An analytical method to predict residual strength based on critical CTOA

Engineering Fracture Mechanics 200 (2018) 31–41

Contents lists available at ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

An analytical method to predict residual strength based on critical CTOA Longkun Lu, Shengnan Wang

T

⁎

School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

A R T IC LE I N F O

ABS TRA CT

Keywords: CTOA Residual strength Stiffened panel Crack growth resistance

An analytical method to predict residual strength based on critical crack tip opening angle (CTOA) is proposed in this paper. According to a previous study, the plastic wake profile and then the stationary crack opening profile can be considered as a kind of crack growth resistance. The core idea of the proposed method is transferring critical CTOA to stationary crack opening profile based crack growth resistance curve. M(T) and C(T) specimens are taken to validate this method. All of these predicted results have an accuracy of 4%. In addition, the proposed method has successfully predicted the residual strength of rivet stiffened panels (the accuracy is 5%) by assuming that stiffeners are only geometry constraints of simple specimens that don’t affect the crack growth resistance. It is worth stressing that no finite element analysis is used in this paper.

1. Introduction Stable crack propagation is a common phenomenon for ductile materials, especially for thin aluminum alloy sheets. These ductile materials are also called elastic plastic materials. When cracks grow in elastic plastic materials, plastic unloading always occurs and will invalidate the deformation theory of plasticity. In this case, J integral will cease to be valid [1]. Thus, another proper crack growth criterion is required. As summarized by Newman et al. [2], the concept crack tip opening angle (CTOA) or displacement (CTOD) at a specified distance from crack tip seems to be most suitable for modeling propagating cracks in elastic plastic materials. In the CTOA method, cracks will grow when a properly defined CTOA value reaches the critical value. And a lot of experiments and finite element analyses have shown that CTOA is nearly a constant during stable crack growth [3–7]. However, the critical CTOA depends on specimen’s thicknesses [8–10], it was found to be independent of in-plane geometry if the crack length and un-cracked ligament size are larger than 4 times thickness [11]. Due to its advantage of transferability, the CTOA concept was widely used to predict stable crack growth of low constraint structures, such as stiffened and unstiffened with or without multiple sites damage during the past three decades [12–20]. In consideration of the great success achieved by the CTOA concept, NASA has already included the CTOA criterion in the STAG shell code [21]. In addition, two standard procedures to measure critical CTOA are established [22,23]. On the basis of this standard, some simple methods to measure critical CTOA are proposed [24–26], which have also prompted the development of the CTOA approach. In the past, the CTOA concept has been extensively used with the elastic plastic finite element analysis. In many cases, the elastic plastic finite element method is time-consuming and requires vast experience. To improve this situation, a simple but reliable engineering method is required to allow the CTOA based fracture criterion applicable to common engineering fracture control issues. This is the purpose of this paper.

⁎

Corresponding author. E-mail address: [email protected] (S. Wang).

https://doi.org/10.1016/j.engfracmech.2018.07.023 Received 15 March 2018; Received in revised form 26 May 2018; Accepted 14 July 2018 0013-7944/ © 2018 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 200 (2018) 31–41

L. Lu, S. Wang

Nomenclature a a0 ae B Bs d C1, C2 E Es f fstif F K Kstif rp W

Ws x, ( x1, x2 ) β Δa δa δH δR

length of crack, mm initial length of crack, mm effective crack length, mm specimen thickness, mm stiffener thickness, mm distance behind the crack tip, mm Material constant Young’s modulus of panel material, MPa Young’s modulus of stiffener material, MPa correction function of specimen boundary correction factor of stiffener applied load, KN stress intensity factor, MPa mm stress intensity factor of stiffener panel, MPa mm Irwin plastic zone size, mm specimen or panel width, mm

δt ρ σ σys ψc CTOA CTOD

stiffener width, mm Cartesian coordinates stiffness ratio between stiffener and panel crack extension, mm the advancing crack opening profile, mm the plastic wake profile, mm resistance to crack extension in terms of crack tip opening profile, mm the stationary crack opening profile, mm a correction length similar to the plastic zone size, mm applied stress, MPa yield stress (0.2% offset), MPa critical crack tip opening angle, degrees crack tip opening angle crack tip opening displacement

A stable crack growth model was proposed in a previous paper [27]. This model regards growing crack opening profile as corresponding stationary crack opening profile minus plastic wake profile. According to dimensional analysis, the plastic wake profile was shown to be independent of in-plane geometry for fixed material and stress state. Combined with the transferability of CTOA, the critical stationary crack opening profile can be considered as a kind of crack growth resistance and thus stable crack growth can be predicted by making stationary crack opening profile equal to this crack growth resistance. In this paper, a novel method to apply critical CTOA to predict the stable crack growth is proposed on the basis of the abovementioned model. The core idea is transferring critical CTOA to stationary crack opening profile-based crack growth resistance curve. In Section 2, the method is presented and simple specimens are used to validate this method. In Section 3, extra boundaries of thinned wall structures are considered as geometry constraints of simple specimens, which do not affect the crack growth resistance. On the foundation of this assumption, the method proposed in Section 2 was found to be appropriate to predict the residual strength of stiffened panels.

2. Method A stable crack growth model, which considers growing crack opening profile in elastic plastic materials as the corresponding stationary crack opening profile minus the plastic wake profile, was proposed in a previous study [27]. The method presented here is based on this model. To make the statement clear, the model is provided in Fig. 1, but the detailed process is omitted here.The stationary and advancing crack opening profiles are denoted as δt and δa respectively, and δH is introduced as the height of the plastic wake. The crack opening angle is defined by the crack opening at a small distance d behind the crack tip, and ψc is the critical CTOA.In addition, δ (x ;a) is introduced as the crack opening profile, the first parameter in the bracket is the coordinate where the crack opening is evaluated, and the second is the current crack length. The crack growth equation for this model is

⎧ δa (a−d;a) = ψc d δ (a−d;a) = δa (a−d;a) + δH (a−d;a) ⎨ t ⎩ δt (a;a) = δH (a;a)

(1)

Since the elastic deformation in plastic wake is much less than the plastic stretch,

Stationary crack profile Plastic wake profile x2

o

x1

Ǔc

Advancing crack profile

d a0

Ʃa

Fig. 1. The stable crack growth model. 32

Engineering Fracture Mechanics 200 (2018) 31–41

L. Lu, S. Wang

δH (a−d;a) = δH (a−d;a−d )

(2)

And then, Eq. (1) can be rewritten as

⎧ δt (a−d;a) = ψc d + δt (a−d;a−d ) δ (Δa) = δt (a;a) ⎨ R ⎩ a = a0 + Δa

(3)

As shown by the previous study [27], δR Δa curves can also be regarded as a kind of crack growth resistance curve. And then the residual strength can be predicted by

δt (a−d;a) = ψc d + δR (Δa−d )

(4)

Eq. (4) is suitable to both brittle and ductile fracture. For brittle fracture, the height of the plastic wake is zero and this method degenerates into K factor method because the stationary crack opening profile can be represented by K factor. For ductile fracture, a lot of experiments [3–11] have already revealed that critical CTOA has transferability and the previous study [27] has proven by dimensional analysis that for a given material and stress state, the δR Δa curve is independent of specimen geometry if the plastic zone stress distribution is invariable (This condition is satisfied for low constraint specimens). The right-hand-side of Eq. (4) becomes the stationary crack opening profile-based crack growth resistance curve. In addition, no plastic unloading occurs for stationary cracks, and then the stationary crack profile can be obtained by deformation plasticity (J integral) or K factor. M(T) and C(T) specimens are recommended by ASTM E2472 [22] to measure critical CTOA. Therefore, three groups of M(T) and C(T) specimens are used to illustrate the residual strength prediction procedure. Table 1 has summarized the detail information. All of these specimens are tested in L-T orientation at room temperature under buckling constraints. In Table 1, B is the thickness, W is the total width, a 0 is the initial crack length and the total crack length of M(T) is 2a 0 . E is the Young’s modulus, σys is the yielding strength. The stationary crack opening profile of M(T) specimen is

( )

⎧ δt (x ;ae ) = 4σ ae2−x 2 sec πae E W ⎨ a = a + r = a + Δa + r e p p 0 ⎩

⎧ rp = 12 ( σσ )2asec ys ⎨ a = a + Δa 0 ⎩

(5)

( ) πa W

(6)

where σ is the applied stress, ae is the effective crack length. The stationary crack opening profile of C(T) specimen is 8K (a )

⎧ δt (x ;ae ) = E e ae2π− x ⎪ ae = a + rp = a0 + Δa + rp ⎨ 2 ⎪ rp = 1 ⎡ K (a) ⎤ 2π ⎣ σys ⎦ ⎩

(7)

Table 1 Specimen information. Material

Type

B, mm

W, mm

a0 , mm

ψc

E, MPa

σys, MPa

Reference

2024-T3 2024-T3 2024-T3 2024-T3 2324-T39 2324-T39 2324-T39 2024-T3 2024-T3 2024-T3 2024-T3 2024-T3 2024-T3 2024-T3 2024-T3

M(T) M(T) M(T) C(T) M(T) M(T) C(T) M(T) M(T) M(T) M(T) M(T) C(T) C(T) C(T)

1.02 1.02 1.02 1.02 7.6 7.6 7.6 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3

450 450 450 200 102 305 152 76.2 305 305 305 610 50.8 102 152

60 70 85 100 16.5 59.7 62 12.7 50.8 63.5 76.2 101.7 20.3 40.8 61

6.0° 6.0° 6.0° 6.0° 4.9° 4.9° 4.9° 5.25° 5.25° 5.25° 5.25° 5.25° 5.25° 5.25° 5.25°

68,000 68,000 68,000 68,000 72,000 72,000 72,000 71,400 71,400 71,400 71,400 71,400 71,400 71,400 71,400

294 294 294 294 455 455 455 345 345 345 345 345 345 345 345

[20] [20] [20] [20] [7] [7] [7] [14] [14] [14] [14] [14] [14] [14] [14]

33

Engineering Fracture Mechanics 200 (2018) 31–41

L. Lu, S. Wang

( )

2+a/W a ⎧ K (a) = F f W B W (1 − a / W )3/2 ⎪ ⎨ f a = ⎡0.886 + 4.64 a −13.32 ⎪ W W ⎣ ⎩

( )

a 2 W

( )

+ 14.72

a 3 W

a 4 W

( ) −5.6 ( ) ⎤⎦

(8)

where F is the applied load. For the stationary crack in elastic plastic materials, a displacement will generate at the crack tip when the external loads apply. And then the effective crack is larger than the true crack length. In Eqs. (5) to (8), we obtain the effective crack length just by adding Irwin plastic zone size to the true crack length. However, this handling method might not be right when the strain hardening effect cannot be ignored. Fig. 2 has presented the generalized stationary crack opening profile. In this figure, ae = a + ρ and ρ is a corrected length. In general, the corrected length ρ is dependent on the applied load, material properties and the specimen geometry (crack length and specimen width). For a fixed material, we can use Young’s modulus and yield strength to represent all material properties. According to dimensional analysis [1],

ρ σ σ a⎞ , = h ⎛⎜ , ⎟ a E σys W ⎠ ⎝

(9)

Therefore, for the case Irwin plastic zone is not applicable, the expression of Eq. (9) needs to be found firstly. A basic approach is provided as follows: Young’s modulus can be omitted because the effective crack length is usually related to plastic deformation around the crack tip. In this case, Eq. (9) turns into

ρ σ a⎞ = h ⎛⎜ , ⎟ a σ ⎝ ys W ⎠

(10)

In Eq. (10), the yielding strength can be replaced with flow stress strength if the strain hardening effect cannot be ignored. In addition, an expression similar to Irwin’s formula is recommended: C2

ρ σ a = C1 ⎛⎜ ⎞⎟ f ⎛ ⎞ a σ W ⎝ ⎠ ys ⎝ ⎠

(11)

C1 and C2 can be assumed to be the material constants, which are dependent on crack tip stress state, because the deformations at the a crack tip should be the same when crack grows for a fixed stress state. f W may be the boundary correction factors in stress intensity factor solutions since finite width plays the same role in effective crack length. In the following, the first group of M(T) and C(T) specimens [20] is taken to verify the transferability of δR Δa curves. The test load-crack extension data is shown in Fig. 3. The δR Δa curve can be calculated as follows:

( )

⎧ δR (Δa) = δt (a;ae ) ⎨ ⎩ a = a0 + Δa

(12)

By substituting the test data into Eq. (12), the δR Δa curve can be obtained. The results are presented in Fig. 4. As shown in Fig. 4, all of δR Δa curves from M(T) specimens coincide with each other, in spite that some discrepancies exist between M(T) and C(T), the fitting curve C(T) is identical to that of M(T). Thus, the δR Δa curve is independent of specimen geometries. Therefore, the right hand side of Eq. (4), i.e., the resistance curve, is determined. Combined Eqs. (5) to (8) with Eq. (4), the load-crack extension curves of these specimens can be predicted. Fig. 5 has provided the specific procedure. In some cases, the iteration process shown in Fig. 5 may be divergent. This phenomenon means that the Irwin plastic zone expression requires corrections on the basis of Eq. (11). And a tested specimen should be used to determine Eq. (11) firstly. For the first group of M(T) and C(T), the Irwin plastic zone expression is used. In this example, d = 0.1 mm . The initial corrected

x2 x1

t ( x ; ae )

x

a a0

a

ae Fig. 2. The stationary crack opening profile. 34

Engineering Fracture Mechanics 200 (2018) 31–41

L. Lu, S. Wang 250

(a)

(b)

4000

CT

200

C(T) specimen B=1.02mm W=200mm a0=100mm

3000

MT-60 MT-70 MT-85

M(T) specimen MT-60:a0=60mm B=1.02mm W=450mm MT-70:a0=70mm E=68000MPa MT-85:a0=85mm ıys=294MPa

100

50

0

F(N)

ı(MPa)

150

0

2

4

6

8

10

12

14

2000

E=68000MPa ıys=294MPa

1000

0

16

Ʃa(mm)

0

5

10

15

20

25

30

Ʃa(mm)

Fig. 3. The test load-crack extension data (a) M(T) and (b) C(T). 1.6

MT-85 MT-70 MT-60 CT fitting

įR(mm)

1.2

0.8

fitting curve: R

0.4

0.0

0

5

( a) 0.2726( a)0.515

10

15

20

25

30

Ʃa(mm) Fig. 4. The δR Δa curve of the first group.

Input

R

R

- a curve, c , d

cd

R

( a d)

Give an initial value of rp .

Replace rp by rp( ) ;

ae a rp ; Determine t (a;ae ) ; Obtain

a curve from R

t

(a;ae );

Get the value of rp( ) ;

rp(

N

)

rp

error

rp Y Output

a curve;

Fig. 5. The residual strength prediction procedure.

length is taken as 0.1a 0 . Figs. 6 and 7 have shown the comparison results. From Fig. 6 3% accuracy is observed between predicted curves and test data. This result means the residual strength can be predicted by critical CTOA on the basis of the flow diagram in Fig. 5. In addition, this result also means the stationary crack opening 35

Engineering Fracture Mechanics 200 (2018) 31–41

L. Lu, S. Wang 250

M(T) B=1.02mm W=450mm

200

test data-MT-60 predicted curve test data-MT-70 predicted curve test data-MT-85 predicted curve

ı(MPa)

150

E=68000MPa ıys=294MPa

100

MT-60:a0=60mm MT-70:a0=70mm

50

0

MT-85:a0=85mm

0

2

4

6

8

10

12

14

16

Ʃa(mm)

Fig. 6. Comparison between predicted result and test data of M(T) specimens. 4500

test data predicted result 3600

C(T) B=1.02mm W=200mm a0=100mm

F(N)

2700

1800

E=68000MPa ıys=294MPa 900

0

0

5

10

15

20

25

30

Ʃa(mm) Fig. 7. Comparison between predicted result and test data of C(T) specimen.

profile of M(T) can be precisely expressed by Eqs. (5) and (6). From Fig. 7, the predicted curve can capture the test data very well up to the maximum load point, and as the crack continues to grow, the predicted curve will underestimate the test applied load. This phenomenon can be explained as follows: The carrying capacity of C(T) will decrease when the plastic zone reaches its back boundary. That means when the maximum load is observed, the plastic zone size formula cannot be used to correct the crack length anymore. The plastic zone formula will underestimate the corrected length and then will underestimate the applied load. The predicted results of the second group [7] are shown in Figs. 8–10. It is noted that Irwin plastic zone size is also used in this example and only 4% accuracy is observed between predicted curves and 0.7

test data fitting

0.6

įR(mm)

0.5

Fitting curve: įR=0.1291(Ʃa)0.5

0.4

M(T) B=7.6mm W=305mm a0=59.7mm

0.3 0.2

E=72000MPa ıys=455MPa

0.1 0.0

0

4

8

12

Ʃa(mm)

16

20

Fig. 8. The δR Δa curve of the second group. 36

24

Engineering Fracture Mechanics 200 (2018) 31–41

L. Lu, S. Wang 320

test data predicted curve

280 240

M(T) B=7.6mm W=102mm a0=16.5mm

ı(MPa)

200 160

Material:2324-T39 Ǔc=4.9°

120

E=72000MPa ıys=455MPa

80 40 0

0

1

2

3

4

Ʃa(mm)

5

6

Fig. 9. The predicted results of M(T) specimen in the second group. 32

test data predicted curve

28 24

F(KN)

20

C(T) B=7.6mm W=152mm a0=62mm

16 12 8

Material:2324-T39 Ǔc=4.9° E=72000MPa ıys=455MPa

4 0

0

4

8

12

16

20

24

28

32

Ʃa(mm)

Fig. 10. The predicted results of C(T) specimen in the second group.

test data. Figs. 9 and 10 have further verified the flow diagram in Fig. 5. Some further explanations are provided as follows: In essence, the method shown in Fig. 5 is a type of R-curve method. However, contrary to conventional R-curve methods, the slope of this resistance curve is not utilized to predict these failure loads. The load-crack extension curve can be directly obtained from Eq. (4) and an iteration process. As proven by the previous study [27], the slopes of δR Δa curves are related to critical CTOA values. The slope of this resistance curve is already taken into consideration when critical CTOA is used. The predicted results of the third group [14] are shown in Figs. 11 and 12. The residual strength plays an important role in damage tolerance analysis and the third group is used to verify the ability of Fig. 5 2a0/W=1/3

1.0

2a0/W=5/12 2a0/W=1/2 fitting curves

įR(mm)

0.8

2024-T3 Ǔc=5.25°

0.6

E=71400MPa ıys=345MPa

0.4

fitting curves: įR=0.2538(Ʃa)0.3845

M(T) B=2.3mm W=305mm

0.2

0.0

0

4

8

12

16

20

24

Ʃa(mm)

Fig. 11. The δR Δa curve of the third group. 37

28

32

Engineering Fracture Mechanics 200 (2018) 31–41

L. Lu, S. Wang 12

test data predicted results

(a)

240

10

(b)

M(T) B=2.3mm 2a0/W=1/3

160

120

test data predicted results

2024-T3 Ǔc=5.25°

80

E=71400MPa ıys=345MPa

40

0

Maximum Load(KN)

Maximum stress(MPa)

200

6

100

200

2024-T3 Ǔc=5.25°

4

E=71400MPa ıys=345MPa

2

0

0

C(T) B=2.3mm a0/W=0.4

8

300

400

500

600

700

0

30

60

90

120

150

180

W(mm)

W(mm) Fig. 12. The predicted results: (a) M(T); (b) C(T) in the third group.

predicting the maximum load or stress of these structures. Fig. 11 provided the δR Δa curves obtained from 305 mm-wide M(T) specimens with different initial crack lengths. All these curves coincide with each other and the fitting curve is utilized in Fig. 5. The predicted results are summarized in Fig. 12. The accuracy is less than 4% for both M(T) and C(T) specimens. In summary, the method proposed here can be used to predict residual strength based on critical CTOA. This method has two advantages as follows: (1) Only the stationary crack opening profile is used in this method. Plastic unloading is avoided in stationary cracks and the deformation plasticity theory can be utilized here. (2) Finite element analysis is not necessary. Once the stationary crack opening profile’ formula is determined, the load-crack extension curve can be predicted according to Fig. 5.

3. Stiffened panels As shown in Ref. [28], CTOA is suitable for low constraint structures, such as stiffened panels. In this section, the method proposed above is applied to stiffened panels. First of all, stiffeners of stiffened panels can be considered as geometry constraints of simple specimens that only affect the left hand side of Eq. (4) (Namely the stationary crack opening profile formula). And the right hand side of Eq. (4) (Crack growth resistance based on CTOA) is independent of these extra geometry constraints. According to this assumption, the crack resistance curves obtained from simple specimens can be used to predict the residual strength of stiffened panels (Material and stress state must keep same). And then if the stationary crack opening formula of stiffened panels is known, the load crack extension curves of stiffened panels can be predicted by Fig. 5. An example was taken from Ref. [15] to validate the assumption above. A group of M(T) specimens and stiffened panels made of 1.6 mm thick 2024-T3 have been tested in L-T orientation at room temperature. These stiffened panels are with 7075-T6 stiffeners (40.6 mm wide, 2.3 mm thick) in L-T orientation. The material properties are shown in Table 2 and the geometry information of the stiffened panels is presented in Fig. 13. These stiffened panels have different initial crack length (2a 0 ) of 50.8, 101.6, 152.4 mm. The panels with the 101.6 mm initial crack length were tested with intact or cut stiffener conditions. The critical CTOA of 1.6 mm 2024-T3 is 5.25° (see Fig. 14). According to Fig. 5, the δR Δa curve from M(T) panels should be determined firstly. And then on account of the assumption geometry constraints don’t affect resistance curves, the δR Δa curve obtained from M(T) specimen can be used to analyze the stiffened panels. To obtain the accurate stiffened panel’s stationary crack opening profile, the elastic plastic finite element analysis should be made. However, an approach without any finite element analysis is searched here. Therefore, the assumption is taken that stress intensity factor can capture stiffened panels’ stationary crack profiles. That is, the linear elastic fracture mechanics (LEFM) is appropriate for stiffened panels. The reason is as follows: The load redistribution, which invalidates LEFM, will take place when large scale yielding occurs. For stiffened panels, stiffeners will provide alternative load paths for load redistributions due to crack tip plasticity. Namely, Table 2 Material properties. Material

Orientation

Young’s modulus, MPa

Poisson’s ratio

Yield strength, MPa

Ultimate strength, MPa

2024-T3 7075-T6

L-T L-T

71,500 71,500

0.3 0.3

345 530

490 575

38

Engineering Fracture Mechanics 200 (2018) 31–41

L. Lu, S. Wang

Fig. 13. Geometry information of stiffened panels.

test-load

(a)

250

3.5

test data fitting

3.0

(b)

200

M(T) 2024-T3 E=71500MPa 2a0/W=1/3 ıys=345MPa W=1016mm

150

100

0

fitting curve:

1.5

B=1.6mm

Ǔc=5.25°

50

2.0

įR(mm)

ı(MPa)

2.5

R

1.0

a)0.6284

0.2446(

0.5

0

10

20

30

40

50

60

70

0.0

80

0

10

20

30

40

50

60

70

80

Ʃa(mm)

Ʃa(mm)

Fig. 14. (a) The test load-crack extension data of M(T) and (b) δR Δa curve. 2.0

(a)

1.0

(b) 1.6

0.8 1.2

fstif

fstif

0.6

0.4

0.2

0.0 0.0

Cut stiffener

0.8

Intact stiffener

0.4

0.1

0.2

0.3

0.4

0.0 0.0

0.5

0.1

0.2

0.3

0.4

0.5

a/W

a/W

Fig. 15. The correction factor of stiffener (a) Intact stiffener and (b) Cut stiffener.

the influence of large scale yielding in stiffened panels is insignificance and then LEFM is applicable. The stationary crack opening profile of stiffened panel can be expressed as 8K

(a )

⎧ δt (x ;ae ) = stif e ae − x E 2π ⎪ ae = a + rp = a0 + Δa + rp ⎨ ⎪ rp = 1 [ K stif (a) ]2 σys 2π ⎩

(13) 39

Engineering Fracture Mechanics 200 (2018) 31–41

L. Lu, S. Wang 250

Intact stiffener 7075-T6 40.6mm wide 2.3mm thick

200

F(KN)

150

7075-T6 E=71500MPa ıys=530MPa

Panel 2024-T3 B=1.6mm W=305mm 2a0=50.8mm

100

50

0

test data predicted curve

2024-T3 E=71500MPa ıys=345MPa

Ǔc=5.25° 0

5

10

15

20

25

30

35

40

Ʃa(mm) Fig. 16. The predicted result of the 2a0 = 50.8 mm panel with intact stiffener.

160 140

F(KN)

100 80 60 40 20 0

test data predicted curve

Intact stiffener 7075-T6 40.6mm wide 2.3mm thick Panel 2024-T3 B=1.6mm W=305mm 2a0=152.4mm

120

7075-T6 E=71500MPa ıys=530MPa 2024-T3 E=71500MPa ıys=345MPa

Ǔc=5.25° 0

5

10

15

20

25

30

Ʃa(mm) Fig. 17. The predicted result of the 2a0 = 152.4 mm panel with intact stiffener. 200 intact stiffener test data

160

intact stiffener predicted curve cut stiffener test data cut stiffener predicted curve

F(KN)

120

80

2024-T3 B=1.6mm W=305mm 2a0=101.6mm

40

Ǔc=5.25°

0 0

5

10

15

20

25

30

35

40

Ʃa(mm) Fig. 18. The predicted result of the 2a0 = 101.6 mm panels (intact and cut stiffener).

where Kstif is the stress intensity factor of stiffened panels.

⎧ Kstif (a) = σ πa fstif ⎨β = ⎩

( ;β) a W

Es Bs Ws E B W

(14)

Where σ is the applied load, fstif is the correction factor of stiffener. E, B, W are Young’s modulus, thickness, width of panel respectively. Es , Bs , Ws are Young’s modulus, thickness, width of stiffener respectively. The correction factor of stiffener is shown in 40

Engineering Fracture Mechanics 200 (2018) 31–41

L. Lu, S. Wang

Fig. 15. Fig. 15 is obtained from Refs. [29,30]. The finite stiffened panels here are regarded as infinite panels with uniformly spaced stiffeners. It is noted that Fig. 15 only provides an approximate value of the correction factor since the stress intensity factor’s formula of stiffened panels is not available. The predicted results are presented in Figs. 16–18. From Figs. 16–18, all of these predicted results showed an accuracy of 5% no matter the stiffener is intact or cut. Namely, the method proposed in Section 2 can be applied to stiffened panels. In addition, several observations can also be obtained: (1) For the first several millimeters crack extension, the estimated load seems to slightly larger than the test data. That means the correction factors of stiffeners shown in Fig. 15 are underestimated. (2) The overall predicted results coincide with the test data quite well. That means the stationary crack profile of stiffened panels can indeed be captured by stress intensity factor. Only rivet stiffened panels data is presented here. For bonded or integral stiffeners, further exploration is required. But the linear elastic fracture mechanics should be appropriate for these type stiffened panels since stiffeners have provided multiple alternative load paths. (3) The predicted results have proven the assumption that stiffeners of stiffened panels can be considered as geometry constraints of simple specimens, or in other words, extra boundaries of these thinned wall structures only affect the crack driving force. 4. Conclusions In a previous paper, a stable crack growth model was proposed that considers growing crack opening profile as corresponding stationary crack opening profile minus the plastic wake profile. On the foundation of this model, an analytical method to predict residual strength based on critical CTOA is proposed. The core idea of this method is transferring critical CTOA to stationary crack opening profile-based crack growth resistance curve. Three groups of M(T) and C(T) specimens are used to validate the proposed method. All of these predicted residual strengths have the accuracy of 4%. In addition, by assuming extra boundaries of these thinned wall structures can be considered as geometry constraints of simple specimens that only affect the stationary crack opening profile formula (Namely crack driving force), the proposed method can also predict the residual strength of stiffened panels. An example was used to verify this assumption and the predicted results have the accuracy of 5%. It is worth stressing that no finite element analysis is used in this paper. References [1] Anderson TL. Fracture mechanics: fundamentals and applications. New York: CSC Press; 2011. [2] Newman JC, James MA, Zerbst U. A review of the CTOA/CTOD fracture criterion. Eng Fract Mech 2003;70:371–85. [3] Newman JC, Dawicke DS, Sutton MA. Finite-element analyses and fracture simulation in thin-sheet aluminum alloy sheet. NASA TM 107662. NASA Langley Res Center; 1992. [4] Dawicke DS, Sutton MA, Newman JC, Bigelow CA. Measurement and Analysis of Critical CTOA for an Aluminum Alloy Sheet NASA TM 109024. NASA Langley Res Center; 1993. [5] Dawicke DS, Sutton MA. Crack tip opening angle measurements and crack tunneling under stable tearing in thin sheet 2024-T3 aluminum alloy. NASA CR 191523. NASA Langley Res Center; 1993. [6] Dawicke DS, Sutton MA. CTOA and crack tunneling measurements in thin sheet 2024–T3 aluminum alloy. Exp Mech 1994;34:357–68. [7] Dawicke DS. Fracture testing of 2324–T39 aluminum alloy. NASA CR 198177. NASA Langley Res Center; 1996. [8] Mahmoud S, Lease K. The effect of specimen thickness on the experimental characterization of critical crack tip opening angle in 2024–T351 aluminum alloy. Engng Fract Mech 2003;70:443–56. [9] Mahmoud S, Lease K. Two dimensional and three dimensional finite element analysis of critical crack tip opening angle in 2024–T351 aluminum alloy at four thicknesses. Eng Fract Mech 2004;71:1379–91. [10] Johnston WM, James MA. A relationship between constraint and the critical crack tip opening angle. NASA CR 215930. NASA Langley Res Center; 2009. [11] Newman JC, Crews JH, Bigelow CA, Dawicke DS. Variations of a global constraint factor in cracked bodies under tension and bending loads. ASTM STP 1995;1244:21–42. [12] Newman JC. Fracture analysis of stiffened panels under biaxial loading with widespread cracking. NASA TM 110197. NASA Langley Res Center; 1995. [13] Seshadri BR, Newman JC. Residual strength analyses of riveted lap-splice joints. NASA TM 209856. NASA Langley Res Center; 2000. [14] Dawicke DS, Newman JC, Starnes JH, Rose CA, Young RD, Seshadri BR. Residual strength analyses methodology: laboratory coupons to structural components. NASA Technical Report, ID 20040086966. NASA Langley Research Center; 2000. [15] Newman JC, Dawicke DS, Seshadri BR. Residual strength analyses of stiffened and un-stiffened panels-Part I: laboratory specimens. Eng Fract Mech 2003;70:493–507. [16] Seshadri BR, Newman JC, Dawicke DS. Residual strength analyses of stiffened and un-stiffened panels-Part II: wide panels. Eng Fract Mech 2003;70:509–24. [17] Hsu CL, Lo J, Yu J, Lee XG, Tan P. Residual strength analysis using CTOA criteria for fuselage structures containing multiple site damage. Eng Fract Mech 2003;70:525–45. [18] Labeas G, Diamantakos J. Residual strength prediction of multiple cracked stiffened panels. Fatigue Fract Eng Mater Struct 2006;29:365–71. [19] Xu W, Wang H, Wu XZ, Zhang XJ, Bai GJ, Huang XL. A novel method for residual strength prediction for sheets with multiple site damage: methodology and experimental validation. Int J Solids Struct 2014;51:551–65. [20] Bai GJ. Residual strength analysis of structures with multiple site damage based on crack tip opening angle criterion: Shanghai Jiao Tong University; 2012 [in Chinese]. [21] Young RD, Rose CA. STAGS developments for residual strength analysis methods for metallic fuselage structures. In: The 55th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, 13–17 January 2014, National Harbor, MD, United States. [22] ASTM E2472. Standard Test Method for determination of resistance to stable crack extension under low constraint conditions. Amer Soc Test Mater; 2012. [23] ASTM E3039. Standard test method for determination of crack-tip-opening angle of pipe steels using DWTT specimens. Amer Soc Test Mater; 2016. [24] Xu S, Petri N, Tyson WR. Evaluation of CTOA from load vs. load-line displacement for C(T) specimen. Eng Fract Mech 2009;76:2126–34. [25] Fang J, Zhang JW, Wang L. An energy based regression method to evaluate critical CTOA of pipeline steels by instrumented drop weight tear tests. Int J Fract 2014;187:123–31. [26] Lu LK, Wang SN. Relationship between crack growth resistance curves and critical CTOA. Eng Fract Mech 2017;173:146–56. [27] Lu LK, Wang SN. A simple model to explain transferability of crack tip opening angle. Eng Fract Mech 2018;193:197–213. [28] Zerbst U, Heinimann M, Donne CD, Steglich D. Fracture and damage mechanics modeling of thin-walled structures–an overview. Eng Fract Mech 2009;76:5–43. [29] China Aviation Academy. Handbook of stress intensity factor. Revised edition Beijing: Science Press; 1993. [In Chinese]. [30] Poe Jr CC. Stress intensity factor for a cracked sheet with riveted and uniformly spaced stringers. NASA Tech Rep R-358 1971.

41