Direct tensor expression for natural strain and fast, accurate approximation

A direct tensor expression for natural strain ln V is developed using the fact that dilatation and distortion strain additively decouple into the spherical and deviatoric parts of ln V, respectively. Upon separating the dilatation and the distortion, our direct expression for the deviator of ln V has scalar coefficients that depend on, at most, two invariants derived from B. By using simple functions for these scalar coefficients, a fast and accurate approximation for the deviatoric part of ln V is obtained. The error in using this approximation diminishes as the strain decreases. For isochoric deformation, the percent error is about 0.2% for uniaxial extension of stretch 2 or for equibiaxial extension with stretches of 1.4. For pure shear, our approximation for ln V is exact. As for any dilatation superimposed on isochoric deformation, the deviator of ln V is unaffected, whereas the spherical part of ln V is obtained exactly and quickly with tr(ln V)=ln(J). Similar results for ln U (Lagrangian log-strain) follow directly from those herein for ln V.