**1. Introduction**

There are several methods of increasing the ductility of metals, such as superimposing hydrostatic pressure [1,2]. In mechanical testing under superimposed hydrostatic pressure, tensile testing of the specimen is carried out in a pressure vessel that applies the desired level of pressure in the load assembly [3]. The effect of superimposed hydrostatic pressure has been studied numerically using the conventional Gurson–Tvergaard–Needleman (GTN) model under tension and bending in previous studies [1–4]. However, in this study, the modified GTN model considering the shear damage growth as an increment in the void volume fraction is used to investigate the effect of superimposed hydrostatic pressure on the shear damage mechanism.

**Citation:** Shahzamanian, M.; Thomsen, C.; Partovi, A.; Xu, Z.; Wu, P. Numerical Study about the Influence of Superimposed Hydrostatic Pressure on Shear Damage Mechanism in Sheet Metals. *Metals* **2021**, *11*, 1193. https://doi.org/ 10.3390/met11081193

Academic Editors: Antonio Mateo, Giovanni Meneghetti and Filippo Berto

Received: 6 June 2021 Accepted: 23 July 2021 Published: 27 July 2021

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The different failure modes shown in Figure 1 can be described in terms of the ratio of shear and normal stresses. Generally, shear failure is dominant for low or zero stress triaxiality, while the process of void nucleation, growth, and coalescence occurs when stress triaxiality is high [5]. Ashby et al. [6] investigated the influence of pressure and temperature on damage in terms of the of brittle, fully plastic, ductile, and shear fracture mechanisms. It was shown that the failure mode is a function of pressure and temperature, with an increase in pressure corresponding to an increase in ductility. It was also demonstrated that a material only fails in a fully plastic manner when all other fracture mechanisms are suppressed. This failure mode is characterized by the onset of necking that progresses to a point of zero area when a material is continuously loaded in tension past its yield point. Kao et. Al [7] used quantitative metallography to determine the effect of hydrostatic pressure on the failure mode of a steel subjected to tensile deformation. It was observed that a superimposed hydrostatic pressure suppressed the nucleation of voids and resulted in a significant increase in ductility. Unlike the void-sheet mechanism, shear decohesion is not strongly influenced by pressure; this causes the latter to be the only valid mechanism to explain the observed failure [7]. Overall, it is generally accepted that a superimposed hydrostatic pressure increases ductility by delaying or completely eliminating void nucleation and growth; this matter has been investigated in other studies [6,8–12].

For many high-strength sheet materials, such as aluminum alloys that contain a significant amount of second phase particles, microvoids often develop in the vicinity of these particles during large plastic deformation. These particle-induced microvoids are known to localize plastic flow and limit the formability of sheet metals [13–16]. One of the well-known models of ductile void growth that is often utilized in analyzing large plastic deformation of ductile metallic materials is the Gurson–Tvergaard–Needleman (GTN) model, proposed by Tvergaard and Needleman [17] as an improvement on the accuracy of the original Gurson model [18]. These models treat voids as spherical cavities and capture their effects on material yield following a modification of the von Mises yield criterion [18]. More recently, the GTN model has been extended to include the effect of shear damage by Nahshon and Hutchinson [19]. Sun et al. [20] used the shear modified GTN model and simulated punch test and identified the parameters using the neural networking. The size effect on damage evolution using the shear modified GTN model under high/low stress triaxiality is performed by Li et al. [21]. Yildiz and Yilmaz [22] used the shear modified GTN model to simulate the plastic deformation for 6061 aluminum alloys. Overall, the shear modified GTN model has been used frequently to simulate various materials for different tests [23–27].

Peng et al. [4] investigated the effect of superimposed hydrostatic pressure on fracture in round bars. It was shown that, because void formation is not significant prior to necking, superimposed pressure has little or no effect on the yield strength of metals.

However, the numerical results showed that due to a suppression in void nucleation and growth by the applied pressure, the fracture strain increased, and the failure process was extended. The effect of superimposed pressure on fracture in sheet metals under tension was studied in [3], where it was again found that the application of hydrostatic pressure increased the ductility in sheet metal. Numerical results showed the transition of fracture surface from planar mode at atmospheric pressure to chisel mode under high pressure as observed experimentally.

The effect of superimposed hydrostatic pressure on the bendability of sheet metals using the GTN model in ABAQUS is investigated in [1]. This study explored how hydrostatic pressure suppresses void growth and leads to an increase in ductility in sheet metals. The pressure and stress triaxiality were shown to decrease with an increase in superimposed hydrostatic pressure. As already mentioned, the void growth decreases, and it causes the fracture strain to increase. In another study [28], the effect of cladding on the ductility of sheet metals was investigated using the GTN model. A softer material with a higher ductility than the substrate metal was applied with perfect bonding. It was demonstrated that the application of the soft ductile layer improved the bendability of the base metal. From these two studies [1,28], it is clear that combining finite element methods (FEM) with the GTN model is a useful and successful approach to perform a range of analyses and to understand various effects on the ductility of metals in three-point bending tests.

The shear damage mechanism is a dominant mode of fracture under pressure as void growth is delayed or completely eliminated. To the best of the authors' knowledge, the effect of superimposed hydrostatic pressure on the shear damage mechanism has not been reported elsewhere. The aim of this paper was to perform a numerical study of the effect of a superimposed hydrostatic pressure on shear fracture in sheet metal under tension. The effect of superimposed pressure is explained in detail and in a step-wise manner, and it is shown what happens when the shear damage mechanism becomes a dominant mode of fracture with increasing pressure. All the simulations presented in this study were performed using ABAQUS/Explicit [29] based on the modified GTN model implemented in a VUMAT subroutine. The effect of hydrostatic pressure on the change in failure mode is explained in detail. The numerical results were found to be in good agreement with experimental observations considering the mixed dimple/shear mode of fractures in a sheet metal. The void growth and shear void growth volume fractions are considered individually in the shear modified GTN model. Therefore, the effect of pressure on void growth and shear void growth volume fractions are studied and compared with each other.
