Introduction
The combined effects of strain, strain rate and temperature on the
behaviour of metals have been widely studied in the literature. Ruggiero
et al. analysed the ductility variations of an ADI JS/1050-6 iron due to
different strain rate and temperatures1, Scapin et al.
made similar investigations on pure copper2 while
Sasso et al.3 studied the strain rate sensitivity of
AA7075 aluminum alloy at different initial temper states. Mirone et
al.4 and Mirone et al.5 highlighted
the interactions between the necking onset, the true strain rate and the
effective dynamic amplification of the material equivalent stress-strain
curve in different metals.
Hart6 and Ghosh7 evaluated the
necking onset strain at high strain rates obtaining different
mathematical relationships due to different starting hypothesis about
the elongation conditions of the specimen. A modified version of the
Hart criterion was proposed by Guan8 while
Lin9 obtained the instability conditions for uniaxial
tension specimens of materials characterized by a significant strain
rate sensitivity. Hart, Ghosh and Lin approaches do not include the grow
of the temperature inevitably occurring during high strain rate
deformations due to the adiabatic development of plastic work inside the
material. The plastic work during a test is proportional to the area
subtended by the equivalent stress-strain curve of the material. Then,
the heat effectively produced is proportional to such plastic work,
through the Taylor-Quinney coefficient. Kapoor &
Nemat-Nasser10 and Walley et al.11obtained that almost all the plastic work developed during high strain
rate tests of different metals is converted to heat while Jovic et
al.12 and Rittel et al.13 calculated
that only lower fractions of it are converted to heat.
The equivalent stress-strain curve of the material, necessary to
correctly evaluate the plastic work and the temperature, can be obtained
from the true curve using the well-known procedure proposed by
Bridgman14. An alternative procedure, simpler and more
accurate, was proposed by Mirone15. For cylindrical
specimens, it is simple to obtain the starting true curve by monitoring
the diameter of the minimum cross-section of the specimen. Instead, for
thick rectangular specimens, the true curve extrapolation is not so
straightforward. To address such issue, Mirone et
al.16 recently proposed a procedure to obtain the true
curve of thick flat specimens at locally necked material points,
starting from the global engineering variables, i.e. experimental force
and elongation of the gage length.
Other methods allow to indirectly obtain the equivalent curve without
the necessity of calculating the true curve. Peroni et
al.17 proposed an equivalent curve extrapolation
procedure based on the monitoring of the entire neck profile; the
inverse method converges when the calibrated material model is able to
reproduce such profile.
Sasso et al.18 proposed an analytical method,
alternative to the classical inverse FEM-based procedure, which gave
results in good agreement with those obtained through finite element
simulations, with a good matching to experimental data.
In this paper, the behavior of A2-70 stainless steel at different
temperatures and under quasi-static and dynamic conditions is
investigated; in particular, the combined effects of strain, strain rate
and temperature on the necking onset have been analyzed both
analytically and experimentally, giving some new insights about the
complex interaction between such variables.
Quasi-static and dynamic instability conditions
The strain for tensile instability onset can be obtained through the
well-known Considère mathematical condition, taking into account also
the eventual effects of true strain rate\({\dot{\varepsilon}}_{\text{True}}\) and temperature \(T\) together
with their variations. In fact, during a general dynamic test, materials
are typically subjected to strain rate increase due to finite rise times
of the loads and to temperature increase due to the adiabatic conversion
of plastic work into heat. The Considère condition is shown in eq. (1)
where \(\sigma_{\text{Eq}}\), \(\varepsilon_{\text{True}}\) and\(\varepsilon_{N-\text{DT}}\) are respectively the equivalent stress,
the true strain and the necking strain affected by both dynamics and
temperature effects.
\(\sigma_{\text{Eq}}-\frac{d\sigma_{\text{Eq}}(\varepsilon_{\text{True}},{\dot{\varepsilon}}_{\text{True}},T)}{d\varepsilon_{\text{True}}}=\sigma_{\text{Eq}}-\frac{\partial\sigma_{\text{Eq}}}{\partial\varepsilon_{\text{True}}}-\frac{\partial\sigma_{\text{Eq}}}{\partial{\dot{\varepsilon}}_{\text{True}}}\bullet\frac{\partial{\dot{\varepsilon}}_{\text{True}}}{\partial\varepsilon_{\text{True}}}-\frac{\partial\sigma_{\text{Eq}}}{\partial T}\bullet\frac{\partial T}{\partial\varepsilon_{\text{True}}}=0\)\(\rightarrow\ \varepsilon_{N-DT}\) (1)
Here a general simple multiplicative material model is assumed, in which
the equivalent stress \(\sigma_{\text{Eq}}\) is the product of the
equivalent stress under quasi-static conditions and room temperature,\(\sigma_{Eq-S}\), by the strain rate amplification function \(R\) and
by the thermal softening function \(S\), as shown in eq. (2).
\(\sigma_{\text{Eq}}\left(\varepsilon_{\text{True}},{\dot{\varepsilon}}_{\text{True}},T\right)=\sigma_{Eq-S}\left(\varepsilon_{\text{True}}\right)\bullet R\left({\dot{\varepsilon}}_{\text{True}}\right)\bullet S\left(T\right)\)(2)
The complete uncoupling between the three relevant variables\(\varepsilon_{\text{True}}\), \({\dot{\varepsilon}}_{\text{True}}\) and\(T\), is tentatively assumed now, meaning that each function is
supposed to only depend on its relevant variable and to be independent
of the remaining two variables.
For simplifying the comparative evaluation of strain rate and
temperature effects on the necking strain, the quasi-static flow curve
at room temperature is assumed here to follow the Hollomon relationship\(\sigma_{Eq-S}=K\bullet{\varepsilon_{\text{True}}}^{n}\). In such
reference condition (static rate and room temperature), the necking
strain is equal to the hardening exponent \(\varepsilon_{N}=n\).
Eq. (2) introduced within eq. (1) yields to eq. (3).
\(\sigma_{Eq-S}\bullet R\bullet S-\frac{\partial\sigma_{Eq-S}}{\partial\varepsilon_{\text{True}}}\bullet R\bullet S-\sigma_{Eq-S}\bullet S\bullet\frac{\partial R}{\partial{\dot{\varepsilon}}_{\text{True}}}\bullet\frac{\partial{\dot{\varepsilon}}_{\text{True}}}{\partial\varepsilon_{\text{True}}}-\sigma_{Eq-S}\bullet R\bullet\frac{\partial S}{\partial T}\bullet\frac{\partial T}{\partial\varepsilon_{\text{True}}}=0\)\(\rightarrow\ \varepsilon_{N-DT}\) (3)
Given the above framework, it is possible to analyze separately the
effects of strain rate and of temperature on the necking inception.
For assessing the effect of strain rate alone on the necking onset, we
can refer to an dynamic test under ideal isothermal conditions at room
temperature (e.g. according to the multiple step procedure by Ashuach et
al. 19), so that\(\frac{\partial T}{\partial\varepsilon_{\text{True}}}=0\) and\(S=1\): the uncoupling in eq. (2) together with the dynamic
isothermal condition deliver the necking strain \(\varepsilon_{N-D}\)in eq. (4).
\(\varepsilon_{N-D}=\frac{n}{1-\frac{1}{R}\ \bullet\ \frac{\partial R}{\partial{\dot{\varepsilon}}_{\text{True}}}\ \bullet\ \frac{\partial{\dot{\varepsilon}}_{\text{True}}}{\partial\varepsilon_{\text{True}}}}\)(4)
To understand the influence of the strain rate effect on the necking
inception we have to compare the necking strain of eq. (4) with the
quasi-static necking strain that is equal to \(n\), i.e. we must
evaluate whether the denominator of the ratio in eq. (4) is lower or
higher than one.
In a standard direct-tension split Hopkinson tension bar (SHTB) test,
true strain rate typically increases with strain during the rise time of
the loading wave, for only becoming constant during the final plateau
phase, until necking onset. Moreover, the strain rate amplification
factor \(R\) of most metals is typically positive and increasing with
respect to strain rate. Therefore, all the factors in the denominator of
eq. (4) are positive, i.e. \(\varepsilon_{N-D}>\varepsilon_{N}\).
It is worth noting that if the strain rate is constant, then\(\frac{\partial{\dot{\varepsilon}}_{\text{True}}}{\partial\varepsilon_{\text{True}}}=0\)and eq. (4) predicts that \(\varepsilon_{N-D}=n\): according to the
assumptions made so far (uncoupled multiplicative hardening), only
variable strain rates can affect the necking by delaying its onset.
Then, for assessing the effect of the temperature alone on the necking
inception, we assume constant quasi-static strain rate so that\(\frac{\partial{\dot{\varepsilon}}_{\text{True}}}{\partial\varepsilon_{\text{True}}}=0\)and \(R=1\): this condition can be easily implemented by static
experiments under controlled temperature and the resulting necking
strain is given by eq. (5).
\(\varepsilon_{N-T}=\frac{n}{1-\frac{1}{S}\ \bullet\ \frac{\partial S}{\partial T}\ \bullet\ \frac{\partial T}{\partial\varepsilon_{\text{True}}}}\)(5)
In real SHTB tests the temperature always increases with strain.
Moreover the thermal softening \(S\) is typically positive and decreases
with temperature. Therefore, the denominator of eq. (5) is always
greater than one and, consequently,\(\varepsilon_{N-T}<\varepsilon_{N-S}\).
Again it is worth noting that if the temperature is constant, then\(\frac{\partial T}{\partial\varepsilon_{\text{True}}}=0\) and eq. (5)
predicts that \(\varepsilon_{N-T}=n\): the uncoupled hardening
assumed so far implies that only variable temperatures can affect the
necking by anticipating its onset.
Summarizing, from eq. (4) and eq. (5) it is possible to see that, if the
uncoupling of eq. (2) is really taking place, constant strain rate or
temperature should not change the necking initiation strain in
comparison to the quasi-static case at room temperature. At the same
time it is understood that, in standard SHTB tests, two opposite
mechanisms, caused by the variation of temperature and strain rate,
compete in respectively anticipating and delaying the necking inception
and it is not possible, a priori, to know which one prevails.
- Experimental Tests on A2-70 steel cylindrical specimens
- Experimental procedures
An experimental campaign on A2-70 steel specimens, including
quasi-static tensile tests by motor driven machines at different
temperatures (room temperature, 80 °C, 140 °C, 200 °C, 300 °C) and
dynamic tensile tests by SHTB at room temperature with incident waves of
15 and 26 kN, has been carried out. All the tests have been conducted on
nominally identical specimens, with a minimum cross section diameter of
3 mm and a gage length of 9 mm. The details of the campaign are shown in
Table 1 where the reference true strain rate is the true strain rate
reached before the necking onset. Such value has been chosen as
representative for the entire test considering that true strain rate
varies greatly during a dynamic tests. The obtained results have been
analyzed with particular attention to the necking phenomenon in order to
evaluate the influences of temperature and strain rate on its onset.
In all the tests, the minimum cross-section diameter is optically
measured during the entire test thanks to standard video camera in the
quasi-static tests and high frame rate camera in the dynamic ones. From
such data it was possible to calculate the true stress, the true strain
and the true strain rate as shown in eqs. (6), (7) and (8).
\(\sigma_{\text{True}}=\frac{F}{\pi/4\bullet d^{2}}\) (6)
\(\varepsilon_{\text{True}}=2\bullet Ln\left(\frac{d_{0}}{d}\right)\)(7)
\({\dot{\varepsilon}}_{\text{True}}=\frac{\partial\varepsilon_{\text{True}}\left(t\right)}{\partial t}\)(8)
Table
1. Summary of the A2-70 Experimental Campaign