Load path effect on fatigue crack propagation in I + II + III mixed mode conditions – Part 1: Experimental investigations

International Journal of Fatigue 62 (2014) 104–112

Contents lists available at SciVerse ScienceDirect

International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

Load path effect on fatigue crack propagation in I + II + III mixed mode conditions – Part 1: Experimental investigations Flavien Fremy a,b, Sylvie Pommier a,⇑, Martin Poncelet a, Bumedijen Raka a, Erwan Galenne c, Stephan Courtin d, Jean-Christophe Le Roux c a

LMT-Cachan (ENS Cachan/CNRS/UPMC/PRES UniverSud Paris), 61, avenue du Président Wilson, 94235 Cachan, France Saint-Gobain R&D, 9 Goddard Road, 01532 Northborough, MA, USA EDF R&D, 1 avenue du Général de Gaulle, 92141 Clamart,France d AREVA NP SAS, Tour Areva.1, Place Jean Millier, 92084 Paris La Défense cedex, France b c

a r t i c l e

i n f o

Article history: Received 30 October 2012 Received in revised form 26 April 2013 Accepted 2 June 2013 Available online 19 June 2013 Keywords: Fatigue Crack propagation Non-proportional Plasticity Mixed mode

a b s t r a c t This paper is devoted to the analysis of the load path effect on mixed mode I + II + III fatigue crack growth in a 316L stainless steel. Experiments were conducted in mode I + II and in mode I + II + III. The loads were applied using the six actuators servo-hydraulic testing machine available at LMT-Cachan and the crack growth rate was measured using digital image correlation. The topographies of the crack paths were determined post-mortem using a numerical optical microscope. The fracture surfaces were also examined at high magnification using a scanning electron microscope (SEM). The load paths used in the experiments were chosen so as to be equivalent with respect to most of the fatigue crack growth criteria, in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi particular with those based on DK eq ¼ n aDK nI þ bDK nII þ cDK nIII since the same maximum, minimum and mean values of the stress intensity factors were used for each loading path. In addition, load paths were constructed by pairs, either so that the extreme values of the stress intensity factors are attained simultaneously, or so as to display the same cumulative ‘‘length’’. The main result of this set of experiments is that very different crack growth rates and crack paths are observed for load paths that are however considered as equivalent in most fatigue criteria. In addition, it was shown that the load path can have a very significant effect on the crack growth rate even if the crack path is not significantly different. The comparison of the results of the experiments conducted in mode I + II and in mode I + II + III, also allowed to show that the addition of mode III loading steps to a mode I + II loading sequence is increasing the fatigue crack growth rate, even when the crack path is not significantly modified. And finally, the SEM observations of the fracture surface showed that in non-proportional mixed mode conditions, a complex system of slip bands is formed at crack tip and is used to promote crack growth. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Applications of fracture mechanics to components loaded in multi-axial conditions are usually dedicated to determine the crack path that maximizes the mode I stress intensity factor amplitude during crack growth [1–8]. Then, the fatigue crack growth rate is usually predicted using the Paris’ law determined in experiments in mode I conditions. The maximum growth rate criterion is also employed to predict shear-mode growth at high DKII [7]. However, when non-proportional loading conditions are encountered, the mode mixity inside the K-dominance area can significantly over the fatigue cycle. In such cases, the determination of the crack path that would maximize the mode I stress

⇑ Corresponding author. E-mail address: [email protected] (S. Pommier). 0142-1123/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2013.06.002

intensity factor or the growth rate in pure mode I or pure mode II become questionable [9]. For example, power shafts are usually subjected to a combination of torsion and bending due to the transmission of the torque, the self-weight of the shaft and its rotation speed. In operating conditions a static torque is applied and a high rotation speed is used (resulting in a low bending moment), when the turbine is disengaged, the torque is null and the rotation speed is reduced (resulting in a higher bending moment). The crack is hence subjected to mixed-mode loading conditions with a mode mixity that varies according to the operating conditions. Researchers have investigated the effect of a static mode III (KIII) or mode II (KII) on fatigue crack growth in mode I conditions (DKI), or of a static mode I (KI) on fatigue crack growth in mode II or mode III (DKII or DKIII) [9–11]. The role of interspersed mode III loading sequences (DKIII) on mode I fatigue crack growth (DKI) was also analyzed. In most cases, a significant effect of the static load was

105

F. Fremy et al. / International Journal of Fatigue 62 (2014) 104–112

found and was associated to crack tip plasticity or to crack closure. Similarly, a significant variation of the crack growth rate in mode I conditions is observed over a ‘‘recovery distance’’ after a mixed mode loading sequence has been applied. The recovery distance was successfully analyzed in terms of plastic zone [12–17]. Mixed mode loading conditions are also encountered when cracks are initiated from surface contact. This problem is encountered in rotating bearings, wheel-rails systems, cams, gears etc.. . . In such applications, the cracks are subjected to a mixed mode fatigue cycle with a continuously varying mode-mixity. Few studies have been devoted to analyze fatigue crack growth under out-ofphase or sequential mixed mode loading conditions but most of them indicate a detrimental effect of the mode-mixity variation and underline the role of crack tip plasticity [18–23]. Most approaches that have been proposed [1,2] to predict the fatigue crack growth rate in mixed mode loading conditions are based on an equivalent stress intensity factor (Eq. 1), whose expression varies according to the authors, but that is usually function of the stress intensity factors amplitudes (Eq. 2).

da ¼ C DK m eq dN

ð1Þ 1=n

DK eq ¼ ðDK nI þ bDK nII þ cDK nIII Þ

ð2Þ

Corrections can be included to account for the closure effect, or for the in-phase or out-of-phase nature of the fatigue cycles, and these approaches are usually successful in correlating the set of data obtained by the authors, but may fail when applied to a different loading scheme. Approaches based on the elastic–plastic behavior of the crack tip region are much more powerful, since they naturally take into account the load-path dependency of fatigue crack growth through the non-linear character of the elasto-plastic behavior of the material [24–29], the plasticity induced crack closure effect and in some cases the roughness induced by crack closure [29–31]. However, these methods are complex and little-used. Further developments are therefore necessary to facilitate their use and these developments require a prioritization of the phenomena to implement so as to introduce simplifying assumptions which must be supported by experimental results. A set of experiments was therefore conducted in order to characterize specifically the importance of the load path effect in mixed mode fatigue crack growth. In these experiments, the stress intensity factor ranges and mean values are kept the same for each mode. A static mode I load is always applied so as to limit the effect of crack closure. In addition, tests were conducted with different loading paths, yet both ‘‘in phase’’ or both ‘‘out of phase’’, in the sense that the extremes values of the stress intensity factors in each mode are attained simultaneously or not. 2. Experimental protocol 2.1. Material The tested material is an austenitic stainless steel (Table 1). This material is highly employed in power plants to produce various components such as pumps, mixing tees and taps because of its excellent resistance to corrosion, its good formability and ductility

Table 1 Chemical content of the EN X2CrNiMo17 12 2, AISI 316L alloy. Element Wt%

Min Max

C

Mn

Si

P

S

Cr

Mo

Ni

M

– 0.03

– 2.

– 1.

– 0.04

– 0.03

16. 19.

2.25 2.75

10. 14.

– 0.1

Table 2 Mechanical properties. Young’s modulus

Fracture toughness

Yield stress

Maximum stress

Elongation

E 193 GPa

KIC 90 MPa m1/2

Rp0.2% 320 MPa

Rm 610 MPa

A% 48

(Table 2). The elastic–plastic behavior of this material has been extensively studied in uniaxial and multi-axial conditions [32–34]. 2.2. Experimental setup The six actuators servo-hydraulic testing machine ASTREE available at LMT-Cachan was used for these experiments (Fig. 1). The horizontal actuators are controlled in pairs so as to keep the location of the intersection of the three loading axes fixed. Each horizontal loading axis is thereafter load controlled. A cruciform specimen is used for these experiments (Fig. 2). The four horizontal actuators are used to apply in-plane loadings FX and FY onto the specimen (Fig. 2) so as to produce mode I and mode II loadings. The upper vertical actuator is used for applying a point load on one of the two lips of the crack in order to produce mode III loadings (Figs. 1a and 2). The second vertical actuator is not used. Two cameras and a lighting system are positioned below the specimen (Fig. 1b) in order to track the crack tip during fatigue testing using stereo digital image correlation [35,36]. As a matter of fact, in this set of experiments, significant out of plane displacements occur, in particular due to mode III loadings. The use of two cameras allows determining the 3D displacement field [35]. In the following, the length 2a measured during the experiments corresponds to the crack length projected along the slit plane. 2.3. Specimen and boundary conditions A cruciform specimen was used for the experiments (Fig. 2). A centered 30 mm long slit is machined in the specimen (Fig. 2b), the slit plane being inclined at 45° with the loading axes of the specimen. Linear elastic finite element analyses were conducted in order to determine the relations between the loads FX, FY and FZ applied along the three axes of the specimen (Fig. 2a) and the stress intensity factors KI, KII and KIII at mid-thickness for a coplanar crack propagating from the slit (Eq. 5).

0

K1 I

1

0

fI ð2aÞ

fI ð2aÞ

0

B 0 B 1 C B fII ð2aÞ fII ð2aÞ @ K II A ¼ B @ 0 1 K III 0 fIII ð2aÞ

1 0 1 C FX CB C C@ F Y A A FZ

ð3Þ

1/2 where K 1 , Fi in kN and 2a in mm.A mode I stress I are in MPa m intensity factor is obtained by applying the same load along the two in-plane axes of the specimen (FX = FY). A mode II stress intensity factor is obtained by applying FX = FY. A mode III stress intensity factor is obtained by applying an out-of-plane load FZ. The mode I + II + III loading cycles used in the experiments do always include a positive mean value of KI, for three main reasons. First of all, a positive mean value of KI allows limiting the crack closure effects. Second, the mean value of KI was determined so that the in-plane loads FX and FY will always remain positive during cycling so as to avoid any buckling of the specimen. Third, when an out-of-plane load FZ is applied onto the specimen, it induces a bending of the specimen and hence it induces a mode I stress intensity factor variation along the crack front. This variation was determined using finite element analyses. It was checked that,

106

F. Fremy et al. / International Journal of Fatigue 62 (2014) 104–112

Fig. 1. (a) Six actuators servo-hydraulics testing machine ASTREE, experimental set-up. Quality of the load control for two complex load paths (b) star load path and (c) cube load path.

for the mixed mode I + II + III fatigue cycles used in these experiments, the mode I stress intensity factor induced by the application of the out-of-plane load FZ could be neglected with respect to the mode I stress intensity factor induced by the application of the in-plane loads FX = FY. 2.4. Loading cases 2.4.1. Preparation of testing Each specimen was pre-cracked in mode I at 10–20 Hz and at R = 0.33 (DFX = DFY = 33.1 kN, FZ = 0 kN) up to a crack length 2a = 34 mm. For this crack length, the stress intensity factors used to pre-crack the specimen corresponds to K min ¼ 5 MPa m1=2 and I 1=2 K max ¼ 15 MPa m . I Each load path considered in this set of experiments is defined by means of evolutions of the stress intensity factors KI(t), KII(t) and KIII(t). For a crack length 2a = 34 mm, the load sequences FX(t), FY(t) and FZ(t) that corresponds to the desired evolutions of the stress intensity factors KI(t), KII(t) and KIII(t), are determined using Eq. (1) and the coefficients in Table 3. Digital image correlation measurements [35]were also used to check the quality of the actual boundary conditions applied on the specimen. The load sequences FX(t), FY(t) and FZ(t) determined for 2a = 34 mm are then applied to grow the crack by fatigue up to a length of about 2a = 38 mm. Their frequencies depend on the ‘‘difficulty’’ of the load path. For example, the star load path, is considered the most difficult one and is performed at about 0.4 Hz Fig. 1b). In any cases, the tests were stopped after 6 h of cycling in mixed mode conditions. It is important to mention that the load sequences FX(t), FY(t) and FZ(t) are not updated during the fatigue

test. Therefore, in the following, when values of the stress intensity factors are given, they correspond to the start of the test when 2a = 34 mm. 2.4.2. Loading cases in mixed mode I + II Four mixed mode I + II loadings cases have been studied (Fig. 3). Each of them is centered around the same mean value for each mode KI = 10 MPa and KI, and has the same stress intensity factor amplitude for each mode DKI = DKII = 10 MPa. These four cases are all equivalent with respect to the criteria based on an equivalent stress intensity factor (Eq. 2) or on a strain energy amplitude (Eq. 4) determined in linear elastic conditions. The load paths were constructed so as to show other similarities: – first, the peak values of KI and KII are attained simultaneously for the ‘‘proportional’’ (Fig. 3C), the ‘‘square’’ (Fig. 3A) and the ‘‘windmill’’ (Fig. 3D) load paths, – second, the cumulative ‘‘lengths’’ of the ‘‘square’’ (Fig. 3A) and the ‘‘cross’’ (Fig. 3B) load paths are the same, and is one half of that of the ‘‘windmill’’ (Fig. 3D) load path, – third, there is one cycle with an amplitude DKI = DKII = 10 MPa in each load path for the ‘‘proportional’’ (Fig. 3C), the ‘‘square’’ (Fig. 3A) and the ‘‘cross’’ (Fig. 3B) load paths. The case of the ‘‘windmill’’ (Fig. 3D) load path is somehow different, since we may either count, per load path, two cycles, with an amplitude DKI = DKII = 10 MPa, or, one cycle with an amplitude DKI = DKII = 10 MPa and two smaller cycles with an amplitude DKI = DKII = 5 MPa. Since the fatigue threshold stress intensity factor in mode I (DK th I ¼ 6 MPa at R = 0.5 [37]) for this material

F. Fremy et al. / International Journal of Fatigue 62 (2014) 104–112

107

Fig. 2. (a) Schematics of the specimen and of the boundary conditions and (b) geometry of the specimen.

Table 3 Values computed for fI (2a), fII (2a) et fIII (2a) in Eq. (5) using linear elastic finite element analyses at mid-thickness of the specimen. 2a (mm)

fI (2a)

fII (2a)

fIII (2a)

30 32 34 36 38 40

– 0.139 0.151 0.158 0.166 0.174

– 0.144 0.175 0.197 0.208 0.222

– 1.51 1.77 1.86 1.95 1.98

 the contribution of mode III to mixed mode fatigue crack growth can be determined by comparing pairs of load cases, namely the two ‘‘proportional’’ load paths (Fig. 3C and Fig. 4B), the ‘‘square’’ and the ‘‘cube’’ load paths (Fig. 3A and Fig. 4A) and finally the ‘‘cross’’ and the ‘‘star’’ load paths (Fig. 3B and Fig. 4C). 3. Results 3.1. Fatigue crack growth in mixed mode I+II conditions

was found to be above 5 MPa and well below10 MPa, the ‘‘windmill’’ (Fig. 3D) load path will be interesting with respect to the counting method to be used in mixed mode conditions. 2.4.3. Loading cases in mixed mode I + II + III As in mixed mode I + II conditions, the loading cases studied in mixed mode I + II + III conditions (Fig. 4) are centered around the same mean value for each mode KI = 10 MPa, KI and KIII = 5 MPa, and have the same stress intensity factor amplitude for each mode DKI = DKIII = DKIII = 10 MPa. They are also equivalent with respect to the criteria in Eqs. (2 and 4). In addition,  the peak values of KI, KII and KIII are attained simultaneously for the ‘‘proportional’’ (Fig. 4B) and the ‘‘cube’’ (Fig. 4A) load paths,  the cumulative ‘‘lengths’’ of the ‘‘cube’’ (Fig. 4A) and the ‘‘star’’ (Fig. 4C) load paths are the same,

The results of the fatigue crack growth experiments conducted in mixed mode I + II conditions indicate that there is a significant load path effect in mixed mode fatigue crack growth. First of all, the crack path is significantly different according to the load path selected. The intersection between the crack and the surface of the specimen was measured periodically during the tests using digital image correlation (Fig. 5a). As a matter of fact, the software used to compute the displacement field assumes a continuous and piecewise linear displacement field along the specimen surface [35,36], while the experimental displacement field is discontinuous over the crack faces. The digital image correlation residual can therefore be used to reveal the crack (Fig. 5a). These observations show a significant crack path bifurcation (40°) for the ‘‘proportional’’ load path, after 1 mm of coplanar crack growth in mixed mode conditions. On the contrary, the bifurcation of the crack path is found to be moderate for the ‘‘square’’ load path (10°) and very small for the two other cases.

108

F. Fremy et al. / International Journal of Fatigue 62 (2014) 104–112

Fig. 3. Loading cases applied in mixed mode I + II conditions. (A) « Square » load path, (B) « Cross » load path, (C) « Proportional »load path and (D) « Windmill »load path.

The evolution of the crack length with the number of cycles is also plotted in Fig. 6 for each load path tested. 3.2. Fatigue crack growth in mixed mode I + II + III conditions

Fig. 4. Loading cases applied in mixed mode I + II + III conditions. (A) « Cube » load path, (B) « Proportional » load path and (C) « Star » load path.

Complementary observations were also performed post-mortem using a numerical optical microscope (by Keyence) available at LMS (Ecole Polytechnique). This microscope allows reconstructing the topography of the fracture surface and was used to confirm the observations performed on the specimen surface (Fig. 5b). In these experiments, below 1 mm of crack propagation, no bifurcation of the crack path is observed even for the ‘‘proportional’’ load path. In this domain, the variation of the crack growth rate from one test to another is solely attributed to a load path effect. The number of cycles required to get a crack extent of 0.65 mm, for instance, is reported in Table 4. It typically varies by a factor 3, according to the load path applied in the experiment, though the extreme values and the mean values of the stress intensity factors are kept the same in each experiment. In addition, if we assume that the growth rate in mixed mode conditions is correlated to that determined in Mode I, using for instance Eq. (1), the values of DKeq required to get the suitable growth rate for each load path (Table 4) are varying from pffiffiffiffiffi pffiffiffiffiffi 11.9 MPa m up to 17.8 MPa m.

The results of the fatigue crack growth experiments conducted in mixed mode I + II + III conditions are plotted in Fig. 7. As in mixed mode I + II, a very significant load path effect is observed. However, the load path effect is limited when the extreme values of the stress intensity factors are attained simultaneously for each mode (‘‘cube’’ and ‘‘proportional’’ load paths). The largest difference is observed between the ‘‘cube’’ and the ‘‘star’’ load paths. As in mixed mode I + II, the number of cycles required to grow the crack by Da = 0.5 mm in each test are gathered in Table 5. According to the loading path the crack growth rate is found to vary by more than a factor 2 (see Table 5). In addition, in mixed mode I + II + III, the crack path is also significantly varying with the loading case. The effects are more pronounced in mixed mode I + II + III than in mixed mode I + II. The overall inclination of the crack path was roughly characterized at the end of the test, by two angles, the tilt angle (a) and the twist angle (b).The results are gathered in Table 6. The ‘‘proportional’’ loading path has promoted the most severe change in the crack path, since the crack plane has twisted by an angle of about 50°. On the contrary, the fatigue crack growth remains more or less coplanar under the ‘‘cube’’ load path. Most surprisingly, the ‘‘star’’ load path is producing a significant tilt and a small twist, while no bifurcation was observed in mode I + II under the ‘‘cross’’ load path. Adding mode III loading steps (‘‘star’’) to a mixed mode I + II loading cycle (‘‘cross’’) did not promote the twisting of the crack path, but induced atilt. 3.3. Comparison of mixed mode I+II and mixed mode I + II + III load paths In Fig. 8, the evolution of the crack length with the number of cycles was plotted for mixed mode I + II and for mixed mode I + II + III loading conditions.

109

F. Fremy et al. / International Journal of Fatigue 62 (2014) 104–112

Fig. 5. (a) Intersection between the crack plane and the surface of the specimen, revealed by the digital image correlation residue. The end of the pre-crack ap is indicated by an empty circle. (b) Topography of the fracture surface of the specimen broken using the « square » load path, determined post-mortem. Left side – height profile determined in the mode I pre-cracking area. Right side – height profile determined after 2 mm of crack propagation in mixed mode conditions using the « square » load path.

Table 4 Number of cycles required to grow the crack by fatigue by Da = 0.65 mm in each pffiffiffiffiffi experiment, equivalent stress intensity factor DKeq (MPa m) required to get a growth rate Da/DNi (m/cycle) assuming Eq. (1), with m = 3 and C = 3.08  1012. Load path

Number of cycles DNi to get Da = 0.65 mm

D N B/ DN i

 m1 DK eq ¼ C DDNai

Square Cross Proportional Windmill (2 cycles per load path) Windmill (1 cycle per load path)

DNA = 44718 DNB = 110211 DNC = 60211 DND2 = 64437

2.46 1.00 1.83 1.71

16.02 11.92 14.53 14.21

DND1 = 32218

3.42

17.84

Fig. 7. Evolutions of the crack lengths with the number of cycles applied for each load case in mixed mode I + II + III conditions.

Table 5 Number of cycles required to grow the crack by fatigue by Da = 0.5 mm in each pffiffiffiffiffi experiment, equivalent stress intensity factor DKeq (MPa m) required to get a growth rate Da/DNi (m/cycle) assuming Eq. (1), with m = 3 and C = 3.08  1012.

Fig. 6. Evolutions of the crack lengths with the number of cycles applied for each load case in mixed mode I + II conditions. Two curves are plotted for the « windmill » load path, the curve (1) was plotted considering 1 cycle per load path, and the curve (2) considering two cycles per load path.

The main result of these experiments is that the addition of a mode III stress intensity factor amplitude to a mixed mode I + II loading case, is significantly increasing the crack growth rate, even when the crack remains coplanar (Fig. 8a).

Load path

Number of cycles DNi to get Da = 0.5 mm

DNC/ DNi

 m1 DK eq ¼ C DDNai

Cube Proportional Star

DNA = 4678 DNB = 6216 DNC = 10920

2.33 1.75 1.00

30.82 28.08 23.34

In these experiments (Fig. 8a), the crack is fully opened in mode I when the mode II and mode III loading steps are applied, which tends to drastically limit the frictional effects and to maximize the effect of the mode II and mode III loading steps. In addition, in the cube loading path, the specimen is first loaded in mode I, then in mode II and finally in mode III. The effect of the mode III loading step is extremely detrimental because it is applied on a specimen that is subjected to a static mode I and mode

110

F. Fremy et al. / International Journal of Fatigue 62 (2014) 104–112

Table 6 Twist angle (b) through the thickness of the specimen (5 mm) and tilt angle (a) after the crack has propagated in mixed mode by Da = 2 mm. Effect of the load path on the crack path, after the crack has propagated in mixed mode I + II + III from 2a = 34 mm up to 2a = 38 mm. Load path

‘‘Prop.’’

‘‘ Cube ’’

‘‘ Star ’’

Tilt angle a

10°

None

40°

Twist angle b

50°

15°

10°

problem is non-linear and the load path has a significant effect on fatigue crack growth. 3.4. Fracture surface analyses by SEM

Fig. 8. Comparison of the evolutions of the crack lengths with the number of cycles in mixed mode I + II and in mixed mode I + II + III. (a) Cube and square loadings. (b) Proportional loadings. (c) Star and cross loadings. The major characteristic of the crack path (coplanar, bifurcation, twisting) is indicated above each curve.

II load, and hence the development of plasticity in mode III is larger, than for instance, for the star load path, or if the cube was constructed by applying the mode I first, then the mode III and finally the mode II. Because the material is elastic–plastic, the

The fracture surfaces of the broken specimens were observed using a scanning electron microscope (SEM). First of all, it is worth to mention that very few traces of friction could be found on the fracture surfaces, expect for the ‘‘proportional’’ load paths. The SEM fracture surfaces of specimen in 316L stainless steel are usually quite characteristic and the crack growth mechanisms are relatively easy to identify. In mode I fatigue crack growth, striations are observed in the Paris’ regime [38] and a faceted fracture surface is found in the near threshold domain. In these experiments, the fracture surfaces display some features that are very typical of a non-proportional loading case. The images obtained for the specimen broken using the ‘‘cube’’ load path are the most typical (Fig. 9) but the fracture surface obtained using the ‘‘square’’ load path shows similar features. First of all, very typical fatigue striations are observed within a thin layer below the surface of the specimen. Within this layer, the overall orientation of the striations indicates a crack growth towards the specimen surfaces (Fig. 9a). In this area, the striation spacing is much larger than the macroscopic growth rate per cycle. However, once projected along the macroscopic growth direction the striation spacing is in the order of magnitude of the macroscopic crack growth. The fracture surface is very different inside the specimen. At the very beginning of the test a faceted fracture surface is found (Fig. 9b).This type of faceted surface differs significantly from to the faceted fracture surfaces observed in mode I in the near threshold regime. In this case, the width of the facets is well below the grain size, and two or three systems of facets appear to co-exist. Then, after a few hundreds microns of crack propagation in mixed mode conditions, there are quite a number of planar facets here and there on the fracture surface (Fig. 9c). Neat traces of the emergence of slip bands are seen on these planar facets. At least three slip systems must have been cyclically activated to leave this type of traces. When non-proportional loading conditions are applied, the slip systems may be active during only certain phases of the fatigue cycle and become inactive while others slip systems take over. A very typical feature, which was observed almost everywhere on the fracture surface of the specimen broken using the ‘‘cube’’ load path, is the complexity of the striations pattern (Fig. 9d–f). In many places, the fracture surfaces display two sets of striations or more, either on the same facet or on different facets. Since nonproportional loading conditions are applied, the different phases of the loading cycle can produce different set of striations, resulting in quite unusual striations pattern as those in Fig. 9e and f. In addition, the striation spacing is well below the macroscopic growth

F. Fremy et al. / International Journal of Fatigue 62 (2014) 104–112

111

Fig. 9. Fracture surface of the specimen broken using the « cube » load path. The overall growth direction is indicated by an arrow. (a) Fatigue striations observed within a thin layer (<100 lm) below the surface of the specimen. (b) Faceted fracture surface observed at mid-thickness of the specimen after less than 0.5 mm of crack propagation in mixed mode. (c) Planar facet with traces of slip bands observed after about 1.5 mm of crack growth, at mid-thickness. (d) A complex striated area very typical of the fracture surface of this specimen. (e) Higher magnification of the same area as in Fig. 9d. (f) Other area displaying complex striations.

rate per cycle. However, there is no evidence of a specific mechanism associated with mode III load steps [39,40]. The main conclusion of these observations is that fatigue crack growth in mixed mode I + II + III conditions is stemming from crack tip plasticity, just as in mode I. However in non-proportional mixed mode conditions, the different phases of the fatigue cycle can produce different systems of plasticity, either by forming slip bands, or by forming striations. The fracture surface combines the results of the plastic activity in each phase of the fatigue cycle.

4. Conclusions Experiments were conducted in mode I + II and in mode I + II + III non-proportional loading conditions in order to characterize the load path effect in fatigue crack propagation in a 316L stainless steel and the contribution to fatigue crack growth of mode III loadings. The loads were applied using the six actuators servo-hydraulic testing machine available at LMT-Cachan and the crack growth rate was measured using digital image correlation. The topographies of the crack paths were determined post-mortem using a numerical optical microscope. Since the same maximum, minimum and mean values of the stress intensity factors were applied

in each experiments, the load paths are all considered as ‘‘equivalent’’ with respect to most of the fatigue crack growth criteria, in particular with those based on DK eq ¼ ðDK nI þ bDK nIII þ cDK III ). Besides, the tested load paths were chosen, either so that the extreme values of the stress intensity factors are attained simultaneously, or so as to display the same cumulative ‘‘length’’. In each test, the mode I loading ratio is positive (R = 0.33) so as to limit the crack closure effects, indeed very few traces of friction and contact could be found on the fracture surface. The main result of this set of experiments is that very different crack growth rates are observed even though the extreme values and the mean values of the stress intensity factors are the same in each test. A variation by up to a factor three of the crack growth rate according to the loading path was observed in these experiments, even when the crack path remains coplanar. In addition, it was shown that the crack path is also significantly dependent of the load path. For instance, the crack path remains coplanar for the ‘‘square’’ load path while a tilt is observed for the ‘‘proportional’’ load path in mixed mode I + II. In these two cases, the extreme values of the mode I and mode II stress intensity factors are attained simultaneously. The comparison of the results of the experiments conducted in mode I + II and in mode I + II + III, allowed also to show that the addition of mode III loading steps to a mode I + II loading sequence

112

F. Fremy et al. / International Journal of Fatigue 62 (2014) 104–112

is increasing the fatigue crack growth rate, even when the crack path is not significantly modified. In addition, it was observed that the addition of a mode III loading sequence to a mode I + II fatigue cycle, was not necessarily promoting at wist of the crack path but could also promote a crack tilt. Finally, the fracture surfaces were also examined at high magnification using a scanning electron microscope (SEM). The main conclusion of these observations is that fatigue crack growth in mixed mode I + II + III conditions is stemming from crack tip plasticity, just as in mode I. However in non-proportional mixed mode conditions, the different phases of the loading cycle produce different systems of plasticity, either by forming slip bands, or by forming striations. The fracture surface combines the results of the plastic activity in each phase of the fatigue cycle. The overall conclusion of this set of experiments is that, in nonproportional mixed mode loading conditions, criteria based solely on the maximum, minimum and mean values of the stress intensity factors, cannot provide accurate predictions of the crack growth rate and of the crack path. The entire load path should be considered so as to predict accurately fatigue crack growth in non-proportional mixed mode conditions. Acknowledgments The authors would like to acknowledge the companies EDF and AREVA for their financial support and for fruitful discussions during the last three years. We would also like to acknowledge Véronique Doquet who kindly lent us the numerical optical microscope Keyence and Florent Mathieu and François Hild for the digital image correlation system that was so useful to track the cracks. References [1] Brown MW. Analysis and design methods in multiaxial fatigue. Adv Fatigue Sci Technol 1989:387–402. [2] Qian J, Fatemi A. Mixed mode fatigue crack growth: a literature survey. Eng Fract Mech 1996;55(6):969–90. [3] Lazarus V, Leblond JB, et al. Crack front rotation and segmentation in mixed mode I + III or I + II + III. Part I: calculation of stress intensity factors. J Mech Phys Solids 2001;49(7):1399–420. [4] Gravouil A, Moës N, Belytschko T. Non-planar 3D crack growth by the extended finite element and level sets—Part II: level set update. Int J Numer Methods Eng 2002;53(11):2569–86. [5] Moës N, Gravouil A, Belytschko T. Non-planar 3D crack growth by the extended finite element and level sets—Part I: mechanical model. Int J Numer Methods Eng 2002;53(11):2549–68. [6] Mroz KP, Mroz Z. On crack path evolution rules. Eng Fract Mech 2010;77(11):1781–807. [7] Hourlier F, Pineau A. Propagation of fatigue cracks under polymodal fatigue. Fatigue Eng Mater Struct 1982;5(4):287–302. [8] Bold PE, Brown MW, et al. A review of fatigue crack-growth in steels under mixed mode-I and mode-II loading. Fatigue Fract Eng Mater Struct 1992;15(10):965–77. [9] Plank R, Kuhn G. Fatigue crack propagation under non-proportional mixed mode loading. Eng Fract Mech 1999;62(2–3):203–29. [10] de Freitas M, Reis L, da Fonte M, Li B. Effect of steady torsion on fatigue crack initiation and propagation under rotating bending, multiaxial and mixed mode cracking. Eng Fract Mech 2011;78:826–35. [11] Akhurst KN, Lindley TC, et al. The effect of mode III loading of fatigue crack growth in a rotating shaft. Fatigue Eng Mater Struct 1983;6(4):345–8.

[12] Brown MWB, Wong SL, et al. Fatigue crack growth rates under sequential mixed-mode I and II loading cycles. Fatigue Fract Eng Mater Struct 2000;23(8):667–74. [13] Choi BH, Lee JM. Experimental observation and modeling of the retardation of fatigue crack propagation under the combination of mixed-mode single overload and constant amplitude loads. Int J Fatigue 2009;31(11– 12):1848–57. [14] Nayeb-Hashemi H, Taslim ME. Effects of the transient mode-II on the steadystate crack-growth in mode-I. Eng Fract Mech 1987;26(6):789–807. [15] Dahlin P, Olsson M. Reduction of mode I fatigue crack growth rate due to occasional mode II loading. Int J Fatigue 2004;26(10):1083–93. [16] Dahlin P, Olsson M. Mode I fatigue crack growth reduction mechanisms after a single Mode II load cycle. Eng Fract Mech 2006;73(13):1833–48. [17] Gao H, Upul S. Effect of non-proportionnal overloading in fatigue life. Fatigue Fract Eng Mater Struct 1996;19:1197–206. [18] Beretta S, Foletti S, Valiullin K. Fatigue crack propagation and threshold for shallow micro-cracks under out-of-phase multiaxial loading in a gear steel. Eng Fract Mech 2010;77:1835–48. [19] Murakami Y, Sakae C, Hamada S. Mechanism of rolling contact fatigue and measurements of DKIIth for steels. Eng Against Fatigue 1999;1999:473–85. [20] Rozulmek D, Marciniak Z. The investigation of crack growth in specimens with rectangular cross sections under out-of-phase bending and torsional loadings. Int J Fatigue 2012;39:81–7. [21] Doquet V, Abbadi M, et al. Influence of the loading path on fatigue crack growth under mixed-mode loading. Int J Fract 2009;159(2):219–32. [22] Doquet V, Pommier S. Fatigue crack growth under non-proportional mixedmode loading in ferritic-pearlitic steel. Fatigue Fract Eng Mater Struct 2004;27(11):1051–60. [23] Tschegg EK. Mode-III and Mode-I fatigue crack-propagation behavior under torsional loading. J Mater Sci 1983;18(6):1604–14. [24] Richard HA, Sander M. Finite element analysis of fatigue crack growth with interspersed mode I and mixed mode overloads. Int J Fatigue 2005;27(8):905–13. [25] Sander M, Richard HA. Experimental and numerical investigations on the influence of the loading direction on the fatigue crack growth. Int J Fatigue 2006;28(5–6):583–91. [26] Pommier S, Lopez-Crespo P, et al. A multi-scale approach to condense the cyclic elastic-plastic behaviour of the crack tip region into an extended constitutive model. Fatigue Fract Eng Mater Struct 2009;32(11):899–915. [27] Decreuse PY, P. S, Poncelet M, Raka B. A novel approach to model mixed mode plasticity at crack tip and crack growth. Experimental validations using velocity fields from digital image correlation. Int J Fatigue; 2011. [28] Doquet V, Bertolino G. Local approach to fatigue cracks bifurcation. Int J Fatigue 2008;30(5):942–50. [29] Doquet V, Bui QH, Constantinescu A. Plasticity and asperity-induced fatigue crack closure under mixed-mode loading. Int J Fatigue 2010;32:1612–9. [30] Mageed AMA, Pandey RK. Fatigue crack closure in kinked cracks and path of crack-propagation. Int J Fracture 1990;44(3):R39–42. [31] Mageed AMA, Pandey RK. Studies on cyclic crack path and the mixed-mode crack closure behavior in Al-alloy. Int J Fatigue 1992;14(1):21–9. [32] Doquet V, Pineau A. Extra hardening due to cyclic nonproportional loading of an austenitic stainless-steel. Scr Metall Et Mater 1990;24(3):433–8. [33] Benallal A, Legallo P, et al. An Experimental investigation of cyclic hardening of 316-stainless steel and of 2024-Aluminum alloy under multiaxial loadings. Nucl Eng a Des 1989;114(3):345–53. [34] Bocher L, Delobelle P, et al. Mechanical and microstructural investigations of an austenitic stainless steel under non-proportional loadings in tensiontorsion-internal and external pressure. Int J Plasticity 2001;17(11):1491–530. [35] Mathieu F, Hild F, Roux S. Identification of a crack propagation law by digital image correlation. Int J Fatigue 2011. [36] Réthoré J, Roux S, Hild F. Hybrid analytical and extended finite element method (HAX-FEM): A new enrichment procedure for cracked solids. Int J Numer Methods Eng 2010;81:269–85. [37] Pickard AC, Ritchie RO, Knott JF. Crack propagation in a type 316 strainless steel weldment. Metal Technol 1975;2:253–63. [38] Pelloux RMN. Mechanisms of formation of ductile fatigue striations. Asm Trans Quart 1969;62(1):281. [39] Pippan R, Pokluda J. Can pure mode III fatigue loading contribute to crack propagation in metallic materials? Fatigue Fract Eng Mater Struct 2005;28(1– 2):179–85. [40] Pippan R, Zelger C, et al. On the mechanism of fatigue crack propagation in ductile metallic materials. Fatigue Fract Eng Mater Struct 2011;34(1):1–16.