Table Of ContentInterdisciplinary 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
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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
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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
