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Numerical investigation on torsional stress concentration factor at the semi elliptical corrosion pit
Corrosion Science 67 (2013) 225–232
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Numerical investigation on torsional stress concentration factor at the semi elliptical corrosion pit Muhammet Cerit ⇑ Mechanical Engineering Department, Sakarya University, Esentepe Campus, 54187 Serdivan, Sakarya, Turkey
a r t i c l e
i n f o
Article history: Received 22 June 2012 Accepted 27 October 2012 Available online 7 November 2012 Keywords: A. Steel B. Modelling studies C. Pitting corrosion
a b s t r a c t Under torsion, stress distribution at semi elliptical corrosion pits, secondary pit nucleated at the bottom of semi elliptical base pits and also orientation of the semi elliptical primary pits have been investigated by conducting a series of three-dimensional pits models to predict stress concentration factor, systematically. Based on the finite element analyses, aspect ratio is the main parameter affecting SCF. Pits, having higher aspect ratio, are the most dangerous form and can cause significant reduction in the load carrying capacity. When a premature pit nucleates at the bottom of primary pit, SCF increases sharply within the secondary pit region. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Pitting corrosion is a localised form of corrosion by which cavities or holes are produced in the material. Pitting is considered to be more dangerous than uniform corrosion damage because it is more difficult to detect and predict. Pitting rate of penetration may be deeper 10–100 times. In some cases, pits may be masked due to general corrosion. A small, narrow pit with minimal overall metal loss can lead to the failure of an entire engineering system. Understanding and predicting corrosion damage is very important for the structural integrity of metal alloys. It is known to be one of the major damage mechanisms affecting the integrity of many materials and structures in civil, nuclear, aerospace and naval engineering [1]. For modern infrastructure systems, the use of mild and high strength low alloy structural steels is extensive and includes ships, offshore platforms, pipelines, tanks, and other containers. Many of these are under saline or marine immersion conditions. In all cases, perforation due to pitting corrosion is important design criterion [2–6]. Pitting corrosion is a difficult phenomenon to overcome because of its complex nature. Corrosion pits generally initiate due to some chemical or physical heterogeneity on the surface, many variables of the metal–environment system, such as alloy composition and microstructure, composition of the surrounding media and temperature, are all involved in the pitting process [1]. Pit nucleation occurs at the microscopic level and some metals show preferential sites of pit nucleation. Fatigue cracks usually initiate from the corrosion pit sites. Under the interaction ⇑ Corresponding author. Tel.: +90 264 295 58 54; fax: +90 264 295 56 01. E-mail address: [email protected] 0010-938X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.corsci.2012.10.028
of cyclic load and the corrosive environment, cyclic loading facilitates the pitting process and corrosion pits, acting as geometrical discontinuities, lead to crack nucleation and propagation and then final failure [7–10]. All of the geometric discontinuities cause a large variation of stress locally, and produce a significant increase in stress. The high stress due to the geometric discontinuity is known as stress concentration factor (SCF) and expressed as mathematically [11].
Kt ¼ smax =snom
ð1Þ
Here snom is the nominal shear stress in the shaft, which occurs at the outer surface. smax is the maximum shear stress which occurs within the pit on the shaft. It is possible to predict the SCF for certain geometric shapes using theory of elasticity approach. For more accurate estimation numerical methods like finite element analysis (FEA) can be employed. FEA calculates the peak stresses directly and nominal stresses may be easily found by integrating stresses in the surrounding material. In the corroded field, a pit is a kind of geometric discontinuities and the maximum stress occurs at this location. One kind of the pitting geometry is nearly semi-elliptical shape. SEM micrograph three of crosssectional shape of the corrosion pits taken from fracture surface of steel specimen failed in the fatigue test were shown in Fig. 1a–c. Crack readily nucleates at the pit region, so the magnification of the stress must be known in engineering analysis and design. Pit formed on the surface of the structure or component tends to intensify the local stress field hence reduce the load carrying capacity of the component. In order to predict the nucleation of cracks resulting from corrosion pits, the stress around pits has to be determined. So it is needed to investigate the pit induced stresses responsible for possible crack initiation by using finite element
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Nomenclature a 2c a/2c b as 2cs as/2cs
snom
smax
deep of the pit (m) diameter of the pit (m) aspect ratio of the base pit minor diameter of ellipse of the base pit (m) deep of the seconder pit (m) diameter of the seconder pit (m) aspect ratio of the seconder pit the nominal shear stress in the shaft (MPa)
(a)
h SCF Kt SEM FEM FEA
the maximum shear stress which occurs around the pit (MPa) orientation angle of the pit measured from shaft axis (°) stress concentration factor value of the stress concentration factor Scanning Electron Microscope Finite Element Method Finite Element Analysis
(c)
(b)
×
μ
×
μ
×
μ
Fig. 1. SEM micrograph of the typical semi elliptical pits: (a) wide–shallow, (b) hemispherical, and (c) narrow–deep taken from fracture surface of steel specimen failed in the fatigue test. Dashed lines demonstrate the shapes of the pits used in the simulations.
analysis [12–19]. It is expected that severity of pits varies with depth, size and shape. The distribution of the pits depth is an important characteristic of the extent of such damage; the deeper the pits are the more dramatic the damage is [20–22]. There are number of studies on fatigue performance of steels and aluminium alloys that are susceptible to pitting corrosion. Although there are some analytic and numerical works on effectiveness of hemispherical pit at the free surface, little or no three-dimensional (3D) and systematic study, which cover different aspect ratios on estimation of SCF for various pit shapes, have been cited in the literature [20–22]. In the present study, a solid shaft under torsional loading, the effect of semi-elliptical pits on stress concentration factor for different depths and diameter has been investigated by conducting a series of 3D stress analyses, systematically. And then, once a secondary (premature) pit nucleates at the bottom of primary (base) pit, secondary pit dimensions (depth and width) on SCF were also investigated. That is, at the bottom of various semi elliptical pits, the contribution of secondary pit formation to value of SCF has
been examined for various aspect ratios (depth to width). Finally, for various orientations and semi elliptical aspect ratio models, stress analyses were performed to predict SCF. A variety of corrosion-like damage states are indicated in the Table 1. 2. Finite element analyses The finite element (FE) method is the most common, powerful and flexible tool in rational structural analysis and makes it possible to predict the strength of complex structures more accurately than existing classical theoretical methods. A circular cylinder having a single semi elliptical pit model based on the SEM micrograph was subjected to torsion, symmetry plane of the solid shaft and finer mesh around the pit were shown in Fig. 2a–c. A series parametric study on stress and SCF was carried out by employing (ANSYSÒ14) finite element software. 3D solid model of the shaft with varying pit diameter (2c) between 100 and 1000 lm and varying pit depth (a) between 25 and 8000 lm were constructed, parametrically.
Table 1 Shapes of the corrosion pits and their parametric variation. Pit shape
2c
2c a
Range
(a/2c): 0.1, 0.25, 0.5, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 3, 4, 5, 6, 7, 8
Constant
c: 0.5 (mm)
Variable parameter
a (mm): 0.1, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 3, 4, 5, 6, 7, 8
a
2cs as
(a/2c): 0.1, 0.25, 0.5, 0.75, 1(as/2cs): 0.1, 0.25, 0.5, 0.75, 1
(a/2c): 0.1, 0.25, 0.5, 0.75, 1
as:cs (mm): 0.001, 0.025, 0.05, 0.075, 0.1
h (°): 0, 10, 15, 20, 30, 40, 45, 50, 60, 75, 90
b:c: 0.5 (mm)
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M. Cerit / Corrosion Science 67 (2013) 225–232
(b)
(a)
(c)
T
T
Plane of Symmetry
Fig. 2. Finite element model used in the analysis; (a) circular cylinder subjected to torsion, (b) half part of the model having plane of symmetry, and (c) finer mesh around the pit.
For 3D numerical analyses, uniform shapes and forms of elements play important role in the sensitivity of the results. So, models have been meshed by using structural solids having 20-node second order hexahedral elements [23]. It can tolerate irregular shapes without much loss of accuracy and can be refined in the region around the pit to enhance the accuracy of stress predictions. When the pit dimensions were changed, the mesh density around the pit was altered to ensure consistency in the size of the elements created around each pit. This means that the meshes created on the different pits will not be identical but should give similar levels of predictive accuracy. SCF was calculated by using the ratio of maximum shear stress to nominal shear stress determined from simple torsion formula. Since the size of pit is very small compared with the diameter of the shaft, it was neglected in the calculation of the nominal shear stress. The solid shaft was subjected to torsional loading at the both ends. In order to read the SCF value directly on the maximum shear stress counter, torsional loading is so determined that maximum shear stress 1 MPa at the outer surface of the shaft. Structure has symmetry with respect to the loading and geometry; thus, symmetry boundary condition was applied to the axially direction in the cutting surface of the shaft. That is, displacement and rotation are zero at the each node in the cutting surface. The solid model of the shaft dimensions were selected 30 mm in diameter and 120 mm length. A linear elastic material model was used. Modulus of elasticity and Poisson’s ratio were taken 200 GPa and 0.3, respectively. 3. Results and discussions In order to assure the mesh model is accurate enough, a mesh convergence study has been carried out to ensure that the SCF in the pit region is convergent before the series of the 3D stress analyses. Several element numbers or mesh size of the shaft model were performed for pit aspect ratio 0.5. The SCF for various mesh number is shown in Fig. 3. The abscissa represents the element number of the model. It shows that the SCF response is almost convergent for mesh number 350,000. In the present study, the simulation model for shaft with the pit was constructed nearly by 380,000 elements. Under torsional loading, stress analyses have been carried out for a solid shaft with semi elliptical corrosion pit. The results were presented for three situations: the effect of single semi elliptical corrosion pit on SCF, then, the contribution of secondary (premature) pit formation to value of SCF at the bottom of semi-elliptical pit, and also, the influence on semi elliptical pit orientations with various pit depth and diameter have been investigated. 3.1. Primary pit Fig. 4a shows stress distribution of semi elliptical wide-shallow pit with 1000 lm in diameter and 100 lm in depth. From the
Fig. 3. Variations of stress concentration factor with element number.
stress contour, it can be seen that critical area is within the pit and its direction 45° ahead of the longitudinal axis. As is well known, in the pure torsion, principal stress is equal to maximum shear stress [24] and it has a maximum value 1.59snom at the pit’s bowl lip or edge. Stress distribution for the nearly hemispherical pit was given in Fig. 4b. Note that, the critical zone lies between bottom and mouth of the pit as a band 45° orientation to the loading direction. It should be expected that location of critical area is affected by loading condition such as torsional, biaxial and threeaxial state. The maximum shear stress 2.36snom takes place slightly below the mouth of the pit. This behaviour can be attributed to the change in maximum stress trajectories around the mouth of the pit. Calculated maximum stress value is consistent with analytic and numeric solution [20,25,26]. Similar findings for distribution of stress at nearly hemispherical pit were reported in numerical studies [27,28]. Both model with 1000 lm in diameter and 2000 lm in depth (see Fig. 4c and d) have the similar stress distribution as mentioned in previous pits model but they have narrowing stress distribution band toward the bottom. The maximum stresses are 3.226snom and 3.729snom and they have located just below the mouth of pits, respectively. Result from parametric study of different pit diameters and depths were plotted in Fig. 5a. The value of the SCF depends on the size of pit dimension. From this plot, it can be seen that increasing the depth value also increases the SCF value. However, for the small diameter of pits, initially this behaviour is more
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Fig. 4. Counter plot of the maximum shear stress distributions in the corrosion pits with various depths.
4.5
4.5
(a)
(b) 4.0 Stress concentration factor, Kt
Stress concentration factor, Kt
4.0
3.5
3.0 Pit diameter 2c 2.5
0.10 mm 0.25 mm
2.0
0.50 mm 1.00 mm
1.5 1.0 0.0
3.5
2c
3.0
a 2.5
2.0 Semi-elliptical pit shape 1.5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
1.0
0
1
2
Pit depth, a [mm]
3 4 5 6 Pit aspect ratio (a/2c)
7
8
Fig. 5. Variations of the SCF; (a) with pit depth for various pit diameters, (b) with pit aspect ratio.
strongly marked and then increment ratio decreases with depth. And also, it reduces with pit diameter for shallow depth. It is noted that certain pair of pit diameter and depth correspond to the specific SCF value. In order to express dimensionless parameter defining pit severity, pit aspect ratio (a/2c) is considered. As it is shown in Fig. 5b value of the SCF is the same value for the same aspect ratio. It is possible to say that there is a reasonable correlation between a/2c and SCF value. For a given semi elliptical pit shape, SCF can be estimated by using pit aspect ratio. When pit aspect ratio reaches up to about eight the value of the SCF behaves asymptotically. It should be recognised that SCF of single semi elliptical
primary pit can reach 4snom if the pit aspect ratio exceeds 3 (see Fig. 5b). Taking into consideration the nature of crack propagation, maximum stress occurs edges or slightly below of the pit having a high potential for crack. 3.2. Secondary pit It is assumed that surface of the pit was smooth and uniform in the FE analysis. However new and small (premature) pit sometimes nucleates at the bottom of the base pit. Generally, corrosive attacks on the metal surface initiate this secondary pit within the
M. Cerit / Corrosion Science 67 (2013) 225–232
229
Fig. 6. Counter plot of the maximum shear stress distributions for secondary pit nucleated at the bottom of semi elliptical base pit; (a,b) shallow and narrow secondary pits within the wide–shallow base pit, (c,d) shallow and narrow secondary pits within the moderate semi elliptical base pit, and (e,f) shallow and narrow secondary pits within the narrow–deep base pit.
base pit. For this reason, from wide-shallow to narrow–deep semi elliptical pits and secondary pits were also analysed, systematically. Results obtained from the FE analyses, in Fig. 6, illustrates stress distributions of the selected some of the base pits with secondary pits. Fig. 6a and b indicates stress distributions in the shallow and narrow secondary pits within the wide–shallow base pit. Then, Fig. 6c and d indicates stress distributions in the wide–shallow and narrow–deep and secondary pits within the hemi–spherical pits. Finally, Fig. 6e and f indicates stress distributions in the shallow and narrow secondary pits within the narrow base pit. It is notable that maximum stress takes place within the secondary pit and its direction 45° ahead of the longitudinal shaft axis. It occurs at the edge of the upper region of the wide–shallow secondary pit. As mentioned above, as the depth of the secondary pit increases, the point moves to below of the pits mouth where the maximum stress occurs. They are very similar as mentioned in the previous corrosion pits. As an illustration, stress distribution of the shallow and narrow secondary pit within the same
hemispherical (2c = 1000 lm and a = 500 lm) base pit are shown in Fig. 6c and d. Depth of the secondary pit raise from as = 10 lm to as = 100 lm, maximum stress value varies from 2.85snom to 5.22snom. Stress contours on the selected pit forms can be seen in Fig. 6. The other models have the similar stress distribution as mentioned above but it has narrowing stress distribution band toward the bottom. For the base pit (2c = 1000 lm) and secondary pit (2cs = 100 lm), the variation of the SCF for various base pit aspect ratios with various secondary pit aspect ratios is plotted in Fig. 7. For secondary pit nucleated at the bottom of hemi-elliptical base pit, maximum stress is always greater than that of without secondary pit in the same size. Shaded marks are indicated that do not have the secondary pit. It is obvious that the maximum stress is a function of base pit aspect ratio and secondary pit aspect ratio. For each aspect ratios of the base pits, when depth of the secondary pit increases gradually, SCF increases by decreasing rate. The maximum stress grows with base pit diameter and depth but its behaviour is not linearly. To be more precise, it rises by decreasing rate.
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5.0
4.0
Basic pit aspect ratio
3.0
Pit aspect ratio a/2c=0.5 (hemisphere) Seconder pit aspect ratio as/2cs=0.5
4.65
a/2c=0.1 a/2c=0.25
Stress concentration factor, Kt
Stress concenration factor, Kt
4.80
Basic pit 2c=1000 μm secondary pit 2cs=100 μm
6.0
2c 4.50
a
2cs
4.35
as
4.20 4.05 3.90
a/2c=0.5
2.0
a/2c=0.75
3.75
a/2c=1.0 1.0 0.0
3.60
0.2
0.4 0.6 0.8 Secondary pit aspect ratio [as/2cs]
1.0
Fig. 7. Variation of the SCF with various both nucleated at the bottom of semi elliptical base pits secondary pit aspect ratio and base pit aspect ratios. (Shaded marks indicate that not have secondary pit).
0
0.05
0.1 0.15 0.2 0.25 0.3 0.35 Secondary pit diameter 2c s [mm]
0.4
0.45
Fig. 9. Variation of the SCF with diameter of hemispherical secondary pit nucleated at the bottom of hemispherical base pit.
3.3. Orientation of the pit Furthermore, this increase is even less for the greater depth pits. In the case of much greater depth, it will lose its effect on the SCF. Based on the nature of pitting corrosion, it is important to note that premature formation, resulting from corrosive attack at the bottom has a high potential for crack nucleation.(see Fig. 8). Other interesting results obtained from both hemispherical base pits and secondary pits which have aspect ratio 0.5. In the investigation, the base pit was taken a constant size (2c = 1000 lm and a = 500 lm) and depth of the secondary pits was changed from 50 lm to 400 lm. To compare the effect of secondary pit dimension on SCF, counter plot of the stress distribution for 2cs = 100 lm, as = 50 lm and 2cs = 400 lm, as = 200 lm respectively. The effect of the secondary pit nucleated at the bottom of hemispherical pit on the SCF for various pit depth is plotted in Fig. 9. From the graph, it is clearly understood that curve of stress decreases nearly as linear with the diameter of the seconder pits. And its reduction reaches to a limit value. This means that the pit severity is moderate for larger diameters of secondary pit. It is expected that for a constant depth, the large diameters of secondary pit tend to neutralize contribution of secondary pit to the severity of pit.
Stress analyses were performed to predict SCF for aspect ratio of the base pit ranges 0.1–1 in the semi-elliptical model. Moreover, for each model of the aspect ratio, finite element analyses were carried out to determine the effect of orientations on SCF by using 11 different directions, from 0 to 90°. Counter plot of the maximum shear stress distributions in a single semi elliptical pit (a = 500 lm, b = 500 lm and 2c = 1000 lm) for various orientation (h = 0°, h = 30°, h = 45° and h = 90°) were shown in Fig. 10a–d respectively. Here, b is the minor diameter of the ellipse of the base pit. For five different aspect ratios, variation of the stress concentration factor Kt with orientation angle of the pit h was plotted in Fig. 11 in the semi elliptical base pit/defect (b = 500 lm and 2c = 1000 lm). The maximum value of SCF reaches at the orientation angle of the pit 0° or 90° and theirs magnitude was nearly same. It is the minimum at 45°. For all aspect ratio, its value decreases from 0° to 45°, then it increases from 45° to 90° with similar rate. The value of the SCF increases with increasing aspect ratio of the pit but not linearly. The maximum stress or SCF increases with aspect ratio of the pit by decreasing rate. The aspect ratio ascends from 0.1 to 1, the value of SCF changes from 1.736 to 3.934 for 0° or 90°. This means that
Fig. 8. Counter plot of the maximum shear stress distributions for hemispherical secondary pits nucleated within the hemispherical base pits having same size; (a,b) for ratio of secondary pit aspect ratio to base pit aspect ratio 0.1 and 0.4 respectively.
M. Cerit / Corrosion Science 67 (2013) 225–232
231
Fig. 10. Counter plot of the maximum shear stress distributions in the semi elliptical pits for various orientation; (a) h = 0°, (b) h = 30°, (c) h = 45°, and (d) h = 90° (h is an orientation angle of the pit measured from the shaft axis to the orientation axis).
4.75
b=500 µm 2c=1000 µm
θ Pit aspect ratio a/2c 0.1
4.25
Stress concentration factor Kt
0.25 0.5 3.75
0.75
2c
1.0
a
3.25
2.75
2.25
1.75
1.25 0
10
20
30 40 50 60 Orientation angle θ [Degree]
70
80
90
Fig. 11. Variation of the SCF with orientation angles for various aspect ratios.
orientation angle h has a strong effect on it. Regardless of the aspect ratio, one of the main parameter is orientation angle on SCF for the semi elliptical pit. 4. Conclusion The numerical simulations clearly showed that the SCF is a function of the depth and diameter of the pit. Both of them cause higher values of the SCF. That is to say that pits aspect ratio (a/2c) is a main parameter affecting the value of SCF. In accordance with pit aspect ratio, the maximum stress occurs either at the bowl
lip or slightly below the mouth. When the value of the aspect ratio is more than eight, SCF continues asymptotically. It obvious that pits, having higher aspect ratio, or narrow pits are the most dangerous form and they can cause significant reduction in the load carrying capacity of the shaft. The aspect ratio of the secondary pit, which is another important factor, should also be considered in the evaluation of stress distribution. Any premature pit formation can cause crack initiation under the torsional loading. When a secondary pit nucleates at the bottom of primary pit, SCF increases sharply within the secondary pit region of all models with various aspect ratios. For both primary and secondary pit aspect ratio 0.5, the hemispherical secondary pit (2cs = 100 lm) nucleates at the bottom of hemispherical primary pit (2c = 1000 lm), SCF is found to be 4.41 which is higher than 1.73 times compared to SCF value of single primary pit. The most critical pit form is the combination of the higher aspect ratio of the base and secondary pit. As for the orientation angle of the semi elliptical pit, SCF increases with increasing aspect ratio of the pit but not linearly. The maximum shear stress or SCF increases with pit aspect ratio by decreasing rate. For all aspect ratio of the pit, its value decreases from 0° to 45°, then it increases from 45° to 90° with similar rate. The value of SCF is the maximum at orientation angle of the pit 0° and 90° and is the minimum at 45°. The orientation angle of the pit has a strong effect on SCF. One of the other main parameter is orientation angle of the pit on SCF for the semi elliptical pit. If the size of the pitting corrosion is known, reduction in load carrying capacity of structure can be predicted by using the value of SCF. References [1] R.M. Pidaparti, Ronak R. Patel, Correlation between corrosion pits and stresses in Al alloys, Mater. Lett. 62 (2008) 4497–4499. [2] P. Marcus, J. Oudar (Eds.), Corrosion Mechanisms in Theory and Practice, Marcel Dekker, Inc., New York, 1995.
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Corrosion Science journal homepage: www.elsevier.com/locate/corsci
Numerical investigation on torsional stress concentration factor at the semi elliptical corrosion pit Muhammet Cerit ⇑ Mechanical Engineering Department, Sakarya University, Esentepe Campus, 54187 Serdivan, Sakarya, Turkey
a r t i c l e
i n f o
Article history: Received 22 June 2012 Accepted 27 October 2012 Available online 7 November 2012 Keywords: A. Steel B. Modelling studies C. Pitting corrosion
a b s t r a c t Under torsion, stress distribution at semi elliptical corrosion pits, secondary pit nucleated at the bottom of semi elliptical base pits and also orientation of the semi elliptical primary pits have been investigated by conducting a series of three-dimensional pits models to predict stress concentration factor, systematically. Based on the finite element analyses, aspect ratio is the main parameter affecting SCF. Pits, having higher aspect ratio, are the most dangerous form and can cause significant reduction in the load carrying capacity. When a premature pit nucleates at the bottom of primary pit, SCF increases sharply within the secondary pit region. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Pitting corrosion is a localised form of corrosion by which cavities or holes are produced in the material. Pitting is considered to be more dangerous than uniform corrosion damage because it is more difficult to detect and predict. Pitting rate of penetration may be deeper 10–100 times. In some cases, pits may be masked due to general corrosion. A small, narrow pit with minimal overall metal loss can lead to the failure of an entire engineering system. Understanding and predicting corrosion damage is very important for the structural integrity of metal alloys. It is known to be one of the major damage mechanisms affecting the integrity of many materials and structures in civil, nuclear, aerospace and naval engineering [1]. For modern infrastructure systems, the use of mild and high strength low alloy structural steels is extensive and includes ships, offshore platforms, pipelines, tanks, and other containers. Many of these are under saline or marine immersion conditions. In all cases, perforation due to pitting corrosion is important design criterion [2–6]. Pitting corrosion is a difficult phenomenon to overcome because of its complex nature. Corrosion pits generally initiate due to some chemical or physical heterogeneity on the surface, many variables of the metal–environment system, such as alloy composition and microstructure, composition of the surrounding media and temperature, are all involved in the pitting process [1]. Pit nucleation occurs at the microscopic level and some metals show preferential sites of pit nucleation. Fatigue cracks usually initiate from the corrosion pit sites. Under the interaction ⇑ Corresponding author. Tel.: +90 264 295 58 54; fax: +90 264 295 56 01. E-mail address: [email protected] 0010-938X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.corsci.2012.10.028
of cyclic load and the corrosive environment, cyclic loading facilitates the pitting process and corrosion pits, acting as geometrical discontinuities, lead to crack nucleation and propagation and then final failure [7–10]. All of the geometric discontinuities cause a large variation of stress locally, and produce a significant increase in stress. The high stress due to the geometric discontinuity is known as stress concentration factor (SCF) and expressed as mathematically [11].
Kt ¼ smax =snom
ð1Þ
Here snom is the nominal shear stress in the shaft, which occurs at the outer surface. smax is the maximum shear stress which occurs within the pit on the shaft. It is possible to predict the SCF for certain geometric shapes using theory of elasticity approach. For more accurate estimation numerical methods like finite element analysis (FEA) can be employed. FEA calculates the peak stresses directly and nominal stresses may be easily found by integrating stresses in the surrounding material. In the corroded field, a pit is a kind of geometric discontinuities and the maximum stress occurs at this location. One kind of the pitting geometry is nearly semi-elliptical shape. SEM micrograph three of crosssectional shape of the corrosion pits taken from fracture surface of steel specimen failed in the fatigue test were shown in Fig. 1a–c. Crack readily nucleates at the pit region, so the magnification of the stress must be known in engineering analysis and design. Pit formed on the surface of the structure or component tends to intensify the local stress field hence reduce the load carrying capacity of the component. In order to predict the nucleation of cracks resulting from corrosion pits, the stress around pits has to be determined. So it is needed to investigate the pit induced stresses responsible for possible crack initiation by using finite element
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Nomenclature a 2c a/2c b as 2cs as/2cs
snom
smax
deep of the pit (m) diameter of the pit (m) aspect ratio of the base pit minor diameter of ellipse of the base pit (m) deep of the seconder pit (m) diameter of the seconder pit (m) aspect ratio of the seconder pit the nominal shear stress in the shaft (MPa)
(a)
h SCF Kt SEM FEM FEA
the maximum shear stress which occurs around the pit (MPa) orientation angle of the pit measured from shaft axis (°) stress concentration factor value of the stress concentration factor Scanning Electron Microscope Finite Element Method Finite Element Analysis
(c)
(b)
×
μ
×
μ
×
μ
Fig. 1. SEM micrograph of the typical semi elliptical pits: (a) wide–shallow, (b) hemispherical, and (c) narrow–deep taken from fracture surface of steel specimen failed in the fatigue test. Dashed lines demonstrate the shapes of the pits used in the simulations.
analysis [12–19]. It is expected that severity of pits varies with depth, size and shape. The distribution of the pits depth is an important characteristic of the extent of such damage; the deeper the pits are the more dramatic the damage is [20–22]. There are number of studies on fatigue performance of steels and aluminium alloys that are susceptible to pitting corrosion. Although there are some analytic and numerical works on effectiveness of hemispherical pit at the free surface, little or no three-dimensional (3D) and systematic study, which cover different aspect ratios on estimation of SCF for various pit shapes, have been cited in the literature [20–22]. In the present study, a solid shaft under torsional loading, the effect of semi-elliptical pits on stress concentration factor for different depths and diameter has been investigated by conducting a series of 3D stress analyses, systematically. And then, once a secondary (premature) pit nucleates at the bottom of primary (base) pit, secondary pit dimensions (depth and width) on SCF were also investigated. That is, at the bottom of various semi elliptical pits, the contribution of secondary pit formation to value of SCF has
been examined for various aspect ratios (depth to width). Finally, for various orientations and semi elliptical aspect ratio models, stress analyses were performed to predict SCF. A variety of corrosion-like damage states are indicated in the Table 1. 2. Finite element analyses The finite element (FE) method is the most common, powerful and flexible tool in rational structural analysis and makes it possible to predict the strength of complex structures more accurately than existing classical theoretical methods. A circular cylinder having a single semi elliptical pit model based on the SEM micrograph was subjected to torsion, symmetry plane of the solid shaft and finer mesh around the pit were shown in Fig. 2a–c. A series parametric study on stress and SCF was carried out by employing (ANSYSÒ14) finite element software. 3D solid model of the shaft with varying pit diameter (2c) between 100 and 1000 lm and varying pit depth (a) between 25 and 8000 lm were constructed, parametrically.
Table 1 Shapes of the corrosion pits and their parametric variation. Pit shape
2c
2c a
Range
(a/2c): 0.1, 0.25, 0.5, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 3, 4, 5, 6, 7, 8
Constant
c: 0.5 (mm)
Variable parameter
a (mm): 0.1, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 3, 4, 5, 6, 7, 8
a
2cs as
(a/2c): 0.1, 0.25, 0.5, 0.75, 1(as/2cs): 0.1, 0.25, 0.5, 0.75, 1
(a/2c): 0.1, 0.25, 0.5, 0.75, 1
as:cs (mm): 0.001, 0.025, 0.05, 0.075, 0.1
h (°): 0, 10, 15, 20, 30, 40, 45, 50, 60, 75, 90
b:c: 0.5 (mm)
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(b)
(a)
(c)
T
T
Plane of Symmetry
Fig. 2. Finite element model used in the analysis; (a) circular cylinder subjected to torsion, (b) half part of the model having plane of symmetry, and (c) finer mesh around the pit.
For 3D numerical analyses, uniform shapes and forms of elements play important role in the sensitivity of the results. So, models have been meshed by using structural solids having 20-node second order hexahedral elements [23]. It can tolerate irregular shapes without much loss of accuracy and can be refined in the region around the pit to enhance the accuracy of stress predictions. When the pit dimensions were changed, the mesh density around the pit was altered to ensure consistency in the size of the elements created around each pit. This means that the meshes created on the different pits will not be identical but should give similar levels of predictive accuracy. SCF was calculated by using the ratio of maximum shear stress to nominal shear stress determined from simple torsion formula. Since the size of pit is very small compared with the diameter of the shaft, it was neglected in the calculation of the nominal shear stress. The solid shaft was subjected to torsional loading at the both ends. In order to read the SCF value directly on the maximum shear stress counter, torsional loading is so determined that maximum shear stress 1 MPa at the outer surface of the shaft. Structure has symmetry with respect to the loading and geometry; thus, symmetry boundary condition was applied to the axially direction in the cutting surface of the shaft. That is, displacement and rotation are zero at the each node in the cutting surface. The solid model of the shaft dimensions were selected 30 mm in diameter and 120 mm length. A linear elastic material model was used. Modulus of elasticity and Poisson’s ratio were taken 200 GPa and 0.3, respectively. 3. Results and discussions In order to assure the mesh model is accurate enough, a mesh convergence study has been carried out to ensure that the SCF in the pit region is convergent before the series of the 3D stress analyses. Several element numbers or mesh size of the shaft model were performed for pit aspect ratio 0.5. The SCF for various mesh number is shown in Fig. 3. The abscissa represents the element number of the model. It shows that the SCF response is almost convergent for mesh number 350,000. In the present study, the simulation model for shaft with the pit was constructed nearly by 380,000 elements. Under torsional loading, stress analyses have been carried out for a solid shaft with semi elliptical corrosion pit. The results were presented for three situations: the effect of single semi elliptical corrosion pit on SCF, then, the contribution of secondary (premature) pit formation to value of SCF at the bottom of semi-elliptical pit, and also, the influence on semi elliptical pit orientations with various pit depth and diameter have been investigated. 3.1. Primary pit Fig. 4a shows stress distribution of semi elliptical wide-shallow pit with 1000 lm in diameter and 100 lm in depth. From the
Fig. 3. Variations of stress concentration factor with element number.
stress contour, it can be seen that critical area is within the pit and its direction 45° ahead of the longitudinal axis. As is well known, in the pure torsion, principal stress is equal to maximum shear stress [24] and it has a maximum value 1.59snom at the pit’s bowl lip or edge. Stress distribution for the nearly hemispherical pit was given in Fig. 4b. Note that, the critical zone lies between bottom and mouth of the pit as a band 45° orientation to the loading direction. It should be expected that location of critical area is affected by loading condition such as torsional, biaxial and threeaxial state. The maximum shear stress 2.36snom takes place slightly below the mouth of the pit. This behaviour can be attributed to the change in maximum stress trajectories around the mouth of the pit. Calculated maximum stress value is consistent with analytic and numeric solution [20,25,26]. Similar findings for distribution of stress at nearly hemispherical pit were reported in numerical studies [27,28]. Both model with 1000 lm in diameter and 2000 lm in depth (see Fig. 4c and d) have the similar stress distribution as mentioned in previous pits model but they have narrowing stress distribution band toward the bottom. The maximum stresses are 3.226snom and 3.729snom and they have located just below the mouth of pits, respectively. Result from parametric study of different pit diameters and depths were plotted in Fig. 5a. The value of the SCF depends on the size of pit dimension. From this plot, it can be seen that increasing the depth value also increases the SCF value. However, for the small diameter of pits, initially this behaviour is more
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Fig. 4. Counter plot of the maximum shear stress distributions in the corrosion pits with various depths.
4.5
4.5
(a)
(b) 4.0 Stress concentration factor, Kt
Stress concentration factor, Kt
4.0
3.5
3.0 Pit diameter 2c 2.5
0.10 mm 0.25 mm
2.0
0.50 mm 1.00 mm
1.5 1.0 0.0
3.5
2c
3.0
a 2.5
2.0 Semi-elliptical pit shape 1.5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
1.0
0
1
2
Pit depth, a [mm]
3 4 5 6 Pit aspect ratio (a/2c)
7
8
Fig. 5. Variations of the SCF; (a) with pit depth for various pit diameters, (b) with pit aspect ratio.
strongly marked and then increment ratio decreases with depth. And also, it reduces with pit diameter for shallow depth. It is noted that certain pair of pit diameter and depth correspond to the specific SCF value. In order to express dimensionless parameter defining pit severity, pit aspect ratio (a/2c) is considered. As it is shown in Fig. 5b value of the SCF is the same value for the same aspect ratio. It is possible to say that there is a reasonable correlation between a/2c and SCF value. For a given semi elliptical pit shape, SCF can be estimated by using pit aspect ratio. When pit aspect ratio reaches up to about eight the value of the SCF behaves asymptotically. It should be recognised that SCF of single semi elliptical
primary pit can reach 4snom if the pit aspect ratio exceeds 3 (see Fig. 5b). Taking into consideration the nature of crack propagation, maximum stress occurs edges or slightly below of the pit having a high potential for crack. 3.2. Secondary pit It is assumed that surface of the pit was smooth and uniform in the FE analysis. However new and small (premature) pit sometimes nucleates at the bottom of the base pit. Generally, corrosive attacks on the metal surface initiate this secondary pit within the
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229
Fig. 6. Counter plot of the maximum shear stress distributions for secondary pit nucleated at the bottom of semi elliptical base pit; (a,b) shallow and narrow secondary pits within the wide–shallow base pit, (c,d) shallow and narrow secondary pits within the moderate semi elliptical base pit, and (e,f) shallow and narrow secondary pits within the narrow–deep base pit.
base pit. For this reason, from wide-shallow to narrow–deep semi elliptical pits and secondary pits were also analysed, systematically. Results obtained from the FE analyses, in Fig. 6, illustrates stress distributions of the selected some of the base pits with secondary pits. Fig. 6a and b indicates stress distributions in the shallow and narrow secondary pits within the wide–shallow base pit. Then, Fig. 6c and d indicates stress distributions in the wide–shallow and narrow–deep and secondary pits within the hemi–spherical pits. Finally, Fig. 6e and f indicates stress distributions in the shallow and narrow secondary pits within the narrow base pit. It is notable that maximum stress takes place within the secondary pit and its direction 45° ahead of the longitudinal shaft axis. It occurs at the edge of the upper region of the wide–shallow secondary pit. As mentioned above, as the depth of the secondary pit increases, the point moves to below of the pits mouth where the maximum stress occurs. They are very similar as mentioned in the previous corrosion pits. As an illustration, stress distribution of the shallow and narrow secondary pit within the same
hemispherical (2c = 1000 lm and a = 500 lm) base pit are shown in Fig. 6c and d. Depth of the secondary pit raise from as = 10 lm to as = 100 lm, maximum stress value varies from 2.85snom to 5.22snom. Stress contours on the selected pit forms can be seen in Fig. 6. The other models have the similar stress distribution as mentioned above but it has narrowing stress distribution band toward the bottom. For the base pit (2c = 1000 lm) and secondary pit (2cs = 100 lm), the variation of the SCF for various base pit aspect ratios with various secondary pit aspect ratios is plotted in Fig. 7. For secondary pit nucleated at the bottom of hemi-elliptical base pit, maximum stress is always greater than that of without secondary pit in the same size. Shaded marks are indicated that do not have the secondary pit. It is obvious that the maximum stress is a function of base pit aspect ratio and secondary pit aspect ratio. For each aspect ratios of the base pits, when depth of the secondary pit increases gradually, SCF increases by decreasing rate. The maximum stress grows with base pit diameter and depth but its behaviour is not linearly. To be more precise, it rises by decreasing rate.
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5.0
4.0
Basic pit aspect ratio
3.0
Pit aspect ratio a/2c=0.5 (hemisphere) Seconder pit aspect ratio as/2cs=0.5
4.65
a/2c=0.1 a/2c=0.25
Stress concentration factor, Kt
Stress concenration factor, Kt
4.80
Basic pit 2c=1000 μm secondary pit 2cs=100 μm
6.0
2c 4.50
a
2cs
4.35
as
4.20 4.05 3.90
a/2c=0.5
2.0
a/2c=0.75
3.75
a/2c=1.0 1.0 0.0
3.60
0.2
0.4 0.6 0.8 Secondary pit aspect ratio [as/2cs]
1.0
Fig. 7. Variation of the SCF with various both nucleated at the bottom of semi elliptical base pits secondary pit aspect ratio and base pit aspect ratios. (Shaded marks indicate that not have secondary pit).
0
0.05
0.1 0.15 0.2 0.25 0.3 0.35 Secondary pit diameter 2c s [mm]
0.4
0.45
Fig. 9. Variation of the SCF with diameter of hemispherical secondary pit nucleated at the bottom of hemispherical base pit.
3.3. Orientation of the pit Furthermore, this increase is even less for the greater depth pits. In the case of much greater depth, it will lose its effect on the SCF. Based on the nature of pitting corrosion, it is important to note that premature formation, resulting from corrosive attack at the bottom has a high potential for crack nucleation.(see Fig. 8). Other interesting results obtained from both hemispherical base pits and secondary pits which have aspect ratio 0.5. In the investigation, the base pit was taken a constant size (2c = 1000 lm and a = 500 lm) and depth of the secondary pits was changed from 50 lm to 400 lm. To compare the effect of secondary pit dimension on SCF, counter plot of the stress distribution for 2cs = 100 lm, as = 50 lm and 2cs = 400 lm, as = 200 lm respectively. The effect of the secondary pit nucleated at the bottom of hemispherical pit on the SCF for various pit depth is plotted in Fig. 9. From the graph, it is clearly understood that curve of stress decreases nearly as linear with the diameter of the seconder pits. And its reduction reaches to a limit value. This means that the pit severity is moderate for larger diameters of secondary pit. It is expected that for a constant depth, the large diameters of secondary pit tend to neutralize contribution of secondary pit to the severity of pit.
Stress analyses were performed to predict SCF for aspect ratio of the base pit ranges 0.1–1 in the semi-elliptical model. Moreover, for each model of the aspect ratio, finite element analyses were carried out to determine the effect of orientations on SCF by using 11 different directions, from 0 to 90°. Counter plot of the maximum shear stress distributions in a single semi elliptical pit (a = 500 lm, b = 500 lm and 2c = 1000 lm) for various orientation (h = 0°, h = 30°, h = 45° and h = 90°) were shown in Fig. 10a–d respectively. Here, b is the minor diameter of the ellipse of the base pit. For five different aspect ratios, variation of the stress concentration factor Kt with orientation angle of the pit h was plotted in Fig. 11 in the semi elliptical base pit/defect (b = 500 lm and 2c = 1000 lm). The maximum value of SCF reaches at the orientation angle of the pit 0° or 90° and theirs magnitude was nearly same. It is the minimum at 45°. For all aspect ratio, its value decreases from 0° to 45°, then it increases from 45° to 90° with similar rate. The value of the SCF increases with increasing aspect ratio of the pit but not linearly. The maximum stress or SCF increases with aspect ratio of the pit by decreasing rate. The aspect ratio ascends from 0.1 to 1, the value of SCF changes from 1.736 to 3.934 for 0° or 90°. This means that
Fig. 8. Counter plot of the maximum shear stress distributions for hemispherical secondary pits nucleated within the hemispherical base pits having same size; (a,b) for ratio of secondary pit aspect ratio to base pit aspect ratio 0.1 and 0.4 respectively.
M. Cerit / Corrosion Science 67 (2013) 225–232
231
Fig. 10. Counter plot of the maximum shear stress distributions in the semi elliptical pits for various orientation; (a) h = 0°, (b) h = 30°, (c) h = 45°, and (d) h = 90° (h is an orientation angle of the pit measured from the shaft axis to the orientation axis).
4.75
b=500 µm 2c=1000 µm
θ Pit aspect ratio a/2c 0.1
4.25
Stress concentration factor Kt
0.25 0.5 3.75
0.75
2c
1.0
a
3.25
2.75
2.25
1.75
1.25 0
10
20
30 40 50 60 Orientation angle θ [Degree]
70
80
90
Fig. 11. Variation of the SCF with orientation angles for various aspect ratios.
orientation angle h has a strong effect on it. Regardless of the aspect ratio, one of the main parameter is orientation angle on SCF for the semi elliptical pit. 4. Conclusion The numerical simulations clearly showed that the SCF is a function of the depth and diameter of the pit. Both of them cause higher values of the SCF. That is to say that pits aspect ratio (a/2c) is a main parameter affecting the value of SCF. In accordance with pit aspect ratio, the maximum stress occurs either at the bowl
lip or slightly below the mouth. When the value of the aspect ratio is more than eight, SCF continues asymptotically. It obvious that pits, having higher aspect ratio, or narrow pits are the most dangerous form and they can cause significant reduction in the load carrying capacity of the shaft. The aspect ratio of the secondary pit, which is another important factor, should also be considered in the evaluation of stress distribution. Any premature pit formation can cause crack initiation under the torsional loading. When a secondary pit nucleates at the bottom of primary pit, SCF increases sharply within the secondary pit region of all models with various aspect ratios. For both primary and secondary pit aspect ratio 0.5, the hemispherical secondary pit (2cs = 100 lm) nucleates at the bottom of hemispherical primary pit (2c = 1000 lm), SCF is found to be 4.41 which is higher than 1.73 times compared to SCF value of single primary pit. The most critical pit form is the combination of the higher aspect ratio of the base and secondary pit. As for the orientation angle of the semi elliptical pit, SCF increases with increasing aspect ratio of the pit but not linearly. The maximum shear stress or SCF increases with pit aspect ratio by decreasing rate. For all aspect ratio of the pit, its value decreases from 0° to 45°, then it increases from 45° to 90° with similar rate. The value of SCF is the maximum at orientation angle of the pit 0° and 90° and is the minimum at 45°. The orientation angle of the pit has a strong effect on SCF. One of the other main parameter is orientation angle of the pit on SCF for the semi elliptical pit. If the size of the pitting corrosion is known, reduction in load carrying capacity of structure can be predicted by using the value of SCF. References [1] R.M. Pidaparti, Ronak R. Patel, Correlation between corrosion pits and stresses in Al alloys, Mater. Lett. 62 (2008) 4497–4499. [2] P. Marcus, J. Oudar (Eds.), Corrosion Mechanisms in Theory and Practice, Marcel Dekker, Inc., New York, 1995.
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