It is clear that infinitesimal strains are obtained from finite strains (e.g. Green-Langragian or Eulerian-Almansi tensor) by removing nonlinear terms which smallness order is greater than that of linear terms.
But it is not clear for me whether such a procedure is justified, if we bear in mind that finite strains are subject to differentiation when plugging their into the equilibrium equation.
After all, the contribution of nonlinear terms in equilibrium equation can be significant, as the derivative of function can be much greater than function itself, even if that function is considered to be sufficiently small.