Inflation and Instabilities of Hyperelastic Membranes Amit Patil

211 Pages·2016·3.72 MB·English

Inflation and Instabilities of Hyperelastic
Membranes
Amit Patil
Doctoral Thesis in Engineering Mechanics
June 2016
Technical Reports from
Royal Institute of Technology
Department of Mechanics
SE-100 44 Stockholm, Sweden
Akademisk avhandling som med tillst˚and av Kungliga Tekniska H¨ogskolan i
Stockholm framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie
doktorsexamen tisdag den 14 juni 2016 kl 13.30 i Kollegiesalen, Brinellv¨agen
8, Kungliga Tekniska H¨ogskolan, Stockholm.
'Amit Patil 2016
Universitetsservice US–AB, Stockholm 2016
Inflation and Instabilities of Hyperelastic Membranes
Amit Patil
Dept. of Mechanics, Royal Institute of Technology
SE-100 44 Stockholm, Sweden
Abstract
The applications of membranes are increasing rapidly in various fields of engi-
neering and science. The geometric, material, force and contact non-linearities
complicate their analysis, which increases the demand for computationally ef-
ficient methods and interpretation of counter-intuitive behaviours. To under-
standthecomplexbehaviourofmembranestructures,itisnecessarytoperform
parametric studies of simple setups and understand their basic mechanical be-
haviour.
The first part of the present work studies the free and constrained infla-
tion of circular and cylindrical membranes. The membranes are assumed to
be in contact with a soft substrate, modelled as a linear spring distribution.
Adhesive and frictionless contact conditions are considered during inflation,
while only adhesive contact conditions are considered during deflation. For a
circular membrane, peeling of the membrane during deflation is studied, and a
numerical formulation of the energy release rate is proposed. The pre-stretch
produces a stiffening effect for the circular membranes, and a softening effect
for the cylindrical membranes, in the pre-limit zone.
The second part of the thesis discusses the instabilities observed for fluid
containing cylindrical membranes. Limit points and bifurcation points are
observed on primary equilibrium branches. The secondary branches emerge
from bifurcation points, with their directions determined by eigenvectors cor-
responding to zero eigenvalues at the bifurcation point. Symmetry has major
implicationsonstabilityanalysisofthestructures,andtherelationshipbetween
eigenvalue analysis and symmetry is highlighted in this part of the thesis. The
occurrence of critical points and the stability of equilibrium branches are de-
termined by perturbation techniques.
In the third part, wrinkling in the pressurized membranes is investigated,
and robustness of the modified membrane theory and tension field theory is
examined. The effect of boundary conditions, thickness variations, and inflat-
ing media on the wrinkling is investigated. It is observed that, with a relaxed
strain energy formulation, the obtained equilibrium solutions are unstable due
to the occurrence of pressure induced instabilities. These instabilities are not
visiblewhenaxisymmetryisassumedorcoarsediscretizationsinthewrinkling
direction are used. A detailed analysis of pressure induced instabilities in the
wrinkled membranes is described in the thesis.
Descriptors: Membranes, Constrained inflation, Energy release rate, Adhe-
sive contact condition, Limit point, Bifurcation point, Wrinkling, Tension field
theory, Pressure induced instability.
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Inflation and Instabilities of Hyperelastic Membranes
Amit Patil
Dept. of Mechanics, Royal Institute of Technology
SE-100 44 Stockholm, Sweden
Sammanfattning
Anv¨andandet av membran ¨okar snabbt inom teknik och naturvetenskap. Icke-
linj¨ariteterp˚agrundavgeometri,material,krafterochkontaktvillkorg¨orf¨orst˚a-
elsenavdessasbeteendensv˚arare.Detta¨okarbehovenaveffektivaber¨aknings-
metoderf¨ortolkningavmembranensbeteendeunderyttrep˚averkningar.Para-
metriskastudieravenklaproblemkan¨okaf¨orst˚aelsenf¨ormembranensmekanik.
Avhandlingensf¨orstadelstuderarfriochbegr¨ansadexpansionavcirkul¨ara
och cylindriska membran. Dessa antas komma i kontakt med ett flexibelt yt-
skikt,modelleratsomenlinj¨arfj¨aderb¨add.Vidh¨aftandeochfriktionsfrikontakt
beaktas i expansionsfasen, medan endast vidh¨aftande villkor beaktas vid av-
lastning. Successiv avskalning av membranet fr˚an ytskiktet studeras f¨or ett
cirkul¨art membran, och en numerisk formulering ges f¨or brottenergibalansen.
F¨orstr¨ackning visas ge ¨okande styvhet f¨or det cirkul¨ara membranet och mins-
kande f¨or cylindriska membran, innan de n˚ar sina max-punkter f¨or tryck.
Avhandlingens andra del diskuterar instabiliteter som intr¨affar f¨or v¨atske-
fyllda cylindriska membran. Gr¨ans- och f¨orgreningspunkter kan existera p˚a de
fundamentala l¨osningsgrenarna. Sekund¨ara l¨osningar utg˚ar fr˚an f¨orgrenings-
punkterna, med riktningar best¨amda av egenvektorerna svarande mot noll-
egenv¨arden i f¨orgreningspunkten. Symmetri visas ha stor betydelse f¨or stabili-
tetsanalyserna, och sambanden mellan symmetri och egenv¨arden p˚avisas i av-
handlingen. Metoderna f¨or att analysera kritiska punkter och stabilitet bygger
p˚a systematiska serieutvecklingar.
Identredjedelenunders¨oksskrynklingidetrycksattamembranen.Tillf¨or-
litligheten hos modifierad membranteori och dragf¨altsteori unders¨oks. Effek-
terna av randvillkor, tjockleksvariation och mediet f¨or trycks¨attning studeras.
En observation ¨ar att de ber¨aknade j¨amviktsl¨osningarna kan vara instabila
p˚a grund av tryckberoende effekter, i en formulering baserad p˚a modifierad
t¨ojningsenergi. Dessa instabiliteter kan f¨orbli osynliga om ett problem analy-
seras med hj¨alp av en axisymmetrisk modell, eller med en modell inneh˚allande
engrovdiskretiseringidenskrynklanderiktningen.Dessatryckgenereradeins-
tabiliteter studeras noggrant i avhandlingen.
Nyckelord: Membran, Begr¨ansad expansion, Brottenergibalans, Vidh¨aftande
kontakt,Gr¨anspunkter,F¨orgreningspunkter,Skrynkling,Dragf¨altsteori,Tryck-
genererad instabilitet.
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Preface
This thesis discusses inflation and instabilities of hyperelastic membranes
with different material models, geometries, boundary conditions, inflating me-
diaandpre-stretches. Semi-analyticalandnumericalformulationsareproposed
in this thesis. It thereby gives the basis for qualitative verifications of results
obtained by other methods and experiments. A brief introduction to the basic
concepts and methods is presented in the first part. The second part contains
five articles. The papers are adjusted to conform with the present thesis for-
mat for consistency, but their contents have not been changed as compared
with their original counterparts.
Paper 1. Amit Patil, Anirvan DasGupta & Anders Eriksson, 2015
ContactMechanicsofaCircularMembraneInflatedagainstaDeformableSub-
strate. International Journal of Solids and Structures, 67–68, 250–262.
Paper 2. Amit Patil, Arne Nordmark & Anders Eriksson, 2014
Free and Constrained Inflation of a Pre-stretched Cylindrical Membrane. Pro-
ceedings of Royal Society A, 470, 20140282.
Paper 3. Amit Patil, Arne Nordmark & Anders Eriksson, 2015
InstabilityInvestigationonFluid-loadedPre-stretchedCylindricalMembranes.
Proceedings of Royal Society A, 471, 20150016.
Paper 4. Amit Patil, Arne Nordmark & Anders Eriksson, 2015
Wrinkling of Cylindrical Membranes with Non-uniform Thickness. European
Journal of Mechanics - A/Solids, 54, 1–10.
Paper 5. Amit Patil, Arne Nordmark & Anders Eriksson, 2016
InstabilitiesofWrinkledMembraneswithPressureLoadings. AcceptedinJour-
nal of Mechanics and Physics of Solids.
The following publications are not included in the thesis, but related to
the presented work.
Paper 1. Amit Patil, Arne Nordmark & Anders Eriksson, 2014
FiniteInflationofFluid-filledPre-stretchedCylindricalMembranes. 27th Nordic
Seminar on Computational Mechanics, Stockholm. 27, 224–227.
vii
Paper 2. Anders Eriksson, Arne Nordmark, Amit Patil & Yang
Zhou, 2015. Parametric Stability Investigations for Hydro-statically Loaded
Membranes. Computers and Structures, (Available Online).
viii
Division of work among authors
ThisprojectwasfirstintroducedbyProfessorAndersEriksson(AE)whoisthe
main supervisor of this work and Arne Nordmark (AN) who has functioned as
assistantsupervisor. ProfessorAnirvanDasGupta(AD)hascontributedinone
research paper. Amit Patil (AP) has discussed the results and progress with
AE, AN and AD for the respective manuscripts.
Paper 1
AP built up the numerical model in Mathematica and performed the simu-
lations with feedback from AD and AE. The paper was written by AP with
inputs from AD and AE.
Paper 2
AP built up the numerical model in Mathematica and performed the simu-
lations with feedback from AN and AE. AN verified some results in Comsol
Multiphysics. The paper was written by AP with inputs from AN and AE.
Paper 3
AP built up the numerical model in Mathematica and performed the simu-
lations with feedback from AN and AE. AN verified some results in Comsol
MultiphysicsandAEinanindependentFEMmodelinMatlab. Thepaperwas
written by AP with inputs from AN and AE.
Paper 4
AP built up the numerical model in Mathematica and performed the simu-
lations with feedback from AN and AE. AN verified some results in Comsol
MultiphysicsandAEinanindependentFEMmodelinMatlab. Thepaperwas
written by AP with inputs from AN and AE.
Paper 5
AP built up the numerical model in Mathematica and performed the simula-
tions with feedback from AN and AE. AP developed theoretical formulations
with feedback from AN and AE. AN verified some results in Comsol Multi-
physics and AE in an independent FEM model in Matlab. The paper was
written by AP with inputs from AN and AE.
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