3.3 | FRACTURE TOUGHNESS UNDER HPB IMPACT CONDITION
3.3.1 | 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 60o 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/m3. 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 | 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
bar16. 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/m3,
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 J -integral21
\(K_{I}=\sqrt{\frac{\text{EJ}}{\left(1-\upsilon^{2}\right)}}\) (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 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.m0.5) and the loading
rate (\({\dot{K}}_{\text{Id}}\)=1.25×106MPa.m0.5/s) are obtained.
4 | FRACTURE BEHAVIOR AND FAILURE ASSESSMENT
4.1 | 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\({\dot{K}}_{\text{Id}}\)=3.78×105MPa.m0.5/s (under instrumented Charpy impact test) and\({\dot{K}}_{\text{Id}}\)=1.25×106MPa.m0.5/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.m0.5), 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.m0.5 and 0.4417,
respectively.
4.2 | 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
collapse22. 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 form22
\(K_{\text{rd}}=\left\{\par
\begin{matrix}\left(1+0.5L_{\text{rd}}^{2}\right)^{-0.5}\left(0.3+0.7e^{-0.6\ L_{\text{rd}}^{6}}\right)\text{\ \ \ \ \ }L_{\text{rd}}\leq\text{L\ }_{\text{rd}}^{\max}\\
\ \ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L_{\text{rd}}>\text{L\ }_{\text{rd}}^{\max}\text{\ \ \ \ \ \ \ }\\
\end{matrix}\text{\ \ \ \ \ \ }\right.\ \) (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\({\dot{K}}_{I}\)=5×105 MPa.m0.5/s
and \({\dot{K}}_{I}\)=1×106MPa.m0.5/s with the same strain rate\(\dot{\varepsilon}\)=1×102 s-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 \(\dot{\varepsilon}\)=1×102s-1 and \(\dot{\varepsilon}\)=5×102s-1 with the same loading rate\({\dot{K}}_{I}\)=1×106 MPa.m0.5/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.
5 | 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.