An incompressible Mooney–Rivlin solid has C_(10)=118 kPa and C_(01)=29.5 kPa, with…

An incompressible Mooney–Rivlin solid has \(C_{10}=118\,\mathrm{kPa}\) and \(C_{01}=29.5\,\mathrm{kPa}\), with \(\sigma=-pI+2C_{10}B-2C_{01}B^{-1}\) and \(B=FF^{\mathrm{T}}\). Write \(F=I+H\) with \(\|H\|\ll 1\), set \(B=I+2\varepsilon+O(\|H\|^2)\) where \(\varepsilon=(H+H^{\mathrm{T}})/2\), and linearize the extra stress \(2C_{10}B-2C_{01}B^{-1}\) to first order in \(\varepsilon\). Identify the infinitesimal shear modulus \(\mu\) in the isotropic incompressible Hooke law \(s=2\mu\varepsilon\) (\(s=\operatorname{dev}\sigma\), \(\operatorname{tr}\varepsilon=0\)) and evaluate \(\mu\) numerically. For homogeneous simple shear \(F=\bigl[[1,\gamma,0],[0,1,0],[0,0,1]\bigr]\) with \(\sigma_{22}=0\), obtain exact expressions for \(\sigma_{12}\) and \(N_1=\sigma_{11}-\sigma_{22}\). Evaluate both at \(\gamma=0.012\) and at \(\gamma=0.75\). Compare each \(\sigma_{12}\) with the linearized prediction \(2\mu(\gamma/2)\), and state the order in \(\gamma\) of the first nonvanishing normal-stress difference. Then, using incompressibility to take \(E=3\mu\), predict the infinitesimal uniaxial Cauchy stress produced by an axial strain \(\varepsilon_{11}=0.0090\) with free lateral surfaces, and compare it with the finite-strain uniaxial nominal stress \(P_{11}=2(C_{10}+C_{01}\lambda^{-1})(\lambda-\lambda^{-2})\) evaluated at \(\lambda=1.0090\).

Invert \(B=I+2\varepsilon+\cdots\) to first order and collect the coefficient of \(\varepsilon\); that coefficient is \(2\mu\).