Abstract
Hill derived a simple component formula for the material time derivative of a generalized Lagrangian strain tensor. We examine Hill's derivation in detail and explain why it is generally valid only when the principal stretches are distinct. We then give a proof of Hill's formula which is valid for any C 2 motion and any C l strain measure. Our proof is based on a component form of the chain rule for a tensor-valued function of a time-dependent symmetric tensor. This result is also used to derive component formulas for the Jaumann rate of a generalized Eulerian strain tensor. Finally, we apply the general formulas to the logarithmic strain tensors.
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