Rapid progress of computer technology and computing technique in recent years has made it possible to apply numerical simulations to the study of response and failure of structures under different loading conditions. As structural failure is usually caused by its material failure, it is, therefore, essential to develop a criterion that can both accurately predict the material failure and be aptly implemented in a numerical model.

Johnson and Cook^[1-2] proposed a constitutive model which included a damage function to estimate the fracture of metallic materials. The fracture criterion is a function of hydrostatic pressure (stress triaxiality), strain rate and temperature but does not depend on the third invariant of the deviatoric stress tensor. Furthermore, the influence of stress triaxiality on the material ductility was also investigated by others^[3-5]. It was found that the material ductility decreases as stress triaxiality increases.

Wierzbicki et al.^[6] assessed the effectiveness and accuracy of 7 fracture criteria for metallic materials. It was demonstrated that the stress triaxiality alone cannot sufficiently characterize the effect of stress state on ductile fracture and that the effect of the Lode angle needs to be taken into account. Xue^[7] further considered the joint effects of pressure and Lode angle on the ductile fracture and defined a fracture envelope in principal stress space and, in particular, suggested two kinds of Lode dependence functions. Bai and Wierzbicki^[8] proposed an uncoupled material model for metal plasticity and ductile fracture in which the failure criterion was of the asymmetric three dimensional fracture locus as compared to the previously suggested symmetric ones. Later on, Bai and Wierzbicki^[9] suggested a general form of asymmetric ductile fracture which represents an extension to the Mohr-Coulomb failure criterion. Meanwhile, other authors^[10-12] also investigated the effect of Lode angle on the ductile fracture.

It should be mentioned here that comments made on the general formulation in terms of interdependence on hydrostatic and deviatoric state are valid for any material. However, there are two problems which need to be addressed. The first one is that there are often too many coefficients which need to be determined from laboratory tests and the second is that the average values of stress triaxiality and Lode angle were used in constructing a fracture locus instead of the actual values of these two parameters.

The objective of the present paper is to propose a new approach for the failure of metallic materials which is dependent on both stress triaxiality and Lode angle. It is applicable not only to metals with ē_f>e_f but also to metals with ē_f≤e_f, here ē_f and e_f are the two parameters defined as the true strains at stress triaxiality of η=1/3 for Lode parameters of ξ=1 (axisymmetric stress state) and ξ=0 (plane strain state) respectively. Furthermore, only two laboratory tests such as smooth bar tension test and pure shear test are needed to calibrate the failure criterion. The new failure criterion is compared with available material test data and discussed.

1. Formulation of a New Approach

A new approach is proposed in the following which can be used to predict the failure of metallic materials under different loading conditions by considering effects of both stress triaxiality and Lode angle. Firstly, some parameters are defined and then the new approach is formulated; finally, a failure criterion is given in which only two parameters are needed for determination from two laboratory tests.

1.1 Definition of ${\bar e_{\rm{f}}},{\gamma _{\rm{f}}},e_{\rm{f}}^ *$ and ${\bar \gamma _{\rm{f}}},{e_{\rm{f}}},\gamma _{\rm{f}}^ *$

For a metallic material, a criterion considering the effects of both stress triaxiality $\eta$ and Lode angle parameter $\xi$ is needed to predict its failure. $\eta$ and $\xi$ are defined by the following equations

where ${\sigma _{\rm{m}}}={{\left( {{\sigma _1} + {\sigma _2} + {\sigma _3}} \right)} / 3}$ , $\bar \sigma=\sqrt {\left[ {{{({\sigma _1} - {\sigma _2})}^2} + {{({\sigma _2} - {\sigma _3})}^2} + {{({\sigma _1} - {\sigma _3})}^2}} \right]/2}$ are mean and equivalent stresses, respectively, with ${\sigma _1}$ , ${\sigma _2}$ and ${\sigma _3}$ as the principal stresses; ${I_1}={\sigma _1} + {\sigma _2} + {\sigma _3}$ represents the first invariant of the stress tensor; ${J_2}=\left[ {{{\left( {{\sigma _1} - {\sigma _2}} \right)}^2} + {{\left( {{\sigma _2} - {\sigma _3}} \right)}^2} + {{\left( {{\sigma _3} - {\sigma _1}} \right)}^2}} \right]/6$ and ${J_3}=\left( {{\sigma _1} - {\sigma _{\rm{m}}}} \right)\left( {{\sigma _2} - {\sigma _{\rm{m}}}} \right)\left( {{\sigma _3} - {\sigma _{\rm{m}}}} \right)$ are the second and third invariants of the stress deviator tensor, respectively. The Lode angle parameter $\xi$ is related to the Lode angle $\theta$ on the π plane through $\xi=- \sin (3\theta )$ and the range of the Lode angle is $- {{\text{π}}/ 6} \leqslant \theta \leqslant {{\text{π}}/ 6}$ . Hence, $\xi$ takes values of $- 1 \leqslant \xi \leqslant 1$ .

Fig.1 shows the definition of various parameters in the space of fracture strain ${\varepsilon _{\rm{f}}}$ versus stress triaxiality $\eta$ where two scenarios are considered, namely, materials with ${\bar e_{\rm{f}}}>{e_{\rm{f}}}$ as shown schematically in Fig.1(a) and materials with ${\bar e_{\rm{f}}} \leqslant {e_{\rm{f}}}$ as shown schematically in Fig.1(b). Here ${\bar e_{\rm{f}}}$ designates the theoretical value of axisymmetric stress state (i.e. tension with $\eta=1/3,\; \xi=1$ ), ${\gamma _{\rm{f}}}$ represents the theoretical value of axisymmetric stress state (i.e. shear with $\eta=0,\;\xi=1$ ) which can be taken as ${\gamma _{\rm{f}}}=\sqrt 3 {\bar e_{\rm{f}}}$ ; ${\bar \gamma _{\rm{f}}}$ represents the theoretical value of plane strain state (i.e. shear with $\eta=0,\;\xi=0$ ), ${e_{\rm{f}}}$ indicates the theoretical value of plane strain state (i.e. tension with $\eta=1/3,\;\xi=0$ ) which can be taken as ${e_{\rm{f}}}{\rm{=}}{{{{\bar \gamma }_{\rm{f}}}} / {\sqrt 3 }}$ ; $e_{\rm{f}}^ *$ is the experimentally obtained true fracture strain in a smooth bar tension test ( $\xi=1$ ) and $\gamma _{\rm{f}}^ *$ is the experimentally measured shear strain in a torsion test ( $\xi=0$ ).

Figure  1.  Definition of parameters in the space of ε_f and η

DownLoad: Full-Size Img PowerPoint

For a constant Lode angle (i.e. $\xi={\rm{constant}}$ ), the relationship between equivalent fracture strain and stress triaxiality can be expressed as an exponential function^[13]. Moreover, the theoretical value of fracture strain in shear ( $\eta=0$ ) is $\sqrt 3$ times that in tension ( $\eta=1/3$ ) (Johnson and Cook^[1]). It can be easily shown that the relationship between equivalent fracture strain ${\varepsilon _{\rm{f}}}$ and stress triaxiality $\eta$ take the form of ${\varepsilon _{\rm{f}}}={C_1}{\rm e^{ - 1.5(\ln 3)\eta }}$ for $\xi=1$ after using ${\gamma _{\rm{f}}}=\sqrt 3 {\bar e_{\rm{f}}}$ and for $\xi=0$ it takes the form of ${\varepsilon _{\rm{f}}}={C_2}{\rm e^{ - 1.5(\ln 3)\eta }}$ after using ${\bar \gamma _{\rm{f}}}=\sqrt 3 {e_{\rm{f}}}$ as shown in Fig.1.

1.2 A New Approach

It has been found experimentally that some more ductile metals such as OFHC cooper have a larger equivalent strain to fracture in shear than that in tension whilst some less ductile metals such as 4340 steel and 2024-T351 aluminum alloy have a smaller equivalent strain to fracture in shear than that in tension (see Johnson and Cook^[2] and Wierzbicki et al.^[6] for more details). In the following, a new approach is suggested for the failure of a metallic material on the basis of the discussion in Section 1.1 where two scenarios are considered depending upon the relative magnitude of ${\bar e_{\rm{f}}}$ and ${e_{\rm{f}}}$ (Fig.2).

Figure  2.  Schematic diagrams of the failure loci for metallic materials in the space of ε_f and ξ for η=1/3

DownLoad: Full-Size Img PowerPoint

1.2.1 Scenario I: ${\bar {e}_{\rm{f}}}>{{e}_{\rm{f}}}$

The failure locus for a metallic material with ${\bar e_{\rm{f}}}>{e_{\rm{f}}}$ in the space of ${\varepsilon _{\rm{f}}}$ and $\xi$ for $\eta=1/3$ can be represented by the lower part of an elliptic curve as shown schematically in Fig.2(a), which can be described by the following equation

Hence, the equivalent fracture strain can be rewritten as

after taking it into consideration that ${\varepsilon _{\rm{f}}}$ is less than ${\bar e_{\rm{f}}}$ .

1.2.2 Scenario II: ${\bar {e}_{\rm{f}}} \leqslant {{e}_{\rm{f}}}$

As shown schematically in Fig.2(b), the failure locus for a metallic material with ${\bar e_{\rm{f}}} \leqslant {e_{\rm{f}}}$ in the space of ${\varepsilon _{\rm{f}}}$ and $\xi$ for $\eta=1/3$ can be represented by the upper part of an elliptic curve, which can be described by the following expression

Hence, the equivalent fracture strain can be recast into the following form

or

It is noted from Eq.(4) and Eq.(7) that no matter what scenario it is the equivalent fracture strain takes the same form. It is also noted here that ${\bar e_{\rm{f}}}$ and ${e_{\rm{f}}}$ are the specific points on the failure curves for $\xi=1$ and $\xi=0$ , respectively and, hence, they must satisfy the corresponding equations, namely, ${\bar e_{\rm{f}}}={C_1}{{\rm {e}}^{ - 1.5(\ln 3)\eta }}$ and ${e_{\rm{f}}}={C_2}{{\rm {e}}^{ - 1.5(\ln 3)\eta }}$ . Substituting ${\bar e_{\rm{f}}}={C_1}{{\rm {e}}^{ - 1.5(\ln 3)\eta }}$ and ${e_{\rm{f}}}={C_2}{{\rm {e}}^{ - 1.5(\ln 3)\eta }}$ into Eq.(4) or Eq.(7) and rearranging yield

where ${C_1},{C_2}$ are constants to be determined from laboratory material tests.

1.3 Determination of ${C_1}, {C_2}$

A criterion for the failure of a metallic material (Eq.(8)) has been developed which is a function of both stress triaxiality and Lode angle parameter. The two constants, i.e. ${C_1},{C_2}$ can be estimated from two laboratory material tests such as smooth bar tension test and pure shear test.

For the smooth bar tension test, the Lode angle parameter $\xi=1$ and the experimentally obtained true fracture strain is ${\varepsilon _{\rm{f}}}=e_{\rm{f}}^*$ and its corresponding stress triaxiality is $\eta={\eta ^*}$ . Substituting these values into Eq.(8) and rearranging it, we arrive at ${C_1}={3^{1.5{\eta ^*}}}e_{\rm{f}}^*$ . Similarly, for the pure shear test, the Lode angle parameter is $\xi=0$ and the experimentally obtained critical shear strain is ${\varepsilon _{\rm{f}}}=\gamma _{\rm{f}}^*$ and its corresponding stress triaxiality can be approximated as $\eta=0$ . Substituting these values into Eq.(8) and rearranging it, we have ${C_2}=\gamma _{\rm{f}}^*$ .

2. Comparison with Test Data and Discussion

The failure criterion developed in Section 1 is compared with the test data for various metallic materials under different loading conditions as reported in the literature. As discussed above only two laboratory material tests are needed to determine the values of $e_{\rm{f}}^*$ (the true fracture strain in tension), $\gamma _{\rm{f}}^*$ (the critical rupture strain in shear) and ${\eta ^*}$ (the stress triaxiality corresponding to $e_{\rm{f}}^*$ ). In the following, the procedures for calculating the values of these 3 parameters are first briefly described and then the newly developed failure criterion (Eq.(8)) is compared with the available experimental results for various metallic materials.

For a smooth bar tensile test, the equivalent true plastic strain to fracture in the necking cross-section of the smooth round bar can be approximately calculated by the following equation

where ${d_0}$ is the initial diameter and ${d_{\rm{f}}}$ is the minimum cross-section diameter at fracture. For a pure shear (i.e. tubular torsion) test, the equivalent true strain to fracture can be calculated by the following expression

where R is the initial radius of tubular torsion specimens, L is the initial length of the gage section and $\varphi$ is the average rotation angle to fracture found in the measured torque-rotation curves.

The value of stress triaxiality ${\eta ^*}$ which corresponds to the equivalent true fracture strain $e_{\rm{f}}^*$ in a smooth bar tension test is obtained by the following method, as shown schematically in Fig.3 which shows variation of stress triaxiality with equivalent true strain in a smooth bar under tensile loading^[14]. As can be seen from Fig.3, the stress triaxiality is not a constant during the entire tensile deformations. Before the necking occurs the deformations in the gauge length are nearly uniform and the stress triaxiality remains almost constant, namely, the theoretical value of 1/3; after the necking occurs the stress triaxiality is no longer a constant but varies with the tensile deformations due to a change in stress state in the necking section. As shown in Fig.3, it is straightforward to determine the value of ${\eta ^*}$ since $e_{\rm{f}}^*$ has been estimated from the smooth bar tension tests using Eq.(9).

Figure  3.  Definition of the stress triaxiality corresponding to the true fracture strain in a smooth bar tension test

DownLoad: Full-Size Img PowerPoint

Comparisons are made in Fig.4–Fig.9 between the newly developed failure criterion (Eq.(8)) and the test data available for different ductile metals under different loading conditions. In Fig.4–Fig.9, the broken line and the dotted line represent Eq.(8) with $\xi=1$ and $\xi=0$ , respectively. The results from the two laboratory tests (i.e. smooth bar tension test and pure shear test) as presented in Fig.4–Fig.9(a) and those (i.e. smooth bar tension test and plane strain flat bar test) as given in Fig.9(b) which are used to determine the values of two constants ${C_1},{C_2}$ in Eq.(8) can be viewed as calibration data and all the other test results as verification data.

Fig.4 shows comparison between the newly developed failure criterion (Eq.(8)) and the experimental data for the 4340 steel as reported by Johnson and Cook^[2] who conducted 4 types of laboratory material tests including the smooth bar tension test, tubular torsion test and two notched tension tests. In the calculation, the value of $\gamma _{\rm{f}}^*$ is obtained directly from the tubular torsion test data in Fig.12 of Johnson and Cook^[2] and it is equal to 0.253 6 (i.e. $\gamma _{\rm{f}}^*=0.253\,6$ ); Similarly, the value of $e_{\rm{f}}^*$ is determined to be $e_{\rm{f}}^*=1$ from the smooth bar tension test data in Fig.6 of Johnson and Cook^[2] and the value of ${\eta ^*}$ is estimated to be ${\eta ^*}=0.722\,0$ from Fig.11 of Johnson and Cook^[2]. It is easy to calculate the values of the constants ${C_1}$ and ${C_2}$ in Eq.(8) and prove them to be ${C_1}={3^{1.5{\eta ^*}}}e_{\rm{f}}^*=3.286\,4$ and ${C_2}=\gamma _{\rm{f}}^*=0.253\,6$ using the values of relevant parameters as obtained above. It can be seen that the present model predictions are in good agreement with the experimental results for the 4340 steel. Furthermore, it is easy to show that ${e_{\rm{f}}}={\rm{0}}{\rm{.146\,4}}$ (Eq.(8) with $\eta=1/3,\xi=0$ ) and ${\bar e_{\rm{f}}}={\rm{1}}{\rm{.897\,4}}$ (Eq.(8) with $\eta=1/3,\xi=1$ ). Hence, by definition the 4340 steel as examined in Johnson and Cook^[2] belongs to Category I material since ${\bar e_{\rm{f}}}>{e_{\rm{f}}}$ as discussed in Section 1.1.

Fig.5 shows comparison of the present model (Eq.(8)) with the experimental data for the Armco iron as reported by Johnson and Cook^[2] who conducted 4 types of laboratory material tests including smooth bar tension test, tubular torsion test and two notched tension tests. In the calculation, the value of $\gamma _{\rm{f}}^*$ is obtained directly from the tubular torsion test data in Fig.12 of Johnson and Cook^[2] and it is equal to 2.915 0 (i.e. $\gamma _{\rm{f}}^*=2.915\,0$ ); Similarly, the value of $e_{\rm{f}}^*$ is determined to be $e_{\rm{f}}^*=1.867\,0$ from the smooth bar tension test data in Fig.6 of Johnson and Cook^[2] and the value of ${\eta ^*}$ is estimated to be ${\eta ^*}=0.815\,0$ from Fig.11 of Johnson and Cook^[2]. It is easy to calculate the values of the constants ${C_1}$ and ${C_2}$ in Eq.(8) to be ${C_1}={3^{1.5{\eta ^*}}}e_{\rm{f}}^*=7.152\,0$ and ${C_2}=\gamma _{\rm{f}}^*=2.915\,0$ using the values of relevant parameters as obtained above. It is evident that the present model predictions are in good agreement with the experimental results for the Armco iron. Moreover, it is easy to show that ${e_{\rm{f}}}={\rm{1}}{\rm{.683\,0}}$ (Eq.(8) with $\eta=1/3, \xi=0$ ) and ${\bar e_{\rm{f}}}=4.129\,2$ (Eq.(8) with $\eta=1/3, \xi=1$ ). By definition the Armco iron as investigated in Johnson and Cook^[2] belongs, therefore, to Category I material since ${\bar e_{\rm{f}}}>{e_{\rm{f}}}$ as discussed in Section 1.1.

Figure  5.  Comparison of the newly developed failure criterion with the test data for Armco iron (Johnson and Cook^[2]) in the space of ε_f and η. Red line: Johnson-Cook failure criterion^[2]; black lines: the present model (Eq.(8))

DownLoad: Full-Size Img PowerPoint

Fig.6 shows comparison between Eq.(8) and the test results for the OFHC copper as reported by Johnson and Cook^[2] who performed 4 types of laboratory material tests including smooth bar tension test, tubular torsion test and two notched tension tests. In the calculation, the value of $\gamma _{\rm{f}}^*$ is taken to be 8.700 0 (i.e. $\gamma _{\rm{f}}^*=8.700\,0$ ) which was obtained from the torsion test on another shorter-gage specimen (Johnson and Cook^[2]). This is because none of the OFHC copper specimens fractured due to rotational limitations of the torsion tester as pointed out by Johnson and Cook. On the other hand, the value of $e_{\rm{f}}^*$ is determined to be $e_{\rm{f}}^*=1.448\,0$ from the smooth bar tension test data in Fig.6 of Johnson and Cook^[2] and the value of ${\eta ^*}$ is estimated to be ${\eta ^*}=0.730\,0$ from Fig.11 of Johnson and Cook^[2]. It is easy to calculate the values of the constants ${C_1}$ and ${C_2}$ in Eq.(8) to be ${C_1}={3^{1.5{\eta ^*}}}e_{\rm{f}}^*=4.821\,9$ and ${C_2}=\gamma _{\rm{f}}^*=8.700\,0$ using the values of relevant parameters as obtained above. It is shown that the present model predictions are in good agreement with the experimental results for the OFHC copper. Furthermore, it is easy to show that ${e_{\rm{f}}}=5.022\,9$ (Eq.(8) with $\eta=1/3,\;\xi=0$ ) and ${\bar e_{\rm{f}}}=2.783\,9$ (Eq. (8) with $\eta=1/3,\;\xi=1$ ). Hence, by definition the Armco iron as examined in Johnson and Cook^[2] belongs to Category II material since ${\bar e_{\rm{f}}} \leqslant {e_{\rm{f}}}$ as discussed in Section 1.1.

Figure  6.  Comparison of the newly developed failure criterion with the test data for OFHC copper (Johnson and Cook^[2]) in the space of ε_f and η. Red line: Johnson-Cook failure criterion^[2]; black lines: the present model (Eq.(8))

DownLoad: Full-Size Img PowerPoint

Fig.7 shows comparison of Eq.(8) with the test data for the 1045 steel as studied by Bai et al.^[15] who carried out 6 types of laboratory material tests including smooth bar tension test, tubular torsion test, notched bar tension test and 3 flat grooved plane strain tests. In the calculation, the value of $\gamma _{\rm{f}}^*$ is obtained from the tubular torsion test and directly taken from Eq.(24) of Bai et al.^[15] to be 0.480 0 (i.e. $\gamma _{\rm{f}}^*=0.480\,0$ ); the value of $e_{\rm{f}}^*$ is taken be $e_{\rm{f}}^*=0.426\,9$ directly from Table 2 of Bai et al.^[15] for the smooth bar tension test which was calculated using Eq.(13) of Bai et al.^[15] and the value of ${\eta ^*}$ is estimated to be ${\eta ^*}=0.599\,0$ from Fig.18 of Bai et al.^[15]. It is easy to calculate the values of the constants ${C_1}$ and ${C_2}$ in Eq.(8) to be ${C_1}=$ ${3^{1.5{\eta ^*}}}e_{\rm{f}}^*=1.145\,6$ and ${C_2}=\gamma _{\rm{f}}^*=0.480\,0$ using the values of relevant parameters as obtained above. It is demonstrated that the present model predictions are in good agreement with the experimental results for the 1045 steel. Furthermore, it is easy to show that ${e_{\rm{f}}}=0.277\,1$ (Eq.(8) with $\eta=1/3, \xi=0$ ) and ${\bar e_{\rm{f}}}=0.661\,4$ (Eq.(8) with $\eta=1/3, \xi=1$ ). Hence, by definition the 1045 steel as studied in Bai et al.^[15] belongs to Category I material since ${\bar e_{\rm{f}}}>{e_{\rm{f}}}$ as discussed in Section 1.1.

Figure  7.  Comparison of the newly developed failure criterion with test data for 1045 steel (Bai et al.^[15]) in the space of ε_f and η. Red line: Johnson-Cook failure criterion^[2]; green lines: model proposed by Xue and Wierzbicki^[6]; black lines: the present model (Eq.(8))

DownLoad: Full-Size Img PowerPoint

Fig.8 shows comparison of the newly developed failure criterion (Eq.(8)) with the test data for 2024-T351 aluminum alloys as reported by Wierzbicki et al.^[6] and Bao^[14] who conducted 15 types of laboratory material tests. In the calculation, the value of $\gamma _{\rm{f}}^*$ is obtained directly from the shear test data in Table 1 of Bai et al.^[9] and it is equal to 0.210 7 (i.e. $\gamma _{\rm{f}}^*=0.210\;7$ ); the value of $e_{\rm{f}}^*$ is determined to be $e_{\rm{f}}^*=0.437\,5$ from the smooth bar tension test data in Fig.8 of Wierzbicki et al.^[6] and the value of ${\eta ^*}$ is estimated to be ${\eta ^*}=0.505\,0$ from Fig.5.18 of Bao^[14]. Hence, the values of the constants ${C_1}$ and ${C_2}$ in Eq. (8) can be easily calculated to be ${C_1}={3^{1.5{\eta ^*}}}e_{\rm{f}}^*=1.005\,5$ and ${C_2}=\gamma _{\rm{f}}^*=0.210\,7$ using the values of relevant parameters as obtained above. It is evident from Fig.8 that the present model predictions are in good agreement with the experimental results for the 2024-T351 aluminum alloy. Moreover, it is easy to show that ${e_{\rm{f}}}={\rm{0}}{\rm{.121\,6}}$ (Eq. (8) with $\eta=1/3,\xi=0$ ) and ${\bar e_{\rm{f}}}=0.580\,5$ (Eq.(8) with $\eta=1/3,\xi=1$ ). By definition the 2024-T351 aluminum as investigated in Wierzbicki et al.^[6] and Bao^[14] belongs, therefore, to Category I material since ${\bar e_{\rm{f}}}>{e_{\rm{f}}}$ as discussed in Section 1.1.

Figure  8.  Comparison of the newly developed failure criterion with the test data for 2024-T351 aluminum alloy (Wierzbicki et al.^[6]; Bao^[14]) in the space of ε_f and η. Red line: Johnson-Cook failure criterion^[2]; green lines: the model proposed by Xue and Wierzbicki^[6]; black lines: the present model (Eq.(8))

DownLoad: Full-Size Img PowerPoint

Fig.9(a) shows comparison of the newly developed failure criterion (Eq.(8)) with the test data for Inconel 718 nickel-base superalloy as reported by Erice et al.^[16-17] who conducted 5 types of laboratory material tests including smooth bar tension test, shear test, plane strain test and two notched tension tests. In the calculation, the value of $\gamma _{\rm{f}}^*$ is obtained directly from the shear test data in Table 3 of Erice et al.^[16] and it is equal to 1.100 0 (i.e. $\gamma _{\rm{f}}^*=1.100\,0$ ); Similarly, the value of $e_{\rm{f}}^*$ is determined to be $e_{\rm{f}}^*=0.320\,0$ from the smooth bar tension test data in Table 3 of Erice et al.^[16] and the value of ${\eta ^*}$ is estimated to be ${\eta ^*}=0.489\,9$ from Fig.14(a) of Erice et al.^[16]. It is easy to calculate the values of the constants ${C_1}$ and ${C_2}$ in Eq. (8) to be ${C_1}={3^{1.5{\eta ^*}}}e_{\rm{f}}^*=0.717\,4$ and ${C_2}=\gamma _{\rm{f}}^*=1.100\,0$ using the values of relevant parameters as obtained above. It can be seen that the present model predictions (Eq.(8)) are in good agreement with the experimental results for the Inconel 718 nickel-base superalloy as examined by Erice et al.^[16]. Furthermore, it is easy to show that ${e_{\rm{f}}}={\rm{0}}{\rm{.635\,1}}$ (Eq.(8) with $\eta=1/3, \xi=0$ ) and ${\bar e_{\rm{f}}}=0.414\,2$ (Eq.(8) with $\eta=1/3, \xi=1$ ). Hence, by definition the Inconel 718 nickel-base superalloy as examined in Erice et al.^[16-17] belongs to Category II material since ${\bar e_{\rm{f}}} \leqslant {e_{\rm{f}}}$ as discussed in Section 1.1.

Comparison is made in Fig.9(b) between the Eq.(8) with the test data for Inconel 718 nickel-base superalloy as reported by Algarni et al.^[18] who performed 4 types of laboratory material tests including smooth bar tension test, plane strain flat bar test and two notched bar tension tests. As no pure shear or tubular torsion test was done and the value of the constant ${C_2}$ needs to be calibrated by other plane strain ( $\xi=0$ ) flat bar test data as reported in Algarni et al.^[18] in the calculation. From Table 8 of Algarni et al.^[18] and Fig.10 of Algarni et al.^[18], it is easy to get the value of the experimental true fracture strain (0.400 4) and the corresponding computed stress triaxiality (0.583 1). Hence, it is easy to show that ${C_2}=1.046\,5$ ; Similarly, from the smooth bar tension test data in Table 8 of Algarni et al.^[18], it is easy to get the value of the true fracture strain (0.680 4) and the corresponding stress triaxiality (0.573 8) from Fig.10 of Algarni et al.^[18]. Hence, It is easy to calculate the values of the constants ${C_1}$ in Eq.(8) to be ${C_1}=1.751\,6$ . It can be seen that the present model predictions are in good agreement with the experimental results for the Inconel 718 nickel-base superalloy as examined by Algarni et al.^[18]. Furthermore, it is easy to show that ${e_{\rm{f}}}={\rm{0}}{\rm{.604\,2}}$ (Eq.(8) with $\eta=1/3, \xi=0$ ) and ${\bar e_{\rm{f}}}=1.011\,3$ (Eq.(8) with $\eta=1/3, \xi=1$ ). Hence, by definition the Inconel 718 nickel-base superalloy as examined in Algarni et al.^[18] belongs to Category I material since ${\bar e_{\rm{f}}}>{e_{\rm{f}}}$ as discussed in Section 1.1.

It appears from Fig.9(a) and Fig.9(b) that the results obtained for the 718 nickel-base superalloy materials investigated by Erice et al.^[16] and Algarni et al.^[18] are contradictory to each other. It should be mentioned here that the Inconel 718 nickel-base superalloy employed by Erice et al.^[16-17] was much more ductile than that used by Algarni et al.^[18]. Closer examination reveals that the discrepancy between these two Inconel 718 nickel-base superalloy materials may be due to different heat treatment (for example, the superalloy used by Erice et al.^[16] was precipitation hardened) and different chemical composition (for instance, the superalloy employed by Algarni et al.^[18] had Fe).

Also shown in Fig.4–Fig.9 are the fracture loci for the plane stress state (as indicated by the black solid line) which are predicted theoretically by the present model with $\xi$ being replaced by the following expression

As can be seen, some of the shapes of the fracture loci for the plane stress state as shown in Fig.4, Fig.5, Fig.7, Fig.8 and Fig.9(b) open upward whilst the others as shown in Fig.6 and Fig.9(a) open downward. This is no surprising since the plane stress state is only a special case of the present model which divides materials into two categories, i.e. Category I material with ${\bar e_{\rm{f}}}>{e_{\rm{f}}}$ and Category II material with ${\bar e_{\rm{f}}} \leqslant {e_{\rm{f}}}$ .

The predictions from the Johnson-Cook failure criterion^[2] and the model suggested by Xue and Wierzbicki^[6] are also shown in Fig.7, Fig.8, Fig.9(a) whilst in Fig.4, Fig.5, Fig.6 and Fig.9(b) only the predictions from the Johnson-Cook failure criterion are given and those from the model proposed by Xue and Wierzbicki are not available due to lack of test data. The values of the empirical constants D₁–D₃ in the Johnson-Cook failure criterion^[2] and c₁–c₄ in the model proposed by Xue and Wierzbicki^[6] are evaluated and listed in Table 1. It is clear from these figures that the Johnson-Cook failure criterion has failed to give consistent results and this is not surprising since it does not taken into account the effect of Lode angle. It is also clear from these figures that the present model (Eq.(8)) and the model proposed by Xue and Wierzbicki^[6] produce more or less similar results for stress triaxiality greater than zero approximately whilst for stress triaxiality less than zero the difference between these two model predictions increases with decreasing stress triaxiality. It should be mentioned here that only two standard laboratory material tests (i.e. smooth bar tensile test and pure shear (tubular torsion) test) are needed to calibrate the two empirical constants in the present model whereas at least two axisymmetric tests (i.e., two notched tensile tests) and two plane strain tests (namely, pure shear test and flat dog-bone tensile test) are required to estimate the four empirical constants in the model suggested by Xue and Wierzbicki^[6].

Table  1.  Values of constants in the Johnson-Cook failure criterion^[2] and the model proposed by Xue and Wierzbicki^[6]

Materials	Johnson-Cook failure criterion				Model proposed by Xue and Wierzbicki
	D1	D2	D3		c1	c2	c3	c4
4340 steel	0.05[2]	3.44[2]	–2.12[2]
Armco iron	–2.20[2]	5.43[2]	–0.47[2]
OFHC copper	0.54[2]	4.89[2]	–3.03[2]
1045 steel	0.06	1.72	–2.58		1.511 0	2.110 2	0.480 0	1.738 8
2024-T351	0.13[6]	0.13[6]	–1.50[6]		1.466 0	2.394 0	0.210 0	0.005 0
Inconel 718[16]	0.04[16]	0.75[16]	–1.45[16]		0.525 1	1.010 9	1.100 0	1.188 7
Inconel 718[18]	0.04	1.54	–1.60

 | Show Table

DownLoad: CSV

Figure  4.  Comparison of the newly developed failure criterion with the test data for 4340 steel (Johnson and Cook^[2]) in the space of ε_f and η. Red line: Johnson-Cook failure criterion^[2]; black lines: the present model (Eq.(8))

DownLoad: Full-Size Img PowerPoint

Figure  9.  Comparison of the newly developed failure criterion with the test data for Inconel 718 nickel-base superalloy in the space of ε_f and η. Red line: Johnson-Cook failure criterion^[2]; green lines: the model proposed by Xue and Wierzbicki^[6]; black lines: the present model (Eq.(8))

DownLoad: Full-Size Img PowerPoint

3. Conclusions

A new approach has been presented in this paper to predict the failure of metallic materials with the main conclusions achieved.

(1) On the basis of the test data metals can be divided into two categories, namely metals with ē_f>e_f and metals with ē_f≤e_f, here ē_f and e_f are the two parameters defined as the true strains at stress triaxiality of η=1/3 for Lode parameters of ξ=1 (axisymmetric stress state) and ξ=0 (plane strain state), respectively.

(2) A new failure criterion, which takes into account the effects of stress triaxiality and Lode parameter, has been obtained.

(3) This newly developed failure criterion has been shown to accord well with the test data for various metals under different loading conditions.

(4) Only two laboratory tests such as the smooth bar tension test and the pure shear (tubular torsion) test are needed to calibrate the failure criterion.