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Advances in Applied Mathematics and Mechanics

School of Mathematics Alan Turing Building University of Manchester Manchester, UK

5th-7th June 2013 Introduction

I am very pleased to welcome you all to the School of Mathematics in the Alan Turing Building at the University of Manchester for this workshop entitled Advances in Applied Mathematics and Mechanics. The workshop will host a number of talks focusing on rather theoretical and more applied aspects of mechanics. Applications range from the prediction of composite material behaviour to aspects of biological materials. In order to ensure variety there is no prescription of the type of mechanics discussed at the workshop although the stress is on the development of the theory and applications of continuum mechanics. Although we have plenty of talks I have benefitted significantly in recent years from the French style of conferences which strives for good break-out time and time for lunch - this ensures good interaction! Although I cannot provide lunch-time wine unfortunately! As you may be aware, the School of Mathematics at the University of Manchester has a rich history. Many famous mathematicians have worked here including Horace Lamb, Lewis Fry Richardson, Sydney Goldstein, Paul Erdos, Max Newman, Frank Adams, James Lighthill and Alan Turing. In 2007 the department moved into the purpose built Alan Turing Building which provides a superb environment in which to teach and do research. The Manchester Institute for Mathematical Sciences (MIMS) hosts numerous visitors throughout the year with dedicated office space. More broadly the University of Manchester has given rise to numerous momentous advances, including the discovery of the nuclear nature of the atom at Manchester by Rutherford and the world’s first stored-program computer coming into being at the university. In total the university counts 25 Nobel Prize winners amongst its current and former staff and students, with its most recent coming in 2010 with the award of two Nobel prizes to Andre Geim and Konstantin Novoselov, for groundbreaking experiments regarding the two-dimensional material graphene. We are assured of a wonderful three days of talks at this workshop. I’d like to thank all of the speakers for their contributions. I sent out a large number of invites and almost all were accepted within a short time frame which made for difficult timetabling but an excellent programme. I’d like to thank the Engineering and Physical Sciences Research Council (EPSRC) for their financial support of this workshop via grant EP/H050779/1. I’d also like to thank Riccardo De Pascalis for his help in producing this short booklet, Dave Abrahams for general discussion and Sue Tizini, Jenny Gradwell and Sebastian Rees for their admin support. The poster session will be on Thursday lunch time. Since many participants of this are PhD students I would urge you to attend and ask questions. Enjoy the meeting and (sunny) Manchester, Regards,

William Parnell Schedule WEDNESDAY 5th JUNE, 2013

10.00-10.40 Arrivals and coffee 10.40-10.45 Introduction and opening (Will Parnell) 10.45-11.15 Richard Craster (Imperial College, UK): “High frequency homogenization” 11.45-12.15 Sasha Movchan (University of Liverpool, UK): “Trapped flexural modes and transmission resonances in structured plates” 12.15-12.45 Vincent Pagneux (University of Maine, France): “Multimodal admittance method in waveguides with improved convergence”

12.45-14.15 Buffet lunch

14.15-14.45 Paul Martin (Colorado School of Mines, USA): “N masses on a string” 14.45-15.15 Karima Khusnutdinova (Loughborough University, UK): “On Boussinesq and Kadomtsev-Petviashvili type models and applications”

15.15-16.00 Coffee, tea and biscuits

16.00-16.30 Ray Ogden (University of Glasgow, UK): “Residual stress and its effect on elastic response” 16.30-17.00 Jacopo Ciambella (University of Bristol, UK): “Large strain viscoelastic effects in short fibre composites” 17.00-17.30 Luigi Vergori (National University of Galway, Ireland): “On the finite element implementation of anisotropic elasticity”

17.30 onwards Wine and nibbles THURSDAY 6th JUNE, 2013

09.30-10.00 Yibin Fu (University of Keele, UK): “Bifurcation and stability of pressurized ellipsoidal membrane shells” 10.00-10.30 Eliot Fried (McGill University, Canada): “Stability and bifurcation in a simple model for shape changes in discoidal high-density lipoproteins”

10.30-11.15 Coffee, tea and biscuits

11.15-11.45 Alain Goriely (University of Oxford, UK): “Magnetic chains: from self-buckling to self-assembly” 11.45-12.15 Oscar Lopez-Paimes (University of Illinois at Urbana Champaign, USA): “The nonlinear elastic response of suspensions of rigid inclusions in rubber” 12.15-12.45 John Berger (Colorado School of Mines, USA): “Stresses due to intercalation and phase transformation in lithium ion battery cathodes”

12.45-14.15 Buffet lunch and poster session

14.15-14.45 Andrew Norris (Rutgers University, USA): “The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces” 14.45-15.15 Daniel Colquitt (University of Liverpool, UK): “Non-singular cloaking of a square inclusion with a microstructured coating”

15.15-16.00 Coffee, tea and biscuits

16.00-16.30 Ian Thompson (University of Liverpool, UK): “Rayleigh wave scattering by a submerged cavity” 16.30-17.00 Diki Porter (University of Bristol, UK): “Trapped modes in simply-supported thin elastic plates” 17.00-17.30 Michele Brun (University of Cagliari, Italy): “Analysis of propagation of a transition flexural wave in a supported beam”

17.30 onwards Wine and nibbles

20.00 Workshop meal - Don Giovanni’s restaurant, Oxford Road FRIDAY 7th JUNE, 2013

09.30-10.00 Oliver Jensen (University of Manchester, UK): “Instabilities of flexible-channel flows” 10.00-10.30 Massimiliano Gei (University of Trento, Italy): “Asymptotic models for the buckling of thin films resting on soft, prestretched substrates” 10.30-11.00 Giuseppe Saccomandi (University of Perugia, Italy): “Weak coaxial transversely isotropic materials”

11.00-11.45 Coffee, tea and biscuits

11.45-12.15 Martine Ben Amar (ENS, France): “Morphogenesis and embryogenesis” 12.15-12.45 Georges Limbert (University of Southampton, UK): “A microstructurally-based anisotropic continuum model of skin”

12.45-14.15 Buffet lunch

14.00-14.30 Salvatore Federico (University of Calgary, Canada): “Porous Materials with Statistical Fibre-Reinforcement” 14.30-15.00 Dave Abrahams (University of Manchester, UK): “Wave propagation in quasi-periodic media”

15.00-16.00 Coffee, tea biscuits and close Participants

David Abrahams University of Manchester, UK Azwani Alias Loughborough University, UK Jean Marc Allain Ecole Polytechnique, France Martine Ben Amar Ecole Normale Superieure, France John Berger Colorado School of Mines, USA Emilie Blanc Aix/Marseille University, France Michele Brun University of Cagliari, Italy Luigi Cabras University of Liverpool, UK Giorgio Carta University of Liverpool, UK John Chapman University of Keele, UK Igor Chernyavsky University of Nottingham, UK Jacopo Ciambella University of Bristol, UK Richard Clift Loughborough University, UK Daniel Colquitt University of Liverpool, UK Simon Cotter University of Manchester, UK Phil Cotterill Thales/University of Manchester, UK Jamie Cowley Strathclyde University, UK Richard Craster Imperial College, UK Riccardo De Pascalis University of Manchester, UK Salvatore Federico University of Calgary, Canada Eliot Fried McGill University, Canada Yibin Fu University of Keele, UK Massimiliano Gei University of Trento, Italy Alain Goriely University of Oxford, UK Art Gower National University of Galway, Ireland Stewart Haslinger Liverpool John Moores University, UK David Harris University of Manchester, UK Oliver Jensen University of Manchester, UK Ian Jones Liverpool John Moores University, UK Lina Joseph Imperial College, UK Savina Joseph University of Oxford, UK Duncan Joyce University of Manchester, UK Karima Khusnutdinova Loughborough University, UK Jane Lawrie Brunel University, UK Georges Limbert University of Southampton, UK Barbara Lynch Ecole polytechnique, France Oscar Lopez-Paimes University of Illinois at Urbana-Champaign Mehul Makwana Imperial College, UK Paul Martin Colorado School of Mines, USA Agnes Maurel ESPCI, France Maureen McIver Loughborough University, UK Kieron Moore Loughborough University, UK Natasha Movchan University of Liverpool, UK Sasha Movchan University of Liverpool, UK Michael Nieves University of Liverpool, UK David Nigro University of Manchester, UK Andrew Norris Rutgers University, USA Ray Ogden University of Glasgow, UK Zuonaki Ongodiebi University of Manchester, UK Jane O’Neill University of Liverpool, UK William Parnell University of Manchester, UK Vincent Pagneux University of Maine, France Phil Pearce University of Manchester, UK Draga Pihler-Puzovic University of Manchester, UK Diki Porter University of Bristol, UK Wassamon Phusakulkajorn University of Manchester, UK Giuseppe Saccomandi University of Perugia, Italy Tom Shearer University of Manchester, UK Rebecca Shipley UCL, UK Ian Thompson University of Liverpool, UK Luigi Vergori National University of Galway, Ireland Feng Xu University of Nottingham, UK Xizheng Zhang Loughborough University, UK WEDNESDAY 5th JUNE, 2013 Richard Craster - High frequency homogenization

It is highly desirable to be able to create continuum equations that embed a known microstructure through effective or averaged quantities such as wavespeeds or shear moduli. The methodology for achieving this at low frequencies and for waves long relative to a microstructure is well-known and such static or quasi-static theories are well developed. However, at high frequencies the multiple scat- tering by the elements of the microstructure, which is now of a similar scale to the wavelength, requires a dynamic homogenization theory. Many interesting features of, say, periodic media: band gaps, local- ization etc occur at these higher frequencies. The materials exhibit effective anisotropy and this leads to topical effects such as cloaking/ invisibility, flat lensing, negative refraction and to inducing direc- tional behaviour of the waves within a structure. A general theory will be described and applications to continuum, discrete and frame lattice structures will be outlined. The results and methodology are confirmed versus various illustrative exact/ numerical calculations showing that theory captures, for instance, all angle negative refraction, ultra refraction and localised defect modes.

Alexander Movchan - Trapped flexural modes and transmission resonances in structured plates

Authors: S. Haslinger, R. McPhedran, N. Movchan, A. Movchan.

We analyse resonance transmission by a regular grating stack and focus on the new phenomenon of elasto-dynamically induced transmission. This is combined with the detailed study of the multipole representations of flexural waves in a structured plate governed by a biharmonic operator. Alongside we address the waveguide problem for trapped modes of different symmetries within a stack of grat- ings. The analytical model is accompanied by numerical simulations and evaluation of quality factors in transmission by structured stacks.

Vincent Pagneux - Multimodal admittance method in waveguides with improved convergence

Co-authors: A. Maurel & J.F. Mercier

Paul A. Martin - N masses on a string

One-dimensional time-harmonic waves interact with N identical scatterers. We solve this classical problem using transfer matrices. For a row of N equally spaced scatterers, the reflection and trans- mission coefficients can be calculated explicitly, using Chebyshev polynomials. Here, the emphasis is placed on problems where the scatterers are perturbed from their equispaced positions. Explicit results are obtained. They are useful in understanding the construction of effective media for one-dimensional random media.

Karima Khusnutdinova - On Boussinesq and Kadomtsev-Petviashvili type models and applications

Boussinesq and Kadomtsev-Petviashvili type models, since their appearance in the context of fluids, have recently emerged in a vast variety of problems describing long nonlinear waves in solids. In this talk I will overview some recent developments concerning this type of models in the context of nonlinear bulk strain waves in layered solid waveguides and surface gravity waves in fluids. The talk is based on the results published in the following papers: [1] K.R. Khusnutdinova, A.M. Samsonov, Fission of a longitudinal strain solitary wave in a delami- nated bar, Phys. Rev. E 77 (2008) 066603. [2] K.R. Khusnutdinova, A.M. Samsonov, A.S. Zakharov, Nonlinear layered lattice model and gener- alized solitary waves in imperfectly bonded structures, Phys. Rev. E 79 (2009) 056606. [3] K.R. Khusnutdinova, K.R. Moore, Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations, Wave Motion 48 (2011) 738-752. [4] G.V. Dreiden, K.R. Khusnutdinova, A.M. Samsonov, I.V. Semenova, Bulk strain solitary waves in bonded layered polymeric bars with delamination, J. Appl. Phys. 112 (2012) 063516. [5] K.R. Khusnutdinova, C. Klein, V.B. Matveev, A.O. Smirnov, On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation, Chaos 23 (2013) 013126.

Ray Ogden - Residual stress and its effect on elastic response

When a material is subject to residual stress, whatever its source, any subsequent elastic response of the material (linear or nonlinear) is strongly influenced by the residual stress. The elastic constitutive law incorporates the residual stress, in effect, as a structure tensor, which generates invariants, the number of which depends on any underlying symmetry in the material. In this talk the associated theory of hyperelasticity is summarized and then applied to simple representative boundary-value problems whose solutions are known in the situation where there is no residual stress in order to show how the residual stress changes the local and global stress-deformation characteristics of the material when it undergoes large strains. The theory is illustrated for two prototype constitutive laws.

Jacopo Ciambella - Large strain viscoelastic effects in short fibres composites

Nanofillers are normally employed to increase mechanical and thermal properties of elastomeric com- pounds. Nanoclays, carbon nanotubes or graphene have large specific surface areas and excellent physical properties that give them the potential to provide significant mechanical improvements. In this talk I will introduce a viscoelastic model at large strain to describe the behaviour of natural rubber reinforced with graphene oxide nanoplatelets. The main aim of the model is to reproduce the hysteresis loops observed during cyclic deformation. The microstructure evolution is analysed and it is shown that time-dependent alignment of fibres in the direction of the stress could explain the strain hardening effect observed in the experiments.

Luigi Vergori - On the finite element implementation of anisoptropic elasticity.

Authors: L. Vergori, M. Destrade, P. McGarry, R. W. Ogden

Incompressible nonlinearly hyperelastic materials are rarely simulated exactly in finite element exper- iments because of the numerical difficulties associated with ideal incompressibility. Most commercial finite element packages therefore assume that the material is slightly compressible. It is then further assumed that the corresponding strain-energy function can be decomposed additively into volumetric and deviatoric parts. We show that this decomposition is not physically realistic for the anisotropic materials of particular interest in soft tissue mechanics. In particular, in the small deformation regime, a finite element code based on the volumetric-deviatoric separation assumption predicts that a sphere made of a compressible anisotropic material deforms into another sphere under hydrostatic pressure loading, instead of the expected ellipsoid. For finite deformations, the commonly adopted assumption that fibres cannot support compression is incorrectly implemented in current finite element codes and leads to the unphysical result that under hydrostatic tension a sphere made of compressible anisotropic material deforms into a larger sphere. THURSDAY 6th JUNE, 2013 Yibin Fu - Bifurcation and stability of pressurized ellipsoidal membrane shells

Authors: Yibin Fu and Yuxin Xie

Pressurized spherical and ellipsoidal membrane shells feature in a variety of situations. For instance, they may be used as actuators in acoustic impedance control [1] or habitat in extra-terrestrial applica- tions [2]. The particular application that motivates our current study is the mathematical modelling of saccular aneurysms in human arteries. Recent studies [3,4] on the inflation of membrane tubes have demonstrated that to have a maximum in uniform inflation is not a necessary condition for the forma- tion of a localized bulge provided the axial stretch is held fixed (which is precisely how arteries are like). This finding re-opened the possibility that the initiation of fusiform aneurysms may be modelled as a mechanical bifurcation phenomenon. The present study is the first step in our exploration of whether stability and bifurcation also play a role in the formation and final rupture of saccular aneurysms. When a spherical balloon is inflated by an internal pressure, the pressure against volume typically takes the form shown in the figure[5]. Previously it has been shown that a pear-shaped solution will bifurcate from the uniform solution when point A further down the curve is reached, and that when B is reached the pear-shaped solution will return to the spherical solution [5,6,7]. Both points A and B correspond to surface tension reaching a maximum [2]. There exist some stability results concerning the various solutions, but they are not complete. We study the bifurcation and stability of a pressurized ellipsoidal membrane shell and show that a pear-shaped configuration may also bifurcate from the axi- symmetrically inflated configuration. The stability of all the possible solutions (including the special case of a spherical shell) is studied, and predications from different stability criteria are compared.

pressure

A

B

volume

Figure 1: Pressure variation when a tube or a spherical balloon is inflated

References

1. Youda, M. and Konishi, S., Acoustic impedance control through structural tuning by pneumatic balloon actuators. Sensors and actuators A 95 (2002), 222-226.

2. Jenkins, C.H.M. (Ed), Gossamer spacecraft: membrane and inflatable structures technology for space applications (vol.191, Progress in Astronautics and Aeronautics), AIAA, 2001.

3. Fu, Y.B., Pearce, S.P., and Liu, K.K., Post-bifurcation analysis of a thin-walled hyperelastic tube under inflation. Int. J. Non-linear Mech. 43 (2008), 697-706.

4. Fu, Y.B. and Xie, Y.X., Effects of imperfections on localized bulging in inflated membrane tubes. Phil. Trans. R. Soc. A 370 (2012), 1896-1911. 5. Haughton, D.M. and Ogden, R.W., Bifurcation of inflated circular cylinders of elastic material under axial loading. I. Membrane theory for thin-walled tubes. J. Mech. Phys Solids 27 (1979), 179-212.

6. Haughton, D.M., Post-bifurcation of perfect and imperfect spherical elastic membranes. Int. J. Solids Structures 16 (1980), 1123-1133.

7. Chen, Y.-C. and Healey, T.J., Bifurcation to pear-shaped equilibria of pressurized spherical membranes. Int. J. Non-Linear Mech. 26 (1991), 279-291.

Eliot Fried - Stability and bifurcation in a simple model for shape changes in discoidal high-density lipoproteins

The packaging and transport of cholesterol in the bloodstream are mediated by nanoparticles called lipoproteins. In a process known as “reverse cholesterol transport,” high-density lipoprotein (HDL) particles scavenge cholesterol from tissues and other lipoproteins and thereafter deliver it to the liver for excretion or other use. During this process, which is consistent with the observed inverse rela- tionship between levels of HDL and the risk of atherosclerosis, HDL particles undergo various shape transitions. The functional properties of these particles are believed to be closely tied to shape. Dis- coidal and spheroidal shapes are observed. For discoidal HDL, experimental evidence and molecular dynamics simulations have led to the proposition of an analog model involving an elastic loop spanned by a soap film, corresponding respectively to the major protein component of the HDL and the lipid molecules. On the basis of a variational formulation, we obtain a boundary-value problem for a vector field that parametrizes both the bounding loop and the spanning surface. Working with the first and second variations of the relevant free-energy functional, we conduct detailed bifurcation and stability analyses. We also discuss the extension of the theory to account for curvature elasticity and other salient effects.

Alain Goriely - Magnetic chains: from self-buckling to self-assembly

Spherical neodymium-iron-boron magnets are marketed as toys as they can be assembled into different shapes due to their high magnetic strength. In particular, we consider two simple structures, chains and cylinders of magnets. By manipulating these structures, it quickly appears that they exhibit an elastic response to small deformations. Indeed, chains buckle on their own weight, rings oscillate, and cylinders resist bending but recover their shape after poking. A natural question is then to understand the response of these structures based on the individual physical properties of the magnets and to understand to what extent, they behave elastically. In this talk, I will show through illustrative exper- iments and simple model calculations that the idea of an effective magnetic bending stiffness is, in fact, an excellent macroscopic characterisation for the mechanical response of magnetic chains. I will then propose a more rigorous approach of the problem by considering discrete-to-continuum asymptotic analysis to derive a continuum model for the energy of a deformed chain of magnets based on the magnetostatic interactions between individual spheres.

Oscar Lopez-Pamies - The nonlinear elastic response of suspensions of rigid inclusions in rubber

In the first part of the talk, I will present an exact solution for the fundamental problem of the overall nonlinear elastic response of ideal (Gaussian/Neo-Hookean) rubber reinforced by a dilute isotropic distribution of rigid particles under arbitrarily large deformations. The derivation makes use of a novel iterative homogenization technique in finite elasticity that allows to construct exact solutions for the homogenization problem of two-phase nonlinear elastic composites with particulate microstructures. The solution is fully explicit for axisymmetric loading, but is otherwise given in terms of an Eikonal partial differential equation in two variables for general loading conditions. In the second part, I will make use of the dilute solution as a fundamental building block together with a new variational pro- cedure to derive a simple explicit approximate solution for the overall nonlinear elastic response of non-Gaussian rubber filled with finite concentrations of particles under arbitrarily large deformations. The key theoretical and practical features of these solutions will be discussed in light of their applica- tion to model emerging electro- and magneto-active filled elastomers.

J. R. Berger - Stresses due to Intercalation and Phase Transformation in Lithium Ion Battery Cath- odes

Authors: V. Malav´e, J. R. Berger

The ever increasing power demands on lithium-ion batteries have forced battery safety into the public focus, as evidenced by recent events with the Boeing Dreamliner. One area of particular interest is how the development of stress, and fracture, can lead to battery degradation. This degradation may be accompanied by local thermal runaway, thereby increasing the risk of fire in the battery. In this talk, we will describe our recent modeling efforts in LiCoO2 cathode materials. We seek to develop numerical models using the actual cathode microstructural geometry extracted from focused ion beam- scanning electron microscope images (FIB-SEM). In the cathode particles, the intercalation of lithium is governed by a stress-dependent diffusion equation,

∂c 2 Ω Ωc 2 = D ∇ c − ∇c ∇σh − ∇ σh ∂t  RT RT  where c is the lithium concentration, D is the diffusivity, R is a gas constant, T is the temperature, 1 and Ω is the partial molar volume. The hydrostatic stress, σh, is given by σh = 3 σkk. The stresses are governed here in the context of linear elasticity by momentum balance and kinematics given by,

∂σij 1 ∂ui ∂uj = 0, εij = + ∂xj 2 ∂xj ∂xi  where εij are the (small) elastic strains and ui are the displacements. Finally, we consider non-isotropic constitutive behavior where εij = Sijklσkl + β∆c β is the Vegard constant and ∆c is the change in concentration of lithium. We will describe our current modeling efforts in this multi-physics problem where the diffusion equation is solved in a computational fluid dynamics framework, and the stress state is computed quasi-statically. A complicating feature of the stress analysis is that the LiCoO2 undergoes a phase transformation from monoclinic to two different, coexisting hexagonal phases, with associated volume changes. We will describe our approach for including these phase transformation related volume changes in the calculations, and show recent results for stresses in the cathode particles under different discharging rates.

Andrew Norris - The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces

The matrix sign function is a simple and direct approach in the theory of surface waves in anisotropic materials. This talk will first review the general topic of surface waves and review the properties of the matrix sign function. We then show how the matrix sign function leads directly to the basic formulas of the Barnett-Lothe integral formalism and to an explicit solution of the algebraic matrix Riccati equation for the surface impedance. The matrix sign function gives a natural basis for finding the surface wave speed in a periodically inhomogeneous half-space with material properties that are independent of depth.

Daniel J. Colquitt - Non-singular cloaking of a square inclusion with a microstructured coating

Authors: DJ Colquitt, IS Jones, NV Movchan, AB Movchan, M Brun, and RC McPhedran.

Using the framework of transformation elastodynamics the talk presents a non-singular cloak for a square inclusion. The cloak is generated by a piecewise smooth deformation leading to a cloak with continuous material parameters. Analysis of wave propagation through the cloak is discussed, together with a range of illustrative simulations for a square inclusion cloaked from a point source. The quality of the cloaking effect is quantified objectively and the cloak is demonstrated to work of a wide range of frequencies. A novel experiment to assess the efficacy of the cloak is proposed and demonstrated by a numerical simulation. Motivated by the continuum cloak, discrete approximate cloaks based on a lattice model are discussed and their effectiveness examined. Numerical illustrations demonstrating the performance of the cloak accompany the analytical work.

Ian Thompson - Rayleigh wave scattering by a submerged cavity

The problem of Rayleigh wave scattering by a submerged cylindrical cavity will be discussed in this presentation. A solution for the case of normal incidence was obtained by R. D. Gregory in 1967, using a technique based on multipole expansions. We will show how Gregory’s method can be generalised to account for oblique incidence. In order to compute the coefficients in the resulting expansion, it is necessary to evaluate certain integrals numerically. These integrals are derived from generalised Mehler–Sonine representations for Hankel functions, and the integrand can have up to six singularities on the real line. However, shifting the integration contour onto a path of steepest descent enables calculation of results to an extraordinary degree of accuracy, using a very simple quadrature method based on the trapezium rule.

Diki Porter - Trapped modes in simply-supported thin elastic plates

A infinitely-long thin elastic plate of constant width is simply supported along its two parallel edges. A cut of finite length is made along the centreline of the strip. We examine the existence of trapped modes (permanent localised oscillations or resonances) in the vicinity of the cut. First, an integral equation is formulated based on a Fourier transforms approach. A rapidly-convergent numerical scheme based on the Galerkin method is used from which a new explicit ‘small-spacing’ approximation is derived. This is shown to be very accurate for short cuts. Next, a wide-spacing approximation is developed using exact results from a scattering problem, solved using the Wiener-Hopf technique. Of particular note is that, in both approaches, knowledge of the location of non-real roots of dispersion relations are not required. Finally an existence proof for all symmetric holes in a simply-supported waveguide is illustrated.

Michele Brun - Analysis of propagation of a transition flexural wave in a supported beam

Authors: M. Brun, A.B. Movchan, L.I. Slepyan

We consider the propagation of a transition flexural wave through a beamlike periodically supported discrete structure. Both propagating and evanescent waves are included in the general solution and influence the interaction between different nodal points within the discrete system. An efficient math- ematical approach based on analysis of a functional equation of the Wiener-Hopf type has led to the expression of the displacement on the interface wave to be written as a product of terms evaluated directly from analysis of the kernel function of the Wiener-Hopf equation. The discrete model has been compared with the simpler continuous approach, and it has been shown that the lattice solution pos- sesses new feature for interfacial flexural waves. The conditions allowing the failure wave to propagate and the characteristic wave speeds are found. FRIDAY 7th JUNE, 2013 Oliver Jensen - Instabilities of flexible-channel flows

Flow driven through a planar channel, having a finite-length membrane inserted in one wall, can be unstable to a variety of self-excited oscillations. Instabilities are global in nature, being sensitive to boundary conditions, in particular to whether the flow is driven by a fixed pressure gradient or by a fixed volume flux. I will describe recent work (with Feng Xu and John Billingham) on a model of the fixed-flux problem, for which a three-parameter unfolding of a bifurcation with four zero eigenvalues reveals at least two physically distinct mechanisms of instability

Massimiliano Gei - Asymptotic models for the buckling of thin films resting on soft, prestretched substrates

Buckling/wrinkling instability of metal or silicon thin films resting on soft elastomeric substrates is a structural phenomenon well exploited in the field of stretchable electronics. Due to the high stiffness contrast between film and substrate, asymptotic models where the top layer is seen as a beam-like structure resting on a compliant bed of springs may give very good predictions in term of buckling stress and wavelength mode. In the talk, this asymptotic model is obtained in a plane-strain setting starting from the exact nonlinear elastic model where both film and supporting medium are seen as two-dimensional domains with initial prestress. The model is able to provide the critical load and wavelength mode very accurately for a wide range of stiffness ratios between the two materials, but, more importantly, it can take into account a possible initial prestretch of the substrate that is a key parameter in the practical design of soft devices based on the wrinkling mechanism.

Giuseppe Saccomandi - Weak coaxial transversely isotropic materials

A material is of coaxial type if the stress tensor T and the strain tensor B are coaxial for all deforma- tions. Clearly a hyperelastic material is of coaxial type if and only if it is isotropic. Here we present a weaker definition of materials of coaxial type. Anisotropic materials may be of a coaxial type in a weak sense if for a given specific B we have that TB = BT for any Q ∈ Rot. We show that for transverse isotropic materials weak coaxial constitutive equations may be characterized using universal relations. Then we discuss the impact of weak coaxial materials in the modeling of soft tissues. Indeed, we show that for many of the usual constitutive equations used in biomechanics considering the above mentioned results very strange mechanical behavior may be observed.

Martine Ben Amar - Morphogenesis and embryogenesis

Authors: Joint work with Julien Dervaux, Fei Jia and Martin Mueller.

The beauty and complexity of living matter begins with the first steps of growth and with the formation of cell clusters and tissues. At the biological level, the processes are complex and often unknown, not measured and quantified, except perhaps in botanics. However quite general ideas can be given using a pragmatic approach based on symmetries following a Landau approach. Using the elasticity of soft tissues, eventually taking into account the existence of fibres, shapes can be explained by a variational treatment where incompressibility of tissues is treated via the existence of a stream function as in hydrodynamics. Despite the complexity of the formalism, especially in 3D, such an approach allows sometimes to answer to open questions, in the biology of development. It is the case for the embryogenesis of villi where the role of the mesenchyme versus the epithelium remains a matter of debate. As examples I will describe the green algae, the sympetalous flowers, the fingerprints and the villi formation. Georges Limbert - A microstructurally-based anisotropic continuum model of skin.

Characterising and modelling the multi-scale mechanical behaviour of biological soft tissues is essential for the development of predictive computational models to assist research in medicine, biology, tissue engineering, pharmaceutics, consumer goods, cosmetics, transport or military safety. It is therefore critical to develop constitutive models that can capture particular rheological mechanisms operating at specific length scales so that these models are adapted for their intended applications. Here, a novel multi-scale constitutive framework for biological soft tissues is developed [1]. One of the key features of the formulation is the full decoupling of deformation modes so that the constitutive parameters are directly linked to physical measurements. Another desirable feature of the constitutive equations is that the response is based on physical geometrical/structural parameters that can be experimentally measured or determined ab initio from molecular dynamic simulations. The constitutive formulation was implemented into a non-linear finite element code using a three- dimensional enhanced strain formulation [2]. A series of direct sensitivity analyses of the constitutive parameters on the shear response was conducted. The constitutive model was shown to reproduce very well the experimental macroscopic multi-axial properties of skin whilst also predicting a posteriori the stiffness of individual collagen molecules as measured experimentally [3] or determined from molecular dynamic simulations [4]. ACKNOWLEDGMENTS Financial support from the Engineering and Physical Sciences Research Council (EPSRC) [Grant EP/F034296/1] is gratefully acknowledged. REFERENCES [1] Limbert, G. 2011. J Mech Behav Biomed Mater, 4, 1637-1657. [2] Korelc, J. et al. 2010. Computational Mechanics, 46, 641-659. [3] Sun, Y. L. et al. 2002. Biochemical and Biophysical Research Communications, 295, 382-386. [4] Gautieri, A. et al. 2011. Nano Lett, 11, 757-766.

Salvatore Federico - Porous Materials with Statistical Fibre-Reinforcement

Authors: S. Federico and A. Grillo

Hydrated soft biological tissues can be well represented by as a porous matrix saturated by a fluid and reinforced by a network of statistically oriented, impermeable fibres. For the case of articular cartilage, the collagen fibres have a location-dependent arrangement, varying from aligned in the direction of the tissue depth in the deep zone, to random in the middle zone, to parallel to the articular surface in the superficial zone [1]. With articular cartilage in mind, this work [2] was aimed at developing a large-deformation framework for elasticity and permeability, which generalises and unifies past work in which elasticity was studied only in the absence of fluid, and permeability was studied only under small deformations (see [2] and references therein). This was achieved in three steps. In the first step, a new form of the elastic potential is proposed for porous materials as the sum of a given “base” potential V and a correction potential U, function of the volumetric deformation J = det F , where F is the deformation gradient. The correction U becomes active only for values of J close to compaction (i.e., when all fluid has escaped) and serves to impose the incompressibility constraint at compaction, without any effect on the material properties in small deformations. In the second step, the matrix and the fibres were assigned the same corrections function U, and then were modelled as isotropic and transversely isotropic, respectively. The statistical arrangement of the fibres was treated by means of averaging integrals over the unit sphere (the set of all possible directions in space) featuring a probability distribution function ψ representing the probability to find a fibre in a given direction. In the third step, the permeability is studied in a representative element of (current) volume comprising a fibre and the surrounding matrix and the integrated over the unit sphere to obtain the overall permeability at one spatial point, similarly as done for the case of elasticity. However, since the fibre orientation distribution in the current configuration is not known a priori, all quantities involved in the expression of the permeability need to be referred to their pulled-back counterparts. The averaging integrals for both the elasticity model (integrals expressing the overall elastic po- tential, the stress and the elasticity tensor) and the permeability model are explicit functions of the deformation, such that the deformation cannot be bought outside of the integral sign. This means that analytical solutions cannot be found in general, and a numerical implementation is imperative. Acknowledgements: Alberta Innovates (AITF, Canada), Natural Sciences and Engineering Research Council (NSERC, Canada) [SF], Federal Ministry of Economics and Technology (Germany) [AG].

References

[1] R. M. Aspden and D. W. L. Hukins. Collagen organization in articular cartilage, determined by X-ray diffraction, and its relationship to tissue function. Proc. R. Soc. Lond. B, 212:299–304, 1981.

[2] S. Federico and A. Grillo. Elasticity and permeability of porous fibre-reinforced materials under large deformations. Mech. Mat., 44:58-71, 2012.

Dave Abrahams - Wave propagation in quasi-periodic media Posters session: abstracts

Azwani Alias - On strongly interacting internal waves in a rotating ocean

Authors: Azwani Alias, Roger Grimshaw, Karima Khusnutdinova

In the weakly nonlinear limit, oceanic internal solitary waves for a single linear long wave mode are described by the KdV equation, extended to the Ostrovsky equation in the presence of background rotation. We consider the scenario when two different linear long wave modes have nearly coincident phase speeds, and show that then the appropriate model is a system of two coupled Ostrovsky equa- tions [1]. These are systematically derived for a density-stratified ocean. Some preliminary numerical simulations are reported which show that, in the generic case, initial solitary-like waves are destroyed and replaced by two coupled nonlinear wave packets, being the counterpart of the same phenomenon in the single Ostrovsky equation. [1] A. Alias, R.H.J. Grimshaw, K.R. Khusnutdinova, On strongly interacting internal waves in a rotating ocean (2013) under consideration in Chaos.

Emilie Blanc - Numerical modeling of 2D poroelastic transient waves with fractional derivatives in anisotropic media

Author: Emilie Blanc, Guillaume Chiavassa, Bruno Lombard

We investigate the propagation of poroelastic waves in anisotropic media, described by the Biot’s model in the time-domain. Most of the existing methods have been developed in the low-frequency range. The aim of our study is to derive some numerical methods in all the domain of validity of the Biot’s model. In the high-frequency range, the effects of the viscous boundary layer inside the pores must be taken into account. We use the model of dynamic permeability of Johnson-Koplik-Dashen (JKD). In this case, some coefficients of the Biot-JKD’s model are proportional to the square root of the frequency. In the time-domain, fractional derivatives are therefore introduced into the evolution partial differential equations. Two strategies exist to calculate these fractional derivatives. The first strategy is to compute the involved convolution integral. However, it requires to store the past of the solution, which is too penalizing in term of computational memory. The second strategy, which we implement, is based on a diffusive representation of the convolution kernel. The latter is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations. The coefficients of the diffusive representation are determined by optimization on the frequency range of interest. We analyze the properties of the Biot-JKD model with diffusive representation: decay of energy, error of the model. We propose a numerical modeling, based on a splitting strategy: a propagative part is discretized by a fourth-order ADER scheme on a Cartesian grid, whereas a diffusive part is solved exactly. An immersed interface method accounts for the jump conditions and for the geometry of the interfaces, preventing from the usual limitations of finite differences on a Cartesian grid. References [1] G. Chiavassa, B. Lombard, Time domain numerical modeling of wave propagation in 2D heteroge- neous porous media, J. Comput. Phys., 230-13 (2011), 5288-5309. [2] E. Blanc, G. Chiavassa, B. Lombard, Biot-JKD model: simulation of 1D transient poroelastic waves with fractional derivatives, J. Comput. Phys., 237 (2013), 1-20. [3] E. Blanc, G. Chiavassa, B. Lombard, A time-domain numerical modeling of two-dimensional waves propagation in porous media with frequency-dependent dynamic permeability, J. Acoust. Soc. Am., (2013).

Igor L. Chernyavsky - History effects in inflammation-driven airway smooth muscle remodelling in asthma: insight from a theoretical model Authors: I.L. Chernyavsky, H. Croisier, L.A.C. Chapman, L.S. Kimpton, J.E. Hiorns, B.S. Brook, O.E. Jensen, C.K. Billington, I.P. Hall, S.R. Johnson

Objectives: Despite a large amount of in vitro data, the dynamics of airway smooth muscle (ASM) hyperplasia in the airways of patients with asthma is not well understood. Here, we present a novel mathematical model that describes the growth dynamics of ASM cells over short and long terms in the normal and inflammatory environments. Methods: Each ASM cell is assumed as being in a non-proliferative or proliferative state. Switching between the states is modulated by a series of inflammatory events that affect the inflammatory status. We use a combination of analytical techniques and Monte Carlo simulations to explore possible ASM growth scenarios. Main Results: Our model encompasses the idea that remodelling due to ASM hyperplasia increases with the frequency and magnitude of inflammatory events. It highlights the importance of inflamma- tion resolution speed by showing that when resolution is slow, even a series of small exacerbation events can result in significant remodelling, which persists after the inflammatory episodes. We demonstrate how the uncertainty in long-term outcome can be quantified and used to design an optimal low-risk individual treatment strategy. Conclusions: The model shows that the rate of clearance of ASM proliferation and recruitment factors after an acute inflammatory event is a potentially important, and hitherto unrecognised, target for anti-remodelling therapy in asthma. It also suggests new ways of quantifying inflammation severity that could improve prediction of the extent of airway remodelling. This ASM growth model should prove useful for designing new experiments or as a building block of more detailed multi-cellular tissue- level models.

A.L. Gower - Counter-Intuitive Acousto-Elasticity

The determination of the direction of greatest tension in a deformed solid is one of the main goals of acoustic non-destructive evaluation because, for isotropic solids, this direction coincides with the direction of greatest stress. In this poster, we present results of acoustic waves on deformed elastic solids, where the wave speed does not have its greatest value along the direction of greatest stretch [1]. This goes against what was commonly accepted. We briefly present the method used and explain that this phenomena occurs because the coupling of acoustics and elasticity is a non-linear phenomenon, even at its lowest order, and it can thus generate counter-intuitive results. [1] A. Gower, M. Destrade, and R. Ogden, Counter-intuitive results in acousto-elasticity, Wave Motion (to appear), (2013).

L.M. Joseph - High-frequency homogenisation for Rayleigh-Bloch waves along line defect in lat- tices

Author: L.M. Joseph and R.V. Craster

Rayleigh-Bloch surface waves are a special case of trapped modes, or edge waves, which represent dis- turbances travelling along an infinite periodic structure. In order to investigate Rayleigh-Bloch surface waves, we will be applying high-frequency homogenisation as classical homogenisation fails to identify wave frequencies away from the origin. This asymptotic procedure is based upon a two-scale approach that captures the microstuctural information of a periodic inhomogeneous medium used in wave prop- agation problems and finds asymptotic homogenised continuum equations valid on the macroscale (J. Kalpunov, R.V. Craster & A.V. Pichugin [Proceedings of The Royal Society A, vol 466, 2341-2362, 2010]). The development of high-frequency homogenisation was motivated by the interest in identify- ing the Bloch spectra at the edges of the irreducible Brillouin zone corresponding to standing waves considered to be in the high-frequency regime. We will be presenting discrete surface wave problems involving infinite lattices with line defects. The two-scale approach is utilised for a simple line defect and demonstrated versus exact solutions for quasi-periodic systems and versus numerical solutions for line defects that are themselves perturbed or altered. In particular, Rayleigh-Bloch states propagat- ing along the line defect, and localised defect states, are identified both asymptotically and numerically.

M. McIver - Hydrodynamic Forces on a Floating Structure

Kieron Moore - Coupled Boussinesq equations and nonlinear waves in layered waveguides

Author: Kieron Moore, Karima Khusnutdinova

A system of coupled regularised Boussinesq equations has been recently rigorously derived as a model describing long nonlinear longitudinal strain waves in a layered waveguide with a soft bonding layer [1]. We construct a weakly nonlinear solution of the initial-value problem for this system on the infinite line, in terms of solutions of various Ostrovsky-type equations [2], and show the striking difference in the behaviour of the solution for different asymptotic regimes. The radiating solitary waves emerging in one of these regimes are described analytically. The solution for a single Boussinesq equation is given in terms of solutions of the Cauchy problems for two Korteweg-de Vries equations [3]. We also perform a number of relevant numerical simulations using finite-difference and pseudo-spectral methods to test our analytical results. [1] K.R. Khusnutdinova, A.M. Samsonov and A. S. Zakharov, Nonlinear layered lattice model and generalized solitary waves in layered elastic structures. Phys. Rev. E 79 (2009) 056606. [2] K.R. Khusnutdinova and K.R. Moore, Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations. Wave Motion 48 (2011). [3] K.R. Khusnutdinova, K.R. Moore, Weakly nonlinear extension of d’Alembert’s formula, IMA J. Appl. Math. 77 (2012) 361-381.

T. Shearer - Hyperelastic modelling of the anterior cruciate ligament and patellar tendon

Authors: T. Shearer, S.H. Cartmell, W.J. Parnell and A. Hazel

The anterior cruciate ligament (ACL) is the most frequently injured knee ligament, and is one of the structures most commonly injured in sport [5]. Due to the fact that it does not heal naturally, the standard treatment for a ruptured ACL is surgical reconstruction [4], to which there are several approaches, the most common being patellar tendon (PT) and hamstring tendon autograft. There is currently no consensus with respect to the choice between these two grafts [4], however, the PT is the most commonly used [1]. In this poster, we mathematically model the ACL and PT in order to compare their mechanical properties within the same framework. Ligaments and tendons have an extremely hierarchical structure [3]. Their main subunit is the fascicle, which is made of fibrils arranged in a crimped pattern. We utilise a strain energy function proposed by Holzapfel et al. [2], which was proposed for the modelling of arteries but is equally applicable to tendons and ligaments, to show that the differing alignments of the fascicles within the ACL and PT have a significant effect on their stress-strain curves.

References

[1] N. Chandrashekar, J. Slauterbeck and J. Hashemi, Effects of cyclic loading on the tensile prop- erties of human patellar tendon, The Knee, 19, (2012), 65–68.

[2] G.A. Holzapfel, T.C. Gasser and R.W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, Journal of Elasticity, 61, (2000), 1–48.

[3] J. Kastelic, A. Galeski and E. Baer, The multicomposite structure of tendon, Connect Tissue Res, 6, (1978), 11–23. [4] N.G.H. Mohtadi, D.S. Chan, K.N. Dainty and D.B. Whelan, Patellar tendon versus hamstring tendon autograft for anterior cruciate ligament rupture in adults (Review), The Cochrane Library, 9, (2011).

[5] S.H. Ryder, R.J. Johnson and B.D. Beynnon, Prevention of ACL injuries, J Sports Rehabil, 6, (1997), 80–96.

Yilang Song - Stop bands in heterogeneous materials

This poster introduces the stop band phenomena and the simulation of wave propagation through heterogeneous materials. We indicates how randomness of the underlying microstructure affects the stop band phenomenon.

Maria Thorpe - Two interacting spherical cavities under hydrostatic compression

We study the interaction effects of two spherical cavities residing inside a linearly elastic medium of infinite extent which is subject to hydrostatic loading in the far field. This elasticity problem is solved analytically via series expansions and applications in the context of complex composites are discussed.

Xizheng Zhang - Cylindrical Korteweg-de Vries type equation for the ring waves in a stratified fluid

Authors: Xizheng Zhang, Karima Khusnutdinova

We study the propagation of a ring wave in a stratified fluid over a prescribed shear flow. A weakly- nonlinear long wave model is derived for internal and surface ring waves from the full set of Euler equations with background stratification and shear flow, subject to the usual free surface and rigid bottom boundary conditions, written in cylindrical coordinates. In the absence of the shear flow, the derived equation reduces to the equation for internal waves obtained by V.D. Lipovskii [1]. In the absence of stratification, the derived equation reduces to the equation for surface waves obtained by R.S. Johnson [2]. The latter is revisited, and it is shown that the equation has the form of the cKdV- type equation for any shear flow, and not just for stationary or constant shear flows, as previously thought. We also discuss the extension of this result for a stratified fluid. References: [1] V.D. Lipovskii, On the nonlinear internal wave theory in fluid of finite depth, Izv. Akad. Nauk SSSR, Ser. Fiz. Atm. Okeana 21 (1985) 864-871 (in Russian). [2] R.S. Johnson, Ring waves on the surface of shear flows: a linear and nonlinear theory, J. Fluid Mech. 215 (1990) 145-160.