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Fracture of notched round-bar NiTi-specimens
Engineering Fracture Mechanics 84 (2012) 1–14
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Fracture of notched round-bar NiTi-specimens J.S. Olsen a, Z.L. Zhang a,⇑, H. Lu b, C. van der Eijk c a b c
Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway Materials Science and Engineering School, Shanghai Jiaotong University, Shanghai 200240, China SINTEF Materials and Chemistry, 7465 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 1 December 2010 Received in revised form 6 December 2011 Accepted 21 December 2011
Keywords: Shape memory alloys Notches Fractography Triaxiality Finite element analysis NiTi SEM
a b s t r a c t In this work fractography has been used to characterize the fracture surfaces of superelastic, notched round-bar NiTi-specimens. Several fractography studies have been performed for conventional materials, but only few studies have been conducted on NiTi-alloys. The aim of this work is to investigate the effect of semi-circular notches on the fracture behavior of NiTi. The main results indicate that the fracture process is a mixture of cleavage and micro-void coalescence, and that decreasing the notch-radius leads to a loss of ductility which manifests in a reduced fracture strain. Within the range of notches studied in this work, we can observe a transition from high scatter to low scatter in the fracture strain when reducing the notch-radius. By combining fractography studies, finite element analyses and the Rice–Tracey void-growth model, it is argued that fracture is most likely initiated close to the notch for all notch-radii studied herein, and that cleavage or quasi cleavage is the dominating fracture mechanism. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Since their discovery in the 1960s, NiTi-alloys have been put to use in a number of applications [1]. It has been particularly successful in the medical industry, due to the material’s bio-compatibility and special mechanical properties [2]. A lot of effort has been put into understanding the mechanisms involved during phase transformation and the superelastic behavior of NiTi-alloys [3,4], and the possibility to reproduce their mechanical behavior numerically through constitutive modeling [5–9]. However, only recently an increasing interest has arisen towards failure of shape memory alloys. The main focus in this regard is found in fatigue failure [10–12], effect of cracks on martensite transformation [13–15] and the effect of martensite transformation on fracture toughness of NiTi-alloys [16–19]. The effect of stress triaxiality is often mentioned as a contributor during fracture of shape memory alloys [11,20]. However, only few attempts have been made to quantify it’s effect [21]. In this work an attempt is made to investigate the effect of triaxiality on the fracture processes in NiTi by studying notched round-bar specimens. Tensile tests until fracture of specimens with different circular notches are conducted. Subsequently SEM has been employed to investigate the fracture surfaces. Also, finite element analyses are used to quantify the effects of notches.
⇑ Corresponding author. E-mail address: [email protected] (Z.L. Zhang). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2011.12.007
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Nomenclature Af As d D R EA EM L Mf Ms r r0 Rf Rf
a
L e epl0 eplc ef depl eq mA mM req
rAf
rm
rMf rAs rMs 0
austenite finish temperature austenite start temperature specimen diameter diameter of minimum cross-section notch-radius elasticity modulus for austenite elasticity modulus for martenite specimen length martensite finish temperature martensite start temperature current void radius initial void radius R-phase start temperature R-phase finish temperature material constant transformation length total equivalent strain initial equivalent plastic strain equivalent plastic strain at fracture fracture strain plastic strain increment Possion ratio for austenite Possion ratio for martensite equivalent von Mises stress critical stress for austenite transformation finish hydrostatic stress critical stress for martensite transformation finish critical stress for austenite transformation start critical stress for martensite transformation start initial value
2. Experimental set-up and procedures In this work, a series of tests have been conducted on superelastic NiTi-alloy round-bars. The specimens have a diameter of d = 5.3 mm and a total length of L = 50 mm. Notches with three different radii were machined on the specimens by latheturning. Due to manufacturing limitations the smallest notch-radius was chosen to be R = 0.75 mm, while two other configurations were set to have radii of R = 1.00 mm and R = 1.25 mm, respectively. The minimum diameter at the notch-root was set to D0 = 2 mm for all specimens. See Fig. 1 for details. The specimens were heat-treated in a furnace at 575 °C and subsequently air-cooled. The heat-treatment was conducted after machining in order to avoid surface stresses that might emerge. Three exemplars of each configuration were tested. 2.1. Tensile testing set-up To understand the effect of the notches on the material behavior, tensile tests were conducted in a Zwick/Roell Z020 tensile testing machine with a 20 kN load capacity. Due to the relatively small size of the specimens, image acquisition and processing was chosen as the preferred tool for strain measurement. The strain was calculated from the change in diameter at
Fig. 1. Schematic drawing of specimen geometry.
J.S. Olsen et al. / Engineering Fracture Mechanics 84 (2012) 1–14
3
Fig. 2. Schematic plot of the image acquisition system.
the minimum cross-section. In order to capture the diametrical changes a Daheng DH-HV31202UC Charged-Coupled Device (CCD) with a 2048 1536 resolution and a MLM-3X-mp lens was used for image acquisition. Images were acquired 1 frame per second using a laptop with an acquisition software provided by the CCD-supplier. The commercial image processing software, Ni-vision, was subsequently used in combination with Matlab to measure and calculate the strains. Fig. 2 schematically shows the image acquisition set-up. The distance between the lens and center of the notch was set to 100 mm under CCD-provider guidelines. Also, a back-light was mounted 60 mm behind the specimen to ensure good contrast in the images. All tests were conducted under deformation control at room temperature (ca. 20 °C), and each specimen was loaded until fracture. The loading speed, 0.05 mm/min, was chosen to ensure quasi-static loading conditions. The strain and stress are calculated by the following relations, respectively:
e ¼ 2 ln
D0 ; D
r¼
F 2
p D4
ð1Þ
where F is the axial force measured by the tensile testing machine. 2.2. Differential scanning calorimetry The differential scanning calorimetry (DSC) was conducted using a Q10 DSC from TA instruments. The specimen was carefully cut using a line-cutter with coolant, to a size of approximately 1 1 mm, not to exceed the maximum allowed weight of 10 mg. Two different specimens were analyzed; one taken from an un-machined NiTi-rod and one taken from a NiTi-rod machined with lathe-turning. Both rods were heat-treated as previously described before line-cutting. The objective of the DSC was to determine the phase transformation temperatures Af, As, Ms and Mf, as well as to investigate if the machining process imposed any effect on the phase transformation temperatures. Austenite finish and start temperatures are denoted Af and As, respectively. Similarly Ms and Mf respectively denote martensite start and finish temperatures. Rs and Rf are the R-phase start and finish temperature. Each of the two specimens was heated from 60 °C to 100 °C, with a temperature increase of 10 °C/ min, and subsequently cooled at the same rate to 60°C. 2.3. Scanning electron microscopy and energy dispersive spectroscopy In this work, the study of the fracture surface is key to get an understanding of how the NiTi-alloy behaves during fracture. Both overview images of the entire fracture surface, and close-up images of particular features have been collected and investigated. All SEM-investigations are conducted post fracture. The scanning electron microscopy (SEM) was conducted using a Zeiss Ultra 55 LE scanning electron microscope. All results presented in this paper are secondary electron images. The working distance (WD) was set to 8 mm, and acceleration voltage 15 keV. An EDAX Genesis v. 5.21 was used for energy dispersive spectroscopy (EDS), to determine the element composition of particularly interesting features (e.g. inclusions) on the fracture surface. It is noted that EDS is not a highly accurate method to determine element composition when employed on inhomogeneous and rough surfaces. However, it serves well as a mean to get a notion of the main elements of a feature. 3. Experimental results and discussion Fig. 3 shows the heatflow-temperature diagram from the DSC-analyses. It can be seen that the lathe-turning does not impose any significant effect on the phase transformation temperatures. The peak heatflow values are slightly higher for the machined specimen. It should be noted, however, that each specimen is taken from a random position in the NiTi rod, i.e. whether there exists any surface effects from the machining process or not is not known. The phase transformation temperatures are summarized in Table 1. With an austenite finish temperature of 14.2 °C the material can be expected to behave superelastic when tested at room temperature. Previous investigations of the ductile fracture behavior of smooth and mildly notched round-bar specimens have showed that for conventional materials (see e.g. Hancock and Mackenzie’s study of HY130 steel [22]), fracture is often initiated by
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6 Un−Machined Machined
Heatflow [mW]
4 2 Heating 0 Cooling −2 −4
−50
0
50
100
Temperature [°C] Fig. 3. Results from DSC analyses of machined and un-machined specimen.
Table 1 Phase transformation temperatures found by DSC analysis. Phase trans. temperatures (°C)
Af
As
Rs
Rf
Ms
Mf
Un-machined Machined
14.2 13.6
4.9 4.8
7.6 7.7
6.4 6.2
36.9 37.7
50.6 49.8
void nucleation, growth and coalescence in the center of the minimum cross-section. This type of fracture behavior is governed by two factors, namely high triaxiality and large plastic strains in the center of the specimen’s minimum cross-section [23]. The process can often be recognized by the characteristic cup-and-cone fracture surface, or a penny-shaped crack in the center of the fracture surface [24]. Fig. 4a–f shows SEM-images of the complete fracture surface, and corresponding stress–strain curves, of notched specimens with notch-radii ranging from R = 0.75 mm to R = 1.25 mm. Stress–strain curves for superelastic–plastic NiTi-alloys differ from those recorded from conventional elastic–plastic materials, in that they have a distinct plateau due to stress induced phase transformation from austenite to martensite, which is a reversible process. If the load is further increased after completed phase transformation, elastic deformation of martensite commences before plastic yielding initiates at some yield stress. For the largest notch, the stress–strain curve (see Fig. 4f) indicates a relatively ductile behavior with plastic strains close to 10%. Given the notion that for ductile materials which are smooth or mildly notched, fracture is prone to initiate at the center of the minimum cross-section (a previous study by Sun [25] showed this for specimen with a minimum cross-section to notch-radius ratio of D0/R = 2), one might expect to observe traces of the afformentioned features which characterize ductile fracture. However, the SEM fractographs show no indications of neither penny-shaped cracks, nor the characteristic cup-and-cone fracture surface. This indicates that the fracture behavior of the investigated NiTi-alloy, although ductile in terms of relatively large fracture strains, is not governed by ductile fracture mechanisms, such as void-growth and coalescence, alone. The main goal of this study is to investigate the effect of notches in general, and whether varying the notch-radius has any effect on the NiTi-alloy fracture behavior. The most evident effect of the latter, is a loss in ductility when decreasing the notch-radius. When notches are introduced in a specimen, notch strengthening will increase the tensile stress needed for yielding, in the case of NiTi-alloys this also include the critical stress for transformation. Notch strengthening is an effect that arise from the triaxial stresses that develops in the notched section of the specimen [23]. It is thought that the loss of ductility is directly connected to the increase in triaxiality [26]. The loss of ductility can be observed in the fracture strain which is plotted as a function of notch-radius in Fig. 5. It can be observed that for the two specimens with the largest notches (R = 1.00 mm and R = 1.25 mm), there is a large scatter in the fracture strain. However, for the specimen with the smallest notch (R = 0.75 mm) the scatter is relatively small. It is noted that sub-notches originating from the machining process were found on some of the notches. As such sub-notches will induce stress-concentrations, this can be one explanation for the scatter. By comparing the fracture surfaces of the different exemplars of each configuration, it was also found that the fracture surface topography of the specimens with the smallest notch was more uniform than the two largest notches (not shown here). Similar findings regarding loss of ductility due to notch size and acuity have been reported for flat dog-bone NiTi-specimens [27]. Fig. 6a shows some details of the various features on the fracture surface of a specimen with notch-radius R = 0.75 mm. Cleavage facets can be observed, which indicate that the fracture mechanism is quasi-cleavage. Quasi-cleavage is considered to be dominated by cleavage, and exhibit small parts with micro-void coalescence [28]. However, there is a strong presence of micro-voids in all the investigated specimens, so it is probably more accurately described as a mixture of cleavage and micro-void coalescence [28]. Similar observations have been made in a previous study by Gall et al. [29] for a poly-crystalline NiTi-alloy.
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True stress [MPa]
1800
1200
600
R=0.75 mm
0
0
0.04
0.08
0.12
0.16
0.2
True strain
(a)
(b) True stress [MPa]
1800
1200
600
R=1.00 mm
0
0
0.04
0.08
0.12
0.16
0.2
True strain
(c)
(d) True stress [MPa]
1800
1200
600
R=1.25 mm
0
0
0.04
0.08
0.12
0.16
0.2
True strain
(e)
(f)
Fig. 4. SEM-images of the fracture surfaces, and corresponding stress–strain curves, from three specimens with notch-radius (a) and (b) R = 0.75 mm, (c) and (d) R = 1.00 mm and (e) and (f) R = 1.25 mm.
0.2
Fracture strain
0.16 0.12 0.08 0.04 0 0.5
0.75
1
1.25
Notch−radius [mm] Fig. 5. Fracture strain as a function of notch-radius.
1.5
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Fig. 6. SEM-images showing (a) voids, cleavage facets and ridges on the fracture surface of a specimen with notch-radius R = 0.75 mm and (b) a shear-lip close to the edge of the fracture surface of a specimen with notch-radius R = 1.25 mm.
3200 Ti
2800
Counts [K]
2400 2000 1600 1200 800 400
C O Ni
0
1
2
3
4
5
6
7
8
9 10
Energy [keV]
(a)
(b)
Fig. 7. SEM-images showing (a) NiTi2 islands with micro-cracks on the fracture surface of a specimen with notch-radius R = 0.75 mm and (b) results from an EDS analysis showing the content of the islands with micro-cracks.
On several specimens, a distinct shear-lip structure can be observed in portions of the circumferential edge (see Fig. 6b). The surface of this shear-lip is relatively smooth indicating that a brittle fracture has occurred in this area. This effect is most profound for the larger notches (R = 1.00 mm and R = 1.25 mm).1 Such brittle behavior indicates that fracture in this portion of the specimen has been relatively fast, and it is reasonable to assume that the shear-lip structure indicates the finalizing stage of fracture. Hence, it can help identify the area of fracture initiation. The topic of fracture initiation will be further discussed below. In addition to the aforementioned features, a number of second phase particles can be found as small islands on the various fracture surfaces (see Fig. 7a). These particles are apparently of a brittle nature as they contain micro-cracks which are limited by the surrounding matrix. An EDS-analysis indicates that the particles consists predominately of Ti (see Fig. 7b); the only Ti-rich compound of the NiTi-system is NiTi2. It is noted that the size of these islands are rather small (2 lm); as a consequence the information volume from the EDS may be larger than the investigated area – resulting in an inaccurate measurement of the particle content. Several inclusions can be observed on the fracture surfaces studied here. Fig. 8a shows the presence of some inclusions with size 5 lm. EDS showed that these inclusions are most likely Ti-oxides (see Fig. 8b). Also, however not investigated by EDS, a number of smaller inclusions can be seen. In a previous study [31] the same material used in this work was metallurgically characterized, and a number of small Ti-carbides (2 lm) was found in the micro-structure – it is possible that the small inclusions in Fig. 8a are of the same character. Conventionally, when studying fracture initiation one usually introduces some crack-like notch in the specimen; either by cyclic-deformation or by machining. This is to better control where fracture initiates, and ease the subsequent investigations. When, as in this study, no such crack-like notch is present, it becomes difficult to establish an exact location for the fracture
1 The reason as to why the shear-lip feature is larger on the blunter notches is not completely understood. It can be argued that blunter notches lead to an increase in elastically stored energy in the specimen causing a larger material volume undergoing unstable fracture at the finale stages of failure [30].
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1800
Ti
Counts [K]
1500 1200 Ni
900
O
600 C
Ni
300 0
1
2
3
4
5
6
7
8
9 10
Energy [keV]
(a)
(b)
Fig. 8. SEM-images showing (a) Ti-oxides and (b) results from an EDS analysis showing the content of the Ti-oxides.
Fig. 9. SEM-images showing fan-like patterns on the fracture surface of specimens with notch-radii (a) R = 0.75 mm, (b) R = 1.00 mm and (c) R = 1.25 mm.
initiation point. However, by carefully studying the fracture surface by SEM, it is still possible to get an idea of where fracture might have occurred. Fig. 9 shows fracture surfaces from three specimens with the different notch-radii investigated herein. On the fracture surfaces a fan-like pattern can be observed (these are elucidated by stippled lines). When fracture occurs at a point, the crack will branch out creating more cracks, following the energetically most favorable direction through the material, and create patterns like the ones showed in Fig. 9. From the figures it can be seen that fan-like patterns all point in the direction of a shear-lip feature on the opposite side of the fracture surface. The combination of these patterns and the aforementioned shear-lip suggests that fracture has initiated close to the notch-surface. A closer study of the circumferential edge of the fracture surface, leads to the discovery of significant cracks on the notch-surface in the vicinity of the projected initiation area (Fig. 10). This is the case for several of the specimens. Several specimens also exhibited cleavage features in the vicinity of the fracture initiation area, close to the notch-root (see Fig. 11). This indicates that fracture initiation is stress controlled, and starts at the notch-root. The fractography studies
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Fig. 10. Cracks on the notch-surface indicating that fracture initiates at the surface or close to the surface.
Fig. 11. SEM-images showing cleavage features in the vicinity of the fracture initiation area for specimens with notch-radii (a) 1.00 mm and (b) 1.25 mm.
do not give a clear answer towards where the fracture initiation point is located for the different specimen configurations. Neither does it completely clarify whether cleavage or ductile fracture mechanisms, as both are significantly present, govern the fracture initiation. Therefore fracture initiation has been further studied through finite element analyses and the Rice– Tracey void-growth model [32]. The results are presented in Section 4.3. 4. Finite element simulations 4.1. Finite element model A fractography study is an excellent tool to investigate which fracture processes are dominant during fracture, and to identify what internal features, such as inclusions, precipitates and micro-cracks, have influenced the process. However, as the fractography study is conducted after fracture it does not yield any information about the material behavior during deformation prior to failure. In order to get a deeper understanding of how notches affect the fracture process, finite element simulations have been conducted using the commercial finite element software Abaqus 6.9–2. Particularly interesting is it to investigate stress, plastic strain and triaxiality distributions over the minimum cross-section at the notch root, as these variables play an important role during fracture. The experimental results have shown that it is difficult to assess exactly the point of fracture initiation, as well as the dominating fracture mechanism at initiation, through fractography studies alone. Therefore, the finite element simulations and the Rice–Tracey model [32] have been employed to assess where in the specimens fracture is most likely to initiate, and establish which fracture mechanism is most likely to be dominating (see Section 4.2 for details). To model the superelastic behavior of the NiTi-alloy, an approach based on the framework developed by Auricchio and coworkers [6,7,33] has been used. The constitutive model is developed and implemented in the commercial finite element software Abaqus at SIMULIA/west by Rebelo et al. It is an extension of Auriccios model that accounts for plastic deformations [34]. For a more thorough outline of the constitutive model used herein see Ref. [35]. The three notch radii investigated experimentally (R = 0.75–1.25), are modeled in the numerical investigations. Due to material and geometrical symmetry, only a quarter of the specimen is modeled using an axis-symmetric approach. Fig. 12 shows the meshed model used in the simulations. Here R and D0 are the notch-radius and minimum diameter at
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Fig. 12. Meshed model used in the finite element analysis.
Table 2 Material parameters used in the simulations. EA, EM (MPa)
mA, mM
M rM s ; rf (MPa)
rAs ; rAf (MPa)
L
47,000, 15.667
0.33
351, 415
250, 175
0.026
the cross-section, respectively. The model consists of 1000 8-node axis-symmetric elements with reduced integration (named CAX8R in Abaqus). The material data were chosen to fit the stress–strain curves from the tested notched specimens. The problem at hand is highly non-linear in nature as it includes non-linear material behavior and geometry effects, and an irregular specimen geometry. This makes convergence an issue during the simulations, hence some of the convergence criteria available in Abaqus was relaxed and the linesearch method was used to stabilize the simulations [36]. Table 22 shows the material data used in the simulations. For more details regarding material parameter fitting and convergence issues in the finite element simulations, the interested reader is referred to [35]. 4.2. The Rice–Tracey void growth model The experimental results presented herein show that the governing fracture mechanism for the NiTi-alloy in question is a mixture of cleavage and ductile fracture. The process of ductile fracture can be summarized by micro-void nucleation, voidgrowth and finally void coalescence leading to failure [24]. For voids to grow, two factors have to be present – namely a certain degree of triaxial stress and plastic deformation. From the experimental results it is difficult to assess the main differences, if any, between the fracture behavior of different notch configurations. Particularly this is true for the fracture initiation point. Rice and Tracey [32] developed a model to describe void-growth in a elasto-plastic material without hardening. The model can be written as
ln
Z epl c r 3rm ¼ depl a exp eq r0 2req epl
ð2Þ
0
2 In Table 2, EA,EM,mA and mM are the elasticity moduli and Possion ratios for the austenite and martensite phase, respectively. rM s and respectively denote the critical stress for start and end of forward and reverse transformation. eL represents the transformation length.
rMf ; rAs and rAf ,
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1800 Experimental Simulation
1500
True Stress [MPa]
True Stress [MPa]
1800
1200 900 600 300
R=1.25mm R=1.00mm R=0.75mm
1500 1200 900 600 300
R=0.75mm
0
0
0.02
0.04
0.06
0.08
0
0.1
0
0.02
0.04
0.06
0.08
True strain
True Strain
(a)
(b)
0.1
Fig. 13. Stress–strain curves comparing (a) experimental and simulations results for a specimen with radius R = 0.75 mm and (b) effect of notch-radius on the stress–strain behavior.
1.4
1.4 R=1.25mm R=1.00mm R=0.75mm
1.2
Triaxiality
Triaxiality
1 0.8 0.6 0.4 0.2 0
R=1.25mm R=1.00mm R=0.75mm
1.2 1 0.8 0.6 0.4 0.2
Node at notch−root
0
0.02
0.04
0.06
0.08
0
Node at center
0
0.02
0.04
True strain
True strain
(a)
(b)
0.06
0.08
Fig. 14. Triaxiality plotted against true strain at (a) the notch-root and (b) the center of the specimen.
where r and r0 is current and initial void radius, respectively. a is a constant related to material behavior. Beremin [37] deterpl mined that a = 0.283 was appropriate to account for hardening. epl c is the equivalent plastic strain at fracture, while e0 is the pl initial equivalent plastic strain. deeq is the equivalent plastic strain increment. Finally, rm and req are the hydrostatic and equivalent von Mises stress, respectively. By taking the exponential on both sides of Eq. (2) we get the void-growth ratio. Since the model does not alter the material behavior [38], it is an excellent tool to perform an uncoupled investigation through finite element simulations of the intrinsic behavior in the material. In this work the Rice–Tracey void-growth model is used to help establish whether ductile fracture or cleavage mechanisms is dominating at fracture initiation. If void-growth and coalescence are dominating, the Rice–Tracey model should predict the same void-growth ratio for all three specimens at the recorded fracture strain, since in the model it is assumed that fracture occur at a critical value of the void-growth ratio.
4.3. Finite element analysis results and discussion A comparison between the experimental and numerical stress–strain curves are shown in Fig. 13a. It is clear that the model used herein manages to capture the particular phase transformation behavior of the tested NiTi-alloy. The simulation results coincide well with the experimental data; both the general stress level and critical values such as the critical stresses L M for transformation (rM s and rf ) and the transformation length ( ) is satisfactorily accurate. Also the part of the stress–strain curve representing plastic deformation shows a reasonably good fit between simulation results and experimental data. When comparing the stress–strain curves from the different specimen configuration (Fig. 13b), it can be observed that reducing the notch-radius slightly elevates the stress level. It can be expected that the stress-level is elevated due to the aforementioned notch-strengthening effect. However, the results show that for the range of notch-sizes investigated herein, the difference in the general stress–strain behavior is not very large. The main goal of introducing finite element analyses in this study is to get a better understanding of the material behavior just prior to fracture, as well as to investigate the effect of changing the notch-radius on the specimen’s mechanical response. Introducing semi-circular notches in round-bar specimen is a simple way to induce a triaxial stress-field in the specimen. Fig. 14a and b shows the triaxiality as a function of strain at the notch root and in the center of the specimen, respectively.
J.S. Olsen et al. / Engineering Fracture Mechanics 84 (2012) 1–14
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1.2 Triaxiality at center Triaxiality at notch
Triaxiality
1 0.8 0.6 0.4 0.2 0 0.5
0.75
1
1.25
1.5
Notch−radius [mm] Fig. 15. Initial triaxiality calculated by finite element analyses as a function of notch-radius.
1.12
R=1.25mm R=1.00mm R=0.75mm
=0.15
r/r0
1.09
1.06
1.03
1
0
0.2
0.4
0.6
0.8
1
Normalized distance from notch Fig. 16. Void-growth ratio over the minimum cross-section for all specimen configurations at e = 0.15.
At the notch root, the triaxiality is close to constant during deformation for all specimen configurations. Also, the triaxiality level varies little for the different notch-radii at the notch root. However, at the center of the specimen, the triaxiality varies a lot during deformation. Comparing Figs. 13 and 14b shows that when there is a ‘‘phase change’’ in the specimen, i.e. change from phase transformation to plastic yielding, there is a drop in the triaxial stress-level. The figure also shows that reducing the notch-radius increases the triaxiality in the center of the specimen. The latter is emphasized in Fig. 15, which shows the initial triaxiality at the notch-root and the center of the specimen as a function of notch-radius. For fracture to occur as a consequence of void nucleation, growth and coalescence, a combination of triaxial stress and plastic deformation has to be present [32]. By employing the Rice–Tracey void-growth model, presented in Section 4.2, over the cross-section, it is possible to evaluate the void-growth ratio at a given point on the cross-section. To show the effect of notch-radius on the void-growth ratio, results are compared for the different specimen configurations at e = 0.15 (see Fig. 16). For all notch-radii investigated, the void-growth is most significant at the notch-root, with the smallest notch (R = 0.75 mm) yielding the highest void-growth ratio. It should be noted that, according to the results shown here, no void-growth is present at the center of the specimens for the two smallest notch-radii (R = 0.75 mm and R = 1.00 mm), while the specimen with the largest notch-radius experience only a small degree of void-growth at the center. A strain level of e = 0.15 far exceeds the maximum measured fracture strain for the specimens with notch-radii R = 0.75 mm and 1.00 mm. So in order to compare the different specimen configurations prior to fracture, results should be extracted at strain levels corresponding to the fracture strain for each specimen. Fracture strains have been chosen to be ef = 0.076, ef = 0.129 and ef = 0.159 for the specimens with notch-radii R = 0.75 mm, 1.00 mm and R = 1.25 mm, respectively. This corresponds to the largest measured fracture strain for each specimen configuration. Fig. 17b shows the voidgrowth ratio for each specimen at e = ef for each notch-radius. The results are similar to the results showed in Fig. 16 by the fact that the largest void-growth is found at the notch-root for all specimens and that no, or only a small degree of void-growth is present at the center, despite that the triaxiality is largest in the center of the specimen (see Fig. 17a). It is noted that at e = ef the specimen with the largest notch-radius exhibit the highest void-growth ratio, and that the voidgrowth ratio at the notch-root is substantially lower for the specimen with notch-radius R = 0.75 mm. In a recent study, Olsen et al. [35] found, when numerically studying a wide range of notched NiTi-specimens, that stress induced martensitic phase transformation is hindered at the center of notched round-bar specimens. As plastic deformation is subsequent to phase transformation in NiTi, it is also hindered. Olsen et al. attributed this to the triaxial stress level in the
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1.5
1.12 R=1.25mm R=1.00mm R=0.75mm
r/r0
0.6 ε = εf
0
0.2
0.4
0.6
0.8
1
1
0.4
0.6
0.8
(a)
(b) R=1.25mm R=1.00mm R=0.75mm
0.12 0.08 0.04 0
0.2
Normalized distance from notch
ε = εf
0.16
0
Normalized distance from notch−root
0.2
0
1.06 1.03
0.3
Equivalent plastic strain
1.09
0.9
0
R=1.25mm R=1.00mm R=0.75mm
ε = εf
Axial Stress [MPa]
Triaxiality
1.2
0.2
0.4
0.6
0.8
1
2000 1750 1500 1250 1000 750 500 250 0
R=1.25mm R=1.00mm R=0.75mm
ε=ε
f
0
1
0.2
0.4
0.6
0.8
Normalized distance from notch−root
Normalized distance from notch−root
(c)
(d)
1
Fig. 17. Cross-sectional plots showing the distribution of (a) the triaxial stress state, (b) the void-growth ratio, (c) the equivalent plastic strain and (d) the axial stress over the cross-section at the notch-root.
2000
σyy=1742MPa
Axial Stress [MPa]
Axial Stress [MPa]
2000 1600 1200
ε=εf
800 400
σyy=1769MPa
1600 1200 ε=εf
800 400
R = 1.25mm 0
0
0.05
0.1
0.15
R = 1.00mm 0
0.2
0
0.05
0.1
ln(A0/A)
ln(A0/A)
(a)
(b)
Axial Stress [MPa]
2000
0.15
0.2
σyy=1720MPa
1600 1200 ε=εf
800 400
R = 0.75mm 0
0
0.05
0.1
0.15
0.2
ln(A0/A)
(c) Fig. 18. Curves showing the axial stress at the notch-root as a function of true strain for (a) R = 1.25 mm, (b) R = 1.00 mm and (c) R = 0.75 mm. The stress and strain at fracture is indicated by the stippled lines.
J.S. Olsen et al. / Engineering Fracture Mechanics 84 (2012) 1–14
13
center of the specimens. From Fig. 17c it can be observed that the distribution of the equivalent plastic strain over the minimum cross-section is very similar to the void-growth ratio. The largest equivalent plastic strain is found at the notch-root pl for all three specimens at the time of fracture, ranging from epl eq ¼ 0:082 to eeq ¼ 0:146 as the notch-radius is increased from R = 0.75 mm to R = 1.25 mm. The degree of plastic deformation over the cross-section also increases with the notch-radius: for the specimen with R = 1.25 mm plastic deformation can be observed over the entire cross-section, while for the specimens with R = 0.75 mm and R = 1.00 mm only 25% and 65% of the cross-section are plastically deformed. It is reasonable to assume that if no, or only limited plastic strain has developed in a material volume, that no void-growth can occur within that volume. Since void-growth is needed to induce ductile fracture it is therefore reasonable to assume that ductile fracture has not initiated at the center of the specimen, but rather on, or close to the notch-surface for the cases investigated herein. The critical void-growth, or void volume fraction, at fracture will vary with varying triaxiality [39]. Given that fracture does initiate at the notch-surface, it is obvious that the triaxiality level is almost the same at the notch-root for the three specimens studied in this work (see Fig. 15). This would make it reasonable to assume that the Rice–Tracey damage parameter would be similar for all three specimens at fracture. However, there is a difference in the results when analyzing the void-growth ratio, at fracture. The results yield a much lower void-growth ratio at fracture for the specimen with notch-radius R = 0.75 mm than for the specimens with notch-radii R = 1.00 mm and R = 1.25 mm. This points in the direction that void-growth and coalescence are not the governing fracture mechanisms at fracture initiation. Further it is noted that at fracture, the stress-level in the axial direction is very similar for the three specimens at the notch-root (see Fig. 17d) which indicates that failure is governed by a critical stress. This is further confirmed by looking at the axial stress at the notch-root plotted as a function of true strain (see Fig. 18): the stress at fracture is quite similar for all three cases investigated herein. The maximum difference in stress at fracture is less than 3% and is most likely due to inaccuracies in the measured strain from the experimental results. This also shows that, under the notion of stress controlled fracture, the ductility reduces as the notch-radius becomes smaller. The aforementioned differences in void-growth ratio, and the coinciding axial stresses at the notch-root makes it reasonable to assume that quasi-cleavage is the governing fracture mechanism at fracture initiation. 5. Summary and conclusions In this work three round-bar NiTi specimen configurations with different notch-radii have been investigated through fractography studies and finite element analyses. For all configurations the fracture surfaces both cleavage facets, tearing ridges and micro-voids can be observed. This is in agreement with other studies [27,29], and indicates that the complete fracture process is a mixture of ductile and brittle fracture. Also, when studying the fracture strains of the various specimen configurations indications were found that reducing the notch-size leads to a loss in ductility, and that it can reduce the scatter in the fracture strain results. By studying patterns on the fracture surface, indications were found that the initiation site is in the vicinity of the notchroot. On the notch-surface, significant cracks were observed for several specimens close to the indicated fracture initiation area. This suggests that fracture initiation occurs at the notch-root. Through numerical analyses in Abaqus, and uncoupled investigations using the Rice–Tracey void-growth model, it has been found that cleavage or quasi-cleavage is most likely to be the dominating fracture mechanism. The numerical results and fractography studies together, strongly indicate that fracture has initiated on or close to the notch surface. Several inclusions of various types could be observed on the fracture surfaces. Inclusions are known to play an important role in fracture initiation and mechanisms. The role of the inclusions on NiTi fracture behavior is not completely understood and should be subject for further research. Acknowledgments We thank the Norwegian Research Council (NFR) for financial support. The authors also thank Willhelm Dall at SINTEF Materials and Chemistry for his expertise and help when conducting SEM investigations. Finally, we would like to thank the anonymous reviewers, who provided insightful comments which proved very helpful in the work with this paper. References [1] Duerig T. Engineering aspect of shape memory alloys. London: Butterworth Heinemann Ltd.; 1990. [2] Morgan NB. Medical shape memory alloy applications – the market and its products. Mater Sci Engng A – Struct Mater Prop Microstruct Process 2004;378(1–2):16–23. [3] Funakubo H. Shape memory alloys. Precision and robotics, vol. 1. Gordon and Breach Science Publishers; 1987. [4] Otsuka K, Ren X. Physical metallurgy of Ti–Ni-based shape memory alloys. Prog Mater Sci 2005;50(5):511–678. [5] Brinson L. One-dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J Intell Mater Syst Struct 1993;4(2):229–42. [6] Auricchio F, Taylor RL. Shape-memory alloys: modelling and numerical simulations of the finite-strain superelastic behavior. Comput Methods Appl Mech Engng 1997;143(1–2):175–94. [7] Auricchio F, Taylor RL, Lubliner J. Shape-memory alloys: macromodelling and numerical simulations of the superelastic behavior. Comput Methods Appl Mech Engng 1997;146(3–4):281–312.
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J.S. Olsen et al. / Engineering Fracture Mechanics 84 (2012) 1–14
[8] Boyd JG, Lagoudas DC. A thermodynamical constitutive model for shape memory materials. Part I. The monolithic shape memory alloy. Int J Plasticity 1996;12(6):805–42. [9] Wang XM, Xu BX, Yue ZF. Micromechanical modelling of the effect of plastic deformation on the mechanical behaviour in pseudoelastic shape memory alloys. Int J Plasticity 2008;24(8):1307–32. [10] McKelvey AL, Ritchie RO. Fatigue-crack propagation in nitinol, a shape-memory and superelastic endovascular stent material. J Biomed Mater Res 1999;47(3):301–8. [11] McKelvey AL, Ritchie RO. Fatigue-crack growth behavior in the superelastic and shape-memory alloy nitinol. Metall Mater Trans A – Phys Metall Mater Sci 2001;32(3A):731–43. [12] Robertson SW, Ritchie RO. In vitro fatigue-crack growth and fracture toughness behavior of thin-walled superelastic nitinol tube for endovascular stents: a basis for defining the effect of crack-like defects. Biomaterials 2007;28(4):700–9. [13] Wang XM, Wang YF, Baruj A, Eggeler G, Yue ZF. On the formation of martensite in front of cracks in pseudoelastic shape memory alloys. Mater Sci Engng A 2005;394(1–2):393–8. [14] Robertson SW, Mehta A, Pelton AR, Ritchie RO. Evolution of crack-tip transformation zones in superelastic nitinol subjected to in situ fatigue: a fracture mechanics and synchrotron X-ray microdiffraction analysis. Acta Mater 2007;55(18):6198–207. [15] Gollerthan S, Young ML, Neuking K, Ramamurty U, Eggeler G. Direct physical evidence for the back-transformation of stress-induced martensite in the vicinity of cracks in pseudoelastic NiTi shape memory alloys. Acta Mater 2009;57(19):5892–7. [16] Yi S, Gao S. Fracture toughening mechanism of shape memory alloys due to martensite transformation. Int J Solids Struct 2000;37(38):5315–27. [17] Yi S, Gao S, Shen L. Fracture toughening mechanism of shape memory alloys under mixed-mode loading due to martensite transformation. Int J Solids Struct 2001;38(24–25):4463–76. [18] Freed Y, Banks-Sills L. Crack growth resistance of shape memory alloys by means of a cohesive zone model. J Mech Phys Solids 2007;55(10):2157–80. [19] Freed Y, Banks-Sills L, Aboudi J. On the transformation toughening of a crack along an interface between a shape memory alloy and an isotropic medium. J Mech Phys Solids 2008;56(10):3003–20. [20] Chen JH, Sun W, Wang GZ. Investigation on the fracture behavior of shape memory alloy NiTi. Metall Mater Trans A – Phys Metall Mater Sci 2005;36(4):941–55. [21] Wang GZ, Xuan FZ, Tu ST, Wang ZD. Effects of triaxial stress on martensite transformation, stress–strain and failure behavior in front of crack tips in shape memory alloy NiTi. Mater Sci Engng A 2010;527(6):1529–36. [22] Hancock JW, Mackenzie AC. On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. J Mech Phys Solids 1976;24(2–3):147–60. [23] Garrison Jr WM, Moody NR. Ductile fracture. J Phys Chem Solids 1987;48(11):1035–74. [24] Anderson T. Fracture mechanics: fundamentals and applications. 2nd ed. New York: CRC press; 1995. [25] Sun DZ, Siegele D, Voss B, Schmitt W. Application of local damage models to the numerical analysis of ductile rupture. Fatigue Fract Engng Mater Struct 1989;12(3):201–12. [26] Agogino AM. Notch effects, stress, state and ductility. J Engng Mater Technol – Trans ASME 1978;100(4):348–55. [27] Wang XM, Lu ZZ, Yue ZF. The effect of notches on the fracture behavior in NiTi shape memory alloys. Int J Solids Struct 2009;46(3–4):557–71. [28] Liu AF. Mechanics and mechanisms of fracture – an introduction. Materials Park (OH): ASM International; 2005. [29] Gall K, Yang N, Sehitoglu H, Chumlyakov YI. Fracture of precipitated NiTi shape memory alloys. Int J Fracture 2001;109(2):189–207. [30] Wang GZ. Effects of notch geometry on stress–strain distribution, martensite transformation and fracture behavior in shape memory alloy NiTi. Mater Sci Engng A 2006;434(1–2):269–79. [31] Olsen JS, Van Der Eijk C, Zhang ZL. Numerical analysis of a new SMA-based seismic damper system and material characterization of two commercial NiTi-alloys. Smart Struct Syst 2008;4(2):137–52. [32] Rice JR, Tracey DM. On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 1969;17(3):201–17. [33] Lubliner J, Auricchio F. Generalized plasticity and shape-memory alloys. Int J Solids Struct 1996;33(7):991–1003. [34] Nuno Rebelo MC, Gong Xiao-Yan. Finite element analysis of plastic behavior in nitnol. In: SMST-2003: proceedings of the international conference on shape memory and superelastic technologies. Pacific Grove (CA, USA): SMST Society, Inc.; 2003. p. 501–7. [35] Olsen JS, Zhang ZL, Hals JK, Lu H. Effect of notches on the behavior of superelastic round-bar NiTi-specimens. Smart Mater Struct 20 (2). Art. no. 025014. [36] SIMULIA. Abaqus user’s manual ver. 6.9; 2009. [37] Beremin F. Experimental and numerical study of the different stages in ductile rupture: application to crack initiation and stable crack growth. In: Nemat-Nasser S, editor. Three-dimensional constitutive relations and ductile fracture, vol. 1. Dordan (France): North-Holland Publishing Company; 1980. p. 185–205. [38] Rakin M, Cvijovic Z, Grabulov V, Putic S, Sedmak A. Prediction of ductile fracture initiation using micromechanical analysis. Engng Fract Mech 2004;71(4–6):813–27. [39] Zhang ZL, Thaulow C, Ødegård J. A complete Gurson model approach for ductile fracture. Engng Fract Mech 2000;67(2):155–68.
Contents lists available at SciVerse ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Fracture of notched round-bar NiTi-specimens J.S. Olsen a, Z.L. Zhang a,⇑, H. Lu b, C. van der Eijk c a b c
Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway Materials Science and Engineering School, Shanghai Jiaotong University, Shanghai 200240, China SINTEF Materials and Chemistry, 7465 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 1 December 2010 Received in revised form 6 December 2011 Accepted 21 December 2011
Keywords: Shape memory alloys Notches Fractography Triaxiality Finite element analysis NiTi SEM
a b s t r a c t In this work fractography has been used to characterize the fracture surfaces of superelastic, notched round-bar NiTi-specimens. Several fractography studies have been performed for conventional materials, but only few studies have been conducted on NiTi-alloys. The aim of this work is to investigate the effect of semi-circular notches on the fracture behavior of NiTi. The main results indicate that the fracture process is a mixture of cleavage and micro-void coalescence, and that decreasing the notch-radius leads to a loss of ductility which manifests in a reduced fracture strain. Within the range of notches studied in this work, we can observe a transition from high scatter to low scatter in the fracture strain when reducing the notch-radius. By combining fractography studies, finite element analyses and the Rice–Tracey void-growth model, it is argued that fracture is most likely initiated close to the notch for all notch-radii studied herein, and that cleavage or quasi cleavage is the dominating fracture mechanism. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Since their discovery in the 1960s, NiTi-alloys have been put to use in a number of applications [1]. It has been particularly successful in the medical industry, due to the material’s bio-compatibility and special mechanical properties [2]. A lot of effort has been put into understanding the mechanisms involved during phase transformation and the superelastic behavior of NiTi-alloys [3,4], and the possibility to reproduce their mechanical behavior numerically through constitutive modeling [5–9]. However, only recently an increasing interest has arisen towards failure of shape memory alloys. The main focus in this regard is found in fatigue failure [10–12], effect of cracks on martensite transformation [13–15] and the effect of martensite transformation on fracture toughness of NiTi-alloys [16–19]. The effect of stress triaxiality is often mentioned as a contributor during fracture of shape memory alloys [11,20]. However, only few attempts have been made to quantify it’s effect [21]. In this work an attempt is made to investigate the effect of triaxiality on the fracture processes in NiTi by studying notched round-bar specimens. Tensile tests until fracture of specimens with different circular notches are conducted. Subsequently SEM has been employed to investigate the fracture surfaces. Also, finite element analyses are used to quantify the effects of notches.
⇑ Corresponding author. E-mail address: [email protected] (Z.L. Zhang). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2011.12.007
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Nomenclature Af As d D R EA EM L Mf Ms r r0 Rf Rf
a
L e epl0 eplc ef depl eq mA mM req
rAf
rm
rMf rAs rMs 0
austenite finish temperature austenite start temperature specimen diameter diameter of minimum cross-section notch-radius elasticity modulus for austenite elasticity modulus for martenite specimen length martensite finish temperature martensite start temperature current void radius initial void radius R-phase start temperature R-phase finish temperature material constant transformation length total equivalent strain initial equivalent plastic strain equivalent plastic strain at fracture fracture strain plastic strain increment Possion ratio for austenite Possion ratio for martensite equivalent von Mises stress critical stress for austenite transformation finish hydrostatic stress critical stress for martensite transformation finish critical stress for austenite transformation start critical stress for martensite transformation start initial value
2. Experimental set-up and procedures In this work, a series of tests have been conducted on superelastic NiTi-alloy round-bars. The specimens have a diameter of d = 5.3 mm and a total length of L = 50 mm. Notches with three different radii were machined on the specimens by latheturning. Due to manufacturing limitations the smallest notch-radius was chosen to be R = 0.75 mm, while two other configurations were set to have radii of R = 1.00 mm and R = 1.25 mm, respectively. The minimum diameter at the notch-root was set to D0 = 2 mm for all specimens. See Fig. 1 for details. The specimens were heat-treated in a furnace at 575 °C and subsequently air-cooled. The heat-treatment was conducted after machining in order to avoid surface stresses that might emerge. Three exemplars of each configuration were tested. 2.1. Tensile testing set-up To understand the effect of the notches on the material behavior, tensile tests were conducted in a Zwick/Roell Z020 tensile testing machine with a 20 kN load capacity. Due to the relatively small size of the specimens, image acquisition and processing was chosen as the preferred tool for strain measurement. The strain was calculated from the change in diameter at
Fig. 1. Schematic drawing of specimen geometry.
J.S. Olsen et al. / Engineering Fracture Mechanics 84 (2012) 1–14
3
Fig. 2. Schematic plot of the image acquisition system.
the minimum cross-section. In order to capture the diametrical changes a Daheng DH-HV31202UC Charged-Coupled Device (CCD) with a 2048 1536 resolution and a MLM-3X-mp lens was used for image acquisition. Images were acquired 1 frame per second using a laptop with an acquisition software provided by the CCD-supplier. The commercial image processing software, Ni-vision, was subsequently used in combination with Matlab to measure and calculate the strains. Fig. 2 schematically shows the image acquisition set-up. The distance between the lens and center of the notch was set to 100 mm under CCD-provider guidelines. Also, a back-light was mounted 60 mm behind the specimen to ensure good contrast in the images. All tests were conducted under deformation control at room temperature (ca. 20 °C), and each specimen was loaded until fracture. The loading speed, 0.05 mm/min, was chosen to ensure quasi-static loading conditions. The strain and stress are calculated by the following relations, respectively:
e ¼ 2 ln
D0 ; D
r¼
F 2
p D4
ð1Þ
where F is the axial force measured by the tensile testing machine. 2.2. Differential scanning calorimetry The differential scanning calorimetry (DSC) was conducted using a Q10 DSC from TA instruments. The specimen was carefully cut using a line-cutter with coolant, to a size of approximately 1 1 mm, not to exceed the maximum allowed weight of 10 mg. Two different specimens were analyzed; one taken from an un-machined NiTi-rod and one taken from a NiTi-rod machined with lathe-turning. Both rods were heat-treated as previously described before line-cutting. The objective of the DSC was to determine the phase transformation temperatures Af, As, Ms and Mf, as well as to investigate if the machining process imposed any effect on the phase transformation temperatures. Austenite finish and start temperatures are denoted Af and As, respectively. Similarly Ms and Mf respectively denote martensite start and finish temperatures. Rs and Rf are the R-phase start and finish temperature. Each of the two specimens was heated from 60 °C to 100 °C, with a temperature increase of 10 °C/ min, and subsequently cooled at the same rate to 60°C. 2.3. Scanning electron microscopy and energy dispersive spectroscopy In this work, the study of the fracture surface is key to get an understanding of how the NiTi-alloy behaves during fracture. Both overview images of the entire fracture surface, and close-up images of particular features have been collected and investigated. All SEM-investigations are conducted post fracture. The scanning electron microscopy (SEM) was conducted using a Zeiss Ultra 55 LE scanning electron microscope. All results presented in this paper are secondary electron images. The working distance (WD) was set to 8 mm, and acceleration voltage 15 keV. An EDAX Genesis v. 5.21 was used for energy dispersive spectroscopy (EDS), to determine the element composition of particularly interesting features (e.g. inclusions) on the fracture surface. It is noted that EDS is not a highly accurate method to determine element composition when employed on inhomogeneous and rough surfaces. However, it serves well as a mean to get a notion of the main elements of a feature. 3. Experimental results and discussion Fig. 3 shows the heatflow-temperature diagram from the DSC-analyses. It can be seen that the lathe-turning does not impose any significant effect on the phase transformation temperatures. The peak heatflow values are slightly higher for the machined specimen. It should be noted, however, that each specimen is taken from a random position in the NiTi rod, i.e. whether there exists any surface effects from the machining process or not is not known. The phase transformation temperatures are summarized in Table 1. With an austenite finish temperature of 14.2 °C the material can be expected to behave superelastic when tested at room temperature. Previous investigations of the ductile fracture behavior of smooth and mildly notched round-bar specimens have showed that for conventional materials (see e.g. Hancock and Mackenzie’s study of HY130 steel [22]), fracture is often initiated by
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6 Un−Machined Machined
Heatflow [mW]
4 2 Heating 0 Cooling −2 −4
−50
0
50
100
Temperature [°C] Fig. 3. Results from DSC analyses of machined and un-machined specimen.
Table 1 Phase transformation temperatures found by DSC analysis. Phase trans. temperatures (°C)
Af
As
Rs
Rf
Ms
Mf
Un-machined Machined
14.2 13.6
4.9 4.8
7.6 7.7
6.4 6.2
36.9 37.7
50.6 49.8
void nucleation, growth and coalescence in the center of the minimum cross-section. This type of fracture behavior is governed by two factors, namely high triaxiality and large plastic strains in the center of the specimen’s minimum cross-section [23]. The process can often be recognized by the characteristic cup-and-cone fracture surface, or a penny-shaped crack in the center of the fracture surface [24]. Fig. 4a–f shows SEM-images of the complete fracture surface, and corresponding stress–strain curves, of notched specimens with notch-radii ranging from R = 0.75 mm to R = 1.25 mm. Stress–strain curves for superelastic–plastic NiTi-alloys differ from those recorded from conventional elastic–plastic materials, in that they have a distinct plateau due to stress induced phase transformation from austenite to martensite, which is a reversible process. If the load is further increased after completed phase transformation, elastic deformation of martensite commences before plastic yielding initiates at some yield stress. For the largest notch, the stress–strain curve (see Fig. 4f) indicates a relatively ductile behavior with plastic strains close to 10%. Given the notion that for ductile materials which are smooth or mildly notched, fracture is prone to initiate at the center of the minimum cross-section (a previous study by Sun [25] showed this for specimen with a minimum cross-section to notch-radius ratio of D0/R = 2), one might expect to observe traces of the afformentioned features which characterize ductile fracture. However, the SEM fractographs show no indications of neither penny-shaped cracks, nor the characteristic cup-and-cone fracture surface. This indicates that the fracture behavior of the investigated NiTi-alloy, although ductile in terms of relatively large fracture strains, is not governed by ductile fracture mechanisms, such as void-growth and coalescence, alone. The main goal of this study is to investigate the effect of notches in general, and whether varying the notch-radius has any effect on the NiTi-alloy fracture behavior. The most evident effect of the latter, is a loss in ductility when decreasing the notch-radius. When notches are introduced in a specimen, notch strengthening will increase the tensile stress needed for yielding, in the case of NiTi-alloys this also include the critical stress for transformation. Notch strengthening is an effect that arise from the triaxial stresses that develops in the notched section of the specimen [23]. It is thought that the loss of ductility is directly connected to the increase in triaxiality [26]. The loss of ductility can be observed in the fracture strain which is plotted as a function of notch-radius in Fig. 5. It can be observed that for the two specimens with the largest notches (R = 1.00 mm and R = 1.25 mm), there is a large scatter in the fracture strain. However, for the specimen with the smallest notch (R = 0.75 mm) the scatter is relatively small. It is noted that sub-notches originating from the machining process were found on some of the notches. As such sub-notches will induce stress-concentrations, this can be one explanation for the scatter. By comparing the fracture surfaces of the different exemplars of each configuration, it was also found that the fracture surface topography of the specimens with the smallest notch was more uniform than the two largest notches (not shown here). Similar findings regarding loss of ductility due to notch size and acuity have been reported for flat dog-bone NiTi-specimens [27]. Fig. 6a shows some details of the various features on the fracture surface of a specimen with notch-radius R = 0.75 mm. Cleavage facets can be observed, which indicate that the fracture mechanism is quasi-cleavage. Quasi-cleavage is considered to be dominated by cleavage, and exhibit small parts with micro-void coalescence [28]. However, there is a strong presence of micro-voids in all the investigated specimens, so it is probably more accurately described as a mixture of cleavage and micro-void coalescence [28]. Similar observations have been made in a previous study by Gall et al. [29] for a poly-crystalline NiTi-alloy.
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True stress [MPa]
1800
1200
600
R=0.75 mm
0
0
0.04
0.08
0.12
0.16
0.2
True strain
(a)
(b) True stress [MPa]
1800
1200
600
R=1.00 mm
0
0
0.04
0.08
0.12
0.16
0.2
True strain
(c)
(d) True stress [MPa]
1800
1200
600
R=1.25 mm
0
0
0.04
0.08
0.12
0.16
0.2
True strain
(e)
(f)
Fig. 4. SEM-images of the fracture surfaces, and corresponding stress–strain curves, from three specimens with notch-radius (a) and (b) R = 0.75 mm, (c) and (d) R = 1.00 mm and (e) and (f) R = 1.25 mm.
0.2
Fracture strain
0.16 0.12 0.08 0.04 0 0.5
0.75
1
1.25
Notch−radius [mm] Fig. 5. Fracture strain as a function of notch-radius.
1.5
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Fig. 6. SEM-images showing (a) voids, cleavage facets and ridges on the fracture surface of a specimen with notch-radius R = 0.75 mm and (b) a shear-lip close to the edge of the fracture surface of a specimen with notch-radius R = 1.25 mm.
3200 Ti
2800
Counts [K]
2400 2000 1600 1200 800 400
C O Ni
0
1
2
3
4
5
6
7
8
9 10
Energy [keV]
(a)
(b)
Fig. 7. SEM-images showing (a) NiTi2 islands with micro-cracks on the fracture surface of a specimen with notch-radius R = 0.75 mm and (b) results from an EDS analysis showing the content of the islands with micro-cracks.
On several specimens, a distinct shear-lip structure can be observed in portions of the circumferential edge (see Fig. 6b). The surface of this shear-lip is relatively smooth indicating that a brittle fracture has occurred in this area. This effect is most profound for the larger notches (R = 1.00 mm and R = 1.25 mm).1 Such brittle behavior indicates that fracture in this portion of the specimen has been relatively fast, and it is reasonable to assume that the shear-lip structure indicates the finalizing stage of fracture. Hence, it can help identify the area of fracture initiation. The topic of fracture initiation will be further discussed below. In addition to the aforementioned features, a number of second phase particles can be found as small islands on the various fracture surfaces (see Fig. 7a). These particles are apparently of a brittle nature as they contain micro-cracks which are limited by the surrounding matrix. An EDS-analysis indicates that the particles consists predominately of Ti (see Fig. 7b); the only Ti-rich compound of the NiTi-system is NiTi2. It is noted that the size of these islands are rather small (2 lm); as a consequence the information volume from the EDS may be larger than the investigated area – resulting in an inaccurate measurement of the particle content. Several inclusions can be observed on the fracture surfaces studied here. Fig. 8a shows the presence of some inclusions with size 5 lm. EDS showed that these inclusions are most likely Ti-oxides (see Fig. 8b). Also, however not investigated by EDS, a number of smaller inclusions can be seen. In a previous study [31] the same material used in this work was metallurgically characterized, and a number of small Ti-carbides (2 lm) was found in the micro-structure – it is possible that the small inclusions in Fig. 8a are of the same character. Conventionally, when studying fracture initiation one usually introduces some crack-like notch in the specimen; either by cyclic-deformation or by machining. This is to better control where fracture initiates, and ease the subsequent investigations. When, as in this study, no such crack-like notch is present, it becomes difficult to establish an exact location for the fracture
1 The reason as to why the shear-lip feature is larger on the blunter notches is not completely understood. It can be argued that blunter notches lead to an increase in elastically stored energy in the specimen causing a larger material volume undergoing unstable fracture at the finale stages of failure [30].
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1800
Ti
Counts [K]
1500 1200 Ni
900
O
600 C
Ni
300 0
1
2
3
4
5
6
7
8
9 10
Energy [keV]
(a)
(b)
Fig. 8. SEM-images showing (a) Ti-oxides and (b) results from an EDS analysis showing the content of the Ti-oxides.
Fig. 9. SEM-images showing fan-like patterns on the fracture surface of specimens with notch-radii (a) R = 0.75 mm, (b) R = 1.00 mm and (c) R = 1.25 mm.
initiation point. However, by carefully studying the fracture surface by SEM, it is still possible to get an idea of where fracture might have occurred. Fig. 9 shows fracture surfaces from three specimens with the different notch-radii investigated herein. On the fracture surfaces a fan-like pattern can be observed (these are elucidated by stippled lines). When fracture occurs at a point, the crack will branch out creating more cracks, following the energetically most favorable direction through the material, and create patterns like the ones showed in Fig. 9. From the figures it can be seen that fan-like patterns all point in the direction of a shear-lip feature on the opposite side of the fracture surface. The combination of these patterns and the aforementioned shear-lip suggests that fracture has initiated close to the notch-surface. A closer study of the circumferential edge of the fracture surface, leads to the discovery of significant cracks on the notch-surface in the vicinity of the projected initiation area (Fig. 10). This is the case for several of the specimens. Several specimens also exhibited cleavage features in the vicinity of the fracture initiation area, close to the notch-root (see Fig. 11). This indicates that fracture initiation is stress controlled, and starts at the notch-root. The fractography studies
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Fig. 10. Cracks on the notch-surface indicating that fracture initiates at the surface or close to the surface.
Fig. 11. SEM-images showing cleavage features in the vicinity of the fracture initiation area for specimens with notch-radii (a) 1.00 mm and (b) 1.25 mm.
do not give a clear answer towards where the fracture initiation point is located for the different specimen configurations. Neither does it completely clarify whether cleavage or ductile fracture mechanisms, as both are significantly present, govern the fracture initiation. Therefore fracture initiation has been further studied through finite element analyses and the Rice– Tracey void-growth model [32]. The results are presented in Section 4.3. 4. Finite element simulations 4.1. Finite element model A fractography study is an excellent tool to investigate which fracture processes are dominant during fracture, and to identify what internal features, such as inclusions, precipitates and micro-cracks, have influenced the process. However, as the fractography study is conducted after fracture it does not yield any information about the material behavior during deformation prior to failure. In order to get a deeper understanding of how notches affect the fracture process, finite element simulations have been conducted using the commercial finite element software Abaqus 6.9–2. Particularly interesting is it to investigate stress, plastic strain and triaxiality distributions over the minimum cross-section at the notch root, as these variables play an important role during fracture. The experimental results have shown that it is difficult to assess exactly the point of fracture initiation, as well as the dominating fracture mechanism at initiation, through fractography studies alone. Therefore, the finite element simulations and the Rice–Tracey model [32] have been employed to assess where in the specimens fracture is most likely to initiate, and establish which fracture mechanism is most likely to be dominating (see Section 4.2 for details). To model the superelastic behavior of the NiTi-alloy, an approach based on the framework developed by Auricchio and coworkers [6,7,33] has been used. The constitutive model is developed and implemented in the commercial finite element software Abaqus at SIMULIA/west by Rebelo et al. It is an extension of Auriccios model that accounts for plastic deformations [34]. For a more thorough outline of the constitutive model used herein see Ref. [35]. The three notch radii investigated experimentally (R = 0.75–1.25), are modeled in the numerical investigations. Due to material and geometrical symmetry, only a quarter of the specimen is modeled using an axis-symmetric approach. Fig. 12 shows the meshed model used in the simulations. Here R and D0 are the notch-radius and minimum diameter at
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Fig. 12. Meshed model used in the finite element analysis.
Table 2 Material parameters used in the simulations. EA, EM (MPa)
mA, mM
M rM s ; rf (MPa)
rAs ; rAf (MPa)
L
47,000, 15.667
0.33
351, 415
250, 175
0.026
the cross-section, respectively. The model consists of 1000 8-node axis-symmetric elements with reduced integration (named CAX8R in Abaqus). The material data were chosen to fit the stress–strain curves from the tested notched specimens. The problem at hand is highly non-linear in nature as it includes non-linear material behavior and geometry effects, and an irregular specimen geometry. This makes convergence an issue during the simulations, hence some of the convergence criteria available in Abaqus was relaxed and the linesearch method was used to stabilize the simulations [36]. Table 22 shows the material data used in the simulations. For more details regarding material parameter fitting and convergence issues in the finite element simulations, the interested reader is referred to [35]. 4.2. The Rice–Tracey void growth model The experimental results presented herein show that the governing fracture mechanism for the NiTi-alloy in question is a mixture of cleavage and ductile fracture. The process of ductile fracture can be summarized by micro-void nucleation, voidgrowth and finally void coalescence leading to failure [24]. For voids to grow, two factors have to be present – namely a certain degree of triaxial stress and plastic deformation. From the experimental results it is difficult to assess the main differences, if any, between the fracture behavior of different notch configurations. Particularly this is true for the fracture initiation point. Rice and Tracey [32] developed a model to describe void-growth in a elasto-plastic material without hardening. The model can be written as
ln
Z epl c r 3rm ¼ depl a exp eq r0 2req epl
ð2Þ
0
2 In Table 2, EA,EM,mA and mM are the elasticity moduli and Possion ratios for the austenite and martensite phase, respectively. rM s and respectively denote the critical stress for start and end of forward and reverse transformation. eL represents the transformation length.
rMf ; rAs and rAf ,
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1800 Experimental Simulation
1500
True Stress [MPa]
True Stress [MPa]
1800
1200 900 600 300
R=1.25mm R=1.00mm R=0.75mm
1500 1200 900 600 300
R=0.75mm
0
0
0.02
0.04
0.06
0.08
0
0.1
0
0.02
0.04
0.06
0.08
True strain
True Strain
(a)
(b)
0.1
Fig. 13. Stress–strain curves comparing (a) experimental and simulations results for a specimen with radius R = 0.75 mm and (b) effect of notch-radius on the stress–strain behavior.
1.4
1.4 R=1.25mm R=1.00mm R=0.75mm
1.2
Triaxiality
Triaxiality
1 0.8 0.6 0.4 0.2 0
R=1.25mm R=1.00mm R=0.75mm
1.2 1 0.8 0.6 0.4 0.2
Node at notch−root
0
0.02
0.04
0.06
0.08
0
Node at center
0
0.02
0.04
True strain
True strain
(a)
(b)
0.06
0.08
Fig. 14. Triaxiality plotted against true strain at (a) the notch-root and (b) the center of the specimen.
where r and r0 is current and initial void radius, respectively. a is a constant related to material behavior. Beremin [37] deterpl mined that a = 0.283 was appropriate to account for hardening. epl c is the equivalent plastic strain at fracture, while e0 is the pl initial equivalent plastic strain. deeq is the equivalent plastic strain increment. Finally, rm and req are the hydrostatic and equivalent von Mises stress, respectively. By taking the exponential on both sides of Eq. (2) we get the void-growth ratio. Since the model does not alter the material behavior [38], it is an excellent tool to perform an uncoupled investigation through finite element simulations of the intrinsic behavior in the material. In this work the Rice–Tracey void-growth model is used to help establish whether ductile fracture or cleavage mechanisms is dominating at fracture initiation. If void-growth and coalescence are dominating, the Rice–Tracey model should predict the same void-growth ratio for all three specimens at the recorded fracture strain, since in the model it is assumed that fracture occur at a critical value of the void-growth ratio.
4.3. Finite element analysis results and discussion A comparison between the experimental and numerical stress–strain curves are shown in Fig. 13a. It is clear that the model used herein manages to capture the particular phase transformation behavior of the tested NiTi-alloy. The simulation results coincide well with the experimental data; both the general stress level and critical values such as the critical stresses L M for transformation (rM s and rf ) and the transformation length ( ) is satisfactorily accurate. Also the part of the stress–strain curve representing plastic deformation shows a reasonably good fit between simulation results and experimental data. When comparing the stress–strain curves from the different specimen configuration (Fig. 13b), it can be observed that reducing the notch-radius slightly elevates the stress level. It can be expected that the stress-level is elevated due to the aforementioned notch-strengthening effect. However, the results show that for the range of notch-sizes investigated herein, the difference in the general stress–strain behavior is not very large. The main goal of introducing finite element analyses in this study is to get a better understanding of the material behavior just prior to fracture, as well as to investigate the effect of changing the notch-radius on the specimen’s mechanical response. Introducing semi-circular notches in round-bar specimen is a simple way to induce a triaxial stress-field in the specimen. Fig. 14a and b shows the triaxiality as a function of strain at the notch root and in the center of the specimen, respectively.
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11
1.2 Triaxiality at center Triaxiality at notch
Triaxiality
1 0.8 0.6 0.4 0.2 0 0.5
0.75
1
1.25
1.5
Notch−radius [mm] Fig. 15. Initial triaxiality calculated by finite element analyses as a function of notch-radius.
1.12
R=1.25mm R=1.00mm R=0.75mm
=0.15
r/r0
1.09
1.06
1.03
1
0
0.2
0.4
0.6
0.8
1
Normalized distance from notch Fig. 16. Void-growth ratio over the minimum cross-section for all specimen configurations at e = 0.15.
At the notch root, the triaxiality is close to constant during deformation for all specimen configurations. Also, the triaxiality level varies little for the different notch-radii at the notch root. However, at the center of the specimen, the triaxiality varies a lot during deformation. Comparing Figs. 13 and 14b shows that when there is a ‘‘phase change’’ in the specimen, i.e. change from phase transformation to plastic yielding, there is a drop in the triaxial stress-level. The figure also shows that reducing the notch-radius increases the triaxiality in the center of the specimen. The latter is emphasized in Fig. 15, which shows the initial triaxiality at the notch-root and the center of the specimen as a function of notch-radius. For fracture to occur as a consequence of void nucleation, growth and coalescence, a combination of triaxial stress and plastic deformation has to be present [32]. By employing the Rice–Tracey void-growth model, presented in Section 4.2, over the cross-section, it is possible to evaluate the void-growth ratio at a given point on the cross-section. To show the effect of notch-radius on the void-growth ratio, results are compared for the different specimen configurations at e = 0.15 (see Fig. 16). For all notch-radii investigated, the void-growth is most significant at the notch-root, with the smallest notch (R = 0.75 mm) yielding the highest void-growth ratio. It should be noted that, according to the results shown here, no void-growth is present at the center of the specimens for the two smallest notch-radii (R = 0.75 mm and R = 1.00 mm), while the specimen with the largest notch-radius experience only a small degree of void-growth at the center. A strain level of e = 0.15 far exceeds the maximum measured fracture strain for the specimens with notch-radii R = 0.75 mm and 1.00 mm. So in order to compare the different specimen configurations prior to fracture, results should be extracted at strain levels corresponding to the fracture strain for each specimen. Fracture strains have been chosen to be ef = 0.076, ef = 0.129 and ef = 0.159 for the specimens with notch-radii R = 0.75 mm, 1.00 mm and R = 1.25 mm, respectively. This corresponds to the largest measured fracture strain for each specimen configuration. Fig. 17b shows the voidgrowth ratio for each specimen at e = ef for each notch-radius. The results are similar to the results showed in Fig. 16 by the fact that the largest void-growth is found at the notch-root for all specimens and that no, or only a small degree of void-growth is present at the center, despite that the triaxiality is largest in the center of the specimen (see Fig. 17a). It is noted that at e = ef the specimen with the largest notch-radius exhibit the highest void-growth ratio, and that the voidgrowth ratio at the notch-root is substantially lower for the specimen with notch-radius R = 0.75 mm. In a recent study, Olsen et al. [35] found, when numerically studying a wide range of notched NiTi-specimens, that stress induced martensitic phase transformation is hindered at the center of notched round-bar specimens. As plastic deformation is subsequent to phase transformation in NiTi, it is also hindered. Olsen et al. attributed this to the triaxial stress level in the
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1.5
1.12 R=1.25mm R=1.00mm R=0.75mm
r/r0
0.6 ε = εf
0
0.2
0.4
0.6
0.8
1
1
0.4
0.6
0.8
(a)
(b) R=1.25mm R=1.00mm R=0.75mm
0.12 0.08 0.04 0
0.2
Normalized distance from notch
ε = εf
0.16
0
Normalized distance from notch−root
0.2
0
1.06 1.03
0.3
Equivalent plastic strain
1.09
0.9
0
R=1.25mm R=1.00mm R=0.75mm
ε = εf
Axial Stress [MPa]
Triaxiality
1.2
0.2
0.4
0.6
0.8
1
2000 1750 1500 1250 1000 750 500 250 0
R=1.25mm R=1.00mm R=0.75mm
ε=ε
f
0
1
0.2
0.4
0.6
0.8
Normalized distance from notch−root
Normalized distance from notch−root
(c)
(d)
1
Fig. 17. Cross-sectional plots showing the distribution of (a) the triaxial stress state, (b) the void-growth ratio, (c) the equivalent plastic strain and (d) the axial stress over the cross-section at the notch-root.
2000
σyy=1742MPa
Axial Stress [MPa]
Axial Stress [MPa]
2000 1600 1200
ε=εf
800 400
σyy=1769MPa
1600 1200 ε=εf
800 400
R = 1.25mm 0
0
0.05
0.1
0.15
R = 1.00mm 0
0.2
0
0.05
0.1
ln(A0/A)
ln(A0/A)
(a)
(b)
Axial Stress [MPa]
2000
0.15
0.2
σyy=1720MPa
1600 1200 ε=εf
800 400
R = 0.75mm 0
0
0.05
0.1
0.15
0.2
ln(A0/A)
(c) Fig. 18. Curves showing the axial stress at the notch-root as a function of true strain for (a) R = 1.25 mm, (b) R = 1.00 mm and (c) R = 0.75 mm. The stress and strain at fracture is indicated by the stippled lines.
J.S. Olsen et al. / Engineering Fracture Mechanics 84 (2012) 1–14
13
center of the specimens. From Fig. 17c it can be observed that the distribution of the equivalent plastic strain over the minimum cross-section is very similar to the void-growth ratio. The largest equivalent plastic strain is found at the notch-root pl for all three specimens at the time of fracture, ranging from epl eq ¼ 0:082 to eeq ¼ 0:146 as the notch-radius is increased from R = 0.75 mm to R = 1.25 mm. The degree of plastic deformation over the cross-section also increases with the notch-radius: for the specimen with R = 1.25 mm plastic deformation can be observed over the entire cross-section, while for the specimens with R = 0.75 mm and R = 1.00 mm only 25% and 65% of the cross-section are plastically deformed. It is reasonable to assume that if no, or only limited plastic strain has developed in a material volume, that no void-growth can occur within that volume. Since void-growth is needed to induce ductile fracture it is therefore reasonable to assume that ductile fracture has not initiated at the center of the specimen, but rather on, or close to the notch-surface for the cases investigated herein. The critical void-growth, or void volume fraction, at fracture will vary with varying triaxiality [39]. Given that fracture does initiate at the notch-surface, it is obvious that the triaxiality level is almost the same at the notch-root for the three specimens studied in this work (see Fig. 15). This would make it reasonable to assume that the Rice–Tracey damage parameter would be similar for all three specimens at fracture. However, there is a difference in the results when analyzing the void-growth ratio, at fracture. The results yield a much lower void-growth ratio at fracture for the specimen with notch-radius R = 0.75 mm than for the specimens with notch-radii R = 1.00 mm and R = 1.25 mm. This points in the direction that void-growth and coalescence are not the governing fracture mechanisms at fracture initiation. Further it is noted that at fracture, the stress-level in the axial direction is very similar for the three specimens at the notch-root (see Fig. 17d) which indicates that failure is governed by a critical stress. This is further confirmed by looking at the axial stress at the notch-root plotted as a function of true strain (see Fig. 18): the stress at fracture is quite similar for all three cases investigated herein. The maximum difference in stress at fracture is less than 3% and is most likely due to inaccuracies in the measured strain from the experimental results. This also shows that, under the notion of stress controlled fracture, the ductility reduces as the notch-radius becomes smaller. The aforementioned differences in void-growth ratio, and the coinciding axial stresses at the notch-root makes it reasonable to assume that quasi-cleavage is the governing fracture mechanism at fracture initiation. 5. Summary and conclusions In this work three round-bar NiTi specimen configurations with different notch-radii have been investigated through fractography studies and finite element analyses. For all configurations the fracture surfaces both cleavage facets, tearing ridges and micro-voids can be observed. This is in agreement with other studies [27,29], and indicates that the complete fracture process is a mixture of ductile and brittle fracture. Also, when studying the fracture strains of the various specimen configurations indications were found that reducing the notch-size leads to a loss in ductility, and that it can reduce the scatter in the fracture strain results. By studying patterns on the fracture surface, indications were found that the initiation site is in the vicinity of the notchroot. On the notch-surface, significant cracks were observed for several specimens close to the indicated fracture initiation area. This suggests that fracture initiation occurs at the notch-root. Through numerical analyses in Abaqus, and uncoupled investigations using the Rice–Tracey void-growth model, it has been found that cleavage or quasi-cleavage is most likely to be the dominating fracture mechanism. The numerical results and fractography studies together, strongly indicate that fracture has initiated on or close to the notch surface. Several inclusions of various types could be observed on the fracture surfaces. Inclusions are known to play an important role in fracture initiation and mechanisms. The role of the inclusions on NiTi fracture behavior is not completely understood and should be subject for further research. Acknowledgments We thank the Norwegian Research Council (NFR) for financial support. The authors also thank Willhelm Dall at SINTEF Materials and Chemistry for his expertise and help when conducting SEM investigations. Finally, we would like to thank the anonymous reviewers, who provided insightful comments which proved very helpful in the work with this paper. References [1] Duerig T. Engineering aspect of shape memory alloys. London: Butterworth Heinemann Ltd.; 1990. [2] Morgan NB. Medical shape memory alloy applications – the market and its products. Mater Sci Engng A – Struct Mater Prop Microstruct Process 2004;378(1–2):16–23. [3] Funakubo H. Shape memory alloys. Precision and robotics, vol. 1. Gordon and Breach Science Publishers; 1987. [4] Otsuka K, Ren X. Physical metallurgy of Ti–Ni-based shape memory alloys. Prog Mater Sci 2005;50(5):511–678. [5] Brinson L. One-dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J Intell Mater Syst Struct 1993;4(2):229–42. [6] Auricchio F, Taylor RL. Shape-memory alloys: modelling and numerical simulations of the finite-strain superelastic behavior. Comput Methods Appl Mech Engng 1997;143(1–2):175–94. [7] Auricchio F, Taylor RL, Lubliner J. Shape-memory alloys: macromodelling and numerical simulations of the superelastic behavior. Comput Methods Appl Mech Engng 1997;146(3–4):281–312.
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J.S. Olsen et al. / Engineering Fracture Mechanics 84 (2012) 1–14
[8] Boyd JG, Lagoudas DC. A thermodynamical constitutive model for shape memory materials. Part I. The monolithic shape memory alloy. Int J Plasticity 1996;12(6):805–42. [9] Wang XM, Xu BX, Yue ZF. Micromechanical modelling of the effect of plastic deformation on the mechanical behaviour in pseudoelastic shape memory alloys. Int J Plasticity 2008;24(8):1307–32. [10] McKelvey AL, Ritchie RO. Fatigue-crack propagation in nitinol, a shape-memory and superelastic endovascular stent material. J Biomed Mater Res 1999;47(3):301–8. [11] McKelvey AL, Ritchie RO. Fatigue-crack growth behavior in the superelastic and shape-memory alloy nitinol. Metall Mater Trans A – Phys Metall Mater Sci 2001;32(3A):731–43. [12] Robertson SW, Ritchie RO. In vitro fatigue-crack growth and fracture toughness behavior of thin-walled superelastic nitinol tube for endovascular stents: a basis for defining the effect of crack-like defects. Biomaterials 2007;28(4):700–9. [13] Wang XM, Wang YF, Baruj A, Eggeler G, Yue ZF. On the formation of martensite in front of cracks in pseudoelastic shape memory alloys. Mater Sci Engng A 2005;394(1–2):393–8. [14] Robertson SW, Mehta A, Pelton AR, Ritchie RO. Evolution of crack-tip transformation zones in superelastic nitinol subjected to in situ fatigue: a fracture mechanics and synchrotron X-ray microdiffraction analysis. Acta Mater 2007;55(18):6198–207. [15] Gollerthan S, Young ML, Neuking K, Ramamurty U, Eggeler G. Direct physical evidence for the back-transformation of stress-induced martensite in the vicinity of cracks in pseudoelastic NiTi shape memory alloys. Acta Mater 2009;57(19):5892–7. [16] Yi S, Gao S. Fracture toughening mechanism of shape memory alloys due to martensite transformation. Int J Solids Struct 2000;37(38):5315–27. [17] Yi S, Gao S, Shen L. Fracture toughening mechanism of shape memory alloys under mixed-mode loading due to martensite transformation. Int J Solids Struct 2001;38(24–25):4463–76. [18] Freed Y, Banks-Sills L. Crack growth resistance of shape memory alloys by means of a cohesive zone model. J Mech Phys Solids 2007;55(10):2157–80. [19] Freed Y, Banks-Sills L, Aboudi J. On the transformation toughening of a crack along an interface between a shape memory alloy and an isotropic medium. J Mech Phys Solids 2008;56(10):3003–20. [20] Chen JH, Sun W, Wang GZ. Investigation on the fracture behavior of shape memory alloy NiTi. Metall Mater Trans A – Phys Metall Mater Sci 2005;36(4):941–55. [21] Wang GZ, Xuan FZ, Tu ST, Wang ZD. Effects of triaxial stress on martensite transformation, stress–strain and failure behavior in front of crack tips in shape memory alloy NiTi. Mater Sci Engng A 2010;527(6):1529–36. [22] Hancock JW, Mackenzie AC. On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. J Mech Phys Solids 1976;24(2–3):147–60. [23] Garrison Jr WM, Moody NR. Ductile fracture. J Phys Chem Solids 1987;48(11):1035–74. [24] Anderson T. Fracture mechanics: fundamentals and applications. 2nd ed. New York: CRC press; 1995. [25] Sun DZ, Siegele D, Voss B, Schmitt W. Application of local damage models to the numerical analysis of ductile rupture. Fatigue Fract Engng Mater Struct 1989;12(3):201–12. [26] Agogino AM. Notch effects, stress, state and ductility. J Engng Mater Technol – Trans ASME 1978;100(4):348–55. [27] Wang XM, Lu ZZ, Yue ZF. The effect of notches on the fracture behavior in NiTi shape memory alloys. Int J Solids Struct 2009;46(3–4):557–71. [28] Liu AF. Mechanics and mechanisms of fracture – an introduction. Materials Park (OH): ASM International; 2005. [29] Gall K, Yang N, Sehitoglu H, Chumlyakov YI. Fracture of precipitated NiTi shape memory alloys. Int J Fracture 2001;109(2):189–207. [30] Wang GZ. Effects of notch geometry on stress–strain distribution, martensite transformation and fracture behavior in shape memory alloy NiTi. Mater Sci Engng A 2006;434(1–2):269–79. [31] Olsen JS, Van Der Eijk C, Zhang ZL. Numerical analysis of a new SMA-based seismic damper system and material characterization of two commercial NiTi-alloys. Smart Struct Syst 2008;4(2):137–52. [32] Rice JR, Tracey DM. On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 1969;17(3):201–17. [33] Lubliner J, Auricchio F. Generalized plasticity and shape-memory alloys. Int J Solids Struct 1996;33(7):991–1003. [34] Nuno Rebelo MC, Gong Xiao-Yan. Finite element analysis of plastic behavior in nitnol. In: SMST-2003: proceedings of the international conference on shape memory and superelastic technologies. Pacific Grove (CA, USA): SMST Society, Inc.; 2003. p. 501–7. [35] Olsen JS, Zhang ZL, Hals JK, Lu H. Effect of notches on the behavior of superelastic round-bar NiTi-specimens. Smart Mater Struct 20 (2). Art. no. 025014. [36] SIMULIA. Abaqus user’s manual ver. 6.9; 2009. [37] Beremin F. Experimental and numerical study of the different stages in ductile rupture: application to crack initiation and stable crack growth. In: Nemat-Nasser S, editor. Three-dimensional constitutive relations and ductile fracture, vol. 1. Dordan (France): North-Holland Publishing Company; 1980. p. 185–205. [38] Rakin M, Cvijovic Z, Grabulov V, Putic S, Sedmak A. Prediction of ductile fracture initiation using micromechanical analysis. Engng Fract Mech 2004;71(4–6):813–27. [39] Zhang ZL, Thaulow C, Ødegård J. A complete Gurson model approach for ductile fracture. Engng Fract Mech 2000;67(2):155–68.