Geodesic
The evolution equations of intensive deformation problems of elastic inhomogeneous medium
Dalʹnevostočnyj matematičeskij žurnal, Tome 15 (2015) no. 1, pp. 76-90

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The motion problems of the plane longitudinal or transverse shock wave for the nonlinear elastic medium model with inhomogeneous properties, which are represented by a continuous changes of the elastic moduli and density are considered. Changing of the medium properties is assumed in the direction of the wave fronts motion. The method of matched asymptotic expansions allows to determine the problems evolution equations, reflecting nonlinear wave processes and the inhomogeneity of the medium. The most interesting variant of the evolution equation occurs when the intensity of the impact process and the small inhomogeneity have the same order. The transition to the limiting inner problem of the small parameter method is dictated by the chain of inner problems for which it is necessary to change all the independent variables and their scales. 
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     author = {V. E. Ragozina and Yu. E. Ivanova},
     title = {The evolution equations of intensive deformation problems of elastic inhomogeneous medium},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {76--90},
     publisher = {mathdoc},
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     number = {1},
     year = {2015},
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     url = {http://geodesic.mathdoc.fr/item/DVMG_2015_15_1_a7/}
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V. E. Ragozina; Yu. E. Ivanova. The evolution equations of intensive deformation problems of elastic inhomogeneous medium. Dalʹnevostočnyj matematičeskij žurnal, Tome 15 (2015) no. 1, pp. 76-90. http://geodesic.mathdoc.fr/item/DVMG_2015_15_1_a7/