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We employ a structurally-motivated phenomenological formulation to identify biomechanical experiments which can be used to determine a vascular constitutive relation directly from data. Large deformations, nonlinear material behavior, load-dependent anisotropy, material heterogeneity and incompressibility are accounted for in the analysis. For purposes of illustration, we outline a procedure for studying elastic arteries wherein the behavior of the media and adventitia is considered separately. This general approach for identifying vascular constitutive relations can be applied to any vessel or airway, however, and should provide certain advantages over previous microstructural or purely phenomenological formulations.