A multi-scale atomistic-continuum modelling of crack propagation in a two-dimensional macroscopic plate

A novel multi-scale seamless model of brittle-crack propagation is proposed and applied to the simulation of fracture growth in a two-dimensional Ag plate with macroscopic dimensions. The model represents the crack propagation at the macroscopic scale as the drift-diffusion motion of the crack tip alone. The diffusive motion is associated with the crack-tip coordinates in the position space, and reflects the oscillations observed in the crack velocity following its critical value. The model couples the crack dynamics at the macroscales and nanoscales via an intermediate mesoscale continuum. The finite-element method is employed to make the transition from the macroscale to the nanoscale by computing the continuum-based displacements of the atoms at the boundary of an atomic lattice embedded within the plate and surrounding the tip. Molecular dynamics (MD) simulation then drives the crack tip forward, producing the tip critical velocity and its diffusion constant. These are then used in the Ito stochastic calculus to make the reverse transition from the nanoscale back to the macroscale. The MD-level modelling is based on the use of a many-body potential. The model successfully reproduces the crack-velocity oscillations, roughening transitions of the crack surfaces, as well as the macroscopic crack trajectory. The implications for a 3-D modelling are discussed.

Multiscale numerical modelling of crack propagation in two-dimensional metal plate

A novel multiscale model of brittle crack propagation in an Ag plate with macroscopic dimensions has been developed. The model represents crack propagation as stochastic drift-diffusion motion of the crack tip atom through the material, and couples the dynamics across three different length scales. It integrates the nanomechanics of bond rupture at the crack tip with the displacement and stress field equations of continuum based fracture theories. The finite element method is employed to obtain the continuum based displacement and stress fields over the macroscopic plate, and these are then used to drive the crack tip forward at the atomic level using the molecular dynamics simulation method based on many-body interatomic potentials. The linkage from the nanoscopic scale back to the macroscopic scale is established via the Ito stochastic calculus, the stochastic differential equation of which advances the tip to a new position on the macroscopic scale using the crack velocity and diffusion constant obtained on the nanoscale. Well known crack characteristics, such as the roughening transitions of the crack surfaces, crack velocity oscillations, as well as the macroscopic crack trajectories, are obtained.

Temporal modelling of short-term rainfall using Cox processes

We study two marked point process models based on the Cox process. These models are used to describe the probabilistic structure of the rainfall intensity process. Mathematical formulation of the models is described and some second-moment characteristics of the rainfall depth, and aggregated processes are considered. The derived second-order properties of the accumulated rainfall amounts at different levels of aggregation are used in order to examine the model fit. A brief data analysis is presented. Copyright © 1998 John Wiley & Sons, Ltd.

Heuristics for short route job shop scheduling problems

We consider two “minimum”NP-hard job shop scheduling problems to minimize the makespan. In one of the problems every job has to be processed on at most two out of three available machines. In the other problem there are two machines, and a job may visit one of the machines twice. For each problem, we define a class of heuristic schedules in which certain subsets of operations are kept as blocks on the corresponding machines. We show that for each problem the value of the makespan of the best schedule in that class cannot be less than 3/2 times the optimal value, and present algorithms that guarantee a worst-case ratio of 3/2.

A multi-scale numerical modelling of crack propagation in a 2D metallic plate

A new multi-scale model of brittle fracture growth in an Ag plate with macroscopic dimensions is proposed in which the crack propagation is identified with the stochastic drift-diffusion motion of the crack-tip atom through the material. The model couples molecular dynamics simulations, based on many-body interatomic potentials, with the continuum-based theories of fracture mechanics. The Ito stochastic differential equation is used to advance the tip position on a macroscopic scale before each nano-scale simulation is performed. Well-known crack characteristics, such as the roughening transitions of the crack surfaces, as well as the macroscopic crack trajectories are obtained.

Finite element modelling of crack detection tests

Four non-destructive tests for determining the length of fatigue cracks within the solder joints of a 2512 surface mount resistor are investigated. The sensitivity of the tests is obtained using finite element analysis with some experimental validation. Three of the tests are mechanically based and one is thermally based. The mechanical tests all operate by applying different loads to the PCB and monitoring the strain response at the top of the resistor. The thermal test operates by applying a heat source underneath the PCB, and monitoring the temperature response at the top of the resistor. From the modelling work done, two of these tests have shown to be sensitive to cracks. Some experimental results are presented but further work is required to fully validate the simulation results.

Assessing the performance of crack detection tests for solder joints

This paper presents both modelling and experimental test data to characterise the performance of four non-destructive tests. The focus is on determining the presence and rough magnitude of thermal fatigue cracks within the solder joints for a surface mount resistor on a strip of FR4 PCB. The tests all operate by applying mechanical loads to the PCB and monitoring the strain response at the top of the resistor. The modelling results show that of the four tests investigated, three are sensitive to the presence of a crack in the joint and its magnitude. Hence these tests show promise in being able to detect cracking caused by accelerated testing. The experimental data supports these results although more validation is required.

Computer simulation of crack propagation in power electronics module solder joints

A numerical modelling method for the analysis of solder joint damage and crack propagation has been described in this paper. The method is based on the disturbed state concept. Under cyclic thermal-mechanical loading conditions, the level of damage that occurs in solder joints is assumed to be a simple monotonic scalar function of the accumulated equivalent plastic strain. The increase of damage leads to crack initiation and propagation. By tracking the evolution of the damage level in solder joints, crack propagation path and rate can be simulated using Finite Element Analysis method. The discussions are focused on issues in the implementation of the method. The technique of speeding up the simulation and the mesh dependency issues are analysed. As an example of the application of this method, crack propagation in solder joints of power electronics modules under cyclic thermal-mechanical loading conditions has been analyzed and the predicted cracked area size after 3000 loading cycles is consistent with experimental results.

Using EMGs and kinematics data to study the take-off technique of experts and novices for a pole vaulting short run-up educational exercise

This study attempts to characterise the electromyographic activity and kinematics exhibited during the performance of take-off for a pole vaulting short run-up educational exercise, for different expertise levels. Two groups (experts and novices) participated in this study. Both groups were asked to execute their take-off technique for that specific exercise. Among the kinematics variables studied, the knee, hip and ankle angles and the hip and knee angular velocities were significantly different. There were also significant differences in the EMG variables, especially in terms of (i) biceps femoris and gastrocnemius lateralis activity at touchdown and (ii) vastus lateralis and gastrocnemius lateralis activity during take-off. During touchdown, the experts tended to increase the stiffness of the take-off leg to decrease braking. Novices exhibited less stiffness in the take-off leg due to their tendency to maintain a tighter knee angle. Novices also transferred less energy forward during take-off due to lack of contraction in the vastus lateralis, which is known to contribute to forward energy transfers. This study highlights the differences in both groups in terms of muscular and angular control according to the studied variables. Such studies of pole vaulting could be useful to help novices to learn expert's technique.

Singular stresses at the tip of a crack terminating at frictional interface of fibre reinforced composites

The stress singularities at the tip of a crack that terminates at a frictional interface between two layers in anisotropic composites are investigated. The order of stress singularities is determined by solving the characteristic equations obtained from the boundary conditions and the frictional interface conditions for the cases concerned. The interface is assumed to be governed by Coulomb's law of friction. Numerical results are presented for the cases with a crack terminating at a frictional interface of a fibre reinforced composite, and it is shown that there is a big difference of stress singularities between cases with and without considering friction along the interface.

Fostering 'habits of reflection, independent study and free inquiry': an analysis of the short-lived phenomenon of General/Liberal Studies in English vocational education and training

In 1957, 12 years after the end of World War II, the Ministry of Education issued Circular 323 to promote the development of an element of ‘liberal studies’ in courses offered by technical and further education (FE) colleges in England. This was perceived to be in some ways a peculiar or uncharacteristic development. However, it lasted over 20 years, during which time most students on courses in FE colleges participated in what were termed General or Liberal Studies classes that complemented and/or contrasted with the technical content of their vocational programmes. By the end of the 1970s, these classes had changed in character, moving away from the concept of a ‘liberal education’ towards a prescribed diet of ‘communication studies’. The steady decline in apprenticeship numbers from the late 1960s onwards accelerated in the late 1970s, resulting in a new type of student (the state-funded ‘trainee’) into colleges whose curriculum would be prescribed by the Manpower Services Commission. This paper examines the Ministry’s thinking and charts the rise and fall of a curriculum phenomenon that became immortalised in the ‘Wilt’ novels of Tom Sharpe. The paper argues that the Ministry of Education’s concerns half a century ago are still relevant now, particularly as fresh calls are being made to raise the leaving age from compulsory education to 18, and in light of attempts in England to develop new vocational diplomas for full-time students in schools and colleges.