Anisotropic ductile failure of a high-strength line pipe steel

Abstract

Fracture properties of a mother plate for API grade X100 line pipe were investigated using tensile notched bars, CT and SENB pre-cracked specimens. The material had an anisotropic plastic behaviour due to the thermo-mechanical control rolling process. In addition, anisotropic rupture properties were also observed. Specimens tested along the rolling direction were more ductile and more crack growth resistant than those tested along the long transverse direction. Unit cell calculations were used to show that this fracture behaviour is not related to plastic anisotropy. Assuming that fracture is controlled by internal necking between anisotropically spaced voids, a model combining GTN and Thomason models is proposed which enables describing rupture anisotropy. A modified phenomenological model is also proposed so as to reduce the computational cost.

Appendix: Simulation of damage growth and crack extension

The models used in this study were implemented in a general purpose object oriented finite element software (Besson and Foerch 1997; Foerch et al. 1997). Ductile rupture is always accompanied by large deformations so that a finite-strain formalism must be used when implementing constitutive equations. This was done using a generic formulation based on a reference frame which facilitates keeping the standard small strain formulation and using an additive strain rate decomposition (i.e. $\underline{\dot{\varepsilon }}\phantom {\varepsilon }=\underline{\dot{\varepsilon }}\phantom {\varepsilon }_e+\underline{\dot{\varepsilon }}\phantom {\varepsilon }_p$ where $\underline{\dot{\varepsilon }}\phantom {\varepsilon }$ is the strain rate tensor and $\underline{\dot{\varepsilon }}\phantom {\varepsilon }_e$ the elastic strain rate tensor) (Sidoroff and Dogui 2001).

In all cases, 8 nodes bricks with full integration (8 Gauss points) were used to perform the finite element (FE) simulations. To avoid pressure fluctuations within the elements, a selective integration technique was used (Hughes 1980).

Usual symmetry conditions were used so that of NT$_{s}$pecimens and of CT and SENB specimens were meshed as exemplified on Fig. 17.

All versions of the damage model proposed in this work lead to a strong material softening with leads to strain and damage localization within one row of elements. The simulation results strongly depend of the chosen geometrical discretization, in particular mesh size. To solve this well known problem, models integrating material internal lengths can be used (see e.g. Feld-Payet et al. 2011; Mediavilla et al. 2006 in the case of ductile failure). They are however still under development and hardly used to simulate actual experimental databases. In this work, the pragmatic solution consisting in using a fixed mesh size along the crack path is used (Liu et al. 1994; Rousselier 1987). In this study, the element height in the direction perpendicular to the crack plane was fixed to 100 $\upmu$m. This dimension controls fracture energy in the case of mesh dependent simulations (Siegmund and Brocks 2000).

Fig. 17

Example of FE meshes

In the case of the GTN (Sect. 5.3) and modified-GTN (Sect. 7) models the material is considered as broken when $f_*$ reaches $1/q_1-\epsilon$ with $\epsilon =10^{-3}$. In the case of the GTN/Thomason model (Sect. 6) coalescence in the T-plane start as soon as $R_\mathrm{L}/L_\mathrm{L}=1$ or $R_\mathrm{S}/L_\mathrm{S}=1$. Similar failure conditions are used for coalescence in the L and S-planes (note that coalescence in the S-plane is never active in this study).

In the case where the material is considered as broken, its behaviour is replaced by an elastic behaviour with a very low stiffness (Young modulus: $E_b=1$ MPa). When the material is considered as broken at four Gauss points within an element, the element is removed from the calculation. To avoid getting a singular global stiffness matrix, displacement increments at nodes belonging only to removed elements are then fixed.

Simulated J values were computed from the simulated load—displacement curve using Eq. 1. Simulated crack advance $\Delta a$ was computed following the ASTM-E1820 procedure for multi-specimen testing applied to the FE results; in particular an average crack advance is computed based on the 9-point method. An example of crack growth simulation is shown in Fig. 18.

Fig. 18

Example of crack growth simulation: CT specimen loaded in the T–L configuration. Plot indicates the values of the opening stress ($\sigma _\mathrm{TT}$) in the ligament at Gauss points. “Broken” elements have been removed