Ed Tarleton

In this paper, we propose a finite element formulation for solving coupled mechanical/diffusion problems. In particular, we study hydrogen diffusion in metals and its impact on their mechanical behaviour (i.e. hydrogen embrittlement). The formulation can be used to model hydrogen diffusion through a material and its accumulation within different microstructural features of the material (dislocations, precipitates, interfaces, etc.). Further, the effect of hydrogen on the plastic response and cohesive strength of different interfaces can be incorporated. The formulation adopts a standard Galerkin method in the discretisation of both the diffusion and mechanical equilibrium equations. Thus, a displacement-based finite element formulation with chemical potential as an additional degree of freedom, rather than the concentration, is employed. Consequently, the diffusion equation can be expressed fundamentally in terms of the gradient in chemical potential, which reduces the continuity requirements on the shape functions to zero degree, $${\mathcal {C}}_{0}$$ C 0 , i.e. linear functions, compared to the $${\mathcal {C}}_{1}$$ C 1 continuity condition required when concentration is adopted. Additionally, a consistent interface element formulation can be achieved due to the continuity of the chemical potential across the interface—concentration can be discontinuous at an interface which can lead to numerical problems. As a result, the coding of the FE equations is more straightforward. The details of the physical problem, the finite element formulation and constitutive models are initially discussed. Numerical results for various example problems are then presented, in which the efficiency and accuracy of the proposed formulation are explored and a comparison with the concentration-based formulations is presented.

Tungsten is the main candidate material for plasma-facing armour components in future fusion reactors. In-service, fusion neutron irradiation creates lattice defects through collision cascades. Helium, injected from plasma, aggravates damage by increasing defect retention. Both can be mimicked using helium-ion-implantation. In a recent study on 3000 appm helium-implanted tungsten (W-3000He), we hypothesized helium-induced irradiation hardening, followed by softening during deformation. The hypothesis was founded on observations of large increase in hardness, substantial pile-up and slip-step formation around nano-indents and Laue diffraction measurements of localised deformation underlying indents. Here we test this hypothesis by implementing it in a crystal plasticity finite element (CPFE) formulation, simulating nano-indentation in W-3000He at 300 K. The model considers thermally-activated dislocation glide through helium-defect obstacles, whose barrier strength is derived as a function of defect concentration and morphology. Only one fitting parameter is used for the simulated helium-implanted tungsten; defect removal rate. The simulation captures the localised large pile-up remarkably well and predicts confined fields of lattice distortions and geometrically necessary dislocation underlying indents which agree quantitatively with previous Laue measurements. Strain localisation is further confirmed through high resolution electron backscatter diffraction and transmission electron microscopy measurements on cross-section lift-outs from centre of nano-indents in W-3000He.

A literature review of studies investigating the topography of nano-indents in ion-implanted materials reveals seemingly inconsistent observations, with reports of both pile-up and sink-in. This may be due to the crystallographic orientation of the measured sample point, which is often not considered when evaluating implantation-induced changes in the deformation response. Here, we explore the orientation dependence of spherical nano-indentation in pure and helium-implanted tungsten, considering grains with ⟨001⟩, ⟨110⟩, and ⟨111⟩ out-of-plane orientations. Atomic force microscopy of indents in unimplanted tungsten shows little orientation dependence. However, in the implanted material, a much larger, more localized pile-up is observed for ⟨001⟩ grains than for ⟨110⟩ and ⟨111⟩ orientations. Based on the observations for ⟨001⟩ grains, we hypothesize that a large initial hardening due to helium-induced defects is followed by localized defect removal and subsequent strain softening. A crystal plasticity finite element model of the indentation process, formulated based on this hypothesis, accurately reproduces the experimentally observed orientation-dependence of the indent morphology. The results suggest that the mechanism governing the interaction of helium-induced defects with glide dislocations is orientation independent. Rather, differences in the pile-up morphology are due to the relative orientations of the crystal slip systems, sample surface, and spherical indenter. This highlights the importance of accounting for crystallographic orientation when probing the deformation behavior of ion-implanted materials using nano-indentation.

Focussed Ion Beam (FIB) milling is a mainstay of nano-scale machining. By manipulating a tightly focussed beam of energetic ions, often gallium (Ga⁺), FIB can sculpt nanostructures via localised sputtering. This ability to cut solid matter on the nano-scale revolutionised sample preparation across the life, earth and materials sciences. Despite its widespread usage, detailed understanding of the FIB-induced structural damage, intrinsic to the technique, remains elusive. Here we examine the defects caused by FIB in initially pristine objects. Using Bragg Coherent X-ray Diffraction Imaging (BCDI), we are able to spatially-resolve the full lattice strain tensor in FIB-milled gold nano-crystals. We find that every use of FIB causes large lattice distortions. Even very low ion doses, typical of FIB imaging and previously thought negligible, have a dramatic effect. Our results are consistent with a damage microstructure dominated by vacancies, highlighting the importance of free-surfaces in determining which defects are retained. At larger ion fluences, used during FIB-milling, we observe an extended dislocation network that causes stresses far beyond the bulk tensile strength of gold. These observations provide new fundamental insight into the nature of the damage created and the defects that lead to a surprisingly inhomogeneous morphology.

Here, we analyse residual stress distributions obtained from various crystal systems using high resolution electron backscatter diffraction (EBSD) measurements. Histograms showing stress probability distributions exhibit tails extending to very high stress levels. We demonstrate that these extreme stress values are consistent with the functional form that should be expected for dislocated crystals. Analysis initially developed by Groma and co-workers for X-ray line profile analysis and based on the so-called “restricted second moment of the probability distribution” can be used to estimate the total dislocation density. The generality of the results are illustrated by application to three quite different systems, namely, face centred cubic Cu deformed in uniaxial tension, a body centred cubic steel deformed to larger strain by cold rolling, and hexagonal InAlN layers grown on misfitting sapphire and silicon carbide substrates.

The fracture behaviour of individual grain boundaries has been studied in order to understand the mechanisms controlling stress corrosion cracking in nuclear reactors. In particular, the role of oxidation in facilitating crack initiation and propagation has been reviewed. Nickel alloys from pressurized water reactors (PWRs) have been tested in simulated primary water conditions to induce grain boundary oxidation. Microcantilevers containing an oxidized grain boundary plane have been prepared and tested for fracture. The brittle nature of the oxide was demonstrated and the required stress to fracture measured.