Definition of Volume Strain

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

The Hencky definition of strains (also called tme or namral or logarithmic ) was used by G Sell et al. (29,30) to alleviate the above difficulty. It gives the expressions 1 = ln(Li/LiJ, 2 = ln(L2/L2, ), and 3 = ln(L3/L3j for the three principal strains. Volume strain can also be written in Hencky formalism as 

The two definitions introduced above are obviously equivalent at infinitesimal deformation, but their difference increases rapidly with strain. For = 20%, the relative discrepancy is already as large as 9%. Consequently, Hencky definitions should always be employed for describing the stretching properties of polymers. 

In the elastic range, the Poisson s ratio, Vei = —/, can be used to relate the amplitude of transverse constriction with axial strain. The volume strain depends on this material coefficient by = (1 — 2vei)e - Similarly, for nonlinear materials, the tangent Poisson s ratio Vt = —d i/d 3 may be introduced at large strain, so that coefficient is also obtained from the instantaneous slope of the volume strain versus axial strain from (1 — 2vt ) = d v/d 3. In this paper, the latter slope will be called dilatation rate (or damage rate ) and denoted by the variable A. 

By adapting an analysis of Bucknall (31) to the Hencky formalism, it was recently shown by one of the authors (32) that volume strain could be usefully decomposed into three contributions  

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