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In this paper, we propose a simple, iteration free and easy-to-implement numerical algorithm for the solution of inverse Cauchy problem in linear or nonlinear elasticity. The bottom of a finite rectangular plate is imposed by overspecified boundary data, and we seek unknown data on the top side. A spring-damping transform method (SDTM) is introduced to the Navier equations, such that after a discretization by the differential quadrature method, we can apply a novel Lie-group integrator, namely the mixed group-preserving scheme (MGPS), to solve them as an initial value problem. Several numerical examples including nonlinear ones are examined to show that the MGPS can overcome the ill-posed behaviour of the inverse Cauchy problem in elasticity, which has good efficiency and stability against the noisy disturbance, even with an intensity large up toand. [ABSTRACT FROM AUTHOR]