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Fundamental processes of craze growth and fracture

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Crazing in Polymers Vol. 2

Part of the book series: Advances in Polymer Science ((POLYMER,volume 91/92))

Abstract

Recent advances in quantitative microscopy and low-angle electron diffraction methods have made it possible to probe the fundamental processes of craze fibril formation and craze fibril breakdown. Both the scale of fibrillation within the craze and the magnitude of the crazing stress may be successfully described by a variant of the Taylor meniscus instability process. Within this framework, the key parameter in governing craze growth is the craze surface energy Γ. In turn Γ reflects the mechanism by which entangled strands are lost (through either chain scission or chain disentanglement) in producing the surfaces of the craze fibrils. A new model, which describes the temperature, strain rate and molecular weight dependence of the crazing stress is presented. This approach provides a clear rationale for the hitherto confusing data on crazing to shear deformation transitions in a wide variety of polymers. Moreover, the modification of the polymer network during craze formation has important implications for craze breakdown. In particular, at low temperatures where chain scission is the dominant process, the molecular weight of the polymer in the fibrils is markedly reduced. A molecular description of craze fibril breakdown based on microscopic measurements of the scale of the fibrillation in the craze and the statistics of craze fibril breakdown is proposed. Satisfactory agreement between the predictions of this model and the experimental data for a variety of glassy polymers is found.

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Abbreviations

D:

average craze fibril diameter

Do :

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

F:

force on the craze fibril

G oN :

shear modulus of the rubbery plateau

Me :

entanglement molecular weight

Mn :

number average molecular weight of polymer before crazing

M′n :

number average molecular weight of polymer in fibrils after crazing

Mo :

molecular weight of mer repeat unit

Mv :

viscosity average molecular weight of polymer before crazing

Mw :

weight average molecular weight of polymer before crazing

M(i):

the number average molecular weight of a fibril in which i strands have survived

Na :

avogadro's number

Pdis(i):

disentanglement probability of a group of i strands in a fibril following the initial chain scission

Psurvival(i):

Probability that i strands in a fibril survive following the initial chain scission in forming the fibril

Ps(0):

probability that all strands in a fibril break

Psd :

probability that a given fibril will fail by scission followed by disentanglement of all of its remaining i strands

Psdf :

fibril failure probability Psd of film unaffected by stress concentrating particles

Psdc :

fibril failure probability Psd in the regions affected by the stress concentration around dust particles

R:

root mean square end-to-end distance of chain with mass M

S:

average tensile stress on craze interface

S1 :

average craze interface stress at a plastic strain of 1

Tg :

the glass transition temperature

U:

energy required to break a single primary bond along polymer backbone

V:

risk volume (i.e. total volume of polymer converted to craze fibrils)

Vb :

volume in which one craze fibril breakdown will be encountered at a reference stress σb

V0 :

initial volume of polymer within one grid square

a:

bond distance

ao :

effective area of polymer chain

ap :

persistence length of polymer strand

d:

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

〈f〉:

average force per effective strand (=F/ne)

fb :

breaking force of polymer strand

fd :

average force required to disengage polymer molecule from its surroundings

fm :

force on a mer in a polymer molecule

h:

thickness of strain softened polymer layer at the active zone

le :

chain contour length between entanglements

lo :

average projected length of mer units along chain

n:

non-Newtonian flow law exponent

vm :

monomer velocity relative to its surroundings

x:

fractional distance along stretched polymer molecule from one of its ends

xc :

critical fractional distance along stretched polymer molecule beyond which the force in the molecule exceeds the breaking force fb

Γ:

total craze fibril surface energy (tension)

Δv:

differential velocity between “upper” and “lower” sets of entanglements during drawing into the fibril

Λ:

density of “intrinsic weak spots” in polymer film

β:

coefficient of proportionality between average hydrostatic stress (σo) and tensile stress S at craze interface

n(x):

number of monomers from end of the chain at x

ne :

mean number of effectively entangled strands that survive craze fibrillation

no :

total number of entangled strands in the “phantom fibril” from which the craze fibril is produced

pc, pb, pf :

cumulative number fraction of grid squares that exhibit craze formation, craze fibril breakdown, and catastrophic fracture, respectively

q:

probability that a given entangled strand survives craze fibril formation

tdis(i):

disentanglement time of i strands in a fibril that survive fibril formation

v:

craze interface velocity

vf :

volume fraction of polymer within craze

vλ :

craze interface velocity required to produce a constant λ (due to chain disentanglement) above the “natural” extension ratio of the craze

γ:

van der Waals (intermolecular) surface energy

ε:

total tensile strain

\(\dot \varepsilon\) :

equivalent tensile strain rate

εc, εb, εf :

median tensile strains for craze formation, craze fibril breakdown, and catastrophic fracture

\(\dot \varepsilon _f\), σfc:

material parameters in non-Newtonian flow law \(\dot \varepsilon = \dot \varepsilon _f (\sigma /\sigma _{fc} )^n\)

εp :

plastic strain in polymer film (=ε − εc)

εw :

Weibull scale parameter

ζo :

monomeric friction coefficient

λ:

craze fibril extension ratio (=1/vf)

〈λ〉:

average extension ratio in stretch zone

λd :

additional extensional ratio above the (low temperature or high strain rate) “natural” extension ratio of the craze

λDZ :

extension ratio within deformation zone

λlt :

low temperature or high strain rate “natural” extension ratio of the craze

λmax :

theoretical maximum extension ratio for a single strand

νe :

density of strands in the entanglement network

νx :

density of crosslinked strands

ξi :

region adjacent to craze interface in which polymer chains (at high T's and low v's) can disentangle during craze widening

ξ:

probability of disentanglement for a given strand

ϱ:

mass density of polymer

ϱw :

Weibull modulus

σ:

equivalent tensile stress

(σo)m :

average hydrostatic stress ahead of craze tip or craze interface

(σo)s :

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

σy :

polymer yield stress

σyo :

polymer yield stress at \(\dot \varepsilon _y\)

τd :

time for complete disentanglement of polymer chain

τres :

effective residence time of strands near the craze matrix interface during which the monomeric friction coefficient is small enough that they can still disentangle (or in the distance ξi just below it)

ϕ:

volume fraction of the polymer film affected by stress concentrating particles

ω:

exponent on the plastic strain εp that describes the dependence of the drawing stress on εp

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Authors and Affiliations

  1. Department of Materials Science and Engineering and the Materials Science Center, Cornell University, 14853, Ithaca, NY, USA

    Edward J. Kramer

  2. Central Research and Development Department, E. I. duPont de Nemours & Co (Inc), Experimental Station, 19898, Wilmington, DE, USA

    Larry L. Berger

Authors
  1. Edward J. Kramer
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  2. Larry L. Berger
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H. -H. Kausch

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© 1990 Springer-Verlag

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Kramer, E.J., Berger, L.L. (1990). Fundamental processes of craze growth and fracture. In: Kausch, H.H. (eds) Crazing in Polymers Vol. 2. Advances in Polymer Science, vol 91/92. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0018018

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  • DOI: https://doi.org/10.1007/BFb0018018

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  • Print ISBN: 978-3-540-51306-3

  • Online ISBN: 978-3-540-46192-0

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