The numerical manifold method surmounting the mesh dependence has successfully solved very complicated problems involving small deformation and large movement but had few applications to large deformation and large rotation problems because the false volume expansion and other issues exist . In this study it is shown that the false volume expansion in NMM can be excellently resolved by using the S R decomposition theorem which can precisely reflect complex physical behaviors occurring in the process of large rotation and large deformation . The numerical methods based on the S R decomposition theorem have been limited to the static analysis of large deformations . To remove this limitation a new formulation taking into account dynamical features is proposed based on the weak form of momentum conservation law . Under the framework of NMM the generalized method is employed to discretize the temporal variables . The updates of variables are described using the updated co moving coordinate system . Thus a new method named S R D based NMM is established . The new formulation can be implemented in any other partition of unity based methods as well so as to improve the performances of such methods in the analysis of dynamic large deformations . Global reference system Co moving coordinate system Position vector of a point before deformation Position vector of a point after deformation Displacement vector of a point Velocity vector of a point Acceleration vector of a point Specified vector of a point Specified traction vector of a point Force per unit volume Basis vector corresponding to initial co moving coordinate system Basis vector corresponding to the co moving coordinate system after deformation Transformation function between and Kronecker delta Displacement derivative Christoffel symbol of the second kind Strain tensor Linear strain tensor Nonlinear strain tensor Local mean rotation tensor Unit vector of the rotation axis Mean rotation angle Area coordinates of any point Partition of unity function Star coordinates Area of a triangular mesh covering one manifold element Parameters in terms of star coordinates Constants with regard to Stress Material tensor Material density Time Time increment or time step size Virtual work of inertia force Virtual work of constraint force of specified displacement Virtual work of external force Penalty parameter Displacement vector of a manifold element Velocity vector of a manifold element Acceleration vector of a manifold element Interpolation matrix Strain vector of a manifold element Linear strain vector of a manifold element Nonlinear strain vector of a manifold element Linear strain displacement matrix Nonlinear strain displacement matrix Element stiffness matrix Mass matrix of a manifold element Equivalent force vector of a manifold element Algorithmic parameters of generalized Spectral radius of generalized 


Summary: A dynamics formulation based on the S R decomposition is deduced. An update scheme for the co moving coordinate used in the S R decomposition is built. A new procedure named S R D based NMM is established for addressing small or large deformation together with impact contact. Some issues such as the false volume expansion in the conventional small deformation based NMM are effectively overcome.