Composites: Part B 56 (2014) 254–262

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Finite element modeling of ballistic impact on multi-layer Kevlar 49 fabrics Deju Zhu a, Aditya Vaidya b, Barzin Mobasher b, Subramaniam D. Rajan b,⇑ a b

College of Civil Engineering, Hunan University, Changsha, Hunan 410082, China School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, AZ 85287, USA

a r t i c l e

i n f o

Article history: Received 8 June 2013 Accepted 12 August 2013 Available online 22 August 2013 Keywords: A. Fabrics/textiles B. Impact behavior C. Finite element analysis (FEA)

a b s t r a c t This paper presents a material model suitable for simulating the behavior of dry fabrics subjected to ballistic impact. The developed material model is implemented in a commercial explicit finite element (FE) software LS-DYNA through a user defined material subroutine (UMAT). The constitutive model is developed using data from uniaxial quasi-static and high strain rate tension tests, picture frame tests and friction tests. Different finite element modeling schemes using shell finite elements are used to study efficiency and accuracy issues. First, single FE layer (SL) and multiple FE layers (ML) were used to simulate the ballistic tests conducted at NASA Glenn Research Center (NASA-GRC). Second, in the multiple layer configuration, a new modeling approach called Spiral Modeling Scheme (SMS) was tried and compared to the existing Concentric Modeling Scheme (CMS). Regression analyses were used to fill missing experimental data – the shear properties of the fabric, damping coefficient and the parameters used in CowperSymonds (CS) model which account for strain rate effect on material properties, in order to achieve close match between FE simulations and experimental data. The difference in absorbed energy by the fabric after impact, displacement of fabric near point of impact, and extent of damage were used as metrics for evaluating the material model. In addition, the ballistic limits of the multi-layer fabrics for various configurations were also determined. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Woven fabrics composed of light weight and high strength continuous filaments are especially useful in a wide-range of applications such as protective materials for military and law enforcement personnel and as well as structural containment of turbine fragments [1]. Their high strength to weight ratio, attractive energy dissipating and penetration resistance ability to resist high speed fragment impact enable them to be very efficient compared to metals. The engine containment system is typically constructed by wrapping multiple layers of Kevlar 49 around a thin aluminum encasement. Designing the containment system consists of determining the type of fabric, the number of fabric layers and fabric width required. Currently the FAA’s certification standards require that a full-scale test be completed to qualify an engine which can cost several million dollars [2]. Consequently, there is a continuing effort to reduce the extent of these experimental test programs by complementing them with the corresponding computation-based engineering analyses and simulations. Some of the analytical, numerical, and experimental approaches that describe the impact

⇑ Corresponding author. Tel.: +1 480 965 1712; fax: +1 480 965 0557. E-mail address: [email protected] (S.D. Rajan). 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.08.051

response of woven fabrics are outlined in the comprehensive review articles by Cheeseman and Bogetti [3], and Tabiei and Nilakantan [4]. Modeling the impact response of woven fabrics is challenging because of their intricate hierarchical architectures, complex material behavior and interactions between the projectile and fabric during transverse impact. There are several modeling techniques used to represent the impact behavior of flexible woven fabrics. The impact behavior can be analyzed by using pin-jointed orthogonal bars in finite element analyses [5–7]. While the pin-jointed orthogonal bars based finite element analyses have proven to be very efficient in approximating the dynamic behavior of woven fabrics, the discrete nature of the yarn models was associated with inherent oversimplifications that significantly limited the predictive capability of the analyses. Unit-cell based approaches have been used extensively in order to derive the equivalent (smeared) continuum-level material models of woven composites from the knowledge of the mesoscale yarn properties, fabric architecture and inter-yarn and inter-ply frictional characteristics [8–11]. More-detailed 3D continuum finite element analyses have proven to be powerful tools for capturing and elucidating the detailed dynamic response of single-layer fabric, however they are associated with large computational requirements in terms of both processing power and

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2. Improvements in constitutive modeling New tensile tests were conducted in both warp and fill directions until the load carrying capacity of the fabric reached zero. The results show that the fabric can deform up to 20% before complete failure [26]. Fig. 1 shows the typical stress–strain curve obtained from quasi-static tensile test using 200 mm  50 mm

500 Experiment Model

400

Stress, MPa

memory requirement because of the large number of degrees of freedom of the model [12–16]. This limits the size of the fabric model that can be simulated within available computational infrastructure and reasonable amount of time. Higher-order membrane/shell finite element analyses are also used to analyze the dynamic response of fabric under ballistic loading conditions and overcome the computational cost associated with the use of full 3D finite element analyses [17]. To provide an acceptable balance between accuracy and the size of the model, multi-scale modeling techniques are being developed [18,19]. Woven solid elements are used to discretize the yarns around the impact region, which transition to more computationally efficient woven shell elements in the surrounding regions [20]. At the moment, it is not feasible to model armor and containment systems which typically contain 30–50 fabric layers by using multi-scale modeling techniques. Yarn crimp is another factor that influences the ballistic performance of woven fabrics. Crimp is the undulation of the yarns due to their interlacing in the woven structure. The early stages of fabric deformation during ballistic impact are due to the straightening of crimped yarns without the yarns stretching; hence, there is little resistance to the projectile. Crimp can be represented by discounting a fraction of the total strain of their linear elements as due to straightening of crimped yarns [21], or by using a bilinear stress– strain relation for the bar elements [5], or by arranging yarn elements in a zigzag manner to accurately reflect the structure of the fabric but discounting some element strain that arises from straightening of the yarns via the constitutive equations of the elements [22]. In idealizing fabrics as a two-dimensional continuum represented by shell elements or as a membrane, the crimp of yarns cannot be physically represented, but can be incorporated into non-linear stress–strain relations in a user defined material model. In our previous study [23,24], two material models (ASU v.1.1 & v.1.2) were developed to include non-linearity in the stress–strain response and strain rate effect on the material response. These models were incorporated into the LS-DYNA through a user defined material subroutine (UMAT). The fabric layers were represented by single and multiple finite element (FE) layers. Both models were built on the experimental data where the fabric was loaded up to the strain of about 4% without complete failure [25], and the complete post-peak behaviors were assumed. New experimental data show that the fabric can undergo large deformations with failure strain of 20%, and the post-peak energy absorption capacity is correspondingly quite high [26]. In the present study, we extend our previous work by using a modified UMAT subroutine with optimized material/model parameters (ASU v.1.3). In section 2 we discuss the improvement of the strain-rate dependent material model from previous work. Verification and validation of the developed material model and finite element modeling are shown in sections 3 and 4. We present both single layer (SL) and multi-layer (ML) models to simulate the ballistic tests, following by a sensitivity analysis of the material model to various parameters, the effects of variation in LS-DYNA version, platform and precision of analysis, and ring boundary conditions on the results. Finally, we summarize the current accomplishments and limitations of the material model, modeling techniques and experimental data.

Linear Pre-peak Region

300 Linear Post-peak Region

200 100

Non-linear Post-peak Region Crimp Region

0 0

0.04

0.08

0.12

0.16

0.2

Strain, mm/mm Fig. 1. General stress–strain curve for Kevlar (experiment and model).

(length  width) swath specimens and the material model used in the FE simulations. There are four distinct regions in the fabric behavior: crimp region, linear pre-peak region, linear post-peak region and non-linear post-peak region. In the crimp region, the stress increase is relative low due to the straightening of the yarns. When the crimp is removed, the straightened yarns exhibit an almost linear load–displacement relationship as evident in the prepeak region. Some of the filaments in the yarns break but the majority of the yarn is intact. When the stress level reaches the tensile strength of the fabric, more filaments begin to break and the stress in the fabric decreases dramatically until reaching a transition point (approximately 70 MPa at the end of the linear postpeak region). After this, the stress decreases gradually to almost zero (approximate strain value of 20%) representing the non-linear post-peak region. Based on the stress–strain curves in both warp and fill directions, it was found that the elastic stiffness in pre-peak region of warp direction is identical to that of fill direction, and the crimp stiffness for warp and fill directions is 0.06 and 0.20 times of the elastic stiffness in pre-peak region, respectively. The absolute value of stiffness in linear post-peak region of warp and fill directions is 2.2 and 5.6 times of the elastic stiffness in pre-peak region. The crimp strain of the warp direction is about 2.6 times larger than that of the fill direction. And the peak stress of the warp direction is 15% lower than that of the fill direction. There is a slight difference in the strain at peak stress and the stiffness of linear postpeak region [26]. The stresses in the non-linear post-peak region for both warp and fill directions were assumed as follows:

rii ¼ rii 1 



eii  eii efail  eii

dfac ! ð1Þ

where rii and eii (i = 1, 2) are the stress and strain values at which the non-linear region begins in each respective direction, efail is the failure strain in each respective direction, and dfac is a factor which specifies the rate of decrease in stress in the non-linear post-peak region. The failure strain in both the warp and the fill directions were chosen to be 0.2 and the factor dfac was 0.30. In the material model, the elastic stiffness and strain at peak stress were assumed to be a function of the strain rate using a Cowper-Symonds type of model. The peak stress was indirectly assumed to be a function of the strain rate. The strain-rate effect on the elastic stiffness in the warp and fill directions is expressed as:

  e_ ii 1=PE Eadj ¼ E 1 þ ii ii CE

ð2Þ

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eiimaxðadjÞ ¼ emax 1þ ii

e_ max ii

1=Pe

Ce

ð3Þ

where Eii (i = 1, 2) is the static elastic stiffness, Eadj ii is the adjusted elastic stiffness considering strain-rate effect, e_ ii is the strain-rate, CE and PE are the Cowper-Symonds factors for elastic stiffness, maxðadjÞ emax is the strain at peak stress, eii is the adjusted strain at peak ii max _ stress, eii is the maximum strain rate experienced by the element in each respective direction, and C e and Pe are the Cowper-Symonds factors for the strain at peak stress. In the previous study [23,24], the C and P values were determined by extrapolating the experimental data obtained by Wang and Xia [27]. However, the fabric under ballistic impact typically experiences a much higher strain rate than the available experimental data. Hence it is necessary to optimize these two values to create a more accurate model. The optimization process is discussed later in the paper. The shear response of the fabric was determined by pictureframe shear tests [26]. The fabric (without any pretension) was sheared at quasi-static loading rate and exhibits very low shear resistance. If the shear properties of the fabric obtained by picture frame test are used directly in FE simulation of high-speed impacts, the fabric behaves like a rubber-like material with very large deformations and low impact resistance. This is because the fabric experiences large tensile stress during shear deformation. The tensile stress in the fabric dramatically influences the shear resistance of the fabric by altering the conditions of the yarn interaction (crimp, yarn compression, normal force at cross-over points), and hence the friction between the fill and warp yarns. Several researchers have investigated the frictional effects on the ballistic impact behavior of plain weave fabric and found that friction contributes to delaying fabric failure and increases the energy absorbing capacity of the fabric [15,28,29]. As the relation between shear properties and tensile stress in fabric is not clear, the shear properties used in the simulation were adjusted until the deformation of the fabric was similar to that of the experiment, and then the properties were optimized to minimize the difference in absorbed energy between the simulation and the experiment. 3. Verification of constitutive model The impact of a projectile with the fabric is associated with the initiation of several phenomena the most important of which are: (a) a resisting force is exerted by the fabric on the projectile which causes a reduction in the projectile velocity; (b) at the same time, the fabric is being deformed and accelerated; and (c) strain waves generated in the impact region propagate along the yarns toward the fabric edges. In general, the energy dissipation due to projectile deformation, fiber intermolecular friction and acoustic losses are all assumed to be negligible [30]. Consequently, any loss in projectile kinetic energy DEpk can be mainly attributed to the following three energy-absorbing mechanisms: yarn strain energy Eys, yarn kinetic energy Eyk and the energy lost due to frictional-sliding Ef. The energy conservation principle requires that:

DEpk ¼ Eys þ Eyk þ Ef

ð4Þ

The loss of projectile kinetic energy DEpk is governed by several factors such as the material properties of the constituent fibers, fabric structure, boundary conditions, projectile geometry, impact velocity and orientation (angle), friction between the projectile and the fabric, the fabric-to-fabric friction, and the yarn-to-yarn and fiber-to-fiber friction within the fabric itself [12]. The DEpk (absorbed energy) is the difference in the kinetic energy of the projectile before and after impact, given by:

 1  DEpk ¼ Ei  Er ¼ m v 2i  v 2r 2

ð5Þ

where m is the mass of the projectile, vi the projectile initial velocity, and vr the projectile residual velocity. If presented as a percentage, it is given by:

DEpk ð%Þ ¼ ðEi  Er Þ=Ei  100 ¼





v 2i  v 2r =v 2i  100

ð6Þ

The difference in absorbed energy between experiments and simulations was computed as: sim D ¼ DEexp pk  DEpk

DEexp pk

where is the absorbed energy in experiment, and sorbed energy in simulation.

ð7Þ DEsim pk

the ab-

3.1. Ballistic tests Ballistic tests were conducted at NASA-GRC. Projectiles were fired at fabric wrapped around a steel ring [31]. Various parameters were varied in the tests such as the initial velocity of the projectile, the orientation of the projectile with respect to the fabric, the number of layers wrapped around the ring, and the type of projectile. Two tip-rounded projectiles were used in the tests – 5 mm  4 mm  5 mm (Old Projectile) and 3.8 mm  3 mm  3.8 mm (New Projectile). For each test the initial and final velocity of the projectile were measured along with the exact orientation of the projectile. A total of twenty-six tests, including eighteen from phases 1 and 2 (LG4xx and LG6xx), and eight from phase 3 (LG9xx), were used for verification of the constitutive model. The tests over the 3 phases were conducted over a period of 8 years and in each phase a fresh set of Kevlar fabrics were ordered and used. In each phase, the experimental tests were carried out to characterize the behavior of the fabric. Table 1 provides the summary of the different test parameters for the 26 tests. 3.2. Finite element modeling To validate the developed material model, finite element models were built to replicate the NASA-GRC ballistic tests described above. A mesh convergence study was conducted to seek a compromise between accuracy of the results and the computational time. In the final model, the fabric was modeled with a uniform mesh containing 6.35  6.35 mm shell elements. The thickness of the shell finite element varies between 0.2% (4 fabric layers) and 1.5% (32 fabric layers) of the element size, justifying the use of the shell elements in modeling the fabric layers. The ring and the projectile are modeled with solid elements. Two types of projectiles were used in the experiments and simulations (Fig. 2a). The tip of projectile was meshed with uniform tetrahedral elements and the body with hexahedral elements. The steel ring was meshed with 6.35  6.35  25.4 mm (length  width  thickness) hexagonal elements. The steel ring is modeled as an elastic material while the projectile is modeled by Johnson–Cook model [32]. Two modeling schemes were used – the Concentric Modeling Scheme (CMS) and the Spiral Modeling Scheme (SMS). In the CMS, the multiple layers are arranged around the ring in the form of concentric layers. In other words, one finite element layer is not connected to the adjacent layer. The region of the fabric just over the cut-out in the ring is modeled as a separate part than the rest of the fabric that goes around the ring. This helps in isolating the region where the projectile makes contact with the fabric so that a more accurate energy balance check can be carried out as a part of the quality assurance checks. The CMS is further sub-divided into SL and ML models. In the SL model (Fig. 2b), one finite element layer is used to represent all the physical layers in the model. This layer is positioned at a radius half way between the ring and the radius where the outermost layer would have been in the actual model. Each fabric layer is

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D. Zhu et al. / Composites: Part B 56 (2014) 254–262 Table 1 Ballistic test data. Model

Fabric layers

Projectile type

Roll, pitch, yaw (degree)

Initial velocity (m/s)

Final velocity (m/s)

Energy absorbed (%)

LG404 LG409 LG424 LG594 LG609 LG610 LG611 LG612 LG618 LG620 LG689 LG692 LG429 LG432 LG411 LG427 LG656 LG657 LG963 LG966 LG965 LG964 LG967 LG971 LG969 LG970

8 8 8 8 8 8 8 8 8 8 8 8 16 16 24 24 32 32 4 8 16 17 24 24 32 32

Old Old Old New New New Old Old New New Old Old Old Old Old Old Old Old Old Old Old Old Old Old Old Old

(0, 0, 0) (0, 0, 0) (0, 0, 0) (27, 6.6, 47.8) (37.4, 0.8, 1.6) (25.3, 0.7, 11.9) (30.9, -1.7, 10.8) (22.8, -3.7, -0.5) (-47, 6.3, 51.6) (-37.8, 0.2, 55) (-12.8, -1.3, 49.7) (38.2, 2.3, 41.5) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (9, 2.3, 10) (22.2, 9.7, 1.4) (7.5,5.5,0.7) (7.6,4.3,5.4) (6.6,37.7,0.9) (4.6,19.9,5.9) (55.7,4.5,54.5) (4.2,6.3,7.2) (2.6,5.4,0.5) (2,3.6,5)

273.1 271.0 253.9 257.3 278.6 270.7 276.1 273.7 264.0 272.5 273.1 269.7 278.9 273.1 270.1 278.9 294.7 253.0 93.9 108.2 169.5 183.2 175.3 171.9 235.0 247.5

249.9 246.0 227.1 147.8 251.5 246.9 243.2 250.9 170.4 177.1 199.6 183.8 219.2 198.1 125.9 185.0 143.0 0.0 53.6 27.7 0.0 25.3 0.0 0.0 0.0 50.3

16.1 13 14.8 67 18.4 16.9 22.4 16.1 58.4 57.8 46.6 53.7 28.3 47.4 78.2 41.3 76.5 100 67.4 93.4 100 98.1 100 100 100 95.9

Old projectile

New projectile

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 2. Finite element models: (a) projectiles FE mesh; (b) CMS SL; (c) CMS ML; (d) close-up view of the fabric in CMS ML model; (e) SMS and (f) close-up view of the fabric in SMS model.

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0.28 mm thick, thus an 8-layer physical model is represented by one FE layer with a thickness of 2.24 mm (8  0.28 mm). A SL model is computationally efficient which makes it beneficial for initial studies. However fabric-to-fabric contact cannot be modeled as well as the extent of the damage cannot be gauged. In the ML model (Fig. 2c and d), one FE layer represents four physical layers. Hence an 8-layer physical model is represented by 2 FE layers each having a thickness of 1.12 mm (4  0.28 mm). The layers are positioned at a radius which is mid-way between the outer and inner layer radii they represent. Though this model runs for a longer time compared to the SL model, it is able to account for the contact and interactions between different fabric layers. This model is also capable of determining the partial failure in the layers when the projectile is contained. In the SMS model, the fabric layers are positioned around the ring in the form of a spiral so that the next layer starts at the point where the first layer ends (Fig. 2e and f). While the SMS model is very close to the manner in which the fabric is actually wrapped around the ring, we found that representing one fabric layer with one FE layer would be computationally expensive especially for 16–32 layer models. Instead similar to the CMS ML models, one FE layer in the SMS model represents four physical layers. Thus the first FE layer starts at half the distance of first four physical layers and goes up to the middle of the next four physical layers where the second FE layer starts. Thus different layers are connected to each other. The wrapping starts at an end diametrically opposite from the point where projectile makes contact with the fabric (Fig. 2e). 3.3. Verification of constitutive model Three metrics were used to verify the constitutive model. (i) The difference in the absorbed energy between experiment and simulation: The experimental data has about a 10% variation in results for the same model tested again. This is most likely due to errors in the measuring systems and inherent variability in the test procedure including material properties; hence any difference greater than 10% between test and simulation is flagged for further investigation. (ii) The temporal evolution and the spatial distribution of fabric deformation: In the latest tests (phase 3), additional cameras were used to track the movement of the region of the fabric

1

2

3

4

POI

5

6

7

8

9

10

11

12

13

14

in the contact zone so as to obtain the full displacement field. A total of 15 points were chosen on a grid where points were separated by 12.7 mm on all sides. Fig. 3a shows the grid of points on the fabric where displacement is obtained and Fig. 3b shows the resolved displacement field of the fabric as processed by the image analysis software. The point with maximum displacement in the test was chosen to compare with the FE simulation and its displacement history was tracked. (iii) Extent damage of fabric: The models tested in the phase 3 were analyzed so that the extent of damage could be found out. The damage could be either in form of complete penetration of fabric through all the layers or partial damage to some layers in the case when the projectile is contained. This is characterized by some yarn breakage or yarn damage in localized region. For the cases of simulation, the case where partial damage occurred was characterized by deletion of some elements while complete damage was when projectile completely penetrated the fabric.

4. Simulation of ballistic impact 4.1. Concentric Modeling Scheme (CMS) A total of eighteen simulations were carried out to model the tests in phases 1 and 2 (see Table 1). The simulations were run using the single precision LS-DYNA version 971 R4.2.1. The FORTRAN compiler used for building the executable was Intel version 10.1 and the computer platform was Windows XP 32-bit single precision. As mentioned earlier, in the absence of experimental results, some of the material properties should be obtained by optimizing their values so as to reduce the error in the energy absorbed metric.

4.1.1. Optimization Five material parameters (GI12 ; cI12 ; cII12 ; C ¼ C E ¼ C e and P ¼ P E ¼ Pe ) were chosen as the two-level design factors to minimize the difference between the experiments and simulations as listed in Table 2. A 25 full factorial design was used in the optimization design which required 32 runs for each simulation. Analysis of variance (ANOVA) was done on this factorial design to find the factors which affect the absorbed energy. The response variable in ANOVA was the percent difference in absorbed energy. It was found that the factors GI12 ; cI12 and cII12 are not significant and two factors C and P and the interactions with the factor P seem to be significant. ANOVA shows that the full factorial model is significant as the model F-value is 468.71. There is only a 0.01% chance that a ‘‘Model F-Value’’ this large could occur due to noise. The final equation in terms of coded factors of the model is given as:

D ¼ 14:98  0:0022GI12 þ 0:34cI12  1:07cII12 þ 3:08C þ 29:21P  3:56cII12 P þ 2:32CP

ð8Þ

Using the optimization function in the software, the optimal values of the five factors are obtained as the following: GI12 ¼ 4:14 MPa ð600 psiÞ; cI12 ¼ 0:25; cII12 ¼ 0:35; C ¼ C E ¼ C e ¼ 0:005 and P ¼ P E ¼ Pe ¼ 40. Table 2 Design factors and their levels.

(a)

(b)

Fig. 3. Displacement field measurement: (a) grid of points and (b) displacement field on the fabric obtained by the image analysis software.

Level

GI12 (MPa)

cI12

cII12

CE

PE

Lower Upper

2.76 6.90

0.25 0.34

0.35 0.65

0.005 0.025

10 50

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D. Zhu et al. / Composites: Part B 56 (2014) 254–262 Table 3 The parameters used in the current material model (unit of stress and stiffness: MPa). Symbol

Material constant

Value

E11 E22 Ecrp x Ecrp y

Warp direction stiffness in elastic region Fill direction stiffness in elastic region Warp direction crimp stiffness factor Fill direction crimp stiffness factor

32300 32300 413 1380

Esoft x

Warp direction post-peak linear region stiffness factor

15200

Esoft y

Fill direction post-peak linear region stiffness factor

38600

Eunl Ecomp G23 G31 GI12

Unloading/reloading stiffness factor Compressive stiffness factor Shear stiffness Shear stiffness Shear stiffness linear region 1

10300 34.5 345 345 4.14

GII12

Shear stiffness linear region 2

41.4

GIII 12

Shear stiffness linear region 3

345

cI12 cII12 ecrp 11 ecrp 22 emax 11 emax 22 r efail 11 emax 22

Shear strain 1 (rad)

0.25

Shear strain 2 (rad)

0.35

Warp direction crimp strain

0.0065

Fill direction crimp strain

0.0025

Warp direction strain at peak stress Fill direction strain at peak stress Stress at post-peak non-linearity Warp direction failure strain

0.0223 0.0201 69 0.2

Fill direction failure strain Cowper-Symonds factor for stiffness Cowper-Symonds factor for stiffness Cowper-Symonds factor for strain Cowper-Symonds factor for strain Post-peak non-linear region factor Failure strain of element

0.2 0.005 40 0.005 40 0.3 0.35

CE PE Ce Pe dfac

efail

The actual shear modulus values at various shear strain values used in the material model are as follows: For: I 12

GI12

c12 < c ¼ 4:14 MPa ð600 psiÞ cI12 < c12 < cII12 GII12 ¼ 41:4 MPa ð6000 psiÞ c12 > cII12 GIII12 ¼ 345 MPa ð50000 psiÞ A conservatively low value of 345 MPa was assumed as the out of plane shear modulus for G23 and G31. Table 3 lists all the parameters used in the current material model after optimization. 4.2. Spiral Modeling Scheme (SMS) In the actual test the innermost layer of the fabric is glued to the ring (using a 24-h epoxy) while the outermost layer is glued to the penultimate layer. This is implemented in the model by using a ⁄CONTACT_TIEBREAK_NODES card. A contact is defined between a set of nodes and a surface by specifying shear and tensile failure forces. These forces define the failure criterion and the connected nodes separate when a failure criterion is satisfied. According to the data provided by NASA-GRC, about 38.1–50.8 cm of the fabric is glued at either end of the continuous wrap. In the FE simulations, an average length of 40.6 cm is used. In the absence of experimental data to characterize the strength of Kevlar–Kevlar glue bond or Kevlar-steel bond, a regression study is done to obtain the force values used in the failure criterion. Four models (LG967, LG971, LG969 and LG970) from phase 3 were chosen for the regression study. In the phase 3 models (LG9XX), the projectile was either contained or uncontained with a very low exit velocity. The portion of the fabric, where the projectile makes contact, has large deformations causing a stress wave to originate from there and travel to the far end. These waves reflect multiple times and cause failure of fabrics. An examination of the

Fig. 4. SMS model: (a) failure and (b) sliding of fabric at far end.

test specimens showed no fabric failure at the back end. To remove the undue vibrations, the damping coefficient value defined for different contacts is increased. The error function which is the basis of the regression function is given as:

f ðEÞ ¼ E21 þ E22 þ E23 þ E24

ð9Þ

where E1 = percentage difference in absorbed energy; E2 = difference in the displacement between the experimental and simulation values at the node close to the point of impact of projectile; E3 = 4 times the failure at back end (failure is taken as either 0 for no failure or 1 if failure occurs); E4 = 4 times sliding of the last layers (sliding is taken as either 0 for no sliding or 1 if the last layer slides). The coefficient 4 is taken so that the contributions from all the four error terms are equal in magnitude. The difference between failure and sliding is shown in Fig. 4a and b. In the regression analysis, five parameters (tensile force and shear force for tiebreak between ring and fabric, fabric and fabric, and the viscous damping coefficient) are varied. Preliminary results showed that the force values at the ring-fabric interface did Table 4 Range of values used for tie-break force regression. Parameter

Low

Mid

High

Shear force between fabric and fabric (N) (A) Tensile force between fabric and fabric (N) (B)

2224 444.8

4448 2224

6672 3558.4

Table 5 Results of tie-break force regression. Run

A

B

E1

E2

E3

E4

f(E)

1 2 3 4 5 6 7 8 9

2224 6672 2224 6672 4448 4448 2224 6672 4448

444.8 444.8 3558.4 3558.4 3558.4 444.8 2224 2224 2224

17.06 17.06 17.06 17.06 17.06 17.06 17.06 17.06 17.06

12.85 12.23 11.8 11.8 11.8 12.23 11.8 12.23 11.8

0 0 4 12 12 0 4 0 4

16 16 16 4 0 16 12 16 4

45.91 45.29 48.86 44.86 40.86 45.29 44.86 45.29 36.86

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Table 6 Comparison of the difference in absorbed energy for SMS models with different damping values. Statistics

Average Minimum Maximum Std. Dev.

Table 8 Summary of results for CMS and SMS models. Model

Number of layers

LG404 LG409 LG424 LG594 LG609 LG610 LG611 LG612 LG618 LG620 LG689 LG692 LG429 LG432 LG411 LG427 LG656 LG657 LG963 LG966 LG965 LG964 LG967 LG971 LG969 LG970

8 8 8 8 8 8 8 8 8 8 8 8 16 16 24 24 32 32 4 8 16 17 24 24 32 32

% Difference in absorbed energy SMS 20–20

SMS 20–10

SMS 20–2

SMS 2–2

9.2 4.1 43.3 11.8

2.6 20.6 24.6 10.7

8.4 42.1 11.7 15.6

9.6 43.8 8.7 15

not affect the results. The waves cause problems in the outer layers. Also a high damping value was required to control the vibrations due to the stress waves. Hence a modified 2-parameter regression analysis is carried out. The values used are shown in Table 4. The value of shear and tensile force between the fabric and ring are held constant at 2224 and 444.8 N. The results of the regression runs are presented in Table 5. Run 9 has the most optimal value. Using these values for tie-break the damping values were varied. The effects of the variation of damping values were done in a separate study. Note that the convention 20–2 for damping means that the damping value was 20 for all contacts for the fabric which is wrapped around the ring while the value was 2 for all contacts of the fabric which were in contact with the projectile. Table 6 summarizes the results for the SMS models for different damping values. Clearly the case with 20–10 is the most optimum value. The average and standard deviation are the least for this case.

Average Minimum Maximum Std. Dev.

% Difference in Absorbed Energy v1.3 CMS SL

v1.3 CMS ML

v1.3 SMS

0.8 2.9 4.9 15.7 4.4 2.8 8.8 2.4 11.2 0.9 13.2 0.7 3.3 9.3 3.7 5.5 21.6 0.2 29.8 6.6 0 1.9 0 0 0 4.1

1.6 2.2 2.1 22.6 4.8 6.9 6.7 3.6 9.0 18.0 5.9 7.1 7.5 10.2 21.7 10.9 14.9 0 32.6 6.6 0 1.9 0 0 0 4.1

5.7 6.4 6.0 24.6 3.2 1.2 2.2 5.3 -6.2 17.5 20.6 8.9 0.9 0.6 21.3 0.5 5.4 0.4 32.6 6.6 0 1.9 0 0 0 4.1

1.0 29.8 21.6 9.4

1.5 32.6 21.7 11.3

3.2 32.6 20.6 10.9

4.3. Ring boundary conditions and material model In the actual NASA ballistic tests, the ring is mounted on a platform which is fixed to the ground. Thus the ring is fixed with respect to the ground. In the analysis described above the ring had no boundary conditions imposed. Simulations were done fixing the nodes on the bottom face of the ring. In the baseline analysis ring was modeled as an elastic material. It was however observed that for some of the cases the ring deformation was noticeable. To ensure that the plastic behavior (if any) is accounted for, simulations were run using Johnson–Cook model for the ring material. Table 7 provides the summary of results for different cases for the CMS and SMS models, respectively. The results do not show any significant difference amongst the models. The fixed-elastic case has the best results for the CMS model while Johnson–Cook model with fixed boundary of the ring has the best result for SMS model.

Model

LG963 LG965 LG964 LG967 LG971 LG969

4.4. Simulation results Once the models were calibrated, the entire suite of ballistic tests were simulated using the developed material and finite element model. Table 8 presents the summary of the results for both

Table 7 Percentage difference in absorbed energy for different ring fixity and material model cases for CMS (ML) and SMS models. Models

Statistics

Free, elastic (baseline model)

Fixed, elastic

Fixed, Johnson–Cook

CMS (ML)

Average Max Min Std. Dev. Average Max Min Std. Dev.

0.3 21.7 22.6 11.1 2.6 20.7 24.6 10.7

0.1 18.8 16.7 10.6 3.4 11.9 30.2 10.7

0.5 19.8 26.3 12.2 1.6 19.9 28.2 10.6

SMS

Table 9 Summary of displacement results for a point on fabric for phase 3 models.

LG970

Resultant displacement of the center point (cm) NASA-GRC Test (Time, ms)

Grid Point

CMS SL (Time, ms)

CMS ML (Time, ms)

SMS (Time, ms)

7.4 (3) 7.1 (2.5) 7.6 (3.2) 6.5 (2.6) 6.5 (2.3) 7.6 (2.4) 9.1 (2.4)

4

17.5 (3) 14.2 (2.2) 13.5 (1.6) 12 (2.4) 12 (1.6) 12.4 (1.5) 12.7 (1.3)

17.5 (3.3) 13.5 (2.3) 13.2 (1.7) 11 (2.6) 11.2 (1.7) 11.4 (1.6) 12.2 (1.5)

18 (3) 13 (2.3) 13.2 (1.6) 12 (2.5) 11.7 (1.7) 11.9 (1.7) 12 (1.5)

7 4 4 POI 10 4

CMS and SMS models with optimal damping values. For all the models the average error is close to zero and the standard deviation is close to 10%. All the three modeling schemes have about the same level of accuracy. The SL models show a slightly lower average and standard deviation but lack the resolution to provide useful damage estimation data. For SMS models the standard deviation is better than CMS-ML but CMS-ML has a better average. The number of models where the error is more than 10% is lesser in the SMS model. In the phase 3 models, the projectile velocities were very low and hence the projectiles had very low exit velocity or were completely contained. Simulation predicted that all the models were contained except for LG963 SL model. Apart from LG963 the errors for other models was less than 10%.

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D. Zhu et al. / Composites: Part B 56 (2014) 254–262 Table 10 Ballistic limit for different models using ASU v1.3.

Table 11 Overall summary for sensitivity analysis.

Number of layers

CMS SL (m/s)

CMS ML (m/s)

SMS (m/s)

8 16 24 32

207.3–216.4 243.6–251.5 271.3–280.4 335.3–342.9

189.0–195.1 228.6–243.8 259.1–266.7 281.9–289.6

204.2–219.3 251.5–259.1 266.7–274.3 274.3–280.4

Fig. 5. Sensitivity analysis of SMS model with different viscous damping coefficient values.

4.4.1. Displacement Displacement of the point on the fabric was the second metric of comparison. From the grid of 15 points (see Fig. 3), the point of maximum displacement was chosen. The displacement of the corresponding point in the simulation was also noted. Table 9 presents the values from the test as well as the simulation. The time at which the maximum displacement occurs is also presented. 4.4.2. Damage comparison When damage is compared for different test cases, we see that the simulation under predicts the damage in the fabric as compared to a test. Damage in simulation is characterized by some element deletions while it represents localized yarn breakage in the case of actual test. It should be noted that the definition of damage in the test and simulation is different. In the actual test, yarn failure and breakage in localized region constitutes damage. As a continuum model is used in the simulation, characterizing yarn damage is not present in the current model. The use of more sophisticated damage models is necessary and is the subject of our current investigations. 4.4.3. Ballistic limit Ballistic limit is defined as the maximum velocity at which the projectile is completely contained by the fabric layers. Table 10 presents the ballistic limits for different models for optimized values of damping. The ballistic limit is presented as a range. The projectile is contained at the lower limit while penetrates the fabric at the upper limit thus the actual ballistic limit is between the two values. 4.4.4. Sensitivity analysis A sensitivity analysis of the material model was performed to determine the multi-layer model’s (SMS) sensitivity to various parameters. The parameters which were used in the sensitivity analysis were the point of impact of the projectile, the roll, pitch, yaw angles, friction coefficient between fabric and steel, CowperSymonds parameters and the damping defined for contacts between fabric and projectile. For point of impact, the projectile was moved left, right, up and down by 6.35 mm. The roll, pitch

Parameter

Average (%)

Std. Dev. (%)

Max (%)

Min (%)

Viscous damping coefficient Cowper-Symonds factor – P Roll Coefficient of friction between fabric and steel Cowper-Symonds factor – C Yaw Pitch Point of impact

4.6 0.7 0.8 2.5

17.4 24.4 22.4 15.7

49.3 42.1 36.7 20.6

33.2 52.6 34.0 40.8

4.5 0.4 2.5 1.9

15.3 18.4 14.6 13.2

24.1 31.4 18.8 19.8

30.8 27.8 28.0 23.1

and yaw were adjusted by ±1°. The coefficient of friction values of 0, 0.05, 0.15 and 0.2 were used. Cowper-Symonds parameter P was varied (30, 35, 45, 50) while C was varied (0.003, 0.004, 0.006, 0.007). The viscous damping coefficient was changed to values 0, 2, 5, 20. The baseline model was v1.3 which included the optimized damping and tie break force values. Fig. 5 shows the plot of sensitivity analysis for viscous damping coefficient. Table 11 summarizes the results. Based on the results obtained the parameters were ranked based on maximum sensitivity to least sensitivity and are as follows: (1) (2) (3) (4) (5) (6) (7) (8)

Viscous damping coefficient Cowper-Symonds factor – P Roll Coefficient of friction between fabric and steel Cowper-Symonds factor – C Yaw Pitch Point of impact

5. Concluding remarks The continuum model developed in previous research for modeling of dry fabrics has been improved on several fronts. The shear properties of the fabric and the parameters (C and P) in Cowper-Symonds (CS) rate dependent model were optimized using regression analysis. The material model was validated by comparing the FE simulation with ballistic test results. Both single and multi-layer models were generated in the modeling configuration. The single layer model is computationally efficient and predicts the ballistic tests with about the same level of accuracy as the multi-layer models. While there are probably a few reasons for this behavior, it would appear that the main reason is that the optimization of the LS-DYNA parameters were carried out using the single layer models. However, the SL model does not have the fidelity required for a detailed analysis of the fabric performance. The SMS model is better than CMS – difference in absorbed energy compared to the experimental results and the number of models having a large difference in absorbed energy. The material model over predicts the displacement of the fabric while under predicting the damage. The SMS predicts a higher ballistic limit than the CMS ML model. The optimized strain rate parameters, shear parameters, tie break forces and damping coefficient bring about an improvement in the ability of models predicting the results more accurately. Sensitivity analysis was done on the material model to find out the effects of varying different parameters on the performance of the material model. The material model, modeling techniques and experimental data have their own errors and limitations. First, the fabric is modeled as a continuum model and hence the damage progress and predictions, in the absence of yarn and filament level modeling,

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are more difficult to capture. Second, in the model, 4 physical fabric layers are represented by one FE layer leading to a loss of modeling resolution. Third, it is not clear what role if any, rate dependency plays in the behavior of the fabric. Fourth, there are several sources of experimental errors. The roll, pitch, yaw and velocity measurements are based on image analysis. Only the translation velocity is accounted for in the FE analyses though an analysis of the test data show that some projectiles also have low rotational velocities. The exact point of impact in the test is difficult to obtain. Last, there are several sources of errors in the explicit finite element analysis including discretization errors, element formulation errors, errors in contact algorithm and calculations etc. Nevertheless, results show that the developed model provides very reasonable predictive capabilities. Acknowledgement The authors wish to thank Mr. William Emmerling, Mr. Donald Altobelli and Mr. Chip Queitzsch of the Federal Aviation Administration for their technical and financial support. References [1] Nilakantan G, Keefe M, Gillespie Jr. J, Bogetti T. Novel multi-scale modeling of woven fabric composites for use in impact studies. In: 10th International LSDYNA users conference. Dearborn, Michigan, USA; 2008. [2] Rajan SD, Mobasher B, Vaidya A. LS-DYNA implemented multi-layer fabric material model development for engine fragment mitigation. In: 11th International LS-DYNA users conference. Dearborn, Michigan, USA; 2010. [3] Cheeseman B, Bogetti T. Ballistic impact into fabric and compliant composite laminates. Compos Struct 2003;61(1–2):161–73. [4] Tabiei A, Nilakantan G. Ballistic impact of dry woven fabric composites: a review. Appl Mech Rev 2008;61:10801–13. [5] Johnson G, Beissel S, Cunniff P. A computational model for fabrics subjected to ballistic impact. In: Proceedings of the 18th international symposium on ballistics; 1999. p. 962–9. [6] Billon H, Robinson D. Models for the ballistic impact of fabric armour. Int J Impact Eng 2001;25(4):411–22. [7] Zohdi T, Powell D. Multiscale construction and large-scale simulation of structural fabric undergoing ballistic impact. Comput Methods Appl Mech Eng 2006;195(1–3):94–109. [8] Ivanov I, Tabiei A. Loosely woven fabric model with viscoelastic crimped fibres for ballistic impact simulations. Int J Numer Meth Eng 2004;61(10):1565–83. [9] King M, Jearanaisilawong P, Socrate S. A continuum constitutive model for the mechanical behavior of woven fabrics. Int J Solids Struct 2005;42(13):3867–96. [10] Shahkarami A, Vaziri R. A continuum shell finite element model for impact simulation of woven fabrics. Int J Impact Eng 2007;34(1):104–19. [11] Grujicic M, Bell W, Arakere G, He T, Cheeseman B. A meso-scale unit-cell based material model for the single-ply flexible-fabric armor. Mater Des 2009;30(9):3690–704.

[12] Duan Y, Keefe M, Bogetti T, Powers B. Finite element modeling of transverse impact on a ballistic fabric. Int J Mech Sci 2006;48(1):33–43. [13] Zhang G, Batra R, Zheng J. Effect of frame size, frame type, and clamping pressure on the ballistic performance of soft body armor. Compos B Eng 2008;39(3):476–89. [14] Borovkov AI, Voinov IB. FE analysis of contact interaction between rigid ball and woven structure in impact process. In: Proceedings of 8th International LS-DYNA users conference. Dearborn, Michigan, USA; May 2–4 2004. [15] Rao M, Duan Y, Keefe M, Powers B, Bogetti T. Modeling the effects of yarn material properties and friction on the ballistic impact of a plain-weave fabric. Compos Struct 2009;89(4):556–66. [16] Chocron S, Figueroa E, King N, Kirchdoerfer T, Nicholls AE, Sagebiel E, et al. Modeling and validation of full fabric targets under ballistic impact. Compos Sci Technol 2010;70(13):2012–22. [17] Scott BR, Yen CF. Analytic design trends of fabric armor. In: 22nd international symposium on ballistics. Vancouver, BC, Canada; 2005. p. 752–60. [18] Barauskas R, Abraitiene A. Computational analysis of impact of a bullet against the multilayer fabrics in LS-DYNA. Int J Impact Eng 2007;34(7):1286–305. [19] Nilakantan G, Keefe M, Bogetti T, Adkinson R, Gillespie Jr J. On the finite element analysis of woven fabric impact using multiscale modeling techniques. Int J Solids Struct 2010;47:2300–15. [20] Nilakantan G, Keefe M, Bogetti TA, Gillespie JW. Multiscale modeling of the impact of textile fabrics based on hybrid element analysis. Int J Impact Eng 2010;37(10):1056–71. [21] Shim V, Tan V, Tay T. Modelling deformation and damage characteristics of woven fabric under small projectile impact. Int J Impact Eng 1995;16(4):585–605. [22] Tan V, Shim V, Zeng X. Modelling crimp in woven fabrics subjected to ballistic impact. Int J Impact Eng 2005;32(1–4):561–74. [23] Stahlecker Z, Mobasher B, Rajan S, Pereira J. Development of reliable modeling methodologies for engine fan blade out containment analysis. Part II: finite element analysis. Int J Impact Eng 2009;36(3):447–59. [24] Bansal S, Mobasher B, Rajan SD, Vintilescu I. Development of fabric constitutive behavior for use in modeling engine fan blade-out events. J Aerospace Eng 2009;22(3):249–59. [25] Naik D, Sankaran S, Mobasher B, Rajan S, Pereira J. Development of reliable modeling methodologies for fan blade out containment analysis-Part I: Experimental studies. Int J Impact Eng 2009;36(1):1–11. [26] Zhu D, Mobasher B, Vaidya A, Rajan SD. Mechanical behaviors of Kevlar 49 fabric subjected to uniaxial, biaxial tension and in-plane large shear deformation. Compos Sci Technol 2013;74(1):121–30. [27] Wang Y, Xia Y. The effects of strain rate on the mechanical behavior of Kevlar fiber bundles: an experimental and theoretical study. Compos. A 1998;29(11):1411–5. [28] Zeng X, Tan V, Shim V. Modelling inter yarn friction in woven fabric armour. Int J Numer Meth Eng 2006;66(8):1309–30. [29] Duan Y, Keefe M, Bogetti T, Cheeseman B, Powers B. A numerical investigation of the influence of friction on energy absorption by a high-strength fabric subjected to ballistic impact. Int J Impact Eng 2006;32(8):1299–312. [30] Duan Y, Keefe M, Bogetti T, Cheeseman B. Modeling the role of friction during ballistic impact of a high-strength plain-weave fabric. Compos Struct 2005;68(3):331–7. [31] Revilock D, Pereira J. Explicit finite element modeling of multilayer composite fabric for gas turbine engine containment systems, Part 2: Ballistic impact testing. Washington, DC: Office of Aviation Research and Development; 2008. [32] Zhu D, Mobasher B, Rajan SD, Peralta P. Characterization of dynamic tensile testing using aluminum alloy 6061 T6 at intermediate strain rates. J Eng Mech 2011;137(10):669–79.

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