Fracture Prediction in Weldox 700E Steel Subjected to High Velocity Impact Using LS-DYNA

Abstract

Analyzing fracture behaviour during high-velocity impacts is critical for designing and developing structures in various scientific and technological fields. This study investigates the fracture behaviour of Weldox 700E steel plates with varying values of fracture parameters and fracture strain criteria using LS-DYNA/explicit. It also discusses the effect of varying plate thicknesses (2 to 6 mm) when impacted by a blunt-nosed projectile with varying masses and varying velocities. The fracture behaviour of flat steel target plates is analyzed with a focus on fracture variables across the top, mid, and bottom elements of the plate thickness. It is observed that the higher value of the fracture parameter results in a smooth fracture surface with Johnson–Cook (JC) fracture strain criteria. For thicker plates, both strain rate and temperature significantly affect the process, while for thinner plates, strain rate predominantly determines element deletion. The results reveal that for the top elements, equivalent plastic strain is the dominant factor causing failure at higher velocities, while temperature plays a significant role at lower velocities. For the mid elements, both plastic strain rate and temperature are critical, whereas for the bottom elements, equivalent plastic strain, plastic strain rate, and temperature are the primary factors. In all cases, shear-type fracture is consistently observed along the mid element of the target plate.

1. Introduction

Impact refers to the brief collision between two objects, a phenomenon that is a significant area of study in engineering and technology. Impact phenomena are relevant across various applications, including terrestrial transportation, aerospace, and aquatic environments. Analyzing impact involves understanding contact interfaces, non-linear deformation, fractures, and fragmentation that occur over short time scales [1]. In recent years, steel plates are increasingly being used in both defense and civil sectors to protect against ballistic threats. Understanding the fracture behaviour of the material under high-velocity impact is crucial for designing components for effective functioning under such protective applications. For simplification, most of the impact studies focus on a bullet as the projectile and a flat plate as the target. Numerous researchers have explored impact-related problems through experimental, numerical, and analytical approaches.

Borvik et al. [2] conducted experimental and numerical studies on the impact of blunt-shaped projectiles on 8 mm thick Weldox 460E steel plates, with velocities ranging from 137 to 298 m/s using LS-DYNA. They found that strain rate, stress, and temperature are critical factors for predicting fracture in the target material. Borvik et al. [3] further investigated the fracture behaviour of 12 mm thick Weldox steel plates impacted by blunt, hemispherical, and conical projectiles at velocities of 181.5–399.6 m/s for blunt projectiles, 278.9–452 m/s for hemispherical, and 206.9–405.7 m/s for conical shapes. They found that the ballistic limit of the blunt-shaped projectile was lower than hemispherical and conical projectiles, with less deformation observed in the plate with the blunt projectile. Borvik et al. [4] also conducted experimental, analytical, and numerical studies on the perforation behaviour of 6–30 mm thick steel target plates impacted by blunt-shaped projectiles at velocities ranging from 150 to 525 m/s. They showed that perforation time remained nearly constant for plates thicker than 8 mm, and deformation increased with higher impact velocity and plate thickness.

Dey et al. [5] studied the perforation strength of three structural steels, viz. Weldox 460E, Weldox 700E, and Weldox 900E, impacted by projectiles with blunt, conical, and ogival nose shapes. The plate thickness was 12 mm, and the velocity range was 150–350 m/s. Dey et al. [6,7] and Borvik et al. [8] explored the perforation resistance of five steel plate variants, 6 mm and 12 mm thick, against 7.62 mm ball-type and 7.62 mm AP-type projectiles at a velocity of 830 m/s. They concluded that, within certain limits, the perforation resistance increased linearly for both types of projectiles.

Iqbal et al. [9] investigated the ballistic limit of 12–16 mm thick mild steel plates impacted by 7.62 mm AP-type projectiles at varying obliquity angles. The study covered impact velocities of 445 to 818 m/s for the 12 mm thick plate and 531 to 819.7 m/s for the 16 mm thick plate. They found the critical ricochet angles for the 12 mm and 16 mm plates to be 58° and 59°, respectively, with ballistic limits of 447.5 m/s for the 12 mm plate and 533 m/s for the 16 mm plate. Gupta et al. [10] investigated the effect of projectile nose shape and impact velocity on fracture behaviour in the target plate. They found that hemispherical-shaped projectiles have the highest ballistic limit. The blunt-nose projectile requires less energy for plug formation. Costas et al. [11] investigated the ballistic resistance of high-strength steel using 7.62 APM2 projectiles under various configurations. They found that the ballistic resistance of the plates improved after heat treatment.

Lou and Huh [12] examined the development of ductile fracture in high-strength steels, where fracture occurs due to stress states transitioning from shear to plane-strain tension. Xue and Wierzbicki [13] found that fracture in ductile materials is primarily caused by the accumulation of plastic strain. Wierzbicki and Xue [14] concluded that the third stress invariant sometimes plays a significant role in crack growth, beyond the influence of mean stress and equivalent stress. In some metals, fracture occurs due to the development of micro-voids [15] and the ductile fracture is also predicted by modified anisotropic yield criteria [16]. Liu et al. [17] performed the investigation on ductile fracture of high-strength steel at low triaxiality. Ghosh and Ghosh [18] performed a metallurgical investigation and found that the strength and toughness increased due to the presence of martensite and ferrite-pearlite in a steel component. The finite element (FE) technique employed to examine the ballistic behaviour of bi-layer Armor 500T steel in different configurations, concludes a better ballistic resistance at 30° angle [19].

Baysal et al. [20] revealed that the impact strength improved with the addition of several UD layers to nonwoven mats. Stamoulis et al. [21] studied the fracture characteristics of laminated composites during low-velocity impact and found that higher impact energy results in greater contact force and a larger fracture area. The performance of an additively manufactured honeycomb structure made of SS 316 plates showed that increased thickness enhances energy absorption and ballistic resistance [22]. Further, the study analyzing in-plane preloaded CFRP plates concluded that the ballistic limit is lower for preloaded specimens [23].

The value of the critical fracture parameter is a crucial factor in predicting the ballistic limit, perforation resistance, and deformation of target plates. Few studies have explored its significance. For instance, Gautam et al. [24] used a value of 0.5 for the fracture parameter with the AISI 1045 steel Taylor rod. Similarly, Indriyantho et al. [25] employed fracture values ranging from 0.90 to 0.99 to model structures under high-velocity impacts, effectively capturing crack growth in the material. Additionally, a few studies have incorporated a fracture value of unity [2,6,26].

Based on the literature, various fracture models and values of critical fracture parameters have been used to study the fracture behaviour, ballistic resistance, and perforation resistance of metallic plates, composite material plates, and CRPF plates. However, comparative studies on fracture strain models and different values of critical fracture parameters in fracture models have been limited in the literature. This highlights the need for a detailed investigation into fracture behaviour due to the variation in critical fracture parameters and fracture strain criteria. The insight into the fracture behaviour of flat target plates with varying thicknesses, when impacted by projectiles at different initial velocities and masses, requires further investigation with critical emphasis on the fracture parameters. This study aims to address these gaps by examining the fracture behaviour of Weldox 700E steel plates, considering varying values of critical fracture parameter and fracture strain criteria across different thicknesses. The plates are subjected to blunt-nosed projectiles of varying velocities and masses.

2. Materials and Methods

2.1. Material Model

The material model formulation incorporating plasticity, plastic strain rate, and temperature is employed to describe the relationship between incremental stress and strain tensors. The formulation incorporates von Mises yield criteria, isotropic hardening effects from plastic strain, plastic strain rate, and temperature-induced softening. In the present work, the Johnson–Cook (JC) material model [27] has been used, given as

$${\sigma }_y=\left(A+B{\left({\epsilon }_{eq}^p\right)}^r\right)\left(1+N\ln {\dot{\epsilon }}_p^{*}\right)\left(1-T^{*q}\right)$$

(1)

where ${\epsilon }_{eq}^p$ is the equivalent plastic strain; ${\dot{\epsilon }}_p^{*}$ is the dimensionless strain which is derived by the ratio of current strain rate ${\dot{\epsilon }}_p$ to the reference strain rate ${\dot{\epsilon }}_0$; the homologous temperature $T^{*}$ is expressed as $T^{*}=\left(T-T_0\right)/\left(T_m-T_0\right)$ where $T$ is the actual temperature; $T_0$ is the room temperature; and $T_m$ is the melting point temperature of the material. The given material model is employed as a user-defined material model for the target material as an external FORTRAN subroutine in LS-DYNA [28].

2.2. Fracture Model

The fracture model [29] is used to incorporate the “fracture” in the target material for analyzing the fracture behaviour. The “fracture” is represented as a scalar parameter to integrate the deteriorating mechanical response under the high-velocity impact. The fracture parameter ($D$) represents the void initiation, coalescence, and growth in the material. The “Element Birth and Death” technique is used to delete the finite elements for modelling fracture. An element is deleted, and the stress value is set to zero for that element when the calculated value for $D$ reaches “unity”. Further, the fracture in the material is calculated when the value of triaxiality exceeds $(-1/3)$ [30]. Additionally, an element is deleted, if the equivalent plastic strain value exceeds a threshold of 10 [26]. The fracture model is also incorporated with the user-defined material model as an external FORTRAN subroutine in LS-DYNA, given as

$$D={\displaystyle \sum \left(\frac{d{\epsilon }_{eq}^p}{{\epsilon }_t}\right)}$$

(2)

where $d{\epsilon }_{eq}^p$ is an incremental equivalent plastic strain and ${\epsilon }_t$ is the fracture strain of ongoing state. The fracture strain is estimated using Johnson–Cook (JC) fracture criteria, given as [27]:

$${\epsilon }_t=(D_1+D_2\exp (D_3\eta ))(1+D_4\ln {\dot{\epsilon }}_p^{*})(1+D_5{T}^{*})$$

(3)

where $D_1, D_2, D_3, D_4 and D_5$ are the material constants and $\eta$ is the triaxiality which is defined as the ratio of mean stress ${\sigma }_m$ to the equivalent von Mises stress ${\sigma }_{eq}$. Mean stress is defined as the average of three principal stresses, i.e., ${\sigma }_m=\left({\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}\right)/3$ where ${\sigma }_{11},{\sigma }_{22} and {\sigma }_{33}$ are the direct Cauchy stress components. The value of the room temperature is taken as $T_0=298 \mathrm{K}$, and the melting point temperature of the material is $T_m=1800 \mathrm{K}$. The reference strain rate is taken as ${\dot{\epsilon }}_0=5\times {10}^{-4} {\mathrm{s}}^{-1}$.

2.3. Material and Fracture Model Constants

The material constants for the material and fracture model are calibrated using experimental data available in the literature [5]. The relevant relationships for the material constants for the material model and fracture model are given in Figure 1 and Figure 2, respectively. The normalized stress to the normalized material parameters is given in Figure 1 and normalized fracture strain to the normalized fracture parameters is presented in Figure 2. Normalized value is the ratio of actual value to the maximum value. The maximum values are given as 1217.3 MPa for yield stress, 1.136 for equivalent strain, 2.1296 for fracture strain, 1.90 for triaxiality, 500 °C for temperature, and 922.5 for strain rate, respectively. The values of material constants for fracture strain are calibrated using a non-linear fitting method using MATLAB R2023b (nlinfit function) by maximizing the R-squared value between the experimental data and the model results. The value of R-squared during non-linear fit in MATLAB for data calibration was 99.9% for equivalent stress vs. equivalent strain curve, 82.5% for equivalent stress vs. strain rate curve, and 97.6% for equivalent stress vs. temperature curve which is shown in Figure 1 in the form of normalized value. In the calibration of fracture parameter during the non-linear fit, the R-squared value for fracture strain vs. triaxiality curve was 98.3%, for fracture strain vs. strain rate was 63%, and for fracture strain vs. temperature was 68% which is shown in Figure 2 in the normalized form. The respective values of material constants for the material model and fracture model are listed in

Table 1. Material properties for Weldox 700E steel.

Material Constants	Notation	Value
Young’s modulus (GPa)	E	210
Poisson’s ratio	ν	0.33
Density (kg/m3)	ρ	7850
Initial Yield Stress (MPa)	A	913
Hardening Coefficient (MPa)	B	275
Hardening exponent	r	0.8215
Strain rate hardening	N	0.0161
Thermal softening constant	q	1.0875
Reference strain rate (s−1)	Ɛ0	5 × 10−4
Reference temperature (K)	T0	298
Melting temperature (K)	Tm	1800
Specific heat (J/kg-K)	cp	452
Fracture strain	D1	0.365
	D2	4.5296
	D3	−5.0397
	D4	−0.0033
	D5	1.3864

. Further, the projectile is assumed to be rigid, and the in-built bi-linear hardening material model is used to model the projectile (${\sigma }_y=A+B{\epsilon }_{eq}^p)$. The relevant properties of the projectile material are given in

Table 2. Material properties for the rigid projectile.

Material Constants	Notation	Value
Young’s modulus (GPa)	E	204
Poisson’s ratio	ν	0.33
Density (kg/m3)	ρ	7750
Initial Yield Stress (MPa)	A	1900
Hardening Coefficient (MPa)	B	15,000

.

2.4. Model Description

In this study, a numerical investigation is conducted to analyze the fracture behaviour of Weldox 700E steel plate using varying critical fracture parameter and fracture strain criteria for varying target plate thickness, varied from 2 to 6 mm. If the ratio of the thickness to the length of a plate is less than or equal to 0.1, it is termed a thin plate, whereas if the ratio is greater than 0.1, it is termed a thick plate [31]. In view of the above, a 2 mm plate is categorized as a thin plate, whereas 4 and 6 mm are categorized as thick plates. The plate is subjected to impact from a blunt-nosed cylindrical projectile at high velocity. Due to the symmetry of the model and loading conditions, the FE study has been performed on one-quarter of the model. The schematic of the quarter model is presented in Figure 3a. The relevant dimensions are as follows: radius of the target plate (R) = 21 mm, radius of the projectile (r) = 2.97 mm, and length of the projectile (L) = 29.80 mm. The mass of the projectile is taken as 6.4 g [26]. The coefficient of friction between the contact surfaces was set to 0.3. The boundary conditions for the quarter model, including the target and projectile, are defined with x-axis and y-axis symmetry. Further, the target plate is assumed to be clamped, and therefore, all degrees of freedom are fixed along the perimeter of the target plate. The projectile is free to move downward along the z-axis to strike the target plate. The FE model is prepared using LS-DYNA/explicit incorporating 8-noded hexahedral elements with one-point quadrature. The contact between the projectile and target plate was modelled using the *contact-automatic-surface-to-surface algorithm in LS-DYNA. The initial gap between the contact surfaces of the target and the projectile is controlled to optimize analysis time. The hourglass control is used using suitably adjusted parameters [28]. The element size in the impact region is also controlled to effectively capture the stress gradients and stress wave propagation [32,33,34]. The element length along the thickness direction is adjusted to account for varying plate thicknesses. The FE mesh used in the analysis is shown in Figure 3b. To analyze the fracture behaviour the representative elements at the “top”, “mid”, and “bottom” locations are identified in the target plate as shown in Figure 3.

3. Results and Discussion

Metallic plates are widely used in the construction of protective shields against fragment impacts and blast loads, making it essential to understand their fracture behaviour under high-velocity impact for applications in the construction industry. In this study, the fracture behaviour of Weldox 700E steel plates is analyzed when impacted by blunt-nosed cylindrical projectiles at high velocities. The finite element analysis is conducted using the LS-DYNA explicit solver to evaluate the fracture response of the steel plates under these conditions.

The present work may be helpful for practical protective designs in various applications. In the case of ballistic limit resistance in the field of armor and the military, this research may provide insight into plastic deformation, fracture, and spallation, enabling the selection of materials with high energy dissipation and delayed failure mechanisms. In the automobile sector, this research contributes to improved safety applications, such as crash energy absorbers, by enhancing the understanding of strain-rate-dependent deformation. The authors believe that the present work strengthens the impact of the research by connecting the findings to real-world engineering challenges in protective design while reducing the need for expensive physical impact tests.

3.1. Mesh Convergence Study

The mesh convergence study is performed to optimize the number of elements, minimize CPU time, and cost-effectiveness with acceptable results. The mesh convergence study has been performed based on the aspect ratio (ratio of largest to smallest edge length) of the element in the target plate. It is known that deformation and stresses have steep gradients around the impact zone under blunt nose-shaped projectiles, i.e., the contact region. Therefore, the mesh size of the elements in the contact region is very fine. The FE mesh for the rest of the plate is coarser with a one-way bias towards the contact region. Furthermore, the elements along the impacting face are kept square and the numbers of the elements along the thickness direction are varied to vary the aspect ratio of the elements. The analysis model with finite element mesh is presented in Figure 3b. The total number of elements and nodes with respect to different values of aspect ratio is given in

Table 3. Number of elements and number of nodes in the target plate for mesh convergence study.

Aspect Ratio	No. of Elements	No. of Nodes
Aspect ratio 1	535,800	568,465
Aspect ratio 2	267,900	291,165
Aspect ratio 3	214,320	235,705
Aspect ratio 4	133,950	152,515
Aspect ratio 5	107,160	124,785

. The results on residual velocity of the projectile with respect to aspect ratio are presented in Figure 4. The initial velocity of the projectile is taken as 275.7 m/s. It has been observed that there is variation in residual velocity at higher values of aspect ratio (greater than 2). Therefore, the subsequent analysis is performed with an aspect ratio value of 2.

3.2. Validation

The finite element model developed in Section 2 is validated before further investigation. The mathematical model is developed in such a way that the fracture model can be associated with the material model as per requirement. The mathematical model without the fracture model is applied first to obtain the deformation in the Taylor rod problem. Later, the model with material and fracture models is applied to analyze the deformation in the Target plate subjected to high-velocity impact using a blunt-shaped projectile.

3.2.1. Validation for Deformation with No Fracture in Taylor Rod

The developed mathematical model is first validated using the Taylor rod impact problem [35]. The Taylor rod, assumed to be of circular shape, is impacted at the rigid flat target. In view of the requirement, the fracture model is not applied with the material model for the present validation work. The dimensions for the problem domain and the values of material constants are taken from the literature [35]. The problem domain is similar to as shown in Figure 3 with the difference that the circular projectile (termed Taylor rod) is deformable, and the flat target is assumed to be rigid. Due to the symmetry of the problem, the quarter model is applied for investigation. The radius of the circular projectile is 3.935 mm, and the initial length is 25.4 mm. The impact velocity is 338 m/s.

Table 4. Validation of the deformed shape for Taylor rod with experimental work [35].

	Length Ratio	Radius Ratio
Experimental	0.66	2.220
Present work	0.65	2.219

shows the comparison of the present work and experimental data [35] in terms of length ratio and radius ratio. The length ratio is defined as the ratio of the final deformed length ($L_f$) to the initial undeformed length ($L_i$), whereas the radius ratio is defined as the ratio of the final radius of the Taylor rod at the impacting face ($r_f$) to the initial radius ($r_i$). It is observed from Table 4 that the result from the present work is in close confirmation to the experimental results from the literature [35]. Further, the final deformed shape of the Taylor rod is also compared with the experimental shape. The deformed shapes of the Taylor rod from experimental work and present work are placed adjacent to each other as shown in Figure 5. The black solid line represents the experimental shape [35], and the red color mesh represents the predicted shape from the present work. It is observed that the deformed shape of the Taylor rod is in close confirmation to the experimental shape [35]. There is some deviation in the results, which may be due to some approximation for material constants and friction conditions at the contact interface.

3.2.2. Validation for Deformation with Fracture in Target Plate

The validated mathematical model with the material model is applied for validation using experimental results from Xiao et al. [26]. Now, the fracture model is associated with the material model. The schematic of the problem domain is given in Figure 3. The circular projectile is assumed to be rigid, and the flat target is assumed to be deformable. The target plate thickness is 4 mm. The results on residual velocity and target deformation along the radial direction are compared. The results for residual velocity with respect to initial velocity are presented in Figure 6. It is observed that the results are in good confirmation with the experimental observations from Xiao et al. [26]. The deviation in the results may be due to numerical errors and approximation of material and fracture constants. Figure 7 presents the experimental results for the cavity after plug formation in target plates under high-velocity impact. Similar results from the present work are also presented, demonstrating the front and rear view of the target plate after plug formation under high-velocity impact. The circular plug is also shown in the present work for a better and more effective presentation of results. It is observed that the deformation in the target plate depicting the plug formation is in close confirmation with the experimental observations [26].

3.3. Investigation of Fracture Behaviour

The validated model is applied to analyze the fracture behaviour of varying target plate thickness under high-velocity impact using a blunt-shaped projectile. The study discusses the fracture behaviour of Weldox 700E steel plates with varying values of critical fracture parameters (Dc) viz. 0.6, 0.8, and 0.99. It also examines the effect of different fracture strain criteria on plates of varying target plate thicknesses viz. 2 mm, 4 mm, and 6 mm. Further, the analysis examines the effects of different projectile velocities 275.7 m/s, 300 m/s, and 400 m/s, while the projectile mass is 6.4 g and 12.8 g.

3.3.1. Fracture Behaviour with Varying Critical Fracture Parameter (Dc)

It is learned that the value of the critical fracture parameter is a crucial factor in the analysis of fracture in a material. The model is applied to evaluate the effect of this parameter. The analysis is performed with a target plate thickness of 4 mm, a projectile velocity of 400 m/s having a mass of 6.4 g, while the Dc values are varied at 0.6, 0.8, and 0.99.

(a)

Plug formation

Figure 8 shows the plug formation in the target plate with varying critical fracture parameters. It is observed that at a higher Dc value, a smoother plug is formed with larger deformation, while at a lower Dc value, a less smooth plug is formed with less deformation.

(b)

Contour plots for von Mises stress

The von Mises stress contour is presented in Figure 9 at different time steps to demonstrate the fracture progression along the thickness in the target plate. The results are presented for 4 mm target plate thickness under high-velocity impact using 400 m/s velocity and 6.4 g mass of projectile.

The blunt-shaped projectile strikes the flat target plate. The elements along the projectile edge at the impacting face experience a rapid increase in von Mises stress (around 1600 MPa) at around 1 μs. The stresses also propagate across the thickness under impact. The high values of von Mises stress affect the values of other material and fracture parameters which are discussed in the subsequent sections. The fracture starts from the top face of the target plate along the periphery of the blunt projectile at around 2 μs. The fracture progresses to mid elements and the bottom elements experience stretching at around 6 μs. It is observed that at around 8 μs, the complete plug is formed in the flat target plate of 4 mm thickness. The von Mises stresses are relieved in the plug as elements are separated from the target (around 1350 MPa). The projectile has completely penetrated the target plate across the thickness at around 16 μs, ejecting the circular plug from the flat target plate.

(c)

Effect on fracture parameters

Figure 10 illustrates that fracture in the “top” elements occurs earliest with a Dc value of 0.6, followed by 0.8 and 0.99. This is due to the rapid increase in triaxiality, which leads to an increase in equivalent plastic strain (Figure 11) and a decrease in load-carrying capacity, as shown in Figure 12. The trend for triaxiality is presented in Figure 13. Therefore, the elements experience the maximum value of fracture parameter and are deleted without appreciable elongation. It is also observed that the temperature does not affect the top element (Figure 14); thus, failure occurs primarily due to an increase in equivalent plastic strain (Equation (2)).

The “mid” elements experience significant elongation before failure due to the considerable effect of plastic strain rate (Figure 11) and temperature (Figure 14). Figure 12 shows that von Mises stress increases as a result of strain hardening in the element. It is observed that the triaxiality value remains either zero or negative with a Dc value of 0.6, whereas for Dc values of 0.8 and 0.99, its value is positive. As a result, the mid element fails due to shearing (with equivalent plastic strain exceeding 12) in the case of Dc = 0.6, while a failure at Dc = 0.8 and 0.99 occurs due to tensile effects.

For the “bottom” elements, triaxiality remains constant until 7 μs (Figure 13), resulting in a stable equivalent strain (Figure 11) and consistent elongation at a constant von Mises stress value during this period (Figure 12). However, the effect of temperature is minimal compared to the mid element (Figure 14). Lower Dc values experience higher temperatures than larger Dc values, leading to greater elongation and delayed failure with lower Dc values compared to higher Dc values. Overall, plug formation in the target plate occurs earlier with a higher Dc value (0.99) than with lower Dc values (0.8 and 0.6) as shown in Figure 10.

It is observed that fracture progression across the thickness of the target plate is better expressed using contour plots of von Mises stress at different time instants. In view of the objective of the present work, fracture behaviour, and progression are completely addressed by observing the trends for the fracture parameters. Therefore, in the subsequent investigation, the trends for von Mises stress at different time instants are not presented, and fracture behaviour is addressed using effects on fracture parameters.

3.3.2. Fracture Behaviour with Varying Fracture Strain Criteria

Now, the study is performed with varying fracture strain criteria for the analysis of fracture behaviour. These include the Johnson–Cook (JC) fracture strain criteria termed “ef-JC” (Equation (3)) [27]. Dey et al. [6] replaced the logarithmic strain rate function in the JC criteria with a power-law function. This modified fracture criterion overcomes the infinite value in the logarithmic function. This modification is termed “ef-DJC”, and the equation for fracture strain criterion reads as follows:

$${\epsilon }_t=\left(D_1+D_2\exp (D_3\eta )\right){\left(1+{\dot{\epsilon }}_p^{*}\right)}^{D_4}\left(1+D_5{T}^{*}\right)$$

(4)

Xiao et al. [26] used the modified JC fracture strain criteria, which alter the thermal softening function by introducing a multiplication factor involving two constants, with one raised to power-law exponent during high-velocity impact. This modified fracture criterion is termed “ef-XJC”, and the equation for fracture strain criterion reads as follows:

$${\epsilon }_t=\left(D_1+D_2\exp \left(D_3\eta \right)\right)\left(1+D_4\ln {\dot{\epsilon }}_p^{*}\right)\left(1+D_5{T}^{*D_6}\right)$$

(5)

The value of material constants for different fracture strain criteria are determined using the methodology discussed in Section 2.3 and are presented in

Table 5. Material constants for varying fracture strain criteria.

Fracture Strain Criterion	D1	D2	D3	D4	D5	D6
ef-JC	0.365	4.5296	−5.0397	−0.0033	1.3864	---
ef-DJC	0.365	4.5296	−5.0397	−0.0022	1.3864	---
ef-XJC	0.365	4.5296	−5.0397	−0.0033	7.5177	2.1484

. Other process and material parameters are kept the same with target thickness at 4 mm, projectile velocity of 400 m/s, and a mass of 6.4 g, while different fracture strain models are considered.

(a)

Plug formation

Figure 15 shows the plug formation using different fracture strain criteria. The ef-JC fracture criteria exhibit a smoother cracked surface compared to the other fracture criteria. A slightly rough cracked surface at the mid-plane is observed in the ef-DJC criteria due to the effect of the plastic strain rate, while with ef-XJC criteria, it is attributed to the effect of temperature.

(b)

Effect on fracture parameters

Figure 16 shows that fracture in the “top” element occurs first with the ef-DJC criteria, followed by the ef-XJC criteria, and lastly with the ef-JC criteria. This is due to the rapid increase in triaxiality, which leads to a corresponding increase in equivalent plastic strain (Figure 17). As a result, the load-carrying capacity of the element decreases, leading to failure, as illustrated in Figure 18. The trend for triaxiality is presented in Figure 19. However, the effect of temperature is not a significant factor for the top element (Figure 20).

For the “mid” element, plastic strain rate (Figure 17) and temperature (Figure 20) play a significant role, leading to greater elongation with higher values of von Mises stress (Figure 18) before fracture occurs. Due to thermal softening, the mid element carries a lower load using ef-XJC criteria. In contrast, higher strain hardening allows it to sustain a higher load with ef-DJC criteria, while the material using ef-JC criteria carries less load compared to both ef-XJC and ef-DJC fracture criteria. Figure 20 shows that the increase in temperature using ef-DJC fracture criteria is greater than with ef-XJC and ef-JC criteria. As a result, the effect of triaxiality is not prominent during this period with its value fluctuating around zero (Figure 19). The increase in temperature leads to an increase in equivalent plastic strain, allowing the element to experience a higher load before failure due to strain hardening. The triaxiality value increases more rapidly using ef-JC fracture criteria than with ef-XJC and ef-DJC criteria after 6 μs (Figure 20). Consequently, failure occurs first with the ef-JC criteria, followed by ef-XJC, and ef-DJC fracture strain criteria, respectively.

For the “bottom” element, plastic strain is the primary factor contributing to failure, as temperature effects are less significant compared to the mid element, as shown in Figure 20. Up to 7 μs, both triaxiality and equivalent plastic strain remain constant (Figure 17 and Figure 19), and during this same period, the von Mises stress also maintains a steady curve as shown in Figure 18. However, after 7 μs, the triaxiality value increases more rapidly using ef-JC fracture criteria compared to ef-DJC and ef-JC fracture criteria. As a result, there is an accumulation of value for fracture parameter, leading to failure occurring first using the ef-JC model, followed by the ef-XJC model, and ef-DJC fracture strain criteria, respectively.

Overall, it can be observed that plug formation in the target plate occurs earlier when using the ef-JC criterion than with the ef-XJC and ef-DJC fracture strain criteria, respectively.

3.3.3. Fracture Behaviour with Varying Target Plate Thickness

This FE model is applied to analyze the fracture behaviour with varying target plate thickness of Weldox 700E steel ranging from 2 mm to 6 mm. The projectile velocity of 400 m/s with a mass of 6.4 g is used, while the other process parameters remain the same. The material and fracture constants are given in Section 2.3.

(a)

Plug formation

Figure 21 shows the plug formation in the target plate with varying plate thickness. It is observed that the greater thickness experienced greater deformation than the lower thickness.

(b)

Effect on fracture parameters

The “top” element experiences higher values of fracture parameter, representing earlier fracture for a 6 mm plate followed by 4 mm and 2 mm plate thickness, respectively, as shown in Figure 22. Figure 23 indicates that the increase in equivalent plastic strain is greater for larger plate thickness. A similar trend is also observed for triaxiality (Figure 24). Therefore, at initial impact, the element in the plate with the lowest thickness experienced greater elongation compared to the plate with the highest thickness. The load-carrying capacity of the larger plate thickness decreases more than that of the lower plate thickness, as shown in Figure 25. However, no significant effect of temperature was observed on the top elements (Figure 26). Therefore, one can deduce that the top elements are primarily deleted due to the prominent effect of equivalent plastic strain.

The “mid” element was deleted earlier for a 2 mm thick plate compared to the 4 mm and 6 mm thick plates (Figure 22). The triaxiality value remains near zero or approaches negative up to 0.9 μs for the 2 mm thick plate, 6.5 μs for the 4 mm plate, and up to 23 μs for the 6 mm plate, as shown in Figure 24. During this time, the von Mises stress remains almost constant (Figure 25), and fracture occurs due to a nominal increase in triaxiality value. Therefore, one can say that the element is deleted due to the shearing effect. Figure 26 shows that the temperature for the 2 mm plate is approximately 311 K, which is close to the reference temperature (298 K), whereas the temperature reaches very high values for 4 mm and 6 mm plates. Hence, for the “mid” element, the prominent factors for fracture are plastic strain rate and temperature in the case of the 4 mm and 6 mm plates, while for the 2 mm plate, it is primarily plastic strain rate.

Figure 23 and Figure 24 show that, for the “bottom” element, the equivalent plastic strain and triaxiality remain almost constant up to 23 μs for the 6 mm plate, whereas it is up to 2.2 μs for 2 mm plate thickness, and for the 4 mm plate, it is up to 6.7 μs. During this period, the von Mises stress also remains almost constant, as shown in Figure 25. Further, there is no significant effect of temperature for lower-thickness plates, but it becomes a prominent factor as the plate thickness increases.

Overall, it can be observed that fracture occurs earliest in the 2 mm plate, followed by the 4 mm and 6 mm plate, respectively. Further, it is found that plastic strain rate and temperature are the prominent factors for thick plates, whereas for thin plates, only the plastic strain rate is the primary factor influencing element deletion.

3.3.4. Fracture Behaviour with Varying Projectile Velocity

This FE model is applied to analyze the fracture behaviour with varying projectile velocity. The target plate thickness is 4 mm, and the projectile mass is kept at 6.4 g. The investigation is performed with varying projectile velocity viz. 275.7 m/s, 300 m/s, and 400 m/s, while the other process parameters remain the same. The material and fracture constants are given in Section 2.3.

(a)

Plug formation

Figure 27 shows the plug formation in a 4 mm thick plate with varying projectile velocities. It is observed that with lower projectile velocity, the plug formation is smoother with higher deformation, whereas with higher velocity, the plug becomes less smooth with reduced deformation in the target plate. This may be due to the fact that higher velocities result in greater kinetic energy, which alters the fracture behaviour.

(b)

Effect on fracture parameters

The “top” element experiences early fracture at higher projectile velocities (Figure 28) due to higher values of equivalent plastic strain (Figure 29). Figure 30 shows that the von Mises stress increases during the initial impact but decreases rapidly at higher velocities due to a rapid increase in equivalent plastic strain and triaxiality. The triaxiality also follows a similar trend to the equivalent plastic strain (Figure 31). Figure 32 demonstrates that the temperature increase is more significant at lower projectile velocities, reaching 300 K at 400 m/s, 353 K at 300 m/s, and 370 K at 275.7 m/s. This indicates that temperature has no significant effect on the top element at higher projectile velocities but has a more pronounced impact at lower velocities. The load-carrying capacity decreases at higher velocities, leading to earlier element deletion (Figure 28).

For the “mid” element, the trend of element deletion is similar to that discussed for the top element, as shown in Figure 28, with the only difference being the time taken for deletion. Figure 29 shows that the equivalent plastic strain remains nearly constant, resulting in elongation of elements. The near-zero value of triaxiality shows that the elements experience fracture due to shear failure. The effect of temperature becomes a more prominent factor for the mid element as projectile velocities decrease, as shown in Figure 32. The increase in temperature is recorded in the range of 650–710 K, with lower temperature values at higher projectile velocity (Figure 32).

The trend for fracture along the “bottom” element is similar to that of the top and mid elements, as shown in Figure 28. Figure 27 shows that the equivalent plastic strain remains constant for approximately 11 μs at 275.7 m/s, 9 μs at 300 m/s, and 6 μs at 400 m/s, with a similar trend observed for triaxiality (Figure 29). During this period, fracture remains constant (Figure 26), and the von Mises stress maintains a constant profile, as shown in Figure 28. Figure 30 demonstrates similar temperature trends to those discussed for the top and mid elements. The increase in temperature is recorded in the range of 370–450 K.

Overall, the order of plug formation is top-bottom-mid at 275.7 m/s and top-mid-bottom at 300 m/s and 400 m/s. The plug formation occurs from higher to lower velocities, specifically first at 400 m/s, then at 300 m/s, and lastly at 275.7 m/s. The effect of temperature is much more noticeable for the mid element than for the bottom and top elements.

3.3.5. Fracture Behaviour with Varying Projectile Mass

Now, the model is applied to analyze the fracture behaviour with varying projectile mass. The target plate thickness is 4 mm, and projectile velocity is kept at 400 m/s. The investigation is performed with varying projectile mass viz. 6.4 g and 12.8 g, while the other process parameters remain the same. The material and fracture constants are given in Section 2.3.

(a)

Plug formation

Figure 33 shows the plug formation in the target plate with varying projectile mass. The plug formation is smoother in the case of the lower mass projectile compared to the higher mass projectile. The kinetic energy is greater with higher projectile mass, and therefore, the fracture behaviour is altered.

(b)

Effect on fracture parameters

The “top” element fails earlier (Figure 34) using higher projectile mass due to a rapid increase in equivalent plastic strain (Figure 35). Triaxiality also increases (Figure 36), leading to a decrease in the load-carrying capacity of the element, as shown in Figure 37. However, no significant effect of temperature is observed, as shown in Figure 38.

For the “mid” element, failure occurs earlier using higher projectile mass (Figure 34) due to a rapid increase in equivalent plastic strain (Figure 35). The von Mises effect has no significant effect on fracture (Figure 37), whereas the triaxiality values are almost zero while using higher projectile mass (Figure 36), indicating a shearing type of failure. In contrast, failure occurs due to the tensile effect, with triaxiality approaching a positive value, with lower projectile mass, as shown in Figure 36. Figure 38 shows that the increase in temperature is greater for the higher projectile mass than for the lower projectile mass due to the higher kinetic energy. In the mid element, due to the effects of plastic strain rate and temperature, the element withstands more load and elongation before failure occurs.

For the “bottom” element, the equivalent plastic strain (Figure 35) and triaxiality (Figure 36) remain constant until 6 μs. Due to the increase in temperature (Figure 38), the element elongates under constant loading during this period, as shown in Figure 37. However, due to the rapid increase in triaxiality while using a higher projectile mass, the element fails earlier than with the lower projectile mass, as shown in Figure 34.

Overall, fracture in the target plate occurs earlier with the higher projectile mass than with the lower projectile mass.

4. Conclusions

The fracture behaviour of Weldox 700E steel plates is investigated using the LS-DYNA/explicit. The study is conducted on flat target plates using varying critical fracture parameters and fracture strain criteria impacted by blunt-nosed cylindrical projectiles. This work also analyzes the fracture behaviour of the target with varying plate thicknesses (2–6 mm) impacted using varying projectile masses (6.4 g and 12.8 g) and impact velocities of 257.5 m/s, 300 m/s, and 400 m/s.

It is concluded that a higher value of the critical fracture parameter results in a smoother plug formation, whereas a lower value of the critical fracture parameter leads to a less smooth plug. Regarding the fracture strain criteria, the plug formed using ef-JC fracture strain criteria is smoother compared to those with ef-DJC and ef-XJC fracture strain criteria. For varying plate thicknesses, the 6 mm plate exhibits more deformation than the 2 mm and 4 mm plates. In the case of varying velocity, a smoother plug is observed at lower projectile velocities (257.5 m/s and 300 m/s). Regarding the effect of projectile mass, a smoother plug with higher deformation is obtained using lower projectile mass. However, no plug formation is observed at 257.5 m/s and 300 m/s for the 6 mm thick plate with a lower projectile mass (6.4 g), whereas a plug is formed at 400 m/s. Plug formation occurs for all thicknesses (2–6 mm) when impacted by a higher projectile mass (12.8 g).

The effect of temperature and equivalent plastic strain is more noticeable for the mid element compared to the top and bottom elements. Overall, the formation of the plug initiates at the top, then progresses to the bottom, and finally to the mid element in the case of a velocity of 275.7 m/s. For higher projectile velocities, the trend follows the order of top-mid-bottom. Further, it is observed that for higher-thickness plates, both plastic strain rate and temperature play significant roles, whereas for lower-thickness plates, the plastic strain rate primarily influences element deletion.