**3. Problem Formulation and Method of Solution**

A sheet metal with length *Lo*, thickness *to*, and width *W<sup>o</sup>* that is under hydrostatic pressure is considered and shown schematically in Figure 2. It is assumed that the sheet is wide enough and that no deformation occurs in the width direction, such that the sheet may be considered to be under plane strain. The shear modified GTN model is not supported in the ABAQUS material behavior library and a VUMAT subroutine was implemented in this study to investigate the effect of pressure on shear damage mechanism. However, the subroutine only supports the three-dimensional elements. The superimposed hydrostatic pressure is represented by small brown arrows directed into the material from all directions. The sequence of tensile strain under superimposed hydrostatic pressure is modeled as two steps. In the first step, the pressure is gradually increased up to a desired level *p* = −*ασ<sup>y</sup>* (*α* defines the value of applied pressure respect with yield stress) without applying any tensile strain. In the second step, tensile strain is applied to the sheet while maintaining the constant pressure value *p* = −*ασy*.

= −<sup>௬</sup>

**Figure 2.** Schematic of a sheet metal under superimposed hydrostatic pressure.

௬⁄ = 0.0033 = 0.3 = 10 <sup>ଵ</sup> = 1.5 <sup>ଶ</sup> = 1.0 <sup>ଵ</sup> <sup>ଶ</sup> ே = 0.04 ே = 0.3 ே = 0.1 0.15 0.25 ) The elastic–plastic properties of the matrix material are specified by *σy*/*E* = 0.0033, *ν* = 0.3 and *n* = 10. It is assumed that the initial void volume fraction is zero and the fit parameters in the GTN model (Equation (7)) are *q*<sup>1</sup> = 1.5 and *q*<sup>2</sup> = 1.0. These values for *q*<sup>1</sup> and *q*<sup>2</sup> were found to be in good agreement in [31] for metals to analyze the bifurcation mode of porous metals. Void nucleation is assumed to be plastic strain controlled, the volume fraction of void nucleating particles *f<sup>N</sup>* = 0.04, the mean strain for void nucleation *ε<sup>N</sup>* = 0.3, and the corresponding standard deviation *S<sup>N</sup>* = 0.1. The parameters related to the final failure, *fc* and *f f* , are assumed to be 0.15 and 0.25, respectively. These values of mechanical properties are taken from Tvergaard and Needleman [17]. It should be emphasized that the main purpose of the present study is to assess the effect of superimposed hydrostatic pressure on the ductility of sheet metals and particularly on the shear damage mechanism, and that the overall results and conclusions are not particularly dependent on the above values of the material parameters. The three-dimensional element C3D8R in ABAQUS/Explicit is used for the sheet. The mass scaling method with a sufficient low target time increment is used and it is carefully attempted to minimize the dynamic effect of the sample. Therefore, a wide sheet with a width (*Wo*) of 100 mm is considered when the length (*Lo*) and thickness (*to*) are 60 mm and 10 mm, respectively. It is to be noted that all nodes in the sheet are constrained in the width direction.

= −<sup>௬</sup>

As the mesh sensitivity is expected in numerical simulations involving localized deformation and fracture, different meshes are considered in this simulation. Figure 3 shows the finite element (FE) configuration of the specimen with a typical mesh for metal sheet consisting of 60 × 110 × 4 elements (60 elements in thickness direction, 110 elements in length direction and 4 elements in width direction) in which the element distribution in the refined area is biased to the middle section of the specimen where fracture is expected to occur. Due to the symmetry, only half of the sheet is investigated and symmetric boundary conditions are imposed in the middle section of specimen. Figure 4 represents the normalized force (*F* ∗ ) as a function of the tensile strain *ε* for fully base material and the effect of mesh sensitivity on this curve is also included. Force is normalized by the

multiplication of the yield stress of the material and the initial cross section of the sheet, and *ε* = ln 1.0 + <sup>∆</sup>*<sup>l</sup> lo* . = ln (1.0 + ௱ ) = ln (1.0 + ௱ )

)

)

 ∗

 ∗

**Figure 4.** Effect of mesh sensitivity on force–tensile strain.
