An improved total Lagrangian SPH method for modeling solid deformation and damage

Total Lagrangian - Smoothed Particle Hydrodynamics (TL-SPH) is a classical algorithm to improve the tensile instability encountered in the conventional Eulerian kernel-based SPH method for solid mechanics. Meanwhile, the gradient correction usually is employed in TL-SPH to enhance the consistency of approximation, which would generate unavoidable time costs. Therefore, a simplified gradient correction under the initial configuration is derived. Aiming at different particle configurations, the corresponding selection modes are provided and verified. Then the simplified gradient correction is introduced into TL-SPH, which is the Improved TL-SPH (ITL-SPH) method. Firstly, a classical example - elastic rubber collision is simulated to verify ITL-SPH has the same ability as TL-SPH to improve tensile instability. Then, the convergence and high efficiency of ITL-SPH is demonstrated via the uniaxial tension of a plate. Moreover, the plate with square and triangle holes also is simulated to show the adaptation and stability of ITL-SPH for the extreme boundary. Under the Total Lagrangian frame, the treatment to model the crack is necessary. So, this paper proposes a damage model and combines it with the ITLSPH method for crack initiation and propagation, which is verified by the plate with holes and the notched beam examples.