# Problem of the Motion of an Elastic Medium Formed at the Solidification Front

- 8 Downloads

## Abstract

The following self-similar problem is considered. At the initial instant of time, a phase transformation front starts moving at constant velocity from a certain plane (which will be called a wall or a piston, depending on whether it is assumed to be fixed or movable); at this front, an elastic medium is formed as a result of solidification from a medium without tangential stresses. On the wall, boundary conditions are defined for the components of velocity, stress, or strain. Behind the solidification front, plane nonlinear elastic waves can propagate in the medium formed, provided that the velocities of these waves are less than the velocity of the front. The medium formed is assumed to be incompressible, weakly nonlinear, and with low anisotropy. Under these assumptions, the solution of the self-similar problem is described qualitatively for arbitrary parameters appearing in the statement of the problem. The study is based on the authors’ previous investigation of solidification fronts whose structure is described by the Kelvin–Voigt model of a viscoelastic medium.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.N. Kh. Arutyunyan and A. V. Manzhirov,
*Contact Problems in Creep Theory*(Akad. Nauk Arm. SSR, Yerevan, 1990) [in Russian].zbMATHGoogle Scholar - 2.A. A. Barmin and A. G. Kulikovskii, “Ionization and recombination fronts in an electromagnetic field,” in
*Itogi Nauki, Gidromekh.*(VINITI, Moscow, 1971), Vol. 5, pp. 5–31; Engl. transl.: Rep. JPRS-56500 (Joint Publ. Res. Serv., Washington, 1972).Google Scholar - 3.A. P. Chugainova, “Special discontinuities in nonlinearly elastic media,” Zh. Vychisl. Mat. Mat. Fiz.
**57**(6), 1023–1032 (2017) [Comput. Math. Math. Phys.**57**, 1013–1021 (2017)].MathSciNetzbMATHGoogle Scholar - 4.A. G. Kulikovskii, “Surfaces of discontinuity separating two perfect media of different properties. Recombination waves in magnetohydrodynamics,” Prikl. Mat. Mekh.
**32**(6), 1125–1131 (1968) [J. Appl. Math. Mech.**32**, 1145–1152 (1968)].zbMATHGoogle Scholar - 5.A. G. Kulikovskii, “Multi-parameter fronts of strong discontinuities in continuum mechanics,” Prikl. Mat. Mekh.
**75**(4), 531–550 (2011) [J. Appl. Math. Mech.**75**, 378–389 (2011)].MathSciNetGoogle Scholar - 6.A. G. Kulikovskii and A. P. Chugainova, “Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory,” Usp. Mat. Nauk
**63**(2), 85–152 (2008) [Russ. Math. Surv.**63**, 283–350 (2008)].CrossRefGoogle Scholar - 7.A. G. Kulikovskii and A. P. Chugainova, “Study of discontinuities in solutions of the Prandtl–Reuss elastoplasticity equations,” Zh. Vychisl. Mat. Mat. Fiz.
**56**(4), 650–663 (2016) [Comput. Math. Math. Phys.**56**, 637–649 (2016)].MathSciNetzbMATHGoogle Scholar - 8.A. G. Kulikovskii and A. P. Chugainova, “A self-similar wave problem in a Prandtl–Reuss elastoplastic medium,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk
**295**, 195–205 (2016) [Proc. Steklov Inst. Math.**295**, 179–189 (2016)].MathSciNetzbMATHGoogle Scholar - 9.A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov,
*Mathematical Aspects of Numerical Solution of Hyperbolic Systems*(Fizmatlit, Moscow, 2001; Chapman & Hall/CRC, Boca Raton, FL, 2001); 2nd ed. (Fizmatlit, Moscow, 2012) [in Russian].zbMATHGoogle Scholar - 10.A. G. Kulikovskii and E. I. Sveshnikova, “On shock wave propagation in stressed isotropic nonlinearly elastic media,” Prikl. Mat. Mekh.
**44**(3), 523–534 (1980) [J. Appl. Math. Mech.**44**, 367–374 (1980)].MathSciNetGoogle Scholar - 11.A. G. Kulikovskii and E. I. Sveshnikova, “A selfsimilar problem on the action of a sudden load on the boundary of an elastic half-space,” Prikl. Mat. Mekh.
**49**(2), 284–291 (1985) [J. Appl. Math. Mech.**49**, 214–220 (1985)].MathSciNetzbMATHGoogle Scholar - 12.A. G. Kulikovskii and E. I. Sveshnikova, “Nonlinear waves arising under the variation of stresses on the boundary of an elastic half-space,” in
*Problems in Nonlinear Continuum Mechanics*(Valgus, Tallinn, 1985), pp. 133–145 [in Russian].Google Scholar - 13.A. G. Kulikovskii and E. I. Sveshnikova, “Non-linear waves in slightly anisotropic elastic media,” Prikl. Mat. Mekh.
**52**(1), 110–115 (1988) [J. Appl. Math. Mech.**52**, 90–93 (1988)].Google Scholar - 14.A. G. Kulikovskii and E. I. Sveshnikova,
*Nonlinear Waves in Elastic Media*(CRC, Boca Raton, FL, 1995; Mosk. Litsei, Moscow, 1998).zbMATHGoogle Scholar - 15.A. G. Kulikovskii and E. I. Sveshnikova, “The formation of an anisotropic elastic medium on the compaction front of a stream of particles,” Prikl. Mat. Mekh.
**79**(6), 739–755 (2015) [J. Appl. Math. Mech.**79**, 521–530 (2015)].MathSciNetGoogle Scholar - 16.A. G. Kulikovskii and E. I. Sveshnikova, “Formation fronts of a nonlinear elastic medium from a medium without shear stresses,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No.
**3**, 48–54 (2017) [Moscow Univ. Mech. Bull.**72**, 59–65 (2017)].Google Scholar - 17.L. D. Landau and E. M. Lifshits,
*Course of Theoretical Physics, Vol. 6: Fluid Mechanics*(Nauka, Moscow, 1986; Pergamon, Oxford, 1987).Google Scholar - 18.E. I. Sveshnikova, “Simple waves in nonlinearly elastic media,” Prikl. Mat. Mekh.
**46**(4), 642–646 (1982) [J. Appl. Math. Mech.**46**, 509–512 (1982)].Google Scholar - 19.Ya. B. Zel’dovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze,
*The Mathematical Theory of Combustion and Explosions*(Nauka, Moscow, 1980; Consultants Bureau [Plenum], New York, 1985).Google Scholar