Low-cycle multiaxial fatigue behavior and life prediction of Q235B steel welded material

International Journal of Fatigue 127 (2019) 417–430

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Low-cycle multiaxial fatigue behavior and life prediction of Q235B steel welded material

T

Qinghua Hana,b,c, Peipeng Wangc, Yan Lua,b,c,

⁎

a

Key Laboratory of Earthquake Engineering Simulation and Seismic Resilience of China Earthquake Administration (Tianjin University), Tianjin 300350, China Key Laboratory of Coast Civil Structure Safety of China Ministry of Education (Tianjin University), Tianjin 300350, China c School of Civil Engineering, Tianjin University, Tianjin 300350, China b

ARTICLE INFO

ABSTRACT

Keywords: Multiaxial fatigue experiment Q235B steel welded material Loading path Energy-based fatigue parameter Life prediction

Low-cycle torsional and multiaxial fatigue experiments are carried out for thin-walled Q235B steel welded specimens under both in-phase and 90° out-of-phase loading paths. A remarkable non-proportional additional hardening effect exists in the Q235B steel welded material under the 90° out-of-phase loading paths. A new energy-based fatigue parameter is proposed that incorporates the shear strain energy on the weighted critical plane and the normal strain energy on both the weighted critical plane and orthogonal plane. The proposed energy-based fatigue parameter provides a superior ability to make life prediction for the Q235B steel welded material under the torsional and multiaxial loading paths.

1. Introduction Steel structures are frequently exposed to complex cyclic or repeated external loads under multiaxial stress/strain fields. Fatigue failure is produced in the critical areas where cracks may initiate and propagate, eventually resulting in fracture [1,2]. In addition, during their service lives, steel structures undergo extreme natural conditions, such as powerful earthquakes, violent typhoons, and strong rainfall. The structural components are prone to plastic strain fatigue and continuous vibrations. Furthermore, steel structural members or joints are very often connected by welding. Due to the stress concentration effects and deteriorative mechanical properties of materials, fatigue failure often occurs at critical locations [3,4]. Thus, the low-cycle multiaxial fatigue phenomenon and fatigue life evaluation of welded materials or related joints remain challenging and are worth investigating [5,6]. Multiaxial fatigue is defined as a failure of load-transferring members and joints due to the detrimental effects of directional time-variable stress/strain components. In general, there are two low-cycle multiaxial fatigue loading modes [7]. When the applied axial and torsional strain components vary simultaneously without a phase-shifting angle, the loading mode is defined as proportional or in-phase loading. In such situations, the principal strain axes remain fixed during loading. Establishing an effective fatigue parameter to assess fatigue damage with the same uniaxial load is easy. However, when the applied axial and torsional strain components vary independently, and the phase

⁎

angle between the components begins to appear, the induced multiaxial fatigue loading mode is known as non-proportional or out-of-phase loading. The major distinction between the above two modes is that both the direction and magnitude of the maximum principal strain are always changeable under out-of-phase loading. Metals exhibit a more complicated fatigue failure mechanism under out-of-phase loading than under in-phase loading [8–10]. To investigate the fatigue performance of metallic materials used in civil, mechanical, aeronautical and chemical engineering, such as S460N structural steel, Ck45 medium carbon steel, hot-rolled 45 steel, 304L stainless steel, and Ti-6Al-4V, immense amounts of multiaxial fatigue experimental research has been performed [11–16]. Due to the high temperature of the welding process, the chemical composition and microstructure of welded metals change greatly, leading to complicated mechanical properties and failure mechanisms under multiaxial loading. Chen et al. [5] found that the 1Cr-18Ni-9Ti welded metal exhibited reduced fatigue life under uniaxial and torsional loading paths, and there was no significant distinction between the fatigue lives of welded metal and base metal under non-proportional conditions. Qu et al. [13] performed multiaxial fatigue experiments on Q235B steel welded material under proportional and circular paths. Their results showed that the welded metal exhibited a non-proportional additional hardening effect and a lower fatigue resistance under the 90° out-ofphase loading conditions than under the uniaxial and in-phase conditions. The above research demonstrated that welded metals exhibited

Corresponding author at: School of Civil Engineering, Tianjin University, Tianjin, China. E-mail address: [email protected] (Y. Lu).

https://doi.org/10.1016/j.ijfatigue.2019.06.027 Received 21 January 2019; Received in revised form 2 June 2019; Accepted 21 June 2019 Available online 22 June 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Fatigue 127 (2019) 417–430

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more degraded multiaxial properties than the base metals. In recent decades, much endeavor has been dedicated to developing reasonable multiaxial fatigue prediction models or modifying parameters based on previous models [17–21]. Generally, fatigue prediction models are divided into four categories: stress-based, strain-based, energy-based and critical plane models [11]. Critical plane models are established based on the experimental observation that the initiation and growth of small microcracks occur along the preferential plane on the surface. In this approach, a specific material plane is taken as the reference plane and combined with certain stress and/or strain quantities extracted from such a plane to make a prediction. The Smith–Watson–Topper (SWT) model [22] for tensile-type failure materials and the Kandil–Brown–Miller (KBM) model [23] for shear-type failure materials are examples of critical plane models. Based on the KBM model, the Fatemi–Socie (FS) model [24] was proposed, which used the normal strain replaced by stress accounting for the additional hardening effect under non-proportional loading paths. The critical plane model has been viewed as an effective method for predicting both fatigue life and crack direction because such a model can reasonably reflect the physical essence of the fatigue damage process [7]. Until now, critical plane models have been prominently used in life prediction and established in conjunction with strain energy models. Many investigations propose fatigue parameters by combining the critical plane concept with an energy-based model. Liu [25] introduced the concept of calculating the virtual strain energy from each candidate plane to measure fatigue damage by considering the effect of mean stress. He regarded the maximum normal work plus shear work as the multiaxial fatigue parameter for tensile-type failure materials and the maximum shear work plus normal work as the damage parameter for shear-type failure materials. Chen et al. [26] proposed combining a damage parameter with normal and shear energy densities when considering different failure mechanisms. Unlike Ref. [25], only the maximum normal strain range was selected for tensile-type failure materials, and the maximum shear strain range was selected for shear-type failure materials. Jiang [27] developed a fatigue parameter with an incremental form of plastic strain energy; this model considered the mean stress and loading sequence and was suitable for variable-amplitude conditions. In consideration of Poisson’s effect, Lu et al. [28] proposed a fatigue parameter consisting of the strain energy from two planes. One part of the parameters was the summation of shear energy from the normal and shear directions of the critical plane, and another part was the strain energy term on the plane that was orthogonal to the critical plane. Although numerous multiaxial life prediction criteria are available in foregoing discussions, acquiring suitable fatigue life prediction criteria for all materials under miscellaneous multiaxial loading paths is difficult. There is still a need to make much effort to properly predict the fatigue lives of different metals. In this paper, torsional and low-cycle multiaxial fatigue experiments are performed for a thin-walled Q235B steel welded material under multiple loading paths. A new fatigue parameter incorporating the shear strain

energy on the weighted critical plane and the normal strain energy on both the weighted critical plane and orthogonal plane is proposed. Finally, the proposed energy-based life prediction approach is applied to assess the multiaxial fatigue life of Q235B steel welded material. 2. Experimental details 2.1. Test specimen The test specimen is a steel welded thin-walled circular tube designed based on ASTM E2207-15 [29]. Q235B steel is typically a lowcarbon steel, and its chemical composition (wt.%) is listed in Table 1. A solid Q235B steel bar is first prepared with a circumferential trapezoidshaped groove. Then, an ER50-6 welding wire with a diameter of 2.5 mm is fused via carbon dioxide welding, and the weld seam is formed. The chemical composition (wt.%) of the welding wire is listed in Table 2. Finally, the welded steel bar is machined to a predesigned test specimen. The length, outer diameter and thickness of the welded tube are 115 mm, 12.5 mm and 1.1 mm, respectively. The weld seam length in the middle of the test specimen is 10 mm. The details of the test specimen are presented in Fig. 1. The outer surface of the specimen is polished by a buff wheel, and the inner surface is machined by lowspeed wire cutting. The fatigue cracks are expected to initiate from the outer surface of the specimen due to torsion. As the thin-walled specimen is machined, the residual stresses created by the welding disappear; the effect of the residual stresses are not studied. The mechanical properties of the specimen are listed in Table 3.

Fig. 1. Scheme drawings of the test specimen. Table 3 Mechanical property of specimen.

Table 1 Chemical composition of Q235B low-carbon steel (wt.%).

Present research Ref. [13]

C

Mn

Si

S

P

0.130 0.160

0.460 0.450

0.130 0.260

0.022 0.025

0.025 0.021

Present research Ref. [13]

Young’s modulus E (GPa)

Shear modulus G (GPa)

Yield strength fy (MPa)

Tensile strength fu (MPa)

185 198

74.1 76.3

283 265

428 371

Table 2 Chemical composition of welding wire (wt.%).

Present research Ref. [13]

C

Mn

Si

S

P

Ni

Cr

Mo

V

Cu

0.073 0.077

1.480 1.540

0.850 0.920

0.019 0.012

0.012 0.011

0.028 0.006

0.048 0.023

0.0062 0.0040

0.001 0.002

0.100 0.126

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2.2. Loading scheme

maximum shear strain range. σn,max is the maximum normal stress on the plane that endures the maximum shear strain range. Nf is the experimental fatigue life. In addition, the stress and strain components on the arbitrary material plane can be calculated using formulas presented in Ref. [31].

The multiaxial fatigue experiments are performed on a computercontrolled testing system with an MTS-370.02 tension-torsion machine that can apply a 25-kN dynamic axial force and a 200-N·m torque. The applied axial and torsional strains are controlled using an MTS biaxial tension-torsional extensometer with a gage length of 25 mm and a diameter of 12.5 mm. Fully and partially reversed axial and torsional strain ranges of triangular and sinusoidal waves are used for the lowcycle fatigue experiments. A constant axial strain rate of 0.006 s−1 is set for all experiments. The load frequency varies with the applied axial strain amplitude. All experiments are carried out in air at room temperature. Experimental fatigue life is defined as the number of repeated loads required to decrease to 90% of the maximum value of either the tensile or shear responsive load [29]. The multiaxial fatigue behavior depends on the material and the state of loading [7,19]. To investigate the multiaxial fatigue performance of Q235B steel welded material under various loading paths, the factors of axial strain amplitude, phase-shifting angle between axial and shear strain components, strain amplitude ratio between shear and axial strains, strain ratio between the minimum and maximum of each strain component and applied waveforms are considered in this paper. The orthogonal test is designed to consider the above five influence factors at various levels [30]. The axial strain amplitude Δε/2 has four levels ranging from 0.2% to 0.5%. The phase-shifting angle between axial and shear strain components φ has two levels: 0° and 90°. The strain amplitude ratios of shear to axial strain λ are set as 1 and 3 . The strain ratios between the minimum and maximum values of each axial strain component Rε are −1 and −0.5, respectively; the same values are set for strain ratios between the minimum and maximum values for shear strain component Rγ. The detailed loading scheme is listed in Table 4. An L16(41 × 212) orthogonal test table is designed to generate 16 loading conditions with 5 different loading paths in the shape of a line, circle, ellipse, square or rhombus. The applied multiaxial loading paths are plotted in Fig. 2. A total of 9 torsional and 34 multiaxial fatigue experiments were carried out. Tables 5 and 6 list the torsional and multiaxial fatigue experimental results, respectively. Δε/2 and Δγ/2 are the axial and torsional strain amplitudes. Δσ/2 and Δτ/2 are the axial and torsional stress amplitudes, respectively. All axial and torsional stress amplitude data are extracted at approximately 50% of the total lifetime as the stable hysteresis loops are observed for the axial and torsional components. Δεeq/2 and Δσeq/2 are the effective strain and stress amplitudes, respectively, calculated by Eqs. (1) and (2) [5,7]. Δγmax is the

eq

2 eq

2

Δε/2 (%)

φ

λ

1 2

0.2

0° 0°

1

3 4 5 6

0.3

7 8 9 10

0.4

11 12 13 14 15 16

0.5

90° 90°

1

0° 0°

1

90° 90°

1

0° 0°

1

90° 90°

1

0° 0°

1

90° 90°

1

3

Rε

Rγ

Waveform

Loading frequency (Hz)

−1 −0.5

−1 −0.5

triangular sinusoidal

0.750

3

−1 −0.5

−1 −0.5

triangular sinusoidal

−1 −0.5

sinusoidal triangular

3

−1 −0.5

−0.5 −1

0.500

3

−0.5 −1

triangular sinusoidal

3

−0.5 −1

−0.5 −1

0.375

3

−0.5 −1

−1 −0.5

sinusoidal triangular

0.300

3 3

−0.5 −1

−0.5 −1

−0.5 −1

−1 −0.5

=

2

2 2

2

+

1 3

+3

2

2

max

(1)

max

(2)

2

2

3. Experimental results and analysis Fig. 3 plots the axial and torsional hysteresis loops of the studied material under stable loading conditions. The maximum and minimum stress values are nearly symmetrically distributed in both the axial and torsional directions. Notably, under the rhombic paths, the maximum cyclic axial stress is approximately 1.19 times that under the proportional path, while the maximum cyclic torsional stress is approximately 1.70 times that under the proportional path (see Fig. 3(a), (b), (i), and (j)). Under the circular paths, the maximum cyclic axial or torsional stress is approximately 1.5 times that under the proportional paths (see Fig. 3(c), (d), (k), and (l)). Under the elliptic paths, the maximum cyclic axial stress is approximately 1.28 times that under the proportional paths, while the maximum cyclic torsional stress is approximately 1.93 times that under the proportional paths (see Fig. 3(e), (f), (m), and (n)). Under the square paths, the maximum cyclic axial or torsional stress is nearly 1.37 times that under the proportional paths (see Fig. 3(g), (h), (o), and (p)). Notably, the maximum cyclic axial stress under the 90° out-of-phase loading paths strongly depends on the strain amplitude ratio between the shear and axial strain amplitudes λ. When the strain amplitude ratio λ equals 1, the maximum cyclic axial stress amplification is smaller than that of the maximum cyclic torsional stress. When λ equals 3 , the maximum cyclic axial stress amplification is the same as that of the maximum cyclic torsional stress. The test specimens undergo a remarkable non-proportional strain-hardening effect under the different loading paths like other metal materials such as S460N structural steel [11] and 304L stainless steel [15]. This finding implies that the Q235B steel welded material tends to absorb more plastic strain energy per cyclic curve under the non-proportional paths than under the proportional paths. Additionally, the stress and strain components reach maximum or minimum values simultaneously under the proportional loading path. However, strain always lags behind the corresponding stress under the circular and elliptic paths. These hysteresis loops show a rotation around the origin under the circular and elliptic paths. The trends of axial/torsional mean stress with cyclic loading under proportional and non-proportional paths are presented in Figs. 4 and 5, respectively. The axial/torsional mean stress keeps stable and nearly equals zero under the fully reversed axial and torsional strain-controlled loading (Rε/Rγ equals −1). However, as Rε/Rγ equals −0.5, the axial or torsional mean stress reaches a large value in the first loading cycle and the mean stresses tend to be zero in the following cycles. It is shown that mean stress relaxation occurs for Q235B steel welded material under the partially reversed axial and torsional strain-controlled loading with a large plastic strain applied. The cyclic equivalent stress-strain curves fitted by the Ramberg–Osgood relation (shown in Eq. (3)) for the Q235B steel welded material are shown in Fig. 6. The material-based constants E, K’ and n’ are the elasticity modulus, cyclic strength coefficient and cyclic strain-hardening exponent, respectively, which are fitted by Eq. (3) and listed in Table 7. The torsional equivalent stress-strain curve is slightly higher than that under the proportional paths. Significant additional hardening under the 90° out-of-phase loading paths was also observed. The hardening extent is the following order: circular path ＞ square

Table 4 Fatigue experiment loading scheme. No.

=

sinusoidal triangular

triangular sinusoidal

sinusoidal triangular

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Fig. 2. Multiaxial loading paths.

Table 5 Torsional fatigue experiment results. No.

Δε/2 (%)

Δγ/2 (%)

Δεeq/2 (%)

Δσ/2 (MPa)

Δτ/2 (MPa)

Δσeq/2 (MPa)

Δγmax/2 (%)

Nf (Cycles)

U1-a U2-a U2-b U3-a U3-b U4-a U4-b U5-a U5-b

0 0

0.200 0.346

0.115 0.200

0 0

0.500

0.289

0

0

0.707

0.408

0

0

0.866

0.500

0

199.08 238.00 239.53 276.04 277.13 303.59 307.46 317.68 313.92

0.200 0.346

0

114.94 137.41 138.29 159.37 160.43 175.28 177.51 183.41 181.24

> 105 28,275 20,695 4222 4638 3113 3447 2553 1226

420

0.500 0.707 0.866

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Table 6 Multiaxial fatigue experiment results. No.

Δε/2 (%)

Δγ/2 (%)

Δεeq/2 (%)

Δσ/2 (MPa)

Δτ/2 (MPa)

Δσeq/2 (MPa)

Δγmax/2 (%)

σn,max (MPa)

Nf (Cycles)

¯ (°)

M1-a M1-b M2-a M2-b M2-c M3-a M3-b M4-a M4-b M5-a M5-b M6-a M6-b M7-a M7-b M7-c M8-a M8-b M8-c M9-a M9-b M9-c M10-a M10-b M11-a M11-b M12-a M12-b M13-a M13-b M14-a M14-b M15-a M16-a

0.200

0.190

0.228

0.346

0.283

0.200

0.200

0.200

0.200

0.346

0.200

0.300

0.300

0.346

0.300

0.519

0.424

0.300

0.300

0.300

0.300

0.519

0.300

0.400

0.400

0.462

0.400

0.690

0.566

0.400

0.400

0.400

0.400

0.690

0.400

0.500

0.500

0.577

0.500

0.866

0.707

0.500 0.500

0.500 0.866

0.500 0.500

78.47 79.04 113.01 116.31 116.20 132.62 126.90 172.49 172.08 86.07 86.43 124.30 125.28 168.00 163.79 165.34 171.75 169.90 169.00 91.81 92.34 90.98 136.82 128.67 156.69 157.58 192.76 196.11 98.92 98.31 136.75 137.09 192.21 185.39

262.38 265.56 244.69 253.30 249.97 280.64 272.56 308.70 308.75 274.74 276.11 275.82 278.24 328.51 323.71 325.48 310.29 310.34 304.56 300.52 297.07 296.08 301.59 287.78 328.05 330.88 353.34 356.91 318.59 317.27 308.15 307.42 360.04 330.04

0.340

0.200

241.20 244.33 194.30 201.81 197.74 280.64 272.56 308.70 308.75 250.31 251.58 222.08 224.12 328.51 323.71 325.48 310.29 310.34 304.56 275.16 271.08 270.79 241.94 232.68 328.05 330.88 353.34 356.91 290.78 289.69 250.13 248.91 360.04 330.04

119.47 121.89 92.23 104.08 100.93 163.69 164.26 298.78 299.35 123.56 125.14 113.14 116.46 264.99 253.10 255.44 299.28 307.71 304.45 131.48 130.97 132.85 116.99 119.24 234.53 236.01 352.77 354.34 137.35 143.17 127.34 125.32 281.62 325.96

11,926 14,855 9364 6765 11,849 9305 7243 3574 4309 3253 3698 2424 2880 1203 1461 1286 1119 1588 1152 1409 1622 1101 841 995 787 753 366 383 961 733 787 607 253 193

152.0 152.0 160.0 160.0 160.0 84.2 85.3 106.4 106.4 152.0 152.0 160.0 160.0 94.5 90.0 89.7 113.6 113.1 113.1 152.0 152.0 152.0 160.0 160.0 105.1 105.8 87.4 85.5 152.0 152.0 160.0 160.0 105.1 89.2

path ＞ elliptic path ＞ rhombic path ＞ torsional path ＞ proportional path. eq

2

=

eq

2E

+

eq

0.271 0.352 0.524 0.673 0.418 0.519 0.700 0.905 0.568 0.693 0.882 1.126 0.714 0.866

material property and external loading condition may have an impact on the selection orientation of the critical plane [19,32]. There are various methods for defining the critical plane according to the multiple types of different kinds of materials [33]. In general, selecting a plane enduring maximum shear strain or stress is appropriate for ductile metals, whereas a representative principal strain or stress component is more suitable for brittle metals [19]. The KBM [23] and FS models [24] proposed prediction criteria with the maximum shear strain range as the critical plane. Fig. 8 shows the variation trend of shear strain with different planes over a certain period of time; such a variable is the actual shear strain extracted from multiaxial fatigue experiments. The shear strain on the arbitrary planes changes with the angle and time. Acquiring the maximum shear strain range on a specified plane is easy due to the regular distribution of shear strain under the proportional conditions (Fig. 8(a), (c), (e), and (g)). Under the non-proportional paths shown in Fig. 8(b), (d), (f), and (h)), the differences are striking. Individual material planes may endure the peak of damage at some instances, but the remaining planes may sustain a relatively high level of damage, especially under a circular loading path. The phenomenon of stripped variation appearing under the circular loading path in Fig. 8(h) demonstrates that fatigue damage attributed to the whole plane would affect the multiaxial fatigue life. Because of the irregularity of the strain component distribution under 90° out-of-phase multiaxial loading, variations in the maximum shear strain plane are taken into account in the fatigue life prediction. Chen et al. [34] presented a new weight function to redefine the direction of the critical plane. They argued that the expected critical plane should be identified by taking the weighted average of every plane that experiences instantaneous maximum shear strain. The

1/ n

2K

0.448

(3)

The observed macroscopic fatigue crack growth patterns under torsional and multiaxial fatigue loading are presented in Fig. 7. Under torsional paths, the shear cracks initiate and propagate early along the longitudinal axis of the specimen (Fig. 7(a)). The reason for this phenomenon is that the maximum shear strain range occurs in the longitudinal axis. Then, the crack continues to grow perpendicularly to the maximum principal strain direction. It follows that the Q235B steel welded material is prone to the shear-type failure mode under cyclic loading. In Fig. 7(b), the macroscopic crack morphology of specimens subjected to multiaxial fatigue exhibits a complex crack growth pattern. As the maximum principal strain direction varies within a certain loading cycle, macrocrack direction is irregular at an angle of approximately 40°–90° from the longitudinal axis. 4. Fatigue life evaluation 4.1. Critical plane determination Many researchers have put forward fatigue life prediction criteria based on the critical plane concept composed of strain and stress items. The critical plane criterion has become a common and effective method for determining the fatigue parameter. Determining the direction of such a plane is key to predicting multiaxial fatigue life. Both the

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critical plane is expediently determined by averaging all the instantaneous values of maximum shear strain with the appropriate weight function. The proposed weight function is suitable for materials that undergo shear-type failure (Eq. (4))

( ti ) =

ti max

min min

strain values within the loading cycle, respectively. By virtue of the weight function, the average critical plane can be redefined by multiplying the weight factors by the transient directions of the potential critical plane, as shown in Eq. (5)

¯= 1 W

(4)

where ti is an arbitrary transient time within one loading cycle, ω(ti) and γti are the weight factor and shear strain at a certain time instant, respectively, and γmax and γmin are the maximum and minimum shear

n

(ti ) (ti ) i=1

(5)

where the weighted critical plane is represented by its normal direction using one angle ¯ , measured counterclockwise from the axial direction of the specimen (shown in Fig. 9). Such a normal direction of the

Fig. 3. Hysteresis loops of Q235B steel welded material.

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Fig. 3. (continued)

critical plane is orthogonal to the normal direction of specimen surface. W is the summation of ω(ti), and θ(ti) is the transient direction of the plane enduring the maximum shear strain range. Under the proportional loading path, the maximum shear strain plane, represented by θ(ti), is fixed at any time. As a result, the weighted orientation of the critical plane ¯ equals θ(ti) at any time under the proportional loading path. The values of ¯ are listed in Table 6.

the parameters are stress-based, strain-based or energy-based and whether the orientations of the reference planes are weighted. These approaches seem reasonable because, in reality, except for existing multiple fatigue sources, multiaxial cracks initiate and propagate early along a preferable direction. However, Lu et al. [28,35] argued that the calculated parameters from a single critical plane cannot sufficiently reflect fatigue damage; furthermore, the parameters derived from a plane orthogonal to the critical plane may contribute to damage. Lu et al. postulated that when the element body is subjected to the normal and shear loads on the critical plane, the corresponding deformations will simultaneously damage both the critical and orthogonal planes. Such a concept can be explained by the fact that the axial deformations

4.2. Fatigue parameters The fatigue parameters are determined only on the critical plane itself using the traditional critical plane method, regardless of whether 423

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Fig. 3. (continued)

of materials are generally accompanied by transverse deformations because of Poisson effects [35]. Verifying the fatigue parameter by considering the additional parameters derived from the orthogonal plane is concluded to be reasonable. Based on the perspective mentioned above, an energy-based fatigue parameter (FP) is proposed as Eq. (6):

FP = S

max

2

max

+

n

2

n,max

+

no

2

no,max

and maximum normal stress on the plane perpendicular to the weighted critical plane, respectively, and S is a material-based constant that depends on material type and reflects the influence of material property on the predicted life. To derive the life prediction formula with Eq. (6), a widely used strain-life relationship, the Manson–Coffin equation, is used [36]; it can be written as Eq. (7):

(6)

2

where Δγmax and τmax are the maximum shear strain range and maximum shear stress on the weighted critical plane, respectively, Δεn and σn,max are the normal strain range and maximum normal stress on the same plane, respectively, Δεno and σno,max are the normal strain range

=

e

2

+

p

2

=

f

E

(2Nf ) b +

f

(2Nf ) c

(7)

where Δε is the applied normal strain range, Δεe and Δεp are the elastic and plastic portions of the normal strain range, respectively, and the parameters f , f , b and c are the fatigue parameters fitted by uniaxial

Fig. 4. Experiment life versus mean stress under proportional paths. 424

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Fig. 5. Experiment life versus mean stress under non-proportional paths.

Young’s modulus, shear modulus, yield strength and tensile strength, are close to each other. Thus, the fitted Manson–Coffin curve in Ref. [13] is used for fatigue life prediction in the paper. From Eq. (7), the elastic portion of the strain amplitude can be presented as Eq. (8). e

2

=

f

E

(2Nf )b

(8)

By multiplying the elastic strain amplitude Δεe/2 (Eq. (8)) by the elasticity modulus E, the elastic stress amplitude Δσ/2 is written as Eq. (9).

=

2

Table 7 Material constants of Q235B weld fitted by Eq. (3). E (GPa)

K’ (MPa)

n’

Torsional Proportional Non-proportional

185

821 865 1001

0.166 0.188 0.174

(2Nf )b

(9)

Furthermore, Fig. 10 plots the strain and stress Mohr’s circle under uniaxial load. In Fig. 10(a), Point A (εn, γmax/2) and Point B (εno, −γmax/2) represent the strain states on the critical plane (1st plane) and the orthogonal plane (2nd plane), respectively. Similarly, in Fig. 10(b), Point A (σn, τmax) and Point B (σno, −τmax) represent the stress states on the 1st and 2nd planes. The half maximum shear strain γmax/2 on the critical plane is the radius of strain Mohr’s circle (Fig. 10(a)), and the maximum shear strain amplitude Δγmax/2 can be expressed as Eq. (10).

Fig. 6. Cyclic equivalent stress-strain curves.

Loading path

f

max

2

1

=

- (- v 2

1)

= (1 + v )

1

(10)

2

Similarly, the normal strain range Δεn is derived as Eq. (11). test data. In this paper, the four abovementioned property parameters are 481 MPa, 0.0375, −0.0755, and −0.315, respectively, based on the uniaxial experimental results of the Q235B steel welded material [13]. Tables 1 and 2 list the chemical compositions of the Q235B base metal and welding wire for the test specimens used in this paper and in Ref. [13] and reveal that the chemical compositions are similar. Table 3 lists the mechanical properties of the test specimens. Based on carbon dioxide gas shielded welding, the mechanical properties, including

n

=

no

=

1

+( v 2

1)

= (1

v)

1

2

(11)

Substituting Eq. (7) into Eqs. (10) and (11), the maximum shear strain amplitude Δγmax/2 and the corresponding normal strain amplitude Δεn/2 on the critical plane can be written as Eqs. (12) and (13), respectively. ve and vp are the elastic and plastic Poisson’s ratios. In this paper, ve and vp are 0.3 and 0.5 [7], respectively. The normal strain amplitude on the orthogonal plane (2nd plane), Δεno, equals Δεn under

Fig. 7. Patterns of crack growth. 425

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Fig. 8. Variation trend of shear strain with different planes over a certain period of time.

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Fig. 9. Two-dimensional strain state within gage section of specimens.

S

max

2

max

+

n

2

n,max

+

no

2

no,max

=

f

2

E

(2Nf )2b +

f

f

(2Nf )b + c (21)

Fig. 11 presents the relationship between the energy components of FP (Eq. (21)) and the experimental fatigue life. This relationship demonstrates that the shear strain energy on the weighted critical plane greatly contributes to fatigue damage for most loading paths. The shear strain energy component under the proportional loading condition is larger than that under the non-proportional loading condition (Fig. 11(a)). However, the normal strain energy on the weighted critical plane is at a relatively low level (Fig. 11(b)), implying that such a component on the weighted critical plane has nearly no significant effect on the multiaxial fatigue life. Furthermore, the normal strain component on the orthogonal plane induces some fatigue damage for non-proportional paths, whereas such a component subjected to proportional loading still has an insignificant impact on fatigue damage (Fig. 11(c)). Fig. 12 presents the percentage of fatigue parameters determined by Eq. (21) with all three strain energy components under all multiaxial loading paths. Here, only the shear energy component on the weighted critical plane (1st plane) is presented because it equals the shear energy component on the orthogonal plane (2nd plane). The shear strain energy on the 1st plane accounts for approximately 80% of the total energy in the proportional paths. However, for the 90° out-of-phase loading paths, the percentage of normal strain energy on the 2nd plane increases significantly. In particular, under the proportional loading paths, when λ equals 1, it accounts for more than 60% of the total strain energy; when λ equals 3 , the normal energy comprises nearly half of the total energy, which is slightly lower than the former energy. The proportions of normal strain energy on the 1st plane under proportional paths (No.1, No.5, No.9 and No.13) are 11.6%, 9.8%, 9.7% and 9.7%, respectively. The proportions of normal strain energy on the 1st plane under proportional paths (No.2, No.6, No.10 and No.14) equal 7.3%. The proportions of normal strain energy on the 1st plane under nonproportional paths (No.4, No.8, No.11 and No.15) are 8.1%, 16.7%, 5.5% and 6.8%, respectively. The proportions of normal strain energy on the 1st plane under non-proportional paths (No.3, No.7, No.12 and No.16) are 1.8%, 0.1%, 1.1% and 0.2%, respectively. Thus, it is suitable to predict the life using Eq. (21) with three strain energy. Particularly, Fig. 12 also shows that the normal strain energy on the 1st plane makes little contribution on the fatigue damage of the averaged critical plane under the non-proportional paths as Rε equals −1. The normal strain energy on the 1st plane can be removed in Eq. (21) for life prediction of Q235B steel welded material under No.3, No.7, No.12 and No.16 loading paths. The correlation between the FP calculated by Eq. (21) and experimental fatigue life when supposing that S equals 0.8 is presented in Fig. 13, which shows that the proposed FP derived from the weighted critical and orthogonal planes possesses an ideal correlation with the fatigue life of Q235B steel welded material. This relationship

Fig. 10. Strain and stress Mohr’s circle under uniaxial load.

fully reversed uniaxial loading, and it can be expressed as Eq. (13). max

= (1 + ve )

2 n

no

=

2

(2Nf ) b + (1 + vp ) f (2Nf ) c

(1

=

2

f

E

ve )

f

2

E

(1

(2Nf ) b +

(12)

vp )

(2Nf ) c

f

2

(13)

The corresponding normal stress range (Δσn, Δσno) and shear stress range Δτn on the two planes are the same and can be calculated using Eq. (14) (Fig. 10(b)). n

=

no

=

n

=

(14)

2

Additionally, the maximum stress components on each plane (σno,max, σn,max, τmax) are presented in Eqs. (15)–(17).

n,max

max

no

=

no,max

n

=

=

(15)

2

(16)

2 n

(17)

2

Thus, each term of the fatigue parameter can be expressed as Eqs. (18)–(20). 2

max

2 n

2 no

2

max

=

n,max

=

no,max

(1 + vp ) (1 + ve ) f (2Nf )2b + 2 E 2

(1

=

ve ) 4

f

2

(2Nf ) 2b +

E

(1

ve ) 4

f

(1

2

E

(2Nf )2b +

f

f

(2Nf ) b+ c

(18)

f

f

(2Nf )b + c

(19)

vp ) 4

(1

vp ) 4

f

f

(2Nf ) b+ c

(20)

Substituting Eqs. (18)–(20) into Eq. (6), the fatigue life prediction model is presented as Eq. (21). 427

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Fig. 11. FP component versus Experiment life.

Fig. 13. Experiment life versus Proposed FP. Fig. 12. Distributed percentage of FP.

torsional data is obtained. Comparisons of the predicted fatigue lives using the SWT [22], KBM [23], FS [24] models (corresponding to Eqs. (22), (23) and (24)) and the proposed model in this paper (Eq. (21)) are shown in Fig. 14. The solid and dashed lines in Fig. 14 represent error factors of 3 and 2 in the predictions, respectively.

demonstrates that the proposed FP can reasonably reflect the fatigue damage mechanism under multiple loading paths. 4.3. Fatigue life prediction Notably, when making life predictions, the proposed FP is determined on the weighed critical plane under 90° out-of-phase loading conditions, but it still relies on the traditional critical plane method for in-phase conditions. Furthermore, the material-based constant S could be obtained by iterating the uniaxial or torsional data that best correlate with the corresponding fatigue life. Here, S is set as 0.8 for the Q235B steel welded material because the best correlation with the

n,max

1

2

=

f

2

E

(2Nf ) 2b +

f f

(2Nf ) b+ c

(22)

where σn,max is the maximum normal stress on critical plane. Δε1 is the maximum normal strain range.

428

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max

2

+S

n

= [1 +

e

+ (1

e )S

]

f

E

(2Nf )b + [1 +

p

+ (1

p)S

experimental life are presented in Fig. 14(c). Most data points fall within an error factor of 3. Although the accuracy of non-proportional data is improved, the FS fatigue damage parameter is not sensitive enough to distinguish torsional and multiple multiaxial loading paths. The fatigue lives predicted using Eq. (21) are presented in Fig. 14(d). The proposed energy-based FP provides a very good correlation for the Q235B steel welded material under the torsional, proportional, and non-proportional loading paths. The error criterion under different levels E(s) has been introduced and expressed by Eq. (25) to further examine the prediction accuracy of the above four models [37]:

] f (2Nf )c

(23) *

where Δεn is the normal strain on critical plane. S is material-based constant and set as 1 for the calculation based on Ref. [13]. max

2

1+k

n,max

fy

=

(1 +

e)

f

E

(2Nf ) b + (1 +

p) f

(2Nf )c

1+k

f

2f y

(2Nf )b

(24) where fy is yield strength. k is material-based constant and set as 1 for the calculation based upon Ref. [13]. Fig. 14(a) shows the fatigue life predicted using the SWT model and the corresponding experimental life. The SWT parameter gives conservative results under proportional loading conditions but overestimates the fatigue life for both torsional and non-proportional conditions. The evaluated points, excluding several torsional and nonproportional data, are within an error factor of 3. This result also indicates that the SWT model, mainly on tensile-type failure, is not appropriate for predicting Q235B steel welded material under torsional and non-proportional conditions. The relationship between the fatigue life predicted using the KBM model and the experimental fatigue life is shown in Fig. 14(b). Most of the evaluated points are distributed within a scattered band of 2, and the rest, other than the individual data under the torsional loading, fall into the error factor of 3. The KBM model, mainly for shear-type failure, better predicts life than SWT for Q235B steel welded material. However, several conservative data exist for the non-proportional path (λ = 3 ), and the overestimated data appear for both the non-proportional path (λ = 1) and the torsional path. The fatigue life predicted using the FS model and the corresponding

E (s ) =

Number of data falling within

1 s

Number of total data

Np Nt

s (25)

where Np and Nt are predicted and experimental lives. The constant s refers to the error factor. The comparisons of error statistics with life prediction models are listed in Table 8. A higher ratio of predicted data falling within the low-error scatter band indicates a higher accuracy of the prediction model. The results show that 73.1% of the torsional and proportional experimental data fall within a scatter band of 2 using the KBM model. The error percentage calculated by other models is slightly lower for the same loading conditions. For the non-proportional loading paths, the percentage of data falling within a scatter band of 2 reaches the maximum value, i.e., 100%, by the proposed model in this paper. When the error factor is 3, E(s) is the highest and equals 92.3% for the torsional and proportional data and 100% for the non-proportional data using proposed model mentioned above. Therefore, the proposed energy-based model is a reliable and accurate approach to fatigue life prediction for Q235B steel welded material.

Fig. 14. Relationship between experiment life and predicted life using prediction models. 429

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Table 8 Comparison of error for the fatigue life prediction. Error

E (2) E (3)

Torsional and proportional loading paths

Non-proportional loading paths

SWT model

KBM model

FS model

Proposed model

SWT model

KBM model

FS model

Proposed model

0.615 0.692

0.731 0.769

0.654 0.769

0.692 0.923

0.250 0.750

0.563 0.938

0.750 0.813

1.000 1.000

5. Conclusion Torsional and multiaxial fatigue experiments are conducted to investigate the fatigue behavior of the Q235B steel welded material. Additionally, the fatigue life is predicted using the proposed energybased method. Several conclusions can be summarized as follows.

[11]

(1) A remarkable additional hardening effect exists in the Q235B steel welded material under the non-proportional loading paths. The hardening extent is in the following order: circular path > square path > elliptic path > rhombic path > torsional path > proportional path. The hysteresis loops of the non-proportional paths rotate to some extent, deviating from the loops of the proportional paths. (2) A new energy-based FP is proposed that incorporates the shear strain energy from the weighted critical plane and the normal strain energy from the weighted critical plane and orthogonal plane. The weighted critical plane is introduced to consider the shear strain contributions on the instantaneous maximum shear strain planes. The proposed FP can effectively reflect the influence of fatigue damage induced by multiple multiaxial loadings. (3) The proposed energy-based FP provides a superior life prediction when compared to other criteria for the Q235B steel welded material under the torsional, proportional, and non-proportional loading paths.

[14]

Acknowledgements

[25]

[12] [13]

[15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

Financial support for this study from the National Natural Science Foundation of China (Nos. 51525803 and 51178307) is gratefully acknowledged. The authors are also sincere thankful for Prof X. Chen of School of Chemical Engineering & Technology in Tianjin University for kindly providing fatigue test facilities and valuable suggestions.

[26] [27] [28]

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[29]

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