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.
  1. Experimental Tests on A2-70 steel cylindrical specimens
  2. 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