Direct tensor expression for natural strain and fast, accurate approximation
A direct tensor expression for natural strain is developed using the fact that dilatation and distortion strain additively decouple into the spherical and deviatoric parts of , respectively. Upon separating the dilatation and the distortion, our direct expression for the deviator of 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 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 is exact. As for any dilatation superimposed on isochoric deformation, the deviator of is unaffected, whereas the spherical part of is obtained exactly and quickly with . Similar results for lnU (Lagrangian log-strain) follow directly from those herein for .
logarithmic strain finite elasticity finite element analysis finite strain