Engineering Fracture Mechanics 158 (2016) 13–22
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Technical Note
Effect of micro-cavities on different plastic zones at the fatigue crack tip of a compact tension specimen Surajit Kumar Paul ⇑ R&D, Tata Steel Limited, Jamshedpur 831007, India
a r t i c l e
i n f o
Article history: Received 17 February 2016 Accepted 21 February 2016 Available online 27 February 2016 Keywords: Cyclic plastic zone Micro-cavities Monotonic plastic zone Ratcheting Crack Chaboche model
a b s t r a c t Knowledge of cyclic plastic deformation response at the fatigue crack tip is crucial to understand the nature of cyclic damage acting on the crack tip. Interaction between crack and defect is equally important to understand the cyclic damage progression at crack tip; however isotropic hardening model was adopted in all existing literatures. Isotropic hardening model is suitable to model only the monotonic plastic zone and unable to model the cyclic/reverse plastic zone. Kinematic hardening model is suitable to model the key cyclic plastic deformation responses like Bauschinger effect, ratcheting, and mean stress relaxation. A non-linear kinematic hardening (Chaboche) model is used in this present investigation to represent the material’s cyclic stress–strain response accurately. Effect of micro-cavity positions (angle and distance from crack tip) and sizes on plastic zones near a crack tip is investigated in this study by two dimensional plane strain finite element model of a compact tension specimen. It is observed that the size and shape of the monotonic and cyclic plastic zones are affected by position and size of the microcavity. During asymmetric loading condition, the ratcheting strain accumulation direction is also affected by the position of micro-cavity. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Progression of plastic deformation and damage at the crack tip is commonly used to study the fracture behaviour of ductile materials. Similarly, advancement of cyclic plastic deformation and damage at the fatigue crack tip is generally used to investigate the fatigue fracture behaviour. Normally three zones are present at the fatigue crack tip: cyclic/reverse plastic zone, monotonic plastic zone and elastic zone. The cyclic plastic zone is surrounded by monotonic plastic zone and these two plastic zones are further surrounded by elastic zone. Among those three zones, cyclic plastic deformation and damage take place only in the cyclic plastic zone. Therefore, size, shape, deformation and damage modes in the cyclic plastic zone always become important in fatigue crack growth study. Recently number of experimental [1] and numerical [2–5] investigation has conducted to understand the cyclic deformation and damage modes in the cyclic plastic zone. Tong et al. [1–4] have concluded from continuum, visco-plastic and crystal plasticity based finite element analysis that ratcheting takes place in the cyclic plastic zone. Ratcheting can be defined as the progressive accumulation of permanent strain during asymmetric stress cycling. Paul and Tarafder [5] have also reported that accumulation of ratcheting strain in the cyclic plastic zone for R ratio – 1 (R ratio is the ratio of minimum ⇑ Present address: School of Engineering, Deakin University, Pigdons Rd, Waurn Ponds, VIC 3217, Australia. Tel.: +61 431362497. E-mail address:
[email protected] http://dx.doi.org/10.1016/j.engfracmech.2016.02.041 0013-7944/Ó 2016 Elsevier Ltd. All rights reserved.
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S.K. Paul / Engineering Fracture Mechanics 158 (2016) 13–22
and maximum stress or stress intensity factor). Recently Paul [6] has reported that low cycle fatigue (LCF) takes place in the cyclic plastic zone for R ratio = 1 and ratcheting for R ratio – 1. Similarly, the effect of applied loading condition on size and shape of cyclic plastic zone also has been studied in details by Paul [6]. The fatigue crack growth and its mechanism are essentially influenced by the presence of micro-defects that are inherent to the materials such as micro-cavities. This type of problem has been investigated by several authors [7–12]. However, most authors have considered an elastic or isotropic hardening behaviour of materials. Bouiadjra et al. [13,14] have conducted non-linear finite element investigation and noticed that the size and shape of crack tip plastic zones are significantly affected by the presence of micro-cavity. On the contrary, Bouiadjra et al. [13,14] have used isotropic hardening law to model cyclic plastic deformation. However, kinematic hardening model is suitable to represent the cyclic plastic deformation behaviour of materials [15]. Basic cyclic plastic response of the material like Bauschinger effect, ratcheting and mean stress relaxation cannot be simulated by isotropic hardening model [15,16], on the other hand kinematic hardening model can simulate successfully those key cyclic plastic deformation responses. The cyclic/reverse plastic zone cannot be modelled by isotropic hardening model, as it is unable to describe the Bauschinger effect. Therefore, the effect of micro-defect presence near fatigue crack tip on the plastic zone sizes should be relooked by advanced kinematic hardening model to properly understand it. Thus, Chaboche kinematic hardening model is used in the present work to understand the effect of micro-defects near fatigue crack tip. 2. Finite element analysis Commercial finite element package ABAQUS [17] is used in this study. Implicit finite element analysis is conducted in the present investigation on a 2D plane strain full compact tension (C(T)) specimen (width (W) = 62.5 mm, height = 60 mm and thickness (B) 20 mm). A stationary fatigue crack with length of 25 mm (notch length 20 mm and 5 mm stationary crack) is considered for the present study (Fig. 1(a)). The semicircular crack tip (Fig. 1) is considered to model crack tip. Similar shape of crack tip is frequently used by number of researchers [18–20]. A 2D finite element mesh is shown in Fig. 1(b), where the meshes are refined in near-tip region to capture the large strain gradients due to the presence of the crack. The 2D plane strain model is meshed by 4-noded quadrilateral element (CPE4R) with reduced integration and enhanced hourglass control to prevent both shear locking and hourglass mode during the analysis. The refined meshes near the crack tip are shown in
(a) 60 20
5
62.5
(b)
Fig. 1. A full C(T) specimen (right hand side pictures are the zoomed version of left hand side pictures near crack tip): (a) stationary fatigue crack dimensions (all dimensions are in mm) and (b) meshing.
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S.K. Paul / Engineering Fracture Mechanics 158 (2016) 13–22 Table 1 Material constants of Chaboche model for SA333 Gr 6 C–Mn steel. Parameter
Value
C1 C2 C3 D1 D2 D3
140.0 GPa 25 GPa 1950 MPa 1750.0 238.0 1
Fig. 1(b). The sizes of element in the fine mesh zone increase linearly along the crack line in the present investigation. The element sizes have been kept accurately small at the crack tip (where the average element size is about 0.5 lm) to reach the required resolution of the stress/strain field. To prevent potential penetration of the crack flanks due to potential crack closure under cyclic loading, a contact pair (i.e. friction less contact between two surfaces) between upper and lower crack surface is assumed. Loading pins are modelled by rigid discrete elements (R2D2). Similarly a contact pair (i.e. friction less contact between two surfaces) is assumed between the loading pin and the CT specimen. At the middle of the upper loading pin, the concentrated force is applied and lower loading pin is kept as fixed in all directions. Applied cyclic load (DP) on top pin can be calculated from Eq. (1) [21].
DPð2 þ aÞ DK ¼ pffiffiffiffiffiffi 0:886 þ 4:64a 13:32a2 þ 14:72a3 5:6a4 3=2 B W ð1 aÞ
ð1Þ
where DK = amplitude of stress intensity factor, B = thickness, a = a/W, a = crack length and W = width of a CT specimen. Triangular waveform is used for the cyclic loading and the cyclic load (DP) is applied on the loading pin. The implicit finite analysis is done up to 50 loading cycles and at least 200 data points are stored for each cycles for further analysis. The plastic behaviour near the tip of stationary crack in engineering materials has been intensively investigated using continuum plasticity theory based on the von-Mises yield criterion and the associative flow rules [22–25]. Similar continuum plasticity theory based finite element model is considered in this study and advance plasticity model (Chaboche kinematic hardening model [26]) is used to represent cyclic stress–strain response of the material. The Chaboche kinematic hardening model [26] can be written in the form
¼ da
3 X daj
ð2Þ
j¼1
j ¼ da
2 j depeq C j dep Dj a 3
ð3Þ
where the total back stress (a) is a summation of three decomposed back stresses (a = a1 + a2 + a3); C1, C2, C3 are the kinematic hardening coefficients and D1, D2, D3 are the kinematic hardening exponents; dep and depeq are the plastic strain increment vector and equivalent plastic strain increment respectively. Loading part of the stress–strain curve can be represented as
r ¼ r0 þ ða1 þ a2 þ a3 Þ
ð4Þ 500 400 300
Stress, MPa
200 100 0 -100 -200 -300 Simulated Experimental
-400 -500 -2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Strain, % Fig. 2. Comparison of hysteresis loops (LCF: strain amplitude of 1.6%) between experimental and simulated by Chaboche model.
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S.K. Paul / Engineering Fracture Mechanics 158 (2016) 13–22
Monotonic plastic zone
Cyclic plastic zone
(a)
Without micro-cavity
2.5 mm
2.5 mm
Monotonic plastic zone
Cyclic plastic zone
(b)
r = 80 μm θ = 0o
Micro-cavity in line with crack 2.5 mm
2.5 mm
Monotonic plastic zone
Cyclic plastic zone
(c)
r = 80 μm θ = 15o
Micro-cavity in 15o to crack tip 2.5 mm
2.5 mm
p p Fig. 3. Monotonic and cyclic/reverse plastic zones at DK of 22 MPa mm and R ratio of 0.5 (Kmax = 29.33 MPa mm) at four different position of microcavity (micro-cavity radius 30 lm) and without micro-cavity. Distance between micro-cavity and crack tip is ‘‘r”, and angle between micro-cavity and crack is ‘‘h”. The equivalent plastic strain up to 10-5 during tensile peak in first cycle is considered as monotonic plastic zone (coloured as red); while the equivalent plastic strain amplitude up to 10-5 is considered as cyclic plastic zone (coloured as red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
where r and r0 are the stress at any point and cyclic yield stress respectively. The detail of material constants determination procedure is available in authors published literatures [5,16]. Material parameters for Chaboche kinematic hardening model are tabulated in Table 1. The ability of Chaboche model to simulate stable LCF hysteresis loop is shown in Fig. 2. The details predictive capability of Chaboche model is discussed in authors published literatures [5,16]. Circular micro-cavity with 30 lm diameter is considered to understand the effect of the location of micro-defect, and to understand the effect of micro-defect size the circular micro-cavity with diameter of 30, 50 and 100 lm are considered. Around 60 elements are selected at the edge of microdefect to model strain gradient accurately. In this study, five different
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S.K. Paul / Engineering Fracture Mechanics 158 (2016) 13–22
Cyclic plastic zone
Monotonic plastic zone
(d)
r = 80 μm θ = 45o
Micro-cavity in 45o to crack tip 2.5 mm
2.5 mm
Cyclic plastic zone
Monotonic plastic zone
(e)
r = 40 μm θ = 90o
Micro-cavity in 90o to crack tip 2.5 mm
2.5 mm Fig. 3 (continued)
positioned of micro-cavity are investigated: (i) no micro-defect (Fig. 3(a)), (ii) single micro-defect positioned horizontally, i.e. in the way of crack propagation (Fig. 3(b)), (iii) single micro-defect positioned 15° angle to horizontal axis (Fig. 3(c)), (iv) double micro-defect positioned 45° angle to horizontal axis (Fig. 3(d)), and (v) double micro-defect positioned 90° angle to horizontal axis (i.e. vertical) (Fig. 3(e)). Similarly, different sizes of micro-cavity and different distances of micro-cavity from crack tip are also investigated in this work. After finite element simulation, monotonic and cyclic plastic zones are determined and ratcheting strain accumulation is measured for all above mentioned conditions.
3. Results and discussion The objective of this investigation is to examine the size and shape of monotonic and cyclic plastic zones and accumulation of strain near crack tip for different position of micro-cavities. For this, implicit plane strain analysis is done on a full p C(T) specimen with a stationary crack length of 25 mm at DK (amplitude of stress intensity factor) 22 MPa mm and R (ratio p of minimum and maximum stress intensity factor) 1.0. The Kmax (maximum stress intensity factor) is 29.33 MPa mm for this loading condition. The monotonic plastic zone is computed on the basis that the equivalent plastic strain is up to 10-5 during tensile peak in first cycle. While, the cyclic plastic zone is calculated on the basis that the equivalent plastic strain amplitude in first cycle is up to 10-5. The size and shape of monotonic and cyclic plastic zones at various positions of micro-cavities are computed and portrayed in Fig. 3. The micro-defect sizes are considered as 30 lm for Fig. 3(b)–(e). The radial distance of 80 lm and angle to horizontal direction of 0°, 15°, and 45° are considered for Fig. 3(b)–(d) respectively. The radial distance of 40 lm and angle to horizontal direction 90° are considered for Fig. 3(e). For, Fig. 3(b) and (c) the size of monotonic and cyclic plastic zones are marginally increased due to slightly increase in stress concentration in the presence of micro-cavity. Whereas, the shape of monotonic and cyclic plastic zones are marginally distorted due to micro-cavity position. High stress field is created around the crack tip because of stress concentration. Similarly, the high stress field is also present around the micro-cavity. The presence of the micro-defect in the vicinity of the crack tip leads to the interaction effect between the two high stress fields. As a consequence, slight alteration of size and shape of monotonic and cyclic plastic zones are observed depending upon the position of the micro-cavity. But the major change in size and huge distortion in shape of monotonic and cyclic plastic zones are not observed. On the contrary, Bouiadjra et al. [13] have noticed 75%
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S.K. Paul / Engineering Fracture Mechanics 158 (2016) 13–22
Ratcheting strain (Δεr22)
Ratcheting strain (Δεr11) (a)
2.5 mm
2.5 mm
Ratcheting strain (Δεr22)
Ratcheting strain (Δεr11)
(b)
2.5 mm
2.5 mm
Ratcheting strain (Δεr22)
Ratcheting strain (Δεr11)
(c)
2.5 mm
2.5 mm
Fig. 4. Increase in ratcheting strain accumulation (Der) from 1 to 50 cycle in parallel (22) and perpendicular (11) to loading direction. The loading p p conditions are: DK of 22 MPa mm and R ratio of 0.5 (Kmax = 29.33 MPa mm) at four different position of micro-cavity (micro-cavity radius 30 lm) and without micro-cavity.
reduction in plastic zone size and huge distortion of plastic zone shape in presence of micro-cavity. Presence of an additional stress concentrator (i.e. micro-cavity) always increases the stress levels. Therefore, reduction of plastic zone size in presence of micro-cavity contradicts with the fundamental of deformation behaviour. Cyclic plastic deformation modes of material are fundamentally three types: LCF, ratcheting and mean stress relaxation. Symmetric stress/strain cycling is normally known as LCF, asymmetric stress and strain cycling are generally known as ratcheting and mean stress relaxation respectively. Cyclic plastic deformation mode ratcheting can be differentiated with other two cyclic plastic deformation modes LCF and mean stress relaxation by observing the progressive accumulation of permanent strain at a certain direction (normally in the direction of mean stress). The ratcheting strain (er) for a particular cycle can be defined as the average of maximum (emax) and minimum (emin) peak strains of that cycle.
er ¼ 1=2ðemax þ emin Þ
ð5Þ
Increase in ratcheting strain accumulation from 1 to 50 cycle can be defined as
Der ¼ er50 er1
ð6Þ
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S.K. Paul / Engineering Fracture Mechanics 158 (2016) 13–22
Ratcheting strain (Δεr22)
Ratcheting strain (Δεr11)
(d)
2.5 mm
2.5 mm
Ratcheting strain (Δεr22)
Ratcheting strain (Δεr11)
(e)
2.5 mm
2.5 mm Fig. 4 (continued)
where er50 and er1 are the ratcheting strain in 50th and 1st cycle respectively. The increment in ratcheting strain accumulation up to 50th cycles in the loading direction (Der22) and perpendicular to loading direction (Der11) are computed in every integration points for above mentioned five conditions and shown in Fig. 4. Author has observed in his previous investigation that ratcheting occurs in the cyclic plastic zone of a fatigue crack tip for asymmetric load cycling [6]. As R ratio is 0.5 for this applied loading condition so, mean stress intensity factor (Kmean) is positive. Therefore, ratcheting is expected in the cyclic plastic zone. Significant amount of ratcheting strain accumulation is noticed for without micro-cavity in Fig. 4(a). However, the ratcheting strain accumulations for both loading (Der22) and perpendicular to loading (Der11) directions are observed at 45° angle to loading direction. As discussed by author’s previous work [6] that this ratcheting strain accumulation plays a role in striation formation during fatigue crack propagation. Depending upon the position of micro-cavity the ratcheting strain accumulation also alter accordingly (Fig. 4). It is noticed that the horizontal position of micro-cavity (Fig. 4(b)) alters the ratcheting strain accumulation direction and it is also directed towards the path of micro-cavity. On the other hand, the micro-cavity position of 15° to horizontal direction (Fig. 4(c)) and radial distance of 80 lm from crack tip does not alter the ratcheting strain accumulation direction. The ratcheting strain accumulation direction alters marginally for the micro-cavity position of 45° (Fig. 4(d)) and 90° (Fig. 4(e)) to horizontal direction. Effect of micro-cavity size on monotonic and cyclic plastic zones is illustrated in Fig. 5. Similar loading condition is also p used in this case, DK of 22 MPa mm and R ratio of 0.5. The different micro-cavity sizes (radius of micro-cavities are 30, 50 and 100 lm) are considered in this current investigation. The micro-cavity position is kept as constant in all three conditions, distance form fatigue crack tip is 80 lm in horizontal direction. Fig. 5 shows that the micro-cavity size also alters the shape of monotonic and cyclic plastic zones. Normally high stress field presents near any stress concentrator. Here, high stress field is observed around both crack and micro-cavity. The stress field also changes with alteration of micro-cavity size. Therefore, the interaction effects between the two high stress fields (crack and micro-cavity) also alter. As a result, shape of monotonic and cyclic plastic zones changes with alteration of micro-cavity size. Effect of distance between crack tip and micro-cavity on monotonic and cyclic plastic zones is demonstrated in Fig. 6. Micro-cavity radius of 100 lm and similar loading conditions are adopted in this case. The two different micro-cavity positions are considered in the current investigation, centres of circular micro-cavities are 80 and 140 lm from the crack tip. The interaction of two stress fields (i.e. around crack and micro-cavity) also changes with the alteration of microcavity position. Therefore, shapes of monotonic and cyclic plastic zones are altered with the alteration of distance between crack tip and micro-cavity. For close distance between crack tip and micro-cavity (Fig. 6(a)), the micro-cavity lies within a
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S.K. Paul / Engineering Fracture Mechanics 158 (2016) 13–22
Monotonic plasc zone
2.5 mm
Monotonic plasc zone
2.5 mm
Monotonic plasc zone
2.5 mm
Cyclic plasc zone
(a)
2.5 mm
Cyclic plasc zone
(b)
2.5 mm
Cyclic plasc zone
(c)
2.5 mm
p p Fig. 5. Monotonic and cyclic/reverse plastic zones at DK of 22 MPa mm and R ratio of 0.5 (Kmax = 29.33 MPa mm) at three different sizes of micro-cavity (80 lm ahead of crack tip in horizontal direction): (a) micro-cavity radius of 30 lm, (b) micro-cavity radius of 50 lm, and (c) micro-cavity radius of 100 lm.
common monotonic and cyclic plastic zones, however the monotonic and cyclic plastic zones are distorted towards the microcavity. For wider distance between crack tip and micro-cavity (Fig. 6(b)), separate monotonic and cyclic plastic zones are visible around the micro-cavity and the monotonic and cyclic plastic zones near the crack tip are in normal shape (not distorted). The position and size of micro-cavity play very important role from fatigue crack growth point of view. A cyclic plastic zone is present around the micro-cavity. This cyclic plastic zone can be the cause of the initiation of micro-crack at the micro-cavity tip. If the intensity of the stresses is sufficient, the two cracks (original fatigue crack and newly form crack at the tip of micro-cavity) can propagate simultaneously and they coalesce rapidly. However, if the intensity of the stresses is not sufficient, the micro-crack initiated at the tip of micro-cavity cannot propagates, but the main crack propagates and their coalescence would be done slowly. 4. Conclusions This investigation has been carried out in order to analyse numerically the effect of presence of micro-cavity on the shape and size of monotonic and cyclic plastic zones at the fatigue crack tip. Two dimensional implicit plane strain finite element
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S.K. Paul / Engineering Fracture Mechanics 158 (2016) 13–22
Monotonic plasc zone
2.5 mm
Monotonic plasc zone
2.5 mm
Cyclic plasc zone
(a)
2.5 mm
Cyclic plasc zone
(b)
2.5 mm
p p Fig. 6. Monotonic and cyclic/reverse plastic zones at DK of 22 MPa mm and R ratio of 0.5 (Kmax = 29.33 MPa mm) at two different positioned of microcavity (micro-cavity radius of 100 lm): (a) 80 lm ahead of crack tip in horizontal direction, and (b) 140 lm ahead of crack tip in horizontal direction.
analysis has been conducted on a compact tension specimen. The Chaboche non-linear kinematic hardening model has been used to represent the cyclic stress–strain response of the material, and the material parameters have been calibrated against strain-controlled experimental data. The following conclusions can be deduced from the obtained results: The obtained results show that the presence of micro-cavity affects the shape and the size of the crack tip plastic zones. Noticeable distortion of shape of monotonic and cyclic plastic zones are observed for certain position of micro-cavity. Micro-cavity position (angle and distance from crack tip) plays a significant role in controlling the shape of monotonic and cyclic plastic zones. The ratcheting strain accumulation direction during asymmetric loading condition is affected by the position of microcavity. The shape of monotonic and cyclic plastic zones alter with the size of micro-cavity. Larger size of micro-cavity results higher stress field and more distortion of shape of monotonic and cyclic plastic zones.
Acknowledgement Author likes to acknowledge Dr. Soumitra Tarafder, National Metallurgical Laboratory, Jamshedpur, India for his valuable suggestions. References [1] Tong J, Lin B, Lu Y-W, Madi K, Tai YH, Yates JR, et al. Near-tip strain evolution under cyclic loading: in situ experimental observation and numerical modelling. Int J Fatigue 2015;71:45–52. [2] Cornet C, Zhao LG, Tong J. Ratchetting strain as a damage parameter in controlling crack growth at elevated temperature. Engng Fract Mech 2009;76:2538–53. [3] Tong J, Zhao LG, Lin B. Ratchetting strain as a driving force for fatigue crack growth. Int J Fatigue 2006;46:49–57. [4] Zhao LG, Tong J. A viscoplastic study of crack-tip deformation and crack growth in a nickel-based superalloy at elevated temperature. J Mech Phys Solids 2008;56:3363–78. [5] Paul SK, Tarafder S. Cyclic plastic deformation response at fatigue crack tips. Int J Press Ves Pip 2013;101:81–90. [6] Paul SK. Numerical models of plastic zones and associated deformations for a stationary crack in a C(T) specimen loaded at different R-ratios. Theor Appl Fract Mech 2015;0:1–10. http://dx.doi.org/10.1016/j.tafmec.2015.10.00 [Available online 3 November 2015].
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