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European Journal of Applied Sciences – Vol. 11, No. 3
Publication Date: June 25, 2023
DOI:10.14738/aivp.113.14670.
Partom, Y. (2023). Calibration of Impact Ignition from Shear-Band Formation on The Mesoscale. European Journal of Applied
Sciences, Vol - 11(3). 110-116.
Services for Science and Education – United Kingdom
Calibration of Impact Ignition from Shear-Band Formation on The
Mesoscale
Y. Partom
18 HaBanim, Zikhron Ya'akov 3094017, Israel
ABSTRACT
Relying on test results in [1], we proposed in [2] a macroscopic impact ignition
model for low velocity impact situations, in terms of the product PD (P=pressure,
D=plastic deformation rate). Here we upgrade this model by taking into account the
time duration to ignition for different PD levels. Our macroscopic impact ignition
model is now based on, and calibrated from, 1D simulations of pure torsion on the
mesoscale. We assume that low velocity impact ignition is invoked by shear
localization and formation of shear bands. We denote by (PD)L the macroscopic
shear localization threshold. When PD>(PD)L in a macroscopic cell, shear bands
start to form there. The shear bands then develop and heat up towards the ignition
temperature. We further assume that the time duration from localization to ignition
h=tig-tL is also dependent on PD. Using 1D simulations of shear band formation in
torsion similar to [3], we calibrate (PD)L and h(PD), which we can then use in
macroscopic hydrocode simulations. Our mesoscale simulations depend on a
realistic strength model for explosives. This model employs the overstress
approach to dynamic viscoplasticity [4], and its main feature here is the pressure
dependence of its plastic flow curve.
INTRODUCTION
Boyle Frey and Blake [1] performed dynamic tests in which they loaded various explosive
samples to characterize their low velocity impact ignition threshold. Under high velocity
impact, explosives would mainly ignite by a pore collapse mechanism, but below some impact
velocity threshold, ignition would come about by a shear-band formation mechanism. In their
low velocity impact tests, Boyle and coauthors were able to change separately pressure and
impact velocity (which can be translated into strain rate). They found that the macroscopic
ignition threshold can be expressed in terms of the product of pressure and strain rate (PD),
where P is pressure and D is strain rate (essentially plastic strain rate). We may deduce from
those tests that low velocity impact occurs when PD is above a threshold value (PD)th. Assuming
that this is an appropriate ignition criterion, we evaluated in [2] the impact ignition threshold
of an HMX based explosive from Steven-test data [5].
Here we try to understand and calibrate low velocity impact ignition from simulations on the
mesoscale. As in [3], we’re able to perform 1D simulations with shear band formation for
torsion of thin-walled tubes. From such simulations with explosives, we learn that low velocity
impact ignition is related to shear localization and shear band formation. Such an ignition is a
two-stage process: 1) shear localization of the plastic flow; and 2) formation of shear bands and
their heating up to the explosive ignition temperature.
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Partom, Y. (2023). Calibration of Impact Ignition from Shear-Band Formation on The Mesoscale. European Journal of Applied Sciences, Vol - 11(3).
110-116.
URL: http://dx.doi.org/10.14738/aivp.113.14670.
In what follows we describe our 1D mesoscopic simulations and show how to extract from
them: 1) the macroscopic localization threshold (PD)L (which replaces (PD)th of our previous
form of the model [2]; and 2) the time duration h from localization to ignition.
TORSION OF A THIN-WALLED TUBE ON THE MESOSCALE
Recently we developed a 1D model for torsion of thin-walled tubes to predict the formation of
adiabatic shear bands on the mesoscale. We run these simulations with a mesh of 1μm, and the
length of the tube is 2 to 5mm. We used this model for metal tubes, and compared the results
to tests on various types of steel [6]. The set of equations that we solve with this model includes:
1) momentum equation; 2) strain rate equation; 3) plastic strain rate equation; 4) hardening
and softening equation; 5) heat balance equation. These five equations are outlined in
references [3] and [7]. We use there the overstress approach to dynamic viscoplasticity [4],
with which we specify: 1) the quasistatic yield stress; 2) a flow curve which gives the effective
plastic deformation rate as function of the overstress; and 3) a failure strain. Comparing runs
with the overstress approach to runs with the traditional flowstress approach (where the
flowstress is specified, and the stress state is determined by what is known as radial return),
we conclude that the overstress approach is superior here in terms of stability and accuracy.
We further discuss this issue in [7].
To solve the system of equations described above we: 1) define as unknowns the values of the
five unknown functions mentioned above at the mesh points; and 2) express the space
derivatives of those functions by finite difference approximations. We obtain a system of 5n
(n=number of cells in the tube) simultaneous rate equations (which are ODEs), which we solve
with a standard ODE solver with the appropriate boundary conditions. We load the tube in
torsion by applying equal, but opposite in sense, tangential velocities at the tube ends.
To create shear localization in the plastic shear flow we need to specify a perturbation. We
specify the perturbation at the tube mid length, and the shear band develops there. There are
several possible ways to specify a perturbation. We choose to specify the perturbation as a 1%
reduction of the initial yield stress at the two cells next to the tube mid plane.
To save space we don’t outline here the flow equations, and those can be found in [3] and [7].
What is different here from our previous work is that the tube material is an explosive, and as
is well known the mechanical response of polymer bonded explosives, and of polymers in
general, is much different from that of metals. As mentioned above, we use the overstress
approach to dynamic viscoplasticity. By this approach we define 1) a quasistatic yield stress qs;
2) a flow curve which defines the effective plastic strain rate in shear as function of the
equivalent stress in shear; and 3) a failure strain in shear. The main issue here is that qs may
depend on the pressure P:
( )
−
= + = − − = −
=
ref
f
P P T m m 0 f
qs 0 P T f
t
t t
F 1 A P F T T T T F exp
F F F
(1)
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European Journal of Applied Sciences (EJAS) Vol. 11, Issue 3, June-2023
where tf=time when a certain cell reaches the strain to failure, tref=a reference time which
determines the rate of strength loss beyond the strain to failure, Tm=melting temperature, and
where the strain to failure is given by:
(1 A P) f f 0
+ f
= (2)
For the flow curve we assume a linear relation:
( )
=
= −
eq
eq qs
p
eff A
(3)
Next, we show results of a typical run of the simulation code, where the end tangential velocities
are 15 and -15m/s, and the average (macroscopic) strain rate is therefore D=15e- 3/μs=15,000/s. Pressure is not part of this pure shear flow. We therefore specify it from the
outset, and in this typical run we use P=0.5GPa, so that PD=7.5e-3GPa/μs. The values of the
parameters that we use in such a run are: Tm=500K, α=1.1, G (shear modulus) =2.5GPa, CV (heat
capacity) =1e-3kJ/g, k (heat conductivity) =5e-10GPa*mm2/μs/K, 0=0.1GPa, f0=0.05,
tref=100μs, A=1.0/GPa/μs, Af=1. /GPa. We show two types of figures: 1) temperature profiles
along the tube every 50μs (Fig. 1); and 2) temperature history in the developing shear band
(Fig. 2). We see from Figs. 1 and 2 that: 1) for the parameters used, the calculation is beyond
the localization threshold; and 2) after about 50μs, the temperature in the shear band is above
the ignition temperature of this explosive (570K).
It’s not difficult to see that the slope of the flow curve A controls what is known with the
flowstress approach as strain rate sensitivity. Lower values of A push the stress needed for full
plastic flow and for a given strain rate to higher values. We see this in Fig. 3, where the value of
A is lowered to 0.3/GPa/μs, and all other parameters stay the same. We see from Fig. 3 that for
this value of A the shear flow in the tube is just at the localization threshold, and that beyond
about 150μs, the excess temperature at the perturbation location hardly changes.
Next, we check how shear velocity and pressure influence the localization process. We find that
shear velocity has little influence, and that pressure has a strong influence. It seems that
pressure influence comes about through its effect on the quasistatic yield stress. Although shear
velocity has little influence, we still adhere to the experimental findings of [1], that localization
threshold depends on the product PD, as stated above. Changing the shear velocity, we have to
change the pressure accordingly to stay at PDL. But changing the pressure would take the
perturbation out of the localization threshold, unless we change the slope A accordingly.