Amplification of nonlinear strain waves in solids PDF
Preview Amplification of nonlinear strain waves in solids
February11,2004 20:10 WSPC/BookTrimSizefor9inx6in ws-book9x6 Publishers’ page i February11,2004 20:10 WSPC/BookTrimSizefor9inx6in ws-book9x6 Publishers’ page ii February11,2004 20:10 WSPC/BookTrimSizefor9inx6in ws-book9x6 Publishers’ page iii February11,2004 20:10 WSPC/BookTrimSizefor9inx6in ws-book9x6 Publishers’ page iv February11,2004 20:10 WSPC/BookTrimSizefor9inx6in ws-book9x6 To my parents v February11,2004 20:10 WSPC/BookTrimSizefor9inx6in ws-book9x6 vi February11,2004 20:10 WSPC/BookTrimSizefor9inx6in ws-book9x6 Preface Itisknownthatthebalancebetweennonlinearityanddispersion mayresult in an appearance of localized long bell-shaped strain waves of permanent form (solitary waves or solitons) which may propagate and transfer energy over the long distance along. Starting with the first documented water surface solitary wave observation, made by J.Scott Russell yet in 1834, solitons in fluids were observed and generated many times. It was the most surprising fact, however, that despite of almost similar description of stressesinfluidsandsolids,bulklongitudinalstrainsolitarywaveshavenot been observed in nonlinearly elastic wave guides. One of reason of the lack of the results on nonlinear wave in solids is that the complete description of a three-dimensional (3-D) nonlinear con- tinuum is a difficult problem. That is why initial 3-D problems are usually reduced to the one-dimensional (1-D) form in order to clarify the simplest but qualitatively new analytical solutions. Certainly the cylindrical elastic rodseemstobeasuitablereal-life1-Dwaveguide. Recently,thetheoryhas been developed to account for long longitudinal strain solitary waves prop- agating in a free lateral surface elastic rod with permanent cross section. The nonlinearity, caused by both the finite stress values and elastic mate- rial properties, and the dispersion resulting from the finite transverse size of the rod, when in balance allow the propagation of the bulk strain soli- tarywaves. Motivatedbyanalyticaltheoreticalpredictions, therehasbeen successfulexperimentalgenerationofstrainsolitarywavesinapolystyrene free lateral surface rod using the holographic interferometry. Hence it was proven that bulk long localized nonlinear strain waves of permanent form really exist. However, presence of a dissipation (accumulation) destroys the balance between nonlinearity and dispersion, and nonlinear strain wave in the rod vii February11,2004 20:10 WSPC/BookTrimSizefor9inx6in ws-book9x6 viii Book Title may attenuate or amplify. One possibility occurs when the radius of the rod varies. Dissipative (active) effects may be caused by internal features of the elastic material, hence, an irreversible part should be included into the stress tensor in addition to the reversible one depending only upon the densityoftheHelmholtzenergy. Dissipation(accumulation)mayalsocome inanelasticwaveguidethroughphenomenaoccurringatitslateralsurface. Among the volume sources of dissipation/accumulation one can mention a microstructure, point moving defects, thermal effects. When dissipation and activation act together there may be another balance resulted in a for- mationofabell-shapedwavewiththeamplitudeandthevelocityprescribed by the condition of the dissipative/active balance. Hence the wave is se- lected. Notethatthereexistanotherkindofnonlinearwaveofapermanent shapesustainedeitherbyabalancebetweennonlinearityanddissipationor by a balance between nonlinearity, dispersion and dissipation. This wave hastheformofashockandisoftencalledkink-shapedwaveorsimplykink. Theamplificationofthewaves(i.e.,growthoftheamplitude)maycause the appearance of plasticity zones or microcracks in a wave guide. This is ofimportanceforanassessmentofdurabilityofelasticmaterialsandstruc- tures, methods of non-destructive testing, determination of the physical properties of elastic materials, particularly, polymeric solids, and ceramics. Bulkwavesprovidebettersuiteddetectionrequirementsthansurfacestrain waves in setting up a valuable non-destructive test for pipelines. Inclusion of dissipation (accumulation) yields nonlinear dispersive- dis- sipative governing equations that are nonintegrable as a rule. Hence, only particular, usually travelling wave, solutions may be obtained analytically. Certainly,thesesolutionsrequirespecificinitialconditions. Moreover,they usually have no free parameters, and additional relationships between the equation coefficientsare required for the existence of the solutions. That is why the obtaining of the exact solutions is often considered as useless by many authors preferring to apply numerical methods only. Iwouldliketoachievetwotasksinthismonograph. First, itisplanned to provide the sequential analytical consideration of the strain waves am- plification/attenuation and selection in solids, mainly in an elastic rod. It maybeofinterestforthoseworkinginthefieldofsolidmechanics. Another taskistodemonstratetheuseofevenparticularanalyticalsolutionsforthe descriptionofunsteadynonlinearwaveprocesses. Itmayattracttheatten- tion of the specialists in various fields since the structure of the governing equationsisratheruniversal. Thecontentisessentiallybasedontheauthor previousresearch. However,manyworksweredoneinacollaboration. The February11,2004 20:10 WSPC/BookTrimSizefor9inx6in ws-book9x6 Chapter Title for Preface ix author thanks a lot Profs. G.V. Dreiden, I.L. Kliakhandler, G.A. Maugin, F.Pastrone,D.F.Parker,A.M.Samsonov,M.G.Velarde;Mr. V.V.Gursky and Mrs. I.V. Semenova for a long time fruitful collaboration. The book preparation has been supported by the INTAS grant 99-0167 and by the RFBR under Grant 2000-01-00482. I dedicate this book to my parents. They always believe in my efforts and expected this book more than whoever it may be. Saint-Petersburg, December, 2002 A.V. Porubov February11,2004 20:10 WSPC/BookTrimSizefor9inx6in ws-book9x6 x Book Title
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