PSI - Issue 43

Volume 43 • 2023

ISSN 2452-3216

ELSEYlER

Stliuctu�al

Material s Structure & Micro m ec h anic s of Fracture

Guest Editors: J aroslav P okluda , P avel Š andera

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10th International Conference on Materials Structure and Micromechanics of Fracture Editorial Jaroslav Pokluda* Brno University of Technology, Brno, Czech Republic This issue of the Procedia Structural Integrity (PSI) contains papers from the 10 th International Conference on Materials Structure and Micromechanics of Fracture (MSMF10) that was held in Brno, Czech Republic, September 12-14, 2022. The series of MSMF conferences was established in Brno, June 1995. The basic idea was to create a periodical international forum for multiscale approaches in fatigue and fracture of materials in the middle Europe. Therefore, sections focused on atomistic models, models based on crystal defects, numerical and statistical continuum models, advanced experimental methods and relationships between microstructure and mechanical properties appeared during the MSMF2 conference in 1998. The power of atomistic, mesoscopic, and multiscale approaches in fracture and fatigue was then demonstrated by MSMF meetings organized in Brno during the next years 2001 - 2019. Many world leading experts in the field of fracture and fatigue attended the MSMF conferences as plenary speakers. The tenth, jubilee conference took place in an extremely complicated period. It was preceded by the global pandemic of the dangerous coronavirus. Although its scope decreased slightly, it still significantly affected the free movement of people at the time of the MSMF. Many important conferences, postponed during the two-year restrictions, filled the calendars of regular participants. In addition, the war in Ukraine badly intervened in European events and disturbed the plans of many scientists. Despite these complications, the MSMF10 conference has successfully carried on the tradition of previous conferences with 127 scientists from 19 countries around the world who presented 122 contributions on fundamental relations between structural and mechanical characteristics of materials. There were five invited plenary talks delivered by Dr. Martin Friák (Brno, CZ), Prof. Shigenobu Ogata

* Corresponding author. Tel.: +420-541-142-827; E-mail address: pokluda@fme.vutbr.cz

2452-3216 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the IC MSMF organizers.

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. 10.1016/j.prostr.2022.12.225

Jaroslav Pokluda / Procedia Structural Integrity 43 (2023) 1–2 Jaroslav Pokluda / Structural Integrity Procedia 00 (2022) 000 – 000

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(Toyonata, JP), Dr. Maxime Sauzay (Gif-sur-Yvette, FR), Prof. Christopher A. Schuh, (Cambridge, USA) and Prof. David J Srolovitz (Hong Kong, HK). After a peer-review procedure, 53 papers based on atomistic, mesoscopic, macroscopic, and multiscale approaches were included in this PSI volume devoted to MSMF10. I would like to thank all the members of the International Advisory Board for their support that was essential to the success of the conference. My thanks also go to all session chairpersons who successfully guided the program as well as to reviewers who thoroughly helped many authors to improve the quality of their manuscripts. Special thanks belong to the local organizing team chaired by Prof. Pavel Šandera and to all participants which contributed to a working and friendly atmosphere during this successful event. It is also my pleasure to thank all the leading personalities of Elsevier and ESIS relevant for production of this volume.

Prof. Jaroslav Pokluda, Chair, On behalf of the organizers of the 10 th International Conference on Materials Structure and Micromechanics of Fracture

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© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. © 20 23 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under the responsibility of MSMF10 organizers. investigation on modeling the cyclic-plastic be aviour of materials under stress-c ntrolled and strain-controlled testing; but amongst these, Chaboch 's isotropic-kinematic hardening (CIKH) model is quite popul r and a frequently used one. A new methodology has been rece tly propos d by some of the present authors to use gen tic algorithm towards opt ization of th parameters of CIKH modeling with a demonstration of its applic bility for a few mat rials. This investigation aims to elucidate th t this approach is pplicable to both cyclically stable materials as well as cyclic soften g or hardening materi ls using several examples on structural materials. The considered metallic materials include several aluminum (like AA7075-T6 alloy), iron CS-1026and Sa333 C-Mn s e l), titanium (like TA16 all y), zirconium (like Zr-4 al oy)-base alloys or superalloys (like INCONEL718 ) . The merit of the analysis of cyclic lastic deform tion behaviour of materials with the cur ent approac is inhere t in its a hieving higher accuracy of fitting o the xperimental data with a single set of material parameters und r both strain- nd stress-controlled cycling. This as been established with a co parative nalysis of the accuracy of itting by th present appro ch with th ones available from the xisti g reports. The pa ameters obtained using the approach for the different materials are compared to get insights into the mechanism of plastic deformation. © 20 23 The Authors. Published by Elsevier B.V. This is an ope access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under the responsibility of MSMF10 organizers. 10th International Conference on Materials Structure and Micromechanics of Fracture Analyses of the model parameters of kinematic-isotropic hardening rule using genetic algorithm approach for predicting cyclic plasticity of metallic structural materials Atri Nath a , Ayan Ray b * and K. K. Ray c a Department of Civil Engineering, IIT Kharagpur India 721302, (currently with IIT Delhi-110016) India b Valeo Autoklimatizace k.s.Sazeˇcsk´a 24 7/2, 108 00 Prague, Czech Republic c Department of Metallurgical and Materials Engineering, IIT Kharagpur, Kharagpur-721302, India Abstract Modeling of engineering components under cyclic loading is complex because cyclic-plastic phenomena like the Bauschinger effect, ratcheting, shakedown, and mean stress relaxation are to be considered in the constitutive modeling. This has led to several investigations on modeling the cyclic-plastic behaviour of materials under stress-controlled and strain-controlled testing; but amongst these, Chaboche's isotropic-kinematic hardening (CIKH) model is quite popular and a frequently used one. A new methodology has been recently proposed by some of the present authors to use genetic algorithm towards optimization of the parameters of CIKH modeling with a demonstration of its applicability for a few materials. This investigation aims to elucidate that this approach is applicable to both cyclically stable materials as well as cyclic softening or hardening materials using several examples on structural materials. The considered metallic materials include several aluminum (like AA7075-T6 alloy), iron (like CS-1026and Sa333 C-Mn steel), titanium (like TA16 alloy), zirconium (like Zr-4 alloy)-base alloys or superalloys (like INCONEL718 ) . The merit of the analysis of cyclic plastic deformation behaviour of materials with the current approach is inherent in its achieving higher accuracy of fitting to the experimental data with a single set of material parameters under both strain- and stress-controlled cycling. This has been established with a comparative analysis of the accuracy of fitting by the present approach with the ones available from the existing reports. The parameters obtained using the approach for the different materials are compared to get insights into the mechanism of plastic deformation. 10th International Conference on Materials Structure and Micromechanics of Fracture Analyses of the model parameters of kinematic-isotropic hardening rule using genetic algorithm approach for predicting cyclic plasticity of metallic structural materials Atri Nath a , Ayan Ray b * and K. K. Ray c a Department of Civil Engineering, IIT Kharagpur India 7213 2, (currently with IIT Delhi-110016) India b Valeo Autoklim tizace k.s.Sazeˇcsk´a 24 7/2, 108 00 P ue, Czech Re blic c Department of Metallurgical and Materials Engineering, IIT Kharagpur, Kharagpur-721302, India Abstract Modeling of e ineering components und r cyclic loading is complex because cy lic-plastic phenomena like the Bauschinger eff ct, r cheting, shak down, and mean stress relaxati n are to b consider d in the stitutive modeling. This has led to several Keywords: Cyclic-plasticity; Isotropic-kinematic hardening; Parameter estimation; Genetic algorithm; Generalized approach Keywords: Cyclic-plasticity; Isotropic-kinematic hardening; Parameter estimation; Genetic algorithm; Generalized approach

* Corresponding author. Tel.: +91-9434230710 E-mail address: ayanray83@gmail.com * Correspon ing autho . Tel.: +91-9434230710 E-mail address: ayanray83@gmail.com

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. 10.1016/j.prostr.2022.12.266 2452-3216 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. 2452-3216 © 2023 The Authors. Published by Elsevier B.V. This is an ope access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers.

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1. Introduction The use of conventional and advanced structural materials for various engineering applications demands an understanding of the material behaviour under different loading conditions, particularly in the elastic-plastic regime. Several phenomena like Bauschinger effect, ratcheting, shakedown, and mean stress relaxation need careful consideration with emphasis on the elastic-plastic analysis subjected to cyclic loading, unlike that for monotonic loading. Chaboche’s isotropic -kinematic hardening (CIKH) model and subsequently modified versions of this model have been used by several researchers to simulate the cyclic plastic behavior of materials, especially to simulate the hysteresis loops and ratcheting characteristics. But a generalized approach to replicate sets of experimental results with a single set of CIKH model parameters is yet to emerge or get established. This study attempts to examine the generalized framework proposed by Nath et al. (2019a) considering Chaboche's (1986) model to analyze the cyclic plastic behaviour of a wide variety of materials; the adopted methodology has been applied to different ferrous and non-ferrous metallic materials exhibiting cyclic hardening or softening or stabilized behaviour. 2. Methodology The mathematical formulation of the CIKH model used in the present study is summarized in Table 1 The adopted methodology considers a non-linear Voce isotropic hardening rule along with Chaboche's (1991) kinematic hardening (KH) as a combination of four nonlinear backstress components ( α ). The evolution of the i th backstress is controlled by the parameters C i , γ i, and a i according to the evolution rules given in Eq.4 (Table 1) and is schematically shown in Figure 1a. Typical responses of the first and the second backstresses are non-linear, the evolution of the third backstress component is linear, and the fourth backstress component has a linear portion depending on the parameter a 4 , followed by a non-linear portion (Bari and Hassan, 2000). The parameter C i is the slope at the initial portion for the i th backstress component, while the parameter γ i is the dynamic recovery parameter, resulting in a stabilized value of the backstress to be C i / γ i , as demonstrated in Figure 1a. Table 1: Mathematical formulation of the combined isotropic-kinematic (CIKH) model used in the present study (Nath et al. 2019b) Von Mises yield criteria = √ 3⁄2 ( − ): ( − ) − ( 0 + ) = 0 (1) Evolution of isotropic hardening ̇ = ( − ) ̇ (2) Plastic strain rate ̇ = ( 2⁄3 ̇ : ̇ ) (3) Evolution of kinematic hardening = ∑ 4 =1 ̇ = 2⁄3 ̇ − ̇ = 1,2,3̇ ̇ = 2⁄3 ̇ − 〈1 − ( ) 〉 ̇ = 4̇ (4) s is the deviatoric part of the stress tensor;  y0 is the initial radius of the yield surface; ̇ is the accumulated equivalent plastic strain rate, ̇ is the plastic strain rate; R, Q, and b are the isotropic hardening parameters;  is the deviatoric backstress tensor related to kinematic hardening described as a combination of individual backstress components ( α i ), C i and γ i are the kinematic hardening parameters, J(α i ) is the second invariant deviatoric of the i th backstress component, a i is the threshold for dynamic recovery of the i th backstress component The CIKH model typically considers the first backstress component ( α 1 ) to be responsible for the prediction of the large modulus at the initial part of the hardening behaviour just after the onset of yielding (Fig 1); α 1 typically stabilizes rapidly. The second ( α 2 ) and the fourth backstress component ( α 4 ) primarily control the simulation of the transient nonlinear part of the stress-strain response in the plastic range of material. The third backstress component ( α 3 ) controls the linear hardening behaviour of the hysteresis loop throughout the entire strain range. The dominant regions of the four backstress components for strain-controlled loading are demonstrated in Figure 1b. A total of 11 parameters of the CIKH model, as summarized in Table 1, are to be determined for the analysis of the cyclic-plastic behavior of a material using the proposed approach. The initial estimate of the CIKH model parameters is obtained from stabilized hysteresis loops using the methodology proposed by Nath et al. (2019a). The

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initial estimates of the model parameters are next calibrated to obtain closer cyclic-plastic response under asymmetric stress-controlled cycles by using a genetic algorithm optimization technique. The optimized parameters thus obtained are then used to simulate the response of CSMs under both monotonic and different cyclic loading conditions. The accuracy of the prediction obtained by the current approach is compared with that of other reported predictions by using a modified root mean square error ( F error ) function given by : = { 1⁄ ∑ [( − )/ ] 2 =1 } 0.5 ℎ (5a) = { 1⁄ ∑ [( , − , )/ ] 2 =1 } 0.5 ℎ / 5(b) = { 1⁄ ∑ [( ∗ − ∗ )/ ∗ ] 2 =1 } 0.5 − (5c) where, n is the number of data points, and ∗ are the stress and modified ratcheting strain recorded from experiments, , and , are the experimental and simulated stress amplitude for the i th cycle , and and ∗ are the predicted stress and modified ratcheting strain using the CIKH model. The ratcheting strain data are modified by applying a 5% offset similar to the study by Nath et al. (2019a). The adopted optimization technique considerably reduces the value of F error . The comparison of the simulation obtained by using the suggested methodology with the reported response for the investigated materials is examined in the following section.

a b Figure 1 Schematic representation of a) contribution of individual backstresses based on the physical meaning of the parameters, and b) tensile half of stabilized hysteresis loop under strain-controlled cycles showing the dominance of individual backstress components in different zones 3. Results and discussions The adopted approach is applied to predict the cyclic-plastic response of two ferrous (CS1026 steel and SA333 C Mn steel) and four non-ferrous materials (viz., TA16 titanium alloy, AA 7075-T6 aluminum alloy, OFHC copper, and Zr- 4 alloy at 400°C ). The final sets of CIKH model parameters obtained by the adopted approach for the investigated materials (summarized in Table 2) are next used to simulate the behavior of the material under different loading conditions as depicted in Fig 2-4 for the different investigated materials. CS1026 (Hassan and Kyriakides, 1992) and SA333 C-Mn (Paul et al., 2010) steels considered in the study typically exhibit a BCC structure and are cyclically stable ((Nath et al., 2019a). The parameters of the CIKH model parameters summarized in Table 2 are used to simulate the stabilized hysteresis loop at 1% strain amplitude (Fig 2a) and ratcheting behavior under three different mean stresses (Fig. 2b). The accuracy of the predictions obtained using the current approach is quantitatively compared with other reported approached in Fig 2c; the adopted methodology provides better simulation for the cyclic-plastic response for CS1026 steel, substantiated by the consistent low magnitude of F error (3%) across all the loading conditions. Similar qualitative and quantitative comparisons between experimental and reported simulations have been carried out for FCC AA7075 alloy (Agius et al., 2018) and OFHC copper (Zhang and Jiang, 2008) in Fig. 3, and for HCP-structured TA16 titanium alloy(Kan et al., 2011) and Zr-4 alloys(Cheng et al., 2015) in Fig. 4. The adopted approach provides a low magnitude of the F error (<3%), for all the investigated materials under different loading condition; this illustrates the superior predictive capability of the suggested approach in comparison to existing approaches, as well as depicts the generalized nature of the adopted approach.

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Table 2: The optimized parameters obtained for the investigated materials using the suggested methodology Materials Elastic Parameters Cyclic plastic parameters Isotropic hardening Kinematic hardening

C i=1,2,3,4 (MPa) = 375754.9 , 15178.7 , 3068.6, 78792.9 γ i=1,2,3,4 = 21902.3 , 279.1 , 14.3 , 4666.6 a 4 = 21.4 MPa C i=1,2,3,4 (MPa) = 129261.5 , 23275 , 2724.8 , 4217.2 γ i=1,2,3,4 = 2397.8 , 284.3 , 10.8 , 3090.9 a 4 =16.6 MPa C i =1,2,3,4 (MPa) = 192694.5 , 22986.6 , 73.2 , 7437.6 γ i =1,2,3,4 = 16250.3 , 1296.8 , 0.2 , 2139.5 a 4 = 32.8 MPa C i =1,2,3,4 (MPa) = 79404.4 , 13311.4 , 565.9 , 87677.2 γ i =1,2,3,4 = 5682.8 , 461.9 , 0.43 , 2739.5 a 4 = 39.4 MPa C i =1,2,3,4 (MPa) = 350531.7 , 80862.8, 1748.5, 28362.9 γ i =1,2,3,4 = 14731.6 , 935.4 , 0.83 , 665.1 a 4 = 234.8 MPa C i =1,2,3,4 (MPa) = 79891.5 , 22521.5 ,4579.5 , 28055.5 γ i =1,2,3,4 = 17328.9 , 3849.1 , 0.86, 1493.6 a 4 = 18.2 MPa

E = 181 GPa  y 0 = 129.6 MPa E = 200 GPa  y 0 = 225 MPa

b = 109.3 Q = 29.4 MPa

CS1026 steel

b = 92.6 Q = 14.5 MPa

SA333 C-Mn steel

E = 69 GPa  y 0 = 500 MPa

b = 862.6 Q = -27.8 MPa

AA7075

E = 117 GPa  y 0 = 8.4 MPa

b = 1.8 Q = 60.6 MPa

OFHC Copper

E = 101 GPa  y 0 = 420 MPa

b = 550.1 Q = -182.5 MPa

TA16

E = 51 GPa  y 0 = 110 MPa

b = 1516.1 Q = -35.4 MPa

Zr-4

b

a

c

d f Figure 2: Prediction of stabilized hysteresis loops and ratcheting response for BCC structured (a-b) CS1026 (Nath et al., 2019a), and (d-e) SA 333 C-Mn (Nath et al., 2019a) steels. (e) and (f) represent the deviation of simulation from experimental for the two materials using F error The results in Fig 2-4 depict that the single set of parameters of the cyclic-plastic model obtained for a material using the adopted approach provides close predictions for the material response under both strain-controlled and stress controlled cyclic loading. However, a marked difference in the contribution of the backstresses to plastic response for materials having different lattice structures is noted and is summarized in Fig. 5. The contribution of the fourth e

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a

b

c

e

d

f Figure 3: Prediction of stabilized hysteresis loops and ratcheting behaviour for FCC structured (a-b) AA7075 alloy(Nath et al., 2019b), and (d-e) OFHC copper (Nath et al., 2022). (e) and (f) represent the deviation of simulation from experimental for the two materials using F error

a

b

c

d f Figure 4: Prediction of stabilized hysteresis loops and ratcheting behaviour for FCC structured (a-b) TA16 alloy (Nath et al., 2019b), and (d-e) Zr-4 alloy at 400 o C (Nath et al., 2019b). (e) and (f) represent the deviation of simulation from experimental for the two materials using F error backstress component, and in particular the threshold term ( a 4 ) of the KH is substantially larger for closely packed HCP materials; this indicates that the backstress evolution in the investigated materials is primarily linear (Fig. 5c). For loosely packed BCC materials, the rapidly stabilizing first backtress component and the primarily non-linear second backstress component are active (Fig 5a-b). The backstress evolution of the investigated FCC alloys is also heavily influenced by the fourth backstress component, similar to HCP materials, although lower in magnitude than the later. The kinematic hardening (KH) component represents the rapid changes in the dislocation structure (Chaboche, 1986); the KH physically represents the monotonic rapid evolutions of the remobilization of dislocation e

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due to the unloading and reverse loading that takes place during each branch in a cyclic loading. In the stress space, this corresponds to the translation of the elastic domain reflecting the description of the fast internal changes during each inelastic transient. The decomposition of the back stress into a rapidly saturated variable (the first backstress component α 1 ) and multiple quasi-linear components ( α i ; i=2,3,4) is associated with the separation of short-range and long-range interactions between defects at the various scales of the microstructure (Chaboche, 1986).

b

a

c Figure 5: Comparison of the stabilized magnitude of the different backstress of the KH component for various microstructures a) first backstress ( α 1 ) component, b) second backstress ( α 2 ) component, and c) fourth backstress ( α 4 ) component with threshold parameter ( a 4 ). 4. Conclusions This study examines a generalized approach to analyze the cyclic-plastic response of a wide variety of metallic materials considering Chaboche’s combined isotropic-kinematic hardening (CIKH) model. The proposed procedure determines the parameters of the cyclic-plastic model from the strain-controlled stabilized loops with a subsequent refinement of the model parameters using the Genetic Algorithm optimization technique. The study encompasses the prediction of symmetric strain-controlled hysteresis loops and ratcheting behaviour for six different ferrous and non ferrous materials having different crystal structures and varied responses in terms of cyclic-hardening/softening or stable behavior. The adopted approach provides a consistently low magnitude of the F error (<3%), for all the different materials under different loading conditions which illustrates the superior predictive capability in comparison to existing approaches, as well as depicts the generalized nature of the adopted approach. Finally, the parameters of the CIKH model obtained for the different materials are compared based on the crystal structure; the contribution of the nonlinear backstresses is more pronounced for the investigated BCC structured materials, while for closely-packed HCP structured material, the backstress evolution is largely linear. References Agius, D., Kourousis, K.I., Wallbrink, C., 2018. A modification of the multicomponent Armstrong – Frederick model with multiplier for the enhanced simulation of aerospace aluminium elastoplasticity. Int. J. Mech. Sci. 144, 118 – 133. Bari, S., Hassan, T., 2000. Anatomy of coupled constitutive models for ratcheting simulation. Int. J. Plast. 16, 381 – 409. Chaboche, J.L., 1991. On some modifications of kinematic hardening to improve the description of ratchetting effects. Int. J. Plast. 7, 661 – 678. Chaboche, J.L., 1986. Time-independent constitutive theories for cyclic plasticity. Int. J. Plast. 2, 149 – 188. Cheng, H., Chen, G., Zhang, Z.,Chen, X., 2015. Uniaxial ratcheting behaviors of Zircaloy- 4 tubes at 400 °c. J. Nucl. Mater. 458, 129– 137. Hassan, T., Kyriakides, S., 1992. Ratcheting in cyclic plasticity, part I: Uniaxial behavior. Int. J. Plast. 8, 91 – 116. Kan, Q.H., Yan, W.Y., Kang, G.Z., Guo, S.J., 2011. Experimental Observation on the Uniaxial Cyclic Deformation Behaviour of TA16 Titanium Alloy. Adv. Mater. Res. 415 – 417, 2318 – 2321. Kourousis, K. I., 2013. A cyclic plasticity model for advanced light metal alloys. App. Mech. Mat. 391, 3-8. Nath, A., Ray, K.K., Barai, S. V., 2019a. Evaluation of ratcheting behaviour in cyclically stable steels through use of a combined kinematic isotropic hardening rule and a genetic algorithm optimization technique. Int. J. Mech. Sci. 152, 138 – 150. Nath, A., Barai, S. V., Ray, K.K., 2019b. Prediction of asymmetric cyclic ‐ plastic behaviour for cyclically stable non ‐ ferrous materials. Fatigue Fract. Eng. Mater. Struct. 42, 2808 – 2822. Nath, A., Barai, S. V., Ray, K.K., 2019b. Estimation of cyclic hardening/softening and ratcheting response of materials through an algorithm to optimize parameters in Chaboche's hardening rule. Fatigue Fract. Eng. Mater. Struct. 45, 1847-1865. Paul, S.K., Sivaprasad, S., Dhar, S., Tarafder, M., Tarafder, S., 2010. Simulation of cyclic plastic deformation response in SA333 C – Mn steel by a kinematic hardening model. Comp. Mat. Sci. 48, 662-671. Zhang, J., Jiang, Y., 2008. Constitutive modeling of cyclic plasticity deformation of a pure polycrystalline copper. Int. J. Plast. 24, 1890 – 1915.

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Procedia Structural Integrity 43 (2023) 77–82

© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. © 20 23 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under the responsibility of MSMF10 organizers. Abstract Now days, n oclay-rei forced polymers ar one of the most wid ly used nanocomposites due to their high aspect ratio, higher contact area and their uniq e properti s. These c mposites are u ed in a number of industrial application , such as construction building sections and structural panels), automotive (gas ank , bumpers, interior and exterior panels), c emical processes (catalysts), pharmaceutical (as carriers of drugs and penetrants), aero p c (flame retar ant pa els and high performance components), f od pack ging and textiles, etc. Their interphase properties a e essential for determ ning their proper design and safety application when applying large mechanical load on them. In thi wo k, a parametric analysis is performed to investigate how the change in the interphase prope tie influ nces the interphas shear str s (ISS) and interphase peel s ress (IPS) n a nanoclay/polymer n ocomposite structure, subjected to axial load. The two-dimensional (2D) stres -function method is applied, which results in obtaining an analytical solution for ISS and IPS. The implemen ed para etric analysis shows how varying of e terphase mechanical properti s (Young ’s modulus and Poisson ’s r tio) and their geom tric pr perties (thickness and length) influence the value o the mod l interphase stress s. It was found, t at increasing the value of Young m dulus f th in erphase does not c ange significantly the value of the model ISS, as well as the value of IPS, n the co sidered nanoclay/polymer structur . On the other hand, increasing he v lue of the interphas layer thickness becomes important for the value of ISS and IPS after 25nm. It turned o t, that the interphase layer length is the most appr ciable parameter, influencing both the odel ISS and IPS. The obtained results could be useful for the proper design and safety application of similar nanoclay/polymer composites in industry. © 20 23 The Authors. Published by Elsevier B.V. This is an ope access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under the responsibility of MSMF10 organizers. 10th International Conference on Materials Structure and Micromechanics of Fracture Analytical and parametric analysis of interface stress behaviour in nanoclay/polymer composite Wilfried Becker a* , Elisaveta Kirilova b , Tatyana Petrova b , Jordanka Ivanova c a TU Darmstadt, Institute of Structural Mechanics, Franziska-Braun-Str. 7, L5|01 347a, 64287 Darmstadt, Germany b Bulgarian Academy of Sciences, Institute of Chemical Engineering, Acad. G. Bonchev Str., Bl.103, 1113 Sofia, Bulgaria c Bulgarian Academy of Sciences, Institute of Mechanics, Acad. G. Bonchev Str., Bl.4, 1113 Sofia, Bulgaria Abstract Nowadays, nanoclay-reinforced polymers are one of the most widely used nanocomposites due to their high aspect ratio, higher contact area and their unique properties. These composites are used in a number of industrial applications, such as construction (building sections and structural panels), automotive (gas tanks, bumpers, interior and exterior panels), chemical processes (catalysts), pharmaceutical (as carriers of drugs and penetrants), aerospace (flame retardant panels and high performance components), food packaging and textiles, etc. Their interphase properties are essential for determining their proper design and safety application when applying large mechanical loads on them. In this work, a parametric analysis is performed to investigate how the change in the interphase properties influences the interphase shear stress (ISS) and interphase peel stress (IPS) in a nanoclay/polymer nanocomposite structure, subjected to axial load. The two-dimensional (2D) stress-function method is applied, which results in obtaining an analytical solution for ISS and IPS. The implemented parametric analysis shows how varying of the interphase mechanical properties (Young ’s modulus and Poisson ’s ratio) and their geometric properties (thickness and length) influence the value of the model interphase stresses. It was found, that increasing the value of Young modulus of the interphase does not change significantly the value of the model ISS, as well as the value of IPS, in the considered nanoclay/polymer structure. On the other hand, increasing the value of the interphase layer thickness becomes important for the value of ISS and IPS after 25nm. It turned out, that the interphase layer length is the most appreciable parameter, influencing both the model ISS and IPS. The obtained results could be useful for the proper design and safety application of similar nanoclay/polymer composites in industry. 10th International Conference on Materials Structure and Micromechanics of Fracture Analytical and parametric analysis of interface stress behaviour in nanoclay/polymer composite Wilfried Becker a* , Elisaveta Kirilova b , Tatyana Petrova b , Jordanka Ivanova c a TU Darmstadt, Institut of Structural Mechanics, Fra z ska-Braun-Str. 7, L5|01 347a, 64287 Darmstadt, Germany b Bulgarian Academy of Sciences, I stitute of Ch mical Engineering, Acad. G. Bonchev Str., Bl. 03, 1113 Sofi , Bulgaria c Bulgarian Academy of Sciences, Institute of Mechanics, Acad. G. Bonchev Str., Bl.4, 1113 Sofia, Bulgaria Keywords: interphase shear and peel stress; nanoclay/interphase/polymer nanocomposite; applied mechanical load; parametric analysis Keywords: interphase shear and peel stress; nanoclay/interphase/polymer nanocomposite; applied mechanical load; parametric analysis

* Corresponding author. Tel.: +0-000-000-0000 E-mail address: becker@fsm.tu-darmstadt.de * Correspon ing author. Tel.: +0 000-000-0000 E-mail address: becker@fsm.tu-darmstadt.de

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. 10.1016/j.prostr.2022.12.238 2452-3216 © 2023 The Authors. Published by Elsevier B.V. This is an ope access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. 2452-3216 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers.

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1. Introduction Nowadays, nanoclay-reinforced polymers are one of the most widely used nanocomposites due to their high aspect ratio, higher contact area and their unique properties. These composites are used in a number of industrial applications, such as construction (building sections and structural panels), automotive (gas tanks, bumpers, interior and exterior panels), chemical processes (catalysts), pharmaceutical (as carriers of drugs and penetrants), aerospace (flame retardant panels and high performance components), food packaging and textiles, (Guo et al., 2018). In recent years the range of research on this type of nanocomposite has considerably increased and shows that more than two phases appear in it. For the proper design and safe application of the latter, it is necessary to study their properties in order to predict the occurrence of delamination in them. For this purpose, a lot of experimental and numerical methods have been developed (Heydari-Meybodi et al., 2015). Some of them are based of application of finite element methods for modeling of elastic modulus or stiffness of the interphase layer with emphasising on the role of inclusion/matrix interphase (Saber ‐ Samandari and Afaghi ‐ Khatibi, 2007). The latter has been applied to determine either the probability of debonding at the interface (Heydari-Meybodi et al., 2015) or for investigation of the effect of various geometrical parameters, such as the change of the nanoclay contact area, the nanoclay angle in the planes, etc. on the elastic modulus of the polymer nanocomposite reinforced with nanoclay (Heydari-Meybodi et al., 2016). There have been developed other methods such as this one of Halpin-Tsai and Mori-Tanaka (Fornes and Paul, 2003) for estimation of the reinforcement in layered aluminosilicates and glass fibers using the composite theories or 3D voxel based model for determination of the damage in nanocomposites with intercalated structures (Mishnaevsky, 2012). The lack of methods in the available literature for investigation of the effect of the interphase properties of considered nanocomposites on the interphase shear stress (ISS) and interphase peel stress (IPS) in them is noteworthy. Тhis study has implemented the two -dimensional stress-function method of Petrova et al. (2022) to obtain the analytical solutions for ISS and IPS in nanoclay/polymer nanocomposite structure, subjected to axial load. The performed parametric analysis shows how varying of the interphase mechanical properties (Young ’s modulus and Poisson ’s ratio) and their geometric properties (thickness and length) influence the value of the model ISS and IPS. The obtained results could be useful for the proper design and safety application of similar nanoclay/polymer composites in industry. 2. Problem statement and obtained analytical solutions Fig. 1 shows a representative volume element (RVE) of a three-layerd nanoclay-interlayer(interphase)-polymer nanocomposite structure. The axial tensile force P (N.m) is applied to the polymer layer. The coordinate system Oxy is placed at the left end of the structure with a length l , the y-coordinates for the layers are: 2 2 2 1 , , a t a b h c h h y h h h = = + = + + For the nanocomposite structure shown in Fig 1 the two - dimensional stress function method provided by Petrova et al (2022) has been applied and as a result, analytical solutions for the ISS and IPS in the middle layer (interphase) of the structure could be obtained The details for the solution development are given in Petrova et al (2022) Here, for brevity, only the most basic equations have been used

y

Nanoclay platelet, h 1 Interphase layer, h a

b = h 2

Polymer matrix, h 2

P

P

x

Fig. 1. RVE of three-layer nanoclay-interlayer(interphase)-polymer nanocomposite structure.

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The initial assumptions for application of 2D stress-function variation method for the considered nanocomposite structure consist in the following: 1) The axial stresses in the layers are assumed to be functions of axial coordinate x only; 2) In the adhesive interface layer the axial stress is neglected, or ( ) 0 a xx  = ; 3) All stresses in the layers (axial, normal (peel) and shear stresses) are determined under the assumption of the plane stress formulation (standard constitutive strain-stress equations from 2D elasticity theory). Of course, the same kind of analysis could be performed with a plane strain state. But this would not change the results significantly. Based on the abovementioned model assumptions, the following Eqs. (1) and (2) represent the model two dimensional stresses ISS - ( ) a xy  and IPS - ( ) a yy  respectively, in the interphase layer of the considered nanocomposite structure. In these equations the ISS and IPS are expressed in terms of a model solution for single stress potential function in Petrova et al. 2022 (the axial stress of the nanoclay layer is noted with 1  , function only of x ) and its first and second derivatives: ( ) ' 1 1 a xy h   = (1)

(  = + −     ) 2 h h c y 1 1 2 

( ) a yy

''  1



(2)

The axial stress 1  in the nanoclay layer (the general analytical solution of four real roots λ i in Petrova et al. (2022)) and its first and second derivative are given by:

( ) 1

( ) 2

( ) 3

( )

1 4 4 C exp x C exp x C exp x C exp x A      = + + + − 1 2 3

(3)

( ) 1

( ) 2

( ) 3

( ) 4

1 1 4 ' C exp x C exp x C exp x C exp x          = + + + 2 2 3 4 1 3

(4)

( ) 1

( ) 2

( ) 3

( )

2 4 4 4 x C exp x C exp x C exp x C exp           = + + + 2 2 2 1 1 1 2 2 3 3

(5)

(

)

The constant 0  and Young ’s modulus of the first and third layer in the structure in Fig.1, C i are the integration constants in the model solution, determined from the boundary conditions. The next section presents a parametric analysis of each of the obtained solutions for the ISS and IPS, when at constant other values of the geometrical parameters and applied load, the following parameters will be varied: ( ) a E - the value of the interphase Young modulus; l - the length of the RVE; a h - the thickness of the interphase layer. The aim of this analysis is to obtain results about the influence of the interphase properties on the interphase shear and peel stresses in it. 3. Results and discussion The geometrical dimensions and mechanical properties of the considered nanocomposite structure (Fig. 1) nanoclay-interphase-polymer are taken from Zhu and Narh, (2004) as: E (1) =178 GPa, E (2) =2.75 GPa, v (1) =0.2, v (a) =0.35, v (2) =0.35, σ 0 =350 MPa, h 1 = 1 nm, h 2 = 1 µ m. Here with superscript indices (1) - nanoclay, (a) - interphase and (2)- polymer, the respective layer number is noted. The interphase layer’s thickness and length as well as the values for interphase Young ’s modulus are varied according to Table 1, as the other properties and dimensions remain unchanged, unless otherwise is said. For the model stresses calculation and graphics representation, Mathcad Prime v.6.0 and Sigma Plot, v.13.0 have been used. ( ) 1 ( ) 1 ( ) 2 0 A E E E   = + in the solution depends on the value of external static load

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Table 1. The values of three parameters for the performed parametric analysis. h a , (nm) l , (nm) Young ’s modulus of the interphase, (GPa) 0.5 50 0.275 1 100 0.688 2 200 1.375 5 300 2.063 10 500 2.75 25 1000

50 70

Firstly, the ISS and IPS values, when varying Young ’s modulus of the interphase according to Table 1, at fixed constant values of the other geometric dimensions of the materials, have been calculated. The obtained results for ISS are presented graphically in Fig. 2 (a) and that for IPS - in Fig.2 (b). For both ISS and IPS the considered length of the nanocomposite structure is fixed as l =200 nm, h a = 0.5 nm. It can be seen, that the influence of interface’s Young ’s modulus is practically insignificant for both ISS and IPS. For simplicity, in Fig. 2(b) only 3 plots are presented. In both graphics the curves (surfaces) at different E (a) values overlap each other. It could be explained theoretically by the type of developed solutions for ISS and IPS – they are both not dependent on E (a) , nor on the values of roots λ i and integration constants C i .

(a)

(b)

Fig. 2. Influence of the Young modulus of the interphase on the ISS (a) and IPS (b) in the interphase layer.

When the thickness h a of the interphase layer is varied according to Table 1, the respective results are shown in Fig.3 (a) for ISS and for IPS – in Fig.3 (b). It has to be noted, that the IPS surface for h a =1nm is seen as a thin dark band to the right-hand side in 3D plot in Fig.3 (b). For better visibility, an additional embedded plot in Fig. 3 (b) in blue represents the IPS for h a =1 nm separately, because 3 IPS surfaces are in different range in respect to the layer thickness. The values of length and E (a) are chosen fixed as l =200 nm and E (a) = 1.375 GPa.

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(a)

(b)

Fig. 3. Influence of the interphase thickness on the ISS (a) and on the IPS (b) in the interphase layer.

It has turned out, that after 25 nm the increase of h a leads to a decrease in the ISS maximal value. For IPS, the increasing in h a leads to slight increasing in IPS maximal values in the boundary between polymer and interface (Fig. 3 (b)). The influence of h a could be explained with the presence of h a in Eq. (2) for IPS. For ISS, the influence of the interphase thickness h a is due to the relationship between the roots λ i in Eq. (1-2) – according to Petrova et al. (2022), they depend on the thickness of the structure layers. Finally, in Fig. 4 (a) and 4 (b), the influence of the interphase length on the ISS and IPS is shown. It could be concluded, that with an increase of the interphase length after 50 nm, the ISS and IPS maximal values near the ends of the structure length also increased.

(a)

(b)

Fig. 4 Influence of the interphase length on the ISS (a) and on the IPS (b) in the interphase layer.

Again, for better visibility, an embedded plot at l=50 nm for IPS is shown in blue (in the whole plot it is seen as a plane, because of the different range in IPS values). In fine, it has to be mentioned, that obtained results for the interphase length and thickness influence on the stresses in the interphase layer are in qualitative agreement with the conclusions of Zare and Rhee (2020), namely: the interphase length and especially its thickness are among the factors,

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that directly influence the interphase strength and improve the degree of stress transfer in nanoclay/polymer composites. The thinnest clay and thickest interphase caused the strongest interphase, the opposite largely reduced the degree of stress transfer between the polymer and the nanoclay particle; the very short length also deteriorates the interphase strength (Zare and Rhee, 2020). 4. Conclusions In this study, an analysis of the influence of the interphase properties of nanoclay / polymer nanocomposite structures subjected to axial load on the interphase shear and peel stresses was performed. For this purpose, 2D stress function method has been applied (Petrova et al., 2022). The latter results from analytical solutions for both ISS and IPS. Based on the data in Zhu and Narh (2004) for nanoclay/polymer nanocomposite with interphase layer, it was found, that increasing the value of Young ’s modulus of the interphase does not change significantly the value of the model ISS and IPS in the considered structure. On the other hand, the interphase layer thickness begins to affect the ISS and IPS with an increasing in its value over 25 nm, while ISS decreases, and IPS - increases. It also turned out, that the interphase layer length is the most important parameter, influencing the model ISS and IPS. The obtained results are in agreement with findings in the literature and could be useful for the proper design and safety application of similar nanoclay/polymer composites in industry. Acknowledgments The authors gratefully acknowledge the support of DFG under the project No. BE 1090/48-1 “Some industrial applications for nanocomposites under mechanical and environmental loading” References Fornes, T. D., Paul, D. R., 2003. Modeling Properties of Nylon 6/clay Nanocomposites using Composite Theories. Polymer 44(17), 4993-5013. Guo, F., Aryana, S., Han, Y., Jiao, Y., 2018. A Review of the Synthesis and Applications of Polymer – Nanoclay Composites. Applied Sciences 8, 1696 (29 pages). Heydari-Meybodi, M., Saber-Samandari, S., Sadighi, M., 2015. A New Approach for Prediction of Elastic Modulus of Polymer/nanoclay Composites by Considering Interfacial Debonding: Experimental and Numerical Investigations. Composites science and technology 117, 379-385. Heydari-Meybodi, M., Saber-Samandari, S., Sadighi, M., 2016. 3D Multiscale Modeling to Predict the Elastic Modulus of Polymer/nanoclay Composites Considering Realistic Interphase Property. Composite Interfaces 23(7), 641-661. Kirilova, E., Petrova, T., Becker W., Ivanova J., 2019. Mathematical Modelling of Stresses in Graphene Polymer Nanocomposites under Static Extension Load, 2019 IEEE 14th Nanotechnology Materials and Devices Conference (NMDC), Stockholm, Sweden, pp. 1-4. Mishnaevsky Jr, L., 2012. Micromechanical analysis of nanocomposites using 3D voxel based material model. Composites Science and Technology 72(10), 1167-1177. Petrova, T., Kirilova, E., Becker, W., Ivanova, J., 2022. Two-dimensional Stress and Strain Analysis for Graphene-polymer Nanocomposite under Axial Load. Journal of Applied and Computational Mechanics 8(3), 1065-1075. Saber‐Samandari, S , Afaghi‐Khatibi, A , 2007. Evaluation of elastic modulus of polymer matrix nanocomposites. Polymer composites 28(3), 405-411. Zare, Y., Rhee, K. Y., 2020. Modeling of interphase strength between polymer host and clay nanoparticles in nanocomposites by clay possessions and interfacial/interphase terms, Applied Clay Science 192, 105644. Zhu, L., Narh, K.A., 2004. Numerical Simulation of the Tensile Modulus of N anoclay‐filled Polymer Composites. Journal of Polymer Science Part B: Polymer Physics 42(12), 2391-2406.