Someone wants to extract the shear modulus of a rubber strip from one tensile measurement

PHYSICS · Subjective · 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

Someone wants to extract the shear modulus of a rubber strip from one tensile measurement. The strip has undeformed cross-sectional area A0=2.0×10−5 m2A_0=2.0\times10^{-5}\,\mathrm{m^2} and is held at a stretch ratio λ=3.0\lambda=3.0 by a force F=6.0 NF=6.0\,\mathrm{N}. Reasoning that the rubber is incompressible, so the deformed area is A(λ)=A0/λA(\lambda)=A_0/\lambda, they compute the true stress σtrue=F/A(λ)\sigma_{\rm true}=F/A(\lambda) and substitute this value directly into the nominal stress-strain law σ=G(λ−λ−2)\sigma=G(\lambda-\lambda^{-2}), solving for GG. Find the error in this reasoning and give the correctly extracted shear modulus.

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