Reproducing kernel particle methods for large deformation analysis of non-linear structures

Large deformation analysis of non-linear elastic and inelastic structures based on Reproducing Kernel Particle Methods (RKPM) is presented. The method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties in large deformation analysis. The current formulation considers hyperelastic and elasto-plastic materials since they represent path-independent and path-dependent material behaviors, respectively. In this paper, a material kernel function and an RKPM material shape function are introduced for large deformation analysis. The support of the RKPM material shape function covers the same set of particles during material deformation and hence no tension instability is encountered in the large deformation computation. The essential boundary conditions are introduced by the use of a transformation method. The transformation matrix is formed only once at the initial stage if the RKPM material shape functions are employed. The appropriate integration procedures for the moment matrix and its derivative are studied from the standpoint of reproducing conditions. In transient problems with an explicit time integration method, the lumped mass matrices are constructed at nodal coordinate so that masses are lumped at the particles. Several hyperelasticity and elasto-plasticity problems are studied to demonstrate the effectiveness of the method. The numerical results indicated that RKPM handles large material distortion more effectively than finite elements due to its smoother shape functions and, consequently, provides a higher solution accuracy under large deformation. Unlike the conventional finite element approach, the nodal spacing irregularity in RKPM does not lead to irregular mesh shape that significantly deteriorates solution accuracy. No volumetric Locking is observed when applying non-linear RKPM to nearly incompressible hyperelasticity and perfect plasticity problems. Further, model adaptivity in RKPM can be accomplished simply by adding more points in the highly deformed areas without remeshing.