Numerical simulations of fabric shields for bulletproof vest: determining the number of sheets to withstand ballistic impacts

Bandeira, Alex Alves

doi:10.1590/1679-78255994

Alex Alves Bandeira ^** Corresponding author

Departamento de Construção e Estruturas (DCE), Escola Politécnica, Universidade Federal da Bahia (UFBA), Rua Aristides Novís, 02, 5º andar, Federação, Salvador, BA, Brasil, CEP: 40.210-630. E-mail: alexbandeira@ufba.br

https://orcid.org/0000-0001-7170-8557

* Corresponding author

Editor:

Marcílio Alves.

Figures (28)
Tables (3)

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Figure 1
Representation of a network model of fabric (based onZohdi (2014cZohdi, T. I. (2014c). Impact and penetration resistance of network models of coated lightweight fabric shielding. GAMM-Mitt., v. 37, n. 1: 124 - 150.)).

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Figure 2
Photography of fabric shield.

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Figure 3
Geometrical interpretation of the force at a yarn-segment.

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Figure 4
Behavior of the damage parameter.

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Figure 5
Graphic of the elastoplastic constitutive law: (

λ = 0

and

P = 10^{4}

).

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Figure 6
Graphic of the elastoplastic constitutive law with rather gradual damage (

λ = 135

and

P = 10^{4}

or

P = 0

).

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Figure 7
Graphic of the elastoplastic constitutive law with sudden damage (

λ = 10^{4}

and

P = 10^{4}

).

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Figure 8
Interpenetration of a fabric node (picture based on Zohdi (2014cZohdi, T. I. (2014c). Impact and penetration resistance of network models of coated lightweight fabric shielding. GAMM-Mitt., v. 37, n. 1: 124 - 150.) and Zohdi (2010aZohdi, T. I. (2010a). High-speed impact of electromagnetically sensitive fabric and induced projectile spin. Comput Mech, vol. 46 : 399-415.)).

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Figure 9
Geometrical interpenetration of a fabric node update.

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Figure 10
Numerical model geometry: impact of a spherical projectile with an incoming velocity of

500 m / s

into a fabric shield (one sheet).

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Figure 11
Motion configurations over time.

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Figure 12
Damage of yarn and coating at time

3.2 \cdot 10^{- 4} s

.

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Figure 13
Cauchy stress of yarn and coating at time

3.2 \cdot 10^{- 4} s

.

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Figure 14
Particle and fabric shield configurations over time (with one layer).

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Figure 15
Front view of the impact of the projectile against a sheet of Zylon.

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Figure 16
Side view of the impact of the projectile against a sheet of Zylon.

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Figure 17
Numerical results for the model of twelve layers at time

7.7 \cdot 10^{- 5} s

.

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Figure 18
Particle and fabric shield configurations over time (with twelve layers).

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Figure 19
Relationship between velocity and time for each model from 1 sheet to 12 sheets.

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Figure 20
Structural fabric configuration at time

v

for the model with 6 sheets.

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Figure 21
Structural fabric configuration at time

1.82 \cdot 10^{- 4} s

for the model with 9 sheets.

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Figure 22
Relationship between velocity and number of sheets at time

7.7 \cdot 10^{- 5} s

and the tendency curve.

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Figure 23
Results obtained numerically and experimentally.

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Figure 24
Front view of the impact of the projectile against multiple sheets of Zylon.

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Figure 25
Uniaxial test of elastic plastic material: a) elastic-perfectly-plastic model, b) elastic-plastic model with hardening.

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Figure 26
Geometrical interpretation of the tensors for elastic-plastic constitutive equations.

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Figure 27
Interpretation of the update procedure for elastic-plastic constitutive equations.

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Algorithm 1
Elastoplasticity with damage.

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Algorithm 2
Numerical solution scheme.

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Algorithm 3
Plasticity algorithmic.

ITERATIVE IMPLICIT SOLUTION METHOD
Step 1	Start a global fixed iteration and set the iteration counter $K = 0$ . Store the variables: $r_{i}^{L} (r_{i}^{L + 1,0})$ , $v_{i}^{L} (v_{i}^{L + 1,0})$ , $r_{p}^{L} (r_{p}^{L + 1,0})$ , $v_{p}^{L} (v_{p}^{L + 1,0})$ , $θ_{p}^{L} (θ_{p}^{L + 1,0})$ and $ω_{p}^{L} (ω_{p}^{L + 1,0})$ . Compute other forces: $F_{i}^{f, o t h, L} (r_{i}^{L}, v_{i}^{L}, r_{p}^{L}, v_{p}^{L}, ω_{p}^{L})$ and $F_{p}^{o t h, L} (r_{i}^{L}, v_{i}^{L}, r_{p}^{L}, v_{p}^{L}, ω_{p}^{L})$ . Compute the total force: $F_{i}^{t o t, L} (r_{i}^{L}, v_{i}^{L}, r_{p}^{L}, v_{p}^{L}, ω_{p}^{L})$ and $F_{p}^{t o t, L} (r_{i}^{L}, v_{i}^{L}, r_{p}^{L}, v_{p}^{L}, ω_{p}^{L})$ . Compute the total momentum: $M_{p}^{t o t, L} (r_{i}^{L}, v_{i}^{L}, r_{p}^{L}, v_{p}^{L}, ω_{p}^{L})$ . Update the iteration counter $K = K + 1$ .
Step 2	Make a loop from $i = 1$ to $i = N_{f}$ and compute for each fabric node $i$ : a) Compute forces: $F_{i}^{f, p r e d, o t h,, L + 1, K - 1} (r_{i}^{L + 1, K - 1}, v_{i}^{L + 1, K - 1}, r_{p}^{L + 1, K - 1}, v_{p}^{L + 1, K - 1}, ω_{p}^{L + 1, K - 1})$ . b) Compute mass predicted position and predicted velocity: $r_{i}^{f, p r e d, L + 1, K} = r_{i}^{f, L} + v_{i}^{f, L} Δ t + \frac{ϕ {(Δ t)}^{2}}{m_{i}} [ϕ F_{i}^{f, p r e d, o t h, L + 1, K - 1} + (1 - ϕ) F_{i}^{f, o t h, L}]$ , $v_{i}^{f, p r e d, L + 1, K} = v_{i}^{f, L} + \frac{Δ t}{m_{i}} [ϕ F_{i}^{f, p r e d, o t h, L + 1, K - 1} + (1 - ϕ) F_{i}^{f, o t h, L}]$ . c) Enforce contact if necessary (if contact occurs do the following procedures): Consider “stick case”. Make $r_{i}^{f, c o r r, L + 1, K} = r_{i}^{p, c p, L + 1, K}$ and $v_{i}^{f, c o r r, L + 1, K} = v_{i}^{p, c p, L + 1, K}$ . Compute $F_{i}^{c o n} = \frac{m_{i}}{Δ t^{2}} [r_{i}^{f, c o r r, L + 1, K} - r_{i}^{f, p r e d, L + 1, K}]$ and $F_{i}^{f r i c} = \frac{m_{i}}{Δ t} (v_{i}^{f, c o r r, L + 1, K} - v_{i}^{f, L}) - F_{i}^{c o n} - F_{i}^{f, o t h, L}$ . IF $F_{i}^{f r i c} \leq μ_{s} F_{i}^{c o n}$ THEN “stick case”. Leave like that. The contact condition is satisfied. ELSE “slip case”. Update the velocity $v_{i}^{f, c o r r, L + 1, K} = v_{i}^{f, p r e d, L + 1, K} + \frac{Δ t}{m_{i}} [F_{i}^{c o n} + F_{i}^{f r i c}]$ and the friction force $F_{i}^{f r i c} = μ_{d} F_{i}^{c o n} \frac{v_{i}^{p, c p, L + 1, K} - v_{i}^{f, c o r r, L + 1, K}}{v_{i}^{p, c p, L + 1, K} - v_{i}^{f, c o r r, L + 1, K}}$ .
Step 3	Make a loop from $p = 1$ to $p = N_{p}$ and compute for each projectile $p$ : a) Compute forces: $F_{p}^{o t h, L + 1, K - 1} (r_{i}^{L + 1, K}, v_{i}^{L + 1, K}, r_{p}^{L + 1, K - 1}, v_{p}^{L + 1, K - 1}, ω_{p}^{L + 1, K - 1})$ . b) Compute momentum: $M_{p}^{t o t, L + 1, K - 1} (r_{i}^{L + 1, K}, v_{i}^{L + 1, K}, r_{p}^{L + 1, K - 1}, v_{p}^{L + 1, K - 1}, ω_{p}^{L + 1, K - 1})$ . c) Compute position, velocity, angular velocity and rotation: $r_{p}^{L + 1, K} = r_{p}^{L} + v_{p}^{L} Δ t + \frac{ϕ {(Δ t)}^{2}}{m_{p}} {- \sum_{i = 1}^{N_{c}} (F_{i}^{c o n} + F_{i}^{f r i c}) + [ϕ F_{p}^{o t h, L + 1, K - 1} + (1 - ϕ) F_{p}^{o t h, L}]}$ , $v_{p}^{L + 1, K} = v_{p}^{L} - \frac{Δ t}{m_{p}} \sum_{i = 1}^{N_{c}} (F_{i}^{c o n} + F_{i}^{f r i c}) + \frac{Δ t}{m_{p}} [ϕ F_{p}^{o t h, L + 1, K - 1} + (1 - ϕ) F_{p}^{o t h, L}]$ . $ω_{p}^{L + 1, K} = ω_{p}^{L} + \frac{Δ t}{{\bar{I}}_{i, s}} [\emptyset M_{p}^{t o t, L + 1, K - 1} + (1 - \emptyset) M_{p}^{t o t, L}]$ . $θ_{p}^{L + 1, K} = θ_{p}^{L} + Δ t [\emptyset ω_{p}^{L + 1, K - 1} + (1 - \emptyset) ω_{p}^{L}]$ .
Step 4	Compute/update forces: $F_{i}^{t o t, L + 1, K}$ (similar to step 2).
Step 5	Compute/update fabric/coating damage: $α_{I}^{y a r n}$ and $α_{I}^{c o a t i n g}$ .
Step 6	Compute/update fabric/coating plastic stretch: $U_{p, I}^{y a r n}$ and $U_{p, I}^{c o a t i n g}$ .
Step 7	Measure normalized error quantities: a) $ϖ_{K, r} ≝ \frac{\sum_{i = 1}^{N_{f} + N_{p}} ‖r_{i}^{L + 1, K} - r_{i}^{L + 1, K - 1}‖}{\sum_{i = 1}^{N_{f} + N_{p}} ‖r_{i}^{L + 1, K} - r_{i}^{L}‖}$ , $ϖ_{K, v} ≝ \frac{\sum_{i = 1}^{N_{f} + N_{p}} ‖v_{i}^{L + 1, K} - v_{i}^{L + 1, K - 1}‖}{\sum_{i = 1}^{N_{f} + N_{p}} ‖v_{i}^{L + 1, K} - v_{i}^{L}‖}$ and $ϖ_{K, ω} ≝ \frac{\sum_{p = 1}^{N_{p}} ‖ω_{p}^{L + 1, K} - ω_{p}^{L + 1, K - 1}‖}{\sum_{p = 1}^{N_{p}} ‖ω_{p}^{L + 1, K} - ω_{p}^{L}‖}$ . b) $Z_{k, r} ≝ \frac{ϖ_{K, r}}{{T O L}_{r}}$ , . $Z_{k, v} ≝ \frac{ϖ_{K, v}}{{T O L}_{v}}$ and $Z_{k, ω} ≝ \frac{ϖ_{K, ω}}{{T O L}_{ω}}$ . c) $Λ_{K, r} ≝ \frac{{(\frac{{T O L}_{r}}{ϖ_{0, r}})}^{\frac{1}{2 K_{d}}}}{{(\frac{ϖ_{K, r}}{ϖ_{0, r}})}^{\frac{1}{2 K}}}$ , $Λ_{K, v} ≝ \frac{{(\frac{{T O L}_{v}}{ϖ_{0, v}})}^{\frac{1}{2 K_{d}}}}{{(\frac{ϖ_{K, v}}{ϖ_{0, v}})}^{\frac{1}{2 K}}}$ and $Λ_{K, ω} ≝ \frac{{(\frac{{T O L}_{ω}}{ϖ_{0, ω}})}^{\frac{1}{2 K_{d}}}}{{(\frac{ϖ_{K, ω}}{ϖ_{0, ω}})}^{\frac{1}{2 K}}}$ .
Step 8	IF $(Z_{k, r} \leq 1 A N D Z_{k, v} \leq 1 A N D Z_{k, ω} \leq 1) A N D K \leq K_{d}$ THEN Increment time: $t = t + Δ t$ , Construct the next time step: ${(Δ t)}^{n e w} = Λ_{K} {(Δ t)}^{o l d}$ , where $Λ_{K} = \min [Λ_{K, r}, Λ_{K, v}, Λ_{K, ω}]$ , Select the minimum size: $Δ t = \min [{(Δ t)}^{\lim}, {(Δ t)}^{n e w}]$ , Go to step 1. ELSE Go to step 9.
Step 9	IF $(Z_{k, r} > 1 O R Z_{k, v} > 1 O R Z_{k, ω} > 1) A N D K < K_{d}$ THEN Update the iteration counter: $K = K + 1$ , Go to step 2. ELSE Go to step 10.
Step 10	IF $(Z_{k, r} > 1 O R Z_{k, v} > 1 O R Z_{k, ω} > 1) A N D K = K_{d}$ THEN Construct a next time step: ${(Δ t)}^{n e w} = Λ_{K} {(Δ t)}^{o l d}$ , where $Λ_{K} = \min [Λ_{K, r}, Λ_{K, v}, Λ_{K, ω}]$ , Select the minimum size: $Δ t = \min [{(Δ t)}^{\lim}, {(Δ t)}^{n e w}]$ , Restart at time $t$ : Set $r_{i}^{L + 1,0} = r_{i}^{L}$ , $v_{i}^{L + 1,0} = v_{i}^{L}$ , $r_{p}^{L + 1,0} = r_{p}^{L}$ , $v_{p}^{L + 1,0} = v_{p}^{L}$ and $ω_{p}^{L + 1,0} = ω_{p}^{L}$ , Go to step 1. ELSE Set a larger number for the variable $K_{d}$ , Restart at time $t$ : Set $r_{i}^{L + 1,0} = r_{i}^{L}$ , $v_{i}^{L + 1,0} = v_{i}^{L}$ , $r_{p}^{L + 1,0} = r_{p}^{L}$ , $v_{p}^{L + 1,0} = v_{p}^{L}$ and $ω_{p}^{L + 1,0} = ω_{p}^{L}$ , Go to step 1.

I) Trial step:
a) $F_{t}^{e} = F_{i + 1} U_{i}^{p}^{- 1}$ , $C_{t}^{e} = {(F_{t}^{e})}^{T} F_{t}^{e}$ ; b) $U_{t}^{e} = {(C_{t}^{e})}^{1 / 2}$ , $E_{t} = \frac{1}{2} (U_{t}^{e}^{2} - I)$ ; c) $R_{t}^{e} = F_{t}^{e} U_{t}^{e - 1}$ , $S_{t}^{e} = D^{e} E_{t}$ ; d) $F_{t} = {\bar{σ}}_{t} - σ_{y} (α_{i})$
II) Radial return algorithm:
If $(F_{t} < 0)$ then Elastic step: 1. $α_{i + 1} = α_{i}$ ; 2. $U_{i + 1}^{p} = U_{i}^{p}$ ; 3. $S_{i + 1} = S_{t}^{e}$ ; 4. $D_{i + 1} = D^{e}$ ; Else if $(F_{t} \geq 0)$ then Elastic-plastic step: 1. $Δ α$ , $α_{i + 1} = α_{i} + Δ α$ ; $σ_{y} (α_{i + 1}) = σ_{y} (α_{i}) + h Δ α$ ; 2. $Δ E^{p} = Δ α N_{i + 1}$ , $Δ U^{p} = U_{i}^{p}^{- 1} Δ E^{p}$ , $F_{i + 1}^{p} = Δ U^{p} U_{i}^{p}$ , $U_{i + 1}^{p} = {[{(F_{i + 1}^{p})}^{T} F_{i + 1}^{p}]}^{\frac{1}{2}}$ 3. $S_{i + 1} = S_{t}^{e} - D^{e} Δ E^{p}$ ; 4. $D_{i + 1} = D^{e p}$ ; End if.

[1] * Corresponding author

1)	Initialize the variables: $U_{i}^{p} (0) = 1$ , $α_{i} (0) = 1$ , $L^{r}$ , $D (α_{i})$ .
2)	Calculate the known variables: $L$ , $U_{i + 1}$ .
3)	Compute the elastic part of the stretch: $U_{t}^{e} = U_{i + 1} {(U_{i}^{p})}^{- 1}$ .
4)	Compute the elastic part of the Green-Lagrange strain: $E_{t}^{e} = \frac{1}{2} [{(U_{t}^{e})}^{2} - 1]$ .
5)	Compute the elastic part of the second Piola-Kirchhoff stress: $S_{t}^{e} = D (α_{i}) E_{t}^{e}$ .
6)	Compute elastic part of the Cauchy stress: $σ_{t}^{e} = U_{t}^{e} S_{t}^{e}$ .
7)	Compute: $F_{t} = σ_{t}^{e} - σ_{y}$ .
8)	Compute the variation of the plastic part of the stretch tensor $Δ U^{p}$ from equation (45).
9)	Update the plastic stretch tensor: IF $F_{t} < 0$ THEN $U_{i + 1}^{p} = U_{i}^{p}$ ELSE IF $U_{i + 1}^{p} = U_{i}^{p} + Δ U^{p}$ .
10)	Compute the elastic part of the stretch: $U_{t}^{e} = U_{i + 1} {(U_{i + 1}^{p})}^{- 1}$ .
11)	Compute the elastic part of the Green-Lagrange strain: $E_{t}^{e} = \frac{1}{2} [{(U_{t}^{e})}^{2} - 1]$ .
12)	Compute the second Piola-Kirchhoff stress: $S_{i + 1} = D (α_{i}) E_{t}^{e}$ .
13)	Compute the Cauchy stress: $σ = U_{t}^{e} S_{i + 1}$ .

Brasil

Brasil

Numerical simulations of fabric shields for bulletproof vest: determining the number of sheets to withstand ballistic impacts

Abstract

Graphical Abstract