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\begin{document}
\title{Experimental study on dynamic fracture behavior of AISI 5140 steel over
a wide range of loading rates}
\author[1]{Qiaoguo Wu}%
\author[2]{Defu Nie}%
\author[1]{Jianhua Pan}%
\author[1]{changzheng cheng}%
\author[1]{Xuan Wang}%
\affil[1]{Hefei University of Technology}%
\affil[2]{National Safety Engineering Technology Research Center for Pressure Vessels and Pipelines}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
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\selectlanguage{english}
\begin{abstract}
Experimental studies were conducted on the dynamic fracture behavior of
AISI 5140 steel over a wide range of loading rates. True stress-strain
relations of the steel were measured under different strain rates, and a
dynamic constitutive model was then expressed with the strain-hardening
term and the strain-rate hardening term. Fracture tests were
investigated under quasi-static condition, instrumented Charpy impact
condition and Hopkinson pressure bar impact condition. Fracture
toughnesses corresponding to each test condition were also determined.
Fracture characteristics were analyzed with observations of the fracture
appearances. It was found that the fracture toughness decreased
significantly with the increasing loading rate, and the fracture
mechanisms of the steel at various loading rates were brittle fractures
characterized by river markings and secondary cracks. Based on the
fracture assessment method of the CEGB R6 procedure, the effects of the
strain rate and the loading rate on the assessment curve were discussed.%
\end{abstract}%
\sloppy
\textbf{1 | INTRODUCTION}
The evaluation of the dynamic fracture behavior of engineering materials
has an important significance for the assurance of the integrity and
safety of structural components subjected to various dynamic loads, such
as impact, explosion, or earthquake. Since the loading rate range of
these dynamic loads is very wide, a more comprehensive understanding of
the relationship between the fracture behavior and the loading rate is
necessary.\textsuperscript{1,2}
Some research has been carried out on the area of dynamic fracture. For
the complexity of the dynamic fracture process, theoretical methods are
limited and tests are the main means of dynamic fracture mechanics
research. Three ranges are divided according to the loading
rate\(\dot{K_{I}}\): low loading rate range (\(10^{-3}\ \text{MPa}\frac{\sqrt{m}}{s}\leq\dot{K_{I}}<10^{3}\ \text{MPa}\frac{\sqrt{m}}{s}\)), at
which it is regarded as quasi-static loading and belongs to quasi-static
fracture; medium loading rate range (\(10^{3}\ \text{MPa}\frac{\sqrt{m}}{s}\leq\dot{K_{I}}<10^{5}\ \text{MPa}\frac{\sqrt{m}}{s}\)), at which the
influence of inertia effect should be considered; and high loading rate
range (\(\dot{K_{I}}{\geq 10}^{5}\ \text{MPa}\frac{\sqrt{m}}{s}\)), at which it must consider the interaction
between stress wave and crack in addition to the influence of inertia
effect. According to different loading rates, different types of loading
test devices can be selected. The experimental techniques for dynamic
fracture often include the drop-weight impact test, instrumented Charpy
impact test, and Hopkinson pressure bar (HPB) impact
test.\textsuperscript{3-6}
Chaouadi and Puzzolante\textsuperscript{7} examined the dynamic fracture
toughness of ferritic steel with an instrumented Charpy impact test. The
dynamic fracture toughness was greater than the quasi-static one.
Similar conclusions were obtained in Foster et al.\textsuperscript{8}and
Prasad et al.\textsuperscript{9}, where fracture toughness of the 4340
steel and Al-Li 8090 alloy increased with the increasing loading rate.
Wu et al.\textsuperscript{10} performed experimental studies on the
dynamic fracture behavior of FV520B steel under quasi-static and dynamic
loading conditions, and the fracture toughness increased linearly with
the increasing loading rate. Galvez et al.\textsuperscript{11} conducted
experimental studies on the fracture behavior of high strength steel
Armox500T, and the static and dynamic fracture toughnesses were quite
similar. The fracture toughness without a marked loading rate effect was
also determined for Al 7075-T651.\textsuperscript{12} Different from
these insensitive ones, the irregular relation of the fracture toughness
and the loading rate was observed experimentally for 685 homogeneous
steel.\textsuperscript{13} When the loading rate was less than 1.8778
MPa.m\textsuperscript{0.5}, the dynamic fracture toughness decreased
with the increasing loading rate. However, when the loading rate was
greater than 1.8778 MPa.m\textsuperscript{0.5}, the dynamic fracture
toughness rose due to the effect of thermal softening near the crack
tip. Additionally, Wu et al.\textsuperscript{14} conducted an
experimental study on fracture behavior of AISI 1045 steel. The fracture
modes exhibited a transition from ductile to brittle fracture with the
increasing loading rate, and the dynamic fracture toughness was less
than the quasi-static one. A similar conclusion was also obtained in
Lorentzon et al.\textsuperscript{15} for the ordinary C-Mn structural
steel. According to the published reports, it was found that when the
loading rate increased, some fracture toughnesses increased, some
decreased, and other fracture toughness levels changed insensitively or
irregularly.
AISI 5140 steel, also named 40Cr steel, has been widely used in
engineering structures, and investigations on the dynamic fracture
behaviors are related to many engineering problems. Xu and
Li\textsuperscript{16} carried out the fracture test of 40Cr steel
loaded by HPB and the fracture toughness of the steel increased with the
increasing loading rate. However, for the same material, a conflicting
result was obtained in Li et al.\textsuperscript{17}, where the fracture
toughness decreased with the increasing loading rate. It should be noted
that in Refs. 16 and 17, the fracture toughnesses were both determined
by the numerical-experimental method, but the dynamic constitutive
relation of the steel was not considered in the numerical simulations.
Besides, as described in the literature, the loading rate range tested
may not be wide enough, resulting in inconsistent conclusions. Few
results can be found on the fracture toughness and fracture mechanism of
the steel over a wide range of loading rates.
In this paper, experimental studies were performed on the fracture
behavior of AISI 5140 steel over a wide range of loading rates. True
stress-strain relations were measured and the dynamic constitutive model
was proposed. Fracture tests under quasi-static condition, instrumented
Charpy impact condition, and HPB impact condition were carried out, and
fracture toughnesses and characteristics were analyzed. Comparisons and
discussions of the fracture behaviors under the above loading rates were
conducted in terms of fracture modes and fracture toughness values.
Finally, based on the fracture assessment method of the CEGB R6
procedure, the effects of the strain rate and the loading rate on the
assessment curve were discussed.
\textbf{2 | TEST MATERIAL AND CONSTITUTIVE RELATION}
The AISI 5140 steel was obtained from a large reciprocating compressor
crankshaft with a diameter of 200 mm. The crankshaft was first forged
with a temperature between 850 and 1150 \textsuperscript{o}C and air
cooled to room temperature. Then it underwent the heat treatment process
with a quenching temperature 850 of\textsuperscript{o}C, a tempering
temperature of 590\textsuperscript{o}C and cooling with oil. The
chemical compositions of the steel were measured and are listed in Table
1. The microstructure of the AISI 5140 steel was mainly composed of fine
pearlites, tempered sorbites and proeutectoid ferrites around prior
austenite grain boundaries, and the grain size was about 20-30 \selectlanguage{greek}μ\selectlanguage{english}m, as
shown in Figure 1.
Quasi-static and dynamic mechanical properties of the AISI 5140 steel
were measured using the electro-hydraulic servo testing machine and HPB.
True stress-strain curves in the plastic stage are presented in Figure
2, and yield strengths of the steel at different strain rates are shown
in Figure 3. The dynamic yield strengths are 847.7 MPa and 892.0 MPa at
the strain rates 3057 s\textsuperscript{-1} and 4700
s\textsuperscript{-1} respectively, which increased by 37.8\% and 45.0\%
compared to the quasi-static value 615.2 MPa. The dynamic yield
strength\(\sigma_{\text{yd}}\) can be expressed with the quasi-static
one\(\sigma_{\text{ys}}\) and the strain rate \(\dot{\varepsilon}\) in
Cowper-Symons form:
\(\sigma_{\text{yd}}=\sigma_{\text{ys}}\left(1+\frac{{\dot{\varepsilon}}_{\text{eq}}}{C}\right)^{\frac{1}{p}}\)(1)
where \(C\)= 257.8, and \(p\) = 8.0 are
material constants related to strain-rate hardening.
As shown in Figure 2, the AISI 5140 steel is sensitive to strain rate
and the flow stress increases with the increasing strain. Therefore, the
dynamic constitutive relation of the steel is expressed with the strain
hardening term and the strain-rate hardening term
\(\sigma_{\text{eq}}={\left(A_{1}+A_{2}\varepsilon_{\text{eq}}^{n}\right)\left(1+\frac{{\dot{\varepsilon}}_{\text{eq}}}{C}\right)}^{\frac{1}{p}}\)(2)
where \(\sigma_{\text{eq}}\) is equivalent stress,\(\varepsilon_{\text{eq}}\) is
equivalent strain, and\({\dot{\varepsilon}}_{\text{eq}}\) is equivalent strain rate.
\(A_{1}\)= 615.2 MPa, \(A_{2}\) = 863.7 MPa, and
\(n\) = 0.45 are material constants related to strain
hardening. It is clear from Figures 2 and 3 that the test data of the
stress-strain relation and the yield strength-strain rate relation can
be well explained by Equations (1) and (2).
\textbf{3 | FRACTURE TOUGHNESS TESTS UNDER DIFFERENT LOADING
CONDITIONS}
\textbf{3.1 | FRACTURE TOUGHNESS UNDER QUASI-STATIC CONDITION}
Quasi-static fracture tests were performed in accordance with GB/T
21143-2007\textsuperscript{18}. Three-point bending specimens were
adopted, and the thickness (\(B\)), width
(\(W\)), length (\(L\)), span
(\(S\)), and initial crack length (\(a_{0}\)) are
10 mm, 20 mm, 100 mm, 80 mm and 12 mm, respectively. The tests were
performed with the electro-hydraulic servo fatigue testing machine.
Three repetitive test results of the load-deflection curves are
presented in Figure 4.
The quasi-static fracture toughness (\(K_{\text{IC}}\)) is determined
by
\(K_{\text{IC}}=\left[\left(\frac{S}{W}\right)\frac{F_{Q}}{\left(B^{2}W\right)^{0.5}}\right]g_{1}\left(\frac{a_{0}}{W}\right)\)(3)
\(g_{1}\left(\frac{a_{0}}{W}\right)=\frac{3\left(\frac{a_{0}}{W}\right)^{0.5}\left[1.99-\left(\frac{a_{0}}{W}\right)\left(1-\frac{a_{0}}{W}\right)\left(2.15-\frac{{3.93a}_{0}}{W}+\frac{2.7a_{0}^{2}}{W^{2}}\right)\right]}{2\left(1+\frac{2a_{0}}{W}\right)\left(1-\frac{a_{0}}{W}\right)^{1.5}}\)(4)
where \(F_{Q}\) is the value of the load at the intersection
point of the load-deflection curve and the line
(\emph{OF\textsubscript{d}} ). The slope of \emph{OF\textsubscript{d}}
is 0.96 times of that of the initial linear part of the load-deflection
curve\textsuperscript{18}. Substituting the \(F_{Q}\) values
5421 N, 5598 N and 5493 N into Equation (3), the fracture toughnesses
(\(K_{\text{IC}}\)) of 63.4 MPa.m\textsuperscript{0.5}, 64.8
MPa.m\textsuperscript{0.5}, and 62.1 MPa.m\textsuperscript{0.5} are
obtained for the three tests, respectively. The average fracture
toughness is 63.4 MPa.m\textsuperscript{0.5}.
\textbf{3.2 | FRACTURE TOUGHNESS UNDER INSTRUMENTED CHARPY IMPACT
CONDITION}
Fracture toughness tests under instrumented Charpy impact condition were
conducted in accordance with GB/T 229-2007\textsuperscript{19}. The
single edge-through cracked Charpy impact specimens were utilized with
the thickness (\(B\)), width (\(W\)), length
(\(L\)), span (\(S\)), and initial crack
length (\(a_{0}\)) of 10 mm, 10 mm, 55 mm, 40 mm and 6 mm,
respectively.
Three repetitive test results of the load-deflection curves are
presented in Figure 5. For the load-deflection curve without yield
stage, the crack initiation time (\(t_{f}\)) was determined as
the time at which the load reached the peak value (\(P_{\max}\))
and the dynamic fracture toughness could be determined by linear elastic
fracture mechanics\textsuperscript{20}
\(K_{\text{Id}}=\frac{P_{\max}S}{BW^{1.5}}f\left(\frac{a_{0}}{W}\right)\)(5)
\(f\left(\frac{a_{0}}{W}\right)=\frac{3\left(\frac{a_{0}}{W}\right)^{0.5}\times\left[1.99-\left(\frac{a_{0}}{W}\right)\times\left(1-\frac{a_{0}}{W}\right)\right]\times\left(2.15-3.93\frac{a_{0}}{W}+2.7\frac{a_{0}^{2}}{W^{2}}\right)}{2\times\left(1+2\frac{a_{0}}{W}\right)\times\left(1-\frac{a_{0}}{W}\right)^{1.5}}\)(6)
The peak loads are 3.16 kN, 3.29 kN and 3.27 kN, with the
corresponding\(t_{f}\) being 0.128 ms, 0.116 ms and 0.120 ms,
respectively. According to Equation (5), the dynamic fracture
toughnesses are 47.7 MPa.m\textsuperscript{0.5}, 49.6
MPa.m\textsuperscript{0.5}, 49.3 MPa.m\textsuperscript{0.5}, and the
loading rates (\({\dot{K}}_{\text{Id}}=K_{\text{Id}}/t_{f}\ \)) are 3.72\selectlanguage{ngerman}×10\textsuperscript{5}
MPa.m\textsuperscript{0.5}/s, 3.88×10\textsuperscript{5}
MPa.m\textsuperscript{0.5}/s and 3.74×10\textsuperscript{5}
MPa.m\textsuperscript{0.5}/s, respectively. The average dynamic fracture
toughness and loading rate are 48.9 MPa.m\textsuperscript{0.5} and
3.78×10\textsuperscript{5}MPa.m\textsuperscript{0.5}/s, respectively.
\selectlanguage{english}\textbf{3.3 | FRACTURE TOUGHNESS UNDER HPB IMPACT CONDITION}
3.3.1 \selectlanguage{english}| FRACTURE TEST
The principle of the dynamic fracture test system loaded by HPB can be
referred to Ref. 12. The projectile and the incident bar are both
cylindrical and 14.5 mm in diameter, and 300 mm and 1000 mm in length.
The incident bar end in contact with the specimen exhibits a wedge shape
with a wedge angle of 60\textsuperscript{o} and a fillet radius of 2 mm.
The specimens utilized in the HPB impact test were the same as the
quasi-static test. The crack initiation time was evaluated by a small
strain gauge mounted on the specimen.
Figure 6 shows the incident and reflected strain waves, and Figure 7
shows the crack initiation signals of the specimens. The crack
initiation time (\(t_{f}\)) can be derived from the crack
initiation signal
\(t_{f}=t_{p}-t_{d}\) (7)
where \(t_{p}\) is the time corresponding to the peak strain
signal and\(t_{d}\) is the propagating time of the strain wave
from the crack tip to the position of the strain gauge. Three repetitive
tests were conducted and the average crack initiation time was
calculated to be 31 us according to Equation (7).
The displacement of the incident bar end initially in contact with the
specimen \(D\left(t\right)\) was calculated from one-dimensional elastic
wave propagation theory
\(D\left(t\right)=\int_{0}^{t}{c\left[\varepsilon_{i}\left(t\right)-\varepsilon_{r}\left(t\right)\right]\text{dt}}\)(8)
where \(\varepsilon_{i}\left(t\right)\) and\(\varepsilon_{r}\left(t\right)\) are incident and
reflected strains,\(c\) is the sound speed in the incident
bar (\(c=\sqrt{\frac{E}{\rho}}\)), and \(\rho\) is the density of the
incident bar. For the steel bar used in the test, the density
(\(\rho\)) is taken to be 7800 kg/m\textsuperscript{3}. Since
the test data of the three specimens were consistent (see Figures 6 and
7), the data of specimen-1 was selected to be analyzed in the following.
Figure 8 shows the displacement of the incident bar end initially in
contact with the specimen.
3.3.2 \selectlanguage{english}| DETERMINATION OF FRACTURE TOUGHNESS
A numerical-experimental method was adopted to determine the fracture
toughness. The finite element model was established with ABAQUS based on
the test parameters. Only one quarter of the incident bar and the
specimen, as well as half of one roller support, was modeled because of
the geometric symmetry, as shown in Figure 9. For simplification, the
incident stress wave calculated from the experimental strain wave was
used as the input load exerted to the free end of the incident
bar\textsuperscript{16}. The C3D8R elements were used for the whole
model. The incident bar model was meshed with 9137 nodes and 6128
elements, and the support model was meshed with 486 nodes and 336
elements. The specimen model was first meshed with 6660 nodes and 5220
elements, and the mesh of the crack tip and adjacent area were then
refined (see Figure 9). The face-to-face contact algorithm was assigned
in the model.
A linear elastic constitutive relation was adopted for the incident bar
and the support. The elastic modulus, the Poisson ratio and the density
were taken to be 210 GPa, 0.3 and 7800 kg/m\textsuperscript{3},
respectively. For the AISI 5140 steel, the dynamic constitutive relation
considering strain hardening and strain-rate hardening (see Equation
(2)) was utilized. The constitutive relation was implemented by
utilizing a user-defined subroutine UMAT.
Since the specimen exhibited a brittle fracture characteristic that
satisfied the small-scale yield condition, the stress intensity factor
(\(K_{I}\)) can be calculated from \emph{J}
-integral\textsuperscript{21}
\(K_{I}=\sqrt{\frac{\text{EJ}}{\left(1-\upsilon^{2}\right)}}\) (9)
where \(E\) is the elastic modulus and \(\upsilon\)
is Poisson's ratio.
Numerical strain history at the strain gauge position of the specimen is
shown in Figure 7 and agrees well with the tests before the crack
initiation time of 31 us. After this time, the numerical strain kept
increasing while the test strain decreased. The reason for this
difference is because the fracture process is not considered in the
simulation. Figure 8 shows a comparison of the numerical and
experimental displacement histories of the bar end initially in contact
with the specimen. The numerical result also agrees well with the test
data before the crack initiation time, and begins to deviate from the
test after this time since the specimen actually fractures. It is clear
from Figures 7 and 8 that the simulation is reliable until the crack
initiates.
The numerical \emph{J-} integral history at the crack tip is shown in
Figure 10. The fracture toughness (\(J_{\text{Id}}\)) is 6.85 MPa.mm
according to \(t_{f}\)= 31 us. From Equation (9), the fracture
toughness (\(K_{\text{Id}}\)= 38.8 MPa.m\textsuperscript{0.5}) and the
loading rate
(\({\dot{K}}_{\text{Id}}\)=1.25x10\textsuperscript{6}MPa.m\textsuperscript{0.5}/s)
are obtained.
\textbf{4} \selectlanguage{english}| \textbf{FRACTURE BEHAVIOR AND FAILURE ASSESSMENT}
\textbf{4.1} \selectlanguage{english}| \textbf{FRACTURE BEHAVIOR UNDER DIFFERENT LOADING RATES}
Macro and micro fracture appearances of the steel tested under different
loading rates are shown in Figure 11. The fracture appearances under the
three loading conditions mentioned above are similar. Brittle fracture
characteristics are exhibited for the material. Macro plastic
deformation near the crack tip and the lateral expansion of the
specimens are not clearly observed. Brittle fracture modes with river
markings and secondary cracks are also revealed from the micro
appearance observations of the fractured specimens.
The fracture toughnesses of the steel under different loading rates are
presented in Figure 12. The fracture toughness decreases with the
increasing loading rate. Compared with the quasi-static one, the
fracture toughnesses
at\({\dot{K}}_{\text{Id}}\)=3.78x10\textsuperscript{5}MPa.m\textsuperscript{0.5}/s
(under instrumented Charpy impact test)
and\({\dot{K}}_{\text{Id}}\)=1.25x10\textsuperscript{6}MPa.m\textsuperscript{0.5}/s
(under HPB impact test) decrease by 22.9\% and 38.8\%, respectively. The
relationship of \(K_{\text{Id}}\) and\({\dot{K}}_{\text{Id}}\) is described
as
\(K_{\text{Id}}=K_{\text{Id}}^{r}-K_{1}\bullet\left(\frac{{\dot{K}}_{\text{Id}}}{{\dot{K}}_{\text{Id}}^{r}}\right)^{c_{1}}\)(10)
where \({\dot{K}}_{\text{Id}}^{r}\) and \(K_{\text{Id}}^{r}\) are respectively the
reference loading rate and reference fracture toughness value
(\({\dot{K}}_{\text{Id}}^{r}\)=1,\(\ K_{\text{Id}}^{r}\)=\(K_{\text{IC}}\)=63.4
MPa.m\textsuperscript{0.5}), and \(K_{1}\) and
\(c_{1}\) are experimental constants. By fitting the
experimental data, \(K_{1}\) and\(c_{1}\) are taken
to be 0.0499 MPa.m\textsuperscript{0.5} and 0.4417, respectively.
\textbf{4.2} \selectlanguage{english}| \textbf{DISCUSSION OF FRACTURE ASSESSMENT CURVE}
The failure assessment of the cracked structure is often implemented
based on the failure assessment curve (FAC) of the CEGB R6 procedure
which considers both the brittle fracture failure and the plastic
collapse\textsuperscript{22}. However, the method is mainly utilized in
static loading condition. To generalize the method to the dynamic
loading conditions, the effects of the strain rate and the loading rate
are introduced into the FAC equation and discussed.
The FAC curve equation based on option 1 in the CEGB R6 procedure is
expressed in the following form\textsuperscript{22}
\(K_{\text{rd}}=\left\{\par
\begin{matrix}\left(1+0.5L_{\text{rd}}^{2}\right)^{-0.5}\left(0.3+0.7e^{-0.6\ L_{\text{rd}}^{6}}\right)\text{\ \ \ \ \ }L_{\text{rd}}\leq\text{L\ }_{\text{rd}}^{\max}\\
\ \ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L_{\text{rd}}>\text{L\ }_{\text{rd}}^{\max}\text{\ \ \ \ \ \ \ }\\
\end{matrix}\text{\ \ \ \ \ \ }\right.\ \) (11)
where \(K_{\text{rd}}\) is the ratio of stress intensity factor at the
crack tip to the dynamic fracture toughness of material
(\(K_{\text{rd}}=\frac{K_{I}}{K_{Id}}\)), \(L_{\text{rd}}\) is defined as the ratio of
the loading condition assessed for the plastic limit load of the
structure (\(L_{\text{rd}}=\frac{\sigma_{\text{ref}}}{\sigma_{\text{yd}}}\)),\(\sigma_{\text{ref}}\) is the reference
stress,\(\text{L\ }_{\text{rd}}^{\max}\) is the cut-off value of
the\(L_{\text{rd}}\) and is taken to be 1.20 here. Corresponding to
the static value \(K_{r}=\frac{K_{I}}{K_{\text{IC}}}\),\(L_{r}=\frac{\sigma_{\text{ref}}}{\sigma_{\text{ys}}}\) and according
to the relationship of dynamic fracture toughness and loading rate, as
well as the relationship of dynamic yield strength and strain rate, the
dynamic FAC equation can be expressed as
\(K_{r}=\left\{\par
\begin{matrix}f_{2}\left({\dot{K}}_{I}\right)\left[1+0.5\left(\frac{L_{r}}{f_{1}\left(\dot{\varepsilon}\right)}\right)^{2}\right]^{-0.5}\left[0.3+0.7e^{-0.6\left(\frac{L_{r}}{f_{1}\left(\dot{\varepsilon}\right)}\right)\ L_{\text{rd}}^{6}}\right]\text{\ \ \ \ \ }L_{r}\leq\text{L\ }_{r}^{\max}\\
\ \ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L_{r}>\text{L\ }_{r}^{\max}\text{\ \ \ \ \ \ \ }\\
\end{matrix}\text{\ \ \ \ \ \ }\right.\ \)(12)
where\(f_{1}\left(\dot{\varepsilon}\right)=\frac{\sigma_{\text{yd}}}{\sigma_{\text{ys}}}=\left(1+\frac{{\dot{\varepsilon}}_{\text{eq}}}{C}\right)^{\frac{1}{p}}\),\(f_{2}\left({\dot{K}}_{I}\right)=\frac{K_{\text{Id}}}{K_{\text{IC}}}=1-\frac{K_{1}}{K_{\text{IC}}}\left(\frac{{\dot{K}}_{\text{Id}}}{{\dot{K}}_{\text{Id}}^{r}}\right)^{c_{1}}\).
The effects of the loading rate and the strain rate on failure
assessment curves are discussed. Figure 13 shows the dynamic failure
assessment diagram (FAD) of the AISI 5140 steel under different loading
rates and strain rates. Figure 13(A) shows the relationship between the
FAC and the loading rate. The black curve is the static FAC, and the red
and blue curves are dynamic FAC at the loading rates
of\({\dot{K}}_{I}\)=5\selectlanguage{ngerman}×10\textsuperscript{5}
MPa.m\textsuperscript{0.5}/s and
\({\dot{K}}_{I}\)=1×10\textsuperscript{6}MPa.m\textsuperscript{0.5}/s
with the same strain rate\(\dot{\varepsilon}\)=1×10\textsuperscript{2}
s\textsuperscript{-1}. The structure is acceptable if the assessment
point (\(L_{r}\), \(K_{r}\)) of a cracked structure
is on or inside the FAD. Otherwise, the structure is unacceptable. It is
clear from Figure 13(A) that the acceptable zone decreases with an
increase in the loading rate. Figure 13(B) shows the relationship
between the FAC and the strain rate. The black curve is static FAC, and
the red and the blue curves are dynamic FAC at the strain rates of
\(\dot{\varepsilon}\)=1×10\textsuperscript{2}s\textsuperscript{-1} and
\(\dot{\varepsilon}\)=5×10\textsuperscript{2}s\textsuperscript{-1} with
the same loading rate\({\dot{K}}_{I}\)=1×10\textsuperscript{6}
MPa.m\textsuperscript{0.5}/s. It is clear from Figure 13(B) that the
acceptable zone is slightly widened with the increasing strain rate, and
the widened zone becomes larger with the increasing \(L_{r}\).
Therefore, it should be noted that fracture assessment of the cracked
structure made of ANSI 5140 steel must consider the effect of the
loading rate, and the direct use of the quasi-static value may lead to
dangerous results.
\textbf{5} \selectlanguage{english}| \textbf{CONCLUSIONS}
The dynamic fracture behavior of AISI 5140 steel was studied over a wide
range of loading rates. The following conclusions are drawn:
(1) True stress-strain relations of AISI 5140 steel at different strain
rates were measured, and a dynamic constitutive model was proposed. The
steel is sensitive to strain rate and the flow stress increases with the
increasing strain rate.
(2) Fracture characteristics and fracture toughnesses of the steel were
studied through the quasi-static test, instrumented Charpy impact test,
and HPB impact test. Fracture toughness decreases with the increasing
loading rate and fracture mechanisms are brittle fractures.
(3) Based on the fracture assessment method of the CEGB R6 procedure,
the effects of the strain rate and the loading rate are discussed. It is
noted that the fracture assessment of the cracked ANSI 5140 steel
structure must consider the effect of the loading rate, and the direct
use of the quasi-static value may lead to dangerous results.
\textbf{ACKNOWLEDGEMENTS}
The authors gratefully acknowledge the support of National Natural
Science Foundation of China (Grant No. 51305122) and National Basic
Research Program of China (Grant No. 2012CB026003).
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\textbf{FIGURE CAPTIONS}
\textbf{FIGURE 1} Microstructure of the AISI 5140 steel
\textbf{FIGURE 2} Experimental true stress-strain curves of the AISI
5140 steel
\textbf{FIGURE 3} Yield strength of the AISI 5140 steel at different
strain rates
\textbf{FIGURE 4} Load-deflection curves of quasi-static specimens
\textbf{FIGURE 5} Load-deflection curves of Charpy impact specimens
\textbf{FIGURE 6} The experimental incident and reflected strain waves
\textbf{FIGURE 7} Crack initiation signals of the test specimens
\textbf{FIGURE 8} Displacement of the incident bar end initially in
contact with the specimen
\textbf{FIGURE 9} Finite element model of the incident bar, the specimen
and the support
\textbf{FIGURE 10} Numerical \emph{J} -integral history at the crack tip
\textbf{FIGURE 11} Fracture appearances of AISI 5140 steel. (A)
quasi-static test, (B) instrumented Charpy impact test, (C) HPB impact
test
\textbf{FIGURE 12} Fracture toughnesses of the AISI 5140 steel under
different loading rates
\textbf{FIGURE 13} Failure assessment diagram of the AISI 5140 steel.
(A) effect of loading rate, (B) effect of strain rate\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\textbf{NOMENCLATURE} & \textbf{NOMENCLATURE}\tabularnewline
\midrule
\endhead
\(a_{0}\) & initial crack length\tabularnewline
\(A_{1}\), \(A_{2}\), \(n\) & material
constants related to strain hardening\tabularnewline
\(B\) & thickness of specimen\tabularnewline
\(c\) & sound speed in incident bar\tabularnewline
\(c_{1}\) & experimental constant\tabularnewline
\(C\)\emph{,} \(p\) & material constant
related to strain-rate hardening\tabularnewline
\(D\) & displacement of incident bar end initially in
contact with specimen\tabularnewline
\(E\) & elastic modulus\tabularnewline
\(F_{Q}\) & load at intersection point of load-deflection
curve and line \emph{OF\textsubscript{d}}\tabularnewline
\(J\) & \emph{J}-integral\tabularnewline
\(J_{\text{Id}}\) & dynamic fracture toughness\tabularnewline
\(K_{1}\) & experimental constant\tabularnewline
\(K_{I}\) & stress intensity factor\tabularnewline
\(K_{\text{IC}}\) & static fracture toughness\tabularnewline
\(K_{\text{Id}}\) & dynamic fracture toughness\tabularnewline
\({\dot{K}}_{\text{Id}}\) & loading rate\tabularnewline
\(K_{\text{Id}}^{r}\) & reference fracture toughness value\tabularnewline
\({\dot{K}}_{\text{Id}}\) & reference loading rate\tabularnewline
\(K_{\text{rd}}\) & ratio of stress intensity factor at crack tip to
dynamic fracture toughness of material\tabularnewline
\(L_{\text{rd}}\) & ratio of loading condition assessed for plastic
limit load of structure \(L_{\text{rd}}=\frac{\sigma_{\text{ref}}}{\sigma_{\text{yd}}}\)\tabularnewline
\(\text{L\ }_{\text{rd}}^{\max}\) & cut-off value of
\emph{L\textsubscript{rd}}\tabularnewline
\(L\) & length of specimen\tabularnewline
\(P_{\max}\) & peak load\tabularnewline
\(S\) & span of specimen\tabularnewline
\(t_{d}\) & propagating time of strain wave from crack tip to
position of strain gauge\tabularnewline
\(t_{f}\) & crack initiation time\tabularnewline
\(t_{p}\) & time corresponding to peak strain
signal\tabularnewline
\(W\) & width of specimen\tabularnewline
\(\varepsilon_{\text{eq}}\) & equivalent strain\tabularnewline
\({\dot{\varepsilon}}_{\text{eq}}\) & equivalent strain rate\tabularnewline
\(\varepsilon_{i}\) & incident strain\tabularnewline
\(\varepsilon_{r}\) & reflected strain\tabularnewline
\(\sigma_{\text{eq}}\) & equivalent stress\tabularnewline
\(\sigma_{\text{ref}}\) & reference stress\tabularnewline
\(\rho\) & density of incident bar\tabularnewline
\(\upsilon\) & Poisson's ratio\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}
\end{center}
\end{figure}
\textbf{FIGURE 1} Microstructure of the AISI 5140 steel\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}
\end{center}
\end{figure}
\textbf{FIGURE 2} Experimental true stress-strain curves of the AISI
5140 steel\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}
\end{center}
\end{figure}
\textbf{FIGURE 3} Yield strength of the AISI 5140 steel at different
strain rates\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}
\end{center}
\end{figure}
\textbf{FIGURE 4} Load-deflection curves of quasi-static specimens\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}
\end{center}
\end{figure}
\textbf{FIGURE 5} Load-deflection curves of Charpy impact specimens\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}
\end{center}
\end{figure}
\textbf{FIGURE 6} The experimental incident and reflected strain waves\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}
\end{center}
\end{figure}
\textbf{FIGURE 7} Crack initiation signals of the test specimens\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}
\end{center}
\end{figure}
\textbf{FIGURE 8} Displacement of the incident bar end initially in
contact with the specimen\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}
\end{center}
\end{figure}
\textbf{FIGURE 9} Finite element model of the incident bar, the specimen
and the support\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}
\end{center}
\end{figure}
\textbf{FIGURE 10} Numerical \emph{J} -integral history at the crack tip\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}
\end{center}
\end{figure}
\textbf{FIGURE 11} Fracture appearances of the AISI 5140 steel. (A)
quasi-static test, (B) instrumented Charpy impact test, (C) HPB impact
test\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}
\end{center}
\end{figure}
\textbf{FIGURE 12} Fracture toughnesses of the AISI 5140 steel under
different loading rates\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image13/image13}
\end{center}
\end{figure}
\textbf{(A)}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image14/image14}
\end{center}
\end{figure}
\textbf{(B)}
\textbf{FIGURE 13} Failure assessment diagram of the AISI 5140 steel.
(A) effect of loading rate, (B) effect of strain rate
\textbf{TABLE 1} Chemical compositions of the AISI 5140 steel (wt. \%)\selectlanguage{english}
\begin{longtable}[]{@{}lllllllll@{}}
\toprule
& C & Si & Mn & P & S & Cr & Ni & Cu\tabularnewline
\midrule
\endhead
ASTM A29/A29M & 0.38-0.43 & 0.15-0.35 & 0.70-0.90 & [?]0.035 & [?]0.040 &
0.70\textasciitilde{}0.90 & [?]0.25 & [?]0.35\tabularnewline
Measured & 0.44 & 0.25 & 0.59 & 0.021 & 0.007 & 0.85 & 0.04 &
0.05\tabularnewline
\bottomrule
\end{longtable}
\selectlanguage{english}
\FloatBarrier
\end{document}