The arterial wall is a complex fiber-reinforced composite. Pathological conditions, such as aneurysms, significantly alter the mechanical response of the arterial wall, resulting in a loss of elasticity, enhanced anisotropy, and increased chances of mechanical failure. Invariant-based models of the healthy and aneurysmal abdominal aorta were constructed based on first principles and published experimental data with implementations for several numerical cases, as well as comparisons to current healthy and aneurysmal tissue data. Inherent limitations of a traditional invariant-based methodology are also discussed and compared to the models’ ability to accurately reproduce experimental trends. The models capture the nonlinear and anisotropic mechanical responses of the two arterial sections and make reasonable predictions regarding the effects of alterations in healthy and diseased tissue histology. Additionally, the new models exhibit convex and anisotropic monotonically increasing energy contours (suggesting numerical stability) but have potentially the inherent limitations of a covariant theoretical framework. Although the traditional invariant framework exhibits significant covariance, the invariant terms utilized in the new models exhibited limited covariance and are able to accurately reproduce experimental trends. A streamlined implementation is also possible for future numerical investigations of fluid-structure interactions in abdominal aortic aneurysms.