Development of a total Lagrangian SPH code for the simulation of solids under dynamioc loading

Abstract

This thesis makes use of an alternative SPH formulation, the Total Lagrangianf ormulation, to characterised ynamic eventsi n solids and to achieve the proposed objectives outlined in Chapter 1. The structure is as follows: Chapter 1, Introduction, describes the motivation for this research and outlines the objectives and the structure of this thesis. Chapter 2, SPH fundamentals, supplies the standard procedure to generate particle equations and provides a comprehensive summary of gradient approximation formulae in SPH. The discretised SPH form of the conservation laws is included here. Chapter 3, SPH drawbacks: describes the limitations of SPH such as particle deficiency, consistency, zero energy modes, treatment of boundaries and the tensile instability problem. A rigorous stability analysis of continua and SPH particle equations is also presented in this chapter. Chapter 4, Total Lagrangian SPH. Continuum Mechanics considerations are discussed here; detailed derivations of SPH equations in a total Lagrangian framework are given together with potential corrections to the total Lagrangian SPH equations. Chapter 5, Total Lagrangian SPH algorithms and their implementation using FORTRAN. This chapter gives a brief introduction to explicit codes. It also provides flow charts describing the Total Lagrangian algorithms and their integration into the MCM code. Chapter 6, Total Lagrangian SPH code validation. This chapter includes problems of varying degrees of complexity. Examples are provided to illustrate how the Total Lagrangian SPH code compares to a conventional collocational SPH code. Cases are supplied for which the analytical solution is known, and the results compared with the SPH approximations in order to show the accuracy of the approximation. Some examples are supplied which provide a direct comparison between SPH and non linear FE results and SPH and experimental results. Chapter 7, Alternative formulation of SPH equations and improvements to the standard MCM code: Various modifications to the standard SPH code are presented. These modifications include the implementation of subroutines that make use of an alternative approach to ensure the conservation of mass law is met locally at every particle. The introduction of XSPH to achieve further stabilisation of the code was also carried out and some examples are provided. The theory behind an alternative form of the conservation of mass equation as proposed by Belytschko [4] is explained and its implementation into the SPH code is assessed through examples. Also, an alternative formulation of SPH equations based on the general theory of mixed Lagrangian-Eulerian formulations [35] is presented: these equations could serve as the foundation for future research in this field. Chapter 8, Conclusions are presented in this chapter. A brief literature review is provided at the beginning of each chapter as a means of introduction to the topic and a concise summary outlines the main points discussed.