  Strain gradient elasticity and nonlocal elasticity are two enhanced elastic
theories intensively used over the last fifty years to explain static and
dynamic phenomena that classical elasticity fails to do. The nonlocal elastic
theory has a clear differentiation from the classical case by considering
stresses in a point of the continuum as an integral of all stresses defined in
the treated elastic body. On the other hand, strain gradient elasticity is
characterized as a non-classical theory because considers both potential and
kinetic energy densities as not only functions of strains and velocities,
respectively but also functions of their gradients. Although the two
considerations seem to be completely different from each other, it is a common
belief that strain gradient elasticity has a lot in common with nonlocal
elasticity. The goal of the present work is to derive all the strain gradient
elastic theories appearing so far in the literature via nonlocal definitions of
the potential and kinetic energy densities. Such a derivation demonstrates the
common roots of the two theories and explains the nature of the involved
intrinsic parameters in strain gradient elastic theories. For the sake of
simplicity and brevity, only one-dimensional wave propagation phenomena are
considered.
