My research falls into the broad area of continuum mechanics. More specifically, most of the problems that I work on are in the areas of solid mechanics, acoustics, linear and nonlinear elasticity, elastodynamics, homogenization and micromechanics. I am also interested more generally in mathematical modelling of physical processes and industrial problems, and the application and development of general mathematical methods for applied problems.

I jointly lead the Waves In Complex Continua (WICC) group in the School of Mathematics at the University of Manchester. This group was launched in September 2010 with significant financial support via grants from the Engineering and Physical Sciences Research Council, The Leverhulme Trust, The Royal Society and industry, primarily from Thales Underwater Systems Ltd .

The group also works with a number of scientific colleagues spread all over the world. In particular, I work with those listed here.

Current projects are:

For more detail on individual papers and links to them see my publications page.

Some background

Recently there has been a great deal of interest surrounding the topic of "cloaking", i.e. making objects invisible to incident waves. Initial investigations related to optics (Lenohardt 2006) and Electromagnetics (Pendry et al 2006) before researchers started to consider applications in acoustics, surface waves and other areas.

All of these applications are driven by the invariance of the governing wave equation under singular transformations, thereby generating "cloaks" with strange material properties. In particular the material properties are required to be both anisotropic and inhomogeneous. In fact a perfect cloak is not possible due to the necessary singularity in the transformation. This sometimes yields unbounded properties on the inner face of the cloak. However, this does not stop approximate cloaks from being generated and recently some experimental validation has been given on this topic.

The difficulty with cloaking in elastodynamics is that the governing equations do not possess similar invariance properties whilst retaining the symmetries of the elastic moduli tensor (Milton et al 2006). This is a big issue! So, although cloaks can be proposed using singular transformations, the predicted material properties are somewhat unphysical. Furthermore if we retain the symmetries of the elastic modulus tensor and invariance, the equations are rather more complicated, being those associated with inhomogeneous materials in the dynamic context, and possessing tensorial density which whilst not being physically implausible, is rather difficult to achieve in practice.

Elastodynamic cloaking via nonlinear pre-stress

The transformation is rather similar (in fact it can almost be thought of as identical!) to the pre-stress which is generated in a nonlinear elastic material when a very small cavity is inflated to a much larger size via internal pressure. Thus it would appear that cloaks be generated via nonlinear pre-stress. This has a number of advantages to classical cloaking:

  • The cloak is generated naturally, no inhomogeneous metamaterials are required
  • The cloaked region can be varied in size
  • The pre-stress naturally generates the elastic moduli without the minor symmetries, as required.
However all of the above would only work together with invariance of the governing equations with respect to the pre-stress. It turns out that in the case of antiplane elastic waves propagating through a neo-Hookean medium (with inflated cavity) this invariance is guaranteed. This was in fact proven in a related problem regarding scattering of antiplane waves - see the prestress section below.

This initial work on elastodynamic cloaking was published in the following paper:

  • Parnell, W.J. "Nonlinear pre-stress for cloaking from antiplane elastic waves" Proceedings of the Royal Society A. (Published online)
The following image exhibits the cloaking effect described in the above paper regarding antiplane elastic waves. On the left we have scattering from a very small cavity due to a line source located at the point indicated by a small white circle. Since the wavelength is larger than the the cavity radius there is no noticable scattered field or shadow. This small cavity is then inflated (middle picture) and we see that waves are bent around the cavity. This is in contrast to the figure on the right which shows scattering from a cavity of equivalent size to the inflated cavity but in an unstressed medium. We see that the shadow region and scattering effects are noticable in the latter case.
This possibility for cloaking from elastic waves opens up a multitude of opportunities, especially with the extension to other wave types (compressional, shear coupled, etc.). Various work will be undertaken in this area in the near future. Also for cloaking we should also read "wave guiding" so that although cloaking has implications regarding stealth, there are many other aspects of applications that are useful as regards guiding waves.
Frequently the stress-strain relationship in materials is NOT linear (i.e. not of the form originally proposed by Hooke). Indeed frequently the induced nonlinearity is useful to us in a practical setting. In some instances the instigation of nonlinearity is unknown. Two specific cases have been of interest recently:

A Microvoid Composite - nonlinearity due to buckling and hysteresis

Influence of nonlinear inclusions in composites

Background - induced inhomogeneity and anisotropy

What happens when a linear wave propagates inside a medium which has been initially nonlinearly stressed, thus altering its anisotropy? This problem can be solved using the so-called theory of "small-on-large". The wavespeed is affected since usually both the density and elastic properties of the medium have been altered by the pre-stress. This prestress also often induces anisotropy, so that waves propagate with different speeds in different directions.

Tunable stop-bands by varying pre-stress

What happens if we now make the underlying material inhomogeneous? We know from the theory of phononics and photonics that inhomogeneous media can give rise to very interesting wave properties such as stop bands where a wave is prohibited to propagate. I investigated the case of a pre-stressed inhomogeneous nonlinear elastic bar (one dimension0 where phases are distributed periodically, in the following paper:
  • Parnell, W.J. 2007 "Effective wave propagation in a pre-stressed nonlinear elastic composite bar", IMA Journal of Applied Mathematics, 72, 223-244. (doi:10.1093/imamat/hxl033)
Of specific interest is the fact that the pre-stress can act as a tuning mechanism to shift the band-gaps of the medium. Similar affects were found by Gei et al (2008), Bertoldi and Boyce (2008) and others.

Scattering from voids in pre-stressed media

What happens if now we put an inclusion or void in an otherwise uniform nonlinear material? The field set up by the pre-stress is now quite complicated and in particular, inhomogeneous equilibrium stress fields arise. Thus the path of the wave through the material will be complicated and in particular how the wave will scatter from the inclusion of void is not at all trivial to determine.

Canonical problems are required to understand this process. The first such problem involving prestress appears to be that solved by Parnell and Abrahams, who considered scattering from a circular cylindrical void in a neo-Hookean material. The initial findings were published in the following paper:

  • Parnell, W.J. and Abrahams, I.D. 2012 "Scattering from a cylindrical void in a pre-stressed incompressible neo-Hookean material", Communications in Computational Physics 11, 367-382 (doi: 10.4208/cicp.111209.160610s)

In this instance we showed that the pre-stress does not affect the scattering coefficients of the scattered antiplane shear wave and so an observer sufficiently far away from the void would not see the effect of the pre-stress! This is quite a remarkable result.

Key to above result was the neo-Hookean host medium. When the material is Mooney-Rivlin, such a result does not hold and the scattering coefficients are affected by the pre-stress. We have also considered the extensions to spherical voids and the coupled compressional/shear wave problems which requires the assumption of a compressible host material. Forthcoming papers on this topic are:

  • Shearer, T., Parnell, W.J., Daros, C.H. and Abrahams, I.D., "Incremental torsional wave propagation in a pre-stressed nonlinear elastic incompressible Mooney-Rivlin cylindrical annulus". In preparation for submission to J. Mech. Phys. Solids.
  • Parnell, W.J. and Abrahams, I.D. "Antiplane response of a pre-stressed nonlinear microvoided composite: analysis via multiple scattering". In preparation for submission to Proc. Roy. Soc. A.
  • Parnell, W.J. and Abrahams, I.D. "Radiation from a cavity in a pre-stressed incompressible nonlinear elastomer". In preparation for submission.
  • Daros, C.H., Parnell, W.J., Abrahams, I.D and Martin, P.A., "Heun's equation governs incremental antiplane deformation superposed on a pressurized incompressible Mooney-Rivlin cylindrical annulus". In preparation for submission to Int. J. Nonlinear Mechanics
  • Daros, C.H., Parnell, W.J., Abrahams, I.D. and Shearer, T., "On the influence of pre-stress on SH wave scattering and radiation from a cylindrical void in an incompressible Mooney-Rivlin material", In preparation for submission to Proc. Roy Soc. A.

Numerous colleagues have contributed to this work, they are:

  • Tom Shearer (PhD student, Manchester)
  • David Abrahams (Faculty, Manchester)
  • Carlos Daros (Faculty, UNICAMP, Brazil)
  • Paul Martin (Faculty, Colorado School of Mines, USA)

This topic is very close to my heart as it was my first foray into research during my MSc thesis (1999-2000) which was entitled "Elastic Wave Scattering From a Strained Region". It detailed how the nonlinear deformation of a medium containing a spherical inclusion affected its scattering properties. Using a simple model, together with my then supervisor David Allwright, we showed that its scattering cross section was significantly diminished by the initial nonlinear deformation.

This work was extended in a recent Study Group (Manchester, 2005) project:

and forms the basis for the work above, mainly the work undertaken in the PhD of Tom Shearer.
All biological materials such as bone, the lungs, tissues, etc. possess microstructure whose properties are driven by their function. Modelling such microstructural properties is an extremely useful thing to do from a medical viewpoint and it is also fascinating from a purely scientific viewpoint. One major application is to try to predict the onset of osteoporosis in elderly patients. This can be predicted by measuring wavespeeds in bone and one major device which has been designed to do this is a piece of equipment measuring wave speed across the heel. It appears that there is a high correlation between the quality of bone in the heel and that elsewhere in patients. With Quentin Grimal and Pascal Laugier from the Laboratoire d'Imagerie Parametrique at the Universite Pierre et Marie Curie in Paris I am currently developing models of wave propagation in cortical (dense) bone in order to compare with experiments and numerical simulations. Initial details of this work can be found in the following paper

  • Parnell, W.J. and Grimal, Q., 2009, "The influence of mesoscopic porosity on cortical bone anisotropy. Investigations via asymptotic homogenization.", Journal of the Royal Society Interface 6, 97-109 (doi:10.1098/rsif.2008.0255)

Follow-up work to this has involved the natural extension to transversely isotropic phases and the following publication has just been accepted:

  • Parnell, W.J., Vu, M.-B., Grimal, Q. and Naili, S. "Analytical methods to determine the effective mesoscopic and macrosopic elastic properties of cortical bone", Journal of Biomechanics and Modelling in Mechanobiology. Accepted (2011)

On a separate but related note, frequently a problem in modelling biological materials is the plethora of parameters that can arise. It is of direct interest to try to reduce the number of input parameters. We did this in the following model of bone, where the model was reduced to only two parameters, these are the ............

  • Grimal, Q., Rus, G., Parnell, W.J. and Laugier, P. 2011, "A two-parameter model of the effective elastic modulus tensor for cortical bone", Journal of Biomechanics 44, 1621-1625 (doi:10.1016/j.jbiomech.2011.03.006 )

Thermoelastic fronts in layered media

Thermal and elastic effects can usually be treated as uncoupled except in certain situations such as regions of rapid variation. As such the propagation of a thermal front through a thermoelastic material is of interest. The homogeneous material problem was studied long ago during the 60's but the inhomogeneous material problem has been left fairly open. In the following recent paper, I analyse the effective properties of a thermoelastic composite and show how the front propagates through a half-space of the material. In particular I discuss the notion of an effective coupling parameter which possesses very interesting behaviour.

  • Parnell, W.J. 2006 "Coupled thermoelasticity in a composite half-space", Journal of Engineering Mathematics, 56, 1-21. (doi:10.1007/s10665-006-9038-1)

Pulses and fronts in heterogeneous media

This is the topic of the PhD of Ellis Barnwell. The question is "how do fronts and pulses travel through complex inhomogeneous media" - this is a very tough problem.

Homogenization and micromechanics are mathematical theories that are able to "upscale" the microstructure of complex materials, in order to provide macroscopic or "homogenized" material properties. This is very useful because frequently we are not interested in the behaviour of stress, displacement, velocity, etc. on a microscale such as a nanometer. We are often only interested in their variation in a much longer lengthscale. A good example is the application of such theories to composite media which are made from distinct materials (often known as phases), combined in such a way as to optimize some physical property such as Youngs modulus, bulk modulus, thermal conductivity, viscosity, etc. Furthermore the composite may often exhibit certain behaviour, not exhibited by the phases from which it is comprised. Here are two pictures of the microstructure of some composite media: the first is a Copper Cobalt composite and the second is a silicon carbide-titanium alloy.

Copper cobalt Silicon Carbide-Titanium

Experimental studies prove extremely expensive and it is therefore of great interest to model these materials mathematically. The subject areas devoted to such studies are those of homogenization and micromechanics. Over the last few decades, these subjects have grown immensely, leading to a much greater understanding of such materials which are used extensively in science, engineering, automotive, aerospace and medical applications. For some references to works in homogenization and micromechanics see my homogenization and micromechanics references page.

Forthcoming textbook

During the course of my PhD I did a great deal of reading around the subject and collected a huge amount of information in surveying the many alternative homogenization methods. This information is now on its way to becoming a textbook, co-authored by David Abrahams and due out in 2012:
  • Parnell, W.J. and Abrahams, I.D., "An Introduction to Homogenization in Continuum Mechanics", Cambridge University Press, due out 2012.

For some really interesting aspects of composites (including piccies and videos), see the brilliant webpage of Rod Lakes at Wisconsin.

An integral equation method of homogenization

My PhD, which was an industrial CASE project, involved trying to develop a model of particulate composites of interest to the industrial collaborator Thales Underwater Systems Ltd (TUSL). I developed a method whose starting point is the integral equation form of the equations of linear elasticity. The method is valid in the quasi-static limit (low-frequency wave propagation, or the so-called separation of scales limit). An application to fibre-reinforced composites with periodic microstructure and the case of antiplane (SH) waves is described in the following paper:

  • Parnell, W.J. and Abrahams, I.D., 2008 "A new integral equation approach to elastodynamic homogenization", Proceedings of the Royal Society A, 464, 1461-1482. (doi:10.1098/rspa.2007.0254)
This was an extension of a method originally proposed by Peter Brazier-Smith from Thales Underwater Systems. In my thesis I formalized Peter's method, understanding its advantages and limitations and gave it context - this work is presented in the following paper, where we compare its predictions with those from Multiple Scattering Theory (e.g. Foldy, Waterman-Truell and QCA)
  • Parnell, W.J. and Abrahams, I.D. and Brazier-Smith, P.R., 2010 "Effective properties of a composite half-space: exploring the relationship between homogenization and multiple scattering theories", Quart. J. Mechanics and Applied Mathematics 63, 145-175 (doi:10.1093/qjmam/hbq002)
Some information about these methods is also given in my PhD thesis.

A current PhD student, Natasha Willoughby is currently developing the so-called "integral equation method" to more complex situations.

Method of Asymptotic Homogenization for Complex Composites

I have also developed the well known method of asymptotic homogenization in order to predict the effective behaviour of fibre reinforced media with complex microstructure. This is discussed in the following papers:

  • Parnell, W.J. and Abrahams, I.D., 2006 "Dynamic homogenization in periodic fibre reinforced media. Quasi-static limit for SH Waves", Wave Motion, 43, 474-498. (doi:10.1016/j.wavemoti.2006.03.003)
  • Parnell, W.J. and Abrahams, I.D., 2008 "Homogenization for wave propagation in periodic fibre reinforced media with complex microstructure", Journal of the Mechanics and Physics of Solids, 56, 2521-2540 (doi:10.1016/j.jmps.2008.02.003)

If a layer is forced on its surface then this will generate vibrations and displacemnent fields to propagate along in layer. Considering time-harmonic forcing gives rise to time-harmonic waves which therefore propagate away from the area of forcing. Choosing a point force is equivalent to deriving the associated Greens function (or tensor) for the problem.

What happens now if the layer is constrained at a finite number of points along either the upper or lower surface? This was a question posed by Thales Underwater Systems Ltd (TUSL), one of our industrial collaborators. If we force the displacement to be zero at those points this will generate a force and will affect the subsequent propagating wave field.

This problem has been worked on by a number of MSc and MMath students:

  • Russ Miller (2009)
  • Ben Poole (2010)
  • Maria Thorpe (2011)
  • Duncan Joyce (2012)
together with Phil Cotterill (Royal Society Industry Fellow and TUSL). We have considered the SH and coupled P/SV wave problems using matched asymptotics and FFT approaches. Recently transverse anisotropy of the layer has been incorporated. The initial findings regarding SH (antiplane) waves are about to be submitted:

  • Miller, R., Thorpe, M., Parnell, W.J., Cotterill, P. and Abrahams, I.D. "The anti-plane elastic response of a constrained layer", In preparation for submission to J. Sound Vibration.

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