A Study of the Dynamic Mechanical Properties of Q460D Steel

Abstract

The dynamic mechanical properties of Q460D steel were studied to facilitate an assessment of the impact resistance of building structures. In the present work, material performance tests of Q460D steel at different temperatures, strain rates, and stress states were conducted. Using a hybrid experimental–numerical approach, a modified Johnson–Cook (JC) constitutive relation, a modified Johnson–Cook (JC) fracture criterion, and a lode-dependent fracture criterion were calibrated. To validate the calibration, Taylor impact tests of Q460D steel rods onto rigid target plates were carried out in a one-stage light-gas gun system. Mushrooming, tensile splitting, and petalling failure modes were obtained as the impact velocity was increased from 191.6 to 422.1 m/s. A three-dimensional finite element model was built for the Taylor impact tests, and FE simulations were run using the material models calibrated. It was found that the FE simulations using the lode-dependent fracture criterion were reasonable in terms of the failure modes of the Taylor rods. In contrast, the fracture behavior of the Taylor rods was significantly underestimated using the lode-independent JC fracture criterion. Finally, the effect of anisotropy, strain rate sensitivity and yield plateau on the Taylor impact FE predictions were explored and discussed.

1. Introduction

In recent years, the impact resistance of building structures has become an active research field due to an increasing number of attacks. Q460 is a commonly used steel type in building structures (such as the Bird’s Nest Olympic Stadium), and there is an urgent need for research on its dynamic material properties. Finite element (FE) simulations and supplementary tailored tests of particular exposed structural elements are usually employed for this purpose. The accuracy of FE simulations of impact events is evidently influenced by the material models utilized [1,2,3,4,5,6,7,8,9,10,11]. Hence, deep understanding and thorough characterization of materials’ mechanical properties through elaborate material models play pivotal roles in research into local dynamic events in structures.

Material models may be distinguished into coupled and uncoupled models. Within coupled material models, it is assumed that the damage [12,13] accumulation process [14] affects the elastic and plastic properties of a material, while in uncoupled models, material yielding and fracturing are treated separately from each other [15]. The latter allows for an independent and therefore simpler calibration of the plasticity and fracture parts of a material model. Depending on a particular problem being solved and on the availability of relevant experimental data, the effects of a material stress state, strain rate, temperature, and other factors may be implemented into the plasticity and/or fracture parts of a material model, no matter whether it is coupled or uncoupled. As a result, despite their less sound physical basis, uncoupled models are much more popular in research and engineering practice [16,17,18,19,20,21,22,23].

Concerning plasticity models, to this date there exist over 200 various criteria for isotropic materials [24,25]. Among them there are elaborate versatile multiparameter three-invariant criteria with affine [26,27,28] and non-affine [29] octahedral cross-sections of a yield surface. Even the very first yield criteria, the so-called strength theories, were mainly two- or three-invariant:(1) maximum principal strain theory by Mariotte [30], Poncelet and de Saint-Venant [31]; (2) maximum principal stress theory by Galilei [32] and Rankine [33]; (3) criterion by Coulomb [34] generalized by Mohr [35]; (4) maximum shear stress theory by Tresca [36,37]; (5) specific elastic strain energy criterion by Beltrami [38]; (6) specific elastic distortion strain energy criterion by Maxwell [39], Huber [40], von Mises [41] and Hencky [42], which is commonly called simply the von Mises criterion, the designation that we will use within the present manuscript for brevity. Using linear transformations of the stress tensor, the technique applied for the first time (according to the authors’ knowledge) by Sobotka [43], any isotropic criterion can be extended to an anisotropic case (see, for instance [44,45,46]). Furthermore, some accepted anisotropic yield criterions contain Hill48 [47], Yld2000-2d [48], BBC2000 [49] and Yld2000-18p [50] are proposed gradually. Applying a concept of the incubation time (see, for instance [51,52]) to the above mentioned yield criteria and, thus, turning from stress invariants in a criterion equation to stress impulse-like measures one can successfully describe plastic flow at very high strain rates in wide temperature ranges. Yet, a quick look through available literature over the past years related to FE simulations of impact interactions [53,54,55,56,57,58,59,60,61,62,63] reveals that, despite existence of all the above-mentioned elaborate criteria and techniques, the most widely used yield criterion has been and remains the von Mises one, in particular in the form proposed by Johnson and Cook (JC) [64]. The reason is not its limitless capabilities, but rather its ease of implementation and calibration.

Considering fracture criteria, one may emphasize two breakthroughs which determined directions of further development of the fracture mechanics in regard to numerical simulations of material fracture and a present state of the art. The first one is theoretical works by McClintock [65] and Rice and Tracey [66] and further experimental verification [67] of their results, which made it evident and universally recognized that an equivalent plastic strain at fracture of ductile materials exponentially decrease with the stress triaxiality increase. The second breakthrough is fracture experiments conducted by Bao [68] and further interpretation [69] of their results, which emphasized that fracture properties of ductile materials are in general dependent onto both the stress triaxiality and the Lode angle. Since that time several elaborate fracture criteria have been developed [70,71,72,73], but as in the case of the aforementioned elaborate yield criteria these and other advanced fracture criteria are not widely employed [61,74,75,76,77,78] even within uncoupled material models. The reason seems to be the same.

In light of the foregoing, great efforts are made to obtain relevant experimental data at different stress states, strain rates, and temperatures and calibrate at least uncoupled material models of various degree of complexity which are widely used in engineering practice. Very large amount of experimental data for 2024-T3(51) aluminum alloy is available in literature [60,64,68,79,80,81,82,83,84,85], In these works samples of various geometry (smooth and notched circular cylinders, thin flat “dog-bone” specimens, thin flat notched specimens, wide smooth and grooved plates, thin-walled tubes, thick-walled tubes, bars with square cross-sections, circular plates, plates with a circular hole, parallelepipeds, specimens with a butterfly-like gauge section, etc.) machined from different blanks (rolled sheets and plates of various thickness, large rolled blocks, extruded bars with circular or square cross-section, etc.) in different directions (through thickness of a plate, in a plane of a plate at 0°, 15°, 30°, 45°, 60°, 75°, 90° with respect to a rolling direction, etc.) are used in quasi-static (strain rates from 10⁻⁴ s⁻¹ to 1 s⁻¹) and dynamic (on Split Hopkinson Bar-type test facilities with strain rates up to 10⁴ s⁻¹) tests of various types (tension, compression, torsion/shear, combined torsion-compression and torsion-tension/shear-tension, bulge test, biaxial compression, etc.) conducted at different temperatures (from −50 °C to 450 °C) and under proportional or non-proportional (pre-tension, pre-compression, pre-torsion, etc.) loading to identify anisotropy of a material yield stress, strain hardening law, strain rate and temperature effects onto material yielding, stress state dependence of the material yielding and construct a material fracture locus in a mixed stress-strain space at different strain rates, temperatures and loading conditions. In most of the mentioned works obtained experimental data has been used to calibrate a particular material model. And, of course, one may simultaneously use several different sources for calibration, like it was done in [86] to calibrate a proposed three-invariant yield criterion, in [87] to calibrate the three-invariant fracture criterion by [71], and in [88] to calibrate a proposed three-invariant fracture criteria. However, most of structural metals and alloys, in particular steels commonly used in civil engineering, are not so “popular” within the research community and not very much experimental data is available for these materials. Yao et al. [89] analyzed the cyclic elastoplastic response of Q235B steel, the effect of Lode dependence of the plasticity response of structural steel Q235 is investigated. Yang et al. [90] investigated a strain rate sensitivity of S690 structural steel, determined parameters of the strain rate hardening laws introduced, respectively, by Cowper and Symonds [91] and Johnson and Cook [92]. Jiang et al. [93] proposed a prediction formula for ultimate bearing capacity of Q690 built-up K joints. Tian et al. [94] provided a novel steel frame joints with corrugated plates to improve the collapse resistance of Q235B steel frame structures. Based on Q345B steel, the “false steel-concrete composite structures” was proposed and the seismic performance of it was analyzed by Huang et al. [95]. Xiao and his coworkers simulated the failure process of 316L austenitic stainless steel [61], Ti-6Al-4V Alloy [96] and 6061-T651 Al [97] target plates that were impacted by blunt rigid projectiles using the Lode-dependent MMC, Lou fracture criterion, and Lode-independent JC fracture criterion and analyzed the effect of the Lode parameter on the prediction of the ballistic limit velocity in FE simulations. Hence, compared to the Lode-independent fracture criteria, the prediction accuracy of FE simulations for Taylor impact or ballistic tests was effectively improved by incorporating the Lode parameter.

At present, research on the dynamic material properties of typical structural steels incorporating the lode parameter is relatively limited. In this study, a modified Johnson–Cook (JC) constitutive relation, a modified Johnson–Cook (JC) fracture criterion, and a lode-dependent fracture criterion were applied to describe, respectively, the flow stress and fracture behavior of Q460D steel. The model parameters were calibrated by conducting material performance tests of Q460D steel at different temperatures, strain rates, and stress states, and using a hybrid experimental-numerical method. To validate the calibration, a Taylor impact test was performed, and the mushrooming, tensile splitting, and petalling failure modes of the projectile were obtained. A three-dimensional finite element model was built for the Taylor impact tests, and FE simulations were run using the material models calibrated. The simulation results of the two fracture criteria were compared with the results observed in the experiment, and the prediction capabilities of the two fracture criteria were compared. It was found that the fracture behavior of the Q460D steel has a significant lode correlation. Then, the effect of the material property representation on the prediction accuracy in FE simulations was examined and discussed.

2. Material Model and Calibration

2.1. Material Model

The JC constitutive relation [64], which is commonly used in engineering practice, was applied to characterize the equivalent plastic strain in the material model. However, in the study of the dynamic material properties of typical structural steels, Lin et al. [98] found that the thermal softening behavior cannot be well captured by the JC constitutive relation. To improve the prediction accuracy, the original temperature item was modified by adding a thermal softening coefficient F. Meanwhile, based on the method inspired by Sung et al. [99] , Xiao et al. [100] modified the strain hardening term of the JC constitutive relation, by substituting the Ludwik law [101] with a linear combination of the Ludwik and Voce laws [102] To include the effect of the thermal softening and strain hardening, a modified Johnson–Cook (MJC) constitutive relation that combined the two types of modifications was adopted, expressed as follows:

$${\sigma }_{eq}=\{\alpha (A+B{\epsilon }_{eq}^n)+(1-\alpha )[A+Q(1-e^{-\beta {\epsilon }_{eq}})]\}(1+C\ln {\dot{\epsilon }}^{*})(1-F{T}^{\ast m}),$$

(1)

where A, B, n, α, Q, and β are the strain hardening coefficients; C is the strain rate sensitivity coefficient; F and m are the thermal softening coefficients; ${\dot{\epsilon }}^{*}=\dot{\epsilon }/{\dot{\epsilon }}_0$, ${\dot{\epsilon }}_0$ and $\dot{\epsilon }$ are the equivalent plastic strain, the dimensionless plastic strain, the reference strain rate, and the strain rate, respectively; and T^* = (T − T_r)/(T_m − T_r), T, T_r, and T_m are the homologous temperature, the material temperature, the reference temperature, and the melting point, respectively. Furthermore, the adiabatic assumption was adopted at a high strain rate, and the temperature rise ΔT can be defined as follows:

$$\Delta T=\frac{\chi }{\rho C_{\mathrm{p}}}{\displaystyle \int {\sigma }_{eq}}\mathrm{d}{\epsilon }_{eq},$$

(2)

where χ is the Taylor–Quinney coefficient, which is usually assumed to be 0.9, and ρ and C_p are the material density and the specific heat, respectively.

The fracture strain was expressed by the JC and the Lode-dependent EJMA fracture criteria. The JC fracture criterion is widely applied in engineering practice due to its comprehensive consideration of the stress triaxiality, strain rate, and temperature. To well characterize the thermal softening effect, Xiao [103] added a thermal softening coefficient D₆ in the original temperature item, and it can be expressed as follows:

$${\epsilon }_f=\left[D_1+D_2\exp (D_3{\sigma }^{*})\right](1+D_4\ln {\dot{\epsilon }}^{*})(1+D_5{T}^{*D_6}),$$

(3)

where D₁, D₂, D₃, D₄, D₅, and D₆ are all material constants, and σ^* is the stress triaxiality defined by σ^* = σ_m/σ_eq, where σ_m is the hydrostatic pressure given by σ_m = (σ₁₁ + σ₂₂ + σ₃₃)/3. The damage indicator can be defined as follows:

$$D={\displaystyle \sum \frac{\Delta {\epsilon }_{eq}}{{\epsilon }_f}},$$

(4)

where Δε_eq is the equivalent plastic strain increment.

In studies of the fracture behavior of metals in different stress states, the effects of the stress triaxiality and the lode parameter, which is determined by the third invariant of the deviatoric stress tensor, should be considered, especially at low stress triaxialities. The lode angle θ is defined as:

$$\xi =\cos (3\theta )=\frac{3\sqrt{3}}{2}\frac{J_3}{J_2^{2/3}},$$

(5)

where J₂ and J₃ are the second and third invariants of the deviatoric stress tensors, 0 < θ < π/3, and −1 < ξ< 1. The relationship between the lode parameter L and lode angle θ can be expressed as follows:

$$\overline{\theta }=1-\frac{6\theta }{\pi }=1-\frac{2}{\pi }\arccos (\xi )=1-\frac{2}{\pi }\arccos {(\frac{r}{q})}^3,$$

(6)

$$r=[\frac{27}{2}({\sigma }_1-{\sigma }_{\mathrm{m}})({\sigma }_2-{\sigma }_{\mathrm{m}})({\sigma }_3-{\sigma }_{\mathrm{m}}){]}^{1/3},$$

(7)

$$q=\sqrt{\frac{1}{2}{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2},$$

(8)

$$L=\frac{3\tan \theta -\sqrt{3}}{\tan \theta +\sqrt{3}},$$

(9)

where σ_i represents the first, second, and third main stresses when i = 1, 2, and 3, respectively.

Thus, the lode parameters θ, $\overline{\theta }$ and L can be inferred when one is known. Due to its more concise expression for L, the EJMA fracture criterion, which is characterized by lode parameter L, was adopted. Combined with the expression of the strain rate and temperature in the MJC fracture criterion, the equivalent plastic strain can finally be defined as follows:

$${\epsilon }_{eq}=\frac{C_3^{{}^{}}}{{[{(\frac{2}{\sqrt{L^2+3}})}^{C_1}+(\eta -\frac{1}{3})]}^{C_2}}(1+D_4\ln {\dot{\epsilon }}^{*})(1+D_5{T}^{*D_6}),$$

(10)

where C_i (i = 1, 2, or 3) and η are the material constants and stress triaxiality, respectively.

2.2. Calibration of the MJC Constitutive Relation

To obtain the strain term parameters of the MJC constitutive relation, quasi-static tensile tests were conducted using a SHIMADZU AG-X Plus electronic universal testing machine at the Nanyang Institute of Technology, and an SHIMADZU SIE-560SA Automatic extensometer was used to record the test data. The tensile speed was 2 mm/min, and the gauge length was 25 mm. The specimen dimensions are shown in Figure 1a. Additionally, all of the specimens in this paper were processed using a Q460D steel plate (length × width × thickness = 500 mm × 500 mm × 30 mm), the chemical components of which are listed in

Table 1. Chemical composition of the Q460D steels (mass fraction, %).

C	Si	Mn	P	S	Al	Nb	V	Ti	Cr	Ni	Cu	Mo	B	N
0.13	0.3	1.4	0.014	0.002	0.03	0.031	0.043	0.014	0.03	0.01	0.02	0.01	0.0002	0.0032

.

The engineering stress–strain curves of each smooth test (ST) are shown in Figure 2. The yield strength was derived from the lower yield point in the yield plateau. The average of each test was A = 434.63 MPa, and the elastic modulus was 210.351 GPa. The curve from the yield plateau to the necking point in the true stress–strain curve was intercepted and fitted using Origin software, based on a linear combination of the Ludwik and Voce laws. The fit parameters were as follows: B = 666.54 MPa, n = 0.57726, Q = 217.61 MPa, and β = 16.819.

To confirm the value of α in Equation (1) and verify the calibration, a 2D axisymmetric model was established using ABAQUS finite element software. The element size in the gauge section was 0.1 mm × 0.1 mm, and that outside the gauge section was 0.1 mm × 0.3 mm. Each strain parameter obtained above was input into the FE simulation using a tabular method [103], and the value of α was varied until the load–displacement curve of the numerically simulated result was similar to the test result, as shown in Figure 2b, and the value of α was 0.85.

The strain rate parameters were calibrated via dynamic tensile tests and split-Hopkinson pressure bar (SHPB) tests. Dynamic tensile tests were conducted using a universal test machine with an automatic extensometer, and the specimen size is shown in Figure 1b. The gauge length was 25 mm. Four tensile speeds from 200 to 800 mm/min were tested, and the engineering stress–strain curves are shown in Figure 3.

The SHPB test was conducted using the SHPB apparatus shown in Figure 4. The apparatus mainly included a striker bar, incident bar, transmitter bar, and absorber bar, all of which were made of 18Ni (350) high-strength steel and had 12 mm diameters. The length of the striker bar was 300 mm, and lengths of the other bars were all 1000 mm. The specimens had an 8 mm diameter and a 6 mm height, as shown in Figure 1c. One or two tests were conducted for each strain rate. The signals from the incident bar and transmitter bar were processed to obtain the engineering stress–strain curves at different pressures, as shown in Figure 3b. Oscillations are evident in stress–strain data at the highest strain rate, which is a very common phenomenon caused by the initial data being obtained while the system is not at equilibrium.

The plastic and elastic sections in the engineering stress–strain curves of SHPB test were fitted, and the value of the intersection point was taken as the yield strength. Meanwhile, this was combined with the yield strength, which was taken from the lower yield point in the yield plateau of the dynamic tensile test, with the reference strain rate as 8.333 × 10⁻⁴ s⁻¹, which derived from the smooth tensile test. The strain rate coefficient was calibrated using the least squares method, as shown in Figure 5, and it was determined that C = 0.0404.

Due to the relatively high melting point of Q460D steel, four temperature gradients from 300 to 900 °C were established to study the material’s properties at different temperatures. The test was conducted using a universal testing machine with a high-temperature furnace and a temperature control system at the Harbin Institute of Technology. The specimen dimensions are shown in Figure 1b. However, at a temperature of 900 °C, the specimen was softened more significantly and became embedded in the fixture. Consequently, the specimen was hard to remove, and damaged the fixture. Hence, only one test was conducted at this temperature. Finally, the engineering stress–strain curves at different temperatures were obtained, as shown in Figure 6.

Figure 6 shows that the yield and tensile strengths decreased gradually as the temperature increased, and the material exhibits a significant thermal softening effect [104]. At temperatures of 20 and 300 °C, the yield strength was derived from the lower yield point of the yield plateau. At temperatures from 500 to 900 °C, the yield strength was determined from the stress, which corresponded to a 0.2% strain. A dimensionless temperature parameter was defined as T* = (T − T_r)/(T_m − T_r), where the melting point of Q460D steel is T_m = 1800 K and the reference temperature was T_r = 293 K. Based on the relationship between the yield strength and T* under different temperature conditions, the parameters of the temperature term of the MJC in Equation (1) were fitted using Origin software, as shown in Figure 7. The values were F = 1.777 and m = 1.228.

2.3. Calibration of the Fracture Criterion

The stress state was studied from various perspectives, including high-stress triaxiality, low-stress triaxiality, and plane strain. Notched tensile tests were performed using radii of 2, 3, and 9 mm. In-plane shear and plane strain tests were conducted on the universal test machine at the tensile speed of 0.5 mm/min. The specimen dimensions are shown in Figure 1d. Matte white undercoats and matte black speckles were printed on the specimen surfaces.

Full-surface deformation was recorded using an optical measurement system provided by Correlated Solution Inc. (Vic2D 2009, Columbia, SC, USA), and the relative elongation between the two gauge points was recorded using the virtual extensometer option in the DIC (digital image correlation) system. A centered single CCD (charged couple device) camera was used to collect test photos in the notched tensile tests, and the gauge length was 25 mm. Front- and rear-facing double CCD cameras were used for the in-plane shear tests to capture the complex deformation behavior, and the gauge length was 30 mm. A three-dimensional (3D) double CCD camera method was used for the in-plane stress tests, as shown in Figure 8. After the tests, MatchID-2D and MatchID-3D software were used to extract the deformation and strain fields from the recorded pictures; the test results are shown in Figure 9 and Figure 10.

Due to the irregularity of the fracture surfaces of the recovered fractured specimens, measurement errors of the fracture area were obtained. The stress triaxiality value was not invariant during the deformation and fracture process. Hence, the fracture properties of each test should be examined through FE simulations. Numerical models were established in ABAQUS. A two-dimensional (2D) axisymmetric model was established to simulate the notched tensile tests. The element size in the gauge section was 0.1 mm × 0.1 mm, and that outside the gauge section was 0.1 mm × 0.3 mm. A 3D model was established for the in-plane shear and plane stress tests. The element size in the gauge section was 0.1 mm × 0.1 mm × 0.1 mm, and increased gradually closer to the edge, as shown in Figure 11.

There was slight error between the experimental and simulated force–displacement curves for the in-plane shear test, which illustrated the effect of the lode angle on the plastic flow behavior of Q460D steel. The correction coefficient was adjusted until the simulated curve overlapped with the experimental curve, and the current α value in Equation (1) was 0.35.

Because the numerical results were similar to the test results, the fracture strain from the numerical simulation, i.e., the equivalent plastic strain (PEEQ) at the moment of fracture, can be used directly, as shown in Figure 12. The equivalent plastic strain at the edge of the gauge section was slightly higher than that at the center. The stress state at the edge section was a pressure condition, and that in the center was a shear condition. Thus, the fracture occurred more easily in the center section, and the edge section was not considered in the analysis.

Meanwhile, it is well known that the stresses and the two stress state parameters, i.e., the stress triaxiality η and the Lode parameter L, in the specimens vary as plastic deformation occurs. Following the typical method in the literature, we adopted averaged stress state parameters, defined as follows:

$${\overline{\eta }}_{\mathrm{av}}=\frac{1}{{\epsilon }_{\mathrm{f}}}{\displaystyle {\int }_0^{{\epsilon }_{\mathrm{f}}}\eta ({\epsilon }_{\mathrm{eq}})\mathrm{d}{\epsilon }_{\mathrm{eq}}},{\overline{\theta }}_{\mathrm{av}}=\frac{1}{{\epsilon }_{\mathrm{f}}}{\displaystyle {\int }_0^{{\epsilon }_{\mathrm{f}}}\overline{\theta }({\epsilon }_{\mathrm{eq}})\mathrm{d}{\epsilon }_{\mathrm{eq}}},$$

(11)

The stress state parameters and fracture strain values for the SRB (smooth round bar), NRB (notched round bar), in-plane shear, and plane stress test are listed in

Table 2. Fracture strain of Q460D steel under different stress states.

Test	Fracture Location	η	L	εf
SRB	Centre	0.575	−1	1.523
NRB, R = 2	Centre	1.091	−1	0.742
NRB, R = 3	Centre	0.999	−1	0.861
NRB, R = 9	Centre	0.755	−1	1.057
In-plane shear	Surface	0.0049	−0.0761	1.480
	Mid-surface	0.6147	−0.2099	0.880

.

Based on the relationship between the fracture strain and the stress triaxiality obtained from the numerical simulations of high-stress triaxiality tests, the stress state term of the MJC fracture criterion was calibrated using Equation (3), and the three parameter values were D₁ = 0.672, D₂ = 9.591, and D₃ = −4.233, as shown in Figure 13.

The fitted curve using the MJC fracture criterion was suitable for the high-stress triaxiality state but significantly deviated for the plane strain state, especially for the in-plane shear state. For Q460D steel, it was not sufficient to only consider the effect of the stress triaxiality on the fracture strain, so the lode-dependent EJMA fracture criterion was fitted to the data. The fitted parameter values in Equation (10) were C₁ = 2.956, C₂ = 1.693, and C₃ = 2.06, as shown in Figure 14. Based on the comparison of the fits, the relationship between the fracture strain, lode parameter L, and stress triaxiality η of Q460D steel in different stress states can be better characterized using the EJMA fracture criterion.

In the calibration of the strain rate parameters, no cracks appeared on the SHPB specimens at different gas pressures, so only the dynamic tensile test results were used. The fractures were clear and easily measured in the recovered specimens in Figure 15, and the fracture strains at different strain rates were calculated as follows:

$${\epsilon }_f=\ln (\frac{A_0}{A_f}),$$

(12)

The reference strain rate was 8.333 × 10⁻⁴ s⁻¹, which was derived from the smooth tensile tests. The relationship between the fracture strain and strain rate was fitted using Origin software, and the strain rate coefficient D₄ = −0.024 was obtained, as shown in Figure 16.

The recovered specimens heated at 300 to 700 °C are shown in Figure 15b. A softening effect occurred at 900 °C. The specimen became embedded in the fixture, and secondary damage occurred when the sample was removed, so it was difficult to measure the fracture area accurately. The fracture diameter of each specimen was measured, the temperature parameters were calibrated based on the relationship between the fracture strain and T*, i.e., the thermal softening coefficients D₅ = 20.3 and D₆ = 3.87 in Equations (3) and (10), as shown in Figure 16b.

Though the nonlinear variation of the Q460D steel with increasing temperature could be well characterized using MJC fracture criterion, the fitting fracture strain at 300 °C was slightly higher than the test value. A blue oxide on the surface of the failure specimen at 300 °C was evident, which indicated the a brittle region formed at this temperature. Consequently, the tensile strength increased, the fracture strain decreased, and the strain became highly time-dependent. Because the strain rate and temperature terms of the MJC fracture criterion are also used in the EJMA fracture criterion, the same coefficients were used. All the material constants are listed in

Table 3. Material parameters of Q460D steel.

Description	Notations	Value
Strain constant of MJC constitutive relation	A/MPa	434.63
	B/MPa	666.54
	n	0.577
	α	0.85
	Q/MPa	217.61
	β	16.819
Strain rate constants	C	0.0404
Temperature constants	F	1.777
	m	1.228
EJMA fracture criterion constants	C1	2.956
	C2	1.693
	C3	2.06
MJC fracture criterion constants	D1	0.672
	D2	9.591
	D3	−4.233
	D4	−0.023
	D5	20.3
	D6	3.87
Others	E/MPa	210
	ν	0.3
	ρ/(kg/m3)	7850
	χ	0.9
	Cp/(J/kg K)	469
	$\dot{\epsilon }$/s−1	8.33 × 10−4
	Tr/K	1800
	Tm/K	293

.

3. Taylor Impact Tests of Q460D Steel

3.1. Test Set-Up

The one-stage light gas gun system of the Nanyang Institute of Technology was used in the tests, as shown in Figure 17. The whole system mainly consisted of a pressurized chamber with a pneumatic device to control the striking velocity, a 1.2 m long and 6 mm caliber diameter launch tube, and a FASTCAMSA-Z high-speed camera that monitored the impact process at 60,000 fps. In addition, it also included flash equipment for the camera, which is not shown in Figure 17. The collection principle is to use the projectile body to pass two laser beams with a known distance to generate an excitation signal to the test device, and to calculate the ejection speed of the projectile through the time difference between the signal generation.

In the Taylor impact tests, the diameter and length of the Q460D Taylor rod were 5.95 and 29.75 mm, respectively. The target plate was Cr12MnV steel with a hardness of 52 HRC, and it had a diameter and thickness of 55 and 20 mm, respectively, as shown in Figure 18.

The test process and results were mainly analyzed using the PLV software, which was connected to the high-speed camera. The pixel size was calibrated using a Vernier caliper. The projectile was filled into the launch tube and the loading pressure was adjusted based on the desired velocity. After the test, no obvious impact-induced indent on the target plate was evident, and the failure mode of the collected projectile was analyzed. The target was replaced with a new plate for the next test to reduce the error.

3.2. Failure Mode Analysis of Taylor Rods

Through the observation and analysis of all the results, mushrooming, tensile splitting, and petalling failure modes were observed in the nine groups, where the projectile velocity ranged from 191.6 to 422.1 m/s. The measured parameters for each projectile before and after the test are summarized in

Table 4. Characteristic size measurements of Q460D steel rod.

Num.	V0/(m/s)	D0/mm	L0/mm	DLf/mm	DSf/mm	Df/mm	Lf/mm	Failure Mode
T1	191.6	5.95	29.76	8.54	8.12	8.33	25.89	Mushrooming
T2	200.0	5.95	29.76	8.75	8.36	8.55	25.59	Mushrooming
T3	238.8	5.95	29.78	9.98	9.38	9.67	24.22	Mushrooming
T4	259.1	5.96	29.75	10.52	9.60	10.09	23.4	Mushrooming
T5	286.1	5.95	29.75	11.28	10.33	10.80	22.68	Mushrooming
T6	326.4	5.94	29.77	/	/	/	21.02	Tensile splitting
T7	372.8	5.95	29.76	/	/	/	19.16	Petalling
T8	378.7	5.95	29.75	/	/	/	18.82	Petalling
T9	422.1	5.95	29.77	/	/	/	16.81	Petalling

. The major and minor axes of the oval-shaped mushrooming failure surface were defined as D_Lf and D_Sf, respectively, and the equivalent diameter of a circle with the same area as the oval was defined as D_f, respectively.

Figure 19 shows the mushrooming failure mode of projectiles with impact velocities from 191.6 to 286.1 m/s. Immediately upon impact, the elastic waves and plastic waves are generated at the impact interface. They reflect and propagate in the projectile from the tip to the back, which leads to deceleration and deformation. Due to the target blocking in the axial direction and the deformation developing in the transverse direction, the front part of the projectile bulges out, and mushrooming failure occurs. As the velocity increased, the deformation of the projectile was more significant, which mainly manifested as an increase in the mushrooming surface area and a decrease in the length after the projectile failure. Because the amount of deformation of the projectile at the impact velocity of 286.1 m/s was greater, images of the whole failure process captured by the high-speed camera at this velocity are shown in Figure 20.

With the increase in striking velocity, the mushroomed projectile head localizes the deformation at the tip of the projectile, which causes high hoop strain, which gradually becomes greater than the fracture strain [105]. As shown in Figure 21, a “stagnation region” in the central area of the projectile and few small axial cracks on the edge of the mushrooming area appeared when the impact velocity was 326.4 m/s, and critical tensile splitting nearly occurred. Petalling was observed when the impact velocity was in the range of 372.8–422.1 m/s, and the cracking degree increased as the impact velocity in-creased. When V = 422.1 m/s, the central area of the mushrooming surface turned blue due to oxidation caused by local adiabatic heating. The whole impact process is shown in Figure 22.

4. Numerical Analysis of Taylor Impact Tests

4.1. FE Model Setup

A 3D FE model of the projectile and target plate was built using ABAQUS/Explicit. The diameter and length of projectile were 5.95 and 29.76 mm, respectively, while the diameter and thickness of the target were 55 and 20 mm, respectively. Both were defined as deformable bodies, and the projectile was positioned 0.2 mm above the target plate center. When setting the relative restraint of contacts, the occurrence of fracture and the internal contact of the elastic material after the fracture are considered. A contact pair was set between the outer surface of the projectile and the top surface of the target plate. The contact between the surface of the projectile element was defined to characterize the mechanical behavior of the material during the impact’s process. Rigid contact that did not account for the friction during the impact was adopted so that the projectile could be separated from the target after impact, which was similar to the rebound phenomenon of the projectile in the experiments.

C3D8R solid elements were applied for both the projectile and target. The bottom 10 mm of the projectile, which had a complex deformation, was observed. The average element size of this region was 0.2 mm × 0.2 mm × 0.25 mm, and the element size gradually increased from 0.25 to 0.6 mm in the length direction to decrease the number of computations for the top area, which only slightly deformed. The target had a higher hardness and deformed imperceptibly during the impact, so the element size in the center was 0.5 mm × 0.5 mm × 0.5 mm, and gradually increased to 2 mm × 2 mm × 1.96 mm at the edge, as shown in Figure 23. Subsequently, failure deletion and torsion control were added to each element of the target and projectile. The material model parameters of the projectile obtained in Section 3 were input using the tabular method [103], and a bilinear hardening model [55], which could be directly input into ABAQUS, was adopted to express the material properties of the target.

4.2. Predicted Result of FE Simulation

In the velocity range from 191.6 to 286.1 m/s, the numerical result of the mushrooming failure projectile was compared with the test specimen, as shown in Figure 24. The failure mode predicted by the MJC and EJMA fracture criteria was mushrooming failure, which agreed with the test results. The prediction accuracies of the two fracture criteria at medium and low velocities were verified. The comparisons between the numerical and experimental results for the characteristic size of the mushrooming surface and the failure projectile length are listed in

Table 5. Comparison of experimental and numerical mushrooming Taylor rods.

Velocity/(m/s)	Test	FE Simulation		Test	FE Simulation
	Df/mm	Df/mm	Error	Lf/mm	Lf/mm	Error
191.6	8.33	7.67	7.88%	25.89	26.39	1.93%
200.0	8.55	7.81	8.65%	25.59	26.04	1.75%
238.8	9.67	8.48	12.35%	24.22	24.91	2.84%
259.1	10.09	8.89	11.89%	23.40	24.27	3.71%
286.1	10.80	9.48	12.03%	22.68	23.34	2.91%

. The numerical result of the characteristic length agreed closely with the experimental values, and all of the errors were less than 5%. The numerical prediction of the characteristic diameter was similar to the test results at low velocities, but the errors were greater than 10% when the impact velocity exceeded 238 m/s. The maximum error reached 12%. Meanwhile, the predicted mushrooming surface was circular, which was different from the elliptical surface of the test specimen. Hence, the influences of anisotropy, the yield plateau, and the strain rate sensitivity should be considered when studying the characteristics of the material plastic flow stress.

The critical tensile splitting predicted by the EJMA fracture criterion occurred at a striking velocity of 336 m/s, as shown in Figure 24. The experimental critical velocity was 326.4 m/s, corresponding to a deviation of 2.94%, which was within the allowable margin of error. However, the predicted failure mode of MJC fracture criterion was still mushrooming failure when the striking velocity was 422.1 m/s, which disagreed with the test results. For the three groups of tests in the velocity range from 372.8 to 422.1 m/s, the numerically simulated results of the EJMA fracture criterion were all those of petalling failure. The crack formation and damage degree were similar to the experimental results, as shown in Figure 24.

To explain the discrepancies between the results predicted using the MJC and EJMA fracture criteria, an analysis was conducted based on the stress state of the Taylor rod impact surface material. The damage and stress state value changed the process of the first tensile splitting failure element at the mushrooming surface edge of the critical tensile splitting failure projectile, which was predicted using the EJMA fracture criterion given in Figure 25. The damage mainly appeared between 2.5 and 20 μs. The lode parameter value at this moment was between 0 and −0.25, and the stress triaxiality was between 0 and 0.3. As shown in Figure 13, the fracture strain predicted using the EJMA fracture criterion was far less than that predicted using the MJC fracture criterion. The damage value predicted using the EJMA criterion was higher than that using the MJC fracture criterion, and the amount of damage was greater. Hence, the predicted accuracy of the cracking failure mode at a high striking velocity of a Q460D steel rod could be effectively improved by incorporating the lode angle. As such, the applicability of the EJMA fracture criterion and the validity of the six-parameter calibration was verified.

5. Discussion

In the previous section, the FE simulation results of the Taylor impact tests of Q460D steel rods onto rigid target plates based on the MJC and EJMA fracture criteria were presented. The MJC constitutive relation and the von Mises yield criterion were also applied in the simulation. By comparing the simulated results and the failure mechanisms of the test results at different velocities, it can be seen that the predicted result using the EJMA fracture criterion was significantly better than that using the MJC fracture criterion. This showed that the numerical accuracy of the predictions for the Q460D steel rod failure mode could be effectively enhanced by incorporating the lode angle. Based on the comparison in Table 5, the error of the numerically predicted characteristic diameter of the mushrooming failure at a low striking velocity was still slightly larger, and the maximum error was 12%. Furthermore, there was a significant discrepancy between the circular mushrooming surface of the numerical predictions and the oval surface of the experimental results, and this difference cannot be ignored.

The first possible explanation is that all the specimens used in the materials test were extracted using a Q460D steel plate from the same orientation, and this orientation was defined as 0°. Complex forging and rolling is usually conducted during steel plate metallurgic forming, and thus the mechanical properties in various orientations of the material may differ, i.e., the material could be anisotropic. Hence, the Hill48 yield criterion, which can account for anisotropy, was adopted to characterize the mechanical properties. The Hill48 criterion is as follows:

$$\begin{array}{c}F({\sigma }_{22}-{\sigma }_{33}{)}^2+G({\sigma }_{33}-{\sigma }_{11}{)}^2+H({\sigma }_{11}-{\sigma }_{22}{)}^2+\\ 2L{\sigma }_{23}^2+2M{\sigma }_{31}^2+2N{\sigma }_{12}^2-1=0\end{array},$$

(13)

where F, G, H, L, M, and N are anisotropic correlation parameters. These parameters can be presented as follows:

$$\{\begin{array}{c}F=\frac{1}{2}\left(\frac{1}{R_{22}^2}+\frac{1}{R_{33}^2}-\frac{1}{R_{11}^2}\right),\begin{array}{ccc}& & \end{array}L=\frac{3}{2}\frac{1}{R_{23}^2}\\ G=\frac{1}{2}\left(\frac{1}{R_{33}^2}+\frac{1}{R_{11}^2}-\frac{1}{R_{22}^2}\right),\begin{array}{ccc}& & \end{array}M=\frac{3}{2}\frac{1}{R_{13}^2}\\ H=\frac{1}{2}\left(\frac{1}{R_{11}^2}+\frac{1}{R_{22}^2}-\frac{1}{R_{33}^2}\right),\begin{array}{ccc}& & \end{array}N=\frac{3}{2}\frac{1}{R_{12}^2}\end{array},$$

(14)

$$\{\begin{array}{l}{R}_{22}=\sqrt{\frac{r_{90}(r_0+1)}{r_0(r_{90}+1)}}\hfill \\ R_{33}=\sqrt{\frac{r_{90}(r_0+1)}{r_0+r_{90}}}\hfill \\ R_{12}=\sqrt{\frac{3r_{90}(r_0+1)}{(r_{45}+1)+(r_0+r_{90})}}\hfill \\ R_{11}=R_{23}=R_{13}=1\hfill \end{array},\{\begin{array}{l}{r}_0=\frac{{\epsilon }_{yy}}{{\epsilon }_{zz}}=\frac{{\epsilon }_{yy}}{-({\epsilon }_{xx}+{\epsilon }_{yy})}\hfill \\ r_{45}=\frac{{\epsilon }_{135}}{{\epsilon }_{zz}}=\frac{{\epsilon }_{135}}{-({\epsilon }_{45}+{\epsilon }_{135})}\hfill \\ r_{90}=\frac{{\epsilon }_{xx}}{{\epsilon }_{zz}}=\frac{{\epsilon }_{xx}}{-({\epsilon }_{xx}+{\epsilon }_{yy})}\hfill \end{array}$$

(15)

where R_ij is the yield-to-stress ratio corresponding to each orientation of the material axes considered, which can be calculated using the anisotropy index r. The anisotropy indices for the orientations of 0°, 45°, and 90° can be obtained based on the ratio of the plastic strains in the width and thickness orientations during unidirectional tensile tests. The relationship between the plastic strains in the different orientations was observed to satisfy the following equation: ε_xx + ε_yy + ε_zz = 0, where ε_xx, ε_yy, and ε_zz are the plastic strains in the x-, y-, and z-directions, respectively.

Because the anisotropy index should be obtained under unidirectional tensile conditions, three groups of flat dog-bone-shaped (Figure 26) test specimens of the Q460D steel were prepared, and tensile tests at 0°, 45°, and 90° orientations were performed. The conventional specimen extracting orientation is 0°, and the other orientations were defined by rotations from this orientation. The test conditions were similar to those of the notched tensile tests.

After the experiments, load–displacement curves were obtained, as shown in Figure 27. The lateral and longitudinal strain of the whole test process were calculated using MatchID-2D software. The analysis was conducted before necking deformation, and the anisotropy indices were r₀ = 0.78, r₄₅ = 0.92, and r₉₀ = 0.91. The yield-to-stress ratio was calculated using Equation (6), and R₁₁ = R₂₃ = R₁₃ = 1, R₂₂ = 1.04, R₃₃ = 0.98, and R₁₂ = 1.01.

However, the adiabatic effect cannot be collocated with the Hill48 yield criterion in ABAQUS FE software, and it was not suitable for the Taylor impact tests, in which the temperature increase is important. An explicit user-defined dynamic material subroutine (VUMAT) based on the Hill48 yield criterion was established. The parameters of the anisotropy yield-to-stress ratio, the constitutive relation, and the fracture criterion were defined in the subroutine, and the specific value was input as an “inp” file. Meanwhile, to facilitate the analysis, the output variables, such as the stress triaxiality, lode angle, and injury were added and output in the form of a solution-dependent variable (SDV). The subroutine was verified by establishing a 3D FE model of the smooth tensile test. The length, width, and thickness directions were defined as the x-, y-, and z-axes. Compared to the experimental results, the numerically predicted oval necking section and output load–displacement curve were similar to the test results, as shown in Figure 28. Thus, the validity of the subroutine was verified.

Based on the numerical model derived in Section 4, the same material directions were defined, the parameters of the Hill48 yield criterion, MJC constitutive relation, and EJMA fracture criterion of the material model were input into the “inp” document using the Hill48 subroutine, and an SDV item was also added as a field variable. The results of the numerical simulations and experiments are listed in

Table 6. Simulation results for an anisotropic yield plateau.

Velocity/(m/s)		191.6	200.0	238.8	259.1	286.1
Test	DLf/mm	8.54	8.75	9.98	10.52	11.28
	DSf/mm	8.12	8.36	9.38	9.60	10.33
	Lf/mm	25.89	25.59	24.22	23.40	22.68
Anisotropy	DLf/mm	7.80	7.96	8.65	9.08	9.68
	Error/%	8.66	9.02	13.32	13.68	14.18
	DSf/mm	7.49	7.63	8.26	8.64	9.20
	Error/%	7.75	8.73	11.94	10.00	10.93
	Lf/mm	26.30	26.02	24.90	24.25	23.32
	Error/%	1.58	1.68	2.81	3.63	2.82
Yield plateau	DLf/mm	7.77	7.93	8.63	9.06	9.67
	Error/%	9.01	9.37	13.52	13.87	14.27
	DSf/mm	7.46	7.60	8.24	8.63	9.19
	Error/%	8.12	9.09	12.15	10.10	11.04
	Lf/mm	26.17	25.89	24.76	24.08	23.14
	Error/%	1.08	1.17	2.22	2.90	2.02
Parameter C	DLf/mm	8.24	8.42	9.28	9.81	10.55
	Error/%	3.51	3.77	7.01	6.74	6.47
	DSf/mm	8.12	8.30	9.12	9.62	10.32
	Error/%	0.00	0.71	2.77	0.21	0.09
	Lf/mm	25.40	25.08	23.68	22.85	21.82
	Error/%	1.89	1.99	2.23	2.35	3.79
Deformation mechanism	DLf/mm	8.25	8.44	9.30	9.83	10.61
	Error/%	3.39	3.54	6.81	6.55	5.93
	DSf/mm	7.91	8.08	8.86	9.32	10.00
	Error/%	2.58	3.34	5.54	2.91	3.19
	Lf/mm	26.07	25.78	24.52	23.78	22.84
	Error/%	0.69	0.74	1.23	1.62	0.70

. The mushrooming surfaces of the numerically predicted result were oval-shaped, similar to the experiments. The predicted error of the characteristic size when V > 238.8 m/s changed slightly but was still higher than 10%. Thus, the prediction accuracy of the Taylor rod mushrooming surface shape could be increased effectively by using the Hill48 yield criterion, which accounts for the anisotropy. However, other factors influenced the prediction accuracy of the characteristic size.

In Figure 28, the load–displacement curve of the test results exhibited yield plateaus at the yield step. However, the yield step curve that was characterized by the strain term showed a rapidly increasing trend, and the characteristic result of the plastic flow stress in this section was significantly higher than the experimental value. Hence, the relationship between the plastic flow stress and the equivalent plastic strain that was obtained using the MJC constitutive relation was discretized using the tabular method. The parameters at the yield section were adjusted and input to the numerical simulation to conduct iterative calculations until the numerical curve coincided with the experimental results, as shown in Figure 28. Isotropic and anisotropic analyses were conducted based on the von Mises and Hill48 yield criteria, respectively. The EJMA fracture criterion and the adjusted constitutive relation were input into the simulations using the tabular method, and the isotropic and anisotropic predictions of the characteristic size are listed in Table 6 and

Table 7. Simulation results for the isotropic yield plateau.

Velocity/(m/s)		191.6	200.0	238.8	259.1	286.1
Test	Df/mm	8.54	8.75	9.98	10.52	11.28
	Lf/mm	25.89	25.59	24.22	23.40	22.68
Yield plateau	DLf/mm	7.63	7.78	8.45	8.87	9.47
	Error/%	8.37	9.00	12.66	12.09	12.31
	Lf/mm	26.16	25.89	24.76	24.07	23.12
	Error/%	1.04	1.17	2.23	2.86	1.94
Deformation mechanism	DLf/mm	8.11	8.28	9.11	9.61	10.35
	Error/%	2.61	3.15	5.84	4.75	4.95
	Lf/mm	26.07	25.77	24.52	23.78	22.81
	Error/%	0.69	0.70	1.23	1.66	0.57

, respectively.

After the yield plateau was characterized reasonably, the predicted mushrooming characteristic sizes for the isotropic and anisotropic materials were similar to the predicted results obtained using the MJC constitutive relation, and the maximum error was also higher than 10%. The predicted characteristic length was better than that obtained using the MJC constitutive relation, the errors decreased by about 30%, and the errors were less than 5%. The characteristic yield plateau was not the main factor that influenced the prediction accuracy of the Taylor impact test.

The strain rate of Taylor impact test can also easily reach 10⁴/s to 10⁵/s at a low impact velocity, which is far above what the SHPB test can reach, and the variation of the plastic flow stress with strain rate at higher strain rates must be explored. Referring to a previously reported method [1], the value of C was derived based on the mushrooming failure characteristic size at low impact velocities. When C_d = 0.015, the deviations between the numerically predicted characteristic diameter, assuming the material is isotropic at velocities from 191.6 to 286.1 m/s, and the equivalent diameter of the experimental results were all less than 5%. The value of C_d decreased by about 60% from the original value of 0.0404. Subsequently, C_d was adopted to replace the value C. and applied to the analysis of the anisotropy using the Hill48 subroutine. The predicted characteristic sizes for the different velocities are listed in Table 6.

From the comparison with the anisotropic conditions, the predicted errors of the characteristic size were all less than 10%, but the characteristic value of the major axis approached that of the minor axis. The predicted error of the major axis was also much higher than that of the minor axis, so the characteristic shape was disproportional, which was inconsistent with the test results. This illustrated that the predicted accuracy of the Taylor impact tests under anisotropy cannot be enhanced effectively by adjusting the value of C.

Based on the analysis of the SHPB and the dynamic tensile tests in Section 3, the sensitivity of the yield strength to the strain rate at a high strain rate was higher. From the analysis of the dislocation dynamics, the plastic deformation transitioned from a thermally activated mechanism to a phonon drag mechanism, but the transformation of the plastic deformation mechanism was not reasonably considered in the JC constitutive relation. Thus, it was re-expressed based on a previously reported method [106].

The strain terms of the MJC constitutive relation are a linear combination of the Ludwik and Voce laws; thus, they have more parameters. This method is more complex and can easily produce errors. To ensure consistency with the original method, the Ludwik law was applied as the strain term. With parameter A and the temperature fixed, ${\sigma }_{eq}=\left(A+B{\epsilon }_{eq}^n\right)\left(1+C\ln {\epsilon }^{*}\right)$ was adopted to fit the engineering stress–strain curves of the SHPB test at high, medium, and low strain rates, respectively, and the average values B_d = 398.61 MPa and n_d = 0.532 were obtained, as shown in Figure 29. After this, the original strain term was replaced and applied in the FE simulations. Analyses for isotropic and anisotropic materials were conducted, and the predicted characteristic sizes are listed in Table 6 and Table 7, respectively.

After considering the effect of the plastic deformation mechanism at high strain rates, the errors of the mushrooming characteristic size for isotropic and anisotropic materials all decreased to about 5%, and the predicted results for the anisotropic conditions were similar to the experiments, as shown in Figure 30. The prediction accuracy increased significantly.

The predicted results in the velocity range from 326.4 to 422.1 m/s are shown in Figure 31. In the predicted result at 326.4 m/s, a few slight axial cracks appeared, which was similar to the experiments. Compared to the predicted result using the MJC constitutive relation, which predicted a critical tensile splitting velocity of 336 m/s, the predicted accuracy increased significantly. For the petalling failure mode at higher velocities, the results were similar to the test results in that the damage area was concentrated along the minor axis of the impact surface. The validity of each parameter in the Hill48 yield criterion was verified, and it illustrated that the predicted accuracy of the Taylor impact tests could be increased effectively by considering the lode angle, anisotropy, and strain rate sensitivity.

6. Conclusions

In the present work, the material characteristics of Q460D steel were examined by conducting performance tests at different temperatures, strain rates, and stress states. Furthermore, a modified Johnson–Cook (JC) constitutive relation, a modified Johnson-Cook fracture criterion, and a lode-dependent fracture criterion were calibrated using a hybrid experimental–numerical approach. Then, Taylor impact tests of Q460D steel rods onto rigid target plates were performed, and a 3D FE simulation model was developed to analyze the influence of the lode angle on the prediction accuracy for the material fracture behavior. Finally, the influences of the material properties on the prediction accuracy were revealed. The important conclusions are as follows:

• Based on the analysis of the material tests, the plastic flow stress and fracture strain varied nonlinearly as the temperature increased. To better characterize the strain hardening and thermal softening behavior, the MJC constitutive relation and MJC fracture criterion were adopted and calibrated. Combined with the research into shear and plane strain tests, the lode correlation of the Q460D steel fracturing was studied. The lode-dependent EJMA fracture criterion was calibrated, while the strain rate and temperature terms from the MJC fracture criterion were used.
• Based on Taylor impact tests of Q460D steel, mushrooming, tensile splitting, and petalling failure modes occurred in the impact velocity range of 191.6–422.1 m/s. Comparing the results predicted using the MJC and EJMA fracture criteria with the experimental results, the failure modes at different striking velocities were well-predicted using the lode-dependent EJMA fracture criterion. Due to a lack of consideration of the Lode angle, the material tenacity was vastly overestimated by the MJC fracture criterion, so that the predicted failure mode at higher velocities did not match that of the experiments. Hence, the prediction accuracy of the cracking failure mode at high striking velocities for the Q460D steel rod could be effectively improved by incorporating the lode angle. Meanwhile, the applicability of the EJMA fracture criterion and the validity of the parameter calibration was verified. However, due to the slightly higher errors of the predicted characteristic sizes at V > 238.8 m/s and a characteristic shape prediction that was inconsistent with the experimental results, the effects of anisotropy, the yield plateau, and the strain rate sensitivity should be considered in the FE simulations.
• In the analysis of the anisotropy, yield plateau, and strain rate sensitivity, the characteristic shape of the projectile impact surface could be better predicted when the Hill48 yield criterion was adopted. The yield plateau was not the main factor. The variation of value C could enhance the predicted characteristic sizes effectively, but was not suitable for anisotropic materials. After considering the effect of the plastic deformation mechanism at high strain rates, the errors of the mushrooming characteristic size under anisotropy all decreased to about 5%, and the predicted elliptical characteristic shape, as well as the failure mode at different impact velocities, was similar to the experimental results. The damaged area of the simulated result was concentrated on the minor axis sides. The validity of each parameter in the Hill48 yield criterion was verified, and it was shown that the predicted accuracy of the Taylor impact test could be increased effectively by considering the lode angle, anisotropy, and strain rate sensitivity.