Micromechanical aspects of the effect of temperature and local plastic strain magnitude on the fracture toughness of ferrite steels

ScienceDirect Available online at www.sciencedirect.com Procedia Structural Integrity 43 (2023) 228–233 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.2022.12.263 10.1016/j.prostr.2022.12.263 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. Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2022) 000–000 www.elsevier.com/locate/procedia 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. 10th International Conference on Materials Structure and Micromechanics of Fracture Micromechanical aspects of the effect of temperature and local plastic strain magnitude on the fracture toughness of ferrite steels Sergiy Kotrechkoa,b,*, Vladislav Kozákc, Oleksandra Zatsarnaa, Galyna Ziminaa, Nataliya Stetsenkoa, and Ivo Dlouhýc,d a G.V. Kurdyumov Institute for Metal Physics, National Academy of Sciences of Ukraine, 36, Academician Vernadsky Blvd., UA-0380 Kyiv, Ukraine b National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” 37, Peremohy Ave. UA-03056 Kyiv, Ukraine c Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, 61600 Brno, Czech Republic d Institute of Materials Science and Engineering, Faculty of Mechanical Engineering, Brno University of Technology, Technicka 2, 61669 Brno, Czech Republic Abstract The paper shows that the influence of plastic strain and temperature on the rate of crack nuclei (CN) formation is a crucial factor controlling the shape of the temperature dependence of fracture toughness and its scatter limits. Within the framework of the microscopic model described, it is explained that the dependence of the incompatibility of microplastic deformation at grain boundaries or interfaces on the value of plastic strain and temperature is the reason for the effect of these factors on the rate of CN formation. The dependencies of the CN bulk density on temperature and plastic strain are given. In the latter case, a nonmonotonic change in CN density is observed. The maximum intensity of CN formation is observed when the critical value of plastic strain is reached. For ferritic structural steels, this strain is about 2%. Using reactor pressure vessel steel and cast manganese steel as examples, it is shown that not taking these effects into account in the local fracture approach leads to considerable errors in the prediction of the temperature dependence of the fracture toughness and its scatter limits. © 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 the responsibility of MSMF10 organizers. Keywords: Fracture toughness; Ductile-to-brittle transition; Local approach to fracture; Crack nuclei, Micromechanism. * Corresponding author. Tel.: +38 044 4241352; fax: +38 044 4242561. E-mail address: [email protected] Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2022) 000–000 www.elsevier.com/locate/procedia 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. 10th International Conference on Materials Structure and Micromechanics of Fracture Micromechanical aspects of the effect of temperature and local plastic strain magnitude on the fracture toughness of ferrite steels Sergiy Kotrechkoa,b,*, Vladislav Kozákc, Oleksandra Zatsarnaa, Galyna Ziminaa, Nataliya Stetsenkoa, and Ivo Dlouhýc,d a G.V. Kurdyumov Institute for Metal Physics, National Academy of Sciences of Ukraine, 36, Academician Vernadsky Blvd., UA-0380 Kyiv, Ukraine b National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” 37, Peremohy Ave. UA-03056 Kyiv, Ukraine c Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, 61600 Brno, Czech Republic d Institute of Materials Science and Engineering, Faculty of Mechanical Engineering, Brno University of Technology, Technicka 2, 61669 Brno, Czech Republic Abstract The paper shows that the influence of plastic strain and temperature on the rate of crack nuclei (CN) formation is a crucial factor controlling the shape of the temperature dependence of fracture toughness and its scatter limits. Within the framework of the microscopic model described, it is explained that the dependence of the incompatibility of microplastic deformation at grain boundaries or interfaces on the value of plastic strain and temperature is the reason for the effect of these factors on the rate of CN formation. The dependencies of the CN bulk density on temperature and plastic strain are given. In the latter case, a nonmonotonic change in CN density is observed. The maximum intensity of CN formation is observed when the critical value of plastic strain is reached. For ferritic structural steels, this strain is about 2%. Using reactor pressure vessel steel and cast manganese steel as examples, it is shown that not taking these effects into account in the local fracture approach leads to considerable errors in the prediction of the temperature dependence of the fracture toughness and its scatter limits. © 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 the responsibility of MSMF10 organizers. Keywords: Fracture toughness; Ductile-to-brittle transition; Local approach to fracture; Crack nuclei, Micromechanism. * Corresponding author. Tel.: +38 044 4241352; fax: +38 044 4242561. E-mail address: [email protected] 2 Author name / Structural Integrity Procedia 00 (2022) 000–000 1. Introduction The Local Approach (LA) to fracture was introduced in the 1980s. It aimed to solve key problems in fracture mechanics, such as the prediction of the specimen geometry and statistical size effect on fracture toughness (transferability problem), as well as the effects of temperature, degradation of the material after neutron irradiation, etc. The possibility of solving these complicated problems was seen in the application of the statistical local criterion for the initiation of fracture in the vicinity of a crack, because only this approach allows to take into account the most important features of the mechanism of fracture initiation on the microscale. This meant the possibility of establishing a permanent physical basis for fracture mechanics. However, as the results of numerous studies show (Wiesner and Goldthorpe (1996), Gao X., Dodds (2000), Pineau (2006), Wasiliuk et al. (2006), Ruggieri and Dodds (2018)), the LA did not fully meet expectations. First of all, it turned out that the parameters of the Weibull distribution - m and u  - are not constants, but their values depend on temperature, specimen geometry and the plastic strain value. In general, it was found that LA can only be used at low temperatures. However, from the application point of view, the ductile to brittle transition region (DBT) is the most important. The main reason for this state of the art is the unjustified simplification of the local quantitative criterion for fracture initiation. Many works have attempted to overcome this shortcoming (e.g. Pluvinage et al. (1999), Bordet et al. (2005), Gao et al. (2005), Ruggieri et al. (2015), Jivkov et al. (2019), Ruggieri and Jivkov (2019)). In general, two key issues have not been addressed and have been identified as needing to be addressed, namely: (i) the need to consider the effects of temperature and the magnitude of local plastic strain on the bulk density ρ of the crack nuclei (CN); (ii) the consideration of the value of the threshold stress th  . As far as the latter is concerned, it is indeed a methodological problem. It consists in developing a technique for the experimental determination of th  . Therefore, a simplified method of th  determination for structural steels has been proposed by Kotrechko et al. (2019). Jivkov et al. (2019), Ruggieri and Jivkov (2019) have tried to take into account the effect of plastic strain and test temperature on the number of CN formed in the local plastic zone in front of a macrocrack tip. Ruggieri and Jivkov (2019) proposed suitable approximations to account for the effects of temperature and plastic strain on crack nuclei density and, accordingly, on Weibull stress magnitude. This allowed the critical values of the IC J integral to be predicted with high accuracy for both low and high temperatures within the DBT region. At the same time, Ruggieri and Jivkov (2019) emphasized the importance of clarifying the physical nature of such a significant effect of temperature on the intensity of CN generation. An attempt to develop a physical version of the Local Approach based on a detailed analysis of the processes of crack nuclei formation and unstable equilibrium in a polycrystalline aggregate was made by Kotrechko (2002, 2013), Kotrechko and Mamedov (2016), Kotrechko et al. (2019). This approach made it possible to determine regularities of the influence of both the metal structure and the conditions of its loading on the probability of fracture. However, it proved to be quite demanding for engineering calculations. At the same time, this approach can be used as a tool to analyze the key effects controlling the initiation of cleavage in the vicinity of the macrocrack, in particular to analyze the effects of temperature and the magnitude of plastic strain on the crack nuclei formation rate within the "process zone". This report examines the physical reasons for the influence of plastic strain and temperature on the intensity of CN formation and shows how this affects both the slope of the temperature dependence and the scatter limits of the fracture toughness of structural steels in the ductile to brittle transition region. 2. Theoretical background Inhomogeneity of microplastic deformation, which gives rise to plastic deformation incompatibility on grain or interphase boundaries, is a general reason for the CN formation in polycrystalline solids. Kotrechko (2013), Kotrechko and Mamedov (2016) proposed a generalised model of CN formation in a polycrystalline aggregate. Adapted version of this model for prediction the brittle fracture of structural steels enables to derive the expression for the bulk density of CN formed at a given value of local plastic strain: Sergiy Kotrechko et al. / Procedia Structural Integrity 43 (2023) 228–233 229 Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2022) 000–000 www.elsevier.com/locate/procedia 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. 10th International Conference on Materials Structure and Micromechanics of Fracture Micromechanical aspects of the effect of temperature and local plastic strain magnitude on the fracture toughness of ferrite steels Sergiy Kotrechkoa,b,*, Vladislav Kozákc, Oleksandra Zatsarnaa, Galyna Ziminaa, Nataliya Stetsenkoa, and Ivo Dlouhýc,d a G.V. Kurdyumov Institute for Metal Physics, National Academy of Sciences of Ukraine, 36, Academician Vernadsky Blvd., UA-0380 Kyiv, Ukraine b National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” 37, Peremohy Ave. UA-03056 Kyiv, Ukraine c Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, 61600 Brno, Czech Republic d Institute of Materials Science and Engineering, Faculty of Mechanical Engineering, Brno University of Technology, Technicka 2, 61669 Brno, Czech Republic Abstract The paper shows that the influence of plastic strain and temperature on the rate of crack nuclei (CN) formation is a crucial factor controlling the shape of the temperature dependence of fracture toughness and its scatter limits. Within the framework of the microscopic model described, it is explained that the dependence of the incompatibility of microplastic deformation at grain boundaries or interfaces on the value of plastic strain and temperature is the reason for the effect of these factors on the rate of CN formation. The dependencies of the CN bulk density on temperature and plastic strain are given. In the latter case, a nonmonotonic change in CN density is observed. The maximum intensity of CN formation is observed when the critical value of plastic strain is reached. For ferritic structural steels, this strain is about 2%. Using reactor pressure vessel steel and cast manganese steel as examples, it is shown that not taking these effects into account in the local fracture approach leads to considerable errors in the prediction of the temperature dependence of the fracture toughness and its scatter limits. © 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 the responsibility of MSMF10 organizers. Keywords: Fracture toughness; Ductile-to-brittle transition; Local approach to fracture; Crack nuclei, Micromechanism. * Corresponding author. Tel.: +38 044 4241352; fax: +38 044 4242561. E-mail address: [email protected] Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2022) 000–000 www.elsevier.com/locate/procedia 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. 10th International Conference on Materials Structure and Micromechanics of Fracture Micromechanical aspects of the effect of temperature and local plastic strain magnitude on the fracture toughness of ferrite steels Sergiy Kotrechkoa,b,*, Vladislav Kozákc, Oleksandra Zatsarnaa, Galyna Ziminaa, Nataliya Stetsenkoa, and Ivo Dlouhýc,d a G.V. Kurdyumov Institute for Metal Physics, National Academy of Sciences of Ukraine, 36, Academician Vernadsky Blvd., UA-0380 Kyiv, Ukraine b National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” 37, Peremohy Ave. UA-03056 Kyiv, Ukraine c Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, 61600 Brno, Czech Republic d Institute of Materials Science and Engineering, Faculty of Mechanical Engineering, Brno University of Technology, Technicka 2, 61669 Brno, Czech Republic Abstract The paper shows that the influence of plastic strain and temperature on the rate of crack nuclei (CN) formation is a crucial factor controlling the shape of the temperature dependence of fracture toughness and its scatter limits. Within the framework of the microscopic model described, it is explained that the dependence of the incompatibility of microplastic deformation at grain boundaries or interfaces on the value of plastic strain and temperature is the reason for the effect of these factors on the rate of CN formation. The dependencies of the CN bulk density on temperature and plastic strain are given. In the latter case, a nonmonotonic change in CN density is observed. The maximum intensity of CN formation is observed when the critical value of plastic strain is reached. For ferritic structural steels, this strain is about 2%. Using reactor pressure vessel steel and cast manganese steel as examples, it is shown that not taking these effects into account in the local fracture approach leads to considerable errors in the prediction of the temperature dependence of the fracture toughness and its scatter limits. © 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 the responsibility of MSMF10 organizers. Keywords: Fracture toughness; Ductile-to-brittle transition; Local approach to fracture; Crack nuclei, Micromechanism. * Corresponding author. Tel.: +38 044 4241352; fax: +38 044 4242561. E-mail address: [email protected] 2 Author name / Structural Integrity Procedia 00 (2022) 000–000 1. Introduction The Local Approach (LA) to fracture was introduced in the 1980s. It aimed to solve key problems in fracture mechanics, such as the prediction of the specimen geometry and statistical size effect on fracture toughness (transferability problem), as well as the effects of temperature, degradation of the material after neutron irradiation, etc. The possibility of solving these complicated problems was seen in the application of the statistical local criterion for the initiation of fracture in the vicinity of a crack, because only this approach allows to take into account the most important features of the mechanism of fracture initiation on the microscale. This meant the possibility of establishing a permanent physical basis for fracture mechanics. However, as the results of numerous studies show (Wiesner and Goldthorpe (1996), Gao X., Dodds (2000), Pineau (2006), Wasiliuk et al. (2006), Ruggieri and Dodds (2018)), the LA did not fully meet expectations. First of all, it turned out that the parameters of the Weibull distribution - m and u  - are not constants, but their values depend on temperature, specimen geometry and the plastic strain value. In general, it was found that LA can only be used at low temperatures. However, from the application point of view, the ductile to brittle transition region (DBT) is the most important. The main reason for this state of the art is the unjustified simplification of the local quantitative criterion for fracture initiation. Many works have attempted to overcome this shortcoming (e.g. Pluvinage et al. (1999), Bordet et al. (2005), Gao et al. (2005), Ruggieri et al. (2015), Jivkov et al. (2019), Ruggieri and Jivkov (2019)). In general, two key issues have not been addressed and have been identified as needing to be addressed, namely: (i) the need to consider the effects of temperature and the magnitude of local plastic strain on the bulk density ρ of the crack nuclei (CN); (ii) the consideration of the value of the threshold stress th  . As far as the latter is concerned, it is indeed a methodological problem. It consists in developing a technique for the experimental determination of th  . Therefore, a simplified method of th  determination for structural steels has been proposed by Kotrechko et al. (2019). Jivkov et al. (2019), Ruggieri and Jivkov (2019) have tried to take into account the effect of plastic strain and test temperature on the number of CN formed in the local plastic zone in front of a macrocrack tip. Ruggieri and Jivkov (2019) proposed suitable approximations to account for the effects of temperature and plastic strain on crack nuclei density and, accordingly, on Weibull stress magnitude. This allowed the critical values of the IC J integral to be predicted with high accuracy for both low and high temperatures within the DBT region. At the same time, Ruggieri and Jivkov (2019) emphasized the importance of clarifying the physical nature of such a significant effect of temperature on the intensity of CN generation. An attempt to develop a physical version of the Local Approach based on a detailed analysis of the processes of crack nuclei formation and unstable equilibrium in a polycrystalline aggregate was made by Kotrechko (2002, 2013), Kotrechko and Mamedov (2016), Kotrechko et al. (2019). This approach made it possible to determine regularities of the influence of both the metal structure and the conditions of its loading on the probability of fracture. However, it proved to be quite demanding for engineering calculations. At the same time, this approach can be used as a tool to analyze the key effects controlling the initiation of cleavage in the vicinity of the macrocrack, in particular to analyze the effects of temperature and the magnitude of plastic strain on the crack nuclei formation rate within the "process zone". This report examines the physical reasons for the influence of plastic strain and temperature on the intensity of CN formation and shows how this affects both the slope of the temperature dependence and the scatter limits of the fracture toughness of structural steels in the ductile to brittle transition region. 2. Theoretical background Inhomogeneity of microplastic deformation, which gives rise to plastic deformation incompatibility on grain or interphase boundaries, is a general reason for the CN formation in polycrystalline solids. Kotrechko (2013), Kotrechko and Mamedov (2016) proposed a generalised model of CN formation in a polycrystalline aggregate. Adapted version of this model for prediction the brittle fracture of structural steels enables to derive the expression for the bulk density of CN formed at a given value of local plastic strain: 230 Sergiy Kotrechko et al. / Procedia Structural Integrity 43 (2023) 228–233 Author name / Structural Integrity Procedia 00 (2022) 000–000 3 ( )  = r C t t cb ttg d2 , (1) where cb  is carbide particles density and function g(t) is given by equation:          =22 1 2 t tg exp)( . (2) In this equation, t is the magnitude of normalised shear microstresses ns  , acting in slip systems: ns D tns   = . (3) Here ns D is the variance of shear microscopic stresses ns  . Then C t and r t are the critical values of normalized microstresses at which the CN forms and which there is relaxation of incompatibilities in intergranular/interphase boundaries. Dependences of these parameters on the plastic strain magnitude and temperature pre-determine the influence of these factors on the density of crack nuclei generating within the “process zone”. In the first approximation, expression for the value of critical microstress of the CN formation, C t , may be represented as follows:                               −+−   +=  1 11 C e C C e e k d e Cd M k t _ * max _ , (4) where  and 𝑒𝑒 are the equivalent macroscopic stresses and strains; d and max d are the average and maximum (with a given probability) ferritic grain sizes; C  is the critical stress of СN formation as a result of the carbide particle cleavage; М is the orientation factor (for -Fe M = 0.36);  k , e k , C, β are the coefficients (for ferritic steels  k = 0.225; e k = 1.52 MPa; С = 0.0336 m/N; β ≈ 2.57 MPa m0.5 (Kotrechko (2013));  is the parameter that takes the value 0 or 1. If the value of the equivalent plastic strain 𝑒𝑒 is less than or equal to the critical value C e , then  = 0 and e*=𝑒𝑒, otherwise  = 1 and e*=eC. The physical meaning of the last two terms in expression (4) lies in the fact that they describe the effect of microstresses arising at grain boundaries on the process of crack nuclei formation. They characterize the value of shear microstresses caused by the incompatibility of microplastic deformations at grain boundaries. The magnitude of this incompatibility changes non-monotonically with an increase in macroplastic strain e . In the early stages of plastic deformation, it grows and then begins to decrease as a result of rearrangements of the crystal structure in the vicinity of facets of grain boundaries (in the near-boundary regions). Transition from growth ( = 0) to a decrease ( = 1) in these incompatibilities occurs at critical strain C e . At the macroscale, the value of C e can be estimated based on the dependence of the cleavage fracture stress of smooth (unnotched) specimens, f , on the value of strain preceding fracture. When the strain C e is achieved, this stress f reaches its minimum value. For common structural steels with basic ferritic microstructure, the value if critical strain C e ≈0.02. With an increase in ferrite grain size by annealing, the critical strain C e can increase to ≈0.05 (Kotrechko et al. (2007)). Generalized expression for the critical value of microstress relaxation due to the incompatibility of microplastic deformations, r t is:                               −+−   +=  1 11 C e b Y re e k d e d r m M k t _ * max _ , (5) where Y  is the critical stress of the relaxation beginning; b m is orientation factor for relaxation slip systems in intergranular boundaries; r is the distance from the grain boundary to the origin of microstress relaxation. 4 Author name / Structural Integrity Procedia 00 (2022) 000–000 The obtained dependences describe the process of CN formation at the microscale, so they contain a significant number of microscopic parameters. The values of these parameters, at best, can be estimated by the order of magnitude, therefore, the obtained dependences can't be used directly in LA. However, the importance of these dependences is that they physically substantiate the effect of plastic strain and temperature on the rate of crack nuclei generation, as well as enable to ascertain the regularities of this effect and evaluate it by the order of ρ magnitude. The non-monotonic dependence of ρ on the plastic strain value is a characteristic feature of the CN formation in polycrystalline metals. This reflects the peculiarity of change in the incompatibility of plastic strains at grain boundaries during the polycrystal deformation. It grows to a certain value of macroplastic strain C e , and then begins to decrease. According to the model proposed, the growth of temperature should cause a monotonic decrease in the value of ρ. This is due to the thermally activated relaxation processes at the grain boundaries. In terms of the current model, this means a decrease in the value r t (dependence (5)) against the background of increasing the critical value of normalized stress of CN formation C t (dependence (4)). Theoretical dependences of ρ on the magnitude of plastic strain and temperature are shown in Fig. 1. The following values of microscopic parameters were used in their building: b m = 0.1; r = 10-6 m; d = 1010-6 m; dmax = 3010-6 m; C  = 7 GPa ( G C 10. , where GPa 70G is the iron carbide shear modulus (Kotrechko (2013)).) In the first approximation, the mean value of may be estimated by the value of thermally activated component of the yield strength YY  : ( )   TeCCC Y  lnexp. 321 50 += , (6) where e  is the plastic strain rate; С1, С2, С3 are the constants, which values for typical ferritic steels are: С1 = 1033 MPa, С2 = 0.0068 K-1, С3 = 0.000415 K-1 (Kotrechko (2002)); e  = 10-4 s1. The equivalent stress value  was calculated as: n e      = 0020 20 . . , (7) where n is strain hardening exponent (value n = 0.05 was used); 20.  is yield strength: ( )   TeCCC a lnexp .32120 ++= , (8) where a  is athermal component of 20.  (in the calculations a typical value a  = 470 MPa was utilized). 3. Experimental verification In accordance with the data shown in Fig. 1, the effect of plastic strain magnitude and temperature on the value of ρ is not additive one. The maximum influence on ρ is observed at strains close to critical one C e . At large strains, sensitivity of ρ to changes in temperature decreases. As noted above, the direct use of the above model in LA implies significant difficulties. This is due to the need to determine the values of microscopic parameters. Therefore, Kotrechko et al. (2021) proposed approximation dependences to describe the effect of temperature and plastic strain on the CN density. The employment of these dependences makes it possible to predict more accurately the slope of the temperature dependence of fracture toughness and its scatter limits (Fig. 2). It should be emphasized that not accounting for the considered effect associated with a change in the CN density, gives rises not only to an error in the absolute value of Jc K , but also to an error in the values of the failure probability. Moreover, if the error in Jc K is tens of percent, the degree of overestimation of the fracture probability may be 2-4 times (Fig. 3, red arrows indicate the error in the value of fracture probability). Sergiy Kotrechko et al. / Procedia Structural Integrity 43 (2023) 228–233 231 Author name / Structural Integrity Procedia 00 (2022) 000–000 3 ( )  = r C t t cb ttg d2 , (1) where cb  is carbide particles density and function g(t) is given by equation:          =22 1 2 t tg exp)( . (2) In this equation, t is the magnitude of normalised shear microstresses ns  , acting in slip systems: ns D tns   = . (3) Here ns D is the variance of shear microscopic stresses ns  . Then C t and r t are the critical values of normalized microstresses at which the CN forms and which there is relaxation of incompatibilities in intergranular/interphase boundaries. Dependences of these parameters on the plastic strain magnitude and temperature pre-determine the influence of these factors on the density of crack nuclei generating within the “process zone”. In the first approximation, expression for the value of critical microstress of the CN formation, C t , may be represented as follows:                               −+−   +=  1 11 C e C C e e k d e Cd M k t _ * max _ , (4) where  and 𝑒𝑒 are the equivalent macroscopic stresses and strains; d and max d are the average and maximum (with a given probability) ferritic grain sizes; C  is the critical stress of СN formation as a result of the carbide particle cleavage; М is the orientation factor (for -Fe M = 0.36);  k , e k , C, β are the coefficients (for ferritic steels  k = 0.225; e k = 1.52 MPa; С = 0.0336 m/N; β ≈ 2.57 MPa m0.5 (Kotrechko (2013));  is the parameter that takes the value 0 or 1. If the value of the equivalent plastic strain 𝑒𝑒 is less than or equal to the critical value C e , then  = 0 and e*=𝑒𝑒, otherwise  = 1 and e*=eC. The physical meaning of the last two terms in expression (4) lies in the fact that they describe the effect of microstresses arising at grain boundaries on the process of crack nuclei formation. They characterize the value of shear microstresses caused by the incompatibility of microplastic deformations at grain boundaries. The magnitude of this incompatibility changes non-monotonically with an increase in macroplastic strain e . In the early stages of plastic deformation, it grows and then begins to decrease as a result of rearrangements of the crystal structure in the vicinity of facets of grain boundaries (in the near-boundary regions). Transition from growth ( = 0) to a decrease ( = 1) in these incompatibilities occurs at critical strain C e . At the macroscale, the value of C e can be estimated based on the dependence of the cleavage fracture stress of smooth (unnotched) specimens, f , on the value of strain preceding fracture. When the strain C e is achieved, this stress f reaches its minimum value. For common structural steels with basic ferritic microstructure, the value if critical strain C e ≈0.02. With an increase in ferrite grain size by annealing, the critical strain C e can increase to ≈0.05 (Kotrechko et al. (2007)). Generalized expression for the critical value of microstress relaxation due to the incompatibility of microplastic deformations, r t is:                               −+−   +=  1 11 C e b Y re e k d e d r m M k t _ * max _ , (5) where Y  is the critical stress of the relaxation beginning; b m is orientation factor for relaxation slip systems in intergranular boundaries; r is the distance from the grain boundary to the origin of microstress relaxation. 4 Author name / Structural Integrity Procedia 00 (2022) 000–000 The obtained dependences describe the process of CN formation at the microscale, so they contain a significant number of microscopic parameters. The values of these parameters, at best, can be estimated by the order of magnitude, therefore, the obtained dependences can't be used directly in LA. However, the importance of these dependences is that they physically substantiate the effect of plastic strain and temperature on the rate of crack nuclei generation, as well as enable to ascertain the regularities of this effect and evaluate it by the order of ρ magnitude. The non-monotonic dependence of ρ on the plastic strain value is a characteristic feature of the CN formation in polycrystalline metals. This reflects the peculiarity of change in the incompatibility of plastic strains at grain boundaries during the polycrystal deformation. It grows to a certain value of macroplastic strain C e , and then begins to decrease. According to the model proposed, the growth of temperature should cause a monotonic decrease in the value of ρ. This is due to the thermally activated relaxation processes at the grain boundaries. In terms of the current model, this means a decrease in the value r t (dependence (5)) against the background of increasing the critical value of normalized stress of CN formation C t (dependence (4)). Theoretical dependences of ρ on the magnitude of plastic strain and temperature are shown in Fig. 1. The following values of microscopic parameters were used in their building: b m = 0.1; r = 10-6 m; d = 1010-6 m; dmax = 3010-6 m; C  = 7 GPa ( G C 10. , where GPa 70G is the iron carbide shear modulus (Kotrechko (2013)).) In the first approximation, the mean value of may be estimated by the value of thermally activated component of the yield strength YY  : ( )   TeCCC Y  lnexp. 321 50 += , (6) where e  is the plastic strain rate; С1, С2, С3 are the constants, which values for typical ferritic steels are: С1 = 1033 MPa, С2 = 0.0068 K-1, С3 = 0.000415 K-1 (Kotrechko (2002)); e  = 10-4 s1. The equivalent stress value  was calculated as: n e      = 0020 20 . . , (7) where n is strain hardening exponent (value n = 0.05 was used); 20.  is yield strength: ( )   TeCCC a lnexp .32120 ++= , (8) where a  is athermal component of 20.  (in the calculations a typical value a  = 470 MPa was utilized). 3. Experimental verification In accordance with the data shown in Fig. 1, the effect of plastic strain magnitude and temperature on the value of ρ is not additive one. The maximum influence on ρ is observed at strains close to critical one C e . At large strains, sensitivity of ρ to changes in temperature decreases. As noted above, the direct use of the above model in LA implies significant difficulties. This is due to the need to determine the values of microscopic parameters. Therefore, Kotrechko et al. (2021) proposed approximation dependences to describe the effect of temperature and plastic strain on the CN density. The employment of these dependences makes it possible to predict more accurately the slope of the temperature dependence of fracture toughness and its scatter limits (Fig. 2). It should be emphasized that not accounting for the considered effect associated with a change in the CN density, gives rises not only to an error in the absolute value of Jc K , but also to an error in the values of the failure probability. Moreover, if the error in Jc K is tens of percent, the degree of overestimation of the fracture probability may be 2-4 times (Fig. 3, red arrows indicate the error in the value of fracture probability). 232 Sergiy Kotrechko et al. / Procedia Structural Integrity 43 (2023) 228–233 Author name / Structural Integrity Procedia 00 (2022) 000–000 5 Fig. 1. The effect of temperature T and plastic strain value _ e on the intensity of CN forming ρ: _ e is the equivalent plastic strain; C e is the critical value of strain ( C e =0.02). Fig. 2. (Color online). The temperature dependences of fracture toughness for RPV steel (a) and cast low-alloyed manganese steel (b)  represent the experimental values; lines are the calculation results (solid dark-yellow lines are the results of calculation taking into account changes in density CN density ρ, dashed black lines are the same at the constant values of ρ)). Fig. 3. Dependence of fracture probability on the magnitude of fracture toughness (for RPV steel, T=-1200C), calculated accounting for change in the crack nuclei bulk density  () and at a constant value of  (). 6 Author name / Structural Integrity Procedia 00 (2022) 000–000 4. Conclusions 1. The dependence of the incompatibilities of microplastic deformations at grain boundaries or other interfacial boundaries on plastic strain and temperature is the main reason for the influence of these factors on the intensity of cleavage nuclei generation, which is a key factor controlling the slope of the temperature dependence and its scatter limits 2. Failure to take this effect into account in conventional models of local approach leads to errors in the prediction not only of the fracture toughness values but also of the fracture probability value. In the latter case, the underestimation of the fracture probability can be about 2-4 times. This error reaches a maximum at low fracture probabilities (lower threshold value for fracture toughness). This is of crucial importance for critical structural elements (RPVs, oil and gas pipelines, etc.). Acknowledgements This work was supported by the National Academy of Sciences of Ukraine (under grant number 0121U107569). References Bordet, S.R., Karstensen, A.D., Knowles, D.M., Wiesner, C.S., 2005. A new statistical local criterion for cleavage fracture in steel. Part I: model presentation. Engineering Fracture Mechanics 72, 435–452. https://doi.org/10.1016/j.engfracmech.2004.02.009 Bordet, S. R., Karstensen, A. D., Knowles, D. M., Wiesner, C. S., 2005. A new statistical local criterion for cleavage fracture in steel. Part II: application to an offshore structural steel. Engineering Fracture Mechanics 72, 453–474. https://doi.org/10.1016/j.engfracmech.2004.02.010 Gao, X., Dodds, R. H., Jr., 2000. Constraint effects on the ductile-to-brittle transition temperature of ferritic steels: a Weibull stress model. International Journal of Fracture 102, 43–69. https://doi.org/10.1023/A:1007526006632 Gao, X., Chang, G., Srivatsan, T.S., 2005, Prediction of cleavage fracture in ferritic steel: A modified Weibull stress model. Materials Science and Engineering: A 394, 210-219. https://doi.org/10.1016/j.msea.2004.11.035 Jivkov, A. P., Burgos, D. S., Ruggieri, C., Beswick, J., Savioli, R. G., James, P., Sherry, A., 2019. Use of local approaches to calculate changes in cleavage fracture toughness due to pre-straining and constraint effects. Theoretical and Applied Fracture Mechanics 104, 102380. https://doi.org/10.1016/j.tafmec.2019.102380 Kotrechko, S., 2002. Physical fundamentals of Local Approach to analysis of cleavage fracture, in: “Transferability of Fracture Mechanical Characteristic, NATO Science Series, Series II, 78”, In: I. Dlouhý (Ed.). Kluwer Academic Publishers, pp. 135-150. Kotrechko, S., Strnadel, B., Dlouhý, I., 2007. Fracture toughness of cast ferritic steel applying local approach. Theoretical and Applied Fracture Mechanics 47, 171-181. https://doi.org/10.1016/j.tafmec.2006.11.008 Kotrechko, S., 2013. The key problems of local approach to cleavage fracture. Journal of Theoretical and Applied Mechanics (Warsaw) 51 (1), 75–89. Kotrechko, S., Mamedov, S., 2016. Multi-scale local approach to cleavage fracture and its applications, 19th European Conference on Fracture (ECF19). - Kazan, Russia, 26-31 Aug 2012. Curran Associates, Inc. 1, pp. 971–982. Kotrechko, S., Zatsarna, O., Kozák, V., Dlouhý, I., 2019. Threshold fracture stress: theory and application. Procedia Structural Integrity 23, 413– 418. https://doi.org/10.1016/j.prostr.2020.01.122 Kotrechko, S., Kozák, V., Zatsarna, O., Zimina, G., Stetsenko, N., Dlouhý, I., 2021. Incorporation of Temperature and Plastic Strain Effects into Local Approach to Fracture. Materials 14, 6224. https://doi.org/10.3390/ma14206224 Pineau A., 2006. Development of the local approach to fracture over the past 25 years: Theory and applications. International Journal of Fracture 138, 139–166. https://doi.org/10.1007/s10704-006-0035-1 Pluvinage, G., Azari, Z., Kadi, N., Dlouhy, I., Kozák, V., 1999. Effect of ferritic microstructure on local damage zone distance associated with fracture near notch, Theoretical and Applied Fracture Mechanics 31, 149-156, https://doi.org/10.1016/S0167-8442(99)00009-9 Ruggieri, C., Savioli, R.G., Dodds, R.H., Jr., 2015. An engineering methodology for constraint corrections of elastic–plastic fracture toughness – Part II: Effects of specimen geometry and plastic strain on cleavage fracture predictions. Engineering Fracture Mechanics 146, 185-209. https://doi.org/10.1016/j.engfracmech.2015.06.087 Ruggieri, C., Dodds, R.H., 2018. A local approach to cleavage fracture modelling: An overview of progress and challenge for engineering applications. Engineering Fracture Mechanics 187, 381–403. https://doi.org/10.1016/j.engfracmech.2017.12.021 Ruggieri, C., Jivkov, A.P., 2019. A local approach incorporating the measured statistics of microcracks to assess the temperature dependence of cleavage fracture for a reactor pressure vessel steel. Procedia Structural Integrity 18, 28–35. https://doi.org/10.1016/j.prostr.2019.08.137 Wasiliuk, B., Petti, J.R., Dodds, R.H., 2006. Temperature dependence of Weibull stress parameters: Studies using euro-material. Engineering Fracture Mechanics 73, 1046-1059. https://doi.org/10.1016/j.engfracmech.2005.11.006 Wiesner, C.S., Goldthorpe, M.R., 1996. The effect of temperature and specimen geometry on the parameters of the ‘Local Approach’ to cleavage fracture. Journal de Physique IV France 06, C6-295 – C6-304. https://doi.org/10.1051/jp4:1996629 Sergiy Kotrechko et al. / Procedia Structural Integrity 43 (2023) 228–233 233 Author name / Structural Integrity Procedia 00 (2022) 000–000 5 Fig. 1. The effect of temperature T and plastic strain value _ e on the intensity of CN forming ρ: _ e is the equivalent plastic strain; C e is the critical value of strain ( C e =0.02). Fig. 2. (Color online). The temperature dependences of fracture toughness for RPV steel (a) and cast low-alloyed manganese steel (b)  represent the experimental values; lines are the calculation results (solid dark-yellow lines are the results of calculation taking into account changes in density CN density ρ, dashed black lines are the same at the constant values of ρ)). Fig. 3. Dependence of fracture probability on the magnitude of fracture toughness (for RPV steel, T=-1200C), calculated accounting for change in the crack nuclei bulk density  () and at a constant value of  (). 6 Author name / Structural Integrity Procedia 00 (2022) 000–000 4. Conclusions 1. The dependence of the incompatibilities of microplastic deformations at grain boundaries or other interfacial boundaries on plastic strain and temperature is the main reason for the influence of these factors on the intensity of cleavage nuclei generation, which is a key factor controlling the slope of the temperature dependence and its scatter limits 2. Failure to take this effect into account in conventional models of local approach leads to errors in the prediction not only of the fracture toughness values but also of the fracture probability value. In the latter case, the underestimation of the fracture probability can be about 2-4 times. This error reaches a maximum at low fracture probabilities (lower threshold value for fracture toughness). This is of crucial importance for critical structural elements (RPVs, oil and gas pipelines, etc.). Acknowledgements This work was supported by the National Academy of Sciences of Ukraine (under grant number 0121U107569). References Bordet, S.R., Karstensen, A.D., Knowles, D.M., Wiesner, C.S., 2005. A new statistical local criterion for cleavage fracture in steel. Part I: model presentation. Engineering Fracture Mechanics 72, 435–452. https://doi.org/10.1016/j.engfracmech.2004.02.009 Bordet, S. R., Karstensen, A. D., Knowles, D. M., Wiesner, C. S., 2005. A new statistical local criterion for cleavage fracture in steel. Part II: application to an offshore structural steel. Engineering Fracture Mechanics 72, 453–474. https://doi.org/10.1016/j.engfracmech.2004.02.010 Gao, X., Dodds, R. H., Jr., 2000. Constraint effects on the ductile-to-brittle transition temperature of ferritic steels: a Weibull stress model. International Journal of Fracture 102, 43–69. https://doi.org/10.1023/A:1007526006632 Gao, X., Chang, G., Srivatsan, T.S., 2005, Prediction of cleavage fracture in ferritic steel: A modified Weibull stress model. Materials Science and Engineering: A 394, 210-219. https://doi.org/10.1016/j.msea.2004.11.035 Jivkov, A. P., Burgos, D. S., Ruggieri, C., Beswick, J., Savioli, R. G., James, P., Sherry, A., 2019. Use of local approaches to calculate changes in cleavage fracture toughness due to pre-straining and constraint effects. Theoretical and Applied Fracture Mechanics 104, 102380. https://doi.org/10.1016/j.tafmec.2019.102380 Kotrechko, S., 2002. Physical fundamentals of Local Approach to analysis of cleavage fracture, in: “Transferability of Fracture Mechanical Characteristic, NATO Science Series, Series II, 78”, In: I. Dlouhý (Ed.). Kluwer Academic Publishers, pp. 135-150. Kotrechko, S., Strnadel, B., Dlouhý, I., 2007. Fracture toughness of cast ferritic steel applying local approach. Theoretical and Applied Fracture Mechanics 47, 171-181. https://doi.org/10.1016/j.tafmec.2006.11.008 Kotrechko, S., 2013. The key problems of local approach to cleavage fracture. Journal of Theoretical and Applied Mechanics (Warsaw) 51 (1), 75–89. Kotrechko, S., Mamedov, S., 2016. Multi-scale local approach to cleavage fracture and its applications, 19th European Conference on Fracture (ECF19). - Kazan, Russia, 26-31 Aug 2012. Curran Associates, Inc. 1, pp. 971–982. Kotrechko, S., Zatsarna, O., Kozák, V., Dlouhý, I., 2019. Threshold fracture stress: theory and application. Procedia Structural Integrity 23, 413– 418. https://doi.org/10.1016/j.prostr.2020.01.122 Kotrechko, S., Kozák, V., Zatsarna, O., Zimina, G., Stetsenko, N., Dlouhý, I., 2021. Incorporation of Temperature and Plastic Strain Effects into Local Approach to Fracture. Materials 14, 6224. https://doi.org/10.3390/ma14206224 Pineau A., 2006. Development of the local approach to fracture over the past 25 years: Theory and applications. International Journal of Fracture 138, 139–166. https://doi.org/10.1007/s10704-006-0035-1 Pluvinage, G., Azari, Z., Kadi, N., Dlouhy, I., Kozák, V., 1999. Effect of ferritic microstructure on local damage zone distance associated with fracture near notch, Theoretical and Applied Fracture Mechanics 31, 149-156, https://doi.org/10.1016/S0167-8442(99)00009-9 Ruggieri, C., Savioli, R.G., Dodds, R.H., Jr., 2015. An engineering methodology for constraint corrections of elastic–plastic fracture toughness – Part II: Effects of specimen geometry and plastic strain on cleavage fracture predictions. Engineering Fracture Mechanics 146, 185-209. https://doi.org/10.1016/j.engfracmech.2015.06.087 Ruggieri, C., Dodds, R.H., 2018. A local approach to cleavage fracture modelling: An overview of progress and challenge for engineering applications. Engineering Fracture Mechanics 187, 381–403. https://doi.org/10.1016/j.engfracmech.2017.12.021 Ruggieri, C., Jivkov, A.P., 2019. A local approach incorporating the measured statistics of microcracks to assess the temperature dependence of cleavage fracture for a reactor pressure vessel steel. Procedia Structural Integrity 18, 28–35. https://doi.org/10.1016/j.prostr.2019.08.137 Wasiliuk, B., Petti, J.R., Dodds, R.H., 2006. Temperature dependence of Weibull stress parameters: Studies using euro-material. Engineering Fracture Mechanics 73, 1046-1059. https://doi.org/10.1016/j.engfracmech.2005.11.006 Wiesner, C.S., Goldthorpe, M.R., 1996. The effect of temperature and specimen geometry on the parameters of the ‘Local Approach’ to cleavage fracture. Journal de Physique IV France 06, C6-295 – C6-304. https://doi.org/10.1051/jp4:1996629