On evolution equations for impact deformation problems with consideration of plane discontinuity surfaces
DOI:
https://doi.org/10.7242/1999-6691/2009.2.3.25Keywords:
nonlinear pressure, shock wave, perturbation method, evolution equationAbstract
The method of constructing approximate solutions of impact deformation problems for front areas of strong discontinuity surfaces is considered. It is shown that application of the method of matched asymptotic decompositions at certain distances from the shock wave front results in first order nonlinear wave equations, known as evolution equations. In the case of shear deformation, the evolution equation differs fundamentally from the volume wave equation (Hopf equation). Some variants of the solution of these equations and their application to definition of displacement field and deformations are offered. One of the variants is based on the additional parametric variable. The basic ideas of this method are illustrated by solving a number of one-dimensional problems on impact loading of a half-space occupied by a nonlinear elastic, isotropic compressible or incompressible medium. It is shown that the asymptotes obtained in this study can be used to develop numerical algorithms for solving the problems of impact deformation of a solid with consideration of discontinuity surfaces.
Downloads
References
Kulikovskij A.G., Svesnikova E.I. Nelinejnye volny v uprugih sredah. - M.: Moskovskij licej. 1998. - 416 s.
Blend D.R. Nelinejnaa dinamiceskaa teoria uprugosti. - M.: Mir, 1972. - 183s.
Godunov S.K., Zabrodin A.V., Ivanov M.A., Krajko A.N., Prokopov G.P. Cislennoe resenie mnogomernyh zadac gazovoj dinamiki. - M.: Nauka, 1976.- 400 s.
Kulikovskij A.G., Pogorelov N.V., Semenov A.U. Matematiceskie voprosy cislennogo resenia giperboliceskih sistem. - M.: Nauka, 2002. - 550s.
Ragozina V.E., Voronin I.I., Vekovsinin E.L. Ob ispol’zovanii prifrontovoj asimptotiki v cislennyh reseniah dinamiceskih zadac teorii uprugosti s udarnymi volnami // Problemy estestvoznania i proizvodstva. - 1995. - Vyp. 115. - S. 25-27.
Burenin A.A., Zinov’ev P.V. K probleme vydelenia poverhnostej razryvov v cislennyh metodah dinamiki deformiruemyh sred // Problemy mehaniki. Sbornik statej k 90-letiu A.U. Islinskogo. - Moskva: Fizmatlit, 2003. - S. 146-155.
Gerasimenko E.A., Zavertan A.V. Rascety dinamiki neszimaemoj uprugoj sredy pri antiploskom i skrucivausem udare // Vycisl. meh. splos. sred. - 2008. - T. 1, No 3. - S. 46-56.
Achenbach J.D., Reddy D.P. Note of wave propagation in lineary viscoelastic media // ZAMP. - 1967. - V. 18, No 1. - P. 141-144. DOI
Babiceva L.A., Bykovcev G.I., Vervejko N.D. Lucevoj metod resenia dinamiceskih zadac v uprugovazkoplasticeskih sredah // PMM. - 1973. - T. 37, vyp. 1. - S. 145-155.
Burenin A.A. Ob odnoj vozmoznosti postroenia priblizennyh resenij nestacionarnyh zadac dinamiki uprugih sred pri udarnyh vozdejstviah // Dal’nevost. matemat. zurnal. - 1999. - Vyp. 8. - S. 49-72.
Koul Dz. Metody vozmusenij v prikladnoj matematike. - M.: Mir, 1972. - 275s.
Van-Dajk M. Metody vozmusenij v mehanike zidkosti. - M.: Mir, 1967. - 239s.
Pelinovskij E.N., Fridman V.E., Engel’breht U.K. Nelinejnye evolucionnye uravnenia. - Tallin.: Valgus, 1984. - 156s.
Burenin A.A., Rossihin U.A. O vlianii vazkosti na harakter rasprostranenia ploskoj prodol’noj volny // PMTF. - 1990. - No 6. - S. 13-17.
Burenin A.A., Ragozina V. E. O prifrontovyh asimptotikah v nelinejnoj dinamiceskoj teorii uprugosti. Problemy mehaniki splosnyh sred i elementov konstrukcij. - Vladivostok: Dal’nauka, 1988. - S. 225-240.
Uizem Dz. Linejnye i nelinejnye volny. M.: Mir, 1977. 622s.
Ivanova U.E., Ragozina V.E. Ob udarnyh osesimmetriceskih dvizeniah neszimaemoj uprugoj sredy pri udarnyh vozdejstviah // Prikl. meh. i tehn. fizika. - Novosibirsk: Izd-vo SO RAN, 2006. - T. 47, No 6. -S. 144-151.
Sedov L.I. Mehanika splosnoj sredy: V 2-h t. - M.: Nauka, 1973. - T.1. - 536s. T.2. - 584s.
Tomas T. Plasticeskoe tecenie i razrusenie v tverdyh telah. - M.: Mir, 1964. - 308s.
Downloads
Published
Issue
Section
License
Copyright (c) 2009 Computational Continuum Mechanics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.