Marianna Ansiliero de Oliveira Coelho Analysis of ... DEPARTAMENTO DE ENGENHARIA CIVIL Programa de...

Marianna Ansiliero de Oliveira Coelho

Analys is of pneumatic s tructures consideringnon linear material models and p ressure–volume

coup ling

TESE DE DOUTORADO

DEPARTAMENTO DE ENGENHARIA CIVILPrograma de Pós-Graduação em

Engenharia Civil

Rio de JaneiroJuly 2012

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PUC-Rio - Certificação Digital Nº 0721425/CA

Marianna Ansili ero de Oliveira Coelho

Analys is of pneumatic structures considering non linearmaterial models and p ressure–volume coup ling

TESE DE DOUTORADO

Thesis presented to the Programa de Pós-Graduação em Engenharia Civil of the Departa-mento de Engenharia Civil, PUC-Rio as partial ful-fillment of the requirements for the degree of Doutorem Engenharia Civil.

Advisor: Prof. Deane de Mesquita RoehlCo–Advisor: Prof. Kai-Uwe Bletzinger

Rio de JaneiroJuly 2012

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Analys is of pneumatic structures considering non linearmaterial models and p ressure–volume coup ling

TESE DE DOUTORADO

Thesis presented to the Programa de Pós-Graduação em Engenharia Civil of the Departa-mento de Engenharia Civil, PUC-Rio as partial ful-fillment of the requirements for the degree of Doutorem Engenharia Civil.

Prof. Deane de Mesquita RoehlAdvisor

Departamento de Engenharia Civil — PUC–Rio

Prof. Kai-Uwe BletzingerCo-Advisor

Lehrstuhl für Statik — Technishe Universität München

Prof. Ruy Marcelo de Oliveira Paulett iDepartamento de Engenharia de Estruturas e Fundações — USP

Prof. Lu iz Eloy VazDepartamento de Engenharia Civil — UFF

Prof. Paulo Batista GonçalvesDepartamento de Engenharia Civil — PUC–Rio

Prof. Raul Rosas e SilvaDepartamento de Engenharia Civil — PUC–Rio

Prof. José Eugenio LealCoordinator of the Centro Técnico Científico — PUC–Rio

Rio de Janeiro, 6th July 2012

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All rights reserved.


The author is graduated in Civil Engineering from UniversidadeEstadual do Oeste do Paraná – UNIOESTE in 2005, she obtainedthe degree of Mestre in Civil Engineering at PUC-Rio in 2007.

Bibliographic data

Coelho, Marianna Ansiliero de Oliveira

Analysis of pneumatic structures considering nonlinear material

models and pressure–volume coupling/ Marianna Ansiliero de

Oliveira Coelho; advisor: Deane de Mesquita Roehl; co–advisor:

Kai-Uwe Bletzinger – 2012.

142 f. il. (color.); 30 cm

Tese (Doutorado em Engenharia Civil) – Pontifícia UniversidadeCatólica do Rio de Janeiro, Departamento de Engenharia Civil,2012. Inclui bibliografia.

1. Engenharia Civil - Tese. 2. Estruturas pneumáticas. 3. Modelosde material. 4. Método dos elementos finitos. 5. Acoplamentopressão-volume. 6. Grandes deformações. 7. Material NURBS.I. Roehl, Deane de Mesquita. II. Kai-Uwe Bletzinger. III. PontifíciaUniversidade Catôlica do Rio de Janeiro.Departamento de Engenharia Civil. IV. Título.

CDD: 624

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Acknowledgements

I would like to thank my mother Lourdes, my father Eduardo, my siblings

Anna Carolina, Edson and Daniele to be always on my side giving me support,

tendernessand love.

I especially thank my husband Alvaro, for patience, care, support, love and

for accepting the challenge to live in Germany.

Furthermore I would like to acknowledge my advisor Deane for the dedica-

tion, incentive, patience, friendship during the development of my thesis and spe-

cially thesupport for me to dothesandwich Ph.D. in Germany.

I also would like to expressmy gratitude to my co-advisor Herr Bletzinger

for the support and wonderful reception in the Lehrstuhl für Statik at Technische

Universität München.

I would liketo expressmy acknowledgement to theprofessorsandemployees

of the Department of Civil Engineering of the Pontifícia Universidade Católicado

Rio de janeiro for thesupport and help.

I also thank the colleagues of the Lehrstuhl für Statik for the pleasant living

and help, and the friendshipsmadeduringmy stay in Germany.

Sincere thanks to my friends in Brazil that however distant they were, they

gavemestrength and support.

Finally I would like to thank DAAD, CNPq and CAPES for the financial

support and specially DAAD for thewelcome in Germany.

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Abstract

Coelho, Marianna Ansili ero de Oliveira; Roehl, Deane de Mesquita; Blet-zinger, Kai-Uwe. Analysis of pneumatic structures considering nonlinearmaterial modelsand pressure–volume coupling. TesedeDoutorado— De-partamento de Engenharia Civil , Pontifícia Universidade Catôlicado Rio deJaneiro.

In thiswork astudy of pneumatic structuresconsidering pressure–volume coupling

under plastic and viscoplastic material behavior is developed. Pneumatic structures

are membrane structures acted on by air or gases stabili zed by tension. These

structures are lighter than conventional structures resulting in economic structural

solutions. They present also some characteristics that contribute to the sustainable

development, such as the utili zation of natural li ghting and ventilation and its

possibilit y of reuse. When pneumatic structures are subjected to external loads

these structures present both internal pressure and volume variation. This coupling

is one of the objects of the present work. Analytical solutions are developed to

describe this coupling. In conventional finite element systems this coupling is not

considered. A formulation for pressure–volume coupling by closed chambers is

included in the framework of a finite element large strain model. The variety of

material models implemented has the purpose to cover the behavior of the many

kinds of membrane materials used in pneumatic structures. In the literature the

study of the membrane materials for pneumatic structures focuses on experimental

analysis. Membrane material models are incorporated in the finite element model

for small andlargestrains. The constitutivematerial modelsconsidered in thiswork

are hyperelastic, elastoplastic and elastoviscoplastic. The onset of large strains is

enclosed. A new material model based onNURBSsurfacesisproposed an validated

on hand of experimental results and classic material models. In thiswork emphasis

is given to the material ETFE (Ethylene tetrafluoroethylene), which is widely

used in pneumatic structures. The models developed here, such as the pressure–

volume coupling and the material models, are implemented in finite elements on

the program used in the Static Chair at TUM (Technische Universität München),

which iscalled CARAT++ (Computer Aided Research AnalysisTool).

KeywordsPneumatic structures; Material models; Finite element method; Pressure–

volume coupling; Largestrains; NURBS material

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Resumo

Coelho, Marianna Ansili ero de Oliveira; Roehl, Deane de Mesquita; Blet-zinger, Kai-Uwe. Análise de estruturas pneumáticas considerando mode-los não lineares do material e o acoplamento pressão–volume. Tese deDoutorado — Departamento de Engenharia Civil , Pontifícia UniversidadeCatôlicadoRio deJaneiro.

Neste trabalho um estudo de estruturas pneumáticas considerando acoplamento

pressão–volume emodelos constitutivos plásticos e viscoplásticos são desenvolvi-

dos. Estruturas pneumáticas são estruturas de membrana sobre as quais atuam

pressão degasesestabili zadaspor tensõesdetração. Essasestruturas são mais leves

que estruturas convencionais resultando em soluções mais econômicas. Elas pos-

suem ainda algumas características que contribuem para um desenvolvimento sus-

tentável, como a utili zação de luz natural e ventilação e apossibili dade de reuti-

li zação. Quandoas estruturas pneumáticas são submetidas a cargas externas, essas

estruturasapresentam variação dapressão internal edo volume. Este acoplamento é

um dos objetos de estudo do presente trabalho. Soluções analíti cas são desenvolvi-

das para descrever este acoplamento. Em programas convencionais de elementos

finitos esse acoplamento não é considerado. Uma formulação para o acoplamento

pressão–volume para câmaras fechadas é incluído no modelo de elementos fini-

tos com grandes deformações. A variedade de modelos de material implementa-

dos tem a finalidade de abranger o comportamento de muitos tipos de materiais

de membrana usados em estruturas pneumáticas. Na literatura o estudo dos mate-

riais de membrana para estruturas pneumáticas tem foco na análise experimental.

Modelos para material de membrana são incorporados no modelo de elementos

finitos para pequenas e grandes deformações. Os modelos constitutivos consider-

ados neste trabalho são hiperelástico, elastoplástico e elastoviscoplástico. A ocor-

rência de grandes deformações é incluída. Um novomaterial baseado em superfí-

cies NURBS é proposto e validadocom base em resultados experimentaise mode-

los clássicos de materiais. Neste trabalho ênfase édada ao material ETFE (Etileno

tetrafluoretileno), o qual é amplamente usado em estruturas pneumáticas. Os mod-

elos desenvolvidos aqui, como o acoplamento pressão–volume e os modelos de

materiais são implementados em elementos finitos no programa usado na cadeira

de estática das construções da TUM (Technische Universität München), chamado

CARAT++ (Computer Aided Research AnalysisTool).

Palavras–chaveEstruturas pneumáticas; Modelo de material; Método dos elementos finitos;

Acoplamento pressão-volume; Grandes deformações; Material NURBS

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Contents

1 Introduction 15

1.1 Membrane structures 15

1.2 Pneumatic structures 16

1.3 Formfinding 24

1.4 Cutting patterns 26

1.5 Wrinkling in membranes 27

1.6 Objective 28

1.7 Thesis outline 29

2 Mechanics of membranes 30

2.1 Kinematics 30

2.2 Strain measure 31

2.3 Stress measure 33

2.4 Membrane formulation 33

3 Membrane Material Models 43

3.1 Small strains — Elastoplasticity 43

3.2 Small strains — Elastoviscoplasticity 51

3.3 Large strains — Hyperelasticity 56

3.4 Large strains — Elastoplasticity 60

3.5 Large strains — Elastoviscoplasticity 65

4 Material model based on NURBS 66

4.1 Nonuniform rational B-Spline curves and surfaces 67

4.2 Linear elastic material model based on NURBS (LE–NURBS) 75

4.3 Material model based on NURBS for principal directions (PD–NURBS) 76

4.4 Data fitting 79

4.5 Validation examples 80

5 Pressure-Volume Coupling 88

5.1 Numerical analysis model for one chamber 88

5.2 Multichambers structures 93

5.3 Analytical solution for a circular inflated membrane 93

5.4 Comparison of analytical and numerical analysis 103

6 Examples of pneumatic structures and material models for membranes 108

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6.1 ETFE–Foils 108

6.2 Uniaxial and biaxial test by ETFE–foils 114

6.3 ETFE-Foil modeled with PD-NURBS 116

6.4 Burst test 119

6.5 Air cushion with single and double chamber 123

6.6 Lyon confluence cushion c©seele 127

7 Conclusions and Suggestions for future works 134

7.1 Membrane material models 134

7.2 Pneumatic structures with pressure–volume coupling 135

7.3 Suggestions for future works 136

8 References 142

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List of Figures

Figure 1.1 - Pneumatic structures in man’s body: (a) Red blood cells, (b)

lung 17

Figure 1.2 - Calceolaria - Inflated flower 18

Figure 1.3 - Inflated cushions (a) 3-D overview of irregular shaped ETFE

cushions used in facade assembly (source: Watts [10]), (b)

Testing of full-scale mock-ups (source: LeCuyer [11]), and

(c) Rigid edge detail (source: Watts [10]) 20

Figure 1.4 - Distant Early Warning (DEW) line (source: Canadian mili-

tary journal [13]) 21

Figure 1.5 - Allianz Arena in Munich 22

Figure 1.6 - Distribution of pneumatic structures with inflatable cushions

in terms of continent and country (source: Moritz [15]) 23

Figure 1.7 - Cutting patterns of six-point tent (source: Linhard [31]) 26

Figure 1.8 - Building process of six-point tent (source: Linhard [31]) 27

Figure 1.9 - Influence of pattern definition on membrane structures

(source: Linhard [31]) 27

Figure 1.10 - Principle states of membranes: (a) reference, (b) taut, (c)

and (d) wrinkle, and (e) slack (source: Jarasjarungkiat et al.

[36]) 28

Figure 1.11 - Wrinkled membrane (source: Wong and Pellegrino [37]) 28

Figure 2.1 - Successive deformations of a continuous body 30

Figure 2.2 - Multiplicative decomposition of the deformation gradient

(source: Souza Neto et al. [40]) 32

Figure 2.3 - Membrane coordinates 34

Figure 2.4 - Triangular elements: (a) linear and (b) quadratic 39

Figure 2.5 - Quadrilateral elements: (a) linear and (b) quadratic 40

Figure 3.1 - Plane stress state (source: Souza Neto et al. [40]) 45

Figure 3.2 - Experimental data from uniaxial and biaxial test of ETFE

and adjusted von Mises yield curve 45

Figure 3.3 - General return mapping schemes. Geometric interpretation:

(a) hardening plasticity and (b) perfect plasticity (source:

Souza Neto et al.[40]) 48

Figure 3.4 - Mesh, geometry and boundary conditions of a perforated

rectangular membrane 51

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Figure 3.5 - Load versus edge displacement 51

Figure 3.6 - Phenomenological aspects: uniaxial tensile tests at high

temperature (a) Strain rate dependence, (b) Creep, and (c)

Relaxation (source: Souza Neto et al. [40]) 52

Figure 3.7 - Force versus displacement curve of a perforated rectangu-

lar membrane: (a) ǫ = 1.0 and (b) ǫ = 0.1. 56

Figure 3.8 - Square sheet with a circular hole (a) undeformed sheet

mesh with applied load (b) diplacement result in y direction

with deformed sheet in a scale of 1:1. 59

Figure 3.9 - Load–displacement curves of stretching of a square sheet 59

Figure 3.10 - Results of the square sheet with a circular hole: (a) normal

stress in x, (b) normal stress in y, (c) shear stress, (d) normal

strain in x, (e) normal strain in y, and (f) shear strain 60

Figure 3.11 - Force versus displacement on the free edge of a perforated

rectangular membrane 65

Figure 3.12 - Stress versus strain for numerical analysis with large and

small strains 65

Figure 4.1 - Example of a B-spline surface (source: Piegl and Tiller [73]) 70

Figure 4.2 - Geometry construction of a NURBS curve (source: Piegl

and Tiller [73]) 72

Figure 4.3 - NURBS surface: (a) Control points net (b) biquadratic

NURBS surface (source: Piegl and Tiller [72]) 73

Figure 4.4 - NURBS surfaces for stresses and strains for LE–NURBS

material: (a) stresses in direction 11 and strains in direc-

tions 11 and 22, (b) stresses in direction 22 and strains in

directions 11 and 22 and, (c) NURBS curve for stresses in

direction 12 and strains in direction 12. 75

Figure 4.5 - Mesh, boundary conditions and applied load for the quadri-

lateral example 81

Figure 4.6 - NURBS surfaces of stresses and strains in principal direc-

tions for NeoHookean material: (a) stresses in direction 1

and (b) stresses in direction 2. 81

Figure 4.7 - Displacement results in y direction: (a) undeformed mem-

brane, (b) conventional material model, and (c) PD-NURBS

material model. 82

Figure 4.8 - NURBS surfaces with stresses and strains in principal direc-

tions for the Mooney-Rivlin material: (a) stresses in direction

1, and (b) stresses in direction 2. 83

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Figure 4.9 - NURBS surfaces for stresses and strains in principal direc-

tions for elastoplastic material: (a) stresses in direction 1,

and (b) stresses in direction 2. 86

Figure 4.10 - Displacements in y direction: (a) conventional material

model and (b) PD-NURBS material model. 87

Figure 4.11 - Stresses in y direction: (a) conventional material model and

(b) PD-NURBS material model. 87

Figure 5.1 - Surface under pressure loading. 89

Figure 5.2 - Radial and circumferential coordinates, vertical deflection,

and radial displacement of a circular membrane 94

Figure 5.3 - Mesh for a circular inflated membrane. 104

Figure 5.4 - Comparison between a mesh with linear and quadratic el-

ements for applied external pressure values of 150kPa and

300kPa. 104

Figure 5.5 - Comparison between Hencky’s and Fichter’s solution for

applied external pressure values of 150kPa and 300kPa. 105

Figure 5.6 - Fichter’s solution and numerical results without pretension

and κ = 0 for applied external pressures values of 150kPa

and 300kPa. 105

Figure 5.7 - Comparison between the numerical solution with a preten-

sion of 1kPa for κ = 0 and κ = 1 for applied external pressure

values of 150kPa and 300kPa. 106

Figure 5.8 - Analytical and numerical solution with a pretension of 1kPa

and κ = 1 for an applied external pressure values of 150kPa

and 300kPa: (a) deformed configuration and (b) pressure

volume curve. 106

Figure 5.9 - Analytical and numerical solution with a pretension of 10kPa

and κ = 1 for the applied external pressure values of 150kPa

and 300kPa: (a) deformed configuration and (b) pressure

volume curve. 107

Figure 5.10 - Analytical and numerical large strains solution without pre-

tension and κ = 1 for applied external pressure values of

150kPa and 300kPa: (a) deformed configuration and (b)

pressure volume curve. 107

Figure 6.1 - Etylene Tetrafluoroetylene chemical structure 108

Figure 6.2 - Eden Project in the United Kingdom 110

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Figure 6.3 - Stress–strain curve of semi–crystalline thermoplastic mate-

rial with schematic representation of the tensile specimen in

different steps (source: Ehrenstein [83]) 110

Figure 6.4 - Stress–strain curve: (a) tensile stress vs. strain and (b)

compressive stress vs. strain (source Properties Handbook

of Tefzel R©[54]) 111

Figure 6.5 - Stress–strain curves for cyclic test: (a) -25C, (b) 0C, (c)

+23C and (d) +35C (source: Moritz [15]) 111

Figure 6.6 - Yield stress and strain versus temperature performed by

Moritz [15] 112

Figure 6.7 - Test curves from DuPONTT M [54]: (a) tensile strength vs.

temperature and (b) ultimate elongation vs. temperature 112

Figure 6.8 - Creep test in DuPONTT M Tefzel 200Flexural [54] 113

Figure 6.9 - Poisson ratio versus stress for different values of tempera-

ture (source: Moritz [15]) 113

Figure 6.10 - Mesh, geometry and boundary conditions for the biaxial test 114

Figure 6.11 - Stress versus strain for small and large strains 115

Figure 6.12 - Stress versus strain for experimental results and numerical

results with small and large strains for the biaxial loading in

the proportion of 1:1 116

Figure 6.13 - Stress versus strain for experimental results and numerical

results with small and large strains for the biaxial loading in

the proportion of 2:1 116

Figure 6.14 - NURBS surface with experimental data 117

Figure 6.15 - NURBS surfaces of stress and strain in principal directions

for von Mises material: (a) stresses in direction 1 and (b)

stresses in direction 2. 118

Figure 6.16 - (a) Burst test and (b) deformation process (source: Schie-

mann [84]) 119

Figure 6.17 - Geometry, mesh and boundary conditions for the burst test

performed by Schiemann 120

Figure 6.18 - Pressure versus displacement results for the specimen V28

[84]; linear (T3) and quadratic (T6) triangular membrane

elements. 120


[84]; step length of 60 and 100. 121


[84]; large strain, and small strain material models. 122

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Figure 6.21 - Deformed configuration of the specimen V28 [84] and nu-

merical model with large strains for pressure states 1 and 2.

122

Figure 6.22 - Stress versus strain curve in y direction 123

Figure 6.23 - Deformed inflated circular membrane with the out of plane

displacement: (a) point 1 and (b) point 2 123

Figure 6.24 - Undeformed cushions: (a) upper and lower membranes of

single chamber cushion and (b) upper, middle and lower

membranes of double chamber cushion 124

Figure 6.25 - Cushion dimensions and formfinding shape 124

Figure 6.26 - Single chamber cushion deformation under external load 125

Figure 6.27 - Volume versus internal pressure for the single chamber

structure 125

Figure 6.28 - Two chambers deformation under external load 126

Figure 6.29 - Volume versus internal pressure for two chambers 127

Figure 6.30 - out of plane displacement versus load 127

Figure 6.31 - Lyon confluence cushion structure: (a) top view and (b)

bottom view 128

Figure 6.32 - Geometry of the triangular cushion 129

Figure 6.33 - Mesh of the cushion structure: (a) and (c) without cutting

patterns (b) and (d) with cutting patterns. 129

Figure 6.34 - Flat patterns of the triangular cushion. 130

Figure 6.35 - Von Mises stress distribution on the cushion structure with

pressure–volume coupling: (a) without cutting patterns, (b)

with cutting patterns. 130

Figure 6.36 - Strain in principal directions 1 on the cushion structure with



Figure 6.37 - Strain in principal directions 2 on the cushion structure with



Figure 6.38 - Stress versus strain for triangular cushion with PD–NURBS

material. 132

Figure 6.39 - Stress versus strain for triangular cushion with elastoplastic

material with small strains. 132

Figure 6.40 - Internal pressure versus volume for the triangular cushion. 133

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List of Tables

Table 1.1 - Membrane materials used in pneumatic structures. (source

Gómez-González et al. [12]) 24

Table 4.1 - Material properties of quadrilateral membrane example 80

Table 4.2 - Maximum error of the PD-NURBS for rectangular membrane 82

Table 4.3 - Displacement residuum for 15x15 to 100x100 control point

net. 84

Table 4.4 - Maximum error of PD-NURBS material with surfaces gener-

ated by control point nets 20x20 to 100x100 for the square

perforated example 85

Table 4.5 - Material properties of the perforated membrane example 85

Table 4.6 - Maximum error of the PD-NURBS for perforated membrane 87

Table 6.1 - Material properties of ETFE–foils 114

Table 6.2 - Relative error of biaxial test for the PD–NURBS material 118

Table 6.3 - Material properties of specimen V28 121

Table 6.4 - Global convergence of the displacement residuum at the crit-

ical pressure for step length values of 60 and 100. 122

Table 6.5 - Material properties of the ETFE–foil 129

Table 6.6 - Maximum result values for the triangular cushion 131

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1Introdu ction

Pneumatic structures are membrane structures that are stabili zed by tension

due to the applied internal pressures. These structures exhibit interesting charac-

teristics from a structural viewpoint, such as a reduced self-weight, which results

in structures that are globally lighter and more economical than conventional ones.

Pneumatic structures also have some characteristics that contribute to sustainable

development, such as their reusabilit y and the use of natural li ghting and ventila-

tion.

Although pneumatic structureshavebeen in use for thepast 30 years, there is

much to bedevelopedandresearched particularly regardingthe employedmaterials,

becausenew materialsarebeing produced for developing pneumatic structureswith

better strength and durabilit y.

1.1Membrane structures

Unlike conventional structures, membrane structures are used as shelter con-

structionsbecauseof oneof their important characteristics: their self–weight.

Otto[1] defines a membrane as a flexible skin stretched in such a way that it

is subjected to tension. Mixed structures that combinetractioncableswith elements

working under compression or bending-compression are also lightweight structure

solutions.

Simple membrane structures such as tents were used as shelter about 40,000

years ago and then were used for transitory activities such as milit ary campaigns

and circus presentations. Because of their easy assembly, tents is also an option for

shelter inemergency situationsdueto natural hazards, suchasfloodsand hurricanes,

or even in war.

Nowadays, aspects related to environment preservation and sustainable de-

velopment are being determined by thedesign of engineeringsolutions. Membrane

structures have some characteristics that contribute to sustainable development,

such as the use of natural li ght and ventilation and its possibilit y of reuse. It is

also shown that this typeof structure enablesarchitectonic flexibilit y andthesearch

for better structural efficiency.

Wakefield [2] hasreported that thedesign of lightweight membranestructures

requires a special approach. As opposed to wood, concrete, and iron structures,

in the case of membrane structures, loads are transfered to the supports by the

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Introduction 16

forces in the structure membrane. Wakefield [2] has reported that these structures

undergosignificant displacementswhen loaded. Therefore, the analysisof this type

of structuremust consider theonset of largedisplacementsandtherefore anonlinear

structural response.

Lightweight structuresare regarded byBletzinger [3] as theproper structures

for optimal material use under extreme load or pretension conditions, i. e., for

maximizing thematerial efficiency under the imposed constraints.

1.2Pneumatic s tructures

According to Dent [4] pneumatic structures refer to structuresacted on byair

or gasand relateparticularly to architecture and construction.

Marcipar et al. [5] defines that pneumatic structures are composed by an

exterior flexiblemembranethat containsafluid inside(in general air or helium). The

function of theinterior fluid isto maintain the exterior membraneunder tension. The

final shapeof the inflatablestructure and its structural resistancedependstrongly of

the deformation of the external membrane, the loads, and the pattern design. The

stiffness of the membrane is directly related to the pressure of the air contained

inside thestructure and the internal volume.

This type of structure has some advantages: with larger volumes and higher

pressures greater spans can be achieved. Furthermore, the pneumatic structures

can be erected or dismantled quickly, are light, portable and reduced material

use. It therefore offers a possible solution to a wide range of problems, both of

social and commercial kinds. For instance, pneumatic constructions can be used

to overcome temporary shortages of warehousing space. It can also be used to

provide shelter for the homeless in times of natural or man-made disaster, and

in these early days of space exploration, it has even been suggested for lunar

shelters. But of more importance than these applications demonstrate, is the fact

that pneumatic construction pointstheway to an architectural revolution. To correct

the environmental deficienciesof rigid traditional structural envelopes, energy must

be supplied to heat and ventilate them, bringing them up to the comfort standards

that are determined by the building’s function; the amount of this applied energy

depends on the insulation characteristics of the structural envelope. The properties

and the different shapes of pneumatic structures are described in the work of

Herzog [6]. Parameters to determine the final shape of these structures are: type

of loading, magnitude of internal pressure, type of boundary conditions, formation

of the membrane, number of membranes, type of utili zation, type of membrane

material, surface curvature, etc.

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Introduction 17

Pneumatic structures are classified on the basis of the pressure applied, as

high– or low–pressure pneumatic structures. A sub-category of pneumatic struc-

tures is defined on the basis of their construction operation: air–controlled con-

struction, air–stabili zed construction, and inflated cushions. In civil engineering,

air–stabili zed constructionsandinflated cushionsare employed. Air–stabili zed con-

structions, as the name suggests, are membrane structures supported by pressure

differentials. Inflated cushions are closed membrane structures having an internal

pressure.

1.2.1Air–stabili sed construction

This is a thin flexible membrane which is supported solely by pressure

differentials. Thesedifferencesin pressureinducetensilestressesinto themembrane

(Dent [4]).

The air–supported structure is made up of four elements according to Dent

[4]: the structural membrane, the means of supporting this membrane, the means

of anchoring it to the ground, and the means of access in and out of the building

structure. The membrane structure is fabricated using fabrics or foils and it is

supported byapressuredifferential maintained bya constant supply of air provided

by simple low pressure fans. The membrane is generally clamped firmly to a

concrete foundation. Air locksare necessary for ease of accessagainst thepressure

differential.

(a) (b)

Figure 1.1: Pneumatic structuresin man’s body: (a) Red bloodcells, (b) lung

Pneumatic structurescan be foundin nature likeflexiblemembranescontain-

ing fluids under pressure. Some examples in human bodyare the red blood cells

(figure 1.1(a)) and the lung(figure 1.1(b)). Another example foundin nature is the

flower calceolaria that is aflower with inflated petalsand it is shown in figure 1.2.

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Introduction 18

Figure 1.2: Calceolaria - Inflated flower

Animal skin was used for water storage and for construction of tents as

shelters. The first attempt to use air pressure in membranes was probably the sail .

Dueto wind, thedifferences in pressure cause inflation of thesail providingamean

of propulsion.

A more recent pneumatic structure development is the balloon, which was

created in 1709 bythe brazili an Bartolomeu Lourenço de Gusmão (Visoni and

Canalle [7]). Dent [4] reports the use of the balloon as air transportation system

in eighteenth century. In 1783the Montgolfier brothers inflated, with hot air a 10

m diameter sphere made of paper and linen, and they observed this sphere rise to a

considerable height before it descended. At the same time as these experiments of

the Montgolfiers, Jean Baptiste Meusnier was suggesting a design for a dirigible

non-rigid airship, which was even more revolutionary. His designs were for a

cigar shaped structure with an inner bag, containing hydrogen as the li fting agent,

surrounded by an outer envelope containingair at a higher pressure than that of the

atmosphere.

According to Dent [4] in one field pneumatic construction has established

itself as the best solution to a particular problem, that of providing motor vehicles

with a smoother ride. Its main advantages over the solid tire are twofold, firstly,

its superior abilit y to absorb road shocksthroughconsiderably greater deformation,

and secondly its unrivaled handling characteristics due to the fact that a greater

surface areaof tire is in contact with the road surface.

Theway that air pressure isused to prestressthemembranedistinguishestwo

typesof pneumatic structures: low–pressure and high–pressure.

DBD


Introduction 19

1.2.2High–pressure infl atable structures

This type of inflatable structure is inflated with high pressure, usually higher

than 1.0kN/m2. According to Kröplin [8] the high–pressure inflatable structures

require the used of reinforced membranematerials. Motro [9] reports that the low–

pressurepneumatic structuresoccur when thewholefunctional spaceispressurized

to the extent required to balance the external applied load. The full structure size

is active and hence structural efficiency is extremely high. Because a substantial

upli ft acts on the membrane, it has to be anchored to the ground or weighted down

along the boundary. Additional architectural drawbacks of this system stem from

the need for the enclosed spaceto be essentially sealed and for air to be pumped

continuously, thus limitingarchitectural flexibilit y and rangeof applications.

Marcipar et al. [5] reported that the use of inflated elements with high–

pressure has often been proposed, but they have rarely been built . One example

istheFuji -Pavill onin Japan. Thereasonsare that thenecessary materials, structural

design and manufacturing techniques have not yet been fully developed. There

are also some disadvantages, such as joints design and execution and their big

vulnerabilit y to air losses. In general, high–pressure inflated structures are difficult

to maintain andrepair and have ahighcost.

1.2.3Low–pressure infl atable structures

Low–pressure inflated structures are the most common type used by civil

engineeringconstructions. Some advantagesof this typeof structuresaredescribed

by Marcipar et al. [5]. Inflatablestructuresformed byan assembly of self–supported

low pressure membrane elements are ideal to cover large space areas. They also

adapt easily to any designshape and haveminimal maintenancerequirements, other

than keeping a constant low internal pressure to account for the air losses through

thematerial poresand theseams.

Kröplin [8] reports that in the case of low pressure, about 0.5kN/m2, an open

wall can beused, whereby thepressure ispermanently imposed byablower, which

iscapable to erect thestructure.

1.2.4Infl ated cushions

Inflated cushions are composed by two or more membranes closed and pres-

surized, with no accessible interior. Rigid elements at the edges or compression

DBD


Introduction 20

rings are required in this structure to give form and to close the envelopes. An ex-

ampleof this rigid elements are shown in figure 1.3(c). Figure1.3(a) isan example

of an ETFE cushions façade and a testing of full -scale mock-up is presented in

Figure1.3(b).

(a) (b) (c)

Figure 1.3: Inflated cushions (a) 3-D overview of irregular shaped ETFE cushions used infacade assembly (source: Watts [10]), (b) Testing of full -scale mock-ups (source: LeCuyer[11]), and (c) Rigid edge detail (source: Watts [10])

Gómez-González et al. [12] reported that althoughthis type of structure first

experienced an important development in the sixties and seventies, it i s in the last

decade that the inflatable system have improved most, allowing new sustainable

strategies in climatic adaptive envelopes.

Theprobablereasonfor theincreasingmembranesuse in constructionsisdue

to the improvement of new materialswith higher resistance and durabilit y.

The study of Gómez-González et al. [12] also shows the geographical situa-

tion of these structures. It is concluded that 40% of the studied projects have been

made in Germany or United Kingdom, where these systems have been more ac-

curately researched and manufactured. Also, the climatic conditionsof these areas,

withsolar radiation gainsin thewinter andmild temperaturesin thesummer, benefit

the application of these systems. However, in the last decade the use of this tech-

nology has been developed in other zones with more extreme summers, li ke south

Europeor someregions in Asia.

Marcipar et al. [5] reported that these structures are used instead of glass

elements due to their lower price, which is 1/3 of a glass covering. So far there

are only two German companies dealing with these inflated cushions in the world

market.

DBD


Introduction 21

1.2.5Pneumatic constructions

The first known architectural attempt to apply the balloon principle to earth-

boundstructures was projected by the English engineer, Frederick Willi am Lanch-

ester (Dent [4]). In his patent of 1917for a field hospital, the basic principles of

air supported construction for buildings were realized. This patent, clearly derived

from balloon and airship construction, is remarkable on two accounts: firstly, he

appears to be fully awareof all thebasic implicationsof buildings supported byair,

and secondly, although his patent concerns a field hospital, he mentions the poten-

tial of air supported buildings for huge spans such as those encountered in air craft

hangarsand sportsarena.

The pneumatic camping structure appeared before the Second World War

consisting of a waterproof membrane stretched between a pair of intersection

air inflated ribs, exempli fying a very feasible form of construction for portable

buildings(Dent [4]).

Thefirst air supported buildingrose in 1946in theU.S.A. to shelter antennae

from the severe climatic conditions: the Distant Early Warning (DEW) line, a

"fence" of radars in the Arctic that would guard against Soviet bomber attacks. It

continues to thisday as theNorth WarningSystem, presented by thenewer radome

on the left in Figure1.4.

Figure 1.4: Distant Early Warning (DEW) line (source: Canadian milit ary journal [13])

According to Dent [4] of prime importance to the whole project for the

radomesweretheLaboratory windtunnel tests, which analyzed thestressesinduced

in the membrane by wind loads. In association with this research, membrane

materials were developed which were able to withstand severe exposure. These

consisted of strongman-madefibers, such asnylon or terylene, which were covered

withasynthetic coating of vinyl, neopreneor hypalon. AsLanchester had predicted,

DBD


Introduction 22

a pressure differential of only 70mm of water pressure was all that was required to

maintain the rigidity of these 15 m diameter radomes in winds of up to 240 km/h.

By the mid 1950’s, the successful performance of these radomes in the extreme

climateof thenorthern frontier of Americahad proved thepracticality of pneumatic

structures.

In the last years the pneumatic structures has improved in conjunction with

the membrane materials and it has been used for temporary or permanent use. An

exampleof temporary use is a low pressure inflatablestructure for conferences and

an example of permanent use is a stadium with inflated cushions as Alli anz Arena

(Figure 1.5 ).

Figure 1.5: Alli anz Arena in Munich

In Brazil , some companiesconstruct pneumatic structuresthat aremainly air–

stabili zed constructions. The material used in this type of construction is generally

fabric. Inflated cushions have several advantages and are the most widely devel-

oped type of pneumatic structure in the last 10 years. Despite the technological

developments and advantages of this type of structure, inflated cushions can be

foundmainly in Europe, and only a few constructions are found outside Europe.

One example is the water cube in Beiji ng, which was constructed for theOlympics

in 2008.

The study of Majorana et al. [14] investigated the numerical and physical

models developed for the design of a membrane roof for the Baptist Church of

Fortaleza as well as the fabrication and construction of the actual membrane; the

results of the models were compared with those of the real structure. The roof

area amounted to about 2,900 m2, which is a national record for flexible border

membranes, andto thebest of the author’sknowledge, this roof is thefirst caseof a

fully computer assisted design processwithin Brazil .

DBD


Introduction 23

The motivation for pursuing this topic goes beyond the advantages of

lightweight structures, which are mentioned earlier. The lack of inflated cushions

in Brazil and their limited use in other countriesof theAmerican continent aremo-

tivations for research onthis topic. The global distribution of pneumatic structures

using inflated cushions, particularly usingethylene tetrafluoroethylene(ETFE) ma-

terial, is shown in figure1.6.

!

"#

$%

Figure 1.6: Distribution of pneumatic structureswith inflatable cushions in terms of conti-nent and country (source: Moritz [15])

The use of inflated cushions is mainly concentrated in Europe, amounting to

95.6% followed by Asia, the Americas, and Oceania. Another interesting pieceof

information is that 59% of the constructions with inflated cushions are located in

Germany.

A material recently used in Europe and for the Olympics in China for

pneumatic structures isETFE. Thismaterial exhibitsa complex behavior. Theyield

stressand viscosity parameters show strong dependenceon temperature.

Despite the extensive use of this material, few studies have been conducted

with focuson constitutivemodels that take into account itscomplex behavior.

Several possibiliti es exit for membrane materials, e.g., reinforced fiber with

glassor plastic, a wooden board, a concreteplate, polyvinyl chloride (PVC) coated

with polyester, tefloncoatedwith glassfiber, fabrics, kevlar R©(para-aramid synthetic

fiber), nylon, polytetrafluoretilene(PTFE), and sili con.

The study of nonlinear material behavior has included the material ETFE,

which iswidely used in pneumaticstructures. Poiraziset al. [16] reported that ETFE

has gained popularity mainly because of its daylight transmittance and its potential

DBD


Introduction 24

for energy conservation. According to Gómez-González et al. [12], sincethedevel-

opment of theETFE foil , it hasbeen used inmost of thestudied proposals(84.75%),

mainly in thelast decade. Self-cleaning, durabilit y, and highlight transmission have

enabled its the use in many permanent envelopes, thus breaking the traditional re-

lation between inflatablesystemsand temporary buildings. Other membranes, such

as polysulfone (PES) or fiberglasscoated with PVC, are also used in large cush-

ions, where high membrane resistance is required. Table 1.1 shows its membrane

materialsused in pneumatic structures.

Table 1.1: Membrane materials used in pneumatic structures. (source Gómez-González etal. [12])

Material Percentage Material Percentage

ETFE 84.75% Fiberglass/PVC 1.53%PES/PVC 4.14% Fiberglass/Sili concoated 1.74%

PVC 3.70% Others 4.14%

Pressure–volume couplingisan important factor in theresponseof pneumatic

structures. This coupling is based onthe fact that an enclosed pneumatic structure

has an internal pressure, and when this structure is subjected to external loads,

the volume decreases (increases) and the internal pressure increases (decreases)

correspondingly. The concept of deformation-dependent forces also exists in this

typeof structure. Theformulationadopted throughout this studyrefersto thestudies

of Hassler and Schweizerhof [17], Rumpel and Schweizerhof [18], Rumpel [19],

Bonet et al. [20], and Berry and Yang [21]. Pressure–volume coupling reveals the

observablefeaturethat thepressureof an enclosed fluid providesadditional stiffness

to the inflatable structure, which is analogous to the behavior of a membrane on

elastic springs. Thiscoupling is not considered in the conventional programsof the

finite element method.

1.3Formfind ing

Formfindingisaprocessof optimizationthat results in an optimal form in the

equili brium configuration for agiven initial topologywith fixed prestressloadsand

boundary conditions.

The computational methods of formfinding are divided into three groups:

simulation of hanging models, numerical simulation of soap films and structural

shapeoptimization.

DBD


Introduction 25

Hanging models are based on experimental models. These models were

improved by Isler [22], and they areused to generate the form of arch–freebending

whensubjected only toan axial compressionload. Theobjectiveof hangingmodels,

as defined by Bletzinger [23, 24, 25], is to achieve the transition from a tension

structure to a membrane structure by minimizing the bending part of the strain

energy. The optimal shape generated by using hanging models is the result of

mechanical deformation for one load case. Stabilit y effects cannot be considered

in hangingmodels.

Plateau demonstrated by numerousexperimentsthat every contour of asingle

closed curve bounds at least one soup film (Lewis [26]). According to Otto [27]

soap films can not be subjected to shear. The biaxial stress state of soap films is

defined asaspherical tensor, in analogywith thestress state induced by hydrostatic

pressure, because the absenceof shear stressesgenerates normal stressesequally in

all directions. Dent [4] has reported that a soap bubble is mounted by the surface

tension forces acting on both sides of the soap film. Because of the uniformity of

these forces, the main characteristic of the film is to form shapes with minimal

surface area, in which the walls are stressed equally at every point and in all

directions, with noconcentration of stressat any one point. Stresses are equalized

by liquid flow in the soap film, and therefore, stresspeaks cannot occur under any

circumstances. Therefore, the analysisof soap films isconsidered important for the

design of membranestructures.

Structural shape optimization has been described by Bletzinger [24] as a

more general tool, which the design variables are the coordinates of the model.

Botkin [28] reported that structural (sizing) optimization has been considered to be

theminimization of structural massby varying member sizes and plate thicknesses

of amodel in which thegeometry remains unchanged.

Bonet and Mahaney [29] used algorithms of formfinding and observed that

in the case of membrane structures it is possible to start with an initial geometry

and determine the surfacegeometry subjected to dead load. In a typical processof

formfinding, it i s conventional to start with a flat initial geometry. The boundary

constraints are set for displacements at points on the boundary mesh. After the

boundary constraintsareset, stepsof formfindingarenecessary to obtain aminimal

surface. For each step, the reference configuration is defined as the final shape of

theprevious step.

DBD


Introduction 26

1.4Cutt ing p atterns

Generally, membrane structures have curved shapes. To build such curved

shapes film strip of a certain width are joined. The smaller the width of the strip,

the closer is the obtained curved surface. This building processis analogous to the

processof manufacturing clothes. The pieces of fabrics are cut in order to obtain

curved forms. The available width size of the membrane fabric also defines the

cutting patterns.

Linhard [30] developed an approach called formfinding via cutting patterns

that adjusts the stress state already during the formfinding process in an iterative

procedure, in such a way that the final form is the equili brium shape for stresses

resultingfrom assemblingthe cutting patterns, whilekeepingthedifferencebetween

the actual and desired stress state as small as possible. Linhard [30] developed an

approachcalled formfinding via cutting patterns, whichemploysan iterativeprocess

to adjust the stress state during the formfinding process. This adjustment is such

that the final form is the equili brium shape for stresses resulting from assembling

the cutting patterns, and the differencebetween the actual and desired stress states

iskept to minimum. Figure1.7 showsthe cutting patternsfor asix-point tent; figure

1.8 shows itsbuilding process.

Figure 1.7: Cutting patterns of six-point tent (source: Linhard [31])

The plane strips can be divided using a geodesic line or a cutting line. The

differencebetween geodesic and cutting lines can be seen in figure 1.9. According

to Ishii [32], in the cases of simple curved surfaces and curved surfaces with low

rise, a cutting pattern can be drawn on a strip without using the geodesic lines.

However, in the cases of complex curved surfaces and thosewith high rise, the use

of geodesic lines is more recommended.

For more details of cutting patterns, refer to thestudies of Linhard [30], Ishii

[32], and Bletzinger et al. [33].

DBD


Introduction 27

Figure 1.8: Building processof six-point tent (source: Linhard [31])

Figure 1.9: Influenceof pattern definition onmembrane structures(source: Linhard [31])

1.5Wrinkling in membranes

Wrinkling in membranes is a widely studied topic because of the large

number of membrane structures that exhibit wrinkling. The concepts of wrinkling

in membranes are briefly described here. The present study does not consider

wrinkling in the implementation, because typically, inflated cushionsdo not exhibit

wrinkling.

According to Schoop et al. [34], membranes cannot carry compressive in-

plane loads because they do not possessany flexural stiffness. In this case, mem-

DBD


Introduction 28

branes wrinkle.

Vázquez[35] reported that at any point on its surface, amembranemust be in

oneof threestates. In theslack state, themembraneisnot stretched in any direction.

In the taut state, the membrane is in tension in all directions. If the membrane is

neither taut not slack, it i s in the wrinkle state corresponding to uniaxial tension.

In the slack or wrinkled state, the real configuration of a membrane is undefined.

Figure1.10showsthe configurations for the threestates.

(a) (b) (c) (d) (e)

Figure 1.10: Principle statesof membranes: (a) reference, (b) taut, (c) and (d) wrinkle, and(e) slack (source: Jarasjarungkiat et al. [36])

Wrinkling experiments on initially flat, thin, linear-elastic isotropic foils

subjected to in-plane loads are presented in the work of Wongand Pellegrino [37];

anda wrinkled membrane is shown in figure 1.11.

Figure 1.11: Wrinkled membrane (source: Wongand Pellegrino [37])

1.6Objective

This study focuseson two main objectives:

– research onmaterial models suitable for membranematerials.

– analysisof theinfluenceof thepressure–volume couplingin inflatedcushions.

DBD


Introduction 29

1.7Thesis outline

The characteristicsandtypesof pneumatic structuresarepresented in chapter

2, which also presents themembraneformulation used in thenumerical analysisby

thefinite element method.

Chapter 3 presents the material models for membranes. Because of the large

variety of membrane materials available for membrane structures, different mate-

rial modelsarepresented andimplemented. Thesematerial modelsare elastoplastic

and elastoviscoplastic for small strains, and hyperelastic, elastoplastic, and elasto-

viscoplastic for largestrains.

A new material model based on non uniform rational basis splines (NURBS)

surfaces is proposed and is presented in chapter 4. NURBS is a mathematical

representation of a 3D geometrical shape and is used for obtaining curves and

surfaces. In this material model, the NURBS surfaces are used to represent the

constitutiverelation between stressesandstrains. Thedefinitionand formulation of

NURBS curves and surfaces and examples of validation for the proposed material

model are also presented in chapter 4.

Chapter 5 presents the discussion on pressure–volume coupling, which is

applied to pneumatic structures. The main objective of this coupling is to take into

account the influences of volume variation, which leads to the change in internal

pressure. The formulationfor thenumerical analysis in thefinite element methodis

presented in conjunction with the analytical analysis that enables the validation of

thenumerical implementation.

Theimplementation of pressure–volume couplingandthematerial modelsfor

inflatablestructureswascarried out in thestructural analysisprogram developed by

the research group at TUM. This program is called CARAT++ (Computer Aided

Research AnalysisTool) andwasinitiated byKai-UweBletzinger, HansStegmüller,

and Stefan Kimmich at the Institut für Baustatik of the University of Stuttgart in

1987.

Examples of application of material models for membranes and pressure–

volume coupling are presented in chapter 6, which also discusses an example of a

real pneumatic structure.

Finally, the conclusions on the material models for membranes, conclusions

of the analysis considering pressure–volume coupling, and suggestions for future

studiesare presented in chapter 7.

DBD


2Mechanics of membranes

In thischapter thebasisfor thenumerical analysisof membranestructuresare

presented.

2.1Kinematics

Kinematics is the study of the deformation and motion of a continuousbody.

This body in an initial state is shown in figure 2.1 with number 1. Successive

deformations are applied in this body represented with the numbers 2 and 3. The

reference configuration in Lagrangian description is defined in the state 1 and the

states 2 and 3 are the current configuration. In Eulerian description the reference

configurationisupdated. For example, in thefirst applied deformationthereference

configuration is thestate1 and the current configuration is thestate2 in thesecond

applied deformation, the state 2 becomes the reference configuration and the state

3 is the current configuration. In the present work the Lagrangian description is

adopted in the implementation.

The deformation gradient F transforms the reference configuration into the

actual configuration.

F =∂x∂X

(2-1)

where x is theposition of apoint in current configuration and X is theposition of a

point in the reference configuration.

X1

X2

X3

1

2

3

x1

x2

x3

F² F³1

2

Figure 2.1: Successive deformations of a continuous body

DBD


Mechanicsof membranes 31

According to Lee and Liu [38], the combination of elastic and plastic strains,

bothfinite, callsfor amore careful study of thekinematicsthan theusual assumption

that the total strain components are simply the sum of the elastic and plastic

components, as for infinitesimal strain theory.

This hypothesis was introduced by Lee and Liu [38] and is defined as the

product:

F = FeFp (2-2)

The transformation from thefirst position to thesecond position isgiven by:

dx2 = F21dx1 (2-3)

wherex1(X1, X2, X3) andx2(X1, X2, X3) are the coordinatesfor theundeformed body

(first position) and deformed body(second position), respectively.

Similarly, the transformation from thefirst position to the third position is:

dx3 = F31dx1 (2-4)

and thesecond position to the third position:

dx3 = F32dx2 = F3

2F21dx1 (2-5)

Substitutingequation2-4 in 2-5 results:

F31 = F3

2F21 (2-6)

According to Lee and Liu [38], such transformations provide a convenient

means of representing elastoplastic deformations in theneighborhood of a particle.

If thestressin thefinal configuration isremoved andthetemperaturereduced to the

uniform initial temperature, the elastic andthermal deformationswill be recovered,

leaving only permanent plastic deformations which provide the secondconfigura-

tion. Therefore, equation 2-6 results in equation 2-2 and it can be represented ac-

cording to Simo and Hughes [39], SouzaNeto et al. [40], Simo and Ortiz [41], and

Simo ([42],[43]) by figure2.2.

2.2Strain measure

Strain express the geometrical deformation and motion of a body. In La-

grangian description theGreen-Lagrange strain tensor isdefined by:

DBD



Fp

Fe

F = FeF

p

initialconfiguration

currentconfiguration

intermediateconfiguration

Figure 2.2: Multiplicative decomposition of the deformation gradient (source: SouzaNetoet al. [40])

E =12

(

FT F − I)

(2-7)

The logarithmic strain measure in Lagrangian description is defined:

EL = ln(U) (2-8)

where U is termed theright stretch tensor.

U =√

C (2-9)

where C is the right Cauchy-Green tensor and its spectral representation is given

by:

C = FT F = U2=

m∑

i=0

λiMi i = 1, 2 (2-10)

whereλi are theprincipal stretchesand Mi are the eigenprojections.

With the eigenprojections, thevaluescos2φ, sin2φ and, cosφsinφ areobtained:

M1 =

cos2φ cosφsinφ

cosφsinφ sin2φ

M2 =

sin2φ −cosφsinφ

−cosφsinφ cos2φ

(2-11)

Equation2-8 is rewritten in spectral representation:

EL =

2∑

i=0

ELiMi =

2∑

i=0

ln(λi)Mi i = 1, 2 (2-12)

DBD



2.3Stress measure

Forceper unit areaphysically express stressmeasure. Thismeasure rise from

the forcesof abody due to the their deformationandmotion. The conjugated stress

pair with Green-Lagrange strain tensor is the secondPiola-Kirchhoff stresstensor,

given by:

S = PF−T (2-13)

where P is the first Piola-Kirchhoff stresstensor, measured with forceper unit area

defined in the reference configuration.

Thefirst Piola-Kirchhoff stresstensor isnot symmetric. Therefore thesecond

Piola-Kirchhoff stresstensor isoften used, which is symmetric but doesnot admit a

physical interpretation in terms of surfacetraction.

The Kirchhoff stress(T) in Lagrangian description conjugate with the loga-

rithmic strain in Lagrangian descriptionand it is related with theKirchhoff stressin

Eulerian description (τ) with:

T = RTτR (2-14)

The relation between the Kirchhoff stresstensor in Eulerian description and

thesecondPiola-Kirchhoff stresstensor isgiven by:

τ = FSFT (2-15)

2.4Membrane formulation

Otto[27, 1] defines a membrane as a flexible skin stretched in such a way to

besubjected to tension.

The membrane formulation presented here is taken from works of Wüchner

andBletzinger [44], Vázquez[35], Holzapfel [45] and Linhard [31].

A point on the surfacein the reference configuration (Ω0) is described by a

position vector X which dependsontwo independent surface coordinatesξ1 andξ2,

presented in Figure 2.3.

X = X(ξ1, ξ2) (2-16)

DBD



Figure 2.3: Membrane coordinates

Theposition vector x in the current configuration isdefined by:

x = x(ξ1, ξ2) (2-17)

The covariant base vectors in the reference and current configuration are

defined respectively by thedifferentiation of X and x with respect to ξ1 and ξ2:

Gα =∂X∂ξα, gα =

∂x∂ξα, α = 1, 2 (2-18)

The covariant base vectors are tangential to the corresponding coordinate

lines. Thesurfacenormalsaredetermined byN or n, defined throughthenormalized

crossproduct:

G3 = G1 ×G2, N =G3

‖G3‖g3 = g1 × g2, n =

g3∥

∥

∥g3

∥

∥

∥

(2-19)

The covariant metric tensorsare

Gαβ = Gα ·Gβ gαβ = gα · gβ (2-20)

The contravariant basevectorsGα and gα aregiven by

Gα = Gαβ ·Gβ gα = gαβ · gβ (2-21)

where the contravariant metric tensorsare

Gαβ = Gαβ−1 gαβ = gαβ

−1 (2-22)

DBD



The relationsbetween the covariant and contravariant basevectorsaregiven

Gα ·Gβ = δαβ gα · gβ = δαβ (2-23)

where theKronecker delta is:

δαβ =

1 i f α = β

0 otherwise(2-24)

Thedeformation gradient F in curvili near coordinates isgiven by:

F = gα ⊗Gα; FT= Gα ⊗ gα; F−1

= Gα ⊗ gα; F−T= gα ⊗Gα (2-25)

The Green-Lagrange strain tensor and the secondPiola-Kirchhoff stressten-

sor are defined as:

E =12

(

gαβ −Gαβ)

Gα ⊗Gβ (2-26)

S = S αβGα ⊗Gβ (2-27)

The second Piola-Kirchhoff stress tensor is obtained from a constitutive

relationwith theGreen-Lagrangestrain tensor.

2.4.1Finite element discretization

The finite element discretization is developed with shape functions in terms

of isoparametric coordinates (ξ1, ξ2) for the total Lagrangian formulation. Hence

theposition vectors for the reference andcurrent configuration are expressed by:

X(ξ1, ξ2) =nnode∑

i

Ni(ξ1, ξ2)Xi x(ξ1, ξ2) =

nnode∑

i

Ni(ξ1, ξ2)xi (2-28)

where Ni are theshape functions.

Replacing equation2-28 in equation2-18gives:

Gα =∂(

∑nnodei Ni(ξ1, ξ2)Xi

)

∂ξα=

nnode∑

i

∂Ni(ξ1, ξ2)∂ξα

Xi (2-29)

gα =∂(

∑nnodei Ni(ξ1, ξ2)xi

)

∂ξα=

nnode∑

i

∂Ni(ξ1, ξ2)∂ξα

xi

DBD



2.4.2Linearization of the virtual work

The virtual work principle is used to establish the equili brium conditions for

the static analysis. This principle will be briefly described. For more details see

Zienkiewicz [46] and Bathe [47].

The virtual work principle states that the equili brium of a bodyrequires that

for any compatible small virtual displacements imposed onthe body in its state of

equili brium, thetotal internal virtual work isequal to thetotal external virtual work:

∫

Vδε · σ dV =

∫

VδU · f B dV +

∫

VδUS · f S dS +

∑

i

δU i · Ric (2-30)

δWint = δWext

where ε are virtual strains corresponding to virtual displacements U, σ are the

stressesinequili briumwithapplied loads, f B are applied bodyforces, f S are applied

surfaceforces and RC are concentrated loads.

The internal virtual work (δWint) is linearized for the solution with a Newton

scheme. Therefore, the left-hand-side of equation 2-30 is expanded into a Taylor

seriesup to thefirst order terms:

δWintlin=

∫

V(δE · S + ∆δE · S + δE · ∆S) dV (2-31)

To obtain approximated solutionsin aform suitablefor finite element analysis

the variation principle is established. The finite element equations derived are

simply thestatementsof thisvariationwith respect to displacements:

∂W∂ui= 0 (2-32)

Substitutingequations2-26 and 2-27 into the equation2-31gives:

δWintlin=

∫

V

(

δ

(

12

(

gαβ −Gαβ)

Gα ⊗Gβ)

· S)

dV+ (2-33)∫

V

(

∆δ

(

12

(

gαβ −Gαβ)

Gα ⊗Gβ)

· S)

dV+

∫

V

(

δ

(

12

(

gαβ −Gαβ)

Gα ⊗Gβ)

· ∆(S)

)

dV

Applying thevariational principle (equation 2-32):

∂W linint

∂ui= h

∫

A

δE∂ui

S dA + h∫

A

(

∂δE∂u j

S + δE∂S∂u j

)

dA = 0 (2-34)

DBD



where h is themembrane thicknessand A is themembranesurface area.

fint = h∫

A

δE∂ui

S dA (2-35)

KT = h∫

A

(

∂δE∂u j

S)

dA + h∫

A

(

δE∂S∂u j

)

dA (2-36)

whereδE isderived w.r.t δui:

δEδui=

δ(

12

(

gαβ −Gαβ)

Gα ⊗Gβ)

δui=

12·(

δgαβδui

)

Gα ⊗Gβ (2-37)

=12·(

δgαgβδui

)

Gα ⊗Gβ =12·(

δgαδui

gβ + gαδgβδui

)

Gα ⊗Gβ

The equation for the internal forces isgiven by:

fint = h ·∫

A

(

12

(

∂gα∂ui

gβ + gα∂gβ∂ui

)

Gα ⊗Gβ)

S αβGα ⊗Gβ dA (2-38)

where δgαδui

and δgβδui

are:

δgαδui=∂gα∂uiδui

δgβδui=∂gβ∂uiδui (2-39)

The first term of the stiffnessmatrix (equation 2-36) is obtained throughthe

equation:

∂δE∂u j=

∂(

12

(

∂gα∂ui· gβ + gα ·

∂gβ∂ui

))

∂u j(2-40)

=12

(

∂2gα∂ui∂u j

+∂gα∂ui

∂gβ∂u j+∂gα∂u j

∂gβ∂ui+∂2gβ∂ui∂u j

)

thesecond derivativesvanish:

∂2gα∂ui∂u j

= 0∂2gβ∂ui∂u j

= 0 (2-41)

Substitutingequation2-40 in thefirst term of equation2-36 gives:

Kg = h ·∫

A

(

∂δE∂u j

S)

dA (2-42)

= h ·∫

A

12

(

∂gα∂ui

∂gβ∂u j+∂gα∂u j

∂gβ∂ui

)

S αβGα ⊗Gβ dA

this is thegeometrical stiffnessmatrix.

DBD



Thesecondterm of equation2-36 is obtained with:

∂S∂u j=∂S∂E∂E∂u j= D :

12

[(

∂gα∂ui


)]

(2-43)

where D isa constitutivematerial tensor.

Km = h ·∫

A

(

δE∂S∂u j

)

dA (2-44)

= h ·∫

A

(

12

(

∂gα∂ui


))

D :12

[(

∂gα∂ui


)]

dA

this is thematerial stiffnessmatrix.

The total stiffnessmatrix is given by:

KT = Kg +Km (2-45)

2.4.3Membrane elements

The membrane elements that will be used in the pneumatic structures ex-

amples will be presented in this section. Quadrilateral and triangular membrane

elementsare implemented to discretizethepneumatic structures.

Shape functions and the derivatives of shape functions w.r.t. to the isopara-

metric coordinates (ξ1 and ξ2) are presented as follows. This equations are applied

to calculate thebasevectors, stiffnessmatrix, internal andexternal forces, displace-

ments, strains, and stresses.

2.4.3.1Triangular elements

Linear and quadratic elements are shown in Figure 2.4 with 3 and 6 nodes

respectively. The number of gausspoints used in the numerical integration is also

represented in Figure 2.4 with one gausspoint for the linear element and 3 gauss

points for thequadratic element.

Theshape functions for the linear triangular element are given from equation

2-46a to 2-46c.

N1 = 1.0− ξ1 − ξ2 (2-46a)

N2 = ξ1 (2-46b)

N3 = ξ2 (2-46c)

DBD



(a)

(b)

Figure 2.4: Triangular elements: (a) linear and (b) quadratic

The derivatives of the shape functions 2-46a, 2-46b, and 2-46c w.r.t ξ1 are

presented in equation2-47ato 2-47c andthederivativesof thesameshapefunctions

w.r.t. ξ2 areshown in equation2-47dto 2-47f.

dN1

dξ1= −1.0 (2-47a)

dN2

dξ1= 1.0 (2-47b)

dN3

dξ1= 0.0 (2-47c)

dN1

dξ2= −1.0 (2-47d)

dN2

dξ2= 0.0 (2-47e)

dN3

dξ2= 1.0 (2-47f)

Equations 2-48a to 2-48f are the shape functions for the quadratic triangular

element.

N1 = 2(ξ1−1+ξ2)(ξ1−12+ξ2) (2-48a)

N2 = 2ξ1ξ1 − ξ1 (2-48b)

N3 = 2ξ2ξ2 − ξ2 (2-48c)

N4 = 4ξ1(1− ξ1 − ξ2) (2-48d)

N5 = 4ξ1ξ2 (2-48e)

N6 = 4ξ2(1− ξ1 − ξ2) (2-48f)

The derivatives of the shape functions 2-48a to 2-48f w.r.t. ξ1 are shown in

equation2-49a to 2-49f and thederivativesof thesameshape functionsw.r.t. ξ2 are

presented in equation2-49gto 2-49l.

DBD



dN1

dξ1= 4ξ1 − 3+ 4ξ2 (2-49a)

dN2

dξ1= 4ξ1 − 1 (2-49b)

dN3

dξ1= 0 (2-49c)

dN4

dξ1= 4− 8ξ1 − 4ξ2 (2-49d)

dN5

dξ1= 4ξ2 (2-49e)

dN6

dξ1= −4ξ2 (2-49f)

dN1

dξ2= 4ξ1 − 3+ 4ξ2 (2-49g)

dN2

dξ2= 0 (2-49h)

dN3

dξ2= 4ξ2 − 1 (2-49i)

dN4

dξ2= −4ξ1 (2-49j)

dN5

dξ2= 4ξ1 (2-49k)

dN6

dξ2= 4− 4ξ1 − 8ξ2 (2-49l)

2.4.3.2Quadrilateral elements

Figure 2.5(a) shows the linear quadrilateral element with 4 nodes and full

gauss point integration and figure 2.5(b) represents the quadratic quadrilateral

element with 9 nodesand reduced gausspoint integration.

(a)

(b)

Figure 2.5: Quadrilateral elements: (a) linear and (b) quadratic

From equation 2-50a to 2-50dthe shape functions of the linear quadrilateral

element are presented.

N1 =14

(1− ξ1)(1− ξ2) (2-50a)

N2 =14

(1+ ξ1)(1− ξ2) (2-50b)

N3 =14

(1+ ξ1)(1+ ξ2) (2-50c)

N4 =14

(1− ξ1)(1+ ξ2) (2-50d)

The derivatives of the shape functions of the linear quadrilateral element are

given byequation 2-51a to 2-51h.

DBD



dN1

dξ1= −

14

(1− ξ2) (2-51a)

dN2

dξ1=

14

(1− ξ2) (2-51b)

dN3

dξ1=

14

(1+ ξ2) (2-51c)

dN4

dξ1= −1

4(1+ ξ2) (2-51d)

dN1

dξ2= −

14

(1− ξ1) (2-51e)

dN2

dξ2= −

14

(1+ ξ1) (2-51f)

dN3

dξ2=

14

(1+ ξ1) (2-51g)

dN4

dξ2=

14

(1− ξ1) (2-51h)

Theshapefunctionsof thequadratic quadrilateral element arepresented from

equation2-52a to 2-52i.

N1 =14ξ1ξ2(ξ2 − 1)(ξ1 − 1) (2-52a)

N2 =14ξ1ξ2(ξ2 − 1)(ξ1 + 1) (2-52b)

N3 =14ξ1ξ2(ξ2 + 1)(ξ1 + 1) (2-52c)

N4 =14ξ1ξ2(ξ2 + 1)(ξ1 − 1) (2-52d)

N5 = −12ξ2(ξ1

2 − 1)(ξ2 − 1) (2-52e)

N6 = −12ξ1(ξ2

2 − 1)(ξ1 + 1) (2-52f)

N7 = −12ξ2(ξ1

2 − 1)(ξ2 + 1) (2-52g)

N8 = −12ξ1(ξ2

2 − 1)(ξ1 − 1) (2-52h)

N9 = (1− ξ12)(1− ξ22

) (2-52i)

The derivatives of the shape functions of the quadratic quadrilateral element

w.r.t. ξ1 are given by equation 2-53a to 2-53i and the derivatives w.r.t. ξ2 are given

by equation2-54a to 2-54i.

dN1

dξ1=

14ξ2(ξ2 − 1)(2ξ1 − 1) (2-53a)

dN2

dξ1=

14ξ2(ξ2 − 1)(2ξ1 + 1) (2-53b)

dN3

dξ1=

14ξ2(ξ2 + 1)(2ξ1 + 1) (2-53c)

dN4

dξ1=

14ξ2(ξ2 + 1)(2ξ1 − 1) (2-53d)

dN5

dξ1= −ξ1ξ2(ξ2 − 1) (2-53e)

dN6

dξ1= −1

2((ξ2)2−1)(2ξ1+1) (2-53f)

dN7

dξ1= −ξ1ξ2(ξ2 + 1) (2-53g)

dN8

dξ1= −

12

((ξ2)2−1)(2ξ1−1) (2-53h)

dN9

dξ1= (2(ξ2)2 − 2)ξ1 (2-53i)

DBD



dN1

dξ2=

14ξ1(ξ1 − 1)(2ξ2 − 1) (2-54a)

dN2

dξ2=

14ξ1(ξ1 + 1)(2ξ2 − 1) (2-54b)

dN3

dξ2=

14ξ1(ξ1 + 1)(2ξ2 + 1) (2-54c)

dN4

dξ2=

14ξ1(ξ1 − 1)(2ξ2 + 1) (2-54d)

dN5

dξ2= −

12

((ξ1)2−1)(2ξ2−1) (2-54e)

dN6

dξ2= −ξ1ξ2(ξ1 + 1) (2-54f)

dN7

dξ2= −

12

((ξ1)2−1)(2ξ2+1) (2-54g)

dN8

dξ2= −ξ1ξ2(ξ1 − 1) (2-54h)

dN9

dξ2= (2(ξ1)2 − 2)ξ2 (2-54i)

DBD


3Membrane Material Models

Membrane structures have a lot of material possibiliti es. Some membrane

materialsarereported in Krishna[48]: reinforced fiber with glassor plastic, wooden

board, concrete plate and a vast variety of fabrics. Lewis[49] shows in his work

that the materials most used are: PVC coated with polyester, teflon coated with

glass fiber and canvas. Elias [50] adds to this list the materials: kevlar R©(para-

aramid synthetic fiber), nylon, polytetrafluoretileno (PTFE) and sili con. A material

that recently finds application specifically to pneumatic structures is the ethylene

tetrafluoroethylene(ETFE).

To comprehend the huge variety of materials available for membrane and

pneumatic structures, several models for material behavior are presented in this

chapter. All the material model formulations presented here were implemented in

theresearch program CARAT++ [51]. Validationexamplesof thesemodelsare also

presented.

3.1Small strains — Elastop lasticity

Small strains or infinitesimal strains theory deals with infinitesimal deforma-

tions of a body. Elastoplastic and elastoviscoplastic material models considering

small strainswill bedescribed.

The formulation used for the elastoplastic material is classic and it is pre-

sented for instancein the studies of Simo and Taylor [52], Simo and Hughes [39],

andSouzaNeto et al.[40].

The total strain E splits into a elastic strain Ee and aplastic strain Ep:

E = Ee + Ep (3-1)

The elastic constitutive law considering linear elasticity is given by the

relation:

S = D : (E − Ep) (3-2)

where D is the elastic moduli tensor. Theyield condition isgiven by the function:

f (S, q) = φ(S) + q(σy,K) ≤ 0 (3-3)

DBD


MembraneMaterial Models 44

where σy is the yield stressand K is the hardening modulus. If K < 0, one

speaksof asoftening response.

The flow rule and the hardening law in associative plasticity models is given

respectively by:

Ep = γ∂ f∂S

(3-4)

α = γ∂ f∂q

(3-5)

where γ is the consistency parameter, ∂ f∂S is a function that defines the direction of

plastic flow, and ∂ f∂q is a function that describes thehardening evolution.

The actual state (S, q) of stress and hardening force is a solution to the

followingconstrained optimization problem:

maximise S : E − q · α (3-6)

sub ject to f (S, q) ≤ 0

Solutionfor theproblem 3-6 satisfies theKuhn-Tucker optimality conditions,

theso called loading/unloadingcondition.

γ ≥ 0, f (S, q) ≤ 0, γ f (S, q) = 0 (3-7)

3.1.1Plane Stress

In the present work membrane structures are analyzed, therefore all material

modelsare implemented considering planestressconditions.

Figure 3.1 shows the plane stress state, where the stresses S 13, S 23, and S 33

are zero. Thestresstensor isgiven by

S =

S 11 S 12 0

S 21 S 22 0

0 0 0

(3-8)

Thestresstensor can bewritten in voigt-notationas:

S =[

S 11 S 22 S 12

]

(3-9)

The components Ei j of the total strain tensor E are correspondingly:

E = [E11 E22 2E12]

DBD



Figure 3.1: Plane stress state (source: SouzaNeto et al. [40])

3.1.2Von Mises y ield criteria - Plane Stress

Figure 3.2 presents the experimental data from uniaxial and biaxial test of

ETFE from works of Moritz [15], Galli ot and Luchsinger [53], and DuPONTT M

Tefzel R© [54] and an adjusted von Mises yield curve. This yield surface was

generated considering an yield stress of 16MPa. Figure 3.2 shows that the von

Mises criteria is a good approximation for the experimental data for the ETFE

material.

Figure 3.2: Experimental data from uniaxial and biaxial test of ETFE and adjusted vonMisesyield curve

DBD



ThevonMisesyieldcriteriasuggeststhat yielding beginswhen J2, thesecond

invariant of thedeviatoric stress, reaches a criti cal value (k) [55].

f (J2) =√

J2− k = 0 ↔ f (J2) = J2− k2 = 0 (3-10)

In vector notation thedeviatoric stresss iswritten:

s = [s11 s22 s12] (3-11)

which can beobtained by theprojection of thestresstensor on thedeviatoric plane.

s = dev[S] = PS P =13

2 −1 0

−1 2 0

0 0 3

(3-12)

J2 is calculated through:

J2 = SPS (3-13)

Similarly the elastic and plastic strain tensors (Ee, Ep) are collected in vector form

as:

Ee =[

Ee11 Ee

22 2Ee12

]

Ep =[

E p11 E p

22 2E p12

]

and thedeviatoric strain isgiven by:

e = dev[E] = PE P =13

2 −1 0

−1 2 0

0 0 6

(3-14)

Linear isotropic hardening isconsidered, for which thescalar hardeningstate

variable is:

q = σy + Kα (3-15)

whereα is the amount of plastic flow and K is ahardeningmaterial parameter.

ThevonMisesyield function for planestressfollowing3-10 is:

f (S, α) =√

ST PS −√

23

q(α) ↔ f (S, α) =12

ST PS −13

(q(α))2 (3-16)

f (S, α) = SPS −√

23·√

ST PS · q(α)

From the above expression, equations3-4 and 3-5 result in:

Ep= γ∂ f∂S= γPS (3-17)

DBD



α = γ∂ f∂q= γ

√

23

ST PS (3-18)

With these equations the J2 plasticity model with isotropic hardening for

planestresscondition is summarized:

E = Ee + Ep

S = DEe

E = γPS

f = 12SPS − 1

3(Kα)2

α = γ

√

23ST PS

(3-19)

where D is the linear elastic constitutivematrix for planestressdefined as:

D =E

1− ν2

1 ν 0

ν 1 0

0 0 1−ν2

(3-20)

whereν is thePoisson ratio and E is the elastic modulus.

The updating scheme for integration of the corresponding rate constitutive

equations requires the formulation of a numerical algorithm. The implicit Euler or

backward scheme is used to discretizethe incremental constitutiveproblem. Based

in equations3-19 the resultingequationswith the implicit Euler follow:

fn+1(∆γ) =12

f −13

R2 (3-21)

f =16

(

S trial11 + S trial

22

)2

(

1+ E∆γ3−3ν

)2+

12

(

S trial11 − S trial

22

)2+ (S trial

21 )2

(

1+ E∆γ1+ν

)2(3-22)

R2 =(

σy + αn+1K)2=

σy +

αn + ∆γ

√

23

√

STn+1PSn+1

K

2

(3-23)

Epn+1 = Ep

n + ∆γPSn+1 (3-24)

αn+1 = αn + ∆γ

√

23

√

STn+1PSn+1 (3-25)

Strialn+1 = D[En+1 − Ep

n ] (3-26)

Sn+1 = Ξ(∆γ)D−1Strialn+1 (3-27)

Ξ(∆γ) =[

D−1 + ∆γP]−1

(3-28)

The consistent elastoplastic tangent moduli i s obtained with equations 3-29

and3-30. For moredetailsof the computation of the consistent elastoplastic tangent

DBD



moduli we refer to Simo andHughes [39].

dSdE

∣

∣

∣

∣

∣

n+1= Ξ −

[ΞPSn+1][ΞPSn+1]Sn+1PΞPSn+1 + βn+1

(3-29)

βn+1 =23

(

KSTn+1PSn+1

)

(

1− 23K∆γ

) (3-30)

The return mapping is the closest point projection (Simo and Hughes [39]).

This return mapping considers a two-step algorithm called the elastic predic-

tor/plastic corrector algorithm. This algorithm assumes that the first step is elastic,

which is called as the elastic trial solution (Strialn+1 ). If this elastic trial stress vio-

lates the yield function (equation 3-16) a new solution must be obtained with the

plastic corrector step. The elastic predictor/plastic corrector algorithm has a geo-

metric interpretationascan beseen in Figure3.3. Theplastic corrector step and the

implementation of the return mappingarepresented in boxes3.1 and3.2. These al-

gorithmsarebased in theworksof Simo andTaylor[52] andSouzaNeto et. ali .[40].

(a)

(b)

Figure 3.3: General return mapping schemes. Geometric interpretation: (a) hardeningplasticity and (b) perfect plasticity (source: SouzaNeto et al.[40])

The plastic multiplier (∆(γ)) is solved using the Newton-Raphson procedure

because of the nonlinear equations in ∆(γ). The Newton-Raphson procedure is

presented in box3.2.

DBD



1. Update the deformation tensor and compute the trial elastic stressand yieldfunction for trial state.

En+1 = En + S u

Strialn+1 = D[En+1 − Ep

n ]

f (∆γ) =12

f −13

R2 ∆γ = 0

2. If f (∆γ) ≤ 0then set (.)n+1 = (.)trial

n+1 andexit

3. Solve f (∆γ) = 0 for ∆γ using theNewton-Raphsonmethod- go to box3.2

4. Compute the algorithmic tangent moduli

Ξ =[

D−1 + ∆γP]−1

5. Update thestressand plastic strain in tn+1

Sn+1 = Ξ(∆γ)D−1Strialn+1

αn+1 = αn + ∆γ

√

23

√

STn+1PSn+1

Epn+1 = Ep

n + ∆γPSn+1

6. Compute the consistent elastoplastic tangent moduli

dSdE

∣

∣

∣

∣

∣

n+1= Ξ −

[ΞPSn+1][ΞPSn+1]Sn+1PΞPSn+1 + βn+1

βn+1 =23

(

KSTn+1PSn+1

)

(

1− 23K∆γ

)

7. UpdateE33

E33n+1 = −ν

E(S 11n+1 + S 22n+1) − (E p

11n+1 + E p22n+1)

Box 3.1: Algorithm for the elastoplastic material

DBD



1. Set initial guessfor ∆γ∆γ = 0

f (∆γ) =12· f (∆γ) −

13

R2(∆γ) = 0

2. Perform Newton-Raphson iteration

f′= −

13

(

S trial11 + S trial

22

)2E

(

1+ E∆γ3−3ν

)3(3− 3ν)

−

(

(


22

)2+ 4S trial

212)

E(

1+ E∆γ1+ν

)3(1+ ν)

R2′

= 2σy

αn + ∆γ

√

23

f

K

√

23

√

f +∆γ f

′

2√

f

f′=

12

f′ −

13

R2′

∆γn+1 = ∆γn −ff ′

3. Check for convergenceif ∆γn+1−∆γn ≤ tol then return to box3.1 elsegoto 1

Box 3.2: Newton–Raphson algorithm to solve∆γ

3.1.3Benchmark Example

The stretching of a perforated rectangular membrane along the longitudinal

axis ispresented asabenchmark exampleto evaluate the implementation described

above. This example is taken from Simo and Hughes [39], Simo and Taylor [56],

and Souza Neto et al. [40] and is modeled in CARAT++ for plane stress with

membrane elements. Thematerial is elastoplastic with isotropic hardeningand von

Misesyield criteria.

Themembranematerial propertiesare: E = 70GPa (membranemodulus), ν =

0.2 (Poisson ratio), K = 0.2GPa (hardening modulus), σy = 0.243GPa (yielding

stress), and membrane thicknessof 1 mm. Thedimensionand boundary conditions

are shown in figure 3.4. The static analysis was carried out with cylindrical arc-

length control of the free edge. The mesh is composed of 531 nodes and 480

quadrilateral li near membrane elements as shown in figure 3.4. Due to symmetry

aquarter of thegeometry ismodeled.

Figure 3.5 presents the results for the total applied forceversus displacement

on the membrane free edge. The results are in accordance with Souza Neto et

al. [40].

DBD



Figure3.4: Mesh, geometry and boundary conditions of aperforated rectangular membrane

Figure 3.5: Load versus edge displacement

3.2Small strains — Elastoviscoplasticity

The elastoviscoplastic material model reflects the plastic deformation depen-

dencewith time. The temperature is often related with thisphenomena.

According to Souza Neto et al. [40], materials such as metals, rubbers,

geomaterials in general, concrete and composites may all present substantial time-

dependent mechanical behavior.

Thephenomenological aspects for elastoviscoplasticity are: strain ratedepen-

dence, creep and relaxation.

The strain rate dependence is observed when a material is subjected to

tests carried out under different prescribed strain rates. According to Souza Neto

et al. [40], the elasticity modulus is mostly independent of the rate of loading

DBD



but, the initial yield limit as well as the hardening curve depend strongly on

the rate of straining. This rate-dependence is also observed at low temperatures,

but usually becomes significant only at higher temperatures. In figure 3.6(a) the

phenomenological aspectsof thestrain ratedependenceis presented.

Creep is the phenomenon by which that at a constant stress condition the

strain increases. For different levelsof stresstheresponsefor strain isalso different.

This is shown in figure 3.6(b). SouzaNeto et al. [40] reports that high strain rates

shown towardsthe end of theschematic curvesfor highandmoderatestressesis the

phenomenon known astertiary creep. Tertiary creep leads to thefinal ruptureof the

material and isassociated with the evolution of internal damage.

e

s

e1

.e2

.e3

.

(a)

(b)

s

time

constant strain

(c)

Figure 3.6: Phenomenological aspects: uniaxial tensile tests at high temperature (a) Strainrate dependence, (b) Creep, and (c) Relaxation (source: SouzaNeto et al. [40])

Relaxation occurs when by a constant strain stress decays in time. This

phenomenonis depicted in figure 3.6(c)

Theviscoplastic flow rule is defined as:

Evp = γ∂ f∂S

(3-31)

The explicit function for γ models how the rate of plastic straining varies

with the level of stress. There are many models to describe γ. Souza Neto et al.

[40] reports that a particular choice should be dictated by its abilit y to model the

dependence of the plastic strain rate on the state of stress for the material under

consideration.

Somemodels for theviscoplastic strain aredescribed next.

DBD



3.2.1Perzyna Model

This model was introduced by Perzyna (apudSouzaNeto et al. [40])) and is

widely used in computational applicationsof viscoplasticity. It isdefined by:

γ(S , σy) =< fn+1 >

µ(3-32)

< fn+1 >=

[

J2(S )q − 1

]1/ǫi f f (S , σy) ≥ 0

0 i f f (S , σy) < 0(3-33)

where µ is the viscosity–related parameter, whose dimension is time and the rate-

sensitivity ǫ is a non-dimensional parameter. Both parameters are strictly positive

and temperature dependent. According to SouzaNeto et al. [40], as a general rule,

as temperature increases (decreases) µ and ǫ increases (decreases).

3.2.2Peric Model

This form has been introduced by Peric (apudSouzaNeto et al. [40]) and is

given by:

< fn+1 >=

[

(

J2(S )q

)1/ǫ− 1

]

i f f (S , σy) ≥ 0

0 i f f (S , σy) < 0(3-34)

Souza Neto et al. [40] reports that in spite of its similarity to Perzyna’s def-

initions, as the rate-independent limit is approached with vanishing rate-sensitivity

ǫ → 0, the Perzyna model does not reproduce the uniaxial stress-strain curve of

the correspondingrate–independent model with yield stressσy. As shown byPeric,

in this limit, the Perzyna model produces a curve with S = 2σy instead. How-

ever, for vanishing viscosity (µ→ 0) or vanishingstrain rates, the responseof both

Perzyna and Peric models coincide with the standard rate-independent model with

yield stressσy.

The implementation of the present elastoviscoplastic material model follows

the algorithm presented in section 3.1 (see boxes 3.1 and 3.2), modifying ∆γ to

include the timeparameter:

∆γ = ∆t · γ =< fn+1 >∆tµ, µ ∈ (0,∞) (3-35)

where∆t is time increment.

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1. Update thedeformation tensor and compute the trial elastic stress.

En+1 = En + S u (3-36)

Strialn+1 = D[En+1 − Evp

n ] (3-37)

f (∆γ) =12

f −13

R2 (3-38)

2. Solve f (∆γ) = 0 for ∆γ using theNewton–Raphsonmethod— go to box3.4

3. Compute the algorithm tangent moduli

Ξ =

[

D−1 + ∆γP +∂∆γ

∂Sn+1⊗ PSn+1

]−1

(3-39)

4. Update thestressand plastic strain in tn+1

Sn+1 = Ξ(∆γ)D−1Strialn+1 (3-40)

αn+1 = αn + ∆γ

√

23

√

Sn+1PSn+1 (3-41)

Evpn+1 = Evp

n + ∆γPSn+1 (3-42)

5. Compute the consistent elastoviscoplastic tangent moduli

Θ =

1K−

√

23∂∆γ

∂q

(

χSn+1P − f)

−1

(3-43)

χ =

(

∆γ

fSn+1P + f

∂∆γ

∂Sn+1

)

Ξ (3-44)

dSdE

∣

∣

∣

∣

∣

n+1= Ξ + Ξ

∂∆γ

∂qSn+1P

Θ

√

23χ

(3-45)

6. Update E33

E33n+1 = −ν

E(S 11n+1 + S 22n+1) − (Evp

11n+1 + Evp22n+1) (3-46)

Box 3.3: Algorithm for the elastoviscoplastic material

In the present work the Peric model is used to describe < fn+1 > (equa-

tion 3-34). This equation was rewritten in a more stable form, according to Peric

apudSouzaNeto et al. [40] as:

φ(∆γ) =

(

∆t∆γµ + ∆t

)ǫ

·(

12

f 2

)

−13

R2 = 0 (3-47)

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Changes in the algorithm of the elastoplastic model, more precisely in equa-

tions 4 to 6, are introduced. Due to internal variables integration in time and to the

viscoplastic parameter. Thismodified algorithm has thework of Simo and Govind-

jee[57] as basisand it is shown in box3.3.

1. Set initial guessfor ∆γ∆γ = 0

φ(∆γ) =

(

∆t∆γµ + ∆t

)ǫ

·(

12

f 2

)

−13

R2 = 0 (3-48)

2. Perform Newton-Raphson iteration

f′= −

13

(

S trial11 + S trial

22

)2E

(

1+ E∆γ3−3ν

)3(3− 3ν)

−

(

(


22

)2+ 4S trial

212)

E(

1+ E∆γ1+ν

)3(1+ ν)

(3-49)

R2′

= 2σy

αn + ∆γ

√

23

f

K

√

23

√

f +∆γ f

′

2√

f

(3-50)

φ′(∆γ) = −

ǫµ

∆γµ + ∆t

(

∆t∆γµ + ∆t

)ǫ

·12

f 2 (3-51)

+

(

∆t∆γµ + ∆t

)ǫ

·(

12

f′)

−13

R2′

∆γn+1 = ∆γn −φ

φ′ (3-52)

3. Check for convergenceif ∆γn+1−∆γn ≤ tol then return to box3.3 elsegoto 1

Box 3.4: Newton–Raphson algorithm to solve∆γ including Peric model


The benchmark example of the viscoplastic material model implementation

is the same presented in section 3.1.3 to validate the implementation of the elasto-

plastic material model. The problem consists of axial stretching at constant rate of

a perforated rectangular strip with the same geometry, mesh, boundary conditions

and the elastic and plastic material properties as in section 3.1.3. The viscosity pa-

rameter is µ = 500s and two values for the rate sensitivity are adopted ǫ = 1 and

0.1.

The results for rate sensitivity of 1.0 and 0.1 are shown in figures 3.7(a) and

3.7(b), respectively.

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Thedeformation rate isdefined by:

vL

(3-53)

where v is the stretching velocity imposed on the free edge and L is the length of

thestrip, which is18 (seefigure3.4).

(a)

(b)

Figure 3.7: Force versus displacement curve of a perforated rectangular membrane:(a) ǫ = 1.0 and (b) ǫ = 0.1.

3.3Large strains — Hyperelasticity

The theory of large strains or finite strains considers that both rotations and

strains of a body are large. As the material of membranes usually present large

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strains, some material models with finite strains are implemented and presented in

this section.

The hyperelasticity theory considers that a material has a nonlinear elastic

responsewith large strains. A hyperelastic material is defined througha Helmholtz

free-energy function (W), often named strain energy.

Somemodelswith their respectivestrain energy functions follow.

3.3.1Moon ey–Rivlin model

Thestrain-energy function for theMooney-Rivlin model is expressed by:

W(I1, I2) = C1(I1 − 3) +C2(I2 − 3) (3-54)

where C1 and C2 are material constants and I1 and I2 are the first and the second

stretch invariantsgiven by:

I1 = det(F)−2/3(

λ21 + λ

22 + λ

23

)

(3-55)

I2 = det(F)−4/3(

λ21λ

22 + λ

22λ

23 + λ

23λ

21

)

(3-56)

3.3.2Neo–Hookean model

The strain-energy function for the Neo-Hookean model is obtained from the

Mooney-Rivlin model by settingC2 = 0

W(I1, I2) = C1(I1 − 3) (3-57)

3.3.3Ogden model

Thestrain-energy for theOgden model [58] isdefined as:

W(λγ) =∑

r

µr

αr[λαr

1 + λαr2 + (λ1λ2)

−αr − 3], γ = 1, 2 (3-58)

In thepresent work theOgden material model ([59],[58]) is implemented, be-

cause it includes thespecial cases of theNeo-Hookean and theMooney-Rivlin ma-

terials. Thisimplementationisbased onthework of GruttmannandTaylor [60]. The

formulation requires the computation and linearization of the principal stretches,

which are the eigenvaluesof the right stretch tensor C.

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In accordance with the deformation energy equation, the second Piola-

Kirchhoff stresstensor is given by:

Sγ = λ−1γ

∂W∂λγ= λ−2

γ

∑

r

µr[λαrγ − (λ1λ2)

−αr ], γ = 1, 2 (3-59)

The tangent material matrix, is determined:

CT = TT CT =

∂S 11

∂E11

∂S 11

∂E22

∂S 11

∂2E12∂S 22

∂E11

∂S 22

∂E22

∂S 22

∂2E12∂S 12

∂E11

∂S 12

∂E22

∂S 12

∂2E12

(3-60)

where:

C =

λ−41

(

λ1∂S 1∂λ1− 2S 1

)

λ−21 λ

−22

(

λ2∂S 1∂λ2

)

0

λ−21 λ−22

(

λ1∂S 2∂λ1

)

λ−42

(

λ2∂S 2∂λ2− 2S 2

)

0

0 0 (S 1−S 2) cos(2φ)C11−C22

(3-61)

T =

cos2φ sin2φ cosφsinφ

sin2φ cos2φ −cosφsinφ

−2cosφsinφ 2cosφsinφ cos2φ − sin2φ

(3-62)

S γ = λ2γS γ =

∑

r

µr[λαrγ − (λ1λ2)

−αr ], γ = 1, 2 (3-63)

λ1∂S 1

∂λ1=

∑

r

µrαr[λαr1 + (λ1λ2)

−αr ] (3-64)

λ2∂S 2

∂λ2=

∑

r

µrαr[λαr2 + (λ1λ2)

−αr ] (3-65)

λ2∂S 1

∂λ2= λ1∂S 2

∂λ1=

∑

r

µrαr[(λ1λ2)−αr ] (3-66)


To validate the implementation of the hyperelastic material model, a bench-

mark example is presented, which consists of the stretching of a square sheet with

a circular hole. This example is foundin Gruttmann and Taylor [60] and in Souza

Neto et al. [40]. The length of the square is 20m, the radius of the circle is 3m and

the thickness is 1m. Due to the symmetry, one quarter of the sheet was analyzed

DBD



and the mesh with 200 linear quadrilateral membrane elements and 231 nodes is

presented in figure 3.8(a). The material used is Mooney-Rivlin with the constant

values of C1 = 25MPa and C2 = 7MPa. Thus the Ogden material constants are

µ1 = 50MPa, µ2 = −14MPa and α1 = 2, α2 = −2. The analysis was performed

under load control conditions in threesteps.

Figure3.9 showsthe load–displacement curveof threepointsonthemesh (A,

B and C highlighted in figure 3.8) compared with the solution of Gruttmann and

Taylor [60].

The results for strains and stresses are shown in figure 3.10. The results

obtained with the present implementation are the same as the results of Gruttmann

andTaylor [60].

(a) (b)

Figure 3.8: Square sheet with a circular hole (a) undeformed sheet mesh with applied load(b) diplacement result in y direction with deformed sheet in a scale of 1:1.

Figure 3.9: Load–displacement curvesof stretching of a square sheet

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(a) (b) (c)

(d) (e) (f)

Figure 3.10: Results of the square sheet with a circular hole: (a) normal stressin x, (b)normal stressin y, (c) shear stress, (d) normal strain in x, (e) normal strain in y, and(f) shearstrain

3.4Large strains — Elastop lasticity

The multiplicative decomposition of the deformation gradient F is the main

hypothesis in the finite strain elastoplasticity [38]. This hypothesis was introduced

in chapter 2 in section2.1 and it ishere rewritten:

F = FeFp

The implementation was carried out in this study preserving the return map-

pingschemesof theinfinitesimal theory presented in section3.1. Simo [42] showed

that using Kirchhoff stress and logarithmic strain, the return mapping algorithm

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takes a format identical to the standard return mappingalgorithms for the infinites-

imal theory.

Takingthe assumptionsdescribed abovetheimplementationfor elastoplastic-

ity with largestrainsare summarized in box3.5.

Souza Neto et al. [40] emphasizes that the simplicity of the integration

algorithm of box 3.5 comes as a result of the assumptions of elastoplastic isotropy

and the particular implicit exponential approximation adopted to discretise the

plastic flow rule.

The present implementation is carried out based onthe works of Peric et al.

[61] and Caminero et al. [62] that present an algorithm for the total Lagrangian

formulation. Caminero et al. [62] developed the large strain theory for anisotropic

elastoplastic material for total and updated Lagrangian formulation. As isotropy

is a particular case of anisotropy, this formulation can be used in the present

implementation. Both works present amodel for finitestrainsbased onlogarithmic

strains.

The logarithmic strain measure and the Kirchhoff stress in Lagrangian de-

scriptionwas introduced in chapter 2 in sections2.2 and 2.3.

The numerical integration of the elastoplastic model is carried out with the

elastic predictor andtheplastic corrector scheme. The elastic predictor iscalculated

based on the multiplicative decomposition presented in equation 2-2 considering

Fpn+1 = Fp

n , the trial elastic deformation gradient isgiven by:

Fetrial

n+1 = Fn+1Fp−1

n+1 (3-67)

The logarithmic trial strain iscalculated with equation2-12andtheKirchhoff

trial stresswith the relation:

Tetrial

n+1 = DELetrial

n+1 (3-68)

where D is the elastic constitutivematrix presented in equation3-20.

With the Kirchhoff trial stress the plastic corrector is calculated with the

algorithmfor small strainspresented in box3.2andtheKirchhoff stressTn+1 andthe

plastic deformation gradient Fpn+1 are updated. Finally the consistent elastoplastic

tangent moduli i s computed.

Simo [63] and Ibrahimbegovic ([64],[65]) computed the elastoplastic tangent

moduli i n spatial description. In the present work the elastoplastic tangent moduli

is considered in material description.

The consistent elastoplastic tangent moduli ∂S∂E iscomputed from thefollowing

equation:

S = F−1τF−T = F−1R−T

τR−1F−T (3-69)

DBD



After some rearrangement and the symmetric tensor property U = UT ,

equation3-69 is rewritten:

S = U−1TU−1→ Si j = U−1im TmnU−1

n j (3-70)

The forth-order tensor dSdE can bewritten as:

∂S∂E=∂S∂C∂C∂E= 2∂S∂C

E =12

(C − I) (3-71)

Thederivativeof equation3-69w.r.t Ckl is given by:

2∂Si j

∂Ckl= 2

∂U−1im

∂CklTmnU−1

n j + U−1im

∂Tmn

∂CklU−1

n j + U−1im Tmn

∂U−1n j

∂Ckl

(3-72)

The fourth-order tensor∂U−1

im∂Ckl

is computed applying the chain rule:

∂U−1im

∂Ckl=∂U−1

im

∂Upq

∂Upq

∂Ckl(3-73)

where∂U−1

im∂Upq

and ∂Upq

∂Cklaccording to Jog [66, 67] aregiven by:

∂U−1

∂U= −U−1

⊠ U−1 ∂U∂C= [(U ⊠ I) + (I ⊠ U)]−1 (3-74)

where A ⊠ B = AikB jl, is defined byJog [66].

The fourth-order tensor ∂Tmn

∂Cklisalso computed applying the chain rule:

∂Tmn

∂Ckl=∂Tmn

∂EL pq

∂EL pq

∂Ckl(3-75)

where ∂Tmn∂EL pq

is the consistent elastoplastic moduli for Kirchhoff stress and

logarithmic strain and∂EL pq

∂Cklis computed with thestudy of Jog [67]:

∂EL

∂C=

12∂ ln(C)

∂C=

12

k∑

i=1

1λi

Pi ⊠ PTi +

k∑

i=1

k∑

j=1 j,i

ln(λi) − ln(λ j)

λi − λ jPi ⊠ PT

j

(3-76)

Box 3.6 summarizes the algorithm to compute the consistent elastoplastic

moduli with largestrains.

DBD



1. Take theplastic deformation gradient for the last converged step

Fpn+1 = Fp

n

2. ComputeFen+1, Ce

n+1, Uetrial

n+1 , Eetrial

n+1 and Ttrialn+1 :

Fen+1 = Fn+1Fp−1

n+1

Cen+1 = FeT

n+1Fen+1 Ce =

2∑

i=0

λ2i Mi i = 1, 2

Uetrial

n+1 =

2∑

i=0

λiMi i = 1, 2

ELetrial

n+1 = ln(Uen+1) =

12

ln(Betrial

n+1 ) =2

∑

i=0

ln(λi)Mi i = 1, 2

Ttrialn+1 = DEL

etrial

n+1

3. Solve f (∆γ) = 0 for ∆γ using theNewton–Raphsonmethod— go to box3.2for elastoplastic material or 3.4 for elastoviscoplastic material (change S toT ) and updateTn+1 and EL

en+1

ELen+1 =

2∑

i=0

ELei Mi i = 1, 2

4. ComputeFpn+1, Ee, and Ep

Fpn+1 = Fp

n+1exp(∆γPTn+1)

Ee =12

(C − Cp) C = FT F Cp = FpT Fp

Ep= E − Ee

5. Compute the consistent elastoplastic tangent moduli dSn+1dEn+1

— go to box3.6

Box 3.5: Algorithm of elastoplastic material with large strain

DBD



1. Compute ∂Tmn∂EL pq

throught the consistent elastoplastic moduli for small strainsfrom box3.1(elastoplastic) or 3.3(elastoviscoplastic)

2. Compute ∂U−1

∂U , ∂U∂C and 1

2∂ln(C)∂

C

∂U−1

∂U= −U−1

⊠ U−1 ∂U∂C= [(U ⊠ I) + (I ⊠ U)]−1

12∂ ln(C)

∂C=

12

k∑

i=1

1λi

Pi ⊠ PTi +

k∑

i=1

k∑

j=1 j,i

ln(λi) − ln(λ j)

λi − λ jPi ⊠ PT

j

3. Compute ∂U−1

∂C and ∂T∂C

∂U−1

∂C=∂U−1

∂U∂U∂C

∂T∂C=∂T∂EL

∂EL

∂C

4. The consistent elastoplastic moduli i s finally obtained

∂Si j

∂Ekl= 2

∂U−1im

∂CklTmnU−1

n j + U−1im

∂Tmn

∂CklU−1

n j + U−1im Tmn

∂U−1n j

∂Ckl

Box 3.6: Algorithm for the consistent elastoplastic or elastoviscoplastic moduli


The benchmark example to validate the formulation implemented for the

elastoplastic material with large strains is the same example presented in section

3.1.3 for the elastoplastic material with small strains. The problem consists of

axial stretching at constant rate of a perforated rectangular strip whose geometry,

mesh, boundary conditions, and material properties are common for both material

behavior and are shown in section 3.1.3. The results obtained with the present

implemented model, the small strains elastoplastic material model and the results

of the literature (SouzaNeto et al. [40]) areshown in figure3.11.

The results obtained with the elastoplastic material model for large strains

are in accordance with the results of the literature. The results of the elastoplastic

material model for small strains are overestimated when the membrane starts to

present large strains.

Figure 3.12 shows the stressversus strain curve for numerical analysis with

large and small strains.

DBD



Figure 3.11: Forceversus displacement on the free edge of a perforated rectangular mem-brane

Figure 3.12: Stressversus strain for numerical analysis with large and small strains

3.5Large strains — Elastoviscoplasticity

The present implementation of elastoviscoplastic material model with large

strains isbased onthe conceptsof elastoviscoplasticity with small strainspresented

in session 3.2 and the concepts of elastoplasticity with large strains presented in

session 3.4. The implementation for this material is shown in box3.5. The change

for this material algorithm compared with the elastoplastic material model is the

solution of ∆γ which is solved with box 3.4 and the constitutive material tensor∂Tmn

∂EL pqwhich is solved with box3.3.

A reference work of elastoviscoplastic material model implementation with

largestrains is thework of Peric [68].

DBD


4Material model based on NURBS

NonUniform Rational BasisSplines (NURBS) isamathematical representa-

tion of a geometry in 3D used for curves and surfaces.

NURBS representation is widely used in Computer-aided design (CAD) to

create and modify designs offering smooth surfaces. Due to the successof the use

of NURBS in CAD, it hasbeen suggested in other applications. An exampleof this

istheisogeometric analysisintroduced byHugheset al. [69], which isanew method

to solveproblems governed by partial differential equations such as, structures and

fluids. This method has many features in common with the finite element method

and some in common with meshlessmethods. However, it i s more geometrically

based and takes inspiration from Computer Aided Design (CAD).

Kiendl et al. [70] reports that in isogeometric analysis the functions from the

geometry descriptionareused asbasis functionsfor the analysis. Thus, the analysis

works on a geometrically exact model and nomeshing is necessary. This offers a

possibilit y to close the existing gap between design and analysis as both use the

samegeometry model.

Another application of NURBS in numerical analysis is the NURBS-

enhanced finite element method(NEFEM). Sevill a et al. [71] reports that the NE-

FEM uses NURBS to accurately describe the boundary of the computational do-

main, but it differs from isogeometric methods in two main facts. First, NURBS

are used to describe the boundary of the computational domain, not the entire do-

main as done in isogeometric methods. Second, the solution is approximated using

polynomials and the approximation is defined with Cartesian coordinates, directly

in the physical space. From a practical point of view, NEFEM considers efficient

strategies for numerical integration onelementsaffected bycurved boundaries.

The proposed NURBS application is for constitutive material modeling.

NURBS surfaces are used to represent the interaction between stresses and strains,

i.e., the NURBS surfaces are used here as response surfaces. These NURBS sur-

faces are based on two axes of strain and one axis of stress. NURBS curves can

also beused with one axisof strain and one axisof stress. The constitutivematerial

tensor iscalculated with thederivativesfrom theNURBS surfaces andcurves.

To the author knowledge, theonly referenceto theuseof NURBSasresponse

surfacefor the proposed material model is the linear elastic plane stressmaterial

model based onNURBS(LE-NURBS) implemented in CARAT++ by A. Widham-

mer [51]. Thismaterial model consistsof two NURBS surfaces and one curve.

DBD


Material model based onNURBS 67

The material model based onNURBS for principal directions (PD-NURBS)

isamodel for materialsusingstressand strain in principal directions.

4.1Nonun iform rational B-Spline curves and surfaces

The concept of NURBS curve and surfaceused in the present study refers to

theworks of Piegl and Till er [72] and L. Piegl [73].

The definition of NURBS curve/surfaceis the rational generalization of the

tensor-product nonrational B-spline curve/surface. Thereforethe conceptsof tensor-

product surfaces and B-spline curve/surfacewill be introduced.

According to Rogers [74], technically, a NURBS surfaceis a special case of

a general rational B-spline surfacethat uses a particular form of knot vector. For a

NURBS surface, the knot vector has multiplicity of duplicate knot values equal to

the order of the basis function at the ends. The knot vector may or may not have

uniform internal knot values.

4.1.1Tensor produ ct surfaces

The curve C(u) is a vector-valued function of one parameter. It is a mapping

of a straight line segment into Euclidean three-dimensional space. A surfaceis a

vector-valued function of two parameters, u and v, and represents a mapping of a

region, of theuv plane into Euclidean three-dimensional space. Thus it hastheform

S (u, v) = (x(u, v), y(u, v), z(u, v)).

The tensor product method is basically a bidirectional curve scheme. It

uses basis functions and geometric coefficients. The basis functions are bivariate

functionsof u andv, which are constructed asproductsof univariatebasisfunctions.

Thegeometric coefficients are arranged in abidirectional, n x m net. Thus, a tensor

product surfacehas the form:

S (u, v) = (x(u, v), y(u, v), z(u, v)) =n∑

i=0

m∑

j=0

fi(u)g j(v)bi, j (4-1)

where bi, j = (xi, j, yi, j, zi, j), 0 ≤ u, and v ≤ 1

S (u, v) can be rewritten in matrix form:

S (u, v) = [ fi(u)]T [bi, j][g j(v)] (4-2)

DBD



where [ fi(u)]T is a (1) x (n+1) row vector, [g j(v)] is a (m+1) x (1) column vector,

and [bi, j] isa (n+1) x (m+1) matrix of three-dimensional points.

4.1.2Defin ition o f B-spline basis functions

Let U = u0, ..., um be a nondecreasing sequence of real numbers, i.e.,

ui ≤ ui+1, i = 0, ...,m − 1. The ui are called knots, and U is the knot vector. The

ith B-splinebasis functionsof p-degree(order p+1), denoted by Ni,p(u), aredefined

as

Ni,0(u) =

1 i f ui ≥ u < ui+1

0 otherwise(4-3)

Ni,p(u) =u − ui

ui+p − uiNi,p−1(u) +

ui+p+1 − u

ui+p+1 − ui+1Ni+1,p−1(u) (4-4)

Ni,p iswritten instead of Ni,p(u) for brevity.

Thederivativeof B-splinebasis functions is given by:

N′

i,p =p

ui+p − uiNi,p−1(u) −

pui+p+1 − ui+1

Ni+1,p−1(u) (4-5)

Theproof of equation4-5 ispresented in Piegl andTill er [72].

4.1.3Defin ition o f B-spline curves

A ph-degreeB-spline isdefined by

C(u) =n∑

i=0

Ni,p(u)CPi a ≤ u ≤ b (4-6)

wheretheCPi arethe control pointsandtheNi,p(u) arethepth-degreeB-splinebasis

functions(equation 4-3) defined onthenonperiodic and nonuniform knot vector

U = a, ..., a︸︷︷︸

p+1

, up+1, ..., um−p−1, b, ..., b︸︷︷︸

p+1

(4-7)

with n + 1 number of control pointsandm + 1 number of knotsare related by:

m = n + p + 1 (4-8)

DBD



Thederivativeof B-spline curve is given by:

C′

(u) =n∑

i=0

N′

i,p(u)CPi (4-9)

Substitutingequation4-5 in equation4-9

C′

(u) =∑n

i=0

(p

ui+p−uiNi,p−1(u) − p

ui+p+1−ui+1Ni+1,p−1(u)

)

CPi (4-10)

= p∑n−1

i=−1 Ni+1,p−1(u) CPi+1ui+p+1−ui+1

− p∑n

i=0 Ni+1,p−1(u) CPiui+p+1−ui+1

= p N0,p−1(u)CP0

up−u0+ p

∑n−1i=0 Ni+1,p−1(u) CPi+1−CPi

ui+p+1−ui+1− p Nn+1,p−1(u)CPn

un+p+1−un+1

The first and last terms yield the quotient 00, which is here set zero. Thus

equation4-10 results:

C′

(u) =n−1∑

i=0

Ni+1,p−1(u)CPi+1 − CPi

ui+p+1 − ui+1=

n−1∑

i=0

Ni+1,p−1(u)Qi (4-11)

where Qi =CPi+1−CPi

ui+p+1−ui+1.

ConsideringU′

theknot obtained by droppingthefirst and last knots from U:

U′

= a, ..., a︸︷︷︸

p

, up+1, ..., um−p−1, b, ..., b︸︷︷︸

p

(4-12)

it has m − 1 knots. Then it is easy to check that the function Ni+1,p−1(u), computed

onU, isequal to Ni,p−1(u) computed onU′

. Thus

C′

(u) =n−1∑

i=0

Ni,p−1(u)Qi (4-13)

andC′

(u) isa p − 1th-degreeB-spline curve.

4.1.4Defin ition o f B-spline surfaces

Takingabidirectional net of control points, two knot vectors, andtheproducts

of theunivariateB-spline functionsaB-splinesurfaceisdefined as:

S (u, v) =n∑

i=0

m∑

j=0

Ni,p(u)N j,q(v)CPi, j (4-14)

with

U = 0, ..., 0︸︷︷︸

p+1

, up+1, ..., ur−p−1, 1, ..., 1︸︷︷︸

p+1

DBD



V = 0, ..., 0︸︷︷︸

q+1

, uq+1, ..., us−q−1, 1, ..., 1︸︷︷︸

q+1

Theknot vector U has r + 1 knots, and knot vector V has s + 1 knots. Equation4-8

takes the form

r = n + p + 1 and s = m + q + 1 (4-15)

Figure4.1 showsan exampleof a B-splinesurface.

Figure 4.1: Example of aB-spline surface(source: Piegl and Till er [73])

Thederivativeof aB-splinesurfacew.r.t. u is given by

S u(u, v) =∂S (u, v)∂u

=

m∑

j=0

N j,q(v)∂∑n

i=0 Ni,p(u)CPi, j

∂u(4-16)

=

m∑

j=0

N j,q(v)∂C j(u)

∂u

where C j(u) =∑n

i=0 Ni,p(u)CPi, j j = 0, ...,m are B-spline curves. Applying equa-

tion4-13 into equation4-16 gives

S u(u, v) =n−1∑

i=0

m∑

j=0

Ni,p−1(u)N j,q(v)CP(1,0)i, j (4-17)

where

CP(1,0)i, j =

CPi+1, j − CPi, j

ui+p+1 − ui+1

U(1) = a, ..., a︸︷︷︸

p

, up+1, ..., ur−p−1, b, ..., b︸︷︷︸

p

V (0) = V

DBD



Analogously S v(u, v) is given by:

S v(u, v) =n∑

i=0

m−1∑

j=0

Ni,p(u)N j,q−1(v)CP(0,1)i, j (4-18)

where

CP(0,1)i, j =

CPi, j+1 − CPi, j

v j+q+1 − v j+1

U(0) = U

V (1) = a, ..., a︸︷︷︸

q

, vq+1, ..., vs−q−1, b, ..., b︸︷︷︸

q

4.1.5Defin ition o f NURBS curves

Based in thepreviousdefinitions, a pth-degreeNURBS curve isdefined by:

CNURBS (u) =

n∑

i=0wiCPiNi,p(u)

n∑

i=0wiNi,p(u)

a ≤ u ≤ b (4-19)

where wi are the weights, CPi are the control points that form a control

polygon, andNi,p(u) arethenormalized B-splinesof degreep in u direction, defined

over theknot vector UNURBS .

UNURBS = [a, ..., a︸︷︷︸

p+1

, up+1, ..., um−p−1, b, ..., b︸︷︷︸

p+1

] (4-20)

We assumethat a = 0, b = 1, and wi > 0 for all i. Setting:

Ri,p(u) =Ni,p(u)wi

n∑

j=0N j,p(u)w j

(4-21)

equation4-19 is rewritten in the form:

CNURBS (u) =n∑

i=0

Ri,p(u)CPi (4-22)

Ri,p(u) are therational basis functions.

For thepropertiesof NURBScurves, we refer to Piegl andTill er [72]. Figure

4.2 ill ustrates the construction of a NURBS curve.

DBD



Figure 4.2: Geometry construction of aNURBScurve (source: Piegl and Till er [73])

4.1.6Derivatives of a NURBS curve

Thederivativesof NURBS curve are computed with thederivativesof nonra-

tional B-spline curves. ConsideringCNURBS (u) as follows:

CNURBS (u) =w(u)CNURBS (u)

w(u)=

A(u)w(u)

(4-23)

whereA(u) isthenumerator of equation4-19. Differentiatingequation4-23, results:

CNURBS′

(u) =w(u)A

′

(u) − w′

(u)A(u)w(u)2

(4-24)

=w(u)A

′

(u) − w′

(u)w(u)C(u)w(u)2

=A′

(u) − w′

(u)C(u)w(u)

where

A′

(u) =n∑

i=0

wiCPiN′

i,p(u) (4-25)

w′

i(u) =n∑

i=0

wiN′

i,p(u) (4-26)

and N′

i,p(u) isgiven by equation4-5

4.1.7Defin ition o f NURBS surfaces

A NURBS surfaceis a bivariate vector-valued piecewise rational function of

the form

DBD



S NURBS (u, v) =

n∑

i=0

m∑

j=0wi, jCPi, jNi,p(u)N j,q(v)

n∑

i=0

m∑

j=0wi, jNi,p(u)N j,q(v)

0 ≤ u, v ≤ 1 (4-27)

wherewi, j are theweights, CPi, j are the control points that form a control net,

and Ni,p(u) and Ni,q(v) are the nonrational B-spline basis functions of degreep and

q in theu and v directions, respectively, defined over theknot vectors:

UNURBS = [0, ..., 0,︸︷︷︸

p+1

up+1, ..., ur−p−1, 1, ..., 1︸︷︷︸

p+1

] (4-28)

VNURBS = [0, ..., 0,︸︷︷︸

q+1

uq+1, ..., us−q−1, 1, ..., 1︸︷︷︸

q+1

] (4-29)

where r = n + p + 1 and s = m + q + 1.

Introducing thepiecewise rational basis functions:

Ri, j(u, v) =Ni,p(u)N j,q(v)wi, j

n∑

k=0

m∑

l=0Nk,p(u)Nl,q(u)wk,l

(4-30)

(a) (b)

Figure4.3: NURBSsurface: (a) Control pointsnet (b) biquadratic NURBSsurface(source:Piegl and Till er [72])

Equation4-27 is rewritten as

S NURBS (u, v) =n∑

i=0

m∑

j=0

Ri, j(u, v)CPi, j (4-31)

DBD



A NURBS surface example is shown in figure4.3

4.1.8Derivatives of a NURBS surface

Thederivativesof aNURBS surface are computed analogously to thederiva-

tivesof aNURBS curve. ConsideringS NURBS (u, v) as follows:

S NURBS (u, v) =w(u, v)S NURBS (u, v)

w(u, v)=

A(u, v)w(u, v)

(4-32)

whereA(u, v) isthenumerator of equation4-27, thederivativesof aNURBSsurface

are calculated:

S NURBSα (u, v) =

Aα(u, v) − wα(u, v)S NURBS (u, v)w(u, v)

(4-33)

andα denotes either u or v. In the above expression Aα(u, v) isgiven by:

Aα(u, v) = w(u, v)∂

∂αS NURBS (u, v) (4-34)

= w(u, v)

∂

∂α

m∑

j=0

N j,q(v)n∑

i=0

Ni,p(u)CPi, j

The final expressions for the derivatives of a NURBS surfacein direction u

follow:∂

∂u

n∑

i=0

Ni,p(u)CPi, j =

n−1∑

i=0

Ni,p−1(u)CP(1,0)i, j

S NURBSu (u, v) =

n−1∑

i=0

m∑

j=0

Ni,p−1(u)N j,q(v)CP(1,0)i, j (4-35)

where

CP(1,0)i, j = p

CPi+1, j − CPi, j

ui+p+1 − ui+1

UNURBS (1)= [0, ..., 0,

︸︷︷︸

p

, up+1, ..., ur−p−1, 1, ..., 1︸︷︷︸

p

]

VNURBS (0)= VNURBS

Analogously for directionv:

S NURBSv (u, v) =

n∑

i=0

m−1∑

j=0

Ni,p(u)N j,q−1(v)CP(0,1)i, j (4-36)

DBD



where

CP(0,1)i, j = q

CPi, j+1 − CPi, j

v j+q+1 − v j+1

UNURBS (0)= UNURBS

VNURBS (1)= [0, ..., 0,

︸︷︷︸

q

, vq+1, ..., vs−q−1, 1, ..., 1︸︷︷︸

q

]

4.2Linear elastic material model based on NURBS (LE–NURBS)

Thismaterial model was developed by A. Widhammer [51].

Considering the plane stresscondition the stressand strain tensors are given

by equations4-37and 4-38, respectively.

S =[

S 11 S 22 S 12

]

(4-37)

E =[

E11 E22 2E12

]

(4-38)

Two NURBS surfaces are defined: one for thestressS 11 and theother for the

stressS 22 with the commonstrain axesE11 andE22. Additionally theNURBScurve

is defined with the shear strain (E12) in one axis and the shear stress(S 12) in the

other. Illustration of theLE–NURBSsurfaces andcurve arepresented in figure4.4.

Because of themodel li nearity, thesurfaces are flat and the curve is linear.

(a) (b)

(c)

Figure 4.4: NURBS surfacesfor stresses and strains for LE–NURBS material: (a) stressesin direction 11and strains in directions 11 and 22, (b) stressesin direction 22and strainsin directions 11 and 22and, (c) NURBS curve for stressesin direction 12and strains indirection 12.

DBD



For given strain input values the corresponding stresses are obtained on the

NURBS surfaces.

The LE-NURBS constitutive material tensor presented in equation 4-39 is

given by the derivatives: dS 11dE11

, dS 11dE22

, dS 22dE11

, dS 22dE22

and dS 122dE12

, which are the derivatives

of the NURBS surfaces and curve. These derivatives are calculated following the

NURBS theory presented in section4.1.

dSdE=

dS 11dE11

dS 11dE22

0dS 22dE11

dS 22dE22

0

0 0 dS 122dE12

(4-39)

where

dS 11dE11dS 11dE22

=

dE11du

dE22du

dE11dv

dE22dv

−T

·

dS 11du

dS 11dv

(4-40)

dS 22dE11dS 22dE22

=

dE11du

dE22du

dE11dv

dE22dv

−T

·

dS 22du

dS 22dv

(4-41)

dS 12

dE12=

dS 12

du·

(

dE12

du

)−1

(4-42)

The derivativesof the NURBS surfacefor S 11 in directions u and v are given

by

S NURBSu11

(u, v) =[

dE11du

dE22du

dS 11du

]

(4-43)

S NURBSv11

(u, v) =[

dE11dv

dE22dv

dS 11dv

]

(4-44)

and analogously for the derivatives of the NURBS surfacefor S 22 in directions u

andv.

S NURBSu22

(u, v) =[

dE11du

dE22du

dS 22du

]

(4-45)

S NURBSv22

(u, v) =[

dE11dv

dE22dv

dS 22dv

]

(4-46)

Thederivativeof theNURBS curve for stressS 12 in directionu is

CNURBS′

(u) =[

dE12du

dS 12du

]

(4-47)

4.3Material model based on NURBS for principal directions (PD–NURBS)

Theproposed material model coversisotropicnonlinear materialsunder plane

stress conditions, consequently the LE–NURBS is also included. The principal

differencebetween this model and the LE–NURBS material model is that the PD–

DBD



NURBS is based on principal directions of stressand strain. Therefore only one

surfaceis required for its definition.

PD–NURBS is valid for isotropic materials because of the use of orthogonal

transformation to calculate the response of the stress. According to Gruttmann and

Taylor [60], for isotropic material response the contravariant components of the

SecondPiola–Kirchhoff stresstensor arerecovered byan orthogonal transformation

of theprincipal stresses.

ThesecondPiola–Kirchhoff stresses and theGreen–Lagrange strains in prin-

cipal directionsare given by:

S =[

S 1 S 2 S 12

]

(4-48)

E =[

E1 E2 E12

]

(4-49)

where S 12 = 0 and E12 = 0.

The constitutive material tensor in general directions is obtained with the

rotationmatrix calculated as follows:

dSdE=

dS 11dE11

dS 11dE22

dS 112dE12

dS 22dE11

dS 22dE22

dS 222dE12

dS 12dE11

dS 12dE22

dS 122dE12

= TT ·dS

dE· T (4-50)

where dSdE

is the constitutivematerial tensor in principal directions

dS

dE=

dS 1dE1

dS 1dE2

dS 1

2dE12dS 2dE1

dS 2dE2

dS 2

2dE12dS 12dE1

dS 12dE2

dS 12

2dE12

=

dS 1dE1

dS 1dE2

0dS 2dE1

dS 2dE2

0

0 0 dS 12

2dE12

(4-51)

and the rotation matrix T is the same matrix introduced for the Ogden material in

chapter 3 section 3.3.3.

T =

cos2φ sin2φ cosφsinφ

sin2φ cos2φ −cosφsinφ

−2cosφsinφ 2cosφsinφ cos2φ − sin2φ

(4-52)

The constitutive material tensor in principal directions is computed with the

DBD



NURBS surfacederivatives introduced in section4.1.8:

dS 1dE1dS 1dE2

=

dE1du

dE2du

dE1dv

dE2dv

−T

·

dS 1du

dS 1dv

(4-53)

dS 2dE1dS 2dE2

=

dE1du

dE2du

dE1dv

dE2dv

−T

·

dS 2du

dS 2dv

(4-54)

Thederivative dS 12

2dE12is calculated as follows:

E =12

(

FT F − I)

=12

(C − I)

C = T · C (4-55)

where C and C are in vector form:

C =[

C11 C22 2C12

]

(4-56)

C =[

C11 C22 2C12

]

(4-57)

Equation4-55gives the constraint:

C12 = C21 = −12

(C11 −C22) sin(2φ) + C12cos(2φ) = 0 (4-58)

S = TT · S (4-59)

Finally the derivative dS 12

2dE12is calculated with the derivatives dS 12

dφ and dφ2dE12

obtained with equation4-59and 4-58:

dS 12

2dE12

=dS 12

dφ·

dφ

2dE12

=−(S 2 − S 1)cos2φ

C11 − C22(4-60)

wheredS 12

dφ= sinφcosφ(S 11− S 22) = S 2 − S 1

dφ

2dE12

=dφ

dC12

=−cos2φ

C11 − C22

andC11 and, C22 arethe componentsof theright stretch tensor C introduced inchap-

ter 2 in section 2.2. The cosine cosφ is calculated with the spectral decomposition

presented in equations2-10and 2-11.

The algorithm of thematerial model based onNURBSfor principal directions

DBD



ispresented in box4.1

1. Update thestrain tensor.En+1 = En +

S u

2. Calculate thestrains in principal directions

En+1 = TT En+1

3. Calculate the local parameter u and v from thestrains.

4. Obtain thestressvaluesS 1(u, v), S 2(u, v).

5. Calculate thederivatives dS 1dE1

, dS 1dE2

, dS 2dE1

, dS 2dE2

, and dS 1

dE1(equations 4-53, 4-54 and

4-60).

6. Constitutivematerial tensor is obtained:

dSdE= TT ·

dS 1dE1

dS 1dE2

0dS 2dE1

dS 2dE2

0

0 0 dS 12

2dE12

· T

7. Calculate thestresstensor.S = TT · S

Box 4.1: Algorithm of the material model based onNURBS

4.4Data fitt ing

Data fitting based onleast-squares aproximation is used to generate NURBS

surfacesfor the experimental data. Thisprocessisbriefly described below. For more

details the reference are the work of Piegl and Till er [73] and L. Piegl [73]. An

alternative approach for the generation of NURBS surfaces is the use of a CAD

software.

4.4.1Curve fitt ing

According to L. Piegl [73] equation4-22can bewritten in matrix form as

CNURBS = R CP (4-61)

DBD



whereCNURBS and CP are (n + 1) x 1 matricesand R isan (n + 1) x (n + 1) matrix.

If there are more data points than control points, equation 4-61 is overdetermined

andcan besolved approximately as follows:

CNURBS f= (RT R)−1RT CP (4-62)

Assigning initial parameters to the data points, as the p-th degree and the

control points, a least-squares fit isgenerated usingequation4-62.

4.4.2Surface fitt ing

The curve-fitting technique can be easily generalized for surfaces yielding:

S NURBS f= (RT R)−1RT CP (4-63)

4.5Validation examples

The PD–NURBS material model is applied to examples with different mate-

rial responses to validate theproposed material model. Attention isgiven to materi-

alswith large strains.

4.5.1Hyperelasticity — NeoHookean

The hyperelastic example is a quadrilateral membrane with dimensions

1m x 1m and the material properties are shown in table 4.1. For this membrane

a finite element model was built for which the mesh, boundary conditions and,

loading are presented in figure 4.5. The mesh is composed by 143 nodes and 100

quadrilateral li near elementswith 2 x 2Gaussintegration. The load was risen upto

89.44MN.

Table4.1: Material propertiesof quadrilateral membrane example

Young’smodulus(E) 250MPaPoisson ratio (ν) 0.3

thickness 1 mm

The validation was carried out comparing the solution with a conventional

formulation for hyperelastic materials and the nonlinear material model based on

DBD



Figure 4.5: Mesh, boundary conditions and applied load for the quadrilateral example

NURBS surfaces. The conventional NeoHookean formulation is obtained with the

Ogden material model presented in section 3.3.3 setting: r = 1, α1 = 2 and

µ1 = G = E2(1+ν) .

Sincenoexperimental datawasavailablefor thisapplication, thePD-NURBS

surfaces were generated with data points from NeoHookean formulation. The

NeoHookean NURBS surfaces are presented in figure 4.6 with stresses and strains

in the principal directions. These surfaces are composed by a control point net

25(u) x 25(v) and degree3 (p = 3 and q = 3).

(a)

(b)

Figure 4.6: NURBSsurfacesof stresses and strains in principal directions for NeoHookeanmaterial: (a) stressesin direction 1and (b) stressesin direction 2.

DBD



4.5.1.1Results

Displacement results in direction y are presented in figure 4.7. The results

demonstrate that the solution with the proposed material model is in accordance

with the conventional NeoHookean material model formulation.

(a) (b) (c)

Figure4.7: Displacement results in y direction: (a) undeformed membrane, (b) conventionalmaterial model, and (c) PD-NURBSmaterial model.

Themaximum error of theresultsareshown in Table4.2 andthisiscalculated

with equation4-64:

Error =NURBS result − Conventional result

Conventional result· 100 (4-64)

Table 4.2: Maximum error of the PD-NURBSfor rectangular membrane

Maximum errordisplacement y Stress y

0.0165% 0.15%

4.5.2Hyperelasticity – Moon ey-Rivlin

Thisexamplewasmodeled in chapter 3 in section3.3.4 consideringa conven-

tional formulation for Mooney-Rivlin material model. The results obtained before

are compared with the nonlinear material model based on NURBS surfaces. Fig-

ure 4.8 shows the NURBS surfaces used in these examples. The degree used in

the NURBS surfaces is 3 (p = 3 and q = 3) and the number of control points is

increased to analyzethe convergence control.

DBD



(a)

(b)

Figure 4.8: NURBS surfaces with stresses and strains in principal directions for theMooney-Rivlin material: (a) stressesin direction 1, and (b) stressesin direction 2.

4.5.2.1Results

NURBS surfaces are generated for different control point nets. The number

of control pointsare: 15x15, 20x20, 40x40, 70x70, and 100x100. Table4.3 presents

the convergencefor each load step comparing thedifferent nets.

For the 15x15 net convergence was achieved by the first step solely. As the

number of control points increases the convergence rate increases as well and the

number of iterations for each step decreases.

Thenumber of iterationschanges substantially for thefirst two stepscompar-

ing the20x20and 40x40 nets.

When the number of control points is increased to the 100x100 net the

convergencerate is improved for the last step andall thesteps have5 iterations.

Table4.4 presents themaximum error for the displacement, stress, and strain

for the analyzed nets. The error is calculated with equation 4-64 presented in the

previousexample.

DBD



Table4.3: Displacement residuum for 15x15to 100x100control point net.

Step it. displacement residuum15x15 20x20 40x40 70x70 100x100

1 9.97E+00 1.24E+01 1.14E+01 1.14E+01 1.14E+012 1.67E+00 1.28E+00 1.06E+00 1.05E+00 1.05E+003 1.22E-01 1.21E-01 1.94E-02 2.15E-02 2.14E-024 9.07E-03 1.37E-02 3.08E-04 8.55E-06 1.55E-05

1 5 1.43E-03 1.48E-03 4.16E-06 2.23E-09 4.44E-116 1.77E-04 1.65E-04 6.45E-087 2.81E-05 1.82E-058 3.92E-06 2.03E-069 6.05E-07 2.25E-071 3.18E+01 3.10E+01 7.10E-01 3.10E+012 8.87E+00 7.82E+00 7.95E+00 7.94E+003 7.14E-01 7.07E-01 7.10E-01 7.14E-014 1.67E-01 1.97E-02 2.61E-03 2.13E-035 3.37E-02 1.21E-03 6.57E-06 1.03E-066 9.84E-03 4.68E-05 2.25E-08 2.69E-09

2 7 N.C. 2.29E-03 2.73E-068 6.54E-04 1.17E-079 1.64E-0410 4.54E-0511 1.18E-0512 3.20E-0613 8.44E-071 4.69E+01 4.84E+01 4.77E+01 4.77E+012 9.71E+00 8.60E+00 8.79E+00 8.78E+003 7.40E-01 2.34E-01 2.58E-01 2.63E-014 1.95E-02 8.72E-03 2.97E-03 2.02E-04

3 5 1.37E-03 2.85E-04 7.03E-05 1.81E-076 8.30E-05 3.10E-05 1.05E-067 7.50E-06 1.17E-06 2.55E-088 6.27E-07 1.21E-07

The improvement in the convergence rate observed with the increase in the

number of control points, isalso observed by the results for displacements, stresses

andstrains.

The maximum error for the stressin direction x–x has an interesting path as

the number of control points increases. The error of 5.558% for the 20x20control

point net, which isquitelarge, decreaseswith thenumber of control pointsreaching

the same results as the conventional material model for the 70x70 control point

net. For the 100x100control point net the value is also the same as the results for

conventional material model or the error isvery small .

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Table4.4: Maximum error of PD-NURBSmaterial with surfacesgenerated bycontrol pointnets 20x20to 100x100for the square perforated example

Maximum error (%)control points 20x20 40x40 70x70 100x100displacement x 1.341 0.111 0.012 0.000displacement y 0.264 0.026 0.000 0.000stress xx 5.558 0.178 0.000 0.010stress yy 0.489 0.074 0.074 0.011stress xy 0.859 0.049 0.012 0.006strain xx 0.743 0.085 0.065 0.011strain yy 0.400 0.030 0.023 0.003strain xy 0.445 0.109 0.016 0.000

4.5.3Comparison with elastop lastic von Mises material model

The von Mises elastoplastic material is used here to investigate the applica-

bilit y of the PD-NURBS to other stress-strain responses. The example consists in

a monotonic stretching of a perforated rectangular membrane modeled in chapter

3 section 3.1.3. The PD-NURBS material is now employed by the constitutive re-

sponse in thefinite element code as presented in box4.1.

It is worth pointing out that a full elastoplastic stress history can not be

obtained with the proposed PD-NURBS since unloading/reloading cycles are not

represented by theNURBS surfaces herein.

The membrane material properties are rewritten in table 4.5 and the mesh

is composed of 531 nodes and 480 quadrilateral membrane elements with linear

discretization and 2 x 2 gausspoints integration. The mesh, geometry, boundary

conditions, and the applied load areshown in figure 3.4 in section 3.1.3.

Table4.5: Material propertiesof the perforated membrane example

Young’smodulus(E) 70 GPaPoisson ratio (ν) 0.2Yield stress(σy) 0.243GPa

Hardeningmodulus(K) 0.2GPathickness 1 mm

The elastoplastic material properties of table4.5 are used to producethe data

points for the generation of the NURBS surfaces in principal directions. These

NURBS surfaces are shown in figure 4.9 and they are composed by a control point

net 70(u) x 70(v) and degree2 (p = 2 and q = 2). The number of control points

for theNURBSsurfaces isdefined with help of thepreviousexample, by which the

convergence andmaximum error for different control point nets are compared.

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(a)

(b)

Figure 4.9: NURBSsurfacesfor stresses and strains in principal directions for elastoplasticmaterial: (a) stressesin direction 1, and (b) stressesin direction 2.

The elastic region can be identified in the NURBS surfaces in Figure 4.9 as

theflat elli pseplane. Outsidethis region nonlinear behavior ispresented. Therefore

the corresponding axis S 11(u, v) and S 22(u, v) values fall i n the elastoplastic range

of theplastic model.

The conventional formulation used for the elastoplastic material model was

presented in section3.1.

4.5.3.1Results

The results obtained with the PD-NURBS material model are compared

with the classical material model. Figures 4.10 and 4.11 show the results for the

conventional elastoplastic and the PD-NURBS material model for displacements

andstressesin direction y, respectively. Themaximum error in theresultsareshown

in table4.6

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(a) (b)

Figure 4.10: Displacements in y direction: (a) conventional material model and (b) PD-NURBSmaterial model.

(a) (b)

Figure 4.11: Stressesin y direction: (a) conventional material model and (b) PD-NURBSmaterial model.

Table 4.6: Maximum error of thePD-NURBSfor perforated membrane

Maximum errorsdisplacement y stress y

0.032% 0.040%

Althoughthe comparison with the elastoplastic models are promising, appli -

cation of PD-NURBS to path dependent problemsrequires further investigation.

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5Pressure-Volume Coupling

Onespecial characteristic of pneumatic structureswhich distinguishesitsme-

chanical behavior from other membranestructures isthepressure-volume coupling.

According to Jarasjarungkiat [75] numerical examples demonstrate not only

the efficiency of the pressure-volume coupling model but also the need to consider

the volume (pressure) variation in addition to the change of surfacenormal vector.

Thestudy of Jarasjarungkiat [75] reveals theobservable feature that thepressureof

an enclosed fluid provides additional stiffnessto the inflatable structure, analogous

to thebehavior of amembraneonelastic springs.

The formulation of the pressure-volume coupling recalls the concept of

deformation-dependent forces. The formulation used in the present study refers

to the works of Hassler and Schweizerhof [17], Rumpel and Schweizerhof [18],

Rumpel [19], Bonet et. al. [20], and Berry andYang [21].

Hassler and Schweizerhof [17] presented a formulation for the static inter-

action of fluid and gas for large deformation in finite element analysis that can be

applied to pneumatic structures. Moreover it provides a realistic and general de-

scription of the interaction of arbitraril y combined fluid and/or gas loaded or filled

multi -chamber systemsundergoing largedeformations.

Theuseof adeformation-dependent forceformulation bringsalongthedraw-

back of afully-populated stiffnessmatrix for which triangular factorizationrequires

large numerical effort. To circumvent this problem Woodbury’s formula was used

to obtain the inverseof thefully-populated stiffnessmatrix asdiscussed in thework

of Hager [76]. The Woodbury’s formula updates the inverse of a matrix with the

update tensorswithout performinganew factorization of thestiffnessmatrix.

To validatethepressure-volume couplingformulation, analytical solutionsal-

ready developed for a circular inflated membrane clamped at its rim is presented.

Since the analytical formulation available in the literature ([77] and [78]) is re-

stricted to small strains conditions, an analytical formulation for large strains is

developed. The results obtained with analytical solutions are compared with the

numerical solutionswith and without pressure-volume coupling.

5.1Numerical analys is model for one chamber

The formulation presented in the work of Hassler and Schweizerhof [17]

concern an enclosed volume filled with combined liquid and gas. Rumpel and

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Pressure-VolumeCoupling 89

Schweizerhof [18] treat the case of structures filled with gas, which is the most

commoncase in civil engineeringand will t hereforebe adopted here.

Taking theprincipleof virtual work as basis for theproblem formulation, the

external virtual work of thepressure load isgiven by:

δWpress =

∫

ap n · δu da (5-1)

ξ

ξ

Figure 5.1: Surfaceunder pressure loading.

where n = x,ξ1 × x,ξ2/∣

∣

∣x,ξ1 × x,ξ2∣

∣

∣ is the surface normal vector,

da =∣

∣

∣x,ξ1 × x,ξ2∣

∣

∣ dξ1dξ2 is the surface element, and p = p(v(x)) is the in-

ternal pressure. The surface position vector x(ξ1, ξ2) is a function of the local

coordinates ξ1 and ξ2 represented in figure 5.1. Substituting these definitions in

equation5-1 gives:

δWpress =

∫

ξ2

∫

ξ1p

x,ξ1 × x,ξ2∣

∣

∣x,ξ1 × x,ξ2∣

∣

∣

· δu∣

∣

∣x,ξ1 × x,ξ2∣

∣

∣ dξ1dξ2 (5-2)

=

∫

ξ2

∫

ξ1p (x,ξ1 × x,ξ2) δudξ1dξ2 =

∫

ξ2

∫

ξ1p n∗ · δu dξ1dξ2

where n∗ = x,ξ1 × x,ξ2.

According to Poisson’s law, the constitutive behavior of the gas is described

by the followingequation:

pivκi = PiV

κi = const (5-3)

whereκ is theisentropyconstant, Pi andVi aretheinitial pressure and volume

and pi and vi are the current pressure and volume for each closed chamber i. This

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equation shows that when the volume decreases (increases) the internal pressure

inside the enclosed volume increases (decreases).

When κ = 1 the adiabatic changesimplifies to Boyle-Mariotte’s law.

Thevolumefor the enclosed chamber vi is computed throughthe equation:

vi =13

∫

ξ2

∫

ξ1x · n∗ dξ1dξ2 (5-4)

The external virtual work islinearized at state t for thesolutionwith aNewton

scheme. Equation5-2 and the constraint 5-3 are expanded into aTaylor seriesupto

thefirst order term:

δW lini,press = δWpress,t + δW

∆ppress,t + δW

∆npress,t (5-5)

δW lini,press =

∫

ξ2

∫

ξ1(pn∗ · δu + ∆pn∗ · δn∗ + p∆n∗ · δu) dξ1dξ2

with

∆n∗ = ∆u,ξ1 × x,ξ2 − ∆u,ξ2 × x,ξ1 (5-6)

∆(pvκ) = 0 (5-7)

∆p · vκt + ∆vκ · pt = 0

where

∆vκ = κvκtvt∆v (5-8)

∆v =13

∫

ξ2

∫

ξ1[∆u · n∗ + x · ∆n∗] dξ1dξ2 = ∆v∆u + ∆v∆n (5-9)

Equation5-7 results in:

∆p +κpt

vt∆v = 0 (5-10)

In thepresent work thefinal results for thepartial integrationsof equation5-5

will bepresented. Thesolutionfor each part of thepartial integration of the external

virtual work are calculated in theworks of Hassler andSchweizerhof [17], Rumpel

and Schweizerhof [18], and Rumpel [19]. The linearized external virtual work due

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to the change in thenormal vector is given by:

δW∆npress,t =

=pt

2

∫

ξ2

∫

ξ1

δu

δu,ξ1

δu,ξ2

·

0 Wξ1 Wξ2

Wξ1T 0 0

Wξ2T 0 0

∆u

∆u,ξ1

∆u,ξ2

dξ1dξ2 (5-11)

where Wξ1 = n ⊗ x,ξ1 − x,ξ1 ⊗ n and Wξ2 = n ⊗ x,ξ2 − x,ξ2 ⊗ n.

The linearized external virtual work due to the change in thepressure is:

δW∆ppress,t = −

κpt

vt

∫

ξ2

∫

ξ1n∗ · ∆u dξ1dξ2

∫

ξ2

∫

ξ1n∗ · δu dξ1dξ2 (5-12)

Replacing equations5-11and 5-12 in equation5-5 gives:

δW∆ppress,t + δW

∆npress,t = −δWpress,t (5-13)

−κpt

vt

∫

ξ2

∫

ξ1n∗ · ∆u dξ1dξ2

∫

ξ2

∫

ξ1n∗ · δu dξ1dξ2

+pt

2

∫

ξ2

∫

ξ1

δu

δu,ξ1

δu,ξ2

·

0 Wξ1 Wξ2

Wξ1T 0 0

Wξ2T 0 0

∆u

∆u,ξ1

∆u,ξ2

dξ1dξ2

= −pt

∫

ξ2

∫

ξ1n∗ · δu dξ1dξ2

The discretization for the finite elements is applied taking the equations 5-13

and the isoparametric representation:

x = Nix, ∆u = Nid and δu = Niδd (5-14)

where Ni are theshape functions.

Theglobal stiffnessmatrix and theglobal load vector are given:

[

KT − (Kpress − ba ⊗ a)]

d = fext + fpress − fint (5-15)

Kpress =

=pt

2

∫

ξ2

∫

ξ1

δN

δN,ξ1

δN,ξ2

T

0 Wξ1 Wξ2

Wξ1T 0 0

Wξ2T 0 0

∆N

∆N,ξ1

∆N,ξ2

dξ1dξ2

(5-16)

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a =∫

ξ2

∫

ξ1NT n∗ dξ1dξ2 (5-17)

fpress = pt

∫

ξ2

∫

ξ1NT n∗ dξ1dξ2 (5-18)

b = κpt

vt(5-19)

where KT is the total stiffnessmatrix containingthegeometrical andmaterial

stiffness, Kpress is the load stiffnessmatrix for each structural element in contact

with gas, a is the coupling vector, fpress is the load vector, fint is the forceresiduum

vector, and fext is the vector of the external forces. According to Rumpel [19] the

symmetric load stiffnessmatrix Kpress reflects the effect of thedirection–dependent

internal pressure and the fully–populated coupling matrix ba ⊗ a is the volume–

dependent internal pressure contribution.

Equation5-15can be rewritten as:

[

K∗ + ba ⊗ a]

d = F (5-20)

where K∗ = KT −Kpress andF = fext + fpress − fint.

The stiffnessmatrix is fully-populated, and therefore triangular factorization

requires great computational effort. To circumvent this problem the Sherman-

Morrison-Woodbury formula is used to solve the fully-populated stiffnessmatrix,

as discussed in thework of Hager [76].

5.1.1Sherman-Morr ison-Woodbu ry formula

Aspresented byHager [76] this formularelates the inverseof amatrix after a

small rank perturbationto theinverseof theoriginal matrix dismissingfactorization.

The focus is on the following result. If both A and I − VA−1U are invertible, then

A − UV is invertible and:

[A − UV]−1 = A−1 + A−1U(I − VA−1U)−1VA−1 (5-21)

whereUV isgiven byequation5-22supposingthat U isn×m with columnsu1, u2,

...,um andV is m × n with rows v1, v2, ...,vm

UV =m

∑

i=1

uivi (5-22)

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In the special case where U is a column vector u and V is a row vector u, equation

5-21simplifies to:

[A − uv]−1 = A−1 + αA−1uvA−1 (5-23)

whereα = 1/(1− vA−1u)

To solve the linear system Bx = b where B = A − UV equation 5-21 is used

to calculate the inverseof B:

x = B−1b

x =[

A−1 + A−1U(I − VA−1U)−1VA−1]

b

x = A−1b + A−1U(I − VA−1U)−1VA−1b

x = y +WC−1Vy

x = y +Wz (5-24)

If V ism×n, wherem ismuch smaller than n, then therank of themodification

UV is small relative to the dimension n of A and the system of m linear equations

z = C−1Vy is solved quickly. If m = 1 then z is a scalar Vy/C. This is the case of a

pneumatic structurewith one chamber.

5.2Multichambers s tructures

Accordingto Hassler andSchweizerhof [17] theprocedurefor single chamber

membrane can be directly expanded to multiple gas filled chambers connected

to each other. Stiffness matrices, coupling vectors and right-hand side vectors in

equation5-15 depicted by index i have to beset up for each chamber i and must be

summed upfor all n chambers:

KT −

n∑

i=1

[

Kpressi + bai ⊗ ai

]

d = fext − fint +

n∑

i=1

[

fpressi

]

(5-25)

5.3Analytical solution for a circular infl ated membrane

A circular inflated membrane clamped at its rim is inflated byauniform pres-

sure. The membrane is supposed to have large displacements. An analytical for-

mulationwas proposed by Hencky (apudFichter [77]), Fichter [77], and Campbell

[78] for membrane under small strain conditions. Fichter [77] considered that the

pressure remains orthogonal to the membrane during the inflation. One the other

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hand, by Hencky the pressure remains vertical to the z–axis (seefigure 5.2) dur-

ing the inflation. Fichter shows that this consideration results in an additive term

in the equation of the radial equili brium. This additional term will be show as fol-

lows. Campbell [78] generalized Hencky’s problem to include the influence of an

arbitrary initial tension.

In the present work an analytical solution is developed for inflated circular

membranes considering that the pressure remains orthogonal to the surfaceduring

the inflation and an arbitrary initial tension in the membrane. The effects of large

strainsare incorporated in thenew analytical solution.

5.3.1Hencky ’s solution

Hencky’s solutionconsidersauniform lateral loading, i.e. , the radial compo-

nent of pressure on the deformed membrane is neglected. The equation for radial

equili brium is:

Nθ =ddr

(r · Nr) (5-26)

and for circumferential equili brium:

Nrddr

(w) = −pr2

(5-27)

r andθ aretheradial andcircumferential coordinatesrespectively andNr andNθ are

the correspondingstressresultants, w is thevertical deflection, and p is theuniform

lateral loading. Figure5.2 showstheradial andcircumferential coordinates, vertical

deflection, and radial displacement of the circular membrane.

Figure 5.2: Radial and circumferential coordinates, vertical deflection, and radial displace-ment of a circular membrane

Linear elastic behavior is assumed for the material, thus the stress-strain

relationsare:

Nθ − µ · Nr = E · h · ǫθ (5-28)

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Nr − µ · Nθ = E · h · ǫr (5-29)

where h is the thicknessof membrane.

Thestrain-displacement relation is given by:

ǫr =ddr

(u) +12·

(

dwdr

)2

(5-30)

ǫθ =ur

(5-31)

where u is the radial displacement andµ is thePoisson ratio.

Theboundary conditionsare:

w(a) = 0 (5-32)

u(a) = 0 (5-33)

where a is themembraneradius.

Combining equations 5-26 through5-31, and defining dimensionlessquanti-

ties W = w/a, N = Nr/(Eh), ρ = r/a and q = pa/(Eh), the resulting equations

are:

ρd

dρ

[

ddρ

(ρN) + N

]

+12

(

dWdρ

)2

= 0 (5-34)

NdWdρ= −

12

qρ (5-35)

Substitution of equation5-35 into equation5-34gives:

N2 ddρ

[

ddρ

(ρN) + N

]

+18

q2ρ = 0 (5-36)

Hencky considered the solution for stress resultant N(ρ) in the form of a power

series:

N(ρ) =14

q2/3∞∑

0

b2nρ2n (5-37)

Substitution of N(ρ) in equation5-36 gives:

(b0+b2ρ2+b4ρ

4+b6ρ6+ ...)2(8b2ρ+24b4ρ

3+48b6ρ5+80b8ρ

7+ ...) = −8ρ (5-38)

Matching the coefficients of equation 5-38, yields the relations between b0, b2,

b4, ... :

b20b2 = −1 (5-39)

2b0b22 + 3b0

2b4 = 0 (5-40)

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...

These equationscan besolved successively for b2, b4, b6 ... in terms of b0:

b2 = −1

b20

, b4 = −2

3b50

, b6 = −13

18b80

, b8 = −17

18b110

, b10 = −37

27b140

,

b12 = −1205

567b170

, b14 = −219241

63504b200

, b16 = −6634069

1143072b230

,

b18 = −51523763

5143824b260

, b20 = −998796305

56582064b−290

(5-41)

The coefficient b0 isobtained byimposingtheremaining boundary conditions,

equation5-33, and combiningequations5-26, 5-28 and5-31:

(

ρ

[

ddρ

(ρN) + N

]

= 0

)

ρ=1

(5-42)

Application of equation5-37 gives:

(−1+ µ) b0 + (−3+ µ) b2 + (−5+ µ) b4 + (−7+ µ) b6 + (−9+ µ) b8

+ (−11+ µ) b10+ (−13+ µ) b12 + (−15+ µ) b14 + (−17+ µ) b16

+ (−19+ µ) b18+ (−21+ µ) b20 = 0

(5-43)

Substitutingequation5-41 in equation5-43, yields the followingequation in b0:

(−1+ µ)b0 −1

b20

(−3+ µ) −2

3b50

(−5+ µ) −13

18b80

(−7+ µ)

−17

18b011

(−9+ µ) −37

27b014

(−11+ µ) −1205

567b017

(−13+ µ)

−219241

63504b020

(−15+ µ) −6634069

1143072b023

(−17+ µ)

−51523763

5143824b026

(−19+ µ) −998796305

56582064b029

(−21+ µ) = 0

(5-44)

Thevalueof b0 can now besolved for aspecified valueof µ.

Thedisplacement W(ρ) is also assumed to be in the form of power series:

W(ρ) = q1/3∞∑

0

a2n(1− ρ2n+2) (5-45)

To obtain the coefficients in the series for W(ρ), expressions 5-37 and 5-45 are

inserted into equation5-35:

(b0+b2ρ2+b4ρ

4+b6ρ6+b8ρ

8+...)(a0+2a2ρ2+3a4ρ

4+4a6ρ6+5a8ρ

8+...) = 1 (5-46)

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Equatingcoefficients in equation5-46yields the relations:

b0a0 = 1 (5-47)

2b0a2 + b2a0 = 0 (5-48)

...

Combinationwith values for bn in equation5-41gives:

a0 =1b0, a2 =

1

2b04, a4 =

5

9b07, a6 =

55

72b010, a8 =

7

6b013

a10 =205

108b016, a12 =

17051

5292b019, a14 =

2864485

508032b022

a16 =103863265

10287648b025, a18 =

27047983

1469664b280

, a20 =42367613873

1244805408b031

(5-49)

The solution of equation 5-44 gives the value for b0 and the coefficients in

5-41and 5-49are also solved. Substitution of these coefficients into equations5-37

and 5-45 gives the dimensionless stress resultant N(ρ) and lateral displacement

W(ρ).

5.3.2Fichter’s solution

The equation of theradial equili briumfor Fichter’s solution hasincomparison

with Hencky’s solution(see equation5-26), an addition term:

Nθ =ddr

(r · Nr) − p · rddr

(w) (5-50)

Thisadditional term isthenormal pressurewhich isneglected in Hencky’s solution.

By Fichter’s solutionthelateral equili brium andthestress-strain relationremain the

same as thoseof Hencky’s solution(see equations5-27 through5-31).

The calculation for Fichter’s solution is analogous to Hencky’s solution,

with equations 5-27 through5-31 and 5-50, and defining dimensionlessquantities

W = w/a, N = Nr/(pa), ρ = r/a and q = pa/(Eh), the resultingequationsare:

N2ρ2 d2

dρ2N +

(

3 N2ρ −12ρ3

)

ddρ

N + αρ2N +18ρ2

q= 0 (5-51)

NdWdρ= −

12ρ (5-52)

whereα = (3+ µ)/2.

DBD



Thesolution for N(ρ) isobtained throughapower series:

N(ρ) =∞∑

0

n2mρ2m (5-53)

Substitutingequation5-53into equation5-51andequatingcoefficientsn2, n4,

n6, n8, ...Thiscoefficients can besolved in terms of n0:

n2 = −1+ 8α qn0

64qn02(5-54)

n4 = −(1+ 8α qn0) (4n0 q + 1+ 4α qn0)

6144n05q2(5-55)

n6 = −(1+ 8α qn0)

4718592q3n08· (5-56)

(

13+ 128α qn0 + 256α2q2n02 + 128n02q2 + 96n0 q + 576n02q2α)

4718592q3n08

...

Thesolution in apower series for W(ρ) isgiven by:

W(ρ) =∞∑

0

w2n(1− ρ2n+2) (5-57)

Substituting the power series 5-57 and 5-53 in equation 5-52 and equating

the coefficients of powers of ρ gives a system of simultaneous equations and the

coefficients w0, w2, w4, w6, ... result in terms of n0:

w0 = 1/4n0−1 (5-58)

w2 =1

5121+ 8α qn0

qn04(5-59)

w4 =1

147456(1+ 8α qn0) (8n0 q + 5+ 32α qn0)

n07q2(5-60)

...

Substituting equations 5-50 and 5-31 into equation 5-28 and applying the

boundary conditions for the radial displacement (5-33), gives:

r

(

ddr

(rNr) − µNr − prdwdr

)

= u (5-61)[

r

(

ddr

Nr − µNr − prdwdr

])

r=a

= 0

DBD



Thedimensionlessform of equation5-61 is given by:

[

ρ

(

ddρ

(ρN) − µN − ρdWdρ

)]

ρ=1

= 0 (5-62)

By specifying values for µ and q, and substituting equations 5-57 and 5-53

in equation 5-62 the value of n0 is obtained. The value n0 is used in the explicit

truncated series for N(ρ) and W(ρ), which are respectively thedimensionless stress

resultant and lateral displacement.

5.3.3Campbell’s solution

Campbell ’s solution is an extension of Hencky’s solution to include the case

of an arbitrary pretension (N0). Therefore the change in the equation 5-27 for the

lateral equili brium considering pretension is:

(N0 + Nr)ddr

(w) = −pr2

(5-63)

Theradial equili brium equation, stress-strain relation, andstrain-displacement rela-

tion, remain thesame as thoseof Hencky’s solution.

Therefore Campbell ’s solution is obtained analogously to Hencky’s solution.

With equations 5-63, 5-26, 5-28 through 5-31, and defining the dimensionless

quantities W = w/a, N = Nr/(Eh), N0 = N0/(Eh), Nθ = Nθ/(Eh), ρ = r/a,

andq = pa/(Eh), the resultingequationsare:

1ρ q2

(N + N0)2 dldρ

(Nθ + N) = −18

(5-64)

(N + N0)dWdρ= −

12

qρ (5-65)

Thesolution for N(ρ) is similar to Hencky’s solution(see equation5-37):

N(ρ) =14

q2/3∞∑

0

b2nρ2n− N0 (5-66)

Substituting N(ρ) in the modified equation for Nθ(ρ) (see equation 5-26) gives

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Nθ(ρ):

Nθ(ρ) =d

dρ(ρN)

Nθ(ρ) = ρd

dρ(N) + N

Nθ(ρ) = ρ14

q2/3(2b2ρ + 4b4ρ3 + 6b6ρ5 + ...)+

14

q2/3(b0 + b2ρ2 + b4ρ4 + ...) − N0 (5-67)

The coefficients b2, b4, b6, b8, ..., can be solved in terms of b0, substituting

equations 5-66 and 5-67 in equation 5-65. The values of the coefficients bn are the

same as the coefficients of Hencky’s solution, which weregiven in equation5-41.

The coefficient b0 is evaluated with equation 5-42 substituting N(ρ), given in

equation5-66, and the coefficients bn presented in equation5-41:

113164128q2/3b030− 339492384q2/3b027

− 377213760q2/3b024

−572107536q2/3b021− 961895088q2/3b018

− 1705844448q2/3b015

−3126483360q2/3b012− 5860311930q2/3b09

− 11165138127q2/3b06

−21536932934q2/3b03− 41949444810q2/3

− 452656512N0b029

−113164128µ q2/3b030 + 113164128µ q2/3b027 + 75442752µ q2/3b024

+81729648µ q2/3b021 + 106877232µ q2/3b018 + 155076768µ q2/3b015

+240498720µ q2/3b012 + 390687462µ q2/3b09 + 656772831µ q2/3b06

+1133522786µ q2/3b03 + 1997592610µ q2/3 + 452656512µN0b029 = 0

(5-68)

The value of b0 can now be solved for specified values of µ, q, and N0 with

equation5-68.

The solution for W(ρ) is the same as the one obtained by Hencky’s solution

(equation 5-45). The coefficients an in the power series equation W(ρ), are solved

with equations5-64 and 5-45 in equation5-65.

With the coefficientsan andbn, the explicit truncated series for N(ρ) andW(ρ)

are calculated.

5.3.4Modified Fichter’s solution

Initial tension or pretension is applied in most cases of membranestructures.

Therefore, an analytical solution with Fichter’s solution considering an initial

tension is developed in thepresent work.

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The equation of radial equili brium is the equation of Fichter’s solution(equa-

tion 5-50) and the equation of lateral equili brium is the one of Campbell ’s solution

(equation5-63). These equationsare rewritten:

Nθ =ddr

(r · Nr) − p · rddr

(w)

(N0 + Nr)ddr

(w) = −pr2

In this case the solution is analogous to Fichter’s solution, with equations

5-63, 5-50, 5-28 through5-31, and defining dimensionlessquantities W = w/a,

N = Nr/(pa), N0 = N0/(pa), ρ = r/a and q = pa/(Eh), the resultingequationsare:

(NR + N0)2

(

ρ2 d2

dρ2(N) + 3ρ

ddρ

(N)

)

+ αρ2 (N + N0) + (5-69)

+ρ2

8q−ρ3

2d

dρ(N + N0) = 0

(N + N0)dWdρ= −

12

qρ (5-70)

Thesolution for N(ρ) is similar to Fichter’s solution(see equation5-53):

N(ρ) =∞∑

0

n2mρ2m− N0 (5-71)

SubstitutingN(ρ) in equation5-69 the coefficients nm are solved in termsof n0:

n2 = −164·

(1+ 8αqn0)

qn20

(5-72)

n4 = −1

6144·

(1+ 8αqn0) (1+ 4αqn0 + 4qn0)

q2n50

(5-73)

n6 = −(1+ 8α qn0)

4718592· (5-74)

(

13+ 128α qn0 + 96qn0 + 256α2q2n02 + 576αq2n0

2 + 128q2n02)

q3n08

...

The solution of W(ρ) is the same solution in power series used in Fichter’s

solution (see equation 5-57). Substituting equations 5-71 and 5-57 into equation

5-70gives the coefficients wm.

w0 =14

n−10 (5-75)

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w2 =1

5121+ 8αqn0

qn04

(5-76)

w4 =1

1474565+ 72αqn0 + 8qn0 + 256α2q2n0

2 + 64αq2n02

q2n07

(5-77)

...

The explicit truncated series N(ρ) and W(ρ) are calculated with the coeffi-

cients nm and wm

5.3.5Finite strain solution

The finite strain solution is obtained throughFichter’s solution (seesection

5.3.2) and the consideration of finite strain term (12 ·

(

dudr

)2) in ǫr. The finite strain

term (12 ·

(

ur

)2) in ǫθ isnot considered.

ǫr =dudr+

12·

(

dwdr

)2

+12·

(

dudr

)2

(5-78)

ǫθ =ur+

12·

(ur

)2

(5-79)

where the terms dudr and u

r account for small strains, the term 12 ·

(

dwdr

)2arises in the

presenceof largedisplacementsandtheterms 12 ·

(

dudr

)2and 1

2 ·

(

ur

)2account for finite

strains.

The calculation for this solution is analogous to the previous solutions, with

equations 5-27, 5-50, 5-28 through5-30, and 5-79, and defining the dimensionless

quantities W = w/a, N = Nr/(pa), ρ = r/a, and q = pa/(Eh), the resulting

equationsare:

pq

(

A +12

A2

)

+ ρ µ pd

dρ(N) + N p (µ − 1) +

12µ pρ2

N+

18

pρ2

qN2= 0 (5-80)

A = qρ2 d2

dρ2(N) + qρ

ddρ

(N)

(

3− µ −12ρ2

N2

)

+ N q (1− µ) +32

qρ2

N(5-81)

The solution of N(ρ) is the same of Fichter’s solution (see equation 5-53).

Substitutingequation 5-53 in equation 5-80 and equating coefficients n2, n4, n6, n8,

... Thiscoefficients are solved in termsof n0:

n2 =18−12q2n0

2− 12qn0 − 1+ 12q2µ n0

2− 4µ qn0

n02q

(

5qµ2n0 + 11qn0 + 8− 16µ qn0) (5-82)

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n4 = −C

8 B(5-83)

B = qn03(

7qµ2n0 − 34µ qn0 + 27qn0 + 24)

(5-84)

C = −648q2µ n22n0

3 + 636q2n03n2

2 + 12µ qn0 n2 + 136q2n02n2 (5-85)

+256qn22n0

2 + 28qn2 n0 − 64q2µ n02n2 + 156q2µ2n0

3n22

+9q2n0 + 2n2

...

Substitutingequations5-53and5-57into equation5-52, givesthe coefficients

w0, w2, w4, ...,of W(ρ):

w0 =1

(4n0)(5-86)

w2 = −12

n2 w0

n0(5-87)

w4 = −16

2n4 w0 + 4n2 w2

n0(5-88)

...

5.4Comparison o f analytical and nu merical analys is

The response of a circular membrane clamped at its rim and inflated by a

uniform pressure is analyzed. Solutions for both small and large strain conditions

obtainedwithanalytical and numerical modelspresented in thiswork are compared.

The data used for the numerical and analytical analysis is from the study of

Bouzidi et. al. [79]. The membrane characteristics are: E = 311488Pa (Young’s

modulus), ν = 0.34 (Poisson ratio) and the radius is 0.1425m. The static analysis

iscarried out in two steps. First the configurationfor an internal pressureof 400kPa

is obtained. After the inflation, external pressures are applied. Bouzidi et. al. [79]

consider the circular membrane initially flat and the inflation for pressures of

100kPa, 250kPa and 400kPa are applied. The mesh for the numerical solution is

composed of 640membrane elements (seefigure5.3).

A comparison between a mesh composed by linear and quadratic elements

is performed and it is presented in figure 5.4. The linear triangular element (T3)

has 3 nodes and 1 gaussintegration point and the linear quadrilateral element (Q4)

has 4 nodes and 2x2 gaussintegration. The quadratic triangular element (T6) has 6

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Figure 5.3: Mesh for a circular inflated membrane.

nodes and 3 gaussintegration points and the quadratic quadrilateral element (Q9)

has 9 nodes and reduced 2x2 gauss integration. The mesh with linear elements

has 641 nodes and the mesh with quadratic elements has 2529 nodes. The results

of the comparison are the same for the mesh with linear and quadratic elements.

Therefore, themesh with linear elements is chosen in these analysisbecause of the

faster performance.

Figure 5.4: Comparison between a mesh with linear and quadratic elements for appliedexternal pressure valuesof 150kPa and 300kPa.

5.4.1Results

Figure 5.5 shows the results of Hencky’s and Fichter’s solutions for the

applied external pressures of 150kPa and 300kPa. The difference between both

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Figure 5.5: Comparison between Hencky’s and Fichter’s solution for applied externalpressure valuesof 150kPa and 300kPa.

solutions is due to the additional term associated to the normal pressure present

only in Fichter’s solution.

Figure5.6: Fichter’s solutionand numerical resultswithout pretension andκ = 0 for appliedexternal pressuresvaluesof 150kPa and 300kPa.

A comparison of Fichter’s solution with the numerical those of FEM is

presented in figure 5.6. Thedifferencein the result obtained with Fichter’s solution

and the numerical solution is accredited to the presence of finite strains, which

are included in the finite element formulation and are precluded in the analytical

solution.

Figure 5.7 presents the results of a numerical solution for the circular mem-

branewith pressure-volume coupling (κ = 1) and without (κ = 0). Pressure-volume

coupling is more noticeable for higher external pressure values, in agreement with

Poisson’s law (see equation 5-3). It is important to observe that according to the

amount of coupling different final configurationsare obtained.

Next, the influence of pretension is investigated. Figure 5.8(a) presents the

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Figure5.7: Comparison between thenumerical solution with apretension of 1kPafor κ = 0and κ = 1 for applied external pressure valuesof 150kPa and 300kPa.

(a) (b)

Figure 5.8: Analytical and numerical solution with a pretension of 1kPa and κ = 1 for anapplied external pressure valuesof 150kPa and 300kPa: (a) deformed configuration and (b)pressure volume curve.

results for the analytical and numerical solutionwith apretension of 1kPa and with

pressure-volume coupling subjected to external pressures of 150kPa and 300kPa.

The analytical solution takes into account both the term from the normal pressure,

which isneglected in Hencky’s solution, and a pretension onthemembrane, which

is considered neither in Hencky’s nor Fichter’s solution. The results obtained with

the analytical solution are in accordance with the numerical results. The relation

between theinternal pressureversusvolume areill ustrated in figure5.8(b), stressing

that when the volume decreases due to the external pressure the internal pressure

increases.

Figure5.9 presents theresults for both analytical and numerical solutionwith

a pretension of 10kPa and pressure-volume couplingsubjected to external pressure

values of 150kPa and 300kPa. Comparing the results of figures 5.9(a) and 5.8(a),

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(a) (b)

Figure 5.9: Analytical and numerical solution with a pretension of 10kPa and κ = 1 for theapplied external pressure valuesof 150kPa and 300kPa: (a) deformed configuration and (b)pressure volume curve.

it i s observed that the deformed configuration and consequently the volume of

the circular membrane decreases for the case with a pretension of 10kPa. This

differenceisaround 10%.

(a) (b)

Figure 5.10: Analytical and numerical large strains solution without pretension and κ = 1for applied external pressure valuesof 150kPa and 300kPa: (a) deformed configuration and(b) pressure volume curve.

The finite strain solution is shown in figure 5.10(a). The results are both for

analytical and numerical solutions.

The results obtained with the analytical solution are similar to the numerical

results, hightlithtingthat thedifferencebetween Fichter’s solutionandthenumerical

solution is due to the presenceof large strains. This analytical solution also shows

the importanceof considering largestrainsby inflated membranes.

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6Examples of pneumatic s tructures and material models formembranes

This chapter is divided in two main topics: material models for membranes

and static analysis of pneumatic structures. Initially uniaxial and biaxial numerical

examples of Ethylene tetrafluoroethylene (ETFE) strips using the material models

described and validated in chapter 3 will be presented. The numerical results are

compared with experimental data. Thesecondexampleisabiaxial test of theETFE

strip modeled with thePD–NURBS material model presented in chapter 4.

A pneumatic structurebased in the experimental analysis of the inflation of a

circular membrane arenumerically analyzed. Thematerial of the circular membrane

isalso theETFE, which is modeled with thematerial modelsof chapter 3.

Analysis of an air cushion with one and two chambers for linear elastic

material and pressure–volume coupling are also presented and the results are

compared. Finally results for a real sizepneumatic structure cushionare presented.

By this model, the PD–NURBS material and the pressure-volume coupling are

considered. Cutting pattern generation isalso performed.

6.1ETFE–Foils

Growing use of ETFE–Foils in pneumatic structures has motivated the appli -

cation of thematerial modelspresented in thiswork to ETFE membranes.

ETFE isapolymer classified asasemi-crystalli nethermoplastic. This typeof

polymer ismore resistant to solventsand other chemicals.

Ethylene tetrafluoroethylene consists of monomers of Ethylene (C2H4) and

Tetrafluorethylene (C2F4). When these monomers are submitted to moderate tem-

peratures, pressures, and in thepresenceof a catalyst, they polymerizes:

Figure 6.1: Etylene Tetrafluoroetylene chemical structure

In 1970an ETFE material was produced for the first time by DuPONTT M

with the name Tefzel R©. The features of Tefzel R© are described in the Properties

Handbook[54].

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Examplesof pneumatic structuresandmaterial models for membranes 109

According to Robinson-Gayle et al. [80], ETFE was first used as a roofing

material in a zoo building in Burgers Zoo, Arnheim in the Netherlands in 1981.

It has subsequently been used in various buildings predominantly in the United

Kingdom and Germany.

The lightweight of the ETFE foil i s one of the most important features that

motivate its use in structural buildings. Moreover, it has been used more often in

roofs, resulting in low cost for the foundation. Beyondthisproperty of lightweight,

ETFE hasmany other advantageousproperties. Tanno[81] listed some:

– Non stick characteristics making it virtually self-cleaning with littl eneed for

maintenance.

– Goodtranslucency and light transmission qualiti es in visible and UV ranges.

– Can be coated to help further in the control of heat and light transmission

properties.

– Excellent thermal control properties can be achieved through multi -layer

foils.

– Extreme resistance to weathering and excellent resistance to solvents and

chemicals.

– Excellent characteristics for fire emergency situations in roofs and atria.

– Linear elastic behavior up to 20MPa and highelongationwithout damage.

The translucency property is advantageous, because it allows the utili zation

of natural li ght, reducing the use of artificial li ght. Another property related with

resource consumption and commented by Robinson-Gayle et al. [80] is the anti-

adhesive nature of ETFE. This property means that roofs and atria need to be

cleaned lessfrequently. This leads to areduction in the cost of detergentsandwater

to maintain thebuilding.

Recycling is other characteristic that is important in terms of sustainabilit y.

Robinson-Gayle et al. [80] points out that once the material is clean it can be

recycled by heating it to its softening temperature. The softening temperature of

an ETFE is low so this is not a very costly operation. The recycled ETFE can be

added into thehopper with virgin ETFE.

Figures 1.5 and 6.2 show some examples of cushion structures with ETFE–

foils. The flexibilit y to create structural forms with this material is highlighted in

these examples.

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Figure 6.2: Eden Project in the United Kingdom

6.1.1Material Behavior

Barthel et al. [82] carried out biaxial experimentswith ETFE–foilsand found

that the results in both directions show a largely matching material mechanical

behavior, in other words, the material behaves almost isotropically. Galli ot and

Luchsinger [53] performed tensile tests at many angles (15o, 30o, 60o and 75o)

and also gave similar results. The curves are identical and small variations appear

in the non–linear domains. They concluded that the extrusion process does not

significantly affect thematerial behavior and that ETFE–foilshave almost isotropic

behaviour. Becauseof this, in thepresent work the assumption of isotropic behavior

will be adopted.

Figure 6.3: Stress–strain curve of semi–crystalli ne thermoplastic material with schematicrepresentation of the tensile specimen in different steps (source: Ehrenstein [83])

Ehrenstein [83] shows in his work a typical stress–strain curve of semi–

crystalli ne thermoplastic material and this curve is presented in figure 6.3. In the

present work two phases are considered: linear elastic and elastoplastic.

Figure6.4 showsthetensile andcompressivestress–strain curvefor theETFE

material at a temperatureof +23C. TheETFE used in buildingsis theTefzel R©200.

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(a) (b)

Figure 6.4: Stress–strain curve: (a) tensile stressvs. strain and (b) compressive stressvs.strain (sourcePropertiesHandbook of Tefzel R©[54])

6.1.1.1Temperature influ ence

Moritz [15] carried out biaxial experiments in the proportion of 3:1 for

different levelsof temperature(-25C, 0C, +23C and+35C). Figure6.5 presents

the results of these experiments. The material is the ASAHI R©FLUON ETFE NJ

(thickness= 250µm). The right side of the curves (positive strain) are the stress

results in axis I and the left sideof the curves (negativestrain) are the stressresults

in axis II .

(a)

(b)

(c)

(d)

Figure6.5: Stress–strain curvesfor cyclic test: (a) -25C, (b) 0C, (c) +23C and(d) +35C(source: Moritz [15])

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Thetemperature influenceby thismaterial isclearly observed in figure6.5. In

figures6.5(a), 6.5(b), and6.5(c) theresultsfor the cyclic loadingtest havereversible

strain and stress, indicating elastic behavior. For the temperature of +35C (figure

6.5(d)) a residual strain is observed, indicating plastic behavior.

Figure 6.6: Yield stress and strain versus temperature performed by Moritz [15]

(a)

(b)

Figure 6.7: Test curvesfrom DuPONTT M [54]: (a) tensile strength vs. temperature and (b)ultimate elongation vs. temperature

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The experiments of Moritz [15] demonstrate that with temperature raise the

yield stressdecreases and the plastic behavior became more evident. These results

areshown in figure 6.6.

Figures6.7(a) and6.7(b) highlight thedependenceof thematerial behavior on

the temperature. The results of interest are those of the Tefzel R©200. Figure 6.7(a)

demonstrates the decrease of the tensile strength as the temperature increases. For

theultimate elongation thevalue increases as the temperature increases.

Figure 6.8: Creep test in DuPONTT M Tefzel 200Flexural [54]

Figure 6.8 presents the creep test for Tefzel R©200 for two values of temper-

ature (+23C and +100C). It is observed that creep deformation increases with

temperature.

Figure 6.9: Poisson ratio versus stress for different values of temperature (source:Moritz [15])

The dependency of the Poisson ratio with stressfor different values of tem-

perature is shown in figure 6.9. For low temperatures the Poisson ratio can be con-

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sidered constant, but for higher temperatures thevariation of thePoisson ratio with

stress should be considered.

6.2Uniaxial and b iaxial test by ETFE–foils

Based on the results and tests described in the previous section, numerical

models based in finite element methodare developed to fit the material parameters

for the constitutivemodel of ETFE.

The mesh used for the uniaxial and biaxial tests is a rectangular membrane

presented in figure 6.10. This mesh has 441 nodes and 400 quadrilateral li near

elements. In figure6.10arepresented theboundary conditionsandthe applied loads

for this model. These examples are symmetric, therefore one quarter is modeled.

The material properties are presented in table 6.1. These properties were extracted

from the work of Galli ot and Luchsinger [53]. The von Mises yield criteria is used

in the elastoplastic model andabili near curve isused in theplastic phasedueto the

significant change in thehardeningmodulusobserved experimentally.

The analysis is carried out with the arclength control and an equivalent nodal

forceis applied ontheboth edges.

Figure 6.10: Mesh, geometry and boundary conditions for thebiaxial test

Table 6.1: Material propertiesof ETFE–foils

Young’smodulus (E) 1100MPaPoissonratio (ν) 0.43

First yield stress(σy1) 16MPaFirst hardeningmodulus(K1) 160MPa

Second yield stress(σy2) 27MPaSecond hardeningmodulus(K2) 80MPa

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6.2.1Uniaxial test

Figure 6.11: Stressversus strain for small and large strains

For the uniaxial test the force in the x direction (Fx) is set to zero and the

force in the y direction is incrementally increased. The results of the uniaxial test

for large and small strainsare presented in figure6.11. The resultsare thesame for

small and large strains in the elastic phase, because the strains are still small . The

difference in the results for small and large strains are large as expected once the

small strains rage has been largely exceeded.

6.2.2Biaxial test

The biaxial test is analyzed for two load path with ratios: 2:1 and 1:1. In

the case of proportion of 2:1, it was applied the double of the force in the y

direction. The results for the numerical models are shown in figures 6.12 and 6.13.

In both figures it is observed that the result with large strain model are closer to

the experimental data. Thedifferencebetween theresults for small strainsand large

strainsare also noticeable as theuniaxial test showed previously.

These results show the importanceof considering large strains in the formu-

lation for this typeof material.

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Figure 6.12: Stressversus strain for experimental results and numerical results with smalland large strains for the biaxial loading in the proportion of 1:1

Figure 6.13: Stressversus strain for experimental results and numerical results with smalland large strains for the biaxial loading in the proportion of 2:1

6.3ETFE-Foil modeled with PD-NURBS

This example shows the application of PD-NURBS presented in chapter 4 to

model a material making use of the available experimental results. The experimen-

tal results used to generate the NURBS surfaces are those of the biaxially loaded

ETFE–foil under two loading programs ratios of applied force: 1:1 and 2:1 pre-

sented in thework of Galli ot and Luchsinger [53]. The available experimental data

is not enoughto generate goodNURBS surfaces. In order to obtain a point cloud

data necessary for the generation of the NURBS surfacedata points based on the

von Mises elastoplatic material formulation will be used. Figure 6.14 shows the

experimental data points represented by the filled circles and the artificial ones by

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hollow squares. In thisfigure the gap between the pointsof the experimental test is

observed. With this data points, NURBS surfaces in principal directions for stress

and strain are generated and figure 6.15 shows the NURBS surfacein conjunction

with the experimental datapoints.

Figure 6.14: NURBSsurfacewith experimental data

There is a dependenceof the material model formulation with the sizeof the

NURBS surfaces, in other words, input strains outside the NURBS surface, do not

generate output stressresults. In these regions artificial data is used to supply the

stressesand strains information.

In figure6.15isobserved that the experimental datapointsareontheNURBS

surfaces.

Thetest iscarried out for two load ratios1:1 and 2:1 as it waspresented in the

previous section. Geometry andmesh are thesameused in thepreviousexample.

6.3.1Results

For both load ratios, the results are compared with the experimental results

of Galli ot and Luchsinger [53]. Table 6.2 shows the relative error of the numerical

model withPD–NURBSmaterial for stressandstrain results. The error iscalculated

taking the experimental resultsas referencebased onthe following

Error =NURBS result − Experimental result

Experimental result· 100 (6-1)

Table 6.2 shows that the error with the PD–NURBS material for the biaxial

test for load ratiosof 1:1 and 2:1 is small compared to the experimental results. We

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(a)

(b)

Figure 6.15: NURBS surfacesof stress and strain in principal directions for von Misesmaterial: (a) stressesin direction 1and (b) stressesin direction 2.

Table6.2: Relative error of biaxial test for thePD–NURBSmaterial

Error (%)Biaxial 1:1 Biaxial 1:1

Strain Stress Strain Stress Strain Stressdirection 2 direction 1

0.42 1.99 0.95 0.32 1.57 1.63

can also conclude that the PD–NURBS material model is suitable for the present

membrane tests.

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6.4Burst test

Schiemann [84] andGalli ot andLuchsinger [53] carried out experiments that

consist in the inflation of an initially flat circular membrane, called burst test.

The burst test was performed with samples of ETFE–foil and were clamped

in abubbleinflationtest devicebetween an aluminium plate andan aluminium ring.

Air was injected between the aluminium plate and the foil , resulting in a spherical

deformation. Tests were performed at room temperature, which corresponds to

about 23 C. Thepressure in thebubblewas recorded with adigital pressuresensor

andthedeformation of thebubblewasmeasured with a3D digital image correlation

system.

(a) (b)

Figure 6.16: (a) Burst test and (b) deformation process(source: Schiemann [84])

The specimens tested by Schiemann [84] have a 53 cm radius and 200µm

thickness. Figures 6.16(a) and 6.16(b) show the apparatus for the experimental

analysisand thedeformation processof theburst test.

The burst test of specimen V28 from Schiemann [84] was carried out at a

constant strain rateof 2.5%/min.

A finite element model is developed to compare with the results of the burst

test of specimenV20 of Schiemann[84]. Figure6.17showsthemesh, geometry and

boundary conditionsused in thenumerical model. Due to symmetry one quarter of

the circular membrane ismodeled.

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Figure 6.17: Geometry, mesh and boundary conditions for the burst test performed bySchiemann

A comparison for linear and quadratic triangular elements is carried out, in

order to evaluate the results for both elements. The mesh is composed of 800

triangular elements for both linear and quadratic. The number of nodes is 441

for linear and 1681for quadratic. The linear triangular element (T3) has 3 nodes

and 1 Gaussintegration point. The quadratic triangular element (T6) has 6 nodes

and 3Gaussintegration points. Figure 6.18 presents the results of pressure versus

displacement results for linear and quadratic triangular membrane elements. These

resultsare thesamefor T3 andT6, therefore the linear triangular element ischosen

to beused in these analysisdue to the faster performance.

Figure 6.18: Pressure versus displacement results for the specimen V28 [84]; linear (T3)and quadratic (T6) triangular membrane elements.

Based on the previous analysis of the uniaxial and biaxial tests of ETFE–

foil i n section 6.2, the elastoplastic material model with vonMises yield criteria is

considered in thenumerical analysis. Thepropertiesof theETFE are extracted from

thework of Schiemann [84] and are presented in table6.3. A bili near curve isused

in theplastic phasedueto thesignificant changein thehardeningmodulusobserved

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Table 6.3: Material propertiesof specimen V28




experimentally.

6.4.1Results

The analysisiscarried out with the cylindrical arc-length method. Figure6.19

shows the pressure versus displacement curve for two different values of the step

length, 60and 100.

Figure6.19: Pressure versus displacement results for the specimen V28 [84]; step length of60and 100.

Table 6.4 presents the global convergencerate of the displacement residuum

at the criti cal pressure for the adopted step length values (60 and 100). A small

differencein the convergenceisobserved.

Figure 6.20 presents the plot of applied pressure versus the out of plane dis-

placementsfor specimenV28, obtainedwith numerical analysisfor the elastoplastic

material model with large andsmall strains. Theresultsobtained with thenumerical

model with largestrainsdemonstrateits suitabilit y to model thisexperiment. On the

other hand, the numerical model with small strains is valid only in thefirst steps of

the analysiswhere thestrains remain small .

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Table6.4: Global convergenceof thedisplacement residuum at the critical pressure for steplength valuesof 60and 100.

step length60 100

1 2.023e+01 2.166e+012 3.261e+00 2.850e+003 8.899e-02 4.937e-024 6.604e-05 4.682e-055 3.684e-09 2.359e-08

Figure 6.20: Pressure versus displacement results for the specimen V28 [84]; large strain,and small strain material models.

Thedeformed configuration of both the experimental and numerical analyses

with large strains are presented in figure 6.21. The resultsare shown for two stages

of the applied load, which are indicated in figure6.20with thenumbers1 (32.9kPa)

and 2(28kPa).

Figure 6.21: Deformed configuration of the specimen V28 [84] and numerical model withlarge strains for pressure states1 and 2.

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Figure 6.22 shows the stress versus strain curve in the y direction for the

numerical analysiswith largestrains. States1 and 2arethesamedepicted in figures

6.20and 6.21. Comparingfigures 6.20and6.22 thenon proportionality of pressure

andstresses isnoticeable. After the criti cal pressure, thestrains increasemightily.

Figure 6.22: Stressversus strain curve in y direction

Deformed configurations of the inflated circular membrane in threedimen-

sionsare shown in figure6.23. The two states1 and 2are again represented.

(a) (b)

Figure 6.23: Deformed inflated circular membrane with the out of plane displacement:(a) point 1 and (b) point 2

6.5Air cushion with sing le and doub le chamber

The objective of this example is to examine the response of the pneumatic

structure considering thepressure–volume coupling formulation presented in chap-

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ter 5.

The single chamber air cushion composed by two membranes was analyzed

in the studies of Jarasjarungkiat [75] and Linhard [31]. This structure is extended

hereto adouble chamber with amembranein themiddle. Cushioncompositionsfor

single chamber and double chamber are represented in figures 6.24(a) and 6.24(b),

respectively.

(a)

(b)

Figure 6.24: Undeformed cushions: (a) upper and lower membranes of single chambercushion and (b) upper, middle and lower membranesof double chamber cushion

Rectangular cushion dimensions are 6 meters length and 3 meters width.

Linhard [31] applies formfinding analysis to this cushion with internal pressure

of 400Pa and prestress of 0.89Pa. Jarasjarungkiat [75] presents a static analysis

after theformfinding processapplyingan external forcein the center of the cushion

distributed on 9elements. The cushion dimensions and the configuration after the

formfindingstage are ill ustrated in figure6.25.

Figure 6.25: Cushion dimensions and formfinding shape

The analysisiscarried out first for theformfindingstagefollowed bythestatic

stage. Both single chamber and double chamber cushion considering the influence

of thepressure–volume couplingare analyzed and presented.

Load control is used in the static stage and the force is applied upto 2.38kN

in 10steps.

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6.5.1Sing le chamber cushion

The single chamber cushion is composed by two membranes, an upper

membrane and a lower membrane. The initial internal pressure is 400Pa and the

initial volume is 9.173m3. The results for the deformation under external load and

volumeversus internal pressure arepresented in figures6.26and6.27, respectively.

Figure 6.26: Single chamber cushion deformation under external load

The deformation of the single chamber cushion (figure 6.26) is for a load

of 2.38kN. Considering the pressure–volume coupling the membrane deforms less

compared to the case without pressure–volume coupling. This is in agreement

with the Boyle–Mariotte law. The analysis with pressure–volume coupling leads

to internal pressure raise as the enclosed volume decreases resulting in smaller

displacementscompared to the analysiswithout coupling.

Figure 6.27: Volume versus internal pressure for the single chamber structure

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Volume versus internal pressure results for the single chamber cushion are

presented in figure 6.27. In this plot it i s observed that by the analysis with no

coupling the internal pressure remains unchanged and the volume decreases more

when pressure-volume coupling ispresented.

6.5.2Doub le chamber cushion

The double chamber cushion under consideration has one additional mem-

branebetween theupper andlower membranes(threemembranes). Theinitial pres-

sure and initial volume for each chamber are respectively: 400Pa and 4.58m3. The

results for thedeformation under external load and volumeversus internal pressure

arepresented respectively in figures 6.28and 6.29.

Figure 6.28: Two chambers deformation under external load

The deformation results of the double chamber cushion correspond to an

external load of 2.38kN. The results for the upper membrane with nocoupling are

thesame astheonesobtained in the analysisof thesingle chamber structure. Figure

6.28 shows that the middle membrane in this case doesn’t introduce any change.

On theother hand themiddlemembranepresents somedeformation bythe coupled

analysis .

The internal pressure of chamber 1 by the uncoupled analysis remains un-

changed, as expected. Chamber 2 has no influence in the results in this case. The

internal pressure of chamber 2 for the coupled analysis increases as the volume

decreases and follows the curveof chamber 1.

Figure6.30presents theout of planedisplacement versusapplied load results

in the center node of the cushions for the single and double cushion structure

with andwithout pressure–volume coupling. Theuncoupled analysis for single and

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Figure 6.29: Volume versus internal pressure for two chambers

Figure 6.30: out of plane displacement versus load

double cushion present the same results. The displacements of the double cushion

obtained by the coupled analysisare smaller than thoseof thesingle cushion.

6.6Lyon conflu ence cushion c©seele

This example explores a pneumatic structure in use. It is a placeof leisure

and shopping center in Lyon (France) and seele is the company responsible for

the cushion roof. According to seele [85] the roof structure is supported by 36m

high steel columnswhich carry the trussed steel arches of circular hollow sections.

Between these, further similar arches run in two diagonal directions. On plane the

roof is therefore anetwork of rhombuses and triangles which determine the shapes

of the two–layer foil cushions from seele. The cushions are framed by aluminium

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sectionsonall sideswhich arefixed to steel channels. Figure6.31showstheoverall

structure.

(a)

(b)

Figure 6.31: Lyon confluence cushion structure: (a) top view and (b) bottom view

The analysisiscarried out for one cushion dueto thedeformation between the

rigid metal frames that surroundthe cushions and the membrane. In other words,

the analysis can be carried out for each cushion separately. Cushion data such as

geometry, membrane properties, internal pressure, and applied load was provided

by seele. Thegeometry of the triangular cushion ispresented in figure 6.32.

Table 6.5 presents the material properties of the triangular cushion. The PD–

NURBS material model is used for the membrane material. Sinceno experimental

data was available for this material, the NURBS surfaces are generated based on

the elastoplastic material with von Mises yield criteria. Its goodaccordance with

theETFE–foil responsewas shown in thepreviousexamples.

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Table 6.5: Material propertiesof theETFE–foil




Theinternal pressureof the cushionis0.3kN/m2 andtheETFE–foil thickness

is250µm. The external load isa upli ft wind pressureof 1.5kN/m2.

Figure 6.32: Geometry of the triangular cushion

The analysis is carried out for load control of the triangular cushion with

and without cutting pattern generation. The meshes for both cases are presented in

figure6.33and theflat patterns in figure6.34.

Formfindinganalysis isperformed, for the internal pressureof 0.3kN/m2 and

prestressof 3.32kN/m2, before the cutting pattern analysis. In other wordsthework

flow for the present pneumatic analysis is first the formfinding, second the cutting

pattern generation, and third thestatic analysis.

(a) (b)

(c) (d)

Figure 6.33: Mesh of the cushion structure: (a) and (c) without cutting patterns (b) and (d)with cutting patterns.

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Figure 6.34: Flat patterns of the triangular cushion.

6.6.1Results

The static analysis has two stages. First, the inflation of the cushion is

performed. Second, the external wind load is applied. The static analysis is run

for both with and without cutting patterns. In each case the effect of the pressure–

volume coupling is presented. Figure 6.35 shows the von Mises stressdistribution

results with pressure–volume coupling. Attention is given to the stressdistribution

onthemembrane. Without cutting pattern generationthemaximum stressis located

on the edge of the membrane depicted with the letter A in Figure 6.35(a). On the

other handfor the casewithcutting pattern generationthemaximumstressislocated

in themiddleof themembranedepicted with the letter B in Figure6.35(b).

(a)

(b)

Figure 6.35: Von Mises stressdistribution onthe cushion structure with pressure–volumecoupling: (a) without cutting patterns, (b) with cutting patterns.

(a)

(b)

Figure 6.36: Strain in principal directions 1 onthe cushion structure with pressure–volumecoupling: (a) without cutting patterns, (b) with cutting patterns.

Figures 6.36 and 6.37 present the results of strain in principal directions for

the cases with and without cutting pattern generation considering pressure–volume

coupling. The distribution of strain values in principal direction 1 is similar for

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both with and without cutting pattern generation but in thepattern unions thestrain

values are smaller. On the other hand the strain distribution in principal direction 2

isdifferent in both cases. The case with cutting pattern presents larger strain values

on thesurfacewhile the casewithout cutting pattern has compressivestrainson the

membraneborder.

(a)

(b)

Figure 6.37: Strain in principal directions 2 onthe cushion structure with pressure–volumecoupling: (a) without cutting patterns, (b) with cutting patterns.

Table6.6: Maximum result valuesfor the triangular cushion

vonMises Strain (%) Stress(MPa)stress(MPa) direc. 1 direc. 2 direc. 1 direc. 2

cpa coupled 14.798 1.61740 0.99754 16.560 14.652uncoupled 14.877 1.65050 1.05760 16.639 14.753

n-cpb coupled 14.346 1.55840 0.85644 16.551 13.931uncoupled 14.553 1.60250 0.87532 16.796 14.229

acp = with cutting pattern bn-cp = without cutting pattern

Table 6.6 presents the maximum result values obtained in the four analysis.

Wind upli ft pressure reduces the internal pressure in the analysis considering

the pressure–volume coupling resulting in smaller values for stress, strain and

displacements.

Larger values are observed for cushion analysis with cutting pattern genera-

tion due to the accumulation of tension onthestrip unions.

The largest valueswere found bythe analysiswith cutting pattern generation,

without pressure–volume coupling.

Figures6.38and6.39present thestressversus strain results for thetriangular

cushion with and without cutting pattern generation. Figure 6.38 shows the results

with PD–NURBS material and figure 6.39 shows the results for the elastoplastic

with small strain. For each case the coupled and uncoupled of pressure–volume

modelsare considered. For the casewithcutting pattern generationthelast twosteps

are in theplastic region. This isobserved throughtheslope changeof the curve. On

the other hand the case without cutting pattern for both coupled and uncoupled are

in the elastic region. The results for the coupled analysispresent smaller values.

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Figure 6.38: Stressversus strain for triangular cushion with PD–NURBSmaterial.

Figure 6.40 shows the results of internal pressure versus volume for the

triangular cushionwith andwithout cutting pattern. The coupled andtheuncoupled

analysis are run for both cases. The uncoupled results are represented with the

hollow symbolsand both have constant internal pressure. Theresultsof the coupled

analysis for both caseswith andwithout cutting pattern, have thesame curveslope.

However, the difference in the initial geometry, due to the cutting pattern, results

in a difference in the initial volume. Last would be to say that seele used cutting

pattern madeof 8 stripsper layer instead of 6 as theinvestigated model based upon.

Figure6.39: Stressversus strain for triangular cushionwith elastoplastic material with smallstrains.

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Figure 6.40: Internal pressure versus volume for the triangular cushion.

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7Conclusions and Sugg estions for future works

In thepresent work membranestructuresmoreprecisely pneumatic structures

for constructions in civil engineering are studied. This type of construction is

quite new and requires new technologies. Therefore, new materials are under

development. Dueto thelarge amount of materialsavailablefor membranes such as

fabrics and polymers, different material models are adopted. The material models

are classified here in two main groups: small and largestrains. In thegroup of small

strainselastoplastic and elastoviscoplastic material modelswere implemented. The

group of large strains comprehend the implementation of the hyperelastic Ogden,

elastoplastic and elastoviscoplastic material models. A new material model is also

proposed and implemented, which is based on NURBS surfaces. Examples are

developed to validate thematerial and implementation.

Emphasis is given to the ETFE material due to its wide use in pneumatic

structures in the last years. The constructions built with ETFE materials show the

efficiency of this material. Numerical analysis with the finite element method are

applied to model theETFE material.

Thepressure–volume couplingincluded in theformulationtakes into account

the variation of the internal pressure in enclosed chambers when the volume is

changed due to the external applied load. Numerical results are compared with

analytical results available in the literature. An analytical formulation for large

strains isalso developed.

Applications of the material models to membranes and pressure–volume

coupling are performed in the present work. The tools developed in the work are

applied to the analysisof astructure in use.

7.1Membrane material models

The elastoplastic material modelsfor small andlargestrainsare considered in

thenumerical modelsof uniaxial and biaxial testsof ETFE–foils. The experimental

results for the biaxial tests for load ratios of 1:1 and 2:1 are compared with

the numerical models. Membrane structures clearly present large deformations by

which thesmall strainmaterial model failsto givegoodresults. Experimental results

validate the implementationandapplicabilit y of the largestrain models.

Theproposed material model based onNURBS (PD–NURBS) was validated

with examples for hyperelastic andelastoplastic material modelswith largestrains.

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ConclusionsandSuggestionsfor futureworks 135

From thesmall error obtained with the resultscomparing thePD–NURBS material

and the conventional material models it can be concluded that the formulation

presented is in accordancewith the resultsof conventional models.

The validation example of the perforated square membrane with Mooney–

Rivlin material model are compared with the variation of the number of control

points for the material model based onNURBS. The results obtained with the dif-

ferent nets of control points are compared with the global convergence of the nu-

merical models. From theresults it i sconcluded that asthenumber of control points

increases the convergencerate also increases and reaches quadratic convergence.

The material behavior is defined with NURBS surfaces with stresses and

strainsin principal directions. TheseNURBSsurfacesaregenerated with theresults

from biaxial tests. The advantage of this material model is that from results of

experimental tests, a material model can describe the material behavior. On the

other hand, the experimental data should provide apoint distribution to generate

goodNURBS surfaces. This point distributioncould result in anecessity of a large

rangeof experimental data.

With respect to timeof the analysisnosignificant differencebetween thePD–

NURBS material and conventional material was observed.

We conclude that this material model is a goodalternative to conventional

material models.

The burst test of a circular membrane clamped at its rim is analyzed. This

test is modeled with finite elements and the numerical models are compared to

experimental results. The elastoplastic material model with vonMisesyield criteria

isconsidered in thenumerical analysis for small and large strains.

The results obtained with the numerical analysis with large strains are in

accordance with the experimental results. On the other hand the results of the

numerical analysiswith small strainsarevalid only in thefirst stepsof the analysis.

These results reinforce the importanceof considering a material model with large

strains to model this typeof material.

7.2Pneumatic s tructures with pressure–volume coup ling

The numerical implementation of pressure–volume coupling for pneumatic

structures was validated with analytical analysis. The analytical analysis in the

literature are for an inflated circular membrane clamped at its rim. Differencewas

observed between the numerical and analytical results and this was accredited to

large strains, which were not considered in the analytical solution available in

the literature. This was confirmed with the development in the present work of

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ConclusionsandSuggestionsfor futureworks 136

an analytical solution with large strain kinematics. The results obtained with this

analytical solutionwere thesame as thoseobtained with thenumerical analysis.

A rectangular inflated cushion of single and double chamber is modeled in

the present work. The results obtained for the case of pressure–volume coupling

andwithout considering thepressure–volume couplingare compared.

For thedouble chamber cushionwith pressure–volume couplingit isobserved

that the displacement result is smaller than the cushion of single chamber, due to

the increase in thedisplacement constraint with the increased chamber.

These analyses show the large difference in the results when the pressure–

volume coupling isconsidered.

An analysis of a pneumatic structure in use is also performed in the present

work. The importance of the pressure–volume coupling is reinforced with this

example by the results of displacement, stresses and strains. The results obtained

for the cases without the pressure–volume coupling are larger than the cases with

pressure–volume coupling.

The analysis with cutting pattern shows the accumulation of the tension on

thestrip unions. Therefore, the cutting patterns should be considered in an analysis

of amembranestructure.

7.3Sugg estions for future works

Based onthepresent work somesuggestions for futureworksare presented:

– Experimental tests for isotropic membranematerials for different stresspaths

should beperformed to generateNURBSsurfacesfor the constitutivematerial

model based onNURBS.

– Extension of the formulation of the material model based on NURBS to

anisotropic materials.

– Consideration of the temperature influence on the material model for pneu-

matic structures.

– Development of experimental analysis of inflated cushions with multi cham-

bers, sincethese experimental analyses arenot available in the literature.

– Wrinklinganalysis in pneumatic structures

– Dynamic analysis in pneumatic structures

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