1. Introduction
An impact is defined as a mechanical process that involves the collision of two or more bodies. The relevant engineering has a wide range of applications, such as the safety assessment of buildings and nuclear reactor vessels, the assessment of the crashworthiness of vehicles, the protection of cargo and barriers, and the design of military vehicles and armor systems. As opposed to static or conventional dynamic loading, forces created by collisions are exerted and removed in an extremely short time duration. Penetration is described as the entrance of an object into a target body without passing through the body, resulting in the embedment of the striker and the formation of a crater, whereas perforation is defined as the complete escape of the target after impact.
Numerous experimental and numerical studies on impact phenomenon have been conducted by many researchers. Backman and Goldsmith studied a comprehensive survey of the mechanics of penetration of projectiles into targets [
1]. An empirical formula for determining projectile penetration into steel barriers is proposed and a method for determining the ballistic limit for penetrating a target from penetration depth is presented [
2]. Johnson et al. conducted research on the quasi-static piercing of metal plates [
3,
4]. Corbett et al. researched the penetration and perforation of plates and cylinders by free-flying projectiles traveling at sub-ordnance velocities [
5]. A numerical analysis of the ballistic perforation of an impactor through steel plates was performed by Lee et al. using the peridynamic method [
6]. Garcia et al. investigated the impact behavior of polymer composites [
7] and Garcia and Trendafilova proposed an approach to predict the damage caused by impact loads in composite structures [
8]. In addition to ballistic impacts, research on aircraft impact that can cause significant damage has also been actively conducted since the 9/11 terrorist attacks. Lee et al. conducted an evaluation of the collision of commercial aircraft [
9], as well as the military aircraft impact [
10].
The dynamic behavior of materials is different from the static behavior because the stiffness and inertia of the material affects the mechanical behavior. In particular, the yield stress and hardening phenomenon depend on the strain rate. To implement the effect of the strain rate in numerical material models, various studies have been carried out. One of the techniques to apply the strain rate effect to the material model is controlling the properties of the plastic region of the material. To apply the strain rate effect to the material model, the technique of scaling the plastic range by the strain rate is employed [
11,
12]. The Johnson-Cook material model introduces the dimensionless parameter of the strain rate to determine the plastic behavior of materials and the behavior of the plastic region is determined by the strain rate [
11]. For performing the impact or blast failure where large deformation occurs during a short period, it is essential to describe the material failure. The material model is used together with the erosion algorithm for describing material failure [
13], and the failure strain is used as a representative failure criterion. It is essential to select the appropriate value of failure strain in order to perform numerical analyses accurately. However, various failure strain values have been applied in other studies [
14,
15], which means that the failure strain values vary depending on the researcher or numerical model. Therefore, the failure strain is not fixed to a specific value but has various values depending on the analysis conditions, such as the strain rate and the element size. The failure strain is known to be dependent on the element size of the finite element model [
15,
16] and small changes in the failure strain cause a substantial effect on the numerical result [
17]. The effect of element size on failure criterion has been investigated in various fields. Alsos et al. investigated the influence of the element size on the failure strain for finite element analysis about resistance to the penetration of stiffened plates [
18]. Gang and Kwak studied the failure criterion that can minimize mesh dependency of blast simulation on reinforced concrete [
19]. Villavicencio et al. confirmed the dependence of the failure strain on the element size for impact simulation on circular aluminum plates [
20]. Grag and Abolmaali performed numerical analyses by varying the failure criterion according to the element size in the analysis of the reinforced concrete box culverts [
21] and Ehlers applied element length-dependent failure strain in fracture of the thin circular plate [
22]. Raimondo et al. proposed a mesh size independent failure model in the impact analysis of composite laminates [
23].
The 7000 series of aluminum alloys, which are investigated in this study have various advantages, such as being lightweight, having high strength, and an excellent machinability, and are used in various fields, such as the commercial, military, and aviation sectors [
24,
25]. 7075 aluminum alloy is the highest strength alloy of aluminum, with high fracture toughness and low fatigue crack growth. The alloy is durable with a strength comparable to steel and has an excellent fatigue strength and machinability. However, 7075 aluminum alloy has a lower resistance to corrosion than many other aluminum alloys. The first 7075 was developed in secret by a Japanese company, Sumitomo Metal, in 1943 [
26]. 7075 was eventually used for airframe production in the Imperial Japanese Navy.
In this study, we propose a damage criterion that minimizes mesh dependency and enables efficient impact simulations. We implement the Johnson-Cook constitutive relationship in the user-defined material model in LS-DYNA and applied the criterion in the subroutine of the material model. Numerical simulations are conducted to observe the effects of element size and failure strain on the perforation response depending on the strain rate. Experimental studies on high-velocity impact against the 7075-T651 aluminum target were carried out. In high-speed impact tests, projectiles were propelled at different initial velocities onto the aluminum plates of two different thicknesses. By comparing the residual velocity after perforation with the test results, the element size and failure strain yielding the same residual velocities with experiment results are inversely calculated. Consequently, the relationship between the element size and failure strain is applied to the damage model, and we verify the efficiency and accuracy of the damage model.
6. Summary and Conclusions
In the finite element analysis, the accuracy of the numerical solution depends on the element size, and significant computational cost is essential to attain sufficient accuracy. For efficient and accurate impact simulation, we propose an enhanced damage criterion. This criterion can alleviate computational costs in impact simulations while providing more accurate results than existing damage criteria, and we verify the criterion by comparing the numerical results with the impact tests. Also, the damage criterion is combined with the Johnson-Cook constitutive relationship and implemented in UMAT of LS-DYNA for the usability of the damage criterion. Using a gas-gun system, impact tests were carried out to investigate the impact response of the 7075-T651 aluminum plate, and the test results are used as the reference data to verify the numerical model and damage criterion. Numerical models are constructed using various element sizes in order to evaluate the effect of the element size on the impact simulation. The correlations among the failure strain, the impact velocity, and the element size are inversely obtained from numerical simulations by comparing the residual velocities with the test results. It is found that the failure strain varies inversely with the element size and impact velocity. In particular, the sensitivity of the failure strain to the impact velocity significantly increases as the element size decreases and, therefore, the failure strain should be carefully determined when a small element size is used. By applying the characteristic of the failure strain, which depends on the impact velocity and element size, we have introduced an element-size dependent failure strain (EDFS) and the results show good agreement with experimental results regardless of the element size. To import EDFS in the subroutine of the material model, the Johnson-Cook constitutive relationship is implemented in UMAT and the numerical results using UMAT are in good agreement with the existing Johnson-Cook material model. Then, EDFS is imported in UMAT to implement a material model that combines EDFS and Johnson-Cook configuration relationships. When using the Johnson-Cook damage model, the accurate solution is obtained if we use a sufficiently small element size, but the computational cost exponentially increases accordingly. The application of the EDFS allows the calculation time to be reduced significantly because the numerical results are in good agreement with the results of experiments even if the large element size is used. Using the damage criterion presented in this study, efficient simulations can be carried out, ending up with a high accuracy as obtained without very fine discretization.