On crack path evolution rules

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Engineering Fracture Mechanics 77 (2010) 1781–1807

Contents lists available at ScienceDirect

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

On crack path evolution rules K.P. Mróz *, Z. Mróz ´ skiego 5b, 02-106 Warsaw, Poland Institute of Fundamental Technological Research, Pawin

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 15 December 2009 Received in revised form 9 March 2010 Accepted 23 March 2010 Available online 27 March 2010

The models of crack growth in mixed mode conditions are reviewed for the plane and three-dimensional (3D) states of stress. Both critical load value and crack path or surface growth are predicted by different criteria in terms of elastic singular stress states and Tstress component. Monotonic and cyclic loading induced crack growth is considered. The energy and critical plane criteria based on local or non-local measure of stress and strain are most useful in developing predictive crack growth simulation. The ﬁnite critical distance from the crack tip should be speciﬁed to provide averaged or local stress and strain states. The application of MK-criterion of crack growth expressed in terms of volumetric and deviatoric strain energies is presented for several speciﬁc cases of monotonic and cyclic loading. The concepts of smooth and rough crack surface are discussed with application to 3D crack surface growth. Ó 2010 Elsevier Ltd. All rights reserved.

Keywords: Fracture criteria T-stress Fatigue crack growth 3D crack Smooth and rough cracks Crack path Critical plane Non-local models

1. Introduction The prediction of crack growth path under imposed loading and of its rate of growth constitutes the most important problem in fracture mechanics. Most papers devoted to this problem are based on the assumption of linear elastic fracture mechanics (LEFM), but recent studies also include the effects of plastic deformation and also damage accumulation at the crack tip. So the term elasto-plastic fracture mechanics (EPFM) can be applied for these studies. The present paper will apply LEFM to predict crack growth but the effects of plasticity and damage will be accounted for by postulating the relative fracture criteria. The crack growth can be simulated as a succession of straight segments in plane cases or linear surface elements in 3D cases. The effects of crack curvature can also be incorporated in the analysis. The stress and strain states near the crack tip (Fig. 1) can be expressed within the assumption of LEFM in terms of the stress intensity factors (SIF) KI, KII, KIII and the asymptotic singularity solution. For the plane strain or stress cases the stress state is expressed in the Cartesian reference system as follows [1]

K

h

h

3h

K

h

h

3h

I II ﬃ cos rxx ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 1 sin sin pﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin 2 þ cos cos þ T; 2 2 2 2 2 2 2pr 2pr KI h h 3h K II h h 3h ﬃ cos 1 þ sin sin þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos sin cos ; ryy ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 2 2 2pr 2pr

K

h

h

3h

K

h

h

3h

I II ﬃ cos sin cos þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos ; rxy ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 1 sin sin 2 2 2 2 2 2 2pr 2pr

* Corresponding author. Tel.: +48 22 826 12 81. E-mail addresses: [email protected] (K.P. Mróz), [email protected] (Z. Mróz). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.03.038

ð1Þ

1782

K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

σ∞

r

θ

α

kσ ∞

kσ ∞

σ∞ Fig. 1. The geometry of the main crack.

and in the polar coordinate system:

K

5

1

h

3h

K

5

3h

3

h

I II ﬃ cos cos þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin þ sin þ T cos2 h; rrr ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2 4 2 4 2 4 2 2pr 4 2p r KI 3 h 1 3h K II 3 h 3 3h 2 ﬃ cos þ cos þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin sin þ T sin h; rhh ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2 4 2 4 2 4 2 2pr 4 2p r

K

1

1

h

3h

K

1

h

3

ð2Þ

3h

I II ﬃ þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ T cos h sin h: sin þ sin cos þ cos rrh ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2 4 2 2 4 2 2pr 4 2p r 4

The nonsingular term T (T-stress) is added in (1) as it affects the crack orientation. When ry0 ¼ r1 and rx0 ¼ kry0 ¼ kr1 are 0 0 0 the principal stresses and the crack tip is inclined at the angle a to the y -axis then T = r(1 k)cos 2a. The x ,y -reference system follows the axes of principal stresses and the x,y-system represents the crack plane and its normal vector, Fig. 1. The higher order terms of the expansion with respect to the distance r from the crack tip are not introduced in the analysis. 2. Two-dimensional (2D) mixed mode fracture criteria The fracture criteria should be based upon physical models accounting for damage processes of the material in front of the crack tip. However, since the exact description of these processes is difﬁcult, the problems are usually treated within the framework of LEFM by using representative values such as stress components or speciﬁc strain energy speciﬁed at some ﬁnite distance from the crack or sharp notch tip. This characteristic distance specifying the core region affected by damage and plastic deformation is an unknown parameter to be speciﬁed experimentally or analytically. 2.1. Monotonic crack growth The ﬁrst fracture criteria related to the angled crack problem and the orientation of crack growth used the simplest assumption of the core region bounded by a circle of radius rC from the crack tip. The criteria such as: maximum circumferential tensile stress (MTS), minimum strain energy density (SED), maximum energy release rate (MERR) and local symmetry (LS) became popular and were analyzed in the literature. The maximum tensile hoop stress criterion (MTS) formulated by Erdogan and Sih [2] in 1963 was one of the ﬁrst conditions predicting critical stress and crack growth orientation. Expressing the hoop stress in the form following from (2) without T-stress effect:

K

3

h

1

3h

K

3

h

3

3h

I II ﬃ cos þ cos þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin sin ; rhh ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 4 2 4 2 4 2 2pr C 4 2pr C

it is postulated that crack will grow when

ð3Þ

K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

@ 2 rhh

@ rhh ¼ 0; @h

@h2

< 0 at h ¼ hPR ;

K

IC ﬃ rhh ðrC ; hPR Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2prC

1783

ð4Þ

and the stationary condition provides

K I sin hPR þ K II ð3 cos hPR 1Þ ¼ 0:

ð5Þ

For mode I there is hPR = 0 and for pure mode II there is hPR = 70.5° and KIIC = 0.866KIC. The SED-criterion by Sih [3] in 1974 expressed the speciﬁc strain energy density at the radius rC in terms of the energy density factor S expressed as follows:

S ¼ a11

K 2I

þ 2a12

p

K I K II

p

þ a22

K 2II

p

ð6Þ

;

where

1 ð1 þ cos hÞðj cos hÞ; 16l 1 ¼ sinð2 cos h ðj 1ÞÞ; 16l 1 ¼ ððj þ 1Þð1 cos hÞ þ ð1 þ cos hÞð3 cos h 1ÞÞ; 16l

a11 ¼ a12 a22

ð7Þ

and l is the elastic shear modulus, j = 3–4m for the plane strain case and j = (3 m)/(1 + m) for the plane stress. The crack growth criteria now are

@S ¼ 0; @h

@2S @h

2

< 0 at h ¼ hPR ;

SðhPR Þ ¼

ð1 2mÞK 2IC 4pl

ð8Þ

and the crack growth orientation hPR corresponds to minimum of S reaching its critical value. For pure mode II there is hPR = 79.2° and KIIC/KIC = 1.054 for m = 0.2, where m is the Poisson ratio. For larger ratios KII/KI this criterion provides predictions differing signiﬁcantly from those of MTS-criterion. The maximum energy release rate criterion (MERR) follows from the Grifﬁth condition and states that crack growth follows the orientation of maximum energy release G = @ P/@s, where P is the potential energy of the element and s denotes the length of the crack path, thus

@G ¼ 0; @h

@2G 2

@h

6 0;

at h ¼ hPR ;

GðhPR Þ ¼

K 2IC ; E

ð9Þ

where E* = E for plane stress cases and E* = E/(1 m2) for plane strain cases. Here GðhÞ ¼ lims!0 ½PðsÞ Pð0Þ where P(s) is the potential energy for the kinked crack of length s and P(0) is the energy for the initial crack. For a straight crack with a kink of inﬁnitesimal length the SIFs at the kink tip are related to the intensity factor of original crack, namely:

h 3 h K II cos sin h; 2 2 2 1 h 1 h K II ¼ K I cos sin h þ K II cos ð3 cos h 1Þ; 2 2 2 2

ð10Þ

K i ¼ F ij K j ;

ð11Þ

K I ¼ K I cos3

or

i; j ¼ 1; 2:

The energy release along the kinked crack now is

G¼

1 2 ðK þ K 2 II Þ ¼ GðK I ; K II ; hÞ ¼ GC : E I

ð12Þ

The MERR-criterion was analyzed by Hussian et al. in 1974 [4] and by Wu [5] for modes I and II loading. For mode II this criterion predicts hPR = 75.6° and KIIC/KIC = 0.817. The local symmetry conditions (LS) proposed by Goldstein and Salganik [6] in 1974 requires that crack propagation occurs along the path of vanishing KII, thus

K II ¼ 0;

on s-path

ð13Þ

and the crack path orientation is the same as that predicted by Eq. (5) resulting from MTS-criterion. To analyze the relation between the conditions MTS, MERR, LS and max KI, let us express the stationary condition of G as follows

@G 1 @K I @K 1 @F 11 @F 12 @F 21 @F 22 ¼ K I þ K II II ¼ K I K I þ K II þ K II K I þ K II ¼ 0: @h E E @h @h @h @h @h @h

ð14Þ

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K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

Assume now that the coefﬁcients of Fij satisfy the conditions

@F 11 =@h @F 12 =@h ¼ ¼ gðhÞ or F 21 F 22

@K I ¼ gðhÞK II : @h

ð15Þ

Then K II ¼ 0 implies @G/@h = 0 and @K I =@h ¼ 0, so these three criteria are equivalent. It can be veriﬁed that the components (10) of Fij satisfy the conditions (15). However, they are obtained for vanishing length of the kink crack. The expression of the SIFs with respect to the crack length s in the form

3 1 pﬃﬃ KðsÞ ¼ K þ K 2 s þ K ð1Þ s þ 0 s2 ;

ð16Þ

developed by Amestoy and Leblond [9,8,7] indicates that conditions (15) are not satisﬁed when the higher order terms are included. The apparent crack extension force criterion (CEF) proposed by Strifors [10] in 1973, next applied by Hellen and Blackburn [11] was expressed in terms of path-independent J-integral vector

Jk ¼

Z

ðWnk ui;k T i Þdl

k ¼ I; II;

ð17Þ

C

where C is an open path connecting two points on opposite crack sides, W is the strain energy density, nk is the k component of the unit outward normal vector to the C, ui,k is the displacement gradient, Ti is the traction on the path C. The J-vector is then projected onto kink growth direction, thus

J p ðhÞ ¼ J 1 cos h þ J2 sin h

ð18Þ

and the crack propagation condition is expressed: max Jp(h) = JPCR. Chang [12] in 1981 proposed the maximum tangential strain criterion (MTSN) specifying the hoop strain ehh ¼ ðrhh v rrr Þ=E along the crack tip core region r = rC. The crack angle then corresponds to a maximum value of ehh reaching its critical value eT = rT/E, where rT is the brittle tensile fracture strength. Papadopoulos [13] in 1987 proposed the stress invariant criterion (Det-criterion), assuming the third stress invariants as a local fracture condition, reaching its maximum at r = rC. Koo and Choy [14] in 1991 postulated that crack propagates along the direction of maximum tangential strain energy density component Sh = 1/2rheh at r = rC (MTSE). Then, the tangential strain energy density factor C is equal:

C ¼ r 0 Sh ¼ b11

K 2I

p

þ b12

K I K II

p

þ b22

K 2II

p

ð19Þ

;

where

1 ðð1 þ cos hÞðj þ 2 þ cos hÞÞ 64l 1 ¼ ðsin h 3=2 j 3 cos hÞÞ; 64l 1 2 ¼ ð3 sin hðj þ 3 cos hÞÞ: 64l

b11 ¼ b12 b22

ð20Þ

Kong et al. [15] in 1995 formulated the Mt-criterion assuming the crack to grow along the line of maximal stress triaxiality ratio M = rH/rEQ, where rH is the hydrostatic stress, and rEQ denotes the Huber–Mises effective stress, thus

1

2ð1 þ v Þ

h

h

rH ¼ ðrxx þ ryy þ rzz Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ K I cos K II sin ; 3 2 2 3 2pr

1=2 1 ðrxx ryy Þ2 þ ðryy rzz Þ2 þ ðrzz rxx Þ2 þ 6r2xy 2 1 3 2 9 2 h 2 ¼ pﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃ K I K II sin h þ 2ð1 2v Þ2 K 2I þ 6K 2II cos2 2 2 2 2pr 2 1=2 2 h þ K I K II ð3 sin 2h 2ð1 2v Þ2 sin hÞ : þ8ð1 v þ v 2 ÞK 2II sin 2

rEQ ¼

ð21Þ

However, there exists the possibility to improve accuracy of the proposed fracture criteria by including the T-stress effect into the criterion. By adding the nonsingular terms to singular stress, the value of r specifying the core region becomes essential. There have been numerous studies of extended fracture criteria accounting for T-stress, starting from the analysis of Ewing and Williams [16] in 1972 related to MTS-criterion. The subsequent paper by Swedlow [17], Maiti and Smith [18,19], Wong [20], Wu and Li [21] demonstrated that the crack growth orientation depends on the radius rC and better agreement with experiment can be attained by assuming the value of rC to depend on the orientation angle h.

K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

1785

A simple extension can be obtained by assuming the process zone to coincide with the localized plastic zone at the crack tip. This idea was applied by Theocaris and Andrianopoulos [22] who in 1982 proposed T-criterion, modifying the earlier SED-criterion. The speciﬁc stress and strain energy S is split into distortional SD and hydrostatic portions SH, thus

S ¼ SD þ SH ; 1 2m ðrxx þ ryy þ rzz Þ2 ; SH ¼ 6E 1 þ m 2 rxx þ r2yy þ r2zz rxx ryy ryy rzz rxx rzz þ 3r2xy : SD ¼ 3E

ð22Þ

As rzz = rzx = rzy = 0 for plane stress and rzz = m(rxx + ryy) for plane strain, there is for the plane stress

SD ¼

1 þ m ðrxx þ ryy Þ2 3 rxx ryy r2xy ; 3E

SH ¼

1 2m ðrxx þ ryy Þ2 6E

ð23Þ

and for the plane strain

1 þ m ð1 m þ m2 Þðrxx þ ryy Þ2 3 rxx ryy r2xy ; 3E ð1 þ mÞð1 2mÞ ðrxx þ ryy Þ2 : SH ¼ 6E

SD ¼

ð24Þ

Assuming the distortional energy to correspond to plastic ﬂow and the hydrostatic stress energy to decohesion and fracture, the T-criterion postulates that the crack propagates along the direction corresponding to a maximum of total speciﬁc stress energy on the perimeter of varying radius, r = rC(h) speciﬁed by the condition of constant distortional energy at the yield point, thus

1þm 2 r ; for the plane stress; 3E YS ð1 þ mÞð1 m þ m2 Þ 2 ¼ rYS for the plane strain; 3E

SD ¼ SYS D ¼

ð25Þ

SD ¼ SYS D

ð26Þ

where rYS is the yield stress in uniaxial tension. The T-criterion can now be expressed as follows

@SH ðr; hÞ ¼ 0 for SD ðr; hÞ ¼ SYS D ¼ const: @h

ð27Þ

This criterion is formulated for the varying core region radius. In the case of brittle materials it tends to the SED-criterion as the size of plastic zone is very small and its shape can be assumed as circular. Wasiluk and Golos [23] in 2000 proposed W-criterion assuming that the crack growth angle is speciﬁed by the minimum value of Z-factor deﬁned as Z = rp(h)/a where rp (h) is radius of plastic zone and a is half the crack length. The Huber–Mises yield conditions has been applied. The W-criterion is based on an assumption of the minimum energy consumed during the fracture process in the plastic zone. A similar criterion formulated by Khan and Khraisheh [24] in 2004 is based on the assumption that the crack growth orientation corresponds to the local or global minimum of radius of plastic core boundary at the crack tip stating that crack growth follows the direction of minimum distance from the crack tip to the plastic core perimeter. Yan et al. [25] in 1992 speciﬁed the plastic core region by applying the yield condition dependent on the ﬁrst stress and second stress deviator invariants, thus

f ¼ aI1 þ ðJ 2 Þ1=2 c ¼ 0;

ð28Þ

where I1 ¼ rkk ; J 2 ¼ sij = rij dijrkk/3, rij and sij are the stress tensor and the deviatoric stress tensor, respectively, I1 and J2 are the ﬁrst stress invariant and the second stress deviator invariant, dij is the Kronecker symbol, a and c are the positive materials constants. In the case a = 0 the Huber–Mises yield condition is obtained. However, in the vicinity of the crack tip we may distinguish three zones of different damage states, presented schematically in Fig. 2. First, the process zone (PZ) of large damage affecting essentially material stiffness and strength. Second, the plastic slip zone and the third zone characterized by growth of microcracks induced by tensile microstress [26]. This damage is assumed to be related to the hydrostatic stress energy density SH. Basing on this assumption Mróz [27] in 2008 proposed the MK-fracture criterion postulating that size of damage zone is speciﬁed by the condition SH ¼ SCH and macrocrack propagation follows the direction of smallest plastic dissipation, that is corresponds to a minimum value of the distortional stress energy SD speciﬁed along the perimeter SH ðr; hÞ ¼ SCH ¼ const:, thus 1 s s , 2 ij ij

@SD ðr; hÞ ¼ 0; @h

for SH ðr; hÞ ¼ SCH ¼ const:

ð29Þ

or

SH j ! hpr SD max

at

SH ¼ SCH ¼ const:

!

SD ðrðSCH ; hÞÞjmin ! hpr

ð30Þ

1786

K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

S (σ ) = const.

PZ

S (σ ) = const.

ρ

ρ (σ ) ρ (σ )

ρ (σ )

0

H

Fig. 2. The schematic plot of damage zones within the crack tip region (where q is the conceptual density of defects).

where

2 rðSCH ; h; aÞ ¼ r H jSH ¼const: ¼

2

p

32

h h 6 K I cos 2 K II sin 2 4qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C 6E SH ð1mÞð1þ Tð m Þ2

7 5

aÞ

ð31Þ

and m* = 0 for plane stress, m* = m for plane strain cases. The term under square root can be replaced by the equivalent stress rC, so we may write

rðrC ; h; aÞ ¼

2 2 K I cos 2h K II sin 2h : p rC TðaÞ

ð32Þ

The crack growth criterion can now be formulated as follows. For the speciﬁed loading the distance deﬁned by (31) provides the size of damage zone associated with the assumed volumetric energy level. The critical value r(rC, hpr, a) = rC is related to the fracture strength parameters KI, KII. The crack growth does not occur when rC P rðrC ; hpr ; aÞ and crack growth occurs when r(rC, hpr, a) P rC with the crack growth onset speciﬁed by r(rC, hpr, a) = rC. This criterion will be discussed in the following. Consider the particular case of mode I when a = p/2 and the crack propagation occurs along the primary crack, h = 0. From (32), it follows that

K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

p 2 rCI ðrC ; 0; Þ ¼ 2 p

2 2 K IC 2 K IC ¼ : rC Tðp2Þ p rC r1 ð1 kÞ

1787

ð33Þ

Assume now, that the critical state is reached on the radius rC = rCI = const. The crack propagation criterion can now be presented as follows

h h rC Tð a Þ rC r1 ð1 kÞ cos 2a ¼ K IC K I cos K II sin ¼ K IC ; 2 2 rC þ r1 ð1 kÞ rC T p2

ð34Þ

where T(p/2) = r1(1 k). This criterion is based on the assumption that mode I is prevailing along the crack path and KIC is the single crack growth parameter. However, in more general cases Eq. (34) should be written in following form:

hPR jK II ¼0 rC TðaÞ h h K I cos K II sin ¼ K IC sin : 2 2 2 rC TjK II ¼0

ð35Þ

The general criterion can be proposed by accounting for two fracture parameters of mode I and II. For pure mode II the critical value rCII equals

rCII ¼

2 2 2 K IIC sinðhPR =2Þ 2 0:76 K IIC ¼ ; p rC TðaÞ p rC

ð36Þ

as T(a) = 0 and sin(hPR/2) is equal to 0.76 for pure mode II loading. In more general cases Eq. (36) should be replaced by

rCII

h j 32 2 PR K I ¼0 2 2 4K IIC sin 5 : ¼ p rC TjK I ¼0

ð37Þ

Assume that for a mixed mode loading the critical value of rC is speciﬁed by the relation

rC ¼ cos c r CI þ sin c r CII ;

ð38Þ

K II =K IIC K I =K IC

ð39Þ

where

tan c ¼

is a measure of mode mixity. Let us note that rC = rCI for mode I (sin c = 0 and cos c = 1) and rC = rCII for mode II (cos c = 0 and sin c = 1). The crack propagation criterion is now stated in the form (Fig. 1)

2 h h rC r1 ð1 kÞ cos 2a 2 rC r1 ð1 kÞ cos 2a 2 K I cos K II sin ¼ cos c K IC þ sin c 0:76 K IIC 2 2 rC þ r1 ð1 kÞ rC

ð40Þ

and in the general form:

!2 !2 2 hPR jK II ¼0 rC TðaÞ hPR jK I ¼0 rC TðaÞ h h K I cos K II sin ¼ cos c K IC cos þ sin c K IIC sin 2 2 2 rC TjK II ¼0 2 rC TjK I ¼0

ð41Þ

or

2 !2 !2 3 2 hPR jK II ¼0 rC TðaÞ hPR jK I ¼0 h h rC TðaÞ 2 4 5; K I cos K II sin ¼ K IC cos c cos þ sin c / sin 2 2 2 rC TjK II ¼0 2 rC uTjK II ¼0

ð42Þ

where KIIC = /KIC and TjK I ¼0 ¼ u TjK II ¼0 . However, in many cases condition (34) which incorporates only KIC is sufﬁciently accurate, especially for brittle materials. In Fig. 3, the analysis of crack growth based on Eq. (34) incorporating only KIC is presented and the critical values of KI and KII are speciﬁed for varying mode mixity. Fig. 3 presents the results according to Eq. (41) in terms of mean values of TjK I;II ¼0 and also KI,II C. Fig. 4 presents the bounding curves based on Eq. (41) and extreme values of TjK I;II ¼0 and KI,II C. The analysis is compared with the experimental results of Ayatollahi et al. [28] who conducted mixed mode I/II fracture tests on polymethylmethacrylate (PMMA) and speciﬁed fracture loads for varying initial crack angles. The numerical analysis of [28] was based on MTS-criterion with account for T-stress. The constant radius r = rC was assumed in specifying the core region for accounting for damage material ahead of the crack tip. Assuming rC varying between 0.065 and 0.25 mm these authors obtained fair correlation between mixed mode fracture toughness prediction and experimental results. The present analysis applying MKcriterion (34) and varying r = rC(h) provides more accurate prediction of critical load locus in the KI, KII plane. Fig. 5 shows the Huber–Mises elastic–plastic boundary and the decohesion boundary for the nonsingular solution (with T-stress term) based on Eq. (31) for a = 90° cf. (Fig. 1), and Figs. 6 and 7 show the similar energy proﬁles for the singular stress ﬁeld for a = 30°, 90°. Particularly in the case a = 90° the decohesion area is considerably larger than the plastic area and for the same value of load-

K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

K II / K IC

1788

K I / K IC

K II / K IC

Fig. 3. Mixed mode fracture locus speciﬁed experimentally in [28]. The solid line represents the average test results, the dashed line represents results according to Eq. (34) and the dotted line represents results obtained from Eq. (41). The results according to Eq. (41) based on the mean values of TjK I ¼0 ; TjK II ¼0 and KIC, KIIC.

K I / K IC Fig. 4. Mixed mode fracture locus speciﬁed experimentally in [28]. The solid line represents the average test results and the dotted and the dashed lines represent results according to Eq. (41) based on the extreme values of TjK I ¼0 ; TjK II ¼0 and KIC, KIIC.

ing the shape and the size of both boundary curves vary, when the T-term is introduced. Fig. 8 presents the energy proﬁles for the plane stress and plane strain conditions in pure shear loading. Next, Fig. 9 presents graphically the MK-criterion: T D ðrðhÞ; T const: Þ calculated along the r ¼ rðT c: H H ; hÞ contour following form (29)–(31) and the nonsingular solution, where the minimum of the radius of T D ðrðT cH ; hÞÞ represents the direction of the crack propagation. Fig. 10 and Table 1 present the angle of crack propagation hpr predicted by MK-criterion for the various main crack inclinations a, for the uniaxial tension (k = 0) and various ratios of applied and critical stress. The most interesting are the differences of the crack propagation angles for the same value of a but different load values. This effect can be associated with the T-stress which affects signiﬁcantly the propagation angles at varying load levels. The T-stress term is also essential in stability control of straight crack path under mode I loading conditions [29]. Also the shape and size of the plastic zone ahead of the crack tip depends on the value of Tstress [30]. The analysis of [31] demonstrates the importance of retaining the second term in asymptotic series representing the local stress.

1789

K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807 90

0.0008

120

90

0.0008

120

60

0.0006 150

150

30

0.0004

30

0.0004

0.0002

0.0002

180

0

180

330

210

0

330

210

300

240

60

0.0006

300

240

270

270

(a)

(b)

Fig. 5. The Mises elastic–plastic boundary, TD = const. (the dotted line) and the decohesion boundary, TV = const. (the solid line) Eq. (31), for Al 7075-T6: pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ a ¼ 90 ; rYS ¼ 538 MPa; rC ¼ 586 MPa;K TH ¼ 1 MPa m; K C ¼ 25 MPa m; E ¼ 71 GPa; m ¼ 0:33; 2a ¼ 20 mm. The nonsingular solution (T – 0). (a) The case of plane stress. (b) The case of plane strain.

90 120

0.0015

90 60

0.0004

120

60

0.001 150

30

0.0002

150

30

0.0005

180

0

330

210

300

240

180

0

330

210

300

240

270

270

(a)

(b)

Fig. 6. The Mises elastic–plastic boundary, TD = const. (the dotted line) and the decohesion boundary, TH = const. (the solid line) Eq. (31), for Al 7075-T6: pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ rYS ¼ 538 MPa; rC ¼ 586 MPa; K TH ¼ 1 MPa m; K C ¼ 25 MPa m; E ¼ 71 GPa; m ¼ 0:33; 2a ¼ 20 mm. The singular solution (T = 0) and the plane strain conditions. (a) a = 90°; (b) a = 30°.

In fact, the T-term introduced the correction to the core contour rC = rC(h), Eq. (31), on which the stress tensor components are speciﬁed. For the singular solution with T = 0 the ratio r/rC does not affect the orientation angle as the stress components are proportionally varying along any radius from the crack tip. Similarly, when the core region is speciﬁed by the constant radius r = rC, there is no effect of r/rC ratio on the crack propagation angle. The scattering of results can be associated with the inhomogeneity of material represented in the MK-criterion by scatter in the values of rC. However, it is interesting to note that the smallest scattering of these results occurs in neighborhood of a = 45° consistently with the MK-criterion predictions. The different predictions of the angle crack propagation by accounting for the T-stress term was observed already in several works. In 1995 Seweryn and Mróz [33,34] discussed the application of the non-local failure criterion to sharp notches and cracks and formulated the R-criterion (see Section 4). They showed that MTS-criterion is a particular case of non-local stress condition. In 1998 Seweryn [35] discussed the effect of nonsingular T-stress term on the crack propagation in R-criterion (Fig. 11) for different ratios d0/a, where d0 is the size parameter representing the size of damage zone generating different values of the propagation angles, but without any physical background. The size parameter d0 representing the size of damage zone can be equivalent to the Grifﬁth condition in the case of tensile crack propagation. This provides

d0 ¼

2 2 K IC

p rc

:

ð43Þ

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0.0015

90

0.0004

120

60

60

0.001 150

0.0002

150

30

30

0.0005

180

0

180

330

210

0

300

240

330

210

300

240

270

270

(a)

(b)

Fig. 7. The Mises elastic–plastic boundary, TD = const. (the dotted line) and the decohesion boundary, TH = const. (the solid line) Eq. (31), for Al 7075-T6: pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ rYS ¼ 538 MPa; rC ¼ 586 MPa; K TH ¼ 1 MPa m; K C ¼ 25 MPa m; E ¼ 71 GPa; m ¼ 0:33; 2a ¼ 20 mm. The singular solution (T = 0) and the plane stress conditions. (a) a = 90°; (b) a = 30°.

90

90

0.0015

120

60

60

0.001

0.001

150

150

30

30

0.0005

0.0005

180

0

330

210

300

240 270

(a)

0.0015

120

180

0

330

210

300

240 270

(b)

Fig. 8. The Mises elastic–plastic boundary, TD = const. (the dotted line) and the decohesion boundary, TH = const. (the solid line) Eq. (31), for Al 7075-T6: pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ rYS ¼ 538 MPa; rC ¼ 586 MPa; K TH ¼ 1 MPa m; K C ¼ 25 MPa m; E ¼ 71 GPa; m ¼ 0:33; 2a ¼ 20 mm. The singular solution (T = 0) and a = 90° for the pure shear loading. (a) Plane stress conditions. (b) Plane strain conditions.

He concluded that the approach applied to the materials with defects should be modiﬁed in order to account for stress redistribution due to plastic strain (or microcracks) in order to predict better crack initiation or growth. In the case of MK-criterion these differences result explicitly from varying geometrical conﬁgurations and loading conditions of the problem. In 2D cases the R-criterion is expressed in terms of the failure stress function averaged over the distance d0, but the distance rC = const. or r(rC, hpr, a) speciﬁes the point ahead of the crack tip where the stress and speciﬁc energy are evaluated. In general there should be d0 P r C . However, for large stress gradient or the singular stress ﬁeld and for a = p/2, k = 0, h = 0 (Fig. 1) r(rC, hpr, a) and d0 values can be equal, when KI = KIC. The present discussion of crack growth conditions and their path orientation in the plane strain or stress cases is based on linear-elastic stress and strain ﬁelds at the crack tip. For the stress or local speciﬁc energy conditions the characteristic distance rC from the crack tip constitutes a new parameter. In fact, the crack propagation condition now has a non-local character. The parallel formulation of critical plane models have the same character as they are expressed in terms of mean stress or strain components over plane face element of speciﬁed ﬁnite area. The use of distortional and volumetric speciﬁc energies to determine rC and the failure condition, such as (34) or (40), provides the simple methodology of prediction of crack path and the limit fracture locus. Let us note that crack growth orientation now depends on the stress ratio r/rC which affects the value of r speciﬁed by (32). Using the formulation of linear fracture mechanics, the size parameter d0, or critical distance r is speciﬁed from the volumetric energy condition as in the MK-criterion, cf. Eqs. (32) and (36). For the non-local critical plane criteria, such as (70),

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0.15

90

0.1

120

60

60

0.08 0.1 150

30

0.06

150

30

0.04

0.05

0.02 180

0

210

330

240

180

0

210

330

300

240

270

300 270

(a)

(b) rðT const: ; hÞ H

Fig. 9. The graphically represented MK-criterion. The decohesion boundary, nonsingular solution (T – 0) and the plane strain conditions: (a) a = 60°; (b) a = 30°.

(the dotted line) and the T D ðrðT cH ; hÞÞ line (the solid line) for the

σ =1 σC

θ [° ]

Experimental results

T =0

β = (π / 2 − α ) [ ° ] Fig. 10. The angle of crack propagation hpr based on the MK-criterion for various initial crack orientations b = p/2 ain uniaxial tension (k = 0) as compared to experimental results of [32] (bounded by dotted lines) and the varying ratios of applied to critical stress (according to Table 1).

Table 1 The angle of crack propagation hpr for varying initial crack orientation a in the case of uniaxial tension (k = 0) and varying ratios of applied and critical stress.

a r/rC

15

30

45

60

80

T=0 0.1 0.2 0.4 0.6 0.8 1

85.7 86.7 87.7 89.7 91.2 92.7 93.7

72.7 73.7 74.7 76.7 78.7 80.2 81.2

59.6 59.6 59.6 59.6 59.6 59.6 59.6

45.1 43.1 40.6 37.6 35.6 34.1 33.1

19.3 15.5 13.5 11.5 11.0 10.5 10.0

the size parameter d0 is speciﬁed from the equivalence condition with the Grifﬁth condition for the tensile crack growth, cf. Eq. (43). The strain energy density criterion for a non-linear Ramberg–Osgood material was assumed by Shlyannikov [36,37] in order to specify the size of fracture damage zone. The combined non-local stress and energy release conditions were proposed by Seweryn [38] and next by Taylor [39], Carpinteri et al. [40] and Leguillon [41]. The averaged stress conditions on a

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θ pr

β = π /2 −α Fig. 11. The angle of crack propagation, based on [35] for different ratio d0/a and experimental results [32].

critical plane element of size d0 is applied and the value of d0 is next speciﬁed from the mean energy release value for a crack growing from a to a + d0, thus

Z

aþd0

GðaÞda ¼ Gc d0 ;

ð44Þ

a

where Gc is the assumed critical value of the energy release. 3. Three-dimensional (3D) cracks and fracture criteria Structural elements are usually subjected to complex stress states inducing singular 3D regimes at sharp defects (cracks, notches, etc.). The analysis of 3D singular regimes is more complex and crack growth modes are not well understood. 3.1. Crack growth modes Any growth of the plane crack surface which is located through the thickness of a plate, can be obtained by superposition of three basic modes, namely: mode I (opening mode) where opposing crack surfaces move directly apart; mode II (sliding mode) where crack surfaces move over each other perpendicularly to the crack front; and mode III (tearing mode), where crack surfaces move over each other parallel to the crack front. Two fundamental concepts can now be assumed the new crack growing surface is smooth and continuous at the edge of existing pre-crack the new crack surface is composed of facets of modes I, II and III oriented according to local critical state conditions. The crack surface is composed of the pattern of facets and its evolution is governed by geometric and mechanical characteristics of the element. We shall refer to the ﬁrst case as smooth crack surface models and to the second as rough crack surface models. To discuss these two approaches, consider the 3D square plane crack, Fig. 12 within the 3D element. It can be expected that new crack surface may grow from each edge. However, satisfying the continuity conditions the new crack surface may evolve in mode I or II or in combined mode I + II. But the mode III loading and the associated crack facets generated locally by the KI max condition violate the continuity at the existing crack edge. Thus the mode III loading may induce the rough crack surface combined with facets developed in mode II, Fig. 13. In fact, brittle materials exhibit rough crack surface in mode III composed of facets of different modes, cf. [42,43]. In ductile materials the shear band evolution may affect the crack roughness due to localized plastic deformation and damage growth. In fact Dyskin and Salganik [44] analyzed 3D cracks in compression and showed that 3D crack growth can result in a more complicated growth mechanism than 2D. They investigated possible mechanisms of wing crack growth induced also by rough crack surface and associated dilatancy effect [45]. A more general case of the 3D crack is the elliptic crack, which was analyzed in uniaxial compression tests by Adams and Sines [46] on PMMA samples. They showed that crack growth generally occurs in wing mode, but in addition at the lateral parts of crack edge a number of microcracks are developed. This can be interpreted as the mode III microcracks in brittle

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1793

a a

I mode

II mode

Fig. 12. 3D crack and pure modes of smooth crack tip deformation.

I mode

II mode θ (s )

III mode

θ (s)

Fig. 13. Pure modes of rough 3D crack tip deformation.

materials, cf. Fig. 18. The extensively investigated 3D elliptical cracks embedded in brittle material under compression loading have also been presented by Dyskin et al. [45,47]. 3.2. Rough (dilatant) crack models The effect of crack surface contact and asperity interaction is essential for modes II and III loading. In fact, the growth of pre-crack generated in mode I and next subjected to shear is associated with evolution of roughness pattern due to process of microcracking and associated inclined facets in mode I with subsequent connecting microcracks, cf. Pook [48]. The interface sliding along formed asperities induced mode I stress and crack dilatancy. Crack tip shielding then occurs due to frictional resistance to sliding and asperity interaction. The cases of closed or partially closed cracks subjected to shear and exhibiting contact shielding are numerous and occur, for instance, in compression or shear induced fracture of rocks or ceramic materials, rolling contact induced sub-surface fatigue cracks, mixed mode fatigue crack growth, etc. The referenced papers [49,50,43,51–56] contain both analytical and experimental studies of asperity interaction modes, speciﬁcation of the effective SIFs and prediction of crack growth rates. The assumption of the contact interface interaction at the crack front or at the whole cracked interface provides different modelling effects. The detailed review of literature is not presented here. A simpliﬁed model of a closed crack interface will only be discussed in the following, cf. Mróz and Seweryn [55]. Consider a crack surface in a form of wedge-shaped asperities, inclined at the angle cz 2 (0, p/2) to the nominal crack l l plane, Fig. 14. Denote the local stresses acting on wedge ﬂanks Cl by rn and sn , so that the local friction condition on Cl is

sln ¼ gl rln ¼ tan cl rln ;

ð45Þ

where gl is the local friction coefﬁcient and cl is the friction angle. The stress rzn and szn acting on the nominal crack segment are expressed from the equilibrium equations for a single asperity, thus

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μ σ nμ τ n

Γ0

γ

Γμ

s1

z

τ nz

Γμ

γz s2

σ nz

Fig. 14. Crack with wedge-shaped asperities and microstresses acting on asperity facets Cl and macrostress acting on the nominal crack plane segment C0.

s s2

s s2

rzn ¼ rln 1 ðcos cz gl sin cz Þ;

szn ¼ sln 1 ðsin cz þ gl cos cz Þ;

ð46Þ

where s1 denotes the wedge ﬂank length and s2 is the asperity length within the plane C0. The conditions on the nominal plane C0 can be expressed as follows

szn ¼ rzn tanðcz þ cl Þ;

u# ¼ jur j tan cz ;

# ¼ p:

ð47Þ K dI ; K dII

These conditions can be expressed in terms of the asymptotic ﬁelds (1) with neglect of T-stress. Denoting by the effecl l tive SIFs for the dilatation crack model and by K I ; K II the SIFs resulting from the stress on the wedge asperity ﬂanks Cl, we obtain l

l

K dI ¼ jK dII j tan cz ; jK II j ¼ K I tanðcz þ cl Þ:

ð48Þ

Denote by KI, KII the SIFs for a smooth and plane crack. Applying the superposition principle for the external loading r, s and the wedge asperity loading rzn ; szn , the effective SIFs are

sin cz ðK I sinðcz þ cl Þ þ jK II j cosðcz þ cl ÞÞ; cos cl cos cz l jK dII j ¼ K II þ K II ¼ ðK I sinðcz þ cl Þ þ jK II j cosðcz þ cl ÞÞ: sin cl l

K dI ¼ K I þ K I ¼

ð49Þ

These relations are valid when the contact occurs on Cl, thus

K I tanðcz þ cl Þ 6 jK II j 6 K I cotðcz Þ:

ð50Þ z

These inequalities specify the sliding domain B, Fig. 15. When jKIIj
K II K

= K cot( γ )

K

= − K tan( γ + γ )

B

A C B

A

γ

KI

B

C

Fig. 15. Domains of crack loading for the dilatant crack model: open crack (A), slip at contact interface (B) and closed crack (C).

K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

KI

KI

K II

K II

K Id

K Id

K IId K K

K K

K IId

K K

K K

γ z = 0° , γ μ = 10°

1795

γ z = 20 °, γ μ = 10°

ψ [° ]

ψ [° ]

Fig. 16. Values of SIFs speciﬁed for the frictional and dilatant crack models, based on [55]. The angle w speciﬁes the combined mode loading, tan w = KII/KI.

z

β −ω

x

z

σ

y

z

Crack plane

B

t

r θ

x

y

n

b A σ Fig. 17. Flat crack with linear front.

Mode III a>b

Mode II a

b Mode II

Fig. 18. 3D elliptic crack growth in compression.

circular or elliptical ﬂat crack, Fig. 18. The case of a linear crack edge can be reduced to a plane case. In fact, assuming the uniaxial loading by the principal stress r oriented at angles b and x to the coordinate axes x, y, z, we can write, cf. Fig. 17

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pﬃﬃﬃﬃﬃﬃ 2 K I ¼ r pb sin b; pﬃﬃﬃﬃﬃﬃ K II ¼ r pb sin b cos b sin x; pﬃﬃﬃﬃﬃﬃ K III ¼ r pb sin b cos b cos x

ð51Þ

and the stress intensity factors (SIF) remain constant on the crack edge. However, they should vary on a curvilinear front AB. In general, SIF depends on the curvature of the crack front. The stress analysis at the elliptical crack front was presented by Sih et al. [61–63]. Firstly in [62,61] it was shown that the three-dimensional stress state (Fig. 18) in a certain plane is identical to the two-dimensional case. However the stress intensity factor in general depends upon the curvature of the crack edge for three-dimensional problems. The same applies to the displacement ﬁeld. Thus introducing the local reference system n, t, z at the crack front, Fig. 19, there is

K ðsÞ

K ðsÞ

K ðsÞ

K ðsÞ

I II ﬃ A1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ A2 rn ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2p r 2pr I II ﬃ B1 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ B2; rz ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2p r 2pr

K ðsÞ

K ðsÞ

I ﬃ C1 pIIﬃﬃﬃﬃﬃﬃﬃﬃﬃ C2 ; rt ﬃ 2m pﬃﬃﬃﬃﬃﬃﬃﬃ 2pr 2pr

K ðsÞ

K ðsÞ

I II ﬃ D1 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ D2; rnz ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ 2pr 2pr

K ðsÞ

rnt ﬃ pIIIﬃﬃﬃﬃﬃﬃﬃﬃﬃ E1; 2pr K ðsÞ

rzt ﬃ pIIIﬃﬃﬃﬃﬃﬃﬃﬃﬃ E2 2p r where

h h 3h h h 3h 1 sin sin ; A2 ¼ sin 2 þ cos cos ; 2 2 2 2 2 2 h h 3h h h 3h 1 þ sin sin ; B2 ¼ cos sin cos ; B1 ¼ cos 2 2 2 2 2 2 h h C1 ¼ cos ; C2 ¼ sin ; 2 2 h h 3h h h 3h D1 ¼ cos sin cos ; D2 ¼ cos 1 sin sin ; 2 2 2 2 2 2 h h E1 ¼ sin ; E2 ¼ cos 2 2 A1 ¼ cos

Fig. 19. The elliptical crack with global and local coordinate systems x, y, z and n, t, z.

ð52Þ

K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

1797

and

12

r sin2 b b

1

2

2

ða2 sin a þ b cos2 aÞ4 ; " # 2 # 12 " r sin b cos b b b a sin a cos x b cos a cos x K II ðsÞ ¼ 1

þ 2 ; 1 2 02 02 02 2 2 a a ðk þ mk ÞEðkÞ mk KðkÞ ðk mÞEðkÞ þ k KðkÞ ða2 sin a þ b cos2 aÞ4 " # " # 2 12 ð1 mÞr sin b cos b b b a sin a cos x b cos a sin x 2 1

K III ðsÞ ¼ : 1 2 02 02 02 2 2 a a 2 2 4 ðk mÞEðkÞ þ k KðkÞ ðk þ mk ÞEðkÞ mk KðkÞ ða sin a þ b cos aÞ

K I ðsÞ ¼

EðkÞ

a

ð53Þ

The coefﬁcients K(k) and E(k) are complete elliptic integrals of the ﬁrst and second kind associated with the argument 02 2 k ¼ ½1 ðb=aÞ2 1=2 where k ¼ 1 k and a speciﬁes the location along the crack border. In the compression case r should be replaced by r. In the subsequent paper Hartranft and Sih [63,64] provided the local singular 3D stress ﬁeld at the crack front for small scale yielding. It can be expressed in the local spherical coordinates (r, h, /) in the form

K I ðsÞ K II ðsÞ ﬃ cos 2h 1 sin 2h sin 3h ﬃ sin 2h 2 þ cos 2h cos 3h rn ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 2 2 2pr cos / 2pr cos / K I ðsÞ K II ðsÞ ﬃ cos 2h 1 þ sin 2h sin 3h ﬃ cos 2h sin 2h cos 3h rz ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2pr cos / 2pr cos /

2m ﬃ K I ðsÞ cos 2h K II ðsÞ sin 2h ; rt ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2pr cos /

K I ðsÞ K II ðsÞ ﬃ cos 2h sin 2h cos 3h ﬃ cos 2h 1 sin 2h sin 3h rnz ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; 2 2 2pr cos / 2pr cos /

ð54Þ

1 ﬃ K III ðsÞ sin 2h ; rnt ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2pr cos / 1 ﬃ K III ðsÞ cos 2h : rzt ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2pr cos /

However, the asymptotic series expansion in 3D can be extended by adding more terms to the singular solution, cf. [64]. The most interesting is the second term, namely the T-stress which for 3D cases was not sufﬁciently analyzed. Based on the linespring method, Wang and Parks [65] evaluated the T-stress distribution along a semi-elliptical surface crack front in 3D plate. By introducing an interaction integral, Nakamura and Parks [66] developed method to obtain T-stress distribution along 3D crack fronts from ﬁnite element solutions. Effects of plate thickness and Poisson’s ratio on the T-stress were also explored in [65]. Though there are numerous fracture criteria proposed for 2D stress states under I/II mixed mode conditions, only several 3D fracture criteria have been proposed and the related experimental work is limited. The ﬁrst approach is based on the assumption that a new increment of fracture surface developing at the crack front is speciﬁed by locus of critical points and linear segments connecting these points to the crack front. The smooth crack surface is then generated. The other approach is based on the critical plane concept by specifying orientation of the critical plane element of a new fracture surface near the crack front. The critical plane approach allows for the rough crack surface evolution. Using the singular stress components of Eq. (54), Sih and Cha [67] extended the S-criterion to 3D by specifying the strain energy density near the crack front as follows

dW 1 Sðh; sÞ ¼ þ Oð1Þ; dV r cos /

ð55Þ

where

Sðh; sÞ ¼ a11

KðsÞ2I

p

þ 2a12

K I ðsÞK II ðsÞ

p

þ a22

K 2II ðsÞ

p

þ a33

K 2III ðsÞ

p

ð56Þ

and

16la11 ¼ ð3 4m cos hÞð1 þ cos hÞ; 16la12 ¼ 2 sin hðcos h 1 þ 2mÞ; 16la22 ¼ 4ð1 mÞð1 cos hÞ þ ð3 cos h 1Þð1 þ cos hÞ;

ð57Þ

16la33 ¼ 4: In the case of / = 0° Eq. (57) are the same as Eq. (7) in the plane case. Similarly, as in the plane case, a minimum of S is searched on a sphere centred at each point on the crack front [67,68]. The crack growth occurs when S = Smin reaches critical value, SC, and the size of rC along a three-dimensional crack front is assumed to vary such that Smin (h,s)/rC (s) = (d W/dV)cr remains constant. The continuous formulation S(h)/cos / exhibits a local minimum at (h0, /0) in the region ðp 6 h 6 pÞ; ðp=2 6 / 6 p=2Þ, provided Sðh; /Þ P Sðh0 ; /0 Þ. It should be noted that SðhÞ= cos / attains a local minimum always in the normal plane to the crack front, so / = 0 in our case and direction of the crack growth does not depend on KIII. So,

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a minimum of S is searched on a circle centered at each point on the crack front. However, as this radius does not affect the value of S-factor, then rC(s) = Smin(hpr, s)/(dW/dV)cr and generates the initial segment of the fracture surface. The extension of stress triaxiality condition (Mt-criterion) to 3D cases was presented by Kong et al. [15]. Introducing the stress triaxiality parameter

M¼

rH F 1 ¼ ; rEQ F 2

ð58Þ

where

pﬃﬃﬃ F 1 ¼ 2 2ð1 þ v Þ K I cos 2h K II sin 2h ; 2 31=2 2 3 2 K I 92 K 2II sin h þ ð2ð1 2v Þ2 K 2I þ 6K 2II Þ cos2 2h 2 5 ; F 2 ¼ 34 2 þ8ð1 v þ v 2 ÞK 2II sin 2h þ K I K II ð3 sin 2h 2ð1 2v Þ2 sin hÞ þ 6K 2III

ð59Þ

the value of the ratio M is maximized with respect to the angle h, as it does not depend on /. The optimal orientation segments on the existing crack front then generate a new incremental crack surface and its size is speciﬁed by setting the critical value of KI = KIC or of the effective SIF measure. In similar way we can extend the MK-criterion to 3D problem using Eqs. (52) and (53) in the following form m ðr þ r þ r Þ; T H ¼ 12 n z t 6E 2 1þm T D ¼ 6E ðrt rz Þ þ ðrz rn Þ2 þ ðrn rt Þ2 þ 6ðr2nz þ r2nt þ r2zt Þ ;

ð60Þ

then

rH ðrC ; h; aÞ ¼ rH jT H ¼const ¼

2 2 K I cos 2h ð1 þ mÞ K II sin 2h ð1 þ mÞ

p

rC

ð61Þ

:

The results of applied MK-criterion to the 3D problem of growth of elliptical crack under mixed loading conditions according to Fig. 19 are presented in Fig. 20. The crack growth for pure tensile and shear loadings is shown in Fig. 21. The sizes of the crack wings are calculated from Eq. (61). It is seen that for pure tensile loading the elliptical crack grows in its plane to a circular shape and for shear loading the wing cracks emanate from two boundary portions along the shear stress direction. 4. Critical plane criteria Let us now present an alternative formulation of crack growth criteria based on the concept of critical plane. Such criteria are expressed in terms of traction or strain components on the material plane element. The unit vector n speciﬁes the orientation of plane on which the normal and shear stresses are speciﬁed as follows:

ω=

π 4

,β=

π

3 π π ω= ,β = 4

4

ω=

π 4

,β=

π 6

Fig. 20. 3D elliptical smooth crack growth based on the MK-criterion for the mixed mode loading.

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rn ¼ rij ni nj ¼ r11 n21 þ r22 n22 þ r33 n23 þ 2r12 n1 n2 þ 2r23 n2 n3 þ 2r31 n3 n1 ; sn ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rij nj rik nk ðrij ni nj Þ2

ð62Þ

and the components of any radius vector r or normal vector n are expressed in the form (Fig. 22)

x ¼ r cos h cos /;

y ¼ r cos h sin /;

z ¼ r sin h:

ð63Þ

Similarly, the normal and shear strain en, cn can be expressed. The simplest critical plane orientation is formulated by assuming that crack will follow along the critical plane on which the normal tensile stress rn is maximal. This criterion was applied by Scholmann [69] and used to specify the orientation angles h0, /0 for the 3D stress state. In fact, formulating the problem

max n

rn ¼ max ðrij ni nj Þ subject to nk nk ¼ 1 n

ð64Þ

and considering the Lagrangian L ¼ rij ni nj kðnk nk 1Þ, the optimality condition is as follows

dL ¼ ðrij nj k ni Þ d ni ¼ 0 or

rij nj ¼ k ni ;

ð65Þ

so the maximal normal stress is the principal stress and the critical plane is the maximal principal stress plane and the angles h0, /0 are speciﬁed by solving the eigenvalue problem (65). A more general critical plane condition can be formulated as follows

F ¼ max f ðrn ; sn ; en ; cn Þ fC ¼ 0;

ð66Þ

n

where fC represents the critical value reached by the failure condition generally depending on both stress and strain components associated with the respective planes. Here en ¼ ðn e nÞn and cn ¼ ð1 n nÞen where 1 is the unit tensor. A particular form (66) applied to fatigue problems is obtained by applying the strain energy density associated with the amplitudes of stress and strain components acting on the critical plane, cf. Glinka et al. [70].

F ¼ W ¼ max n

Dcn Dsn Den Drn þ wC ¼ 0: 2 2 2 2

ð67Þ

This parameter represents only a fraction of the strain energy. However, it does not account for the effect of mean stress. An alternative energy condition was proposed by Chu [71] who combined maximum normal and shear stress with the corresponding strain amplitudes, thus max F ¼ W ¼ max ð2smax Den Þ wC ¼ 0: n Dcn þ 2rn

ð68Þ

n

We shall not provide the review of the critical plane models proposed by numerous authors in order to simulate fatigue damage and crack growth. The comprehensive review was recently presented by Karolczuk and Macha [72]. In this section, the local critical plane models will be extended by introducing non-local failure criteria applicable to both regular and singular stress regimes and also for monotonic and cyclic loading cases. Consider an arbitrary physical plane D and the local coordinate system (n1, n2, n3), Fig. 23. In the global coordinate system (x1, x2, x3) the origin of the local system is speciﬁed by the position vector x0(x01, x02, x03) and the unit normal vector n(n1, n2, n3) speciﬁes the plane orientation, where ni = cos(n3, xi). The resulting shear stress and strain in the plane D are exqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pressed as follows: sn ¼ s2n1 þ s2n2 , cn ¼ c2n1 þ c2n2 . Assume that crack initiation and propagation process depends on the 0.4 0.3 0.2 0.1 0

x 10 1.5 1 0.5 0 −0.5

−0.1 −1 −0.2 −0.3

0.2

0.1

0

(a)

−0.1

−0.4 −0.2

−1.5 0.1

0.05

0

−0.05

−0.1

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(b)

Fig. 21. 3D elliptical smooth crack growth based on the MK-criterion: (a) Pure tensile loading (b = p/2), (b) shear loading (x = 0, b?0).

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K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

z

n MC r

θ

y

φ

x Fig. 22. Orientation of the critical plane MC.

x3

x0

x2

x1

Fig. 23. The 3D system in R-criterion, from [73].

contact stress and strain components and also on the damage accumulation on the physical plane. Consider an elliptical condition for rn > 0 and the Coulomb condition for rn 6 0, thus

8 > rn 2

9 >

2 1=2 þ ssnc 1;

rn > 0; =

sc ðjsn j þ rn tan uÞ 1;

> rn < 0; ;

< rn sn 1 ¼ max F ¼ Rr ; n > rc sc :1

rc

ð69Þ

where rc, sc denote the failure stress of material in tension and shear. For large stress gradients or singular stress regimes such as those occurring at vertices of wedge shaped notches, the non-local stress failure conditions is applied by averaging the failure stress function over an area d0 d0, thus

" F ¼ Rr 1 ¼ max ðn;x0 Þ

Z

1 2

d0

0

d0

Z

#

d0

Rr dn1 dn2 1 ¼ 0:

ð70Þ

0

An alternating, non-local condition can be obtained by averaging normal and shear stress components on the plane D, thus

r n ¼

1 2 d0

Z

d0 0

Z

d0 0

1 rn dn1 dn2 ; sn ¼ 2

Z

d0

0

d0

Z

d0

sn dn1 dn2

ð71Þ

0

and substituting to (70)

F ¼ Rr 1 ¼ max Rr n

r n sn 1 ¼ 0: ; rc sc

ð72Þ

The size parameter d0 representing the size of damage zone can be speciﬁed by requiring the non-local conditions (72) to be equivalent to the Grifﬁth condition in the case of tensile crack propagation (43). The extensive application of the non-local failure criterion to monotonically loaded elements witch sharp notches and cracks was discussed by Seweryn and Mróz [33,34]. The application to fatigue crack initiation and growth was discussed in [74,75] by applying the non-local criterion with account for local damage growth. To formulate the non-local criterion in the case of fatigue crack growth, introduce the damage initiation function Rr 0(rn/ r0, sn/s0) which is assumed to have the same form as the failure function Rr(rn/r0, sn/s0), thus

K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

r0 s0 ¼ ¼ f: rc sc

1801

ð73Þ

Assume the stress values r0 and s0 to depend on the accumulated damage, so that

r0 ¼ r00 ð1 xn Þq ;

s0 ¼ s00 ð1 xn Þq ;

ð74Þ

where 0 6 xn 6 1 is the damage measure on the physical plane, and q is the material parameter. Consider the domain of damage accumulation bounded by the curves F 0 ¼ 0 and F ¼ 0. The actual stress point lies on the loading surface n =r0 ; s n =s0 Þ f ¼ 0. The damage growth can now be expressed as follows: F l ¼ Rr ðr

( dxn ¼

for dRn > 0 and f P f ; for dRn < 0 or f < f ;

Wr dRn ; 0;

ð75Þ

r ds and W is the damage growth function assumed in the form where dRr ¼ @@Rrrn drn þ @R n r @ sn

Wr ðRr Þ ¼ A

f f 1f

!n

dRr : 1f

ð76Þ

The term in brackets represents the ratios PP0/PCP0 for the stress state corresponding to point P in Fig. 24, and A, n are the material parameters. The accumulated damage is now expressed as follows

xn ¼

Z

Rr

WðrÞ dRr

ð77Þ

0

and the crack initiation condition is satisﬁed when xn = 1. The crack propagation condition in Mode I due to cyclic loading can now be generated from (76), namely

da ¼C dN

" nþ1 nþ1 # K max DK th K min DK th ; K C DK th K C DK th

ð78Þ

where Kmax and Kmin are the maximal and minimal values of the stress intensity factor in consecutive loading cycles, DKth is the threshold value of DK corresponding to the onset of fatigue and KC is the critical value of SIF, C and n are the material parameters. The application of the non-local critical plane approach to modelling crack initiation stress and orientation of crack growth was presented in papers by Seweryn et al. [27,74], and the application to fatigue crack growth simulation in [43,75]. The evolution of damage zones at the crack tip can then be speciﬁed for any loading history, and the orientation of crack growth can be predicted. The polar diagram of damage distribution in a tubular element subjected to combined bending and torsion with varying stresses rz(t) = ra sin(x t) + rm, szh(t) = sa sin(xt d) + sm is shown in Fig. 25, cf. [34]. When the maximal tensile conditions is used in (69), then the Novozhilov [76] condition is obtained. Fig. 26 presents the angle of the crack pﬃﬃﬃ propagation hpr for the various main crack orientation angle in the uniaxial tension (k = 0) and the various values of s ¼ 3rc =sc between the tension and shear critical failure stress. The most interesting are the differences of the angle of the crack propagation for the same value of w but different s values. In fact, for s = 0 the mode I crack grows at vary-

σ τ Rσ n n = σc τc

τn ∂Rσ ∂σ

τc τ

σf

P P

σu Rσ

σn τn = σ τ

Pc

σ

σc σn

Fig. 24. Damage initiation F0 = 0, stress failure, F = 0 and loading curve Fl = 0 in the plane rn, sn.

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σc τc = R fσ =

σm σa = k τm τa = k δ= f = σ a = τ a nσ =

ω nω θ

θ=

Rd

θ

θ= θ= k=-1

k=-0.5 k=0-1

Fig. 25. Polar diagram of damage distribution for proportional bending and torsion, after [34].

θpr

ψ Fig. 26. The angle of crack propagation, based on [33] for different values of the ratio s ¼

pﬃﬃﬃ r 3 scc and KII/KI = tan w.

ing orientations depending on the mode mixity factor. Similarly, for s = 1, the mode II crack grows. Between the lines representing growth orientation in modes I and II shown in Fig. 26 there are discontinuity points represented by vertical segments of transition from mode I to mode II (cf. paths for s = 1.0, 1.5, 2.0) and bifurcation points corresponding to continuous transition of orientation between these modes (cf. paths for s = 2.5, 2.75). 5. Fatigue crack growth In fact, the mixed mode fracture models are unable to predict accurately the crack growth rate or the initiation angle, especially for mode II dominant loading (low value of a, k = 0, Fig. 1 ). Usually, this prediction is difﬁcult for several initial cycles for which mode II condition dominates (DKII > DKI). Similar effect is observed also for monotonic loading cases. The other important factor is prediction of the fatigue crack growth rate. The history of rate of crack growth modeling starts from the Paris law [77,78] formulated in the form

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K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

da ¼ CðDKÞn ; dN

ð79Þ

where C and n are the material quasi-parameters, and DK = Kmax Kmin. This equation predicts the fatigue crack growth in one cycle for the case of small scale yielding in terms of the amplitude DK in mode I. In actuality, the rate of crack growth depends on many factors, such as mean and maximal stress, crack closure effect, and mode mixity. There have been numerous extensions of (79) to account for other effects. Tanaka [79] in 1974 introduced the concept of the effective stress intensity factor DKeff for mixed mode conditions, thus

da ¼ CðDK eff Þn ; dN

ð80Þ

DK eff ¼ ðDK I þ 8DK II Þ0:25 :

ð81Þ

where

Another form of the DKeff resulting from the MTS-criterion was proposed by Yan et al.[80], namely

DK eff ¼ 0:5 cos

hpr ðDK I ð1 þ cos hpr Þ 3DK II sin hpr Þ; 2

ð82Þ

where hpr is obtained from MTS-criterion. There are also other parameters used to correlate fatigue crack growth under mixed mode loading. Sih and Barthelemy [81] used the strain energy density factors DS replacing DK in the Paris type equation. They compared the predicted crack path used S parameter with experimental data [82] for specimen made of Ti–6Al–4V with inclined cracks cf. Table 2. However, these predictions are unsatisfactory, especially for a = 30°. The other parameter used frequently to predict the crack growth rate is the crack tip opening displacement (CTOD). It is assumed that the crack growth in the loading cycle is proportional to CTOD [83]. This parameter is often combined with other factors affecting the crack growth, such as the crack closure parameter [84]. In 1989 Li [85] developed the method of simulation of crack growth in terms of crack tip displacement (CTD) treated as a vector composed of CTOD and CTSD associated with mode I and II, thus allowing for speciﬁcation of the effective CTD. Usually, the fatigue crack growth rate conventionally is correlated with SIFs by Paris type equation. However, experimental results show that specimen geometry seems to play an important inﬂuence on fatigue behaviour. Picard et al. [86] and Tong et al. [87] showed that fatigue crack growth rates differ signiﬁcantly under the same testing conditions for different specimens, CN (corner notch) and CT (compact tension). Analysis based on the DK alone does not provide satisfactory results. Therefore, the T-stress could also be used to explain this phenomenon with satisfactory results [88]. The application of MK-fracture criterion to the case of cyclic loading was presented in 2008 by Mróz [27]. This criterion predicts the crack growth orientation depending on the load level. For the stress cycle with stress level varying between rmin and rmax, the crack growth initiation stress rpr was introduced with the assumption that rpr = rop, where rop is the crack opening stress associated with crack closure effect [89]. Then it is assumed that rpr = rmin if rmin > rop when T-stress effect is associated. The following relation specifying the crack growth per one cycle was proposed in [27]:

da ¼ C DðSD ðr pr Þ rpr ðSconst: ÞÞn ; V dN

ð83Þ

where rrp is the length of decohesion zone along the crack growth direction and C, n are the fatigue parameters obtained from uniaxial tests (a = 90°). The relation (83) can be also written in the following form

C ðSD Dr pr þ DSD r pr Þn ;

ð84Þ

where DSD = SD max SD min, Drpr = rD pr max rD pr min and SD ; rpr are the mean values of SD and rpr on one cycle, respectively. Let us note that the crack growth rate now depends on both deviatoric and volumetric energy amplitudes. In view of (25), (26) and (31) the growth rule (83) can be presented in the following form expressed in terms of the singular stress ﬁeld (1) (T = 0):

0 1n DK 2I ððm2 þ m þ 1ÞðC þ AÞ2 3ðAC F 2 ÞÞþ da B C ¼ C 2 @ þDK 2II ððm2 þ m þ 1ÞðB DÞ2 þ 3ðBD G2 ÞÞþ A ; dN þDK I DK II ð2ðm2 þ m þ 1ÞðC þ AÞðB DÞ 3ðCB ADÞÞ

ð85Þ

Table 2 Characterization of specimen A and B. Initial crack length a0 (mm)

Specimen A: a0 = 7.11 mm

Specimen B: a0 = 6.73 mm

a (°) rmin (MPa) rmax (MPa)

a = 30

a = 43 17.24 172.38

20.69 206.85

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Table 3 Mixed mode fatigue results for specimen A. Prediction [81]

Prediction Eq. (83)

Exp. results According to [82]

N

da/dN

N

da/dN

Cycle numb.

[10–7 m]

Cycle numb.

7.05 7.01 8.52 8.53 10.1 10 11 13 16 16

1390 2611 3340 4260 4771 5208 5682 6141 6648 7111

16,000

22,000

25,000

Prediction Eq. (87)

N

da/dN

N

da/dN

[10–7 m]

Cycle numb.

6.77 6.25 7.4 8.53 9.78 10.86 12.14 13.71 15.78 18.38

1390 2430 3110 4000 4550 4900 5400 5900 6400 7000

[10–7 m]

Cycle numb.

[10–7 m]

6.54 7.59 7.35 9.1 8 11.1 12 12.2 16 18

3583 5416 6352 7365 7872 8279 8695 9075 9471 9813

2.64 4.15 5.77 7.74 9.87 16.67 13.83 16.55 20.19 24.89

Table 4 Mixed mode fatigue results for specimen B. Prediction [81]

Prediction Eq. (83)

Exp. results According to [82]

Prediction Eq. (87)

N

da/dN

N

da/dN

N

da/dN

N

da/dN

Cycle numb.

[10–7 m]

Cycle numb.

[10–7 m]

Cycle numb.

[10–7 m]

Cycle numb.

[10–7 m]

0.64 2.84 3.76 3.82 5.43 6.27 5.43 7.59 8.46 10

2954 4331 6523 8106 9804 10,569 11,268 12,988 13,605 14,319

2.91 3.12 3.6 4.17 4.91 5.62 6.15 7.29 8.58 9.54

4660 6100 8250 10,000 11,490 12,080 13,000 14,660 15,310 15,960

1.85 3.12 3.58 3.71 5.77 8.3 3.91 7.53 7.84 10.92

3546 5732 8859 10,873 12,862 13,711 14,462 16,192 16,780 17,436

1.64 1.96 2.53 3.27 4.19 5.06 5.72 7.25 9.0 10.36

10,000 12,080

19,100

where

; D ¼ sin 2h 2 þ cos 2h cos 3h ; A ¼ cos 2h 1 þ sin 2h sin 3h 2 2 n 1þm 1 h h 3h h h 3h C2 ¼ C ; B ¼ sin 2 cos 2 cos 2 ; F ¼ sin 2 cos 2 cos 2 ; 3E 2p

C ¼ cos 2h 1 sin 2h sin 3h ; G ¼ cos 2h 1 sin 2h sin 3h 2 2

ð86Þ

and m* = 0 for plane stress, m* = m for plane strain cases. When T-stress is included, the crack growth rule is expressed in terms of both stress intensity amplitudes and their mean values over loading cycle. Following the rule (69), the simpliﬁed equation to determine the crack increment in mixed mode per cycle can be proposed

2sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3n1 2 2 da rn sn 5 ¼ C 1 D4 þ ; dN rf sf

rn > 0;

ð87Þ

pﬃﬃﬃ where C1, n1 are the fatigue constants for the uniaxial test (a = 90°), rf is the fatigue limit and rf =sf ¼ 3. The comparison of results is presented in Figs. 27 and 28, also in Tables 3 and 4, where the last two columns present the results according to Eq. (87), the second the results according to MK-criterion and Eq. (83), and the third the experimental results of Pustejowsky [82] conducted for titanium alloy specimens. The comparison with predictions of Sih model [81] is also presented in tables in the ﬁrst two columns. It is seen that the relation (83) provides much more accurate prediction of the growth rate. It should be noted that T-stress should be retained in the analysis of stress state near the crack tip and in prediction of rate and orientation of crack growth.

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K.P. Mróz, Z. Mróz / Engineering Fracture Mechanics 77 (2010) 1781–1807

3110

Ti-6Al-4V

4900

5900 7000

3340 5208 6141 7111

Prediction Exp.results

0

R=

= 7 11 mm

σ min 20.6 = = 0.1 σ max 206

[m] Fig. 27. Mixed mode fatigue trajectory for specimen A.

8250

Ti-6Al-4V 6523

12080

15960

10569 14319

Prediction Exp. results

R=

0

σ min 17.2 = = 0.1 σ max 172

= 6 73 mm

[m] Fig. 28. Mixed mode fatigue trajectory for specimen B.

6. Concluding remarks The present paper provides a synthetic review of different criteria of crack growth for 2D and 3D cases. The criteria are based on the elastic singular stress distribution and the regular T-stress component. The stress criteria, energy concepts and critical plane approaches are discussed and their prediction compared for several speciﬁc cases. The objective of this paper is

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to indicate also the major topics requiring further study, such as speciﬁcation of size parameters at the crack front, generation of rough crack patterns, and crack growth in 3D stress condition. The non-local character of the criteria is incorporated by specifying a critical distance segment from the crack tip on which the averaged stresses and strain states or their local values at the end segment point are determined and used in formulating the damage and crack growth criteria. Such critical segment length can be interpreted as representing the fracture or localized damage zone. The forms and sizes of plastic and distributed damages zones are illustrated in Fig. 2. Considering only elastic singular stress ﬁeld with T-stress, it can be assumed that the deviatoric and volumetric portions of the speciﬁc strain energy may represent the forms and sizes of plastic and distributed damage zones. The crack growth criterion can then be formulated and the critical distance rc speciﬁed in terms of the stress intensity factors and the critical stress value rc representing the critical volumetric energy level. The critical plane approaches require also speciﬁcation of the averaging size parameter d0 which can be expressed by considering non-local energy release for a crack growth through the ﬁnite distance d0. For modes II and III the interaction of rough crack interfaces with account for frictional slip and dilatancy effects is of fundamental importance. For compressive normal stress the crack interface slip then induces shielding of KII due to friction and growth of KI due to dilatancy. The generation of rough crack pattern in mode III can be modeled by assuming the facets of mode I developing periodically at the existing crack edge and creating the crack microsurface with facets of mode II and III. The rough crack pattern violates the continuity condition at the existing crack edge on the local scale of facets size but preserves the continuity for a crack macrosurface. For 3D-cracks both smooth and rough crack surfaces can be generated at different locations of the initial crack boundary. The predictive models of rough cracks are not yet available and require further development. The growth of 3D cracks is still not well investigated as more complex crack patterns can grow at the existing crack edge. Both experimental observations and analytical models are limited. It is expected that in a near future the substantial progress in this area will be achieved. Acknowledgments The results presented in this paper have been obtained within the project KomCerMet (Contract No. POIG.01.03.01-00013/08 with the Polish Ministry of Science and Higher Education) in the framework of the Operational Programme Innovative Economy 2007–2013. The authors wish to thank reviewers for their critical remarks and improvement suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

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On crack path evolution rules

On crack path evolution rules

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