I just realized that the problem comes from that part of the code that 
applies relative displacement between bottom and top faces, since there is 
not corner element distortion when having u_x (0,y)=u_x(L,y)+delta_x and I 
uncomment the second part of the code namely,

 {
                IndexSet selected_dofs_y;
                std::set< types::boundary_id > boundary_ids_y= 
std::set<types::boundary_id>();
                        boundary_ids_y.insert(2);

                DoFTools::extract_boundary_dofs(dof_handler,
                                                                
 fe.component_mask(y_displacement),
            selected_dofs_y,
     boundary_ids_y);
                unsigned int nb_dofs_face_y = selected_dofs_y.n_elements();
                IndexSet::ElementIterator dofs_y = selected_dofs_y.begin();

                double relative_displacement_y;
                if (timestep <10)
                  relative_displacement_y = 0.05e-9;
                else
                 relative_displacement_y = 0.005e-9;

                for(unsigned int i = 0; i < nb_dofs_face_y; i++)
                {
               constraints.set_inhomogeneity(*dofs_y, (apply_dirichlet_bc ? 
relative_displacement_y : 0.0));
                    dofs_y++;
                }

I cant realize what is the difference between relative displacement 
implementation in x and y direction that the code works for x direction but 
not for y direction while for both the code is identical !!!

Thanks

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