Ductile extension of 3-D external circumferential cracks in pipe structures

Abstract

This study investigates the ductile fracture resistance of 3-D external circumferential cracks in the wall of a steel pipe under remote tension, using a damage-mechanism model originally proposed by Gurson and Tvergaard. The ductile crack extension utilizes an element extinction technique implemented in the computational cell framework. The key parameter for the computational cell method, i.e., the initial porosity ratio f₀, is calibrated using both the fracture resistance and the load-deformation responses obtained from fracture tests of multiple single-edge bend [SE(B)] specimens made of high-strength steel, HY80, which has a yield strength of 630 MPa. The fracture resistance along the 3-D semi-elliptical crack front is computed from the calibrated cell model. Based on the similarity concept in the near-tip stress-strain fields, this study demonstrates that an equivalent 2-D axi-symmetric model provides conservative estimations of the fracture resistance for 3-D circumferential cracks in pipes.

Keywords

ductile fracture / computational cell method / G-T model / J-R curve

Introduction

The safety and reliability of critical engineering structures, such as oil and gas pipelines, rely strongly on the fracture resistance of the material against ductile tearing (often characterized by the J-R curve). The fracture resistance of such materials usually exhibits a significant increase beyond the crack initiation and depends on the increasing plastic deformation in the material ahead of the crack tip. Laboratory tests adopt scaled specimens to predict the increased fracture resistance, and experimental results usually reveal that the J-R curve for a material demonstrates pronounced sensitivity to the specimen size, geometry, crack depth ratio (a₀/W) and loading mode (tension vs. bending) [1,2]. These effects are caused by the strong interaction between the micro-structural features of the material that govern the actual material separation process and the loss of stress tri-axiality in the crack-front region due to large-scale yielding [3].

To understand these complex interactions and provide a reliable engineering approach for predicting the fracture resistance of ductile materials, Xia and Shih [4,5] proposed the computational cell method for evaluating the J-R curve for structural components based on the calibration of model parameters, using results from laboratory-scale specimens. Xia et al. [6] applied this method to reproduce the load-displacement and crack growth histories of five typical laboratory-scale specimens that exhibited differences in the crack-tip constraints. Ruggieri et al. [3] successfully predicted J-R curves, load-displacement records, and crack extensions over the thickness of SE(B) specimens by employing the computational cell method with 3-D finite element (FE) models.

Numerical evaluations of the ductile fracture resistance in pipes have drawn tremendous research efforts over the last few decades. Rahman and Brust [7], Ahamed et al. [8], Kumar and German [9] developed procedures for evaluating the fracture resistance of cracked pipes using a load-based method. Jayadevan et al. [10] studied the effect of crack and pipe configurations on fracture responses. Chattopadhyay et al. [11], using nonlinear FE simulations, reproduced load-displacement and crack initiation results for tests on cracked pipes. However, the above numerical simulations were focused on the fracture resistance of stationary cracks in a pipe. Hippert et al. [12] utilized the computational cell method to predict the burst pressure of cracked pipes made of API 5L X70 steel. Dotta and Ruggieri [13] further extended the same method to correctly forecast the burst pressure of high-pressure pipelines that contained cracks and were made of API 5L X60 steel. Qian [14] reported an out-of-plane length scale for 3-D ductile crack extensions simulated by the Gurson-Tvergaard model for X65 pipeline steel.

This paper begins with an introduction on the computational cell method. The following section presents the calibration and validation of the key material parameter, f₀, for HY80 steel based on the experimental results of multiple SE(B) specimens. The fracture resistance along the 3-D semi-elliptical crack front is then computed from the calibrated cell model, and an equivalent 2-D axi-symmetric model is deployed to estimate the fracture resistance for 3-D circumferential cracks in pipes. The last section summarizes the main conclusions drawn from this study.

Computational cell method for ductile fracture resistance

Xia and Shih [4,5] proposed the computational cell methodology, which provides an approach to describe the ductile crack extension process using a micro-structural length scale physically coupled with the size of the fracture process zone. Figure 1 shows a simplified model of mode I crack growth in ductile metals, which involves the growth, nucleation and final coalescence of microscopic voids initiated near the secondary inclusions of the material. Experimental evidence has revealed that the formation, growth and coalescence of voids occur within a narrow strip of material ahead of the crack tip. The height of this strip of material is approximately equal to the mean spacing between the inclusions. Negligible void growth occurs in the material beyond this strip of material. Xia and Shih [4] further simplified the ductile tearing layer of materials as a single layer of void-containing cell elements, as shown in Fig. 1(b).

Void growth is confined to one layer of material symmetrically located along the crack plane with thickness D, which depends on the mean spacing of the void-initiating inclusions. This layer consists of cube-shaped cell elements of length D on each side. Each cell contains an idealized spherical cavity of initial volume fraction f₀ (the initial void volume divided by the cell volume). D provides a length scale that couples the microscopic features of the material with the macroscopic crack growth behavior. The material beyond this strip, referred to as the background material, remains undamaged by the void growth. The computational cell method adopts the Gurson [15] and Tvergaard [16] (G-T) model to describe the macroscopic softening behavior that occurs due to void growth. The potential function of the G-T model is as follows:

(1)\mathrm{\Phi }\left({\sigma }_e,p,\overline{\sigma },f\right)={\left({\displaystyle \frac{{\sigma }_e}{\overline{\sigma }}}\right)}^{\mathrm{2}}+\mathrm{2}{q}_{\mathrm{1}}f\cosh \left(-{\displaystyle \frac{\mathrm{3}{q}_{\mathrm{2}}{\sigma }_m}{\mathrm{2}\overline{\sigma }}}\right)-[\mathrm{1}+{(q_{\mathrm{1}}f)}^{\mathrm{2}}]=\mathrm{0},

where σ_e denotes the effective Mises stress, σ_m refers to the hydrostatic stress, \overline{\sigma } represents the current flow stress of the cell material and f specifies the current void volume fraction. A value of f = 0 returns Eq. (1) to the classical Von-Mises potential function. Tvergaard [16] introduced the factors q₁ and q₂ to improve the model predictions for periodic arrays of cylindrical and spherical voids.

The flow properties of the G-T material and the background material are described by the uniaxial tension, true stress-logarithmic strain curve following a power-hardening model:

(2){\displaystyle \frac{ϵ}{{ϵ}_{\mathrm{0}}}}={\displaystyle \frac{\overline{\sigma }}{{\sigma }_{\mathrm{0}}}}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}ϵ\le {ϵ}_{\mathrm{0}};{\displaystyle \frac{ϵ}{{ϵ}_{\mathrm{0}}}}={\left({\displaystyle \frac{\overline{\sigma }}{{\sigma }_{\mathrm{0}}}}\right)}^n\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}ϵ>{ϵ}_{\mathrm{0}},

where σ₀ and ϵ₀ are the yield stress and the yield strain, respectively, and n is the strain hardening exponent. Faleskog et al. [17] calibrated the q₁ and q₂ values based on material properties, including the ratio of the yield stress over the Young modulus, σ₀/E, and the strain hardening exponent, n. Faleskog’s [17] conclusion leads to the values of q₁ = 1.375 and q₂ = 0.98 for the HY80 steel considered in the current study. The porosity ratio, f, which is a damage parameter, determines the remaining volume of the effective material. Chu and Needleman suggested a modified void growth rate [18],

(3)df=(\mathrm{1}-f)d{ϵ}_{kk}^p+\Re \left({\overline{ϵ}}^p\right)d{\overline{ϵ}}^p,

where the first term defines the growth rate of the existing voids, and the second term illustrates the evolution of the porosity ratio due to the nucleation of new voids, which describes the cell response at low-stress tri-axiality [3]. The current study excludes the effect of void nucleation, i.e., \Re = 0. The critical porosity ratio, f_E, triggers element extinction as the void volume fraction approaches f_E. Figure 1(c) shows a typical FE mesh near the crack tip for the computational cell approach. The first element at the crack tip is slightly smaller than the adjacent elements.

The computational cell method requires the input of the typical material properties needed for the incremental plasticity of the material, including the Young modulus E, the Poisson ratio v, the yield strength σ₀ and the strain hardening exponent n for the background material. Furthermore, two additional material parameters, D and f₀, are required for the G-T model. The background material and the G-T material have identical flow properties. The crack tip-opening displacement (CTOD) at the crack initiation provides a good estimate of D [19]. Previous research has shown that f_E ranges from 0.10 to 0.20 and imposes negligible effects on the crack extension process [18,19].The current study, therefore, fixes the f_E value at 0.15. Consequently, f₀ is the only material variable that needs to be determined by calibrating the experimental results, typically using the J-R curve and the force versus load-line displacement (LLD) curve.

The FE simulations in this study employ the implicit FE code WARP3D [20], which adopts the modified element extinction procedure originally proposed by Tvergaard [21]. Element extinction occurs as the f value (averaged over Gaussian points in a computational cell) reaches the critical value f_E. The remaining nodal forces applied on the extinct cell taper to zero following a linear traction-separation model [20], as shown in Fig. 1(d). WARP3D also introduces a step-size control parameter, α_c ( = Δf_max/f_E), to avoid rapid void growth and severe element extinctions in a single load increment. Gullerud et al. showed that λ = 0.05 and α_c = 0.01provide convergent, stable numerical results [22]. The current study employs these values of λ and α_c.

WARP3D [20] computes the energy release rate, the J-integral, through a domain-integral approach:

(4)J=\underset{r_{\mathrm{0}}\to \mathrm{0}}{\lim }{\displaystyle \underset{S_{\mathrm{0}}}{\int }\left[\overline{W}{n}_{\mathrm{1}}-P_{ji}{\displaystyle \frac{\partial u_i}{\partial X_{\mathrm{1}}}}\right]}q{n}_j\mathrm{d}S,

where S₀ denotes the surface area of the domain surrounding a segment of the crack front with a specified length and a radius r₀. The unit vector n_j refers to the outward normal to S₀. \overline{W} defines the stress-work density per unit of un-deformed volume. P_ji and u_i are Cartesian components of the first (unsymmetric) Piola-Kirchoff stress tensor and the displacement vector, respectively. The scalar q is a weighting function interpreted as a normalized virtual displacement. For crack fronts that experience ductile crack extension, the evaluation of J utilizes domains well outside the highly non-proportional histories of the near-front fields [22] and thus retains a path-independent value. With the material damage properly represented and calibrated against the experimental observations, the J-values thus calculated represent the fracture resistance as the crack extension occurs.

Calibration of the computational cell method

This section presents the FE procedure for calibrating the material-dependent parameters (D and f₀) in the G-T model. The derivation of a unique pair of D and f₀ values has become a challenging task for many researchers. In some engineering applications, a large D value may be compensated for by a large f₀ value [13,14]. Nevertheless, previous researchers have confirmed that the length scale D lies in the range of 50 to 300 μm [19,22]. In addition, the D value, which represents the width of the fracture process zone, should be a material-dependent parameter approximately equal to the critical crack tip-opening displacement (CTOD) [19], which can be derived from a fracture toughness (J_IC) test using standard fracture specimens. The J_IC values for HY80 specimens tested under varying crack tip constraint levels range from 200 MPa to 250 MPa [23]. The CTOD, therefore, can be computed from the well-established relationship [23]:

(5)\mathrm{C}\mathrm{T}\mathrm{O}\mathrm{D}=d_n{J}_{IC}/{\sigma }_{\mathrm{0}},

where d_n = 0.507 for HY80 steel, as tabulated by Shih [23]. This yields an in-plane length scale of HY80 steel of approximately 0.2 mm, i.e., D = 0.2 mm.

The only parameter that requires calibration is therefore the initial void volume fraction, f₀. The calibration of f₀ utilizes both the experimental J-R and P-LLD curves [24] for a shallow-crack SE(B) specimen with a₀/W = 0.186. Subsequent numerical analyses validate the calibrated f₀ value through the J-R and P-LLD curves for SE(B) specimens fabricated using the same HY80 steel with different crack-front constraints, i.e., different crack depth ratios (a₀/W = 0.286, 0.393 and 0.549).

Figure 2 shows the 2-D plane-strain FE model adopted in the current FE analyses. The half-symmetric model contains 719 8-node 3-D isoperimetric elements and 1573 nodes, with one layer of elements though the thickness. The displacement degree of freedom along the thickness direction is constrained for all nodes to enforce a plain-strain condition. One layer of 60 computational cell elements lies along the plane of symmetry. The numerical analysis used the following G-T material parameters for the computational cells: D/2= 0.1mm, f_E = 0.15, q₁ = 1.375 and q₂ = 0.98, based on the discussion in the above section.

Figure 3 shows the uniaxial true stress versus the true strain curve for HY80 steel, which has a yield strength of σ₀ = 630 MPa, a tensile strength of σ_u = 735 MPa, a Young modulus of E = 207 GPa, and a Poisson ratio of υ = 0.3. The experimental specimen consists of 1T (i.e., 1 inch or 25 mm thick) SE(B) specimens as outlined in ASTM E1820-01 [25]. All specimens undergo a fatigue pre-cracking process followed by side-grooving, which reduces the specimen thickness by 20% to maintain an approximately plane-strain condition along the crack front. The experimental procedure applies three-point bending on the specimen with a span of S = 203 mm, a span to width ratio of S/W = 4 and a thickness to width ratio of B/W = 0.5.

Figure 4 compares the experimental J-R and P-LLD curves with those computed from the FE analyses. Both the J-R curve and the P-LLD curves depend strongly on the f₀ values. A larger f₀ value yields a lower J-R curve and a correspondingly lower P-LLD curve as well. The value f₀ = 0.0075 for HY80 steel provides not only a good fit to the J-R curve but also to the P-LLD response.

Because f₀ is a material-dependent constant, the calibrated f₀ values should provide accurate predictions of the ductile fracture resistance as the crack extends, independent of the crack-tip constraints. Additional analyses of SE(B) specimens with various crack depth ratios (a₀/W = 0.286, 0.393 and 0.549) confirmed that the same f₀ value provides close agreement with the experimental J-Rand P-LLD curves, as illustrated in Fig. 5. In Fig. 5, the discrete symbols represent the experimental data, while the continuous curves represent the FE results.

Extension of external circumferential cracks in pipes

This section describes the numerical prediction of the ductile fracture resistance, or the J-R curves, for circumferential cracks in the walls of pipe structures, which experience remote axial tension. The material parameters follow the set of parameters selected based on the 2-D plane-strain calibration, with f_{\mathrm{0}}=\mathrm{0.0075}. The numerical investigation also attempts to estimate the ductile fracture resistance for 3-D circumferential cracks in the wall of a pipe through a computationally more efficient 2-D axi-symmetric model. The 2-D representation of the 3-D crack driving force has been shown to be feasible for the assessment of brittle fracture failure in pressure vessels [26].

Circumferential cracks often occur under fatigue loading near the weld toe of the girth welds in, e.g., a pipeline or a pressure vessel structure. The crack plane of a circumferential crack remains perpendicular to the longitudinal axis of the pipe. The external circumferential crack opens on the outer surface of the pipe. This study focuses on the fracture resistance of the base material of the pipe, excluding the effects of residual stresses, heat-affected zones and the weld material on the fracture resistance.

Figure 6(a) shows a typical, one-quarter FE model for the cracked pipe, built from 3-D 8-node brick elements. The quarter pipe has a length of L = 1000 mm, with an inner radius of R_i=\mathrm{250}\mathrm{m}\mathrm{m} and a thickness of t=\mathrm{25}\mathrm{m}\mathrm{m}. The displacement loading is imposed at the remote end of the pipe. The presence of two planes of symmetry (as shown in Fig. 6(a)) enables a one-quarter model. The total number of nodes in the 3-D FE models with different crack sizes is approximately 24000, with the number of elements being approximately 20000. The meshes of all of the FE models were generated by FEA Crack [27]. Figure 6(b) shows the geometrical configurations of a circumferential crack with a length of c and a depth of a. Table 1 summarizes the various crack dimensions considered in the current study, which includes shallow (a/t = 0.2), medium (a/t = 0.4) and deep (a/t = 0.6) cracks. The parametric study includes a few a/c ratios for each of the a/c ratios considered.

The 3-D FE model consists of one layer of G-T elements at the crack front, as shown in Fig. 6(b). Figure 6(c) provides a close-up view for the layout of the G-T elements. There are 45 elements along the crack front and 30 G-T elements along the direction of the crack extension ahead of each crack-front location. The total number of G-T elements in each 3-D FE model is thus 1350. The out-of-plane height of the shaded cell elements (as shown in Fig. 6(c)) is D at the deepest crack-front location. As we move away from the deepest crack-front location, the computational cells have a gradual increase in cell height. This meshing scheme does not compromise the ability of these cells to capture the ductile crack extension, assuming that statistical variations of the microstructure along the crack front are insignificant [19]. A recent study on the out-of-plane length scale for X65 steel by Qian [14] showed that the ductile fracture resistance exhibits very little dependence on the element size in the thickness direction for plane-sided models with L_{\min }/D\le \mathrm{2}, where L_min is the minimum length of the cell elements along the thickness direction The above observation supports the meshing scheme used in the current study, as shown in Fig. 6(c). Along the direction of the crack extension, the cell width is fixed at D/2, except for the element at the crack tip, which has a width of D/4. The G-T model predicts a rapid loss of loading capacity in regions under high-stress triaxiality. In the current 3-D FE model, the crack extension at the deepest location of the semi-elliptical crack front therefore initiates earlier than the extensions at other regions along the crack front. The amount of crack extension that occurs is computed from the number of extinct computational cells at the deepest crack-front location, as shown in Fig. 6(c). The J-\mathrm{int}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l} value is computed from a far-field domain of elements, as shown in Fig. 6(d).

Figure 7(a) illustrates the extraction of a one-degree slice of a 2-Daxi-symmetric model from the 3-D pipe with a semi-elliptical circumferential crack. Figure 7(b) shows a typical 2-D axi-symmetric FE model, built from 3-D 8-node brick elements. This model has a length of 75 mm and a width of W = t = 25 mm. Figure 7(c) demonstrates the out-of-plane boundary conditions for the current 2-D FE models. Plane A refers to a global symmetric plane, while plane B represents a circumferentially symmetric plane. The out-of-plane dimension of the one-degree slice has a length of 4.4 mm, corresponding to the inner surface of the pipe. The layout of the G-T elements in the current 2-D models follows the same approach used in the previous SE(B) models. The domain for the J-integral computation has the same size as that of the 3-D models, as shown in Fig. 7(d). The 2-D models consist of a total of approximately 800 nodes and approximately 400 elements.

Figure 8 compares the J-R curves obtained from the 3-D analyses with those determined from the 2-D FE analyses for a/t = 0.2, a/t = 0.4 or a/t = 0.6. The initial ductile crack extension in the 3-D model involves extinction of the computational cells at a very small distance () away from the crack front. The location of the first cell extinction corresponds to the position of the maximum opening stresses ahead of the crack tip. The maximum opening stress ahead of the crack often occurs in the region J/{\sigma }_{\mathrm{0}}~\mathrm{2}J/{\sigma }_{\mathrm{0}} based on elastic-plastic fracture mechanics theory [28]. The extinction of the first cell immediately ahead of the crack tip, therefore, does not occur until the fourth cell away from the crack front is extinct, causing a significantly increased crack driving stress on the first cell. Once the first four G-T elements are extinct, the crack extension occurs progressively ahead of the crack front.

Figure 8 demonstrates that the 2-D J-R curve provides a lower bound for the 3-D FE fracture resistance for the three different crack depth ratios considered. For shallow- and medium-depth cracks, i.e., a/t = 0.2 and a/t = 0.4, crack extensions for larger a/c ratios show a higher J-R curve. For the crack with a/t = 0.2, the 2-D result agrees well with the 3-D result for a/c. The 2-D axi-symmetric model thus closely represents shallow and long3-D circumferential cracks. The 2-D axi-symmetric model, compared with the 3-D model, has a stronger out-of-plane constraint, which reduces the plastic zone size ahead of the crack tip, therefore yielding a lower bound for the fracture toughness. For deeper cracks, the plane-strain condition imposes slightly higher crack-front constraints on the 2-D axi-symmetric model in comparison to the crack-front constraints experienced by the semi-elliptical 3-D crack front. In addition, the redistribution of the very high crack-tip stress field for the 3-D crack front occurs not only in the material ahead of the deepest crack tip but also in the adjacent crack-front material away from the deepest point. Figure 8 also demonstrates that the J-R curves obtained from the 3-D analyses are independent of the a/c ratio for deep cracks and are slightly higher than the corresponding 2-D predictions. In addition, the variation of the crack length has a correspondingly smaller effect on the predicted J-R curve as the crack depth increases.

Summary and conclusions

This study examined the ductile crack extension in circumferential cracks in pipe structures made of high-strength HY80 steel using a material damage model implemented in the computational cell framework. The calibration of the Gurson-Tvergaard material damage model employed a plane-strain simulation of the fracture toughness test performed on SE(B) specimens. The subsequent numerical analysis compared the ductile fracture resistance computed from the 3-D pipe model with a circumferential crack and that from a 2-D axi-symmetric model. The following conclusions can be made from this study.

1) The ductile crack extension simulated for a material length scale of D = 0.2 mm and an initial void volume fraction of f_{\mathrm{0}}=\mathrm{0.0075} provides close agreement with the experimental J-R and P-LLD curves for SE(B) specimens made of high-strength steel, HY80, for four different crack depth ratios with differing crack-front constraints. This result demonstrates that both the material length scale and the initial void volume fraction are independent of the crack-front constraint conditions.

2) 2-Daxi-symmetricmodels provide a convenient and highly efficient alternative for predicting the ductile fracture resistance for 3-D circumferential cracks in pipe structures. For all crack geometries considered in this study, the 2-D axi-symmetric model provided conservative lower-bound predictions of the J-R curve as compared to the fracture resistance computed from the 3-D models. The prediction of the fracture resistance for an external circumferential crack in a pipe structure obtained using 2-D axi-symmetric models remains applicable to other material types with the computation cell parameters calibrated based on the fracture resistance curve.

3) The effect of the crack length on the predicted fracture resistance in 3-D pipes is less significant for deep cracks with a/t = 0.6. For shallow and medium cracks, pipes with a long surface crack exhibit a fracture resistance closer to the 3-D predictions as compared to the estimation made by the 2-D axi-symmetric model.

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