Mechanics of Soft MaterialsTuesday and Thursday

L13, 2:00-3:30 PM

What Are Soft Materials?

Rubber

Skin

Tissue

Paper

Gel … Modulus < 10 MPa

Deformation when subjected to load

Type of load

Geometry: micro-nano structures

Rheological properties

Interfacial properties

Tissue scafold

Soft Materials have modulus of the order of few Pa to few MPa

Shear Modulus(109 N/m2, GPa )

Acrylic 3.2Aluminum 69

Bone 9Brasses 100 - 125Bronzes 100 - 125

Reinforced Plastic 150Concrete 30Diamond 1,050 - 1,200

Glass 50 - 90Magnesium 45

grain) 11Polycarbonate 2.6

Polyethylene HDPE 0.8 Terephthalate PET 2 - 2.7

Polyimide 2.5Polypropylene 1.5 - 2

Polystyrene 3 - 3.5Silicon Carbide 450Titanium Alloy 105 - 120

Tungsten 400 - 410Tungsten Carbide 450 - 650

Wrought Iron 190 - 210Nylon 2 - 4

Rubber 0.01 - 0.1Gels 10-6 - 10-3

Material

ElasticViscoelasticPoroelasticDuctileViscoplastic

GasketsSealantsAdhesivesSkinSoft PatterningSolid lubricantsProsthetic devicesDrug delivery devicesPackaging MaterialsComponents of automobileSoft robotic componentsArtificial TissueFunctional surfacesExtracellular Matrix

Hierarchical structure

Strong and reusable adhesion

Adheres to almost all surfaces

Self-cleaning

Does not leave any residue

Easy release during locomotionGorb et al, J. Micromech. Microeng. 2000, 10, 359–364

Smooth adhesive

Patterned adhesive

Bio-inspired Patterned Adhesives:

Enhancement of adhesion by ~10 times

Ghatak et al, Proc. Roy. Soc. London, A, 460, 2725 (2004)

Crack arrest, crack initiation

Adhesive suitable for dry and wet adhesion and delivery of drugs and nutrients

• Reversible adhesion: Clean adhesion• Good mechanical strength: ability to sustain large deformation• Multi-functionality: adhesion and drug delivery• Biocompatibility and biodegradability• Resistance against particulate contamination• Amenable for easy wash or cleaning

Transdermal Patches Cosmetic patches

Tissue Scar

Tissue adhesive

Gecko inspired Adhesive

• Adhesion on dry and wet substrate• van der Waals interaction in dry state• Swelling of keratin protein of setae

in moist environment

Under water adhesion of mussel

• Release of water repellent proteinmolecules at interface

• Crosslinking of these molecules leadingto a sticky glue

Inspiration from naturally occurring Adhesives

70 gm 0

50

100

150

200

250

300

1:5 2:5 3:524

hr

24 h

r24

hr

24 h

r24

hr

24 h

r

4 da

ys

24 hr

Vitamin C solution-gelatin volume ratio

Rate ofVitamin C

release (μg/cm2/day)

c

A novel adhesivepatch satisfies theserequirements

Rhodnius Prolixus(kissing bug)

Blood filled vesselsWigglesworth, et al, Proc. R. Soc. Lond. B 111, 364-376 (1932).

Climbing organ

Adult Rhodnius could climb the glass walls of the jars … ability to climb smooth surfaces was due to the

existence in the adult insects of a flashy pad situated at the lower end of tibia of

the first two pair of legs.

Blood Vessels in Climbing Organ of Insects:

Attachment pad of Tettigonia viridissima

Air Pockets at the Adhesive Pads of Insects

J. Comp. Physiology A, 2006, 186, 821-831

AS: Air sackCL: Epidermal cell layerEXO: Rod containing exo- cuticle of the padHM: HaemolymphTD: Tendon of the claw

flexor mussleTK: Tanned cuticle

Peeling off a Microfluidic Adhesive

Majumder et al, Science, 318, 258-261, 2007

a

0 1 2 3 405

10152025

(X 10-

2)

M,

Nm/m

∆, mm

a

G,

J/m2

7 83 54 6210.0

0.3

0.6

0.9

1.2

1.5

1.8

9 10

1: smooth adhesive

2: 120 cp

3: 1000 cp

4: 5000 cp

5-10: 380 cp

mh µ, md µ,

Peeling Torque:

570:5 530

750:7 710800:8 7101200:9 800

600:6 530

1200:10 1090

h = 300 µmd = 50 µm

25-30 timesenhancement inadhesion

5

6

7

peeledAdFG ∫ ∆= .

F

Flexible contacting plate

1dh2d1s

Adhesive film

Substrate

1d 2d1s

s

t

200 μm

airair airoiloil oil

Asymmetry Induced by Pair of Embedded Channels:Differently filled with wetting liquid

t = 50 μm, s1 = 15 μm, d = 550 μmMajumder et al Soft Mat., 2012

01020

0 200 400 600 800 1000 1200

01020

0 200 400 600 800 1000 1200

010

0 200 400 600 800 1000 1200

01020

0 200 400 600 800 1000 1200( )μmx

( )μmδ

Dynamic Change in Surface Profile During Separation of Adherent:

Case 1

peel direction

1s1200.0

0.6

1.2

1.8

2.4

3.0

3.6

4.2

150( ) 60μm1 =s

( )2J/mG

Case 2

Case 3

Case 4

Adhesion Between a Flexible Plate on a Layer ofAdhesive Bonded to a Rigid Substrate:

Elastic film

Rigid substrate

Spacer

x

z

Flexible plate

a

Contact line

∆

Adhesive: Elastic, Incompressible, ThinAdherent: Thin, Flexible &

Rigid Plate

)( zzyyxxx uuup ++= µ)( zzyyxxy vvvp ++= µ

)( zzyyxxz wwwp ++= µ

0=++ zyx wvu

Stress Equilibrium Relations:

Incompressibility relation:

u, v, w are displacements in the x, y and z direction respectively

0=+zw

xu

∂∂

∂∂

Plane Strain Approximation:

Stress equilibrium relation

Incompressibility relation

0=== zyxyyy eee

∂∂

+∂∂

=∂∂

∂∂

+∂∂

=∂∂

2

2

2

2

2

2

2

2

zw

xw

zp

zu

xu

xp

µ

µ

( )

( )

∂∂

+∂∂

=∂∂

∂∂

+∂∂

=∂∂

2

2

2

2

22

2

2

22

2

2

11

11

ZW

hXW

Lh

ZP

hLh

ZU

hL

XU

LXP

LLh

µµ

µµ

02

22

2

24

2

2

2

2

2

22

≈∂∂

+∂∂

=∂∂

∂∂

≈∂∂

+∂∂

=∂∂

ZW

XW

ZP

ZU

ZU

XU

XP

εε

ε

2

.,.,.,.

εµ

ε

=

=====

pLh

hWwLUuhZzLXxDimensionlessQuantities:

DimensionlessStress-Equilibriumrelation:

Lubrication Approximation:

1<<ε

0==== hzzzhzxz σσat 0 < x < a

Boundary Conditions for Film:

ψ========)(,0)(0)0()0(

hzwhzuzwzu

4

4

)(dxdDhzp ψ

==at x<0

Elastic film

x

z ( )zxw ,

( )zxu ,0=z

hz =

0

2

2

=∂∂

∂∂

=∂∂

zp

zu

xp µ

xphAB

BAzzxpu

∂∂

−==

++∂∂

=

µ

µ

2,0

21 2

( )zhzxpu −

∂∂

= 2

21µ

( )zhzxp

xu

zw

−∂∂

−=∂∂

−=∂∂ 2

2

2

21µ

Chzzxpw +

−

∂∂

−=232

1 23

2

2

µ

6

63

12 xDh

∂∂

=ψ

µψ

Integration

Boundary Condition

Incompressibility relation Integration

Boundary Condition

Equation for Plate:

6

63

12 xDh

∂∂

=ψ

µψ

4

4

0x∂

∂=

ψ ax <<0

0<x

Boundary Conditions for Plate:(i) (ii)

(iii) (iv)

(v)

(vii) or

(vi)00

=∂∂

=xxp

+=−==

00 xxψψ

FdxdD

ax

=−=

3

3ψ

+=−=∂∂

=∂∂

02

2

02

2

xx xxψψ

+=−= ∂∂

=∂∂

00 xx xxψψ

+=−=∂∂

=∂∂

03

3

03

3

xx xxψψ

∆==ax

ψ

02

2

=∂∂

=axxψ

0===−∞=−∞=−∞= xxxxxx

ψψψ(viii-x)

( ) ( ) ( )

( ) ( ) ( )xh

Fzhzzxw

xkh

Fhzzzxu

23

2

13

'23,

'6,

φ

φ

−=

−=

( )

( ) ( )

++

++=

−

++=

23cos2

23sin

323

23cos

23sin

343

222

221

kxakkxakeakex

kxakkxakeakex

kxkx

kxkx

φ

φ

Displacement Field in Adhesive Film:

1−k 'Fand have dimension of length and are constants

Displacement of Plate and Normal Stress at Interface:

( ) ( )

++

++=

23cos2

23sin

323

'22 kxakkxakeake

Fx kxkxψ

( ) ( ),

291263' 32 akakak

F+++

∆=

6131

12

=−

µDhk

'Fψ

adhesive film

rigid substrate

flexible adherentcontact line

(x = 0)

0h

a

F

( )zxw ,

( )zxu ,

xy

z

z = 0

0.4

1.0

1.6

0.99

1.00

1.01

0 5 10 15 20 25 30x (mm)

0hh

0µµ

O

0.4

1.0

1.6

0.99

1.00

1.01

0 5 10 15 20 25 30x (mm)

0hh

0µµ

O’

Adhesive with SpatiallyVarying Thickness andModulus

In-phase variation of thickness and modulus

Out-of-phase variation of thickness and modulus

( ) ( ) ( )( )xkhxhxh hh sin100 δφ +==

( ) ( ) ( )( )xkxfx µµδµµµ sin100 +==

Ghatak, Phys. Rev. E, 2010

0.35

0.40

0.45

0.50

2π− 0 2π ππ−lφ

Mmax(Nm/m)

(b)

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 5 10 15 20

M(Nm/m)

∆ (mm)

Mmax(a)

1

2

3

4

lφ

0µµ

0hh

0.4

1.0

1.6

0.99

1.00

1.01

Adhesive with Phase Lag Between Thickness andModulus Variation:

Non-monotonic variation in torque with phase lag

∆Mmax

0.0=lφ

2ππ

2π−1234

,25.1 qkk h ==µ 9.0=µδ,005.0=hδ

1.Brief Introduction: Total 3Definition of strain, strain tensor, stress, stress tensor, Saint Venant’s principle 1Hooke’s law, stress equilibrium relations 1One dimensional stretching of a rod 1

2.Solid bodies in contact with and without interactions: Total 9Line loading of an elastic half space, distributed loading 1Axisymmetric loading of an elastic half space 1Normal contact of elastic solids: 2Hertzian theory: 1Contact with adhesion, JKR theory 3Compression of an elastic layer between two parallel plates 1

3.Equilibrium of rods and plates: Total 12Equations of equilibrium of rods 2Euler’s buckling instability 1Twisting instability of rods 1Equation of equilibrium for a thin bent plate 1longitudinal deformation of plates 1large deflection of plates 1Contact of two rigid or flexible Adherents 3Analysis of wrinkling instability 1Elasticity of an interfacial particle raft 1

Typical Content of Course:

4.Nonlinear elasticity: Total 8Molecular approach to rubber elasticity 1Neo-Hookean elasticity 1Analysis of large deformation of an incompressible elastic material 4Inflation of a balloon 1Cavitation in crosslinked networks 1

5.Mechanics of cell wall: Total 8Entropic elasticity-stretching, bending and twisting, persistence length 2Mechanics of cellular filaments; 2D and 3D networks in cell 2Polymerization and the generated force 2biomembranes, membrane undulations 2.

Text book:

• Theory of Elasticity, 3rd edition, by Landau and Lifshitz. Course of theoretical physics, vol-7.• A treatise on the mathematical theory of elasticity by A. E. H. Love.• Contact Mechanics by K. L. Johnson.• Stability problems in applied mechanics by A. K. Mallik and J. K. Bhattacharjee.• Mechanics of the cell by David Boal.• Research Papers published in variety of journals.

Thank You