Appendix C — Derivation of the neo-Hookean material tangent

This appendix derives the stress variation used in Section 9.3. Let \(\bm H\) be the increment of the deformation gradient \(\bF\), as in Equation 9.14.

C.1 Inverse and determinant variations

The neo-Hookean stress in Equation 4.69 contains \(\bF^{-T}\) and \(\ln J\). To differentiate it, take \(\delta\bF=\bm H\) and vary the identity \(\bF\bF^{-1}=\bI\):

\[ \bm H\bF^{-1}+\bF\delta(\bF^{-1})=\bm0. \]

Multiplying on the left by \(\bF^{-1}\) and then transposing gives

\[ \begin{aligned} \delta(\bF^{-1})&=-\bF^{-1}\bm H\bF^{-1},\\ \delta(\bF^{-T})&=-\bF^{-T}\bm H^T\bF^{-T}. \end{aligned} \tag{C.1}\]

The derivative of the determinant can be expressed through the cofactor tensor \(J\bF^{-T}\):

\[ \delta J=(J\bF^{-T}):\bm H =J\tr(\bF^{-1}\bm H). \tag{C.2}\]

Here \(\bF^{-T}:\bm H=\tr(\bF^{-1}\bm H)\) follows from the definition of the double contraction. Applying the scalar chain rule to \(\ln J\) yields

\[ \delta(\ln J)=\frac{\delta J}{J} =\tr(\bF^{-1}\bm H). \tag{C.3}\]

All these formulas refer to the same perturbation \(\delta\bF=\bm H\) and require an invertible \(\bF\); the logarithm also requires \(J>0\).

C.2 The neo-Hookean material tangent

Rewrite Equation 4.69 as

\[ \bP(\bF)=\mu\bF+(\lambda\ln J-\mu)\bF^{-T}. \]

The product rule first gives

\[ \delta\bP(\bF;\bm H) =\mu\bm H+\lambda\delta(\ln J)\bF^{-T} +(\lambda\ln J-\mu)\delta(\bF^{-T}). \]

Substitution of Equation C.1 and Equation C.3 gives

\[ \boxed{\begin{aligned} \delta\bP(\bF;\bm H) ={}&\mu\bm H +\lambda\tr(\bF^{-1}\bm H)\bF^{-T}\\ &+(\mu-\lambda\ln J)\bF^{-T}\bm H^T\bF^{-T}. \end{aligned}} \]

This gives the material tangent in Equation 9.17.