Chaboche Model for Fatigue by Ratcheting Phenomena of Austenitic Stainless Steel under Biaxial Sinusoidal Loading

This study deals with the investigation of the cyclic behaviour of 316L and 304L austenitic stainless steels in oligocyclic fatigue under biaxial loading. As a first step, we investigated the prediction of the character of 316L steel under imposed stress, by the fixation of a stress and the evolution of another, forming a cross-proportional loading path in a range of stresses. In addition, the analysis of the behavior of steel 304L with respect to the bi-axial union (primary and secondary loadings) was studied in order to produce the ratcheting phenomenon induced by the non-zero mean stress, governing the structure to damage in two opposite directions, diagonally symmetrical. An appreciable confrontation of the intrinsic characters of the two steels under the same loading conditions was discussed in the last intervention, controlled in strain, generating the phenomenon of cross-hardening and imposed stress. Producing the progressive strain that manifests itself at each loading cycle will make it possible to quantify the degree of plasticity of each material and optimize the most relevant steel. In this numerical study, the Chaboche model is selected, which is based mainly on perfect predictions and robust constitutive laws capable of reproducing observed macroscopic phenomena. All the simulations were carried out using the ZéBulon computation code. A lot of work on the behavior of 304L and 316L stainless steel has been carried out by several researchers in recent years. The results of previous experiments and numerical simulations have been compared to the results of this study, and a good match has been found.


Introduction
Multiaxially loaded oligocyclic fatigue is encountered in many structures and can result in the failure of material and structures. Numerical modeling techniques must then take into consideration the complexity of the material, mechanisms, and phenomena observed by an adequate model. The modeling of the behavior of austenitic stainless steels is still an open subject, despite the diversity of models or criteria existing in this field. The complexity is mainly based on the penalizing interaction of different stresses (mechanical, thermal, or chemical), the dispersions on the properties of the material, the ignorance of the geometry and of the loadings, the randomness of the material, the plastic deformations Cyclic cycles generated by the repetition of stresses, even below the elastic limit of the material, play a major role in oligocyclic fatigue and can lead to the failure of parts in service [1,2]. All these factors increase the difficulty of modeling [3,4]. Uniaxial tests are inexpensive, easy to perform, and provide useful information about material behavior [5,6]. Even so, most parts in service are subject to complicated loads and aren't properly checked and tested [7].

506
The study of the behavior of a material is a process of observing, describing, and analyzing the phenomena involved and obtaining numerical answers. This last approach is exploited in order to draw cyclic curves, giving rise to different phenomena: hardening, softening, accommodation, adaptation, and ratcheting [8][9][10]. This last phenomenon has been widely studied by Taleb et al. (2006) [11], Hassan et al. (2008) [12], Taleb et al. (2009) [13], Taleb et al. (2011) [14], Halama et al. (2012) [15], Meggiolaro et al. (2015Meggiolaro et al. ( , 2016 [16,17]. The sensitivity and foresight of the comportment of the 304L steel have been an numerical field of investigation led by Boussalih et al. (2019) [18]. In addition to some experimental investigations and simulations, the work of Belattar and Taleb (2021) [19], was carried out on both 304L and 316L steels, the results made it possible to predict several cyclic phenomena, under imposed stress and strain, under cross loading, experimentally and numerically using the Multi-Mechanism (MM) model.
In this study, the behavior of the two steels, 304L and 316L, under biaxial alternating loading (traction-torsion) will be predicted. By opting for the model of Chaboche, we then diagnose their phenomenological and macroscopic behavior. Biaxial fatigue is encountered in the oil and gas industries, in energy installations, and in the chemical industries [19,20]. It was a motivating subject for several researchers [21][22][23]. Other researchers look into biaxial fatigue in [24][25][26][27][28]. The notoriety of austenitic stainless steels is based primarily on their excellent mechanical attributes such as ductility, impact resistance, and toughness [29][30][31][32][33][34][35][36]. This study aims to simulate and correctly predict the nonlinear behavior of the two 316L and 304L steels. The industrial inventory brings into play macroscopic phenomena such as: elastic shakedown, plastic shakedown, ratcheting, softening, hardening, creep, all related to the plasticity of the material, which leads to the damage of structural components subjected to a wide variety of stresses of mechanical origin. The cyclical nature of this behavior threatens the safety of structures and is induced by the phenomenon of fatigue, which appears as a consequence of this formalism.
The prediction of the discrete character of the two steels studied requires the control of their behavior based primarily on the understanding of the macroscopic phenomenological observations during cyclic plasticity, allowing for numerical simulations by the application of the sinusoidal bi-axial stresses in oligocyclic fatigue, which depends only on the number of cycles and not on the frequency. It is driven by two decisive factors: cyclic stress and strain, and proliferating plasticity, which progresses and subsequently produces damage. The ratcheting phenomenon represents the progressive strain which leads to sudden and catastrophic failure caused by the non-zero mean stress propagating rapidly in the structure. This subject will be discussed further in this study.
The first step explores a discussion between the two interacting loading scenarios in order to identify the predictive ratchet at failure by the application of bi-axial tensile-compression stresses followed by a non-zero stress torsional loading applied to 316L steel, by opting for the Chaboche model using the Zebulon calculation code. The second part of this work is concerned with the prediction of the elastoplastic character of steel 304L under an associated proportional cyclic stress field, generating a biaxial ratcheting phenomenon, diagonally symmetrical, shown by the evolution of the stress as a function of deformation. The last part is devoted to the comparison of the two steels 304L and 316L, showing the phenomenon of over hardening called "cross-hardening effect", induced by the application of the loading: primary in the axial direction, followed by a secondary loading in the direction of torsion at imposed strain shown by hysteresis loops. An estimate of the fastest ratcheting between the two steels studied, which led to failure, generated almost the same level of stress over several successive cycles and was studied. In the conclusion section, a summary of the results will be provided along with recommendations for future studies.  The model chosen uses material parameters corresponding to the two steels studied, resulting from the literature ( Table 2).

Methodology
 A validation on the first hysteresis loops of the uni-axial tension-compression tests, in order to ensure the adequate choice of the model chosen and its capacity to suitably describe the elastoplastic behavior (Figures 1 and 2).
 The concordance between the results of the numerical simulation and the experimental data allows the performance of various simulations on steels.
This strategy requires a complete hierarchy, which is shown in the flowchart of Figure 1.

Chaboche Material Model
The Chaboche formulation was presented for the cyclic behavior of metals using nonlinear hardening models. The Chaboche material model can handle tension-compression cyclic loadings where the hardening properties change. Depending on what type of material, temperature, or initial states are used, the material may soften or harden. The field of pure reversibility is delimited in the space of the stresses by a surface of load described by von Mises from which plastic flow can occur [15,37]. This surface is represented in the constraint space by a load function of the following form: where, is the second invariant of the deviatoric tensor of the stress. The material studied is characterized by its resistance to rupture and its ability to work hard. Consequently the work hardening is manifested by the expansion of the load surface corresponds to an isotropic work hardening ( ) and the displacement of its center corresponds to a kinematic work hardening ( ). The evolution of R is given by: where, and are constants of the material which have the effect of introducing a progressive hardening or softening. The evolution of kinematic hardening ( ) is given by: where, and being characteristic coefficients of each material, is the increment of the cumulated plastic deformation: The evolution of the kinematic hardening variable ( ) is given by the model proposed initially by Armstrong and Frederick (1966) introducing a recall term, called dynamic recovery:

Verification of the Model
The Chaboche model adopted in this study is based on the association of two nonlinear isotropic and kinematic hardening, in order to describe the behaviour of 316L and 304L stainless steel alloys, the chemical composition and mechanical properties of these material are presented in Tables 1, and [38], which will be implemented in the Zebulon calculation code [39], see Table 2. The first step before starting the simulation, it is necessary to carry out the comparison between the first hysteresis loops of the simulation and experimental cyclic curve, carried out between ± 0.5% at deformation imposed with a tensilecompression cycle, in order to ensure the reliability of the parameters materials chosen before their use. Figure 2-a shows the comparison between the simulation curve, performed by the computer code ABAQUS and the experimental test on 316L steel, by cyclic loading carried out by Roya et al. [40]. On the other hand, the simulation of the present study is carried out by the finite element calculator code ZéBulon (Figure 2-b). Figure 3-a is a cyclic confrontation curve, between simulation calculated by the ZéBulon code relating to 304L steel, led by Bouusalih et al. [18]. In the present study, we see that there is a good agreement between simulation and experiment ( Figure 3

Ratcheting Experiments on Stainless Steel 316L
In order to predict the cyclic behavior of 316L steel pipe under biaxial cyclic loading performed by Moslemi et al. [41], using the finite element calculation code ANSYS (Figure 4-a). The Chaboche model has been exploited for its ability to correctly simulate the different plastic deformation curves obtained as a function of the number of cycles, and compared with the experimental test (Figure 4-b) uniaxial ratcheting with non-zero mean stress occupied a key place, in this study shown by the evolution hysteresis loops (Figure 4-c).

Impact of Increasing Tensile Loading at Constant Torsion
In order to perceive the influence of the tension loading on the work hardening, we carried out four simulations on the steel 316L with increasing amplitude stress and a loading in constant torsion. The loading conditions are shown in the Table 3.  Figure 5 shows stress-strain hysteresis curves, revealing a decreasing hardening proportional, when the loading in torsion is constant. The curves obtained, see Figure 6 translate the plastic deformation peaks, resulting in an apparent progressive ratcheting rate, as a function of the number of cycles.

Impact of Increasing Torsional Loading at Constant Tension
The present simulation was carried out with a reverse approach to the previous test. Four simulations were performed on 316L steel at constant stress amplitude with increasing torsional load. The loading conditions are shown in Table 4. The phenomenological analysis of the behaviour of the two alloys in cyclic loading (Figure 7), shows stress-strain hysteresis curves describing the fatigue of steel by the ratcheting phenomenon under various loading cases. One observes in particular a ratcheting growing in the opposite direction of the intensity of the loading in torsion. The curves obtained ( Figure 8) the plastic deformation peaks resulting in an apparent progressive ratcheting rate as a function of the number of cycles.

Ratcheting Experiments on Stainless Steel 304L
According to the experiments carried out on the behavior of 304L steel by Hassan Tasnim &al, under stress cyclic, of average stress = 50 MPa in the uniaxial direction (Figure 9-a), and an equivalent stress of = 200 in the direction of torsion (Figure 9-b). The combination of the two loads constitutes a sinusoidal biaxial loading and a loading path proportional, which subsequently produces a biaxial pawl, represented by the evolution of the axial stress as a function of the plastic deformation illustrated by hysteresis loops (Figure 9-c). The evolution of the strain angular as a function of the axial strain produced by the same cross loading, shown in Figure 9-d. This experimental database will be used later to perform simulations on 304L steel and to ensure the performance of the model chosen and its ability to describe the nonlinear behavior of the steel studied. c) Cross ratcheting experimental responses of SS304L d) Axial and shear strain ratcheting from experiment  (Figure10-a). The numerical study consists in modeling the plastic deformation at the level of the hinges by a model which is based on the finite elements (Figure 10-b). The numerical results reveal the ratchet phenomenon, controlled in non-zero mean stress, which manifests itself by the evolution of hysteresis loops in the negative direction (Figure 10-c).

Simulation of Biaxial Stress-Controlled Ratcheting Response
In order to understand the ratcheting sensitivity to the mean stress for the 304L steel, some uniaxial cyclic tension simulations were performed with constant stress amplitude. Two strain controlled tests have been performed with the same stresses amplitudes ( = + 125 , i): The first history represents a classical uniaxial ratcheting test where 20 cycles of tension-compression is applied about an axial mean stress ( = 75 ), ii): the second history 20 cycles of tension-compression is applied about negative mean stress ( = − 75 ) , iii): the third story represents 20 cycles of a torsional loading ( = ±150 ) ( see Table 5).  In order to test the sensitivity of 304L steel, and to analyze its behavior under repeated multiaxial loading, which was broken down into three simulations. The first two simulations are with loading uni axial asymmetric in mean stress ,developing a positive ratcheting in the direction of traction (Figure11-a) and a mean stress = − 75 , offering a negative ratcheting in the direction of compression, Figure 11-b), the third simulation with alternating torsional loading =± 150 (Figure 11-c). The combination of the three simulations ( Figures  11-a, 11-b, 11-c) shows a multiaxial loading with crossed loading path. There is an accumulation of progressive strain which manifests as a biaxial ratcheting phenomenon diagonally symmetrical with respect to zero stress and strain, and which has evolved in the direction of tension and compression, by hysteresis loops (Figure 11-d). (Figure 11-e) reveals the relationship between the two deformations: angular and axial following multiaxial loading (Figure 11-e).

Cross Path Effect in Strain Controlled Tests
The experiment carried out by Taleb and Hauet (2009) [13], consists in carrying out a cyclic test with biaxial loading on steel 304L controlled in strain of 10 cycles = 1 % applied in the direction of traction-compression, followed by 10 cycles of the alternating equivalent strain of torsion between = (±√3). The present simulation was carried out with 50 cycles of traction -compression in the axial direction with imposed strain of = 0.5% followed by 50 cycles of the equivalent strain = 0.86 % in the direction of torsion. The two loads constitute a cyclic biaxial loading with crossed loading path, carried out on the two steels 304L and 316L (see Table 6).  Figure 12 represents a cyclic biaxial test carried out on 304L steel performed by Taleb and Hauet (2009) [13], shows the phenomenon of "cross hardening" caused by the additional torsional loading. On the other hand, over hardening is observed on the cyclic curves of 304L and 316L steel in the present study. In addition, the cyclic curve of 304L steel identifies the curve of 316L steel Figure 13.

Ratcheting Simulation
The two 304 L and 316L steels will be simulated under bidirectional imposed stress in order to analyze their cyclic behaviors, achieved for a load ratio R = -0.58 for 100 cycles. The primary loading is applied in the direction of tensioncompression around an average stress and an alternating secondary loading applied symmetrically in the direction of torsion. A summary of the simulation conditions is reported in Table 7. show a biaxial ratcheting of the two 304L and 316L steels, with repeated loading cross loading path. The ratcheting phenomenon is governed by the non-zero mean stress, illustrated by the evolution of the hysteresis loops in the axial direction and represents an excess of progressive strain, which leads to damage. The goal of this simulation is to estimate the optimal biaxial ratcheting, which corresponds to the most relevant progressive strain having a rapid plastic strain intensity, which leads to failure. This simulation was carried out on the two 304L and 316L steels by the superposition of the two ratcheting with the same conditions. It can be seen from Figure 15-a that the optimal biaxial ratcheting corresponds to 316L steel. Note, in the case of using two steels in industry with the same loading conditions, 316L steel will experience rapid failure compared to 304L steel. This estimate has been confirmed by Figure  15-b which represents the evolution of plastic strain as a function of the number of cycles, where we notice that the plastic strain of 316L steel is higher compared to 304L steel.

Conclusion
The aim of this work is thus to make predictions on the behaviour of the two 316L and 304L stainless steels under biaxial stresses. These two alloys are widely used in several fields of industry for their quality and characteristics. The numerical models offer a great capacity to the researchers and specialists in materials and structures for the study and the exploration of different scenarios of loading and multiphysical stresses in order to predict the performances of the alloys and materials distinguished at the shaping of the structures and rooms. The simulated loading consists of two cycles of tensile loading on the uniaxial and torsion on the vertical direction. The results of the simulation obtained on 316L steel under cross loading reveal an optimized ratcheting, which leads to sudden and rapid fracture in two cases: the torsional stress is constant and the axial stress amplitude is low, and the amplitude of the axial stress is constant and the torsional stress is low. The 304L steel investigation illustrates a radially symmetrical rock, for an average stress imposed in two axially opposite directions. Various studies on the behavior of 304L and 316L stainless steel have been carried out by several researchers in recent years. The results presented in this study are discussed with other previous experimental and numerical studies. This comparison shows that the presented results are in good agreement with the results of other previous studies. A comparison was made between the two steels, and it was concluded that the cyclic curve of 304L steel envelops that of 316L steel, for an identical load ratio to imposed strain. On the other hand, the progressive deformation of 316L steel is more accelerated than that of 304L steel in the case where the stress is imposed. As for recommendations, it is suggested to conduct a series of experiments on the two alloys, 304L and 316L, with the same boundary conditions used in this study. This will allow us to build a database that can be used to find out about finite element calculation codes like ZéBulon.