On the cut-off value of negative triaxiality for fracture

Abstract

Although yield condition of metals does not strongly depend on it, mean stress plays an important role in fracture of metals. A cut-off value of the stress triaxiality equal to −1/3, below which fracture never occurs, was derived analytically from the fracture locus in the principal strain space experimentally reported from upsetting tests. It was found that this result is consistent with tensile tests under hydrostatic pressure (Bridgman tests). Numerical simulations performed in this study with the cut-off value in fracture loci successfully captured the main features observed in tensile tests under hydrostatic pressure by Bridgman and others, and simulated the tests on 1045 steel performed by Kao et al.

Introduction

For isotropic materials, plastic yielding depends only on the magnitudes of three principal stresses or three invariants of the stress tensor. To a first approximation, the yielding of a metal is assumed by the community to be independent of the first invariant, based on the experimental fact that is unaffected or weakly affected by hydrostatic pressure. The well-known J₂ plasticity suggested by von Mises is among those simplest criteria.

However, there is overwhelming evidence that ductile crack formation strongly depends on the stress triaxiality. The stress triaxiality parameter is defined as ${\sigma }_{\mathrm{m}}/\overline{\sigma }$, where σ_m is the mean stress or hydrostatic stress. McClintock [1] has shown that fracture of ductile metals are strongly dependent of hydrostatic stress by studying growth of long cylindrical voids. Another important study on void growth is the one by Rice and Tracey [2]. They found by studying the growth of a spherical void in a general remote field that for any remote strain rate field, the enlargement rate of spherical voids was amplified over the remote strain rate by a factor of an exponential function of the stress triaxiality. Atkins [3], [4] also pointed out that ductile fracture should depend on hydrostatic stress.

Experimental studies for high positive stress triaxialities were mainly conducted on pre-notched bars. For example, Hancock and Mackenzie [5] carried out a series of tensile tests on pre-notched steel specimens. Recently, Hopperstad et al. [6] and Borvik et al. [7] obtained a relationship of stress triaxiality and fracture strain of Weldox 460 E steel. It was found that ductility depended markedly on the triaxiality of stress states. In their study, stress triaxiality was calculated using the Bridgman’s [8] formula based on initial geometry and the strain was assumed constant across the minimum cross-section. Mirza et al. [9] performed an experimental and numerical study on three different materials (pure iron, mild steel and aluminum alloy BS1474) over a wide range of strain-rates (10⁻³–10⁴ s⁻¹). Equivalent strain to fracture ${\overline{\epsilon }}_{\mathrm{f}}$ for all the three materials was found to be strongly dependent of the level of stress triaxiality. The dependence was different for different materials.

Round bars under tensile loading with superposed hydrostatic pressures experience negative stress triaxialites. The most comprehensive experimental program in the above test configuration was performed by Bridgman and described in his famous book [10]. A follow up was described in a series of papers by French and Weinrich [11], [12], [13], [14] on copper, aluminum and brass. The natural strain to fracture calculated was found to increase with the hydrostatic pressure. In French and Weinrich’s work on ductile copper [11] with 223 MPa ultimate tensile strength, classical void nucleation, growth, and linkage were found in the fracture surface from the SEM examination in the case with no pressure. As the pressure increased, the relative area of the rough portion due to void nucleation, growth and linkage decreased. In more recent studies performed by Margevicius and Lewandowski [15], Liu and Lewandowski [16], [17] and Kao et al. [18], similar phenomena were observed.

Axial compression of short cylinders (so-called upsetting tests) also provides clues of ductile crack formation in the range of negative stress triaxiality. The barreling of the cylindrical surface furnishes considerable flexibility because barrel severity changes by altering the die contact friction conditions and cylinder aspect ratios. This leads to a variation of the tensile hoop stress at the bulge surface. Ductile fracture in upsetting tests was studied experimentally first by Kudo and Aoi [19]. A fracture locus in the space of principal strains was found. The results were later confirmed and extended by a number of others (e.g. Kuhn and Dieter [20], Thomason [21], [22], Ganser et al. [23]).

In upsetting tests, fracture would never occur if there were no friction between the die platens and specimens. In tensile tests on round bars under different hydrostatic pressures, fracture will also not occur if the pressure is sufficiently high. In fact, in about half of the over 350 tests carried out by Bridgman [10], fracture was not observed. French and Weinrich [11] pointed out that the natural strain to fracture at pressures in excess of 300 MPa approached infinity for a ductile copper with 223 MPa ultimate tensile strength. Therefore, there must be a cut-off value below which fracture does not come into the picture. The objective of the present paper is to determine a cut-off value of stress triaxiality by investigating the upsetting tests and the tensile tests on round bars with different hydrostatic pressures.