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A model for the compressive buckling of extended chain polymers

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Abstract

A model for the compressive buckling of an extended polymer chain is presented. The application of classical elastic instability analysis to an idealized polymer chain reveals that the bending rigidity and critical buckling loads for a chain are proportional to the force constants for valence bond angle bending and torsion. Highly oriented polymer fibres are treated as a collection of elastic chains that interact laterally. The critical stresses to buckle this collection of chains are calculated following a procedure developed to predict the compressive strengths of fibre-reinforced composites. This buckling stress is predicted to be equal to the shear modulus of the fibres and is the limiting value of compressive strength. Comparison of experimental and predicted values shows that the theory overestimates the compressive strength, but that there is a correlation of shear modulus with axial compressive strength. Consideration of flaws in both the theory and the material indicate that the compressive strength should be proportional to either the shear modulus or shear strength of the fibres.

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Abbreviations

P :

axial compressive load (force)

P cr :

critical buckling load (force)

M,M i :

bending moments

l :

length of a link

p :

number of links

k :

elastic hinge constant

α,α i :

angular rotation of hinges

L :

overall chain or column length

v,v i :

lateral deflection of buckled chain or column

x, y, z :

Cartesian coordinate axes

E :

Young's modulus of isotropic column

I :

moment of inertia

a ij :

matrix coefficients

A p :

coefficient for exact buckling loads of chains

ΔT :

energy change due to work of external load on buckled column or chain

ΔU 1 :

bending strain energy change of buckled column or chain

ΔU 2, ΔU e2 , ΔU s2 :

strain energy changes in elastic foundation, where e refers to extension mode buckling and s refers to shear mode buckling

E t :

transverse modulus

G :

longitudinal shear modulus

b :

dimension associated with chain packing

A :

cross-sectional area per chain (=b 2)

f(x):

curve fitted to shape of buckled chain

m,n,r :

integers

a n :

coefficients of trigonometric series

ε y :

normal strain iny-direction

σ y :

normal stress iny-direction

γxy :

shear strain inxy plane

τxy :

shear stress inxy plane

u x :

displacement inx-direction

u y :

displacement iny-direction

V :

volume

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Authors and Affiliations

  1. Materials Research Laboratory, Polymer Science and Engineering Department, University of Massachusetts, 01003, Amherst, Massachusetts, USA

    Steven J. DeTeresa, Roger S. Porter & Richard J. Farris

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  1. Steven J. DeTeresa
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  2. Roger S. Porter
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DeTeresa, S.J., Porter, R.S. & Farris, R.J. A model for the compressive buckling of extended chain polymers. J Mater Sci 20, 1645–1659 (1985). https://doi.org/10.1007/BF00555268

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