4.1. Setup of Numerical Model
To predict the dynamic response and obtain the ballistic limit of UHMWPE composite armor under the ballistic impact of the A–P core, three–dimensional numerical models are carried out using the AUTODYN nonlinear software. The version of AUTODYN is v11.0 in the software of ANSYS 11.0, located in Nanjing, China.
As shown in
Figure 10, the 3D Lagrange algorithm is adopted for all of the components in numerical simulation. The half 3D model is carried out with a mesh size of about 1.2 mm per grid. A hexahedral structured grid is used to model both the projectile and the composite armor. The numerical simulation model is composed of about 810 thousand nodes and 800 elements. On the edge of the target, fixed boundaries are used to constrain the movement of the armor. The boundary conditions are applied on the edges of both the face and back sheets. Different initial velocities are applied to the ogive–nose head penetrator to simulate the dynamic penetration behavior with different impact velocities. The material models and the parameters will be described below.
As presented in
Table 3, the material models for the penetrator, face sheet, and UHMWPE laminate are listed. For steel, the shock equation of state, also called Grüneisen, is employed in conjunction with the Johnson–Cook constitutive model to simulate the dynamic response under ballistic impact. The Grüneisen EOS [
27] can be used to describe how the materials interact with the shock wave and are based on Hugoniot’s relation between the v
s. and the
vp, as
vs =
c0 +
svp, where v
s. is the shock wave velocity,
vp is the material particle velocity,
c0 is the wave speed, and
s is a material–related coefficient. The expression of the equation of state of Grüneisen for the compressed state is:
In the expanded state,
where
C is the intercept of the velocity curve between the shock wave and particle;
S1,
S2, and
S3 represent the slope of the
vs −
vp curve;
γ0 is the coefficient of the Grüneisen;
a is the one–order correction of
γ0.
μ =
ρ/
ρ0 − 1 is a non–dimensional coefficient based on initial and instantaneous material densities. The parameters of the Grüneisen equation of state are listed in
Table 4.
The Johnson–Cook model [
28,
29] incorporates the effect of strain rate–dependent work hardening and thermal softening, which is given by:
where
ε is the plastic strain, and the temperature factor is expressed as:
where
Tr is the room temperature, and
Tm is the melt temperature of the material.
A,
B,
n,
C, and
m are material–related parameters. The material parameters of S-7 tool steel and Q235 steel are presented in
Table 5.
The orthotropic material model proposed by Long H. Nguyen et al. [
14] was used for modeling the dynamic behavior of the UHMWPE layer subjected to ballistic impact. The material models consist of a nonlinear equation of the state of orthotropic, a strength model, and a failure model. The constitutive response of the material in the elastic regime is described as the orthotropic EOS composed of volumetric and deviatoric components. The pressure is defined by:
where
Cij are the coefficients of the stiffness matrix,
refers to the deviatoric strains in the principal directions, and the volumetric component
is defined by the Mie–Grüneisen EOS:
where
v,
e, and Γ(
v) represent the volume, internal energy, and the Grüneisen coefficient, respectively.
Pr(
v) is the reference pressure, and
er(
v) is the reference internal energy. The quadratic yield surface was adopted as the material strength model to describe the nonlinear, irreversible hardening behavior of the composite laminate:
where
aij are the plasticity coefficients, and
σij represent the stresses in the principal directions of the material. In addition, the state variable,
k, is used to define the border of the yield surface. It is described with a master and stress–effective plastic strain curve defined by ten piecewise points to consider the effect of strain hardening.
In the numerical models, the failure model of the orthotropic material is based on a combined stress criterion given as follows:
where
S is the failure strength in the respective directions of the material, and
D is the damage parameter following a linear relationship with stress and strain, as shown below:
where
L is the characteristic cell length,
εcr refers to the crack strain, and
Gii,f presents the fracture energy in the direction of damage.
The constants for the orthotropic equation of state are presented in
Table 6, and the parameters for orthotropic yield strength are shown in
Table 7.
4.3. Perforation Models and Analysis
- (1)
Principle of energy conservation
The energy balance for the perforation is given by
where
m is the mass of the projectile,
vi is the impact velocity,
vr is the residual velocity, and
W is the work performed during perforation. The mass of the A–P core was set at 40.4 g, then the work conducted during the perforation of the composite armor could be calculated, as listed in
Table 9. The value of
W stayed stable from 5.05 kJ to 5.09 kJ, which means that dissipated energy in the petaling stays stable at around 5 kJ. At the ballistic limit from the numerical results, 500 m/s, the dissipated energy is the same as the work performed at a higher velocity after perforation. So, the principle of energy conservation can be applied here.
- (2)
Lambert–Jonas model
The Lambert–Jonas model [
26,
30,
31,
32] can provide a reasonable fit to predict the residual velocity of the penetrator after perforation. The model can be expressed as
where
vi,
vr, and
vbl are the impact, residual, and ballistic limit velocity in normal impact.
α and
p are the coefficients, where 0 ≤
α ≤ 1 and
p > 1. Based on the numerical simulation results, the Lambert–Jonas model can be established to predict the residual velocity of the A–P core after perforating the PE composite armor.
When the model with
p = 2, the coefficient
α can be set as 1, and the model can be justified based on the energy conservation law [
33]. This model can be written as
the predicted
vr −
vs curve and the simulation results are presented below. As shown in
Figure 12, the Lambert–Jonas model can be an effective method in predicting the residual velocity of the A–P core after perforation. In addition, the perforation process can be regarded as a rigid body penetration.
- (3)
Cavity–Expansion Model
As the A–P core has a diameter of 12.48 mm and a length of 53.4 mm, the composite armor with a thickness of 53 mm can be considered an intermediate target. The square armor has a width of 300 mm, which is about 24 times the diameter of the A–P core. Thus, the cylindrical cavity expansion can be used to predict the ballistic limit of the A–P core.
Figure 13 shows the dimensions of the A–P core. The caliber–radius–head (CRH) is 3.05, which is also denoted as
ψ.
Coefficient
k1 is expressed as
The radial stress
σr at the cavity surface versus cavity expansion velocity
V is given by [
34]
where
σs is the quasi–static radial stress required to open the cylindrical cavity,
ρt is the density of the target, and
B is a dimensionless constant.
σs,
b, and
B are obtained from [
23]
where
Y is, the yield stress and
ν is Poisson’s ratio of the target.
α and
γ are given by
Furthermore, a rigid ogive–nosed projectile, with the impact velocity of
vi, the ballistic limit of
vbl and the residual velocity
vr, is given by
where
C is a small parameter related to the target inertia. When target inertia is neglected, the ballistic limit of
vbl and the residual velocity
vr can be simplified as [
23,
25,
35] as
where the residual velocity
vr is the same as the Lambert–Jones model in Equation (13).
Based on the constitutive models of the target materials, the quasi–static radial stress
σs can be expressed as [
36]
where
E and
H are Young’s modulus and the constant tangent modulus in the plastic region if the stress versus strain curve of the target can be expressed as
Thus, the value of
σs for the Q235 face sheets can be calculated. For UHMWPE laminates, there may not be a mature model to predict the quasi–static radial stress required to open the cylindrical cavity, but the range of the
σs can be estimated from the empirical formula [
37,
38] below,
When the coefficient is set as the minimum value of 1.33, the value at a relatively low level can be obtained, as listed in
Table 10.
For the composite armor composed of Q235 face sheets and UHMWPE laminates, the effective value of σs can range from 2.76 GPa to 4.26 GPa. When the value of effective σs is set as 3.08 GPa, the ballistic limit of the composite armor calculated from Equation (23) is 467 m/s, which is consistent with the value obtained from the numerical simulation results.
In conclusion, the principle of energy conservation and the Lambert–Jonas model can be applied to calculate the work performed during the perforation and the residual velocities of the A–P core after perforation. In addition, the quasi–static radial stress σs required to open the cylindrical cavity can be estimated from the cavity–expansion model. With the value of 3.08 GPa, the predicted ballistic limit is consistent with the numerical simulation results.