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Inflation and Instabilities of Hyperelastic Membranes
KTH, School of Engineering Sciences (SCI), Mechanics. (Structural Mechanics)ORCID iD: 0000-0003-3716-8520
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The applications of membranes are increasing rapidly in various fields of engineering and science. The geometric, material, force and contact non-linearities complicate their analysis, which increases the demand for computationally efficient methods and interpretation of counter-intuitive behaviours.

The first part of the present work studies the free and constrained inflation 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 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 corresponding to zero eigenvalues at the bifurcation point. Symmetry has major implications on stability analysis of the structures, and the relationship between eigenvalue analysis and symmetry is highlighted in this part of the thesis.

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 inflating 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. A detailed analysis of pressure induced instabilities in the wrinkled membranes is described in the thesis.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. , 197 p.
Series
TRITA-MEK, ISSN 0348-467X ; 2016-09
Keyword [en]
Membranes, Constrained inflation, Energy release rate, Adhesive contact condition, Limit point, Bifurcation point, Wrinkling, Tension field theory, Pressure induced instability.
National Category
Applied Mechanics
Research subject
Engineering Mechanics
Identifiers
URN: urn:nbn:se:kth:diva-187041ISBN: 978-91-7595-989-4 (print)OAI: oai:DiVA.org:kth-187041DiVA: diva2:928684
Public defence
2016-06-14, Kollegiesalen,, Brinellvagen 8, Stockholm, 13:25 (English)
Opponent
Supervisors
Funder
Swedish Research Council
Note

QC 20160518

Available from: 2016-05-18 Created: 2016-05-16 Last updated: 2016-06-17Bibliographically approved
List of papers
1. Contact mechanics of a circular membrane inflated against a deformable substrate
Open this publication in new window or tab >>Contact mechanics of a circular membrane inflated against a deformable substrate
2015 (English)In: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146, Vol. 67-68, 250-262 p.Article in journal (Refereed) Published
Abstract [en]

Finite inflation of a hyperelastic flat circular membrane against a deformable adhesive substrate and peeling upon deflation are analyzed. The membrane material is considered to be a homogeneous, isotropic and incompressible Mooney-Rivlin solid. The deformable substrate is assumed to be a distributed linear stiffness in the direction normal to the undeformed surface. The adhesive contact is considered to be perfectly sticking with no tangential slip between the dry surfaces of the membrane and the substrate. The inflation mechanics problem in the variational form yields the governing equations and boundary conditions, which are transformed to a nonlinear two-point boundary value problem by a careful choice of field variables for efficient computation. It is found that during inflation (deflation) with adhesive contact, the meridional stretch exhibits continuity up to C0 (C-1) at the contact junction, while the circumferential stretch remains continuous up to C1 (C0). Interestingly, stretch locking in an adhesive contact is found to give a higher indentation on the substrate than in a frictionless contact. Peeling at the contact junction has been studied, and numerical formulations for the energy release rate are proposed.

Keyword
Adhesive contact, Constrained inflation, Contact peeling, Deformable substrate, Energy release rate
National Category
Other Engineering and Technologies
Identifiers
urn:nbn:se:kth:diva-170316 (URN)10.1016/j.ijsolstr.2015.04.025 (DOI)000357243900020 ()2-s2.0-84930757627 (Scopus ID)
Note

QC 20150629

Available from: 2015-06-29 Created: 2015-06-29 Last updated: 2017-12-04Bibliographically approved
2. Free and constrained inflation of a pre-stretched cylindrical membrane
Open this publication in new window or tab >>Free and constrained inflation of a pre-stretched cylindrical membrane
2014 (English)In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 470, no 2169, UNSP 20140282- p.Article in journal (Refereed) Published
Abstract [en]

This paper presents the free and constrained inflation of a pre-stretched hyperelastic cylindrical membrane and a subsequent constrained deflation. The membrane material is assumed as a homogeneous and isotropic Mooney-Rivlin solid. The constraining soft cylindrical substrate is assumed to be a distributed linear stiffness normal to the undeformed surface. Both frictionless and adhesive contact are modelled during the inflation as an interaction between the dry surfaces of the membrane and the substrate. An adhesive contact is modelled during deflation. The free and constrained inflation yields governing equations and boundary conditions, which are solved by a finite difference method in combination with a fictitious time integration method. Continuity in the principal stretches and stresses at the contact boundary is dependent on the contact conditions and inflation-deflation phase. The pre-stretch has a counterintuitive softening effect on free and constrained inflation. The variation of limit point pressures with pre-stretch and the occurrence of a cusp point is shown. Interesting trends are observed in the stretch and stress distributions after the interaction of the membrane with soft substrate, which underlines the effect of material parameters, pre-stretch and constraining properties.

Keyword
contact conditions, soft substrate, Mooney-Rivlin, limit points, deflation, continuity
National Category
Applied Mechanics
Identifiers
urn:nbn:se:kth:diva-148597 (URN)10.1098/rspa.2014.0282 (DOI)000338717800012 ()
Note

QC 20140811

Available from: 2014-08-11 Created: 2014-08-11 Last updated: 2017-12-05Bibliographically approved
3. Instability investigation on fluid-loaded pre-stretched cylindrical membranes
Open this publication in new window or tab >>Instability investigation on fluid-loaded pre-stretched cylindrical membranes
2015 (English)In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 471, no 2177, 20150016Article in journal (Refereed) Published
Abstract [en]

This paper discusses the evaluation of instabilities on the quasi-static equilibrium path of fluid-loaded pre-stretched cylindrical membranes and the switching to a secondary branch at a bifurcation point. The membrane is represented by only the in-plane stress components, for which an incompressible, isotropic hyperelastic material model is assumed. The free inflation problem yields governing equations and boundary conditions, which are discretized by finite differences and solved by a Newton-Raphson method. An incremental arclength-cubic extrapolation method is used to find generalized equilibrium paths, with different parametrizations. Limit points and bifurcation points are observed on the equilibrium path when fluid level is seen as the controlling parameter. An eigen-mode injection method is employed to switch to a secondary equilibrium branch at the bifurcation point. A limit point with respect to fluid level is observed for a partially filled membrane when the aspect ratio (length/radius) is high, whereas for smaller aspect ratios, the limit point with respect to fluid level is observed at over-filling. Pre-stretch is observed to have a stiffening effect in the pre-limit zone and a softening effect in the post-limit zone.

Keyword
hydrostatic loading, limit point, bifurcation point, softening, eigen-mode injection method, finite differences
National Category
Applied Mechanics
Identifiers
urn:nbn:se:kth:diva-160730 (URN)10.1098/rspa.2015.0016 (DOI)000353352400016 ()2-s2.0-84929207245 (Scopus ID)
Note

QC 20150813. Updated from submitted to published.

Available from: 2015-02-27 Created: 2015-02-27 Last updated: 2017-12-04Bibliographically approved
4. Wrinkling of cylindrical membranes with non-uniform thickness
Open this publication in new window or tab >>Wrinkling of cylindrical membranes with non-uniform thickness
2015 (English)In: European journal of mechanics. A, Solids, ISSN 0997-7538, E-ISSN 1873-7285, Vol. 54, 1-10 p.Article in journal (Refereed) Published
Abstract [en]

Thin membranes are prone to wrinkling under various loading, geometric and boundary conditions, affecting their functionality. We consider a hyperelastic cylindrical membrane with non-uniform thickness pressurized by internal gas or fluid. When pre-stretched and inflated, the wrinkles are generated in a certain portion of the membrane depending on the loading medium and boundary conditions. The wrinkling is determined through a criterion based on kinematic conditions obtained from non-negativity of Cauchy principal stresses. The equilibrium solution of a wrinkled membrane is obtained by a specified combination of standard and relaxed strain energy function. The governing equations are discretized by a finite difference approach and a Newton-Raphson method is used to obtain the solution. An interesting relationship between stretch induced softening/stiffening with the wrinkling phenomenon has been discovered. The effects of pre-stretch, inflating medium, thickness variations and boundary conditions on the wrinkling patterns are clearly delineated. (C) 2015 Elsevier Masson SAS. All rights reserved.

Keyword
Wrinkling, Pressure loadings, Relaxed strain energy
National Category
Applied Mechanics
Identifiers
urn:nbn:se:kth:diva-174910 (URN)10.1016/j.euromechsol.2015.05.015 (DOI)000361581400001 ()2-s2.0-84947943118 (Scopus ID)
Note

QC 20151028

Available from: 2015-10-28 Created: 2015-10-09 Last updated: 2017-12-01Bibliographically approved
5. Instabilities of wrinkled membranes with pressure loadings
Open this publication in new window or tab >>Instabilities of wrinkled membranes with pressure loadings
2016 (English)In: Journal of the mechanics and physics of solids, ISSN 0022-5096, E-ISSN 1873-4782, Vol. 94, 298-315 p.Article in journal (Refereed) Published
Abstract [en]

Wrinkling can affect the functionality of thin membranes subjected to various loadings or boundary conditions. The concept of relaxed strain energy was studied for isotropic, hyperelastic, axisymmetric membranes pressurized by gas or fluid. Non-intuitive instabilities were observed when axisymmetric wrinkled membranes were perturbed with angle dependent displacement fields. A linearized theory showed that static equilibrium states of pressurized membranes, modelled by a relaxed strain energy formulation, are unstable, when the wrinkled surface is subjected to pressure loadings. The theory is extended to the non-axisymmetric membranes and it is shown that these instabilities are local phenomena. Simulations for the pressurized cylindrical membranes with non-uniform thickness and hemispherical membranes support the claims in both theoretical and numerical contexts including finite element simulations.

Place, publisher, year, edition, pages
Elsevier, 2016
Keyword
Wrinkling; Relaxed strain energy; Instability; Wave number; Tension field theory
National Category
Applied Mechanics
Research subject
Engineering Mechanics
Identifiers
urn:nbn:se:kth:diva-187039 (URN)10.1016/j.jmps.2016.05.014 (DOI)000382342300017 ()2-s2.0-84973340702 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20160518

Available from: 2016-05-16 Created: 2016-05-16 Last updated: 2017-11-30Bibliographically approved

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