Mechanics of Soft Materials Tuesday and Thursday L13, 2:00-3:30 PM - - PowerPoint PPT Presentation

Mechanics of Soft Materials

Tuesday 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 3.2 Aluminum 69 Bone 9 Brasses 100 - 125 Bronzes 100 - 125 Reinforced Plastic 150 Concrete 30 Diamond 1,050 - 1,200 Glass 50 - 90 Magnesium 45 grain) 11 Polycarbonate 2.6 Polyethylene HDPE 0.8 Terephthalate PET 2 - 2.7 Polyimide 2.5 Polypropylene 1.5 - 2 Polystyrene 3 - 3.5 Silicon Carbide 450 Titanium Alloy 105 - 120 Tungsten 400 - 410 Tungsten Carbide 450 - 650 Wrought Iron 190 - 210 Nylon 2 - 4 Rubber 0.01 - 0.1 Gels 10-6 - 10-3 Material

Elastic Viscoelastic Poroelastic Ductile Viscoplastic Gaskets Sealants Adhesives Skin Soft Patterning Solid lubricants Prosthetic devices Drug delivery devices Packaging Materials Components of automobile Soft robotic components Artificial Tissue Functional surfaces Extracellular Matrix

 Hierarchical structure  Strong and reusable adhesion  Adheres to almost all surfaces  Self-cleaning  Does not leave any residue  Easy release during locomotion

Gorb 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 protein

molecules at interface

• Crosslinking of these molecules leading

to a sticky glue

Inspiration from naturally occurring Adhesives

70 gm

50 100 150 200 250 300 1:5 2:5 3:5

24 hr 24 hr 24 hr 24 hr 24 hr 24 hr 4 days 24 hr

Vitamin C solution-gelatin volume ratio Rate of Vitamin C release (μg/cm2/day) c

A novel adhesive patch satisfies these requirements

Rhodnius Prolixus (kissing bug)

Blood filled vessels

Wigglesworth, 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 I nsects:

Attachment pad of Tettigonia viridissima

Air Pockets at the Adhesive Pads of I nsects

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

AS: Air sack CL: Epidermal cell layer EXO: Rod containing exo- cuticle of the pad HM: Haemolymph TD: Tendon of the claw flexor mussle TK: Tanned cuticle

Peeling off a Microfluidic Adhesive

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

1 2 3 4 5 10 15 20 25

(X 10-

2)

M, Nm/m ∆, mm

a

G, J/m2

7 8 3 5 4 6 2 1 0.0 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

m h µ , m d µ ,

Peeling Torque:

570 : 5

530

750 : 7

710

800 : 8

710

1200 : 9

800

600 : 6

530 1200 : 10

1090

h = 300 µm d = 50 µm

25-30 times enhanceme nt in adhesion 5 6 7

peeled

A d F G ∫ ∆ = .

F

Flexible contacting plate

1

d h

2

d

1

s

Adhesive film Substrate

1

d

2

d

1

s s t

200 μm

air air air

• il
• il
• il

Asymmetry I nduced by Pair of Embedded Channels:

Differently filled with wetting liquid

t = 50 μm, s1 = 15 μm, d = 550 μm

Majumder et al Soft Mat., 2012

10 20 200 400 600 800 1000 1200 10 20 200 400 600 800 1000 1200 10 200 400 600 800 1000 1200 10 20 200 400 600 800 1000 1200

( )

μm x

( )

μm δ

Dynamic Change in Surface Profile

During Separation of Adherent:

Case 1 peel direction

1

s

120

0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2

150

( )

60 μm

1

= s

( )

2

J/m G

Case 2 Case 3 Case 4

Adhesion Between a Flexible Plate on a Layer of Adhesive Bonded to a Rigid Substrate: Elastic film Rigid substrate Spacer

x z

Flexible plate a Contact line

∆ Adhesive: Elastic, Incompressible, Thin Adherent: Thin, Flexible & Rigid Plate

) (

zz yy xx x

u u u p + + = µ ) (

zz yy xx y

v v v p + + = µ ) (

zz yy xx z

w w w p + + = µ

= + +

z y x

w v u

Stress Equilibrium Relations: I ncompressibility relation: u, v, w are displacements in the x, y and z direction respectively

= + z w x u ∂ ∂ ∂ ∂

Plane Strain Approximation:

Stress equilibrium relation Incompressibility relation

= = =

zy xy yy

e e e

        ∂ ∂ + ∂ ∂ = ∂ ∂         ∂ ∂ + ∂ ∂ = ∂ ∂

2 2 2 2 2 2 2 2

z w x w z p z u x u x p µ µ

( ) ( )

        ∂ ∂ + ∂ ∂ = ∂ ∂         ∂ ∂ + ∂ ∂ = ∂ ∂

2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 1 Z W h X W L h Z P h L h Z U h L X U L X P L L h µ µ µ µ

2 2 2 2 2 4 2 2 2 2 2 2 2

≈ ∂ ∂ + ∂ ∂ = ∂ ∂ ∂ ∂ ≈ ∂ ∂ + ∂ ∂ = ∂ ∂ Z W X W Z P Z U Z U X U X P ε ε ε

2

. , . , . , . ε µ ε = = = = = = p L h h W w L U u h Z z L X x

Dimensionless Quantities: Dimensionless Stress-Equilibrium relation: Lubrication Approximation:

1 << ε

= =

= = h z zz h z xz

σ σ

at 0 < x < a

Boundary Conditions for Film:

ψ = = = = = = = = ) ( , ) ( ) ( ) ( h z w h z u z w z u

4 4

) ( dx d D h z p ψ = =

at x<0

Elastic film

x

z

( )

z x w ,

( )

z x u ,

= z h z =

2 2

= ∂ ∂ ∂ ∂ = ∂ ∂ z p z u x p µ

x p h A B B Az z x p u ∂ ∂ − = = + + ∂ ∂ = µ µ 2 , 2 1

2

( )

zh z x p u − ∂ ∂ =

2

2 1 µ

( )

zh z x p x u z w − ∂ ∂ − = ∂ ∂ − = ∂ ∂

2 2 2

2 1 µ C h z z x p w +         − ∂ ∂ − = 2 3 2 1

2 3 2 2

µ

6 6 3

12 x Dh ∂ ∂ = ψ µ ψ

Integration Boundary Condition Incompressibility relation Integration Boundary Condition

Equation for Plate:

6 6 3

12 x Dh ∂ ∂ = ψ µ ψ

4 4

x ∂ ∂ = ψ a x < < < x

Boundary Conditions for Plate: (i) (ii) (iii) (iv) (v) (vii)

• r

(vi)

= ∂ ∂

= x

x p

+ = − =

=

x x

ψ ψ

F dx d D

a x

= −

= 3 3ψ

+ = − =

∂ ∂ = ∂ ∂

2 2 2 2 x x

x x ψ ψ

+ = − =

∂ ∂ = ∂ ∂

x x

x x ψ ψ

+ = − =

∂ ∂ = ∂ ∂

3 3 3 3 x x

x x ψ ψ

∆ =

=a x

ψ

2 2

= ∂ ∂

=a x

x ψ

= = =

−∞ = −∞ = −∞ = x xx x x x

ψ ψ ψ

(viii-x)

( ) ( ) ( ) ( ) ( ) ( )

x h F z h z z x w x kh F h z z z x u

2 3 2 1 3

' 2 3 , ' 6 , φ φ − = − =

( ) ( ) ( )

                + +         + + =                 −         + + = 2 3 cos 2 2 3 sin 3 2 3 2 3 cos 2 3 sin 3 4 3

2 2 2 2 2 1

kx ak kx ak e ak e x kx ak kx ak e ak e x

kx kx kx kx

φ φ

Displacement Field in Adhesive Film:

1 −

k ' F

and have dimension of length and are constants

Displacement of Plate and Normal Stress at I nterface: ( ) ( )

                + +         + + = 2 3 cos 2 2 3 sin 3 2 3 '

2 2

kx ak kx ak e ak e F x

kx kx

ψ

( ) ( )

, 2 9 12 6 3 '

3 2

ak ak ak F + + + ∆ =

6 1 3 1

12         =

−

µ Dh k

'

F ψ

adhesive film rigid substrate flexible adherent contact line (x = 0) h a F

( )

z x w ,

( )

z x u , x y z z = 0 0.4 1.0 1.6 0.99 1.00 1.01 5 10 15 20 25 30

x (mm) h h µ µ

O

0.4 1.0 1.6 0.99 1.00 1.01 5 10 15 20 25 30

x (mm) h h µ µ

O’

Adhesive with Spatially Varying Thickness and Modulus

In-phase variation of thickness and modulus Out-of-phase variation of thickness and modulus

( ) ( ) ( ) ( )

x k h x h x h

h h sin

1 δ φ + = =

( ) ( )

( ) ( )

x k x f x

µ µ

δ µ µ µ sin 1 + = =

Ghatak, Phys. Rev. E, 2010

0.35 0.40 0.45 0.50 2 π − 2 π π π −

l

φ

Mmax (Nm/m) (b)

0.20 0.25 0.30 0.35 0.40 0.45 0.50 5 10 15 20 M (Nm/m) ∆ (mm)

Mmax (a) 1 2 3 4

l

φ

µ µ h h

0.4 1.0 1.6 0.99 1.00 1.01

Adhesive with Phase Lag Between Thickness and Modulus Variation:

Non-monotonic variation in torque with phase lag

∆Mmax

. =

l

φ 2 π π 2 π −

1 2 3 4

, 25 . 1 q k k

h =

=

µ

9 . =

µ

δ , 005 . =

h

δ

1.Brief Introduction: Total 3 Definition of strain, strain tensor, stress, stress tensor, Saint Venant’s principle 1 Hooke’s law, stress equilibrium relations 1 One dimensional stretching of a rod 1 2.Solid bodies in contact with and without interactions: Total 9 Line loading of an elastic half space, distributed loading 1 Axisymmetric loading of an elastic half space 1 Normal contact of elastic solids: 2 Hertzian theory: 1 Contact with adhesion, JKR theory 3 Compression of an elastic layer between two parallel plates 1 3.Equilibrium of rods and plates: Total 12 Equations of equilibrium of rods 2 Euler’s buckling instability 1 Twisting instability of rods 1 Equation of equilibrium for a thin bent plate 1 longitudinal deformation of plates 1 large deflection of plates 1 Contact of two rigid or flexible Adherents 3 Analysis of wrinkling instability 1 Elasticity of an interfacial particle raft 1

Typical Content of Course:

4.Nonlinear elasticity: Total 8 Molecular approach to rubber elasticity 1 Neo-Hookean elasticity 1 Analysis of large deformation of an incompressible elastic material 4 Inflation of a balloon 1 Cavitation in crosslinked networks 1 5.Mechanics of cell wall: Total 8 Entropic elasticity-stretching, bending and twisting, persistence length 2 Mechanics of cellular filaments; 2D and 3D networks in cell 2 Polymerization and the generated force 2 biomembranes, 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.

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