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An out-of-plane length scale for ductile crack extensions in 3-D SSY models for X65 pipeline materials

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Abstract

This paper investigates the ductile crack extension in the API 5L X65 pipeline steels in 3-D small-scale yielding (SSY) models using the Gurson-Tvergaard (GT) dilatational plasticity model implemented in the “computational cell” framework. The objective of the study targets at determining an out-of-plane length scale of the computational cell for 3-D crack extensions in fracture models with a through-thickness, straight crack front. The basic Gurson material parameters: the initial void volume fraction f 0 and the in-plane size of the computational cell D, calibrate from a set of notched tension specimens fabricated using API X65 steels. Cell extinctions based on a critical void volume fraction facilitates the process of the void growth and void coalescence which leads to the final failure of the “cell” and thus crack extensions. This study examines two types of 3-D SSY models: the side-grooved model and the plane-sided model. The element size in the thickness direction imposes significant effects on the computed fracture resistance and crack extension. Converged predictions of the J-Δa curve requires the out-of-plane length scale near the side groove equal to the in-plane length scale D. Plane-sided models converge faster than does the side-grooved model and requires the out-of-plane length scale to be 2D near the free surface. The required out-of-plane length scale does not indicate strong dependence on the material hardening exponent and the initial void volume fraction of the material.

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Abbreviations

A :

Scaling coefficient for void nucleation

B :

Thickness of the small-scale yielding Model

B N :

Net thickness of the small-scale yielding model, excluding the side groove

D :

In-plane length scale of the computational cell

E :

Elastic modulus

J :

Elastic-plastic energy release rate

K I :

Mode I stress-intensity factor

L :

Size of the element in the thickness Direction

R :

Radius of the small-scale yielding model

da :

Incremental crack extension

d :

Height of the undeformed cell

\({\overline{d}}\) :

Current height of the cell

\({\overline{d}_0}\) :

Height of cell at f = f E

f :

Void volume fraction

f 0 :

Initial void volume fraction

f E :

Void volume fraction at the element extinction

f N :

Void volume fraction for the nucleating particles

n :

Strain hardening exponent

q i (i = 1, 2, 3):

Material parameters in the Gurson model introduction by Tvergaard

r :

Radius of the notch in the tension specimens

s ij :

Deviatoric stress tensor

s N :

Standard deviation of the nucleation strain

u :

Displacement in the x-direction

v :

Displacement in the y-direction

(x, y, z):

Cartesian coordinate system with the origin at the crack tip

Φ:

Yield surface and flow potential

α g :

Groove angle

\({\varepsilon_{eq}^p}\) :

Equivalent plastic strain

ε ij :

Strain tensor

ε N :

Nucleating strain

ε y :

Yield strain

γ :

Remaining force factor in the killed cell

λ:

Force release factor

θ :

Counter-clockwise angle measured from the horizontal x-axis

σ 0 :

Flow stress

σ e :

Mises effective stress

σ m :

Mean stress

σ y :

Yield stress

υ :

Poisson’s ratio

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Authors and Affiliations

  1. Department of Civil Engineering, National University of Singapore, 1 Engineering Drive 2, Singapore, 117576, Singapore

    Xudong Qian

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Qian, X. An out-of-plane length scale for ductile crack extensions in 3-D SSY models for X65 pipeline materials. Int J Fract 167, 249–265 (2011). https://doi.org/10.1007/s10704-010-9550-1

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