Chapter 19
Elastic Deformation
Stress and strain
Elastic deformation of polymers
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
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
brittleductile yield
elastomeric
0.1 1
Ch 19 sl 3
Chapt 21
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
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
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
e and e: in this textbook pp471-473
e and e: more generally
engineering [nominal] strain e
= DL/L
0true strain de = dL/L
e = ln [L/L0] = ln [(L0+DL)/L0] = ln [1+(DL/L0)]
engineering
[nominal] stress s = F/A
0true 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
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 ex2
x1 x3
Ch 19 sl 8
plane s
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
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
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
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
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
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
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
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
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
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
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
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