Many treatments for cardiovascular diseases include an endovascular insertion of stents or stent grafts into arteries, a procedure which may cause high tissue stresses and even damage in the arterial wall. In order to study such problems by using finite element methods, both appropriate constitutive models and experimental data on human tissue samples are required. Layer-specific experimental data for human tissue tested up to the supra-physiological loading range are rare in the literature. In this study, intact and layer-separated experimental data from uniaxial extension tests are presented for human thoracic and abdominal aortas with non-atherosclerotic intimal thickening undergoing supra-physiological loading. A novel pseudo-elastic damage model, proposed to describe discontinuous softening in aortic arterial tissues, is fit to the obtained experimental data. Fitting of the model with and without consideration of damage accumulation in the non-collagenous matrix material reveals that tissue damage is primarily related to the collagen fiber fabric. By employing the fit model, the effect of aortic tissue pre-conditioning on the material parameters from the resulting data fits is evaluated. Histological examination of the collagen fibers under different applied stretches is used to gain more insights into the structural changes of the tissue under supra-physiological loading.