Combined effect of prestrain history and strength mismatch on crack tip constraints in welded specimen

2.1 Crack tip constraints

From traditional elastic-plastic fracture mechanic prospective, the fundamental crack tip constraint could be characterized by a single parameter, J integral (HRR field) or CTOD. Actually, a single parameter characterization of crack tip stress field may lose its validity due to the effects of geometry, loading, initial stress distribution, and mechanical mismatch on the crack tip constraints. In general, the crack tip stress field is expressed by the following equation with two terms in a cylindrical coordinate system (r, θ):

(1)

The first nonsingular term origins from the Williams's series for linear elastic crack-tip fields. Where K_Ι represents the mode I elastic stress intensity factor. O′ Dowd and Shih¹¹ employed a dimensionless constraint parameter Q, which could show the fully yielded conditions to quantity the effect of crack tip constraint. Therefore, the two-parameter formula could be written as follow:

(2)

where σ₀ and n are the yield stress and plastic strain hardening exponent, is the reference stress field control by the loading, and is another field. The first term is determined to describe the J-dominant HRR stress field by the modified boundary layer (MBL) model.

The stress distribution of interface crack is affected not only by the geometry condition but also by the material mechanical mismatch between the weldment and base metal. Zhang et al¹² extended the J-Q two-parameter approach to examine the effect of material mismatch on crack tip constraint. A three-parameter J-Q-M formulation was proposed to characterize the mismatch constraint effect on the basis of J-Q formulation:

(3)

where f_ij donates the angular function of the mismatch-induced difference field, β = 0 for overmatch, and β = 1 for undermatch. Except the material mismatch effect, the crack tip stress field in welded joint is more complicated due to different prestrain history states and residual stress. Therefore, different parameters are defined to describe the corresponding constraints. For example, a three-parameter equation (CTOD, Q, R) is used to characterize the crack tip field with residual stress. The three-parameter equation including prestrain effect is similar with two-parameter formulations as follows:

(5)

In this paper, the effects of material mismatch and prestrain history on fracture toughness, crack tip stress fields, and constraints are investigated simultaneously by SENT models.

2.2 Finite element framework

The typical 2-D finite element model of SENT specimens with weld strength mismatch is presented in Figure 1. Half plane of models for analyses have 6500, 2-D bilinear plane strain quadrilateral elements with line constraints imposed on nodes considering the symmetry of geometry. The idealized weld zone (fusion line is vertical to specimen length) is considered into the ductile fracture models. The crack with an initial blunt distance 0.02 mm is assumed as the crack-tip located on the center of the model, described in Figure 2A. The initial crack size of welded zone is set as a/W = 0.17. In addition, surface-to-surface contact between a rigid plane and symmetrical plane is defined in the interaction module of ABAQUS in order to realize the compression process of half model. It is noticed that the heat-affected zone is not taken idealized welded joints into account.

In the MBL model, a displacement loading field is imposed on the outer boundary, as shown in Figure 2. The values of displacement loading are controlled by the elastic stress asymptotic field equations (Equations (6) and (7)) under mode I plane strain condition.

(6)

(7)

By using these boundary conditions, the combined prestrain and mismatch constraints of SENT specimens is investigated.

2.3 Materials

High-strength low alloyed pipeline steel X70 is studied in this paper, and the basic mechanical properties is showed in Fagerholt et al.¹³ The material stress-strain behavior can be described by the von Mises flow theory with a power-law hardening relationship:

(8)

where represents the yield stress, ε_P stands for the equivalent plastic strain, ε_y is the yield strain, and n is the hardening exponent. The strain hardening exponent of base material and mismatched welds is assumed as 0.05. In our study, the mismatch factor M between weld filler materials and base material is defined as

(9)

where σ_yw and σ_yb are 0.2% proof stress of these two materials. The characteristic material parameters are given in Table 1. Meanwhile, isotropic hardening law is used in ductile fracture calculations. The initial void volume fraction in the welds and base metal is all assumed as 0.005 on Gurson constitutive model.¹⁴

3.1 Nominal stress-strain curves of different conditions

During actual reeling processing in industries, it may occur many prestrain cycles or nonproportional tension-compression loading for the steel pipelines. In this study, a typical symmetrical prestrain cycle history was conducted in FE models. More specifically, the prior tension loading was applied and then the same displacement was compressed. Prestrain history could be analyzed by characterizing the remote displacement varieties (as nominal strain). Four prestrain cycle levels were taken the SENT models into consideration, which were 0%, 0.2%, 0.3%, and 0.4%. In addition, the mismatch effect was studied by various mismatch ratios (M), which were 1, 1.17, 1.33, and 1.5. It should be noticed that the geometry effect on crack tip constraint is not considered in this research. For all the analyses, the prestrain cycle history started with tension loading. Figure 3A shows that the hysteresis loop increases gradually with the increase of prestrain in. Figure 3B shows the results from four cases of different mismatched ratios under 0.4% prestrain value. With the increase of mismatch ratio, the hysteresis loop tended to narrow down with a same prestrain amplitude. The phenomenon could be explained by the improvement of fracture resistance in overmatched welds.

Figures 3C and 3D show CTOD and nominal stress relationships by different prestrain levels and mismatched ratios. At the elastic stage, the CTOD increased gradually with the increase of nominal stress. After the stress exceeded the yield stress, the CTOD rapidly grew at the retension stage. The compression stress over the yielding points could lead to the decrease of CTOD, which was similar with the tension stage. In the last loading stage, the increase of CTOD values demonstrates that the ductile crack growth occurs. The CTOD response on nominal stress curves has less sensitivity than on nominal strain as the material plastic phase in Figure 3D.

3.2 Effects of prestrain and strength mismatch on ductile crack growth

To address the effect prestrain history on crack growth resistance, this section presents the results of 2D analyses conducted on SENT specimens with mismatch ratio M = 1.17, joints width 2H = 12.8 mm, and initial crack size to width ratios a/W = 0.17. Four cases of the symmetrical prestrain levels, which are 0%, 0.2%, 0.3%, and 0.4%, are computed. Figure 4 provides a summary plot, which emphasizes the effects of symmetrical prestrain levels and mismatched on the crack growth behavior for the SENT specimens. Figures 4A and 4B compare the ductile fracture resistance of different prestrain values and mismatch ratios. The increase of prestrain values can result in the decreases of CTOD values in the retension stage. In Figure 4B, the resistance curves depend strongly on the mismatch ratio in the range of 1 ≤ M ≤ 1.5. The increase of mismatch ratio can elevate the slope of resistance curves.

Generally, these effects on crack-tip fields in SENT specimen incorporating the evolution of near tip stress with following stable crack growth can be studied based on the two-parameter or three-parameter approach. However, the combined effect of these factors is still obscured. In this section, the crack tip stress distribution in the process of ductile fracture with combined effect of prestrain and mismatch is discussed. The constraint parameter Q is used to characterize the geometry effect of crack tip stress distributions in SENT specimens. Considering the CTOD values as the crack driving force in this study, the Q parameter is defined as follows¹⁵:

(10)

where the σ_θθ is the opening stress component in the polar coordinate and is the reference stress component, which can be obtained from the MBL model solution with T = 0.x/CTOD = 4 represents the result output location ahead of the crack tip. Figure 4C shows the typical opening stress distribution with increased amounts of ductile fracture. Figure 4D provides the evolution of Q with increased values of CTOD in SENT specimens.

3.3 Crack tip stress distributions and constraints with prestrain effect

Figures 5A to 5C show the different opening stress fields at different crack extension, which can be characterized by CTOD from 0.1 to 0.5 mm considering the prestrain effect. It can be found that the maximum opening stress increases with the increase of prestrain amplitude at fixed CTOD values. An interesting observation from the evolution with the crack growth (CTOD) is the weak correlation between the maximum opening stress and prestrain cycle. While the elevation of opening stress without prestrain history leads to less significance of prestrain effect.

The plastic prestrain constraint of crack tip stress field has been quantitatively developed by Eikrem et al.¹⁶ To make this point, a new parameter P was introduced to assess the prestrain effect considering the single prestrain cycle. It can be expressed as follows:

(11)

where represents the case with prestrain cycle and the denotes the typical monotonic loading case. What should be emphasized on is the normalized crack tip distance given by x/CTOD = 4.

3.4 Crack tip stress distributions and constraints with combined effect of prestrain and weld strength mismatch

In the section, the combined constraint effect was examined by the J-Q-P-M methodology, which was extended from the J-Q-P formula. Therefore, the J-Q-P-M formulation is whited according to the J-Q-P formula as follows:

(12)

The coupled model was applicable to describe the crack tip opening stress fields in terms of the combined effect of prestrain and strength mismatch. The coupled constraint parameter of P and M could be expressed according to the aforementioned definition of prestrain constraint as following:

(13)

The combined constraint parameter was investigated at the crack-tip distance given by x/CTOD = 4 for a fixed prestrain history with the variation of mismatched ratio as following in Figure 6A. This plot exhibited clearly trends on the variation of combined constraint with the increase of mismatch ratios, which was consistent with the results of exclusive prestrain constraint. The combined constraint significantly elevated the constraint level with the increased crack extension. It revealed that the weld strength mismatch constraint for 1≦M≦1.5 was more sensitive than the prestrain constraint with the symmetrical loading from 0 to 0.4%, which was compared by the separate result of constraints from Figure 6B.