A.V. Metrikine, H. Askes

An isotropic dynamically consistent gradient
elasticity model derived from a 2D lattice 
Philosophical
Magazine 86 (2122) (2006) 32593286 DOI:
10.1080/14786430500197827 
Summary. This
paper presents a derivation of a secondorder isotropic continuum from a 2D
lattice. The derived continuum is isotropic and dynamically consistent in the
sense that it is unconditionally stable and prohibits the infinite speed of
energy propagation. The Lagrangian density of the continuum is obtained from
the Lagrange function of the underlying lattice. This density is used to
obtain the expressions for standard and higherorderstresses in direct
correspondence with the equations of the continuum motion. The derived
continuum is characterized by two additional parameters relative to the
classical elastic continuum. These are the characteristic lengthscale and a
dimensionless continualization parameter, which characterizes indirectly the
timescale of the derived continuum. The margins for the latter parameters are
found from the stability analysis. It is envisaged that the continualization
parameter could be measured employing a highfrequency pulse propagating
along the surface of the continuum. Excitation and propagation of such pulse
is studied theoretically in this paper. 