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Elastic Deformation

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(1)

Chapter 19

Elastic Deformation

Stress and strain

Elastic deformation of polymers

(2)

Mechanical behavior of polymers

As structural

(not functional)

materials, polymers are

resilient but weak to heat

Metals are strong but weak to corrosion/fatigue: Ceramics stiff but brittle.

Polymers are light, cheap, high strength/wt, tough, corrosion- resistant, insulating, low friction, ---

mechanical property

mechanical response of a material to the applied stress [load]

response of a polymer depends on

chemical structure ~ PE, PS, ---

physical structure ~ crystallinity, aged, ---

magnitude of stress

state of stress ~ uniaxial, bending, ---

temperature

time [rate of loading]

Ch 19 sl 2

(3)

response to the magnitude of stress

upon small stresses

elastic deformation

viscous deformation [flow] after Chapt 21

viscoelastic deformation Chapt 20

upon large stresses

plastic deformation Chapt 22

yielding [ductile] or crazing [brittle]

failure [fracture] Chapt 23

response of the polymer chain

deformations of the bond lengths and angles

uncoiling of the chains

slippage of the chains

scission of the chains

e s

brittle

ductile yield

elastomeric

0.1 1

Ch 19 sl 3

Chapt 21

(4)

Stress

stress

load [force] / area

s = F/A [N/m2 = Pa]

stress components

i = plane ㅗ to i-axis

j = direction

3 normal stresses [수직응력]

6 shear stresses [비틀림응력, t]

by equilibrium (no rotation), 6  3

Fig 19.1 p470

Ch 19 sl 4

sign of stress?

(+) for tensile

(−) for compressive

(5)

principal stresses and principal axes

if all t’s are 0

normal stresses are principal stresses

x1, x2, and x3 are principal axes

two axes on free surface are principal

hydrostatic and deviatoric stress

hydrostatic pressure

mean normal stress sm = (−)p

deviatoric stress (component)

Ch 19 sl 5

s12 = 0

(6)

Strain

strain

displacement by load

3 normal strains

de11 = dL/L

3 shear strains

g21 = dx1/dx2 + dx2/dx1

g21 = g12 for equilibrium

tensor strain e12 = ½ g12

L0 dL

g

Ch 19 sl 6

x2

x1 x3

gyx

x y

(7)

e and e: in this textbook pp471-473

e and e: more generally

engineering [nominal] strain e

= DL/L

0

true strain de = dL/L

e = ln [L/L0] = ln [(L0+DL)/L0] = ln [1+(DL/L0)]

engineering

[nominal] stress s = F/A

0

true stress s = F/A

true ≈ engineering for small s and e (< 1%)

Ch 19 sl 7

general strain = rotation + pure strain

L0 DL

ee ss

(8)

State of stress [testing geometry]

uniaxial tension

s11 = s > 0 , all others = 0

uniaxial compression

s11 = s < 0, all others = 0

simple shear ~ torsion

s12 [s6]= t, all others = 0

plane stress

s33 = s31 = s23 = 0

plane strain

s22 = n (s11 + s33)  e22 = 0

bending or flexure

plane e

x2

x1 x3

Ch 19 sl 8

plane s

(9)

Stress-strain relationship

constitutive equation

s = c e c ~ stiffness [modulus]

e = s s s ~ compliance

81  36  21  13  9  5  2

2 constants for isotropic solids

Ch 19 sl 9

equilibrium symmetry in material

p474

(10)

Elastic constants

uniaxial tension test [UTT]

s11 = s > 0 , all other stresses = 0

s1 = E e1

Hooke’s law (for UTT)

E = Young’s modulus [영탄성률, 引張彈性率]

modulus = resistance to deformation, stiffness

e = D s

D = (tensile) compliance [순응도]

Poisson’s ratio n

definition ~ n = – e2 / e1 > 0

for rubbers, n = 0.5 ~ no volume change

for plastics, n ~ 0.4

for metals, n < 0.4 (~ 0.33)

x2

x1 x3

L0 DL

s s

Ch 19 sl 10

(11)

shear deformation

s6 [s12] = t, all other stresses = 0

t6 = G g6

Hooke’s law (for simple shear)

G = shear modulus [비틀림탄성률]

g = J t

J ~ shear compliance

g

t

Ch 19 sl 11

x2 x1 x3

(12)

when all stresses are present

e1 = s1/E – n e2 – n e3

= s1/E – n s2/E – n s3/E

= (1/E) [s1 – n (s2 + s3)]

e2 =

e3 =

g6 = t6/G

g4 =

g5 =

generalized Hooke’s law

for isotropic materials (with one E and one n)

no interaction betw normal and shear

e2 = s2/E

Ch 19 sl 12

p476

(13)

volume change

hydrostatic pressure (−)p

= (−) mean normal stress sm (more common)

dilatation or volume strain D

p = K D K = bulk modulus (= resistance to volume change)

D = b p b = compressibility

relations between elastic constants

Only 2 of 4 (E, G, K, n) are independent.

e.g., E = 3G, K =  for elastomers

s2

s1 s3

Ch 19 sl 13

p471

p475

x2

x1 x3

p475-476 or

(14)

Stress-strain behavior

linear elastic response of isotropic solid in UTT

s(1) = E e(1) (Hooke’s law)

e2 = − n e1

viscoelastic ~ time-dependent

s(t) = E(t) eo

nonlinear ~ strain-level-dependent

s(eo) = E(eo) eo

s(t, eo) = E(t, eo) eo ~ nonlinear viscoelastic

anisotropic solid (fiber, composite) ~ orientation-dependent

E11 ≠ E22; n12 ≠ n31

It is hard to express the real mechanical response with a constitutive equation.

s

e

linear elastic

linear viscoelastic nonlinear elastic

Ch 19 sl 14

(15)

Deformation of polymers

amorphous [glassy] polymers

below Tg ~ glassy (PS, PC)

above Tg ~ rubbery (not useful)

semicrystalline polymers

below Tg ~ glassy + crystal (PET)

above Tg ~ rubbery + crystal (PE)

crosslinked polymers

below Tg ~ glassy (epoxy)

above Tg ~ elastomer (PBD)

crosslinked

Tm Tg

E (Pa)

106 109

Temp semicrystalline amorphous

Fig 19.4 ~ wrong!!

brittle ductile

Table19.1

Ch 19 sl 15

(16)

Deformation of polymer single chain

a (zigzag) chain in crystal

change in length

k ~ force constant (of bond length and angle)

estimated by raman, IR, --

A ~ area/chain

Young’s modulus

estimated for PE, E = 180 GPa

rather low, due to intermolecular interaction? should be small

Ch 19 sl 16

Fig 19.5

errors p479

(17)

Deformation of polymer crystal

chain-direction modulus of crystal

tested mechanically

for single crystal

measured with XRD (of fiber)

c = lattice parameter

by < 50%

due to amorphous regions

Ch 19 sl 17

Fig 19.7 before and after deform’n

drawn fiber Fig 19.8

Fig 19.6

460 GPa

(18)

E

c

zigzag > helix

PE > POM, PTFE

linear > side-group

PE > PP (larger A)

polymers are weak?

metals strong?

no, comparable

Kevlar® , Spectra® ~ 150 GPa

PPTA, PBO > 200 GPa

E of steel ~ 150 GPa

specific modulus [E/wt]

is higher!

Ch 19 sl 18

(19)

Deformation of semicrystalline polymers

semicrystalline polymer

= composite (of amorphous and crystallite) material

two groups

at above Tg

PE, PP, -- at room temp

crystallites in the rubbery matrix

E(crystal) >> E(matrix)

E increases with Xc

at below Tg

PET, nylon, -- at room temp

crystallites in the glassy matrix

E(crystal) ≈ E(matrix)

Fig 19.9 for PE

Ch 19 sl 19

(20)

modulus

serial [Reuss] model

s same ~ e additive

lower bound

parallel [Voigt] model

e same ~ s additive

upper bound [rule of mixture]

Ch 19 sl 20

p, m

- polymer, monomer for polydiacetylene - crystal, amorphous

for semicrystalline

Fig 19.9

PE

Fig 19.10

polydiacetylene

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