Three Homogenization Models for a Nonlinear Heterogeneous Rod with Microscale Dead Loading

A discrete chain of one-dimensional elements with concentrated elastic stiffness is analyzed to investigate the effect of self-equilibrated dead loads acting at the microscale. Three homogenization schemes are formulated for a finite chain whose total length is much larger than the length of an individual cell. Model (i) homogenizes the elastic properties and the microscopic loading simultaneously, yielding an equivalent rod with homogeneous axial and flexural stiffnesses. The effect of the microforces is incorporated through an effective prestress state and a distribution of nonlinear rotational springs accounting for their smeared contribution. Two additional models are developed in which the microforces are not homogenized but remain periodically distributed: Model (ii), an equivalent rod with homogeneous axial and flexural stiffnesses, and Model (iii), a flexurally homogeneous equivalent rod with piecewise-constant axial stiffness. Although Model (iii) most closely resembles the original discrete mechanical system and Model (ii) may appear the most intuitive from an engineering standpoint, Model (i) is shown to recover the continuum limit of the fully loaded discrete system and to provide the most accurate, fully homogenized description. Moreover, the explicit incorporation of the microforces into the nonlinear homogenization procedure of Model (i) gives rise to a form of loading and distributed constraint in a continuous rod that has no direct counterpart in conventional mechanical systems. In particular, it can produce an effective distribution of rotational springs with negative stiffness. The nonlinear equilibrium equation of Model (i) also admits a closed-form solution in terms of elliptic functions, which is used to analyze the large-deflection response and the initial post-buckling behavior of a cantilever. The model predicts both tensile and compressive buckling, as well as restabilization of the straight configuration under suitable loading conditions. Finally, the combined control of the end load and the microforces is explored as a mechanism for soft-robotic actuation. Suitable loading paths allow the system to switch between straight and bent configurations and to modify the bifurcation character from supercritical to subcritical.