# Fatigue Crack Growth from Notches: A Numerical Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material Model

_{0}, Y

_{Sat}, and C

_{Y}are material parameters and ${\overline{\epsilon}}^{p}$ is the equivalent plastic strain.

## 3. Numerical Model

_{0}, equal to 8.096 mm. Different notch radii, r, equal to 8, 4, 2, and 1 mm were modeled, as shown in Figure 2c–f, respectively. As Figure 2 indicates, only ¼ of the specimen was modeled considering adequate boundary conditions. A small thickness of 0.1 mm was considered in the simulations, in order to reduce the numerical effort. Both plane stress and plane strain states were studied, applying proper boundary conditions, as indicated in Figure 2g,h, respectively. In the case of plane strain, an additional boundary condition is necessary to eliminate out-of-plane deformation.

_{max}= 400 N and F

_{min}= 4 N, leading to a stress ratio R = 0.01 and to a maximum remote stress of 80 MPa.

^{2}elements, to accurately quantify stress and strain gradients. Out of the crack tip region, a coarser mesh was introduced to reduce the computational cost. The finite element mesh comprised 7175 3D linear isoparametric elements and 7359 nodes. Two load cycles were applied between crack propagations, each one conducting to a crack increment of 8 μm that took place at minimum load. The simulations stopped when the total crack propagation reached 1272 μm, i.e., after 159 crack increments. The contact of the crack flanks was modeled considering a rigid flat surface aligned with the crack symmetry plane. Some simulations were conducted with the possibility to overlap the crack flanks, i.e., for these simulations the flat surface was removed enabling the interference of the crack flanks and disabling crack closure.

_{0}equal to 8.096 mm.

## 4. Results

#### 4.1. Typical CTOD-F Curves

_{p}, versus F curve. CTOD

_{p}is obtained from CTOD by subtracting the elastic CTOD, CTOD

_{e}. As said before, at stretches B–C and D–G, the material experiences elastic deformation and therefore CTOD

_{p}remains constant. In the elastic–plastic regimes, C–D and G–H, there is a quick increase and decrease, respectively, of the plastic deformation, as shown in Figure 3b. Figure 3a,b show the elastic and plastic CTOD ranges, respectively, δ

_{e}and δ

_{p}. A linear elastic relation was found between the plastic CTOD range and da/dN [25,28]; therefore, the trends observed of δ

_{p}are those expected for da/dN.

#### 4.2. Unnotched-Cracked Specimen in No Contact Simulations

_{p}versus crack propagation, Δa, for the three material models studied and both plane stress and plane strain conditions. A great influence of material is evident. Higher values of δ

_{p}are achieved for AA6082-T6, which may be attributed to the lower yield stress of this alloy (see Table 1). Regarding the AA7050-T6, the pure kinematic hardening model provides greater resistance to plastic deformation than the isotropic model, since the level of the stress–strain curve (in uniaxial tension) is higher for the pure kinematic hardening model than for the isotropic hardening model [34]. The effect of stress state varies significantly with material. It has a more pronounced effect in the AA6082-T6 model, where the plane stress case offers less resistance to plastic deformation. For the AA7050-T6 models, there is some influence for small crack increments, which disappears as the crack propagates. The lower values of δ

_{p}observed for plane strain state compared with plane stress state could be expected considering that stress triaxiality due to plane strain state promotes lower levels of plastic deformation. Concerning the effect of crack increment, there is an initial peak of deformation that results from the material not having previous hardening when first loaded. The application of the following loads produces hardening, which explains the progressive reduction of δ

_{p}. The minimum value indicates the end of this initial transient regime. On the other hand, the increase of crack length increases the stress level ahead of crack tip, which explains the progressive increase of δ

_{p}with Δa. The variation is linear with a slope that depends on material, being higher for the AA6082-T6. Figure 4b plots the slope of the linear increase, m, versus δ

_{p}(measured for a crack increment of 800 μm). The increase of δ

_{p}promotes a greater influence of crack length, and a linear relation exists between m and δ

_{p}.

#### 4.3. Insertion of Contact at the Crack Flanks in the Unnotched-Cracked Specimen

_{p}against Δa with and without contact at the crack flanks for the three materials in study, and plane strain and plane stress conditions. As can be seen, in plane strain conditions (Figure 5a,c,e), the contact simulations have a slight reduction of δ

_{p}, but maintaining the trends observed for the no contact simulations. For the plane stress cases (Figure 5b,d,f), the insertion of contact at the crack flanks causes a major reduction of δ

_{p}with Δa. The reduction of δ

_{p}is more pronounced for AA7050-T6 with pure isotropic behavior, and the decrease rate is higher for pure kinematic behavior. On the other hand, the AA6082-T6 shows a relatively slow reduction of δ

_{p}and the stabilization occurs for higher values of Δa. The increase of δ

_{p}with Δa observed for the no contact simulations is almost eliminated with the contact of crack flanks. Anyway, there is a small increase of Δa with δ

_{p}, which is evident in Figure 5d. The minimum values, indicated by small arrows in Figure 5d,f, indicate the end of a transient regime associated with the formation of residual plastic wake. This transient regime is much more extensive than the transient behavior associated with the stabilization of cyclic plastic deformation, observed in Figure 4a.

_{open}is the crack opening load and represents the percentage of load cycle during which the crack is closed. The variations of δ

_{p}observed in Figure 5a,d,f for the simulations with contact, are perfectly symmetric to those observed in Figure 6. This means that the decreases seen in Figure 5 have exactly the same upward trends seen in Figure 6. U* increases with Δa and stabilizes for higher values of Δa, which explains the reduction of δ

_{p}with Δa and the consequent stabilization. The growth rate of U* with Δa is higher for AA7050-T6 with pure kinematic behavior, and higher values of U* are achieved for AA7050 with pure isotropic behavior, which are in accordance with Figure 5d,f.

_{p}observed in Figure 5d,f, compared to the no contact simulations. Under plane strain conditions, the level of crack closure is significantly lower, less than 10%, which explains the small variation observed in Figure 5a,c,e.

_{p}is evident when the contact at the crack flanks is introduced. The effective load range has a major effect on plastic CTOD range.

#### 4.4. Insertion of a Notch with Different Radius in No Contact Simulations

_{p}with Δa without contact, at the crack flanks, for different notch radii, r = 1, 2, 4, and 8 mm, for the three materials in study assuming both plane strain and plane stress states. The inclusion of the notch reduces the plastic CTOD range, and this effect increases with the radius. This means that the notch has a protective effect on δ

_{p}and therefore on da/dN, which results from a smoothing effect of stresses around crack tip. There is only one exception for the case of the AA6082-T6 in plane strain state, Figure 7c. The increase in r decreases δ

_{p}, promoting a higher fatigue life as expected since stress-concentration factors decrease with the increase of r. The protective effect of the notch may sound strange. However, note that the crack length of the unnotched crack is 8.096 mm, as illustrated in Figure 8a. The protective effect of the notch is observed comparing the results of cracked geometries exhibited in Figure 8a,b. On the other hand, the comparison between cracked geometries in Figure 8b,c indicates that the notch accelerates da/dN, and this is the comparison typically made in literature. The dashed lines added to Figure 7, indicate the behavior expected for the cracked geometry of Figure 8c, i.e., assuming that the crack length is measured from the notch.

_{p}with Δa, being the growth rate higher for lower values of Δa. The effect of Δa is more pronounced for notched specimens than for unnotched specimens. These changes are due to the propagation of the crack tip through the stress field generated by the notch. For relatively large crack increments, the curves start to approach. This means that the notch is losing its effect and also that the difference between the curves is due to the notch. Anyway, for the maximum crack length studied Δa = 1.272 mm) the effect did not disappear totally because the curves still have some separation. This means that the effect of notch is larger than the 1.272 mm studied, particularly for the higher values of notch radius. On the other hand, for the smallest radius studied (r = 1 mm), the curves tend to the results of the unnotched specimens, which means that the boundary of the influence of notch was reached. Without contact at the crack flanks, the trends observed for δ

_{p}are not affected by stress state or material.

_{p}against r for three values of Δa, in plane stress state and for the AA7050 with isotropic behavior. The increase of r and Δa reduces δ

_{p}, as shown in Figure 7f. The trends observed in Figure 9 suggest an increase of linearity between δ

_{p}and r, as the crack becomes longer. The same relationships were found under plane strain state and for the AA6082 and AA7050 with kinematic behavior. A correlation coefficient squared, R

^{2}, of 0.99 was found when a linear trendline was fitted to Δa equal to 1272 μm for all cases studied, proving the linearity referred to above.

#### 4.5. Insertion of Contact at the Crack Flanks in the Notched-Cracked Specimen

_{p}against Δa for r = 1 mm and r = 8 mm in contact and no contact simulations. As observed for the unnotched specimens, there is a great influence of crack closure for the plane stress state and a lower effect for the plane strain state. The contact at the crack flanks reduces δ

_{p}, as expected, because there is a reduction of the effective load range. The decrease of notch radius produces a significant increase of crack closure phenomenon in both plane strain and plane stress conditions, being this effect more prominent in plane stress state. This dependence of crack closure on notch radius reduces, apparently, the zone affected by the notches, promoting a faster convergence of the curves for different radius.

#### 4.6. Comparison of Unnotched and Notched Specimens in Simulations with Contact

_{p}versus Δa for simulations with and without notch for the three material models studied here assuming both plane strain and plane stress states. Most of the trends observed without contact (Figure 7) also exist with contact. The notch reduces δ

_{p}, i.e., has a protective effect on da/dN. Once again, the length of unnotched crack used for comparison included the size of the notch. The protective effect increases with notch radius but tends to disappear as the crack propagates from the notch. There is a convergence to the curve corresponding to the unnotched sample, which occurs when the notch effect disappears. However, with contact, there is a marked influence of stress state. For plane strain conditions, there is a sharp increase of δ

_{p}with Δa. A relatively large propagation is required to converge the notched and unnotched curves, particularly for the larger values of notch radius (Figure 10a,c,e). Under plane stress conditions, the increase of notch radius reduces substantially δ

_{p}, which means that it has a protective effect. The convergence of the curves is now much faster than for plane strain state, i.e., the notch affected zone seems to reduce substantially. The material does not seem to affect these trends. The V-shaped behavior, typically observed for short cracks growing from notches, is not observed here, with the exception of AA7050-T6 with pure isotropic behavior and plane stress behavior with notch radius r = 1mm. The increase of load level is likely to promote this behavior.

_{p}is significantly higher than the percentage variation of U*, because δ

_{p}increases nonlinearly with effective load range.

## 5. Discussion

- -
- Two transient effects, one associated with the stabilization of cyclic plastic deformation and the other associated with the built of plasticity induced crack closure. The former, which is very evident in unnotched cracked specimens without contact, needs a very short crack propagation to disappear. The stabilization of crack closure level is much longer, depending on material and increasing with the decrease of notch radius;
- -
- The influence of the notch fields. The reduction of notch radius increases δ
_{p}, and therefore FCG rate, and substantially reduces the extension affected by the notch. Without contact, the limit of notch influence zone has only been reached for the smallest radius studied (r = 1 mm); - -
- The crack closure phenomenon, which has a dramatic effect under plane stress conditions and a limited effect under plane strain conditions. The formation of residual plastic wake is responsible for a progressive increase of U* towards a stabilized value, after which a slow but steady increase is observed. The long stabilization of the closure values means that in many cases it occupies a significant part of the notch affected zone. The level of crack closure is defined by two opposite mechanisms, which are the elongation of plastic wedges behind crack tip and crack tip blunting. This complexity explains the great influence of material on crack closure level and stabilization distance. The introduction of contact in notched samples has a major effect on plastic CTOD. The decrease of notch radius produces a significant increase of crack closure phenomenon, which apparently attenuates the effect of notch radius and reduces the zone affected by the notch;
- -
- After the notch affected zone there is a linear increase of δ
_{p}with Δa, linked to the increase of crack tip stress fields. This effect is clearly observed in unnotched cracked specimens without contact of crack flanks. The slope of the linear variation increases linearly with δ_{p}.

## 6. Conclusions

_{p}, was followed to study fatigue crack growth (FCG) from notches. The identification of fundamental mechanisms was made considering notched and unnotched models, with and without contact of crack flanks. Different parameters were studied, namely, notch radius, crack length, and stress state. The main conclusions are:

- -
- in unnotched cracked specimens, there is an initial transient regime associated with the stabilization of cyclic plastic deformation, which slows down the FCG rate to a minimum. After this minimum, which defines the end of initial regime, the crack growth increases the strength of crack tip fields, producing a linear increase of δ
_{p}. The slope increases with the reduction of yield stress, and a linear trend was observed between the slope and the level of δ_{p}. The linear increase of da/dN with Δa is expected to occur in notched samples after the notch affected zone; - -
- the introduction of contact produced a major effect on δ
_{p}in unnotched cracks under plane stress conditions. There is a progressive decrease of δ_{p}to a minimum value associated with the formation of residual plastic wake. The minimum value of δ_{p}defines the end of the transient regime, which is much more extensive than the transient regime associated with the stabilization of cyclic plastic deformation. The effect of crack closure is very limited under plane strain conditions; - -
- the notches were found to increase the plastic deformation level at the crack tip. The reduction of notch radius increases the notch effect and reduces significantly the notch affected zone. The limit of this zone was reached only for r = 1 mm in this numerical study, being about 0.8 mm for the load level studied. The stress state and material do not affect these trends. As the crack propagates ahead of the notch, a linear relation is observed between δ
_{p}and notch radius; - -
- the introduction of contact in notched samples attenuates the effect of notch radius and reduces the notch affected zone;
- -
- concerning the stress state, lower values of δ
_{p}were obtained for plane strain state compared with plane stress state. This could be expected since the stress triaxiality tends to reduce plastic deformation. Additionally, the effect of stress state is greatly linked with crack closure variations. - -
- Concerning the material, the global trends related to the influence of notch were found to be independent of material parameters. However, there is a significant influence on crack closure phenomenon. Additionally, there is a global trend for the increase of δ
_{p}with the reduction of yield stress.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Overview of the geometry, finite element mesh, and boundary conditions of the notched specimens. (

**a**) Loading, boundary conditions, and main dimensions, in mm. (

**b**) Detail of refined mesh. (

**c**) ρ = 8 mm. (

**d**) ρ = 4 mm. (

**e**) ρ = 2 mm. (

**f**) ρ = 1 mm. (

**g**) Plane stress boundary. (

**h**) Plane strain boundary conditions.

**Figure 3.**Typical plots of crack tip opening displacement (CTOD) versus load, F. (

**a**) Total CTOD vs. F; (

**b**) Plastic CTOD vs. F (contact).

**Figure 4.**Representation of the evolution of (

**a**) δ

_{p}versus Δa; (

**b**) m versus δ

_{p}, with δ

_{p}evaluated at Δa = 800 μm (no contact).

**Figure 5.**Effect of contact of crack flanks on δ

_{p}versus Δa data (unnotched cracked specimen). (

**a**) AA6082-T6; plane strain; (

**b**) AA6082-T6; plane stress; (

**c**) AA7050-T6 with pure kinematic behavior; plane strain; (

**d**) AA7050-T6 with pure kinematic behavior; plane stress; (

**e**) AA7050-T6 with pure isotropic behavior; plane strain; and (

**f**) AA7050-T6 with pure isotropic behavior; plane stress.

**Figure 6.**(

**a**) Evolution of U* with Δa, for AA6082-T6 and AA7050-T6, with pure isotropic and kinematic hardening behaviors (unnotched-cracked specimen; contact); (

**b**) Effect of U* on CTOD vs. F curves (AA7050-T6 isotropic; unnotched-cracked specimen; plane stress).

**Figure 7.**Influence of the notch in δ

_{p}versus Δa curves (no contact). (

**a**) AA6082-T6 in plane strain; (

**b**) AA6082-T6 in plane stress; (

**c**) AA7050-T6 with pure kinematic behavior in plane strain; (

**d**) AA7050-T6 with pure kinematic behavior in plane stress; (

**e**) AA7050-T6 with pure isotropic behavior in plane strain; and (

**f**) AA7050-T6 with pure isotropic behavior in plane stress.

**Figure 8.**Cracked geometries: (

**a**) unnotched crack with a

_{0}= 8.096 mm; (

**b**) crack with notch; and (

**c**) unnotched crack with a

_{0}= 0.096 mm.

**Figure 9.**Evolution of δ

_{p}versus r, for different crack lengths (AA7050 isotropic; notched-cracked specimen; plane stress).

**Figure 10.**Influence of the presence of the notch and the contact at the crack flanks in the δ

_{p}versus Δa curves (notched cracked specimen): (

**a**) AA6082-T6 under plane strain; (

**b**) AA6082-T6 under plane stress; (

**c**) AA7050-T6 with pure kinematic behavior under plane strain; (

**d**) AA7050-T6 with pure kinematic behavior under plane stress; (

**e**) AA7050-T6 with pure isotropic behavior under plane strain; and (

**f**) AA7050-T6 with pure isotropic behavior under plane stress.

**Figure 11.**Comparison of notched and unnotched geometries regarding the evolution of δ

_{p}versus Δa (notched cracked specimen): (

**a**) AA6082-T6 under plane strain; (

**b**) AA6082-T6 under plane stress; (

**c**) AA7050-T6 with pure kinematic behavior under plane strain; (

**d**) AA7050-T6 with pure kinematic behavior under plane stress; (

**e**) AA7050-T6 with pure isotropic behavior under plane strain; and (

**f**) AA7050-T6 with pure isotropic behavior under plane stress.

**Figure 12.**Evolution of U* with Δa for different notch radii: (

**a**) AA6082-T6; (

**b**) AA7050-T6 with kinematic hardening; and (

**c**) AA7050-T6 with isotropic hardening (notched cracked specimen, contact; plane stress).

**Figure 13.**Plastic zone size versus notch radius, considering ε

^{p}= 0% as criterium for the end of the plastic zone (AA6082-T6; contact). (

**a**) Plane strain and (

**b**) plane stress.

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## Share and Cite

**MDPI and ACS Style**

Borges, M.; Caldas, M.; Antunes, F.; Branco, R.; Prates, P.
Fatigue Crack Growth from Notches: A Numerical Analysis. *Appl. Sci.* **2020**, *10*, 4174.
https://doi.org/10.3390/app10124174

**AMA Style**

Borges M, Caldas M, Antunes F, Branco R, Prates P.
Fatigue Crack Growth from Notches: A Numerical Analysis. *Applied Sciences*. 2020; 10(12):4174.
https://doi.org/10.3390/app10124174

**Chicago/Turabian Style**

Borges, Micael, Manuel Caldas, Fernando Antunes, Ricardo Branco, and Pedro Prates.
2020. "Fatigue Crack Growth from Notches: A Numerical Analysis" *Applied Sciences* 10, no. 12: 4174.
https://doi.org/10.3390/app10124174