Analytical homogenization for equivalent in-plane elastic moduli of prestressed lattices based on the micropolar elasticity model

Highly compressible and stretchable lattice materials have gained significant attention due to their advanced structural characteristics. However, most existing research on homogenization relies on linear classical elasticity theory, omitting nonlinear deformation and/or the micropolar elasticity rotation. This paper addresses this gap by introducing a novel analytical homogenization method for the in-plane equivalent elastic moduli of prestressed two-dimensional lattices that incorporates both nonlinear elastic deformation and micropolar elastic effects. First, the stiffness matrices of a lattice cell wall under axial force is formulated. Based on the micropolar elasticity theory, the micropolar elastic constitutive relations for four lattice materials under prestresses were established, i.e., rectangular, diamond, equilateral triangular, and mixed diamond lattices. Then, the micropolar elastic constants for different prestressed lattices are formulated. The closed-form expressions for the equivalent elastic moduli of nonlinear micropolar elastic bodies were derived from these micropolar elastic constants by their physical significance. The analytical expressions for the lattice elastic moduli are validated by using independent nonlinear finite element simulation in ANSYS with relative errors less than 4%. The proposed analytical method and new closed-form expressions provide a framework with high computational efficiency and accuracy for the analysis and parametric design of lattice materials under external stress.