Crack path evaluation on HC and BCC microstructures under multiaxial cyclic loading

International Journal of Fatigue 58 (2014) 102–113

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Crack path evaluation on HC and BCC microstructures under multiaxial cyclic loading V. Anes, L. Reis ⇑, B. Li, M. Freitas Instituto Superior Técnico, ICEMS & Dept. of Mechanical Engineering, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

a r t i c l e

i n f o

Article history: Received 22 December 2012 Received in revised form 22 March 2013 Accepted 24 March 2013 Available online 4 April 2013 Keywords: Crack path Multiaxial fatigue Microstructure Fatigue life Experimental tests

a b s t r a c t In this paper the multiaxial loading path effect on the fatigue crack initiation, fatigue life and fracture surface topology are evaluated for two different crystallographic microstructures (bcc and hc): high strength low-alloy 42CrMo4 steel and the extruded Mg alloy AZ31B-F, respectively. A series of multiaxial loading paths were carried out in load control, smooth specimens were used. Experimental fatigue life and fractographic results were analyzed to depict the mechanical behavior regarding the different microstructures. A theoretical analysis was performed with various critical plane models such as the Fatemi–Socie, SWT and Liu in order to correlate the theoretical estimations with the experimental data. A new approach based on maximum stress concentration factors is proposed to estimate the crack initiation plane, estimations from this new approach were compared with the measured ones with acceptable results. To implement this new approach a virtual micro-notch was considered using FEM. Moreover, the multiaxial loading path effect on stress concentration factors is also studied. The obtained results clearly show the effect of the applied load conditions on local microstructures response. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Structural failure is often caused by fatigue cracks which frequently initiate and propagate in the critical regions, generally due to complex geometrical shapes and/or multiaxial loading conditions. Fatigue crack initiation and crack growth orientation have been paid growing research attentions, since it is a crucial issue for an accurate assessment of fatigue crack propagation lives and for the final fracture modes of cracked components and structures [1–3]. Although there are a lot of publications regarding multiaxial fatigue behavior of steels, there are very few studies regarding multiaxial fatigue of magnesium alloys; Bentachfine et al. [4] studied a lithium–magnesium alloy under proportional and non-proportional loading paths under low-cycle and high cycle fatigue regime observing the deformation mode evolution and plasticity behavior. It was stated that the phase shift angle in the non-proportional loading paths decreases the material fatigue strength. The comparative parameter used to correlate experimental data was the von Mises equivalent stress/strain. However with this approach the material under non-proportional loadings keeps a constant equivalent stress and in this way no change in the material occurs along each loading cycle. However, the constant change in the direction of principal stress during the loading period due to ⇑ Corresponding author. Tel.: +351 966415585; fax: +351 218417954. E-mail address: [email protected] (L. Reis). 0142-1123/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijfatigue.2013.03.014

the phase shift increases the anisotropy on the plastic deformation at grain level and causes, in certain cases, the decrease on fatigue life [5]. Biaxial fatigue studies were performed by Ito and Shimamoto [6] with cruciform specimens made of a magnesium alloy. It was analyzed the fatigue crack propagation as well as the effect of microstructure on the material fatigue strength. From the biaxial low cycle fatigue tests, it was concluded that the twinning density evolution is strictly related with crack initiation and slip band’s formation on wrought magnesium alloys. Recently, Yu et al. [7], have also studied in-phase and out-phase behavior under strain controlled tests on AZ61A extruded magnesium alloy using tubular specimens. The conclusions were similar to Bentachfine et al. [4], the presence of the phase shift angle leads to decrease the fatigue strength compared with in-phase cases for the same equivalent strain amplitude. At low-cycle fatigue regime, it was reported a kink in the strain life curve which is a typical behavior for uniaxial fatigue regime in magnesium alloys. Furthermore, the effect of compressive mean stress was evaluated, concluding that a compressive mean stress enhances fatigue life [8]. There are mainly three types of shear transformations beyond slip mechanisms namely deformation twinning, stress induced at martensitic transformations and kinking. The twinning deformation occurs in hc metals deformed at ambient temperature and at bcc metals when they are deformed at lower temperatures. Twinning mechanism occurs when is created a boundary on the material lattice defining a symmetric region due to shear strain at

V. Anes et al. / International Journal of Fatigue 58 (2014) 102–113

atomic level. This twin boundary defines a mirror image between deformed and undeformed lattice grid [9,10]. At wrought Mg alloys the crack initiation is also associated with material inclusions but in most cases the twinning deformation and slip bands inherent to the twinning density flow are the main cause for the crack initiation. Crack propagation follows in general along the deformation twin’s fields [11–13]. The goal of this paper is to evaluate the mechanical behavior of two different microstructures, bcc and hc, subjected to the same multiaxial loading paths and point out the main differences concerning the multiaxial loading effect on the fatigue crack initial path, fatigue life and fracture surface topology. In this work a critical plane study was performed comparing the agreement between experimental data and the theoretical one. Critical plane models such as Fatemi–Socie, SWT, Liu1 and Liu2 were applied and a new approach based on the highest stress concentration factor to determine the critical plane direction was performed. Regarding the two different microstructures the applicability of the studied models and the corresponding obtained results are discussed. 2. Materials and methodology

processes [7,8,15–18]. On the other hand 42CrMo4 material tends to cyclically softening. Identifying the cyclic material behavior is of prime importance to accurately interpret the material local stress states. The softening behavior leads to have a local stress lower than the one estimated with the monotonic curve, however, if the monotonic curve is used as reference to fatigue experiments then the cyclic total strains will be greater than the monotonic ones. At cyclic hardening the opposite occurs, the local stresses are greater than the monotonic ones for the same total strain value. Under these assumptions it can be concluded that fatigue models used in order to estimate crack initiation planes and fatigue life estimations must be implemented taking in to account the cyclic behavior of the material once they are based on stress and strain amplitudes [5,18,19]. Material cyclic curve can be used as a reference to fatigue loadings in order to implement elastic–plastic numeric simulations, however the fatigue crack nucleation process induces micro-notches which in turn induces stress risers, with great probability of local plasticity. Besides, the fatigue crack opening occurrence involves plasticity in the fatigue process which leads to conclude that the plasticity effect on the stress and strains states cannot be neglected on the determination of the crack initiation plane [12,20,21]. In this

In this work two materials were studied, one is the low-alloy steel DIN 42CrMo4 (AISI 4140), the other one is the extruded Mg alloy AZ31B-F with 3% of aluminum and 1% zinc. The mechanical properties of both materials were determined by the authors following the standard procedures, namely following ASTM E8 and ASTM E606 standards, and are presented in Table 1. 2.1. Material cyclic behavior Structural materials have different properties regarding cyclic and monotonic regimes, for instance the cyclic yield stress can be quite different from monotonic one, depending on the material behavior. Some materials can soften or harden or even maintain similar to monotonic properties under a cyclic regime [14,15]. Another category is point out by Lopez and Fatemi, where cyclic softening at low strain amplitudes is followed by cyclic hardening at higher strain amplitudes [16]. When a cyclic softening occurs the cyclic yield stress is lower than the monotonic one, in this case, usually, the cyclic curve is under the monotonic curve for all strains. Material hardening occurs when the cyclic regime induce material plasticity in such way that the cyclic yield stress appears above the monotonic yield stress, as well as the cyclic curve appears above the monotonic curve. Magnesium alloys tends to have a cyclic hardening behavior being highly dependent on the grain refinement, purity, lattice intrinsic behavior like twinning or foundry transformation

Fig. 1. 42CrMo4 monotonic and cyclic behavior.

Table 1 Mechanical properties of the materials studied, 42CrMo4 and AZ31B-F. 42CrMo4

AZ31B-F

Microstructure type Poisson’s ratio Density (Kg/m3) Hardness (HV) Tensile strength (MPa) Yield strength (MPa) Elongation (%) Young’s modulus (GPa) r0f Fatigue strength coefficient (MPa)

bcc 0.3 7830 362 1100 980 16 206 1154

hc 0.35 1770 86 290 203 14 45 450

b Fatigue strength coefficient

e0f Fatigue ductility coefficient

0.061 0.180

0.12 0.26

c Fatigue ductility exponent

0.53

0.71

103

Fig. 2. AZ31 alloy monotonic and cyclic behavior.

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study the elastic–plastic behavior of the material is based on the experiments with bulk materials, and consequently it is used, as approximation, for evaluating the local plasticity. In multiaxial fatigue analysis, generally it is considered firstly the load or strain histories and then it is used a cyclic plasticity model to determine the stress or strain tensors to perform a critical plane search in order to evaluate the inherent fatigue damage parameter [5]. Fig. 1 shows the cyclic and monotonic results for the high strength steel 42CrMo4, this material shows a cyclic softening, stating from 0.3% of strain. Fig. 2 shows the AZ31 magnesium alloy monotonic and cyclic behavior, the magnesium alloy shows a cyclic hardening from 0.6% of strain. Curves plotted in Figs. 1 and 2 were determined based on standards, ASTM E8 and ASTM 606 requirements. The AZ31 alloy cyclic response at different strain hysteresis loops shows different yield stress in tension and compression. The compression yield stress has a value close to the monotonic value compared with the tensile one which is about 26% greater than the monotonic yield stress, 203 MPa and 256 MPa, respectively. Currently the commercial FEM softwares regarding stress/strain analyses do not take into account different material behaviors, i.e. different values for the yield stress at tensile and at compression states of a material, excepting for cast iron case [22]. In this work to overcome this issue concerning the AZ31 alloy behavior, it was considered, as a conservative approach, the tensile branch of the AZ31 cyclic response. 2.2. Chaboche plasticity model In general, plasticity models have three main parts to follow the material response under plastic deformation. One of them is the yield function, which determines when a material yielding occurs. The most common yield criterion is the von Mises function, where a combination of principal stresses on the octahedral plane determines the yield boundary. The second part of plasticity models is the flow rule, which is based on constitutive equations where the stresses and strains are computed on incremental plasticity procedures, where the next plastic deformation is dependent on the prior one. This rule is generally based on the Drucker postulate [23], where the plastic strains increments are normal to the yield surface defined by the yield function. The third part is the hardening rule, which establishes the changes on the yield surface during the plastic deformation [24]. Chaboche plasticity model is a nonlinear kinematic hardening model, the yield function F is given through the follow equation:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðs  aÞ : ðs  aÞ  k ¼ 0 F¼ 2

ð1Þ

where s is the deviatoric stress, a is the back stress, and k is the yield stress [25,26]. The kinematic hardening is governed through the back stress tensor; this model component contribution is related with the yield surface translation. The back stress tensor calculations are present in the following equations:

fag ¼

n X fai g

2 1 dC i C i fDepl g  ci fai gD^epl þ Dhfag 3 C i dh

Chaboche plasticity model allows to use various kinematic models and material constants; C1 and c1 are model inputs for one kinematic model, however extra sets of C and c can be added. In this study the C1 and c1 are determined based on the material cyclic response, present on previous subsection. 2.3. Chaboche plasticity calibration Chaboche material parameters can be determined through stress–strain tests under stabilized hysteresis loops. Having the plastic strain values and the inherent recall term, which is the difference between the stress amplitude Dr/2 applied and the material cyclic yield stress (k), for a fixed total strain, the relation between these two values can be fitted with the Eq. (4). From a fitting procedure it is possible to obtain the material constants C1 and c1.

  Dr C1 Depl  k ¼ tanh c1 2 c1 2

ð4Þ

Fig. 3 shows the data fitting curves obtained for 42CrMo4 and AZ31 materials, regarding Eq. (4). Table 2 presented the Chaboche material constants determined for both materials. 2.4. Loading paths and specimen test To evaluate the microstructure´s influence, four biaxial cyclic loading paths were selected, see Fig. 4. The first one is a pure uniaxial cyclic tension test, case PT, and the second one is a pure shear loading, case PS. Case PP is a proportional biaxial loading and the OP case is a 90° out of phase loading case. In Fig. 5 is presented the specimen geometry and dimensions. All tests were performed at ambient temperature and ended when the specimens were totally separated. 3. Theoretical analysis

ð2Þ

i¼1

fDagj ¼

Fig. 3. 42CrMo4 and AZ31Chaboche plasticity model calibration.

ð3Þ

where Depl is the accumulated plastic strain, h is temperature, and Ci and ci are the Chaboche material parameters. In Eq. (3) the hardening modulus and back stress variation (recall term) are represented by the first and second terms respectively. The third term is related with temperature variation. In commercial software Ansys, the

To estimate the critical plane orientation and multiaxial fatigue life it is often used critical plane models, where the multiaxial Table 2 Chaboche plasticity input parameters for 42CrMo4 and AZ31B-F. Material

C1

c1

AZ31 42CrMo4

25,000 19,281

90 69

V. Anes et al. / International Journal of Fatigue 58 (2014) 102–113

105

Fig. 4. Loading paths: (a) case PT, (b) case PS, (c) case PP and (d) case OP.

where Dc/2 is the maximum shear strain amplitude on a h plane, rn,max is the maximum normal stress on that plane, ry is the material monotonic yield strength and k is a material constant, k = 1.0 in this case [27]. 3.2. SWT criterion Smith et al. [14] proposed a model which was later extended to multiaxial fatigue situations in terms of maximum normal strain by Socie and Marquis [5]. This model is based on the principal strain range plane, and maximum stress on that plane:

maxðrn Þ h

stress tensor is projected in several planes in order to determine the one which maximizes the inherent damage parameter, i.e. the critical plane. These damage parameters are then associated with the material Coffin–Mason equation to estimate fatigue life. Next subsections present some critical plane models and a new approach to estimate the critical plane orientation. 3.1. Fatemi–Socie criterion Fatemi–Socie [19] proposed a model which predicts that the critical plane is the plane orientation h with the maximum F–S damage parameter:

  Dc rn;max 1 þ kFS 2 ry max h

ð6Þ

where rn is the normal stress on a plane h, and De1 is the principal strain range on that plane.

Fig. 5. Specimen test geometry and dimensions.



De1 2

ð5Þ

3.3. Liu criterion Liu [28] proposed an energy method to estimate the fatigue life, based on virtual strain energy (VSE). This model considers two parameters associated with two different modes of fatigue cracks, a tensile failure mode (Mode I), DWI, and a shear failure mode (Mode II), DWII. Failure is expected to occur on the plane h in the material, having the maximum VSE quantity. According to Mode I fracture, the parameter, DWI is:

DW I ¼ maxðDrn Den Þ þ DsDc h

ð7Þ

For Mode II fracture, the parameter, DWII is, see the following equation:

DW II ¼ Drn Den þ maxDsDc h

Fig. 6. (a) Identification of nominal and local stress points and (b) micro-notch geometry and / angle.

ð8Þ

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where Ds and Dc are the shear stress range and shear strain range, respectively, Drn and Den are the normal stress range and normal strain range, respectively. 3.4. New maximum kt approach regarding critical plane orientation As early stated [5,7] loading paths have a strong influence on the crack nucleation and initiation process, as a nucleation result intrusion and extrusion occurs at material surface leading to create micro-notches where plasticity mechanisms, at micro-notch root or at surface, govern the early crack opening process. Mode I governs the crack opening process. One characteristic of this mode is that the opening direction is perpendicular to the cracked surface. The present methodology aims at being used mainly for wellknown geometries where nominal stress can be calculated. In this approach the main concept is based on the assumption that the crack opening front has the biggest stress values. These values can be determined using numerical tools such as Ansys [29], Abaqus [30] or even an in-house FEA software. One issue related with this approach is the crack geometry to adopt on numeric simulations, since the crack initiation direction is the missing variable and the crack geometry has a strong influence on the final result; with this in mind a virtual micro-notch was implemented using FEM, a spherical cap geometry with depth equal to half diameter was adopted. This geometry does not have a favorable direction to induce particular stress risers since it is equal in all directions. In this way the stress riser’s locations are strictly dependent on the loading path type and stress levels. In Fig. 6a is shown the nominal and local regions in the specimen test, the nominal region is sufficiently far from the local region. Fig. 6b shows the spherical cap (micro-notch) numerically implemented, the spherical radius is 100 lm and spherical cap depth is also 100 lm. These values are in agreement with magnesium alloys grain size which has around 50 lm as average value depending on the heat treatment and grain growth. Concerning the mesh, a convergence study was conducted to assure no influence in the number of nodes used. The control nodes, around the cap edge, at micro-notch outer, and on specimen surface are equally separated, in order to optimize the critical plane estimations, also an additional node at micro-notch root

(depth) is monitored. Thus, the critical plane orientation at initiation stage will be estimated by the highest kt value, between the maximum axial kt and the maximum shear kt, determined at spherical cap (micro-notch) considering one period of loading path applied for the different loading paths. In this work the kt values, in tension and shear, are obtained by comparing the local stresses at notch cap for the several nodes and the nominal stresses, in tension and shear, at a location apart from the notch cap, considering the same direction for both situations, respectively, see Fig. 6a. 4. Results and discussion 4.1. Fatigue life analysis Fig. 7 presents the experimental fatigue life results obtained for both materials concerning the nominal equivalent von Mises stress. Results show that the relative damage between the loading paths considered in this study tends to have a different relative arrangement in the selected materials. For the AZ31 material the PS, PT and PP loading cases almost have the same trend line, leading to conclude that von Mises equivalent stress is a good candidate to perform fatigue life correlation for this material, excepting in the situation of non-proportional loading cases. On 42CrMo4 material the PS loading case is the less damage and the PT and PP SN curves are between OP and PS damage limits. Concerning Mg alloy, the OP loading case seems to present a bilinear tendency. Due to the OP loading nature the strain energy involved is higher than the other loading cases on same fatigue life region, i.e., the activation of twinning mechanisms leads to a different mechanical behavior. Moreover, regarding SN curves trends, the damage rate between loading paths is similar for the same material excepting for the loading case OP where the damage rate is bigger. In Tables 3 and 4 are presented the fatigue data shown in Fig. 7. 4.2. Loading path effect on kt In Table 5 are shown numerical results for the stress concentration factor inherent to the studied loading paths and considering the 42CrMo4 material with and without the Chaboche plasticity model implemented on Ansys software, respectively.

Fig. 7. SN fatigue life considering nominal stresses for: (a) 42CrMo4 and (b) Mg AZ31B-F.

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V. Anes et al. / International Journal of Fatigue 58 (2014) 102–113 Table 3 Fatigue life for 42CrMo4 material.

Table 4 Fatigue life for AZ31 material.

Loading case

Normal stress (MPa)

Shear stress (MPa)

Nf

Loading case

Normal stress (MPa)

Shear stress (MPa)

Nf

PT

700 600 560 550 495 485 480

0 0 0 0 0 0 0

6040 19,951 53,752 56,929 247,953 269,178 1,000,000 (runout)

PT

140 135 130 120 105 100

0 0 0 0 0 0

131,64 22,873 38,102 62,352 721,573 1,000,000 (runout)

PS

PS

0 0 0 0 0 0

546 484 440 402 395 391

2088 11,302 70,610 159,854 315,668 1,000,000 (runout)

0 0 0 0 0

75 69 64 59 53

88,871 128,769 227,808 388,236 1,000,000 (runout)

OP

510 495 490 485 475 465 450

294 286 283 280 274 269 260

56411 97,548 107,374 197,548 316,712 618,128 1,000,000 (runout)

106 95 78 74 73 71

61 55 45 43 42 41

7182 8595 11,986 167,525 576,336 1000,000 (runout)

PP

610 520 495 470 465 445 440 435 425

352 300 286 271 269 257 254 251 245

4114 27,204 48,740 97,366 109,087 239,600 311,401 564,088 1,000,000 (runout)

106 92 78 74 71 67

61 53 45 43 41 39

16,800 46,878 69,169 242,685 353,718 1,000,000 (runout)

OP

PP

The stress analysis monitoring was performed at micro-notch root (spherical cap depth) and at the intersection boundary between micro-notch cap and specimen test surface. The calculus of kt(s) are based on the shear and axial stress components at micro-notch root and micro-notch cap boundary nodes, divided by the nominal shear and nominal axial stress components, respectively. This approach based on stress concentration factor determination was implemented to avoid to use the concept of equivalent stress where the relation between axial and shear stress components remains under discussion. Therefore, in this study, it was determined a kt for axial loading and another one for shear loading. The nominal axial and shear stresses considered are about 30% lower than the material cyclic yielding stress, in order to ensure no plasticity at nominal stress control point, see Fig. 6a. Comparing the results with and without plasticity in Table 5, it can be concluded that the results from plasticity numerical approach are lower. This result is justified through the Chaboche plasticity input, whereas the material cyclic curve instead of monotonic one was applied, regarding the 42CrMo4 cyclic softening behavior. Excepting the PP loading case, the highest kt concerning the axial and shear components occurs at micro-notch root with and without plasticity simulation. The average results for axial kt on the loading cases OP and PP are greater than for shear kt, under numeric plasticity simulation the axial kt results are about 25% greater at micro-notch root and 35% at micro-notch surface. Under elastic simulation the axial kt values are 7% greater at micro-notch root and 45 % greater for micro-notch surface. In Table 6 are shown the mean values of axial and shear kt at micro-notch cap surface under plasticity simulation, the greatest values are obtained for OP loading case and the lowest for the PP loading case. This means that for OP loading case, during one cycle,

Table 5 Kt variation due to loading path at micro-notch root and surface with and without plasticity model, maximum value in a period of time (1 cycle). With plasticity

Without plasticity

Loading path

Location

Kt axial

Kt shear

Kt axial

Kt shear

PT

Root Surface Root Surface Root Surface Root Surface

1.66 1.46 NAN NAN 1.66 1.47 1.34 1.54

NAN NAN 1.53 1.36 1.53 1.36 1.15 1.36

1.79 1.64 NAN NAN 1.79 1.65 1.67 2.14

NAN NAN 1.74 1.34 1.79 1.40 1.78 1.60

PS OP PP

Table 6 Kt mean values with plasticity on micro-notch surface along one loading period (1 cycle). With plasticity model – 42CrMo4 Loading path

Location

Kt axial

Kt shear

PT PS OP PP

Surface Surface Surface Surface

0.85 NAN 0.85 0.65

NAN 0.85 0.93 0.70

this loading has in average, the highest stress level on micro-notch compared with the PP loading case. Regarding elastic and plastic simulations it is clearly shown that the obtained values for stress concentration factors are not only dependent on the micro-notch geometry and nominal stress but also on the loading path type. 4.3. Critical plane orientations 4.3.1. Critical plane models Fig. 8 shows the critical plane results for the loading paths studied for the 42CrMo4, the AZ31 results are very similar from these

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Fig. 8. Fatigue damage parameter and critical plane orientation for 42CrMo4.

Fig. 9. Critical plane estimation based on maximum kt for PT loading case with plasticity model: (a) axial and (b) shear.

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109

Fig. 10. Critical plane estimation based on maximum kt for PS loading case with plasticity model: (a) axial and (b) shear.

Fig. 11. Critical plane estimation based on maximum kt for OP loading case with plasticity model: (a) axial and (b) shear.

ones only changing on the damage parameter values apart that the critical plane estimations are equal for both materials. Critical plane orientation is determined for each loading path through the maximum damage parameter obtained along the loading period. The biaxial stress components for each critical plane model represented in Fig. 8 were determined for the same equivalent von Mises stress. From the results can be concluded that SWT and Liu1 damage parameters have the same values along all projections, moreover, for the same equivalent stress, the Fatemi–Socie damage parameter for the OP loading case is bigger than the other cases, i.e, PT, PS and PP loading cases. 4.3.2. Maximum kt approach In Figs. 9–12 are shown the results for the maximum kt approach on the estimation of the crack initiation plane orientation. On this approach the maximum value for axial and shear kt is determined along the loading period and assigned to the respective micro-notch cap interface with specimen surface node and micronotch root node. Linking the graph origin with a point in the respective kt curve will give the kt magnitude for a specific direction. Linking the graph origin to the maximum kt in axial will give the crack initiation plane estimated by the axial kt, the same procedure is used to

estimate the crack initiation plane using the shear kt. From these two kt estimations the one with the greatest kt value will determine the crack initiation plane orientation inherent to that loading path. Fig. 9 shows the kt gradient in the notch cap regarding the axial and shear computed stress components for the PT loading case. In Fig. 9a the highest kt value in axial is achieved on the nodes located at 0° and 180°, see U referential at Fig. 6b, whereas in Fig. 9b the highest kt shear values were obtained at ±45°. From this two results the most high kt value will indicate the critical plane estimation. For the PT loading case, Fig. 9a, the maximum axial kt value indicates the critical plane at 0°, and the highest axial kt value occurs at notch root. In Fig. 10a is shown the axial kt gradient for the PS loading case, here the maximum axial kt was found at ±45°, however for this loading case the maximum kt is the shear one. In Fig. 10b, the highest shear kt on micro-notch cap interception with specimen surface occurs at ±45° and the highest shear kt is found on the micro-notch root. It is expected that the crack initiation process occurs at maximum kt location propagating at ±45° through notch cap reaching the specimen surface at same direction. In Fig. 11a is shown the OP loading case results, the highest kt is the axial one which suggests that the crack initiation plane occurs at 0°. On this loading case the kt at micro-notch root is almost

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Fig. 12. Critical plane estimation based on maximum kt for PP loading case with plasticity model: (a) axial and (b) shear.

Fig. 13. Loading case PT: fracture surface for (a and b) 42CrMo4 and (d and e), Mg AZ31B-F. Crack initiation angle for (c) 42CrMo4 and (f) Mg AZ31B-F.

equal to the one verified at notch cap interception with specimen surface. The axial kt is greater than the shear one, see Fig. 11b, thus the maximum kt estimation for the critical plane in this loading path is given through the axial kt. In the PP loading case the axial kt, Fig. 12a, estimates the critical plane on the 13° direction, moreover in this loading case the surface micro-notch axial kt is greater than the one verified on the micro-notch root, the same was found for the shear kt i.e. the surface results are greater than the ones verified at notch root.

4.4. Experimental fractographic analysis In this study only early crack propagation was considered, which occurs in a very thin layer of the surface, where the gradient effect is relatively small. Regarding the fracture surface topology analysis it was studied the identification of crack initiation local to measure the crack initiation angle and the different crack propagation zones of fracture surfaces to analyze the influence of the different loading paths on fracture topology.

V. Anes et al. / International Journal of Fatigue 58 (2014) 102–113

Fig. 14. Loading case PS: fracture surface for (a and b) 42CrMo4 and (d and e), Mg AZ31B-F. Crack initiation angle for (c) 42CrMo4 and (f) Mg AZ31B-F.

Fig. 15. Loading case PP: fracture surface for (a and b) 42CrMo4 and (d and e), Mg AZ31B-F. Crack initiation angle for (c) 42CrMo4 and (f) Mg AZ31B-F.

111

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Fig. 16. Loading case OP: fracture surface for (a and b) 42CrMo4 and (d and e), Mg AZ31B-F. Crack initiation angle for (c) 42CrMo4 and (f) Mg AZ31B-F.

Table 7 Critical planes measured and estimated for 42CrMo4 and AZ31B-F. Case PT

Measured FS SWT Liu 1 Liu 2 Max kt

Case PS

Case PP

Case OP

AZ31

42CrMo4

AZ31

42CrMo4

AZ31

42CrMo4

AZ31

42CrMo4

0 ±40 0 0 ±45 0

0 ±40 0 0 ±45 0

45 ±3; ±87 ±45 ±45 ±90; 0 ±45

45 ±3; ±87 ±45 ±45 ±90; 0 ±45

40 13 25 25 20; 70 13

16 13 25 25 20; 70 13

5 0 0 0 ±90; 0 0

0 0 0 0 ±90; 0 0

Fracture surfaces and crack initiation angles for loading paths studied in this work are presented in Figs. 13–16. These results, measured angle and fracture surface topology are representative of the different loading path and material, respectively. Regarding Fig. 13 and PT loading case, both fracture surfaces suggest a ductile fatigue failure mechanism with two different zones and roughness: a fatigue zone (FZ), with smooth area and an instantaneous zone (IZ), with rough area. In the smoothest part of the fatigue zone area is possible to identify the crack initiation local, as in the rough part, in the IZ, the final fracture. The roughness change in fracture surface indicates a different crack growth speed proving that the failure did not happen suddenly. Fatigue life is spent mainly in mode I crack growth and finishes with rough and crystalline appearance. No expressive propagation marks were observed in the high strength steel, however in Mg alloy were observed a slight river marks pointing the crack growth and the initiation spot. Initiation angles measured in both materials were 0°. Fatigue crack results for loading case pure shear PS, are presented in Fig. 14. The steel specimen fracture surface shows a unique initiation source with an initiation plane oriented at 45°; this is a typical result for twisting loads on ductile materials [31]. It is

expected that a reduction of this angle value occurs if there are other stresses than the shear ones involved in the fatigue process, i.e., if mixed mode crack propagation is present. The FZ and IZ show a strong granulated surface in the steel specimen, however in the Mg specimen fracture surface is smoother. In general both fracture surfaces are similar, the FZ have a fracture plane equally oriented and the IZ in both cases have a similar arrangement. At Mg specimen it is observed ratchet marks with two initiation spots growing toward the center of the specimen, and in the IZ can be seen a material riffling like progression marks. In Fig. 15 is shown the fracture surfaces results for the proportional loading case, PP. In this situation the obtained fracture surfacés topology are a little different. The steel specimen fracture surface shows one crack origin and three distinct zones: the usual FZ and IZ zones and a surface wear region. In Mg specimen fracture surface can be identified two crack origins and many river marks pointing out to first crack origin. The crack initiation plane angle measured for the steel specimen was 16° and 40° for Mg specimen. The results for the loading case OP are shown in Fig. 16. Fracture surfaces are also similar in this loading case, supporting the idea on the topology convergence in both materials in high cycle regime. At Mg specimen IZ can be seen diagonal riffles starting from the

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FZ throughout the end of IZ. Diagonal riffles in instantaneous zone indicate a biaxial loading at instant fracture time. In this case the crack initiation plane angle measured for the steel specimen was 0° and 5° for Mg specimen. 4.5. Critical plane analysis Table 7 presents the measured crack initiation angles and also the critical planes estimations by the FS, SWT, Liu I and Liu II critical plane models and maximum kt criterion for the studied loading paths and materials. The SWT and Liu1 models estimate well the crack initiation plane in both materials for the loading cases PT and PS. The first principal strain has a key role on these results, however, the FS and Liu2 models have poor results for this two loading cases. For the PP loading case, the results are satisfactory in the 42CrMo4, regarding FS and Liu2 criteria, on contrary, for the Mg alloy, the estimated results differ from the experimental data, the minor deviation is 20° considering Liu2 model. In the OP loading case the estimated critical plane angle by all models agree well with the experimental results. From the maximum kt approach results can be concluded that the crack orientation planes were very well estimated for the PT and OP loading cases, and for the PS and PP loading cases the results are relatively close to the experimental trends. These results show that maximum kt approach is a sensitive approach to the stress amplitude ratio variations, being sensitive to axial or shear predominance on a multiaxial loading, in contrast with other approaches which are only sensitive to axial or shear stresses. 5. Conclusions This paper studies the influence of multiaxial loading conditions on the fatigue crack initial path, fatigue life and fracture surface topology on two different crystallographic microstructures. From the experimental and theoretical work carried out with two materials, a low-alloy steel and a Mg alloy, some remarks can be drawn: Regarding fatigue life analysis the damage between the applied loading paths tends to have different relative arrangement in both materials. The damage rate between loading paths is similar for the same material excepting the OP loading case where the damage rate is more pronounced. Concerning fractographic analysis in loading cases PT and PS, the fracture surface topology in both 42CrMo4 and Az31B-F specimens are similar and independent on the equivalent stress level. Under multiaxial loading regime the loading path and equivalent stress level have a huge influence on the AZ31B-F surface topology. In high cycle fatigue regime the fracture surface is strongly dependent on the loading path type; for the same loading path the 42CrMo4 and AZ31B-F fracture surface tends to be similar. Regarding critical plane analysis the crack initiation angle in pure axial and pure torsional loading cases do not change with the equivalent stress level. Moreover, at uniaxial loading cases the initiation angles do not vary for the studied materials, however in multiaxial loadings that is not true. The influence of the loading path trajectories on stress concentration factors was clearly shown and the maximum kt approach proved to be sensitive to the stress amplitude ratio and local stress states, achieving very good results with the loading conditions applied in this study.

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