Numerical prediction of fatigue threshold of metallic materials in vacuum

Engineering Fracture Mechanics xxx (xxxx) xxxx

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Numerical prediction of fatigue threshold of metallic materials in vacuum ⁎

F.V. Antunesa, , P.A. Pratesa, D. Camasb, C. Sarrazin-Baudouxc, C. Gardinc a b c

Department of Mechanical Engineering, University of Coimbra, Portugal Departamento de Ingeniería Civil, de Materiales y Fabricación, Universidad de Málaga, Spain PPrime Institute, ISAE-ENSMA, Poitiers, France

A R T IC LE I N F O

ABS TRA CT

Keywords: Fatigue crack growth Effective fatigue threshold Crack tip opening displacement (CTOD) Environmental effect

The fatigue threshold is determined experimentally following international standards, but its measurement is laborious and time consuming. This paper proposes a numerical approach to determine the fatigue threshold based on the analysis of the crack tip opening displacement, CTOD, evaluated using the finite element method. It assumes that fatigue crack propagation only occurs with plastic deformation at the crack tip. However, a factor of about two was found between the numerical predictions and experimental results of effective fatigue threshold from literature. This difference was attributed to a different mechanism of fatigue crack growth at relatively low load ranges, resulting from environmental effects at the crack tip. Therefore, the approach proposed is adequate to predict the fatigue threshold in vacuum, which is supposed to be controlled by crack tip plastic deformation.

1. Introduction Engineering analysis of fatigue crack growth is usually performed by relating da/dN to the stress intensity factor range, ΔK. Three regimes can be identified in da/dN-ΔK curves, as is well known. In log-log scales a linear relationship is generally observed at intermediate values of ΔK, named Paris law (da/dN = C(ΔK)m). At relatively low ΔK values there is a fast decrease of da/dN with the reduction of load range, and below a certain ΔK there is no measurable crack growth. This limit is called fatigue threshold, ΔKth, and its knowledge is important for a proper design of components. In fact, most of fatigue life is spent at relatively low values of ΔK, therefore accurate threshold data is fundamental for the safety of design based on damage tolerance approach. ASTM E647 [1] and ISO 12108 [2] standards define the experimental procedure that must be followed in order to obtain valid and comparable values of fatigue threshold. The load range is gradually reduced until there is no crack propagation. In each loading step, some crack propagation is required to eliminate the influence of crack closure phenomena resulting from previous load range. The reduction of load range may be done with a fixed stress ratio, R (=Kmin/Kmax), or fixing the maximum stress. In the constant R approach, the maximum and minimum loads are successively reduced such that the stress ratio is kept constant. However, the measured values of ΔKth are affected by stress ratio, which is explained by crack closure phenomena. For relatively high stress ratios an effective threshold stress intensity, ΔKth,eff, is obtained, free of crack closure, which is often referred to as the intrinsic measurement of fatigue crack growth resistance. The constant Kmax test procedure proposed in the ASTM E647 standard also eliminates the effect of stress ratio, therefore gives an effective fatigue threshold. In this approach, Kmin is progressively increased in order to eliminate the effect of crack closure. Values of fatigue crack propagation threshold for metallic alloys may be found in literature [3–7], which has been largely derived ⁎

Corresponding author. E-mail address: [email protected] (F.V. Antunes).

https://doi.org/10.1016/j.engfracmech.2019.106491 Received 20 February 2019; Received in revised form 8 May 2019; Accepted 22 May 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: F.V. Antunes, et al., Engineering Fracture Mechanics, https://doi.org/10.1016/j.engfracmech.2019.106491

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Nomenclature a a0 AA BMF Cx Cy C(T) da/dN CTOD CTODp E F K Kmax Kmin

L1 M(T) R SS SLM W Xsat Ysat Y Y0 YK δp σ Δa ΔK ΔKth ΔKth,eff

crack length initial crack length Aluminium Alloy Brittle Micro Fracture material constant of kinematic hardening material constant of isotropic hardening compact Tension (specimen) fatigue crack growth rate Crack Tip Opening Displacement Plastic CTOD Young’s modulus remote force stress intensity factor maximum value of stress intensity factor minimum value of stress intensity factor

size of near crack tip elements Middle-Tension (specimen) stress ratio Stainless Steel Selective Laser Melting specimen’s width material constant of kinematic hardening material constant of isotropic hardening yield stress initial yield stress geometric factor for K calculation range of plastic CTOD remote stress total increment of crack growth stress intensity factor range (Kmax-Kmin) fatigue threshold effective fatigue threshold

from standard test geometries, i.e., containing large (more than few mm) through-thickness cracks. Jones et al. [5] made a literature review showing ΔKth values in the range 0.9–3.8 MPa·m0.5 for aluminum alloys, in the range 1.6–10 MPa·m0.5 for titanium alloys and in the range 2.6–16 MPa·m0.5 for steels. Li and Rosa [7] in their literature review presented effective threshold values in the ranges of 2.3–3.7; 1.0–1.4; 1.3–3.9 and 2.9–3.2 for ferrous alloys, aluminium alloys, copper alloys for nickel alloys, respectively. Zersbst et al. [8] identified the main factors affecting the fatigue threshold, which are: (i) the geometry and size of the crack; (ii) the size of specimens; (iii) the material properties (iv) the loading parameters such as R ratio and variable amplitude loading parameters, and (v) the environment. There is a great influence of stress ratio, namely when the constant R approach is being used. Borrego et al. [9] presented values of ΔKth for 6082-T6 aluminum alloy, showing that the increase of stress ratio decreases the fatigue threshold. A similar trend was observed by Ritchie et al. [10] in the Ti–6Al–4 V alloy. On the contrary, when the constant Kmax approach is followed, only one value of fatigue threshold is expected (ΔKth,eff), as already mentioned. Even so, there is an influence of stress ratio not linked to the crack closure phenomenon. Several mechanisms were proposed to explain this effect of stress ratio, namely the interaction between monotonic fracture modes and fatigue crack propagation mechanisms, sustained-load cracking, crack closure at a microscopic level and macro-residual stresses [8]. Sunder [11] stated that ΔKth is extremely sensitive to load history, therefore is not a material constant. According to him, this sensitivity to load history has nothing to do with crack closure. Dependency of the fatigue threshold on specimen geometry may also occur, linked to crack closure, which reduces for high R ratio [3]. Finally, according to Alkan and Sehitoglu [12], the threshold value is not unique, depending on the microstructure around the crack tip, namely, the grain size, the grain boundary types and the initial dislocation density. According to them, near-threshold fatigue crack growth remains one of the most challenging fields in fatigue research. The experimental measurement of fatigue threshold is particularly laborious and time consuming. Due to the dependence relatively to stress ratio, several tests are needed to obtain ΔKth versus R. Additionally, there is a significant difference between ASTM E647 [1] and ISO 12108 [2] in terms of critical crack propagation rate, which may affect the values obtained for ΔKth. The former sets the limit to (da/dN)th = 10−7 mm/cycle whereas the latter specifies it at (da/dN)th = 10−8 mm/cycle [8]. In fact, these limits are just a working definition, having no scientific reason. Crack closure is a major source of problems for determining the fatigue threshold, since it affects the experimental results in a complex and generally unknown way. Therefore, alternative approaches have been used to study fatigue threshold, like the study of dislocation movement at the crack tip [13]. It is assumed that the crack progresses by shear mechanisms along the critical slip system with maximum resolved shear stress. The advancing crack may be arrested at a grain level obstacle such as a grain or a twin boundary. The numerical prediction of ΔKth will be exploited in this work. The proposed numerical approach uses the crack tip opening displacement (CTOD), and is based on two fundamental assumptions: (i) that fatigue crack growth only occurs if there is plastic deformation at the crack tip. (ii) that the CTOD is sensitive to the elastic and plastic deformation regimes happening at the crack tip. Therefore, it can be used to quantify the load range below which there is no plastic deformation, i.e., the fatigue threshold. According to ASTM E647 [1], ΔKth is the asymptotic value of ΔK at which fatigue crack growth rate approaches zero. According to Ritchie et al. [10], a sound approach is to model the intrinsic threshold in terms of the applied driving force below which dislocations can no longer be emitted from the crack tip [14,15]. According to Wasén and Heier [16], “The growth of a fatigue crack in a ductile metallic material requires cyclic microplastic activity at the crack tip …”. Therefore the first assumption seems to be acceptable. On the other hand, the capabilities of CTOD to study crack closure and fatigue crack growth were fully demonstrated in a previous work of the authors [17]. Zerbst et al. [8] also say that “the CTOD would be a far more meaningful parameter for describing fatigue crack propagation and its threshold than K or ΔK, …”. The crack opening displacement is in fact a classical parameter in elastic-plastic fracture mechanics, still widely used nowadays [18]. It has a physical meaning and can be measured directly in experiments. In the finite element analysis, the displacement of the first node behind the crack tip is generally used as an operational CTOD [19]. The numerical prediction of ΔKth would have several advantages, namely, the ability to determine effective values free of crack closure, the adequacy to develop parametric studies isolating the effect of each parameter (e.g. specimen geometry, material properties, etc.), and the 2

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ability to study the mechanisms behind the variations observed. The present paper has therefore a main objective: to propose a numerical approach able to predict fatigue threshold. As will be seen, this methodology gives values of fatigue threshold in vacuum, since in air the environmental mechanisms are expected to dominate. The numerical model included the modeling of material’s elastic-plastic behavior, the application of a cyclic loading and the simulation of crack propagation. Two specimen geometries and six materials were considered in this analysis. 2. Numerical model Different materials were studied here, namely five aluminium alloys (AA6016-T4, AA6082-T6, AA7050-T6, AA2050-T8 and A2024-T351), one selective laser sintered steel (18Ni300) and one stainless steel (304L). The constitutive material model considered that the elastic behaviour is isotropic and is described by the generalised Hooke’s law, with material parameters E (Young’s modulus) and ν (Poisson ratio); the plastic behaviour is described by a mixed isotropic-kinematic hardening law coupled with von Mises yield criterion under an associated flow rule. In this work, the isotropic hardening model described by Voce law [20],

Y = Y0 + (Ysat − Y0)[1 − exp(−CY ε¯p)]

(1)

was combined with a non-linear kinematic hardening model described by Armstrong-Frederick law [21]:

(σ ′ − X) Ẋ = CX ⎡XSat − X⎤ ε¯ ṗ , ⎢ ⎥ σ¯ ⎣ ⎦

X (0) = 0

(2)

In the previous equations, Y is the flow stress, ε¯ p is the equivalent plastic strain, Ẋ is the back stress tensor rate, ε¯ ṗ is the equivalent plastic strain rate, σ ′ is the deviatoric Cauchy stress tensor, X is the back stress tensor and σ¯ is the equivalent stress; Y0, YSat, CY, CX and XSat are material constants. The von Mises isotropic yield criterion is written as follows:

(Σ yy − Σzz )2 + (Σzz − Σ xx )2 + (Σxx − Σ yy )2 + 6(Σ2yz + Σ2xz + Σ2xy) = 2σ¯ 2

(3)

where Σxx , Σ yy , Σzz , Σxy , Σxz and Σ yz are the components of the effective stress tensor (Σ = σ ′ − X ). Table 1 presents the properties considered for the different materials studied. The identification of these constitutive material parameters was carried out by minimizing the difference between experimental stress-strain results and numerical predictions. The fitting was performed for about 50 load cycles, therefore the material models include the cyclic behavior. An excellent agreement was found between the experimental results of low-cycle fatigue tests and the elastic-plastic models, therefore these models are considered adequate for describing the cyclic behavior occurring at the crack tip [17,22–25]. Standard Compact-Tension (C(T)) and Middle-Tension (M(T)) specimen geometries were considered in the prediction of fatigue threshold. Fig. 1 shows the geometry and relevant dimensions in the sheet plane of the M(T) and C(T) specimens used to study the 6082-T6 aluminium alloy and the 18Ni300 steel, respectively. Table 1 indicates the geometries considered for the different materials. In both specimen geometries, the thickness was assumed to be 0.2 mm in the numerical model. A straight crack was modelled, with an initial size, ao, of 5 mm (ao/W = 0.083). Due to geometrical and material symmetries, only one eight of the M(T) specimen and one quarter of the C(T) specimen were considered in the numerical simulation models, considering proper boundary conditions. Note that 3D models are being used, therefore there is a symmetry condition along the thickness. The finite element model of the M(T) specimen had a total number of 6639 3D solid linear isoparametric elements and 13,586 nodes. Similarly, the C(T) specimen was modelled with 7163 elements and 14,654 nodes. The traditional tri-linear eight-node hexahedral finite element was associated with a selective reduced integration scheme [26]. The finite element meshes were refined near the crack tip, having 8 × 8 μm2 elements in this region. The size permits a good balance between accuracy and numerical effort. Only one layer of elements was considered along the thickness. Crack propagation was simulated by successive debonding of nodes at minimum load. Each crack increment corresponded to one finite element size (8 µm) and two load cycles were applied between increments. In each cycle, the crack propagated uniformly over the thickness by simultaneously releasing both current crack front nodes. A total number of 320 load cycles were applied, corresponding to a total crack propagation of Δa = (160 − 1) × 8 μm = 1272 μm. Note that the first two load cycles were applied for the initial crack length, i.e., without crack increment. The objective of this propagation was the stabilization of crack tip deformation, and therefore of the fatigue threshold, Table 1 Elastic-plastic parameters for the materials under study. Material

AA6016-T4 [22] AA6082-T6 [17] AA7050-T6 [23] AA2050-T8 [24] 18Ni300 304L SS [25] AA2024-T351

Hooke’s law parameters

Isotropic hardening (Voce)

Kinematic hardening (Armstrong-Frederick)

E [GPa]

ν [–]

Y0 [MPa]

YSat [MPa]

CY [–]

CX [–]

XSat [MPa]

70.0 70.0 71.7 77.4 160.0 196.0 72.3

0.29 0.29 0.33 0.30 0.30 0.30 0.29

124.00 238.15 420.50 383.85 683.62 117.00 288.96

415.00 487.52 420.50 383.85 683.62 204.00 381.55

9.5 0.01 0 0 0 9 6.9

146.50 244.44 228.91 97.38 728.34 300.00 138.80

34.90 83.18 198.35 265.41 402.06 176.00 111.84

3

Geometry

MT (W = 50) MT (W = 60) MT (W = 50) MT (W = 150) CT (W = 36) CT (W = 50) CT (W = 50)

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2a

60

300

R4.5

43.2 19.8

W=36

Fig. 1. Geometry and dimensions (in mm) of the M(T) and C(T) specimens used to study the 6082-T6 aluminium alloy and the 18Ni300 steel, respectively.

since a transient behavior is observed at the beginning of crack propagation. The numerical studies were made with and without contact between crack flanks. The numerical removal of contact has no physical sense, but it may be interesting to understand the effect of crack closure. Note that, in order to study the influence of material and loading parameters, all numerical parameters were kept constant, namely the number of load cycles between crack increments and the size of crack tip elements. The numerical simulations were performed with the Three-Dimensional Implicit Elasto-plastic Finite Element program (DD3IMP). This software was originally developed to model deep-drawing processes [27–29], and was used in here due to its great competence in the modeling of plastic deformation. The evolution of the deformation process is described by an updated Lagrangian scheme, assuming a hypoelasticplastic model. Thus, the mechanical model takes into account large elastoplastic strains and rotations and assumes that the elastic strains are negligibly small with respect to unity. The elastic behaviour is assumed to be isotropic and the plastic behaviour is modelled considering the yield surface and its evolution and an associated flow rule. The contact of the crack flanks is modeled considering a rigid body (plane surface) aligned with the crack symmetry plane. A master–slave algorithm is adopted and the contact problem is treated using an augmented Lagrangian approach [28]. Further details of the numerical procedure may be found in previous publications of the authors [22]. The CTOD was measured at the first node behind crack tip, i.e., 8 μm from crack tip. This point was selected because it has the greatest sensitivity to crack tip phenomena. Anyway, the analysis of other nodes behind crack tip showed that there is a relatively low influence of measurement point on the values predicted for fatigue threshold (results not presented). Fig. 2a shows a typical curve of CTOD versus load, obtained in the 6082-T6 aluminium alloy after 159 crack propagations (Δa = 1.272 mm). The load is presented in the form of σ/Y0, being σ the remote stress and Y0 the initial yield stress. The remote stresses were obtained dividing the loads by the area of cross section, i.e., σ = F/Area, being Area = 30 × 0.1 mm2. Fig. 2b presents the plastic CTOD, which was obtained subtracting the elastic deformation from the total CTOD presented in Fig. 2a. The crack is closed for relatively low loads, i.e., between points A and B. The increase of load above point B opens the crack, and the variation of CTOD is linear up to point C, where plastic deformation starts. The elastic variation of CTOD is linked to the elastic behaviour of the material; therefore it is intimately related with the Young’s modulus, E. Materials with high values of E deform less than those with lower E values, i.e., the variation of CTOD with load is relatively small. The transition between the elastic and the elastic-plastic regimes (point C) was assumed to occur for CTODp = 0.001 μm. This limit is totally empirical. Note that the definition of fatigue threshold proposed in standards is also empirical. Plastic deformation starts above point C and reaches its maximum value at point D, which corresponds to the plastic CTOD range, δp. The application of the load range between points A and C is not expected to produce plastic deformation, therefore can be use to predict fatigue threshold, ΔKth. On the other hand, the the load range between points B and C can be used to predict the 4

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F.V. Antunes, et al. 1.2 D

1.0

CTOD [μm]

CTOD 0.8

0.6

0.4

0.2

C

(a)

A

0.0

0.0

0.1

B

0.2

0.3

0.4

σ/Y0 0.35 D

0.30

CTODp [μm]

0.25 0.20 δp

0.15 0.10 0.05 B

A

0.00

0.0

(b)

C

0.1

0.2

0.4

0.3

σ/ Y0

Fig. 2. Typical results. (a) CTOD versus load. (b) Plastic CTOD versus load. (M(T) specimen; plane stress; 6082-T6; Fmin = 0; Fmax = 240 N; with contact between crack flanks).

effective fatigue threshold ΔKth,eff, free of crack closure. The stress intensity factor for the M(T) specimen was calculated using a solution for the geometric factor YK, developed by the authors [22]:

YK =

K a a a = 1.187086( )3 - 0.068016( )2 + 0.113481( ) + 1.009325 σ πa W W W

(4)

For the C(T) specimen, the solution presented in the standards was used [1,2]. Fig. 3 plots both CTOD and CTODp predictions obtained removing the contact between crack flanks. The total CTOD is plotted on the left axis while the plastic CTOD is plotted on 2.0

3 CTOD CTODp

2.5

1.0

2

0.5

1.5

0.0

1 0.0

0.1

0.2

0.3

0.4 δp

-0.5

-1.0

CTODp [mm]

CTOD [μm]

1.5

C

σ/Y0

0.5

0

Fig. 3. CTOD without contact between crack flanks (M(T) specimen; plane stress; 6082-T6; Fmin = 0 N; Fmax = 240 N). 5

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the right axis. Negative values of total CTOD can be observed, which means that the crack flanks are overlapping. The elastic deformation starts at the minimum load since there is no crack closure. The load range between point C and minimum load can be used to define fatigue threshold. 3. Numerical results 3.1. Constant Kmax approach Two approaches were followed to predict the fatigue threshold. In the first approach, the maximum load was kept constant while the minimum load was increased progressively. Fig. 4a presents the variations of CTOD with load for the 304L stainless steel. The increase of minimum load, i.e., the reduction of ΔK, reduces progressively the plastic CTOD range, δp. The increase of stress ratio also eliminates progressively the crack closure phenomenon, as is well known. Fig. 4b presents results of applied ΔK versus δp for the 7050-T6 aluminium alloy. The extrapolation to a null plastic deformation, as indicated, gives the effective fatigue threshold, ΔKth,eff. Fig. 5 presents results for different materials, while Table 2 presents the values predicted using this approach. As already mentioned, different aluminium alloys were studied, along with the 304L stainless steel and the 18Ni300 steel obtained by Selective Laser Melting (SLM). The values predicted for the threshold are relatively high, as will be discussed later. 3.2. Constant R approach A second approach was followed, keeping the stress ratio constant and using the elastic portion of the CTOD. Looking to Figs. 2 and 3, the load range corresponding to fatigue threshold is assumed to be the difference between the limit of elastic regime (point C) and the crack opening load (point B). Fig. 6 presents the results obtained for the 18Ni300 steel, for different load cases. The threshold defined this way is plotted versus the range of plastic CTOD, δp, and the extrapolation to zero gives the proposed value for the effective fatigue threshold. There is a convergence to a nearly constant value, independently of stress state, stress ratio and even specimen size, which indicates that the approach is robust. The position of the measurement point behind crack tip was also studied 0.8 R=0.1

0.7 0.6 R=0.1

CTOD [μm]

0.5 0.4

R=0.56

0.3

R=0.66 R=0.85

0.2 0.1

(a)

0 0

10

20

30

40

50

60

Load [N] 14 12

ΔK [MPa.m1/2 ]

10 8 6 4 2 0 0.00

(b) 0.01

0.02

0.03

0.04

0.05

0.06

δp [μm] Fig. 4. Constant Kmax approach: (a) CTOD versus load (C(T) specimen; plane stress; 304L stainless steel; W = 36 mm, a = 17.272 mm). (b) ΔK versus δp (7050-T6 Aluminium Alloy). 6

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25

ΔK [MPa.m1/2 ]

20

15

7050-T6 AA

10

304L SS 6082-T6

5

2050-T8 18Ni300 SLM

0 0.0

0.1

0.2

0.3

δp [μm] Fig. 5. ΔK versus plastic CTOD range for different materials. Table 2 Effective fatigue thresholds predicted using the constant Kmax approach. Material

ΔKth,eff [MPa·m0.5]

AA6082-T6 AA7050-T6 AA2050-T8 304L SS 18Ni300 SLM AA2024-T351

6.48 7.64 7.57 4.26 11.16 3.7

30

ΔKth [MPa.m0.5 ]

25 20 15 10

Plane stress; R=0.05; CT36 Plane stress; R=0.3; CT36

5

Plane stress; R=0.6; CT36

Experimental

Plane strain; R=0.1; CT50 Numerical; Kmax

0 0

0.02

0.04

0.06

0.08

δp [μm] Fig. 6. Numerical prediction of fatigue threshold using the constant R approach (18Ni300 steel).

and no influence was found, which reinforces the robustness of the procedure. The value obtained with the constant Kmax approach is also presented, being slightly higher that the values obtained using the constant R approach. However, the experimental value is 5.2 MPa·m0.5 [30], therefore is significantly lower than the numerical predictions. Fig. 7 compares experimental results with numerical predictions of effective fatigue threshold for three materials. As can be seen, the numerical results are nearly the double of the experimental values, which is a significant difference. 4. Discussion Note that the present numerical model assumes that the material is continuum, homogeneous and isotropic. Li and Rosa [7] also used continuous mechanics properties to predict intrinsic fatigue threshold in metal alloys. On the other hand, several authors studied fatigue threshold using models of dislocation movement, which assume that the onset of plastic deformation occurs at a micro level along the slip planes of individual grains [13]. In fact, according to Pippan et al. [31] it is not possible to study fatigue threshold with 7

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10 Numerical Experimental

ΔKth,eff

8 6 4 2 0

18Ni300 SLM

2024 AA

6082 AA

Fig. 7. Comparison between experimental results and numerical predictions of effective fatigue threshold.

continuum mechanics models therefore they recommend the use of dislocation models. They say that at small intensity ranges, the continuum mechanics solution is too soft since it does not account for the constraint of plasticity due to its discrete nature. The application of a continuum mechanics model is expected to be more problematic for coarse-grained materials. The orientation of the grains at each position of crack tip may therefore be expected to be relevant for the fatigue threshold [12]. The use of micromechanical models [32] and the comparison with continuum mechanics models for the same material would be interesting to check this possibility. Tong et al. [33] found significant differences in terms of strain range at four grain sizes with different orientation. A second explanation for the mismatch observed in Fig. 7 is the existence of a different crack growth mechanism at relatively low load ranges. Sunder [34] proposed the Brittle Micro-Fracture (BMF), as an alternative to the cyclic plastic deformation assumed in here. This brittle fracture is linked to the surface diffusion of hydrogen, released by the reaction of moisture with the crack tip surface, resulting in oxygen and hydrogen formation. Each load rising increases near-tip stress, acting as a diffusion pump to promote embrittlement of surface atomic layers. The intensity of this surface diffusion driven mechanism at the beginning of the rising half of the load cycle is controlled by near tip stress, which is a measure of residual elastic strain and therefore of initial inter-atomic spacing of atoms in the surface layers. The increase of this spacing enhances the adsorption of air components. A well-defined relation between near tip stress and ΔKth, with two linear regions, was obtained by Sunder [11]. A competition is expected between BMF and cyclic slip. The BMF is restricted to crack tip surface atomic layers and therefore becomes insignificant for relatively large crack growth rates. According to Sunder [11], slip-driven fatigue dominates for low-cycle fatigue and crack growth rates exceeding 0.1 μm/ cycle. In vacuum the environmental effect is eliminated therefore the threshold is only linked to cyclic slip. This vacuum threshold was proposed to be three times greater than the effective ΔKth in air [11]. In here, the ratios observed between the numerical and experimental values of effective threshold were 1.8, 2.1 and 2.7 for the SLM steel, the AA2024-T351 and the AA6082-T6, respectively. Therefore, are of the same order of magnitude of the ratio obtained by Sunder [11]. Additionally, this seems to indicate that the numerical approach based on the onset of plastic deformation is particularly interesting for the prediction of fatigue threshold in vacuum. Zerbst et al. [8] indicated two main environmental mechanisms: stress assisted anodic material dissolution and hydrogen embrittlement. In the first case, the protective layer is locally destroyed at the crack tip and the exposed material dissolves electrochemically. In the second case, the hydrogen atoms penetrate into the process zone and modify the deformation and the damage

Fig. 8. Experimental and numerical results of fatigue threshold. The squares represent experimental values of ΔKth in vacuum and air obtained in literature. 8

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mechanisms at the crack tip [35–37]. The corrosion damage is a time-dependent process which will occur when the medium is in contact with the crack tip. In the near-threshold regime the crack is almost stationary, reason why the environment has a major influence. Karr et al. [38] studied near-threshold fatigue in AZ61 magnesium alloy tested in air and vacuum. They observed that the fracture surfaces were smoother in vacuum than in air. In ambient air, the fracture surfaces showed facets and fine steps typical of brittle fracture. Fig. 8 presents experimental values of fatigue threshold versus numerical predictions. Therefore, the circles in Fig. 8 repeat the information of Fig. 7. There is a reasonable agreement with the dashed line, which represents a factor of about two between numerical and experimental results of effective fatigue threshold. Therefore, considering the results in Table 2, effective threshold values of 3.8, 3.8 and 2.1 MPa·m0.5 may be predicted for the AA7050-T6, for the AA2050-T8 and for the 304L SS, respectively. The squares indicate experimental results from literature, obtained in air and vacuum [38–41]. These results also fit very well the dashed line, indicating that the environment may be the reason for the difference between numerical predictions based on plastic deformation, and the experimental results. Additionally, this reinforces that the model proposed here, based on the analysis of crack tip plastic deformation using continuum mechanics models, can be used to predict effective fatigue threshold values in vacuum. The numerical prediction of fatigue threshold opens the opportunity to develop parametric studies, which in fact is one of the main advantages of the numerical studies. A study focused on the effect of stress ratio was developed just to show this possibility. Fig. 9 presents the results obtained for the 2024-T351 aluminium alloy considering the constant stress ratio approach. The extrapolation to a null plastic deformation gives threshold values of about 3.4 MPa, independently of stress ratio. This seems to confirm that the effective threshold is a material property. Finally, a numerical study was developed in order to check the effect of mesh refinement on the predictions of fatigue threshold. Three meshes were considered with elements of 8, 16 and 32 μm near the crack tip. Fig. 10 presents the variation of fatigue threshold predicted for the 6082-T6 aluminium alloy with the size of near crack tip elements, L1. There is a decrease of fatigue threshold with mesh refinement, as could be expected, however the variation is not dramatic. This reinforces the robustness of the approach proposed. An extrapolation to elements with zero size may be proposed, which gives value of about three for the fatigue threshold of this material. 5. Conclusions A numerical procedure was developed to predict fatigue threshold based on the prediction of Crack Tip Opening Displacement (CTOD). It is assumed that fatigue crack growth only occurs in the presence of crack tip plastic deformation and that this deformation is proportional to plastic CTOD range, δp. δp was quantified for different load ranges, under constant maximum load or constant R, and an extrapolation to δp equal zero predicts the load range corresponding to fatigue threshold. This value is effective since is free of crack closure. However, the predictions of the effective threshold were found to be too high when compared with experimental values obtained in literature. A factor of about two was found between the numerical predictions and the experimental results. This difference was attributed to a different mechanism of fatigue crack growth at relatively low load ranges, which is associated with environmental effect at the crack tip. Therefore, the numerical approach proposed here to predict fatigue threshold is adequate to predict the fatigue threshold in vacuum. Acknowledgements The authors would like to acknowledge the sponsoring under the project no. 028789, financed by the European Regional 6

ΔKth [MPa.m0.5 ]

5 4 3 2 R=0.1 R=0.35

1

R=0.6 0 0.0

0.2

0.4

0.6

0.8

δp [μm] Fig. 9. Effect of stress ratio (AA2024-T351, C(T) specimen; plane strain). 9

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ΔKth [MPa.m1/2 ]

8 7 6 5 4 3 2 1 0 0

5

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15

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L1 [μm] Fig. 10. Effect of finite element mesh on fatigue threshold (MT, W = 60 mm, plane strain).

Development Fund (FEDER), through the Portugal-2020 program (PT2020), under the Regional Operational Program of the Center (CENTRO-01-0145-FEDER-028789) and the Foundation for Science and Technology IP/MCTES through national funds (PIDDAC). This work was also supported by the project EZ-SHEET, co-funded by Portuguese Foundation for Science and Technology, by FEDER, through the program Portugal-2020 (PT2020), and by POCI, with reference POCI-01-0145-FEDER-031216. All supports are gratefully acknowledged. The authors would also like to thank the DD3IMP in-house code developer team for providing the code and all the support services. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

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