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Microscopic and molecular fundamentals of crazing

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Crazing in Polymers

Part of the book series: Advances in Polymer Science ((POLYMER,volume 52-53))

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Abbreviations

A′, mc :

prefactor and exponent of K in crack growth equation

\(\overline D\) :

average craze fibril diameter \(\overline D\)=〈D2〉/〈D〉

〈D〉:

mean fibril diameter

〈D2〉:

mean square fibril diameter

D0 :

average fibril spacing as well as average fibril diameter before deformation (diameter of the “phantom” fibril cylinder)

D0*, ∇σ0*:

fibril spacing of fastest growing of craze interface and gradient of hydrostatic stress corresponding to this spacing

E*:

an effective Young's modulus of the polymer which equals Young's modulus for plane stress and E/(1−ν2) for plane strain

G, G 0N :

shear modulus, shear modulus level of the rubbery plateau

GIc :

critical strain energy release rate of crack (fracture toughness)

K:

stress intensity factor for crack

M:

axis of molecular orientation

M0 :

molecular weight of stiff units along chain

Mc :

critical molecular weight below which crazes do not form

Me :

entanglement molecular weight

Mn, Mn′:

number average molecular weights of chains in the glass before crazing and in the craze fibrils respectively

Mw′:

weight average molecular weight of chains in the craze fibrils

NA :

Avogadro's number

P:

probability that there is at least one entangled chain in each entanglement length along a fibril

Q:

SAXS invarient

R:

radius of curvature at craze front

R:

universal gas constant

S, St :

tensile stress on craze surface and at the craze tip respectively averaged over several fibrils

ΔS(x):

self stress of craze necessary to produce its measured displacement profile w(x)

TA :

absolute temperature

T(x):

craze thickness profile

Tg :

glass transition temperature

U*:

activation energy for chain scission

U:

energy needed to break a single primary polymer chain bond

V:

crack growth velocity

χ:

volume fraction of polymer in a blend

Y:

tensile yield stress

Z:

number of entanglement transfer lengths along a fibril

a:

half length of craze (or of crack plus craze)

ai :

initial radius of a penny shaped crack

c:

crack half length

d:

root mean square end-to-end distance of a chain of molecular weight Me, entanglement mesh size

f:

force on a single entangled chain in a fibril

i(s):

scattered X-ray intensity

k′:

Porod's law constant from high angle tail of SAXS intensity

k0 :

rate constant prefactor

kb :

rate of bond scission

l:

entanglement transfer length along fibril

l0 :

average projected length of stiff units along chain

lact :

activation distance

le :

chain contour length between entanglements

n:

non Newtonian flow law exponent

nc :

refractive index of craze

p:

probability that at least one entangled chain is left umbroken in a given entanglement transfer length

\(\bar p\)(ζ), ζ:

Fourier transform of craze selfstress and the transform variable (spatial frequency) ζ

q:

the ratio of entangled chain density before to that after crazing (=ν′E/νE)

r:

radius of curvature of steady state finger at craze tip

rd :

chain disentanglement rate

s:

magnitude of the X-ray scattering vector

s0 :

deviatoric stress

v:

velocity of craze — bulk polymer interface in craze thickening

w(x):

displacement profile of craze interfaces

x, z:

coordinates in craze plane normal to and parallel to the craze front

2θ:

X-ray scattering angle

Λ:

wave length of sinusoidal perturbation of craze front

Λc :

fastest growing wave length of craze front perturbation

Λm :

minimum wavelength of sinusoidal craze front perturbation that can grow

Λs :

fastest growing spacing of steady state finger structure at craze tip

Φcraze, Φfilm, Φhole :

optical density on TEM micrograph of craze, solid film and hole through film

Γ:

energy to create new surface at craze tip or craze-bulk interface including an energy of primary chain rupture

α:

proportionality constant between finger spacing and finger radius of curvature (r=αΛs)

α(x):

linear dislocation density of continuously distributed dislocations of Burger's vector b necessary to give the same displacement profile as the craze

γ:

van der Waal's surface energy

\(\dot \varepsilon\) :

equivalent tensile strain rate

\(\dot \varepsilon\)F, σF:

material parameter in non-Newtonian flow law \(\dot \varepsilon\)=\(\dot \varepsilon\) F(σ/σF)n

εc :

critical uniaxial strain for crazing

ζ:

amplitude of sinusoidal perturbation of craze front

λ:

extension ratio of craze fibrils relative to undeformed polymer glass

λDZ :

extension ratio of shear deformation zone

(λmax)∥, (λmax)⊥ :

maximum extension ratio of a single entangled chain parallel and prependicular to the preorientation direction

λmax :

maximum extension ratio of a single entangled chain (=le/d)

λnet :

maximum extension ratio of entangled chain network

λp :

extension ratio achieved in uniaxial preorientation above Tg

λx :

X-ray wavelength

λ∥, λ⊥ :

extension ratio measured in craze fibrils grown by applying tensile stress parallel, and perpendicular, to the axis of molecular orientation M

ν:

Poisson's ratio

νE :

density of chains between entanglement junction points

ν′E :

density of chains between entanglement junction points after geometrically necessary entanglement loss

ν sE , ν dE :

density of entangled chains lost by scission or disentanglement respectively when forming a fibril surface

ϱ:

polymer density

β, β*:

coefficient of proportionality between average hydrostatic stress (σ0) and tensile stress S or St at craze interface or craze tip respectively (i.e. (σ0)m=βS; (σ0)m=β*St)

σ:

equivalent tensile stress

σ0 :

hydrostatic stress (negative pressure)

δσ0 :

increment of hydrostatic stress

∇σ0, s*:

gradient of hydrostatic stress and its sign

(σ0)m :

average hydrostatic stress ahead of craze tip or craze interface

(σ0)s :

hydrostatic stress at the surface of the void “ceiling” between fibrils at craze interface

(σ0)t :

hydrostatic stress at craze void finger tip

σD :

lower yield stress

σf :

true tensile stress in the craze fibrils

σij :

components of stress tensor

σ∞ :

tensile stress normal to the craze at large distances from the craze plane

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  1. Department of Materials Science & Engineering and the Materials Science Center, Cornell University, 14853, Ithaca, NY, U.S.A.

    Edward J. Kramer

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Kramer, E.J. (1983). Microscopic and molecular fundamentals of crazing. In: Kausch, H.H. (eds) Crazing in Polymers. Advances in Polymer Science, vol 52-53. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0024055

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