite Structures 214 (2019) 73–82 Q. Zhu et al. Fig. 3. Comparison of numerical and experimental impact initial-residual velocity curves [16] under three projectiles (a) CRALL2/1-F (b) CRALL2/1-H (c) CRALL2/ 1-S. plate and the extrusion of projectile, which results in the tensile tear perforation of the target plate. In the neighboring zone near the perforation hole, a large local bending forms and the deformation gradually recovers after the projectile completely penetrates the target. the composite layers and splashes out as displayed in Fig. 5(c). The thickness of the target is thin and the bending stiﬀness is small, then under the lateral impact loading, a large bending deformation happens and results in greater tensile stress on the back of the target, as shown in Fig. 5(d). Afterwards, since the projectile no longer contacts with the target plate, the deformation gradually recovers, and the perforated area of the target no longer increases, as shown in Fig. 5(e, f). Fig. 6 demonstrates the penetration process of CRALL2/1 target under the ballistic impact of a hemisphere-nosed projectile. In Fig. 6(a), after CRALL2/1 is impacted by the hemisphere-nosed projectile, at t = 50 μs, a spherical concave shape is formed on the front of the target plate and a local convex is appeared on the back of the target. As shown in Fig. 6(b), at t = 100 μs, the failure in the contact zone of the front aluminum sheet is mainly shear failure and a small plug column is formed. With the ballistic penetration to continue, the contact area between the projectile and the target increases gradually. The composite layers and the back aluminum sheet are deformed simultaneously. Since the tensile strain at which the stress-based ﬁber tensile failure criterion satisﬁed is much less than that of aluminum alloy, tensile fracture failure happens earlier in composite layers than that in back aluminum sheet. However, due to the coupling eﬀect of such ﬁber tensile failure, similar tensile failure shape occurs on the back aluminum sheet, as displayed in Fig. 6(c). After that, the local tensile fracture expands outward along the target plate resulting in the tensile fracture perforation of the target plate, as shown in Fig. 6(d–f). Fig. 7 displays the penetration process of CRALL2/1 target under the ballistic impact of a sharp-nosed projectile. It can be found that, under impact of sharp-nose projectile, the failure of both the front and back aluminum sheets are tension failure. As seen in Fig. 7(a), at t = 50 s, the contact area between the projectile and the target plate is small but the stress is very large. That is why some tensile fracture failure occurs in the front aluminum sheet. During the continuous penetration of projectile, the failure modes of the ﬁber composite layers and back aluminum sheet are basically similar to those under the ballistic penetration of hemisphere-nosed projectile. That is, both of them are tensile fracture failures, as given in Fig. 7(b–f). Under ballistic impact, the stress wave propagation velocity of the composite layers is much faster than that of aluminum sheet. Therefore, the ﬁber composite layers can eﬀectively transform the stress state, disperse the concentrated stress, and then increase the deformation and energy absorption of the back aluminum sheet. By comparing the material damage modes induced by these three projectiles, it can be found that for ﬂat-nosed projectile, the contact area between the projectile and target is large and the damage is distributed around the penetration hole. Moreover, the damaged area is very large and distributed with trumpet shape along the thickness direction. Compared with ﬂat-nosed projectile, hemisphere and sharpnosed projectiles have stronger penetration abilities and the phenomenon of local extrusion perforation is more obvious. After local tensile fracture perforation occurs, the damage spreads rapidly along the initial cracking direction under the actions of bending deformation of target 4.3. Analysis of impact delamination damage Delamination is a critical failure mode of FMLs under impact loading, which inﬂuences the residual mechanical properties of FMLs signiﬁcantly. Delamination is mainly caused by the inter-laminar tension and shear stress in the impact zone. The distributions of delamination of the typical layers of CRALL2/1 impacted by three projectiles at the velocity of 150 m/s are shown in Fig. 8. Here, F, H and S refer to ﬂat, hemisphere and sharp-nosed projectiles respectively. As can be seen from Fig. 8, the delamination caused by ﬂat-nosed projectile is more serious than those of hemisphere and sharp-nosed projectiles, and the delamination areas of each layer for the hemisphere and sharp-nosed projectile are similar. From the above analysis, it is known that the shear and tensile failure caused by the ballistic impact of ﬂat-nosed projectile on the target plate is relatively larger, while the failure modes caused by hemisphere and sharp-nosed projectiles are basically similar. However, the trends of delamination caused by these three projectiles are consistent. The ﬁrst interface (Al/0) has the largest critical delamination because the plastic deformation of aluminum alloy is much bigger than that of carbon ﬁber composites, which results in shear failure of the interface during large deformation. Since the middle composite layer is an orthogonal layer, there is no signiﬁcant diﬀerence of delamination in the longitudinal and transverse mechanical properties, as seen from the second layer (0/90), ﬁfth layer (90/0) and eighth layer (0/90). Hereafter, the scale of delamination damage is basically similar and the delamination shape is close to a circle with the diameter is approximate to the diameter of the projectile. For the ninth interface (90/Al), it is bent and deformed along with the back aluminum sheet, thus it has the same failure shape with the back aluminum sheet. 4.4. Analysis of velocity and acceleration curves In this section, for the high velocity impact simulation of CRALL2/1 penetrated by ﬂat, hemisphere and sharp-nosed projectiles, the initial impact velocity is set as 150 m/s. The velocity-time curves and acceleration-time curves for diﬀerent projectiles in ballistic penetration process are given in Fig. 9. It can be found that the acceleration has a large peak at t = 2 μs when the ﬂat-nosed projectile contacts the target plate. Once the contact is happened, because the contact area between the ﬂat-nosed projectile and the target plate is large, the bending deformation of the aluminum sheet becomes large suddenly and more energy is absorbed. The velocity of the ﬂat-nosed projectile decreases drastically. At about t = 130 μs, the projectile passes through the target plate completely and then the velocity will not change any more. 78 Composite Structures 214 (2019) 73–82 Q. Zhu et al. (a) CRALL2/1-F (b) CRALL2/1-H (c) CRALL2/1-S Fig. 4. Comparison of numerical and experimental fracture modes [16] of CRALL2/1 under ballistic impact by diﬀerent projectiles (a) CRALL2/1-F (b) CRALL2/1-H (c) CRALL2/1-S. 79 Composite Structures 214 (2019) 73–82 Q. Zhu et al. Fig. 5. Ballistic penetration process of CRALL2/1 with ﬂat-nosed projectile. Compared with the ﬂat-nosed projectile, the decreasing velocity is relatively stable of the hemisphere-nosed projectile. The acceleration has no particularly large peaks and the velocity of the projectile does not change any more at approximately t = 100 μs. For the sharp-nosed projectile, the velocity decreases slowly at the beginning. With the impact continues, the contact area between the projectile and the target plate increases and the velocity decreases quickly. At last, when t = 90 μs, the velocity-time curve tends to be straight. Due to the similar failure modes caused by the ballistic impact of hemisphere and sharpnosed projectiles, the values of residual velocity are relatively near to each other. The numerical results indicate that the changes of projectile’s velocity and acceleration during high velocity impact are determined by many factors: contact states, material damage modes, material failure time and elements deletion, etc. Under the same velocity impact of the three projectiles, CRALL2/1 has the strongest impact resistance and the highest energy absorption to ﬂat-nosed projectile, while those are similar to hemisphere and sharp-nosed projectiles. 5. Conclusions We have developed a nonlinear ﬁnite element model for simulating the ballistic penetration process of FMLs impacted by projectiles with diﬀerent shapes. The present investigation again shows that the ﬁnite element simulation is the most economical and eﬀective approach to investigate the high velocity impact problems of FMLs. In this paper, the strain rate eﬀect of metal sheets and ﬁber composites are considered and the delamination damage between adjacent layers is involved. The predicted ballistic limit velocities, initial-residual velocity curves and failure modes are shown to be in good agreement with the experimental data, which validates the proposed ﬁnite element model. Numerical analyses are carried out to study the inﬂuence of projectile shapes on the ballistic performance and damage characteristics Fig. 6. Ballistic penetration process of CRALL2/1 with hemisphere- nosed projectile. 80 Composite Structures 214 (2019) 73–82 Q. Zhu et al. Fig. 7. Ballistic penetration process of CRALL2/1 with sharp- nosed projectile. eﬀect of projectile shape on the impact results is closely related to the initial impact velocity. When the initial velocity is not very high, the projectile shape has a great inﬂuence on the ballistic penetration performance. The penetration capability of the ﬂat-nosed projectile is the lowest while those of the hemisphere-nosed projectile and the sharpnosed projectile are similar. As the initial impact velocity increases, the inﬂuence of the projectile’s shape becomes not obvious. Since the current numerical model can accurately analyze the intralayer and inter-layer ballistic damage response of FMLs and the proposed damage formula is naturally universal, it can be suitable for of CRALL2/1. The target plate forms cylindrical plug under the ballistic impact of ﬂat-headed projectile. The main failure modes are shear failure on the front aluminum sheet, tensile failure on the back aluminum sheet and tensile and shear ﬁber failure in the composite layers. Under ballistic impact of hemisphere-nosed projectile, the CRALL2/1 produces a plug on the front aluminum sheet, while on the back is mainly extrusion tensile failure, accompanied by a few broken ﬁber strips and debris. The main failure phenomenon under sharp-nosed projectile is similar to that under hemisphere-nosed projectile, ﬁber fragments are also observed on the back of CRALL2/1. In addition, the Fig. 8. Distributions of delamination in the typical layers of CRALL2/1 impacted by three projectiles at the velocity of 150 m/s. 81 Composite Structures 214 (2019) 73–82 Q. Zhu et al. Fig. 9. Velocity-time and acceleration-time curves for diﬀerent projectiles in the ballistic penetration process (a) Velocity-time curves (b) Acceleration-time curves. studying the damage behavior of other types FMLs under high velocity impact. [13] Acknowledgments [14] This work was supported by the National Natural Science Foundation of China (11832014), Natural Science Foundation of Jiangsu Province (BK20180855), China Postdoctoral Science Foundation (2018M640459) and Natural Science Research Project of Colleges and Universities in Jiangsu Province (17KJB130004). [15] [16] [17] Appendix A. Supplementary data [18] Supplementary data to this article can be found online at https:// doi.org/10.1016/j.compstruct.2019.02.009. [19] [20] References [21] [1] Reyes-Villanueva G, Cantwell WJ. 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