The Ghost Solid Methods for the Solid-Solid Interface

Ph.DDOCTOR OF PHILOSOPH

On the stability of the μ(I)\mu(I)-rheology for granular flow

This article deals with the Hadamard instability of the so-called $\mu(I)$ model of dense rapidly-sheared granular flow, as reported recently by Barker et al. (2015,this journal, ${\bf 779}$, 794-818). The present paper presents a more comprehensive study of the linear stability of planar simple shearing and pure shearing flows, with account taken of convective Kelvin wave-vector stretching by the base flow. We provide a closed form solution for the linear stability problem and show that wave-vector stretching leads to asymptotic stabilization of the non-convective instability found by Barker et al. We also explore the stabilizing effects of higher velocity gradients achieved by an enhanced-continuum model based on a dissipative analog of the van der Waals-Cahn-Hilliard equation of equilibrium thermodynamics. This model involves a dissipative hyper-stress, as the analog of a special Korteweg stress, with surface viscosity representing the counterpart of elastic surface tension. Based on the enhanced continuum model, we also present a model of steady shear bands and their non-linear stability against parallel shearing. Finally, we propose a theoretical connection between the non-convective instability of Barker et al. and the loss of generalized ellipticity in the quasi-static field equations. Apart from the theoretical interest, the present work may suggest stratagems for the numerical simulation of continuum field equations involving the $\mu(I)$ rheology and variants thereof.Comment: 30 pages, 13 figure

A Moving Boundary Flux Stabilization Method for Cartesian Cut-Cell Grids using Directional Operator Splitting

An explicit moving boundary method for the numerical solution of time-dependent hyperbolic conservation laws on grids produced by the intersection of complex geometries with a regular Cartesian grid is presented. As it employs directional operator splitting, implementation of the scheme is rather straightforward. Extending the method for static walls from Klein et al., Phil. Trans. Roy. Soc., A367, no. 1907, 4559-4575 (2009), the scheme calculates fluxes needed for a conservative update of the near-wall cut-cells as linear combinations of standard fluxes from a one-dimensional extended stencil. Here the standard fluxes are those obtained without regard to the small sub-cell problem, and the linear combination weights involve detailed information regarding the cut-cell geometry. This linear combination of standard fluxes stabilizes the updates such that the time-step yielding marginal stability for arbitrarily small cut-cells is of the same order as that for regular cells. Moreover, it renders the approach compatible with a wide range of existing numerical flux-approximation methods. The scheme is extended here to time dependent rigid boundaries by reformulating the linear combination weights of the stabilizing flux stencil to account for the time dependence of cut-cell volume and interface area fractions. The two-dimensional tests discussed include advection in a channel oriented at an oblique angle to the Cartesian computational mesh, cylinders with circular and triangular cross-section passing through a stationary shock wave, a piston moving through an open-ended shock tube, and the flow around an oscillating NACA 0012 aerofoil profile.Comment: 30 pages, 27 figures, 3 table

Modelling Plasticity in Nanoscale Contact

The problem of mechanical contact is a truly multiscale one. Atomistic effects that violate continuum theory dominate the deformations of contacting asperities, while the interactions between distant asperities occur through long-range elasticity. This thesis concentrates on the numerical modelling of nanoscale frictional contact between crystalline metals by using both single-scale atomistic methods and improving concurrent multiscale methods. A novel approach to quantify frictional work and the energy associated with plastic activity in \md simulations is presented. In combination with a statistical criterion to determine the significance of simulation box size, microstructure and sliding rate effects on the frictional quantities such as the friction coefficient and stored plastic energies, the method is used in a large parametric molecular dynamics study of single-asperity nanoscratch on monocrystalline and polycrystalline aluminium substrates. Some fundamental differences in the friction mechanisms between monocrystalline and polycrystalline substrates are presented. The study shows the limitations of single-scale modelling and highlights the importance of developing appropriate multiscale methods for nanoscale plasticity. One such method is the Coupled Atomistics and Discrete Dislocations (CADD), which previously only existed for two-dimensional problems. A three-dimensional version of the CADD method is presented theoretically as well as a detailed practical road map for its efficient implementation. The foundations of three-dimensional CADD are presented using practical test cases. CADD avoids ghost forces at the coupling interfaces through displacement-coupling. I reveal that such displacement-coupling methods generally exhibit an inherent dynamic instability which makes them particularly ill suited for finite temperature calculations, despite their wide use. The instability is analysed in detail. Multiple remedies to manage it are discussed and a fundamental solution to the underlying problem is presented in the form of a new coupling method

A specialised finite element for simulating self-healing quasi-brittle materials

A new specialised finite element for simulating the cracking and healing behaviour of quasi-brittle materials is presented. The element employs a strong discontinuity approach to represent displacement jumps associated with cracks. A particular feature of the work is the introduction of healing into the element formulation. The healing variables are introduced at the element level, which ensures consistency with the internal degrees freedom that represent the crack; namely, the crack opening, crack sliding and rotation. In the present work, the element is combined with a new cohesive zone model to simulate damage-healing behaviour and implemented with a crack tracking algorithm. To demonstrate the performance of the new element and constitutive models, a convergence test and two validation examples are presented that consider the response of a vascular self-healing cementitious material system for three different specimens. The examples show that the model is able to accurately capture the cracking and healing behaviour of this type of self-healing material system with good accuracy

High performance simulations of yield stress fluids in a structured adaptive mesh refinement framework with embedded boundaries

Viscoplastic fluids are a class of non-Newtonian liquids characterised by their yield stress. Unless an external stress is applied which is larger than this threshold value, the fluid does not flow, but exhibits rigid body behaviour. Above the yield stress, applied forces cause viscous deformation. Such fluids play important roles in a range of fields, notably in wellbore drilling, which is the application that motivated this project. One aspect of this operation requires displacement of drilling fluid by cement in the annulus between casing and geological surroundings, and both of these fluids are viscoplastics. Ensuring that this is done properly is of utmost importance to the overall safety of the drilling operation. Often, numerical simulations are the only viable way of experimenting with the effect of drilling parameters and fluid properties on the flow configuration and resulting behaviour. Unfortunately, the presence of a yield stress leads to a singularity in the apparent viscosity at zero strain. This causes substantial computational expense for the algorithms used to simulate fluid flow numerically, even when regularisation techniques are employed to alleviate the problem. Consequently, most published results in the literature on computational viscoplasticity has been restricted to two-dimensional and steady-state flows. In an attempt to address this, we have applied state-of-the-art techniques from high-performance computational fluid dynamics to the viscoplastic flow problem. Specifically, we utilise spatio-temporal adaptive mesh refinement on structured meshes in this context for the first time. This is achieved through the software framework AMReX, which includes state-of-the-art numerical tools for solving partial differential equations with optimal parallel scaling. The ability to rapidly simulate unsteady viscoplastic flow problems in three dimensions is demonstrated by novel numerical experiments in a lid-driven cavity. In order to investigate flows in more interesting domain geometries and around objects, an embedded boundary algorithm has been developed which works alongside the viscoplastic flow solver. We show how this methodology can be utilised to simulate flow inside non-rectangular objects, and investigate fully three-dimensional viscoplastic flow past several shapes of bodies for the first time.EPSRC Centre for Doctoral Training in Computational Methods for Materials Science grant number EP/L015552/1 BP International Centre for Advanced Materials (BP-ICAM) Extra support towards living expenses through the Aker Scholarshi

Computational Approaches to Localized Deformation Within the Lithosphere and for Crust-Mantle Interactions

The thesis addresses selected problems related to localized deformation of the solid Earth’s lithosphere that stem from non-uniform strengths or emerge from non-linear rheologies. A new code has been developed to model the spontaneous localization through strain-weakening plasticity. A code coupling technique is introduced as an attempt to efficiently solve multi-material and multi-physics problems like crust-mantle interactions. We first address a problem of localized deformation that is caused by pre-existing heterogeneities. Specifically, the effects of laterally varying viscous strength on the Cenozoic extension of the northern Basin and Range are investigated using numerical models. Three-dimensional viscous flow models with imposed plate motions and localized zones of low viscosity show that strain rates are concentrated in weak zones with adjacent blocks experiencing little deformation. This result can explain the geodetically discovered concentrated strain in the eastern part of the northern Basin and Range as the high strains are a response to far field plate motions within a locally less viscous mantle. The low viscosity of mantle is consistent with the low seismic velocities in the region. As an instance of spontaneously emergent localized deformations, brittle deformations in oceanic lithosphere are investigated next. We developed a Lagrangian finite difference code, SNAC, to investigate this class of problems. Brittle deformations are modeled as localized plastic strain. The detailed algorithm of SNAC is presented in Appendix A. The spacing of fracture zones in oceanic lithosphere is numerically explored. Numerical models represent a ridge-parallel cross-section of young oceanic lithosphere. An elasto-visco-plastic rheology can induce brittle deformation or creep according to the local temperature. The spacing of localized plastic zones, corresponding to fracture zones, decreases as crustal thickness increases. The stronger creep strength raises the threshold value of crustal thickness: If the crust is thinner than the threshold, the brittle deformation can evolve into primary cracks. Plastic flow rules are parametrized by the dilation angle. If the dilatational deformation is allowed in the plastic flow rules (dilation angle>0°), the primary cracks tend to be vertical; otherwise, a pair of primary cracks form a graben. The modeling results are compatible with the correlation between crustal thickness and the spacing of fracture zones found in different regions such as the Reykjanes ridge and the Australian Antarctic Discordance. Three-dimensional (3D) numerical models are used to find the mechanics responsible for the various patterns made by the segments of the mid-ocean ridges and the structures connecting them. The models are initially loaded with thermal stresses due to the cooling of oceanic lithosphere and prescribed plate motions. The two driving forces are comparable in magnitude and the thermal stresses can exert ridge-parallel forces when selectively released by ridges and ridge-parallel structure. Represented by localized plastic strain, ridge segments interact in two different modes as they propagate towards each other: An overlapping mode where ridge segments overlap and bend toward each other and a connecting mode where two ridge segments are connected by a transform-like fault. As the ratio of thermal stress to spreading-induced stress (γ) increases, the patterns of localized plastic strain change from the overlapping to connecting mode. Rate effects are taken into account by the spreading rate normalized by a reference-cooling rate (Pe′) and the ratio of thermal stress to the reference spreading-induced stresses (γ′). The stability fields of the two modes are unambiguously defined by Pe′ paired with γ'. Crust and mantle are distinct in terms of composition and rheology. To study the combined response of crust and mantle, it is necessary to solve multi-material and multi-physics problems that are numerically challenging. As an efficient way of solving such a problem, we introduce a code coupling technique. We adapt Pyre, a framework allowing distinct codes to exchange variables through shared interfaces, to the coupling of SNAC, a Lagrangian code for lithospheric dynamics, and CitcomS, an Eulerian code for mantle convection. The continuity of velocities and tractions and no-slip conditions are imposed on the interfaces. The benchmarks against analytic solutions to the bending of a thin plate verifies that SNAC gives an accurate solution for the given traction boundary condition. It is also shown that Pyre can correctly handle the data exchanges at the interfaces. In a preliminary high-resolution model, an elasto-visco-plastic lithosphere is coupled to a Newtonian viscous mantle. This coupled model shows a steady growth of dome in the lithosphere directly above a hot sphere placed in the mantle. However, the two coupled codes incur unnecessarily high numerical costs because they use different methods for time integration.</p

Upscaling from Atomistic Models to Higher Order Gradient Continuum Models for Crystalline Solids

In this work a new upscaling scheme for the derivation of a continuum mechanical model from an atomistic model for crystalline solids is developed. The scheme, called the inner expansion technique, is based on a Taylor series expansion of the deformation function and leads to a continuum mechanical model which involves higher order derivatives. It provides an approximation of the atomistic model within the quasi-continuum regime and allows to capture the microscopic material properties and the discreteness effects of the underlying atomistic system up to an arbitrary order. The quality of approximation is investigated for the model problem of an atomic chain with different types of potentials, including many-body potentials. The outcome of the inner expansion technique is numerically compared to other upscaling techniques, namely the classical thermodynamic limit and the direct expansion technique. It is shown that our technique carries over certain properties such as convexity from the atomistic to the continuum mechanical level, which results in well-posed problems on the continuum mechanical level. Furthermore, macroscopic approximation techniques are discussed to reduce the complexity of the continuum model. The upscaling technique is applied to the Stillinger-Weber potential for crystalline silicon and to the potential given by the Embedded-Atom Method (EAM) for shape memory alloys (SMA). Numerical simulations of the dynamic response of a silicon crystal and of one-way and two-way SMA micro-actuators are performed