In this paper, we study the stability of the nonsymmetric version of the Nitsche method without penalty for compressible and incompressible elasticity. For the compressible case we prove convergence of the error in the <f>$H^1$</f>- and <f>$L^2$</f>-norms. In the incompressible case we use a Galerkin least squares pressure stabilization and we prove convergence in the <f>$H^1$</f>-norm for the velocity and convergence of the pressure in the <f>$L^2$</f>-norm.