An approximate method for calculating elastic-plastic stress and strain on notched specimens

  Citation: Lutovinov, M.; Halama, R.; Papuga, J.; Bartošák, M.; Kuželka, J.; R˚užiˇcka, M. An Approximate Method for Calculating Elastic–Plastic Stress and Strain on Notched Specimens. Materials 2022, 15, 1432. https://doi.org/10.3390/ ma15041432 Academic Editor: Wojciech Borek Received: 30 December 2021 Accepted: 10 February 2022 Published: 15 February 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). materials Article An Approximate Method for Calculating Elastic–Plastic Stress and Strain on Notched Specimens Maxim Lutovinov 1, Radim Halama 2,* , Jan Papuga 1, Michal Bartošák1, JiˇríKuželka 1and Milan R˚užiˇcka 1 1Faculty of Mechanical Engineering, Czech Technical University in Prague, Technická 4, 16607 Prague, Czech Republic; [email protected] (M.L.); [email protected] (J.P.); [email protected] (M.B.); [email protected] (J.K.); [email protected] (M.R.) 2Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 17.listopadu 2172/15, 708 00 Ostrava, Czech Republic *Correspondence: [email protected] Abstract: The paper deals with an approximate method for calculating elastic–plastic stresses and strains on the surface of notched samples. The method is based on the Abdel–Karim–Ohno cyclic plasticity model. The plane stress condition is considered within the evaluation. The output of the approximation on several multiaxial axial–torsion load paths is compared to our own experimental results. Experiments were carried out on samples of two notch types manufactured from the 2124-T851 aluminum alloy. Strain distribution in the notch area was measured by digital image correlation. The comparison between computational solution and measured response shows that the new method allows for obtaining reasonably good approximation, even for relatively complicated multiaxial load cases. Keywords: plasticity; multiaxial loading; pseudostress; stress–strain estimation; 2124-T851 1. Introduction Most initially isotropic engineering materials exhibit elastic–plastic behavior. However, the finite-element calculation of plasticity is time-consuming and requires more input data. Engineers dealing with fatigue life estimation, therefore, often use approaches that consider only elastic behavior to assess the durability of structures. Such a simplification is possible in the domain of high-cycle fatigue, but it is unacceptable if the low-cycle regime is evaluated, where the scale of plasticity is much more substantial. In order to assess the elastic–plastic stress state, approximate methods that take elastic–plastic material behavior into consideration might be a good alternative to time–consuming finite-element elastic–plastic analyses whenever a longer load history should be analyzed. Many methods have already been suggested for the estimation of elastic–plastic stresses and strains. The first group of methods contains those intended for monotonic loading only [ 1 – 5 ]. The methods do not take cyclic hardening or cyclic softening into account, and they do not describe the movement of the yield surface. Therefore, this group of methods is not suitable for cyclic loading. The investigation of these methods can be found in [6]. The second group of methods [ 7 – 16 ] deals with cycling loading and incorporates plasticity models to describe cyclic hardening or softening and yield surface movement. Unlike finite-element analyses (FEA), approximate methods do not deal with elastic–plastic stiffness matrices to obtain a solution. Instead, they use an elastic solution that they convert into an elastic–plastic solution by using a relation either between pseudomaterial and real material or between linear–elastic and elastic–plastic strain energies. Barkey introduced one of the first elastic–plastic stress–strain approximation methods for cyclic loading [ 7 ]. Nominal stresses and plastic strains were related in his work in order to retrieve the approximation. The kinematic work-hardening model of Mróz [ 17 ] was used Materials 2022,15, 1432. https://doi.org/10.3390/ma15041432 https://www.mdpi.com/journal/materials Materials 2022,15, 1432 2 of 22 to describe elastic–plastic behavior. Barkey reported good correlation between estimates with the experiments and FEA. Two approaches, pseudonotch stress and pseudonotch strain, were presented in [ 8 ]. Both approaches incorporated the Mróz model of plasticity. The essence of the methods is in relating entities from a purely elastic solution to real elastic–plastic ones, e.g., the equivalent pseudostress to real plastic equivalent strain, creating a pseudomaterial with it. The behavior of the pseudomaterial is simultaneously characterized by stresses from the elastic solution and by plastic strains from the real elastic–plastic material response. In conjunction with Koettgen’s model, Langlais [ 9 ] used the infinite surface hardening rule, which is a modification of Mróz’s model presented by Chu [ 18 ]. Instead of the flow rule, Langlais used Drucker’s equation, which related plastic strain rate and generalized plastic modulus. Langlais reported that the same level of precision was achieved as in [ 7 , 8 ]. Firat [ 10 ] used a pseudostress method similar to the method presented by Koettgen. The approach was combined with the rate-independent plasticity model by Chaboche [ 19 ]. The author reported a high accuracy of the predicted values. Ince et al. [ 11 ] combined the Prandtl–Reuss flow rule [ 20 , 21 ], an assumption about the equivalence of increments of the total distortional strain energy density, and the Garud multisurface plasticity model [ 22 ]. The authors reported nonconservative estimates, as strain ranges were predicted to be 4–15% smaller than the experimental results that had been used for comparison. Regarding energy approaches, an overview of the equivalent strain energy density (ESED) approach development by Glinka and coworkers is described in [23]. Ye et al. [ 12 ] proposed a new unified expression based on the thermodynamic analysis of cyclic plastic deformation. The authors used this solution with the material constitutive model proposed by Jiang and Sehitoglu [ 24 ] to estimate notch stresses and strains. The mean relative errors were reported to be − 3.5% for the axial strain component, and − 3.9% for the shear strain component. The authors indicated that the unified expression developed in the paper had the range of applicability limited to chosen geometries and loading conditions, and that further verifications of the proposed approximate method were needed. One of the recent works on the topic of pseudocurve approaches was conducted by Li et al. [13] . They combined the pseudostrain method with the Jiang–Sehitoglu plasticity model. The authors reported reasonable results under multiaxial cyclic axial–torsion loading . The approach in [ 14 ] used tangent moduli of pseudo and real curves to calculate the real stress history. The Garud plasticity model was used to describe the behavior of the material. Experimental notch strain data were presented for samples from the TC21 titanium alloy. To the authors’ knowledge, it is the first work in which notch stress–strain correction method estimates were validated on other material than steel. Similarly to [ 14 ], Li et al. [ 15 ] used tangent moduli for stress estimation, but their method also took into account the influence of temperature. This was achieved by incorporating the Ramberg–Osgood equation in high-temperature form [ 25 ]. Estimates were validated against FEA. Kraft and Vormwald [ 16 ] combined the unified expression of [ 12 ] with the Ohno– Wang plasticity model [ 26 ]. The integration algorithm used to calculate the elastic–plastic variables was described in depth. One drawback of some of the methods mentioned above, also noted by other authors [ 13 ], is that the methods are not sufficiently described for replication by other research teams. The second drawback of the current state of such methods is that experimental verification was only carried out on a limited number of materials and test paths (Table 1). Table 1shows that most of the materials used for the validation of the methods were steel types, and only four sources of experimental data were used. This proves that a wider experimental investigation is necessary to explore the range of applicability of such methods. Materials 2022,15, 1432 3 of 22 Table 1. Materials used in other papers for the validation of the methods for calculating the elastic– plastic stress and strain. Author Material Used Data Barkey [7] 1070 steel [7] Koettgen et al. [8] steel only FEA Langlais [9] 1070 steel [7] Firat [10] 1070 steel [7] Ince et al. [11] 1070 steel [7] Ye et al. [12] S460N steel [12] Li et al. [13] 1070 steel, S460N steel [7,12] Tao et al. [14] TC21 titanium alloy, 1070 steel [7,14] Li et al. [15] GH4169 superalloy only FEA Kraft [16] steel only FEA The present paper introduces a new combination of the pseudostress method and the Abdel–Karim–Ohno (AKO) plasticity model [ 27 ]. The article describes the pseudostress– plastic strain approach. Validation was performed on the own new experimental data measured on notched samples manufactured from the 2124-T851 aluminum alloy and on available experimental data of 1070 steel [ 7 ] and TC21 titanium alloy [ 14 ]. The AKO model has its advantages in an additional parameter that influences the ratcheting response. It can be enhanced by a memory surface introduction [ 28 ] in a future application of the pseudostress–plastic strain approach. 2. Materials and Methods 2.1. Approximate Method Pseudomaterial approaches are based on material behavior that couples either elastic stress with elastic–plastic strain or elastic strain with elastic–plastic stress. The behavior of the material can be represented by a pseudocurve that is analogous to the static/cyclic stress–strain curve (Figure 1). In the case of the pseudostress–real plastic strain curve, which is used in this work, the pseudomaterial experiences stresses that correspond to the stresses related to a purely elastic solution, while it plasticizes according to its real material response at the same time. Materials 2022, 15, x FOR PEER REVIEW 3 of 24 Table 1. Materials used in other papers for the validation of the methods for calculating the elastic– plastic stress and strain. Author Material Used Data Barkey [7] 1070 steel [7] Koettgen et al. [8] steel only FEA Langlais [9] 1070 steel [7] Firat [10] 1070 steel [7] Ince et al. [11] 1070 steel [7] Ye et al. [12] S460N steel [12] Li et al. [13] 1070 steel, S460N steel [7,12] Tao et al. [14] TC21 titanium alloy, 1070 steel [7,14] Li et al. [15] GH4169 superalloy only FEA Kraft [16] steel only FEA Table 1 shows that most of the materials used for the validation of the methods were steel types, and only four sources of experimental data were used. This proves that a wider experimental investigation is necessary to explore the range of applicability of such methods. The present paper introduces a new combination of the pseudostress method and the Abdel–Karim–Ohno (AKO) plasticity model [27]. The article describes the pseudostress– plastic strain approach. Validation was performed on the own new experimental data measured on notched samples manufactured from the 2124-T851 aluminum alloy and on available experimental data of 1070 steel [7] and TC21 titanium alloy [14]. The AKO model has its advantages in an additional parameter that influences the ratcheting response. It can be enhanced by a memory surface introduction [28] in a future application of the pseudostress–plastic strain approach. 2. Materials and Methods 2.1. Approximate Method Pseudomaterial approaches are based on material behavior that couples either elastic stress with elastic–plastic strain or elastic strain with elastic–plastic stress. The behavior of the material can be represented by a pseudocurve that is analogous to the static/cyclic stress–strain curve (Figure 1). In the case of the pseudostress–real plastic strain curve, which is used in this work, the pseudomaterial experiences stresses that correspond to the stresses related to a purely elastic solution, while it plasticizes according to its real material response at the same time. Figure 1. Pseudostress–real plastic strain curve and cyclic stress–strain curve; points represent discrete versions of curves based on the Hollomon parameters. Figure 1. Pseudostress–real plastic strain curve and cyclic stress–strain curve; points represent discrete versions of curves based on the Hollomon parameters. There are two types of pseudostress approaches. In the first approach, pseudostress is paired with the total strain [ 13 ]; in the second type, pseudostress is paired with the plastic strain [ 8 , 10 ]. The main difference between the two approaches is in the strain component that is obtained when a plasticity model is applied to pseudostress history, namely, if it is a total strain tensor or a plastic strain tensor. In the present paper, the second type of solution is used. Materials 2022,15, 1432 4 of 22 2.1.1. Establishing Pseudostress Material Curve A pseudocurve was established by combining elastic stress with plastic strain. The plastic strain values were the same as the plastic strain values of the real cyclic stress–strain (CSS) curve. The CSS curve could be obtained by the Ramberg–Osgood expression using Hollomon parameters (Section 2.3). Because the number of selected plastic strain values was finite, both curves were discrete. Elastic stress was calculated on the basis of the modification of the Neuber rule: σe=rσεp+σ EE, (1) where σ and εp are real stress and plastic strain described by the cyclic stress–strain curve, respectively, and Eis Young’s modulus. When the curve was established, the parameters of plasticity model C i and γi were calculated. The number of intervals between discrete points iof pseudo or real curves affected the number of backstresses used in the approximation because pairs of C i and γi parameters were calculated here for each discrete interval of either a pseudo or real curve, and the number of pairs of Ciand γigave the number of backstresses. During the approximation process, curves were represented solely by the C i and γi parameters. They are not referenced in any other way. 2.1.2. Getting Real Strain and Real Stress When parameters C i and γi representing pseudo and real curves had been defined, the plasticity model was applied to the elastic stress history. Elastic stress history can be obtained, e.g., for a chosen loading path from an elastic FEA. Because of the way the pseudocurve was built, this step provided a real plastic stress tensor and accumulated plastic strain as its outputs. Detailed analyses of this property of pseudomaterial are presented in Section 2.1.3. Once the plastic strain tensor had been obtained, the plasticity model was applied again, and the real stress and real total strain were estimated. However, this time, the incremental algorithm to acquire the accumulated plastic strain was not involved because it had been calculated in the previous step. 2.1.3. Equivalence of Pseudo and Real Plastic Strain Tensors The key part in calculating the real response from the pseudovariables is the equivalence of the accumulated plastic strain dp of the pseudocurve and of the real stress–strain curve. This was ensured by the way the pseudocurve had been established (Section 2.1.1). Unlike the case of accumulated plastic strain, the equivalence of the pseudoplastic strain tensor and of the real plastic strain tensor was not explicitly stated in [ 8 ] or in [ 10 ], where similar approaches were used. However, it was stated in [ 8 ] that applying the plasticity model to the pseudostress history results in a real plastic strain tensor. This statement supports the claim of plastic strain tensor equivalence. Justification can be found when analyzing the widely used relationship between accumulated plastic strain increment dp and plastic strain tensor increment dεp dp =r2 3dεp:dεp, (2) and the flow rule: dεp=3 2dps−a σy(3) where s is the deviatoric part of the stress tensor, a is the deviatoric part of the backstress, and σyis yield strength. dp and σywere identical for the real curve and the pseudocurve. Due to the intrinsic difference between the elastic and elastic–plastic material behavior of isotropic materials, the real stress was smaller than the pseudostress under the same Materials 2022,15, 1432 5 of 22 load. Backstress α followed the stress while maintaining the radius of the yield sphere during the kinematic hardening. Because of this, it was safe to assume that tensors s−a for the real and the pseudomaterial similarly changed; more specifically, the corresponding components of s−a changed in a similar manner for both materials. If one component increased for the pseudomaterial, the corresponding component of the real material also increased. In other words, the increments should have at least the same sign. The increment of plastic strain was obtained by multiplying s−a . Therefore, the same assumptions as for s−a tensor were valid for the increment of plastic strain. However, then, if all the corresponding components of the real/pseudoplastic strain changed in a similar manner by either increasing or decreasing, they could not provide the same dp in Equation (2) unless they were equal. Hence, the components of pseudo and real plastic strains had to be the same. 2.1.4. Approximation Method Step by Step This section summarizes the approximation method described in Sections 2.1.1 and 2.1.2: 1. Pseudomaterial curve is established. 2. Pseudostress history is obtained either by elastic FEA or using stress concentration factors [10]. 3. The plasticity model is applied to the pseudostress history. In this step, plasticity parameters Ci and γ iobtained for the pseudomaterial are used. The plastic strain tensor and the accumulated strain are calculated. 4. The plasticity model is applied to the plastic strain tensor obtained and to the accumulated strain. In this step, the plasticity parameters Ci and γ ifor the real material are used. Real stress and real backstress are calculated. 2.2. Abdel–Karim–Ohno Plasticity Model The time-independent theory of plasticity was considered because the aluminum alloy is not sensitive to the strain rate. It was assumed that the total strain tensor was composed of elastic and plastic strain tensors εe,εpaccording to the additive rule: ε=εe+εp. (4) Stresses are computed using Hooke’s law: σ=D:εe, (5) where D is the elastic stiffness tensor of the fourth order. The yield surface is introduced in the deviatoric space based on the von Mises condition: f(σ)=r3 2(s−a):(s−a)–σY=0, (6) where σY represents the radius of the yield surface (and yield strength). The flow rule and the accumulated plastic strain were defined in Equations (2) and (3). According to Chaboche [ 29 ], the backstress is constructed by superposing relevant parts of the backstresses: a=∑M i=1a(i). (7) The kinematic hardening rule is introduced in accordance with Abdel–Karim and Ohno theory ([27], AKO), i.e., the evolution of backstress is defined by the differential equation: da(i)=2 3Cidεp−µiγia(i)dp −γiH(fi)hdλiia(i), (8) Materials 2022,15, 1432 6 of 22 where fi=3 2a(i):a(i)−Ci γi2 , (9) dλi=dεp:a(i) Ci/(γi)–µidp. (10) In Equations (8)–(10), Ci , γi are basic material parameters, µi is the ratcheting parameter, the symbol hxi represents Macaulay brackets ( hxi=(x+|x|)/ 2) and H(fi) is the Heaviside step function. Under uniaxial loading, the model gives plastic shakedown for µi= 0 for all i . Then, more precisely, the multilinear model of Ohno and Wang [ 26 ] is obtained. This option is called OWI hereafter (Ohno-Wang model I, see [ 26 ]). The maximal ratcheting rate is given by µi= 1 for all i (Armstrong and the Frederick rule [ 30 ]), and the corresponding option is marked as CHAB (Chaboche model [19]). The same value of the ratcheting parameter is usually set for all backstress parts, µi=µ for all i , and its value is usually small. Therefore, the AKO model is calibrated in the same way as the OWI model [ 26 ]. The selected points of the cyclic stress–strain curve directly define the values of the basic material parameters by relations Ci=σa(i)−σa(i−1) εap(i)−εap(i−1)−σa(i+1)−σa(i) εap(i+1)−εap(i) for i6=M, (11) CM=σa(M)−σa(M−1) εap(M)−εap(M−1) , (12) γi=1 εap(i) for all i, (13) where σa(i) , εap(i) are the stress amplitude and the plastic strain amplitude of the given i -th point, respectively. The resulting basic material parameters for the 2124-T851 aluminum alloy are listed in Table 2. Table 2. Material parameters of plasticity model obtained for 2124-T851 aluminum alloy. Real Pseudo Parameter Value Parameter Value C1(MPa) 16,401 C1(MPa) 51,760 γ1(-) 500 γ1(-) 500 C2(MPa) 4561 C2(MPa) 7205 γ2(-) 250 γ2(-) 250 C3(MPa) 1948 C3(MPa) 3908 γ3(-) 166.7 γ3(-) 166.7 C4(MPa) 1097 C4(MPa) 2637 γ4(-) 125 γ4(-) 125 C5(MPa) 4216 C5(MPa) 27,589 γ5(-) 100 γ5(-) 100 2.3. Experimental Data To verify the approximation method, fatigue experiments were carried out on two types of notched samples (Figures 2and 3) manufactured from aluminum alloy 2124-T851 . The Young’s modulus of the material was 73,100 MPa [31], Poisson number was 0.33, and Hollomon parameters for the Ramberg–Osgood curve were K= 646 MPa and n= 0.089 . Cyclic yield strength σy of 330 MPa was used for the FEA simulation and for the approximation method. Materials 2022,15, 1432 7 of 22 Materials 2022, 15, x FOR PEER REVIEW 7 of 24 Cyclic yield strength 𝜎 of 330 MPa was used for the FEA simulation and for the approximation method. Figure 2. Specimen with U-notch. A and B in frames indicate surfaces based on which a datum axis for geometrical run-out tolerance is defined. Figure 3. Specimen with fillet. A and B in frames indicate surfaces based on which a datum axis for geometrical run-out tolerance is defined. The aim of the experiments was to measure the notch tip strains, specifically the axial and shear strain components. These components are commonly used to compare estimates with experimental results [7–16]. Experiments were carried out under force and moment control. The first reason for the load-controlled experiments is the possibility of recalculating the loading forces and moments into the local elastic notch stress history using the stress concentration factors as described in [10]. However, in this work, the elastic FEA was used to obtain the local stress history. The second reason for using force control is that the strain control of notched specimens would require complex real-time notch strain measurement and processing. The stress ratio of nominal axial stress to shear stress was 1 for paths Square, NV shape, X, and path 7 (see Figure 4). In the case of path Circle, results corresponding to stress ratios 1 and 1.73 are presented. For stress ratio 1, maximal force was 65.8 kN, and maximal moment was 329 Nm. For the 1.73 ratio, maximal force was 100.5 kN, and maximal moment was 290 Nm. Experiments were carried out at room temperature. Figure 2. Specimen with U-notch. A and B in frames indicate surfaces based on which a datum axis for geometrical run-out tolerance is defined. Materials 2022, 15, x FOR PEER REVIEW 7 of 24 Cyclic yield strength 𝜎 of 330 MPa was used for the FEA simulation and for the approximation method. Figure 2. Specimen with U-notch. A and B in frames indicate surfaces based on which a datum axis for geometrical run-out tolerance is defined. Figure 3. Specimen with fillet. A and B in frames indicate surfaces based on which a datum axis for geometrical run-out tolerance is defined. The aim of the experiments was to measure the notch tip strains, specifically the axial and shear strain components. These components are commonly used to compare estimates with experimental results [7–16]. Experiments were carried out under force and moment control. The first reason for the load-controlled experiments is the possibility of recalculating the loading forces and moments into the local elastic notch stress history using the stress concentration factors as described in [10]. However, in this work, the elastic FEA was used to obtain the local stress history. The second reason for using force control is that the strain control of notched specimens would require complex real-time notch strain measurement and processing. The stress ratio of nominal axial stress to shear stress was 1 for paths Square, NV shape, X, and path 7 (see Figure 4). In the case of path Circle, results corresponding to stress ratios 1 and 1.73 are presented. For stress ratio 1, maximal force was 65.8 kN, and maximal moment was 329 Nm. For the 1.73 ratio, maximal force was 100.5 kN, and maximal moment was 290 Nm. Experiments were carried out at room temperature. Figure 3. Specimen with fillet. A and B in frames indicate surfaces based on which a datum axis for geometrical run-out tolerance is defined. The aim of the experiments was to measure the notch tip strains, specifically the axial and shear strain components. These components are commonly used to compare estimates with experimental results [7–16]. Experiments were carried out under force and moment control. The first reason for the load-controlled experiments is the possibility of recalculating the loading forces and moments into the local elastic notch stress history using the stress concentration factors as described in [ 10 ]. However, in this work, the elastic FEA was used to obtain the local stress history. The second reason for using force control is that the strain control of notched specimens would require complex real-time notch strain measurement and processing. The stress ratio of nominal axial stress to shear stress was 1 for paths Square, NV shape, X, and path 7 (see Figure 4). In the case of path Circle, results corresponding to stress ratios 1 and 1.73 are presented. For stress ratio 1, maximal force was 65.8 kN, and maximal moment was 329 Nm. For the 1.73 ratio, maximal force was 100.5 kN, and maximal moment was 290 Nm. Experiments were carried out at room temperature. Path Square was achieved by multiaxial loading by trapezoidal waveforms of force and moment signals with a mutual phase shift of 90 ◦ . The common load frequency for the entire test was 0.0417 Hz. Other authors used a similar path to validate their estimates [7,9–16]. For the NV path, both channels had sinusoidal waveforms. The loading frequency of the torsion channel was five times faster than that of the tension compression channel. For the digital image correlation (DIC) measurement, which was carried out for the first Materials 2022,15, 1432 8 of 22 49 cycles , frequencies were 0.004 Hz for the torque channel, and 0.02 Hz for the force channel. Other authors used this path to validate other approximation methods [7–9,11,14,15]. Materials 2022, 15, x FOR PEER REVIEW 8 of 24 Figure 4. Loading paths: (top left) Square; (top middle) NV; (top right) X; (bottom left and bottom middle) Circle; (bottom right) 7. Path Square was achieved by multiaxial loading by trapezoidal waveforms of force and moment signals with a mutual phase shift of 90°. The common load frequency for the entire test was 0.0417 Hz. Other authors used a similar path to validate their estimates [7,9–16]. For the NV path, both channels had sinusoidal waveforms. The loading frequency of the torsion channel was five times faster than that of the tension compression channel. For the digital image correlation (DIC) measurement, which was carried out for the first 49 cycles, frequencies were 0.004 Hz for the torque channel, and 0.02 Hz for the force channel. Other authors used this path to validate other approximation methods [7–9,11,14,15]. Path Circle consisted of two sinusoidal waveforms of axial force and torque with a phase shift of 90°. The loading frequency was 0.1 Hz. In contrast to paths Square and NV, which appeared in several publications by other authors, path Circle has so far only been presented in [7,12,16]. Path 7 represents loading by a constant torque in one channel and a sinusoidal waveform of tension–compression in the other. The loading frequency when measuring the strains was 0.1 Hz. Path 7 was used for validation in [7,8] but only in the form of FEA simulations. Loading frequencies were set according to the ability of the testing machine used to maintain the loading paths on each channel without distortions. The sampling frequency of the DIC cameras was also taken into account, as the loading frequencies had to be 20 times smaller than the sampling frequency of the used cameras to avoid aliasing. The testing machine used for the experiments was INOVA FU 250 (distributed by Inova Praha s.r.o.), multiaxial tension–compression and torsion load frame with hydraulic actuator for dynamic loading. The maximal value of the axial force channel of the machine is 250 kN, and the maximal moment is 2000 Nm. The Dantec Dynamics 3D Q-450 high-speed image correlation system was used for the DIC measurement. The system consists of MKII-NanoSense cameras with a CCD sensor with resolution of 1024 × 1280 pixels and Istra 4-D software (version 4.4.3.414). The software was used for calibration, measurement, displacement evaluation, and displacement export. For calculating axial and shear strains on the basis of exported displacements from Istra 4D , a program was written in MATLAB language. The reason for processing the Figure 4. Loading paths: ( top left ) Square; ( top middle ) NV; ( top right ) X; ( bottom left and bottom middle) Circle; (bottom right) 7. Path Circle consisted of two sinusoidal waveforms of axial force and torque with a phase shift of 90 ◦ . The loading frequency was 0.1 Hz. In contrast to paths Square and NV, which appeared in several publications by other authors, path Circle has so far only been presented in [7,12,16]. Path 7 represents loading by a constant torque in one channel and a sinusoidal waveform of tension–compression in the other. The loading frequency when measuring the strains was 0.1 Hz. Path 7 was used for validation in [ 7 , 8 ] but only in the form of FEA simulations. Loading frequencies were set according to the ability of the testing machine used to maintain the loading paths on each channel without distortions. The sampling frequency of the DIC cameras was also taken into account, as the loading frequencies had to be 20 times smaller than the sampling frequency of the used cameras to avoid aliasing. The testing machine used for the experiments was INOVA FU 250 (distributed by Inova Praha s.r.o.), multiaxial tension–compression and torsion load frame with hydraulic actuator for dynamic loading. The maximal value of the axial force channel of the machine is 250 kN, and the maximal moment is 2000 Nm. The Dantec Dynamics 3D Q-450 high-speed image correlation system was used for the DIC measurement. The system consists of MKII-NanoSense cameras with a CCD sensor with resolution of 1024 × 1280 pixels and Istra 4-D software (version 4.4.3.414). The software was used for calibration, measurement, displacement evaluation, and displacement export. For calculating axial and shear strains on the basis of exported displacements from Istra 4D, a program was written in MATLAB language. The reason for processing the data outside the DIC system was the possibility of applying a higher level of automation and more control over displacement smoothing. 3. Finite Element Analyses FE analyses with purely elastic and elastic–plastic material data were carried out in Abaqus v6.14-5. The elastic material model aimed to achieve notch tip stress histories to use them as inputs for the approximations. Analyses with elastic–plastic material data were performed to verify the correspondence of the material data and the experimental results. Materials 2022,15, 1432 9 of 22 Specimens were modeled as axisymmetric. The same mesh was used for both elastic and elastic–plastic analyses. The final mesh size of the quadratic axisymmetric stress elements CGAX8R (8-node biquadratic, reduced integration) was 0.1 mm (Figure 5). Attempts to further decrease the element size did not affect the results by more than by 0.007%. All elements passed the mesh quality check without errors and warnings. Materials 2022, 15, x FOR PEER REVIEW 9 of 24 data outside the DIC system was the possibility of applying a higher level of automation and more control over displacement smoothing. 3. Finite Element Analyses FE analyses with purely elastic and elastic–plastic material data were carried out in Abaqus v6.14-5. The elastic material model aimed to achieve notch tip stress histories to use them as inputs for the approximations. Analyses with elastic–plastic material data were performed to verify the correspondence of the material data and the experimental results. Specimens were modeled as axisymmetric. The same mesh was used for both elastic and elastic–plastic analyses. The final mesh size of the quadratic axisymmetric stress elements CGAX8R (8-node biquadratic, reduced integration) was 0.1 mm (Figure 5). Attempts to further decrease the element size did not affect the results by more than by 0.007%. All elements passed the mesh quality check without errors and warnings. Figure 5. Axisymmetric model of U-notched specimen in Abaqus. For elastic–plastic analyses, combined hardening behavior was chosen with the stabilized data type. The number of backstresses was set to 5. The cyclic stress–strain curve used in the model was calculated on the basis of the Hollomon parameters presented in Section 2.3. Its values are shown in Table 3. The coincidence of the responses from the experiments and from FEA (Figure 6) at the initial loading and at the beginning of cyclic loading suggests that the elastic data were valid. There were small differences in cyclic regions that could have been caused by the absence of a ratcheting parameter in the combined hardening model of plasticity in Abaqus. Figure 5. Axisymmetric model of U-notched specimen in Abaqus. For elastic–plastic analyses, combined hardening behavior was chosen with the stabilized data type. The number of backstresses was set to 5. The cyclic stress–strain curve used in the model was calculated on the basis of the Hollomon parameters presented in Section 2.3. Its values are shown in Table 3. Table 3. Cyclic stress–strain curve used for elastic–plastic simulation. Stress (MPa) Plastic Strain (-) 330 0.000 371.56 0.002 395.19 0.004 409.72 0.006 420.35 0.008 428.78 0.010 435.79 0.012 441.81 0.014 447.09 0.016 451.81 0.018 The coincidence of the responses from the experiments and from FEA (Figure 6) at the initial loading and at the beginning of cyclic loading suggests that the elastic data were Materials 2022,15, 1432 16 of 22 %% %%%%%%%%%%%%%%%%%%%%%%%%% THE ESTIMATION PROCEDURE ii=1;% index of estimation points, all matrices at 1 consist of zeros. for i = 2:1:(length(sigYyFI)) % i . . . index through loading steps SPsStart = SPs; % deviatoric pseudostress at the beginning of the step alfaPsI = alfaPs; % I as initial ii = ii+1; %% find the start of the plasticization sigPs = [0 0 0; 0 sigYyFI(i sigYzFI(i); 0 sigYzFI(i) sigZzFI(i);]; SPs = sigPs - trace(sigPs)/3*eye(3); % pseudostress deviator % find start of the plasticization tol = 1; %(MPa), tolerance beginAt = 0.01; for i2 = beginAt:beginAt/10:1 SInt = SPsStart+(SPs-SPsStart) * i2; sEq = calc_equiv_stress(SInt-alfaPsI); if (sEq - yieldStrength) < tol && (sEq - yieldStrength) > 0 elastic_part_instep = 1; break; elseif (sEq - yieldStrength) > 0 % no purely elastic loading during the step elastic_part_instep = 0; break; end end % check if elastic loading/unloading has occurred if i2 ~= beginAt %% PURELY ELASTIC region solution dSigYPs =(sigYyFI(i)-sigYyFI(i-1))*i2;dSigZPs =(sigZzFI(i)-sigZzFI(i-1))*i2; dSigYzPs = (sigYzFI(i)-sigYzFI(i-1))*i2; elastic_region_solution; % >> function epsXxPl(ii) = epsXxPl(ii-1);epsYyPl(ii) = epsYyPl(ii-1); epsZzPl(ii) = epsZzPl(ii-1);epsYzPl(ii) = epsYzPl(ii-1); else % no purely elastic loading during the step sigYPs( ii)=sigYyFI(i-1);sigZPs( ii)=sigZzFI(i-1); sigYzPs(ii)=sigYzFI(i-1); ii= ii-1; % in order to not skip index when elastic variables weren’t calculated end %% plastic region solution (elastic+plastic stresses and strains) if i2 < 1 % plasticization has occurred % pseudonotch stresses increments till the end of current step dSigYPs =(sigYyFI(i) - sigYPs( ii))/incPerPlR; dSigZPs =(sigZzFI(i) - sigZPs( ii))/incPerPlR; dSigYzPs = (sigYzFI(i) - sigYzPs(ii))/incPerPlR; for j = 1:1:incPerPlR %% cycle trough increments of plastic range ii=ii+1;% index of calculated variables – stresses and strains sigYPs(ii) = sigYPs(ii-1)+dSigYPs;sigZPs(ii) = sigZPs(ii-1)+dSigZPs; sigYzPs(ii) = sigYzPs(ii-1)+dSigYzPs;% pseudostress sigPs = [0 0 0; 0 sigYPs(ii) sigYzPs(ii); 0 sigYzPs(ii) sigZPs(ii)]; SPs = sigPstrace(sigPs)/3*eye(3);% deviator of pseudostress Materials 2022,15, 1432 17 of 22 [dp,theta] = calc_dp_AKO(gPC,cPC,SPs,alfa_partPs); [alfaPs, alfa_partPs, dEpsPlPs] = ... calc_alfa_and_dEpsPl(dp,gPC,cPC,alfa_partPs,SPs, theta); dEpsPl = dEpsPlPs; [alfa, alfa_part, SReal] = ... calc_alfa_and_Snp1(dp,gRC,cRC,alfa_part,dEpsPl); %% non-deviatoric real stress components sigRealH = -SReal(1,1);%% plain stress condition sigReal = SReal + sigRealH*eye(3); sigY(ii) = sigReal(2,2);sigZ(ii) = sigReal(3,3);sigYz(ii) = sigReal(2,3); dSigY = sigY(ii) - sigY(ii-1);dSigZ = sigZ(ii) - sigZ(ii-1); dSigYz = sigYz(ii) - sigYz(ii-1); %% storing calculated strains % plastic strain increments - tensorial shear strain form! x(1) = dEpsPl(1,1); x(2) = dEpsPl(2,2); x(3) = dEpsPl(3,3); x(4) = dEpsPl(2,3); % elastic strain dEpsXxE = 1/E * (-ny) * (dSigY + dSigZ); dEpsYyE = 1/E * (dSigY + (-ny) * dSigZ); dEpsZzE = 1/E * (dSigZ + (-ny) * dSigY);dGammaYzE = 1/G* dSigYz; dEpsYzE = dGammaYzE/ 2; epsXxE(ii) = epsXxE(ii-1) + dEpsXxE; epsYyE(ii) = epsYyE(ii-1) + dEpsYyE; epsZzE(ii) = epsZzE(ii-1) + dEpsZzE;epsYzE(ii) = epsYzE(ii-1) + dEpsYzE; % plastic strain epsXxPl(ii) = epsXxPl(ii-1) + x(1); epsYyPl(ii) = epsYyPl(ii-1) + x(2); epsZzPl(ii) = epsZzPl(ii-1) + x(3);epsYzPl(ii) = epsYzPl(ii-1) + x(4); % total strain (elastic total new + plastic total) epsXx(ii) = epsXxE(ii) + epsXxPl(ii); epsYy(ii) = epsYyE(ii) + epsYyPl(ii); epsZz(ii) = epsZzE(ii) + epsZzPl(ii);epsYz(ii) = epsYzE(ii) + epsYzPl(ii); end %j = 1:1:(incPerPlR) end end graphs; % function for plotting results %% %%%%%%%%%%%%%%%%%% FUNCTIONS: %% %%%%%%%%%%%%%%% function [ePlRC, sRC, ePlPC, sPC] = material_curves(E, K, nRO) % RC . . . real curve (cyclic stress strain curve); Pl . . . plastic % PC . . . pseudocurve;e . . . epsilon, s . . . sigma %% REAL CURVE % 4 intervals for gamma and C and 1 corresponding to the yield % strength - added later -> 5 intervals == 5 backstress parts ePlRC = [0.002 0.004 0.006 0.008 0.01]0; n = size(ePlRC,1); %number of discrete points on the curve for i = 1:n sRC(i,1) = K*ePlRC(i)ˆnRO; end %% PSEUDOCURVE ePlPC= ePlRC; for i = 1:n %sig_e = sig *eps_tot*E sPC(i,1) = sqrt( sRC(i)* (ePlRC(i)+sRC(i)/E) *E ); end end%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION function [K, n] = ramberg_osgood_coefficients(eps, sig) f = fit(eps,sig,‘power1’); regCs = coeffvalues(f); Materials 2022,15, 1432 18 of 22 K = regCs(1); %RO with Hollomon parameters: eps = sig/E + (sig/K)ˆ(1/n) n = regCs(2); end%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION function [gamma, cMatrix] = calc_C_gamma(epsPl,stress,K,n) global yieldStrength slope = zeros(size(epsPl,1),1);gamma = zeros(size(epsPl,1)-1,1); cMatrix = zeros(size(epsPl,1)-1,1);epsPlAtYS = (yieldStrength/K)ˆ(1/n); nn = size(epsPl,1); for i = 1:nn if i == 1 slope(1) = (stress(1)-yieldStrength)/(epsPl(i)-epsPlAtYS); else slope(i) = (stress(i)-stress(i-1))/(epsPl(i)-epsPl(i-1)); end end slope(nn+1) = 0; for i = 1:nn gamma(i) = 1/epsPl(i); cMatrix(i) = slope(i) - slope(i+1); end end%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION %% %%%%%%%%%%%%% PROCEDURE define_notch_stress_imputs.m if strcmp(path_name,‘square’) % Path Square,F = 66 kN, M = 329.133 Nm, D 26 mm test_spcm = ‘u3’; num_cycles_text = ‘1st 100 cycles’; sigYyFI = [0112 109 109 -3-116-120-120-7 112 ]; sigZzFI = [0427 428 428 0 -429-427-4271427]; sigYzFI = [0-291 1 292 292 292 1 -291-291 -291]; elseif strcmp(path_name,‘NV’) % Path NV;F = 66 kN, M = 329.133 Nm, D 26 mm test_spcm = ‘u4’; num_cycles_text = ‘1st 49 cycles’; % coarse input - multilinear sigYyFI = [0366591105 112 104 906333-4-37 -71 -95 ... -113 -117-114-96 -73 -41 -8]; sigZzFI = [0132 252 346 407 428 408 346 253 133 1 -131-250 . . . -345 -405-427-405-345-249-1303]; sigYzFI = [0-2920 292 0 -2920 292 0 -2920 ... 2910 -2910 291 0 -2910 291 0]; elseif strcmp(path_name,‘circle’) % Path Circle;F = 66 kN, M = 329.133 Nm, D 26 mm test_spcm = ‘u73’; num_cycles_text = ‘1st 100 cycles’; sigYyFI = [0112 106 9167361 -34 -66 -91 -108-114 ... -108 -92 -67 -35 0 356590105 110]; sigZzFI = [0427 406 345 251 131 -1-133-253-347-408 ... -429 -408-347-253-133-1131 251 345 406 427]; sigYzFI = [00 90172 236 277 292 277 236 171 900 -90 -171 . . . -236 -277-291-277-236-171-90 0]; elseif strcmp(path_name,‘circle_1p73’) % F = 101 kN, M = 290 Nm, D 26 mm sigYyFI = [0170 170 162 138 101 531 -53 -101-140-165-174-165-140-102-53 0 53100 137 161 169]; Materials 2022,15, 1432 19 of 22 sigZzFI = [0651 651 619 527 382 200 -2-204-386-531-624-656-624-531-386-204-2200 382 526 619 651]; sigYzFI = [00 0 80151 208 245 257 244 208 151 790 -79 -151-208-244-257-245-208-151-80 0]; elseif strcmp(path_name,‘7’) % Path “7”; F = 66 kN, M = 329.133 Nm, D 26 mm test_spcm = ‘f9’; num_cycles_text = ‘1st 100 cycles’; sigYyFI = [080.8 -78.280.7];% tangential direction sigZzFI = [0373.3-371.7 372.9]; % axial direction, sigYzFI = [0 -271.4-270.5-271.5]; % FI as for incrementation elseif strcmp(path_name,‘X’) % F = 66 kN, M = 329.133 Nm, D 26 mm sigYyFI = [0112-4109-7-120-10-123-14]; sigZzFI = [0427 04281-428 1 -428 1]; sigYzFI = [0 -292 12920 292 1 -291 0]; end %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF PROCEDURE %% %%%%%%%%%%%%% PROCEDURE prealocate_variables.m n = size(gPC,1);% number of backstress parts alfa = zeros(3,3);% total backstress alfa_part = zeros(3,3,n); % backstress parts alfaPs = zeros(3,3);% total pseudobackstress alfa_partPs = zeros(3,3,n); % parts of the pseudobackstress alfa_partPsI = zeros(3,3,n); % parts of the pseudobackstress at the beginning of the step theta = ones(1,n); SPs = zeros(3,3);SPsStart = zeros(3,3);SReal = zeros(3,3); dEpsPl = zeros(3,3);dEpsPlPs = zeros(3,3); dp = 0; if sigYyFI(length(sigYyFI)) == 0 % m = number of points in elastic range + number of points in plastic %range + one initial zero state m = length(sigYyFI)-1 + incPerPlR*(length(sigYyFI)-2) + 1; else m = length(sigYyFI)-1 + incPerPlR*(length(sigYyFI)-1) + 1; end epsXx(1) = 0; epsYy(1) = 0; epsZz(1) = 0; epsYz(1) = 0; epsXxE(1) = 0;epsYyE(1) = 0;epsZzE(1) = 0;epsYzE(1) = 0; epsXxPl(1) = 0; epsYyPl(1) = 0; epsZzPl(1) = 0; epsYzPl(1) = 0; sigY = zeros(m, 1); sigZ = zeros(m, 1); sigYz = zeros(m, 1); sigYPs(1) = 0; sigYPs(2) = 0; sigZPs = zeros(m, 1); sigYzPs = zeros(m, 1); %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF PROCEDURE function equivalent_stress = calc_equiv_stress(A) aMatrix = A.*A; equivalent_stress = sqrt(3/2* sum(aMatrix(:)) ); end %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION %% %%%%%%%%%%%%% PROCEDURE elastic_region_solution.m sigYPs(ii)= sigYPs(ii-1) + dSigYPs;sigZPs(ii)= sigZPs(ii-1) + dSigZPs; sigYzPs(ii) = sigYzPs(ii-1)+ dSigYzPs;sigY(ii)= sigY(ii-1) + dSigYPs; % in elastic regime increments of real and pseudostresses are the same sigZ(ii)= sigZ(ii-1) + dSigZPs;sigYz(ii) = sigYz(ii-1)+ dSigYzPs; sigReal = [0 0 0; 0 sigY(ii)sigYz(ii); 0 sigYz(ii) sigZ(ii)]; SReal = sigReal - trace(sigReal)/3*eye(3); Materials 2022,15, 1432 20 of 22 dEpsXxE = 1/E * (-ny) * (dSigYPs + dSigZPs);dEpsYyE = 1/E * (dSigYPs + (-ny) * dSigZPs); dEpsZzE = 1/E * (dSigZPs + (-ny) * dSigYPs);dGammaYzE = 1/G *dSigYzPs; dEpsYzE = dGammaYzE/ 2; epsXxE(ii) = epsXxE(ii-1) + dEpsXxE;epsYyE(ii) = epsYyE(ii-1) + dEpsYyE; epsZzE(ii) = epsZzE(ii-1) + dEpsZzE;epsYzE(ii) = epsYzE(ii-1) + dEpsYzE; epsXx(ii) = epsXx(ii-1) + dEpsXxE;epsYy(ii) = epsYy(ii-1) + dEpsYyE; epsZz(ii) = epsZz(ii-1) + dEpsZzE;epsYz(ii) = epsYz(ii-1) + dEpsYzE; %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF PROCEDURE function [dp,theta] = calc_dp_AKO(gamma,cMatrix,S,alfa_part) global yieldStrength n = size(gamma,1);theta = ones(1,n);% vectors prealocation mu1 = 0; % ratcheting parameter; 0 . . . OWI; 0.1 . . . AKO; 1 . . . CHAB dpkm1 = 0;dpkm2 = 0;% variables to check convergence for k =1:100 aMatrix = zeros(3,3); % a supportive variable for the calculation aNum = 0; for i2 = 1:n aMatrix = aMatrix + theta(i2)*alfa_part(:,:,i2); aNum = aNum + cMatrix(i2)*theta(i2); end aNum2 = calc_equiv_stress(S-aMatrix); dp = (aNum2 - yieldStrength)/aNum; % dp from the first iteration % The Aitken’s deltaˆ2 process to shorten the convergence —– if mod(k,3) == 0 con = dp-(dp-dpkm1)*(dp-dpkm1)/(dp-2*dpkm1+dpkm2); if con > 0 dp = con; end end % ——————————————————— SminusA = yieldStrength/(yieldStrength+aNum*dp)*(S-aMatrix); dEpsPl = 3/2*dp*SminusA/yieldStrength; if abs(1-dpkm1/dp) < 10ˆ(-4) break;% solution found end for i2 = 1:n aMatrix = zeros(3,3);alnp1_st(:,:,i2) = alfa_part(:,:,i2)+2/3*cMatrix(i2)*dEpsPl; alnp1_dash(i2) = calc_equiv_stress(alnp1_st(:,:,i2)); mu(i2) = mu1;c(i2) = 1/(1+mu(i2)*gamma(i2)*dp); alnp1_hash(:,:,i2) = c(i2)*(alnp1_st(:,:,i2)); fnp1_hash = calc_equiv_stress(alnp1_hash(:,:,i2))ˆ2-(cMatrix(i2)/gamma(i2))ˆ2; theta(i2)=c(i2)+heaviside(fnp1_hash)* ... (cMatrix(i2)/gamma(i2)/alnp1_dash(i2)-c(i2)); end dpkm2 = dpkm1;dpkm1 = dp; if k == 100 error(‘Error: number of iterations has exceeded the allowed value’); k end end end %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION function [alfa, alfa_part, dEpsPl] = ... calc_alfa_and_dEpsPl(dp,gamma,cMatrix,alfa_part,S,theta) global yieldStrength m2 = size(gamma,1); % number of backstress parts Materials 2022,15, 1432 21 of 22 aMatrix = zeros(3,3);alfa = zeros(3,3);aNum = 0; for i = 1:m2 aMatrix =aMatrix + theta(i)*alfa_part(:,:,i);aNum = aNum + cMatrix(i)*theta(i); end % plastic strain tensor SminusA = yieldStrength/(yieldStrength+aNum*dp)*(S-aMatrix); dEpsPl = 3/2*dp*SminusA/yieldStrength;% calculate backstress parts (for the next iteration) for i = 1:m2 alfa_part(:,:,i) = (alfa_part(:,:,i)+2/3*cMatrix(i)*dEpsPl)*theta(i); alfa = alfa + alfa_part(:,:,i); end end %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION function [alfa, alfa_part, Snp1] = ... calc_alfa_and_Snp1(dp,gamma,cMatrix,alfa_part,dEpsPl) global yieldStrength m2 = size(gamma,1); % number of backstress parts alfa = zeros(3,3); % calculate backstress parts for i = 1:m2 alfa_part(:,:,i) = (alfa_part(:,:,i)+2/3*cMatrix(i)*dEpsPl)/(1+gamma(i)*dp); alfa = alfa + alfa_part(:,:,i); end Snp1 = dEpsPl*yieldStrength*2/3/dp + alfa; end %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END OF FUNCTION References 1. 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