In this research, newly developed assumed stress hybrid elements are presented. The elements are applicable for geometrically as well as materially nonlinear problems with or without volume constraints. Variational principles using unsymmetric stresses and rotations are adopted as the basis of the formulations. The variational principles are modified through a regularization term, to eliminate every stress term at element level. Two types of four-noded quadrilateral plane element formulations are derived through direct discretizations of the variational principles with proper suggestions for the shape functions. The first one has three independent fields-displacements (velocities), rotations (spins), and unsymmetric (Biot) stress fields. In the second one, the hydrostatic pressure field is added as an independent field, to account for volume constraints. It is confirmed that better solutions can be obtained by rather small numbers of stress parameters. The elements are also employed to analyze strain softening incompressible hyperelastic materials, in which the ellipticity of the constitutive relation fails and discontinuous deformation gradients may occur. The numerical examples of the shear band problems are demonstrated in the paper.