CC BY
51
УДК 539.421
Multiaxial fatigue crack orientation and early growth investigation considering the nonproportional loading
W. Song1, X. Liu1, F. Berto2
1 State Key Laboratory of Advanced Welding and Joining, Harbin Institute of Technology, Harbin, 150001, China
2 Department of Engineering Design and Materials, Norwegian University of Science and Technology, Trondheim, 7491, Norway
The paper presents a comprehensive investigation of fatigue cracking behaviors under various nonproportional multiaxial cycle loading paths based on the critical plane approach. The maximum normal and shear stress/strain fields are presented to analyze the crack orientation and early growth directions in polar diagrams. The experimental observed crack paths and directions were compared with maximum strain loci of each angle to determine multiaxial fatigue failure mode. The results show that crack orientation and growth paths appear in the maximum shear and normal strain plane, respectively. Likelihood cracking regions of various loading paths are predicted according the determined failure mode. Besides, nonproportionality factor is introduced to characterize the degree of multiaxiality on these loading paths.
Keywords: multiaxial fatigue, cracking orientation, nonproportional loading, critical plane, polar representation
Исследование ориентации и начального направления роста трещин при многоосном усталостном растрескивании с учетом непропорционального нагружения
W. Song1, X. Liu1, F. Berto2
1 Харбинский технический университет, Харбин, 150001, Китай 2 Норвежский университет естественных и технических наук, Тронхейм, 7491, Норвегия
В статье представлено комплексное исследование усталостного растрескивания для различныж траекторий непропорционального многоосного циклического нагружения в рамках подхода критической плоскости. С использованием полей максимальныж нормальныж и сдвиговых напряжений и деформаций проведен анализ ориентации трещин и начального направления их распространения на основе полярных диаграмм. Экспериментально наблюдаемые траектории и направление роста трещин сравнивали с очагами максимальной деформации для каждого угла с целью определения режима многоосного усталостного разрушения. Показано, что трещины ориентированы и распространяются в плоскостях максимального сдвига и нормальной деформации соответственно. Предсказаны области возможного растрескивания для различньж путей нагружения в соответствии с определенным режимом разрушения. Введен показатель непропорциональности, характеризующий степень многоосности для данных путей нагружения.
Ключевые слова: многоосная усталость, направление роста трещины, непропорциональное нагружение, критическая плоскость, полярное представление
1. Introduction
Engineering structures are often subjected to the complex time-variable loading resulting in local multiaxial fatigue damage of material. Although fatigue failure has been investigated for many decades, the scientific community has not yet agreed on a specific methodology to perform the multiaxial fatigue assessment of components. Partly due to the diversity of loading types, the essential reason is the
difference of fatigue damage mechanisms. Therefore, a reliable damage quantitation model for robust multiaxial fatigue life estimation need to capture accurately the damage mechanisms.
Generally, material fatigue damage consists of crack nucleation and crack growth. Different types of material have different damage mechanisms. In ductile-behaving materials, cracks nucleate along slip systems, which are origi-
© Song W., Liu X., Berto F., 2017
nated from the maximum shear planes. In brittle-behaving materials, cracks often nucleate directly at inherent defeats or discontinuities, such as voids and inclusions. Once crack nucleation occurs, the growth of these crack orientation can be developed into two stages. The crack early growth stage (stage I) is microcrack growth along maximum shear strain plane, and subsequent growth stage (stage II) proceeds along maximum tensile strain plane [1]. Therefore, the crack nuc-leation and growth prefer to specific direction plane rather than random orientation, and the potential orientation depends on the material and loading states. To gain a better understanding of the multiaxial fatigue damage, critical plane concept is introduced to study these issues combined with actual damage mechanisms [2, 3]. From the research of Fatemi [1], the microcrack observations of fracture plane in a scanning electron microscope can show clearly the orientation of crack nucleation and growth plane which provides the physical basis for critical plane approaches.
So far, many researchers have widely investigated the cracking behaviors under multiaxial fatigue loading systematically and proposed various fatigue damage models to predict the multiaxial fatigue life according to different critical plane based on reasonable assumptions. They may be categorized into four main groups, (i) stress-based criteria [3]; (ii) strain-based criteria [4]; (iii) energy-based criteria [5]; (iv) fracture mechanics criteria. Due to the neglection of plastic deformation, stress-based critical plane models, such as Findely model [3], are only suitable for the high cycle fatigue. For low cycle fatigue, cycle plastic deformation plays an important role in the damage process. Although strain-based critical models, such as Brown and Miller's model [6], can predict some cases of low cycle fatigue, only the strain term cannot reflect nonproportional effect of a material under some complex loading. The typical strain-stress-based critical plane model proposed by Fatemi and Socie is widely applied for multiaxial high and low cycle fatigue, which takes the additional nonproportional cycle hardening into consideration [5]. In this model, the driving strain parameter, shear strain on the critical plane, is good at predicting the fatigue life of shear failure material. While Smith-Watson-Topper model based on the stress and strain of critical plane tends to estimate the tension failure mode fatigue life [7]. Additionally, there are so many extension models from aforementioned types of critical plane theories, such as Carpinteri model [8], MWCM model by Susmel [9], Shang model [10]. These models often take different influence factors into consideration of critical plane concept to improve the accuracy of estimation. Part of them can also predict the notched specimens and components considering the notch stress concentration. On the other hand, strain energy density based approach can availably estimate medium/high cycle multiaxial fatigue life for notched specimens [11].
Actually, critical plane approaches have proved to be successful for an engineering description of fatigue dam-
age in a wide range of various materials, including different levels of steel [12], aluminum alloy, titanium alloy and other metallic materials [13]. Experimental evaluations have also exhibited the capability of critical models to predict the crack initiation under different loading conditions [1, 14-19]. Moreover, reliable fatigue damage models based on critical plane approaches can not only accurately predict the fatigue life under multiaxial loading, but also fatigue cracking orientation. Available information of crack nucleation and growth is benefit to determine failure degree of components and predict fatigue remaining life. Nevertheless, there is still no unified definition of fatigue crack orientation so far. In general, the crack length less than 102 ^m is considered as the crack orientation critical values, which marks the initiation size. Subsequently, the early crack growth stage occurs in the length range of 102 to 103 ^m. The final observed microscopic cracks can be predicted by fracture mechanics theory. Due to the differences of material microstructure, it exhibits difference failure behaviors on whether the crack initiation or crack growth dominates the major portion of fatigue life. It will further affect the selection of fatigue damage model for multiaxial fatigue life estimation. However, we can often find that different critical plane assumptions are proposed to predict the similar fatigue lives [10, 20]. How to connect the fatigue damage model with the critical plane assumptions is the essential to obtain accurate fatigue life.
Jiang et al. [21] compared the experimentally observed cracking orientation with three different fatigue criteria. They assumed the possible cracking material plane angle in a range of 10% the maximum fatigue damage. The results revealed that Fatemi-Socie maximum shear strain-based plane model only predicted correctly the cracking angle with a ratio of 20%. The Jiang criterion was found to give satisfactory fatigue life estimations and a reasonable description of cracking behavior [22]. Ohkawa et al. [23] reported that crack initiation and early propagation occur on the maximum shear plane for in-phase cycle loading, while the crack growth transforms along the direction of the maximum normal strain plane. Recently, the plane maximum stress and strain terms of different loading paths are investigated by Albinmousa and Jahed [24]. They found that fatigue cracks fundamentally initiate from the maximum shear strain plane, while the cracking direction will be changed with the crack growth. Latterly, Albinmousa measured the crack paths by employing the polar diagrams to explore the crack growth behaviors based on the critical plane concept [25]. The polar representation can directly reveal crack angles by the stress or strain fields surrounding an infinitesimal element on the surface of specimen. It concluded that crack orientation had the equal chance for the maximum shear or normal strains. The potential crack orientation and growth regions of different loading paths are predicted. However, the compute results of these maximum shear and normal strains seem to be inaccuracy to
Fig. 1. Strain state of thin-tubular specimen
determine the critical plane angles so that it cannot illustrate the actual crack orientation. On the other hand, the variations of normal strain and shear strain on the critical plane of a material element with loading paths can reflect the nonproportional degree of additional hardening in the polar coordinate system [13, 26]. A nonproportionality factor according to a geometrical ratio of maximum shear strain area and corresponding full circle area was proposed by Chen et al. [27]. Therefore, it is necessary to obtain accurately the maximum shear and normal strain and stress responses to investigate the crack initiation and early propagation based on the critical plane approach.
The aim of this paper is to study crack initiation and early growth behaviors of multiaxial fatigue based on critical plane concept. This experimental results of multiaxial stress and strain responses from researches of Hoffmeyer [28], Jiang [21], Reis et al. [29] were reanalyzed. Comparison of analytical results of stress and strain terms using polar representation is conducted by two critical plane algorithms. Full maximum normal and shear loci of an infinitesimal element in polar diagrams were presented to determine the fatigue failure mode and potential cracking regions according to experimental cracking paths under different loading conditions. Besides, the nonproportional additional hardening degree of loading paths is also evaluated by the nonproportionality factor obtained from the full circle area and swept area of the maximum shear strain in polar diagrams. A general discussion of Fatemi-Soice damage model for these loading paths is presented, and further studies are recommended.
2. The critical plane approach
The critical plane concept has been considered as one of the most successful methods in the assessment of multiaxial fatigue and damage parameter based on the critical
plane can accurately estimate the multiaxial fatigue life. Most of the critical plane based damage parameters are formulated in the form of stresses and strains components or a combined of stresses and strains on the critical plane. Generally, the computation procedure of critical plane firstly needs to obtain the stress or strain histories. The strain state of thin-walled tubular specimen subjected by multiaxial loading is schematically shown in Fig. 1. Then the maximum normal and shear stress/strain results on an arbitrary plane with angle 9 related to the specimen axial are transformed by the plane transformation relationship, shown as Fig. 2.
For a given strain controlled tubular specimen, the strain tensor is expressed as
Ae, AYxy/2 0
=
AY xy/ 2 0
*eff 0
Ae,
0
-V eff Ae -
(1)
(2)
(3)
If the applied strains are sinusoidal, Ae
e x = ysin(wi) + em,
Y xy sin(wt-9) + Y m,
where a is the phase angle between the tensional strain and torsion strain, and the Ae, Ay are the applied tension and torsion strains, respectively, e m and y m are the mean tension and torsion strains, respectively. These equations are input to Matlab software to obtain the loading histories.
Since the material can give rise to plastic damage under low-cycle loading, the effective Poisson ratio veff is changed due to the presence of localized plastic deformations. Carpinteri et al. [8] further evaluated the effect of Poisson ratio on multiaxial fatigue life by employing three different strategies, including analytical approach, numerical formulation and a constant value assumption. The total normal strain is decomposed into plastic strain and elastic strain in the analytical Eq. (4). The assumption of a con-= 0.5:
stant value is v ve
V eff ="
eff +v Pe P
e
(4)
where ve is the elastic Poisson ratio, and vp is the plastic ratio. The comparison between experimental data and fa-
+ a -a,.
a. =-
- cos (2a) + Tsin (2a),
a. - a .
- sin (2a) - t^ cos (2a)
2 —v—, = ^ + ey ^cos (2a) + ^ sin (2a),
e x + e y e-e ea =——y + a 2 2
Ya = sin (2a) - ^ cos (2a )
2 2 2
Fig. 2. Plane stress-strain transformation relationship
Monotonie and cycle properties of investigated materials
E, GPa G, GPa os, MPa K n' f MPa Ef b c Tf, MPa Yf b> Co
S460N 208.5 80.2 500 1115 0.161 969.6 0.28 -0.086 -0.493 463.2 0.224 -0.071 -0.422
30CrNiMo8HH 210 80 870 1617 0.1345 946.1 1.05 -0.0404 -0.732 964 0.11 -0.43 -0.0558
CK45 206 410 1206 0.2 340 0.17 -0.102 -0.44
AISI303 178 330 2450 0.35 310 0.05 -0.07 -0.292
42CrMo4 206 980 1420 0.12 640 0.18 -0.061 -0.53
tigue life prediction based on critical plane approach proved that Poisson ratio has only a slight influence on these strategies. Therefore, we assume the Poisson ratio veff = 0.5 in our study.
Substituting Eqs. (2), (3) and the transformation relationship equations in Fig. 2, the stress and strain terms for any plane can be expressed by following equations: Ymax (t) = £a { H1 + Veff ) sin (2a) + A cos9 cos (2a)f + + [A sin ^cos(2a)]2}1/'2 sin (rot -n),
En (t) = -f{ [(1 - Veff) + (1 + Veff )cos(2a ) +
+ Xcos 9 sin (2a)] + [Xsin 9 sin (2a)]2 }12 x x sin (rot -£), where
n = tan
-1
X sin 9 cos (2a)
-(1 + veff ) sin (2 a) + X cos 9 cos (2 a)
(5)
(6) (7)
^ = tan 1[Asin ^sin(2a)(1 -veff +
+ (1 + veff) cos (2a) + A cos 9 sin (2a))-1 ]. (8)
The phase angle between Ymax and en is (n + the direct computation equations of shear strain on critical plane will be verified by strain transformation equations of Fig. 2. After that, this approach is used to calculate other loading paths orientation of critical plane and damage parameters.
3. Material and experiments
In order to check the crack paths by polar diagram based on the critical plane concept, experimental crack measurements of orientation and growth have been obtained from the work of Hoffmeyer [28], Jiang [21] and Reis [29]. Besides, the multiaxial cycle loading paths of specimens in [30] are investigated by polar coordinate system. The test thin-wall tubular smooth specimens are made by the struc-
0
/ \ Y/VJ ^TjWTW,
! Ë i i i i i i i i i i t " r~7~ 1 1 1 1 1 1 T~T~ 1 1 1 i 1 1 1 1 1 1 T 2T
y/-/ ■ñ: 1 ^
\ ¡ E
Y/Vß
rasa
,0
0
y/Vß
i i i i i i i i i i i i i i i i i i i i
i i i i i\i 1/1 i i i i i\i 1/1 i i i i
1" 1" 1" T T Tl"ITITlTiT r r TT1 " I I I I I I I I I I I I I I I I I I I I
T 2T
1 1 1 --¿r - \---h- 1 1 -^-H- - —
1 1 1 1 1 1 till _ _ 1_ _ J___L _ J /\ ' ' 1 1 1 1 1 1 _ JL_ J_ _ 7\ i __ ! ! 't 1 1 1 1 __L _ 1 _ A 1
i i\ 1 y --1--- ¥ - • 1 1 1 i —1 —1- -1 1 — ! ! t — «— -1— ■ 1
E,
Y/Vß
2T
f
I I I I I I I I II
2T
Fig. 3. Loading path and time histories: proportional loading (a), 90° nonproportional loading (b), square loading (c), nonproportional loading with mean stress (d), two square loading (e), butterfly shape loading (f
E
E
Loading path Axial strain, % Shear strain, % Strain ratio aa, MPa Ta, MPa Nf
Proportional (I) 216.5 147.3 130 000
90° nonproportional (II) 0.144 284.3 195.5 47140
Square (III) 0.25 1.74 276.8 187.9 18 300
Synchronous (IV) Amp.: 0.144 Mean: 0.144 242.0 150.5 106 300
Two square (V) 0.35 0.68 1.93 1265
Butterfly shape (VI) 0.66 1.90 789
Loading path
Axial strain, %
Shear strain, %
Strain ratio
aa, MPa
Ta, MPa
N
Proportional (I)
90° nonproportional (II)
0.144
Square (III)
0.25
Synchronous (IV)
Amp.: 0.144 Mean: 0.144
216.5
284.3
1.74
276.8
242.0
147.3
195.5
187.9
150.5
130 000
47140
18 300
106 300
Two square (V)
Butterfly shape (VI)
0.35
0.68
0.66
1.93
1.90
1265
789
tural steel S460N, 30CrNiMo8HH [31], CK45, AISI303, 42CrMo4 steel, which the monotonic and fatigue properties are listed Table 1. The proportional and nonproportional loading paths and time histories prescribed in experiments are shown in Fig. 3. Fatigue test data were performed by both the stress and strain cycle loading, which are shown in Table 2. It should be noted that the cracking path length was traced in the range from 5 to 500 ¡im using plastic replica method [32]. These cracking loci will be compared with the predicted potential crack orientation and growth prediction zone in the section of analysis.
4. Critical plane damage model
For biaxial loading conditions in our study, both the fatigue life and the potential cracking orientation are analyzed by critical plane approaches. The Fatemi-Socie model is designed for the material exhibiting the shear cracking behavior according to the critical plane approaches. The damage parameter is obtained by combining the shear strain and the normal stress on the maximum shear critical plane as shown in following equation:
i ^.max A
1 (9)
DFS =Ym
1 + -
where y max is the maximum shear strain, amax is the maximum normal stress on ymax plane, k is a material constant which can be calculated by fitting the uniaxial data against the pure torsion data. We assume k = 1 in this study, a y represents the yield stress. In this study, the Fatemi-Socie damage model is selected to quantify damage degree of multiaxial fatigue under different loading conditions. Further, the damage results are presented in polar diagrams, which are clearly shown to predict the crack orientation.
5. Results and analysis
The maximum normal and shear stress-strain terms of different loading paths on any plane in tests are calculated using the transformation equations shown in Fig. 2. On the searching process of critical plane, an increment of a = 1° is conducted to calculate the stress-strain terms from 0o to
360° in algorithms. The variations in normal strain and shear strain for arbitrary directions of a material element during one cycle were investigated and the maximum values of stress and strain components in each direction were plotted in polar coordinate graphs. The corresponding fatigue damage parameter can also be evaluated on each plane to determine the critical plane that is maximized in all data.
Firstly, the critical plane algorithm of transformation relationship for any loading paths is validated by Eqs. (2)-(8) for sinusoidal loading paths. Afterwards, the available equations can be used to calculate stress and strain components. Figure 4 shows the normal and shear strain ranges on various planes based on different algorithms. The results of these two algorithms are completely identified for the normal and shear strain components. It is further proved that it is accuracy for the calculation of critical plane.
Figure 5 shows the maximum normal and shear stressstrain amplitudes vary on each angle for pure axial and pure shear loading paths, which are indicated by red and blue solid lines, respectively. The maximum normal strain planes of these two paths present in the direction of (0°) and (45°, 135°) as shown in Fig. 5, a. Similarly, the maximum shear planes of these two loading paths are shown in (45°, 135°) and (0°, 90°), respectively, as shown in Fig. 5, b. Thus, the potential cracking direction for various materials
Fig. 4. Normal and shear strain ranges on various planes based on different algorithms
180'
210
270
0° 180
210
^max
Ymax 240
Emax Plane Ymax Plane
270°
Fig. 5. Polar representation of strain critical plane: pure axial (a) and pure shear cycle loading (b)
will appear in aforementioned angles, which are plotted using gray solid lines (maximum normal strain plane) and black dashed lines (maximum shear strain plane).
An examination of the cracking behavior for the tested specimens and the comparison with prediction cracking direction based on critical plane approaches may give an avenue to assess the fatigue damage. Using the polar representations of stress and strain responses to arbitrary plane could better understand fatigue crack orientation and early growth by comparison with observed crack paths. Two crack paths shown in Fig. 6 stand for different cracking stages. The black solid line with a length of 55 ¡xm represents the orientation stage, while the gray dashed lines represent the crack growth stages with a size range between 230 and 870 ¡m. It is should be noted that the crack paths are not drawn on the basis of actual scale. Due to the uncertain definition of fatigue crack length between initiation and growth, the crack
states should be determined to illustrate cracking direction. According to Ref. [25], crack initiation can be characterized by shortest uni-oriented observed microcrack, the crack growth can be deduced from the long determined cracking direction. The difference of material anisotropy and texture leads to the different fatigue failure modes. Therefore, it is beneficial to capture the fundamental fatigue mechanism through the observations of crack initiation and growth. It is should be noticed that the maximum normal and shear strain planes in Figs. 6-9 show the identical directions with corresponding normal and shear stress plane. The analysis of cracking behaviors can be conducted by employing the strain planes.
Figure 6 shows the stress-strain responses and cracking paths in polar diagram under in-phase cycle loading. The results from the figure suggest that fatigue crack initiates from the maximum shear strain plane at the angle of 60°.
Lc = 50 ¡m 240° 270° — — — Lc = 345 ¡m 270°
Fig. 6. Polar diagrams of strain (a) and stress (b) loci of crack orientation and growth for proportional loading path (I)
Song W., Liu X., BertoF. / &u3uuecmH Me30MexaHum 20 6 (2017) 86-97 90° _ ct_ 90°
Fig. 7. Polar diagrams of strain (a) and stress (b) loci of crack orientation and growth for 90° out of phase nonproportional loading path
(II)
As the crack grow to 25 M-m, the orientation direction turns to the maximum normal strain plane. This result is different from the conclusion of [25] that the crack initiated from the maximum normal strain plane and grows toward the maximum shear strain direction. The essential reason is the discrepancy of stress and strain fields used in the evaluations. The cracking behavior of 90° nonproportional loading path in terms of stress-strain fields is illustrated by Fig. 7. Similar with proportional path cracking law, the crack orientation with a length of 28 M-m is started from the maximum shear plane and grow toward to the maximum normal plane.
The crack growth behavior of the square loading path is described with stress and strain loci in Fig. 8 by the polar representation. With the same normal and shear strain am-
plitudes of path (I) and path (II), however, the crack initiates from the angle surrounding (-30°, 150°) which is the maximum normal or shear strain plane. As the maximum normal strain plane angle is identified with the maximum shear strain plane, it is difficult to judge the crack initiate plane. From the test observations of crack path about the square cycle loading, the crack direction is completely agreed with the maximum strain plane. It is further proved that S460N steel displays typical mixed cracking behaviors. Figure 9 shows the crack growth behavior with respect to stress and strain loci in the polar diagram under synchronous loading path. This loading path normal and shear amplitudes of this loading path are identical to that previous loading paths. Additionally, a mean normal strain with a same value with normal strain is loaded on the specimen.
^max 90
Ymax 120
180
210
270° ---Lc = 875 Mm 270°
Fig. 8. Polar diagrams of strain (a) and stress (b) loci of crack orientation and growth for square loading path (III)
max 90!
Tmax 120°
150
0° 180
210
270° ---Lc = 351 ¡m 270°
Fig. 9. Polar diagrams of strain (a) and stress (b) loci of crack orientation and growth for synchronous loading path (IV)
The crack is also initiated from the maximum shear strain plane and grows towards the maximum normal strain.
Taking the consistency of maximum normal and shear plane under the square loading path into consideration, the crack initiation of S460N steel of various loading paths seems to appear in the maximum shear strain plane and the crack early growth will turn to the maximum normal strain based on the experimental observed cracking behaviors under different loading paths. The crack initiation and growth planes under six loading paths have been summarized in Table 3. Therefore, it is demonstrated that the torsion loading can nucleate small cracks on planes and further lead to the growth of crack, and the normal strain dominates the final crack direction.
6. Discussion
Multiaxial fatigue damage parameters based on critical plane approach cannot only satisfactorily estimate the fatigue life [13, 33-35], but also the critical plane cracking orientation under different loading conditions [21, 29, 36]. Generally, the cracking orientation and crack growth stage can be observed within a range of 50-1000 ¡xm. For a tubu-
Table 3
Crack initiation and growth planes
Loading path Initiation plane Early growth plane
Proportional (I) Y max E max
90° nonproportional (II) Y max E max
Square (III) Y or E max max E or Y max max
Synchronous (IV) Y max E max
Two square (V) Y max E max
Butterfly shape (VI) Ymax E max
lar specimen, the crack orientation stage is determined by the observed uni-oriented microcrack, and the crack early growth direction is defined by the final surface cracking angle, which can be caused by the combined axial-torsion loading. The definition of cracking state is shown in the Fig. 10.
Figures 11, a -f summarize the potential cracking region of each loading path according to aforementioned critical plane analysis and compare with observed cracking orientations. The gray areas represent the predicted regions of cracking directions in the figures. The black solid lines with arrow in the figures are the observed crack orientations based on the Hoffmeyer [28] and Jiang [21] work about S460N steel. Table 4 summaries observed cracking
Crack early growth
Fig. 10. Definition of the surface cracking orientation for the tubular specimen
^max 90° - ^max 90°
£max 90° - Emax 90°
Fig. 11. Polar diagrams of potential crack orientation and growth zone for various loading condition: proportional (I) (a), 90° nonproportional (II) (b), square (III) (c), synchronous (IV) (d), two square (V) (e), butterfly shape (VI) f)
orientation angles of four loading paths from the Jiang re- and path (IV) are selected to make comparison of cracking search [21]. It is noticed that four loading paths of crack- behaviors, while other four loading paths from Jiang, path ing trajectory from Hoffmeyer, path (I), path (II), path (III) (II), path (III), path (IV) and path (IV) are used to verify
The experimental observed cracking orientation angles of different loading paths [21, 29]
Proportional (I) 90° nonproportional (II) Square (III) Butterfly shape (VI) Synchronous (IV)
Material Angle Material Angle Material Angle Material Angle Material Angle
S460N 2° S460N 0° S460N 1.6° S460N -29° S460N -1 °
S460N 10° S460N -42° S460N 4° S460N 10° S460N 3.3°
42CrM04 -16° S460N 9.2° S460N 22° S460N 3° S460N 11°
Ck45 -20° 42CrM04 0° 42CrM04 15° S460N -27°
AISI330 -25° Ck45 0° Ck45 -5°
AISI 330 24° AISI 330 32°
the cracking direction with the predicted cracking range. The determination of these regions is dependent on the relationship of the maximum shear strain plane and the maximum normal plane.
Figure 11, a shows two likelihood cracking regions of proportional loading path. Since it exists four maximum shear strain values in all directions, which show the symmetrical relation in the polar diagram. The maximum normal strain planes are lied in the range of the intersection angle of two maximum shear strain planes. Therefore, the potential cracking region can be defined between the two maximum shear strain planes. According to the analysis of cracking behaviors under various cycle loading paths in section of results, a crack initiates from the maximum shear strain plane, and then turn towards to the maximum normal strain plane with the crack growth. It can be concluded that the cracking path considering the mixed fatigue failure mode of S460N are in the predicted cracking regions. The same trends with the proportional loading path show in Fig. 11, e, which characterizes the two squares loading path. On the other hand, 90° nonproportional, square and butterfly sharp cycle loading paths, shown as in Figs. 11, b, c and e, present the same characteristics that the direc-
tions of the maximum shear plane approximately coincide with the maximum normal strain plane. The likelihood regions are surrounding the maximum shear or normal strain plane. For the synchronous loading path, as shown in Fig. 11, d, it has a phase difference between the maximum shear and normal strain planes. The predicted cracking paths are also lied in the predicted region based on the mixed fatigue failure law. Yet the direction lines of the synchronous loading path in Fig. 11, d are not in the predicted range of cracking directions, the potential cracking region can precisely predict the crack orientation direction of loading paths (II), (III) and (IV) by black lines with arrow in figures.
Since other two paths have not the stress loading amplitude, the maximum stress values at critical plane, the Fa-temi-Soice damage loci of four loading paths in polar diagrams were plotted in Fig. 12. Such curves are similar to the loci of maximum shear strain with the increases of phase angle, although the maximum damage parameters of each loading paths in polar representation have slightly changed in the range of 10°.
For the nonproportional strain loading, a rotation of the principal strain axes can result in the additional cyclic hard-
Fig. 12. Fatemi-Socie damage model of S460N steel
1.2 "I-;---
^ -Proportional
g 1.0~ i -----45° nonproportional
g 0 g _ A ------60° nonproportional
^ I \ 90° nonproportional
g 0.6 - I \
1 0.4- / \ ^ 0.2- /
| ---
-0.2- , , , ,
0 2 4 6 8 10 Strain ratio A
Fig. 13. Relationship between the nonproportionality factor and the strain ratio [37]
ening behaviors, and decrease fatigue life comparing to the stable proportional loading conditions. Since many slip systems are activated by continuously principal axis rotations, an additional parameter should be introduced to represent the hardening behaviors for predicting the fatigue life accurately. The maximum shear strain loci can demonstrate various cycle loading conditions according to critical plane strain analysis in polar diagrams. Chen proposed an equivalent factor considering the relationship between the geometry of maximum shear strain path and the hardening mechanism of the path, as shown the equation [27]:
^ = 2^^-1, (10)
^max
where the Amax is the area of the circle with radius equal to the maximum shear strain in one cycle, 4 max is the swept area of yamax - a polar coordinate space. Therefore, the correlation between the nonproportionality factor O and the strain ratio A was shown in Fig. 13 [37]. In this figure, the nonproportionality factor is equal to 0 for proportional loading path, while the parameter reaches the maximum value 1 for 90o nonproportional loading path with strain ratio A = 1.5. The hardening level decreases with the decrease of phase angles, while the tendency is accordance with the 90o phase angle. In order to investigate the nonproportionality of various loading paths, the in-phase, 90o out of phase, square, two squares and butterfly loading paths
Path VI
Path V
UlU
Path III
Path II
0.0 0.5 1.0 1.5
Nonproportionality factor Fig. 14. Nonproportionality factor under different loading paths
were performed at the same strain amplitude ea = 0.144% and y a = 0.25%. The hardening extent is in the order 90o nonproportional > butterfly > square > two squares > proportional loading paths, as shown in Fig. 14. Therefore, the polar representation of maximum shear strain fields can effectively illustrate the nonproportional additional hardening degree by calculating the relationship between the full circle area and swept area of the maximum shear strain under nonproportional loading.
In our study, it has been demonstrated that the most crack orientations and growth directions are associated with the maximum normal and shear plane. This suggests that the critical approach could illustrate the multiaxial fatigue damage behavior by combining the different failure mode of materials. It is also proved that polar representation can be an effective expression for analyzing the cracking behavior of multiaxial fatigue. However, there is limited mic-roscale images information in terms of fatigue crack orientation and early growth characteristics in the literature. The cracking behavior under multiaxial loading conditions, therefore, has still not been studied sufficiently connecting with critical plane approaches. A better critical plane model considering the actual fatigue failure mode, including the crack path, sizes and surface of multiaxial fatigue, may estimate a satisfactory fatigue life.
7. Conclusion
To determine the fatigue crack orientation and early growth of different loading paths, polar representation of full stress-strain fields based on the critical plane approach have been investigated. The crack initiation and early growth stage are defined by the shortest uni-oriented observed crack morphology and the crack evolutional direction. The variations of stress-strain loci in polar coordinates determine the fatigue failure mode by combining experimental observed cracking paths under cycle loading paths. The potential cracking regions of these cases are further predicted according to analyzed fatigue failure mode. Therefore, the cracking behavior under cycle loading can provide a view to understand and evaluate the critical plane approach. The results obtained are presented here and the following conclusions are drawn:
The comparison between the maximum strain loci on polar diagram based on critical plane approach and the experimental observed cracking path shows that cracks orientate from the maximum shear strain planes.
The S460N steel demonstrates the mixed fatigue failure mode in the crack orientation and growth stages. Cracks firstly initiate from the maximum shear plane and grow towards to the maximum normal strain plane.
The predicted latent cracking regions have a good agreement with the observed cracking directions for different nonproportional loading paths under strain polar representation. It provides available information for evaluating the multiaxial fatigue life.
-Proportional
----45° nonproportional
----- 60° nonproportional
90° nonproportional
/ / \ \
T
The polar representation of maximum shear strain fields could be employed to conclude the nonproportional additional hardening degree by calculating the relationship between the full circle area and swept area of the maximum shear strain under nonproportional loading.
References
1. Fatemi A., Shamsaei N. Multiaxial fatigue: An overview and some approximation models for life estimation // Int. J. Fatigue. - 2011. -V. 33. - No. 8. - P. 948-958.
2. Stulen F.B., Cummings H.N. A failure criterion for multiaxial fatigue stresses // Proc. ASTM. - 1954. - V. 54. - P. 822-835.
3. Findley W.N. A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending // J. Eng. Industry. - 1959. - V. 81. - No. 4. - P. 301-306.
4. Wang C.H., Brown M. W. Life prediction techniques for variable amplitude multiaxial fatigue. Part 1: Theories // J. Eng. Mater. Tech. Trans. ASME. - 1996. - V. 118. - No. 3. - P. 367-370.
5. Fatemi A., Socie D.F. A critical plane approach to multiaxial fatigue damage including out-of-phase loading // Fatigue Fract. Eng. Mater. Struct. - 1988. - V. 11. - No. 3. - P. 149-165.
6. Brown M.W., Miller K.J. A theory for fatigue failure under multiaxial stress-strain conditions // Proc. Inst. Mech. Eng. - 1973. - P. 745755.
7. Smith K.N., Watson P., Topper T.H. Stress-strain function for the fatigue of metals // J. Mater. - 1970. - V. 5. - No. 4. - P. 767-778.
8. Carpinteri A., Ronchei C., Scorza D., VantadoriS. Fatigue life estimation for multiaxial low-cycle fatigue regime: The influence of the effective Poisson ratio value // Theor. Appl. Fract. Mech. - 2015. - V. 79. -P. 77-83.
9. Susmel L. Four stress analysis strategies to use the modified Wohler curve method to perform the fatigue assessment of weldments subjected to constant and variable amplitude multiaxial fatigue loading // Int. J. Fatigue. - 2014. - V. 67. - P. 38-54.
10. Wang X.W., Shang D.G. Determination of the critical plane by a weight-function method based on the maximum shear stress plane under multiaxial high-cycle loading // Int. J. Fatigue. - 2016. - V. 90. -P. 36-46.
11. Berto F. Fatigue and fracture assessment of notched components by means of the strain energy density // Eng. Fract. Mech. - 2016. -V. 167. - P. 176-187.
12.Li J., Sun Q, Zhang Z.P., Li C.W., Zhang D.W. A new multiaxial fatigue life prediction model under proportional and nonproportional loading // J. Eng. Mater. Tech. Trans. ASME. - 2010. - V. 132. -No. 2. - P. 0210161-0210168.
13.Li J., Li C.W., Qiao Y.J., Zhang Z.P. Fatigue life prediction for some metallic materials under constant amplitude multiaxial loading // Int. J. Fatigue. - 2014. - V. 68. - P. 10-23.
14. Socie D., Bannantine J. Bulk deformation fatigue damage models // Mater. Sci. Eng. - 1988. - V. 103. - No. 1. - P. 3-13.
15. Zhao T., Jiang Y Fatigue of 7075-T651 aluminum alloy // Int. J. Fatigue. - 2008. - V. 30. - No. 5. - P. 834-849.
16. Socie D. Critical plane approaches for multiaxial fatigue damage assessment // ASTM Spec. Tech. Publ. - 1993. - P. 7-36.
17. Kim K.S., Park J.C., Lee J.W. Multiaxial fatigue under variable amplitude loads // J. Eng. Mater. Technol. Trans. ASME. - 1999. -V. 121.- No. 3. - P. 286-293.
18.Kim K.S., Lee B.L., Park J.C. Biaxial fatigue of stainless steel 304 under irregular loading // ASTM Spec. Tech. Publ. - 2000. -No. 1389. - P. 79-93.
19. Gao Z., Zhao T., Wang X., Jiang Y. Multiaxial fatigue of 16MnR steel // J. Pressure Vessel Technol. Trans. ASME. - 2009. - V. 131. - No.2.
20. Carpinteri A., Ronchei C., Spagnoli A., Vantadori S. On the use of the Prismatic Hull method in a critical plane-based multiaxial fatigue criterion // Int. J. Fatigue. - 2014. - V. 68. - P. 159-167.
21. Jiang Y., Hertel O., Vormwald M. An experimental evaluation of three critical plane multiaxial fatigue criteria // Int. J. Fatigue. - 2007. -V. 29. - No. 8. - P. 1490-1502.
22. Castro F., Jiang Y. Fatigue of extruded AZ31B magnesium alloy under stress- and strain-controlled conditions including step loading // Mech. Mater. - 2017. - V. 108. - P. 77-86.
23. Ohkawa I., Takahashi H., Moriwaki M., Misumi M. A study on fatigue crack growth under out-of-phase combined loadings // Fatigue Fract. Eng. Mater. Struct. - 1997. - V. 20. - No. 6. - P. 929-940.
24.Albinmousa J., Jahed H. Multiaxial effects on LCF behaviour and fatigue failure of AZ31B magnesium extrusion // Int. J. Fatigue. -2014. - V. 67. - P. 103-116.
25.Albinmousa J. Investigation on multiaxial fatigue crack path using polar stress-strain representation // Int. J. Fatigue. - 2016. - V. 92. -P. 406-414.
26. Chen X., Gao Q, Sun X.F. Low-cycle fatigue under non-proportional loading // Fatigue Fract. Eng. Mater. Struct. - 1996. - V. 19. - No. 7. -P. 839-854.
27. Chen X., Xu S., Huang D. Critical plane-strain energy density criterion for multiaxial low-cycle fatigue life under non-proportional loading // Fatigue Fract. Eng. Mater. Struct. - 1999. - V. 22. - No. 8. -P. 679-686.
28.Hoffmeyer J., DoringR., Seeger T., VormwaldM. Deformation behaviour, short crack growth and fatigue lives under multiaxial nonproportional loading // Int. J. Fatigue. - 2006. - V. 28. - No. 5-6. - P. 508520.
29.Reis L., Li B., de Freitas M. Crack initiation and growth path under multiaxial fatigue loading in structural steels // Int. J. Fatigue. - 2009. -V. 31. - No. 11-12. - P. 1660-1668.
30.Noban M, Jahed H., Ibrahim E., Ince A. Load path sensitivity and fatigue life estimation of 30CrNiMo8HH // Int. J. Fatigue. - 2012. -V. 37. - P. 123-133.
31. Noban M., Jahed H., Winkler S., Ince A. Fatigue characterization and modeling of 30CrNiMo8HH under multiaxial loading // Mater. Sci. Eng. A. - 2011. - V. 528. - No. 6. - P. 2484-2494.
32.Doring R., Hoffmeyer J., Seeger T., VormwaldM. Short fatigue crack growth under nonproportional multiaxial elastic-plastic strains // Int. J. Fatigue. - 2006. - V. 28. - No. 9. - P. 972-982.
33.Li J., Zhang Z., Sun Q., Li C. Low-cycle fatigue life prediction of various metallic materials under multiaxial loading // Fatigue Fract. Eng. Mater. Struct. - 2011. - V. 34. - No. 4. - P. 280-290.
34.Fatemi A., Molaei R., Sharifimehr S., Shamsaei N., Phan N. Tor-sional fatigue behavior of wrought and additive manufactured Ti-6Al-4V by powder bed fusion including surface finish effect // Int. J. Fatigue. - 2017. - V. 99. - P. 187-201.
35.Shamsaei N., Fatemi A., Socie D.F. Multiaxial cyclic deformation and non-proportional hardening employing discriminating load paths // Int. J. Plasticity. - 2010. - V. 26. - No. 12. - P. 1680-1701.
36. Castro F., Jiang Y Fatigue life and early cracking predictions of extruded AZ31B magnesium alloy using critical plane approaches // Int. J. Fatigue. - 2016. - V. 88. - P. 236-246.
37.Li J., Zhang Z.P., Sun Q., Li C.W. Multiaxial fatigue life prediction for various metallic materials based on the critical plane approach // Int. J. Fatigue. - 2011. - V. 33. - No. 2. - P. 90-101.
Поступила в редакцию 24.04.2017 г.
Сведения об авторах
Wei Song, Dr., Harbin Institute of Technology, China, swingways@hotmail.com Xuesong Liu, Prof., Harbin Institute of Technology, China, xuesongliuhit@gmail.com Filippo Berto, Prof., NTNU, Norway, berto@gestunipd.it
