Evaluation of Electric Current-Induced Improvement of Fracture Characteristics in SUS316

Abstract

The application of high-density pulsed electric current (HDPEC) is one of the effective methods for the modification of material properties in metals. To evaluate fracture behavior modified by HDPEC, critical fracture parameters such as fracture strength, fracture toughness, and fracture profile of crack tip are important criteria. This work investigates the finite element analysis (FEA) based evaluation of improved fracture characteristics by the application of HDPEC in a SUS 316 austenite stainless steel. Tensile tests were first conducted to deduce the modified material properties with different conditions of HDPEC. A series of theoretical considerations was employed to estimate the modified fracture toughness. The relationship between critical fracture strength and critical crack length was numerically determined based on the estimated fracture toughness. The results in FEA showed that critical von Mises stress on the singularity at the crack tip increases as the effect of HDPEC increases. The evolution of increased fracture toughness with respect to conditions of HDPEC was specified. Crack opening profiles were simulated to assist the explanation. The evaluation of fracture parameters in this study proposes that the modified material properties by HDPEC play a positive role to resist crack propagation.

1. Introduction

The prediction of fracture threshold in crack initiation and propagation is crucial for the reliability of materials because most of the mechanical components are broken by the progress of a pre-existing crack in the material.^1–3) For the sake of the evaluation of fracture limits, understanding elastic-plastic behavior on stress singularity at the crack tip is important because crack propagation is determined by stress distribution near the singularity. Although it is still difficult to enhance the resistance of crack initiation and propagation, there exist several methods that have been broadly used. For example, crack resistance by the geometrical blunting of the crack tip can slow crack propagation in certain environmental loading conditions,^4–6) and improve fracture toughness due to elemental reorganization.^7,8)

Recently, there have been reports that the fracture behavior of metallic materials can be modified by the high-density pulsed electric current (HDPEC).^9–13) It leads to the resistance of crack propagation with crack tip healing and modification of microstructures. It has been verified as a kind of method to improve fracture characteristics. Because of the scientific proof, the HDPEC has been considered being a promising method used for industry due to the simplicity in the maintenance of mechanical components. The advantage is that it does not require much time and effort to completely dissemble the mechanical parts of a running machine. Based on the advantages of use, it has recently become interesting to optimize conditions of HDPEC for a given material. The modified material properties depend on the conditions of HDPEC, which leads to different fracture characteristics. To predict the degree of modification quality, finite element analysis (FEA) is one of the possible methods because it is broadly used as a low-cost assessment to describe a model of cracking in a material.¹⁴⁾

In this study, we present the prediction of fracture parameters through the HDPEC-induced modification of material properties such as yield stress, elastic modulus, and strain hardening exponent. Fracture parameters were evaluated using a combination between the elastic-plastic fracture theory and the FEA. In tensile tests, the modified material properties were obtained with the various conditions of HDPEC. The obtained material properties were used to evaluate the modification of fracture toughness and critical applied stress given by critical crack length. FEA was used to predict the fracture strength on singularity at the crack tip. The evolution of increased fracture toughness by HDPEC was achieved. In addition, crack profiles such as crack opening displacements (CODs) and crack tip opening displacements (CTODs) were considered to evaluate the fracture threshold improved with the application of HDPEC.

2. Simulation Methodology

The flowchart of the implemented process in this study is illustrated in Fig. 1. It starts with the application of HDPEC on specimens to conduct the tensile test. The elastic modulus E, yield stress σ_y, and strain hardening exponent n depending on coefficient m were obtained from the tensile test. A series of elastic-plastic fracture theories was used to estimate the critical fracture toughness K_IC. Based on the analytical outputs for critical applied stress σ_ac, length of the plastic yielding zone r_y, and critical crack length a_c, FEA with the isotropic hardening model was performed. Each calculation was repeatably performed to quantify data-sets. It calculates the local fracture stress distribution in terms of von Mises stress σ_vm and their critical value σ_cm on singularity at the crack tip. Evolution of the increased ratio $\sigma _{\text{cm}}/\sigma _{\text{cm}}^{\text{o}}$ was verified by comparison with the value $\sigma _{\text{cm}}^{\text{o}}$ of the original state. The ratio $K_{\text{IC}}/K_{\text{IC}}^{\text{o}}$ between enhanced K_IC and $K_{\text{IC}}^{\text{o}}$ of the original state of critical fracture toughness was determined. CODs and CTODs were additionally calculated to explain the enhanced fracture threshold. Consequently, the comparison between the original and the enhanced fracture parameters with the application of HDPEC was demonstrated.

Fig. 1

Schematic flowchart of the investigated process.

2.1 Tensile test

A commercial SUS316 stainless steel (JIS G 4305) was used for tensile tests to obtain modified material properties by HDPEC. The chemical composition of the material is listed in Table 1. All specimens were cut with electro-discharge machining (EDM) for dumbbell-shaped geometry as shown in Fig. 2. After cutting the specimens, annealing heat treatment was applied to release the residual stress induced by EDM and to generalize the microstructure. The temperature was gradually increased to 900°C with 4 h, keeping for 10 min, and cooled to room temperature in a furnace.

Table 1 Chemical composition (mass%) of SUS316. (Fe: balanced)

Fig. 2

Illustration of the specimen for tensile test and application of HDPEC.

In order to distinguish the effect of HDPEC, different conditions were applied. Untreated one was denoted by case A. The treated conditions were denoted as case B (two pulses of 150 A/mm² for 5 ms), case C (one pulse of 150 A/mm² for 10 ms), case D (two pulses of 300 A/mm² for 5 ms), and case E (one pulse of 300 A/mm² for 10 ms), respectively. The current densities j, applied time t and the number of pulses p were summarized in Table 2.

Table 2 The condition of HDPEC for the tensile test.

2.2 Simulation analysis and elastic-plastic fracture

In this simulation of elastic-plastic fracture, we briefly introduce the governing equations. In a material domain Ω given by a force f, surface force T on the boundary ∂Ω_T and a displacement d on the boundary ∂Ω_d can be imposed. Mechanical equilibrium of continuous material is defined as div σ + f = 0 on Ω, σ.n = T on ∂Ω_T, and u = d on ∂Ω_d. Where σ and u are stress and displacement field, respectively. The constitutive law can be expressed with elastic and plastic strains ($\varepsilon _{\text{ij}}^{\text{e}} + \varepsilon _{\text{ij}}^{\text{p}}$) in the following equations:   

\begin{equation} \varepsilon_{\text{ij}} = \varepsilon_{\text{ij}}^{\text{e}} + \varepsilon_{\text{ij}}^{\text{p}}, \end{equation} (1)

  

\begin{equation} \sigma = E_{\text{ijkl}}\varepsilon_{\text{kl}}^{\text{e}} = \mathbb{E}{:}\nabla_{\text{s}}u + \sigma_{\text{p}}, \end{equation} (2)

where $\mathbb{E}$ is the elasticity tensor of the matrix E_ijkl, ∇_su the symmetric gradient of u related to elastic strain $\varepsilon _{\text{kl}}^{\text{e}}$, and σ_p plastic stress. The von Mises yield criterion is expressed by eq. (3), let σ₁, σ₂, and σ₃ denote the principal stresses:   

\begin{align} f(\sigma_{\text{ij}},\varepsilon_{\text{ij}}^{\text{p}}) &= \sqrt{\frac{1}{2}[(\sigma_{1} - \sigma_{2})^{2} + (\sigma_{1} - \sigma_{3})^{2} + (\sigma_{2} - \sigma_{3})^{2}]} \\ &\quad - \sigma_{\text{y}}\varepsilon_{\text{ij}}^{\text{p}} = 0. \end{align} (3)

Discretization for finite elements in a mesh domain Ω^h of Ω is given by eq. (4):   

\begin{equation} \nu^{\text{h}}(x) = \sum\nolimits_{\text{i}}\nu_{\text{i}}N_{\text{i}}(x), \end{equation} (4)

mesh displacement field ν^h of Ω^h is discretized to that of elements ν_i with the interpolation function N_i(x). The finite element formulation is given as follows:   

\begin{align} &\int_{\partial \Omega^{\text{h}}} TN\,dS + \int_{\partial \Omega_{\text{d}}^{\text{h}}}TN\,dS + \int_{\Omega^{\text{h}}}fN\,dV - \int_{\Omega^{\text{h}}}\sigma_{\text{p}}\nabla N\,dV \\ &\quad = \int_{\Omega^{\text{h}}}\mathbb{E}u_{\text{j}}\nabla N_{\text{i}}\nabla N_{\text{j}}\,dV, \end{align} (5)

  

\begin{equation} K^{\text{e}}.U = F^{\text{S}} + F^{\text{R}} + F^{\text{V}} - B.\sigma_{\text{p}}. \end{equation} (6)

Where K^e is the elastic stiffness, and U the displacement of mesh nodes. Equivalent nodal forces are composed of terms such as surface nodal force F^S, reaction nodal force F^R of imposed displacement, volumetric nodal force F^V and interior term B.σ_p.

In the plastic yielding theory, a radius of the plastic yielding zone proposed by Kuyawski and Ellyin^3,15–17) can be defined by the following equation:   

\begin{equation} r_{\text{y}} = \frac{1}{6\pi (1 + n)}\left(\frac{K_{\text{I}}}{\sigma_{\text{y}}} \right)^{2}, \end{equation} (7)

where K_I is the stress intensity factor in the mode I fracture. The plastic zone mainly depends on the modification of n and σ_y in the true stress-strain relation. To identify a series of plasticity-corrected crack lengths $a^{*}$, the eq. (8) was used with a length of initial crack a₀:^18,19)   

\begin{equation} a^{*} = a_{0} + r_{\text{y}}, \end{equation} (8)

then it yields the elastic-plastic fracture toughness. Since it assumes r_y ≪ a₀, the relationship between the given applied stress σ_a and $a^{*}$ gives $K_{\text{I}} = \sigma _{a}\sqrt{2\pi a^{*}} $, and it can be re-expressed from eqs. (7) and (8) as the follows:   

\begin{equation} K_{\text{I}} = \sqrt{\frac{2\pi a_{0}\sigma_{a}{}^{2}}{1 - (\sigma_{a}/\sigma_{\text{y}})^{2}/[3(1 + n)]}}. \end{equation} (9)

The σ_ac can be calculated with the plasticity-corrected critical crack length $a_{\text{c}}^{*}$ based on the critical relation $K_{\text{IC}} =\sigma _{a\text{c}}\sqrt{2\pi a_{\text{c}}^{*}} $. Thus, it can be expressed as the following equation:   

\begin{equation} \sigma_{a\text{c}} = \frac{K_{\text{IC}}}{\sqrt{2\pi (a_{\text{c}} + r_{\text{y}})}}. \end{equation} (10)

Equation (10) is used to determine the relationship between σ_ac and a given a_c in FEA. Figure 3(a) represents half-cut meshes of the investigated model for FEA, in which the same geometry of the tensile specimen was designed. At the right upper side (center of the full model), a single notch was made to start a crack propagation. The displacement boundary conditions (BCs) were applied as u(y) = 0 to define symmetric boundaries and u(x) = 0 for directional fixation. The deformed model and the calculated stress contour are shown in Fig. 3(b). A mesh refinement area in Fig. 3(c) shows the example contour of σ_vm distribution when a given σ_ac and a_c were applied. We have designed the FEA model with the mesh refinement area which exists at the crack tip. If the crack length varies, it will move in the corresponding location of the crack tip. We have used the finest mesh available to our computational resources. This simulation assumes the plane stress condition using the two-dimensional von Mises yield criterion on account of the thickness of geometry.

Fig. 3

Investigated model for FEA with a single notch, (a) finite element mesh with half-cut of geometry, (b) stress distribution of the half-cut, and (c) mesh refinement area near the crack tip.

3. Results

3.1 Evaluation of fracture toughness

Based on the true stress-strain relation as shown in Fig. 4, the relationship between the modified material properties and the conditions of HDPEC was determined. In Table 3, a series of σ_y was deduced from the 0.2% proof stress. Plasticity constants (m and n) were predicted from fittings with typical working hardening equation, $\varepsilon = \frac{\sigma }{E} + ( \frac{\sigma }{m} )^{1/n}$.²⁾ The Young’s modulus E = 195 GPa was determined from elastic rule σ = Eε_e. The Poisson’s ratio ν = 0.295 was used based on the reference value.⁹⁾ A slight mismatch of the fitting in the elastic domain revealed, which has been reported in materials having certain ductility.²⁰⁾ Nevertheless, a prediction of the plasticity constants by curve fitting provides practical application to avoid the time-consuming of FEA. Based on the summarized variables (E, ν, σ_y and n depending on m), the analytic and simulation inputs were determined. A series of numerically predicted K_IC for each HDPEC condition was plotted in the inset bar chart of Fig. 5. The fracture toughness of case A was calculated based on the reference value of this material, approximately 3200 MPa·mm^1/2.²¹⁾ To estimate the modified K_IC by the application of HDPEC, the same crack length in the evaluation of case A was substituted to the eq. (9). Figure 5 shows the prediction of the relationship between σ_ac and a_c with eq. (10) and FEA estimation. Where the symbols indicate the predicted stress in FEA as the critical values. The value of K_IC increases as the increase of HDPEC density, i.e., higher σ_ac at a given a_c is needed to reach the K_IC. The higher current density (cases D and E) has the better effect than the lower one (cases B and C). It suggests that, in terms of critical fracture, higher density of HDPEC induces higher σ_ac in the cracking threshold. Moreover, the result also shows that the HDPEC condition of two pulses (case D) is more effective than that of one pulse (case E) even they have the same applied time (10 ms). The analytic results mean that the modified material properties play a determinative role in the relationship between σ_ac and a_c.

Fig. 4

True stress-strain relation obtained from tensile tests with (cases B, C, D, and E) and without (case A) the application of HDPEC.

Table 3 Material properties deduced from the tensile test.

Fig. 5

Predicted results between critical applied stress and crack length. Inset: predicted fracture toughness.

3.2 Local fracture strength

Based on the predicted critical fracture threshold values, the local fracture stress distribution was calculated with the FEA. Figure 6 gives the calculated σ_vm as a function of the distance r from the crack tip on the propagation direction (θ = 0) in the constant crack length (a_c = 1 mm). The values of σ_vm for cases B, C, D, and E are higher than that of case A for any distance. The σ_vm was calculated and plotted simultaneously in the range of r ≥ r_y. The r_y was estimated by eq. (7). The singularity-dominated assumption was used based on the following equation:³⁾   

\begin{equation} \sigma_{\text{vm}} = \frac{\sigma_{a}(a_{0} + r)}{\sqrt{2a_{0}r + r^{2}}},\quad (r \geq r_{\text{y}}). \end{equation} (11)

Fig. 6

Predicted distribution of von Mises stress versus the distance r from the crack tip when θ = 0.

The σ_cm is predicted by the values of σ_vm when r = r_y. Figure 7 shows the determination of σ_cm normalized by the value of the original state (case A), $\sigma _{\text{cm}}/\sigma _{\text{cm}}^{\text{o}}$. Based on the repeat calculations for different crack lengths, the enhanced values were determined, where a/w is the crack length per width in the geometry of FEA. To quantify the effectiveness of fracture resistance, a simple expression can be considered as the inset equation in Fig. 7. The inset figure shows the ratio of increased σ_cm for each HDPEC condition. The positive effect of HDPEC is verified as expected. The ratio of cases B, C, D, and E is higher than that of case A. Notably, case D shows an increased ratio of 17.2%, it can be thought as the best condition of HDPEC in this study.

Fig. 7

FEA results of normalized local fracture strength with crack length. Inset: increased ratio (%) of local fracture strength for each case.

To quantify the degree of modified K_IC, we propose as follows:   

\begin{equation} K_{\text{I}} = \sigma_{a}\sqrt{\pi a} f\left(\frac{a}{w} \right), \end{equation} (12)

and a critical condition of eq. (12) gives eq. (13):   

\begin{equation} K_{\text{IC}} \propto \sigma_{\text{cm}}\sqrt{a_{\text{c}}} f\left(\frac{a}{w} \right), \end{equation} (13)

where the function f(a/w) is to assume a periodic array of colinear cracks, f(a/w) = [2w/(πa) tan(πa/(2w))]^1/2. The inset figure of Fig. 8 shows the evolution of K_I as a function of the crack length in analytic and FEA results. It has an agreement with both two approaches. Since fracture occurs when the K_I reaches the critical one K_IC, K_I/K_IC = 1, eq. (14) can be proposed to determine the increased ratio by the effect of HDPEC:   

\begin{equation} \frac{K_{\text{IC}}}{K_{\text{IC}}^{\text{o}}} = \frac{\sigma_{\text{cm}}}{\sigma_{\text{cm}}^{\text{o}}}\sqrt{\frac{a_{\text{c}}}{a_{\text{c}}^{\text{o}}}}. \end{equation} (14)

Fig. 8

FEA results of normalized critical fracture toughness with crack length. Inset: evolution of K_I with FEA and analytic results.

Where $K_{\text{IC}}^{\text{o}}$, $\sigma _{\text{cm}}^{\text{o}}$, and $a_{\text{c}}^{\text{o}}$ are the values of original state (case A) of K_IC, σ_cm, and a_c. Figure 8 indicates the calculated values of $K_{\text{IC}}/K_{\text{IC}}^{\text{o}}$. The increased ratio is shown in Fig. 9. As expected, the enhancement of $K_{\text{IC}}/K_{\text{IC}}^{\text{o}}$ in cases B, C, D, and E was confirmed. The increased ratio of case D was evaluated to be 16.26%. It can be thought that the significant enhancement of K_IC can be expected by applying multiple pulses of HDPEC.

Fig. 9

Increased ratio (%) of modified fracture toughness for each case with compared to original fracture toughness.

3.3 Estimation by crack profiles

In addition to the stress intensity concept, the CTODs (δ) can be also considered in terms of fracture parameters. Equation (15) describes generally the relationship between K_I and δ:³⁾   

\begin{equation} \delta = \frac{K_{\text{I}}{}^{2}}{\alpha E\sigma_{\text{y}}}, \end{equation} (15)

where α is a dimensionless constant. The critical CTOD (δ_c) can be formulated with the strip-yield assumption in eq. (16):³⁾   

\begin{equation} \delta_{\text{c}} = \frac{K_{\text{IC}}{}^{2}}{\alpha E\sigma_{\text{y}}} = \frac{8\sigma_{\text{y}}a_{\text{c}}}{\alpha \pi E}\ln \sec \left(\frac{\pi \sigma_{a}}{2\sigma_{\text{y}}} \right). \end{equation} (16)

It is a fracture criterion which proposes that fracture occurs when δ reached δ_c. Figure 10 shows the simulated COD profiles of cases A, B, C, D, and E when the a_c of 4 mm was used. The values of cases B, C, D, and E are larger than that of case A at any given crack position. In the inset of Fig. 10, the comparison of CTODs between the analytic and FEA estimations was shown. The values of δ_c in the theory-axis of inset were calculated by eq. (16). Where α = 2 was used. In Fig. 11, the simulated values of CTOD were plotted with the predicted lines of cases A and D by eq. (16). The evolution of critical CTODs for each condition is plotted. The values of CTOD obtained in cases B, C, D, and E are larger than that obtained in case A. Generally, a crack tip can be blunted due to the dominant plasticity. The simulation result shows also the blunting of the crack tip. The prediction in Fig. 11 provides an alternative evaluation of fracture parameters. The FEA results show also a gradual increase in δ_c. When the CODs are determined by equivalent state between σ_ac and a_c, the increase of σ_y provides higher crack tip blunting. It can be interpreted by the resistance of crack propagation.

Fig. 10

COD profiles of each case obtained by FEA when the constant crack length is posed (a_c = 4 mm). Inset: comparison of critical CTODs between FEA and theory.

Fig. 11

Critical CTOD obtained by FEA versus critical crack length. Lines indicate the predicted plots of cases A and D.

4. Discussion

This study elaborates on the numerical evaluation of the modified fracture parameters by HDPEC in SUS316 stainless steel. The experimentally obtained material properties are used to predict the critical fracture parameters such as σ_cm, K_IC, and CTOD. Based on the FEA results, the following issues were considered.

4.1 Role of HDPEC in strengthening mechanism

Getting a higher σ_y and lesser n was confirmed as the effect of HDPEC treatment. The density of HDPEC is high, the effect is strong. The effects of 150 A/mm² were not influenced by a different number of pulses, whereas the significant different effect was obtained in cases of 300 A/mm² even if the total applied time is the same. One reason for the different results induced by different density of HDPEC can be considered due to Joule heating-induced temperature rising ΔT_J of material, ΔT_J = ρj²t/(C_pd_m).²²⁾ Where ρ is the electrical resistivity, C_p the specific heat capacity, and d_m the density of a material, respectively. Based on the constants of the material (ρ = 7.4 × 10⁻⁷ Ω·m, C_p = 502 J/(kg K), and d_m = 7.87 × 10³ kg/m³) and t = 10 ms, the values of ΔT_J for 150 A/mm² and 300 A/mm² were calculated to be 42.1 and 168.6°C, respectively. Higher temperature could often more energy thereby promote the modification of materials. On the other hand, the high-density electric current induces the collective drift electrons which can push atoms as a force to promote their motion, which can be defined as electron wind force (EWF),²³⁾ F_EW = (ρ_d/N_d)en_ej. Where F_EW is the EWF, ρ_d the specific resistivity due to dislocation, N_d the dislocation density, e the electron charge, and n_e the electron density, respectively. It has been reported that the microstructure of materials such as dislocation, crystal texture, grain size, and phase structure could be modified by the high-density electric current.^{9–13,22,24,25)} Dislocation motion induced by EWF could result in the accumulation of dislocations around the grain boundary, thereby form some new grain boundaries and lead to the formation of small grains. This partial grain refinement can decrease the average size of grain, thus increase the strength of the material. Therefore, the most important reason of the effect induced by HDPEC is considered to be due to EWF. The increased EWF with the increase of HDPEC density increases the effect of the enhancement of material strength. Furthermore, two pulses could increase the EWF comparing with one pulse due to the abruptly increased voltage during the application of pulsed current for the same application time.

4.2 Evaluation of modified fracture parameters

To judge the quality of modified fracture resistance affected by HDPEC, a series of the following equations can be employed as a criterion:   

\begin{equation} \frac{K_{\text{IC}}}{K_{\text{IC}}^{\text{o}}} > 1, \end{equation} (17)

  

\begin{equation} \frac{K_{\text{IC}}}{K_{\text{IC}}^{\text{o}}} = \frac{\sigma_{\text{cm}}}{\sigma_{\text{cm}}^{\text{o}}}\sqrt{\frac{a_{\text{c}}}{a_{\text{c}}^{\text{o}}}} = 1, \end{equation} (18)

and   

\begin{equation} \frac{K_{\text{IC}}}{K_{\text{IC}}^{\text{o}}} < 1. \end{equation} (19)

It depicts whether the material has better fracture resistance or not. Based on $K_{\text{IC}}/K_{\text{IC}}^{\text{o}} = 1$ for the original state, if $K_{\text{IC}}/K_{\text{IC}}^{\text{o}} > 1$, the fracture toughness of the material is enhanced by the HDPEC. On the other hand, $K_{\text{IC}}/K_{\text{IC}}^{\text{o}} < 1$ indicates that damage could be induced. As shown in Fig. 9, the positive modification of fracture behavior has been demonstrated. It provides a method to evaluate the degree of improved fracture parameters by HDPEC.

5. Conclusion

In this study, the enhancement of fracture characteristics of SUS316 stainless steel by HDPEC was analyzed with the combination between fracture theory and FEA of cracking. From the analytical and simulation results, the following conclusion can be obtained,

1. (1)    The yield stress increases and the strain hardening exponent decreases with the increase of HDPEC density. The improvement of mechanical property can be considered to be due to the EWF-induced modification of microstructures. The modification of dislocations induced by EWF leads to partly grain refinement, thereby decreases the average size of grain at certain conditions of HDPEC.
2. (2)    The evaluation of K_I values for each HDPEC condition provides a way to predict the modified fracture toughness which is used to determine the relationship between the critical applied stress and critical crack length. Results in FEA show that the distribution of critical von Mises stress on singularity at the crack tip is increased with the increase of HDPEC density. The modified quality by higher density and more pulses shows the higher fracture strength in this study. Cases treated by HDPEC need more stress to propagate a crack than that without HDPEC treatment.
3. (3)    The enhancement of fracture toughness was also confirmed in FEA. The enhancement shows the best effect under the high density and multiple pulses of HDPEC.
4. (4)    The results in CTODs show that the plasticity-induced crack tip blunting at a given crack length. In this model, an agreement between the theory and FEA was verified.

Consequently, the application of HDPEC has the advantage to resist crack propagation with the modification of material properties. The enhancement of local fracture strength on the singularity at the crack tip, fracture toughness, and critical CTOD was realized.

Acknowledgments

This work was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (S) No. 17H06146.

REFERENCES

• 1)   A.A.  Griffith: Philos. Trans. R. Soc. A 221 (1921) 163–198.
• 2)  R.I. Stephens, A. Fatemi, R.R. Stephens and H.O. Fuchs: Metal Fatigue in Engineering, (John Wiley & Sons, New York, 2000).
• 3)  T.L. Anderson: Fracture Mechanics Fundamentals and Applications, (CRC press Taylor&Francis, Florida, 2017).
• 4)   S.  Suresh and  R.A.  Ritchie: Metall. Trans. A 13 (1982) 1627–1631.
• 5)   Ilhamdi,  T.  Kakiuchi,  H.  Miura,  T.  Fukihara and  Y.  Uematsu: Mater. Trans. 59 (2018) 1296–1303.
• 6)   A.  Evans: Int. J. Fract. 10 (1974) 251–259.
• 7)   H.  Somekawa: Mater. Trans. 61 (2020) 1–13.
• 8)   H.Y.  Liu,  G.T.  Wang,  Y.W.  Mai and  Y.  Zeng: Compos. B Eng. 42 (2011) 2170–2175.
• 9)   A.  Hosoi,  T.  Nagahama and  Y.  Ju: Mater. Sci. Eng. A 533 (2012) 38–42.
• 10)   Y.  Tang,  A.  Hosoi,  Y.  Morita and  Y.  Ju: Int. J. Fatigue 56 (2013) 69–74.
• 11)   Y.  Tang,  A.  Hosoi,  Y.  Iwase and  Y.  Ju: Mater. Trans. 54 (2013) 2085–2092.
• 12)   J.  Jung,  Y.  Ju,  Y.  Morita,  Y.  Toku and  Y.  Uematsu: Mater. Trans. 57 (2016) 2104–2109.
• 13)   J.  Jung,  Y.  Ju,  Y.  Morita and  Y.  Toku: Int. J. Fatigue 103 (2017) 419–425.
• 14)   K.  Toishi,  Y.  Miki and  N.  Kikuchi: ISIJ Int. 59 (2019) 865–871.
• 15)   D.  Kujawski and  F.  Ellyin: Eng. Fract. Mech. 25 (1986) 229–236.
• 16)   K.H.  Benrahou,  M.  Benguediab,  M.  Belhouari,  M.  Nait-Abdelaziz and  A.  Imad: Comput. Mater. 38 (2007) 595–601.
• 17)   D.S.  Antolovich and  R.W.  Armstrong: Prog. Mater. Sci. 59 (2014) 1–160.
• 18)   J.R.  Rice: J. Appl. Mech. 35 (1968) 379–386.
• 19)   D.E.  Hauf,  D.M.  Parks and  H.  Lee: Mech. Mater. 20 (1995) 273–289.
• 20)   M.  Kamaya: Int. J. Press. Vessels Piping 137 (2016) 1–12.
• 21)   W.J.  Mills: Int. Mater. Rev. 42 (1997) 45–82.
• 22)   D.  Waryoba,  Z.  Islam,  B.  Wang and  A.  Haque: J. Alloy. Compd. 820 (2020) 153409.
• 23)   H.  Conrad: Mater. Sci. Eng. A 287 (2000) 276–287.
• 24)   W.  Dai,  X.  Wang,  H.  Zhao and  X.  Zhao: Mater. Trans. 53 (2012) 229–233.
• 25)   Y.  Sakaida,  K.  Iwai and  S.  Asai: Mater. Trans. 48 (2007) 2872–2876.

Nomenclature

a₀

initial crack length

a_c

critical crack length

$a_{\text{c}}^{\text{o}}$

original a_c

$a^{*}$

effective crack lengths

$a_{\text{c}}^{*}$

critical $a^{*}$

C_p

specific heat capacity

d

displacement

d_m

density of a material

E, E_ijkl, $\mathbb{E}$

elastic modulus

F^S

surface nodal force

F^R

reaction nodal force

F^V

volumetric nodal force

f

force

K_I

stress intensity factor in mode I

K_IC

fracture toughness

$K_{\text{IC}}^{\text{o}}$

original K_IC

K^e

elastic stiffness

m

hardening coefficient

N_i(x)

interpolation function

n

strain hardening exponent

r_y

length of plastic yielding zone

T

surface force

U

displacement of mesh nodes

u

displacement field

δ

CTOD

δ_c

critical CTOD

$\varepsilon _{\text{ij}}^{\text{e}}$

elastic strain

$\varepsilon _{\text{ij}}^{\text{p}}$

plastic strain

ν

Poisson’s ratio

ν^h

mesh displacement field

ν_i

discretized displacement field

ρ

electrical resistivity

σ, σ_ij

stress field

σ_1,2,3

principal stresses

σ_a

applied stress

σ_ac

critical σ_a

σ_cm

critical σ_vm

$\sigma _{\text{cm}}^{\text{o}}$

original σ_cm

σ_p

plastic stress

σ_vm

von Mises stress

σ_y

yield stress

Ω

material domain

Ω^h

mesh domain

∂Ω_d

displacement boundary

∂Ω_T

surface boundary