The solution of solid mechanics problems in large displacement and large deformation regime, dealing with incompressible or nearly incompressible materials, is a topic of paramount importance in the computational mechanics community. Many engineering problems require the simulation of complex materials under such conditions. In particular, simulation of granular flows remains a challenge under the mathematical and computational perspective. In some specific cases, the standard Galerkin displacement-based formulation fails, because the formulation is unable to evaluate the strain field. For this reason, in this work a mixed formulation is chosen. However, this formulation does not satisfy the well-known inf-sup condition and therefore a stabilization technique is required to obtain stabilized results. This work presents the development of a mixed formulation for a nonlinear solid mechanics framework considering both a weakly-compressible and an incompressible regime in a Material Point Method approach. Two different stabilization techniques are employed for solving the dynamic problem in mixed formulation. Both are based on the variational multiscale (VMS) method. The MPM is extremely useful for dealing with large material deformation. This framework has many advantages that allows to avoid the classical limitations of the Finite Element Method, such as element tangling and extreme mesh distortion. The proposed mixed formulations, with displacement and pressure as primary variables, are tested through classical benchmarks such as the Cook’s membrane, and in three-dimensional problems which present extreme deformations as the twisting column. Further, the stabilized mixed formulation based in the VMS method is compared with other stabilization techniques such as the Polynomial Pressure Projection [1], to assess its accuracy.