An asymptotic formula for displacement field in triangular lattice with vacancy

V. Tsaplin, V. Kuzkin


Simple approximate expression for the displacement field in harmonic triangular lattice under imposed volumetric mean strain is derived.An infinite harmonic triangular lattice with a vacancy under imposed volumetric strain is investigated. Displacement field around the vacancy is considered. In previous works of the authors, an exact displacement field is obtained in the integral form. The integral contains large parameter, notably particle index proportional to distance from the vacancy. In the present paper, behavior of the displacement field far from the vacancy is considered. Asymptotic expansion of the exact solution yields simple expression for the displacement field. The expression has reasonable accuracy for all lattice nodes and it rapidly converges to the exact solution with increasing distance from the vacancy. Strain concentration factor, defined as the ratio of the maximal deformation of the bonds adjacent to the vacancy to the deformations of bonds at infinity, is calculated. It is shown that the asymptotic formula predicts the strain concentration factor with 4% accuracy. The results are compared with predictions of continuum elasticity theory. In continuum formulation, the lattice with vacancy is modelled by an elastic plate with circular hole. It is shown that the continuum displacement field has the same form as the asymptotic expression for the discrete displacement field. Comparison of discrete and continuum results yields an effective diameter of the vacancy. The comparison also shows that influence of vacancies on strength of the lattice can not be accurately modeled using continuum theory.

References (21)

R. V. Goldshtein, N. F. Morozov, Phys. Mesomech. 10 (5), 17 (2007)
I. F. Golovnev, E. I. Golovneva, V. M. Fomin, Comp. Mat. Sci. 36, 176 (2006)
S. N. Korobeynikov, V. V. Alyokhin, B. D. Annin, A. V. Babichev. Arch. Mech. 64, 367 (2012)
C. Lee, X. Wei, J. W. Kysar, J. Hone, Science, 321, 385 – 388 (2008)
I. E. Berinskii, D. A. Indeitsev, N. F. Morozov, D. Y. Skubov, L. V. Shtukin, Mech. Sol. 50 (2), 127 – 134 (2015)
J. A. Baimova, E. A. Korznikova, S. V. Dmitriev, B. Liu, K. Zhou, Rev. Adv. Mat. Sci., 39 (1) 69 – 83 (2014)
J. A. Baimova, R. T. Murzaev, S. V. Dmitriev, Phys. Solid State, 56 (10), 2010 – 2016 (2014)
A. M. Iskandarov, Y. Umeno, Lett. Mat., 4 (2), 121 – 123 (2014)
J. D. Eshelby, Solid State Phys., 3, 79 – 144 (1956)
A. M. Krivtsov, N. F. Morozov, Phys. Solid State, 44, 2260 – 2265 (2002)
A. A. Maradudin, G. H. Weiss, E. W. Montroll, Theory of lattice dynamics in the harmonic approximation. New York: Academic Press, 1963.
G. S. Mishuris, A. B. Movchan, L. I. Slepyan, J. Mech. Phys. Sol., 57, 1958 (2009).
A. M. Krivtsov, Dokl. Phys., 60 (9), 407 (2015)
V. A. Kuzkin, A. M. Krivtsov, Phys. Solid State, 59 (5), 1051 – 1062 (2017).
H. Kanzaki, J. Phys. Chem. Solids, 2, 24 – 36 (1957)
V. K. Tewary, Adv. Phys. 22, 757 – 810 (1973)
V. A. Kuzkin, A. M. Krivtsov, E. A. Podolskaya, M. L. Kachanov, Phil. Mag., 96 (15), 1538 – 1555 (2016)
M. V. Fedoryuk, The stationary phase method and pseudodifferential operators. Russian Math. Surv., 6 (1), 65 – 115 (1971)
R. Wong, Asymptotic approximations of integrals, Academic Press, (1989)
S. V. Dmitriev, E. A. Korznikova, Yu. A. Baimova, M. G. Velarde, Phys. Usp., 59, 446 – 461 (2016)
V. A. Kuzkin, A. M. Krivtsov, R. E. Jones, J. A. Zimmerman, Phys. Mesomech. 18(13), 13-23 (2015)