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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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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