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Dynamics of steps along a martensitic phase boundary. II: Numerical simulations. (English) Zbl 1171.74402
Summary: We investigate the dynamics of steps along a phase boundary in a cubic lattice undergoing antiplane shear deformation. The phase transition is modeled by assuming piecewise linear stress-strain law with respect to one component of the shear strain, while the material response to the other component is linear. In the first part of the paper [J. Mech. Phys. Solids 56, No. 2, 496–520 (2008; Zbl 1171.74401)] we have constructed semi-analytical solutions featuring sequential propagation of steps. In this work we conduct a series of numerical simulations to investigate stability of these solutions and study other phenomena associated with step nucleation. We show that sequential propagation of sufficiently small number of steps can be stable, provided that the velocity of the steps is below a certain critical value that depends on the material parameters and the step configuration. Above this value we observe a cascade nucleation of multiple steps which then join sequentially moving groups. Depending on material anisotropy, the critical velocity can be either subsonic or supersonic, resulting in subsonic step nucleation in the first case and steady supersonic sequential motion in the second. The numerical simulations are facilitated with an exact non-reflecting boundary condition and a fast algorithm for its implementation, which are developed to eliminate the possible artificial wave reflection from the computational domain boundary.

MSC:
74N05 Crystals in solids
74S30 Other numerical methods in solid mechanics (MSC2010)
Software:
Algorithm 682
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[1] Adelman, S.; Doll, J., Generalized Langevin equation approach for atom/solid-surface scattering: collinear atom/harmonic chain model, J. chem. phys., 61, 4242-4245, (1974)
[2] Bayliss, A.; Turkel, E., Radiation boundary conditions for wave-like equations, Commun. pure appl. math., 33, 707-725, (1980) · Zbl 0438.35043
[3] Cai, W.; Koning, M.; Bulatov, V.; Yip, S., Minimizing boundary reflections in coupled-domain simulations, Phys. rev. lett., 85, 15, 3213-3216, (2000)
[4] Cserti, J., Application of the lattice Green’s function for calculating the resistance of infinite networks of resistors, Am. J. phys., 68, 896-906, (2000)
[5] Engquist, B.; Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. comput., 31, 629-651, (1977) · Zbl 0367.65051
[6] Engquist, B.; Majda, A., Absorbing boundary conditions for acoustic and elastic wave calculations, Commun. pure appl. math., 32, 313-357, (1979) · Zbl 0387.76070
[7] Flytzanis, N.; Celli, V.; Nobile, A., Motion of two screw dislocations in a lattice, J. appl. phys., 45, 12, 5176-5181, (1974)
[8] Givoli, D., Non-reflecting boundary conditions, J. comput. phys., 94, 1-29, (1991) · Zbl 0731.65109
[9] Givoli, D., Exact representations on artificial interfaces and applications in mechanics, Appl. mech. rev., 52, 333-349, (1999)
[10] Glasser, M.; Boersma, J., Exact values for the cubic lattice Green functions, J. phys. A math. gen., 33, 5017-5023, (2000) · Zbl 0973.82010
[11] Grote, M.; Keller, J., Exact nonreflecting boundary conditions for the time dependent wave equation, SIAM J. appl. math., 55, 2, 280-297, (1995) · Zbl 0817.35049
[12] Grote, M.; Keller, J., On nonreflecting boundary conditions, J. comput. phys., 122, 231-243, (1995) · Zbl 0841.65099
[13] Hagstrom, T., Radiation boundary conditions for the numerical simulation of waves, (), 47-106 · Zbl 0940.65108
[14] Higdon, R.L., Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation, Math. comput., 47, 176, 437-459, (1986) · Zbl 0609.35052
[15] Higdon, R.L., Radiation boundary conditions for elastic wave propagations, SIAM J. numer. anal., 27, 4, 831-870, (1990) · Zbl 0718.35058
[16] Ishioka, S., Stress field around a high speed screw dislocation, J. phys. chem. solids, 36, 427-430, (1975)
[17] Karpov, E.; Wagner, G.; Liu, W., A Green’s function approach to deriving non-reflecting boundary conditions in molecular dynamics simulations, Int. J. numer. methods eng., 62, 9, 1250-1262, (2005) · Zbl 1080.74050
[18] Katsura, S.; Inawashiro, S., Lattice Green’s functions for the rectangular and the square lattices at arbitrary points, J. math. phys., 12, 8, 1622-1630, (1971) · Zbl 0224.33025
[19] Katsura, S.; Morita, T.; Inawashiro, S.; Horiguchi, T.; Abe, Y., Lattice Green’s function. introduction, J. math. phys., 12, 5, 892-895, (1971) · Zbl 0272.33006
[20] Koizumi, H.; Kirchner, H.; Suzuki, T., Lattice waves emission from a moving dislocation, Phys. rev. B, 65, 214104, (2002)
[21] Li, X.; E, W., Variational boundary conditions for molecular dynamics simulations of solids at low temperature, Commun. comput. phys., 1, 1, 135-175, (2006) · Zbl 1114.74326
[22] Lubich, C.; Schädle, A., Fast convolution for nonreflecting boundary conditions, SIAM J. sci. comput., 24, 1, 161-182, (2002) · Zbl 1013.65113
[23] Morita, T., Useful procedure for computing the lattice Green’s function—square, tetragonal, and bcc lattices, J. math. phys., 12, 8, 1744-1747, (1971)
[24] Murli, A.; Rizzardi, M., ALGORITHM 682: Talbot’s method for the Laplace inversion problem, ACM trans. math. software, 16, 2, 158-168, (1990) · Zbl 0900.65374
[25] Park, H.; Karpov, E.; Liu, W.; Klein, P., The bridging scale for two-dimensional atomistic/continuum coupling, Philos. mag., 85, 79-113, (2005)
[26] Ryaben’kii, V.; Tsynkov, S.; Turchaninov, V., Global discrete artificial boundary conditions for time-dependent wave propagation, J. comput. phys., 174, 712-758, (2001) · Zbl 0991.65100
[27] Teng, Z.-H., Exact boundary condition for time-dependent wave equation based on boundary integral, J. comput. phys., 190, 398-418, (2003) · Zbl 1031.65098
[28] Tewary, V., Computationally efficient representation for elastostatic and elastodynamic Green’s functions for anisotropic solids, Phys. rev. B, 51, 22, 15695-15702, (1995)
[29] Tsynkov, S., Numerical solution of problems on unbounded domains. A review, Appl. numer. math., 27, 465-532, (1998) · Zbl 0939.76077
[30] Wagner, G.; Karpov, E.; Liu, W., Molecular dynamics boundary conditions for regular crystal lattice, Comput. method appl. mech. eng., 193, 1579-1601, (2004) · Zbl 1079.74526
[31] Zhen, Y., Vainchtein, A., 2007. Dynamics of steps along a martensitic phase boundary I: Semi-analytical solution. J. Mech. Phys. Solids, doi:10.1016/j.jmps.2007.05.017. · Zbl 1171.74401
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