PhD Theses from the Institute of Mathematics & Physics
http://hdl.handle.net/2160/1859
Fri, 06 May 2016 15:09:18 GMT2016-05-06T15:09:18ZWeight function approach to studying perfect and imperfect interfaces in anisotropic and piezoelectric bimaterials
http://hdl.handle.net/2160/42808
Weight function approach to studying perfect and imperfect interfaces in anisotropic and piezoelectric bimaterials
Pryce, Lewis
The focus of the thesis is interfacial crack problems in anisotropic
and piezoelectric bimaterials. We seek to solve a variety of problems
using weight function techniques and singular integral equations.
We begin by studying a dynamic crack along a perfectly bonded
interface in an anisotropic bimaterial. Using a weight function de-
rived from a mirrored problem it is possible to derive important ma-
terial parameters which govern the crack propagation. Following this
a static crack is considered. However, in this case the materials are not
bonded perfectly, an imperfect interface is present instead. A method
is derived where singular integral equations for the imperfect interface
problem are derived through use of perfect interface weight functions.
The weight functions are then extended to fracture in piezoelectric
bimaterials which allows equivalent integral equations to be derived
relating the mechanical and electrical elds. In past literature a num-
ber of results have been found which can only be used when consider-
ing a symmetric load system on the crack faces. All of the problems
considered here have asymmetric loading.
Firstly, a steady-state formulation is used to derive asymptotic
coe cients of the crack displacement and interfacial tractions for a
dynamic crack along a perfect interface. The method can be used to
nd many asymptotic coe cients but the one of most importance here
is the stress intensity factor which therefore enables the calculation
of energy release rate at the crack tip. As an example an orthotropic
bimaterial with two di erent loading con gurations is used to examine
the importance of crack speed and load asymmetry on the properties
of the crack propagation.
iv
We proceed to study imperfect interface conditions for an anisotropic
bimaterial. Usually when looking at such a problem it is necessary to
derive new weight functions which correspond to the imperfect inter-
face. An innovative method which makes use of the Betti formula and
existing weight functions for the analogous perfect interface problem
is derived. This procedure is used to obtain singular integral equations
which relate the crack loading, which is assumed to be known, to the
displacement jump over both the crack and interface and tractions
along the bonded area between the materials. Examples of the results
obtained through solving the integral equations numerically are given.
Finally, we extend the weight functions used previously in the the-
sis to a piezoelectric setting. The general form of the weight function
for any piezoelectric bimaterial is given before two speci c examples
are studied in depth. The examples are chosen in such a way to illus-
trate the e ect that the poling direction of the bimaterial can have on
both the mechanical and electrical elds. For both examples explicit
expressions are derived for the weight functions which are then used
to derive singular integral equations which can be used to study the
e ect of both mechanical loading and electrical charges being applied
to the crack faces. To nish we present some examples for both poling
directions to illustrate the use of the derived equations.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2160/428082015-01-01T00:00:00ZMathematical and numerical modeling of hydraulic fractures for non-Newtonian fluids
http://hdl.handle.net/2160/42786
Mathematical and numerical modeling of hydraulic fractures for non-Newtonian fluids
Perkowska, Monika
The aim of this thesis is to construct an accurate and effective numerical algorithm to solve a problem of hydraulic fracturing for non--Newtonian fluids of power--law type rheology. Also, to improve the existing semi--analytical solutions using the recent advances in the area.
The theoretical background is presented, along with the practical applications and physical processes driving the fracture growth. Equations used to mathematically formulate the problem are derived. A full mathematical description is supplemented by boundary and initial conditions. Basic 1D models are presented for various crack propagation regimes.
Mechanisms used extensively throughout this work are thoroughly analysed, amongst which are: (i) numerical integration, (ii) regularisation techniques, (iii) implementation of boundary conditions, (iv) utilisation of appropriate dependent variables, (v) asymptotic behaviour of solution, (vi) appropriate meshing strategy. Each of these methods is described in detail and investigated.
A universal particle velocity based algorithm for simulating hydraulic fractures is proposed. The computations are based on two dependent variables: the crack opening and the reduced particle velocity. The application of the latter facilitates utilisation of the local condition of Stefan type (speed equation) to trace the fracture front. The condition is given in a general explicit form which relates the crack propagation speed (and the crack length) to the solution tip asymptotics.
A number of analytical benchmark solutions are derived. They are employed to validate the computational accuracy of the proposed algorithm. Moreover, the performance of the numerical scheme is tested against other solutions available in the literature.
Following the analysis of the performance of the algorithm, new improved approximations are provided for each model. They can be used as benchmark solutions for testing other algorithms.
The extensive computations prove that the numerical scheme is stable and efficient. It provides accurate results for various hydraulic fracturing models and regimes, as well as fracturing fluids. The algorithm works equally well for time--independent and transient versions of the problem. The utilisation of a modular structure and the adaptive character of its basic blocks result in a flexible numerical scheme.
The numerical code has been provided as a supplementary attachment to the electronic version of this thesis, and is also available on request from the author.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/2160/427862016-01-01T00:00:00ZStudies of the Solar Wind with Interplanetary Scintillation
http://hdl.handle.net/2160/42783
Studies of the Solar Wind with Interplanetary Scintillation
Hardwick, Stuart A.
The Sun is at the centre of our solar system and it is constantly ejecting a flow of plasma outward into interplanetary space. This flow can take the form of two primary types; the fast solar wind; and the slow solar wind. The Sun may also eject plasma with more dynamic events which cause high density clouds of plasma to travel alongside the solar wind. In the modern world, a number of different techniques have been developed to study the solar wind from the Earth and also in interplanetary space. Interplanetary scintillation (IPS) is a technique that has been applied at Aberystwyth University to observe the continual density variations in the solar wind by observing a distant astronomical radio source and the scintillation of its radio waves as they pass through the solar wind. To perform this, a number of radio sites around the world have been used such as the
European Incoherent SCATter radar (EISCAT) in northern Scandinavia, and the ever growing LOw Frequency ARray (LOFAR) based in the Netherlands. The IPS technique has also proved to be highly complementary with Heliospheric Imagers (HIs) onboard the Solar TErrestrial RElations Observatory (STEREO) spacecraft which detect the white light which has been scattered by the solar wind particles.
This thesis investigates the solar wind using a wide variety of techniques to observe the small- and large-scale structures which form in the solar wind. A study of IPS in 2007 and 2008 detected numerous events of rapid velocity variations which occurred in the slow solar wind. This showed evidence that the slow solar wind may have a ‘blobular’ nature formed of many small-scale transients as well as the large-scale transients which are already well known to us.
In November 2011 LOFAR observed its first coronal mass ejection (CME) while observing IPS. This event proved to become of interest as it was successfully detected by a number of different techniques and could modeled and tracked accurately as it propagated outward to Earth’s orbit and even interacted with the Earth’s magnetosphere.
A technique which could be applied to coronagraph observation separated the dynamic and
quiescent solar wind structures from the images. The quiescent component became the basis for the development of a new solar wind model which could quickly and accurately map the background quiescent solar wind free from any of the complexities cause by CMEs.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2160/427832015-01-01T00:00:00ZNumerical simulation of hydraulic fractures: various leak-off regimes and multifracturing
http://hdl.handle.net/2160/30574
Numerical simulation of hydraulic fractures: various leak-off regimes and multifracturing
Kusmierczyk, Piotr
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2160/305742015-01-01T00:00:00Z