A stretch-based hyperelastic material formulation for isogeometric Kirchhoff-Love shells is presented , particularly useful when experimental material data fitting is involved to capture the model parameters, for instance using the Ogden material model. Complementing the existing invariant-based formulation, the stretch-based stress and material tensors are expressed in spectral form and transformed to the covariant curvilinear bases for consistency with the variational formulation. For both compressible and incompressible material formulations, analytical and numerical benchmarks show excellent results. In particular, the response of a collapsing conical shell-a complex collapsing mechanism-was revealed when using the arc-length method.