Lagrangian strain

To describe large deformations, the Lagrangian strains rjy are defined by... 

For both mathematical and physical reasons, there are many instances in which the spatial variations in the field variables are sufficiently gentle to allow for an approximate treatment of the geometry of deformation in terms of linear strain measures as opposed to the description including geometric nonlinearities introduced above. In these cases, it suffices to build a kinematic description around a linearized version of the deformation measures discussed above. Note that in component form, the Lagrangian strain may be written as... 

In the case of uniaxial tension, early authors introduced the nominal definition of strains (fuUy Lagrangian strains) for describing elastic deformation (27),... 

Care must be taken to formulate expressions that will lead to stresses that behave properly. For this reason it is convenient to formulate the strain energy density in terms of the Lagrangian strains... 

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

We can introduce the following family of n-th order Lagrangian strain measures ... 

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

Second Piola-Kirchhoff stress tensor and related traction vector Lagrangian strain tensor and displacement vector... 

In order to explain thoroughly elastomers behavior under oscillatory deformation, let be the longitudinal displacement in an uniaxial deformation from which the nonlinear Lagrangian strain follows as ... 

Leaderman showed that if (A. - 1/X ) is used as a measure of the deformation, both creep and recovery, and creep curves at different load levels, can be described by a single time-dependent function. This is shown in Figures 11.5(a) and (b). The quantity (A. — 1 /X ) is the equivalent quantity to the Lagrangian strain measme in the theory of finite elasticity. 

In addition to the trne strain definition in Equation 2.2, there are other large strain definitions that are used in nonlinear continuum mechanics. These are the Lagrangian and Eulerian definitions depending on whether the original coordinates or the deformed coordinates jc, are used as the reference coordinates. The Lagrangian strain is defined by... 

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