Nonlocal continuum analysis of a nonlinear uniaxial elastic lattice system under non-uniform axial load

• The static behavior of a nonlinear axial chain under distributed loading is examined.
• Exact analytical solutions based on Hurwitz zeta functions are presented.
• The nonlinear lattice possesses scale effects and possible localization properties in the absence of energy convexity.
• A nonlinear continuum elasticity model is developed to capture the main phenomena observed regarding the discrete axial problem.
• This associated continuum is an enriched gradient-based or nonlocal axial medium.

The static behavior of the Fermi-Pasta-Ulam (FPU) axial chain under distributed loading is examined. The FPU system examined in the paper is a nonlinear elastic lattice with linear and quadratic spring interaction. A dimensionless parameter controls the possible loss of convexity of the associated quadratic and cubic energy. Exact analytical solutions based on Hurwitz zeta functions are developed in presence of linear static loading. It is shown that this nonlinear lattice possesses scale effects and possible localization properties in the absence of energy convexity. A continuous approach is then developed to capture the main phenomena observed regarding the discrete axial problem. The associated continuum is built from a continualization procedure that is mainly based on the asymptotic expansion of the difference operators involved in the lattice problem. This associated continuum is an enriched gradient-based or nonlocal axial medium. A Taylor-based and a rational differential method are both considered in the continualization procedures to approximate the FPU lattice response. The Padé approximant used in the continualization procedure fits the response of the discrete system efficiently, even in the vicinity of the limit load when the non-convex FPU energy is examined. It is concluded that the FPU lattice system behaves as a nonlocal axial system in dynamic but also static loading.