A second-degree micromorphic continuum model for 1D granular structures

This paper introduces a second-degree micromorphic continuum model for one-dimensional (1D) granular structures. The proposed framework aims to capture the mechanical behavior of materials with a distinct microstructure, such as granular rods or layered composites, by incorporating internal degrees of freedom. The kinematics are described by both macro-scale displacement and micro-scale deformation fields, where the micro-displacement within a representative line element (LE) is approximated by a third-order polynomial expansion. This approach introduces higher-order kinematic variables (micro-stretch and micro-curvature) and their gradients, accounting for non-local effects and size-dependent responses. The governing equilibrium equations and the corresponding boundary conditions (BCs) are obtained using the principle of minimum potential energy. The resulting system of sixth-order differential equations is solved using the state-space approach. A systematic investigation of 12 BC scenarios (Dirichlet and Neumann types) reveals the critical role of microstructural constraints in energy distribution. It is demonstrated that Dirichlet-type geometric clamping leads to pronounced boundary layers and energy localization peaks, whereas Neumann-type conditions promote field homogenization. The results highlight that micro-curvature and higher-order gradients are indispensable for predicting internal redistribution and potential failure zones in microstructured materials, providing a foundation for the design of architected metamaterials with tailored properties.