Atomistic simulations of Bauschinger effects of metals with high angle and low angle grain boundaries

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1 Comput. Methods Appl. Mech. Engrg. 193 (2004) Atomistic simulations of Bauschinger effects of metals with high angle and low angle grain boundaries H. Fang a, *, M.F. Horstemeyer a,b, *, M.I. Baskes c, K. Solanki a a Center for Advanced Vehicular Systems, P.O. BOX 9627, Mississippi State University, Mississippi State, MS 39762, USA b Department of Mechanical Engineering, Mississippi State University, 206 Carpenter Building, Mississippi State, MS 39762, USA c MST-8, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 5 May 2003; received in revised form 20 August 2003; accepted 2 December 2003 Abstract In this paper, we examined Bauschinger effects in nickel single crystals and nickel containing arrays of high angle or low angle grain boundaries under shear deformation using molecular dynamics with embedded atom method (EAM) potentials. In order to take into account dislocation nucleation under different boundary conditions and their effects on the stress strain relationship, two limiting constraints were used to both high angle and low angle grain boundaries: fixed end on all sides and free ends on all sides. Stress strain curves were then compared under these two boundary conditions for three cases: single crystal, high angle grain boundary arrays, and low angle grain boundary arrays. In each of the three cases, loading was reversed at different strain levels after yield and Bauschinger effects were examined on all the scenarios. The simulation results were also compared with macroscopic mechanics ideas for both high angle and low angle grain boundaries. The Bauschinger effect was found to be the largest for the case of high angle boundaries and the lowest for the single crystal. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Atomistic simulations; Nanoscale; Bauschinger; Embedded atom method 1. Introduction Atomistic simulation studies have recently focused on multiscale issues related to helping characterize continuum quantities such as stress [1], strain [2], resolved shear stress [3], and damage nucleation [4]. Other studies have focused on multiscale aspects in an effort to bridge the atomic scale with continuum concepts [5 8]. Most of these previous studies have focused on single crystal effects. This particular study focuses on samples with grain boundary (GB) arrays and the relationship between the aforementioned single crystal studies and effects of GBs. For example, in the aforementioned studies, dislocation nucleation was a * Corresponding authors. Fax: (M.F. Horstemeyer). address: mfhorst@me.msstate.edu (M.F. Horstemeyer) /$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi: /j.cma

2 1790 H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) dominant mechanism upon the volume averaged stress strain response. This study seeks to compare the dislocation nucleation from free surfaces and GBs. Furthermore, we analyze the effect of rigid constraints versus more flexible constraints on the samples with GBs. Finally, we also perform some reverse straining conditions in order to evaluate the Bauschinger effect at the atomic scale. Stout and Rollett [18], have performed numerous experiments to examine the Bauschinger. Chuna et al. [19,20] developed a new material model and used it in finite element method to simulate the Bauschinger effect. However, to the authors knowledge, a study to understand the Bauschinger effect at the atomic scale has not been performed before. 2. Embedded atom method (EAM) The premise for the atomistic simulations starts from embedded atom method (EAM) potential cast in a molecular dynamics framework since metals are the focus. Daw and Baskes [9] developed EAM, which employs a pair potential augmented by a function of another pair-wise sum, for metals. We use EAM in the atomistic calculations for study of finite deformations of nickel. The notion of embedding energy was first suggested by Freidel [10] and further developed by Stott and Zaremba [11]. Daw and Baskes [9] proposed a numerical method for calculating atomic energetics. Daw et al. [12] summarize many applications of EAM. Essentially, EAM comprises a cohesive energy of an atom determined by the local electron density into which that atom is placed. A function, qðrþ, is viewed as the contribution to the electron density at a site due to the neighboring atoms. The embedding energy F is associated with placing an atom in that electron environment. The functional form of the total energy is given by E ¼ X i F i! X q i ðr ij Þ þ 1 X / ij ðr ij Þ; ð1þ 2 i6¼j ij where i refers to the atom in question and j refers to the neighboring atom, r ij is the separation distance between atoms i and j, and / ij is the pair potential. Because each atom is counted, contra-variant and covariant index notation is not used here. Subscripts denote the rank of the tensor, for example, one subscript denotes a vector, two subscripts denote a second rank tensor, and so on. Superscripts identify the atom of interest. In molecular dynamics, the energy is used to determine the forces on each atom. At each atom the dipole force tensor, b, is given by b i km ¼ 1 X i X N jð6¼iþ f i k ðrij Þr ij m ; ð2þ where i refers to the atom in question and j refers to the neighboring atom, f k is the force vector between atoms, r m is a displacement vector between atoms i and j, N is the number of nearest neighbor atoms, and X i is the undeformed atomic volume. If stress could be defined at an atom, then b would be the stress tensor at that point. Since stress is defined at a continuum point, we determine the stress tensor as a volume average over the block of material, r mk ¼ 1 X N b i N mk ; ð3þ i in which the stress tensor is defined in terms of the total number of atoms, N, in the block of material.

3 3. Description of model set-up 3.1. Geometry and constraints In this study, we performed molecular dynamics, simple shear simulations on three atomic models: single crystal, low angle GB arrays, and high angle GB arrays. For simplicity, these three models are referred to as single crystal, low angle, and high angle in this paper. Fig. 1 illustrates the geometry and orientation of the three models. The single crystal model had 5800 atoms, the low angle model had 5860 atoms, and the high angle model had 5840 atoms. The single crystal computational block of material had free surfaces in the x-direction [1 0 0] and was periodic only in the z-direction [011]. The low angle and high angle cases had periodic conditions in the x-direction [100] and z-direction [011] representing an array of planar GBs. Because we desired to study the highest and lowest constraints for limiting cases, the y-direction [0 1 1] constraints were treated as fixed-end shear and flexible-end shear for all the three models. For the fixed-end boundary condition, atoms on top and bottom surfaces were fixed in the y and z-directions, and a constant velocity of A/ps in the x-direction was applied to the top row of atoms (top means the upper surfaces in Fig. 1). For the flexibleend (free-end) boundary condition, the bottom row of atoms were treated the same way as in the fixed-end case, but the top row of atoms was allowed to move in the y-direction. A velocity of A/ps was applied to the top row of atoms in the x-direction. Atoms between the top and bottom rows were thermostatted to a temperature of 300 K [21] Simulation conditions H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) The analyses were performed using a constant number of atoms, constant periodic lengths, and constant temperature. The applied velocity of A/ps resulted in a strain rate of 10 9 s 1 because the y- dimensions for the three models were approximately 35 A. The single crystal nickel was oriented in a quadruple slip orientation (011)[1 00] with a resolved shear stress of as shown in Fig. 2 in which (011) is the shear plane and [100] is the shear direction. The GBs included the same orientation for the crystal to the right. The crystal to the left of this one was oriented in two different configurations: one with a low angle (8 ) boundary and one with a high angle (90 ) boundary. Fig. 3 illustrates the simulations and orientations. The EAM functions used are described in [13,14]. The Nordsieck integrator and a time step of ps were used for all the simulations. Fig. 1. Geometry for atomistic calculations.

4 1792 H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) Fig. 2. Crystal orientation for multiple slips. Fig. 3. Various Microstructures are examined: single crystal, high angle grain boundaries, and low angle grain boundaries. Computation times on a single processor (Sun 750 MHz UltraSPARC III) for these three models ranged from 20 to 40 h, depending on what level of strain were to be achieved. The Bauschinger effects were examined for single crystals, low angle GBs, and high angle GBs under the aforementioned two boundary conditions. Comparisons are made for all the three models under the same boundary conditions. The reverse load points were chosen such that the three models passed their yield points and have the same total strain. The plastic strains for the two boundary conditions may not necessarily be the same, because the yield stresses may not be the same for these two boundary conditions.

5 H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) Results and discussions 4.1. Grain boundary and dislocation effects on yield stresses Fig. 4 shows the volume averaged stress strain response for the single crystal nickel and the correlating dislocation nucleation, motion, and interaction throughout the history. The dislocation motion is demonstrated by use of the centrosymmetry parameter [15]. For FCC metals, the centrosymmetry parameter is defined as follows: CSP ¼ X6 i¼1 ðr i þ r iþ6 Þ 2 ; where r i and r iþ6 are vectors corresponding to the six pairs of opposite nearest neighbors in the FCC lattice. The six-pair vectors for each atom are first determined in the undistorted bulk FCC lattice. The analogous six-pair vectors for each atom in the distorted lattice are then calculated by finding neighbors in the distorted lattice with vectors closest in distance to the undistorted nearest-neighbor vectors. The centrosymmetry parameter is zero for atoms in a perfect FCC lattice. Clearly, in this initially pristine material, the dislocations nucleate from the corners of the material block. As first described in [16], in an effort to reach dynamic equilibrium the internal structure responds with a dislocation. Fig. 5 shows simulation results from the fixed-end shearing case in which the relative displacements of the atoms are plotted in order to illustrate the kinematic response of the single crystal versus the low and high angle GBs. This snapshot occurs at 10% strain after plasticity has started. The volume averaged stress strain response in Fig. 5 shows that global yielding occurred at approximately 8% strain. The low angle GBs and single crystal gave very similar relative displacement results, but the high angle GBs gave a much different result. Furthermore, the stress strain response of the single crystal showed yielding at the same strain as the low angle GBs in this fixed-end highly constrained case. ð4þ Fig. 4. Evolution of dislocations relative to volume averaged stress strain response.

6 1794 H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) Fig. 5. Inhomogeneous deformation observed in all cases with fixed-end boundaries. Fig. 6. Flexible-end surface reduces yield stress. Now that we have observed the grain boundary effects for the fixed-end shearing case, we need to analyze the flexible-end case. This case will provide a more flexible constraint upon the GBs. Fig. 6 shows the relative displacements of the single crystal and two GB cases along with the volume averaged stress strain curves. When comparing with Fig. 5 (fixed-end case), we can see that yielding occurs sooner. Furthermore, when viewing differences for the high angle boundary case, one can see that although the large

7 displacements occur at the grain boundary for both flexible-end and fixed-end cases, the directions of displacements in both cases are different. Not much difference is observed at the low angle grain boundaries Bauschinger effects analyses H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) The Bauschinger effects can be observed qualitatively from the stress strain response curves. In order to quantitatively describe the Bauschinger effects for nanoscale simulations, we use two types of definitions, the Bauschinger stress parameter (BSP) and the Bauschinger effect parameters (BEP). The definitions are given as follows [17]: BSP ¼ jrf j jr r j ; ð5þ jr f j BEP ¼ jrf j jr r j jr f j jr y j ; ð6þ where r f is the stress in the forward load path at reverse point, r r is the yield stress in the reverse load path, and r y is the initial forward yield stress. We first examine the Bauschinger effects for the fixed-end boundary condition. Fig. 7 shows the volume averaged stress strain responses including both forward and reverse loading path for single crystal, low angle GB, and high-angle GB. The reverse load points were chosen at about 9% strain. Fig. 7 shows that a strong Bauschinger effects existed in all the three simulations. Table 1 gives the stresses and strains at the reverse load points. The BSPs and BEPs are calculated and given in Table 2 to quantify the result. The definitions of BSP and BEP indicate that the larger the BSP and BEP, the stronger the Bauschinger effect. The results in Table 2 shows that grain boundaries do have an effect on the stress strain responses Fig. 7. Volume averaged stress strain response for fixed-end, reverse loading at 9% strain.

8 1796 H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) Table 1 Reverse load points for fixed-end boundary condition Forward loading time (ps) Yield stress (GPa) Stress at reverse (GPa) Strain at reverse Single crystal Low angle GB High angle GB Table 2 Bauschinger parameters BSP and BEP for fixed-end, reverse load at 9% strain r f (GPa) r r (GPa) r y (GPa) BSP BEP Single crystal Low angle GB High angle GB Fig. 8. Volume averaged stress strain response flexible-end, reverse loading at 8% strain. including reverse loading. Single crystal, which has the smallest BSP (0.475) and BEP (2.105), has the weakest Bauschinger effect among the three models. High angle GB has the largest BSP (0.969) and BEP (5.734) and shows the strongest Bauschinger effect. The Bauschinger for low angle GB is between the single crystal and the high angle GB. Therefore, under fixed-end constraint, the larger angle grain boundary exhibits the stronger Bauschinger effect. Now let us examine the results for the flexible-end boundary conditions. Fig. 8 gives the volume averaged stress strain responses including both forward and reverse loading path. As the y-direction constraint

9 H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) Table 3 Reverse load points for flexible-end boundary condition Forward loading time (ps) Yield stress (GPa) Stress at reverse (GPa) Strain at reverse Single crystal Low angle GB High angle GB Table 4 Bauschinger parameters BSP and BEP for flexible-end, reverse load at 8% strain r f (GPa) r r (GPa) r y (GPa) BSP BEP Single crystal Low angle GB 3.92 ) High angle GB for the top row of atoms is removed, the yield stresses are reduced as compared with the fixed-end case. The Bauschinger effect can be seen from Fig. 8. Stresses and strains for the three models at reverse load points are listed in Table 3. Table 4 gives the calculated Bauschinger parameters BSPs and BEPs. The results here demonstrate the same trend as that of the fixed-end boundary condition. Single crystal, which has the smallest BSP (0.517) and BEP (1.244), has the weakest Bauschinger effect. The high angle GB has the largest BSP (0.797) and BEP (3.222) and therefore shows the strongest Bauschinger effect. The results from both fixed-end and flexible-end boundary conditions show that grain boundary does have an effect on the Bauschinger effect, that is, the larger angle the grain boundary has, the stronger the Bauschinger effect. Fig. 9. Dislocations relative to volume averaged stress strain response of single crystal with flexible-end.

10 1798 H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) Fig. 10. Dislocations relative to volume averaged stress strain response of low angle GB arrays with flexible-end. Fig. 11. Dislocations relative to volume averaged stress strain response of high angle GB arrays with flexible-end.

11 H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) Fig. 12. Dislocations relative to volume averaged stress strain response of single crystal with fixed-end. Fig. 13. Dislocations relative to volume averaged stress strain response of low angle GB arrays with fixed-end.

12 1800 H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) Fig. 14. Dislocations relative to volume averaged stress strain response of high angle GB arrays with fixed-end. Another interesting finding is that in the flexible-end simulations, the low angle GB has a much higher apparent yield stress than the single crystal and high-angle GB. This seems to contradict to what was expected; that is, the yield stress of the low angle GB bi-crystal would be closer to the other two models. By closely examining the dislocations in these models, this apparent inconsistency becomes clear. Figs illustrate the dislocations along the forward and reverse loading path for single crystal, low angle GB, and high angle GB, respectively. Fig. 10 shows that for the low angle GB, a dislocation nucleated at an earlier stage than the other two models and material hardening actually occurred before the apparent yield point due to dislocation nucleation. This means, a dislocation nucleated the easiest at the low angle GB without the constraint in the y-direction on the top row atoms. Even so, the Bauschinger effect for the low angle GB is still smaller than that of the high angle GB, according to the BSP and BEP definitions. Figs illustrate the dislocations under fixed-end boundary conditions for single crystal, low angle GBs, and high angle GBs, respectively. We can see that dislocation nucleation occurs earlier (before apparent yield) at the low angle GB. In the single crystal and high angle GB cases, dislocations do not occur until the apparent yield occurs. 5. Conclusions In this study, we examined the volume averaged stress strain responses of three nickel models, single crystal, low angle GB arrays, and high angle GB arrays, using molecular dynamics with the embedded atom method. Forward and reverse simple shear load was applied to the three models under two types of

13 H. Fang et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) boundary condition, fixed-end and flexible-end. The stress strain responses and Bauschinger effect were then compared to show the effects of grain boundaries as well as constraints. The analyses showed that less constrained materials (e.g., flexible-end) exhibited reduced yield stresses compared to highly constrained materials (e.g., fixed-end). The single crystal had the smallest Bauschinger effect and high angle GB had the strongest Bauschinger effect, for both fixed-end and flexible-end boundary conditions. An interesting finding was that dislocation nucleation occurred earlier in low angle GB than the single crystal and high angle GB cases. This finding was true for both fixed-end and flexible-end boundary conditions. Acknowledgements The work by Fang, Horstemeyer, and Solanki was sponsored by the Mississippi State University Center for Advanced Vehicular Systems, and the work by Baskes has been sponsored by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences. We would like to thank Dr. B. Zhang, Department of Mechanical Engineering, Mississippi State University, for his valuable comments and discussions on the work presented in this paper. References [1] P.M. Gullett, M.F. Horstemeyer, D.J. Bammann, and M.I. Baskes, A comparison of atomistic and elastic continuum based shear stress distributions near an edge dislocation, in: Proceedings of the International Conference on Computational Engineering and Science, August 19 25, 2001, Puerta Vallarta, Mexico. [2] M.F. Horstemeyer, and M.I. Baskes, Strain tensors at the atomic scale, MRS in: Proceedings of Multiscale Phenomena in Materials: Experiments and Modeling, vol. 578, Pittsburgh, PA, [3] M.F. Horstemeyer, M.I. Baskes, D.A. Hughes, A. Godfrey, Orientation effects on the stress state of molecular dynamics large deformation simulations, Int. J. Plasticity 18 (2002) [4] K.A. Gall, M.F. Horstemeyer, M. Van Schilfgaarde, M.I. Baskes, Atomistic simulations on the tensile debonding of an aluminum silicon interface, J. Mech. Phys. Solids 48 (2000) [5] W.W. Gerberich, N.I. Tymak, J.C. Grunlan, M.F. Horstemeyer, M.I. Baskes, Interpretations of indentation size effects, J. Appl. Mech. 69 (2002) [6] M.F. Horstemeyer, S.J. Plimpton, M.I. Baskes, Size scale and strain rate effects on yield and plasticity of metals, Acta Mater. 49 (2001) [7] M.F. Horstemeyer, M.I. Baskes, S.J. Plimpton, Computational nanoscale plasticity simulations using embedded atom potentials, Theor. Appl. Fract. Mech. 37 (2001) [8] M.F. Horstemeyer, T.J. Lim, W.Y. Lu, D.A. Mosher, M.I. Baskes, V.C. Prantil, S.J. Plimpton, Torsion/simple shear of single crystal copper, J. Engrg. Mater. Technol. 124 (2002) [9] M.S. Daw, M.I. Baskes, Embedded-tom method: derivation and application to impurities, surfaces, and other defects in metals, Phys. Rev. B (1984) [10] J. Freidel, Phil. Mag. 43 (1952) 153. [11] M.J. Stott, E. Zaremba, Quasi atoms: An approach to atoms in nonuniform electronic systems, Phys. Rev. B 22 (1980) [12] M.S. Daw, S.M. Foiles, M.I. Baskes, The embedded-atom method: a review of theory and applications, Materials Science Reports, A Review Journal 9 (1993) [13] J.E. Angelo, N.R. Moody, M.I. Baskes, Trapping of hydrogen to lattice defects in nickel, J. Model. Simul. Mater. Sci. Engrg. 3 (1995) [14] M.I. Baskes, X. Sha, J.E. Angelo, N.R. Moody, Comment: trapping of hydrogen to lattice defects in nickel, Model. Simul. Mater. Sci. Engrg. 5 (1997) [15] C. Kelchner, S. Plimpton, Hamilton, Dislocation nucleation and defect structure during surface indentation, J. Phys. Rev. B 58 (1998) [16] M.F. Horstemeyer, M.I. Baskes, Atomistic finite deformation simulations: a discussion on length scale effects in relation to mechanical stresses, J. Engrg. Matls. Techn. Trans. ASME 121 (1998) [17] M.F. Horstemeyer, Damage influence on Bauschinger effect of a CAST A356 alluminum alloy, Scripta Mater. 39 (1998)

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