Experimental and Numerical Investigation on Dynamic Shear Behavior of 30CrMnSiNi2A Steel Using Flat-Hat Specimens

Abstract

An absolutely conflicting value for the incorporation of the Lode parameter into a fracture criterion was reported in the literature when predicting the ballistic resistance of metallic plates failing through shear plugging. In this study, a combined experimental–numerical investigation was conducted to understand the dynamic shear fracture behavior under compression–shear stress states. Flat-hat-shaped specimens of 30CrMnSiNi2A high-strength steel were loaded using a Split Hopkinson Pressure Bar apparatus, combining the ultra-high-speed photography technique, digital image correlation method, and microstructure observation. Parallel finite element simulations were performed using both a modified Johnson–Cook (MJC) fracture criterion or an extended Xue–Wierzbicki (EXW) fracture criterion with Lode dependence to reveal the value of the Lode parameter incorporation. It was found that deformed shear bands with a width of approximately 0.14 mm form at a critical impact velocity. The EXW criterion correctly predicts the critical fracture velocity and estimates the fracture initiation instants within an error of 5.3%, whereas the MJC fracture criterion overestimates the velocity by 14.3%. Detailed analysis shows that the EXW criterion predicts a combined failure mechanism involving ductile fracture and material instability, while the MJC fracture criterion attributes the failure exclusively to material instability. The improved accuracy achieved by employing the Lode-dependent EXW fracture criterion may be attributed to the compression–shear stress state and the accurate prediction of the failure mechanism of the dynamic shear fracture.

1. Introduction

Ultra-high-strength steels are extensively utilized in the aerospace, marine, and construction industries. During service, these steel structures may be exposed to impact loads from foreign objects, potentially leading to structural failure and disrupting the normal operation of the systems. To date, substantial research has been conducted to elucidate the failure mechanisms of metal plates subjected to cylindrical projectile impacts within the sub-ordnance velocity range. Depending on factors such as projectile nose shape, target thickness, impact velocity, and the material strengths of both the target and projectile, various failure mechanisms may emerge, including shear plugging, petalling, dishing, discing, and ductile hole growth [1,2,3,4,5].

Among these failure mechanisms, shear plugging has drawn significant attention because the ballistic limit of targets that fail via shear plugging is typically much lower compared to those failing by other mechanisms [1,3,5]. It has been shown that the lower ballistic performance is related to the localized shear deformation in a narrow width during an extremely short time [1,2,3]. One challenge associated with shear plugging failure in metal targets is that the ballistic limit is difficult to accurately predict using numerical simulations. Dey et al. [1] found that the experimentally observed decreasing trend of the ballistic limit with increasing target strength is predicted in the completely opposite direction using finite element (FE) simulations. To the best of the authors’ knowledge, the experimental trend has not been accurately reproduced by any numerical simulations.

High plastic deformation, temperature rise, and strain rates, as well as fracture initiation and growth, develop within the interior localized shear band of the target plates over a very short time period [6]. To predict such a complex process, accurate material models, namely, the plasticity model and fracture criterion, are essential. The plasticity model characterizes the flow stress under various conditions of plastic strain, the strain rate, and temperature, while the fracture criterion determines the critical fracture strain. Regarding the fracture strain of metals, the effects of the stress state, the strain rate, and temperature are typically taken into account. In the past, it has been widely acknowledged that stress triaxiality plays a pivotal role in the fracture behavior of ductile metals [7,8]. However, recent studies have also identified the influence of the deviatoric stress parameter or Lode angle, particularly at low and negative stress triaxialities [8,9,10,11,12]. The Lode parameter, defined as a function of the third invariant of the stress deviator, serves to distinguish among various three-dimensional stress states, ranging from axisymmetric tension to in-plane shear and biaxial tension with axisymmetric compression. A significant reduction in material ductility under plane strain stress states, as characterized by the Lode angle, has been documented for certain metals, particularly at low and negative stress triaxialities, as reported in Refs. [13,14,15]. It has been shown that the mechanism for the Lode parameter dependence is correlated to the effect of the maximum shear stress on the coalescence of voids [16]. Following the finding on the Lode angle effect in metal ductility and in an effort to enhance the numerical simulation accuracy of the ballistic performance of metallic targets that fail by shear plugging, Xiao and his co-workers [5,17,18,19,20] employed fracture criteria incorporating both stress triaxiality and Lode angle parameters (i.e., Lode-dependent fracture criterion) and compared the simulation results obtained using Lode-dependent and Lode-independent (only stress triaxiality-dependent in terms of stress state parameters) fracture criteria. They found that for 2024-T351 [5,17] aluminum alloy and 316L austenitic stainless steel [19] target plates, Lode-dependent fracture criteria predicted significantly improved ballistic performance compared to the results obtained from simulations using Lode-independent fracture criteria. However, for Weldox 700 E [18] and Weldox 900 E [19] steel plate targets, the predicted ballistic performance was not significantly affected by incorporating the Lode angle into the fracture criteria, despite the fact that the ductile fracture of the three steels exhibited an obvious Lode angle dependence and the dominant stress state along the shear plugging path was within the stress states with strong Lode angle effect.

The aforementioned findings regarding the opposite value of incorporating the Lode angle (or the deviatoric state parameter) into a fracture criterion in predicting ballistic performance of metallic targets motivated us to compare more simulation results with the experimental observations and recordings. However, apart from the initial residual velocity data and ballistic limit velocity, very limited additional information can be extracted from ballistic impact tests, which restricts effective comparisons.

In recent years, flat-hat-shaped (FHS) specimens, originally proposed by Meyer and Manwaring [21], have been adopted to enhance the understanding of the dynamic shear and adiabatic shear localization behavior of metals [22,23,24,25]. In the tests, an FHS specimen is sandwiched between an incident bar and a transmitter bar, and dynamic shear deformation occurs in the specimen due to the impact of a striker bar on the opposite end of the incident bar. Three characteristics of the dynamic shear behavior in FHS specimens are noteworthy and may contribute to a deeper understanding of the shear plugging of metal plates. Firstly, a compression–shear stress state develops in the shear deformation if the width of the hat is larger than that of the hole at the bottom of the specimen [24]. Such a stress state may be analogous to the one developed during the shear plugging behavior of a metal target struck by a blunt projectile. Secondly, the deformation and fracture process of the dynamic shear behavior can be directly and in situ observed using ultra-high-speed photography techniques. Finally, digital image correlation (DIC) analysis can be applied to the recorded ultra-high-speed images. It is well-established that the DIC method has been extensively utilized for capturing strain localization prior to fracture initiation [26], identifying fracture nucleation sites in high-strength alloys under ballistic loading [27], and determining the size of fatigue process zones attributed to the three-dimensional distribution of plastic strains near the stress concentrator vertex [28]. DIC analysis may further elucidate the characteristics of the dynamic shear and fracture behavior.

It is important to highlight that the double-shear specimens introduced by Miyauchi [29], Gary and Nowacki [30], Klepaczko [31], and Xu et al. [32], as well as the single-shear specimens developed by Jia et al. [33] and Lee and Huh [34], represent valuable alternative configurations for our research. Nevertheless, given the constraints imposed by material availability and experimental equipment, flat-hat-shaped (FHS) specimens were prioritized in this study.

In this study, the dynamic shear behavior of 30CrMnSiNi2A high-strength steel was investigated using flat-hat-shaped specimens via a hybrid experimental–numerical method. A conventional Split Hopkinson Pressure Bar was employed for loading, while an ultra-high-speed camera was employed to capture the surface deformation of the specimens during the tests. DIC analysis was subsequently conducted to quantify the distribution and evolution of the shear deformation throughout the loading process. Microstructure observations were performed to elucidate the underlying failure mechanisms. Finite element simulations were carried out using both Lode-dependent and Lode-independent fracture criteria. The results from two series of simulations were compared with experimental counterparts, highlighting the advantages of the incorporation of the Lode parameter into a fracture criterion. Finally, the mechanism behind the improved prediction accuracy achieved by using the deviatoric state parameter-dependent fracture criterion was analyzed and discussed in detail.

2. Materials and Methods

2.1. Materials and Specimen Geometry

The material utilized in this study is 30CrMnSiNi2A steel, a low-alloy steel that contains chromium, manganese, silicon, and nickel. Its chemical composition is summarized in

Table 1. Chemical composition of 30CrMnSiNi2A steel (wt.%).

C	Si	Mn	P	S	Cr	Ni	Mo	Cu	W	V	Ti
0.28	1.02	1.12	0.014	0.001	1.01	1.50	0.08	0.03	0.01	0.01	0.0065

. Owing to its superior comprehensive properties, including high strength, fracture toughness, and excellent corrosion resistance, this steel is extensively employed in the aerospace and national defense industries. The heat treatment process involved austenitization at 890 °C for 30 min, followed by oil quenching. Subsequently, the steel was tempered at 600 °C for 180 min and cooled to ambient temperature in air. The resultant hardness achieved was 38 HRC.

The FHS specimen was employed to gain an in situ and direct insight into the dynamic deformation and fracture response. The geometry and dimensions of the FHS specimen are illustrated in Figure 1a. The actual shape and dimensions of the specimen were meticulously verified using a digital microscope (VHX-6000, Keyence, Osaka, Japan), confirming that the specimens were manufactured with high precision, as shown in Figure 1b.

2.2. Experimental Work

The schematic illustration of the loading and the measuring apparatus is presented in Figure 2. A standard Split Hopkinson Pressure Bar (SHPB) facility, installed at Nanyang Institute of Technology, China, was employed to impose the dynamic loading. All bars had a diameter of 11.89 mm and were fabricated from hardened 18Ni(350) maraging steel, which has a yield strength of 2420 MPa and a Young’s modulus of 196 GPa. The SHPB facility primarily comprises a gas gun launch system, a striker bar, an incident bar, a transmitter bar, and an absorber. The lengths of the striker, incident, and transmitter bars are 200 mm, 1400 mm, and 1400 mm, respectively. The FHS specimens were sandwiched between the incident and transmitter bars.

Strain gauges were attached to the midpoints of the incident and transmitter bars to record the deformation waves. The waves in the incident and transmitter bars were recorded by a data acquisition system, which comprises Wheatstone bridges, dynamic strain amplifiers, and a digital oscilloscope. Specifically, the Wheatstone bridge converts the deformation of a strain gauge into a voltage signal, which is subsequently amplified by the dynamic strain amplifier and finally passed to the oscilloscope for recording. During the tests, the data acquisition system was triggered by wave signals in the incident bar. Based on the one-dimensional elastic wave propagation theory, the axial loading F carried by the FHS specimens was estimated as [25,35]:

$$F=E{A}_{\mathrm{t}}{\epsilon }_{\mathrm{t}}(t)$$

(1)

where E and A_t are the Young’s modulus and cross-section area of the transmitter bar and ${\epsilon }_{\mathrm{t}}(t)$ is the transmitted waves recorded by the strain gauge attached in the transmitter bar. The loading displacement of the incident bar on the FHS specimen can be calculated by [25,35]:

$$\delta =-2C_0{\displaystyle {\int }_0^t{\epsilon }_{\mathrm{r}}(t)\mathrm{d}t}$$

(2)

where ${\epsilon }_{\mathrm{r}}(t)$ is the reflected wave measured by the strain gauge at the incident bar and C₀ is the elastic wave velocity of the bar. According to our tests, C₀ = 4930 m/s.

For the present Half-Bridge Configuration, the relationship between the measured electrical signals (voltage output from the Wheatstone bridge) and strain values was established through:

$$\epsilon ={\displaystyle \frac{2\mathrm{\Delta }U(t)}{K_1{K}_2{U}_0}}$$

(3)

where K₁ represents the strain gauge sensitivity coefficient (K₁ = 2.15 ± 0.5% as provided by the manufacturer), K₂ is the magnification factor of the dynamic strain amplifiers, and $\mathrm{\Delta }U(t)$ and U₀ are, respectively, the Wheatstone bridge’s output and excitation voltage.

It is worth noting that some recorded signals were accompanied by non-physical disturbances originating from electronic noise within the acquisition system and potential slight misalignment of the bars. To address this issue, a baseline correction method was applied to all experimental signals prior to analysis, ensuring that these non-physical components would not influence the quantified dynamic responses. Specifically, zero-point alignment was conducted using pre-trigger data for the transmitted wave and data between the incident and reflected waves for the signals in the incident bar. In addition, as will be seen in Section 3.2, despite the occurrence of non-physical disturbances in some tests, the experimentally obtained incident waves exhibited excellent consistency with the simulated signals in terms of duration, amplitude, peak value, and envelopes. This further substantiates the reliability of the experimental signals.

An optical barrier was installed at the muzzle of the launch tube to accurately measure the striker bar velocity. The measurement apparatus consists of two parallel laser curtains separated by a calibrated distance of 50.00 ± 0.05 mm, combined with a high-resolution chronograph featuring 1 μs temporal resolution. This configuration achieves an impact velocity measurement accuracy of 0.3% for impact velocities below 100 m/s.

The deformation and potential fracture of the FHS specimens were captured by an ultra-high-speed camera (Kirana-05M, Specialized Image Ltd., Tring, UK) operating at a frame rate of 2,000,000 fps with a resolution of 924 × 768 pixels. The camera was meticulously positioned to ensure a perpendicular view of the specimen surface, which is a critical prerequisite for performing an accurate 2D DIC analysis. The camera was synchronized with the incident stress wave using the oscilloscope as a trigger. However, a delay time was introduced to ensure that image recording commenced precisely when the stress wave reached the FHS specimen.

To visualize the evolution of the distribution of the local deformation and strain fields, DIC analysis was performed using the images recorded by the Kirana-05M camera. An artificial speckle pattern was created on the surface of the FHS specimens by spaying black paint onto a white painted surface. Special attention was paid to the application of the black paint, as shear deformation in the FHS specimen can be highly localized, requiring a fine speckle pattern for high-magnification DIC measurements. In this study, the black paint was created using an airbrush, resulting in an average speckle size of approximately 24 μm and a spatial resolution of about 0.006 mm/pixel. The deformation and strain fields were calculated using MatchID-2D commercial software (v2022.2.3, MatchID, Ghent, Belgium) with the DIC settings listed in

Table 2. Digital image correlation setting using MatchID.

Processing Parameters	Specification
Matching criteria	ZNSSD
Shape function	Affine
Interpolation function	Local bicubic splines
Progress history	Spatial + update reference
Subset size SS [px]	27
Step size (ST) [px]	7
Strain window (SW)	9
Strain tensor—polynomial	Hencky-Q4
Virtual strain gauge (VSG) [px]	41

.

In addition, to identify the fracture mechanism underlying the dynamic fracture behavior, post-test FHS specimens were prepared in accordance with standard mechanical polishing procedures, subsequently etched in a 5% nitric acid solution, and finally observed using an optical microscope (OM, OLS4100-SAF, Olympus, Tokyo, Japan).

2.3. Finite Element Simulation Set-Up

A quarter 3D FE model was built to simulate the dynamic shear behavior of the FHS specimens using the symmetry condition twice using the commercial finite element software ABAQUS/Explicit (2022.HF6, Dassault Systèmes Simulia Corp., Johnston, RI, USA). The striker, incident, and transmitter bars, as well as the FHS specimen, were all created in the model and discretized with C3D8R (8-node linear brick, reduced integration) solid elements. A relatively large and uniform mesh size of 0.5 mm × 0.5 mm × 2.0 mm was employed for the striker, incident, and transmitter bars due to their small deformation. In contrast, a gradually varied mesh strategy was implemented for the FHS specimen, as illustrated in Figure 3. Extremely fine elements with an average mesh size of 0.02 mm × 0.02 mm × 0.1 mm were generated in the shear deformation region, where large deformation and potential fractures are expected to occur. The element size was gradually increased in other regions while maintaining a constant element size of 0.1 mm in the thickness direction of the FHS specimen.

Contacts between the striker and incident bar, the incident bar and FHS specimen, and the FHS specimen and transmitter bar were established using the surface-to-surface contact algorithm provided by ABAQUS 2022.HF6. Considering the possible self-contact among different parts of the FHS specimen during the fracture process, the general contact algorithm was employed by defining interior surfaces within the FHS specimen. Normal contact behavior was modeled as a hard contact, while tangential friction was neglected.

All the bars were treated as elastic, whereas the FHS specimen was assumed to exhibit isotropic elastic–plastic behavior and was modeled using a modified Johnson–Cook (MJC) plasticity [36]. According to the MJC plasticity, the von Mises flow stress ${\sigma }_{\mathrm{eq}}$ reads:

$${\sigma }_{\mathrm{eq}}=\left(\alpha \left(A+B{\epsilon }_{\mathrm{eq}}^n\right)+\left(1-\alpha \right)\left(A+Q\left(1-\exp \left(-\beta {\epsilon }_{\mathrm{eq}}\right)\right)\right)\right)\left(1+C\ln \left({\displaystyle \frac{{\dot{\epsilon }}_{\mathrm{eq}}}{{\dot{\epsilon }}_0}}\right)\right)\left(1-p{\left({\displaystyle \frac{T-T_0}{T_{\mathrm{m}}-T_0}}\right)}^m\right)$$

(4)

where $A, B, n, Q, \beta$, and α are strain hardening parameters; ${\epsilon }_{\mathrm{eq}}$ is the equivalent plastic strain; C is the parameter reflecting the strain rate hardening effect; p and m are temperature softening parameters; ${\dot{\epsilon }}_{\mathrm{eq}}$ and ${\dot{\epsilon }}_0$ are, respectively, the current strain rate and reference strain rate; and T, T_0, and T_m are the actual, refence and melting temperature, respectively. $A, B, n, Q, \beta$, α, C, p, and m are nine material parameters.

Two ductile fracture criteria were employed to model the potential initiation and evolution of fractures. The first criterion was a slightly modified Johnson–Cook (MJC) criterion [19], which relates the fracture strain of a material with strain rate, temperature, and stress triaxiality. The second criterion was the extended Xue–Wierzbicki (EXW) fracture criterion [9]. In contrast to the MJC criterion, the EXW additionally accounts for the influence of the Lode parameter in terms of the stress state. The MJC fracture criterion reads:

$${\epsilon }_{\mathrm{f}}=\left(D_1+D_2\exp \left(D_3\eta \right)\right)\left(1+D_4\ln \left({\displaystyle \frac{{\dot{\epsilon }}_{\mathrm{eq}}}{{\dot{\epsilon }}_0}}\right)\right)\left(1+D_5{\left({\displaystyle \frac{T-T_0}{T_{\mathrm{m}}-T_0}}\right)}^{D_6}\right)$$

(5)

where D₁~D₃ are parameters related to stress triaxiality in terms of the stress state; D₄ is the parameter related to the strain rate; D₅ and D₆ are parameters reflecting the effect of temperature; and $\eta$ is the stress triaxiality defined as $\eta =\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right)/{\sigma }_{\mathrm{eq}}$, where ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$ are the three principle stresses. The EXW fracture criterion reads:

$${\epsilon }_{\mathrm{f}}=\left(C_1{e}^{-C_2\eta }-\left(C_1{e}^{-C_2\eta }-C_3{e}^{-C_4\eta }\right){\left(1-{\xi }^{C_5}\right)}^{1/C_5}\right)\left(1+D_4\ln \left({\displaystyle \frac{{\dot{\epsilon }}_{\mathrm{eq}}}{{\dot{\epsilon }}_0}}\right)\right)\left(1+D_5{\left({\displaystyle \frac{T-T_0}{T_{\mathrm{m}}-T_0}}\right)}^{D_6}\right)$$

(6)

where C₁~C₅ are parameters associated with the stress state and $\xi$ is the deviatoric state parameter, defined as:

$$\xi ={\displaystyle \frac{27}{2}}{\displaystyle \frac{s_1{s}_2{s}_3}{{\sigma }_{\mathrm{eq}}^3}}$$

(7)

where $s_1$, $s_2$, and $s_3$ are three principal values of the stress deviator tensor. This deviatoric state parameter is correlated with the frequently adopted normalized Lode angle $\overline{\theta }$ by:

$$\overline{\theta }=1-{\displaystyle \frac{2}{\pi }}\mathrm{acos}\left(\xi \right)$$

(8)

In addition to the aforementioned fracture criteria, a linear damage accumulation rule was employed to model the onset of fracture in the calculations. It reads:

$$D={\displaystyle \sum \frac{\mathrm{\Delta }{\epsilon }_{\mathrm{eq}}}{{\epsilon }_{\mathrm{f}}}}$$

(9)

where $\mathrm{\Delta }{\epsilon }_{\mathrm{eq}}$ is the increment of the equivalent plastic strain in a numerical simulation increment, ${\epsilon }_{\mathrm{f}}$ is the fracture strain of a material at the current stress state, strain rate, and temperature, and D is the damage indicator. The fracture strain ${\epsilon }_{\mathrm{f}}$ was predicted by either the MJC criterion or the EXW fracture criterion. In the simulations, an element whose damage indicator reached 1 was removed to represent the initiation and propagation of a crack.

An adiabatic process was assumed for the present simulations, as the deformation of the FHS specimen developed in an extremely short time (~80 μs). In such a process, the local temperature rise resulting from the convention of plastic work must be accounted for. This was calculated as:

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

(10)

where $\mathrm{\Delta }T$ is the temperature rise, r is the material density, C_p is the specific heat, and χ is the Taylor–Quinney (TQ) coefficient. The TQ coefficient defines the fraction of plastic work converted to heat. In the present paper, we simply adopted a value of 0.9.

A series of mechanical tests, covering a wide range of stress states, strain rates, and temperatures, was conducted by Wu et al. [36]. Through the application of a hybrid experimental–numerical method, the parameters of the MJC plasticity model, as well as those of the two fracture criteria, were calibrated and are presented in

Table 3. Material parameters of 30CrMnSiNi2A steel [36].

Description	Notation	Value
Modulus of elasticity	E (GPa)	200.4
Poisson’s ratio	V	0.3
Density	r (g/m3)	7790
Yield stress constant	A (MPa)	1095.6
Strain hardening constants	B (MPa)	1020.8
	n	0.601
	Q (MPa)	204.4
	β	33.455
	α	0.23
Strain rate constant	C	0.014
Reference strain rate	${\dot{\epsilon }}_0$(/s)	4.17 × 10−4
Thermal softening constants	p	2.880
	m	1.677
Reference temperature	T0 (K)	293
Melting temperature	Tm (K)	1764.4
Taylor–Quinney coefficient	χ	0.9
Specific heat	Cp(J/kg/K)	530
MJC fracture criterion	D1	0.215
	D2	125.205
	D3	−7.049
	D4	−0.037
	D5	149.17
	D6	5.48
EXW fracture criterion	C1	16.7
	C2	3.898
	C3	0.7342
	C4	0.8425
	C5	4
	D4	−0.037
	D5	149.17
	D6	5.48

.

In the initial step of the finite element model, an initial velocity was assigned to the striker bar, and an initial temperature of 293 K was applied to the FHS specimen. The stress history of the elements corresponding to the strain gauges on the incident and transmitter bar was monitored for further comparison with the corresponding experimental data.

In addition, the element sizes in the shear deformation region were also varied to investigate the mesh sensitivity. The element length in the thickness direction was maintained at 0.1 mm, while the other two dimensions were varied using values of 0.04 mm, 0.06 mm, 0.08 mm, and 0.12 mm.

3. Results

3.1. Experimental Results

The initial velocity of the striker bar was varied to achieve different deformation and fracture responses of the FHS specimen. The achieved velocity ranged from 8.0 to 11.8 m/s. According to the ultra-high-speed images, deformation was exclusively observed in the specimen at impact velocities below 10.0 m/s, whereas evident fracture occurred at impact velocities exceeding 11.1 m/s. At an impact velocity of 10.5 m/s, fracture initiation was detected toward the end of the loading process, indicating the critical fracture velocity is 10.5 m/s. According to the elastic wave theory, the total loading duration was approximately 80 μs.

3.1.1. Mechanical Response

Two typical signals recorded by the data acquisition system are shown in Figure 4. As shown, the duration of the transmitted pulse at an impact velocity of V = 11.8 m/s is shorter than that at V = 9.0 m/s, and the reflected pulse at V = 11.8 m/s exhibits a sudden increase during the end stage of the loading process. These observations are attributed to the distinct mechanical responses of the FHS specimen under increasing impact velocities; specifically, fracture initiation and propagation occurred during the final loading phase at V = 11.8 m/s, whereas only deformation was induced at V = 9.0 m/s. It is anticipated that the loading capacity of the FHS specimen will decrease once a crack initiates.

Using Equations (1) and (2), the force and displacement histories, as well as the combined force–displacement curves, were obtained, as illustrated in Figure 5. In this figure, the displacement curves are represented using dashed lines, whereas the load curves are depicted using solid lines for clear distinction. As shown, the transmitted force oscillated during the loading of the plastic stage. However, considering the inherent oscillatory nature of the SHPB signals and the envelope of the incident pulse, it can be inferred that the transmitted force in the plastic deformation stage increased with deformation during the plastic deformation stage for impacts at lower impact velocities (V ≤ 10.0 m/s). Conversely, for the impact at V = 10.5 m/s, the transmitted force gradually decreased at the end of the loading process (between 60 and 80 μs). At even higher impact velocities, the transmitted force began to decrease earlier in the loading process because of fracture initiation and the potential occurrence of adiabatic shear localization.

The recorded ultra-high-speed images, along with the DIC results of equivalent strain distribution in the shear deformation region for impacts at impact velocities of 10.0, 10.5, and 11.1 m/s, are shown in Figure 6 to further elucidate the mechanical responses. It is worth noting that the region captured in the ultra-high-speed images corresponds to the area marked in green in Figure 3b. As illustrated, the equivalent strain exhibits a non-uniform distribution pattern within the shear deformation zone during the entire loading process. Notably, the equivalent plastic strain is higher near the two corners, especially in the vicinity of the inner corner.

According to the ultra-high-speed images, a surface fracture initiated at about 70.5 μs and 66.5 μs for the impact at V = 10.5 m/s and 11.1 m/s, respectively. At the fracture initiation instant, the transmitted force had an obvious drop, which is in accordance with Figure 5a. It is crucial to emphasize that the aforementioned fracture initiation instants should be regarded as late approximations. This conclusion arises from two primary factors: first, interior cracks might have initiated earlier; second, despite the employment of ultra-high-speed imaging, accurately identifying the precise fracture initiation instants remains challenging. This is attributed to the slight compressive stress state within the shear region (stress triaxiality < 0), which inhibits an initiated fracture from opening under such conditions and thereby prevents the clear observation of fracture initiation under this stress state.

For the impact at V = 11.8 m/s, the speckles on the specimen surface peeled off during the loading, and, therefore, the fracture instant was not identified from the ultra-high-speed images.

It is worth noting that 0 µs corresponds to the instant at which the stress wave arrives at the top of the FHS specimens.

Figure 6c also indicates that the highest level of the accumulated equivalent plastic strain is approximately 0.4. At a lower impact velocity, the accumulated equivalent plastic strain decreased, as shown in Figure 6a,b.

For the FHS specimens under dynamic loading, the shear deformation is typically localized within a narrow region. Figure 7 illustrates the distribution of the equivalent strain along a path perpendicular to the dominant shear region at increasing stages of the loading for an impact at V = 10.0 m/s. This path, originally measuring 0.698 mm in length, is marked as a red line in Figure 7a. As observed, the equivalent strain is approximately uniform during the early stages (≤10 μs) of the loading process and gradually becomes localized with increasing loading stages. At the end of the loading, the deformation is localized within the normalized distance range of 0.4 to 0.6, corresponding to a width of approximately 0.14 mm.

For a more comprehensive understanding of the deformation and fracture behavior of the FHS specimens, supplementary material includes several videos captured from ultra-high-speed imaging. Interested readers are encouraged to refer to these resources for further insight.

3.1.2. Microstructural Observation

The specimen impacted at a velocity of 10.5 m/s was selected for microstructural observation. This specific velocity was prioritized as it represents the critical fracture threshold in our experimental framework. At velocities exceeding 10.5 m/s, complete specimen fracture consistently occurred, which precluded reliable microstructural analysis of fracture mechanisms. Conversely, impact events below this critical velocity manifested solely as localized deformation within shear zones, devoid of observable crack formation. Figure 8 illustrates the microstructure within one of the shear deformation paths. As observed, localized shear deformation developed along the shear deformation path, and cracks initiated at and near the two corners of the specimen. Specifically, discontinuous cracks were detected near the internal corner. However, the deformation and fracture extent depicted in Figure 8 were more severe than those observed in the ultra-high-speed images shown in Figure 6b. Consequently, some of the cracks presented in Figure 8 may be caused by multiple loadings resulting from the reflection of the loading pulse in the bar system.

Nevertheless, the so-called white etching bands were not observed in the localized shear deformation path, suggesting that only deformed shear bands formed without any phase transformation.

To conduct a precise investigation into the shear fracture mechanism of FHS specimens based on microstructural observations, it is recommended to utilize the so-called “stoper ring” [25] as a means to prevent repeated loading resulting from the propagation and reflection of stress waves in the incident bar. However, it should be noted that while this technique represents a potential solution, it was not implemented in the current study.

3.2. Numerical Results

Finite element simulations were conducted utilizing the MJC plasticity (Equation (4)). Regarding the fracture criterion, either the MJC model (Equation (5)) or the EXW model (Equation (6)) was employed to examine the validity of the two fracture criteria and the effect of incorporating the Lode angle parameter into a fracture criterion in predicting the dynamic shear behavior of the FHS specimens.

3.2.1. Mesh Size Sensitivity

The simulation results obtained using various element sizes indicate that the wave profiles exhibit minimal sensitivity to mesh size at impact velocities below the critical fracture velocity. In contrast, the behavior of the critical fracture velocity is more complex. Specifically, when using the MJC model, the simulated critical fracture velocity exhibits a high degree of sensitivity to the element size, with a decreasing trend as the element size decreases. Conversely, for simulations based on the EXW fracture criterion, the critical fracture velocity remains within a narrow range of 10.5–10.9 m/s, as depicted in Figure 9.

By considering the trade-off between prediction accuracy and computational resources, this study adopted a mesh size of 0.02 mm in the following sections. This element size has also been utilized in similar studies, as evidenced by Ref. [37].

3.2.2. Critical Fracture Velocity

According to the simulation results, deformation was found in the FHS specimens, whereas no fracture was predicted, even at the highest impact velocity achieved during the tests (see Figure 10), using the MJC fracture criterion. This indicates that simulations using the MJC fracture criterion cannot accurately predict the deformation and fracture behavior of the FHS specimens under the investigated range of increasing impact velocities.

In contrast, the EXW fracture criterion yielded generally good predictions regarding the transition of the deformation and fracture behavior of the FHS specimens at increasing impact velocities. Specifically, deformation was predicted for impact velocities below 10.5 m/s (see Figure 11a), whereas a complete fracture was predicted for impact velocities of 11.1 m/s (see Figure 11c) and 11.8 m/s. At V = 10.5 m/s, cracks were predicted at the two corners of the specimen, as shown in Figure 11b, which is consistent with the experimental observations presented in Figure 8. However, minor discrepancies between the simulations and tests were also identified; for instance, the surface crack observed in the ultra-high-speed images (see Figure 6b) was not predicted.

To identify the critical fracture velocity predicted by the MJC fracture criterion, additional simulations were conducted at even higher impact velocities (with an increment of 0.1 m/s) using the MJC fracture criterion. It was finally found that the critical fracture velocity predicted is 12.0 m/s. Given that the experimentally identified critical fracture velocity is 10.5 m/s, it can be concluded that the FE simulations using the MJC fracture criterion overpredict the critical fracture velocity by 14.3%, whereas those using the EXW fracture criterion predict an identical critical fracture velocity, as shown in Figure 10b. In other words, the FE simulations employing the MJC fracture criterion predict a critical fracture velocity that is 14.3% higher than that predicted using the EXW fracture criterion.

3.2.3. Loading History

The simulation results for the pulse in the incident bar and the transmitter bar were compared with the experimental recordings. Figure 12 shows the comparison at lower impact velocities, where only deformation was predicted by both fracture criteria. At these impact velocities, both fracture criteria predicted identical pulses in the two bars; therefore, only the simulation results using the MJC fracture criterion are shown in Figure 12.

It is worth noting that at V = 10.5 m/s, the predicted fracture using the EXM fracture criterion is rather limited and localized to the corners of the specimen, as shown in Figure 11b. Consequently, the predicted pulse in the transmitter bar is nearly identical to that predicted using the MJC fracture criterion.

According to Figure 12, the general trend of the simulated transmitted and reflected pulses is in good agreement with the experimental results, thereby validating the effectiveness of the MJC plasticity model in predicting the dynamic mechanical response of FHS specimens. However, at V = 10.0 m/s, both the transmitted and reflected loads show slight deviations from the experimental data. Notably, the predicted transmitted load is higher than the experimental value, which suggests the flow stress of the FHS specimen in the plastic hardening state is overestimated by the MJC plasticity model. This overestimation could potentially arise from an overestimated strain rate sensitivity parameter in the MJC plasticity model.

At higher impact velocities, as shown in Figure 13, the predicted results of the transmitted and reflected loads using the MJC and EXW fracture criteria exhibit significant differences during the last stage of loading because simulations employing these two fracture criteria yield distinct deformation and fracture behaviors of the specimens, as previously discussed. Notably, the predicted duration of the transmitted load using the EXW fracture criterion aligns well with the experimental results, whereas that predicted by the MJC fracture criterion matches the duration of the incident pulse, deviating from the test findings. These observations indicate that the EXW fracture criterion accurately predicts the fracture initiation instants of the FHS specimens. Conversely, the MJC fracture criterion overestimates the ductility of the steel, failing to predict any fractures at the investigated impact velocities.

Regarding the fracture initiation instants, simulations using the EXW fracture criterion predicted fracture initiation instants of 67 μs and 63 μs for impacts at V = 10.5 m/s and 11.1 m/s, respectively. When compared with the experimentally identified values (70.5 and 66.5 μs), the predicted results were all 3.5 μs ahead, with an error within 5.3%.

3.2.4. Strain Distribution

The predicted equivalent plastic strain distribution and evolution using the EXW fracture criterion were compared with those obtained from DIC analysis, as shown in Figure 6. It is evident that the non-uniform distribution within the shear deformation region and the strain concentration near the corners of the specimens were accurately predicted.

A comparison of the maximum equivalent strain between the FE and DIC results reveals that the simulation errors at V = 10.5 m/s and 11.1 m/s are within 6%. However, the maximum error reaches 16% at V = 10.0 m/s. This discrepancy might be attributed to a slightly finer speckle size for the FHS specimen in this particular test.

4. Discussion

As previously outlined, the primary aim of the present paper is to understand the dynamic shear fracture of 38CrMnSiNi2A high-strength steel under compression–shear stress states. The motivation of the objective stems from the observation that incorporating the Lode parameter into a fracture criterion exhibits limited sensitivity when simulating the ballistic limit velocity of some high-strength steel plates impacted by blunt rigid projectiles, as reported in Refs. [18,19]. According to Xiao et al. [18], the discrepancy in the predicted ballistic limit velocities of Weldox 700 E steel plates impacted by blunt projectiles, when using the MJC and a Lode-dependent fracture criterion, is within 3.9%. This indicates that incorporating the Lode parameter into a fracture criterion has a limited effect on improving the simulation accuracy of the ballistic limit velocity or may even have no significant impact.

In the present paper, an ultra-high-speed camera and a digital image correlation (DIC)-assisted experimental study were used to identify the critical fracture velocity, strain distribution, and evolution in the shear deformation region, as well as the fracture mechanism. FE simulations were performed using both the MJC and EXW fracture criteria. It was finally demonstrated that the simulations using the Lode-dependent EXW fracture criterion can reasonably predict the shear fracture responses of FHS specimens at increasing impact velocities in terms of critical fracture velocity, fracture instants, and strain distribution, whereas those using the MJC fracture criterion fail to do so. Furthermore, compared with the results using the EXW fracture criterion, the simulations employing the MJC fracture criterion overpredicted the critical fracture velocity by 14.3%. Such a difference is significantly greater than the aforementioned discrepancy observed in the prediction of the ballistic limit velocity, as reported in Ref. [18], i.e., 3.9%. For the case reported in the present paper, we may conclude that the effect of incorporating the Lode parameter into a fracture criterion is significant in improving the accuracy of the predicted critical fracture velocity and fracture response of FHS specimens.

To better interpret the significant effect of incorporating the Lode angle in the present study, additional details of the simulations may provide further insights.

4.1. Stress State

As previously stated, the MJC fracture criterion relates exclusively the fracture strain of a metal to stress triaxiality, while the EXW also incorporates the effect of the deviatoric state parameter in terms of stress state parameters. Thus, the effect of the Lode parameter incorporation into a fracture criterion should be related to the involved stress state during the loading process.

Figure 14 shows the distribution of the averaged stress state parameters for the elements that failed during the loading process of the FHS specimen predicted using the EXW fracture criterion at V = 11.1 m/s. The averaged stress state parameters are as follows:

$${\eta }_{\mathrm{a}v}={\displaystyle \frac{1}{\epsilon }}{\displaystyle {\int }_0^{\epsilon }\eta \left({\epsilon }_{\mathrm{eq}}\right)}\mathrm{d}{\epsilon }_{\mathrm{eq}}$$

(11)

$${\xi }_{\mathrm{a}v}={\displaystyle \frac{1}{\epsilon }}{\displaystyle {\int }_0^{\epsilon }\xi \left({\epsilon }_{\mathrm{eq}}\right)}\mathrm{d}{\epsilon }_{\mathrm{eq}}$$

(12)

where ${\eta }_{\mathrm{a}v}$ and ${\xi }_{\mathrm{a}v}$ are, respectively, the average stress triaxiality and average deviatoric state parameter and $\epsilon$ is the equivalent plastic strain at the investigated increment. The two average parameters were adopted herein because the stress state parameters usually vary in the loading process and an estimation of the stress state is impossible without such average parameters. Similar average parameters were usually adopted in estimating the stress state; see Refs. [12,14,15,17].

As shown in Figure 14, the average stress triaxiality ${\eta }_{\mathrm{a}v}$ is below zero for all failed elements and concentrates between −0.8 and 0, which suggests that the failed elements were in compression. As for the average deviatoric state parameter ${\xi }_{\mathrm{a}v}$, its primary range lies between −0.3 and 0, which aligns with expectations given that the dominant deformation mode of the FHS specimens is shear. It is well known that a zero deviatoric state parameter corresponds to a shear or a generalized plane–strain stress state. Under such stress states, the fracture strain of the metal is lower and can be well predicted by the EXW fracture criterion. However, unfortunately, the Lode-independent MJC fracture criterion cannot follow such a trend but predicts extremely high fracture strain values, as shown in Figure 15. A higher fracture strain will subsequently give a lower damage according to the damage accumulation rule in Equation (8). In such a way, the MJC fracture criterion will underpredict the fracture behavior of the material. This may be the reason for the enhanced prediction accuracy by incorporating the deviatoric state parameter into a fracture criterion when simulating the dynamic behavior of 38CrMnSiNi2A high-strength FHS specimens.

4.2. Fracture Mechanism

According to the current simulation strategy, particularly the damage accumulation rule, an element is removed from the simulation only when its damage value reaches 1 (as defined in Equation (9)); thus, crack initiation is exclusively associated with the attainment of this damage threshold. This appears to suggest that ductile fracture is the sole possible fracture mechanism. However, failure caused by adiabatic shear instability remains a possibility. As indicated by Equations (4) and (10), adiabatic shear localization and instability may emerge in the shear deformation region once the thermal softening overwhelms the strain and strain rate hardening. It is well known that thermal softening can gradually reduce the strain-hardening capacity of a material and cause material instability.

By combining Equations (4) and (10), the flow curve as well as the equivalent plastic at which the thermal softening effect overwhelms the strain and strain rate hardening effect (material instability strain) under adiabatic conditions can be calculated at elevating strain rates, as depicted in Figure 16 and summarized in

Table 4. Material instability strain of 30CrMnSiNi2A steel at various strain rates.

Strain Rate (/s)	Material Instability Strain
102	0.239
103	0.236
104	0.235
105	0.239
106	0.210

. It is evident that under strain rates less than 10⁶/s, the material instability strains are all quite low. In detail, at strain rates of 10²–10⁶/s, which should cover the dominant range of the strain rate developed during the deformation and fracture behavior of the FHS specimens, the material instability strains are all below 0.239.

Figure 17 presents the predicted equivalent plastic and temperature distribution of the failed elements from the simulations using the EXW fracture criterion at V = 11.1 m/s, and Figure 18 shows the results using the MJC criterion at V = 12.0 m/s. As shown, rather than a complete fracture, only a partial fracture of the specimen was predicted, as illustrated in Figure 18a.

As illustrated in Figure 17b, the temperature of the majority of the failed elements predicted by the FE simulations using the EXW fracture criterion is within the range of 375 to 500 °C, whereas their equivalent plastic strains vary between 0.2 and 0.5. In contrast, as depicted in Figure 18b, the FE results based on the MJC fracture criterion indicate that the temperatures of the failed elements are primarily concentrated between 700 and 1100 °C, with equivalent plastic strains generally exceeding 1. Consequently, the temperatures and equivalent plastic strains of the failed elements predicted using the MJC fracture criterion are significantly higher than those predicted using the EXW fracture criterion.

Recalling that the material instability strains were below 0.239, we may conclude that the failure predicted by the EXW fracture criterion is a combination of ductile fracture and material instability because the failure strain of the failed elements is within 0.2 and 0.5. However, the failure strain of the failed elements predicted using the MJC fracture criterion is generally greater than 1.0, which is far beyond the material instability values. Accordingly, the failure of the FHS specimens predicted by MJC is mainly attributed to material instability.

This may be the other reason for the improved prediction accuracy achieved by incorporating the deviatoric state parameter into a fracture criterion in simulating the dynamic behavior of 38CrMnSiNi2A high-strength FHS specimens.

5. Conclusions

A combined experimental–numerical investigation was conducted to explore the dynamic shear behavior of 30CrMnSiNi2A steel using flat-hat-shaped specimens in a standard Split Hopkinson Pressure Bar facility. Ultra-high-speed photography and digital image correlation techniques were employed to identify the critical fracture velocity, strain distribution, and evolution along the shear path, while microstructure analysis was conducted to elucidate the fracture mechanism. Finite element simulations were carried out using either the modified Johnson–Cook (MJC) criterion or the extended Xue–Wierzbicki (EXW) fracture criterion to evaluate the value of using a Lode parameter-dependent fracture criterion in predicting such dynamic response. A detailed examination of the simulation results was performed to better interpret the effect of incorporating the Lode parameter in the present study. According to this work, we can draw the following conclusions:

• The critical fracture velocity leading to partial fracture of the flat-hat-shaped specimens is 10.5 m/s, and deformed shear bands with a width of approximately 0.14 mm form at this critical velocity.
• The strain along the shear path exhibits a non-uniform distribution with localized concentrations near the corners of the specimen, especially the inner corner.
• The EXW fracture criterion predicts an identical critical fracture velocity and estimates the fracture initiation instants within an error of 5.3%.
• The MJC fracture criterion overestimates the critical fracture velocity by 14.3%.
• The EXW fracture criterion attributes the shear fracture behavior to a combination of ductile fracture and material instability, whereas the MJC criterion attributes it exclusively to material instability.
• The improved accuracy achieved by employing the Lode-dependent EXW fracture criterion may be attributed to the compress–shear stress state and the accurate prediction of the failure mechanism of the dynamic shear fracture.

However, it is worth noting that, in this study, no pulse shaper was employed, resulting in a non-monotonic waveform. In future work, we plan to utilize a pulse shaper to achieve a monotonic waveform and further validate the value of the incorporation of the Lode parameter into a fracture criterion in predicting dynamic shear fractures.