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Strain distribution and fatigue life estimation for steel plate weld joint low cycle fatigue based on DIC
Optics and Lasers in Engineering 124 (2020) 105839
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Strain distribution and fatigue life estimation for steel plate weld joint low cycle fatigue based on DIC Xiangyun Ren, Xiangyang Xu, Congxiao Jiang, Zhen Huang, Xiaoyuan He∗ Jiangsu Key Laboratory of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing 211189, PR China
a r t i c l e Keywords: Steel plate weld joint Fatigue life prediction Strain distribution DIC
i n f o
a b s t r a c t Welded steel plates are widely used in several engineering applications. Usually, the material properties of the parent material are prior to the welded structure, which frequently causes the welding joint to fail due to fatigue. Predicting the fatigue life of a material is not a simple task, especially, when considering the welding joint with various constituents. In this study, a strain distribution for a steel plate weld joint was investigated using the digital image correlation technology to estimate the low cycle fatigue life. Using the reconstructed strain, a nonlinear model of strain was presented. Moreover, the rate change of strain was used as a fatigue failure criterion to predict the lifetime. The proposed model considers the strain-rules from two aspects: maximum value and change rate. With the information of the initial strain value, we can achieve the desired prediction for the fatigue life of the steel plate weld joint under the low cycle fatigue loading.
1. Introduction Welding is a widely utilized method of joining steel structures. Estimating the fatigue life of a weld joint is a difficult task, and requires the development and association of several factors. The global strength of any weld depends on the properties across the zones affected by the welding operation [1]. Although different operations induce different defects that often include slag inclusions, incomplete fusion, and gas pores Lee et al. [2], the fatigue damage research has found that damage accumulation depends on loading parameters, such as the stress ratio, mean stress, and loading sequence [3]. Furthermore, with the development of robot technology and welding skills, the automatic welding technology using laser-arc hybrid or CO2 has been developed with high welding speed, precise heat input, and low accumulated stress, thereby increasing the welding stability and quality [4]. However, under this condition, the traditional method design and prediction of welded structures based on the nominal stress or hot spot stress approach with a series of classified weld S-N curves [5,6] is still slightly effective. This is because no precise relations were established to determine constitutive relations [7]. Besides, Lockwood et al. proved that it is still difficult to describe the stress–strain due to the limitations of the isostress load assumption in mechanical characterization [8]. The constitutive relations of the weld are difficult to establish not only because of the steep microstructural gradients and thickness variations, but also due to the inhomogeneity of the weld joint [9]. Consequently, many new technolo-
∗
gies and solutions have been researched to improve the life prediction accuracy. The hardness test is widely used as an essential determination method of mechanical properties. Some researchers mapped the hardness of steel spot weld in the base metal (BM) zone, fusion zone (FZ), and heat-affected zone (HAZ). The hardness values were measured at the intervals of 0.25 mm in the transverse direction to establish the hardness profiles [10]. Cheng et al. identified the limitations and possibilities of deducing stress–strain relations from load–displacement curves through the indentation in elastic–plastic solids [11]. This provided an accuracy relation between the local mechanical properties and welds; however, it was complicated and difficult to apply in the engineering case. Additionally, the traditional Paris law and Manson-Coffin statistics theory are still applicable, especially the statistics method. Meggiolaro et al. combined with the Manson-Coffin and extensive statistical data presented a new estimation method [12]. With the support of 845 different metals’ tensile and fatigue data, this approach obtained good estimation results; however, it was limited by the large data. Meanwhile, as McClintock, Rice, and Tracey clarified the key role of microvoids [13] and the development of continuum mechanics, the nonlinear damage models [14] were still the main methods used to estimate the fatigue life. Fierro et al. proposed a nonlinear ultrasound modulation method to estimate the residual fatigue life, in which an ultrasonic probe signal was generated by modulating two optimized waves coupled to analytical models to find the relationship [15]. Given the effect caused by the low-amplitude loading cycles below the fatigue limit on the damage accumulation, Zhang et al. modified the damage model and
Corresponding author. E-mail addresses: [email protected] (X. Ren), [email protected] (X. He).
https://doi.org/10.1016/j.optlaseng.2019.105839 Received 11 June 2019; Received in revised form 19 August 2019; Accepted 19 August 2019 0143-8166/© 2019 Published by Elsevier Ltd.
X. Ren, X. Xu and C. Jiang et al.
Optics and Lasers in Engineering 124 (2020) 105839
Table 1 Parameters of specimens. Plate thickness
Weld width
gripping end distance
7 mm
8 mm
80 mm
obtained a satisfying precision in life prediction [16]. The nonlinear damage models [17–19] were proposed to correlate the damage accumulation with loading parameters. Therefore, it was important to select the loading parameters to construct the models. Consequently, in this study, the full-field strain distribution of a steel plate weld joint was investigated using the digital image correlation (DIC) method to construct the strain models and estimate the fatigue life. The DIC technology was one of the non-contact optical measuring methods used to acquire full-field displacement and strain [20], which had been widely employed in many engineering projects [21]. Leitão et al. researched the local constitutive properties of aluminum friction stir welds and acquired the strain value using the DIC technology [1]. He et al. investigated the tensile and fatigue behaviors in a laser-arc hybrid welded aluminum alloy joint using the DIC technology [22]. Corigliano et al. analyzed the marine welded joints by means of DIC and IR images during the static and fatigue tests [23]. The strain tracking and distribution measured using the DIC technology not only enable the estimation of the low-cycle fatigue and ductile fracture [24] using the strain energy [25–30] or space average of strain criteria [31,32], but also provide a true value of the loading parameter to modify the model. In addition, a new strain-model was proposed and proved by tracking the rate change and distribution of the strain combined with the nonlinear damage theory [33]. A criterion as the rate change of strain was presented to predict the life of low cycle fatigue of the welded steel plate.
the motion and deformed information were stored between adjacent photos. The full-field strain of the weld was calculated using the DIC system. Due to the large deformation, the strain gauge system was only correlated and compared to the data obtained from the DIC system, which avoided the limitation of a single system and improved the reliability of the DIC system. Fig. 5 shows the data comparison of strain gauge NO.2 and the DIC system under 0.05 Hz loading. When analyzing the deformation, these images were divided into subsets containing a group of pixels of size M × M, with each pixel having a gray intensity ranging from 0 to 255. Furthermore, the cross correlation function was chosen to evaluate this correlation of the images, and was defined as follows: ( ) ⎡ 𝑀 ∑ 𝑓 𝑥𝑖 , 𝑦𝑗 − 𝑓𝑚 ⎢ ⎢ √∑ ( ( ) )2 ∑𝑀 𝑀 𝑥=−𝑀 𝑦=−𝑀 ⎢ 𝑥=−𝑀 𝑦=−𝑀 𝑓 𝑥𝑖 , 𝑦𝑗 − 𝑓𝑚 ⎣ 𝑀 ∑
F (𝐩 ) =
( ) 𝑔 𝑥′𝑖 , 𝑦′𝑗 − 𝑔𝑚
where: f(x, y) is the gray level value of the reference image, g(x′, y′) is the gray level value of the deformed image, 𝑀 𝑀 ∑ ∑ 1 𝑓𝑚 = 𝑓 (𝑥, 𝑦) is the average gray level value of the 2 (2𝑀+1) 𝑥=−𝑀 𝑦−𝑀
reference image, and 𝑀 𝑀 ∑ ∑ 1 𝑔𝑚 = 𝑔(𝑥′ , 𝑦′ ) is the average gray level value of the 2 (2𝑀+1) 𝑥=−𝑀 𝑦−𝑀
deformed image.
2. Specimens
4. Results and discussion
CO2 welding was run along with the butt between two Q345B steel plates, which formed the specimens. Considering the weld joint and stability of the welding process, it was proposed to weld two large steel plates first, and then cut out the required components, ensuring the consistency of each group of components. Fig. 1a and b show the manufacturing process. Welding guaranteed the perpendicularity through the given base line. The required components were then cut, and each of the cutting sides was grinded. Next, the samples were annealed at 500 °C for an extension of 3 h to eliminate the residual stresses generated during the cutting process. The specimens are shown in Fig. 2, and Table 1 illuminates the detailed sizes. The 50 t MTS fatigue machine was used for this experiment. The triangular wave control was used at a frequency of 0.1 Hz to ensure that the strain or stress was consistent with the time waveform throughout the whole process.
4.1. Static tests
3. Measurement system Two sets of measurement systems were applied on the weld joint, strain gauges system and DIC system, installed on both sides of the plate. The DIC system worked throughout the whole loading process until the specimen broke to acquire the strain distribution of the whole field. Meanwhile, the strain gauges system kept running and collecting data that contrasted with DIC results. Fig. 3 shows the distribution of strain gauges. Six strain gauges were attached to the surface of the specimen. Four were distributed along the weld direction, and the other two were distributed along the loading direction. On the other side of the plate, a speckle was created using black and white spray paints. It was important to make sure that the speckle patterns were random and had an appropriate gray distribution. Two cameras and a lamp were set in front of the plate, as shown in Fig. 4, capturing and recording the changes in the speckle patterns. Thereby,
2
⎤ ⎥ ⎥ −√ ( ( ) ) 2⎥ ∑𝑀 ∑𝑀 ′ ′ ⎥ 𝑥=−𝑀 𝑦=−𝑀 𝑔 𝑥𝑖 , 𝑦𝑗 − 𝑔𝑚 ⎦
The tensile test was applied to the weld in the transverse direction before the fatigue experiment was performed in the testing machine, operating at 25°C, to obtain the yield strength and displacement. Before testing, the specimens were prepared by applying a random black speckle pattern over the previous mat white painted surface of the transverse samples to enable data acquisition using DIC. The tensile curve of the steel plate welded joint was calculated as shown in Fig. 6. In addition, the corresponding local strain fields throughout the weld at the displacement of 1 mm are also presented in the inset. Based on the isostress configuration, the material behavior was described completely by local strain fields measured using the DIC; however, it was not accurate and was difficult to predict. On the contrary, the strain was the true value measured; hence, the mapping and analyses of the strain fields should be more significant for estimation. The local strains of three points located in the BM, FZ, and HAZ are extracted and plotted in Fig. 7. The constitutive relations of these three points are almost the same when the loading is below their limitation. Furthermore, it is evident that the amplified strain in the BM zone is much lower than others. Consequently, the failure location rarely occurred in this zone. Meanwhile, the FZ and HAZ zones demonstrate a large strain and good ductility, and the HAZ zone exhibits the highest strain because of the material softening. Therefore, it can be concluded that with the increase in loading, the maximum strain would generally act on the HAZ zone, causing the fracture to occur. 4.2. Fatigue tests The axial low cycle fatigue was conducted at 25 °C in the air on base. The MTS servohydraulic fatigue machine was used to determine the fa-
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Optics and Lasers in Engineering 124 (2020) 105839
Fig. 1. (a) Manufacturing of steel plates. (b) Heat treatment of steel plates.
Fig. 2. Details of specimens.
tigue behavior of the welded steel plate under displacement ∆u control. ∆u was obtained in tensile tests corresponding to yield strain. A triangular waveform was used as the input signal for all the tests. Tests were carried out under the ambient laboratory conditions with R = −1, where R is the proportion of minimum load to maximum load. To obtain more data to evaluate hysteresis loops and analyze the strain distribution, the frequency of the DIC system should be higher than the tests. For the above reasons, the test and DIC frequencies were 0.1 Hz and 2 Hz, respectively. A couple of CCD cameras of resolution 2048 × 2048 pixels were used to
record the un-deformed and deformed specimen surfaces during the test. The strain fields with a grid of 29 × 29 pixels (1 pixel = 0.06 mm) were calculated. When there was a distinct crack on the surface, which indicated that the limit of the sample was exceeded as a failure, the loading mechanism was stopped, and the loading process was completed. The values of the maximum and minimum loads versus the number of cycles carried out at displacement amplitude of 1.8 mm were recorded in the MTS machine, as shown in Fig. 8a. The hysteresis cycles were plotted, as shown in Fig. 8b.
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Optics and Lasers in Engineering 124 (2020) 105839
Fig. 3. Distribution of strain gauges. Fig. 4. MTS machine and measurement systems.
4.3. Strain results
Fig. 5. Strain values measured using the DIC and strain gauges.
The results of the strain measured using DIC are shown in Fig. 9. From the start to the end, the tensile process was plotted every 100 cycles. The three figures in each horizontal row in Fig. 9 form a group. Each group stands for the strain distribution in the weld under a tensile process. The first, second, and third figures in each group demonstrate the corresponding strain distributions when the values of extension are 0 mm, 0.9 mm, and 1.8 mm respectively. With the increase in the loading cycles, the maximum strain zone extended continuously over the steel weld joint zone. It was evident that in the weld not only most of the accumulated plastic strain contributed much more to the HAZ zone than the BM and FZ zones, but it also had a transfer effect throughout the loading process, which was similar to the transfer of the lattice in fracture mechanics. Besides, during the low cycle fatigue, this high plastic strain accumulation in the HAZ was responsible for the majority of failures in welded joints, strongly influencing the fatigue. Thereby, a characterization of the plastic deformation strain had a strong effect on the weld’s fatigue life. Therefore, based on the distribution of the measured strain, the whole lifetime was divided into four stages: the start, transfer, reconstruction, and failure. At the first stage, an initial strain existed because of many reasons such as materials and machining. Next, with the continual increase in loading cycles, the strain also increased continually and transferred along the transverse direction. When the transfer was over, the distribution of strain would reconstruct due to the incompatibility on the weld. Consequently, a surface crack occurred, which propagated until failure. The evolution process and characteristics of strain are shown in Fig. 10. 4.4. Fatigue evaluation
Fig. 6. Tensile curve of steel plate welded joint.
Based on the above-mentioned results, a relationship diagram between the calculation unit at the weld and the number of cycles loaded were plotted to discuss the process of strain. As shown in Fig. 11, the calculation unit at the weld (weld position) is the x-axis, the number of cycles loaded is the y-axis, and the strain value is the z-axis. The four stages defined were easy to distinguish, besides there was no difference
X. Ren, X. Xu and C. Jiang et al.
Optics and Lasers in Engineering 124 (2020) 105839
Fig. 7. Local strains of three points located in the BM, FZ, and HAZ under tensile test.
Fig. 8. (a) Maximum and minimum loads. (b) Hysteresis cycles.
in each position about the strain trend, indicating that the change in the strain in the steel plate weld joint under this condition was consistent during the whole period. Thereby, most of the characteristics of the weld joint were similar but differed in terms of rate. Furthermore, the analysis of the strain from the diagram at the x-z axis started with analyzing the range of value, and the continuity characteristics of the strain were investigated. It should be noted that although the strain varied continuously until the weld was broken, fluctuating the value dramatically and resulting in peak, it was still continuous. The peak could be considered as an extremum value for the broken characteristics, and the position of the steel plate weld joint is encircled in Fig. 11. The results revealed that the strain rule of the low cycle fatigue of the weld steel plate could be described using a type of continuous functions in which the extremum value represented the broken state as a fatigue failure criterion. Furthermore, changing the vision of the diagram to the y-z axis, as shown in Fig. 12, the trend of the measured strain value following the change in the control wave given by the MTS machine not only ex-
plained the validity of the DIC systems, but also illustrated a type of exponent rule that worked on the strain change. Due to the continuous character and change trend during the four states, the strain law of the low cycle fatigue of the weld steel plate was described as the following function with position x and cycles times t: 𝜀 = 𝐹 (𝑥, 𝑡) + 𝐷(𝑥, 𝑡) + 𝐶 (𝑥, 𝑡)
(1)
Furthermore, considering the consistency of the function, the maximum strain could be simplified as: 𝜀𝑚𝑎𝑥 = 𝐹 (𝑡) + 𝐷(𝑡) + 𝐶 (𝑡) 1 𝜆𝑒𝜆𝑡 𝐴
(2)
Here, 𝐹 (𝑡) = is defined as the configuration function that describes the trend of the strain change. 𝐷(𝑡) = 𝑎1 𝑡𝑛 + 𝑎2 𝑡𝑛−1 + … + 𝑎𝑛−1 𝑡 + 𝑎𝑛 is defined as the transfer function indicating the transfer of strain in the weld joint. The initial value of the strain after the first loading is illustrated using a constant C(t). A and ai (i=1,2…n) are the constant coefficients. Mathematically, 𝜆 stands for the change rate of function F. It can be observed from Fig. 9 that with the increase in loading cycles, the
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Optics and Lasers in Engineering 124 (2020) 105839
Fig. 9. Strain distribution during the tensile process in every 100 cycles.
Fig. 10. Four stages of fatigue process.
Fig. 11. Relation of the strain distribution with loading cycles and position.
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Optics and Lasers in Engineering 124 (2020) 105839
Fig. 12. Relation between strain and loading cycles.
Fig. 13. (a) Measured values of maximum strain (b) Fitting values of maximum strain.
maximum value of the strain is changing slightly over time; however, the maximum zone in the weld will not transfer in the transverse direction until a sudden maximum value burst occurs due to the failure. Thus, 𝜆 was used as an evaluation metric of the lifetime. The functions D and C represent the initial and transmission states, respectively. F is the description and synthesis of the whole process. In addition, as stated, the extremum value of function (2) was originated from fatigue failure. Fig. 13a shows the measured value of the maximum strain, and Fig. 13b shows the results given by function (2). Besides, based on the extremum value fatigue failure criterion, an estimation method of steel plate weld joint under low cycle fatigue was presented. However, with time, the weld was broken before the extremum value was obtained. On the other hand, considering that the strain changed sharply at the last moment, the position of the tangent with 45° angle of function (2) was used to predict the fatigue failure. It is as following. 𝑡=
1 𝐴 𝐼𝑛 𝜆 𝜆2
(3)
Moreover, compared to the classical low cycle fatigue Manson-Coffin formula Eq. (4), Eq. (3) can be rewritten as Eq. (5). Δ𝜀𝑝 2 (
𝐴 𝜆2
( )𝑐 = 𝜀′𝑓 2𝑁𝑓 (
)1
𝜆
=𝑒
(4) )1
Δ𝜀𝑝 2𝑐+1 𝜀𝑓 ′
𝑐
(5)
Here, Δɛp is the strain given by the experiment. 𝜀′𝑓 and c both are the fatigue plasticity coefficients, and Nf is the fatigue life. It was shown that 𝜆 was related to Δɛp , c, and 𝜀′𝑓 . Therefore, 𝜆 was defined as the strain plasticity ductile characteristic parameter of low cycle fatigue. Meantime, it was noticed that the initial value C must be determined by Δɛp , c and 𝜀′𝑓 , which means that the parameter value 𝜆 could be estimated for the fatigue life using only locally measurable values C. However, this mapping process was too complex between C and 𝜆 to provide an analytic expression. A sequence value of 𝜆 could be
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Optics and Lasers in Engineering 124 (2020) 105839
Fig. 14. Range of the predicted life and test life value for each sample.
calculated using a linear model with C as a sequence coefficient, and then a range of fatigue life could be estimated using Eq. (3). Under this condition, given the coefficient range of 0.14–0.16, the predicted results are shown in Fig. 14. The actual failure points are all located in the box, above and below the median of the estimated range, and are accurately predicted. Based on C, which could be measured conveniently in the initial state, the estimated method for low cycle fatigue of the weld steel plate was implemented easily and more accurately. 5. Conclusion In this study, a strain distribution for low cycle fatigue of a weld steel plate was proposed. As the parameter to quantify the fatigue life, the maximum strain change rate was evaluated using the derivative of strain model function. Based on the continuity and consistency of the strain and strain change rate, an exponent function was applied to describe the change rules. The process of the strain distribution was divided into four stages: the start, transfer, reconstruction, and failure. By analyzing the characters of the four phenomena, a criterion of strainrate was presented, and fatigue life was predicted using the tangent at 45° of the model function. The life prediction of the weld steel plate was completed based on the strain and strain change rate. Moreover, most of the predicted lifetimes were scattered well in the estimated box. Acknowledgments This research was funded by the National Natural Science Foundation of China (NSFC) 11532005 and 11827801. References [1] Leitão C, Galvão I, Leal RM, et al. Determination of local constitutive properties of aluminium friction stir welds using digital image correlation. Mater Des 2012;33:69–74. [2] Lee HK, Kim KS, Kim CM. Fracture resistance of a steel weld joint under fatigue loading. Eng Fract Mech 2000;66(4):403–19. [3] Fatemi A, Yang L. Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials. I J fatigue 1998;20(1):9–34.
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Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Strain distribution and fatigue life estimation for steel plate weld joint low cycle fatigue based on DIC Xiangyun Ren, Xiangyang Xu, Congxiao Jiang, Zhen Huang, Xiaoyuan He∗ Jiangsu Key Laboratory of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing 211189, PR China
a r t i c l e Keywords: Steel plate weld joint Fatigue life prediction Strain distribution DIC
i n f o
a b s t r a c t Welded steel plates are widely used in several engineering applications. Usually, the material properties of the parent material are prior to the welded structure, which frequently causes the welding joint to fail due to fatigue. Predicting the fatigue life of a material is not a simple task, especially, when considering the welding joint with various constituents. In this study, a strain distribution for a steel plate weld joint was investigated using the digital image correlation technology to estimate the low cycle fatigue life. Using the reconstructed strain, a nonlinear model of strain was presented. Moreover, the rate change of strain was used as a fatigue failure criterion to predict the lifetime. The proposed model considers the strain-rules from two aspects: maximum value and change rate. With the information of the initial strain value, we can achieve the desired prediction for the fatigue life of the steel plate weld joint under the low cycle fatigue loading.
1. Introduction Welding is a widely utilized method of joining steel structures. Estimating the fatigue life of a weld joint is a difficult task, and requires the development and association of several factors. The global strength of any weld depends on the properties across the zones affected by the welding operation [1]. Although different operations induce different defects that often include slag inclusions, incomplete fusion, and gas pores Lee et al. [2], the fatigue damage research has found that damage accumulation depends on loading parameters, such as the stress ratio, mean stress, and loading sequence [3]. Furthermore, with the development of robot technology and welding skills, the automatic welding technology using laser-arc hybrid or CO2 has been developed with high welding speed, precise heat input, and low accumulated stress, thereby increasing the welding stability and quality [4]. However, under this condition, the traditional method design and prediction of welded structures based on the nominal stress or hot spot stress approach with a series of classified weld S-N curves [5,6] is still slightly effective. This is because no precise relations were established to determine constitutive relations [7]. Besides, Lockwood et al. proved that it is still difficult to describe the stress–strain due to the limitations of the isostress load assumption in mechanical characterization [8]. The constitutive relations of the weld are difficult to establish not only because of the steep microstructural gradients and thickness variations, but also due to the inhomogeneity of the weld joint [9]. Consequently, many new technolo-
∗
gies and solutions have been researched to improve the life prediction accuracy. The hardness test is widely used as an essential determination method of mechanical properties. Some researchers mapped the hardness of steel spot weld in the base metal (BM) zone, fusion zone (FZ), and heat-affected zone (HAZ). The hardness values were measured at the intervals of 0.25 mm in the transverse direction to establish the hardness profiles [10]. Cheng et al. identified the limitations and possibilities of deducing stress–strain relations from load–displacement curves through the indentation in elastic–plastic solids [11]. This provided an accuracy relation between the local mechanical properties and welds; however, it was complicated and difficult to apply in the engineering case. Additionally, the traditional Paris law and Manson-Coffin statistics theory are still applicable, especially the statistics method. Meggiolaro et al. combined with the Manson-Coffin and extensive statistical data presented a new estimation method [12]. With the support of 845 different metals’ tensile and fatigue data, this approach obtained good estimation results; however, it was limited by the large data. Meanwhile, as McClintock, Rice, and Tracey clarified the key role of microvoids [13] and the development of continuum mechanics, the nonlinear damage models [14] were still the main methods used to estimate the fatigue life. Fierro et al. proposed a nonlinear ultrasound modulation method to estimate the residual fatigue life, in which an ultrasonic probe signal was generated by modulating two optimized waves coupled to analytical models to find the relationship [15]. Given the effect caused by the low-amplitude loading cycles below the fatigue limit on the damage accumulation, Zhang et al. modified the damage model and
Corresponding author. E-mail addresses: [email protected] (X. Ren), [email protected] (X. He).
https://doi.org/10.1016/j.optlaseng.2019.105839 Received 11 June 2019; Received in revised form 19 August 2019; Accepted 19 August 2019 0143-8166/© 2019 Published by Elsevier Ltd.
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Optics and Lasers in Engineering 124 (2020) 105839
Table 1 Parameters of specimens. Plate thickness
Weld width
gripping end distance
7 mm
8 mm
80 mm
obtained a satisfying precision in life prediction [16]. The nonlinear damage models [17–19] were proposed to correlate the damage accumulation with loading parameters. Therefore, it was important to select the loading parameters to construct the models. Consequently, in this study, the full-field strain distribution of a steel plate weld joint was investigated using the digital image correlation (DIC) method to construct the strain models and estimate the fatigue life. The DIC technology was one of the non-contact optical measuring methods used to acquire full-field displacement and strain [20], which had been widely employed in many engineering projects [21]. Leitão et al. researched the local constitutive properties of aluminum friction stir welds and acquired the strain value using the DIC technology [1]. He et al. investigated the tensile and fatigue behaviors in a laser-arc hybrid welded aluminum alloy joint using the DIC technology [22]. Corigliano et al. analyzed the marine welded joints by means of DIC and IR images during the static and fatigue tests [23]. The strain tracking and distribution measured using the DIC technology not only enable the estimation of the low-cycle fatigue and ductile fracture [24] using the strain energy [25–30] or space average of strain criteria [31,32], but also provide a true value of the loading parameter to modify the model. In addition, a new strain-model was proposed and proved by tracking the rate change and distribution of the strain combined with the nonlinear damage theory [33]. A criterion as the rate change of strain was presented to predict the life of low cycle fatigue of the welded steel plate.
the motion and deformed information were stored between adjacent photos. The full-field strain of the weld was calculated using the DIC system. Due to the large deformation, the strain gauge system was only correlated and compared to the data obtained from the DIC system, which avoided the limitation of a single system and improved the reliability of the DIC system. Fig. 5 shows the data comparison of strain gauge NO.2 and the DIC system under 0.05 Hz loading. When analyzing the deformation, these images were divided into subsets containing a group of pixels of size M × M, with each pixel having a gray intensity ranging from 0 to 255. Furthermore, the cross correlation function was chosen to evaluate this correlation of the images, and was defined as follows: ( ) ⎡ 𝑀 ∑ 𝑓 𝑥𝑖 , 𝑦𝑗 − 𝑓𝑚 ⎢ ⎢ √∑ ( ( ) )2 ∑𝑀 𝑀 𝑥=−𝑀 𝑦=−𝑀 ⎢ 𝑥=−𝑀 𝑦=−𝑀 𝑓 𝑥𝑖 , 𝑦𝑗 − 𝑓𝑚 ⎣ 𝑀 ∑
F (𝐩 ) =
( ) 𝑔 𝑥′𝑖 , 𝑦′𝑗 − 𝑔𝑚
where: f(x, y) is the gray level value of the reference image, g(x′, y′) is the gray level value of the deformed image, 𝑀 𝑀 ∑ ∑ 1 𝑓𝑚 = 𝑓 (𝑥, 𝑦) is the average gray level value of the 2 (2𝑀+1) 𝑥=−𝑀 𝑦−𝑀
reference image, and 𝑀 𝑀 ∑ ∑ 1 𝑔𝑚 = 𝑔(𝑥′ , 𝑦′ ) is the average gray level value of the 2 (2𝑀+1) 𝑥=−𝑀 𝑦−𝑀
deformed image.
2. Specimens
4. Results and discussion
CO2 welding was run along with the butt between two Q345B steel plates, which formed the specimens. Considering the weld joint and stability of the welding process, it was proposed to weld two large steel plates first, and then cut out the required components, ensuring the consistency of each group of components. Fig. 1a and b show the manufacturing process. Welding guaranteed the perpendicularity through the given base line. The required components were then cut, and each of the cutting sides was grinded. Next, the samples were annealed at 500 °C for an extension of 3 h to eliminate the residual stresses generated during the cutting process. The specimens are shown in Fig. 2, and Table 1 illuminates the detailed sizes. The 50 t MTS fatigue machine was used for this experiment. The triangular wave control was used at a frequency of 0.1 Hz to ensure that the strain or stress was consistent with the time waveform throughout the whole process.
4.1. Static tests
3. Measurement system Two sets of measurement systems were applied on the weld joint, strain gauges system and DIC system, installed on both sides of the plate. The DIC system worked throughout the whole loading process until the specimen broke to acquire the strain distribution of the whole field. Meanwhile, the strain gauges system kept running and collecting data that contrasted with DIC results. Fig. 3 shows the distribution of strain gauges. Six strain gauges were attached to the surface of the specimen. Four were distributed along the weld direction, and the other two were distributed along the loading direction. On the other side of the plate, a speckle was created using black and white spray paints. It was important to make sure that the speckle patterns were random and had an appropriate gray distribution. Two cameras and a lamp were set in front of the plate, as shown in Fig. 4, capturing and recording the changes in the speckle patterns. Thereby,
2
⎤ ⎥ ⎥ −√ ( ( ) ) 2⎥ ∑𝑀 ∑𝑀 ′ ′ ⎥ 𝑥=−𝑀 𝑦=−𝑀 𝑔 𝑥𝑖 , 𝑦𝑗 − 𝑔𝑚 ⎦
The tensile test was applied to the weld in the transverse direction before the fatigue experiment was performed in the testing machine, operating at 25°C, to obtain the yield strength and displacement. Before testing, the specimens were prepared by applying a random black speckle pattern over the previous mat white painted surface of the transverse samples to enable data acquisition using DIC. The tensile curve of the steel plate welded joint was calculated as shown in Fig. 6. In addition, the corresponding local strain fields throughout the weld at the displacement of 1 mm are also presented in the inset. Based on the isostress configuration, the material behavior was described completely by local strain fields measured using the DIC; however, it was not accurate and was difficult to predict. On the contrary, the strain was the true value measured; hence, the mapping and analyses of the strain fields should be more significant for estimation. The local strains of three points located in the BM, FZ, and HAZ are extracted and plotted in Fig. 7. The constitutive relations of these three points are almost the same when the loading is below their limitation. Furthermore, it is evident that the amplified strain in the BM zone is much lower than others. Consequently, the failure location rarely occurred in this zone. Meanwhile, the FZ and HAZ zones demonstrate a large strain and good ductility, and the HAZ zone exhibits the highest strain because of the material softening. Therefore, it can be concluded that with the increase in loading, the maximum strain would generally act on the HAZ zone, causing the fracture to occur. 4.2. Fatigue tests The axial low cycle fatigue was conducted at 25 °C in the air on base. The MTS servohydraulic fatigue machine was used to determine the fa-
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Optics and Lasers in Engineering 124 (2020) 105839
Fig. 1. (a) Manufacturing of steel plates. (b) Heat treatment of steel plates.
Fig. 2. Details of specimens.
tigue behavior of the welded steel plate under displacement ∆u control. ∆u was obtained in tensile tests corresponding to yield strain. A triangular waveform was used as the input signal for all the tests. Tests were carried out under the ambient laboratory conditions with R = −1, where R is the proportion of minimum load to maximum load. To obtain more data to evaluate hysteresis loops and analyze the strain distribution, the frequency of the DIC system should be higher than the tests. For the above reasons, the test and DIC frequencies were 0.1 Hz and 2 Hz, respectively. A couple of CCD cameras of resolution 2048 × 2048 pixels were used to
record the un-deformed and deformed specimen surfaces during the test. The strain fields with a grid of 29 × 29 pixels (1 pixel = 0.06 mm) were calculated. When there was a distinct crack on the surface, which indicated that the limit of the sample was exceeded as a failure, the loading mechanism was stopped, and the loading process was completed. The values of the maximum and minimum loads versus the number of cycles carried out at displacement amplitude of 1.8 mm were recorded in the MTS machine, as shown in Fig. 8a. The hysteresis cycles were plotted, as shown in Fig. 8b.
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Fig. 3. Distribution of strain gauges. Fig. 4. MTS machine and measurement systems.
4.3. Strain results
Fig. 5. Strain values measured using the DIC and strain gauges.
The results of the strain measured using DIC are shown in Fig. 9. From the start to the end, the tensile process was plotted every 100 cycles. The three figures in each horizontal row in Fig. 9 form a group. Each group stands for the strain distribution in the weld under a tensile process. The first, second, and third figures in each group demonstrate the corresponding strain distributions when the values of extension are 0 mm, 0.9 mm, and 1.8 mm respectively. With the increase in the loading cycles, the maximum strain zone extended continuously over the steel weld joint zone. It was evident that in the weld not only most of the accumulated plastic strain contributed much more to the HAZ zone than the BM and FZ zones, but it also had a transfer effect throughout the loading process, which was similar to the transfer of the lattice in fracture mechanics. Besides, during the low cycle fatigue, this high plastic strain accumulation in the HAZ was responsible for the majority of failures in welded joints, strongly influencing the fatigue. Thereby, a characterization of the plastic deformation strain had a strong effect on the weld’s fatigue life. Therefore, based on the distribution of the measured strain, the whole lifetime was divided into four stages: the start, transfer, reconstruction, and failure. At the first stage, an initial strain existed because of many reasons such as materials and machining. Next, with the continual increase in loading cycles, the strain also increased continually and transferred along the transverse direction. When the transfer was over, the distribution of strain would reconstruct due to the incompatibility on the weld. Consequently, a surface crack occurred, which propagated until failure. The evolution process and characteristics of strain are shown in Fig. 10. 4.4. Fatigue evaluation
Fig. 6. Tensile curve of steel plate welded joint.
Based on the above-mentioned results, a relationship diagram between the calculation unit at the weld and the number of cycles loaded were plotted to discuss the process of strain. As shown in Fig. 11, the calculation unit at the weld (weld position) is the x-axis, the number of cycles loaded is the y-axis, and the strain value is the z-axis. The four stages defined were easy to distinguish, besides there was no difference
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Optics and Lasers in Engineering 124 (2020) 105839
Fig. 7. Local strains of three points located in the BM, FZ, and HAZ under tensile test.
Fig. 8. (a) Maximum and minimum loads. (b) Hysteresis cycles.
in each position about the strain trend, indicating that the change in the strain in the steel plate weld joint under this condition was consistent during the whole period. Thereby, most of the characteristics of the weld joint were similar but differed in terms of rate. Furthermore, the analysis of the strain from the diagram at the x-z axis started with analyzing the range of value, and the continuity characteristics of the strain were investigated. It should be noted that although the strain varied continuously until the weld was broken, fluctuating the value dramatically and resulting in peak, it was still continuous. The peak could be considered as an extremum value for the broken characteristics, and the position of the steel plate weld joint is encircled in Fig. 11. The results revealed that the strain rule of the low cycle fatigue of the weld steel plate could be described using a type of continuous functions in which the extremum value represented the broken state as a fatigue failure criterion. Furthermore, changing the vision of the diagram to the y-z axis, as shown in Fig. 12, the trend of the measured strain value following the change in the control wave given by the MTS machine not only ex-
plained the validity of the DIC systems, but also illustrated a type of exponent rule that worked on the strain change. Due to the continuous character and change trend during the four states, the strain law of the low cycle fatigue of the weld steel plate was described as the following function with position x and cycles times t: 𝜀 = 𝐹 (𝑥, 𝑡) + 𝐷(𝑥, 𝑡) + 𝐶 (𝑥, 𝑡)
(1)
Furthermore, considering the consistency of the function, the maximum strain could be simplified as: 𝜀𝑚𝑎𝑥 = 𝐹 (𝑡) + 𝐷(𝑡) + 𝐶 (𝑡) 1 𝜆𝑒𝜆𝑡 𝐴
(2)
Here, 𝐹 (𝑡) = is defined as the configuration function that describes the trend of the strain change. 𝐷(𝑡) = 𝑎1 𝑡𝑛 + 𝑎2 𝑡𝑛−1 + … + 𝑎𝑛−1 𝑡 + 𝑎𝑛 is defined as the transfer function indicating the transfer of strain in the weld joint. The initial value of the strain after the first loading is illustrated using a constant C(t). A and ai (i=1,2…n) are the constant coefficients. Mathematically, 𝜆 stands for the change rate of function F. It can be observed from Fig. 9 that with the increase in loading cycles, the
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Optics and Lasers in Engineering 124 (2020) 105839
Fig. 9. Strain distribution during the tensile process in every 100 cycles.
Fig. 10. Four stages of fatigue process.
Fig. 11. Relation of the strain distribution with loading cycles and position.
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Optics and Lasers in Engineering 124 (2020) 105839
Fig. 12. Relation between strain and loading cycles.
Fig. 13. (a) Measured values of maximum strain (b) Fitting values of maximum strain.
maximum value of the strain is changing slightly over time; however, the maximum zone in the weld will not transfer in the transverse direction until a sudden maximum value burst occurs due to the failure. Thus, 𝜆 was used as an evaluation metric of the lifetime. The functions D and C represent the initial and transmission states, respectively. F is the description and synthesis of the whole process. In addition, as stated, the extremum value of function (2) was originated from fatigue failure. Fig. 13a shows the measured value of the maximum strain, and Fig. 13b shows the results given by function (2). Besides, based on the extremum value fatigue failure criterion, an estimation method of steel plate weld joint under low cycle fatigue was presented. However, with time, the weld was broken before the extremum value was obtained. On the other hand, considering that the strain changed sharply at the last moment, the position of the tangent with 45° angle of function (2) was used to predict the fatigue failure. It is as following. 𝑡=
1 𝐴 𝐼𝑛 𝜆 𝜆2
(3)
Moreover, compared to the classical low cycle fatigue Manson-Coffin formula Eq. (4), Eq. (3) can be rewritten as Eq. (5). Δ𝜀𝑝 2 (
𝐴 𝜆2
( )𝑐 = 𝜀′𝑓 2𝑁𝑓 (
)1
𝜆
=𝑒
(4) )1
Δ𝜀𝑝 2𝑐+1 𝜀𝑓 ′
𝑐
(5)
Here, Δɛp is the strain given by the experiment. 𝜀′𝑓 and c both are the fatigue plasticity coefficients, and Nf is the fatigue life. It was shown that 𝜆 was related to Δɛp , c, and 𝜀′𝑓 . Therefore, 𝜆 was defined as the strain plasticity ductile characteristic parameter of low cycle fatigue. Meantime, it was noticed that the initial value C must be determined by Δɛp , c and 𝜀′𝑓 , which means that the parameter value 𝜆 could be estimated for the fatigue life using only locally measurable values C. However, this mapping process was too complex between C and 𝜆 to provide an analytic expression. A sequence value of 𝜆 could be
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Optics and Lasers in Engineering 124 (2020) 105839
Fig. 14. Range of the predicted life and test life value for each sample.
calculated using a linear model with C as a sequence coefficient, and then a range of fatigue life could be estimated using Eq. (3). Under this condition, given the coefficient range of 0.14–0.16, the predicted results are shown in Fig. 14. The actual failure points are all located in the box, above and below the median of the estimated range, and are accurately predicted. Based on C, which could be measured conveniently in the initial state, the estimated method for low cycle fatigue of the weld steel plate was implemented easily and more accurately. 5. Conclusion In this study, a strain distribution for low cycle fatigue of a weld steel plate was proposed. As the parameter to quantify the fatigue life, the maximum strain change rate was evaluated using the derivative of strain model function. Based on the continuity and consistency of the strain and strain change rate, an exponent function was applied to describe the change rules. The process of the strain distribution was divided into four stages: the start, transfer, reconstruction, and failure. By analyzing the characters of the four phenomena, a criterion of strainrate was presented, and fatigue life was predicted using the tangent at 45° of the model function. The life prediction of the weld steel plate was completed based on the strain and strain change rate. Moreover, most of the predicted lifetimes were scattered well in the estimated box. Acknowledgments This research was funded by the National Natural Science Foundation of China (NSFC) 11532005 and 11827801. References [1] Leitão C, Galvão I, Leal RM, et al. Determination of local constitutive properties of aluminium friction stir welds using digital image correlation. Mater Des 2012;33:69–74. [2] Lee HK, Kim KS, Kim CM. Fracture resistance of a steel weld joint under fatigue loading. Eng Fract Mech 2000;66(4):403–19. [3] Fatemi A, Yang L. Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials. I J fatigue 1998;20(1):9–34.
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