3Dept. of Theoretical and Applied Mechanics, Cornell University, 321 ..... et al., A short course on penetration mechani
Modelling of ballistic impact on fiber composites H. van der Werff1, U. Heisserer2, S.L. Phoenix3 1
DSM Dyneema, P.O. Box 1163, 6160 BD Geleen, The Netherlands,
[email protected] 2 DSM Performance Materials, Materials Science Centre, P.O. Box 18, 6160 MD Geleen, The Netherlands,
[email protected] 3 Dept. of Theoretical and Applied Mechanics, Cornell University, 321 Thurston Hall, Ithaca, NY 14853, USA,
[email protected] Abstract. A numerical model is presented for a projectile impacting on a fiber-composite target. The projectile is rigid with a flat circular nose (e.g., a right circular cylinder). Sole material parameters are fiber strength, tensile modulus and density appropriately normalized for matrix content and fiber or ply orientation. While quantitative agreement between calculated and experimental ballistic limits is very reasonable, the numerical model also corroborates very well empirical and theoretical results in literature. A single master curve has been found, relating ballistic limits to the ratio of the target and projectile areal density and the fiber material parameters. The anticipated power-law dependence of the ballistic limit on fiber strength and modulus were quantitatively reproduced. The results clarify the large potential of ultra high molecular weight polyethylene fibers in fiber based armor materials.
1. Introduction High performance polymeric fibers play an important role in light-weight armor materials used to protect personnel either through armor worn on the body (vests, inserts, helmets) or through armor attached to the interior of personnel transport vehicles used in air, on land or water. It is the intention of this paper to identify the essential fiber and geometric properties that largely determine the ballistic performance of fiber based composite targets. Knowledge of these relations is necessary to find new opportunities for improved armor materials.
2. Existing model and theories for ballistic impact on fiber based materials A broad overview on this subject has been given by Phoenix and Porwal [1]. Empirically, it had already been reported by Cunniff [2] that, through dimensionless analysis, a single curve could be used to relate normalized experimental V50 values of a wide variety of fiber based armor systems impacted by standardized steel right circular cylinder (RCC) projectiles to the system areal density ratio. The dimensionless fiber quantity of relevance was found to be the product of fiber specific toughness (i.e. elastic energy to break in a tension test per kg of material) and the velocity of sound in the fiber, to be denoted here by (having units m3/s3):
E 2
(1)
where is the fiber strength (N/m2), the elongation at break of a fiber, the fiber density (kg/m3) and E the fiber modulus. Basically, the following relation was found by Cunniff [2]:
V50 3
where AD is areal density, and f
f
ADtarget ADprojectile
(2)
is the empirically determined functional relationship, i.e., master
curve, from experiments on a wide variety of material systems.
1
A similar relation was theoretically derived by Phoenix and Porwal [1] for an initially untensioned, infinite, in-plane isotropic target membrane with a certain density, thickness and Young’s modulus, impacted with a projectile with a certain radius and areal density However, these authors were also able to derive the functional form of the relation, showing excellent agreement with Cunniff’s master curve. Rearrangement of eq. (1) using textile based units for strength (cN/dtex) and modulus (N/tex) gives the following result for the cube-root of , which is then linearly related theoretically, via the function f, to the V50 of any combination of a fiber based composite target and an RCC projectile: 3
(m/s)
171
(cN / dtex) E (N / tex)
1
2
3
(3) 6
The basic conclusion of eq. (3) is that the combination of strength, modulus and density of a fiber (appropriately averaging in the corresponding properties of any matrix) will determine the ballistic performance of any target made from the fiber. For instance, at a constant fiber modulus, increase of the fiber strength will increase ballistic performance, whereas at constant fiber strength, decrease of the fiber modulus will increase the ballistic performance, following power laws as indicated. It is the intention of the work reported here to investigate and possibly corroborate this relation between fiber properties and ballistic performance of a fiber composite through numerical simulations using a fully descriptive and simple physical model.
3. The numerical model of ballistic impact on fiber based composites. The model used in the work reported here is constructed as follows: The target panel to be simulated has a square size of 40 cm by 40 cm in the x,y-direction, and is impacted by a cylindrical projectile (RCC) in exactly the center of the panel, moving with a velocity purely in the z-direction. Due to symmetry considerations, the simulations can then be restricted to one quadrant. For the case studied in this paper, this quadrant is then assumed to consist of 2001 x 2001 dimensionless point masses, situated before impact at the crossover points of the rectangular grid which has a spacing of 0.1 mm in both the x- and y directions. These point masses are each given a mass corresponding to the areal density of the target times the surface area of one grid square (i.e., 0.01 mm2). The point masses are connected to each other with springs. The force-displacement curves of these springs are taken to have Hookean elastic behaviour in extension, characterized by an effective modulus of the panel material times the cross-sectional area of a grid square (depth times width) and an elongation at break beyond which the elastic force becomes zero. Also since significant in plane compressive loads cannot be supported in the panel (as fibers will locally buckle), zero modulus is assumed to apply in compression. In fact, this omission of compressive loads was found to be necessary to obtain stable and energy conserving simulations. Furthermore, once the dynamics is stabilized, strains in the elements resolved along the main axes are virtually never compressive. The basic setup of the physical model for the panel is schematically depicted in Figure 1. The projectile is incorporated into the model in the following manner: The RCC projectile is a cylinder, which has a certain mass and radius in the x,y-plane. Prior to the start of the simulation, the point masses in the target that are covered directly by the projectile are identified, and additional mass from the projectile, being the projectile areal density (mass divided by cross-sectional area) times the initial surface area of a grid square, is added to the mass of each target point covered by the projectile. Furthermore, during the simulation, the coordinates of these target point masses covered by the projectile are fixed in the x- and y-direction and all forces applied by the panel to the projectile in the z-direction are averaged out over all the target mass points. In this way, a non-deforming projectile has been implemented, and a ‘noslip’ condition applies to the panel material under the projectile. The equations of motion for this multi-particle system are solved using a leap-frog algorithm, very commonly used in molecular dynamics [3]. The conditions for the starting situation (at t = 0 s) for the impact process are derived in the same way as done by Phoenix and Porwal [1]: it is assumed that the starting velocity of the point masses covered by the projectile are governed by simple momentum exchange (as in blunt impact) between the target point masses and the projectile, which approaches with its original projectile velocity. Thus, the starting velocity of the point masses is the original projectile velocity divided
2
by one plus the areal density ratio in eq. (2), so is a few percent lower than the original projectile velocity. The time-steps in the simulations were typically 0.4-0.8 x 10-8 s, which happens to result in a stable dynamic response. The edges of the panel were taken to be unclamped or free moving. Rigid clamping of the edges can also be done, but typically leads to fracture processes at these edges due to reflection of the longitudinal sound wave. Typically it has been found that clamping effects lead to small deviations in the calculated ballistic limits (if at all) and only for very low areal density ratios and correspondingly low impact velocities necessary to avoid penetration. Free moving edges resemble most the conditions of normal use of body armor. Clamping is often used in experimental testing of panels, but even there it is often difficult to avoid slip when the tensile wave arrives, since the tensile stresses are so large. The software for this system was written in Fortran 90 and run in double precision on normal desktop computers.
break
Figure 1. Schematic representation of the target and the stress-strain curve of the springs connecting the point masses. 4. Results The following material properties were used in order to simulate the ballistic performance of unidirectional composites based on Dyneema® fibers. The strength of these Dyneema® fibers typically is 3.43 GPa with a modulus of 122 GPa (defining a strain at break of 0.0281). Density is set to 980 kg/m3. Unidirectional Dyneema® composites consist of 0°/90° stacked fiber layers containing a matrix. For the simulations, the matrix content was set to 17 wt.% and the matrix density also to 980 kg/m3. Using these boundary conditions, it is straightforward to calculate that the modulus of the springs should be equal to (10.17) x 122 GPa / 2 =50.6 GPa. The strain at break of the springs is still 0.0281. A set of simulations was then carried out using the target and projectile parameters given in Table 1, together with the resulting calculated ballistic limits. The series of simulations is in fact an exploration around a reference situation of a fragment simulating projectile (FSP) with a mass of 1.1 gram and a radius of 2.73 mm, impacting a Dyneema® uni-directional composite with an areal density of 4.89 kg/m3.For this reference situation, a ballistic limit was found of 568 m/s. The experimental V50 for this condition was around 573 m/s, so agreement is extremely good (for this condition, more experimental data will be given later). At a projectile velocity of 569 m/s, the projectile will perforate the target and was found to have a residual velocity of 419 m/s. At 568 m/s, the projectile will not perforate and will be stopped. The projectile velocity versus time after impact is given in Figure 2 for these two situations. The observation then is, that the numerical model implies a step-function for the residual velocity as a function of projectile impact velocity. This reason for this becomes clear if one looks at the maximal strains in the springs directly adjacent to the point masses covered by the projectile (Figure 3). Figure 3 shows that the strain in the target, at the perimeter of the projectile, goes through a maximum very quickly and then decreases gradually again with the time. The critical moment is the time after impact where these strains reach their maximum. For the reference situation, this is at 1.94 s after impact. The projectile velocity has then only dropped to about 419 m/s, which will be approximately the residual velocity when perforation occurs, since no further work is done on the projectile to decelerate it. Figure 3 shows some spikes at the critical moment, both under and above the ballistic limit. This is due to the fact that at the ballistic limit a few springs can break without leading to a total perforation. In simulations with a few m/s less impact velocity, these spikes disappear.
3
Table 1. Target and projectile parameter variations used for simulations, together with the calculated ballistic limit. The reference situation is in bold face.
Projectile mass kg 0.0011 0.0011 0.0011 0.0011 0.0011 0.0044 0.0022 0.0011 0.0006 0.0003 0.0011 0.0011 0.0011 0.0011 0.0011
Projectile radius m 0.00273 0.00273 0.00273 0.00273 0.00273 0.00273 0.00273 0.00273 0.00273 0.00273 0.00137 0.00195 0.00273 0.0039 0.00546
Projectile Target AD AD kg/m2 kg/m2 47.0 1.23 47.0 2.45 47.0 4.89 47.0 9.78 47.0 19.56 187.9 4.89 94.0 4.89 47.0 4.89 25.6 4.89 12.8 4.89 186.6 4.89 92.1 4.89 47.0 4.89 23.0 4.89 11.7 4.89
=AD target / AD proj 0.026 0.052 0.104 0.208 0.416 0.026 0.052 0.104 0.191 0.382 0.026 0.053 0.104 0.212 0.416
Calculated ballistic limit m/s 405 490 568 675 798 408 485 568 658 777 459 529 568 668 783
600 V impact = 568 m/s
Projectile velocity (m/s)
500 V impact = 569 m/s 400
Residual velocity = 419 m/s
300
200
100
0 0.0E+00
1.0E-05
2.0E-05
3.0E-05
4.0E-05
5.0E-05
6.0E-05
Time after impact (s)
Figure 2. Projectile velocity versus time after impact in seconds for the reference situation at projectile impact velocities of 568 and 569 m/s.
4
Maximum target strain around projectile
0.030
0.025
Target strain at break V impact = 568 m/s
0.020
V impact = 569 m/s
0.015
0.010
0.005
0.000 0.0E+00
1.0E-05
2.0E-05
3.0E-05
4.0E-05
5.0E-05
6.0E-05
Time after impact (s)
Figure 3. Maximum strain in the target directly around the projectile versus time after impact in seconds for the reference situation at impact velocities of 568 m/s and 569 m/s. The critical moment for perforation is at 1.94 x 10-6 s after impact.
45
U-kin proj (t
