Discrete and continuous models of linear elasticity: history and connections

This paper tracks the development of lattice models that aim to describe linear elasticity solids and field equations which converge asymptotically toward those isotropic continua, thus showing connection between discrete continuum. In 1759, Lagrange used strings/rod dynamics show link mixed differential-difference equation a one-dimensional (1D) partial differential associated A consistent three-dimensional (3D) generalization this model was given much later: Poincaré Voigt reconciled molecular continuum approaches at end nineteenth century, but only in 1912 Born von Kármán presented elasticity. Their is 3D Lagrange’s 1D considers longitudinal, diagonal shear elastic springs among particles, so characterized by three constants. proved Navier’s ones asymptotically, being formulation continuous terms spatial finite differences, as for lattice. Neglecting Born–Kármán’s equals assumption pure central forces molecules: limit, behaves one-parameter solid (“rari-constant” theory: equal Lamé parameters, or, equivalently, Poisson’s ratio $$\upsilon =1/4$$ ). Hrennikoff McHenry revisited approach with interactions using plane truss; equivalent stress limit tends = 1/3$$ . Contrary McHenry–Hrennikoff’s truss, leads “free” bounded its “limit’ bound ( strain or elasticity; =1/3$$ elasticity). Unfortunately, does not comply rotational invariance principle, non-central forces. The achieved Gazis et al. considering an energy depends on changes both lengths angles An alternative three-parameter Hrennikoff’s, additional structure cell. We also discuss capability nonlocal bridge gap low frequencies anisotropic high frequencies.