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.
4FRACTURE BEHAVIOR AND FAILURE ASSESSMENT
4.1FRACTURE 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.
4.2DISCUSSION 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.
5CONCLUSIONS
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.