Sage Journals: Discover world-class research

Abstract

High-performance ultra-high molecular weight polyethylene (UHMWPE) soft ballistic sub-laminates ([0/90]_n, SBSL) are stacked to build a soft body armor pack (SBAP) that can defeat handgun projectiles. Transverse impact on single-layer [0/90] SBSL of different size is modeled with shell elements and is solved using LS-DYNA composite material model MAT54. The finite element (FE) model is validated using 1D and 2D theories for transverse impact. The validated FE models are then used to study the perforation behavior of a [0/90] SK76/PU SBSL under constant and variable velocity impact. Results show that the basal shape of the transverse deformation cone has a diamond shape; the cone wave speed along primary material direction agrees well with 2D membrane theory, there exists a minimum perforation velocity below which the SBSL will not perforate, the peak perforation force reduces with the size of the SBSL, and the work of perforation decreases with increasing speed. Detail perforation mechanics of [0/90] SK76/PU SBSL is presented for the first time.

Keywords

UHMWPE transverse impact FE modeling perforation mechanics LS-DYNA MAT54

Introduction

Ultra-high molecular weight polyethylene (UHMWPE) fibers, for example, Dyneema^® and Spectra^® fibers are used with compliant polyurethane (PU) matrix material to make [0/90] soft ballistic sub-laminates (SBSLs) for soft and hard body armor applications.^1
–3 Dyneema-Shield and Spectra-Shield are the commercial SBSLs used in soft ballistic armor applications^4
–6 for protection against handgun threats. Multiple SBSLs or shields are stacked in a soft ballistic armor pack (SBAP) that provides higher ballistic protection per unit areal weight. SBSL/shields can also be consolidated in a hot press under compression to fabricate hard ballistic body armor.^7,8 In this article, we will discuss the transverse impact on a single-layer [0/90] Dyneema SK76/PU SBSL under constant and variable velocity projectile impact in order to understand the mechanics of SBSL deformation and associated energy dissipation mechanisms.

During the transverse impact of a projectile onto a SBAP, failure of the first few sub-laminates occurs due to high stress concentration under and around the projectile followed by reduction in projectile kinetic energy and transfer of projectile momentum to the sub-laminates. During the deformation process, a dynamic cone is developed in the SBAP subjecting all the sub-laminates in the pack under membrane tension. At a certain point of time, failure of the sub-laminates stops and the dynamic cone of the remaining sub-laminates continues to grow. The sub-laminates stretch under tension providing additional resistance forces to decelerate the projectile, the projectile kinetic energy is transferred to the remaining sub-laminates, the total momentum is conserved, and the projectile is finally arrested.⁹

This description of projectile defeat mechanism is qualitative and the detail mechanics is generally not well understood. The goal of this research is to study the perforation mechanics of a single [0/90] Dyneema SK76/PU SBSL that can be extended to the study of transverse impact on multilayer SBAP. In achieving the goal, we will first discuss the available material properties of UHMWPE fibers and composites, and then we will briefly review the 1D and 2D theories of constant velocity transverse impact on fibers and membranes, and finally conduct the finite element (FE) modeling of constant velocity transverse impact on a single-layer SBSL to validate the FE model. In the FE analysis section, we will develop a FE model of an isotropic membrane and validate the model with Phoenix and Porwal (P&P)⁹ membrane theory for isotropic materials under constant velocity transverse impact without any boundary effects. We will then extend the validated FE model for isotropic membrane to [0/90] UHMWPE sub-laminate under constant velocity transverse impact to show the difference between an isotropic membrane and an orthotropic thin sub-laminate in a non-perforating scenario. The FE model of the [0/90] UHMWPE sub-laminate will then be extended to study the effect of laminate stacking sequence under constant velocity transverse impact. In reality, the projectile’s rigid body velocity (RBV) is decreased by the momentum transfer to the target, a theory of which does not exist in literature. In this article, we will use our validated FE model of [0/90] UHMWPE sub-laminate to study the perforation mechanics of infinite and finite size sub-laminates under variable velocity transverse impact conditions for the first time.

Properties of UHMWPE fibers, unidirectional laminas, and [0/90] sub-laminates

UHMWPE fibers are transversely isotropic in nature, however, the commercial manufactures only provide tensile modulus, strength, and tensile failure strains of these fibers (Table 1A).^3,10
–12 Recently, transversely orthotropic properties of some selected UHMWPE fibers (Table 1B) are available in literature,^12

–15 and we have calculated the transversely isotropic properties of unidirectional (UD; Table 1D) laminas using continuous fiber micromechanics. Limited experimental data on the properties of [0/90] sub-laminates are available in literature (Table 1E).^13,16 In the present analysis, we will use the properties of unidirectional Dyneema SK76/PU lamina presented in Table 1.

Table 1.

Transversely isotropic properties of unidirectional Dyneema SK76/PU SBSL (Table 1D; Levi-Sasson et al.).

Density, (g/cm³)	Fiver volume fraction (v_f)	a_0,11(m/s)	$X_{1}^{T}$ (MPa)	$ε_{1}^{T}$
0.970	0.80	9781.85	2880	0.03103
E₁₁(GPa)	E₂₂(GPa)	E₃₃(GPa)	G₁₂(GPa)	G₂₃(GPa)	G₃₁(GPa)
92.814	0.507	0.507	0.188	0.169	0.188
ν ₁₂	ν ₁₃	ν ₂₃	ν ₂₁	ν ₃₁	ν ₃₂
0.303	0.303	0.4993	0.0017	0.0017	0.4993

SBSL: soft ballistic sub-laminate; PU: polyurethane.

Theory of constant velocity transverse impact on isotropic elastic 1D fiber and 2D membrane

Theories of constant velocity transverse impact on 1D fiber had been developed between 1945 to 1960 by Rakhmatulin and Smith et al.^{17

–24} Following these classic theories, P&P⁹ has developed an axisymmetric membrane theory for constant velocity transverse impact on an isotropic elastic membrane with a cylindrical projectile (Figure 1). This article provides a good literature review on related past works on this subject including the related works on transverse impact on isotropic elastic 1D fiber.^{17

–24} In the membrane theory, 2D equations are often reduced to 1D equations in P&P⁹ to compare with the 1D fiber theory (Smith et al.^21,22). In the following sections, the classic 1D fiber theory is discussed first, followed by the 2D membrane theory. The presented derivations of key quantities and equations herein will be compared later with the FE calculations.

Figure 1.

Nomenclature for Phoenix and Porwal⁹ 2D membrane theory for isotropic materials.

Constant velocity transverse impact on isotropic elastic 1D fiber (Smith et al)

The classic work of Rakhmatulin^17

–20 and Smith et al.^21,22 on the transverse impact on isotropic elastic 1D fiber considers the velocity of the projectile to be constant. Upon the transverse impact of a projectile on the 1D fiber at a constant velocity V_P, an implosion or axial tensile strain wave of amplitude ${\tilde{ε}}_{t} = ε_{0}$ (where, the subscript “0” is used to express the 1D quantities) propagates along the fiber (in region “k − s”) with the wave-front (located at point “s”) velocity $a_{0} = \sqrt{E / ρ}$ . A transverse deformation cone develops with a constant downward particle velocity $ẇ = - V_{P}$ at each point between points “p” and “k” while the point of discontinuity k (also known as “kink”) propagates along the fiber axis with the “cone wave” or “kink wave” velocity ${\bar{c}}_{k}$ in the laboratory frame of coordinates. The shape of the deformed fiber between points p and k is assumed to be straight and the cone angle γ between line segment “p − k” and the radial axis is constant. In this classic 1D problem, the two unknowns $ε_{0} (\tilde{r})$ and ${\bar{c}}_{k, 0} (r_{k})$ are constant for all values of $\tilde{r}$ and r_k, respectively. The implosion strain ε₀ can be expressed in terms of V_P and a₀ as follows (Smith et al.,²² equation (8)):

V_{P} = a_{0} \sqrt{ε_{0} (1 + ε_{0}) - {[\sqrt{ε_{0} (1 + ε_{0})} - ε_{0}]}^{2}}

The 1D cone/kink wave velocity in laboratory frame of coordinates ( ${\bar{c}}_{k, 0}$ ) can be expressed as (Smith et al.,²² equation (9)):

{\bar{c}}_{k, 0} = a_{0} [\sqrt{ε_{0} (1 + ε_{0})} - ε_{0}]

And in material coordinate (c_k,0) as (Smith et al.,²² equation (7)):

\frac{c_{k, 0}}{a_{0}} = \sqrt{\frac{ε_{0}}{1 + ε_{0}}} = α_{0}

Equation (1) can be simplified to determine the approximate value of ε₀ as follows (Rakhmatulin,¹⁷ Ringleb,²⁴ and P&P,⁹ equations (129) to (133)):

ε_{0} \approx {(\frac{V_{P}}{a_{0} \sqrt{2}})}^{\frac{4}{3}}

Once the value of of ε₀ as a function of impact velocity (V_P) and axial wave speed (a₀) is known, then equations (2) and (3) can be evaluated. The 1D cone angle γ can then be calculated from P&P,⁹ equation (144).

\sin γ = \frac{V_{P}}{a_{0} \sqrt{ε_{0} (1 + ε_{0})}}

Equations (1) to (5) have been used to determine the ballistic behavior of high-performance fibers (e.g., Aramid, UHMWPE, Twaron, Zylon, Vectran, M5, etc.) by experimentally determining the fracture strain (∊_f) of a fiber type and then assigning ε₀ = ∊_f in determining the critical impact speed (equation (1)), cone wave velocity (equations (2) and (3)), and the cone angle (equation (5)). These equations however cannot be used for SBAP and hard ballistic armor pack and were the main motivations for the P&P 2D membrane theory and are described next.

Constant velocity transverse impact on isotropic elastic membrane (P&P)

Motivated from the classic work of Rakhmatulin and Smith, P&P⁹ have considered the transverse impact on an axisymmetric isotropic elastic membrane under constant impact velocity, developed necessary governing differential equations, and provided many theoretical solutions based on logical approximations. Like the 1D fiber impact model, the unknown quantities of interest are the tensile strain distribution ${\tilde{ε}}_{t} (\tilde{r})$ in the implosion/axial wave region and the same ε_t(r) in the cone wave region, the cone/kink wave velocity ${\bar{c}}_{k} (r_{k})$ with respect to laboratory fame of coordinate, the deformed shape of the cone region w(r,t), and the distribution of particle velocity $ẇ (r, t)$ in the cone region. With rigorous mathematical analysis and simplified assumptions, the membrane strain in the implosion/axial wave region is found to be (P&P,⁹ equations (46) to (55)):

{\tilde{ε}}_{t} (\tilde{r}) = \frac{\partial \tilde{u}}{\partial \tilde{r}} = \frac{μ}{(\tilde{r} / r_{s})} r_{k} < \tilde{r} < r_{s}

Where μ = ε₀ for 1D model. P&P has not provided an exact solution for implosion strain in their manuscript.⁹ However, the membrane strain in the cone wave and implosion/axial wave regions is expressed together by one simple equation (P&P,⁹ equations (46), (80), and (139)):

ε_{t} (r) \approx ε_{t} (r_{k}) \frac{r_{k}}{r} \approx α^{2} \frac{r_{k}}{r} r_{p} < r < r_{s}

Where α = c_k/a₀. The cone/kink wave velocity in laboratory frame of coordinates ( ${\bar{c}}_{k}$ ) can be expressed as (P&P,⁹ equation (142), misprint in equation (142) in P&P⁹ has been corrected in the present equation (8)):

{\bar{c}}_{k} = c_{k} + {\dot{u}}_{k} = a_{0} {α + \frac{α^{2} [ln α - ln {1 - \frac{r_{p}}{r_{k}} (1 - α)} - 1]}{1 + \frac{(1 - α) (r_{k} - r_{p})}{α r_{k}}}}

where c_k is the kink wave velocity in materials coordinate and ${\dot{u}}_{k} = {\dot{\tilde{u}}}_{k}$ is the particle velocity at the kink point k. The value of α can be determined iteratively using the following formula (P&P,⁹ equations (135) to (138)):

α_{n + 1} \approx α_{0} {(\frac{r_{p}}{r_{k}})}^{\frac{1}{3}} \frac{{\frac{1}{α_{n}} (\frac{r_{k}}{r_{p}} - 1)}^{\frac{1}{3}}}{{\ln [1 + \frac{1}{α_{n}} (\frac{r_{k}}{r_{p}} - 1)]}^{\frac{1}{3}}}

Where n = 0, 1, 2, … and $α_{0} = c_{k, 0} / a_{0} \approx \sqrt{ε_{0}} \approx {(V_{P} / a_{0} \sqrt{2})}^{2 / 3}$ . P&P⁹ showed that only two iterations provide quite accurate result, that is, at α ≈ α₂. Once the value of α can be determined using equations (9), (3), and (4), then equations (7) and (8) can be evaluated. Equation (8) has been further reduced for 1D case in P&P,⁹ equation (143):

{\bar{c}}_{k, 0} \approx c_{k, 0} + {\dot{u}}_{k, 0} = α_{0} a_{0} (1 - α_{0} + α_{0}^{2})

Equation (10) provides a slightly smaller value of ${\bar{c}}_{k, 0}$ as compared to equation (2). P&P⁹ showed that the cone/kink wave velocity for the 2D membrane, c_k, is equal to c_k,0 at r_k = r_p, increases to a maximum at around r_k = 2r_p, and then reduces to a plateau value (higher than c_k,0) for r_k ≫ 2r_p for a constant value of ε₀ (Figure 3, P&P⁹). It is also important to mention that the 2D membrane cone/kink wave speed increases with the increase in ε₀, that is, with impact speed V_P. We will use these 1D and 2D equations to validate the FE models of transverse impact on 1D ribbons and 2D membranes.

Figure 2.

Geometric dimensions of the UHMWPE sub-laminate and coordinate definitions.

Figure 3.

Constant velocity transverse impact on an isotropic 1D ribbon. L = 1000 mm, V_P = 150 m/s, ε₀ = 0.00381, [a₀]_1D = 6916.81 m/s, [a₀]_FEA = 6947.09 m/s, γ_1D = 20.53°, and γ_FEA = 20.50°. (a) Axial strain $(\tilde{ε_{t}} = ε_{X})$ in the implosion wave region at different time; (b) shape of the cone wave region at different time.

FE modeling

In the 1D and 2D theories, radial direction extends to infinity; however, in FE analysis (FEA), we will consider a 2D square membrane of large in-plane dimensions (e.g., lenght (L) = width (W) = 1000 mm) and terminate the computation before the implosion wave reaches the boundary of the membrane to mimic infinite boundary conditions.

FE model of a 2D membrane/shell

A single [0/90] SBSL of L = 1000 mm and W = 1000 mm is defined in the XY plane with the center of the sub-laminate located at the origin. Total thickness of all sub-laminates analyzed is taken as h = 0.127 mm. The sub-laminate is discretized with uniform shell elements (type 16) of dimensions l_E = w_E. A single-element study has been conducted for the fully integrated layered shell element type 16 with different hourglass types (1, 2, 4, 6, and 7) to determine the hourglass type which produces minimum hourglass energy. Hourglass type 7, which is the linear total strain form of Belytschko-Bindeman (1993) assumed strain co-rotational stiffness form for 2D shell and 3D solid elements, is found to generate minimum hourglass energy and is used in the present analysis. A mesh sensitivity analysis is performed with four mesh sizes, that is, l_E = w_E = 0.50, 0.75, 1.00 and 2.00 mm for the sub-laminate with dimensions L = W = 360 mm. Results presented in Appendix 2 show that a mesh size of dimension l_E = w_E = 1.00 mm provides comparable solution to 0.50 mm mesh, and thus all subsequent analyses are performed using 1.00 mm mesh size. Since the SBSL is composed of two UD laminas in a [0/90] stacking sequence, the *PART_COMPOSITE option of LS-DYNA has been used to define the thickness (h/2) and material angles for both the through thickness [0] and [90] laminas. Properties of the transversely isotropic UD soft ballistic lamina made from Dyneema SK76 fiber and PU matrix material is used as the input to the *MAT_ENHANCED_COMPOSITE_DAMAGE model (MAT54) for both the [0] and [90] laminas of the [0/90] SBSL and is presented in Table 1. A right circular cylinder projectile of diameter D_P = 12.7 mm is used in all analyses, the mass is kept constant, and the material is considered rigid.

No boundary condition is imposed on the single-layer [0/90] SBSL model (Figure 2). The segment-based (SOFT = 2) eroding single surface contact definition with warped segment checking and sliding option (SBOPT = 5) is used between the RCC projectile and the SBSL. Considering the friction between the [0/90] SBSL and the projectile to be very low, the static and dynamic contact friction coefficients are taken as FS = 0.05 and FD = 0.03, respectively. In order to ensure accuracy of the computations, contact is searched at every time step while the time step is automatically calculated in LS-DYNA, and a time step scale factor of 0.95 is used. Since the shear modulus of the PU matrix is several order of magnitude lower than the axial modulus of the fiber, only tensile failure in the fiber direction of the UD lamina is considered. The axial tensile strength of the UD lamina ( $X_{1}^{T}$ ) is estimated by multiplying the strength of the SK76 fiber with the fiber volume fraction (v_f) of the UD lamina. Accordingly, the axial tensile failure strain ( $ε_{1}^{T}$ ) is calculated by dividing the estimated axial strength of the UD lamina with its axial modulus (Table 1).

FEM of 1D ribbon

One row of elements (1 mm mesh) along the geometric X axis of dimensions $L \times w_{E} \times h = 1000 \times 1 \times 0.127 {mm}^{3}$ have been used to perform 1D transverse impact on a ribbon to validate the FE models with Smith et al.^21,22 equations.

Transverse impact on isotropic ribbon and membrane

In order to validate the suitability of the present FE analyses, the FE models of 1D ribbon and 2D membrane/shell presented in Figure 2 are considered as a single-isotropic layer of membrane elements and assigned with isotropic material properties equivalent to an equal thickness [0/90] SBSL, that is, $E = {[E_{11}]}_{UD} / 2 = 46.407 GPa$ , ν = 0.00, ρ = 0.970 g/cc. The simulation results are compared with the 1D Smith equations and 2D membrane theory P&P.⁹ The 1D axial implosion wave velocity of an isotropic material with these properties is calculated to be ${[a_{0}]}_{1 D} = \sqrt{E / ρ} = 6916.81 m / s$ .

The characteristic time (τ_L) of an isotropic 1D ribbon or 2D membrane of length L is defined by the time taken by the implosion wave to propagate from the impact center at x = 0, to the edge of the ribbon at x = L/2 and reflecting back from the edge to the impact center, that is:

τ_{L} = \frac{L}{a_{0}}

In order to exclude reflections from free boundaries, computations need to be terminated at time $t = τ_{L} / 2$ . In order to study the perforation behavior of a [0/90] SBSL, the perforation should occur at time t < τ_L to ensure that the first wave reflection does not reach the projectile and contribute on the perforation resistance force (F_P). This concept of characteristic time will be further discussed in the results and discussion section.

Results and discussion

Constant velocity transverse impact on 1D isotropic ribbon

Smith equations consider a constant velocity transverse impact on an isotropic fiber (infinitely long flexible filament²¹). In the present analyses, a constant impact velocity of V_P = 150 m/s on an isotropic ribbon of L = 1000 mm is simulated. The characteristic time of the 1D ribbon is calculated to be τ₁₀₀₀ = 144.58 μs. To avoid the effect of wave reflections from the edge/boundary, computations are terminated at time t = 70 μs. The axial strain ( ${\tilde{ε}}_{t} = ε_{X}$ ) in the implosion wave region of the 1D ribbon at different time is presented in Figure 3(a). The 1D implosion strain (ε₀) given by Smith’s equation (1) is estimated using equation (4) and is found to be ε₀ = 0.00381.

The axial strain predicted from the simulation matches well with the Smith’s equation. The implosion wave front predicted by the simulation has a finite thickness (≈ 23 mm); however, in the 1D theory, the implosion wave front is a discontinuity and is the fundamental difference between the FEA and 1D theory. The average implosion wave-front velocity from the FEA is calculated by measuring the front location x at $ε_{X} = 0.002$ and dividing by the time in the time range $t = 10 \sim 70 μ s$ at an interval of $5 μ s$ and is found to be ${[a_{0}]}_{FEA} = 6947.09 m / s$ , which is 0.44% higher than the 1D theory. The shape of the cone wave at 5 μs time interval is presented in Figure 3(b), which shows that the cone angle at the projectile root (p) is almost constant and the average value (in the time range $t = 15 \sim 70 μ s$ ) is calculated to be $γ_{FEA} = 20.50 °$ which is almost equal to the 1D cone angle given by equation (5) $γ_{1 D} = 20.53 °$ . These results validate that the choice of 1-mm mesh is good enough to capture the 1D Smith theoretical results using the present FEA except the fact that the axial wave fronts have a finite thickness instead of a theoretical discontinuity.

Constant velocity transverse impact on linear elastic 2D isotropic membrane

Using the same isotropic properties as in the case of the 1D ribbon ( $E = 46.407 GPa$ , ν = 0.00, $ρ = 0.970 gm / cc$ ), constant velocity transverse impact on isotropic 2D membrane of dimension L = W = 1000 mm is conducted. FEA results of 2D isotropic membrane are compared with the membrane theory P&P,⁹ which ignores the effect of Poisson’s ratio. Properties of the UD SK76/PU SBSL presented in Table 1 is used to simulate the constant velocity transverse impact on a [0/90] SK76/PU SBSL without failure. Similar to the 1D ribbon case, the constant impact velocity is chosen to be V_P = 150 m/s for both the isotropic membrane and the [0/90] SK76/PU SBSL.

The shape of the cone wave region for the isotropic membrane and [0/90] SBSL is presented in Figure 4 for the time range t = 5 ∼ 70 μs at an interval of 5 μs. Clearly the cone wave shapes are membrane like while the cone angles at the projectile root (p) and at the kink location (k) are different. While the shape of the cone wave region for isotropic membrane is convex (outward), the shape for the [0/90] SBSL is concave (inward) with several nonlinear segments in the profile (one near the projectile root p and the other near the kink k is clearly distinguishable). While the kink location at time t = 70 μs for isotropic membrane is close to x = 50 mm, the same for the [0/90] SBSL is at x = 40 mm; which tells us that the kink wave velocity of the [0/90] SBSL is less than the isotropic membrane. Figure 5 shows the shape of the cone wave in the basal plane (XY Plane) at time t = 70 μs. The basal shape of the cone wave is circular for the 2D isotropic membrane, while the 2D [0/90] SBSL is diamond shaped (attributed to the orthotropic properties of the SBSL).

Figure 4.

z-Displacement along X-axis on Y = 0 plane. Constant velocity transverse impact on 2-D isotropic membrane and [0/90] SK76/PU SBSL. L = W = 1000 mm, V_P = 150 m/s. (a) 2D isotropic membrane; (b) 2D [0/90] SK76/PU SBSL shell. SBSL: soft ballistic sub-laminate; PU: polyurethane.

Figure 5.

Basal shape of the cone wave at time t = 70 μs under constant velocity (V_P = 150 m/s) transverse impact. L = W = 1000 mm. Snip size: 200 × 200 mm². Contours of z-displacements: red, 0.0 mm and blue, 10.5 mm. The waviness in the circular basal cone shape may be related to the finite mesh size (1 × 1 mm²) which is not further studied for the L = W = 1000 mm isotropic membrane. (a) 2D isotropic membrane; (b) 2D [0/90] SK76/PU SBSL shell.

The cone wave velocities with respect to laboratory frame of coordinate ( ${\bar{c}}_{k}$ ) as a function of the kink location (r_k) obtained from the theory (equation (8)) and FEA are presented in Figure 6(a). P&P⁹ showed that the kink wave velocity for an isotropic membrane (solid blue line) is equal to the constant 1D kink wave velocity ${\bar{c}}_{k, 0} = 401.4 m / s$ (equation (2), dotted red line) at the time of impact (at r_k = r_p) and increases to a peak value at about r_k/r_p = 2 given by equations (8) and (9). FEA of the 2D isotropic membrane shows that the predicted kink wave velocity is higher than the P&P membrane theory and the plateau value at r_k ≈ 48 mm (r_k/r_p = 7.5) is about 10% higher. Kink wave velocity predicted from the FEA of 1D ribbon (presented earlier) has a rising behavior and approaches the 1D theoretical value. The kink wave velocity predicted from the FEA of [0/90] SBSL is significantly lower than the isotropic membrane but higher than the 1D theory, which is related to the orthotropy-induced basal cone wave shape. Figure 6(a) provides a quantitative comparison of kink wave velocities between theories and FEA under constant velocity impact, and using these values as the baseline, the FE model of [0/90] SK76/PU SBSL will further be used for the perforation studies of the SBSL.

Figure 6.

Constant velocity transverse impact on 2D isotropic membrane and [0/90] SK76/PU SBSL: Comparison between theory and FEA. L = W = 1000 mm, V_P = 150 m/s. (a) Cone wave velocity $(\bar{c_{k}})$ ; (b) membrane strain $(ε_{t})$ . SBSL: soft ballistic sub-laminate; PU: polyurethane.

The membrane strain along X-axis in the cone wave and implosion wave region is calculated using Lagrangian description.

ε_{t} (t) = \frac{l_{E} (t) - l_{E, 0}}{l_{E, 0}}

Where $l_{E} (t)$ is the length of a membrane element at any time t and $l_{E, 0}$ is the same at time t = 0. Figure 6(b) shows the 2D theoretical membrane strain (equation (7)) by the solid green line at time t = 70 μs. Figure 6(b) also shows the location of the projectile edge at x = r_p, and the location of the cone wave (point k) x = r_k at time t = 70 μs. Clearly, the membrane strain $ε_{t} (r_{k})$ at the kink point (k) is higher than the 1D strain ε₀ and increases asymptotically at the root of the projectile, that is, at x = r_p. FEA of the isotropic 2D membrane shows qualitatively similar membrane strain as the theory with higher strain at the kink point (k) and shows that the present 1 mm mesh size can capture the theoretical shape and attributes of the membrane strain. The [0/90] SK76/PU SBSL shows a very different membrane strain and invalidates the isotropic membrane theory for the analyses of [0/90] SBSL under transverse impact.

Constant velocity transverse impact on 1D isotropic fiber assumes constant particle velocity in the cone wave region; however, the 1D FEA shows (Figure 7(a)) that the particle velocity is oscillating in nature indicating bending wave propagation in the cone wave region. Similar bending wave propagation in the cone wave region of the 2D isotropic membrane is also evident in Figure 7(b). Presence of this high frequency bending wave in the cone region may be responsible for the nonuniform cone angles presented in the cone wave profiles in Figure 4(a) for isotropic membrane and in Figure 4(b) for [0/90] SBSL. Note that the 1 mm mesh size used in the present analysis is sufficient to capture the spatial distribution of high frequency bending waves as outlined in chapter 4 of Graff’s book²⁵ titled “Waves in membranes, thin plates, and shells.” No global damping has been used to suppress these oscillations, because of the underlying physics described by Graff.²⁵

Figure 7.

z-Velocity distribution of the cone wave at two different times. L = W = 1000 mm, Constant velocity transverse impact, V_P = 150 m/s. (a) 1D isotropic ribbon; (b) 2D isotropic membrane.

FEA of constant velocity transverse impact on isotropic 1D ribbon and 2D membrane provided important insight on the mechanics of implosion wave and cone wave propagation, on the spatial distribution of the membrane strain at different times, and on the particle velocities in the cone wave region. The FE results correlated well with the 1D and 2D theories with limitations of neglecting bending wave propagation. It has also been shown that the 2D membrane theory is not applicable to the [0/90] orthotropic SBSL. Further FEA with failure of [0/90] SBSL under constant velocity impact has been conducted to understand the perforation mechanics of a single-layer [0/90] SBSL shell and is presented next.

Constant velocity transverse impact on 2D [0/90] SK76/PU SBSL with failure

Constant velocity transverse impacts on [0/90] SK76/PU SBSL (L = W = 1000 mm) have been conducted in the impact velocity range $100 m / s < V_{P} < 175 m / s$ with failure and element erosion to model the perforation behavior. As defined by equation (11), the characteristic time for this 1000 mm [0/90] SBSL is calculated to be $τ_{1000} = 144.58 μ s$ assuming the implosion wave velocity to be equal to ${[a_{0}]}_{1 D} = 6916.81 m / s$ . Following Table 1, failure of a [0] layer in a [0/90] SBSL will occur if the tensile strain of the layer exceeds the failure strain, that is, $ε_{[0]} \geq ε_{1}^{T} = 0.031$ . Similarly, the [90] layer will fail if $ε_{[90]} \geq ε_{1}^{T} = 0.031$ . An element (shell) will erode if both the [0] and [90] layers of the layered shell fail. Time history of the F_P on the constant velocity projectiles are presented in Figure 8(a) for all impact velocities studied. For the impact velocity case V_P = 150 m/s, a drop in F_P at t = τ₁₀₀₀ signifies that the tensile implosion wave is reflected back from the free edge of the membrane and reached the impact center reducing the resistant force. In the impact velocity range, $155 m / s < V_{P} < 200 m / s$ , perforation occurs in the time range $τ_{1000} / 2 < t < τ_{1000}$ while the peak resistance force or perforation force is about 1927 N.

Figure 8.

Perforation of a [0/90] SK76/PU SBSL under constant velocity transverse impact. L = W = 1000 mm, $τ_{1000} = 144.58 μ s$ . (a) Force–time; (b) force–displacement. SBSL: soft ballistic sub-laminate; PU: polyurethane.

From these observations, it is obvious that there exist a minimum impact velocity at which perforation will occur at t = τ₁₀₀₀ when the implosion wave reaches the boundary of the membrane, reflects, and travels back to the impact center. We will define this impact velocity as the minimum perforation velocity, $V_{Perf}^{min}$ , under constant velocity transverse impact on a membrane. The minimum perforation velocity $V_{Perf}^{min}$ for the 1000 mm [0/90] SBSL membrane is found to be approximately equal to 155 m/s. Because the definition of $V_{Perf}^{min}$ is tied to characteristic time τ_L, the minimum perforation velocity may increase with the decrease in characteristic length L.

Figure 8(b) shows the F_P versus projectile displacement under constant velocity transverse impact cases presented in Figure 8(a). Force–displacement plot up to a projectile displacement z ≈ r_p is somehow oscillatory in nature which is attributed to the loss and reestablishment of contacts between the membrane and the projectile in the early stage of cone wave formation. Stable growth of the cone wave and continuous contact beyond z ≈ r_p yields almost a linear force–displacement response till the perforation of the [0/90] SBSL membrane at the peak force $F_{P}^{max}$ and at corresponding displacement $z_{P}^{max}$ . We define the structural membrane stiffness (K_P) by the slope of the linear force–displacement plot in the displacement range $r_{p} < z < z_{P}^{\max}$ .

K_{P} = \frac{d}{d z} F_{P} (z) r_{p} < z < z_{P}^{\max}

The membrane stiffness defined by equation (13) increases with impact velocity, and for impact velocities at 155 m/s and 200 m/s are calculated to be approximately 71 and 95 N/mm, respectively, indicating the fact that structural stiffness of the membrane increases with the increase of impact velocity due to dynamic effects (the mechanics is not formulated here). The area under the force–displacement curve represents the work of perforation, W_P, and is listed for each impact velocity in the legend.

W_{P} (z) = \int_{0}^{z} F_{P} (z) d z

Work of perforation decreases with increase in impact velocity; however, the detail mechanics is not known/ formulated. Time sequence of perforation of the SBSL at V_P = 150 m/s is presented in Figure 9 along with the perforation force time history. A sharp drop in the force time history represents the initiation of failure in the [0/90] SBSL and is found to occur at time t ≈ 142 μs, signifying the initiation of the failure process. Load drops continuously revealing the first element erosion (mimicking fiber tensile fracture) between 150 μs and 160 μs and most probably at around t ≈ 156 μs. For time greater than 160 μs, Figure 9(d) to (f), dynamic propagation of the fiber fracture continues yielding complete perforation (CP) of the SBSL at time t = 200 μs, where the resistance force drops to zero.

Figure 9.

Perforation behavior of a single-layer [0/90] SK-76/PU SBSL under constant velocity impact at V_P = 150 m/s. L = W = 1000 mm, $τ_{1000} = 144.58 μ s$ . Snipped picture size $250 \times 250 {mm}^{2}$ . (a) Force–time; (b) t = 150 μs; (c) t = 160 μs; (d) t = 170 μs; (e) t = 180 μs; and (f) t = 190 μs. SBSL: soft ballistic sub-laminate; PU: polyurethane.

Constant velocity transverse impact on 2D SK76/PU SBSL with different stacking sequence

The cone wave of a [0/90] SBSL is found to have a diamond shape in the basal plane (Figure 5(b)) as compared to the isotropic circular shape (Figure 5(a)). Also, the z-displacement profile of the [0/90] SBSL is significantly different than the isotropic membrane (Figure 4). The difference in cone shape at the basal plane and the z-displacement profile is thought to be the main reason of lower kink/cone wave velocity of the [0/90] SBSL (Figure 6(a)) than the isotropic membrane. In order to understand the effect on stacking sequence of the SBSL, we have modified the composite laminate architecture from two to four layers while keeping the total SBSL thickness constant, that is, h = 0.127 mm. Four SBSL stacking sequences have been studied under a constant velocity transverse impact of V_P = 150 m/s, and they are (a) $[0^{°} / 0^{°} / 90^{°} / 90^{°}]$ , (b) $[0^{°} / + 45^{°} / - 45^{°} / 90^{°}]$ , (c) $[+ 45^{°} / + 45^{°} / - 45^{°} / - 45^{°}]$ , and (d) $[+ {22.5}^{°} / + {22.5}^{°} / - {22.5}^{°} / - {22.5}^{°}]$ .

Figure 10 shows the shape of the cone wave at the basal plane of these SBSLs at time t = 70 μs. z-Displacements along X- and Y-axes are plotted (Figures 3A to 3D, Appendix 3) to determine the kink/cone wave velocities and the cone angles for different SBSL architecture and are presented in Figure 11(a) and (b), respectively. From the FE simulation results, insignificant differences in quantities along X- and Y-directions have been observed for all laminates (except $[\pm 22.5 °]$ ) for which average quantities are presented in Figure 11. Figure 11(a) shows that the highest cone/kink wave velocity is obtained for isotropic membrane followed by $[\pm 22.5 ° : X]$ along X-direction (→, Figure 10(d)), quasi-isotropic $[0 / + 45 / - 45 / 90]$ (Figure 10(b)), $[0 / 90]$ (Figure 10(a)), 1D theory $[\pm 45 °]$ (Figure 10(c)), and $[\pm 22.5 ° : Y]$ along Y-direction (↑, Figure 10(d)). Cone wave velocities above 1D theory ( ${\bar{c}}_{k, 0} = 401.4 m / s$ ) have an exponential trend starting from the 1D theory at time t = 0 and rises to a plateau value for time t > 70 μs. On the other hand, kink/cone wave velocities for $[\pm 45 °]$ (Figure 10(c)) and $[\pm 22.5 ° : Y]$ (Figure 10(d)) are much smaller than the 1D theory and can be assumed constant over time. From the 1D and 2D theories, we know that the cone wave velocity increases with impact velocity and can be found in P&P’s study⁹ for isotropic membranes. Study of the dependence between impact velocity with the cone wave speed for [0/90] SBSL is out of scope and will be presented somewhere else. Figure 11(b) shows the cone angle at the root of the projectile for all laminate stacking sequence studied which varies over a wide range. It is obvious that for lower cone/kink wave speed, the cone angle will be larger and vice versa. For SBSL applications, one might prefer higher cone/kink wave velocities along all directions for quicker momentum transport and might chose the quasi-isotropic sequence (lowest cone angle) over cross-ply [0/90] sequence. However, from ease of fabrication perspective, [0/90] stacking sequence is the choice of commercial SBSLs.

Figure 10.

Basal shape of the cone wave at time t = 70 μs under constant velocity (V_P = 150 m/s) transverse impact for different laminate stackings. L = W = 1000 mm. Snip size: 120 × 120 mm². Horizontal (→) direction is the X-direction and vertical (↑) direction is the Y-direction. (a) [0°/0°/90°/90°]; (b) [0°/+45°/−45°/90°]; (c) [+45°/+45°/−45°/−45°]; and (d) [+22.5°/+22.5°/−22.5°/−22.5°].

Figure 11.

Average cone/kink wave velocity and cone angle along X-direction (→) as a function of time for different laminate architecture. Constant velocity (V_P = 150 m/s). Transverse impact, L = W = 1000 mm. (a) Cone/kink wave velocity; (b) cone angle.

Variable velocity transverse impact on 2D [0/90] SK76/PU SBSL with failure

As mentioned in the introduction, projectile velocity continuously changes during the impact on a target and the cone wave velocity may also continuously change with the changing velocity of the projectile. A theoretical model for projectile with continuously changing velocity impacting a membrane does not exist in literature, and FEA is thus essential. With the fundamental understanding of constant velocity transverse impact on the perforation mechanics of a [0/90] SBSL, we will now present the variable velocity impact on the [0/90] SBSL of dimension L = W = 1000 mm and then extend the study to SBSL of finite dimensions, that is, L = W = 500, 360, 250 mm. In case of variable velocity transverse impact, we will denote the initial impact velocity at time t = 0 by V_I and the instantaneous RBV of the projectile at any time t by V(t) and will present the projectile dynamics by plotting the time history of projectile RBV and F_P and cross-plot the resistance force versus projectile rigid body displacement (RBD) to gain insight of the perforation mechanics. A right circular cylinder (RCC) projectile of diameter, D_P = 12.7, and of mass, m_P = 12.61 g is used for the variable velocity impact analyses.

Figure 12 shows the time history of projectile RBV and F_P of a 1000 mm SBSL under a wide range of impact velocities ( $150 m / s < V_{I} < 200 m / s$ ).CP of the SBSL is observed for impact velocities V_I ≥ 165 m/s in the time range, $τ_{1000} / 2 < t < τ_{1000}$ , where $τ_{1000} = 144.58 μ s$ . Since no perforation (NP) is observed for $V_{I} \leq 160 m / s$ , the minimum perforation velocity of 1000 mm SBSL under variable velocity impact can be approximated as: $V_{Perf}^{min} = 165 m / s$ . After the CP, as the projectile loses contact with the SBSL, the RBV of the projectile become constant while the resistance force progressively drops from the maximum/peak perforation force, $F_{P}^{max}$ , to zero. It is interesting to note that the projectile RBV versus time plots for all impact velocities have similar decay behavior up to perforation, which may be useful in self-similar theoretical modeling purpose. From the time history of RBV and resistance force data, the residual velocity (V_R) of the projectile, the maximum perforation force ( $F_{P}^{\max}$ ), and the corresponding time of perforation (t_f) at $F_{P}^{\max}$ for the perforating impacts are determined and are presented in Table 2.

Figure 12.

Time history of projectile velocity and perforation resistance force. Variable velocity impact ( $150 m / s < V_{I} < 200 m / s$ ), L = W = 1000 mm. $τ_{1000} = 144.58 μ s$ . (a) Rigid body velocity (V(t)); (b) perforation resistance force (F_p).

Table 2.

Variable velocity perforation data of [0/90] SK76/PU SBSL, L = W = 1000 mm, $τ_{1000} = 144.58 μ s$ .

$V_{I} (m / s #x0029;$	CP/NP	V_R (m/s#x0029;	t_f (μs)	$F_{P}^{max} (N)$	$\frac{t_{f}}{τ_{1000}}$	E_I (J)	E_R (J)	E_T(J)	$W_{P} (J)$	$\frac{(E_{T} - W)}{E_{T}} \times 100$	$E_{KE}^{SBSL} (J)$	$E_{IE}^{SBSL} (J)$	$E_{TE}^{SBSL} (J)$	$E_{Eroded} (J)$
150	NP	—	—	1617.3	—	141.82	—	—	—	—	—	—	—	—
160	NP	—	—	1832.1	—	161.36	—	—	—	—	—	—	—	—
165	CP	149.5	140.18	1908.2	0.969	171.59	140.85	30.74	30.74	0.00	20.03	9.84	29.87	0.87
170	CP	155.3	131.85	1904.7	0.912	182.14	151.99	30.15	30.11	0.13	19.02	10.11	29.13	1.02
175	CP	160.4	127.13	1965.5	0.879	193.02	162.19	30.83	30.77	0.19	18.97	10.67	29.64	1.19
180	CP	167.3	113.98	1978.3	0.788	204.20	176.47	27.73	27.66	0.25	16.34	10.42	26.76	0.97
185	CP	173.0	113.96	1940.8	0.788	215.70	188.67	27.03	26.94	0.33	15.55	10.48	26.03	1.00
190	CP	177.8	102.26	1965.5	0.707	227.54	199.44	28.10	27.94	0.57	15.52	11.19	26.71	1.39
195	CP	183.6	98.05	2019.9	0.678	239.66	212.59	27.06	26.91	0.56	14.95	10.84	25.79	1.27
200	CP	190.8	79.11	1839.6	0.547	252.08	229.35	22.73	22.63	0.44	11.61	9.68	21.29	1.44
Average				1940 ± 55^a						0.31		10.40

SBSL: soft ballistic sub-laminate; PU: polyurethane; NP: no perforation; CP: complete perforation.

^aAverage of complete perforation (CP) values.

Figure 13(a) shows the F_P plotted with projectile RBD ( $| w_{p} |$ ). The resistance force versus RBD plot is essential in calculating the work done (W_P, area under the curve) using equation (13). The impact energy ( $E_{I} = (1 / 2) m_{P} V_{I}^{2}$ ) and the residual energy ( $E_{R} = (1 / 2) m_{P} V_{R}^{2}$ ) of the projectile and the energy transferred to the [0/90] SBSL ( $E_{T} = E_{I} - E_{R}$ ) are also calculated and presented in Table 2 along with the work done (W_P) by the projectile on the SBSL. Energies and work done are made dimensionless by dividing with the impact energy at minimum perforation velocity, $E_{I}^{Perf}$ . The impact velocities are made dimensionless by dividing with the 1D cone wave velocity at $165 m / s$ impact velocity, ${[{\bar{c}}_{k, 0}]}_{165 m / s} = 426 m / s$ . Figure 13(b) shows the dimensionless energies versus impact velocities, from where we can see that minimum perforation velocity ( $V_{Perf}^{min}$ ) is a little less than 44% of the cone wave velocity and the energy transferred is less than 20% of the impact energy.

Figure 13.

Time history of projectile velocity and perforation resistance force. Variable velocity impact ( $150 m / s < V_{I} < 200 m / s$ ), L = W = 1000 mm. $τ_{1000} = 144.58 μ s$ . (a) resistance force versus rigid body displacement; (b) dimensionless energies versus impact velocity.

Table 2 shows the perforation data of all perforating impacts. In obtaining the perforation in the time range $0.50 < t / τ_{1000} < 1.00$ to avoid reflection effects from the free edge/boundary of the SBSL, impact velocity or energy used was very high. This is obvious from the energy plots as a function of impact velocities which shows that the energy transferred (E_T) or work done W_P is much smaller (less than 20%) than the impact energy (E_I) yielding a high residual energy (E_R). Ideally, the transferred energy to the SBSL (E_T) should be equal to the work done (W_P). Very small difference (0.31%) between the transferred energy (E_T) and the work done (W_P) reveals that most of the transferred energy is being dissipated by the deformation mechanisms of the [0/90] SBSL. It is also clear that with the increase in initial impact energy, the work needed to perforate the [0/90] SBSL decreases, because the membrane strain ε_t in the cone wave region is proportional to the square of $α = c_{k} / a_{0}$ (equations (7), (8), and (9)), c_k is an increasing function of ε₀, and ε₀ is an increasing function of impact velocity (equation (4)). Thus with the increase in impact velocity, membrane strain at the root of the projectile will quickly reach the failure strain ∊_f of the SBSL layers and failure will occur early in time and thereby reducing the work of perforation. Figures 12 and 13 and Table 2 present the perforation mechanics of a 1000-mm [0/90] SBSL under variable velocity impact with no boundary effects. The kinetic energy of the SBSL, $E_{KE}^{SBSL}$ , and the internal energy of the SBSL, $E_{IE}^{SBSL}$ , at end of the perforation process have been calculated from the global energy statistics of LS-DYNA output. The total energy of the SBSL is then calculated by adding the two, that is, $E_{TE}^{SBSL} = E_{KE}^{SBSL} + E_{IE}^{SBSL}$ . Ideally the projectile energy transferred to the SBSL should be equal to the total energy of the SBSL, that is, $E_{T} = E_{TE}^{SBSL}$ ; however, the difference between them is the eroded energy of the elements eroded during the perforation process, that is, $E_{Eroded} = E_{T} - E_{TE}^{SBSL}$ . Table 2 shows that irrespective of the impact velocity, the internal energy is almost constant and an average value has been calculated to be ${[E_{IE}^{SBSL}]}_{Avg} = 10.40 J$ , and the eroded energy is about 3% to 6% of the transferred energy (E_T).

According to the 2D membrane theory for isotropic materials (P&P⁹), the membrane strain is (equation (7)) is a function of c_k, which is a function of constant impact velocity V_P. For a variable velocity impact, projectile velocity is decreasing at every instant and thus c_k is also changing at every instant. P&P further modified the 2D membrane theory in the study of woven fabrics where they have derived few equations (equations (179) to (182), (213), and (214) in P&P) for strain concentration, K, value of which is 1.00 at $ψ = r_{k} / r_{p} = 1$ , increases to the maximum value K_max at $ψ_{max} > 1$ and then exponentially reduces to 0.00 for ψ ≫ 1 (Figure 4(a) and (b) in P&P). These equations reveals the fact that in 2D, the membrane strain is not maximum at r_k/r_p = 1 rather is maximum at ψ_max > 1 where failure will occur (note that for 1D, there is no strain concentration, i.e.,K = 1 at ψ = 1). Since an exact expression for membrane strain for non-isotopic materials is not derived as a function of radial locations, we would like to study the membrane strain of the [0/90] SBSL along X-axis (radial locations along [0] direction) for the minimum perforating impact velocity, that is, at V = 165 m/s (Figure 14).

Figure 14.

Membrane strain along X-axis of [0/90] SBSL. Variable velocity impact, V_I = 165 m/s. L = W = 2000 mm, $τ_{2000} = 289.16 μ s$ . SBSL: soft ballistic sub-laminate.

In order to avoid axial/implosion wave reflection from the free edges, simulations are conducted on a larger [0/90] SBSL of dimension L = W = 2000 mm. Considering symmetry, data along the positive X-axis are presented in Figure 14 which shows the membrane strain along X-axis of a [0/90] SBSL under the variable velocity perforating impact of V_I = 165 m/s, at two different time, that is, at t = 72 and 144 μs (t/τ₂₀₀₀ = 0.249 and 0.498), respectively. At time t = 72 μs, the axial/implosion wave front reaches at x = r = 500; and at time t = 144 μs, the axial wave front reaches x = r = 1000 mm. Thus the membrane strains plotted in Figure 14 do not have any boundary effects. The membrane strain profile for the [0/90] SBSL is very similar to the strain concentration profile of P&P (Figure 4(a) and (b) in P&P) and thus qualitatively agrees well with the P&P membrane theory. The exact axial/radial locations where the strain concentrations occur in the membrane strain profiles are identified in Figure 14.

Variable velocity transverse impact on finite size 2D [0/90] SK76/PU SBSL

In order to study the effect of boundaries on the perforation mechanics of finite size [0/90] SBSLs, we have performed similar analysis on three different finite size SBSLs, namely, L = W = 500, 360, and 250 mm.. Time history of RBV of the projectile and the F_P of L = W = 360 mm [0/90] SBSL is presented in Figure 15 and the same for L = W = 500 and 250 mm [0/90] SBSLs are presented in Figures 4A and 4B in Appendix 4.

Figure 15.

Time history of projectile velocity and perforation resistance force. Variable velocity impact ( $125 m / s < V_{I} < 225 m / s$ ), L = W = 360 mm. $τ_{360} = 52.05 μ s$ . (a) Rigid body velocity (V(t)); (b) perforation resistance force (F_p).

For finite size [0/90] SBSLs, perforation occurs at $t_{f} / τ_{L} > 1$ . This means that the axial/implosion wave reverberate couple of times in the SBSL causing large deformation modes such as: (i) edge-pull (t = 200 μs), (ii) crease formation (t = 400 μs), (iii) folding (t = 600 μs), and (iv) dishing (t = 800 μs) which are presented in Figure 16. While the minimum perforation velocity of 360 mm [0/90] SBSL is approximately ${[V_{min}^{Perf}]}_{360 mm} = 150 m / s$ , the same for 500 and 250 mm SBSLs are found to be 200 and 145 m/s, respectively.

Figure 16.

Large deformation modes of finite size SBSL. Variable velocity impact, V_I = 125 m/s, L = W = 360 mm. $τ_{360} = 52.05 μ s$ . SBSL: soft ballistic sub-laminate.

In case of impact velocities lower than the perforation velocities, finite size SBSBs deform like a cone and move at a constant velocity with the projectile for time $t / τ_{L} ≫ 1$ . Perforation data of all finite size SBSLs (similar to Table 2) are presented in Tables 4A to 4C in Appendix 4 for completeness, which show that the average maximum perforation force of all perforating SBSLs are almost constant and their average values, ${[F_{P}^{max}]}_{avg}$ , decreases with the decrease in SBSL size (Table 3).

Table 3.

Average maximum perforation resistance force of [0/90] SK76/PU SBSLs of different size.

[0/90] SBSL size	1000 × 1000 mm²	400 × 500 mm²	360 × 360 mm²	250 × 250 mm²
${[F_{P}^{\max}]}_{a v g} (N)$	1940 ± 55	1721 ± 114	1626 ± 117	1315 ± 94

SBSL: soft ballistic sub-laminate; PU: polyurethane.

Understanding the large deformation modes and the perforation mechanics of finite size SBSLs are important in understanding the variable velocity transverse impact on multilayer SBAP. Preliminary studies on the perforation mechanics of a SBAP made from 32 layers of 360 mm [0/90] SBSLs have been performed by the authors²⁶ and further detail will be presented elsewhere.

Summary

Constant velocity transverse impacts on a 1D isotropic ribbon, on a 2D isotropic membrane, and on a 2D [0/90] SK76/PU soft ballistic sub-laminate (SBSL) have been presented using LS-DYNA FEA. Theories of 1D fiber and 2D membrane are compared with the FEA results to validate the suitability of the FE model for further analysis. It is well known that both 1D Smith equations and 2D membrane theory (P&P⁹) assumes perfect contact between the fiber/membrane and the impactor and ignores bending wave propagation. However, the present FEA shows that both the ribbon and membrane under the projectile have oscillatory contact behavior, and the particle velocities in the cone wave region along the direction of impact are oscillatory in nature prevailing bending wave propagation in the cone wave region. While the basal cone shape is circular for isotropic membrane, the same for the [0/90] SK76/PU SBSL is found to be of diamond (or square) shape and not cylindrically symmetric. Thus the cylindrically symmetric 2D isotropic membrane theory cannot be used in the analysis of [0/90] SBSL. It has also been identified that the cone wave speed of [0/90] SK76/PU SBSL along the primary material axes ([0], 1) starts at the 1D value and shows a nonlinear increase with time reaching a plateau value for long time. The effect of laminate stacking sequence on the cone wave velocity and cone angle has been quantified. A characteristic time and minimum perforation velocity of a SBSL of finite dimension is defined and determined for a 1000-mm [0/90] SK76/PU SBSL under constant velocity impact.

Perforation mechanics study of a [0/90] SBSL where the projectile velocity continuously changes was one of the main goal of this research. Following the lessons learned from the constant velocity impact problems, F_P of a 1000-mm [0/90] SBSL have been predicted using the FE model at different impact velocities such that the perforation occurs in the time range $τ_{L} / 2 < t_{f} < τ_{L}$ . The peak value of F_P remains approximately constant with impact velocities; however, the work done or energy dissipation during the perforation process is found to decrease with increasing speed. Membrane strains calculated along primary material axis at minimum perforating impact velocity reveals that the [0/90] SBSL shows strain concentrations at radial locations r/r_p > 1, which qualitatively agrees well with the P&P (2003) approximate equations for woven fabrics.

In gaining insight into the perforation mechanics of finite size SBSLs, three different finite sizes are chosen, that is, L = W = 500, 360, and 250 mm. Time history of projectile velocity, displacement, and F_P have been analyzed to determine the minimum perforation velocity and dissipated energy. Because of finite size, failure of the SBSLs have been observed at $t_{f} / τ_{L} > 1$ , that is, axial wave reverberates couple of times in the SBSL before perforation. Large deformation of the SBSL showing several deformation mechanisms have been identified, that is, (i) edge-pull, (ii) crease formation, (iii) folding, and (iv) dishing. The perforation mechanics of finite size [0/90] SK76/PU SBSL under constant and variable velocity impact presented in this article is probably the first time in literature.

Footnotes

Appendix 1. Properties of UHMWPE fibers,unidirectional soft-ballistic lamina,and [0/90] sub-laminate

Table 1E.

Orthotropic properties of [0/90] SK 76/PU SBSL.

Properties (unit)	HB26-[0/90] composite	HB26-[0/90] composite	HB26-[0/90] composite
v_f (%)	—	72	37.2
ρ_C (g/cm³)	—	—	—
E₁₁ (GPa)	26.9	43.9	22.31
E₂₂ (GPa)	26.9	43.9	22.31
E₃₃ (GPa)	3.62	3.65	1.116
ν ₂₁	—	—	—
ν ₃₁	—	—	—
ν ₃₂	—	—	—
ν ₁₂	0.0	0.0238	0.0133
ν ₁₃	0.1	0.0188	0.0144
ν ₂₃	0.5	0.0188	0.0144
G₁₂ (GPa)	0.0423	1.118	0.4044
G₂₃ (GPa)	0.0307	8.828	0.3682
G₃₁ (GPa)	0.0307	8.828	0.3682
Reference	Lassig et al.¹⁶	Levi-Sasson et al.¹³	Levi-Sasson et al.¹³

SBSL: soft ballistic sub-laminate; PU: polyurethane.

Appendix 2: Mesh sensitivity analysis

A mesh sensitivity analysis is performed with four different mesh sizes, that is, $l_{E} = w_{E} = 0.50, 0.75, 1.00, and 2.00 mm$ for the [0/90] SK76/PU SBSL with dimensions L = W = 360 mm (Figure 2A) without any boundary conditions at the edges. An impact velocity of V_I = 50 m/s is chosen to monitor (i) the axial displacement U_X of an axial edge node B, (ii) the axial displacement U_X of a corner node C, (iii) the RBV of the projectile at the impact center A, and (iv) the penetration resistance force between the projectile and the SBSL (Figures 2B and 2C).

The solution time for mesh sensitivity analyses is chosen to be t = 1000 μs which is 19 times higher than the characteristic time ( $τ_{360} = 52.05 μ s$ ) for 360 mm wide SBSL. Transverse impact induced axial wave or “implosion wave” (P&P) takes about $τ / 2 = 26 μ s$ to reach the edge node location B and then reflects back toward the impact center. Multiple stress wave reversals pull the free edge of the SBSL inward and monitoring the axial displacement at axial edge node location B (Figure 2B(a)) is thus a good choice for mesh sensitivity analysis. Similar argument can be made for corner node location C (Figure 2B(b)). Time history of axial displacement U_X at node locations B and C shows that the mesh size $l_{E} = w_{E} = 1.00 mm$ is the largest mesh size that can be used with good engineering accuracy to capture the dynamic axial deformation behavior of the SBSL under transverse impact. Time history of the RBV of the projectile (Figure 2C(a)) and the penetration resistance force (Figure 2C(b)) also support this statement.

Finer mesh predicts better folding and pull-in behavior in the primary material directions, however, the choice of 1 mm mesh preserves almost all deformation behavior that can be seen for 0.50 mm mesh (Figure 2D(a) and (b)).

Appendix 3: Constant velocity transverse impact on SBSLs with different laminate architecture

Appendix 4: Perforation mechanics of finite size SBSL

Table 4C.

Variable velocity perforation data of [0/90] SK76/PU SBSL, L = W = 250 mm, $τ_{250} = 36.14 μ s$ .

$V_{I} (m / s)$	CP/NP	$V_{R} (m / s #x0029;$	t_f (μs)	$F_{P}^{max} (N)$	$\frac{t_{f}}{τ_{250}}$	E_I (J)	E_R (J)	E_T (J)	W_P (J)
100	NP	—	—	680.7	—	63.0	—	—	—
125	NP	—	—	963.0	—	98.5	—	—	—
130	NP	—	—	1008.8	—	106.5	—	—	—
135	NP	—	—	1061.8	—	114.9	—	—	—
140	NP	—	—	1124.8	—	123.5	—	—	—
145	CP	119.7	297.71	1163.3	8.238	132.5	90.4	42.2	42.3
150	CP	126.7	282.70	1212.2	7.822	141.8	101.2	40.6	40.6
160	CP	134.6	231.92	1332.8	6.417	161.4	114.3	47.1	46.9
170*	CP	137.8	198.62	1339.0	5.496	182.2	119.6	62.5	60.2
180	CP	163.4	152.39	1353.6	4.217	204.2	168.3	36.0	35.9
190	CP	174.2	150.31	1377.0	4.159	227.5	191.2	36.3	36.1
200	CP	185.0	105.70	1430.5	2.925	252.1	215.8	36.4	36.1
Average				$1315 \pm 94$ ^b

SBSL: soft ballistic sub-laminate; PU: polyurethane; NP: no perforation; CP: complete perforation.

^aA ribbon was moving with the projectile, and thus the energy transfer and work done is an outlier in the sequence.

^bAverage of complete perforation (CP) values.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

http://www.dyneema.com/americas/applications.aspx (September 15, 2014).

http://www.honeywell-advancedfibersandcomposites.com/products/fibers/ (September 15, 2014).

http://www.polyco.co.uk/the-polyco-difference/technical-library/dyneema%28r%29-fibre (September 15, 2014).

Jacobs

MJN

Dingenen

. Ballistic protection mechanisms in personal armour. J Mater Sci 2001; 36: 3137–3142.

Cheeseman

Bogetti

. Ballistic impact into fabric and compliant composite laminates. Compos Struct 2003; 61: 161–173.

David

Gao

X-L

Zheng

. Ballistic resistant body armor: contemporary and prospective materials and related protection mechanisms. Appl Mech Rev 2009; 62: 1–20.

Ayotte

Gama

Adkinson

. Quasi-Static Penetration Behavior of UHMWPE Soft Composite Laminates. In CD Proceedings, SAMPE 2011, Long Beach, CA, 23–26 May, 2011.

Ayotte

Gama

Adkinson

. Ballistic penetration behavior of UHMWPE soft composite laminates. In CD Proceedings, SAMPE 2011, Long Beach, CA, 23–26 May, 2011.

Phoenix

Porwal

. A new membrane model for the ballistic impact response and V50 performance of multi-ply fibrous systems. Int J Solids Struct 2003; 40: 6723–6765.

10.

Afshari

Sikkema

Lee

. High performance fibers based on rigid and flexible polymers. Polym Rev 2008; 48: 230–274.

11.

Karbalaie

Yazdanirad

Mirhabibi

. High performance dyneema fiber laminate for impact resistance/macro structural composites. J Thermoplast Compos 2011; 25(4): 403–414.

12.

Chocron

Nicholls

Brill

. Modeling unidirectional composites by bundling fibers into strips with experimental determination of shear and compression properties at high pressures. Compos Sci Technol 2014; 101: 32–40.

13.

Levi-Sasson

Meshi

Mustachi

. Experimental determination of linear and nonlinear mechanical properties of laminated soft composite material system. Compos B 2014; 57: 96–104.

14.

Grujicic

Arakere

. A ballistic material model for cross-plied unidirectional ultra-high molecular-weight polyethylene fiber-reinforced armor-grade composites. Mater Sci Eng A 2008; 498: 231–241.

15.

Chocron

Kirchdoerfer

King

. Modeling of fabric impact with high speed imaging and nickel-chromium wires validation. J Appl Mech 2011; 78: 1–13.

16.

Lassig

Nguyen

May

. A non-linear orthropic hydrocode model model for ultra-high molecular weight polyethylene in impact simulations. Int J Impact Eng 2015; 75: 110–122.

17.

Rakhmatulin

KHA

. Oblique impact at a large velocity on a flexible fiber in the presence of friction. Prikl Mat Mekh 1945; 9: 449–462. (in Russian)

18.

Rakhmatulin

KHA

. Impact on a flexible fiber. Prikl Mat Mekh 1947; 11: 379–382. (in Russian)

19.

Rakhmatulin

KHA

. Normal impact at a varying velocity on a flexible fiber. Uchenye Zapiski Moskovosk gos Univ 1951; 4: 154. (in Russian)

20.

Rakhmatulin

KHA

. Normal impact on a flexible fiber by a body of given shape. Prikl Mat Mekh 1952; 16: 23–24. (in Russian)

21.

Smith

McCrackin

Schiefer

. Stress-strain relationships in yarns subjected to rapid impact loading: 5. Wave propagation in long textile yarns impacted transversely. J Res Natl Bur Stand 1958; 60(5): 517–534.

22.

Smith

Blandford

Schiefer

. Stress-strain relationships in yarns subjected to rapid impact loading: part VI: Velocities of strain waves resulting from impact. Text Res J 1960; 30: 752–760.

23.

Roylance

Wilde

Tocci

. Ballistic impact of textile structures. Text Res J 1973; 43: 34–41.

24.

Ringleb

. Motion and stress of an elastic cable due to impact. J Appl Mech 1957; 24: 417–425.

25.

Graff

. Wave motions in elastic solids. New York: Dover Publications, 1975.

26.

(Gama) Haque

Ali

Gillespie

Modeling transverse impact on multi-layer UHMWPE soft ballistic armor pack (SBAP). In CD Proceeding, American Society for Composites 30th Technical Conference, Michigan State University, East Lancing, MI 48824, 28–30 September, 2015.

Modeling transverse impact on UHMWPE soft ballistic sub-laminate

Abstract

Keywords

Introduction

Properties of UHMWPE fibers, unidirectional laminas, and [0/90] sub-laminates

Theory of constant velocity transverse impact on isotropic elastic 1D fiber and 2D membrane

Constant velocity transverse impact on isotropic elastic 1D fiber (Smith et al)

Constant velocity transverse impact on isotropic elastic membrane (P&P)

FE modeling

FE model of a 2D membrane/shell

FEM of 1D ribbon

Transverse impact on isotropic ribbon and membrane

Results and discussion

Constant velocity transverse impact on 1D isotropic ribbon

Constant velocity transverse impact on linear elastic 2D isotropic membrane

Constant velocity transverse impact on 2D [0/90] SK76/PU SBSL with failure

Constant velocity transverse impact on 2D SK76/PU SBSL with different stacking sequence

Variable velocity transverse impact on 2D [0/90] SK76/PU SBSL with failure

Variable velocity transverse impact on finite size 2D [0/90] SK76/PU SBSL

Summary

Footnotes

Appendix 1. Properties of UHMWPE fibers,unidirectional soft-ballistic lamina,and [0/90] sub-laminate

Appendix 2: Mesh sensitivity analysis

Appendix 3: Constant velocity transverse impact on SBSLs with different laminate architecture

Appendix 4: Perforation mechanics of finite size SBSL

Declaration of Conflicting Interests

Funding

References