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European Journal of Applied Sciences – Vol. 11, No. 3
Publication Date: June 25, 2023
DOI:10.14738/aivp.113.14959
Partom, Y. (2023). Ductile Materials Stress Upturn at High Strain Rates. European Journal of Applied Sciences, Vol - 11(3). 641-
648.
Services for Science and Education – United Kingdom
Ductile Materials Stress Upturn at High Strain Rates
Y. Partom
18 HaBanim, Zikhron Ya'akov 3094017, Israel
ABSTRACT
Ductile materials (mainly metals) exhibit a sharp stress upturn, which is not
specific to a certain type of material, at strain rates around 103 to 104/s. It is
important to consider such stress high-rate upturn when dealing with shock
loading and unloading. Using classical strength models, usually calibrated at not so
high rates, may lead to errors when considering high-rate loading under not so high
pressures. Here we model high-rate upturn on the macroscale. We assume that the
upturn mechanism is also the mechanism responsible for the 4th power law
mechanism, put forward by Swegle and Grady. In the past we calibrated our
overstress dynamic viscoplasticity model for aluminium from 4th power law data.
Here we use this calibration to predict the high-rate stress upturn response.
INTRODUCTION
Ductile materials (mainly metals) exhibit a sharp upturn of stress at strain rates around 103 to
104/s [1-5], which is not specific to a certain type of metal [5, 6]. It is important to consider
stress high-rate upturn when dealing with high-rate loading, such as shock loading and
unloading. Using classical strength models, usually calibrated at not so high rates, may lead to
inaccurate results with high-rate loading at not so high pressure.
Initially it was believed that stress upturn is caused by a transition from thermal activation of
dislocation motion to viscous drag [7, 8]. But later it was proposed that stress upturn is caused
by an increase of dislocation generation rate [9]. An entirely different approach to the
understanding of stress upturn has been put forward by Fan et al. [6]. They derive their theory
analytically on the mesoscale, and verify it with molecular dynamics (MD) simulations. They
find that: 1) at high temperature, stress approaches zero (melting), and at very low
temperature, stress increases dramatically; 2) strain rate influences stress in an opposite way
to that of temperature, similar to time-temperature equivalence of polymers. High-rate loading
is equivalent to low temperature, and hence the upturn; and 3) stress upturn results don’t
depend on details of a specific type of material, which is in general agreement with
experimental data.
Here we model stress upturn on the macroscale. First, we distinguish between total strain rate
and plastic strain rate. In the literature, when plotting stress versus log strain rate, they usually
don’t specify which strain rate they mean, total or plastic strain rate. But total strain rate is not
a state variable, while effective plastic strain rate may be regarded as a hidden (internal) state
variable. Effective plastic strain rate represents the effect of mobile dislocation density and of
average dislocation velocity, as in Orowan’s equation. This leads us to replace the usual Y(deff)
(flow stress as function of effective deformation rate, or strain rate) equation by the flow curve
dp
eff (seq-Yqs), as in our overstress approach to dynamic viscoplasticity [10].
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European Journal of Applied Sciences (EJAS) Vol. 11, Issue 3, June-2023
Next, we assume that the high-rate stress upturn mechanism is also responsible for the 4th
power law mechanism, put forward by Swegle and Grady (SG) [11]. Hints to this effect are made
in [12, 13]. In [14] we calibrated the flow curve of our overstress viscoplastic model to
reproduce the 4th power law data for aluminum given in [15]. Similarly, we show here that the
same overstress flow curve that reproduces the 4th power law response, reproduces the stress
upturn response as well.
In section 2 we revisit our 4th power law modeling.
In section 3 we use our overstress flow curve to reproduce the high-rate stress upturn
response.
REVISITING FOURTH POWER LAW RESPONSE
Since the works of Barker [15] and Grady [11], it has been known that for a steady structured
shock in a viscoplastic material, the following power law holds for many materials:
( )
max
= dmax
= B (1)
where dmax and are defined in Fig. 1.
Figure 1. Schematics of a steady structured viscoplastic wave.
This unique response, known as the 4th power law, is usually shown as straight lines (for
different materials) on a log-log plot. In [14] we reproduce the 4th power law response for 6061-
T6 aluminum, using our overstress approach to dynamic viscoplasticity [10]. We solve there a
system of three ODEs, describing the steady viscoplastic wave from a planar impact (uniaxial
strain), as developed by Duvall and coworkers [16]. The system of ODEs is:
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Partom, Y. (2023). Ductile Materials Stress Upturn at High Strain Rates. European Journal of Applied Sciences, Vol - 11(3). 641-648.
URL: http://dx.doi.org/10.14738/aivp.113.14959
( )
( )
p
2 2
L
2
0
p
2 2
0 L
p
2 2
L
d
U c / U 1
2G
d
U c / U 1
2G
u
d
c / U 1
2G
−
=
−
=
−
=
(2)
where =longitudinal stress component, u=particle velocity, U=viscoplastic wave velocity,
cL=longitudinal sound speed, G=shear modulus, =longitudinal strain component, 0=initial
density, P=pressure, and dp=longitudinal plastic deformation rate component.
By our overstress approach to dynamic viscoplasticity [10], dp is given by:
s P
0
Y
q s 1.5s Y
d A
p
= −
−
=
(3)
and it can be shown that for the uniaxial strain state:
s s s s
d d d d
2
3
2 ij ij
3
eq
p p
ij
p
3 ij
p 2
eff
= =
= =
(4)
where dp
eff=effective plastic strain rate and seq=equivalent stress. We integrate the system of
equations (3) with a standard ODE solver. We use standard material parameters of aluminum,
and we run the integration code for different levels of the incoming shock. For a given set of the
parameters A, α we obtain a straight line on the log-log plot of versus the maximum strain
rate, as in the experiments. Changing these parameters, we find that the exponent α controls
the slope of this line, and the coefficient A controls its vertical position B. In Fig. 2 we show
as function of α from these computations.