Isn't that u the symmetrised infinitesimal strain, based on the second definition? I found this book (www.cambridge.org/core/books/c...) very instructive, and tried to put into MOF context here: books.rsc.org/books/edited... (although I didn't go into MC algorithms)
cambridge.org
Continuum Mechanics and Thermodynamics
Cambridge Core - Mathematical Methods - Continuum Mechanics and Thermodynamics