Continua with microstructure: second gradient theory.

Second-gradient theories represent a frequently used subset of theories of continua with microstructure. This paper presents an extended overview of second-gradient theories, starting from a simple one-dimensional example, proceeding with a thorough description of gradient elasticity and additionally briefly describing some other theories of this kind. A series of characteristic examples is presented to demonstrate the main aspects and applications of second-gradient theories. Finally, the complications in the finite-element implementation of second-gradient theories are presented, along with a review of the finite elements that have been developed for this purpose