Allan F. Bower

Indentation of a power law creeping solid

The aim of this paper is to establish a rigorous theoretical basis for interpreting the results of hardness tests on creeping specimens. We investigate the deformation of a creeping half-space with uniaxial stress-strain behaviour ⋵ = ⋵0 (σ/σ0)m, which is indented by a rigid punch. Both axisymmetric and plane indenters are considered. The shape of the punch is described by a general expression which includes most indenter profiles of practical importance. Two methods are used to solve the problem. The main results are found using a transformation method suggested by R. Hill. It is shown that the creep indentation problem may be reduced to a form which is independent of the geometry of the punch, and depends only on the material properties through m. The reduced problem consists of a nonlinear elastic half-space, which is indented to a unit depth by a rigid flat punch of unit radius (in the axisymmetric case), or unit semi-width (in the plane case). Exact solutions are given for m = 1 and m = ∞. For m between these two limits, the reduced problem has been solved using the finite element method. The results enable the load on the indenter and the contact radius to be calculated in terms of the indentation depth and rate of penetration. The stress, strain and displacement fields in the half-space may also be deduced. The accuracy of the solution is demonstrated by comparing the results with full-field finite element calculations. The predictions of the theory are shown to be consistent with experimental observations of hardness tests on creeping materials reported in the literature.

A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a lithium-ion half-cell

We formulate the continuum field equations and constitutive equations that govern deformation, stress, and electric current flow in a Li-ion half-cell. The model considers mass transport through the system, deformation and stress in the anode and cathode, electrostatic fields, as well as the electrochemical reactions at the electrode/electrolyte interfaces. It extends existing analyses by accounting for the effects of finite strains and plastic flow in the electrodes, and by exploring in detail the role of stress in the electrochemical reactions at the electrode–electrolyte interfaces. In particular, we find that that stress directly influences the rest potential at the interface, so that a term involving stress must be added to the Nernst equation if the stress in the solid is significant. The model is used to predict the variation of stress and electric potential in a model 1-D half-cell, consisting of a thin film of Si on a rigid substrate, a fluid electrolyte layer, and a solid Li cathode. The predicted cycles of stress and potential are shown to be in good agreement with experimental observations.

A simple technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces

Numerical simulations of crack initiation which use a cohesive zone law to model a weak interface in the solid are often limited by the occurrence of an elastic snap-back instability. At the point of instability, quasi-static finite element computations are unable to converge to an equilibrium solution, which usually terminates the calculation and makes it impossible to follow the post-instability behaviour. In this paper, we show that such numerical difficulties can easily be avoided by introducing a small viscosity in the constitutive equations for the cohesive interface. Simple boundary value problems are used to develop guidelines for selecting appropriate values of viscosity in numerical simulations involving crack nucleation and growth. As a representative application, we model crack nucleation at the interface between an elastic thin film and an elastic–plastic substrate, which is subjected to contact loading.

In Situ Measurements of Stress-Potential Coupling in Lithiated Silicon

An analysis of the dependence of electric potential on the state of stress of a lithiated-silicon electrode is presented. Based on the Larche and Cahn chemical potential for a solid solution, a thermodynamic argument is made for the existence of the stresspotential coupling in lithiated silicon; based on the known properties of the material, the magnitude of the coupling is estimated to be 60 mV/GPa in thin-film geometry. An experimental investigation is carried out on silicon thin-film electrodes in which the stress is measured in situ during electrochemical lithiation and delithiation. By progressively varying the stress through incremental delithiation, the relation between stress change and electric-potential change is measured to be 100–120 mV/GPa, which is of the same order of magnitude as the prediction of the analysis. The importance of the coupling is discussed in interpreting the hysteresis observed in the potential vs state-of-charge plots and the role of stress in modifying the maximum charge capacity of a silicon electrode under stress.

The influence of strain hardening on cumulative plastic deformation in rolling and sliding line contact

Abstract The influence of strain hardening on the cumulative plastic deformation (ratchetting) which takes place in repeated rolling and sliding contacts has been assessed by the use of a non-linear kinematic hardening law proposed and tested by B ower (J. Mech. Phys. Solids37,455, 1989). Both the sub-surface flow, which occurs at low traction coefficients ( 0.25), have been investigated. Two materials have been studied: hard-drawn copper and rail steel. Good correlation was found for copper between the theory and rolling contact experiments.

Cyclic hardening properties of hard-drawn copper and rail steel

Abstract A cyclic hardening law due to Armstrong and Frederick (CEGB Report RD/B/N731, 1966) has been extended to describe plastic strain accumulation (ratchetting) in hard-drawn copper and rail steel. The four parameters of the theoretical model were determined from a single uniaxial test on each material, in which unequal tension and compression were applied. Using these parameters the model was found to give good predictions of the ratchetting rate measured in non-proportional cycles of tension-torsion-compression, which are representative of the stress cycles experienced by surface elements in rolling and sliding contact.

A three-dimensional analysis of crack trapping and bridging by tough particles

The toughness of a brittle material may be substantially improved by adding small quantities of tough particles to the solid. Three mechanisms may be responsible. Firstly, the front of a crack propagating through the solid can be trapped by the particles, causing it to bow out between them. Secondly, the particles may remain intact in the wake of the crack, thereby pinning its faces and reducing the crack tip stress intensity factors. Finally, the toughness may be enhanced by frictional energy dissipation as particles are pulled out in the wake of the crack. This paper estimates the improvement in toughness that might be expected due to these mechanisms, by means of a three-dimensional model. The analysis considers a semi-infinite crack propagating through a brittle matrix material, which contains a regular distribution of tough particles. Particles in the wake of the crack are modelled by finding an appropriate distribution of point forces that pin the crack faces; and the effect of the crack bowing between obstacles is included by means of an incremental perturbation method based on work byRice [J. Appl. Mech.56, 619 (1985)]. The calculation predicts the shape of the crack as it propagates through the solid; the resulting R-curve behaviour; and the length of the bridged zone in the wake of the crack.

Diffuse interface model for electromigration and stress voiding

We report a diffuse interface or phase field model for simulating electromigration and stress-induced void evolution in interconnect lines. Our approach is based on the introduction of an order parameter field to characterize the damaged state of the interconnect. The order parameter takes on distinct uniform values within the material and the void, varying rapidly from one to the other over narrow interfacial layers associated with the void surface. The evolution of this order parameter is shown to be governed by a form of the Cahn–Hilliard equation. An asymptotic analysis of the equation demonstrates, as intended, that the zero contour of the order parameter tracks the motion of a void evolving by diffusion under the coupled effects of stresses and the “electron wind” force. An implicit finite element scheme is used to solve the modified Cahn–Hilliard equation, together with equations associated with the accompanying mechanical and electrical problems. The diffuse interface model is applied to simulate ...

A finite element analysis of the motion and evolution of voids due to strain and electromigration induced surface diffusion

Abstract Microelectronic circuits often fail because cracks and voids cause open circuits in their interconnects. Many of the mechanisms of failure are believed to be associated with diffusion of material along the surfaces, interfaces or grain boundaries in the line; material may also flow through the lattice of the crystal. The diffusion is driven by variations in elastic strain energy and stress in the solid, and by the flow of electric current. To predict the conditions necessary for failure to occur in an interconnect, one must account for the influence of both deformation and electric current flow through the interior of the solid, and also for the effects of mass flow. To this end, we describe a two dimensional finite element method for computing the motion and evolution of voids by surface diffusion in an elastic, electrically conducting solid. Various case studies are presented to demonstrate the accuracy and capabilities of the method, including the evolution of a void towards a circular shape due to diffusion driven by surface energy, the migration and evolution of a void in a conducting strip due to electromigration induced surface diffusion, and the evolution of a void in an elastic solid due to strain energy driven surface diffusion.

Plastic flow and shakedown of the rail surface in repeated wheel-rail contact

Abstract Recent progress in modelling the near-surface plastic deformation caused by repeated wheel-rail contact is reported. A simple non-linear kinematic hardening law is described which is capable of predicting the response of rail steel to sliding contact loading. This hardening law is incorporated into a theory of elastic-plastic sliding contact. Particular attention is paid to the deformation at the surface caused by high tractive loads, since this appears to be the most common form of deformation leading to wear and fatigue of railway track. Shakedown limits are calculated for various combinations of contact loading relevant to railway practice, and an approximate technique is used to calculate the plastic flow that occurs when the shakedown limit is exceeded. The predictions of the theory are compared with experimental measurements of plastic flow in disks.