Theoretical and Applied Fracture Mechanics 41 (2004) 9 14 www.elsevier.com/locate/tafmec Numerical simulation of deformation and fracture in low-carbon steel coated by diffusion borating R.R. Balokhonov a, *, S.V. Panin a, V.A. Romanova a, S. Schmauder b, P.V. Makarov a a Institute of Strength Physics and Materials Science, Russian Academy of Sciences, Siberian Branch, 2/1 Av. Academicheskii, Tomsk 634021, Russia b Staatliche Materialpruefungsanstaalt (MPA), University of Stuttgart, Germany Abstract Numerical simulation of tensile mechanical behavior of low-carbon steel test piece surface-hardened by diffusion borating is performed to investigate the effect of stress concentration on the mesoscale plastic-deformation pattern. The computer simulations under review employ low-carbon steel as a substrate material with FeB 2 serving as a surfacehardened layer. The computations are performed for representative mesovolumes based on real experimentally revealed structure with an intermediate layer of different configurations. A model taking into account crack formation is applied to investigate coating fracture. The results obtained are presented and discussed. Ó 2003 Elsevier Ltd. All rights reserved. 1. Introduction Coating deposition technologies of different types are widely used in industry: strengthening treatment, alloying treatment, renewing treatment of worn surfaces etc. The technologies are applied to increase lifetime of machine parts and structure elements. Due to essential difference between mechanical and physical properties of coating and substrate, surface-hardened materials are the most convenient object to be investigated in the framework of physical mesomechanics [1 3]. Occurrence of * Corresponding author. Tel.: +7-3822-286876; fax: +7-3822- 259576. E-mail address: rusy@ms.tsc.ru (R.R. Balokhonov). powerful stress concentrations near the base material-coating interface allows us to most completely and complexly investigate physical aspects of deformation at the mesolevel. Earlier plastic deformation in structural steel specimens surface-hardened by ion-nitriding was numerically investigated [4]. Calculations were carried out for representative mesovolumes, with the grain structure of the structural steel as a base material being taken into account in an explicit form. Coating was assumed to be uniform with plane interface profile. Along with, the diffusion borating technique provides high-strength coatings to be produced with a toothed shape of the borated layer steel substrate interface. Findings of experimental investigations carried out with the use of television optical measuring technique TOMSC [5,6] show 0167-8442/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2003.11.002
10 R.R. Balokhonov et al. / Theoretical and Applied Fracture Mechanics 41 (2004) 9 14 that diffusion borated test pieces under tension exhibit a non-uniform strain distribution within the surface layer. This kind of strain pattern makes it possible to change the performance properties of the entire test piece. In this case, no extended shear bands are formed in the base material and the value of ultimate tensile strain strongly depends on the both coating depth and structure as well as configuration of the coating substrate interface. The main targets of this work are to simulate deformation of a local mesovolume containing coating of complex structure and investigate the influence of certain stress concentrations near the base material coating interface of different configurations on the plastic flow character. 2. Mathematical formulation Mesoscopic deformation processes were simulated within the two-dimensional formulation of the problem (plane strain). Numerical solutions are performed in terms of Lagrangian variables using the finite-difference method. The total system of equations includes: equations of motion and continuity, expressions for components of strain rate tensor and the constitutive equation [4,7]. To describe plastic strain use was made of the plasticity theory law _e p ij ¼ _ ks ij associated with the yield criterion of the following form r eff ¼ /ð R de p eff Þ. Here k is a scalar parameter, _ep ij and S ij are the plastic strain rate and stress deviator components, r eff and e p eff are the equivalent stress and plastic strain, respectively: Function /ð R de p eff Þ of the certain type allows us to prescribe strain hardening for the low-carbon steel substrate. 3. Calculation results 3.1. Elasto-plastic formulation Fig. 1. Schematic of the mesovolumes under study. Fig. 1 shows a map of the test section for the cases of toothed (a) and smoothed (b) structure of the low-carbon steel substrate intermediate layer interface. The images correspond to the real structure of the mesovolumes, which were experimentally investigated early [5,6]. Boundary conditions on the right and left surfaces of the area under calculation determine the grip displacement velocity (Fig. 1(a)), while on the top and bottom surfaces they correspond to the free surface and symmetry conditions, respectively. The following model parameters for the materials containing the mesovolumes were used. For the coating: shear modulus l ¼ 140 GPa, bulk modulus j ¼ 200 GPa and density q ¼ 7:13 g/cm 3 correspond to the characteristics of FeB 2. The model does not imply strain hardening in the coating that is why function /ð R de p eff Þ¼1GPais assumed to be constant. For the low-carbon steel substrate the corresponding values are: l ¼ 80 GPa, j ¼ 133 GPa, q ¼ 7:9 g/cm 3, while the strain hardening function is described by the relation Z / ¼ A þ Be p eff Cðep eff Þ2 de p eff with A ¼ 300 MPa: The averaged characteristics were assigned for the intermediate layer material: l ¼ 120 GPa, j ¼ 170 GPa, q ¼ 7:5 g/cm 3, A ¼ 700 MPa. For both the coating and the intermediate material: B ¼ 2:52, C ¼ 12 GPa.
R.R. Balokhonov et al. / Theoretical and Applied Fracture Mechanics 41 (2004) 9 14 11 Fig. 2. Equivalent stress (a) and equivalent plastic strain (b) patterns for the mesovolume presented in Fig. 5. Total strain is 0.2%. Fig. 2 shows stress and plastic strain distribution patterns at the initial stages of loading. It is easily seen on the figures that stress distribution is of essentially heterogeneous nature (Fig. 2(a)). Plastic shears originate in local areas in the substrate, while the most specimen volume continues to keep elastic state (Fig. 2(b)). Occurrence of the local centers of plastic deformation can be explained by acting of the most powerful stress concentrators. The pattern of further plastic deformation development is presented in Fig. 3(a). The highest level of strain localization arises along the substrate intermediate layer interface that results from incompatibility of plastic strains. The circles in Fig. 3(a) mark the areas of initial shear origination presented in Fig. 2(b). At later stages of plastic deformation development just near these local areas cracks and shear bands will probably form in the coating. In contrast to this a quite different strain pattern is observed for simulating tension of the mesovolume presented in Fig. 1(b). In this case the low-carbon steel substrate intermediate layer Fig. 3. Equivalent plastic strain patterns for the mesovolumes presented in Fig. 5. Total strain is 0.4%. interface has a smoothed structure. The absence of powerful stress concentrators near the interface results in more uniform distribution of stress strain state parameters. In this case material continues to be in elastic state even under total strain of 0.2% that is contrary to the case of the toothed interface structure (Fig. 2). The result obtained is explained by the fact that average stress level is not high enough to exceed the current yield point in any local area of the mesovolume. With the further loading average stress level grows but stress concentrators near the FeB 2 intermediate layer interface are involved into origination of plastic flow in local areas (Fig. 3(b)). These areas of strain localization are located along the interface. The highest degrees of strain localization occur closer to the free surface of material near the bases of FeB 2 teeth. They are marked by circles in Fig. 3. 3.2. Simulation of cracking The experimental data testify that during tension of the steel specimens coated by diffusion
12 R.R. Balokhonov et al. / Theoretical and Applied Fracture Mechanics 41 (2004) 9 14 Fig. 4. Top view of the experimentally investigated test piece fractured under tension [8]. borating multiple quasi-periodic cracking of borated layers happen (see Fig. 4). It is obvious, that for an adequate simulation of the mechanical behaviour of such a composite material the use of elasto-plastic description only is insufficient. For studying the coating cracking processes an energetic fracture criterion of the following kind was employed. ði 2 ; P; C 1 ; C 2 Þ¼0; where I 2 is the second invariant of the stress deviator tensor, P is the pressure, C 1, C 2 are constants that characterize yield strength of FeB 2 under tension and compression. Fracture criterion means that the following conditions are associated with any local coating material region: if bulk deformation e kk takes on a positive value and r eff reaches its critical value of C 1 ¼ 1 GPa than pressure and all components of the stress deviator tensor in the region considered tend to zero. In the case if e kk < 0 and r eff P C 2 ¼ 3:6 GPa the pressure does not tend to zero. In so doing, if fracture criterion is fulfilled, coating material demonstrates the reaction that is similar to one of an uncompressed liquid. The material density maintains to be constant and corresponds to the density of FeB 2. A schematic sketch of the structure of the mesovolume consisting of two layers is shown in Fig. 5. Mechanical properties of coating and substrate materials as well as boundary conditions are the same as described in Section 3.1. The boundary conditions are set to determine the mesovolume tension at a constant velocity and have the following form: Fig. 5. Initial structure of the mesovolume. Black low-carbon steel substrate, white FeB 2 coating, gray FeB 2 elastic regions. BC1: U x ¼ U ¼ const:; BC3: U x ¼ U; BC2: r ij n j ¼ 0; BC4: U y ¼ 0: Fig. 6(a) (e) illustrates cracking pattern, equivalent plastic strain and stress distributions over the mesovolume under different degrees of total strain. Black colour corresponds to the regions where the abovementioned parameters have maximum values and as well marks fractured regions of the coating. Initially the local fracture area is formed near the most powerful stress concentrator at the FeB 2 coating substrate interface. This gives rise to a significant increase of the strain intensity of the neighboring regions transversely to the loading direction. As a consequence a new stress concentrator nucleates in the coating and the crack propagates towards the free surface. This process is accompanied by stress relaxation along the stretching direction: coating unloading happens. In so doing, the steel substrate continues to deform plastically according to a straining law given. The newly emerged stress concentration in the vicinity of crack nucleation relaxes by shear band formation in the substrate. These shear bands orient towards the maximum tangential stresses direction. With further loading the total stress level in the interface region begins to arise due to intensive substrate straining. This results in formation of a new region with increased stress concentration located in the vicinity of a tooth base. When in this region the equivalent stress
R.R. Balokhonov et al. / Theoretical and Applied Fracture Mechanics 41 (2004) 9 14 13 Fig. 6. Calculated plastic strain, cracking and stress pattern at different total strains (a) (e) in comparison with experimentally observed surface [9] (f). reaches the critical value, generation of a new crack in the coating and strain intensification in the substrate occur. Then the process consequently repeats. The calculation results are in a good agreement with one revealed experimentally (see Fig. 6, where the micrograph of lateral face of a borated specimen is shown). The material area under observation corresponds physically to the calculated area, since the simulation was performed at plane strain statement. The integral stress strain diagram of the mesovolume under study is presented in Fig. 7. The stress was calculated as an average value of equivalent stress over the mesovolume: Stress ¼ X, X r k eff vk v k ; k¼1;n k¼1;n where N the number of computational mesh points, v k local volume. Deformation represents Fig. 7. Stress strain curve for the local mesovolume of lowcarbon steel coated with FeB 2. relative mesovolume elongation towards X -direction. It is seen that the dependence has a pronounced step pattern. Formation of every crack results in
14 R.R. Balokhonov et al. / Theoretical and Applied Fracture Mechanics 41 (2004) 9 14 descending portion on the stress strain curve. Circles mark the mesovolume deformation stages shown in Fig. 6 in the form of plastic strain and stress distribution patterns. 4. Conclusion remarks The model taking into account both strain hardening and crack formation is applied to numerically investigate the mechanical behavior of FeB 2 coated low-carbon steel test-pieces. The simulation results obtained lead us to conclude that: The areas of highest stress concentrations are located near the bases of FeB 2 teeth, where cracks in the FeB 2 coating and shear bands in the substrate begin to form as loading continues. The low-carbon steel substrate intermediate layer interface plays the dominant role in the localization of plastic flow. If the interface has a toothed structure, it is the place where the most powerful stress concentrations occur. A crack will later nucleate in these areas. In the case of a smoothed structure of the interface, stress concentrators of less power emerge. In this case the interface between the coating and the intermediate layer begins to play the determining role in strain localization and cracking. Cracks nucleate in the coating in the vicinity of its interface with the substrate with the former propagating towards the free surface of the specimen. This results in average stress level descending in the mesovolume under study. Acknowledgements The supports from INTAS through the project YSF02-159/D and from the Russian Foundation for Basic Research through the Grant No. NSH- 2324.2003.1 (Leading Scientific School of Academician V.E. Panin) are gratefully acknowledged. References [1] V.E. Panin (Ed.), Physical Mesomechanics of Heterogeneous Media and Computer-Aided Design of Materials, Cambridge International Science Publishing, 1998, p. 339. [2] V.E. Panin, Synergetic principles of physical mesomechanics, Phys. Mesomech. (6) (2000) 5 36. [3] V.E. Panin, A.D. Korotayev, P.V. Makarov, V.M. Kuznetsov, Russ. Phys. J. 9 (1998) 8 36. [4] R.R. Balokhonov, Yu.P. Stefanov, P.V. Makarov, I.Yu. Smolin, Deformation and fracture of surface-hardened materials at meso- and macroscale levels, Theor. Appl. Fract. Mech. 33 (2000) 9 15. [5] A.V. Koval, S.V. Panin, Mesoscale deformation and cracking of surface-hardened low carbon steel, Theor. Appl. Fract. Mech. (34) (2000) 117 121. [6] A.V. Koval, S.V. Panin, Formation of fractal mesostructure in structural steels with heterogeneous hardening layers under tension, in: Proceedings of an International Conference of Role of Mechanics for Development of Science and Technology, held at XiÕan, China, vol. 2, 13 16 June 2000, pp. 585 592. [7] R.R. Balokhonov, P.V. Makarov, V.A. Romanova, I.Yu. Smolin, Simulation of crystal plasticity under dynamic loading, J. Comput. Mater. Sci. 16 (1 4) (1999) 237 243. [8] A.V. Koval, S.V. Panin, Influence of hardened layer structure and coating-substrate interface geometry on plastic deformation pattern of structural steels at mesolevel, in: Proceedings of the 4th Korea Russia International Symposium on Science and Technology, held at Ulsan, Republic of Korea, 27 June 1 July 2000, pp. 375 380. [9] S.V. Panin, A.V. Koval, G.V. Trusova, et al., Effects of the interface geometry and structure on the character of plastic deformation of borated structural steel specimens at the mesolevel, Phys. Mesomech. 3 (2) (2000) 93 108.