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A new viscoelastic constitutive model for subaqueous clay-rich gravity flows is presented. It is explained that for the materials which exhibit a minimum in their strain controlled flow curves the structure parameter must be a symmetric function of strain rate and stress. Therefore, the destruction of structure within the material is modeled using the dissipation energy. An expression for the elastic strain of the flowing structure is derived. The final set of equations can reproduce the viscosity bifurcation that clay suspensions may exhibit under a given load. This is explained to be important for prediction of the run-out distance of clay-rich gravity flows. The ability of the model to reproduce the general response of pasty materials to step stress and step shear rate tests is examined. The model requires four empirical parameters. A methodology is presented for obtaining these parameters and power law functions are given for their calculations for a limited rest time of 3000 seconds. The ability of the model to reproduce the rheological behavior that clay-rich suspensions exhibit under both stress and strain controlled conditions is examined using rheometry tests.