Modelling elastic dynamics and fracture with coupled mixed correction Eulerian Total Lagrangian SPH
Young, James Roger
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In this thesis, the Smoothed Particle Hydrodynamics (SPH) method is applied to elastic dynamics and fracture. More specifically, two coupling methods are presented which make use of both the Eulerian and Total Lagrangian formulations. These coupling methods are intended for problems whereby SPH particles, which constitute the domain, are required to convert from a Total Lagrangian kernel to an Eulerian kernel once a damage criterion is activated. The conservation equations are derived for the Eulerian and Total Lagrangian formulations, in a consistent manner which naturally presents the conditions required for the conservation of momentum and energy. These derivations are written such that they make no use of the symmetrical nature of the kernel function or the anti-symmetrical nature of the kernel function gradient. The conservation of momentum and energy is then enforced, along with improving the consistency of the formulations, by implementing the mixed kernel-and-gradient correction. This mixed correction can be applied to both the Eulerian and Total Lagrangian formulations without detracting from the energy and momentum preserving properties provided that the kernel gradient anti-symmetry property is not exploited. The symmetry terms, which are often found in SPH, are included in the derivation of the conservation equations. This is done both to reduce the number of calculations required and to simplify the first coupling procedure. Both coupled formulations are further expanded by highlighting how artificial viscosity can be introduced. A disadvantage of the first coupling method, this being the incompatibility with artificial stress, is also detailed. The equations of state and the plasticity and damage models used in this work are outlined. Additionally, a number of practical details concerning numerical implementation are given. These include the coupled implementations of ghost particle boundary conditions, memory storage, OpenMP implementation, and the Predict, Evaluate, Correct (PEC) form of leapfrog time integration used. Lastly, the proposed formulations and models are verified and validated. This is done by modelling progressively more complex simulations that verify individual aspects of the formulations. Either analytical or experimental results are used for validation where possible. The final simulations highlight how high velocity impacts can be modelled using the proposed coupled mixed correction Eulerian Total Lagrangian SPH method.