An incompressible neo-Hookean tensile bar has stored energy W=(μ/2)(I_1-3), μ=226 kPa…

An incompressible neo-Hookean tensile bar has stored energy \(W=(\mu/2)(I_1-3)\), \(\mu=226\,\mathrm{kPa}\), and undeformed cross-sectional area \(A_0=31.5\,\mathrm{mm}^2\). Homogeneous uniaxial extension \(\mathbf{F}=\operatorname{diag}(\lambda,\lambda^{-1/2},\lambda^{-1/2})\) is imposed, with the lateral faces Cauchy-traction-free. Using \(\boldsymbol{\sigma}=-p\mathbf{I}+\mu\mathbf{B}\) and \(p\) from \(\sigma_{22}=0\), derive closed-form expressions for the axial Cauchy stress \(\sigma_{11}(\lambda)\) and the axial first Piola–Kirchhoff stress \(P_{11}(\lambda)=\sigma_{11}/\lambda\), valid for all \(\lambda>0\). Evaluate \(\sigma_{11}\), \(P_{11}\), \(S_{11}=P_{11}/\lambda\), \(W\), and the axial force \(f=P_{11}A_0\) at \(\lambda=1.48\), \(\lambda=1.90\), and \(\lambda=2.55\). Compute the axial nominal tangent \(\mathrm{d}P_{11}/\mathrm{d}\lambda\) at each of those three stretches and determine whether \(\mathrm{d}P_{11}/\mathrm{d}\lambda\) remains positive throughout \(\lambda>0\). Linearize \(P_{11}(\lambda)\) about \(\lambda=1\) and identify the infinitesimal Young’s modulus \(E_0\) in terms of \(\mu\). Report the current cross-sectional area at \(\lambda=1.90\).

Lateral traction-free faces give \(p=\mu/\lambda\), hence \(\sigma_{11}=\mu(\lambda^2-\lambda^{-1})\) and \(P_{11}=\mu(\lambda-\lambda^{-2})\).