Influence of welding residual stresses on the ductile crack growth resistance of circumferentially cracked pipe

Abstract

Welding residual stress is one of the main concerns for fabrication and operation of steel structures due to its potential effect on structural integrity. This paper focuses on the effect of welding residual stress on the ductile crack growth resistance of circumferentially cracked steel pipes. Two-dimensional axi-symmetry model has been used to simulate the pipe. Residual stresses were introduced into the model by using so-called eigenstrain method. The complete Gurson model has been employed to calculate the ductile crack growth resistance. Results show that residual stresses reduce the ductile crack growth resistance. However, the effect of residual stresses on ductile crack growth resistance decreases with the increase of crack growth. The effect of residual stress has also been investigated for cases with different initial void volume fraction, material hardening and crack sizes.

Keywords

residual stress / ductile crack growth resistance / complete Gurson model / eigenstrain method

Introduction

It is widely accepted that the presence of residual stresses can influence the failure characteristics of components and structures. Tensile residual stresses reduce the load-bearing capacity of the structures while compressive residual stress may cause local buckling of thin wall structures. Therefore, it is important to investigate the effect of residual stresses on various aspects of fracture modes.

Ductile crack growth plays an important role in the analysis of the fracture behavior of steel structures. The near tip stress/strain fields may be changed due to ductile crack extension, which can trigger the transition to unstable cleavage fracture. The ductile fracture process is influenced by the conditions of stress triaxiality and plastic strain within the vicinity of a stress concentrator such as a notch or a crack-tip [1]. It has been demonstrated that residual stress can affect the near-tip stress field and induce an additional crack-tip constraint [2,3]. It is thus interesting to investigate how residual stresses influence the ductile crack growth resistance. Based on the theoretical modified boundary layer model, Ren et al. [4] studied the effect of residual stresses on ductile crack growth resistance. Their results showed that residual stresses have significant effect on the ductile crack initiation toughness while with the increase of the crack extension the effect of residual stresses tends to diminish. There are also several experimental studies to investigate the effect of residual stresses on ductile tearing behavior and resistance [1,5,6].

This paper focuses on the effect of welding residual stresses on the ductile crack growth resistance of circumferentially cracked steel pipes. Two-dimensional axi-symmetry models have been utilized to simulate the pipe. Welding residual stresses were introduced into the model using the so-called eigenstrain or “inherent strain” method [7]. The complete Gurson model [8] has been utilized to calculate the ductile crack growth resistance in ABAQUS. The effect of residual stresses on ductile crack growth resistance for cases with different initial void volume fraction, material hardening and crack sizes has been studied.

Methodology

Problem description

The case studied in this paper is single-pass hyperbaric gas metal arc welding (GMAW) under pressure 3 bar. The inner radius of the pipe is 150 mm, and thickness is 20 mm, as illustrated in Fig. 1. The computational procedure is as follows: (a) residual stresses were introduced into the model by using the eigenstrain method; (b) a sharp crack was then inserted, and residual stresses were redistributed; (c) external loading was applied and the crack growth resistance was calculated.

Complete Gurson model

Ductile fracture is a common failure mechanism of metallic materials. A typical ductile fracture process can be divided into three stages: the micro void nucleation, growth and coalescence. Microvoids nucleate at inclusions and second-phase particles when sufficient stress or strain is reached to break the interfacial bonds between the particle and the matrix or break the inclusions. Further plastic strain and hydrostatic stress cause the voids to grow and coalesce. Damage mechanics models have been widely used in numerical simulation of ductile crack growth. Gurson model [9] is one of the most popular models used to assess ductile tearing behavior under primary loading, where the effect of the void growth is taken into account in the constitutive model. The Gurson model has later been modified by Tvergaard and Needleman [10-12], and it is most often referred to as the Gurson–Tvergaard–Needleman (GTN) model. The GTN model is in fact a void growth model, and coalescence of voids is considered by introducing so-called critical void volume fraction. The softening effect due to the presence of voids is reflected in a yield function:

(1)\mathrm{\Phi }\left(q,\overline{\sigma },f,{\sigma }_m\right)={\displaystyle \frac{q^{\mathrm{2}}}{{\overline{\sigma }}^{\mathrm{2}}}}+\mathrm{2}{q}_{\mathrm{1}}f\mathit{c}\mathit{o}\mathit{s}\mathit{h}\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 f is the void volume fraction, which is the average measure of a void-matrix aggregate, σ_m is the mean normal stress, q is the conventional von Mises equivalent stress, \overline{\sigma } is the flow stress of the matrix material, q₁, q₂ are constants introduced by Tvergaard [10,11]. It should be noted that the yield surface of the Gurson model decreases with the increase of damage void volume fraction until the complete loss of load-carrying capacity.

Void nucleation can be stress controlled or strain controlled, and the strain controlled nucleation can be written as:

(2)d{f}_{nucleation}=f_{ϵ}({ϵ}^p)d{ϵ}^p,

where f_e is the void nucleation intensity and ϵ^p is the equivalent plastic strain. Two types of void nucleation models may be used for engineering materials, the cluster nucleation model and continuous void nucleation. Due to the incompressible nature of the matrix material the growth of existing voids can be expressed as:

(3)d{f}_{growth}=(\mathrm{1}-f)d{ϵ}^p:\mathrm{I},

where ϵ^p is the plastic strain tensor and I is the second-order unite tensor. Once the void coalescence criterion is satisfied, the post-coalescence deformation behavior of the Gurson model is numerically simulated by an artificial acceleration of void growth, as suggested by Tvergaard and Needleman [12]:

(4)f^{\ast }=\{\begin{array}{ll}f& \mathrm{f}\mathrm{o}\mathrm{r}f\le f_c\\ f_c+{\displaystyle \frac{f_u^{\ast }-f_c}{f_F-f_c}}(f-f_c)& \mathrm{f}\mathrm{o}\mathrm{r}f>f_c\end{array},

where f_u^{*}=1/q_1, f_c is critical void volume fraction, and f_F is the void volume fraction at the end of void coalescence. However, the void coalescence criterion of original Gurson model is not a physical mechanism-based criterion. As is known, fracture of ductile materials displays two distinct phases, the homogeneous phase and localized phase. Thomason [13] proposed a so-called dual dilatational constitutive equation theory for ductile fracture, and argued that two fracture phases are in competition for a void containing material. Both deformation modes are dilatational, by which plastic deformation will result in change of material volume and the material will always follow the deformation mode that needs less energy. In the early stage of deformation, the voids are small and it is easier to follow the homogeneous deformation mode. With the advance of plastic deformation and increase of void volume fraction, the stress required for localized deformation decreases. When the stress for localized deformation is equal to the stress for homogeneous deformation, the void coalescence occurs. The plastic limit criterion by Thomason states that no coalescence will occur as long as the following condition is satisfied [13,14]:

(5)

where σ₁ is the current maximum principal stress, r is the void space ratio,

r=\sqrt[\mathrm{3}]{(\mathrm{3}f/\mathrm{4}\pi )e^{{ϵ}_{\mathrm{1}}+{ϵ}_{\mathrm{2}}+{ϵ}_{\mathrm{3}}}}/(\sqrt{e^{{ϵ}_{\mathrm{2}}+{ϵ}_{\mathrm{3}}}}/\mathrm{2})

and ϵ₁ is the maximum principal strain, ϵ₂ and ϵ₃ are the two other principal strains, α = 0.1 and β = 1.2 are constants fitted by Thomason [13].

By combining the GTN model for void growth and Thomason’s plastic limit load model for coalescence, a so-called “complete Gurson model” has been proposed by Zhang et al. [8], which can predict the complete ductile fracture process. The complete Gurson model has been implemented into the commercial finite element package ABAQUS through material user subroutine UMAT using the generalized midpoint algorithms [15-19]. A free copy of the ABAQUS UMAT code can be obtained from the corresponding author.

Eigenstrain method

Mapping the residual stress distribution in structures is essential for assessing its safety and durability. However, in practice, it is often very difficult to characterize the residual stress state completely. It can also be difficult to construct the patterns of the residual stresses based on limited measurements for the whole welded structures. Numerical prediction of welding residual stresses is also challenging due to the complexity of welding process, e.g., localized heating, temperature dependence of the material properties, and moving the heat source, restraint conditions, number of weld pass, and welding sequence.

Residual stress is generally self-balanced in structural components, and the source of residual stress is an incompatible strain field that could be produced by plastic deformation, thermal strain, phase-transformation or other means. Ueda et al. [7] refer to the sum total of all such incompatible strain as “inherent strain,” which can also be referred to as eigenstrain. The idea of so-called “eigenstrain method” is that the residual stresses can be computed once the distributions of eigenstrain are known. In practice, the distribution of eigenstrain can be measured. However, the experimental requirements of the eigenstrain method make its application expensive. Therefore, some simplifications have been made to improve the applicability of such a method [20,21]. In this study, the eigenstrain for the base metal was assumed to be zero, and isotropic nonzero value for the weld was assigned. The residual stress was introduced into the model through the following steps: (a) assume different eigenstrain distributions in base and weld metal respectively. Set the eigenstrain values equal to the thermal expansion coefficients of different regions; (b) load the model by a unit temperature changes; (c) insert the crack and residual stress redistributions.

Finite element model

Two-dimensional axi-symmetry model has been utilized to model the pipe section. Due to the symmetry, only upper-half of the model has been modeled. The model was meshed by first order axi-symmetric elements. To capture the high temperature gradient and microstructure evolution within the weld and heat affected zone (HAZ), a fine mesh was used in this region. Close to the crack tip, mesh size of 0.1 mm × 0.05 mm for the first layer and 0.1 mm × 0.1 mm for the rest of the layers are utilized. The global and local meshes are shown in Fig. 2.

Material properties

The weld metal and base metal were assumed to have the same elastic properties and plastic properties. The rate independent power law strain hardening materials were assumed to have the following form:

(6){\sigma }_f={\sigma }_{\mathrm{0}}{\left(\mathrm{1}+{\displaystyle \frac{\overline{ϵ}}{{ϵ}_{\mathrm{0}}}}\right)}^n,

where σ_f is the flow stress; {\overline{ϵ}}^p is the equivalent plastic strain; the yield stress σ₀ = 581 MPa. {ϵ}_{\mathrm{0}}={\sigma }_{\mathrm{0}}/E is the yield strain; n is the plastic strain hardening exponent. Different thermal expansion coefficients for the base metal (α_b) and weld metal (α_w) are assumed to introduce the residual stresses into the model by eigenstrain method.

Results and discussion

In this section, the distribution of welding residual stresses introduced by eigenstrain method will be investigated. Also, the effect of the introduced residual stresses on the ductile crack growth resistance will be studied.

Residual stress field

As mentioned in the previous section, the eigenstrain method has been used to introduce residual stresses into the model. The eigenstrain for the base metal was assumed to be zero, i.e., α_b = 0. However, the eigenstrain for the weld metal is assumed to be α_w = 0.005. It should be noted that the eigenstrain value selected here are based on our previous study [3,4]. Figure 3 shows the distribution of welding residual stresses along the path from the crack tip (as illustrated in Fig. 2) before and after the crack was inserted.

It can be seen that tensile residual stresses have been introduced at the crack tip. The magnitude of radial and axial stress is about 0.5σ_y before the crack was inserted, and hoop stress has the level about 1.5σ_y. It can also be observed that introduction of the crack enhances the magnitude of residual stresses at the crack tip significantly. Especially, the axial residual stress has increased over 1.5σ_y. In the following sections, the effect of introduced residual stresses on the ductile crack growth resistance will be investigated.

Effect of residual stresses on crack growth resistance

In this section, the effect of residual stresses (as shown in Fig. 3) on the ductile crack growth resistance was investigated. For this part of study, the initial void volume fraction is set to f₀ = 0.005, and material hardening exponent n = 0.05. On the top of the finite element model, external tension has been applied (2 mm displacement). The crack tip opening displacement (CTOD, δ) has been selected as the fracture parameter to describe the crack growth resistance. The CTOD was defined as the displacement of the first node to the crack tip on the crack surface. The ductile crack growth resistance has been calculated for both with and without residual stresses, as shown in Fig. 4.

It can be seen that the introduced residual stresses decrease the ductile crack growth resistance, which corresponds to the results reported in Ref. [4]. To investigate the relative contribution of residual stresses on the ductile crack growth resistance with the crack propagation, the normalized ductile crack growth resistance has also been calculated, as shown in Fig. 5. Here, the resistance with residual stresses was normalized by the resistance at the same crack growth without residual stresses.

As shown in Fig. 5, the effect of residual stresses is significant at crack initiation stage and decreases with the crack propagation, which is also corresponding to the finding shown in Ref. [4].

Effect of initial void volume fraction

The initial void volume fraction f₀ represents the degree of damage in the material. The larger the initial void volume fraction, the larger damage the material has. It is thus interesting to investigate how residual stresses can affect the ductile crack growth resistance for different inherent damage. In this section, two initial void volume fractions have been investigated, i.e., f₀ = 0.001 and f₀ = 0.005 for material hardening exponent n = 0.05. The normalized ductile crack growth resistance has been plotted in Fig. 6.

It can be seen that residual stress has stronger effect on larger initial void volume fraction, which is in accordance with the results shown in Ref. [4]. The ductility becomes better when the initial void volume fraction is lower, and the effect of welding residual stresses is thus weaker. It can also be observed that with the increase of the crack growth, the effect of residual stresses on the ductile crack growth resistance become less dependent on f₀.

Effect of material strain hardening

It is also interesting to investigate the effect of residual stress on ductile crack growth resistance for materials with different hardening. Figure 7 shows the effect of residual stresses on two material hardening exponents, i.e.,n = 0.05 andn = 0.1 with the same initial void volume fraction f₀ = 0.001.

It can be seen that effect of residual stress on the ductile crack growth resistance is stronger for stronger hardening materials. With the increase of the crack growth, the reduction of the crack growth resistance decreases and tends to converge to the case without residual stresses.

Effect of crack size

It has been shown that the fracture toughness and crack growth resistance curve are strongly geometry-dependent [22,23]. Investigation of the effect of constraint on ductile crack growth resistance is important for structural integrity. In this study, effect of residual stresses on ductile crack growth resistance for two crack sizes has been studied, i.e., crack depth a/w= 0.335 and a/w= 0.485. The initial void volume fraction f₀ = 0.005 and material strain hardening exponent n= 0.05. The normalized ductile crack growth resistance curves are shown in Fig. 8.

It can be seen that effect of residual stresses is weaker for deeper crack size case (a/w = 0.485). However, with the increase of ductile crack growth, the effect of residual stress on the ductile crack growth resistance becomes less dependent on the crack size.

Concluding remarks

This paper studied the effect of hyperbaric welding residual stress on ductile crack growth resistance of circumferentially cracked pipe. Two-dimensional axi-symmetry model has been utilized to simulate the pipe section. Eigenstrain method has been employed to introduce welding residual stresses. The ductile crack growth resistance was calculated using complete Gurson model. Based on the results obtained, the following conclusions can be drawn:

• Residual stresses reduce the ductile crack growth resistance. However, the effect of residual stress diminishes with the increase of the crack size.

• Effect of residual stresses on the ductile crack growth resistance is stronger for material with larger initial void volume fraction and stronger hardening.

• Influence of residual stresses on the ductile crack growth resistance is weaker for deeper circumferentially cracked pipe.

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