\documentclass[10pt]{article}
\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}
\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
linkcolor = blue,
urlcolor = blue,
citecolor = blue,
anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
\errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother
\usepackage{natbib}
\renewenvironment{abstract}
{{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
{\bigskip}
\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}
\usepackage{authblk}
\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%
\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}
\usepackage[utf8]{inputenc}
\usepackage[ngerman,english]{babel}
\usepackage{float}
\begin{document}
\title{Quasi-static and dynamic hardening and fracture strain of A2-70
stainless steel under different temperatures}
\author[1]{Giuseppe Mirone}%
\author[1]{Raffaele Barbagallo}%
\affil[1]{University of Catania}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
In this paper, the behaviour of A2-70 stainless steel is investigated by
performing tensile tests on round specimens at different temperatures
under quasi-static and dynamic conditions. The relationship between
thermal softening and strain is firstly investigated, highlighting that
the variability of the necking strain under different temperatures is a
proof of such interaction. The competing effects of strain rate and
temperature in respectively delaying and anticipating the necking onset
are also quantified analytically, referring to multiplicative hardening
models with and without the coupling of strain and temperature. Then,
the comparison of necking strains from static and dynamic (Hopkinson
bar) tests at different temperatures is analysed, for understanding
which effect among thermal softening and dynamic stress amplification
prevails in anticipating/delaying the necking. Fracture strains and the
shapes of specimens at failure are finally related to the respective
strain rates and temperatures.%
\end{abstract}%
\sloppy
Quasi-static and dynamic hardening and fracture strain of A2-70
stainless steel under different temperatures
Giuseppe Mirone11* Corresponding author. Tel.: +39 095 738
2418.\emph{E-mail address:} gmirone@dii.unict.it, Raffaele Barbagallo
University of Catania, DICAR - Department of Civil Engineering and
Architecture, Via Santa Sofia 64 -- 95125 -- Catania, Italy
Abstract
In this paper, the behaviour of A2-70 stainless steel is investigated by
performing tensile tests on round specimens at different temperatures
under quasi-static and dynamic conditions. The relationship between
thermal softening and strain is firstly investigated, highlighting that
the variability of the necking strain under different temperatures is a
proof of such interaction. The competing effects of strain rate and
temperature in respectively delaying and anticipating the necking onset
are also quantified analytically, referring to multiplicative hardening
models with and without the coupling of strain and temperature. Then,
the comparison of necking strains from static and dynamic (Hopkinson
bar) tests at different temperatures is analysed, for understanding
which effect among thermal softening and dynamic stress amplification
prevails in anticipating/delaying the necking. Fracture strains and the
shapes of specimens at failure are finally related to the respective
strain rates and temperatures.
\emph{Keywords:} Necking; Tensile instability; Steel; Temperature;
Strain Rate; Thermal Softening; Fracture.\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
\textbf{Nomenclature} & \textbf{Nomenclature} & \textbf{Nomenclature} &
\textbf{Nomenclature}\tabularnewline
\midrule
\endhead
\(d\) & Diameter of the specimen & \(\varepsilon_{N-D}\) &
Dynamic necking onset true strain in isothermal
conditions\tabularnewline
\(d_{0}\) & Initial diameter of the specimen &
\(\varepsilon_{N-DT}\) & Dynamic necking onset true strain at variable
T\tabularnewline
\(F\) & Force acting on the specimen & \(\varepsilon_{N-S}\) &
Necking onset true strain in quasi-static conditions and constant
temperature\tabularnewline
\(K,n\) & Hollomon equation parameters & \(\varepsilon_{N-T}\) &
Necking onset true strain in quasi-static conditions and variable
temperature\tabularnewline
\(R\) & Strain rate amplification of the equivalent stress
& \(\varepsilon_{\text{true}}\) & True strain\tabularnewline
\(S\) & Thermal softening & \({\dot{\varepsilon}}_{\text{true}}\) & True
strain rate\tabularnewline
\(T\) & Temperature & \(\sigma_{\text{Eq}}\) & Equivalent
stress\tabularnewline
& & \(\sigma_{Eq-S}\) & Equivalent stress in a static
test\tabularnewline
& & \(\sigma_{\text{true}}\) & True stress\tabularnewline
& & \(\sigma_{\text{yield}}\) & Yield stress\tabularnewline
\bottomrule
\end{longtable}
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 temperatures\textsuperscript{1}, Scapin et al.
made similar investigations on pure copper\textsuperscript{2} while
Sasso et al.\textsuperscript{3} studied the strain rate sensitivity of
AA7075 aluminum alloy at different initial temper states. Mirone et
al.\textsuperscript{4} and Mirone et al.\textsuperscript{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.
Hart\textsuperscript{6} and Ghosh\textsuperscript{7} 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 Guan\textsuperscript{8} while
Lin\textsuperscript{9} 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-Nasser\textsuperscript{10} and Walley et
al.\textsuperscript{11}obtained that almost all the plastic work
developed during high strain rate tests of different metals is converted
to heat while Jovic et al.\textsuperscript{12} and Rittel et
al.\textsuperscript{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
Bridgman\textsuperscript{14}. An alternative procedure, simpler and more
accurate, was proposed by Mirone\textsuperscript{15}. 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.\textsuperscript{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.\textsuperscript{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.\textsuperscript{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 Consid\selectlanguage{ngerman}ère 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. \textsuperscript{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.
\begin{enumerate}
\tightlist
\item
Experimental Tests on A2-70 steel cylindrical specimens
\item
Experimental procedures
\end{enumerate}
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)
\begin{quote}
Table 1. Summary of the A2-70 Experimental Campaign
\end{quote}\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
\textbf{Test Series} & \textbf{Test Name} & \textbf{Reference True
Strain Rate {[}s\textsuperscript{-1}{]}} & \selectlanguage{ngerman}\textbf{Test Environment
Temperature {[}°C{]}}\tabularnewline
\midrule
\endhead
Static T\textsubscript{ROOM} & S-T\textsubscript{ROOM}-01 & 0.003 &
19\tabularnewline
& S-T\textsubscript{ROOM}-02 & 0.003 & 22\tabularnewline
& S-T\textsubscript{ROOM}-03 & 0.003 & 22\tabularnewline
Static T80 & S-T80-01 & 0.003 & 80\tabularnewline
& S-T80-02 & 0.003 & 80\tabularnewline
Static T140 & S-T140-01 & 0.003 & 140\tabularnewline
& S-T140-02 & 0.003 & 140\tabularnewline
Static T200 & S-T200-01 & 0.003 & 200\tabularnewline
& S-T200-02 & 0.003 & 200\tabularnewline
Static T300 & S-T300-01 & 0.003 & 300\tabularnewline
& S-T300-02 & 0.003 & 300\tabularnewline
& S-T300-03 & 0.003 & 300\tabularnewline
Dynamic T\textsubscript{ROOM} & D-01 (15 kN) & 700 & 20\tabularnewline
& D-02 (15 kN) & 890 & 20\tabularnewline
& D-03 (26 kN) & 1800 & 20\tabularnewline
& D-04 (26 kN) & 1850 & 20\tabularnewline
\bottomrule
\end{longtable}
Analysis of the necking inception in the quasi-static tests at different
temperatures
In order to assess the influence of the thermal softening on the necking
inception, the quasi-static curves at different temperatures have been
analysed. The obtained quasi-static experimental true curves are shown
in Fig. 1 while the mean yield stress and necking strains, obtained for
each group of tests by the maximum load condition, are reported in Table
2.
Such results clearly demonstrate that despite the temperature is
maintained constant during each test, it has a great effect in reducing
the necking inception strain, the latter varying from 0.44 at room
temperature to 0.18 at 300 \selectlanguage{ngerman}°C. This is in contrast with eq. (5),
predicting that whatever temperature, if constant, cannot affect the
necking onset.
Such fact means that the thermal softening cannot only depend on the
temperature as postulated in eq. (5), but it must also include a direct
dependence on the strain.
In other words, different constant temperatures do not simply scale the
quasi-static flow curves by constant factors leaving them homothetic to
each other, but such different temperatures also change the shapes of
the stress-strain curves during the straining history thanks to a
certain degree of strain-dependency; otherwise the necking inception
strain would not have changed.
\begin{quote}
Table 2. Quasi-static Yield Stress and Necking Strain at different
temperatures
\end{quote}\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
& \textbf{S-T\textsubscript{ROOM} (20)} & \textbf{S-T80} &
\textbf{S-T140} & \textbf{S-T200} & \textbf{S-T300}\tabularnewline
\midrule
\endhead
\(\sigma_{\text{yield}}[MPa]\) & 447.5 & 460 & 420 & 420 & 400\tabularnewline
\(\varepsilon_{N-T}\) & 0.44 & 0.35 & 0.19 & 0.18 & 0.18\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}
\end{center}
\end{figure}
Fig. 1. True stress-true strain experimental data of the quasi-static
tests at different temperature
Then, still assuming a multiplicative hardening function like eq. (2)
but now discarding any assumption about uncoupling of variables, the
experimental values of the thermal softening are derived at different
instants of the tests by calculating\({S=\sigma}_{\text{Eq}}\left(\varepsilon_{\text{True}},T\right)/\sigma_{\text{Eq}}\left(\varepsilon_{\text{True}},T_{\text{Room}}\right)\). The resulting
softening values are then related to the corresponding current values of
strain and temperature, so that a general two-variables function
\(S\left(T,\varepsilon\right)\) is then derived as a best-fit of such experimental
data associated in triplets of the kind \(T,\varepsilon,S\).
The equivalent stress-strain functions at all temperatures, for
calculating the above softening values, are derived from each experiment
by correcting the respective true curve through the MLR
function\textsuperscript{15}.
The bestfit function\(\text{\ S}\left(T,\varepsilon\right)\) is plotted in Fig. 2 and clearly
shows a remarkable dependence of \(S\) on both
\(T\) and \(\varepsilon\). The coupling of temperature
and strain is frequently neglected in the literature, but for materials
like the A2-70 steel it is clear that it cannot be neglected.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}
\end{center}
\end{figure}
Fig. 2. 3D Thermal softening function
Assuming now a multiplicative hardening with the coupled effects of
strain and temperature included within the softening function, the
general Consid\selectlanguage{ngerman}ère condition of eq. (3) must be updated; for the
quasistatic case with temperature effect
(\(\frac{\partial{\dot{\varepsilon}}_{\text{True}}}{\partial\varepsilon_{\text{True}}}=0\),\({\dot{\varepsilon}}_{\text{True}}=0\) and \(R=1\)) it takes
the following form:
\(\sigma_{Eq-S}\bullet S-\frac{\partial\sigma_{Eq-S}}{\partial\varepsilon_{\text{True}}}\bullet S-\sigma_{Eq-S}\bullet\frac{\partial S}{\partial T}\bullet\frac{\partial T}{\partial\varepsilon_{\text{True}}}-\sigma_{Eq-S}\bullet\frac{\partial S}{\partial\varepsilon_{\text{True}}}=0\)\(\rightarrow\ \varepsilon_{N-DT}\) (9)
then eq. (5) only valid if \(S=S\left(T\right)\), is substituted by the
more realistic eq. (10) which is based on\(S=S\left(T,\varepsilon\right)\).
\(\varepsilon_{N-T}=\frac{n}{1-\frac{1}{S}\ \bullet\ \left(\frac{\partial S}{\partial\varepsilon_{\text{True}}}+\frac{\partial S}{\partial T}\ \bullet\ \frac{\partial T}{\partial\varepsilon_{\text{True}}}\right)}\)(10)
The significant improvement of eq. (10) with respect to eq. (5) is that,
thanks to the incorporation of the strain-temperature coupling
within\(S\left(T,\varepsilon\right)\) and to the related term\(\frac{\partial S}{\partial\varepsilon_{\text{True}}}\), the
former equation recognizes the necking onset anticipation due to
constant temperatures higher than \(T_{\text{Room}}\), which eq.(5) was
not able to capture.
The obtained quasi-static thermal softening is now applied to the
dynamic tests, for also predicting the corresponding changes of the
necking strain. The anticipation effect due to the temperature will be
in contrast to the delay effect caused by the strain rate variation, and
the analysis of dynamic experiments will show which one of the two
effects will prevail.
Analysis of the necking phenomenon in dynamic tests with combined strain
rate and temperature effects
During dynamic SHTB tests, both temperature and strain rate undergo
significant variations. In fact the temperature greatly increases due to
the plastic work generated in almost adiabatic conditions, while the
plastic strain rate evolves from zero at first yield up to the regime
value at plateau.
Then, also the necking inception strain can remarkably change with
respect to \(\varepsilon_{N}\) from static tests at \(T_{\text{Room}}\),
because of the combined effect from the thermal softening and from the
dynamic amplification of the stress.
Such combined effect is very clear in the left part of Fig. 3, where the
dynamic true curves are compared to the quasi-static ones at room
temperature.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}
\end{center}
\end{figure}
Fig. 3. True Stress-True strain curves from SHTB experiments compared to
the quasi-static one at room temperature (left) and dynamic effective
diameter-based true strain rate vs. true strain curves (right)
In the first part of the dynamic tests, the temperature is very close to
the room one and it is possible to see a clear strain rate effect, with
both dynamic curves higher than the static ones and, among the dynamics,
the 26 kN curves higher than the 15 kN ones. Then, the temperature rises
with the strain and, at the end of all the dynamic tests, it is so high
that the corresponding thermal softening has a greater effect than the
strain rate amplification; in fact, at late strains the dynamic true
curves become lower than the quasi-static ones at room temperature.
To complete the description of the dynamic tests, in the right part of
Fig. 3 also the strain rate vs strain curves are shown for all the
dynamic tests.
Classical strain rate histories from SHTB tests are supposed to remain
constant from the end of the rise time up to failure. Instead optical
measurements of the current specimen diameter show that the necking
induces an intense spontaneous increase of the effective strain rate, so
that the strain rate histories in Figure 3 exhibit a plateau just
limited to the necking onset, followed by a steep increasing ramp
extending up to failure.
Such intense spontaneous increase of the strain rate after necking onset
was firstly evidenced by Mirone\textsuperscript{20}, confirmed by Mirone
et al.\textsuperscript{21} and recently acknowledged by Zhang et
al.\textsuperscript{22}.
In order to calculate the temperature increase due to plastic work
conversion during dynamic tests, it is necessary to evaluate the
equivalent stress-strain curves; these are obtained here by correcting
the experimental true curves all over their postnecking range, through
the MLR function\textsuperscript{15}. The plastic work is then converted
into heat via the Taylor-Quinney Coefficient (TQC) assumed to be equal
to 1, according to the findings of Kapoor \&
Nemat-Nasser\textsuperscript{10} and Walley et al.\textsuperscript{11}.
The calculated temperature histories for all the dynamic tests are shown
in Fig. 4. The necking strains identified from experiments are then
reported in Table 3 together with the corresponding temperature at that
instant.
Fig. 5 shows the necking strains against the temperature from
quasi-static and dynamic tests. The dynamic necking strains (around 0.2
at nearly 70 \selectlanguage{ngerman}°C) lie below the fitting curve of the quasi-static necking
strains, delivering a value of about 0.35 at 70 °C.
This means that the anticipation effect induced by\(\frac{\partial S}{\partial T}\ \bullet\ \frac{\partial T}{\partial\varepsilon_{\text{True}}}\)in
eq.(8) is greater than the delaying effect of the strain rate, resulting
in lower dynamic necking strains with respect to the quasi-static
counterparts at the same temperature.
It is important to underline that, without the coupling between strain
and temperature within the thermal softening function, no explanation
could have been provided for the dynamic necking strains being lower
than their quasi-static counterparts at the same temperature.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
Fig. 4. Material temperature histories in the dynamic test & Fig. 5.
Quasi-static and dynamic necking inception strains\tabularnewline
\bottomrule
\end{longtable}
\begin{quote}
Table 3. Dynamic necking strain and corresponding temperature
\end{quote}\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
& \textbf{D-01 (15 kN)} & \textbf{D-02 (15 kN)} & \textbf{D-03 (26 kN)}
& \textbf{D-04 (26 kN)}\tabularnewline
\midrule
\endhead
\(\varepsilon_{N-D}\) & 0.24 & 0.24 & 0.19 & 0.23\tabularnewline
\(T_{N}\) & 75 & 76 & 65 & 75\tabularnewline
\bottomrule
\end{longtable}
It is highly expectable that a coupling of strain rate and temperature
also occurs under dynamic conditions, although its determination
requires further tests not available for this work. However, the overall
predominance of the temperature effects on the dynamic ones in affecting
the necking inception is ascertained here independently of the possible
further coupling above.
Failure strains under combined strain rate and temperature effects
Failure is a typically local phenomenon, so the local diameter-based
true strains at fracture should be much more appropriate than the
elongation-based engineering strains frequently adopted in the
literature, which only reflect a global volume-averaged strain
indicator.
Both types of fracture strains are reported in Table 4 for each test
family (average of two or three test repetitions), together with the
corresponding values of nominal strain rates and of estimated specimen
temperatures at fracture. In the same table are reported also the
effective / engineering strains ratio and the max postnecking true
strain (Fail -- Neck).
Table 4 Strain-related variables at fracture\selectlanguage{english}
\begin{longtable}[]{@{}llllllll@{}}
\toprule
& \textbf{S-T20} & \textbf{S-T80} & \textbf{S-T140} & \textbf{S-T200} &
\textbf{S-T300} & \textbf{D 15 kN} & \textbf{D 26 kN}\tabularnewline
\midrule
\endhead
Reference true strain rate & 0.003 & 0.003 & 0.003 & 0.003 & 0.003 & 800
& 1800\tabularnewline
Temperature at fracture & 20 & 80 & 140 & 200 & 300 & 324 &
299\tabularnewline
Fracture true strain & 1.48 & 1.53 & 1.32 & 1.27 & 1.06 & 1.07 &
0.93\tabularnewline
Fracture engineering strain & 0.67 & 0.66 & 0.45 & 0.35 & 0.30 & 0.45 &
0.35\tabularnewline
Effective / Engineering strains ratio & 1.27 & 2.32 & 2.93 & 3.63 & 3.53
& 2.38 & 2.66\tabularnewline
Max Postnecking true strain (Fail -- Neck) & 1.04 & 1.18 & 1.13 & 1.09 &
0.88 & 0.83 & 0.715\tabularnewline
\bottomrule
\end{longtable}
The same data are arranged in Fig. 6 as a 3D plot with the reference
true strain rate and the temperature at fracture as the X and Y axes and
the true strain at fracture as the Z axis.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}
\end{center}
\end{figure}
Fig. 6 3D plot of the true fracture strain versus temperature at
fracture and reference true strain rate of the test
The fracture true strain at static rates clearly decreases with the
temperature, from 1.48 at room temperature to 1.06 at 300 \selectlanguage{ngerman}°C. This can
also be indirectly caused by the temperature anticipating the necking
onset, which, in turn, causes the necking-induced stress triaxialty to
evolve much sooner and much largely than it does for tests at room
temperature. In fact, the postnecking strain range up to failure is
nearly identical for static tests from Troom up to 200 °C and slowly
decreases at 300 °C.
Dynamic tests exhibit a material temperature at fracture close to 300
°C, similar to the quasi-static tests at the highest temperature, but
the true strains at failure are lower, close to 0.93 for the tests at
nominal strain rate of 1800 s\textsuperscript{-1}. This means that the
strain rate too tends to anticipate failure of the A270 steel together
with temperature, by further decreasing the fracture strain.
The dynamic tests at nominal 800 s\textsuperscript{-1}, progressively
heating from Troom up to 300 °C at incipient failure, exhibits a failure
strain close to that of static 300 °C: the fracture-delaying effect of
the initially low temperature of the dynamic test with respect to the
static one (at 300 °C since first yield) is compensated by the
fracture-anticipating strain rate effect. Therefore, for the A2-70 steel
at hand, both temperature and strain rate have a decreasing effect on
the fracture true strain.
Comparing the necked specimen shapes from dynamic tests to those from
the quasi-static tests at 300 °C, it is also possible to see that they
show different degrees of strain localization at fracture, fully
reflecting the maximum postnecking strain in the last row of Table 4.
Fig. 7 shows the comparison between the last frames before fracture of
the S-T300 and the D-26, where the difference between the two diameters
is highlighted. The two tests series show a similar fracture engineering
strain (0.3 and 0.35 respectively) but different fracture true strain
(1.06 and 0.93 respectively), i.e. different diameter. In other words,
the test S-T300 shows a greater strain localization than the dynamic
test at 1800 s\textsuperscript{-1}, despite a comparable overall
engineering deformation.
Similarly, in Fig. 8 it is shown the comparison between the last frames
before fracture of the S-T300 and the D-15, in which is highlighted the
difference between the two gage lengths. In this case, the two group of
tests show a similar fracture true strain (1.06 and 1.07 respectively),
i.e. similar diameter, but different fracture engineering strain (0.3
and 0.45 respectively), i.e. different overall gage length. Therefore,
the S-T300 shows a similar fracture true strain in respect to the D-15
with a lower overall engineering deformation.
Both comparisons highlight that the high temperature quasi-static tests
show a greater strain localization with respect to the dynamic tests.
This is due to the earlier necking onset of static high temperature
tests with respect to the dynamic tests at room temperature, leading to
larger postnecking strains at the same overall true strain which either
means more pronounced shrinking at given elongation or lower elongation
at given diameter contraction.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}
\end{center}
\end{figure}
Fig. 7 Comparison at fracture between S-T300 and D-26 kN with
highlighted the difference between the diameters, i.e. between the true
strains, with a similar gage length, i.e. similar engineer strain\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}
\end{center}
\end{figure}
Fig. 8 Comparison at fracture between S-T300 and D-15 kN with
highlighted the difference between the gage lengths, i.e. between the
engineering strains, with a similar diameter, i.e. similar true strain
Conclusions
In the present work, an in-depth analysis of the necking onset under
quasi-static and dynamic conditions has been carried out for the A2-70
stainless steel.
Firstly the influences of temperature and strain rate together with
their variability have been evaluated on the necking inception strain,
by assuming that strain, strain rate and temperature are uncoupled to
each other. From such qualitative analysis it was shown that, in
standard SHTB tests, the increasing temperature and strain rate should
respectively cause a decrease and an increase of the necking initiation
strain.
The experimental campaign on the A2-70 steel included quasi-static tests
at different temperatures and dynamic SHTB test at room temperatures,
all with cylindrical specimens. The true stress -- true strain curves,
derived from the tests by means of fast camera acquisitions of the
evolving diameter, are translated into equivalent stress-strain curves
by using the MLR function.
The necking anticipation found from quasi-static experiments at constant
high temperatures demonstrated the existence of a coupling between
strain and temperature within the thermal softening function. This
necking anticipation was then mathematically demonstrated by simply
introducing a general coupling between strain and temperature within the
thermal softening function.
The evolving temperatures of all the tests, due to the fast adiabatic
conversion of plastic work into heat, were evaluated; then, the triplets
of strain, temperature and thermal softening values were best-fitted by
a general polynomial.
The dynamic necking strains lower than their quasi-static counterparts
at similar temperatures evidenced that the anticipating effect caused by
the growing temperature is greater than the delaying effect of the
growing strain rate.
Lastly, the analysis of fracture strains showed that, for the A2-70
steel at hand, both temperature and strain rate have a decreasing effect
on the fracture true strain. Moreover, comparing the necked specimens
from dynamic tests to those from quasi-static tests at high temperature,
an influence of temperature on the strain localization and on the
maximum postnecking strain range was evidenced and addressed to earlier
necking strains from higher temperatures.
Author contribution statement
All persons who meet authorship criteria are listed as authors, and all
authors certify that they have participated sufficiently in the work to
take public responsibility for the content, including participation in
the concept, design, analysis, writing, or revision of the manuscript.
References
{[}1{]} Ruggiero, Andrew, et al. Strain rate effects on fracture
behavior of austempered ductile irons. In: AIP Conference Proceedings.
AIP Publishing LLC, 2018. p. 070028.
{[}2{]} Scapin, M.; Peroni, L.; Fichera, C. Investigation of dynamic
behaviour of copper at high temperature. Materials at High Temperatures,
2014, 31.2: 131-140.
{[}3{]} Sasso, Marco, et al. High strain rate behaviour of AA7075
aluminum alloy at different initial temper states. In: Key Engineering
Materials. Trans Tech Publications Ltd, 2015. p. 114-119.
{[}4{]} Mirone, G., et al. Static and dynamic response of titanium alloy
produced by electron beam melting. Procedia Structural Integrity, 2016,
2: 2355-2366.
{[}5{]} Mirone, G.; Barbagallo, R.; Giudice, F. Locking of the strain
rate effect in Hopkinson bar testing of a mild steel. International
Journal of Impact Engineering, 2019, 130: 97-112.
{[}6{]} Hart, E. W. Theory of the tensile test. Acta metallurgica, 1967,
15.2: 351-355.
{[}7{]} Ghosh, A. K. Tensile instability and necking in materials with
strain hardening and strain-rate hardening. Acta Metallurgica, 1977,
25.12: 1413-1424.
{[}8{]} Guan, Zhiping. Quantitative analysis on the onset of necking in
rate-dependent tension. Materials \& Design (1980-2015), 2014, 56:
209-218.
{[}9{]} Lin, E. I. H. Plastic instability criteria for necking of bars
and ballooning of tubes. 1977.
{[}10{]} Kapoor, Rajeev; Nemat-Nasser, Sia. Determination of temperature
rise during high strain rate deformation. Mechanics of Materials, 1998,
27.1: 1-12.
{[}11{]} Walley, S. M., et al. Comparison of two methods of measuring
the rapid temperature rises in split Hopkinson bar specimens. Review of
scientific instruments, 2000, 71.4: 1766-1771.
{[}12{]} Jovic, C., et al. Mechanical behaviour and temperature
measurement during dynamic deformation on split Hopkinson bar of 304L
stainless steel and 5754 aluminium alloy. In: Journal de Physique IV
(Proceedings). EDP sciences, 2006. p. 1279-1285.
{[}13{]} Rittel, D.; Zhang, L. H.; Osovski, S. The dependence of the
Taylor--Quinney coefficient on the dynamic loading mode. Journal of the
Mechanics and Physics of Solids, 2017, 107: 96-114.
{[}14{]} Bridgman, Percy Williams. Studies in large plastic flow and
fracture. New York: McGraw-Hill, 1952.
{[}15{]} Mirone, Giuseppe. Approximate model of the necking behaviour
and application to the void growth prediction. International Journal of
Damage Mechanics, 2004, 13.3: 241-261.
{[}16{]} Mirone, G.; Verleysen, Patricia; Barbagallo, R. Tensile testing
of metals: Relationship between macroscopic engineering data and
hardening variables at the semi-local scale. International Journal of
Mechanical Sciences, 2019, 150: 154-167.
{[}17{]} Peroni, Lorenzo; Scapin, Martina; Fichera, Claudio. An advanced
identification procedure for material model parameters based on image
analysis. In: 10th European LS-DYNA Conference tenutosi a W\selectlanguage{ngerman}ürzburg
(Germania). 2015.
{[}18{]} Sasso, M., et al. High speed imaging for material parameters
calibration at high strain rate. The European Physical Journal Special
Topics, 2016, 225.2: 295-309.
{[}19{]} Ashuach Y., Avinadav C., and Rosenberg Z. Eliminating the
thermal softening of dynamically loaded specimens in the Kolsky bar
system by multi-step loading, EPJ Web of Conferences, 2012, 26, 01047.
{[}20{]} Mirone, Giuseppe. The dynamic effect of necking in Hopkinson
bar tension tests. Mechanics of materials, 2013, 58: 84-96.
{[}21{]} Mirone, G.; Corallo, D.; Barbagallo, R. Experimental issues in
tensile Hopkinson bar testing and a model of dynamic hardening.
International Journal of Impact Engineering, 2017, 103: 180-194.
{[}22{]} Zhang, Longhui, et al. Rate dependent behaviour and dynamic
strain localisation of three novel impact resilient titanium alloys:
Experiments and modelling. Materials Science and Engineering: A, 2020,
771: 138552.
\selectlanguage{english}
\FloatBarrier
\end{document}