Arteries are not elastic bodies in the usual sense. But when it is subjected to cyclic loading and unloading, the stress-strain relationship is well defined after preconditioning. For either the loading branch or the unloading branch, a pseudo strain energy function can be used to express the stress-strain relationship. We show that the following expression is the most practical approximation for an artery subjected to internal pressure and longitudinal stretching within the physiologic range:​

ρ₀W⁽²⁾ = C/2 exp[a¹(E²_θθ - E^*2_θθ) + a₂(E²_zz - E^*2_zz) + 2a₄(E²_θθ - E^*2_θθ) + a₄(E_θθE_zz - E^*_θθE^*_zz)

where C, a₁, a₂, a₄ are material constants and E^*_θθ, E^*_zz are strains corresponding to a standard pair of stresses S^*_θθ, S^*_zz. Differentiation of ρ₀W⁽²⁾ with respect to E_θθ and E_zz yields the stresses S_θθ and S_θθ, respectively. The radial stress is ignored in comparison with See, Szz. The application of this formula is illustrated by the variation of the mechanical properties of rabbit arteries along the arterial tree.