  This article presents a $P_0$ finite element method for boundary value
problems for linear elasticity equations. The new method makes use of piecewise
constant approximating functions on the boundary of each polytopal element, and
is devised by simplifying and modifying the weak Galerkin finite element method
based on $P_1/P_0$ approximations for the displacement. This new scheme
includes a tangential stability term on top of the simplified weak Galerkin to
ensure the necessary stability due to the rigid motion. The new method involves
a small number of unknowns on each element; it is user-friendly in computer
implementation; and the element stiffness matrix can be easily computed for
general polytopal elements. The numerical method is of second order accurate,
locking-free in the nearly incompressible limit, ease polytopal partitions in
practical computation. Error estimates in $H^1$, $L^2$, and some negative norms
are established for the corresponding numerical displacement. Numerical results
are reported for several 2D and 3D test problems, including the classical
benchmark Cook's membrane problem in two dimensions as well as some three
dimensional problems involving shear loaded phenomenon. The numerical results
show clearly the simplicity, stability, accuracy, and the efficiency of the new
method.
