Study of anisotropic crack growth behavior for aluminum alloy 7050-T7451

Engineering Fracture Mechanics xxx (xxxx) xxx–xxx

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Study of anisotropic crack growth behavior for aluminum alloy 7050-T7451 ⁎

Jun Cao, Fuguo Li , Xinkai Ma, Zhankun Sun State Key Laboratory of Solidification Processing, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Anisotropy Normalized plastic zone Crack growth Compact-tension-shear specimen

Hill's normalized plastic zone at crack tip under mode-I and mixed-mode loadings were studied in compact-tension-shear specimen. Effects of the parameters of Hill’s criterion on the normalized plastic zone were analyzed. Moreover, the shapes of normalized plastic zones for distortional and total strain-energy-density were investigated. To analyze the crack growth behavior by different normalized plastic zones, fatigue crack growth tests of AA7050-T7451 were performed using digital image correlation technique. The initiation angle predicted by anisotropic R-criterion under mixed-mode loading is closest to experimental data.

1. Introduction Plastic anisotropy is an inevitable phenomenon due to some material processing techniques, such as rolling and extruding processes. Effect of plastic anisotropy on crack growth behavior under complex loading in the domain of fracture mechanics is significant. Furthermore, investigations of crack initiation and propagation considering the effect of plastic anisotropy are meaningful to predict cracks of aircraft structure parts in the damage tolerance and the design of full-life. Gdoutos and Meletis [1] used strain-energy-density (SED) theory to analyze the fracture behavior of anisotropic plate and revealed the dependence of crack growth angle and crack direction on material properties. Theocaris and Philippidis [2] applied Tcriterion to predict the mixed-mode fracture characteristics of anisotropic plates, and the predictions for dimensionless fracture stress of a plate with an inclined crack fit with experimental data. Khan and Khraisheh [3] employed the Hill's criterion to obtain normalized variable-radius crack tip plastic zone, and the study shows a significant effect of plastic anisotropy on the normalized plastic zone and the crack initiation angle. Tvergaard and Legarth [4] applied an elastic-viscoplastic material model to account for plastic anisotropy on mixed-mode interface crack growth and concluding that the resistance to crack growth is sensitive to anisotropy. Makas [5] studied the influence of rolling-induced anisotropy on fatigue crack initiation and short crack propagation for AA2024T351 and finding that fractured particles for longitudinal cruciform samples are responsible for crack nucleation while nucleated cracks in transverse samples are caused by deboned and fractured particles. Jin et al. [6] carried out four-point bend fatigue tests on L-T (Rolling-Transverse), L-S (Rolling-thickness), and T-S planes of AA7075-T651 respectively to study effects of preceding fractured particles and drawing that the anisotropy of fatigue strength results from the difference in grain structure and the particle between on these planes. Atzori et al. [7] studied fatigue behavior of several notched carbon steels under multi-axial loading by using SED theory, and the energy-based approach makes all the fatigue data obtained from the notched specimens to be concentrated in a single scatter band. Tavares et al. [8] reported the comparison of the strain field of numerical modeling against experimental data from digital image correlation (DIC) method under Mode-II loading of fatigue test, and the obtained results are an important step towards a ⁎

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

https://doi.org/10.1016/j.engfracmech.2018.04.011 Received 1 November 2016; Received in revised form 2 April 2018; Accepted 6 April 2018 0013-7944/ © 2018 Elsevier Ltd. All rights reserved.

Please cite this article as: Cao, J., Engineering Fracture Mechanics (2018), https://doi.org/10.1016/j.engfracmech.2018.04.011

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Nomenclature

RM radius of Hill's plastic zone RH radius of Hill's normalized plastic zone r0,r45,r90,rθ plastic strain ratio at 0°, 45°, 90°, and θ directions for rolling direction W width of the CTS specimen α linear weight factor between Swift and Voce Equations material coefficient of Voce Equation β θ polar angle at the crack tip κ material constant depending upon stress states ν Poisson’s ratio σsa saturated stress of Voce Equation σapp applied stress around crack tip σM , σH Mises equivalent stress and Hill equivalent stress σx , σy , τxy normal and shear stress components dεx , dεy strain increment along x and y directions shear strain increment on the xy plane dγxy plastic strain εp ε0 prestrain of Swift Equation ψ loading angle F, G, H, Nparameters of Hill’s criterion

a crack length a11, a12 , a22 coefficients of function for normalized plastic zone radius B thickness of the compact-tension-shear specimen C material constant of Paris Equation fKi (ψ) function of loading angle ψ E elastic modulus stress-intensity-factors of mode-I or mode-I Ki m material constant of Paris Equation Nu number of fatigue cycles P applied load Q material coefficient of Voce Equation radius of plastic zone for distortional strain-enrd ergy-density (SED) radius of normalized plastic zone for distortional rd SED radius of plastic zone for total SED rs rs radius of normalized plastic zone for total SED RM radius of Mises's plastic zone RM radius of Mises's normalized plastic zone

development of a practical tool for crack behavior prediction in fatigue dominated events. The experimental data shown in Refs. [9,10] were not fitted well with the yield surfaces of the Von Mises type including Hill's criterion of anisotropy. To describe anisotropic yielding behavior preferably, Barlat et al. [11–13] proposed and developed specific yield surfaces for aluminum sheets. Moreover, Bron and Besson [14] extended the yield functions given by Barlat et al. [11] and Karafillis and Boyce [15] to represent plastic anisotropy of aluminum sheets. Although Hill's criterion may have some limitations to describe yield surface of 7050-T7451 aluminum alloy, the purpose is to focus on providing a heuristic thinking for anisotropic crack growth behavior. Efforts were made to reveal the law of crack growth behavior of aluminum alloy 7050-T7451 [16–19]. However, rare reports considered the effect of plastic anisotropy on crack growth behavior. Since the anisotropic R-criterion [3] was rarely applied to predict crack initiation angle of compact-tension-shear (CTS) specimens, Hill's normalized plastic zones under mixed-mode loading in CTS specimens were investigated to analyze fatigue crack growth behavior. To comprehensively analyze the crack growth behavior the normalized plastic zone for energy density was investigated. Crack initiation angle predicted by three criteria, i.e. anisotropic Rcriterion, the minimum strain energy density criterion (S-criterion), and maximum tangential stress criterion (MTS-criterion) were compared with experimental data under mode-I and mixed-mode loadings in CTS specimen. Crack growth rate was investigated to analyze the relation between crack growth behavior and normalized plastic zones under mode-I and mixed-mode loadings of CTS specimen as well. Then, the crack initiation mechanism under mode-I and mixed-mode loadings was discussed based on the analyses of fracture appearance.

Fig. 1. Stress field at crack tip. 2

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2. Theoretical method For stress state at crack tip under mixed-mode loading, as shown in Fig. 1, the elastic stress field at the crack tip on plane stress condition is expressed as

σx =

1 ⎡⎧ θ θ 3θ θ θ 3θ KIcos ⎛1−sin sin ⎞ ⎫−⎧KIIsin ⎛2 + cos cos ⎞ ⎫ ⎤ 2⎝ 2 2 ⎠⎬ 2⎝ 2 2 ⎠⎬ 2πr ⎢ ⎩ ⎭ ⎨ ⎩ ⎭⎥ ⎣⎨ ⎦

σy =

1 ⎡⎧ θ θ 3θ θ θ 3θ ⎤ KIcos ⎛1 + sin sin ⎞ ⎫ + KIIsin cos cos 2⎝ 2 2 ⎠⎬ 2 2 2 ⎥ 2πr ⎢ ⎩ ⎭ ⎣⎨ ⎦

τxy =

3θ 1 ⎡ θ θ KIcos sin sin 2 2 2 2πr ⎢ ⎣

{

{

}

(1a)

}

(1b)

3θ θ θ + ⎧KIIcos ⎛1−sin sin ⎞ ⎫ ⎤ ⎨ 2 2 2 ⎠⎬ ⎝ ⎩ ⎭⎥ ⎦

(1c)

σz = 0

(1d)

where polar radius r, polar angle θ are the parameters of polar coordinate, KI , and KII are the stress–intensity–factors (SIFs) under mode I and mode II loadings, respectively. For the loading in CTS specimen, the SIFs, KI and KII on the condition of 0.5 ⩽ a/ W ⩽ 0.7 being considered as in the form [20,21]:

KI =

P πa cosψ WB 1−a/ W

0.26 + 2.65(a/(W −a)) , 1 + 0.55(a/(W −a))−0.08(a/(W −a))2

(2)

KII =

P πa sinψ WB 1−a/ W

−0.23 + 1.40(a/(W −a)) , 1 + 0.67(a/(W −a)) + 2.08(a/(W −a))2

(3)

where W and B are the width and the thickness of the CTS specimen respectively, P is the applied load, and ψ is the applied load angle between the longitudinal axis of the CTS specimen and the loading direction. The fatigue pre-crack length is 25 mm and the four loading angles ψ are 0°, 15°, 45°, and 60°, respectively. As the SIFs are similarly expressed in Eqs. (2) and (3), the equations were simplified as

P πa fKi (ψ),i = I,II WB (1−a/ W )

Ki =

(4)

Here fKi (ψ) is a function of the loading angle ψ . To calculate the radii of plastic zone, Mises’s criterion and Hill’s criterion are applied. The Mises's criterion and Hill's criterion [22] on plane stress condition are expressed as

σM =

2 σx2 + σy2−σx σy + 3τxy

(5)

σH =

2 (G + H ) σx2 + (H + F ) σy2−2Hσx σy + 2Nτxy

(6)

where F, G, H, and N are the parameters of Hill's criterion. The yield function f of Hill’s criterion is expressed as 2 2f = (G + H ) σx2 + (H + F ) σy2−2Hσx σy + 2Nτxy =1

(7)

For a uniaxial tensile specimen cut along α angle concerning rolling direction, the relation between stress components (σx ,σy,and τxy ) and σα which is yield stress along α direction under uniaxial tension condition can be known based on equilibrium of forces. To relate stress and strain components, these equations are followed based on Ducker postulate:

dεx = [(G + H )cos2 α−H sin2 α ] σα dλ

(8a)

dεy = [(F + H )sin2 α−H cos2 α ] σα dλ

(8b)

dγxy = (N sinα cosα ) σα dλ

(8c)

where dεx , dεy , and dγxy are normal strain increments along x and y directions and shear strain increment on the xy plane, respectively. The plastic strain ratio rα which is ratio of transverse strain to thickness strain along α direction concerning rolling direction can be expressed as

rα =

(dεx sin2 α + dεy cos2 α−2dγxy sinα cosα ) dεα + π /2 H + (2N −F −G−4H )sin2 α cos2 α = = dεz −(dεx + dεy ) F sin2 α + cos2 α

(9)

Then, substituting 0°, 45°, and 90° into Eq. (9) and then three r-values (r0,r45 , and r90 ) are expressed as

r0 =

H G

(10a) 3

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r45 =

1 ⎛ 2N −1⎞ 2 ⎝F + G ⎠

(10b)

r90 =

H F

(10c)

Since rolling direction is assumed to coincide with x direction and σ0 is reference yield stress, then σx = σ0 , σy = 0 ,τxy = 0, and σ0 = σH . Substituting these equations into Eq. (6), then (11)

G+H=1

Furthermore, according to Eqs. (10a)–(10c) and Eq. (11) and then the parameters F, G, H, and N of Hill's criterion [22] can be expressed as

F=

r0 r90 (1 + r0)

(12a)

G=

1 1 + r0

(12b)

H=

r0 1 + r0

(12c)

N=

(r0 + r90)(2r45 + 1) 2r90 (1 + r0)

(12d)

where F, G, H, and N can be determined from the values of r0 , r45 , and r90 . Then, the three rα -values can be calculated by this equation [23,24],

rα = −

kα 1 + kα

(13)

where k α is ratio of transverse strain to longitudinal strain along α direction concerning rolling direction. The measuring positions are chosen in the different locations of gauge for uniaxial tensile specimens. The choices of measuring position are the center and two sides of gauge to avoid the accidental error. As the transverse strain and longitudinal strain are easily measured by DIC technique, the plastic strain ratios r0 , r45 , and r90 were measured as 0.4143, 1.2792, and 0.7694, respectively. Therefore, the parameters F, G, H, and N of AA7050-T7451 were calculated as 0.3807, 0.7071, 0.2929, and 1.9354. Substituting Eqs. (1a)–(1c) into Eqs. (5) and (6), and then the radii RM [25] and RH [3] of normalized plastic zone can be obtained as

RM =

RM σapp 2

a⎡ σ ⎤ ⎣ M⎦ RH =

RH σapp 2

a⎡ σ ⎤ ⎣ H⎦

= g1 f K2I (ψ) + g2 f K2II (ψ) + g12 fKI (ψ) fKII (ψ) (14)

= h1 f K2I (ψ) + h2 f K2II (ψ) + h12 fKI (ψ) fKII (ψ) (15)

where RM and RH are the actual radii of the plastic zones boundaries by using Mises’s criterion and Hill’s criterion, σapp is the applied stress at crack tip and is equal to P /(WB (1−a/ W )) . g1, g2 , g12 , h1, h2 , and h12 are function of θ , then these expressions of the parameters are given in Refs. [25,3], respectively. The two radii RM [25] and RH [3] were applied in the mixed-mode loadings of CTS specimen rather than the loadings of the plate cracked in the center . The radii rS and rd (see Appendix A) of normalized plastic zone using distortional and total SED were derived as

rd =

rS =

rd 2

σapp / 2E a ⎡ (dW / dV ) ⎤ d ⎣ ⎦

rS 2 / 2E σapp

a ⎡ (dW / dV ) ⎤ ⎣ ⎦

= g′1 f K2I (ψ) + g′2 f K2II (ψ) + g′12 fKI (ψ) fKII (ψ) (16)

= a1 f K2I (ψ) + a2 f K2II (ψ) + a12 fKI (ψ) fKII (ψ) (17)

on plane stress condition, where rd and rS are actual radii of the plastic zones at crack tip by using distortional SED (dW / dV )d and total SED dW / dV . E is elastic modulus, a1, a2 , and a12 are functions regarding θ and ν , which are defined in Appendix A. The radii are suitable for all loading conditions. Stress field at crack tip is based on linear elastic fracture mechanics (LEFM). However, plastic deformation at crack tip makes the crack tip blunt and then the stress relaxation phenomenon exists at crack tip under fatigue crack growth testing. For instance, the method of Irwin’s plastic zone correction analogy is an equivalent integration to approach the plastic zone at crack tip. If the method is applied in the correction of normalized plastic zone, the radius will have a correction factor. However, the shape of normalized plastic zone will not change, so the crack initiation angle under complex loading does not change. Besides, the plastic zone for energy 4

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density is normalized, which does not compare with the actual plastic zone for energy density.The size and shape of normalized plastic zone are related to two angles, θ and ψ under mixed-mode loading of CTS specimen. In Refs. [25,3], the actual size and shape of the plastic zones are closely interrelated to the square of the ratios (σapp/ σM and σapp/ σH ). However, the actual size and shape of the plastic zone for energy density depend on two factors, one of which is the initial crack length a and the other one is the ratio of the remote applied energy to near SED surrounding crack tip. The size and shape of the normalized plastic zone for energy density will 2 2 /2E )/(dW / dV ) when the crack length gradually increases. /2E )/(dW / dV )d and (σapp change based on the ratios of (σapp

3. Experiments and simulations 3.1. Uniaxial tensile tests and simulations The nominal composition of the aluminum alloy 7050-T7451 is listed in Table 1. The uniaxial tensile specimens with gauge length 50 mm and thickness 2.5 mm at 0°, 45°, and 90° directions were performed using Instron 3382 testing equipment. The uniaxial tensile experiments with DIC system in three directions (0°, 45°, and 90°) were conducted to obtain the plastic strain ratios, r0 , r45 , and r90 . The thickness of the uniaxial tensile specimen is consistent with the thickness of fatigue crack growth tests. The yield strengths of the aluminum alloy 7050-T7451 at 0°, 45°, and 90° directions are 429 MPa, 409 MPa, and 410 MPa, respectively. To verify the significance of these parameters values, the finite element simulations of uniaxial tensile specimens at 0°, 45°, and 90° directions were performed with ABAQUS/Explicit software. The Hill's criterion was embedded in the user material subroutine (VUMAT). The uniaxial tensile specimens were meshed using reduced-integration eight-node solid elements (C3D8R). In this simulation, a linear combination [26] between Swift and Voce constitutive equations was used in these models, and the special form is expressed as

Eεe ,forσ < σs σ=⎧ q (K (εp + ε0)n) + (1−q)(σsa−Q exp(−βεp)),forσ > σs ⎨ ⎩

(18)

where εe is elastic strain, σs is yield stress, q is weight factor, K is strength coefficient, εp is plastic strain, ε0 is pre-strain, n is strain hardening index, σsa is saturated stress, Q , and β are material coefficients for Voce Equation. These hardening parameters are shown in Table 2 and q is variable so that the stress-strain curve in larger deformation is accurate. The comparisons of engineering stressstrain curves at 0°, 45°, and 90° directions between experiment and simulation were shown in Fig. 2. Then, the simulation results are in accordance with experimental results at 0°, 45°, and 90° directions, which verify the parameters of Hill's criterion.

3.2. Fatigue crack growth tests The fatigue crack growth tests of CTS specimens for anisotropic aluminum alloy 7050-T7451 were conducted on Instron 8801 (dynamic fatigue tensile machine) with a frequency of 10 Hz under constant amplitude loading. Moreover, the thickness of all specimens is 2.5 mm. The load ratio of minimum stress and maximum stress, R = σmin/ σmax is 0.1. The CTS specimens (see Fig. 3(a)) in the loading device (see Fig. 3(b)) were applied to carry out different loading modes (mode-I, mixed-mode, and mode-II) for fracture and fatigue experiments. Meanwhile, the mode-I, mode-II, and mixed-mode loadings are realized by changing the loading angle ψ which is between initial notch direction and loading direction. During pre-cracking, decreasing the load range is not more than 10% to reach the SIF range. The crack grows to a length several times at every step to minimize possible effect of residual stresses. Besides the crack length was measured by a digital microscope, as shown in Fig. 4(a). The initial notch depth of CTS specimens is 23 mm, and the notches were introduced to the specimen by wire-electrode cutting machine, as shown in Fig. 3(a). The loading ranges are 470–4700 N for CTS specimens. The extended crack length calculation [27] was used to calculate the crack growth length. Tanka [28] expressed the fatigue crack propagation rate (da/dNu) by using equivalent stress intensity factor K eq in Paris Equation:

da = C Δ(K eq)m dNu

(19)

where C and m are material constants that were fitted by experimental data, and ΔK eq [28] is expressed by

ΔK eq =

4

ΔKI4 + 8ΔKII4

(20)

The crack growth rates were calculated by da/dNu = (a2−a1)/(Nu2−Nu1) , and the data was not further manipulated to make a smoothing. Table 1 Nominal chemical compositions of aluminum alloy 7050-T7451 (wt.%). Zn

Mg

Cu

Zr

Si

Fe

Mn

Cr

Al

6.02

2.31

2.04

0.10

0.03

0.11

0.003

0.009

Balance

5

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Table 2 The hardening parameters of AA7050-T7451.

AA7050-T7451

K (MPa)

n

ε0

σsa (MPa)

Q (MPa)

β

672

0.09118

0.00423

530

112.7

39.3

Fig. 2. Comparisons of engineering stress-strain curves of aluminum alloy 7050-T7451 at 0°, 45°, and 90° directions.

3.3. Digital image correlation tests DIC is a contactless optical strain measuring approach to extract the whole-field strain data by calculating local strain on surface of specimen. The local strain was realized by measuring the displacement of scattered spots in a pair of digital images regarding specimen surface. The CTS specimens were prepared to spray black speckles on the white paint of the surface evenly, and then provide a random speckle for the DIC program to perform. The DIC system provides a method to measure strain in local field of specimen surface, so the strain surrounding crack tip under fatigue loading can be measured by the method. Images obtained from two CCD cameras with 6 megapixel (2736 × 2192) resolutions were processed by using XJTUDIC_VS software. The subset is a square element and the size of the element edge is (6 × 2 + 1) pixel. The strain filter uses 3 × 3 square elements and then calculates the strain value of the center point to reduce noise. Fig. 4(b) shows the fatigue crack growth tests with DIC device. The specimens and testing parameters are the same to those of the fatigue crack growth tests with a digital microscope (see Fig. 4(a)). 4. Results and discussion 4.1. Analysis of normalized plastic zones at crack tip Fig. 5 shows the Hill's normalized plastic zones with F = 1/2 G = 1/2, H = 1/2, and N = 3/2 under mode-I and mixed-mode loadings. In this case, the Hill's normalized plastic zones are equivalent to that of Mises. As the assumption of material orientation for Mises criterion is isotropic, there are some unavoidable predictive errors for anisotropic materials. Figs. 6–9 show the effects of the parameters (F, G, H, and N) of Hill's criterion on normalized plastic zones. F and H have a similar effect on the size of Hill's normalized 6

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Fig. 3. Crack growth experiment scheme: (a) CTS specimen; (b) schematic representation of loading device (All units are in [mm]).

Fig. 4. (a) Crack growth rate testing device and (b) fatigue crack growth testing with DIC devices for the CTS specimen.

plastic zone rather than the shape. Also, the two parameters reflect the weights of σy and σx −σy on Hill equivalent stress. H has a more significant effect than F on deflection angle of the normalized plastic zone and the deflection angle between loading angles 0° and 15° is larger. That means the weight changes easily induce the variation of size for Hill's normalized plastic zone at crack tip. G and N have a similar effect on the shape of Hill's normalized plastic zone. The Hill's normalized plastic zone is a heart shape under the effect of G. Besides, the radii along the rolling direction under the loading angles of 0° and 15° are longer than the other normalized plastic zones. G and N reflect that the weights of the σx and τxy on Hill equivalent stress. That means the weight changes easily induce the variation of shape for normalized plastic zone, which explains that the radius along rolling direction is longer. In addition, the Hill's normalized plastic zones with the effect of N exhibit different shapes under the loading angle of 45° and 60°, reflecting that shear stress is dominated in the shear-dominated loading angles. Fig. 10 shows the normalized plastic zones with distortional and total SED under mode-I and mixed-mode loadings. As can be observed from Fig. 10, the size and shape of normalized plastic zones for energy density change with the variation of loading angle. The normalized plastic zones with distortional SED are always less than those with total SED, which reflects the effects of distortional SED and total SED on normalized plastic zones are different. Due to the distortional SED is related to the Mises yield criterion and 7

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Fig. 5. Hill's normalized plastic zone: F = 1/2, G = 1/2, H = 1/2, N = 3/2 (Mises criterion) under mode-I and mixed-mode loadings of CTS specimen.

Fig. 6. The effect of F on Hill's normalized plastic zone under mode-I and mixed mode loadings of CTS specimen.

Fig. 7. The effect of G on Hill's normalized plastic zone under mode-I and mixed-mode loadings for CTS specimen.

total SED is related to the S-criterion, the shapes of normalized plastic zones for the two criteria are associated with the yielding and fracture at crack tip under mixed-mode loading for CTS specimen. To predict crack initiation angles under mode-I and mixed-mode loadings, it is significant to find the relation between the crack 8

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Fig. 8. The effect of H on Hill's normalized plastic zone under mode-I and mixed-mode loadings for CTS specimen.

Fig. 9. The effect of N on Hill's normalized plastic zone under mode-I and mixed-mode loading for CTS specimen.

Fig. 10. The normalized plastic zones with distortional SED and total SED under mode-I and mixed-mode loading for CTS specimen.

initiation angle and loading angle for the CTS specimen. The maximum tangential stress (MTS) criterion [29] states that the crack initiation angle between the initial crack direction and the direction of the maximum tangential stress along a constant radius around the crack tip. The MTS criterion is expressed as

θ 1 K θ 1 tan2 − ⎛ I ⎞ tan − = 0, 2 2 ⎝ KII ⎠ 2 2 ⎜

⎟

(21) 9

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3 1 θ θ θ K θ 7 θ θ − ⎡ ⎛ cos3 −cos sin2 ⎞ + ⎛ II ⎞ ⎛sin3 − sin cos3 ⎞ ⎤ < 0 2⎢ 2 2 2 ⎠ ⎝ KI ⎠ ⎝ 2 2 2 2 ⎠⎥ ⎦ ⎣⎝ 2 ⎜

⎟

(22)

The minimum strain energy density criterion (S-criterion) [30] states that the initiation angle between the initial crack direction and the direction of minimum SED along a constant radius around the crack tip. The S-criterion is expressed as 2

2

2

2

⎡2(1 + κ ) KI ⎤ tan4 θ + ⎡2κ ⎜⎛1−⎛ KI ⎞ ⎟⎞−2 ⎛ KI ⎞ + 10⎤ tan3 θ −24 ⎛ KI ⎞ tan2 θ + ⎡2κ ⎜⎛1−⎛ KI ⎞ ⎟⎞ + 6 ⎛ KI ⎞ −14⎤ tan θ + 2(3−κ ) = 0, ⎢ ⎥ ⎢ ⎥ ⎢ KII ⎥ K K K 2 2 2 2 ⎣ ⎦ ⎝ KII ⎠ ⎝ KII ⎠ ⎣ ⎝ ⎝ II ⎠ ⎠ ⎝ II ⎠ ⎦ ⎣ ⎝ ⎝ II ⎠ ⎠ ⎦ ⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

(23) 2

⎡2(κ−1) KI ⎤ sinθ−8 ⎛ KI ⎞ sin2θ + ⎡(κ−1) ⎜⎛1−⎛ KI ⎞ ⎢ ⎢ ⎥ KII ⎦ ⎣ ⎝ KII ⎠ ⎝ ⎝ KII ⎠ ⎣ ⎜

⎟

⎜

⎟

2

⎞ ⎤ cosθ + ⎡2 ⎛ KI ⎞ −3⎤ cos2θ > 0, ⎟⎥ ⎢ KII ⎥ ⎠⎦ ⎣ ⎝ ⎠ ⎦ ⎜

⎟

(24)

where κ is a material constant related to stress states, and it can be expressed as

κ=

(3−ν ) for plane stress, (1 + ν )

(25)

where ν is Poisson’s ratio. In this paper the plane stress state is performed as the thickness of the CTS specimen is very thin, and ν is 0.33 for aluminum alloy 7050-T7451. The anisotropic R-criterion [3] states that the crack initiation angle is between the initial crack direction and the direction of the minimum radius of the Hill's normalized plastic zone, and it is expressed as:

θ θ θ θ θ θ θ θ θ ω10tan10 ⎛ ⎞ + ω9tan9 ⎛ ⎞ + ω8tan8 ⎛ ⎞ + ω7tan7 ⎛ ⎞ + ω6tan6 ⎛ ⎞ + ω5tan5 ⎛ ⎞ + ω4 tan4 ⎛ ⎞ + ω3tan3 ⎛ ⎞ + ω2tan2 ⎛ ⎞ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ θ + ω1tan ⎛ ⎞ + ω0 = 0 (26) ⎝2⎠

w1cos(θ) + w2cos(2θ) + w3cos(3θ) + w4 cos(5θ) + w5sin(θ) + w6sin(2θ) + w7sin(3θ) + w8sin(5θ) = 0

(27)

where ωi (i = 1–10) and wi (i = 1–10) are function of Hill's criterion parameters and KI/ KII . The expressions of ωi (i = 1–10) and ωi (i = 1–10) are given in Ref. [3]. The curves of theoretic prediction for crack initiation angle are illustrated in Fig. 11. Crack propagation directions under mode-I and mixed-mode loadings depend on the ratio of KI/ KII which is a function of loading angle ψ and crack length a. The ratio of KI/ KII is different under different loading conditions, such as uniaxial tension, pure shear, uniaxial compression, and proportional tension torsion loading. However, this paper emphasizes the mode-I loading and mixed-mode loading in CTS specimen. The crack initiation angles (θ ) at crack tip under the loading angles of ψ= 0°, 15°, 45°, and 60° are 3.6°, 14.7°, 41.6°, and 53°, respectively. Fig. 11 shows the comparison between the predictions of three criteria and the experimental data. The prediction accuracy of anisotropic R-criterion is the best and the S-criterion takes the second place. Therefore, the anisotropic R-criterion is suitable for predicting the crack initiation angle of CTS specimen under mode-I and mixed mode-loading. Fig. 12 shows the comparisons between DIC contour maps and Hill's normalized plastic zones of CTS specimen at the crack initiation stage under mode-I and mixed-mode loadings. These major strains were all measured at the time of Kmax . With the increase of the loading angle the major strain at crack tip increases, which indicates that the lager shear deformation, the more of strain

Fig. 11. Comparison between theoretical prediction curves of three criteria and experimental data. 10

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Fig. 12. Comparison between DIC contour maps at crack tip and Hill's normalized plastic zones at loading angles of (a) 0°, (b) 15°, (c) 45°, and (d) 60°.

around the crack tip is. Although the major strain under mode-I loading is less than that under mixed-mode loadings, the high strain region under the mode-I loading is large and symmetric. Due to the effect of the rolling process, the high major strain region is long along the direction of crack tip propagation under the mode-I loading, which is in accord with the predictive Hill's normalized plastic zone. Under the mixed-mode loading the high strain region located at crack tip shrinks with the increase of loading angles, which is consistent with the predictions. 4.2. Analysis of crack growth rate Fatigue crack growth tests of CTS specimens for AA7050-T7451 were performed under loading angles of ψ = 0°, 15°, 45°, and 60°. Fig. 13 shows the relation between extended crack length (a) and the number of fatigue cycles (Nu) under different loading angles. As can be observed from Fig. 13, the fatigue cycle number increases and the steady propagation stage becomes longer with the increase of loading angle. It can be explained by the Hill's normalized plastic zones of AA7050-T7451, as shown in Fig. 12. The crack growth length decreases with the increase of the ratio of Mode-II to the Mode-I at initial slow propagation stage, as known from the Hill's normalized plastic zone. Since the crack growth length under shear-dominated loading is less than that under tension-dominated loading, the fatigue cycle number under shear-dominated loading is more. To analyze the crack growth rate under mode-I and mixed-mode loadings, the relations between da/dNu and ΔK eq under different loading angles were plotted in Fig. 14. The relations were converted by the data points in the Fig. 13 using extended crack length calculation [27]. According to the fitting result, the crack growth rate of the steady crack propagation stage at ψ 2.17 = 0° was determined as da/dN u = 2.69 × 10−7ΔK eq , and it is slight larger than that of the hole specimen geometry (da/dN u = 3 ) [31]. For the difference of the da/dNu between loading angles, da/dNu under loading angles of ψ = 0° and 15° 3.47× 10−8ΔK eq

Fig. 13. Crack lengths versus load cycles with different loading angles. 11

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Fig. 14. Fatigue crack growth rate versus equivalent stress intensity factor under different loading angles.

are higher than those under loading angles of ψ = 45° and 60° at crack initiation stage. It reflects that the ratio of tensile stress to shear stress is directly proportional to the crack growth rate. 4.3. Fatigue fracture appearances of crack initiation Fig. 15 shows the scanning electron microscope (SEM) images under the loading angles of ψ = 0°, 15°, 45°, and 60° at crack initiation stage. Fig. 16 shows the enlarged SEM images of Fig. 15 under the four different loading angles and the SEM scale is 20 μm. The fatigue fractured surfaces present many small striations rather than large and continuous striations at loading angles of 0° and 15°. The crack tip continuously blunts and sharps, which are responsible for the existing of striations. The grains in the anisotropic plate along rolling direction have been elongated, so small striations exist in the lathy regions. Some small shear planes exist on fractured surfaces under shear-dominated loadings (loading angles of 45° and 60°). Moreover, the fractured surfaces of the sheardominated loadings are rough and cover many shear planes. As can be observed from the enlarged SEM images in Fig. 16, a large number of particles are existed under the loading angle of 0°. Besides, the shear plane is large and smooth under the loading angle of

Fig. 15. The SEM images of fractured surface at crack initiation stages at loading angles of (a) 0°, (b) 15°, (c) 45°, and (d) 60° (SEM scale is 100 μm). 12

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Fig. 16. The SEM images of more amplification times at crack initiation stages at loading angles of (a) 0°, (b) 15°, (c) 45°, and (d) 60° (SEM scale is 20 μm).

60°. It proves that the strain at crack tip increases with the increase of loading angle, as shown in Fig. 12. The relative smooth fracture surfaces under tension-dominated loadings and the rougher fracture surfaces under shear-dominated loadings at initiation stage reflect that the rate of released energy is different under the two loadings. 5. Conclusions Fatigue crack growth behaviors of anisotropic aluminum alloy 7050-T7451 on plane stress condition were studied. The following conclusions can be obtained from the results of the study: (1) F and H have a significant effect on size of Hill's normalized plastic zones. Additionally, G and N have a significant effect on shape of Hill's normalized plastic zones. (2) According to the comparison of initiation angles between experimental data and theoretical prediction curves of three criteria, which concludes that the prediction accuracy of anisotropic R-criterion is the best and the prediction accuracy of S-criterion is the second. (3) The DIC contours at crack tip are consistent with the Hill's normalized plastic zones of AA7050-T7451 under mode-I and mixedmode loading. In addition, the strain at crack tip increases with the increase of loading angle, which can be proved from the fracture appearances. (4) The crack growth lengths of Hill's normalized plastic zones with total SED exhibit a decreasing tendency with the increase of loading angle at the initial stage, which proves that the fatigue crack growth life decreases with the increase of loading angle. Acknowledgements The authors are very grateful for the support received from the National Natural Science Foundation of China (Grant No. 51275414), the Fundamental Research Funds for the Central Universities with Grant No. 3102015BJ (II) ZS007 and the Research Fund of the State Key Laboratory of Solidification Processing (NWPU), China (Grant No. 130-QP-2015). Appendix A Substituting Eq. (4) into Eq. (1), then

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σ app a fx (θ,fKi )

σx =

(A.1)

2r

σ app a f y (θ,fKi )

σy =

(A.2)

2r σ app a fxy (θ,fKi )

τxy =

(A.3)

2r

where fx (θ,fKi ) , f y (θ,fKi ) , and fxy (θ,fKi ) are defined as:

θ θ 3θ θ θ 3θ fx (θ,fKi ) = ⎧fKI (ψ)cos ⎛1−sin sin ⎞ ⎫−⎧fKII (ψ)sin ⎛2 + cos cos ⎞ ⎫ ⎨ 2⎝ 2 2 ⎠⎬ 2⎝ 2 2 ⎠⎬ ⎩ ⎭ ⎨ ⎩ ⎭

{

θ θ 3θ θ θ 3θ fx (θ,fKi ) = ⎧fKI (ψ)cos ⎛1 + sin sin ⎞ ⎫ + fKII (ψ)sin cos cos ⎨ 2⎝ 2 2 ⎠⎬ 2 2 2 ⎩ ⎭

{

θ θ 3θ fxy (θ,fKi ) = fKI (ψ)cos sin cos 2 2 2

}

}

θ θ 3θ + ⎧fKII (ψ)cos ⎛1−sin sin ⎞ ⎫ ⎨ 2⎝ 2 2 ⎠⎬ ⎩ ⎭

(A.4)

(A.5)

(A.6)

Here, we use the distortional SED and total SED to determine the variable radii of the normalized plastic zones for energy density at crack tip, the distortional SED and total SED [32] are defined on the plane stress state as: 2 ⎛ dW ⎞ = ⎛ 1 + ν ⎞ [σx2 + σy2−σx σy + 3τxy ] ⎝ dV ⎠d ⎝ 3E ⎠

(A.7)

dW 1+ν ⎡ 2 ν ⎞ 2⎤ ⎞ σx + σy2−⎛ (σx + σy )2 + 2τxy =⎛ ⎥ dV 1 ν⎠ + ⎝ 2E ⎠ ⎢ ⎝ ⎦ ⎣

(A.8)

Substituting Eqs. (A.1)–(A.3) and Eqs. (A.4)–(A.6) into Eqs. (A.7) and (A.8), respectively, then the radii rd and rS of normalized plastic zone for mixed mode loading are obtained

rd

rd =

2 / 2E σapp

a ⎡ (dW / dV ) ⎤ d ⎣ ⎦

rS

rS =

2 / 2E σapp

a ⎡ (dW / dV ) ⎤ ⎣ ⎦

= g′1 f K2I (ψ) + g′2 f K2II (ψ) + g′12 fKI (ψ) fKII (ψ) (A.9)

= a1 f K2I (ψ) + a2 f K2II (ψ) + a12 fKI (ψ) fKII (ψ) (A.10)

for plane stress. where g′1, g′2 and g′12 are functions of the angle θ , a1, a2 and a12 are functions of the angle θ and ν , defined as:

g′1 = (1 + ν )

[7 + 4cos(θ)−3cos(2θ)] 24

(A.11)

g′2 = (1 + ν )

[19−4cos(θ ) + 9cos(2θ)] 24

(A.12)

[−2sin(θ) + 3sin(2θ)] 6

(A.13)

g′12 = (1 + ν )

a1 =

1+ν [(1 + cosθ )(κ−cosθ)] 4

(A.14)

a2 =

1+ν [(κ + 1)(1−cosθ) + (1 + cosθ)(3cosθ−1)] 4

(A.15)

a12 =

κ=

1+ν sinθ [2cosθ−(κ−1)] 2

(A.16)

3−ν for plane stress. 1+ν

(A.17)

where κ is a constant depending upon stress states.

References [1] Gdoutos EE, Meletis DAZI. Mixed-mode crack growth in anisotropic media. Eng Fract Mech 1989;34:337–46.

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