C. Atkinson

On the stress intensity factors associated with cracks interacting with an interface between two elastic media

Abstract Closed form expressions are obtained for the stresses at a crack tip when a crack is approaching a welded boundary (or a free surface) and when it has just passed through the interface. The solutions which are obtained in terms of a small parameter, the distance from or through the interface, are given in explicit form for the mode 3 situation and for some mode 1 and 2 cases. The importance of the change of stress singularity when the crack meets the interface is demonstrated.

Some ribbon-like inclusion problems

Abstract A model of a thin ribbon inclusion is discussed and problems involving two inclusions and the interaction between an inclusion and a free boundary are also considered. For the problem of an inclusion intersecting a free boundary it is shown that analytical results can be obtained via the Wiener-Hopf technique. A numerical method is used for the ribbon in a half-plane problem and a useful comparison made between these numerical results and the analytical ones obtained when the ribbon actually meets the boundary.

Some calculations of the energy-release rate G for cracks in micropolar and couple-stress elastic media

A few years ago Sternberg and Muki [I] considered the effect of couple stresses on the stress concentration at the tip of a crack. They treated the problem of a finite crack in an infinite medium under conditions of plane strain with a uniform tension acting at infinity. The main conclusions were that at the crack tip the stress and couple stress fields had singularities of the same order, the order of the stress singularities being the same as those of the classical elastic problem. It was found that the limit of the stress intensity factor as 1 (the couple stress parameter) tended to zero was different to the usual elastic result (1 identically equal to zero). However, their approach which involved the numerical solution of integral equations did not give a precise evaluation of the coefficients involved in the stress and couple stress intensity factors. The couple stress theory has been criticised by Eringen [2] who replaces it by the micropolar theory of elasticity (see [2] for a review). In this note we consider the problem of a semi-infinite crack in a strip using both theories. These solutions which are accomplished by the use of a path independent integral demonstrate that G, the energy release rate, does tend to the elastic result as 1 + 0 even though the stress intensity factor may not.

Influence of pore pressure on the drilling response in low-permeability shear-dilatant rocks

This paper examines the dependence of the drilling specific energy E (the amount of energy required to drill a unit volume of rock) on the virgin pore pressure p0 in low-permeability fluid-saturated rocks, through a two-dimensional analysis of the cutting process. For this analysis, we consider a single cutter advancing at steady speed v and removing material over a constant depth of cut d from a fluid-saturated half-plane subjected to a surface pressure pm and a far-field pore pressure p0. On the basis of a simple failure mechanism involving a moving single shear plane (shear shock), we first establish that the specific energy E (here equal to the ratio of the cutting force parallel to the cutter velocity over the depth of cut) depends linearly on the difference between the surface pressure pm and the average pore pressure pb on the shear plane. Next, we address the dependence of pb on the virgin pore pressure p0 and the conditions for which there is cavitation on the shear plane (pb=0). The relationship between pb and p0 is determined by considering fluid mass balance across the shear plane. Different pore pressure regimes are identified that are controlled by a dimensionless number λ: low-speed, transient, and high-speed regime. In the high-speed regime, the rock in the shear zone is undrained and pressure drop induced by shear-induced dilatancy leads to cavitation. It is shown that the drilling conditions in low-permeability shear-dilatant rocks, such as shales, are in the high-speed regime, leading therefore to the conclusion that the specific energy E does not depend on the virgin pore pressure p0 in these rocks.

Multi‐asset portfolio optimization with transaction cost

The inclusion of transaction costs in the optimal portfolio selection and consumption rule problem is accomplished via the use of perturbation analyses. The portfolio under consideration consists of more than one risky asset, which makes numerical methods impractical. The objective is to establish both the transaction and the no‐transaction regions that characterize the optimal investment strategy. The optimal transaction boundaries for two and three risky assets portfolios are solved explicitly. A procedure for solving the N risky assets portfolio is described. The formulation used also reduces the restriction on the functional form of the utility preference.

The Growth Kinetics of Ledges During Phase Transformations: Multistep Interactions

An analysis of the growth characteristics of a train of ledges is presented, where volume diffusion in the parent phase is assumed to be the rate- controlling factor. First a train of steps of unequal height is considered where the step heights are assumed to be consistent with a steady-state motion so that each step moves with the same speed. It is possible to analyse this situation by asymptotic methods when the steps are either far apart or close together. Explicit results are given for both two- and three-step trains and it is shown how the step heights must vary if a given train is to move steadily at a specified speed. Trains of steps of equal height are also considered and an analysis is made of the relative velocities of such steps due to their interaction.

A crack at the interface between two power-law materials under plane strain loading

The stress fields at the tip of a crack lying along the interface between two incompressible power-law materials is considered for the case of plane strain loading. An asymptotic solution is constructed which has the property that the energy density W is of the order of 1/r, as r→0, in the material with the largest hardening parameter (material 1), where r denotes the radial distance from the crack tip. The solution in the other material (material 2) is such that W = o(1/r) as r→0. The solution in material 1 behaves, asymptotically, as if the other material is rigid. It is shown numerically that the near-tip fields are functions containing only one arbitrary constant, and this is corroborated analytically by using a small perturbation technique.

The interaction between the wellbore and pressure-induced fractures

A method based on modeling a crack by a continuous distribution of dislocations has been used to study the interaction between the wellbore and pressure-induced fractures. This method allows the stress intensity factor and opening of the fractures to be calculated as a function of fracture length, in-situ state of stress, fracture pressure and rock properties. It also allows one to determine the wellbore pressure during quasi-static propagation. Two configurations have been considered in this report; a fracture in which acts a pressure equal to the pressure in the hole (hydraulic fracturing using a perfectly penetrating fluid) and a stress free fracture (i.e. sleeve fracturing or hydraulic fracturing using a non penetrating fluid). These two cases give respectively the lower bound and upper bound for the pressure during a micro-hydraulic fracture experiment.The pressure behaviour has been found to be strongly dependent on fracture toughness and σ2 for short cracks, but also on the mode of loading (i.e. pressurized crack vs. stress free crack). The influence of fracture toughness and σ2 on the opening has been found to increase with fracture length when the fluid pressurizes the crack and to be relatively small during sleeve fracturing. These various behaviours could be used to determine the value of σ2, σ3, G/(1-ν) and fracture toughness if the crack opening at the wellbore is measured during a pressure test: the model is efficient enough to be used in an inversion procedure.The model has also shown that the use of Kirschs solution to predict breakdown pressure is only valid if the fluid does not penetrate the fracture.

On the dynamic stress and displacement field associated with a crack propagating across the interface between two media

Abstract Two half-spaces with different elastic constants are wellled together and subjected to a longitudinal shear strain at infinity so that the whole system is in a state of anti-plane strain. Suddenly the weld breaks and a crack begins to propagate at right angles to the interface into each of the two media simultaneously with a different velocity in each. This paper attempts to calculate the resulting stress and displacement fields.

Rational Solutions for the Time-Fractional Diffusion Equation

A uniform rational approximation of the Mittag-Leffler function is derived which serves as a global approximant, accounting for both the Taylor series for small arguments and asymptotic series for ...