Simulations of crack extensions in arc-shaped tension specimens of uncharged and hydrogen-charged 21-6-9 austenitic stainless steels using cohesive zone modeling with varying cohesive parameters
Introduction
The cohesive zone modeling (CZM) approach has been used to model crack extensions in laboratory fracture specimens of ductile materials [1], [2], [3], [4], [5], [6]. In order to obtain accurate computational results with the CZM approach for practical applications, the two most important cohesive parameters, cohesive energy and cohesive strength, should be carefully calibrated. Different approaches were used to calibrate the cohesive energy and cohesive strength for the exponential and trapezoidal traction-separation laws for different fracture specimens of different ductile materials, depending on the availability of the experimental load–displacement-crack extension data for the fracture specimens and the experimental data for tensile or notched specimens as, for example, in Roychowdhury et al. [1], Cornec et al. [2], Scheider and Brocks [3], [4], Sung et al. [5], and Wu et al. [6]. In the cohesive zone modeling approach, cohesive parameters are usually assumed as fixed values [1], [2], [3], [4], [5], [6]. The CZM approach with fixed cohesive parameters can be adopted with only two important cohesive parameters, cohesive energy and cohesive strength, to be calibrated.
For a fracture specimen, where the size requirements of the plane strain conditions of the linear elastic fracture mechanics are satisfied at crack initiation and during crack extension, the selection of the cohesive energy can be a fixed value related to the fracture toughness at crack initiation and during crack extension. However, for a cracked thin structure or a fracture specimen cut from the actual applications, the majority of the crack front may not be subjected to the plane strain conditions at crack initiation and during crack extension. Then the cohesive energy should vary to reflect the change from the flat fracture to the slant fracture mode or the change of the constraint conditions due to the change of the plastic zone size and shape along the crack front for a two-dimensional plane stress or plane strain finite element analysis of crack extension with the CZM approach. Schwalbe et al. [7] discussed the possibility of splitting the finite element model for thin cracked structures into two regions where the first region close to crack initiation is governed by the cohesive parameters for a flat crack extension and the second region is governed by the cohesive parameters for a slant crack extension. Simonsen and Törnqvist [8] conducted experiments for crack propagation in large thin fracture specimens of normal strength steel, high strength steel and aluminum A5083 H116 and H321 alloys. Woelke et al. [9] later conducted finite element analyses using the CZM approach with the varying cohesive parameters as functions of the crack extension for crack extension of many thicknesses in the large thin edge-notched specimen of aluminum A5083 H116 of Simonsen and Törnqvist [8]. The width, initial notch length and thickness of the large thin edge-notched specimen are 500 mm, 100 mm and 10 mm, respectively. Woelke et al. [9] used the shell elements with the element size of 12.7 mm, slightly larger than the specimen thickness, in the finite element analyses to model the crack extension in the thin specimen.
Woelke et al. [9] used the experimental load–displacement curve of Simonsen and Törnqvist [8] to calibrate the varying cohesive parameters and demonstrated the accuracy of the plane stress shell finite element analyses with the cohesive zone modeling of crack extension. The computational load–displacement curve of the large thin specimen from the plane stress shell finite element analysis is in good agreement with the experimental data based on the identified varying cohesive parameters. In Simonsen and Törnqvist [8], the final crack extension in the large thin edge-notched specimen is about 35 times of the specimen thickness. Woelke et al. [9] selected the cohesive energy increased from a lower value close to the one corresponding to the fracture toughness for crack initiation to a large value of the cohesive energy for plane strain tension for the crack extension of about 7 times of the specimen thickness. It should be emphasized again that in Woelke et al. [9], the element size in their shell finite element model is scaled with the specimen thickness and the steady-state cohesive energy is based on the work per unit area under the normalized force–elongation curve for plane strain tension from the onset of necking to the onset of shear localization as presented in Nielsen and Hutchinson [10]. The transition from the low cohesive energy for the initial flat fracture to the high cohesive energy for the slant fracture requires the crack to grow about 7 times of the thickness. The results of the plane stress shell finite element analysis catch the trends of the increasing cohesive energy and cohesive strength for the first shell element with the size of about the specimen thickness for the initial flat fracture supposedly under near plane strain conditions.
Andersen et al. [11] conducted three-dimensional finite element analysis of the crack extension in the same thin edge-notched specimen of Simonsen and Törnqvist [8] based on the GTN material model (Gurson [12], Tvergaard [13], and Tvergaard and Needleman [14]). The cohesive energy and cohesive strength as functions of the radial distance to the original notched tip were extracted from the results of the three-dimensional finite element analysis. Then, the extracted varying cohesive energy and varying cohesive strength were adopted in two-dimensional plane stress finite element analyses to simulate the crack extension. They found that in order to use the varying cohesive energy and varying cohesive strength in the two-dimensional plane stress finite element analysis to fit the simulation results from the three-dimensional finite element analysis with the GTN material model, some fitting processes such as increases of the cohesive strength and cohesive energy were still needed. However, the fitted varying cohesive strength and varying cohesive energy for the two-dimensional plane stress finite element analysis follow the general trends of those extracted from the results of the three-dimensional finite element analysis with the GTN material model. It should be noted that the transition crack extension corresponding to the varying cohesive strength from a low value to a steady-state value is about 2 times of the specimen thickness and the transition crack extension corresponding to the varying cohesive energy from a low value to a steady-state value is about 7 times of the specimen thickness. Similarly, the results of the two-dimensional plane stress finite element analysis also catch the trends of the increasing cohesive energy and cohesive strength for the first plane stress element with the size of about the specimen thickness for the initial flat fracture supposedly under near plane strain conditions.
For thin structures such as pressure tubes in nuclear reactors, modeling of crack extensions less than the specimen thickness by a two-dimensional finite element analysis with the CZM approach is an important fracture mechanics application. Sung et al. [15] conducted two-dimensional finite element analyses with the varying cohesive energy to simulate the crack extension in a small thin curved compact tension (CCT) specimen of Zr-2.5Nb pressure tube material. The CCT specimen was cut from an ex-service pressure tube. The width, initial crack length and thickness of the CCT specimen are 17 mm, 8.75 mm and 4.2 mm, respectively. In Sung et al. [15], the final crack extension of 3.88 mm in the as-removed CCT specimen is smaller than the specimen thickness of 4.2 mm. Therefore, the two-dimensional shell or plane stress finite element analyses with the CZM approach of Woelke et al. [9] and Andersen et al. [11] with the crack extension of many thicknesses are not applicable. It should be noted that Sung et al. [16] found that a two-dimensional elastic plane stress finite element analysis of the CCT specimen can be used to model the initial elastic stiffness of the load–displacement curve. Therefore, the CCT specimen can be modeled as a thin plane stress specimen. However, Sung et al. [17] conducted three-dimensional elastic–plastic finite element analyses of the CCT specimen. Their computational results indicated that the majority of the crack front is under plane strain conditions at the maximum load. Therefore, a two-dimensional plane strain finite element analysis to simulate crack extension in the CCT specimen is appropriate to be used with the elastic stiffness adjustment method to account for the initial plane stress load–displacement response of the specimen in Sung et al. [15].
Sung et al. [15] selected the finite element size scaled with the crack tip opening displacement, estimated from the J integral divided by the yield stress, for the simulation of the crack extension in the CCT specimen. The finite element size was selected to catch the maximum opening stress ahead of the crack tip in the two-dimensional plane strain finite element analysis with consideration of the finite deformation of the crack tip under plane strain conditions. Sung et al. [15] selected the maximum opening stress ahead of the initial crack tip as the cohesive strength. Sung et al. [15] selected the bilinear varying cohesive energy as a function of the crack extension by following the trend of the experimental J-R curve of the specimen. The computational load–displacement curve in Sung et al. [15] can fit the experimental data quite well. Also, the computational load-crack extension, crack extension-displacement and J-R curves can fit the general trends of the experimental data. The increase of the varying cohesive energy, following the trends of the experimental J integrals and the trend of the computational separation work rate [18] (or the negative work to open the crack per unit area), can be used to account for the decrease of the plane strain portion of the crack front from the increase of the plastic zone size as the crack extension increases. The two-dimensional plane strain finite element analysis of the initial flat crack extension with the element size scaled with the crack tip opening displacement cannot be modeled by the shell or plane stress finite element analyses with the element size scaled with the specimen thickness as in Woelke et al. [9] and Andersen et al. [11].
It should be noted that when the constraint conditions decrease for the portion of the crack front due to the effect of the free surfaces, the cohesive energy to extend the entire crack front is higher as shown in Sigmund et al. [19], [20] based on the GTN material model. It should be mentioned that the plane strain conditions of the linear elastic fracture mechanics (LEFM) for the CCT specimen are satisfied at crack initiation. However, after a small amount of crack extension, the plane strain conditions of the LEFM are not satisfied anymore. The increase of the cohesive energy for the entire crack front is due to the higher cohesive energy from the decrease of the constraint conditions for the portion of the crack front close to the free surfaces as the crack extension increases. The results of the two-dimensional plane strain finite element analysis in Sung et al. [15] catch the trends of the increasing cohesive energy for the crack extension less than the specimen thickness with the element size scaled with the crack tip opening displacement for the initial flat fracture under near plane strain conditions.
For small thin stainless steel specimens with different hydrogen contents, modeling and testing for crack extension less than the specimen thickness is also an important fracture mechanics application. Wu et al. [6] conducted two-dimensional plane strain finite element analyses with the CZM approach to simulate the crack extension in small side-grooved arc-shaped tension A(T) specimens of uncharged and tritium-charged-and-decayed CF 21–6-9 stainless steels. The side grooves were cut to promote the flat fracture. The width and net thickness of the A(T) specimen are 9.22 mm and 3.81 mm, respectively. The initial crack lengths in the A(T) specimens are in the range of 4.32 mm to 5.75 mm. In Wu et al. [6], the final crack extensions are in the range of 1.61 mm to 2.48 mm which are smaller than the net specimen thickness of 3.81 mm. With the side grooves, the two-dimensional plane strain finite element analyses can be used to model the initial elastic stiffnesses of the experimental load–displacement curves of the A(T) specimens. It should be mentioned that the A(T) specimens are small and the size requirements of the J integral testing are not satisfied as discussed in Kim et al. [21].
Wu et al. [6] selected the fixed cohesive parameters based on the deviation load approach by comparing the computational load–displacement curves from the stationary crack models with the experimental data as in Sung et al. [5] for small single edge bend specimens of additively manufactured 304 stainless steels. In Wu et al. [6], the experimental J integrals at the deviation loads of the A(T) specimens were taken as the reference cohesive energies, and the maximum opening stresses ahead of the initial crack tips in the A(T) specimens were taken as the reference cohesive strengths for the uncharged and tritium-charged-and-decayed A(T) specimens to start the calibration processes. Wu et al. [6] calibrated the fixed cohesive parameters based on fitting the maximum loads of the experimental load–displacement curves. The computational load–displacement curves agreed reasonably well with the experimental data for both uncharged and tritium-charged-and-decayed A(T) specimens. However, the computational load-crack extension and crack extension-displacement curves showed some differences from the experimental data. For example, the computational maximum loads occurred much earlier than the corresponding experimental data.
Sung et al. [22] conducted the two-dimensional plane strain finite element analyses to simulate the crack extensions in the A(T) specimens with the nodal release method. Sung et al. [22] obtained the separation work rates (or the negative work to open the crack per unit crack area) and the maximum opening stresses ahead of crack tips as functions of the crack extension in the A(T) specimens. The computational results indicated that the separation work rates and maximum opening stresses ahead of the crack tips vary as the cracks extend from the original crack tips to the final crack lengths in the small A(T) specimens. The varying separation work rates and maximum opening stresses indicated that the cohesive energy and cohesive strength may vary as the crack extension increases due to the varying constraint conditions along the crack front. It should be mentioned that since the size requirements of the J integral testing are not satisfied for the small A(T) specimens, the cohesive energy may increase as the crack extension increases due to gradual loss of the plane strain conditions along the portions of the crack front from the increase of the plastic zone size.
In this investigation, finite element analyses with fixed cohesive parameters are first conducted to determine the cohesive strengths and cohesive energies to fit the maximum loads of the experimental load–displacement curves of uncharged and hydrogen-charged A(T) specimens of CF 21–6-9 stainless steels. It should be noted that the finite element size near the crack tips for the simulations of crack extensions is scaled with the crack tip opening displacement to catch the maximum opening stresses ahead of the crack tips with consideration of the finite deformation of the crack tips under plane strain conditions as in Sung et al. [22]. The procedures to select the fixed cohesive parameters for uncharged and hydrogen-charged A(T) specimens are similar to those reported in Wu et al. [6] based on the deviation loads and the maximum opening stresses ahead of the initial crack tips determined from the stationary crack models with the initial crack lengths. Next, finite element analyses with varying cohesive parameters are conducted to calibrate the cohesive parameters by following the general trends of the separation work rates and maximum opening stresses ahead of the crack tips in Sung et al. [22] to improve the computational load-crack extension and crack extension-displacement curves for uncharged and hydrogen-charged A(T) specimens. In this investigation, the cohesive energies, cohesive strengths and softening ratios are selected as functions of the crack extension to fit the experimental load–displacement, load-crack extension, and crack extension-displacement curves of uncharged and hydrogen-charged A(T) specimens. The selection processes of the varying cohesive parameters are presented in detail in Appendix A. After the varying cohesive parameters are calibrated for one uncharged A(T) specimen, finite element analyses with the selected varying cohesive parameters are also conducted for two other uncharged A(T) specimens with different initial crack lengths to obtain the computational load–displacement, load-crack extension, crack extension-displacement and J-R curves for comparison with the experimental data in Appendix B. Then, the separation work rate and maximum opening stress ahead of the crack tip in Sung et al. [22] are presented and compared with the selected varying cohesive energy and varying cohesive strength, respectively, for the uncharged A(T) specimen. The plastic zone sizes and shapes at different crack extensions from crack initiation to a large crack extension under fully plastic conditions for the uncharged A(T) specimen are also presented for a possible connection to the varying cohesive energy. Finally, some conclusions are made.
Section snippets
Two-dimensional finite element models of A(T) specimens without and with cohesive elements
Fig. 1 shows schematics of side and front views of an A(T) specimen with side grooves. As shown in the figure, the A(T) specimen has side grooves on the free surfaces. The initial crack length , the specimen thickness , the net specimen thickness , and the initial crack mouth opening displacement (COD) for the uncharged H94-1, H94-2 and H94-4 and hydrogen-charged H94-54 A(T) specimens from Morgan [23] are listed in Table 1. The hydrogen concentration in the hydrogen-charged H94-54 A(T)
Cohesive zone model
The smooth trapezoidal traction-separation law [2], [3], [4] was used to simulate crack extensions in finite element analyses of fracture specimens of ductile metals. Fig. 4 shows a schematic of the normalized smooth trapezoidal traction-separation law used in this study. The smooth traction-separation law has the initial stiff part so that the initial responses of the experimental load–displacement curves of fracture specimens can be fitted well [2], [3], [4], [5], [6]. The smooth
Computational results based on fixed cohesive parameters
In this investigation, finite element analyses without cohesive elements are conducted first to estimate the cohesive parameters to fit the maximum loads of the experimental load–displacement curves for uncharged and hydrogen-charged A(T) specimens. The crack initiations are identified at the deviation loads where the computational load–displacement curves of the finite element analyses with the initial crack lengths without cohesive elements start to deviate from the experimental data. Fig. 5
Computational results based on varying cohesive parameters
The results of the finite element analyses with the fixed cohesive parameters were shown to have reasonable computational load–displacement curves when compared with the experimental data for uncharged and tritium-charged-and-decayed A(T) specimens in Wu et al. [6]. However, the computational load-crack extension and crack extension-displacement curves showed some differences from the experimental data for uncharged and tritium-charged-and-decayed A(T) specimens in Wu et al. [6]. Similarly, the
Comparisons of fixed and varying cohesive parameters approaches
The computational load–displacement, load-crack extension, crack extension-displacement and J-R curves with the fixed and varying cohesive parameters for the uncharged H94-2 specimen are compared with the experimental data in Figs. 14(a), (b), (c) and (d), respectively. As shown in Fig. 14, the computational results with the fixed cohesive parameters can fit the experimental load–displacement curve well while the computational load-crack extension and crack extension-displacement curves show
Conclusions
Two-dimensional plane strain finite element analyses with the cohesive zone modeling approach are adopted to model crack extensions in uncharged and hydrogen-charged arc-shaped tension A(T) specimens of conventionally forged 21–6-9 austenitic stainless steels. Finite element analyses with fixed cohesive parameters are conducted first to select the cohesive parameters to fit the maximum loads of the experimental load–displacement curves for uncharged and hydrogen-charged A(T) specimens. The
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The support of this research by the U.S. Department of Energy is appreciated. Helpful discussions with Dr. Shin-Jang Sung of University of Michigan are appreciated.
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