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The influence of deformation waves on impact energy absorption and heat release in multi-layer woven fabric ballistic body armor
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The influence of deformation waves on impact energy absorption and heat release in multi-layer woven fabric ballistic body armor O.N. Budadina, V.O. Kaledinb, V.P. Vavilovc,d, A.E. Gilevab, S.O. Kozelskayaa, M.V. Kuimovac,∗ a
Central Research Institute for Special Machinery, Zavodskaya St., Khotkovo, Russia Kemerovo State University, Krasnaya St., 6, Kemerovo, Russia c National Research Tomsk Polytechnic University, Lenin Av., 30, Tomsk, Russia d National Research Tomsk State University, Lenin Av., 36, Tomsk, Russia b
A R T I C LE I N FO
A B S T R A C T
Keywords: Armor protection Composite material Infrared thermography Heat release Energy absorption Mathematical model
The mechanical and thermo-mechanical damage mechanisms of a multi-layer woven polymeric fabric ballistic body armor are analyzed. A proposed two-stage model takes into account the wave interaction between an impacting element and an armor package at the initial stage of the collision. The model allows the prediction of initial fabric compaction and the estimation of velocities and displacements. These can then be used to predict the response of a layered flexible barrier and the contribution of stresses and frictional forces in causing permanent deformation.
1. Introduction The modern textile industry has developed textile composite materials and protective apparel made with them. These have been thoroughly discussed in Ref. [1]. The development, designs, materials and manufacturing methods of soft body armors is documented in Ref. [2] where the effects of ballistic impact on single- and multi-layer fabrics are also presented, and numerical models that predict stress and transverse wave propagations, projectile deformation, and fabric penetration are included. Ballistic performance of ceramics as function of their structure and properties, armor system design and type of projectile has been discussed by Medvedovski [3,4]. Mathematical modeling of fiber slippage was presented by Sueki et al. [5]. The estimation of energy absorption in woven layered fabrics is a challenging task in designing lightweight ballistic body armor. The scattering of energy takes a significant part in ensuring armor material quality due to friction between fabric fibers. According to known estimates, this phenomenon is responsible for absorbing up to 30% of impact energy. The experimental estimation of the amount of absorbed energy is possible by the indirect effect – the change in the temperature of the fabric during the mutual collision process. The corresponding measurements can be performed thermographically, but only on the package surface. Therefore, to interpret the experimental data, it is necessary to use a mathematical model of thermo-mechanical processes occurring in layered fabrics [6–11].
∗
A simplified mathematical model proposed in Ref. [12] and later improved in Ref. [13] represented a woven fabric as a layered continuous medium thus allowing the calculation of non-stationary temperature distributions by integrating the equations of motion at the stage of deformation. However, to analyze the differences in heat release in different layers of a package, a reasonable formulation of the initial conditions for the equations of motion is necessary. In fact, in the simplified models the initial moment of time is characterized by the singularity when a damage agent (DA) moving with a high velocity comes into contact with a motionless target. This problem can be solved by introducing a model of a membrane-like body subjected to deformation. Such realistic model should take into account the difference in the initial velocities in layers which are located at different depths in regard to the front layer. For this, one may estimate velocities and displacements near the point of impact that can be obtained by considering the wave stage of collision. The corresponding model is presented in this paper. 2. Experimental basis and model presentation The initial stage of the collision between a DA with a multi-layer woven fabric barrier has been considered elsewhere [6,11,12]. Some models of the mechanical processes occurring have been analyzed in details in Ref. [12]. It is noted that the velocity of the sound wave is commensurable with the initial velocity of the DA, and, as a result, the
Corresponding author. E-mail address: [email protected] (M.V. Kuimova).
https://doi.org/10.1016/j.ceramint.2019.08.149 Received 31 July 2019; Accepted 15 August 2019 0272-8842/ © 2019 Elsevier Ltd and Techna Group S.r.l. All rights reserved.
Please cite this article as: O.N. Budadin, et al., Ceramics International, https://doi.org/10.1016/j.ceramint.2019.08.149
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deformation of the composite barrier remains localized for a certain time within the region where the wave front arrives. Based on the difference in the dynamic elastic moduli of tension (along the fibers) and shear, the velocity of the longitudinal and transverse waves has been estimated. It has been assumed that the area of damage is limited to the zone where the growth rate of the radius of the contact spot exceeds the velocity of transverse acoustic waves while a DA is penetrating through the sample. The theoretical evaluation of this radius shows a good match with the experiment. It is worth mentioning that the velocity of the longitudinal wave propagating in the transversal direction depends on the transverse elastic modulus, which is substantially less than the longitudinal one being comparable to the dynamic shear modulus. One of the proposed models is as follows [11]. Irreversible deformations of the normal compression are due to the tightening of fabric layers having a certain initial porosity. Deflections of the fibers near the surface of the contact with the DA lead to their elongation, and the deformation is greater with a smaller radius of DA curvature. Despite the shear nature of the initial destruction, the mechanism of damage involves the breakage of fibers occurring due to longitudinal stresses. Thus, it can be assumed that at the initial stage of impact, the displacements in a small-sized contact zone are directed along the normal to the package surface, and the dominant deformation presents compaction along the normal. Fabric, of which volume consists of 50–60% pores, becomes compact due to irreversible deformation of the normal compression. These phenomena should be supplemented by taking into account the heat release occurring due to irreversible compression. The fabric in the contact zone is being essentially heated, and its properties change significantly. According to Refs. [2,12–15], the temperature rise in the zone of contact with an impacting element may exceed 100°С (see the example in Fig. 1). In combination with a high compressive stress, this leads to the sealing of polymer fibers. Since both the thermo-physical properties of fabrics at high temperatures are known with a high degree of uncertainty, a detailed modeling of all processes at the micro-structure level seems to be difficult if computing resources are limited. Acceptable adequacy of the model can be monitored by checking the momentum conservation law.
Fig. 2. Layer deformation scheme at initial stage of impact.
where rп is the DA radius, V is the initial velocity of the impacting element, a is the velocity of shear waves. Consider the deformation of the fabric layer closest to the DA (Fig. 2) and limit ourselves to the time interval during which the shock wave passes a distance equal to the thickness of this layer. The initial porosity of the fabric is μ0, the thickness of the layer is h, the density of the fabric is ρ and the modulus of elasticity in the transverse direction is Ez. At the moment of contact of the DA, whose center is at the point О1, with the front surface of the armor fabric barrier, the sound wave starts to pass through the thickness of the package in the direction of the z axis. During the time Δt, the wave front reaches the rear surface of the layer z = h, and the center of the DA moves over the distance of V⋅Δt to the O2 position. The surface z = 0 moves over the distance of V⋅Δt, and the displacement of the surface z = h equals zero. Thus, the layer thickness becomes equal to h = V⋅Δt. Taking into account the fact that the volumic fraction of pores μ in the layer is non-negative and cannot exceed μ0, its thickness after a complete compaction becomes h (1-μ0), and with incomplete compacting it can be greater than this value. Then one of two situations may occur: if V⋅Δt < μ0 h, the distance covered by the DA is not enough for a complete compaction, thereby only crumple deformation of the fibers occurs; if V⋅Δt > μ0h, the closure of all pores is not enough to reduce the thickness, and further elastic compression deformation of the compacted fabric occurs. Since the deformations reach μ0 = 50–60% [6], we take the logarithmic measure of compressive deformations:
3. Approximate description of the deformation at the initial stage of impact Assume that the displacements of the fabric of the woven package are significant in the local zone, of which radius can be determined as follows [6]:
ra =
rп V V 2 + a2
,
∂u ⎞ ε = ln ⎛1 + , ∂z ⎠ ⎝
(2)
where u is displacement in the direction of impact along the axis z. The governing equations can be formulated by considering the deformation as elastic-plastic. The closure of pores occurs at a stress equal to the stress limit of the fabric during transverse compression; this limit, i.e. crumpling stress, is determined experimentally, and some measurement results are reported in Refs. [15–17]. The current porosity of the fabric layer μ and the irreversible deformation of the crumpling of the normal εcr < 0 (“cr” specifies crumpling) sum up to the initial porosity: − εcr = μ0 − μ . When simultaneously meeting the conditions ε˙ < 0 and μ < μ0, one may consider the loading to be active (henceforward, the points denote time derivatives). With active loading, the growth of irreversible deformation of compression occurs at the constant stress:
(1)
σz = σcr , ε˙ cr = ε ˙,
εcr < 0 .
(3)
When the zero porosity is reached, or when the compression deformation is reduced, a status of unloading appears being described by linear equation:
Fig. 1. Example of the temperature distribution on the surface of a ballistic armor vest (projectile velocity 506 m/s, infrared image acquisition 146 Hz, FLIR X6530sc infrared imager).
σz = Ez (ε − εcr ) 2
(4)
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Fig. 3b shows the change of velocity in each layer at different times. The solid line 4 corresponds to initial time when, for the 1st layer, the velocity was 300 m/s and zero for other layers. At time point 4 μs (dashed line 5), a non-zero velocity is observed from the 1st to the 7th layer. This indicates that the shock front reaches the rear surface of the 7th layer. At the last time point (dash-dotted line 6), the wave front reaches the last layer causing a change in velocity in all layers of the package. Fig. 4a shows the change in displacements for the 1st, 2nd and last 15th layer (note a wave character of displacements). The largest values of displacement are observed in the first layer of the package. The displacements in the last layer slightly increase in time until 10 μs; afterwards, when the wave front reaches this layer, the displacements start to increase sharply and reach a maximum value of 1.35 mm. In their turn, the curves in Fig. 4b show that, in spite of the wave character of motion in the process of compaction, the distribution of displacements across the package seems to be linear immediately from the time of impact and until at least 10 μs (curve 4 corresponds to zero displacement at the very beginning of the wave process). This is an argument in favor of a membrane shell model when describing displacements after the wave stage of the process. Fig. 5 shows crumpling deformation curves for five layers of the package: 1, 2, 5, 10 and 15. If deformation reaches the value of 0.5, equal to the initial porosity, this means that all layer pores become closed. This appears at the time of 0.375 μs (1st layer), 1.15 μs (2nd layer), 3.175 μs (5th layer), 6.6 μs (10th layer) and 10.375 μs (15th) layer. Thus, in the given example, the complete package compaction is achieved at 10.375 μs. Displacements at this time are distributed through the sample according to a law close to linear, and the velocity varies irregularly through the sample thickness. These calculations are easily generalized in the case of axisymmetric deformation, in which the governing equations contain, along with the transverse elastic modulus, a dynamic shearing modulus, which can be expressed through the velocity of the transverse wave. The axisymmetric problem is set and solved in a cylindrical region whose axis coincides with the axis of impact, and the boundary radius is equal to the distance that the transverse wave travels during the wave process. Taking the found displacements and velocities as initial ones, one can further obtain the variables of the layer state at the shell stage of deformation.
Stresses are determined by Eq. (4) even in the case where, after unloading, the compressive deformation increases to the value of irreversible deformation; after that, the loading becomes active. Due to fabric compaction, the modulus of elasticity increases in proportion to the degree of compaction:
Ez =
Ezin , 1 + εcr
(5)
Ezin
is the initial value of the elasticity modulus (at low dewhere formation). The displacements u (z , t ) and velocity u˙ (z , t ) are determined by solving the equation of motion under the action of stresses defined by Eqs. (3) and (4):
ρu¨ =
∂σz ∂z
(6)
under the initial conditions: u (z , 0) = 0 , u˙ (z , 0) = 0 at z > 0, u˙ (0,0) = V , and stress boundary conditions on the front and back surface of the package:
πra2 σz (0, t ) = Mu¨ (0, t ), σz (N ⋅h, t ) = 0,
(7)
where M is the mass of the DA, N is the number of package layers. The solution is obtained numerically using an implicit difference scheme. Computation errors were controlled by changing the full moment of momentum and applying the Runge rule. The period of integration was chosen by 50% longer than the time needed to fully compact all layers of the package. It is worth noting that, during this time, the calculated amount of movement computationally changed by no more than 10−13 of the initial DA moment of momentum. Figs. 3–5 show the graphs of the dependencies calculated. In Fig. 3a, the solid line 1 shows the evolution of velocity in time for the 1st layer of the package. This curve has the form of a smooth descending curve, which means that no wave phenomena occur in this layer after the impacting element penetrates through the layer. The dotted line 2 indicates the curve corresponding to the velocity distribution for the 2nd layer: at the time t = 0.025 μs, the velocity increases to the value of 407 m/s, afterwards, the oscillations decay. For the last 15th layer (dash-dotted line 3), the velocity gradually increases until the time point t = 10 μs, then there is a sharp increase in speed to 375 m/s, following which the velocity decreases with slight fluctuations.
Fig. 3. Fabric velocity relationships: a - velocity vs. time in the 1st, 2nd and 15th layers (curve numbers: 1 – 1st, 2 - 2nd layer, 3–15th layer from impacted surface), b –velocity vs. layer number (4 - initial time point, 5 - time point 4 μs, 6 - last time point 15 μs). 3
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Fig. 4. Fabric displacement relationships: a - displacement vs. time in the 1st, 2nd and 15th layers (curve numbers: 1 – 1st, 2 - 2nd layer, 3–15th layer from impacted surface),b –displacement vs. layer number (4 - initial time point, 5 - time point 4 μs, 6 - last time point 15 μs).
to take the front surface of the package contacting with the DA as a baseline. The transportation movement of the initial surface occurs in the direction of the Z axis of the Euler coordinate system located in parallel with the velocity vector of the DA. In the initial position, when laying the warp fibers along the edge of the sample, the base vectors of the Lagrangian coordinate system eα and eβ coincide with the base vectors of the Eulerian coordinate system ex and ey. If the layup angle in relation to the edge ϕ is non-zero, then the following transformation of the basis is applied:
eα0 = e X ⋅ cos φ − eY ⋅sin φ , eβ0 = eX ⋅sin φ + eY ⋅cos φ .
(8) (eα0,
eβ0,
en0 )
Initial bases (eX, eY, eZ) and are orthogonal. In the process of movement, the axes of the Lagrangian coordinates are curved forming the surface defined by the vector equation in the form of the dependence of the radius-vector on two coordinates:
⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ M = M (α, β ), or M = M (x , y )
(9)
The surface defined by equation (9) is curvilinear, and the α and β or ⎯⎯⎯→ ⎯⎯⎯→ x and y lines are its coordinates. The vectors Mα and Mβ which are tangent to the coordinate lines α and β can be found by differentiating the surface equation by the corresponding coordinate, taking into account the fact that the transportation motion occurs only along the Eulerian axis Z [11]:
Fig. 5. Crumpling deformation vs. time (curve numbers correspond to layer numbers counting from impacted surface).
4. Thermo-mechanical processes at the shell stage of deformation
⎯→ ⎯ Mα =
The model of coupled thermo-mechanical processes at the shell stage of deformation is based on the approximate model [12], in which the movement of a layer package is considered as combined, transitive movement of the package as a whole, as well as relative movement of fibers in the layers of fabric. Considering different initial conditions in the layers, one may introduce two coordinate systems for each layer: the Lagrangian system whose basis changes as it moves, and the Eulerian system which coincides with the initial position of the Lagrangian system. The kinematics of the transportation motion of a package as a whole is determined by the hypotheses of the Reddy shell theory [7,17]: the material normal to the original middle surface during deformation remains rectilinear, but rotates, forming an angle of transverse shear with the geometric normal to the deformed middle surface, and its length changes by the magnitude of normal compression. It is more convenient
⎯→ ⎯ Mβ =
⎯→ ⎯ ∂M ∂α ⎯→ ⎯ ∂M ∂β
= eα0 +
∂uz 0 ⋅e , ∂α z
= eβ0 +
∂uz 0 ⋅e . ∂β z
(10)
Introduce the following notations:
uα′ =
∂uz ∂uz , uβ′ = ∂α ∂β
(11)
Omitting detailed calculations, the final results are as follows. The differential of an arbitrary arc γ 2
ds γ =
2
∂β ∂α ∂α ∂β (1 + uα′ 2) ⎜⎛ ⎞⎟ + 2uα′ uβ′ + (1 + u′β2) ⎛⎜ ⎞⎟ dγ ∂γ ∂γ ⎝ ∂γ ⎠ ⎝ ∂γ ⎠
(12)
can be used to determine the extension of an arbitrary line, in particular, elongations along the warp and weft. The curvature and torsion 4
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of the surface in coordinates (α, β) are characterized vy the following expressions:
−
−
2
1 + εr
L 1 = 112 2 = − , Rα Aα dα 1 + uα′ + uβ′ ⋅(1 + u′α2)
1 L12 = = R αβ A α Aβ
(13)
1+
(1 + u′α2)(1 + u′β2)
Here, Rα and Rβ are the radii of the curvature along the warp and weft, and Rαβ is the reciprocal to the torsion of the initial surface. Deformations of fiber elongation on the base surface are: 2
2
∂uz ⎞ 1+⎛ , ⎝ ∂α ⎠
1 + εβ =
∂uz ⎞ 1 + ⎛⎜ ⎟ ⎝ ∂β ⎠
(14)
The coefficients of the distance increase between the fibers on the base surface are as follows:
1 + εˆα =
∂uz 2 ∂β
∂uz 2 ∂α
( ) +( ) 1+
∂uz 2 ∂β
1+ ,
( )
1 + εˆβ =
∂uz 2 ∂α
∂uz 2 ∂β
( ) +( ) 1+( ) ∂uz 2 ∂α
(15) At the distance n from the base surface, the displacements in the transportation motion additionally depend on the deformation of compression of the normal and angles of the transverse shear. Assuming that the compression deformation of the normal εn is constant over the thickness of the package, one may find the displacement of a point with the Lagrangian coordinates (α, β, n) in relation to a point located on the base surface:
uz (α, β, n) = u z (α, β , 0) + εn
n = uz (α, β , 0) + εn n 1 + u′α2 + u′β2 nz (16)
Then the elongations of the layer fibers separated from the base surface by a distance n = Z are determined by the following expressions:
1 + εα (Z ) =
∂ (uz + uz⁎) ⎤2 1+⎡ , ⎥ ⎢ ∂α ⎦ ⎣
1 + εβ (Z ) =
∂ (uz + uz⁎) ⎤2 1+⎡ ⎥ ⎢ ∂α ⎦ ⎣ (17)
The transverse shear angle γn can be found by considering the transfer displacement of each layer that is directed along the Z axis of the Euler coordinate system, as the angle between the geometric and material normals:
tg2γn = u′α2 + u′β2, γn = arctg u′α2 + u′β2 ,
(18)
then the mutual displacement of the layers can be calculated as follows:
Δu = h⋅tgγ = h uα′ 2 + uβ′ 2
2 ∂uβ(β ) ⎞ ∂u (β ) ⎛ 1+ + ⎜⎛ α ⎟⎞ ⎜ ∂β ⎟ ⎝ ∂β ⎠ ⎝ ⎠
(19)
The relative movement of the fibers within each layer is uniquely determined by four kinematic parameters – relative displacements of the warp fibers ur(α ) and weft fibers ur(β ) [13]:
ur(α ) = uα(α ) eˆα + uβ(α ) eˆβ , ur(β ) = uα(β ) eˆα + uβ(β ) eˆβ
(21)
Thus, at the shell stage, the kinematics of deformation is determined by the following displacement components: two transfer components, namely, displacement of the front surface along the axis Z and normal compression of the normal, and four relative components of fiber displacement, see Eq. (20). Through these functions, each of which depends on two coordinates, one may obtain the deformations of the fibers and relative displacements of the layer fibers. Unlike the models discussed in Ref. [13], in this case, the conditions for the deformation of the layers include both the normal compression and differences in the normal displacements in the fabric layers. The deformations and rates of deformation determine the stresses in the fibers and tangential stresses between the fibers and the fabric when the fibers are pulled out of the fabric, as well as the tangential stresses between the layers of the fabric. In fibers, tensile stresses non-linearly depend on elongation. The governing equation for tensile stresses is obtained from stress-strain diagrams under loading of samples with subsequent unloading. The tests were carried out on flat rectangular samples, which were a strip of fabric of ten fibers connected by a binding. The sample width was 3.5 mm and thickness – 0.5 mm. The distance between the grips of the tension testing machine was 87 mm. In total, two batches of flat rectangular samples of the same type were tested, the first batch included 20 samples, the second - 6 samples. The experiments were carried out by applying uniaxial tension, the loading speed varied from 4 to 60 mm/min. In the first session of experiments, the samples were subjected to a tensile load to reach the limit value at which the fibers broke. Fig. 6 shows a typical view of the sample after fracture accompanied with fiber breakage. In the second session of experiments, the samples were first loaded to ~2000 N, afterwards, they were unloaded and reloaded. In the process of loading, the values of the actual load and grip travels were recorded at the stage of both active loading and unloading. Fig. 7 shows a typical stretch curve which illustrates the linear relationship between actual tensile load and grip travel. The interpretation of the experimental results included recalculating curves of deformation into the “stress – strain” coordinates, then approximating them with the second-degree polynomials and determining the initial and final moduli of elasticity, ultimate deformation and ultimate stress. The results of processing are presented in Table 1. The maximum value of the coefficient of variation in all parameters is 7.45% thus indicating a weak variability of the data and no need of excluding extreme values. The friction of the fibers is due to their slippage. Following [11], one may take the dimensionless quantity, namely, the ratio of the displacement of a fiber along its direction to the thickness of the layer, as the kinematic parameter of slippage:
u″αβ u′α2 ⋅
(α ) 2
(α ) ⎛1 + ∂uα ⎞ + ⎛ ∂uβ ⎞ , ⎜ ⎟ ⎜ ∂α ⎟ ∂α ⎠ ⎝ ⎝ ⎠ 2
1 + εr (β ) =
′′ uββ L dβ 2 1 = 222 2 = − , Rβ Aβ dβ 1 + u′α + u′β ⋅(1 + u′β2)
1+
=
′′ uαα
dα 2
1 + εα =
(α )
(20)
where, eˆα , eˆβ are the Lagrangian basis vectors directed tangentially to the warp and weft respectively. Here the subscript denotes the components of the displacement, and the superscript is related to the family of fibers. Then the elongation of the fibers in relative motion can be calculated as follows:
Fig. 6. Photo of the thin strip-like sample after fracture. 5
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Fig. 7. Experimental stretch curve.
γα(α ) =
uβ(β ) uα(α ) , γβ(β ) = h h
relative displacements, it turns out to be expedient to use the Lagrangian spline on the same finite elements. When integrating by time, a splitting scheme has been applied to save calculation time. Each step of time integration consists of two stages: a transport one, in which relative displacements are considered fixed, and a relative one, in which only relative displacements with fixed transfer displacements vary. At each stage, this numerical scheme is brought to the form, which provides absolute stability [12,18]. As a result of solving the motion equations, the velocities, displacements, deformations, and stresses have been calculated thus making possible to calculate the work of stresses on irreversible deformations. Assuming that the work of the friction forces completely converts into heat, and the work of the destruction of the fibers transits into heat partially, one may determine the heat spent for increasing the fabric temperature. Despite the temperature difference between layers, the temperature change due to heat conduction after the impact is insignificant because of a short duration of collision. Thus, the suggested model of the thermomechanical processes in a woven barrier under collision with a damage agent allows accounting for the different conditions of energy absorption in layers located at different distances from the front surface of the barrier.
(22)
The friction forces, applied to the fibers, do work on relative displacements; the friction forces applied along the warp fibers do the work on the displacements uα(α ) , and those applied to the weft fibers – on the displacements uβ(β ) . By attributing the friction forces to a small area in the fabric plane, one may obtain the values of τα and τβ expressed in terms of stress. The work related to the unit volume of fabric will be equal to the product of the obtained stresses and the deformations defined by Eq. (22). While the fibers are at rest, the friction forces are either absent or do no work. Therefore, the expression linking the stresses τα and τβ with deformations may be defined as follows:
τα = η⋅γ˙α(α ), τβ = η⋅γ˙ β(β )
(23)
In the same form, the equation describing the friction forces between the layers can be written as follows:
τn = ηn⋅γ˙n,
(24)
where τn is the friction force per unit area of a contact zone between layers, and γn is determined by Eq. (18). In this way, the internal forces are expressed in terms of the kinematic parameters of the layer transport motion and the relative motion of the fibers thus making possible to derive the motion equations. The boundary conditions for the transport motion are the conditions for fixing the edges, and the initial conditions are taken from the calculation of the wave stage of the mutual collision. A numerical scheme for solving the problem above can be described as follows. To approximate displacements in the transport motion, the Ermit spline has been used. This allows one to obtain both linear displacements of the front surface of a layer package and their derivatives. For
5. Conclusion Current models of the thermo-mechanical interactions in an impact between a projectile and a simple membrane shell, do not adequately represent a projectile impact with a more complex multi-layer woven fabric ballistic body armor structure. The simple models do not take into account a singularity that occurs at the initial projectile impact. They also do not take into account the friction and resultant heat generation that occurs between the individual layers of the multi-layer structure as the projectile deforms the layers. In this study, a proposed two-stage model considers a wave interaction between an impacting
Table 1 Calculated physico-mechanical parameters. Sample parameter
Maximum deformation ε, %
Maximum stress σ, GPa
Initial modulus of elasticity E0, GPa
Final module elasticity Ef, GPa
Mean value Root-mean-square deviation Minimal value Maximal value Range Relative range, % Coefficient of variation, %
5.53 0.410 4.48 6.54 2.06 37.2 7.45
1.08 0.0740 0.929 1.24 0.314 29.0 6.86
21.8 0.745 20.9 23.25 2.32 10.6 3.41
16.2 0.969 15.3 17.20 1.93 11.8 3.29
6
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body and a multi-layer fabric structure. The model accounts for the initial compression of the fabric and it estimates initial velocities and displacements. These are used for subsequent modeling of the motion of a multi-layer flexible barrier and the contributions of stresses and frictional forces in the case of irreversible deformations.
[8] [9] [10]
Acknowledgment
[11]
This study was conducted within Tomsk Polytechnic University Competitiveness Enhancement Program.
[12]
References
[13]
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7
under elastic-plastic deformation of materials of composition components, PNRPU Mechanics Bulletin 2 (2018) 133–146, https://doi.org/10.15593/perm. mech/ 2018.2.12 (in Russian). A.L. Gol'Denveizer, Theory of Elastic Thin Shells, first ed., Pergamon Press, U.S.A., 1961, p. 680. W. Olszak (Ed.), Thin Shell Theory: New Trends and Applications, Springer-Verlag, Wien, Austria, 1980, p. 310. G. Wempner, D. Talaslidis, Mechanics of Solids and Shells: Theory and Approximations, CRS Press, Boca Raton, 2002, p. 497. N. Feito, J.A. Loya, A. Muńoz-Sánchez, R. Das, Numerical modeling of ballistic impact response at low velocity in aramid fabrics, Materials 12 (13) (2019) 2087–2098. V.O. Kaledin, O.N. Budadin, S.O. Kozelskaya, A.E. Gileva, Quality control of armor fabric by modeling thermomechanical processes under projectile impact, Russ. J. Nondestruct. Test. 54 (5) (May 2018) 363–371. O.N. Budadin, S.O. Kozelskaya, V.O. Kaledin, V.P. Vavilov, M.V. Kuimova, Evaluating impact damage to fabric-based personal armor by infrared NDT, Int. J. Damage Mech. (2019) 1–25 Published online 8 January 2019 https://doi.org/10. 1177/1056789518823880. A.G. Nilakantan, Virtual ballistic testing of Kevlar soft armor: predictive and validated modeling of the V0-V-100 probabilistic penetration response using LS-DYNA, Proc. 15th Intern. LS-DYNA Users Conf., 10-12 June 2018, Detroit, Michigan, U.S.A, 2018, pp. 1–12. W. Świderski, P. Hlosta, Non-destructive evaluation of impacted CFRP by IR thermography, Materials 12 (6) (2019) 956–967. C.L. Sorrentino, C. Bellini, A. Corrado, W. Polini, R. Aricò, Ballistic performance evaluation of composite laminates in Kevlar 29, Procedia Engineering 88 (2014) 255–262. J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, second ed., CRC Press, Boca Raton, FL, U.S.A, 2004, p. 812. S. Blanes, F. Casas, A. Murua, On the linear stability of splitting methods, Foundations of Computational Mathematics, June 8 (3) (2008) 357–393.
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Ceramics International journal homepage: www.elsevier.com/locate/ceramint
The influence of deformation waves on impact energy absorption and heat release in multi-layer woven fabric ballistic body armor O.N. Budadina, V.O. Kaledinb, V.P. Vavilovc,d, A.E. Gilevab, S.O. Kozelskayaa, M.V. Kuimovac,∗ a
Central Research Institute for Special Machinery, Zavodskaya St., Khotkovo, Russia Kemerovo State University, Krasnaya St., 6, Kemerovo, Russia c National Research Tomsk Polytechnic University, Lenin Av., 30, Tomsk, Russia d National Research Tomsk State University, Lenin Av., 36, Tomsk, Russia b
A R T I C LE I N FO
A B S T R A C T
Keywords: Armor protection Composite material Infrared thermography Heat release Energy absorption Mathematical model
The mechanical and thermo-mechanical damage mechanisms of a multi-layer woven polymeric fabric ballistic body armor are analyzed. A proposed two-stage model takes into account the wave interaction between an impacting element and an armor package at the initial stage of the collision. The model allows the prediction of initial fabric compaction and the estimation of velocities and displacements. These can then be used to predict the response of a layered flexible barrier and the contribution of stresses and frictional forces in causing permanent deformation.
1. Introduction The modern textile industry has developed textile composite materials and protective apparel made with them. These have been thoroughly discussed in Ref. [1]. The development, designs, materials and manufacturing methods of soft body armors is documented in Ref. [2] where the effects of ballistic impact on single- and multi-layer fabrics are also presented, and numerical models that predict stress and transverse wave propagations, projectile deformation, and fabric penetration are included. Ballistic performance of ceramics as function of their structure and properties, armor system design and type of projectile has been discussed by Medvedovski [3,4]. Mathematical modeling of fiber slippage was presented by Sueki et al. [5]. The estimation of energy absorption in woven layered fabrics is a challenging task in designing lightweight ballistic body armor. The scattering of energy takes a significant part in ensuring armor material quality due to friction between fabric fibers. According to known estimates, this phenomenon is responsible for absorbing up to 30% of impact energy. The experimental estimation of the amount of absorbed energy is possible by the indirect effect – the change in the temperature of the fabric during the mutual collision process. The corresponding measurements can be performed thermographically, but only on the package surface. Therefore, to interpret the experimental data, it is necessary to use a mathematical model of thermo-mechanical processes occurring in layered fabrics [6–11].
∗
A simplified mathematical model proposed in Ref. [12] and later improved in Ref. [13] represented a woven fabric as a layered continuous medium thus allowing the calculation of non-stationary temperature distributions by integrating the equations of motion at the stage of deformation. However, to analyze the differences in heat release in different layers of a package, a reasonable formulation of the initial conditions for the equations of motion is necessary. In fact, in the simplified models the initial moment of time is characterized by the singularity when a damage agent (DA) moving with a high velocity comes into contact with a motionless target. This problem can be solved by introducing a model of a membrane-like body subjected to deformation. Such realistic model should take into account the difference in the initial velocities in layers which are located at different depths in regard to the front layer. For this, one may estimate velocities and displacements near the point of impact that can be obtained by considering the wave stage of collision. The corresponding model is presented in this paper. 2. Experimental basis and model presentation The initial stage of the collision between a DA with a multi-layer woven fabric barrier has been considered elsewhere [6,11,12]. Some models of the mechanical processes occurring have been analyzed in details in Ref. [12]. It is noted that the velocity of the sound wave is commensurable with the initial velocity of the DA, and, as a result, the
Corresponding author. E-mail address: [email protected] (M.V. Kuimova).
https://doi.org/10.1016/j.ceramint.2019.08.149 Received 31 July 2019; Accepted 15 August 2019 0272-8842/ © 2019 Elsevier Ltd and Techna Group S.r.l. All rights reserved.
Please cite this article as: O.N. Budadin, et al., Ceramics International, https://doi.org/10.1016/j.ceramint.2019.08.149
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deformation of the composite barrier remains localized for a certain time within the region where the wave front arrives. Based on the difference in the dynamic elastic moduli of tension (along the fibers) and shear, the velocity of the longitudinal and transverse waves has been estimated. It has been assumed that the area of damage is limited to the zone where the growth rate of the radius of the contact spot exceeds the velocity of transverse acoustic waves while a DA is penetrating through the sample. The theoretical evaluation of this radius shows a good match with the experiment. It is worth mentioning that the velocity of the longitudinal wave propagating in the transversal direction depends on the transverse elastic modulus, which is substantially less than the longitudinal one being comparable to the dynamic shear modulus. One of the proposed models is as follows [11]. Irreversible deformations of the normal compression are due to the tightening of fabric layers having a certain initial porosity. Deflections of the fibers near the surface of the contact with the DA lead to their elongation, and the deformation is greater with a smaller radius of DA curvature. Despite the shear nature of the initial destruction, the mechanism of damage involves the breakage of fibers occurring due to longitudinal stresses. Thus, it can be assumed that at the initial stage of impact, the displacements in a small-sized contact zone are directed along the normal to the package surface, and the dominant deformation presents compaction along the normal. Fabric, of which volume consists of 50–60% pores, becomes compact due to irreversible deformation of the normal compression. These phenomena should be supplemented by taking into account the heat release occurring due to irreversible compression. The fabric in the contact zone is being essentially heated, and its properties change significantly. According to Refs. [2,12–15], the temperature rise in the zone of contact with an impacting element may exceed 100°С (see the example in Fig. 1). In combination with a high compressive stress, this leads to the sealing of polymer fibers. Since both the thermo-physical properties of fabrics at high temperatures are known with a high degree of uncertainty, a detailed modeling of all processes at the micro-structure level seems to be difficult if computing resources are limited. Acceptable adequacy of the model can be monitored by checking the momentum conservation law.
Fig. 2. Layer deformation scheme at initial stage of impact.
where rп is the DA radius, V is the initial velocity of the impacting element, a is the velocity of shear waves. Consider the deformation of the fabric layer closest to the DA (Fig. 2) and limit ourselves to the time interval during which the shock wave passes a distance equal to the thickness of this layer. The initial porosity of the fabric is μ0, the thickness of the layer is h, the density of the fabric is ρ and the modulus of elasticity in the transverse direction is Ez. At the moment of contact of the DA, whose center is at the point О1, with the front surface of the armor fabric barrier, the sound wave starts to pass through the thickness of the package in the direction of the z axis. During the time Δt, the wave front reaches the rear surface of the layer z = h, and the center of the DA moves over the distance of V⋅Δt to the O2 position. The surface z = 0 moves over the distance of V⋅Δt, and the displacement of the surface z = h equals zero. Thus, the layer thickness becomes equal to h = V⋅Δt. Taking into account the fact that the volumic fraction of pores μ in the layer is non-negative and cannot exceed μ0, its thickness after a complete compaction becomes h (1-μ0), and with incomplete compacting it can be greater than this value. Then one of two situations may occur: if V⋅Δt < μ0 h, the distance covered by the DA is not enough for a complete compaction, thereby only crumple deformation of the fibers occurs; if V⋅Δt > μ0h, the closure of all pores is not enough to reduce the thickness, and further elastic compression deformation of the compacted fabric occurs. Since the deformations reach μ0 = 50–60% [6], we take the logarithmic measure of compressive deformations:
3. Approximate description of the deformation at the initial stage of impact Assume that the displacements of the fabric of the woven package are significant in the local zone, of which radius can be determined as follows [6]:
ra =
rп V V 2 + a2
,
∂u ⎞ ε = ln ⎛1 + , ∂z ⎠ ⎝
(2)
where u is displacement in the direction of impact along the axis z. The governing equations can be formulated by considering the deformation as elastic-plastic. The closure of pores occurs at a stress equal to the stress limit of the fabric during transverse compression; this limit, i.e. crumpling stress, is determined experimentally, and some measurement results are reported in Refs. [15–17]. The current porosity of the fabric layer μ and the irreversible deformation of the crumpling of the normal εcr < 0 (“cr” specifies crumpling) sum up to the initial porosity: − εcr = μ0 − μ . When simultaneously meeting the conditions ε˙ < 0 and μ < μ0, one may consider the loading to be active (henceforward, the points denote time derivatives). With active loading, the growth of irreversible deformation of compression occurs at the constant stress:
(1)
σz = σcr , ε˙ cr = ε ˙,
εcr < 0 .
(3)
When the zero porosity is reached, or when the compression deformation is reduced, a status of unloading appears being described by linear equation:
Fig. 1. Example of the temperature distribution on the surface of a ballistic armor vest (projectile velocity 506 m/s, infrared image acquisition 146 Hz, FLIR X6530sc infrared imager).
σz = Ez (ε − εcr ) 2
(4)
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Fig. 3b shows the change of velocity in each layer at different times. The solid line 4 corresponds to initial time when, for the 1st layer, the velocity was 300 m/s and zero for other layers. At time point 4 μs (dashed line 5), a non-zero velocity is observed from the 1st to the 7th layer. This indicates that the shock front reaches the rear surface of the 7th layer. At the last time point (dash-dotted line 6), the wave front reaches the last layer causing a change in velocity in all layers of the package. Fig. 4a shows the change in displacements for the 1st, 2nd and last 15th layer (note a wave character of displacements). The largest values of displacement are observed in the first layer of the package. The displacements in the last layer slightly increase in time until 10 μs; afterwards, when the wave front reaches this layer, the displacements start to increase sharply and reach a maximum value of 1.35 mm. In their turn, the curves in Fig. 4b show that, in spite of the wave character of motion in the process of compaction, the distribution of displacements across the package seems to be linear immediately from the time of impact and until at least 10 μs (curve 4 corresponds to zero displacement at the very beginning of the wave process). This is an argument in favor of a membrane shell model when describing displacements after the wave stage of the process. Fig. 5 shows crumpling deformation curves for five layers of the package: 1, 2, 5, 10 and 15. If deformation reaches the value of 0.5, equal to the initial porosity, this means that all layer pores become closed. This appears at the time of 0.375 μs (1st layer), 1.15 μs (2nd layer), 3.175 μs (5th layer), 6.6 μs (10th layer) and 10.375 μs (15th) layer. Thus, in the given example, the complete package compaction is achieved at 10.375 μs. Displacements at this time are distributed through the sample according to a law close to linear, and the velocity varies irregularly through the sample thickness. These calculations are easily generalized in the case of axisymmetric deformation, in which the governing equations contain, along with the transverse elastic modulus, a dynamic shearing modulus, which can be expressed through the velocity of the transverse wave. The axisymmetric problem is set and solved in a cylindrical region whose axis coincides with the axis of impact, and the boundary radius is equal to the distance that the transverse wave travels during the wave process. Taking the found displacements and velocities as initial ones, one can further obtain the variables of the layer state at the shell stage of deformation.
Stresses are determined by Eq. (4) even in the case where, after unloading, the compressive deformation increases to the value of irreversible deformation; after that, the loading becomes active. Due to fabric compaction, the modulus of elasticity increases in proportion to the degree of compaction:
Ez =
Ezin , 1 + εcr
(5)
Ezin
is the initial value of the elasticity modulus (at low dewhere formation). The displacements u (z , t ) and velocity u˙ (z , t ) are determined by solving the equation of motion under the action of stresses defined by Eqs. (3) and (4):
ρu¨ =
∂σz ∂z
(6)
under the initial conditions: u (z , 0) = 0 , u˙ (z , 0) = 0 at z > 0, u˙ (0,0) = V , and stress boundary conditions on the front and back surface of the package:
πra2 σz (0, t ) = Mu¨ (0, t ), σz (N ⋅h, t ) = 0,
(7)
where M is the mass of the DA, N is the number of package layers. The solution is obtained numerically using an implicit difference scheme. Computation errors were controlled by changing the full moment of momentum and applying the Runge rule. The period of integration was chosen by 50% longer than the time needed to fully compact all layers of the package. It is worth noting that, during this time, the calculated amount of movement computationally changed by no more than 10−13 of the initial DA moment of momentum. Figs. 3–5 show the graphs of the dependencies calculated. In Fig. 3a, the solid line 1 shows the evolution of velocity in time for the 1st layer of the package. This curve has the form of a smooth descending curve, which means that no wave phenomena occur in this layer after the impacting element penetrates through the layer. The dotted line 2 indicates the curve corresponding to the velocity distribution for the 2nd layer: at the time t = 0.025 μs, the velocity increases to the value of 407 m/s, afterwards, the oscillations decay. For the last 15th layer (dash-dotted line 3), the velocity gradually increases until the time point t = 10 μs, then there is a sharp increase in speed to 375 m/s, following which the velocity decreases with slight fluctuations.
Fig. 3. Fabric velocity relationships: a - velocity vs. time in the 1st, 2nd and 15th layers (curve numbers: 1 – 1st, 2 - 2nd layer, 3–15th layer from impacted surface), b –velocity vs. layer number (4 - initial time point, 5 - time point 4 μs, 6 - last time point 15 μs). 3
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Fig. 4. Fabric displacement relationships: a - displacement vs. time in the 1st, 2nd and 15th layers (curve numbers: 1 – 1st, 2 - 2nd layer, 3–15th layer from impacted surface),b –displacement vs. layer number (4 - initial time point, 5 - time point 4 μs, 6 - last time point 15 μs).
to take the front surface of the package contacting with the DA as a baseline. The transportation movement of the initial surface occurs in the direction of the Z axis of the Euler coordinate system located in parallel with the velocity vector of the DA. In the initial position, when laying the warp fibers along the edge of the sample, the base vectors of the Lagrangian coordinate system eα and eβ coincide with the base vectors of the Eulerian coordinate system ex and ey. If the layup angle in relation to the edge ϕ is non-zero, then the following transformation of the basis is applied:
eα0 = e X ⋅ cos φ − eY ⋅sin φ , eβ0 = eX ⋅sin φ + eY ⋅cos φ .
(8) (eα0,
eβ0,
en0 )
Initial bases (eX, eY, eZ) and are orthogonal. In the process of movement, the axes of the Lagrangian coordinates are curved forming the surface defined by the vector equation in the form of the dependence of the radius-vector on two coordinates:
⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ ⎯→ ⎯ M = M (α, β ), or M = M (x , y )
(9)
The surface defined by equation (9) is curvilinear, and the α and β or ⎯⎯⎯→ ⎯⎯⎯→ x and y lines are its coordinates. The vectors Mα and Mβ which are tangent to the coordinate lines α and β can be found by differentiating the surface equation by the corresponding coordinate, taking into account the fact that the transportation motion occurs only along the Eulerian axis Z [11]:
Fig. 5. Crumpling deformation vs. time (curve numbers correspond to layer numbers counting from impacted surface).
4. Thermo-mechanical processes at the shell stage of deformation
⎯→ ⎯ Mα =
The model of coupled thermo-mechanical processes at the shell stage of deformation is based on the approximate model [12], in which the movement of a layer package is considered as combined, transitive movement of the package as a whole, as well as relative movement of fibers in the layers of fabric. Considering different initial conditions in the layers, one may introduce two coordinate systems for each layer: the Lagrangian system whose basis changes as it moves, and the Eulerian system which coincides with the initial position of the Lagrangian system. The kinematics of the transportation motion of a package as a whole is determined by the hypotheses of the Reddy shell theory [7,17]: the material normal to the original middle surface during deformation remains rectilinear, but rotates, forming an angle of transverse shear with the geometric normal to the deformed middle surface, and its length changes by the magnitude of normal compression. It is more convenient
⎯→ ⎯ Mβ =
⎯→ ⎯ ∂M ∂α ⎯→ ⎯ ∂M ∂β
= eα0 +
∂uz 0 ⋅e , ∂α z
= eβ0 +
∂uz 0 ⋅e . ∂β z
(10)
Introduce the following notations:
uα′ =
∂uz ∂uz , uβ′ = ∂α ∂β
(11)
Omitting detailed calculations, the final results are as follows. The differential of an arbitrary arc γ 2
ds γ =
2
∂β ∂α ∂α ∂β (1 + uα′ 2) ⎜⎛ ⎞⎟ + 2uα′ uβ′ + (1 + u′β2) ⎛⎜ ⎞⎟ dγ ∂γ ∂γ ⎝ ∂γ ⎠ ⎝ ∂γ ⎠
(12)
can be used to determine the extension of an arbitrary line, in particular, elongations along the warp and weft. The curvature and torsion 4
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of the surface in coordinates (α, β) are characterized vy the following expressions:
−
−
2
1 + εr
L 1 = 112 2 = − , Rα Aα dα 1 + uα′ + uβ′ ⋅(1 + u′α2)
1 L12 = = R αβ A α Aβ
(13)
1+
(1 + u′α2)(1 + u′β2)
Here, Rα and Rβ are the radii of the curvature along the warp and weft, and Rαβ is the reciprocal to the torsion of the initial surface. Deformations of fiber elongation on the base surface are: 2
2
∂uz ⎞ 1+⎛ , ⎝ ∂α ⎠
1 + εβ =
∂uz ⎞ 1 + ⎛⎜ ⎟ ⎝ ∂β ⎠
(14)
The coefficients of the distance increase between the fibers on the base surface are as follows:
1 + εˆα =
∂uz 2 ∂β
∂uz 2 ∂α
( ) +( ) 1+
∂uz 2 ∂β
1+ ,
( )
1 + εˆβ =
∂uz 2 ∂α
∂uz 2 ∂β
( ) +( ) 1+( ) ∂uz 2 ∂α
(15) At the distance n from the base surface, the displacements in the transportation motion additionally depend on the deformation of compression of the normal and angles of the transverse shear. Assuming that the compression deformation of the normal εn is constant over the thickness of the package, one may find the displacement of a point with the Lagrangian coordinates (α, β, n) in relation to a point located on the base surface:
uz (α, β, n) = u z (α, β , 0) + εn
n = uz (α, β , 0) + εn n 1 + u′α2 + u′β2 nz (16)
Then the elongations of the layer fibers separated from the base surface by a distance n = Z are determined by the following expressions:
1 + εα (Z ) =
∂ (uz + uz⁎) ⎤2 1+⎡ , ⎥ ⎢ ∂α ⎦ ⎣
1 + εβ (Z ) =
∂ (uz + uz⁎) ⎤2 1+⎡ ⎥ ⎢ ∂α ⎦ ⎣ (17)
The transverse shear angle γn can be found by considering the transfer displacement of each layer that is directed along the Z axis of the Euler coordinate system, as the angle between the geometric and material normals:
tg2γn = u′α2 + u′β2, γn = arctg u′α2 + u′β2 ,
(18)
then the mutual displacement of the layers can be calculated as follows:
Δu = h⋅tgγ = h uα′ 2 + uβ′ 2
2 ∂uβ(β ) ⎞ ∂u (β ) ⎛ 1+ + ⎜⎛ α ⎟⎞ ⎜ ∂β ⎟ ⎝ ∂β ⎠ ⎝ ⎠
(19)
The relative movement of the fibers within each layer is uniquely determined by four kinematic parameters – relative displacements of the warp fibers ur(α ) and weft fibers ur(β ) [13]:
ur(α ) = uα(α ) eˆα + uβ(α ) eˆβ , ur(β ) = uα(β ) eˆα + uβ(β ) eˆβ
(21)
Thus, at the shell stage, the kinematics of deformation is determined by the following displacement components: two transfer components, namely, displacement of the front surface along the axis Z and normal compression of the normal, and four relative components of fiber displacement, see Eq. (20). Through these functions, each of which depends on two coordinates, one may obtain the deformations of the fibers and relative displacements of the layer fibers. Unlike the models discussed in Ref. [13], in this case, the conditions for the deformation of the layers include both the normal compression and differences in the normal displacements in the fabric layers. The deformations and rates of deformation determine the stresses in the fibers and tangential stresses between the fibers and the fabric when the fibers are pulled out of the fabric, as well as the tangential stresses between the layers of the fabric. In fibers, tensile stresses non-linearly depend on elongation. The governing equation for tensile stresses is obtained from stress-strain diagrams under loading of samples with subsequent unloading. The tests were carried out on flat rectangular samples, which were a strip of fabric of ten fibers connected by a binding. The sample width was 3.5 mm and thickness – 0.5 mm. The distance between the grips of the tension testing machine was 87 mm. In total, two batches of flat rectangular samples of the same type were tested, the first batch included 20 samples, the second - 6 samples. The experiments were carried out by applying uniaxial tension, the loading speed varied from 4 to 60 mm/min. In the first session of experiments, the samples were subjected to a tensile load to reach the limit value at which the fibers broke. Fig. 6 shows a typical view of the sample after fracture accompanied with fiber breakage. In the second session of experiments, the samples were first loaded to ~2000 N, afterwards, they were unloaded and reloaded. In the process of loading, the values of the actual load and grip travels were recorded at the stage of both active loading and unloading. Fig. 7 shows a typical stretch curve which illustrates the linear relationship between actual tensile load and grip travel. The interpretation of the experimental results included recalculating curves of deformation into the “stress – strain” coordinates, then approximating them with the second-degree polynomials and determining the initial and final moduli of elasticity, ultimate deformation and ultimate stress. The results of processing are presented in Table 1. The maximum value of the coefficient of variation in all parameters is 7.45% thus indicating a weak variability of the data and no need of excluding extreme values. The friction of the fibers is due to their slippage. Following [11], one may take the dimensionless quantity, namely, the ratio of the displacement of a fiber along its direction to the thickness of the layer, as the kinematic parameter of slippage:
u″αβ u′α2 ⋅
(α ) 2
(α ) ⎛1 + ∂uα ⎞ + ⎛ ∂uβ ⎞ , ⎜ ⎟ ⎜ ∂α ⎟ ∂α ⎠ ⎝ ⎝ ⎠ 2
1 + εr (β ) =
′′ uββ L dβ 2 1 = 222 2 = − , Rβ Aβ dβ 1 + u′α + u′β ⋅(1 + u′β2)
1+
=
′′ uαα
dα 2
1 + εα =
(α )
(20)
where, eˆα , eˆβ are the Lagrangian basis vectors directed tangentially to the warp and weft respectively. Here the subscript denotes the components of the displacement, and the superscript is related to the family of fibers. Then the elongation of the fibers in relative motion can be calculated as follows:
Fig. 6. Photo of the thin strip-like sample after fracture. 5
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Fig. 7. Experimental stretch curve.
γα(α ) =
uβ(β ) uα(α ) , γβ(β ) = h h
relative displacements, it turns out to be expedient to use the Lagrangian spline on the same finite elements. When integrating by time, a splitting scheme has been applied to save calculation time. Each step of time integration consists of two stages: a transport one, in which relative displacements are considered fixed, and a relative one, in which only relative displacements with fixed transfer displacements vary. At each stage, this numerical scheme is brought to the form, which provides absolute stability [12,18]. As a result of solving the motion equations, the velocities, displacements, deformations, and stresses have been calculated thus making possible to calculate the work of stresses on irreversible deformations. Assuming that the work of the friction forces completely converts into heat, and the work of the destruction of the fibers transits into heat partially, one may determine the heat spent for increasing the fabric temperature. Despite the temperature difference between layers, the temperature change due to heat conduction after the impact is insignificant because of a short duration of collision. Thus, the suggested model of the thermomechanical processes in a woven barrier under collision with a damage agent allows accounting for the different conditions of energy absorption in layers located at different distances from the front surface of the barrier.
(22)
The friction forces, applied to the fibers, do work on relative displacements; the friction forces applied along the warp fibers do the work on the displacements uα(α ) , and those applied to the weft fibers – on the displacements uβ(β ) . By attributing the friction forces to a small area in the fabric plane, one may obtain the values of τα and τβ expressed in terms of stress. The work related to the unit volume of fabric will be equal to the product of the obtained stresses and the deformations defined by Eq. (22). While the fibers are at rest, the friction forces are either absent or do no work. Therefore, the expression linking the stresses τα and τβ with deformations may be defined as follows:
τα = η⋅γ˙α(α ), τβ = η⋅γ˙ β(β )
(23)
In the same form, the equation describing the friction forces between the layers can be written as follows:
τn = ηn⋅γ˙n,
(24)
where τn is the friction force per unit area of a contact zone between layers, and γn is determined by Eq. (18). In this way, the internal forces are expressed in terms of the kinematic parameters of the layer transport motion and the relative motion of the fibers thus making possible to derive the motion equations. The boundary conditions for the transport motion are the conditions for fixing the edges, and the initial conditions are taken from the calculation of the wave stage of the mutual collision. A numerical scheme for solving the problem above can be described as follows. To approximate displacements in the transport motion, the Ermit spline has been used. This allows one to obtain both linear displacements of the front surface of a layer package and their derivatives. For
5. Conclusion Current models of the thermo-mechanical interactions in an impact between a projectile and a simple membrane shell, do not adequately represent a projectile impact with a more complex multi-layer woven fabric ballistic body armor structure. The simple models do not take into account a singularity that occurs at the initial projectile impact. They also do not take into account the friction and resultant heat generation that occurs between the individual layers of the multi-layer structure as the projectile deforms the layers. In this study, a proposed two-stage model considers a wave interaction between an impacting
Table 1 Calculated physico-mechanical parameters. Sample parameter
Maximum deformation ε, %
Maximum stress σ, GPa
Initial modulus of elasticity E0, GPa
Final module elasticity Ef, GPa
Mean value Root-mean-square deviation Minimal value Maximal value Range Relative range, % Coefficient of variation, %
5.53 0.410 4.48 6.54 2.06 37.2 7.45
1.08 0.0740 0.929 1.24 0.314 29.0 6.86
21.8 0.745 20.9 23.25 2.32 10.6 3.41
16.2 0.969 15.3 17.20 1.93 11.8 3.29
6
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body and a multi-layer fabric structure. The model accounts for the initial compression of the fabric and it estimates initial velocities and displacements. These are used for subsequent modeling of the motion of a multi-layer flexible barrier and the contributions of stresses and frictional forces in the case of irreversible deformations.
[8] [9] [10]
Acknowledgment
[11]
This study was conducted within Tomsk Polytechnic University Competitiveness Enhancement Program.
[12]
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