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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