  In this paper, we propose a multiphysics finite element method for a
nonlinear poroelasticity model. To better describe the processes of deformation
and diffusion, we firstly reformulate the nonlinear fluid-solid coupling
problem into a fluid-fluid coupling problem by a multiphysics approach. Then we
design a fully discrete time-stepping scheme to use multiphysics finite element
method with $P_2-P_1-P_1$ element pairs for the space variables and backward
Euler method for the time variable, and we adopt the Newton iterative method to
deal with the nonlinear term. Also, we derive the discrete energy laws and the
optimal convergence order error estimates without any assumption on the
nonlinear stress-strain relation. Finally, we show some numerical examples to
verify the rationality of theoretical analysis and there is no "locking
phenomenon".
