Adhesion-Detachment Mechanics of Thin Shells in the Presence of Intrinsic Surface Forces

Dissertation Title

Adhesion-detachment Mechanics of Thin Shells in the Presence of Intrinsic Surface Forces

PhD Student: Jiayi Shi

Advisor: Professor Kai-tak Wan

Co Advisor: Professor Sinan Müftü

Department of Mechanical and Industrial Engineering

Northeastern University

Boston, Massachusetts

NORTHEASTERN UNIVERSITY

Graduate School of Engineering

Dissertation Title: Adhesion-detachment Mechanics of Thin Shells in the Presence of Intrinsic Surface Forces

Author: Jiayi Shi

Department: Mechanical and Industrial Engineering

Approved for Dissertation Requirement for the Doctor of Philosophy Degree

Dissertation Advisor: Prof. Kai-Tak Wan Date

Dissertation Co Advisor: Prof. Sinan Müftü Date

Dissertation Reader: Prof. Ashkan Vaziri Date

Dissertation Reader: Prof. Ferdi Hellweger Date

Department Chair: Prof. Jacqueline Isaacs Date

Graduate School Notified of Acceptance:

Director of the Graduate School Date

ACKNOWLEDGEMENT

I would like to thank all people who have helped me during my graduate study. I would

like to express my sincere gratitude to my advisor Professor Kai-Tak Wan for his continuous guidance and support through my Ph.D studies. Working with him for the last four years has been a great honor and an invaluable learning experience. His valuable advices and influences will benefit me for lifetime. I would also like to gratefully thank my co-advisor Professor Sinan

Müftü for his patient guidance and great support. This work would not have been possible without his help. I would like to give special thanks to Professor Ashkan Vaziri and Professor

Ferdi Hellweger for their valuable time to review this thesis. I am thankful to my lab mates

Guangxu Li, Xin Wang, Zong Zong, Michael Robitaille, Masoud Khabiry and Nazli Caner for

creating a warm and friendly atmosphere in the lab. Last but not least, I would like to thank my

parents and my husband for their unconditional love and for being a driving force behind me

throughout my life. The financial support from National Science Foundation Grant CMMI-

0757140 is gratefully acknowledged.

TABLE OF CONTENTS

1. INTRODUCTION...... 1 1.1 Introduction of adhesion ...... 1 1.1.1 Hydrogel contact lens adhesion ...... 4 1.1.2 Cell adhesion ...... 5 1.2 Literature review on adhesion theory ...... 6 1.2.1 Solid adhesion theory ...... 6 1.2.2 Shell adhesion theory ...... 8 1.3 Thesis structure ...... 12 2. ADHESIVE INTERACTIONS ...... 15 2.1 Introduction ...... 15 2.2 Van der Waals interactions between atoms ...... 16 2.3 Van der Waals interactions between surfaces and solids ...... 18 2.4 Surface Energy and classic adhesive models ...... 20 2.4.1 Surface energy...... 20 2.4.2 Classical adhesion models ...... 20 2.4.3 The DLVO potential ...... 25 2.5 Summary ...... 27 3. ADHESIONAL CONTACT OF SPHERICAL SHELL ...... 29 3.1 Introduction ...... 29 3.2 Theory ...... 30 3.2.1 General equation of Reissner’s thin shell theory ...... 30 3.2.2 Two deformation modes ...... 35 3.2.3 Numerical method ...... 39 3.2.4 Thin shell with adhesion ...... 42 3.3 Results and discussion ...... 45 3.4 Summary ...... 52 4. EXPERIMENTAL CHARACTERIZATION OF SOFT CONVEX SHELL ...... 53 4.1 Introduction ...... 53 4.2 Contact lens characterization...... 55 4.2.1 Sample preparation ...... 55

4.2.2 Tensile test and indentation ...... 56 4.2.3 Parallel plate and central load compression test ...... 58 4.3 Preliminary adhesion measurement ...... 67 4.4 Discussion ...... 70 4.5 Summary ...... 73 5. MECHANICAL CHARACTERIZATION OF MOUSE OOCYTE ...... 74 5.1 Introduction ...... 74 5.2 Indentation on single oocytes ...... 77 5.3 Solid sphere (SS) model ...... 80 5.3.1 Model ...... 80 5.3.2 Results ...... 83 5.4 Shell model ...... 86 5.4.1 Hollow spherical shell (HSS) model ...... 86 5.4.2 Liquid-filled spherical shell (LFSS) model ...... 88 5.4.3 Results ...... 89 5.5 Summary ...... 93 6. ADHESION OF CYLINDRICAL THIN SHELL ...... 94 6.1 Introduction ...... 94 6.2 Theory ...... 95 6.2.1 Intersurface forces ...... 96 6.2.2 Mechanical energy ...... 98 6.2.3 Numerical methods ...... 100 6.3 Results and discussion ...... 104 6.3.1 Convergence ...... 104 6.3.2 Deformed profile and stress distribution ...... 105 6.3.3 Mechanical response ...... 108 6.4 Summary ...... 114 7. MECHANICS OF A CYLINDRICAL SHELL IN PRESENCE OF DLVO POTENTIAL ...... 115 7.1 Introduction ...... 115 7.2 Theory ...... 117 7.2.1 Intersurface force ...... 118 7.2.2 Mechanical energy ...... 120

7.2.3 Numerical Method ...... 121 7.3 Results and discussion ...... 125 7.3.1 Equilibrium configuration ...... 125 7.3.2 Mechanical response ...... 127 7.4 Discussion ...... 135 7.5 Summary ...... 137 8. CONCLUSION AND FUTURE WORK ...... 139 8.1 Conclusions ...... 139 8.2 Future work ...... 140 9. REFERENCES ...... 142 10. APPENDIX 1 Matlab program for spherical shell adhesion ...... 157 11. APPENDIX 2 Matlab program for cylindrical shell adhesion ...... 165 12. APPENDIX 3 Fortran program for cylindrical shell within DLVO potential ...... 176 13. VITA ...... 200

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LIST OF FIGURES

Figure 1.1. Hierarchical microstructures of gecko’s foot [4]...... 2

Figure 1.2. SEM of aligned CNTs (Argonne national laboratory)...... 3

Figure 1.3. An elephant is fitted with a contact lens. (News from Radio Netherland Worldwide)...... 4

Figure 1.4. SEM of contaminants on the surface of a contact lens after 3 weeks of use. A clump of dust

and epithelial (skin) cells can be seen at lower left, along with two bacteria (blue)...... 5

Figure 2.1. Van der Waals interaction free energies between bodies for different geometries [23], in which

A is Hamaker constant, C is the coefficient in the atom-atom potential...... 19

Figure 2.2. Lennard-Jones potential as a function of distance, with (a) energy and (b) corresponding disjoining pressure...... 21

Figure 2.3. Comparison between a Herzian contact and a JKR contact under same load [28]...... 23

Figure 2.4. Relation between radius of contact and applied load for various λ [28]...... 24

Figure 2.5. A schematic drawing of Lennard-Jones potential, JKR, DMT, Maugis...... 25

Figure 2.6. Schematic energy versus distance profile of DLVO interaction [23]. (a) Surface repel strongly; small colloidal particles remain “stable”. (b) Surface come into stable equilibrium at secondary minimum if it is deep enough; colloids remain “stable”. (c) Surface come into secondary minimum; colloids coagulate slowly. (d) Surface may remain in secondary minimum or adhere; colloids coagulate rapidly.

(e) Surface and colloids coalesce rapidly...... 27

Figure 3.1. Side view of element of shell in undeformed and in deformed state. Variable stress resultants and couples and load intensity component is also shown...... 31

Figure 3.2. Sketch of a spherical shell being compressed by two parallel plates...... 35

Figure 3.3. Sketch of coordinate system for parallel plate compression. Meridional angle, φ, is taken as the only independent variable...... 36

Figure 3.4. Sketch of coordinate system for central load compression...... 38

Figure 3.5. Algorithm for finite difference computation ...... 42

Figure 3.6. Load-displacement relation with E = 2.0MPa, t = 200µm, and R = 15mm. Elastic energy UE is

the area under loading curve (in grey). Area ABDEA is the elastic energy associated with elastic recovery

due to relaxation...... 43

Figure 3.7. Deformed profiles of points A, B ,C in Figure 3.6...... 43

Figure 3.8. Total potential energy as a function of contact area with E = 2.0MPa, t = 200µm, and R =

15mm. The thick gray curve represents the energy balance at equilibrium. Under fixed load, UT possesses

local minimum or stable equilibrium until “pull-off” where UT shows an inflexion point and the

equilibrium becomes neutral (inset). Under fixed grips, stable equilibrium is always maintained till the

contact radius virtually vanishes...... 44

Figure 3.9. (a) Contact radius as a function of applied load with E = 2.0MPa, t = 200µm, R = 15mm, and a

range of adhesion energy as shown. Shell detachment follows path PGHJ. Grey curve and symbols shows

the trajectory of fixed load “pull-off”. (b) Deformed shell profiles at P, G, H and J for adhesion energy γ =

200mJ/m2...... 46

Figure 3.10. Universal shell adhesion behavior in terms of normalized quantities. Error bars denote

uncertainties due to limited number of elements used in computation. (a) Contact radius as a function of

external load. (b) Approach distance as a function of contact radius. Fixed grips “pull-off” at d† occurs at the terminal end of the energy balance curve. (c) External load as a function of approach distance. “Pull- off” under fixed load occurs at the most negative tensile force at F*, while “pull-off” under fixed grips occurs at the most negative d†...... 48

Figure 4.1. Loading configurations: (a) parallel plate compression, (b) central load compression via a rigid sphere...... 54

Figure 4.2. Stress-strain relation for tensile test ...... 56

Figure 4.3. Load-displacement relations for contact lens indentation test, with dashed line showing the upper and lower bound of tests...... 57

Figure 4.4. Experimental setup showing an immersed lens contained in a square container sitting on a

manual x-y stage...... 58

Figure 4.5. Experimental setup showing (a) Typical side view of a sample lens subject to parallel plate

compression and (b) Typical side profile of sample lens subject to central load compression via a ball

bearing...... 60

Figure 4.6. Typical mechanical response under parallel plate compression. Load as a function of top plate

displacement in linear-linear and log-log scales...... 61

Figure 4.7. Increasing contact radius as function of approach distance as compression proceeds...... 62

Figure 4.8. Contact radius as a monotonic function of applied load...... 62

Figure 4.9. Snapshots of deformed profiles with displacements of 0, 0.5, and 1.0 mm...... 63

Figure 4.10. Superimposed snapshots of deformed profiles compare with theoretical fittings...... 63

Figure 4.11. Load as a function of approach distance...... 64

Figure 4.12. Increasing contact radius as compression proceeds...... 65

Figure 4.13. Contact radius as a monotonic function of applied load...... 65

Figure 4.14. Superimposed snapshots of deformed profile when approach distance is 0µm, 500µm, and

1000µm...... 66

Figure 4.15. Superimposed snapshots of deformed profile compare with theoretical fittings...... 66

Figure 4.16. Schematic drawing of adhesion test configuration ...... 67

Figure 4.17. Loading- unloading curve for contact lens adhesion measurement...... 68

Figure 4.18. In-situ images for Loading- unloading process, corresponding to Figure 4.17...... 68

Figure 4.19. “pull-off” force F* as function of central thickness t, with symbols showing experiment

measurement and solid line curve fitting in (a) linear scale and (b) log scale...... 69

Figure 5.1. Mouse oocyte microinjection system used for force measurement...... 77

Figure 5.2. Image during microinjection of mouse oocytes...... 78

Figure 5.3. Stress-relaxation indentation test. (a) Displacement of the micromanipulator as a function of

time. The micromanipulator controlled the micropipette to indent the oocyte at a constant speed of 60

µm/s (stage I), and maintained a vertical displacement of 40 µm for 45s (stage II). (b) Schematic of the expected force response as a function of time...... 79

Figure 5.4. A two-step solid sphere model of the large-deformation oocyte. (a) Schematic diagram of the two-step deformation, including an initial plate compression (Step 1) and a subsequent flat-punch indentation with the half-space assumption. (b) A meshed 2D axisymmetric model of the oocyte for finite element analysis. (c) The distribution of displacement (left) and Von-Mises stress (right) in an oocyte after compression...... 81

Figure 5.5. Force-deformation data from the young (green) and aged (grey) oocytes at Stage I...... 84

Figure 5.6. Force-time data from the young (green) and aged (grey) oocytes at Stage II. The black lines are typical curve fits to single sets of experimental data...... 84

Figure 5.7. Apparent elastic modulus as a function of time for a sample young oocyte...... 85

Figure 5.8. A 3-D HSS model of mouse oocyte. (a) Displacement distribution of deformed shell with depth of 30 µm. (b) von Mises distribution of deformed shell with depth of 30 µm...... 87

Figure 5.9. A 3-D LFSS model of mouse oocyte. The cytoplasm is modeled as an incompressible fluid.

(a) Displacement distribution of deformed shell. (b) von Mises stress distribution of deformed shell with deep penetration of the indenter...... 88

Figure 5.10. Force measurement during indentation. A typical curve-fit to experimental data by using three models is shown, with (a) Young oocyte and (b) Aged oocyte. Red solid line is LFSS model fit, blue long dashed line is SS model fit and black short dashed line is HSS model fit...... 90

Figure 5.11. A Comparison between experimental deformed profile and theoretical models with blue line

SS model, black line HSS model and red LFSS model(depth of 0, 10, 20, 30 µm)...... 91

Figure 6.1. Schematic of a cylindrical shell deformed by an external load coupled with intersurface attraction or disjoining pressure...... 95

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Figure 6.2. Coordinate system for the deformed shell. Note that ρ and ρ′ indicate the positions vector of a point P before and after deformation. The position vector is referenced with respect to the center of the initially undeformed cylindrical configuration...... 96

Figure 6.3. Lennard-Jones potential for the disjoining pressure and the Dugdale – Barenblatt – Maugis cohesive zone approximation...... 97

Figure 6.4. Incremental deformation of the shell...... 100

Figure 6.5. Flow chart for the solution algorithm...... 103

Figure 6.6. Convergence study with total element number increasing from 100 to 600 at z* = 1...... 105

Figure 6.7. Cylindrical shell adhering to a rigid substrate under zero external load, F = 0: (a) deformed profile in the vicinity of the contact edge and inset showing the global deformation, and (b) pressure at equilibrium. The adhesion force range z*/R is allowed to vary as indicated, while the adhesion energy is maintained at p*z*/ER = 1×10-8...... 106

9 10 Figure 6.8. Effect of repulsive slope ms, ranging from 2x10 to 2x10 : (a) deformed profile in the vicinity

of the contact edge, and (b) corresponding pressure at equilibrium...... 108

Figure 6.9. Delamination trajectory for different adhesion force range z*/R values and fixed adhesion

energy p*z*/ER = 1×10-8. (a) Contact width as a function of applied load for fixed adhesion energy and a

range of disjoining pressure. (b) Applied load as a function of approach displacement. (c) Diminishing

contact width as the external load turns tensile. Delamination follows trajectory ABCDGHP and “pull-

off” occurs at P when the contact reduces to a centerline...... 109

Figure 6.10. Shell profiles for a range of external loads F/γR as indicated, and for z*/R = 1/25 and fixed adhesion energy p*z*/ER = 10-8...... 110

Figure 6.11. Shell profiles close to the contact edge for z*/R = 1/25 and fixed adhesion energy p*z*/ER =

10-8...... 111

Figure 7.1. Schematic of a cylindrical shell deformed by an external load coupled with intersurface force

to find equilibrium at 2omin or 1omin...... 117

viii

Figure 7.2. Disjoining pressure according to DLVO potential and the Dugdale-Barenblatt-Maugis cohesive zone approximation. “Pull-off” from 1omin is shown at the energy barrier peak (c.f. “K” in

Figure 7.7)...... 118

Figure 7.3. Finite element implementation of shell model...... 122

Figure 7.4. Flow chart for the solution algorithm...... 124

Figure 7.5. Deformed profile and contact pressure of a shell resting in 2omin: (a) Equilibrium profile and

(b) corresponding pressure distribution...... 125

Figure 7.6. Deformed profile and contact pressure of a shell resting in 1omin: (a) Equilibrium profile and

(b) corresponding pressure distribution...... 126

Figure 7.7. Contact width as a function of applied load for fixed adhesion energy. The dashed lines show

2omin to 1omin transition for different energy barriers...... 127

Figure 7.8. Snapshots of deformed profile of A, B, C and D in Figure 7.7...... 128

Figure 7.9. Contact pressure of A, B, C and D in Figure 7.7...... 129

Figure 7.10. Snapshots of deformed profile of C, D, H and K in Figure 7.7...... 130

Figure 7.11. Contact pressure of C, D, H and K in Figure 7.7...... 131

Figure 7.12. Eccentricity, or degree of deformation, of the adhering shell as a function of external load.

...... 132

Figure 7.13. Schematic of a rigid shell and deformable shell within effect of intersurface force...... 133

Figure 7.14. External force as function of displacement for a rigid cylinder in presence of DLVO

potential...... 134

LIST OF TABLES

Table 3.1. Comparison between JKR adhesion theory and the shell model ...... 50

Table 5.1. Summary materials properties of young and aged oocytes using SS model...... 85

Table 5.2. Comparison of Elastic modulus of mouse oocyte obtained from three models ...... 92

ABSTRACT

Adhesion is the force that holds particles or surfaces together, which has significant impacts in life sciences such as: cell adhesion, bacterial aggregation and drug delivery vesicles in

micro-scale, and contact lens adhesion in macro-scale. In order to quantify adhesion, several

classical models for solid adhesion were constructed and widely adopted by the scientific

community over the past decades, such as the celebrated Johnson-Kendall-Roberts (JKR),

Derjaguin-Muller-Toporov (DMT), and the Tabor-Muller-Maugis’s JKR-DMT transition models. However, nano-shells / particles, vesicles and biological cells are shells instead of solid spheres, rendering the classical models invalid. A new mechanics model is necessary for shell adhesion.

Adhesion models for two extreme shell configurations are considered in this thesis:

spherical (R1=R2=R) and cylindrical shell (R1=R, R2=∞). The constitutive relation without

adhesion for a spherical shell compressed by a rigid planar plate or point load at the apex is

constructed based on the classical Reissner’s theory. Short range intersurface forces are then

modeled by introducing an additional surface energy to the thermodynamic energy balance.

Mechanical equilibrium is achieved by minimizing the total energy comprising the elastic energy

stored in the shell, the potential energy input from the external load, and the surface energy to

create/destroy new surfaces. Interrelationships between applied load (F), approach displacement

(d), contact radius (a), as well as the deformed shell profile and stress distribution are

generalized for ranges of elastic modulus E, shell curvature R, shell thickness t and adhesion

energy γ. Several model and practical systems are investigated here in this thesis.

Spherical shells. Soft hydrogel based contact lenses are mechanically deformed by an

external load as well as adhesion surface forces with a substrate. The coupled deformation and

adhesion is studied via thermodynamic energy balance and computational mechanics. Another

model is that of a spherical murine oocyte (mouse egg) under a point load of micropipette during

sperm injection. The constitutive relation and deformed geometry are investigated by assuming

the cell to be (i) a classical solid sphere, (ii) a hollow spherical shell, and (iii) a liquid-filled

spherical shell. Liquid-filled spherical shell model yields the best consistency with both force

and profile measurements, which further verifies that cell is a shell instead of a solid sphere.

Cylindrical shell. An adhesion model is also derived for a thin-walled cylindrical shell adhering to a rigid planar substrate in the presence of long range intersurface interactions.

Application is found in cigar-shaped bacterial adhesion. Two distinct surface force potentials are discussed, namely, (i) the classic Lennard-Jones (LJ) potential and (ii) Derjaguin-Landau-

Verwey-Overbeek (DLVO) potential with a secondary energy minimum due to combined van

der Waals attraction and electrostatic double layer repulsion. A Dugdale-Barenblatt-Maugis

(DBM) cohesive zone approximation is adapted such that the surface interaction is modeled as

step functions with finite magnitudes and ranges. The shell is subjected to mixed plate-bending

and membrane-stretching deformation as well as an internal pressure. The nonlinear problem is

solved numerically to generate the pressure distribution within the contact, the deformed

membrane profiles, and the adhesion-delamination mechanics.

The spherical and cylindrical shell models show unique mechanical and adhesion-

detachment behavior distinctly different from their solid counterparts. These computational tools

are indispensable in nano-science and nano-technology in nano-/micro-scale biological,

electronic, and materials structures.

xii

Glossary

φ0 Angle between axis normal and axis of symmetry of undeformed shell

φ Angle between axis normal and axis of symmetry of deformed shell

* φ0 Denotes value of φ0 at the contact edge

β Rotation of normal

r0 Distance from axis of symmetry before deformation

R Radius of undeformed shell

E Young’s modulus

v Poisson’s ratio

t Thickness of shell

K Tensile stiffness

D Bending rigidity

F Total load on shell

εφ, εθ Meridional and circumferential strain

κφ, κθ Meridional and circumferential bending curvature

w, u Deflection parallel and perpendicular to axis of symmetry

V, H Stress resultants parallel and perpendicular to axis of symmetry

pV, pH Surface loads parallel and perpendicular to axis of symmetry

Nφ, Nθ Meridional and circumferential stress resultants

Mφ, Mθ Meridional and circumferential moment resultant

Q Transverse shear resultants

xiii b Unit length

κs, κb Extensional and bending stiffness

ϑ Angular displacement p Disjoining pressure z Surface separation z0 Equilibrium atomic spacing

US Surface energy

UF Mechanical energy

UE Elastic energy

Uint Work due to internal pressure

UT Total energy of the system

Ni Shape function

T Transformation matrix

J Jacobian matrix r Residual vector

ρ Position vector pint Internal pressure ms Repulsive slope

xiv

1 INTRODUCTION

1.1 Introduction of adhesion

This thesis investigates the effect of intersurface forces on the mechanics of very compliant thin shell structures encountered in life sciences and industrial applications.

Adhesion is defined as the force that holds particles or surfaces together in opposition to stresses exerted to pull apart [1], which has significant impacts in life sciences, such as, cell adhesion, bacterial aggregation and drug delivery vesicles in micro-scale, and contact lens adhesion in macro-scale. The forces that cause adhesion can be divided into several categories, including mechanical, chemical, dispersive, electrostatic adhesion. Mechanical adhesion, or mechanical interlocking, describes adhesive materials physically fill surface voids or pores and hold surfaces together by interlocking, for example, rubber bonding to textiles and paper.

Chemical adhesion occurs when the surface atoms of two separate surfaces form ionic, covalent, or hydrogen bonds, for example, most of glue bonding is chemical bonding where the compound is formed at join [2]. Dispersive adhesion arises when two materials are held together by molecular forces, more specifically, van der Waals forces. Although van der Waals force is one of the weakest forces at the atomic level, when acting collectively, van der Waals force can provide significant adhesive force. This effect is skillfully exploited by Geckos when they move on vertical surfaces and ceilings. As shown in Figure 1.1, Gecko’s feet comprise highly hierarchical microstructures, which require minimal attachment force, leave no residue, are directional, detach without measurable forces, are self-cleaning, and work underwater, in a vacuum, and on nearly every surface material and profile. Inspired by the adhesion mechanism of gecko foot hair, researchers developed various biomimic devices [3]. In many situations, surface forces between two objects are due to combination of different mechanisms. 1

Figure 1.1. Hierarchical microstructures of gecko’s foot [4].

Adhesion has significant impacts in life sciences. At the macro level, adhesion can cause

tissues and organs stick together. Normally, internal tissues and organs have slippery surfaces,

which allow them to shift easily as the body moves. However, adhesion could be a major cause

of intestinal obstruction and pelvic pain [5]. At the cell level, all aspects of life and disease are

fundamentally connected to the adhesion of cells. Cancer cells have less adhesion than normal

cells, which allow them to disobey the “social order”, and invade other tissues, through the blood

and lymph systems [6]. Adhesion can also be advantageous when it is used in drug delivery. The

capsules coated with functional molecules can target and interact with specific cells, tissues and

organs. By changing the surface composition, the adhesion of capsules can be controlled for drug

delivery applications [7]. Bacteria adhesion can have beneficial and harmful effects. Beneficial

adhesion allows bacteria to adhere to the intestines, to help digestion and destroying of harmful

organisms; harmful adhesion is an essential step in bacterial pathogenesis or infection.

Adhesion is also important in nano-science. Recent experiments show that the adhesion effects in multi-walled carbon nanotubes (CNTs) is larger than the microstructures in the gecko

feet. Carbon nanotubes have diameters ranging from a few nanometers for single-walled CNTs

to a few tens of nanometers for multi-walled CNTs. Due to their unique mechanical, thermal and

electronic properties, CNTs have many potential applications such as field emitters for displays,

electronic biosensors and as reinforcing elements in composites. The problems related to

adhesion arise in the many situations. For example, CNTs are extremely difficult to separate once they form bundles. Figure 1.2 gives a Scanning Electron Micrograph (SEM) picture of

aligned CNTs, with many of them deforming and adhering with each other due to adhesion.

Figure 1.2. SEM of aligned CNTs (Argonne national laboratory).

There have been many studies on the mechanical behavior of CNTs, including

experimental study, continuum modeling and numerical simulation using molecular dynamics

(MD) and the finite element (FE) method [8-10]. Robertson et al. showed that individual tubes can

be modeled as shells, where specific elastic properties and mechanical thickness of tube wall has

to be identified. Good agreement is shown between MD and FE methods with this method [11].

The present study mainly focuses on modeling mechanical deformation of thin shell structures in presence of intersurface forces. The models are used to characterize the mechanical and adhesion phenomena of (i) hydrogel contact lens adhesion in macro- scale and (ii)

cell/bacteria adhesion in micro- scale.

1.1.1 Hydrogel contact lens adhesion

Hydrogel soft contact lenses have been used for vision correction for over 30 years, and have become a successful and preferred choice of vision correction for millions of people worldwide. Not only human, elephant wears contact lens, too. Figure 1.3 shows specialists fitted an elephant with contact lens, which can help the elephant release the pain from damaged cornea.

Soft contact lenses are composed of polymeric hydrogels, which consist of a water-swollen network of crosslinked polymer chains [12]. The porous structure allows for transportation of oxygen, water and other nutrients through the lens to the cornea.

Figure 1.3. An elephant is fitted with a contact lens. (News from Radio Netherland Worldwide).

One major problem associated with contact lenses is microbial contamination of the lens surface [13]. Adhesion of bacteria to contact lenses is considered a primary risk factor of serious corneal problems [14]. The contact lenses provide a suitable substrate for bacterial adherence and biofilm formation. Figure 1.4 shows the bacteria contamination on used contact lens.

The adhesion between contact lenses and human cornea can cause serious eye health problem. The corneal interaction with the contact lens can overwhelm the protective mechanisms

4 of the cornea, increasing the ability of microbial cells to adhere to the cornea and progress to microbial keratitis (MK). The development of contact lens related MK is believed to follow a sequence of events beginning with bacterial contamination of the lens and corneal compromise culminating in corneal infection. About 10 to 15% of associate patients have some loss of vision.

Several studies have examined the ability of various bacteria to adhere to different contact lenses and have found that adhesion depends on a variety of factors including lens material, duration of wear, and surface roughness [15]. The surface properties of contact lenses including friction, adhesion, and structural arrangement of polymer chains are not well understood [16]. Therefore it is critical to establish the relationships between the chemical and physical properties and the biological response of the tissue as observed in wear comfort, protein adhesion and bacterial infection [17].

Figure 1.4. SEM of contaminants on the surface of a contact lens after 3 weeks of use. A clump of dust and epithelial (skin) cells can be seen at lower left, along with two bacteria (blue).

1.1.2 Cell adhesion

All aspects of life and disease are fundamentally connected to the adhesion of cells, both with each other and to the extracellular matrix (ECM) surrounding them. In some cases, relatively strong adhesion is required for preserving and maintaining the integrity and function of

living tissues, while in other cases, relatively weak adhesion is required for cell movement such

as migration and rolling [18]. Adhesion of cells to ECM through focal adhesion complexes

provides both signaling and structural functions. ECM regulates cell and tissue morphogenesis

by altering the structure of the intracellular cytoskeleton (CSK) [19]. Cell migration can be guided

by the rigidity of ECM substrate [20]. Stem cells, in particular, can sense the stiffness of their

substrate and modify their chondrogenesis. The matrix elasticity could direct their lineage to

brain cells, muscle cells, or bone cells [21, 22].

The adhesive interactions between cell and substrate generally comprise specific and non-specific interactions. The non-specificity includes all generic interactions between cell and substrate, such as van der Waals interactions, electrostatic interactions, hydrostatic forces, and membrane undulation associated with thermodynamic fluctuations of the cell membrane [23]. The

attraction can also be highly specific to the interacting materials, as is the case for biological

materials when certain proteins (ligands) on one surface bond exclusively with certain proteins

(receptors) on other surface. The ligand-receptor bonds are influenced and mainly dominated by

a background field of electrostatic, hydrogen bonding, steric, and hydrophobic interactions [24].

However, in the extreme, if the non-specific interactions are strong and attractive, cell adhesion

may occur without ligand–receptor interactions [25].

In this thesis, only non-specific interactions between cell-cell and cell-substrate are

considered.

1.2 Literature review on adhesion theory

1.2.1 Solid adhesion theory

The mechanics of contact between solids has been investigated extensively in studying

the surface energy of materials. The classical models to study solid sphere adhesion are the

Johnson-Kendall-Roberts (JKR) [26], Derjaguin-Muller-Toporov (DMT) [27], and the Tabor-

Muller-Maugis’s JKR-DMT transition [28] models. These are widely adopted in virtually all engineering and even biological adhesion models, such as, MEMS stiction [29], adhesion of living

cells [30], nanoindentation [31], atomic force microscopy (AFM) [32], gecko feet adhesion [33, 34].

JKR theory extends the classical Hertz theory, which discusses the mechanical contact

between a solid sphere and a planar rigid substrate or two solid spheres in the absence of

adhesion, by including intersurface attraction due to an ideal force with an ideal infinite

magnitude and zero range [26]. Here, a finite sized contact circle arises even in the absence of the

external load, and an external “pull-off” force is required to pull the adhering sphere out of

contact. Another classical adhesion model for an infinite intersurface force range is due to

Derjaguin-Muller-Toporov (DMT) which predicts in a distinctly different mechanical response

[27]. For quite some time, it is not clear which theory is correct since they are based on different

sets of assumptions. The dilemma is resolved by Muller [35], followed by Greenwood [36], and

Feng [37], using a realistic Lennard-Jones surface force law. To make it convenient, Tabor [28] and

then Maugis [38] adopt the Dugdale cohesive zone model by assuming a constant adhesive stress

p* and a cut off adhesive distance z*. The transition between JKR and DMT model is governed

3 by the Tabor parameter λ = (Rγ 2 / Ez * )1/ 3 ,which depends on the radii R, a combination of two

sphere radii R1 and R2, elastic modulus of sphere E, a combination of E1 and E2, work of

adhesion γ and cohesive range z*. It is shown that λ > 3 in the JKR limit and λ < 0.25 in the DMT

limit. These classical models are empirically shown to be valid in many systems spanning a wide

range of materials, interfaces and dimensions.

Chaudhury et al. [39], Johnson and Greenwood [40] and Wu [41] further extend the model to

adhesion of a solid cylinder lying on its curved side to a rigid planar substrate based on the JKR-

DMT assumptions. Johnson [40] and Morrow [42] both deduce a similar Tabor/ Maugis parameter,

3 λ = (Rγ 2 / Ez * )1/ 3 , which governs the transition from a stiff cylinder with long surface force

range (DMT like) behavior to a soft cylinder with short force range (JKR-like) with same critical

value. Chaudhury perform experiments to measure the surface energy of polymers using long

cylinders [39]. Unlike the pull-off force being independent of elastic modulus for sphere contacts,

the pull-off force is found to be dependent on elastic constants for cylindrical contacts. Barthel

[43] review the methods developed in the past decade to account for adhesive interactions in

elastic contacts, and emphasize the connection between the local, physical mechanism of

adhesion and the macroscopic mechanical loading.

Other researchers also have done extensive work on this topic. The extension of the

elastic JKR theory to a linear viscoelastic material is addressed by Lin [44]. Here, the stress

intensity is no longer dependent on the current contact area and applied load, but the entire

loading history. Johnson and Greenwood further extend the JKR theory to contact between

elliptical solids, where the contact area remains elliptical, and the eccentricity varies with

external load [45].

1.2.2 Shell adhesion theory

The JKR theory has been used with much success for assessing solid/solid interfacial

interactions for over four decades. However, many nanoparticles and biological vesicles are

hollow shells of various shapes, such as spherical and cylindrical shape. In the presence of strong

interfacial adhesion, a shell deforms and conforms to the substrate geometry, a concentrated

compressive stress distribution forms at the contact edge, and attenuates rapidly within contact region. The JKR theory is inappropriate to model such problem. A new mechanics model is

needed for shells. Adhesion studies of thin shell structure help to understand adhesion behavior

of cells, vesicles, carbon nanotubes (CNTs) and many other micro- and nano- structure.

The mechanics of adhesion and separation of a 1-D membrane thin film with no intrinsic

curvature in contact with a substrate is studied widely [46]. Wan [47, 48] derive the adhesion- delamination mechanics of 1-D rectangular and 2-D axisymmetric planar membranes in the presence of zero range intersurface forces with mixed plate-bending and membrane-stretching deformation. Li [49, 50] further extends the study of thin film with substrate to long-range

1/ 4 intersurface forces. A new dimensionless parameter λ = [6(1−ν )R 4 /γ 3 Et] p* , similar to Tabor

parameter, relating the membrane dimensions (radius, R and thickness, t), material properties

(elastic modulus, E and Poisson’s ratio, v) and adhesion properties (interface energy, γ and

disjoining pressure, p*) is derived to account for the transition from JKR (large λ) to DMT limit

(small λ) for thin film adhesion.

When a circular thin film is pressurized with a uniform hydrostatic pressure, the planar

film bulges up and gives rise to a similar shell configuration. Contact is formed between bulged

up film and flat rigid substrate. The attractive forces between thin film and the rigid surface

affect the radius of circular contact area. The so called “pressurized blister test” is widely used to

measure the thin film adhesion bonded to a substrate [48, 49, 51]. The relation between the pressure,

the adhesion forces and contact radius is investigated by Plaut and Dillard [52], with both JKR and

DMT approaches to adhesion. A similar type of problem is treated by Shanahan [53, 54] with JKR

assumption only, in which a pressurized spherical membrane, more like a “balloon”, comes into

contact with a flat substrate. Free energy changes due to stretching, bending, mechanical contact

force and adhesion are addressed and minimized to obtain equilibrium. Shull and colleagues [30,

55] also investigate the adhesion of an inflated membrane on a rigid planar substrate, and model the mechanical deformation and adhesion mechanics by rubber elasticity and energy balance.

Unlike thin film adhesion model with no intrinsic curvature, classical shell theory

addresses problems of surfaces with intrinsic curvatures. Schwarz investigate the mechanical

properties of multi-walled hollow nanoparticles based on continuum elasticity theory and

pairwise summation of van der Waals interactions [56]. The effects of adhesion, internal pressure

and shear flow on the deformation and mechanical stability of hollow nanoparticles are also

investigated. Springman and Bassani consider a shallow spherical cap adhering to a substrate

with flat or sinusoidal topography using Reissner’s shell theory with mechano-chemical coupling

effect [57, 58]. Tamura studies the adhesion induced buckling of spherical shells onto a rigid

substrate by means of numerical minimization of the sum of the elastic and adhesion energies

[59].

Cylindrical shell is another interesting geometry to study. Tang et al. investigate the

theoretical adhesion between two identical single-walled CNTs using the linear elastica theory based on JKR only [29]. Effective elastic properties are established by comparing with MD

simulations. Glassmaker and Hui theoretically study the self adhesive contact of a nanotube folded by 180o [60]. The tube is modeled as a 2-D elastica with a large range of material

properties. The analysis shows how the tube dimensions and shape depend upon material

properties and the film thickness. Mockensturm and Mahadavi model self-collapse phenomenon of single-walled CNTs due to van der Waals interaction using elastica [61]. Majidi and Wan

construct a model for two dissimilar thin-walled cylindrical shells with a range of stiffness and

radius values in the JKR spirit [62].

In biological context, adhesion of vesicles, which consist of a bilayer membrane of

amphiphilic lipid molecules, represent an essential step for many processes such as endocytosis

and exocytosis. Adhesion is also important in biotechnological process such as drug delivery by

artificially-prepared vesicles. Neglecting the details of the molecular structure within the

membrane, vesicles can be treated as continuum shell structures. The process of cell/vesicle

adhesion and detachment from a substrate is usually modeled in two ways in literature: non-

specific interactions, denoting accumulated effects due to van der Waals interactions,

electrostatic interactions, hydrostatic forces and specific ligand-receptor interactions. Seifert

theoretically studies adhesion of vesicles to intersurface and membranes, with non-specific

interactions [63-65]. Cheng et al. [66, 67] and Liu et al. [66, 67] perform simulated the micropipette mediated process of cell spreading and cell detachment from a flat substrate with specific receptor-ligand bindings. Boulbitch [68] present a model for detachment of a vesicle from a

substrate by means of a time-dependent detachment force. Lin and Freund [69] study the time-

dependent detachment of a vesicle from a substrate in presence of specific ligand-receptor

interactions.

Many experimental techniques have been developed to quantify cell adhesion, including

atomic force microscopy [6], optical traps [70], micropipette manipulations [71]. With the

advancement of high resolution equipments, it is possible to measure various quantities of

adhesive contacts at the micro- and nano- levels. Flattened contact radius as a function of the applied force is measured for a hemisphere interacting with a variety of materials by using optical microscope [72] and the surface forces apparatus [73]. Another common way to test the

strength of an adhesive contact of a cell or vesicle is to measure the force of detachment. De

Gennes and his collages [71] measure the pulling force applied on the pipette to detach vesicle

from flat plate and further derive the adhesion energy. This model assumes that the contact

radius between the cell and the substrate remains constant during unbinding, and that the

deformation is small. Evans and his coworkers experimentally measure the rupture strength

between two bonded red blood cells [74]. Colbert investigate the separation process of two living

cells using micropipette aspiration techniques [75].

1.3 Thesis structure

Although the solid-solid adhesion models have been extensively studied, variable models

of thin shell adhesion are urgently needed. In this thesis, four models (Chapter 3, 5, 6, 7) of shell

structures are constructed to tackle technological relevant adhesion problem. The various types

of basic interface interactions are summarized in Chapter 2. Adhesion models for two extreme shell configurations are considered in this thesis: (i) spherical shell (R1=R2=R) and (ii) cylindrical

shell (R1=R, R2=∞). Spherical shell problems are investigated in Chapters 3 - 5.

In Chapter 3, a geometrically nonlinear shell, deformed with two loading configurations

without adhesion is discussed first, specifically (i) parallel plate compression and (ii) central load

compression. Reissner’s nonlinear shell theory is employed to solve for axisymmetric

deformation for either closed spherical shells or open caps. Surface energy is then taken into

account by following the classical JKR framework.

In Chapter 4, mechanical and adhesive properties of hydrogel contact lens are

characterized by the shell model proposed in Chapter 3. The elastic modulus of the materials is

deduced from different loading configurations. Initial lens- to- substrate adhesion is also measured by using parallel plate compression mode.

In Chapter 5, from macro to micro scale, finite element models are constructed to

simulate indentation of mouse oocyte test by using finite element software ABAQUS 6-9.2.

Single mouse oocytes is treated as (i) a spherical solid according to the classical Hertz contact

theory as widely adopted by the biomedical community, (ii) a hollow spherical shell where

bending and stretching are the dominant deformation mode, and (iii) a liquid filled spherical

shell, respectively. The three methods yield vastly different elastic moduli from the force- displacement response. Chapter 5 further improves the concept that biological cell behaves more like a shell than a solid.

Cylindrical shell problems are investigated in Chapters 6 - 7. In Chapter 6, adhesion of a

cylindrical shell to a rigid substrate in single energy minimum potential is studied by using the finite element method. A Lennard-Jones like potential is introduced to mimic the convoluted surface force potentials such as electrostatic and van der Waals. The highly nonlinear problem is numerically solved by the Newton-Raphson method to generate the deformed membrane profiles, the adhesion-delamination trajectory and mechanical responses. Relations between external load, contact area, vertical displacement, and internal pressure to achieve mechanical equilibrium and the associated membrane stress and strain are derived.

Chapter 7, extended from Chapter 6, investigates adhesion behavior of cylindrical shell in presence of oscillatory energy potential according to the Derjaguin-Landau-Verwey-Overbeek

(DLVO) theory. The corresponding intersurface forces comprise a short-range and a long-range attraction separated by an intermediate repulsion. An equilibrium solution is only found at either primary (1o) minima or secondary (2o) minima by an explicit dynamic method. Interrelationships between external load, contact area, approach distance, and deformed shell profile, combined with the interplay of 1o and 2o minima are derived.

In Chapter 8, conclusion and suggestions for future work are summarized. The following

publications have resulted from the work contained in this thesis:

1. J. Shi, S. Müftü, K.T. Wan. “Adhesion of a Compliant Cylindrical Shell onto a Rigid

Substrate”, Journal of Applied Mechanics, 2012, 79(4): 041015.

2. X. Liu, J. Shi, Z. Zong, K.T. Wan, Y. Sun, “Elastic and Viscoelastic Characterization of

Mouse Oocytes Using Micropipette Indentation”, Annals of Biomedical Engineering, 2012,

DOI: 10.1007/s10439-012-0595-3.

3. J. Shi, M. Robitaille. S. Müftü, K.T. Wan, “Deformation of a Convex Shell by Parallel Plate

and Central Compression”, Experimental Mechanics, 2012, 52(5): 539-549.

4. J. Shi, S. Müftü, K.T. Wan, “Adhesion of an Elastic Convex Shell onto a Rigid Plate”,

Journal of Adhesion. 2011, 87(6): 579-594.

5. J. Shi, S. Müftü, A. Gu, K.T. Wan. “Adhesion of a Cylindrical Glycoprotein Shell in the

Presence of DLVO Surface Potential”, submitted, 2012.

6. J. Shi, X. Liu, Y. Sun, K.T. Wan. “Indentation of a single cell: spherical solid or shell”, To be

submitted, 2012.

2 ADHESIVE INTERACTIONS

In this dissertation, we consider adhesive contacts of elastic systems for which continuum

assumption holds the adhesive contact interaction between two bodies originates from the

interaction of individual atoms. These interatomic interactions are characterized by weak, long

range van der Waals interactions. Although van der Waals interactions are one of the weakest

forces between atoms and molecules, when acting collectively, van der Waals forces can provide

significant adhesive forces between two solids. In this Chapter, van der Waals interactions that

give rise to adhesive interaction is introduced. Various types of van der Waals potentials are

studied, integrating from atom-to-atom, atom-to-surface, to surface-to-surface. Several classic

adhesive models and the assumptions used for simplification are introduced. These models are

found to be important for studies of adhesion in shells.

2.1 Introduction

It is well established that there are four distinct forces in nature, which are, in decreasing

order of intensity: Strong interaction, electromagnetic forces, weak interaction, and gravitational

interactions [23]. All the bonds between atoms or molecules can also be divided into strong bonds

(around one hundred kcal/mole) and weak bonds (a few kcal/mole). Strong bonds include the

covalent bonds, the ionic bonds and the metallic bonds. Weak bonds include the hydrogen bonds

and van der Waals forces. In this thesis, we are interested in non-specific, weak interactions,

such as long range van der Waals forces, with the reason being that van der Waals forces are

always present. Van der Waals forces determine most of the properties of liquids, viscosity, surface tension and also play a role in the stability and the coagulation of colloids.

2.2 Van der Waals interactions between atoms

In this section, the van der Waals interactions between particles (atoms, molecules, or ions) are introduced. In contrast to other types of forces that may or may not be present depending on the properties of the molecules, van der Waals forces between atoms and molecules are always present. They differ from covalent and ionic bonding in that they are caused by correlations in the fluctuating polarizations of nearby particles [23]. Attractive

intermolecular interactions consist of four types: Dipole–dipole interactions (known as Keesom

forces), Ion–dipole interactions, Dipole-induced dipole interactions (known as Debye forces),

and Instantaneous dipole-induced dipole interactions (known as London dispersion forces).

Among these, the combined of Keesom, Debye and London dispersion interactions are known

collectively as the van der Waals attraction. The total long range van der Waals interactions

between two particles collect contribution from three distinct types of interactions:

6 U VDW (z) = U Keesom + U Debye + U London = −CVDW / z (2-1)

Keesom interactions

Dipole–dipole interactions are electrostatic interactions of permanent dipoles in

molecules. These interactions tend to align the molecules to increase the attraction and reduce

potential energy. An example of a dipole–dipole interaction can be seen in hydrogen chloride

(HCl). The positive end of a polar molecule will attract the negative end of another molecule and

cause them to be arranged in a specific arrangement. Keesom interactions are attractive

interactions of dipoles that are Boltzmann-averaged over different rotational orientations of the

dipoles, with expression of [23]

 u 2u 2  1 U (z) = − 1 2  (2-2) Keesom  πε ε 2  6  3(4 0 ) kT  z

where u1 and u2 are dipole moment, ε0 is the dielectric permittivity in vacuum, ε is the relative permittivity, k is Boltzmann constant, T is the temperature, and z is the intermolecular distance. It

is one of the three important interactions, that contributes to total van del Waals interaction.

Debye interactions

Debye interactions describe the interaction between a polar molecule and a non-polar

molecule, which is analogous to the ion-dipole force except that the polarizing field comes from

a permanent dipole rather than a charge. The dipole-induced dipole interaction is weaker than

dipole–dipole interaction, but stronger than London dispersion interaction. The energy is

described as [23]

 u 2α + u 2α  1 U (z) = − 1 2 2 1  (2-3) Debye  πε ε 2  6  (4 0 )  z

with α1 and α2 are dipole polarizabilities of the respective atoms.

London dispersion interactions

The London dispersion force is a temporary attractive force that results when the

electrons in two adjacent atoms occupy positions that cause the atoms to form temporary dipoles.

Dispersion forces are present between all molecules. London’s famous expression for the

dispersion interaction energy between two identical atoms or molecules is [23]

 3α α I I  1 U (z) =  1 2 1 2  (2-4) London  πε 2 +  6  2(4 0 ) (I1 I 2 )  z

6 here I1 and I2 are the first ionization potentials of the atoms. It is noted that the 1/z distance

dependence is the same as that for the Keesom and Debye interactions.

Lennard-Jones potential

The Lennard-Jones potential (LJ potential) is a general mathematically model proposed by John Lennard-Jones in 1924. The potential approximates the interaction between a pair of

neutral atoms or molecules, combining the effect of der Waals attractive interaction and the

electron cloud overlap interaction by Pauli Exclusion Principle [76]. The famous “exclusion

principle repulsion” or “Pauli repulsion” states that the main repulsion between atoms is not due

to repulsion between the nuclei, but due to the repulsion to keep electrons apart. The most

common form of the LJ potential is given by

 12 6   z0   z0  U LJ (z) = 4U 0   −    (2-5)  z   z  

where U0 denotes the potential energy minimum, z0 is the distance at which the inter-particle

potential is zero, z is the distance between atoms. Potential minimum is reached by dULJ/dz = 0,

1/6 −12 at distance of z = 2 z0. The z term, which is the repulsive term, describes the repulsion at

short ranges due to overlapping electron orbitals and the z−6 term, which is the attractive long

range term, describes the van der Waals attraction at long ranges. Due to its computational

simplicity, the Lennard-Jones potential is widely used in computer simulations.

2.3 Van der Waals interactions between surfaces and solids

The long-range van der Waals interactions between two solids or surfaces are considered

in this section. Since the interactions potentials between a pair of atoms are already derived in

previous section, the energies of all the atoms in one body can be integrated first, to obtain an

overall potential between an atom and a surface, and further integrated in other body to obtain

potential between two surfaces. De Boer in 1936 derived the interaction potential between two

half spaces by integration of the van der Waals pair potential [77]. Surfaces with different

18 geometries result in different potential. The van der Waals interaction potentials for some common geometries are shown in Figure 2.1. The potential form is expected to be more complicated if other long-range interactions are considered, for example, electrostatic and steric- polymer force in liquid environment, and other types of steric forces.

Figure 2.1. Van der Waals interaction free energies between bodies for different geometries [23], in which A is Hamaker constant, C is the coefficient in the atom-atom potential.

2.4 Surface Energy and classic adhesive models

2.4.1 Surface energy

For the various interaction potentials between molecules discussed above, the energy minimum is taken as the adhesion energy, or work of adhesion. Work of adhesion W between two media surfaces (media 1 and media 2) is defined as free energy change, or reversible work done, to separate two media from contact to full separation at infinity, which is given in unites of energy per unit area: J/m2. Noting that the process of creating unit area of surface is equivalent to separating two half-unit areas from contact, therefore surface energy change γ can be written as

1 γ = W (2-6) 1 2 11

Where the subscript 1 indicates the contacting material. When two media surfaces are in contact, the free energy change in expanding their interfacial area by unit area is known as interfacial energy, or surface energy. The energy associated with expansion process can be understood by two hypothetical steps: unit areas of media 1 and 2 are first created, and then brought into contact. The total free energy change γ12 is therefore

1 1 γ = W + W −W = γ + γ −W (2-7) 12 2 11 2 22 12 1 2 12

Where W11, W22 are the internal interactions of units of each material component, while W12 express the interaction between units of the two components.

2.4.2 Classical adhesion models

Although the original derivations only account for the van der Waals attraction, the integration can be repeated with the Lennard-Jones potential as shown by Maugis [28]. The general potential between two adhering surfaces is given by [28]

z z U (z) = C [( 0 ) a1 − 0 ) a2 ] (2-8) LJ s z z

The choice of the constant Cs, and the power indices a1 and a2 depends on the material properties, surface geometries and environment.

The disjoining pressure is given from differentiating of Equation (2-8) with respect to z

dU (z) z z p(z) = LJ = C ' [( 0 ) a3 − 0 ) a4 ] (2-9) dz s z z

Figure 2.2 shows schematic drawing of Lennard-Jones potential as a function of spacing z and the corresponding disjoining pressure, with negative pressure in short range indicating repulsion and attractive pressure in long range decaying with increasing separation. Equilibrium is reached at z = z0. Several classical solid adhesive models are introduced here by making different assumptions.

Figure 2.2. Lennard-Jones potential as a function of distance, with (a) energy and (b) corresponding disjoining pressure.

Hertz theory

The classical Hertz theory [78] discusses the problem of a sphere pressing into an elastic

half-space, or of the contact of two elastic spheres in absence of adhesion. Hertz gives the stress

distribution in the area of contact and the interrelationship of the measurable quantities of applied

compressive load FH, approach distance dH and contact radius aH as follows:

3 FH R aH = (2-10) K

1/ 3 a  F 2  = H =  H  d H  2  (2-11) R  K R 

1− v 2 1− v 2  1 3  1 2  where =  +  , with vi, Ei (i =1, 2) indicate material properties Poisson’s ratio K 4  E1 E2  and Elastic moduli of two materials, respectively.

Johnson-Kendall-Roberts (JKR) theory

Around the contact region, the gap between contacting surfaces is usually comparable to the size of the atoms, therefore, adhesive surface force due to long-range intermolecular interactions become significant. Johnson et al. presented a model that includes intersurface attraction, by modeling an ideal force with infinite magnitude and zero range and established the

now celebrated Johnson-Kendall-Roberts (JKR) theory [26]. This work showed that a non-zero

contact circle arises even in the absence of an external load, and a “pull-off” force is required to

spontaneously pull the adhering sphere out of contact. The surface energy causes an infinite

tensile stress to act at the contact edge. The equilibrium relation between radius of contact aJ and

applied load FJ is given as follows,

 2  F R 3πγR 3πγR  3πγR  a3 = J 1+ + 2 +    (2-12) J K  F F  F    J J  J   where γ is work of adhesion, with unit of [J/m2], related to the energy needed per area to create new surface. A comparison between JKR and Hertz contact is shown in Figure 2.3. In particular, at the critical “pull-off” point, a non-zero contact radius and a critical “pull-off” force are derived as follows

3 Fc = − πγR (2-13) 2

1/ 3  3πγR 2  =   amin   (2-14)  2K  Note that the critical “pull-off” force is independent of Young’s modulus E.

Figure 2.3. Comparison between a Herzian contact and a JKR contact under same load [28].

Derjaguin-Muller-Toporov (DMT) theory

Another adhesion model for an infinite intersurface force range is due to Derjaguin-

Muller-Toporov (DMT) which results in a distinctly different mechanical response [27]. In the

DMT theory, adhesion forces act in an annular zone around the contact but without deforming

the profile, which remains Herzian. The “pull-off” force obtained is incompatible with the JKR

theory with magnitude of

Fc = −2πγR (2-15)

Maugis’ cohesive zone theory

To reconcile the two conflicting models, Tabor [79] points out that the main defect of the

DMT theory is to neglect the deformations due to adhesion forces around the contact ,whereas

that of JKR theory is to neglect the adhesion forces outside the contact area. He and later Maugis

[38] adopt the Dugdale cohesive zone model to derive the JKR-DMT transition for an intermediate finite force magnitude p* and range z*. The transition from JKR theory toward DMT

theory is governed by Tabor parameter, λ= (Rγ2/E z*3)1/3, where JKR theory applies for λ >3 and

that of DMT theory for λ<0.25. The relation between radius of contact and applied load for

transition from JKR-DMT is shown in Figure 2.4.

Figure 2.4. Relation between radius of contact and applied load for various λ [28].

The transition from JKR theory to DMT theory is also shown in Figure 2.5, in terms of disjoining pressure p* and intersurface separation z*. Though the Lennard-Jones potential (solid

line) is good to describe the interfacial interaction, the mathematical function is too complex to

solve real problems. In order to simplify the calculation, the Dugdale cohesive zone model is

used where the adhesive forces are inside the cohesive zone (0< z ≤ z*) and zero outside (z > z*).

The product of attractive force magnitude and cohesive zone range equals the adhesion energy,

and this quantity is made to equal the area under the Lennard-Jones potential curve. When the intersurface force is long range and force magnitude is small, the DMT limit is valid. The JKR

model is best suited for short range intersurface force with infinite magnitude. Both of the limits

are shown in Figure 2.5.

Figure 2.5. A schematic drawing of Lennard-Jones potential, JKR, DMT, Maugis.

2.4.3 The DLVO potential

In case the particles are dispersed in a liquid, electrostatic forces develop due to charge

exchange at the liquid-particle boundary. This can either be due to ionization of surface groups, or adsorption of ions from solution onto a previously uncharged surface. Whatever the charging mechanism is, the surface charge is balanced by an equal number of irons with opposite charge.

This forms a layer near the solid surface, known as the electrical double layer. In the presence of

an electrolyte, interaction of two surfaces with electrostatic double layers gives rise to a potential

energy, Us, as a function of intersurface separation, z. This potential energy is described by

Derjaguin, Landau, Verwey and Overbeek as follows [80, 81],

−κz A d p πε 0ε r d p  1+ e  − κ  = − + ψ ψ   + ψ 2 +ψ 2 − 2 z (2-16) U s (z) 2 p c log −κz  ( p c ) log(1 e ) 12z(1+14z / λl ) 2  1− e  

where dp a characteristic dimension of the particle or equivalent diameter, A is the Hamaker constant, λl is the characteristic wave length of the dielectric, ε0 is the dielectric permittivity in vacuum, εr is the dielectric constant of water, κ is the inverse Debye length, ψp and ψc are the

surface potentials of the particle and substrate, respectively. The first term in Equation (2-16)

represents the van der Waals attraction between two particles or between a particle and a

substrate. The second term represents the typical electrostatic double layer repulsion. The double

layer interaction between surfaces or particles decays exponentially with distance. The

characteristic decay length is known as the Debye length.

Figure 2.6 shows schematically the various types of interaction potentials that can occur

between two surfaces under the combined action of these two forces. Depending on the

electrolyte concentration and surface charge density, one of the following situations may occur:

(i) With highly charged surface in dilute electrolyte (i.e., long Debye length), there is a strong

long-range repulsion that peaks at some distance (curve (a) in Figure 2.6). (ii) In more

concentrated electrolyte, there is a significant secondary minimum before the energy barrier.

(curve (b) in Figure 2.6). (iii) As the surface charge approaches zero, the interaction curve

approaches pure van der Waals curve and two surfaces attract each other strongly at all

separations (curve (e) in Figure 2.6).

For a colloidal system, even though the thermodynamical equilibrium state may be with the particles in contact in the deep primary minimum, the energy barrier may be too high for the particles to overcome. When this happens, the particles will either sit in the weaker secondary minimum or remain fully dispersed in solution.

2.5 Summary

This chapter introduces the forces that drive adhesion in both dry and aqueous environment. The mathematical model, Lennard- Jones potential captures the basic mechanical

27 characteristics of adhesion. Several classic adhesive models that introduced in this chapter are adopted for shell adhesion analysis in following chapters.

3 ADHESIONAL CONTACT OF SPHERICAL SHELL

In this chapter, an elastic spherical shell with uniform thickness is compressed in two

loading configurations and undergoes large geometrical deformation. The shell profile and

contact stresses under large deformation are obtained numerically by using the Reissner’s shell theory [82]. For parallel plate compression, a thermodynamic energy balance following the classical JKR model is established to construct the adhesion mechanics, such that the sum of potential energy of the external load, elastic energy stored in the elastic shell, and surface energy to create new surface is minimized. The relationships between the applied load, approach distance, contact radius, and deformed profile, as well as the “pull-off” phenomenon, are investigated.

3.1 Introduction

The classical solid sphere adhesion models of Johnson-Kendall-Roberts (JKR) [26],

Derjaguin-Muller-Toporov (DMT) [27], and the Tabor-Muller-Maugis’s JKR-DMT transition [28]

are widely adopted in virtually all engineering and even biological adhesion. Despite their

successful application, these models are inadequate to account for adhering shells. In the

presence of strong interfacial adhesion, a shell deforms and conforms to the substrate geometry,

yet the compressive stress concentrates at the contact edge while the stress is quite negligible

within the contact.

Literature devoted to shell analysis without adhesion is quite extensive. The earlier

models of shells formed by the full revolution of a curve are dated back to the 1960’s. Reissner

[82] derived the general solutions for shallow spherical shells with small deformation.

Timoshenko derived the general linear equations for small deformation of spherical shells [83].

Taber [84, 85] examined spherical shells under concentrated load including the effect of an internal

incompressible fluid, and further investigated indentation with a range of indenter radii and the

associated transition from point-load type to flat-plate type compression [86]. Ashwell [87], Ranjan

and Steele [88] presented analytical solutions to the problem of large deflections of spherical

shells with a concentrated load applied at the apex. Updike and Kalnins [89] employed Reissner’s

theory [90] to analyze a spherical shell compressed between rigid plates. Adhesion of thin-walled

shells is by and large rare in the literature until recently.

In this chapter, we first extend Updike’s work to parallel compression and shaft load on a

convex shell, and then consider adhesion contact of a spherical shell being compressed by two

parallel plates in the presence of adhesion at the shell-substrate interface. Mixed plate bending

and membrane stretching is considered, and the contact mechanics is established based on the

JKR assumptions.

3.2 Theory

In order to account for large deflection of an axisymmetric shell, Reissner’s theory [91] is

employed. It is noted that despite the large geometrical deformation, the mechanical strain

remains small, therefore linear elastic material behavior is employed. The geometry and

materials parameters are chosen to fit a typical contact lens of hydrogel: radius of curvature, R =

5 – 17 mm, elastic modulus, E = 0.5 – 2.0 MPa, Poisson’s ratio, v = 0.5, shell thickness t = 10 –

200 µm, adhesion energy γ = 10 – 200 mJ/m2.

3.2.1 General equation of Reissner’s thin shell theory

A theory for axisymmetric deformations of thin shells of revolution has been developed

by E. Reissner [91] by assuming the Love-Kirchhoff hypotheses. Therefore in this work the

assumptions are made as following:

i. The undeformed middle-surface normals deform without stretching into the deformed

middle-surface normals.

ii. The transverse normal stress is neglected in the stress-strain relations. iii. The principle radii of the shell middle surface are much larger than the shell thickness.

Figure 3.1. Side view of element of shell in undeformed and in deformed state. Variable stress

resultants and couples and load intensity component is also shown.

Assumption (i) leads to zero transverse deformation (i.e. plane strain), which requires the

transverse shear to vanish. A shell of revolution is generated by rotation of a regular plane curve

about the z- axis. The curve is parameterized by the radial coordinate r(φ) and vertical coordinate

z(φ), where φ is the meridian angle. Figure 3.1 shows the side view of element of shell in

undeformed and in deformed state in r-z coordinates. The tangent angle φ0 along this curve in the

undeformed configuration is defined by as follows:

dz 1 dz 1 dz tanφ = cosφ = sinφ = (3-1) 0 0 α φ 0 α φ dr 0 d 0 d

where the metric α0 is the undeformed radius of curvature, which is defined as follows:

2 2 1/ 2  dr(φ)  dz(φ)  α 0 =   +    (3-2)  dφ   dφ     Subscript 0 denotes the undeformed configuration.

Strain displacement relationship

The middle surface strain for axisymmetric deformation is defined as follows:

(α −α 0 ) cosφ0  du / dφ0  ε φ = = 1+  − 1 (3-3) α cosφ  dr / dφ   0 0 

r − r0 u εθ = = (3-4) r0 r0 where ε is the elastic strain and the subscripts φ and θ denote the meridional and circumferential

components respectively, u is the shell deflection component along a normal to the symmetry axis, β is the rotation of the normal to the shell surface given by

φ = φ0 – β (3-5) r is the distance from the shell mid-surface to the symmetry axis, relating to the horizontal displacement as follows:

r = r0 + u (3-6)

The meridional strain εφ is related with the vertical displacement w as follows,

dw =Rr(εκφθ − 0 ) (3-7) dφ 0 where R is the shell radius.

The bending curvatures in the meridional and circumferential directions are defined as

follows:

dβ / dφ0 κφ = (3-8) R

sinφ0 − sinφ κθ = (3-9) r0 Equilibrium state

The internal forces acting in the shell are expressed by using the membrane stress

resultants Nφ and Nθ, the stress-couples Mφ and Mθ, and the resultant shear force Q. These

quantities are defined by integration of the stresses σφ, σθ, and σφθ acting on a cross-section of the shell, along the shell thickness direction χ as follows:

t / 2 t / 2 Nφ = σ φ dχ Nθ = σθ dχ ∫− ∫− t / 2 t / 2

t / 2 t / 2 M φ = χσ φ dχ Mθ = χσθ dχ (3-10) ∫− ∫− t / 2 t / 2

t / 2 Q = σ φθ dχ ∫− t / 2 Note that the thin shell approximation is used to reduce the scale factors associated with integration through the thickness to unity. The coordinates (φ, θ, χ) define an orthogonal,

curvilinear coordinate system. The vertical and horizontal components of the shell force, denoted

by V and H, respectively, are defined by the following relationships,

Nφ = H cosφ + V sinφ (3−11)

Q = – H sinφ + V cosφ (3−12) The externally applied load, which acts on the middle shell surface, has vertical and

horizontal contributions denoted by pV and pH (per unit area), respectively. Force balance on an

infinitesimal area of the middle-surface of the shell in the deformed configuration gives the

following equilibrium equations,

d(rV ) + r α p = 0 (3-13) V dφ0

d(rH ) − α N + r α p = 0 (3-14) θ H dφ0

d(rM φ ) dr + (M − M ) − r α Q = 0 (3-15) φ θ dφ0 dφ0

1 dr where radius of curvature α has expression of α = . Equations (3-13) and (3-14) cosφφd 0 represent the force equilibrium along directions parallel and normal to the axis of symmetry, respectively, and Equation (3-15) represents the moment equilibrium.

Stress-strain relations

For a shell with linear-elastic material behavior, characterized by Young’s modulus E and

Poisson’s ratio ν, the stress-strain relations are given by the following plane-srain relationships,

E E σ φ = (ε φ +νε θ ) σ θ = (ε θ +νε φ ) (3-16) (1−ν 2 ) (1−ν 2 )

By using Equation (3-10) and Equation (3-16) along with strain-curvature relationship, the following relationships are established,

N = K (ε + νε ) N = K (ε + νε ) (3-17) φ φ θ θ θ φ

M = D (κ + νκ ) M = D (κ + νκ ) (3-18) φ φ θ θ θ φ where K = Et/(1- ν2) is the extensional stiffness and D = Et3/12(1−ν2 ) is the bending rigidity of the shell.

In summary, the governing equations of Reissner’s shell theory are given by (3-3), (3-4),

(3-8), (3-9), (3-11), (3-12), (3-13), (3-14), (3-15), (3-17) and (3-18), with other related equations.

All symbols are listed in glossary.

3.2.2 Two deformation modes

Mechanical models for the following two loading configurations based on Reissner’s theory [82, 91] are described in this section: (i) Compression between parallel plates and (ii)

Central load compression.

3.2.2.1 Parallel Plate Compression

Figure 3.2. Sketch of a spherical shell being compressed by two parallel plates.

Figure 3.2 shows an isotropic spherical shell with thickness of t, shell radius of R, compressed by two parallel rigid plates, causing an approach distance of 2d and contact radius of a. Figure 3.3 shows the corresponding spherical coordinate system, though only a quadrant is considered due to symmetry. When the top plate presses against the shell, the shell deforms symmetrically and conforms to the planar plate geometry at the poles. A similar problem is solved by Kalnins and Lestingi [92]. The shell meridian angle, φ, is the only independent variable in the model. Each point on the shell, is described by three displacement/rotation variables, deflection in horizontal direction u, vertical direction v, and rotation β, and three stress/moment

variable, stress in horizontal direction H, vertical direction V, and meridional moment resultant

Mφ.

The deformed shell is divided into two regions separated by the contact edge: (i) the inner

planar contact region with φ = 0 or β = φ0, and (ii) the outer freestanding traction-free annular shell with periphery allowed to expand, that is, H = w = β = 0. It is known that once the applied load exceeds a critical threshold, central buckling occurs where the contact center lifts off from the substrate forming an axisymmetric dimple [89, 93]. Updike predicts contact region would

buckle from the rigid plate when contact angle φ0 beyond 9 degrees, thus, only limited

deformation is considered here such that the shell maintains full contact with the top plate

throughout loading. The classical plate theory according to Essenberg predicts zero pressure

within the contact circle [94]. The applied load is represented by a line force at the boundary of

the contact circle with a magnitude of Q* = F / 2 π a.

Figure 3.3. Sketch of coordinate system for parallel plate compression. Meridional angle, φ, is

taken as the only independent variable.

* To solve the nonlinear equations, contact angle φ 0 is pre-assigned to separate the shell

* * into two regions: contact region (φ0 < φ 0 ) and free region (φ0 > φ 0 ). Equations for contact region

are partially solved analytically, to provide three continuity conditions for outer free region. In contact, the planar contact requires φ = 0, and thus β = φ0. Equation (3-8), (3-9) requires κφ = κθ

= 1/R, and thus (3-18) leads to Mφ = (1 + ν) D / R. Since moments are constant and equal, V = 0

and pV = 0. Therefore, (3-14) yields the following,

3− ν 3 u = A φ0 + R φ0 (3-19) 16

2 K A (1− ν ) 2 H = (1+ ν) + K φ0 (3-20) R 16 with A an arbitrary constant. At the contact edge, by eliminating constant A in Equation (3-19)

and (3-20), three continuity conditions are obtained

* β = φ 0 (3-21) D M φ = (1+ ν) (3-22) R

K u 1+ ν * 2 H − (1+ ν) = − K φ 0 (3-23) R α 8

* where φ denotes value of φ at the contact edge. Governing equations are solved numerically in 0 0 the outer freestanding shell. These equations are discretized using second order accurate finite difference approximations of the derivatives. The solution algorithm described by Aktas and

Stetter for first-order, non-linear, two-point boundary-value problems is used [95]. The quarter symmetric model shown in Figure 3.3 is divided into 100 segments. The solution of {w u β V H

T Mφ} is obtained at any point on the shell. Once H and u in the traction free annular shell and the

T constant A are found, {w u β V H Mφ} in the contact circle can be determined. In this chapter,

37 the contact is shallow and thus the circumferential strain is sufficiently small such that the

* o gradient at the contact edge is constrained by φ0 < 15 . The approach distance d is found to be

* = + − φ d w R (1 cos 0 ) (3-24) and the applied force

* = − π φ F 2 RV sin 0 (3-25) and contact radius

* = φ + a a( 0 ) u (3-26) Under zero external load (F = 0), the convex shell touches the substrate at one point. It is remarked that during initial compressive loading between two parallel plates, d < t, and plate bending dominates. As d exceeds t, membrane stretching becomes dominant. Our calculation considers mixed bending-stretching deformation.

3.2.2.2 Central Load Compression

Figure 3.4. Sketch of coordinate system for central load compression.

Procedures for parallel plate compression remain valid for central load compression, in the case that the whole shell is considered as a freestanding region, i.e. region (ii), shown in

Figure 3.4. Upon loading, the shell conforms to a concave geometry. The new boundary

* conditions become d = w0, u = 0, and β = φ0 at the center, where w0 is the central displacement.

3.2.3 Numerical method

The finite difference method is employed to solve this boundary boundary-value problem

that is governed by a system of first order ordinary differential equations (ODEs) [95]. A two

point boundary value problem in the interval I = [c1, c2] can be defined by a system of first-order

ODEs and boundary conditions as follows.

dy(φ0 ) = f (φ , y(φ )) and r (y(c ), y(c )) = 0 (3-27) φ 0 0 B 1 2 d 0 The interval I is divided into M segments uniformly and the choice of M should be a sufficiently

large number to guarantee the accuracy of the solution. Im = [cm-1, cm] denotes one specific

segment where m = 1…M. For the shell problem described by equations above, the boundary-

value problem is stated in terms of the six unknowns, y ={w u β V H Mφ}. Governing equations

are rearranged in the form in Equation (3-27) as follows.

dw = R (εφ sin φ − κθr0 ) dφ0

du = R ( εφ cosφ + 2 sin(β/ 2) sin(φ0 −β/ 2) ) dφ0

dβ = R κ (3-28) φ φ d 0

dV r1 = − V − α pV dφ0 r

dH r1 α Nθ = − H + − α pH dφ0 r r

dM φ r1 α cos φ = − M φ + M θ − α (H sin φ −V cos φ) dφ0 r r

Using the (implicit) trapezoidal rule, the residual vector ψ m for a segment Im is defined as

follows.

ym − ym−1 1 ψ (y , y − ) = − [ f (φ − , y − )+ f (φ , y )] = 0 (3-29) m m m 1 ∆φ m 1 m 1 m m 0 2

where φm represents the terms on the right hand side of Equation (3-28). Considering the finite difference segmentation of the entire shell one would get (M + 1) vector equations for the (M +

1) vectors y0, y1, …, yM each having 6 unknowns. For the solution of the large system of nonlinear Equation (3-29), Newton’s method is employed. A multivariable Taylor series expansion

of the residual form of Equation (3-29) is used to linearize the system. When all of the segments

are considered the system of equations is expressed in the following form.

(i) (i+1) (i) [Dψ ] {∆y} = −{ψ} (3-30) y where the superscript i indicates solution iteration level, and {∆y}(i+1) is the correction to the dof

vector between the iterations. The Jacobian matrix Dψy is defined as

  ∆φ  ∂    ∆φ  ∂     1 f 1 f  − I +    I −      2  ∂y0 φ   2  ∂y1 φ     0   1     ∆φ  ∂f    ∆φ  ∂f    − I + 2    I − 2          (3-31)  2  ∂y1    2  ∂y2     φ1   φ2   ψ = D y            ∆φM  ∂f  ∆φM  ∂f    − I +    I −      2  ∂yM −1 φ   2  ∂yM φ    m−1   m    ∂r ∂r     ∂y0 ∂yM 

To calculate Jacobian matrix (3-31), Equation (3-28) is differentiated with respect to each

T of the unknown variables, i.e. {w u β V H Mφ} . The prime below denotes derivative respect to

any one of the six variables. Thus,

dw' = α' sin φ + α φ' cosφ dφ0

du' = α' cosφ + α φ' sin φ dφ0

dβ' = R κφ ' dφ0

dV ' r ' r r r' = − 1 V − 1 V ' + 1 V − α' p (3-32) φ 2 V d 0 r r r dH ' r ' r r r' α' N αN ' αr' N 1 1 1 θ θ θ = − H − H ' + 2 H + + − 2 − α' pH dφ0 r r r r r r

dM φ ' r1'M φ r1M φ ' r1r' = − − + 2 M φ dφ0 r r r α'cosφ α φ'sin φ αcosφ α r'cosφ + M − M + M ' − M r θ r θ r θ r 2 θ − α'H sin φ − α H 'sin φ − α H φ'cosφ + α'V cosφ + α V 'cosφ + α V φ'sin φ

where εθ '= u'/ r0 , φ'= −β', κθ '= − φ'cos φ / r0 , Nφ = H 'cos φ − H φ'sin φ + V 'sin φ + V φ'cos φ,

εφ '= Nφ '/ K − ν εθ ', κφ '= M φ '/ D − ν κθ ', Nθ '= K (εθ '+νεφ '), M θ '= D (κθ '+νκφ '), α'= R εφ ' , r' = u' , and

r1'= α'cosφ − α φ'sin φ.

The solution algorithm is shown in Figure 3.5, a constant relaxation coefficient C, with a value between 0 and 1, is employed in some cases to help converge.

Assume a trial solution {y}(1), i=1

Else Compute residual Ψm

i=i+1 Ψm > εtor

Compute Jacobian DΨy

Update the solution {y}(i+1) ={y}(i)+C{Δy}(i+1)

Stop

Figure 3.5. Algorithm for finite difference computation

3.2.4 Thin shell with adhesion

To account for shell adhesion, we adopt the classical JKR framework [26]. In the absence of an external load (F = 0), the shell comes into adhesion contact with a rigid plate, producing a non-zero contact radius a. Total energy of the system, UT, is a sum of the potential energy of applied load, UP, elastic energy stored in the shell, UE, and surface energy to create new surfaces,

US. Under fixed load configuration, F moves a vertical distance, Δd, such that (dUP)F=constant = F

Δd.

Figure 3.6. Load-displacement relation with E = 2.0MPa, t = 200µm, and R = 15mm. Elastic energy UE is the area under loading curve (in grey). Area ABDEA is the elastic energy

associated with elastic recovery due to relaxation.

Figure 3.7. Deformed profiles of points A, B ,C in Figure 3.6. The elastic energy is found by computing the area under the force-displacement curve,

F(d), as shown in Figure 3.6 for R = 15 mm, E = 2.0 MPa, and t of 200 µm. The overall load

acting on the shell, F1, is the sum of the applied load and the virtual adhesion force. If there is no adhesion, the loading curve according to Equation (3-28) traverses along path OA and the contact radius increases to a1. Adhesion now keeps a1 constant, and elastic relaxation is allowed

to reduce the load from F1 to F along path AB until equilibrium at B. The corresponding deformed profiles of A, B, C points are shown in Figure 3.7. It is important to note that the path

AB is here nonlinear because of the shell behavior, unlike the linear relaxation in JKR solids.

The total elastic energy is thus the area enclosed by OABDO.

Figure 3.8. Total potential energy as a function of contact area with E = 2.0MPa, t = 200µm, and

R = 15mm. The thick gray curve represents the energy balance at equilibrium. Under fixed load,

UT possesses local minimum or stable equilibrium until “pull-off” where UT shows an inflexion

point and the equilibrium becomes neutral (inset). Under fixed grips, stable equilibrium is always

maintained till the contact radius virtually vanishes.

2 The surface energy is given by US = πa γ, where γ is the adhesion energy or surface

energy per unit contact area. The disjoining pressure is taken to be infinite in magnitude but with

zero range, so that adhesion becomes a line force at the contact edge and the intersurface

attraction vanishes beyond the contact circle (r > a). Shear at the contact interface is ignored.

The above formulation can be repeated for fixed load, but the total energy is rewritten as UT = –

UE + US, since (dUP) d=constant = 0 as Δd = 0.

2 Figure 3.8 shows UT as a function of the contact area πa . Equilibrium is achieved at

2 ∂UT/∂(πa ) = 0, or, ∂UT/∂a = 0. Under a fixed load (dark solid curve), shell detachment is gradual

2 2 since (∂ UT/∂a )F > 0 corresponding to stable equilibrium. Onset of instability or “pull-off”

* 2 2 occurs when the tensile load increases to F where (∂ UT/∂a )F = 0 at the point of inflexion,

indicating neutral equilibrium, as shown on the inset of Figure 3.8. Further increase in tensile

2 2 load leads to (∂ UT/∂a )F < 0, where the shell spontaneously detaches from the substrate in an unstable manner. The energy balance curve joins the loci of the local extrema (thick gray curve).

On the other hand, under fixed grips (dashed curve), only local minima can be found,

2 2 corresponding to stable equilibrium as (∂ UT/∂a )d > 0. Stability is maintained untill the smallest

possible d† is reached, then fixed grips “pull-off” occurs. Note that these local minima coincide with the extrema of the fixed load configuration at any contact area. In the range of d* < d < d†

fixed grips leads to stable equilibrium but fixed load is unstable.

3.3 Results and discussion

The mechanical adhesion-detachment of a shell can now be established by setting up

relations between the measurable quantities F, d and a. Figure 3.9(a) and (b) show the relation,

a(F), and deformed profile, Y(X), for γ as stated, contrasting the Hertz-like contact (γ = 0).

Figure 3.9. (a) Contact radius as a function of applied load with E = 2.0MPa, t = 200µm, R =

15mm, and a range of adhesion energy as shown. Shell detachment follows path PGHJ. Grey curve and symbols shows the trajectory of fixed load “pull-off”. (b) Deformed shell profiles at P,

G, H and J for adhesion energy γ = 200mJ/m2.

For all γ, compressive load enlarges the contact area and the shell is squashed into a pseudo ellipsoid rendering the major axis horizontal. Tensile force, F < 0, reduces the contact area and stretches the shell in the vertical direction. Note that a > 0 at G where F = 0. Along path PGHJ on the energy balance curve, the external load turns from compression to tension and reaching a maximum tensile value F* at J. Further increase of tensile force beyond F* can no

longer satisfy energy balance. The shell thus spontaneously detaches from the substrate at a non- zero contact radius a* (gray circles). Weaker adhesion leads to smaller F* and a* as shown. The

corresponding relations d(a) and F(d) for fixed γ can also be obtained (not shown). The

deformed profile resembles a truncated spherical cap with a planar contact circle, besides the

small annulus near the contact edge.

A universal shell behavior can be derived similar to Maugis’s normalization scheme [96],

where F, d, and a can also be made dimensionless in terms of the materials and geometrical parameters E, t, R and γ. All normalized symbols are shown as bold hereafter. The Buckingham

Π Theory [97] requires F = F × En ti Rq γm with the variables n, i, q, m chosen to ensure force and

length are eliminated. The relations a(F), d(a) and F(d) then collapse into single universal curves, respectively, independent of adhesion energy, with the following definitions,

1/ 3 1/ 3 1/ 3  Et 2   E 2t 4   Et 2  F = F   d = d   a = a   (3-33) γ 4 R 4 (1− v 2 ) γ 2 R 5 (1− v 2 ) 2 γR 4 (1− v 2 )       The universal mechanical response is shown in Figure 3.10. The curves are obtained by

averaging the combinations of the aforementioned upper and lower limits of E, t, R and γ.

Should more elements be added for our computational model, the error bars and uncertainties are

believed to further diminish. Figure 3.10(a) shows a(F) for any γ, which is consistent with Figure

3.9(a). Under fixed load, “pull-off” occurs at

4 4 2 1/ 3 * γ R (1− v ) F = − ( 13.2 ± 0.6 )   (3-34) Et 2  

Figure 3.10. Universal shell adhesion behavior in terms of normalized quantities. Error bars denote uncertainties due to limited number of elements used in computation. (a) Contact radius as a function of external load. (b) Approach distance as a function of contact radius. Fixed grips

“pull-off” at d† occurs at the terminal end of the energy balance curve. (c) External load as a function of approach distance. “Pull-off” under fixed load occurs at the most negative tensile force at F*, while “pull-off” under fixed grips occurs at the most negative d†.

“Pull-off” under fixed grips is illustrated in d(a) as shown in Figure 3.10(b). Here

positive d corresponds to shell squashing. As d grows more negative, a diminishes

monotonically until “pull-off” or the terminal end of the energy curve. With limited number of

elements in our computation, it is unclear at this stage whether the “pull-off” occurs at a = 0 or a small but nonzero value. The distinction between the fixed load and fixed grips configurations is further illustrated by F(d) as shown in Figure 3.10(c). As d is lowered, external tensile force

(negative F) increases until a local minimum in the energy curve, d*, or maximum tensile load,

F*, is reached. “Pull-off” occurs here under fixed load. On the other hand, under fixed grips, d is

able to proceed further with stable equilibrium. As d lowers further from d* to d †, tensile force decreases to F†, but stable equilibrium maintains until “pull-off” at the most negative d.

It is worthwhile to compare the present model with the classical JKR theory. Maugis

shows that the universal JKR behavior for sphere-substrate adhesion can be expressed in terms of

a set of dimensionless parameters given in Table 3.1 contrasting the shell parameters. “Pull-off”

parameters under fixed load and fixed grips are also shown. A few distinct features are noted.

Firstly, the shell thickness, t, does not appear in the JKR normalization constants, but plays a

crucial role in shells. For instance, Equation (3-34) gives a ∝ t-2/3 such that the thinner the shell,

the more compliant it is, and the larger the contact area. Secondly, the JKR normalized load does

not depend on the elastic modulus, since the contact area is taken to be negligible compared to

the sphere radius (a << R). Such assumption is invalid in thin shells because a is comparable

with R. A direct consequence is the JKR “pull-off” load being independent of both E and t. In a

shell, F* ∝ (Et2)-3 such that a thinner and softer shell requires a larger “pull-off” force. In JKR, a

large compressive stress is present in the contact area because of geometrical incompatibility

between the sphere and substrate, but a shell conforms to the planar substrate and the compress-

Table 3.1. Comparison between JKR adhesion theory and the shell model

JKR solid sphere Spherical shell

Normalized parameters 1/ 3  1   Et 2  Compressive F = F =   F F  4 4 2  load 3π R γ  γ R (1− v ) 1/ 3 1/ 3  16E 2   E 2t 4  Approach = = d d  2 2 2 2  d d  2 5 2 2  distance 3π γ R (1− v )  γ R (1− v ) 

1/3 1/ 3  4E   Et 2  Contact radius = = a a  2 2  a a  4 2  9π γ R (1− v ) γ R (1− v )

Fixed load “Pull-off”

4 4 2 1/ 3 Applied load * 3 γ R (1− v ) F = − πRγ * = − ± F ( 13.2 0.6 )  2  (tensile) 2  Et  1/ 3 1/ 3 3π 2γ 2 R (1− v 2 ) 2  γ 2 R 5 (1− v 2 ) 2  Approach * = − * = ± d   d ( 0.13 0.02 )  2 4  distance 2 E t  64E    1/3 1/ 3 9π γ R2 (1− v2 ) γ R 4 (1− v 2 ) Contact radius * = * = ± a   a ( 0.55 0.03 )  2   8E   Et  Fixed grips “Pull-off”

4 4 2 1 / 3 Applied load † 5 γ R (1− v ) F = − πRγ F † = − ( 3.5 ± 2.5 ) (tensile)  2  6  Et  1/ 3 1/ 3 27π 2γ 2 R (1− v 2 ) 2  γ 2 R 5 (1− v 2 ) 2  Approach † = − † = − ± d   d ( 0.11 0.02 )  2 4  distance E 2 E t  64    2 2 1/3 † π γ R (1− v ) † Contact radius a =   a ≈ 0 8E  

-ion thus becomes insignificant within the contact circle. The local deformation in JKR solids is

more pronounced than the small perturbation in shells (c.f. Figure 3.9(b)).

Another remark is that both the JKR model and the present work do not consider shear at

the interface, though the inclusion, depending on its magnitude can lead to significant

modifications of the F-d-a relation [98].

It is worthwhile to note two related works in thin shell literature. de Gennes [71] considers

the detachment of two adhering thin-walled vesicles using micropipette aspiration, and deduces

F* = –πRγ. Shanahan [53, 54] investigates the theoretical adhesion of a fluid filled spherical membrane capsule in contact with a planar rigid substrate based on the JKR model and an incompressible fluid. An energy balance leads to F* = –2πRγ. There are a few common features

contrasting the present work as follows: (i) “pull-off” force varies with the factor (R γ) as in

JKR-solids but remains independent of E and t, (ii) no distinction is found between fixed load

and fixed grips configurations, and (iii) contact circle reduces to a* = 0 and d* = 0 at “pull-off”.

Notwithstanding the different loading geometry, the present model bears some

resemblance to Shull and Hui’s studies in adhesion of an inflated membrane on rigid substrate

[55]. In their experiment, a uniform hydrostatic pressure presses against a planar membrane clamped at its edge until the spherical cap makes adhesion contact with the substrate that is a fixed distance beneath. The instantaneous membrane curvature thus resembles a shell and the membrane-substrate gap the approach distance in our model. In fact, a “pull-off” event at a

critical reduced pressure with a non-zero contact radius is experimentally proven. It is evident

that such experiment can also be conducted by adjusting the membrane-substrate gap while

keeping a zero applied pressure. These results and predictions are consistent with our linear

elastic and large deformation model.

3.4 Summary

Adhesion-detachment mechanics of a spherical shell is established based on the JKR style energy balance. Relations derived for the applied load, approach distance and contact radius are useful for experimentalists to characterize the elastic deformation and adhesion behavior of convex shells. The model predicts behaviors distinctly different from the classical JKR theory, and finds a number of engineering and biological applications.

4 EXPERIMENTAL CHARACTERIZATION OF SOFT CONVEX SHELL

In this chapter, mechanical properties of a macro-scale soft convex shell – hydrogel contact lens is characterized. A sample lens is mechanically deformed by two loading configurations: (i) compression between two parallel plates, and (ii) central load applied by a

shaft with a spherical tip. A universal testing machine with nano-Newton and submicron

resolutions is used to measure the applied force, F, as a function of vertical displacement of the

plate/shaft, d, while a homemade laser aided topography system records the in-situ deformed shell profile and the contact radius or central dimple, a. Reissener’s nonlinear shell theory described in Chapter 3 is used to extract the material properties of hydrogel contact lens.

Preliminary adhesion measurement is also performed using plate compression configuration. Part of the measurement work is done by Michael Robitaille in our lab.

4.1 Introduction

The mechanical properties of hydrogel contact lenses are critical to the comfort and clinical performance of the lens, e.g. on–eye movement, fitting, and wettability. The stiffness of the lens is related to the comfort and “foreign body” sensation felt by user, and how well the lens stores elastic energy contributes to its lifetime. The mechanical properties of lenses can also lead to ocular complications such as superior epithelial arcuate lesions (SEALS) and contact lens- related papillary conjunctivitis (CLPC) [99]. The mechanical properties are also important to the design and fabrication lenses [100]. The elastic modulus of hydrogel contact lenses has been

measured using a variety of methods over the years. The conventional standard tensile testing has been adopted by cutting a rectangular strip from sample contact lens, to extract the modulus for contact lens materials [101], but runs into complications since small flaws introduced to the

sample during preparation can easily compromise the data obtained [102]. The intrinsic curvature 53

and non-uniform thickness of most lenses also complicates sample preparation and geometry.

Even with ideal sample preparation, there is a possibility of introducing the residual stress by clamping the soft hydrogels. Also, due to the anisotropic and otherwise complex nature hydrogels exhibit, there is concern of the usefulness of obtaining a tensile modulus for lenses which undergo little tension during actual use. Therefore, there is a need to investigate detail material properties and adhesion properties of a contact lens, since millions of people wear

contact lenses worldwide on a daily basis.

Figure 4.1. Loading configurations: (a) parallel plate compression, (b) central load compression

via a rigid sphere.

In this chapter, contact lens are mechanically loaded in two new loading configuration configurations described in Chapter 3: (i) parallel plate compression where the sample lens is compressed between two parallel plates perpendicular to the optical axis, and (ii) central load applied to the apex of the convex surface, as shown in Figure 4.1. The main difference between the two methods is that the former can be treated as ring force at the contact edge according to classic plate theory by Essenberg [94], while the latter by a concentrated load at the center. The

quantities of applied load, F, approach distance or central displacement, d, deformed profile,

w(r,φ), and contact radius, a, in case of parallel plate compression, or, the dimple radius, a, in

case of central load, are measured simultaneously using a universal testing machine and a

homemade optical device to monitor the in-situ shell geometry. Materials properties such as

elastic modulus, E, can be deduced from F(d) measurement alone, which is further checked for

consistency with the predicted deformed profile. The material parameters are also compared with

standard tensile test of a rectangular strip and indentation.

4.2 Contact lens characterization

4.2.1 Sample preparation

Commercially available contact lenses PureVision from Bausch & Lomb are

characterized at an ambient temperature of 22oC. The samples had base diameter, 2c = 14 ± 0.10 mm, and vertical height, h = 3.70 ± 0.10 mm. The shell thickness is assumed to be uniform and measured by optical coherence tomography (OCT) and found to be t = 94 ± 5 µm. The radius of curvature is measured from optical images obtained by a side camera and found to be R = 8.46 ±

0.10 mm. A fresh sample is taken from a new package for every measurement. The sample lenses are soaked in an isotonic solution in a glass container to prevent dehydration and swelling.

4.2.2 Tensile test and indentation

Standard mechanical characterization methods are used to measure the elastic modulus to

be compared with the parallel plate compression and central load. Strips with width 4.4 mm,

length 16.3 mm, and thickness t = 94 µm are cut from the center of the contact lens, similar to the method by Tranoudis [102]. The strips are clamped onto an Agilent T150 Universal Testing

Machine (UTM) with force and displacement resolutions of 30 nN and 10 nm respectively.

Isotonic solution is manually applied via a pipette to prevent dehydration. The lenses are strained at a rate of 12% per minute. A typical stress- strain curve is shown in Figure 4.2 and the slope of the initial linear portion of the stress-strain curve is used to calculate the elastic modulus (dash line with strain up to 0.04), which is found to be E = 1.41 ± 0.11 MPa for the sample lenses. No yield stress is observed in the experiment with maximum strain of 17%. Full elastic recovery is observed for all our samples. This is in good agreement with previous tensile test studies on

Balafilcon A (the hydrogel backbone of PureVision), which reports a bulk tensile modulus of E =

1.5 ± 0.15 MPa [103].

Figure 4.2. Stress-strain relation for tensile test

Figure 4.3. Load-displacement relations for contact lens indentation test, with dashed line showing the upper and lower bound of tests.

Indentation is also performed. A sample lens is inverted with the convex surface sitting on a rigid substrate. A stainless steel spherical tip (rb = 1 mm) is used to indent at the leeside of

the shell apex, shown in Figure 4.3. The indentation depth is limited up to 1/10 of total thickness,

thus we assume there is no effect of substrate coming into play. Assuming a linear elasticity and

a semi-infinite elastic half-space [104], the gradient of the initial loading curve is given by F’ = 2 [

E/(1 – v2 ) ] R1/2 d1/2. Small loading-unloading hysteresis and loading rate dependency due to viscoelasticity are observed in our measurements [105], but are neglected in our analyses. Figure

4.3 shows the applied force as a function of indentation depth, with only loading curve showing.

Slope of the gradient is indicated in grey solid line. It is found that E = 0.078 ± 0.062 MPa. The

significant error is the result of difficulty in identifying when the sphere first comes into contact

with the lens. These numbers can be compared to a similar hydrogel lens (Acuvue Advance)

where microindentation yields E = 0.13-0.17 MPa in ambient air [105] and E = 0.05-0.06 MPa in

57 lens solution [106]. Noting that Young’s modulus obtained from indentation is very different from tensile test. This is attributed to the porous nature of hydrogel, consisting of a network of matrix filled with fluid media. It is likely that in indentation test, instead of characterizing the material as a whole, it is actually squeezing fluid out of solid matrix. Therefore, the E obtained from indentation test is inaccurate.

4.2.3 Parallel plate and central load compression test

Figure 4.4. Experimental setup showing an immersed lens contained in a square container sitting on a manual x-y stage.

Figure 4.4 shows the experimental setup of an Agilent (Santa Clara, CA, USA) T150

Universal Testing Machine with force and displacement resolutions of 30 nN and 10 nm

respectively. External load is applied by an Agilent nano-bionix UTM. A laser beam illuminates

the sample, while an orthogonal long focal microscope captures the deformed profile. All

experiments are displacement-controlled with the shaft staying fixed and the moveable platform

moving upwards. An aluminum cylindrical plate with radius rp = 10 mm is fabricated for parallel

plate compression, and a shaft with a stainless steel ball with radius rb = 1 mm at the tip for

central load. For convenience and consistency, the loading speed is kept constant at the standard

of value v = 1.00 mm.min-1. The maximum approach distance is confined to the range of 50 µm

< d < 1200 µm.

In the experiment, contact lens is fully immersed in solution. Since water has surface

energy density as low as 0.072 J/m2, interfacial adhesion between lens and indenter is ignored.

The lens is placed on an aluminum sample holder with a shoulder of radius 7.25 mm fitting to

the base radius c. A hole is bored through the sample holder axis below the shell to allow free

flow of medium liquid and thus preventing hydrostatic pressure buildup during loading-

unloading. The isotonic solution container sat on an x-y stage to align the loading shaft with the shell apex. The entire setup is placed on an anti-vibration table. Measurements start with the plate/shaft held back some small distance from the sample so that the buoyancy forces and surface tension are measured and subtracted from all force measurements. Great care is taken to accurately determine the instant when the probe first makes contact with the sample. It is taken to be the point when the load cell records a sudden jump of ~10 µN, which is also confirmed by a long-focal digital camera that captured a complete side view of the sample (see in Figure 4.5).

The camera is made horizontal and in the line of sight of the lens. To raise the optical

contrast, a laser beam is used to illuminate the shell from the side so that the lens profile is

highlighted. Images are then processed using software ImageJ (National Institutes of Health,

Bethesda, Maryland, USA), and the deformed profile extracted by pixel intensity differentiation.

Dimensions are measured by pixel scaling.

Figure 4.5. Experimental setup showing (a) Typical side view of a sample lens subject to parallel plate compression and (b) Typical side profile of sample lens subject to central load compression via a ball bearing.

Parallel plate compression

Figure 4.6 shows the mechanical response, F(d), of five samples, and the theoretical solution. The applied load lies in the range of 0 – 7 mN. The logarithmic plot of F(d) in Figure

4.6 shows a transition from F ∝ d to F ∝ d 3/2 at roughly the shell thickness t = 94 µm as indicated. In the initial loading, the contact radius increases rapidly from zero but remains small

compared to the base radius (a << c) as shown in Figure 4.7, and F(d) is therefore linear as

expected by the classical elastic shell model. When the central deflection approaches the shell

thickness (d ~ t), the contact region gradually increases and becomes significant compared to the

base radius, F(d) deviates from the linear relationship. As d exceeds 500 µm, the measured load

deviates further from the model. The inconsistency is likely the validity of shallow shell

assumption, as (d)max = 800 µm represents roughly 10% of the lens curvature radius and 22% of

the shell height. The best curve-fit yields E = 1.26 ± 0.06 MPa.

Figure 4.6. Typical mechanical response under parallel plate compression. Load as a function of top plate displacement in linear-linear and log-log scales.

Figure 4.7. Increasing contact radius as function of approach distance as compression proceeds.

Figure 4.8. Contact radius as a monotonic function of applied load. 62

Figure 4.8 shows a(F) and the vertical line denotes d ~ t, where a change in slope is

expected. Deviation of measured a(d) and a(F) from the model is expected at small d (d <= t) is because the contact radius measurement is based on the outer surface, rather than inner surface of contact lens. When approach distance is comparable or smaller than shell thickness t, difference between outer surface contact radius and inner surface contact radius is significant. The difference becomes negligible when approach distance is large (d >> t).

Figure 4.9. Snapshots of deformed profiles with displacements of 0, 0.5, and 1.0 mm.

Figure 4.10. Superimposed snapshots of deformed profiles compare with theoretical fittings.

Figure 4.9 shows the superimposed deformed profiles for several approach distances as

indicated, and the theoretical curve fitting in Figure 4.10 using the above E. Within the contact

edge, the shell makes full contact with the top plate. The deformed shell resembles a truncated

spherical shell with a small deviation at the contact edge for large d. Most elastic energy of the

sample is stored at the contact edge where the smallest radius of curvature occurs.

Central load compression

Figure 4.11 - Figure 4.13 show the mechanical response, F(d), a(d), and a(F), and the theoretical curve-fit. The measured force range of 0.0 – 0.6 mN is an order of magnitude smaller than that in parallel plate compression. The best curve-fit yields E = 1.20 ± 0.10 MPa, which is consistent with values obtained from previous Section. A transition of the slope F’ from 1 to 1/2 occurs as the approach distance reaches roughly the shell thickness, shown in Figure 4.11.

Despite the reasonable fit of the model a(d) and a(F) to the experimental data, the measured F(d) resembles the form but falls short of the predicted values. The discrepancy is essentially due to the non-zero contact area at the shaft-shell junction.

Figure 4.11. Load as a function of approach distance.

Figure 4.12. Increasing contact radius as compression proceeds.

Figure 4.13. Contact radius as a monotonic function of applied load. 65

Figure 4.14 shows the superimposed deformed profiles, and indicates the monotonic increasing contact area during the course of loading. Careful observation shows that the dimple radius grows in sync with the contact radius until d ≈ 0.3 mm when the dimple finally decoupled

from the contact edge and is pushed further out into the freestanding annular shell. Contrasting

the parallel plates, most elastic energy here is stored at the dimple ridge and the contact region.

Figure 4.14. Superimposed snapshots of deformed profile when approach distance is 0µm,

500µm, and 1000µm.

Figure 4.15. Superimposed snapshots of deformed profile compare with theoretical fittings.

4.3 Preliminary adhesion measurement

Experimental setup

Adhesion between contact lenses and the outer surface of contact lens package is

measured with the experimental setup shown in Figure 4.16. A contact lens is clamped at its

periphery up-side-down and attached to the load cell of UTM 150 machine. Contact lens

container is fixed at lower stage with double side tape. To prevent contact lens from dehydration,

homemade plastic bag serves the purpose of environmental chamber to keep ~100% humidity.

Testing is performed at room temperature (22oC).

Contact lens

Load cell Environmental chamber

Contact lens Container container (water)

Double sided sticky tape

Figure 4.16. Schematic drawing of adhesion test configuration The typical loading- unloading curve is shown in Figure 4.17, with lens diopter D = -0.5.

As the lens container approaches contact lens, a negative “pull-in” force is observed due to

adhesion at point A, indicating the contact moment. Further compression between lens and container leads to increase of applied force, until a maximum compressive force of F = 0.3 mN is reached at point C. The follow-up unloading curve along CD gives rise to a hysteresis loop due

to viscoelastic properties of hydrogel contact lens. A significant lager “pull-off” force is required

67 at D point to detach the contact lens from container surface. The area of the loop is equal to the energy lost during the loading circle. A long focal high-speed video is used to camera capture the lens-substrate interface, during the experiment, as shown in Figure 4.18.

Figure 4.17. Loading- unloading curve for contact lens adhesion measurement.

1. Approach 2. Contact formed: A 3. Maximum contact: C

4. Unloading: C-D 5. Capillary 6. Detachment: E

Figure 4.18. In-situ images for Loading- unloading process, corresponding to Figure 4.17.

Initial adhesion analysis

Figure 4.19. “pull-off” force F* as function of central thickness t, with symbols showing experiment measurement and solid line curve fitting in (a) linear scale and (b) log scale.

Contact lenses with seven different diopters of -12, -9.5, -7.5, -6, -0.5, +3 and +5 are used for adhesion measurement. Consistent adhesion energy γ for all diopters are expected to extract, since surface properties are the same. Each diopter is tested repeatedly for 10 cycles to obtain consistent results. “pull-off” forces are further extracted to characterize the adhesive properties of the various lenses according to model introduced in Chapter 3. Uniform thickness is assumed for all diopters of lenses. The apex thicknesses of freestanding lenses are measured via Optical

Coherence Tomography (OCT), showing that lenses with negative diopters have roughly same magnitude of thickness at center, while positive diopters have much thicker center thickness.

Assuming the JKR assumption remains valid for adhesion measurement, based on Equation

* * −2 / 3 (3-34), “pull-off” force F has a power law relationship with thickness t as F = C1t , where

C1 is function of shell radius R, Poisson’s ratio v, elastic modulus E, and work of adhesion γ.

Since the rest parameters can be either directly measured (t, R) or determined from previous section (E), work of adhesion γ is the only parameter to be determined.

Figure 4.19 shows the “pull-off” force as a function of lens thickness. The best yield work of adhesion is γ = 0.0512 J/m2, which is close to surface tension of water at room

temperature to be 0.072 J/m2. The offset between measurement and theory is due to (i) JKR

adhesion assumption, (ii) uniform thickness simplifications and (iii) substrate is curved surface

instead of planar surface. Further modification to the model is needed to account for shell with

varying thickness and various geometric substrates.

4.4 Discussion

Modern engineering and nano-technology using soft polymer and biological materials require extensive mechanical characterization of broad range of micro-/nano-structures. Many

excellent classical models for spherical shell or spherical cap deformation essentially aim at

architectural structures, macro-scale metallic engineering parts and stiff shells such as a tennis

ball, but whether they remain valid in soft and thin shells is yet to be verified. The present work

constructs the needed elastic models for large deformation, and provides experimental data for

demonstration in pragmatic contact lenses. It is nonetheless worthwhile to investigate the validity

of the classical models of small strain and small deformation.

For parallel plate compression, Taber considered a hemispherical shell with R/t ≈ 100 and

derived a linear mechanical response, F ~ (E/10t) d, at initial loading or small F. A transition of

the gradient F’ from 1 to 3/2 when the central deflection exceeds the shell thickness is expected

by theory and verified by experiment. At higher load, central buckling is expected to occur at

(F*/2πREt) ≈ 120 that is equivalent to a critical force of F* ≈ 0.7 mN. In our experiments, the

shell remains in full contact with the top plate throughout the loading process and no buckling is

observed.

For central load compression on a shallow spherical cap with the edge subtended by an

† angle, φ0 , from the shell axis, the shell profile and force-deflection relationship can be obtained for small strain approximation, though only a numerical solution is possible. An exact solution is available for a full spherical shell as follows [107-110],

Fχ 2 d = kei ( 2 χ φ0 ) (4-1) πEt

4 Et  F =   d (4-2)  χ 2 

1/ 4 1  12R 2   χ = − 2  +  − since 1/ 4 (1 v )  2 1 1 4   t  

and kei(x) is the Bessel-Kevin function. For t << R, Equation (4-2) becomes

 4 E t 2  F ≈ d (4-3)  2 1/ 2   3 R (1− v ) 

† In our sample lenses, 2 χφ0 ≈ 15. Since the full spherical approximation is valid when 2

† χφ0 > 10, Equation (4-1) and (4-3) are thus valid for our analysis. Linear curve fit to the initial

loading (d < t) as shown in Figure 4.11 yields an elastic modulus of 1.15 ± 0.10MPa, which is

consistent with other calculated results stated in previous Sections. For d > t, membrane

stretching gradually becomes dominant and the central dimple disengages from the contact

circle. Threshold load leading to central buckling of a shallow shell (c << R) is governed by [111]

µ = 0.093 (α +11.5) − 0.94 (4-4)

with α = c4 / R 2t 2 , µ = F *R / Et 3 and 100 < α < 500. Our sample lenses do not fall in the valid range as c ≈ 0.83R and α = 3796, but Equation (4-4) nonetheless gives F* ≈ 2.1mN,

roughly 10 times larger than our experimental value. For large deformation involving mixed

bending-stretching, Landau derived the post-buckling F(d) based on virtual work principle and

elastic energy stored mainly at the circular dimple shoulder [110],

5 / 2  E t  1/ 2 F ~   d (4-5)  R 

A transition of the gradient F’ from 1 to 1/2 is therefore expected close to d ≈ t, which is

consistent with our experimental data (Figure 4.11).

It is important to emphasize that the present work does not intend to provide new

methods merely to measure the elastic modulus of the shell materials. In certain situations, the

deformed profile is more crucial than the F(d) measurement. For instance, atomic force microscopy and nano-indentation are oftentimes used to characterize the mechanical behavior of single cells and thereby to deduce the elastic modulus of cell membranes. When a spherical cell, bacteria, or liposome microcapsule is subjected to physiological mechanical load (e.g. van der

Waals and ligand-receptor interactions), whether the apposing capsules are geometrically compatible to adhere, to aggregate and to form biofilms or multi-cell tissues depends essentially on the largely deformed profile instead of F(d) [112]. Though the present study considers

oversimplified configurations like point load on a spherical cap, it lays a foundation for general

shell deformation and sets new guidelines for geometry design and materials selection for

artificial drug delivery capsules.

4.5 Summary

An axisymmetric convex shell is characterized by parallel plate and central compression.

Mechanical response and the in-situ deformed profiles are captured by a novel experimental set

up. By employing the classical shell theory, a numerical approach is adopted to account for

shells with large ratio of base to curvature radii, shell deformation and central buckling.

Consistency is found between modeling and experiments as well as classical models in the

literature. The method is useful in characterizing soft polymer shells in general.

5 MECHANICAL CHARACTERIZATION OF MOUSE OOCYTE

Mechanial properties of micro-scale biological cells are investigated in this chapter. Two

groups of mouse oocytes (young and aged) are indented by a micropipette during sperm

injection. The displacement controlled experiment is done by research group in University of

Toronto, which deforms the cell to a desirable extent, while the changing load is measured as a

function of time in a stress-relaxation mode. The simultaneous measurement of applied load,

indentation displacement, and deformed profile, is analyzed by treating the cell as (i) a spherical

solid according to the classical Hertz contact theory as widely adopted by the biomedical

community, (ii) a hollow spherical shell where bending and stretching are the dominant

deformation mode, and (iii) a liquid filled spherical shell where the internal pressure is non-zero.

The three methods yield vastly different elastic moduli from the force-displacement response. It

is found that 40-week old aged cells are more compliant but more viscous compared with 10-

week old young cells.

5.1 Introduction

Mechanical characterization of single biological cells is crucial in understanding cellular

structures and physiological functions, predicting cellular responses to mechanical cues and

stimuli, and correlating mechanical and pathophysiological behaviors etc [113]. In the context of

embryology and reproduction research, mechanical characterization of mammalian oocytes

represents a promising technique to detect cellular defects in pre-implantation oocytes [114], and

to assess quality and bio-viability of processes such as cryopreservation [115].

The experimental techniques include atomic force microscopy (AFM) [116], optical tweezers [117], and micropipette aspiration [118], where nano-indentation is by far the most

convenient and widely accepted technique. To extract the materials properties such as elastic 74

modulus of the cell from the force measurement, a rigorous solid-mechanics model is necessary

to relate the simultaneous applied load, indenter displacement and deformed profile. For the

sake of simplicity, most investigators measured only the force-displacement relation and deduced the elastic modulus by fitting their data to some theoretical model, but neglect the deformed profile. The AFM tip is treated as a rigid solid with a spherical cap of finite radius, and the sample is taken as an elastic half space. The force response is therefore governed by a pseudo

Hertz contact theory where the substrate, rather than the AFM tip, is deformed [78]. The model is

widely adopted to analyze materials responses in polymers [119] , thin films [120] and cells [121-123].

Li characterized single breast cancer cells using AFM indentation and deduced the elastic

modulus using Hertz model [124]. Significant differences were observed in cell elasticity when

comparing healthy and malignant cell samples. Wojcikiewicz et al. [125] also measured elastic

behavior, as well as adhesion behavior, of single cells using AFM and analyzed their data

according to Hertz model.

A number of shortcomings arise from the assumption of solid spheres. In general, cells

are not solid spheres, but comprise a lipid membrane encapsulating an incompressible cytoplasm.

When deformed by a large external load, deformation is no longer confined to the surface area,

but the global structure of the cell. Dao characterize human erythrocyte using optical tweezers

[117] and show that the cell should be properly modeled as membrane containing a fluid of fixed

volume of cytosol. Schape [126] investigate the mechanical behavior of amphibian oocyte nuclei

using AFM but analyze the data by two vastly different mechanical models, namely, solid

spheres according to Hertz and a thin elastic shell. It is claimed that unless the effect due to the

nucleus envelop is negligible, the shell model is more appropriate than the Hertz model.

Experiments indeed show that the envelop takes up most of the external load. There are

situations when the Hertz model is obviously invalid. For instance, Fritz [127] perform AFM

indentation on rod shape bacteria with characteristic height of 500 nm and the indentation depth

go as far as 200 nm. The small deformation assumption in Hertz thus fails. Sun [128] report intracytoplasmic sperm injection (ICSI) of mouse oocytes, where a homemade force device indented a single cell using a glass micro-pipette. The force measurement is analyzed using a primitive thin shell model, where the elastic modulus is found to be almost 3 times stiffer in the post-fertilized stage compared with the fresh oocyte.

It is worthwhile to note that the any theoretical model can be made to fit any set of experimental data provided there are sufficient variables to adjust. For instance, should linear elasticity fail, progressively more sophisticated nonlinear models with more adjustable parameters can be adopted until the curve fitting routine is optimized. It is virtually impossible to verify these models by merely scrutinizing the force-displacement relation. The deformed cell profile simultaneously recorded with the applied load might serve as an ultimate proof. In this chapter, we present comprehensive models including both force and profile measurements, and verify the theory experimentally. Mouse oocytes from young and aged female mice are characterized by a vision-based indentation method where the force, displacement and profile are captured. Three models based on (i) classical solid sphere (SS), (ii) hollow spherical shell (HSS), and (iii) liquid-filled spherical shell (LFSS) are constructed based on linear elasticity, and used to analyze the experimental data. The most appropriate model is expected to show consistencies in both force and profile.

5.2 Indentation on single oocytes

Figure 5.1. Mouse oocyte microinjection system used for force measurement.

A vision-based technique [129, 130] is reported recently to simultaneously measure the applied mechanical force and the deformation of a single mouse oocyte during microinjection.

Figure 5.1 shows a microinjection system for in-situ measurement of mechanical load applied to an oocyte. It consists of a Polydimethylsiloxane (PDMS) cell holding device, an inverted microscope (TE2000-S, Nikon) with a CMOS digital camera (A601f, Basler), a three-degree-of- freedom (3-DOF) micromanipulator (MP-285, Sutter) for motion control of the injection micropipette, a motorized X-Y stage (ProScan II, Prior) for positioning oocytes, and a temperature-controlled chamber (Solent Scientific) to maintain oocytes at 37 °C.

Figure 5.2. Image during microinjection of mouse oocytes.

Figure 5.2 shows a cell being immobilized and contained in a micro-fabricated PDMS fixture and indented by a glass micropipette. Radius of the sample cell, R ≈ 45 µm, and thickness of the zona pellucida (glycoprotein shell of the oocyte), t ≈ 5 µm, are measured from optical micrographs. The 100 µm wide planar circular platform of the fixture is bounded by seven micro-posts with height of 45 µm and diameter of 12 µm to contain the cell and to measure the applied load. A micro-pipette passes through the inter-pillar gap to reach the cell surface. The external load creates an indentation dimple in the cell and is balanced by the deflection of the three backing posts in the leeside as shown. The applied load is deduced by the magnitude of post deflection and is calibrated to within a resolution of 2 nN. The loading process and the continual deformation are monitored in-situ by an optical microscope to determine (i) micropipette tip displacement, d, and thus the applied force, F, (ii) dimple profile, (iii) deflection of the 3 backing micro-posts, and (iv) overall deformed profile in conventional polar coordinate system, w(r,θ,φ).

Figure 5.3. Stress-relaxation indentation test. (a) Displacement of the micromanipulator as a

function of time. The micromanipulator controlled the micropipette to indent the oocyte at a

constant speed of 60 µm/s (stage I), and maintained a vertical displacement of 40 µm for 45s

(stage II). (b) Schematic of the expected force response as a function of time.

Force-displacement relation, F(d), as well as the profile at any given F, are recorded.

Figure 5.3 show the schematic d(t) and F(t) of the two consecutive loading stages. In Stage I, displacement-controlled loading is performed at a constant speed of 60 µm/s until d reaches approximately dmax = 30-40 µm, which is comparable to the cell size of 2R ~ 90 µm. The

instantaneous indentation load is monitored by tracking the micro-post deflections. The

mechanical response, F(d), is used to deduce E, using a theoretical model to be discussed later.

In Stage II, the micropipette tip stayed fixed for the subsequent ~ 45 s, while the external load,

F(t), is monitored as a function of time, with t the time elapsed from the instant when maximum

d is reached. Viscoelastic properties of the cell are deduced from F(t), namely, the instantaneous

modulus, E(t=0) = E0, the relaxed modulus E(t→∞) = ER, and the stress-relaxation time constant,

τ. Only SS model is adopted to deduce these parameters, which do not differentiate the zona

pellucida (ZP) from the cytoplasm, but the overall apparent behavior of the cell. A total of 16

oocytes with 8 young cells (8-12 weeks old) and 8 aged cells (40-45 weeks old) are investigated.

All mechanical tests are performed at 37°C inside a standard environmental chamber to prevent

dehydration.

5.3 Solid sphere (SS) model

An appropriate theoretical model for the indentation process should yield rigorous

relation between (i) materials properties such as elastic modulus and Poisson’s ratio of cell wall,

Ew and vw respectively, and that of the cytoplasm, Ec and vc etc., (ii) geometry such as the cell

radius and wall thickness etc, (iii) deformation mode such as plate-bending and membrane-

stretching of the cell wall, and (iv) mechanical responses such as F(d) and subsequent profile w(r,θ,φ). Given all the measurable quantities from experiments, the elastic modulus can be quantified using the model. Three linear elastic models are proposed here based on different assumptions, and the predicted F(d) and w(r,θ,φ) are compared with experimental observations.

Computation is performed using a commercial finite element analysis (FEA) software ABAQUS

(Standard version 6.9-2).

5.3.1 Model

The classical Hertz theory is based on three fundamental assumptions: (i) sample cell is treated as isotropic elastic half space where all deformation resides, (ii) AFM tip is a rigid spherical cap of finite radius of curvature, and (iii) viscoelastic behavior is ignored [131]. The mechanical response is given by

4 R F = ind E*d 3/ 2 3 (5-1)

* 2 where E = E/(1 – v ) is the effective modulus, and E = Ew = Ec of the overall cell assuming uniform deformation of the cell wall and cytoplasm composite. Equation (5-1) requires F ∝ d3/2 for a spherical indenter based on linear elasticity. Other indenter geometry (e.g. circular cone) might require a slightly modified relation, F ∝ d2 [132].

Here, since dmax ~ R, large global deformation of the sample cell is expected, and the

validity of Hertz model is doubtful. Nevertheless, for the sake of comparison, a solid sphere

model is constructed. The cellular materials is taken to be incompressible such that vw = vc =

0.50. Our FEA code works quite well at small d, but fails at large d, especially when dmax

approaches ~ 40% of the cell dimension leading to global deformation.

To circumvent the convergence problem and to simulate the cell resting in the fixture, a

semi-rigorous 2-step method is constructed as shown in Figure 5.4(a). Firstly, being supported by

the 3 frictionless backing posts, the sphere is first compressed by a rigid planar plate in the

vertical direction with a load of F, resulting in an approach distance, d1, and a planar contact

circle with radius, c. Since the micro-posts experience small deflection compared to the cell dimension, they are assumed rigid in the FEA calculation. Secondly, the top plate is then removed and replaced by a central load with the same magnitude, F, distributed over the micropipette rim. The original planar region is treated as an infinite half continuum and the central load a cylindrical rod with radius, a, thus forming an indentation dimple with depth, d2.

The overall displacement of the micropipette tip is therefore d = d1 + d 2 . The constitutive

relation in step 1, F(d) for d ≤ d1, is found by FEA based on large elastic deformation. The stress- displacement analysis is performed using an 8-node linear brick, reduced integration elements

(C3D8R). To promote computational efficiency, the number density of elements is chosen to be the highest near the vicinity of contact and reduced further away. Figure 5.4(b) shows the 3D structured element mesh and Figure 5.4(c) a typical deform-stress field. In step 2, the classical

problem of indenting an infinite half space by a cylindrical punch is given by [133]

1− ν 2 F d 2 = 2E a (5-2) provided a << c. Profile of the indentation dimple is given by

 1−ν 2 F <  r a 2E a w(r) =  2 (5-3) 1−ν F − a  sin 1 ( ) r ≥ a  πE a r for 0 < r < c. A small mismatch of deformed profile at the dimple edge is thus inevitable in the current scheme, but Equation (5-3) by and large gives the first approximation of profile in the vicinity of the indenter contact. As a last remark, the SS model does not preserve the cell volume. The dimple serves as a small perturbation to the deformed body where the overall spherical geometry is maintained throughout the loading process. The dimple has a negligibly small volume compared to the cell volume, and essentially conforms to the micropipette tip geometry.

To account for the viscoelastic behavior, the elastic model is slightly modified. During relaxation, the external load decreases as a function of time though the deformed profile does not show any measurable change. It is therefore assumed that the only the Young’s modulus decreases while all other parameters remain constants. In a 3-parameter standard linear solid, the apparent elastic modulus is given by

 T  E(t) = (E − E ) exp−  + E (5-4) 0 R  τ  R

such that E(T=0) = E0, E(T →∞) = ER.

5.3.2 Results

Figure 5.5 shows the instantaneous mechanical responses of the young (green, n = 8) and old (gray, n = 8) oocytes during Stage I loading. Elastic properties of the indented oocytes are extracted by fitting the theoretical model to individual data sets. The solid curve shows fit to one particular set of data. Figure 5.6 shows F(t) during relaxation. Figure 5.7 shows softening of young oocytes during relaxation E(t) fitted to Equation(5-4). The elastic and viscoelastic

parameters of the two types of occytes are summarized in Table 5.1. Both E0 and ER of the aged

cells are about half that of the young, and τaged ≈ 2 τyoung. The aged oocytes appear to be

significantly more compliant but more viscous than the young counterpart.

Figure 5.5. Force-deformation data from the young (green) and aged (grey) oocytes at Stage I.

Figure 5.6. Force-time data from the young (green) and aged (grey) oocytes at Stage II. The black lines are typical curve fits to single sets of experimental data.

Figure 5.7. Apparent elastic modulus as a function of time for a sample young oocyte.

Table 5.1. Summary materials properties of young and aged oocytes using SS model.

Elastic Property Viscoelastic Properties

Instantaneous Relaxed Oocyte Sample Young’s Modulus Relaxation Time Modulus Modulus

E (kPa) τ (s) E0 (kPa) ER (kPa)

Young Oocytes 5.2±0.5 5.2±0.5 2.7±0.4 2.3±0.2

Aged Oocytes 2.2±0.6 2.2±0.6 1.0±0.3 4.0±0.6

No data are found in the literature regarding viscoelastic properties of murine oocytes, let alone comparison between young and aged. Viscoelastic characterization of articular

[134] chondrocytes using nano-indentation yields viscoelastic parameters (E0, ER and τ) in the same order of magnitude as in the present study.

5.4 Shell model

To simulate the actual cellular structure with cell wall encapsulating an incompressible cytoplasm, two shell models are derived.

5.4.1 Hollow spherical shell (HSS) model

The first approximation is a hollow spherical shell in the absence of an internal pressure.

The shell is assumed to be made of isotropic linear elastic material. A 4-node reduced integration shell element (S4R) is used in the computation. Though the micropipette is in fact a cylindrical shell, it is here taken to be a rigid conical indenter with spherical tip without loss of generality.

To facilitate computational convergence, the tip radius is taken to be Rind = 5 µm even though the

actual dimension is 3 ± 1 µm as estimated from the optical micrograph. Our calculation shows

that such difference does not cause any significant deviation in the mechanical response.

Moreover, as will be shown later, the resulting deformed profile fits better to the experimental

data. To avoid nonphysical skewness and asymmetric loads, the following conditions are

reinforced: (i) zero rotation in the azimuthal direction, (ii) zero translation motion of both the

indenter and the shell, and (iii) the backing posts are fixed in all degrees of freedoms. The shell is

structured-meshed into ~2400 elements.

Figure 5.8(a) shows the 3-dimensional displacement field for a hollow shell with radius R

= 45 µm, thickness t = 5 µm, Poisson ratio v = 0.5 and an indentation depth of d = 30 µm. Large

displacement is present within the indentation dimple and vanishes towards the stationary

backing posts. Figure 5.8(b) shows the corresponding von Mises stress field. The stress

distribution is quite involved. Highly concentrated stress is present immediately under the

indenter, but decreases radially outwards. A ring of local high stress is present at the dimple

crater ridge, which diminishes along the meridianal direction towards the backing posts.

(a)

(b)

Figure 5.8. A 3-D HSS model of mouse oocyte. (a) Displacement distribution of deformed shell with depth of 30 µm. (b) von Mises distribution of deformed shell with depth of 30 µm.

The deformed profile is expected to be significantly different from that of the SS model.

Since the shell is not supported by an elastic continuum as in SS, large deformation occurs around the indenter and the dimple extends to a larger area. Since there is no constraint on the shell volume here similar to SS, the shell profile outside the dimple is expected to remain largely unchanged or spherical.

5.4.2 Liquid-filled spherical shell (LFSS) model (a)

(b)

Figure 5.9. A 3-D LFSS model of mouse oocyte. The cytoplasm is modeled as an incompressible

fluid. (a) Displacement distribution of deformed shell. (b) von Mises stress distribution of

deformed shell with deep penetration of the indenter.

To better simulate an actual cell that encloses a cytoplasm, the elastic shell is filled with an incompressible liquid. In the presence of an applied load, the liquid maintains the internal volume of the shell by exerting a uniform hydrostatic pressure. Leakage of the liquid via the

shell is strictly prohibited. Hydrostatic fluid element (F3D4) is employed in the model.

Cytoplasm density of ~ 103 kg/m3 is chosen as the first approximation. Figure 5.9(a) shows the

3-D displacement field for a shell with radius R = 45 µm, thickness t = 5 µm, and an indentation depth of d = 30 µm, similar to HSS. Here the shell portion having a minimal displacement change significantly retreats towards the backing posts. Figure 5.9(b) shows the corresponding von Mises stress field, which is quite distinct from HSS. Despite the stress concentration within the dimple, local maximal stress at the dimple crater ridge disappears or not discernible in the

FEA computation. Instead, a larger portion of the shell is now under minimal uniform stress close to the backing posts.

The corresponding deformed shell profile differs from SS but to a lesser extent compared to HSS, because the shell volume is now constrained. Buildup of the internal pressure helps to push the dimple wall closer to the indenter, resulting in a smaller dimple volume compared to

HSS. A small lateral bulging of the shell is expected immediately outside the dimple as shown.

5.4.3 Results

Figure 5.10 shows force-displacement relation, F(d), for two typical oocytes, along with the best fitting curves based on the SS, HSS, and LFSS models. The terminal point of each curve denotes the last converged solution available from the FEA. The mechanical responses F(d) are similar in the two oocyte types, though the young cell is significantly stiffer than aged counterpart, independent of the model chosen. The SS model leads to a linear F(d) in initial loading until d ≈ 20% × 2R, where the load deviates positively from the linear trend. A homogenous cell neglecting the shell structure is thus insufficient to account for the overall mechanical behavior. The HSS model leads to a linear F(d) in initial loading until d ≈ 10% × 2R, where the load falls below the linear trend, contradictory to the experimental data. The LFSS

89 curve apparently yields the best fit to the experimental data among the three models. In order to quantify the indentation deformation of a spherical cell, it is necessary to consider the structure of shell and cytoplasm in a single cell and their combined mechanical properties. The cell membrane in fact takes up the majority of external load, while the loaded cytoplasm gives rise to an internal pressure.

d=0 µ m d=10 µ m

25 µ m 25 µ m

d=20 µ m d=30 µ m

25 µ m 25 µ m

Figure 5.11. A Comparison between experimental deformed profile and theoretical models with blue line SS model, black line HSS model and red LFSS model(depth of 0, 10, 20, 30 µm).

Vella and Vaziri also theoretically and experimentally investigated the indentation of

pressurized elastic shells. Two linear regimes at small and large indentation depths in force-

displacement relation are proposed, which is function of material properties (E, v), shell geometry (R, h), and internal pressure (p). However, it is not quite possible to check the consistence of two slopes since in Vella’s model, constant internal pressure is maintained throughout the indentation process, while in our LFSS shell model, internal pressure increases as indentation depth increases.

Another verification of the LFSS model is to check for consistency in the deformed profile. Figure 5.11 shows the progressive loading of an oocyte and the fitted profile according to the three models. The dimple follows quite closely the micro-pipette geometry close to the indenter, indicating a shell-like structure. The SS profile conspicuously misses the deformed cell geometry to a large extent in the indentation dimple as expected. Also, the theoretical profile does not show the observed lateral bulging. On the other hand, both HSS and LFSS fit quite well to the cell profile in terms of the dimple and lateral bulging. In fact, the predicted profiles fit better for larger indentation depth due to the large global deformation.

Table 5.2. Comparison of Elastic modulus of mouse oocyte obtained from three models

Oocyte E (kPa) E (kPa) E (kPa) Sample SS HSS LFSS

Young 5.2 ± 0.5 26.4±3.4 11.6±2.5

Aged 2.2 ± 0.6 10.4±1.5 4.7±1.3

Table 5.2 summarizes the average elastic moduli of the 8 young and 8 aged mouse

oocytes based on the three theoretical models. The former is roughly twice that of the latter for

92 the same model. SS yields the lowest value of E because more elastic materials including the cell wall and cytoplasm are assumed to support the external load. HSS generates the highest value of

E because only the thin shell is deformed and the original spherical geometry is distorted the most. LFSS yields an intermediate stiffness because of the additional internal pressure supporting the applied load. Elastic moduli obtained from the two shell models are more consistent with the literature where E ranges from to 8.26 kPa to 17.9 kPa [135, 136].

5.5 Summary

The architectural structure of a cell with encapsulating wall and internal cytoplasm holds the key to the overall mechanical behavior, which cannot be accounted by the solid sphere model. Indentation of single oocyte shows that shell models must be used to analyze the force and profile measurements in a consistent manner. The shell thickness is a key parameter in the theoretical model that is missing in the solid sphere model. Our experimental work in micropipette indentation of mouse oocytes verifies such conclusion. It is important to note that the conventional analysis of AFM indentation data according to the Hertz contact theory significantly underestimates the actual cell wall stiffness.

6 ADHESION OF CYLINDRICAL THIN SHELL

In this chapter, the mechanical deformation of an ideal thin-walled cylindrical shell is investigated in the presence of intersurface interactions with a planar rigid substrate. Dugdale-

Barenblatt-Maugis (DBM) cohesive zone approximation is introduced to simulate the convoluted

surface force potential. Without loss of generality, the repulsive component of the surface forces

is approximated by a linear soft-repulsion, and the attractive component is described by two

essential variables, namely, surface force range and magnitude, which are allowed to vary. The nonlinear problem is solved numerically to generate the pressure distribution within the contact, the deformed membrane profiles, and the adhesion-delamination mechanics, which are distinctly

different from the classical solid cylinder adhesion models. The model has applications in cell

adhesion and nano-structures.

6.1 Introduction

Adhesion of a biological cell to an apposing cell or a bio-substrate is essential in

biological sciences in terms of many cell signaling and physiological functions. When multiple

cells are involved, adhesion leads to aggregation as in tissue and biofilm formation. In either

case, adhesion is strongly coupled to the mechanical deformation of the soft lipid membrane of a

cell or the stiff glycoprotein shell of a bacterium.

In this chapter, we consider the adhesion of a thin-walled cylindrical shell to a rigid

substrate in the presence of a van der Waals like intersurface attraction with finite magnitude and

range, and also repulsion in case of surface interpenetration. The shell is allowed to deform by

mixed bending and stretching. The pressure distribution within and without the contact area,

deformed profile, mechanical response in terms of applied load, contact width and approach

distance, as well as the characteristic “pull-off” event, are investigated, and compared with the 94

classical modified JKR and DMT models for solid cylinders. Distinct characteristics unique to

shells are found.

6.2 Theory

Figure 6.1. Schematic of a cylindrical shell deformed by an external load coupled with

intersurface attraction or disjoining pressure.

Figure 6.1 shows the schematic of a cylindrical shell lying flat on a rigid substrate. The

cylinder has unit length b, radius, R, thickness, t (<< R), elastic modulus E, Poisson’s ratio, ν,

3 2 bending stiffness κb = Et /12(1 – ν ), extensional stiffness κs = Et. In the presence of intersurface

attraction, the cylinder comes to an intimate contact with the substrate with width, 2a, and the

cohesive zone (where the effective surface force is present) extends further out to 2c as shown in

Figure 6.1. The arc length s is measured from the lower apex and is given by s = Rϑ in a non- deformed cylinder, with ϑ the angular displacement. The tangent and normal unit vectors t and n

are fixed to the deforming shell (Figure 6.2). Upon deformation, an arbitrary position on the

shell with an original position vector ρ, moves to a new equilibrium position P’ with position vector ρ'. Deflections along t and n are denoted by u and v, respectively. The local slope of the shell changes by an amount φ with respect to the axial direction t × n. Here φ also corresponds to the rotation of the normal n as the shear strains γnt are neglected.

Figure 6.2. Coordinate system for the deformed shell. Note that ρ and ρ′ indicate the positions

vector of a point P before and after deformation. The position vector is referenced with respect to

the center of the initially undeformed cylindrical configuration.

6.2.1 Intersurface forces

Figure 6.3 shows a Lennard-Jones (LJ) like intersurface potential between two adhering

surfaces with a separation, z. The planar substrate is taken to be infinitely rigid and non-

deformable. The interaction force, or disjoining pressure, is given by, [28]

− − p (z) = C [(z / z ) a1 − (z / z ) a2 ] (6-1) s s 0 0

where the subscript s hereafter denotes variables related to surface force, the constants CS, a1 and

a2 define the potential and work of adhesion, z is the distance between the outer surface of the

shell and the top of the flat substrate, and z0 is the equilibrium atomic spacing between the two adhering surfaces where the attraction vanishes. The contact half-width a is here defined to be

the border where the intersurface separation reaches z0, or ps(z0) = 0, and its computation is

described in the following section. It is assumed that z0 = 4 Å in the present work. Shaded area

under the positive curve of ps(z) is defined to be the adhesion energy, γ. In order to investigate

the effect of surface force range for a fixed work of adhesion, γ, a modified form of the Dugdale-

Barenblatt-Maugis (DBM) cohesive zone approximation [28, 38] is adopted,

 ms (z − z0 ) for z ≤ z0 + δz   p* for z + δz ≤ z ≤ z − δz p (z) =  0 1 (6-2) s − ≤ ≤ − ms (z − z1 ) for z1 δz z z1   0 for z ≥ z1

as shown in Figure 6.3. Here the surface force is taken to be constant, p*, its effective range, z*

= z1 – z0. The work of adhesion is given by γ = p* × (z* – δz), which is equal to the shaded area.

The repulsive (negative) component is approximated by a linear force in the range z ≤ z0 + δz, and the slope ms is chosen such that δz / z* → 0, and thus γ ≈ p* × z*. Classical adhesion models

represent limiting cases: (i) JKR requires ms → ∞, p* → ∞, and z* → 0, (ii) DMT requires ms →

∞, p* → 0, and z* → ∞, and (iii) DBM requires m→ ∞s, z*, and p* being finite and non-zero.

Figure 6.3. Lennard-Jones potential for the disjoining pressure and the Dugdale – Barenblatt –

Maugis cohesive zone approximation.

As the shell adheres, free surfaces are destroyed and the corresponding surface energy,

US, can be computed as follows. The net surface force is expressed as ps = pst t + psn n with

respect to the (t, n) coordinate system. Since the surface traction is assumed to act perpendicular

to the adhering surfaces, the tangential component is negligible, rendering ps = ps n. Defining d

as the deformation due to surface force, the surface energy can be expressed as follows,

= − ⋅ U S ∫ ps d ds (6-3)

Note that, in this work, the substrate-to-cylinder spacing z is measured along the normal to the

surface of the shell. As a result we are able to neglect the tangential component of the surface

traction pst acting on the cylinder. In fact, z can also be measured normal to the substrate surface,

in which case pst cannot be neglected. The difference between the two approaches should be

minimal for short ranged adhesion force (i.e. small z*). In case the adhesion forces are not short

ranged, and z is measured with respect to the substrate, the magnitude of the surface force ps will be larger for a given spot on the cylinder. Considering that the out-of-plane deformation of the shell is primarily governed by the normal component psn, it is conceivable to find the contact area

to be smaller than the case considered in this work. More work is necessary to determine these

lower and upper bounds of the surface energy.

6.2.2 Mechanical energy

While the model developed in this work is applicable to generic micro-scale shell configurations with applications in (bacterial and plant) cell adhesion, carbon nano-tubes, and the like, it is worthwhile to note that in the case of biological cells the concentration differences between various molecules, contained in the cytoplasm of a cell and the liquid environment in which it resides, is cause for osmotic pressure. Therefore, the effect of a small net (osmotic) pressure in the shell is included in the model. In vesicles, whose membranes consist of a bi-layer

of phosphor-lipids, such a pressure difference typically causes the vesicle to adjust its volume to

dissipate the osmotic pressure [65]. However, the metabolic activity inside biological cells (such as bacteria), result in substantial concentration differences leading to a sustained osmotic pressure, which can be handled effectively by the stiffer walls of such cells [137]. In addition to

osmotic effects, additional internal pressure in the cell can develop due to quasi static motion of

the cells in a flowing liquid medium, such as the motion of red or white blood cells, or suction of

a cell into a pipette in an experiment [69, 138].

Therefore, when a cylindrical shell is loaded by an external load with or without

interfacial adhesion, it is deformation is influenced by a bending and stretching of its shell, as

well as an internal pressure [65, 139]. In this work, internal pressure is assumed small and constant.

The energy involved in the deformation of such a shell has therefore three separate components:

(i, ii) energy input due to the external load, and the internal pressure, and (iii) the elastic strain

energy stored in the shell. The external compressive/tensile load, Fi acting on the cylinder

surface at si moves a distance (or causes a deformation) of di . The associated potential energy is

therefore given by UF = –Σi Fi · di Here we consider a single force (per unit length) acting along

the upper apex of the shell, thus UF becomes,

UF = – F · d (6-4)

The deformed shell induces an internal pressure, pint (with the subscript int denoting

internal pressure hereafter), in the encapsulated liquid, leading to an energy term [65, 139],

U = − p v ds (6-5) int ∫ int

with v transverse deflection. For linear-elastic material behavior the strain energy stored in the

shell is given as follows [65, 139],

U = ( ½κ H 2 + ½κ ε2 ) ds (6-6) E ∫ b s

99 where H(s) = d2v/ds2 is the local change of curvature from the circular cross-section and ε(s) = du/ds is the membrane strain along the arc length. Large deformation is allowed in (6) as reference state of the shell is updated during solution iterations as described in the following numerical method session. Total energy of the system, UT , is sum of all the four energy terms,

UT = US + UF + Uint + UE (6-7)

The principle of minimum total potential energy (δUT = 0) provides the equation of equilibrium of the system. Owing to the general nature of the model, constancy of volume and/or area of the shell is not imposed, but it is noted that for the chosen parameter range reported below the change in these quantities is on the order of 0.05 - 0.5%.

6.2.3 Numerical methods

Figure 6.4. Incremental deformation of the shell.

(e) The shell is discretized in to M two-node elements defined by a vector dnt = {uj, vj, φj,

T uk, vk, φ k} with u and v being the in-plane and transverse deflection components, φ the rotation of the normal to the mid-plane of the shell due to deformation, and subscripts j and k the element

100

nodes (Figure 6.4). The deflection components are interpolated for each element, in the (t, n)

coordinates, as follows,

u = Nd()e v = Nd()e (6-8) u nt v nt

where Nu = [NN1200 00], Nv = [00NN34 NN 56] and N1 and N2 are C0-

[140] continuous, and N3, N4, N5, and N6 are C1-continuous shape functions . The equilibrium

equation is found by substituting Equation (6-7) into (6-8) and letting δUT = 0. The equilibrium

is expressed as follows,

(e) (e) = f (e) + f (e) + f (e) = f (e) (6-9) k nt dnt nts nt F ntint nt

where (e) is the stiffness matrix, and f (e) , f (e) , f (e) are the force vectors for surface force, k nt nt s nt F ntint

external load, and internal pressure, respectively, given by,

L L (e) e T T e T k nt = (B v EIB v + B u EAB u ) ds and f = N q ds (6-10) ∫0 ∫0

22= with Le the element length, Bvv= dN / ds , BuvdN / ds , N = Nu + Nv and q being one of pint ,

ps and F. The stiffness matrix represents linear-elastic material behavior. Equation (6-9) is transformed into the global coordinate system (x, y) by a coordinate transformation through a coordinate transformation matrix T which depends on the angle of the elements’ orientation

[140] θ N with respect to the global coordinate axes , is given as

 cos(θ N ) sin(θ N ) 0 = − θ θ  (6-11) T  sin( N ) cos( N ) 0    0 0 1

Global equilibrium is then found by summing the forces acting on the mesh nodes,

K d = f (6-12)

101

where K is the global stiffness matrix , d is the global degree of freedom vector, and f is the

global force vector.

Boundary Conditions

Only a half cylinder is modeled due to symmetry. Surface of the shell, Γ, is composed of

three regions Γ = Γr U Γa U Γ0 with Γr and Γa being the segments of the boundary under the

effects of repulsive and attractive forces, respectively, and Γ0 the segment free of surface forces.

Involvement of the surface force in this problem modifies the classical conditions applied on the

boundary of this problem. In fact, we impose no specific conditions on the penetration of the

boundary in the y-direction, but let the equilibrium decide how the shell positions itself with

respect to the equilibrium distance z0 near the rigid wall. The rigid wall is considered at x/R = – 1

position parallel to x-axis. In this work we use symmetry boundary conditions which are

expressed as follows,

u (0) = 0, φ(0) = 0 (6-13)

u(Ls) = 0, φ(Ls) = 0 (6-14)

Equation (6-12) is nonlinear as the surface forces depend on the shell-substrate spacing and as the extent of the contact region is not known a priori. The residual r of Equation (6-12) is

linearized around the equilibrium level as follows,

(i+1) (i) (i) (i +1) r = r +J Δd (6-15)

the tangent-stiffness matrix which depends on both shell-stiffness and intersurface forces is

found by ϑ (i) = (∂r/∂d) (i). The Newton-Raphson iterative scheme is then used to solve for the

degree of freedom vector,

(i+1) (i) (i +1) (i +1) (i) d = d + C Δd and J ()i Δd = – r (6-16)

102

where the superscript i indicates solution iteration level, and C is a constant relaxation coefficient

with 0 ≤ C ≤ 1, being employed to facilitate convergence. The strain free reference state is updated after each iteration. This requires the position vector ρ and slope θ of each point of the

mesh to be updated. For node-j this update is computed as follows,

(i+ 1) ( i ) ( i ++ 1) ( i 1) ( ii ) ( + 1) ρρj= j +∆C ρ j and θ j= θϕ jj + (6-17)

(ii++ 1) ( 1)ˆˆ ( i + 1) ˆˆ where, ∆=ρjjjuvi+ jwith i and j representing the unit vectors of the global (x,y)

coordinates (see Figure 6.4). The coordinate transformation matrix T is then computed by using

(i+ 1) θ j rather than the original undeformed (circular) configuration of the shell. Figure 6.5 shows

the flow chart of solution algorithm.

Figure 6.5. Flow chart for the solution algorithm.

103

Detection of Contact Half-Width, a

Once a converged solution is found, each element is tested for the case of having one

node below (zj < z0) and one node above (zj+1 > z0) the equilibrium spacing z0, with subscripts j

and j+1 indicating the nodes attached to a particular element. The exact location of the half- contact width a is then obtained by using the normal displacement components vi and vi+1 and

searching for v(a) = 0 in Equation (6-8).

6.3 Results and discussion

The range of values chosen for analysis is motivated by liposomes and biological cells,

based on the literature as follows: wall thickness of rod-like bacteria is 6 – 40 nm [141], R = 5 µm

[142] for typical cells, E = 100 MPa for typical wild-type typhae fungus , ν = 0.50, pint = 5 Pa for

typical deformation induced pressure in cells [143], adhesion energy for weak vesicle adhesion γ =

-3 2 [65] 10 5×10 mJ/m . The repulsive pressure is determined using ms = 10 Pa/m and wall thickness

is t = 10 nm. Non-dimensional concentrated force F = F/γR, pressure p = p/E, distance

parameters a = a/R, z* = z*/R, and work of adhesion γ = p*z* are used to represent the results.

Note that the constant work of adhesion used throughout this work corresponds to non-

dimensional adhesion energy of γ = p*z* = 10-8. In what follows, the short and long range

interactions are investigated by modifying the z* value. Thus a larger p* corresponds to a short z*.

-8 The non-dimensional internal pressure is pint = pint/E = 5×10 .

6.3.1 Convergence

In this study, the undeformed circular shell is discretized into numbers of beam elements.

The approximate profile is approaching the ideal circular profile by increasing the number of

beam elements. In order to investigate the convergence of the solution, the number of elements

for half circular shell is varied from 100 to 600, while keeping other parameters unchanged.

104

Figure 6.6 shows the external force applied on the apex node versus the contact length with

10 Young’s modulus of 1000 MPa and repulsive slope ms of 10 Pa/m. Except for the case of 100

elements, all the cases converge to same results. For R = 5 μm, M = 600 elements distributed evenly along the half-circle translates to individual element length being 25 nm, which is sufficiently small in the molecular scale to approximate a circular cross section. Therefore, M =

600 is chosen in the present study.

Figure 6.6. Convergence study with total element number increasing from 100 to 600 at z* = 1.

6.3.2 Deformed profile and stress distribution

Figure 6.7(a) shows the deformed profiles close to the contact plane for a range of

disjoining pressure and force range, with the inset showing the global profile. Figure 6.7(b)

shows the corresponding mechanical pressure exerted on the shell. As the shell comes to contact

with the substrate, the contact edge ( x = a ) marks the demarcation of both stress and profile.

Immediately within the contact edge ( x < a ), the intersurface separation falls below the

equilibrium spacing z0 leading to interpenetration of the two adhering surfaces. The profile then gradually flattens towards the contact centerline ( x = 0). Surface interpenetration leads to strong

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repulsion, and the compressive stress rises to a sharp spike which decays in an oscillatory

manner, settling to a small negative value. The magnitude of such limiting compression depends

on p * much like the repulsion spike which is more pronounced at increasing p *. The limiting

stress discontinuity is expected as the shell geometry transitions from planar within the contact

edge to finite curvature without the edge. In the JKR limit, the spike magnitude is expected to

approach infinity [144].

Figure 6.7. Cylindrical shell adhering to a rigid substrate under zero external load, F = 0: (a)

deformed profile in the vicinity of the contact edge and inset showing the global deformation,

and (b) pressure at equilibrium. The adhesion force range z*/R is allowed to vary as indicated, while the adhesion energy is maintained at p*z*/ER = 1×10-8.

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The alternating repulsion-attraction close to the contact edge is a distinct characteristic of

a membrane detaching from an adhering substrate. In fact, such stress field is long predicted for

thin films being peeled from a rigid substrate as a result of stress singularity at the contact edge

[145]. The repulsion is experimentally observed in the interaction between an advancing straight

crack with a penny crack at a membrane-substrate interface in muscovite mica flakes [146]. The

convoluted stress field in the vicinity of the contact edge is a remarkable feature unique to shells.

There is a stark difference in solid cylinder behavior [144], repulsion is a maximum at the contact

centerline due to geometrical incompatibility, and diminishes radially outward until attraction

takes over and approaches infinity at the contact edge.

Beyond the contact edge ( x > a ), the long range disjoining pressure gives rise to a fairly

uniform attraction that extends to the outer edge of the cohesive zone. The shell-substrate

separation finally exceeds the surface force range ( z = z * and y ≈ z *) and the shell

becomes traction free. In case of strong surface force (large p *), the shell profile is distorted to

the largest extent at the contact edge (c.f. Figure 6.7(a)) alluding to the JKR “neck” formation

[28]. Such behavior is consistent with Barenblatt’s crack [147] where the mechanical stress

vanishes at the crack tip (contact edge) but the trailing cohesive zone behind is under traction.

By changing the repulsive slope factor ms, one could incorporate the soft-repulsion between the shell and the substrate, which encountered in various cell-surface interactions.

Figure 6.8(a) shows the deformed profiles of cylindrical shell in contact region, both x and y coordinates are normalized by radius R. Equilibrium position is plotted out with dotted line, which is z0 spacing above the substrate. Mechanical equilibrium is established slightly below the

atomic equilibrium spacing z0 due to finite shell stiffness, and the shell thus inter-penetrates the contact plane that gives rise to the repulsion within the contact edge. With increasing the

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repulsive slope ms, the cylinder senses more repulsive force with same amount of penetration,

which would push the shell toward the equilibrium position z0, shown in the figure Figure 6.8(a).

The corresponding reaction pressure distribution is shown in Figure 6.8(b). Strong repulsive reaction pressure is established at contact edge. With further increasing the repulsive slope ms, a

stronger concentrated stress is formed at the edge, which is consistent with fracture mechanics

theory—infinite stress is occurred at crack tip area.

9 10 Figure 6.8. Effect of repulsive slope ms, ranging from 2x10 to 2x10 : (a) deformed profile in

the vicinity of the contact edge, and (b) corresponding pressure at equilibrium.

6.3.3 Mechanical response

In the presence of an external force F applied to the upper apex, the shell adheres to or

delaminates from the substrate. There are three measurable quantities, namely, the applied load, the central displacement, and the contact radius. Figure 6.9(a) shows the mechanical response

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Figure 6.9. Delamination trajectory for different adhesion force range z*/R values and fixed adhesion energy p*z*/ER = 1×10-8. (a) Contact width as a function of applied load for fixed

adhesion energy and a range of disjoining pressure. (b) Applied load as a function of approach

displacement. (c) Diminishing contact width as the external load turns tensile. Delamination

follows trajectory ABCDGHP and “pull-off” occurs at P when the contact reduces to a

centerline. 109

aF() for fixed adhesion energy with positive F denoting compression. It should be noted that, in case the membrane wall exhibits stiffening upon deformation the contact half-width would be smaller all other parameters being equal. At F = 0 (point C), the contact width is finite and non-

zero due to adhesion. In the presence of a progressive compression along path CBA, the contact

widens until an upper bound is reached because of the finite shell dimension. In such limit, the

cylindrical shell is squashed into a pseudo-biconcave shape with an indentation dimple at the

upper apex and a flattened planar contact with the substrate (c.f. curve A in Figure 6.10). On the

other hand, should a tensile load be applied, the contact gradually diminishes along path CDGHP

and shell elongates gradually along the vertical direction.

F/γR ×109 a / R d / R

A 0.5 0.557 0.950 B 0.2 0.493 0.411 C 0.0 0.373 0.144 D − 0.2 0.228 −0.073 P − 0.7 0.000 −0.401

Figure 6.10. Shell profiles for a range of external loads F/γR as indicated, and for z*/R = 1/25

and fixed adhesion energy p*z*/ER = 10-8.

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Figure 6.10 shows A-P snapshots of the deformed shell profile as a function of external

loading for p * = 25×10-8, and z * = 1/25 ( z * = 0.20 μm). At P where F †×109 = –0.7 (the

† † superscript denotes “pull-off” hereafter), the contact reduces to a line with a = 0, and the shell

pinches off from the substrate. The phenomenon is known as “pull-off” in the literature. For a constant adhesion energy, variation of p * and z * does not change the general adhesion- delamination trajectory, though F † and aF(= 0) increase with p *.

Figure 6.11. Shell profiles close to the contact edge for z*/R = 1/25 and fixed adhesion energy

p*z*/ER = 10-8.

It is remarkable that aF() shows two distinct segments where the gradient ∂∂aF/ turns

sharply from ABCDGH to HP. To explain the eccentric behavior, it is noted that as the tensile F

increases while the repulsion spikes behind the two contact edges translate in an invariant

manner (Figure 6.7(a)) [146]. At large loads along ABC, the repulsion spikes at the two opposite contact edges are virtually unchanged and thus maintaining the shell profile. This is an allusion to linear elastic fracture mechanics where the crack profile and the associated stress field ahead of a crack front are invariant as the crack propagates into a continuum solid. As the contact width

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shrinks further to H, the two spikes merge, leading to interpenetration of the adhering surface

and merely repulsive pressure within the contact area. Here the repulsion is thinly balanced by the applied tensile load and the attraction in the cohesive zone beyond the contact edge. At equilibrium, the shell deforms towards the substrate with a non-zero curvature in the contact region (Figure 6.11). Further increase in tensile applied load reduces the contact width to zero leading to “pull-off” at P.

Figure 6.9 shows the other two relations between F , d and a . Figure 6.9(b) shows

Fd(), external force as a function of approach distance, or vertical displacement of the upper

apex where F acts. Positive d corresponds to shell squashing. Note that d > 0 at F = 0 due to

adhesion. “Pull-off” occurs at d † at P. Figure 6.9(c) shows ad(). The drastic change in ∂∂ad/

is the consequence of the aforementioned non-zero curvature.

Comparing with the other existing models in the adhesion literature, there are several distinct features unique to cylindrical shells. For solid cylinder adhesion [40, 144], it is predicted that the “pull-off” width and applied load are respectively given by

1/ 3  E  1/ 3 0≤ †   ≤  2  ≈ a  2    0.8603 (6-18)  R γ   π 

1/ 3  1  3 1/ 3 0≤ †   ≤ π ≈ F  2  (4 ) 1.7437 (6-19)  ER γ  4

with the lower and upper bounds corresponding to the DMT and JKR limits respectively. The

JKR-DMT transition is governed by the Tabor parameter, λ = (Rγ 2 / Ez 3 )1/ 3 [38], where λ > 3 in 0

the JKR limit and λ < 0.25 in the DMT limit [40, 148, 149]. Cylindrical shells possess radically

different behavior with a† = 0 regardless of intersurface force range and F† is given by Figure

6.9. The non-zero “pull-off” width (a† > 0) in solid spheres and cylinders is a direct consequence

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of the geometrical incompatibility of the adherends, the subsequent Hertz compression, and the

elastic energy stored mainly within the contact with a maximum at the contact center. In contrary, the adhering shell virtually conforms to the planar substrate within the contact such that the geometry is compatible, and the elastic energy is stored mainly at the contact edge, leading to a† = 0.

The Majidi-Wan model [62] for cylindrical shell adhesion based on the JKR limit ignores

the repulsive component of the Lennard-Jones potential (c.f. Figure 6.3) and shows the adhesion- detachment mechanics being by and large consistent with the present model. In fact, a† = 0 is also predicted, though the change in shell curvature prior to “pull-off” and the subtle profile at the contact edge due to the repulsion spikes are absent.

Thin membranes have a different definition of λ because it has to account for the membrane thickness, t. For a planar thin film clamped at its circular periphery being adhered to

m n a rigid plane, we define λ film = (z0 / a).(γ / Et) .(a / t) with the exponents ¼ ≤ m ≤ ½ and 0 ≤ n

≤ 1[49, 50]. The lower limits of m and n correspond to thin and flexible membranes under pure

stretching, the upper limits refer to thick and stiff plates under pure bending, and the intermediate

values are for a compliant film under mixed bending-stretching. The JKR-limit is approached as

* * λfilm > 5 and a > 0, while the DMT-limit requires small λfilm, and a = 0. Based on these results,

it is possible to derive yet another λshell for a cylindrical shell with radius of curvature, R, and

thickness, t. Since a shell is essentially a curved plate, it is expected that λshell → λsolid when t →

∞, and λshell → λfilm when R → ∞.

It is interesting to note that whether the contact area at “pull-off” is zero has serious consequences in thin-walled cells and bacteria. When two adhering cells detach from each other, a non-zero “pull-off” contact area will cause severe damage (e.g. tearing) to the cell wall or the

113 lipid bilayers. The fact that cells are thin membrane encapsulated microcapsules instead of a continuum solid minimizes damages naturally. Another characteristic of the present model is that it can be scaled up or down to any dimension as long as the assumptions are valid. For novel nano-structures such as the single wall carbon nanotubes, it is uncertain whether change in profile curvature prior to “pull-off” will occur because the membrane is only one monolayer thick and plate-bending can take on a very different meaning.

6.4 Summary

A new linear elastic mechanical model is constructed for the adhesion-delamination mechanics of a compliant 2D thin-walled shell adhered to a rigid planar substrate in the presence of an interfacial attraction with finite effective range. The interrelationship between the applied load, approach displacement and contact radius are derived, which can be generalized to other shell dimension and stiffness. The new model is qualitatively compared to the JKR and DMT theories for solid cylinders, and is shown indispensable as the classical models fail to capture many distinct features unique to cylindrical shells.

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7 MECHANICS OF A CYLINDRICAL SHELL IN PRESENCE OF DLVO

POTENTIAL

Chapter 7 is an extension from Chapter 6. In Chapter 7, a theoretical model is built for a

single cylindrical bacterium adhering to rigid surface. In the presence of surface electrostatic

double layers and van der Waals attraction according to the Derjaguin-Landau-Verwey-

Overbeek (DLVO) theory, the bacterium glycoprotein shell deforms and may settle in either a

primary (1omin) or secondary (2omin) energy minimum depending on whether it has sufficient energy to overcome the repulsive energy barrier. The adhesion-detachment mechanics is

constructed and solved computationally, yielding the relations between applied load, deformed

profile, and mechanical stress distribution in the shell. Critical compressive load needed for

transition from 2omin to 1omin is found for several repulsive barrier heights. Upon critical

tension, shell in the 1omin detaches spontaneously at a non-zero contact area, but the one in the

2omin detaches smoothly with the contact shrinking to a line contact. The model leads to better

understanding of bacteria adhesion-aggregation-transportation, and has significant relevance to

environmental science and engineering.

7.1 Introduction

Transportation and migration of microorganisms (e.g. protozoa, bacteria) through a

porous medium of collectors (e.g. sand or sediment) in an aquatic environment are strongly

influenced by the attractive surface forces at the particle-collector interfaces. The retention and

migration of microbes in subsurface environments (e.g. in-situ or enhanced bioremediation sites)

or sand filtration processes (e.g. sand or multiple media filters for water and wastewater

treatment) have significant implications in and impacts on the performance of these systems. The

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classical colloidal filtration theory (CFT) has been widely employed to model microbial transport

in porous media, in which the microbes are assumed to be hard non-deformable colloidal

particles [150]. Rates of attachment and detachment are taken to be stochastically based and are

experimentally determined through flow-through column tests. The physics and mechanics of

adhesion-detachment of deformable particles are by and large ignored.

Elimelech et al. [151-153] elegantly showed the inconsistency in CFT and experimental

measurements, and suggested the necessity to introduce the full surface potential according to the

Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. The classical DLVO theory accounts for the combined repulsion between electrostatic double layers and attraction due to van der Waals interactions in the presence of an electrolyte solution. The resulting surface potential comprises a short-range but strong primary energy minimum (1omin) and a long-range but weak secondary minimum (2omin) separated by an intermediate energy barrier. When a colloidal particle

approaches a rigid surface, it will sense the presence of the weak 2omin and will be trapped

provided the repulsive barrier is sufficiently large and the thermal agitation is sufficiently low to

prevent escape. Alternatively, a particle may overcome a low barrier and thus settle in the 1omin,

which provides a stronger attachment. Despite the sound conceptual description, details of the

coupled adhesion mechanics and mechanical deformation of the particle have not been properly

accounted for so far. Although the model provides better description of microbial transport, the

coupled adhesion and deformation of the elastic microbe is yet to be investigated by rigorous

solid-mechanics. Other significant factors such as cell nature [154, 155], microbial phenotypes [155-

157], and surface forces [158] have been scarcely considered. This chapter aims to build a rigorous

mechanical adhesion model consistent with the DLVO theory, which can be used to make

predictions of bacterial adhesion behavior.

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In this chapter, we investigate the adhesion-detachment mechanics of a cylindrical shell

(e.g. E. coli) with a rigid substrate (e.g. sand grain) in presence of a typical DLVO surface potential and an external normal load. An applied tension is to imitate the external load such as liquid flow that tends to pull the particle off the adhering collectors. Only quasi-static mechanical

equilibrium is considered here.

7.2 Theory

Figure 7.1. Schematic of a cylindrical shell deformed by an external load coupled with

intersurface force to find equilibrium at 2omin or 1omin.

Figure 7.1 shows the schematic of a cylinder with length, b, radius, R, thickness, t (<< R),

elastic modulus E’, Poisson’s ratio, ν, and E = E’/(1–v2) making an adhesive contact with a rigid

planar substrate. An external compressive force per unit length, F (positive for compression),

coupled with the surface forces at the shell-substrate interface, deforms the shell to conform to the planar substrate, resulting in a contact area with width, 2a, and a cohesive zone of width (c –

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a) immediately outside the intimate contact edge where the surface forces act. An internal

pressure pint presses against the shell interior to maintain constant volume of the incompressible

liquid content. Depending on the nature of the surface potential, the shell might find equilibrium either at the 2omin with small deformation, or at the 1omin with relatively large deformation.

7.2.1 Intersurface force

Figure 7.2. Disjoining pressure according to DLVO potential and the Dugdale-Barenblatt-

Maugis cohesive zone approximation. “Pull-off” from 1omin is shown at the energy barrier peak

(c.f. “K” in Figure 7.7).

In the presence of an electrolyte, counterions build up electrostatic double layers on the

immersed surfaces. Interaction of a particle with an infinite rigid plane leads to potential energy,

U, as a function of intersurface separation, z, given by the DLVO theory in typical forms [154, 155]

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discussed in Chapter 2. Figure 7.2(a) shows U(z) as blue curve. A very short-range repulsion due

to electron cloud overlap based on Pauli’s exclusion principle is artificially introduced in U(z).

The two energy minima are separated by an energy barrier. Figure 7.2(b) shows the corresponding disjoining pressure, or surface force per unit area, ps(z) = –dU(z)/dz, as gray curve.

The 1omin corresponds to strong attraction with a short range, an intermediate energy barrier of

repulsion, and the 2omin weak attraction with long range. To highlight the underlying physics, a

modified form of the Dugdale-Barenblatt-Maugis (DBM) cohesive zone approximation is

[28, 38] adopted , where ps(z) assumes a multiple step function with linear transition shown as gray

curve in Figure 7.2(b). Disjoining pressure is taken to be a constant pi within each

o attractive/repulsive regions with a finite range zi with the subscript i = 1 and 2 for 1 min and

2omin respectively, and i = r for repulsion. The short range electron cloud overlap repulsion is

taken as a linear force with slope ms in the range z ≤ 0 to allow partial intersurface penetration.

Thus,

 − ms z for z ≤ p1 / ms  − < ≤ −  p1 for p1 / ms z z1 p1 / ms  m (z − z ) for z − p / m < z ≤ z + p / m  s 1 1 1 s 1 r s  pr for z1 + pr / ms < z ≤ z1 + zr − pr / ms (7-1) ps (z) =   − ms (z − z1 − zr ) for z1 + zr − pr / ms < z ≤ z1 + zr + p2 / ms  − p for z + z + p / m < z ≤ z + z + z − p / m  2 1 r 2 s 1 r 2 2 s ms (z − z1 − zr − z2 ) for z1 + zr + z2 − p2 / ms < z ≤ z1 + zr + z2   0 for z ≤ z1 + zr + z2

The work of adhesion, γ, is therefore the area enclosed by ps(z), or, γ ≈ p1 × z1 + p2 × z2 – pr × zr.

It is interesting to note here that in the classical adhesion model with a fixed and finite γ, the

JKR-limit requires zi → 0 and p → ∞, while the DMT-limit requires zi → ∞ and p → 0.

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When a shell approaches the substrate and falls in the range of adhesion (z ≤ z1 + zr + z2), intersurface force is sensed. For flat rigid substrate, z is always measured vertical to the substrate.

The total intersurface energy can be expressed as follows

U = − p ⋅d ds (7-2) S ∫ s with d being the displacement of shell.

7.2.2 Mechanical energy

Deformation of the shell is not only determined by intersurface energy, but also the

external load F, bending and stretching stiffnesses of the shell materials, κb and κs, respectively, and the internal pressure pint. For a linear elastic material, the strain energy stored within a deformed shell is given as follows,

U = ( ½κ H 2 + ½κ ε 2 ) ds (7-3) E ∫ b s where the first term denotes bending and the second term stretching, H(s) = d2v/ds2 is the local

change of curvature from the circular cross-section, and the nonlinear strain ε along the arc

length s is defined as follows,

2 du 1  dv  ε = +   (7-4) ds 2  ds  with u and v the tangential and transverse deflections respectively. The external load acting on

the cylinder apex F traverses a vertical displacement of d. The associated potential energy is

therefore given by,

UF = – F · d (7-5)

Constant volume of the shell in the presence of external load is maintained by introducing an

internal pressure,

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∆V p = K (7-6) int V b

where Kb is the bulk modulus of the encapsulating liquid and ∆V/V is the fractional change in

volume. In the computation routines, Kb is assigned a large number to ensure ∆V/V < 1%.

Internal energy due to hydrostatic pressure is found by integration over the shell surface,

= − U int ∫ pint v ds (7-7)

Total energy of the system, UT , is therefore the sum of all the four aforementioned energy terms,

UT = US + UE + UF + Uint (7-8)

Equation of equilibrium is derived by the principle of minimum total potential energy

(δUT = 0).

7.2.3 Numerical Method

Since the solution is highly nonlinear due to the unknown instantaneous contact width,

transient dynamic analysis is employed to deduce the steady-state equilibrium. Two more terms

are introduced into the governing equation to account for mass and damping. A finite element

code has been developed that solves for equilibrium by using the explicit Newmark’s algorithm.

Stiffness matrix K

Being subjected to mixed plate bending and membrane stretching, a shell is unevenly

discretized into a number of two-node elements in the non-contact region and condensed nodes

within the contact region as shown in Figure 7.3. Each node has three degrees of freedom such

(e) that the displacement vector for each element d = {ui, vi, φi, uj, vj, φj}, where u and v are

respectively the in-plane and transverse deflection components, φ is the rotation of the normal to

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the mid-plane of the shell, i and j are the first and second nodes in each element, and the

superscript e denotes vector for single element.

Figure 7.3. Finite element implementation of shell model.

The stiffness matrix K is derived from an energy consideration. For single element, the

strain energy of a linear elastic beam in its local coordinates system (x, y), with element length,

L, cross sectional area, A, second moment of area, I = bt3/12, can be written as

2 L L (e) EI  dϕ  EA U =   dx + ε 2dx (7-9) E 2 ∫ 0  dx  2 ∫ 0

(e) (e) (e) Taking u = Nu d and v = Nv d as polynomial expansions, k is derived by the principle of

virtual work,

(e) (e) (e) ∂U ∂U δU = E δu + E δv = 0 (7-10) E ∂u ∂v

The mathematical manipulation is straight forward but tedious, and will not be explicitly

given here. The resulting element stiffness matrix k (e) is asymmetric due to the nonlinear mixed

(e) (e) deformation mode, i.e. k 12 ≠ k 21 . Using a transformation matrix T based on the direction

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(e) cosines of individual elements with respect to the global coordinate axes, k is transformed

into the global k in new global coordinate system (X,Y) using a mathematical routine reported

in our previous work [159].

Transient dynamic analysis

Shell deformation is governed by the general equation of motion of transient dynamic analysis,   M d +C d + K d = Ftot (7-11)

with M being the global mass matrix, C the Rayleigh global damping matrix, Ftot the total external force, and d and d the acceleration and velocity vectors, respectively.

Diagonal ‘lumped’ mass matrix is adopted here for the explicit dynamic method,

mAL 1 1 1 1  M (e) = λL2 λL2 (7-12) 2 2 2 2 2 

where m is the mass density of an element and λ = 1/24 is derived from moment of inertia. Since

only the final equilibrium state, rather than time dependent response, is relevant here, m = 1010 is

chosen for mass scaling purpose. C (e) is chosen as a linear combination of mass matrix and

stiffness matrix such that C (e) = ηM (e) + ξ K (e) , where η = 103, ξ = 0 are two arbitrary numerical

constants chosen to stabilize the system.

Explicit Newmark’s method

The governing equation is highly nonlinear due to the unknown instantaneous contact

width. Equilibrium is found by the explicit Newmark’s method, with time increments of ∆t. The

velocity and displacement at time step n are given as follows,

    d n = d n−1 + ∆t [ ς d n + (1− ς) d n−1 ] (7-13)

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1 d = d + ∆t d + ∆t 2 [ 2β d + (1− 2β) d ] (7-14) n n-1 n−1 2 n n−1

The two numerical factors ς = – 1/2 and β = 1/4 are chosen for average acceleration such that variables on the right hand side of (7-13) and (7-14) are all known besides the acceleration

  d n . Equations (7-13) and (7-14) are substituted into (7-11) to solve for the only unknown d n .

 An iterative method is implemented to find d , d and the deformed profile ρ at each iteration step. Figure 7.4 shows the algorithmic flow chart.

At time frame t = n∆t d d d Knowns: n−1 , n−1 , n − 1 d d d n−2 , n−2 , n − 2 …

Compute shell geometry in (X, Y) coordinate system

Compute K , C , M matrixes

Compute separation z based on current profile

Compute total external surface forces, Ftot t = t +∆t

 Calculate acceleration d n by Newmark’s method

 Update velocity and displacement dn , dn .

Update profile ρ = ρ0 + dn

Figure 7.4. Flow chart for the solution algorithm.

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7.3 Results and discussion

To model actual bacterial adhesion, realistic ranges of numerical parameters based on the literature are chosen for the following analysis. For a typical rod-shape bacteria, t = 10 nm [141],

R = 5 µm, E = 10 MPa [142], ν = 0.50 for most polymeric and biomaterials, and γ ~ 2 × 10-4 mJ/m2

that is consistent with the measured range of 10-4 to 1 mJ/m2 reported in the literature [62, 65, 160].

A typical surface potential is chosen for solution with ionic strength of roughly 30 mM. The

typical parameters are taken to be p1 = 2.0 Pa, pr = 0.45 Pa, p2 = 0.15 Pa, z1 = 100 nm, zr = 10

10 -1 nm, z2 = 200 nm, and ms = 10 Pa.m .

7.3.1 Equilibrium configuration

Figure 7.5. Deformed profile and contact pressure of a shell resting in 2omin: (a) Equilibrium

profile and (b) corresponding pressure distribution.

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When the shell approaches the substrate with insufficient kinetic energy, it is unable to

overcome the large energy barrier and is influenced only by the 2omin. Figure 7.5(a) shows the deformed shell profile at the 2omin when it is not subjected to any external force or F = 0. The

contact edge (r = a) is defined at z = z1 + zr , and the cohesive edge (r = c) at z = z1 + zr+ z2.

Profile within the contact (r ≤ a) is fairly planar and at a distance from the substrate surface.

Figure 7.5(b) shows the traction on the shell normal to the substrate. Maximum compression pr is

present within the contact. Beyond the contact edge, attraction of the 2omin leads to a uniform

attraction that extends to the cohesive zone edge. The shell is traction free at r > c.

Figure 7.6. Deformed profile and contact pressure of a shell resting in 1omin: (a) Equilibrium

profile and (b) corresponding pressure distribution.

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Should the shell acquire sufficient kinetic energy to overcome the energy barrier, the

mechanical equilibrium is influenced by the full surface potential stated in Equation (7-1). Figure

7.6(a) shows the deformed profile with F = 0. The coupled 1omin and 2omin creates larger contact and cohesive zone widths than the 2omin alone. Figure 7.6(b) shows the normal traction.

The compressive stress is minimal within the contact but reaches a maximum at the contact edge,

contrasting the parabolic compression profile in contacting solid spheres in the classical Hertz

contact theory [104]. The cohesive zone comprises the 1omin and 2omin attractions as well as the

intermediate repulsive barrier.

7.3.2 Mechanical response

Figure 7.7. Contact width as a function of applied load for fixed adhesion energy. The dashed lines show 2omin to 1omin transition for different energy barriers.

127

In the presence of an external force, F, the adhesion-detachment trajectory, a(F), is obtained as shown in Figure 7.7. At F = 0, the shell is influenced only by the 2omin at B, with a

= 0.458 µm. External tension reduces the contact area until “pull-off” occurs at A when the shell completely detaches from the substrate with F* = – 0.4 µN/m and a* = 0 indicating a line contact.

On the other hand, compression causes the contact to grow along BC. Loading-unloading along

ABC is reversible.

Figure 7.8. Snapshots of deformed profile of A, B, C and D in Figure 7.7.

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Figure 7.9. Contact pressure of A, B, C and D in Figure 7.7.

At a sufficiently large F, the shell overcomes the energy barrier, and a sudden transition

to the 1omin occurs at C reaching D. The adhesion-detachment trajectory hereafter shifts to

DHK. As F is reduced along DH, the contact width shrinks. At H, the external load is removed

(F = 0) and aH > aB as expected. Loading-unloading along BCDH thus leads to hysteresis.

Tensile force of F* = –1.6 µN/m leads to “pull-off” at K and the shell spontaneously snaps from the substrate with a non-zero “pull-off” contact area a* =1.13µm.

129

Figure 7.10. Snapshots of deformed profile of C, D, H and K in Figure 7.7.

Among the many adjustable parameters in our model, the repulsive disjoining pressure,

o o pr, plays a critical role in determining whether the shell stays at 2 min or transits to 1 min.

Figure 7.7 shows such transitions for a range of pr. A large barrier requires a large applied

compression to make the transition, but a sufficiently small barrier leads to a spontaneous

transition as in case of pr = 0.43 Pa as shown. It is noted that a changing pr does not lead to measurable changes in a(F), since the spatial width of repulsion within the cohesive zone is so small that it does not affect the total work of adhesion.

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Figure 7.8 - Figure 7.9 show snapshots of the deformed shell profiles and the corresponding normal tractions along the adhesion-detachment trajectories ABCD (c.f. Figure

7.7). When “pull-off” occurs at A, the top pole is raised significantly such that the shell is largely distorted and the contact reduces to a line (a = 0). This is consistent with our previous work for both cylindrical solid and cylindrical shell subjected to an ideal zero-range or long- range surface attraction [62, 160].

Figure 7.11. Contact pressure of C, D, H and K in Figure 7.7.

Along AB, exclusive compression is felt within the contact rendering attraction outside the contact. Along BC, the applied load causes the shell to expand laterally and shrink vertically.

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Central buckling occurs in the contact region when repulsion dominates close to the edge (r = a) and central 2omin attraction maintains the planar contact. The contact repulsion at B now splits into two spikes at C, which are pushed towards the opposite contact edges. Profile D serves a reference after the shell transits to the 1omin and the contact suddenly expands. The disjoining pressure outside the contact edge manifests the combined 1omin and 2omin as well as repulsion due to the intermediate energy barrier. Figure 7.10 - Figure 7.11show the effects of the full surface potential along CDHK. Because of the strong 1omin attraction, the profile is severely distorted especially at “pull-off”. Note that the two compression spikes at the contact edges never merge even at “pull-off”, indicating the influence of the repulsive barrier.

Figure 7.12. Eccentricity, or degree of deformation, of the adhering shell as a function of external load.

Figure 7.12 shows the eccentricity e (ratio of shell height to equator width) as a function of applied load for the two adhesion-detachment trajectories. Note that e = 1 does not necessarily

132

refer to a perfectly circular cross-section, but also to a distorted geometry having the same height and width. In the initial loading in either tension or compression close to F = 0, the gradients de/dF and da/dF (Figure 7.7) reach their highest values, indicating a sharp increase in the contact area and the associated sudden and significant distortion. Increasing compression reduces both de/dF and da/dF. On the other hand, increasing tension leads to a plateau in a(F) with da/dF ≈ 0

but an increasing e(F), indicating that the external load is by and large transferred to the shell

distortion. The largest geometrical distortion occurs at pull-off.

Figure 7.13. Schematic of a rigid shell and deformable shell within effect of intersurface force.

The case of a rigid cylinder with same dimension in presence of DLVO potential is also

investigated for comparison purpose. As expected, compliant cylindrical shell deforms within

effect of intersurface force and forms finite contact radius, while rigid cylinder does not deform

and remains line contact with substrate, shown in Figure 7.13. All the potential parameters are

chosen same as before. For rigid body, equilibrium is achieved when external force balanced

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with intersurface force. External force, F, as function of displacement, d, can be formulated as follows:

 p1L1 (d) for 0 ≤ d ≤ z1   p L (d) + p L (d) for z < d ≤ z + z F(d) =  1 1 r r 1 1 r (7-15)  p1L1 (d) + pr Lr (d) + p2 L2 (d) for z1 + zr < d ≤ z1 + zr + z2   0 for d > z1 + zr + z2

where d = 0 indicates cylinder is in contact with rigid substrate, L1, Lr, L2 are the arc length in

presence of 1omin attraction, barrier, 2omin attraction, respectively, which are function of

displacement d (see Figure 7.14). p1, pr, p2 are disjoinging pressures as mentioned before.

Figure 7.14. External force as function of displacement for a rigid cylinder in presence of DLVO

potential.

Figure 7.14 shows “pull-off” tensile forces needed to detach cylinder from 2omin and

1omin are 0.21µN/m and 2.08µN/m, comparing with 0.4 µN/m and 1.6 µN/m for deformable

o cylinder. Interestingly, with pr = 0.45 Pa, deformable shell finds equilibrium at 2 min (c.f. Figure

7.7), unlikely, for rigid cylinder, F is always greater than zero (grey curve in Figure 7.14),

134

o indicating cylinder fall into 1 min with no external force required. When repulsion barrier pr is

increased to 0.85 Pa, the increase of barrier repulsion prevents cylinder from falling into 1omin, a

compression force is needed to overcome the barrier. Therefore, deformation of shell structure

would largely affect its mechanical adhesion-detachment behavior.

7.4 Discussion

Adhesion of a cylindrical shell in the presence of a single long-range surface force (i.e. in

the absence of a 2omin) has been recently reported [160]. The present model consistently shows

that a shell influenced by the 2omin alone possesses virtually the same adhesion-detachment behavior and the contact area is expected to shrink to a line contact along the shell axis prior to detachment, similar to the DMT behavior. The introduction of a 1omin to be coupled with the

2omin leads to a remarkable abrupt “pull-off” with a non-zero contact area, similar to the JKR behavior. The repulsive barrier here acts as a virtual “latch”. Unless the shell acquires sufficient

energy to overcome the barrier, the latch is shut and the adhesion-detachment behavior remains

status quo as in 2omin only. But if the barrier is sufficiently lowered by raising the ionic concentration, the latch is thrown open, and the shell is locked and trapped inside the 1omin

energy well. As the external tension gradually increases (c.f. Figure 7.7), the shell climbs up the

energy well reaching the peak of the barrier that represents the latch threshold. An incremental

increase in F thus pushes the shell off the energy cliff. Surpassing the shallow 2omin, the shell

spontaneously detaches from the substrate. Depending on the barrier height and the ultimate

strength of the shell materials, the sudden shrinkage of the non-zero contact area might lead to

severe damage of the shell. Compared with the classical CFT where an individual colloidal like

cell is has a single probability of being adhered to a collector, the new model renders a physical

meaning to the attachment-detachment mechanism that is based on the actual cell-surface

135

interactions and its coupling with mechanical deformation of the cells. The distinct “pull-off”

mechanics in a surface potential with double minima compared to one with single well further

demonstrates the necessity of assigning two probabilities of attachment for individual cells. It

also implies that environmental variables such as ion strength or temperature that can potentially

alter the interactive forces and mechanical properties of cells can consequently change the

adhesion-aggregation-transportation of microorganisms in porous medium.

Microbial transport in porous medium is governed by many factors including convection-

diffusion, growth, death/decay, aggregation, as well as adhesion to the collector. When aquatic

environment have relatively higher ionic strengths, the cell attachment rate depends

predominantly on ionic strength of the liquid medium, surface charges on the cells, and

temperature etc [152]. Tufenkji et al. [151] postulated the presence of a 2omin in the surface

potential that causes drastic change in colloidal behavior and transport as manifested by the dual

“unfavorable” and “favorable” deposition rates. Particle deposition is termed “favorable” in the

absence of repulsive interaction energies, whereas “unfavorable” deposition refers to the case

where repulsive colloidal interactions predominate. When the energy barrier height, pr, is large, the cells are influenced only by the weak 2omin energy well. At sufficiently high flow rate or

high temperature, thermal fluctuation alone is capable to shrug off the weak attraction and the

shell leaves the substrate. The temporary retention of the cell thus leads to an unfavorable

deposition. On the other hand, a small pr due to increased ionic strength in the electrolyte

medium enables the cells to fall easily into the strong 1omin attraction, leading to a favorable

deposition. Now, if, for instance, the liquid flow increases and produces a stronger shear on the

trapped cell, detachment from the 1omin can possibly lead to “pull-off”, but the non-zero “pull-

off” contact area is possible to cause damage to the cell such as bursting of the glycoprotein

136

[151, 152] shell. In calculating the single-collector contact efficiency (η0), Elimelech suggested five

most dominant mechanisms of bacteria approaching the collector surface, namely, interception,

gravitational sedimentation, Brownian diffusion, hydrodynamic (viscous) interactions, and van

der Waals forces, and suggested a phenomenological form of single–collector removal efficiency

based on experimental measurements and curve-fitting routines,

1/ 3 −0.081 −0.715 0.052 1.675 0.125 −0.24 1.11 0.053 η0 = 2.4As N R N Pe NvdW + 0.55As N R N A + 0.22N R NG NvdW (7-16)

with As is the porosity number, NR = dp/dc the aspect ratio of the colloidal particle and collector

dimensions, NPe the Peclet number of diffusion-convection, NG the gravity number related to

buoyancy of particle in liquid, NA the combined van der Waals attraction and fluid flow rate, and

NvdW the ratio of van der Waals attraction to thermal energy. Additional factors include fluid

viscosity, geometrical packing of collector and the resulting inter-granular percolation channels and their dimensions etc. The present work demonstrates yet another critical aspect of coupled

elastic shell deformation and intersurface forces. It is obvious that the pull-off force F* plays a

significant role in colloid filtration and is closely related to α. Some of the aforementioned

factors that are presumed to be mutually exclusive in classical CFT turn out to have significant

* influence on one another. For instances, F depends on both the surface potential ps(z) and size R,

mutual interactions between deformed cell and liquid flow, diffusion-convection influenced by

intersurface forces etc. should be incorporated into the grand theory of CFT.

7.5 Summary

A new adhesion model is constructed for a cylindrical shell exposed to the full DLVO

surface potential. Interrelations are established between the measurable quantities of applied

compression/tension, approach displacement, contact area, deformed shell profile, critical

137 parameters at “pull-off”, and transition from first to second minima etc. The detailed surface potential profile can thus be extracted from direct experimental measurements of the above quantities and trends. Such physical insights shed lights on the conventional stochastic description of cell attachment-detachment in porous medium, and bear significant consequences in building a comprehensive model for microbial transport in porous media.

138

8 CONCLUSION AND FUTURE WORK

8.1 Conclusions

This thesis proposes new adhesion-delamination mechanics of thin spherical and cylindrical shell structures with ranges of (i) shell radius, (ii) wall thickness and (iii) material parameters.

Based on the Reissener’s nonlinear shell theory with no adhesion, the shell adhesion model incorporates JKR energy balance, and derives the relations between four measurable quantities: applied load, approach distance, contact radius, and deformed profile. Features unique to shells are revealed and shown to be distinctly different from the solid-solid adhesion. In particular, shell thickness plays a crucial role in defining the adhesion mechanics and the critical

“pull-off” forces at which the shells spontaneously detaches from the substrate, which is

1/ 3 γ 4 R 4 (1− v 2 ) * = − ± generalized as F ( 13.2 0.6 )  2  . The model is useful for experimental  Et  characterization of the coupled elastic deformation and adhesion behavior of convex shells applicable to hydrogel contact lens and biological cells. Better understanding of the mechanical performance of contact lenses and its relation to bio-chemo-optical properties are necessary to improve the lens quantities, especially in terms of users’ degree of comfort. The mechanical indentation model for sperm injection in mouse oocytes leads to better understanding of embryology, developmental biology, cellular structure, aging and pathophysiology.

To understand cell / bacterium aggregation and transportation behavior through a porous medium, an adhesion model of a thin-walled cylindrical shell adhering to a rigid substrate is investigated in two long- range surface force potentials: (i) the classic Lennard-Jones potential and (ii) the DLVO potential with a secondary minimum. The new model predicts behaviors of

139

shells being distinctly different from the classical solid adhesion theory, in terms of pressure distribution within contact area, deformed profile, and adhesion-delamination mechanics. The discovered zero “pull-off” width for cylindrical shells, unlike non-zero “pull-off” for solid cylinders, is the results of the compatible geometry within contact, leaving elastic energy mainly stored at contact edge. The extended model with DLVO potential consistently shows that if shell is influenced by the 2omin alone, same adhesion-detachment behavior is expected as LJ potential

and the contact area shrinks to a line contact prior to detachment. The introduction of a 1omin to be coupled with the 2omin leads to a remarkable abrupt “pull-off” with a non-zero contact area.

The sudden shrinkage of the non-zero contact area might lead to severe damage of the cell membrane. Compared with the classical CFT, the new model renders a physical meaning to the attachment-detachment mechanism that is based on the actual cell-surface interactions and its coupling with mechanical deformation of the cells. Such physical insights shed lights on the conventional stochastic description of cell/bacteria attachment-detachment in porous medium,

and bear significant consequences in building a comprehensive model for microbial transport in

porous media.

8.2 Future work

Further investigations are needed in several aspects. In terms of surface force potential

law, the spherical shell adhesion model only considers short range adhesive intersurface force

(JKR), but not long range intersurface forces. Similar Maugis’ cohesive zone model in Chapter 6

can introduce long range intersurface force with finite range and magnitude. Transition from

long range to short range intersurface force should be investigated. In solid adhesion, the

transition is governed by Tabor’s parameter λsolid, discussed in Chapter 2. Planar membrane

[50] adhesion has a different form of λfilm, which is studied by Li . It is possible to derive a new

140

Tabor’s parameter for shell λshell, which is function of curvature, R, and thickness, t. It is

expected that λshell approaches λsolid when thickness t approaches infinite, and λshell approaches

λfilm when R approaches infinite.

Further extension of current shell model with varying thickness along the meridional direction has the potential application in charactering contact lens with different diopters. It is believed that thickness profiles of shell plays a crucial role in determining the mechanical behavior, and further adhesion-detachment behavior. We are currently working on the related work, and expect new breakthroughs in the near future.

In addition, the universal behavior of spherical shell, (e.g., Equation (3-33)) is proved to be valid in certain range so far, whether the behavior can be scale down to nano-scale, or extended to macro-scale remains unknown. As far as geometry’s concern, this thesis only considers elastic shell adhering to a rigid flat substrate. The model extended to curved substrate has potential applications for the ongoing contact lens and lens container adhesion measurement and lens-lens adhesion measurement. Finally, further comparison with experiments should be pursued in future work, particularly for cylindrical adhesion model.

141

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APPENDIX 1 Matlab program for spherical shell adhesion

% ‘main_parallelplate.m’ % Clear work space, command window and breakpoints before start clc;clear; % dbclear all %------% Making a choice from 'Uniform thickness' or 'Varied thickness' %------% Construct a questdlg with two options choice = questdlg('What would you like to do?', ... 'Thickness Options', ... 'Uniform thickness','Varied thickness','Varied thickness'); % Handle response switch choice case 'Uniform thickness' disp([choice ' You choose "Uniform thicknes".']) flag = 1; case 'Varied thickness' disp([choice ' You choose "Varied thicknes".']) flag = 2; end %------% Using dialogbox, ask for "Input parameters" %------if flag==1 % In the case of uniform thickness prompt = {'Enter Radius of curvature R in [m]:', ... 'Enter Radian in [degree]:',... 'Enter Elastic modulus E in [Pa]:', ... 'Enter Lens thickness t in [m]:', }; dlg_title = 'Input for Contact lens parameters'; num_lines = 1; % Default input values def = {'8.66e-3','42.02','1.275e6','71.2e-6'}; options.Resize = 'on'; % Optional sets options.WindowStyle = 'normal'; options.Interpreter = 'tex';

% Creat a dialogbox and returns user input for multiple prompts in the cell array. promp is a cell array containing prompt strings answer = inputdlg(prompt,dlg_title,num_lines,def,options); %------% Assign the answer to R,Total_angle,E,h, stop if inputs are wrong %------R = str2double(answer{1}); % Use curly bracket for subscript Total_angle_degree = str2double(answer{2}); Total_angle = Total_angle_degree/180*pi; E = str2double(answer{3}); h0 = str2double(answer{4}); elseif flag == 2 prompt = {'Enter Radius of curvature in [m]:', ... 'Enter Radian in [degree]:',... 'Enter Elastic modulus in [Pa]:' }; dlg_title = 'Input for Contact lens parameters'; num_lines = 1; 157

% Default input values def = {'8.5e-3','70.25','0.68e6'}; options.Resize = 'on'; % Optional sets options.WindowStyle = 'normal'; options.Interpreter = 'tex'; % A warning dialog box as a reminder warndlg(sprintf('Make sure you have thickness data \n saved in current directory with file name \n "LensThickness.txt"')) pause(2); % Creat a dialogbox and returns user input for multiple prompts in the cell % array. promp is a cell array containing prompt strings

answer = inputdlg(prompt,dlg_title,num_lines,def,options); %------% Assign the answer to R,Total_angle,E,h, stop if inputs are wrong %------R = str2double(answer{1}); % Use curly bracket for subscript Total_angle_degree = str2double(answer{2}); Total_angle = Total_angle_degree/180*pi; E = str2double(answer{3}); end %------% Numerical input setting, including discritized element number, % Maximum iteration, relax coefficient, range of phi_star, etc %------prompt = {'Enter Number of elements :', ... 'Enter Maximum iterations :',... 'Enter Relax coefficient:', ... 'Enter Phi star start in [degree]:',... 'Enter Phi star end in [degree]:',... 'Enter Phi star increment in [degree]:' }; dlg_title = 'Numerical input settings'; num_lines = 1; % Default input values def = {'100','200','0.3','1','20','1'}; options.Resize = 'on'; % Optional sets options.WindowStyle = 'normal'; options.Interpreter = 'tex';

% Creat a dialogbox and returns user input for multiple prompts in the cell % array. promp is a cell array containing prompt strings answer = inputdlg(prompt,dlg_title,num_lines,def,options); %------% Assign the answer to R,Total_angle,E,h, stop if inputs are wrong %------M = str2double(answer{1}); % Number of segments maxiter = str2double(answer{2}); Relax_coef = str2double(answer{3}); Phi_star_start = str2double(answer{4}); Phi_star_end = str2double(answer{5}); Phi_star_increment = str2double(answer{6}); %------% Processing thickness

158

%------if flag == 1 % flag =1 refers to uniform thickness h = h0*ones(M,1); end if flag == 2 data = load('LensThickness.txt'); Theta_load = data(:,1); h_load = data(:,2); p = polyfit(Theta_load,h_load,8); % figure(10) % plot(Theta_load,h_load,'bo-') % hold on end

%------% Calculating starts here %------fid2= fopen('Profile.txt','wt'); counting =1; for mm = Phi_star_start:Phi_star_increment:Phi_star_end Contact_angle=mm/180*pi; Angle=Total_angle-Contact_angle; % Total Arc mu=0.5; % Possion's ratio Node=M+1; % M numbers with M+1 nodes d_Phi=Angle/M; % Length of each segment ph0_initial=Contact_angle; %------% Varied thickness %------if (flag == 2) polyfit_theta = linspace(mm,max(Theta_load),M); h = polyval(p,polyfit_theta); % figure(10) % plot(polyfit_theta,h,'ro-') end %------iter=1; y=zeros(6,Node); % trial solution at y=[w u beta V H M_phi] while(iter<= maxiter) S_global=zeros(Node*6,Node*6); R_global=zeros(Node*6,1); for i=1:M K(i)=E*h(i)/(1-mu^2); % K,D are elastic constants D(i)=E*h(i).^3/12/(1-mu^2);

for j=i:i+1 phi0 = ph0_initial+Angle/M*(j-1); % phi0 might be in denominator,... % set up a samll number to make ... % the code work r0 = R*sin(phi0); e_theta = y(2,j)/r0; phi = -y(3,j)+phi0; k_theta = (2*sin(phi0)*sin(y(3,j)/2)^2+cos(phi0)*sin(y(3,j)))/r0;

159

N_phi = y(5,j)*cos(phi)+y(4,j)*sin(phi); e_phi = N_phi/K(i)-mu*e_theta; k_phi = y(6,j)/D(i)-mu*k_theta; N_theta = K(i)*(e_theta+mu*e_phi); M_theta = D(i)*(k_theta+mu*k_phi); alpha = R*(1+e_phi); r = r0+ y(2,j); r1 = alpha*cos(phi); dYdx(1,1) = 0; dYdx(1,2) = R*sin(phi)*(-mu/r0); dYdx(1,3) = R*sin(phi)/K(i)*(y(5,j)*sin(phi)- y(4,j)*cos(phi))-alpha*cos(phi); dYdx(1,4) = R*sin(phi)^2/K(i); dYdx(1,5) = R*sin(phi)/K(i)*cos(phi); dYdx(1,6) = 0; dYdx(2,1) = 0; dYdx(2,2) = R*cos(phi)*(-mu/r0); dYdx(2,3) = R*cos(phi)/K(i)*(y(5,j)*sin(phi)- y(4,j)*cos(phi))+alpha*sin(phi); dYdx(2,4) = R*cos(phi)/K(i)*sin(phi); dYdx(2,5) = R*cos(phi)^2/K(i); dYdx(2,6) = 0; dYdx(3,1) = 0; dYdx(3,2) = 0; dYdx(3,3) = -mu*cos(phi)*R/r0; dYdx(3,4) = 0; dYdx(3,5) = 0; dYdx(3,6) = R/D(i); dYdx(4,1) = 0; dYdx(4,2) = y(4,j)/r*R*cos(phi)*mu/r0+y(4,j)*r1/r^2; dYdx(4,3) = -y(4,j)/r*R*cos(phi)/K(i)*(y(5,j)*sin(phi)- y(4,j)*cos(phi))-y(4,j)/r*alpha*sin(phi); dYdx(4,4) = -y(4,j)/r*R*cos(phi)/K(i)*sin(phi)-r1/r; dYdx(4,5) = -y(4,j)/r*R*cos(phi)/K(i)*cos(phi); dYdx(4,6) = 0; dYdx(5,1) = 0; dYdx(5,2) = y(5,j)/r*R*cos(phi)*mu/r0+y(4,j)*r1/r^2+N_theta/r*R*mu/ r0+alpha/r*K(i)*(1-mu^2)/r0-alpha*N_theta/r^2; dYdx(5,3) = -y(5,j)/r*(R*cos(phi)/K(i)*(y(5,j)*sin(phi)- y(4,j)*cos(phi))+alpha*sin(phi))+(N_theta/r*R/K(i)+alpha*mu/r) * (y(5,j)*sin(phi)-y(4,j)*cos(phi)); dYdx(5,4) = - y(5,j)/r*R*cos(phi)/K(i)*sin(phi)+N_theta/r*R/K(i)*sin(phi)+al pha/r*mu*sin(phi); dYdx(5,5) = -y(5,j)/r*R*cos(phi)/K(i)*cos(phi)- r1/r+N_theta/r*R/K(i)*cos(phi)+alpha/r*mu*cos(phi); dYdx(5,6) = 0; dYdx(6,1) = 0;

160

dYdx(6,2) =1/(r^2*r0)*(y(6,j)*r0*r1+(y(6,j)*mu*r*R- M_theta*mu*r*R-alpha*M_theta*r0- mu*r^2*R*y(4,j))*cos(phi)+y(5,j) *mu*r^2*R*sin(phi)); dYdx(6,3) =1/(K(i)*r*r0)*(-(alpha*D(i)*K(i)*(-1+ mu^2)+R*r0*y(4,j)*(-y(6,j) +M_theta+r*y(4,j)))*cos(phi)^2+r0*sin(phi)*(alpha*K(i)* (-y(6,j)+M_theta+r*y(4,j))-y(5,j)^2*r*R*sin(phi))+ y(5,j)*r0*cos(phi)*(alpha*K(i)*r-R*(y(6,j)-M_theta- 2*r*y(4,j))*sin(phi))); dYdx(6,4) =(- y(5,j)*r*R*sin(phi)^2+cos(phi)*(alpha*K(i)*r+R*(-y(6,j) +M_theta+r*y(4,j))*sin(phi)))/(K(i)*r);

dYdx(6,5) = (R*(-y(6,j)+M_theta+r*y(4,j))*cos(phi)^2- alpha*K(i)*r*sin(phi)-y(5,j)*r*R*cos(phi)*sin(phi))/(K(i)*r); dYdx(6,6) = -r1/r+alpha*cos(phi)*mu/r;

f(1)=R*(e_phi*sin(phi)-k_theta*r0); f(2)=R*(e_phi*cos(phi)+2*sin(y(3,j)/2)*sin(phi0-y(3,j)/2)); f(3)=R*k_phi; f(4)=-r1*y(4,j)/r; f(5)=-r1*y(5,j)/r+alpha*N_theta/r; f(6)=-r1*y(6,j)/r+alpha*cos(phi)*M_theta/r- alpha*(y(5,j)*sin(phi)-y(4,j)*cos(phi));

%------% Calculate residual %------

if j==i A1=-(eye(6)+dYdx.*(d_Phi/2)); f1=f;

elseif j==i+1 A2=eye(6)-dYdx.*(d_Phi/2); f2=f; end end Residual=-(y(:,j)-y(:,i)-d_Phi/2.*(f1+f2)'); %calculate residual %------% Get Y and z for each element %------for i1=1:6 k=(i-1)*6+i1; R_global(k,1)=R_global(k,1)+Residual(i1); for j1=1:6 l=(i-1)*6+j1; S_global(k,l)=S_global(k,l)+A1(i1,j1); S_global(k,l+6)=S_global(k,l+6)+A2(i1,j1); end end end

%------161

% Apply boundary condition %------S_global(Node*6-5,3)=1; % beta_1= contact angle S_global(Node*6-4,6)=1; % M_1 = Constant S_global(Node*6-3,5)=1; % H & u S_global(Node*6-3,2)=-(1+mu)*K(1)/R/ph0_initial;

S_global(Node*6-2,Node*6-5)=1; % w_N+1=0 S_global(Node*6-1,Node*6-4)=1; % u_N+1=0 S_global(Node*6 ,Node*6-0)=1; % M_N+1=0 R_global(Node*6-5)=-(y(3,1)-ph0_initial); R_global(Node*6-4)=-(y(6,1)-(1+mu)*D(1)/R); R_global(Node*6-3)=-(y(5,1)- (1+mu)*K(1)*y(2,1)/R/ph0_initial+(1+mu)*K(1)*ph0_initial^2/8);

R_global(Node*6-2)=-(y(1,Node)); R_global(Node*6-1)=-(y(2,Node)); R_global(Node*6) =-(y(6,Node));

err(iter)=max(max(abs(R_global))); tol=err(1)/1000; fprintf('iter : %4i, Res: %12.3e, \n',iter, err(iter)) if err(iter)>1e20 break elseif (err(iter)<=tol ) fprintf('Newton`s method converges after %3.0f iterations \n',iter) break end

dy=S_global\R_global; for i=1:Node for j=1:6 dy_new(j,i)=dy((i-1)*6+j); end end y=y+dy_new*Relax_coef; iter=iter+1; end

[Nodal_position_orig]=configuration(Node,R,M,Angle,ph0_initial); for i=1:Node Pos_x(i)= Nodal_position_orig(i*3-2); Pos_y(i)= Nodal_position_orig(i*3-1); end figure(1) plot(Pos_x,Pos_y,'b') hold on axis equal Pos_x_new=Pos_x+y(2,:); %y(2,:) is delta u Pos_y_new=Pos_y-y(1,:); %y(1,:) is delta w %Notice: minus sign is dut to convention difference. plot(Pos_x_new,Pos_y_new,'k')

%------162

%Contact region %------ele=50; %------% Varied thickness %------if (flag == 1) h_c = h(1)*ones(1,ele+1); K_c = K(1)*ones(1,ele+1); D_c = D(1)*ones(1,ele+1); elseif (flag == 2) theta_c = linspace(0,Contact_angle,ele+1); h_c = polyval(p,theta_c); K_c = E*h_c./(1-mu^2); % K,D are elastic constants D_c = E*h_c.^3./12./(1-mu^2); end %------Constant_A=(y(2,1)-(3-mu)*R*ph0_initial^3/16)/ph0_initial; angle_c =0:Contact_angle/ele: Contact_angle; u_c = Constant_A.*angle_c+(3-mu).*R.*angle_c.^3./16; M_c = (1+mu).*D_c./R; beta_c = angle_c; V_c = zeros(1,ele+1); H_c = (1+mu).*K_c.*Constant_A./R+(1-mu.^2).*K_c.*angle_c.^2./16; %H_c = (1+mu).*K.*u_c./R./angle_c-(1+mu).*K.*angle_c.^2/16;

Info(counting,1)= (-2*pi*R*y(4,1)*sin(Contact_angle)); % Load in [N] Info(counting,2) = (y(1,1)+R*(1-cos(Contact_angle))); % Depth in [m] Info(counting,3)= (Pos_x_new(1)); % Radius in [m]

%------%Caculate the Contact configuration %------Nodal_position_orig_c=zeros((ele+1)*3,1); Nodal_position_orig_c(1)=0; Nodal_position_orig_c(2)=R; Nodal_position_orig_c(3)=0; %Normal vector for i=1:ele Nodal_position_orig_c(i*3+1)=R*cos(pi/2-Contact_angle/ele*i); Nodal_position_orig_c(i*3+2)=R*sin(pi/2-Contact_angle/ele*i); Nodal_position_orig_c(i*3+3)=Contact_angle/ele*i; end for i=1:ele+1 Pos_x_c(i)= Nodal_position_orig_c(i*3-2); Pos_y_c(i)= Nodal_position_orig_c(i*3-1); end figure(1) plot(Pos_x_c,Pos_y_c,'r') Pos_x_new_c=Pos_x_c+u_c; %y(2,:) is delta u Pos_y_new_c=Pos_y_new(1)*ones(1,ele+1); %y(1,:) is delta w %Notice: minus sign is dut to convention difference. plot(Pos_x_new_c,Pos_y_new_c,'m') xlabel('X (m)') ylabel('Y (m)') fprintf(fid2,'#------\n');

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fprintf(fid2,'Contact_angle = %10.1e [degree] \n',mm); fprintf(fid2,'displacement = %10.4e [m] \n',Info(counting,2)); fprintf(fid2,'#------\n'); for i=1:ele fprintf(fid2,'%10.4e %10.4e \n',Pos_x_new_c(i),Pos_y_new_c(i)); end for i=1:Node fprintf(fid2,'%10.4e %10.4e \n',Pos_x_new(i),Pos_y_new(i)); end counting=counting+1; end

P=Info(:,1); Delta_c= Info(:,2); Contact_radius=Info(:,3); figure(2) plot(Delta_c,P,'b-o') hold on xlabel('Displacement (m)') ylabel('External force (N)')

%------% Write P, Delta_c, Contact_radius into output file %------fid1= fopen('Output.txt','wt'); fprintf(fid1,'Force, Displacement, Contact radius \n'); for i=1:length(P) fprintf(fid1,'%10.4e %10.4e %10.4e\n',P(i),Delta_c(i),Contact_radius(i)); end fclose(fid1);

% ‘configuration.m’ function [Nodal_position_orig]=configuration(Node,Radius,Numel,angle... ,ph0_initial) %------%Caculate the configuration %------Nodal_position_orig=zeros(Node*3,1); Nodal_position_orig(1)=Radius*cos(pi/2-ph0_initial); Nodal_position_orig(2)=Radius*sin(pi/2-ph0_initial); Nodal_position_orig(3)=0; %Normal vector for i=1:Numel Nodal_position_orig(i*3+1)=Radius*cos(pi/2-angle/Numel*i-ph0_initial); Nodal_position_orig(i*3+2)=Radius*sin(pi/2-angle/Numel*i-ph0_initial); Nodal_position_orig(i*3+3)=angle/Numel*i+ph0_initial; end

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APPENDIX 2 Matlab program for cylindrical shell adhesion %‘main.m’ %------%FEM Analysis of cylindrical shell- rigid substrate Contact Problem %------% Clean the workspace variables and command window in Matlab clear;clc; % Results are written into three files: % 'Profile.dat' records deformed profile % 'P_D_A.dat' records force - displacement - contact radius relationships % 'Internal_stress.dat' record the bending and stretching stress within shell fid1=fopen('Profile.dat','wt'); fid2=fopen('P_D_A.dat','wt'); fid3=fopen('Internal_stress.dat','wt');

% The outer loop runs cases with different surface stress and surface % cut-off radius % F_slope: soft repulsion slope ms % r : surface cut-off radius % stress : surface stress

F_slope_vec = [1e10 1e10 1e10 1e10]; r_vec = [1 0.1 0.04 0.014286]; Stress_vec = [1 10 25 70]; % Disjoining pressure for im = 1:1:1 F_slope = F_slope_vec(im); r = r_vec(im); Stress = Stress_vec(im); fprintf('%12.4e %12.4e %12.4e \n',F_slope,r,Stress); %------%Assign input parameters %------Numel =200; % Number of elements Node =Numel+1; % Number of nodes E =100e6; % Young's modulus Radius_orig =5e-6; % Radius of circle is R=5 Radius =Radius_orig/Radius_orig; % Normalization of R b_orig =1; % Width of cross section b =b_orig/Radius_orig; % Normalization of b h_orig =10e-9; % Thickness h =h_orig/Radius_orig; % Normalization of h p0_orig =5; % Internal pressure p0 =p0_orig*b; % Normalization of p0 Relax_coef_orig=0.01; % Relax coefficient of Newton Method Relax_coef = Relax_coef_orig; maxiter =5000; % Maxmum iteration of Newton Method % External force applied on the top of shell % Loop starts from "pulloff_start", end at "pulloff_end", with increment of % "pulloff_inc" pulloff_start = 0e5; pulloff_end = 0e5; pulloff_inc = 1e5;

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%------% Maugis cohesive zone % z0 : atomic equilibrium distance %------z0_orig =4e-10; z0 =z0_orig/Radius_orig; %------% Parameter of rigid wall, y= Wall_A*x+Wall_B % The shell is placed at coordinates center (0.0,0.0) %------Wall_A =0 ; Wall_B =-1; %------%Funtion input var_nor write inputs into inputdata.txt %------inputvar_nor(Numel,Node,b,h,Radius,E); Input_var=load('inputdata.txt'); Numel=Input_var(1,1); % Number of elements Node=Input_var(1,2); % Number of nodes %------%Plot rigid wall %------figure(1) Xp=0:0.1:Radius; Yp=Wall_A*Xp+Wall_B; plot(Xp,Yp,'k') hold on %------%Record input parameters into output files %------fprintf(fid1,'#------\n'); fprintf(fid1,'# Numel,Node,E,Radius_orig,b_orig\n'); fprintf(fid1,'#%12.4e %12.4e %12.4e %12.4e %12.4e \n',Numel,Node,E,Radius_orig,b_orig); fprintf(fid1,'# h_orig,p0_orig,Relax_coef,maxiter\n'); fprintf(fid1,'#%12.4e %12.4e %12.4e %12.4e \n',h_orig,p0_orig,Relax_coef,maxiter); fprintf(fid1,'# pulloff_start,pulloff_end\n'); fprintf(fid1,'# %12.4e %12.4e \n',pulloff_start,pulloff_end); fprintf(fid1,'# z0_orig,z0,r,Stress,F_slope\n'); fprintf(fid1,'#%12.4e %12.4e %12.4e %12.4e %12.4e \n',z0_orig,z0,r,Stress,F_slope); fprintf(fid1,'# Wall_A,Wall_B\n'); fprintf(fid1,'#%12.4e %12.4e \n',Wall_A,Wall_B); fprintf(fid1,'#------\n'); fprintf(fid2,'#------\n'); fprintf(fid2,'#F_pulloff_start = %16.4e, F_pulloff_end = %16.4e \n',pulloff_start,pulloff_end); fprintf(fid2,'#F_slope = %12.4e, r = %12.4e, p = %12.4e \n',F_slope,r,Stress); fprintf(fid2,'#------\n');

166 nn=1; %------% Loop with increase of external force %------for F_pulloff = pulloff_start : pulloff_inc : pulloff_end

fprintf('F_pulloff = %12.2e \n',F_pulloff); fprintf(fid1,'#------\n'); fprintf(fid1,'#F_pulloff = %16.4e \n',F_pulloff); fprintf(fid1,'#F_slope = %12.4e, r = %12.4e, p = %12.4e \n',F_slope,r,Stress); fprintf(fid1,'#------\n'); %------% Read input parameters from input_var %------for i=2:Numel+1; Nod(1,i-1)=Input_var(i,2); Nod(2,i-1)=Input_var(i,3); A(i-1)=Input_var(i,4); E(i-1)=Input_var(i,5); L(i-1)=Input_var(i,6); I(i-1)=Input_var(i,7); angle(i-1)=Input_var(i,8); RE(i-1,1)=Input_var(i,9); RE(i-1,2)=Input_var(i,10); RE(i-1,3)=Input_var(i,11); RE(i-1,4)=Input_var(i,12); RE(i-1,5)=Input_var(i,13); RE(i-1,6)=Input_var(i,14); end %------% Read boundary conditions %------Boundary_input(Node); Input_BCs=load('input_bc.txt'); Total_imposed_BCs=Input_BCs(1,1); %------% Initial guess Uc, Uc1 % Uc: Total displacement accumulated from 1st iteration % Uc1: Deformation in ith step %------Uc=zeros(Node*3,1); Uc1=zeros(Node*3,1); %------% Loops srarts %------iter=1; while(iter<=maxiter) %------% increase Relax_coef to accelerate converge speed %------if iter==250; Relax_coef=2*Relax_coef; end if iter==350; Relax_coef=2*Relax_coef; end 167 if iter==450; Relax_coef=2*Relax_coef; end %------% Initial internal pressure is zero %------PE=zeros(Numel,6); %------% Function 'configuration.m' caculate the inital configuration %------[Nodal_position_orig]=configuration(Node,Radius,Numel); %------% compute element properties $ Assemble the structure % Function 'Angle_modify' updated the angles each iteration. % Function 'Length_modify' updated the element length each iteration. % Function 'Assembly' assemble the global stiffness matrix % S_before :Global stiffness matrix % R : Right hand side external force vector (S_before * Uc == R ) %------angle=Angle_modify(angle,Uc1,Numel); L=Length_modify(Numel,Uc,Nodal_position_orig,Node); [S_before,R]=Assembly(Numel,Node,angle,E,I,A,L,Nod,RE,PE); %------% read boundary conditions %------for i=2:Total_imposed_BCs+1 BC_node(i-1)=Input_BCs(i,1); BC_dof(i-1)=Input_BCs(i,2); BC_magnitude(i-1)=Input_BCs(i,3); end %------% Apply BCs by using penalty method % Function 'BC_penalty.m' apply penalty method to S_before matrix %------[S,R]=BC_penalty(Total_imposed_BCs,BC_node,BC_dof,BC_magnitude,S_befor e,R); %------%Calculate adhesive force %------G=zeros(Node*3,1); % Clear space of Gap Fc=zeros(Node*3,1); % Clear space of Contact force dFc=zeros(Node*3,1); % Clear space of slope of Contact force for i=1:Node x0=Nodal_position_orig(i*3-2)+Uc(i*3-2); % X Coordinates of nodes on the circular membrane y0=Nodal_position_orig(i*3-1)+Uc(i*3-1); % Y Coordinates of nodes on the circular membrane Theta(i)=Nodal_position_orig(i*3)+Uc(i*3);

if i>(Node+1)/2 % If it is the upper half the circle F_vector(i)=0; % there is no need to calculate F dF_vector(i)=0; else m(i)=tan(Theta(i)); x(i)=(y0-Wall_B-m(i)*x0)/(Wall_A-m(i)); % X Coordinates of nodes on the substrate y(i)=Wall_A*x(i)+Wall_B; 168

% Y Coordinates of nodes on the substrate G(i*3-2)=x(i)-x0; % Gap between the two nodes G(i*3-1)=y(i)-y0; if y0>y(i) % y0y(i) means penetration end F_magnitude=Stress*b*L(i); % Turn stress into force Peak_point=F_magnitude/F_slope; if z(i)<= Peak_point+z0 F_vector(i)=F_slope*(z(i)-z0); dF_vector(i)=F_slope; elseif z(i)>Peak_point+z0 && z(i)< r F_vector(i)=F_magnitude; dF_vector(i)=0; elseif z(i)>=r && z(i)<= r+Peak_point F_vector(i)=-F_slope*(z(i)-(r+Peak_point)); dF_vector(i)=-F_slope; else F_vector(i)=0; dF_vector(i)=0; end end Fc(i*3-2)=F_vector(i)*cos(Theta(i)); Fc(i*3-1)=F_vector(i)*sin(Theta(i)); dFc(i*3-2)=dF_vector(i)*cos(Theta(i)); dFc(i*3-1)=dF_vector(i)*sin(Theta(i)); %------% caculate internal pressure, noting that theta is change every % iteration %------R(i*3-2)=p0*L(i)*cos(Theta(i)); R(i*3-1)=p0*L(i)*sin(Theta(i)); end R(Node*3-1)=R(Node*3-1)+F_pulloff; % Add up the external force applied on the top of the shell X=S*Uc-R-Fc; % Residual X of Newton's method err(iter)=max(abs(X)); % Error of Newton's method tol=err(1)/5000; % Making the tolerant 5000 times smaller fprintf('iter : %4i, Res: %12.3e, \n',iter, err(iter)) if err(iter)>1e20 break end if (err(iter)<=tol ) fprintf('Newton`s method converges after %3.0f iterations \n',iter) break; else %------% Calculate Jacobian matrix. The differece between Jacobian % matrix and stiffness matrix is that Jacobian has dFc terms %------[J]=Jacobian(Numel,Node,A,E,L,I,angle,Nod,dFc,Total_imposed_BCs,B C_node,BC_dof,BC_magnitude,R); dU=-inv(J)*X; 169

Uc1=Relax_coef*dU; Uc=Uc+Relax_coef*dU; end iter=iter+1; Nodal_position_new=Nodal_position_orig+Uc; end if iter==maxiter+1 disp('Newton`s method does not converge') continue elseif err(iter)>1e20 continue end for i=1:Node x_pos(i)=Nodal_position_new(i*3-2); y_pos(i)=Nodal_position_new(i*3-1); end figure(1) plot(x_pos,y_pos,'k-'); hold on %------% Calculate displacement %------Displacement(nn)=Wall_B-1; %------% Calculate contact radius %------for i=Node-1:-1:1 if Nodal_position_new(i*3-1)< (Wall_B+z0) && Nodal_position_new((i+1)*3-1)> (Wall_B+z0) Contact_node=i; u1_coord=Nodal_position_new(Contact_node*3-2); v1=Uc(Contact_node*3-1); theta1=Uc(Contact_node*3); u2_coord=Nodal_position_new((Contact_node+1)*3-2); v2=Uc((Contact_node+1)*3-1); theta2=Uc((Contact_node+1)*3); f = @(x)(Nodal_position_new(Contact_node*3- 1)+(Nodal_position_new((Contact_node+1)*3-1)- Nodal_position_new(Contact_node*3-1))/L(i)*x)-(Wall_B+z0); zero_y_coord = fzero(f,0); Contact_Radius(nn)= (L(i)- zero_y_coord)/L(i)*u1_coord+zero_y_coord/L(i)*u2_coord; break else continue end end if i==Node Contact_Radius(nn)=0; end F_pulloff1(nn)=F_pulloff; R_max(nn)=max(x_pos); Height_y(nn)=max(y_pos); HR_ratio(nn)=(Height_y(nn)+1)/2/R_max(nn); fprintf(fid2,'%16.6e %16.6e %16.6e \n',- F_pulloff1(nn),Contact_Radius(nn),HR_ratio(nn)); 170

nn=nn+1;

for kk=1:length(x_pos) fprintf(fid1,'%18.9e %18.9e %18.9e \n',x_pos(kk),y_pos(kk),F_vector(kk)); end end end % End of the outside parameter loop

%------%Plot internal stress %------for i=1:Numel Phi=angle(i); u1(i)=Uc(i*3-2)*cos(Phi)+Uc(i*3-1)*sin(Phi); u2(i)=Uc((i+1)*3-2)*cos(Phi)+Uc((i+1)*3-1)*sin(Phi); Sigma_streching(i)=(u2(i)-u1(i))/L(i)*E(1); v1(i)=-Uc(i*3-2)*sin(Phi)+Uc(i*3-1)*cos(Phi); v2(i)=-Uc((i+1)*3-2)*sin(Phi)+Uc((i+1)*3-1)*cos(Phi); Mz(i)=E(1)*I(1)*(6/L(i)^2*v1(i)+2/L(i)*Uc(i*3)- 6/L(i)^2*v2(i)+4/L(i)*Uc((i+1)*3)); %x=L % all these three expression (x=0,L/2,L)are consistent Sigma_bending(i)=-Mz(i)/I(1)*(h/2); Sigma_shear(i)=E(1)*I(1)*(12/L(i)^3*v1(i)+6/L(i)^2*Uc(i*3)- 12/L(i)^3*v2(i)+6/L(i)^2*Uc((i+1)*3)); Sigma(i)=Sigma_streching(i)+Sigma_bending(i); end ss=0:pi/Numel*Radius:pi/Numel*(Numel-1)*Radius; figure(3) plot(ss,Sigma_streching,'k-') hold on figure(4) plot(ss, Sigma_bending,'k-') hold on for kk=1:length(ss) fprintf(fid3,'%18.9e %18.9e %18.9e \n',ss(kk),Sigma_streching(kk),Sigma_bending(kk) ); end fclose(fid1); fclose(fid2); fclose(fid3);

%‘Angle_modify.m’ %------% Function 'Angle_modify.m' updated angle at each iteration %------function angle=Angle_modify(angle,Uc1,Numel) for i=1:Numel angle(i)=angle(i)+Uc1(i*3); end

%‘Assembly.m’ %------171

%This function is to compute element properties $ Assemble the structure %------function [S,R]=Assembly(Numel,No_node,angle,E,I,A,L,Nod,RE,PE)

S=zeros(No_node*3,No_node*3); R=zeros(No_node*3,1); for N=1:Numel Trans=[cos(angle(N)) sin(angle(N)) 0 0 0 0 -sin(angle(N)) cos(angle(N)) 0 0 0 0 0 0 1 0 0 0 0 0 0 cos(angle(N)) sin(angle(N)) 0 0 0 0 -sin(angle(N)) cos(angle(N)) 0 0 0 0 0 0 1];

Ke= [A(N)*E(N)/L(N) 0 0 -A(N)*E(N)/L(N) 0 0 0 12*E(N)*I(N)/L(N)^3 6*E(N)*I(N)/L(N)^2 0 -12*E(N)*I(N)/L(N)^3 6*E(N)*I(N)/L(N)^2 0 6*E(N)*I(N)/L(N)^2 4*E(N)*I(N)/L(N) 0 -6*E(N)*I(N)/L(N)^2 2*E(N)*I(N)/L(N) -A(N)*E(N)/L(N) 0 0 A(N)*E(N)/L(N) 0 0 0 -12*E(N)*I(N)/L(N)^3 -6*E(N)*I(N)/L(N)^2 0 12*E(N)*I(N)/L(N)^3 -6*E(N)*I(N)/L(N)^2 0 6*E(N)*I(N)/L(N)^2 2*E(N)*I(N)/L(N) 0 -6*E(N)*I(N)/L(N)^2 4*E(N)*I(N)/L(N)]; SE=Trans'*Ke*Trans; kk=zeros(6,1); kk(3)=3*Nod(1,N); kk(2)=kk(3)-1; kk(1)=kk(2)-1; kk(6)=3*Nod(2,N); kk(5)=kk(6)-1; kk(4)=kk(5)-1; %Assemble into global stiffness matrix for i=1:6 k=kk(i); R(k,1)=R(k,1)+RE(N,i)+PE(N,i); for j=1:6 l=kk(j); S(k,l)=S(k,l)+SE(i,j); end end end

%‘BC_penalty.m’ %------% Function 'BC_penalty.m' apply penalty method to S and R %------function [S,R]=BC_penalty(Total_imposed_BCs,BC_node,BC_dof,BC_magnitude,S,R) penalty_number=1e5; for i=1:Total_imposed_BCs if BC_dof(i)==1 j=BC_node(i)*3-2; else if BC_dof(i)==2 j=BC_node(i)*3-1; else

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j=BC_node(i)*3; end end S(j,j)=S(j,j)+S(j,j)*penalty_number; R(j)=BC_magnitude(i)*penalty_number; end

%‘Boundary_input.m’ function Boundary_input(Node) fid = fopen('input_bc.txt', 'wt'); fprintf( fid,'%% Input boundary condition'); fprintf( fid,'\n %12.0f %12.0f %12.0f',4,0,0); fprintf( fid,'\n %12.0f %12.0f %12.0f',1,1,0); %fprintf( fid,'\n %12.0f %12.0f %12.0f',1,2,0); fprintf( fid,'\n %12.0f %12.0f %12.0f',1,3,0); fprintf( fid,'\n %12.0f %12.0f %12.0f',Node,1,0); %fprintf( fid,'\n %12.0f %12.0f %12.0f',Node,2,0); fprintf( fid,'\n %12.0f %12.0f %12.0f',Node,3,0); fclose(fid)

%‘configuration.m’ %------% Function 'configuration' calculates the profile %------function [Nodal_position_orig]=configuration(Node,Radius,Numel) Nodal_position_orig=zeros(Node*3,1); Nodal_position_orig(1)=0; Nodal_position_orig(2)=-Radius; Nodal_position_orig(3)=-pi/2; %Normal vector for i=1:Numel Nodal_position_orig(i*3+1)=Radius*cos(pi/Numel*i-pi/2); Nodal_position_orig(i*3+2)=Radius*sin(pi/Numel*i-pi/2); Nodal_position_orig(i*3+3)=pi/Numel*i-pi/2; end

%‘inputvar.m’ %------%Input and initialization %------function inputvar(Numel,Node,b,h,Radius,E) A=ones(Numel,1)*b*h; %Cross-sectional areas E=ones(Numel,1)*E; %Young's modulus L=ones(Numel,1)*pi*Radius/Numel; %Length of each element I=ones(Numel,1)*b*h^3/12; %Second moment of area %------%Assign global node numbers of each element %------Nod=zeros(Numel,2); Nod(1,1)=1; Nod(1,2)=2; for i=1:2 for j=2:Numel Nod(j,i)=Nod(j-1,i)+1; end end

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%------%Compute the angle of each element %------angle=zeros(Numel,1); angle_star=(pi-pi/Numel)/2; angle(1)=pi/2-angle_star; for i=2:Numel angle(i)=angle(i-1)+pi-2*angle_star; end %------%Compute concentrated forces %------RE=zeros(Numel,6); %------%Wirte data into 'input.txt' %------fid = fopen('inputdata.txt', 'wt'); fprintf( fid,'%% Input and initialization Numel Node(i) Node(j) A E L I angle RE'); fprintf( fid,'\n %12.0f %12.0f %12.0f %16.4e %16.4e %16.4e %16.4e %16.9e %16.4e %16.4e %16.4e %16.4e %16.4e %16.4e',Numel,Node,0,0,0,0,0,0,0,0,0,0,0,0); for i=1:Numel fprintf( fid,'\n %12.0f %12.0f %12.0f %16.4e %16.4e %16.4e %16.4e %16.9e %16.4e %16.4e %16.4e %16.4e %16.4e %16.4e',i, Nod(i,1),Nod(i,2),A(i),E(i),L(i),I(i),angle(i),RE(i,1),RE(i,2),RE(i,3),RE(i,4 ),RE(i,5),RE(i,6)); end fclose(fid);

%‘Jacobian.m’ %------%This function is to compute element properties $ Assemble the structure, %calculate Jacobian matrix [J] %------function [J]=Jacobian(Numel,Node,A,E,L,I,angle,Nod,dFc,Total_imposed_BCs,BC_node,BC_do f,BC_magnitude,R) J=zeros(Node*3,Node*3); for N=1:Numel JE=[ A(N)*E(N)*cos(angle(N))^2/L(N)+12*E(N)*I(N)*sin(angle(N))^2/(L(N)^3) …… 4*E(N)*I(N)/L(N)]; kk=zeros(6,1); kk(3)=3*Nod(1,N); kk(2)=kk(3)-1; kk(1)=kk(2)-1; kk(6)=3*Nod(2,N); kk(5)=kk(6)-1; kk(4)=kk(5)-1; for i=1:6 k=kk(i); for j=1:6 l=kk(j); J(k,l)=J(k,l)+JE(i,j); end

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end end J=BC_penalty(Total_imposed_BCs,BC_node,BC_dof,BC_magnitude,J,R); J=J-diag(dFc);

%‘Length_modify.m’ function L=Length_modify(Numel,Uc,Nodal_position_orig,Node) for i=1:Numel x=Nodal_position_orig(3*(i+1)-2)+Uc(3*(i+1)-2)- Nodal_position_orig(3*i- 2)-Uc(3*i-2); y=Nodal_position_orig(3*(i+1)-1)+Uc(3*(i+1)-1)- Nodal_position_orig(3*i- 1)-Uc(3*i-1); L(i)=(x.^2+y.^2)^(1/2); end L(Node)=L(Numel);

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APPENDIX 3 Fortran program for cylindrical shell within DLVO potential

! main.f90 ! !**************************************************************************** ! PROGRAM: Cylindrical shell adhesion in DLVO potential ! PURPOSE: Calculate a cylindrical shell interacting with a rigid substrate ! in a potential with secondary minimum. ! ! Importance of this program: ! 1, Modeling interaction between deformable shell and substrate. ! 2, Stretching and bending effects of beam element are coupled. ! 3, Maugis's approximation for DLVO potential ! !****************************************************************************

program Shell

implicit none ! ! Fixed parameters ! To avoid unnecessary errors, we keep more than 20 digits ! DOUBLE PRECISION , parameter :: pi= 3.14159265358d0 integer Ntimes,cntr,Nprint ! Ntimes: Total iterations. cntr: Current iteration. ! Nprint: Data is written into .txt files every Nprint iterations integer Numel_1,Numel_2,Node1,Node2,Dof1,Dof2 ! Numel: Number of elements in beam ! Node: Number of nodes in beam (for closed geometry , Node = Numel, for open geometry ,Node = Numel+1 ) integer bound1,bound2 ! Number of boundary conditions applied integer i,j integer Ele_RT1,Ele_RT2 ! Ele_RT: Ratio of Largest element size and smallest element size ! Condensed nodes are located closed to contact area integer ml,mu,mm,lda integer node_i, node_j integer round_digit ! ! Declear double precision variables ! DOUBLE PRECISION E1,b1,h1,Area1,Inertia1,m_density1,Radius1,R1_orig ! Material and geometrical parameters for beam1 (Top) ! h: Thickness of beam element. ! Area: Cross sectional area. ! Inertia: bh^3/12. ! dx: length of each element. ! m_density: density of beam element. ! Radius1: Radius of shell after normalization. (Radius = 1) ! R1_orig: Radius of shell originally DOUBLE PRECISION E2,b2,h2,Area2,Inertia2,Length2,m_density2 ! Material and geometrical parameters for beam2 (Bottom)

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DOUBLE PRECISION Structure_damp_coef1,Mass_damp_coef1,Penalty_parameter1 ! Dynamic parameters for beam1 ! [Hg]= Structure_damp_coef*[Kg]+ Mass_damp_coef*[Mg] DOUBLE PRECISION Structure_damp_coef2,Mass_damp_coef2,Penalty_parameter2 ! Dynamic parameters for beam2 DOUBLE PRECISION Gamma,beta,alpha,dt ! Gamma, beta are coefficients used in Newton's method ! alpha: Parameter used to assemble mass matrix ! dt: time increment DOUBLE PRECISION w_freq_max1,w_freq_max2,w_freq_min,temp ! Parameter temperarily used. ! w_freq_max,w_freq_min: used to calculate critical dt DOUBLE PRECISION F_str1,F_str2,center1(2),pos_beam2,P_offset2 ! F_str : Stretching force applied on beam, positive denotes tensile ! Center: Center coordiates of circular shell ! pos_beam: Original position in global coordinate system. DOUBLE PRECISION stress_1st,stress_rep,stress_2nd,r_1st,r_rep,r_2nd,F_slope,z0,initial_vo ! Maugis' parameter. ! stress: disjoining pressure (Maugis' parameter) ! z: range (Maugis' parameter) ! F_slope: Slope of replusion force in Maugis approximation ! z0: Atomic equilibrium distance DOUBLE PRECISION x_RT2,dx_RT2 DOUBLE PRECISION Vol, Vol_0, Ev ,p0 ! Vol, Vol_0: changed volume and initial volumn ! Ev: bulk modulus, indicating compresibility of liquid ! p0: internal pressure (Pa)

!------! Allocatable arrays !------integer, ALLOCATABLE:: Nod1(:,:),Nod2(:,:) ! Nod: A matrix to record node-node connect relation ! [ 1 2 3 4 .... ! 2 3 4 5 .... ] DOUBLE PRECISION , ALLOCATABLE:: Kg1(:,:),Hg1(:,:),Mg1(:,:),Rg1(:),dn1(:),vn1(:),an1(:),angle1(:),dx1(:) ! Kg: Global stiffness matrix ! Hg: Global damping matrix ! Mg: Global mass matrix ! Rg: External force vector ! dn,vn,an : Displacement, velocity, and acceleration ! angle: angle of each beam element respect to Global X axis DOUBLE PRECISION , ALLOCATABLE:: Kg2(:,:),Hg2(:,:),Mg2(:,:),Rg2(:),dn2(:),vn2(:),an2(:),angle2(:),dx2(:) DOUBLE PRECISION , ALLOCATABLE:: position_orig_1(:),position_orig_2(:),position_1(:),position_2(:) ! position_orig: Original position ! position: Deformed postion DOUBLE PRECISION , ALLOCATABLE:: w_freq_square1(:),w_freq_square2(:) ! w_freq_square: Frequency related parameter to calculate critical time increment 'dt' DOUBLE PRECISION , ALLOCATABLE:: BC_nodes_1(:),BC_nodes_2(:) ! BC_nodes: A vector to record boundary nodes

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DOUBLE PRECISION , ALLOCATABLE:: Sigma_s1(:),Sigma_b1(:),Sigma1(:),Sigma_s2(:),Sigma_b2(:),Sigma2(:) ! Sigma: Internal stress (stretching, bending and sum of two) DOUBLE PRECISION , ALLOCATABLE:: vector_temp(:) DOUBLE PRECISION , ALLOCATABLE:: Kg1_round(:,:),Hg1_round(:,:),Mg1_round(:,:),Rg1_round(:),dn1_round(:) DOUBLE PRECISION , ALLOCATABLE:: Kg2_round(:,:),Hg2_round(:,:),Mg2_round(:,:),Rg2_round(:),dn2_round(:)

! ! fixed arrays ! real(4) time_begin, time_end CALL CPU_TIME ( time_begin ) ! To calculate compute time

!------! Lapack functions 'dgbco','dgbsl' are adopted to inverse matrices ! ml,mu,mm,lda are parameters used in 'dgbco','dgbsl', see details in these functions !------ml = 5 mu = 5 mm = ml + mu + 1 lda = 2*ml + mu + 1 !------! Read inputs matlab version !------open(503,file = 'inputs.dat') read(503,"(A1)") !text (jump one line) read(503,"(A1)") !text read(503,"(A1)") !text read(503,*) Numel_1, Numel_2,Node1,Node2,Ele_RT1,Ele_RT2

! Degrees of freedom. Each node has three degrees of freedom Dof1 = 3*Node1 Dof2 = 3*Node2

!------! array dimention allocations !------ALLOCATE(Nod1(2,Numel_1),Nod2(2,Numel_2)) ALLOCATE(Kg1(Dof1,Dof1),Hg1(Dof1,Dof1),Mg1(Dof1,Dof1),Rg1(Dof1)) ALLOCATE(dn1(Dof1),vn1(Dof1),an1(Dof1),angle1(Numel_1),dx1(Numel_1)) ALLOCATE(Kg2(Dof2,Dof2),Hg2(Dof2,Dof2),Mg2(Dof2,Dof2),Rg2(Dof2)) ALLOCATE(dn2(Dof2),vn2(Dof2),an2(Dof2),angle2(Numel_2),dx2(Numel_1))

ALLOCATE(position_orig_1(Dof1),position_orig_2(Dof2),position_1(Dof1),posi tion_2(Dof2)) ALLOCATE(w_freq_square1(Dof1),w_freq_square2(Dof2)) ALLOCATE(Sigma_s1(Numel_1),Sigma_b1(Numel_1),Sigma1(Numel_1)) ALLOCATE(Sigma_s2(Numel_2),Sigma_b2(Numel_2),Sigma2(Numel_2)) ALLOCATE(vector_temp(Numel_2))

ALLOCATE(Kg1_round(Dof1,Dof1),Hg1_round(Dof1,Dof1),Mg1_round(Dof1,Dof1),Rg 1_round(Dof1),dn1_round(Dof1))

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ALLOCATE(Kg2_round(Dof2,Dof2),Hg2_round(Dof2,Dof2),Mg2_round(Dof2,Dof2),Rg 2_round(Dof2),dn2_round(Dof2)) read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,*) E1,b1,h1,Radius1,R1_orig read(503,*) E2,b2,h2,Length2 !------! Normalize b,h and length2 (not in use now, since R1_orig = 1 ) !------b1 =b1/R1_orig h1 =h1/R1_orig b2 =b2/R1_orig h2 =h2/R1_orig Length2 =Length2/R1_orig

Area1 =b1*h1 Inertia1 =b1*h1**3/12.0 Area2 =b2*h2 Inertia2 =b2*h2**3/12.0 read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,*) m_density1,Structure_damp_coef1,Mass_damp_coef1,F_str1,center1 read(503,*) m_density2,Structure_damp_coef2,Mass_damp_coef2,F_str2,pos_beam2,P_offset2 read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,*) Gamma,Beta read(503,*) Ntimes, Nprint, alpha read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,*) stress_1st,stress_rep,stress_2nd,r_1st,r_rep,r_2nd read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,*) F_slope,z0 read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,*) bound1,bound2

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!------! BC_nodes dimention allocations !------if (bound1 /= 0) then ALLOCATE(BC_nodes_1(bound1)) read(503,*) BC_nodes_1 else ALLOCATE(BC_nodes_1(1)) read(503,"(A1)") endif if (bound2 /= 0) then ALLOCATE(BC_nodes_2(bound2)) read(503,*) BC_nodes_2 else ALLOCATE(BC_nodes_2(1)) read(503,"(A1)") endif read(503,"(A1)") read(503,"(A1)") read(503,"(A1)") read(503,*) initial_vo, Ev,round_digit

!------!end reading !------

!------! Subroutine 'Nod_angle_assign': ! Calculate original angles (angles are used to assemble global matrix) ! Assign global node numbers of each element for beam element !------angle1(:) = 0.0 Nod1(:,:) = 0.0 call Nod_angle_assign(Node1,Numel_1,Nod1,angle1,Ele_RT1) angle2(:) = 0.0 Nod2(:,:) = 0.0 call Nod_angle_assign(Node2,Numel_2,Nod2,angle2,Ele_RT2)

!------!Record output into '**.txt' file !------open (unit=500, file='Profile.txt', status='unknown') open (unit=501, file='central_w.txt', status='unknown') open (unit=502, file='Internal_stress.txt', status='unknown') open (unit=504, file='AnVnDn.txt', status='unknown')

!------! Initial configuraion and matrix assembly ! (position_orig_1 for cylinder 1 and position_orig_2 for membrane 2) ! Originally, the code was made in such a way that both cylinder ! and membrane are deformable. However, current version does not allow ! membrane to deform, i.e. it is rigid substrate in this case. !------! 180

! Initial configuraion for cylinder 1 ! position_orig_1 = 0.0d0 dx1 = 0.0d0 call configuration(Node1,Radius1,Numel_1,position_orig_1,center1,Ele_RT1,dx1) !------! Calculate initial volume Vol_0, used to calculate internal P !------call AREA_computation (Numel_1,Node1,position_orig_1,Vol_0)

! ! Initial configuraion for membrane 2 ! !------! Uneven distribution !------dx2 = 0.0d0 x_RT2 = Length2 /(Numel_2/2)/(1+Ele_RT2) !Numel/2 is for half circle dx_RT2 = (Ele_RT2-1)*x_RT2/(Numel_2/2-1) do i = 1, Numel_2/2 vector_temp(i) = x_RT2 + (Numel_2/2-i)*dx_RT2 vector_temp(Numel_2+1-i) = x_RT2 + (Numel_2/2-i)*dx_RT2 end do position_orig_2 = 0.0d0 position_orig_2(1)= -Length2/2+P_offset2 position_orig_2(2)= pos_beam2 position_orig_2(3) = pi/2.0 do i=2,Node2 position_orig_2(i*3-2)= position_orig_2(i*3-5)+ vector_temp(i-1) !x coordinate of beam 2 position_orig_2(i*3-1)= pos_beam2 !y coordinate of beam 2 position_orig_2(i*3) = pi/2.0 !Normal vector

dx2(i-1) = sqrt ((position_orig_2(i*3-1)-position_orig_2(i*3-4))**2 +(position_orig_2(i*3-2)-position_orig_2(i*3-5))**2) end do

position_1= position_orig_1 position_2= position_orig_2 !------! Write initial configuration into output file !------write (500,*) '#------' write (500,*) "Time", 0 , "cntr", 0 write (500,*) '#------' write (500,*) '#Shell 1 pos_x, pos_y' do i=1,Node1 write(500,9001) position_1(i*3-2),position_1(i*3-1) end do write (500,*) '#------' write (500,*) '#Beam 2 pos_x, pos_y' 181

do i=1,Node2 write(500,9001) position_2(i*3-2),position_2(i*3-1) end do

!------! Calculate critical dt based on Kg, Mg matrix. (Based on FEM text book) ! w_max^2 <= max(sum[Kij]/Mii), dt<=2/ w_max !------call Assembly(Numel_1,Node1,Dof1,angle1,E1,Inertia1,Area1,Nod1,Kg1,Hg1,Mg1,Rg1, m_density1,dx1,Structure_damp_coef1,Mass_damp_coef1,alpha) call Assembly(Numel_2,Node2,Dof2,angle2,E2,Inertia2,Area2,Nod2,Kg2,Hg2,Mg2,Rg2, m_density2, dx2,Structure_damp_coef2,Mass_damp_coef2,alpha)

do i=1,Dof1 temp = 0.0d0 do j=1,Dof1 temp = temp + Kg1(i,j) end do w_freq_square1(i)=temp/Mg1(i,i) end do

w_freq_max1 = Maxval(w_freq_square1)

dt = 2.0d0/w_freq_max1**(0.5) dt = dt/5.0d0 write(*,*) "dt=", dt

!------! Compute Penalty_parameter 1e4 times as the max value of [Kg],[Hg],[Mg] !------temp = Kg1(1,1) do i =1, Dof1 if (temp < Kg1(i,i)) then temp = Kg1(i,i) elseif (temp < Hg1(i,i)) then temp = Hg1(i,i) elseif (temp < Mg1(i,i)) then temp = Mg1(i,i) endif end do Penalty_parameter1 =1.0d4*temp temp = Kg2(1,1) do i =1, Dof2 outer1: if (temp < Kg2(i,i)) then temp = Kg2(i,i) elseif (temp < Hg2(i,i)) then temp = Hg2(i,i) elseif (temp < Mg2(i,i)) then temp = Mg2(i,i) endif outer1 end do Penalty_parameter2 =1.0d4*temp

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!------! Implement BCs to Kg,Mg,Hg matrices !------if (bound1 /= 0) then do i = 1, size(BC_nodes_1) Kg1(BC_nodes_1(i),BC_nodes_1(i)) = (0.0+ Penalty_parameter1) Hg1(BC_nodes_1(i),BC_nodes_1(i)) = (0.0+ Penalty_parameter1) Mg1(BC_nodes_1(i),BC_nodes_1(i)) = (0.0+ Penalty_parameter1) end do endif if (bound2 /= 0) then do i = 1, size(BC_nodes_2) Kg2(BC_nodes_2(i),BC_nodes_2(i)) = (0.0+ Penalty_parameter2) Hg2(BC_nodes_2(i),BC_nodes_2(i)) = (0.0+ Penalty_parameter2) Mg2(BC_nodes_2(i),BC_nodes_2(i)) = (0.0+ Penalty_parameter2) end do endif

!------! Putting everything into Explicit method's format !------! ! Time integration,Initial conditions ! dn1 = 0.0d0 vn1 = 0.0d0 do i = 1, Node1 vn1(i*3-1) = -initial_vo end do an1 = 0.0d0 dn2 = 0.0d0 vn2 = 0.0d0 an2 = 0.0d0

Loop1: do cntr = 1,Ntimes

!------! Assemble matrix and calculate surface force Rg1 for cylinder 1 !------

Rg1(:) = 0.0

if (cntr == 1) then call Assembly_coupled_matrix(Numel_1,Node1,Dof1,angle1,E1,Inertia1,Are a1,Nod1,Kg1,Hg1,Mg1,Rg1,m_density1,dx1,Structure_damp_coef1,Mass_ damp_coef1,alpha,dn1)

!------! Write and reload to avoid round-off errors !------do j=1,Dof1 do i = 1, Dof1 call r8_roundx(round_digit, Kg1(i,j), Kg1_round(i,j)) 183

call r8_roundx(round_digit, Hg1(i,j), Hg1_round(i,j)) call r8_roundx(round_digit, Mg1(i,j), Mg1_round(i,j)) end do end do Kg1 = Kg1_round Hg1 = Hg1_round Mg1 = Mg1_round end if !------! Calculate surface force Rg1 for cylinder 1 !------call surface_force_maugis(Numel_1,Node1,Numel_2,Node2,position_1,position _2,Nod2,dx1,Rg1,stress_1st,stress_rep,stress_2nd,r_1st,r_rep,r_2nd,b 1,F_slope,z0)

! Initial acceleration if (cntr == 1) then do i=1,Dof1 an1(i) = Rg1(i)/Mg1(i,i) enddo endif !------! Add up external force if there is any !------Rg1(Numel_1/2.0*3.0+2) = Rg1(Numel_1/2.0*3.0+2)+F_str1 !------! Calculate internal pressure !------call AREA_computation (Numel_1,Node1,position_1,Vol) call internal_pressure (Numel_1,Node1,Nod1,Vol,Vol_0,position_1,p0,b1,dx1,Ev,Rg1) !------! Write and reload to avoid round-off errors !------do i = 1, Dof1 call r8_roundx(round_digit, Rg1(i), Rg1_round(i)) end do Rg1 = Rg1_round

!------! Both implicit and explicit method are coded ! if no restriction applied on shell, run 'compute an explicit noboundary' ! if there is restriction applied on shell, run 'compute an explicit' !------call compute_an_explicit_noboundary(Dof1,dn1,vn1,an1,Gamma,beta,dt,Kg1,Hg 1,Mg1,Rg1)

!------! Write and reload to avoid round-off errors !------do i = 1, Dof1 184

call r8_roundx(round_digit, dn1(i), dn1_round(i)) end do dn1 = dn1_round

!------! Updated deformed profiles !------position_1= position_orig_1 + dn1

!------!Calculate Rg2 for stretched beam2 based on modified position 1 !------position_2= position_orig_2 + dn2

!------! Write output into txt files !------if (mod(cntr,Nprint).eq.0) then ! write into 500: deformed profiles write (500,*) '#------' write (500,*) "Time", cntr*dt, "cntr",cntr write (500,*) '#------' CALL CPU_TIME ( time_end ) write(500,*) 'Program has been running for', time_end - time_begin, 'seconds.' write (500,*) '#------' write (500,*) '#Shell 1 pos_x, pos_y, normal direction, p_x, p_y' do i=1,Node1 if (i < Node1/2) then temp = dx1(i) else temp = dx1(i-1) endif write(500,9005) position_1(i*3-2),position_1(i*3- 1),position_1(i*3),Rg1(i*3-2)/temp/b1,Rg1(i*3-1)/temp/b1 end do write (500,*) '#------' write (500,*) '#Beam 2 pos_x, pos_y, normal direction, p_x, p_y' do i=1,Node2 write(500,9005) position_2(i*3-2),position_2(i*3- 1),position_2(i*3),Rg2(i*3-2),Rg2(i*3-1) end do ! write into 504: displacement, velocity, acceleration write (504,*) '#------' write (504,*) "Time", cntr*dt, "cntr",cntr write (504,*) '#------' CALL CPU_TIME ( time_end ) write(504,*) 'Program has been running for', time_end - time_begin, 'seconds.' write (504,*) '#------' write (504,*) '#Shell 1 an, vn, dn' do i=1,Dof1 write(504,9003) an1(i),vn1(i),dn1(i) 185

end do write (504,*) '#------' write (504,*) '#beam an, vn, dn' do i=1,Dof2 write(504,9003) an2(i),vn2(i),dn2(i) end do

! write into 502: internal stresses Sigma_s1 = 0.0 Sigma_b1 = 0.0 Sigma1 = 0.0 Sigma_s2 = 0.0 Sigma_b2 = 0.0 Sigma2 = 0.0 call Internal_stress(Numel_1,Dof1,angle1,Nod1,dn1,dx1,E1,Inertia1,h1, Sigma_s1,Sigma_b1,Sigma1) call Internal_stress(Numel_2,Dof2,angle2,Nod2,dn2,dx2,E2,Inertia2,h2, Sigma_s2,Sigma_b2,Sigma2) write (502,*) '#------' write (502,*) "Time", cntr*dt, "cntr",cntr write (502,*) '#------' write (502,*) 'Program has been running for', time_end - time_begin, 'seconds.' write (502,*) '#------' write (502,*) '#Shell 1 pos_x, Sigma_s, Sigma_b, Sigma'

do i=1,Numel_1 node_i = Nod1(1,i) node_j = Nod1(2,i) write(502,9002) (position_1(node_i*3-2)+position_1(node_j*3- 2))/2.0d0, Sigma_s1(i),Sigma_b1(i),Sigma1(i) end do write (502,*) '#------' write (502,*) '#Beam 2 pos_x, Sigma_s, Sigma_b, Sigma' do i=1,Numel_2 node_i = Nod2(1,i) node_j = Nod2(2,i) write(502,9002) (position_2(node_i*3-2)+position_2(node_j*3- 2))/2.0d0,Sigma_s2(i),Sigma_b2(i),Sigma2(i) end do endif ! write into 501: Central displacement, check convergence if (mod(cntr,Nprint).eq.0) then write(501,9002) cntr*dt,position_1(Numel_1/2.0*3.0+2),position_1(2),p0 !write(*,*) "Loop", cntr,"An", an1(5) write(*,*) "Loop", cntr,"Central D", position_1(Numel_1/2.0*3.0+2), position_1(2),p0 !write(501,9001) cntr*dt,position_1(Numel_1/2.0*3.0+2) !write(*,*) "Loop", cntr, "Dn1", position_1(Numel_1/2.0*3.0+2) end if end do Loop1

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! ------! End of integration ! ------

9001 format(2(1X,E16.10)) 9002 format(4(1X,E16.10)) 9003 format(3(1X,E16.10)) 9004 format((E16.10)) 9005 format(5(1X,E16.10)) ! ! Release variables ! DEALLOCATE (Nod1,Nod2) DEALLOCATE (Kg1,Hg1,Mg1,Rg1,dn1,vn1,an1,angle1) DEALLOCATE (Kg2,Hg2,Mg2,Rg2,dn2,vn2,an2,angle2) DEALLOCATE (position_orig_1,position_orig_2,position_1,position_2) DEALLOCATE (w_freq_square1,w_freq_square2) DEALLOCATE (BC_nodes_1,BC_nodes_2) DEALLOCATE (Kg1_round,Hg1_round,Mg1_round,Rg1_round,dn1_round) ! ! closing files ! close(500) close(501) close(502) close(504)

end program Shell

! PROGRAM AREA_computation.f90 !*********************************************************************** ! Program to approximate the integral of a function over the interval ! Node : total element nodes ! Numel: total element numbers ! x_pos: position vecter ! y_pos: position vecter ! position: has normal vecter included ! SUM : the approximating sum ! !* Input: position, Node, Numel ! !* Output: Approximation to integral of SUM ! !************************************************************************

SUBROUTINE AREA_computation (Numel,Node,position,SUM) INTEGER Numel,Node,i REAL(8) position(Node*3),x_pos(Node),y_pos(Node) real(8) SUM

do i = 1,Node x_pos(i) = position(i*3-2) y_pos(i) = position(i*3-1) end do

SUM = 0 187

! Compute closed area for circular shell do i = 1,Node if (i /= Node) then SUM = SUM + (y_pos(i)+y_pos(i+1))/2*(x_pos(i+1)-x_pos(i)) else SUM = SUM + (y_pos(i)+y_pos(1))/2*(x_pos(1)-x_pos(i)) endif end do

END SUBROUTINE AREA_computation

! Assembly.f90 !------!This function is to compute element properties $ Assemble the structure !------!------! Input: ! Numel: Number of elements ! Node: Number of nodes ! Dof: Total degrees of freedom. ! angle: angle of each beam element respect to Global X axis ! E: Elastic modulus. ! Inertia: bh^3/12. ! A: Cross sectional area. ! Nod: A matrix to record node-node connect relation ! m_density: density of beam element. ! length_dx: length of each element. ! Cost1: structual demping coef ! Cost2: mass demping coef ! alpha: A small constant used in lumped mass matrix ! ! Output: ! Kg: Global stiffness matrix ! Hg: Global damping matrix ! Mg: Global mass matrix ! Rg: External force vector !------SUBROUTINE Assembly(Numel,Node,Dof,angle,E,Inertia,A,Nod,Kg,Hg,Mg,Rg,m_density,length_dx ,Cost1,Cost2,alpha)

integer N,i,j,m,k,Numel,Node,Dof,Nod(2,Numel) DOUBLE PRECISION Trans(6,6),Ke(6,6),Me(6,6),He(6,6),kk(6),angle(Numel),E,m_density,c_density DOUBLE PRECISION length_dx(Numel),A,Inertia DOUBLE PRECISION Kg(Dof,Dof),Hg(Dof,Dof),Mg(Dof,Dof),Rg(Dof),Cost1,Cost2,alpha

Mg(:,:)=0.0d0 Hg(:,:)=0.0d0 Kg(:,:)=0.0d0 Rg(:) =0.0d0 do N=1,Numel Trans(:,:) =0 188

Trans(1,1) =dcos(angle(N)) Trans(2,1) =-dsin(angle(N)) Trans(1,2) =dsin(angle(N)) Trans(2,2) =dcos(angle(N)) Trans(3,3) =1 Trans(4,4) =dcos(angle(N)) Trans(5,4) =-dsin(angle(N)) Trans(4,5) =dsin(angle(N)) Trans(5,5) =dcos(angle(N)) Trans(6,6) =1

Ke(:,:) = 0.0 Ke(1,1) = A*E/length_dx(N) Ke(4,1) =-A*E/length_dx(N) Ke(2,2) = 12.0*E*Inertia/length_dx(N)**3 Ke(3,2) = 6.0*E*Inertia/length_dx(N)**2 Ke(5,2) =-12.0*E*Inertia/length_dx(N)**3 Ke(6,2) = 6.0*E*Inertia/length_dx(N)**2 Ke(2,3) = 6.0*E*Inertia/length_dx(N)**2 Ke(3,3) = 4.0*E*Inertia/length_dx(N) Ke(5,3) =-6.0*E*Inertia/length_dx(N)**2 Ke(6,3) = 2.0*E*Inertia/length_dx(N) Ke(1,4) =-A*E/length_dx(N) Ke(4,4) = A*E/length_dx(N) Ke(2,5) =-12.0*E*Inertia/length_dx(N)**3 Ke(3,5) =-6.0*E*Inertia/length_dx(N)**2 Ke(5,5) = 12.0*E*Inertia/length_dx(N)**3 Ke(6,5) =-6.0*E*Inertia/length_dx(N)**2 Ke(2,6) = 6.0*E*Inertia/length_dx(N)**2 Ke(3,6) = 2.0*E*Inertia/length_dx(N) Ke(5,6) =-6.0*E*Inertia/length_dx(N)**2 Ke(6,6) = 4.0*E*Inertia/length_dx(N)

Ke=Matmul(Matmul(Transpose(Trans),Ke),Trans)

Me(:,:) =0.0 Me(1,1) =(m_density*A*length_dx(N)/2.0d0)*1.0d0 Me(2,2) =(m_density*A*length_dx(N)/2.0d0)*1.0d0 Me(3,3) =(m_density*A*length_dx(N))*alpha*length_dx(N)**2 Me(4,4) =(m_density*A*length_dx(N)/2.0d0)*1.0d0 Me(5,5) =(m_density*A*length_dx(N)/2.0d0)*1.0d0 Me(6,6) =(m_density*A*length_dx(N))*alpha*length_dx(N)**2

Me = Matmul(Matmul(Transpose(Trans),Me),Trans) He =Cost1*Ke+Cost2*Me

! Assemble into global matrix

kk(3)=3*Nod(1,N) kk(2)=kk(3)-1 kk(1)=kk(2)-1 kk(6)=3*Nod(2,N) kk(5)=kk(6)-1 kk(4)=kk(5)-1

189

do i=1,6 k=kk(i) do j=1,6 m=kk(j) Kg(k,m)=Kg(k,m)+Ke(i,j) Mg(k,m)=Mg(k,m)+Me(i,j) Hg(k,m)=Hg(k,m)+He(i,j) end do end do end do

END SUBROUTINE Assembly

! Compute_an_explicit.f90 !------! This function is to compute An, Vn, Dn using Newmark's method !------!------! Input: ! Dof: Total degrees of freedom. ! dn,vn,an: Displacement, velocity, and acceleration. ! Gamma, beta: coefficients used in Newton's method. ! dt: Time increment. ! Kg: Global stiffness matrix ! Hg: Global damping matrix ! Mg: Global mass matrix ! Rg: External force vector ! BC_nodes: A vector to record boundary nodes ! bound: Number of boudary nodes ! Penalty_parameter: A large parameter used to apply boundary conditons ! ! Outputs: ! an,vn,dn for the next time increment. !------SUBROUTINE compute_an_explicit(Dof,dn,vn,an,Gamma,beta,dt,Kg,Hg,Mg,Rg,BC_nodes,bound,Pen alty_parameter) integer Dof,i,temp_int,bound real(8) dn(Dof),vn(Dof),an(Dof),Gamma,beta,dt,Kg(Dof,Dof),Hg(Dof,Dof) real(8) Mg(Dof,Dof),Rg(Dof),V_tilde(Dof),d_tilde(Dof),temp,XX(Dof),BC_nodes(bound) real(8) Penalty_parameter

V_tilde = vn + (1.0d0-Gamma)*dt*an d_tilde = dn + dt* vn + (0.5d0 - beta)*dt**2*an XX = 0 do i = 1, Dof temp = 0.0d0 do j = 1,Dof temp = temp -Kg(i,j)*d_tilde(j)-Hg(i,j)*V_tilde(j) end do XX(i) = Rg(i) + temp end do outer: if (bound /= 0) then do i=1,size(BC_nodes) 190

temp_int = BC_nodes(i) inner: if (temp_int/=0) then XX(temp_int) = 0.0* Penalty_parameter endif inner end do endif outer do i=1, Dof an(i)= XX(i)/Mg(i,i) end do dn = d_tilde + beta *dt**2* an vn = v_tilde + gamma * dt * an

END SUBROUTINE compute_an_explicit

! Compute_an_explicit.f90 !------!configuration subroutine is to caculate the original configuration !------!------! Input: ! Node: Number of nodes ! Numel: Number of elements ! Radius: Radius of circular shell ! Center: Center coordinates of circular shell ! ! Output: ! Nodal_position_orig: original configuration !------SUBROUTINE configuration(Node,Radius,Numel,Nodal_position_orig,center,Ele_RT,dx) integer i,Node,Numel,Ele_RT real(8) Radius,Nodal_position_orig(Node*3),center(2),dx(Numel) real(8) Theta,delta_theta,vector_temp(Numel) real(8), parameter :: pi= 3.1415926535897932384626433832795028841971d0 ! real(8) has a precision of 16 decimal digits

! Another way to define pi: pi= 2.0d0*dacos(0d0) !------! Uneven distribution !------Theta = 2*pi/(Numel/2)/(1+Ele_RT) !Numel/2 is for half circle. Theta is angle(1) delta_theta = (Ele_RT-1)*Theta/(Numel/2-1) do i = 1, Numel/2 vector_temp(i) = Theta + (i-1)*delta_theta vector_temp(Numel+1-i) = Theta + (i-1)*delta_theta end do !------

Nodal_position_orig(1)=0 Nodal_position_orig(2)=-Radius Nodal_position_orig(3)=-pi/2 !Normal vector in arange of [-pi/2 3/2Pi]

191

do i=1,Numel-1 Nodal_position_orig(i*3+3)=Nodal_position_orig(i*3)+ vector_temp(i) Nodal_position_orig(i*3+1)=Radius*dcos(Nodal_position_orig(i*3+3))+ center(1) ! configuration of "whole shell" Nodal_position_orig(i*3+2)=Radius*dsin(Nodal_position_orig(i*3+3))+ center(2) end do do i = 1,Numel/2 dx(i) = dsqrt ((Nodal_position_orig(i*3+2)- Nodal_position_orig(i*3-1))**2 & +(Nodal_position_orig(i*3+1)- Nodal_position_orig(i*3-2))**2) dx(Numel+1-i) = dx(i) end do end SUBROUTINE configuration

! DistanceFromLine.90 !------! The funtion of this subroutine is to find the distance from the point (cx,cy) to the line ! determined by the points (ax,ay) and (bx,by) ! distanceSegment = distance from the point to the line segment ! distanceLine = distance from the point to the line (assuming infinite extent in both directions !------! Let the point be C (Cx,Cy) and the line be AB (Ax,Ay) to (Bx,By).Let P be the point of ! perpendicular projection of C on AB. The parameter ratio_dis, which indicates P's position ! along AB, is computed by the dot product of AC and AB divided by the square of the length of AB: ! ! (1) AC dot AB ! ratio_dis= ------! ||AB||^2 ! ! ratio_dis has the following meaning: ! r=0 P = A ! r=1 P = B ! r<0 P is on the backward extension of AB ! r>1 P is on the forward extension of AB ! 0

! ! The point P can then be found: ! Px = Ax + r(Bx-Ax) ! Py = Ay + r(By-Ay) ! And the distance from A to P = r*L. ! ! Use another parameter s to indicate the location along PC, with the ! following meaning: ! s<0 C is left of AB ! s>0 C is right of AB ! s=0 C is on AB ! ! Compute s as follows: ! ! (Ay-Cy)(Bx-Ax)-(Ax-Cx)(By-Ay) ! s = ------! L^2 ! ! Then the distance from C to P = |s|*L. !------SUBROUTINE DistanceFromLine(cx,cy,ax,ay,bx,by,xx,yy,distanceSegment)

Real(8) cx,cy,ax,ay,bx,by,distanceSegment,distanceLine,ratio_dis Real(8) r_numerator,r_denomenator,px,py,s,xx,yy Real(8) dist1,dist2

r_numerator = (cx-ax)*(bx-ax) + (cy-ay)*(by-ay) r_denomenator = (bx-ax)*(bx-ax) + (by-ay)*(by-ay) ratio_dis = r_numerator / r_denomenator

px = ax + ratio_dis*(bx-ax) py = ay + ratio_dis*(by-ay)

s = ((ay-cy)*(bx-ax)-(ax-cx)*(by-ay) ) / r_denomenator

distanceLine = dabs(s)*dsqrt(r_denomenator)

! (xx,yy) is the point on the lineSegment closest to (cx,cy)

xx = px yy = py

if ( (ratio_dis >= 0) .AND. (ratio_dis <= 1) ) then

distanceSegment = distanceLine

else

dist1 = (cx-ax)*(cx-ax) + (cy-ay)*(cy-ay) dist2 = (cx-bx)*(cx-bx) + (cy-by)*(cy-by) if (dist1 < dist2) then xx = ax yy = ay distanceSegment = dsqrt(dist1) else xx = bx yy = by 193

distanceSegment = dsqrt(dist2) endif endif

END SUBROUTINE DistanceFromLine

! internal_pressure.90 !*********************************************************************** ! Program to calculate internal pressure based on volume change ! Node : total element nodes ! Numel: total element numbers ! x_pos: position vecter ! y_pos: position vecter ! position: has normal vecter included ! SUM : the approximating sum ! ! Input: position, Node, Numel ! ! Output: Approximation to integral of SUM ! !************************************************************************

SUBROUTINE internal_pressure (Numel,Node,Nod,Vol,Vol_0,position,p0,b,L,Ev,Rg) INTEGER Numel,Node,i,j,N1,N2,Nod(2,Numel) real(8) p0,b,Vol,Vol_0,Ev real(8) position(Node*3),L(Numel),Rg(Node*3),Pg(Node*3)

Pg(:) = 0 p0 = -(Vol-Vol_0)/Vol_0*Ev do i = 1,Numel N1 = Nod(1,i) N2 = Nod(2,i) Pg(N1*3-2) = Pg(N1*3-2)+ p0*dcos(position(N1*3))*b*L(i)/2 Pg(N1*3-1) = Pg(N1*3-1)+ p0*dsin(position(N1*3))*b*L(i)/2 Pg(N1*3 ) = Pg(N1*3 )+ p0*b*L(i)**2/12 Pg(N2*3-2) = Pg(N2*3-2)+ p0*dcos(position(N2*3))*b*L(i)/2 Pg(N2*3-1) = Pg(N2*3-1)+ p0*dsin(position(N2*3))*b*L(i)/2 Pg(N2*3 ) = Pg(N2*3 )- p0*b*L(i)**2/12 end do ! Make sure Pg is symmetrical do i = 2,Numel/2 j = Numel+2-i Pg(j*3-2) =-Pg(i*3-2) Pg(j*3-1) = Pg(i*3-1) Pg(j*3 ) =-Pg(i*3 ) end do Rg = Rg+ Pg;

END SUBROUTINE internal_pressure

! internal_pressure.90 !------! The subroutine is to calculate the internal stress of equilibeium !------194

!------! Input: ! Numel: Number of elements ! Dof: Total degrees of freedom ! angle: angle of each beam element respect to Global X axis ! Nod: A matrix to record node-node connect relation ! dn: Displacement vector ! dx: Length of each element. dx = Total length/ Numel. ! E: Elastic modulus. ! Inertia: bh^3/12. ! h: thickness.

! Outputs: ! Sigma_s: stress due to stretching ! Sigma_b: stress due to bending ! Sigma: stretching + bending !------SUBROUTINE Internal_stress(Numel,Dof,angle,Nod,dn,dx,E,Inertia,h,Sigma_s,Sigma_b,Sigma)

integer i,Numel,Dof,Node1,Node2,Nod(2,Numel) real(8) angle(Numel),dn(Dof),dx(Numel),E,Inertia,h real(8) Phi,u1,u2,v1,v2,Mz,Sigma_s(Numel),Sigma_b(Numel),Sigma(Numel) do i=1,Numel Phi = angle(i) node1 = Nod(1,i) node2 = Nod(2,i)

u1 = dn(node1*3-2)*dcos(Phi)+dn(node1*3-1)*dsin(Phi) u2 = dn(node2*3-2)*dcos(Phi)+dn(node2*3-1)*dsin(Phi) Sigma_s(i)=E*((u2-u1)/dx(i)+ 1.0/2.0*dabs((dn(node2*3)-dn(node1*3)))**2) ! For nonlinear beam : Ex= du/dx+1/2*(dw/dx)**2

v1 = -dn(node1*3-2)*dsin(Phi)+dn(node1*3-1)*dcos(Phi) v2 = -dn(node2*3-2)*dsin(Phi)+dn(node2*3-1)*dcos(Phi)

Mz = E*Inertia*(6.0/dx(i)**2*v1+2.0/dx(i)*dn(node1*3)- 6.0/dx(i)**2*v2+4.0/dx(i)*dn(node2*3)) ! at x=L, all these three expression (x=0,L/2,L)are consistent

Sigma_b(i)=-Mz/Inertia*(h/2) Sigma(i)=Sigma_s(i)+Sigma_b(i) end do

END SUBROUTINE Internal_stress

! Nod_angle_assign.90 !------! This subroutine is to assign 'Nod' and 'angle' to ! curcle or beam. If (Node=Numel), it is a circle, ! if (Node=Numel+1), it is a flat beam. !------!------! Input: ! Node: Number of nodes 195

! Numel: Number of elements ! Output: ! Nod: A matrix to record node-node connect relation ! angle: angle of each beam element respect to Global X axis !------

SUBROUTINE Nod_angle_assign(Node,Numel,Nod,angle,Ele_RT)

integer i,j,Nod(2,Numel),Node,Numel,Ele_RT DOUBLE PRECISION angle(Numel) DOUBLE PRECISION Theta,delta_theta,vector_temp(Numel) DOUBLE PRECISION, parameter :: pi= 3.1415926d0

!------!Assign global node numbers of each element !------Nod(:,:)=0 Nod(1,1)=1 Nod(2,1)=2 do i=1,2 do j=2,Numel Nod(i,j)=Nod(i,j-1)+1 end do end do if (Numel==Node) then Nod(1,Numel) = 1 Nod(2,Numel) = Node endif

!------!Compute the angle of each element !------Theta = 2.d0*pi/(Numel/2)/(1+Ele_RT) !Numel/2 is for half circle delta_theta = (Ele_RT-1)*Theta/(Numel/2-1) do i = 1, Numel/2 vector_temp(i) = Theta + (i-1)*delta_theta vector_temp(Numel+1-i) = Theta + (i-1)*delta_theta end do angle(:)=0.0d0 if (Numel==Node) then ! Modified to keep the shell symmetrical angle(1) = pi/2-(pi- vector_temp(1))/2 do i=2,Numel-1 angle(i) = angle(i-1)+ (vector_temp(i-1)+vector_temp(i))/2 end do angle(Numel) = pi - angle(1) endif end SUBROUTINE Nod_angle_assign

! surface_force_maugis.90

!------196

!This function is to compute surface force between two deformable beams !by using Maugis's approximation. (Stress * r_cir) !------!------! Input: ! Node: Number of nodes ! Numel: Number of elements ! position: Deformed configuration ! Nod: A matrix to record node-node connect relation ! dx: Length of each element. dx = Total length/ Numel. ! stress: disjoining pressure (Maugis' parameter) ! r_cir: range (Maugis' parameter) ! b: Width of beam element. ! F_slope: Slope of replusion force in Maugis approximation ! z0: Atomic equilibrium distance ! Outputs: ! Rg: External force vector !------SUBROUTINE surface_force_maugis(Numel_master,Node_master,Numel_slave,Node_slave,position _master,position_slave,Nod_slave,dx_master,Rg_master,stress_1st,stress_rep,st ress_2nd,r_1st,r_rep,r_2nd,b_master,F_slope,z0) integer i,j,counter integer Numel_master,Node_master,Numel_slave,Node_slave integer nodal_a,nodal_b,Nod_slave(2,Numel_slave)

Real(8) position_master(Node_master*3),position_slave(Node_slave*3),dx_master(Numel_m aster) Real(8) stress_1st,stress_rep,stress_2nd,r_1st,r_rep,r_2nd,temp1,temp2,temp3 Real(8) gx,gy,cx,cy,ax,ay,bx,by,xx,yy,distanceSegment,dist_min,Gap,Indicator Real(8) unit_vec_1_x(2),unit_vec_1_y(2),unit_vec_2(2),vec_globle_x(2),vec_globle_y(2) Real(8) Rg_master(Node_master*3) Real(8) F_maugis,b_master,F_slope,z0

!------! For each node on master, search all the elements on slave, find the shortest ! distance dist_min. --> Based on 'dist_min', calculate intersurface force F_maugis ! --> Distribute F_maugis into global coordinate system Rg. !------

Loop1: do i = 1,Node_master,1 !------counter = 0 ! Increment dist_min = 0 !------

cx = position_master(3*i-2) ! (cx,cy) is coordinates of molecule on the master element cy = position_master(3*i-1)

Loop2: do j= 1, Numel_slave 197

nodal_a = Nod_slave(1,j) nodal_b = Nod_slave(2,j) ax = position_slave(3*nodal_a-2) ay = position_slave(3*nodal_a-1) bx = position_slave(3*nodal_b-2) by = position_slave(3*nodal_b-1)

call DistanceFromLine(cx,cy,ax,ay,bx,by,xx,yy,distanceSegment) !DistanceFromLine subroutine gives the shortest distance and the correspond

!node position(gx,gy) if (distanceSegment <= r_1st+r_rep+r_2nd) then counter = 1 if (dist_min == 0) then dist_min = distanceSegment gx = xx gy = yy elseif (distanceSegment < dist_min) then dist_min = distanceSegment gx = xx gy = yy endif endif end do Loop2

!------! Calculate force based on the shortest distance between two surfaces using Maugis approximation !------if (counter == 1) then unit_vec_1_y = (/dcos(position_master(i*3)),dsin(position_master(i*3))/) ! Unit vector, local y coordinate unit_vec_1_x = (/-unit_vec_1_y(2),unit_vec_1_y(1)/) ! Unit vector, local x is rotate y coordinate

! couter clockwise 90 degrees Gap = dsqrt((gx-cx)**2+(gy-cy)**2) unit_vec_2 = (/(gx-cx)/Gap,(gy-cy)/Gap/)

Indicator = dot_product(unit_vec_2,unit_vec_1_y) ! Projection to local y axis if (Indicator < 0) then dist_min = - dist_min endif

!------! Maugis approximation !------if (i < Node_master/2) then dx_size = dx_master(i) else dx_size = dx_master(i-1) endif 198

temp1=Stress_1st/F_slope temp2=Stress_rep/F_slope temp3=Stress_2nd/F_slope

if (dist_min <= temp1+z0) then F_maugis=dabs(F_slope*(dist_min-z0)); !F_maugis in stress unit [Pa], turn into Force later elseif (dist_min>temp1+z0.and.dist_min<=z0+r_1st-temp1) then F_maugis=Stress_1st elseif (dist_min > z0 + r_1st-temp1 .and. dist_min <= z0+r_1st+temp2) then F_maugis=-F_slope*(dist_min-(r_1st+z0)); elseif (dist_min > z0+r_1st+temp2 .and. dist_min <= z0+r_1st+r_rep-temp2) then F_maugis=-Stress_rep elseif (dist_min > z0+r_1st+r_rep-temp2 .and. dist_min <=z0+r_1st+r_rep+temp3 ) then F_maugis= F_slope*(dist_min -(z0+r_1st+r_rep)) elseif (dist_min> z0+r_1st+r_rep+temp3 .and. dist_min <=z0+r_1st+r_rep+r_2nd-temp3) then F_maugis= Stress_2nd elseif (dist_min > z0+r_1st+r_rep+r_2nd-temp3 .and. dist_min <= z0+r_1st+r_rep+r_2nd) then F_maugis=-F_slope*(dist_min -(z0+r_1st+r_rep+r_2nd)) else F_maugis=0 endif

F_maugis=F_maugis*b_master*dx_size ! Turn F_maugis into force

vec_globle_x = (/1,0/) vec_globle_y = (/0,1/) Rg_master(i*3-2) = dot_product(F_maugis*unit_vec_2,vec_globle_x) ! Projection to global x axis Rg_master(i*3-1) = dot_product(F_maugis*unit_vec_2,vec_globle_y) ! Projection to global y axis end if end do loop1

END SUBROUTINE surface_force_maugis

199

VITA

Jiayi Shi was born on December 26th 1983 in Hangzhou, Zhejiang, P. R. China. She

studied in Harbin Institute of Technology and received bachelor and master degree in

Engineering Mechanics in 2006 and 2008, respectively. During the period, she remained on the

department’s honor list, with scholarships awarded each academic year. She joined Northeastern

University in Boston, Massachusetts, USA to pursue Ph.D degree in 2008. She is a member of

American Society of Mechanical Engineers (ASME), Material Research Society (MRS) and

American Physical Society (APS). She received her PhD degree in Mechanical Engineering from

Northeastern University in September 2012.

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