The mixed mode crack growth rate in cruciform specimens subject to biaxial loading

Theoretical and Applied Fracture Mechanics 73 (2014) 68–81

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The mixed mode crack growth rate in cruciform specimens subject to biaxial loading V.N. Shlyannikov ⇑, A.V. Tumanov, A.P. Zakharov Research Center for Rower Engineering Problems of the Russian Academy of Science, Russia

a r t i c l e

i n f o

Article history: Available online 15 July 2014 Keywords: Mixed mode Crack growth rate Biaxial loading Cruciform specimens Plastic stress intensity factor

a b s t r a c t Cruciform specimens of two configurations with an inclined crack subject to a system of biaxial loads are used to study the fatigue crack growth rate. A method for infiltrating the mixed mode displacement of cracks in the deformed state is suggested. For the particular specimen geometries considered, the T-stress and the geometry dependent correction factors, as well as the numerical constant of the plastic stress distributions In, are obtained as a function of the dimensionless crack length, load biaxiality and mode mixity. The combined effect of load biaxiality and crack orientation on the crack growth rate for low-strength and high-strength steels is made explicit. Additionally, a comparative study of a cruciform specimen with a working area thinned with respect to a flat cruciform specimen is performed through experiments and numerical computations under various mixed mode biaxial loading conditions. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction According to Brown and Miller [1,2] the fatigue lifetime may be roughly divided into four phases: (i) nucleation of fatigue cracks (defined as initiation); (ii) growth on a plane of maximum shear; (iii) propagation normal to the tensile stress; and (iv) final rupture of the specimen. Although, in the general fatigue problems, considerable attention has been paid to the uniaxial test, there has been comparatively little study into the load biaxiality effects on fatigue crack propagation. The multiaxial fatigue results are of particular importance to practical application, cumulative damage studies and many situations where the principal stress axes can rotate. There are a number of theories concerning the cause of the phase (ii) to phase (iii) transition which depends on multiaxial loading conditions. Some of these may be disproved by the occurrence of a phase (iii) to phase (ii) transition. Modern multiaxial fatigue failure criteria is based on the critical plane approach which is specified according to the dominating fracture mechanism or crack growth phases mentioned above. Importance of these criteria has increased during last two decades due to of its effectiveness in the assessment of multiaxial fatigue life and application possibilities. The general purpose of critical plane concept is the transformation of a multiaxial stress–strain state to some equivalent iniaxial one and determination of lifetime as well as dominating ⇑ Corresponding author. E-mail address: [email protected] (V.N. Shlyannikov). http://dx.doi.org/10.1016/j.tafmec.2014.06.016 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.

fatigue crack plane position. In multiaxial fatigue, Carpinteri et al. [3–6] using the critical plane approach have introduced the C–S criterion and highlighted its application to fatigue life estimation and structural integrity assessment of welded joints and components. The authors reported the influence of both proportional (in-phase) and non-proportional (out-of-phase) cyclic loading and proposed the modification of the Carpinteri and Spagnoli (C–S) criterion. These modifications are related to the weighting procedure of the principal stress axes; the definition of the equivalent normal stress by taking into account the mean normal stress effect; the expression of the quadratic combination of stresses. Karolczuk and Macha [7] have published extensive observations of the critical plane positions as a function of multiaxial stress–strain state, combined cyclic or random loading, stress ratio and mean stress. The analysis was concerned to multiaxial failure criteria and the different methods of their formulation (the damage accumulation, the weight function and the variance method). Systematic studies of both the Shear Stress-Maximum Variance Method and the Theory of Critical Distances (TCDs) have been described by Susmel et al. [8,9]. The orientation of the critical plane is used to estimate fatigue lifetime of both plain and notched engineering components under constant as well as variable amplitude uniaxial/multiaxial fatigue loading. The TCDs is treated as a material property whose length increases as the number of cycles to failure decreases. Shlyannikov [10] considered the behavior of the characteristic distance or damage zone size under both static and cyclic mixed mode fracture on the base of strain energy density theory.

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Nomenclature a E G In J K1, K2 K PM MP n Px,Py R r,h S t T

crack length Young’s modulus shear modulus numerical constant J-integral mode I and mode II elastic stress intensity factors plastic stress intensity factor mode mixity parameter strain hardening exponent applied to the specimen arm loads in the X-axis and Yaxis direction cyclic stress ratio polar coordinates strain energy density factor specimen thickness dimensionless non-singular T-stresses

Many different test specimens have been used to investigation of crack growth rate under multiaxial loading on both phase (ii) and phase (iii) crack extention. The cruciform specimen is one of the most used samples for static and cyclic biaxial and mixed mode tests. A review of literature devoted to the application of biaxially loaded specimens in fracture mechanics tests is given by Smith and Pascoe [11]. The cruciform specimen (CS) is very suitable for mixed mode fracture experiments because it reproduces the complete range of mode mixities from pure mode I to pure mode II and has the largest area of uniform nominal stress distributions in working area of specimen. By changing both initial crack angle and the nominal stress biaxiality, different combinations of modes I and II can be achieved. Miller and Brown [12,13] investigated fatigue crack propagation in biaxially loaded plates. They found that fatigue crack growth is related to two parameters in the plane of plate, the maximum shear stress range and the stress normal to the plane of maximum shear. Both these parameters affected the crack-tip opening displacement and hence fatigue crack growth rate. Shlyannikov et al. [14–16] have published extensive observations of crack growth in compact tension and dual-cruciform specimens. They have shown that the plastic material properties first of all influenced mixed mode crack deviation angle, crack paths and crack growth rate. It should also noted that subject for tests were ten type of aluminum alloys and three type of steels of wide range of elastic–plastic properties. Dalle Donne and Doker [17], testing two cruciform specimen types found that predominant mode II loading drove the stable crack in the direction according to maximum shear strain criterion while subsequent mode I crack-tip loading caused a crack path deviation, that is, the stable crack grew normal to the maximum tensile stress. A number of studies of fatigue crack propagation have been conducted for biaxial stress conditions with the observations of the T-stress effect. Howard [18] was the first to analyze the crack growth rate by taking into account the T-stress. Kitagawa et al. [19] and Gao et al. [20] have shown that the application of higher stresses and a negative T-stress increases the fatigue crack propagation rates. The generalization of dimensionless T-stress (normalized by nominal stress T ¼ T=r) effects for small-scale yield conditions and mixed-mode loading on crack path, fatigue crack growth direction and crack growth rates is given by Shlyannikov [21–23] for materials different properties and specimens of various geometries. To study the material cyclic fracture resistance characteristics, the mixed mode crack growth rate is calculated as the averaged

u, v  ; v u ~ ; v~ u w Y1, Y2

a a g k

m 1 r1 xx ; ryy

re r0

displacement components normalized displacement components dimensionless angular functions of displacement components specimen width geometry dependent correction factors crack angle strain hardening coefficient biaxial nominal stress ratio applied to the specimen arm load ratio Poisson’s ratio nominal stress in the X-axis the Y-axis direction von Mises equivalent stress yield stress

crack extension per one load cycle versus certain equivalent fracture parameters, namely, the strain energy density (SED) factor. In this paper, the plastic stress intensity factor (SIF) approach, originally proposed to describe fracture toughness for pure mode I under monotonic/static loading, is employed to study the crack growth rate under cyclic mixed mode fracture. The elastic mode I and II elastic stress intensity factors KI and K2 calculations are supplemented by plastic SIF’s determination for a full range of mixed mode conditions, crack length and crack angle combinations. To obtain a plastic stress intensity factor formulation under plane stress and plane strain, the load biaxiality influence is considered and is applied to quantify the crack growth rate in two cruciform specimen types.

2. Specimens and material properties In the present study, two cruciform specimen (CS) types under biaxial loading were considered. One of them is flat specimen of constant thickness (CS-1, Fig. 1a), the other is a cruciform specimen with a thinned working area (CS-2, Fig. 1b). Different degrees of mode mixity, from pure mode I to pure mode II, are obtained by combinations of the far-field stress level r ¼ r1 yy in the Y-axis direction, remote biaxial stress ratio g and inclination crack angle a. In the current notation, the magnitude of the applied load biax1 iality is described by the remote nominal stress ratio g ¼ r1 xx =ryy . By changing a, different combinations of modes I and II can be achieved. For example, it is clear that for the biaxially loaded CS, a = 0° or a = 90° corresponds to pure mode I, whereas pure mode II can be obtained when a = 45° and g = 1. The flat cruciform specimens of constant thickness (CS-1) were machined from 3 mm-thick sheets of a low strength structural steel 3, and the cruciform specimens with thinned working area (CS-2) were manufactured from 20 mm-thick plates of finegrained high-strength steel 34XH3MA. The thickness of the working area was also 3 mm. The most important properties of the tested materials are given in Table 1. The linear elastic FEM calculations were performed using FEmeshes of both specimen configurations considered (Fig. 2) to 1 determine nominal stress r1 xx , ryy distributions and a uniform strain field size as well as to determine the relationship between the applied arm load ratio k and the nominal stress biaxial ratio g under different loading conditions. The commercial finite element code, ANSYS [24], has been used to calculate the displacement and stress distributions ahead of crack tips. 2D plane stress

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Fig. 1. (a) Flat cruciform specimen (CS-1) and (b) cruciform specimen with thinning working area (CS-2).

Table 1 Main mechanical properties. Specimen configuration

Material

Young modulus E (MPa)

Yield stress r0 (MPa)

Ultimate stress rf (MPa)

Strain hardening exponent n

Strain hardening  coefficient a

CS-1 CS-2

Steel 3 Steel 34XH3MA

215655 194871

295 788

487.5 978.9

4.418 7.300

15.482 2.457

eight-node isoparametric elements have been used for the 2D flat under biaxial loading of CS-1 configurations, and the twenty-node quadrilateral brick isoparametric three-dimensional solid elements have been used to model for the 3D biaxially loaded CS-2 in its thin central part. Typical finite element meshes for the cruciform specimens are illustrated in Fig. 2. The relations between load and nominal stress biaxiality ratio listed in Table 2 correspond to the central part of the working area of the CS-1 and CS-2 specimens. These relations are the same for each biaxial loading condition within the area of 50% w, where w is specimen width. It should be noted that a uniform strain field in the center of the CS-2 cruciform specimen was achieved through the slits in the loading arms. A rigid rim located on the perimeter of the working area of the CS-2 specimen prevents buckling by applying compressive loads. By fitting the numerical calculations, the biaxial stress ratio g as a function of the applied load ratio k for the particular cruciform specimen geometry considered has been represented in the form of polynomial equations as follows: for CS-1 g ¼ 0:1998k2 þ 0:9885k  0:2013 and for CS-2 g ¼ 0:2114k2 þ k  0:2117. Since the crack-tip region contains steep displacement and high stress gradients, the mesh needs to be very refined at the crack tip. For this purpose, a corresponding mesh topology having a focused ring of elements surrounding the crack front was used to enhance convergence of the numerical solutions (Fig. 3). To model in CS-2 the 3D stress field correctly and because of the strong variations of the stress gradients, the thicknesses of the successive element layers are gradually reduced toward the free surface with respect to the crack-front line. A finite element mesh consisting of 162 elements placed along crack front. In the circumferential direction, 40 equally sized elements are defined in the angular region from 0 to p. The size of each ring increases gradually with the radial distance from the crack tip. Radial sizes of elements are varied according to the geometric progression. A finite element mesh consisting of

1,172,034 nodes and 281,616 twenty-node quadrilateral brick elements was used. To consider the details of a large deformation of the crack tip, a typical mesh was used to model the region near the notch tip (Fig. 3). In all the FEA calculations the material is assumed to be linear elastic and characterized by E = 200 GPa and m = 0.3. Ref. [22] contain more details about the procedure for computing of T-stress and geometry dependent correction factors Yi(a/w) for the SIF K1 and K2 by using the crack flank displacements method. For pure mode I at crack angle equal a = 90°, four biaxial load ratios for CS-1, k equal to +1.0, +0.5, 0.0 (uniaxial), 0.2 and three ratios for CS-2, k equal to +1.0, 0.0 (uniaxial), 1.0, were considered, respectively. Mixed mode crack growth analysis with the variation of an initial crack angle was performed for cruciform specimens CS-1 under uniaxial tension with k = 0.0 and for the CS-2 at k = 1.0 under equibiaxial tension–compression. 3. Infiltrate mixed mode crack-tip displacements For the structural materials of different properties the crack growth rate characteristics is usually presented in the form of a fatigue fracture diagram (FFD). In essence, the FFD is a plot of the crack growth rate variation versus an amplitude factor of singular elastic or elastic–plastic stress strain fields. It can include stress intensity factors, J-integral and strain energy density factor (SEDF). Few methods have been proposed for calculating these fracture mechanics parameters, one of them explores the direct application of FEM analysis by using the crack flank nodal displacements [25]. In practical situations, the loading experienced at a crack tip could be very complex, resulting in a mixed-mode fracture. In general, when solving asymmetrical problems of the fracture mechanics, displacements can be divided into two parts. The first of them is displacements as a rigid body (without deformation) and the second part is displacements causing strain. The question arises for

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Fig. 2. FE meshes and nominal stress distributions of (a, b) CS-1 and (c, d) CS-2.

Table 2 Relations between load and nominal stress biaxiality ratios. k ¼ P x =P y

g

1.0

0.7

0.5

0.3

0.0

0.3

0.5

0.7

1.0

1.0 1.0

0.58 0.59

0.33 0.34

0.11 0.107

0.20 0.21

0.47 0.49

0.64 0.66

0.79 0.808

1.0 1.0

1 ¼ rrx1 y

CS-1 CS-2

an inclined cracks how to determine the true crack tip displacement components without displacements as a rigid body. This problem has been solved as follows [26]. Fig. 4 gives a schematic overview of macroscopic fracture appearance of the cracked body under mixed mode loading. To determination of the magnitude of a crack-tip displacement vector, two points (2 and 4) were positioned on both fatigue crack flanks behind the crack tip. Intermediate point 3 is always located in the middle of the distance between opposite points 2 and 4 in the undeformed and deformed state. The position of the line connecting points 1 and 3 determines the angle of crack rotation b in the deformed state. For the purpose of crack tip displacement, filtering is necessary to reconcile (combine) the position of the points 1 and 10 . In Fig. 3, u and v are the crack-tip displacement

in the Cartesian coordinate system, whereas ur and uh are the displacement in the polar coordinate system with r, h centered at the crack tip. The original coordinates of point A in the Cartesian coordinate system are x0, y0 for the crack tip with a finite curvature radius R (Fig. 5). These coordinates are related to the polar coordinate system r, h as follows



x0 ¼ r cosðhÞ y0 ¼ r sinðhÞ

ð1Þ

After mixed mode loading history the coordinates for point A are



x ¼ x0 þ uFEM y ¼ y0 þ v FEM

ð2Þ

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Fig. 3. FE meshes of CS-2 containing through-thickness crack.

Fig. 4. The crack tip behavior: (a) original, (b) deformed and (c) infiltrate.

8 þ  < u ¼ MX þMX 0 2 : : v ¼ MþY þMY 0

ð5Þ

2

Using the coordinates of points on the opposite crack flanks X 0N and Y 0N , which are located behind the crack tip (Fig. 5), the angle of crack rotation can be calculated in the deformed state

b ¼ arctgðY 0N =X 0N Þ

ð6Þ

where

(

X 0N ¼ ðNþX 0 þ NX 0 Þ=2 Y 0N ¼ ðNþY 0 þ NY 0 Þ=2

The relative rotation of the Cartesian coordinate systems is determined by the following formulae

Fig. 5. The crack tip displacement under mixed mode loading.

 where uFEM, vFEM are calculated by the FEM displacement components. The new position of the Cartesian coordinate system source is given by



0

X ¼ x  u0 Y0 ¼ y  v0

ð3Þ

where u0, and v0 are the displacement components. The coordinates of opposite points on the upper and lower crack flanks are

(

Mþ ð0; RÞ M ð0; RÞ

ð7Þ

:

ð4Þ

where in the origin of the coordinate system in a deformed state may be defined as

X 00 ¼ X 0 cosðbÞ þ Y 0 sinðbÞ Y 00 ¼ Y 0 cosðbÞ  X 0 sinðbÞ

:

ð8Þ

The resulting displacement components in the local Cartesian system can be represented as

(

uFEM ¼ X 00  X 0 X 00

v FEM ¼ Y 00  Y 0 Y

ð9Þ

00

The displacement calculations contained in Eq. (9) may be facilitated by the use of polar coordinates originating at the crack tip by means of transformation from a Cartesian to a polar coordinate system in terms of the complex variable of Muskhelishvili

u þ iv ¼ ður þ iuh Þeih ur þ iuh ¼ ðu þ iv Þeih

ð10Þ

V.N. Shlyannikov et al. / Theoretical and Applied Fracture Mechanics 73 (2014) 68–81

Using Eq. (10), the expressions for the polar components of displacement can be rearranged into the following form

ur ¼ u cos h þ v sin h

ð11Þ

uh ¼ u sin h þ v cos h

Finally, substituting Eqs. (8) and (9) into (11), we can obtain the infiltrate displacement components in the local polar coordinate system

uFEM ¼ ðX 00  x0 Þ cos h þ ðY 00  y0 Þ sin h r uFEM ¼ ðX 00  x0 Þ sin h þ ðY 00  y0 Þ cos h h

:

ð12Þ

Eq. (12) will be further used for calculations of the elastic and plastic fracture parameters under mixed mode loading. In the present study, the analysis and discussion of the experimental results of the crack growth rate under biaxial loading will be given in two parts, the first one related to linear elastic fracture mechanics (LEFM) approaches and the second one to elastic–plastic parameters. 4. Elastic fracture parameters The general peculiarity of the mixed-mode cyclic fracture is that the crack growth is along a curvilinear path and does not follow a universal trajectory. For in-plane mixed mode fracture conditions, when the direction of the applied loading does not coincide with the initial crack plane, two forms of crack tip displacement occur, characterized by the corresponding stress intensity factors K1 and K2. In other words, the fracture under mixed mode loading is complex when two elastic SIFs K1 and K2 are acting at the same time. This fact predestines the need to use the effective stress intensity factor Keff, which is a function both K1 and K2, for the mixed mode fatigue experimental data interpretation. Our method of mixed mode fatigue interpretation of the experimental results has been performed based on the effective stress intensity factor in the form of the strain energy density factor S introduced by Sih [27]

S ¼ b11 K 21 þ 2b12 K 1 K 2 þ b22 K 22

9 1 b11 ¼ 16G ðj  cos hÞð1 þ cos hÞ > > > 2 > b12 ¼ 16G ðcos h  j þ 1Þ sin h > > = 1 b22 ¼ 16G ½ðj þ 1Þð1  cos hÞþ ; > > > þð3 cos h  1Þð1 þ cos hÞ > > > ; E G ¼ 2ð1þ mÞ ; j ¼ 3  4m ð13Þ

where the elastic stress intensity factors are pffiffiffiffi   K 1 ¼ r 2pa ½1 þ g  ð1  gÞ cos 2a  Y 1 wa pffiffiffiffi   K 2 ¼ r 2pa ð1  gÞ sin 2a  Y 2 wa :

ð14Þ

In Eq. (14), a is the half-crack length, a is the inclined angle of the crack with respect to the y-axis and g is the biaxial nominal stress ratio, whereas Y1(a/w) and Y2(a/w) are the geometry dependent correction factors, E is Young’s modulus, and m is Poisson’s ratio. Shlyannikov [22] introduced the flow chart of the procedure for computing the geometry dependent correction factors Y1 and Y2 for the stress intensity factors (SIF) KI and K2 and the T-stress for cruciform specimens under different biaxial loading. This study explores the direct use of FEM analysis for calculating KI and K2 and the T-stress by using Eq. (12) for the crack flank nodal displacements. Calculation algorithm is made so that the first step is the Tstress determination. Once the T-stress is known for each type of cruciform specimen CS-1 and CS-2 geometry and crack length and crack angle combination, the geometry dependent correction factors Y1 and Y2 for the SIF K1 and K2 may be obtained as a function of the T-stress. To this end, the commercial finite element code,

73

ANSYS 14.0 [24], has been used in [22] to calculate the displacement and stress distributions ahead of crack tips. Fig. 6 is a plot of the normalized (T ¼ T=r) T-stress (Fig. 6a and c) as well as the geometry dependent correction factor Y1 (Fig. 6b and d) as a function of the relative crack length a/w for CS specimens with both configurations under different remote load biaxiality values. The pure mode I distributions of these parameters in Fig. 6 ahead of the crack-tip (h = 0°) for the CS-1 is related to the plane stress of the thin, flat specimen. For the cruciform specimen CS-2, Fig. 6 shows the distributions of the T-stress and Y1 for two crack front position, denoted as 2D on the free surface of the specimen (z/t = 0) and 3D on the mid-section (z/t = 0.5), where t is the specimen thickness. For the CS-1, the deviation of the current value of T from the corresponding value for a/w = 0.1 increases by increasing the relative crack length at the fixed loading biaxiality. The distributions of the T-stress and Y1 for CS-2 are almost independent of the relative crack length a/w. Moreover, the behavior of the geometry dependent correction factor Y1 in Fig. 6d differs for z/t = 0 (2D) and z/t = 0.5 (3D). It can be seen from Fig. 6a and c that in pure mode I under equibiaxial tension g = +1.0, for any relative crack length a/w the value of T is close to zero. This result is in good agreement with the theoretical prediction for an infinite cracked body. The normalized T-stress and geometry dependent correction factors Y1 and Y2 distributions for various cruciform specimen geometries under mixed mode loading conditions are determined from the displacement fields of finite element calculations and are presented in Fig. 7. For the CS-1, the numerical results are obtained under a load biaxiality k = 0.0 with an initial crack angle variation a e (25–90°), whereas for the CS-2, k = 1.0 and a e (45–90°). The choice of combinations of k = 0.0, a = 25° (CS-1) and k = 1.0, a = 45° (CS-2) is not accidental because both variants correspond to the pure mode II. Again, it can be seen from Fig. 7d–f that the distributions of T-stress and Y1, Y2 for the CS-2 are minimally sensitive to changes of the relative crack length a/w. The numerical results reveal the main variation pattern of mode I and mode II of both T-stress and geometry dependent correction factors Y1 and Y2 for the particular geometry considered. The variation pattern also provides a useful tool for analyzing and explaining the behavior of the fatigue crack growth rate under mixed mode biaxial fracture. By substituting the calculated values of the geometry dependent correction factors Y1 and Y2 into Eqs. (14), the current values of the strain energy density factors S, which are used for plotting of the fatigue fracture diagram, can be determined. In this paper the interpretation of the biaxial crack growth rate as mentioned above will be given on the basis of both the elastic and plastic fracture mechanics parameters. For this purpose, it is necessary to consider the formulation of the plastic stress intensity factor for in-plane mixed mode loading.

5. Plastic fracture parameter To interpret the fracture resistance material characteristics as a function of plastic fracture parameters, Shlyannikov and Tumanov [28] reconsidered the classical Hutchinson [29] solution for both plane strain and plane stress conditions. They supposed that under small-scale yielding, the expression for the governing parameters of the elastic–plastic solution in the form of In-integral depends implicitly on the dimensionless crack length and specimen thickness and configuration. This suggestion leads the authors [28] to the new formulation of the amplitude factor governing singular stress–strain fields for the mixed-mode fracture of a power-low hardening material, which continued the small-scale yielding analysis given by Shih [30]. It is shown [28] that the amplitude of the

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Fig. 6. Dimensionless T -stress and Y1-factor distributions under biaxial loading of (a, b) CS-1 and (c, d) CS-2 for pure mode I.

first term in the mixed-mode small-scale yielding in the form of the plastic stress intensity factor K PM can be expressed in terms of the corresponding elastic SIFs K1 and K2, but the expression depends implicitly on the additional mode mixity parameter MP. In accordance with the approach of Shih [30], the plastic stress intensity factor K pM in the mixed-mode small-scale yielding can be directly expressed in terms of the corresponding elastic stress intensity factor using Rice’s J-integral

ðK 21 þ K 22 Þ ar20 nþ1 ¼ 0 In ðK pM Þ E0 E " #1=ðnþ1Þ ðK 21 þ K 22 Þ ¼ ; a r20 In

J¼

and K pM ð15Þ

where K1 and K2 are the elastic stress intensity factors (Eq. (14)) for the CS and E0 = E for plane stress and E0 = E/(1  m2) for plane strain. Substituting Eq. (14) into Eq. (15) gives the following formula for the plastic SIF for the cruciform specimen with an inclined crack under biaxial loading K PM

" ¼

r r0

2

p  a ½1 þ g  ð1  gÞ cos 2a2 Y 21 þ ½ð1  gÞ sin 2a2 Y 22  w 4a In ðh; n; M P ; ða=wÞÞ

#1=ðnþ1Þ : ð16Þ

 and n are the strain hardening In the above equation, a parameters and r0 is the yield stress. It has been shown by Hilton  ¼ r=r0 , the and Sih [31] that even though at half the yield stress r elastic–plastic boundary has been distorted considerably from that

associated with the small-scale yielding conditions and that the plastic SIF still gives accurate predictions for the fracture initiation at this level of applied stress. In the framework of the present work, the governing parameter of the elastic–plastic stress–strain fields in the form of the numerical constant In(h, n, MP, (a/w)) for the cruciform specimen of both configurations is obtained by means of the elastic–plastic FEanalysis of the near crack-tip fields. The analysis was extended to the In-integral behavior in an infinitely sized cracked body to treat the test specimen’s specified geometries. The use of Hutchinson’s [29] theoretical definition for the In-factor directly adopted in the numerical finite element analyses leads to the following expression for the test specimen’s specified geometries

IFEM n ðh; M p ; n; ða=wÞÞ ¼

Z p

XFEM ðh; MP ; n; ða=wÞÞdh

ð17Þ

p

where XFEM ðh; MP ; n; ða=wÞÞ ¼

n FEM ~ nþ1 Þ cosh ðr nþ1 e

    ~ FEM ~ FEM du du ~ FEM ~ FEM ~ FEM ~ FEM  r  r þ h r sinh u u rr h rh r dh dh 1 ~ FEM u ~ FEM þ r ~ FEM ~ FEM Þcosh: ðr  rh uh n þ 1 rr r

In such a case, the numerical integral of the crack tip field In not only changes with the strain hardening exponent n and mode mixity MP but also changes with the relative crack length a/w and specimen configuration. It is found by Shlyannikov and Tumanov [28],

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75

Fig. 7. Dimensionless T -stress and Y1, Y2-factor distributions under mixed mode loading of (a, b, c) CS-1 and (d, e, f) CS-2.

from the stress–strain relations for plane strain and plane stress and the geometrical equations, that in cylindrical coordinates originating at the crack tip, the numerically determined dimensionless  FEM displacement components u ¼ ðuFEM =aÞ contained in Eq. (12) are i i

  ½ðK PM Þ  FEM ¼a u r   ½ðK PM Þ  FEM ¼a u h ¼

FEM n

~ FEM   r nðS2Þþ1 u ; r

FEM n

~ FEM   r nðS2Þþ1 u ; h

n  FEM ~ FEM ðr Þ u h e ; n  FEM r a ðr e Þ

~ FEM ¼ u r

n  FEM ~ FEM ðr Þ u r e n  FEM r a ðr e Þ

~ FEM u h ð18Þ

where ðK PM Þ

FEM

r FEM

¼ rs2er~ FEM . e

 e ¼ re =r0 is the von Mises equivalent stress, In these equations r and r ¼ r=a is the dimensionless crack tip distance, s = (2n + 1)/ (n + 1). The values of the numerical constant of the elastic–plastic stress field in the form of the In-integral are calculated by substitution of Eq. (18) into expression (17), keeping in mind that the ~ FEM dimensionless angular stress functions r are directly deterij mined from the FEA distributions. More details concerning the Infactor determination for the test specimen’s specified geometries is given by the authors [23,28]. In all numerical elastic–plastic FEM calculations, the material behavior is described by the Ramberg–Osgood equation, and its constants are listed in Table 1.

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Fig. 8 presents, for the CS-1 and CS-2 geometries, a clear illustration of the influence of the biaxial loading under pure mode I on the In-factor distribution along increasing relative crack length. It can be observed that the In-curves as a function of load biaxiality differ from each other for the same specimen geometries. As expected, the computed values of the In-integral are sensitive to a general state of either plane stress or plane strain conditions for biaxially loaded cruciform specimens. Fig. 9 shows the relationship between the In-integral and the mode mixity parameter Mp as well as the relative crack length a/ w under plane stress and plane strain for the CS-1 and CS-2 fracture specimen configurations. A comparison of the numerically obtained In-factor distributions demonstrates that the influence of the specimen geometry characteristics under mixed-mode loading is significant for the parameters of elastic–plastic stress–strain fields. Again, it can be seen from Fig. 9c and d that the In-factor for the CS-2 are minimally sensitive to changes of the relative crack length a/w. It is interesting to note in Fig. 9a and b that uniaxial tension with the load ratio k = 0.0 and nominal stress biaxial ratio g ffi 0.2 allows for the realization, in the CS-1 cruciform specimen, of the full range of mode mixity from pure mode II (MP = 0.0) to pure mode I (MP = 1.0) due to crack angle variation a e (25–90°). The same range of mode mixity can be achieved in the CS-2 geometry under equibiaxial tension–compression k = 1.0 with crack angle variation of a e (45–90°). By substituting the calculated values of the In-factor into Eq. (16), the current values of the plastic stress intensity factor K pM , which are used for plotting the fatigue fracture diagram, can be determined.

6. Experimental results and discussion The CS fatigue crack growth tests have been performed with servohydraulic biaxial test equipment at a frequency of 5 Hz at a stress ratio R = 0.1. The equipment has four independent loading arms with load actuators, which exert up to 50 kN on the both axes. Tensile or compressive loads are applied to each pair of arms of the cruciform specimens (Fig. 1), developing a biaxial stress field in the working section. The loads are controlled such that the specified forces are produced on opposing arms of the CS according to the given load biaxiality. For pure mode I at crack angle equal a = 90°, four biaxial load ratios for CS-1, k equal to +1.0, +0.5, 0.0 (uniaxial), and 0.2 and three ratios for CS-2, k equal to +1.0, 0.0 (uniaxial), and 1.0, were considered, respectively. Mixed mode crack growth tests were performed for cruciform specimens CS-1 under uniaxial tension with k = 0.0 with the initial crack angle equal a = 25° and for the CS-2 at k = 1.0 under equibiaxial tension–compression with the initial crack angle a = 45°. The cruciform specimens were fatigue pre-cracked with a load biaxiality ratio of k = +1.0, that is, predominantly in mode I. The tests were conducted at room temperature under main loading axis displacement control. The load output signal of the main axis load cell was multiplied by the load biaxiality ratio k and fed as the controlling signal into the secondary axis. The secondary axis was thus under load control. The main mechanical properties for the tested materials are listed in Table 1. The experimental study of the fatigue crack growth rate in lowstrength Steel 3 was performed on biaxially loaded flat cruciform

Fig. 8. In-factor as a function of relative crack length for pure mode I in (a, b) CS-1 and (c, d) CS-2 under (a, c) plane stress and (b, d) plane strain.

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Fig. 9. In-factor as a function of mode mixity in (a, b) CS-1 and (c, d) CS-2 under (a, c) plane stress and (b, d) plane strain.

specimen CS-1 with a thickness of 3 mm. All specimens for biaxial loading contained through-thickness central cracks with initial size equal to 20 mm. A series of tests were performed to study the effect of the tension–compression loading biaxiality on the crack growth rate in the same configuration of cruciform specimens under pure mode I. The range of nominal stress r1 yy was kept constant throughout a test series, whereas r1 was varied for four difxx ferent forms of the test. The stress state designated through g, 1 equal to the ratio of r1 xx and ryy , (where the latter two terms are the maximum values of stress in a cycle), has been assumed equal to +1.0, +0.34, 0.2 and 0.4. In the four considered cases, the maximum applied stress and the R-ratio were the same with only the biaxiality ratio g being varied. The crack growth results are shown in Fig. 10a in terms of the crack growth rate plotted against the maximum elastic strain energy density factor SEDmax ¼ SEDmax  G for the different biaxialities corresponding to the same CS-1 configuration of the low strength Steel 3. The elastic SEDmax is used only as a convenient parameter because it provides a suitable comparison of pure mode I and mixed mode cyclic fracture. For the Steel 3, the crack growth rate (da/dN) plotted against the elastic SED showed that for both positive (k = + 0.5, k = 0.0) and negative (k ¼ 0:2) values of biaxial load ratios, the growth lines fall to the right of the equibiaxial tension line k = +1. Fig. 10c represents the pure mode I fatigue fracture diagram in the coordinates of the crack growth rate versus the maximum

elastic strain energy density factor SEDmax for different biaxialities corresponding to the cruciform specimen CS-2 configuration of the low strength Steel 3. In contrast to Fig. 10a, for the steel 34XH3MA, the crack growth rate (da/dN) plotted against elastic SED in Fig. 10c showed that for both positive (k = 0.0) and negative (k = 1.0) values of biaxial load ratio, the growth lines fall to the left of the equibiaxial tension line k = +1. The fatigue fracture diagram corresponding to uniaxial tension (k = 0.0) lies between the experimental curves of the equibiaxial tension (k = 1.0) and the equibiaxial tension–compression (k = 1.0) loading. A similar situation is observed in Fig. 10b and d in the interpretation of the crack growth rate experimental data for different biaxial loading of the structural materials considered in terms of the plastic stress intensity factor. The necessity to present a multiaxial fatigue failure diagram in such coordinates is due to the obvious advantages of using the plastic SIF for fracture mechanics problems. More detailed discussion of these matters is given by the review of Shlyannikov and Tumanov [28], as well as the study by the authors [23]. For instance, practical structural components have finite thicknesses, and the stress–strain state changes between plane stress and plane strain. However, elastic SIFs have the same values for these situations at the specified nominal stress level. In contrast, as expected, the computed values of the plastic SIFs are sensitive to a general state of either the plane stress or plane strain conditions. Furthermore, K pM may vary along the three-dimensional crack front.

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Fig. 10. Crack growth rate versus (a, c) elastic SED and (b, d) plastic SIF in (a, b) CS-1 and (c, d) CS-2 under pure mode I biaxial loading.

It is well known that the in-plane constrain is related to the influence of cracked body geometry and loading conditions while the source of a change of the out-of-plane constraint is associated with the body size in the z-direction or the specimen thickness. Fatigue crack growth rate data are not always geometry-independent in the strict sense since thickness effects sometimes occur. Under some circumstance it can be found that substitution of appropriate values of plastic stress intensity factor into a fatigue growth equation is sufficient to account for the effect of specimen thickness. Notably, plane stress and plane strain can be considered special cases for out-of-plane or thickness constraints. The numerical results from the present study concerning the In-distributions in Fig. 8 for the CS both configuration in the range between the plane stress and the plane strain, as mentioned in [23,28], indicates the obvious potential of the plastic SIF to describe the influence of the in-plane and out-of-plane constraint interactions on the crack growth rate as a function of the tested specimen configuration. The decisive argument for the choice of the fracture parameter is the sensitivity of K PM through In values to the influence of cracked body geometry and biaxial/mixed mode loading conditions.

Going back to the elastic strain energy density, it should be noted that the elastic solutions do not coincide with any of the biaxial elastic–plastic solutions because the elastic solutions used only the elastic material properties of the Young’s modulus and Poisson’s ratio without taking into account the yield stress and strain hardening exponent. Strictly speaking, elastic stress intensity factor in the expression for the SED is only a function of the loading conditions, the size of the defect and the configuration of a cracked body. Nevertheless, the elastic stress intensity factor is not sensitive to load biaxiality for pure Mode I. Looking at the present experimental results in Fig. 10 and considering changes in the fatigue fracture diagrams of the cruciform specimens under pure Mode I loading, the foregoing effect of the stress range on the magnitude of the biaxial influence is evident. Nevertheless, a qualitative comparison between the results for CS-1 and CS-2, which were produced from two types of steel, would lead us to the opposite conclusions, based not only on the different properties of the various materials being studied. A number of studies of Mode I fatigue crack propagation under biaxial stress conditions have been reported in the literature

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[1,2,12,14,15,18–20]. Some of the results show that the crack propagation rate decreases as the biaxial stress ratio g increases in its positive range. This fact is confirmed by the results of Miller [12] who, when considering the effect of the stress state on crack growth by using a cruciform specimen of the CS-2 configuration, showed that for mode I cracks, the fatigue fracture diagrams for equibiaxial tension, uniaxial tension and equibiaxial tension–compression are the same order as is shown in Fig. 9c for the steel 34XH3MA. Other studies [19] claim that compressive stress parallel to the plane of the mode I crack decreases the rate of crack growth under biaxial cyclic loading, which coincides with the results presented in Fig. 10a for the steel 3. The reasons for such discrepancies can be ascertained when comparing different configurations of biaxially loaded cruciform specimens of CS-1 and CS-2. Turning to the geometry dependent correction factor distributions in Fig. 6d and c, it should be noted that the CS-1 values of Y1(a/w) at the considered load biaxiality increased with increasing relative crack length (a/w), whereas the Y1(a/w) behavior in CS-2 is almost independent of the crack size. The CS-1 specimen is more constrained in comparison with the CS-2 because the absolute values of the T-stress in Fig. 6a increase as the relative crack length increases. Due to the weak dependence of both the geometry dependent correction factor Y1(a/w) and the T-stress on the relative crack length, the cruciform specimen CS-2 with the thinned working area is more convenient to study the crack growth rate under biaxial loading because material property effects appeared in the pure form. The state of stress will affect the plastic response of a material, and it can be assumed that crack growth is a function of the extent of the crack tip plastic zone, which is a consideration that emphasizes the importance of plasticity on fatigue crack growth, supposing a possible correlation between crack-tip plastic zone size and propagation rate. The authors of [12,19,20] have attempted to explain the effect of stress biaxiality on fatigue crack growth rate in terms of the distribution of plastic strain near the crack tip. However, the same trend of changes for the plastic zone size as a function of load biaxiality leads to the opposite effect on the crack growth rate, as shown in Fig. 10. Hence, neither solely the elastic stress intensity factor nor the plastic zone size provide unambiguous one-parameter descriptions of the crack growth rate under pure mode I biaxial loading. The typical experimental fatigue fracture diagrams represented through the crack growth rate (da/dN) versus the maximum elastic strain energy density factor (SED), defined by Eqs. (13), (14) are shown in Fig. 11a and b for both the investigated CS-1 and CS-2 specimens for the mixed mode loading. Pure mode I fatigue fracture diagrams, for a comparison purpose, are also presented in these figures. Mode I crack growth corresponds to a = 90°, whereas a = 25°, k = 0.0 for CS-1 and a = 45°, k = 1.0 for CS-2 represent mixed mode crack growth. Fig. 11a shows that mixed mode-loading causes an increase in the crack growth rate in terms of the LEFM parameters with respect to pure mode I for flat CS-1 specimens. In contrast, the crack growth rate decreases in the cruciform specimen CS-2 (Fig. 11b) with the thinned working area and decreases when the mode mixity changes from pure mode I to pure mode II. In addition to the traditional interpretation of the cyclic fracture resistance characteristic of the materials as (da/dN) versus the elastic SED, in Fig. 12a and b, the biaxial mixed mode crack growth results are presented in terms of the crack growth rate plotted against the maximum plastic stress intensity factor K PM in the formulation of Eq. (16) for both tested specimen geometries. A similar trend of the influence of mode mixity on the crack growth rate is observed for CS-1 and CS-2 in terms of the elastic SED (Fig. 11a and b) and plastic SIF (Fig. 12a and b). Namely, in Fig. 11 a = 90° at k = 0 and k = 1 represent pure mode I whereas a = 25° at k = 0

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and a = 45° at k = 1.0 relates to mixed mode fatigue fracture. Accordingly, in Fig. 12 the combination of a = 90° and k = +1.0 is pure mode I, and combination of a = 45° and k = 1.0 is mixed mode loading conditions. The tested materials have the same elastic properties but differ significantly from each other on the characteristics of plasticity, as shown in Table 1. Interpretation of the experimental results through the elastic fracture parameters does not allow for determining the influence of the plastic properties of the material on the crack growth rate under mixed mode loading. Furthermore, the elastic stress intensity factor in the presence of appreciable plastic deformations around the crack tip no longer represents the true mechanical conditions, and it fails to distinguish between two basically different phenomena that may have the same numerical values for SIF’s, i.e., plane strain and plane stress. Using the plastic stress intensity factor in the interpretation of the fatigue failure diagram removes this limitation. Unlike the elastic SED, the plastic stress intensity factor K PM in the formulation of Eq. (16) is sensitive enough to account for the influence of the plastic properties of the tested materials. Moreover, the plastic SIF, in contrast to the LEFM models, provides the ability to account for the effect of load biaxiality under pure mode I. In recent years, observations of the T-stress effect have been reported in the literature [18–20,22,23]. Brown and Miller [13] have shown that the application of higher stresses and a negative T-stress increases the fatigue crack propagation rate. A similar dependence of the crack growth rate on the T-stress through the plastic zone size has been observed by Kitagawa et al. [19] and Gao et al. [20] for small-scale yield conditions and mixedmode loading. According to the T-stress distributions given by Shlyannikov [22], the fatigue crack growth rates obtained from specimens of various geometries may differ if the nonsingular component of stress parallel to the crack direction. Biaxial fatigue crack propagation tests have shown [23] that Mode I cracks are accelerated by positive T-stress, and the magnitude of this effect is dependent on the T-stress distributions as a function of both the load biaxiality and the relative crack length. However, experimental data on the influence of the thickness on the fatigue crack growth rate are contrasted. In support of this assertion can be considered two loading cases, namely, equibiaxial tension at k = 1.0, a = 90° and equibiaxial tension–compression at k = 1.0, a = 45°. The T-stress theoretical expression for a plate under biaxial loading with an inclined crack is defined by the Eftis and Subramonian [32] equation

T ¼ rð1  gÞ cos 2a:

ð19Þ

For both cases of biaxial loading, the T-stress is equal to zero, T = 0. This theoretical conclusion is confirmed by the numerical results presented in Figs. 6c and 7d. Nevertheless, for the same values of the T-stress, the corresponding fatigue failure diagrams do not coincide with each other as shown in Fig. 12a and b. Therefore, the T-stress is not an unambiguous characteristic of the material fracture resistance, and different values of T-stress may be used only as an additional parameter characterizing the crack growth rate under mixed modes of loading. The practical application of the fracture mechanics to real structure elements requires appropriate parameters to quantify the material cyclic fracture resistance characteristics in the form of a fatigue fracture diagram. In the well-known equation of Paris, describing a linear part of such diagrams represented through the crack growth rate da/dN versus the elastic SIF, the parameters C and m can be considered as constants to characterize the material resistance to crack growth under cyclic loading. Analogously, in the

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Fig. 11. Crack growth rate versus elastic SED in (a) CS-1 and (b) CS-2 under pure mode I and mixed mode loading.

Fig. 12. Crack growth rate versus (a) elastic SED and (b) plastic SIF in CS-2 under pure mode I and mixed mode loading.

small-scale yielding range, when the plastic SIF is the governing parameter of cyclic fracture, the predictions based on the plastic SIFs are similar to those based on the elastic SIFs in the form of the fatigue fracture diagram (da/dN vs. plastic SIF) with parameters CP and mP. Using a simple definition by Eq. (16), which requires no additional assumptions, the elastic–plastic stress intensity factor can also characterize the crack growth rate under biaxial loading and mixed mode fracture. The decisive argument for the choice of the fracture parameter is the sensitivity of K PM through the In values to the constraint effect and plastic properties of structural materials. It is further assumed that for moderate large-scale yielding conditions or plastic deformations, the multiaxial fracture process can be controlled by the single parameter K PM based on the elastic– plastic numerical solutions and, therefore, is reflected in the influence of the cruciform specimen configuration, as well as the biaxial and mixed mode loading conditions.

7. Conclusions The distributions of the governing parameter of the elastic– plastic stress fields, the In-factor, have been determined in two cruciform specimen geometries from numerical calculations. For both CS configurations, the In-factor variation pattern for plane stress and plane strain has been obtained as a function of the material plastic properties, load biaxiality, mode mixity and relative crack length. Biaxial and mixed mode fatigue crack propagation tests have been performed on cruciform specimens. Discrepancies in the fatigue crack growth rate have been observed in various biaxial loading conditions. It is stated that using specimens of different configurations can lead to opposite conclusions about the crack growth rate under biaxial loading due to the influence of the specimen geometries and not a manifestation of the material properties. In this sense, the cruciform specimen CS-2 with thinned

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working area is more convenient to study the crack growth rate under biaxial loading because material property effects appeared in the pure form. The experimental results of the present study provided the opportunity to explore the suggestion that biaxial and mixed mode fatigue crack propagation may be governed more strongly by the plastic stress intensity factor rather than the magnitude of the elastic SIFs alone. One advantage of the plastic SIF is its sensitivity to pure mode I biaxial loading due to accounting for the plastic properties of the material. Acknowledgments The authors gratefully acknowledges the financial support of the Russian Foundation for Basic Research under the Project 1308-92699 IND_a. References [1] M.W. Brown, K.J. Miller, Initiation and growth of cracks in biaxial fatigue, Fatigue Eng. Mater. Struct. 1 (1979) 231–246. [2] M.W. Brown, K.J. Miller, Two decades of progress in the assessment of multiaxial low-cycle fatigue life. Low-Cycle Fatigue and Life Prediction, ASTM Spec. Tech. Publ. 770 (1982) 482–499. [3] A. Carpinteri, A. Spagnoli, S. Vantadori, Multiaxial fatigue life estimation in welded joints using the critical plane approach, Int. J. Fatigue 31 (2009) 188– 196. [4] A. Carpinteri, A. Spagnoli, S. Vantadori, Multiaxial assessment using a simplified critical plane-based criterion, Int. J. Fatigue 33 (2011) 969–976. [5] A. Carpinteri, A. Spagnoli, S. Vantadori, C. Bagni, Structural integrity assessment of metallic components under multiaxial fatigue: the C–S criterion and its evolution, Fatigue Fract. Eng. Mater. Struct. 36 (2013) 870– 883. [6] A. Carpinteri, A. Spagnoli, A. Vantadori, Reformulation in the frequency domain of a critical plane-based multiaxial fatigue criterion. Int. J. Fatigue (2014) . [7] A. Karolczuk, E. Macha, Critical Planes in Multiaxial Fatigue of Materials. Fortschritt-Berichte VDI, Reihe 18 Mechanik/Bruchmechanik, N298, VDI Verlag, Dusseldorf, 2005, 204 s. [8] L. Susmel, R. Tovo, D.F. Socie, Estimating the orientation of Stage I crack paths through the direction of maximum variance of the resolved shear stress, Int. J. Fatigue 58 (2014) 94–101. [9] L. Susmel, D. Taylor, A critical distance/plane method to estimate finite life of notched components under variable amplitude uniaxial/multiaxial fatigue loading, Int. J. Fatigue 38 (2012) 7–24. [10] V.N. Shlyannikov, Modelling of crack growth by fracture damage zone, Theoret. Appl. Fract. Mech. 25 (1996) 187–201. [11] E.W. Smith, K.J. Pascoe, The behaviour of fatigue cracks subject to applied biaxial stress: a review of experimental evidence, Fatigue Fract. Eng. Mater. Struct. 6 (1983) 201–224.

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