Embedding 1D Euler beam in 2D second-gradient continua

This work introduces a novel approach to modeling one-dimensional (1D) reinforcements within a two-dimensional (2D) second-gradient elastic matrix, suitable for describing various engineering structures undergoing small displacements and strains. The matrix obeys the first strain gradient elasticity in the Mindlin formulation and incorporates reinforcements, which are represented as zero-thickness interfaces with the elastic properties of one-dimensional (1D) extensional Euler–Bernoulli beams. The core innovation lies in the variational deduction of the generalized boundary conditions at these interfaces, which effectively capture the behavior of the reinforcements without requiring their full geometric representation. The proposed methodology is validated through finite element simulations of a reinforced structural element subjected to uniform bending.