An automatic algorithm for mixed mode crack growth rate based on drop potential method

International Journal of Fatigue 81 (2015) 227–237

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

An automatic algorithm for mixed mode crack growth rate based on drop potential method A.V. Tumanov a,⇑, V.N. Shlyannikov a, J.M. Chandra Kishen b a b

Kazan Scientific Center of the Russian Academy of Sciences, Russia Dept. of Civil Engineering, Indian Institute of Science, Bangalore, India

a r t i c l e

i n f o

Article history: Received 31 March 2015 Received in revised form 7 August 2015 Accepted 9 August 2015 Available online 13 August 2015 Keywords: Mixedmode cyclic fracture Drop potential method Crack growth rate Test automation algorithm Equivalent straightline crack

a b s t r a c t A new automatic algorithm for the assessment of mixed mode crack growth rate characteristics is presented based on the concept of an equivalent crack. The residual ligament size approach is introduced to implementation this algorithm for identifying the crack tip position on a curved path with respect to the drop potential signal. The automatic algorithm accounting for the curvilinear crack trajectory and employing an electrical potential difference was calibrated with respect to the optical measurements for the growing crack under cyclic mixed mode loading conditions. The effectiveness of the proposed algorithm is confirmed by fatigue tests performed on ST3 steel compact tension–shear specimens in the full range of mode mixities from pure mode I to pure mode II. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The concept of the mode I elastic stress intensity factor is the basis of most contemporary standards for the fracture resistance characteristic determination [1,2]. In accordance with these standards, it is necessary to know the dependence of the crack length on the number of loading cycles for the determination of the crack growth rate characteristics. The combination of experimental, numerical and analytical methods for pure mode I cracks makes it possible to automatically obtain all the necessary parameters. However, in industrial practice, mixed-mode fracture and crack growth are more likely to be the rule than the exception. It is well known that mixed-mode conditions appear when the direction of the applied loading does not coincide with the orthogonal KI–KII–KIII space. The main feature of the mixed-mode fracture is that the crack growth would no longer take place in a self-similar manner and does not follow a universal trajectory, that is, it will grow on a curvilinear path [3]. Criteria that cover the initial branch crack direction [4–7] and crack growth rate models [8–12] under mixed mode I and II loading conditions in both brittle and ductile materials have been extensively discussed in the literature, and a wide range of experiments have been performed. The traditional formulations of most crack reorientation criteria and crack growth ⇑ Corresponding author. E-mail addresses: [email protected] (A.V. Tumanov), [email protected] (V.N. Shlyannikov), [email protected] (J.M. Chandra Kishen). http://dx.doi.org/10.1016/j.ijfatigue.2015.08.005 0142-1123/Ó 2015 Elsevier Ltd. All rights reserved.

rate models are connected with the use of the straight line crack concept in common with elastic singular solutions. The crack growth from an inclined crack illustrates mixed-mode crack behavior on the initial crack. In previous studies [13,14], it was shown that the effect of mixed mode parameters on the fracture resistance characteristics was significant. Currently, there are no guidelines in the literature to determine the curvilinear crack size for cases wherein visual observation is not available. In most cases involving cyclic mixed mode fracture, the crack deviation angle is not constant and changes continuously as a function of the number of loading cycles. Different curvilinear crack paths are observed for the same specimen geometry depending on the applied mode mixities. In this connection, establishment of the correlation between the crack size and the observed parameters becomes more important. On that score, it is necessary to solve the problem of identifying the position of the crack tip on a curved path with respect to the signal of a measurement tool. For mixed mode I and II crack propagation, the crack front continuously changes shape and direction with each loading cycle. As a result, the angle of crack propagation h
continuously changes. At each successive position of the crack front, the stress intensity factors in a plate, K1 and K2, must be calculated. However, for the actual ‘‘bent” crack geometry, the expressions for K1 and K2 cannot be easily determined. To overcome this difficulty, an approximate procedure has been independently proposed by Sih and Barthelemy [15] and Shlyannikov and Ivanishin [16] and has been used by Au [17], Pandey and Patel [18], Gdoutos [19] and Xua [20].

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Nomenclature

a

b x; y r; h h
/ a0 a aeqv ax ars W t F R E

t r0 rf


n a


angle between load line and initial crack angle between initial crack and equivalent crack global Cartesian coordinate system centered on initial notch tip polar coordinate system centered on crack tip crack propagation angle additional parameter for crack propagation angle determination initial crack length crack size along curvilinear path equivalent crack length projection of crack tip on OX axis residual ligament specimen width specimen thickness load applied to the specimen stress ratio Young modulus Poisson’s ratio yield stress ultimate stress strain hardening exponent hardening amplitude

Essentially, the procedure involves replacing the bent crack with a straight line crack approximation that is a function of the curvilinear crack path. Despite the existence of several experimental methods for the investigation of pure mode I crack extension problems, for instance [21–23], the automation of mixed-mode cyclic fracture phenomena is far from complete, as few procedures have been proposed in the literature [24–26]. It should be noted that these methods do not, in general, take into account the curvilinear crack path under cyclic mixed mode fracture. However, one essential part of evaluating the behavior and modeling crack growth under mixed mode loading is the actual crack length determination. There are several ways to measure the crack length that are supported by the manufacturers of standard test systems: 1. optical method – in this case, the crack measurement is directly on the surface of the specimen; 2. crack opening displacement measurement – for this method, we need to know the calibration function between displacements on a load line (or another fixed point on the specimen) and the average crack size; 3. drop potential method – this method is based on the change in electrical resistance of the specimen due to changing the ligament between the crack tip and the specimen border. The purpose of an automation process for measuring the crack length is important, especially when direct observation of the specimen surface is not possible, such as under extremely high (or low) temperatures or surface crack tests. Of the above mentioned measurement methods, the drop potential method is preferred. This method does not have any restrictions on the ambient temperature, in contrast to the use of the extensometer, making it a versatile method for the determination of the fatigue crack size. The literature concerning this aspect of experimental fracture mechanics is reasonably comprehensive, and several numerical–experimental methods have been proposed to determine the relationship between the drop potential signal and the current crack length. Such dependences with respect to problems of mixed mode cyclic fracture are restricted to only special cases

rn the nominal stress applied to the specimen rij ði; j ¼ x; y; zÞ stress fields near the crack tip eij ði; j ¼ x; y; zÞ strain fields near the crack tip ui ði ¼ x; y; zÞ displacement fields near the crack tip

rFEM ij ði; j ¼ x; y; zÞ stresses obtained from the finite element

solution K i ði ¼ 1; 2Þ stress intensity factors K i ði ¼ 1; 2Þ dimensionless stress intensity factors T nonsingular T-stress T dimensionless T-stress Y i ði ¼ 1; 2Þ geometry dependent stress intensity N number of the load cycle Ai ; Bi ; C i ði ¼ 1; 2; . . . ; nÞ coefficients for polynomial approximation Si ði ¼ 1; 2; 3Þ coefficients for strain energy density value of the strain energy density where crack growth S
rate equal to 104 mm/cycle C constant of the Paris type law V potential drop V0 potential drop for initial crack V a=w¼0:7 potential drop when the residual ligament is equal to 30% of the specimen width D ratio between dimensionless projection of crack tip on OX axis and dimensionless potential difference

of pure mode II and bending of straight inclined cracks. Recently, a variation of the potential drop technique has been proposed by the authors [25] for the real time assessment of the length of a straight crack growing from in-plane shear using a specially designed shear specimen. The study [26] considers straight in-shape propagating cracks inclined to the transverse axis of symmetry of the bending specimen in the frame of the automation process based on the potential drop technique. In the present paper, a new automatic algorithm for the assessment of mixed mode crack growth rate characteristics is presented, based on the concept of an equivalent crack. The residual ligament size approach is introduced to implement this algorithm for identifying the crack tip position on a curved path with respect to the drop potential signal. The automatic algorithm accounting for the curvilinear crack trajectory and employing an electrical potential difference was calibrated with respect to optical measurements for the growing crack under cyclic mixed mode loading conditions. To represent the material cyclic fracture resistance characteristics, the mixed mode crack growth rate is calculated as the average crack extension per unit load cycle versus certain equivalent fracture parameters in the form of the strain energy density (SED) factor. The proposed algorithm was experimentally verified by fatigue tests performed on ST3 steel compact tension-shear specimens. The effectiveness of the proposed algorithm is confirmed by testing the CTS specimens over the full range of mode mixities from pure mode I to pure mode II.

2. Theoretical background It is known that a ‘‘bent” crack does not propagate in its initial orientation direction. A mixed mode crack propagates along a definite trajectory that is determined by the stress state, the previous crack orientation angle and the material properties. Under these conditions, to predict the fatigue crack propagation rate, it is necessary to determine it by means of calculations. An approximate procedure based on an equivalent straight line crack was applied by Shlyannikov [6,27] for crack path prediction for geometric configurations containing a single-edge crack of length and oblique-

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229

increment of crack growth, the crack angle changes from the original angle, as does the effective length of the crack. For the next increment of crack growth, one has to consider the new crack length and crack angle. As shown in Fig. 1, OA is the initial crack length a0 oriented to the load line at an angle a. Let r 0 ¼ AB be the crack growth increment for the first growth step. The value r0 is then extended along AB with an angle h
0 . For the single-edge crack geometry (Fig. 1), the first step of crack growth is obtained as /0 ¼ h
0 and x0 ¼ r0 cos h
0 , y0 ¼ r 0 sin h
0 . The next step is plotting r 1 along BC oriented at an pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 P 2 P P angle h
1 , while AC ¼ x þ y ; /1 ¼ tan1 ð y= xÞ and P x1 ¼ r1 cos c1 , y1 ¼ r1 sin c1 , where c1 ¼ Db1 þ h
1 , x ¼ x0 þ x1 , P y ¼ y0 þ y1 and so on. The values of ai and bi can be determined using the vectorial method. The application of the procedure introduced through the experimental study of mixed mode curved crack trajectories in a wide range of both brittle and ductile materials confirms the effectiveness of the proposed approach based on an equivalent straight line crack [6,27,28]. Fig. 1. Curvilinear crack growth trajectory approximation for single-edge crack geometry.

3. Processing algorithm ness, as shown in Fig. 1. Crack path prediction for a mixed mode I and II initial crack involves replacing a bent crack with a straight line crack approximation, as shown in Fig. 1. A crack may be assumed to grow in a number of discrete steps. After each

Fig. 2 shows a flow chart for the automatic mixed mode crack growth rate characteristic determination. The principal feature of the proposed algorithm is the evaluation of the crack growth rate

Fig. 2. Flowchart for the automatic mixed mode crack growth rate characteristics determination.

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Fig. 3. Mixed mode loading for compact tension–shear specimen.

along curvilinear crack paths. This algorithm is based on the general, for all mode mixities, linear relationship between the ligament size and electrical potential difference in dimensionless coordinates. The ligament size is a distance measured from the crack tip to the specimen border during a fatigue test. Although the algorithm is relatively simple, the analysis can be very timeconsuming, as a large number of crack length and crack angle combinations is required. It should be noted that the sequence, which defines the order of the mixed mode fracture parameter determination, is the same for different test specimen configurations using experimental fracture mechanics. There are three primary components of this algorithm: the analytical description of the crack path, a complementary finite element analysis and the evaluation of the experimentally measured potential signals. These three notional parts can be divided into four interconnected blocks: test data, numerical analysis, approximation and material property determination. The blocks in this algorithm can be varied depending on the purpose of the test and the constitutive equations underlying the tests, as these equations will affect the set of defined parameters. The consistent implementation of the proposed algorithm is as follows. 3.1. Test data The subjects for the mixed mode crack growth rate study are compact tension-shear (CTS) specimens (Fig. 3). The specimen width (W) is 80 mm. The total length of the notch after precracking (a0) was 41 mm for all specimens. The CTS specimens of constant thickness were machined from 3 mm-thick sheets of low-strength structural steel ST3. The most important properties

of the tested material are given in Table 1. Different degrees of mode mixity from pure mode I to pure mode II are given by variations of the inclined crack angle a with respect to the applied force direction. Tests were carried out on a servo-hydraulic test system with a maximum capacity of 100 kN. The CTS fatigue crack growth rate tests were performed at a frequency of 10 Hz at a stress ratio R = 0.1. The reference resistance for the drop potential measurement was set in the measuring units. The curvilinear trajectory and crack size along the path was monitored on the surface of the specimen using an optical microscope. The test was stopped when the residual ligament (ars) reached a value of 0.3 W or lower. This condition limits the scope of the results obtained. The mixed mode loading was achieved by changing the load line angle using special test equipment [29]. This tool set makes it possible to obtain seven different mixed mode loading cases, including pure mode I and pure mode II (Fig. 4), in load steps of 15°. It has been demonstrated by the authors [21] that by attaching four electrodes at selected positions near the tip of the pre-notch of compact tension (CT) and single-edge-notch bending (SENB) specimens, it is possible to obtain electrical measurements that can be correlated to the actual length of the propagating crack. According to this technique, the two electrodes 1 and 2 are welded at a distance 0.35 W along a line starting at the tip of the artificial prenotch and running along the long axis of symmetry of the compact tension–shear specimen (Fig. 5). A constant electrical current passes between these two points, and this creates a distribution of electrical potentials at the surface of the specimen that is symmetrical across the transverse axis, resulting in a measurable potential drop between two other points 3 and 4 placed on the border of the specimen (Fig. 5). The continuous measurements of the electrical potential difference are associated with the actual position of the crack tip on the curvilinear trajectory. During the experiment, a relationship between the potential drops on the crack edges and load cycles is obtained, as shown in Fig. 6. To verify the applicability of experimental mixed mode calibrations, the relationship between the potential drop and crack size along the curvilinear crack path is obtained using optical control, as shown in Fig. 7. Because the crack growth trajectories obtained in the full range of mixed modes (Fig. 4) are arranged between the two lines of pure shear and pure mode I, other mixed mode cases are omitted for submission, except for a = 30°, without loss of generality. Thus, the drop potential method has been calibrated by optical measurements under mixed mode cyclic tests using an instrumental microscope. It can be observed from Fig. 7 that for particular cases of cyclic mixed mode fracture, the registered parameters during the fatigue test, namely, the current drop potential signal and corresponding crack size along the curvilinear crack path, are functions of the mixed mode loading conditions. To apply the described procedure for determining the crack size during the test as a function of fatigue cycles, it is necessary to perform the following actions. According to the proposed algorithm, the initial V0 and final Vf potential difference readings correspond to the initial ax0 and final axf crack lengths along the curved path projection on the X-axis, respectively, during the cyclic test. For the intermediate points, the crack size at any instant may be determined by a direct linear

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

CTS

Steel ST3

215,655

295

487.5

4.418

15.482

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231

Fig. 4. Experimental (a) test equipment and (b) crack growth trajectories.

Fig. 7. Relationships between the drop potential and crack length along curved crack path.

Fig. 5. Electrode position in CTS specimen.

  ðV  V 0 Þ þ ax0 ax ¼ ðaxf  ax0 Þ ðV f  V 0 Þ

ð1Þ

where V0 and Vf are the initial and final potential difference readings, respectively, and V is the instantaneous potential difference corresponding to the crack size, ax. The experimental results conveniently lead to the following dimensionless form:

Fig. 6. Potential drop on crack edges as a function of accumulated loading cycles.

interpolation of the potential difference data corresponding to the measured initial crack size, axo, and final measured crack size, axf, provided that both axo and axf can be precisely measured on the fracture surface of the specimen at the end of the test. Thus, the crack size at any instant, ax, is given by:


xi ¼ a

axi ; W  ða0 þ ars Þ

ð2Þ

Vi ¼

Vi  V0 : V a=w¼0:7  V 0

ð3Þ

where axi is the projection of a crack tip on the X-axis, a0 is the initial crack length, ars is the residual ligament size, W is the specimen width, V i is the current potential drop value, V 0 is the initial potential drop value, and V a=w¼0:7 is the potential drop value when ða0 þ ax Þ=w ¼ 0:7. Fig. 8 is a plot of the dimensionless crack length projection on the X-axis as a function of the dimensionless drop potential voltage for the CTS specimen geometry. It should be noted that the experimental data confirm that there is, for the full range of mode mixities from pure mode I to pure mode II, a common relationship
x and the between the crack length projection on the X-axis a potential difference V represented in dimensionless form (Eqs. (2) and (3)). The deviation of the experimental results from the

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3.2. Approximation At the end of the test, using the Cartesian coordinate system centered at the initial crack tip, the experimental mixed mode crack trajectories shown in Fig. 4 are plotted in Fig. 9 and described by a polynomial equation as

yi ðNÞ ¼ f ½xi ðNÞ ¼ An ½xi ðNÞn þ An1 ½xi ðNÞn1 þ    þ A0 ½xi ðNÞ0

ð8Þ

The parameters of these mathematical curves describing the crack path on the surface of the broken samples are generally determined from a best fit to the experimental data. The relationship between the crack size along a curvilinear path and its projection on the X-axis can be written in the form

Z Fig. 8. Projection of crack path on X-axis versus current value potential drop.

straight line shown in Fig. 8 is caused by to the complexity of the optical control of the curvilinear crack size. The potential difference scatter is much less than the accumulated error for the measurements by the microscope. Usually, this error is less than 5%. An
x and V as a function of the accuempirical relationship between a mulated loading cycles N that have been commonly used for description of mixed mode fatigue crack growth data is given by the following linear equation:


xi ðNÞ ¼ DV i ðNÞ; a

ð4Þ

ai ðNÞ ¼

axi ðNÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 1 þ ½f ðxÞdx

ð9Þ

a0

Knowing the crack size along a curvilinear path ai(N) and the equation describing it (Eq. (9)), we can find the relationship between the crack length along a curved crack path ai(N) and projection axi(N), as shown in Fig. 10. Hence, we can determine the dependence of the residual ligament size between the crack tip projection on the X-axis and specimen border as a function of the number of loading cycles. Relationship (9) will be used to determine the crack growth rate per cycle, da/dN, as shown in Fig. 2.

where coefficient D is determined through the initial V0 and final Vf potential differences and the corresponding initial ao and final af crack sizes. Based on this dependency, the relationship between the projection of the crack path on the X-axis and the current value of the relative potential drop is determined. Accordingly, we can introduce the following notation for the current projection of the crack size in the global Cartesian coordinate system centered at the crack tip

xi ðNÞ ¼ axi ðNÞ:

ð5Þ

A separate result of the mixed mode test is a set of points xi and yi of a curved crack path on the surface of the broken CTS sample. Eq. (5) enables their use to obtain the accumulated number of loading cycles N, i.e., xi(N) and yi(N). The main feature of cyclic mixed mode fracture is that the crack growth would no longer take place in a self-similar manner and does not follow a universal trajectory, that is, it will grow on a curvilinear path. For mixed mode crack propagation, the crack front continuously changes shape and direction with each loading cycle. As a result, the angle of crack propagation h⁄ (Fig. 1) continuously changes. At each successive position of the crack tip, the elastic stress intensity factors in the CTS specimen, K1 and K2, must be calculated. According to the proposed algorithm discussed above, the approach of equivalent crack length can be used to this end. This concept involves replacing a bent crack with a straight line approximation, as shown in Figs. 1 and 3. The crack may be assumed to grow in a number of discrete steps. After each increment of crack growth, the crack angle changes from the original angle b0 and so does the equivalent length of the crack in the following form

aeqv ;i ðNÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½a0 þ axi ðNÞ2 þ ½Ryi ðNÞ2 

bi ðNÞ ¼ arctg

Ryi ðNÞ a0 þ axi ðNÞ

 :

Fig. 9. Experimental crack paths for different mode mixity.

ð6Þ ð7Þ

Each pair of values of aeqv,i and bi position the ith point on the experimental trajectory of crack propagation.

Fig. 10. Relationship between crack size along curvilinear path and projection on Xaxis.

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The fatigue crack growth is generally controlled by the stress/ strain parameters near the crack tip. Usually, the crack growth per cycle for pure mode I loading conditions is correlated with the range of the elastic stress intensity factor K1. To study the influence of the in-plane mixed mode loading conditions on the material fatigue fracture resistance characteristics, unlike pure mode I, it is necessary to calculate the two fracture parameters, namely, the mode I and II stress intensity factors KI and K2 (SIF). Shlyannikov [30] generalized the numerical method for calculation of the geometry dependent correction factors Y1 and Y2 for the SIF KI and K2 under mixed mode fracture. The generalization consists of accounting for the T-stress distribution along a curvilinear crack path. The T-stress has been recognized to present a measure of constraint for the mixed mode small-scale yielding conditions and is a function of the dimensionless crack length and tested specimen configuration. The present study explores the direct use of FEM analysis for calculating the T-stress on the basis of the crack flank nodal displacements method [31]. According to this method, it is necessary to know the displacements on the crack edges with the coordinate system centered on the crack tip and the T-stress, which is computed as

T¼

     1 dux dux þ E 2 dx h¼p dx h¼þp

ð10Þ

Taking these considerations into account, the solutions for the mode I and mode II stress intensity factors KI and K2 for each mixed mode loading condition of the CTS specimen geometry can be obtained based on the theoretical elastic stress distribution in the crack tip. For a particular 2D case of an arbitrary oriented inclined crack, the asymptotic stress expansion formula is given by Eftis and Subramonian [32]. For in-plane mixed mode crack problems, consider the first two terms of formulae [32] as a particular case, i.e., the terms proportional to r1/2 and r0, and use the following stress expansion in the form of equations

  K1 h h 3h ffi cos 1  sin sin rxx ¼ pffiffiffiffiffiffiffiffi 2 2 2 2p r   K2 h h 3h  pffiffiffiffiffiffiffiffiffi sin 2 þ cos cos þT 2 2 2 2p r   K1 h h 3h K2 h h 3h ffi cos 1 þ sin sin þ pffiffiffiffiffiffiffiffiffi sin cos cos ryy ¼ pffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2p r 2pr   K1 h h 3h K2 h h 3h ffi sin cos cos þ pffiffiffiffiffiffiffiffiffi cos 1  sin sin rxy ¼ pffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2p r 2pr

K

K

K

2 ffi rxy ¼ pKffiffiffiffiffi 2pr

ð12Þ

pffiffiffiffiffiffi where K i ¼ K i =rn pa are the dimensionless stress intensity factors, T ¼ T=rn is the dimensionless T-stress, and rn is the nominal stress applied to the specimen. The T-stress, and subsequently Y1 and Y2, can be determined using Eq. (11) along several directions that correspond to different angular positions around the crack tip. It is convenient to use both the upper (h = +p) and lower (h = p) edges of the crack together with the position ahead of the crack-tip (h = 0°). A set of formulae to calculate the geometry-dependent correction factors Y1 and Y2 at these points accounting for the T-stress are the following:

ð13Þ

ð14Þ 2 ffi sffiffiffiffiffiffiffiffiffi þT rxx ¼ 2 pKffiffiffiffiffi a ðr  TÞ 2r 1 2pr eqv xx h ¼ p ryy ¼ 0 Y2 ¼ 2r aeqv cos a w rxy ¼ 0

ð15Þ During the post-processing of the results of mixed mode fracture problems through finite element analysis, the original method of displacements filtering near the crack tip, as described in Ref. [33], is applied. The stress intensity factors for the compact tension–shear specimen (Fig. 3) contained in Eqs. (13–15) could be calculated through finite element analysis using an applied load value and the specified geometric parameters of the CTS specimen

Ki ¼

F pffiffiffiffiffiffiffiffiffiffiffiffi paeqv Y i ; Wt

ð16Þ

where W is the specimen width, t is the specimen thickness, F is the þax Þ is the geometryload applied to the specimen, Y i ¼ f ða; b; a0W dependent SIF correction function, aeqv is the distance between a specimen edge and the crack tip, a is the angle between the load line and the initial crack, b is the angle between the equivalent crack and the initial crack, and a0 ¼ 0:5 W is the size of the initial crack. Then, as the particular case of Eqs. (13–15), the geometrydependent correction factors can be written as

Y1 ¼ r

ð11Þ

w

aeqv 

2 ffi sffiffiffiffiffiffiffiffiffi þT rxx ¼ 2 pKffiffiffiffiffi a ðT  r Þ 2r 1 2pr eqv xx h ¼ þp ryy ¼ 0 Y2 ¼ 2r aeqv cos a w rxy ¼ 0

FEM y

where T is a nonsingular term, and ðr; hÞ are polar coordinates in a coordinate system centered at the crack tip. When the x-axis coincides with the position of the crack ðh ¼ 0Þ, Eq. (11) can be simplified to dimensionless form 1 1 2
yy ¼ pffiffiffiffiffiffiffiffi
xy ¼ pffiffiffiffiffiffiffiffi ffi þ T; r ffi; r ffi: r
xx ¼ pffiffiffiffiffiffiffiffi 2pr 2p r 2p r

1 ffi ryy ¼ pKffiffiffiffiffi 2pr

h¼0

aeqv 

1 ¼ ðrxxrTÞ a2r eqv sin a qffiffiffiffiffiffi r 1 Y 1 w ¼ ryy a2r eqv sin a a  r qffiffiffiffiffiffi 1 Y 2 eqwv ¼ rxy a2r eqv cos a

1 ffi rxx ¼ pKffiffiffiffiffi þ T Y1 2pr

3.3. Numerical analysis

qffiffiffiffiffiffi

Wt F

sffiffiffiffiffiffiffiffiffi 2r ; aeqv

Y2 ¼ r

FEM xy

Wt F

sffiffiffiffiffiffiffiffiffi 2r aeqv

ð17Þ

where rFEM are the stresses obtained from the finite element solui tion. For all calculations in this study, the ratio between the polar radius and equivalent crack length is fixed atr=aeqv ¼ 0:01: More details on the calculations using Eqs. (13–17) are given by Shlyannikov [30]. The commercial finite element code ANSYS [34] has been used to calculate the displacement and stress distributions ahead of the crack tips. 2D plane stress eight-node isoparametric elements have been used for the 2D flat CTS configurations. Because 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 the convergence of the numerical solutions. In all the FEA calculations the material is assumed to be linear elastic and characterized by E = 216 GPa and m = 0.3. As mentioned above, the present work explores the direct use of FEM analysis for calculating T-stress by using the of crack flank nodal displacements. According to the elaborated algorithm, once the T-stress is known for each type of specimen geometry and crack length and crack angle combination, the geometrydependent correction factors Y1 and Y2 for the SIF K1 and K2 may be obtained as a function of the T stress by using Eqs. (13–17). Numerical results, converted into normalized mixed mode stress intensity factors in the forms of Y1 and Y2, are shown in Fig. 11, where the variation of the Y1 (or Y2) is a function of the relative

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Fig. 11. SIF correction functions and T-stresses for compact tension–shear specimen.

crack length for different values of the crack inclination angle in CTS specimens. These data reveal the main variation patterns of mode I and mode II geometry-dependent correction factors for an inclined crack under the full range of in-plane mode mixities. The variation pattern also provides a useful tool for analyzing and explaining the behavior of both the crack path and fatigue crack growth rate under mixed mode fracture. Fig. 11 represents the distributions of the normalized T-stress and the geometry-dependent correction factors Y1 and Y2 for the SIF K1 and K2 CTS under mixed mode loading, where the deviation of the current value of T from the corresponding original value for a/w = 0.5 for CTS increases upon increasing the relative crack length at a fixed crack angle position. The minimum negative T-stress is attained at a/w = 0.5 and a = 90° in the compact tension–shear specimen. Furthermore, for the CTS, there is a greater variation of the T-stress along the crack under mixed mode loading when the T-stress rapidly decreases with an increase in the relative crack length. It is shown in Fig. 11 for the CTS that under the pure mode II condition (a = 0°), the T-stress varies with the crack length and deviates from its well-established value equal to zero for an infinite-plate. The geometry-dependent correction factors Y1 and Y2 for the SIF K1 and K2 for the CTS calculated by using the proposed algorithm (Fig. 2) are shown in Fig. 11; they coincide with the known results that have been derived otherwise by Richard [29].

By fitting the numerical calculations, the constraint parameter T-stress and the mixed mode geometry-dependent correction factors as functions of the crack length and crack angle for the particular geometry considered have been represented in the form of polynomial equations. The length, angle and geometry correction factors for the CTS specimen configuration analyzed in this study were obtained by curve fitting. The equations for these factors in terms of geometric parameters are as follows:

Y 1 ðaw ; bÞ ¼ A1 ðaw Þb2 þ A2 ðaw Þb þ A3 ðaw Þ; Y 2 ðaw ; bÞ ¼ B1 ðaw Þb2 þ B2 ðaw Þb þ B3 ðaw Þ;

ð18Þ

2

Tðaw ; bÞ ¼ C 1 ðaw Þb þ C 2 ðaw Þb þ C 3 ðaw Þ: In these equations, coefficients Ai ; Bi ; C i depend on the relative crack size aw ¼ ða0 þ ax Þ=W. The polynomials that describe these relationships are presented in Appendix A. In the method suggested by the author [6] for the experimental data interpretation, based on the concept of a straight line crack, the crack growth is connected to a variation in the inclined crack angle. The principal feature of such an interpretation is that a crack may be assumed to grow in a number of discrete steps. After each increment of crack growth, the crack angle b changes and so does the effective length of the crack. Generally, the position of the crack tip may not coincide with the line of specified values of

A.V. Tumanov et al. / International Journal of Fatigue 81 (2015) 227–237

Y 1 ðaw Þja¼const and Y 2 ðaw Þja¼const and placed between two adjacent lines. This fact predestines the need to transfer from one function Y 1 jai ¼const (or Y 2 jai ¼const ) to another one. Therefore, when calculating the stress intensity factors K1 and K2, double interpolation was performed to find the Y1 and Y2 functions, by using both the crack length and its inclined angle through the Lagrange polynomial. Approximation functions for the T-stress for the CTS considered in the present work reflect the influence of the relative crack length, crack angle, specimen geometry and loading conditions. Ref. [30] contains more details on the calculation of the T-stress and geometry-dependent correction factors Y1, Y2 for the general case of mixed-mode elastic–plastic fracture. Finally, by substituting Eq. (18) into formula (16), we obtain the value of the SIF as a function of the mixed mode loading conditions and combination of crack length and crack angle.

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 of both K1 and K2, for the mixed mode fatigue experimental data interpretation. Our algorithm 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 [5]. A comprehensive review of the equations for the effective stress intensity factors and interrelated formulae for the mixed crack growth rate law can be found in Ref. [14]. Because the thickness of the CTS specimens in our study is of an order of magnitude less than the specimen width, plane stress conditions can be applied. Under plane stress conditions, the relationship between the elastic stress and strains is given by

1 E 1 ¼ ðryy  mrxx Þ E 2ð1 þ mÞ ¼ rxy E

exx ¼ ðrxx  mryy Þ; eyy exy

the strain energy density factors Si ðK i ; T; hÞ ahead of the crack-tip when h = 0°. A set of formulae to calculate the SED factors at this point are the following:

1 2 K 1 ð1  tÞ þ 4K 22 ð1 þ tÞ 2 1  S2 ¼ pffiffiffi ð1  tÞ cos 2aK 1 þ sin 2a cos aK 2 2 S3 ¼ 4½cos 2a2 S1 ¼

ð19Þ

Without loss of generality of the described algorithm, we can

energy density factors S1 , which are used for plotting the fatigue fracture diagram, can be determined. Thus, following the proposed algorithm, we can obtain the crack growth rate diagrams shown in Fig. 12, where first term of the strain energy density parameter S1 (Eq. (22)) is the analog of the mode I stress intensity factor for mixed mode conditions [5,6]. 4. Results and discussion The crack growth results are presented in Fig. 12 in terms of the crack growth rate plotted against the elastic SED S1 for different mode mixities corresponding to the CTS configuration. The elastic SED S1 is used only as a convenient parameter because it provides suitable correction factors Y1(a/w) and Y2(a/w) for the effects of the finite plate width in the cases of the SIF K1 and K2 combinations. The typical experimental fatigue fracture diagrams represented through the crack growth rate (da/dN) versus the elastic strain energy density factor (SED), defined by the first Eq. (22), are shown in Fig. 12 for all the investigated steel CTS specimens for the full range of mixed mode loading. Mode I crack growth corresponds to a = 90°, while a = 0° is pure mode II and 0° < a < 90° represents mixed mode crack growth. Fig. 12 shows that mixed modeloading causes a significant increase in the crack growth rate in

ð20Þ

which yields

dW 1 1 ¼ S1 ðK i ; hÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffi S2 ðK i ; T; hÞ þ S3 ðTÞ dV r=aeqv r=aeqv

ð21Þ

The singular part of Eq. (21) contributes only the first two terms, while the remaining second and third terms are necessary to take full account of the T-stress on the stress field. The specific forms of Si ðK i ; T; hÞ (i = 1, 2, 3) are given by the authors [5,13]. It is convenient for mixed mode crack growth rate interpretation to use

ð23Þ

which is derived on the basis of the SIF K1 and K2, is of limited generally, holding only the singular part of Eq. (21) for the particular circumstance of a very small crack tip distance r/a  1. By substituting the calculated values of the geometry-dependent correction factors Y1 and Y2 into Eq. (22), the current values of the strain

where m is Poisson’s ratio. The strain energy density (SED) at points close to the tip of the crack can be calculated by substitution of Eqs. (11) and (19) into the expressions

" # dW 1 X ¼ rij eij ; dV 2 i;j¼x;y

ð22Þ

use only the first singular term S1 of the total SED as an equivalent function of the SIF K1 and K2 in the case of in-plane mixed mode loading conditions. This implies that the relation for strain energy density when h = 0

dW 1 ¼ S1 ðK i ; mÞ dV r c =aeqv

3.4. Material property determination

235

Fig. 12. Mixed mode crack growth rate diagrams for ST3 steel.

236

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terms of the LEFM parameters with respect to pure mode I for all tested CTS specimens. Moreover, the crack growth rate increases when the mode mixity changes from pure mode I to pure mode II, and the proposed algorithm allows reproducing this effect. A number of experimental results show that the crack growth rate, da/dN, may be described by a sigmoidal curve in a log(da/ dN) versus log(K1,max) coordinate system, bounded at the two extremeties by the threshold stress intensity factor Kth and the critical value Kfc. In the intermediate DK range, the crack growth rate is almost linearly related to the elastic SIF, so in the semi-empirical relation proposed by Paris and Erdogan

da ¼ CK m 1;max dN

ð24Þ

there exists a strong correlation between parameters C and m (correlation coefficient not less than 0.96)

log C ¼ am þ b; ða; b < 0Þ:

ð25Þ

Constants C and m are generally determined from the best fit to the experimental data. It is recognized that the coefficients are interrelated (as shown by Eq. (25)) and are also functions of the loading conditions. Therefore, they cannot be considered independent constants to describe the material’s resistance to crack growth under cyclic loading. Some authors [35] have proposed the use of a different equation to describe the rate of crack propagation

m  
 da da K 1;max ¼ ; dN dN K
1

ð26Þ

where the exponent m is the same as in formula (24); K* is a parameter with the dimension of the stress intensity factor; and (da/dN)* is a coefficient with the dimension of the crack growth rate. It is assumed that (da/dN)* = 107 [m/cycle] because that is the crack growth rate, which is always located within the linear part of the fatigue failure diagram, for most of structural materials. There is no correlation between parameters m and K*, so parameter K*, along with constant m, is an independent characteristic of the cyclic crack resistance of the material. Parameter K* has a clear physical meaning: it is the greatest stress intensity factor at a value of the crack growth rate (da/dN)* = 107 [m/cycle]. From the compatibility solutions of Eqs. (24) and (26), we can find the relationship between parameters C and K⁄

 C¼

da dN






K
1

m

:

ð27Þ

Table 2 Mixed mode crack growth rate characteristics.

C n S⁄1

a = 0°

a = 30°

a = 90°

9.046E5 2.35 2.78

2.598E4 1.96 1.989

1.592E5 2.52 5.17

not coincide with each other and are different for the cases of pure mode I and the mixed mode loading of CTS specimens. The presented data shows that the proposed algorithm (Fig. 2) can be used to automate the experimental study of the full range of cyclic mixed mode fracture from pure mode I to pure mode II, realized in a CTS specimen. The experimental data confirm that the dependence between the residual ligament size and the relative potential drop at the crack edges has a linear character. This dependence leads to the conclusion that a direct visual observation of the crack growth is not needed. Only two points of the crack tip position are required to obtain the relationship between the crack tip position and the loading cycle. In other words, if we have the information about the crack size along the curvilinear front at the start and end of the test, we can obtain all the needed information from the relationship between the drop potential and the accumulated number of loading cycles. 5. Conclusions An algorithm for the automatic determination of crack growth rate characteristics under mixed mode loading is proposed based on the concept of an equivalent crack. The principal moment of the elaborated algorithm is the linear dependence between the residual ligament size and the relative potential drop at the crack edges. This relationship makes it possible to identify the position of a crack tip on a curvilinear path on the basis of the potential drop signal. The effectiveness of the automatic algorithm is demonstrated by the results of tests conducted on compact tension–shear specimens made from low carbon steel in a full range of mixed-mode loading from pure mode I to pure mode II. The proposed algorithm for fatigue crack growth automation can be implemented on all known standard test machines equipped with a drop potential unit without changing the source code of the controller. One feature of the proposed algorithm is the possibility to realize the crack growth rate tests without directly observing the specimen, which is especially relevant for tests carried out under extremely high or low temperatures.

In the case of mixed mode fracture, instead of stress intensity factor K1, one can use a particular equivalent parameter. In the present algorithm, a such parameter is the strain energy density function S. Then, the crack growth rate Eqs. (24) and (26) can be rewritten as

This research was partially supported by Russian Foundation for Basic Research under the Project No. 13-08-92699.

da ¼ CSn1;max dN

ð28Þ

Appendix A

 
 n da da S1;max ; ¼ dN dN S
1

ð29Þ

where n ¼ m=2. Table 2 shows the experimental values of the constants of Eqs. (28) and (29) for compact tension shear specimens tested under pure mode I and mixed mode loading. There are obvious advantages when using Eqs. (28) and (29) to characterize the material resistance to cyclic crack growth. The quantitative values of parameter S⁄1 are also evident from Fig. 12; in particular, the minimum value of S⁄ is related to the minimum specimen durability. The results of Table 2 indicate that the constants C, n and S⁄1 do

Acknowledgment

For pure mode II, when a ¼ 0

A1 ¼ 0:052788a2w  0:048149aw þ 0:011611; A2 ¼ 0:619660a2w  0:322710aw þ 0:108421; A3 ¼ 0:880393a2w  0:829345aw þ 0:199532; B1 ¼ 2:9242E  04a2w þ 1:8793E  03aw  1:6376E  03; B2 ¼ 8:0003E  02a2w þ 6:7749E  02aw  1:8379E  02; B3 ¼ 0:411580a2w  1:372477aw þ 0:628948;

A.V. Tumanov et al. / International Journal of Fatigue 81 (2015) 227–237

C 1 ¼ 0:031554a2w þ 0:062410a2w  0:021248; C 2 ¼ 17:037898a2w þ 21:204861aw  6:473520; C 3 ¼ 3:961398a2w  1:988222aw  0:713584; for mixed mode, when a ¼ p=6

A1 ¼ 6:07889129E  04a2w þ 9:07120902E  02aw þ 1:3845; A2 ¼ 1:62947842E  03a2w þ 1:15368478E  01aw þ 2:0122; A3 ¼ 3:50468374E  03a2w þ 1:52865121E  01aw þ 3:15936; B1 ¼ 4:474671E  05a3w  1:87622E  03a2w  1:9168E  02aw þ 1:542832; B2 ¼ 2:3078194E  05a3w  1:28176E  03a2w  1:31979E  02aw þ 1:37889; B3 ¼ 5:676861E  06a3w  7:954275E  04a2w  1:30156E  02aw þ 1:2264;

C 1 ¼ 2:5504E  03a2w  1:075E  01a2w  6:1698E  01; C 2 ¼ 6:34757E  03a2w þ 1:048E  01aw  2:0597E  02; C 3 ¼ 1:0504E  02a2w  7:1014E  03aw þ 1:2417; for pure mode I, when a ¼ p=2

A1 ¼ 0;

A2 ¼ 0;

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