Generalized Lagrangian strain measure

Following Hill (1978) the generalized Lagrangian strain measure is defined by... 

The result (3.34) can be extended by introducing the rate of the generalized Lagrangian strain measure E ( ) ... 

In Eq. (46), the x(t) are the spatial (deformed) coordinates and the partial differentiation is performed with respect to the material (undeformed) coordinates. Expression (45) was introduced by Blatz et al. and Ogden, independently, who adopted the idea of a generalized strain measure to predict stress-strain relations of crosslinked samples of elastomers under various types of deformation. Representation in the principal axes system yields for the components of the Lagrangian stress tensor... 

We can generalize this procedure If a function

Lagrangian description, and if

Eulerian description. The choice of the form is arbitrary but will be influenced by any advantage of a problem formulation in either description. For example, in solid mechanics, the Lagrangian description is commonly used, while in fluid mechanics the Eulerian description is popular. This is because in solid mechanics we can attach labels (e.g., visualize strain gauges at various points) on the surface of a solid body, and each material point can be easily traced from the reference state to the current state. On the other hand for a fluid we measure the velocity V or pressure p at the current position jc, therefore the Eulerian description better represents the fluid (note that for a fluid it is difficult to know the exact reference point X corresponding to all the current points jr). 

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