Short cracks growth in low cycle fatigue under multiaxial in-phase loading

Accepted Manuscript Short cracks growth in low cycle fatigue under multiaxial in-phase loading S. Foletti, F. Corea, S. Rabbolini, S. Beretta PII: DOI: Reference:

S0142-1123(17)30403-6 https://doi.org/10.1016/j.ijfatigue.2017.10.010 JIJF 4484

To appear in:

International Journal of Fatigue

Received Date: Revised Date: Accepted Date:

9 June 2017 16 October 2017 18 October 2017

Please cite this article as: Foletti, S., Corea, F., Rabbolini, S., Beretta, S., Short cracks growth in low cycle fatigue under multiaxial in-phase loading, International Journal of Fatigue (2017), doi: https://doi.org/10.1016/j.ijfatigue. 2017.10.010

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Short cracks growth in low cycle fatigue under multiaxial in-phase loading S. Foletti∗, F. Corea1 , S. Rabbolini2 , S. Beretta Politecnico di Milano, Dipartimento di Meccanica, Via La Masa 1, 20156 Milan, Italy

Abstract Crack propagation in full plastic regions is one of the main aspects of fatigue life design for components subjected to high strain concentrations. Residual life assessment for those components, in which high stress concentrations cause cyclic yielding of the material, can be considered as a crack propagation problem by assuming crack growth from the first load cycle. The aim of this paper is to study the crack growth behaviour of short cracks in low cycle fatigue under a multiaxial loading condition. In particular, a series of experiments in LCF regime at room temperature was performed to determine crack growth during axial, torsional and axial-torsional tests. Crack advancement was checked with the plastic replica technique, during test interruptions. Experimental results were compared, in terms of crack growth rates and fatigue life assessment, with those analytically calculated, considering different multiaxial fatigue parameters introduced in an exponential crack growth law and an approach based on the multiaxial cyclic J-Integral concept. Keywords: Multiaxial Low Cycle Fatigue, short crack, J-Integral range, multiaxial fatigue criteria

1. Introduction Multiaxial low cycle fatigue life assessment is important for several components like rotor disks for turbines, which are subjected to stress cycles induced by centrifugal loads and by differential temperatures during startups and shutdowns [1, 2], or pipelines employed in challenging harsh environments, where they have to sustain severe loading conditions [3]. ∗ Corresponding

author Email address: [email protected] (S. Foletti) URL: www.polimi.it (S. Foletti) 1 Present address: LPE SpA, Via Falzarego, 8 - 20021 Baranzate (MI) - Italy 2 Present address: Exergy SpA, via Santa Rita, 14, 21057 Olgiate Olona (VA), Italy

Preprint submitted to Int. J. Fatigue

October 19, 2017

Fatigue life of a component working in Low Cycle Fatigue (LCF) regime can be assessed considering a crack propagation problem, in which residual life is calculated as the number of cycles necessary for a crack to propagate and to cause component failure [4]. The accuracy of this method strongly depends on the adopted crack propagation model: traditional approaches based on Linear Elastic Fracture Mechanics (LEFM) cannot be applied in LCF, since cracks usually nucleate and propagate in regions with high stress concentrations, such as near notches, in which cyclic yielding of the material occurs [5]. Crack growth rates in uniaxial LCF can be described as a function of the applied plastic strain amplitude. Such an approach, originally proposed by Tomkins [6], relates crack growth rates with the plastic strain amplitude through an exponential law, [7, 8]. Several applications of Tomkins’ model on notched specimens can be found in [9, 10], in which model parameters were derived from the coefficients of the Manson-Coffin curve. Skelton [11] modified Tomkins’ model by substituting the applied plastic strain amplitude with the total applied strain range: this model was included in the British R5 procedure [12]. Crack growth rates in uniaxial LCF regime can be also described as a function of ∆J, the cyclic J-Integral. The concept of ∆J was originally postulated by Dowling [13], who extended the J-integral concept [14] to fatigue, proposing to modify Paris relationship [15] by substituting the stress intensity factor range with ∆J. Polak [16] successfully applied a ∆J−based model to describe short crack propagation in an austenitic-ferritic duplex steel, whereas in [17] the cyclic J-Integral was employed to describe fatigue lives of notched specimens subjected to fully reversed loadings. Experimental observations [18, 19] showed that plasticity induced crack closure plays an important role in LCF. McClung and Sehitoglu [20], together with Seeger and Vormwald [21], proposed to calculate the cyclic J-Integral considering only the portion of the load cycle in which the crack stays open. In their works, crack opening levels were computed with the set of equations proposed by Newman [22] for long cracks, in which opening stress for a given strain ratio depends on the constraint factor and the ratio between the maximum stress and the flow stress. Recently, Wu et al. [23] proposed a fatigue crack growth approach based on low cycle fatigue properties. By using a proper crack opening function for closure effects the model captures both long and physically short crack growth behaviour. However, cracks usually propagate in regions in which stress and strain fields are multiaxial, generated by multi-directional loads, residual stresses or geometrical effects. Several criteria for the prediction of crack growth rate and direction under mixed-mode loadings

2

have been proposed over the last 50 years, [24, 25]. Tschegg et al. [26] demostrated that in Mode III fatigue crack growth the crack surfaces in contact glide each other reducing the effective stress at the crack tip. This phenomenon , known as ”sliding mode crack closure”, determines the resulting crack growth rates in a mixed mode loading. However, applications of these criteria are limited to the linear elastic fracture mechanics regime. In order to account for multiaxiality in the elasto-palstic fracture mechanics regime, Vormwald et al. [27, 28, 29] extended ∆J concepts to multiaxial fatigue, accounting not only for Mode I opening, but also for the effects of sliding and friction. In their framework, Newman’s model was redefined to account for mixed mode propagation, whereas ∆J reduction due to friction was computed trough a modified Dugdale’s model [30], which considered crack flanks roughness. Alternatively, crack growth of short cracks under mixed mode propagation has been described by the definition of an effective strain [31, 32] or effective stress [33] intensity factor. Crack growth life can then be calculated by integrating a Paris-type equation. Socie et al. [31] proposed an effective strain-based intensity factor for mode I and II fatigue crack growth under axial-torsional strain-controlled loading: q 2 2√ ∆Keff = (E∆n ) + (G∆γmax ) πa

(1)

where ∆γmax is the maximum shear strain range, ∆n is the maximum normal strain range on the maximum shear plane, E is the Young modulus, G is the shear modulus, and a is the surface crack half-length. Reddy and Fatemi [32] proposed an effective strain intensity factor based on the Fatemi-SOCIE (FS) critical plane fatigue damage parameter [34]:   σn,max √ ∆KCP A = G∆γmax 1 + k πa (2) σy where σy is the material monotonic yield strength, k is a material constant found by fitting fatigue data from uniaxial tests to fatigue data from torsion tests, and σn,max is the maximum normal stress acting on the maximum shear strain plane. Recently, Shamsaei and Fatemi [35] showed that the Reddy-Fatemi (RF) model correlated crack growth rate data for several low and high strength steels under a variety of multiaxial loading condition reasonably well. Crack growth rate correlations were improved by using a modified version of the RF parameter to account for the high percentage of damage in out of phase loading:  ∆KM CP A = G∆γmax

σn,max 1+k σy 3

s

Θ95,OP πa Θ95,IP

(3)

where Θ95,IP and Θ95,IP are the range of plane orientation angles experiencing 95% of fatigue damage for in phase (IP) and out of phase (OP) loading, respectively. This study investigates short crack growth behaviour of an high strength quenched and tempered steel under multiaxial low cycle fatigue condition. In particular, a series of tests was performed on tubular and axial specimens to determine crack growth rates under different multiaxial loading conditions, by applying the plastic replica technique during fatigue tests interruptions. Experimental results were compared, in terms of crack growth rates, with the analytical results provided by an exponential model, modified to account for a generic multiaxial state of stress, and those provided by the model based on the multiaxial cyclic J-Integral.

2. Experiments Fatigue crack growth behaviour in LCF regime was investigated on a high strength quenched and tempered 30NiCrMoV12 steel adopted for manufacturing railway axles. Material tensile properties and cyclic properties are reported in Table 1. Monotonic properties Young’s modulus E [GPa]

196

Yield strength σy (0.2%) [MPa]

878

Ultimate tensile strength U T S [MPa]

1045

Elongation at fracture A%

21.6

Cyclic properties Cyclic strength coefficient K 0 [MPa] 0

Cyclic hardening exponent n Cyclic yield strength

1175 0.0607

σy0 (0.2%)

[MPa]

806

σf0

[MPa]

1162

Fatigue strength coefficient

Fatigue strength exponent b Fatigue ductility coefficient

0f

Fatigue ductility exponent c

-0.0533 [mm/mm]

3.56 -0.8780

Table 1: Monotonic and cyclic properties of 30NiCrMoV12 steel.

Three different experimental campaigns were carried out during the activities presented in this paper: initially, fatigue crack growth was investigated in specimens subjected to cyclic axial loads. Cylindrical specimens with a diameter of 8 mm and a uniform gage length

4

of 20 mm were tested under constant strain amplitude loading at different amplitude levels. An extensometer with a 10 mm gage length was employed to control the strain applied to the specimens. Two different strain ratios were investigated in this phase: the first series of tests was carried out under fully reversed loads, whereas the effects of an applied mean strain were investigated by performing tests at R = 0.25. Tests were performed on a servo-hydraulic load frame, a 100 kN MTS 810 testing machine. A second experimental campaign was performed to study short crack propagation under torsional loads. Experiments were performed on a multiaxial servo-hydraulic load frame, model MTS 809. In this case, thin walled tubular specimens were employed for testing. Specimens were designed according to ASTM E 2207 standard: the outer diameter was set to 16 mm, with a wall thickness of 2.5 mm and a uniform gage length of 35 mm. A multiaxial extensometer, with a gage length of 25 mm, was employed in this phase, since it allowed to measure both axial and torsional strains. The third and final series of experiments was carried out on the same geometry employed for torsional testing. In this phase, axial-torsional tests were performed. All the tests were performed under in-phase loading conditions. The ratio between normal strain amplitude, a , and shear strain amplitude, γa , was kept constant for all the axial-torsional tests and √ was set equal to 1/ 3. All the specimens contained a 200 µm deep micro-notch, obtained by drilling specimen surface with a drill bit, whose diameter was 200 µm. Crack advancement was measured, during fatigue tests interruptions, using the plastic replicas technique with a thin acetate foil. An example of a so-obtained replica is presented in Fig. 1, where, starting from the drilled micro-notch, the surface crack propagation can be observed. Experimental half crack length, a, was plotted against cycles. Crack growth rate data, da/dN , were then derived by the secant method and models parameters calculation was based on these growth rate data. The averaged crack propagation angle, α was measured considering as positive a clockwise rotation with respect to the horizontal axis. A summary of the experimental results is shown in Table 2 and Fig. 2.

5

Specimen Axis Specimen Axis

Specimen Axis

2a a (+ 36°)

2a

2a

(a) Axial (A-9)

a (+ 60°)

a (0°)

(b) Torsional (T-5)

(c) Axial Torsional (AT-1)

Figure 1: Example of plastic replica and definition of crack propagation angle (positive in clockwise direction).

Axial (R = -1)

10 -4

a

da/dN [m/cycle]

a a

10

-6

10

-7

10

-8

a

a

= 0.0040 mm/mm

a

= 0.0050 mm/mm

10 -6

= 0.0060 mm/mm

a

= 0.00300 mm/mm = 0.00375 mm/mm = 0.00450 mm/mm

= 0.0080 mm/mm

da/dN [m/cycle]

a

10 -5

Axial (R = 0.25)

10 -5

= 0.0035 mm/mm a

= 0.0100 mm/mm

10 -7

10 -8

10 -9 -5 10

10

-4

10

-3

10

10 -9 10 -5

-2

10 -4

10 -3

a [m]

(a)

(b)

Torsional

10 -5

= 0.00225 (R=-1), a = 0.00390 (R=-1) mm/mm a = 0.00225 (R=-1), a = 0.00390 (R=0) mm/mm a

= 0.0070 mm/mm = 0.0080 mm/mm

a

10 -5

10 -6

da/dN [m/cycle]

da/dN [m/cycle]

a

10 -7

10 -3

10 -2

a a a

= 0.00225 (R=0), = 0.00325 (R=-1), = 0.00325 (R=-1), = 0.00325 (R=0),

a

= 0.00390 (R=0) mm/mm

a a a

= 0.00563 (R=-1) mm/mm = 0.00563 (R=0) mm/mm

= 0.00563 (R=0) mm/mm

10 -6

10

10 -8 10 -4

Axial-torsional

10 -4

= 0.0060 mm/mm a a

10 -2

a [m]

-7

10 -8 -4 10

10

-3

10

-2

a [m]

a [m]

(c)

(d)

Figure 2: Crack growth rate versus crack length: a) Axial tests R = −1, b) Axial tests R = 0.25, c) Torsional tests R = −1, d) Axial-torsional tests. Symbols and colors refer to different test conditions: strain amplitudes in a), b) and c) - strain amplitudes and strain ratios in d) (see Table 2 for details).

6

Specimen

a

γa

R

Rγ

[mm/mm]

f

σa

σm

τa

τm

Nf (75%)

α

[Hz]

[MPa]

[MPa]

[MPa]

[MPa]

[cicli]

[deg]

A-1

0.0035

-

-1

-

0.5

660

-7

-

-

21385

0

A-2

0.0035

-

-1

-

0.5

645

-9

-

-

18500

0

A-3

0.0040

-

-1

-

0.5

690

-22

-

-

7118

-19

A-4

0.0050

-

-1

-

0.5

689

-12

-

-

4713

24

A-5

0.0050

-

-1

-

0.5

673

-11

-

-

3585

-31

A-6

0.0060

-

-1

-

0.5

700

-11

-

-

2162

-23

A-7

0.0060

-

-1

-

0.5

683

-8

-

-

1597

-28

A-8

0.0080

-

-1

-

0.5

701

-5

-

-

1200

33

A-9

0.0100

-

-1

-

0.5

700

-13

-

-

575

36

A-10

0.00300

-

0.25

-

0.5

567

246

-

-

16758

9

A-11

0.00300

-

0.25

-

0.5

543

241

-

-

13000

27

A-12

0.00375

-

0.25

-

0.5

613

177

-

-

6718

13

A-13

0.00375

-

0.25

-

0.5

655

122

-

-

8529

-9

A-14

0.00450

-

0.25

-

0.5

618

26

-

-

4250

27

A-15

0.00450

-

0.25

-

0.5

626

-20

-

-

5000

30

T-1

-

0.00600

-

-1

0.5

-

-

386

-24

15501

90

T-2

-

0.00700

-

-1

0.5

-

-

421

0

10302

2

T-3

-

0.00800

-

-1

0.5

-

-

413

-3

6000

1

T-4

-

0.00800

-

-1

0.5

-

-

417

-3

5105

0

T-5

-

0.00800

-

-1

0.5

-

-

404

0

4592

2

AT-1

0.00225

0.00390

-1

-1

0.5

400

-100

286

22

17801

60

AT-2

0.00225

0.00390

-1

-1

0.5

410

-23

290

-8

13902

61

AT-3

0.00225

0.00390

-1

0

0.5

435

-46

270

97

14634

58

AT-4

0.00225

0.00390

0

0

0.5

460

93

263

-16

8703

59

AT-5

0.00325

0.00563

-1

-1

0.5

477

20

298

-22

4943

-5

AT-6

0.00325

0.00563

-1

-1

0.5

457

10

285

3

4000

62

AT-7

0.00325

0.00563

-1

0

0.5

470

-11

280

19

4400

60

AT-8

0.00325

0.00563

0

0

0.5

457

59

271

-12

3170

57

Table 2: Experimental results in low cycle fatigue crack propagation tests.

7

3. The elasto-plastic response The cyclic elasto-plastic behaviour of the material is modelled by using the Jiang and Sehitoglu plasticity model [36]. The elastic stress-strain relations can be used until the stresses satisfy the yield condition following the von Mises criterion: f = (S − α) : (S − α) − 2k 2 = 0

(4)

with S representing the deviatoric stress vector containing the 9 components of the deviatoric stress tensor, α the backstress vector in the deviatoric space and k the yield strength in shear. The symbol ” : ” denotes the inner (dot or scalar) product of two vectors. For the plastic strain increment, the normality flow rule is considered: dp =

1 (dS : n) n h

(5)

where h is the plastic modulus function and n is the unit exterior normal vector to the yield surface for a given deviatoric stress state: S−α n= √ 2k

(6)

The plastic modulus function, h, can be obtained by imposing the consistency condition that requires that the stress state should lie on the yield surface during elastic-plastic deformation: h=

n : dα √ dk + 2 dp dp

(7)

where dp is the equivalent plastic strain increment: dp =

p dp : dp

(8)

In the present study the isotropic hardening is neglected, however the yield shear stress is allowed to change in order to correctly predict the non-Masing behaviour, see Fig. 3-a: k = k1 1 + ak eck RM




(9)

where k1 , ak and ck are material parameters that can be obtained by fitting the yield stress of the stabilised hysteresis loop for different strain amplitudes and RM is the radius of a memory surface. The kinematic hardening rule is expressed as: α=

M X i=1

8

α(i)

(10)

(i) !χ(i) +1 α

 dα(i) = c(i) r(i) n −

r(i)

 L(i)  dp

(11)

where the total backstress vector α has been divided into M parts α(i) (i = 1, 2, · · · , M ), c(i) , r(i) and χ(i) are three sets of material parameters and L(i) is defined as: α(i) L(i) = (i) α

(12)

In order to incorporate the stress level and non-proportionality effects on the transient behaviour, i.e. ratchetting and mean stress relaxation, the following relation is introduced:  
χ(i) = Q(i) 2 − n : L(i) 1 + aχ ebχ RM

(13)

where Q(i) , aχ and bχ are constants, p is the accumulated equivalent plastic strain and RM is the radius of the memory surface: g = |α| − RM ≤ 0

(14)

where |α| is the magnitude of the backstress vector. In the original model, [36], the maximum stress is used to measure the memory effect and a recovery term is introduced in order to mirror experimental results where the memory of prior events decays with additional hardening. Following these assumptions the evolution for the radius of the memory surface can be defined as:   α dp dRM = H (g) hL : dαi − cM 1 − RM

(15)

where H is the Heaviside step function (H(x) = 1 if x ≥ 0 and H(x) = 0 if x < 0), the
symbol hi denotes the MacCauley bracket hxi = 12 (x + |x|) and the vectorial quantity L is defined as: L=

α |α|

(16)

The first term in Eq. 15 allows the memory surface to expand when the total backstress is moving outward. The second term, called the recovery term, is introduced to allow the memory surface contraction when the backstress is moving within the surface. The radius of the memory surface, when the material reach the stabilised condition, is equal to the maximum |α| of the α locus. However, the choice of considering the maximum stress as a measure of the memory effect does not correctly reflect the experimental results obtained in the fatigue tests with 9

a positive strain ratio, where the mean stress relaxation seems to be more influenced by the stress range than the maximum value. Two hysteresis loops, for two uniaxial tests with different maximum stresses but similar stress amplitudes, are shown in Fig. 3-b. It can be concluded that the stress range is a more appropriate measure for the memory surface, because at the same stress amplitude corresponds the same plastic strain range. Following this assumption, the memory surface can be defined in the deviatoric stress space considering both the expansion, by changing the radius RM , and the translation, by changing the centre β: g = α − β − RM ≤ 0

(17)

Neglecting the recovery term in Eq. 15 and considering that when the total backstress α is on the memory surface and is moving outward the condition g = 0 of Eq. 17 has to be satisfied, it is possible to define the translation of the memory surface and its expansion: dβ = H (g) (1 − η) hdα : n∗ in∗

(18)

dRM = ηH (g) hdα : n∗ i

(19)

where η is a material parameter and n∗ is the unit normal vector to the memory surface: n∗ =

α−β RM

(20)

All the material constants in Eqs. 4-20 have been calculated from the stabilised hysteresis loops obtained of the strain controlled uniaxial fatigue tests conducted with different strain amplitudes and different strain ratios. The results are reported in Table 3. The prediction of the cyclic plasticity model is showed in Fig. 4. All the results refer to the stabilised hysteresis loop. About the axial fatigue tests, the plasticity model tends to slightly overestimate the stress amplitude when the applied strain amplitude is increasing, with a maximum error of 5%. For the torsional fatigue tests as well as for the axial-torsional ones, the model’s prediction is satisfactory.

4. Multiaxial low cycle fatigue criteria approach In uniaxial fatigue the crack growth of short cracks can be described by the following model: da = B 0 aQ dN 10

(21)

Kinematic hardening M r

(i)

10 r(1) = 28.0 r(2) = 41.6 r(3) = 48.1 r(4) = 51.3 r(5) = 52.8

[MPa]

r(6) = 53.5 r(7) = 53.9 r(8) = 54.1 r(9) = 54.2 r(10) = 104.9 c(i)

c(1) = 12170 c(2) = 3965 c(3) = 1655 c(4) = 751 c(5) = 352 c(6) = 168 c(7) = 81 c(8) = 39 c(9) = 19 c(10) = 9

Q(i)

Q(1) = 20 Q(2) = 20 Q(3) = 10 Q(4) = 10 Q(5) = Q(6) = 2 Q(7) = Q(8) = Q(9) = Q(10) = 1

aχ

1

bχ [MPa

−1

]

-0.005

Non-Masing behaviour k1 [MPa]

0.3424

ak

996.4

ck [MPa

−1

]

-0.0017

Memory surface η

0.5 Table 3: Material constants used in the plasticity model.

1400

1500

1200 1000

Stress [MPa]

Stress [MPa]

1000 800 600

500 400 Sa = 666 MPa Smax = 603 MPa

200

Sa = 636 MPa Smax = 729 MPa

0

0 0

0.005

0.01

0.015

0.02

0

1

2

3

4

5

6

7

Strain [mm/mm]

Strain [mm/mm]

(a)

(b)

Figure 3: a) Experimental Non-Masing Behaviour; b) Effect of stress amplitude on plastic strain range.

where Q is a constant, generally assumed equal to one, giving exponential growth and the term B 0 depends on the plastic, [6], or the total strain, [37, 38]. In the present paper, in order to easily extend the applicability of the model to a multiaxial state of stress, the

11

8 10

-3

Stress Amplitude [MPa]

800

Axial R = -1

Axial R = 0.25

Tors R = -1

Ax-Tors

600

400

200

Axial Stress - Exp. Axial Stress - Num. Shear Stress - Exp. Shear Stress - Num.

200

100

0

-100

A-1 A-2 A-3 A-4 A-5 A-6 A-7 A-8 A-9 A-10 A-11 A-12 A-13 A-14 A-15 T-1 T-2 T-3 T-4 T-5 AT-1 AT-2 AT-3 AT-4 AT-5 AT-6 AT-7 AT-8

Mean Stress [MPa]

0 300

Figure 4: Cyclic plasticity model prediction. Specimen’s labels on X-axis (A: axial tests, T: torsional tests, AT: axial torsional tests). Stress amplitude on Y-axis (first row). Mean stress on Y-axis (second row). Symbols: open square and cross refer to experimental and predicted shear stress respectively, open circle and plus refer to experimental and predicted axial stress respectively.

term B 0 become a function of a generic fatigue parameter: da d = K0 (F P ) 0 a dN

(22)

where F P is the fatigue parameter defined by the use of a multiaxial low cycle fatigue criterion, K0 and d0 are two material constants and the constant Q is assumed equal to one. By integrating Eq. 22 from an initial crack length ai to a final crack length af it is possible to obtain the number of cycle to failure Nf : Nf =

ln (af /ai )

(23)

d0

K0 (F P )

where ai = 100µm is half the length of the initial defect and af = 3mm is assumed to be half the crack length at failure. Equation 23 can be used to obtain the material constants K0 and d0 by linear fitting the fully reversed uniaxial experimental results in a double logarithmic scale:  log Nf = −d0 log (F P ) + log

1 ln K0



af a0

 (24)

4.1. Low cycle fatigue criteria All the multiaxial fatigue criteria used in the present study are based on the definition of a critical plane. An elementary material plane ∆ is completely defined by the spherical 12

coordinates, θ and φ, of its unit normal vector:    sin(θ) cos(φ)   n∆ = sin(θ) sin(φ)     cos(θ)

    

(25)

   

Starting from the knowledge of the stress tensor, [σ (t)] at instant t, it is possible to obtain the normal stress acting on the plane: σn (t, θ, φ) = nT∆ · [σ (t)] · n∆

(26)

and the corresponding amplitude, mean value or maximum value in time: i 1h σa (θ, φ) = max (σn (t, θ, φ)) − min (σn (t, θ, φ)) t t 2

σm (θ, φ) =

i 1h max (σn (t, θ, φ)) + min (σn (t, θ, φ)) t t 2

σmax (θ, φ) = max (σn (t, θ, φ))

(27)

(28)

(29)

t

In order to define the shear stress and the shear strain acting on the plane it is useful to introduce two unit vector acting on the ∆ plane:         − sin(φ)  − cos(θ) cos(φ)       l∆ = r∆ = cos(φ) − cos(θ) sin(φ)             0 sin(θ)

    

(30)

   

The shear stress can be expressed in the (n∆ l∆ r∆ ) frame where only the components along l and r exist: τl (t, θ, φ) = lT∆ · [σ (t)] · n∆ τr (t, θ, φ) = rT∆ · [σ (t)] · n∆

(31)

The mean value of the shear stress can now be obtained as the centre of the minimum circumscribed circle to the curve described by the shear stress vector on the ∆ plane, [39]: n o 0 τ m (θ, φ) : min max ||τ (t, θ, φ) − τ || 0 t

τ

(32)

where τ = [τl τr ] is the shear stress vector acting on the elementary material plane, τ m is the mean shear stress vector and the symbol ||·|| denotes the norm of a vector. Once the centre τ m has been found, the amplitude of the shear stress is obtained as: τa (θ, φ) = max||τ (t, θ, φ) − τ m || t

13

(33)

Finally, the maximum shear stress can be obtained as: τmax = ||τ m || + τa

(34)

The same procedure from Eq. 25 to Eq. 34 can be used in order to obtain the amplitude, mean value and maximum value for the normal and shear strain acting on the elementary material plane ∆. 4.1.1. Tomkins model The model originally proposed by Tomkins, [6], is based on the plastic strain amplitude:

where pl a


d0 da = K0 pl a (35) a dN is the strain amplitude in the uniaxial condition. In the present study the model

has been extended to a multiaxial state, considering a critical plane approach and looking for the plane where the normal plastic strain amplitude is maximum. This plane is defined by the spherical coordinates (θ∗ , φ∗ ), of the unit vector n∆ normal to the critical plane. Following this approach the couple (θ∗ , φ∗ ) is determined as: (θ∗ , φ∗ ) : max pl a (θ, φ)




θ,φ

(36)

where pl a (θ, φ) is the plastic normal strain amplitude computed by Eq. 27 using the plastic strain tensor instead of the stress tensor . Once the critical plane has been defined the fatigue parameter is written as: ∗ ∗ F PT OM = pl a (θ , φ )

(37)

4.1.2. Smith-Watson-Topper Criterion The Smith Watson and Topper (SWT) model, [40], is based on a combination of the cyclic strain range and the maximum stress. The SWT parameter can be used for materials that fail primarily due to Mode I tensile cracking. The critical plane, defined by the spherical coordinates (θ∗ , φ∗ ), is the plane experiencing the maximum normal strain range: (θ∗ , φ∗ ) : max (∆ (θ, φ)) θ,φ

(38)

where ∆ (θ, φ) can be calculated for each elementary plane as double the value in Eq. 27 considering the strain tensor instead of the stress tensor. Once the critical plane has been fixed, the SWT fatigue parameter is defined as: F PSW T = σn,max (θ∗ , φ∗ ) 14

∆ (θ∗ , φ∗ ) 2

(39)

where σn,max (θ∗ , φ∗ ) is the maximum stress on the critical plane and it can be obtained using Eq. 29. 4.1.3. Fatemi and Socie model The Fatemi and Socie (FS) model, [34], has been developed for materials that fail by the nucleation and early growth of shear cracks. It is based on a combination of the shear strain range and the maximum normal stress: ∆γeq (θ, φ) ∆γ (θ, φ) = 2 2



σn,max (θ, φ) 1+k σy0

 (40)

where ∆γ (θ, φ) /2 and σn,max (θ, φ) can be calculated using Eq. 33 and Eq. 29 using the strain tensor and the stress tensor, respectively. The sensitivity of a material to normal stress is reflected in the value k/σy0 , where k is a material parameter and σy is the cyclic yield stress of the material. In the present study a value k = 1 has been assumed. The introduction of the normal stress term in the FS parameter allows to include the crack closure effects and to describe the mean stress and the non-proportional hardening effects. The critical plane is defined as the plane experiencing the maximum value of the damage parameter:  ∆γeq (θ, φ) (θ , φ ) : max θ,φ 2 Once the critical plane has been identified, the fatigue parameter is: ∗



∗

F PF S =

∆γeq (θ∗ , φ∗ ) 2

(41)

(42)

4.1.4. Chu, Conle and Bonnen model The Chu, Conle and Bonnen (CCB) model, [41, 42], is a combined critical plane and energy model. In a fracture mechanics context, the J-Integral for Mode I and Mode II can be added to obtain the total driving force: J = JI + JII = f (n0 )Y πa (∆σ∆ + ∆τ ∆γ)

(43)

where Y is a dimensionless parameter depending on geometry, loading condition and crack shape, f (n0 ) is a function of the strain hardening exponent of the cyclic curve. Equation 43 suggests that a damage parameter may be written as a combination of the shear and normal work. Starting from these calculations Chu et al. proposed a combined critical plane energy model substituting the stress ranges with the maximum stress in order to include the mean stress effect: ∆W (θ, φ) = τn,max (θ, φ)

∆γ (θ, φ) ∆ (θ, φ) + σn,max (θ, φ) 2 2 15

(44)

where the values of the maximum shear and normal stresses as well as the values of the normal and shear strain ranges can be obtained, for each elementary material plane, by using the equations from Eq. 26 to Eq. 34. The critical plane is the plane with the maximum damage parameter ∆W : (θ∗ , φ∗ ) : max (∆W (θ, φ))

(45)

θ,φ

and the corresponding fatigue parameter is: F PCCB = ∆W (θ∗ , φ∗ )

(46)

4.2. Comparison with experimental results The parameters K0 and d0 of Eq. 22 have been obtained by fitting Eq. 24 with the axial fatigue tests results with a strain ratio R = −1. The results are reported in Table 4. LCF Criterion

K0

Tomkins

1.25

SWT

1.09 −5

2.33 · 10

2.85

245.75

2.74

Fatemi - Socie Chu

d0

−5

2.26 · 10

2.82

Table 4: Material constants used in the low cycle fatigue crack propagation models.

The comparison between the experimental fatigue life and the model prediction is showed in Fig. 5 together with the standard error, σ ˆlog10 (da/dN ) , of log10 (da/dN ) data assumed as a measure of the scatter. Tomkins’ model is able to correctly predict the fatigue life when the applied strain is high. Decreasing the applied strain, the plastic part of the total strain tends to vanish and the model prediction become unsafe. Moreover, the torsional fatigue data are overestimated leading to a safe prediction. Even considering the crack propagation angles, the results are not consistent. For the axial tests the predicted crack propagation angle is α = 0◦ , for the torsional tests α = ±45◦ and finally for the axial-torsional tests α = −26◦ . All these results are in discrepancy with the experimental values reported in Table 2. The SWT model gives good results for the axial test due to its ability to correctly predict the mean stress effect. On the other hand, the prediction for the torsional tests and the axial-torsional tests are strongly unsafe increasing the overall scatter. The crack

16

propagation angles are the same as predicted by the Tomkins model and reflect the property of the model of representing materials that fail primarily due to Mode I tensile cracking. Concerning the FS model and the CCB model the fatigue life prediction is very accurate with a lower scatter. Not only the fatigue life is well predicted but even the crack propagation angle is very close to the experimental observed one as reported in Table 2. For the FS model the crack propagation angle in axial tests is in the range α = ± (37◦ − 38◦ ), in torsional fatigue tests is α = ±10◦ , ±80◦ , and finally for the axial-torsional tests is α = +63◦ , −10◦ . The values predicted by the CCB model are very similar and in details are equal to: α = ± (20◦ − 30◦ ) for axial tests, α = ±0◦ , ±90◦ for torsional tests and α = +57◦ , −5◦ for the axial torsional tests. The only drawback in the use of the FS model is the need to define the additional material parameter k with the consequence of run some additional fatigue tests, axial tests with mean strain or torsional tests, to identify it.

17

TOMKINS model

10 -4

-5

Axial R = -1 Axial R = 0 Torsional Axial-Torsional

TOMKINS model Axial R = -1 Axial R = 0 Torsional Axial-Torsional

10

4

UNSAFE

N f predicted

da/dN [m/cycle]

10

10 5

10 -6

10 -7

10 3 10

SAFE

-8

10 -9 -8 10

10

-7

10

-6

10

10

-5

2

10 2

10 3

10 4

10 5

N f experimental

FP a(1/d 0)

(a) SWT model

10 -4

-5

10

4

UNSAFE

N f predicted

da/dN [m/cycle]

10

SWT model

10 5

10 -6

10 -7

10 3 10

SAFE

-8

10 -9 10

-1

10

10

0

2

10 2

FP a

10 3

10 4

10 5

N f experimental

(1/d ) 0

(b) 10 -4

FS model

10 5

-5

10

4

UNSAFE

N f predicted

da/dN [m/cycle]

10

FS model

10 -6

10 -7

10 3 10

SAFE

-8

10 -9 -4 10

10

10

-3

10 2

10 3

10 4

N f experimental

(1/d )

FP a

2

0

(c)

18

10 5

CCB model

10

-5

10

-6

CCB model

10 5

10

4

UNSAFE

N f predicted

da/dN [m/cycle]

10 -4

10 -7

10 3 10

SAFE

-8

10 -9 10

-1

10

10

0

10 2

10 3

10 4

10 5

N f experimental

(1/d )

FP a

2

0

(d) Figure 5: Crack growth rate and life prediction: a) Tomkins model, b) SWT model, c) Fatemi and Socie model, d) Chu, Conle, and Bonnen model.

19

5. Fracture mechanics approach 5.1. Cyclic J-integral formulation for mixed-mode loaded cracks Fatigue crack growth of short cracks in the LCF regime can be described as a function of the cyclic J-integral, ∆J, considering a Paris-type equation [13, 43]: da m = C (∆J) dN

(47)

The concept of ∆J was originally introduced in the work of Dowling [13, 44], who extended to cyclic loads the line integral proposed by Rice [14]. Cyclic J-Integral was computed by substituting remote stresses and strains with the remote stress and strain amplitudes applied to components with defects. In this work, the value of ∆J is calculated in the experimentally observed crack growth direction, see Fig. 6a and Table 2, following the approach originally proposed by Wuthrich [45].

𝜎"

𝜏"#

x 10

−3

x 10

−3

4

6

8

10

0

12

6

8

10

12

200

200

300 400 500

500

𝛾"#

600

600

700

700

800

800

900

900

1000

1000

1100

1100

(b)

400

4

100

100

Δ𝑊"#,)

300

2

0

Δ𝑊"#,(

𝜖"

(a)

2

Δ𝑊",(

0 0

Δ𝑊",'

(c)

Figure 6: Stress and strain components for ∆JI and ∆JII calculations. a) Crack tip coordinate system. b) and c) Elastic and plastic strain energy components for cyclic J-integral calculations.

It is assumed that only Mode I and Mode II contribute to crack advancement, as reported in Eq. 48, where ∆JI and ∆JII are the cyclic J-integral associated to Mode I opening and Mode II sliding, respectively. ∆J = ∆JI + ∆JII

(48)

Accordingly, ∆JI can be computed as [46]: ∆JI = 2YI2 πa




 1 − ν 2 ∆Wy,e + f (n0 ) ∆Wy,p 20

(49)

where YI is a factor that accounts for crack geometry, a is crack length, ν is Poisson’s ratio and ∆Wy,e and ∆Wy,p are, respectively, the elastic and plastic strain energy densities, calculated as reported in Fig. 6b. The function f (n0 ) takes into account crack geometry and material elastic-plastic behaviour. For a crack under plane strain conditions, f (n0 ) can be calculated as [44]: f (n0 ) =

3 (1 + n0 ) √ 8 n0

(50)

with only a slight difference in the case of the plane stress condition. ∆JII component can be calculated as [27, 28, 29]: ∆JII =

π Y 2 a (∆Wxy,e + ∆Wxy,p ) 1 + ν II

(51)

where ∆Wxy,e and ∆Wxy,p are, respectively, the elastic and plastic strain energy densities, calculated as reported in Fig. 6c. Accordingly, both elastic and plastic terms provide the same contribution to ∆JII . In this work, Eq. 51 has been rewritten, introducing a plastic correction function, g (n0 ), following the same calculation scheme of Eq. 49: ∆JII =

π Y 2 a [∆Wxy,e + g (n0 ) ∆Wxy,p ] 1 + ν II

(52)

The plastic correction function associated to Mode II sliding, g (n0 ), has been numerically calculated by considering a finite element model of a cracked panel subjected to monotonic loading. Material cyclic elasto-plastic behaviour has been described with the deformation plasticity model available in Abaqus. The crack angle θ, see Fig. 7a, has been taken equal to 54.7◦ corresponding to JI = 0. Mesh was refined in the crack tip region, with an average element size of 0.05 mm. The numerical J value has been obtained at different applied remote strain values, see Fig. 7a. The comparison between numerical results and those calculated according to Vormwald’s model (Eq. 51) is reported in Fig. 7a, in which it can be noted that Vormwald’s equation underestimates JII . In order to improve the analytical prediction the numerical results were fitted with Eq. 52: it was found that a good agreement between FEM and theory can be obtained with the following formulation: 1 + n0 g (n0 ) = √ 2n0

(53)

Eventually, equations 49 and 52 were validated by performing several simulations of the panel with differently oriented cracks: in Fig. 7b numerical results are compared to those analytically obtained, showing excellent agreement.

21

300

Present formulation Numerical Results

700 600

\theta

# = 54.7o a = 1 mm

J [MPa mm]

J [MPa mm]

200

800

L = 100 mm

250

!"

150

100

500 400 300 200

Numerical results Vormwald's formulation Present formulation

50

0

0

0.02

0.04

0.06

0.08

100

0.1

0 0

10

20

30

40

[mm/mm]

50

60

70

80

[°]

(a)

(b)

Figure 7: FE model of a slant crack in a square panel for g (n0 ) calculations: a) comparison between numerical and analytical results (θ = 54.7◦ ). Vormwald’s formulation (Eq. 51) in red and the model presented in this work (Eq. 52) in blue; b) Comparison between numerical and analytical results (Eqns. 49 and 52) J.

5.2. Crack closure effects during LCF propagation The effects of crack closure in LCF were initially introduced by Vormwald and Seeger [21] together with McClung and Sehitoglu [20]. In these works, the authors only dealt with cracks subjected to pure mode I opening: therefore, it was found that crack growth rates were a function of the mode I effective cyclic J-Integral, ∆JI,ef f , computed considering only the part of the fatigue cycle in which the crack stays open. Accordingly, Eq. 49 has been modified, as proposed in Eq. 54, in which ∆Wy,e,ef and ∆Wy,p,ef are the effective elastic and plastic strain energy densities. Experimental observations [19, 18] showed that crack opening and closing, during LCF propagation, occur at the same strain level. Accordingly, ∆Wy,e,ef and ∆Wy,p,ef were computed as reported in Fig. 8. ∆JI,ef f = 2YI2 πa




 1 − ν 2 ∆Wy,e,ef + f (n0 ) ∆Wy,p,ef

(54)

It was found [18] that an accurate estimation of crack opening stress, σy,open , can be calculated with the set of equations proposed by Newman [22], considering plane stress conditions and a flow stress equal to the average of the cyclic yield stress and the ultimate tensile strength: σF =

0 Rp,0.2 + Rm 2

0 where Rp,0.2 is the cyclic yield stress and Rm is the ultimate tensile strength.

22

(55)

90

𝜎" x 10

−3

0

2

4

6

8

10

12

0

Δ𝑊",','(

100 200 300 400 500

Δ𝑊",),'(

𝜖"

600

𝜎",*)'+

700 800 900

𝜎",,-*.'

1000 1100

Figure 8: Effective elastic and plastic strain energy densities for ∆JI,ef f calculation.

In this work, the extension of this approach to cracks subjected to multiaxial loading conditions is employed. Vormwald et al. [27, 28, 29] modified Newman’s model, suggesting the following set of equations:     σy,max A0 + A1 · Ref f + A2 · R2 + A3 · R3 ef f ef f σy,open =  σy,max (A0 + A1 · Ref f )

if Ref f ≥ 0

(56)

if Ref f < 0

where:  A0 = 0.535 · cos A1 = 0.344 ·

π σmax,ef f 2 σF



σmax,ef f σF

A2 = 1 − A0 − A1 − A3 A3 = 2 · A0 + A1 − 1   R if R ≥ 0 Ref f =  R · σeqv,max /σmax,ef f if R < 0 q
2 2 + τ2 σmax,ef f = σy,max + 3 τxy xz r h i
1 2 2 2 2 + τ2 + τ2 σeqv,max = (σy,max − σx ) + (σy,max − σz ) + (σz − σx ) + 3 τxy xz yz 2 (57) The effect of crack closure on ∆JII calculations were introduced following a different frame. Vormwald introduced a reduction factor, Uef f , so that: ∆JII,ef f = Uef f · ∆JII The reduction factor can be calculated as [29]:  −1 ! ,  −1 ! π ∆τ /2 − τf ric π ∆τ /2 Uef f = ln cos · ln cos · 2 τF − τf ric 2 τF 23

(58)

(59)

in which ∆τ is the applied shear stress range and τF is the flow stress in shear, defined as: σF τF = √ 3

(60)

The friction stress on the crack surface, τf ric , is computed as: τf ric = hτact − µ · σn,max i

(61)

where σn,max is the maximum normal stress over one whole shear strain cycle, τact is the shear stress threshold value according to crack face indentation, and µ is the friction coefficient. 5.3. Experimental results Experimental results have been analysed following the procedure discussed in the previous section. A value of τact = 200 MPa has been considered. It has been calculated by fitting experimental results of the torsional tests with a reference da/dN - ∆Jef f curve, obtained testing a compact tension C(t) specimen with a high stress ratio (σmin /σmax = 0.7). A high stress ratio was chosen in order to remove crack closure effects from crack growth data. Two different values of the friction coefficient have been taken into account. Initially, a value of µ = 0.3 was taken into account: in Fig. 9a, short crack propagation experiments are compared to the reference da/dN − ∆Jef f curve. The reference curve represents an upper bound for the experimental data from torsional testing. The same trend can be observed when axial-torsional tests are taken into account. On the other hand, experimental data from mono-axial tests lie on the reference curve, meaning that the model can accurately predict crack growth under these loading conditions. An increase in the scatter with respect to the FS criterion, σ ˆlog10 (da/dN ) = 0.3043, can be observed and all the assessments of axial specimens tend to be nonconservative, lying on the upper side of the diagram. On the other hand, conservative assessments are obtained for pure torsional and axial-torsional tests. For these loading conditions, the assessments are very conservative, with an average ratio between experimental and predicted fatigue life of 2. A friction coefficient of 0.7 was also taken into account, following the experimental observations reported in [47]: the results of this analysis are reported in Fig. 9b. It can be noted that a higher value of µ implies an increase in the value of the applied ∆Jef f for the torsional and axial-torsional tests. The increase of the friction coefficient implies an increase in the scatter, σ ˆlog10 (da/dN ) = 0.3711, and a more conservative life assessment,

24

with all the data-points shifting towards the safe side of the diagram and the maximum ratio Npredicted /Nexperimental tending to 0.3.

10

-5

10

-6

10 5 Axial R = -1 Axial R = 0 Torsional R = -1 Axial-Torsional R = -1 C(t) specimen R=0.7

10

4

UNSAFE

N f predicted

da/dN [m/cycle]

10 -4

10 -7

10 3

SAFE

10 -8

10

-9

10 -4

10 -3

10 -2

10 -1

10

10 0

2

10 2

J tot,eff [MPa m]

10 3

10 4

10 5

N f experimental

(a)

10

-5

10

-6

10 5 Axial R = -1 Axial R = 0 Torsional R = -1 Axial-Torsional R = -1 C(t) specimen R=0.7

10

4

UNSAFE

N f predicted

da/dN [m/cycle]

10 -4

10 -7

10 3

SAFE

10 -8

10

-9

10 -4

10 -3

10 -2

10 -1

10 0

J tot,eff [MPa m]

10

2

10 2

10 3

10 4

N f experimental

(b) Figure 9: Crack growth rate and life prediction based on ∆Jef f . a) friction coefficient µ = 0.3; b) friction coefficient µ = 0.7.

6. Conclusions The results of three different experimental campaigns were employed to study short crack propagation in multiaxial fatigue. Initially, experimental results were analysed by introducing different critical plane LCF criteria into an exponential crack growth law. It was found that FS and CCB models both provide very accurate results, in terms of crack 25

10 5

growth rates, fatigue life prediction and crack propagation angle. On the other hand, assessments based on Tomkins model are conservative only when a high plastic strain is present, coherently with model main assumptions. Predictions based on SWT model showed that the model is strongly nonconservative, in particular when torsional and axialtorsional tests are considered. A different analysis was performed considering a model based on the cyclic J-integral. It was found that the model can accurately predict crack closure effects in uniaxial tests, since all the experimental data-points lie on the reference closure-free long crack growth curve. A conservative assessment can be obtained with a friction coefficient of 0.7, a value typically found in the literature for cracks loaded under pure Mode II. The main problem in the use of the fracture mechanics approach is that the direction of crack growth must be known as an a priori. For this reason, the approach based on a multiaxial low cycle fatigue criterion seems to be more promising for the implementation in a finite element framework.

Acknowledgements The authors acknowledge support by the Grant CTN01-00236-494934 (MIUR, Italy).

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Highlights (1) The results of several crack growth experimental tests in LCF regime under axial, torsional and axial/torsional loading condition are reported. (2) Two approaches to predict fatigue life are proposed: i) exponential crack growth law based on multiaxial fatigue parameters, ii) multiaxial cyclic J-Integral approach. (3) Fatemi-Socie and Chu-Conle-Bonnen models both provide very accurate results, in terms of crack growth rates, fatigue life prediction and crack propagation angle. (4) A conservative assessment can be obtained with the cyclic J-Integral approach (5) The approach based on a multiaxial low cycle fatigue criterion seems to be more promising for the implementation in a finite element framework.