Table Of ContentInterdisciplinary Applied Mathematics 51
Merab Svanadze
Potential Method
in Mathematical
Theories of
Multi-Porosity
Media
Interdisciplinary Applied Mathematics
Volume 51
Editors
AnthonyBloch,UniversityofMichigan,AnnArbor,MI,USA
CharlesL.Epstein,UniversityofPennsylvania,Philadelphia,PA,USA
AlainGoriely,UniversityofOxford,Oxford,UK
LeslieGreengard,NewYorkUniversity,NewYork,USA
Advisors
L.Glass,McGillUniversity,Montreal,QC,Canada
R.Kohn,NewYorkUniversity,NewYork,NY,USA
P.S.Krishnaprasad,UniversityofMaryland,CollegePark,MD,USA
AndrewFowler,UniversityofOxford,Oxford,UK
C.Peskin,NewYorkUniversity,NewYork,NY,USA
S.S.Sastry,UniversityofCaliforniaBerkeley,CA,USA
J.Sneyd,UniversityofAuckland,Auckland,NewZealand
RickDurrett,DukeUniversity,Durham,NC,USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/1390
Merab Svanadze
Potential Method
in Mathematical Theories
of Multi-Porosity Media
MerabSvanadze
IliaStateUniversity
Tbilisi,Georgia
ISSN0939-6047 ISSN2196-9973 (electronic)
InterdisciplinaryAppliedMathematics
ISBN978-3-030-28021-5 ISBN978-3-030-28022-2 (eBook)
https://doi.org/10.1007/978-3-030-28022-2
MathematicsSubjectClassification:74F05,74F10,74G25,74G30,74H45,74M25,35E05
©SpringerNatureSwitzerlandAG2019
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG.
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Dedicatedto
Maia,Lika,andGiorgi
“Natureiswritteninmathematicallanguage”
GalileoGalilei,TheAssayer,1623
Preface
The goal of this book is to investigate the three-dimensional problems of the
mathematicaltheoriesofelasticityandthermoelasticityformulti-porositymaterials.
Thepurposesofthisworkaretwofold.Thefirstistodeveloptheclassicalpotential
method(boundaryintegralequationmethod)inthesetheories,andthesecondisto
generalize the basic results of the classical mathematical theories of elasticity and
thermoelasticity.
The theory of porous media plays a central role in various fields of applied
mathematics, continuum mechanics, and engineering and has already been dealt
withinnumerousbooksandpapers.Inthelastthreedecades,therehasbeeninterest
ininvestigationofproblemsofelasticityandthermoelasticityforsolidswithdouble
andtripleporosities.Moreover,intherecentresearchpapers,thereareindications
thattheconstructionandinvestigationofthemathematicalmodelsofquadrupleand
quintupleporositystructuresofgeo-andbiomaterialsareimportantandnecessary
forvariousfieldsofengineeringandtechnology.
For clarity, in this book, we investigate the linear mathematical theories of
elasticity and thermoelasticity for quadruple porosity materials. On the one hand,
each result of this work can be reduced to the corresponding result for the triple
(double) porosity materials neglecting fourth level (third and fourth levels) of
porosity of materials. On the other hand, in a quite similar manner, the research
methods and results of this book can be generalized to quintuple and more stage
hierarchical porosity systems. Furthermore, quadruple porosity structures are the
ones most likely to represent realistic systems occurring in nature, for example,
bone,humanbrain,oil,gas,andthermalreservoirs.
The quadruple porosity model represents a new possibility for the study of
importantproblemsofengineeringandmechanics.Theintendedapplicationsofthe
mathematicaltheoriesofelasticityandthermoelasticityforsolidswithaquadruple
porosity structure are to geological materials such as oil, gas and geothermal
reservoirs, rocks and soils, manufactured porous materials such as ceramics and
pressedpowders,andbiomaterialssuchasboneandhumanbrain.
ix
x Preface
All the results of this book are unpublished contributions of the author. In this
work, the boundary value problems (BVPs) of the linear mathematical theories
of elasticity and thermoelasticity for quadruple porosity materials and the BVPs
of rigid bodies with quadruple porosity are investigated, and in each theory, the
followingresultsareobtained:
• The fundamental and singular solutions of the systems of equations of equi-
librium, quasi-static, steady vibrations, and in the Laplace transform space are
expressed in explicit form by means of elementary (harmonic, biharmonic,
metaharmonic)functions,andtheirbasicpropertiesareestablished.
• TherepresentationsofGalerkin-typesolutionsofthesesystemsofequationsare
obtained.
• Green’s formulas and integral representations of Somigliana type of regular
vectorandclassicalsolutionsareestablished.
• AwideclassoftheinternalandexternalBVPsofequilibrium,quasi-static,steady
vibrations,andintheLaplacetransformspaceareformulated.
• Theradiationconditionsareestablishedandtheuniquenesstheoremsofregular
(classical)solutionsoftheBVPsareproved.
• The surface (single-layer and double-layer) and volume potentials are con-
structed,andtheirbasicpropertiesarestudied.
• Onthebasisofthesepotentials,theBVPsarereducedtotheequivalentintegral
equationsforwhichFredholm’stheoremsarevalid.
• Theexistencetheoremsforclassicalsolutionsoftheabove-mentionedBVPsare
provedbyusingthepotentialmethodandthetheoriesofsingularandFredholm
integralequations.
This book is addressed to a large circle of readers: mathematicians (especially
those involved in applied mathematics), physicists (particularly those interested in
mechanicsanditsconnections),engineersofvariousspecialities(civil,mechanical,
etc., who are scientific researchers), graduate students of these professions, etc.
I have assumed that the reader is conversant with the basic elements of the
mathematicaltheoryofelasticity,butotherwise,thebookisself-contained.
The text is written in the moderately clear form and structured in a stepwise
hierarchicalfashionfrombasictomoreadvancedtopics.
This book contains 13 chapters. The text begins with an introductory chapter.
Chapters2–12canberoughlydividedintotwoparts.Thefirstpart(Chaps.2–8)and
thesecondpart(Chaps.9–12)dealwithproblemsofthelineartheoriesofelasticity
and thermoelasticity of quadruple porosity materials, respectively. The objective
of the final chapter is to mention several interesting topics in the theory of multi-
porositymediaforfutureresearch.
The author warmly thanks Professor Brian Straughan for his many helpful dis-
cussionsonthemodelingquestionsofmulti-porositymaterialsandwhoprompted
the author to write this book. I am also grateful to three anonymous referees for
usefulcommentsandsuggestions.
Preface xi
Finally, I would like to thank Christopher Tominich for his help with editorial
matters and the Springer production team for their professional competence and
fruitfulcooperationinpreparingthefinalversionofthisbook.
Tbilisi,Georgia MerabSvanadze
September2019
