The anisotropic R-criterion for crack initiation

Engineering Fracture Mechanics 75 (2008) 4257–4278

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The anisotropic R-criterion for crack initiation Shafique M.A. Khan a,*, Marwan K. Khraisheh b a b

Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506, USA

a r t i c l e

i n f o

Article history: Received 7 December 2007 Received in revised form 28 March 2008 Accepted 7 April 2008 Available online 14 April 2008 Keywords: Anisotropy Crack initiation Crack tip plasticity Mixed mode fracture Stress intensity factor

a b s t r a c t The anisotropic nature of mixed modes I–II crack tip plastic core region and crack initiation is investigated in this study using an angled crack plate problem under various loading conditions. Hill’s anisotropic yield criterion along with singular elastic stress field at the crack tip is employed to obtain the non-dimensional variable-radius crack tip plastic core region. In addition, the R-criterion for crack initiation proposed by the authors for isotropic materials is also extended to include anisotropy. The effect of Hill’s anisotropic constants on the shape and size of the crack tip plastic core region and crack initiation angle is presented for both plane stress and plane strain conditions at the crack tip. The study shows a significant effect of anisotropy on the crack tip core region and crack initiation angle and calls for further development of anisotropic crack initiation theory. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The issue of crack initiation prediction has been at the heart of all crack propagation studies. It is well established that the crack initiation direction depends on the conditions at the crack tip. Several crack initiation criteria have been developed primarily for the isotropic materials and Khan and Khraisheh [1] presented a detailed analysis of some of these criteria. Many engineering materials can be assumed to be isotropic and results within reasonable accuracy are obtained using these criteria. However, a large number of engineering materials have anisotropic properties. Obviously, the fracture behavior of such materials under mixed mode loading cannot be predicted by the criteria that are developed for the isotropic materials. Therefore, these criteria must be modified to account for anisotropy. The conventional methodology to include anisotropic effects is to employ the anisotropic stress field at the crack tip and incorporate it into one of the known fracture criteria. For example, Saouma et al. [2] modified the maximum tangential stress (MTS [3]) theory and showed that anisotropy has a profound effect on the crack propagation direction. Gdoutos et al. [4] modified the minimum strain energy density criterion (Scriterion [5]) and analyzed the problem of an anisotropic plate with a crack inclined with respect to the principal axes of material symmetry. The S-criterion was also modified by Ye and Ayari [6]. Instead of minimizing the strain energy density factor S (as done by Gdoutos and Zacharopoulos [4]), they defined the S-criterion by minimizing the ratio of S to critical value (Sc), to account for the polar variation of the fracture toughness; they analyzed mixed modes I–II and I–III crack growth for solids with material symmetries. Theorcaris and Philippidis [7] have modified the T-criterion [8], by decomposing the anisotropic elastic potential into distinct components, for homogenous anisotropic materials. It is claimed in their paper that even though the T-criterion is modified assuming homogenously anisotropic media, the results compare satisfactorily with the experimental data for unidirectional glass/epoxy cracked plates. Ayari and Ye [9] modified the maximum strain theory for anisotropic solids. Pan and Shih [10] have investigated the near-tip stress and strain fields for power-law hardening

* Corresponding author. Tel.: +966 3 860 7225; fax: +966 3 860 2949. E-mail address: [email protected] (S.M.A. Khan). 0013-7944/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2008.04.002

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Nomenclature K stress intensity factor I, II subscripts denoting the mode of loading x, y Cartesian co-ordinates at the crack tip x0 , y0 Cartesian co-ordinates for the cracked plate r, h polar co-ordinates at the crack tip S strain energy density factor rx, ry, sxy stresses at the crack tip rx0 ; ry0 ; sx0 y0 stresses applied to the plate with angled crack b crack inclination angle function of angle b fKI h0 crack initiation angle k loading ratio for biaxial loading a loading ratio for proportional tension torsion loading m Poisson’s ratio l ratio of stress intensity factors (=KI/KII) rP,H radius of Hill’s elastic–plastic boundary non-dimensionalized radius of Hill’s elastic–plastic boundary RP,H a half crack length f(rij) yield potential F, G, H, N Hill’s anisotropic constants

orthotropic materials under plane strain conditions. Khraisheh and Khan [11] modified the maximum stress triaxiality ratio criterion [12] by incorporating an anisotropic yield function. All of these studies demonstrate that the anisotropic behavior of materials strongly affects the crack initiation angles. The approach followed by Khraisheh and Khan [11] is different from the conventional approach. In this approach, a generalized anisotropic yield function is incorporated in the formulation using the isotropic singular elastic stress field. This approach offers an advantage over the conventional approach of using an anisotropic stress field. The crack initiation criterion developed using this new approach is more flexible and can be applied to different anisotropic materials simply by selecting appropriate values for the yield function parameters. The conventional approach, however, requires the formulation of the anisotropic stress field at the crack tip for each new anisotropic material prior to applying a fracture criterion (for example, see [13]). It is important to mention that for the crack initiation criteria that assume a constant radius core region, the only choice is to introduce anisotropy in the crack tip stress field. However, for the crack initiation criteria based on a variable region core region, the anisotropy characterizing constants can be introduced either in the crack tip stress field or in the yield criterion used to define the elastic–plastic boundary. The use of an anisotropic stress field at the crack tip along with an anisotropic yield criterion will duplicate the anisotropy characterizing parameters. The current study presents this new approach of introducing anisotropy in the yield criterion as an alternative to the conventional practice. We have proposed a new criterion (R-criterion) based on the characteristics of the crack tip core region for isotropic materials [14]. This work has been referenced by several researchers [15–22] to investigate the prediction of fatigue crack initiation angles. Bian and Kim [15] applied the R-criterion (referred to as minimum plastic zone radius criterion) to inclined surface crack and through-crack specimens under mixed mode loading. They concluded that comparison with the test data indicates that the R-criterion provides better overall prediction of the crack initiation angle than the other criteria considered in the study. Recently, Carpinteri et al. [22] have extended the R-criterion to include the temperature dependence and then applied the modified R-criterion to predict the crack path and the crack extension force for an edge-cracked finite plate under tension. In the present work, we extend the R-criterion for application to anisotropic materials. Since the R-criterion is based on crack tip plastic core region, first the shape and size of the anisotropic crack tip core region is investigated. Anisotropic effects are incorporated into the R-criterion formulation by employing Hill’s yield function. The resulting anisotropic R-criterion is used to investigate the crack initiation angles under various loading conditions. A part of this work was presented at the ASME conference [23]. In the present paper, detailed results are presented with two additional loading conditions, and for both plane stress and plane strain conditions at the crack tip. The current study focuses on small scale yielding at the crack tip with global elastic behavior. 2. Analysis The basic angled crack problem is shown in Fig. 1a. The axes of material symmetry coincide with the x0 and y0 axes (Cartesian coordinates for the cracked plate, Fig. 1a). The crack is oriented at an angle b (crack inclination angle) measured clockwise to the vertical center line. The length of the crack is 2a. The singular elastic stress field at the crack tip for an isotropic cracked body under mixed mode I/II is given by

S.M.A. Khan, M.K. Khraisheh / Engineering Fracture Mechanics 75 (2008) 4257–4278

a

y′

4259

σ y′

2a β

− θo

τ x′y′ σ x′ x′

For uniaxial loading σ y′ = σ, σ x ′ = 0, τ x ′y′ = 0

For pure shear loading σ y′ = 0, σ x ′ = 0, τ x ′y′ = τ

For biaxial loading σ y′ = σ, σ x ′ = λσ, τ x ′y′ = 0

For proportional tension torsion loading σ y′ = σ, σ x ′ = 0, τ x ′y′ = ασ

b

y

σy τ xy σx r

θ Crack

x

Fig. 1. (a) General loading at the cracked plate and (b) stresses at the crack tip.

      1 h h 3h h h 3h 1  sin sin  K II sin 2 þ cos cos ; rx ¼ pffiffiffiffiffiffiffiffi K I cos 2 2 2 2 2 2 2pr      1 h h 3h h h 3h ry ¼ pffiffiffiffiffiffiffiffi K I cos 1 þ sin sin þ K II sin cos cos ; 2 2 2 2 2 2 2pr      1 h h 3h h h 3h sxy ¼ pffiffiffiffiffiffiffiffi K I cos sin cos þ K II cos 1  sin sin ; 2 2 2 2 2 2 2pr

rz ¼ m rx þ ry For plane stress For plane strain; rz ¼ 0

ð1Þ

where r and h are the polar co-ordinates with origin at the crack tip (Fig. 1b) and m is the Poisson’s ratio. When this stress field is used to apply any crack initiation criterion based on linear elastic fracture mechanics, the angle h0 defines the crack initiation angle, and r defines the radius of the core region. The above stress field assumes a global elastic behavior with small scale yielding as represented by the crack tip plastic core region. A crack initiation criterion either assumes a core region of constant radius or uses a yield criterion to define the shape of the elastic–plastic boundary, thus giving a variable radius crack tip core region. The isotropic R-criterion [14] is based on a variable radius for the core region and utilizes the isotropic von Mises yield function to define the extent of the crack tip plastic zone. To extend the isotropic R-criterion to include anisotropy and develop the anisotropic R-criterion, we employ Hill’s anisotropic yield function [24] to define the variable radius of the core region for anisotropic materials. This yield function is based on the assumption that the material is homogenous and is characterized by three orthogonal axes of anisotropy, about which the properties have twofold symmetry. It also assumes that the tensile and compressive strengths are equal in any given direction. For a 2D state of stress, the Hill’s anisotropic yield criterion has the form 2f ðrij Þ ¼ Fðry  rz Þ2 þ Gðrz  rx Þ2 þ Hðrx  ry Þ2 þ 2Ns2xy

ð2Þ

where f(rij) is the yield potential and F, G, H, and N are constants, which characterize the anisotropy. F, G, and H define the degree of anisotropy in the normal directions and N defines the degree of anisotropy in the shear direction. For F = G = H = 1 and N = 3F, Hill’s criterion reduces to the isotropic von Mises criterion. The anisotropic constants F, G and H can be evaluated from simple tension tests, performed to obtain the yield stress in each direction. N can be evaluated from shear test.

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Based on the arguments and technique explained earlier [14] for obtaining the non-dimensional mixed mode I/II isotropic crack tip plastic core regions and the isotropic R-criterion, we split and represent the mode I and mode II stress intensity factors as pffiffiffiffiffiffi K I ¼ rapp pafKI ðbÞ ð3Þ pffiffiffiffiffiffi K II ¼ rapp pafK II ðbÞ where rapp is the applied stress, which may be tensile, compressive, shear stress, or a combination of these, fKi ðbÞ is a function of the crack inclination angle, b, and the loading condition (will be defined later for different loading conditions). Using Eq. (3) in Eq. (1), we get pffiffiffi rapp a rx ¼ pffiffiffiffiffi fx ðh; fKi Þ 2r pffiffiffi rapp a ð4Þ ry ¼ pffiffiffiffiffi fy ðh; fKi Þ 2r pffiffiffi rapp a sxy ¼ pffiffiffiffiffi fxy ðh; fKi Þ 2r where fx ðh; fKi Þ; fy ðh; fKi Þ; and f xy ðh; fKi Þ are defined as       h h 3h h h 3h 1  sin sin  fKII ðbÞ sin 2 þ cos cos fx ðh; fKi Þ ¼ fKI ðbÞ cos 2 2 2 2 2 2      h h 3h h h 3h fy ðh; fKi Þ ¼ fKI ðbÞ cos 1 þ sin sin þ fKII ðbÞ sin cos cos 2 2 2 2 2 2      h h 3h h h 3h fxy ðh; fKi Þ ¼ fKI ðbÞ cos sin cos þ fKII ðbÞ cos 1  sin sin 2 2 2 2 2 2

ð5Þ

Substituting Eq. (4) in Eq. (2), we get 2f ðrij Þ ¼

o r2app a n 2 Fðfy  fz Þ2 þ Gðfz  fx Þ2 þ Hðfx  fy Þ2 þ 2Nf xy 2r

ð6Þ

Substituting Eq. (5) in Eq. (6), we get the expression for the non-dimensional variable core region radius for anisotropic materials, RP,H, based on Hill’s anisotropic yield function. For plane stress: i rp;H ðh; fKi Þ 1 h 2 f ðbÞg 1 ðhÞ þ fK2II ðbÞg 2 ðhÞ þ fKI ðbÞfKII ðbÞg 12 ðhÞ RP;H ðh; fKi Þ ¼ h ð7aÞ i2 ¼ 4 KI rapp a f ðr Þ ij

and for plane strain RP;H ðh; fKi Þ ¼

rp;H ðh; fKi Þ 1 2 2 h i2 ¼ ½fKI ðbÞðg 1 ðhÞ þ h1 ðh; mÞÞ þ fKII ðbÞðg 2 ðhÞ þ h2 ðh; mÞÞ þ fKI ðbÞfKII ðbÞðg 12 ðhÞ þ h12 ðh; mÞÞ 4 rapp a f ðrij Þ

ð7bÞ

where g1, g2, g12 are functions of angle h, and h1, h2, h12 are functions of h and m. Both g’s and h’s are also functions of the anisotropic constants F, G, H and N as given in Appendix A. Eq. (7) can be used to study the effect of anisotropy on the crack tip core region shape and size for any cracked plate loading case for which the stress intensity factors are available. These equations show that the shape of the core region is a function of two angles, h and b under mixed mode I/II loading. The function of the angle b in turn depends on the cracked plate loading conditions, for example, uniaxial, pure shear, biaxial, or proportional tension–torsion loading are considered in this study. The shape of the core region is also a function of the Poisson’s ratio in the case of plane strain condition at the crack tip. The shape of the core region strongly depends on the strength of anisotropy in various directions. The two parameters used to non-dimensionalize the core region radius namely, the initial half crack length a, and the ratio of the applied stress to the yield potential f(rij) control the actual size of the core region. The above formulation is general and valid for all cracked plate loading conditions. Four different types of loading conditions are investigated in this study. 3. Loading conditions and core regions 3.1. Uniaxial loading Referring to the angled crack problem under uniaxial loading (ry0 ¼ r; rx0 ¼ 0; sx0 y0 ¼ 0, Fig. 1a), the stress intensity factors are given as 2 pffiffiffiffiffiffi K I ¼ r sin b pa ð8Þ pffiffiffiffiffiffi K II ¼ r sin b cos b pa

S.M.A. Khan, M.K. Khraisheh / Engineering Fracture Mechanics 75 (2008) 4257–4278

4261

It follows from Eqs. (3) and (8): 2

fKI ðbÞ ¼ sin b fKII ðbÞ ¼ sin b cos b

ð9Þ

rapp ¼ r where r is the applied stress in y0 direction (Fig. 1a). Using Eq. (9) with Eq. (7), the non-dimensional Hill’s anisotropic elastic– plastic radius of the core region is obtained for uniaxial loading as RP;H ðh; fKi Þ ¼

r p;H ðh; fKi Þ 1 4 2 3 2  2 ¼ ½sin bg 1 ðhÞ þ sin b cos bg 2 ðhÞ þ sin b cos bg 12 ðhÞ 4 rapp a f r ð ij Þ

ð10aÞ

for plane stress condition at the crack tip, and RP;H ðh; fKi Þ ¼

¼

r p;H ðh; fKi Þ h i2 r a f ðrappij Þ 1 4 2 3 ½sin bðg 1 ðhÞ þ h1 ðh; mÞÞ þ sin b cos2 bðg 2 ðhÞ þ h2 ðh; mÞÞ þ sin b cos bðg 12 ðhÞ þ h12 ðh; mÞÞ 4

ð10bÞ

for plane strain condition at the crack tip. 3.2. Pure shear loading Referring to the angled crack problem under pure shear loading (ry0 ¼ 0; rx0 ¼ 0; sx0 y0 ¼ s, Fig. 1a), the stress intensity factors are given as pffiffiffiffiffiffi K I ¼ s sin 2b pa ð11Þ pffiffiffiffiffiffi K II ¼ s cos 2b pa It follows from Eqs. (3) and (11): fKI ðbÞ ¼  sin 2b fKII ðbÞ ¼  cos 2b rapp ¼ s

ð12Þ

where s is the applied shear stress in the direction shown in Fig. 1a. Combining Eq. (12) and Eq. (7), the non-dimensional Hill’s anisotropic elastic–plastic radius of the core region is obtained for pure shear loading as RP;H ðh; fKi Þ ¼

r p;H ðh; fKi Þ 1 2 2 h i2 ¼ ½sin 2bg 1 ðhÞ þ cos 2bg 2 ðhÞ þ sin 2b cos 2bg 12 ðhÞ 4 r a f ðrappÞ

ð13aÞ

ij

for plane stress condition at the crack tip, and RP;H ðh; fKi Þ ¼

r p;H ðh; fKi Þ 1 2 2 h i2 ¼ ½sin 2bðg 1 ðhÞ þ h1 ðh; mÞÞ þ cos 2bðg 2 ðhÞ þ h2 ðh; mÞÞ þ sin 2b cos 2bðg 12 ðhÞ þ h12 ðh; mÞÞ 4 rapp a f ðrij Þ

ð13bÞ

for plane strain condition at the crack tip. 3.3. Biaxial loading Referring to the angled crack problem under proportional biaxial loading (ry0 ¼ r; rx0 ¼ kr; sx0 y0 ¼ 0, Fig. 1a), the stress intensity factors are given as [14] pffiffiffiffiffiffi 2 K I ¼ rðsin b þ k cos2 bÞ pa pffiffiffiffiffiffi K II ¼ rð1  kÞðsin b cos bÞ pa

ð14Þ

It follows from Eqs. (3) and (14): 2

fKI ðb; kÞ ¼ ðsin b þ k cos2 bÞ fKII ðb; kÞ ¼ ð1  kÞðsin b cos bÞ

ð15Þ

rapp ¼ r where k is the biaxial loading ratio and r is the applied normal stress in the y0 direction shown in Fig. 1a. Using Eq. (15) in Eq. (7), the non-dimensional Hill’s elastic–plastic radius of the core region is obtained for biaxial loading as

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Elastic/Plastic Boundary Plastic region

Elastic region

1

Crack

2

∫γ 1

p

dA < ∫ γ p dA 2

Fig. 2. Plastic work done per unit surface area created for two different paths.

F=1,G=1 H=1,N=3

75 60

0.6 r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

165

0.8

120

135 150

β = 90 o

90 105

2

β = 60 o

45 30

0.4 0.2

15

0

0

σ

180

−θo

β

Crack

-165

-15 σ

-150

-30 -135

β = 30 o

-45 -120

-60 -105

-75 -90 90

F=1,G=1 H=1,N=3

105

β = 90 o

60 0.6 r

⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

165

75

120

135 150

0.8

2

45

β = 60 o

30

0.4 0.2

15

0

0

σ

180

−θo

β

Crack

-165

-15

-150

σ

-30 -135

-45 -120

β = 30 o

-60 -105

-75 -90

Fig. 3. Hill’s elastic–plastic core region: uniaxial loading F = 1, G = 1, H = 1, N = 3 (isotropic case): (a) plane stress and (b) plane strain.

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RP;H ðh; fKi Þ ¼

rp;H ðh; fKi Þ 1 2 2 2 2 2 2 2 2 h i2 ¼ ½ðsin b þ k cos bÞ g 1 ðhÞ þ ð1  kÞ sin b cos bg 2 ðhÞ þ ð1  kÞ sin b cos bðsin b þ k cos bÞg 12 ðhÞ 4 r a f ðrappÞ ij

ð16aÞ for plane stress condition at the crack tip, and RP;H ðh; fKi Þ ¼

r p;H ðh; fKi Þ 1 2 2 2 2 2 2 h i2 ¼ ½ðsin b þ k cos bÞ ðg 1 ðhÞ þ h1 ðh; mÞÞ þ ð1  kÞ sin b cos bðg 2 ðhÞ þ h2 ðh; mÞÞ 4 r a f ðrappÞ ij

2

þ ð1  kÞ sin b cos bðsin b þ k cos2 bÞðg 12 ðhÞ þ h12 ðh; mÞÞ

ð16bÞ

for plane strain condition at the crack tip. 3.4. Proportional tension torsion loading Referring to the angled crack problem under proportional tension–torsion loading (ry0 ¼ r; rx0 ¼ 0; sx0 y0 ¼ ar, Fig. 1a), the stress intensity factors are given as [14] F=1,G=1 H=1,N=8

75 60

1.6

β = 60 o

45

1.2

r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

165

2

120

135 150

β = 90 o

90 105

30

2

0.8 15

0.4

σ

0

180

0

−θo

β

Crack

-165

-15 σ

-150

-30 -135

β = 30 o

-45 -120

-60 -105

-75 -90 90

F=1,G=1 H=1,N=8

105

β = 90 o

60 1.6

45

β = 60 o

1.2

r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

165

75

120

135 150

2

30

2

0.8 15

0.4

σ

180

0

0

−θo

β

Crack

-165

-15

-150

σ

-30 -135

-45 -120

β = 30 o

-60 -105

-75 -90

Fig. 4. Hill’s elastic–plastic core region: Uniaxial loading F = 1, G = 1, H = 1, N = 8 (effect of N): (a) plane stress and (b) plane strain.

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pffiffiffiffiffiffi 2 K I ¼ rðsin b  a sin 2bÞ pa

ð17Þ

pffiffiffiffiffiffi K II ¼ rðsin b cos b  a cos 2bÞ pa

It follows from Eqs. (3) and (17): 2

fKI ðb; aÞ ¼ ðsin b  a sin 2bÞ ð18Þ

fKII ðb; aÞ ¼ ðsin b cos b  a cos 2bÞ rapp ¼ r

where a is the proportional tension torsion loading ratio and r is the applied normal stress in the y0 direction shown in Fig. 1a. Using Eq. (18) in Eq. (7), the non-dimensional Hill’s elastic–plastic radius of the core region is obtained for proportional tension torsion loading as RP;H ðh; fKi Þ ¼

rp;H ðh; fKi Þ 1 2 2 2 h i2 ¼ ½ðsin b  a sin 2bÞ g 1 ðhÞ þ ðsin b cos b  a cos 2bÞ g 2 ðhÞ 4 rapp a f ðrij Þ 2

þ ðsin b  a sin 2bÞðsin b cos b  a cos 2bÞg 12 ðhÞ

F=4,G=1 H=1,N=3

2

75

120

60 1.6

β = 60 o

45

1.2

r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

165

β = 90 o

90 105

135 150

ð19aÞ

30

2

0.8 15

0.4

σ

0

180

0

−θo

β

Crack

-165

-15 σ

-150

-30 -135

β = 30 o

-45 -120

-60 -105

-75 -90 90

F=4,G=1 H=1,N=3

105

β = 90 o

60 1.6

45

β = 60 o

1.2

r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

165

75

120

135 150

2

30

2

0.8 15

0.4

σ

180

0

0 −θo

β

Crack

-165

-15

-150

σ

-30 -135

-45 -120

β = 30 o

-60 -105

-75 -90

Fig. 5. Hill’s elastic–plastic core region: Uniaxial loading F = 4, G = 1, H = 1, N = 3 (effect of F): (a) plane stress and (b) plane strain.

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for plane stress condition at the crack tip, and RP;H ðh; fKi Þ ¼

r p;H ðh; fKi Þ 1 2 2 2 h i2 ¼ ½ðsin b  a sin 2bÞ ðg 1 ðhÞ þ h1 ðh; mÞÞ þ ðsin b cos b  a cos 2bÞ ðg 2 ðhÞ þ h2 ðh; mÞÞ 4 r a f ðrappÞ ij

2

þ ðsin b  a sin 2bÞðsin b cos b  a cos 2bÞðg 12 ðhÞ þ h12 ðh; mÞÞ

ð19bÞ

for plane strain condition at the crack tip. 4. Anisotropic R-criterion The physical basis for the R-criterion was presented in the authors’ previous publication [14] and it will be included in this paper for the sake of completeness. The core region presents a highly strained area. The crack tip has to propagate through this highly strained area to reach the elastically loaded material outside the core radius. The crack will follow the ‘‘easiest” path to the outside material. It makes sense to assume that the easiest path is the shortest distance to the

F=1,G=4 H=1,N=3

75 60

1.6

β = 60 o

45

1.2

r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

165

2

120

135 150

β = 90 o

90 105

30

2

0.8 15

0.4

σ

0

180

0

−θo

β

Crack

-165

-15 σ

-150

-30 -135

β = 30 o

-45 -120

-60 -105

-75 -90 90

F=1,G=4 H=1,N=3

105

β = 90 o

60 1.6

45

β = 60 o

1.2

r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

165

75

120

135 150

2

30

2

0.8 15

0.4

σ

180

0

0 −θo

β

Crack

-165

-15

-150

σ

-30 -135

-45

β = 30 o

-60

-120 -105

-75 -90

Fig. 6. Hill’s elastic–plastic core region: uniaxial loading F = 1, G = 4, H = 1, N = 3 (effect of G): (a) plane stress and (b) plane strain.

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‘‘outside” material. In addition, it is also based on the modified Griffith Criterion [25,26] as presented by Anderson [27] which pffiffiffiffiffiffi states that the fracture stress rf is proportion to the square root of the fracture energy wf ðrf / wf Þ. The fracture energy wf for a quasi-brittle elastic–plastic material depends on the surface energy cs and the plastic work done per unit surface area created cp, essentially wf = cs + cp. Typically, cp is much larger than cs and therefore dictates the magnitude of wf. The shortest distance from the crack tip to the elastic–plastic boundary represents the minimum plastic work needed to create the crack surfaces as illustrated in Fig. 2. Then the shortest distance from the crack tip to the elastic–plastic boundary represents the minimum fracture energy and stress. Therefore, for anisotropic materials, the R-criterion can be stated as: the direction of crack initiation coincides with the direction of the local or global minimum of the core region radius defined by the Hill’s anisotropic yield function. Mathematically, it can be stated as oRP;H ¼0 oh 2 o RP;H >0 oh2

ð20Þ

Applying the anisotropic R-criterion to Eq. (7), we get

F=1,G=1 H=4,N=3

75 60

1.6

β = 60 o

45

1.2

r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

165

2

120

135 150

β = 90 o

90 105

30

2

0.8 15

0.4

σ

0

180

0

−θo

β

Crack

-165

-15 σ

-150

-30 -135

β = 30 o

-45 -120

-60 -105

-75 -90 90

F=1,G=1 H=4,N=3

105

β = 90 o

60 1.6

45

β = 60 o

1.2

r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

165

75

120

135 150

2

30

2

0.8 15

0.4

σ

180

0

0

−θo

β

Crack

-165

-15

-150

σ

-30 -135

-45 -120

β = 30 o

-60 -105

-75 -90

Fig. 7. Hill’s elastic–plastic core region: uniaxial loading F = 1, G = 1, H = 4, N = 3 (effect of H): (a) plane stress and (b) plane strain.

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h h h h h h h þ x9 tan9 þ x8 tan8 þ x7 tan7 þ x6 tan6 þ x5 tan5 þ x4 tan4 2 2 2 2 2 2 2 h h h þ x3 tan3 þ x2 tan2 þ x1 tan þ x0 ¼ 0 2 2 2 w1 cos h þ w2 cos 2h þ w3 cos 3h þ w4 cos 5h þ w5 sin h þ w6 sin 2h þ w7 sin 3h þ w8 sin 5h > 0

x10 tan10

ð21aÞ

for plane stress condition at the crack tip, and h h h h h h þ ðx9 þ g9 Þ tan9 þ ðx8 þ g8 Þ tan8 þ ðx7 þ g7 Þ tan7 þ ðx6 þ g6 Þ tan6 þ ðx5 þ g5 Þ tan5 2 2 2 2 2 2 h h h h þ ðx4 þ g4 Þ tan4 þ ðx3 þ g3 Þ tan3 þ ðx2 þ g2 Þ tan2 þ ðx1 þ g1 Þ tan þ ðx0 þ g0 Þ ¼ 0 2 2 2 2 ðw1 þ n1 Þ cos h þ ðw2 þ n2 Þ cos 2h þ ðw3 þ n3 Þ cos 3h þ ðw4 þ n4 Þ cos 5h þ ðw5 þ n5 Þ sin h þ ðw6 þ n6 Þ sin 2h

ðx10 þ g10 Þ tan10

þ ðw7 þ n7 Þ sin 3h þ ðw8 þ n8 Þ sin 5h > 0 for plane strain condition at the crack tip. The coefficients are defined in Appendix B.

β = 90 o

90 105

5

75

120

60 4

135

β = 60 o

45

3 150

30 2

165

15

1

τ

0

180

0

β

Crack

-165

− θo

τ

-15

-150 F=1,G=1 H=4,N=3 -135

-30

r ⎛ τ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

2

-45

β = 30 o

-60

-120 -105

-75 -90

90

β = 90

105

o

5

75

120

60 4

135

β = 30 o 45

3 150

30 2

165

15

1 0

180

β = 60 o

0

τ

Crack

-165

-15

β

− θo

τ

r ⎛ τ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

F=1,G=1 -150 H=4,N=3

2

-135

-30 -45

-120

-60 -105

-75 -90

Fig. 8. Hill’s elastic–plastic core region: pure shear loading F = 1, G = 1, H = 4, N = 3 (effect of H): (a) plane stress and (b) plane strain.

ð21bÞ

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5. Results and discussion The anisotropic crack tip core regions are plotted in Fig. 3 through Fig. 10. It is important to mention here that in Eq. (2) F, G, and H define the degree of anisotropy in normal direction in yz-, xz-, and xy-planes respectively. The constant N defines the degree of shear anisotropy in xy-plane. The comparison of the crack tip core regions presented in this study with earlier studies will be done wherever possible, since not all of the studies use the same methodology to include anisotropy and therefore a direct quantitative comparison is not possible. Figs. 3–7 are the polar plots, which show the core regions for uniaxial loading, for both plane stress and plane strain conditions at the crack tip, for different values of anisotropic factors F, G, H, and N. Fig. 3 shows the core regions for the isotropic case (F = G = H = 1, N = 3F) and these have the same shape and size as those for the isotropic von Mises yield function [14] as expected. Fig. 4 shows that the anisotropic shear constant N affects the entire shape of the core region and deforms it proportionally as compared to the isotropic case. Theocaris and Philippidis [7] have shown the characteristic contours of the strain energy density at the crack tip for various fiber composites. When these contours are compared with the shape of

90 105

r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

2.5

β = 90 o

75

120

2

60 2

135

β = 60 o

45

1.5

β = 30 o

30

150 1 165

15

0.5 0

180

σ

0

β

− θo

0.5σ

Crack

-15

-165 -150

-30 -135

F=1,G=1 H=4,N=3

-45 -120

-60 -105

-75 -90

β = 90 o

90 105

2.5

75

120

60 2

135

β = 60 o

45

1.5 150

β = 30 o

30 1

165 180

σ

15

0.5 0

0

β

− θo

0.5σ

Crack

-165

-15 r

-150 F=1,G=1 H=4,N=3

-30

⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

-135

2

-120

-45 -60

-105

-75 -90

Fig. 9. Hill’s elastic–plastic core region: biaxial loading k = 0.5 F = 1, G = 1, H = 4, N = 3 (effect of H): (a) plane stress and (b) plane strain.

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the crack tip core regions for different values of N, a reasonable resemblance is observed for b = 30°, 60°, 90°. It can be concluded that if the exact values of the anisotropic constants in the Hill’s yield criterion are known for these fiber composites, or in general any anisotropic material, then Eq. (7) carries the capability to predict the exact shape and size of the crack tip core region. Figs. 5–7 show the effect of the normal anisotropic constants F, G, and H respectively, for both plane stress and plane strain conditions at the crack tip under uniaxial loading. The factor F affects the core region strongly in the half that contains the crack (left half on the polar plot), whereas the factor G affects strongly in the other half ahead of the crack (right half on the polar plot). The factor H affects the overall core region shape. In all the cases, the actual size of the core region is greater, roughly twice, than the case of isotropic core region. Analyzing Figs. 3–7, it can be observed that the effect of the Hill’s anisotropic constants on the size of the core region depends on the crack inclination angle. Crack inclination angle controls the strength of each fracture mode in the mixed mode loading, so in effect how the anisotropic constants influence the size

90 105

r ⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

120

2

3

60

2.5

135

β = 60 o

75

150

β = 30 o

45

2

30

1.5 1

165

15

0.5 0

180

0

-165

-15

-150

β

− θo

-30 -135

F=1,G=1 H=4,N=3

σ

0.5σ

Crack

-45 -120

-60 -105

-75

β = 90 o

-90

90

β = 90 o

105 120

3

75 60

2.5

135

2

150

β = 30 o

30

1.5 1

165

β = 60 o

45

15

0.5 r

0

180

⎛ σ ⎞ ⎟ a⎜ ⎜ f (σ ) ⎟ ij ⎠ ⎝

Crack

-165 -150 F=1,G=1 H=4,N=3

0

2

-15 -30

-135

0.5σ

σ β

− θo

-45 -120

-60 -105

-75 -90

Fig. 10. Hill’s elastic–plastic core region: Proportional tension–torsion loading a = 0.5 F = 1, G = 1, H = 4, N = 3 (effect of H): (a) plane stress and (b) plane strain.

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of the core region depends on the ratio of stress intensity factors. Keeping in mind that KI increases and KII decreases from b = 30° to b = 90°, lets analyze Fig. 5. Comparing to the size for isotropic case, for plane stress condition the maximum radius of the core region increases by a factor of 1.3 for b = 30°, 2 for b = 60°, and 2.7 for b = 90°; for plane strain condition the maximum radius of the core region increases by a factor of 1.3 for b = 30°, 1.5 for b = 60°, and 2 for b = 90°. Based on this information, we can conclude that for anisotropic constant F, the size of the core region is affected dominantly by KI as compared to KII and this effect is more pronounced for plane stress condition at the crack tip. Similar arguments can be arrived at for the effect of the other anisotropic constants. For all the cases, the size of the core regions for plane strain condition is less than the size for plane stress condition, and the core region is symmetric about horizontal for pure mode-I (b = 90°). Both of these observations are the same as for the isotropic case. Zhang and Venugopalan [28] have investigated the effect of anisotropy on the crack tip core regions only for pure mode-I, for which the shapes of the crack tip core region in this study compare favorably with those presented in their study. While investigating the effects of texture induced anisotropy on crack tip plasticity in textured aluminum alloys under pure modeI, Potirniche et al. [29] have shown the shapes of core region obtained from finite element simulations for different textures. A qualitative comparison between this study and Potriniche et al. [29] reveals similar shapes of the crack tip core regions (e.g., similar shape as that of Fig. 7, b = 90° appears frequently in Ref. [29]) and it can be concluded that if exact values

100 90

Isotropic R Pl. Stress F = 1, G = 1, H = 1, N = 1

80

F = 1, G = 1, H = 1, N = 2

Crack initiation angle, −θo

o

F = 1, G = 1, H = 1, N = 4

70

F = 1, G = 1, H = 1, N = 8

60 50 σ

40 β

30

− θo

20 σ

10 0 0

10

20

30

40

50

60

70

80

90

60

70

80

90

Crack inclination angle, βo 100 90

Crack initiation angle, −θo

o

80 70 60 50

σ

40

β

− θo

30 20

Isotropic R Pl. Strain F = 1, G = 1, H = 1, N = F = 1, G = 1, H = 1, N = F = 1, G = 1, H = 1, N = F = 1, G = 1, H = 1, N =

1 2 4 8

σ

10 0 0

10

20

30

40

50

Crack inclination angle, βo Fig. 11. Uniaxial tension: Effect of N: (a) plane stress and (b) plane strain.

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for the Hill’s anisotropic constants are available then Eq. (7) can predict the exact shape and size of the crack tip core region for the textured aluminum or any other anisotropic material. Figs. 8–10 show the shape of the core regions for pure shear, biaxial (k =  0.5), and proportional tension torsion (a =  0.5) loading, for different crack inclination angles, for both plane stress and plane strain conditions at the crack tip. All of these plots show significant shape and size change for the crack tip core regions as compared to the isotropic case (please see Ref. [14]). Another important feature is that the maximum core region radius for pure shear loading for b = 90° (Fig. 8) is increased by a factor of approximately 2.5 (for H = 4; the factor differs for different values of anisotropic constants) as compared to the corresponding isotropic case [14]. Ye [30] has shown crack tip core region shapes for biaxial loading, however the paper severely lacks in proper nomenclature. Therefore, the results shown by Ye [30] appear somewhat ambiguous, since there is no mention of the loading case geometry and these are plotted for different arbitrary values of KII/KI ratios. Moreover, there is no explanation as to why the shape of the core region for KII/KI = 1 is not symmetric, while it is symmetric for other ratios. Therefore no comparison of the present study with the work of Ye [30] is possible. In general, analyzing all the loading cases considered in this study, the reduction in the size of crack tip plastic core region between plane stress and plane strain conditions is dependent on the Hill’s anisotropic constants. It can be concluded from Figs. 3–10, that F and G strongly affect the reduction in the size of the core region from plane stress to plane strain condition

180 160

Crack initiation angle, θo

o

140 120 100 80

σ

60

θo

β

Isotropic R Pl. Stress F = 1, G = 1, H = 1, N = 1 F = 1, G = 1, H = 1, N = 2 F = 1, G = 1, H = 1, N = 4 F = 1, G = 1, H = 1, N = 8

40 σ

20 0 0

10

20

30

40

50

60

70

80

90

Crack inclination angle, βo 180 160

Crack initiation angle, θo

o

140 120 100 80 σ

60 β

θo

Isotropic R Pl. Strain F = 1, G = 1, H = 1, N = F = 1, G = 1, H = 1, N = F = 1, G = 1, H = 1, N = F = 1, G = 1, H = 1, N =

40 20

σ

1 2 4 8

0 0

10

20

30

40

50

60

70

Crack inclination angle, βo Fig. 12. Uniaxial compression: Effect of N: (a) plane stress and (b) plane strain.

80

90

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at the crack tip, while H and N have a weak effect in this respect. This reduction in the size of the core region between plane stress and plane strain condition is also dependent on the crack inclination angle (or the strength of each mode of fracture), e.g., in Fig. 5, b = 90° core region has the maximum reduction in size and b = 30° core region has the minimum reduction in size from plane stress to plane strain condition at the crack tip. The core regions show one global minimum and one local minimum, and for some values of crack inclination angle b, one weak local minimum is also shown, which is ignored as a possible value for crack initiation angle. The results of anisotropic R-criterion are present in Fig. 11 through Fig. 17. The isotropic R-criterion is also plotted with each figure for comparison. Figs. 11 and 12 show the effect of N on the crack initiation angles under uniaxial tension and uniaxial compression for both plane stress and plane strain conditions at the crack tip. The effect of N is more pronounced for plane strain case than for plane stress case. Overall, N shows little effect on the crack initiation angles, from which we can conclude that although the core region shape and size is affected considerably (Fig. 4) by the value of N, yet the orientation of minima do not change significantly. Figs. 13 and 14 show the effect of the normal anisotropic constants (F, G, H) for uniaxial loading for both plane stress and plane strain conditions. Normal anisotropic constants affect the predicted crack initiation angles for the case of uniaxial compression for plane stress condition at the crack tip whereas there seems to be no effect on the crack initiation

100

Isotropic R Pl. Stress F = 1, G = 4, H = 1, N = F = 1, G = 8, H = 1, N = F = 4, G = 1, H = 1, N = F = 8, G = 1, H = 1, N = F = 1, G = 1, H = 4, N = F = 1, G = 1, H = 8, N =

90

Crack initiation angle, −θo

o

80

3 3 3 3 3 3

70 60 50 σ

40 β

30

− θo

20 σ

10 0 0

10

20

30

40

50

60

70

80

90

80

90

Crack inclination angle, βo 100 Isotropic R Pl. Strain F = 1, G = 4, H = 1, N = F = 1, G = 8, H = 1, N = F = 4, G = 1, H = 1, N = F = 8, G = 1, H = 1, N = F = 1, G = 1, H = 4, N = F = 1, G = 1, H = 8, N =

90

Crack initiation angle, −θo

o

80 70

3 3 3 3 3 3

60 50

σ

40

β

− θo

30 20

σ

10 0 0

10

20

30

40

50

60

70

Crack inclination angle, βo Fig. 13. Uniaxial tension: Effect of F, G, H: (a) plane stress and (b) plane strain.

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180 160

Crack initiation angle, θo

o

140 120 100 σ

80 β

60

Isotropic R Pl. Stress F = 1, G = 4, H = 1, N = 3 F = 1, G = 8, H = 1, N = 3 F = 4, G = 1, H = 1, N = 3 F = 8, G = 1, H = 1, N = 3 F = 1, G = 1, H = 4, N = 3 F = 1, G = 1, H = 8, N = 3

θo

40 σ

20 0 0

10

20

30

40

50

60

70

80

90

Crack inclination angle, βo 180 160

Crack initiation angle, θo

o

140 120 100 80 σ

60 β

Isotropic R Pl. Strain F = 1, G = 4, H = 1, N = F = 1, G = 8, H = 1, N = F = 4, G = 1, H = 1, N = F = 8, G = 1, H = 1, N = F = 1, G = 1, H = 4, N = F = 1, G = 1, H = 8, N =

θo

40 20

σ

3 3 3 3 3 3

0 0

10

20

30

40

50

60

70

80

90

Crack inclination angle, βo Fig. 14. Uniaxial compression: Effect of F, G, H: (a) plane stress and (b) plane strain.

angles for the case of uniaxial compression for plane strain condition at the crack tip. Keeping in mind the small effect of N on the crack initiation angles, we can conclude that for uniaxial compression, F, G, and H affect the most under plane stress condition at the crack tip. The normal anisotropic constants (F, G, and H) affect the crack initiation angles more strongly (as can be seen from Fig. 13) in comparison to the effect of N, for uniaxial tension for both plane stress and plane strain conditions at the crack tip. The crack initiation angles for different values of F show lower values than the isotropic case, while for different values of H, the predicted crack initiation angles are higher than the isotropic case for b < 60°. The shape of the curves for different values of F can be compared with those shown by Theocaris and Phillippidis [7] for different fiber composites. It can be concluded that with the correct values of the anisotropic constants for a particular material, the anisotropic R-criterion can accurately predict the crack initiation angles. The results for different values of G show a different behavior for uniaxial tension case especially for plane stress condition at the crack tip. This could be explained by considering the shape of the core region (Fig. 6); for plane stress case, the core region for G = 4 and b = 90° lacks a local minimum, which is the direction of crack initiation under uniaxial tension (Reminder: Under uniaxial loading, the local minimum of the core region is the direction of crack initiation for tensile loading and the global minimum of the core region is the direction of the crack initiation for compressive loading). This indicates that for an anisotropic material with specific anisotropic constants’ values, the crack tip can be made stagnant (non-

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propagating) by choosing the appropriate loading axis. For the case of uniaxial tension under plain strain condition at the crack tip (note that the core region for this case does show a local minimum, Fig. 6b), crack initiation angle has a non-zero value for b = 90°; which increases with increasing value of G. Figs. 15–17 show the effect of F, G, and H on crack initiation angles for pure shear, biaxial (k = 0.5), proportional tension torsion (a = 0.5) loading, for both plane stress and plane strain condition at the crack tip. These results can be explained by comparing with the results for uniaxial loading cases as explained in Khan and Khraisheh [1]. For example, in the case of pure shear, the behavior is like uniaxial tension between b = 0–45°, and the behavior is like mirrored uniaxial compression between b = 45–90°. Similarly for biaxial loading case, for b = 0–35°, the behavior is like mirrored uniaxial compression, and for b = 35–90°, it is like uniaxial tension. The results for proportional tension–torsion loading can be explained in a similar fashion as the pure shear case. Due to the inclusion of a normal stress, the shape is shifted towards the left with reversal of resultant stress sign occurring at 67.5° instead of 45°. The purpose of presenting results from parametric study for various loading conditions is to show the flexibility and applicability of the new approach of introducing anisotropy in the crack initiation criterion through the yield criterion. The results of current study show qualitative similarity to previous studies. Therefore, it is shown that the new approach and analysis presented, which is applicable to a wide range of materials and loading conditions, can easily be used to char-

180 τ

160 β

− θo

τ

Crack initiation angle, −θo

o

140 120 100 80

Isotropic R Pl. Stress F = 1, G = 4, H = 1, N = F = 1, G = 8, H = 1, N = F = 4, G = 1, H = 1, N = F = 8, G = 1, H = 1, N = F = 1, G = 1, H = 4, N = F = 1, G = 1, H = 8, N =

60 40 20

3 3 3 3 3 3

0 0

10

20

30

40

50

60

70

80

90

80

90

Crack inclination angle, βo 180 τ

160 β

− θo

τ

Crack initiation angle, −θo

o

140 120 100 80

Isotropic R Pl. Strain F = 1, G = 4, H = 1, N = 3 F = 1, G = 8, H = 1, N = 3 F = 4, G = 1, H = 1, N = 3 F = 8, G = 1, H = 1, N = 3 F = 1, G = 1, H = 4, N = 3 F = 1, G = 1, H = 8, N = 3

60 40 20 0 0

10

20

30

40

50

60

70

Crack inclination angle, βo Fig. 15. Pure shear loading: Effect of F, G, H: (a) plane stress and (b) plane strain.

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180 Isotropic R Pl. Stress F = 1, G = 4, H = 1, N = 3 F = 1, G = 8, H = 1, N = 3 F = 4, G = 1, H = 1, N = 3 F = 8, G = 1, H = 1, N = 3 F = 1, G = 1, H = 4, N = 3 F = 1, G = 1, H = 8, N = 3

160

Crack initiation angle, −θo

o

140 120 100 80

σ

60

β

− θo

0.5σ

40 20 0 0

10

20

30

40

50

60

70

80

90

Crack inclination angle, βo 180 σ

160 β

Isotropic R Pl. Strain F = 1, G = 4, H = 1, N = 3 F = 1, G = 8, H = 1, N = 3 F = 4, G = 1, H = 1, N = 3 F = 8, G = 1, H = 1, N = 3 F = 1, G = 1, H = 4, N = 3 F = 1, G = 1, H = 8, N = 3

0.5σ

Crack initiation angle, −θo

o

140

− θo

120 100 80 60 40 20 0 0

10

20

30

40

50

60

70

80

90

Crack inclination angle, βo Fig. 16. Biaxial loading k = 0.5: Effect of F, G, H: (a) plane stress and (b) plane strain.

acterize crack initiation behavior of any material by determining the values of Hill’s anisotropic constants either directly by experiments or by parametric study to fit particular experimental data.

6. Conclusions Hill’s anisotropic yield criterion is used with linear elastic stress field to obtain non-dimensional variable crack tip core region radius. In addition, the R-criterion proposed earlier by the authors is also modified to include anisotropy. The relations developed are used to analyze mixed modes I–II cracks using the angled crack plate problem and the results are presented for various loading configurations of the plate and for both plane stress and plane strain conditions at the crack tip. Following conclusions can be made based on the results and discussion presented: 1. The R-criterion is based on a solid physical foundation and several studies have positively cited it [15–22] and have used it in the analyses [15,22].

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180 0.5σ

160

σ β

− θo

Crack initiation angle, −θo

o

140

Isotropic R Pl. Stress F = 1, G = 4, H = 1, N = F = 1, G = 8, H = 1, N = F = 4, G = 1, H = 1, N = F = 8, G = 1, H = 1, N = F = 1, G = 1, H = 4, N = F = 1, G = 1, H = 8, N =

3 3 3 3 3 3

120 100 80 60 40 20 0 0

10

20

30

40

50

Crack inclination angle, β

60

70

80

90

70

80

90

o

180 0.5σ

160

σ β

Isotropic R Pl. Strain F = 1, G = 4, H = 1, N = 3 F = 1, G = 8, H = 1, N = 3 F = 4, G = 1, H = 1, N = 3 F = 8, G = 1, H = 1, N = 3 F = 1, G = 1, H = 4, N = 3 F = 1, G = 1, H = 8, N = 3

− θo

Crack initiation angle, −θo

o

140 120 100 80 60 40 20 0 0

10

20

30

40

50

60

Crack inclination angle, βo Fig. 17. Proportional tension torsion loading a = 0.5: Effect of F, G, H: (a) plane stress and (b) plane strain.

2. The Hill’s anisotropic constants can be determined by simple uniaxial tests and then the core region characteristics and the crack initiation angles can be determined using techniques presented in this study. On the other hand, parametric studies can also be performed to determine the values for anisotropic constants to fit experimental data for any anisotropic material. Therefore the methodology presented can be widely and easily applied to various anisotropic materials. 3. The methodology used in this study to include anisotropy for crack initiation criterion presents an advantage over the conventional approach. This new approach uses an anisotropic yield criterion to introduce anisotropy along with singular elastic stress field and therefore does not require determination of anisotropic crack tip stress field for every anisotropic material. 4. The effect of each anisotropic constant on the crack tip core region shape is different and it varies from modifying the shape of a part of the core region to modifying the entire shape of the core region. 5. The effect of anisotropic constants on the crack tip core region size depends on the mode of fracture, e.g., the effect of normal anisotropic constant F on the core region size becomes dominant as the strength of mode-I is increased by increasing the crack inclination angle, b. 6. It is well established that the size of the core region in plane strain condition at the crack tip is smaller than the size in the case of plane stress condition at the crack tip. In anisotropic materials, this reduction in size from plane stress to plane strain condition at the crack tip strongly depends on and varies widely for each anisotropic constant.

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7. For uniaxial loading case, the effect of F, G, and H is the strongest for uniaxial tension case. The effect of G presents a possible way of arresting a crack or making it dormant in an anisotropic material by selecting the proper loading axis. 8. Comparison with previous studies is made wherever possible. However limited literature available on anisotropic crack initiation, particularly experimental results, calls for a great need for development of theoretical models and experimental investigation into anisotropic crack initiation. The issue of experimental investigation is currently under consideration by the authors.

Appendix A. Coefficients for the non-dimensional Hill’s anisotropic core region radius       13F þ 5G þ 4H þ 2N 25F þ 9G þ 4H  2N 5F þ 3G  4H  2N þ cos h þ cos 2h 16 32 16     5F þ 3G  4H þ 2N F þ G þ 4H  2N þ cos 3h þ cos 5h 16 32       F þ 25G þ 20H þ 10N F  49G  36H þ 18N F þ 7G þ 12H þ 6N g 2 ðhÞ ¼ þ cos h þ cos 2h 16 32 16     F  7G  12H þ 6N F  G  4H þ 2N þ cos 3h þ cos 5h 16 32       5F  21G þ 12H  6N F þ G þ 4H þ 2N 3F  5G  4H þ 2N g 12 ðhÞ ¼ sin h þ sin 2h þ sin 3h 16 4 8   F  G  4H þ 2N þ sin 5h 16 g 1 ðhÞ ¼

    mð4m  5ÞF þ mð4m  3ÞG mðF  GÞ ð1 þ cos hÞ þ ðcos 2h þ cos 3hÞ 2 2     mð4m  1ÞF þ mð4m  7ÞG mðF  GÞ h2 ðhÞ ¼ ð1  cos hÞ þ ðcos 2h  cos 3hÞ 2 2

ðA-1Þ

h1 ðhÞ ¼

ðA-2Þ

h12 ðhÞ ¼ ðmð3  4mÞF þ mð5  4mÞGÞ sin h þ mðG  FÞ sin 3h

Appendix B. Coefficients for the anisotropic R-criterion x10 ¼ 4lðG þ HÞ x9 ¼ 4½ð4G þ 7H  2NÞ  l2 ðG þ HÞ x8 ¼ 12lð3G  7H þ 3NÞ x7 ¼ 2½ð9F  5G  36H þ 24NÞ þ l2 ð12G þ 32H  18NÞ x6 ¼ lð75F þ 35G þ 212H  134NÞ x5 ¼ 3½ð15F þ 7G þ 36H  18NÞ þ l2 ð25F  9G  60H þ 30NÞ

ðB-1Þ

x4 ¼ lð89F  49G  268H þ 106NÞ x3 ¼ 2½ð8F  12G  48H þ 18NÞ þ l2 ð5F þ 9G þ 36H  16NÞ x2 ¼ 3lðF þ 11G þ 24H  14NÞ x1 ¼ ½ðF þ 9G þ 16H  14NÞ þ l2 ð5F  7G þ 2NÞ x0 ¼ lðF  3G þ 2NÞ g10 ¼ 4lm½2G þ ðF þ GÞm g9 ¼ 4m½ð3F  5GÞ þ 2Gl2 þ mðF þ GÞð1  l2 Þ g8 ¼ 12lm½2ð2F þ GÞ þ ðF þ GÞm g7 ¼ 8m½ðF  3GÞ þ ð5F  GÞl2 þ 2mðF þ GÞð1  l2 Þ g6 ¼ 8lm½2ð3F þ 2GÞ þ ðF þ GÞm g5 ¼ 8m½2F þ ð3F  GÞl2 þ mðF þ GÞð1  l2 Þ g4 ¼ 8lm½2ð3F  2GÞ  ðF þ GÞm g3 ¼ 8m½ð3F  GÞ þ ð3F þ GÞl2 þ 2mðF þ GÞð1  l2 Þ g2 ¼ 12lm½2ð2F  GÞ  ðF þ GÞm g1 ¼ 4m½ðF  3GÞ þ 2ðF þ 2GÞl2 þ mðF þ GÞð1  l2 Þ g0 ¼ 4lm½2G  ðF þ GÞm

ðB-2Þ

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w1 ¼ ½ðF þ 49G þ 36H  18NÞ þ l2 ð25F  9G  4H þ 2NÞ w2 ¼ 8½ðF  7G  12H  6NÞ þ l2 ð5F  3G þ 4H þ 2NÞ w3 ¼ 18½ðF þ 7G þ 12H  6NÞ þ l2 ð5F  3G þ 4H  2NÞ w4 ¼ 25½ðF þ G þ 4H  2NÞ þ l2 ðF  G  4H þ 2NÞ w5 ¼ 2½ð5F þ 24G  12H þ 6NÞl w6 ¼ 32½ðF  G  4H  2NÞl

ðB-3Þ

w7 ¼ 18½ð6F þ 10G þ 8H  4NÞl w8 ¼ 50½ðF þ G þ 4H  2NÞl n1 ¼ 16m½ðF þ 7GÞ þ l2 ð5F þ 3GÞ þ 4mðF þ GÞð1  l2 Þ n2 ¼ 64m½ðF þ GÞð1 þ l2 Þ n3 ¼ 144m½ðF  GÞð1  l2 Þ n4 ¼ 0 n5 ¼ 32lm½ð3F þ 5GÞ þ 4mðF þ GÞ

ðB-4Þ

n6 ¼ 0 n7 ¼ 288lmðF  GÞ n8 ¼ 0 where l is defined as l¼

KI fKI ðbÞ ¼ K II fKII ðbÞ

ðB-5Þ

fKI(b) and fK II(b) can be found for different loading conditions from Eqs. 9, 12, 15, 18, (Refer to Fig. 1a). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

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