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The numerical and experimental analysis of ratcheting behavior for 304 steel sheets under cyclic axial loading

The numerical and experimental analysis of ratcheting behavior for 304 steel sheets under cyclic... The present study examines the effect of the thickness, cutouts numbers, and different cutouts shapes on the ratcheting behavior of 304 steel sheets under cyclic axial loading. The cutout shapes considered here are circular, triangular, and square cutouts. The effect of the number of circular cutouts in 304 steel sheets on ratcheting behavior is investigated. The Instron 8502 device is used to perform the experiments in which the cyclic axial loading is applied to six specimens with different cutouts and thicknesses at ambient temperature. The obtained results highlight the fact that reducing the thickness and increasing the number of cutouts in the sheets are attributed to the rise in the ratcheting displacement. Accordingly, the ratcheting displacement is more striking in the sheets with triangular cutouts. Notably, a numerical anal- ysis is considered using the isotropic/kinematic nonlinear hardening model and FEM in ABAQUS software. Since a good agreement is seen between the numerical and experimental results, the analysis conducted in this study is reliable in terms of accuracy and authenticity. Keywords Ratcheting, cyclic loading, FEM, SS304, ABAQUS Date received: 29 November 2022; accepted: 20 March 2023 Handling Editor: Chenhui Liang still considerable ambiguity regarding this topic because Introduction it is a secondary plastic deformation process and progres- 10–12 Due to the recent development in mechanical and civil sively occurs cycle by cycle. Thus, the study in this engineering, the structures and mechanical components field is of great importance in predicting the materials’ are significantly exposed to randomly variable loads lifetime under characteristic cyclic loading. 1–4 nowadays. On the other hand, the engineering struc- The study of ratcheting dates back many years tures like pressure vessels, piping systems, and offshore 13–18 ago, but it is still a challenging issue that needs to structures are constantly subjected to cyclic axial loading be studied in depth. The number of cycles that lead to or cyclic thermal loading. The stress cycles’ values applied to the structures are sometimes more than their yield 5–8 strength and bring unwanted damages. These stress Mechanical Engineering, Malayer University, Malayer, Hamadan Province, Iran cycles, along with the mean stress, create remarkable accumulations of plastic deformation, namely ratcheting, Corresponding author: and finally lead to the failure phenomenon. However, Ali Shahrjerdi, Mechanical Engineering, Malayer University, Arak - Malayer the previous decades have witnessed a growing interest in Hwy, Malayer, Hamadan Province 65719-9586, Iran. the study of ratcheting behavior in the material; there is Email: shahrjerdi.mail@gmail.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 Advances in Mechanical Engineering ratcheting can be significantly effective in terms of another study in this regard, the author did some intensification and amount of ratcheting strain. In experiments on cylindrical shells with different lengths 19 37 2003, Chen et al. aimed to simulate the ratcheting under combined and cyclic axial loading. According strain considering a high number of cycles subjected to to the results obtained in this study, a rise in the length biaxial loadings. This study has considered a corrected of cylindrical shells due to increasing flexural torque form of the Ohno–Wang kinematic hardening rule to and increasing vertical stresses at different sections of reach more accurate numerical results. The question oblique cylindrical shells with longer lengths leads to here arises of what factors are effective in ratcheting increasing ratcheting displacement. One of the signifi- behavior to what extent. In 2004, Weiß et al. ratchet- cant factors that have an essential factor on the ratchet- ing was introduced as a secondary phenomenon of cyc- ing behavior of the cylindrical sells is the diameter lic plasticity that intensifies fatigue damage. Also, investigated by Shariati et al. in 2016. In this study, deformation behaviors and microstructure evolution of the effect of this factor was experimented with by con- a hot-rolled AZ31B magnesium alloy were examined sidering two samples of cylindrical shells produced of 18 21 by Wang et al. After that, in 2021, Chen et al. stud- SS304L steel with different diameters and under cyclic ied the issue of cyclic lithium-ion diffusion-induced pure bending loading. The effect of cutout shape on the stress within the charging-discharging process. In this ratcheting angle on torsional loading on cylindrical study, the plastic behavior of lithium-ion battery cath- shells has also been studied by Shariati et al. ode has been systematically examined, and the effect of According to this research, the cylindrical shells of the cycle number has been analyzed. These studies SS316L with circular, square, and triangular cutouts mainly investigated the effect of cycle number on ratch- were subjected to cyclic torsional loading, and the eting behavior and did not address the effect of shape ratcheting angle diagram was obtained based on the dimensions. Notably, the geometrical dimensions and number of loading cycles. material are also important to reach reliable results. One of the industry’s most functional components is A large number of existing studies in the broader lit- a pipe that transfers liquid, gas, slurries, and other solids erature have examined the ratcheting phenomenon for and fluids from one area. Since these components are 22–28 various materials. Besides, the different geometries sometimes subjected to severe cyclic loadings, the study have been investigated to analyze the obtained results in this regard is also important. Concerning the ratchet- 39,40 so far, among which the cylindrical specimens have ing phenomenon in the pipes under cyclic loadings, 29,30 6 been widely employed. For instance, Kobayashi Zeinoddini et al. considered the ratcheting behavior in and Ohno examined the ratcheting behavior for the the low-alloy, high-strength steel pipes (API-5L X80) SS304 cylindrical shells and presented valuable results. subjected to cyclic bending. Additionally, Zakavi et al. Lee et al. also conducted an identical study for the assessed the combined stiffness model in the elbow cylindrical shells of SS316 in 2003. Hamidinejad and ratcheting behavior of pressurized pipe under in-plane Varvani-Farahani conducted an experimental study torque. The ratcheting phenomenon for the elbow under for 1045 and 1Cr18Ni9Ti steel in 2015. Shariati et al. pressurized bending after thermal aging was investigated investigated the ratcheting behavior of cantilever speci- experimentally and numerically by Liu et al. in 2019. mens of SS316L steel cylindrical shells under cyclic In another study, the uniaxial and biaxial ratcheting bending loads. The authors indicated that plastic defor- behavior of pressurized AISI 316L pipe under cyclic mation accumulates in the specimen after each cycle, loading was examined by Moslemi et al. and presented and as the cycles increase, the diagram loops become acceptable results. In addition, Kang and Karimi and 34 35 44 closer. Shariati et al. also conducted a numerical Shariati studied experimental and numerical analysis study in 2011 concerning the investigation of ratcheting of the ratcheting behavior of SS316L thin-walled pipes in cylindrical shells. The authors investigated the effect under cyclic internal pressure. There exists a consider- of frequency and thickness on ratcheting behavior and able body of literature on ratcheting behavior in other fatigue life of polystyrene pipe under cyclic axial load- geometries. The previous studies aimed to examine the ing. Zhu investigated the effect of increasing force effect of this phenomenon in a variety of geometries like 45,46 17,47 48 amplitude on the ratcheting behavior of cylindrical pipes, circular components, axial pipes, 49–51 shells produced of CK20 steel under cyclic loading, and sheets, and so on. To mention a few, Kolasangiani accumulation of ratcheting strain increases with et al. studied the ratcheting behavior of plates with a increasing stress amplitude at constant average stress. circular cutout under axial cyclic loading. The ratcheting It was also revealed that the increased average stress behavior of notched steel samples subjected to asym- could also be one of the factors affecting this behavior. metric loading cycles through coupled kinematic hard- 36 53 Shariati et al. investigated the effect of this factor on ening-Neuber rules was also examined. The authors SS316L steel and concluded that in the constant ampli- also investigated the ratcheting progress at the notch tude of force, with increasing average force, the displa- root of 1045 steel samples over asymmetric loading 54 55 cement accumulates, and its rate increases. Also, in cycles. Chen et al. also analyzed the effect of circular Shahrjerdi and SafariFard 3 holes on the ratchet limit and cracked tip plastic strain examined under cyclic axial loading with various cut- range in a center cracked plate. Dong et al. examined outs and dimensions. the low cycle fatigue and ratcheting failure behavior of The rest of this study is organized as follows: Section AH32 steel under uniaxial cyclic loading. Luo et al. 2 outlines the methods and material considered to con- conducted an experimental study on the heterogeneous duct the experiment and validate the results. The infor- ratcheting behavior of SUS301L stainless steel butt weld mation regarding the ratcheting concept and the issue joint during uniaxial cyclic loading. An experimental of hardening and softening is presented in the third sec- study was conducted on the uniaxial ratcheting-fatigue tion. The fourth section analyzes and discusses the interaction of polyamide-6 by Yang et al. Further, results obtained for the sheet specimens under cyclic 20 59 Weiß et al. and Liu et al. investigated ratcheting- axial loading. In addition, a careful comparison is made fatigue behavior and damage mechanism of GH4169 at between the numerical and experimental results in the 650C. The ratcheting fatigue behavior of 42CrMo4 fifth section to prove the authenticity and accuracy of steel under different heat treatment conditions was also the analysis conducted in this research. The main con- examined by Kreethi et al. in 2017. One of the most clusions obtained from the experiments and numerical important components in engineering is sheet metal due analysis, as well as the suggestions for future study, are to its remarkable capabilities in modern buildings, man- drawn in the sixth section. ufacturing, and construction sectors. In fact, sheet metal is widely employed in various industries such as car Preliminaries manufacturing, aircraft parts, tools, agriculture, mining, catering, shipping, medicine, and electronics. Thus, most Since nowadays many structures are under cyclic exter- early studies, as well as current work, focus on the ratch- nal forces which lead to fracture, the issue of ratcheting 50,61–63 eting behavior of the steel sheets, Z2CND18.12 is very important. These cyclic loads gradually increase 64 65 steel sheets, and molybdenum sheets. For example, the probability of failure due to the creation of more De et al. experimentally studied the ratcheting phe- strains by small values per cycle. This type of loading is nomenon on steel sheets in 2017. Concerning the results called cyclic plasticity or elasticity, a low-cycle fatigue obtained in this study, the plastic strain accumulation type. However, to observe the sample’s ratcheting phe- rate turned out to be high in the initial cycles and nomenon, the load needs to be in the form of control decreased in the subsequent cycles. Shahrjerdi et al. stress, and the amount of stress amplitude must be obtained the Bree diagram to specify the ratcheting and beyond its elastic domain. In general, the following non-ratcheting areas for a functionally graded beam conditions need to be applied to examine the ratcheting under cyclic thermal and axial loading. In light of recent behavior in the sample: events in the ratcheting phenomenon for the various material and geometries, there is now some considerable The sample is subjected to cyclic loading. concern about the influential factors affecting the inten- Plastic deformation occurs in each cycle. sification of this behavior. The mean stress is non-zero. The majority of studies mentioned above emphasize the effects of dimensions and cycle number of loading The plastic strain (average of each cycle) increases on the amount of ratcheting strain and the correspond- when the cycle rises. One of the significant effects of ing results. Besides, the importance of study in the field the ratcheting phenomenon on a component is that the of sheet ratcheting has been clarified. Despite such crack growth is observed earlier, and failure occurs ear- interest, we have yet to study the effect of sheets’ thick- lier in the ratcheting zone. For this reason, in designing ness on ratcheting the factors affecting this behavior to parts and industrial structures, it is essential to evaluate the best of our knowledge. Considering the widespread the ratcheting phenomenon as an important and effec- use of 304 steel sheets with different thicknesses in the tive factor for the failure phenomenon. industry, it is important to study their ratcheting beha- In general, three types of ratcheting can be observed vior considering the effect of thickness. On the other in material behavior. Accordingly, the first type is hand, examining the effect of the shape and number of related to the mitigation of the ratcheting rate, which cutouts on sheet ratcheting is essential because cutouts causes elastic/plastic shakedown. This result is mainly with different shapes are sometimes created in the obtained when the material is highly cyclic hardening. sheets for many purposes. A more detailed look at the The second type is related to the condition in which the literature reveals a number of gaps and shortcomings, strain rate is constant, which leads to the accumulation which are as follows. As a novelty, the effect of shape, of plastic strains in the material. The third type of cutout number, thickness, and dimension on 304 steel sheets are simultaneously discussed in the current ratcheting is associated with a rise in strain rate. Such paper. Also, the numerical simulation and experiments conditions are presented in Figure 1 to understand their regarding the ratcheting behavior of 304 steel sheets are effects entirely. 4 Advances in Mechanical Engineering with a diameter of 8 mm are created. It should be noted that there is an equal distance between the cutouts exist- ing in the center of the samples. Since the effect of the cutout shape on the ratcheting behavior is considerable, the SS304 sheet samples with circular, square, and tri- angular (equilateral) cutouts and the same area (about 50 mm ) are examined in the center of the samples. According to the prepared samples shown in Figures 3 and 4, the diameter of the circular cutout is 8 mm, and the lengths of the square and triangular cutouts are 7.09 and 10.75 mm, respectively. One specimen was tested for each case. However, two or three samples were Figure 1. The schematic of various types of ratcheting. tested before the primary analysis to reach a suitable loading. Furthermore, these samples’ geometric and loading Table 1. The mechanical properties of SS304. specifications are reported in Table 3. Since the loading Poisson’s Modulus Yield Ultimate is force-control and the ratcheting phenomenon of ratio elasticity (GPa) stress (MPa) stress (Mpa) SS304 is considered in this study, the applied force needs to result in ratcheting behavior and simultane- 0.33 196 260 798 ously prevent failing the specimens in the initial cycles. The stress-strain curve obtained by the standard tensile test is illustrated in Figure 4. The simple tensile test is uniaxial according to which Methodology and material s ¼ s; s ¼ s ¼ 0 and for the plastic state, 1 2 3 Overall, the present study consists of the experimental 1 1 e ¼ e; n ¼ ; e ¼ e ¼ ne ¼ e. The relationship and numerical analyses considered for the samples of 1 2 3 1 2 2 steel sheet 304. In this section, these methods are between the equivalent stress and equivalent strain is explained in detail, and the mechanical properties of s;e $ s ; e. Where the von Mises stress is : SS304 based on ASTM E8 are highlighted in Table 1. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Also, Table 2 shows the chemical properties of SS304 1 2 2 2 s  ¼ pffiffiffi ðÞ s  s +ðÞ s  s +ðÞ s  s ð1Þ 1 2 2 3 3 1 used in this study. Figure 2 gives the necessary information regarding For plane stress, the von Mises stress can be represented the process of the analyses conducted in this study. In by equation (2) : order to thoroughly understand the steps that need to be taken in this study, devoting attention to this figure pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 is of great importance. s  ¼ pffiffiffi s  s s + s ð2Þ 1 1 2 2 In the beginning, some specimens were considered for The experimental method the tensile test and stability to specify the same range of The experimental analysis is conducted through the load for applying the whole specimens. In fact, their Instron 8502 device that is able to apply dynamic loads capability to bring ratcheting results without failure in up to 250 kN, and the main experiments are performed the primary cycles is guaranteed in this test. As seen on six samples of SS304. The samples are subjected to from Table 3, according to the tensile and stability tests, cyclic axial loading, and the force-displacement dia- the average force of 6550 N and amplitude of 5750 N gram of each sample up to 200 cycles is extracted and were suitable for applying to the specimens periodically examined. The experiment investigates the effect of (sinusoidal) with a frequency of 0.2. This average and sheet thickness, shape, and number of cutouts on the amplitude produce a maximum force of 12,300 N and a rectangular specimens prepared for the experiment. The minimum force of 800 N. In this study, the highest sheet specimens considered in the experiments are 100 mm displacement in each cycle is considered ratcheting. long, and their width is 50 mm. In order to examine the Figure 4 outlines one of these specimens subjected to effect of sheet thickness on the ratcheting behavior, the the average load. samples with thicknesses of 0.93 and 1.93 mm and a cir- cular cutout in the center of the samples are employed. The numerical method Also, the samples with a thickness of 0.93 mm are con- sidered to examine the effect of the cutouts on the In order to prove the reliability and authenticity of the ratcheting behavior. One, two, and four circular cutouts results obtained in this study, the simulation method is Shahrjerdi and SafariFard 5 Table 2. The chemical properties of SS304. Fe C Si Mn P S Cr Mo Ni Co Cu V N 7.6 0.055 0.6 1.14 0.037 0.01 18.7 0. 1 8.05 0.17 0.39 0.056 0.085 Figure 2. The steps need to be taken to reach the final conclusion. also considered to make a careful comparison between directions, the yield surface decreases or increases the numerical and experimental analyses. For the simu- equally. Nevertheless, the yield surface in kinematic lation, the finite element method is employed using hardening is transferred in the stress space and does not ABAQUS software and considering a combination of change size. This displacement moves in proportion to isotropic and kinematic hardening methods. In general, the back stress in the yield space, but it does not deform. three types of isotropic, kinematic, and combined hard- Combining two isotropic and kinematic hardening meth- ening are defined in the simulation. Isotropic hardening ods, that is, isotropic/kinematic nonlinear hardening, the means that the level of yield changes equally in all material’s behavior can be modeled accurately under directions. When the plastic stress appears in all cyclic loading. This model is a combination of two 6 Advances in Mechanical Engineering Using this model, phenomena such as ratcheting can be modeled. This model is based on the equation provided by Chaboche, which is shown below. According to this equation, the movement of the yield surface is propor- tional to the value of a as the back stress. Also, the change in the size of the yield surface is proportional to pl 70 the value of  e as the plastic strain. pl pl a _ ¼ C 0 s  a  e  ga e + a C ð3Þ ij ij ij Where C and g denote the constant of the material and c is the modulus of the material hardening. Besides, s pl is the current yield stress, and  e is the plastic strain. The value of c is also calculated by equation (4) in the ABAQUS software : s  s^ c ¼ ð4Þ pl Where s is yield stress for zero plastic strain. s is known as an exponential function in equation (5), and the increase of the yield surface is determined : Figure 3. The circular, triangle, and rectangular specimens. pl 0 be s ¼ s^ +Q ð1  e Þð5Þ According to equation (5), Q and b represent the material constants. The isotropic and kinematic hard- ening behavior parameters are inserted directly into the ABAQUS software, and then the ratcheting phenom- enon is simulated. Results and discussion This section outlines the results obtained by applying the average force to the specimens and considering the force amplitude (N) and other geometric conditions illustrated in Table 3. Figure 4. Sample P1 under loading condition. The effect of thickness on ratcheting caused by cyclic axial loading isotropic and kinematic hardening, in which the yield surface is transferred by kinematic hardening, and the Figure 5 indicates the force-displacement diagram in yield surface size is changed by isotropic hardening. 200 load cycles for Samples P1 and P2. As shown in Table 3. The loading and geometric characteristics of the SS304 sheet samples. Force Average Dimensions Shape and Thickness Width Length Sample amplitude force (N) of cutouts number of (mm) (mm) (mm) (N) (mm) cutouts 5750 6550 D = 8 Circular-1 0.93 50 100 P1 5750 6550 D = 8 Circular-1 1.935 50 100 P2 5750 6550 D = 8 Circular-2 0.93 50 100 P3 5750 6550 D = 8 Circular-4 0.93 50 100 P4 5750 6550 L = 7.09 Square-1 0.93 50 100 P5 5750 6550 L = 10.75 Triangular-1 0.93 50 100 P6 Shahrjerdi and SafariFard 7 Figure 5. Force-displacement diagram of the Sample: (a) P1 and (b) P2. Figure 6. Ratcheting displacement diagram versus the number of cycles of Samples P1 and P2 in (a) 200 cycles and (b) 30 cycles. both diagrams, after each cycle, the loops of the dia- after a certain number of cycles, which depends on the gram move, and the plastic strain starts accumulating. applied load and the material and geometric conditions Notably, the distance between the diagram loops of the samples appear. Also, the increase in the sheet becomes slighter as the cycles increase. Hence, a thickness reduces the ratcheting displacement, which is remarkable reduction in the ratcheting displacement due to increasing the cross-section area and thus reducing rate in the higher cycles is observed. the stress created in the sheet.Notably,increasingthe Figure 6 shows the ratcheting displacement diagram sheet thickness up to two times in size leads to decreasing versus the number of cycles for Samples P1 and P2. the accumulated ratcheting displacement by 14.47%. The According to the sheet thicknesses, as the number of reason for the reduction of the ratcheting displacement cycles increases, the ratcheting displacement rate rate to zero and the cessation of the accumulation of accumulated in the sheet decreases. This result is more plastic deformation is the regular and stable dislocations accurately illustrated in Figure 6(b), where the ratcheting after a certain number of cycles, which depends on the displacement diagram versus the number of cycles up to applied loading, the material, and the geometric condi- 30 based on the applied force is shown. The diagram tions of the sheets. The increase in thickness increases the slope reaches zero after a characteristic cycle, and the cross-section, and subsequently, the stress reduces. accumulated ratcheting displacement is saturated. Hence, the strain values created in the thick areas are Accordingly, the increase of the ratcheting displacement insignificant. Notably, the ultimate strength of the speci- almost stops from the 30th cycle onward and tends to men increases with the elevation of thickness, that leads become zero since the plastic deformation accumulation stops and the regular and stable formation of dislocations to reducing the strain. 8 Advances in Mechanical Engineering Figure 7. Force-displacement diagram of Samples: (a) P1, (b) P3, and (c) P4. Figure 8. Hysteresis loops of the force-displacement diagram of: (a) Sample P3 and (b) P4. The effect of circular cutouts on the ratcheting samples, respectively. The initial loop is related to the first cycle of the diagram, the second loop is related to behavior caused by cyclic axial loading the 80th cycle of the diagram, and the final loop is Three samples, namely P1, P3, and P4, were subjected related to the 160th cycle force-displacement diagram. to cyclic loading considering the control force with an Clearly, the rate of displacement accumulation reduces average force of 6550 N and a force amplitude of after a while. It can be seen that from the first cycle to 5750 N. Then, the effect of the number of cutouts on the 80th cycle, the ratcheting displacement accumula- the ratcheting behavior of 304 steel sheets was exam- tion in the sample is significantly greater than the ratch- ined. Figure 7(a)–(c) highlight the force-displacement eting displacement existing from the 80th to the 160th diagrams of Samples P1, P3, and P4, respectively. cycle. Regarding Figure 7(a)–(c), the number of cycles The ratcheting displacement diagram versus the directly relates to the ratcheting displacement, and the number of cycles for Samples P1, P3, and P4 are shown diagram loops move forward with increasing the cycles. in Figure 9. According to the diagrams of all three sam- Also, the diagram loops become closer to each other, ples, it is clear that as the number of cycles increases, indicating a decrease in the rate of ratcheting displace- the ratcheting displacement increases. Also, the ratchet- ment accumulation. The same findings were obtained ing displacement rate in Sample P1, which has a circu- in the study of Shariati and Hatami in 2012. In the lar cutout, decreases to zero from cycle 30 onward. research, the authors indicated that the rate of ratchet- This could also be observed in the force-displacement ing rises with the higher force amplitude. The authors diagrams of all three samples. In contrast, the ratchet- concluded that the cutout effect causes softening and ing displacement rate does not reach zero for the other ratcheting behaviors in a cylindrical shell. two sheets, and the accumulation process continues Figure 8(a) and (b) show the three hysteresis loops of until the ratcheting displacement rate in Sample 4 the force-displacement diagram of the P3 and P4 increases from the 150 cycle onward. The main reason Shahrjerdi and SafariFard 9 reduction of the cross-section that brings more accu- mulation of plastic strains. Furthermore, as regards Figures 7(b) and 8(a), the range of ratcheting displacement variations for the min- imum force with the value of 0.2166 starts from the first cycle and continues to 0.4333 in 160 cycles, based on which the total of displacement accumulation is 0.2167. This displacement starts from 0.36041 in the first cycle and continues to 0.5833 in 160 cycles according to the maximum force. Notably, the total displacement accu- mulation is 0.2229. According to such difference, the accumulation created for the minimum and maximum forces shows that the slope of the hysteresis loops in Sample P3 is decreasing, and the softening behavior occurs in the sample. The results of Figures 7(c) and Figure 9. Ratcheting displacement diagram versus the number 8(b) highlight that the ratcheting displacement varia- of cycles for Samples P1, P3, and P4. tions for the minimum force are 0.3998, which starts from the first cycle and continues up to 0.6581 within 160 cycles. In total, the accumulated displacement is for this increase in sheets P3 and P4 compared to sheet 0.2583. The maximum value of displacement force for P1 is the presence of more cutouts and the stress con- the first cycle starts from 0.5623 and continues until centration in these areas, which leads to creating dislo- 0.8519 in 160 cycles. In this case, the displacement force cations and small cracks around them. It should be is 0.2896 in total. This difference is based on the maxi- noted that by increasing the cutouts from a circular mum and minimum forces, and the slope of the hyster- cutout to two circular cutouts whose diameters are esis loops in Sample P4 is decreasing, which leads to 8 mm and are the same, the ratcheting displacement softening behavior in the sample. increases about 180% – also, increasing two circular cutouts up to four circular cutouts with an 8 mm dia- meter results in a 49% increase in ratcheting displace- The effect of cutout shape on the ratcheting behavior ment. The ratcheting in the sheets causes micro-cracks under cyclic axial loading in the holes, eventually leading to a failure by increas- ing the intensification of ratcheting over time. As men- In order to examine the effect of cutout shape on the tioned by Chen et al. in 2011, the growth of cracks in ratcheting behavior, a cyclic load with an average force the manufacturing of the structures affects the load of 6550 and a force amplitude of 5750 was applied to capacity, residual strength, life, and integrity of the Samples P1, P5, and P6 as a controlled force. Due to structures. The local stresses increase with the rise in the similarity of the other conditions in the three sam- the hole area, leading to increased ratcheting displace- ples, a comparison can be made between the effect of ments and strains. The local stresses depend on the three circular, triangular, and square cutouts with the stress concentration factor and the stresses created in same area on the sheet ratcheting behavior. Figure the cut cross-section. However, the stress concentration 10(a)–(c) indicate Samples P1, P5, and P6’s force- factor is reduced by the increase of the hole diameters, displacement diagrams. As can be concluded from the and the stresses in this area increase due to the diagrams, the number of cycles is in direct proportion Figure 10. Force-displacement diagram of Sample: (a) P1, (b) P5, and (c) P6. 10 Advances in Mechanical Engineering Figure 11. Hysteresis loops of the force-displacement diagram of Sample: (a) P5 and (b) P6. to the ratcheting displacement in terms of quantity. Hence, a rise in the cycles constitutes the proliferation in the ratcheting displacement. Notably, at the higher cycles, the loops approach each other further and fur- ther so that a decline in the ratcheting displacement rate is observed in this condition. According to Figure 10(b) and (c), the ratcheting dis- placement variations for the minimum force is 0.0312, which starts from the first cycle and continues up to 0.1770 within 160 cycles. Notably, the accumulated dis- placement is 0.1458 in total. The maximum value of displacement force for the first cycle is 0.1645 in the beginning and continues up to 0.3270 in 160 cycles. In this situation, the total displacement force is 0.1625. This difference is illustrated according to the maximum Figure 12. Ratcheting displacement diagram versus the and minimum forces, and the slope of the hysteresis number of cycles for Samples P1, P5, and P6. loops in Sample P5 decreases, which leads to softening behavior in the sample. The analysis of this research bears a close resemblance to the one conducted by the slope becomes stable, and subsequently, the ratchet- Kolasangiani and Shariati. This study examined the ing displacement increases in Samples P5 and P6, with cutout effects on cylindrical shells exposed to cyclic the increasing number of cycles. loadings, raising the plastic deformation and its rate. As regards Figure 12, the ratcheting displacement of Figure 11(a) and (b) correspond to square and trian- Sample P6 in each cycle is generally greater than those gular diagrams; respectively, a decrease in the ratchet- of the rest. Accordingly, the ratcheting displacement of ing displacement rate can be seen. It is clear that in Sample P5 is greater than Sample P1 due to the pres- both samples, the accumulated ratcheting displacement ence of triangular and square cutouts in these samples in the sample from cycles 1 to 80 is much higher than and plastic deformations caused by sharp roots in these the accumulated ratcheting displacement from cycles cutouts, which leads to creating and propagating the 80 to 160. Overall, in the higher cycles, the hysteresis cracks around them. Since the stress concentration is loops get closer to each other since the ratcheting dislo- more remarkable in the triangular cutouts, the defor- cations form stably, and the plastic deformation accu- mation in Sample P6 is considerably more than in mulation decreases in higher cycles. Samples P1 and P5. Concerning the presence of four According to Figure 11(a) and (b), the ratcheting sharp roots in Sample P5, the ratcheting displacement displacement rate rapidly approaches zero, and the rate of this sample is higher than in Sample P6, where accumulation of the ratcheting displacement stops in the diagram slope is stable. It should be noted that the Sample P1, which has a circular cutout. In samples accumulated ratcheting displacement in the initial with square and triangular cutouts, the increase in cycles for the specimens with rectangular and triangular cycles leads to decrement the ratcheting displacement cutouts is 38% and 63% more than the specimen with rate in the beginning. Then, this trend continues until circular cutouts. Shahrjerdi and SafariFard 11 is 0.1937 at first and reaches 0.3479 within 160 cycles. Accordingly, the displacement force is 0.1542 in total. This difference is obtained based on the maximum and minimum forces, and the slope of the hysteresis loops in Sample P6 decreases, leading to softening behavior in the sample. Validation In this section, a careful comparison is made between the numerical and experimental results of Samples P1– P6 to prove the reliability of the analysis conducted here in terms of accuracy and authenticity. Numerical and experimental Force-displacement diagrams and numerical and experimental ratcheting displacement diagrams are also compared. As stated earlier, the FEM method and ABAQUS software were used. The specimens were considered based on the boundary con- ditions and loading applied to the specimens to simulate the experiments in ABAQUS software. Figure 13 indi- cates the meshed structure of the specimen in which the CPS4R element, a quadrilateral element with plane Figure 13. The meshed structure of Sample P5 in ABAQUS. stress, is employed. No code has been used for simula- tion. The specimens were designed in the beginning using the plain stress condition. The mechanical proper- Table 4. The mechanical properties of SS304. ties of SS304 were applied to the simulation processes. Then, the coefficients of the hardening techniques were C C C g g g Qb 1 2 3 1 2 3 (MPa) (MPa) (MPa) inserted into the software. The analysis was conducted based on the whole conditions of the experiments. The 63000 41000 1650 8950 500 6 215 15 equations presented in the manuscript are essential to present the theoretical information of the study. Also, the properties of SS304 and the parameters of the hard- ening behavior are determined in the software. The isotropic/kinematic hardening parameters employed in Abaqus software are characterized in Table 4. The comparison between the FEM results and the experimental results of Samples P1–P6 based on the ratcheting displacement and number of cycles is illu- strated in Figures 14–19. Figure 14 shows the numerical and experimental results of Sample P1 under loading with an average force of 6550 N and an amplitude of 5750 N. There is a good agreement between the numerical and experimen- tal results, and the comparison results represent the Figure 14. Numerical and experimental diagram of unique capability of the isotropic/kinematic nonlinear ratcheting displacement based on the number of cycles for hardening model to simulate the ratcheting behavior of Sample P1. the sample. The FEM method and Abaqus software results show more ratcheting displacement based on cycles. Notably, the error of the numerical results based According to Figures 10(c) and 11(b), the ratcheting on the experimental ones is 15%. In Sample P2, the displacement variations for the minimum force is error of numerical results is higher than the experimen- 0.0562, which starts from the first cycle and continues tal results. By increasing the thickness of the sample, up to 0.2 during 160 cycles. It is noteworthy that the the numerical results in simulating the sampling beha- accumulated displacement is a total of 0.1438. The vior of the sample need to show more accuracy. Figures maximum value of displacement force for the first cycle 16 and 17 compare the numerical and experimental 12 Advances in Mechanical Engineering Figure 17. Numerical and experimental diagram of ratcheting Figure 15. Numerical and experimental diagram of ratcheting displacement based on the number of cycles for Sample P4. displacement based on the number of cycles for Sample P2. Figure 18. Numerical and experimental diagram of ratcheting displacement based on the number of cycles for Sample P5. Figure 16. Numerical and experimental diagram of ratcheting displacement based on the number of cycles for Sample P3. results of samples P3 and P4. There is a minor differ- ence between the experimental and numerical results in the sample with two circular cutouts. The isotropic/ kinematic nonlinear hardening model’s capability to accurately simulate this sample’s ratcheting behavior is represented. However, the difference between the numerical and experimental diagrams widens based on the rise in the number of cycles. The error in the numer- ical results based on the experimental results is 15%. But this error value is about 8% in Figure 17, which is acceptable in terms of accuracy. In samples P5 and P6, which have non-circular cutouts, the rate of ratcheting Figure 19. Numerical and experimental diagram of ratcheting displacement accumulation in the experimental analysis displacement based on the number of cycles for Sample P6. is higher than numerical analysis, and numerical results do not correspond to experimental results with a desir- able accuracy (Figures 18 and 19). Other results are Samples P1, P3, and P4 with acceptable accuracy. On highlighted in Figures 20–23. the other hand, there is no good agreement between the Figure 20 indicates that the number of elements does numerical and experimental results for Samples P2, P5, not profoundly impact the obtained results. Overall, and P6, in which the accumulated ratcheting displace- using the isotropic/kinematic combined hardening ment has been low in the experimental analysis. model of Abaqus software and considering the harden- A more detailed look at Figures 21–23 reveals that, ing parameters of Table 4 can simulate the behaviors of similar to the experimental results, the accumulation of Shahrjerdi and SafariFard 13 Figure 23. Numerical diagram of ratcheting displacement in Figure 20. Numerical diagram of ratcheting displacement terms of the cycle number for Samples P1, P5, and P6. based on the cycle number of Sample P3 meshed with 2262 and 4956 elements. increasing the number of cutouts from one circular cut- out with a diameter of 8 mm to two circular cutouts, each with a diameter of 8 mm, increases the ratcheting displacement by about 120%. Also, increasing the number of cutouts from two circular cutouts with a diameter of 8 mm to four circular cutouts, each with a diameter of 8 mm, increases the ratcheting displacement by about 65%. Moreover, more ratcheting occurs in the samples with square and triangular cutouts than in circular cutouts. Notably, the samples with square and triangular cutouts in the initial cycles have about 16 and 26% more ratcheting displacement than those with circular cutouts. The nonlinear isotropic and kinematic hardening were used for simulating the specimens in ABAQUS. The software could have performed better Figure 21. Numerical diagram of ratcheting displacement in in the specimens in which the ratcheting was insignifi- terms of the cycle number for Samples P1 and P2. cant. Also, the used coefficients of hardening used in the simulation were the same, which led to some differ- ences in the analysis. Conclusion In summary, the effect of cutout numbers, cutout shapes, and thickness on the ratcheting behavior of SS304 sheets subjected to cyclic axial loading was investigated in the current study. The experiments were performed on six specimens with different shapes and cutouts numbers using Instron 8502 device. ABAQUS software conducted a numerical analysis for the specimens under cyclic axial loadings. The Isotropic/Kinematic nonlinear hardening model and the FEM method were considered to obtain Figure 22. Numerical diagram of ratcheting displacement in results in the simulation. Besides, a careful comparison terms of the cycle number for Samples P1, P3, and P4. was also made between the numerical and experimental results. It was revealed that a rise in cycles causes more ratcheting displacement in the sample decreased with accumulation in the ratcheting displacement. Then, the increasing thickness. The numerical and experimental diagram slope reaches zero after a characteristic cycle, results show identical mechanical behaviors since and the accumulated ratcheting displacement is satu- increasing the number of cutouts in the selection rated. The slight difference between the diagram loops in increases the displacement of accumulated ratcheting in the higher cycles represented a decrease in the ratcheting both analyses. Accordingly, in the numerical results, displacement rate. As the thickness of the sheet increases, 14 Advances in Mechanical Engineering the ratcheting displacement decreases due to the increase ORCID iD in the cross-section area and reduces the stress created in Ali Shahrjerdi https://orcid.org/0000-0002-2525-132X the sheet. Increasing the sheet thickness up to two times in size led to decreasing the accumulated ratcheting dis- References placement by 14.47%. The ratcheting displacement var- iation for the minimum force was 0.3998, which started 1. 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The numerical and experimental analysis of ratcheting behavior for 304 steel sheets under cyclic axial loading

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Abstract

The present study examines the effect of the thickness, cutouts numbers, and different cutouts shapes on the ratcheting behavior of 304 steel sheets under cyclic axial loading. The cutout shapes considered here are circular, triangular, and square cutouts. The effect of the number of circular cutouts in 304 steel sheets on ratcheting behavior is investigated. The Instron 8502 device is used to perform the experiments in which the cyclic axial loading is applied to six specimens with different cutouts and thicknesses at ambient temperature. The obtained results highlight the fact that reducing the thickness and increasing the number of cutouts in the sheets are attributed to the rise in the ratcheting displacement. Accordingly, the ratcheting displacement is more striking in the sheets with triangular cutouts. Notably, a numerical anal- ysis is considered using the isotropic/kinematic nonlinear hardening model and FEM in ABAQUS software. Since a good agreement is seen between the numerical and experimental results, the analysis conducted in this study is reliable in terms of accuracy and authenticity. Keywords Ratcheting, cyclic loading, FEM, SS304, ABAQUS Date received: 29 November 2022; accepted: 20 March 2023 Handling Editor: Chenhui Liang still considerable ambiguity regarding this topic because Introduction it is a secondary plastic deformation process and progres- 10–12 Due to the recent development in mechanical and civil sively occurs cycle by cycle. Thus, the study in this engineering, the structures and mechanical components field is of great importance in predicting the materials’ are significantly exposed to randomly variable loads lifetime under characteristic cyclic loading. 1–4 nowadays. On the other hand, the engineering struc- The study of ratcheting dates back many years tures like pressure vessels, piping systems, and offshore 13–18 ago, but it is still a challenging issue that needs to structures are constantly subjected to cyclic axial loading be studied in depth. The number of cycles that lead to or cyclic thermal loading. The stress cycles’ values applied to the structures are sometimes more than their yield 5–8 strength and bring unwanted damages. These stress Mechanical Engineering, Malayer University, Malayer, Hamadan Province, Iran cycles, along with the mean stress, create remarkable accumulations of plastic deformation, namely ratcheting, Corresponding author: and finally lead to the failure phenomenon. However, Ali Shahrjerdi, Mechanical Engineering, Malayer University, Arak - Malayer the previous decades have witnessed a growing interest in Hwy, Malayer, Hamadan Province 65719-9586, Iran. the study of ratcheting behavior in the material; there is Email: shahrjerdi.mail@gmail.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 Advances in Mechanical Engineering ratcheting can be significantly effective in terms of another study in this regard, the author did some intensification and amount of ratcheting strain. In experiments on cylindrical shells with different lengths 19 37 2003, Chen et al. aimed to simulate the ratcheting under combined and cyclic axial loading. According strain considering a high number of cycles subjected to to the results obtained in this study, a rise in the length biaxial loadings. This study has considered a corrected of cylindrical shells due to increasing flexural torque form of the Ohno–Wang kinematic hardening rule to and increasing vertical stresses at different sections of reach more accurate numerical results. The question oblique cylindrical shells with longer lengths leads to here arises of what factors are effective in ratcheting increasing ratcheting displacement. One of the signifi- behavior to what extent. In 2004, Weiß et al. ratchet- cant factors that have an essential factor on the ratchet- ing was introduced as a secondary phenomenon of cyc- ing behavior of the cylindrical sells is the diameter lic plasticity that intensifies fatigue damage. Also, investigated by Shariati et al. in 2016. In this study, deformation behaviors and microstructure evolution of the effect of this factor was experimented with by con- a hot-rolled AZ31B magnesium alloy were examined sidering two samples of cylindrical shells produced of 18 21 by Wang et al. After that, in 2021, Chen et al. stud- SS304L steel with different diameters and under cyclic ied the issue of cyclic lithium-ion diffusion-induced pure bending loading. The effect of cutout shape on the stress within the charging-discharging process. In this ratcheting angle on torsional loading on cylindrical study, the plastic behavior of lithium-ion battery cath- shells has also been studied by Shariati et al. ode has been systematically examined, and the effect of According to this research, the cylindrical shells of the cycle number has been analyzed. These studies SS316L with circular, square, and triangular cutouts mainly investigated the effect of cycle number on ratch- were subjected to cyclic torsional loading, and the eting behavior and did not address the effect of shape ratcheting angle diagram was obtained based on the dimensions. Notably, the geometrical dimensions and number of loading cycles. material are also important to reach reliable results. One of the industry’s most functional components is A large number of existing studies in the broader lit- a pipe that transfers liquid, gas, slurries, and other solids erature have examined the ratcheting phenomenon for and fluids from one area. Since these components are 22–28 various materials. Besides, the different geometries sometimes subjected to severe cyclic loadings, the study have been investigated to analyze the obtained results in this regard is also important. Concerning the ratchet- 39,40 so far, among which the cylindrical specimens have ing phenomenon in the pipes under cyclic loadings, 29,30 6 been widely employed. For instance, Kobayashi Zeinoddini et al. considered the ratcheting behavior in and Ohno examined the ratcheting behavior for the the low-alloy, high-strength steel pipes (API-5L X80) SS304 cylindrical shells and presented valuable results. subjected to cyclic bending. Additionally, Zakavi et al. Lee et al. also conducted an identical study for the assessed the combined stiffness model in the elbow cylindrical shells of SS316 in 2003. Hamidinejad and ratcheting behavior of pressurized pipe under in-plane Varvani-Farahani conducted an experimental study torque. The ratcheting phenomenon for the elbow under for 1045 and 1Cr18Ni9Ti steel in 2015. Shariati et al. pressurized bending after thermal aging was investigated investigated the ratcheting behavior of cantilever speci- experimentally and numerically by Liu et al. in 2019. mens of SS316L steel cylindrical shells under cyclic In another study, the uniaxial and biaxial ratcheting bending loads. The authors indicated that plastic defor- behavior of pressurized AISI 316L pipe under cyclic mation accumulates in the specimen after each cycle, loading was examined by Moslemi et al. and presented and as the cycles increase, the diagram loops become acceptable results. In addition, Kang and Karimi and 34 35 44 closer. Shariati et al. also conducted a numerical Shariati studied experimental and numerical analysis study in 2011 concerning the investigation of ratcheting of the ratcheting behavior of SS316L thin-walled pipes in cylindrical shells. The authors investigated the effect under cyclic internal pressure. There exists a consider- of frequency and thickness on ratcheting behavior and able body of literature on ratcheting behavior in other fatigue life of polystyrene pipe under cyclic axial load- geometries. The previous studies aimed to examine the ing. Zhu investigated the effect of increasing force effect of this phenomenon in a variety of geometries like 45,46 17,47 48 amplitude on the ratcheting behavior of cylindrical pipes, circular components, axial pipes, 49–51 shells produced of CK20 steel under cyclic loading, and sheets, and so on. To mention a few, Kolasangiani accumulation of ratcheting strain increases with et al. studied the ratcheting behavior of plates with a increasing stress amplitude at constant average stress. circular cutout under axial cyclic loading. The ratcheting It was also revealed that the increased average stress behavior of notched steel samples subjected to asym- could also be one of the factors affecting this behavior. metric loading cycles through coupled kinematic hard- 36 53 Shariati et al. investigated the effect of this factor on ening-Neuber rules was also examined. The authors SS316L steel and concluded that in the constant ampli- also investigated the ratcheting progress at the notch tude of force, with increasing average force, the displa- root of 1045 steel samples over asymmetric loading 54 55 cement accumulates, and its rate increases. Also, in cycles. Chen et al. also analyzed the effect of circular Shahrjerdi and SafariFard 3 holes on the ratchet limit and cracked tip plastic strain examined under cyclic axial loading with various cut- range in a center cracked plate. Dong et al. examined outs and dimensions. the low cycle fatigue and ratcheting failure behavior of The rest of this study is organized as follows: Section AH32 steel under uniaxial cyclic loading. Luo et al. 2 outlines the methods and material considered to con- conducted an experimental study on the heterogeneous duct the experiment and validate the results. The infor- ratcheting behavior of SUS301L stainless steel butt weld mation regarding the ratcheting concept and the issue joint during uniaxial cyclic loading. An experimental of hardening and softening is presented in the third sec- study was conducted on the uniaxial ratcheting-fatigue tion. The fourth section analyzes and discusses the interaction of polyamide-6 by Yang et al. Further, results obtained for the sheet specimens under cyclic 20 59 Weiß et al. and Liu et al. investigated ratcheting- axial loading. In addition, a careful comparison is made fatigue behavior and damage mechanism of GH4169 at between the numerical and experimental results in the 650C. The ratcheting fatigue behavior of 42CrMo4 fifth section to prove the authenticity and accuracy of steel under different heat treatment conditions was also the analysis conducted in this research. The main con- examined by Kreethi et al. in 2017. One of the most clusions obtained from the experiments and numerical important components in engineering is sheet metal due analysis, as well as the suggestions for future study, are to its remarkable capabilities in modern buildings, man- drawn in the sixth section. ufacturing, and construction sectors. In fact, sheet metal is widely employed in various industries such as car Preliminaries manufacturing, aircraft parts, tools, agriculture, mining, catering, shipping, medicine, and electronics. Thus, most Since nowadays many structures are under cyclic exter- early studies, as well as current work, focus on the ratch- nal forces which lead to fracture, the issue of ratcheting 50,61–63 eting behavior of the steel sheets, Z2CND18.12 is very important. These cyclic loads gradually increase 64 65 steel sheets, and molybdenum sheets. For example, the probability of failure due to the creation of more De et al. experimentally studied the ratcheting phe- strains by small values per cycle. This type of loading is nomenon on steel sheets in 2017. Concerning the results called cyclic plasticity or elasticity, a low-cycle fatigue obtained in this study, the plastic strain accumulation type. However, to observe the sample’s ratcheting phe- rate turned out to be high in the initial cycles and nomenon, the load needs to be in the form of control decreased in the subsequent cycles. Shahrjerdi et al. stress, and the amount of stress amplitude must be obtained the Bree diagram to specify the ratcheting and beyond its elastic domain. In general, the following non-ratcheting areas for a functionally graded beam conditions need to be applied to examine the ratcheting under cyclic thermal and axial loading. In light of recent behavior in the sample: events in the ratcheting phenomenon for the various material and geometries, there is now some considerable The sample is subjected to cyclic loading. concern about the influential factors affecting the inten- Plastic deformation occurs in each cycle. sification of this behavior. The mean stress is non-zero. The majority of studies mentioned above emphasize the effects of dimensions and cycle number of loading The plastic strain (average of each cycle) increases on the amount of ratcheting strain and the correspond- when the cycle rises. One of the significant effects of ing results. Besides, the importance of study in the field the ratcheting phenomenon on a component is that the of sheet ratcheting has been clarified. Despite such crack growth is observed earlier, and failure occurs ear- interest, we have yet to study the effect of sheets’ thick- lier in the ratcheting zone. For this reason, in designing ness on ratcheting the factors affecting this behavior to parts and industrial structures, it is essential to evaluate the best of our knowledge. Considering the widespread the ratcheting phenomenon as an important and effec- use of 304 steel sheets with different thicknesses in the tive factor for the failure phenomenon. industry, it is important to study their ratcheting beha- In general, three types of ratcheting can be observed vior considering the effect of thickness. On the other in material behavior. Accordingly, the first type is hand, examining the effect of the shape and number of related to the mitigation of the ratcheting rate, which cutouts on sheet ratcheting is essential because cutouts causes elastic/plastic shakedown. This result is mainly with different shapes are sometimes created in the obtained when the material is highly cyclic hardening. sheets for many purposes. A more detailed look at the The second type is related to the condition in which the literature reveals a number of gaps and shortcomings, strain rate is constant, which leads to the accumulation which are as follows. As a novelty, the effect of shape, of plastic strains in the material. The third type of cutout number, thickness, and dimension on 304 steel sheets are simultaneously discussed in the current ratcheting is associated with a rise in strain rate. Such paper. Also, the numerical simulation and experiments conditions are presented in Figure 1 to understand their regarding the ratcheting behavior of 304 steel sheets are effects entirely. 4 Advances in Mechanical Engineering with a diameter of 8 mm are created. It should be noted that there is an equal distance between the cutouts exist- ing in the center of the samples. Since the effect of the cutout shape on the ratcheting behavior is considerable, the SS304 sheet samples with circular, square, and tri- angular (equilateral) cutouts and the same area (about 50 mm ) are examined in the center of the samples. According to the prepared samples shown in Figures 3 and 4, the diameter of the circular cutout is 8 mm, and the lengths of the square and triangular cutouts are 7.09 and 10.75 mm, respectively. One specimen was tested for each case. However, two or three samples were Figure 1. The schematic of various types of ratcheting. tested before the primary analysis to reach a suitable loading. Furthermore, these samples’ geometric and loading Table 1. The mechanical properties of SS304. specifications are reported in Table 3. Since the loading Poisson’s Modulus Yield Ultimate is force-control and the ratcheting phenomenon of ratio elasticity (GPa) stress (MPa) stress (Mpa) SS304 is considered in this study, the applied force needs to result in ratcheting behavior and simultane- 0.33 196 260 798 ously prevent failing the specimens in the initial cycles. The stress-strain curve obtained by the standard tensile test is illustrated in Figure 4. The simple tensile test is uniaxial according to which Methodology and material s ¼ s; s ¼ s ¼ 0 and for the plastic state, 1 2 3 Overall, the present study consists of the experimental 1 1 e ¼ e; n ¼ ; e ¼ e ¼ ne ¼ e. The relationship and numerical analyses considered for the samples of 1 2 3 1 2 2 steel sheet 304. In this section, these methods are between the equivalent stress and equivalent strain is explained in detail, and the mechanical properties of s;e $ s ; e. Where the von Mises stress is : SS304 based on ASTM E8 are highlighted in Table 1. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Also, Table 2 shows the chemical properties of SS304 1 2 2 2 s  ¼ pffiffiffi ðÞ s  s +ðÞ s  s +ðÞ s  s ð1Þ 1 2 2 3 3 1 used in this study. Figure 2 gives the necessary information regarding For plane stress, the von Mises stress can be represented the process of the analyses conducted in this study. In by equation (2) : order to thoroughly understand the steps that need to be taken in this study, devoting attention to this figure pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 is of great importance. s  ¼ pffiffiffi s  s s + s ð2Þ 1 1 2 2 In the beginning, some specimens were considered for The experimental method the tensile test and stability to specify the same range of The experimental analysis is conducted through the load for applying the whole specimens. In fact, their Instron 8502 device that is able to apply dynamic loads capability to bring ratcheting results without failure in up to 250 kN, and the main experiments are performed the primary cycles is guaranteed in this test. As seen on six samples of SS304. The samples are subjected to from Table 3, according to the tensile and stability tests, cyclic axial loading, and the force-displacement dia- the average force of 6550 N and amplitude of 5750 N gram of each sample up to 200 cycles is extracted and were suitable for applying to the specimens periodically examined. The experiment investigates the effect of (sinusoidal) with a frequency of 0.2. This average and sheet thickness, shape, and number of cutouts on the amplitude produce a maximum force of 12,300 N and a rectangular specimens prepared for the experiment. The minimum force of 800 N. In this study, the highest sheet specimens considered in the experiments are 100 mm displacement in each cycle is considered ratcheting. long, and their width is 50 mm. In order to examine the Figure 4 outlines one of these specimens subjected to effect of sheet thickness on the ratcheting behavior, the the average load. samples with thicknesses of 0.93 and 1.93 mm and a cir- cular cutout in the center of the samples are employed. The numerical method Also, the samples with a thickness of 0.93 mm are con- sidered to examine the effect of the cutouts on the In order to prove the reliability and authenticity of the ratcheting behavior. One, two, and four circular cutouts results obtained in this study, the simulation method is Shahrjerdi and SafariFard 5 Table 2. The chemical properties of SS304. Fe C Si Mn P S Cr Mo Ni Co Cu V N 7.6 0.055 0.6 1.14 0.037 0.01 18.7 0. 1 8.05 0.17 0.39 0.056 0.085 Figure 2. The steps need to be taken to reach the final conclusion. also considered to make a careful comparison between directions, the yield surface decreases or increases the numerical and experimental analyses. For the simu- equally. Nevertheless, the yield surface in kinematic lation, the finite element method is employed using hardening is transferred in the stress space and does not ABAQUS software and considering a combination of change size. This displacement moves in proportion to isotropic and kinematic hardening methods. In general, the back stress in the yield space, but it does not deform. three types of isotropic, kinematic, and combined hard- Combining two isotropic and kinematic hardening meth- ening are defined in the simulation. Isotropic hardening ods, that is, isotropic/kinematic nonlinear hardening, the means that the level of yield changes equally in all material’s behavior can be modeled accurately under directions. When the plastic stress appears in all cyclic loading. This model is a combination of two 6 Advances in Mechanical Engineering Using this model, phenomena such as ratcheting can be modeled. This model is based on the equation provided by Chaboche, which is shown below. According to this equation, the movement of the yield surface is propor- tional to the value of a as the back stress. Also, the change in the size of the yield surface is proportional to pl 70 the value of  e as the plastic strain. pl pl a _ ¼ C 0 s  a  e  ga e + a C ð3Þ ij ij ij Where C and g denote the constant of the material and c is the modulus of the material hardening. Besides, s pl is the current yield stress, and  e is the plastic strain. The value of c is also calculated by equation (4) in the ABAQUS software : s  s^ c ¼ ð4Þ pl Where s is yield stress for zero plastic strain. s is known as an exponential function in equation (5), and the increase of the yield surface is determined : Figure 3. The circular, triangle, and rectangular specimens. pl 0 be s ¼ s^ +Q ð1  e Þð5Þ According to equation (5), Q and b represent the material constants. The isotropic and kinematic hard- ening behavior parameters are inserted directly into the ABAQUS software, and then the ratcheting phenom- enon is simulated. Results and discussion This section outlines the results obtained by applying the average force to the specimens and considering the force amplitude (N) and other geometric conditions illustrated in Table 3. Figure 4. Sample P1 under loading condition. The effect of thickness on ratcheting caused by cyclic axial loading isotropic and kinematic hardening, in which the yield surface is transferred by kinematic hardening, and the Figure 5 indicates the force-displacement diagram in yield surface size is changed by isotropic hardening. 200 load cycles for Samples P1 and P2. As shown in Table 3. The loading and geometric characteristics of the SS304 sheet samples. Force Average Dimensions Shape and Thickness Width Length Sample amplitude force (N) of cutouts number of (mm) (mm) (mm) (N) (mm) cutouts 5750 6550 D = 8 Circular-1 0.93 50 100 P1 5750 6550 D = 8 Circular-1 1.935 50 100 P2 5750 6550 D = 8 Circular-2 0.93 50 100 P3 5750 6550 D = 8 Circular-4 0.93 50 100 P4 5750 6550 L = 7.09 Square-1 0.93 50 100 P5 5750 6550 L = 10.75 Triangular-1 0.93 50 100 P6 Shahrjerdi and SafariFard 7 Figure 5. Force-displacement diagram of the Sample: (a) P1 and (b) P2. Figure 6. Ratcheting displacement diagram versus the number of cycles of Samples P1 and P2 in (a) 200 cycles and (b) 30 cycles. both diagrams, after each cycle, the loops of the dia- after a certain number of cycles, which depends on the gram move, and the plastic strain starts accumulating. applied load and the material and geometric conditions Notably, the distance between the diagram loops of the samples appear. Also, the increase in the sheet becomes slighter as the cycles increase. Hence, a thickness reduces the ratcheting displacement, which is remarkable reduction in the ratcheting displacement due to increasing the cross-section area and thus reducing rate in the higher cycles is observed. the stress created in the sheet.Notably,increasingthe Figure 6 shows the ratcheting displacement diagram sheet thickness up to two times in size leads to decreasing versus the number of cycles for Samples P1 and P2. the accumulated ratcheting displacement by 14.47%. The According to the sheet thicknesses, as the number of reason for the reduction of the ratcheting displacement cycles increases, the ratcheting displacement rate rate to zero and the cessation of the accumulation of accumulated in the sheet decreases. This result is more plastic deformation is the regular and stable dislocations accurately illustrated in Figure 6(b), where the ratcheting after a certain number of cycles, which depends on the displacement diagram versus the number of cycles up to applied loading, the material, and the geometric condi- 30 based on the applied force is shown. The diagram tions of the sheets. The increase in thickness increases the slope reaches zero after a characteristic cycle, and the cross-section, and subsequently, the stress reduces. accumulated ratcheting displacement is saturated. Hence, the strain values created in the thick areas are Accordingly, the increase of the ratcheting displacement insignificant. Notably, the ultimate strength of the speci- almost stops from the 30th cycle onward and tends to men increases with the elevation of thickness, that leads become zero since the plastic deformation accumulation stops and the regular and stable formation of dislocations to reducing the strain. 8 Advances in Mechanical Engineering Figure 7. Force-displacement diagram of Samples: (a) P1, (b) P3, and (c) P4. Figure 8. Hysteresis loops of the force-displacement diagram of: (a) Sample P3 and (b) P4. The effect of circular cutouts on the ratcheting samples, respectively. The initial loop is related to the first cycle of the diagram, the second loop is related to behavior caused by cyclic axial loading the 80th cycle of the diagram, and the final loop is Three samples, namely P1, P3, and P4, were subjected related to the 160th cycle force-displacement diagram. to cyclic loading considering the control force with an Clearly, the rate of displacement accumulation reduces average force of 6550 N and a force amplitude of after a while. It can be seen that from the first cycle to 5750 N. Then, the effect of the number of cutouts on the 80th cycle, the ratcheting displacement accumula- the ratcheting behavior of 304 steel sheets was exam- tion in the sample is significantly greater than the ratch- ined. Figure 7(a)–(c) highlight the force-displacement eting displacement existing from the 80th to the 160th diagrams of Samples P1, P3, and P4, respectively. cycle. Regarding Figure 7(a)–(c), the number of cycles The ratcheting displacement diagram versus the directly relates to the ratcheting displacement, and the number of cycles for Samples P1, P3, and P4 are shown diagram loops move forward with increasing the cycles. in Figure 9. According to the diagrams of all three sam- Also, the diagram loops become closer to each other, ples, it is clear that as the number of cycles increases, indicating a decrease in the rate of ratcheting displace- the ratcheting displacement increases. Also, the ratchet- ment accumulation. The same findings were obtained ing displacement rate in Sample P1, which has a circu- in the study of Shariati and Hatami in 2012. In the lar cutout, decreases to zero from cycle 30 onward. research, the authors indicated that the rate of ratchet- This could also be observed in the force-displacement ing rises with the higher force amplitude. The authors diagrams of all three samples. In contrast, the ratchet- concluded that the cutout effect causes softening and ing displacement rate does not reach zero for the other ratcheting behaviors in a cylindrical shell. two sheets, and the accumulation process continues Figure 8(a) and (b) show the three hysteresis loops of until the ratcheting displacement rate in Sample 4 the force-displacement diagram of the P3 and P4 increases from the 150 cycle onward. The main reason Shahrjerdi and SafariFard 9 reduction of the cross-section that brings more accu- mulation of plastic strains. Furthermore, as regards Figures 7(b) and 8(a), the range of ratcheting displacement variations for the min- imum force with the value of 0.2166 starts from the first cycle and continues to 0.4333 in 160 cycles, based on which the total of displacement accumulation is 0.2167. This displacement starts from 0.36041 in the first cycle and continues to 0.5833 in 160 cycles according to the maximum force. Notably, the total displacement accu- mulation is 0.2229. According to such difference, the accumulation created for the minimum and maximum forces shows that the slope of the hysteresis loops in Sample P3 is decreasing, and the softening behavior occurs in the sample. The results of Figures 7(c) and Figure 9. Ratcheting displacement diagram versus the number 8(b) highlight that the ratcheting displacement varia- of cycles for Samples P1, P3, and P4. tions for the minimum force are 0.3998, which starts from the first cycle and continues up to 0.6581 within 160 cycles. In total, the accumulated displacement is for this increase in sheets P3 and P4 compared to sheet 0.2583. The maximum value of displacement force for P1 is the presence of more cutouts and the stress con- the first cycle starts from 0.5623 and continues until centration in these areas, which leads to creating dislo- 0.8519 in 160 cycles. In this case, the displacement force cations and small cracks around them. It should be is 0.2896 in total. This difference is based on the maxi- noted that by increasing the cutouts from a circular mum and minimum forces, and the slope of the hyster- cutout to two circular cutouts whose diameters are esis loops in Sample P4 is decreasing, which leads to 8 mm and are the same, the ratcheting displacement softening behavior in the sample. increases about 180% – also, increasing two circular cutouts up to four circular cutouts with an 8 mm dia- meter results in a 49% increase in ratcheting displace- The effect of cutout shape on the ratcheting behavior ment. The ratcheting in the sheets causes micro-cracks under cyclic axial loading in the holes, eventually leading to a failure by increas- ing the intensification of ratcheting over time. As men- In order to examine the effect of cutout shape on the tioned by Chen et al. in 2011, the growth of cracks in ratcheting behavior, a cyclic load with an average force the manufacturing of the structures affects the load of 6550 and a force amplitude of 5750 was applied to capacity, residual strength, life, and integrity of the Samples P1, P5, and P6 as a controlled force. Due to structures. The local stresses increase with the rise in the similarity of the other conditions in the three sam- the hole area, leading to increased ratcheting displace- ples, a comparison can be made between the effect of ments and strains. The local stresses depend on the three circular, triangular, and square cutouts with the stress concentration factor and the stresses created in same area on the sheet ratcheting behavior. Figure the cut cross-section. However, the stress concentration 10(a)–(c) indicate Samples P1, P5, and P6’s force- factor is reduced by the increase of the hole diameters, displacement diagrams. As can be concluded from the and the stresses in this area increase due to the diagrams, the number of cycles is in direct proportion Figure 10. Force-displacement diagram of Sample: (a) P1, (b) P5, and (c) P6. 10 Advances in Mechanical Engineering Figure 11. Hysteresis loops of the force-displacement diagram of Sample: (a) P5 and (b) P6. to the ratcheting displacement in terms of quantity. Hence, a rise in the cycles constitutes the proliferation in the ratcheting displacement. Notably, at the higher cycles, the loops approach each other further and fur- ther so that a decline in the ratcheting displacement rate is observed in this condition. According to Figure 10(b) and (c), the ratcheting dis- placement variations for the minimum force is 0.0312, which starts from the first cycle and continues up to 0.1770 within 160 cycles. Notably, the accumulated dis- placement is 0.1458 in total. The maximum value of displacement force for the first cycle is 0.1645 in the beginning and continues up to 0.3270 in 160 cycles. In this situation, the total displacement force is 0.1625. This difference is illustrated according to the maximum Figure 12. Ratcheting displacement diagram versus the and minimum forces, and the slope of the hysteresis number of cycles for Samples P1, P5, and P6. loops in Sample P5 decreases, which leads to softening behavior in the sample. The analysis of this research bears a close resemblance to the one conducted by the slope becomes stable, and subsequently, the ratchet- Kolasangiani and Shariati. This study examined the ing displacement increases in Samples P5 and P6, with cutout effects on cylindrical shells exposed to cyclic the increasing number of cycles. loadings, raising the plastic deformation and its rate. As regards Figure 12, the ratcheting displacement of Figure 11(a) and (b) correspond to square and trian- Sample P6 in each cycle is generally greater than those gular diagrams; respectively, a decrease in the ratchet- of the rest. Accordingly, the ratcheting displacement of ing displacement rate can be seen. It is clear that in Sample P5 is greater than Sample P1 due to the pres- both samples, the accumulated ratcheting displacement ence of triangular and square cutouts in these samples in the sample from cycles 1 to 80 is much higher than and plastic deformations caused by sharp roots in these the accumulated ratcheting displacement from cycles cutouts, which leads to creating and propagating the 80 to 160. Overall, in the higher cycles, the hysteresis cracks around them. Since the stress concentration is loops get closer to each other since the ratcheting dislo- more remarkable in the triangular cutouts, the defor- cations form stably, and the plastic deformation accu- mation in Sample P6 is considerably more than in mulation decreases in higher cycles. Samples P1 and P5. Concerning the presence of four According to Figure 11(a) and (b), the ratcheting sharp roots in Sample P5, the ratcheting displacement displacement rate rapidly approaches zero, and the rate of this sample is higher than in Sample P6, where accumulation of the ratcheting displacement stops in the diagram slope is stable. It should be noted that the Sample P1, which has a circular cutout. In samples accumulated ratcheting displacement in the initial with square and triangular cutouts, the increase in cycles for the specimens with rectangular and triangular cycles leads to decrement the ratcheting displacement cutouts is 38% and 63% more than the specimen with rate in the beginning. Then, this trend continues until circular cutouts. Shahrjerdi and SafariFard 11 is 0.1937 at first and reaches 0.3479 within 160 cycles. Accordingly, the displacement force is 0.1542 in total. This difference is obtained based on the maximum and minimum forces, and the slope of the hysteresis loops in Sample P6 decreases, leading to softening behavior in the sample. Validation In this section, a careful comparison is made between the numerical and experimental results of Samples P1– P6 to prove the reliability of the analysis conducted here in terms of accuracy and authenticity. Numerical and experimental Force-displacement diagrams and numerical and experimental ratcheting displacement diagrams are also compared. As stated earlier, the FEM method and ABAQUS software were used. The specimens were considered based on the boundary con- ditions and loading applied to the specimens to simulate the experiments in ABAQUS software. Figure 13 indi- cates the meshed structure of the specimen in which the CPS4R element, a quadrilateral element with plane Figure 13. The meshed structure of Sample P5 in ABAQUS. stress, is employed. No code has been used for simula- tion. The specimens were designed in the beginning using the plain stress condition. The mechanical proper- Table 4. The mechanical properties of SS304. ties of SS304 were applied to the simulation processes. Then, the coefficients of the hardening techniques were C C C g g g Qb 1 2 3 1 2 3 (MPa) (MPa) (MPa) inserted into the software. The analysis was conducted based on the whole conditions of the experiments. The 63000 41000 1650 8950 500 6 215 15 equations presented in the manuscript are essential to present the theoretical information of the study. Also, the properties of SS304 and the parameters of the hard- ening behavior are determined in the software. The isotropic/kinematic hardening parameters employed in Abaqus software are characterized in Table 4. The comparison between the FEM results and the experimental results of Samples P1–P6 based on the ratcheting displacement and number of cycles is illu- strated in Figures 14–19. Figure 14 shows the numerical and experimental results of Sample P1 under loading with an average force of 6550 N and an amplitude of 5750 N. There is a good agreement between the numerical and experimen- tal results, and the comparison results represent the Figure 14. Numerical and experimental diagram of unique capability of the isotropic/kinematic nonlinear ratcheting displacement based on the number of cycles for hardening model to simulate the ratcheting behavior of Sample P1. the sample. The FEM method and Abaqus software results show more ratcheting displacement based on cycles. Notably, the error of the numerical results based According to Figures 10(c) and 11(b), the ratcheting on the experimental ones is 15%. In Sample P2, the displacement variations for the minimum force is error of numerical results is higher than the experimen- 0.0562, which starts from the first cycle and continues tal results. By increasing the thickness of the sample, up to 0.2 during 160 cycles. It is noteworthy that the the numerical results in simulating the sampling beha- accumulated displacement is a total of 0.1438. The vior of the sample need to show more accuracy. Figures maximum value of displacement force for the first cycle 16 and 17 compare the numerical and experimental 12 Advances in Mechanical Engineering Figure 17. Numerical and experimental diagram of ratcheting Figure 15. Numerical and experimental diagram of ratcheting displacement based on the number of cycles for Sample P4. displacement based on the number of cycles for Sample P2. Figure 18. Numerical and experimental diagram of ratcheting displacement based on the number of cycles for Sample P5. Figure 16. Numerical and experimental diagram of ratcheting displacement based on the number of cycles for Sample P3. results of samples P3 and P4. There is a minor differ- ence between the experimental and numerical results in the sample with two circular cutouts. The isotropic/ kinematic nonlinear hardening model’s capability to accurately simulate this sample’s ratcheting behavior is represented. However, the difference between the numerical and experimental diagrams widens based on the rise in the number of cycles. The error in the numer- ical results based on the experimental results is 15%. But this error value is about 8% in Figure 17, which is acceptable in terms of accuracy. In samples P5 and P6, which have non-circular cutouts, the rate of ratcheting Figure 19. Numerical and experimental diagram of ratcheting displacement accumulation in the experimental analysis displacement based on the number of cycles for Sample P6. is higher than numerical analysis, and numerical results do not correspond to experimental results with a desir- able accuracy (Figures 18 and 19). Other results are Samples P1, P3, and P4 with acceptable accuracy. On highlighted in Figures 20–23. the other hand, there is no good agreement between the Figure 20 indicates that the number of elements does numerical and experimental results for Samples P2, P5, not profoundly impact the obtained results. Overall, and P6, in which the accumulated ratcheting displace- using the isotropic/kinematic combined hardening ment has been low in the experimental analysis. model of Abaqus software and considering the harden- A more detailed look at Figures 21–23 reveals that, ing parameters of Table 4 can simulate the behaviors of similar to the experimental results, the accumulation of Shahrjerdi and SafariFard 13 Figure 23. Numerical diagram of ratcheting displacement in Figure 20. Numerical diagram of ratcheting displacement terms of the cycle number for Samples P1, P5, and P6. based on the cycle number of Sample P3 meshed with 2262 and 4956 elements. increasing the number of cutouts from one circular cut- out with a diameter of 8 mm to two circular cutouts, each with a diameter of 8 mm, increases the ratcheting displacement by about 120%. Also, increasing the number of cutouts from two circular cutouts with a diameter of 8 mm to four circular cutouts, each with a diameter of 8 mm, increases the ratcheting displacement by about 65%. Moreover, more ratcheting occurs in the samples with square and triangular cutouts than in circular cutouts. Notably, the samples with square and triangular cutouts in the initial cycles have about 16 and 26% more ratcheting displacement than those with circular cutouts. The nonlinear isotropic and kinematic hardening were used for simulating the specimens in ABAQUS. The software could have performed better Figure 21. Numerical diagram of ratcheting displacement in in the specimens in which the ratcheting was insignifi- terms of the cycle number for Samples P1 and P2. cant. Also, the used coefficients of hardening used in the simulation were the same, which led to some differ- ences in the analysis. Conclusion In summary, the effect of cutout numbers, cutout shapes, and thickness on the ratcheting behavior of SS304 sheets subjected to cyclic axial loading was investigated in the current study. The experiments were performed on six specimens with different shapes and cutouts numbers using Instron 8502 device. ABAQUS software conducted a numerical analysis for the specimens under cyclic axial loadings. The Isotropic/Kinematic nonlinear hardening model and the FEM method were considered to obtain Figure 22. Numerical diagram of ratcheting displacement in results in the simulation. Besides, a careful comparison terms of the cycle number for Samples P1, P3, and P4. was also made between the numerical and experimental results. It was revealed that a rise in cycles causes more ratcheting displacement in the sample decreased with accumulation in the ratcheting displacement. Then, the increasing thickness. The numerical and experimental diagram slope reaches zero after a characteristic cycle, results show identical mechanical behaviors since and the accumulated ratcheting displacement is satu- increasing the number of cutouts in the selection rated. The slight difference between the diagram loops in increases the displacement of accumulated ratcheting in the higher cycles represented a decrease in the ratcheting both analyses. Accordingly, in the numerical results, displacement rate. As the thickness of the sheet increases, 14 Advances in Mechanical Engineering the ratcheting displacement decreases due to the increase ORCID iD in the cross-section area and reduces the stress created in Ali Shahrjerdi https://orcid.org/0000-0002-2525-132X the sheet. Increasing the sheet thickness up to two times in size led to decreasing the accumulated ratcheting dis- References placement by 14.47%. The ratcheting displacement var- iation for the minimum force was 0.3998, which started 1. 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Journal

Advances in Mechanical Engineering – SAGE

Published: May 1, 2023

Keywords: Ratcheting; cyclic loading; FEM; SS304; ABAQUS

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