Closed-cell foams are lightweight cellular materials widely used in applications requiring high mechanical strength, low density, and excellent energy absorption capacity. These materials are used in impact absorption systems in the automotive industry, thermal and acoustic insulation in civil construction, structural components in the aerospace sector, biomedical prosthetics and implants, and functional elements in sensors, microactuators, and precision engineering devices. At the micro- and nanoscale, they are particularly relevant in the design of microbeams for emerging technologies. Due to scale effects that significantly influence the mechanical behavior of such structures, classical models become inadequate, necessitating the use of higher-order continuum theories for a more accurate representation. This study proposes a high-order laminated beam model based on the Modified Strain Gradient Theory, incorporating thickness-direction deformation (quasi-3D effect) and three distinct porosity distributions. The Halpin-Tsai model is employed to estimate the effective material properties. The governing differential equations and boundary conditions are derived through a variational formulation, and the analytical solution is obtained using the Navier method. The results demonstrated excellent agreement with data from the literature, confirming the accuracy and robustness of the proposed model.
