Composite Structures 214 (2019) 73–82
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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: zhangchao@ujs.edu.cn (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.
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Composite Structures 214 (2019) 73–82
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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.
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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.
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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.
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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.
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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]
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