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On the propagation of acceleration waves in thermoelastic micropolar medias
Sobre la propagación de ondas aceleradas en medios micropolares termoelásticos
Schlagworte:
Acceleration waves, Micropolar continuum, Cosserat continuum, Nonlinear elasticity, 2000 Mathematics Subject Classification. 74A35, 74B20, 74J40. (en)Ondas de aceleración, Continuo micropolar, Continuo de Cosserat, Elasticidad no-lineal (es)
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Abstract. The conditions for propagation of accelerating waves in a general nonlinear thermoelastic micropolar media are established. Deformation of micropolar media is described by the time-varying displacement vector r(t) and tensor of microrotation r(t) at each point. We call a surface S(t) an accelerating wave (or a singular surface for a solution of the dynamic problem for the medium) if the points are points of continuity of both r(t) and ℍ(t) and their first spatial and time derivatives while the second spatial and time derivatives (acceleration) of r(t) and ℍ(t) have jumps on S(t) (meaning that their one-sided limits at S(t) differ). So S(t) carries jumps in the acceleration fields as it propagat es through the body. In the thermomechanics of a micropolar continuum, similar propagating surfaces of singularities can exist for the fields of temperature, heat flux, etc. We establish the kinematic and dynamic compatibility relations for the singular surface S(t) in a nonlinear micropolar thermoelastic medium. An analog of Fresnel-Hadamard-Duhem theorem and an expression for the acoustic tensor are derived.
The work was supported by the Russian Science Support Foundation and the Russian Foundation of Basic Research under grants 07-01-00525, 07-08-13589-ofi_c and by Universidad Nacional de Colombia, project of investigation No. 8003061.
Se establecen las condiciones de propagación de ondas aceleradas en un medio no lineal micropolar termoelástico. Las deformaciones del medio micropolar son descritas por las variaciones temporales del vector de desplazamiento r(t) y del tensor de microrotación r(t) en cada punto. Llamamos una superficie S(t) a una onda acelerada (o superficie singular para la solución del problema dinámico del medio) si los puntos son puntos de continuidad de r(t), ℍ(t) y sus primeras derivas espaciales y temporales, mientras que las segundas derivadas espaciales y temporales tienen saltos en S(t). Entonces S(t) transporta los saltos en los campos acelerados cuando se propagan en el cuerpo. En la termomecánica de un continuo micropolar, superficies de propagación similares pueden existir para los campos de temperatura y de flujo de calor. Establecemos las relaciones de compatibilidad cinética y dinámica para las superficies singulares en un medio micropolar termoelástico no lineal. Un análogo del teorema Fresnel-Hadamard-Duhem y una expresión para el tensor acústico son establecidos.
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