Sea ice is a complex material that displays composite structure on length scales ranging over many orders of magnitude. Finding the effective or homogenized behavior relevant to modeling large-scale sea ice dynamics and thermodynamics is a central challenge in making long-term predictions. Stieltjes integral representations for effective properties of composites have opened up new theoretical and computational approaches to sea ice modeling. We’ll give an overview of how these representations have been used to study sea ice electromagnetics, wave-ice interactions, and advection diffusion processes. They also allow us to make unexpected connections to random matrix theory, spectral analysis, uncertainty quantification, and exotic metamaterials.