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To my parents
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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
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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
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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
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