Modeling flat to slant fracture transition using the computational cell methodology

Highlights

• A 3-D method for studying flat to slant rupture transition in ductile metal sheet is proposed.

• Results show that a final tilt angle equal to 45° corresponds to a minimum energy dissipation rate.

• Local stress and strain states along the crack path do not depend on the assumed crack tilt angle.

• In the slant crack growth region, a plane strain state prevails along the crack propagation direction.

• The flat to slant rupture transition is essentially controlled by plane strain localization at the macroscopic scale.

Abstract

Macroscopic mode I ductile crack propagation in metallic sheets or plates often starts in mode I as a flat triangle (coplanar with the precrack) whose normal corresponds to the loading direction. After some limited extension, the crack becomes slanted and propagates under local mixed mode I/III. Modeling and understanding this phenomenon is challenging. In this work, the “computational cell” methodology proposed in [1], which uses a predefined crack path, is used to study flat to slant fracture transition. The energy dissipation rate is studied as a function of the assumed crack tilt angle. It is shown that a minimum is always reached for an angle equal to 45°. This correlates well with the variation of the crack tip opening angle (CTOA) or the mean plastic deformation along the crack path. Stress and strain states in the stable tearing region hardly depend on the assumed tilt angle. A parametric study shows that flat to slant fracture transition is less likely to occur in materials having high work hardening and favored if additional damage is caused by the local stress/strain state (plane strain, low Lode parameter) in the stable tearing region.

Introduction

Ductile crack extension in sheet/plates specimens often starts with an initially flat crack coplanar with the precrack which becomes slanted as soon as maximum crack extension is about 1 to 2 times larger than the sheet thickness. This phenomenon was often observed in aluminum alloys [2], [3], [4], [5], [6] and steels [7], [8]. Pure flat fracture was observed in Al6082 alloy in O temper [9] which has a high hardening capability. Similar results were reported by Pardoen et al. [10] on various materials having a high hardening capacity. In the case of large plates made of grade X70 pipeline steel [8] it was found that flat fracture occurred under quasi-static loading conditions whereas slant fracture was observed for dynamic loading. This was interpreted as an effect of adiabatic heating which lowers the hardening capacity thus promoting slant fracture. In higher grades with a lower work hardening, such as X100, slant fracture is always observed unless significant delamination occurs [11]. These results clearly show that slant fracture is promoted in materials with low hardening capacity.

Modeling the flat to slant transition using continuum damage mechanics remains a difficult task in particular if the crack path is not prescribed. In 2D plane strain or axisymmetric cases, crack path changes and slant fracture were simulated using continuum damage mechanics using the Gurson [12] model and its extensions. Axisymmetric cup-cone fracture was successfully simulated [13], [14] as well as plane strain slant fracture [15]. This method was recently extended [16], [17] by introducing discontinuities based on a bifurcation analysis. Note that cup-cone fracture was also reproduced using a cohesive zone model [18].

The actual flat to slant transition is a fully 3D problem which was first addressed by Mathur et al. [19] in the case of dynamic crack growth but no attempt was made to compare results with actual tests. The transition was also modeled by Besson et al. [20] but simultaneously matching crack paths and load–displacement curves was not possible. The flat to slant transition was obtained using an explicit simulation by Xu and Wierzbicki [21], [22] but comparison with experiments was missing. The authors used a damage growth law depending on the Lode parameter of the stress tensor to favor slant fracture. More recently the flat to slant transition was reproduced using an implicit simulation algorithm in [23] using strain controlled damage nucleation depending on the Lode parameter of the strain rate tensor; a satisfactory comparison with experiment was obtained. The main problem encountered while performing such FE simulations, is that a large number of elements must be introduced to allow for crack path change from flat to slant. The number of elements (as well as the model parameters) influences the result of the simulation and it was shown that using too few elements leads to flat crack growth [20], [14], [22]. This can be partially attributed to the mesh size dependence observed in the case of strain softening materials.

In order to solve the problem of crack path dependence within the continuum damage mechanics approach, it is possible to use the so called “computational cell” technique proposed by Xia et al. [24], [1]. Following this method, ductile fracture is confined to a material layer ahead of the initial crack. The layer’s thickness is prescribed as a model parameter. It is modeled by a single row of uniformly sized cells represented by one single finite element. The material outside of this strip remains undamaged. The method was first used for 2D plane strain problems and rapidly extended to 3D cases [25], [26], [27], [28]. An alternative approach consists in using cohesive zone models. It is however known that the stress triaxiality dependence which is characteristic of ductile fracture is not well captured by such models so that cohesive parameters should depend on the local stress state [29].

In this work the “computational cell” methodology is used to simulate flat to slant transition in a grade X100 line pipe steel. A slanted crack path is meshed using different tilt angles. Local strain and stress states as well as macroscopic quantities such as the energy dissipation rate, the crack tip opening angle (CTOA), crack front shape and area reduction are studied as function of the tilt angle. The numerical tool is then used to study the effect of work hardening and secondary void nucleation.