Page 194 - The Combined Finite-Discrete Element Method

NUMERICAL DEMONSTRATION OF FINITE ROTATION ELASTICITY 177

Figure 4.29 Dancing triangles: Dancing away.

of the material is not large, the rotations of individual finite elements is significant.
Thus, assumptions employed in formulation of finite-rotation elasticity are valid, including
the approximate validity of the strain tensor definition and stress-strain relationship. For
simulations involving large stretches (say over 20%), the Green–St. Venant strain tensor
is not applicable, nor is the constitutive law described earlier in this chapter. Thus, in
such cases both strain and stress tensors and the constitutive law need modification. One
possible solution is to employ a logarithmic strain tensor.
Unlike the Cauchy–St. Venant strain tensor, a logarithmic strain tensor involves log-
arithmic functions. The only way to resolve these is to employ the frame of reference
coinciding with the principal directions of the stretch tensor in either the initial or deformed
configurations. The stress-strain relationship is also best resolved using these principal
directions. This requires spectral decomposition of the stretch tensor. The CPU overheads
are not significant, and implementation is relatively simple.