On the convolutive development of elastic substrate media as nano foundation

This study introduces a novel framework for developing models of elastic substrate foundations using an integral convolution approach. The proposed methodology systematically breaks down the applied load function into integral components and employs a multiplicative kernel transformation to derive the governing equations for substrate behavior. By extending traditional foundation models like the Winkler and Pasternak models, this formulation incorporates shear interactions and spatial variations in material properties, thereby addressing limitations in conventional approaches. The resulting equations effectively capture both local and global effects of applied loads, providing a more accurate representation of substrate behavior in heterogeneous, anisotropic, and non-uniform systems. The validity of the proposed model is verified through comparisons with established theories, demonstrating its precision and broader applicability to complex structural scenarios. The convolution-based formulation also enhances the analysis of advanced loading conditions and nonlinear material responses, making it highly adaptable to real-world engineering applications. The analytical and numerical results of this study contribute significantly to structural mechanics, especially in the design and analysis of beams, plates, and other structural elements interacting with elastic substrates. The findings have potential applications in nano- and micro-scale engineering, geotechnical studies, and advanced material modeling, highlighting the importance of nonlocal elasticity in contemporary structural analysis.