In this paper, we analyze one specific assumption used in many cyclic plasticity models: The depen- dence of the yield surface evolution on the accumulated plasticity, i.e. the time-integrated plastic multiplier. In carefully designed experiments, we measure this scalar variable and track the yield sur- face evolution. Our results show that this variable is insufficient to describe the isotropic or distortional hardening of the investigated steel, invalidating the commonly used assumption. The results highlight requirements for developing and verifying new isotropic and distortional hardening laws. Keywords: Cyclic plasticity, isotropic hardening, distortional hardening, yield surface 1. Introduction Modeling the plastic material response is nec- essary for accurate analyses in many industrial applications, such as sheet metal forming, rolling contact loading, and other applications where low cycle fatigue loading occur. Cyclic plasticity has, therefore, been a subject of intense research for many decades. Most cyclic plasticity models are extensions of the general framework presented in Chaboche (1986). In terms of isotropic harden- ing, however, the rule denoted ”Classical isotropic hardening” is used by most authors. In these cases, the isotropic hardening evolution is only a function of the accumulated plasticity (See Sec- tion 3 for an exact definition). Some models, e.g. El Abdi et al. (1999); Zhou et al. (2018), include the so-called ”strain-range memorization” effect introduced by Chaboche et al. (1979). This exten- sion allows the saturated yield strength to depend on the plastic strain amplitude. In Chaboche (1986); Novak et al. (2019), approximate anal- yses based on peak stresses were conducted to ∗Corresponding authorshow how the yield strength evolves due to the accumulated plasticity. In the present work, we explicitly measure the accumulated plasticity and then investigate how the yield strength evolves for a range of different loading conditions. Between 1980 and 2000, both isotropic and kinematic hardening rules were researched inten- sively. However, in the same period, many ex- perimental works showed that anisotropic yield surfaces evolve due to plastic deformations, cf. e.g. Ishikawa and Sasaki (1988). Bari and Hassan (2002) fitted the material response dur- ing non-proportional loading with advanced kine- matic hardening laws. However, an anisotropic yield criteria is required to model some load cases. Many papers in the last decades have therefore considered distortional hardening, see e.g. Feigen- baum and Dafalias (2007); Harrysson and Ristin- maa (2007); Pietryga et al. (2012); Bartels and Mosler (2017). In their most basic form, these models include self- and cross-hardening. Many of these models predict that these hardening laws only depend on the accumulated plasticity during proportional loading. However, to the authors’ knowledge, no study has been performed to vali- date this assumption. Hence, in the present work,Email address: k.a.meyer@tu-bs.de (Knut Andreas Meyer) Preprint September 16, 2022 we will also measure how the yield strength in a direction that is perpendicular to the loading evolves during cyclic loading. The paper is organized as follows: Section 2 describes the material used for testing: the microstructure, heat treatment, and the tensile stress-strain curve. Section 3 introduces the ma- terial modeling framework we use to analyze the experiments. The specific experimental setup is described and motivated in Section 4. Finally, Section 5 presents the experimental results and discusses their implications for material model- ing. For readability and simplicity, the major part of this paper considers a small strain framework. However, the experimental results are compen- sated such that the true stresses and strains are used. This compensation is outlined in Appendix A. Furthermore, a generalization of the analysis for a specific finite strain plasticity model is given in Appendix B. 2. Material The material chosen for this study is pearlitic steel, a microstructure found in many industrial applications. Examples include railway rails and wheels, as well as high strength wires used as steel cords in tires and in ropes for suspension bridges. The pearlitic microstructure with its arrangement of alternating ferrite and cementite (Fe 3C) lamel- las has a strong but saturating work hardening, cf. Taleff et al. (2002). In the present study, we start with seamless precision tubes of grade E-235 (EN10305-1 (2016)) with a 20 mm outer diameter and a 1 .5 mm wall thickness. The next subsection describes the heat treatment procedure used to obtain a well-defined pearlitic microstructure. In Subsection 2.2, the overall stress-strain behavior of the normalized microstructure is discussed and compared to other materials. 2.1. Heat treatment The goal of the heat treatment was to achieve an isotropic and homogeneous test material. In particular, the chemical composition, grain sizes, and lamellar spacing should have a low spatial variance. After cutting into 150 mm long pieces, 0.650.700.75 0.75 0.50 0.25 0.00Figure 1: Carbon concentration and microstructure after heat treatments. the tubes were carburized at 950◦C for about eight hours and dwelled at 860◦C for about three hours. During the carburization cycle, the car- bon activity varied with time to acquire an even chemical composition throughout the wall thick- ness. A high initial carbon activity gave sufficient driving force to increase the carbon concentra- tion at greater depths, while a lower activity in the next step ensured an even carbon level. Fig- ure 1a shows that the resulting carbon levels were between 0.68 and 0 .73 %. The average chemical composition is given in Table 1. After carburization, the grain sizes were larger in the middle of the tube wall compared to at the surface, shown in Figure 1b. A subsequent normalization at 900◦C for 45 minutes in argon, followed by air cooling, lead to an even grain size wt % C Si Mn S P Fe Tubes 0.71 0.2 0.5 0.003 0.03 98.4 Rails 0.76 0.2 0.9 0.02 0.02 98.0 Table 1: Mean chemical composition of heat-treated tubes in comparison to R260 rails. 2 0 1 2 3 4 5 6 705001000 ϵ[%]σ[MPa] Normalized tubes Normalized R260 As-received R260Figure 2: Stress strain behavior of railway steel R260 and heat treated cold-drawn tubes. throughout the wall thickness (Figure 1c). The two heat treatments yielded a slight deviation in roundness, meaning that exact cross sections were difficult to measure. However, based on the well- documented Young’s modulus of pearlitic steel of 210 GPa, the cross-sectional area could be easily obtained from the slope of the elastic range of the uniaxial loading curve. 2.2. Stress-strain behavior Upon yielding, the normalized test tubes ex- perience a drop in the flow stress, followed by a plateau phase, see Figure 2 (the regular unloading is nearly elastic and can be disregarded). While such behavior is well known for ferritic steels, cf. e.g. Lomer (1952), it is less common for pearlitic steels. Figure 2 shows that the pearlitic R260 steel (EN13674-1 (2011), as received) behaves very dif- ferent upon yielding. The R260 samples were ex- tracted from manufactured rails, see Meyer et al. (2018) for details. Rail straightening during the manufacturing induces some plastic deformations, which could explain the more gradual elastic-to- plastic transition. Normalizing the R260 sample causes stress plateauing, although it is less pro- nounced than for the normalized tubes. This ob- servation is in accordance with the findings of Wang et al. (2020), who found that annealing pearlitic steels only results in stress plateauing after cold working. Table 1 shows that the rail material has more manganese and slightly higher carbon content. Manganese gives a solid solutionstrengthening of the ferrite phase, while more car- bon results in a higher cementite fraction. Con- sequently, the normalized rail material has higher flow stress than the tube material. Furthermore, a higher cooling rate than during the rail pro- duction explains the increased flow stress after normalization of the rail material. Despite these differences in composition, cooling rate, and flow stresses, all test samples show similar hardening behavior after the initial plateau. 3. Material modeling This section presents the material modeling framework and how to connect it to the experi- mental results. Since many different approaches for modeling finite strain plasticity exist in the literature, we consider only the small strain no- tation in this paper. The geometrical changes are accounted for by calculating the true stresses and strains from the experiments, see Appendix A. However, the analysis is also applicable to fi- nite strain models, exemplified for a specific finite strain model in Appendix B. 3.1. General material modeling framework The models assume a quadratic yield criterion of the form Φ=p σred:C:σred−(Y0+κ)≤0 (1) where Y0is the initial yield limit, κthe isotropic hardening stress, and Ca 4th order anisotropy tensor that may evolve during plastic deforma- tion.σredis the reduced stress, i.e. σred=σ−β (2) where βis the back-stress due to kinematic hard- ening (Bauschinger effect) and σthe Cauchy stress. An additive decomposition of the strain, ϵ, into elastic, ϵe, and plastic, ϵp, parts is assumed, along with a linear elastic response, i.e. ϵ=ϵe+ϵp (3) σ=E:ϵe (4) 3 where Eis the 4th order elasticity tensor. Fur- thermore, we assume associative evolution of the plastic flow according to ˙ϵp=˙λ∂Φ ∂σ(5) Here, the plastic multiplier, ˙λ, is the rate of the accumulated plasticity, given by the standard KKT-conditions for rate-independent plasticity, i.e. Φ≤0,˙λ≥0, Φ ˙λ= 0 (6) 3.2. Isotropic hardening laws The two most common isotropic hardening laws for cyclic metal plasticity are the Voce-law and the Swift power-law. These can be formulated as Voce: κ=κ∞ 1−expHiso κ∞λ (7) Swift: κ=K[e0+λ]n(8) Some authors also combine these two laws, and even further generalizations are possible, cf. Ristinmaa et al. (2007). However, the common denominator for all these models, is that they postulate that κis a function of the accumulated plasticity λ, λ=Zt 0˙λdt (9) In Chaboche (1986), such models are denoted ”Classical isotropic hardening”. The concept of ”Maximum plastic strain memory” was intro- duced in Chaboche et al. (1979), introducing a so- called ”non-hardening surface in the plastic strain space”. This modification was motivated by the need to obtain different saturated peak stresses for different strain amplitudes. However, this cor- relates only approximately with isotropic harden- ing; see Equation 118 in Chaboche et al. (1979). Although the results show significant improve- ments, the suggested approach is rarely applied in modern literature. Some exceptions are El Abdi et al. (1999); Zhou et al. (2018). Some authors, e.g. Park et al. (2019), have adopted the alter- native of using the plastic work in place of the accumulated plasticity, λ, suggested in Lemaitre and Chaboche (1990).3.3. Distortional hardening laws Many models for distortional hardening em- ploy a yield surface described by Equation (1), e.g. Feigenbaum and Dafalias (2007); Barthel et al. (2008); Noman et al. (2010); Pietryga et al. (2012); Shi et al. (2014). Typically, the evolution of the Cis governed by n⊗nandβ, where n=σdev red σdev red (10) Terms involving only the back-stress βare typ- ically responsible for the high curvature in the loading direction observed in many experiments, e.g. Naghdi et al. (1958); Phillips and Juh-Ling (1972). However, as shown by Khan and Wang (1993), this effect is only present when a small plastic strain offset is used to define the yield points. As discussed in Section 4, such low off- sets are not relevant for most engineering appli- cations and are not employed in the present study. Barthel et al. (2008); Noman et al. (2010) do not include the high curvature in the loading direc- tion. In Pietryga et al. (2012); Shi et al. (2014) it can be removed by setting a parameter to zero (cHandbCrespectively). The terms involving n are typically split into two parts, N=n⊗nand N⊥=Idev−N, e.g. ˙C=˙λ h∥(C,N)N+h⊥(C,N)N⊥ (11) C(t= 0) =3 2Idev where Idev=I−I⊗I/3.IandIare the 4th and 2nd order identity tensors. While some works consider an initial anisotropy, the samples consid- ered herein are initially isotropic, and the initial condition for Cis chosen accordingly. The first term, along N, is responsible for self-hardening, i.e. the hardening in the current loading direc- tion. The second term, along N⊥, results in cross- hardening, i.e. hardening in directions perpendic- ular to the current loading direction. Self- and cross-hardening are also denoted by dynamic and latent hardening in the literature. If proportional loading along the fixed stress direction n0is con- sidered, i.e. σ(t) =f(t)n0 (12) 4 the back-stress, β, is co-linear with the deviatoric stress, σdev=Idev:σ, for all kinematic hardening laws known by the authors. Hence, the reduced stress may be expressed as σdev red(t) =fred(t)ndev 0 (13) causing NandN⊥to be constant. For such a case, the anisotropy tensor Cdepend only on the accumulated plasticity, λ. Thus, for proportional loading the distortional hardening in many mod- els is also governed by λ. 3.4. Measuring the plastic multiplier In the models considered herein, Cis not a di- rect function of σ, although its evolution may be. This implies that∂C ∂σ= 0, and the evolution of the plastic strains may be calculated as ˙ϵp=˙λ∂Φ ∂σ=˙λC:σred√σred:C:σred(14) Assuming that we only have proportional load- ing according to Equation (13) and by using Equation (11), Ccan be expressed as C=3 2Idev+k∥(λ)N+k⊥(λ)N⊥ (15) where the functions k∥(λ) and k⊥(λ) also depend on the material parameters. Insertion into Equa- tion (14) yields ˙ϵp=˙λsign(fred(t))r 3 2+k∥(λ)ndev 0 (16) Meyer and Menzel (2021) obtained better re- sults by replacing the self-hardening contribution with one additional isotropic hardening stress. As seen here, this choice affects the model behavior regarding the plasticity accumulation, λ. Letting k∥(λ) = 0, following Meyer and Menzel (2021), enables a direct calculation of λfrom the exper- imental results. For the analysis in the present paper, we assume k∥(λ) = 0, but explain at the end of Section 5.1 how the general expression in Equation (16) when k∥(λ)̸= 0 does not affect our conclusions. Finally, note that for axial loading, ndev 0= [2/3,−1/3,−1/3,0,0,0]. Hence, the increment ofplasticity is ∆ λ=p 3/2|[∆ϵp]11|during a mono- tonic change of [ ϵp]11. Using the additive decom- position, Equation (3), and the linear elastic law, Equation (4), yields [ϵp]11=ϵ11−σ11 E(17) where Eis the axial stiffness modulus (Young’s modulus in the case of isotropy). For improved accuracy, the stiffness Eis calibrated for the start of the specific load segment (up to a stress change of 100 MPa). The plastic strain increment is then calculated as [∆ϵp]11= [ϵp]11(ti+1)−[ϵp]11(ti) (18) Here, tiandti+1are the times at the start and end of the load segment, respectively. 4. Experimental setup We use an axial-torsion testing machine with transducer capacities 100 kN and 1100 Nm. The biaxial extensometer can measure an axial strain ϵ∈[−0.04,0.10] and a surface shear strain γ= ±0.04. The tubular test bars with outer diameter 20 mm and thickness 1 .5 mm are described in Sec- tion 4.1. While the results are compensated for large strains according to Appendix A, the load- ing parameters are set in terms of the engineering stresses and strains. Each experiment consists of a probing sequence followed by a loading sequence. The prob- ing sequence measures the axial and shear yield strengths, and is described below in Section 4.2. As discussed in Section 3, the identification pro- cedure relies on proportional loading. Therefore, we ensure that the plastic shear strains are small during the probing sequence. The individual ex- periments are characterized by their specific load- ing sequence. It can be either cyclic loading with different strain amplitudes or monotonic loading. The specific parameters are described in Section 4.3. 4.1. Test bar geometry As previously discussed in Section 2, we use heat-treated, cold-drawn tubes as test bars. Fig- ure 3 shows how the tubes are mounted in the 5 testing machine. The outer sleeve is made from hardened steel and is slotted on the rear side to improve the grip. To avoid stress concentrations at the end of the sleeve, it is not slotted next to the gauge section and extends outside the ma- chine grips. The green plug in the center prevents excessive deformations due to the gripping force. 24 Ø17 Ø20 Sym Figure 3: Tube specimen (gray) with inner plugs (green), outer sleeves (red), and machine grips (light blue). The grip force exerted by the machine is shown by blue arrows. All dimensions in mm. 4.2. Yield probing sequence Most papers on yield surface measurements use deviation from linearity to detect yielding, see e.g. Naghdi et al. (1958); Phillips et al. (1972); More- ton et al. (1978). While these studies had the op- erator deciding when to stop, more modern stud- ies include automated yield detection, cf. Sung et al. (2011). However, these studies also rely on an offset from linearity. One potential draw- back of this criterion is the assumption of linear elastic deformations. As shown by Sommer et al. (1991), this assumption may lead to incorrect re- sults for high-strength steels. Additionally, the regression of the stress-strain response in an as- sumed linear region introduces additional uncer- tainties. We avoid these issues by measuring the residual plastic strains via loading and unloading during probing. However, upon unloading after cyclic loading the material can plasticize slightly. To prevent this phenomenon, we increment the stress unloading level. Figure 4 shows the beginning of a probing se- quence in the compressive axial direction. Duringthe last 3 s of the 4 s time intervals mij, the aver- age strains are measured. The first index, i, repre- sents the current reference (unloading) level. The second index, j, counts the measurement num- ber at the current reference level. Letting the material relax for 1 s, and then taking the aver- age strains during 3 s, decreases the measurement scatter. mϵ ijandmγ ijdenote these averaged ax- ial and shear strains. The peak axial and shear stresses prior to mijare recorded at pijand de- noted pσ ijandpτ ij, respectively. The first measure- ment interval, m00, is used as a reference point where the plastic strain increment is zero by defi- nition. After Nsteps, corresponding to a 25 MPa increase in probing stress, the reference level is changed. Hence, the plastic strain ϵij patmijcan be calculated as ϵij p=mϵ ij−mϵ i0+i−1X k=0[mϵ kN−mϵ k0] (19) Each term in the sum in Equation (19) adds to the uncertainty of the strain measurements. There- fore, we did not choose a lower reference step than 25 MPa. In other loading directions, the von Mises effective stress is used. The plastic shear strain, γij pis calculated equivalently to Equation (19). 20 40 60 80 100 120 0 -5 -10 -15 -20 -25 -30 -35 σ[MPa] time [s]m00 m02 m0N m10 m12 p01 p0N··· m0j··· i= 0 i= 1 Figure 4: Probing sequence. Many measured yield surfaces exclude the ori- gin due to a translation along the axial stress di- rection. In such cases, we observed some ratch- eting in the axial direction when measuring the 6 shear yield limit. To prevent these strains from influencing the offset strain definition, the plastic strain is projected onto the stress direction (which was controlled). The stress during probing may be described as σ=σvM(t) cos( α) +σ0 (20) τ=σvM(t) sin(α)/√ 3 +τ0 (21) for a probe direction αand with a load origin (σ0, τ0). Then, the effective plastic offset strain is calculated as ϵoffset ,ij p =ϵij pcos(α) +γij psin(α)/√ 3 (22) Choosing the limit value of the offset strain ϵoffset pis governed by contradictory requirements. A lower value reduces the influence on subsequent measurements and by causing lower plastic shear strains. The latter is vital for testing the validity of the accumulated plasticity’s effect on the shear yield strength. When studying microstructural yielding processes, the onset of yielding may be of primary interest. Figure 5 shows yield points calculated by different offset definitions. In the first cycle, the 0 .001 % offset yield point occurs at a much lower stress than when the bulk material plasticizes. When considering the cyclic plastic behavior during, e.g., low cycle fatigue analyses, the plasticizing of the bulk material has more in- fluence than the onset of yielding. In the second cycle, the material has undergone some plastic de- formation. This history causes a smoother elas- tic to plastic transition without a distinct yield point. However, in the author’s experience, cal- ibrated models in the literature typically do not capture the non-linearity prior to the 0 .01 % offset point. Hence, we consider this point a reasonable compromise for the present study. The first probe is along α=−πto measure σp1, with τ0= 0 and σ0=σp0. Here, σp0is the current stress at the end of the previous loading cycle. The center of the yield surface is then given by τ0= 0 and σ0= [σp0+σp1]/2, and then we probe in directions α=±π/2 to find the shear yield strength. −1−0.5 0 0.5 1−500−2500250500 Axial strain, ϵ[%]Axial stress, σ[MPa]Cycle 1 Cycle 2 Elastic 0.001 % 0.01 % 0.1 %Figure 5: Stress-strain curve with different offset strain yield points. 4.3. Proportional loading sequences To test the modeling hypotheses identified in Section 3 during cyclic loading, we require a range of strain amplitudes. The results in Figure 5 show the material’s distinct yield at about 528 MPa, corresponding to a strain of 0 .27 %. However, some plasticity occurs before this point. Due to the significant cyclic softening of the material, a hysteresis loop developed after the first cycle atϵ=±0.25 %, see Figure 9a. In addition to ±0.25 %, we chose strains ±0.40 %,±0.80 %, and ±1.20 %. Furthermore, the modeling hypothesis can also be tested for monotonic loading. Dur- ing this loading, yield surface measurements occur each 0 .4 % strain increment (see Figure 8). 40 60 80 100560580 11.21.41.6 time [s]Axial stress, σ[MPa] ϵ[%]Stress Strain Figure 6: Stress versus time during the hold time after the first loading cycle. Even at room temperature, the material ex- hibits some relaxation after cyclic loading, as shown by Figure 6. Therefore, a hold time of 1 min follows each loading before starting the 7 /epsilon1=±0.25 %/epsilon1=±0.40 %/epsilon1=±0.80 %/epsilon1=±1.20 % Monotonic 10−310−210−1100101100200300400 Accumulated plasticity, λ[-]σy[MPa](a) Axial yield stress. 10−310−210−1100101100200300400 Accumulated plasticity, λ[-] √ 3τy[MPa] (b) Shear yield stress. Figure 7: Yield surface evolution as a function of the accumulated plasticity for monotonic and different cyclic load amplitudes. yield surface measurements. The cyclic loading rate was 0 .025 % /s, and the stress loading rate was 10 MPa /s during the probing. 5. Results and discussion The main objective of the present paper is to investigate yield surface evolution during propor- tional cyclic and monotonic loading. Hence, these results are presented first. To further understand the results and the material behavior, the cyclic stress-strain responses are discussed thereafter. Finally, we discuss how our findings should in- fluence future cyclic plasticity models. 5.1. Yield evolution Figures 7a and 7b show how the yield stress evolves with the accumulated plastic deformation λ. According to the standard modeling hypothe- sis, the isotropic yield stress should evolve equally for all loading types. Without distortional hard- ening, the axial yield stress in Figure 7a shows the isotropic yield stress evolution. And as dis- cussed in Section 3, quite some distortional hard- ening models predict the same. Such models also assume that the cross-hardening response, hereshown by the shear yield stress evolution, should only depend on λfor proportional loading. Figure 7a shows that λalone cannot describe the axial yield stress evolution. For cyclic loading and an accumulated plasticity λ < 0.1, the ax- ial yield stress evolution depends strongly on the strain amplitude. While the differences are large for amplitudes between 0 .25 % and 0 .8 %, no sig- nificant difference is observed between 0 .8 % and 1.2 %. At higher λvalues, the differences between the strain amplitudes are much smaller. Still, there is a tendency that the high strain ampli- tudes (0.8 and 1 .2 %) show a cyclic hardening for λ > 1. The lower amplitudes (0.25 and 0 .4 %) show a continued cyclic softening, although many orders of magnitudes less than the initial soften- ing (note the logarithmic λ-axis). The shear yield stress evolution in Figure 7b shows a similar trend to the axial yield stress. Scaling the τ-axis by√ 3 results in the equiva- lent von Mises stress, enabling comparisons to the axial yield stress. After the first cycle of 0 .25 % strain amplitude, the von Mises yield stresses are similar in the axial and shear directions. As for the axial results, the shear yield stresses are sim- ilar after many cycles for different strain ampli- tudes. However, the von Mises yield stress is 8 much higher than the corresponding axial yield stress, indicating an anisotropic yield surface. A notable difference to the axial response, is that the 1.2 % strain amplitude results in a lower yield stress than the 0 .8 % amplitude. The monotonic responses in Figure 7 are dif- ferent from the cyclic responses: The initial yield limit is about 500 MPa. Until λ= 0.01, the yield stress decreases for all loading cases. However, after this, the monotonic response show an in- creasing yield stress. The high amplitude load cases have a fairly constant yield stress, while the low amplitude cases keep reducing the yield limit. For shear stresses, the behavior is similar, but less pronounced. The extensometer’s strain range lim- its the maximum accumulated plasticity during monotonic loading. The accumulated plasticity, λ, in Figure 7, was calculated by assuming k∥(λ) = 0 in Equation (16). Without this assumption, it is impossible to calculate λfrom the experiments as it depends on material parameters. However, as k∥(λ) only depends on λand material parameters for propor- tional loading, k∥(λ)̸= 0 would only result in a non-uniform scaling of the x-axis. Therefore, the differences between the yield stresses would not change for such models. 5.2. Monotonic response As discussed in Section 2, the material exhibits a stress plateau after an initial peak during mono- tonic loading. Figure 8 shows the two tests for monotonic loading. The second test, shown with a dotted line, was restarted due to a computer failure during the first axial probing sequence. The restart resulted in slightly more accumulated compressive plastic strains before the first cycle. These strains explain why the behavior during the first cycle differs from the first test (solid line). After this first cycle, both tests show similar re- sults on the overall stress-strain response. The solid markers with the axial yield limits, σy, show the initial isotropic softening followed by hardening. The start point of each step shows the current yield center, i.e. β, governed by the kine- matic hardening. The stress curve is the sum of these two contributions. Hence, the initial plateau 0 2 4 6 802004006008001000 Axial strain, ϵ[%]Axial stress [MPa]Stress, σ Stress, σ Yield stress, σy Back-stress, βFigure 8: Two monotonic experiments, where each new step starts from the identified yield surface center, σ=β. seem to occur due to the competing effects of a rapid isotropic softening and kinematic harden- ing. The kinematic hardening continues to harden with decreasing rate. When isotropic softening decreases, the peak stress begin increasing. While isotropic hardening can explain a small part of the stress increase, most is due to the continued kine- matic hardening. 5.3. Hysteresis loops The first hysteresis loop in Figure 9a is almost fully elastic. At the end of the cycle, it seems to pass over the initial peak observed in e.g. Figure 8. In the second cycle, the wider loop shape shows more plasticity. The lack of any stress plateau is very different from the monotonic stress-strain response, even though the amount of plasticity is similar. Hence, it seems that the load reversal strongly impacts the material response. The be- havior with an initial peak followed by a rapid softening makes the experiments sensitive to any heterogeneities in the test bars. If some parts yield before others, the global stress-strain re- sponse might not exhibit such a peak. There- fore, results at small plastic deformations before this peak must be analyzed with care, as this sen- sitivity introduces additional uncertainty in the 9 −0.2−0.10 0.10.20.30.4−600−400−2000200400600 Axial strain, ϵ[%]Axial stress, σ[MPa](a) First 4 cycles for ϵ=±0.25 %. Each cycle is shifted 0 .06 % from the previous for better visibility (1 test bar). −0.4 −0.2 0 0.2 0.4−600−400−2000200400600 Axial strain, ϵ[%]Axial stress, σ[MPa](b) First cycle for ϵ=±0.40 % (4 test bars). −0.8−0.6−0.4−0.200.20.40.60.8−600−400−2000200400600 Axial strain, ϵ[%]Axial stress, σ[MPa]Compression Tension (c) First cycle for ϵ=±0.80 % (3 test bars). −1−0.5 0 0.5 1−600−400−2000200400600 Axial strain, ϵ[%]Axial stress, σ[MPa] (d) First 2 cycles for ϵ=±1.20 % (1 test bar). Figure 9: Initial hysteresis loops for different strain amplitudes. experimental results. However, after some plas- tic deformations, the material responses show less spread. This observation is illustrated clearly by Figure 7a, where the measurement uncertainty decrease with more plastic deformations. For 0.4 and 0 .8 % strain amplitude, Figures 9b and 9c, there are significant plastic strains already in the first cycle. Some experiments show similar behavior to the monotonic experiment in Figure 8 exhibiting an initial peak. However, the uncer- tainties during this initial loading phase are high- lighted by Figures 9b and 9c. While all experi- ments show the plateau phase, not all show a clear initial peak. In some experiments, the machine grips were not perfectly aligned, causing small bending stresses in the samples. These stresses affect the initial plasticity, but the results after the first cycle do not seem to have been affected.The tensile-first experiment in Figure 9c was con- ducted before alignment, while the monotonic ex- periments in Figure 8 were conducted after align- ment. Overall, at least one experiment after align- ment was performed for each test condition. No clear effect was observed on the yield evolution. Therefore, we also include the results from exper- iments conducted before the alignment. The stress during the first cycle for the 1 .2 % strain amplitude plateaus after yielding until about −0.9 % strain, and then hardening occurs. This response is similar to that in Figure 8. Af- ter this initial phase, a smooth elastic-to-plastic transition occurs in each cycle. 5.4. Peak stress evolution For the monotonic stress-strain response in Figure 8, we previously concluded that the Bauschinger effect caused most of the hardening. 10 10−410−310−210−1100101400500600 Accumulated plasticity, λ[-]Axial stress, σ[MPa]ϵ=±0.25 % ϵ=±0.40 % ϵ=±0.80 % ϵ=±1.20 %Figure 10: Peak (dashed) and valley (absolute value, solid) axial stress evolution. The peak and valley stress evolutions in Figure 10 support this conclusion as no large increase in stress ranges occurs. Based on the axial yield stress evolution, we could observe a slight hard- ening tendency for the high amplitudes. The re- sults in Figure 10 reinforces this finding. Fur- thermore, these results also show the previously observed continued softening for low strain am- plitudes. A similar behavior is also observed in Fig. 4 in Ahlstr¨ om and Karlsson (2005) for the pearlitic rail grade 900A. 5.5. Discussion of implication for future models The results show that the common assumption that the yield surface size depends only on the ac- cumulated plasticity leads to incorrect model pre- dictions. Of course, there may exist materials for which this assumption holds, as our results only consider one pearlitic steel. Previously, we men- tioned the rarely used concept of ”Maximum plas- tic strain memory” introduced by Chaboche et al. (1979). This extension should account for the dif- ferent stabilized isotropic hardening for different plastic strain amplitudes. However, our results show similar stabilized isotropic hardening for dif- ferent amplitudes. Instead, the evolution before an approximate stable cycle depends on the strainamplitude. Hence, the ”Maximum plastic strain memory” strategy does not capture this material behavior. One important observation from our results is the qualitative similarity between the axial and shear yield stress evolution. This find- ing indicates that similar evolution laws may be used, although with different parameters. As mentioned in Section 5.2, the plateau ef- fect can be seen as a competing effect of isotropic softening and kinematic hardening. Such a re- sponse can be approximated by a standard ma- terial model with rapid isotropic softening bal- anced by an approximately equally strong kine- matic hardening with higher saturation stress. However, as previously stated, isotropic harden- ing cannot be specified as a function of only the accumulated plasticity, λ, when modeling differ- ent load cases. Adding multiple isotropic hard- ening stresses as in Qin et al. (2018); Meyer and Menzel (2021) cannot solve this problem. Other state variables are required in the evolution laws for isotropic and distortional hardening. 6. Contributions The yield surface evolution during cyclic and monotonic loading has been studied. In particu- lar, we question how many cyclic plasticity mod- els assume that the isotropic yield strength only depends on the accumulated plasticity. Our re- sults reveal that this assumption does not hold for the investigated steel. Furthermore, we show that many distortional hardening models predict the same behavior for cross-hardening during propor- tional loading. However, according to our results, the accumulated plasticity cannot capture cross- hardening for different loading cases. Adding mul- tiple hardening stresses cannot solve the identi- fied discrepancies. Instead, our findings clarify the need for new evolution laws to obtain more accurate models. 7. Acknowledgements The major part of this work was conducted within the CHARMEC Centre of Excellence at Chalmers University of Technology. Parts of the 11 study have been funded within the Shift2Rail Joint Undertaking in European Union’s Horizon 2020 research and innovation programme in the project In2Track2 and In2Track3 under grant agreements nos. 826255 and 10102456. K. A. Meyer has moved from Chalmers University of Technology to TU Braunschweig during the final- ization of this paper. 8. CRediT K.A.Meyer: Conceptualization, Methodol- ogy, Formal analysis, Investigation, Writing - Original Draft, Visualization J.Ahlstr¨ om: In- vestigation, Writing - Original Draft A. True stress and strain calculation In order to estimate the true stresses, σand τ, linear elasticity wrt. to the logarithmic ax- ial strain, ϵ, using constant parameters, E= 210 GPa and ν= 0.3, is assumed. We measure the stresses Pσ=F/A andPτ=T/(AR) based on the force, F, torque T, initial area A, and initial mean radius R. The engineering strains ˜ ϵ= ∆L/L and γ=r∆ϕ/Lare also measured. ϵ= ln(1 + ˜ ϵ) (23) ϵe≈Pσ/E, ϵ =ϵe+ϵp (24) ϵrr=ϵθθ=−νϵe−0.5ϵp (25) The elastic strain ϵeis small. Hence, the error in- troduced by the approximation in Equation (24) on the transverse strains in Equation (25) is ac- ceptable. Note that the compensated stress is used when calculating the plastic strains in Equa- tion (17) for measuring the accumulated plastic- ity. Equation (25) uses the plastic incompressibil- ity following the associative plastic evolution law in Equation (5) The updated area acan then be calculated from the undeformed area A as a=A 1−νϵe−0.5ϵp2(26) and the updated mean radius rfrom the original radius Ras r=R 1−νϵe−0.5ϵp (27)This gives the updated stresses σ=Pσh 1 + [0 .5−ν]Pσ E−0.5ϵi2(28) τ=Pτh 1 + [0 .5−ν]Pσ E−0.5ϵi3(29) B. Finite strain formulation In this paper we presented method for identify- ing the accumulated plastic deformation directly from experimental data. However, the analysis was based on a linear kinematics, or a small strain analysis. For large plastic deformations, a finite strain model may be required. In this appendix, we show that the same approach can be used in the case of a finite strain model, although the derivations are more involved. To this end, we use the model proposed in Meyer and Menzel (2021) as an example. B.1. Model description First, we briefly describe the model denoted Hr= 0 in Meyer and Menzel (2021). It has a multiplicative split of the deformation gradient; F=FeFp. Based on a Neo-Hookean formula- tion, the Mandel stress Mis M=GCdev e I1/3 3Ce+Kh I3Ce−q I3Cei I (30) where Ce=Ft eFeand I 3Ce= det ( Ce). The yield criterion, Φ= 0, is based on linear transforma- tions to introduce anisotropy, Φ=r 3 2[C:Mred]t: [C:Mred]−[Y0+κ] =q Mred:ˆC:Mred−[Y0+κ] (31) ˆC=3 2CT: [I⊗I] :C 12 Mredis the Mandel stress reduced by the back- stresses Mkin,i, i.e. Mred=M−NbackX i=1Mkin,i (32) Mkin,i=Hkin,icdev kin,i det ckin,i1/3(33) where ckin,i=F−t kin,iF−1 kin,i. This introduces the kinematic deformation gradients Fkin,i. The non- standard open product ⊗is defined by [ a⊗b] : c=actbtwhere a,b, and care 2nd order ten- sors. The associative flow rule, ˙FpF−1 p=:Lp=˙λν,ν=∂Φ ∂M(34) and the non-associative kinematic hardening via Lkin,i:=˙Fkin,iF−1 kin,i: Lkin,i=˙λ −ν+ν∗ kin,i ν∗ kin,i=δhp νt:νiMt kin,i B∞,i+ [1−δ]Mkin,i:ˆν B∞,i ν (35) ˆν=ν√ νt:ν govern the evolution of FpandFkin,i. Isotropic hardening, κ=κ1+κ2, has the form of Equation (8) for κ1andκ2. Similar to in Equation (14), νbecomes ν=ˆC:Mred+Mred:ˆC 2q Mred:ˆC:Mred(36) The evolution of Cgoverns the anisotropy, and does not include self-hardening, i.e. C= [1−b⊥]Idev+b⊥C⊥ (37) ˙C⊥=−˙λc⊥h NT ⊥::C⊥i N⊥ (38) N⊥=Idev−Mdev red⊗h Mdev redit Mdev red:h Mdev reditB.2. Identification of the accumulated plasticity To identify the accumulated plasticity directly from experiments, we must be able to obtain ˙λ using Equation (34). Hence, we must be able to calculate Fpandνin the experiment. In fi- nite strains, it is important that the proportional loading occurs in a normal loading direction to avoid any rotations. We will therefore consider a uniaxial experiment. The Kirchhoff stress τand deformation gradient Fare then τ= 0 0 0 0 0 0 0 0 s ,F= y0 0 0y0 0 0 x  (39) where xis measured in the experiment, and yand sare not directly measured. We can measure a nominal axial stress, S, which is related to the Kirchhoff axial stress, s=Sx. Henceforth, we will denote a tensor of the form of F; diagonal with equal 11 and 22 components, a 2-1-diagonal tensor. Such a tensor has two properties that we will use: Consider that aandbare 2-1 diagonal. Then, abis also 2-1 diagonal, and there exists a scalar ksuch that adev=kbdev. In the elastic state, since Fe=F, M=Ft eτF−t e=FtτF−tτ (40) follows from Equation (39) Next, we perform one explicit integration step into the plastic regime from the onset of yielding. In this case, M=τ,C⊥=Idev, andMkin,i=0, yielding ν= sign( s) −0.5 0 0 0−0.5 0 0 0 1  (41) Using this ν, we obtain the new Fpfrom Equa- tion (34) as Fp=oldFp+ ∆λνoldFp (42) Fpis thus a 2-1-diagonal tensor ifoldFpis a 2-1- diagonal tensor. By induction (initial value I),Fp will then remain 2-1-diagonal assuming νretains this property. Hence, we conclude that Fe= ye0 0 0ye0 0 0 xe  (43) 13 resulting in τ=Mduring proportional uniaxial loading. To investigate whether νremains a 2-1- diagonal tensor, we must also consider the evolu- tion of Mkin,iandˆC. Following Equation (35), we have Fkin,i=oldFkin,i−∆λ[ν−ν∗]oldFkin,i(44) and we obtain that Fkin,iretains its 2-1-diagonal property if νandMkin,ialso does. Subsequently ckin,iis 2-1-diagonal and thus also Mkin,i(Equa- tion (33)). The back-stress can then be expressed asMkin,i=bMdevwhere bis some scalar value. The evolution of C⊥becomes C⊥=Idev+aN⊥, yielding ˆC=3 2h Idev+aNT ⊥i : [I⊗I] :h Idev+aN⊥i (45) With these updated tensors, we can now eval- uateνto investigate its form. To that end, we first evaluate ˆC:Mred=ˆC:Mdev red=3 2h Mdev redit (46) and hence ν= sign( s[1−b]) −0.5 0 0 0−0.5 0 0 0 1  (47) is a 2-1-diagonal, known, tensor. Finally, it remains to identify Fpfrom the mea- sured Sandx. This is possible, by using the plas- tic incompressibility, det( Fp) = 1 and the elas- tic law in Equation (30) assuming known elastic constants. A non-linear equation system must be solved. In conclusion, we can also in the case of a finite strain formulation use the proposed calibra- tion strategy and investigation of the role of the accumulated plasticity λ.