Different kinds of hot mix asphalt mixtures are used in highway and runway constructions. Each of these mixtures cater to specific needs and differ from each other in the type and percentage of aggregates and asphalt used, and their response can be markedly different. Constitutive models used in the literature do not differentiate between these different kinds of mixtures and use models which treat them as if they are one and the same. In this study, we propose constitutive models for two different kinds of hot mix asphalt, viz., asphalt concrete and sand asphalt. We use a framework for materials that possess multiple natural configurations for deriving the constitutive equations. While asphalt concrete is modeled as a two constituent mixture, sand asphalt is modeled as a single constituent mixture due to the peculiarity in its makeup. In this study, we present a unified approach for deriving models for these different kind of mixtures. In a companion paper, we compare the predictions of the model for a compressive creep test with available experimental results.