A rubber band cut from a sheet has an unstretched cross-section of and a natural length of

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Soft Matter at the Thermal Scale, Ch Rubber elasticity, swelling and gelation

Problem

A rubber band cut from a sheet has an unstretched cross-section of

2.0\,\mathrm{mm}\times1.0\,\mathrm{mm}

and a natural length of

8.0\,\mathrm{cm}

. Held stretched by hand to a total length of

20\,\mathrm{cm}

, it pulls back with a force of

1.5\,\mathrm{N}

, read from a spring balance. Treat the band as an ideal affine Gaussian network (incompressible, isotropic) and take

T=298\,\mathrm{K}

. ((a)) Estimate the shear modulus

G

of the rubber. ((b)) Estimate the number density

n_s

of elastically effective strands, and say whether it is a plausible value for a lightly cross-linked elastomer (typical strand densities lie between

10^{25}

and

10^{27}\,\mathrm{m^{-3}}