Hyperbolicity, convexity and shock waves in one-dimensional crystalline solids. (English) Zbl 1070.74025

Ruggeri, Tommaso; Sugiyama, Masaru

J. Phys. A, Math. Gen. 38, No. 20, 4337-4347 (2005).

Summary: For a continuum model of one-dimensional anharmonic crystal lattices at finite temperatures, which was derived from a statistical-mechanical model, we clarify the classification of its differential system. That is, we determine not only strict hyperbolicity and convexity regions, but also elliptic and parabolic regions in the space of the state. The melting point is found to be on the boundary of the convexity region. Then we derive Rankine-Hugoniot relations, and we prove that the admissible shocks are always in the stable region of convexity.

MSC:

74J40	Shocks and related discontinuities in solid mechanics
74E15	Crystalline structure
35L67	Shocks and singularities for hyperbolic equations

Keywords:

anharmonic crystal lattices; Rankine-Hugoniot relations