Stochastic Elasticity: A Nondeterministic Approach to the Nonlinear Field Theory PDF
Preview Stochastic Elasticity: A Nondeterministic Approach to the Nonlinear Field Theory
Interdisciplinary Applied Mathematics 55 L. Angela Mihai Stochastic Elasticity A Nondeterministic Approach to the Nonlinear Field Theory Interdisciplinary Applied Mathematics Volume 55 SeriesEditors AnthonyBloch,UniversityofMichigan,AnnArbor,MI,USA CharlesL.Epstein,UniversityofPennsylvania,Philadelphia,PA,USA AlainGoriely,UniversityofOxford,Oxford,UK LeslieGreengard,NewYorkUniversity,NewYork,USA AdvisoryEditors RickDurrett,DukeUniversity,Durham,NC,USA AndrewFowler,UniversityofOxford,Oxford,UK L.Glass,McGillUniversity,Montreal,QC,Canada R.Kohn,NewYorkUniversity,NewYork,NY,USA P.S.Krishnaprasad,UniversityofMaryland,CollegePark,MD,USA C.Peskin,NewYorkUniversity,NewYork,NY,USA S.S.Sastry,UniversityofCalifornia,Berkeley,CA,USA J.Sneyd,UniversityofAuckland,Auckland,NewZealand Problems in engineering, computational science, and the physical and biological sciences are using increasingly sophisticated mathematical techniques. Thus, the bridgebetweenthemathematicalsciencesandotherdisciplinesisheavilytraveled. The correspondingly increased dialog between the disciplines has led to the establishmentoftheseries:InterdisciplinaryAppliedMathematics. The purpose of this series is to meet the current and future needs for the interaction between various science and technology areas on the one hand and mathematics on the other. This is done, firstly, by encouraging the ways that mathematics may be applied in traditional areas, as well as point towards new andinnovativeareasofapplications;andsecondly,byencouragingotherscientific disciplines to engage in a dialog with mathematicians outlining their problems to both access new methods as well as to suggest innovative developments within mathematicsitself. The series will consist of monographs and high-level texts from researchers working on the interplay between mathematics and other fields of science and technology. L. Angela Mihai Stochastic Elasticity A Nondeterministic Approach to the Nonlinear Field Theory L.AngelaMihai Mathematics CardiffUniversity Cardiff,UK ISSN0939-6047 ISSN2196-9973 (electronic) InterdisciplinaryAppliedMathematics ISBN978-3-031-06691-7 ISBN978-3-031-06692-4 (eBook) https://doi.org/10.1007/978-3-031-06692-4 Mathematics Subject Classification: 33B15, 34C15, 60G50, 60G60, 60H15, 60H30, 70K50, 74B20, 74C15,74E05,74E10,76A15,82D60,94A15 ©SpringerNatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland InmemoryofIulianBeju,mathematical mechanicianandprofessoratBucharest University Preface Set at the interface between analytical continuum mechanics and advances in probabilitytheory,thisbookcombinesinacoherentandunifiedmannernonlinear elasticity and applied probability to develop a nondeterministic approach for the quantification of uncertainties in the macroscopic elastic responses of materials at largestrains. At a continuum level, the elasticity of many solid materials can be described by phenomenological relations with the input parameters derived from carefully designedexperiments.Thesearetheso-called“hyperelasticmodels”characterised bystrain-energydensities.TheyincludetheclassicalHooke’slawoflinearelasticity at small strains and may take different forms at large strains. When performing a physical experiment multiple times on a test sample or on different samples of a material, the measurements, although qualitatively similar, are quantitatively different,andmoresounderdifferenttestingprotocols.Traditionally,deterministic approaches have been employed to obtain model parameters from average values of the experimental measurements. However, as predictions rely on constitutive modelling, it may not be adequate for a model to depend on a single set of fixed parameters, regardless of how well they reproduce certain experiments. Recently, the use of information about uncertainties in the elastic behaviour of materials, stemmingfromuncertaintiesintheacquireddata,beguntobeexplored. The nondeterministic methods in this book place random variables at the foundationofstochastichyperelasticmodelswheretheinputparametersaredefined by probability densities. Such models rely on the notion of entropy and on the maximum entropy principle, and are able to propagate uncertainties from input datatooutputphysicalquantities.Specificgeometriesincludecuboids,spheres,and cylindricaltubes,whicharecapableofuniversaldeformationsthatcanbesustained for a wide range of materials. However, under particular loading conditions, a nonlinearhyperelasticmaterialmaycausethedeformationtobecomeunstable,and these instabilities must be characterised. Although synthetic and natural structures aretypicallyirregular,regulargeometriescanbestudiedsystematicallytoidentify the independent influence of material properties on the elastic response. For each problem, the elastic solution is firstly derived analytically, then the effect of ix x Preface fluctuating model parameters is quantified in terms of probabilities. In this way, the propagation of uncertainties from input data to output quantities of interest is mathematically tractable, uncovering key aspects of how probabilistic approaches are incorporated into the nonlinear field theory. As a by-product, addressing well- knownproblemsfromanondeterministicperspectivecreatesopportunitiesforfresh insightsandrefinementsofpreviouslyestablishedresults. Stochasticelasticityisafast-developingfieldthatcombinesnonlinearelasticity and stochastic theories in order to significantly improve model predictions by accountingforuncertaintiesinthemechanicalresponsesofmaterials.However,in contrast to the tremendous development of computational methods for large-scale problems,whichhavebeenproposedandimplementedextensivelyinrecentyears, at the fundamental level, there is very little understanding of the uncertainties in the behaviour of elastic materials under large strains. In this case, small sample tests often present situations that might be overlooked by large-scale stochastic techniques. Based on the idea that every large-scale project starts as a small-scale data problem, this book aims to combine fundamental aspects of large-strain finite elasticity and probability theories, which are prerequisites for the quantification of uncertainties in the elastic responses of soft materials. In practice, this type of material can be found under both load-carrying and non-load-carrying conditions, andformthesubjectofintensiveresearchinindustrialandbiomedicalapplications where the advent of additive manufacturing has led to increased interest in the optimal design of controlled reproducible geometries. For these materials, the deformations are inherently nonlinear, and their elastic properties, which change withthedeformation,canbeindependentlyoptimisedforimprovedserviceability. Acentralchallengeinpredictingmechanicalperformanceofmanyengineeringand natural materials is the lack of quantitative characterisation of the uncertainties inherent in the experimental data and in the mathematical models derived from them.Byrecastingspecificproblems,forwhichthedeformationisknownexplicitly, within the nondeterministic framework, crucial insights can be gained into the effectsoffluctuatingparametersatacontinuumlevel,whichcannotbecapturedby adeterministicapproach.Sucheffectshaveimportantimplicationsfortheoptimal designofsoftelasticmaterialsandtheirmodelpredictioninvariousapplications. Therearemanygreatbooksonnonlinearelasticity,including:Biot[61],Ciarlet [101–103],GreenandZerna[218],Ogden[415]andTruesdellandNoll[558]where thefundamentaltheoryispresentedandexplainedindetail;Treloar[550]focusing on phenomenological models for rubber-like materials; Antman [23], Atkin and Fox [31], Bigoni [60], Holzapfel [256], Landau & Lifshitz [315], Marsden and Hughes [343], and Steigmann [525] where classical and modern results are made accessible to a wider audience; Goriely [205] advancing the phenomenological modellingofbiologicalstructures;DorfmannandOgden[146]wherethenonlinear elasticity framework is extended to electroelasticity and magnetoelasticity; Anand andGovindjee[16]andVolokh[575]providingamodernintroductiontoelasticity, plasticity, viscoelasticity and coupled field theories; Warner & Terentjev [594] where the theory of rubber elasticity is adapted to liquid crystal elastomers; and Preface xi Belytschkoetal.[55],BonetandWood[66]andOden[407]onthefiniteelement representationandnumericalapproximationofsoftsolids. Fewerbookshavebeenwrittenonuncertaintyquantificationinsolidmechanics, namely:Wiener[607]onstatisticalmechanicsinelasticity;Soize[510]settingthe foundation for stochastic modelling and uncertainty quantification of large-scale computational models for elastic solids; Elishakoff [150, 151] where probability theory is applied to problems in structural mechanics underpinned by classical linear elasticity; Sobcczyk and Kirkner [505] presenting a concise introduction to stochasticmodellingofmaterialmicrostructures;Ostoja-Starzewski[420]offering a comprehensive analysis of stochastic models and methods in the mechanics of randommedia;andMalyarenkoandOstoja-Starzewski[338]andMalyarenkoetal. [339] introducing stochastic continuum models for elastic materials coupled with electricormagneticfields. StochasticElasticityisthefirstbooktocombinefundamentallargestrainnonlin- earelasticityandprobabilitytheories.Theaimistomakeboththeoriesaccessibleto thosewishingtoincorporateuncertaintyquantificationinphenomenologicalstudies of soft materials. In this sense, the book should be of interest to workers and researchstudentsinappliedmathematics,engineering,biomechanicsandmaterials sciencewhoneedtogainaninsightintofundamentalaspectsofnonlinearelasticity from a continuum point of view, within a nondeterministic framework. This is the more obvious audience, as the book is written on an advanced and detailed level, with the more elementary background deferred to appendices. There is also an expectation that the book should reach further to those with no previous exposure to the subject, who would like to adopt a similar nondeterministic mind-set and incorporateuncertaintiesintheirownscientificwork.Whetherwiththeintentionto keepthemundercontrol,asisthecaseofmanufacturedandfunctionalmaterials,or justtoobservethem,asinnaturalandlivingbiologicalmatter,takinguncertainties intoaccountcanbothincreasetheimmediatesignificanceofmanyneworexisting results, and render them more likely in the future. Of course, there is always an element of surprise when dealing with uncertainties: One never quite knows what moretheycanbring.Thatmakeseverystudyofuncertaintyquantificationunique. Cardiff,UK L.AngelaMihai March21,2022
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