Table Of ContentAdvanced Structured Materials
Holm Altenbach · Svetlana Bauer ·
Victor A. Eremeyev · Gennadi I. Mikhasev ·
Nikita F. Morozov Editors
Recent Approaches
in the Theory
of Plates and
Plate-Like Structures
Advanced Structured Materials
Volume 151
Series Editors
Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of
Applied Sciences, Esslingen, Germany
Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of
Engineering, University of Porto, Porto, Portugal
Holm Altenbach , Faculty of Mechanical Engineering, Otto von Guericke
University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany
Common engineering materials reach in many applications their limits and new
developments are required to fulfil increasing demands on engineering materials.
The performance ofmaterials can beincreasedby combiningdifferent materials to
achieve better properties than a single constituent or by shaping the material or
constituents in a specific structure. The interaction between material and structure
mayariseondifferentlengthscales,suchasmicro-,meso-ormacroscale,andoffers
possible applications in quite diverse fields.
Thisbookseriesaddressesthefundamentalrelationshipbetweenmaterialsandtheir
structure on the overall properties (e.g. mechanical, thermal, chemical or magnetic
etc.) and applications.
The topics of Advanced Structured Materials include but are not limited to
• classical fibre-reinforced composites (e.g. glass, carbon or Aramid reinforced
plastics)
• metal matrix composites (MMCs)
• micro porous composites
• micro channel materials
• multilayered materials
• cellular materials (e.g., metallic or polymer foams, sponges, hollow sphere
structures)
• porous materials
• truss structures
• nanocomposite materials
• biomaterials
• nanoporous metals
• concrete
• coated materials
• smart materials
Advanced Structured Materials is indexed in Google Scholar and Scopus.
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Holm Altenbach Svetlana Bauer
• •
Victor A. Eremeyev Gennadi I. Mikhasev
• •
Nikita F. Morozov
Editors
Recent Approaches
in the Theory of Plates
and Plate-Like Structures
123
Editors
HolmAltenbach SvetlanaBauer
Fakultät für Maschinenbau Faculty of Mathematics andMechanics
Otto-von-Guericke-Universität St.Petersburg State University
Magdeburg St.Petersburg, Russia
Magdeburg, Germany
Gennadi I.Mikhasev
Victor A.Eremeyev Faculty of MechanicsandMathematics
DICAAR Belarusian State University
University of Cagliari Minsk,Belarus
Cagliari, Italy
Faculty of Civil andEnvironmental
Engineering
Gdansk University of Technology
Gdansk,Poland
Research Institute for Mechanics
National Research LobachevskyState
University of NizhniNovgorod
NizhniNovgorod, Russia
Nikita F. Morozov
Faculty of Mathematics andMechanics
St.Petersburg State University
St.Petersburg, Russia
ISSN 1869-8433 ISSN 1869-8441 (electronic)
AdvancedStructured Materials
ISBN978-3-030-87184-0 ISBN978-3-030-87185-7 (eBook)
https://doi.org/10.1007/978-3-030-87185-7
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ToPetrEvgenievichTovstik,
aprominentRussianscientist
inthefieldofthin-walledstructures
—ourteacher,colleague,collaborator,
andfriend.
Preface
On December 30, 2020 in only two days after his 85th anniversary professor P.E.
Tovstik,outstandingresearcherandteacher,diedasaresultofCOVID-19.
All his life was inseparably linked with the Faculty of Mathematics and Me-
chanicsofLeningrad,andthenSt.PetersburgStateUniversity.In1958P.E.Tovstik
graduatedwithdistinctionfromthefacultyandstartedthepostgraduatestudy.Af-
ter defense in 1963 his Candidate (PhD) thesis he worked in laboratory of vibra-
tionsoftheScientificResearchInstituteofMathematicsandMechanics(NIIMM)
at Leningrad State University. Having defended in 1968 his doctoral dissertation
(Graduation),hebecametheAssociateProfessor,andthenProfessoroftheDepart-
mentofTheoreticalandAppliedMechanics.Since1978heheadedthisdepartment.
The first research projects made by P.E. Tovstik in the Laboratory of Vibra-
tionsatNIIMMundersupervisionoftheheadofthelaboratoryAssociateProfessor
G.N.Bukharinovweredevotedtoasymptoticanalysisofequationsofvibrationsof
Fig. 1 Petr Evgenievich
Tovstik(1935-2020).
vii
viii Preface
screw springs. The high scientific level of the young scientist research was recog-
nizedbythemathematicalcommunityinFominandShilov(1970).Furtherhebe-
cameinterestedinverydifficultproblemsofthinshelltheory,includingproblemson
freevibrationsofthinshellsofrevolution,whicharedescribedbyordinarydifferen-
tialequationswithvariablecoefficientscontainingasmallparameteratthehighest
derivative.Forsolutionsofthewholeclassofsuchsingularlyperturbeddifferential
equationscontainingturningpoint(s)hedevelopedanewasymptoticmethodofin-
tegration.Theobtainedasymptoticexpansionswereusedtofindnaturalfrequencies
andmodesofvibrationsofshellsofrevolution.In1968,theseresultswereincluded
intohisdoctoralthesis.ManyofresultsofthisresearchwerelaterincludedinGold-
enveizeretal(1979).In1970,foracycleofworksonthetheoryofshellshegotthe
firstprizeofLeningradStateUniversityforscientificworks.
Inthemid-eightiesP.E.Tovstikdevelopedanewasymptoticmethodofapprox-
imate evaluation of the lower frequencies of vibrations and critical buckling loads
for noncircular cylindrical and conic shells. By means of this method, simple ap-
proximatesolutionsforalargenumberofproblemsofthetheoryofvibrationsand
bucklingofshells,forwhichvibrationalandbucklingmodesarelocalizednearso-
calledweakestgeneratrices,wereobtained.Earlierfortheseproblemsonlynumeri-
calresultsforspecialcaseswereknown.Inthedoctoraldissertations(Dr.Sc.)ofhis
pupils,G.I.MikhasevandS.B.Filippov,thismethodwasappliedanddeveloped.
Alsoothertypesoflocalizationofvibrationswerestudied,includingcases,when
small pits are concentrated near the weakest point on a median surface or in the
neighborhood of weakly supported edge. Main results of the research on the shell
bucklingtheory,localizedbucklingandvibrationalmodesofthinelasticshellswere
included in Tovstik (1995); Tovstik and Smirnov (2001); Mikhasev and Tovstik
(2009).In1998,foracycleofworks"Fundamentalproblemsofthetheoryofthin-
walledstructures"P.E.Tovstik,togetherwithhisco-authors,wasawardedtheState
PrizeoftheRussianFederationinScienceandTechnology.Inthesameyearhewas
awardedthetitle“HonoredScientistoftheRussianFederation”.
Theerroroftheshelltheoryincreasesinnonlinearproblemsasdeformationsgo
up,andthemaininaccuracyisassociatedwithconstitutiverelationsbetweenstress-
resultants/stresscouplesanddeformations.InP.E.Tovstikworks,refinedelasticity
relations were derived from 3D equations of the theory of elasticity. When these
relationsareused,theerrorofthenon-linearshelltheoryhasanorderofthedimen-
sionlessthicknessofashell.Theformulationofrefinedelasticityrelationsisanew
fundamentalresultofthenon-linearshelltheory,whichwasincludedinKabritsetal
(2002).ForthisbooktheauthorsgottheFirstPrizeforScientificResearchfromSt.
PetersburgStateUniversityin2002.
Inrecentyears,P.E.Tovstikpaidattentiontononclassicalproblemsofstaticsand
dynamicsofthinshellsandplates:
1. problemsofbucklingandvibrationsofmultilayeredshellsandplates,
2. shellsandplateslyingonelasticfoundations,and
3. beams, plates and shells, made of material with anisotropy of the general type,
andothers.
Preface ix
AlargeseriesofpapersbyP.E.Tovstikwasdevotedtobeams,platesandshells
madeoflinearlyelastichomogeneousanisotropicmaterialsofageneraltype,which
elasticity relations contain 21 elastic modules. It is well known that theories of
plates and shells are either based on some assumptions (hypotheses) or obtained
by means of asymptotic methods permitted to derive equations for 2D structures
from equations of 3D elasticity theory. P.E. Tovstik was one of the leading spe-
cialistsamongthose,whousedthelastapproachtoderiveasymptoticallyconsistent
equationsforplatesandshells.Foranisotropic,inhomogeneousinthickness(inpar-
ticular,multilayer)beams,platesandshells,theproblemofconstructing2Dmodels
becomesmuchmorecomplicated.Hypothesis-basedmethodsbecomeinapplicable
or the area of their applicability narrowed significantly. At the same time, meth-
odsofasymptoticintegrationgivereliableresultsconfirmingbytestexamples,for
which3Dproblemsadmitexactsolutions.
BasedonthegeneralizedTymoshenko–Reissnerhypotheses,freevibrationsand
buckling under uniform external pressure of thin elastic cylindrical shells made
of material with general type anisotropy were studied and then using asymptotic
methodP.E.Tovstikderivedapproximateformulasforvibrationalfrequenciesand
modes,critical externalpressureand bucklingmodes. Theaccuracyof asymptotic
resultswasconfirmedbynumericalanalysiswiththehelpoffiniteelementmethod.
P.E.TovstiktogetherwithN.F.Morozovsolvedtheproblemsoffreevibrationsand
bucklingofthecompressedtransverselyisotropicspace,half-spaceandplate.Spe-
cial attention was paid to analysis of buckling modes both a plate and its support-
ing foundation. The non-linear analysis of the energy of post-critical deformation
reultedinconclusionthatthebucklingmodehadthechess-boardlikeform,which
agreedwiththeexperimentalresults.
Research interests of P.E. Tovstik was never limited to the field of thin-walled
structures. For many years he headed numerous and varied industrial projects.
Amazingintuitionhelpedhimtocatchimmediatelythemainfeaturesofthestudied
mechanical system and develop an adequate mathematical model for it. Here we
mentiononlysomeoftheseprojects.
The large cycle of applied research on analysis of lightweight metal mirrors of
telescopeshasbeencarriedoutinlong-termcooperationwiththeStateOpticalIn-
stitute.Thetelescopemirrorwassimulatedasanannularlaminatedplateofvariable
thickness.MainresultsofthesestudieswereincludedinBaueretal(1997).
Since1994,P.E.Tovstiktookanactivepartintheapplicationoftheoryofshells
andplatestoproblemsofophthalmology.Modelsproposedbyhimhelpindescrib-
ing pathological changes in the sclera of an eye, as well as changes in the stress-
strain state of eye shells after surgery. Some of the results he obtained were pub-
lishedinBaueretal(2000).
In2013–2015,P.E.TovstikanalyzedtheclassicalIshlinsky–Lavrentievproblem
ofdynamicbehaviorofarodaxiallycompressedbyanappliedforce.Applicationof
the two-scale asymptotic decomposition method allowed studying main and com-
binationresonancesfortransverserodvibrationscausedbyitsaxialvibrations,es-
timatethemaximumdeflectionvalue,examinebeatmodesassociatedwithtransfer
ofenergyfromaxialtotransversevibrationsandback.
x Preface
P.E. Tovstik also considered the influence of the form of initial imperfections
on emergence and development of transverse vibrations. With the use of integral
transforms he revealed the connection between the most dangerous imperfections
andthefastestgrowingdeflectionmodesfoundintheworkbyM.A.Lavrentievand
A.Yu.Ishlinsky.In2015fortheseriesofpapersentitled“RodDynamicsunderAxial
Compression: the development of ideas by M.A. Lavrentiev and A.Yu. Ishlinsky”
he (together with co-authors N.F. Morozov and A.K. Belyaev) was awarded the
Lavrentiev prize of Russia Academy of Sciences for outstanding achievements in
mathematicsandmechanics.
Until his last days, P.E. Tovstik retained very high scientific potential, which
manyyoungercolleaguesmayjustenvy.In2016–2020,hepublishedmorethan40
scientific paper including 27 publications in the most prestigious Russian and Eu-
ropean journals. In addition, together with G.I. Mikhasev he published a book on
localizeddynamicsofthin-walledstructures(MikhasevandTovstik,2020).Intotal
heistheauthorofmorethan270scientificpublications,including11monographs.
Thesepublicationscontainstudiesofactualanddifficultproblemsinthemostgen-
eralformulationsandtheirgracefulsolutionswiththehelpofasymptoticmethods,
to which P.E. Tovstik made a great contribution. In all his researches, the general
fundamental theoretical analyses are followed by considerations of actual exam-
ples, which he solved numerically writing and debugging computer codes. Since
1994,P.E.Tovstikwaspermanentlyaprincipalinvestigatorintheteamofhispupils
working on a grant on thin-walled structures presented by Russian Foundation for
BasicResearch.Mainresultsincludedingrantreportsusuallybelongedtohim.
P.E. Tovstik spared no time and efforts for the numerous pupils, among which
therewere9Doctors(Dr.Sc.)andmorethan30CandidatesofScience(PhD)with
whomhegenerouslysharedideas.P.E.Tovstikfoundedascientificschoolonappli-
cationofasymptoticmethodsintheoriesofthin-walledstructures.Formanyyears
P.E.Tovstikdeliveredacourseonasymptoticmethodsinmechanics.Basedonthese
lecturesheandhispupilspublishedtextbooksonapplicationofasymptoticmethods
inmechanicsofsolids(VaillancourtandSmirnov,1993;Baueretal,2007,2015),
whicharewidelyusedinRussianuniversitiesandabroad.
HeheadedtheDepartmentofTheoreticalandAppliedMechanicsofSt.Peters-
burg State University for more than 42 years, longer, than any of his prominent
predecessors. When taken the office he was the only Full Professor of the Depart-
ment,butwithtimefivemoreFullProfessorsjointtheDepartment.Thebreadthof
scientific interests allowed P.E. Tovstik to supervise research in all scientific areas
those were studied under his predecessor Professor N.N. Polyakhov, who headed
thedepartmentfrom1952till1978.FordecadesP.E.Tovstikdeliveredacourseon
theoreticalmechanics.Hewasoneoftheauthorsandtheeditorofthetwo-volume
textbookontheoreticalmechanics,thatisplannedtobepublishedin2021bytheSt.
PetersburgUniversityPressinRussianand,simultaneously,bySpringerinEnglish.
P.E. Tovstik always successfully combined research and scientific and profes-
sional social work. He was the member of the Russian National Committee on
Theoretical and Applied Mechanics, expert of the Russian Foundation for Basic
Research,Editor-in-chiefofthejournals“VestnikSt.PetersburgUniversity.Math-
