Đang tải... (xem toàn văn)
The model is coupled with irradiation effect via tracking dislocation loop evolution on each slip system. On the basis of the model, uniaxial tensile tests of unirradiated and irradiated RPV steel(take Chinese A508-3 as an example) at different temperatures are simulated, and the simulation results agree well with the experimental results. Furthermore, crystal plasticity damage is introduced into the model. Then the damage behavior before and after irradiation is studied using the model. The results indicate that the model is an effective tool to study the effect of irradiation and temperature on the mechanical properties and damage behavior.
Nuclear Engineering and Technology 51 (2019) 501e509 Contents lists available at ScienceDirect Nuclear Engineering and Technology journal homepage: www.elsevier.com/locate/net Original Article Study on the irradiation effect of mechanical properties of RPV steels using crystal plasticity model Junfeng Nie a, *, Yunpeng Liu a, Qihao Xie c, Zhanli Liu b a Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing, 100084, China b Applied Mechanics Lab., School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China c Data Science and Information Technology Research Center, Tsinghua-Berkeley Shenzhen Institute, Shenzhen, 518055, China a r t i c l e i n f o a b s t r a c t Article history: Received August 2018 Received in revised form October 2018 Accepted 21 October 2018 Available online 30 October 2018 In this paper a body-centered cubic(BCC) crystal plasticity model based on microscopic dislocation mechanism is introduced and numerically implemented The model is coupled with irradiation effect via tracking dislocation loop evolution on each slip system On the basis of the model, uniaxial tensile tests of unirradiated and irradiated RPV steel(take Chinese A508-3 as an example) at different temperatures are simulated, and the simulation results agree well with the experimental results Furthermore, crystal plasticity damage is introduced into the model Then the damage behavior before and after irradiation is studied using the model The results indicate that the model is an effective tool to study the effect of irradiation and temperature on the mechanical properties and damage behavior © 2018 Korean Nuclear Society, Published by Elsevier Korea LLC This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Keywords: Crystal plasticity Dislocation evolution Irradiation effect Damage RPV steel Introduction Reactor pressure vessel(RPV) is an important component of nuclear power plant It will be subjected to extreme conditions such as high temperature, high pressure and high energy neutron during its operation This will lead to the effect of temperature and radiation on the mechanical properties of RPV steel, which are the critical factors for operating safely a nuclear power plant or for extending its lifetime [1] Irradiation effect is related to many factors such as irradiation dose [2], temperature [3], microstructure and defects [4,5] Formation and evolution of microstructure have a more substantial impact among all factors [6] Crystal plasticity theory for metal reveals that macroscopic deformation of crystals is closely related to dislocation slip Taylor [7] formulated a model for the relationship between texture and mechanics A mathematical description of crystal plastic deformation and kinematics was derived rigorously by Hill and Rice [8] Asaro [9] and Peirce [10] further developed and improved the plastic constitutive theory of crystals Combining crystal plasticity theory and finite element method can be used to introduce microstructure defects to the study of the mechanical response of crystal materials [11] A Ma [12] established a constitutive model based on dislocation density for face-centered cubic(FCC) crystals and implemented the model under crystal plasticity finite element framework There are some studies on the application of crystal plasticity theory to the research of irradiation effect of materials Vincent [13] carried out the modeling of RPV steel brittle fracture using crystal plasticity computations on polycrystalline aggregates, and the modeling attempted to predict the brittle fracture by calculating and discussing the largest values of the stresses Fabien Onimus [14] provided a polycrystalline modeling to study the mechanical behavior of neutron irradiated zirconium alloys, and the model described the effects of the dislocation channeling mechanism on the mechanical behavior of irradiated zirconium alloys Xiao [15] studied the strain softening in BCC iron induced by irradiation used crystal plasticity, and it was also indicated that the flow stress increases under neutron irradiation Generally, crystal plasticity is an effective approach to study the mechanical properties and microstructure evolution induced by irradiation for metal * Corresponding author E-mail address: niejf@tsinghua.edu.cn (J Nie) https://doi.org/10.1016/j.net.2018.10.020 1738-5733/© 2018 Korean Nuclear Society, Published by Elsevier Korea LLC This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/) 502 J Nie et al / Nuclear Engineering and Technology 51 (2019) 501e509 materials In this paper, a body-centered cubic(BCC) crystal plasticity model comprising irradiation effect and crystal plasticity damage based on dislocation evolution is proposed and numerically implemented under finite element framework Influences of irradiation and temperature on the mechanical properties of Chinese A508-3 steel which has a BCC structure are studied based on the model Furthermore, crystal plasticity damage is introduced to the model, and the damage behavior of the material is also simulated The simulation results agree well with the experimental results BCC crystal plasticity model based on microscopic dislocation mechanism 2.1 Kinematical theory of crystals F ¼ F * $F P (1) Where F * and F P are the elastic part and plastic part of the deformation gradient, respectively Velocity gradient tensor is related to the deformation gradient, and can be decomposed into elastic part and plastic part as: L ẳ F $F ẳ L* ỵ LP (2) L* ¼ F * $F *À1 (3) n X a¼1 ga s*a 5m*a (4) vectors s*a and m*a are the slip direction and normal to the slip plane of the slip system a in the intermediate configuration, respectively ga is the slipping rate of the slip system a 2.2 Basic crystal plasticity model formulation 2.2.1 Constitutive relation Following the elastic constitutive relation proposed by Hill and Rice, the relation describing the distortion and rotation is given by V s Ki ¼ C : D À V n X ma ẳ (6) *a s 5m*a ỵ m*a 5s*a (7) · This section is a simple summary of the kinematical theory for the mechanics of crystals, following the work of Taylor [7], Hill [8], Rice [8] and Asaro [9] In order to facilitate the analysis, crystal deformation is assumed to be divided into two processes in crystal plasticity theory Firstly, the crystal translates from the original reference configuration to the intermediate configuration via dislocation slip; then the crystal undergoes lattice distortion and torsion and reaches the current configuration The two processes respectively correspond to plastic deformation and elastic deformation of the crystal Deformation gradient of the crystal is decomposed as: 2.2.2 Dislocation slip formulation and flow rules In the dislocation motion theory, it is argued that plastic deformation of the material is accomplished via the slipping of dislocations on the slip plane The dislocation slip is assumed to obey Schmid's law [16] When the resolved shear stress on the slip plane exceeds the corresponding slip resistance, the dislocations start to slip resulting in the plastic deformation in the crystal 48 potential slip systems should be considered in the BCC crystal which is 12 in the FCC crystal In the process of finite deformation, Schmid resolved shear stress is given by ta ¼ sKi : ma BCC crystal plasticity model based on dislocation evolution will be described in the following chapter LP ¼ F P $F P1 ẳ deformation rate à ẵC : ma ỵ ua $sKi À sKi $ua ga The slipping rate ga is generally expressed as the following thermal activation form [17] a > > >g ¼ ; < jta j < g a !p #q ) " ( a a > a ¼ ga exp À Q0 À jt j À g > > g : a _ kT t sgnðta Þ ; jta j > g a (8) _a _a t ¼ t0 G G0 (9) where Q0 denotes the activation energy of sliding without extern force, p and q are flow rules related parameters, G and G0 are the elastic shear modulus at current temperature T and 0K, respec_a tively, t is the maximum of slip resistance without thermal activation energy, g a is the slip resistance caused by dislocations in the material 2.2.3 Hardening based on dislocations Dislocation mechanism of crystal plastic deformation proposed by Orowan [18] has been generally acknowledged With the dislocation density increasing, interactions among dislocations are stronger and the slip resistance increases A hardening formula based on dislocation density according to Taylor's hardening law [7] is raised so as to represent the hardening behavior of the crystal: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N h i u X Aab rbM ỵ rbI ga ¼ Gbtqr (10) b¼1 where b is the magnitude of the Burgers vector, qr is a statistical parameter denoting the deviation between real distribution and hypothetical regular distribution of dislocations, Aaa (no sum) and Aab ðasbÞ are self and latent hardening coefficients respectively and denote the contribution of each slip system to the slip resistance of the current slip system, rbM is the mobile dislocation den- (5) a¼1 Where s Ki denotes the Jaumann derivative of the Kirchoff stress tensor sKi in the original configuration, C is the stiffness tensor, D is the deformation rate tensor, u is the spin rate and m is the sity and rbI is the immobile dislocation density of slip system b The evolution of dislocations in a BCC crystal mainly has the following patterns: multiplication and annihilation of dislocations [19,20], mobile dislocations being trapped as immobile dislocations [21], dynamic recovery of immobile dislocations [22] The evolution formulas of mobile and immobile dislocations are integrated and J Nie et al / Nuclear Engineering and Technology 51 (2019) 501e509 given by raM ¼ raI ¼ Table Parameters of the elastic modulus(GPa) kmul 2Rc a À rM À a ga bld b bl (11) a ga À k r dyn I bla (12) 2.3 Evolution of dislocation loops induced by irradiation A lot of researches have shown that collisions between incident particles and atoms of the crystal lattice induce irradiation defects during the irradiation For BCC crystals, dislocation loops are the primary defect structure Reactions between dislocation loops and mobile dislocations inhibit the motion of dislocations and result in the irradiation hardening A term with respect to dislocation loops is introduced into Eq (10) to reflect the contribution of irradiation to slip resistance The hardening formula coupling with irradiation effect is (13) where Ni denotes the dislocation loop density of slip system a, di is the mean size of dislocation loops in slip system a, qi is the hardening strength of dislocation loops Deo [23] indicated that the density and mean size of dislocation loops at the initial stage of defect accumulation are respectively proportional to the square root of the irradiation dose on the basis of the experiments N ¼ A,dpa2 (14) dai ¼ B,dpa2 (15) A and B are material constants, the scope of application is 0.0001~10dpa It is widely considered that the interactions between dislocation loops and dislocations raise the evolution of dislocation loops while the real process remains unknown Similar to the evolution of dislocations, results of the interactions between dislocation loops and dislocations can be described by the following patterns: annihilation of dislocation loops, translation from dislocation loops to mobile dislocations, mobile dislocations cut by dislocation loops Patra [24] proposed a phenomenological model to describe the annihilation of dislocation loops In order to make the model more universal, we change the exponential constant in the origin formula to a variable The final annihilation formula of dislocation loops is given as: Ác À a Á1Àc a Ra À a rM N_ i;ann dai ¼ i N dai jg_ j b C12 C44 G0 G 236 134 119 87.6 82.534 g_ ðsÀ1 Þ p q b t ðMPaÞ 107 0.47 1.1 390 Q0 ð10À19 JÞ Tð CÞ unirr irr À100 1.85 2.41 20 2.15 2.81 288 3.98 4.71 equivalent to increasing the multiplication rate of mobile dislocations, the cutting process between dislocation loops and mobile dislocations decreases the density of mobile dislocations Both of the processes are fitted by adjusting kmul in Eq (11) Due to the introduction of dislocation loops, the ld and la have changed into new forms: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,v u N uX b t ld z1 rM ỵ Nbi dbi (16) where Rai is the critical size of the annihilation of dislocation loops, c is used as the annihilate index reflecting the influence caused by dislocation loop density and mobile dislocation density on the annihilation rate of dislocation loops The translation from dislocation loops to mobile dislocations is (17) b¼1 b¼1 C11 Table Parameters of the plastic flow law where kmul is the multiplication coefficient of mobile dislocations, ld is the mean length of mobile dislocation fragments, ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN a b ld z1= bÀ1 rM , Rc denotes the critical size of the annihilation, l p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the mean free path of the trapping process, la ¼ 1=br raM þ raI , kdyn is the dynamic recovery coefficient of immobile dislocations vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N h i u X a Aab rbM ỵ rbI ỵ qi N dai g ẳ Gbtqr 503 1 ẳ ỵ ẳ la lar lai q q br raM ỵ raI ỵ bi Nai dai (18) Up to now the model has been completely described Moreover, the BCC crystal plasticity model based on dislocation evolution coupling with irradiation effect is implemented into ABAQUS as an interface of user material (UMAT) [25] for numerically computation Material parameters and modeling The RPV material used in this study is Chinese A508-3 steel The material properties were characterized via tensile testing carried out by Lin Yun [26] on the unirradiated and irradiated conditions Parameter selection mainly refers to the tests It is assumed that 48 slip systems in A508-3 steel have the same initial value of critical resolved slip resistance, 390MPa [27] The dislocation density of steel up to 107/mm2 and increases as the deformation increasing Assuming an uniform initial mobile and immobile dislocation density for each slip system,  107/mm2 The elastic constants can be obtained from Refs [28,29] Irradiation damage is estimated at 0.1 dpa [30] Ashby [31] suggested the activation energy for dislocation glide Q0 to be of the order of 0:5Gb3 for irradiated materials The optimized values of the material parameters are obtained by minimizing the difference between experimental results and the macroscopic response of the polycrystalline Finite Element computations The optimum set of parameters is given in Tables 1e4 Voronoi [32] method is generally used to construct a Table Parameters of the hardening law qr qi bðnmÞ kdyn Aaa 0.06 1.0 0.248 275 1.0 Rc ðnmÞ br Aab kmul unirr irr 1.5 0.074 0.2 0.0735 0.0955 504 J Nie et al / Nuclear Engineering and Technology 51 (2019) 501e509 Table Parameters of the irradiation effect bi Að1013 mmÀ3 Þ Bð10À6 mmÞ c 0.1 3:7 0.8 Fig Simulation stress-strain curves of the models with sets of grid numbers Fig The 3D voronoi model consists of 40 grains with random distributed crystlline orientations polycrystalline model A 3D voronoi polycrystalline model is shown in Fig To be consistent with the actual grain size, the model is assigned a length of 5mm for each side, containing 40 grains The distribution of the grain orientations is random and periodic boundary conditions are set for all directions It is a quasi-tension, and the constant stretch rate is 0.2 mm=min which is the same as the experimental condition [21] The element type is C3D10 The validity of the model is analyzed from the perspective of grain orientations and grid numbers Owing to the randomness of grain orientations, the stress and strain of the polycrystal are calculated by volume averaging the values at all integral points of each grain According to Ref [33], sets of grain orientations are tested, including [111], [110] and sets of random orientations The simulation results agree with theoretical analysis [34] that the uniaxial tensile stress-strain curves of the models with random Fig Simulation stress-strain curves of the models with sets of grain orientations orientations are between the stress-strain curves of the models with orientation [111] and orientation [110] The stress-strain curves of the models with random orientations are very close and have the same trend (Fig 2), indicating that 40 grains can better reflect the tensile property of the material In order to be more representative, the model with the second set of random orientations is used The models are divided into 10589, 48123 and 71897 grids separately As the number of the grids increases, the stress is basically unchanged (Fig 3), which proves that the results are convergent and the model can represent the real stress level of the structure For the purpose of saving computing time, the model is divided into 48123 grids The comparison of the stress-strain curves between experimental and simulation results at À100 C, 20 C, and 288 C are plotted in Fig and Fig Crystal plasticity damage Generally, metal materials exhibit better ductility and can support a large plastic deformation before fracture In the process of Fig Experimental and simulation stress-strain curves for samples at different temperatures before irradiation J Nie et al / Nuclear Engineering and Technology 51 (2019) 501e509 505 Table Parameters of the reference stress B0 B0 ðMPaÞ Tð CÞ À100 20 288 unirr 1420 1300 1165 irr 1650 1320 1230 initiation term of the damage corresponds to a specific plastic strain threshold, below which no damage by microcraking occurs The initiation term of the damage is described by cumulative slip strain and the model is validated in Ref [37] The final damage evolution law is " # ( ) b m s f ẳ Hg gsth ị bI5I ỵ bị I : B0 · Fig Experimental and simulation stress-strain curves for samples at different temperatures after irradiation Table Parameters of the initial value of cumulative slip strain at the beginning of damage gsth gsth Tð CÞ À100 20 unirr 0.4 0.3 0.3 irr 0.4 0.25 0.32 288 ð4Þ (21) 〈〉 in Eq (20) or (21) is the McCauley symbol When the value in parentheses is greater than 0, the value is the same; instead, the value is The guidelines deems that only the material being stretched may appear damage HðÞ is the Heaviside function, controlling that the damage occurs when the cumulative slip strain in all slip systems reaches the threshold Effective stress and damage factor is introduced into the UMAT Coupled with the damage, parameters selection and the calculation results are listed in Tables and The values b ¼ 0:5; n ¼ 20 are adopted to fit the experimental data The simulation results with damage model are shown in Fig and Fig and agree well with the experimental results of breaking process of the tensile test Results and discussion the growth of plastic strain, the damage begins to occur and accumulate, resulting in the degradation of mechanical properties The material will lose the carrying capacity when the damage reaches a certain extent Lemaitre [35] put forward the principle of strain equivalence and concluded that the constitutive model of the material with damage could be derived from the constitutive relationship of the material without damage by simply replacing the stress with the effective stress The definition of effective stress tensor is b ẳ I fịn $ s $I fịn ¼ s X s s bin bi b i i5 n s (19) The stress and strain contours at room temperature are shown in Fig and Fig Fig is the Mises stress contour and Fig is the maximum principal strain contour Both figures show obvious inhomogeneous distribution The values of stress and strain are larger around the grain boundary than interior of the grain, which indicate that stress concentration and non-uniform large deformation occur at the grain boundary Actually, the grain boundary will absorb the point defects induced by irradiation and reduce the defect density inside the grain In this study, the grain boundary is treated as pure geometric surface, the process of the absorption is i¼1 where I is the unit tensor, f is the damage factor tensor, n is a b i and n b b si are the ith eigenvector of the stress s material constant, s respectively In order to study the damage behavior of materials in the case of large deformation, anisotropic damage model improved by Lu Feng [36] Defining a damage factor tensor f, then the rate of damage is à " 4ị f ẳ bI5I ỵ bÞ I # : ( )m X b si b si n b si n B i¼1 (20) ð4Þ where I is four-order identity tensor, b and B0 are material parameters, b ¼ is corresponding to the isotropic damage and b ¼ is corresponding to the complete anisotropic damage Ref [35] has explained the introduction of the initiation term of the damage, and it described that before the microcracks are initiated, they must nucleate by the accumulation of microstresses accompanying incompatibilities of microstrains or by the accumulation of dislocations in metals For the pure tension case, the Fig Experimental and simulation stress-strain curves comprising damage for samples at different temperatures before irradiation 506 J Nie et al / Nuclear Engineering and Technology 51 (2019) 501e509 Fig Experimental and simulation stress-strain curves comprising damage for samples at different temperatures after irradiation Fig 10 The error bars associated with the experimental data compared with numerical results before irradiation Fig The contour of Mises stress at room temperature after irradiation Fig The contour of the maximum principle strain at room temperature after irradiation J Nie et al / Nuclear Engineering and Technology 51 (2019) 501e509 507 Table Experimental and simulation results of the yield stress before and after irradiation at different temperatures Yield Stress(MPa) Relative error(%) T ð CÞ Unirr(exp/sim) Irr(exp/sim) Incre-Mental (%) unirr irr À100 À40 20 100 200 288 555/547 N/447 409/417 N/408 N/405 389/395 638/655 N/575 517/534 N/510 N/475 441/472 14.9 28.7 26.4 25.1 17.3 13.3 1.4 2.7 1.9 3.3 1.5 7.0 Table Experimental and simulation results of the tensile strength before and after irradiation at different temperatures Tensile Strength(MPa) Relative error(%) T ð CÞ Unirr (exp/sim) Irr (exp/sim) Incre-mental (%) unirr irr À100 À40 20 100 200 288 708/691 N/642 573/607 N/599 N/597 556/584 766/768 N/696 639/635 N/625 N/625 605/612 8.1 8.4 11.5 4.3 4.7 8.8 2.4 0.3 5.9 0.6 5.0 1.1 ignored though it will cause the defects to converge at the grain boundary and enhance the resistance to the motion of dislocations The calculation results of stress-strain curve (Figs 6e7) agree well with the experimental results and the validity of the model is verified The error bars associated with the experimental data [26] before irradiation are plotted in Fig 10 The results of yield strength are within the margin of error The error bars of tensile strengths are small so the computed results are not in the error range, the actual deviation is not large The error values are in Table and Table To analysis the trends of mechanical properties, additional temperature conditions are calculated and the data of critical points are extracted into Tables and The variations of the yield stress and tensile strength with temperature are shown in Fig 11 and Fig 12 The results indicate that temperature has a great influence on the mechanical properties of the material The effect of thermal activation on the dislocation motion is enhanced with the increasing of temperature, which reduces the resistance of dislocation slip Therefore, the increasing temperature entails decrease of the yield strength and tensile strength of Chinese A508-3 steel Temperature will also influence the interactions between dislocation loops and dislocations and the irradiation hardening effect is also affected At room temperature, the yield stress and tensile strength after irradiation increase more significantly, which are 26.4% and 11.5% respectively That means the irradiation effect is more obvious at room temperature On the whole, the irradiation hardening effect increases first and then decreases from low temperature to high temperature Irradiation damage can be reflected by parameters gsth and B0 gsth is the slip strain when the damage of the material occurs Reference stress B0 controls the rate and degree of the damage According to the fitted values of gsth and B0 , the two parameters also show a temperature dependence The influence of irradiation on the damage is more pronounced at room temperature for that the damage occurs much earlier after irradiation at a cumulative slip strain of 0.25 The influence is inhibited and the values of gsth are larger at low temperature and high temperature Fig 11 The variations of the yield stress and tensile strength with temperature before and after irradiation Fig 12 The incremental of the yield stress and tensile strength after irradiation with temperature 508 J Nie et al / Nuclear Engineering and Technology 51 (2019) 501e509 Dta and Dsij are written in the function of Dεij Conclusion In this study, a crystal plasticity model based on dislocation evolution for BCC crystals is constructed and the model is coupled with the irradiation effect via introducing irradiation defects evolution The model is numerically implemented and then the mechanical properties of Chinese A508-3 steel with irradiation at different temperatures are simulated The conclusions have been presented as followed: $ The model can better describe the irradiation hardening of the material in a certain temperature range $ The model considering damage evolution can better describe the degradation of mechanical properties of RPV steel with irradiation and unirradiation $ The model can reflect the variations of the mechanical properties and damage behavior with temperature Future work will focus on developing a more detailed classification of irradiation defects and a numerical simulation method to capture the mechanical properties of the material Then the model can easily predict the mechanical properties of other BCC crystal materials after irradiation with a higher accuracy Acknowledgements The support of the National Natural Science Foundation of China under Grant No 11202114, Beijing Higher Education Young Elite Teacher Project under Grant No YETP0156 and National Science and Technology Major Project of China, Chnia, Grant No 2017ZX06902012 are gratefully acknowledged Appendix Incremental form is beneficial to the implementation of subroutines The tangent modulus method for rate dependent solid developed by Peirce [38] is used in the subroutine It is assumed that the increment of ga within the time increment Dt is defined as Dga ¼ ga ðt þ DtÞ À ga ðtÞ (22) the linear form is Dga ẳ Dt qịg_ at ỵ qg_ atỵDt (23) q is the integral parameter whose value between and Seeing that g_ a is a function of ta and g a , we can substitute the Taylor expansion of g_ atỵDt into Eq (22) and get Dga ẳ Dt g_ a ỵ q vg_ a vta vg_ a a vg_ a Dt ỵ q a Dga a vt vg ! _a and vvgga can be obtained according to Eq (8) (24) i h Dta ẳ Lijkl makl ỵ uaik sjk ỵ uajk sik $4Dij À Dsij ¼ Lijkl Dεkl À sij Dεkk À X b b mij Dg (27) b i Xh Lijkl makl ỵ uaik sjk ỵ uajk sik Dga a (28) Dga is written in the funtion of dislocation density Dga ẳ N ! Gbị2 X ab kmul À 2Rc À k a Dgb A q r r dyn I 2ga bld b M (29) b¼1 References [1] E Meslin, M Lambrecht, M Hern andez-Mayoral, et al., Characterization of neutron-irradiated ferritic model alloys and a RPV steel from combined APT, SANS, TEM and PAS analyses, J Nucl Mater 406 (1) (2009) 73e83 [2] T Takeuchi, A Kuramoto, J Kameda, et al., Effects of chemical composition and dose on microstructure evolution and hardening of neutron-irradiated reactor pressure vessel steels, J Nucl Mater 402 (2/3) (2010) 93e101 [3] S A Maloy, M R James, G Willcutt, et al., The mechanical properties of 316L/ 304L stainless steels, Alloy 718 and Mod 9Cre1Mo after irradiation in a spallation environment, J Nucl Mater 296 (1e3) (2001) 119e128 [4] M.K Miller, M.G Burke, An atom probe field ion microscopy study of neutronirradiated pressure vessel steels, J Nucl Mater 195 (1992) 68e82 camps, A Barbu, et al., Dual-beam irradiation of a-iron: [5] D Brimbal, B De heterogeneous bubble formation on dislocation loops, J Nucl Mater 418 (1) (2011) 313e315 [6] C Domain, C.S Becquart, L Maplerba, Simulation of radiation damage in Fe alloys: an object kinetic Monte Carlo approach, J Nucl Mater 335 (1) (2004) 121e145 [7] G.I Taylor, Plastic strain in metals, J Inst Met 62 (1938) 307e324 [8] R Hill, J.R Rice, Constitutive analysis of elastic-plastic crystals at arbitrary strain, J Mech Phys Solid 20 (6) (1972) 401e403 [9] R.J Asaro, Micro mechanics of crystals and polycrystals, Adv Appl Mech 23 (8) (1983) 11e15 [10] D Peirce, C.F Shih, A Needleman, A tangent modulus method for rate dependent solids, Comput Struct 18 (5) (1984) 875e887 [11] F.T Meissornnier, E.P Busso, N.P O'Dowd, Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains, Int J Plast 17 (2001) 601e640 [12] A Ma, F Roters, D Raabe, A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations, Acta Mater 54 (8) (2006) 2169e2179 [13] L Vincent, M Libert, B Marini, et al., Towards a modelling of RPV steel brittle fracture using crystal plasticity computations on polycrystalline aggregates, J Nucl Mater 406 (2010) 91e96 chade, A polycrystalline modeling of the me[14] Fabien Onimus, Jean-Luc Be chanical behavior of neutron irradiated zirconium alloys, J Nucl Mater 384 (2009) 163e174 [15] Xiazi Xiao, Dmitry Terentyev, Long Yu, et al., Modelling irradiation-induced softening in BCC iron by crystal plasticity approach, J Nucl Mater 466 (2015) 312e315 [16] E Schmid, W Boas, Plasticity of crystals, Aeronaut J 54 (479) (1950) 353e719 [17] E.P Busso, Cyclic Deformation of Monocrystalline Nickel Aluminide and High Temperature Coatings, Doctoral Thesis, Massachusetts Institute of Technology, Cambridge, 2005 [18] E Orowan, Zur Kristallogr, Z Phys 89 (3/4) (1934) 327e343 a !q ' a !qÀ1 a & vg_ a Q0 pq a Q0 jt j À g a p jt j À g a p jt j À ga pÀ1 _ À $ À ¼ g exp À vta kT b b b kT b ta ta ta ta (25) a !q ' a !qÀ1 a & vg_ a Q0 pq a Q0 jt j À g a p jt j À g a p jt j À g a pÀ1 _ À $ À ¼ À g exp À sgnðta Þ a a a a vga kT b b b kT b t t t t (26) J Nie et al / Nuclear Engineering and Technology 51 (2019) 501e509 [19] U Essmann, H Mughrabi, Annihilation of dislocations during tensile and cyclic deformation and limits of dislocation densities, Philos Mag A 40 (1979) 731 [20] A Arsenlis, D.M Parks, Modeling the evolution of crystallographic dislocation density in crystal plasticity, J Mech Phys Solid 50 (2002) 1979e2009 [21] U.F Kocks, A statistical theory of flow stress and work-hardening, Philos Mag 13 (1966) 541e566 [22] Y Estrin, H Mecking, A unified phenomenological description of work hardening and creep based on one-parameter models, Acta Metall 32 (1984) 57e70 , R Lebensohn, et al., Modeling and simulation of irradiation [23] C Deo, C Tome hardening in structural ferritic steels for advanced nuclear reactors, J Nucl Mater 377 (1) (2008) 136e140 [24] Anirban Patra, L David, McDowell, Crystal plasticity-based constitutive modelling of irradiated bcc structures, Philos Mag A 92 (7) (2011) 1e27 [25] Hibbitt Karlsson, Sorensen, ABAQUS/Standard User's Manuals., v6.5, 2005 [26] Yun Lin, Guang-sheng Ning, Chang-yi Zhang, et al., Mechanical property of China A508-3 steel after neutron irradiation, Energy Sci Technol 50 (02) (2016) 204e207 [27] R.A Johnson, D.J Oh, Analytic embedded atom method model for BCC metals, J Mater Res (5) (1989) 1195e1201 [28] D Brunner, J Diehl, Strain-rate and temperature dependence of the tensile flow stress of high-purity a-iron above 250 K (regime I) studied by means of stress-relaxation tests, Phys Status Solidi Appl Res 124 (1) (1991) 155e170 [29] W.A Spitzig, A.S Keh, The role of internal and effective stresses in the plastic 509 flow of iron single crystals, Metall Mater Trans B (12) (1970) 3325e3331 [30] X.M Bai, H Ke, Y Zhang, et al., Modeling copper precipitation hardening and embrittlement in a dilute Fe-0.3at.%Cu alloy under neutron irradiation, J Nucl Mater 495 (2017) 442e454 [31] M.F Ashby, Mechanisms of deformation and fracture, Adv Appl Mech 23 (1983) 117e177 [32] P Zhang, M Karimpour, D Balint, et al., A controlled Poisson Voronoi tessellation for grain and cohesive boundary generation applied to crystal plasticity analysis[J], Comput Mater Sci 64 (2012) 84e89 [33] Junfeng Nie, Zhenrui Tang, et al., Crystal plasticity constitutive model for BCC based on the dislocation density[J], J Tsinghua Univ 57 (2017) 780e784 [34] J.W Hutchinson, Plastic stress-strain relations of F.C.C polycrystalline metals hardening according to Taylor's rule[J], J Mech Phys Solid 12 (1) (1964) 11e24 [35] Lemaitre, A Course on Damage Mechanics, Springer-Verlag, 1996 [36] Feng Lu, Ke-shi Zhang, Guang Zhang, et al., Anisotropic damage model under continuum slip crystal plasticity theory for single crystals, Int J Solid Struct 39 (20) (2002) 5279e5293 [37] Zhao-liang Wang, The Research of Inhomogenous Plastic Deformation and Damage of Metal Materials, Master's Thesis, Guangxi University, Nanning, 2016 [38] Y Estrin, H Mecking, A unified phenomenological description of work hardening and creep based on one-parameter models, Acta Metall 32 (1) (1984) 57e70 ... classification of irradiation defects and a numerical simulation method to capture the mechanical properties of the material Then the model can easily predict the mechanical properties of other BCC crystal. .. Dεij Conclusion In this study, a crystal plasticity model based on dislocation evolution for BCC crystals is constructed and the model is coupled with the irradiation effect via introducing irradiation. .. Fig 12 The results indicate that temperature has a great influence on the mechanical properties of the material The effect of thermal activation on the dislocation motion is enhanced with the increasing