Derivation of SPH equations in a moving referential coordinate system

The conventional SPH method uses kernel interpolation to derive the spatial semi-discretisation of the governing equations. These equations, derived using a straight application of the kernel interpolation method, are not used in practice. Instead the equations, commonly used in SPH codes, are heuristically modified to enforce symmetry and local conservation properties. This paper revisits the process of deriving these semi-discrete SPH equations. It is shown that by using the assumption of a moving referential coordinate system and moving control volume, instead of the fixed referential coordinate system and fixed control volume used in the conventional SPH method, a set of new semi- discrete equations can be rigorously derived. The new forms of semi-discrete equations are similar to the SPH equations used in practice. It is shown through numerical examples that the new rigorously derived equations give similar results to those obtained using the conventional SPH equations

Non-linear idealisation error analysis of an aerospace stiffened panel loaded in compression

The SAFE Structural Analysis procedure is an idealisation error control methodology devised for linear static finite element analysis. This study examines the applicability of this process to non-linear problems. The studied case is the collapse analysis of an aircraft stiffened panel loaded in compression. This article presents the critical investigation of important modelling assumptions, including the joint modelling, boundary conditions, geometrical imperfections and scattering in material parameters. Potential error sources are identified and then analysed using the non-linear finite element solver ABAQUS. The analysis derived an improved finite element model and concrete idealisation error estimates. The finally simulated failure behaviour corresponds well to the data measured in the test

Advisory system development for reliable FEM modelling in aerospace

This paper aims to describe the development of an advisory system that helps building sound finite element (FE) models from computer-aided design data, with actual uncertainty levels expressed by error values in per cent, as today there is no widely accepted tool for FE idealisation error control

The nonlocal, local and mixed forms of the SPH method

Publisher Copyright: © 2021 The Author(s)From its early days the SPH method has been criticised for its shortcomings namely tensile instability and consistency. Without thorough understanding of the method attempts were made to make the classical SPH method consistent and stable which resulted in the local and Total Lagrangian forms of SPH similar to the finite element method. In this paper we derived and analysed a consistent nonlocal SPH which has similarity with Bazant's imbricate continuum. In addition, the paper provides comparison and discussion of different SPH forms including: Classical SPH, Nonlocal, Local and Mixed SPH. The partition of unity approach was used to define the following two mixed forms: Local–Nonlocal and Local–Classical SPH. These mixed forms were intended for modelling of physical processes characterised with local and nonlocal effects (local and nonlocal constitutive equations), e.g. progressive damage and failure. The stabilising effect of the Local form on the Classical SPH, which is inherently unstable (tensile instability), are also illustrated. The stability analysis, presented in appendices A and B, demonstrate stability of the continuous and discrete form of the nonlocal SPH based on Eulerian kernels for elastic continuum.Peer reviewe

Modelling of behaviour of metals at high strain rates

The aim of the work presented in this thesis was to produce the improvement of the existing simulation tools used for the analysis of materials and structures, which are dynamically loaded and subjected to the different levels of temperatures and strain rates. The main objective of this work was development of tools for modelling of strain rate and temperature dependant behaviour of aluminium alloys, typical for aerospace structures with pronounced orthotropic properties, and their implementation in computer codes. Explicit finite element code DYNA3D has been chosen as numerical test-bed for implementation of new material models. Constitutive model with an orthotropic yield criterion, damage growth and failure mechanism has been developed and implemented into DYNA3D. Second important aspect of this work was development of relatively simple experimental methods for characterization of engineering materials, and extensive experimental work has been undertaken. Tensile test has been used for the characterisation of two aluminium alloys, at different levels of the strain rates and temperatures, and for three different orientations of materials. The results from these tests allowed derivation of material constants for constitutive models and lead to a better understanding of aluminium alloy behaviour. Procedures for derivation of parameters for temperature and strain rate dependant strength models were developed and parameters for constitutive relations were derived on the basis of uniaxial tensile tests. Taylor cylinder impact test was used as a validation experiment. This test was used to validate the implementation, and accuracy of material model in computer code. At the end of each incremental development, validation of the constitutive material model has been performed through numerical simulations of Taylor cylinder impact test, where simulation results have been compared with the experimental post-test geometries in terms of major and minor side profiles and impact-interface footprints. Plate impact test has been used to determine the material properties at high strain rate, and to investigate damage evolution in impact-loaded material. Initially the material model has been designed as a temperature and strain rate dependant strength model in a simple isotopic form, which then has been tested and verified against the experimental results. Coupling of the Hill’s orthotropic yield criterion with isotropic, temperature and strain rate dependant, hardening material model, has been chosen to suit the orthotropic behaviour. Method for calibration of orthotropic yield criterion has been developed and parameters have been identified for the orthotropic model under the associated flow rule assumption and in case of plane stress on the basis of tensile and cylinder impact tests. The complexity of the model has been further increased through coupling of hardening model with orthotropic yield criterion including damage evolution and failure criteria. The constitutive model was developed within the general framework of continuum thermodynamics for irreversible processes, and plate impact test and tensile tests have been used for determination of parameters for damage part of the new material model.EThOS - Electronic Theses Online ServiceAirbus UKGBUnited Kingdo

Towards better understanding of the Smoothed Particle Hydrodynamic Method

Numerous approaches have been proposed for solving partial differential equations; all these methods have their own advantages and disadvantages depending on the problems being treated. In recent years there has been much development of particle methods for mechanical problems. Among these are the Smoothed Particle Hydrodynamics (SPH), Reproducing Kernel Particle Method (RKPM), Element Free Galerkin (EFG) and Moving Least Squares (MLS) methods. This development is motivated by the extension of their applications to mechanical and engineering problems. Since numerical experiments are one of the basic tools used in computational mechanics, in physics, in biology etc, a robust spatial discretization would be a significant contribution towards solutions of a number of problems. Even a well-defined stable and convergent formulation of a continuous model does not guarantee a perfect numerical solution to the problem under investigation. Particle methods especially SPH and RKPM have advantages over meshed methods for problems, in which large distortions and high discontinuities occur, such as high velocity impact, fragmentation, hydrodynamic ram. These methods are also convenient for open problems. Recently, SPH and its family have grown into a successful simulation tools and the extension of these methods to initial boundary value problems requires further research in numerical fields. In this thesis, several problem areas of the SPH formulation were examined. Firstly, a new approach based on ‘Hamilton’s variational principle’ is used to derive the equations of motion in the SPH form. Secondly, the application of a complex Von Neumann analysis to SPH method reveals the existence of a number of physical mechanisms accountable for the stability of the method. Finally, the notion of the amplification matrix is used to detect how numerical errors propagate permits the identification of the mechanisms responsible for the delimitation of the domain of numerical stability. By doing so, we were able to erect a link between the physics and the numerics that govern the SPH formulation.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

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

This thesis makes use of an alternative SPH formulation, the Total Lagrangian formulation, to characterise dynamic events in 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.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

MATERIAL FAILURE MODELLING IN METALS AT HIGH STRAIN RATES

To account for the physical mechanisms of failure, the concept of thermal activation of damage and failure has been adopted as basis for this material model development. This basic assumption makes the proposed approach compatible with the Mechanical Threshold Stress (MTS) model, which was used as the strength part of the proposed constitutive model. The developments were incorporated into public domain DYNA3D. In order to validate the model, a series of FE simulations of plate impact experiments were performed for OFHC Cu. The numerical analysis results clearly demonstrate the ability of the model to predict the spall process and experimentally observed tensile damage and failure. The model allows simulation of high strain rate deformation processes and dynamic failure in tension for wide range of temperatures. The model is able to reproduce typical longitudinal stress reloading observed in plate impact tests, which is caused by the creation of the internal free surface. Plate impact tests used for model validation were performed on a single-stage gas gun. Longitudinal stresses were measured with stress gauges

Modelling of Ductile Failure in Metals under High Velocity Impact Loading

The objective of the work presented in this paper was to generate the thermodynamically consistent coupled thermo-elastic-plastic damage model of solid media at a macroscopic level applicable to hypervelocity impacts. The model is based on the thermodynamics of irreversible processes and the assumption that damage within a continuum can be represented as a damage tensor ωij [1], [4]. This allows for definition of two scalars that are ω =ωkk/3 (the volume damage) [2], [3] and α = SQR[ω′ijω′ij] (a norm of the damage tensor deviator ω′ij =ωij −ωδij ) [4]. The parameter ω describes the accumulation of micro-pore type damage (which may disappear under compression) and the parameter α describes the shear related damage. The parameter ω may be considered as a volume content of micro-pores in the material. In the damage-free material we have ω =α = 0 ; if damage is accumulated, ω and α increase in such a manner that they remain less than one. This damage evolution is then coupled to a strain, strain-rate and temperature dependent plasticity model. The initiation of failure is based on a critical value of a specific dissipation function. The performance of the model in modelling high velocity impacts is illustrated by few numerical examples