The molecular stress function theory for polymer melts is extended to include a new, dissipative convective constraint release process. First the Helmholtz free energy of tube segments with strain-dependent tube diameter is established neglecting constraint release, and it is demonstrated that the molecular stress is a function of the average logarithmic stretch under these conditions. Then convective constraint release is introduced as a dissipative process in the energy balance of tube deformation, which leads to a strain-dependent evolution equation for the molecular stress function. Constraint release is considered to be the consequence of different convection mechanisms for tube orientation and tube cross section. Our new, dissipative constraint release model emphasizes that tube kinematics are fundamentally different for rotational and nonrotational flows, and therefore distinguishes explicitly between simple shear and pure shear (planar extension). For the startup of simple shear and extensional flows, the predictions of our set of constitutive equations consisting of a history integral for the stress tensor and a differential evolution equation for the molecular stress function with only two nonlinear material parameters are in excellent agreement with experimental data of a polydisperse high-density polyethylene (HDPE) and a polydisperse low-density polyethylene (LDPE) melt. Also, stress relaxation after step-shear strain is described for both the HDPE and the LDPE melt.